Short questions and Problem (ch 4)

macro economics question and need guidance to help me learn.

(All answers in Chapter 4) the book in the attachment (Use easy word and sentences)
Short questions:
1. Define “Utility Maximization” and discuss the two limitations of utility maximization with examples.

2. What is the first-order function for a maximum utility? Draw a graph demonstrating utility Maximization and explain underlying concepts.

3. Explain the Lump Sum Principle, give an example of it, and draw the related graph.

4. Discuss the three properties of expenditure functions.

Problems:
1. On a given morning, Tom enjoys the consumption of cake (c) and bread (b) according to the following function:
U (c, b) = 20c- c² + 18b-3b²
How many pieces of cakes and breads does he consume during a morning? (Cost is no object to Tom)
Requirements: 1 d
PowerPoint Slides prepared by: V. Andreea CHIRITESCUEastern Illinois UniversityWalter Nicholson | Christopher Snyder 12th editionCHAPTER Utility Maximization 4 and Choice© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.1
Utility Maximization and Choice•Complaints about the Economic Approach–Do individuals make the “lightning calculations” required for utility maximization?•The utility-maximization model predicts many aspects of behavior•Economists assume that people behave as if they made such calculations© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.2
Utility Maximization and Choice•Complaints about the Economic Approach–The economic model of choice is extremely selfish•Nothing in the model prevents individuals from getting satisfaction from “doing good”© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.3
An Initial Survey•Optimization principle, Utility maximization–To maximize utility, given a fixed amount of income to spend–An individual will buy those quantities of goods •That exhaust his or her total income •And for which the MRS is equal to the rate at which the goods can be traded one for the other in the marketplace– MRS (of x for y) = the ratio of the price of x to the price of y (px/py)© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.4
The Two-Good Case•Assumptions –Budget: I dollars to allocate between good x and good y– px – price of good x – py – price of good y•Budget constraint: pxx + pyy ≤ I –Slope = -px/py–If all of I is spent on good x, buy I/px units of good x© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.5
4.1 The Individual’s Budget Constraint for Two GoodsThose combinations of x and y that the individual can afford are shown in the shaded triangle. If, as we usually assume, the individual prefers more rather than less of every good, the outer boundary of this triangle is the relevant constraint where all the available funds are spent either on x or on y. The slope of this straight-line boundary is given by –px/py.© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.6I=pxx+pyyQuantity of xQuantity of y
The Two-Good Case•First-order conditions for a maximum–Point of tangency between the budget constraint and the indifference curve:© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.7
4.2 A Graphical Demonstration of Utility MaximizationPoint C represents the highest utility level that can be reached by the individual, given the budget constraint. Therefore, the combination x*,y* is the rational way for the individual to allocate purchasing power. Only for this combination of goods will two conditions hold: All available funds will be spent, and the individual’s psychic rate of trade-off (MRS) will be equal to the rate at which the goods can be traded in the market ( px/py).© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.8I=pxx+pyyQuantity of xQuantity of yU1U3DU2ABCy*x*
The Two-Good Case•The tangency rule –Is necessary but not sufficient unless we assume that MRS is diminishing•If MRS is diminishing, then indifference curves are strictly convex•If MRS is not diminishing, we must check second-order conditions to ensure that we are at a maximum© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.9
4.3 Example of an Indifference Curve Map for Which the Tangency Condition Does Not Ensure a MaximumIf indifference curves do not obey the assumption of a diminishing MRS, not all points of tangency (points for which MRS = px/py) may truly be points of maximum utility. In this example, tangency point C is inferior to many other points that can also be purchased with the available funds. In order that the necessary conditions for a maximum (i.e., the tangency conditions) also be sufficient, one usually assumes that the MRS is diminishing; that is, the utility function is strictly quasi-concave.© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.10Quantity of xQuantity of yU1U3U2CAB
The Two-Good Case•Corner solutions–Individuals may maximize utility by choosing to consume only one of the goods–At the optimal point the budget constraint is flatter than the indifference curve•The rate at which x can be traded for y in the market is lower than the MRS© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.11
4.4 Corner Solution for Utility MaximizationWith the preferences represented by this set of indifference curves, utility maximization occurs at E, where 0 amounts of good y are consumed. The first-order conditions for a maximum must be modified somewhat to accommodate this possibility.© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.12Quantity of xQuantity of yU2U3U1x*
The n-Good Case•The individual’s objective is to maximizeutility = U(x1,x2,…,xn)– subject to the budget constraintI = p1x1 + p2x2 +…+ pnxn•Set up the Lagrangian:ℒ = U(x1,x2,…,xn) + (I – p1x1 – p2x2 -…- pnxn)© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.13
The n-Good Case•First-order conditions for an interior maximum ℒ/x1 = U/x1 – p1 = 0 ℒ /x2 = U/x2 – p2 = 0 … ℒ /xn = U/xn – pn = 0 ℒ / = I – p1x1 – p2x2 – … – pnxn = 0© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.14
The n-Good Case•Implications of first-order conditions–For any two goods, xi and yj:© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.15
The n-Good Case•Interpreting the Lagrange multiplier– is the marginal utility of an extra dollar of consumption expenditure•The marginal utility of income© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.16
The n-Good Case•At the margin, the price of a good–Represents the consumer’s evaluation of the utility of the last unit consumed–How much the consumer is willing to pay for the last unit© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.17
The n-Good Case•Corner solutions–Means that the first-order conditions must be modified:ℒ/xi = U/xi – pi  0 (i = 1,…,n)–If ℒ/xi = U/xi – pi < 0, then xi = 0–This means that© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.18–any good whose price exceeds its marginal value to the consumer will not be purchased 4.1 Cobb–Douglas Demand Functions© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.19•Cobb-Douglas utility function:U(x,y) = xy, where α+β=1•Setting up the Lagrangian:ℒ = xy + (I - pxx - pyy)•First-order conditions:ℒ/x = x-1y - px = 0ℒ/y = xy-1 - py = 0ℒ/ = I - pxx - pyy = 0 4.1 Cobb–Douglas Demand Functions© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.20•First-order conditions imply: y/x = px/py–Since  +  = 1: pyy = (/)pxx = [(1- )/]pxx•Substituting into the budget constraint: I = pxx + [(1- )/]pxx = (1/)pxx•Solving: x*=I/px and y*=I/py–The individual will allocate  percent of his income to good x and  percent of his income to good y 4.1 Cobb–Douglas Demand Functions© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.21•Cobb-Douglas utility function–Is limited in its ability to explain actual consumption behavior–The share of income devoted to a good often changes in response to changing economic conditions•A more general functional form might be more useful 4.2 CES Demand© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.22•Assume that  = 0.5U(x,y) = x0.5 + y0.5•Setting up the Lagrangian:ℒ = x0.5 + y0.5 + (I - pxx - pyy)•First-order conditions for a maximum:ℒ/x = 0.5x -0.5 - px = 0ℒ/y = 0.5y -0.5 - py = 0ℒ/ = I - pxx - pyy = 0 4.2 CES Demand© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.23•This means that: (y/x)0.5 = px/py–Substituting into the budget constraint, we can solve for the demand functions•The share of income spent on either x or y is not a constant•Depends on the ratio of the two prices •The higher is the relative price of x, the smaller will be the share of income spent on x 4.2 CES Demand© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.24•If  = -1,U(x,y) = -x -1 - y -1•First-order conditions imply thaty/x = (px/py)0.5•The demand functions are 4.2 CES Demand© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.25•If  = -, U(x,y) = min(x,4y)–The person will choose only combinations for which x = 4y–This means thatI = pxx + pyy = pxx + py(x/4)I = (px + 0.25py)x•The demand functions are Indirect Utility Function•It is often possible to manipulate first-order conditions to solve for optimal values of x1,x2,…,xn–These optimal values will bex*1 = x1(p1,p2,…,pn,I)x*2 = x2(p1,p2,…,pn,I)…x*n = xn(p1,p2,…,pn,I)© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.26 Indirect Utility Function•We can use the optimal values of the x’s to find the indirect utility functionmaximum utility = U[x*1(p1,p2,…,pn,I), x*2(p1,p2,…,pn,I),…,x*n(p1,p2,…,pn,I)] = = V(p1,p2,…,pn,I)–The indirect utility function is an example of a value function•The optimal level of utility will depend indirectly on prices and income© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.27 The Lump Sum Principle•Taxes on an individual’s general purchasing power –Are superior to taxes on a specific good•An income tax allows the individual to decide freely how to allocate remaining income•A tax on a specific good will reduce an individual’s purchasing power and distort his choices© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.28 4.5 The Lump Sum Principle of TaxationA tax on good x would shift the utility-maximizing choice from x*, y* to x1, y1. An income tax that collected the same amount would shift the budget constraint to I’. Utility would be higher (U2) with the income tax than with the tax on x alone (U1).© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.29 4.3 Indirect Utility and the Lump Sum Principle© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.30•Cobb-Douglas utility function–With  =  = 0.5, –We know that x*=I/2px and y*=I/2py•The indirect utility function 4.3 Indirect Utility and the Lump Sum Principle© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.31•Fixed proportions x*=I/[px + 0.25py] and y*=I/[4px+py]•The indirect utility function 4.3 Indirect Utility and the Lump Sum Principle© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.32•The lump sum principle•Cobb-Douglas–If a tax of $1 was imposed on good x•The individual will purchase x* = 2•Indirect utility will fall from 2 to 1.41–An equal-revenue tax will reduce income to $6•Indirect utility will fall from 2 to 1.5 4.3 Indirect Utility and the Lump Sum Principle© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.33•The lump sum principle•Fixed-proportions–If a tax of $1 was imposed on good x•Indirect utility will fall from 4 to 8/3–An equal-revenue tax will reduce income to $16/3•Indirect utility will fall from 4 to 8/3•Since preferences are rigid, the tax on x does not distort choices Expenditure Minimization•Dual minimization problem for utility maximization–Allocate income to achieve a given level of utility with the minimal expenditure•The goal and the constraint have been reversed© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.34 4.6 The Dual Expenditure-Minimization ProblemThe dual of the utility-maximization problem is to attain a given utility level (U2) with minimal expenditures. An expenditure level of E1 does not permit U2 to be reached, whereas E3 provides more spending power than is strictly necessary. With expenditure E2, this person can just reach U2 by consuming x and y.© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.35E2Quantity of xQuantity of yU2E1E3BCAy*x* Expenditure Minimization•The individual’s problem is to choose x1,x2,…,xn to minimizetotal expenditures = E = p1x1 + p2x2 +…+ pnxn subject to the constraintutility = Ū = U(x1,x2,…,xn)–The optimal amounts of x1,x2,…,xn will depend on the prices of the goods and the required utility level© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.36 Expenditure Minimization•Expenditure function–The individual’s expenditure function –Shows the minimal expenditures–Necessary to achieve a given utility level –For a particular set of prices –Also a value function•minimal expenditures = E(p1,p2,…,pn,U)© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.37 Expenditure Minimization•The expenditure function and the indirect utility function –Are inversely related–Both depend on market prices –But involve different constraints (income or utility) © 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.38 4.4 Two Expenditure Functions© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.39•Cobb-Douglas –The indirect utility function in the two-good case:–If we interchange the role of utility and income (expenditure), we will have the expenditure function E(px,py,U) = 2px0.5py0.5U 4.4 Two Expenditure Functions© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.40•Fixed-proportions case–The indirect utility function:•If we interchange the role of utility and income (expenditure), we will have the expenditure function E(px,py,U) = (px + 0.25py)U Properties of Expenditure Functions•Homogeneity–A doubling of all prices will precisely double the value of required expenditures•Homogeneous of degree one•Nondecreasing in prices–E/pi  0 for every good, i•Concave in prices –Functions that always lie below tangents to them© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.41 4.7 Expenditure Functions Are Concave in PricesAt p1 this person spends E(p1* , . . .). If he or she continues to buy the same set of goods as p1 changes, then expenditures would be given by Epseudo. Because his or her consumption patterns will likely change as p1 changes, actual expenditures will be less than this.© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.42E(p1,…)Epseudop1E(p1,…)E(p*1,…)E(p*1, …) Budget Shares© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.43•Engel’s law –Fraction of income spent on food decreases as income increases•Budget shares, si=pixi / I•Recent budget share data–Engel’s law is clearly visible•Cobb–Douglas utility function –Is not useful for detailed empirical studies of household behavior E4.1 Budget shares of U.S. households, 2008© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.44Annual Income$10,000−$14,999$40,000−$49,999Over $70,000Expenditure ItemFood15.713.411.8Shelter23.121.219.3Utilities, fuel, and public services11.28.65.8Transportation14.117.816.8Health insurance5.34.02.6Other health-care expenses2.62.82.3Entertainment (including alcohol)4.65.25.8Education2.31.22.6Insurance and pensions2.28.514.6Other (apparel, personal care, other housing expenses, and misc.)18.917.318.4 Linear expenditure system© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.45•Generalization of the Cobb–Douglas function–Incorporates the idea that certain minimal amounts of each good must be bought by an individual (x0, y0)U(x,y)=(x-x0)(y-y0) •For x ≥ x0 and y ≥ y0, •Where +=1 Linear expenditure system© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.46•Supernumerary income (I*)–Amount of purchasing power remaining after purchasing the minimum bundle I*=I-pxx0-pyy0 •The demand functions are: x = (pxx0+I*)/px and y = (pyy0+I*)/py –The share equations: sx= +(pxx0-pyy0)/I sy= +(pyy0 -pxx0)/I•Not homothetic CES utility© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.47•CES utility function•Budget shares:sx=1/[1+(py/px)K] and sy=1/[1+(px/py)K]•Where K = /(-1)•Homothetic The almost ideal demand system© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.48•Expenditure functions–Logarithmic differentiation The almost ideal demand system© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.49•Almost ideal demand system–Expenditure function•Almost ideal demand system–Expenditure function •Homogeneous of degree one in the prices – a1+a2=1, b1+b2=0, b2+b3=0, and c1+c2=0 The almost ideal demand system© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.50 MICROECONOMICTHEORYBASICPRINCIPLESANDEXTENSIONSTENTHEDITION This page intentionally left blank MICROECONOMICTHEORYBASICPRINCIPLESANDEXTENSIONSTENTHEDITIONWalterNicholsonAmherstCollegeChristopherSnyderDartmouthCollege VP/EditorialDirector:JackW.CalhounEditor-in-Chief:AlexvonRosenbergExecutiveEditor:MikeRocheSr.DevelopmentalEditor:SusanSmartSr.ContentProjectManager:CliffKallemeynProductionTechnologyAnalyst:AdamGrafaExecutiveMarketingManager:BrianJoynerSr.MarketingManager:JohnCareyArtDirector:MichelleKunklerSr.FirstPrintBuyer:SandeeMilewskiPrinter:WestGroupEagan,MNMicroeconomicTheoryBasicPrinciplesandExtensionsTenthEditionWalterNicholsonChristopherSnyderCOPYRIGHT©2008,2005ThomsonSouth-Western,apartofTheThomsonCorporation.Thomson,theStarlogo,andSouth-Westernaretrademarksusedhereinunderlicense.PrintedintheUnitedStatesofAmerica1234510090807ISBN13:978-0-324-42162-0ISBN10:0-324-42162-1ALLRIGHTSRESERVED.Nopartofthisworkcoveredbythecopyrighthereonmaybereproducedorusedinanyformorbyanymeans—graphic,elec-tronic,ormechanical,includingphotocopying,recording,taping,Webdistributionorinformationstorageandretrievalsystems,orinanyothermanner—withoutthewrittenpermissionofthepublisher.Forpermissiontousematerialfromthistextorproduct,submitarequestonlineathttp://www.thomsonrights.com.LibraryofCongressControlNumber:2007921464ThomsonHigherEducation5191NatorpBoulevardMason,OH45040USAFormoreinformationaboutourproducts,contactusat:ThomsonLearningAcademicResourceCenter1-800-423-0563 ToBeth,Sarah,David,Sophia,andAbbyToMaura This page intentionally left blank AbouttheAuthorsWalterNicholsonistheWardH.PattonProfessorofEconomicsatAmherstCollege.HereceivedhisB.A.inmathematicsfromWilliamsCollegeandhisPh.D.ineconomicsfromMIT.ProfessorNicholson’sprincipalresearchinterestsareintheeconometricanalysesoflabormarketproblemsincludingunemployment,jobtraining,andtheimpactofinter-nationaltrade.Heisalsotheco-author(withChrisSnyder)ofIntermediateMicroeconomicsandItsApplication,TenthEdition(Thomson/South-Western,2007).ProfessorNicholsonandhiswife,Susan,liveinAmherst,Massachusetts,andNaples,Florida.Whatwaspreviouslyaverybusyhousehold,withfourchildreneverywhere,isnowratherempty.Butanever-increasingnumberofgrandchildrenbreathesomelifeintotheseplaceswhenevertheyvisit,whichseemsfartooseldom.ChristopherM.SnyderisaProfessorofEconomicsatDartmouthCollege.HereceivedhisB.A.ineconomicsandmathematicsfromFordhamUniversityandhisPh.D.ineconomicsfromMIT.BeforecomingtoDartmouthin2005,hetaughtatGeorgeWashingtonUniversityforoveradecade,andhehasbeenavisitingprofessorattheUniversityofChicagoandMIT.HeiscurrentlyPresidentoftheIndustrialOrganizationSocietyandAssociateEditoroftheInternationalJournalofIndustrialOrganizationandReviewofIndustrialOrganization.Hisresearchcoversvarioustheoreticalandempiricaltopicsinindustrialorganization,contracttheory,andlawandeconomics.ProfessorSnyderandhiswifeMauraDoyle(whoalsoteacheseconomicsatDartmouth)livewithinwalkingdistanceofcampusinHanover,NewHampshire,withtheirthreeelementary-school-ageddaughters. This page intentionally left blank BriefContentsPart1IntroductionChapter1:EconomicModels3Chapter2:MathematicsforMicroeconomics19Part2ChoiceandDemandChapter3:PreferencesandUtility87Chapter4:UtilityMaximizationandChoice113Chapter5:IncomeandSubtitutionEffects141Chapter6:DemandRelationshipsamongGoods182Chapter7:UncertaintyandInformation202Chapter8:StrategyandGameTheory236Part3ProductionandSupplyChapter9:ProductionFunctions295Chapter10:CostFunctions323Chapter11:ProﬁtMaximization358Part4CompetitiveMarketsChapter12:ThePartialEquilibriumCompetitiveModel391Chapter13:GeneralEquilibriumandWelfare441Part5MarketPowerChapter14:Monopoly491Chapter15:ImperfectCompetition521Part6PricinginInputMarketsChapter16:LaborMarkets573Chapter17:CapitalandTime595Part7MarketFailureChapter18:AsymmetricInformation627Chapter19:ExternalitiesandPublicGoods670ix BriefAnswerstoQueries701SolutionstoOdd-NumberedProblems711GlossaryofFrequentlyUsedTerms721Index727xBriefContents ContentsPrefacexixPART1INTRODUCTION1CHAPTER1EconomicModels3TheoreticalModels3VeriﬁcationofEconomicModels3GeneralFeaturesofEconomicModels5DevelopmentoftheEconomicTheoryofValue8ModernDevelopments16Summary17SuggestionsforFurtherReading18CHAPTER2MathematicsforMicroeconomics19MaximizationofaFunctionofOneVariable19FunctionsofSeveralVariables23MaximizationofFunctionsofSeveralVariables28ImplicitFunctions31TheEnvelopeTheorem32ConstrainedMaximization36EnvelopeTheoreminConstrainedMaximizationProblems42InequalityConstraints43Second-OrderConditions45HomogeneousFunctions53Integration56DynamicOptimization60MathematicalStatistics64Summary74Problems75SuggestionsforFurtherReadings79Extensions:Second-OrderConditionsandMatrixAlgebra81xi PART2CHOICEANDDEMAND85CHAPTER3PreferencesandUtility87AxiomsofRationalChoice87Utility88TradesandSubstitution91AMathematicalDerivation97UtilityFunctionsforSpeciﬁcPreferences100TheMany-GoodCase104Summary105Problems106SuggestionsforFurtherReading109Extensions:SpecialPreferences110CHAPTER4UtilityMaximizationandChoice113AnInitialSurvey114TheTwo-GoodCase:AGraphicalAnalysis114Then-GoodCase118IndirectUtilityFunction124TheLumpSumPrinciple125ExpenditureMinimization127PropertiesofExpenditureFunctions130Summary132Problems132SuggestionsforFurtherReading136Extensions:BudgetShares137CHAPTER5IncomeandSubstitutionEffects141DemandFunctions141ChangesinIncome143ChangesinaGood’sPrice144TheIndividual’sDemandCurve148CompensatedDemandCurves151AMathematicalDevelopmentofResponsetoPriceChanges155DemandElasticities158ConsumerSurplus165RevealedPreferenceandtheSubstitutionEffect169Summary172Problems173xiiContents SuggestionsforFurtherReading176Extensions:DemandConceptsandtheEvaluationofPriceIndices178CHAPTER6DemandRelationshipsAmongGoods182TheTwo-GoodCase182SubstitutesandComplements184NetSubstitutesandComplements186SubstitutabilitywithManyGoods188CompositeCommodities188HomeProduction,AttributesofGoods,andImplicitPrices191Summary195Problems195SuggestionsforFurtherReading199Extensions:SimplifyingDemandandTwo-StageBudgeting200CHAPTER7UncertaintyandInformation202MathematicalStatistics202FairGamesandtheExpectedUtilityHypothesis203ThevonNeumann–MorgensternTheorem205RiskAversion207MeasuringRiskAversion209ThePortfolioProblem214TheState-PreferenceApproachtoChoiceunderUncertainty216TheEconomicsofInformation221PropertiesofInformation221TheValueofInformation222FlexibilityandOptionValue224AsymmetryofInformation225Summary226Problems226SuggestionsforFurtherReading231Extensions:PortfoliosofManyRiskAssets232CHAPTER8StrategyandGameTheory236BasicConcepts236Prisoners’Dilemma237NashEquilibrium240Contentsxiii MixedStrategies247Existence251ContinuumofActions252SequentialGames255RepeatedGames259IncompleteInformation268SimultaneousBayesianGames268SignalingGames273ExperimentalGames281EvolutionaryGamesandLearning282Summary283Problems284SuggestionsforFurtherReading287Extensions:ExistenceofNashEquilibrium288PART3PRODUCTIONANDSUPPLY293CHAPTER9ProductionFunctions295MarginalProductivity295IsoquantMapsandtheRateofTechnicalSubstitution298ReturnstoScale302TheElasticityofSubstitution305FourSimpleProductionFunctions306TechnicalProgress311Summary315Problems315SuggestionsforFurtherReading319Extensions:Many-InputProductionFunctions320CHAPTER10CostFunctions323DeﬁnitionsofCost323Cost-MinimizingInputChoices325CostFunctions330CostFunctionsandShiftsinCostCurves334Shephard’sLemmaandtheElasticityofSubstitution344Short-Run,Long-RunDistinction344Summary350Problems351xivContents SuggestionsforFurtherReading354Extensions:TheTranslogCostFunction355CHAPTER11ProﬁtMaximization358TheNatureandBehaviorofFirms358ProﬁtMaximization359MarginalRevenue361Short-RunSupplybyaPrice-TakingFirm365ProﬁtFunctions369ProﬁtMaximizationandInputDemand374Summary380Problems381SuggestionsforFurtherReading385Extensions:ApplicationsoftheProﬁtFunction386PART4COMPETITIVEMARKETS389CHAPTER12ThePartialEquilibriumCompetitiveModel391MarketDemand391TimingoftheSupplyResponse395PricingintheVeryShortRun395Short-RunPriceDetermination396ShiftsinSupplyandDemandCurves:AGraphicalAnalysis401MathematicalModelofMarketEquilibrium403Long-RunAnalysis406Long-RunEquilibrium:ConstantCostCase407ShapeoftheLong-RunSupplyCurve410Long-RunElasticityofSupply412ComparativeStaticsAnalysisofLong-RunEquilibrium413ProducerSurplusintheLongRun416EconomicEfﬁciencyandWelfareAnalysis419PriceControlsandShortages422TaxIncidenceAnalysis423TradeRestrictions427Summary431Problems432SuggestionsforFurtherReading436Extensions:DemandAggregationandEstimation438Contentsxv CHAPTER13GeneralEquilibriumandWelfare441PerfectlyCompetitivePriceSystem441ASimpleGraphicalModelofGeneralEquilibriumwithTwoGoods442ComparativeStaticsAnalysis451GeneralEquilibriumModelingandFactorPrices453ExistenceofGeneralEquilibriumPrices455GeneralEquilibriumModels462WelfareEconomics466EfﬁciencyinOutputMix469CompetitivePricesandEfﬁciency:TheFirstTheoremofWelfareEconomics471DepartingfromtheCompetitiveAssumptions475DistributionandtheSecondTheoremofWelfareEconomics476Summary481Problems482SuggestionsforFurtherReading486Extensions:ComputableGeneralEquilibriumModels487PART5MARKETPOWER489CHAPTER14Monopoly491BarrierstoEntry491ProﬁtMaximizationandOutputChoice493MonopolyandResourceAllocation497Monopoly,ProductQuality,andDurability501PriceDiscrimination503Second-DegreePriceDiscriminationthroughPriceSchedules508RegulationofMonopoly510DynamicViewsofMonopoly513Summary513Problems514SuggestionsforFurtherReading518Extensions:OptimalLinearTwo-partTariffs519CHAPTER15ImperfectCompetition521Short-RunDecisions:PricingandOutput521BertrandModel523xviContents CournotModel524CapacityConstraints531ProductDifferentiation531TacitCollusion537Longer-RunDecisions:Investment,Entry,andExit541StrategicEntryDeterrence547Signaling551HowManyFirmsEnter?554Innovation558Summary560Problems561SuggestionsforFurtherReading565Extensions:StrategicSubstitutesandComplements566PART6PRICINGININPUTMARKETS571CHAPTER16LaborMarkets573AllocationofTime573AMathematicalAnalysisofLaborSupply576MarketSupplyCurveforLabor580LaborMarketEquilibrium581MonopsonyintheLaborMarket584LaborUnions586Summary589Problems590SuggestionsforFurtherReading594CHAPTER17CapitalandTime595CapitalandtheRateofReturn595DeterminingtheRateofReturn597TheFirm’sDemandforCapital604PresentDiscountedValueApproachtoInvestmentDecisions606NaturalResourcePricing611Summary614Problems614SuggestionsforFurtherReading618Appendix:TheMathematicsofCompoundInterest619Contentsxvii PART7MARKETFAILURE625CHAPTER18AsymmetricInformation627ComplexContractsasaResponsetoAsymmetricInformation627Principal-AgentModel629HiddenActions630Owner-ManagerRelationship632MoralHazardinInsurance637HiddenTypes642NonlinearPricing642AdverseSelectioninInsurance650MarketSignaling657Auctions659Summary663Problems663SuggestionsforFurtherReading666Extensions:NonlinearPricingwithaContinuumofTypes667CHAPTER19ExternalitiesandPublicGoods670DeﬁningExternalities670ExternalitiesandAllocativeInefﬁciency672SolutionstotheExternalityProblem675AttributesofPublicGoods679PublicGoodsandResourceAllocation680LindahlPricingofPublicGoods684VotingandResourceAllocation687ASimplePoliticalModel690VotingMechanisms692Summary694Problems694SuggestionsforFurtherReading698Extensions:PollutionAbatement699BriefAnswerstoQueries701SolutionstoOdd-NumberedProblems711GlossaryofFrequentlyUsedTerms721Index727xviiiContents PrefaceThe10theditionofMicroeconomicTheory:BasicPrinciplesandExtensionsrepresentsbothacontinuationofahighlysuccessfultreatmentofmicroeconomicsatarelativelyadvancedlevelandamajorchangefromthepast.Thischange,ofcourse,isthatChrisSnyderhasjoinedmeasaco-author.Hisinsightshaveimprovedallsectionsofthebook,especiallywithrespecttoitscoverageofgametheory,industrialorganization,andmodelsofimperfectinformation.Henceinmanywaysthisisanewbook,althoughonmattersofstyleandpedagogyitretainsmuchofwhathasmadeitsuccessfulformorethan35years.Thisbasicapproachistofocusonbuildingintuitionabouteconomicmodelswhileprovidingstudentswiththemathematicaltoolsneededtogofurtherintheirstudies.Thetextalsoseekstofacilitatethatlinkagebyprovidingmanynumericalexamples,advancedproblems,andextendeddiscussionsofempiricalimplementation—allofwhichareintendedtoshowstudentshowmicroeconomictheoryisusedtoday.Newdevelopmentshavemadetheﬁeldmoreexcitingthanever,andIhopethiseditionmanagestocapturethatexcitement.NEWTOTHETENTHEDITIONTheprimarychangetothiseditionhasbeentheinclusionofthreeentirelynewchapterswrittenbyChrisSnyder:anextendedandmoreadvancedtreatmentofbasicgametheoryconcepts(Chapter8);athoroughlyreworkedandexpandedchapteronmodelsusedinindustrialorgani-zationtheory(Chapter15);andacompletelynewchapteronasymmetricinformationthatfocusesontheprincipal–agentproblemandmoderncontracttheory(Chapter18).Theimportanceoftheseadditionstotheoverallqualityofthetextcannotbeoverstated.Becausethetopicscoveredinthesenewchaptersconstitutesomeofthemostimportantgrowthareasinmicroeconomics,thebookisnowwellpositionedformanyyearsintothefuture.Severalotherchaptersofthebookhaveundergonemajorrevisionsforthisedition.Asigniﬁcantamountofmaterialhasbeenaddedtothechapteronmathematicalbackground(Chapter2);newtopicsinclude:anexpandedcoverageofintegration,basicmodelsofdynamicoptimization,andabriefintroductiontomathematicalstatistics.Thematerialonuncertaintyandriskaversionhasbeenthoroughlyrevisedandupdated(Chapter7).Muchofthetheoryoftheﬁrm,especiallyoftheﬁrm’sdemandsforinputs,hasbeenexpanded(Chapters9–11).xix Thechapterongeneralequilibriummodeling(Chapter13)hasbeenthoroughlyreworkedwiththegoalofprovidingstudentswithmoredetailsabouthowcompu-tablegeneralequilibriummodelsactuallywork.Thechapteroncapitalandtime(Chapter17)hasbeensigniﬁcantlyexpandedtoincludemoreonoptimalsavingsbehaviorandonresourceallocationovertime.Numerousminorchangeshavealsobeenmadeinthecoverageandorganizationofthebooktoensurethatitcontinuestoprovideclearandup-to-datecoverageofallofthetopicsexamined.Twomodiﬁcationshavebeenmadetothetexttoenhanceitslinkagetomoregeneraleconomicliterature.First,theproblemshavebeencategorizedintotwotypes:basicproblemsandanalyticalproblems.Whereasthebasicproblemsareintendedtoreinforceconceptsfromthetext,theanalyticalproblemsareintendedtoallowthestudenttogofurtherbyshowingthemhowtoobtainresultsontheirown.Thenumberofsuchproblemshasbeensigniﬁcantlyexpandedinthisedition.Manyoftheanalyticalproblemsprovidereferencessothatstudentswhowishtopursuethetopiccanreadmore.Asecondmodiﬁcationofthetexthasbeentoexpandandrewritemanyoftheend-of-chapterExtensions.ThecommongoaloftheserevisedExtensionsistoprovidestudentsbetterlinkagebetweenthetheoreticalmaterialinthetextandthatmaterial’suseinactualempiricalapplications.Therefore,manyoftheExtensionsintroducethefunctionalformscustomarilyusedaswellassomeoftheeconometricissuesfacedbyresearcherswhenusingavailabledata.TheExtensionsarethusintendedtoshowstudentstheimportanceofjoiningmicroeconomictheoryandeconometricpractice.SUPPLEMENTSTOTHETEXTThethoroughlyrevisedancillariesforthiseditionincludethefollowing.TheSolutionsManualandTestBank(bythetextauthors).TheSolutionsManualcontainscommentsandsolutionstoallproblemsandisavailabletoalladoptinginstructorsinbothprintandelectronicversions.TheSolutionsManualandTestBankmaybedownloadedonlybyqualiﬁedinstructorsatthetextbooksupportWebsite(www.thomsonedu.com/economics/nicholson).PowerPointLecturePresentationSlides(byLindaGhent,EasternIllinoisUniversity).PowerPointslidesforeachchapterofthetextprovideathoroughsetofoutlinesforclassroomuseorforstudentsasastudyaid.Instructorsandstudentsmaydown-loadtheseslidesfromthebook’sWebsite(www.thomsonedu.com/economics/nicholson).ONLINERESOURCESThomsonSouth-Westernprovidesstudentsandinstructorswithasetofvaluableonlineresourcesthatareaneffectivecomplementtothistext.EachnewcopyofthebookcomeswitharegistrationcardthatprovidesaccesstoEconomicApplicationsandInfoTracCollegeEdition.EconomicApplicationsThepurchaseofthisnewtextbookincludescomplimentaryaccesstoSouth-Western’sEconomicApplications(EconApps)Website.TheEconAppsWebsiteincludesasuiteofxxPreface regularlyupdatedWebfeaturesforeconomicsstudentsandinstructors:EconDebateOnline,EconNewsOnline,EconDataOnline,andEconLinksOnline.Theseresourcescanhelpstudentsdeepentheirunderstandingofeconomicconceptsbyanalyzingcurrentnewsstories,policydebates,andeconomicdata.EconAppscanalsohelpinstructorsdevelopassignments,casestudies,andexamplesbasedonreal-worldissues.EconDebatesOnlineprovidescurrentcoverageofeconomicspolicydebates;itincludesaprimerontheissues,linkstobackgroundinformation,andcommentaries.EconNewsOnlinesummarizesrecenteconomicsnewsstoriesandoffersquestionsforfurtherdiscussion.EconDataOnlinepresentscurrentandhistoricaleconomicdatawithaccompanyingcom-mentary,analysis,andexercises.EconLinksOnlineoffersanavigationpartnerforexploringeconomicsontheWebviaalistofkeytopiclinks.StudentsbuyingausedbookcanpurchaseaccesstotheEconAppssiteathttp://econapps.swlearning.com.InfoTracCollegeEditionThepurchaseofthisnewtextbookalsocomeswithfourmonthsofaccesstoInfoTrac.Thispowerfulandsearchableonlinedatabaseprovidesaccesstofulltextarticlesfrommorethanathousanddifferentpublicationsrangingfromthepopularpresstoscholarlyjournals.Instructorscansearchtopicsandselectreadingsforstudents,andstudentscansearcharticlesandreadingsforhomeworkassignmentsandprojects.Thepublicationscoveravarietyoftopicsandincludearticlesthatrangefromcurrenteventstotheoreticaldevelopments.InfoTracCollegeEditionoffersinstructorsandstudentstheabilitytointegratescholarshipandapplicationsofeconomicsintothelearningprocess.ACKNOWLEDGMENTSInpreparationforundertakingthisrevision,wereceivedveryhelpfulreviewsfrom:TiborBesedes,LouisianaStateUniversityElaineP.Catilina,AmericanUniversityYiDeng,SouthernMethodistUniversitySilkeForbes,UniversityofCalifornia–SanDiegoJosephP.Hughes,RutgersUniversityQihongLiu,UniversityofOklahomaRaganPetrie,GeorgiaStateUniversityWehaveusuallytriedtofollowtheirgoodadvice,butofcoursenoneoftheseindividualsbearsanyresponsibilityfortheﬁnaloutcome.Thiseditionofthebookistheﬁrstthatwaswrittenwithmyco-author,ChrisSnyderofDartmouthCollege.IhavebeenverypleasedwiththeworkingrelationshipwehavedevelopedandwithChris’sfriendship.Ihopemanymoreeditionswillfollow.IamalsoindebtedtotheteamatThomsonSouth-WesternandespeciallytoSusanSmartforonceagainbringingherorganizingandcajolingskillstothisedition.Duringhertemporaryabsencefromtheproject,wewerecompletelylost.Prefacexxi Copyeditingthismanuscriptwas,Iknow,arealchore.ThoseatNewgen-Austindidagreatjobofpenetratingourmessymanuscriptstoobtainsomethingthatactuallymakessense.ThedesignofthetextbyMichelleKunklersucceededinachievingtwoseeminglyirreconcilablegoals—makingthetextbothcompactandeasytoread.CliffKallemeyndidaﬁnejobofkeepingtheproductionontrack;Iespeciallyappreciatedthewayhecoordinatedthecopyeditingandpageproductionprocesses.Asalways,myAmherstCollegecolleaguesandstudentsdeservesomeofthecreditforthisnewedition.FrankWesthoffhasbeenmymostfaithfuluserofthistextovermanyyears.Thistime(withhispermission,Ithink)Iactuallyliftedsomeofhisworkongeneralequilibriumtosigniﬁcantlyimprovethatportionofthetext.Tothelistofformerstudents—MarkBruni,EricBudish,AdrianDillon,DavidMacoy,TatyanaMamut,KatieMerrill,JordanMilev,DougNorton,andJeffRodman—whoseeffortsarestillevidentIcannowaddthenameofAnoopMenon,whohelpedmesolveproblemswhenIranoutofpatiencewiththealgebra.Asalways,specialthanksagaingotomywifeSusan;afterseeingtwentyeditionsofmymicroeconomicstextscomeandgo,shemustsurelyhopethateventhisgoodthingmusteventuallycometoanend.Mychildren(Kate,David,Tory,andPaul)allseemtobelivinghappyandproductivelivesdespiteaseverelackofmicroeconomiceducation.Asthenextgeneration(Beth,Sarah,David,Sophia,andAbby)growsolder,perhapstheywillseekenlightenment—atleasttotheextentofwonderingwhatthebooksdedicatedtothemareallabout.WalterNicholsonAmherst,MassachusettsJune2007ItwasaprivilegetocollaboratewithWalteronthistenthedition.IusedthistextbookintheﬁrstcourseIevertaught,asagraduateinstructoratMIT,andIhaveenjoyedusingitinmymicroeconomicscoursesinthethirteenyearssince.Ihavealwaysappreciatedthetext’sambitiouscoverageoftheconceptsandmethodsusedbyprofessionaleconomistsaswellasitsaccessibilitytostudents,whichisenhancedbynumerouselegantexamplestogetherwithWalter’slucidprose.Itwasachallengetomaintainthishighstandardwithmycon-tribution—althoughthiswasmadeeasierbyWalter’ssuggestions,patience,andexample,forwhichIamgrateful.Iencourageteachersandstudentstoe-mailmewithanycommentsonthetext(Christopher.M.Snyder@dartmouth.edu).IwouldliketoaddmywholeheartedthankstothosewhomWalteracknowledgedforcontributingtothebook.IalsothankGretchenOttoandhercolleaguesatNewgen–AustinaswellasMattDarnellforcarefullycopyeditingmyportionoftherevision.IthankDartmouthCollegeforprovidingtheresourcesandenvironmentthatgreatlyfacilitatedwritingthebook.Ithankmycolleaguesintheeconomicsdepartmentforhelpfuldiscussionsandunderstanding.Committingtosuchanextensiveprojectisinsomesenseafamilydecision.Iamindebtedtomywife,Maura,foraccommodatingthemanylatenightsthatwererequiredandforlisteningtomymonotonousprogressreports.Ithankmydaughters,Clare,Tess,andMeg,fortheirgoodbehavior,whichexpeditedthewritingprocess.ChristopherSnyderHanover,NewHampshireJune2007xxiiPreface PART1IntroductionCHAPTER1EconomicModelsCHAPTER2MathematicsforMicroeconomicsThispartcontainsonlytwochapters.Chapter1examinesthegeneralphilosophyofhoweconomistsbuildmodelsofeconomicbehavior.Chapter2thenreviewssomeofthemathematicaltoolsusedintheconstructionofthesemodels.ThemathematicaltoolsfromChapter2willbeusedthroughouttheremainderofthisbook. 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CHAPTER1EconomicModelsThemaingoalofthisbookistointroduceyoutothemostimportantmodelsthateconomistsusetoexplainthebehaviorofconsumers,ﬁrms,andmarkets.Thesemodelsarecentraltothestudyofallareasofeconomics.Therefore,itisessentialtounderstandboththeneedforsuchmodelsandthebasicframeworkusedtodevelopthem.Thegoalofthischapteristobeginthisprocessbyoutliningsomeoftheconceptualissuesthatdeterminethewaysinwhicheconomistsstudypracticallyeveryquestionthatintereststhem.THEORETICALMODELSAmoderneconomyisacomplicatedentity.Thousandsofﬁrmsengageinproducingmillionsofdifferentgoods.Manymillionsofpeopleworkinallsortsofoccupationsandmakedecisionsaboutwhichofthesegoodstobuy.Let’susepeanutsasanexample.Peanutsmustbeharvestedattherighttimeandshippedtoprocessorswhoturnthemintopeanutbutter,peanutoil,peanutbrittle,andnumerousotherpeanutdelicacies.Theseprocessors,inturn,mustmakecertainthattheirproductsarriveatthousandsofretailoutletsintheproperquantitiestomeetdemand.Becauseitwouldbeimpossibletodescribethefeaturesofeventhesepeanutmarketsincompletedetail,economistshavechosentoabstractfromthecomplexitiesoftherealworldanddeveloprathersimplemodelsthatcapturethe“essentials.”Justasaroadmapishelpfuleventhoughitdoesnotrecordeveryhouseoreverystore,economicmodelsof,say,themarketforpeanutsarealsousefuleventhoughtheydonotrecordeveryminutefeatureofthepeanuteconomy.Inthisbookwewillstudythemostwidelyusedeconomicmodels.Wewillseethat,eventhoughthesemodelsoftenmakeheroicabstractionsfromthecomplexitiesoftherealworld,theynonethelesscaptureessentialfeaturesthatarecommontoalleconomicactivities.Theuseofmodelsiswidespreadinthephysicalandsocialsciences.Inphysics,thenotionofa“perfect”vacuumoran“ideal”gasisanabstractionthatpermitsscientiststostudyreal-worldphenomenainsimpliﬁedsettings.Inchemistry,theideaofanatomoramoleculeisactuallyasimpliﬁedmodelofthestructureofmatter.Architectsusemock-upmodelstoplanbuildings.Televisionrepairersrefertowiringdiagramstolocateproblems.Economists’modelsperformsimilarfunctions.Theyprovidesimpliﬁedportraitsofthewayindividualsmakedecisions,thewayﬁrmsbehave,andthewayinwhichthesetwogroupsinteracttoestablishmarkets.VERIFICATIONOFECONOMICMODELSOfcourse,notallmodelsprovetobe“good.”Forexample,theearth-centeredmodelofplanetarymotiondevisedbyPtolemywaseventuallydiscardedbecauseitprovedincapableofaccuratelyexplaininghowtheplanetsmovearoundthesun.Animportantpurposeofscientiﬁcinvestigationistosortoutthe“bad”modelsfromthe“good.”Twogeneralmethodshave3 beenusedforverifyingeconomicmodels:(1)adirectapproach,whichseekstoestablishthevalidityofthebasicassumptionsonwhichamodelisbased;and(2)anindirectapproach,whichattemptstoconﬁrmvaliditybyshowingthatasimpliﬁedmodelcorrectlypredictsreal-worldevents.Toillustratethebasicdifferencesbetweenthetwoapproaches,let’sbrieﬂyexamineamodelthatwewilluseextensivelyinlaterchaptersofthisbook—themodelofaﬁrmthatseekstomaximizeproﬁts.Theproﬁt-maximizationmodelThemodelofaﬁrmseekingtomaximizeproﬁtsisobviouslyasimpliﬁcationofreality.Itignoresthepersonalmotivationsoftheﬁrm’smanagersanddoesnotconsiderconﬂictsamongthem.Itassumesthatproﬁtsaretheonlyrelevantgoaloftheﬁrm;otherpossiblegoals,suchasobtainingpowerorprestige,aretreatedasunimportant.Themodelalsoassumesthattheﬁrmhassufﬁcientinformationaboutitscostsandthenatureofthemarkettowhichitsellstodiscoveritsproﬁt-maximizingoptions.Mostreal-worldﬁrms,ofcourse,donothavethisinformationreadilyavailable.Yet,suchshortcomingsinthemodelarenotnecessarilyserious.Nomodelcanexactlydescribereality.Therealquestioniswhetherthissimplemodelhasanyclaimtobeingagoodone.TestingassumptionsOnetestofthemodelofaproﬁt-maximizingﬁrminvestigatesitsbasicassumption:Doﬁrmsreallyseekmaximumproﬁts?Someeconomistshaveexaminedthisquestionbysendingques-tionnairestoexecutives,askingthemtospecifythegoalstheypursue.Theresultsofsuchstudieshavebeenvaried.Businesspeopleoftenmentiongoalsotherthanproﬁtsorclaimtheyonlydo“thebesttheycan”toincreaseproﬁtsgiventheirlimitedinformation.Ontheotherhand,mostrespondentsalsomentionastrong“interest”inproﬁtsandexpresstheviewthatproﬁtmaximizationisanappropriategoal.Testingtheproﬁt-maximizingmodelbytestingitsassumptionshasthereforeprovidedinconclusiveresults.TestingpredictionsSomeeconomists,mostnotablyMiltonFriedman,denythatamodelcanbetestedbyinquiringintothe“reality”ofitsassumptions.1Theyarguethatalltheoreticalmodelsarebasedon“unrealistic”assumptions;theverynatureoftheorizingdemandsthatwemakecertainab-stractions.Theseeconomistsconcludethattheonlywaytodeterminethevalidityofamodelistoseewhetheritiscapableofpredictingandexplainingreal-worldevents.Theultimatetestofaneconomicmodelcomeswhenitisconfrontedwithdatafromtheeconomyitself.Friedmanprovidesanimportantillustrationofthatprinciple.Heaskswhatkindofatheoryoneshouldusetoexplaintheshotsexpertpoolplayerswillmake.Hearguesthatthelawsofvelocity,momentum,andanglesfromtheoreticalphysicswouldbeasuitablemodel.Poolplayersshootshotsasiftheyfollowtheselaws.Butmostplayersaskedwhethertheypreciselyunderstandthephysicalprinciplesbehindthegameofpoolwillundoubtedlyanswerthattheydonot.Nonetheless,Friedmanargues,thephysicallawsprovideveryaccuratepredictionsandthereforeshouldbeacceptedasappropriatetheoreticalmodelsofhowexpertsplaypool.Atestoftheproﬁt-maximizationmodel,then,wouldbeprovidedbypredictingthebehaviorofreal-worldﬁrmsbyassumingthattheseﬁrmsbehaveasiftheyweremaximizingproﬁts.(SeeExample1.1laterinthischapter.)Ifthesepredictionsarereasonablyinaccordwithreality,wemayaccepttheproﬁt-maximizationhypothesis.However,wewouldreject1SeeM.Friedman,EssaysinPositiveEconomics(Chicago:UniversityofChicagoPress,1953),chap.1.Foranalternativeviewstressingtheimportanceofusing“realistic”assumptions,seeH.A.Simon,“RationalDecisionMakinginBusinessOrganizations,”AmericanEconomicReview69,no.4(September1979):493–513.4Part1Introduction themodelifreal-worlddataseeminconsistentwithit.Hence,theultimatetestofeithertheoryisitsabilitytopredictreal-worldevents.ImportanceofempiricalanalysisTheprimaryconcernofthisbookistheconstructionoftheoreticalmodels.Butthegoalofsuchmodelsisalwaystolearnsomethingabouttherealworld.Althoughtheinclusionofalengthysetofappliedexampleswouldneedlesslyexpandanalreadybulkybook,2theEx-tensionsincludedattheendofmanychaptersareintendedtoprovideatransitionbetweenthetheorypresentedhereandthewaysinwhichthattheoryisactuallyappliedinempiricalstudies.GENERALFEATURESOFECONOMICMODELSThenumberofeconomicmodelsincurrentuseis,ofcourse,verylarge.Speciﬁcassumptionsusedandthedegreeofdetailprovidedvarygreatlydependingontheproblembeingaddressed.ThemodelsemployedtoexplaintheoveralllevelofeconomicactivityintheUnitedStates,forexample,mustbeconsiderablymoreaggregatedandcomplexthanthosethatseektointerpretthepricingofArizonastrawberries.Despitethisvariety,however,practicallyalleconomicmodelsincorporatethreecommonelements:(1)theceterisparibus(otherthingsthesame)assumption;(2)thesuppositionthateconomicdecisionmakersseektooptimizesomething;and(3)acarefuldistinctionbetween“positive”and“normative”questions.Becausewewillencountertheseelementsthroughoutthisbook,itmaybehelpfulattheoutsettobrieﬂydescribethephilosophybehindeachofthem.TheceterisparibusassumptionAsinmostsciences,modelsusedineconomicsattempttoportrayrelativelysimplerela-tionships.Amodelofthemarketforwheat,forexample,mightseektoexplainwheatpriceswithasmallnumberofquantiﬁablevariables,suchaswagesoffarmworkers,rainfall,andconsumerincomes.Thisparsimonyinmodelspeciﬁcationpermitsthestudyofwheatpricinginasimpliﬁedsettinginwhichitispossibletounderstandhowthespeciﬁcforcesoperate.Althoughanyresearcherwillrecognizethatmany“outside”forces(presenceofwheatdiseases,changesinthepricesoffertilizersoroftractors,orshiftsinconsumerattitudesabouteatingbread)affectthepriceofwheat,theseotherforcesareheldconstantintheconstructionofthemodel.Itisimportanttorecognizethateconomistsarenotassumingthatotherfactorsdonotaffectwheatprices;rather,suchothervariablesareassumedtobeunchangedduringtheperiodofstudy.Inthisway,theeffectofonlyafewforcescanbestudiedinasimpliﬁedsetting.Suchceterisparibus(otherthingsequal)assumptionsareusedinalleconomicmodeling.Useoftheceterisparibusassumptiondoesposesomedifﬁcultiesfortheveriﬁcationofeconomicmodelsfromreal-worlddata.Inothersciences,suchproblemsmaynotbesoseverebecauseoftheabilitytoconductcontrolledexperiments.Forexample,aphysicistwhowishestotestamodeloftheforceofgravityprobablywouldnotdosobydroppingobjectsfromtheEmpireStateBuilding.Experimentsconductedinthatwaywouldbesubjecttotoomanyextraneousforces(windcurrents,particlesintheair,variationsintemperature,andsoforth)topermitaprecisetestofthetheory.Rather,thephysicistwouldconductexperimentsinalaboratory,usingapartialvacuuminwhichmostotherforcescouldbecontrolledorelim-inated.Inthisway,thetheorycouldbeveriﬁedinasimplesetting,withoutconsideringalltheotherforcesthataffectfallingbodiesintherealworld.2Foranintermediate-leveltextcontaininganextensivesetofreal-worldapplications,seeW.NicholsonandC.Snyder,IntermediateMicroeconomicsandItsApplication,10thed.(Mason,OH:Thomson/Southwestern,2007).Chapter1EconomicModels5 Withafewnotableexceptions,economistshavenotbeenabletoconductcontrolledexperimentstotesttheirmodels.Instead,economistshavebeenforcedtorelyonvariousstatisticalmethodstocontrolforotherforceswhentestingtheirtheories.Althoughthesestatisticalmethodsareasvalidinprincipleasthecontrolledexperimentmethodsusedbyotherscientists,inpracticetheyraiseanumberofthornyissues.Forthatreason,thelimitationsandprecisemeaningoftheceterisparibusassumptionineconomicsaresubjecttogreatercon-troversythaninthelaboratorysciences.OptimizationassumptionsManyeconomicmodelsstartfromtheassumptionthattheeconomicactorsbeingstudiedarerationallypursuingsomegoal.Webrieﬂydiscussedsuchanassumptionwheninvestigatingthenotionofﬁrmsmaximizingproﬁts.Example1.1showshowthatmodelcanbeusedtomaketestablepredictions.Otherexampleswewillencounterinthisbookincludeconsumersmaxi-mizingtheirownwell-being(utility),ﬁrmsminimizingcosts,andgovernmentregulatorsattemptingtomaximizepublicwelfare.Although,aswewillshow,alloftheseassumptionsareunrealistic,allhavewonwidespreadacceptanceasgoodstartingplacesfordevelopingeconomicmodels.Thereseemtobetworeasonsforthisacceptance.First,theoptimizationassumptionsareveryusefulforgeneratingprecise,solvablemodels,primarilybecausesuchmodelscandrawonavarietyofmathematicaltechniquessuitableforoptimizationproblems.Manyofthesetechniques,togetherwiththelogicbehindthem,arereviewedinChapter2.Asecondreasonforthepopularityofoptimizationmodelsconcernstheirapparentempiricalvalidity.AssomeofourExtensionsshow,suchmodelsseemtobefairlygoodatexplainingreality.Inall,then,opti-mizationmodelshavecometooccupyaprominentpositioninmoderneconomictheory.EXAMPLE1.1ProﬁtMaximizationTheproﬁt-maximizationhypothesisprovidesagoodillustrationofhowoptimizationas-sumptionscanbeusedtogenerateempiricallytestablepropositionsabouteconomicbehavior.Supposethataﬁrmcansellalltheoutputthatitwishesatapriceofpperunitandthatthetotalcostsofproduction,C,dependontheamountproduced,q.Then,proﬁtsaregivenbyprofits¼π¼pqCðqÞ:(1:1)Maximizationofproﬁtsconsistsofﬁndingthatvalueofqwhichmaximizestheproﬁtex-pressioninEquation1.1.Thisisasimpleproblemincalculus.DifferentiationofEquation1.1andsettingthatderivativeequalto0givethefollowingﬁrst-orderconditionforamaximum:dπdq¼pC0ðqÞ¼0orp¼C0ðqÞ:(1:2)Inwords,theproﬁt-maximizingoutputlevel(q)isfoundbyselectingthatoutputlevelforwhichpriceisequaltomarginalcost,C0ðqÞ.Thisresultshouldbefamiliartoyoufromyourintroductoryeconomicscourse.Noticethatinthisderivationthepricefortheﬁrm’soutputistreatedasaconstantbecausetheﬁrmisapricetaker.Equation1.2isonlytheﬁrst-orderconditionforamaximum.Takingaccountofthesecond-orderconditioncanhelpustoderiveatestableimplicationofthismodel.Thesecond-orderconditionforamaximumisthatatqitmustbethecasethatd2πdq2¼C00ðqÞ<0orC00ðqÞ>0:(1:3)6Part1Introduction
Thatis,marginalcostmustbeincreasingatqforthistobeatruepointofmaximumproﬁts.Ourmodelcannowbeusedto“predict”howaﬁrmwillreacttoachangeinprice.Todoso,wedifferentiateEquation1.2withrespecttoprice(p),assumingthattheﬁrmcontinuestochooseaproﬁt-maximizinglevelofq:d½pC0ðqÞ¼0dp¼1C00ðqÞdqdp¼0:(1:4)Rearrangingtermsabitgivesdqdp¼1C00ðqÞ>0:(1:5)Heretheﬁnalinequalityagainreﬂectsthefactthatmarginalcostmustbeincreasingifqistobeatruemaximum.Thisthenisoneofthetestablepropositionsoftheproﬁt-maximizationhypothesis—ifotherthingsdonotchange,aprice-takingﬁrmshouldrespondtoanincreaseinpricebyincreasingoutput.Ontheotherhand,ifﬁrmsrespondtoincreasesinpricebyreducingoutput,theremustbesomethingwrongwithourmodel.Althoughthisisaverysimplemodel,itreﬂectsthewaywewillproceedthroughoutmuchofthisbook.Speciﬁcally,thefactthattheprimaryimplicationofthemodelisderivedbycalculus,andconsistsofshowingwhatsignaderivativeshouldhave,isthekindofresultwewillseemanytimes.QUERY:Ingeneralterms,howwouldtheimplicationsofthismodelbechangedifthepriceaﬁrmobtainsforitsoutputwereafunctionofhowmuchitsold?Thatis,howwouldthemodelworkiftheprice-takingassumptionwereabandoned?Positive-normativedistinctionAﬁnalfeatureofmosteconomicmodelsistheattempttodifferentiatecarefullybetween“positive”and“normative”questions.Sofarwehavebeenconcernedprimarilywithpositiveeconomictheories.Suchtheoriestaketherealworldasanobjecttobestudied,attemptingtoexplainthoseeconomicphenomenathatareobserved.Positiveeconomicsseekstodeterminehowresourcesareinfactallocatedinaneconomy.Asomewhatdifferentuseofeconomictheoryisnormativeanalysis,takingadeﬁnitestanceaboutwhatshouldbedone.Undertheheadingofnormativeanalysis,economistshaveagreatdealtosayabouthowresourcesshouldbeallocated.Forexample,aneconomistengagedinpositiveanalysismightinvestigatehowpricesaredeterminedintheU.S.health-careeconomy.Theeconomistalsomightwanttomeasurethecostsandbeneﬁtsofdevotingevenmoreresourcestohealthcare.Butwhenheorshespeciﬁcallyadvocatesthatmoreresourcesshouldbeallocatedtohealthcare,theanalysisbecomesnormative.Someeconomistsbelievethattheonlypropereconomicanalysisispositiveanalysis.Drawingananalogywiththephysicalsciences,theyarguethat“scientiﬁc”economicsshouldconcernitselfonlywiththedescription(andpossiblyprediction)ofreal-worldeconomicevents.Totakemoralpositionsandtopleadforspecialinterestsareconsideredtobeoutsidethecompetenceofaneconomistactingassuch.Othereconomists,however,believestrictapplicationofthepositive-normativedistinctiontoeconomicmattersisinappropriate.Theybelievethatthestudyofeconomicsnecessarilyinvolvestheresearchers’ownviewsaboutethics,morality,andfairness.Accordingtotheseeconomists,searchingforscientiﬁc“objectivity”insuchcircumstancesishopeless.Despitesomeambiguity,thisbookadoptsamainlypositivisttone,leavingnormativeconcernsforyoutodecideforyourself.Chapter1EconomicModels7
DEVELOPMENTOFTHEECONOMICTHEORYOFVALUEBecauseeconomicactivityhasbeenacentralfeatureofallsocieties,itissurprisingthattheseactivitieswerenotstudiedinanydetailuntilrecently.Forthemostpart,economicphenomenaweretreatedasabasicaspectofhumanbehaviorthatwasnotsufﬁcientlyinterestingtodeservespeciﬁcattention.Itis,ofcourse,truethatindividualshavealwaysstudiedeconomicactivitieswithaviewtowardmakingsomekindofpersonalgain.Romantraderswerenotabovemakingproﬁtsontheirtransactions.Butinvestigationsintothebasicnatureoftheseactivitiesdidnotbegininanydepthuntiltheeighteenthcentury.3Becausethisbookisabouteconomictheoryasitstandstoday,ratherthanthehistoryofeconomicthought,ourdiscussionoftheevolutionofeconomictheorywillbebrief.Onlyoneareaofeconomicstudywillbeexaminedinitshistoricalsetting:thetheoryofvalue.EarlyeconomicthoughtsonvalueThetheoryofvalue,notsurprisingly,concernsthedeterminantsofthe“value”ofacommodity.Thissubjectisatthecenterofmodernmicroeconomictheoryandiscloselyintertwinedwiththefundamentaleconomicproblemofallocatingscarceresourcestoalternativeuses.Thelogicalplacetostartiswithadeﬁnitionoftheword“value.”Unfortunately,themeaningofthistermhasnotbeenconsistentthroughoutthedevelopmentofthesubject.Todayweregardvalueasbeingsynonymouswiththepriceofacommodity.4Earlierphilosopher-economists,however,madeadistinctionbetweenthemarketpriceofacommodityanditsvalue.Theterm“value”wasthenthoughtofasbeing,insomesense,synonymouswith“importance,”“essentiality,”or(attimes)“godliness.”Because“price”and“value”wereseparateconcepts,theycoulddiffer,andmostearlyeconomicdiscussionscenteredonthesedivergences.Forexample,St.ThomasAquinasbelievedvaluetobedivinelydetermined.Sincepricesweresetbyhumans,itwaspossibleforthepriceofacommoditytodifferfromitsvalue.Apersonaccusedofchargingapriceinexcessofagood’svaluewasguiltyofchargingan“unjust”price.Forexample,St.Thomasbelievedthe“just”rateofinteresttobezero.Anylenderwhodemandedapaymentfortheuseofmoneywascharginganunjustpriceandcouldbe—andsometimeswas—prosecutedbychurchofﬁcials.ThefoundingofmoderneconomicsDuringthelatterpartoftheeighteenthcentury,philosophersbegantotakeamorescientiﬁcapproachtoeconomicquestions.The1776publicationofTheWealthofNationsbyAdamSmith(1723–1790)isgenerallyconsideredthebeginningofmoderneconomics.Inhisvast,all-encompassingwork,Smithlaidthefoundationforthinkingaboutmarketforcesinanorderedandsystematicway.Still,Smithandhisimmediatesuccessors,suchasDavidRicardo(1772–1823),continuedtodistinguishbetweenvalueandprice.ToSmith,forexample,thevalueofacommoditymeantits“valueinuse,”whereasthepricerepresentedits“valueinexchange.”Thedistinctionbetweenthesetwoconceptswasillustratedbythefamouswater-diamondparadox.Water,whichobviouslyhasgreatvalueinuse,haslittlevalueinexchange(ithasalowprice);diamondsareoflittlepracticalusebuthaveagreatvalueinexchange.Theparadoxwithwhichearlyeconomistsstruggledderivesfromtheobservationthatsomeveryusefulitemshavelowpriceswhereascertainnonessentialitemshavehighprices.3Foradetailedtreatmentofearlyeconomicthought,seetheclassicworkbyJ.A.Schumpeter,HistoryofEconomicAnalysis(NewYork:OxfordUniversityPress,1954),pt.II,chaps.1–3.4Thisisnotcompletelytruewhen“externalities”areinvolvedandadistinctionmustbemadebetweenprivateandsocialvalue(seeChapter19).8Part1Introduction
LabortheoryofexchangevalueNeitherSmithnorRicardoeversatisfactorilyresolvedthewater-diamondparadox.Thecon-ceptofvalueinusewasleftforphilosopherstodebate,whileeconomiststurnedtheirattentiontoexplainingthedeterminantsofvalueinexchange(thatis,toexplainingrelativeprices).Oneobviouspossibleexplanationisthatexchangevaluesofgoodsaredeterminedbywhatitcoststoproducethem.Costsofproductionareprimarilyinﬂuencedbylaborcosts—atleastthiswassointhetimeofSmithandRicardo—andthereforeitwasashortsteptoembracealabortheoryofvalue.Forexample,toparaphraseanexamplefromSmith,ifcatchingadeertakestwicethenumberoflaborhoursascatchingabeaver,thenonedeershouldexchangefortwobeavers.Inotherwords,thepriceofadeershouldbetwicethatofabeaver.Similarly,diamondsarerelativelycostlybecausetheirproductionrequiressubstantiallaborinput.Tostudentswithevenapassingknowledgeofwhatwenowcallthelawofsupplyanddemand,Smith’sandRicardo’sexplanationmustseemincomplete.Didn’ttheyrecognizetheeffectsofdemandonprice?Theanswertothisquestionisbothyesandno.Theydidobserveperiodsofrapidlyrisingandfallingrelativepricesandattributedsuchchangestodemandshifts.However,theyregardedthesechangesasabnormalitiesthatproducedonlyatemporarydivergenceofmarketpricefromlaborvalue.Becausetheyhadnotreallydevelopedatheoryofvalueinuse,theywereunwillingtoassigndemandanymorethanatransientroleindeter-miningrelativeprices.Rather,long-runexchangevalueswereassumedtobedeterminedsolelybylaborcostsofproduction.ThemarginalistrevolutionBetween1850and1880,economistsbecameincreasinglyawarethattoconstructanadequatealternativetothelabortheoryofvalue,theyhadtocometodeviseatheoryofvalueinuse.Duringthe1870s,severaleconomistsdiscoveredthatitisnotthetotalusefulnessofacommoditythathelpstodetermineitsexchangevalue,butrathertheusefulnessofthelastunitconsumed.Forexample,wateriscertainlyveryuseful—itisnecessaryforalllife.But,becausewaterisrelativelyplentiful,consumingonemorepint(ceterisparibus)hasarelativelylowvaluetopeople.These“marginalists”redeﬁnedtheconceptofvalueinusefromanideaofoverallusefulnesstooneofmarginal,orincremental,usefulness—theusefulnessofanadditionalunitofacommodity.TheconceptofthedemandforanincrementalunitofoutputwasnowcontrastedtoSmith’sandRicardo’sanalysisofproductioncoststoderiveacomprehensivepictureofpricedetermination.5Marshalliansupply-demandsynthesisThecleareststatementofthesemarginalprincipleswaspresentedbytheEnglisheconomistAlfredMarshall(1842–1924)inhisPrinciplesofEconomics,publishedin1890.Marshallshowedthatdemandandsupplysimultaneouslyoperatetodetermineprice.AsMarshallnoted,justasyoucannottellwhichbladeofascissorsdoesthecutting,sotooyoucannotsaythateitherdemandorsupplyalonedeterminesprice.ThatanalysisisillustratedbythefamousMarshalliancrossshowninFigure1.1.Inthediagramthequantityofagoodpurchasedperperiodisshownonthehorizontalaxisanditspriceappearsontheverticalaxis.ThecurveDDrepresentsthequantityofthegooddemandedperperiodateachpossibleprice.Thecurveisnegativelyslopedtoreﬂectthemarginalistprinciplethatasquantityincreases,peopleare5Ricardohadearlierprovidedanimportantﬁrststepinmarginalanalysisinhisdiscussionofrent.Ricardotheorizedthatastheproductionofcornincreased,landofinferiorqualitywouldbeusedandthiswouldcausethepriceofcorntorise.InhisargumentRicardoimplicitlyrecognizedthatitisthemarginalcost—thecostofproducinganadditionalunit—thatisrelevanttopricing.NoticethatRicardoimplicitlyheldotherinputsconstantwhendiscussingdiminishinglandproductivity;thatis,heemployedoneversionoftheceterisparibusassumption.Chapter1EconomicModels9
willingtopaylessforthelastunitpurchased.Itisthevalueofthislastunitthatsetsthepriceforallunitspurchased.ThecurveSSshowshow(marginal)productioncostsriseasmoreoutputisproduced.Thisreﬂectstheincreasingcostofproducingonemoreunitastotaloutputexpands.Inotherwords,theupwardslopeoftheSScurvereﬂectsincreasingmarginalcosts,justasthedownwardslopeoftheDDcurvereﬂectsdecreasingmarginalvalue.Thetwocurvesintersectatp,q.Thisisanequilibriumpoint—bothbuyersandsellersarecontentwiththequantitybeingtradedandthepriceatwhichitistraded.Ifoneofthecurvesshouldshift,theequilibriumpointwouldshifttoanewlocation.Thuspriceandquantityaresimultaneouslydeterminedbythejointoperationofsupplyanddemand.ParadoxresolvedMarshall’smodelresolvesthewater-diamondparadox.Pricesreﬂectboththemarginaleval-uationthatdemandersplaceongoodsandthemarginalcostsofproducingthegoods.Viewedinthisway,thereisnoparadox.Waterislowinpricebecauseithasbothalowmarginalvalueandalowmarginalcostofproduction.Ontheotherhand,diamondsarehighinpricebecausetheyhavebothahighmarginalvalue(becausepeoplearewillingtopayquiteabitforonemore)andahighmarginalcostofproduction.Thisbasicmodelofsupplyanddemandliesbehindmuchoftheanalysispresentedinthisbook.GeneralequilibriummodelsAlthoughtheMarshallianmodelisanextremelyusefulandversatiletool,itisapartialequilibriummodel,lookingatonlyonemarketatatime.Forsomequestions,thisnarrowingofperspectivegivesvaluableinsightsandanalyticalsimplicity.Forother,broaderquestions,suchanarrowviewpointmaypreventthediscoveryofimportantrelationshipsamongmarkets.Toanswermoregeneralquestionswemusthaveamodelofthewholeeconomythatsuitablymirrorstheconnectionsamongvariousmarketsandeconomicagents.TheFrencheconomistLeonWalras(1831–1910),buildingonalongContinentaltraditioninsuchanalysis,createdthebasisformoderninvestigationsintothosebroadquestions.HismethodofrepresentingtheFIGURE1.1TheMarshallianSupply-DemandCrossMarshalltheorizedthatdemandandsupplyinteracttodeterminetheequilibriumprice(p)andthequantity(q)thatwillbetradedinthemarket.Heconcludedthatitisnotpossibletosaythateitherdemandorsupplyalonedeterminespriceorthereforethateithercostsorusefulnesstobuyersalonedeterminesexchangevalue.Quantity per periodPriceSSDDq*p*10Part1Introduction
economybyalargenumberofsimultaneousequationsformsthebasisforunderstandingtheinterrelationshipsimplicitingeneralequilibriumanalysis.Walrasrecognizedthatonecannottalkaboutasinglemarketinisolation;whatisneededisamodelthatpermitstheeffectsofachangeinonemarkettobefollowedthroughothermarkets.EXAMPLE1.2Supply-DemandEquilibriumAlthoughgraphicalpresentationsareadequateforsomepurposes,economistsoftenusealgebraicrepresentationsoftheirmodelstobothclarifytheirargumentsandmakethemmoreprecise.Asanelementaryexample,supposewewishedtostudythemarketforpeanutsand,onthebasisofstatisticalanalysisofhistoricaldata,concludedthatthequantityofpeanutsdemandedeachweek(q,measuredinbushels)dependedonthepriceofpeanuts(p,measuredindollarsperbushel)accordingtotheequationquantitydemanded¼qD¼1,000100p:(1:6)BecausethisequationforqDcontainsonlythesingleindependentvariablep,weareimplicitlyholdingconstantallotherfactorsthatmightaffectthedemandforpeanuts.Equation1.6indicatesthat,ifotherthingsdonotchange,atapriceof$5perbushelpeoplewilldemand500bushelsofpeanuts,whereasatapriceof$4perbusheltheywilldemand600bushels.ThenegativecoefﬁcientforpinEquation1.6reﬂectsthemarginalistprinciplethatalowerpricewillcausepeopletobuymorepeanuts.Tocompletethissimplemodelofpricing,supposethatthequantityofpeanutssuppliedalsodependsonprice:quantitysupplied¼qS¼125þ125p:(1:7)Herethepositivecoefﬁcientofpricealsoreﬂectsthemarginalprinciplethatahigherpricewillcallforthincreasedsupply—primarilybecause(aswesawinExample1.1)itpermitsﬁrmstoincurhighermarginalcostsofproductionwithoutincurringlossesontheadditionalunitsproduced.Equilibriumpricedetermination.Equation1.6and1.7thereforereﬂectourmodelofpricedeterminationinthemarketforpeanuts.Anequilibriumpricecanbefoundbysettingquantitydemandedequaltoquantitysupplied:qD¼qS(1:8)or1,000100p¼125þ125p(1:9)or225p¼1,125,(1:10)sop¼5:(1:11)Atapriceof$5perbushel,thismarketisinequilibrium:atthispricepeoplewanttopurchase500bushels,andthatisexactlywhatpeanutproducersarewillingtosupply.ThisequilibriumispicturedgraphicallyastheintersectionofDandSinFigure1.2.Amoregeneralmodel.Inordertoillustratehowthissupply-demandmodelmightbeused,let’sadoptamoregeneralnotation.Supposenowthatthedemandandsupplyfunctionsaregivenby(continued)Chapter1EconomicModels11
EXAMPLE1.2CONTINUEDFIGURE1.2ChangingSupply-DemandEquilibriaTheinitialsupply-demandequilibriumisillustratedbytheintersectionofDandS(p¼5,q¼500).WhendemandshiftstoqD0¼1,450100p(denotedasD0),theequilibriumshiftstop¼7,q¼750.0Quantity perperiod (bushels)Price($)SSD′D′DD14.5107550075010001450qD¼aþbpandqS¼cþdp(1:12)whereaandcareconstantsthatcanbeusedtoshiftthedemandandsupplycurves,respectively,andb(<0)andd(>0)representdemanders’andsuppliers’reactionstoprice.EquilibriuminthismarketrequiresqD¼qSoraþbp¼cþdp:(1:13)So,equilibriumpriceisgivenby6p¼acdb:(1:14)6Equation1.14issometimescalledthe“reducedform”forthesupply-demandstructuralmodelofEquations1.12and1.13.Itshowsthattheequilibriumvaluefortheendogenousvariablepultimatelydependsonlyontheexogenousfactorsinthemodel(aandc)andonthebehavioralparametersbandd.Asimilarequationcanbecalculatedforequilibriumquantity.12Part1Introduction
Noticethat,inourpriorexample,a¼1,000,b¼100,c¼125,andd¼125,sop¼1,000þ125125þ100¼1,125225¼5:(1:15)Withthismoregeneralformulation,however,wecanposequestionsabouthowtheequi-libriumpricemightchangeifeitherthedemandorsupplycurveshifted.Forexample,differentiationofEquation1.14showsthatdpda¼1db>0,dpdc¼1db<0:(1:16)Thatis,anincreaseindemand(anincreaseina)increasesequilibriumpricewhereasanin-creaseinsupply(anincreaseinc)reducesprice.Thisisexactlywhatagraphicalanalysisofsupplyanddemandcurveswouldshow.Forexample,Figure1.2showsthatwhenthecon-stantterm,a,inthedemandequationincreasesto1450,equilibriumpriceincreasestop¼7½¼ð1,450þ125Þ=225.QUERY:HowmightyouuseEquation1.16to“predict”howeachunitincreaseintheconstantaaffectsp?Doesthisequationcorrectlypredicttheincreaseinpwhentheconstantaincreasesfrom1,000to1,450?Forexample,supposethatthedemandforpeanutsweretoincrease.Thiswouldcausethepriceofpeanutstoincrease.Marshalliananalysiswouldseektounderstandthesizeofthisincreasebylookingatconditionsofsupplyanddemandinthepeanutmarket.Generalequilibriumanalysiswouldlooknotonlyatthatmarketbutalsoatrepercussionsinothermarkets.Ariseinthepriceofpeanutswouldincreasecostsforpeanutbuttermakers,whichwould,inturn,affectthesupplycurveforpeanutbutter.Similarly,therisingpriceofpeanutsmightmeanhigherlandpricesforpeanutfarmers,whichwouldaffectthedemandcurvesforallproductsthattheybuy.Thedemandcurvesforautomobiles,furniture,andtripstoEuropewouldallshiftout,andthatmightcreateadditionalincomesfortheprovidersofthoseproducts.Consequently,theeffectsoftheinitialincreaseindemandforpeanutseventuallywouldspreadthroughouttheeconomy.Generalequilibriumanalysisattemptstodevelopmodelsthatpermitustoexaminesucheffectsinasimpliﬁedsetting.SeveralmodelsofthistypearedescribedinChapter13.ProductionpossibilityfrontierHerewebrieﬂyintroducesomegeneralequilibriumideasbyusinganothergraphyoushouldrememberfromintroductoryeconomics—theproductionpossibilityfrontier.Thisgraphshowsthevariousamountsoftwogoodsthataneconomycanproduceusingitsavailableresourcesduringsomeperiod(say,oneweek).Becausetheproductionpossibilityfrontiershowstwogoods,ratherthanthesinglegoodinMarshall’smodel,itisusedasabasicbuildingblockforgeneralequilibriummodels.Figure1.3showstheproductionpossibilityfrontierfortwogoods,foodandclothing.Thegraphillustratesthesupplyofthesegoodsbyshowingthecombinationsthatcanbeproducedwiththiseconomy’sresources.Forexample,10poundsoffoodand3unitsofclothingcouldbeproduced,or4poundsoffoodand12unitsofclothing.Manyothercombinationsoffoodandclothingcouldalsobeproduced.Theproductionpossibilityfrontiershowsallofthem.Combinationsoffoodandclothingoutsidethefrontiercannotbeproducedbecausenotenoughresourcesareavailable.TheproductionpossibilityfrontierChapter1EconomicModels13 remindsusofthebasiceconomicfactthatresourcesarescarce—therearenotenoughresourcesavailabletoproduceallwemightwantofeverygood.Thisscarcitymeansthatwemustchoosehowmuchofeachgoodtoproduce.Figure1.3makesclearthateachchoicehasitscosts.Forexample,ifthiseconomyproduces10poundsoffoodand3unitsofclothingatpointA,producing1moreunitofclothingwould“cost”12poundoffood—increasingtheoutputofclothingby1unitmeanstheproductionoffoodwouldhavetodecreaseby12pound.So,theopportunitycostof1unitofclothingatpointAis12poundoffood.Ontheotherhand,iftheeconomyinitiallyproduces4poundsoffoodand12unitsofclothingatpointB,itwouldcost2poundsoffoodtoproduce1moreunitofclothing.Theopportunitycostof1moreunitofclothingatpointBhasincreasedto2poundsoffood.BecausemoreunitsofclothingareproducedatpointBthanatpointA,bothRicardo’sandMarshall’sideasofincreasingincrementalcostssuggestthattheopportunitycostofanadditionalunitofclothingwillbehigheratpointBthanatpointA.ThiseffectisshownbyFigure1.3.TheproductionpossibilityfrontierprovidestwogeneralequilibriuminsightsthatarenotclearinMarshall’ssupplyanddemandmodelofasinglemarket.First,thegraphshowsthatproducingmoreofonegoodmeansproducinglessofanothergoodbecauseresourcesarescarce.Economistsoften(perhapstoooften!)usetheexpression“thereisnosuchthingasafreelunch”toexplainthateveryeconomicactionhasopportunitycosts.Second,theproductionpossibilityfrontiershowsthatopportunitycostsdependonhowmuchofeachgoodisproduced.Thefrontierislikeasupplycurvefortwogoods:itshowstheopportunitycostofproducingmoreofonegoodasthedecreaseintheamountofthesecondgood.Theproductionpossibilityfrontieristhereforeaparticularlyusefultoolforstudyingseveralmarketsatthesametime.FIGURE1.3ProductionPossibilityFrontierTheproductionpossibilityfrontiershowsthedifferentcombinationsoftwogoodsthatcanbeproducedfromacertainamountofscarceresources.Italsoshowstheopportunitycostofproducingmoreofonegoodastheamountoftheothergoodthatcannotthenbeproduced.TheopportunitycostattwodifferentlevelsofclothingproductioncanbeseenbycomparingpointsAandB.Quantityof foodper weekBA0249.510341213Quantityof clothingper weekOpportunity cost ofclothing = 2 poundsof foodOpportunity cost ofclothing = pound of food1214Part1Introduction EXAMPLE1.3TheProductionPossibilityFrontierandEconomicIneﬃciencyGeneralequilibriummodelsaregoodtoolsforevaluatingtheefﬁciencyofvariouseconomicarrangements.AswewillseeinChapter13,suchmodelshavebeenusedtoassessawidevarietyofpoliciessuchastradeagreements,taxstructures,andenvironmentalregulation.Inthissimpleexample,weexploretheideaofefﬁciencyinitsmostelementaryform.Supposethataneconomyproducestwogoods,xandy,usinglaborastheonlyinput.Theproductionfunctionforgoodxisx¼l0:5x(wherelxisthequantityoflaborusedinxproduction)andtheproductionfunctionforgoodyisy¼2l0:5y.Totallaboravailableisconstrainedbylxþly200.Constructionoftheproductionpossibilityfrontierinthiseconomyisextremelysimple:lxþly¼x2þ0:25y2200(1:17)iftheeconomyistobeproducingasmuchaspossible(which,afterall,iswhyit’scalleda“frontier”).Equation1.17showsthatthefrontierherehastheshapeofaquarterellipse—itsconcavityderivesfromthediminishingreturnsexhibitedbyeachproductionfunction.Opportunitycost.Assumingthiseconomyisonthefrontier,theopportunitycostofgoodyintermsofgoodxcanbederivedbysolvingforyasy2¼8004x2ory¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ8004x2p¼½8004x20:5(1:18)andthendifferentiatingthisexpression:dydx¼0:5½8004x20:5ð8xÞ¼4xy:(1:19)Suppose,forexample,laborisequallyallocatedbetweenthetwogoods.Thenx¼10,y¼20,anddy=dx¼4ð10Þ=20¼2.Withthisallocationoflabor,eachunitincreaseinxoutputwouldrequireareductioninyof2units.Thiscanbeveriﬁedbyconsideringaslightlydifferentallocation,lx¼101andly¼99.Nowproductionisx¼10:05andy¼19:9.MovingtothisalternativeallocationwouldhaveDyDx¼ð19:920Þð10:0510Þ¼0:10:05¼2,whichispreciselywhatwasderivedfromthecalculusapproach.Concavity.Equation1.19clearlyillustratestheconcavityoftheproductionpossibilityfrontier.Theslopeofthefrontierbecomessteeper(morenegative)asxoutputincreasesandyoutputfalls.Forexample,iflaborisallocatedsothatlx¼144andly¼56,thenoutputsarex¼12andy15andsody=dx¼4ð12Þ=15¼3:2.Withexpandedxproduction,theopportunitycostofonemoreunitofxincreasesfrom2to3.2unitsofy.Ineﬃciency.Ifaneconomyoperatesinsideitsproductionpossibilityfrontier,itisoperatinginefﬁciently.Movingoutwardtothefrontiercouldincreasetheoutputofbothgoods.Inthisbookwewillexploremanyreasonsforsuchinefﬁciency.Theseusuallyderivefromafailureofsomemarkettoperformcorrectly.Forthepurposesofthisillustration,let’sassumethatthelabormarketinthiseconomydoesnotworkwellandthat20workersarepermanentlyunemployed.Nowtheproductionpossibilityfrontierbecomesx2þ0:25y2¼180,(1:20)(continued)Chapter1EconomicModels15 EXAMPLE1.3CONTINUEDandtheoutputcombinationswedescribedpreviouslyarenolongerfeasible.Forexample,ifx¼10thenyoutputisnowy17:9.Thelossofabout2.1unitsofyisameasureofthecostofthelabormarketinefﬁciency.Alternatively,ifthelaborsupplyof180wereallocatedevenlybetweentheproductionofthetwogoodsthenwewouldhavex9:5andy19,andtheinefﬁciencywouldshowupinbothgoods’production—moreofbothgoodscouldbepro-ducedifthelabormarketinefﬁciencywereresolved.QUERY:Howwouldtheinefﬁciencycostoflabormarketimperfectionsbemeasuredsolelyintermsofxproductioninthismodel?Howwoulditbemeasuredsolelyintermsofyproduction?Whatwouldyouneedtoknowinordertoassignasinglenumbertotheefﬁciencycostoftheimperfectionwhenlaborisequallyallocatedtothetwogoods?WelfareeconomicsInadditiontotheiruseinexaminingpositivequestionsabouthowtheeconomyoperates,thetoolsusedingeneralequilibriumanalysishavealsobeenappliedtothestudyofnormativequestionsaboutthewelfarepropertiesofvariouseconomicarrangements.Althoughsuchquestionswereamajorfocusofthegreateighteenth-andnineteenth-centuryeconomists(Smith,Ricardo,Marx,Marshall,andsoforth),perhapsthemostsigniﬁcantadvancesintheirstudyweremadebytheBritisheconomistFrancisY.Edgeworth(1848–1926)andtheItalianeconomistVilfredoPareto(1848–1923)intheearlyyearsofthetwentiethcentury.Theseeconomistshelpedtoprovideaprecisedeﬁnitionfortheconceptof“economicefﬁciency”andtodemonstratetheconditionsunderwhichmarketswillbeabletoachievethatgoal.Byclarifyingtherelationshipbetweentheallocationpricingofresources,theyprovidedsomesupportfortheidea,ﬁrstenunciatedbyAdamSmith,thatproperlyfunctioningmarketsprovidean“invisiblehand”thathelpsallocateresourcesefﬁciently.Latersectionsofthisbookfocusonsomeofthesewelfareissues.MODERNDEVELOPMENTSResearchactivityineconomicsexpandedrapidlyintheyearsfollowingWorldWarII.Amajorpurposeofthisbookistosummarizemuchofthisresearch.Byillustratinghoweconomistshavetriedtodevelopmodelstoexplainincreasinglycomplexaspectsofeconomicbehavior,thisbookseekstohelpyourecognizesomeoftheremainingunansweredquestions.ThemathematicalfoundationsofeconomicmodelsAmajorpostwardevelopmentinmicroeconomictheorywastheclariﬁcationandformalizationofthebasicassumptionsthataremadeaboutindividualsandﬁrms.Theﬁrstlandmarkinthisdevelopmentwasthe1947publicationofPaulSamuelson’sFoundationsofEconomicAnalysis,inwhichtheauthor(theﬁrstAmericanNobelPrizewinnerineconomics)laidoutanumberofmodelsofoptimizingbehavior.7Samuelsondemonstratedtheimportanceofbasingbehav-ioralmodelsonwell-speciﬁedmathematicalpostulatessothatvariousoptimizationtechniquesfrommathematicscouldbeapplied.Thepowerofhisapproachmadeitinescapablyclearthatmathematicshadbecomeanintegralpartofmoderneconomics.InChapter2ofthisbookwereviewsomeofthemathematicalconceptsmostoftenusedinmicroeconomics.7PaulA.Samuelson,FoundationsofEconomicAnalysis(Cambridge,MA:HarvardUniversityPress,1947).16Part1Introduction NewtoolsforstudyingmarketsAsecondfeaturethathasbeenincorporatedintothisbookisthepresentationofanumberofnewtoolsforexplainingmarketequilibria.Theseincludetechniquesfordescribingpricinginsinglemarkets,suchasincreasinglysophisticatedmodelsofmonopolisticpricingormodelsofthestrategicrelationshipsamongﬁrmsthatusegametheory.Theyalsoincludegeneralequilibriumtoolsforsimultaneouslyexploringrelationshipsamongmanymarkets.Asweshallsee,allofthesenewtechniqueshelptoprovideamorecompleteandrealisticpictureofhowmarketsoperate.TheeconomicsofuncertaintyandinformationAﬁnalmajortheoreticaladvanceduringthepostwarperiodwastheincorporationofuncertaintyandimperfectinformationintoeconomicmodels.Someofthebasicassumptionsusedtostudybehaviorinuncertainsituationswereoriginallydevelopedinthe1940sinconnectionwiththetheoryofgames.Laterdevelopmentsshowedhowtheseideascouldbeusedtoexplainwhyindividualstendtobeadversetoriskandhowtheymightgatherinformationinordertoreducetheuncertaintiestheyface.Inthisbook,problemsofuncertaintyandinformationentertheanalysisonmanyoccasions.ComputersandempiricalanalysisOneﬁnalaspectofthepostwardevelopmentofmicroeconomicsshouldbementioned—theincreasinguseofcomputerstoanalyzeeconomicdataandbuildeconomicmodels.Ascomputershavebecomeabletohandlelargeramountsofinformationandcarryoutcomplexmathematicalmanipulations,economists’abilitytotesttheirtheorieshasdramaticallyimproved.Whereaspreviousgenerationshadtobecontentwithrudimentarytabularorgraphicalanalysesofreal-worlddata,today’seconomistshaveavailableawidevarietyofsophisticatedtechniquestogetherwithextensivemicroeconomicdatawithwhichtotesttheirmodels.Toexaminethesetech-niquesandsomeoftheirlimitationswouldbebeyondthescopeandpurposeofthisbook.But,Extensionsattheendofmostchaptersareintendedtohelpyoustartreadingaboutsomeoftheseapplications.SUMMARYThischapterprovidedbackgroundonhoweconomistsap-proachthestudyoftheallocationofresources.Muchofthematerialdiscussedhereshouldbefamiliartoyoufromintro-ductoryeconomics.Inmanyrespects,thestudyofeconomicsrepresentsacquiringincreasinglysophisticatedtoolsforad-dressingthesamebasicproblems.Thepurposeofthisbook(and,indeed,ofmostupper-levelbooksoneconomics)istoprovideyouwithmoreofthesetools.Asastartingplace,thischapterremindedyouofthefollowingpoints:•Economicsisthestudyofhowscarceresourcesareal-locatedamongalternativeuses.Economistsseektodevelopsimplemodelstohelpunderstandthatprocess.Manyofthesemodelshaveamathematicalbasisbe-causetheuseofmathematicsoffersapreciseshorthandforstatingthemodelsandexploringtheirconsequences.•Themostcommonlyusedeconomicmodelisthesupply-demandmodelﬁrstthoroughlydevelopedbyAlfredMarshallinthelatterpartofthenineteenthcentury.Thismodelshowshowobservedpricescanbetakentorepresentanequilibriumbalancingoftheproductioncostsincurredbyﬁrmsandthewillingnessofdemanderstopayforthosecosts.•Marshall’smodelofequilibriumisonly“partial”—thatis,itlooksonlyatonemarketatatime.Tolookatmanymarketstogetherrequiresanexpandedsetofgeneralequilibriumtools.•Testingthevalidityofaneconomicmodelisperhapsthemostdifﬁculttaskeconomistsface.Occasionally,amodel’svaliditycanbeappraisedbyaskingwhetheritisbasedon“reasonable”assumptions.Moreoften,however,modelsarejudgedbyhowwelltheycanexplaineconomiceventsintherealworld.Chapter1EconomicModels17 SUGGESTIONSFORFURTHERREADINGOnMethodologyBlaug,Mark,andJohnPencavel.TheMethodologyofEconom-ics:OrHowEconomistsExplain,2nded.Cambridge:Cam-bridgeUniversityPress,1992.Arevisedandexpandedversionofaclassicstudyoneconomicmeth-odology.Tiesthediscussiontomoregeneralissuesinthephilosophyofscience.Boland,LawrenceE.“ACritiqueofFriedman’sCritics.”JournalofEconomicLiterature(June1979):503–22.Goodsummaryofcriticismsofpositiveapproachestoeconomicsandoftheroleofempiricalveriﬁcationofassumptions.Friedman,Milton.“TheMethodologyofPositiveEconom-ics.”InEssaysinPositiveEconomics,pp.3–43.Chicago:UniversityofChicagoPress,1953.BasicstatementofFriedman’spositivistviews.Harrod,RoyF.“ScopeandMethodinEconomics.”EconomicJournal48(1938):383–412.Classicstatementofappropriateroleforeconomicmodeling.Hausman,DavidM.,andMichaelS.McPherson.EconomicAnalysis,MoralPhilosophy,andPublicPolicy,2nded.Cambridge:CambridgeUniversityPress,2006.Theauthorsstresstheirbeliefthatconsiderationofissuesinmoralphilosophycanimproveeconomicanalysis.McCloskey,DonaldN.IfYou’reSoSmart:TheNarrativeofEconomicExpertise.Chicago:UniversityofChicagoPress,1990.DiscussionofMcCloskey’sviewthateconomicpersuasiondependsonrhetoricasmuchasonscience.Foraninterchangeonthistopic,seealsothearticlesintheJournalofEconomicLiterature,June1995.Sen,Amartya.OnEthicsandEconomics.Oxford:BlackwellReprints,1989.Theauthorseekstobridgethegapbetweeneconomicsandethicalstudies.Thisisareprintofaclassicstudyonthistopic.PrimarySourcesontheHistoryofEconomicsEdgeworth,F.Y.MathematicalPsychics.London:KeganPaul,1881.Initialinvestigationsofwelfareeconomics,includingrudimentarynotionsofeconomiceﬃciencyandthecontractcurve.Marshall,A.PrinciplesofEconomics,8thed.London:Mac-millan&Co.,1920.Completesummaryofneoclassicalview.Along-running,populartext.Detailedmathematicalappendix.Marx,K.Capital.NewYork:ModernLibrary,1906.Fulldevelopmentoflabortheoryofvalue.Discussionof“transforma-tionproblem”providesa(perhapsfaulty)startforgeneralequilibriumanalysis.Presentsfundamentalcriticismsofinstitutionofprivateproperty.Ricardo,D.PrinciplesofPoliticalEconomyandTaxation.London:J.M.Dent&Sons,1911.Veryanalytical,tightlywrittenwork.Pioneerindevelopingcarefulanalysisofpolicyquestions,especiallytrade-relatedissues.Discussesﬁrstbasicnotionsofmarginalism.Smith,A.TheWealthofNations.NewYork:ModernLibrary,1937.Firstgreateconomicsclassic.Verylonganddetailed,butSmithhadtheﬁrstwordonpracticallyeveryeconomicmatter.Thiseditionhashelpfulmarginalnotes.Walras,L.ElementsofPureEconomics.TranslatedbyW.Jaffé.Homewood,IL:RichardD.Irwin,1954.Beginningsofgeneralequilibriumtheory.Ratherdiﬃcultreading.SecondarySourcesontheHistoryofEconomicsBackhouse,RogerE.TheOrdinaryBusinessofLife:TheHistoryofEconomicsfromtheAncientWorldtothe21stCentury.Princeton,NJ:PrincetonUniversityPress,2002.Aniconoclastichistory.Quitegoodontheearliesteconomicideas,butsomeblindspotsonrecentusesofmathematicsandeconometrics.Blaug,Mark.EconomicTheoryinRetrospect,5thed.Cambridge:CambridgeUniversityPress,1997.Verycompletesummarystressinganalyticalissues.Excellent“Readers’Guides”totheclassicsineachchapter.Heilbroner,RobertL.TheWorldlyPhilosophers,7thed.NewYork:Simon&Schuster,1999.Fascinating,easy-to-readbiographiesofleadingeconomists.Chap-tersonUtopianSocialistsandThorsteinVeblenhighlyrecommended.Keynes,JohnM.EssaysinBiography.NewYork:W.W.Norton,1963.Essaysonmanyfamouspersons(LloydGeorge,WinstonChurchill,LeonTrotsky)andonseveraleconomists(Malthus,Marshall,Edgeworth,F.P.Ramsey,andJevons).ShowsthetruegiftofKeynesasawriter.Schumpeter,J.A.HistoryofEconomicAnalysis.NewYork:OxfordUniversityPress,1954.Encyclopedictreatment.Coversallthefamousandmanynot-so-famouseconomists.Alsobrieﬂysummarizesconcurrentdevelopmentsinotherbranchesofthesocialsciences.18Part1Introduction CHAPTER2MathematicsforMicroeconomicsMicroeconomicmodelsareconstructedusingawidevarietyofmathematicaltechniques.Inthischapterweprovideabriefsummaryofsomeofthemostimportanttechniquesthatyouwillencounterinthisbook.Amajorportionofthechapterconcernsmathematicalproceduresforﬁndingtheoptimalvalueofsomefunction.Becausewewillfrequentlyadopttheassumptionthataneconomicactorseekstomaximizeorminimizesomefunction,wewillencountertheseprocedures(mostofwhicharebasedondiﬀerentialcalculus)manytimes.Afterourdetaileddiscussionofthecalculusofoptimization,weturntofourtopicsthatarecoveredmorebrieﬂy.First,welookatafewspecialtypesoffunctionsthatariseineconomicproblems.Knowledgeofpropertiesofthesefunctionscanoftenbeveryhelpfulinsolvingeconomicproblems.Next,weprovideabriefsummaryofintegralcalculus.Althoughintegrationisusedinthisbookfarlessfrequentlythanisdiﬀerentiation,wewillneverthelessencounterseveralsituationswherewewillwanttoemployintegralstomeasureareasthatareimportanttoeconomictheoryortoaddupoutcomesthatoccurovertimeoracrossmanyindividuals.Oneparticularuseofintegrationistoexamineproblemsinwhichtheobjectiveistomaximizeastreamofoutcomesovertime.Ourthirdaddedtopicfocusesontechniquestobeusedforsuchproblemsindynamicoptimization.Finally,Chapter2concludeswithabriefsummaryofmathematicalstatistics,whichwillbeparticularlyusefulinourstudyofeconomicbehaviorinuncertainsituations.MAXIMIZATIONOFAFUNCTIONOFONEVARIABLELet’sstartourstudyofoptimizationwithasimpleexample.Supposethatamanagerofaﬁrmdesirestomaximize1theproﬁtsreceivedfromsellingaparticulargood.SupposealsothattheproﬁtsðπÞreceiveddependonlyonthequantityðqÞofthegoodsold.Mathematically,π¼fðqÞ:(2.1)Figure2.1showsapossiblerelationshipbetweenπandq.Clearly,toachievemaximumproﬁts,themanagershouldproduceoutputq,whichyieldsproﬁtsπ.IfagraphsuchasthatofFigure2.1wereavailable,thiswouldseemtobeasimplemattertobeaccomplishedwitharuler.Suppose,however,asismorelikely,themanagerdoesnothavesuchanaccuratepictureofthemarket.Heorshemaythentryvaryingqtoseewhereamaximumproﬁtisobtained.Forexample,bystartingatq1,proﬁtsfromsaleswouldbeπ1.Next,themanagermaytryoutputq2,observingthatproﬁtshaveincreasedtoπ2.Thecommonsenseideathatproﬁtshaveincreasedinresponsetoanincreaseinqcanbestatedformallyasπ2π1q2q1>0or∆π∆q>0,(2.2)1Herewewillgenerallyexploremaximizationproblems.Avirtuallyidenticalapproachwouldbetakentostudyminimiza-tionproblemsbecausemaximizationoffðxÞisequivalenttominimizingfðxÞ.19
wherethe∆notationisusedtomean“thechangein”πorq.Aslongas∆π=∆qispositive,proﬁtsareincreasingandthemanagerwillcontinuetoincreaseoutput.Forincreasesinoutputtotherightofq,however,∆π=∆qwillbenegative,andthemanagerwillrealizethatamistakehasbeenmade.DerivativesAsyouprobablyknow,thelimitof∆π=∆qforverysmallchangesinqiscalledthederivativeofthefunction,π¼fðqÞ,andisdenotedbydπ=dqordf=dqorf0ðqÞ.Moreformally,thederivativeofafunctionπ¼fðqÞatthepointq1isdeﬁnedasdπdq¼dfdq¼limh!0fðq1þhÞfðq1Þh:(2.3)Noticethatthevalueofthisratioobviouslydependsonthepointq1thatischosen.ValueofthederivativeatapointAnotationalconventionshouldbementioned:Sometimesonewishestonoteexplicitlythepointatwhichthederivativeistobeevaluated.Forexample,theevaluationofthederivativeatthepointq¼q1couldbedenotedbydπdqq¼q1:(2.4)Atothertimes,oneisinterestedinthevalueofdπ=dqforallpossiblevaluesofqandnoexplicitmentionofaparticularpointofevaluationismade.IntheexampleofFigure2.1,dπdqq¼q1>0,whereasdπdqq¼q3<0:Whatisthevalueofdπ=dqatq?Itwouldseemtobe0,becausethevalueispositiveforvaluesofqlessthanqandnegativeforvaluesofqgreaterthanq.Thederivativeistheslopeofthecurveinquestion;thisslopeispositivetotheleftofqandnegativetotherightofq.Atthepointq,theslopeoffðqÞis0.FIGURE2.1HypotheticalRelationshipbetweenQuantityProducedandProﬁtsIfamanagerwishestoproducethelevelofoutputthatmaximizesproﬁts,thenqshouldbeproduced.Noticethatatq,dπ=dq¼0.π=f(q)πQuantityq1q2q*q3π*π2π3π120Part1Introduction First-orderconditionforamaximumThisresultisquitegeneral.Forafunctionofonevariabletoattainitsmaximumvalueatsomepoint,thederivativeatthatpoint(ifitexists)mustbe0.Hence,ifamanagercouldestimatethefunctionfðqÞfromsomesortofreal-worlddata,itwouldtheoreticallybepossibletoﬁndthepointwheredf=dq¼0.Atthisoptimalpoint(say,q),dfdqq¼q¼0:(2.5)Second-orderconditionsAnunsuspectingmanagercouldbetricked,however,byanaiveapplicationofthisﬁrst-derivativerulealone.Forexample,supposethattheproﬁtfunctionlookslikethatshownineitherFigure2.2aor2.2b.IftheproﬁtfunctionisthatshowninFigure2.2a,themanager,byproducingwheredπ=dq¼0,willchoosepointqa.Thispointinfactyieldsminimum,notmaximum,proﬁtsforthemanager.Similarly,iftheproﬁtfunctionisthatshowninFigure2.2,themanagerwillchoosepointqb,which,althoughityieldsaproﬁtgreaterthanthatforanyoutputlowerthanqb,iscertainlyinferiortoanyoutputgreaterthanqb.Thesesituationsillustratethemathematicalfactthatdπ=dq¼0isanecessaryconditionforamaximum,butnotasufﬁcientcondition.Toensurethatthechosenpointisindeedamaximumpoint,asecondconditionmustbeimposed.Intuitively,thisadditionalconditionisclear:Theproﬁtavailablebyproducingeitherabitmoreorabitlessthanqmustbesmallerthanthatavailablefromq.Ifthisisnottrue,themanagercandobetterthanq.Mathematically,thismeansthatdπ=dqmustbegreaterFIGURE2.2TwoProﬁtFunctionsThatGiveMisleadingResultsIftheFirstDerivativeRuleIsAppliedUncriticallyIn(a),theapplicationoftheﬁrstderivativerulewouldresultinpointqabeingchosen.Thispointisinfactapointofminimumproﬁts.Similarly,in(b),outputlevelqbwouldberecommendedbytheﬁrstderivativerule,butthispointisinferiortoalloutputsgreaterthanqb.Thisdemonstratesgraphicallythatﬁndingapointatwhichthederivativeisequalto0isanecessary,butnotasufﬁcient,conditionforafunctiontoattainitsmaximumvalue.q*aπ*bπ*aq*bπQuantity(a)(b)πQuantityChapter2MathematicsforMicroeconomics21 than0forqq.Therefore,atq,dπ=dqmustbedecreasing.Anotherwayofsayingthisisthatthederivativeofdπ=dqmustbenegativeatq.SecondderivativesThederivativeofaderivativeiscalledasecondderivativeandisdenotedbyd2πdq2ord2fdq2orfqðÞ:Theadditionalconditionforqtorepresenta(local)maximumistherefored2πdq2q¼q¼f″qðÞq¼q<0,(2.6)wherethenotationisagainareminderthatthissecondderivativeistobeevaluatedatq.Hence,althoughEquation2.5ðdπ=dq¼0Þisanecessaryconditionforamaximum,thatequationmustbecombinedwithEquation2.6ðd2π=dq2<0Þtoensurethatthepointisalocalmaximumforthefunction.Equations2.5and2.6togetherarethereforesufﬁcientconditionsforsuchamaximum.Ofcourse,itispossiblethatbyaseriesoftrialsthemanagermaybeabletodecideonqbyrelyingonmarketinformationratherthanonmathematicalreasoning(rememberFriedman’spool-playeranalogy).Inthisbookweshallbelessinterest-edinhowthepointisdiscoveredthaninitspropertiesandhowthepointchangeswhenconditionschange.Amathematicaldevelopmentwillbeveryhelpfulinansweringthesequestions.RulesforﬁndingderivativesHereareafewfamiliarrulesfortakingderivatives.Wewillusetheseatmanyplacesinthisbook.1.Ifbisaconstant,thendbdx¼0:2.Ifbisaconstant,thend½bfðxÞdx¼bf0xðÞ:3.Ifbisaconstant,thendxbdx¼bxb1:4.dlnxdx¼1xwherelnsigniﬁesthelogarithmtothebaseeð¼2:71828Þ.5.daxdx¼axlnaforanyconstantaAparticularcaseofthisruleisdex=dx¼ex.NowsupposethatfðxÞandgðxÞaretwofunctionsofxandthatf0ðxÞandg0ðxÞexist.Then:6.d½fðxÞþgðxÞdx¼f0xðÞþg0xðÞ:22Part1Introduction 7.d½fðxÞ⋅gðxÞdx¼fxðÞg0xðÞþf0xðÞgxðÞ:8.d½fðxÞ=gðxÞdx¼f0ðxÞgðxÞfðxÞg0ðxÞ½gðxÞ2,providedthatgðxÞ6¼0.Finally,ify¼fðxÞandx¼gðzÞandifbothf0ðxÞandg0ðzÞexist,then9.dydz¼dydx⋅dxdz¼dfdx⋅dgdz:Thisresultiscalledthechainrule.ItprovidesaconvenientwaytostudyhowonevariableðzÞaffectsanothervariableðyÞsolelythroughitsinﬂuenceonsomeintermediatevariableðxÞ.Someexamplesare10.deaxdx¼deaxdðaxÞ⋅dðaxÞdx¼eax⋅a¼aeax:11.d½lnðaxÞdx¼d½lnðaxÞdðaxÞ⋅dðaxÞdx¼1ax⋅a¼1x:12.d½lnðx2Þdx¼d½lnðx2Þdðx2Þ⋅dðx2Þdx¼1x2⋅2x¼2x:FUNCTIONSOFSEVERALVARIABLESEconomicproblemsseldominvolvefunctionsofonlyasinglevariable.Mostgoalsofinteresttoeconomicagentsdependonseveralvariables,andtrade-offsmustbemadeamongthesevariables.Forexample,theutilityanindividualreceivesfromactivitiesasaconsumerdependsontheamountofeachgoodconsumed.Foraﬁrm’sproductionfunction,theamountproduceddependsonthequantityoflabor,capital,andlanddevotedtoproduction.Inthesecircumstances,thisdependenceofonevariableðyÞonaseriesofothervariablesðx1,x2,…,xnÞisdenotedbyy¼fðx1,x2,…,xnÞ:(2.7)PartialderivativesWeareinterestedinthepointatwhichyreachesamaximumandinthetrade-offsthatmustbemadetoreachthatpoint.Itisagainconvenienttopicturetheagentaschangingthevariablesathisorherdisposal(thex’s)inordertolocateamaximum.Unfortunately,forafunctionofseveralvariables,theideaofthederivativeisnotwell-deﬁned.Justasthesteepnessofascentwhenclimbingamountaindependsonwhichdirectionyougo,sodoestheslope(orderivative)ofthefunctiondependonthedirectioninwhichitistaken.Usually,theonlydirectionalslopesofinterestarethosethatareobtainedbyincreasingoneofthex’swhileholdingalltheothervariablesconstant(theanalogyformountainclimbingmightbetomeasureslopesonlyinanorth-southoreast-westdirection).Thesedirectionalslopesarecalledpartialderivatives.Thepartialderivativeofywithrespectto(thatis,inthedirectionof)x1isdenotedby∂y∂x1or∂f∂x1orfx1orf1:Itisunderstoodthatincalculatingthisderivativealloftheotherx’sareheldconstant.Againitshouldbeemphasizedthatthenumericalvalueofthisslopedependsonthevalueofx1andonthe(preassigned)valuesofx2,…,xn.Chapter2MathematicsforMicroeconomics23 EXAMPLE2.1ProﬁtMaximizationSupposethattherelationshipbetweenproﬁtsðπÞandquantityproducedðqÞisgivenbyπðqÞ¼1,000q5q2:(2.8)AgraphofthisfunctionwouldresembletheparabolashowninFigure2.1.Thevalueofqthatmaximizesproﬁtscanbefoundbydifferentiation:dπdq¼1,00010q¼0,(2.9)soq¼100:(2.10)Atq¼100,Equation2.8showsthatproﬁtsare50,000—thelargestvaluepossible.If,forexample,theﬁrmoptedtoproduceq¼50,proﬁtswouldbe37,500.Atq¼200,proﬁtsareprecisely0.Thatq¼100isa“global”maximumcanbeshownbynotingthatthesecondderivativeoftheproﬁtfunctionis10(seeEquation2.9).Hence,therateofincreaseinproﬁtsisalwaysdecreasing—uptoq¼100thisrateofincreaseisstillpositive,butbeyondthatpointitbecomesnegative.Inthisexample,q¼100istheonlylocalmaximumvalueforthefunctionπ.Withmorecomplexfunctions,however,theremaybeseveralsuchmaxima.QUERY:Supposethataﬁrm’soutputðqÞisdeterminedbytheamountoflaborðlÞithiresaccordingtothefunctionq¼2ﬃﬃlp.Supposealsothattheﬁrmcanhireallofthelaboritwantsat$10perunitandsellsitsoutputat$50perunit.ProﬁtsarethereforeafunctionoflgivenbyπlðÞ¼100ﬃﬃlp10l.Howmuchlaborshouldthisﬁrmhireinordertomaximizeproﬁts,andwhatwillthoseproﬁtsbe?Asomewhatmoreformaldeﬁnitionofthepartialderivativeis∂f∂x1_x2,…,_xn¼limh→0fðx1þh;_x2,…,_xnÞfðx1,_x2,…,_xnÞh,(2.11)wherethenotationisintendedtoindicatethatx2,…,xnareallheldconstantatthepreassignedvalues_x2,…,_xnsotheeffectofchangingx1onlycanbestudied.Partialderivativeswithrespecttotheothervariablesðx2,…,xnÞwouldbecalculatedinasimilarway.CalculatingpartialderivativesItiseasytocalculatepartialderivatives.Thecalculationproceedsasfortheusualderivativebytreatingx2,…,xnasconstants(whichindeedtheyareinthedeﬁnitionofapartialderivative).Considerthefollowingexamples.1.Ify¼fx1,x2¼ax21þbx1x2þcx22,then∂f∂x1¼f1¼2ax1þbx2and∂f∂x2¼f2¼bx1þ2cx2:24Part1Introduction Noticethat∂f=∂x1isingeneralafunctionofbothx1andx2andthereforeitsvaluewilldependontheparticularvaluesassignedtothesevariables.Italsodependsontheparametersa,b,andc,whichdonotchangeasx1andx2change.2.Ify¼fðx1,x2Þ¼eax1þbx2,then∂f∂x1¼f1¼aeax1þbx2and∂f∂x2¼f2¼beax1þbx2:3.Ify¼fðx1,x2Þ¼alnx1þblnx2,then∂f∂x1¼f1¼ax1and∂f∂x2¼f2¼bx2:Noticeherethatthetreatmentofx2asaconstantinthederivationof∂f=∂x1causesthetermblnx2todisappearupondifferentiationbecauseitdoesnotchangewhenx1changes.Inthiscase,unlikeourpreviousexamples,thesizeoftheeffectofx1onyisindependentofthevalueofx2.Inothercases,theeffectofx1onywilldependonthelevelofx2.PartialderivativesandtheceterisparibusassumptionInChapter1,wedescribedthewayinwhicheconomistsusetheceterisparibusassumptionintheirmodelstoholdconstantavarietyofoutsideinﬂuencessotheparticularrelationshipbeingstudiedcanbeexploredinasimpliﬁedsetting.Partialderivativesareaprecisemathe-maticalwayofrepresentingthisapproach;thatis,theyshowhowchangesinonevariableaffectsomeoutcomewhenotherinﬂuencesareheldconstant—exactlywhateconomistsneedfortheirmodels.Forexample,Marshall’sdemandcurveshowstherelationshipbetweenpriceðpÞandquantityðqÞdemandedwhenotherfactorsareheldconstant.Usingpartialderiva-tives,wecouldrepresenttheslopeofthiscurveby∂q=∂ptoindicatetheceterisparibusassumptionsthatareineffect.Thefundamentallawofdemand—thatpriceandquantitymoveinoppositedirectionswhenotherfactorsdonotchange—isthereforereﬂectedbythemathematicalstatement“∂q=∂p<0.”Again,theuseofapartialderivativeservesasare-minderoftheceterisparibusassumptionsthatsurroundthelawofdemand.PartialderivativesandunitsofmeasurementInmathematicsrelativelylittleattentionispaidtohowvariablesaremeasured.Infact,mostoftennoexplicitmentionismadeoftheissue.Butthevariablesusedineconomicsusuallyrefertoreal-worldmagnitudesandthereforewemustbeconcernedwithhowtheyaremeasured.Perhapsthemostimportantconsequenceofchoosingunitsofmeasurementisthatthepartialderivativesoftenusedtosummarizeeconomicbehaviorwillreﬂecttheseunits.Forexample,ifqrepresentsthequantityofgasolinedemandedbyallU.S.consumersduringagivenyear(measuredinbillionsofgallons)andprepresentsthepriceindollarspergallon,then∂q=∂pwillmeasurethechangeindemand(inbillionsofgallonsperyear)foradollarpergallonchangeinprice.Thenumericalsizeofthisderivativedependsonhowqandparemeasured.AdecisiontomeasureconsumptioninmillionsofgallonsperyearwouldmultiplyChapter2MathematicsforMicroeconomics25 thesizeofthederivativeby1,000,whereasadecisiontomeasurepriceincentspergallonwouldreduceitbyafactorof100.Thedependenceofthenumericalsizeofpartialderivativesonthechosenunitsofmea-surementposesproblemsforeconomists.Althoughmanyeconomictheoriesmakepredic-tionsaboutthesign(direction)ofpartialderivatives,anypredictionsaboutthenumericalmagnitudeofsuchderivativeswouldbecontingentonhowauthorschosetomeasuretheirvariables.Makingcomparisonsamongstudiescouldprovepracticallyimpossible,especiallygiventhewidevarietyofmeasuringsystemsinusearoundtheworld.Forthisreason,econ-omistshavechosentoadoptadifferent,unit-freewaytomeasurequantitativeimpacts.Elasticity—AgeneraldeﬁnitionEconomistsuseelasticitiestosummarizevirtuallyallofthequantitativeimpactsthatareofinteresttothem.Becausesuchmeasuresfocusontheproportionaleffectofachangeinonevariableonanother,theyareunit-free—theunits“cancelout”whentheelasticityiscalculated.Suppose,forexample,thatyisafunctionofxand,possibly,othervariables.Thentheelasticityofywithrespecttox(denotedasey,x)isdeﬁnedasey,x¼∆yy∆xx¼∆y∆x⋅xy¼∂y∂x⋅xy:(2.12)Noticethat,nomatterhowthevariablesyandxaremeasured,theunitsofmeasurementcanceloutbecausetheyappearinbothanumeratorandadenominator.Noticealsothat,becauseyandxarepositiveinmosteconomicsituations,theelasticityey,xandthepartialderivative∂y=∂xwillhavethesamesign.Hence,theoreticalpredictionsaboutthedirectionofcertainderivativeswillalsoapplytotheirrelatedelasticities.Speciﬁcapplicationsoftheelasticityconceptwillbeencounteredthroughoutthisbook.Theseincludeoneswithwhichyoushouldbefamiliar,suchasthemarketpriceelasticityofdemandorsupply.Butmanynewconceptsthatcanbeexpressedmostclearlyinelasticitytermswillalsobeintroduced.EXAMPLE2.2ElasticityandFunctionalFormThedeﬁnitioninEquation2.12makesclearthatelasticityshouldbeevaluatedataspeciﬁcpointonafunction.Ingeneralthevalueofthisparameterwouldbeexpectedtovaryacrossdifferentrangesofthefunction.Thisobservationismostclearlyshowninthecasewhereyisalinearfunctionofxoftheformy¼aþbxþotherterms:Inthiscase,ey,x¼∂y∂x⋅xy¼b⋅xy¼b⋅xaþbxþ…,(2.13)whichmakesclearthatey,xisnotconstant.Hence,forlinearfunctionsitisespeciallyimpor-tanttonotethepointatwhichelasticityistobecomputed.Ifthefunctionalrelationshipbetweenyandxisoftheexponentialformy¼axbthentheelasticityisaconstant,independentofwhereitismeasured:ey,x¼∂y∂x⋅xy¼abxb1⋅xaxb¼b:26Part1Introduction Alogarithmictransformationofthisequationalsoprovidesaveryconvenientalternativedeﬁnitionofelasticity.Becauselny¼lnaþblnx,wehaveey,x¼b¼∂lny∂lnx:(2.14)Hence,elasticitiescanbecalculatedthrough“logarithmicdifferentiation.”Asweshallsee,thisisfrequentlytheeasiestwaytoproceedinmakingsuchcalculations.QUERY:Arethereanyfunctionalformsinadditiontotheexponentialthathaveaconstantelasticity,atleastoversomerange?Second-orderpartialderivativesThepartialderivativeofapartialderivativeisdirectlyanalogoustothesecondderivativeofafunctionofonevariableandiscalledasecond-orderpartialderivative.Thismaybewrittenas∂ð∂f=∂xiÞ∂xjormoresimplyas∂2f∂xj∂xi¼fij:(2.15)Fortheexamplesabove:1.∂2f∂x1∂x1¼f11¼2af12¼bf21¼bf22¼2c:2.f11¼a2eax1þbx2f12¼abeax1þbx2f21¼abeax1þbx2f22¼b2eax1þbx23.f11¼ax21f12¼0f21¼0f22¼bx22:Chapter2MathematicsforMicroeconomics27 Young’stheoremTheseexamplesillustratethemathematicalresultthat,underquitegeneralconditions,theorderinwhichpartialdifferentiationisconductedtoevaluatesecond-orderpartialderivativesdoesnotmatter.Thatis,fij¼fji(2.16)foranypairofvariablesxi,xj.Thisresultissometimescalled“Young’stheorem.”Foranintuitiveexplanationofthetheorem,wecanreturntoourmountain-climbinganalogy.Inthisexample,thetheoremstatesthatthegaininelevationahikerexperiencesdependsonthedirectionsanddistancestraveled,butnotontheorderinwhichtheseoccur.Thatis,thegaininaltitudeisindependentoftheactualpathtakenaslongasthehikerproceedsfromonesetofmapcoordinatestoanother.Heorshemay,forexample,goonemilenorth,thenonemileeastorproceedintheoppositeorderbyﬁrstgoingonemileeast,thenonemilenorth.Ineithercase,thegaininelevationisthesamesinceinbothcasesthehikerismovingfromonespeciﬁcplacetoanother.Inlaterchapterswewillmakegooduseofthisresultbecauseitprovidesaveryconvenientwayofshowingsomeofthepredictionsthateconomicmodelsmakeaboutbehavior.2Usesofsecond-orderpartialsSecond-orderpartialderivativeswillplayanimportantroleinmanyoftheeconomictheoriesthataredevelopedthroughoutthisbook.Probablythemostimportantexamplesrelatetothe“own”second-orderpartial,fii.Thisfunctionshowshowthemarginalinﬂuenceofxionyði:e:,∂y=∂xiÞchangesasthevalueofxiincreases.Anegativevalueforfiiisthemathematicalwayofindicatingtheeconomicideaofdiminishingmarginaleffectiveness.Similarly,thecross-partialfijindicateshowthemarginaleffectivenessofxichangesasxjincreases.Thesignofthiseffectcouldbeeitherpositiveornegative.Young’stheoremindicatesthat,ingeneral,suchcross-effectsaresymmetric.Moregenerally,thesecond-orderpartialderivativesofafunctionprovideinformationaboutthecurvatureofthefunction.Laterinthischapterwewillseehowsuchinformationplaysanimportantroleindeterminingwhethervarioussecond-orderconditionsforamaximumaresatisﬁed.MAXIMIZATIONOFFUNCTIONSOFSEVERALVARIABLESUsingpartialderivatives,wecannowdiscusshowtoﬁndthemaximumvalueforafunctionofseveralvariables.Tounderstandthemathematicsusedinsolvingthisproblem,ananalogytotheone-variablecaseishelpful.Inthisone-variablecase,wecanpictureanagentvaryingxbyasmallamount,dx,andobservingthechangeiny,dy.Thischangeisgivenbydy¼f0ðxÞdx:(2.17)TheidentityinEquation2.17recordsthefactthatthechangeinyisequaltothechangeinxtimestheslopeofthefunction.Thisformulaisequivalenttothepoint-slopeformulausedforlinearequationsinbasicalgebra.Asbefore,thenecessaryconditionforamaximumisthatdy¼0forsmallchangesinxaroundtheoptimalpoint.Otherwise,ycouldbeincreasedbysuitablechangesinx.Butbecausedxdoesnotnecessarilyequal0inEquation2.17,dy¼0mustimplythatatthedesiredpoint,f0ðxÞ¼0.Thisisanotherwayofobtainingtheﬁrst-orderconditionforamaximumthatwealreadyderived.2Young’stheoremimpliesthatthematrixofthesecond-orderpartialderivativesofafunctionissymmetric.Thissymmetryoffersanumberofeconomicinsights.Forabriefintroductiontothematrixconceptsusedineconomics,seetheExtensionstothischapter.28Part1Introduction Usingthisanalogy,let’slookatthedecisionsmadebyaneconomicagentwhomustchoosethelevelsofseveralvariables.Supposethatthisagentwishestoﬁndasetofx’sthatwillmaximizethevalueofy¼fðx1,x2,…,xnÞ.Theagentmightconsiderchangingonlyoneofthex’s,sayx1,whileholdingalltheothersconstant.Thechangeiny(thatis,dy)thatwouldresultfromthischangeinx1isgivenbydy¼∂f∂x1dx1¼f1dx1:Thissaysthatthechangeinyisequaltothechangeinx1timestheslopemeasuredinthex1direction.Usingthemountainanalogyagain,thegaininaltitudeaclimberheadingnorthwouldachieveisgivenbythedistancenorthwardtraveledtimestheslopeofthemountainmeasuredinanorthwarddirection.TotaldiﬀerentialIfallthex’sarevariedbyasmallamount,thetotaleffectonywillbethesumofeffectssuchasthatshownabove.Thereforethetotalchangeinyisdeﬁnedtobedy¼∂f∂x1dx1þ∂f∂x2dx2þ…þ∂f∂xndxn¼f1dx1þf2dx2þ…þfndxn:(2.18)Thisexpressioniscalledthetotaldifferentialoffandisdirectlyanalogoustotheexpressionforthesingle-variablecasegiveninEquation2.17.Theequationisintuitivelysensible:Thetotalchangeinyisthesumofchangesbroughtaboutbyvaryingeachofthex’s.3First-orderconditionforamaximumAnecessaryconditionforamaximum(oraminimum)ofthefunctionfðx1,x2,…,xnÞisthatdy¼0foranycombinationofsmallchangesinthex’s.Theonlywaythiscanhappenisif,atthepointbeingconsidered,f1¼f2¼…¼fn¼0:(2.19)ApointwhereEquations2.19holdiscalledacriticalpoint.Equations2.19arethenecessaryconditionsforalocalmaximum.Toseethisintuitively,notethatifoneofthepartials(say,fi)weregreater(orless)than0,thenycouldbeincreasedbyincreasing(ordecreasing)xi.Aneconomicagentthencouldﬁndthismaximalpointbyﬁndingthespotwhereydoesnotrespondtoverysmallmovementsinanyofthex’s.Thisisanextremelyimportantresultforeconomicanalysis.Itsaysthatanyactivity(thatis,thex’s)shouldbepushedtothepointwhereits“marginal”contributiontotheobjective(thatis,y)is0.Tostopshortofthatpointwouldfailtomaximizey.3ThetotaldifferentialinEquation2.18canbeusedtoderivethechainruleasitappliestofunctionsofseveralvariables.Supposethaty¼fðx1,x2Þandthatx1¼gðzÞandx2¼hðzÞ.Ifallofthesefunctionsaredifferentiable,thenitispossibletocalculatetheeffectsofachangeinzony.Thetotaldifferentialofyisdy¼f1dx1þf2dx2:Dividingthisequationbydzgivesdydz¼f1dx1dzþf2dx2dz¼f1dgdzþf2dhdz:Hence,calculatingtheeffectofzonyrequirescalculatinghowzaffectsbothofthedeterminantsofy(thatis,x1andx2).Ifydependsonmorethantwovariables,ananalogousresultholds.Thisresultactsasaremindertoberathercarefultoincludeallpossibleeffectswhencalculatingderivativesoffunctionsofseveralvariables.Chapter2MathematicsforMicroeconomics29 EXAMPLE2.3FindingaMaximumSupposethatyisafunctionofx1andx2givenbyy¼ðx11Þ2ðx22Þ2þ10(2.20)ory¼x21þ2x1x22þ4x2þ5:Forexample,ymightrepresentanindividual’shealth(measuredonascaleof0to10),andx1andx2mightbedailydosagesoftwohealth-enhancingdrugs.Wewishtoﬁndvaluesforx1andx2thatmakeyaslargeaspossible.Takingthepartialderivativesofywithrespecttox1andx2andapplyingthenecessaryconditionsgivenbyEquations2.19yields∂y∂x1¼2x1þ2¼0,∂y∂x2¼2x2þ4¼0(2.21)orx1¼1,x2¼2:Thefunctionisthereforeatacriticalpointwhenx1¼1,x2¼2.Atthatpoint,y¼10isthebesthealthstatuspossible.Abitofexperimentationprovidesconvincingevidencethatthisisthegreatestvalueycanhave.Forexample,ifx1¼x2¼0,theny¼5,orifx1¼x2¼1,theny¼9.Valuesofx1andx2largerthan1and2,respectively,reduceybecausethenegativequadratictermsinEquation2.20becomelarge.Consequently,thepointfoundbyapplyingthenecessaryconditionsisinfactalocal(andglobal)maximum.4QUERY:Supposeytookonaﬁxedvalue(say,5).Whatwouldtherelationshipimpliedbetweenx1andx2looklike?Howaboutfory¼7?Ory¼10?(Thesegraphsarecontourlinesofthefunctionandwillbeexaminedinmoredetailinseverallaterchapters.SeealsoProblem2.1.)Second-orderconditionsAgain,however,theconditionsofEquations2.19arenotsufﬁcienttoensureamaximum.Thiscanbeillustratedbyreturningtoanalreadyoverworkedanalogy:Allhilltopsare(moreorless)ﬂat,butnoteveryﬂatplaceisahilltop.Asecond-orderconditionsimilartoEquation2.6isneededtoensurethatthepointfoundbyapplyingEquations2.19isalocalmaximum.Intuitively,foralocalmaximum,yshouldbedecreasingforanysmallchangesinthex’sawayfromthecriticalpoint.Asinthesingle-variablecase,thisnecessarilyinvolveslookingatthesecond-orderpartialderivativesofthefunctionf.Thesesecond-orderpartialsmustobeycertainrestrictions(analogoustotherestrictionthatwasderivedinthesingle-variablecase)ifthecriticalpointfoundbyapplyingEquations2.19istobealocalmaximum.Laterinthischapterwewilllookattheserestrictions.4Moreformally,thepointx1¼1,x2¼2isaglobalmaximumbecausethefunctiondescribedbyEquation2.20isconcave(seeourdiscussionlaterinthischapter).30Part1Introduction IMPLICITFUNCTIONSAlthoughmathematicalequationsareoftenwrittenwitha“dependent”variable(y)asafunc-tionofoneormoreindependentvariables(x),thisisnottheonlywaytowritesucharela-tionship.Asatrivialexample,theequationy¼mxþb(2.22)canalsobewrittenasymxb¼0(2.23)or,evenmoregenerally,asfðx,y,m,bÞ¼0,(2.24)wherethisfunctionalnotationindicatesarelationshipbetweenxandythatalsodependsontheslope(m)andintercept(b)parametersofthefunction,whichdonotchange.Functionswrittenintheseformsaresometimescalledimplicitfunctionsbecausetherelationshipsbetweenthevariablesandparametersareimplicitlypresentintheequationratherthanbeingexplicitlycalculatedas,say,yasafunctionofxandtheparametersmandb.Oftenitisasimplemattertotranslatefromimplicitfunctionstoexplicitones.Forexample,theimplicitfunctionxþ2y4¼0(2.25)caneasilybe“solved”forxasx¼2yþ4(2.26)orforyasy¼x2þ2:(2.27)DerivativesfromimplicitfunctionsInmanycircumstancesitishelpfultocomputederivativesdirectlyfromimplicitfunctionswithoutsolvingforoneofthevariablesdirectly.Forexample,theimplicitfunctionfðx,yÞ¼0hasatotaldifferentialof0¼fxdxþfydy,sodydx¼fxfy:(2.28)Hence,theimplicitderivativedy=dxcanbefoundasthenegativeoftheratioofthepartialderivativesoftheimplicitfunction,providingfy6¼0.EXAMPLE2.4AProductionPossibilityFrontier—AgainInExample1.3weexaminedaproductionpossibilityfrontierfortwogoodsoftheformx2þ0:25y2¼200(2.29)or,writtenimplicitly,fðx,yÞ¼x2þ0:25y2200¼0:(2.30)Hence,(continued)Chapter2MathematicsforMicroeconomics31 EXAMPLE2.4CONTINUEDfx¼2x,fy¼0:5y,and,byEquation2.28,theopportunitycosttrade-offbetweenxandyisdydx¼fxfy¼2x0:5y¼4xy,(2.31)whichispreciselytheresultweobtainedearlier,withconsiderablylesswork.QUERY:Whydoesthetrade-offbetweenxandyheredependonlyontheratioofxtoyandnotonthesizeofthelaborforceasreﬂectedbythe200constant?ImplicitfunctiontheoremItmaynotalwaysbepossibletosolveimplicitfunctionsoftheformgðx,yÞ¼0foruniqueexplicitfunctionsoftheformy¼fðxÞ.Mathematicianshaveanalyzedtheconditionsunderwhichagivenimplicitfunctioncanbesolvedexplicitlywithonevariablebeingafunctionofothervariablesandvariousparameters.Althoughwewillnotinvestigatetheseconditionshere,theyinvolverequirementsonthevariouspartialderivativesofthefunctionthataresufﬁcienttoensurethatthereisindeedauniquerelationshipbetweenthedependentandindependentvariables.5Inmanyeconomicapplications,thesederivativeconditionsarepreciselythoserequiredtoensurethatthesecond-orderconditionsforamaximum(oraminimum)hold.Hence,inthesecases,wewillassertthattheimplicitfunctiontheoremholdsandthatitisthereforepossibletosolveexplicitlyfortrade-offsamongthevariablesinvolved.THEENVELOPETHEOREMOnemajorapplicationoftheimplicitfunctiontheorem,whichwillbeusedmanytimesinthisbook,iscalledtheenvelopetheorem;itconcernshowtheoptimalvalueforaparticularfunctionchangeswhenaparameterofthefunctionchanges.Becausemanyoftheeconomicproblemswewillbestudyingconcerntheeffectsofchangingaparameter(forexample,theeffectsthatchangingthemarketpriceofacommoditywillhaveonanindividual’spurchases),thisisatypeofcalculationwewillfrequentlymake.Theenvelopetheoremoftenprovidesaniceshortcut.AspeciﬁcexamplePerhapstheeasiestwaytounderstandtheenvelopetheoremisthroughanexample.SupposeyisafunctionofasinglevariableðxÞandaparameterðaÞgivenbyy¼x2þax:(2.32)Fordifferentvaluesoftheparametera,thisfunctionrepresentsafamilyofinvertedparab-olas.Ifaisassignedaspeciﬁcvalue,Equation2.32isafunctionofxonly,andthevalueofxthatmaximizesycanbecalculated.Forexample,ifa¼1,thenx¼12and,forthesevaluesofxanda,y¼14(itsmaximalvalue).Similarly,ifa¼2,thenx¼1andy¼1.Hence,anincrease5Foradetaileddiscussionoftheimplicitfunctiontheoreminvariouscontexts,seeCarlP.SimonandLawrenceBlume,MathematicsforEconomists(NewYork:W.W.Norton,1994),chap.15.32Part1Introduction of1inthevalueoftheparameterahasincreasedthemaximumvalueofyby34.InTable2.1,integralvaluesofabetween0and6areusedtocalculatetheoptimalvaluesforxandtheassociatedvaluesoftheobjective,y.Noticethatasaincreases,themaximalvalueforyalsoincreases.ThisisalsoillustratedinFigure2.3,whichshowsthattherelationshipbetweenaandyisquadratic.Nowwewishtocalculateexplicitlyhowychangesastheparameterachanges.TABLE2.1OptimalValuesofyandxforAlternativeValuesofainy¼x2þaxValueofaValueofxValueofy0001121421133294424552254639FIGURE2.3IllustrationoftheEnvelopeTheoremTheenvelopetheoremstatesthattheslopeoftherelationshipbetweeny(themaximumvalueofy)andtheparameteracanbefoundbycalculatingtheslopeoftheauxiliaryrelationshipfoundbysubstitutingtherespectiveoptimalvaluesforxintotheobjectivefunctionandcalculating∂y=∂a.ay*063152412345678910y* = f(a)Chapter2MathematicsforMicroeconomics33 Adirect,time-consumingapproachTheenvelopetheoremstatesthattherearetwoequivalentwayswecanmakethiscalculation.First,wecancalculatetheslopeofthefunctioninFigure2.3directly.Todoso,wemustsolveEquation2.32fortheoptimalvalueofxforanyvalueofa:dydx¼2xþa¼0;hence,x¼a2:SubstitutingthisvalueofxinEquation2.32givesy¼ðxÞ2þaðxÞ¼a22þaa2¼a24þa22¼a24,andthisispreciselytherelationshipshowninFigure2.3.Fromthepreviousequation,itiseasytoseethatdyda¼2a4¼a2(2.33)and,forexample,ata¼2,dy=da¼1.Thatis,neara¼2themarginalimpactofincreasingaistoincreaseybythesameamount.Neara¼6,anysmallincreaseinawillincreaseybythreetimesthischange.Table2.1illustratesthisresult.TheenvelopeshortcutArrivingatthisconclusionwasabitcomplicated.Wehadtoﬁndtheoptimalvalueofxforeachvalueofaandthensubstitutethisvalueforxintotheequationfory.Inmoregeneralcasesthismaybequiteburdensomesinceitrequiresrepeatedlymaximizingtheobjectivefunction.Theenvelopetheorem,providinganalternativeapproach,statesthatforsmallchangesina,dy=dacanbecomputedbyholdingxconstantatitsoptimalvalueandsimplycalculating∂y=∂afromtheobjectivefunctiondirectly.Proceedinginthiswaygives∂y∂a¼x,(2.34)andatxwehave∂y∂a¼x¼a2:(2.35)Thisispreciselytheresultobtainedearlier.ThereasonthatthetwoapproachesyieldidenticalresultsisillustratedinFigure2.3.Thetangentsshownintheﬁgurereportvaluesofyforaﬁxedx.Thetangents’slopesare∂y=∂a.Clearly,atythisslopegivesthevalueweseek.Thisresultisquitegeneral,andwewilluseitatseveralplacesinthisbooktosimplifyouranalysis.Tosummarize,theenvelopetheoremstatesthatthechangeintheoptimalvalueofafunctionwithrespecttoaparameterofthatfunctioncanbefoundbypartiallydifferentiatingtheobjectivefunctionwhileholdingxconstantatitsoptimalvalue.Thatis,34Part1Introduction dyda¼∂y∂ax¼xaðÞ,(2.36)wherethenotationprovidesareminderthat∂y=∂amustbecomputedatthatvalueofxthatisoptimalforthespeciﬁcvalueoftheparameterabeingexamined.Many-variablecaseAnanalogousenvelopetheoremholdsforthecasewhereyisafunctionofseveralvariables.Supposethatydependsonasetofx’sðx1,…,xnÞandonaparticularparameterofinterest,say,a:y¼fðx1,…,xn,aÞ:(2.37)Findinganoptimalvalueforywouldconsistofsolvingnﬁrst-orderequationsoftheform∂y∂xi¼0i¼1,…,nðÞ,(2.38)andasolutiontothisprocesswouldyieldoptimalvaluesforthesex’sx1,x2,…,xnthatwouldimplicitlydependontheparametera.Assumingthesecond-orderconditionsaremet,theimplicitfunctiontheoremwouldapplyinthiscaseandensurethatwecouldsolveeachxiasafunctionoftheparametera:x1¼x1ðaÞ,x2¼x2ðaÞ,...xn¼xnðaÞ:(2.39)Substitutingthesefunctionsintoouroriginalobjective(Equation2.37)yieldsanexpressioninwhichtheoptimalvalueofy(say,y)dependsontheparameterabothdirectlyandindirectlythroughtheeffectofaonthex’s:y¼f½x1ðaÞ,x2ðaÞ,…,xnðaÞ,a:Totallydifferentiatingthisexpressionwithrespecttoayieldsdyda¼∂f∂x1⋅dx1daþ∂f∂x2⋅dx2daþ…þ∂f∂xn⋅dxndaþ∂f∂a:(2.40)But,becauseoftheﬁrst-orderconditionsallofthesetermsexceptthelastareequalto0ifthex’sareattheiroptimalvalues.Hence,againwehavetheenveloperesult:dyda¼∂f∂a,(2.41)wherethisderivativeistobeevaluatedattheoptimalvaluesforthex’s.EXAMPLE2.5TheEnvelopeTheorem:HealthStatusRevisitedEarlier,inExample2.3,weexaminedthemaximumvaluesforthehealthstatusfunctiony¼ðx11Þ2ðx22Þ2þ10(2.42)andfoundthatx1¼1,x2¼2,(2.43)(continued)Chapter2MathematicsforMicroeconomics35 EXAMPLE2.5CONTINUEDandy¼10:Supposenowweusethearbitraryparameterainsteadoftheconstant10inEquation2.42.Hereamightrepresentameasureofthebestpossiblehealthforaperson,butthisvaluewouldobviouslyvaryfrompersontoperson.Hence,y¼fðx1,x2,aÞ¼ðx11Þ2ðx22Þ2þa:(2.44)Inthiscasetheoptimalvaluesforx1andx2donotdependona(theyarealwaysx1¼1,x2¼2),soatthoseoptimalvalueswehavey¼a(2.45)anddyda¼1:(2.46)Peoplewith“naturallybetterhealth”willhaveconcomitantlyhighervaluesfory,providingtheychoosex1andx2optimally.Butthisispreciselywhattheenvelopetheoremindicates,becausedyda¼∂f∂a¼1(2.47)fromEquation2.44.Increasingtheparameterasimplyincreasestheoptimalvalueforybyanidenticalamount(again,assumingthedosagesofx1andx2arecorrectlychosen).QUERY:Supposewefocusedinsteadontheoptimaldosageforx1inEquation2.42—thatis,supposeweusedageneralparameter,sayb,insteadof1.Explaininwordsandusingmathematicswhy∂y=∂bwouldnecessarilybe0inthiscase.CONSTRAINEDMAXIMIZATIONSofarwehavefocusedourattentiononﬁndingthemaximumvalueofafunctionwithoutrestrictingthechoicesofthex’savailable.Inmosteconomicproblems,however,notallvaluesforthex’sarefeasible.Inmanysituations,forexample,itisrequiredthatallthex’sbepositive.Thiswouldbetruefortheproblemfacedbythemanagerchoosingoutputtomaximizeproﬁts;anegativeoutputwouldhavenomeaning.Inotherinstancesthex’smaybeconstrainedbyeconomicconsiderations.Forexample,inchoosingtheitemstoconsume,anindividualisnotabletochooseanyquantitiesdesired.Rather,choicesareconstrainedbytheamountofpurchasingpoweravailable;thatis,bythisperson’sbudgetconstraint.Suchconstraintsmaylowerthemaximumvalueforthefunctionbeingmaximized.Becausewearenotabletochoosefreelyamongallthex’s,ymaynotbeaslargeasitcouldbe.Theconstraintswouldbe“nonbinding”ifwecouldobtainthesamelevelofywithorwithoutimposingtheconstraint.LagrangianmultipliermethodOnemethodforsolvingconstrainedmaximizationproblemsistheLagrangianmultipliermethod,whichinvolvesaclevermathematicaltrickthatalsoturnsouttohaveausefuleconomicinterpretation.Therationaleofthismethodisquitesimple,althoughnorigorous36Part1Introduction presentationwillbeattemptedhere.6Inapriorsection,thenecessaryconditionsforalocalmaximumwerediscussed.Weshowedthatattheoptimalpointallthepartialderivativesoffmustbe0.Therearethereforenequations(fi¼0fori¼1,…,n)innunknowns(thex’s).Generally,theseequationscanbesolvedfortheoptimalx’s.Whenthex’sareconstrained,however,thereisatleastoneadditionalequation(theconstraint)butnoadditionalvariables.Thesetofequationsthereforeisoverdetermined.TheLagrangiantechniqueintroducesanadditionalvariable(theLagrangianmultiplier),whichnotonlyhelpstosolvetheproblemathand(becausetherearenownþ1equationsinnþ1unknowns),butalsohasaninterpre-tationthatisusefulinavarietyofeconomiccircumstances.TheformalproblemMorespeciﬁcally,supposethatwewishtoﬁndthevaluesofx1,x2,…,xnthatmaximizey¼fðx1,x2,…,xnÞ,(2.48)subjecttoaconstraintthatpermitsonlycertainvaluesofthex’stobeused.Ageneralwayofwritingthatconstraintisgðx1,x2,…,xnÞ¼0,(2.49)wherethefunction7grepresentstherelationshipthatmustholdamongallthex’s.First-orderconditionsTheLagrangianmultipliermethodstartswithsettinguptheexpressionℒ¼fðx1,x2,…,xnÞþλgðx1,x2,…,xnÞ,(2.50)whereλisanadditionalvariablecalledtheLagrangianmultiplier.Laterwewillinterpretthisnewvariable.First,however,noticethatwhentheconstraintholds,ℒandfhavethesamevalue[becausegðx1,x2,…,xnÞ¼0].Consequently,ifwerestrictourattentiononlytovaluesofthex’sthatsatisfytheconstraint,ﬁndingtheconstrainedmaximumvalueoffisequivalenttoﬁndingacriticalvalueofℒ.Letusproceedthentodoso,treatingλalsoasavariable(inadditiontothex’s).FromEquation2.50,theconditionsforacriticalpointare:∂ℒ∂x1¼f1þλg1¼0,∂ℒ∂x2¼f2þλg2¼0,...∂ℒ∂xn¼fnþλgn¼0,∂ℒ∂λ¼gðx1,x2,…,xnÞ¼0:(2.51)Equations2.51arethentheconditionsforacriticalpointforthefunctionℒ.Noticethattherearenþ1equations(oneforeachxandaﬁnaloneforλ)innþ1unknowns.Theequationscangenerallybesolvedforx1,x2,…,xn,andλ.Suchasolutionwillhavetwo6Foradetailedpresentation,seeA.K.Dixit,OptimizationinEconomicTheory,2nded.(Oxford:OxfordUniversityPress,1990),chap.2.7Aswepointedoutearlier,anyfunctionofx1,x2,…,xncanbewritteninthisimplicitway.Forexample,theconstraintx1þx2¼10couldbewritten10x1x2¼0.Inlaterchapters,wewillusuallyfollowthisprocedureindealingwithconstraints.Oftentheconstraintsweexaminewillbelinear.Chapter2MathematicsforMicroeconomics37 properties:(1)thex’swillobeytheconstraintbecausethelastequationin2.51imposesthatcondition;and(2)amongallthosevaluesofx’sthatsatisfytheconstraint,thosethatalsosolveEquations2.51willmakeℒ(andhencef)aslargeaspossible(assumingsecond-orderconditionsaremet).TheLagrangianmultipliermethodthereforeprovidesawaytoﬁndasolutiontotheconstrainedmaximizationproblemweposedattheoutset.8ThesolutiontoEquations2.51willusuallydifferfromthatintheunconstrainedcase(seeEquations2.19).Ratherthanproceedingtothepointwherethemarginalcontributionofeachxis0,Equations2.51requireustostopshortbecauseoftheconstraint.Onlyiftheconstraintwereineffective(inwhichcase,asweshowbelow,λwouldbe0)wouldthecon-strainedandunconstrainedequations(andtheirrespectivesolutions)agree.Theserevisedmarginalconditionshaveeconomicinterpretationsinmanydifferentsituations.InterpretationoftheLagrangianmultiplierSofarwehaveusedtheLagrangianmultiplier(λ)onlyasamathematical“trick”toarriveatthesolutionwewanted.Infact,thatvariablealsohasanimportanteconomicinterpretation,whichwillbecentraltoouranalysisatmanypointsinthisbook.Todevelopthisinterpreta-tion,rewritetheﬁrstnequationsof2.51asf1g1¼f2g2¼…¼fngn¼λ:(2.52)Inotherwords,atthemaximumpoint,theratiooffitogiisthesameforeveryxi.ThenumeratorsinEquations2.52arethemarginalcontributionsofeachxtothefunctionf.Theyshowthemarginalbeneﬁtthatonemoreunitofxiwillhaveforthefunctionthatisbeingmaximized(thatis,forf).AcompleteinterpretationofthedenominatorsinEquations2.52isprobablybestleftuntilweencountertheseratiosinactualeconomicapplications.Therewewillseethattheseusuallyhavea“marginalcost”interpretation.Thatis,theyreﬂecttheaddedburdenontheconstraintofusingslightlymorexi.Asasimpleillustration,supposetheconstraintrequiredthattotalspendingonx1andx2begivenbyaﬁxeddollaramount,F.Hence,theconstraintwouldbep1x1þp2x2¼F(wherepiistheperunitcostofxi).Usingourpresentterminology,thisconstraintwouldbewritteninimplicitformasgðx1,x2Þ¼Fp1x1p2x2¼0:(2.53)Inthissituation,then,gi¼pi(2.54)andthederivativegidoesindeedreﬂecttheperunit,marginalcostofusingxi.PracticallyalloftheoptimizationproblemswewillencounterinlaterchaptershaveasimilarinterpretationforthedenominatorsinEquations2.52.Lagrangianmultiplierasabeneﬁt-costratioNowwecangiveEquations2.52anintuitiveinterpretation.Theyindicatethat,attheoptimalchoicesforthex’s,theratioofthemarginalbeneﬁtofincreasingxitothemarginalcostofincreasingxishouldbethesameforeveryx.Toseethatthisisanobviouscondition8Strictlyspeaking,thesearethenecessaryconditionsforaninteriorlocalmaximum.Insomeeconomicproblems,itisnecessarytoamendtheseconditions(infairlyobviousways)totakeaccountofthepossibilitythatsomeofthex’smaybeontheboundaryoftheregionofpermissiblex’s.Forexample,ifallofthex’sarerequiredtobenonnegative,itmaybethattheconditionsofEquations2.51willnotholdexactly,becausethesemayrequirenegativex’s.Welookatthissituationlaterinthischapter.38Part1Introduction foramaximum,supposethatitwerenottrue:Supposethatthe“beneﬁt-costratio”werehigherforx1thanforx2.Inthiscase,slightlymorex1shouldbeusedinordertoachieveamaximum.Considerusingmorex1butgivingupjustenoughx2tokeepg(theconstraint)constant.Hence,themarginalcostoftheadditionalx1usedwouldequalthecostsavedbyusinglessx2.Butbecausethebeneﬁt-costratio(theamountofbeneﬁtperunitofcost)isgreaterforx1thanforx2,theadditionalbeneﬁtsfromusingmorex1wouldexceedthelossinbeneﬁtsfromusinglessx2.Theuseofmorex1andappropriatelylessx2wouldthenincreaseybecausex1providesmore“bangforyourbuck.”Onlyifthemarginalbeneﬁt–marginalcostratiosareequalforallthex’swilltherebealocalmaximum,oneinwhichnosmallchangesinthex’scanincreasetheobjective.Concreteapplicationsofthisbasicprinciplearedevelopedinmanyplacesinthisbook.Theresultisfundamentalforthemicroeconomictheoryofoptimizingbehavior.TheLagrangianmultiplier(λ)canalsobeinterpretedinlightofthisdiscussion.λisthecommonbeneﬁt-costratioforallthex’s.Thatis,λ¼marginalbenefitofximarginalcostofxi(2.55)foreveryxi.Iftheconstraintwererelaxedslightly,itwouldnotmatterexactlywhichxischanged(indeed,allthex’scouldbealtered),because,atthemargin,eachpromisesthesameratioofbeneﬁtstocosts.TheLagrangianmultiplierthenprovidesameasureofhowsuchanoverallrelaxationoftheconstraintwouldaffectthevalueofy.Inessence,λassignsa“shadowprice”totheconstraint.Ahighλindicatesthatycouldbeincreasedsubstantiallybyrelaxingtheconstraint,becauseeachxhasahighbeneﬁt-costratio.Alowvalueofλ,ontheotherhand,indicatesthatthereisnotmuchtobegainedbyrelaxingtheconstraint.Iftheconstraintisnotbindingatall,λwillhaveavalueof0,therebyindicatingthattheconstraintisnotrestrictingthevalueofy.Insuchacase,ﬁndingthemaximumvalueofysubjecttotheconstraintwouldbeidenticaltoﬁndinganunconstrainedmaximum.Theshadowpriceoftheconstraintis0.Thisinterpretationofλcanalsobeshownusingtheenvelopetheoremasdescribedlaterinthischapter.9DualityThisdiscussionshowsthatthereisaclearrelationshipbetweentheproblemofmaximizingafunctionsubjecttoconstraintsandtheproblemofassigningvaluestoconstraints.Thisreﬂectswhatiscalledthemathematicalprincipleof“duality”:Anyconstrainedmaximizationproblemhasanassociateddualprobleminconstrainedminimizationthatfocusesattentionontheconstraintsintheoriginal(primal)problem.Forexample,tojumpabitaheadofourstory,economistsassumethatindividualsmaximizetheirutility,subjecttoabudgetconstraint.Thisistheconsumer’sprimalproblem.Thedualproblemfortheconsumeristominimizetheexpenditureneededtoachieveagivenlevelofutility.Or,aﬁrm’sprimalproblemmaybetominimizethetotalcostofinputsusedtoproduceagivenlevelofoutput,whereasthedualproblemistomaximizeoutputforagivencostofinputspurchased.Manysimilarexampleswillbedevelopedinlaterchapters.Eachillustratesthattherearealwaystwowaystolookatanyconstrainedoptimizationproblem.Sometimestakingafrontalattackbyanalyzingtheprimalproblemcanleadtogreaterinsights.Inotherinstances,the“backdoor”approachofexaminingthedualproblemmaybemoreinstructive.Whicheverrouteistaken,theresultswillgenerally,thoughnotalways,beidentical,sothechoicemadewillmainlybeamatterofconvenience.9Thediscussioninthetextconcernsproblemsinvolvingasingleconstraint.Ingeneral,onecanhandlemconstraintsðm0,andthischoicevariablemeetstheprecisebeneﬁt-costtestthatf1þλ1g1¼0.Alternatively,solutionswhereb¼0havex1¼0,andalsorequirethatλ2>0.So,suchsolutionsdonotinvolveanyuseofx1becausethatvariabledoesnotmeetthebeneﬁt-costtestasshownbytheﬁrstlineofEquation2.79,whichimpliesthatf1þλ1g1<0.Anidenticalresultholdsforthechoicevariablex2.Theseresults,whicharesometimescalledKuhn-Tuckerconditionsaftertheirdiscoverers,showthatthesolutionstooptimizationproblemsinvolvinginequalityconstraintswilldifferfromsimilarproblemsinvolvingequalityconstraintsinrathersimpleways.Hence,wecannotgofarwrongbyworkingprimarilywithconstraintsinvolvingequalitiesandassumingthatwecanrelyonintuitiontostatewhatwouldhappeniftheproblemsactuallyinvolvedinequal-ities.Thatisthegeneralapproachwewilltakeinthisbook.13SECOND-ORDERCONDITIONSSofarourdiscussionofoptimizationhasfocusedprimarilyonnecessary(ﬁrst-order)condi-tionsforﬁndingamaximum.Thatisindeedthepracticewewillfollowthroughoutmuchofthisbookbecause,asweshallsee,mosteconomicproblemsinvolvefunctionsforwhichthesecond-orderconditionsforamaximumarealsosatisﬁed.Inthissectionwegiveabriefanalysisoftheconnectionbetweensecond-orderconditionsforamaximumandtherelatedcurvatureconditionsthatfunctionsmusthavetoensurethatthesehold.Theeconomicexplanationsforthesecurvatureconditionswillbediscussedthroughoutthetext.FunctionsofonevariableFirstconsiderthecaseinwhichtheobjective,y,isafunctionofonlyasinglevariable,x.Thatis,y¼fðxÞ:(2.80)12Wewillnotexaminethedegeneratecasewherebothofthesevariablesare0.13Thesituationcanbecomemuchmorecomplexwhencalculuscannotbereliedupontogiveasolution,perhapsbecausesomeofthefunctionsinaproblemarenotdifferentiable.Foradiscussion,seeAvinaskK.Dixit,OptimizationinEconomicTheory,2nded.(Oxford:OxfordUniversityPress,1990).Chapter2MathematicsforMicroeconomics45 Anecessaryconditionforthisfunctiontoattainitsmaximumvalueatsomepointisthatdydx¼f0xðÞ¼0(2.81)atthatpoint.Toensurethatthepointisindeedamaximum,wemusthaveydecreasingformovementsawayfromit.Wealreadyknow(byEquation2.81)thatforsmallchangesinx,thevalueofydoesnotchange;whatweneedtocheckiswhetheryisincreasingbeforethat“plateau”isreachedanddecliningthereafter.WehavealreadyderivedanexpressionforthechangeinyðdyÞ,whichisgivenbythetotaldifferentialdy¼f0ðxÞdx:(2.82)Whatwenowrequireisthatdybedecreasingforsmallincreasesinthevalueofx.ThedifferentialofEquation2.82isgivenbyddyðÞ¼d2y¼d½f0ðxÞdxdx⋅dx¼f00xðÞdx⋅dx¼f00xðÞdx2:(2.83)Butd2y<0impliesthatf00ðxÞdx2<0,(2.84)andsincedx2mustbepositive(becauseanythingsquaredispositive),wehavef00ðxÞ<0(2.85)astherequiredsecond-ordercondition.Inwords,thisconditionrequiresthatthefunctionfhaveaconcaveshapeatthecriticalpoint(contrastFigures2.1and2.2).Similarcurvatureconditionswillbeencounteredthroughoutthissection.EXAMPLE2.8ProﬁtMaximizationAgainInExample2.1weconsideredtheproblemofﬁndingthemaximumofthefunctionπ¼1,000q5q2:(2.86)Theﬁrst-orderconditionforamaximumrequiresdπdq¼1,00010q¼0(2.87)orq¼100:(2.88)Thesecondderivativeofthefunctionisgivenbyd2πdq2¼10<0,(2.89)andhencethepointq¼100obeysthesufﬁcientconditionsforalocalmaximum.QUERY:Herethesecondderivativeisnegativenotonlyattheoptimalpoint;itisalwaysnegative.Whatdoesthatimplyabouttheoptimalpoint?Howshouldthefactthatthesecondderivativeisaconstantbeinterpreted?46Part1Introduction FunctionsoftwovariablesAsasecondcase,weconsideryasafunctionoftwoindependentvariables:y¼fðx1,x2Þ:(2.90)Anecessaryconditionforsuchafunctiontoattainitsmaximumvalueisthatitspartialderivatives,inboththex1andthex2directions,be0.Thatis,∂y∂x1¼f1¼0,∂y∂x2¼f2¼0:(2.91)Apointthatsatisﬁestheseconditionswillbea“ﬂat”spotonthefunction(apointwheredy¼0)andthereforewillbeacandidateforamaximum.Toensurethatthepointisalocalmaximum,ymustdiminishformovementsinanydirectionawayfromthecriticalpoint:Inpictorialtermsthereisonlyonewaytoleaveatruemountaintop,andthatistogodown.AnintuitiveargumentBeforedescribingthemathematicalpropertiesrequiredofsuchapoint,anintuitiveapproachmaybehelpful.Ifweconsideronlymovementsinthex1direction,therequiredconditionisclear:Theslopeinthex1direction(thatis,thepartialderivativef1)mustbediminishingatthecriticalpoint.Thisisadirectapplicationofourdiscussionofthesingle-variablecase.Itshowsthat,foramaximum,thesecondpartialderivativeinthex1directionmustbenegative.Anidenticalargumentholdsformovementsonlyinthex2direction.Hence,bothownsecondpartialderivativesðf11andf22Þmustbenegativeforalocalmaximum.Inourmountainanalogy,ifattentionisconﬁnedonlytonorth-southoreast-westmovements,theslopeofthemountainmustbediminishingaswecrossitssummit—theslopemustchangefrompositivetonegative.Theparticularcomplexitythatarisesinthetwo-variablecaseinvolvesmovementsthroughtheoptimalpointthatarenotsolelyinthex1orx2directions(say,movementsfromnortheasttosouthwest).Insuchcases,thesecond-orderpartialderivativesdonotprovidecompleteinformationabouthowtheslopeischangingnearthecriticalpoint.Conditionsmustalsobeplacedonthecross-partialderivativeðf12¼f21Þtoensurethatdyisdecreasingformovementsthroughthecriticalpointinanydirection.Asweshallsee,thoseconditionsamounttorequiringthattheownsecond-orderpartialderivativesbesufﬁcientlynegativesoastocounterbalanceanypossible“perverse”cross-partialderivativesthatmayexist.Intuitively,ifthemountainfallsawaysteeplyenoughinthenorth-southandeast-westdirections,relativelyminorfailurestodosoinotherdirectionscanbecompensatedfor.AformalanalysisWenowproceedtomakethesepointsmoreformally.Whatwewishtodiscoveraretheconditionsthatmustbeplacedonthesecondpartialderivativesofthefunctionftoensurethatd2yisnegativeformovementsinanydirectionthroughthecriticalpoint.Recallﬁrstthatthetotaldifferentialofthefunctionisgivenbydy¼f1dx1þf2dx2:(2.92)Thedifferentialofthatfunctionisgivenbyd2y¼ðf11dx1þf12dx2Þdx1þðf21dx1þf22dx2Þdx2(2.93)ord2y¼f11dx21þf12dx2dx1þf21dx1dx2þf22dx22:(2.94)Chapter2MathematicsforMicroeconomics47 Because,byYoung’stheorem,f12¼f21,wecanarrangetermstogetd2y¼f11dx21þ2f12dx1dx2þf22dx22:(2.95)ForEquation2.95tobeunambiguouslynegativeforanychangeinthex’s(thatis,foranychoicesofdx1anddx2),itisobviouslynecessarythatf11andf22benegative.If,forexample,dx2¼0,thend2y¼f11dx21(2.96)andd2y<0impliesf11<0:(2.97)Anidenticalargumentcanbemadeforf22bysettingdx1¼0.Ifneitherdx1nordx2is0,wethenmustconsiderthecrosspartial,f12,indecidingwhetherornotd2yisunambiguouslynegative.Relativelysimplealgebracanbeusedtoshowthattherequiredconditionis14f11f22f212>0:(2.98)ConcavefunctionsIntuitively,whatEquation2.98requiresisthattheownsecondpartialderivativesðf11andf22Þbesufﬁcientlynegativesothattheirproduct(whichispositive)willoutweighanypossibleperverseeffectsfromthecross-partialderivativesðf12¼f21Þ.Functionsthatobeysuchaconditionarecalledconcavefunctions.Inthreedimensions,suchfunctionsresembleinvertedteacups(foranillustration,seeExample2.10).Thisimagemakesitclearthataﬂatspotonsuchafunctionisindeedatruemaximumbecausethefunctionalwaysslopesdownwardfromsuchaspot.Moregenerally,concavefunctionshavethepropertythattheyalwaysliebelowanyplanethatistangenttothem—theplanedeﬁnedbythemaximumvalueofthefunctionissimplyaspecialcaseofthisproperty.EXAMPLE2.9Second-OrderConditions:HealthStatusfortheLastTimeInExample2.3weconsideredthehealthstatusfunctiony¼fx1,x2¼x21þ2x1x22þ4×2þ5:(2.99)Theﬁrst-orderconditionsforamaximumaref1¼2×1þ2¼0,f2¼2×2þ4¼0(2.100)orx1¼1,×2¼2:(2.101)14Theproofproceedsbyaddingandsubtractingthetermðf12dx2Þ2=f11toEquation2.95andfactoring.Butthisapproachisonlyapplicabletothisspecialcase.AmoreeasilygeneralizedapproachthatusesmatrixalgebrarecognizesthatEquation2.95isa“QuadraticForm”indx1anddx2,andthatEquations2.97and2.98amounttorequiringthattheHessianmatrixf11f12f21f22be“negativedeﬁnite.”Inparticular,Equation2.98requiresthatthedeterminantofthisHessianbepositive.Foradiscussion,seetheExtensionstothischapter.48Part1Introduction
Thesecond-orderpartialderivativesforEquation2.99aref11¼2,f22¼2,f12¼0:(2.102)ThesederivativesclearlyobeyEquations2.97and2.98,sobothnecessaryandsufﬁcientconditionsforalocalmaximumaresatisﬁed.15QUERY:Describetheconcaveshapeofthehealthstatusfunctionandindicatewhyithasonlyasingleglobalmaximumvalue.ConstrainedmaximizationAsanotherillustrationofsecond-orderconditions,considertheproblemofchoosingx1andx2tomaximizey¼fðx1,x2Þ,(2.103)subjecttothelinearconstraintcb1x1b2x2¼0(2.104)(wherec,b1,b2areconstantparametersintheproblem).Thisproblemisofatypethatwillbefrequentlyencounteredinthisbookandisaspecialcaseoftheconstrainedmaximumproblemsthatweexaminedearlier.Thereweshowedthattheﬁrst-orderconditionsforamaximummaybederivedbysettinguptheLagrangianexpressionℒ¼fðx1,x2Þþλðcb1x1b2x2Þ:(2.105)Partialdifferentiationwithrespecttox1,x2,andλyieldsthefamiliarresults:f1λb1¼0,f2λb2¼0,cb1x1b2x2¼0:(2.106)Theseequationscaningeneralbesolvedfortheoptimalvaluesofx1,x2,andλ.Toensurethatthepointderivedinthatwayisalocalmaximum,wemustagainexaminemovementsawayfromthecriticalpointsbyusingthe“second”totaldifferential:d2y¼f11dx21þ2f12dx1dx2þf22dx22:(2.107)Inthiscase,however,notallpossiblesmallchangesinthex’sarepermissible.Onlythosevaluesofx1andx2thatcontinuetosatisfytheconstraintcanbeconsideredvalidalternativestothecriticalpoint.Toexaminesuchchanges,wemustcalculatethetotaldifferentialoftheconstraint:b1dx1b2dx2¼0(2.108)ordx2¼b1b2dx1:(2.109)15NoticethatEquations2.102obeythesufﬁcientconditionsnotonlyatthecriticalpointbutalsoforallpossiblechoicesofx1andx2.Thatis,thefunctionisconcave.Inmorecomplexexamplesthisneednotbethecase:Thesecond-orderconditionsneedbesatisﬁedonlyatthecriticalpointforalocalmaximumtooccur.Chapter2MathematicsforMicroeconomics49
Thisequationshowstherelativechangesinx1andx2thatareallowableinconsideringmovementsfromthecriticalpoint.Toproceedfurtheronthisproblem,weneedtousetheﬁrst-orderconditions.Theﬁrsttwooftheseimplyf1f2¼b1b2,(2.110)andcombiningthisresultwithEquation2.109yieldsdx2¼f1f2dx1:(2.111)Wenowsubstitutethisexpressionfordx2inEquation2.107todemonstratetheconditionsthatmustholdford2ytobenegative:d2y¼f11dx21þ2f12dx1f1f2dx1þf22f1f2dx12¼f11dx212f12f1f2dx21þf22f21f22dx21:(2.112)Combiningtermsandputtingeachoveracommondenominatorgivesd2y¼ðf11f222f12f1f2þf22f21Þdx21f22:(2.113)Consequently,ford2y<0,itmustbethecasethatf11f222f12f1f2þf22f21<0:(2.114)Quasi-concavefunctionsAlthoughEquation2.114appearstobelittlemorethananinordinatelycomplexmassofmathematicalsymbols,infacttheconditionisanimportantone.Itcharacterizesasetoffunctionstermedquasi-concavefunctions.Thesefunctionshavethepropertythatthesetofallpointsforwhichsuchafunctiontakesonavaluegreaterthananyspeciﬁcconstantisaconvexset(thatis,anytwopointsinthesetcanbejoinedbyalinecontainedcompletelywithintheset).Manyeconomicmodelsarecharacterizedbysuchfunctionsand,aswewillseeinconsiderabledetailinChapter3,inthesecasestheconditionforquasi-concavityhasarelativelysimpleeconomicinterpretation.Problems2.9and2.10examinetwospeciﬁcquasi-concavefunctionsthatwewillfrequentlyencounterinthisbook.Example2.10showstherelationshipbetweenconcaveandquasi-concavefunctions.EXAMPLE2.10ConcaveandQuasi-ConcaveFunctionsThedifferencesbetweenconcaveandquasi-concavefunctionscanbeillustratedwiththefunction16y¼fðx1,x2Þ¼ðx1⋅x2Þk,(2.115)wherethex’stakeononlypositivevalues,andtheparameterkcantakeonavarietyofpositivevalues.16ThisfunctionisaspecialcaseoftheCobb-Douglasfunction.SeealsoProblem2.10andtheExtensionstothischapterformoredetailsonthisfunction.50Part1Introduction Nomatterwhatvaluektakes,thisfunctionisquasi-concave.Onewaytoshowthisistolookatthe“levelcurves”ofthefunctionbysettingyequaltoaspeciﬁcvalue,sayc.Inthiscasey¼c¼ðx1x2Þkorx1x2¼c1=k¼c0:(2.116)Butthisisjusttheequationofastandardrectangularhyperbola.Clearlythesetofpointsforwhichytakesonvalueslargerthancisconvexbecauseitisboundedbythishyperbola.Amoremathematicalwaytoshowquasi-concavitywouldapplyEquation2.114tothisfunction.Althoughthealgebraofdoingthisisabitmessy,itmaybeworththestruggle.ThevariouscomponentsofEquation2.114are:f1¼kxk11xk2,f2¼kxk1xk12,f11¼kðk1Þxk21xk2,f22¼kðk1Þxk1xk22,f12¼k2xk11xk12:(2.117)So,f11f222f12f1f2þf22f21¼k3ðk1Þx3k21x3k222k4x3k21x3k22þk3ðk1Þx3k21x3k22¼2k3x3k21x3k22ð1Þ,(2.118)whichisclearlynegative,asisrequiredforquasi-concavity.Whetherornotthefunctionfisconcavedependsonthevalueofk.Ifk<0:5thefunctionisindeedconcave.Anintuitivewaytoseethisistoconsideronlypointswherex1¼x2.Forthesepoints,y¼ðx21Þk¼x2k1,(2.119)which,fork<0:5,isconcave.Alternatively,fork>0:5,thisfunctionisconvex.AmoredeﬁnitiveproofmakesuseofthepartialderivativesfromEquation2.117.Inthiscasetheconditionforconcavitycanbeexpressedasf11f22f212¼k2ðk1Þ2x2k21x2k22k4x2k21x2k22¼x2k21x2k22½k2ðk1Þ2k4¼x2k11x2k12½k2ð2kþ1Þ,(2.120)andthisexpressionispositive(asisrequiredforconcavity)forð2kþ1Þ>0ork<0:5:Ontheotherhand,thefunctionisconvexfork>0:5.Agraphicillustration.Figure2.4providesthree-dimensionalillustrationsofthreespeciﬁcexamplesofthisfunction:fork¼0:2,k¼0:5,andk¼1.Noticethatinallthreecasesthelevelcurvesofthefunctionhavehyperbolic,convexshapes.Thatis,foranyﬁxedvalueofythefunctionsarequitesimilar.Thisshowsthequasi-concavityofthefunction.Theprimarydifferencesamongthefunctionsareillustratedbythewayinwhichthevalueofyincreasesas(continued)Chapter2MathematicsforMicroeconomics51
EXAMPLE2.10CONTINUEDbothx’sincreasetogether.InFigure2.4a(whenk¼0:2),theincreaseinyslowsasthex’sincrease.Thisgivesthefunctionarounded,teacuplikeshapethatindicatesitsconcavity.Fork¼0:5,yappearstoincreaselinearlywithincreasesinbothofthex’s.Thisistheborderlinebetweenconcavityandconvexity.Finally,whenk¼1(asinFigure2.4c),simultaneousincreasesinthevaluesofbothofthex’sincreaseyveryrapidly.Thespineofthefunctionlooksconvextoreﬂectsuchincreasingreturns.FIGURE2.4ConcaveandQuasi-ConcaveFunctionsInallthreecasesthesefunctionsarequasi-concave.Foraﬁxedy,theirlevelcurvesareconvex.Butonlyfork¼0:2isthefunctionstrictlyconcave.Thecasek¼1:0clearlyshowsnonconcavitybecausethefunctionisnotbelowitstangentplane.(a) k= 0.2(b) k= 0.5(c) k= 1.052Part1Introduction
AcarefullookatFigure2.4asuggeststhatanyfunctionthatisconcavewillalsobequasi-concave.YouareaskedtoprovethatthisisindeedthecaseinProblem2.8.Thisexampleshowsthattheconverseofthisstatementisnottrue—quasi-concavefunctionsneednotnecessarilybeconcave.Mostfunctionswewillencounterinthisbookwillalsoillustratethisfact;mostwillbequasi-concavebutnotnecessarilyconcave.QUERY:ExplainwhythefunctionsillustratedbothinFigure2.4aand2.4cwouldhavemaxi-mumvaluesifthex’sweresubjecttoalinearconstraint,butonlythegraphinFigure2.4awouldhaveanunconstrainedmaximum.HOMOGENEOUSFUNCTIONSManyofthefunctionsthatarisenaturallyoutofeconomictheoryhaveadditionalmathemat-icalproperties.Oneparticularlyimportantsetofpropertiesrelatestohowthefunctionsbehavewhenall(ormost)oftheirargumentsareincreasedproportionally.Suchsituationsarisewhenweaskquestionssuchaswhatwouldhappenifallpricesincreasedby10percentorhowwouldaﬁrm’soutputchangeifitdoubledalloftheinputsthatituses.Thinkingaboutthesequestionsleadsnaturallytotheconceptofhomogeneousfunctions.Speciﬁcally,afunctionfðx1,x2,…,xnÞissaidtobehomogeneousofdegreekiffðtx1,tx2,…,txnÞ¼tkfðx1,x2,…,xnÞ:(2.121)Themostimportantexamplesofhomogeneousfunctionsarethoseforwhichk¼1ork¼0.Inwords,whenafunctionishomogeneousofdegreeone,adoublingofallofitsargumentsdoublesthevalueofthefunctionitself.Forfunctionsthatarehomogeneousofdegree0,adoublingofallofitsargumentsleavesthevalueofthefunctionunchanged.Functionsmayalsobehomogeneousforchangesinonlycertainsubsetsoftheirarguments—thatis,adoublingofsomeofthex’smaydoublethevalueofthefunctioniftheotherargumentsofthefunctionareheldconstant.Usually,however,homogeneityappliestochangesinalloftheargumentsinafunction.HomogeneityandderivativesIfafunctionishomogeneousofdegreekandcanbedifferentiated,thepartialderivativesofthefunctionwillbehomogeneousofdegreek1.Aproofofthisfollowsdirectlyfromthedeﬁnitionofhomogeneity.Forexample,differentiatingEquation2.121withrespecttoitsﬁrstargumentgives∂fðtx1,…,txnÞ∂x1⋅t¼tk∂fðx1,…,xnÞ∂x1orf1ðtx1,…,txnÞ¼tk1f1ðx1,…,xnÞ,(2.122)whichshowsthatf1meetsthedeﬁnitionforhomogeneityofdegreek1.Becausemarginalideasaresoprevalentinmicroeconomictheory,thispropertyshowsthatsomeimportantpropertiesofmarginaleffectscanbeinferredfromthepropertiesoftheunderlyingfunctionitself.Chapter2MathematicsforMicroeconomics53
Euler’stheoremAnotherusefulfeatureofhomogeneousfunctionscanbeshownbydifferentiatingthedeﬁnitionforhomogeneitywithrespecttotheproportionalityfactor,t.Inthiscase,wedifferentiatetherightsideofEquation2.121ﬁrst:ktk1f1ðx1,…,xnÞ¼x1f1ðtx1,…,txnÞþ…þxnfnðtx1,…,txnÞ:Ifwelett¼1,thisequationbecomeskfðx1,…,xnÞ¼x1f1ðx1,…,xnÞþ…þxnfnðx1,…,xnÞ:(2.123)ThisequationistermedEuler’stheorem(afterthemathematicianwhoalsodiscoveredtheconstante)forhomogeneousfunctions.Itshowsthat,forahomogeneousfunction,thereisadeﬁniterelationshipbetweenthevaluesofthefunctionandthevaluesofitspartialderivatives.Severalimportanteconomicrelationshipsamongfunctionsarebasedonthisobservation.HomotheticfunctionsAhomotheticfunctionisonethatisformedbytakingamonotonictransformationofahomogeneousfunction.17Monotonictransformations,bydeﬁnition,preservetheorderoftherelationshipbetweentheargumentsofafunctionandthevalueofthatfunction.Ifcertainsetsofx’syieldlargervaluesforf,theywillalsoyieldlargervaluesforamonotonictransfor-mationoff.Becausemonotonictransformationsmaytakemanyforms,however,theywouldnotbeexpectedtopreserveanexactmathematicalrelationshipsuchasthatembodiedinhomogeneousfunctions.Consider,forexample,thefunctionfðx,yÞ¼x⋅y.Clearlythisfunctionishomogeneousofdegree2—adoublingofitstwoargumentswillmultiplythevalueofthefunctionby4.Butthemonotonictransformation,F,thatsimplyadds1tof[thatis,FðfÞ¼fþ1¼xyþ1]isnothomogeneousatall.Hence,exceptinspecialcases,homo-theticfunctionsdonotpossessthehomogeneitypropertiesoftheirunderlyingfunctions.Homotheticfunctionsdo,however,preserveonenicefeatureofhomogeneousfunctions.Thispropertyisthattheimplicittrade-offsamongthevariablesinafunctiondependonlyontheratiosofthosevariables,notontheirabsolutevalues.Hereweshowthisforthesimpletwo-variable,implicitfunctionfðx,yÞ¼0.Itwillbeeasiertodemonstratemoregeneralcaseswhenwegettotheeconomicsofthematterlaterinthisbook.Equation2.28showedthattheimplicittrade-offbetweenxandyforatwo-variablefunctionisgivenbydydx¼fxfy:Ifweassumefishomogeneousofdegreek,itspartialderivativeswillbehomogeneousofdegreek1andtheimplicittrade-offbetweenxandyisdydx¼tk1fxðtx,tyÞtk1fyðtx,tyÞ¼fxðtx,tyÞfyðtx,tyÞ:(2.124)Nowlett¼1=yandEquation2.124becomesdydx¼fxðx=y,1Þfyðx=y,1Þ,(2.125)whichshowsthatthetrade-offdependsonlyontheratioofxtoy.Nowifweapplyanymonotonictransformation,F(withF0>0),totheoriginalhomogeneousfunctionf,wehave17Becausealimitingcaseofamonotonictransformationistoleavethefunctionunchanged,allhomogeneousfunctionsarealsohomothetic.54Part1Introduction
dydx¼F0fxðx=y,1ÞF0fyðx=y,1Þ¼fxðx=y,1Þfyðx=y,1Þ,(2.126)andthisshowsboththatthetrade-offisunaffectedbythemonotonictransformationandthatitremainsafunctiononlyoftheratioofxtoy.InChapter3(andelsewhere)thispropertywillmakeitveryconvenienttodiscusssometheoreticalresultswithsimpletwo-dimensionalgraphs,forwhichweneednotconsidertheoveralllevelsofkeyvariables,butonlytheirratios.EXAMPLE2.11CardinalandOrdinalPropertiesInappliedeconomicsitissometimesimportanttoknowtheexactnumericalrelationshipamongvariables.Forexample,inthestudyofproduction,onemightwishtoknowpreciselyhowmuchextraoutputwouldbeproducedbyhiringanotherworker.Thisisaquestionaboutthe“cardinal”(i.e.,numerical)propertiesoftheproductionfunction.Inothercases,onemayonlycareabouttheorderinwhichvariouspointsareranked.Inthetheoryofutility,forexample,weassumethatpeoplecanrankbundlesofgoodsandwillchoosethebundlewiththehighestranking,butthattherearenouniquenumericalvaluesassignedtotheserankings.Mathematically,ordinalpropertiesoffunctionsarepreservedbyanymonotonictransformationbecause,bydeﬁnition,amonotonictransformationpre-servesorder.Usually,however,cardinalpropertiesarenotpreservedbyarbitrarymono-tonictransformations.ThesedistinctionsareillustratedbythefunctionsweexaminedinExample2.10.Therewestudiedmonotonictransformationsofthefunctionfðx1,x2Þ¼ðx1x2Þk(2.127)byconsideringvariousvaluesoftheparameterk.Weshowedthatquasi-concavity(anordinalproperty)waspreservedforallvaluesofk.Hence,whenapproachingproblemsthatfocusonmaximizingorminimizingsuchafunctionsubjecttolinearconstraintsweneednotworryaboutpreciselywhichtransformationisused.Ontheotherhand,thefunctioninEquation2.127isconcave(acardinalproperty)onlyforanarrowrangeofvaluesofk.Manymonotonictransformationsdestroytheconcavityoff.ThefunctioninEquation2.127alsocanbeusedtoillustratethedifferencebetweenhomogeneousandhomotheticfunctions.Aproportionalincreaseinthetwoargumentsoffwouldyieldfðtx1,tx2Þ¼t2kx1x2¼t2kfðx1,x2Þ:(2.128)Hence,thedegreeofhomogeneityforthisfunctiondependsonk—thatis,thedegreeofhomogeneityisnotpreservedindependentlyofwhichmonotonictransformationisused.Alternatively,thefunctioninEquation2.127ishomotheticbecausedx2dx1¼f1f2¼kxk11xk2kxk1xk12¼x2x1:(2.129)Thatis,thetrade-offbetweenx2andx1dependsonlyontheratioofthesetwovariablesandisunaffectedbythevalueofk.Hence,homotheticityisanordinalproperty.Asweshallsee,thispropertyisquiteconvenientwhendevelopinggraphicalargumentsabouteconomicpropositions.QUERY:Howwouldthediscussioninthisexamplebechangedifweconsideredmonotonictransformationsoftheformfðx1,x2,kÞ¼x1x2þkforvariousvaluesofk?Chapter2MathematicsforMicroeconomics55
INTEGRATIONIntegrationisanotherofthetoolsofcalculusthatﬁndsanumberofapplicationsinmicroeco-nomictheory.Thetechniqueisusedbothtocalculateareasthatmeasurevariouseconomicoutcomesand,moregenerally,toprovideawayofsummingupoutcomesthatoccurovertimeoracrossindividuals.Ourtreatmentofthetopicherenecessarilymustbebrief,soreadersdesiringamorecompletebackgroundshouldconsultthereferencesattheendofthischapter.Anti-derivativesFormally,integrationistheinverseofdifferentiation.Whenyouareaskedtocalculatetheintegralofafunction,fðxÞ,youarebeingaskedtoﬁndafunctionthathasfðxÞasitsderivative.Ifwecallthis“anti-derivative”FðxÞ,thisfunctionissupposedtohavethepropertythatdFðxÞdx¼F0ðxÞ¼fðxÞ:(2.130)IfsuchafunctionexiststhenwedenoteitasFðxÞ¼∫fðxÞdx:(2.131)Theprecisereasonforthisratherodd-lookingnotationwillbedescribedindetaillater.First,let’slookatafewexamples.IffðxÞ¼xthenFðxÞ¼∫fðxÞdx¼∫xdx¼x22þC,(2.132)whereCisanarbitrary“constantofintegration”thatdisappearsupondifferentiation.Thecorrectnessofthisresultcanbeeasilyveriﬁed:F0ðxÞ¼dðx2=2þCÞdx¼xþ0¼x:(2.133)Calculatinganti-derivativesCalculationofanti-derivativescanbeextremelysimple,ordifﬁcult,oragonizing,orimpossi-ble,dependingontheparticularfðxÞspeciﬁed.Herewewilllookatthreesimplemethodsformakingsuchcalculations,but,asyoumightexpect,thesewillnotalwayswork.1.Creativeguesswork.Probablythemostcommonwayofﬁndingintegrals(anti-derivatives)istoworkbackwardsbyasking“whatfunctionwillyieldfðxÞasitsderivative?”Hereareafewobviousexamples:FðxÞ¼∫x2dx¼x33þC,FðxÞ¼∫xndx¼xnþ1nþ1þC,FðxÞ¼∫ðax2þbxþcÞdx¼ax33þbx22þcxþC,FðxÞ¼∫exdx¼exþC,FðxÞ¼∫axdx¼axlnaþC,FðxÞ¼∫1xdx¼lnðjxjÞþC,FðxÞ¼∫ðlnxÞdx¼xlnxxþC:(2.134)56Part1Introduction
YoushouldusedifferentiationtocheckthatalloftheseobeythepropertythatF0ðxÞ¼fðxÞ.Noticethatineverycasetheintegralincludesaconstantofintegrationbecauseanti-deriva-tivesareuniqueonlyuptoanadditiveconstantwhichwouldbecomezeroupondifferentia-tion.Formanypurposes,theresultsinEquation2.134(ortrivialgeneralizationsofthem)willbesufﬁcientforourpurposesinthisbook.Nevertheless,herearetwomoremethodsthatmayworkwhenintuitionfails.2.Changeofvariable.Acleverredeﬁnitionofvariablesmaysometimesmakeafunctionmucheasiertointegrate.Forexample,itisnotatallobviouswhattheintegralof2x=ð1þx2Þis.But,ifwelety¼1þx2,thendy¼2xdxand∫2×1þx2dx¼∫1ydy¼lnðjyjÞ¼lnðj1þx2jÞ:(2.135)Thekeytothisprocedureisinbreakingtheoriginalfunctionintoaterminyandatermindy.Ittakesalotofpracticetoseepatternsforwhichthiswillwork.3.Integrationbyparts.Asimilarmethodforﬁndingintegralsmakesuseofthedifferen-tialexpressionduv¼udvþvduforanytwofunctionsuandv.Integrationofthisdifferentialyields∫duv¼uv¼∫udvþ∫vduor∫udv¼uv∫vdu:(2.136)Herethestrategyistodeﬁnefunctionsuandvinawaythattheunknownintegralontheleftcanbecalculatedbythedifferencebetweenthetwoknownexpressionsontheright.Forexample,itisbynomeansobviouswhattheintegralofxexis.Butwecandeﬁneu¼x(sodu¼dx)anddv¼exdx(sov¼ex).Hencewenowhave∫xexdx¼∫udv¼uv∫vdu¼xex∫exdx¼ðx1ÞexþC:(2.137)Again,onlypracticecansuggestusefulpatternsinthewaysinwhichuandvcanbedeﬁned.DeﬁniteintegralsTheintegralswehavebeendiscussingsofarare“indeﬁnite”integrals—theyprovideonlyageneralfunctionthatistheanti-derivativeofanotherfunction.Asomewhatdifferent,thoughrelated,approachusesintegrationtosumuptheareaunderagraphofafunctionoversomedeﬁnedinterval.Figure2.5illustratesthisprocess.WewishtoknowtheareaunderthefunctionfðxÞfromx¼atox¼b.Onewaytodothiswouldbetopartitiontheintervalintonarrowsliversofxð∆xÞandsumuptheareasoftherectanglesshownintheﬁgure.Thatis:areaunderfðxÞXifðxiÞ∆xi,(2.138)wherethenotationisintendedtoindicatethattheheightofeachrectangleisapproximatedbythevalueoffðxÞforavalueofxintheinterval.Takingthisprocesstothelimitbyshrinkingthesizeofthe∆xintervalsyieldsanexactmeasureoftheareawewantandisdenotedby:areaunderfðxÞ¼∫x¼bx¼afðxÞdx:(2.139)Thisthenexplainstheoriginoftheoddlyshapedintegralsign—itisastylizedS,indicating“sum.”Asweshallsee,integratingisaverygeneralwayofsummingthevaluesofacontinuousfunctionoversomeinterval.Chapter2MathematicsforMicroeconomics57
FundamentaltheoremofcalculusEvaluatingtheintegralinEquation2.139isverysimpleifweknowtheanti-derivativeoffðxÞ,say,FðxÞ.InthiscasewehaveareaunderfðxÞ¼∫x¼bx¼afðxÞdx¼FðbÞFðaÞ:(2.140)Thatis,allweneeddoiscalculatetheanti-derivativeoffðxÞandsubtractthevalueofthisfunctionatthelowerlimitofintegrationfromitsvalueattheupperlimitofintegration.Thisresultissometimestermedthe“fundamentaltheoremofcalculus”becauseitdirectlytiestogetherthetwoprincipaltoolsofcalculus,derivativesandintegrals.InExample2.12,weshowthatthisresultismuchmoregeneralthansimplyawaytomeasureareas.Itcanbeusedtoillustrateoneoftheprimaryconceptualprinciplesofeconomics—thedistinctionbetween“stocks”and“ﬂows.”EXAMPLE2.12StocksandFlowsThedeﬁniteintegralprovidesausefulwayforsummingupanyfunctionthatisprovidingacontinuousﬂowovertime.Forexample,supposethatnetpopulationincrease(birthsminusdeaths)foracountrycanbeapproximatedbythefunctionfðtÞ¼1,000e0:02t.Hence,thenetpopulationchangeisgrowingattherateof2percentperyear—itis1,000newpeopleinyear0,1,020newpeopleintheﬁrstyear,1,041inthesecondyear,andsoforth.Supposewewishtoknowhowmuchintotalthepopulationwillincreasewithin50years.Thismightbeatediouscalculationwithoutcalculus,butusingthefundamentaltheoremofcalculusprovidesaneasyanswer:FIGURE2.5DeﬁniteIntegralsShowtheAreasundertheGraphofaFunctionDeﬁniteintegralsmeasuretheareaunderacurvebysummingrectangularareasasshowninthegraph.ThedimensionofeachrectangleisfxðÞdx.f(x)f(x)abx58Part1Introduction
increaseinpopulation¼∫t¼50t¼0fðtÞdt¼∫t¼50t¼01,000e0:02tdt¼FðtÞ500¼1,000e0:02t0:02500¼1,000e0:0250,000¼85,914(2:141)[wherethenotationjbaindicatesthattheexpressionistobeevaluatedasFðbÞFðaÞ].Hence,theconclusionisthatthepopulationwillgrowbynearly86,000peopleoverthenext50years.Noticehowthefundamentaltheoremofcalculustiestogethera“ﬂow”concept,netpopulationincrease(whichismeasuredasanamountperyear),witha“stock”concept,totalpopulation(whichismeasuredataspeciﬁcdateanddoesnothaveatimedimension).Notealsothatthe86,000calculationrefersonlytothetotalincreasebetweenyearzeroandyearﬁfty.Inordertoknowtheactualtotalpopulationatanydatewewouldhavetoaddthenumberofpeopleinthepopulationatyearzero.Thatwouldbesimilartochoosingaconstantofintegrationinthisspeciﬁcproblem.Nowconsideranapplicationwithmoreeconomiccontent.SupposethattotalcostsforaparticularﬁrmaregivenbyCðqÞ¼0:1q2þ500(whereqrepresentsoutputduringsomeperiod).Heretheterm0:1q2representsvariablecosts(coststhatvarywithoutput)whereasthe500ﬁgurerepresentsﬁxedcosts.Marginalcostsforthisproductionprocesscanbefoundthroughdifferentiation—MC¼dCðqÞ=dq¼0:2q—hence,marginalcostsareincreasingwithqandﬁxedcostsdropoutupondifferentiation.Whatarethetotalcostsassociatedwithproducing,say,q¼100?Onewaytoanswerthisquestionistousethetotalcostfunctiondirectly:Cð100Þ¼0:1ð100Þ2þ500¼1,500.Analternativewaywouldbetointegratemarginalcostovertherange0to100togettotalvariablecost:variablecost¼∫q¼100q¼00:2qdq¼0:1q21000¼1,0000¼1,000,(2.142)towhichwewouldhavetoaddﬁxedcostsof500(theconstantofintegrationinthisproblem)togettotalcosts.Ofcourse,thismethodofarrivingattotalcostismuchmorecumbersomethanjustusingtheequationfortotalcostdirectly.Butthederivationdoesshowthattotalvariablecostbetweenanytwooutputlevelscanbefoundthroughintegrationastheareabelowthemarginalcostcurve—aconclusionthatwewillﬁndusefulinsomegraphicalapplications.QUERY:Howwouldyoucalculatethetotalvariablecostassociatedwithexpandingoutputfrom100to110?Explainwhyﬁxedcostsdonotenterintothiscalculation.DiﬀerentiatingadeﬁniteintegralOccasionallywewillwishtodifferentiateadeﬁniteintegral—usuallyinthecontextofseekingtomaximizethevalueofthisintegral.Althoughperformingsuchdifferentiationscansome-timesberathercomplex,thereareafewrulesthatshouldmaketheprocesseasier.1.Differentiationwithrespecttothevariableofintegration.Thisisatrickquestion,butinstructivenonetheless.Adeﬁniteintegralhasaconstantvalue;henceitsderivativeiszero.Thatis:d∫bafðxÞdxdx¼0:(2.143)Thesummingprocessrequiredforintegrationhasalreadybeenaccomplishedoncewewritedownadeﬁniteintegral.ItdoesnotmatterwhetherthevariableofintegrationisxortorChapter2MathematicsforMicroeconomics59
anythingelse.Thevalueofthisintegratedsumwillnotchangewhenthevariablexchanges,nomatterwhatxis(butseerule3below).2.Differentiationwithrespecttotheupperboundofintegration.Changingtheupperboundofintegrationwillobviouslychangethevalueofadeﬁniteintegral.Inthiscase,wemustmakeadistinctionbetweenthevariabledeterminingtheupperboundofintegration(say,x)andthevariableofintegration(say,t).Theresultthenisasimpleapplicationofthefundamentaltheoremofcalculus.Forexample:d∫xafðtÞdtdx¼d½FðxÞFðaÞdx¼fðxÞ0¼fðxÞ,(2.144)whereFðxÞistheantiderivativeoffðxÞ.ByreferringbacktoFigure2.5wecanseewhythisconclusionmakessense—weareaskinghowthevalueofthedeﬁniteintegralchangesifxincreasesslightly.Obviously,theansweristhatthevalueoftheintegralincreasesbytheheightoffðxÞ(noticethatthisvaluewillultimatelydependonthespeciﬁedvalueofx).Iftheupperboundofintegrationisafunctionofx,thisresultcanbegeneralizedusingthechainrule:d∫gðxÞafðtÞdtdðxÞ¼d½FðgðxÞÞFðaÞdx¼d½FðgðxÞÞdx¼fdgðxÞdx¼fg0ðxÞ,(2.145)where,again,thespeciﬁcvalueforthisderivativewoulddependonthevalueofxassumed.Finally,noticethatdifferentiationwithrespecttoalowerboundofintegrationjustchangesthesignofthisexpression:d∫bgðxÞfðtÞdtdx¼d½FðbÞFðgðxÞÞdx¼dFðgðxÞÞdx¼fg0ðxÞ:(2.146)3.Differentiationwithrespecttoanotherrelevantvariable.Insomecaseswemaywishtointegrateanexpressionthatisafunctionofseveralvariables.Ingeneral,thiscaninvolvemultipleintegralsanddifferentiationcanbecomequitecomplicated.Butthereisonesimplecasethatshouldbementioned.Supposethatwehaveafunctionoftwovariables,fðx,yÞ,andthatwewishtointegratethisfunctionwithrespecttothevariablex.Thespeciﬁcvalueforthisintegralwillobviouslydependonthevalueofyandwemightevenaskhowthatvaluechangeswhenychanges.Inthiscase,itispossibleto“differentiatethroughtheintegralsign”toobtainaresult.Thatis:d∫bafðx,yÞdxdy¼∫bafyðx,yÞdx:(2.147)Thisexpressionshowsthatwecanﬁrstpartiallydifferentiatefðx,yÞwithrespecttoybeforeproceedingtocomputethevalueofthedeﬁniteintegral.Ofcourse,theresultingvaluemaystilldependonthespeciﬁcvaluethatisassignedtoy,butoftenitwillyieldmoreeconomicinsightsthantheoriginalproblemdoes.SomefurtherexamplesofusingdeﬁniteintegralsarefoundinProblem2.8.DYNAMICOPTIMIZATIONSomeoptimizationproblemsthatariseinmicroeconomicsinvolvemultipleperiods.18Weareinterestedinﬁndingtheoptimaltimepathforavariableorsetofvariablesthatsucceedsinoptimizingsomegoal.Forexample,anindividualmaywishtochooseapathoflifetime18Throughoutthissectionwetreatdynamicoptimizationproblemsasoccurringovertime.Inothercontexts,thesametechniquescanbeusedtosolveoptimizationproblemsthatoccuracrossacontinuumofﬁrmsorindividualswhentheoptimalchoicesforoneagentaffectwhatisoptimalforothers.60Part1Introduction
consumptionsthatmaximizeshisorherutility.Oraﬁrmmayseekapathforinputandoutputchoicesthatmaximizesthepresentvalueofallfutureproﬁts.Theparticularfeatureofsuchproblemsthatmakesthemdifﬁcultisthatdecisionsmadeinoneperiodaffectoutcomesinlaterperiods.Hence,onemustexplicitlytakeaccountofthisinterrelationshipinchoosingoptimalpaths.Ifdecisionsinoneperioddidnotaffectlaterperiods,theproblemwouldnothavea“dynamic”structure—onecouldjustproceedtooptimizedecisionsineachperiodwithoutregardforwhatcomesnext.Here,however,wewishtoexplicitlyallowfordynamicconsiderations.TheoptimalcontrolproblemMathematiciansandeconomistshavedevelopedmanytechniquesforsolvingproblemsindynamicoptimization.Thereferencesattheendofthischapterprovidebroadintroductionstothesemethods.Here,however,wewillbeconcernedwithonlyonesuchmethodthathasmanysimilaritiestotheoptimizationtechniquesdiscussedearlierinthischapter—theoptimalcontrolproblem.Theframeworkoftheproblemisrelativelysimple.AdecisionmakerwishestoﬁndtheoptimaltimepathforsomevariablexðtÞoveraspeciﬁedtimeinterval½t0,t1.Changesinxaregovernedbyadifferentialequation:dxðtÞdt¼g½xðtÞ,cðtÞ,t,(2.148)wherethevariablecðtÞisusedto“control”thechangeinxðtÞ.Ineachperiodoftime,thedecisionmakerderivesvaluefromxandcaccordingtothefunctionf½xðtÞ,cðtÞ,tandhisorhergoaltooptimize∫t1t0f½xðtÞ,cðtÞ,tdt.Oftenthisproblemwillalsobesubjectto“endpoint”constraintsonthevariablex.Thesemightbewrittenasxðt0Þ¼x0andxðt1Þ¼x1.Noticehowthisproblemis“dynamic.”Anydecisionabouthowmuchtochangexthisperiodwillaffectnotonlythefuturevalueofx,itwillalsoaffectfuturevaluesoftheoutcomefunctionf.TheproblemthenishowtokeepxðtÞonitsoptimalpath.Economicintuitioncanhelptosolvethisproblem.Supposethatwejustfocusedonthefunctionfandchosexandctomaximizeitateachinstantoftime.Therearetwodifﬁcultieswiththis“myopic”approach.First,wearenotreallyfreeto“choose”xatanytime.Rather,thevalueofxwillbedeterminedbyitsinitialvaluex0andbyitshistoryofchangesasgivenbyEquation2.148.Asecondproblemwiththismyopicapproachisthatitdisregardsthedynamicnatureoftheproblembynotaskinghowthisperiod’sdecisionsaffectthefuture.Weneedsomewaytoreﬂectthedynamicsofthisprobleminasingleperiod’sdecisions.Assigningthecorrectvalue(price)toxateachinstantoftimewilldojustthat.BecausethisimplicitpricewillhavemanysimilaritiestotheLagrangianmultipliersstudiedearlierinthischapter,wewillcallitλðtÞ.Thevalueofxistreatedasafunctionoftimebecausetheimportanceofxcanobviouslychangeovertime.ThemaximumprincipleNowlet’slookatthedecisionmaker’sproblematasinglepointintime.Heorshemustbeconcernedwithboththecurrentvalueoftheobjectivefunctionf½xðtÞ,cðtÞ,tandwiththeimpliedchangeinthevalueofxðtÞ.BecausethecurrentvalueofxðtÞisgivenbyλðtÞxðtÞ,theinstantaneousrateofchangeofthisvalueisgivenby:d½λðtÞxðtÞdt¼λðtÞdxðtÞdtþxðtÞdλðtÞdt,(2.149)Chapter2MathematicsforMicroeconomics61
andsoatanytimetacomprehensivemeasureofthevalueofconcern19tothedecisionmakerisH¼f½xðtÞ,cðtÞ,tþλðtÞg½xðtÞ,cðtÞ,tþxðtÞdλðtÞdt:(2.150)Thiscomprehensivevaluerepresentsboththecurrentbeneﬁtsbeingreceivedandtheinstantaneouschangeinthevalueofx.NowwecanaskwhatconditionsmustholdforxðtÞandcðtÞtooptimizethisexpression.20Thatis:∂H∂c¼fcþλgc¼0orfc¼λgc;∂H∂x¼fxþλgxþ∂λðtÞdt¼0orfxþλgx¼∂λðtÞ∂t:(2.151)Thesearethenthetwooptimalityconditionsforthisdynamicproblem.Theyareusuallyreferredtoasthe“maximumprinciple.”ThissolutiontotheoptimalcontrolproblemwasﬁrstproposedbytheRussianmathematicianL.S.Pontryaginandhiscolleaguesintheearly1960s.Althoughthelogicofthemaximumprinciplecanbestbeillustratedbytheeconomicapplicationswewillencounterlaterinthisbook,abriefsummaryoftheintuitionbehindthemmaybehelpful.Theﬁrstconditionasksabouttheoptimalchoiceofc.Itsuggeststhat,atthemargin,thegainfromcintermsofthefunctionfmustbebalancedbythelossesfromcintermsofthevalueofitsabilitytochangex.Thatis,presentgainsmustbeweighedagainstfuturecosts.ThesecondconditionrelatestothecharacteristicsthatanoptimaltimepathofxðtÞshouldhave.Itimpliesthat,atthemargin,anynetgainsfrommorecurrentx(eitherintermsofforintermsoftheaccompanyingvalueofchangesinx)mustbebalancedbychangesintheimpliedvalueofxitself.Thatis,thenetcurrentgainfrommorexmustbeweighedagainstthedecliningfuturevalueofx.EXAMPLE2.13AllocatingaFixedSupplyAsanextremelysimpleillustrationofthemaximumprinciple,assumethatsomeonehasinherited1,000bottlesofwinefromarichuncle.Heorsheintendstodrinkthesebottlesoverthenext20years.Howshouldthisbedonetomaximizetheutilityfromdoingso?Supposethatthisperson’sutilityfunctionforwineisgivenbyu½cðtÞ¼lncðtÞ.Hencetheutilityfromwinedrinkingexhibitsdiminishingmarginalutilityðu0>0,u00<0Þ.Thisper-son’sgoalistomaximize∫200u½cðtÞdt¼∫200lncðtÞdt:(2.152)LetxðtÞrepresentthenumberofbottlesofwineremainingattimet.Thisseriesiscon-strainedbyxð0Þ¼1,000andxð20Þ¼0.Thedifferentialequationdeterminingtheevolu-tionofxðtÞtakesthesimpleform:2119WedenotethiscurrentvalueexpressionbyHtosuggestitssimilaritytotheHamiltonianexpressionusedinformaldynamicoptimizationtheory.UsuallytheHamiltoniandoesnothavetheﬁnalterminEquation2.150,however.20Noticethatthevariablexisnotreallyachoicevariablehere—itsvalueisdeterminedbyhistory.Differentiationwithrespecttoxcanberegardedasimplicitlyaskingthequestion:“IfxðtÞwereoptimal,whatcharacteristicswouldithave?”21Thesimpleformofthisdifferentialequation(wheredx=dtdependsonlyonthevalueofthecontrolvariable,c)meansthatthisproblemisidenticaltooneexploredusingthe“calculusofvariations”approachtodynamicoptimization.Insuchacase,onecansubstitutedx=dtintothefunctionfandtheﬁrst-orderconditionsforamaximumcanbecompressedinto62Part1Introduction dxðtÞdt¼cðtÞ:(2.153)Thatis,eachinstant’sconsumptionjustreducesthestockofremainingbottles.ThecurrentvalueHamiltonianexpressionforthisproblemisH¼lncðtÞþλ½cðtÞþxðtÞdλdt,(2.154)andtheﬁrst-orderconditionsforamaximumare∂H∂c¼1cλ¼0,∂H∂x¼dλdt¼0:(2.155)Thesecondoftheseconditionsrequiresthatλ(theimplicitvalueofwine)beconstantovertime.Thismakesintuitivesense:becauseconsumingabottleofwinealwaysreducestheavailablestockbyonebottle,anysolutionwherethevalueofwinedifferedovertimewouldprovideanincentivetochangebehaviorbydrinkingmorewinewhenitischeapandlesswhenitisexpensive.CombiningthissecondconditionforamaximumwiththeﬁrstconditionimpliesthatcðtÞitselfmustbeconstantovertime.IfcðtÞ¼k,thenumberofbottlesremainingatanytimewillbexðtÞ¼1,000kt.Ifk¼50,thesystemwillobeytheendpointconstraintsxð0Þ¼1000andxð20Þ¼0.Ofcourse,inthisproblemyoucouldprobablyguessthattheoptimumplanwouldbetodrinkthewineattherateof50bottlesperyearfor20yearsbecausediminishingmarginalutilitysuggestsonedoesnotwanttodrinkexcessivelyinanyperiod.Themaximumprincipleconﬁrmsthisintuition.Morecomplicatedutility.Nowlet’stakeamorecomplicatedutilityfunctionthatmayyieldmoreinterestingresults.Supposethattheutilityofconsumingwineatanydate,t,isgivenbyu½cðtÞ¼½cðtÞγ=γifγ6¼0,γ<1;lncðtÞifγ¼0:(2.156)Assumealsothattheconsumerdiscountsfutureconsumptionattherateδ.Hencethisperson’sgoalistomaximize∫200u½cðtÞdt¼∫200eδt½cðtÞγγdt(2.157)subjecttothefollowingconstraints:dxðtÞdt¼cðtÞ,xð0Þ¼1,000,xð20Þ¼0:(2.158)SettingupthecurrentvalueHamiltonianexpressionyieldsH¼eδt½cðtÞγγþλðcÞþxðtÞdλðtÞdt,(2.159)andthemaximumprinciplerequiresthat(continued)thesingleequationfx¼dfdx=dt=dt,whichistermedthe“Eulerequation.”InChapter17wewillencountermanyEulerequations.Chapter2MathematicsforMicroeconomics63 EXAMPLE2.13CONTINUED∂H∂c¼eδt½cðtÞγ1λ¼0and∂H∂x¼0þ0þdλdt¼0:(2.160)Hence,wecanagainconcludethattheimplicitvalueofthewinestock(λ)shouldbeconstantovertime(callthisconstantk)andthateδt½cðtÞγ1¼korcðtÞ¼k1=ðγ1Þeδt=ðγ1Þ:(2.161)So,optimalwineconsumptionshouldfallovertimeinordertocompensateforthefactthatfutureconsumptionisbeingdiscountedintheconsumer’smind.If,forexample,weletδ¼0:1andγ¼1(“reasonable”values,aswewillshowinlaterchapters),thencðtÞ¼k0:5e0:05t(2.162)Nowwemustdoabitmoreworkinchoosingktosatisfytheendpointconstraints.Wewant∫200cðtÞdt¼∫200k0:5e0:05tdt¼20k0:5e0:05t200¼20k0:5ðe11Þ¼12:64k0:5¼1,000:(2.163)Finally,then,wehavetheoptimalconsumptionplanascðtÞ79e0:05t:(2.164)Thisconsumptionplanrequiresthatwineconsumptionstartoutfairlyhighanddeclineatacontinuousrateof5percentperyear.Becauseconsumptioniscontinuouslydeclining,wemustuseintegrationtocalculatewineconsumptioninanyparticularyearðxÞasfollows:consumptioninyearx∫xx1cðtÞdt¼∫xx179e0:05tdt¼1,580e0:05txx1¼1,580ðe0:05ðx1Þe0:05xÞ:(2.165)Ifx¼1,consumptionisabout77bottlesinthisﬁrstyear.Consumptionthendeclinessmoothly,endingwithabout30bottlesbeingconsumedinthe20thyear.QUERY:Ourﬁrstillustrationwasjustanexampleofthesecondinwhichδ¼γ¼0.Explainhowalternativevaluesoftheseparameterswillaffectthepathofoptimalwineconsumption.Explainyourresultsintuitively(formoreonoptimalconsumptionovertime,seeChapter17).MATHEMATICALSTATISTICSInrecentyearsmicroeconomictheoryhasincreasinglyfocusedonissuesraisedbyuncertaintyandimperfectinformation.Tounderstandmuchofthisliterature,itisimportanttohaveagoodbackgroundinmathematicalstatistics.Thepurposeofthissectionis,therefore,tosummarizeafewofthestatisticalprinciplesthatwewillencounteratvariousplacesinthisbook.64Part1Introduction RandomvariablesandprobabilitydensityfunctionsArandomvariabledescribes(innumericalform)theoutcomesfromanexperimentthatissubjecttochance.Forexample,wemightﬂipacoinandobservewhetheritlandsheadsortails.Ifwecallthisrandomvariablex,wecandenotethepossibleoutcomes(“realizations”)ofthevariableas:x¼1ifcoinisheads,0ifcoinistails:Noticethat,priortotheﬂipofthecoin,xcanbeeither1or0.Onlyaftertheuncertaintyisresolved(thatis,afterthecoinisﬂipped)doweknowwhatthevalueofxis.22DiscreteandcontinuousrandomvariablesTheoutcomesfromarandomexperimentmaybeeitheraﬁnitenumberofpossibilitiesoracontinuumofpossibilities.Forexample,recordingthenumberthatcomesuponasingledieisarandomvariablewithsixoutcomes.Withtwodice,wecouldeitherrecordthesumofthefaces(inwhichcasethereare12outcomes,someofwhicharemorelikelythanothers)orwecouldrecordatwo-digitnumber,oneforthevalueofeachdie(inwhichcasetherewouldbe36equallylikelyoutcomes).Theseareexamplesofdiscreterandomvariables.Alternatively,acontinuousrandomvariablemaytakeonanyvalueinagivenrangeofrealnumbers.Forexample,wecouldviewtheoutdoortemperaturetomorrowasacontinuousvariable(assumingtemperaturescanbemeasuredveryﬁnely)rangingfrom,say,50°Cto+50°C.Ofcourse,someofthesetemperatureswouldbeveryunlikelytooccur,butinprinciplethepreciselymeasuredtemperaturecouldbeanywherebetweenthesetwobounds.Similarly,wecouldviewtomorrow’spercentagechangeinthevalueofaparticularstockindexastakingonallvaluesbetween100%and,say,+1,000%.Again,ofcourse,percentagechangesaround0%wouldbeconsiderablymorelikelytooccurthanwouldbetheextremevalues.ProbabilitydensityfunctionsForanyrandomvariable,itsprobabilitydensityfunction(PDF)showstheprobabilitythateachspeciﬁcoutcomewilloccur.Foradiscreterandomvariable,deﬁningsuchafunctionposesnoparticulardifﬁculties.Inthecoinﬂipcase,forexample,thePDF[denotedbyfðxÞ]wouldbegivenbyfðx¼1Þ¼0:5,fðx¼0Þ¼0:5:(2.166)Fortherollofasingledie,thePDFwouldbe:fðx¼1Þ¼1=6,fðx¼2Þ¼1=6,fðx¼3Þ¼1=6,fðx¼4Þ¼1=6,fðx¼5Þ¼1=6,fðx¼6Þ¼1=6:(2.167)22Sometimesrandomvariablesaredenotedbyxetomakeadistinctionbetweenvariableswhoseoutcomeissubjecttorandomchanceand(nonrandom)algebraicvariables.Thisnotationaldevicecanbeusefulforkeepingtrackofwhatisrandomandwhatisnotinaparticularproblemandwewilluseitinsomecases.Whenthereisnoambiguity,however,wewillnotemploythisspecialnotation.Chapter2MathematicsforMicroeconomics65 NoticethatinbothofthesecasestheprobabilitiesspeciﬁedbythePDFsumto1.0.Thisisbecause,bydeﬁnition,oneoftheoutcomesoftherandomexperimentmustoccur.Moregenerally,ifwedenotealloftheoutcomesforadiscreterandomvariablebyxifori¼1,…,n,thenwemusthave:Xni¼1fðxiÞ¼1:(2.168)ForacontinuousrandomvariablewemustbecarefulindeﬁningthePDFconcept.Becausesucharandomvariabletakesonacontinuumofvalues,ifweweretoassignanynon-zerovalueastheprobabilityforaspeciﬁcoutcome(i.e.,atemperatureof+25.53470°C),wecouldquicklyhavesumsofprobabilitiesthatareinﬁnitelylarge.Hence,foracontinuousrandomvariablewedeﬁnethePDFfðxÞasafunctionwiththepropertythattheprobabilitythatxfallsinaparticularsmallintervaldxisgivenbytheareaoffðxÞdx.Usingthisconvention,thepropertythattheprobabilitiesfromarandomexperimentmustsumto1.0isstatedasfollows:∫þ∞∞fðxÞdx¼1:0:(2.169)AfewimportantPDFsMostanyfunctionwilldoasaprobabilitydensityfunctionprovidedthatfðxÞ0andthefunctionsums(orintegrates)to1.0.Thetrick,ofcourse,istoﬁndfunctionsthatmirrorrandomexperimentsthatoccurintherealworld.Herewelookatfoursuchfunctionsthatwewillﬁndusefulinvariousplacesinthisbook.GraphsforallfourofthesefunctionsareshowninFigure2.6.1.Binomialdistribution.Thisisthemostbasicdiscretedistribution.Usuallyxisassumedtotakeononlytwovalues,1and0.ThePDFforthebinomialisgivenby:fðx¼1Þ¼p,fðx¼0Þ¼1p,where0b:(2.171)Noticethatheretheprobabilitiesintegrateto1.0:∫þ∞∞fðxÞdx¼∫ba1badx¼xbaba¼bbaaba¼baba¼1:0:(2.172)3.Exponentialdistribution.Thisisacontinuousdistributionforwhichtheprobabilitiesdeclineatasmoothexponentialrateasxincreases.Formally:fðxÞ¼λeλxifx>0,0ifx0,(2.173)66Part1Introduction
whereλisapositiveconstant.Again,itiseasytoshowthatthisfunctionintegratesto1.0:∫þ∞∞fðxÞdx¼∫∞0λeλxdx¼eλx∞0¼0ð1Þ¼1:0:(2.174)4.Normaldistribution.TheNormal(orGaussian)distributionisthemostimportantinmathematicalstatistics.It’simportancestemslargelyfromthecentrallimittheorem,whichstatesthatthedistributionofanysumofindependentrandomvariableswillincreasinglyFIGURE2.6FourCommonProbabilityDensityFunctionsRandomvariablesthathavethesePDFsarewidelyused.EachgraphindicatestheexpectedvalueofthePDFshown.f(x)f(x)f(x)1/ 2π√–––x0f(x)P0(a) Binomial(c) Exponential(d) Normal(b) Uniform1–PPab1xxb–aa+b2λ1/λ1xChapter2MathematicsforMicroeconomics67
approximatetheNormaldistributionasthenumberofsuchvariablesincrease.Becausesampleaveragescanberegardedassumsofindependentrandomvariables,thistheoremsaysthatanysampleaveragewillhaveaNormaldistributionnomatterwhatthedistributionofthepopulationfromwhichthesampleisselected.Hence,itmayoftenbeappropriatetoassumearandomvariablehasaNormaldistributionifitcanbethoughtofassomesortofaverage.ThemathematicalformfortheNormalPDFisfðxÞ¼1ﬃﬃﬃﬃﬃﬃﬃ2πpex2=2,(2.175)andthisisdeﬁnedforallrealvaluesofx.Althoughthefunctionmaylookcomplicated,afewofitspropertiescanbeeasilydescribed.First,thefunctionissymmetricaroundzero(becauseofthex2term).Second,thefunctionisasymptotictozeroasxbecomesverylargeorverysmall.Third,thefunctionreachesitsmaximalvalueatx¼0.Thisvalueis1=ﬃﬃﬃﬃﬃﬃ2πp0:4.Finally,thegraphofthisfunctionhasageneral“bellshape”—ashapeusedthroughoutthestudyofstatistics.Integrationofthisfunctionisrelativelytricky(thougheasyinpolarcoordinates).Thepresenceoftheconstant1=ﬃﬃﬃﬃﬃﬃ2πpisneededifthefunctionistointegrateto1.0.ExpectedvalueTheexpectedvalueofarandomvariableisthenumericalvaluethattherandomvariablemightbeexpectedtohave,onaverage.23Itisthe“centerofgravity”oftheprobabilitydensityfunction.Foradiscreterandomvariablethattakesonthevaluesx1,x2,…,xn,theexpectedvalueisdeﬁnedasEðxÞ¼Xni¼1xifðxiÞ:(2.176)Thatis,eachoutcomeisweightedbytheprobabilitythatitwilloccurandtheresultissummedoverallpossibleoutcomes.Foracontinuousrandomvariable,Equation2.176isreadilygeneralizedasEðxÞ¼∫þ∞∞xfðxÞdx:(2.177)Again,inthisintegration,eachvalueofxisweightedbytheprobabilitythatthisvaluewilloccur.Theconceptofexpectedvaluecanbegeneralizedtoincludetheexpectedvalueofanyfunctionofarandomvariable[say,gðxÞ].Inthecontinuouscase,forexample,wewouldwriteE½gðxÞ¼∫þ∞∞gðxÞfðxÞdx:(2.178)23Theexpectedvalueofarandomvariableissometimesreferredtoasthemeanofthatvariable.Inthestudyofsamplingthiscansometimesleadtoconfusionbetweentheexpectedvalueofarandomvariableandtheseparateconceptofthesamplearithmeticaverage.68Part1Introduction
Asaspecialcase,consideralinearfunctiony¼axþb.ThenEðyÞ¼EðaxþbÞ¼∫þ∞∞ðaxþbÞfðxÞdx¼a∫þ∞∞xfðxÞdxþb∫þ∞∞fðxÞdx¼aEðxÞþb:(2:179)Sometimesexpectedvaluesarephrasedintermsofthecumulativedistributionfunction(CDF)FðxÞ,deﬁnedasFðxÞ¼∫x∞fðtÞdt:(2.180)Thatis,FðxÞrepresentstheprobabilitythattherandomvariabletislessthanorequaltox.Withthisnotation,theexpectedvalueofgðxÞisdeﬁnedasE½gðxÞ¼∫þ∞∞gðxÞdFðxÞ:(2.181)Becauseofthefundamentaltheoremofcalculus,Equation2.181andEquation2.178meanexactlythesamething.EXAMPLE2.14ExpectedValuesofaFewRandomVariablesTheexpectedvaluesofeachoftherandomvariableswiththesimplePDFsintroducedearlierareeasytocalculate.Alloftheseexpectedvaluesareindicatedonthegraphsofthefunctions’PDFsinFigure2.6.1.Binomial.Inthiscase:EðxÞ¼1⋅fðx¼1Þþ0⋅fðx¼0Þ¼1⋅pþ0⋅ð1pÞ¼p:(2.182)Forthecoinﬂipcase(wherep¼0:5),thissaysthatEðxÞ¼p¼0:5—theexpectedvalueofthisrandomvariableis,asyoumighthaveguessed,onehalf.2.Uniform.Forthiscontinuousrandomvariable,EðxÞ¼∫baxbadx¼x22ðbaÞba¼b22ðbaÞa22ðbaÞ¼bþa2:(2.183)Again,asyoumighthaveguessed,theexpectedvalueoftheuniformdistributionispreciselyhalfwaybetweenaandb.3.Exponential.Forthiscaseofdecliningprobabilities:EðxÞ¼∫∞0xλeλxdx¼xeλx1λeλx∞0¼1λ,(2.184)wheretheintegrationfollowsfromtheintegrationbypartsexampleshownearlierinthischapter(Equation2.137).Noticeherethatthefastertheprobabilitiesdecline,theloweristheexpectedvalueofx.Forexample,ifλ¼0:5thenEðxÞ¼2,whereasifλ¼0:05thenEðxÞ¼20.(continued)Chapter2MathematicsforMicroeconomics69
EXAMPLE2.14CONTINUED4.Normal.BecausetheNormalPDFissymmetricaroundzero,itseemsclearthatEðxÞ¼0.Aformalproofusesachangeofvariableintegrationbylettingu¼x2=2ðdu¼xdxÞ:∫þ∞∞1ﬃﬃﬃﬃﬃﬃﬃ2πpxex2=2dx¼1ﬃﬃﬃﬃﬃﬃﬃ2πp∫þ∞∞eudu¼1ﬃﬃﬃﬃﬃﬃﬃ2πp½ex2=2þ∞∞¼1ﬃﬃﬃﬃﬃﬃﬃ2πp½00¼0:(2.185)Ofcourse,theexpectedvalueofanormallydistributedrandomvariable(orofanyrandomvariable)maybealteredbyalineartransformation,asshowninEquation2.179.QUERY:Alineartransformationchangesarandomvariable’sexpectedvalueinaverypredictableway—ify¼axþb,thenEðyÞ¼aEðxÞþb.Hence,forthistransformation[say,hðxÞ]wehaveE½hðxÞ¼h½EðxÞ.Supposeinsteadthatxweretransformedbyaconcavefunction,saygðxÞwithg0>0andg00<0.HowwouldE½gðxÞcomparetog½EðxÞ?Note:ThisisanillustrationofJensen’sinequality,aconceptwewillpursueindetailinChapter7.SeealsoProblem2.13.VarianceandstandarddeviationTheexpectedvalueofarandomvariableisameasureofcentraltendency.Ontheotherhand,thevarianceofarandomvariable[denotedbyσ2xorVarðxÞ]isameasureofdispersion.Speciﬁcally,thevarianceisdeﬁnedasthe“expectedsquareddeviation”ofarandomvariablefromitsexpectedvalue.Formally:VarðxÞ¼σ2x¼E½ðxEðxÞÞ2¼∫þ∞∞ðxEðxÞÞ2fðxÞdx:(2.186)Somewhatimprecisely,thevariancemeasuresthe“typical”squareddeviationfromthecentralvalueofarandomvariable.Inmakingthecalculation,deviationsfromtheexpectedvaluearesquaredsothatpositiveandnegativedeviationsfromtheexpectedvaluewillbothcontributetothismeasureofdispersion.Afterthecalculationismade,thesquaringprocesscanbereversedtoyieldameasureofdispersionthatisintheoriginalunitsinwhichtherandomvariablewasmeasured.Thissquarerootofthevarianceiscalledthe“standarddeviation”andisdenotedasσxð¼ﬃﬃﬃﬃﬃσ2xpÞ.Thewordingofthetermeffectivelyconveysitsmeaning:σxisindeedthetypical(“standard”)deviationofarandomvariablefromitsexpectedvalue.Whenarandomvariableissubjecttoalineartransformation,itsvarianceandstandarddeviationwillbechangedinafairlyobviousway.Ify¼axþb,thenσ2y¼∫þ∞∞½axþbEðaxþbÞ2fðxÞdx¼∫þ∞∞a2½xEðxÞ2fðxÞdx¼a2σ2x:(2.187)Hence,additionofaconstanttoarandomvariabledoesnotchangeitsvariance,whereasmultiplicationbyaconstantmultipliesthevariancebythesquareoftheconstant.Itisclearthereforethatmultiplyingavariablebyaconstantmultipliesitsstandarddeviationbythatconstant:σax¼aσx.70Part1Introduction EXAMPLE2.15VariancesandStandardDeviationsforSimpleRandomVariablesKnowingthevariancesandstandarddeviationsofthefoursimplerandomvariableswehavebeenlookingatcansometimesbequiteusefulineconomicapplications.1.Binomial.Thevarianceofthebinomialcanbecalculatedbyapplyingthedeﬁnitioninitsdiscreteanalog:σ2x¼Xni¼1ðxiEðxÞÞ2fðxiÞ¼ð1pÞ2⋅pþð0pÞ2ð1pÞ¼ð1pÞðpp2þp2Þ¼pð1pÞ:(2:188)Hence,σx¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpð1pÞp.Oneimplicationofthisresultisthatabinomialvariablehasthelargestvarianceandstandarddeviationwhenp¼0:5,inwhichcaseσ2x¼0:25andσx¼0:5.Becauseoftherelativelyﬂatparabolicshapeofpð1pÞ,modestdeviationsofpfrom0.5donotchangethisvariancesubstantially.2.Uniform.Calculatingthevarianceoftheuniformdistributionyieldsamildlyinterest-ingresult:σ2x¼∫baxaþb221badx¼xaþb2313ðbaÞba¼13ðbaÞðbaÞ38ðabÞ38"#¼ðbaÞ212:(2:189)Thisisoneofthefewplaceswherethenumber12hasanyuseinmathematicsotherthaninmeasuringquantitiesoforangesordoughnuts.3.Exponential.Integratingthevarianceformulafortheexponentialisrelativelylaborious.Fortunately,theresultisquitesimple;fortheexponential,itturnsoutthatσ2x¼1=λ2andσx¼1=λ.Hence,themeanandstandarddeviationarethesamefortheexponentialdistribu-tion—itisa“one-parameterdistribution.”4.Normal.Inthiscasealso,theintegrationcanbeburdensome.Butagaintheresultissimple:fortheNormaldistribution,σ2x¼σx¼1.AreasbelowtheNormalcurvecanbereadilycalculatedandtablesoftheseareavailableinanystatisticstext.TwousefulfactsabouttheNormalPDFare:∫þ11fðxÞdx0:68and∫þ22fðxÞdx0:95:(2.190)Thatis,theprobabilityisabouttwothirdsthataNormalvariablewillbewithin1standarddeviationoftheexpectedvalueand“mostofthetime”(i.e.,withprobability0.95)itwillbewithin2standarddeviations.StandardizingtheNormal.IftherandomvariablexhasastandardNormalPDF,itwillhaveanexpectedvalueof0andastandarddeviationof1.However,asimplelineartransformationcanbeusedtogivethisrandomvariableanydesiredexpectedvalue(μ)andstandarddeviation(σ).Considerthetransformationy¼σxþμ.NowEðyÞ¼σEðxÞþμ¼μandVarðyÞ¼σ2y¼σ2VarðxÞ¼σ2:(2.191)Reversingthisprocesscanbeusedto“standardize”anyNormallydistributedrandomvariable(y)withanarbitraryexpectedvalue(μ)andstandarddeviation(σ)(thisissometimesdenoted(continued)Chapter2MathematicsforMicroeconomics71 EXAMPLE2.15CONTINUEDasy∼Nðμ,σÞ)byusingz¼ðyμÞ=σ.Forexample,SATscores(y)aredistributedNormallywithanexpectedvalueof500pointsandastandarddeviationof100points(thatis,y∼Nð500,100Þ).Hence,z¼ðy500Þ=100hasastandardNormaldistributionwithexpectedvalue0andstandarddeviation1.Equation2.190showsthatapproximately68percentofallscoresliebetween400and600pointsand95percentofallscoresliebetween300and700points.QUERY:Supposethattherandomvariablexisdistributeduniformlyalongtheinterval[0,12].Whatarethemeanandstandarddeviationofx?Whatfractionofthexdistributioniswithin1standarddeviationofthemean?Whatfractionofthedistributioniswithin2standarddeviationsoftheexpectedvalue?Explainwhythisdiffersfromthefractionscom-putedfortheNormaldistribution.CovarianceSomeeconomicproblemsinvolvetwoormorerandomvariables.Forexample,aninvestormayconsiderallocatinghisorherwealthamongseveralassetsthereturnsonwhicharetakentoberandom.Althoughtheconceptsofexpectedvalue,variance,andsoforthcarryovermoreorlessdirectlywhenlookingatasinglerandomvariableinsuchcases,itisalsonecessarytoconsidertherelationshipbetweenthevariablestogetacompletepicture.Theconceptofcovarianceisusedtoquantifythisrelationship.Beforeprovidingadeﬁnition,however,wewillneedtodevelopsomebackground.Consideracasewithtwocontinuousrandomvariables,xandy.Theprobabilitydensityfunctionforthesetwovariables,denotedbyfðx,yÞ,hasthepropertythattheprobabilityassociatedwithasetofoutcomesinasmallarea(withdimensionsdxdy)isgivenbyfðx,yÞdxdy.TobeaproperPDF,itmustbethecasethat:fðx,yÞ0and∫þ∞∞∫þ∞∞fðx,yÞdxdy¼1:(2.192)Thesingle-variablemeasureswehavealreadyintroducedcanbedevelopedinthistwo-variablecontextby“integratingout”theothervariable.Thatis,EðxÞ¼∫þ∞∞∫þ∞∞xfðx,yÞdydxandVarðxÞ¼∫þ∞∞∫þ∞∞½xEðxÞ2fðx,yÞdydx:(2.193)Inthisway,theparametersdescribingtherandomvariablexaremeasuredoverallpossibleoutcomesforyaftertakingintoaccountthelikelihoodofthosevariousoutcomes.Inthiscontext,thecovariancebetweenxandyseekstomeasurethedirectionofassociationbetweenthevariables.Speciﬁcallythecovariancebetweenxandy[denotedasCovðx,yÞ]isdeﬁnedasCovðx,yÞ¼∫þ∞∞∫þ∞∞xEðxÞyEðyÞfðx,yÞdxdy:(2.194)72Part1Introduction Thecovariancebetweentworandomvariablesmaybepositive,negative,orzero.IfvaluesofxthataregreaterthanEðxÞtendtooccurrelativelyfrequentlywithvaluesofythataregreaterthanEðyÞ(andsimilarly,iflowvaluesofxtendtooccurtogetherwithlowvaluesofy),thenthecovariancewillbepositive.Inthiscase,valuesofxandytendtomoveinthesamedirection.Alternatively,ifhighvaluesofxtendtobeassociatedwithlowvaluesfory(andviceversa),thecovariancewillbenegative.Tworandomvariablesaredeﬁnedtobeindependentiftheprobabilityofanyparticularvalueof,say,xisnotaffectedbytheparticularvalueofythatmightoccur(andviceversa).24Inmathematicalterms,thismeansthatthePDFmusthavethepropertythatfðx,yÞ¼gðxÞhðyÞ—thatis,thejointprobabilitydensityfunctioncanbeexpressedastheproductoftwosingle-variablePDFs.Ifxandyareindependent,theircovariancewillbezero:Covðx,yÞ¼∫þ∞∞∫þ∞∞½xEðxÞ½yEðyÞgðxÞhðyÞdxdy¼∫þ∞∞½xEðxÞgðxÞdx⋅∫þ∞∞½yEðyÞhðyÞdy¼0⋅0¼0:(2.195)Theconverseofthisstatementisnotnecessarilytrue,however.Azerocovariancedoesnotnecessarilyimplystatisticalindependence.Finally,thecovarianceconceptiscrucialforunderstandingthevarianceofsumsordifferencesofrandomvariables.Althoughtheexpectedvalueofasumoftworandomvariablesis(asonemightguess)thesumoftheirexpectedvalues:EðxþyÞ¼∫þ∞∞∫þ∞∞ðxþyÞfðx,yÞdxdy¼∫þ∞∞xfðx,yÞdydxþ∫þ∞∞yfðx,yÞdxdy¼EðxÞþEðyÞ,(2.196)therelationshipforthevarianceofsuchasumismorecomplicated.UsingthedeﬁnitionswehavedevelopedyieldsVarðxþyÞ¼∫þ∞∞∫þ∞∞½xþyEðxþyÞ2fðx,yÞdxdy¼∫þ∞∞∫þ∞∞½xEðxÞþyEðyÞ2fðx,yÞdxdy¼∫þ∞∞∫þ∞∞½xEðxÞ2þ½yEðyÞ2þ2½xEðxÞ½yEðyÞfðx,yÞdxdy¼VarðxÞþVarðyÞþ2Covðx,yÞ:(2.197)Hence,ifxandyareindependentthenVarðxþyÞ¼VarðxÞþVarðyÞ.Thevarianceofthesumwillbegreaterthanthesumofthevariancesifthetworandomvariableshaveapositivecovarianceandwillbelessthanthesumofthevariancesiftheyhaveanegativecovariance.Problems2.13and2.14providefurtherdetailsonstatisticalissuesthatariseinmicroeconomictheory.24Aformaldeﬁnitionreliesontheconceptofconditionalprobability.TheconditionalprobabilityofaneventBgiventhatAhasoccurred(writtenPðBjAÞisdeﬁnedasPðBjAÞ¼PðAandBÞ=PðAÞ;BandAaredeﬁnedtobeindependentifPðBjAÞ¼PðBÞ.Inthiscase,PðAandBÞ¼PðAÞ⋅PðBÞ.Chapter2MathematicsforMicroeconomics73 SUMMARYDespitetheformidableappearanceofsomepartsofthischapter,thisisnotabookonmathematics.Rather,theintentionherewastogathertogetheravarietyoftoolsthatwillbeusedtodevelopeconomicmodelsthroughouttheremainderofthetext.Materialinthischapterwillthenbeusefulasahandyreference.Onewaytosummarizethemathematicaltoolsintro-ducedinthischapterisbystressingagaintheeconomiclessonsthatthesetoolsillustrate:•Usingmathematicsprovidesaconvenient,shorthandwayforeconomiststodeveloptheirmodels.Implica-tionsofvariouseconomicassumptionscanbestudiedinasimpliﬁedsettingthroughtheuseofsuchmathe-maticaltools.•Themathematicalconceptofthederivativesofafunc-tioniswidelyusedineconomicmodelsbecauseecono-mistsareofteninterestedinhowmarginalchangesinonevariableaffectanothervariable.Partialderivativesareespeciallyusefulforthispurposebecausetheyaredeﬁnedtorepresentsuchmarginalchangeswhenallotherfactorsareheldconstant.•Themathematicsofoptimizationisanimportanttoolforthedevelopmentofmodelsthatassumethateconomicagentsrationallypursuesomegoal.Intheunconstrainedcase,theﬁrst-orderconditionsstatethatanyactivitythatcontributestotheagent’sgoalshouldbeexpandeduptothepointatwhichthemarginalcontributionoffurtherexpansioniszero.Inmathematicalterms,theﬁrst-orderconditionforanoptimumrequiresthatallpartialderiva-tivesbezero.•Mosteconomicoptimizationproblemsinvolveconstraintsonthechoicesagentscanmake.Inthiscasetheﬁrst-orderconditionsforamaximumsuggestthateachactivitybeoperatedatalevelatwhichtheratioofthemarginalbeneﬁt–oftheactivitytoitsmarginalcostisthesameforallactivitiesactuallyused.Thiscommonmarginalbeneﬁt–marginalcostratioisalsoequaltotheLagrangianmulti-plier,whichisoftenintroducedtohelpsolveconstrainedoptimizationproblems.TheLagrangianmultipliercanalsobeinterpretedastheimplicitvalue(orshadowprice)oftheconstraint.•Theimplicitfunctiontheoremisausefulmathematicaldeviceforillustratingthedependenceofthechoicesthatresultfromanoptimizationproblemontheparametersofthatproblem(forexample,marketprices).Theenve-lopetheoremisusefulforexamininghowtheseoptimalchoiceschangewhentheproblem’sparameters(prices)change.•Someoptimizationproblemsmayinvolveconstraintsthatareinequalitiesratherthanequalities.Solutionstotheseproblemsoftenillustrate“complementaryslackness.”Thatis,eithertheconstraintsholdwithequalityandtheirrelatedLagrangianmultipliersarenonzero,ortheconstraintsarestrictinequalitiesandtheirrelatedLagrang-ianmultipliersarezero.AgainthisillustrateshowtheLagrangianmultiplierimpliessomethingaboutthe“im-portance”ofconstraints.•Theﬁrst-orderconditionsshowninthischapterareonlythenecessaryconditionsforalocalmaximumormini-mum.Onemustalsochecksecond-orderconditionsthatrequirethatcertaincurvatureconditionsbemet.•Certaintypesoffunctionsoccurinmanyeconomicprob-lems.Quasi-concavefunctions(thosefunctionsforwhichthelevelcurvesformconvexsets)obeythesecond-orderconditionsofconstrainedmaximumorminimumproblemswhentheconstraintsarelinear.Homotheticfunctionshavetheusefulpropertythatimplicittrade-offsamongthevariablesofthefunctiondependonlyontheratiosofthesevariables.•Integralcalculusisoftenusedineconomicsbothasawayofdescribingareasbelowgraphsandasawayofsum-mingresultsovertime.Techniquesthatinvolvevariouswaysofdifferentiatingintegralsplayanimportantroleinthetheoryofoptimizingbehavior.•Manyeconomicproblemsaredynamicinthatdecisionsatonedateaffectdecisionsandoutcomesatlaterdates.ThemathematicsforsolvingsuchdynamicoptimizationproblemsisoftenastraightforwardgeneralizationofLagrangianmethods.•Conceptsfrommathematicalstatisticsareoftenusedinstudyingtheeconomicsofuncertaintyandinformation.Themostfundamentalconceptisthenotionofaran-domvariableanditsassociatedprobabilitydensityfunc-tion.Parametersofthisdistribution,suchasitsexpectedvalueoritsvariance,alsoplayimportantrolesinmanyeconomicmodels.74Part1Introduction PROBLEMS2.1SupposeUðx,yÞ¼4x2þ3y2.a.Calculate∂U=∂x,∂U=∂y.b.Evaluatethesepartialderivativesatx¼1,y¼2.c.WritethetotaldifferentialforU.d.Calculatedy=dxfordU¼0—thatis,whatistheimpliedtrade-offbetweenxandyholdingUconstant?e.ShowU¼16whenx¼1,y¼2.f.InwhatratiomustxandychangetoholdUconstantat16formovementsawayfromx¼1,y¼2?g.Moregenerally,whatistheshapeoftheU¼16contourlineforthisfunction?Whatistheslopeofthatline?2.2Supposeaﬁrm’stotalrevenuesdependontheamountproducedðqÞaccordingtothefunctionR¼70qq2:Totalcostsalsodependonq:C¼q2þ30qþ100:a.Whatlevelofoutputshouldtheﬁrmproduceinordertomaximizeproﬁts(RC)?Whatwillproﬁtsbe?b.Showthatthesecond-orderconditionsforamaximumaresatisﬁedattheoutputlevelfoundinpart(a).c.Doesthesolutioncalculatedhereobeythe“marginalrevenueequalsmarginalcost”rule?Explain.2.3Supposethatfðx,yÞ¼xy.Findthemaximumvalueforfifxandyareconstrainedtosumto1.Solvethisproblemintwoways:bysubstitutionandbyusingtheLagrangianmultipliermethod.2.4ThedualproblemtotheonedescribedinProblem2.3isminimizexþysubjecttoxy¼0:25:SolvethisproblemusingtheLagrangiantechnique.ThencomparethevalueyougetfortheLagrangianmultipliertothevalueyougotinProblem2.3.Explaintherelationshipbetweenthetwosolutions.2.5Theheightofaballthatisthrownstraightupwithacertainforceisafunctionofthetime(t)fromwhichitisreleasedgivenbyfðtÞ¼0:5gt2þ40t(wheregisaconstantdeterminedbygravity).Chapter2MathematicsforMicroeconomics75 a.Howdoesthevalueoftatwhichtheheightoftheballisatamaximumdependontheparameterg?b.Useyouranswertopart(a)todescribehowmaximumheightchangesastheparametergchanges.c.Usetheenvelopetheoremtoanswerpart(b)directly.d.OntheEarthg¼32,butthisvaluevariessomewhataroundtheglobe.Iftwolocationshadgravitationalconstantsthatdifferedby0.1,whatwouldbethedifferenceinthemaximumheightofaballtossedinthetwoplaces?2.6Asimplewaytomodeltheconstructionofanoiltankeristostartwithalargerectangularsheetofsteelthatisxfeetwideand3xfeetlong.Nowcutasmallersquarethatistfeetonasideoutofeachcornerofthelargersheetandfoldupandweldthesidesofthesteelsheettomakeatraylikestructurewithnotop.a.ShowthatthevolumeofoilthatcanbeheldbythistrayisgivenbyV¼tðx2tÞð3x2tÞ¼3tx28t2xþ4t3:b.HowshouldtbechosensoastomaximizeVforanygivenvalueofx?c.Isthereavalueofxthatmaximizesthevolumeofoilthatcanbecarried?d.Supposethatashipbuilderisconstrainedtouseonly1,000,000squarefeetofsteelsheettoconstructanoiltanker.Thisconstraintcanberepresentedbytheequation3x24t2¼1,000,000(becausethebuildercanreturnthecut-outsquaresforcredit).Howdoesthesolutiontothisconstrainedmaximumproblemcomparetothesolutionsdescribedinparts(b)and(c)?2.7Considerthefollowingconstrainedmaximizationproblem:maximizey¼x1þ5lnx2subjecttokx1x2¼0,wherekisaconstantthatcanbeassignedanyspeciﬁcvalue.a.Showthatifk¼10,thisproblemcanbesolvedasoneinvolvingonlyequalityconstraints.b.Showthatsolvingthisproblemfork¼4requiresthatx1¼1.c.Ifthex’sinthisproblemmustbenonnegative,whatistheoptimalsolutionwhenk¼4?d.Whatisthesolutionforthisproblemwhenk¼20?Whatdoyouconcludebycomparingthissolutiontothesolutionforpart(a)?Note:Thisprobleminvolveswhatiscalleda“quasi-linearfunction.”Suchfunctionsprovideimportantexamplesofsometypesofbehaviorinconsumertheory—asweshallsee.2.8SupposethataﬁrmhasamarginalcostfunctiongivenbyMCðqÞ¼qþ1.a.Whatisthisﬁrm’stotalcostfunction?Explainwhytotalcostsareknownonlyuptoaconstantofintegration,whichrepresentsﬁxedcosts.b.Asyoumayknowfromanearliereconomicscourse,ifaﬁrmtakesprice(p)asgiveninitsdecisionsthenitwillproducethatoutputforwhichp¼MCðqÞ.Iftheﬁrmfollowsthisproﬁt-maximizingrule,howmuchwillitproducewhenp¼15?Assumingthattheﬁrmisjustbreakingevenatthisprice,whatareﬁxedcosts?76Part1Introduction c.Howmuchwillproﬁtsforthisﬁrmincreaseifpriceincreasesto20?d.Showthat,ifwecontinuetoassumeproﬁtmaximization,thenthisﬁrm’sproﬁtscanbeexpressedsolelyasafunctionofthepriceitreceivesforitsoutput.e.Showthattheincreaseinproﬁtsfromp¼15top¼20canbecalculatedintwoways:(i)directlyfromtheequationderivedinpart(d);and(ii)byintegratingtheinversemarginalcostfunction½MC1ðpÞ¼p1fromp¼15top¼20.Explainthisresultintuitivelyusingtheenvelopetheorem.AnalyticalProblems2.9Concaveandquasi-concavefunctionsShowthatiffðx1,x2Þisaconcavefunctionthenitisalsoaquasi-concavefunction.DothisbycomparingEquation2.114(deﬁningquasi-concavity)toEquation2.98(deﬁningconcavity).Canyougiveanintuitivereasonforthisresult?Istheconverseofthestatementtrue?Arequasi-concavefunctionsnecessarilyconcave?Ifnot,giveacounterexample.2.10TheCobb-DouglasfunctionOneofthemostimportantfunctionswewillencounterinthisbookistheCobb-Douglasfunction:y¼ðx1Þαðx2Þβ,whereαandβarepositiveconstantsthatareeachlessthan1.a.Showthatthisfunctionisquasi-concaveusinga“bruteforce”methodbyapplyingEqua-tion2.114.b.ShowthattheCobb-Douglasfunctionisquasi-concavebyshowingthatanycontourlineoftheformy¼c(wherecisanypositiveconstant)isconvexandthereforethatthesetofpointsforwhichy>cisaconvexset.c.Showthatifαþβ>1thentheCobb-Douglasfunctionisnotconcave(therebyillustratingagainthatnotallquasi-concavefunctionsareconcave).Note:TheCobb-DouglasfunctionisdiscussedfurtherintheExtensionstothischapter.2.11ThepowerfunctionAnotherfunctionwewillencounterofteninthisbookisthe“powerfunction”:y¼xδ,where0δ1(attimeswewillalsoexaminethisfunctionforcaseswhereδcanbenegative,too,inwhichcasewewillusetheformy¼xδ=δtoensurethatthederivativeshavethepropersign).a.Showthatthisfunctionisconcave(andthereforealso,bytheresultofProblem2.9,quasi-concave).Noticethattheδ¼1isaspecialcaseandthatthefunctionis“strictly”concaveonlyforδ<1.b.Showthatthemultivariateformofthepowerfunctiony¼fðx1,x2Þ¼ðx1Þδþðx2Þδisalsoconcave(andquasi-concave).Explainwhy,inthiscase,thefactthatf12¼f21¼0makesthedeterminationofconcavityespeciallysimple.c.Onewaytoincorporate“scale”effectsintothefunctiondescribedinpart(b)istousethemonotonictransformationgðx1,x2Þ¼yγ¼½ðx1Þδþðx2Þδγ,whereγisapositiveconstant.Doesthistransformationpreservetheconcavityofthefunction?Isgquasi-concave?Chapter2MathematicsforMicroeconomics77 2.12TaylorapproximationsTaylor’stheoremshowsthatanyfunctioncanbeapproximatedinthevicinityofanyconvenientpointbyaseriesoftermsinvolvingthefunctionanditsderivatives.Herewelookatsomeapplicationsofthetheoremforfunctionsofoneandtwovariables.a.Anycontinuousanddifferentiablefunctionofasinglevariable,fðxÞ,canbeapproximatednearthepointabytheformulafðxÞ¼fðaÞþf0ðaÞðxaÞþ0:5f00ðaÞðxaÞ2þtermsinf000,f0000,…:UsingonlytheﬁrstthreeofthesetermsresultsinaquadraticTaylorapproximation.UsethisapproximationtogetherwiththedeﬁnitionofconcavitygiveninEquation2.85toshowthatanyconcavefunctionmustlieonorbelowthetangenttothefunctionatpointa.b.ThequadraticTaylorapproximationforanyfunctionoftwovariables,fðx,yÞ,nearthepointða,bÞisgivenbyfðx,yÞ¼fða,bÞþf1ða,bÞðxaÞþf2ða,bÞðybÞþ0:5½f11ða,bÞðxaÞ2þ2f12ða,bÞðxaÞðybÞþf22ðybÞ2:Usethisapproximationtoshowthatanyconcavefunction(asdeﬁnedbyEquation2.98)mustlieonorbelowitstangentplaneat(a,b).2.13MoreonexpectedvalueBecausetheexpectedvalueconceptplaysanimportantroleinmanyeconomictheories,itmaybeusefultosummarizeafewmorepropertiesofthisstatisticalmeasure.Throughoutthisproblem,xisassumedtobeacontinuousrandomvariablewithprobabilitydensityfunctionfðxÞ.a.(Jensen’sinequality)SupposethatgðxÞisaconcavefunction.ShowthatE½gðxÞg½EðxÞ.Hint:ConstructthetangenttogðxÞatthepointEðxÞ.ThistangentwillhavetheformcþdxgðxÞforallvaluesofxandcþdEðxÞ¼g½EðxÞwherecanddareconstants.b.Usetheprocedurefrompart(a)toshowthatifgðxÞisaconvexfunctionthenE½gðxÞg½EðxÞ.c.Supposextakesononlynonnegativevalues—thatis,0x∞.UseintegrationbypartstoshowthatEðxÞ¼∫∞0½1FðxÞdx,whereFðxÞisthecumulativedistributionfunctionforx[thatis,FðxÞ¼∫x0fðtÞdt].d.(Markov’sinequality)Showthatifxtakesononlypositivevaluesthenthefollowinginequalityholds:PðxtÞEðxÞt:Hint:EðxÞ¼∫∞0xfðxÞdx¼∫t0xfðxÞdxþ∫∞txfðxÞdx:e.ConsidertheprobabilitydensityfunctionfðxÞ¼2x3forx1.(1)ShowthatthisisaproperPDF.(2)CalculateFðxÞforthisPDF.(3)Usetheresultsofpart(c)tocalculateEðxÞforthisPDF.(4)ShowthatMarkov’sinequalityholdsforthisfunction.f.Theconceptofconditionalexpectedvalueisusefulinsomeeconomicproblems.Wedenotetheexpectedvalueofxconditionalontheoccurrenceofsomeevent,A,asEðxjAÞ.TocomputethisvalueweneedtoknowthePDFforxgiventhatAhasoccurred[denotedbyfðxjAÞ].Withthis78Part1Introduction notation,EðxjAÞ¼∫þ∞∞xfðxjAÞdx.Perhapstheeasiestwaytounderstandtheserelationshipsiswithanexample.LetfðxÞ¼x23for1x2:(1)ShowthatthisisaproperPDF.(2)CalculateEðxÞ.(3)Calculatetheprobabilitythat1x0.(4)Considertheevent0x2,andcallthiseventA.WhatisfðxjAÞ?(5)CalculateEðxjAÞ.(6)Explainyourresultsintuitively.2.14MoreonvariancesandcovariancesThisproblempresentsafewusefulmathematicalfactsaboutvariancesandcovariances.a.ShowthatVarðxÞ¼Eðx2Þ½EðxÞ2.b.Showthattheresultinpart(a)canbegeneralizedasCovðx,yÞ¼EðxyÞEðxÞEðyÞ.Note:IfCovðx,yÞ¼0,thenEðxyÞ¼EðxÞEðyÞ.c.ShowthatVarðaxbyÞ¼a2VarðxÞþb2VarðyÞ2abCovðx,yÞ.d.Assumethattwoindependentrandomvariables,xandy,arecharacterizedbyEðxÞ¼EðyÞandVarðxÞ¼VarðyÞ.ShowthatEð0:5xþ0:5yÞ¼EðxÞ.Thenusepart(c)toshowthatVarð0:5xþ0:5yÞ¼0:5VarðxÞ.Describewhythisfactprovidestherationalefordiversiﬁcationofassets.SUGGESTIONSFORFURTHERREADINGDadkhan,Kamran.FoundationsofMathematicalandComputationalEconomics.Mason,OH:Thomson/South-Western,2007.Thisisagoodintroductiontomanycalculustechniques.ThebookshowshowmanymathematicalquestionscanbeapproachedusingpopularsoftwareprogramssuchasMatlaborExcel.Dixit,A.K.OptimizationinEconomicTheory,2nded.NewYork:OxfordUniversityPress,1990.Acompleteandmoderntreatmentofoptimizationtechniques.Usesrelativelyadvancedanalyticalmethods.Hoy,Michael,JohnLivernois,ChrisMcKenna,RayRees,andThanasisStengos.MathematicsforEconomists,2nded.Cambridge,MA:MITPress,2001.Acompleteintroductiontomostofthemathematicscoveredinmicroeconomicscourses.Thestrengthofthebookisitspresentationofmanyworked-outexamples,mostofwhicharebasedonmicro-economictheory.Mas-Colell,Andreu,MichaelD.Whinston,andJerryR.Green.MicroeconomicTheory.NewYork:OxfordUniversityPress,1995.Encyclopedictreatmentofmathematicalmicroeconomics.Extensivemathematicalappendicescoverrelativelyhigh-leveltopicsinanalysis.Samuelson,PaulA.FoundationsofEconomicAnalysis.Cambridge,MA:HarvardUniversityPress,1947.Mathe-maticalAppendixA.Abasicreference.MathematicalAppendixAprovidesanadvancedtreatmentofnecessaryandsuﬃcientconditionsforamaximum.Silberberg,E.,andW.Suen.TheStructureofEconomics:AMathematicalAnalysis,3rded.Boston:Irwin/McGraw-Hill,2001.Amathematicalmicroeconomicstextthatstressestheobservablepredictionsofeconomictheory.Thetextmakesextensiveuseoftheenvelopetheorem.Simon,CarlP.,andLawrenceBlume.MathematicsforEconomists.NewYork:W.W.Norton,1994.Averyusefultextcoveringmostareasofmathematicsrelevanttoeconomists.Treatmentisatarelativelyhighlevel.Twotopicsdis-cussedbetterherethanelsewherearediﬀerentialequationsandbasicpoint-settopology.Sydsaeter,K.,A.Strom,andP.Berck.Economists’MathematicalManual,3rded.Berlin:Springer-Verlag,2000.Anindispensabletoolformathematicalreview.Contains32chapterscoveringmostofthemathematicaltoolsthateconomistsuse.Chapter2MathematicsforMicroeconomics79 Discussionsareverybrief,sothisisnottheplacetoencounternewconceptsfortheﬁrsttime.Taylor,AngusE.,andW.RobertMann.AdvancedCalculus,3rded.NewYork:JohnWiley,1983,pp.183–95.AcomprehensivecalculustextwithagooddiscussionoftheLagrang-iantechnique.Thomas,GeorgeB.,andRossL.Finney.CalculusandAnalyticGeometry,8thed.Reading,MA:Addison-Wesley,1992.Basiccalculustextwithexcellentcoverageofdiﬀerentiationtechniques.80Part1Introduction EXTENSIONSSecond-OrderConditionsandMatrixAlgebraThesecond-orderconditionsdescribedinChapter2canbewritteninverycompactwaysbyusingmatrixalgebra.Inthisextension,welookbrieﬂyatthatnota-tion.Wereturntothisnotationatafewotherplacesintheextensionsandproblemsforlaterchapters.MatrixalgebrabackgroundTheextensionspresentedhereassumesomegeneralfamiliaritywithmatrixalgebra.Asuccinctreminderoftheseprinciplesmightinclude:1.Annkmatrix,A,isarectangulararrayoftermsoftheformA¼aijhi¼a11a12…a1ka21a22…a2k...an1an2…ank2666437775:Herei¼1,n;j¼1,k.Matricescanbeadded,subtracted,ormultipliedprovidingtheirdimen-sionsareconformable.2.Ifn¼k,thenAisasquarematrix.Asquarematrixissymmetricifaij¼aji.Theidentityma-trix,In,isannþnsquarematrixwhereaij¼1ifi¼jandaij¼0ifi6¼j.3.Thedeterminantofasquarematrix(denotedbyjAj)isascalar(i.e.,asingleterm)foundbysuitablymultiplyingtogetherallofthetermsinthematrix.IfAis22,jAj¼a11a22a21a12:Example:IfA¼1352thenjAj¼215¼13:4.Theinverseofannnsquarematrix,A,isanothernnmatrix,A1,suchthatAA1¼In:Noteverysquarematrixhasaninverse.AnecessaryandsufﬁcientconditionfortheexistenceofA1isthatjAj6¼0.5.TheleadingprincipalminorsofannnsquarematrixAaretheseriesofdeterminantsoftheﬁrstprowsandcolumnsofA,wherep¼1,n.IfAis22,thentheﬁrstleadingprincipalminorisa11andthesecondisa11a22a21a12.6.Annnsquarematrix,A,ispositivedeﬁniteifallofitsleadingprincipalminorsarepositive.Thematrixisnegativedeﬁniteifitsprincipalminorsalternateinsignstartingwithaminus.17.AparticularlyusefulsymmetricmatrixistheHessianmatrixformedbyallofthesecond-orderpartialderivativesofafunction.Iffisacontinuousandtwicedifferentiablefunctionofnvariables,thenitsHessianisgivenbyHðfÞ¼f11f12…f1nf21f22…f2n...fn1fn2…fnn2666437775:Usingthesenotationalideas,wecannowexam-ineagainsomeofthesecond-orderconditionsderivedinChapter2.E2.1ConcaveandconvexfunctionsAconcavefunctionisonethatisalwaysbelow(oron)anytangenttoit.Alternatively,aconvexfunctionisalwaysabove(oron)anytangent.Theconcavityorconvexityofanyfunctionisdeterminedbyitssecondderivative(s).Forafunctionofasinglevariable,fðxÞ,therequirementisstraightforward.UsingtheTaylorapproximationatanypoint(x0)fðx0þdxÞ¼fðx0Þþf0ðx0Þdxþf00ðx0Þdx22þhigher-orderterms:Assumingthatthehigher-ordertermsare0,wehavefðx0þdxÞfðx0Þþf0ðx0Þdxiff00ðx0Þ0andfðx0þdxÞfðx0Þþf0ðx0Þdxiff00ðx0Þ0.Becausetheexpressionsontherightoftheseinequalitiesareinfacttheequationofthetangenttothefunctionatx0,itisclearthatthe1Ifsomeofthedeterminantsinthisdeﬁnitionare0thenthematrixissaidtobepositivesemideﬁniteornegativesemideﬁnite.Chapter2MathematicsforMicroeconomics81 functionis(locally)concaveiff00ðx0Þ0and(locally)convexiff00ðx0Þ0.Extendingthisintuitiveideatomanydimensionsiscumbersomeintermsoffunctionalnotation,butrela-tivelysimplewhenmatrixalgebraisused.ConcavityrequiresthattheHessianmatrixbenegativedeﬁnitewhereasconvexityrequiresthatthismatrixbepositivedeﬁnite.Asinthesinglevariablecase,theseconditionsamounttorequiringthatthefunctionmoveconsis-tentlyawayfromanytangenttoitnomatterwhatdirectionistaken.2Iffðx1,x2Þisafunctionoftwovariables,theHes-sianisgivenbyH¼f11f12f21f22:Thisisnegativedeﬁniteiff11<0andf11f22f21f12>0,whichispreciselytheconditiondescribedinEqua-tion2.98.Generalizationstofunctionsofthreeormorevariablesfollowthesamematrixpattern.Example1ForthehealthstatusfunctioninChapter2(Equa-tion2.20),theHessianisgivenbyH¼2002,andtheﬁrstandsecondleadingprincipalminorsareH1¼2<0andH2¼ð2Þð2Þ0¼4>0:Hence,thefunctionisconcave.Example2TheCobb-Douglasfunctionxaybwherea,b2ð0,1Þisusedtoillustrateutilityfunctionsandproductionfunctionsinmanyplacesinthistext.Theﬁrst-andsecond-orderderivativesofthefunctionarefx¼axa1yb,fy¼bxayb1,fxx¼aða1Þxa2yb,fyy¼bðb1Þxayb2:Hence,theHessianforthisfunctionisH¼aða1Þxa2ybabxa1yb1abxa1yb1bðb1Þxayb2:TheﬁrstleadingprincipalminorofthisHessianisH1¼aða1Þxa2yb<0andsothefunctionwillbeconcave,providingH2¼aða1ÞðbÞðb1Þx2a2y2b2a2b2x2a2y2b2¼abð1abÞx2a2y2b2>0:Thisconditionclearlyholdsifaþb<1.Thatis,inproductionfunctionterminology,thefunctionmustexhibitdiminishingreturnstoscaletobeconcave.Geometrically,thefunctionmustturndownwardasbothinputsareincreasedtogether.E2.2MaximizationAswesawinChapter2,theﬁrst-orderconditionsforanunconstrainedmaximumofafunctionofmanyvariablesrequiresﬁndingapointatwhichthepartialderivativesarezero.Ifthefunctionisconcaveitwillbebelowitstangentplaneatthispointandthereforethepointwillbeatruemaximum.3Becausethehealthstatusfunctionisconcave,forexample,theﬁrst-orderconditionsforamaximumarealsosufﬁcient.E2.3ConstrainedmaximaWhenthex’sinamaximizationorminimizationprob-lemaresubjecttoconstraints,theseconstraintshavetobetakenintoaccountinstatingsecond-ordercondi-tions.Again,matrixalgebraprovidesacompact(ifnotveryintuitive)wayofdenotingtheseconditions.ThenotationinvolvesaddingrowsandcolumnsoftheHessianmatrixfortheunconstrainedproblemandthencheckingthepropertiesofthisaugmentedmatrix.Speciﬁcally,wewishtomaximizefðx1,…,xnÞsubjecttotheconstraint4gðx1,…,xnÞ¼0:2AproofusingthemultivariableversionofTaylor’sapproximationisprovidedinSimonandBlume(1994),chap.21.3Thiswillbea“local”maximumifthefunctionisconcaveonlyinaregion,or“global”ifthefunctionisconcaveeverywhere.4Herewelookonlyatthecaseofasingleconstraint.Generalizationtomanyconstraintsisconceptuallystraightforwardbutnotationallycom-plex.ForaconcisestatementseeSydsaeter,Strom,andBerck(2000),p.93.82Part1Introduction WesawinChapter2thattheﬁrst-orderconditionsforamaximumareoftheformfiþλgi¼0,whereλistheLagrangianmultiplierforthisproblem.Second-orderconditionsforamaximumarebasedontheaugmented(“bordered”)Hessian5Hb¼0g1g2…gng1f11f12f1ng2f21f22f2n...gnfn1fn2…fnn26666643777775:Foramaximum,(1)Hbmustbenegativedeﬁnite—thatis,theleadingprincipalminorsofHbmustfollowthepattern++andsoforth,startingwiththesecondsuchminor.6Thesecond-orderconditionsforminimumrequirethat(1)Hbbepositivedeﬁnite—thatis,allofthelead-ingprincipalminorsofHb(excepttheﬁrst)shouldbenegative.ExampleTheLagrangianfortheconstrainedhealthstatusprob-lem(Example2.6)isℒ¼x21þ2x1x22þ4x2þ5þλð1x1x2Þ,andtheborderedHessianforthisproblemisHb¼0111201022435:ThesecondleadingprincipalminorhereisHb2¼0112¼1,andthethirdisHb3¼011120102264375¼0þ0þ0ð2Þ0ð2Þ¼4,sotheleadingprincipalminorsoftheHbhavetherequiredpatternandthepointx2¼1,x1¼0,isaconstrainedmaximum.ExampleIntheoptimalfenceproblem(Example2.7),thebor-deredHessianisHb¼0222012102435andHb2¼4,Hb3¼8,soagaintheleadingprincipalminorshavethesignpatternrequiredforamaximum.E2.4Quasi-concavityIftheconstraintgislinear,thenthesecond-orderconditionsexploredinExtension2.3canberelatedsolelytotheshapeofthefunctiontobeoptimized,f.Inthiscasetheconstraintcanbewrittenasgðx1,…,xnÞ¼cb1x1b2x2…bnxn¼0,andtheﬁrst-orderconditionsforamaximumarefi¼λbi,i¼1,…,n:Usingtheconditions,itisclearthattheborderedHessianHbandthematrixH0¼0f1f2…fnf1f11f12f1nf2f21f22f2nfnfn1fn2…fnn26643775havethesameleadingprincipalminorsexceptfora(positive)constantofproportionality.7Thecondi-tionsforamaximumoffsubjecttoalinearconstraintwillbesatisﬁedprovidedH0followsthesamesignconventionsasHb—thatis,(1)H0mustbenegativedeﬁnite.AfunctionfforwhichH0doesfollowthispatterniscalledquasi-concave.Asweshallsee,fhasthepropertythatthesetofpointsxforwhichfðxÞc(wherecisanyconstant)isconvex.Forsuchafunction,thenecessaryconditionsforamaximumarealsosufﬁcient.ExampleForthefencesproblem,fðx,yÞ¼xyandH0isgivenby5Noticethat,ifgij¼0foralliandj,thenHbcanberegardedasthesimpleHessianassociatedwiththeLagrangianexpressiongiveninEquation2.50,whichisafunctionofthenþ1variablesλ,x1,…,xn.6NoticethattheﬁrstleadingprincipalminorofHbis0.7Thiscanbeshownbynotingthatmultiplyingarow(oracolumn)ofamatrixbyaconstantmultipliesthedeterminantbythatconstant.Chapter2MathematicsforMicroeconomics83 H0¼0yxy01x102435:SoH02¼y2<0,H03¼2xy>0,andthefunctionisquasi-concave.8ExampleMoregenerally,iffisafunctionofonlytwovariables,thenquasi-concavityrequiresthatH02¼ðf1Þ2<0andH03¼f11f22f22f21þ2f1f2f12>0,whichispreciselytheconditionstatedinEqua-tion2.114.Hence,wehaveafairlysimplewayofdeterminingquasi-concavity.ReferencesSimon,C.P.,andL.Blume.MathematicsforEconomists.NewYork:W.W.Norton,1994.Sydsaeter,R.,A.Strom,andP.Berck.Economists’Math-ematicalManual,3rded.Berlin:Springer-Verlag,2000.8Sincefðx,yÞ¼xyisaformofaCobb-Douglasfunctionthatisnotconcave,thisshowsthatnoteveryquasi-concavefunctionisconcave.Noticethatamonotonicfunctionoff(suchasf1=3)wouldbeconcave,however.84Part1Introduction
PART2ChoiceandDemandCHAPTER3PreferencesandUtilityCHAPTER4UtilityMaximizationandChoiceCHAPTER5IncomeandSubstitutionEffectsCHAPTER6DemandRelationshipsamongGoodsCHAPTER7UncertaintyandInformationCHAPTER8StrategyandGameTheoryInPart2wewillinvestigatetheeconomictheoryofchoice.Onegoalofthisexaminationistodevelopthenotionofdemandinaformalwaysothatitcanbeusedinlatersectionsofthetextwhenweturntothestudyofmarkets.Amoregeneralgoalofthispartistoillustratethetheoryeconomistsusetoexplainhowindividualsmakechoicesinawidevarietyofcontexts.Part2beginswithadescriptionofthewayeconomistsmodelindividualpreferences,whichareusuallyreferredtobytheformaltermutility.Chapter3showshoweconomistsareabletoconceptualizeutilityinamathematicalway.Thispermitsthedevelopmentof“indiﬀerencecurves,”whichshowthevariousex-changesthatindividualsarewillingtomakevoluntarily.TheutilityconceptisusedinChapter4toillustratethetheoryofchoice.Thefundamentalhypothesisofthechapteristhatpeoplefacedwithlimitedincomeswillmakeeconomicchoicesinsuchawayastoachieveasmuchutilityaspossible.Chapter4usesmathematicalandintuitiveanalysestoindicatetheinsightsthatthishypothesisprovidesabouteconomicbehavior.Chapters5and6usethemodelofutilitymaximizationtoinvestigatehowindividualswillrespondtochangesintheircircumstances.Chapter5isprimarilyconcernedwithresponsestochangesinthepriceofacommodity,ananalysisthatleadsdirectlytothedemandcurvenotion.Chapter6appliesthistypeofanalysistodevelopinganunderstandingofdemandrelationshipsamongdiﬀerentgoods.Theﬁnaltwochaptersinthispartlookatindividualbehaviorinuncertainsituations.InChapter7wedescribewhypeoplegenerallydislikerisksandarewillingtopaysomethingtoavoidtakingthem.Chapter8thenlooksatuncertaintiesthatarisewhentwoormorepeopleﬁndthemselvesina“game”inwhichtheymustmakestrategicchoices.Theequilibriumnotionswedevelopinstudyingsuchgamesarewidelyusedthroughouteconomics.
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CHAPTER3PreferencesandUtilityInthischapterwelookatthewayinwhicheconomistscharacterizeindividuals’preferences.Webeginwithafairlyabstractdiscussionofthe“preferencerelation,”butquicklyturntotheeconomists’primarytoolforstudyingindividualchoices—theutilityfunction.Welookatsomegeneralcharacteristicsofthatfunctionandafewsimpleexamplesofspeciﬁcutilityfunctionswewillencounterthroughoutthisbook.AXIOMSOFRATIONALCHOICEOnewaytobeginananalysisofindividuals’choicesistostateabasicsetofpostulates,oraxioms,thatcharacterize“rational”behavior.Thesebeginwiththeconceptof“preference”:Anindividualwhoreportsthat“AispreferredtoB”istakentomeanthatallthingscon-sidered,heorshefeelsbetteroffundersituationAthanundersituationB.Thepreferencerelationisassumedtohavethreebasicpropertiesasfollows.I.Completeness.IfAandBareanytwosituations,theindividualcanalwaysspecifyexactlyoneofthefollowingthreepossibilities:1.“AispreferredtoB,”2.“BispreferredtoA,”or3.“AandBareequallyattractive.”Consequently,peopleareassumednottobeparalyzedbyindecision:Theycompletelyunderstandandcanalwaysmakeuptheirmindsaboutthedesirabilityofanytwoalternatives.TheassumptionalsorulesoutthepossibilitythatanindividualcanreportboththatAispreferredtoBandthatBispreferredtoA.II.Transitivity.Ifanindividualreportsthat“AispreferredtoB”and“BispreferredtoC,”thenheorshemustalsoreportthat“AispreferredtoC.”Thisassumptionstatesthattheindividual’schoicesareinternallyconsistent.Suchanassumptioncanbesubjectedtoempiricalstudy.Generally,suchstudiesconcludethataperson’schoicesareindeedtransitive,butthisconclusionmustbemodiﬁedincaseswheretheindividualmaynotfullyunderstandtheconsequencesofthechoicesheorsheismaking.Because,forthemostpart,wewillassumechoicesarefullyinformed(butseethediscussionofuncertaintyinChapter7andelsewhere),thetransitivitypropertyseemstobeanappropriateassumptiontomakeaboutpreferences.III.Continuity.Ifanindividualreports“AispreferredtoB,”thensituationssuitably“closeto”AmustalsobepreferredtoB.Thisrathertechnicalassumptionisrequiredifwewishtoanalyzeindividuals’responsestorelativelysmallchangesinincomeandprices.Thepurposeoftheassumptionistoruleoutcertainkindsofdiscontinuous,knife-edgepreferencesthatposeproblemsforamathematicaldevelopmentofthetheoryofchoice.Assumingcontinuitydoes87
notseemtoriskmissingtypesofeconomicbehaviorthatareimportantintherealworld.UTILITYGiventheassumptionsofcompleteness,transitivity,andcontinuity,itispossibletoshowformallythatpeopleareabletorankallpossiblesituationsfromtheleastdesirabletothemost.1Followingtheterminologyintroducedbythenineteenth-centurypoliticaltheoristJeremyBentham,economistscallthisrankingutility.2WealsowillfollowBenthambysayingthatmoredesirablesituationsoffermoreutilitythandolessdesirableones.Thatis,ifapersonpreferssituationAtosituationB,wewouldsaythattheutilityassignedtooptionA,denotedbyUðAÞ,exceedstheutilityassignedtoB,UðBÞ.NonuniquenessofutilitymeasuresWemightevenattachnumberstotheseutilityrankings;however,thesenumberswillnotbeunique.Anysetofnumberswearbitrarilyassignthataccuratelyreﬂectstheoriginalprefer-enceorderingwillimplythesamesetofchoices.ItmakesnodifferencewhetherwesaythatUðAÞ¼5andUðBÞ¼4,orthatUðAÞ¼1,000,000andUðBÞ¼0:5.InbothcasesthenumbersimplythatAispreferredtoB.Intechnicalterms,ournotionofutilityisdeﬁnedonlyuptoanorder-preserving(“monotonic”)transformation.3Anysetofnumbersthataccuratelyreﬂectsaperson’spreferenceorderingwilldo.Consequently,itmakesnosensetoask“howmuchmoreisApreferredthanB?”sincethatquestionhasnouniqueanswer.Surveysthataskpeopletoranktheir“happiness”onascaleof1to10couldjustaswelluseascaleof7to1,000,000.Wecanonlyhopethatapersonwhoreportsheorsheisa“6”onthescaleonedayanda“7”onthenextdayisindeedhappieronthesecondday.Utilityrankingsarethereforeliketheordinalrankingsofrestaurantsormoviesusingone,two,three,orfourstars.Theysimplyrecordtherelativedesirabilityofcommoditybundles.Thislackofuniquenessintheassignmentofutilitynumbersalsoimpliesthatitisnotpossibletocompareutilitiesofdifferentpeople.Ifonepersonreportsthatasteakdinnerprovidesautilityof“5”andanotherreportsthatthesamedinneroffersautilityof“100,”wecannotsaywhichindividualvaluesthedinnermorebecausetheycouldbeusingverydifferentscales.Similarly,wehavenowayofmeasuringwhetheramovefromsituationAtosituationBprovidesmoreutilitytoonepersonoranother.Nonetheless,aswewillsee,economistscansayquiteabitaboututilityrankingsbyexaminingwhatpeoplevoluntarilychoosetodo.TheceterisparibusassumptionBecauseutilityreferstooverallsatisfaction,suchameasureclearlyisaffectedbyavarietyoffactors.Aperson’sutilityisaffectednotonlybyhisorherconsumptionofphysicalcommod-ities,butalsobypsychologicalattitudes,peergrouppressures,personalexperiences,andthe1ThesepropertiesandtheirconnectiontorepresentationofpreferencesbyautilityfunctionarediscussedindetailinAndreuMas-Colell,MichaelD.Whinston,andJerryR.Green,MicroeconomicTheory(NewYork:OxfordUniversityPress,1995).2J.Bentham,IntroductiontothePrinciplesofMoralsandLegislation(London:Hafner,1848).3WecandenotethisideamathematicallybysayingthatanynumericalutilityrankingðUÞcanbetransformedintoanothersetofnumbersbythefunctionFprovidingthatFðUÞisorderpreserving.ThiscanbeensuredifF0ðUÞ>0.Forexample,thetransformationFðUÞ¼U2isorderpreservingasisthetransformationFðUÞ¼lnU.Atsomeplacesinthetextandproblemswewillfinditconvenienttomakesuchtransformationsinordertomakeaparticularutilityrankingeasiertoanalyze.88Part2ChoiceandDemand
generalculturalenvironment.Althougheconomistsdohaveageneralinterestinexaminingsuchinﬂuences,anarrowingoffocusisusuallynecessary.Consequently,acommonpracticeistodevoteattentionexclusivelytochoicesamongquantiﬁableoptions(forexample,therelativequantitiesoffoodandshelterbought,thenumberofhoursworkedperweek,orthevotesamongspeciﬁctaxingformulas)whileholdingconstanttheotherthingsthataffectbehavior.Thisceterisparibus(otherthingsbeingequal)assumptionisinvokedinalleco-nomicanalysesofutility-maximizingchoicessoastomaketheanalysisofchoicesmanageablewithinasimpliﬁedsetting.UtilityfromconsumptionofgoodsAsanimportantexampleoftheceterisparibusassumption,consideranindividual’sproblemofchoosing,atasinglepointintime,amongnconsumptiongoodsx1,x2,…,xn:Weshallassumethattheindividual’srankingofthesegoodscanberepresentedbyautilityfunctionoftheformutility¼Uðx1,x2,…,xn;otherthingsÞ,(3.1)wherethex’srefertothequantitiesofthegoodsthatmightbechosenandthe“otherthings”notationisusedasareminderthatmanyaspectsofindividualwelfarearebeingheldconstantintheanalysis.QuiteoftenitiseasiertowriteEquation3.1asutility¼Uðx1,x2,…,xnÞ(3.2)or,ifonlytwogoodsarebeingconsidered,asutility¼Uðx,yÞ,(3.20)whereitisclearthateverythingisbeingheldconstant(thatis,outsidetheframeofanalysis)exceptthegoodsactuallyreferredtointheutilityfunction.Itwouldbetedioustoremindyouateachstepwhatisbeingheldconstantintheanalysis,butitshouldberememberedthatsomeformoftheceterisparibusassumptionwillalwaysbeineffect.ArgumentsofutilityfunctionsTheutilityfunctionnotationisusedtoindicatehowanindividualrankstheparticularargumentsofthefunctionbeingconsidered.Inthemostcommoncase,theutilityfunction(Equation3.2)willbeusedtorepresenthowanindividualrankscertainbundlesofgoodsthatmightbepurchasedatonepointintime.Onoccasionwewilluseotherargumentsintheutilityfunction,anditisbesttoclearupcertainconventionsattheoutset.Forexample,itmaybeusefultotalkabouttheutilityanindividualreceivesfromrealwealthðWÞ.Therefore,weshallusethenotationutility¼UðWÞ.(3.3)Unlesstheindividualisaratherpeculiar,Scrooge-typeperson,wealthinitsownrightgivesnodirectutility.Rather,itisonlywhenwealthisspentonconsumptiongoodsthatanyutilityresults.Forthisreason,Equation3.3willbetakentomeanthattheutilityfromwealthisinfactderivedbyspendingthatwealthinsuchawayastoyieldasmuchutilityaspossible.Twootherargumentsofutilityfunctionswillbeusedinlaterchapters.InChapter16wewillbeconcernedwiththeindividual’slabor-leisurechoiceandwillthereforehavetocon-siderthepresenceofleisureintheutilityfunction.Afunctionoftheformutility¼Uðc,hÞ(3.4)willbeused.Here,crepresentsconsumptionandhrepresentshoursofnonworktime(thatis,leisure)duringaparticulartimeperiod.Chapter3PreferencesandUtility89
InChapter17wewillbeinterestedintheindividual’sconsumptiondecisionsindifferenttimeperiods.Inthatchapterwewilluseautilityfunctionoftheformutility¼Uðc1,c2Þ,(3.5)wherec1isconsumptioninthisperiodandc2isconsumptioninthenextperiod.Bychangingtheargumentsoftheutilityfunction,therefore,wewillbeabletofocusonspecificaspectsofanindividual’schoicesinavarietyofsimplifiedsettings.Insummarythen,westartourexaminationofindividualbehaviorwiththefollowingdeﬁnition.DEFINITIONUtility.Individuals’preferencesareassumedtoberepresentedbyautilityfunctionoftheformUðx1,x2,…,xnÞ,(3.6)wherex1,x2,…,xnarethequantitiesofeachofngoodsthatmightbeconsumedinaperiod.Thisfunctionisuniqueonlyuptoanorder-preservingtransformation.EconomicgoodsInthisrepresentationthevariablesaretakentobe“goods”;thatis,whatevereconomicquan-titiestheyrepresent,weassumethatmoreofanyparticularxiduringsomeperiodispreferredtoless.Weassumethisistrueofeverygood,beitasimpleconsumptionitemsuchasahotdogoracomplexaggregatesuchaswealthorleisure.Wehavepicturedthisconventionforatwo-goodutilityfunctioninFigure3.1.There,allconsumptionbundlesintheshadedareaareFIGURE3.1MoreofaGoodIsPreferredtoLessTheshadedarearepresentsthosecombinationsofxandythatareunambiguouslypreferredtothecombinationx,y.Ceterisparibus,individualsprefermoreofanygoodratherthanless.Combina-tionsidentiﬁedby“?”involveambiguouschangesinwelfarebecausetheycontainmoreofonegoodandlessoftheother.Quantity of xQuantityof y??Preferredtox*, y*Worsethanx*, y*y*x*90Part2ChoiceandDemand
preferredtothebundlex,ybecauseanybundleintheshadedareaprovidesmoreofatleastoneofthegoods.Byourdeﬁnitionof“goods,”then,bundlesofgoodsintheshadedareaarerankedhigherthanx,y.Similarly,bundlesintheareamarked“worse”areclearlyinferiortox,y,sincetheycontainlessofatleastoneofthegoodsandnomoreoftheother.Bundlesinthetwoareasindicatedbyquestionmarksaredifﬁculttocomparetox,ybecausetheycontainmoreofoneofthegoodsandlessoftheother.Movementsintotheseareasinvolvetrade-offsbetweenthetwogoods.TRADESANDSUBSTITUTIONMosteconomicactivityinvolvesvoluntarytradingbetweenindividuals.Whensomeonebuys,say,aloafofbread,heorsheisvoluntarilygivinguponething(money)forsomethingelse(bread)thatisofgreatervaluetothatindividual.Toexaminethiskindofvoluntarytransaction,weneedtodevelopaformalapparatusforillustratingtradesintheutilityfunctioncontext.IndifferencecurvesandthemarginalrateofsubstitutionTodiscusssuchvoluntarytrades,wedeveloptheideaofanindifferencecurve.InFigure3.2,thecurveU1representsallthealternativecombinationsofxandyforwhichanindividualisequallywelloff(rememberagainthatallotherargumentsoftheutilityfunctionarebeingFIGURE3.2ASingleIndifferenceCurveThecurveU1representsthosecombinationsofxandyfromwhichtheindividualderivesthesameutility.Theslopeofthiscurverepresentstherateatwhichtheindividualiswillingtotradexforywhileremainingequallywelloff.Thisslope(or,moreproperly,thenegativeoftheslope)istermedthemarginalrateofsubstitution.Intheﬁgure,theindifferencecurveisdrawnontheassumptionofadiminishingmarginalrateofsubstitution.Quantity of xQuantityof yx2x1y1U1U1y2Chapter3PreferencesandUtility91
heldconstant).Thispersonisequallyhappyconsuming,forexample,eitherthecombinationofgoodsx1,y1orthecombinationx2,y2.Thiscurverepresentingalltheconsumptionbundlesthattheindividualranksequallyiscalledanindifferencecurve.DEFINITIONIndifferencecurve.Anindifferencecurve(or,inmanydimensions,anindifferencesurface)showsasetofconsumptionbundlesaboutwhichtheindividualisindifferent.Thatis,thebundlesallprovidethesamelevelofutility.TheslopeoftheindifferencecurveinFigure3.2isnegative,showingthatiftheindividualisforcedtogiveupsomey,heorshemustbecompensatedbyanadditionalamountofxtoremainindifferentbetweenthetwobundlesofgoods.Thecurveisalsodrawnsothattheslopeincreasesasxincreases(thatis,theslopestartsatnegativeinﬁnityandincreasestowardzero).Thisisagraphicalrepresentationoftheassumptionthatpeoplebecomeprogressivelylesswillingtotradeawayytogetmorex.Inmathematicalterms,theabsolutevalueofthisslopediminishesasxincreases.Hence,wehavethefollowingdeﬁnition.DEFINITIONMarginalrateofsubstitution.ThenegativeoftheslopeofanindifferencecurveðU1Þatsomepointistermedthemarginalrateofsubstitution(MRS)atthatpoint.Thatis,MRS¼dydxU¼U1,(3.7)wherethenotationindicatesthattheslopeistobecalculatedalongtheU1indifferencecurve.TheslopeofU1andtheMRSthereforetellussomethingaboutthetradesthispersonwillvoluntarilymake.Atapointsuchasx1,y1,thepersonhasquitealotofyandiswillingtotradeawayasigniﬁcantamounttogetonemorex.Theindifferencecurveatx1,y1isthereforerathersteep.Thisisasituationwherethepersonhas,say,manyhamburgersðyÞandlittletodrinkwiththem(x).Thispersonwouldgladlygiveupafewburgers(say,5)toquenchhisorherthirstwithonemoredrink.Atx2,y2,ontheotherhand,theindifferencecurveisﬂatter.Here,thispersonhasquiteafewdrinksandiswillingtogiveuprelativelyfewburgers(say,1)togetanothersoftdrink.Consequently,theMRSdiminishesbetweenx1,y1andx2,y2.ThechangingslopeofU1showshowtheparticularconsumptionbundleavailableinﬂuencesthetradesthispersonwillfreelymake.IndifferencecurvemapInFigure3.2onlyoneindifferencecurvewasdrawn.Thex,yquadrant,however,isdenselypackedwithsuchcurves,eachcorrespondingtoadifferentlevelofutility.Becauseeverybundleofgoodscanberankedandyieldssomelevelofutility,eachpointinFigure3.2musthaveanindifferencecurvepassingthroughit.Indifferencecurvesaresimilartocontourlinesonamapinthattheyrepresentlinesofequal“altitude”ofutility.InFigure3.3severalindifferencecurvesareshowntoindicatethatthereareinﬁnitelymanyintheplane.Thelevelofutilityrepresentedbythesecurvesincreasesaswemoveinanortheastdirection;theutilityofcurveU1islessthanthatofU2,whichislessthanthatofU3.ThisisbecauseoftheassumptionmadeinFigure3.1:Moreofagoodispreferredtoless.Aswasdiscussedearlier,thereisnouniquewaytoassignnumberstotheseutilitylevels.ThecurvesonlyshowthatthecombinationsofgoodsonU3arepreferredtothoseonU2,whicharepreferredtothoseonU1.92Part2ChoiceandDemand
IndifferencecurvesandtransitivityAsanexerciseinexaminingtherelationshipbetweenconsistentpreferencesandtherepresenta-tionofpreferencesbyutilityfunctions,considerthefollowingquestion:Cananytwoofanindividual’sindifferencecurvesintersect?TwosuchintersectingcurvesareshowninFigure3.4.Wewishtoknowiftheyviolateourbasicaxiomsofrationality.Usingourmapanalogy,therewouldseemtobesomethingwrongatpointE,where“altitude”isequaltotwodifferentnumbers,U1andU2.Butnopointcanbeboth100and200feetabovesealevel.Toproceedformally,letusanalyzethebundlesofgoodsrepresentedbypointsA,B,C,andD.Bytheassumptionofnonsatiation(i.e.,moreofagoodalwaysincreasesutility),“AispreferredtoB”and“CispreferredtoD.”ButthispersonisequallysatisﬁedwithBandC(theylieonthesameindifferencecurve),sotheaxiomoftransitivityimpliesthatAmustbepreferredtoD.Butthatcannotbetrue,becauseAandDareonthesameindifferencecurveandarebydeﬁnitionregardedasequallydesirable.Thiscontradictionshowsthatindifferencecurvescannotintersect.ThereforeweshouldalwaysdrawindifferencecurvemapsastheyappearinFigure3.3.ConvexityofindifferencecurvesAnalternativewayofstatingtheprincipleofadiminishingmarginalrateofsubstitutionusesthemathematicalnotionofaconvexset.Asetofpointsissaidtobeconvexifanytwopointswithinthesetcanbejoinedbyastraightlinethatiscontainedcompletelywithintheset.Theas-sumptionofadiminishingMRSisequivalenttotheassumptionthatallcombinationsofxandyFIGURE3.3ThereAreInfinitelyManyIndifferenceCurvesinthex–yPlaneThereisanindifferencecurvepassingthrougheachpointinthex–yplane.Eachofthesecurvesrecordscombinationsofxandyfromwhichtheindividualreceivesacertainlevelofsatisfaction.Movementsinanortheastdirectionrepresentmovementstohigherlevelsofsatisfaction.Quantity of xQuantityof yIncreasing utilityU1U1U2U3U2U3Chapter3PreferencesandUtility93
thatarepreferredorindifferenttoaparticularcombinationx,yformaconvexset.4ThisisillustratedinFigure3.5a,whereallcombinationspreferredorindifferenttox,yareintheshadedarea.Anytwoofthesecombinations—say,x1,y1andx2,y2—canbejoinedbyastraightlinealsocontainedintheshadedarea.InFigure3.5bthisisnottrue.Alinejoiningx1,y1andx2,y2passesoutsidetheshadedarea.Therefore,theindifferencecurvethroughx,yinFigure3.5bdoesnotobeytheassumptionofadiminishingMRS,becausethesetofpointspreferredorindifferenttox,yisnotconvex.ConvexityandbalanceinconsumptionByusingthenotionofconvexity,wecanshowthatindividualsprefersomebalanceintheirconsumption.Supposethatanindividualisindifferentbetweenthecombinationsx1,y1andx2,y2.Iftheindifferencecurveisstrictlyconvex,thenthecombinationðx1þx2Þ=2,ðy1þy2Þ=2willbepreferredtoeitheroftheinitialcombinations.5Intuitively,“well-balanced”bundlesofcommoditiesarepreferredtobundlesthatareheavilyweightedtowardonecommodity.ThisisillustratedinFigure3.6.Becausetheindifferencecurveisassumedtobeconvex,allpointsonthestraightlinejoiningðx1,y1Þandðx2,y2Þarepreferredtotheseinitialpoints.Thisthereforewillbetrueofthepointðx1þx2Þ=2,ðy1þy2Þ=2,whichliesatthemidpointofsuchaline.FIGURE3.4IntersectingIndifferenceCurvesImplyInconsistentPreferencesCombinationsAandDlieonthesameindifferencecurveandthereforeareequallydesirable.ButtheaxiomoftransitivitycanbeusedtoshowthatAispreferredtoD.Hence,intersectingindifferencecurvesarenotconsistentwithrationalpreferences.Quantity of xQuantityof yDCEABU2U14Thisdefinitionisequivalenttoassumingthattheutilityfunctionisquasi-concave.SuchfunctionswerediscussedinChapter2,andweshallreturntoexaminetheminthenextsection.Sometimesthetermstrictquasi-concavityisusedtoruleoutthepossibilityofindifferencecurveshavinglinearsegments.Wegenerallywillassumestrictquasi-concavity,butinafewplaceswewillillustratethecomplicationsposedbylinearportionsofindifferencecurves.5Inthecaseinwhichtheindifferencecurvehasalinearsegment,theindividualwillbeindifferentamongallthreecombinations.94Part2ChoiceandDemand
FIGURE3.5TheNotionofConvexityasanAlternativeDefinitionofaDiminishingMRSIn(a)theindifferencecurveisconvex(anylinejoiningtwopointsaboveU1isalsoaboveU1).In(b)thisisnotthecase,andthecurveshownheredoesnoteverywherehaveadiminishingMRS.Quantityof xQuantityof xQuantityof yQuantityof y(b)(a)U1U1U1U1y1y2y2x1x1x2x*x2x*y*y1y*FIGURE3.6BalancedBundlesofGoodsArePreferredtoExtremeBundlesIfindifferencecurvesareconvex(iftheyobeytheassumptionofadiminishingMRS),thenthelinejoininganytwopointsthatareindifferentwillcontainpointspreferredtoeitheroftheinitialcombinations.Intuitively,balancedbundlesarepreferredtounbalancedones.Quantity of xQuantityof y2x1+ x22y1+ y2U1U1y1x1x2y2
Indeed,anyproportionalcombinationofthetwoindifferentbundlesofgoodswillbepreferredtotheinitialbundles,becauseitwillrepresentamorebalancedcombination.Thus,strictconvexityisequivalenttotheassumptionofadiminishingMRS.Bothassumptionsruleoutthepossibilityofanindifferencecurvebeingstraightoveranyportionofitslength.EXAMPLE3.1UtilityandtheMRSSupposeaperson’srankingofhamburgersðyÞandsoftdrinksðxÞcouldberepresentedbytheutilityfunctionutility¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃx⋅yp.(3.8)Anindifferencecurveforthisfunctionisfoundbyidentifyingthatsetofcombinationsofxandyforwhichutilityhasthesamevalue.Supposewearbitrarilysetutilityequalto10.Thentheequationforthisindifferencecurveisutility¼10¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃx⋅yp.(3.9)Becausesquaringthisfunctionisorderpreserving,theindifferencecurveisalsorepre-sentedby100¼x⋅y,(3.10)whichiseasiertograph.InFigure3.7weshowthisindifferencecurve;itisafamiliarrectangularhyperbola.OnewaytocalculatetheMRSistosolveEquation3.10fory,y¼100=x,(3.11)FIGURE3.7IndifferenceCurveforUtility¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃx⋅ypThisindifferencecurveillustratesthefunction10¼U¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃx⋅yp.AtpointA(5,20),theMRSis4,implyingthatthispersoniswillingtotrade4yforanadditionalx.AtpointB(20,5),however,theMRSis0.25,implyingagreatlyreducedwillingnesstotrade.Quantity of xQuantityof y2012.55200512.5ACBU= 1096Part2ChoiceandDemand
andthenusethedeﬁnition(Equation3.7):MRS¼dy=dxðalongU1Þ¼100=x2.(3.12)ClearlythisMRSdeclinesasxincreases.AtapointsuchasAontheindifferencecurvewithalotofhamburgers(say,x¼5,y¼20),theslopeissteepsotheMRSishigh:MRSatð5,20Þ¼100=x2¼100=25¼4.(3.13)Herethepersoniswillingtogiveup4hamburgerstoget1moresoftdrink.Ontheotherhand,atBwheretherearerelativelyfewhamburgers(herex¼20,y¼5),theslopeisflatandtheMRSislow:MRSatð20,5Þ¼100=x2¼100=400¼0:25.(3.14)Nowheorshewillonlygiveuponequarterofahamburgerforanothersoftdrink.NoticealsohowconvexityoftheindifferencecurveU1isillustratedbythisnumericalexample.PointCismidwaybetweenpointsAandB;atCthispersonhas12.5hamburgersand12.5softdrinks.Hereutilityisgivenbyutility¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃx⋅yp¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃð12:5Þ2q¼12:5,(3.15)whichclearlyexceedstheutilityalongU1(whichwasassumedtobe10).QUERY:Fromourderivationhere,itappearsthattheMRSdependsonlyonthequantityofxconsumed.Whyisthismisleading?HowdoesthequantityofyimplicitlyenterintoEquations3.13and3.14?AMATHEMATICALDERIVATIONAmathematicalderivationprovidesadditionalinsightsabouttheshapeofindifferencecurvesandthenatureofpreferences.Inthissectionweprovidesuchaderivationforthecaseofautilityfunctioninvolvingonlytwogoods.Thiswillallowustocomparethemathematicstothetwo-dimensionalindifferencecurvemap.Thecaseofmanygoodswillbetakenupattheendofthechapter,butitwillturnoutthatthismorecomplicatedcasereallyaddsverylittle.TheMRSandmarginalutilityIftheutilityapersonreceivesfromtwogoodsisrepresentedbyUðx,yÞ,wecanwritethetotaldifferentialofthisfunctionasdU¼∂U∂x⋅dxþ∂U∂y⋅dy.(3.16)AlonganyparticularindifferencecurvedU¼0,asimplemanipulationofEquation3.16yieldsMRS¼dydxU¼constant¼∂U=∂x∂U=∂y.(3.17)Inwords,theMRSofxforyisequaltotheratioofthemarginalutilityofx(thatis,∂U=∂x)tothemarginalutilityofyð∂U=∂yÞ.Thisresultmakesintuitivesense.Supposethataperson’sutilitywereactuallymeasurablein,say,unitscalled“utils.”Assumealsothatthispersonconsumesonlytwogoods,foodðxÞandclothing(y),andthateachextraunitoffoodprovides6utilswhereaseachextraunitofclothingprovides2utils.ThenEquation3.17wouldmeanthatChapter3PreferencesandUtility97
MRS¼dydxU¼constant¼6utils2utils¼3,sothispersoniswillingtotradeaway3unitsofclothingtoget1moreunitoffood.Thistradewouldresultinnonetchangeinutilitybecausethegainsandlosseswouldbepreciselyoffsetting.Noticethattheunitsinwhichutilityismeasured(whatwehave,forlackofabetterword,called“utils”)canceloutinmakingthiscalculation.Althoughmarginalutilityisobviouslyaffectedbytheunitsinwhichutilityismeasured,theMRSisindependentofthatchoice.6TheconvexityofindifferencecurvesInChapter1wedescribedhowtheassumptionofdiminishingmarginalutilitywasusedbyMarshalltosolvethewater-diamondparadox.Marshalltheorizedthatitisthemarginalvaluationthatanindividualplacesonagoodthatdeterminesitsvalue:Itistheamountthatanindividualiswillingtopayforonemorepintofwaterthatdeterminesthepriceofwater.Becauseitmightbethoughtthatthismarginalvaluedeclinesasthequantityofwaterthatisconsumedincreases,Marshallshowedwhywaterhasalowexchangevalue.Intuitively,itseemsthattheassumptionofadecreasingmarginalutilityofagoodisrelatedtotheassump-tionofadecreasingMRS;bothconceptsseemtorefertothesamecommonsenseideaofanindividualbecomingrelativelysatiatedwithagoodasmoreofitisconsumed.Unfortunately,thetwoconceptsarequitedifferent.(SeeProblem3.3.)Technically,theassumptionofadiminishingMRSisequivalenttorequiringthattheutilityfunctionbequasi-concave.Thisrequirementisrelatedinarathercomplexwaytotheassumptionthateachgoodencountersdiminishingmarginalutility(thatis,thatfiiisnegativeforeachgood).7Butthatistobeexpectedbecausetheconceptofdiminishingmarginalutilityisnotindependentofhow6Moreformally,letFðUÞbeanyarbitraryorder-preservingtransformationofU(thatis,F0ðUÞ>0).Then,forthetransformedutilityfunction,MRS¼∂F=∂x∂F=∂y¼F0ðUÞ∂U=∂xF0ðUÞ∂U=∂y¼∂U=∂x∂U=∂y,whichistheMRSfortheoriginalfunctionU.ThattheF0ðUÞtermscanceloutshowsthattheMRSisindependentofhowutilityismeasured.7WehaveshownthatifutilityisgivenbyU¼fðx,yÞ,thenMRS¼fxfy¼f1f2¼dydx.TheassumptionofadiminishingMRSmeansthatdMRS=dx<0,butdMRSdx¼f2ðf11þf12⋅dy=dxÞf1ðf21þf22⋅dy=dxÞf22:Usingthefactthatf1=f2¼dy=dx,wehavedMRSdx¼f2½f11f12ðf1=f2Þf1½f21f22ðf1=f2Þf22:Combiningtermsandrecognizingthatf12¼f21yieldsdMRSdx¼f2f112f1f12þðf22f21Þ=f2f22or,multiplyingnumeratoranddenominatorbyf2,dMRSdx¼f22f112f1f2f12þf21f22f32.98Part2ChoiceandDemand utilityitselfismeasured,whereastheconvexityofindifferencecurvesisindeedindependentofsuchmeasurement.EXAMPLE3.2ShowingConvexityofIndifferenceCurvesCalculationoftheMRSforspeciﬁcutilityfunctionsisfrequentlyagoodshortcutforshowingconvexityofindifferencecurves.Inparticular,theprocesscanbemuchsimplerthanapplyingthedeﬁnitionofquasi-concavity,thoughitismoredifﬁculttogeneralizetomorethantwogoods.HerewelookathowEquation3.17canbeusedforthreedifferentutilityfunctions(formorepractice,seeProblem3.1).1.Uðx,yÞ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃx⋅yp.ThisexamplejustrepeatsthecaseillustratedinExample3.1.OneshortcuttoapplyingEquation3.17thatcansimplifythealgebraistotakethelogarithmofthisutilityfunction.Becausetakinglogsisorderpreserving,thiswillnotaltertheMRStobecalculated.So,letUðx,yÞ¼ln½Uðx,yÞ¼0:5lnxþ0:5lny.(3.18)ApplyingEquation3.17yieldsMRS¼∂U=∂x∂U=∂y¼0:5=x0:5=y¼yx,(3.19)whichseemstobeamuchsimplerapproachthanweusedpreviously.8ClearlythisMRSisdiminishingasxincreasesandydecreases.Theindifferencecurvesarethereforeconvex.2.Uðx,yÞ¼xþxyþy.Inthiscasethereisnoadvantagetotransformingthisutilityfunction.ApplyingEquation3.17yieldsMRS¼∂U=∂x∂U=∂y¼1þy1þx.(3.20)Again,thisratioclearlydecreasesasxincreasesandydecreases,sotheindifferencecurvesforthisfunctionareconvex.3.Uðx,yÞ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃx2þy2p.ForthisexampleitiseasiertousethetransformationUðx,yÞ¼½Uðx,yÞ2¼x2þy2.(3.21)Becausethisistheequationforaquarter-circle,weshouldbegintosuspectthatthere(continued)Ifweassumethatf2>0(thatmarginalutilityispositive),thentheMRSwilldiminishaslongasf22f112f1f2f12þf21f22<0.Noticethatdiminishingmarginalutility(f11<0andf22<0)willnotensurethisinequality.Onemustalsobeconcernedwiththef12term.Thatis,onemustknowhowdecreasesinyaffectthemarginalutilityofx.Ingeneralitisnotpossibletopredictthesignofthatterm.TheconditionrequiredforadiminishingMRSispreciselythatdiscussedinChapter2toensurethatthefunctionfisstrictlyquasi-concave.Theconditionshowsthatthenecessaryconditionsforamaximumoffsubjecttoalinearconstraintarealsosufficient.WewillusethisresultinChapter4andelsewhere.8InExample3.1welookedattheU¼10indifferencecurve.So,forthatcurve,y¼100=xandtheMRSinEquation3.19wouldbeMRS¼100=x2ascalculatedbefore.Chapter3PreferencesandUtility99 EXAMPLE3.2CONTINUEDmightbesomeproblemswiththeindifferencecurvesforthisutilityfunction.ThesesuspicionsareconfirmedbyagainapplyingthedefinitionoftheMRStoyieldMRS¼∂U=∂x∂U=∂y¼2x2y¼xy.(3.22)Forthisfunction,itisclearthat,asxincreasesandydecreases,theMRSincreases!Hencetheindifferencecurvesareconcave,notconvex,andthisisclearlynotaquasi-concavefunction.QUERY:DoesadoublingofxandychangetheMRSineachofthesethreeexamples?Thatis,doestheMRSdependonlyontheratioofxtoy,notontheabsolutescaleofpurchases?(SeealsoExample3.3.)UTILITYFUNCTIONSFORSPECIFICPREFERENCESIndividuals’rankingsofcommoditybundlesandtheutilityfunctionsimpliedbytheserankingsareunobservable.Allwecanlearnaboutpeople’spreferencesmustcomefromthebehaviorweobservewhentheyrespondtochangesinincome,prices,andotherfactors.Itisneverthelessusefultoexamineafewoftheformsparticularutilityfunctionsmighttake,becausesuchanexaminationmayofferinsightsintoobservedbehaviorand(moretothepoint)understandingthepropertiesofsuchfunctionscanbeofsomehelpinsolvingproblems.Herewewillexaminefourspeciﬁcexamplesofutilityfunctionsfortwogoods.IndifferencecurvemapsforthesefunctionsareillustratedinthefourpanelsofFigure3.8.Asshouldbevisuallyapparent,thesecoverquiteafewpossibleshapes.Evengreatervarietyispossibleoncewemovetofunctionsforthreeormoregoods,andsomeofthesepossibilitiesarementionedinlaterchapters.Cobb-DouglasutilityFigure3.8ashowsthefamiliarshapeofanindifferencecurve.Onecommonlyusedutilityfunctionthatgeneratessuchcurveshastheformutility¼Uðx,yÞ¼xαyβ,(3.23)whereαandβarepositiveconstants.InExamples3.1and3.2,westudiedaparticularcaseofthisfunctionforwhichα¼β¼0:5.ThemoregeneralcasepresentedinEquation3.23istermedaCobb-Douglasutilityfunction,aftertworesearcherswhousedsuchafunctionfortheirdetailedstudyofproductionrelationshipsintheU.S.economy(seeChapter7).Ingeneral,therelativesizesofαandβindicatetherelativeimportanceofthetwogoodstothisindividual.Sinceutilityisuniqueonlyuptoamonotonictransformation,itisoftenconvenienttonormalizetheseparameterssothatαþβ¼1.PerfectsubstitutesThelinearindifferencecurvesinFigure3.8baregeneratedbyautilityfunctionoftheformutility¼Uðx,yÞ¼αxþβy,(3.24)100Part2ChoiceandDemand where,again,αandβarepositiveconstants.Thattheindifferencecurvesforthisfunctionarestraightlinesshouldbereadilyapparent:AnyparticularlevelcurvecanbecalculatedbysettingUðx,yÞequaltoaconstantthat,giventhelinearformofthefunction,clearlyspeciﬁesastraightline.Thelinearnatureoftheseindifferencecurvesgaverisetothetermperfectsubstitutestodescribetheimpliedrelationshipbetweenxandy.BecausetheMRSisconstant(andequaltoα=β)alongtheentireindifferencecurve,ourpreviousnotionsofadiminishingMRSdonotapplyinthiscase.Apersonwiththesepreferenceswouldbewillingtogiveupthesameamountofytogetonemorexnomatterhowmuchxwasbeingconsumed.Suchasituationmightdescribetherelationshipbetweendifferentbrandsofwhatisessentiallythesameproduct.Forexample,manypeople(includingtheauthor)donotcarewheretheybuygasoline.AgallonofgasisagallonofgasinspiteofthebesteffortsFIGURE3.8ExamplesofUtilityFunctionsThefourindifferencecurvemapsillustratealternativedegreesofsubstitutabilityofxfory.TheCobb-DouglasandCESfunctions(drawnhereforrelativelylowsubstitutability)fallbetweentheextremesofperfectsubstitution(panelb)andnosubstitution(panelc).Quantity of x(a) Cobb-DouglasQuantityof yQuantityof yQuantityof yQuantityof yQuantity of x(b) Perfect substitutesQuantity of x(c) Perfect complementsQuantity of x(d) CESU2U2U2U2U1U0U1U0U1U1U0U0Chapter3PreferencesandUtility101 oftheExxonandShelladvertisingdepartmentstoconvincemeotherwise.Giventhisfact,Iamalwayswillingtogiveup10gallonsofExxoninexchangefor10gallonsofShellbecauseitdoesnotmattertomewhichIuseorwhereIgotmylasttankful.Indeed,aswewillseeinthenextchapter,oneimplicationofsucharelationshipisthatIwillbuyallmygasfromtheleastexpensiveseller.BecauseIdonotexperienceadiminishingMRSofExxonforShell,IhavenoreasontoseekabalanceamongthegasolinetypesIuse.PerfectcomplementsAsituationdirectlyoppositetothecaseofperfectsubstitutesisillustratedbytheL-shapedindifferencecurvesinFigure3.8c.Thesepreferenceswouldapplytogoodsthat“goto-gether”—coffeeandcream,peanutbutterandjelly,andcreamcheeseandloxarefamiliarexamples.TheindifferencecurvesshowninFigure3.8cimplythatthesepairsofgoodswillbeusedintheﬁxedproportionalrelationshiprepresentedbytheverticesofthecurves.Apersonwhoprefers1ounceofcreamwith8ouncesofcoffeewillwant2ouncesofcreamwith16ouncesofcoffee.Extracoffeewithoutcreamisofnovaluetothisperson,justasextracreamwouldbeofnovaluewithoutcoffee.Onlybychoosingthegoodstogethercanutilitybeincreased.TheseconceptscanbeformalizedbyexaminingthemathematicalformoftheutilityfunctionthatgeneratestheseL-shapedindifferencecurves:utility¼Uðx,yÞ¼minðαx,βyÞ.(3.25)Hereαandβarepositiveparameters,andtheoperator“min”meansthatutilityisgivenbythesmallerofthetwotermsintheparentheses.Inthecoffee-creamexample,ifweletouncesofcoffeeberepresentedbyxandouncesofcreambyy,utilitywouldbegivenbyutility¼Uðx,yÞ¼minðx,8yÞ.(3.26)Now8ouncesofcoffeeand1ounceofcreamprovide8unitsofutility.But16ouncesofcoffeeand1ounceofcreamstillprovideonly8unitsofutilitybecausemin(16,8)¼8.Theextracoffeewithoutcreamisofnovalue,asshownbythehorizontalsectionoftheindifferencecurvesformovementawayfromavertex;utilitydoesnotincreasewhenonlyxincreases(withyconstant).Onlyifcoffeeandcreamarebothdoubled(to16and2,respectively)willutilityincreaseto16.Moregenerally,neitherofthetwogoodswillbeinexcessonlyifαx¼βy.(3.27)Hencey=x¼α=β,(3.28)whichshowsthefixedproportionalrelationshipbetweenthetwogoodsthatmustoccurifchoicesaretobeattheverticesoftheindifferencecurves.CESutilityThethreespeciﬁcutilityfunctionsillustratedsofararespecialcasesofthemoregeneralconstantelasticityofsubstitutionfunction(CES),whichtakestheformutility¼Uðx,yÞ¼xδδþyδδ,(3.29)whereδ1,δ6¼0,andutility¼Uðx,yÞ¼lnxþlny(3.30)102Part2ChoiceandDemand whenδ¼0.Itisobviousthatthecaseofperfectsubstitutescorrespondstothelimitingcase,δ¼1,inEquation3.29andthattheCobb-Douglas9casecorrespondstoδ¼0inEquation3.30.Lessobviousisthatthecaseofﬁxedproportionscorrespondstoδ¼∞inEquation3.29,butthatresultcanalsobeshownusingalimitsargument.Theuseoftheterm“elasticityofsubstitution”forthisfunctionderivesfromthenotionthatthepossibilitiesillustratedinFigure3.8correspondtovariousvaluesforthesubstitutionparameter,σ,whichforthisfunctionisgivenbyσ¼1=ð1δÞ.Forperfectsubstitutes,then,σ¼∞,andtheﬁxedproportionscasehasσ¼0.10BecausetheCESfunctionallowsustoexploreallofthesecases,andmanycasesinbetween,itwillprovequiteusefulforillustratingthedegreeofsubstitutabilitypresentinvariouseconomicrelationships.ThespeciﬁcshapeoftheCESfunctionillustratedinFigure3.8aisforthecaseδ¼1.Thatis,utility¼x1y1¼1x1y.(3.31)Forthissituation,σ¼1=ð1δÞ¼1=2and,asthegraphshows,thesesharplycurvedin-differencecurvesapparentlyfallbetweentheCobb-Douglasandfixedproportioncases.Thenegativesignsinthisutilityfunctionmayseemstrange,butthemarginalutilitiesofbothxandyarepositiveanddiminishing,aswouldbeexpected.ThisexplainswhyδmustappearinthedenominatorsinEquation3.29.IntheparticularcaseofEquation3.31,utilityincreasesfrom∞(whenx¼y¼0)toward0asxandyincrease.Thisisanoddutilityscale,perhaps,butperfectlyacceptable.EXAMPLE3.3HomotheticPreferencesAlloftheutilityfunctionsdescribedinFigure3.8arehomothetic(seeChapter2).Thatis,themarginalrateofsubstitutionforthesefunctionsdependsonlyontheratiooftheamountsofthetwogoods,notonthetotalquantitiesofthegoods.Thisfactisobviousforthecaseoftheperfectsubstitutes(whentheMRSisthesameateverypoint)andthecaseofperfectcomplements(wheretheMRSisinﬁnitefory=x>α=β,undeﬁnedwheny=x¼α=β,andzerowheny=x<α=β).ForthegeneralCobb-Douglasfunction,theMRScanbefoundasMRS¼∂U=∂x∂U=∂y¼αxα1yββxαyβ1¼αβ⋅yx,(3.32)whichclearlydependsonlyontheratioy=x.ShowingthattheCESfunctionisalsohomo-theticisleftasanexercise(seeProblem3.12).Theimportanceofhomotheticfunctionsisthatoneindifferencecurveismuchlikeanother.Slopesofthecurvesdependonlyontheratioy=x,notonhowfarthecurveisfromtheorigin.Indifferencecurvesforhigherutilityaresimplecopiesofthoseforlowerutility.Hence,wecanstudythebehaviorofanindividualwhohashomotheticpreferencesbylookingonlyatoneindifferencecurveoratafewnearbycurveswithoutfearingthatourresultswouldchangedramaticallyatverydifferentlevelsofutility.(continued)9TheCESfunctioncouldeasilybegeneralizedtoallowfordifferingweightstobeattachedtothetwogoods.Sincethemainuseofthefunctionistoexaminesubstitutionquestions,weusuallywillnotmakethatgeneralization.InsomeoftheapplicationsoftheCESfunction,wewillalsoomitthedenominatorsofthefunctionbecausetheseconstituteonlyascalefactorwhenδispositive.Fornegativevaluesofδ,however,thedenominatorisneededtoensurethatmarginalutilityispositive.10TheelasticityofsubstitutionconceptisdiscussedinmoredetailinconnectionwithproductionfunctionsinChapter9.Chapter3PreferencesandUtility103 EXAMPLE3.3CONTINUEDQUERY:Howmightyoudeﬁnehomotheticfunctionsgeometrically?WhatwouldthelocusofallpointswithaparticularMRSlooklikeonanindividual’sindifferencecurvemap?EXAMPLE3.4NonhomotheticPreferencesAlthoughalloftheindifferencecurvemapsinFigure3.8exhibithomotheticpreferences,thisneednotalwaysbetrue.Considerthequasi-linearutilityfunctionutility¼Uðx,yÞ¼xþlny.(3.33)Forthisfunction,goodyexhibitsdiminishingmarginalutility,butgoodxdoesnot.TheMRScanbecomputedasMRS¼∂U=∂x∂U=∂y¼11=y¼y.(3.34)TheMRSdiminishesasthechosenquantityofydecreases,butitisindependentofthequantityofxconsumed.Becausexhasaconstantmarginalutility,aperson’swillingnesstogiveupytogetonemoreunitofxdependsonlyonhowmuchyheorshehas.Contrarytothehomotheticcase,then,adoublingofbothxandydoublestheMRSratherthanleavingitunchanged.QUERY:WhatdoestheindifferencecurvemapfortheutilityfunctioninEquation3.33looklike?Whymightthisapproximateasituationwhereyisaspeciﬁcgoodandxrepresentseverythingelse?THEMANY-GOODCASEAlloftheconceptswehavestudiedsofarforthecaseoftwogoodscanbegeneralizedtosituationswhereutilityisafunctionofarbitrarilymanygoods.Inthissection,wewillbrieﬂyexplorethosegeneralizations.Althoughthisexaminationwillnotaddmuchtowhatwehavealreadyshown,consideringpeoples’preferencesformanygoodscanbequiteimportantinappliedeconomics,aswewillseeinlaterchapters.TheMRSwithmanygoodsSupposeutilityisafunctionofngoodsgivenbyutility¼Uðx1,x2,…,xnÞ.(3.35)ThetotaldifferentialofthisexpressionisdU¼∂U∂x1dx1þ∂U∂x2dx2þ…þ∂U∂xndxn(3.36)and,asbefore,wecanfindtheMRSbetweenanytwogoodsbysettingdU¼0.Inthisderivation,wealsoholdconstantquantitiesofallofthegoodsotherthanthosetwo.Hencewehave104Part2ChoiceandDemand dU¼0¼∂U∂xidxiþ∂U∂xjdxj;(3.37)aftersomealgebraicmanipulation,wegetMRSðxiforxjÞ¼dxjdxi¼∂U=∂xi∂U=∂xj,(3.38)whichispreciselywhatwegotinEquation3.17.Whetherthisconceptisasusefulasitisintwodimensionsisopentoquestion,however.Withonlytwogoods,askinghowapersonwouldtradeonefortheotherisaninterestingquestion—atransactionwemightactuallyobserve.Withmanygoods,however,itseemsunlikelythatapersonwouldsimplytradeonegoodforanotherwhileholdingallothergoodsconstant.Rather,itseemsmorelikelythatanevent(suchasapriceincrease)thatcausedapersontowanttoreduce,say,thequantityofcornflakesðxiÞconsumedwouldalsocausehimorhertochangethequantitiesconsumedofmanyothergoodssuchasmilk,sugar,Cheerios,spoons,andsoforth.AsweshallseeinChapter6,thisentirereallocationprocesscanbestbestudiedbylookingattheentireutilityfunctionasrepresentedinEquation3.35.Still,thenotionofmakingtrade-offsbetweenonlytwogoodswillproveusefulasawayofconceptualizingtheutilitymaximizationprocessthatwewilltakeupnext.MultigoodindifferencesurfacesGeneralizingtheconceptofindifferencecurvestomultipledimensionsposesnomajormathe-maticaldifﬁculties.WesimplydeﬁneanindifferencesurfaceasbeingthesetofpointsinndimensionsthatsatisfytheequationUðx1,x2,…,xnÞ¼k,(3.39)wherekisanypreassignedconstant.Iftheutilityfunctionisquasi-concave,thesetofpointsforwhichUkwillbeconvex;thatis,allofthepointsonalinejoininganytwopointsontheU¼kindifferencesurfacewillalsohaveUk.Itisthispropertythatwewillfindmostusefulinlaterapplications.Unfortunately,however,themathematicalconditionsthatensurequasi-concavityinmanydimensionsarenotespeciallyintuitive(seetheExtensionstoChapter2),andvisualizingmanydimensionsisvirtuallyimpossible.Hence,whenintuitionisrequired,wewillusuallyreverttotwo-goodexamples.SUMMARYInthischapterwehavedescribedthewayinwhichecono-mistsformalizeindividuals’preferencesaboutthegoodstheychoose.Wedrewseveralconclusionsaboutsuchpreferencesthatwillplayacentralroleinouranalysisofthetheoryofchoiceinthefollowingchapters:•Ifindividualsobeycertainbasicbehavioralpostulatesintheirpreferencesamonggoods,theywillbeabletorankallcommoditybundles,andthatrankingcanberepresentedbyautilityfunction.Inmakingchoices,individualswillbehaveasiftheyweremaximizingthisfunction.•Utilityfunctionsfortwogoodscanbeillustratedbyanindifferencecurvemap.Eachindifferencecurvecontouronthismapshowsallthecommoditybundlesthatyieldagivenlevelofutility.•Thenegativeoftheslopeofanindifferencecurveisdeﬁnedtobethemarginalrateofsubstitution(MRS).Thisshowstherateatwhichanindividualwouldwill-inglygiveupanamountofonegood(y)ifheorshewerecompensatedbyreceivingonemoreunitofanothergood(x).•TheassumptionthattheMRSdecreasesasxissubsti-tutedforyinconsumptionisconsistentwiththenotionthatindividualsprefersomebalanceintheirconsump-tionchoices.IftheMRSisalwaysdecreasing,individualsChapter3PreferencesandUtility105 PROBLEMS3.1Graphatypicalindifferencecurveforthefollowingutilityfunctionsanddeterminewhethertheyhaveconvexindifferencecurves(thatis,whethertheMRSdeclinesasxincreases).a.Uðx,yÞ¼3xþy.b.Uðx,yÞ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃx⋅yp:c.Uðx,yÞ¼ﬃﬃﬃxpþy:d.Uðx,yÞ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃx2y2q:e.Ux,yðÞ¼xyxþy.3.2Infootnote7weshowedthat,inorderforautilityfunctionfortwogoodstohaveastrictlydiminishingMRS(thatis,tobestrictlyquasi-concave),thefollowingconditionmusthold:f22f112f1f2f12þf21f22<0:UsethisconditiontochecktheconvexityoftheindifferencecurvesforeachoftheutilityfunctionsinProblem3.1.Describeanyshortcutsyoudiscoverinthisprocess.3.3Considerthefollowingutilityfunctions:a.Uðx,yÞ¼xy.b.Uðx,yÞ¼x2y2.c.Uðx,yÞ¼lnxþlny.ShowthateachofthesehasadiminishingMRSbutthattheyexhibitconstant,increasing,anddecreasingmarginalutility,respectively.Whatdoyouconclude?3.4AswesawinFigure3.5,onewaytoshowconvexityofindifferencecurvesistoshowthat,foranytwopointsðx1,y1Þandðx2,y2ÞonanindifferencecurvethatpromisesU¼k,theutilityassociatedwiththepointx1þx22,y1þy22isatleastasgreatask.Usethisapproachtodiscusstheconvexityoftheindifferencecurvesforthefollowingthreefunctions.Besuretographyourresults.a.Uðx,yÞ¼minðx,yÞ.b.Uðx,yÞ¼maxðx,yÞ.c.Uðx,yÞ¼xþy.willhavestrictlyconvexindifferencecurves.Thatis,theirutilityfunctionwillbestrictlyquasi-concave.•Afewsimplefunctionalformscancaptureimportantdifferencesinindividuals’preferencesfortwo(ormore)goods.HereweexaminedtheCobb-Douglasfunction,thelinearfunction(perfectsubstitutes),theﬁxedpro-portionsfunction(perfectcomplements),andtheCESfunction(whichincludestheotherthreeasspecialcases).•Itisasimplemattermathematicallytogeneralizefromtwo-goodexamplestomanygoods.And,asweshallsee,studyingpeoples’choicesamongmanygoodscanyieldmanyinsights.Butthemathematicsofmanygoodsisnotespeciallyintuitive,sowewillprimarilyrelyontwo-goodcasestobuildsuchintuition.106Part2ChoiceandDemand 3.5ThePhilliePhanaticalwayseatshisballparkfranksinaspecialway;heusesafoot-longhotdogtogetherwithpreciselyhalfabun,1ounceofmustard,and2ouncesofpicklerelish.Hisutilityisafunctiononlyofthesefouritemsandanyextraamountofasingleitemwithouttheotherconstituentsisworthless.a.WhatformdoesPP’sutilityfunctionforthesefourgoodshave?b.HowmightwesimplifymattersbyconsideringPP’sutilitytobeafunctionofonlyonegood?Whatisthatgood?c.Supposefoot-longhotdogscost$1.00each,bunscost$0.50each,mustardcosts$0.05perounce,andpicklerelishcosts$0.15perounce.Howmuchdoesthegooddeﬁnedinpart(b)cost?d.Ifthepriceoffoot-longhotdogsincreasesby50percent(to$1.50each),whatisthepercentageincreaseinthepriceofthegood?e.Howwoulda50percentincreaseinthepriceofabunaffectthepriceofthegood?Whyisyouranswerdifferentfrompart(d)?f.Ifthegovernmentwantedtoraise$1.00bytaxingthegoodsthatPPbuys,howshoulditspreadthistaxoverthefourgoodssoastominimizetheutilitycosttoPP?3.6Manyadvertisingslogansseemtobeassertingsomethingaboutpeople’spreferences.Howwouldyoucapturethefollowingsloganswithamathematicalutilityfunction?a.Promisemargarineisjustasgoodasbutter.b.ThingsgobetterwithCoke.c.Youcan’teatjustonePringle’spotatochip.d.KrispyKremeglazeddoughnutsarejustbetterthanDunkin’.e.MillerBrewingadvisesustodrink(beer)“responsibly.”[Whatwould“irresponsible”drinkingbe?]3.7a.Aconsumeriswillingtotrade3unitsofxfor1unitofywhenshehas6unitsofxand5unitsofy.Sheisalsowillingtotradein6unitsofxfor2unitsofywhenshehas12unitsofxand3unitsofy.Sheisindifferentbetweenbundle(6,5)andbundle(12,3).Whatistheutilityfunctionforgoodsxandy?Hint:Whatistheshapeoftheindifferencecurve?b.Aconsumeriswillingtotrade4unitsofxfor1unitofywhensheisconsumingbundle(8,1).Sheisalsowillingtotradein1unitofxfor2unitsofywhensheisconsumingbundle(4,4).Sheisindifferentbetweenthesetwobundles.AssumingthattheutilityfunctionisCobb-DouglasoftheformUðx,yÞ¼xαyβ,whereαandβarepositiveconstants,whatistheutilityfunctionforthisconsumer?c.Wastherearedundancyofinformationinpart(b)?Ifyes,howmuchistheminimumamountofinformationrequiredinthatquestiontoderivetheutilityfunction?3.8Findutilityfunctionsgiveneachofthefollowingindifferencecurves[deﬁnedbyU(⋅)¼C]:a.z¼C1=δxα=δyβ=δ.b.y¼0:5ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃx24ðx2CÞq0:5x:c.z¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃy44xðx2yCÞp2xy22x:Chapter3PreferencesandUtility107 AnalyticalProblems3.9InitialendowmentsSupposethatapersonhasinitialamountsofthetwogoodsthatprovideutilitytohimorher.Theseinitialamountsaregivenby_xand_y.a.Graphtheseinitialamountsonthisperson’sindifferencecurvemap.b.Ifthispersoncantradexfory(orviceversa)withotherpeople,whatkindsoftradeswouldheorshevoluntarilymake?Whatkindswouldnotbemade?Howdothesetradesrelatetothisperson’sMRSatthepoint(_x,_y)?c.Supposethispersonisrelativelyhappywiththeinitialamountsinhisorherpossessionandwillonlyconsidertradesthatincreaseutilitybyatleastamountk.Howwouldyouillustratethisontheindifferencecurvemap?3.10Cobb-DouglasutilityExample3.3showsthattheMRSfortheCobb-DouglasfunctionUðx,yÞ¼xαyβisgivenbyMRS¼αβyx.a.Doesthisresultdependonwhetherαþβ¼1?Doesthissumhaveanyrelevancetothetheoryofchoice?b.Forcommoditybundlesforwhichy¼x,howdoestheMRSdependonthevaluesofαandβ?Developanintuitiveexplanationofwhy,ifα>β,MRS>1.Illustrateyourargumentwithagraph.c.Supposeanindividualobtainsutilityonlyfromamountsofxandythatexceedminimalsub-sistencelevelsgivenbyx0,y0.Inthiscase,Uðx,yÞ¼ðxx0Þαðyy0Þβ.Isthisfunctionhomothetic?(Forafurtherdiscussion,seetheExtensionstoChapter4.)3.11IndependentmarginalutilitiesTwogoodshaveindependentmarginalutilitiesif∂2U∂y∂x¼∂2U∂x∂y¼0.Showthatifweassumediminishingmarginalutilityforeachgood,thenanyutilityfunctionwithindependentmarginalutilitieswillhaveadiminishingMRS.Provideanexampletoshowthattheconverseofthisstatementisnottrue.3.12CESutilitya.ShowthattheCESfunctionαxδδþβyδδishomothetic.HowdoestheMRSdependontheratioy=x?b.Showthatyourresultsfrompart(a)agreewithourdiscussionofthecasesδ¼1(perfectsubstitutes)andδ¼0(Cobb-Douglas).c.ShowthattheMRSisstrictlydiminishingforallvaluesofδ<1.d.Showthatifx¼y,theMRSforthisfunctiondependsonlyontherelativesizesofαandβ.108Part2ChoiceandDemand e.CalculatetheMRSforthisfunctionwheny=x¼0:9andy=x¼1:1forthetwocasesδ¼0:5andδ¼1.WhatdoyouconcludeabouttheextenttowhichtheMRSchangesinthevicinityofx¼y?Howwouldyouinterpretthisgeometrically?3.13Thequasi-linearfunctionConsiderthefunctionUðx,yÞ¼xþlny.Thisisafunctionthatisusedrelativelyfrequentlyineco-nomicmodelingasithassomeusefulproperties.a.FindtheMRSofthefunction.Now,interprettheresult.b.Conﬁrmthatthefunctionisquasi-concave.c.Findtheequationforanindifferencecurveforthisfunction.d.Comparethemarginalutilityofxandy.Howdoyouinterpretthesefunctions?Howmightconsumerschoosebetweenxandyastheytrytoincreasetheirutilityby,forexample,consum-ingmorewhentheirincomeincreases?(Wewilllookatthis“incomeeffect”indetailintheChapter5problems.)e.Consideringhowtheutilitychangesasthequantitiesofthetwogoodsincrease,describesomesituationswherethisfunctionmightbeuseful.3.14UtilityfunctionsandpreferencesImaginetwogoodsthat,whenconsumedindividually,giveincreasingutilitywithincreasingamountsconsumed(theyareindividuallymonotonic)butthat,whenconsumedtogether,detractfromtheutilitythattheotheronegives.(Onecouldthinkofmilkandorangejuice,whichareﬁneindividuallybutwhich,whenconsumedtogether,yieldconsiderabledisutility.)a.Proposeafunctionalformfortheutilityfunctionforthetwogoodsjustdescribed.b.FindtheMRSbetweenthetwogoodswithyourfunctionalform.c.Which(ifany)ofthegeneralassumptionsthatwemakeregardingpreferencesandutilityfunctionsdoesyourfunctionalformviolate?SUGGESTIONSFORFURTHERREADINGAleskerov,Fuad,andBernardMonjardet.UtilityMaximiza-tion,Choice,andPreference.Berlin:Springer-Verlag,2002.Acompletestudyofpreferencetheory.Coversavarietyofthresholdmodelsandmodelsof“context-dependent”decisionmaking.Jehle,G.R.,andP.J.Reny.AdvancedMicroeconomicTheory,2nded.Boston:AddisonWesley/Longman,2001.Chapter2hasagoodproofoftheexistenceofutilityfunctionswhenbasicaxiomsofrationalityhold.Kreps,DavidM.ACourseinMicroeconomicTheory.Princeton,NJ:PrincetonUniversityPress,1990.Chapter1coverspreferencetheoryinsomedetail.Gooddiscussionofquasi-concavity.Kreps,DavidM.NotesontheTheoryofChoice.London:WestviewPress,1988.Gooddiscussionofthefoundationsofpreferencetheory.Mostofthefocusofthebookisonutilityinuncertainsituations.Mas-Colell,Andrea,MichaelD.Whinston,andJerryR.Green.MicroeconomicTheory.NewYork:OxfordUniversityPress,1995.Chapters2and3provideadetaileddevelopmentofpreferencerela-tionsandtheirrepresentationbyutilityfunctions.Stigler,G.“TheDevelopmentofUtilityTheory.”JournalofPoliticalEconomy59,pts.1–2(August/October1950):307–27,373–96.Alucidandcompletesurveyofthehistoryofutilitytheory.Hasmanyinterestinginsightsandasides.Chapter3PreferencesandUtility109 EXTENSIONSSpecialPreferencesTheutilityfunctionconceptisaquitegeneralonethatcanbeadaptedtoalargenumberofspecialcircum-stances.Discoveryofingeniousfunctionalformsthatreﬂecttheessentialaspectsofsomeproblemcanpro-videanumberofinsightsthatwouldnotbereadilyapparentwithamoreliteraryapproach.Herewelookatfouraspectsofpreferencesthateconomistshavetriedtomodel:(1)thresholdeffects;(2)quality;(3)habitsandaddiction;and(4)second-partypreferences.InChapters7and17,weillustrateanumberofaddi-tionalwaysofcapturingaspectsofpreferences.E3.1ThresholdeffectsThemodelofutilitythatwedevelopedinthischapterimpliesanindividualwillalwaysprefercommoditybun-dleAtobundleB,providedUðAÞ>UðBÞ.TheremaybeeventsthatwillcausepeopletoshiftquicklyfromconsumingbundleAtoconsumingB.Inmanycases,however,suchalightning-quickresponseseemsunlikely.Peoplemayinfactbe“setintheirways”andmayrequirearatherlargechangeincircumstancestochangewhattheydo.Forexample,individualsmaynothaveespeciallystrongopinionsaboutwhatprecisebrandoftoothpastetheychooseandmaystickwithwhattheyknowdespiteaproliferationofnew(andperhapsbetter)brands.Simi-larly,peoplemaystickwithanoldfavoriteTVshoweventhoughithasdeclinedinquality.Onewaytocapturesuchbehavioristoassumeindividualsmakedecisionsasiftheyfacedthresholdsofpreference.Insuchasituation,commoditybundleAmightbechosenoverBonlywhenUðAÞ>UðBÞþε,(i)whereεisthethresholdthatmustbeovercome.Withthisspecification,then,indifferencecurvesmayberatherthickandevenfuzzy,ratherthanthedistinctcontourlinesshowninthischapter.Thresholdmodelsofthistypeareusedextensivelyinmarketing.ThetheorybehindsuchmodelsispresentedindetailinAleskerovandMonjardet(2002).There,theauthorsconsideranum-berofwaysofspecifyingthethresholdsothatitmightdependonthecharacteristicsofthebundlesbeingcon-sideredoronothercontextualvariables.AlternativefuelsVedenov,Dufﬁeld,andWetzstein(2006)usethethresholdideatoexaminetheconditionsunderwhichindividualswillshiftfromgasolinetootherfuels(primarilyethanol)forpoweringtheircars.Theauthorspointoutthatthemaindisadvantageofusinggasolineinrecentyearshasbeentheexcessivepricevolatilityoftheproductrelativetootherfuels.Theyconcludethatswitchingtoethanolblendsisefﬁcient(especiallydur-ingperiodsofincreasedgasolinepricevolatility),pro-videdthattheblendsdonotdecreasefuelefﬁciency.E3.2QualityBecausemanyconsumptionitemsdifferwidelyinqual-ity,economistshaveaninterestinincorporatingsuchdifferencesintomodelsofchoice.Oneapproachissimplytoregarditemsofdifferentqualityastotallyseparategoodsthatarerelativelyclosesubstitutes.Butthisapproachcanbeunwieldybecauseofthelargenumberofgoodsinvolved.Analternativeap-proachfocusesonqualityasadirectitemofchoice.Utilitymightinthiscasebereﬂectedbyutility¼Uðq,QÞ,(ii)whereqisthequantityconsumedandQisthequalityofthatconsumption.Althoughthisapproachpermitssomeexaminationofquality-quantitytrade-offs,itencountersdifficultywhenthequantityconsumedofacommodity(e.g.,wine)consistsofavarietyofqualities.Qualitymightthenbedefinedasanaverage(seeTheil,11952),butthatapproachmaynotbeappropriatewhenthequalityofnewgoodsischangingrapidly(asinthecaseofpersonalcomputers,forexample).Amoregeneralapproach(originallysuggestedbyLancaster,1971)focusesonawell-definedsetofattributesofgoodsandassumesthatthoseattributesprovideutility.Ifagoodqprovidestwosuchattributes,a1anda2,thenutilitymightbewrittenasutility¼U½q,a1ðqÞ,a2ðqÞ,(iii)andutilityimprovementsmightariseeitherbecausethisindividualchoosesalargerquantityofthegoodorbecauseagivenquantityyieldsahigherlevelofvaluableattributes.PersonalcomputersThisisthepracticefollowedbyeconomistswhostudydemandinsuchrapidlychangingindustriesaspersonal1Theilalsosuggestsmeasuringqualitybylookingatcorrelationsbe-tweenchangesinconsumptionandtheincomeelasticitiesofvariousgoods.110Part2ChoiceandDemand
computers.Inthiscaseitwouldclearlybeincorrecttofocusonlyonthequantityofpersonalcomputerspur-chasedeachyear,sincenewmachinesaremuchbetterthanoldones(and,presumably,providemoreutility).Forexample,Berndt,Griliches,andRappaport(1995)ﬁndthatpersonalcomputerqualityhasbeenrisingabout30percentperyearoverarelativelylongperiodoftime,primarilybecauseofimprovedattributessuchasfasterprocessorsorbetterharddrives.Apersonwhospends,say,$2,000forapersonalcomputertodaybuysmuchmoreutilitythandidasimilarconsumer5yearsago.E3.3HabitsandaddictionBecauseconsumptionoccursovertime,thereisthepossibilitythatdecisionsmadeinoneperiodwillaffectutilityinlaterperiods.Habitsareformedwhenindivid-ualsdiscovertheyenjoyusingacommodityinoneperiodandthisincreasestheirconsumptioninsubse-quentperiods.Anextremecaseisaddiction(beittodrugs,cigarettes,orMarxBrothersmovies),wherepastconsumptionsigniﬁcantlyincreasestheutilityofpres-entconsumption.Onewaytoportraytheseideasmath-ematicallyistoassumethatutilityinperiodtdependsonconsumptioninperiodtandthetotalofallpriorconsumptionofthehabit-forminggood(say,X):utility¼Utðxt,yt,stÞ,(iv)wherest¼X∞i¼1xti.Inempiricalapplications,however,dataonallpastlevelsofconsumptionusuallydonotexist.Itisthere-forecommontomodelhabitsusingonlydataoncurrentconsumption(xt)andonconsumptioninthepreviousperiod(xt1).Acommonwaytoproceedistoassumethatutilityisgivenbyutility¼Utðxt,ytÞ,(v)wherextissomesimplefunctionofxtandxt1,suchasxt¼xtxt1orxt¼xt=xt1.Suchfunctionsimplythat,ceterisparibus,thehigherisxt1,themorextwillbechoseninthecurrentperiod.ModelinghabitsTheseapproachestomodelinghabitshavebeenap-pliedtoawidevarietyoftopics.StiglerandBecker(1977)usesuchmodelstoexplainwhypeopledevelopa“taste”forgoingtooperasorplayinggolf.Becker,Grossman,andMurphy(1994)adaptthemodelstostudyingcigarettesmokingandotheraddictivebehav-ior.Theyshowthatreductionsinsmokingearlyinlifecanhaveverylargeeffectsoneventualcigarettecon-sumptionbecauseofthedynamicsinindividuals’util-ityfunctions.Whetheraddictivebehavioris“rational”hasbeenextensivelystudiedbyeconomists.Forexam-ple,GruberandKoszegi(2001)showthatsmokingcanbeapproachedasarational,thoughtime-incon-sistent,2choice.E3.4Second-partypreferencesIndividualsclearlycareaboutthewell-beingofotherindividuals.Phenomenasuchasmakingcharitablecon-tributionsormakingbequeststochildrencannotbeunderstoodwithoutrecognizingtheinterdependencethatexistsamongpeople.Second-partypreferencescanbeincorporatedintotheutilityfunctionofpersoni,say,byutility¼Uiðxi,yi,UjÞ,(vi)whereUjistheutilityofsomeoneelse.If∂Ui=∂Uj>0thenthispersonwillengageinaltruisticbehavior,whereasif∂Ui=∂Uj<0thenheorshewilldemonstratethemalevolentbehaviorasso-ciatedwithenvy.Theusualcaseof∂Ui=∂Uj¼0isthensimplyamiddlegroundbetweenthesealternativepreferencetypes.GaryBeckerhasbeenapioneerinthestudyofthesepossibilitiesandhaswrittenonavarietyoftopics,includingthegeneraltheoryofsocialinteractions(1976)andtheimportanceofaltruisminthetheoryofthefamily(1981).EvolutionarybiologyandgeneticsBiologistshavesuggestedaparticularformfortheutilityfunctioninEquationiv,drawnfromthetheoryofgenetics.Inthiscaseutility¼Uiðxi,yiÞþXjrjUj,(vii)whererjmeasuresclosenessofthegeneticrelation-shipbetweenpersoniandpersonj.Forparentsandchildren,forexample,rj¼0:5,whereasforcousinsrj¼0:125.Bergstrom(1996)describesafewoftheconclusionsaboutevolutionarybehaviorthatbiolo-gistshavedrawnfromthisparticularfunctionalform.2Formoreontimeinconsistency,seeChapter17.Chapter3PreferencesandUtility111 ReferencesAleskerov,Fuad,andBernardMonjardet.UtilityMaximi-zation,Choice,andPreference.Berlin:Springer-Verlag,2002.Becker,GaryS.TheEconomicApproachtoHumanBehavior.Chicago:UniversityofChicagoPress,1976.———.ATreatiseontheFamily.Cambridge,MA:HarvardUniversityPress,1981.Becker,GaryS.,MichaelGrossman,andKevinM.Murphy.“AnEmpiricalAnalysisofCigaretteAddiction.”Ameri-canEconomicReview(June1994):396–418.Bergstrom,TheodoreC.“EconomicsinaFamilyWay.”JournalofEconomicLiterature(December1996):1903–34.Berndt,ErnstR.,ZviGriliches,andNealJ.Rappaport.“EconometricEstimatesofPriceIndexesforPersonalComputersinthe1990s.”JournalofEconometrics(July1995):243–68.Gruber,Jonathan,andBotondKoszegi.“IsAddiction‘Rational’?TheoryandEvidence.”QuarterlyJournalofEconomics(November2001):1261–1303.Lancaster,KelvinJ.ConsumerDemand:ANewApproach.NewYork:ColumbiaUniversityPress,1971.Stigler,GeorgeJ.,andGaryS.Becker.“DeGustibusNonEstDisputandum.”AmericanEconomicReview(March1977):76–90.Theil,Henri.“Qualities,Prices,andBudgetEnquiries.”ReviewofEconomicStudies(April1952):129–47.Vedenov,DmitryV.,JamesA.Duffield,andMichaelE.Wetzstein.“EntryofAlternativeFuelsinaVolatileU.S.GasolineMarket.”JournalofAgriculturalandResourceEconomics(April2006):1–13.112Part2ChoiceandDemand CHAPTER4UtilityMaximizationandChoiceInthischapterweexaminethebasicmodelofchoicethateconomistsusetoexplainindividuals’behavior.Thatmodelassumesthatindividualswhoareconstrainedbylimitedincomeswillbehaveasiftheyareusingtheirpurchasingpowerinsuchawayastoachievethehighestutilitypossible.Thatis,individualsareas-sumedtobehaveasiftheymaximizeutilitysubjecttoabudgetconstraint.Althoughthespeciﬁcapplicationsofthismodelarequitevaried,aswewillshow,allofthemarebasedonthesamefundamentalmathematicalmodel,andallarriveatthesamegeneralconclusion:Tomaximizeutility,individualswillchoosebundlesofcommoditiesforwhichtherateoftrade-offbetweenanytwogoods(theMRS)isequaltotheratioofthegoods’marketprices.Marketpricesconveyinformationaboutopportunitycoststoindividuals,andthisin-formationplaysanimportantroleinaffectingthechoicesactuallymade.UtilitymaximizationandlightningcalculationsBeforestartingaformalstudyofthetheoryofchoice,itmaybeappropriatetodisposeoftwocomplaintsnoneconomistsoftenmakeabouttheapproachwewilltake.Firstisthechargethatnorealpersoncanmakethekindsof“lightningcalculations”requiredforutilitymaximization.Accordingtothiscomplaint,whenmovingdownasupermarketaisle,peoplejustgrabwhatisavailablewithnorealpatternorpurposetotheiractions.Economistsarenotpersuadedbythiscomplaint.Theydoubtthatpeoplebehaverandomly(everyone,afterall,isboundbysomesortofbudgetconstraint),andtheyviewthelightningcalculationchargeasmisplaced.Recall,again,Friedman’spoolplayerfromChapter1.Thepoolplayeralsocannotmakethelightningcalculationsrequiredtoplanashotaccordingtothelawsofphysics,butthoselawsstillpredicttheplayer’sbehavior.Sotoo,asweshallsee,theutility-maximizationmodelpredictsmanyaspectsofbehavioreventhoughnoonecarriesaroundacomputerwithhisorherutilityfunctionprogrammedintoit.Tobeprecise,economistsassumethatpeoplebehaveasiftheymadesuchcalculations,sothecomplaintthatthecalculationscannotpossiblybemadeislargelyirrelevant.AltruismandselfishnessAsecondcomplaintagainstourmodelofchoiceisthatitappearstobeextremelyselﬁsh;noone,accordingtothiscomplaint,hassuchsolelyself-centeredgoals.Althougheconomistsareprobablymorereadytoacceptself-interestasamotivatingforcethanareother,moreUtopianthinkers(AdamSmithobserved,“Wearenotreadytosuspectanypersonofbeingdeﬁcientinselﬁshness”1),thischargeisalsomisplaced.Nothingintheutility-maximizationmodelpreventsindividualsfromderivingsatisfactionfromphilanthropyorgenerally“doinggood.”Theseactivitiesalsocanbeassumedtoprovideutility.Indeed,economistshaveusedtheutility-maximizationmodelextensivelytostudysuchissuesasdonatingtimeandmoney1AdamSmith,TheTheoryofMoralSentiments(1759;reprint,NewRochelle,NY:ArlingtonHouse,1969),p.446.113 tocharity,leavingbequeststochildren,orevengivingblood.Oneneednottakeapositiononwhethersuchactivitiesareselﬁshorselﬂesssinceeconomistsdoubtpeoplewouldunder-takethemiftheywereagainsttheirownbestinterests,broadlyconceived.ANINITIALSURVEYThegeneralresultsofourexaminationofutilitymaximizationcanbestatedsuccinctlyasfollows.OPTIMIZATIONPRINCIPLEUtilitymaximizationTomaximizeutility,givenaﬁxedamountofincometospend,anindividualwillbuythosequantitiesofgoodsthatexhausthisorhertotalincomeandforwhichthepsychicrateoftrade-offbetweenanytwogoods(theMRS)isequaltotherateatwhichthegoodscanbetradedonefortheotherinthemarketplace.Thatspendingallone’sincomeisrequiredforutilitymaximizationisobvious.Becauseextragoodsprovideextrautility(thereisnosatiation)andbecausethereisnootheruseforincome,toleaveanyunspentwouldbetofailtomaximizeutility.Throwingmoneyawayisnotautility-maximizingactivity.Theconditionspecifyingequalityoftrade-offratesrequiresabitmoreexplanation.Becausetherateatwhichonegoodcanbetradedforanotherinthemarketisgivenbytheratiooftheirprices,thisresultcanberestatedtosaythattheindividualwillequatetheMRS(ofxfory)totheratioofthepriceofxtothepriceofyðpx=pyÞ.Thisequatingofapersonaltrade-offratetoamarket-determinedtrade-offrateisaresultcommontoallindividualutility-maximizationproblems(andtomanyothertypesofmaximizationproblems).Itwilloccuragainandagainthroughoutthistext.AnumericalillustrationToseetheintuitivereasoningbehindthisresult,assumethatitwerenottruethatanin-dividualhadequatedtheMRStotheratioofthepricesofgoods.Speciﬁcally,supposethattheindividual’sMRSisequalto1andthatheorsheiswillingtotrade1unitofxfor1unitofyandremainequallywelloff.Assumealsothatthepriceofxis$2perunitandofyis$1perunit.Itiseasytoshowthatthispersoncanbemadebetteroff.Supposethispersonreducesxconsumptionby1unitandtradesitinthemarketfor2unitsofy.Only1extraunitofywasneededtokeepthispersonashappyasbeforethetrade—thesecondunitofyisanetadditiontowell-being.Therefore,theindividual’sspendingcouldnothavebeenallocatedoptimallyintheﬁrstplace.AsimilarmethodofreasoningcanbeusedwhenevertheMRSandthepriceratiopx=pydiffer.Theconditionformaximumutilitymustbetheequalityofthesetwomagnitudes.THETWO-GOODCASE:AGRAPHICALANALYSISThisdiscussionseemseminentlyreasonable,butitcanhardlybecalledaproof.Rather,wemustnowshowtheresultinarigorousmannerand,atthesametime,illustrateseveralotherimportantattributesofthemaximizationprocess.Firstwetakeagraphicanalysis;thenwetakeamoremathematicalapproach.BudgetconstraintAssumethattheindividualhasIdollarstoallocatebetweengoodxandgoody.Ifpxisthepriceofgoodxandpyisthepriceofgoody,thentheindividualisconstrainedby114Part2ChoiceandDemand pxxþpyyI.(4.1)Thatis,nomorethanIcanbespentonthetwogoodsinquestion.ThisbudgetconstraintisshowngraphicallyinFigure4.1.Thispersoncanaffordtochooseonlycombinationsofxandyintheshadedtriangleofthefigure.IfallofIisspentongoodx,itwillbuyI=pxunitsofx.Similarly,ifallisspentony,itwillbuyI=pyunitsofy.Theslopeoftheconstraintiseasilyseentobepx=py.Thisslopeshowshowycanbetradedforxinthemarket.Ifpx¼2andpy¼1,then2unitsofywilltradefor1unitofx.First-orderconditionsforamaximumThisbudgetconstraintcanbeimposedonthisperson’sindifferencecurvemaptoshowtheutility-maximizationprocess.Figure4.2illustratesthisprocedure.TheindividualwouldbeirrationaltochooseapointsuchasA;heorshecangettoahigherutilityleveljustbyspendingmoreofhisorherincome.Theassumptionofnonsatiationimpliesthatapersonshouldspendallofhisorherincomeinordertoreceivemaximumutility.Similarly,byreallocatingexpenditures,theindividualcandobetterthanpointB.PointDisoutofthequestionbecauseincomeisnotlargeenoughtopurchaseD.ItisclearthatthepositionofmaximumutilityisatpointC,wherethecombinationx,yischosen.ThisistheonlypointonindifferencecurveU2thatcanbeboughtwithIdollars;nohigherutilitylevelcanbeFIGURE4.1TheIndividual’sBudgetConstraintforTwoGoodsThosecombinationsofxandythattheindividualcanaffordareshownintheshadedtriangle.If,asweusuallyassume,theindividualprefersmoreratherthanlessofeverygood,theouterboundaryofthistriangleistherelevantconstraintwherealloftheavailablefundsarespenteitheronxorony.Theslopeofthisstraight-lineboundaryisgivenbypx=py.Quantity of x0Quantityof ypyIpxII= pxx+ pyyChapter4UtilityMaximizationandChoice115 bought.Cisapointoftangencybetweenthebudgetconstraintandtheindifferencecurve.Therefore,atCwehaveslopeofbudgetconstraint¼pxpy¼slopeofindifferencecurve¼dydxU¼constant(4.2)orpxpy¼dydxU¼constant¼MRSðofxforyÞ.(4.3)Ourintuitiveresultisproved:forautilitymaximum,allincomeshouldbespentandtheMRSshouldequaltheratioofthepricesofthegoods.Itisobviousfromthediagramthatifthisconditionisnotfulfilled,theindividualcouldbemadebetteroffbyreallocatingexpenditures.Second-orderconditionsforamaximumThetangencyruleisonlyanecessaryconditionforamaximum.Toseethatitisnotasufﬁcientcondition,considertheindifferencecurvemapshowninFigure4.3.Here,apointFIGURE4.2AGraphicalDemonstrationofUtilityMaximizationPointCrepresentsthehighestutilitylevelthatcanbereachedbytheindividual,giventhebudgetconstraint.Thecombinationx,yisthereforetherationalwayfortheindividualtoallocatepurchasingpower.Onlyforthiscombinationofgoodswilltwoconditionshold:Allavailablefundswillbespent,andtheindividual’spsychicrateoftrade-off(MRS)willbeequaltotherateatwhichthegoodscanbetradedinthemarketðpx=pyÞ.Quantity of xQuantity of yU1U1U2U3U2U30I= pxx+ pyyBDCAy*x*116Part2ChoiceandDemand oftangencyðCÞisinferiortoapointofnontangencyðBÞ.Indeed,thetruemaximumisatanotherpointoftangencyðAÞ.Thefailureofthetangencyconditiontoproduceanunam-biguousmaximumcanbeattributedtotheshapeoftheindifferencecurvesinFigure4.3.IftheindifferencecurvesareshapedlikethoseinFigure4.2,nosuchproblemcanarise.Butwehavealreadyshownthat“normally”shapedindifferencecurvesresultfromtheassumptionofadiminishingMRS.Therefore,iftheMRSisassumedtobediminishing,theconditionoftangencyisbothanecessaryandsufﬁcientconditionforamaximum.2Withoutthisassump-tion,onewouldhavetobecarefulinapplyingthetangencyrule.CornersolutionsTheutility-maximizationproblemillustratedinFigure4.2resultedinan“interior”maxi-mum,inwhichpositiveamountsofbothgoodswereconsumed.Insomesituationsindivid-uals’preferencesmaybesuchthattheycanobtainmaximumutilitybychoosingtoconsumeFIGURE4.3ExampleofanIndifferenceCurveMapforWhichtheTangencyConditionDoesNotEnsureaMaximumIfindifferencecurvesdonotobeytheassumptionofadiminishingMRS,notallpointsoftangency(pointsforwhichMRSpx=pyÞmaytrulybepointsofmaximumutility.Inthisexample,tangencypointCisinferiortomanyotherpointsthatcanalsobepurchasedwiththeavailablefunds.Inorderthatthenecessaryconditionsforamaximum(thatis,thetangencyconditions)alsobesufﬁcient,oneusuallyassumesthattheMRSisdiminishing;thatis,theutilityfunctionisstrictlyquasi-concave.Quantity of xQuantity of yU1U3U2U1U2U3ACBI= pxx+ pyy2Inmathematicalterms,becausetheassumptionofadiminishingMRSisequivalenttoassumingquasi-concavity,thenecessaryconditionsforamaximumsubjecttoalinearconstraintarealsosufficient,asweshowedinChapter2.Chapter4UtilityMaximizationandChoice117 noamountofoneofthegoods.Ifsomeonedoesnotlikehamburgersverymuch,thereisnoreasontoallocateanyincometotheirpurchase.ThispossibilityisreﬂectedinFigure4.4.There,utilityismaximizedatE,wherex¼xandy¼0,soanypointonthebudgetconstraintwherepositiveamountsofyareconsumedyieldsalowerutilitythandoespointE.NoticethatatEthebudgetconstraintisnotpreciselytangenttotheindifferencecurveU2.Instead,attheoptimalpointthebudgetconstraintisﬂatterthanU2,indicatingthattherateatwhichxcanbetradedforyinthemarketislowerthantheindividual’spsychictrade-offrate(theMRS).Atprevailingmarketpricestheindividualismorethanwillingtotradeawayytogetextrax.Becauseitisimpossibleinthisproblemtoconsumenegativeamountsofy,however,thephysicallimitforthisprocessistheX-axis,alongwhichpurchasesofyare0.Hence,asthisdiscussionmakesclear,itisnecessarytoamendtheﬁrst-orderconditionsforautilitymaximumabittoallowforcornersolutionsofthetypeshowninFigure4.4.Followingourdiscussionofthegeneraln-goodcase,wewillusethemathematicsfromChapter2toshowhowthiscanbeaccomplished.THEn-GOODCASETheresultsderivedgraphicallyinthecaseoftwogoodscarryoverdirectlytothecaseofngoods.Againitcanbeshownthatforaninteriorutilitymaximum,theMRSbetweenanytwogoodsmustequaltheratioofthepricesofthesegoods.Tostudythismoregeneralcase,however,itisbesttousesomemathematics.FIGURE4.4CornerSolutionforUtilityMaximizationWiththepreferencesrepresentedbythissetofindifferencecurves,utilitymaximizationoccursatE,where0amountsofgoodyareconsumed.Theﬁrst-orderconditionsforamaximummustbemodiﬁedsomewhattoaccommodatethispossibility.Quantity of xQuantityof yU3U1Ex*U2118Part2ChoiceandDemand First-orderconditionsWithngoods,theindividual’sobjectiveistomaximizeutilityfromthesengoods:utility¼Uðx1,x2,…,xnÞ,(4.4)subjecttothebudgetconstraint3I¼p1x1þp2x2þ…þpnxn(4.5)orIp1x1p2x2…pnxn¼0.(4.6)FollowingthetechniquesdevelopedinChapter2formaximizingafunctionsubjecttoaconstraint,wesetuptheLagrangianexpressionℒ¼Uðx1,x2,…,xnÞþλðIp1x1p2x2…pnxnÞ.(4.7)Settingthepartialderivativesofℒ(withrespecttox1,x2,…,xnandλ)equalto0yieldsnþ1equationsrepresentingthenecessaryconditionsforaninteriormaximum:∂ℒ∂x1¼∂U∂x1λp1¼0,∂ℒ∂x2¼∂U∂x2λp2¼0,...∂ℒ∂xn¼∂U∂xnλpn¼0,∂ℒ∂λ¼Ip1x1p2x2…pnxn¼0.(4.8)Thesenþ1equationscan,inprinciple,besolvedfortheoptimalx1,x2,…,xnandforλ(seeExamples4.1and4.2tobeconvincedthatsuchasolutionispossible).Equations4.8arenecessarybutnotsufﬁcientforamaximum.Thesecond-ordercondi-tionsthatensureamaximumarerelativelycomplexandmustbestatedinmatrixterms(seetheExtensionstoChapter2).However,theassumptionofstrictquasi-concavity(adimin-ishingMRSinthetwo-goodcase)issufﬁcienttoensurethatanypointobeyingEquations4.8isinfactatruemaximum.Implicationsoffirst-orderconditionsTheﬁrst-orderconditionsrepresentedbyEquations4.8canberewritteninavarietyofinterestingways.Forexample,foranytwogoods,xiandxj,wehave∂U=∂xi∂U=∂xj¼pipj.(4.9)InChapter3weshowedthattheratioofthemarginalutilitiesoftwogoodsisequaltothemarginalrateofsubstitutionbetweenthem.Therefore,theconditionsforanoptimalallo-cationofincomebecomeMRSðxiforxjÞ¼pipj.(4.10)Thisisexactlytheresultderivedgraphicallyearlierinthischapter;tomaximizeutility,theindividualshouldequatethepsychicrateoftrade-offtothemarkettrade-offrate.3Again,thebudgetconstrainthasbeenwrittenasanequalitybecause,giventheassumptionofnonsatiation,itisclearthattheindividualwillspendallavailableincome.Chapter4UtilityMaximizationandChoice119 InterpretingtheLagrangianmultiplierAnotherresultcanbederivedbysolvingEquations4.8forλ:λ¼∂U=∂x1p1¼∂U=∂x2p2¼…¼∂U=∂xnpn(4.11)orλ¼MUx1p1¼MUx2p2¼…¼MUxnpn.Theseequationsstatethat,attheutility-maximizingpoint,eachgoodpurchasedshouldyieldthesamemarginalutilityperdollarspentonthatgood.Eachgoodthereforeshouldhaveanidentical(marginal)benefit-to-(marginal)-costratio.Ifthiswerenottrue,onegoodwouldpromisemore“marginalenjoymentperdollar”thansomeothergood,andfundswouldnotbeoptimallyallocated.Althoughthereaderisagainwarnedagainsttalkingveryconﬁdentlyaboutmarginalutility,whatEquation4.11saysisthatanextradollarshouldyieldthesame“additionalutility”nomatterwhichgooditisspenton.ThecommonvalueforthisextrautilityisgivenbytheLagrangianmultiplierfortheconsumer’sbudgetconstraint(thatis,byλ).Conse-quently,λcanberegardedasthemarginalutilityofanextradollarofconsumptionexpendi-ture(themarginalutilityof“income”).Oneﬁnalwaytorewritethenecessaryconditionsforamaximumispi¼MUxiλ(4.12)foreverygoodithatisbought.Tointerpretthisequation,considerasituationwhereaperson’smarginalutilityofincome(λ)isconstantoversomerange.ThenvariationsinthepriceheorshemustpayforgoodiðpiÞaredirectlyproportionaltotheextrautilityderivedfromthatgood.Atthemargin,therefore,thepriceofagoodreflectsanindividual’swillingnesstopayforonemoreunit.Thisisaresultofconsiderableimportanceinappliedwelfareeconomicsbecausewillingnesstopaycanbeinferredfrommarketreactionstoprices.InChapter5wewillseehowthisinsightcanbeusedtoevaluatethewelfareeffectsofpricechangesand,inlaterchapters,wewillusethisideatodiscussavarietyofquestionsabouttheefficiencyofresourceallocation.CornersolutionsTheﬁrst-orderconditionsofEquations4.8holdexactlyonlyforinteriormaximaforwhichsomepositiveamountofeachgoodispurchased.AsdiscussedinChapter2,whencornersolutions(suchasthoseillustratedinFigure4.4)arise,theconditionsmustbemodiﬁedslightly.4Inthiscase,Equations4.8become∂ℒ∂xi¼∂U∂xiλpi0ði¼1,…,nÞ(4.13)and,if∂ℒ∂xi¼∂U∂xiλpi<0,(4.14)thenxi¼0.(4.15)4Formally,theseconditionsarecalledthe“Kuhn-Tucker”conditionsfornonlinearprogramming.120Part2ChoiceandDemand Tointerprettheseconditions,wecanrewriteEquation4.14aspi>∂U=∂xiλ¼MUxiλ.(4.16)Hence,theoptimalconditionsareasbefore,exceptthatanygoodwhosepriceðpiÞexceedsitsmarginalvaluetotheconsumer(MUxi=λ)willnotbepurchased(xi¼0).Thus,themathematicalresultsconformtothecommonsenseideathatindividualswillnotpurchasegoodsthattheybelievearenotworththemoney.Althoughcornersolutionsdonotprovideamajorfocusforouranalysisinthisbook,thereadershouldkeepinmindthepossibilitiesforsuchsolutionsarisingandtheeconomicinterpretationthatcanbeattachedtotheoptimalconditionsinsuchcases.EXAMPLE4.1Cobb-DouglasDemandFunctionsAsweshowedinChapter3,theCobb-DouglasutilityfunctionisgivenbyUðx,yÞ¼xαyβ,(4.17)where,forconvenience,5weassumeαþβ¼1.Wecannowsolvefortheutility-maximizingvaluesofxandyforanyprices(px,py)andincome(I).SettinguptheLagrangianexpressionℒ¼xαyβþλðIpxxpyyÞ(4.18)yieldsthefirst-orderconditions∂ℒ∂x¼αxα1yβλpx¼0,∂ℒ∂y¼βxαyβ1λpy¼0,∂ℒ∂λ¼Ipxxpyy¼0.(4.19)Takingtheratioofthefirsttwotermsshowsthatαyβx¼pxpy,(4.20)orpyy¼βαpxx¼1ααpxx,(4.21)wherethefinalequationfollowsbecauseαþβ¼1.Substitutionofthisfirst-orderconditioninEquation4.21intothebudgetconstraintgivesI¼pxxþpyy¼pxxþ1ααpxx¼pxx1þ1αα¼1αpxx;(4.22)solvingforxyieldsx¼αIpx,(4.23)(continued)5NoticethattheexponentsintheCobb-Douglasutilityfunctioncanalwaysbenormalizedtosumto1becauseU1=ðαþβÞisamonotonictransformation.Chapter4UtilityMaximizationandChoice121
EXAMPLE4.1CONTINUEDandasimilarsetofmanipulationswouldgivey¼βIpy.(4.24)TheseresultsshowthatanindividualwhoseutilityfunctionisgivenbyEquation4.17willalwayschoosetoallocateαproportionofhisorherincometobuyinggoodx(i.e.,pxx=I¼α)andβproportiontobuyinggoodyðpyy=I¼βÞ.AlthoughthisfeatureoftheCobb-Douglasfunctionoftenmakesitveryeasytoworkoutsimpleproblems,itdoessuggestthatthefunctionhaslimitsinitsabilitytoexplainactualconsumptionbehavior.Becausetheshareofincomedevotedtoparticulargoodsoftenchangessignificantlyinresponsetochangingeconomicconditions,amoregeneralfunctionalformmayprovideinsightsnotprovidedbytheCobb-Douglasfunction.WeillustrateafewpossibilitiesinExample4.2,andthegeneraltopicofbudgetsharesistakenupinmoredetailintheExtensionstothischapter.Numericalexample.First,however,let’slookataspeciﬁcnumericalexamplefortheCobb-Douglascase.Supposethatxsellsfor$1andysellsfor$4andthattotalincomeis$8.Succinctlythen,assumethatpx¼1,py¼4,I¼8.Supposealsothatα¼β¼0:5sothatthisindividualsplitshisorherincomeequallybetweenthesetwogoods.NowthedemandEquations4.23and4.24implyx¼αI=px¼0:5I=px¼0:5ð8Þ=1¼4,y¼βI=py¼0:5I=py¼0:5ð8Þ=4¼1,(4.25)and,attheseoptimalchoices,utility¼x0:5y0:5¼ð4Þ0:5ð1Þ0:5¼2.(4.26)NoticealsothatwecancomputethevaluefortheLagrangianmultiplierassociatedwiththisincomeallocationbyusingEquation4.19:λ¼αxα1yβ=px¼0:5ð4Þ0:5ð1Þ0:5=1¼0:25.(4.27)Thisvalueimpliesthateachsmallchangeinincomewillincreaseutilitybyaboutone-fourthofthatamount.Suppose,forexample,thatthispersonhad1percentmoreincome($8.08).Inthiscaseheorshewouldchoosex¼4:04andy¼1:01,andutilitywouldbe4:040:5⋅1:010:5¼2:02.Hence,a$0.08increaseinincomeincreasesutilityby0.02,aspredictedbythefactthatλ¼0:25.QUERY:WouldachangeinpyaffectthequantityofxdemandedinEquation4.23?Explainyouranswermathematically.Alsodevelopanintuitiveexplanationbasedonthenotionthattheshareofincomedevotedtogoodyisgivenbytheparameteroftheutilityfunction,β.EXAMPLE4.2CESDemandToillustratecasesinwhichbudgetsharesareresponsivetoeconomiccircumstances,let’slookatthreespeciﬁcexamplesoftheCESfunction.Case1:δ¼0:5.Inthiscase,utilityisUðx,yÞ¼x0:5þy0:5.(4.28)122Part2ChoiceandDemand
SettinguptheLagrangianexpressionℒ¼x0:5þy0:5þλðIpxxpyyÞ(4.29)yieldsthefollowingfirst-orderconditionsforamaximum:∂ℒ=∂x¼0:5×0:5λpx¼0,∂ℒ=∂y¼0:5y0:5λpy¼0,∂ℒ=∂λ¼Ipxxpyy¼0.(4.30)Divisionofthefirsttwooftheseshowsthatðy=xÞ0:5¼px=py.(4.31)Bysubstitutingthisintothebudgetconstraintanddoingsomemessyalgebraicmanipulation,wecanderivethedemandfunctionsassociatedwiththisutilityfunction:x¼I=px½1þðpx=pyÞ,(4.32)y¼I=py½1þðpy=pxÞ.(4.33)Priceresponsiveness.Inthesedemandfunctionsnoticethattheshareofincomespenton,say,goodx—thatis,pxx=I¼1=½1þðpx=pyÞ—isnotaconstant;itdependsonthepriceratiopx=py.Thehigheristherelativepriceofx,thesmallerwillbetheshareofincomespentonthatgood.Inotherwords,thedemandforxissoresponsivetoitsownpricethatariseinthepricereducestotalspendingonx.ThatthedemandforxisverypriceresponsivecanalsobeillustratedbycomparingtheimpliedexponentonpxinthedemandfunctiongivenbyEquation4.32(2)tothatfromEquation4.23(1).InChapter5wewilldiscussthisobservationmorefullywhenweexaminetheelasticityconceptindetail.Case2:δ¼1.Alternatively,let’slookatademandfunctionwithlesssubstitutability6thantheCobb-Douglas.Ifδ¼1,theutilityfunctionisgivenbyUðx,yÞ¼x1y1,(4.34)anditiseasytoshowthatthefirst-orderconditionsforamaximumrequirey=x¼ðpx=pyÞ0:5.(4.35)Again,substitutionofthisconditionintothebudgetconstraint,togetherwithsomemessyalgebra,yieldsthedemandfunctionsx¼I=px½1þðpy=pxÞ0:5,y¼I=py½1þðpx=pyÞ0:5.(4.36)Thatthesedemandfunctionsarelesspriceresponsivecanbeseenintwoways.First,nowtheshareofincomespentongoodx—thatis,pxx=I¼1=½1þðpy=pxÞ0:5—respondspositivelytoincreasesinpx.Asthepriceofxrises,thisindividualcutsbackonlymodestlyongoodx,sototalspendingonthatgoodrises.ThatthedemandfunctionsinEquations4.36arelesspriceresponsivethantheCobb-Douglasisalsoillustratedbytherelativelysmallexponentsofeachgood’sownpriceð0:5Þ.(continued)6Onewaytomeasuresubstitutabilityisbytheelasticityofsubstitution,whichfortheCESfunctionisgivenbyσ¼1=ð1δÞ.Hereδ¼0:5impliesσ¼2,δ¼0(theCobb-Douglas)impliesσ¼1,andδ¼1impliesσ¼0:5.SeealsothediscussionoftheCESfunctioninconnectionwiththetheoryofproductioninChapter9.Chapter4UtilityMaximizationandChoice123
EXAMPLE4.2CONTINUEDCase3:δ¼∞.Thisistheimportantcaseinwhichxandymustbeconsumedinﬁxedproportions.Suppose,forexample,thateachunitofymustbeconsumedtogetherwithexactly4unitsofx.TheutilityfunctionthatrepresentsthissituationisUðx,yÞ¼minðx,4yÞ.(4.37)Inthissituation,autility-maximizingpersonwillchooseonlycombinationsofthetwogoodsforwhichx¼4y;thatis,utilitymaximizationimpliesthatthispersonwillchoosetobeatavertexofhisorherL-shapedindifferencecurves.SubstitutingthisconditionintothebudgetconstraintyieldsI¼pxxþpyy¼pxxþpyx4¼ðpxþ0:25pyÞx.(4.38)Hencex¼Ipxþ0:25py,(4.39)andsimilarsubstitutionsyieldy¼I4pxþpy.(4.40)Inthiscase,theshareofaperson’sbudgetdevotedto,say,goodxrisesrapidlyasthepriceofxincreasesbecausexandymustbeconsumedinfixedproportions.Forexample,ifweusethevaluesassumedinExample4.1(px¼1,py¼4,I¼8),Equations4.39and4.40wouldpredictx¼4,y¼1,and,asbefore,halfoftheindividual’sincomewouldbespentoneachgood.Ifweinsteadusepx¼2,py¼4,andI¼8thenx¼8=3,y¼2=3,andthispersonspendstwothirds½pxx=I¼ð2⋅8=3Þ=8¼2=3ofhisorherincomeongoodx.Tryingafewothernumberssuggeststhattheshareofincomedevotedtogoodxapproaches1asthepriceofxincreases.7QUERY:DochangesinincomeaffectexpendituresharesinanyoftheCESfunctionsdiscussedhere?Howisthebehaviorofexpendituresharesrelatedtothehomotheticnatureofthisfunction?INDIRECTUTILITYFUNCTIONExamples4.1and4.2illustratetheprinciplethatitisoftenpossibletomanipulatetheﬁrst-orderconditionsforaconstrainedutility-maximizationproblemtosolvefortheoptimalvaluesofx1,x2,…,xn.Theseoptimalvaluesingeneralwilldependonthepricesofallthegoodsandontheindividual’sincome.Thatis,x1¼x1ðp1,p2,…,pn,IÞ,x2¼x2ðp1,p2,…,pn,IÞ,…xn¼xnðp1,p2,…,pn,IÞ.(4.41)Inthenextchapterwewillanalyzeinmoredetailthissetofdemandfunctions,whichshowthedependenceofthequantityofeachxidemandedonp1,p2,…,pnandI.Hereweuse7TheserelationshipsfortheCESfunctionarepursuedinmoredetailinProblem4.9andinExtensionE4.3.124Part2ChoiceandDemand
theoptimalvaluesofthex’sfromEquations4.42tosubstituteintheoriginalutilityfunctiontoyieldmaximumutility¼Uðx1,x2,…,xnÞ(4.42)¼Vðp1,p2,…,pn,IÞ.(4.43)Inwords:becauseoftheindividual’sdesiretomaximizeutilitygivenabudgetconstraint,theoptimallevelofutilityobtainablewilldependindirectlyonthepricesofthegoodsbeingboughtandtheindividual’sincome.ThisdependenceisreﬂectedbytheindirectutilityfunctionV.Ifeitherpricesorincomeweretochange,thelevelofutilitythatcouldbeattainedwouldalsobeaffected.Sometimes,inbothconsumertheoryandmanyothercontexts,itispossibletousethisindirectapproachtostudyhowchangesineconomiccircumstancesaffectvariouskindsofoutcomes,suchasutilityor(laterinthisbook)ﬁrms’costs.THELUMPSUMPRINCIPLEManyeconomicinsightsstemfromtherecognitionthatutilityultimatelydependsontheincomeofindividualsandonthepricestheyface.Oneofthemostimportantoftheseistheso-calledlumpsumprinciplethatillustratesthesuperiorityoftaxesonaperson’sgeneralpurchasingpowertotaxesonspeciﬁcgoods.Arelatedinsightisthatgeneralincomegrantstolow-incomepeoplewillraiseutilitymorethanwillasimilaramountofmoneyspentsubsidizingspeciﬁcgoods.Theintuitionbehindthisresultderivesdirectlyfromtheutility-maximizationhypothesis;anincometaxorsubsidyleavestheindividualfreetodecidehowtoallocatewhateverﬁnalincomeheorshehas.Ontheotherhand,taxesorsubsidiesonspeciﬁcgoodsbothchangeaperson’spurchasingpoweranddistorthisorherchoicesbecauseoftheartiﬁcialpricesincorporatedinsuchschemes.Hence,generalin-cometaxesandsubsidiesaretobepreferredifefﬁciencyisanimportantcriterioninsocialpolicy.ThelumpsumprincipleasitappliestotaxationisillustratedinFigure4.5.InitiallythispersonhasanincomeofIandischoosingtoconsumethecombinationx,y.Ataxongoodxwouldraiseitsprice,andtheutility-maximizingchoicewouldshifttocombinationx1,y1.Taxcollectionswouldbet⋅x1(wheretisthetaxrateimposedongoodx).Alternatively,anincometaxthatshiftedthebudgetconstraintinwardtoI0wouldalsocollectthissameamountofrevenue.8ButtheutilityprovidedbytheincometaxðU2ÞexceedsthatprovidedbythetaxonxaloneðU1Þ.Hence,wehaveshownthattheutilityburdenoftheincometaxissmaller.Asimilarargumentcanbeusedtoillustratethesuperiorityofincomegrantstosubsidiesonspeciﬁcgoods.EXAMPLE4.3IndirectUtilityandtheLumpSumPrincipleInthisexampleweusethenotionofanindirectutilityfunctiontoillustratethelumpsumprincipleasitappliestotaxation.Firstwehavetoderiveindirectutilityfunctionsfortwoillustrativecases.(continued)8BecauseI¼ðpxþtÞx1þpyy1,wehaveI0¼Itx1¼pxx1þpyy1,whichshowsthatthebudgetconstraintwithanequal-sizeincometaxalsopassesthroughthepointx1,y1.Chapter4UtilityMaximizationandChoice125
EXAMPLE4.3CONTINUEDCase1:Cobb-Douglas.InExample4.1weshowedthat,fortheCobb-Douglasutilityfunctionwithα¼β¼0:5,optimalpurchasesarex¼I2px,y¼I2py⋅(4.44)SotheindirectutilityfunctioninthiscaseisVðpx,py,IÞ¼Uðx,yÞ¼ðxÞ0:5ðyÞ0:5¼I2p0:5xp0:5y.Noticethatwhenpx¼1,py¼4,andI¼8wehaveV¼8=ð2⋅1⋅2Þ¼2,whichistheutilitythatwecalculatedbeforeforthissituation.Case2:Fixedproportions.InthethirdcaseofExample4.2wefoundthatx¼Ipxþ0:25py,y¼I4pxþpy⋅(4.46)So,inthiscaseindirectutilityisgivenbyVðpx,py,IÞ¼minðx,4yÞ¼x¼Ipxþ0:25py¼4y¼44pxþpy¼Ipxþ0:25py;(4.47)withpx¼1,py¼4,andI¼8,indirectutilityisgivenbyV¼4,whichiswhatwecalcu-latedbefore.Thelumpsumprinciple.ConsiderﬁrstusingtheCobb-Douglascasetoillustratethelumpsumprinciple.Supposethatataxof$1wereimposedongoodx.Equation4.45showsthatindirectutilityinthiscasewouldfallfrom2to1:41½¼8=ð2⋅20:5⋅2Þ.Becausethispersonchoosesx¼2withthetax,totaltaxcollectionswillbe$2.Anequal-revenueincometaxwouldthereforereducenetincometo$6,andindirectutilitywouldbe1:5½¼6=ð2⋅1⋅2Þ.Sothein-cometaxisaclearimprovementoverthecasewherexaloneistaxed.Thetaxongoodxreducesutilityfortworeasons:itreducesaperson’spurchasingpoweranditbiaseshisorherchoicesawayfromgoodx.Withincometaxation,onlytheﬁrsteffectisfeltandsothetaxismoreefﬁcient.9Theﬁxed-proportionscasesupportsthisintuition.Inthatcase,a$1taxongoodxwouldreduceindirectutilityfrom4to8=3½¼8=ð2þ1Þ.Inthiscasex¼8=3andtaxcollectionswouldbe$8=3.Anincometaxthatcollected$8=3wouldleavethisconsumerwith$16=3innetincome,andthatincomewouldyieldanindirectutilityofV¼8=3½¼ð16=3Þ=ð1þ1Þ.Henceafter-taxutilityisthesameunderboththeexciseandincometaxes.Thereasonthelumpsumresultdoesnotholdinthiscaseisthatwithﬁxed-proportionsutility,theexcisetaxdoesnotdistortchoicesbecausepreferencesaresorigid.QUERY:Bothoftheindirectutilityfunctionsillustratedhereshowthatadoublingofincomeandallpriceswouldleaveindirectutilityunchanged.Explainwhyyouwouldexpectthistobeapropertyofallindirectutilityfunctions.9Thisdiscussionassumesthattherearenoincentiveeffectsofincometaxation—probablynotaverygoodassumption.126Part2ChoiceandDemand
EXPENDITUREMINIMIZATIONInChapter2wepointedoutthatmanyconstrainedmaximumproblemshaveassociated“dual”constrainedminimumproblems.Forthecaseofutilitymaximization,theassociateddualminimizationproblemconcernsallocatingincomeinsuchawayastoachieveagivenutilitylevelwiththeminimalexpenditure.Thisproblemisclearlyanalogoustotheprimaryutility-maximizationproblem,butthegoalsandconstraintsoftheproblemshavebeenreversed.Figure4.6illustratesthisdualexpenditure-minimizationproblem.There,theindividualmustattainutilitylevelU2;thisisnowtheconstraintintheproblem.Threepossibleexpenditureamounts(E1,E2,andE3)areshownasthree“budgetconstraint”linesintheﬁgure.ExpenditurelevelE1isclearlytoosmalltoachieveU2,henceitcannotsolvethedualproblem.WithexpendituresgivenbyE3,theindividualcanreachU2(ateitherofthetwopointsBorC),butthisisnottheminimalexpenditurelevelrequired.Rather,E2clearlyprovidesjustenoughtotalexpenditurestoreachU2(atpointA),andthisisinfactthesolutiontothedualproblem.BycomparingFigures4.2and4.6,itisobviousthatboththeprimaryutility-maximizationapproachandthedualexpenditure-minimizationapproachyieldthesamesolutionðx,yÞ;theyaresimplyalternativewaysofviewingthesameprocess.Oftentheexpenditure-minimizationapproachismoreuseful,however,becauseexpendituresaredirectlyobservable,whereasutilityisnot.FIGURE4.5TheLumpSumPrincipleofTaxationAtaxongoodxwouldshifttheutility-maximizingchoicefromx,ytox1,y1.AnincometaxthatcollectedthesameamountwouldshiftthebudgetconstrainttoI0.UtilitywouldbehigherðU2ÞwiththeincometaxthanwiththetaxonxaloneðU1Þ.Quantity of xQuantityof yy1U1U2U3l′lx1x2x*y2y*Chapter4UtilityMaximizationandChoice127
AmathematicalstatementMoreformally,theindividual’sdualexpenditure-minimizationproblemistochoosex1,x2,…,xnsoastominimizetotalexpenditures¼E¼p1x1þp2x2þ…þpnxn,(4.48)subjecttotheconstraintutility¼_U¼Uðx1,x2,…,xnÞ.(4.49)Theoptimalamountsofx1,x2,…,xnchoseninthisproblemwilldependonthepricesofthevariousgoodsðp1,p2,…,pnÞandontherequiredutilitylevel_U2.Ifanyofthepricesweretochangeoriftheindividualhadadifferentutility“target,”thenanothercommoditybundlewouldbeoptimal.Thisdependencecanbesummarizedbyanex-penditurefunction.DEFINITIONExpenditurefunction.Theindividual’sexpenditurefunctionshowstheminimalexpendi-turesnecessarytoachieveagivenutilitylevelforaparticularsetofprices.Thatis,minimalexpenditures¼Eðp1,p2,…,pn,UÞ.(4.50)Thisdefinitionshowsthattheexpenditurefunctionandtheindirectutilityfunctionareinversefunctionsofoneanother(compareEquations4.43and4.50).BothdependonFIGURE4.6TheDualExpenditure-MinimizationProblemThedualoftheutility-maximizationproblemistoattainagivenutilitylevelðU2Þwithminimalexpenditures.AnexpenditurelevelofE1doesnotpermitU2tobereached,whereasE3providesmorespendingpowerthanisstrictlynecessary.WithexpenditureE2,thispersoncanjustreachU2byconsumingxandy.Quantity of xQuantityof yBE3E2U2E1CAx*y*128Part2ChoiceandDemand
marketpricesbutinvolvedifferentconstraints(incomeorutility).Inthenextchapterwewillseehowthisrelationshipisquiteusefulinallowingustoexaminethetheoryofhowindividualsrespondtopricechanges.First,however,let’slookattwoexpenditurefunctions.EXAMPLE4.4TwoExpenditureFunctionsTherearetwowaysonemightcomputeanexpenditurefunction.Theﬁrst,moststraight-forwardmethodwouldbetostatetheexpenditure-minimizationproblemdirectlyandapplytheLagrangiantechnique.Someoftheproblemsattheendofthischapteraskyoutodopreciselythat.Here,however,wewilladoptamorestreamlinedprocedurebytakingadvantageoftherelationshipbetweenexpenditurefunctionsandindirectutilityfunc-tions.Becausethesetwofunctionsareinversesofeachother,calculationofonegreatlyfaci-litatesthecalculationoftheother.WehavealreadycalculatedindirectutilityfunctionsfortwoimportantcasesinExample4.3.Retrievingtherelatedexpenditurefunctionsissimplealgebra.Case1:Cobb-Douglasutility.Equation4.45showsthattheindirectutilityfunctioninthetwo-good,Cobb-DouglascaseisVðpx,py,IÞ¼I2p0:5xp0:5y.(4.51)Ifwenowinterchangetheroleofutility(whichwewillnowtreatasaconstantdenotedbyU)andincome(whichwewillnowterm“expenditures,”E,andtreatasafunctionoftheparametersofthisproblem),thenwehavetheexpenditurefunctionEðpx,py,UÞ¼2p0:5xp0:5yU.(4.52)Checkingthisagainstourformerresults,nowweuseautilitytargetofU¼2with,again,px¼1andpy¼4.Withtheseparameters,Equation4.52predictsthattherequiredminimalexpendituresare$8ð¼2⋅10:5⋅40:5⋅2Þ.Notsurprisingly,boththeprimalutility-maximizationproblemandthedualexpenditure-minimizationproblemareformallyidentical.Case2:Fixedproportions.Fortheﬁxed-proportionscase,Equation4.47gavetheindirectutilityfunctionasVðpx,py,IÞ¼Ipxþ0:25py.(4.53)Ifweagainswitchtheroleofutilityandexpenditures,wequicklyderivetheexpenditurefunction:Eðpx,py,UÞ¼ðpxþ0:25pyÞU.(4.54)AcheckofthehypotheticalvaluesusedinExample4.3ðpx¼1,py¼4,U¼4Þagainshowsthatitwouldcost$8½¼ð1þ0:25⋅4Þ⋅4toreachtheutilitytargetof4.Compensatingforapricechange.Theseexpenditurefunctionsallowustoinvestigatehowapersonmightbecompensatedforapricechange.Speciﬁcally,supposethatthepriceofgoodyweretorisefrom$4to$5.Thiswouldclearlyreduceaperson’sutility,sowemightaskwhatamountofmonetarycompensationwouldmitigatetheharm.Becausetheexpendi-turefunctionallowsutilitytobeheldconstant,itprovidesadirectestimateofthisamount.Speciﬁcally,intheCobb-Douglascase,expenditureswouldhavetobeincreasedfrom$8to(continued)Chapter4UtilityMaximizationandChoice129
EXAMPLE4.4CONTINUED$8:94ð¼2⋅1⋅50:5⋅2Þinordertoprovideenoughextrapurchasingpowertopreciselycompensateforthispricerise.Withﬁxedproportions,expenditureswouldhavetobeincreasedfrom$8to$9tocompensateforthepriceincrease.Hence,thecompensationsareaboutthesameinthesesimplecases.Thereisoneimportantdifferencebetweenthetwoexamples,however.Intheﬁxed-proportionscase,the$1ofextracompensationsimplypermitsthispersontoreturntohisorherpriorconsumptionbundleðx¼4,y¼1Þ.ThatistheonlywaytorestoreutilitytoU¼4forthisrigidperson.IntheCobb-Douglascase,however,thispersonwillnotusetheextracompensationtoreverttohisorherpriorconsumptionbundle.Instead,utilitymaximizationwillrequirethatthe$8.94beallocatedsothatx¼4:47andy¼0:894.ThiswillstillprovideautilitylevelofU¼2,butthispersonwilleconomizeonthenowmoreexpensivegoody.QUERY:Howshouldapersonbecompensatedforapricedecline?Whatsortofcompensa-tionwouldberequiredifthepriceofgoodyfellfrom$4to$3?PROPERTIESOFEXPENDITUREFUNCTIONSBecauseexpenditurefunctionsarewidelyusedinappliedeconomics,itisusefultounder-standafewofthepropertiessharedbyallsuchfunctions.Herewelookatthreesuchproperties.Allofthesefollowdirectlyfromthefactthatexpenditurefunctionsarebasedonindividualutilitymaximization.1.Homogeneity.ForbothofthefunctionsillustratedinExample4.4,adoublingofallpriceswillpreciselydoublethevalueofrequiredexpenditures.Technically,theseexpenditurefunctionsare“homogeneousofdegreeone”inallprices.10Thisisaquitegeneralpropertyofexpenditurefunctions.Becausetheindividual’sbudgetconstraintislinearinprices,anyproportionalincreaseinbothpricesandpurchasingpowerwillpermitthepersontobuythesameutility-maximizingcommoditybundlethatwaschosenbeforethepricerise.InChapter5wewillseethat,forthisreason,demandfunctionsarehomogenousofdegree0inallpricesandincome.2.Expenditurefunctionsarenondecreasinginprices.Thispropertycanbesuccinctlysummarizedbythemathematicalstatement∂E∂pi0foreverygoodi.(4.55)Thisseemsintuitivelyobvious.Becausetheexpenditurefunctionreportsthemini-mumexpenditurenecessarytoreachagivenutilitylevel,anincreaseinanypricemustincreasethisminimum.Moreformally,supposep1takesontwovalues:pa1andpb1withpb1>pa1,whereallotherpricesareunchangedbetweenstatesaandb.Also,letxbethebundleofgoodspurchasedinstateaandythebundlepurchasedinstateb.Bythedefinitionoftheexpenditurefunction,bothofthesebundlesofgoodsmust10AsdescribedinChapter2,thefunctionfðx1,x2,…,xnÞissaidtobehomogeneousofdegreekiffðtx1,tx2,…,txnÞ¼tkfðx1,x2,…,xnÞ.Inthiscase,k¼1.130Part2ChoiceandDemand
yieldthesametargetutility.Clearlybundleycostsmorewithstate-bpricesthanitwouldwithstate-aprices.Butweknowthatbundlexisthelowest-costwaytoachievethetargetutilitylevelwithstate-aprices.Hence,expendituresonbundleymustbegreaterthanorequaltothoseonbundlex.Similarly,adeclineinapricemustnotincreaseexpenditures.3.Expenditurefunctionsareconcaveinprices.InChapter2wediscussedconcavefunctionsasfunctionsthatalwaysliebelowtangentstothem.Althoughthetechnicalmathematicalconditionsthatdescribesuchfunctionsarecomplicated,itisrelativelysimpletoshowhowtheconceptappliestoexpenditurefunctionsbyconsideringthevariationinasingleprice.Figure4.7showsanindividual’sexpendituresasafunctionofthesingleprice,p1.Attheinitialprice,p1,thisperson’sexpendituresaregivenbyEðp1,…Þ.Nowconsiderpriceshigherorlowerthanp1.Ifthispersoncontinuedtobuythesamebundleofgoods,expenditureswouldincreaseordecreaselinearlyasthispricechanged.ThiswouldgiverisetothepseudoexpenditurefunctionEpseudointheﬁgure.Thislineshowsalevelofexpendituresthatwouldallowthispersontobuytheoriginalbundleofgoodsdespitethechangingvalueofp1.If,asseemsmorelikely,thispersonadjustedhisorherpurchasesasp1changed,weknow(becauseofexpenditureminimization)thatactualexpenditureswouldbelessthanthesepseudoFIGURE4.7ExpenditureFunctionsAreConcaveinPricesAtp1thispersonspendsEðp1,…Þ.Ifheorshecontinuestobuythesamesetofgoodsasp1changes,thenexpenditureswouldbegivenbyEpseudo.Becausehisorherconsumptionpatternswilllikelychangeasp1changes,actualexpenditureswillbelessthanthis.E(p1, . . .)E(p1, . . .)p1E(p1*, . . .)EpseudoE(p1*, . . .)Chapter4UtilityMaximizationandChoice131
amounts.Hence,theactualexpenditurefunction,E,willlieeverywherebelowEpseudoandthefunctionwillbeconcave.11Theconcavityoftheexpenditurefunctionisausefulpropertyforanumberofapplications,especiallythoserelatedtotheconstruc-tionofindexnumbers(seetheExtensionstoChapter5).PROBLEMS4.1EachdayPaul,whoisinthirdgrade,eatslunchatschool.HelikesonlyTwinkies(t)andsoda(s),andtheseprovidehimautilityofutility¼Uðt,sÞ¼ﬃﬃﬃﬃtsp.a.IfTwinkiescost$0.10eachandsodacosts$0.25percup,howshouldPaulspendthe$1hismothergiveshiminordertomaximizehisutility?b.IftheschooltriestodiscourageTwinkieconsumptionbyraisingthepriceto$0.40,byhowmuchwillPaul’smotherhavetoincreasehislunchallowancetoprovidehimwiththesamelevelofutilityhereceivedinpart(a)?4.2a.Ayoungconnoisseurhas$600tospendtobuildasmallwinecellar.Sheenjoystwovintagesinparticular:a2001FrenchBordeaux(wF)at$40perbottleandalessexpensive2005Californiavarietalwine(wC)pricedat$8.Ifherutilityis11Oneresultofconcavityisthatfii¼∂2E=∂p2i0.ThisispreciselywhatFigure4.7shows.SUMMARYInthischapterweexploredthebasiceconomicmodelofutilitymaximizationsubjecttoabudgetconstraint.Althoughweapproachedthisprobleminavarietyofways,alloftheseapproachesleadtothesamebasicresult.•Toreachaconstrainedmaximum,anindividualshouldspendallavailableincomeandshouldchooseacom-moditybundlesuchthattheMRSbetweenanytwogoodsisequaltotheratioofthosegoods’marketprices.Thisbasictangencywillresultintheindividualequatingtheratiosofthemarginalutilitytomarketpriceforeverygoodthatisactuallyconsumed.Sucharesultiscommontomostconstrainedoptimizationproblems.•Thetangencyconditionsareonlytheﬁrst-ordercondi-tionsforauniqueconstrainedmaximum,however.Toensurethattheseconditionsarealsosufﬁcient,theindi-vidual’sindifferencecurvemapmustexhibitadiminish-ingMRS.Informalterms,theutilityfunctionmustbestrictlyquasi-concave.•Thetangencyconditionsmustalsobemodiﬁedtoallowforcornersolutionsinwhichtheoptimallevelofcon-sumptionofsomegoodsiszero.Inthiscase,theratioofmarginalutilitytopriceforsuchagoodwillbebelowthecommonmarginalbeneﬁt–marginalcostratioforgoodsactuallybought.•Aconsequenceoftheassumptionofconstrainedutilitymaximizationisthattheindividual’soptimalchoiceswilldependimplicitlyontheparametersofhisorherbudgetconstraint.Thatis,thechoicesobservedwillbeimplicitfunctionsofallpricesandincome.Utilitywillthereforealsobeanindirectfunctionoftheseparameters.•Thedualtotheconstrainedutility-maximizationprob-lemistominimizetheexpenditurerequiredtoreachagivenutilitytarget.Althoughthisdualapproachyieldsthesameoptimalsolutionastheprimalconstrainedmax-imumproblem,italsoyieldsadditionalinsightintothetheoryofchoice.Speciﬁcally,thisapproachleadstoex-penditurefunctionsinwhichthespendingrequiredtoreachagivenutilitytargetdependsongoods’marketprices.Expenditurefunctionsaretherefore,inprinciple,measurable.132Part2ChoiceandDemand
UðwF,wCÞ¼w2=3Fw1=3C,thenhowmuchofeachwineshouldshepurchase?b.Whenshearrivedatthewinestore,ouryoungoenologistdiscoveredthatthepriceoftheFrenchBordeauxhadfallento$20abottlebecauseofadeclineinthevalueofthefranc.IfthepriceoftheCaliforniawineremainsstableat$8perbottle,howmuchofeachwineshouldourfriendpurchasetomaximizeutilityunderthesealteredconditions?c.Explainwhythiswinefancierisbetteroffinpart(b)thaninpart(a).Howwouldyouputamonetaryvalueonthisutilityincrease?4.3a.Onagivenevening,J.P.enjoystheconsumptionofcigars(c)andbrandy(b)accordingtothefunctionUðc,bÞ¼20cc2þ18b3b2.Howmanycigarsandglassesofbrandydoesheconsumeduringanevening?(CostisnoobjecttoJ.P.)b.Lately,however,J.P.hasbeenadvisedbyhisdoctorsthatheshouldlimitthesumofglassesofbrandyandcigarsconsumedto5.Howmanyglassesofbrandyandcigarswillheconsumeunderthesecircumstances?4.4a.Mr.OddeBallenjoyscommoditiesxandyaccordingtotheutilityfunctionUðx,yÞ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃx2þy2q:MaximizeMr.Ball’sutilityifpx¼$3,py¼$4,andhehas$50tospend.Hint:ItmaybeeasierheretomaximizeU2ratherthanU.Whywon’tthisalteryourresults?b.GraphMr.Ball’sindifferencecurveanditspointoftangencywithhisbudgetconstraint.WhatdoesthegraphsayaboutMr.Ball’sbehavior?Haveyoufoundatruemaximum?4.5Mr.Aderivesutilityfrommartinis(m)inproportiontothenumberhedrinks:UðmÞ¼m.Mr.Aisveryparticularabouthismartinis,however:Heonlyenjoysthemmadeintheexactproportionoftwopartsgin(g)toonepartvermouth(v).Hence,wecanrewriteMr.A’sutilityfunctionasUðmÞ¼Uðg,vÞ¼ming2,v.a.GraphMr.A’sindifferencecurveintermsofgandvforvariouslevelsofutility.Showthat,regardlessofthepricesofthetwoingredients,Mr.Awillneveralterthewayhemixesmartinis.b.Calculatethedemandfunctionsforgandv.c.Usingtheresultsfrompart(b),whatisMr.A’sindirectutilityfunction?d.CalculateMr.A’sexpenditurefunction;foreachlevelofutility,showspendingasafunctionofpgandpv.Hint:Becausethisprobleminvolvesaﬁxed-proportionsutilityfunction,youcannotsolveforutility-maximizingdecisionsbyusingcalculus.4.6Supposethatafast-foodjunkiederivesutilityfromthreegoods—softdrinks(x),hamburgers(y),andicecreamsundaes(z)—accordingtotheCobb-DouglasutilityfunctionChapter4UtilityMaximizationandChoice133
Uðx,y,zÞ¼x0:5y0:5ð1þzÞ0:5.Supposealsothatthepricesforthesegoodsaregivenbypx¼0:25,py¼1,andpz¼2andthatthisconsumer’sincomeisgivenbyI¼2.a.Showthat,forz¼0,maximizationofutilityresultsinthesameoptimalchoicesasinExample4.1.Showalsothatanychoicethatresultsinz>0(evenforafractionalz)reducesutilityfromthisoptimum.b.Howdoyouexplainthefactthatz¼0isoptimalhere?c.Howhighwouldthisindividual’sincomehavetobeinorderforanyztobepurchased?4.7ThelumpsumprincipleillustratedinFigure4.5appliestotransferpolicyaswellastaxation.Thisproblemexaminesthisapplicationoftheprinciple.a.UseagraphsimilartoFigure4.5toshowthatanincomegranttoapersonprovidesmoreutilitythandoesasubsidyongoodxthatcoststhesameamounttothegovernment.b.UsetheCobb-DouglasexpenditurefunctionpresentedinEquation4.52tocalculatetheextrapurchasingpowerneededtoraisethisperson’sutilityfromU¼2toU¼3.c.UseEquation4.52againtoestimatethedegreetowhichgoodxmustbesubsidizedinordertoraisethisperson’sutilityfromU¼2toU¼3.Howmuchwouldthissubsidycostthegovernment?Howwouldthiscostcomparetothecostcalculatedinpart(b)?d.Problem4.10asksyoutocomputeanexpenditurefunctionforamoregeneralCobb-DouglasutilityfunctionthantheoneusedinExample4.4.Usethatexpenditurefunctiontore-solveparts(b)and(c)hereforthecaseα¼0:3,aﬁgureclosetothefractionofincomethatlow-incomepeoplespendonfood.e.Howwouldyourcalculationsinthisproblemhavechangedifwehadusedtheexpenditurefunctionfortheﬁxedproportionscase(Equation4.54)instead?4.8Mr.Carrderivesalotofpleasurefromdrivingunderthewideblueskies.Forthenumberofmilesxthathedrives,hereceivesutilityUðxÞ¼500xx2.(Oncehedrivesbeyondacertainnumberofmiles,wearinesskicksinandtheridebecomeslessandlessenjoyable.)Now,hiscargiveshimadecenthighwaymileageof25milestothegallon.Butpayingforgas,representedbyy,inducesdisutilityforMr.Carr,shownbyUðyÞ¼1,000y.Mr.Carriswillingtospendupto$25forleisurelydrivingeveryweek.a.FindtheoptimumnumberofmilesdrivenbyMr.Carreveryweek,giventhatthepriceofgasis$2.50pergallon.b.Howdoesthatvaluechangewhenthepriceofgasrisesto$5.00pergallon?c.Now,furtherassumethatthereisaprobabilityof0.001thatMr.Carrwillgetaﬂattireeverymilehedrives.ThedisutilityfromaﬂattireisgivenbyUðzÞ¼50,000z(wherezisthenumberofﬂattiresincurred),andeachﬂattirecosts$50toreplace.FindthedistancedriventhatmaximizesMr.Carr’sutilityaftertakingintoaccounttheexpectedlikelihoodofﬂattires(assumethatthepriceofgasis$2.50pergallon).4.9SupposethatwehaveautilityfunctioninvolvingtwogoodsthatislinearoftheformUðx,yÞ¼axþby.Calculatetheexpenditurefunctionforthisutilityfunction.Hint:Theexpenditurefunctionwillhavekinksatvariouspriceratios.134Part2ChoiceandDemand
AnalyticalProblems4.10Cobb-DouglasutilityInExample4.1welookedattheCobb-DouglasutilityfunctionUðx,yÞ¼xαy1α,where0α1.Thisproblemillustratesafewmoreattributesofthatfunction.a.CalculatetheindirectutilityfunctionforthisCobb-Douglascase.b.Calculatetheexpenditurefunctionforthiscase.c.Showexplicitlyhowthecompensationrequiredtooffsettheeffectofariseinthepriceofxisrelatedtothesizeoftheexponentα.4.11CESutilityTheCESutilityfunctionwehaveusedinthischapterisgivenbyUðx,yÞ¼xδδþyδδ.a.Showthattheﬁrst-orderconditionsforaconstrainedutilitymaximumwiththisfunctionrequireindividualstochoosegoodsintheproportionxy¼pxpy !1=ðδ1Þ.b.Showthattheresultinpart(a)impliesthatindividualswillallocatetheirfundsequallybetweenxandyfortheCobb-Douglascase(δ¼0),aswehaveshownbeforeinseveralproblems.c.Howdoestheratiopxx=pyydependonthevalueofδ?Explainyourresultsintuitively.(Forfurtherdetailsonthisfunction,seeExtensionE4.3.)d.Derivetheindirectutilityandexpenditurefunctionsforthiscaseandcheckyourresultsbydescribingthehomogeneitypropertiesofthefunctionsyoucalculated.4.12Stone-GearyutilitySupposeindividualsrequireacertainleveloffood(x)toremainalive.Letthisamountbegivenbyx0.Oncex0ispurchased,individualsobtainutilityfromfoodandothergoods(y)oftheformUðx,yÞ¼ðxx0Þαyβ,whereαþβ¼1:a.ShowthatifI>pxx0thentheindividualwillmaximizeutilitybyspendingαðIpxx0Þþpxx0ongoodxandβðIpxx0Þongoody.Interpretthisresult.b.Howdotheratiospxx=Iandpyy=Ichangeasincomeincreasesinthisproblem?(SeealsoExtensionE4.2formoreonthisutilityfunction.)4.13CESindirectutilityandexpenditurefunctionsInthisproblem,wewilluseamorestandardformoftheCESutilityfunctiontoderiveindirectutilityandexpenditurefunctions.SupposeutilityisgivenbyUðx,yÞ¼ðxδþyδÞ1=δ[inthisfunctiontheelasticityofsubstitutionσ¼1=ð1δÞ].a.ShowthattheindirectutilityfunctionfortheutilityfunctionjustgivenisV¼IðprxþpryÞ1=r,wherer¼δ=ðδ1Þ¼1σ.Chapter4UtilityMaximizationandChoice135
b.Showthatthefunctionderivedinpart(a)ishomogeneousofdegree0inpricesandincome.c.Showthatthisfunctionisstrictlyincreasinginincome.d.Showthatthisfunctionisstrictlydecreasinginanyprice.e.ShowthattheexpenditurefunctionforthiscaseofCESutilityisgivenbyE¼VðprxþpryÞ1=r.f.Showthatthefunctionderivedinpart(e)ishomogeneousofdegree1inthegoods’prices.g.Showthatthisexpenditurefunctionisincreasingineachoftheprices.h.Showthatthefunctionisconcaveineachprice.SUGGESTIONSFORFURTHERREADINGBarten,A.P.,andVolkerBöhm.“ConsumerTheory.”InK.J.ArrowandM.D.Intriligator,Eds.,HandbookofMathe-maticalEconomics,vol.II.Amsterdam:North-Holland,1982.Sections10and11havecompactsummariesofmanyofthecon-ceptscoveredinthischapter.Deaton,A.,andJ.Muelbauer.EconomicsandConsumerBehavior.Cambridge:CambridgeUniversityPress,1980.Section2.5providesanicegeometrictreatmentofdualityconcepts.Dixit,A.K.OptimizationinEconomicTheory.Oxford:OxfordUniversityPress,1990.Chapter2providesseveralLagrangiananalysesfocusingontheCobb-Douglasutilityfunction.Hicks,J.R.ValueandCapital.Oxford:ClarendonPress,1946.ChapterIIandtheMathematicalAppendixprovidesomeearlysug-gestionsoftheimportanceoftheexpenditurefunction.Mas-Colell,A.,M.D.Whinston,andJ.R.Green.Microeco-nomicTheory.Oxford:OxfordUniversityPress,1995.Chapter3containsathoroughanalysisofutilityandexpenditurefunctions.Samuelson,PaulA.FoundationsofEconomicAnalysis.Cambridge,MA:HarvardUniversityPress,1947.ChapterVandAppendixAprovideasuccinctanalysisoftheﬁrst-orderconditionsforautilitymaximum.Theappendixprovidesgoodcover-ageofsecond-orderconditions.Silberberg,E.,andW.Suen.TheStructureofEconomics:AMathematicalAnalysis,3rded.Boston:Irwin/McGraw-Hill,2001.Auseful,thoughfairlydifﬁcult,treatmentofdualityinconsumertheory.Theil,H.TheoryandMeasurementofConsumerDemand.Amsterdam:North-Holland,1975.Goodsummaryofbasictheoryofdemandtogetherwithimplicationsforempiricalestimation.136Part2ChoiceandDemand
EXTENSIONSBudgetSharesThenineteenth-centuryeconomistErnstEngelwasoneoftheﬁrstsocialscientiststointensivelystudypeople’sactualspendingpatterns.Hefocusedspeciﬁcallyonfoodconsumption.HisﬁndingthatthefractionofincomespentonfooddeclinesasincomeincreaseshascometobeknownasEngel’slawandhasbeenconﬁrmedinmanystudies.Engel’slawissuchanempiricalregularitythatsomeeconomistshavesug-gestedmeasuringpovertybythefractionofincomespentonfood.Twootherinterestingapplicationsare:(1)thestudybyHayashi(1995)showingthattheshareofincomedevotedtofoodsfavoredbytheelderlyismuchhigherintwo-generationhouseholdsthaninone-generationhouseholds;and(2)ﬁndingsbyBehr-man(1989)fromless-developedcountriesshowingthatpeople’sdesiresforamorevarieddietastheirincomesrisemayinfactresultinreducingthefractionofincomespentonparticularnutrients.Intheremain-derofthisextensionwelookatsomeevidenceonbudgetshares(denotedbysi¼pixi=I)togetherwithabitmoretheoryonthetopic.E4.1ThevariabilityofbudgetsharesTableE4.1showssomerecentbudgetsharedatafromtheUnitedStates.Engel’slawisclearlyvisibleinthetable:asincomerisesfamiliesspendasmallerpropor-tionoftheirfundsonfood.Otherimportantvariationsinthetableincludethedecliningshareofincomespentonhealth-careneedsandthemuchlargershareofincomedevotedtoretirementplansbyhigher-incomepeople.Interestingly,thesharesofincomedevotedtoshelterandtransportationarerelativelyconstantovertherangeofincomeshowninthetable;apparently,high-incomepeoplebuybiggerhousesandcars.ThevariableincomesharesinTableE4.1illustratewhytheCobb-Douglasutilityfunctionisnotusefulfordetailedempiricalstudiesofhouseholdbehavior.WhenutilityisgivenbyUðx,yÞ¼xαyβ,theimplieddemandequationsarex¼αI=pxandy¼βI=py.Therefore,sx¼pxx=I¼αandsy¼pyy=I¼β,(i)andbudgetsharesareconstantforallobservedin-comelevelsandrelativeprices.Becauseofthisshort-coming,economistshaveinvestigatedanumberofotherpossibleformsfortheutilityfunctionthatpermitmoreflexibility.E4.2LinearexpendituresystemAgeneralizationoftheCobb-Douglasfunctionthatincorporatestheideathatcertainminimalamountsofeachgoodmustbeboughtbyanindividualðx0,y0ÞistheutilityfunctionUðx,yÞ¼ðxx0Þαðyy0Þβ(ii)forxx0andyy0,whereagainαþβ¼1.DemandfunctionscanbederivedfromthisutilityfunctioninawayanalogoustotheCobb-Douglascasebyintroducingtheconceptofsupernumeraryin-comeðIÞ,whichrepresentstheamountofpurchas-ingpowerremainingafterpurchasingtheminimumbundleI¼Ipxx0pyy0.(iii)Usingthisnotation,thedemandfunctionsarex¼ðpxx0þαIÞ=px,y¼ðpyy0þβIÞ=py.(iv)Inthiscase,then,theindividualspendsaconstantfractionofsupernumeraryincomeoneachgoodoncetheminimumbundlehasbeenpurchased.Manipula-tionofEquationivyieldstheshareequationssx¼αþðβpxx0αpyy0Þ=I,sy¼βþðαpyy0βpxx0Þ=I,(v)whichshowthatthisdemandsystemisnothomothetic.InspectionofEquationvshowstheunsurprisingresultthatthebudgetshareofagoodispositivelyrelatedtotheminimalamountofthatgoodneededandneg-ativelyrelatedtotheminimalamountoftheothergoodrequired.Becausethenotionofnecessarypurchasesseemstoaccordwellwithreal-worldobservation,thislinearexpendituresystem(LES),whichwasfirstdevel-opedbyStone(1954),iswidelyusedinempiricalstud-ies.TheutilityfunctioninEquationiiisalsocalledaStone-Gearyutilityfunction.TraditionalpurchasesOneofthemostinterestingusesoftheLESistoexaminehowitsnotionofnecessarypurchaseschangesasconditionschange.Forexample,OczkowskiandChapter4UtilityMaximizationandChoice137
Philip(1994)studyhowaccesstomodernconsumergoodsmayaffecttheshareofincomethatindividualsintransitionaleconomiesdevotetotraditionallocalitems.TheyshowthatvillagersofPapua,NewGuinea,reducesuchsharessigniﬁcantlyasoutsidegoodsbe-comeincreasinglyaccessible.Hence,suchimprove-mentsasbetterroadsformovinggoodsprovideoneoftheprimaryroutesbywhichtraditionalculturalpracticesareundermined.E4.3CESutilityInChapter3weintroducedtheCESutilityfunctionUðx,yÞ¼xδδþyδδ(vi)forδ1,δ6¼0.Theprimaryuseofthisfunctionistoillustratealternativesubstitutionpossibilities(asreflectedinthevalueoftheparameterδ).Budgetsharesimpliedbythisutilityfunctionprovideanumberofsuchinsights.Manipulationofthefirst-orderconditionsforaconstrainedutilitymaximumwiththeCESfunctionyieldstheshareequationssx¼1=½1þðpy=pxÞK,sy¼1=½1þðpx=pyÞK,(vii)whereK¼δ=ðδ1Þ.ThehomotheticnatureoftheCESfunctionisshownbythefactthattheseshareexpressionsdependonlyonthepriceratio,px=py.BehaviorofthesharesinresponsetochangesinrelativepricesdependsonthevalueoftheparameterK.FortheCobb-Douglascase,δ¼0andsoK¼0andsx¼sy¼1=2.Whenδ>0;substitutionpossibilitiesaregreatandK<0.Inthiscase,Equationviishowsthatsxandpx=pymoveinoppositedirections.Ifpx=pyrises,theindividualsub-stitutesyforxtosuchanextentthatsxfalls.Alterna-tively,ifδ<0,thensubstitutionpossibilitiesarelimited,K>0,andsxandpx=pymoveinthesameTABLEE4.1BudgetSharesofU.S.Households,2004AnnualIncome$10,000–$14,999$40,000–$49,999Over$70,000ExpenditureItemFood15.314.311.8Shelter21.818.517.6Utilities,fuel,andpublicservices10.27.75.4Transportation15.418.417.6Healthinsurance4.93.82.3Otherhealth-careexpenses4.42.92.4Entertainment(includingalcohol)4.44.65.4Tobacco1.20.90.4Education2.51.12.6Insuranceandpensions2.79.614.7Other(apparel,personalcare,otherhousingexpenses,andmisc.)17.218.219.8SOURCE:ConsumerExpenditureReport,2004,BureauofLaborStatisticswebsite:http://www.bls.gov.138Part2ChoiceandDemand
direction.Inthiscase,anincreaseinpx=pycausesonlyminorsubstitutionofyforx,andsxactuallyrisesbe-causeoftherelativelyhigherpriceofgoodx.NorthAmericanfreetradeCESdemandfunctionsaremostoftenusedinlarge-scalecomputermodelsofgeneralequilibrium(seeChapter13)thateconomistsusetoevaluatetheimpactofmajoreconomicchanges.BecausetheCESmodelstressesthatsharesrespondtochangesinrelativeprices,itisparticularlyappropriateforlookingatinno-vationssuchaschangesintaxpolicyorininternationaltraderestrictions,wherechangesinrelativepricesarequitelikely.OneimportantrecentareaofsuchresearchhasbeenontheimpactoftheNorthAmericanFreeTradeAgreementforCanada,Mexico,andtheUnitedStates.Ingeneral,thesemodelsﬁndthatallofthecountriesinvolvedmightbeexpectedtogainfromtheagreement,butthatMexico’sgainsmaybethegreatestbecauseitisexperiencingthegreatestchangeinrelativeprices.KehoeandKehoe(1995)presentanumberofcomputableequilibriummodelsthatecon-omistshaveusedintheseexaminations.1E4.4ThealmostidealdemandsystemAnalternativewaytostudybudgetsharesistostartfromaspeciﬁcexpenditurefunction.Thisapproachisespe-ciallyconvenientbecausetheenvelopetheoremshowsthatbudgetsharescanbederiveddirectlyfromexpen-diturefunctionsthroughlogarithmicdifferentiation:∂lnEðpx,py,VÞ∂lnpx¼1Eðpx,py,VÞ⋅∂E∂px⋅∂px∂lnpx¼xpxE¼sx.(viii)DeatonandMuellbauer(1980)makeextensiveuseofthisrelationshiptostudythecharacteristicsofapar-ticularclassofexpenditurefunctionsthattheytermanalmostidealdemandsystem(AIDS).Theirexpendi-turefunctiontakestheformlnEðpx,py,VÞ¼a0þa1lnpxþa2lnpyþ0:5b1ðlnpxÞ2þb2lnpxlnpyþ0:5b3ðlnpyÞ2þVc0pc1xpc2y.(ix)Thisformapproximatesanyexpenditurefunction.Forthefunctiontobehomogeneousofdegree1intheprices,theparametersofthefunctionmustobeytheconstraintsa1þa2¼1,b1þb2¼0,b2þb3¼0,andc1þc2¼0.UsingtheresultsofEquationviiishowsthat,forthisfunction,sx¼a1þb1lnpxþb2lnpyþc1Vc0pc1xpc2y,sy¼a2þb2lnpxþb3lnpyþc2Vc0pc1xpc2y⋅(x)Noticethat,giventheparameterrestrictions,sxþsy¼1.Makinguseoftheinverserelationshipbetweenindirectutilityandexpenditurefunctionsandsomeadditionalalgebraicmanipulationwillputthesebudgetshareequationsintoasimpleformsuitableforeconometricestimation:sx¼a1þb1lnpxþb2lnpyþc1ðE=pÞ,sy¼a2þb2lnpxþb3lnpyþc2ðE=pÞ,(xi)wherepisanindexofpricesdefinedbylnp¼a0þa1lnpxþa2lnpyþ0:5b1ðlnpxÞ2þb2lnpxlnpyþ0:5b3ðlnpyÞ2.(xii)Inotherwords,theAIDSshareequationsstatethatbudgetsharesarelinearinthelogarithmsofpricesandintotalrealexpenditures.Inpractice,simplerpriceindicesareoftensubstitutedfortherathercom-plexindexgivenbyEquationxii,althoughthereissomecontroversyaboutthispractice(seetheExten-sionstoChapter5).BritishexpenditurepatternsDeatonandMuellbauerapplythisdemandsystemtothestudyofBritishexpenditurepatternsbetween1954and1974.Theyﬁndthatbothfoodandhousinghavenegativecoefﬁcientsofrealexpenditures,imply-ingthattheshareofincomedevotedtotheseitemsfalls(atleastinBritain)aspeoplegetricher.Theauthorsalsoﬁndsigniﬁcantrelativepriceeffectsinmanyoftheirshareequations,andpriceshaveespe-ciallylargeeffectsinexplainingtheshareofexpendi-turesdevotedtotransportationandcommunication.InapplyingtheAIDSmodeltoreal-worlddata,theauthorsalsoencounteravarietyofeconometricdifﬁ-culties,themostimportantofwhichisthatmanyoftheequationsdonotappeartoobeytherestrictionsnecessaryforhomogeneity.Addressingsuchissueshasbeenamajortopicforfurtherresearchonthisdemandsystem.1TheresearchontheNorthAmericanFreeTradeAgreementisdiscussedinmoredetailintheExtensionstoChapter13Chapter4UtilityMaximizationandChoice139
ReferencesBehrman,JereR.“IsVarietytheSpiceofLife?ImplicationsforCaloricIntake.”ReviewofEconomicsandStatistics(November1989):666–72.Deaton,Angus,andJohnMuellbauer.“AnAlmostIdealDemandSystem.”AmericanEconomicReview(June1980):312–26.Hyashi,Fumio.“IstheJapaneseExtendedFamilyAltruis-ticallyLinked?ATestBasedonEngelCurves.”JournalofPoliticalEconomy(June1995):661–74.Kehoe,PatrickJ.,andTimothyJ.Kehoe.ModelingNorthAmericanEconomicIntegration.London:KluwerAcademicPublishers,1995.Oczkowski,E.,andN.E.Philip.“HouseholdExpenditurePatternsandAccesstoConsumerGoodsinaTransi-tionalEconomy.”JournalofEconomicDevelopment(June1994):165–83.Stone,R.“LinearExpenditureSystemsandDemandAnalysis.”EconomicJournal(September1954):511–27.140Part2ChoiceandDemand
CHAPTER5IncomeandSubstitutionEffectsInthischapterwewillusetheutility-maximizationmodeltostudyhowthequantityofagoodthatanindividualchoosesisaffectedbyachangeinthatgood’sprice.Thisexaminationallowsustoconstructtheindividual’sdemandcurveforthegood.Intheprocesswewillprovideanumberofinsightsintothenatureofthispriceresponseandintothekindsofassumptionsthatliebehindmostanalysesofdemand.DEMANDFUNCTIONSAswepointedoutinChapter4,inprincipleitwillusuallybepossibletosolvethenecessaryconditionsofautilitymaximumfortheoptimallevelsofx1,x2,…,xn(andλ,theLagrangianmultiplier)asfunctionsofallpricesandincome.Mathematically,thiscanbeexpressedasndemandfunctionsoftheformx1¼x1ðp1,p2,…,pn,IÞ,x2¼x2ðp1,p2,…,pn,IÞ,…xn¼xnðp1,p2,…,pn,IÞ.(5.1)Ifthereareonlytwogoods,xandy(thecasewewillusuallybeconcernedwith),thisnotationcanbesimplifiedabitasx¼xðpx,py,IÞ,y¼yðpx,py,IÞ.(5.2)Onceweknowtheformofthesedemandfunctionsandthevaluesofallpricesandincome,wecan“predict”howmuchofeachgoodthispersonwillchoosetobuy.Thenotationstressesthatpricesandincomeare“exogenous”tothisprocess;thatis,theseareparametersoverwhichtheindividualhasnocontrolatthisstageoftheanalysis.Changesinthepa-rameterswill,ofcourse,shiftthebudgetconstraintandcausethispersontomakedifferentchoices.Thatquestionisthefocusofthischapterandthenext.Specifically,inthischapterwewillbelookingatthepartialderivatives∂x=∂Iand∂x=∂pxforanyarbitrarygoodx.Chapter6willcarrythediscussionfurtherbylookingat“cross-price”effectsoftheform∂x=∂pyforanyarbitrarypairofgoodsxandy.HomogeneityAﬁrstpropertyofdemandfunctionsrequireslittlemathematics.Ifweweretodoubleallpricesandincome(indeed,ifweweretomultiplythemallbyanypositiveconstant),thentheoptimalquantitiesdemandedwouldremainunchanged.Doublingallpricesandincomechangesonlytheunitsbywhichwecount,notthe“real”quantityofgoodsdemanded.This141
resultcanbeseeninanumberofways,althoughperhapstheeasiestisthroughagraphicapproach.ReferringbacktoFigures4.1and4.2,itisclearthatdoublingpx,py,andIdoesnotaffectthegraphofthebudgetconstraint.Hence,x,ywillstillbethecombinationthatischosen.Further,pxxþpyy¼Iisthesameconstraintas2pxxþ2pyy¼2I.Somewhatmoretechnically,wecanwritethisresultassayingthat,foranygoodxi,xi¼xiðp1,p2,…,pn,IÞ¼xiðtp1,tp2,…,tpn,tIÞ(5.3)foranyt>0.FunctionsthatobeythepropertyillustratedinEquation5.3aresaidtobehomogeneousofdegree0.1Hence,wehaveshownthatindividualdemandfunctionsarehomogeneousofdegree0inallpricesandincome.Changingallpricesandincomeinthesameproportionswillnotaffectthephysicalquantitiesofgoodsdemanded.Thisresultshowsthat(intheory)individuals’demandswillnotbeaffectedbya“pure”inflationduringwhichallpricesandincomesriseproportionally.Theywillcontinuetodemandthesamebundleofgoods.Ofcourse,ifaninflationwerenotpure(thatis,ifsomepricesrosemorerapidlythanothers),thiswouldnotbethecase.EXAMPLE5.1HomogeneityHomogeneityofdemandisadirectresultoftheutility-maximizationassumption.Demandfunctionsderivedfromutilitymaximizationwillbehomogeneousand,conversely,demandfunctionsthatarenothomogeneouscannotreﬂectutilitymaximization(unlesspricesenterdirectlyintotheutilityfunctionitself,astheymightforgoodswithsnobappeal).If,forexample,anindividual’sutilityforfoodðxÞandhousingðyÞisgivenbyutility¼Uðx,yÞ¼x0:3y0:7,(5.4)thenitisasimplematter(followingtheprocedureusedinExample4.1)toderivethedemandfunctionsx¼0:3Ipx,y¼0:7Ipy.(5.5)Thesefunctionsobviouslyexhibithomogeneity,sinceadoublingofallpricesandincomewouldleavexandyunaffected.Iftheindividual’spreferencesforxandywerereﬂectedinsteadbytheCESfunctionUðx,yÞ¼x0:5þy0:5,(5.6)then(asshowninExample4.2)thedemandfunctionsaregivenbyx¼11þpx=py !⋅Ipx,y¼11þpy=px !⋅Ipy.(5.7)Asbefore,bothofthesedemandfunctionsarehomogeneousofdegree0;adoublingofpx,py,andIwouldleavexandyunaffected.1Moregenerally,aswesawinChapters2and4,afunctionfðx1,x2,…,xnÞissaidtobehomogeneousofdegreekiffðtx1,tx2,…,txnÞ¼tkfðx1,x2,…,xnÞforanyt>0.Themostcommoncasesofhomogeneousfunctionsarek¼0andk¼1.Iffishomogeneousofdegree0,thendoublingallofitsargumentsleavesfunchangedinvalue.Iffishomogeneousofdegree1,thendoublingallofitsargumentswilldoublethevalueoff.142Part2ChoiceandDemand
QUERY:Dothedemandfunctionsderivedinthisexampleensurethattotalspendingonxandywillexhausttheindividual’sincomeforanycombinationofpx,py,andI?Canyouprovethatthisisthecase?CHANGESININCOMEAsaperson’spurchasingpowerrises,itisnaturaltoexpectthatthequantityofeachgoodpurchasedwillalsoincrease.ThissituationisillustratedinFigure5.1.AsexpendituresincreasefromI1toI2toI3,thequantityofxdemandedincreasesfromx1tox2tox3.Also,thequantityofyincreasesfromy1toy2toy3.NoticethatthebudgetlinesI1,I2,andI3areallparallel,reﬂectingthatonlyincomeischanging,nottherelativepricesofxandy.Becausetheratiopx=pystaysconstant,theutility-maximizingconditionsalsorequirethattheMRSstayconstantastheindividualmovestohigherlevelsofsatisfaction.TheMRSisthereforethesameatpoint(x3,y3)asat(x1,y1).NormalandinferiorgoodsInFigure5.1,bothxandyincreaseasincomeincreases—both∂x=∂Iand∂y=∂Iarepositive.Thismightbeconsideredtheusualsituation,andgoodsthathavethispropertyarecallednormalgoodsovertherangeofincomechangebeingobserved.FIGURE5.1EffectofanIncreaseinIncomeontheQuantitiesofxandyChosenAsincomeincreasesfromI1toI2toI3,theoptimal(utility-maximizing)choicesofxandyareshownbythesuccessivelyhigherpointsoftangency.Observethatthebudgetconstraintshiftsinaparallelwaybecauseitsslope(givenby−px=py)doesnotchange.Quantity of xQuantityof yy3U1U2U3U3I3U2I2U1I1y2y1x1x2x3Chapter5IncomeandSubstitutionEffects143
Forsomegoods,however,thequantitychosenmaydecreaseasincomeincreasesinsomeranges.Examplesofsuchgoodsarerotgutwhiskey,potatoes,andsecondhandclothing.Agoodzforwhich∂z=∂Iisnegativeiscalledaninferiorgood.ThisphenomenonisillustratedinFigure5.2.Inthisdiagram,thegoodzisinferiorbecause,forincreasesinincomeintherangeshown,lessofzisactuallychosen.Noticethatindifferencecurvesdonothavetobe“oddly”shapedinordertoexhibitinferiority;thecurvescorrespondingtogoodsyandzinFigure5.2continuetoobeytheassumptionofadiminishingMRS.Goodzisinferiorbecauseofthewayitrelatestotheothergoodsavailable(goodyhere),notbecauseofapeculiarityuniquetoit.Hence,wehavedevelopedthefollowingdeﬁnitions.DEFINITIONInferiorandnormalgoods.Agoodxiforwhich∂xi=∂I<0oversomerangeofincomechangesisaninferiorgoodinthatrange.If∂xi=∂I0oversomerangeofincomevariationthenthegoodisanormal(or“noninferior”)goodinthatrange.CHANGESINAGOOD’SPRICETheeffectofapricechangeonthequantityofagooddemandedismorecomplextoanalyzethanistheeffectofachangeinincome.Geometrically,thisisbecausechangingapriceinvolveschangingnotonlytheinterceptsofthebudgetconstraintbutalsoitsslope.Con-sequently,movingtothenewutility-maximizingchoiceentailsnotonlymovingtoanotherindifferencecurvebutalsochangingtheMRS.Therefore,whenapricechanges,twoanalyticallydifferenteffectscomeintoplay.Oneoftheseisasubstitutioneffect:evenifFIGURE5.2AnIndifferenceCurveMapExhibitingInferiorityInthisdiagram,goodzisinferiorbecausethequantitypurchasedactuallydeclinesasincomeincreases.Here,yisanormalgood(asitmustbeifthereareonlytwogoodsavailable),andpurchasesofyincreaseastotalexpendituresincrease.Quantity of zQuantityof yy3U3I3U2U1I2I1y2y1z3z2z1144Part2ChoiceandDemand theindividualweretostayonthesameindifferencecurve,consumptionpatternswouldbeallocatedsoastoequatetheMRStothenewpriceratio.Asecondeffect,theincomeeffect,arisesbecauseapricechangenecessarilychangesanindividual’s“real”income.Theindivid-ualcannotstayontheinitialindifferencecurveandmustmovetoanewone.Webeginbyanalyzingtheseeffectsgraphically.Thenwewillprovideamathematicaldevelopment.GraphicalanalysisofafallinpriceIncomeandsubstitutioneffectsareillustratedinFigure5.3.Thisindividualisinitiallymaximizingutility(subjecttototalexpenditures,I)byconsumingthecombinationx,y.FIGURE5.3DemonstrationoftheIncomeandSubstitutionEffectsofaFallinthePriceofxWhenthepriceofxfallsfromp1xtop2x,theutility-maximizingchoiceshiftsfromx,ytox,y.Thismovementcanbebrokendownintotwoanalyticallydifferenteffects:ﬁrst,thesubstitutioneffect,involvingamovementalongtheinitialindifferencecurvetopointB,wheretheMRSisequaltothenewpriceratio;andsecond,theincomeeffect,entailingamovementtoahigherlevelofutilitybecauserealincomehasincreased.Inthediagram,boththesubstitutionandincomeeffectscausemorextobeboughtwhenitspricedeclines.NoticethatpointI=pyisthesameasbeforethepricechange;thisisbecausepyhasnotchanged.PointI=pythereforeappearsonboththeoldandnewbudgetconstraints.Quantity of xQuantityof yU2U2U1U1x*y*xBx**y**SubstitutioneffectIncomeeffectTotal increasein xBI= px1x+ pyyI= p2xx+ pyyIpyChapter5IncomeandSubstitutionEffects145 TheinitialbudgetconstraintisI¼p1xxþpyy.Nowsupposethatthepriceofxfallstop2x.ThenewbudgetconstraintisgivenbytheequationI¼p2xxþpyyinFigure5.3.Itisclearthatthenewpositionofmaximumutilityisatx,y,wherethenewbudgetlineistangenttotheindifferencecurveU2.Themovementtothisnewpointcanbeviewedasbeingcomposedoftwoeffects.First,thechangeintheslopeofthebudgetconstraintwouldhavemotivatedamovetopointB,evenifchoiceshadbeenconﬁnedtothoseontheoriginalindifferencecurveU1.ThedashedlineinFigure5.3hasthesameslopeasthenewbudgetconstraint(I¼p2xxþpyy)butisdrawntobetangenttoU1becauseweareconceptuallyholding“real”income(thatis,utility)constant.Arelativelylowerpriceforxcausesamovefromx,ytoBifwedonotallowthisindividualtobemadebetteroffasaresultofthelowerprice.Thismovementisagraphicdemonstrationofthesubstitutioneffect.ThefurthermovefromBtotheoptimalpointx,yisanalyticallyidenticaltothekindofchangeexhibitedearlierforchangesinincome.Becausethepriceofxhasfallen,thispersonhasagreater“real”incomeandcanaffordautilitylevel(U2)thatisgreaterthanthatwhichcouldpreviouslybeattained.Ifxisanormalgood,moreofitwillbechoseninresponsetothisincreaseinpurchasingpower.Thisobservationexplainstheoriginofthetermincomeeffectforthemovement.Overallthen,theresultofthepricedeclineistocausemorextobedemanded.Itisimportanttorecognizethatthispersondoesnotactuallymakeaseriesofchoicesfromx,ytoBandthentox,y.WeneverobservepointB;onlythetwooptimalpositionsarereﬂectedinobservedbehavior.However,thenotionofincomeandsubstitutioneffectsisanalyticallyvaluablebecauseitshowsthatapricechangeaffectsthequantityofxthatisdemandedintwoconceptuallydifferentways.Wewillseehowthisseparationoffersmajorinsightsinthetheoryofdemand.GraphicalanalysisofanincreaseinpriceIfthepriceofgoodxweretoincrease,asimilaranalysiswouldbeused.InFigure5.4,thebudgetlinehasbeenshiftedinwardbecauseofanincreaseinthepriceofxfromp1xtop2x.Themovementfromtheinitialpointofutilitymaximization(x,y)tothenewpoint(x,y)canbedecomposedintotwoeffects.First,evenifthispersoncouldstayontheinitialin-differencecurve(U2),therewouldstillbeanincentivetosubstituteyforxandmovealongU2topointB.However,becausepurchasingpowerhasbeenreducedbytheriseinthepriceofx,heorshemustmovetoalowerlevelofutility.Thismovementisagaincalledtheincomeeffect.NoticeinFigure5.4thatboththeincomeandsubstitutioneffectsworkinthesamedirectionandcausethequantityofxdemandedtobereducedinresponsetoanincreaseinitsprice.EffectsofpricechangesforinferiorgoodsSofarwehaveshownthatsubstitutionandincomeeffectstendtoreinforceoneanother.Forapricedecline,bothcausemoreofthegoodtobedemanded,whereasforapriceincrease,bothcauselesstobedemanded.Althoughthisanalysisisaccurateforthecaseofnormal(noninferior)goods,thepossibilityofinferiorgoodscomplicatesthestory.Inthiscase,incomeandsubstitutioneffectsworkinoppositedirections,andthecombinedresultofapricechangeisindeterminate.Afallinprice,forexample,willalwayscauseanindividualtotendtoconsumemoreofagoodbecauseofthesubstitutioneffect.Butifthegoodisinferior,theincreaseinpurchasingpowercausedbythepricedeclinemaycauselessofthegoodtobebought.Theresultisthereforeindeterminate:thesubstitutioneffecttendstoincreasethequantityoftheinferiorgoodbought,whereasthe(perverse)incomeeffecttendstoreducethisquantity.Unlikethesituationfornormalgoods,itisnotpossibleheretopredicteventhedirectionoftheeffectofachangeinpxonthequantityofxconsumed.146Part2ChoiceandDemand Giffen’sparadoxIftheincomeeffectofapricechangeisstrongenough,thechangeinpriceandtheresultingchangeinthequantitydemandedcouldactuallymoveinthesamedirection.LegendhasitthattheEnglisheconomistRobertGiffenobservedthisparadoxinnineteenth-centuryIre-land:whenthepriceofpotatoesrose,peoplereportedlyconsumedmoreofthem.Thispeculiarresultcanbeexplainedbylookingatthesizeoftheincomeeffectofachangeinthepriceofpotatoes.Potatoeswerenotonlyinferiorgoods,theyalsousedupalargeportionoftheIrishpeople’sincome.Anincreaseinthepriceofpotatoesthereforereducedrealincomesubstantially.TheIrishwereforcedtocutbackonotherluxuryfoodconsumptioninordertobuymorepotatoes.Eventhoughthisrenderingofeventsishistoricallyimplausible,theFIGURE5.4DemonstrationoftheIncomeandSubstitutionEffectsofanIncreaseinthePriceofxWhenthepriceofxincreases,thebudgetconstraintshiftsinward.Themovementfromtheinitialutility-maximizingpoint(x,y)tothenewpoint(x,y)canbeanalyzedastwoseparateeffects.ThesubstitutioneffectwouldbedepictedasamovementtopointBontheinitialindifferencecurve(U2).Thepriceincrease,however,wouldcreatealossofpurchasingpowerandaconsequentmovementtoalowerindifferencecurve.Thisistheincomeeffect.Inthediagram,boththeincomeandsubstitutioneffectscausethequantityofxtofallasaresultoftheincreaseinitsprice.Again,thepointI=pyisnotaffectedbythechangeinthepriceofx.Total reductionin xSubstitutioneffectIncomeeffectQuantity of xQuantityof yx*xBx**y*y**IpyU2U2U1U1BI= px2x+ pyyI= px1x+ pyyChapter5IncomeandSubstitutionEffects147 possibilityofanincreaseinthequantitydemandedinresponsetoanincreaseinthepriceofagoodhascometobeknownasGiffen’sparadox.2LaterwewillprovideamathematicalanalysisofhowGiffen’sparadoxcanoccur.AsummaryHence,ourgraphicalanalysisleadstothefollowingconclusions.OPTIMIZATIONPRINCIPLESubstitutionandincomeeffects.Theutility-maximizationhypothesissuggeststhat,fornormalgoods,afallinthepriceofagoodleadstoanincreaseinquantitypurchasedbecause:(1)thesubstitutioneffectcausesmoretobepurchasedastheindividualmovesalonganindifferencecurve;and(2)theincomeeffectcausesmoretobepurchasedbecausethepricedeclinehasincreasedpurchasingpower,therebypermittingmovementtoahigherindiffer-encecurve.Whenthepriceofanormalgoodrises,similarreasoningpredictsadeclineinthequantitypurchased.Forinferiorgoods,substitutionandincomeeffectsworkinoppositedirections,andnodeﬁnitepredictionscanbemade.THEINDIVIDUAL’SDEMANDCURVEEconomistsfrequentlywishtographdemandfunctions.Itwillcomeasnosurprisetoyouthatthesegraphsarecalled“demandcurves.”Understandinghowsuchwidelyusedcurvesrelatetounderlyingdemandfunctionsprovidesadditionalinsightstoeventhemostfunda-mentalofeconomicarguments.Tosimplifythedevelopment,assumethereareonlytwogoodsandthat,asbefore,thedemandfunctionforgoodxisgivenbyx¼xðpx,py,IÞ.Thedemandcurvederivedfromthisfunctionlooksattherelationshipbetweenxandpxwhileholdingpy,_I,andpreferencesconstant.Thatis,itshowstherelationshipx¼xðpx,_py,_IÞ,(5.8)wherethebarsoverpyandIindicatethatthesedeterminantsofdemandarebeingheldconstant.ThisconstructionisshowninFigure5.5.Thegraphshowsutility-maximizingchoicesofxandyasthisindividualispresentedwithsuccessivelylowerpricesofgoodx(whileholdingpyandIconstant).Weassumethatthequantitiesofxchosenincreasefromx0tox00tox000asthatgood’spricefallsfromp0xtop00xtop000x.Suchanassumptionisinaccordwithourgeneralconclusionthat,exceptintheunusualcaseofGiffen’sparadox,∂x=∂pxisnegative.InFigure5.5b,informationabouttheutility-maximizingchoicesofgoodxistrans-ferredtoademandcurvewithpxontheverticalaxisandsharingthesamehorizontalaxisasFigure5.5a.Thenegativeslopeofthecurveagainreﬂectstheassumptionthat∂x=∂pxisnegative.Hence,wemaydeﬁneanindividualdemandcurveasfollows.DEFINITIONIndividualdemandcurve.Anindividualdemandcurveshowstherelationshipbetweenthepriceofagoodandthequantityofthatgoodpurchasedbyanindividual,assumingthatallotherdeterminantsofdemandareheldconstant.2AmajorproblemwiththisexplanationisthatitdisregardsMarshall’sobservationthatbothsupplyanddemandfactorsmustbetakenintoaccountwhenanalyzingpricechanges.IfpotatopricesincreasedbecauseofthepotatoblightinIreland,thensupplyshouldhavebecomesmaller,sohowcouldmorepotatoespossiblyhavebeenconsumed?Also,sincemanyIrishpeoplewerepotatofarmers,thepotatopriceincreaseshouldhaveincreasedrealincomeforthem.Foradetaileddiscussionoftheseandotherfascinatingbitsofpotatolore,seeG.P.DwyerandC.M.Lindsey,“RobertGiffenandtheIrishPotato,”AmericanEconomicReview(March1984):188–92.148Part2ChoiceandDemand FIGURE5.5ConstructionofanIndividual’sDemandCurveIn(a),theindividual’sutility-maximizingchoicesofxandyareshownforthreedifferentpricesofx(p0x,p00x,andp000x).In(b),thisrelationshipbetweenpxandxisusedtoconstructthedemandcurveforx.Thedemandcurveisdrawnontheassumptionthatpy,I,andpreferencesremainconstantaspxvaries.Quantity of x per periodQuantityof yperperiodI= px′ x+ pyyI= px″ x+ pyyI= px″‴ x+ pyyx‴x″x′U2U3U1I/py(a) Individual’s indifference curve mapQuantity of x per periodpxx(px,py,I)(b) Demand curvex‴x″x′px′ px″px‴ ThedemandcurveillustratedinFigure5.5staysinaﬁxedpositiononlysolongasallotherdeterminantsofdemandremainunchanged.Ifoneoftheseotherfactorsweretochangethenthecurvemightshifttoanewposition,aswenowdescribe.ShiftsinthedemandcurveThreefactorswereheldconstantinderivingthisdemandcurve:(1)income;(2)pricesofothergoods(say,py);and(3)theindividual’spreferences.Ifanyoftheseweretochange,theentiredemandcurvemightshifttoanewposition.Forexample,ifIweretoincrease,thecurvewouldshiftoutward(providedthat∂x=∂I>0,thatis,providedthegoodisa“normal”goodoverthisincomerange).Morexwouldbedemandedateachprice.Ifanotherprice(say,py)weretochangethenthecurvewouldshiftinwardoroutward,dependingpreciselyonhowxandyarerelated.Inthenextchapterwewillexaminethatrelationshipindetail.Finally,thecurvewouldshiftiftheindividual’spreferencesforgoodxweretochange.AsuddenadvertisingblitzbytheMcDonald’sCorporationmightshiftthedemandforham-burgersoutward,forexample.Asthisdiscussionmakesclear,onemustrememberthatthedemandcurveisonlyatwo-dimensionalrepresentationofthetruedemandfunction(Equation5.8)andthatitisstableonlyifotherthingsdostayconstant.Itisimportanttokeepclearlyinmindthedifferencebetweenamovementalongagivendemandcurvecausedbyachangeinpxandashiftintheentirecurvecausedbyachangeinincome,inoneoftheotherprices,orinpreferences.Traditionally,thetermanincreaseindemandisreservedforanoutwardshiftinthedemandcurve,whereasthetermanincreaseinthequantitydemandedreferstoamovementalongagivencurvecausedbyachangeinpx.EXAMPLE5.2DemandFunctionsandDemandCurvesTobeabletographademandcurvefromagivendemandfunction,wemustassumethatthepreferencesthatgeneratedthefunctionremainstableandthatweknowthevaluesofincomeandotherrelevantprices.IntheﬁrstcasestudiedinExample5.1,wefoundthatx¼0:3Ipx(5.9)andy¼0:7Ipy.Ifpreferencesdonotchangeandifthisindividual’sincomeis$100,thesefunctionsbecomex¼30px,y¼70py,(5.10)orpxx¼30,pyy¼70,whichmakesclearthatthedemandcurvesforthesetwogoodsaresimplehyperbolas.Ariseinincomewouldshiftbothofthedemandcurvesoutward.Noticealso,inthiscase,thatthedemandcurveforxisnotshiftedbychangesinpyandviceversa.150Part2ChoiceandDemand
ForthesecondcaseexaminedinExample5.1,theanalysisismorecomplex.Forgoodx,weknowthatx¼11þpx=py !⋅Ipx,(5.11)sotographthisinthepx–xplanewemustknowbothIandpy.IfweagainassumeI¼100andletpy¼1,thenEquation5.11becomesx¼100p2xþpx,(5.12)which,whengraphed,wouldalsoshowageneralhyperbolicrelationshipbetweenpriceandquantityconsumed.InthiscasethecurvewouldberelativelyflatterbecausesubstitutioneffectsarelargerthanintheCobb-Douglascase.FromEquation5.11,wealsoknowthat∂x∂I¼11þpx=py !⋅1px>0(5.13)and∂x∂py¼IðpxþpyÞ2>0,soincreasesinIorpywouldshiftthedemandcurveforgoodxoutward.QUERY:HowwouldthedemandfunctionsinEquations5.10changeifthispersonspenthalfofhisorherincomeoneachgood?Showthatthesedemandfunctionspredictthesamexconsumptionatthepointpx¼1,py¼1,I¼100asdoesEquation5.11.UseanumericalexampletoshowthattheCESdemandfunctionismoreresponsivetoanincreaseinpxthanistheCobb-Douglasdemandfunction.COMPENSATEDDEMANDCURVESInFigure5.5,thelevelofutilitythispersongetsvariesalongthedemandcurve.Aspxfalls,heorsheismadeincreasinglybetter-off,asshownbytheincreaseinutilityfromU1toU2toU3.Thereasonthishappensisthatthedemandcurveisdrawnontheassumptionthatnominalincomeandotherpricesareheldconstant;hence,adeclineinpxmakesthispersonbetteroffbyincreasinghisorherrealpurchasingpower.Althoughthisisthemostcommonwaytoimposetheceterisparibusassumptionindevelopingademandcurve,itisnottheonlyway.Analternativeapproachholdsrealincome(orutility)constantwhileexaminingreactionstochangesinpx.ThederivationisillustratedinFigure5.6,whereweholdutilityconstant(atU2)whilesuccessivelyreducingpx.Aspxfalls,theindividual’snominalincomeiseffectivelyreduced,thuspreventinganyincreaseinutility.Inotherwords,theeffectsofthepricechangeonpurchasingpowerare“compensated”soastoconstraintheindividualtoremainonU2.Reactionstochangingpricesincludeonlysubstitutioneffects.Ifwewereinsteadtoexamineeffectsofincreasesinpx,incomecompensationwouldbepositive:Thisindividual’sincomewouldhavetobeincreasedtopermithimorhertostayontheU2indifferencecurveinresponsetothepricerises.Wecansummarizetheseresultsasfollows.DEFINITIONCompensateddemandcurve.Acompensateddemandcurveshowstherelationshipbe-tweenthepriceofagoodandthequantitypurchasedontheassumptionthatotherpricesandutilityareheldconstant.Thecurve(whichissometimestermeda“Hicksian”demandcurveChapter5IncomeandSubstitutionEffects151
aftertheBritisheconomistJohnHicks)thereforeillustratesonlysubstitutioneffects.Mathe-matically,thecurveisatwo-dimensionalrepresentationofthecompensateddemandfunctionx¼xcðpx,py,UÞ.(5.14)RelationshipbetweencompensatedanduncompensateddemandcurvesThisrelationshipbetweenthetwodemandcurveconceptsisillustratedinFigure5.7.Atp00xthecurvesintersect,becauseatthatpricetheindividual’sincomeisjustsufﬁcienttoattainFIGURE5.6ConstructionofaCompensatedDemandCurveThecurvexcshowshowthequantityofxdemandedchangeswhenpxchanges,holdingpyandutilityconstant.Thatis,theindividual’sincomeis“compensated”soastokeeputilityconstant.Hence,xcreﬂectsonlysubstitutioneffectsofchangingprices.px′px″x‴x″x*Quantity of xQuantity of xQuantityof ypySlope= – pySlope= – px‴pySlope= – U2pxxc(px,py,U)(a) Individual’s indifference curve map(b) Compensated demand curvex″x*x**px′ px″px‴152Part2ChoiceandDemand
utilitylevelU2(compareFigures5.5andFigure5.6).Hence,x00isdemandedundereitherdemandconcept.Forpricesbelowp00x,however,theindividualsuffersacompensatingreductioninincomeonthecurvexcthatpreventsanincreaseinutilityfromthelowerprice.Hence,assumingxisanormalgood,itfollowsthatlessxisdemandedatp000xalongxcthanalongtheuncompensatedcurvex.Alternatively,forapriceabovep00x(suchasp0x),incomecompensationispositivebecausetheindividualneedssomehelptoremainonU2.Hence,againassumingxisanormalgood,atp0xmorexisdemandedalongxcthanalongx.Ingeneral,then,foranormalgoodthecompensateddemandcurveissomewhatlessresponsivetopricechangesthanistheuncompensatedcurve.Thisisbecausethelatterreﬂectsbothsubstitutionandincomeeffectsofpricechanges,whereasthecompensatedcurvereﬂectsonlysubstitutioneffects.Thechoicebetweenusingcompensatedoruncompensateddemandcurvesineconomicanalysisislargelyamatterofconvenience.Inmostempiricalwork,uncompensatedcurves(whicharesometimescalled“Marshalliandemandcurves”)areusedbecausethedataonpricesandnominalincomesneededtoestimatethemarereadilyavailable.IntheExtensionstoChapter12wewilldescribesomeoftheseestimatesandshowhowtheymightbeem-ployedforpracticalpolicypurposes.Forsometheoreticalpurposes,however,compensateddemandcurvesareamoreappropriateconceptbecausetheabilitytoholdutilityconstantofferssomeadvantages.Ourdiscussionof“consumersurplus”laterinthischapteroffersoneillustrationoftheseadvantages.FIGURE5.7ComparisonofCompensatedandUncompensatedDemandCurvesThecompensated(xc)anduncompensated(x)demandcurvesintersectatp00xbecausex00isdemandedundereachconcept.Forpricesabovep00x,theindividual’sincomeisincreasedwiththecompensateddemandcurve,somorexisdemandedthanwiththeuncompensatedcurve.Forpricesbelowp00x,incomeisreducedforthecompensatedcurve,solessxisdemandedthanwiththeuncompensatedcurve.Thestandarddemandcurveisﬂatterbecauseitincorporatesbothsubstitu-tionandincomeeffectswhereasthecurvexcreﬂectsonlysubstitutioneffects.Quantity of xx*x**pxx(px,py,I)xc(px,py,U)x′x″x‴px′ px″px‴Chapter5IncomeandSubstitutionEffects153
EXAMPLE5.3CompensatedDemandFunctionsInExample3.1weassumedthattheutilityfunctionforhamburgers(y)andsoftdrinks(x)wasgivenbyutility¼Uðx,yÞ¼x0:5y0:5,(5.15)andinExample4.1weshowedthatwecancalculatetheMarshalliandemandfunctionsforsuchutilityfunctionsasx¼αIpx¼I2px,y¼βIpy¼I2py.(5.16)Also,inExample4.3wecalculatedtheindirectutilityfunctionbycombiningEquations5.15and5.16asutility¼VðI,px,pyÞ¼I2p0:5xp0:5y.(5.17)Toobtainthecompensateddemandfunctionsforxandy,wesimplyuseEquation5.17tosolveforIandthensubstitutethisexpressioninvolvingVintoEquations5.16.Thispermitsustointerchangeincomeandutilitysowemayholdthelatterconstant,asisrequiredforthecompensateddemandconcept.Makingthesesubstitutionsyieldsx¼Vp0:5yp0:5x,y¼Vp0:5xp0:5y.(5.18)Thesearethecompensateddemandfunctionsforxandy.Noticethatnowdemanddependsonutility(V)ratherthanonincome.Holdingutilityconstant,itisclearthatincreasesinpxreducethedemandforx,andthisnowreflectsonlythesubstitutioneffect(seealsoExample5.4).Althoughpydidnotenterintotheuncompensateddemandfunctionforgoodx,itdoesenterintothecompensatedfunction:increasesinpyshiftthecompensateddemandcurveforxoutward.Thetwodemandconceptsagreeattheassumedinitialpointpx¼1,py¼4,I¼8,andV¼2;Equations5.16predictx¼4,y¼1atthispoint,asdoEquations5.18.Forpx>1orpx<1,thedemandsdifferunderthetwoconcepts,however.If,say,px¼4,thentheuncompensatedfunctions(Equations5.16)predictx¼1,y¼1,whereasthecompensatedfunctions(Equations5.18)predictx¼2,y¼2.Thereductioninxresultingfromtheriseinitspriceissmallerwiththecompensateddemandfunctionthanitiswiththeuncompensatedfunctionbecausetheformerconceptadjustsforthenegativeeffectonpurchasingpowerthatcomesaboutfromthepricerise.Thisexamplemakesclearthedifferentceterisparibusassumptionsinherentinthetwodemandconcepts.Withuncompensateddemand,expendituresareheldconstantatI¼2andsotheriseinpxfrom1to4resultsinalossofutility;inthiscase,utilityfallsfrom2to1.Inthecompensateddemandcase,utilityisheldconstantatV¼2.Tokeeputilityconstant,expendituresmustrisetoE¼1ð2Þþ1ð2Þ¼4inordertooffsettheeffectsofthepricerise(seeEquation5.17).QUERY:ArethecompensateddemandfunctionsgiveninEquations5.18homogeneousofdegree0inpxandpyifutilityisheldconstant?Wouldyouexpectthattobetrueforallcompensateddemandfunctions?154Part2ChoiceandDemand AMATHEMATICALDEVELOPMENTOFRESPONSETOPRICECHANGESUptothispointwehavelargelyreliedongraphicaldevicestodescribehowindividualsrespondtopricechanges.Additionalinsightsareprovidedbyamoremathematicalapproach.Ourbasicgoalistoexaminethepartialderivative∂x=∂px—thatis,howachangeinthepriceofagoodaffectsitspurchase,ceterisparibus.Inthenextchapter,wetakeupthequestionofhowchangesinthepriceofonecommodityaffectpurchasesofanothercommodity.DirectapproachOurgoalistousetheutility-maximizationmodeltolearnsomethingabouthowthedemandforgoodxchangeswhenpxchanges;thatis,wewishtocalculate∂x=∂px.Thedirectapproachtothisproblemmakesuseoftheﬁrst-orderconditionsforutilitymaximization(Equations4.8).Differentiationofthesenþ1equationsyieldsanewsystemofnþ1equations,whicheventuallycanbesolvedforthederivativeweseek.3Unfortunately,obtain-ingthissolutionisquitecumbersomeandthestepsrequiredyieldlittleinthewayofeconomicinsights.Hence,wewillinsteadadoptanindirectapproachthatreliesontheconceptofduality.Intheend,bothapproachesyieldthesameconclusion,buttheindirectapproachismuchricherintermsoftheeconomicsitcontains.IndirectapproachTobeginourindirectapproach,4wewillassume(asbefore)thereareonlytwogoods(xandy)andfocusonthecompensateddemandfunction,xcðpx,py,UÞ,introducedinEquation5.14.Wenowwishtoillustratetheconnectionbetweenthisdemandfunctionandtheordinarydemandfunction,xðpx,py,IÞ.InChapter4weintroducedtheexpenditurefunction,whichrecordstheminimalexpenditurenecessarytoattainagivenutilitylevel.Ifwedenotethisfunctionbyminimumexpenditure¼Eðpx,py,UÞ(5.19)then,bydefinition,xcðpx,py,UÞ¼x½px,py,Eðpx,py,UÞ.(5.20)ThisconclusionwasalreadyintroducedinconnectionwithFigure5.7,whichshowedthatthequantitydemandedisidenticalforthecompensatedanduncompensateddemandfunctionswhenincomeisexactlywhatisneededtoattaintherequiredutilitylevel.Equation5.20isobtainedbyinsertingthatexpenditurelevelintothedemandfunction,xðpx,py,IÞ.NowwecanproceedbypartiallydifferentiatingEquation5.20withrespecttopxandrecognizingthatthisvariableentersintotheordinarydemandfunctionintwoplaces.Hence∂xc∂px¼∂x∂pxþ∂x∂E⋅∂E∂px,(5.21)andrearrangingtermsyields∂x∂px¼∂xc∂px∂x∂E⋅∂E∂px.(5.22)3See,forexample,PaulA.Samuelson,FoundationsofEconomicAnalysis(Cambridge,MA:HarvardUniversityPress,1947),pp.101–3.4ThefollowingproofisadaptedfromPhillipJ.Cook,“A‘OneLine’ProofoftheSlutskyEquation,”AmericanEconomicReview62(March1972):139.Chapter5IncomeandSubstitutionEffects155 ThesubstitutioneffectConsequently,thederivativeweseekhastwoterms.Interpretationoftheﬁrsttermisstraightforward:Itistheslopeofthecompensateddemandcurve.Butthatsloperepresentsmovementalongasingleindifferencecurve;itis,infact,whatwecalledthe“substitutioneffect”earlier.TheﬁrsttermontherightofEquation5.22isamathematicalrepresentationofthateffect.TheincomeeffectThesecondterminEquation5.22reﬂectsthewayinwhichchangesinpxaffectthedemandforxthroughchangesinnecessaryexpenditurelevels(thatis,changesinpurchasingpower).Thistermthereforereﬂectstheincomeeffect.ThenegativesigninEquation5.22showsthedirectionoftheeffect.Forexample,anincreaseinpxincreasestheexpenditurelevelthatwouldhavebeenneededtokeeputilityconstant(mathematically,∂E=∂px>0).ButbecausenominalincomeisheldconstantinMarshalliandemand,theseextraexpendituresarenotavailable.Hencex(andy)mustbereducedtomeetthisshortfall.Theextentofthere-ductioninxisgivenby∂x=∂E.Ontheotherhand,ifpxfalls,theexpenditurelevelrequiredtoattainagivenutilityalsofalls.Thedeclineinxthatwouldnormallyaccompanysuchafallinexpendituresispreciselytheamountthatmustbeaddedbackthroughtheincomeeffect.Noticethatinthiscasetheincomeeffectworkstoincreasex.TheSlutskyequationTherelationshipsembodiedinEquation5.22wereﬁrstdiscoveredbytheRussianeconomistEugenSlutskyinthelatenineteenthcentury.AslightchangeinnotationisrequiredtostatetheresultthewaySlutskydid.First,wewritethesubstitutioneffectassubstitutioneffect¼∂xc∂px¼∂x∂pxU¼constant(5.23)toindicatemovementalongasingleindifferencecurve.Fortheincomeeffect,wehaveincomeeffect¼∂x∂E⋅∂E∂px¼∂x∂I⋅∂E∂px,(5.24)becausechangesinincomeorexpendituresamounttothesamethinginthefunctionxðpx,py,I).Thesecondtermintheincomeeffectcanbestudiedmostdirectlybyusingtheenvelopetheorem.Rememberthatexpenditurefunctionsrepresentaminimizationprobleminwhichtheexpenditurerequiredtoreachaminimumlevelofutilityisminimized.TheLagrangianexpressionforthisminimizationisℒ¼pxxþpyyþλ½_UUðx,yÞ.Applyingtheenvelopetheoremtothisproblemyields∂E∂px¼∂ℒ∂px¼x.(5.25)Inwords,theenvelopetheoremshowsthatpartialdifferentiationoftheexpenditurefunc-tionwithrespecttoagood’spriceyieldsthedemandfunctionforthatgood.Becauseutilityisheldconstantintheexpenditurefunction,thisdemandfunctionwillbeacompensatedone.Thisresult,andasimilaroneinthetheoryofthefirm,isusuallycalledShephard’slemmaaftertheeconomistwhofirststudiedthisapproachtodemandtheoryindetail.Theresultisextremelyusefulinboththeoreticalandappliedmicroeconomics;partialdif-ferentiationofmaximizedorminimizedfunctionsisoftentheeasiestwaytoderivedemand156Part2ChoiceandDemand
functions.5Noticealsothattheresultmakesintuitivesense.Ifweaskhowmuchextraexpenditureisnecessarytocompensateforariseinthepriceofgoodx,asimpleappro-ximationwouldbegivenbythenumberofunitsofxcurrentlybeingconsumed.BycombiningEquations5.23–5.25,wecanarriveatthefollowingcompletestatementoftheresponsetoapricechange.OPTIMIZATIONPRINCIPLESlutskyequation.Theutility-maximizationhypothesisshowsthatthesubstitutionandincomeeffectsarisingfromapricechangecanberepresentedby∂x∂px¼substitutioneffectþincomeeffect,(5.26)or∂x∂px¼∂x∂pxU¼constantx∂x∂I.(5.27)TheSlutskyequationallowsamoredeﬁnitivetreatmentofthedirectionandsizeofsubstitu-tionandincomeeffectsthanwaspossiblewithagraphicanalysis.First,thesubstitutioneffectð∂x=∂pxjU¼constantÞisalwaysnegativeaslongastheMRSisdiminishing.Afall(rise)inpxreduces(increases)px=py,andutilitymaximizationrequiresthattheMRSfall(rise)too.Butthiscanoccuralonganindifferencecurveonlyifxincreases(or,inthecaseofariseinpx,ifxdecreases).Hence,insofarasthesubstitutioneffectisconcerned,priceandquantityalwaysmoveinoppositedirections.Equivalently,theslopeofthecompensateddemandcurvemustbenegative.6Wewillshowthisresultinasomewhatdifferentwayintheﬁnalsectionofthischapter.Thesignoftheincomeeffect(x∂x=∂I)dependsonthesignof∂x=∂I.Ifxisanormalgood,then∂x=∂Iispositiveandtheentireincomeeffect,likethesubstitutioneffect,isnegative.Thus,fornormalgoods,priceandquantityalwaysmoveinoppositedirections.Forexample,afallinpxraisesrealincomeand,becausexisanormalgood,purchasesofxrise.Similarly,ariseinpxreducesrealincomeandsopurchasesofxfall.Overall,then,aswedescribedpreviouslyusingagraphicanalysis,substitutionandincomeeffectsworkinthesamedirectiontoyieldanegativelyslopeddemandcurve.Inthecaseofaninferiorgood,∂x=∂I<0andthetwotermsinEquation5.27wouldhavedifferentsigns.Itisatleasttheoreticallypossiblethat,inthiscase,thesecondtermcoulddominatetheﬁrst,leadingtoGiffen’sparadox(∂x=∂px>0).EXAMPLE5.4ASlutskyDecompositionThedecompositionofapriceeffectthatwasﬁrstdiscoveredbySlutskycanbenicelyillus-tratedwiththeCobb-Douglasexamplestudiedpreviously.InExample5.3,wefoundthattheMarshalliandemandfunctionforgoodxwasxðpx,py,IÞ¼0:5Ipx(5.28)(continued)5Forinstance,inExample4.4,forexpenditurewefoundasimpleCobb-DouglasutilityfunctionoftheformEðpx,py,VÞ¼2Vp0:5xp0:5y.Hence,fromShephard’slemmaweknowthatx¼∂E=∂px¼Vp0:5xp0:5y,whichisthesameresultweobtainedinExample5.3.6ItispossiblethatsubstitutioneffectswouldbezeroifindifferencecurveshaveanL-shape(implyingthatxandyareusedinfixedproportions).SomeexamplesareprovidedintheChapter5problems.Chapter5IncomeandSubstitutionEffects157
EXAMPLE5.4CONTINUEDandthattheHicksian(compensated)demandfunctionwasxcðpx,py,VÞ¼Vp0:5yp0:5x.(5.29)TheoveralleffectofapricechangeonthedemandforgoodxcanbefoundbydifferentiatingtheMarshalliandemandfunction:∂x∂px¼0:5Ip2x.(5.30)NowwewishtoshowthatthiseffectisthesumofthetwoeffectsthatSlutskyidentified.Asbefore,thesubstitutioneffectisfoundbydifferentiatingthecompensateddemandfunction:substitutioneffect¼∂xc∂px¼0:5Vp0:5yp1:5x.(5.31)Wecaneliminateindirectutility,V,bysubstitutionfromEquation5.17:substitutioneffect¼0:5ð0:5Ip0:5xp0:5yÞp0:5yp1:5x¼0:25Ip2x.(5.32)Calculationoftheincomeeffectinthisexampleisconsiderablyeasier.ApplyingtheresultsfromEquation5.27,wehaveincomeeffect¼x∂x∂I¼0:5Ipx⋅0:5px¼0:25Ip2x.(5.33)AcomparisonofEquation5.30withEquations5.32and5.33showsthatwehaveindeeddecomposedthepricederivativeofthisdemandfunctionintosubstitutionandincomecomponents.Interestingly,thesubstitutionandincomeeffectsareofpreciselythesamesize.This,aswewillseeinlaterexamples,isoneofthereasonsthattheCobb-Douglasisaveryspecialcase.Thewell-wornnumericalexamplewehavebeenusingalsodemonstratesthisdecomposi-tion.Whenthepriceofxrisesfrom$1to$4,the(uncompensated)demandforxfallsfromx¼4tox¼1butthecompensateddemandforxfallsonlyfromx¼4tox¼2.Thatdeclineof50percentisthesubstitutioneffect.Thefurther50percentfallfromx¼2tox¼1rep-resentsreactionstothedeclineinpurchasingpowerincorporatedintheMarshalliandemandfunction.Thisincomeeffectdoesnotoccurwhenthecompensateddemandnotionisused.QUERY:Inthisexample,theindividualspendshalfofhisorherincomeongoodxandhalfongoody.HowwouldtherelativesizesofthesubstitutionandincomeeffectsbealterediftheexponentsoftheCobb-Douglasutilityfunctionwerenotequal?DEMANDELASTICITIESSofarinthischapterwehavebeenexamininghowindividualsrespondtochangesinpricesandincomebylookingatthederivativesofthedemandfunction.Formanyanalyticalquestionsthisisagoodwaytoproceedbecausecalculusmethodscanbedirectlyapplied.However,aswepointedoutinChapter2,focusingonderivativeshasonemajordisadvantageforempiricalwork:thesizesofderivativesdependdirectlyonhowvariablesaremeasured.158Part2ChoiceandDemand
Thatcanmakecomparisonsamonggoodsoracrosscountriesandtimeperiodsverydifﬁcult.Forthisreason,mostempiricalworkinmicroeconomicsusessomeformofelasticitymeasure.Inthissectionweintroducethethreemostcommontypesofdemandelasticitiesandexploresomeofthemathematicalrelationsamongthem.Again,forsimplicitywewilllookatasituationwheretheindividualchoosesbetweenonlytwogoods,thoughtheseideascanbeeasilygeneralized.MarshalliandemandelasticitiesMostofthecommonlyuseddemandelasticitiesarederivedfromtheMarshalliandemandfunctionxðpx,py,IÞ.Speciﬁcally,thefollowingdeﬁnitionsareused.DEFINITION1.Priceelasticityofdemandðex,pxÞ.Thismeasurestheproportionatechangeinquantitydemandedinresponsetoaproportionatechangeinagood’sownprice.Math-ematically,ex,px¼∆x=x∆px=px¼∆x∆px⋅pxx¼∂x∂px⋅pxx.(5.34)2.Incomeelasticityofdemandðex,IÞ.Thismeasurestheproportionatechangeinquan-titydemandedinresponsetoaproportionatechangeinincome.Inmathematicalterms,ex,I¼∆x=x∆I=I¼∆x∆I⋅Ix¼∂x∂I⋅Ix.(5.35)3.Cross-priceelasticityofdemandðex,pyÞ.Thismeasurestheproportionatechangeinthequantityofxdemandedinresponsetoaproportionatechangeinthepriceofsomeothergood(y):ex,py¼∆x=x∆py=py¼∆x∆py⋅pyx¼∂x∂py⋅pyx.(5.36)Noticethatallofthesedeﬁnitionsusepartialderivatives,whichsigniﬁesthatallotherdeterminantsofdemandaretobeheldconstantwhenexaminingtheimpactofaspeciﬁcvariable.Intheremainderofthissectionwewillexploretheown-priceelasticitydeﬁnitioninsomedetail.Examiningthecross-priceelasticityofdemandistheprimarytopicofChapter6.PriceelasticityofdemandThe(own-)priceelasticityofdemandisprobablythemostimportantelasticityconceptinallofmicroeconomics.Notonlydoesitprovideaconvenientwayofsummarizinghowpeoplerespondtopricechangesforawidevarietyofeconomicgoods,butitisalsoacentralconceptinthetheoryofhowﬁrmsreacttothedemandcurvesfacingthem.Asyouprobablyalreadylearnedinearliereconomicscourses,adistinctionisusuallymadebetweencasesofelasticdemand(wherepriceaffectsquantitysigniﬁcantly)andinelasticdemand(wheretheeffectofpriceissmall).Onemathematicalcomplicationinmakingtheseideaspreciseisthatthepriceelasticityofdemanditselfisnegative7because,exceptintheunlikelycaseofGiffen’spara-dox,∂x=∂pxisnegative.Thedividinglinebetweenlargeandsmallresponsesisgenerallyset7Sometimeseconomistsusetheabsolutevalueofthepriceelasticityofdemandintheirdiscussions.Althoughthisismathematicallyincorrect,suchusageisquitecommon.Forexample,astudythatfindsthatex,px¼1:2maysometimesreportthepriceelasticityofdemandas“1.2.”Wewillnotdosohere,however.Chapter5IncomeandSubstitutionEffects159
at1.Ifex,px¼1,changesinxandpxareofthesameproportionatesize.Thatis,a1percentincreaseinpriceleadstoafallof1percentinquantitydemanded.Inthiscase,demandissaidtobe“unit-elastic.”Alternatively,ifex,px<1,thenquantitychangesareproportionatelylargerthanpricechangesandwesaythatdemandis“elastic.”Forexample,ifex,px¼3,each1percentriseinpriceleadstoafallof3percentinquantitydemanded.Finally,ifex,px>1thendemandisinelasticandquantitychangesarepropor-tionatelysmallerthanpricechanges.Avalueofex,px¼0:3,forexample,meansthata1percentincreaseinpriceleadstoafallinquantitydemandedof0.3percent.InChapter12wewillseehowaggregatedataareusedtoestimatethetypicalindividual’spriceelasticityofdemandforagoodandhowsuchestimatesareusedinavarietyofquestionsinappliedmicroeconomics.PriceelasticityandtotalspendingThepriceelasticityofdemanddetermineshowachangeinprice,ceterisparibus,affectstotalspendingonagood.Theconnectionismosteasilyshownwithcalculus:∂ðpx⋅xÞ∂px¼px⋅∂x∂pxþx¼xðex,pxþ1Þ.(5.37)So,thesignofthisderivativedependsonwhetherex,pxislargerorsmallerthan1.Ifdemandisinelastic(0>ex,px>1),thederivativeispositiveandpriceandtotalspendingmoveinthesamedirection.Intuitively,ifpricedoesnotaffectquantitydemandedverymuch,thenquantitystaysrelativelyconstantaspricechangesandtotalspendingreflectsmainlythosepricemovements.Thisisthecase,forexample,withthedemandformostagriculturalproducts.Weather-inducedchangesinpriceforspecificcropsusuallycausetotalspendingonthosecropstomoveinthesamedirection.Ontheotherhand,ifdemandiselastic(ex,px<1),reactionstoapricechangearesolargethattheeffectontotalspendingisreversed:ariseinpricecausestotalspendingtofall(becausequantityfallsalot)andafallinpricecausestotalspendingtorise(quantityincreasessignificantly).Fortheunit-elasticcase(ex,px¼1),totalspendingisconstantnomatterhowpricechanges.CompensatedpriceelasticitiesBecausesomemicroeconomicanalysesfocusonthecompensateddemandfunction,itisalsousefultodeﬁneelasticitiesbasedonthatconcept.SuchdeﬁnitionsfollowdirectlyfromtheirMarshalliancounterparts.DEFINITIONLetthecompensateddemandfunctionbegivenbyxcðpx,py,UÞ.Thenwehavethefollowingdeﬁnitions.1.Compensatedown-priceelasticityofdemand(exc,px).Thiselasticitymeasuresthepro-portionatecompensatedchangeinquantitydemandedinresponsetoaproportionatechangeinagood’sownprice:exc,px¼∆xc=xc∆px=px¼∆xc∆px⋅pxxc¼∂xc∂px⋅pxxc.(5.38)2.Compensatedcross-priceelasticityofdemand(exc,px).Thismeasurestheproportionatecompensatedchangeinquantitydemandedinresponsetoaproportionatechangeinthepriceofanothergood:exc,py¼∆xc=xc∆py=py¼∆xc∆py⋅pyxc¼∂xc∂py⋅pyxc.(5.39)160Part2ChoiceandDemand WhetherthesepriceelasticitiesdiffermuchfromtheirMarshalliancounterpartsdependsontheimportanceofincomeeffectsintheoveralldemandforgoodx.ThepreciseconnectionbetweenthetwocanbeshownbymultiplyingtheSlutskyresultfromEquation5.27bythefactorpx=x:pxx⋅∂x∂px¼ex,px¼pxx⋅∂xc∂pxpxx⋅x⋅∂x∂I¼exc,pxsxex,I,(5.40)wheresx¼pxx=Iistheshareoftotalincomedevotedtothepurchaseofgoodx.Equation5.40showsthatcompensatedanduncompensatedown-priceelasticitiesofdemandwillbesimilarifeitheroftwoconditionshold:(1)TheshareofincomedevotedtogoodxðsxÞissmall;or(2)theincomeelasticityofdemandforgoodxðex,IÞissmall.Eitheroftheseconditionsservestoreducetheimportanceoftheincomecompensationemployedintheconstructionofthecompensateddemandfunction.Ifgoodxisunimportantinaperson’sbudget,thentheamountofincomecompensationrequiredtooffsetapricechangewillbesmall.Evenifagoodhasalargebudgetshare,ifdemanddoesnotreactstronglytochangesinincomethentheresultsofeitherdemandconceptwillbesimilar.Hence,therewillbemanycircumstanceswhereonecanusethetwopriceelasticityconceptsmoreorlessinterchange-ably.Putanotherway,therearemanyeconomiccircumstancesinwhichsubstitutioneffectsconstitutethemostimportantcomponentofpriceresponses.RelationshipsamongdemandelasticitiesThereareanumberofrelationshipsamongtheelasticityconceptsthathavebeendevelopedinthissection.Allofthesearederivedfromtheunderlyingmodelofutilitymaximization.Herewelookatthreesuchrelationshipsthatprovidefurtherinsightonthenatureofin-dividualdemand.Homogeneity.Thehomogeneityofdemandfunctionscanalsobeexpressedinelasticityterms.Becauseanyproportionalincreaseinallpricesandincomeleavesquantitydemandedunchanged,thenetsumofallpriceelasticitiestogetherwiththeincomeelasticityforaparticulargoodmustsumtozero.AformalproofofthispropertyreliesonEuler’stheorem(seeChapter2).Applyingthattheoremtothedemandfunctionxðpx,py,IÞandremember-ingthatthisfunctionishomogeneousofdegree0yields0¼px⋅∂x∂pxþpy⋅∂x∂pyþI⋅∂x∂I.(5.41)IfwedivideEquation5.41byxthenweobtain0¼ex,pxþex,pyþex,I,(5.42)asintuitionsuggests.Thisresultshowsthattheelasticitiesofdemandforanygoodcannotfollowacompletelyflexiblepattern.Theymustexhibitasortofinternalconsistencythatreflectsthebasicutility-maximizingapproachonwhichthetheoryofdemandisbased.Engelaggregation.IntheExtensionstoChapter4wediscussedtheempiricalanalysisofmarketsharesandtookspecialnoteofEngel’slawthattheshareofincomedevotedtofooddeclinesasincomeincreases.Fromanelasticityperspective,Engel’slawisastatementoftheempiricalregularitythattheincomeelasticityofdemandforfoodisgenerallyfoundtobeconsiderablylessthan1.Becauseofthis,itmustbethecasethattheincomeelasticityofallnonfooditemsmustbegreaterthan1.IfanindividualexperiencesanincreaseinhisorherincomethenwewouldexpectfoodexpenditurestoincreasebyasmallerproportionalChapter5IncomeandSubstitutionEffects161 amount,buttheincomemustbespentsomewhere.Intheaggregate,theseotherexpendi-turesmustincreaseproportionallyfasterthanincome.Aformalstatementofthispropertyofincomeelasticitiescanbederivedbydifferentiatingtheindividual’sbudgetconstraint(I¼pxxþpyy)withrespecttoincomewhiletreatingthepricesasconstants:1¼px⋅∂x∂Iþpy⋅∂y∂I.(5.43)Abitofalgebraicmanipulationofthisexpressionyields1¼px⋅∂x∂I⋅xIxIþpy⋅∂y∂I⋅yIyI¼sxex,Iþsyey,I;(5.44)here,asbefore,sirepresentstheshareofincomespentongoodi.Equation5.44showsthattheweightedaverageonincomeelasticitiesforallgoodsthatapersonbuysmustbe1.Ifweknew,say,thatapersonspentafourthofhisorherincomeonfoodandtheincomeelasticityofdemandforfoodwere0.5,thentheincomeelasticityofdemandforeverythingelsemustbeapproximately1:17½¼ð10:25⋅0:5Þ=0:75.Becausefoodisanimportant“necessity,”everythingelseisinsomesensea“luxury.”Cournotaggregation.Theeighteenth-centuryFrencheconomistAntoineCournotpro-videdoneoftheﬁrstmathematicalanalysesofpricechangesusingcalculus.Hismostimportantdiscoverywastheconceptofmarginalrevenue,aconceptcentraltotheproﬁt-maximizationhypothesisforﬁrms.Cournotwasalsoconcernedwithhowthechangeinasinglepricemightaffectthedemandforallgoods.Ourﬁnalrelationshipshowsthatthereareindeedconnectionsamongallofthereactionstothechangeinasingleprice.Webeginbydifferentiatingthebudgetconstraintagain,thistimewithrespecttopx:∂I∂px¼0¼px⋅∂x∂pxþxþpy⋅∂y∂px.Multiplicationofthisequationbypx=Iyields0¼px⋅∂x∂px⋅pxI⋅xxþx⋅pxIþpy⋅∂y∂px⋅pxI⋅yy,0¼sxex,pxþsxþsyey,px,(5.45)sothefinalCournotresultissxex,pxþsyey,px¼sx.(5.46)Thisequationshowsthatthesizeofthecross-priceeffectofachangeinthepriceofxonthequantityofyconsumedisrestrictedbecauseofthebudgetconstraint.Direct,own-priceeffectscannotbetotallyoverwhelmedbycross-priceeffects.Thisisthefirstofmanycon-nectionsamongthedemandsforgoodsthatwewillstudymoreintensivelyinthenextchapter.Generalizations.Althoughwehaveshowntheseaggregationresultsonlyforthecaseoftwogoods,theyareactuallyeasilygeneralizedtothecaseofmanygoods.YouareaskedtodojustthatinProblem5.11.Amoredifﬁcultissueiswhethertheseresultsshouldbeexpectedtoholdfortypicaleconomicdatainwhichthedemandsofmanypeoplearecombined.Ofteneconomiststreataggregatedemandrelationshipsasdescribingthebehav-iorofa“typicalperson,”andtheserelationshipsshouldinfactholdforsuchaperson.Butthesituationmaynotbequitethatsimple,aswewillshowwhendiscussingaggregationlaterinthebook.162Part2ChoiceandDemand EXAMPLE5.5DemandElasticities:TheImportanceofSubstitutionEffectsInthisexamplewecalculatethedemandelasticitiesimpliedbythreeoftheutilityfunctionswehavebeenusing.Althoughthepossibilitiesincorporatedinthesefunctionsaretoosimpletoreﬂecthoweconomistsactuallystudydemandempirically,theydoshowhowelasticitiesultimatelyreﬂectpeople’spreferences.Oneespeciallyimportantlessonistoshowwhymostofthevariationindemandelasticitiesamonggoodsprobablyarisesbecauseofdifferencesinthesizeofsubstitutioneffects.Case1:Cobb-Douglasðσ¼1Þ.Uðx,yÞ¼xαyβ,whereαþβ¼1.Thedemandfunctionsderivedfromthisutilityfunctionarexðpx,py,IÞ¼αIpx,yðpx,py,IÞ¼βIpy¼ð1αÞIpy.Applicationoftheelasticitydefinitionsshowsthatex,px¼∂x∂px⋅pxx¼αIp2x⋅pxαI=px¼1,ex,py¼∂x∂py⋅pyx¼0⋅pyx¼0,ex,I¼∂x∂I⋅Ix¼αpx⋅IαI=px¼1.(5.47)Theelasticitiesforgoodytakeonanalogousvalues.Hence,theelasticitiesassociatedwiththeCobb-Douglasutilityfunctionareconstantoverallrangesofpricesandincomeandtakeonespeciallysimplevalues.Thattheseobeythethreerelationshipsshownintheprevioussectioncanbeeasilydemonstratedusingthefactthatheresx¼αandsy¼β:Homogeneity:ex,pxþex,pyþex,I¼1þ0þ1¼0.Engelaggregation:sxex,Iþsyey,I¼α⋅1þβ⋅1¼αþβ¼1.Cournotaggregation:sxex,pxþsyey,px¼αð1Þþβ⋅0¼α¼sx.WecanalsousetheSlutskyequationinelasticityform(Equation5.40)toderivethecompensatedpriceelasticityinthisexample:exc,px¼ex,pxþsxex,I¼1þαð1Þ¼α1¼β.(5.48)Here,then,thecompensatedpriceelasticityforxdependsonhowimportantothergoods(y)areintheutilityfunction.Case2:CESðσ¼2;δ¼0:5Þ.Uðx,yÞ¼x0:5þy0:5.InExample4.2weshowedthatthedemandfunctionsthatcanbederivedfromthisutilityfunctionarexðpx,py,IÞ¼Ipxð1þpxp1yÞ,yðpx,py,IÞ¼Ipyð1þp1xpyÞ.(continued)Chapter5IncomeandSubstitutionEffects163 EXAMPLE5.5CONTINUEDAsyoumightimagine,calculatingelasticitiesdirectlyfromthesefunctionscantakesometime.Herewefocusonlyontheown-priceelasticityandmakeuseoftheresult(fromProblem5.6)thatthe“shareelasticity”ofanygoodisgivenbyesx,px¼∂sx∂px⋅pxsx¼1þex,px.(5.49)Inthiscase,sx¼pxxI¼11þpxp1y,sotheshareelasticityismoreeasilycalculatedandisgivenbyesx,px¼∂sx∂px⋅pxsx¼p1yð1þpxp1yÞ2⋅pxð1þpxp1yÞ1¼pxp1y1þpxp1y.(5.50)Becausetheunitsinwhichgoodsaremeasuredareratherarbitraryinutilitytheory,wemightaswelldefinethemsothatinitiallypx¼py,inwhichcase8wegetex,px¼esx,px1¼11þ11¼1.5.(5.51)Hence,demandismoreelasticinthiscasethanintheCobb-Douglasexample.ThereasonforthisisthatthesubstitutioneffectislargerforthisversionoftheCESutilityfunction.ThiscanbeshownbyagainapplyingtheSlutskyequation(andusingthefactsthatex,I¼1andsx¼0:5):exc,px¼ex,pxþsxex,I¼1.5þ0.5ð1Þ¼1,(5.52)whichistwicethesizeofthesubstitutioneffectfortheCobb-Douglas.Case3.CESðσ¼0:5;δ¼1Þ:Uðx,yÞ¼x1y1.ReferringbacktoExample4.2,wecanseethattheshareofgoodximpliedbythisutilityfunctionisgivenbysx¼11þp0:5yp0:5x,sotheshareelasticityisgivenbyesx,px¼∂sx∂px⋅pxsx¼0:5p0:5yp1:5xð1þp0:5yp0:5xÞ2⋅pxð1þp0:5yp0:5xÞ1¼0:5p0:5yp0:5x1þp0:5yp0:5x..(5.53)Ifweagainadoptthesimplificationofequalprices,wecancomputetheown-priceelas-ticityasex,px¼esx,px1¼0:521¼0:75(5.54)andthecompensatedpriceelasticityasexc,px¼ex,pxþsxex,I¼0:75þ0:5ð1Þ¼0:25.(5.55)So,forthisversionoftheCESutilityfunction,theown-priceelasticityissmallerthaninCase1andCase2becausethesubstitutioneffectissmaller.Hence,themainvariationamongthecasesisindeedcausedbydifferencesinthesizeofthesubstitutioneffect.8Noticethatthissubstitutionmustbemadeafterdifferentiationbecausethedefinitionofelasticityrequiresthatwechangeonlypxwhileholdingpyconstant.164Part2ChoiceandDemand Ifyouneverwanttoworkoutthiskindofelasticityagain,itmaybehelpfultomakeuseofthequitegeneralresultthatexc,px¼ð1sxÞσ.(5.56)Youmaywishtocheckoutthatthisformulaworksinthesethreeexamples(withsx¼0:5andσ¼1,2,0.5,respectively),andProblem5.9asksyoutoshowthatthisresultisgen-erallytrue.BecauseallofthesecasesbasedontheCESutilityfunctionhaveaunitaryincomeelasticity,theown-priceelasticitycanbecomputedfromthecompensatedpriceelasticitybysimplyaddingsxtothefigurecomputedinEquation5.56.QUERY:Whyisitthatthebudgetshareforgoodsotherthanxð1sxÞentersintothecompensatedown-priceelasticitiesinthisexample?CONSUMERSURPLUSAnimportantprobleminappliedwelfareeconomicsistodeviseamonetarymeasureofthegainsandlossesthatindividualsexperiencewhenpriceschange.Oneuseforsuchameasureistoplaceadollarvalueonthewelfarelossthatpeopleexperiencewhenamarketismonopo-lizedwithpricesexceedingmarginalcosts.Anotherapplicationconcernsmeasuringthewelfaregainsthatpeopleexperiencewhentechnicalprogressreducesthepricestheypayforgoods.Relatedapplicationsoccurinenvironmentaleconomics(measuringthewelfarecostsofincorrectlypricedresources),lawandeconomics(evaluatingthewelfarecostsofexcessprotectionstakeninfearoflawsuits),andpubliceconomics(measuringtheexcessburdenofatax).Inordertomakesuchcalculations,economistsuseempiricaldatafromstudiesofmarketdemandincombinationwiththetheorythatunderliesthatdemand.Inthissectionwewillexaminetheprimarytoolsusedinthatprocess.ConsumerwelfareandtheexpenditurefunctionTheexpenditurefunctionprovidestheﬁrstcomponentforthestudyoftheprice/welfareconnection.Supposethatwewishedtomeasurethechangeinwelfarethatanindividualexperiencesifthepriceofgoodxrisesfromp0xtop1x.Initiallythispersonrequiresexpendi-turesofEðp0x,py,U0ÞtoreachautilityofU0.Toachievethesameutilityoncethepriceofxrises,heorshewouldrequirespendingofatleastEðp1x,py,U0Þ.Inordertocompensateforthepricerise,therefore,thispersonwouldrequireacompensation(formallycalledacompen-satingvariationorCV)ofCV¼Eðp1x,py,U0ÞEðp0x,py,U0Þ.(5.57)ThissituationisshowngraphicallyinthetoppanelofFigure5.8.Initially,thispersonconsumesthecombinationx0,y0andobtainsutilityofU0.Whenthepriceofxrises,heorshewouldbeforcedtomovetocombinationx2,y2andsufferalossinutility.IfheorshewerecompensatedwithextrapurchasingpowerofamountCV,heorshecouldaffordtoremainontheU0indifferencecurvedespitethepricerisebychoosingcombinationx1,y1.ThedistanceCV,therefore,providesamonetarymeasureofhowmuchthispersonneedsinordertobecompensatedforthepricerise.UsingthecompensateddemandcurvetoshowCVUnfortunately,individuals’utilityfunctionsandtheirassociatedindifferencecurvemapsarenotdirectlyobservable.ButwecanmakesomeheadwayonempiricalmeasurementbydetermininghowtheCVamountcanbeshownonthecompensateddemandcurveintheChapter5IncomeandSubstitutionEffects165 FIGURE5.8ShowingCompensatingVariationIfthepriceofxrisesfromp0xtop1x,thispersonneedsextraexpendituresofCVtoremainontheU0indifferencecurve.IntegrationshowsthatCVcanalsoberepresentedbytheshadedareabelowthecompensateddemandcurveinpanel(b).Quantity of xQuantityofyCVy1U0U1y2y0x2x1x0E(px0, . . . ,U0)E(px0, . . . ,U0)E(px1, . . . ,U0)E(px1, . . . ,U0)Quantity of xPricepx0x1x0xc(px, . . . ,U0)BApx1px2(a) Indifference curve map(b) Compensated demand curve bottompanelofFigure5.8.Shephard’slemmashowsthatthecompensateddemandfunc-tionforagoodcanbefounddirectlyfromtheexpenditurefunctionbydifferentiation:xcðpx,py,UÞ¼∂Eðpx,py,UÞ∂px.(5.58)Hence,thecompensationdescribedinEquation5.57canbefoundbyintegratingacrossasequenceofsmallincrementstopricefromp0xtop1x:CV¼∫p1xp0xdE¼∫p1xp0xxcðpx,py,U0Þdpx(5.59)whileholdingpyandutilityconstant.TheintegraldefinedinEquation5.59hasageometricinterpretation,whichisshowninthelowerpanelofFigure5.9:itistheshadedareatotheleftofthecompensateddemandcurveandboundedbyp0xandp1x.Sothewelfarecostofthispriceincreasecanalsobeillustratedusingchangesintheareabelowthecompensatedde-mandcurve.TheconsumersurplusconceptThereisanotherwaytolookatthisissue.Wecanaskhowmuchthispersonwouldbewillingtopayfortherighttoconsumeallofthisgoodthatheorshewantedatthemarketpriceofp0xratherthandoingwithoutthegoodcompletely.ThecompensateddemandcurveinthebottompanelofFigure5.8showsthatifthepriceofxrosetop2x,thisperson’sconsumptionwouldfalltozeroandheorshewouldrequireanamountofcompensationequaltoareap2xAp0xinordertoacceptthechangevoluntarily.Therighttoconsumex0atapriceofp0xisthereforeworththisamounttothisindividual.Itistheextrabeneﬁtthatthispersonreceivesbybeingabletomakemarkettransactionsattheprevailingmarketprice.Thisvalue,givenbytheareabelowthecompensateddemandcurveandabovethemarketprice,istermedconsumersurplus.Lookedatinthisway,thewelfareproblemcausedbyariseinthepriceofxcanbedescribedasalossinconsumersurplus.Whenthepricerisesfromp0xtop1xtheconsumersurplus“triangle”decreasesinsizefromp2xAp0xtop2xBp1x.Astheﬁguremakesclear,thatissimplyanotherwayofdescribingthewelfarelossrepresentedinEquation5.59.WelfarechangesandtheMarshalliandemandcurveSofarouranalysisofthewelfareeffectsofpricechangeshasfocusedonthecompensateddemandcurve.Thisisinsomewaysunfortunatebecausemostempiricalworkondemandactuallyestimatesordinary(Marshallian)demandcurves.InthissectionwewillshowthatstudyingchangesintheareabelowaMarshalliandemandcurvemayinfactbequiteagoodwaytomeasurewelfarelosses.ConsidertheMarshalliandemandcurvexðpx,…ÞillustratedinFigure5.9.Initiallythisconsumerfacesthepricep0xandchoosestoconsumex0.ThisconsumptionyieldsautilitylevelofU0,andtheinitialcompensateddemandcurveforx[thatis,xcðpx,py,U0Þ]alsopassesthroughthepointx0,p0x(whichwehavelabeledpointA).Whenpricerisestop1x,theMarshalliandemandforgoodxfallstox1(pointConthedemandcurve)andthisperson’sutilityalsofallsto,say,U1.Thereisanothercompensateddemandcurveassociatedwiththislowerlevelofutility,anditalsoisshowninFigure5.9.BoththeMarshalliandemandcurveandthisnewcompensateddemandcurvepassthroughpointC.ThepresenceofasecondcompensateddemandcurveinFigure5.9raisesanintriguingconceptualquestion.ShouldwemeasurethewelfarelossfromthepriceriseaswedidinFigure5.8usingthecompensatingvariation(CV)associatedwiththeinitialcompensateddemandcurve(areap1xBAp0x)orshouldwe,perhaps,usethisnewcompensateddemandcurveChapter5IncomeandSubstitutionEffects167 andmeasurethewelfarelossasareap1xCDp0x?Apotentialrationaleforusingtheareaunderthesecondcurvewouldbetofocusontheindividual’ssituationafterthepricerise(withutilitylevelU1).Wemightaskhowmuchheorshewouldnowbewillingtopaytoseethepricereturntoitsold,lowerlevels.9Theanswertothiswouldbegivenbyareap1xCDp0x.Thechoicebetweenwhichcompensateddemandcurvetousethereforeboilsdowntochoosingwhichlevelofutilityoneregardsastheappropriatetargetfortheanalysis.Luckily,theMarshalliandemandcurveprovidesaconvenientcompromisebetweenthesetwomeasures.BecausethesizeoftheareabetweenthetwopricesandbelowtheMarshalliancurve(areap1xCAp0x)issmallerthanthatbelowthecompensateddemandcurvebasedonU0butlargerthanthatbelowthecurvebasedonU1,itdoesseemanattractivemiddleground.Hence,thisisthemeasureofwelfarelosseswewillprimarilyusethroughoutthisbook.DEFINITIONConsumersurplus.ConsumersurplusistheareabelowtheMarshalliandemandcurveandabovemarketprice.Itshowswhatanindividualwouldpayfortherighttomakevoluntarytransactionsatthisprice.Changesinconsumersurpluscanbeusedtomeasurethewelfareeffectsofpricechanges.WeshouldpointoutthatsomeeconomistsuseeitherCVorEVtocomputethewelfareeffectsofpricechanges.Indeed,economistsareoftennotveryclearaboutwhichmeasureofwelfarechangetheyareusing.Ourdiscussionintheprevioussectionshowsthatifincomeeffectsaresmall,itreallydoesnotmakemuchdifferenceinanycase.FIGURE5.9WelfareEffectsofPriceChangesandtheMarshallianDemandCurveTheusualMarshallian(nominalincomeconstant)demandcurveforgoodxisxðpx,…Þ.Further,xcð…,U0Þandxcð…,U1Þdenotethecompensateddemandcurvesassociatedwiththeutilitylevelsexperiencedwhenp0xandp1x,respectively,prevail.Theareatotheleftofxðpx,…Þbetweenp0xandp1xisboundedbythesimilarareastotheleftofthecompensateddemandcurves.Hence,forsmallchangesinprice,theareatotheleftoftheMarshalliandemandcurveisagoodmeasureofwelfareloss.pxpx0px1x1x0Quantity of x per periodABCDx(px,. . . )xc(. . . ,U0)xc(. . . ,U1)9Thisalternativemeasureofcompensationissometimestermedthe“equivalentvariation”(EV).168Part2ChoiceandDemand EXAMPLE5.6WelfareLossfromaPriceIncreaseTheseideascanbeillustratednumericallybyreturningtoouroldhamburger/softdrinkexample.Let’slookatthewelfareconsequencesofanunconscionablepriceriseforsoftdrinks(goodx)from$1to$4.InExample5.3,wefoundthatthecompensateddemandforgoodxwasgivenbyxcðpx,py,VÞ¼Vp0:5yp0:5x.(5.60)Hence,thewelfarecostofthepriceincreaseisgivenbyCV¼∫41xcðpx,py,VÞdpx¼∫41Vp0:5yp0:5xdpx¼2Vp0:5yp0:5xpx¼4px¼1.(5.61)Ifweusethevalueswehavebeenassumingthroughoutthisgastronomicfeast(V¼2,py¼4),thenCV¼2⋅2⋅2⋅ð4Þ0:52⋅2⋅2⋅ð1Þ0:5¼8.(5.62)Thisfigurewouldbecutinhalf(to4)ifwebelievedthattheutilitylevelafterthepricerise(V¼1)werethemoreappropriateutilitytargetformeasuringcompensation.IfinsteadwehadusedtheMarshalliandemandfunctionxðpx,py,IÞ¼0:5Ip1x,thelosswouldbecalculatedasloss¼∫41xðpx,py,IÞdpx¼∫410:5Ip1xdpx¼0:5Ilnpx41.(5.63)So,withI¼8,thislossisloss¼4lnð4Þ4lnð1Þ¼4lnð4Þ¼4ð1:39Þ¼5:55,(5.64)whichseemsareasonablecompromisebetweenthetwoalternativemeasuresbasedonthecompensateddemandfunctions.QUERY:Inthisproblem,noneofthedemandcurveshasaﬁnitepriceatwhichdemandgoestopreciselyzero.Howdoesthisaffectthecomputationoftotalconsumersurplus?Doesthisaffectthetypesofwelfarecalculationsmadehere?REVEALEDPREFERENCEANDTHESUBSTITUTIONEFFECTTheprincipalunambiguouspredictionthatcanbederivedfromtheutility-maximationmodelisthattheslope(orpriceelasticity)ofthecompensateddemandcurveisnegative.TheproofofthisassertionreliesontheassumptionofadiminishingMRSandtherelatedobservationthat,withadiminishingMRS,thenecessaryconditionsforautilitymaximumarealsosufﬁcient.Tosomeeconomists,therelianceonahypothesisaboutanunobservableutilityfunctionrepresentedaweakfoundationindeedonwhichtobaseatheoryofdemand.Analternativeapproach,whichleadstothesameresult,wasﬁrstproposedbyPaulSamuelsoninthelate1940s.10Thisapproach,whichSamuelsontermedthetheoryofrevealedpreference,deﬁnesaprincipleofrationalitythatisbasedonobservedbehaviorand10PaulA.Samuelson,FoundationsofEconomicAnalysis(Cambridge,MA:HarvardUniversityPress,1947).Chapter5IncomeandSubstitutionEffects169 thenusesthisprincipletoapproximateanindividual’sutilityfunction.Inthissense,apersonwhofollowsSamuelson’sprincipleofrationalitybehavesasifheorsheweremaximizingaproperutilityfunctionandexhibitsanegativesubstitutioneffect.BecauseSamuelson’sapproachprovidesadditionalinsightsintoourmodelofconsumerchoice,wewillbrieﬂyexamineithere.GraphicalapproachTheprincipleofrationalityinthetheoryofrevealedpreferenceisasfollows:Considertwobundlesofgoods,AandB.If,atsomepricesandincomelevel,theindividualcanaffordbothAandBbutchoosesA,wesaythatAhasbeen“revealedpreferred”toB.Theprincipleofrationalitystatesthatunderanydifferentprice-incomearrangement,BcanneverberevealedpreferredtoA.IfBisinfactchosenatanotherprice-incomeconﬁguration,itmustbebecausetheindividualcouldnotaffordA.TheprincipleisillustratedinFigure5.10.Supposethat,whenthebudgetconstraintisgivenbyI1,pointAischoseneventhoughBalsocouldhavebeenpurchased.ThenAhasbeenrevealedpreferredtoB.If,forsomeotherbudgetconstraint,Bisinfactchosen,thenitmustbeacasesuchasthatrepresentedbyI2,whereAcouldnothavebeenbought.IfBwerechosenwhenthebudgetconstraintisI3,thiswouldbeaviolationoftheprincipleofrationalitybecause,withI3,bothAandBcanbebought.WithbudgetconstraintI3,itislikelythatsomepointotherthaneitherAorB(say,C)willbebought.Noticehowthisprincipleusesobservablereactionstoalternativebudgetconstraintstorankcommoditiesratherthanassumingtheexistenceofautilityfunctionitself.AlsonoticeFIGURE5.10DemonstrationofthePrincipleofRationalityintheTheoryofRevealedPreferenceWithincomeI1theindividualcanaffordbothpointsAandB.IfAisselectedthenAisrevealedpreferredtoB.ItwouldbeirrationalforBtoberevealedpreferredtoAinsomeotherprice-incomeconﬁguration.Quantity of xQuantityof yyaI1I3I2ybxaxbABC170Part2ChoiceandDemand howtheprincipleoffersaglimpseofwhyindifferencecurvesareconvex.Nowweturntoaformalproof.NegativityofthesubstitutioneffectSupposethatanindividualisindifferentbetweentwobundles,C(composedofxCandyC)andD(composedofxDandyD).LetpCx,pCybethepricesatwhichbundleCischosenandpDx,pDythepricesatwhichbundleDischosen.BecausetheindividualisindifferentbetweenCandD,itmustbethecasethatwhenCwaschosen,DcostatleastasmuchasC:pCxxCþpCyyCpCxxDþpCyyD.(5.65)AsimilarstatementholdswhenDischosen:pDxxDþpDyyDpDxxCþpDyyC.(5.66)RewritingtheseequationsgivespCxðxCxDÞþpCyðyCyDÞ0,(5.67)pDxðxDxCÞþpDyðyDyCÞ0.(5.68)AddingthesetogetheryieldsðpCxpDxÞðxCxDÞþðpCypDyÞðyCyDÞ0.(5.69)Nowsupposethatonlythepriceofxchanges;assumethatpCy¼pDy.ThenðpCxpDxÞðxCxDÞ0.(5.70)ButEquation5.70saysthatpriceandquantitymoveintheoppositedirectionwhenutilityisheldconstant(remember,bundlesCandDareequallyattractive).Thisispreciselyastatementaboutthenonpositivenatureofthesubstitutioneffect:∂xcðpx,py,VÞ∂px¼∂x∂pxU¼constant0.(5.71)WehavearrivedattheresultbyanapproachthatrequiresneithertheexistenceofautilityfunctionnortheassumptionofadiminishingMRS.MathematicalgeneralizationGeneralizingtherevealedpreferenceideatongoodsisstraightforward.Ifatpricesp0i,bundlex0iischoseninsteadofx1iandifbundlex1iisalsoaffordable,thenXni¼1p0ix0iXni¼1p0ix1i;(5.72)thatis,bundle0hasbeen“revealedpreferred”tobundle1.Consequently,atthepricesthatprevailwhenbundle1isbought(say,p1i),itmustbethecasethatx0iismoreexpensive:Xni¼1p1ix0i>Xni¼1p1ix1i.(5.73)Althoughthisinitialdeﬁnitionofrevealedpreferencefocusesontherelationshipbetweentwobundlesofgoods,themostoftenusedversionofthebasicprinciplerequiresadegreeofChapter5IncomeandSubstitutionEffects171
transitivityforpreferencesamonganarbitrarilylargenumberofbundles.Thisissummarizedbythefollowing“strong”axiom.DEFINITIONStrongaxiomofrevealedpreference.Thestrongaxiomofrevealedpreferencestatesthatifcommoditybundle0isrevealedpreferredtobundle1,andifbundle1isrevealedpreferredtobundle2,andifbundle2isrevealedpreferredtobundle3,…,andifbundleK1isrevealedpreferredtobundleK,thenbundleKcannotberevealedpreferredtobundle0(whereKisanyarbitrarynumberofcommoditybundles).Mostotherpropertiesthatwehavedevelopedusingtheconceptofutilitycanbeprovedusingthisrevealedpreferenceaxiominstead.Forexample,itisaneasymattertoshowthatdemandfunctionsarehomogeneousofdegree0inallpricesandincome.Itthereforeisapparentthattherevealedpreferenceaxiomandtheexistenceof“well-behaved”utilityfunctionsaresomehowequivalentconditions.ThatthisisinfactthecasewasﬁrstshownbyH.S.Houthakkerin1950.Houthakkershowedthatasetofindifferencecurvescanalwaysbederivedforanindividualwhoobeysthestrongaxiomofrevealedpreference.11Hence,thisaxiomprovidesaquitegeneralandbelievablefoundationforutilitytheorybasedonsimplecomparisonsamongalternativebudgetconstraints.Thisapproachiswidelyusedintheconstructionofpriceindicesandforavarietyofotherappliedpurposes.11H.S.Houthakker,“RevealedPreferenceandtheUtilityFunction,”Economica17(May1950):159–74.SUMMARYInthischapter,weusedtheutility-maximizationmodeltostudyhowthequantityofagoodthatanindividualchoosesrespondstochangesinincomeortochangesinthatgood’sprice.Theﬁnalresultofthisexaminationisthederivationofthefamiliardownward-slopingdemandcurve.Inarrivingatthatresult,however,wehavedrawnawidevarietyofinsightsfromthegeneraleconomictheoryofchoice.•Proportionalchangesinallpricesandincomedonotshifttheindividual’sbudgetconstraintandthereforedonotchangethequantitiesofgoodschosen.Informalterms,demandfunctionsarehomogeneousofdegree0inallpricesandincome.•Whenpurchasingpowerchanges(thatis,whenincomeincreaseswithpricesremainingunchanged),budgetcon-straintsshiftandindividualswillchoosenewcommoditybundles.Fornormalgoods,anincreaseinpurchasingpowercausesmoretobechosen.Inthecaseofinferiorgoods,however,anincreaseinpurchasingpowercauseslesstobepurchased.Hencethesignof∂xi=∂Icouldbeeitherpositiveornegative,although∂xi=∂I0isthemostcommoncase.•Afallinthepriceofagoodcausessubstitutionandincomeeffectsthat,foranormalgood,causemoreofthegoodtobepurchased.Forinferiorgoods,however,substitutionandincomeeffectsworkinoppositedirec-tionsandnounambiguouspredictionispossible.•Similarly,ariseinpriceinducesbothsubstitutionandincomeeffectsthat,inthenormalcase,causelesstobedemanded.Forinferiorgoodsthenetresultisagainambiguous.•TheMarshalliandemandcurvesummarizesthetotalquantityofagooddemandedateachpossibleprice.Changesinpriceinducebothsubstitutionandincomeeffectsthatpromptmovementsalongthecurve.Foranormalgood,∂xi=∂pi0alongthiscurve.Ifincome,pricesofothergoods,orpreferenceschange,thenthecurvemayshifttoanewlocation.•Compensateddemandcurvesillustratemovementsalongagivenindifferencecurveforalternativeprices.Theyareconstructedbyholdingutilityconstantandexhibitonlythesubstitutioneffectsfromapricechange.Hence,theirslopeisunambiguouslynegative.•Demandelasticitiesareoftenusedinempiricalworktosummarizehowindividualsreacttochangesinpricesandincome.Themostimportantsuchelasticityisthe(own-)priceelasticityofdemand,ex,px.Thismea-surestheproportionatechangeinquantityinresponsetoa1percentchangeinprice.Asimilarelasticitycanbedeﬁnedformovementsalongthecompensatedde-mandcurve.•Therearemanyrelationshipsamongdemandelasticities.Someofthemoreimportantonesare:(1)own-price172Part2ChoiceandDemand
PROBLEMS5.1ThirstyEddrinksonlypurespringwater,buthecanpurchaseitintwodifferent-sizedcontainers:0.75literand2liter.Becausethewateritselfisidentical,heregardsthesetwo“goods”asperfectsubstitutes.a.AssumingEd’sutilitydependsonlyonthequantityofwaterconsumedandthatthecontainersthemselvesyieldnoutility,expressthisutilityfunctionintermsofquantitiesof0.75Lcontain-ers(x)and2Lcontainers(y).b.StateEd’sdemandfunctionforxintermsofpx,py,andI.c.Graphthedemandcurveforx,holdingIandpyconstant.d.HowdochangesinIandpyshiftthedemandcurveforx?e.Whatwouldthecompensateddemandcurveforxlooklikeinthissituation?5.2DavidN.gets$3perweekasanallowancetospendanywayhepleases.Becausehelikesonlypeanutbutterandjellysandwiches,hespendstheentireamountonpeanutbutter(at$0.05perounce)andjelly(at$0.10perounce).Breadisprovidedfreeofchargebyaconcernedneighbor.Davidisaparticulareaterandmakeshissandwicheswithexactly1ounceofjellyand2ouncesofpeanutbutter.Heissetinhiswaysandwillneverchangetheseproportions.a.HowmuchpeanutbutterandjellywillDavidbuywithhis$3allowanceinaweek?b.Supposethepriceofjellyweretoriseto$0.15anounce.Howmuchofeachcommoditywouldbebought?c.ByhowmuchshouldDavid’sallowancebeincreasedtocompensatefortheriseinthepriceofjellyinpart(b)?d.Graphyourresultsinparts(a)to(c).e.Inwhatsensedoesthisprobleminvolveonlyasinglecommodity,peanutbutterandjellysandwiches?Graphthedemandcurveforthissinglecommodity.f.Discusstheresultsofthisproblemintermsoftheincomeandsubstitutioneffectsinvolvedinthedemandforjelly.5.3AsdeﬁnedinChapter3,autilityfunctionishomotheticifanystraightlinethroughtheorigincutsallindifferencecurvesatpointsofequalslope:TheMRSdependsontheratioy=x.a.Provethat,inthiscase,∂x=∂Iisconstant.b.Provethatifanindividual’stastescanberepresentedbyahomotheticindifferencemapthenpriceandquantitymustmoveinoppositedirections;thatis,provethatGiffen’sparadoxcannotoccur.elasticitiesdeterminehowapricechangeaffectstotalspendingonagood;(2)substitutionandincomeeffectscanbesummarizedbytheSlutskyequationinelasticityform;and(3)variousaggregationrelationsholdamongelasticities—theseshowhowthedemandsfordifferentgoodsarerelated.•Welfareeffectsofpricechangescanbemeasuredbychangingareasbeloweithercompensatedorordinarydemandcurves.Suchchangesaffectthesizeofthecon-sumersurplusthatindividualsreceivefrombeingabletomakemarkettransactions.•Thenegativityofthesubstitutioneffectisthemostbasicconclusionfromdemandtheory.Thisresultcanbeshownusingrevealedpreferencetheoryandsodoesnotrequireassumingtheexistenceofautilityfunction.Chapter5IncomeandSubstitutionEffects173
5.4AsinExample5.1,assumethatutilityisgivenbyutility¼Uðx,yÞ¼x0:3y0:7.a.UsetheuncompensateddemandfunctionsgiveninExample5.1tocomputetheindirectutilityfunctionandtheexpenditurefunctionforthiscase.b.Usetheexpenditurefunctioncalculatedinpart(a)togetherwithShephard’slemmatocomputethecompensateddemandfunctionforgoodx.c.Usetheresultsfrompart(b)togetherwiththeuncompensateddemandfunctionforgoodxtoshowthattheSlutskyequationholdsforthiscase.5.5Supposetheutilityfunctionforgoodsxandyisgivenbyutility¼Uðx,yÞ¼xyþy.a.Calculatetheuncompensated(Marshallian)demandfunctionsforxandyanddescribehowthedemandcurvesforxandyareshiftedbychangesinIorthepriceoftheothergood.b.Calculatetheexpenditurefunctionforxandy.c.Usetheexpenditurefunctioncalculatedinpart(b)tocomputethecompensateddemandfunctionsforgoodsxandy.Describehowthecompensateddemandcurvesforxandyareshiftedbychangesinincomeorbychangesinthepriceoftheothergood.5.6Overathree-yearperiod,anindividualexhibitsthefollowingconsumptionbehavior:Isthisbehaviorconsistentwiththestrongaxiomofrevealedpreference?5.7Supposethatapersonregardshamandcheeseaspurecomplements—heorshewillalwaysuseonesliceofhamincombinationwithonesliceofcheesetomakeahamandcheesesandwich.Supposealsothathamandcheesearetheonlygoodsthatthispersonbuysandthatbreadisfree.a.Ifthepriceofhamisequaltothepriceofcheese,showthattheown-priceelasticityofdemandforhamis0.5andthatthecross-priceelasticityofdemandforhamwithrespecttothepriceofcheeseisalso0.5.b.Explainwhytheresultsfrompart(a)reﬂectonlyincomeeffects,notsubstitutioneffects.Whatarethecompensatedpriceelasticitiesinthisproblem?c.Usetheresultsfrompart(b)toshowhowyouranswerstopart(a)wouldchangeifasliceofhamcosttwicethepriceofasliceofcheese.d.Explainhowthisproblemcouldbesolvedintuitivelybyassumingthispersonconsumesonlyonegood—aham-and-cheesesandwich.pxpyxyYear13374Year24266Year35173174Part2ChoiceandDemand
5.8Showthattheshareofincomespentonagoodxissx¼dlnEdlnpx,whereEistotalexpenditure.AnalyticalProblems5.9ShareelasticitiesIntheExtensionstoChapter4weshowedthatmostempiricalworkindemandtheoryfocusesonincomeshares.Foranygood,x,theincomeshareisdeﬁnedassx¼pxx=I.Inthisproblemweshowthatmostdemandelasticitiescanbederivedfromcorrespondingshareelasticities.a.Showthattheelasticityofagood’sbudgetsharewithrespecttoincomeðesx,I¼∂sx=∂I⋅I=sxÞisequaltoex,I1.Interpretthisconclusionwithafewnumericalexamples.b.Showthattheelasticityofagood’sbudgetsharewithrespecttoitsownpriceðesx,px¼∂sx=∂px⋅px=sxÞisequaltoex,pxþ1.Again,interpretthisﬁndingwithafewnumericalexamples.c.Useyourresultsfrompart(b)toshowthatthe“expenditureelasticity”ofgoodxwithrespecttoitsownprice½ex⋅px,px¼∂ðpx⋅xÞ=∂px⋅1=xisalsoequaltoex,pxþ1.d.Showthattheelasticityofagood’sbudgetsharewithrespecttoachangeinthepriceofsomeothergoodðesx,py¼∂sx=∂py⋅py=sxÞisequaltoex,py.e.IntheExtensionstoChapter4weshowedthatwithaCESutilityfunction,theshareofincomedevotedtogoodxisgivenbysx¼1=ð1þpkypkxÞ,wherek¼δ=ðδ1Þ¼1σ.UsethisshareequationtoproveEquation5.56:exc,px¼ð1sxÞσ.Hint:Thisproblemcanbesimpliﬁedbyassumingpx¼py,inwhichcasesx¼0:5.5.10MoreonelasticitiesPart(e)ofProblem5.9hasanumberofusefulapplicationsbecauseitshowshowpriceresponsesdependultimatelyontheunderlyingparametersoftheutilityfunction.Speciﬁcally,usethatresulttogetherwiththeSlutskyequationinelasticitytermstoshow:a.IntheCobb-Douglascaseðσ¼1Þ,thefollowingrelationshipholdsbetweentheown-priceelasticitiesofxandy:ex,pxþey,py¼2.b.Ifσ>1thenex,pxþey,py<2,andifσ<1thenex,pxþey,py>2.Provideanintuitiveexplanationforthisresult.c.Howwouldyougeneralizethisresulttocasesofmorethantwogoods?Discusswhethersuchageneralizationwouldbeespeciallymeaningful.5.11AggregationofelasticitiesformanygoodsThethreeaggregationrelationshipspresentedinthischaptercanbegeneralizedtoanynumberofgoods.Thisproblemasksyoutodoso.Weassumethattherearengoodsandthattheshareofincomedevotedtogoodiisdenotedbysi.Wealsodeﬁnethefollowingelasticities:ei,I¼∂xi∂I⋅Ixi,ei,j¼∂xi∂pj⋅Ixi.Usethisnotationtoshow:a.Homogeneity:Pnj¼1ei,jþei,I¼0.b.Engelaggregation:Pni¼1siei,I¼1.c.Cournotaggregation:Pni¼1siei,j¼sj.Chapter5IncomeandSubstitutionEffects175
5.12Quasi-linearutility(revisited)Considerasimplequasi-linearutilityfunctionoftheformUðx,yÞ¼xþlny.a.Calculatetheincomeeffectforeachgood.Alsocalculatetheincomeelasticityofdemandforeachgood.b.Calculatethesubstitutioneffectforeachgood.Alsocalculatethecompensatedown-priceelasticityofdemandforeachgood.c.ShowthattheSlutskyequationappliestothisfunction.d.ShowthattheelasticityformoftheSlutskyequationalsoappliestothisfunction.Describeanyspecialfeaturesyouobserve.5.13ThealmostidealdemandsystemThegeneralformofthealmostidealdemandsystem(AIDS)isgivenbylnEð!p,UÞ¼a0þXni¼1αilnpiþ12Xni¼1Xnj¼1γijlnpilnpjþUβ0∏ki¼1pβkk,where!pisthevectorofprices,Eistheexpenditurefunction,andUisthelevelofutilityrequired.Foranalyticalease,assumethatthefollowingrestrictionsapply:γij¼γji,Xni¼1αi¼1,andXnj¼1γij¼Xnk¼1βk¼0.a.DerivetheAIDSfunctionalformforatwo-goodscase.b.Giventhepreviousrestrictions,showthatEð!p,UÞishomogeneousofdegree1inallprices.This,alongwiththefactthatthisfunctionresemblescloselytheactualdata,makesitan“ideal”function.c.Usingthefactthatsx¼dlnEdlnpx(seeProblem5.8),calculatetheincomeshareofeachofthetwogoods.5.14PriceindifferencecurvesPriceindifferencecurvesareiso-utilitycurveswiththepricesoftwogoodsonthex-andy-axes,respectively.Thus,theyhavethefollowinggeneralform:ðp1,p2Þjvðp1,p2,IÞ¼v0.a.DerivetheformulaforthepriceindifferencecurvesfortheCobb-Douglascasewithα¼β¼0:5.Sketchoneofthem.b.Whatdoestheslopeofthecurveshow?c.Whatisthedirectionofincreasingutilityinyourgraph?SUGGESTIONSFORFURTHERREADINGCook,P.J.“A‘OneLine’ProofoftheSlutskyEquation.”AmericanEconomicReview62(March1972):139.CleveruseofdualitytoderivetheSlutskyequation;usesthesamemethodasinChapter5butwithrathercomplexnotation.Fisher,F.M.,andK.Shell.TheEconomicTheoryofPriceIndices.NewYork:AcademicPress,1972.Complete,technicaldiscussionoftheeconomicpropertiesofvariouspriceindexes;describes“ideal”indexesbasedonutility-maximizingmodelsindetail.Mas-Colell,Andreu,MichaelD.Whinston,andJerryR.Green.MicroeconomicTheory.NewYork:OxfordUniversityPress,1995.Chapter3coversmuchofthematerialinthischapteratasomewhathigherlevel.SectionIonmeasurementofthewelfareeffectsofpricechangesisespeciallyrecommended.Samuelson,PaulA.FoundationsofEconomicAnalysis.Cambridge,MA:HarvardUniversityPress,1947,Chap.5.Providesacompleteanalysisofsubstitutionandincomeeffects.Alsodevelopstherevealedpreferencenotion.176Part2ChoiceandDemand
Silberberg,E.,andW.Suen.TheStructureofEconomics:AMathematicalAnalysis,3rded.Boston:Irwin/McGraw-Hill,2001.ProvidesanextensivederivationoftheSlutskyequationandalengthypresentationofelasticityconcepts.Sydsaetter,K.,A.Strom,andP.Berck.Economist’sMathematicalManual.Berlin:Springer-Verlag,2003.Providesacompactsummaryofelasticityconcepts.Thecoverageofelasticityofsubstitutionnotionsisespeciallycomplete.Varian,H.MicroeconomicAnalysis,3rded.NewYork:W.W.Norton,1992.Formaldevelopmentofpreferencenotions.Extensiveuseofexpendi-turefunctionsandtheirrelationshiptotheSlutskyequation.AlsocontainsaniceproofofRoy‘sidentity.Chapter5IncomeandSubstitutionEffects177
EXTENSIONSDemandConceptsandtheEvaluationofPriceIndicesInChapters4and5weintroducedanumberofrelateddemandconcepts,allofwhichwerederivedfromtheunderlyingmodelofutilitymaximization.Relation-shipsamongthesevariousconceptsaresummarizedinFigureE5.1.Wehavealreadylookedatmostofthelinksinthetableformally.WehavenotyetdiscussedthemathematicalrelationshipbetweenindirectutilityfunctionsandMarshalliandemandfunctions(Roy’sidentity),andwewilldothatbelow.Alloftheentriesinthetablemakeclearthattherearemanywaystolearnsomethingabouttherelationshipbetweenin-dividuals’welfareandthepricestheyface.Inthisex-tensionwewillexploresomeoftheseapproaches.Speciﬁcally,wewilllookathowtheconceptscanshedlightontheaccuracyoftheconsumerpriceindex(CPI),theprimarymeasureofinﬂationintheUnitedStates.Wewillalsolookatafewotherpriceindexconcepts.TheCPIisa“marketbasket”indexofthecostofliving.Researchersmeasuretheamountsthatpeopleconsumeofasetofgoodsinsomebaseperiod(inthetwo-goodcasethesebase-periodconsumptionlevelsmightbedenotedbyx0andy0)andthenusecurrentpricedatatocomputethechangingpriceofthismar-ketbasket.Usingthisprocedure,thecostofthemarketbasketinitiallywouldbeI0¼p0xx0þp0yy0andthecostinperiod1wouldbeI1¼p1xx0þp1yy0.ThechangeinthecostoflivingbetweenthesetwoperiodswouldthenbemeasuredbyI1=I0.Althoughthispro-cedureisanintuitivelyplausiblewayofmeasuringin-ﬂationandmarketbasketpriceindicesarewidelyused,suchindiceshavemanyshortcomings.E5.1ExpenditurefunctionsandsubstitutionbiasMarket-basketpriceindicessufferfrom“substitutionbias.”Becausetheindicesdonotpermitindividualstomakesubstitutionsinthemarketbasketinresponsetochangesinrelativeprices,theywilltendtooverstateFIGUREE5.1RelationshipsamongDemandConceptsPrimalDualInversesShephard’s lemmaRoy’s identityMaximize U(x,y)s.t. I=Pxx+PyyIndirect utility functionU*=V(px,py,I)Minimize E(x,y)s.t. U=U(x, y)x(px, py, I)= –Marshallian demandxc(px, py, U)=∂E∂pxCompensated demandExpenditure functionE* = E(px, py, U)∂px∂I∂V∂V178Part2ChoiceandDemand
thewelfarelossesthatpeopleincurfromrisingprices.ThisexaggerationisillustratedinFigureE5.2.ToachievetheutilitylevelU0initiallyrequiresexpendi-turesofE0,resultinginapurchaseofthebasketx0,y0.Ifpx=pyfalls,theinitialutilitylevelcannowbeob-tainedwithexpendituresofE1byalteringtheconsumptionbundletox1,y1.Computingtheexpen-diturelevelneededtocontinueconsumingx0,y0exaggerateshowmuchextrapurchasingpowerthispersonneedstorestorehisorherlevelofwell-being.Economistshaveextensivelystudiedtheextentofthissubstitutionbias.AizcorbeandJackman(1993),forexample,ﬁndthatthisdifﬁcultywithamarketbasketindexmayexaggeratethelevelofinﬂationshownbytheCPIbyabout0.2percentperyear.E5.2Roy’sidentityandnewgoodsbiasWhennewgoodsareintroduced,ittakessometimeforthemtobeintegratedintotheCPI.Forexample,Hausman(1999,2003)statesthatittookmorethan15yearsforcellphonestoappearintheindex.Theproblemwiththisdelayisthatmarketbasketindiceswillfailtoreﬂectthewelfaregainsthatpeopleexperi-encefromusingnewgoods.Tomeasurethesecosts,Hausmansoughttomeasurea“virtual”price(p)atwhichthedemandfor,say,cellphoneswouldbezeroandthenarguedthattheintroductionofthegoodatitsmarketpricerepresentedachangeinconsumersurplusthatcouldbemeasured.Hence,theauthorFIGUREE5.2SubstitutionBiasintheCPIInitiallyexpendituresaregivenbyE0andthisindividualbuysx0,y0.Ifpx=pyfalls,utilitylevelU0canbereachedmostcheaplybyconsumingx1,y1andspendingE1.Purchasingx0,y0atthenewpriceswouldcostmorethanE1.Hence,holdingtheconsumptionbundleconstantimpartsanupwardbiastoCPI-typecomputations.E0U0E1x0y0x1Quantity of xQuantityof yChapter5IncomeandSubstitutionEffects179
wasfacedwiththeproblemofhowtogetfromtheMarshalliandemandfunctionforcellphones(whichheestimatedeconometrically)totheexpenditurefunction.TodosoheusedRoy’sidentity(seeRoy,1942).Rememberthattheconsumer’sutility-maxi-mizingproblemcanberepresentedbytheLagrangianexpressionℒ¼Uðx,yÞþλðIpxxpyyÞ.Ifweapplytheenvelopetheoremtothisexpression,weknowthat∂U∂px¼∂ℒ∂px¼λxðpx,py,IÞ,∂U∂I¼∂ℒ∂I¼λ.(i)HencetheMarshalliandemandfunctionisgivenbyxðpx,py,IÞ¼∂U=∂px∂U=∂I.(ii)UsinghisestimatesoftheMarshalliandemandfunction,HausmanintegratedEquationiitoobtain-theimpliedindirectutilityfunctionandthencalculat-editsinverse,theexpenditurefunction(checkFigureE5.1toseethelogicoftheprocess).Thoughthiscertainlyisaroundaboutscheme,itdidyieldlargeestimatesforthegaininconsumerwelfarefromcellphones—apresentvaluein1999ofmorethan$100billion.DelaysintheinclusionofsuchgoodsintotheCPIcanthereforeresultinamisleadingmeasureofconsumerwelfare.E5.3OthercomplaintsabouttheCPIResearchershavefoundseveralotherfaultswiththeCPIascurrentlyconstructed.Mostofthesefocusontheconsequencesofusingincorrectpricestocomputetheindex.Forexample,whenthequalityofagoodimproves,peoplearemadebetter-off,thoughthismaynotshowupinthegood’sprice.Throughoutthe1970sand1980sthereliabilityofcolortelevisionsetsimproveddramatically,butthepriceofasetdidnotchangeverymuch.Amarketbasketthatincluded“onecolortelevisionset”wouldmissthissourceofimprovedwelfare.Similarly,theopeningof“bigbox”retailerssuchasCostcoandHomeDepotduringthe1990sundoubtedlyreducedthepricesthatconsumerspaidforvariousgoods.ButincludingthesenewretailoutletsintothesampleschemefortheCPItooksev-eralyears,sotheindexmisrepresentedwhatpeoplewereactuallypaying.AssessingthemagnitudeoferrorintroducedbythesecaseswhereincorrectpricesareusedintheCPIcanalsobeaccomplishedbyusingthevariousdemandconceptsinFigureE5.1.Forasum-maryofthisresearch,seeMoulton(1996).E5.4ExactpriceindicesInprinciple,itispossiblethatsomeoftheshortcom-ingsofpriceindicessuchastheCPImightbeamelio-ratedbymorecarefulattentiontodemandtheory.Iftheexpenditurefunctionfortherepresentativecon-sumerwereknown,forexample,itwouldbepossibletoconstructan“exact”indexforchangesinpurchas-ingpowerthatwouldtakecommoditysubstitutionintoaccount.Toillustratethis,supposethereareonlytwogoodsandwewishtoknowhowpurchasingpowerhaschangedbetweenperiod1andperiod2.IftheexpenditurefunctionisgivenbyEðpx,py,UÞthentheratioI1,2¼Eðp2x,p2y,_UÞEðp1x,p1y,_UÞ(iii)showshowthecostofattainingthetargetutilitylevelU_haschangedbetweenthetwoperiods.If,forexample,I1,2¼1:04,thenwewouldsaythatthecostofattainingtheutilitytargethadincreasedby4per-cent.Ofcourse,thisanswerisonlyaconceptualone.Withoutknowingtherepresentativeperson’sutilityfunction,wewouldnotknowthespecificformoftheexpenditurefunction.ButinsomecasesEquationiiimaysuggesthowtoproceedinindexconstruction.Suppose,forexample,thatthetypicalperson’spref-erencescouldberepresentedbytheCobb-DouglasutilityfunctionUðx,yÞ¼xαy1α.Inthiscaseitiseasytoshowthattheexpenditurefunctionisagen-eralizationoftheonegiveninExample4.4:Eðpx,py,UÞ¼pαxp1αyU=ααð1αÞ1α¼kpαxp1αyU.InsertingthisfunctionintoEquationiiiyieldsI1,2¼kðp2xÞαðp2yÞ1α_Ukðp1xÞαðp1yÞ1α_U¼ðp2xÞαðp2yÞ1αðp1xÞαðp1yÞ1α.(iv)So,inthiscase,theexactpriceindexisarelativelysimplefunctionoftheobservedprices.Theparticularlyusefulfeatureofthisexampleisthattheutilitytargetcancelsoutintheconstructionofthecost-of-livingindex(asitwillanytimetheexpenditurefunctionishomogeneousinutility).Noticealsothattheexpendi-tureshares(αand1α)playanimportantroleintheindex—thelargeragood’sshare,themoreimpor-tantwillchangesbeinthatgood’spriceintheﬁnalindex.180Part2ChoiceandDemand
E5.5DevelopmentofexactpriceindicesTheCobb-Douglasutilityfunctionis,ofcourse,averysimpleone.Muchrecentresearchonpriceindiceshasfocusedonmoregeneraltypesofutilityfunctionsandonthediscoveryoftheexactpriceindicestheyimply.Forexample,FeenstraandReinsdorf(2000)showthatthealmostidealdemandsystemdescribedintheExtensionstoChapter4impliesanexactpriceindex(I)thattakesa“Divisia”form:lnðIÞ¼Xni¼1wi∆lnpi(v)(herethewiareweightstobeattachedtothechangeinthelogarithmofeachgood’sprice).OftentheweightsinEquationvaretakentobethebudgetsharesofthegoods.Interestingly,thisispreciselythepriceindeximpliedbytheCobb-DouglasutilityfunctioninEquationiv,sincelnðI1;2Þ¼αlnp2xþð1αÞlnp2yαlnp1xð1αÞlnp1y¼α∆lnpxþð1−αÞ∆lnpy.(vi)Inactualapplications,theweightswouldchangefromperiodtoperiodtoreﬂectchangingbudgetshares.Similarly,changesoverseveralperiodswouldbe“chained”togetherfromanumberofsingle-periodpricechangeindices.ChangingdemandsforfoodinChina.Chinahasoneofthefastestgrowingeconomiesintheworld:itsGDPpercapitaiscurrentlygrowingatarateofabout8percentperyear.Chineseconsumersalsospendalargefractionoftheirincomesonfood—approximately38percentoftotalexpendituresinrecentsurveydata.OneimplicationoftherapidgrowthinChineseincomes,however,isthatpatternsoffoodconsump-tionarechangingrapidly.Purchasesofstaples,suchasriceorwheat,aredeclininginrelativeimportance,whereaspurchasesofpoultry,ﬁsh,andprocessedfoodsaregrowingrapidly.ArecentpaperbyGouldandVillarreal(2006)studiesthesepatternsindetailusingtheAIDSmodel.Theyidentifyavarietyofsub-stitutioneffectsacrossspeciﬁcfoodcategoriesinre-sponsetochangingrelativeprices.Suchchangingpatternsimplythataﬁxedmarketbasketpriceindex(suchastheU.S.ConsumerPriceIndex)wouldbeparticularlyinappropriateformeasuringchangesinthecostoflivinginChinaandthatsomealternativeapproachesshouldbeexamined.ReferencesAizcorbe,AnaM.,andPatrickC.Jackman.“TheCom-moditySubstitutionEffectinCPIData,1982–91.”MonthlyLaborReview(December1993):25–33.Feenstra,RobertC.,andMarshallB.Reinsdorf.“AnExactPriceIndexfortheAlmostIdealDemandSystem.”EconomicsLetters(February2000):159–62.Gould,BrainW.,andHectorJ.Villarreal.“AnAssessmentoftheCurrentStructureofFoodDemandinUrbanChina.”AgriculturalEconomics(January2006):1–16.Hausman,Jerry.“CellularTelephone,NewProducts,andtheCPI.”JournalofBusinessandEconomicStatistics(April1999):188–94.Hausman,Jerry.“SourcesofBiasandSolutionstoBiasintheConsumerPriceIndex.”JournalofEconomicPerspectives(Winter2003):23–44.Moulton,BrentR.“BiasintheConsumerPriceIndex:WhatIstheEvidence?”JournalofEconomicPerspectives(Fall1996):159–77.Roy,R.Del’utilité,contributionàlathéoriedeschoix.Paris:Hermann,1942.Chapter5IncomeandSubstitutionEffects181
CHAPTER6DemandRelationshipsamongGoodsInChapter5weexaminedhowchangesinthepriceofaparticulargood(say,goodx)affectthequantityofthatgoodchosen.Throughoutthediscussion,weheldthepricesofallothergoodsconstant.Itshouldbeclear,however,thatachangeinoneoftheseotherpricescouldalsoaffectthequantityofxchosen.Forexample,ifxweretakentorepresentthequantityofautomobilemilesthatanindividualdrives,thisquantitymightbeexpectedtodeclinewhenthepriceofgasolinerisesorincreasewhenairandbusfaresrise.Inthischapterwewillusetheutility-maximizationmodeltostudysuchrelationships.THETWO-GOODCASEWebeginourstudyofthedemandrelationshipamonggoodswiththetwo-goodcase.Unfortunately,thiscaseprovestoberatheruninterestingbecausethetypesofrelationshipsthatcanoccurwhenthereareonlytwogoodsarequitelimited.Still,thetwo-goodcaseisusefulbecauseitcanbeillustratedwithtwo-dimensionalgraphs.Figure6.1startsourex-aminationbyshowingtwoexamplesofhowthequantityofxchosenmightbeaffectedbyachangeinthepriceofy.Inbothpanelsoftheﬁgure,pyhasfallen.ThishastheresultofshiftingthebudgetconstraintoutwardfromI0toI1.Inbothcases,thequantityofgoodychosenhasalsoincreasedfromy0toy1asaresultofthedeclineinpy,aswouldbeexpectedifyisanormalgood.Forgoodx,however,theresultsshowninthetwopanelsdiffer.In(a)theindifferencecurvesarenearlyL-shaped,implyingafairlysmallsubstitutioneffect.AdeclineinpydoesnotinduceaverylargemovealongU0asyissubstitutedforx.Thatis,xdropsrelativelylittleasaresultofthesubstitution.Theincomeeffect,however,reﬂectsthegreaterpurchasingpowernowavailable,andthiscausesthetotalquantityofxchosentoincrease.Hence,∂x=∂pyisnegative(xandpymoveinoppositedirections).InFigure6.1bthissituationisreversed:∂x=∂pyispositive.TherelativelyﬂatindifferencecurvesinFigure6.1bresultinalargesubstitutioneffectfromthefallinpy.ThequantityofxdeclinessharplyasyissubstitutedforxalongU0.AsinFigure6.1a,theincreasedpurchasingpowerfromthedeclineinpycausesmorextobebought,butnowthesubstitutioneffectdominatesandthequantityofxdeclinestox1.Inthiscase,then,xandpymoveinthesamedirection.AmathematicaltreatmentTheambiguityintheeffectofchangesinpycanbefurtherillustratedbyaSlutsky-typeequation.ByusingproceduressimilartothoseinChapter5,itisfairlysimpletoshowthat∂xðpx,py,IÞ∂py¼substitutioneffectþincomeeffect¼∂x∂pyU¼constanty⋅∂x∂I,(6.1)182
or,inelasticityterms,ex,py¼exc,pysyex,I.(6.2)Noticethatthesizeoftheincomeeffectisdeterminedbytheshareofgoodyinthisperson’spurchases.Theimpactofachangeinpyonpurchasingpowerisdeterminedbyhowim-portantyistothisperson.Forthetwo-goodcase,thetermsontherightsideofEquations6.1and6.2havedifferentsigns.Assumingthatindifferencecurvesareconvex,thesubstitutioneffect∂x=∂pyjU¼constantispositive.Ifweconﬁneourselvestomovesalongoneindifferencecurve,increasesinpyincreasexanddecreasesinpydecreasethequantityofxchosen.But,assumingxisanormalgood,theincomeeffect(y∂x=∂Iorsyex,I)isclearlynegative.Hence,thecombinedeffectisambiguous;∂x=∂pycouldbeeitherpositiveornegative.Eveninthetwo-goodcase,thedemandrelationshipbetweenxandpyisrathercomplex.EXAMPLE6.1AnotherSlutskyDecompositionforCross-PriceEffectsInExample5.4weexaminedtheSlutskydecompositionfortheeffectofachangeinthepriceofx.Nowlet’slookatthecross-priceeffectofachangeinypricesonxpurchases.Rememberthattheuncompensatedandcompensateddemandfunctionsforxaregivenbyxðpx,py,IÞ¼0:5Ipx(6.3)andxcðpx,py,VÞ¼Vp0:5yp0:5x.(6.4)(continued)FIGURE6.1DifferingDirectionsofCross-PriceEffectsInbothpanels,thepriceofyhasfallen.In(a),substitu-tioneffectsaresmallsothequantityofxconsumedincreasesalongwithy.Because∂x=∂py<0,xandyaregrosscomplements.In(b),substitutioneffectsarelargesothequantityofxchosenfalls.Because∂x=∂py>0,xandywouldbetermedgrosssubstitutes.Quantityof xQuantityof xQuantity of yQuantity of y(a) Gross complements(b) Gross substitutesx0y0y1x1y0y1x1x0I0I0I1I1U0U1U0U1Chapter6DemandRelationshipsamongGoods183
EXAMPLE6.1CONTINUEDAswehavepointedoutbefore,theMarshalliandemandfunctioninthiscaseyields∂x=∂py¼0;thatis,changesinthepriceofydonotaffectxpurchases.Nowweshowthatthisoccursbecausethesubstitutionandincomeeffectsofapricechangearepreciselycounterbalancing.Thesubstitutioneffectinthiscaseisgivenby∂x∂pyU¼constant¼∂xc∂py¼0:5Vp0:5yp0:5x.(6.5)SubstitutingforVfromtheindirectutilityfunction(V¼0:5Ip0:5yp0:5x)givesafinalstate-mentforthesubstitutioneffect:∂x∂pyU¼constant¼0:25Ip1yp1x.(6.6)ReturningtotheMarshalliandemandfunctionforyðy¼0:5Ip1y)tocalculatetheincomeeffectyieldsy∂x∂I¼½0:5Ip1y⋅½0:5p1x¼0:25Ip1yp1x,(6.7)andcombiningEquations6.6and6.7givesthetotaleffectofthechangeinthepriceofyas∂x∂py¼0:25Ip1yp1x0:25Ip1yp1x¼0.(6.8)ThismakesclearthatthereasonthatchangesinthepriceofyhavenoeffectonxpurchasesintheCobb-Douglascaseisthatthesubstitutionandincomeeffectsfromsuchachangearepreciselyoffsetting;neitheroftheeffectsalone,however,iszero.Returningtoournumericalexample(px¼1,py¼4,I¼8,V¼2),supposenowthatpyfallsto2.ThisshouldhavenoeffectontheMarshalliandemandforgoodx.Thecompen-sateddemandfunctioninEquation6.4showsthatthepricechangewouldcausethequantityofxdemandedtodeclinefrom4to2.83(¼2ﬃﬃﬃ2p)asyissubstitutedforxwithutilityunchanged.However,theincreasedpurchasingpowerarisingfromthepricedeclinepreciselyreversesthiseffect.QUERY:Whywoulditbeincorrecttoarguethatif∂x=∂py¼0,thenxandyhavenosubstitutionpossibilities—thatis,theymustbeconsumedinﬁxedproportions?Isthereanycaseinwhichsuchaconclusioncouldbedrawn?SUBSTITUTESANDCOMPLEMENTSWithmanygoods,thereismuchmoreroomforinterestingrelationsamonggoods.ItisrelativelyeasytogeneralizetheSlutskyequationforanytwogoodsxi,xjas∂xiðp1,…,pn,IÞ∂pj¼∂xi∂pjU¼constantxj∂xi∂I,(6.9)andagainthiscanbereadilytranslatedintoanelasticityrelation:ei,j¼eci,jsjei,I.(6.10)Thissaysthatthechangeinthepriceofanygood(here,goodj)inducesincomeandsubstitutioneffectsthatmaychangethequantityofeverygooddemanded.Equations6.9and6.10canbeusedtodiscusstheideaofsubstitutesandcomplements.Intuitively,theseideasare184Part2ChoiceandDemand
rathersimple.Twogoodsaresubstitutesifonegoodmay,asaresultofchangedconditions,replacetheotherinuse.Someexamplesareteaandcoffee,hamburgersandhotdogs,andbutterandmargarine.Complements,ontheotherhand,aregoodsthat“gotogether,”suchascoffeeandcream,fishandchips,orbrandyandcigars.Insomesense,“substitutes”substituteforoneanotherintheutilityfunctionwhereas“complements”complementeachother.Therearetwodifferentwaystomaketheseintuitiveideasprecise.Oneofthesefocusesonthe“gross”effectsofpricechangesbyincludingbothincomeandsubstitutioneffects;theotherlooksatsubstitutioneffectsalone.Becausebothdeﬁnitionsareused,wewillexamineeachindetail.GrosssubstitutesandcomplementsWhethertwogoodsaresubstitutesorcomplementscanbeestablishedbyreferringtoobservedpricereactionsasfollows.DEFINITIONGrosssubstitutesandcomplements.Twogoods,xiandxj,aresaidtobegrosssubsti-tutesif∂xi∂pj>0(6.11)andgrosscomplementsif∂xi∂pj<0.(6.12)Thatis,twogoodsaregrosssubstitutesifariseinthepriceofonegoodcausesmoreoftheothergoodtobebought.Thegoodsaregrosscomplementsifariseinthepriceofonegoodcauseslessoftheothergoodtobepurchased.Forexample,ifthepriceofcoffeerises,thedemandforteamightbeexpectedtoincrease(theyaresubstitutes),whereasthedemandforcreammightdecrease(coffeeandcreamarecomplements).Equation6.9makesitclearthatthisdeﬁnitionisa“gross”deﬁnitioninthatitincludesbothincomeandsubstitutioneffectsthatarisefrompricechanges.Becausetheseeffectsareinfactcombinedinanyreal-worldobservationwecanmake,itmightbereasonablealwaystospeakonlyof“gross”substitutesand“gross”complements.AsymmetryofthegrossdefinitionsThereare,however,severalthingsthatareundesirableaboutthegrossdeﬁnitionsofsub-stitutesandcomplements.Themostimportantoftheseisthatthedeﬁnitionsarenotsymmetric.Itispossible,bythedeﬁnitions,forx1tobeasubstituteforx2andatthesametimeforx2tobeacomplementofx1.Thepresenceofincomeeffectscanproduceparadoxicalresults.Let’slookataspeciﬁcexample.EXAMPLE6.2AsymmetryinCross-PriceEffectsSupposetheutilityfunctionfortwogoods(xandy)hasthequasi-linearformUðx,yÞ¼lnxþy.(6.13)SettinguptheLagrangianexpressionℒ¼lnxþyþλðIpxxpyyÞ(6.14)(continued)Chapter6DemandRelationshipsamongGoods185 EXAMPLE6.2CONTINUEDyieldsthefollowingﬁrst-orderconditions:∂ℒ∂x¼1xλpx¼0,∂ℒ∂y¼1λpy¼0,∂ℒ∂λ¼Ipxxpyy¼0.(6.15)Movingthetermsinλtotherightanddividingthefirstequationbythesecondyields1x¼pxpy,(6.16)pxx¼py.(6.17)SubstitutionintothebudgetconstraintnowpermitsustosolvefortheMarshalliandemandfunctionfory:I¼pxxþpyy¼pyþpyy.(6.18)Hence,y¼Ipypy.(6.19)Thisequationshowsthatanincreaseinpymustdecreasespendingongoody(thatis,pyy).Therefore,sincepxandIareunchanged,spendingonxmustrise.So∂x∂py>0,(6.20)andwewouldtermxandygrosssubstitutes.Ontheotherhand,Equation6.19showsthatspendingonyisindependentofpx.Consequently,∂y∂px¼0(6.21)and,lookedatinthisway,xandywouldbesaidtobeindependentofeachother;theyareneithergrosssubstitutesnorgrosscomplements.Relyingongrossresponsestopricechangestodefinetherelationshipbetweenxandywouldthereforerunintoambiguity.QUERY:InExample3.4,weshowedthatautilityfunctionoftheformgivenbyEquation6.13isnothomothetic:theMRSdoesnotdependonlyontheratioofxtoy.Canasymmetryariseinthehomotheticcase?NETSUBSTITUTESANDCOMPLEMENTSBecauseofthepossibleasymmetriesinvolvedinthedeﬁnitionofgrosssubstitutesandcomplements,analternativedeﬁnitionthatfocusesonlyonsubstitutioneffectsisoftenused.DEFINITIONNetsubstitutesandcomplements.Goodsxiandxjaresaidtobenetsubstitutesif∂xi∂pjU¼constant>0(6.22)186Part2ChoiceandDemand
andnetcomplementsif∂xi∂pjU¼constant<0.(6.23)Thesedeﬁnitions,1then,lookonlyatthesubstitutiontermstodeterminewhethertwogoodsaresubstitutesorcomplements.Thisdeﬁnitionisbothintuitivelyappealing(becauseitlooksonlyattheshapeofanindifferencecurve)andtheoreticallydesirable(becauseitisunambig-uous).Oncexiandxjhavebeendiscoveredtobesubstitutes,theystaysubstitutes,nomatterinwhichdirectionthedeﬁnitionisapplied.Asamatteroffact,thedeﬁnitionsareperfectlysymmetric:∂xi∂pjU¼constant¼∂xj∂piU¼constant.(6.24)Thesubstitutioneffectofachangeinpiongoodxjisidenticaltothesubstitutioneffectofachangeinpjonthequantityofxichosen.Thissymmetryisimportantinboththeoreticalandempiricalwork.2ThedifferencesbetweenthetwodeﬁnitionsofsubstitutesandcomplementsareeasilydemonstratedinFigure6.1a.Inthisﬁgure,xandyaregrosscomplements,buttheyarenetsubstitutes.Thederivative∂x=∂pyturnsouttobenegative(xandyaregrosscomplements)becausethe(positive)substitutioneffectisoutweighedbythe(negative)incomeeffect(afallinthepriceofycausesrealincometoincreasegreatly,and,consequently,actualpurchasesofxincrease).However,astheﬁguremakesclear,ifthereareonlytwogoodsfromwhichtochoose,theymustbenetsubstitutes,althoughtheymaybeeithergrosssubstitutesorgrosscomplements.BecausewehaveassumedadiminishingMRS,theown-pricesub-stitutioneffectmustbenegativeand,consequently,thecross-pricesubstitutioneffectmustbepositive.SUBSTITUTABILITYWITHMANYGOODSOncetheutility-maximizingmodelisextendedtomanygoods,awidevarietyofdemandpatternsbecomepossible.Whetheraparticularpairofgoodsarenetsubstitutesornetcomplementsisbasicallyaquestionofaperson’spreferences,soonemightobserveallsortsofoddrelationships.Amajortheoreticalquestionthathasconcernedeconomistsiswhethersubstitutabilityorcomplementarityismoreprevalent.Inmostdiscussions,wetendtoregardgoodsassubstitutes(apriceriseinonemarkettendstoincreasedemandinmostothermarkets).Itwouldbenicetoknowwhetherthisintuitionisjustiﬁed.1Thesearesometimescalled“Hicksian”substitutesandcomplements,namedaftertheBritisheconomistJohnHicks,whooriginallydevelopedthedefinitions.2ThissymmetryiseasilyshownusingShephard’slemma.Compensateddemandfunctionscanbecalculatedfromex-penditurefunctionsbydifferentiation:xciðp1,…,pn,VÞ¼∂Eðp1,…,pn,VÞ∂pi.Hence,thesubstitutioneffectisgivenby∂xi∂pjU¼constant¼∂xci∂pj¼∂2E∂pj∂pi¼Eij.ButnowwecanapplyYoung’stheoremtotheexpenditurefunction:Eij¼Eji¼∂xcj∂pi¼∂xj∂piU¼constant,whichprovesthesymmetry.Chapter6DemandRelationshipsamongGoods187 TheBritisheconomistJohnHicksstudiedthisissueinsomedetailabout50yearsagoandreachedtheconclusionthat“most”goodsmustbesubstitutes.Theresultissummarizedinwhathascometobecalled“Hicks’secondlawofdemand.”3Amodernproofstartswiththecompensateddemandfunctionforaparticulargood:xciðp1,…,pn,VÞ.Thisfunctionishomogeneousofdegree0inallprices(ifutilityisheldconstantandpricesdouble,quantitiesdemandeddonotchangebecausetheutility-maximizingtangenciesdonotchange).Apply-ingEuler’stheoremtothefunctionyieldsp1⋅∂xci∂p1þp2⋅∂xci∂p2þ…þpn⋅∂xci∂pn¼0.(6.25)WecanputthisresultintoelasticitytermsbydividingEquation6.25byxi:eci1þeci2þ…þecin¼0.(6.26)Butweknowthatecii0becauseofthenegativityoftheown-substitutioneffect.HenceitmustbethecasethatXj≠iecij0.(6.27)Inwords,thesumofallthecompensatedcross-priceelasticitiesforaparticulargoodmustbepositive(orzero).Thisisthesensethat“most”goodsaresubstitutes.Empiricalevidenceseemsgenerallyconsistentwiththistheoreticalfinding:instancesofnetcom-plementaritybetweengoodsareencounteredrelativelyinfrequentlyinempiricalstudiesofdemand.COMPOSITECOMMODITIESOurdiscussionintheprevioussectionshowedthatthedemandrelationshipsamonggoodscanbequitecomplicated.Inthemostgeneralcase,anindividualwhoconsumesngoodswillhavedemandfunctionsthatreﬂectnðnþ1Þ=2differentsubstitutioneffects.4Whennisverylarge(asitsurelyisforallthespeciﬁcgoodsthatindividualsactuallyconsume),thisgeneralcasecanbeunmanageable.Itisoftenfarmoreconvenienttogroupgoodsintolargeraggregatessuchasfood,clothing,shelter,andsoforth.Atthemostextremelevelofaggregates,wemightwishtoexamineonespeciﬁcgood(say,gasoline,whichwemightcallx)anditsrelationshipto“allothergoods,”whichwemightcally.Thisistheprocedurewehavebeenusinginsomeofourtwo-dimensionalgraphs,andwewillcontinuetodosoatmanyotherplacesinthisbook.Inthissectionweshowtheconditionsunderwhichthisprocedurecanbedefended.IntheExtensionstothischapter,weexploremoregeneralissuesinvolvedinaggregatinggoodsintolargergroupings.CompositecommoditytheoremSupposeconsumerschooseamongngoodsbutthatweareonlyinterestedspeciﬁcallyinoneofthem—say,x1.Ingeneral,thedemandforx1willdependontheindividualpricesoftheothern1commodities.Butifallthesepricesmovetogether,itmaymakesenseto3SeeJohnHicks,ValueandCapital(Oxford:OxfordUniversityPress,1939),mathematicalappendices.ThereissomedebateaboutwhetherthisresultshouldbecalledHicks’“second”or“third”law.Infact,twootherlawsthatwehavealreadyseenarelistedbyHicks:(1)∂xci=∂pi0(negativityoftheown-substitutioneffect);and(2)∂xci=∂pj¼∂xcj=∂pi(symmetryofcross-substitutioneffects).Butherefersexplicitlyonlytotwo“properties”inhiswrittensummaryofhisresults.4Toseethis,noticethatallsubstitutioneffects,sij,couldberecordedinannnmatrix.However,symmetryoftheeffects(sij¼sji)impliesthatonlythosetermsonandbelowtheprincipaldiagonalofthismatrixmaybedistinctlydifferentfromeachother.Thisamountstohalfthetermsinthematrix(n2=2)plustheremaininghalfofthetermsonthemaindiagonalofthematrix(n=2).188Part2ChoiceandDemand lumpthemintoasingle“compositecommodity,”y.Formally,ifweletp02,…,p0nrepresenttheinitialpricesofthesegoods,thenweassumethatthesepricescanonlyvarytogether.Theymightalldouble,oralldeclineby50percent,buttherelativepricesofx2,…,xnwouldnotchange.Nowwedeﬁnethecompositecommodityytobetotalexpendituresonx2,…,xn,usingtheinitialpricesp02,…,p0n:y¼p02x2þp03x3þ…þp0nxn.(6.28)Thisperson’sinitialbudgetconstraintisgivenbyI¼p1x1þp02x2þ…þp0nxn¼p1x1þy.(6.29)Byassumption,allofthepricesp2,…,pnchangeinunison.Assumeallofthesepriceschangebyafactoroftðt>0Þ.NowthebudgetconstraintisI¼p1x1þtp02x2þ…þtp0nxn¼p1x1þty.(6.30)Consequently,thefactorofproportionality,t,playsthesameroleinthisperson’sbudgetconstraintasdidthepriceofyðpyÞinourearliertwo-goodanalysis.Changesinp1ortinducethesamekindsofsubstitutioneffectswehavebeenanalyzing.Solongasp2,…,pnmovetogether,wecanthereforeconfineourexaminationofdemandtochoicesbetweenbuyingx1orbuying“everythingelse.”5Simplifiedgraphsthatshowthesetwogoodsontheiraxescanthereforebedefendedrigorouslysolongastheconditionsofthis“compositecommoditytheorem”(thatallotherpricesmovetogether)aresatisfied.Notice,however,thatthetheoremmakesnopredictionsabouthowchoicesofx2,…,xnbehave;theyneednotmoveinunison.Thetheoremfocusesonlyontotalspendingonx2,…,xn,notonhowthatspendingisallocatedamongspecificitems(althoughthisallocationisassumedtobedoneinautility-maximizingway).GeneralizationsandlimitationsThecompositecommoditytheoremappliestoanygroupofcommoditieswhoserelativepricesallmovetogether.Itispossibletohavemorethanonesuchcommodityifthereareseveralgroupingsthatobeythetheorem(i.e.,expenditureson“food,”“clothing,”andsoforth).Hence,wehavedevelopedthefollowingdeﬁnition.DEFINITIONCompositecommodity.Acompositecommodityisagroupofgoodsforwhichallpricesmovetogether.Thesegoodscanbetreatedasasingle“commodity”inthattheindividualbehavesasifheorshewerechoosingbetweenothergoodsandtotalspendingontheentirecompositegroup.Thisdeﬁnitionandtherelatedtheoremareverypowerfulresults.Theyhelpsimplifymanyproblemsthatwouldotherwisebeintractable.Still,onemustberathercarefulinapplyingthetheoremtotherealworldbecauseitsconditionsarestringent.Findingasetofcom-moditieswhosepricesmovetogetherisrare.Slightdeparturesfromstrictproportionalitymaynegatethecompositecommoditytheoremifcross-substitutioneffectsarelarge.IntheExtensionstothischapter,welookatwaystosimplifysituationswherepricesmoveindependently.5Theideaofa“compositecommodity”wasalsointroducedbyJ.R.HicksinValueandCapital,2nded.(Oxford:OxfordUniversityPress,1946),pp.312–13.Proofofthetheoremreliesonthenotionthattoachievemaximumutility,theratioofthemarginalutilitiesforx2,…,xnmustremainunchangedwhenp2,…,pnallmovetogether.Hence,then-goodproblemcanbereducedtothetwo-dimensionalproblemofequatingtheratioofthemarginalutilityfromxtothatfromytothe“priceratio”p1=t.Chapter6DemandRelationshipsamongGoods189
EXAMPLE6.3HousingCostsasaCompositeCommoditySupposethatanindividualreceivesutilityfromthreegoods:food(x),housingservices(y)measuredinhundredsofsquarefeet,andhouseholdoperations(z)asmeasuredbyelec-tricityuse.Iftheindividual’sutilityisgivenbythethree-goodCESfunctionutility¼Uðx,y,zÞ¼1x1y1z,(6.31)thentheLagrangiantechniquecanbeusedtocalculateMarshalliandemandfunctionsforthesegoodsasx¼Ipxþﬃﬃﬃﬃﬃﬃﬃﬃﬃpxpypþﬃﬃﬃﬃﬃﬃﬃﬃﬃpxpzp,y¼Ipyþﬃﬃﬃﬃﬃﬃﬃﬃﬃpypxpþﬃﬃﬃﬃﬃﬃﬃﬃﬃpypzp,z¼Ipzþﬃﬃﬃﬃﬃﬃﬃﬃﬃpzpxpþﬃﬃﬃﬃﬃﬃﬃﬃﬃpzpyp.(6.32)IfinitiallyI¼100,px¼1,py¼4,andpz¼1,thenthedemandfunctionspredictx¼25,y¼12:5,z¼25:(6.33)Hence,25isspentonfoodandatotalof75isspentonhousing-relatedneeds.Ifweassumethathousingserviceprices(py)andhouseholdoperationprices(pz)alwaysmovetogether,thenwecanusetheirinitialpricestodefinethe“compositecommodity”housing(h)ash¼4yþ1z.(6.34)Here,wealso(arbitrarily)definetheinitialpriceofhousing(ph)tobe1.Theinitialquantityofhousingissimplytotaldollarsspentonh:h¼4ð12:5Þþ1ð25Þ¼75.(6.35)Furthermore,becausepyandpzalwaysmovetogether,phwillalwaysberelatedtothesepricesbyph¼pz¼0:25py.(6.36)Usingthisinformation,wecanrecalculatethedemandfunctionforxasafunctionofI,px,andph:x¼Ipxþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ4pxphpþﬃﬃﬃﬃﬃﬃﬃﬃﬃpxphp¼Ipyþ3ﬃﬃﬃﬃﬃﬃﬃﬃﬃpxphp.(6.37)Asbefore,initiallyI¼100,px¼1,andph¼1,sox¼25.Spendingonhousingcanbemosteasilycalculatedfromthebudgetconstraintash¼75,becausespendingonhousingrepresents“everything”otherthanfood.Anincreaseinhousingcosts.Ifthepricesofyandzweretoriseproportionallytopy¼16,pz¼4(withpxremainingat1),thenphwouldalsoriseto4.Equation6.37nowpredictsthatthedemandforxwouldfalltox¼1001þ3ﬃﬃﬃ4p¼1007(6.38)190Part2ChoiceandDemand
andthathousingpurchaseswouldbegivenbyphh¼1001007¼6007,(6.39)or,becauseph¼4,h¼1507.(6.40)NoticethatthisispreciselythelevelofhousingpurchasespredictedbytheoriginaldemandfunctionsforthreegoodsinEquation6.32.WithI¼100,px¼1,py¼16,andpz¼4,theseequationscanbesolvedasx¼1007,y¼10028,z¼10014,(6.41)andsothetotalamountofthecompositegood“housing”consumed(accordingtoEqua-tion6.34)isgivenbyh¼4yþ1z¼1507.(6.42)Hence,weobtainedthesameresponsestopricechangesregardlessofwhetherwechosetoexaminedemandsforthethreegoodsx,y,andzortolookonlyatchoicesbetweenxandthecompositegoodh.QUERY:HowdoweknowthatthedemandfunctionforxinEquation6.37continuestoensureutilitymaximization?WhyistheLagrangianconstrainedmaximizationproblemunchangedbymakingthesubstitutionsrepresentedbyEquation6.36?HOMEPRODUCTION,ATTRIBUTESOFGOODS,ANDIMPLICITPRICESSofarinthischapterwehavefocusedonwhateconomistscanlearnabouttherelationshipsamonggoodsbyobservingindividuals’changingconsumptionofthesegoodsinreactiontochangesinmarketprices.Insomewaysthisanalysisskirtsthecentralquestionofwhycoffeeandcreamgotogetherorwhyﬁshandchickenmaysubstituteforeachotherinaperson’sdiet.Todevelopadeeperunderstandingofsuchquestions,economistshavesoughttoexploreactivitieswithinindividuals’households.Thatis,theyhavedevisedmodelsofnonmarkettypesofactivitiessuchasparentalchildcare,mealpreparation,ordo-it-yourselfconstructiontounderstandhowsuchactivitiesultimatelyresultindemandsforgoodsinthemarket.Inthissectionwebrieﬂyreviewsomeofthesemodels.Ourprimarygoalistoillustratesomeoftheimplicationsofthisapproachforthetraditionaltheoryofchoice.HouseholdproductionmodelThestartingpointformostmodelsofhouseholdproductionistoassumethatindividualsdonotreceiveutilitydirectlyfromgoodstheypurchaseinthemarket(aswehavebeenassumingsofar).Instead,itisonlywhenmarketgoodsarecombinedwithtimeinputsbytheindividualthatutility-providingoutputsareproduced.Inthisview,then,rawbeefanduncookedChapter6DemandRelationshipsamongGoods191
potatoesyieldnoutilityuntiltheyarecookedtogethertoproducestew.Similarly,marketpurchasesofbeefandpotatoescanbeunderstoodonlybyexaminingtheindividual’spreferencesforstewandtheunderlyingtechnologythroughwhichitisproduced.Informalterms,assumeasbeforethattherearethreegoodsthatapersonmightpurchaseinthemarket:x,y,andz.Purchasingthesegoodsprovidesnodirectutility,butthegoodscanbecombinedbytheindividualtoproduceeitheroftwohome-producedgoods:a1ora2.Thetechnologyofthishouseholdproductioncanberepresentedbytheproductionfunc-tionsf1andf2(seeChapter9foramorecompletediscussionoftheproductionfunctionconcept).Therefore,a1¼f1ðx,y,zÞ,a2¼f2ðx,y,zÞ,(6.43)andutility¼Uða1,a2Þ.(6.44)Theindividual’sgoalistochoosex,y,zsoastomaximizeutilitysubjecttotheproductionconstraintsandtoafinancialbudgetconstraint:6pxxþpyyþpzz¼I.(6.45)Althoughwewillnotexamineindetailtheresultsthatcanbederivedfromthisgeneralmodel,twoinsightsthatcanbedrawnfromitmightbementioned.First,themodelmayhelpclarifythenatureofmarketrelationshipsamonggoods.Becausetheproductionfunc-tionsinEquations6.43areinprinciplemeasurableusingdetaileddataonhouseholdoperations,householdscanbetreatedas“multi-product”firmsandstudiedusingmanyofthetechniqueseconomistsusetostudyproduction.Asecondinsightprovidedbythehouseholdproductionapproachisthenotionofthe“implicit”or“shadow”pricesassociatedwiththehome-producedgoodsa1anda2.Becauseconsumingmorea1,say,requirestheuseofmoreofthe“ingredients”x,y,andz,thisactivityobviouslyhasanopportunitycostintermsofthequantityofa2thatcanbeproduced.Toproducemorebread,say,apersonmustnotonlydivertsomeﬂour,milk,andeggsfromusingthemtomakecupcakesbutmayalsohavetoaltertherelativequantitiesofthesegoodspurchasedbecauseheorsheisboundbyanoverallbudgetconstraint.Hence,breadwillhaveanimplicitpriceintermsofthenumberofcupcakesthatmustbeforgoneinordertobeabletoconsumeonemoreloaf.Thatimplicitpricewillreﬂectnotonlythemarketpricesofbreadingredientsbutalsotheavailablehouseholdproductiontechnologyand,inmorecomplexmodels,therelativetimeinputsrequiredtoproducethetwogoods.Asastartingpoint,however,thenotionofimplicitpricescanbebestillustratedwithaverysimplemodel.ThelinearattributesmodelAparticularlysimpleformofthehouseholdproductionmodelwasﬁrstdevelopedbyK.J.Lancastertoexaminetheunderlying“attributes”ofgoods.7Inthismodel,itistheattributesofgoodsthatprovideutilitytoindividuals,andeachspeciﬁcgoodcontainsaﬁxedsetofattributes.If,forexample,wefocusonlyonthecalories(a1)andvitamins(a2)thatvariousfoodsprovide,Lancaster’smodelassumesthatutilityisafunctionoftheseattributesandthatindividualspurchasevariousfoodsonlyforthepurposeofobtainingthecaloriesandvitaminstheyoffer.Inmathematicalterms,themodelassumesthatthe“production”6Oftenhouseholdproductiontheoryalsofocusesontheindividual’sallocationoftimetoproducinga1anda2ortoworkinginthemarket.InChapter16welookatafewsimplemodelsofthistype.7SeeK.J.Lancaster,“ANewApproachtoConsumerTheory,”JournalofPoliticalEconomy74(April1966):132–57.192Part2ChoiceandDemand
equationshavethesimpleforma1¼a1xxþa1yyþa1zz,a2¼a2xxþa2yyþa2zz,(6.46)wherea1xrepresentsthenumberofcaloriesperunitoffoodx,a2xrepresentsthenumberofvitaminsperunitoffoodx,andsoforth.Inthisformofthemodel,then,thereisnoactual“production”inthehome.Rather,thedecisionproblemishowtochooseadietthatprovidestheoptimalmixofcaloriesandvitaminsgiventheavailablefoodbudget.IllustratingthebudgetconstraintsTobeginourexaminationofthetheoryofchoiceundertheattributesmodel,weﬁrstillustratethebudgetconstraint.InFigure6.2,theray0xrecordsthevariouscombinationsofa1anda2availablefromsuccessivelylargeramountsofgoodx.Becauseofthelinearproductiontechnologyassumedintheattributesmodel,thesecombinationsofa1anda2liealongsuchastraightline,thoughinmorecomplexmodelsofhomeproductionthatmightnotbethecase.Similarly,raysof0yand0zshowthequantitiesoftheattributesa1anda2providedbyvariousamountsofgoodsyandzthatmightbepurchased.Ifthispersonspendsallofhisorherincomeongoodx,thenthebudgetconstraint(Equation6.45)allowsthepurchaseofx¼Ipx,(6.47)andthatwillyielda1¼a1xx¼a1xIpx,a2¼a2xx¼a2xIpx.(6.48)Thispointisrecordedaspointxonthe0xrayinFigure6.2.Similarly,thepointsyandzrepresentthecombinationsofa1anda2thatwouldbeobtainedifallincomewerespentongoodyorgoodz,respectively.Bundlesofa1anda2thatareobtainablebypurchasingbothxandy(withaﬁxedbudget)arerepresentedbythelinejoiningxandyinFigure6.2.8Similarly,thelinexzrepresentsthecombinationsofa1anda2availablefromxandz,andthelineyzshowscombinationsavailablefrommixingyandz.Allpossiblecombinationsfrommixingthethreemarketgoodsarerepresentedbytheshadedtriangularareaxyz.CornersolutionsOnefactisimmediatelyapparentfromFigure6.2:Autility-maximizingindividualwouldneverconsumepositivequantitiesofallthreeofthesegoods.Onlythenortheastperimeterofthexyztrianglerepresentsthemaximalamountsofa1anda2availabletothispersongivenhisorherincomeandthepricesofthemarketgoods.Individualswithapreferencetowarda1willhaveindifferencecurvessimilartoU0andwillmaximizeutilitybychoosingapointsuchasE.Thecombinationofa1anda2speciﬁedbythatpointcanbeobtainedby8Mathematically,supposeafractionαofthebudgetisspentonxand(1α)ony;thena1¼αa1xxþð1αÞa1yy,a2¼αa2xxþð1αÞa2yy.Thelinexyistracedoutbyallowingαtovarybetween0and1.Thelinesxzandyzaretracedoutinasimilarway,asisthetriangularareaxyz.Chapter6DemandRelationshipsamongGoods193
consumingonlygoodsyandz.Similarly,apersonwithpreferencesrepresentedbytheindifferencecurveU00willchoosepointE0andconsumeonlygoodsxandy.Theattributesmodelthereforepredictsthatcornersolutionsatwhichindividualsconsumezeroamountsofsomecommoditieswillberelativelycommon,especiallyincaseswhereindividualsattachvaluetofewerattributes(here,two)thantherearemarketgoodstochoosefrom(three).Ifincome,prices,orpreferenceschange,thenconsumptionpatternsmayalsochangeabruptly.Goodsthatwerepreviouslyconsumedmayceasetobeboughtandgoodspreviouslyne-glectedmayexperienceasigniﬁcantincreaseinpurchases.Thisisadirectresultofthelinearassumptionsinherentintheproductionfunctionsassumedhere.Inhouseholdproductionmodelswithgreatersubstitutabilityassumptions,suchdiscontinuousreactionsarelesslikely.FIGURE6.2UtilityMaximizationintheAttributesModelThepointsx,y,andzshowtheamountsofattributesa1anda2thatcanbepurchasedbybuyingonlyx,y,orz,respectively.Theshadedareashowsallcombinationsthatcanbeboughtwithmixedbundles.SomeindividualsmaymaximizeutilityatE,othersatE0.a*2a2a1a*10U′0U0x*y*z*E′zEyxSUMMARYInthischapter,weusedtheutility-maximizingmodelofchoicetoexaminerelationshipsamongconsumergoods.Althoughtheserelationshipsmaybecomplex,theanalysispresentedhereprovidedanumberofwaysofcategorizingandsimplifyingthem.•Whenthereareonlytwogoods,theincomeandsubsti-tutioneffectsfromthechangeinthepriceofonegood(say,py)onthedemandforanothergood(x)usuallyworkinoppositedirections.Thesignof∂x=∂pyisthere-foreambiguous:itssubstitutioneffectispositivebutitsincomeeffectisnegative.•Incasesofmorethantwogoods,demandrelationshipscanbespeciﬁedintwoways.Twogoods(xiandxj)are“grosssubstitutes”if∂xi=∂pj>0and“grosscomple-ments”if∂xi=∂pj<0.Unfortunately,becausethesepriceeffectsincludeincomeeffects,theyneednotbesym-metric.Thatis,∂xi=∂pjdoesnotnecessarilyequal∂xj=∂pi.•Focusingonlyonthesubstitutioneffectsfrompricechangeseliminatesthisambiguitybecausesubstitutioneffectsaresymmetric;thatis,∂xci=∂pj¼∂xcj=∂pi.Nowtwogoodsaredeﬁnedasnet(orHicksian)substitutesif∂xci=∂pj>0andnetcomplementsif∂xci=∂pj<0.Hicks’194Part2ChoiceandDemand PROBLEMS6.1Heidireceivesutilityfromtwogoods,goat’smilk(m)andstrudel(s),accordingtotheutilityfunctionUðm,sÞ¼m⋅s.a.Showthatincreasesinthepriceofgoat’smilkwillnotaffectthequantityofstrudelHeidibuys;thatis,showthat∂s=∂pm¼0.b.Showalsothat∂m=∂ps¼0.c.UsetheSlutskyequationandthesymmetryofnetsubstitutioneffectstoprovethattheincomeeffectsinvolvedwiththederivativesinparts(a)and(b)areidentical.d.Provepart(c)explicitlyusingtheMarshalliandemandfunctionsformands.6.2HardTimesBurtbuysonlyrotgutwhiskeyandjellydonutstosustainhim.ForBurt,rotgutwhiskeyisaninferiorgoodthatexhibitsGiffen’sparadox,althoughrotgutwhiskeyandjellydonutsareHicksiansubstitutesinthecustomarysense.Developanintuitiveexplanationtosuggestwhyariseinthepriceofrotgutmustcausefewerjellydonutstobebought.Thatis,thegoodsmustalsobegrosscomplements.6.3Donald,afrugalgraduatestudent,consumesonlycoffee(c)andbutteredtoast(bt).Hebuystheseitemsattheuniversitycafeteriaandalwaysusestwopatsofbutterforeachpieceoftoast.Donaldspendsexactlyhalfofhismeagerstipendoncoffeeandtheotherhalfonbutteredtoast.a.Inthisproblem,butteredtoastcanbetreatedasacompositecommodity.Whatisitspriceintermsofthepricesofbutter(pb)andtoast(pt)?b.Explainwhy∂c=∂pbt¼0.c.Isitalsotrueherethat∂c=∂pband∂c=∂ptareequalto0?6.4Ms.SarahTravelerdoesnotownacarandtravelsonlybybus,train,orplane.Herutilityfunctionisgivenbyutility¼b⋅t⋅p,whereeachletterstandsformilestraveledbyaspecificmode.Supposethattheratioofthepriceoftraintraveltothatofbustravel(pt=pb)neverchanges.a.Howmightonedeﬁneacompositecommodityforgroundtransportation?b.PhraseSarah’soptimizationproblemasoneofchoosingbetweenground(g)andair(p)transportation.c.WhatareSarah’sdemandfunctionsforgandp?d.OnceSarahdecideshowmuchtospendong,howwillsheallocatethoseexpendituresbetweenbandt?“secondlawofdemand”showsthatnetsubstitutesaremoreprevalent.•Ifagroupofgoodshaspricesthatalwaysmoveinunison,thenexpendituresonthesegoodscanbetreatedasa“compositecommodity”whose“price”isgivenbythesizeoftheproportionalchangeinthecompositegoods’prices.•Analternativewaytodevelopthetheoryofchoiceamongmarketgoodsistofocusonthewaysinwhichmarketgoodsareusedinhouseholdproductiontoyieldutility-providingattributes.Thismayprovideadditionalinsightsintorelationshipsamonggoods.Chapter6DemandRelationshipsamongGoods195 6.5Supposethatanindividualconsumesthreegoods,x1,x2,andx3,andthatx2andx3aresimilarcommodities(i.e.,cheapandexpensiverestaurantmeals)withp2¼kp3,wherek<1—thatis,thegoods’priceshaveaconstantrelationshiptooneanother.a.Showthatx2andx3canbetreatedasacompositecommodity.b.Supposebothx2andx3aresubjecttoatransactioncostoftperunit(forsomeexamples,seeProblem6.6).Howwillthistransactioncostaffectthepriceofx2relativetothatofx3?Howwillthiseffectvarywiththevalueoft?c.Canyoupredicthowanincome-compensatedincreaseintwillaffectexpendituresonthecompositecommodityx2andx3?Doesthecompositecommoditytheoremstrictlyapplytothiscase?d.Howwillanincome-compensatedincreaseintaffecthowtotalspendingonthecompositecommodityisallocatedbetweenx2andx3?6.6ApplytheresultsofProblem6.5toexplainthefollowingobservations:a.Itisdifﬁculttoﬁndhigh-qualityapplestobuyinWashingtonStateorgoodfreshorangesinFlorida.b.Peoplewithsigniﬁcantbaby-sittingexpensesaremorelikelytohavemealsoutatexpensive(ratherthancheap)restaurantsthanarethosewithoutsuchexpenses.c.IndividualswithahighvalueoftimearemorelikelytoﬂytheConcordethanthosewithalowervalueoftime.d.Individualsaremorelikelytosearchforbargainsforexpensiveitemsthanforcheapones.Note:Observations(b)and(d)formthebasesforperhapstheonlytwomurdermysteriesinwhichaneconomistsolvesthecrime;seeMarshallJevons,MurderattheMarginandTheFatalEquilibrium.6.7Ingeneral,uncompensatedcross-priceeffectsarenotequal.Thatis,∂xi∂pj6¼∂xj∂pi.UsetheSlutskyequationtoshowthattheseeffectsareequaliftheindividualspendsaconstantfractionofincomeoneachgoodregardlessofrelativeprices.(ThisisageneralizationofProblem6.1.)6.8Example6.3computesthedemandfunctionsimpliedbythethree-goodCESutilityfunctionUðx,y,zÞ¼−1x−1y−1z.a.UsethedemandfunctionforxinEquation6.32todeterminewhetherxandyorxandzaregrosssubstitutesorgrosscomplements.b.Howwouldyoudeterminewhetherxandyorxandzarenetsubstitutesornetcomplements?AnalyticalProblems6.9ConsumersurpluswithmanygoodsInChapter5,weshowedhowthewelfarecostsofchangesinasinglepricecanbemeasuredusingexpenditurefunctionsandcompensateddemandcurves.Thisproblemasksyoutogeneralizethistopricechangesintwo(ormany)goods.196Part2ChoiceandDemand a.Supposethatanindividualconsumesngoodsandthatthepricesoftwoofthosegoods(say,p1andp2)rise.Howwouldyouusetheexpenditurefunctiontomeasurethecompensatingvariation(CV)forthispersonofsuchapricerise?b.Awaytoshowthesewelfarecostsgraphicallywouldbetousethecompensateddemandcurvesforgoodsx1andx2byassumingthatonepricerosebeforetheother.Illustratethisapproach.c.Inyouranswertopart(b),woulditmatterinwhichorderyouconsideredthepricechanges?Explain.d.Ingeneral,wouldyouthinkthattheCVforapriceriseofthesetwogoodswouldbegreaterifthegoodswerenetsubstitutesornetcomplements?Orwouldtherelationshipbetweenthegoodshavenobearingonthewelfarecosts?6.10SeparableutilityAutilityfunctioniscalledseparableifitcanbewrittenasUðx,yÞ¼U1ðxÞþU2ðyÞ,whereU0i>0,U00i<0,andU1,U2neednotbethesamefunction.a.Whatdoesseparabilityassumeaboutthecross-partialderivativeUxy?Giveanintuitivediscus-sionofwhatwordthisconditionmeansandinwhatsituationsitmightbeplausible.b.Showthatifutilityisseparablethenneithergoodcanbeinferior.c.Doestheassumptionofseparabilityallowyoutoconcludedeﬁnitivelywhetherxandyaregrosssubstitutesorgrosscomplements?Explain.d.UsetheCobb-Douglasutilityfunctiontoshowthatseparabilityisnotinvariantwithrespecttomonotonictransformations.Note:SeparablefunctionsareexaminedinmoredetailintheExten-sionstothischapter.6.11GraphingcomplementsGraphingcomplementsiscomplicatedbecauseacomplementaryrelationshipbetweengoods(undertheHicksdeﬁnition)cannotoccurwithonlytwogoods.Rather,complementaritynecessarilyinvolvesthedemandrelationshipsamongthree(ormore)goods.Inhisreviewofcomplementarity,Samuelsonprovidesawayofillustratingtheconceptwithatwo-dimensionalindifferencecurvediagram(seetheSuggestedReadings).Toexaminethisconstruction,assumetherearethreegoodsthataconsumermightchoose.Thequantitiesofthesearedenotedbyx1,x2,x3.Nowproceedasfollows.a.Drawanindifferencecurveforx2andx3,holdingthequantityofx1constantatx01.Thisindifferencecurvewillhavethecustomaryconvexshape.b.Nowdrawasecond(higher)indifferencecurveforx2,x3,holdingx1constantatx01h.Forthisnewindifferencecurve,showtheamountofextrax2thatwouldcompensatethispersonforthelossofx1;callthisamountj.Similarly,showthatamountofextrax3thatwouldcompensateforthelossofx1andcallthisamountk.c.Supposenowthatanindividualisgivenbothamountsjandk,therebypermittinghimorhertomovetoanevenhigherx2=x3indifferencecurve.Showthismoveonyourgraphanddrawthisnewindifferencecurve.d.Samuelsonnowsuggeststhefollowingdeﬁnitions:•Ifthenewindifferencecurvecorrespondstotheindifferencecurvewhenx1¼x012h,goods2and3areindependent.•Ifthenewindifferencecurveprovidesmoreutilitythanwhenx1¼x012h,goods2and3arecomplements.•Ifthenewindifferencecurveprovideslessutilitythanwhenx1¼x012h,goods2and3aresubstitutes.Showthatthesegraphicaldefinitionsaresymmetric.Chapter6DemandRelationshipsamongGoods197 e.DiscusshowthesegraphicaldeﬁnitionscorrespondtoHicks’moremathematicaldeﬁnitionsgiveninthetext.f.Lookingatyourﬁnalgraph,doyouthinkthatthisapproachfullyexplainsthetypesofrelation-shipsthatmightexistbetweenx2andx3?6.12ShippingthegoodapplesoutDetailsoftheanalysissuggestedinProblems6.5and6.6wereoriginallyworkedoutbyBorcherdingandSilberberg(seetheSuggestedReadings)basedonasuppositionﬁrstproposedbyAlchianandAllen.Theseauthorslookathowatransactionchargeaffectstherelativedemandfortwocloselysubstitutableitems.Assumethatgoodsx2andx3areclosesubstitutesandaresubjecttoatransactionchargeoftperunit.Supposealsothatgood2isthemoreexpensiveofthetwogoods(i.e.,“goodapples”asopposedto“cookingapples”).Hencethetransactionchargelowerstherelativepriceofthemoreexpensivegood[thatis,ðp2þtÞ=ðp3þtÞfallsastincreases].Thiswillincreasetherelativedemandfortheexpensivegoodif∂ðxc2=xc3Þ=∂t>0(whereweusecompensateddemandfunctionsinordertoeliminatepeskyincomeeffects).BorcherdingandSilberbergshowthisresultwillprobablyholdusingthefollowingsteps.a.Usethederivativeofaquotientruletoexpand∂ðxc2=xc3Þ=∂t.b.Useyourresultfrompart(a)togetherwiththefactthat,inthisproblem,∂xci=∂t¼∂xci=∂p2þ∂xci=∂p3fori¼2,3,toshowthatthederivativeweseekcanbewrittenas∂ðxc2=xc3Þ∂t¼xc2xc3s22x2þs23x2s32x3s33x3,wheresij¼∂xci=∂pj.c.Rewritetheresultfrompart(b)intermsofcompensatedpriceelasticities:ecij¼∂xci∂pj⋅pjxci.d.UseHicks’thirdlaw(Equation6.26)toshowthattheterminbracketsinparts(b)and(c)cannowbewrittenas½ðe22e32Þð1=p21=p3Þþðe21e31Þ=p3.e.Developanintuitiveargumentaboutwhytheexpressioninpart(d)islikelytobepositiveundertheconditionsofthisproblem.Hints:Whyistheﬁrstproductinthebracketspositive?Whyisthesecondterminbracketslikelytobesmall?f.ReturntoProblem6.6andprovidemorecompleteexplanationsforthesevariousﬁndings.198Part2ChoiceandDemand
SUGGESTIONSFORFURTHERREADINGBorcherding,T.E.,andE.Silberberg.“ShippingtheGoodApplesOut—TheAlchian-AllenTheoremReconsidered,”JournalofPoliticalEconomy(February1978):131–38.Gooddiscussionoftherelationshipsamongthreegoodsindemandtheory.SeealsoProblems6.5and6.6.Hicks,J.R.ValueandCapital,2nded.Oxford:OxfordUniversityPress,1946.SeeChaps.I–IIIandrelatedappendices.Proofofthecompositecommoditytheorem.Alsohasoneoftheﬁrsttreatmentsofnetsubstitutesandcomplements.Mas-Colell,A.,M.D.Whinston,andJ.R.Green.MicroeconomicTheory.NewYork:OxfordUniversityPress,1995.Explorestheconsequencesofthesymmetryofcompensatedcross-priceeffectsforvariousaspectsofdemandtheory.Rosen,S.“HedonicPricesandImplicitMarkets.”JournalofPoliticalEconomy(January/February1974):34–55.Nicegraphicalandmathematicaltreatmentoftheattributeap-proachtoconsumertheoryandoftheconceptof“markets”forattributes.Samuelson,P.A.“Complementarity—AnEssayonthe40thAnniversaryoftheHicks-AllenRevolutioninDemandTheory.”JournalofEconomicLiterature(December1977):1255–89.Reviewsanumberofdeﬁnitionsofcomplementarityandshowstheconnectionsamongthem.Containsanintuitive,graphicaldiscussionandadetailedmathematicalappendix.Silberberg,E.,andW.Suen.TheStructureofEconomics:AMathematicalAnalysis,3rded.Boston:Irwin/McGraw-Hill,2001.Gooddiscussionofexpenditurefunctionsandtheuseofindirectutilityfunctionstoillustratethecompositecommoditytheoremandotherresults.Chapter6DemandRelationshipsamongGoods199
EXTENSIONSSimplifyingDemandandTwo-StageBudgetingInChapter6wesawthatthetheoryofutilitymaximi-zationinitsfullgeneralityimposesratherfewrestric-tionsonwhatmighthappen.Otherthanthefactthatnetcross-substitutioneffectsaresymmetric,practicallyanytypeofrelationshipamonggoodsisconsistentwiththeunderlyingtheory.Thissituationposesprob-lemsforeconomistswhowishtostudyconsumptionbehaviorintherealworld—theoryjustdoesnotpro-videverymuchguidancewhentherearemanythou-sandsofgoodspotentiallyavailableforstudy.Therearetwogeneralwaysinwhichsimpliﬁcationsaremade.TheﬁrstusesthecompositecommoditytheoremfromChapter6toaggregategoodsintocate-gorieswithinwhichrelativepricesmovetogether.Forsituationswhereeconomistsarespeciﬁcallyinterestedinchangesinrelativepriceswithinacategoryofspend-ing(suchaschangesintherelativepricesofvariousformsofenergy),thisprocesswillnotdo,however.Analternativeistoassumethatconsumersengageinatwo-stageprocessintheirconsumptiondecisions.Firsttheyallocateincometovariousbroadgroupingsofgoods(food,clothing,andsoforth)andthen,giventheseexpenditureconstraints,theymaximizeutilitywithineachofthesubcategoriesofgoodsusingonlyinformationaboutthosegoods’relativeprices.Inthatway,decisionscanbestudiedinasimpliﬁedsettingbylookingonlyatonecategoryatatime.Thisprocessiscalled“two-stage”budgeting.Intheseextensions,weﬁrstlookatthegeneraltheoryoftwo-stagebudgetingandthenturntoexaminesomeempiricalexamples.E6.1Theoryoftwo-stagebudgetingTheissuethatarisesintwo-stagebudgetingcanbestatedsuccinctly:Doesthereexistapartitionofgoodsintomnonoverlappinggroups(denotedbyr¼1,m)andaseparatebudget(lr)devotedtoeachcategorysuchthatthedemandfunctionsforthegoodswithinanyonecategorydependonlyonthepricesofgoodswithinthecategoryandonthecategory’sbudgetallo-cation?Thatis,canwepartitiongoodssothatdemandisgivenbyxiðp1,…,pn,IÞ¼xi2rðpi2r,IrÞ(i)forr¼1,m,?Thatitmightbepossibletodothisissuggestedbycomparingthefollowingtwo-stagemaximizationproblem,Vðp1,…,pn,I1,…,ImÞ¼maxx1,…,xnhUðx1,…,xnÞs:t:Xi2rpixiIr,r¼1,mi(ii)andmaxI1,…,ImVs.t.Xmr¼1Ir¼I,totheutility-maximizationproblemwehavebeenstudying,maxxiUðx1,…,xnÞs.t.Xni¼1pixiI.(iii)Withoutanyfurtherrestrictions,thesetwomaxi-mizationprocesseswillyieldthesameresult;thatis,EquationiiisjustamorecomplicatedwayofstatingEquationiii.So,somerestrictionshavetobeplacedontheutilityfunctiontoensurethatthedemandfunctionsthatresultfromsolvingthetwo-stageprocesswillbeoftheformspecifiedinEquationi.Intuitively,itseemsthatsuchacategorizationofgoodsshouldworkprovidingthatchangesinthepriceofagoodinonecategorydonotaffecttheallocationofspendingforgoodsinanycategoryotherthanitsown.InProblem6.9weshowedacasewherethisistrueforan“additivelyseparable”utilityfunction.Unfortunately,thisprovestobeaveryspecialcase.Themoregeneralmathematicalrestrictionsthatmustbeplacedontheutilityfunctiontojustifytwo-stagebudgetinghavebeenderived(seeBlackorby,Primont,andRussell,1978),butthesearenotespeciallyintuitive.Ofcourse,economistswhowishtostudydecentralizeddecisionsbyconsumers(or,perhapsmoreimportantly,byfirmsthatoperatemanydivisions)mustdosomethingtosimplifymatters.Nowwelookatafewappliedexamples.E6.2RelationtothecompositioncommoditytheoremUnfortunately,neitherofthetwoavailabletheoreticalapproachestodemandsimpliﬁcationiscompletelysat-isfying.Thecompositecommoditytheoremrequiresthattherelativepricesforgoodswithinonegroupremainconstantovertime,anassumptionthathasbeenrejectedduringmanydifferenthistoricalperiods.200Part2ChoiceandDemand
Ontheotherhand,thekindofseparabilityandtwo-stagebudgetingindicatedbytheutilityfunctioninEquationialsorequiresverystrongassumptionsabouthowchangesinpricesforagoodinonegroupaffectspendingongoodsinanyothergroup.Theseassumptionsappeartoberejectedbythedata(seeDiewertandWales,1995).Economistshavetriedtodeviseevenmoreelabo-rate,hybridmethodsofaggregationamonggoods.Forexample,Lewbel(1996)showshowthecompos-itecommoditytheoremmightbegeneralizedtocaseswherewithin-grouprelativepricesexhibitconsiderablevariability.HeusesthisgeneralizationforaggregatingU.S.consumerexpendituresintosixlargegroups(food,clothing,householdoperation,medicalcare,transportation,andrecreation).Usingtheseaggre-gates,heconcludesthathisprocedureismuchmoreaccuratethanassumingtwo-stagebudgetingamongtheseexpenditurecategories.E6.3HomotheticfunctionsandenergydemandOnewaytosimplifythestudyofdemandwhentherearemanycommoditiesistoassumethatutilityforcertainsubcategoriesofgoodsishomotheticandmaybeseparatedfromthedemandforothercom-modities.ThisprocedurewasfollowedbyJorgenson,Slesnick,andStoker(1997)intheirstudyofenergydemandbyU.S.consumers.Byassumingthatdemandfunctionsforspeciﬁctypesofenergyareproportionaltototalspendingonenergy,theauthorswereabletoconcentratetheirempiricalstudyonthetopicthatisofmostinteresttothem:estimatingthepriceelasticitiesofdemandforvarioustypesofenergy.Theyconcludethatmosttypesofenergy(thatis,electricity,naturalgas,gasoline,andsoforth)havefairlyelasticdemandfunctions.Demandappearstobemostresponsivetopriceforelectricity.ReferencesBlackorby,Charles,DanielPrimont,andR.RobertRussell.Duality,SeparabilityandFunctionalStructure:TheoryandEconomicApplications.NewYork:NorthHolland,1978.Diewert,W.Erwin,andTerrenceJ.Wales.“FlexibleFunctionalFormsandTestsofHomogeneousSepara-bility.”JournalofEconometrics(June1995):259–302.Jorgenson,DaleW.,DanielT.Slesnick,andThomasM.Stoker.“Two-StageBudgetingandConsumerDemandforEnergy.”InDaleW.Jorgenson,Ed.,Welfare,vol.1:AggregateConsumerBehavior,pp.475–510.Cam-bridge,MA:MITPress,1997.Lewbel,Arthur.“AggregationwithoutSeparability:AStandardizedCompositeCommodityTheorem.”Amer-icanEconomicReview(June1996):524–43.Chapter6DemandRelationshipsamongGoods201
CHAPTER7UncertaintyandInformationInthischapterwewillexploresomeofthebasicelementsofthetheoryofindividualbehaviorinuncertainsituations.Ourgeneralgoalistoshowwhyindividualsdonotlikeriskandhowtheymayadoptstrategiestoreduceit.Moregenerally,thechapterisintendedtoprovideabriefintroductiontoissuesraisedbythepossibilitythatinformationmaybeimperfectwhenindividualsmakeutility-maximizingdecisions.Someofthethemesdevelopedherewillrecurthroughouttheremainderofthebook.MATHEMATICALSTATISTICSManyoftheformaltoolsformodelinguncertaintyineconomicsituationswereoriginallydevelopedintheﬁeldofmathematicalstatistics.SomeofthesetoolswerereviewedinChapter2andinthischapterwewillbemakingagreatdealofuseoftheconceptsintroducedthere.Speciﬁcally,fourstatisticalideaswillrecurthroughoutthischapter.•Randomvariable:Arandomvariableisavariablethatrecords,innumericalform,thepossibleoutcomesfromsomerandomevent.1•Probabilitydensityfunction(PDF):Afunctionthatshowstheprobabilitiesassoci-atedwiththepossibleoutcomesfromarandomvariable.•Expectedvalueofarandomvariable:Theoutcomeofarandomvariablethatwilloccur“onaverage.”TheexpectedvalueisdenotedbyEðxÞ.IfxisadiscreterandomvariablewithnoutcomesthenEðxÞ¼Pni¼1xi¼fðxiÞ,wherefðxÞisthePDFfortherandomvariablex.Ifxisacontinuousrandomvariable,thenEðxÞ¼∫þ∞∞xfðxÞdx.•Varianceandstandarddeviationofarandomvariable:Theseconceptsmea-surethedispersionofarandomvariableaboutitsexpectedvalue.Inthediscretecase,VarðxÞ¼σ2x¼Pni¼1½xiEðxÞ2fðxiÞ;inthecontinuouscase,VarðxÞ¼σ2x¼∫þ∞∞½xEðxÞ2fðxÞdx.Thestandarddeviationisthesquarerootofthevariance.Asweshallsee,alloftheseconceptswillcomeintoplaywhenwebeginlookingatthedecision-makingprocessofapersonfacedwithanumberofuncertainoutcomesthatcanbeconceptuallyrepresentedbyarandomvariable.1Whenitisnecessarytodifferentiatebetweenrandomvariablesandnonrandomvariables,wewillusethenotation∼xtodenotethefactthatthevariablexisrandominthatittakesonanumberofpotentialrandomlydeterminedoutcomes.Often,however,itwillnotbenecessarytomakethedistinctionbecauserandomnesswillbeclearfromthecontextoftheproblem.202
FAIRGAMESANDTHEEXPECTEDUTILITYHYPOTHESISA“fairgame”isarandomgamewithaspeciﬁedsetofprizesandassociatedprobabilitiesthathasanexpectedvalueofzero.Forexample,ifyouﬂipacoinwithafriendforadollar,theexpectedvalueofthisgameiszerobecauseEðxÞ¼0:5ðþ$1Þþ0:5ð$1Þ¼0,(7.1)wherewinsarerecordedwithaplussignandlosseswithaminussign.Similarly,agamethatpromisedtopayyou$10ifacoincameupheadsbutwouldcostyouonly$1ifitcameuptailswouldbe“unfair”becauseEðxÞ¼0:5ðþ$10Þþ0:5ð$1Þ¼$4:50.(7.2)Thisgamecaneasilybeconvertedintoafairgame,however,simplybychargingyouanentryfeeof$4.50fortherighttoplay.2Ithaslongbeenrecognizedthatmostpeoplewouldprefernottoplayfairgames.Althoughpeoplemaysometimeswillinglyﬂipacoinforafewdollars,theywouldgenerallybalkatplayingasimilargamewhoseoutcomewas+$1millionor$1million.OneoftheﬁrstmathematicianstostudythereasonsforthisunwillingnesstoengageinfairbetswasDanielBernoulliintheeighteenthcentury.3HisexaminationofthefamousSt.Petersburgparadoxprovidedthestartingpointforvirtuallyallstudiesofthebehaviorofindividualsinuncertainsituations.St.PetersburgparadoxIntheSt.Petersburgparadox,thefollowinggameisproposed:Acoinisﬂippeduntilaheadappears.Ifaheadﬁrstappearsonthenthﬂip,theplayerispaid$2n.Thisgamehasaninﬁnitenumberofoutcomes(acoinmightbeﬂippedfromnowuntildoomsdayandnevercomeupahead,althoughthelikelihoodofthisissmall),buttheﬁrstfewcaneasilybewrittendown.Ifxirepresentstheprizeawardedwhentheﬁrstheadappearsontheithtrial,thenx1¼$2,×2¼$4,×3¼$8,…,xn¼$2n.(7.3)Theprobabilityofgettingaheadforthefirsttimeontheithtrialis12i;itistheprobabilityofgetting(i1)tailsandthenahead.HencetheprobabilitiesoftheprizesgiveninEquation7.3areπ1¼12,π2¼14,π3¼18,…,πn¼12n.(7.4)TheexpectedvalueoftheSt.Petersburgparadoxgameisthereforeinfinite:EðxÞ¼X∞i¼1πixi¼X∞i¼12ið1=2iÞ¼1þ1þ1þ…þ1þ…¼∞.(7.5)Someintrospection,however,shouldconvinceanyonethatnoplayerwouldpayverymuch(muchlessthaninfinity)toplaythisgame.IfIcharged$1billiontoplaythegame,Iwouldsurelyhavenotakers,despitethefactthat$1billionisstillconsiderablylessthantheexpectedvalueofthegame.This,then,istheparadox:Bernoulli’sgameisinsomesensenotworthits(infinite)expecteddollarvalue.2Thegamesdiscussedhereareassumedtoyieldnoutilityintheirplayotherthantheprizes;hence,theobservationthatmanyindividualsgambleat“unfair”oddsisnotnecessarilyarefutationofthisstatement.Rather,suchindividualscanreasonablybeassumedtobederivingsomeutilityfromthecircumstancesassociatedwiththeplayofthegame.Itisthereforepossibletodifferentiatetheconsumptionaspectofgamblingfromthepureriskaspect.3TheoriginalBernoullipaperhasbeenreprintedasD.Bernoulli,“ExpositionofaNewTheoryontheMeasurementofRisk,”Econometrica22(January1954):23–36.Chapter7UncertaintyandInformation203
ExpectedutilityBernoulli’ssolutiontothisparadoxwastoarguethatindividualsdonotcaredirectlyaboutthedollarprizesofagame;rather,theyrespondtotheutilitythesedollarsprovide.Ifweassumethatthemarginalutilityofwealthdeclinesaswealthincreases,theSt.Petersburggamemayconvergetoaﬁniteexpectedutilityvaluethatplayerswouldbewillingtopayfortherighttoplay.Bernoullitermedthisexpectedutilityvaluethemoralvalueofthegamebecauseitrepresentshowmuchthegameisworthtotheindividual.Becauseutilitymayriselessrapidlythanthedollarvalueoftheprizes,itispossiblethatagame’smoralvaluewillfallshortofitsmonetaryexpectedvalue.Example7.1looksatsomeissuesrelatedtoBernoulli’ssolution.EXAMPLE7.1Bernoulli’sSolutiontotheParadoxandItsShortcomingsSuppose,asdidBernoulli,thattheutilityofeachprizeintheSt.PetersburgparadoxisgivenbyUðxiÞ¼lnðxiÞ.(7.6)Thislogarithmicutilityfunctionexhibitsdiminishingmarginalutility(thatis,U0>0butU00<0),andtheexpectedutilityvalueofthisgameconvergestoafinitenumber:expectedutility¼X∞i¼1πiUðxÞi¼X∞i¼112ilnð2iÞ.(7.7)Somemanipulationofthisexpressionyields4theresultthattheexpectedutilityvalueofthisgameis1.39.Anindividualwiththistypeofutilityfunctionmightthereforebewillingtoinvestresourcesthatotherwiseyieldupto1.39unitsofutility(acertainwealthofabout$4providesthisutility)inpurchasingtherighttoplaythisgame.AssumingthattheverylargeprizespromisedbytheSt.Petersburgparadoxencounterdiminishingmarginalutilitythere-forepermittedBernoullitoofferasolutiontotheparadox.Unboundedutility.Bernoulli’ssolutiontotheSt.Petersburgparadox,unfortunately,doesnotcompletelysolvetheproblem.Solongasthereisnoupperboundtotheutilityfunction,theparadoxcanberegeneratedbyredeﬁningthegame’sprizes.Forexample,withthelogarithmicutilityfunction,prizescanbesetasxi¼e2i,inwhichcaseUðxiÞ¼ln½e2i¼2i(7.8)andtheexpectedutilityvalueofthegamewouldagainbeinfinite.Ofcourse,theprizesinthisredefinedgameareverylarge.Forexample,ifaheadfirstappearsonthefifthflip,apersonwouldwine25¼e32¼$7:9⋅1013,thoughtheprobabilityofwinningthiswouldbeonly1=25¼0:031.Theideathatpeoplewouldpayagreatdeal(say,billionsofdollars)toplaygameswithsmallprobabilitiesofsuchlargeprizesseems,tomanyobservers,tobeunlikely.Hence,inmanyrespectstheSt.Petersburggameremainsaparadox.4Proof:expectedutility¼X∞i¼1i2i⋅ln2¼ln2X∞i¼1i2i:Butthevalueofthisfinalinfiniteseriescanbeshowntobe2.0.Hence,expectedutility¼2ln2¼1:39.204Part2ChoiceandDemand QUERY:HerearetwoalternativesolutionstotheSt.Petersburgparadox.Foreach,calculatetheexpectedvalueoftheoriginalgame.1.Supposeindividualsassumethatanyprobabilitylessthan0.01isinfactzero.2.SupposethattheutilityfromtheSt.PetersburgprizesisgivenbyUðxiÞ¼xiifxi1,000,000,1,000,000ifxi>1,000,000.THEVONNEUMANN–MORGENSTERNTHEOREMIntheirbookTheTheoryofGamesandEconomicBehavior,JohnvonNeumannandOscarMorgensterndevelopedmathematicalmodelsforexaminingtheeconomicbehaviorofin-dividualsunderconditionsofuncertainty.5Tounderstandtheseinteractions,itwasnecessaryﬁrsttoinvestigatethemotivesoftheparticipantsinsuch“games.”Becausethehypothesisthatindividualsmakechoicesinuncertainsituationsbasedonexpectedutilityseemedintuitivelyreasonable,theauthorssetouttoshowthatthishypothesiscouldbederivedfrommorebasicaxiomsof“rational”behavior.Theaxiomsrepresentanattemptbytheauthorstogeneralizethefoundationsofthetheoryofindividualchoicetocoveruncertainsituations.Althoughmostoftheseaxiomsseememinentlyreasonableatﬁrstglance,manyimportantquestionsabouttheirtenabilityhavebeenraised.Wewillnotpursuethesequestionshere,however.6ThevonNeumann–MorgensternutilityindexTobegin,supposethattherearenpossibleprizesthatanindividualmightwinbyparticipat-inginalottery.Lettheseprizesbedenotedbyx1,x2,…,xnandassumethatthesehavebeenarrangedinorderofascendingdesirability.Therefore,x1istheleastpreferredprizefortheindividualandxnisthemostpreferredprize.Nowassignarbitraryutilitynumberstothesetwoextremeprizes.Forexample,itisconvenienttoassignUðx1Þ¼0,UðxnÞ¼1,(7.9)butanyotherpairofnumberswoulddoequallywell.7Usingthesetwovaluesofutility,thepointofthevonNeumann–Morgensterntheoremistoshowthatareasonablewayexiststoassignspecificutilitynumberstotheotherprizesavailable.Supposethatwechooseanyotherprize,say,xi.Considerthefollowingexperiment.Asktheindividualtostatetheprobability,say,πi,atwhichheorshewouldbeindifferentbetweenxiwithcertainty,andagambleofferingprizesofxnwithprobabilityπiandx1withprobabilityð1πiÞ.Itseemsreasonable(althoughthisisthemostproblematicassumptioninthevonNeumann–Morgensternapproach)thatsuchaprobabilitywillexist:Theindividualwillalwaysbeindifferentbetweenagambleandasurething,providedthatahighenoughprobabilityofwinningthebestprizeisoffered.Italsoseemslikelythatπiwillbehigherthemoredesirablexiis;thebetterxiis,the5J.vonNeumannandO.Morgenstern,TheTheoryofGamesandEconomicBehavior(Princeton,NJ:PrincetonUniversityPress,1944).Theaxiomsofrationalityinuncertainsituationsarediscussedinthebook’sappendix.6ForadiscussionofsomeoftheissuesraisedinthedebateoverthevonNeumann–Morgensternaxioms,especiallytheassumptionofindependence,seeC.Gollier,TheEconomicsofRiskandTime(Cambridge,MA:MITPress,2001),chap.1.7Technically,avonNeumann–Morgensternutilityindexisuniqueonlyuptoachoiceofscaleandorigin—thatis,onlyuptoa“lineartransformation.”Thisrequirementismorestringentthantherequirementthatautilityfunctionbeuniqueuptoamonotonictransformation.Chapter7UncertaintyandInformation205
betterthechanceofwinningxnmustbetogettheindividualtogamble.Theprobabilityπithereforemeasureshowdesirabletheprizexiis.Infact,thevonNeumann–Morgensterntechniqueistodefinetheutilityofxiastheexpectedutilityofthegamblethattheindividualconsidersequallydesirabletoxi:UðxiÞ¼πi⋅UðxnÞþð1πiÞ⋅Uðx1Þ.(7.10)BecauseofourchoiceofscaleinEquation7.9,wehaveUðxiÞ¼πi⋅1þð1πiÞ⋅0¼πi.(7.11)Byjudiciouslychoosingtheutilitynumberstobeassignedtothebestandworstprizes,wehavebeenabletodeviseascaleunderwhichtheutilitynumberattachedtoanyotherprizeissimplytheprobabilityofwinningthetopprizeinagambletheindividualregardsasequivalenttotheprizeinquestion.Thischoiceofutilitynumbersisarbitrary.Anyothertwonumberscouldhavebeenusedtoconstructthisutilityscale,butourinitialchoice(Equation7.9)isaparticularlyconvenientone.ExpectedutilitymaximizationInlinewiththechoiceofscaleandoriginrepresentedbyEquation7.9,supposethatprobabilityπihasbeenassignedtorepresenttheutilityofeveryprizexi.Noticeinparticularthatπ1¼0,πn¼1,andthattheotherutilityvaluesrangebetweentheseextremes.Usingtheseutilitynumbers,wecanshowthata“rational”individualwillchooseamonggamblesbasedontheirexpected“utilities”(thatis,basedontheexpectedvalueofthesevonNeumann–Morgensternutilityindexnumbers).Asanexample,considertwogambles.Onegambleoffersx2,withprobabilityq,andx3,withprobability(1q).Theotheroffersx5,withprobabilityt,andx6,withprobability(1t).Wewanttoshowthatthispersonwillchoosegamble1ifandonlyiftheexpectedutilityofgamble1exceedsthatofgamble2.Nowforthegambles:expectedutilityð1Þ¼q⋅Uðx2Þþð1qÞ⋅Uðx3Þ,expectedutilityð2Þ¼t⋅Uðx5Þþð1tÞ⋅Uðx6Þ.(7.12)Substitutingtheutilityindexnumbers(thatis,π2isthe“utility”ofx2,andsoforth)givesexpectedutilityð1Þ¼q⋅π2þð1qÞ⋅π3,expectedutilityð2Þ¼t⋅π5þð1tÞ⋅π6.(7.13)Wewishtoshowthattheindividualwillprefergamble1togamble2ifandonlyifq⋅π2þð1qÞ⋅π3>t⋅π5þð1tÞ⋅π6.(7.14)Toshowthis,recallthedefinitionsoftheutilityindex.Theindividualisindifferentbetweenx2andagamblepromisingx1withprobability(1π2)andxnwithprobabilityπ2.Wecanusethisfacttosubstitutegamblesinvolvingonlyx1andxnforallutilitiesinEquation7.13(eventhoughtheindividualisindifferentbetweenthese,theassumptionthatthissubstitutioncanbemadeimplicitlyassumesthatpeoplecanseethroughcomplexlotterycombinations).Afterabitofmessyalgebra,wecanconcludethatgamble1isequivalenttoagamblepromisingxnwithprobabilityqπ2þð1qÞπ3,andgamble2isequivalenttoagamblepromisingxnwithprobabilitytπ5þð1tÞπ6.Theindividualwillpresumablypreferthegamblewiththehigherprobabilityofwinningthebestprize.Consequently,heorshewillchoosegamble1ifandonlyifqπ2þð1qÞπ3>tπ5þð1tÞπ6.(7.15)Butthisispreciselywhatwewantedtoshow.Consequently,wehaveprovedthatanindivid-ualwillchoosethegamblethatprovidesthehighestlevelofexpected(vonNeumann–Morgenstern)utility.Wenowmakeconsiderableuseofthisresult,whichcanbesummarizedasfollows.206Part2ChoiceandDemand
OPTIMIZATIONPRINCIPLEExpectedutilitymaximization.IfindividualsobeythevonNeumann–Morgensternaxiomsofbehaviorinuncertainsituations,theywillactasiftheychoosetheoptionthatmaximizestheexpectedvalueoftheirvonNeumann–Morgensternutilityindex.RISKAVERSIONTwolotteriesmayhavethesameexpectedmonetaryvaluebutmaydifferintheirriskiness.Forexample,ﬂippingacoinfor$1andﬂippingacoinfor$1,000arebothfairgames,andbothhavethesameexpectedvalue(0).However,thelatterisinsomesensemore“risky”thantheformer,andfewerpeoplewouldparticipateinthegamewheretheprizewaswinningorlosing$1,000.Thepurposeofthissectionistodiscussthemeaningofthetermriskyandexplainthewidespreadaversiontorisk.Thetermriskreferstothevariabilityoftheoutcomesofsomeuncertainactivity.8Ifvariabilityislow,theactivitymaybeapproximatelyasurething.Withnomoreprecisenotionofvariabilitythanthis,itispossibletoshowwhyindividuals,whenfacedwithachoicebetweentwogambleswiththesameexpectedvalue,willusuallychoosetheonewithasmallervariabilityofreturn.Intuitively,thereasonbehindthisisthatweusuallyassumethatthemarginalutilityfromextradollarsofprizemoney(thatis,wealth)declinesastheprizesgetlarger.Aﬂipofacoinfor$1,000promisesarelativelysmallgainofutilityifyouwinbutalargelossofutilityifyoulose.Abetofonly$1is“inconsequential,”andthegaininutilityfromawinapproximatelycounterbalancesthedeclineinutilityfromaloss.9RiskaversionandfairbetsThisargumentisillustratedinFigure7.1.HereWrepresentsanindividual’scurrentwealthandUðWÞisavonNeumann–Morgensternutilityindexthatreﬂectshowheorshefeelsaboutvariouslevelsofwealth.10Intheﬁgure,UðWÞisdrawnasaconcavefunctionofWtoreﬂecttheassumptionofadiminishingmarginalutility.Itisassumedthatobtaininganextradollaraddslesstoenjoymentastotalwealthincreases.Nowsupposethispersonisofferedtwofairgambles:a50–50chanceofwinningorlosing$hora50–50chanceofwinningorlosing$2h.TheutilityofpresentwealthisUðWÞ:Theexpectedutilityifheorsheparticipatesingamble1isgivenbyUhðWÞ:UhðWÞ¼12UðWþhÞþ12UðWhÞ,(7.16)andtheexpectedutilityofgamble2isgivenbyU2hðWÞ:U2hðWÞ¼12UðWþ2hÞþ12UðW2hÞ.(7.17)Itisgeometricallyclearfromthefigurethat11UðWÞ>UhðWÞ>U2hðWÞ.(7.18)8Oftenthestatisticalconceptsofvarianceandstandarddeviationareusedtomeasurerisk.Wewilldosoatseveralplaceslaterinthischapter.9Technically,thisresultisadirectconsequenceofJensen’sinequalityinmathematicalstatistics.TheinequalitystatesthatifxisarandomvariableandfðxÞisaconcavefunctionofthatvariable,thenE½fðxÞf½EðxÞ.Intheutilitycontext,thismeansthatifutilityisconcaveinarandomvariablemeasuringwealth(i.e.,ifU0ðWÞ>0andU00ðWÞ<0Þ,thentheexpectedutilityofwealthwillbelessthantheutilityassociatedwiththeexpectedvalueofW.10Technically,UðWÞisanindirectutilityfunctionbecauseitistheconsumptionallowedbywealththatprovidesdirectutility.InChapter17wewilltakeuptherelationshipbetweenconsumption-basedutilityfunctionsandtheirimpliedindirectutilityofwealthfunctions.11Toseewhytheexpectedutilitiesforbethandbet2harethoseshown,noticethattheseexpectedutilitiesaretheaverageoftheutilitiesfromafavorableandanunfavorableoutcome.BecauseWishalfwaybetweenWþhandWh,UisalsohalfwaybetweenUðWþhÞandUðWhÞ.Chapter7UncertaintyandInformation207 Thispersonthereforewillpreferhisorhercurrentwealthtothatwealthcombinedwithafairgambleandwillpreferasmallgambletoalargeone.Thereasonforthisisthatwinningafairbetaddstoenjoymentlessthanlosinghurts.Althoughinthiscasetheprizesareequal,winningprovideslessthanlosingcostsinutilityterms.RiskaversionandinsuranceAsamatteroffact,thispersonmightbewillingtopaysomeamounttoavoidparticipatinginanygambleatall.NoticethatacertainwealthofWprovidesthesameutilityasdoesparticipatingingamble1.ThispersonwouldbewillingtopayuptoWWinordertoavoidparticipatinginthegamble.Thisexplainswhypeoplebuyinsurance.Theyaregivingupasmall,certainamount(theinsurancepremium)toavoidtheriskyoutcometheyarebeinginsuredagainst.Thepremiumapersonpaysforautomobilecollisioninsurance,forexample,providesapolicythatagreestorepairhisorhercarshouldanaccidentoccur.Thewidespreaduseofinsurancewouldseemtoimplythataversiontoriskisquiteprevalent.Hence,weintroducethefollowingdeﬁnition.DEFINITIONRiskaversion.Anindividualwhoalwaysrefusesfairbetsissaidtoberiskaverse.Ifindividualsexhibitadiminishingmarginalutilityofwealth,theywillberiskaverse.Asaconsequence,theywillbewillingtopaysomethingtoavoidtakingfairbets.EXAMPLE7.2WillingnesstoPayforInsuranceToillustratetheconnectionbetweenriskaversionandinsurance,considerapersonwithacurrentwealthof$100,000whofacestheprospectofa25percentchanceoflosinghisorher$20,000automobilethroughtheftduringthenextyear.Supposealsothatthisperson’svonNeumann–Morgensternutilityindexislogarithmic;thatis,UðWÞ¼lnðWÞ:FIGURE7.1UtilityofWealthfromTwoFairBetsofDifferingVariabilityIftheutility-of-wealthfunctionisconcave(i.e.,exhibitsadiminishingmarginalutilityofwealth),thenthispersonwillrefusefairbets.A50–50betofwinningorlosinghdollars,forexample,yieldslessutility½UhðWÞthandoesrefusingthebet.Thereasonforthisisthatwinninghdollarsmeanslesstothisindividualthandoeslosinghdollars.UtilityU(W)U(W*)Uh(W*)U2h(W*)W*− 2hW*+2hWealth (W)W*− hW*+hW*W208Part2ChoiceandDemand Ifthispersonfacesnextyearwithoutinsurance,expectedutilitywillbeexpectedutility¼0.75Uð100,000Þþ0.25Uð80,000Þ¼0.75ln100,000þ0.25ln80,000¼11.45714.(7.19)Inthissituation,afairinsurancepremiumwouldbe$5,000(25percentof$20,000,assumingthattheinsurancecompanyhasonlyclaimcostsandthatadministrativecostsare$0).Consequently,ifthispersoncompletelyinsuresthecar,hisorherwealthwillbe$95,000regardlessofwhetherthecarisstolen.Inthiscase,then,expectedutility¼Uð95,000Þ¼lnð95,000Þ¼11.46163.(7.20)Thispersonismadebetter-offbypurchasingfairinsurance.Indeed,wecandeterminethemaximumamountthatmightbepaidforthisinsuranceprotection(x)bysettingexpectedutility¼Uð100,000xÞ¼lnð100,000xÞ¼11.45714.(7.21)Solvingthisequationforxyields100,000x¼e11.45714.(7.22)Therefore,themaximumpremiumisx¼5,426.(7.23)Thispersonwouldbewillingtopayupto$426inadministrativecoststoaninsurancecompany(inadditiontothe$5,000premiumtocovertheexpectedvalueoftheloss).Evenwhenthesecostsarepaid,thispersonisaswell-offasheorshewouldbewhenfacingtheworlduninsured.QUERY:Supposeutilityhadbeenlinearinwealth.Wouldthispersonbewillingtopayanythingmorethantheactuariallyfairamountforinsurance?Howaboutthecasewhereutilityisaconvexfunctionofwealth?MEASURINGRISKAVERSIONInthestudyofeconomicchoicesinriskysituations,itissometimesconvenienttohaveaquantitativemeasureofhowaversetoriskapersonis.ThemostcommonlyusedmeasureofriskaversionwasinitiallydevelopedbyJ.W.Prattinthe1960s.12Thisriskaversionmeasure,rðWÞ,isdeﬁnedasrðWÞ¼U00ðWÞU0ðWÞ.(7.24)Becausethedistinguishingfeatureofrisk-averseindividualsisadiminishingmarginalutilityofwealth½U00ðWÞ<0,Pratt’smeasureispositiveinsuchcases.Themeasureisinvariantwithrespecttolineartransformationsoftheutilityfunction,andthereforenotaffectedbywhichparticularvonNeumann–Morgensternorderingisused.12J.W.Pratt,“RiskAversionintheSmallandintheLarge,”Econometrica(January/April1964):122–36.Chapter7UncertaintyandInformation209 RiskaversionandinsurancepremiumsAusefulfeatureofthePrattmeasureofriskaversionisthatitisproportionaltotheamountanindividualwillpayforinsuranceagainsttakingafairbet.Supposethewinningsfromsuchafairbetaredenotedbytherandomvariableh(thisvariablemaybeeitherpositiveornegative).Becausethebetisfair,EðhÞ¼0.Nowletpbethesizeoftheinsurancepremiumthatwouldmaketheindividualexactlyindifferentbetweentakingthefairbethandpayingpwithcertaintytoavoidthegamble:E½UðWþhÞ¼UðWpÞ,(7.25)whereWistheindividual’scurrentwealth.WenowexpandbothsidesofEquation7.25usingTaylor’sseries.13Becausepisafixedamount,alinearapproximationtotheright-handsideoftheequationwillsuffice:UðWpÞ¼UðWÞpU0ðWÞþhigher-orderterms.(7.26)Fortheleft-handside,weneedaquadraticapproximationtoallowforthevariabilityinthegamble,h:E½UðWþhÞ¼EUðWÞþhU0ðWÞþh22U00ðWÞþhigher-orderterms(7.27)¼UðWÞþEðhÞU0ðWÞþEðh2Þ2U00ðWÞþhigher-orderterms.(7.28)IfwerecallthatEðhÞ¼0andthendropthehigher-ordertermsandusetheconstantktorepresentEðh2Þ=2,wecanequateEquations7.26and7.28asUðWÞpU0ðWÞ≅UðWÞþkU00ðWÞ(7.29)orp≅kU00ðWÞU0ðWÞ¼krðWÞ.(7.30)Thatis,theamountthatarisk-averseindividualiswillingtopaytoavoidafairbetisapproximatelyproportionaltoPratt’sriskaversionmeasure.14Becauseinsurancepremiumspaidareobservableintherealworld,theseareoftenusedtoestimateindividuals’riskaversioncoefficientsortocomparesuchcoefficientsamonggroupsofindividuals.Itisthereforepossibletousemarketinformationtolearnquiteabitaboutattitudestowardriskysituations.RiskaversionandwealthAnimportantquestioniswhetherriskaversionincreasesordecreaseswithwealth.Intuitively,onemightthinkthatthewillingnesstopaytoavoidagivenfairbetwoulddeclineaswealthincreases,becausediminishingmarginalutilitywouldmakepotentiallosseslessseriousforhigh-wealthindividuals.Thisintuitiveanswerisnotnecessarilycorrect,however,becausediminishingmarginalutilityalsomakesthegainsfromwinninggambleslessattractive.Sothe13Taylor’sseriesprovidesawayofapproximatinganydifferentiablefunctionaroundsomepoint.IffðxÞhasderivativesofallorders,itcanbeshownthatfðxþhÞ¼fðxÞþhf0ðxÞþðh2=2Þf00ðxÞþhigher-orderterms.Thepoint-slopeformulainalgebraisasimpleexampleofTaylor’sseries.14Inthiscase,thefactorofproportionalityisalsoproportionaltothevarianceofhbecauseVarðhÞ¼E½hEðhÞ2¼Eðh2Þ.Foranillustrationwherethisequationfitsexactly,seeExample7.3.210Part2ChoiceandDemand netresultisindeterminate;italldependsonthepreciseshapeoftheutilityfunction.Indeed,ifutilityisquadraticinwealth,UðWÞ¼aþbWþcW2,(7.31)whereb>0andc<0,thenPratt’sriskaversionmeasureisrðWÞ¼U00ðWÞU0ðWÞ¼2cbþ2cW,(7.32)which,contrarytointuition,increasesaswealthincreases.Ontheotherhand,ifutilityislogarithmicinwealth,UðWÞ¼lnðWÞðW>0Þ,(7.33)thenwehaverðWÞ¼U00ðWÞU0ðWÞ¼1W,(7.34)whichdoesindeeddecreaseaswealthincreases.TheexponentialutilityfunctionUðWÞ¼eAW¼expðAWÞ(7.35)(whereAisapositiveconstant)exhibitsconstantabsoluteriskaversionoverallrangesofwealth,becausenowrðWÞ¼U00ðWÞU0ðWÞ¼A2eAWAeAW¼A.(7.36)Thisfeatureoftheexponentialutilityfunction15canbeusedtoprovidesomenumericalestimatesofthewillingnesstopaytoavoidgambles,asthenextexampleshows.EXAMPLE7.3ConstantRiskAversionSupposeanindividualwhoseinitialwealthisW0andwhoseutilityfunctionexhibitsconstantabsoluteriskaversionisfacinga50–50chanceofwinningorlosing$1,000.Howmuch(f)wouldheorshepaytoavoidtherisk?Toﬁndthisvalue,wesettheutilityofW0fequaltotheexpectedutilityfromthegamble:exp½AðW0fÞ¼0.5exp½AðW0þ1,000Þ0.5exp½AðW01,000Þ.(7.37)BecausethefactorexpðAW0ÞiscontainedinallofthetermsinEquation7.37,thismaybedividedout,therebyshowingthat(fortheexponentialutilityfunction)thewillingnesstopaytoavoidagivengambleisindependentofinitialwealth.TheremainingtermsexpðAfÞ¼0.5expð1,000AÞþ0.5expð1,000AÞ(7.38)cannowbeusedtosolveforfforvariousvaluesofA.IfA¼0:0001,thenf¼49:9;apersonwiththisdegreeofriskaversionwouldpayabout$50toavoidafairbetof$1,000.Alternatively,ifA¼0:0003,thismorerisk-aversepersonwouldpayf¼147:8toavoidthegamble.Becauseintuitionsuggeststhatthesevaluesarenotunreasonable,valuesoftheriskaversionparameterAintheserangesaresometimesusedforempiricalinvestigations.(continued)15Becausetheexponentialutilityfunctionexhibitsconstant(absolute)riskaversion,itissometimesabbreviatedbythetermCARAutility.Chapter7UncertaintyandInformation211
EXAMPLE7.3CONTINUEDAnormallydistributedrisk.Theconstantriskaversionutilityfunctioncanbecombinedwiththeassumptionthatapersonfacesarandomthreattohisorherwealththatfollowsanormaldistribution(seeChapter2)toarriveataparticularlysimpleresult.Speciﬁcally,ifaperson’sriskywealthfollowsanormaldistributionwithmeanμWandvarianceσ2W,thentheprobabilitydensityfunctionforwealthisgivenbyfðWÞ¼ð1=ﬃﬃﬃﬃﬃﬃ2πpÞez2=2,wherez¼½ðWμWÞ=σW:IfthispersonhasautilityfunctionforwealthgivenbyUðWÞ¼eAW,thenexpectedutilityfromhisorherriskywealthisgivenbyE½UðWÞ¼∫∞∞UðWÞfðWÞdW¼1ﬃﬃﬃﬃﬃﬃﬃ2πp∫eAWe½ðWμWÞ=σW2=2dW.(7.39)Perhapssurprisingly,thisintegrationisnottoodifficulttoaccomplish,thoughitdoestakepatience.PerformingthisintegrationandtakingavarietyofmonotonictransformationsoftheresultingexpressionyieldsthefinalresultthatE½UðWÞ≅μWA2⋅σ2W.(7.40)Hence,expectedutilityisalinearfunctionofthetwoparametersofthewealthprobabilitydensityfunction,andtheindividual’sriskaversionparameter(A)determinesthesizeofthenegativeeffectofvariabilityonexpectedutility.Forexample,supposeapersonhasinvestedhisorherfundssothatwealthhasanexpectedvalueof$100,000butastandarddeviationðσWÞof$10,000.WiththeNormaldistribution,heorshemightthereforeexpectwealthtodeclinebelow$83,500about5percentofthetimeandriseabove$116,500asimilarfractionofthetime.Withtheseparameters,expectedutilityisgivenbyE½UðWÞ¼100,000ðA=2Þð10,000Þ2:IfA¼0:0001¼104,expectedutilityisgivenby100,00000:5⋅104⋅ð104Þ2¼95,000:Hence,thispersonreceivesthesameutilityfromhisorherriskywealthaswouldbeobtainedfromacertainwealthof$95,000.Amorerisk-aversepersonmighthaveA¼0:0003andinthiscasethe“certaintyequivalent”ofhisorherwealthwouldbe$85,000.QUERY:Supposethispersonhadtwowaystoinvesthisorherwealth:Allocation1,μW¼107,000andσW¼10,000;Allocation2,μW¼102,000andσW¼2,000:Howwouldthisperson’sattitudetowardriskaffecthisorherchoicebetweentheseallocations?16RelativeriskaversionItseemsunlikelythatthewillingnesstopaytoavoidagivengambleisindependentofaperson’swealth.AmoreappealingassumptionmaybethatsuchwillingnesstopayisinverselyproportionaltowealthandthattheexpressionrrðWÞ¼WrðWÞ¼WU00ðWÞU0ðWÞ(7.41)mightbeapproximatelyconstant.FollowingtheterminologyproposedbyJ.W.Pratt,17therrðW)functiondefinedinEquation7.41isameasureofrelativeriskaversion.Thepowerutilityfunction16Thisnumericalexample(veryroughly)approximateshistoricaldataonrealreturnsofstocksandbonds,respectively,thoughthecalculationsareillustrativeonly.17Pratt,“RiskAversion.”212Part2ChoiceandDemand
UðWÞ¼WRRðR<1,R6¼0Þ(7.42)andUðWÞ¼lnWðR¼0Þexhibitsdiminishingabsoluteriskaversion,rðWÞ¼U00ðWÞU0ðWÞ¼ðR1ÞWR2WR1¼ðR1ÞW,(7.43)butconstantrelativeriskaversion:rrðWÞ¼WrðWÞ¼ðR1Þ¼1R.(7.44)Empiricalevidence18isgenerallyconsistentwithvaluesofRintherangeof–3to–1.Hence,individualsseemtobesomewhatmoreriskaversethanisimpliedbythelogarithmicutilityfunction,thoughinmanyapplicationsthatfunctionprovidesareasonableapprox-imation.ItisusefultonotethattheconstantrelativeriskaversionutilityfunctioninEquation7.42hasthesameformasthegeneralCESutilityfunctionwefirstdescribedinChapter3.Thisprovidessomegeometricintuitionaboutthenatureofriskaversionthatwewillexplorelaterinthischapter.EXAMPLE7.4ConstantRelativeRiskAversionAnindividualwhosebehaviorischaracterizedbyaconstantrelativeriskaversionutilityfunctionwillbeconcernedaboutproportionalgainsorlossofwealth.Wecanthereforeaskwhatfractionofinitialwealth(f)suchapersonwouldbewillingtogiveuptoavoidafairgambleof,say,10percentofinitialwealth.First,weassumeR¼0,sothelogarithmicutilityfunctionisappropriate.Settingtheutilityofthisindividual’scertainremainingwealthequaltotheexpectedutilityofthe10percentgambleyieldsln½ð1fÞW0¼0:5lnð1:1W0Þþ0:5lnð0:9W0Þ.(7.45)BecauseeachtermcontainslnW0,initialwealthcanbeeliminatedfromthisexpression:lnð1fÞ¼0:5½lnð1:1Þþlnð0:9Þ¼lnð0:99Þ0:5;henceð1fÞ¼ð0:99Þ0:5¼0:995andf¼0:005.(7.46)Thispersonwillthussacrificeupto0.5percentofwealthtoavoidthe10percentgamble.AsimilarcalculationcanbeusedforthecaseR¼2toyieldf¼0:015.(7.47)Hencethismorerisk-aversepersonwouldbewillingtogiveup1.5percentofhisorherinitialwealthtoavoida10percentgamble.QUERY:Withtheconstantrelativeriskaversionfunction,howdoesthisperson’swillingnesstopaytoavoidagivenabsolutegamble(say,of1,000)dependonhisorherinitialwealth?18SomeauthorswritetheutilityfunctioninEquation7.42asUðWÞ¼W1a=ð1aÞandseektomeasurea¼1R.Inthiscase,aistherelativeriskaversionmeasure.TheconstantrelativeriskaversionfunctionissometimesabbreviatedasCRRA.Chapter7UncertaintyandInformation213 THEPORTFOLIOPROBLEMOneoftheclassicproblemsinthetheoryofbehaviorunderuncertaintyistheissueofhowmuchofhisorherwealtharisk-averseinvestorshouldinvestinariskyasset.Intuitively,itseemsthatthefractioninvestedinriskyassetsshouldbesmallerformorerisk-averseinvestors,andonegoalofouranalysiswillbetoshowthatformally.Togetstarted,assumethataninvestorhasacertainamountofwealth,W0,toinvestinoneoftwoassets.Theﬁrstassetyieldsacertainreturnofrf,whereasthesecondasset’sreturnisarandomvariable,er.Ifwelettheamountinvestedintheriskyassetbedenotedbyk,thenthisperson’swealthattheendofoneperiodwillbeW¼ðW0kÞð1þrfÞþkð1þerÞ¼W0ð1þrfÞþkðerrfÞ.(7.48)Noticethreethingsaboutthisend-of-periodwealth.First,Wisarandomvariablebecauseitsvaluedependsoner.Second,kcanbeeitherpositiveornegativeheredependingonwhetherthispersonbuystheriskyassetorsellsitshort.Asweshallsee,however,intheusualcaseEðerrfÞ>0andthiswillimplyk0.Finally,noticealsothatEquation7.48allowsforasolutioninwhichk>W0.Inthiscase,thisinvestorwouldleveragehisorherinvestmentintheriskyassetbyborrowingattherisk-freeraterf.IfweletUðWÞrepresentthisinvestor’sutilityfunction,thenthevonNeumann–Mor-gensterntheoremstatesthatheorshewillchoosektomaximizeE½UðWÞ.Theﬁrst-orderconditionforsuchamaximumis19∂E½UðWÞ∂k¼∂E½UðW0ð1þrfÞþkðerrfÞÞ∂k¼E½U0⋅ðerrfÞ¼0.(7.49)Becausethisfirst-orderconditionliesattheheartofmanyproblemsinthetheoryofuncer-tainty,itmaybeworthwhilespendingsometimetounderstanditintuitively.Equation7.49islookingattheexpectedvalueoftheproductofmarginalutilityandthetermerrf.Bothofthesetermsarerandom.Whethererrfispositiveornegativewilldependonhowwelltheriskyassetsperformoverthenextperiod.Butthereturnonthisriskyassetwillalsoaffectthisinvestor’send-of-periodwealthandthuswillaffecthisorhermarginalutility.Iftheinvestmentdoeswell,Wwillbelargeandmarginalutilitywillberelativelylow(becauseofdiminishingmarginalutility).Iftheinvestmentdoespoorly,wealthwillberelativelylowandmarginalutilitywillberelativelyhigh.Hence,intheexpectedvaluecalculationinEquation7.49,negativeoutcomesforerrfwillbeweightedmoreheavilythanpositiveoutcomestotaketheutilityconsequencesoftheseoutcomesintoaccount.IftheexpectedvalueinEquation7.49werepositive,apersoncouldincreasehisorherexpectedutilitybyinvestingmoreintheriskyasset.Iftheexpectedvaluewerenegative,heorshecouldincreaseexpectedutilitybyreducingtheamountoftheriskyassetheld.Onlywhenthefirst-orderconditionholdswillthispersonhaveanoptimalportfolio.TwootherconclusionscanbedrawnfromtheoptimalityconditioninEquation7.49.First,solongasEðerrfÞ>0,aninvestorwillchoosepositiveamountsoftheriskyasset.Toseewhy,noticethatmeetingEquation7.49willrequirethatfairlylargevaluesofU0beattachedtosituationswhereerrfturnsouttobenegative.Thatcanonlyhappeniftheinvestorownspositiveamountsoftheriskyassetsothatend-of-periodwealthislowinsuchsituations.Asecondconclusionfromtheﬁrst-orderconditioninEquation7.49isthatinvestorswhoaremoreriskaversewillholdsmalleramountsoftheriskyassetthanwillinvestorswhoaremoretolerantofrisk.Again,thereasonrelatestotheshapeoftheU0function.Forveryrisk-averseinvestors,marginalutilityrisesrapidlyaswealthfalls.Hence,theyneedrelativelylittleexposuretopotentialnegativeoutcomesfromholdingtheriskyassettosatisfy19Incalculatingthisfirst-ordercondition,wecandifferentiatethroughtheexpectedvalueoperator.SeeChapter2foradiscussionofdifferentiatingintegrals.214Part2ChoiceandDemand
Equation7.49.InvestorswhoaremoretolerantofriskwillﬁndthatU0riseslessrapidlywhentheriskyassetperformspoorly,sotheywillbewillingtoholdmoreofit.Insummary,then,aformalstudyoftheportfolioproblemconﬁrmssimpleintuitionsabouthowpeoplechoosetoinvest.Tomakefurtherprogressonthequestionrequiresthatwemakesomespeciﬁcassumptionsabouttheinvestor’sutilityfunction.InExample7.5,welookatatwoexamples.EXAMPLE7.5ThePortfolioProblemwithSpecificUtilityFunctionsInthisproblemweshowtheimplicationsofassumingeitherCARAorCRRAutilityforthesolutiontotheportfolioallocationproblem.1.CARAUtility.IfUðWÞ¼expðAWÞthenthemarginalutilityfunctionisgivenbyU0ðWÞ¼AexpðAWÞ;substitutingforend-of-periodwealth,wehaveU0ðWÞ¼Aexp½AðW0ð1þrfÞþkðerrfÞÞ¼Aexp½AW0ð1þrfÞexp½AkðerrfÞ.(7.50)Thatis,themarginalutilityfunctioncanbeseparatedintoarandompartandanonrandompart(bothinitialwealthandtherisk-freeratearenonrandom).Hence,theoptimalityconditionfromEquation7.49canbewrittenasE½U0⋅ðerrfÞ¼Aexp½AW0ð1þrfÞE½expðAkðerrfÞÞ⋅ðerrfÞ¼0:(7.51)Nowwecandividebytheexponentialfunctionofinitialwealth,leavinganoptimalityconditionthatinvolvesonlytermsink,A,anderrf.Solvingthisconditionfortheoptimallevelofkcaningeneralbequitedifficult(butseeProblem7.14).Regardlessofthespecificsolution,however,Equation7.51showsthatthisoptimalinvestmentamountwillbeaconstantregardlessofthelevelofinitialwealth.Hence,theCARAfunctionimpliesthatthefractionofwealththataninvestorholdsinriskyassetsshoulddeclineaswealthincreases—aconclusionthatseemspreciselycontrarytoempiricaldata,whichtendtoshowthefractionofwealthheldinriskyassetsrisingwithwealth.2.CRRAUtility.IfUðWÞ¼WR=RthenthemarginalutilityfunctionisgivenbyU0ðWÞ¼WR1.SubstitutingtheexpressionforﬁnalwealthintothisequationyieldsU0ðWÞ¼½W0ð1þrfÞþkðerrfÞR1¼½W0ð1þrfÞR11þkW0ð1þrfÞ⋅ðerrfÞ.(7.52)InsertingthisexpressionintotheoptimalityconditioninEquation7.49showsthattheterm½W0ð1þrfÞR1canbecanceledout,implyingthattheoptimalsolutionwillnotinvolvetheabsolutelevelofinitialwealthbutonlytheratiok=W0ð1þrfÞ.Inwords,theCRRAutilityfunctionimpliesthatallindividualswiththesamerisktolerancewillholdthesamefractionofwealthinriskyassets,regardlessoftheirabsolutelevelsofwealth.ThoughthisconclusionisslightlymoreinaccordwiththefactsthanistheconclusionfromtheCARAfunction,itstillfallsshortofexplainingwhythefractionofwealthheldinriskyassetstendstorisewithwealth.QUERY:CanyousuggestareasonwhyinvestorsmightincreasetheproportionoftheirportfoliosinvestedinriskyassetsaswealthincreaseseventhoughtheirpreferencesarecharacterizedbytheCRRAutilityfunction?Chapter7UncertaintyandInformation215
THESTATE-PREFERENCEAPPROACHTOCHOICEUNDERUNCERTAINTYAlthoughouranalysisinthischapterhasofferedinsightsonanumberofissues,itseemsratherdifferentfromtheapproachwetookinotherchapters.Thebasicmodelofutilitymaximizationsubjecttoabudgetconstraintseemstohavebeenlost.Inordertomakefurtherprogressinthestudyofbehaviorunderuncertainty,wewillthereforedevelopsomenewtechniquesthatwillpermitustobringthediscussionofsuchbehaviorbackintothestandardchoice-theoreticframework.StatesoftheworldandcontingentcommoditiesWestartbyassumingthattheoutcomesofanyrandomeventcanbecategorizedintoanumberofstatesoftheworld.Wecannotpredictexactlywhatwillhappen,say,tomorrow,butweassumethatitispossibletocategorizeallofthepossiblethingsthatmighthappenintoaﬁxednumberofwell-deﬁnedstates.Forexample,wemightmaketheverycrudeapproxima-tionofsayingthattheworldwillbeinonlyoneoftwopossiblestatestomorrow:Itwillbeeither“goodtimes”or“badtimes.”Onecouldmakeamuchﬁnergradationofstatesoftheworld(involvingevenmillionsofpossiblestates),butmostoftheessentialsofthetheorycanbedevelopedusingonlytwostates.Aconceptualideathatcanbedevelopedconcurrentlywiththenotionofstatesoftheworldisthatofcontingentcommodities.Thesearegoodsdeliveredonlyifaparticularstateoftheworldoccurs.Asanexample,“$1ingoodtimes”isacontingentcommoditythatpromisestheindividual$1ingoodtimesbutnothingshouldtomorrowturnouttobebadtimes.Itisevenpossible,bystretchingone’sintuitiveabilitysomewhat,toconceiveofbeingabletopurchasethiscommodity:Imightbeabletobuyfromsomeonethepromiseof$1iftomorrowturnsouttobegoodtimes.Becausetomorrowcouldbebad,thisgoodwillprobablysellforlessthan$1.Ifsomeonewerealsowillingtosellmethecontingentcommodity“$1inbadtimes,”thenIcouldassuremyselfofhaving$1tomorrowbybuyingthetwocontingentcommodities“$1ingoodtimes”and“$1inbadtimes.”UtilityanalysisExaminingutility-maximizingchoicesamongcontingentcommoditiesproceedsformallyinmuchthesamewayweanalyzedchoicespreviously.Theprincipaldifferenceisthat,afterthefact,apersonwillhaveobtainedonlyonecontingentgood(dependingonwhetheritturnsouttobegoodorbadtimes).Beforetheuncertaintyisresolved,however,theindividualhastwocontingentgoodsfromwhichtochooseandwillprobablybuysomeofeachbecauseheorshedoesnotknowwhichstatewilloccur.WedenotethesetwocontingentgoodsbyWg(wealthingoodtimes)andWb(wealthinbadtimes).Assumingthatutilityisindependentofwhichstateoccurs20andthatthisindividualbelievesthatgoodtimeswilloccurwithprobabilityπ,theexpectedutilityassociatedwiththesetwocontingentgoodsisVðWg,WbÞ¼πUðWgÞþð1πÞUðWbÞ.(7.53)Thisisthemagnitudethisindividualseekstomaximizegivenhisorherinitialwealth,W.20Thisassumptionisuntenableincircumstanceswhereutilityofwealthdependsonthestateoftheworld.Forexample,theutilityprovidedbyagivenlevelofwealthmaydifferdependingonwhetheranindividualis“sick”or“healthy.”Wewillnotpursuesuchcomplicationshere,however.Formostofouranalysis,utilityisassumedtobeconcaveinwealth:U0ðWÞ>0,U00ðWÞ<0.216Part2ChoiceandDemand PricesofcontingentcommoditiesAssumingthatthispersoncanpurchaseadollarofwealthingoodtimesforpgandadollarofwealthinbadtimesforpb,hisorherbudgetconstraintisthenW¼pgWgþpbWb.(7.54)Thepriceratiopg=pbshowshowthispersoncantradedollarsofwealthingoodtimesfordollarsinbadtimes.If,forexample,pg¼0:80andpb¼0:20,thesacrificeof$1ofwealthingoodtimeswouldpermitthispersontobuycontingentclaimsyielding$4ofwealthshouldtimesturnouttobebad.Whethersuchatradewouldimproveutilitywill,ofcourse,dependonthespecificsofthesituation.Butlookingatproblemsinvolvinguncertaintyassituationsinwhichvariouscontingentclaimsaretradedisthekeyinsightofferedbythestate-preferencemodel.FairmarketsforcontingentgoodsIfmarketsforcontingentwealthclaimsarewelldevelopedandthereisgeneralagreementaboutthelikelihoodofgoodtimes(π),thenpricesfortheseclaimswillbeactuariallyfair—thatis,theywillequaltheunderlyingprobabilities:pg¼π,pb¼ð1πÞ:(7.55)Hence,thepriceratiopg=pbwillsimplyreflecttheoddsinfavorofgoodtimes:pgpb¼π1π.(7.56)Inourpreviousexample,ifpg¼π¼0:8andpb¼ð1πÞ¼0:2thenπ=ð1πÞ¼4.Inthiscasetheoddsinfavorofgoodtimeswouldbestatedas“4-to-1.”Fairmarketsforcontingentclaims(suchasinsurancemarkets)willalsoreflecttheseodds.Ananalogyisprovidedbythe“odds”quotedinhorseraces.Theseoddsare“fair”whentheyreflectthetrueprobabilitiesthatvarioushorseswillwin.RiskaversionWearenowinapositiontoshowhowriskaversionismanifestedinthestate-preferencemodel.Speciﬁcally,wecanshowthat,ifcontingentclaimsmarketsarefair,thenautility-maximizingindividualwilloptforasituationinwhichWg¼Wb;thatis,heorshewillarrangematterssothatthewealthultimatelyobtainedisthesamenomatterwhatstateoccurs.Asinpreviouschapters,maximizationofutilitysubjecttoabudgetconstraintrequiresthatthisindividualsettheMRSofWgforWbequaltotheratioofthese“goods”prices:MRS¼∂V=∂Wg∂V=∂Wb¼πU0ðWgÞð1πÞU0ðWbÞ¼pgpb.(7.57)Inviewoftheassumptionthatmarketsforcontingentclaimsarefair(Equation7.56),thisfirst-orderconditionreducestoU0ðWgÞU0ðWbÞ¼1or21Wg¼Wb.(7.58)21ThissteprequiresthatutilitybestateindependentandthatU0ðWÞ>0.Chapter7UncertaintyandInformation217
Hencethisindividual,whenfacedwithfairmarketsincontingentclaimsonwealth,willberiskaverseandwillchoosetoensurethatheorshehasthesamelevelofwealthregardlessofwhichstateoccurs.AgraphicanalysisFigure7.2illustratesriskaversionwithagraph.Thisindividual’sbudgetconstraint(I)isshowntobetangenttotheU1indifferencecurvewhereWg¼Wb—apointonthe“certaintyline”wherewealthðWÞisindependentofwhichstateoftheworldoccurs.AtWtheslopeoftheindifferencecurve½π=ð1πÞispreciselyequaltothepriceratiopg=pb.Ifthemarketforcontingentwealthclaimswerenotfair,utilitymaximizationmightnotoccuronthecertaintyline.Suppose,forexample,thatπ=ð1πÞ¼4butthatpg=pb¼2becauseensuringwealthinbadtimesprovesquitecostly.InthiscasethebudgetconstraintwouldresemblelineI0inFigure7.2andutilitymaximizationwouldoccurbelowthecertaintyline.22InthiscasethisindividualwouldgambleabitbyoptingforWg>Wb,becauseclaimsonWbarerelativelycostly.Example7.6showstheusefulnessofthisapproachinevaluatingsomeofthealternativesthatmightbeavailable.FIGURE7.2RiskAversionsintheState-PreferenceModelThelineIrepresentstheindividual’sbudgetconstraintforcontingentwealthclaims:W¼pgWgþpbWb.Ifthemarketforcontingentclaimsisactuariallyfair½pg=pb¼π=ð1−πÞ,thenutilitymaxi-mizationwilloccuronthecertaintylinewhereWg¼Wb¼W.Ifpricesarenotactuariallyfair,thebudgetconstraintmayresembleI0andutilitymaximizationwilloccuratapointwhereWg>Wb.CertaintylineWbWgl′lW*W*U122Because(asEquation7.58shows)theMRSonthecertaintylineisalwaysπ=ð1−πÞ,tangencieswithaflatterslopethanthismustoccurbelowtheline.218Part2ChoiceandDemand
EXAMPLE7.6InsuranceintheState-PreferenceModelWecanillustratethestate-preferenceapproachbyrecastingtheautoinsuranceillustrationfromExample7.2asaprobleminvolvingthetwocontingentcommodities“wealthwithnotheft”ðWgÞand“wealthwithatheft”ðWbÞ.If,asbefore,weassumelogarithmicutilityandthattheprobabilityofatheft(thatis,1π)is0.25,thenexpectedutility¼0.75UðWgÞþ0:25UðWbÞ¼0.75lnWgþ0:25lnWb.(7.59)Iftheindividualtakesnoactionthenutilityisdeterminedbytheinitialwealthendowment,Wg¼100,000andWb¼80,000,soexpectedutility¼0:75ln100,000þ0.25ln80,000¼11.45714.(7.60)Tostudytradesawayfromtheseinitialendowments,wewritethebudgetconstraintintermsofthepricesofthecontingentcommodities,pgandpb:pgWgþpbWb¼pgWgþpbWb.(7.61)Assumingthatthesepricesequaltheprobabilitiesofthetwostatesðpg¼0:75,pb¼0:25Þ,thisconstraintcanbewritten0:75ð100,000Þþ0:25ð80,000Þ¼95,000¼0:75Wgþ0:25Wb;(7.62)thatis,theexpectedvalueofwealthis$95,000,andthispersoncanallocatethisamountbetweenWgandWb.NowmaximizationofutilitywithrespecttothisbudgetconstraintyieldsWg¼Wb¼95,000.Consequently,theindividualwillmovetothecertaintylineandreceiveanexpectedutilityofexpectedutility¼ln95,000¼11.46163,(7.63)aclearimprovementoverdoingnothing.Toobtainthisimprovement,thispersonmustbeabletotransfer$5,000ofwealthingoodtimes(notheft)into$15,000ofextrawealthinbadtimes(theft).Afairinsurancecontractwouldallowthisbecauseitwouldcost$5,000butreturn$20,000shouldatheftoccur(butnothingshouldnotheftoccur).Noticeherethatthewealthchangespromisedbyinsurance—dWb=dWg¼15,000=5,000¼3—exactlyequalthenegativeoftheoddsratioπ=ð1πÞ¼0:75=0:25¼3.Apolicywithadeductibleprovision.Anumberofotherinsurancecontractsmightbeutilityimprovinginthissituation,thoughnotallofthemwouldleadtochoicesthatlieonthecertaintyline.Forexample,apolicythatcost$5,200andreturned$20,000incaseofatheftwouldpermitthispersontoreachthecertaintylinewithWg¼Wb¼94,800andexpectedutility¼ln94,800¼11.45953,(7.64)whichalsoexceedstheutilityobtainablefromtheinitialendowment.Apolicythatcosts$4,900andrequirestheindividualtoincurthefirst$1,000ofalossfromtheftwouldyieldWg¼100,0004,900¼95,100,Wb¼80,0004,900þ19,000¼94,100;(7.65)thenexpectedutility¼0:75ln95,100þ0:25ln94,100¼11:46004:(7.66)Althoughthispolicydoesnotpermitthispersontoreachthecertaintyline,itisutilityimproving.Insuranceneednotbecompleteinordertoofferthepromiseofhigherutility.(continued)Chapter7UncertaintyandInformation219
EXAMPLE7.6CONTINUEDQUERY:Whatisthemaximumamountanindividualwouldbewillingtopayforaninsurancepolicyunderwhichheorshehadtoabsorbtheﬁrst$1,000ofloss?RiskaversionandriskpremiumsThestate-preferencemodelisalsoespeciallyusefulforanalyzingtherelationshipbetweenriskaversionandindividuals’willingnesstopayforrisk.Considertwopeople,eachofwhomstartswithacertainwealth,W.EachpersonseekstomaximizeanexpectedutilityfunctionoftheformVðWg,WbÞ¼πWRgRþð1πÞWRbR.(7.67)Heretheutilityfunctionexhibitsconstantrelativeriskaversion(seeExample7.4).NoticealsothatthefunctioncloselyresemblestheCESutilityfunctionweexaminedinChapter3andelsewhere.TheparameterRdeterminesboththedegreeofriskaversionandthedegreeofcurvatureofindifferencecurvesimpliedbythefunction.Averyrisk-averseindividualwillhavealargenegativevalueforRandhavesharplycurvedindifferencecurves,suchasU1showninFigure7.3.ApersonwithmoretoleranceforriskwillhaveahighervalueofRandflatterindifferencecurves(suchasU2).23FIGURE7.3RiskAversionandRiskPremiumsIndifferencecurveU1representsthepreferencesofaveryrisk-averseperson,whereasthepersonwithpreferencesrepresentedbyU2iswillingtoassumemorerisk.Whenfacedwiththeriskoflosinghinbadtimes,person2willrequirecompensationofW2−Wingoodtimeswhereasperson1willrequirealargeramountgivenbyW1−W.CertaintylineWbWgW*W*− hW*W1W2U1U223TangencyofU1andU2atWisensured,becausetheMRSalongthecertaintylineisgivenbyπ=ð1πÞregardlessofthevalueofR.220Part2ChoiceandDemand
Supposenowtheseindividualsarefacedwiththeprospectoflosinghdollarsofwealthinbadtimes.Suchariskwouldbeacceptabletoindividual2ifwealthingoodtimesweretoincreasefromWtoW2.Fortheveryrisk-averseindividual1,however,wealthwouldhavetoincreasetoW1tomaketheriskacceptable.ThedifferencebetweenW1andW2thereforeindicatestheeffectofriskaversiononwillingnesstoassumerisk.Someoftheproblemsinthischaptermakeuseofthisgraphicdeviceforshowingtheconnectionbetweenpreferences(asreﬂectedbytheutilityfunctioninEquation7.67)andbehaviorinriskysituations.THEECONOMICSOFINFORMATIONInformationisavaluableeconomicresource.Peoplewhoknowwheretobuyhigh-qualitygoodscheaplycanmaketheirbudgetsstretchfurtherthanthosewhodon’t;farmerswithaccesstobetterweatherforecastingmaybeabletoavoidcostlymistakes;andgovernmentenvironmentalregulationcanbemoreefﬁcientifitisbasedongoodscientiﬁcknowledge.Althoughtheseobservationsaboutthevalueofinformationhavelongbeenrecognized,formaleconomicmodelingofinformationacquisitionanditsimplicationsforresourceallocationarefairlyrecent.24Despiteitslatestart,thestudyofinformationeconomicshasbecomeoneofthemajorareasincurrentresearch.Inthischapterwebrieﬂysurveysomeoftheissuesraisedbythisresearch.FarmoredetailontheeconomicsofinformationisprovidedinChapter18.PROPERTIESOFINFORMATIONOnedifﬁcultyencounteredbyeconomistswhowishtostudytheeconomicsofinformationisthat“information”itselfisnoteasytodeﬁne.Unliketheeconomicgoodswehavebeenstudyingsofar,the“quantity”ofinformationobtainablefromvariousactionsisnotwelldeﬁned,andwhatinformationisobtainedisnothomogeneousamongitsusers.Theformsofeconomicallyusefulinformationaresimplytoovariedtopermitthekindsofprice-quantitycharacterizationswehavebeenusingforbasicconsumergoods.Instead,economistswhowishtostudyinformationmusttakesomecaretospecifywhattheinformationalenvironmentisinaparticulardecisionproblem(thisissometimescalledtheinformationset)andhowthatenvironmentmightbechangedthroughindividualactions.Asmightbeexpected,thisapproachhasresultedinavastnumberofmodelsofspeciﬁcsituationswithlittleoverallcommonalityamongthem.Asecondcomplicationinvolvedinthestudyofinformationconcernssometechnicalpropertiesofinformationitself.Mostinformationisdurableandretainsvalueafterithasbeenused.Unlikeahotdog,whichisconsumedonlyonce,knowledgeofaspecialsalecanbeusednotonlybythepersonwhodiscoversitbutalsobyanyfriendswithwhomtheinformationisshared.Thefriendsthenmaygainfromthisinformationeventhoughtheydon’thavetospendanythingtoobtainit.Indeed,inaspecialcaseofthissituation,informationhasthecharacteristicofapurepublicgood(seeChapter19).Thatis,theinformationisbothnonrivalinthatothersmayuseitatzerocostandnonexclusiveinthatnoindividualcanpreventothersfromusingtheinformation.Theclassicexampleofthesepropertiesisanewscientiﬁcdis-covery.Whensomeprehistoricpeopleinventedthewheel,otherscoulduseitwithoutdetractingfromthevalueofthediscovery,andeveryonewhosawthewheelcouldcopyitfreely.Thesetechnicalpropertiesofinformationimplythatmarketmechanismsmayoftenope-rateimperfectlyinallocatingresourcestoinformationprovisionandacquisition.Standard24Theformalmodelingofinformationissometimesdatedfromthepath-breakingarticlebyG.J.Stigler,“TheEconomicsofInformation,”JournalofPoliticalEconomy(June1961):213–25.Chapter7UncertaintyandInformation221
modelsofsupplyanddemandmaythereforebeofrelativelylimiteduseinunderstandingsuchactivities.Ataminimum,modelshavetobedevelopedthataccuratelyreﬂectthepropertiesbeingassumedabouttheinformationalenvironment.Throughoutthelatterportionsofthisbook,wewilldescribesomeofthesituationsinwhichsuchmodelsarecalledfor.Here,however,wewillpayrelativelylittleattentiontosupply-demandequilibriaandwillinsteadfocusprimarilyoninformationissuesthatariseinthetheoryofindividualchoice.THEVALUEOFINFORMATIONDevelopingmodelsofinformationacquisitionnecessarilyrequiresusingtoolsfromourstudyofuncertaintyearlierinthischapter.Lackofinformationclearlyrepresentsaproblemin-volvinguncertaintyforadecisionmaker.Intheabsenceofperfectinformation,heorshemaynotbeabletoknowexactlywhattheconsequencesofaparticularactionwillbe.Betterinformationcanreducethatuncertaintyandthereforeleadtobetterdecisionsthatprovideincreasedutility.InformationandsubjectivepossibilitiesThisrelationshipbetweenuncertaintyandinformationacquisitioncanbeillustratedusingthestate-preferencemodel.Earlierweassumedthatanindividualformssubjectiveopinionsabouttheprobabilitiesofthetwostatesoftheworld,“goodtimes”and“badtimes.”Inthismodel,informationisvaluablebecauseitallowstheindividualtorevisehisorherestimatesoftheseprobabilitiesandtotakeadvantageoftheserevisions.Forexample,informationthatforetoldthattomorrowwoulddeﬁnitelybe“goodtimes”wouldcausethispersontorevisehisorherprobabilitiestoπg¼1,πb¼0andtochangehisorherpurchasesaccordingly.Whentheinformationreceivedislessdeﬁnitive,theprobabilitiesmaybechangedonlyslightly,butevensmallrevisionsmaybequitevaluable.IfyouasksomefriendsabouttheirexperienceswithafewbrandsofDVDplayersyouarethinkingofbuying,youmaynotwanttheiropinionstodictateyourchoice.Thepricesoftheplayersandothertypesofinformation(say,obtainedfromconsultingConsumerReports)willalsoaffectyourviews.Ultimately,however,youmustprocessallofthesefactorsintoadecisionthatreﬂectsyourassessmentoftheprobabilitiesofvarious“statesoftheworld”(inthiscase,thequalityobtainedfrombuyingdifferentbrands).AformalmodelToillustratewhyinformationhasvalue,assumethatanindividualfacesanuncertainsituationinvolving“good”and“bad”timesandthatheorshecaninvestina“message”thatwillyieldsomeinformationabouttheprobabilitiesoftheseoutcomes.Thismessagecantakeontwopotentialvalues,1or2,withprobabilitiespandð1pÞ,respectively.Ifthemessagetakesthevalue1,thenthispersonwillbelievethattheprobabilityofgoodtimesisgivenbyπ1g[andtheprobabilityofbadtimesbyπ1b¼ð1π1gÞ].Ifthemessagetakesthevalue2,ontheotherhand,theprobabilitiesareπ2gandð1π2gÞ.Oncethemessageisreceived,thispersonhastheoppor-tunitytomaximizeexpectedutilityonthebasisoftheseprobabilities.Ingeneral,itwouldbeexpectedthatheorshewillmakedifferentdecisionsdependingonwhatthemessageis.LetV1bethe(indirect)maximalexpectedutilitywhenthemessagetakesthevalue1andV2bethismaximalutilitywhenthemessagetakesthevalue2.Hence,whenthispersonisconsideringpurchasingthemessage(thatis,beforeitisactuallyreceived),expectedutilityisgivenby:Ewithm¼pV1þð1pÞV2.(7.68)Nowlet’sconsiderthesituationofthispersonwhenheorshedecidesnottopurchasethemessage.Inthiscase,asingledecisionmustbemadethatisbasedontheprobabilitiesof222Part2ChoiceandDemand
goodandbadtimes,π0gandð1π0gÞ.Becausetheindividualknowsthevariousprobabilitiesinvolved,consistencyrequiresthatπ0g¼pπ1gþð1pÞπ2g.NowletV0representthemaximalexpectedutilitythispersoncanobtainwiththeseprobabilities.Hence,wecanwriteexpectedutilitywithoutthemessageasEwithoutm¼V0¼pV0þð1pÞV0.(7.69)AcomparisonofEquations7.68and7.69showsthatthispersoncanalwaysachieveEwithoutmwhenheorshehastheinformationprovidedbythemessage.Thatis,heorshecanjustchoosetodisregardwhatthemessagesays.Butifheorshechoosestomakenew,differentdecisionsbasedontheinformationinthemessage,itmustbethecasethatthisinformationhasvalue.Thatis:EwithmEwithoutm.(7.70)Presumably,then,thispersonwillbewillingtopaysomethingforthemessagebecauseofthebetterdecision-makingopportunitiesitprovides.25Example7.7providesasimpleillustration.EXAMPLE7.7TheValueofInformationonPricesToillustratehownewinformationmayaffectutilitymaximization,let’sreturntooneoftheﬁrstmodelsweusedinChapter4.ThereweshowedthatifanindividualconsumestwogoodsandutilityisgivenbyUðx,yÞ¼x0:5y0:5,thentheindirectutilityfunctionisVðpx,py,IÞ¼I2p0:5xp0:5y.(7.71)Asanumericalexample,weconsideredthecasepx¼1,py¼4,I¼8,andcalculatedthatV¼I=2⋅1⋅2¼2.Nowsupposethatgoodyrepresents,say,acanofbrand-nametennisballs,andthisconsumerknowsthatthesecanbeboughtatapriceofeither$3or$5fromtwostoresbutdoesnotknowwhichstorechargeswhichprice.Becauseitisequallylikelythateitherstorehasthelowerprice,theexpectedvalueofthepriceis$4.But,becausetheindirectutilityfunctionisconvexinprice,thispersonreceivesanexpectedvalueofgreaterthanV¼2fromshoppingbecauseheorshecanbuymoreifthelow-pricedstoreisencountered.Beforeshopping,expectedutilityisE½Vðpx,py,IÞ¼0:5⋅Vð1,3,8Þþ0:5⋅Vð1,5,8Þ¼1:155þ0:894¼2:049.(7.72)Iftheconsumerknewwhichstoreofferedthelowerprice,utilitywouldbeevengreater.Ifthispersoncouldbuyatpy¼3withcertainty,thenindirectutilitywouldbeV¼2:309andwecanusethisresulttocalculatewhatthevalueofthisinformationis.Thatis,wecanaskwhatlevelofincome,I,wouldyieldthesameutilitywhenpy¼3,asisobtainedwhenthispersonmustchoosewhichstoretopatronizebychance.HenceweneedtosolvetheequationVðpx,py,IÞ¼I2p0:5xp0:5y¼I2⋅1⋅30:5¼2:049.(7.73)SolvingthisyieldsavalueofI¼7:098.Hence,thispersonwouldbewillingtopayupto0.902ð¼87.098Þfortheinformation.Noticethatavailabilityofthepriceinformation(continued)25Amoregeneralwaytostatethisresultistoconsiderthepropertiesoftheindividual’sindirectexpectedutilityfunction(V)asdependentontheprobabilitiesintheproblem.Thatis,VðπgÞ¼max½πgUðWgÞþð1πgÞUðWbÞ.Compar-ingEquations7.68and7.69amountstocomparingpVðπ1gÞþð1pÞVðπ2gÞtoVðπ0gÞ¼V½pπ1gþð1pÞπ2g.BecausetheVfunctionisconvexinπg,theinequalityinEquation7.70necessarilyholds.Chapter7UncertaintyandInformation223
EXAMPLE7.7CONTINUEDhelpsthispersonintwoways:(1)itincreasestheprobabilityheorshewillpatronizethelow-pricestorefrom0.5to1.0;and(2)itpermitsthispersontotakeadvantageofthelowerpriceofferedbybuyingmore.QUERY:Itseemsoddinthisproblemthatexpectedutilitywithpriceuncertainty(V¼2.049)isgreaterthanutilitywhenpricetakesitsexpectedvalue(V¼2).Doesthisviolatetheassumptionofriskaversion?FLEXIBILITYANDOPTIONVALUETheavailabilityofnewinformationallowsindividualstomakebetterdecisionsinsituationsinvolvinguncertainty.Itmaythereforebebeneﬁcialtotrytopostponemakingdecisionsuntiltheinformationarrives.Ofcourse,ﬂexibilitymaysometimesinvolvecostsofitsown,sothedecision-makingprocesscanbecomecomplex.Forexample,someoneplanningatriptotheCaribbeanwouldobviouslyliketoknowwhetherheorshewillhavegoodweather.Avacationerwhocouldwaituntilthelastminuteindecidingwhentogocouldusethelatestweatherforecasttomakethatdecision.Butwaitingmaybecostly(perhapsbecauselast-minuteairfaresaremuchhigher),sothechoicecanbeadifﬁcultone.Clearlytheoptiontodelaythedecisionisvaluable,butwhetherthis“optionvalue”exceedsthecostsinvolvedindelayisthecrucialquestion.Modelingtheimportanceofﬂexibilityindecisionmakinghasbecomeamajortopicinthestudyofuncertaintyandinformation.“Realoptiontheory”hascometobeanimportantcomponentofﬁnancialandmanagementtheory.Otherapplicationsarebeginningtoemergeinsuchdiverseﬁeldsasdevelopmenteconomics,naturalresourceeconomics,andlawandeconomics.Becausethisbookfocusesongeneraltheory,however,wecannotpursuetheseinterestinginnovationshere.Rather,ourbrieftreatmentwillfocusonhowquestionsofﬂexibilitymightbeincorporatedintosomeofthemodelswehavealreadyexamined,followedbyafewconcludingremarks.FlexibilityintheportfoliomodelSomeofthebasicprinciplesofrealoptiontheorycanbeillustratedbycombiningtheportfoliochoicemodelthatweintroducedearlierinthischapterwiththeideaofinformationmessagesintroducedintheprevioussection.Supposethataninvestorisconsideringputtingsomeportionofhisorherwealth(k)intoariskyasset.Thereturnontheassetisrandomanditscharacteristicswilldependonwhetherthereare“goodtimes”or“badtimes.”Thereturnsunderthesetwosituationsaredesignatedbyer1ander2,respectively.First,considerasituationwherethispersonwillgetamessagetellinghimorherwhetheritisgoodorbadtimes,butthemessagewillarriveaftertheinvestmentdecisionismade.Theprobabilitythatthemessagewillindicategoodtimesisgivenbyp.Inthiscase,thispersoncanbeviewedasinvestinginariskyassetwhosereturnisgivenbyer0¼per1þð1pÞer2.Followingtheprocedureoutlinedearlier,associatedwiththisassetwillbeanoptimalinvestment,k0,andtheexpectedutilityassociatedwiththisportfoliowillbeU0.Suppose,alternatively,thatthispersonhastheﬂexibilitytowaituntilafterthemessageisreceivedtodecideonhowhisorherportfoliowillbeallocated.Ifthemessagerevealsgoodtimes,thenheorshewillchoosetoinvestk1intheriskyassetandexpectedutilitywillbeU1.224Part2ChoiceandDemand
Ontheotherhand,ifthemessagerevealsbadtimes,thenheorshewillchoosetoinvestk2intheriskyassetandexpectedutilitywillbeU2.Hence,theexpectedutilityprovidedbytheoptionofwaitingbeforechoosingkwillbeU¼pU1þð1pÞU2.(7.74)Asbefore,itisclearthatUU0.Theinvestorcouldalwayschoosetoinvestk0nomatterwhatthemessagesays,butifheorshechoosesdifferingk’sdependingontheinformationinthemessage,itmustbebecausethatstrategyprovidesmoreexpectedutility.WhenU>U0,theoptiontowaithasrealvalueandthispersonwillbewillingtopaysomething(say,inforgoneinterestreceipts)forthatpossibility.FinancialoptionsInsomecasesoptionvaluescanbeobservedinactualmarkets.Forexample,ﬁnancialoptionsprovideabuyertheright,butnottheobligation,toconductaneconomictransaction(typicallybuyingorsellingastock)atspeciﬁedtermsatacertaindateinthefuture.AnoptiononMicrosoftCorporationshares,forinstance,mightgivethebuyertheright(butnottheobligation)tobuythestockinsixmonthsatapriceof$30pershare.Oraforeignexchangeoptionmightprovidethebuyerwiththerighttobuyeurosatapriceof$1.30pereurointhreemonths.Allsuchoptionshavevaluebecausetheypermittheownertoeithermakeordeclinethespeciﬁedtransactiondependingonwhatnewinformationbecomesavailableovertheoption’sduration.Suchbuilt-inﬂexibilityisusefulinawidevarietyofinvestmentstrategies.OptionsembeddedinothertransactionsManyothertypesofeconomictransactionshaveoptionsembeddedinthem.Forexample,thepurchaseofagoodthatcomeswitha“money-backguarantee”givesthebuyeranoptiontoreversethetransactionshouldhisorherexperiencewiththegoodbeunfavorable.Similarly,manymortgagesprovidethehomeownerwiththeoptiontopayofftheloanwithoutpenaltyshouldconditionschange.Allsuchoptionsareclearlyvaluable.Acarbuyerisnotrequiredtoreturnhisorherpurchaseifthecarrunswellandthehomeownerneednotpayoffthemortgageifinterestratesrise.Hence,embeddingabuyer’soptioninatransactioncanonlyincreasethevalueofthattransactiontothebuyer.Contractswithsuchoptionswouldbeexpectedtohavehigherprices.Ontheotherhand,transactionswithembeddedselleroptions(forexample,therighttorepurchaseahouseatastatedprice)willhavelowerprices.Examiningpricedifferencescanthereforebeonewaytoinferthevalueofsomeembeddedoptions.ASYMMETRYOFINFORMATIONOneobviousimplicationofthestudyofinformationacquisitionisthatthelevelofinforma-tionthatanindividualbuyswilldependontheper-unitpriceofinformationmessages.Unlikethemarketpriceformostgoods(whichweusuallyassumetobethesameforeveryone),therearemanyreasonstobelievethatinformationcostsmaydiffersigniﬁcantlyamongindividuals.Someindividualsmaypossessspeciﬁcskillsrelevanttoinformationacquisition(theymaybetrainedmechanics,forexample)whereasothersmaynotpossesssuchskills.Someindividualsmayhaveothertypesofexperiencethatyieldvaluableinformation,whereasothersmaylackthatexperience.Forexample,thesellerofaproductwillusuallyknowmoreaboutitslimitationsthanwillabuyer,becausethesellerwillknowpreciselyhowthegoodwasmadeandwherepossibleproblemsmightarise.Similarly,large-scalerepeatbuyersofagoodmayhavegreateraccesstoinformationaboutitthanwouldﬁrst-timebuyers.Finally,someChapter7UncertaintyandInformation225
individualsmayhaveinvestedinsometypesofinformationservices(forexample,byhavingacomputerlinktoabrokerageﬁrmorbysubscribingtoConsumerReports)thatmakethemarginalcostofobtainingadditionalinformationlowerthanforsomeonewithoutsuchaninvestment.Allofthesefactorssuggestthatthelevelofinformationwillsometimesdifferamongtheparticipantsinmarkettransactions.Ofcourse,inmanyinstances,informationcostsmaybelowandsuchdifferencesmaybeminor.Mostpeoplecanappraisethequalityoffreshvegetablesfairlywelljustbylookingatthem,forexample.Butwheninformationcostsarehighandvariableacrossindividuals,wewouldexpectthemtoﬁnditadvantageoustoacquiredifferentamountsofinformation.WewillpostponeadetailedstudyofsuchsituationsuntilChapter18.PROBLEMS7.1Georgeisseentoplaceaneven-money$100,000betontheBullstowintheNBAFinals.IfGeorgehasalogarithmicutility-of-wealthfunctionandifhiscurrentwealthis$1,000,000,whatmusthebelieveistheminimumprobabilitythattheBullswillwin?7.2Showthatifanindividual’sutility-of-wealthfunctionisconvexthenheorshewillpreferfairgamblestoincomecertaintyandmayevenbewillingtoacceptsomewhatunfairgambles.Doyoubelievethissortofrisk-takingbehavioriscommon?Whatfactorsmighttendtolimititsoccurrence?7.3Anindividualpurchasesadozeneggsandmusttakethemhome.Althoughmakingtripshomeiscostless,thereisa50percentchancethatalloftheeggscarriedonanyonetripwillbebrokenduringthetrip.Theindividualconsiderstwostrategies:(1)takeall12eggsinonetrip;or(2)taketwotripswith6eggsineachtrip.SUMMARYThegoalofthischapterwastoprovidesomebasicmaterialforthestudyofindividualbehaviorinuncertainsituations.Thekeyconceptscoveredmaybelistedasfollows.•Themostcommonwaytomodelbehaviorunderuncer-taintyistoassumethatindividualsseektomaximizetheexpectedutilityoftheiractions.•Individualswhoexhibitadiminishingmarginalutilityofwealthareriskaverse.Thatis,theygenerallyrefusefairbets.•Risk-averseindividualswillwishtoinsurethemselvescompletelyagainstuncertaineventsifinsurancepre-miumsareactuariallyfair.Theymaybewillingtopaymorethanactuariallyfairpremiumsinordertoavoidtakingrisks.•Twoutilityfunctionshavebeenextensivelyusedinthestudyofbehaviorunderuncertainty:theconstantabsoluteriskaversion(CARA)functionandthecon-stantrelativeriskaversion(CRRA)function.Neitheriscompletelysatisfactoryontheoreticalgrounds.•Oneofthemostextensivelystudiedissuesintheeco-nomicsofuncertaintyisthe“portfolioproblem,”whichaskshowaninvestorwillsplithisorherwealthbetweenriskyandrisk-freeassets.Insomecasesitispossibletoobtainprecisesolutionstothisproblem,dependingonthenatureoftheriskyassetsthatareavailable.•Thestate-preferenceapproachallowsdecisionmakingunderuncertaintytobeapproachedinafamiliarchoice-theoreticframework.Theapproachisespeciallyusefulforlookingatissuesthatariseintheeconomicsofinformation.•Informationisvaluablebecauseitpermitsindividualstomakebetterdecisionsinuncertainsituations.Informa-tioncanbemostvaluablewhenindividualshavesomeﬂexibilityintheirdecisionmaking.226Part2ChoiceandDemand
a.Listthepossibleoutcomesofeachstrategyandtheprobabilitiesoftheseoutcomes.Showthat,onaverage,6eggswillremainunbrokenafterthetriphomeundereitherstrategy.b.Developagraphtoshowtheutilityobtainableundereachstrategy.Whichstrategywillbepreferable?c.Couldutilitybeimprovedfurtherbytakingmorethantwotrips?Howwouldthispossibilitybeaffectedifadditionaltripswerecostly?7.4Supposethereisa50–50chancethatarisk-averseindividualwithacurrentwealthof$20,000willcontractadebilitatingdiseaseandsufferalossof$10,000.a.Calculatethecostofactuariallyfairinsuranceinthissituationanduseautility-of-wealthgraph(suchasshowninFigure7.1)toshowthattheindividualwillpreferfairinsuranceagainstthislosstoacceptingthegambleuninsured.b.Supposetwotypesofinsurancepolicieswereavailable:(1)afairpolicycoveringthecompleteloss;and(2)afairpolicycoveringonlyhalfofanylossincurred.Calculatethecostofthesecondtypeofpolicyandshowthattheindividualwillgenerallyregarditasinferiortothefirst.7.5Ms.Foggisplanninganaround-the-worldtriponwhichsheplanstospend$10,000.TheutilityfromthetripisafunctionofhowmuchsheactuallyspendsonitðYÞ,givenbyUðYÞ¼lnY.a.Ifthereisa25percentprobabilitythatMs.Foggwilllose$1,000ofhercashonthetrip,whatisthetrip’sexpectedutility?b.SupposethatMs.Foggcanbuyinsuranceagainstlosingthe$1,000(say,bypurchasingtraveler’schecks)atan“actuariallyfair”premiumof$250.Showthatherexpectedutilityishigherifshepurchasesthisinsurancethanifshefacesthechanceoflosingthe$1,000withoutinsurance.c.WhatisthemaximumamountthatMs.Foggwouldbewillingtopaytoinsureher$1,000?7.6Indecidingtoparkinanillegalplace,anyindividualknowsthattheprobabilityofgettingaticketispandthattheﬁneforreceivingtheticketisf.Supposethatallindividualsareriskaverse(thatis,U00ðWÞ<0,whereWistheindividual’swealth).Willaproportionalincreaseintheprobabilityofbeingcaughtoraproportionalincreaseintheﬁnebeamoreeffectivedeterrenttoillegalparking?Hint:UsetheTaylorseriesapproximationUðWfÞ¼UðWÞfU0ðWÞþðf2=2ÞU00ðWÞ.7.7Afarmerbelievesthereisa50–50chancethatthenextgrowingseasonwillbeabnormallyrainy.Hisexpectedutilityfunctionhastheformexpectedutility¼12lnYNRþ12lnYR,whereYNRandYRrepresentthefarmer’sincomeinthestatesof“normalrain”and“rainy,”respectively.Chapter7UncertaintyandInformation227 a.Supposethefarmermustchoosebetweentwocropsthatpromisethefollowingincomeprospects:Whichofthecropswillheplant?b.Supposethefarmercanplanthalfhisﬁeldwitheachcrop.Wouldhechoosetodoso?Explainyourresult.c.Whatmixofwheatandcornwouldprovidemaximumexpectedutilitytothisfarmer?d.Wouldwheatcropinsurance—whichisavailabletofarmerswhogrowonlywheatandwhichcosts$4,000andpaysoff$8,000intheeventofarainygrowingseason—causethisfarmertochangewhatheplants?7.8InEquation7.30weshowedthattheamountanindividualiswillingtopaytoavoidafairgamble(h)isgivenbyp¼0:5Eðh2ÞrðWÞ,whererðWÞisthemeasureofabsoluteriskaversionatthisperson’sinitiallevelofwealth.Inthisproblemwelookatthesizeofthispaymentasafunctionofthesizeoftheriskfacedandthisperson’slevelofwealth.a.Considerafairgamble(v)ofwinningorlosing$1.Forthisgamble,whatisEðv2Þ?b.Nowconsidervaryingthegambleinpart(a)bymultiplyingeachprizebyapositiveconstantk.Leth¼kv.WhatisthevalueofEðh2Þ?c.SupposethispersonhasalogarithmicutilityfunctionUðWÞ¼lnW.WhatisageneralexpressionforrðWÞ?d.Computetheriskpremium(p)fork¼0:5,1,and2andforW¼10and100.Whatdoyouconcludebycomparingthesixvalues?AnalyticalProblems7.9HARAUtilityTheCARAandCRRAutilityfunctionsarebothmembersofamoregeneralclassofutilityfunctionscalledharmonicabsoluteriskaversion(HARA)functions.ThegeneralformforthisfunctionisUðWÞ¼θðμþW=γÞ1γ,wherethevariousparametersobeythefollowingrestrictions:•γ1,•μþW=γ>0,•θ½ð1γÞ=γ>0.Thereasonsforthefirsttworestrictionsareobvious;thethirdisrequiredsothatU0>0.a.CalculaterðWÞforthisfunction.ShowthatthereciprocalofthisexpressionislinearinW.Thisistheoriginoftheterm“harmonic”inthefunction’sname.b.Showthat,whenμ¼0andθ¼½ð1γÞ=γγ1,thisfunctionreducestotheCRRAfunctiongiveninChapter7(seefootnote17).c.Useyourresultfrompart(a)toshowthatifγ!∞thenrðWÞisaconstantforthisfunction.d.Lettheconstantfoundinpart(c)berepresentedbyA.ShowthattheimpliedformfortheutilityfunctioninthiscaseistheCARAfunctiongiveninEquation7.35.e.Finally,showthataquadraticutilityfunctioncanbegeneratedfromtheHARAfunctionsimplybysettingγ¼1.f.DespitetheseeminggeneralityoftheHARAfunction,itstillexhibitsseverallimitationsforthestudyofbehaviorinuncertainsituations.Describesomeoftheseshortcomings.CropYNRYRWheat$28,000$10,000Corn19,00015,000228Part2ChoiceandDemand
7.10TheresolutionofuncertaintyInsomecasesindividualsmaycareaboutthedateatwhichtheuncertaintytheyfaceisresolved.Suppose,forexample,thatanindividualknowsthathisorherconsumptionwillbe10unitstoday(c1)butthattomorrow’sconsumption(c2)willbeeither10or2.5,dependingonwhetheracoincomesupheadsortails.Supposealsothattheindividual’sutilityfunctionhasthesimpleCobb-DouglasformUðc1,c2Þ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃc1c2p.a.Ifanindividualcaresonlyabouttheexpectedvalueofutility,willitmatterwhetherthecoinisﬂippedjustbeforeday1orjustbeforeday2?Explain.b.Moregenerally,supposethattheindividual’sexpectedutilitydependsonthetimingofthecoinﬂip.Speciﬁcally,assumethatexpectedutility¼E1½fE2½Uðc1,c2Þgα,whereE1representsexpectationstakenatthestartofday1,E2representsexpectationsatthestartofday2,andαrepresentsaparameterthatindicatestimingpreferences.Showthatifα¼1,theindividualisindifferentaboutwhenthecoinisflipped.c.Showthatifα¼2,theindividualwillpreferearlyresolutionoftheuncertainty—thatis,ﬂippingthecoinatthestartofday1.d.Showthatifα¼0.5,theindividualwillpreferlaterresolutionoftheuncertainty(ﬂippingatthestartofday2).e.Explainyourresultsintuitivelyandindicatetheirrelevanceforinformationtheory.Note:Thisproblemisanillustrationof“resolutionseeking”and“resolutionaverse”behavior;seeD.M.KrepsandE.L.Porteus,“TemporalResolutionofUncertaintyandDynamicChoiceTheory,”Econometrica(January1978):185–200.7.11MoreontheCRRAfunctionFortheconstantrelativeriskaversionutilityfunction(Equation7.42),weshowedthatthedegreeofriskaversionismeasuredbyð1RÞ.InChapter3weshowedthattheelasticityofsubstitutionforthesamefunctionisgivenby1=ð1RÞ.Hence,themeasuresarereciprocalsofeachother.Usingthisresult,discussthefollowingquestions.a.Whyisriskaversionrelatedtoanindividual’swillingnesstosubstitutewealthbetweenstatesoftheworld?Whatphenomenonisbeingcapturedbybothconcepts?b.HowwouldyouinterpretthepolarcasesR¼1andR¼∞inboththerisk-aversionandsubstitutionframeworks?c.Ariseinthepriceofcontingentclaimsin“bad”timesðPbÞwillinducesubstitutionandincomeeffectsintothedemandsforWgandWb.Iftheindividualhasaﬁxedbudgettodevotetothesetwogoods,howwillchoicesamongthembeaffected?WhymightWgriseorfalldependingonthedegreeofriskaversionexhibitedbytheindividual?d.Supposethatempiricaldatasuggestanindividualrequiresanaveragereturnof0.5percentbeforebeingtemptedtoinvestinaninvestmentthathasa50–50chanceofgainingorlosing5percent.Thatis,thispersongetsthesameutilityfromW0asfromanevenbeton1.055W0and0.955W0.(1)WhatvalueofRisconsistentwiththisbehavior?(2)Howmuchaveragereturnwouldthispersonrequiretoaccepta50–50chanceofgainingorlosing10percent?Note:Thispartrequiressolvingnonlinearequations,soapproximatesolutionswillsufﬁce.Thecomparisonoftherisk-rewardtrade-offillustrateswhatiscalledthe“equitypremiumpuzzle”inthatriskyinvestmentsseemactuallytoearnmuchmorethanisconsistentwiththedegreeofriskaversionsuggestedbyotherdata.SeeN.R.Kocherlakota,“TheEquityPremium:It’sStillaPuzzle,”JournalofEconomicLiterature(March1996):42–71.Chapter7UncertaintyandInformation229
7.12GraphingriskyinvestmentsInvestmentinriskyassetscanbeexaminedinthestate-preferenceframeworkbyassumingthatWdollarsinvestedinanassetwithacertainreturnrwillyieldWð1þrÞinbothstatesoftheworld,whereasinvestmentinariskyassetwillyieldWð1þrgÞingoodtimesandWð1þrbÞinbadtimes(whererg>r>rb).a.Graphtheoutcomesfromthetwoinvestments.b.Showhowa“mixedportfolio”containingbothrisk-freeandriskyassetscouldbeillustratedinyourgraph.Howwouldyoushowthefractionofwealthinvestedintheriskyasset?c.Showhowindividuals’attitudestowardriskwilldeterminethemixofrisk-freeandriskyassetstheywillhold.Inwhatcasewouldapersonholdnoriskyassets?d.Ifanindividual’sutilitytakestheconstantrelativeriskaversionform(Equation7.42),explainwhythispersonwillnotchangethefractionofriskyassetsheldashisorherwealthincreases.267.13TaxingrisksassetsSupposetheassetreturnsinProblem7.12aresubjecttotaxation.a.Show,undertheconditionsofProblem7.12,whyaproportionaltaxonwealthwillnotaffectthefractionofwealthallocatedtoriskyassets.b.Supposethatonlythereturnsfromthesafeassetweresubjecttoaproportionalincometax.Howwouldthisaffectthefractionofwealthheldinriskyassets?Whichinvestorswouldbemostaffectedbysuchatax?c.Howwouldyouranswertopart(b)changeifallassetreturnsweresubjecttoaproportionalincometax?Note:Thisproblemasksyoutocomputethepre-taxallocationofwealththatwillresultinpost-taxutilitymaximization.7.14TheportfolioproblemwithaNormallydistributedriskyassetInExample7.3weshowedthatapersonwithaCARAutilityfunctionwhofacesaNormallydistributedriskwillhaveexpectedutilityoftheformE½UðWÞ¼μWðA=2Þσ2W,whereμWistheexpectedvalueofwealthandσ2Wisitsvariance.UsethisfacttosolvefortheoptimalportfolioallocationforapersonwithaCARAutilityfunctionwhomustinvestkofhisorherwealthinaNormallydistributedriskyassetwhoseexpectedreturnisμrandvarianceinreturnisσ2r(youranswershoulddependonA).Explainyourresultsintuitively.26ThisproblemandthenextaretakenfromJ.E.Stiglitz,“TheEffectsofIncome,Wealth,andCapitalGainsTaxationinRiskTaking,”QuarterlyJournalofEconomics(May1969):263–83.230Part2ChoiceandDemand
SUGGESTIONSFORFURTHERREADINGArrow,K.J.“TheRoleofSecuritiesintheOptimalAllocationofRiskBearing.”ReviewofEconomicStudies31(1963):91–96.Introducesthestate-preferenceconceptandinterpretssecuritiesasclaimsoncontingentcommodities.———.“UncertaintyandtheWelfareEconomicsofMedicalCare.”AmericanEconomicReview53(1963):941–73.Excellentdiscussionofthewelfareimplicationsofinsurance.Hasaclear,concise,mathematicalappendix.ShouldbereadinconjunctionwithPauly’sarticleonmoralhazard(seeChapter18).Bernoulli,D.“ExpositionofaNewTheoryontheMeasurementofRisk.”Econometrica22(1954):23–36.ReprintoftheclassicanalysisoftheSt.Petersburgparadox.Dixit,A.K.,andR.S.Pindyck.InvestmentunderUncer-tainty.Princeton:PrincetonUniversityPress,1994.Focusesmainlyontheinvestmentdecisionbyﬁrmsbuthasagoodcoverageofoptionconcepts.Friedman,M.,andL.J.Savage.“TheUtilityAnalysisofChoice.”JournalofPoliticalEconomy56(1948):279–304.Analyzeswhyindividualsmaybothgambleandbuyinsurance.Veryreadable.Gollier,Christian.TheEconomicsofRiskandTime.Cam-bridge,MA:MITPress,2001.Containsacompletetreatmentofmanyoftheissuesdiscussedinthischapter.Especiallygoodontherelationshipbetweenallocationunderuncertaintyandallocationovertime.Mas-Colell,Andreu,MichaelD.Whinston,andJerryR.Green.MicroeconomicTheory.NewYork:OxfordUniversityPress,1995,chap.6.Providesagoodsummaryofthefoundationsofexpectedutilitytheory.Alsoexaminesthe“stateindependence”assumptionindetailandshowsthatsomenotionsofriskaversioncarryoverintocasesofstatedependence.Pratt,J.W.“RiskAversionintheSmallandintheLarge.”Econometrica32(1964):122–36.Theoreticaldevelopmentofrisk-aversionmeasures.Fairlytechnicaltreatmentbutreadable.Rothschild,M.,andJ.E.Stiglitz.“IncreasingRisk:1.ADefinition.”JournalofEconomicTheory2(1970):225–43.Developsaneconomicdeﬁnitionofwhatitmeansforonegambletobe“riskier”thananother.AsequelarticleintheJournalofEconomicTheoryprovideseconomicillustrations.Silberberg,E.,andW.Suen.TheStructureofEconomics:AMathematicalAnalysis,3rded.Boston:Irwin/McGraw-Hill,2001.Chapter13providesaniceintroductiontotherelationshipbetweenstatisticalconceptsandexpectedutilitymaximization.AlsoshowsindetailtheintegrationmentionedinExample7.3.Chapter7UncertaintyandInformation231
EXTENSIONSPortfoliosofManyRiskyAssetsTheportfolioproblemwestudiedinChapter7lookedataninvestor’sdecisiontoinvestaportionofhisorherwealthinasingleriskyasset.IntheseExtensionswewillseehowthismodelcanbegeneralizedtoconsiderportfolioswithmanysuchassets.Throughoutourdis-cussionwewillassumethattherearenriskyassets.Thereturnoneachassetisarandomvariabledenotedbyriði¼1,nÞ.Theexpectedvaluesandvariancesoftheseassets’returnsaredenotedbyEðriÞ¼μiandVarðriÞ¼σ2i,respectively.AninvestorwhoinvestsaportionofhisorherwealthinaportfoliooftheseassetswillobtainarandomreturnðrPÞgivenbyrP¼Xni¼1αiri,(i)whereαið0ÞisthefractionoftheriskyportfolioheldinassetiandwherePni¼1αi¼1.Inthissit-uation,theexpectedreturnonthisportfoliowillbeEðrPÞ¼μP¼Xni¼1αiμi.(ii)Ifthereturnsofeachassetareindependent,thenthevarianceoftheportfolio’sreturnwillbeVarðrPÞ¼σ2P¼Xni¼1α2iσ2i.(iii)Ifthereturnsarenotindependent,Equationiiiwouldhavetobemodifiedtotakecovariancesamongthereturnsintoaccount.Usingthisgeneralnotation,wenowproceedtolookatsomeaspectsofthisportfolioallocationproblem.E7.1DiversificationwithtworiskyassetsEquationiiiprovidesthebasicrationaleforholdingmanyassetsinaportfolio:sothatdiversiﬁcationcanreducerisk.Suppose,forexample,thatthereareonlytwoindependentassetsandthattheexpectedreturnsandvariancesofthosereturnsforeachoftheassetsareidentical.Thatis,assumeμ1¼μ2andσ21¼σ22.ApersonwhoinvestshisorherriskyportfolioinonlyoneoftheseseeminglyidenticalassetswillobtainμP¼μ1¼μ2andσ2P¼σ21¼σ22.Bymixingtheassets,however,thisinvestorcandobetterinthesensethatheorshecangetthesameexpectedyieldwithlowervariance.Noticethat,nomatterhowthispersoninvests,theexpectedreturnontheportfoliowillbethesame:μP¼α1μ1þð1α1Þμ2¼μ1¼μ2.(iv)Butthevariancewilldependontheallocationbe-tweenthetwoassets:σ2P¼α21σ21þð1α1Þ2σ22¼ð12α1þ2α21Þσ21.(v)Choosingα1tominimizethisexpressionyieldsα1¼0:5andσ2P¼0:5σ21.(vi)Hence,holdinghalfofone’sportfolioineachassetyieldsthesameexpectedreturnasholdingonlyoneasset,butitpromisesavarianceofreturnthatisonlyhalfaslarge.AsweshowedearlierinChapter7,thisistheprimarybenefitofdiversification.E7.2EfficientportfoliosWithmanyassets,theoptimaldiversiﬁcationproblemistochooseassetweightings(theα’s)soastomini-mizethevariance(orstandarddeviation)oftheport-folioforeachpotentialexpectedreturn.Thesolutiontothisproblemyieldsan“efﬁciencyfrontier”forriskyassetportfoliossuchasthatrepresentedbythelineEEinFigureE7.1.Portfoliosthatliebelowthisfrontierareinferiortothoseonthefrontierbecausetheyofferlowerexpectedreturnsforanydegreeofrisk.Portfolioreturnsabovethefrontierareunattainable.Sharpe(1970)discussesthemathematicsassociatedwithcon-structingtheEEfrontier.MutualfundsThenotionofportfolioefﬁciencyhasbeenwidelyappliedtothestudyofmutualfunds.Ingeneral,mu-tualfundsareagoodanswertosmallinvestors’diver-siﬁcationneeds.Becausesuchfundspoolthefundsofmanyindividuals,theyareabletoachieveeconomiesofscaleintransactionsandmanagementcosts.Thispermitsfundownerstoshareinthefortunesofamuchwidervarietyofequitiesthanwouldbepossibleifeachactedalone.Butmutualfundmanagershaveincen-tivesoftheirown,sotheportfoliostheyholdmaynotalwaysbeperfectrepresentationsoftheriskattitudesoftheirclients.Forexample,ScharfsteinandStein(1990)developamodelthatshowswhymutualfund232Part2ChoiceandDemand
managershaveincentivesto“followtheherd”intheirinvestmentpicks.Otherstudies,suchastheclassicinves-tigationbyJensen(1968),ﬁndthatmutualfundman-agersareseldomabletoattainextrareturnslargeenoughtooffsettheexpensestheychargeinvestors.Inrecentyearsthishasledmanymutualfundbuyerstofavor“index”fundsthatseeksimplytoduplicatethemarketaverage(asrepresented,say,bytheStan-dardandPoor’s500stockindex).Suchfundshaveverylowexpensesandthereforepermitinvestorstoachievediversiﬁcationatminimalcost.E7.3PortfolioseparationIfthereexistsarisk-freeassetwithexpectedreturnμfandσf¼0,thenoptimalportfolioswillconsistofmixturesofthisassetwithriskyones.Allsuchportfo-lioswillliealongthelinePPinFigure7.1,becausethisshowsthemaximumreturnattainableforeachvalueofσforvariousportfolioallocations.Theseallocationswillcontainonlyonespeciﬁcsetofriskyassets:thesetrepresentedbypointM.Inequilibriumthiswillbethe“marketportfolio”consistingofallcapitalassetsheldinproportiontotheirmarketvaluations.ThismarketportfoliowillprovideanexpectedreturnofμMandastandarddeviationofthatreturnofσM.TheequationforthelinePPthatrepresentsanymixedportfolioisgivenbythelinearequationμP¼μfþμMμfσM⋅σP.(vii)ThisshowsthatthemarketlinePPpermitsindivid-ualinvestorsto“purchase”returnsinexcessoftherisk-freereturnðμMμfÞbytakingonproportionallymoreriskðσP=σMÞ.ForchoicesonPPtotheleftofthemarketpointM,σP=σM<1andμf<μP<μM.High-riskpointstotherightofM—whichcanbeobtainedbyborrowingtoproducealeveragedport-folio—willhaveσP=σM>1andwillpromiseanexpectedreturninexcessofwhatisprovidedbythemarketportfolioðμP>μMÞ.Tobin(1958)wasoneoftheﬁrsteconomiststorecognizetherolethatrisk-freeassetsplayinidentifyingthemarketportfolioandinsettingthetermsonwhichinvestorscanobtainreturnsaboverisk-freelevels.E7.4IndividualchoicesFigureE7.2illustratestheportfoliochoicesofvariousinvestorsfacingtheoptionsofferedbythelinePP.FIGUREE7.1EfficientPortfoliosThefrontierEErepresentsoptimalmixturesofriskyassetsthatminimizethestandarddeviationoftheportfolio,σP,foreachexpectedreturn,μP.Arisk-freeassetwithreturnμfoffersinvestorstheopportunitytoholdmixedportfoliosalongPPthatmixthisrisk-freeassetwiththemarketportfolio,M.EEPPMPMMPfChapter7UncertaintyandInformation233
Thisﬁgureillustratesthetypeofportfoliochoicemodelpreviouslydescribedinthischapter.Individualswithlowtoleranceforrisk(I)willoptforportfoliosthatareheavilyweightedtowardtherisk-freeasset.Investorswillingtoassumeamodestdegreeofrisk(II)willoptforportfoliosclosetothemarketportfo-lio.High-riskinvestors(III)mayoptforleveragedportfolios.Noticethatallinvestorsfacethesame“price”ofriskðμMμfÞwiththeirexpectedreturnsbeingdeterminedbyhowmuchrelativeriskðσP=σMÞtheyarewillingtoincur.Noticealsothattheriskas-sociatedwithaninvestor’sportfoliodependsonlyonthefractionoftheportfolioinvestedinthemarketportfolioðαÞ,sinceσ2P¼α2σ2Mþð1αÞ2⋅0.Hence,σP=σM¼αandsotheinvestor’schoiceofportfolioisequivalenttohisorherchoiceofrisk.E7.5CapitalassetpricingmodelAlthoughtheanalysisofE7.4showshowaportfoliothatmixesarisk-freeassetwiththemarketportfoliowillbepriced,itdoesnotdescribetherisk-returntrade-offforasingleasset.Because(assumingtransactionsarecostless)aninvestorcanalwaysavoidriskunrelatedtotheoverallmarketbychoosingtodiversifywitha“marketportfolio,”such“unsystematic”riskwillnotwarrantanyexcessreturn.Anassetwill,however,earnanexcessreturntotheextentthatitcontributestooverallmarketrisk.Anassetthatdoesnotyieldsuchextrareturnswouldnotbeheldinthemarketportfolio,soitwouldnotbeheldatall.Thisisthefundamentalinsightofthecapitalassetpricingmodel(CAPM).Toexaminetheseresultsformally,consideraport-foliothatcombinesasmallamountðαÞofanassetwitharandomreturnofxwiththemarketportfolio(whichhasarandomreturnofM).Thereturnonthisport-folio(z)wouldbegivenbyz¼αxþð1αÞM.(viii)Theexpectedreturnisμz¼αμxþð1αÞμM(ix)FIGUREE7.2InvestorBehaviorandRiskAversionGiventhemarketoptionsPP,investorscanchoosehowmuchrisktheywishtoassume.Veryrisk-averseinvestors(UI)willholdmainlyrisk-freeassets,whereasrisktakers(UIII)willoptforleveragedportfolios.MPUIIUIIIUIPPfP234Part2ChoiceandDemand
withvarianceσ2z¼α2σ2xþð1αÞ2σ2Mþ2αð1αÞσx,M,(x)whereσx,Misthecovariancebetweenthereturnonxandthereturnonthemarket.Butourpreviousanalysisshowsμz¼μfþðμMμfÞ⋅σzσM.(xi)SettingEquationixequaltoxianddifferentiatingwithrespecttoαyields∂μz∂α¼μxμM¼μMμfσM∂σz∂α.(xii)Bycalculating∂σz=∂αfromEquationxandtakingthelimitasαapproacheszero,wegetμxμM¼μMμfσMσx,Mσ2MσM !,(xiii)or,rearrangingterms,μx¼μfþðμMμfÞ⋅σx,Mσ2M.(xiv)Again,riskhasarewardofμMμf,butnowthequantityofriskismeasuredbyσx,M=σ2M.Thisratioofthecovariancebetweenthereturnxandthemarkettothevarianceofthemarketreturnisreferredtoasthebetacoefficientfortheasset.Estimatedbetaco-efficientsforfinancialassetsarereportedinmanypublications.StudiesoftheCAPMThisversionofthecapitalassetpricingmodelcarriesstrongimplicationsaboutthedeterminantsofanyasset’sexpectedrateofreturn.Becauseofthissimplic-ity,themodelhasbeensubjecttoalargenumberofempiricaltests.Ingeneraltheseﬁndthatthemodel’smeasureofsystemicrisk(beta)isindeedcorrelatedwithexpectedreturns,whilesimplermeasuresofrisk(forexample,thestandarddeviationofpastreturns)arenot.PerhapsthemostinﬂuentialearlyempiricaltestthatreachedsuchaconclusionwasFamaandMacBeth(1973).ButtheCAPMitselfexplainsonlyasmallfractionofdifferencesinthereturnsofvariousassets.And,contrarytotheCAPM,anumberofauthorshavefoundthatmanyothereconomicfactorssigniﬁcantlyaffectexpectedreturns.Indeed,apromi-nentchallengetotheCAPMcomesfromoneofitsoriginalfounders—seeFamaandFrench(1992).ReferencesFama,E.F.,andK.R.French.“TheCrossSectionofExpectedStockReturns.”JournalofFinance47(1992):427–66.Fama,E.F.,andJ.MacBeth.“Risk,Return,andEquilib-rium.”JournalofPoliticalEconomy8(1973):607–36.Jensen,M.“ThePerformanceofMutualFundsinthePeriod1945–1964.”JournalofFinance(May1968):386–416.Scharfstein,D.S.,andJ.Stein.“HerdBehaviorandInvestment.”AmericanEconomicReview(June1990):465–89.Sharpe,W.F.PortfolioTheoryandCapitalMarkets.NewYork:McGraw-Hill,1970.Tobin,J.“LiquidityPreferenceasBehaviorTowardsRisk.”ReviewofEconomicStudies(February1958):65–86.Chapter7UncertaintyandInformation235
CHAPTER8StrategyandGameTheoryThischapterprovidesanintroductiontononcooperativegametheory,atoolusedtounderstandthestrategicinteractionsamongtwoormoreagents.Therangeofapplicationsofgametheoryhasbeengrowingconstantly,includingallareasofeconomics(fromlaboreconomicstomacroeconomics)andotherﬁeldssuchaspoliticalscienceandbiology.Gametheoryisparticularlyusefulinunderstandingtheinteractionbetweenﬁrmsinanoligopoly,sotheconceptslearnedherewillbeusedextensivelyinChapter15.WebeginwiththecentralconceptofNashequilibriumandstudyitsapplicationinsimplegames.WethengoontostudyreﬁnementsofNashequilibriumthatareusedingameswithmorecomplicatedtimingandinformationstructures.BASICCONCEPTSSofarinPartIIofthistext,wehavestudiedindividualdecisionsmadeinisolation.Inthischapterwestudydecisionmakinginamorecomplicated,strategicsetting.Inastrategicsetting,apersonmaynolongerhaveanobviouschoicethatisbestforhimorher.Whatisbestforonedecisionmakermaydependonwhattheotherisdoingandviceversa.Forexample,considerthestrategicinteractionbetweendriversandthepolice.Whetherdriversprefertospeedmaydependonwhetherthepolicesetupspeedtraps.Whetherthepoliceﬁndspeedtrapsvaluabledependsonhowmuchdriversspeed.Thisconfusingcircularitywouldseemtomakeitdifﬁculttomakemuchheadwayinanalyzingstrategicbehavior.Infact,thetoolsofgametheorywillallowustopushtheanalysisnearlyasfar,forexample,asouranalysisofconsumerutilitymaximizationinChapter4.Therearetwomajortasksinvolvedwhenusinggametheorytoanalyzeaneconomicsituation.Theﬁrstistodistillthesituationintoasimplegame.Becausetheanalysisinvolvedinstrategicsettingsquicklygrowsmorecomplicatedthaninsimpledecisionproblems,itisimportanttosimplifythesettingasmuchaspossiblebyretainingonlyafewessentialelements.Thereisacertainarttodistillinggamesfromsituationsthatishardtoteach.Theexamplesinthetextandproblemsinthischaptercanserveasmodelsthatmayhelpinapproachingnewsituations.Thesecondtaskisto“solve”thegivengame,whichresultsinapredictionaboutwhatwillhappen.Tosolveagame,onetakesanequilibriumconcept(Nashequilibrium,forex-ample)andrunsthroughthecalculationsrequiredtoapplyittothegivengame.Muchofthechapterwillbedevotedtolearningthemostwidelyusedequilibriumconcepts(includingNashequilibrium)andtopracticingthecalculationsnecessarytoapplythemtoparticulargames.Agameisanabstractmodelofastrategicsituation.Eventhemostbasicgameshavethreeessentialelements:players,strategies,andpayoffs.Incomplicatedsettings,itissometimesalsonecessarytospecifyadditionalelementssuchasthesequenceofmovesandtheinformationthatplayershavewhentheymove(whoknowswhatwhen)todescribethegamefully.236
PlayersEachdecisionmakerinagameiscalledaplayer.Theseplayersmaybeindividuals(asinpokergames),ﬁrms(asinmarketswithfewﬁrms),orentirenations(asinmilitaryconﬂicts).Aplayerischaracterizedashavingtheabilitytochoosefromamongasetofpossibleactions.Usually,thenumberofplayersisﬁxedthroughoutthe“play”ofthegame.Gamesaresometimescharacterizedbythenumberofplayersinvolved(two-player,three-player,orn-playergames).Asdoesmuchoftheeconomicliterature,thischapteroftenfocusesontwo-playergamesbecausethisisthesimpleststrategicsetting.Wewilllabeltheplayerswithnumbers,soinatwo-playergamewewillhaveplayers1and2.Inann-playergamewewillhaveplayers1,2,…,n,withthegenericplayerlabeledi.StrategiesEachcourseofactionopentoaplayerduringthegameiscalledastrategy.Dependingonthegamebeingexamined,astrategymaybeasimpleaction(driveoverthespeedlimitornot)oracomplexplanofactionthatmaybecontingentonearlierplayinthegame(say,speedingonlyifthedriverhasobservedspeedtrapslessthanaquarterofthetimeinpastdrives).Manyaspectsofgametheorycanbeillustratedingamesinwhichplayerschoosebetweenjusttwopossibleactions.LetS1denotethesetofstrategiesopentoplayer1,S2thesetopentoplayer2,and(moregenerally)Sithesetopentoplayeri.Lets12S1beaparticularstrategychosenbyplayer1fromthesetofpossibilities,s22S2theparticularstrategychosenbyplayer2,andsi2Siforplayeri.Astrategyproﬁlewillrefertoalistingofparticularstrategieschosenbyeachofagroupofplayers.PayoffsTheﬁnalreturnstotheplayersattheconclusionofagamearecalledpayoffs.Payoffsaremeasuredinlevelsofutilityobtainedbytheplayers.Forsimplicity,monetarypayoffs(say,proﬁtsforﬁrms)areoftenused.Moregenerally,payoffscanincorporatenonmonetaryout-comessuchasprestige,emotion,riskpreferences,andsoforth.Playersareassumedtopreferhigherpayoffsthanlowerpayoffs.Inatwo-playergame,u1ðs1,s2Þdenotesplayer1’spayoffgiventhatheorshechoosess1andtheotherplayerchoosess2andsimilarlyu2ðs2,s1Þdenotesplayer2’spayoff.Thefactplayer1’spayoffmaydependon2’sstrategy(andviceversa)iswherethestrategicinterdependenceshowsup.Inann-playergame,wecanwritethepayoffofagenericplayeriasuiðsi,siÞ,whichdependsonplayeri’sownstrategysiandtheproﬁlesi¼ðs1,…,si1,siþ1,…,snÞofthestrategiesofallplayersotherthani.PRISONERS’DILEMMAThePrisoners’Dilemma,introducedbyA.W.Tuckerinthe1940s,isoneofthemostfamousgamesstudiedingametheoryandwillservehereasaniceexampletoillustrateallthenotationjustintroduced.Thetitlestemsfromthefollowingsituation.Twosuspectsarear-restedforacrime.Thedistrictattorneyhaslittleevidenceinthecaseandiseagertoextractaconfession.Sheseparatesthesuspectsandtellseach:“Ifyouﬁnkonyourcompanionbutyourcompaniondoesn’tﬁnkonyou,Icanpromiseyouareduced(one-year)sentence,whereasyourcompanionwillgetfouryears.Ifyoubothﬁnkoneachother,youwilleachgetathree-yearsentence.”Eachsuspectalsoknowsthatifneitherofthemﬁnksthenthelackofevidencewillresultinbeingtriedforalessercrimeforwhichthepunishmentisatwo-yearsentence.Chapter8StrategyandGameTheory237
Boileddowntoitsessence,thePrisoners’Dilemmahastwostrategicplayers,thesuspects,labeled1and2.(Thereisalsoadistrictattorney,butsinceheractionshavealreadybeenfullyspeciﬁed,thereisnoreasontocomplicatethegameandincludeherinthespeciﬁcation.)Eachplayerhastwopossiblestrategiesopentohim:ﬁnkorremainsilent.WethereforewritetheirstrategysetsasS1¼S2ffink,silentg.Toavoidnegativenumberswewillspecifypayoffsastheyearsoffreedomoverthenextfouryears.Forexample,ifsuspect1ﬁnksand2doesnot,suspect1willenjoythreeyearsoffreedomand2none,thatis,u1ðfink,silentÞ¼3andu2ðsilent,finkÞ¼0.ExtensiveformThereare22¼4combinationsofstrategiesandtwopayoffstospecifyforeachcombi-nation.Soinsteadoflistingallthepayoffs,itwillbeclearertoorganizetheminagametreeoramatrix.Thegametree,alsocalledtheextensiveform,isshowninFigure8.1.Theactionproceedsfromlefttoright.Eachnode(shownasadotonthetree)representsadecisionpointfortheplayerindicatedthere.Theﬁrstmoveinthisgamebelongstoplayer1;hemustchoosewhethertoﬁnkorbesilent.Thenplayer2makeshisdecision.Thedottedovaldrawnaroundthenodesatwhichplayer2movesindicatesthatthetwonodesareinthesameinformationset,thatis,player2doesnotknowwhatplayer1haschosenwhen2moves.Weputthetwonodesinthesameinformationsetbecausethedistrictattorneyapproacheseachsuspectseparatelyanddoesnotrevealwhattheotherhaschosen.Wewilllaterlookatgamesinwhichthesecondmoverdoeshaveinformationabouttheﬁrstmover’schoiceandsothetwonodesareinseparateinformationsets.Payoffsaregivenattheendofthetree.Theconventionisforplayer1’spayofftobelistedﬁrst,thenplayer2’s.FIGURE8.1ExtensiveFormforthePrisoners’DilemmaInthisgame,player1choosestoﬁnkorbesilent,andplayer2hasthesamechoice.Theovalsurrounding2’snodesindicatesthattheysharethesame(lackof)information:2doesnotknowwhatstrategy1haschosenbecausethedistrictattorneyapproacheseachplayerinsecret.Payoffsarelistedattheright.Fink122Finku1=1, u2=1u1=3, u2=0u1=0,u2=3u1=2, u2=2FinkSilentSilentSilent238Part2ChoiceandDemand
NormalformAlthoughtheextensiveforminFigure8.1offersausefulvisualpresentationofthecompletestructureofthegame,sometimesitismoreconvenienttorepresentgamesinmatrixform,calledthenormalformofthegame;thisisshownforthePrisoners’DilemmainTable8.1.Player1istherowplayer,and2isthecolumnplayer.Eachentryinthematrixliststhepayoffsﬁrstforplayer1andthenfor2.ThinkingstrategicallyaboutthePrisoners’DilemmaAlthoughwehavenotdiscussedhowtosolvegamesyet,itisworththinkingaboutwhatwemightpredictwillhappeninthePrisoners’Dilemma.StudyingTable8.1,onﬁrstthoughtonemightpredictthatbothwillbesilent.Thisgivesthemosttotalyearsoffreedomforboth(four)comparedtoanyotheroutcome.Thinkingabitdeeper,thismaynotbethebestpredictioninthegame.Imagineourselvesinplayer1’spositionforamoment.Wedon’tknowwhatplayer2willdoyetsincewehaven’tsolvedoutthegame,solet’sinvestigateeachpossibility.Suppose2chosetoﬁnk.Byﬁnkingourselveswewouldearnoneyearoffreedomversusnoneifweremainedsilent,soﬁnkingisbetterforus.Suppose2chosetoremainsilent.Finkingisstillbetterforusthanremainingsilentsincewegetthreeratherthantwoyearsoffreedom.Regardlessofwhattheotherplayerdoes,ﬁnkingisbetterforusthanbeingsilentsinceitresultsinanextrayearoffreedom.Sinceplayersaresymmetric,thesamereasoningholdsifweimagineourselvesinplayer2’sposition.Therefore,thebestpredictioninthePrisoners’Dilemmaisthatbothwillﬁnk.Whenweformallyintroducethemainsolutionconcept—Nashequilibrium—wewillindeedﬁndthatbothﬁnkingisaNashequilibrium.Thepredictionhasaparadoxicalproperty:bybothﬁnking,thesuspectsonlyenjoyoneyearoffreedom,butiftheywerebothsilenttheywouldbothdobetter,enjoyingtwoyearsoffreedom.Theparadoxshouldnotbetakentoimplythatplayersarestupidorthatourpredictioniswrong.Rather,itrevealsacentralinsightfromgametheorythatpittingplayersagainsteachotherinstrategicsituationssometimesleadstooutcomesthatareinefﬁcientfortheplayers.(Whenwesaytheoutcomeisinefﬁcient,wearefocusingjustonthesuspects’utilities;ifthefocuswereshiftedtosocietyatlarge,thenbothﬁnkingmightbequiteagoodoutcomeforthecriminaljusticesystem—presumablythemotivationbehindthedistrictattorney’soffer.)Thesuspectsmighttrytoavoidtheextraprisontimebycomingtoanagreementbeforehandtoremainsilent,perhapsreinforcedbythreatstoretaliateafterwardsifoneortheotherﬁnks.IntroducingagreementsandthreatsleadstoagamethatdiffersfromthebasicPrisoners’Dilemma,agamethatshouldbeanalyzedonitsowntermsusingthetoolswewilldevelopshortly.SolvingthePrisoners’Dilemmawaseasybecausetherewereonlytwoplayersandtwostrategiesandbecausethestrategiccalculationsinvolvedwerefairlystraightforward.Itwouldbeusefultohaveasystematicwayofsolvingthisaswellasmorecomplicatedgames.Nashequilibriumprovidesuswithsuchasystematicsolution.TABLE8.1NormalFormforthePrisoners’DilemmaSuspect2FinkSilentSuspect1Finku1¼1,u2¼1u1¼3,u2¼0Silentu1¼0,u2¼3u1¼2,u2¼2Chapter8StrategyandGameTheory239
NASHEQUILIBRIUMIntheeconomictheoryofmarkets,theconceptofequilibriumisdevelopedtoindicateasituationinwhichbothsuppliersanddemandersarecontentwiththemarketoutcome.Giventheequilibriumpriceandquantity,nomarketparticipanthasanincentivetochangehisorherbehavior.Inthestrategicsettingofgametheory,wewilladoptarelatednotionofequilibrium,formalizedbyJohnNashinthe1950s,calledNashequilibrium.1Nashequilib-riuminvolvesstrategicchoicesthat,oncemade,providenoincentivesfortheplayerstoaltertheirbehaviorfurther.ANashequilibriumisastrategyforeachplayerthatisthebestchoiceforeachplayergiventheothers’equilibriumstrategies.Nashequilibriumcanbedeﬁnedverysimplyintermsofbestresponses.Inann-playergame,strategysiisabestresponsetorivals’strategiessiifplayericannotobtainastrictlyhigherpayoffwithanyotherpossiblestrategys0i2Sigiventhatrivalsareplayingsi.DEFINITIONBestresponse.siisabestresponseforplayeritorivals’strategiessi,denotedsi2BRiðsiÞ,ifuiðsi,siÞuiðs0i,siÞforalls0i2Si.(8.1)Atechnicalityembeddedinthedeﬁnitionisthattheremaybeasetofbestresponsesratherthanauniqueone;thatiswhyweusedthesetinclusionnotationsi2BRiðsiÞ.Theremaybeatieforthebestresponse,inwhichcasethesetBRiðsiÞwillcontainmorethanoneelement.Ifthereisn’tatie,thentherewillbeasinglebestresponsesiandwecansimplywritesi¼BRiðsiÞ.WecannowdeﬁneaNashequilibriuminann-playergameasfollows.DEFINITIONNashequilibrium.ANashequilibriumisastrategyproﬁleðs1,s2,…,snÞsuchthat,foreachplayeri¼1,2,…,n,siisabestresponsetotheotherplayers’equilibriumstrategiessi.Thatis,si2BRiðsiÞ.Thesedeﬁnitionsinvolvealotofnotation.Thenotationisabitsimplerinatwo-playergame.Inatwo-playergame,ðs1,s2ÞisaNashequilibriumifs1ands2aremutualbestresponsesagainsteachother:u1ðs1,s2Þu1ðs1,s2Þforalls12S1(8.2)andu2ðs2,s1Þu2ðs2,s1Þforalls22S2.(8.3)ANashequilibriumisstableinthat,evenifallplayersrevealedtheirstrategiestoeachother,noplayerwouldhaveanincentivetodeviatefromhisorherequilibriumstrategyandchoosesomethingelse.Nonequilibriumstrategiesarenotstableinthisway.IfanoutcomeisnotaNashequilibrium,thenatleastoneplayermustbenefitfromdeviating.HyperrationalplayerscouldbeexpectedtosolvetheinferenceproblemanddeducethatallwouldplayaNashequilibrium(especiallyifthereisauniqueNashequilibrium).Evenifplayersarenothyper-rational,overthelongrunwecanexpecttheirplaytoconvergetoaNashequilibriumastheyabandonstrategiesthatarenotmutualbestresponses.1JohnNash,“EquilibriumPointsinn-PersonGames,”ProceedingsoftheNationalAcademyofSciences36(1950):48–49.Nashistheprincipalfigureinthe2001filmABeautifulMind(seeProblem8.7foragame-theoryexamplefromthefilm)andco-winnerofthe1994NobelPrizeineconomics.240Part2ChoiceandDemand
Besidesthisstabilityproperty,anotherreasonNashequilibriumisusedsowidelyineconomicsisthatitisguaranteedtoexistforallgameswewillstudy(allowingformixedstrategies,tobedeﬁnedbelow;Nashequilibriainpurestrategiesdonothavetoexist).Nashequilibriumhassomedrawbacks.TheremaybemultipleNashequilibria,makingithardtocomeupwithauniqueprediction.Also,thedeﬁnitionofNashequilibriumleavesunclearhowaplayercanchooseabest-responsestrategybeforeknowinghowrivalswillplay.NashequilibriuminthePrisoners’DilemmaLet’sapplytheconceptsofbestresponseandNashequilibriumtotheexampleofthePrisoners’Dilemma.Oureducatedguesswasthatbothplayerswillendupﬁnking.WewillshowthatbothﬁnkingisaNashequilibriumofthegame.Todothis,weneedtoshowthatﬁnkingisabestresponsetotheotherplayers’ﬁnking.RefertothepayoffmatrixinTable8.1.Ifplayer2ﬁnks,weareintheﬁrstcolumnofthematrix.Ifplayer1alsoﬁnks,hispayoffis1;ifheissilent,hispayoffis0.Sinceheearnsthemostfromﬁnkinggivenplayer2ﬁnks,ﬁnkingisplayer1’sbestresponsetoplayer2’sﬁnking.Sinceplayersaresymmetric,thesamelogicimpliesthatplayer2’sﬁnkingisabestresponsetoplayer1’sﬁnking.Therefore,bothﬁnkingisindeedaNashequilibrium.Wecanshowmore:thatbothﬁnkingistheonlyNashequilibrium.Todoso,weneedtoruleouttheotherthreeoutcomes.Considertheoutcomeinwhichplayer1ﬁnksand2issilent,abbreviated(ﬁnk,silent),theupperrightcornerofthematrix.ThisisnotaNashequilibrium.Giventhatplayer1ﬁnks,aswehavealreadysaid,player2’sbestresponseistoﬁnk,nottobesilent.Symmetrically,theoutcomeinwhichplayer1issilentand2ﬁnksinthelowerleftcornerofthematrixisnotaNashequilibrium.Thatleavestheoutcomeinwhichbotharesilent.Giventhatplayer2issilent,wefocusourattentiononthesecondcolumnofthematrix:thetworowsinthatcolumnshowthatplayer1’spayoffis2frombeingsilentand3fromﬁnking.Therefore,silentisnotabestresponsetoﬁnkandsobothbeingsilentcannotbeaNashequilibrium.ToruleoutaNashequilibrium,itisenoughtoﬁndjustoneplayerwhoisnotplayingabestresponseandsowouldwanttodeviatetosomeotherstrategy.Consideringtheoutcome(ﬁnk,silent),althoughplayer1wouldnotdeviatefromthisoutcome(heearns3,whichisthemostpossible),player2wouldprefertodeviatefromsilenttoﬁnk.Symmetrically,consider-ingtheoutcome(silent,ﬁnk),althoughplayer2doesnotwanttodeviate,player1preferstodeviatefromsilenttoﬁnk,sothisisnotaNashequilibrium.Consideringtheoutcome(silent,silent),bothplayersprefertodeviatetoanotherstrategy.HavingtwoplayersprefertodeviateismorethanenoughtoruleoutaNashequilibrium.Underliningbest-responsepayoffsAquickwaytoﬁndtheNashequilibriaofagameistounderlinebest-responsepayoffsinthematrix.TheunderliningprocedureisdemonstratedforthePrisoners’DilemmainTable8.2.Theﬁrststepistounderlinethepayoffscorrespondingtoplayer1’sbestresponses.Player1’sTABLE8.2UnderliningProcedureinthePrisoners’DilemmaSuspect2FinkSilentSuspect1Finku1¼1,u2¼1u1¼3,u2¼0Silentu1¼0,u2¼3u1¼2,u2¼2Chapter8StrategyandGameTheory241
bestresponseistoﬁnkifplayer2ﬁnks,soweunderlineu1¼1intheupperleftbox,andtoﬁnkifplayer2issilent,soweunderlineu1¼3intheupperleftbox.Next,wemovetoun-derliningthepayoffscorrespondingtoplayer2’sbestresponses.Player2’sbestresponseistoﬁnkifplayer1ﬁnks,soweunderlineu2¼1intheupperleftbox,andtoﬁnkifplayer1issilent,soweunderlineu2¼3inthelowerleftbox.Nowthatthebest-responsepayoffshavebeenunderlined,welookforboxesinwhicheveryplayer’spayoffisunderlined.TheseboxescorrespondtoNashequilibria.(TheremaybeadditionalNashequilibriainvolvingmixedstrategies,deﬁnedlaterinthechapter.)InTable8.2,onlyintheupperleftboxarebothpayoffsunderlined,verifyingthat(ﬁnk,ﬁnk)—andnoneoftheotheroutcomes—isaNashequilibrium.DominantStrategies(Fink,ﬁnk)isaNashequilibriuminthePrisoners’Dilemmabecauseﬁnkingisabestresponsetotheotherplayer’sﬁnking.Wecansaymore:ﬁnkingisthebestresponsetoalloftheotherplayer’sstrategies,ﬁnkandsilent.(Thiscanbeseen,amongotherways,fromtheunderliningprocedureshowninTable8.2:allplayer1’spayoffsareunderlinedintherowinwhichheplaysﬁnk,andallplayer2’spayoffsareunderlinedinthecolumninwhichheplaysﬁnk.)Astrategythatisabestresponsetoanystrategytheotherplayersmightchooseiscalledadominantstrategy.Playersdonotalwayshavedominantstrategies,butwhentheydothereisstrongreasontobelievetheywillplaythatway.Complicatedstrategicconsiderationsdonotmatterwhenaplayerhasadominantstrategybecausewhatisbestforthatplayerisindepen-dentofwhatothersaredoing.DEFINITIONDominantstrategy.Adominantstrategyisastrategysiforplayerithatisabestresponsetoallstrategyproﬁlesofotherplayers.Thatis,si2BRiðsiÞforallsi.NotethedifferencebetweenaNashequilibriumstrategyandadominantstrategy.AstrategythatispartofaNashequilibriumneedonlybeabestresponsetoonestrategyproﬁleofotherplayers—namely,theirequilibriumstrategies.AdominantstrategymustbeabestresponsenotjusttotheNashequilibriumstrategiesofotherplayersbuttoallthestrategiesofthoseplayers.Ifallplayersinagamehaveadominantstrategy,thenwesaythegamehasadominantstrategyequilibrium.AswellasbeingtheNashequilibriumofthePrisoners’Dilemma,(ﬁnk,ﬁnk)isadominantstrategyequilibrium.Asisclearfromthedeﬁnitions,inanygamewithadominantstrategyequilibrium,thedominantstrategyequilibriumisaNashequilibrium.Problem8.4willshowthatwhenadominantstrategyexists,itistheuniqueNashequilibrium.BattleoftheSexesThefamousBattleoftheSexesgameisanotherexamplethatillustratestheconceptsofbestresponseandNashequilibrium.Thestorygoesthatawife(player1)andhusband(player2)wouldliketomeeteachotherforaneveningout.Theycangoeithertotheballetortoaboxingmatch.Bothprefertospendtimetogetherthanapart.Conditionalonbeingtogether,thewifepreferstogototheballetandthehusbandtoboxing.TheextensiveformofthegameispresentedinFigure8.2andthenormalforminTable8.3.Forbrevitywedispensewiththeu1andu2labelsonthepayoffsandsimplyre-emphasizetheconventionthattheﬁrstpayoffisplayer1’sandthesecondplayer2’s.Wewillworkwiththenormalform,examiningeachofthefourboxesinTable8.3anddeterminingwhichareNashequilibriaandwhicharenot.Startwiththeoutcomeinwhichbothplayerschooseballet,written(ballet,ballet),theupperleftcornerofthepayoffmatrix.Giventhatthehusbandplaysballet,thewife’sbestresponseistoplayballet(thisgivesherher242Part2ChoiceandDemand
highestpayoffinthematrixof2).Usingnotation,ballet¼BR1(ballet).[Wedon’tneedthefancyset-inclusionsymbolasin“ballet2BR1ðballetÞ”becausethehusbandhasonlyonebestresponsetothewife’schoosingballet.]Giventhatthewifeplaysballet,thehusband’sbestresponseistoplayballet.Ifhedeviatedtoboxingthenhewouldearn0ratherthan1,sincetheywouldendupnotcoordinating.Usingnotation,ballet¼BR2(ballet).So(ballet,ballet)isindeedaNashequilibrium.Symmetrically,(boxing,boxing)isaNashequilibrium.Considertheoutcome(ballet,boxing)intheupperleftcornerofthematrix.Giventhehusbandchoosesboxing,thewifeearns0fromchoosingballetbut1fromchoosingboxing,soballetisnotabestresponseforthewifetothehusband’schoosingboxing.Innotation,ballet62BR1ðboxingÞ.Hence(ballet,boxing)cannotbeaNashequilibrium.[Thehusband’sstrategyofboxingisnotabestresponsetothewife’splayingballeteither,soinfactbothplayerswouldprefertodeviatefrom(ballet,boxing),althoughweonlyneedtoﬁndoneplayerwhowouldwanttodeviatetoruleoutanoutcomeasaNashequilibrium.]Symmetri-cally,(boxing,ballet)isnotaNashequilibrium,either.FIGURE8.2ExtensiveFormfortheBattleoftheSexesInthisgame,player1(wife)andplayer2(husband)choosetoattendtheballetoraboxingmatch.Theyprefertocoordinatebutdisagreeonwhicheventtocoordinate.Becausetheychoosesimulta-neously,thehusbanddoesnotknowthewife’schoicewhenhemoves,sohisdecisionnodesareconnectedinthesameinformationset.122BoxingBalletBoxing1, 20, 00, 02, 1BoxingBalletBalletTABLE8.3NormalFormfortheBattleoftheSexesPlayer2(Husband)BalletBoxingPlayer1ðWifeÞBallet2,10,0Boxing0,01,2Chapter8StrategyandGameTheory243
TheBattleoftheSexesisanexampleofagamewithmorethanoneNashequilibrium(infact,ithasthree—athirdinmixedstrategies,aswewillsee).Itishardtosaywhichofthetwowehavefoundsofarismoreplausible,sincetheyaresymmetric.Itisthereforedifﬁculttomakeaﬁrmpredictioninthisgame.TheBattleoftheSexesisalsoanexampleofagamewithnodominantstrategies.Aplayerpreferstoplayballetiftheotherplaysballetandboxingiftheotherplaysboxing.Table8.4appliestheunderliningprocedure,usedtoﬁndNashequilibriaquickly,totheBattleoftheSexes.TheprocedureveriﬁesthatthetwooutcomesinwhichtheplayerssucceedincoordinatingareNashequilibriaandthetwooutcomesinwhichtheydon’tcoordinatearenot.Examples8.1,8.2,and8.3provideadditionalpracticeinﬁndingNashequilibriainmorecomplicatedsettings(agamethathasmanytiesforbestresponsesinExample8.1,agamewiththreestrategiesforeachplayerinExample8.2,andagamewiththreeplayersinExample8.3).EXAMPLE8.1ThePrisoners’DilemmaReduxInthisvariationonthePrisoners’Dilemma,asuspectisconvictedandreceivesasentenceoffouryearsifheisﬁnkedonandgoesfreeifnot.Thedistrictattorneydoesnotrewardﬁnking.Table8.5presentsthenormalformforthegamebeforeandafterapplyingtheprocedureforunderliningbestresponses.Payoffsareagainrestatedintermsofyearsoffreedom.Tiesforbestresponsesarerife.Forexample,givenplayer2ﬁnks,player1’spayoffis0whetherheﬁnksorissilent.Sothereisatiefor1’sbestresponseto2’sﬁnking.Thisisanexampleofthesetofbestresponsescontainingmorethanoneelement:BR1ðfinkÞ¼ffink,silentg.TABLE8.5ThePrisoners’DilemmaRedux(a)NormalformSuspect2FinkSilentSuspect1Fink0,01,0Silent0,11,1(b)UnderliningprocedureSuspect2FinkSilentSuspect1Fink0,01,0Silent0,11,1TABLE8.4UnderliningProcedureintheBattleoftheSexesPlayer2(Husband)BalletBoxingPlayer1ðWifeÞBallet2,10,0Boxing0,01,2244Part2ChoiceandDemand
TheunderliningprocedureshowsthatthereisaNashequilibriumineachofthefourboxes.Giventhatsuspectsreceivenopersonalrewardorpenaltyforﬁnking,theyarebothindifferentbetweenﬁnkingandbeingsilent;thusanyoutcomecanbeaNashequilibrium.QUERY:Doesanyplayerhaveadominantstrategy?Canyoudrawtheextensiveformforthegame?EXAMPLE8.2Rock,Paper,ScissorsRock,Paper,Scissorsisachildren’sgameinwhichthetwoplayerssimultaneouslydisplayoneofthreehandsymbols.Table8.6presentsthenormalform.Thezeropayoffsalongthediagonalshowthatifplayersadoptthesamestrategythennopaymentsaremade.Inothercases,thepayoffsindicatea$1paymentfromlosertowinnerundertheusualhierarchy(rockbreaksscissors,scissorscutpaper,papercoversrock).Asanyonewhohasplayedthisgameknows,andastheunderliningprocedurereveals,noneofthenineboxesrepresentsaNashequilibrium.Anystrategypairisunstablebecauseitoffersatleastoneoftheplayersanincentivetodeviate.Forexample,(scissors,scissors)providesanincentiveforeitherplayer1or2tochooserock;(paper,rock)providesanincentivefor2tochoosescissors.ThegamedoeshaveaNashequilibrium—notanyofthenineboxesinTable8.6butinmixedstrategies,deﬁnedinthenextsection.QUERY:Doesanyplayerhaveadominantstrategy?Whyisn’t(paper,scissors)aNashequilibrium?(b)UnderlyingprocedurePlayer2RockPaperScissorsPlayer1Rock0,01,11,1Paper1,10,01,1Scissors1,11,10,0TABLE8.6Rock,Paper,Scissors(a)NormalformPlayer2RockPaperScissorsPlayer1Rock0,01,11,1Paper1,10,01,1Scissors1,11,10,0Chapter8StrategyandGameTheory245
EXAMPLE8.3Three’sCompanyThree’sCompanyisathree-playerversionoftheBattleoftheSexesbasedona1970ssitcomofthesamenameaboutthemisadventuresofaman(Jack)andtwowomen(JanetandChrissy)whosharedanapartmenttosaverent.ModifythepayoffsfromtheBattleoftheSexesasfollows.Playersgetone“util”fromattendingtheirfavoriteevent(Jack’sisboxingandJanetandChrissy’sisballet).Playersgetanadditionalutilforeachoftheotherplayerswhoshowsupattheeventwiththem.Table8.7presentsthenormalform.Foreachofplayer3’sstrategies,thereisaseparatepayoffmatrixwithallcombinationsofplayer1and2’sstrategies.Eachboxliststhethreeplayers’payoffsinorder.Forplayers1and2,theunderliningprocedureisthesameasinatwo-playergameexceptthatitmustberepeatedforthetwopayoffmatrices.Tounderlineplayer3’sbest-responsepayoffs,comparethetwoboxesinthesamepositioninthetwodifferentmatrices.Forexample,givenJanetandChrissybothplayballet,wecomparethethirdpayoffintheupper-leftboxinbothmatrices:Jack’spayoffis2intheﬁrstmatrix(inwhichheplaysballet)and1inthesecond(inwhichheplaysboxing).Soweunderlinethe2.AsintheBattleoftheSexes,Three’sCompanyhastwoNashequilibria,oneinwhichallgotoballetandoneinwhichallgotoboxing.QUERY:WhatpayoffsmightmakeThree’sCompanyevencloserinspirittotheBattleoftheSexes?WhatwouldthenormalformlooklikeforFour’sCompany?(Four’sCompanyissimilartoThree’sCompanyexceptwithtwomenandtwowomen.)(b)UnderliningProcedurePlayer3(Jack)playsBalletPlayer2(Chrissy)BalletBoxingPlayer1ðJanetÞBallet3,3,22,0,1Boxing0,2,11,1,0TABLE8.7Three’sCompany(a)NormalformPlayer3(Jack)playsBalletPlayer2(Chrissy)BalletBoxingPlayer1ðJanetÞBallet3,3,22,0,1Boxing0,2,11,1,0Player3(Jack)playsBoxingPlayer2(Chrissy)BalletBoxingPlayer1ðJanetÞBallet2,2,11,1,2Boxing1,1,22,2,3Player3(Jack)playsBoxingPlayer2(Chrissy)BalletBoxingPlayer1ðJanetÞBallet2,2,11,1,2Boxing1,1,22,2,3246Part2ChoiceandDemand
MIXEDSTRATEGIESPlayers’strategiescanbemorecomplicatedthansimplychoosinganactionwithcertainty.Inthissectionwestudymixedstrategies,whichhavetheplayerrandomlyselectfromseveralpossibleactions.Bycontrast,thestrategiesconsideredintheexamplessofarhaveaplayerchooseoneactionoranotherwithcertainty;thesearecalledpurestrategies.Forexample,intheBattleoftheSexes,wehaveconsideredthepurestrategiesofchoosingeitherballetorboxingforsure.Apossiblemixedstrategyinthisgamewouldbetoﬂipacoinandthenattendtheballetifandonlyifthecoincomesupheads,yieldinga50–50chanceofshowingupateitherevent.Althoughatﬁrstglanceitmayseembizarretohaveplayersﬂippingcoinstodeterminehowtheywillplay,therearegoodreasonsforstudyingmixedstrategies.First,somegames(suchasRock,Paper,Scissors)havenoNashequilibriainpurestrategies.Aswewillseeinthesectiononexistence,suchgameswillalwayshaveaNashequilibriuminmixedstrategies,soallowingformixedstrategieswillenableustomakepredictionsinsuchgameswhereitwasimpossibletodosootherwise.Second,strategiesinvolvingrandomizationarequitenaturalandfamiliarincertainsettings.Studentsarefamiliarwiththesettingofclassexams.Classtimeisusuallytoolimitedfortheprofessortoexaminestudentsoneverytopictaughtinclass,butitmaybesufﬁcienttoteststudentsonasubsetoftopicstoinducethemtostudyallofthematerial.Ifstudentsknewwhichtopicswereonthetestthentheymightbeinclinedtostudyonlythoseandnottheothers,sotheprofessormustchoosethetopicsatrandominordertogetthestudentstostudyeverything.Randomstrategiesarealsofamiliarinsports(thesamesoccerplayersometimesshootstotherightofthenetandsometimestotheleftonpenaltykicks)andincardgames(thepokerplayersometimesfoldsandsome-timesbluffswithasimilarlypoorhandatdifferenttimes).Third,itispossibleto“purify”mixedstrategiesbyspecifyingamorecomplicatedgameinwhichoneortheotheractionisbetterfortheplayerforprivatelyknownreasonsandwherethatactionisplayedwithcertainty.2Forexample,ahistoryprofessormightdecidetoaskanexamquestionaboutWorldWarIbecause,unbeknownsttothestudents,sherecentlyreadaninterestingjournalarticleaboutit.Tobemoreformal,supposethatplayerihasasetofMpossibleactionsAi¼fa1i,…,ami,…,aMig,wherethesubscriptreferstotheplayerandthesuperscripttothedifferentchoices.AmixedstrategyisaprobabilitydistributionovertheMactions,si¼ðσ1i,…,σmi,…,σMiÞ,whereσmiisanumberbetween0and1thatindicatestheprobabilityofplayeriplayingactionami.Theprobabilitiesinsimustsumtounity:σ1iþ…þσmiþ…þσMi¼1.IntheBattleoftheSexes,forexample,bothplayershavethesametwoactionsofballetandboxing,sowecanwriteA1¼A2¼fballet,boxingg.Wecanwriteamixedstrategyasapairofprobabilitiesðσ,1σÞ,whereσistheprobabilitythattheplayerchoosesballet.Theprobabilitiesmustsumtounityandso,withtwoactions,oncetheprobabilityofoneactionisspeciﬁed,theprobabilityoftheotherisdetermined.Mixedstrategy(1=3,2=3)meansthattheplayerplaysballetwithprobability1=3andboxingwithprobability2=3;(1=2,1=2)meansthattheplayerisequallylikelytoplayballetorboxing;(1,0)meansthattheplayerchoosesballetwithcertainty;and(0,1)meansthattheplayerchoosesboxingwithcertainty.Inourterminology,amixedstrategyisageneralcategorythatincludesthespecialcaseofapurestrategy.Apurestrategyisthespecialcaseinwhichonlyoneactionisplayedwith2JohnHarsanyi,“GameswithRandomlyDisturbedPayoffs:ANewRationaleforMixed-StrategyEquilibriumPoints,”InternationalJournalofGameTheory2(1973):1–23.Harsanyiwasaco-winner(alongwithNash)ofthe1994NobelPrizeineconomics.Chapter8StrategyandGameTheory247
positiveprobability.Mixedstrategiesthatinvolvetwoormoreactionsbeingplayedwithpos-itiveprobabilityarecalledstrictlymixedstrategies.ReturningtotheexamplesfromthepreviousparagraphofmixedstrategiesintheBattleoftheSexes,allfourstrategies(1=3,2=3),(1=2,1=2),(1,0),and(0,1)aremixedstrategies.Theﬁrsttwoarestrictlymixedandthesecondtwoarepurestrategies.Withthisnotationforactionsandmixedstrategiesbehindus,wedonotneednewdeﬁnitionsforbestresponse,Nashequilibrium,anddominantstrategy.Thedeﬁnitionsintroducedwhensiwastakentobeapurestrategyalsoapplytothecaseinwhichsiistakentobeamixedstrategy.Theonlychangeisthatthepayofffunctionuiðsi,siÞ,ratherthanbeingacertainpayoff,mustbereinterpretedastheexpectedvalueofarandompayoff,withprobabilitiesgivenbythestrategiessiandsi.Example8.4providessomepracticeincomputingexpectedpayoffsintheBattleoftheSexes.EXAMPLE8.4ExpectedPayoffsintheBattleoftheSexesLet’scomputeplayers’expectedpayoffsifthewifechoosesthemixedstrategy(1=9,8=9)andthehusband(4=5,1=5)intheBattleoftheSexes.Thewife’sexpectedpayoffisU119,89,45,15¼1945U1ðballet,balletÞþ1915U1ðballet,boxingÞþ8945U1ðboxing,balletÞþ8915U1ðboxing,boxingÞ¼1945ð2Þþ1915ð0Þþ8945ð0Þþ8915ð1Þ¼1645.(8.4)TounderstandEquation8.4,itishelpfultoreviewtheconceptofexpectedvaluefromChapter2.Equation(2.176)indicatesthatanexpectedvalueofarandomvariableequalsthesumoveralloutcomesoftheprobabilityoftheoutcomemultipliedbythevalueoftherandomvariableinthatoutcome.IntheBattleoftheSexes,therearefouroutcomes,correspondingtothefourboxesinTable8.3.Sinceplayersrandomizeindependently,theprobabilityofreachingaparticularboxequalstheproductoftheprobabilitiesthateachplayerplaysthestrategyleadingtothatbox.So,forexample,theprobabilityof(boxing,ballet)—thatis,thewifeplaysboxingandthehusbandplaysballet—equalsð8=9Þð4=5Þ.Theprobabilitiesofthefouroutcomesaremultipliedbythevalueoftherelevantrandomvariable(inthiscase,player1’spayoff)ineachoutcome.Nextwecomputethewife’sexpectedpayoffifsheplaysthepurestrategyofgoingtoballet[thesameasthemixedstrategy(1,0)]andthehusbandcontinuestoplaythemixedstrategyð4=5,1=5Þ.Nowthereareonlytworelevantoutcomes,givenbythetwoboxesintherowinwhichthewifeplaysballet.Theprobabilitiesofthetwooutcomesaregivenbytheprobabilitiesinthehusband’smixedstrategy.Therefore,U1ballet,45,15¼45U1ðballet,balletÞþ15U1ðballet,boxingÞ¼45ð2Þþ15ð0Þ¼85.(8.5)Finally,wewillcomputethegeneralexpressionforthewife’sexpectedpayoffwhensheplaysmixedstrategyðw,1wÞandthehusbandplaysðh,1hÞ:ifthewifeplaysballetwithprobabilitywandthehusbandwithprobabilityh,then248Part2ChoiceandDemand
u1ððw,1wÞ,ðh,1hÞÞ¼ðwÞðhÞU1ðballet,balletÞþðwÞð1hÞU1ðballet,boxingÞþð1wÞðhÞU1ðboxing,balletÞþð1wÞð1hÞU1ðboxing,boxingÞ¼ðwÞðhÞð2ÞþðwÞð1hÞð0Þþð1wÞðhÞð0Þþð1wÞð1hÞð1Þ¼1hwþ3hw.(8.6)QUERY:Whatisthehusband’sexpectedpayoffineachcase?Showthathisexpectedpayoffis22h2wþ3hwinthegeneralcase.Giventhehusbandplaysthemixedstrategyð4=5,1=5Þ,whatstrategyprovidesthewifewiththehighestpayoff?ComputingNashequilibriumofagamewhenstrictlymixedstrategiesareinvolvedisquiteabitmorecomplicatedthanwhenpurestrategiesareinvolved.Beforewadingin,wecansavealotofworkbyaskingwhetherthegameevenhasaNashequilibriuminstrictlymixedstrategies.Ifitdoesnotthen,havingfoundallthepure-strategyNashequilibria,onehasﬁnishedanalyzingthegame.ThekeytoguessingwhetheragamehasaNashequilibriuminstrictlymixedstrategiesisthesurprisingresultthatalmostallgameshaveanoddnumberofNashequilibria.3Let’sapplythisinsighttosomeoftheexamplesconsideredsofar.Wefoundanoddnumber(one)ofpure-strategyNashequilibriainthePrisoners’Dilemma,suggestingweneednotlookfurtherforoneinstrictlymixedstrategies.IntheBattleoftheSexes,wefoundanevennumber(two)ofpure-strategyNashequilibria,suggestingtheexistenceofathirdoneinstrictlymixedstrategies.Example8.2—Rock,Paper,Scissors—hasnopure-strategyNashequilibria.ToarriveatanoddnumberofNashequilibria,wewouldexpecttoﬁndoneNashequilibriuminstrictlymixedstrategies.EXAMPLE8.5Mixed-StrategyNashEquilibriumintheBattleoftheSexesAgeneralmixedstrategyforthewifeintheBattleoftheSexesisðw,1wÞandforthehusbandisðh,1hÞ;wherewandharetheprobabilitiesofplayingballetforthewifeandhusband,respectively.WewillcomputevaluesofwandhthatmakeupNashequilibria.Bothplayershaveacontinuumofpossiblestrategiesbetween0and1.Therefore,wecannotwritethesestrategiesintherowsandcolumnsofamatrixandunderlinebest-responsepayoffstoﬁndtheNashequilibria.Instead,wewillusegraphicalmethodstosolvefortheNashequilibria.Givenplayers’generalmixedstrategies,wesawinExample8.4thatthewife’sexpectedpayoffisu1ððw,1wÞ,ðh,1hÞÞ¼1hwþ3hw.(8.7)AsEquation8.7shows,thewife’sbestresponsedependsonh.Ifh<1=3,shewantstosetwaslowaspossible:w¼0.Ifh>1=3,herbestresponseistosetwashighaspossible:w¼1.Whenh¼1=3,herexpectedpayoffequals2=3regardlessofwhatwshechooses.Inthiscasethereisatieforthebestresponse,includinganywfrom0to1.(continued)3JohnHarsanyi,“OddnessoftheNumberofEquilibriumPoints:ANewProof,”InternationalJournalofGameTheory2(1973):235–50.GamesinwhichtherearetiesbetweenpayoffsmayhaveanevenorinfinitenumberofNashequilibria.Example8.1,thePrisoners’DilemmaRedux,hasseveralpayoffties.Thegamehasfourpure-strategyNashequilibriaandaninfinitenumberofdifferentmixedstrategyequilibria.Chapter8StrategyandGameTheory249
EXAMPLE8.5CONTINUEDInExample8.4,westatedthatthehusband’sexpectedpayoffisU2ððh,1hÞ,ðw,1wÞÞ¼22h2wþ3hw.(8.8)Whenw<2=3,hisexpectedpayoffismaximizedbyh¼0;whenw>2=3,hisexpectedpayoffismaximizedbyh¼1;andwhenw¼2=3,heisindifferentamongallvaluesofh,obtaininganexpectedpayoffof2=3regardless.ThebestresponsesaregraphedinFigure8.3.TheNashequilibriaaregivenbytheintersectionpointsbetweenthebestresponses.Attheseintersectionpoints,bothplayersarebestrespondingtoeachother,whichiswhatisrequiredfortheoutcometobeaNashequilibrium.TherearethreeNashequilibria.ThepointsE1andE2arethepure-strategyNashequilibriawefoundbefore,withE1correspondingtothepure-strategyNashequilibriuminwhichbothplayboxingandE2tothatinwhichbothplayballet.PointE3isthestrictlymixed-strategyNashequilibrium,whichcancanbespelledoutas“thewifeplaysballetwithprobability2=3andboxingwithprobability1=3andthehusbandplaysballetwithprobability1=3andboxingwithprobability2=3.”Moresuccinctly,havingdeﬁnedwandh,wemaywritetheequilibruimas“w¼2=3andh¼1=3.”QUERY:Whatisaplayer’sexpectedpayoffintheNashequilibriuminstrictlymixedstrate-gies?Howdoesthispayoffcomparetothoseinthepure-strategyNashequilibria?WhatargumentsmightbeofferedthatoneoranotherofthethreeNashequilibriamightbethebestpredictioninthisgame?FIGURE8.3NashEquilibriainMixedStrategiesintheBattleoftheSexesBalletischosenbythewifewithprobabilitywandbythehusbandwithprobabilityh.Players’bestresponsesaregraphedonthesamesetofaxes.ThethreeintersectionpointsE1,E2,andE3areNashequilibria.TheNashequilibriuminstrictlymixedstrategies,E3,isw¼2=3andh¼1=3.Husband’sbest response,BR212/31/31/32/310Wife’s bestresponse,BR1E2E3E1hw250Part2ChoiceandDemand
Example8.5runsthroughthelengthycalculationsinvolvedinﬁndingalltheNashequi-libriaoftheBattleoftheSexes,thoseinpurestrategiesandthoseinstrictlymixedstrategies.Thestepsinvolveﬁndingplayers’expectedpayoffsasfunctionsofgeneralmixedstrategies,usingthesetoﬁndplayers’bestresponses,andthengraphingplayers’bestresponsestoseewheretheyintersect.AshortcuttoﬁndingtheNashequilibriuminstrictlymixedstrategiesisbasedontheinsightthataplayerwillbewillingtorandomizebetweentwoactionsinequilibriumonlyifheorshegetsthesameexpectedpayofffromplayingeitheractionor,inotherwords,isindifferentbetweenthetwoactionsinequilibrium.Otherwise,oneofthetwoactionswouldprovideahigherexpectedpayoff,andtheplayerwouldprefertoplaythatactionwithcertainty.Supposethehusbandisplayingmixedstrategyðh,1hÞ;thatis,playingballetwithprobabilityhandboxingwithprobability1h.Thewife’sexpectedpayofffromplayingballetisU1ðballet,ðh,1hÞÞ¼ðhÞð2Þþð1hÞð0Þ¼2h.(8.9)HerexpectedpayofffromplayingboxingisU1ðboxing,ðh,1hÞÞ¼ðhÞð0Þþð1hÞð1Þ¼1h.(8.10)Forthewifetobeindifferentbetweenballetandboxinginequilibrium,Equations8.9and8.10mustbeequal:2h¼1h,implyingh¼1=3.Similarcalculationsbasedonthehusband’sindifferencebetweenplayingballetandboxinginequilibriumshowthatthewife’sprobabilityofplayingballetinthestrictlymixedstrategyNashequilibriumisw¼2=3.(Workthroughthesecalculationsasanexercise.)Noticethatthewife’sindifferenceconditiondoesnot“pindown”herequilibriummixedstrategy.Thewife’sindifferenceconditioncannotpindownherownequilibriummixedstrat-egybecause,giventhatsheisindifferentbetweenthetwoactionsinequilibrium,heroverallexpectedpayoffisthesamenomatterwhatprobabilitydistributionsheplaysoverthetwoactions.Rather,thewife’sindifferenceconditionpinsdowntheotherplayer’s—thehusband’s—mixedstrategy.Thereisauniqueprobabilitydistributionhecanusetoplayballetandboxingthatmakesherindifferentbetweenthetwoactionsandthusmakesherwillingtorandomize.Givenanyprobabilityofhisplayingballetandboxingotherthanð1=3,2=3Þ,itwouldnotbeastableoutcomeforhertorandomize.Thus,twoprinciplesshouldbekeptinmindwhenseekingNashequilibriainstrictlymixedstrategies.Oneisthataplayerrandomizesoveronlythoseactionsamongwhichheorsheisindifferent,givenotherplayers’equilibriummixedstrategies.Thesecondisthatoneplayer’sindifferenceconditionpinsdowntheotherplayer’smixedstrategy.EXISTENCEOneofthereasonsNashequilibriumissowidelyusedisthataNashequilibriumisguaranteedtoexistinawideclassofgames.Thisisnottrueforsomeotherequilibriumconcepts.Considerthedominantstrategyequilibriumconcept.ThePrisoners’Dilemmahasadominantstrategyequilibrium(bothsuspectsﬁnk),butmostgamesdonot.Indeed,therearemanygames—including,forexample,theBattleoftheSexes—inwhichnoplayerhasadominantstrategy,letalonealltheplayers.Insuchgames,wecan’tmakepredictionsusingdominantstrategyequilibriumbutwecanusingNashequilibrium.TheExtensionssectionattheendofthischapterwillprovidethetechnicaldetailsbehindJohnNash’sproofoftheexistenceofhisequilibriuminallﬁnitegames(gameswithaﬁnitenumberofplayersandaﬁnitenumberofactions).Theexistencetheoremdoesnotguaranteetheexistenceofapure-strategyNashequilibrium.Wealreadysawanexample:Rock,Paper,Chapter8StrategyandGameTheory251
ScissorsinExample8.2.However,ifaﬁnitegamedoesnothaveapure-strategyNashequilibrium,thetheoremguaranteesthatitwillhaveamixed-strategyNashequilibrium.TheproofofNash’stheoremissimilartotheproofinChapter13oftheexistenceofpricesleadingtoageneralcompetitiveequilibrium.TheExtensionssectionincludesanexistenceproofforgameswithacontinuumofactions,asstudiedinthenextsection.CONTINUUMOFACTIONSMostoftheinsightfromeconomicsituationscanoftenbegainedbydistillingthesituationdowntoafeworeventwoactions,aswithallthegamesstudiedsofar.Othertimes,additionalinsightcanbegainedbyallowingacontinuumofactions.Tobeclear,wehavealreadyencounteredacontinuumofstrategies—inourdiscussionofmixedstrategies—butstilltheprobabilitydistributionsinmixedstrategieswereoveraﬁnitenumberofactions.Inthissectionwefocusoncontinuumofactions.Somesettingsaremorerealisticallymodeledviaacontinuousrangeofactions.InChapter15,forexample,wewillstudycompetitionbetweenstrategicﬁrms.Inonemodel(Bertrand),ﬁrmssetprices;inanother(Cournot),ﬁrmssetquantities.Itisnaturaltoallowﬁrmstochooseanynonnegativepriceorquantityratherthanartiﬁciallyrestrictingthemtojusttwoprices(say,$2or$5)ortwoquantities(say,100or1,000units).Continuousactionshaveseveralotheradvan-tages.Withcontinuousactions,thefamiliarmethodsfromcalculuscanoftenbeusedtosolveforNashequilibria.Itisalsopossibletoanalyzehowtheequilibriumactionsvarywithchangesinunderlyingparameters.WiththeCournotmodel,forexample,wemightwanttoknowhowequilibriumquantitieschangewithasmallincreaseinaﬁrm’smarginalcostsorademandparameter.TragedyoftheCommonsExample8.6illustrateshowtosolvefortheNashequilibriumwhenthegame(inthiscase,theTragedyoftheCommons)involvesacontinuumofactions.Theﬁrststepistowritedownthepayoffforeachplayerasafunctionofallplayers’actions.Thenextstepistocomputetheﬁrst-orderconditionassociatedwitheachplayer’spayoffmaximum.Thiswillgiveanequa-tionthatcanberearrangedintothebestresponseofeachplayerasafunctionofallotherplayers’actions.Therewillbeoneequationforeachplayer.Withnplayers,thesystemofnequationsforthenunknownequilibriumactionscanbesolvedsimultaneouslybyeitheralgebraicorgraphicalmethods.EXAMPLE8.6TragedyoftheCommonsTheterm“TragedyoftheCommons”hascometosignifyenvironmentalproblemsofover-usethatarisewhenscarceresourcesaretreatedascommonproperty.4Agame-theoreticillustrationofthisissuecanbedevelopedbyassumingthattwoherdersdecidehowmanysheeptograzeonthevillagecommons.Theproblemisthatthecommonsisquitesmallandcanrapidlysuccumbtoovergrazing.Inordertoaddsomemathematicalstructuretotheproblem,letqibethenumberofsheepthatherderi¼1,2grazesonthecommons,andsupposethattheper-sheepvalueofgrazingonthecommons(intermsofwoolandsheep-milkcheese)is4ThistermwaspopularizedbyG.Hardin,“TheTragedyoftheCommons,”Science162(1968):1243–48.252Part2ChoiceandDemand
vðq1,q2Þ¼120ðq1þq2Þ:(8.11)Thisfunctionimpliesthatthevalueofgrazingagivennumberofsheepislowerthemoresheeparearoundcompetingforgrass.Wecannotuseamatrixtorepresentthenormalformofthisgameofcontinuousactions.Instead,thenormalformissimplyalistingoftheherders’payofffunctionsu1ðq1,q2Þ¼q1vðq1,q2Þ¼q1ð120q1q2Þ,u2ðq1,q2Þ¼q2vðq1,q2Þ¼q2ð120q1q2Þ:(8.12)TofindtheNashequilibrium,wesolveherder1’svalue-maximizationproblem:maxq1fq1ð120q1q2Þg:(8.13)Thefirst-orderconditionforamaximumis1202q1q2¼0(8.14)or,rearranging,q1¼60q22¼BR1ðq2Þ:(8.15)Similarstepsshowthatherder2’sbestresponseisq2¼60q12¼BR2ðq1Þ:(8.16)TheNashequilibriumisgivenbythepairðq1,q2ÞthatsatisfiesEquations8.15and8.16simultaneously.Takinganalgebraicapproachtothesimultaneoussolution,Equation8.16canbesubstitutedintoEquation8.15,whichyieldsq1¼601260q12;(8.17)uponrearranging,thisimpliesq1¼40.Substitutingq1¼40intoEquation8.17impliesq2¼40aswell.Thus,eachherderwillgraze40sheeponthecommon.Eachearnsapayoffof1,600,ascanbeseenbysubstitutingq1¼q2¼40intothepayofffunctioninEquation8.13.Equations8.15and8.16canalsobesolvedsimultaneouslyusinggraphicalmethods.Figure8.4plotsthetwobestresponsesonagraphwithplayer1’sactiononthehorizontalaxisandplayer2’sontheverticalaxis.Thesebestresponsesaresimplylinesandsoareeasytographinthisexample.(Tobeconsistentwiththeaxislabels,theinverseofEquation8.15isactuallywhatisgraphed.)ThetwobestresponsesintersectattheNashequilibriumE1.ThegraphicalmethodisusefulforshowinghowtheNashequilibriumshiftswithchangesintheparametersoftheproblem.Supposetheper-sheepvalueofgrazingincreasesfortheﬁrstherderwhilethesecondremainsasinEquation8.11,perhapsbecausetheﬁrstherderstartsraisingmerinosheepwithmorevaluablewool.Thischangewouldshiftthebestresponseoutforherder1whileleaving2’sthesame.Thenewintersectionpoint(E2inFigure8.4),whichisthenewNashequilibrium,involvesmoresheepfor1andfewerfor2.TheNashequilibriumisnotthebestuseofthecommons.Intheoriginalproblem,bothherders’per-sheepvalueofgrazingisgivenbyEquation8.11.Ifbothgrazedonly30sheeptheneachwouldearnapayoffof1,800,ascanbeseenbysubstitutingq1¼q2¼30intoEquation8.13.Indeed,the“jointpayoffmaximization”problemmaxq1fðq1þq2Þvðq1,q2Þg¼maxq1fðq1þq2Þð120q1q2Þg(8.18)issolvedbyq1¼q2¼30or,moregenerally,byanyq1andq2thatsumto60.(continued)Chapter8StrategyandGameTheory253
EXAMPLE8.6CONTINUEDQUERY:HowwouldtheNashequilibriumshiftifbothherders’beneﬁtsincreasedbythesameamount?Whataboutadecreasein(only)herder2’sbeneﬁtfromgrazing?AsExample8.6shows,graphicalmethodsareparticularlyconvenientforquicklydeter-mininghowtheequilibriumshiftswithchangesintheunderlyingparameters.Theexampleshiftedthebeneﬁtofgrazingtooneofherders.Thisexercisenicelyillustratesthenatureofstrategicinteraction.Herder2’spayofffunctionhasn’tchanged(onlyherder1’shas),yethisequilibriumactionchanges.Thesecondherderobservestheﬁrst’shigherbeneﬁt,anticipatesthattheﬁrstwillincreasethenumberofsheephegrazes,andreduceshisowngrazinginresponse.TheTragedyoftheCommonsshareswiththePrisoners’DilemmathefeaturethattheNashequilibriumislessefﬁcientforallplayersthansomeotheroutcome.InthePrisoners’Dilemma,bothﬁnkinequilibriumwhenitwouldbemoreefﬁcientforbothtobesilent.IntheTragedyoftheCommons,theherdersgrazemoresheepinequilibriumthanisefﬁ-cient.Thisinsightmayexplainwhyoceanﬁshinggroundsandothercommonresourcescanendupbeingoverusedeventothepointofexhaustioniftheiruseisleftunregulated.Moredetailonsuchproblems—involvingwhatwewillcallnegativeexternalities—isprovidedinChapter19.FIGURE8.4Best-ResponseDiagramfortheTragedyoftheCommonsTheintersection,E1,betweenthetwoherders’bestresponsesistheNashequilibrium.Anincreaseintheper-sheepvalueofgrazingintheTragedyoftheCommonsshiftsoutherder1’sbestresponse,resultinginaNashequilibriumE2inwhichherder1grazesmoresheep(andherder2,fewersheep)thanintheoriginalNashequilibrium.120601204004060E1E2BR2(q1)BR1(q2)q2q1254Part2ChoiceandDemand
SEQUENTIALGAMESInsomegames,theorderofmovesmatters.Forexample,inabicycleracewithastaggeredstart,itmayhelptogolastandthusknowthetimetobeat.Ontheotherhand,competitiontoestablishanewhigh-deﬁnitionvideoformatmaybewonbytheﬁrstﬁrmtomarketitstechnology,therebycapturinganinstalledbaseofconsumers.Sequentialgamesdifferfromthesimultaneousgameswehaveconsideredsofarinthataplayerthatmoveslaterinthegamecanobservehowothershaveplayeduptothatmoment.Theplayercanusethisinformationtoformmoresophisticatedstrategiesthansimplychoos-inganaction;theplayer’sstrategycanbeacontingentplanwiththeactionplayeddependingonwhattheotherplayershavedone.Toillustratethenewconceptsraisedbysequentialgames—and,inparticular,tomakeastarkcontrastbetweensequentialandsimultaneousgames—wetakeasimultaneousgamewehavediscussedalready,theBattleoftheSexes,andturnitintoasequentialgame.SequentialBattleoftheSexesConsidertheBattleoftheSexesgameanalyzedpreviouslywithallthesameactionsandpayoffs,butnowchangethetimingofmoves.Ratherthanthewifeandhusbandmakingasimultaneouschoice,thewifemovesﬁrst,choosingballetorboxing;thehusbandobservesthischoice(say,thewifecallshimfromherchosenlocation)andthenthehusbandmakeshischoice.Thewife’spossiblestrategieshavenotchanged:shecanchoosethesimpleactionsballetorboxing(orperhapsamixedstrategyinvolvingbothactions,althoughthiswillnotbearelevantconsiderationinthesequentialgame).Thehusband’ssetofpossiblestrategieshasexpanded.Foreachofthewife’stwoactions,hecanchooseoneoftwoactions,sohehasfourpossiblestrategies,whicharelistedinTable8.8.Theverticalbarinthehusband’sstrategiesmeans“conditionalon”andso,forexample,“boxing|ballet”shouldbereadas“thehusbandchoosesboxingconditionalonthewife’schoosingballet”.Giventhatthehusbandhasfourpurestrategiesratherthanjusttwo,thenormalform(giveninTable8.9)mustnowbeexpandedtoeightboxes.Roughlyspeaking,thenormalformistwiceascomplicatedasthatforthesimultaneousversionofthegameinTable8.3.Bycontrast,theextensiveform,giveninFigure8.5,isnomorecomplicatedthantheextensiveformforthesimultaneousversionofthegameinFigure8.2.TheonlydifferencebetweentheTABLE8.8Husband’sContingentStrategiesContingentstrategyWritteninconditionalformatAlwaysgototheballet(ballet|ballet,ballet|boxing)Followhiswife(ballet|ballet,boxing|boxing)Dotheopposite(boxing|ballet,ballet|boxing)Alwaysgotoboxing(boxing|ballet,boxing|boxing)TABLE8.9NormalFormfortheSequentialBattleoftheSexesHusband(Ballet|BalletBallet|Boxing)(Ballet|BalletBoxing|Boxing)(Boxing|BalletBallet|Boxing)(Boxing|BalletBoxing|Boxing)WifeBallet2,12,10,00,0Boxing0,01,20,01,2Chapter8StrategyandGameTheory255
extensiveformsisthattheovalaroundthehusband’sdecisionnodeshasbeenremoved.Inthesequentialversionofthegame,thehusband’sdecisionnodesarenotgatheredtogetherinthesameinformationsetbecausethehusbandobserveshiswife’sactionandsoknowswhichnodeheisonbeforemoving.Wecanbegintoseewhytheextensiveformbecomesmoreusefulthanthenormalformforsequentialgames,especiallyingameswithmanyroundsofmoves.TosolvefortheNashequilibria,considerthenormalforminTable8.9.Applyingthemethodofunderliningbest-responsepayoffs—beingcarefultounderlinebothpayoffsincasesoftiesforthebestresponse—revealsthreepure-strategyNashequilibria:1.wifeplaysballet,husbandplays(ballet|ballet,ballet|boxing);2.wifeplaysballet,husbandplays(ballet|ballet,boxing|boxing);3.wifeplaysboxing,husbandplays(boxing|ballet,boxing|boxing).AswiththesimultaneousversionoftheBattleoftheSexes,hereagainwehavemultipleequi-libria.Yetnowgametheoryoffersagoodwaytoselectamongtheequilibria.ConsiderthethirdNashequilibrium.Thehusband’sstrategy(boxing|ballet,boxing|boxing)involvestheimplicitthreatthathewillchooseboxingevenifhiswifechoosesballet.Thisthreatissufﬁcienttodeterherfromchoosingballet.Giventhatshechoosesboxinginequilibrium,hisstrategyearnshim2,whichisthebesthecandoinanyoutcome.SotheoutcomeisaNashequilibrium.Butthehus-band’sthreatisnotcredible—thatis,itisanemptythreat.Ifthewifereallyweretochooseballetﬁrst,thenhewouldbegivingupapayoffof1bychoosingboxingratherthanballet.Itisclearwhyhewouldwanttothreatentochooseboxing,butitisnotclearthatsuchathreatshouldbebelieved.Similarly,thehusband’sstrategy(ballet|ballet,ballet|boxing)intheﬁrstNashequilib-riumalsoinvolvesanemptythreat:thathewillchooseballetifhiswifechoosesboxing.(Thisisanoddthreattomakesincehedoesnotgainfrommakingit,butitisanemptythreatnonetheless.)Anotherwaytounderstandemptyversuscrediblethreatsisbyusingtheconceptoftheequilibriumpath,theconnectedpaththroughthegametreeimpliedbyequilibriumstrategies.FIGURE8.5ExtensiveFormfortheSequentialBattleoftheSexesInthesequentialversionoftheBattleoftheSexes,thehusbandmovessecondafterobservinghiswife’smove.Thehusband’sdecisionnodesarenotgatheredinthesameinformationset.Ballet20, 01, 20, 02, 121BalletBalletBoxingBoxingBoxing256Part2ChoiceandDemand
Figure8.6usesadashedlinetoillustratetheequilibriumpathforthethirdofthelistedNashequilibriainthesequentialBattleoftheSexes.ThethirdoutcomeisaNashequilibriumbecausethestrategiesarerationalalongtheequilibriumpath.However,followingthewife’schoosingballet—aneventthatisofftheequilibriumpath—thehusband’sstrategyisirrational.Theconceptofsubgame-perfectequilibriuminthenextsectionwillruleoutirrationalplaybothonandofftheequilibriumpath.Subgame-perfectequilibriumGametheoryoffersaformalwayofselectingthereasonableNashequilibriainsequentialgamesusingtheconceptofsubgame-perfectequilibrium.Subgame-perfectequilibriumisareﬁnementthatrulesoutemptythreatsbyrequiringstrategiestoberationalevenforcon-tingenciesthatdonotariseinequilibrium.Beforedeﬁningsubgame-perfectequilibriumformally,weneedafewpreliminarydeﬁni-tions.Asubgameisapartoftheextensiveformbeginningwithadecisionnodeandincludingeverythingthatbranchesouttotherightofit.Apropersubgameisasubgamethatstartsatadecisionnodenotconnectedtoanotherinaninformationset.Conceptually,thismeansthattheplayerwhomovesﬁrstinapropersubgameknowstheactionsplayedbyothersthathaveleduptothatpoint.Itiseasiertoseewhatapropersubgameisthantodeﬁneitinwords.Figure8.7showstheextensiveformsfromthesimultaneousandsequentialversionsoftheBattleoftheSexeswithboxesdrawnaroundthepropersubgamesineach.InthesimultaneousBattleoftheSexes,thereisonlyonedecisionnode—thetopmostmode—thatisnotconnectedtoanotherinthesameinformationset;hencethereisonlyonepropersubgame,thegameitself.InthesequentialBattleoftheSexes,therearethreepropersubgames:thegameitselfandtwolowersubgamesstartingwithdecisionnodeswherethehusbandgetstomove.FIGURE8.6EquilibriumPathInthethirdoftheNashequilibrialistedforthesequentialBattleoftheSexes,thewifeplaysboxingandthehusbandplays(boxing|ballet,boxing|boxing),tracingoutthebranchesindicatedwiththicklines(bothsolidanddashed).Thedashedlineistheequilibriumpath;therestofthetreeisreferredtoasbeing“offtheequilibriumpath.”Ballet20, 01, 20, 02, 121BalletBalletBoxingBoxingBoxingChapter8StrategyandGameTheory257
DEFINITIONSubgame-perfectequilibrium.Asubgame-perfectequilibriumisastrategyproﬁleðs1,s2,…,snÞthatconstitutesaNashequilibriumforeverypropersubgame.Asubgame-perfectequilibriumisalwaysaNashequilibrium.Thisistruebecausethewholegameisapropersubgameofitselfandsoasubgame-perfectequilibriummustbeaNashequilibriumforthewholegame.InthesimultaneousversionoftheBattleoftheSexes,thereisnothingmoretosaybecausetherearenosubgamesotherthanthewholegameitself.InthesequentialversionoftheBattleoftheSexes,subgame-perfectequilibriumhasmorebite.StrategiesmustnotonlyformaNashequilibriumonthewholegameitself,theymustalsoformNashequilibriaonthetwopropersubgamesstartingwiththedecisionpointsatwhichthehusbandmoves.Thesesubgamesaresimpledecisionproblems,soitiseasytocomputethecorrespondingNashequilibria.ForsubgameC,beginningwiththehusband’sdecisionnodefollowinghiswife’schoosingballet,hehasasimpledecisionbetweenballet(whichearnshimapayoffof1)andboxing(whichearnshimapayoffof0).TheNashequilibriuminthissimpledecisionsubgameisforthehusbandtochooseballet.Fortheothersubgame,D,hehasasimpledecisionbetweenballet,whichearnshim0,andboxing,whichearnshim2.TheNashequi-libriuminthissimpledecisionsubgameisforhimtochooseboxing.Thehusbandthereforehasonlyonestrategythatcanbepartofasubgame-perfectequilibrium:(ballet|ballet,boxing|boxing).AnyotherstrategyhashimplayingsomethingthatisnotaNashequilibriumforsomepropersubgame.ReturningtothethreeenumeratedNashequilibria,onlythesecondissub-gameperfect;theﬁrstandthethirdarenot.Forexample,thethirdequilibrium,inwhichthehusbandalwaysgoestoboxing,isruledoutasasubgame-perfectequilibriumbecausetheFIGURE8.7ProperSubgamesintheBattleoftheSexesThesimultaneousBattleoftheSexesin(a)hasonlyonepropersubgame:thewholegameitself,labeledA.ThesequentialBattleoftheSexesin(b)hasthreepropersubgames,labeledB,C,andD.122BoxingBalletABoxing(b) Sequential1, 20, 00, 02, 1BoxingBalletBallet122BoxingBalletBCDBoxing1, 20, 00, 02, 1BoxingBalletBallet(a) Simultaneous258Part2ChoiceandDemand
husband’sstrategy(boxing|boxing)isnotaNashequilibriuminpropersubgameC:Subgame-perfectequilibriumthusrulesouttheemptythreat(ofalwaysgoingtoboxing)thatwewereuncomfortablewithearlier.Moregenerally,subgame-perfectequilibriumrulesoutanysortofemptythreatinasequentialgame.Ineffect,Nashequilibriumrequiresbehaviortoberationalonlyontheequilibriumpath.Playerscanchoosepotentiallyirrationalactionsonotherpartsofthegametree.Inparticular,oneplayercanthreatentodamagebothinordertoscaretheotherfromchoosingcertainactions.Subgame-perfectequilibriumrequiresrationalbehaviorbothonandofftheequilibriumpath.Threatstoplayirrationally—thatis,threatstochoosesomethingotherthanone’sbestresponse—areruledoutasbeingempty.Subgame-perfectequilibriumisnotausefulreﬁnementforasimultaneousgame.Thisisbecauseasimultaneousgamehasnopropersubgamesbesidesthegameitselfandsosubgame-perfectequilibriumwouldnotreducethesetofNashequilibria.BackwardinductionOurapproachtosolvingfortheequilibriuminthesequentialBattleoftheSexeswastoﬁndalltheNashequilibriausingthenormalformandthentoseekamongthoseforthesubgame-perfectequilibrium.Ashortcutforﬁndingthesubgame-perfectequilibriumdirectlyistousebackwardinduction,theprocessofsolvingforequilibriumbyworkingbackwardsfromtheendofthegametothebeginning.Backwardinductionworksasfollows.Identifyallofthesubgamesatthebottomoftheextensiveform.FindtheNashequilibriaonthesesubgames.Replacethe(potentiallycomplicated)subgameswiththeactionsandpayoffsresultingfromNashequilibriumplayonthesesubgames.Thenmoveuptothenextlevelofsubgamesandrepeattheprocedure.Figure8.8illustratestheuseofbackwardinductiontosolveforthesubgame-perfectequilibriumofthesequentialBattleoftheSexes.First,wecomputetheNashequilibriaofFIGURE8.8ApplyingBackwardInductionThelastsubgames(whereplayer2moves)arereplacedbytheNashequilibriaonthesesubgames.Thesimplegamethatresultsatrightcanbesolvedforplayer1’sequilibriumaction.122BoxingBalletBoxing1, 20, 00, 02, 1BoxingBalletBalletBoxingBallet12 playsboxing | boxingpayoff 1, 22 playsballet | balletpayoff 2, 1Chapter8StrategyandGameTheory259
thebottommostsubgamesatthehusband’sdecisionnodes.Inthesubgamefollowinghiswife’schoosingballet,hewouldchooseballet,givingpayoffs2forherand1forhim.Inthesubgamefollowinghiswife’schoosingboxing,hewouldchooseboxing,givingpayoffs1forherand2forhim.Next,substitutethehusband’sequilibriumstrategiesforthesubgamesthemselves.Theresultinggameisasimpledecisionproblemforthewife(drawninthelowerpaneloftheﬁgure):achoicebetweenballet,whichwouldgiveherapayoffof2,andboxing,whichwouldgiveherapayoffof1.TheNashequilibriumofthisgameisforhertochoosetheactionwiththehigherpayoff,ballet.Insum,backwardinductionallowsustojumpstraighttothesubgame-perfectequilibriuminwhichthewifechoosesballetandthehusbandchooses(ballet|ballet,boxing|boxing),bypassingtheotherNashequilibria.Backwardinductionisparticularlyusefulingamesthatfeaturemultipleroundsofsequen-tialplay.Asroundsareadded,itquicklybecomestoohardtosolveforalltheNashequilibriaandthentosortthroughwhicharesubgame-perfect.Withbackwardinduction,anadditionalroundissimplyaccommodatedbyaddinganotheriterationoftheprocedure.REPEATEDGAMESInthegamesexaminedsofar,eachplayermakesonechoiceandthegameends.Inmanyreal-worldsettings,playersplaythesamegameoverandoveragain.Forexample,theplayersinthePrisoners’DilemmamayanticipatecommittingfuturecrimesandthusplayingfuturePrisoners’Dilemmastogether.Gasolinestationslocatedacrossthestreetfromeachother,whentheysettheirpriceseachmorning,effectivelyplayanewpricinggameeveryday.Thesimpleconstituentgame(e.g.,thePrisoners’Dilemmaorthegasoline-pricinggame)thatisplayedrepeatedlyiscalledthestagegame.AswesawwiththePrisoners’Dilemma,theequilibriuminoneplayofthestagegamemaybeworseforallplayersthansomeother,morecooperative,outcome.Repeatedplayofthestagegameopensupthepossibilityofcooperationinequilibrium.Playerscanadopttriggerstrategies,wherebytheycontinuetocooperateaslongasallhavecooperateduptothatpointbutreverttoplayingtheNashequilibriumifanyonedeviatesfromcooperation.Wewillinvestigatetheconditionsunderwhichtriggerstrategiesworktoincreaseplayers’payoffs.Asisstandardingametheory,wewillfocusonsubgame-perfectequilibriaoftherepeatedgames.FinitelyrepeatedgamesFormanystagegames,repeatingthemaknown,ﬁnitenumberoftimesdoesnotincreasethepossibilityforcooperation.Toseethispointconcretely,supposethePrisoners’DilemmawererepeatedforTperiods.Usebackwardinductiontosolveforthesubgame-perfectequilibrium.ThelowestsubgameisthePrisoners’DilemmastagegameplayedinperiodT:Regardlessofwhathappenedbefore,theNashequilibriumonthissubgameisforbothtoﬁnk.FoldingthegamebacktoperiodT1,triggerstrategiesthatconditionperiod-TplayonwhathappensinperiodT1areruledout.AlthoughaplayermightliketopromisetoplaycooperativelyinperiodTandsorewardtheotherforplayingcooperativelyinperiodT1,wehavejustseenthatnothingthathappensinperiodT1affectswhathappenssubsequentlybecauseplayersbothﬁnkinperiodTregardless.ItisasifperiodT1werethelast,andtheNashequilibriumofthissubgameisagainforbothtoﬁnk.Workingbackwardinthisway,weseethatplayerswillﬁnkeachperiod;thatis,playerswillsimplyrepeattheNashequilibriumofthestagegameTtimes.ReinhardSelten,winneroftheNobelPrizeineconomicsforhiscontributionstogametheory,showedthatthesamelogicappliesmoregenerallytoanystagegamewithauniqueNashequilibrium.5ThisresultiscalledSelten’stheorem:IfthestagegamehasauniqueNashequilibrium,thentheuniquesubgame-perfectequilibriumoftheﬁnitelyrepeatedgameistoplaytheNashequilibriumeveryperiod.260Part2ChoiceandDemand
IfthestagegamehasmultipleNashequilibria,itmaybepossibletoachievesomecooperationinaﬁnitelyrepeatedgame.Playerscanusetriggerstrategies,sustainingcoopera-tioninearlyperiodsonanoutcomethatisnotanequilibriumofthestagegame,bythreaten-ingtoplayinlaterperiodstheNashequilibriumthatyieldsaworseoutcomefortheplayerwhodeviatesfromcooperation.Example8.7illustrateshowsuchtriggerstrategiesworktosustaincooperation.EXAMPLE8.7CooperationinaFinitelyRepeatedGameThestagegamegiveninnormalforminTable8.10hastwopure-strategyNashequilibria.Inthe“bad”pure-strategyequilibrium,eachplaysBandearnsapayoffof1;inthe“good”equilibrium,eachplaysCandearnsapayoffof3.Playerswouldearnstillmore(i.e.,4)ifbothplayedA,butthisisnotaNashequilibrium.IfoneplaysA,thentheotherwouldprefertodeviatetoBandearn5.Thereisathird,mixed-strategyNashequilibriuminwhicheachplaysBwithprobability3=4andCwithprobability1=4.ThepayoffsaregraphedassolidcirclesinFigure8.9.Ifthestagegameisrepeatedtwice,awealthofnewpossibilitiesariseinsubgame-perfectequilibria.Thesameper-periodpayoffs(1or3)fromthestagegamecanbeobtainedsimplybyrepeatingthepure-strategyNashequilibriafromthestagegametwice.Per-periodaveragepayoffsof2.5canbeobtainedbyalternatingbetweenthegoodandthebadstage-gameequilibria.Amorecooperativeoutcomecanbesustainedwiththefollowingstrategy:beginbyplayingAintheﬁrstperiod;ifnoonedeviatesfromA,playCinthesecondperiod;ifaplayerdeviatesfromA,thenplayBinthesecondperiod.Backwardinductioncanbeusedtoshowthatthesestrategiesformasubgame-perfectequilibrium.ThestrategiesformaNashequilib-riuminsecond-periodsubgamesbyconstruction.ItremainstocheckwhetherthestrategiesformaNashequilibriumonthegameasawhole.Inequilibriumwiththesestrategies,playersearn4þ3¼7intotalacrossthetwoperiods.BydeviatingtoBintheﬁrstperiod,aplayercanincreasehisorherﬁrst-periodpayofffrom4to5,butthisleadstobothplayingBinthesecondperiod,reducingthesecond-periodpayofffrom3to1.Thetotalpayoffacrossthetwoperiodsfromthisdeviationis5þ1¼6,lessthanthe7earnedintheproposedequilibrium.Theaverageper-periodpayoffinthissubgame-perfectequilibriumis7=2¼3.5foreachplayer.Asymmetricequilibriaarealsopossible.Inone,player1beginsbyplayingBandplayer2byplayingA;ifnoonedeviatesthenbothplaythegoodstage-gameNashequilibrium(bothplayC),andifsomeonedeviatesthenbothplaythebadequilibrium(bothplayB).Player2(continued)TABLE8.10StageGameforExample8.7Player2ABCPlayer1A4,40,50,0B5,01,10,0C0,00,03,35R.Selten,“ASimpleModelofImperfectCompetition,Where4AreFewand6AreMany,”InternationalJournalofGameTheory2(1973):141–201.Chapter8StrategyandGameTheory261
EXAMPLE8.7CONTINUEDdoesnotwanttodeviatetoplayingBintheﬁrstperiodbecauseheorsheearns1fromthisdeviationintheﬁrstperiodand1inthesecondwhentheyplaythebadequilibriumforatotalof1þ1¼2,whereasheorsheearnsmore,0þ3¼3;inequilibrium.Theaverageper-periodpayoffinthissubgame-perfectequilibriumisð5þ3Þ=2¼4forplayer1and3=2¼1.5forplayer2.Thereversepayoffscanbeobtainedbyreversingthestrategies.Theaverageper-periodpayoffsfromtheadditionalsubgame-perfectequilibriawecomputedforthetwice-repeatedgamearegraphedassquaresinFigure8.9.Ifthegameisrepeatedthreetimes(T¼3),thenadditionalpayoffcombinationsarepossibleinsubgame-perfectequilibria.PlayerscancooperateonplayingAfortwoperiodsandCinthelast,astrategythatissustainedbythethreatofimmediatelymovingtothebadequilibrium(bothplayB)ifanyonedeviatesintheﬁrsttwoperiods.Thissubgame-perfectequilibriumgiveseachaper-periodaveragepayoffofð4þ4þ3Þ=23:7,morethanthe3.5thatwasthemostbothcouldearnintheT¼2game.AsymmetricequilibriaintheT¼3gameincludethepossibilitythat1playsBand2playsAfortheﬁrsttwoperiodsandthenbothplayC,withthethreatofimmediatelymovingtothebadequilibriumifanyonedeviates.Player1’sper-periodaveragepayoffinthissubgame-perfectequilibriumisð5þ5þ3Þ=34:3,andplayer2’spayoffisð0þ0þ3Þ=3¼1:Thereversestrategiesandpayoffsalsoconstituteapossiblesubgame-perfectequilibrium.Thepayoffsfromtheaddi-tionalsubgame-perfectequilibriaoftheT¼3gamearegraphedastrianglesinFigure8.9.FIGURE8.9Per-PeriodAveragePayoffsinExample8.7SolidcirclesindicatepayoffsinNashequilibriaofthestagegame.Squares(inadditiontocircles)indicateper-periodaveragepayoffsinsubgame-perfectequilibriaforT¼2repetitionsofthestagegame.Triangles(inadditiontocirclesandsquares)indicateper-periodaveragepayoffsforT¼3.54321102345u1u2262Part2ChoiceandDemand
QUERY:Therearemanyothersubgame-perfectequilibriumpayoffsfortherepeatedgamethanareshowninFigure8.9.FortheT¼2game,canyouﬁndatleasttwoothercombina-tionsofaverageper-periodpayoffsthatcanbeattainedinasubgame-perfectequilibrium?Forcooperationtobesustainedinasubgame-perfectequilibrium,thestagegamemustberepeatedoftenenoughthatthepunishmentfordeviation(repeatedlyplayingtheless-preferredNashequilibrium)issevereenoughtodeterdeviation.ThemorerepetitionsofthestagegameT,themoreseverethepossiblepunishmentandthusthegreaterthelevelofcooperationandthehigherthepayoffsthatcanbesustainedinasubgame-perfectequilib-rium.InExample8.7,themostbothplayerscanearninasubgame-perfectequilibriumincreasesfrom3to3.5toabout3.7asTincreasesfrom1to2to3.Example8.7suggeststhattherangeofsustinablepayoffsinasubgame-perfectequilib-riumexpandsasthenumberofrepetitionsTincreases.Infact,theassociatedFigure8.9understatestheexpansionbecauseitdoesnotgraphallsubgame-perfectequilibriumpayoffsforT¼2andT¼3(theQueryinExample8.7asksyoutoﬁndtwomore,forexample).WearelefttowonderhowmuchthesetofpossibilitiesmightexpandforyethigherT:JeanPierreBenoitandVijayKrishnaanswerthisquestionwiththeirfolktheoremforﬁnitelyrepeatedgames:6SupposethatthestagegamehasmultipleNashequilibriaandnoplayerearnsaconstantpayoffacrossallequilibria.Anyfeasiblepayoffinthestagegamegreaterthantheplayer’spure-strategyminmaxvaluecanbeapproachedarbitrarilycloselybytheplayer’sper-periodaveragepayoffinsomesubgame-perfectequilibriumoftheﬁnitelyrepeatedgameforlargeenoughT:7Wewillencounterotherfolktheoremsinlatersectionsofthischapter.Generallyspeak-ing,afolktheoremisaresultthat“anythingispossible”inthelimitwithrepeatedgames.Suchresultsarecalled“folk”theoremsbecausetheywereunderstoodinformallyandthuswerepartofthe“folkwisdom”ofgametheorywellbeforeanyonewrotedownformalproofs.Tounderstandthefolktheoremfully,weneedtounderstandwhatfeasiblepayoffsandminmaxvaluesare.Afeasiblepayoffisonethatcanbeachievedbysomemixed-strategyproﬁleinthestagegame.Graphically,thefeasiblepayoffsetappearsastheconvexhullofthepure-strategystage-gamepayoffs.Theconvexhullofasetofpointsistheborderandinteriorofthelargestpolygonthatcanbeformedbyconnectingthepointswithlinesegments.Forexample,Figure8.10graphsthefeasiblepayoffsetforthestagegamefromExample8.7astheupward-hatchedregion.Toderivethisset,oneﬁrstgraphsthepure-strategypayoffsfromthestagegame.ReferringtothenormalforminTable8.10,thedistinctpure-strategypayoffsare(4,4),(0,5),(0,0),(5,0),(1,1),and(3,3).Theconvexhullisthepolygonformedbylinesegmentsgoingfrom(0,0)to(0,5)to(4,4)to(5,0),andbackto(0,0).Eachpointintheconvexhullcorrespondstotheexpectedpayoffsfromsomecombinationofmixedstrategiesforplayers1and2overactionsA,B,andC:Forexample,thepoint(3,0)ontheboundaryoftheconvexhullcorrespondstoplayers’expectedpayoffsif1playsthemixedstrategy(0,3=5,2=5)and2playsA:Aminmaxvalueistheleastthatplayericanbeforcedtoearn.6J.P.BenoitandV.Krishna,“FinitelyRepeatedGames,”Econometrica53(1985):890–904.7Anadditional,technicalconditionisthatthedimensionofthefeasiblesetofpayoffsmustequalthenumberofplayers.Inthetwo-playergameinExample8.7,thisconditionwouldrequirethefeasiblepayoffsettobearegion(whichisthecase,asshowninFigure8.12)ratherthanalineorapoint.Chapter8StrategyandGameTheory263
DEFINITIONMinmaxvalue.Theminmaxvalueisthefollowingpayoffforplayeri:minsihmaxsiuiðsi,siÞi,(8.19)thatis,thelowestpayoffplayericanbeheldtoifallotherplayersworkagainsthimorherbutplayeriisallowedtochooseabestresponsetothem.InExample8.7,if2playsthemixedstrategy(0,3=4,1=4)thenthemostplayer1canearninthestagegameis3=4(byplayinganymixedstrategyinvolvingonlyactionsBandC).Alittleworkshowsthat3=4isindeedplayer1’sminmaxvalue:anyotherstrategyfor2besides(0,3=4,1=4)wouldallow1toearnahigherpayoffthan3=4.Thefolktheoremforﬁnitelyrepeatedgamesinvolvesthepure-strategyminmaxvalue—thatis,theminmaxvaluewhenplayersarerestrictedtousingonlypurestrategies.Thepure-strategyminmaxvalueiseasiertocomputethanthegeneralminmaxvalue.Thelowestthatplayer2canhold1toinExample8.7isapayoffof1;player2doesthisbyplayingBandthen1respondsbyplayingB:Figure8.10graphsthepayoffsexceedingbothplayers’pure-strategyminmaxvaluesasthedownward-hatchedregion.Thefolktheoremforﬁnitelyrepeatedgamesassuresusthatanypayoffsinthecross-hatchedregionofFigure8.10—payoffsthatarefeasibleandabovebothplayers’purestrategyFIGURE8.10FolkTheoremforFinitelyRepeatedGamesinExample8.7ThefeasiblepayoffsforthestagegameinExample8.7areintheupward-hatchedregion;payoffsgreaterthaneachplayer’sminmaxvaluesareinthedownward-hatchedregion.Theirintersection(thecross-hatchedregion)constitutestheper-periodaveragepayoffsthatcanbeapproachedbysomesubgame-perfectequilibriumoftherepeatedgame,accordingtothefolktheoremforﬁnitelyrepeatedgames.RegionsaresuperimposedontheequilibriumpayoffsfromFigure8.9.54321102345u1u2264Part2ChoiceandDemand
minmaxvalues—canbeapproachedastheper-periodaveragepayoffsinasubgame-perfectequilibriumifthestagegameinExample8.7isrepeatedoftenenough.Payoffs(4,4)canbeapproachedbyhavingplayerscooperateonplayingAforhundredsofperiodsandthenplayingCinthelastperiod(threateningthebadequilibriuminwhichbothplayBifanyonedeviatesfromcooperation).Theaverageofhundredsofpayoffsof4withonepayoffof3comesarbitrarilycloseto4.Therefore,aconsiderableamountofcooperationispossibleifthegameisrepeatedoftenenough.Figure8.10alsoshowsthatmanyoutcomesotherthanfullcooperationarepossibleifthenumberofrepetitions,T,islarge.Althoughsubgame-perfectequilibriumwasselectiveinthesequentialversionoftheBattleoftheSexes,allowingustoselectoneofthreeNashequilibria,weseethatsubgameperfectionmaynotbeselectiveinrepeatedgames.ThefolktheoremstatesthatifthestagegamehasmultipleNashequilibriathenalmostanythingcanhappenintherepeatedgameforTlargeenough.8InfinitelyrepeatedgamesWithﬁnitelyrepeatedgames,thefolktheoremappliesonlyifthestagegamehasmultipleequilibria.If,likethePrisoners’Dilemma,thestagegamehasonlyoneNashequilibrium,thenSelten’stheoremtellsusthattheﬁnitelyrepeatedgamehasonlyonesubgame-perfectequilibrium:repeatingthestage-gameNashequilibriumeachperiod.BackwardinductionstartingfromthelastperiodTunravelsanyotheroutcomes.Withinﬁnitelyrepeatedgames,however,thereisnodeﬁniteendingperiodTfromwhichtostartbackwardinduction.AfolktheoremwillapplytoinﬁnitelyrepeatedgameseveniftheunderlyingstagegamehasonlyoneNashequilibrium.Therefore,whilebothplayersﬁnkeveryperiodintheuniquesubgame-perfectequilibriumoftheﬁnitelyrepeatedPrisoners’Dilemma,playersmayendupcooperating(beingsilent)intheinﬁnitelyrepeatedversion.Onedifﬁcultywithinﬁnitelyrepeatedgamesinvolvesaddinguppayoffsacrossperiods.Withﬁnitelyrepeatedgames,wecouldfocusonaveragepayoffs.Withinﬁnitelyrepeatedgames,theaverageisnotwell-deﬁnedbecauseitinvolvesaninﬁnitesumofpayoffsdividedbyaninﬁnitenumberofperiods.Wewillcircumventthisproblemwiththeaidofdiscounting.Letδbethediscountfactor(discussedintheChapter17Appendix)measuringhowmuchapayoffunitisworthifreceivedoneperiodinthefutureratherthantoday.InChapter17weshowthatδisinverselyrelatedtotheinterestrate.Iftheinterestrateishighthenapersonwouldmuchratherreceivepaymenttodaythannextperiodbecauseinvestingtoday’spaymentwouldprovideareturnofprincipalplusalargeinterestpaymentnextperiod.Besidestheinterestrate,δcanalsoincorporateuncertaintyaboutwhetherthegamecontinuesinfutureperiods.Thehighertheprobabilitythatthegameendsafterthecurrentperiod,thelowertheexpectedreturnfromstagegamesthatmightnotactuallybeplayed.Factoringinaprobabilitythattherepeatedgameendsaftereachperiodmakesthesettingofaninﬁnitelyrepeatedgamemorebelievable.Thecrucialissuewithaninﬁnitelyrepeatedgameisnotthatitgoesonforeverbutthatitsendisindeterminate.Interpretedinthisway,thereisasenseinwhichinﬁnitelyrepeatedgamesaremorerealisticthanﬁnitelyrepeatedgameswithlargeT:Supposeweexpecttwoneighboringgasolinestationstoplayapricinggameeachdayuntilelectriccarsreplacegasoline-poweredones.ItisunlikelythegasolinestationswouldknowthatelectriccarswerecominginexactlyT¼2,000days.Morerealisti-cally,thegasolinestationswillbeuncertainabouttheendofgasoline-poweredcarsandsotheendoftheirpricinggameisindeterminate.8Thefolktheoremforfinitelyrepeatedgamesdoesnotnecessarilycaptureallsubgame-perfectequilibria.InFigure8.12,thepoint(3=4,3=4)liesoutsidethecross-hatchedregion;nonetheless,itcanbeachievedinasubgame-perfectequilibriuminwhich,eachperiod,bothplayersplaytheNashequilibriumofthestagegameinstrictlymixedstrategies.Payoffs(3=4,3=4)areina“grayarea”betweenplayer’spure-strategyandmixed-strategyminmaxvalues.Chapter8StrategyandGameTheory265
Playerscansustaincooperationininﬁnitelyrepeatedgamesbyusingtriggerstrategies:playerscontinuecooperatingunlesssomeonehasdeviatedfromcooperation,andthisdevia-tiontriggerssomesortofpunishment.Inorderfortriggerstrategiestoformanequilibrium,thepunishmentmustbesevereenoughtodeterdeviation.SupposebothplayersusethefollowingtriggerstrategyinthePrisoners’Dilemma:con-tinuebeingsilentifnoonehasdeviatedbyplayingﬁnk;ﬁnkforeverafterwardifanyonehasdeviatedtoﬁnkinthepast.Toshowthatthistriggerstrategyformsasubgame-perfectequilibrium,weneedtocheckthataplayercannotgainfromdeviating.Alongtheequilibriumpath,bothplayersaresilenteveryperiod;thisprovideseachwithapayoffof2everyperiodforapresentdiscountedvalueofVeq¼2þ2δþ2δ2þ2δ3þ…þ2ð1þδþδ2þδ3þ…Þ¼21δ:(8.20)Aplayerwhodeviatesbyfinkingearns3inthatperiod,butthenbothplayersfinkeveryperiodfromthenon—eachearning1perperiodforatotalpresenteddiscountedpayoffofVdev¼3þð1ÞðδÞþð1Þðδ2Þþð1Þðδ3Þþ…þ3þδð1þδþδ2þ…Þ¼3þδ1δ:(8.20)Thetriggerstrategiesformasubgame-perfectequilibriumifVeqVdev;implyingthat21δ3þδ1δ;(8.22)aftermultiplyingthroughby1δandrearranging,weobtainδ1=2:Inotherwords,playerswillfindcontinuedcooperativeplaydesirableprovidedtheydonotdiscountfuturegainsfromsuchcooperationtoohighly.Ifδ<1=2,thennocooperationispossibleintheinfinitelyrepeatedPrisoners’Dilemma;theonlysubgame-perfectequilibriuminvolvesfink-ingeveryperiod.Thetriggerstrategyweconsideredhasplayersreverttothestage-gameNashequilibriumofﬁnkingeachperiodforever.Thisstrategy,whichinvolvestheharshestpossiblepunishmentfordeviation,iscalledthegrimstrategy.Lessharshpunishmentsincludetheso-calledtit-for-tatstrategy,whichinvolvesonlyoneroundofpunishmentforcheating.Sinceitinvolvestheharshestpunishmentpossible,thegrimstrategyelicitscooperationforthelargestrangeofcases(thelowestvalueofδ)ofanystrategy.Harshpunishmentsworkwellbecause,ifplayerssucceedincooperating,theyneverexperiencethelossesfromthepunishmentinequilibrium.9ThediscountfactorδiscrucialindeterminingwhethertriggerstrategiescansustaincooperationinthePrisoners’Dilemmaor,indeed,inanystagegame.Asδapproaches1,grim-strategypunishmentsbecomeinﬁnitelyharshbecausetheyinvolveanunendingstreamofundiscountedlosses.Inﬁnitepunishmentscanbeusedtosustainawiderangeofpossibleoutcomes.Thisisthelogicbehindthefolktheoremforinﬁnitelyrepeatedgames:109NobelPrize–winningeconomistGaryBeckerintroducedarelatedpoint,themaximalpunishmentprincipleforcrime.Theprinciplesaysthatevenminorcrimesshouldreceivedraconianpunishments,whichcandetercrimewithminimalexpenditureonpolicing.Thepunishmentsarecostlesstosocietybecausenocrimesarecommittedinequilibrum,sopunishmentsneverhavetobecarriedout.SeeG.Becker,“CrimeandPunishment:AnEconomicApproach,”JournalofPoliticalEconomy76(1968):169–217.Lessharshpunishmentsmaybesuitableinsettingsinvolvinguncertainty.Forexample,citizensmaynotbecertainaboutthepenalcode;policemaynotbecertaintheyhavearrestedtheguiltyparty.10ThisfolktheoremisduetoD.FudenbergandE.Maskin,“TheFolkTheoreminRepeatedGameswithDiscountingorwithIncompleteInformation,”Econometrica54(1986):533–56.266Part2ChoiceandDemand Anyfeasiblepayoffinthestagegamegreaterthantheplayer’sminmaxvaluecanbeobtainedastheplayer’snormalizedpayoff(normalizedbymultiplyingby1δ:)insomesubgame-perfectequilibriumoftheinﬁnitelyrepeatedgameforδcloseenoughto1.11Afewdifferenceswiththefolktheoremforﬁnitelyrepeatedgamesareworthemphasizing.First,thelimitinvolvesincreasesinδratherthaninthenumberofperiodsT:Thetwolimitsarerelated.Interpretingδascapturingtheprobabilitythatthegamecontinuesintothenextperiod,anincreaseinδincreasestheexpectednumberofperiodsthegameisplayedintotal—similartoanincreaseinTwiththedifferencethatnowtheendofthegameisindeﬁnite.AnotherdifferencebetweenthetwofolktheoremsisthattheoneforinﬁnitelyrepeatedgamesholdsevenifthestagegamehasjustasingleNashequilibriumwhereasthetheoremforﬁnitelyrepeatedgamesrequiresthestagegametohavemultipleNashequilibria.Aﬁnaltechnicalityisthatcomparingstage-gamepayoffswiththepresentdiscountedvalueofastreamofpayoffsfromtheinﬁnitelyrepeatedgameislikecomparingappleswithoranges.Tomakethetwocomparable,we“normalize”thepayofffromtheinﬁnitelyrepeatedgameviamultiplyingby1δ:Thisnormalizationallowsustothinkofallpayoffsinper-periodtermsforeasycomparison.12FIGURE8.11Folk-TheoremPayoffsintheInfinitelyRepeatedPrisoners'DilemmaFeasiblepayoffsareintheupward-hatchedregion;payoffsgreaterthaneachplayer’sminmaxvaluesareinthedownward-hatchedregion.Theirintersection(thecross-hatchedregion)constitutestheachievablepayoffsaccordingtothefolktheoremforinﬁnitelyrepeatedgames.3211023u1u211Asinfootnote9,anadditionaltechnicalconditiononthedimensionofthefeasiblepayoffsetisalsorequired.12Forexample,supposeaplayerearns$1atthebeginningofeachperiod.Thepresentdiscountedvalueofthestreamofthese$1payoffsforaninfinitenumberofperiodsis$1þ$1δþ$1δ2þ$1δ3þ…¼$11δ:Multiplyingthroughby1δconvertsthisstreamofpaymentsbackintotheper-periodpayoffof$1.TheChapter17Appendixprovidesmoredetailonthecalculationofpresentdiscountedvaluesofannuitystreams(thoughbewaretheChapter8StrategyandGameTheory267 Figure8.11illustratesthefolktheoremforinﬁnitelyrepeatedgamesinthecaseofthePrisoners’Dilemma.Theﬁgureshowstherangeofnormalizedpayoffsthatarepossibleinsomesubgame-perfectequilibriumoftheinﬁnitelyrepeatedPrisoners’Dilemma.Againweseethatsubgameperfectionmaynotbeparticularlyselectiveincertainrepeatedgames.INCOMPLETEINFORMATIONInthegamesstudiedsofar,playerskneweverythingtherewastoknowaboutthesetupofthegame,includingeachothers’strategysetsandpayoffs.Mattersbecomemorecomplicated,andpotentiallymoreinteresting,ifsomeplayershaveinformationaboutthegamethatothersdonot.Pokerwouldbequitedifferentifallhandswereplayedfaceup.Thefunofplayingpokercomesfromknowingwhatisinyourhandbutnotothers’.Incompleteinformationarisesinmanyotherreal-worldcontextsbesidesparlorgames.Asportsteammaytrytohidetheinjuryofastarplayerfromfutureopponentstopreventthemfromexploitingthisweakness.Firms’productiontechnologiesmaybetradesecrets,andthusﬁrmsmaynotknowwhethertheyfaceefﬁcientorweakcompetitors.Thissection(andthenexttwo)willintroducethetoolsneededtoanalyzegamesofincompleteinformation.Theanalysisinte-gratesthematerialongametheorydevelopedsofarinthischapterwiththematerialonuncertaintyandinformationfromthepreviouschapter.Gamesofincompleteinformationcanquicklybecomeverycomplicated.Playersthatlackfullinformationaboutthegamewilltrytousewhattheydoknowtomakeinferencesaboutwhattheydonot.Theinferenceprocesscanbequiteinvolved.Inpoker,forexample,knowingwhatisinyourhandcantellyousomethingaboutwhatisinothers’.Aplayerthatholdstwoacesknowsthatothersarelesslikelytoholdacesbecausetwoofthefouracesarenotavailable.Informationonothers’handscanalsocomefromthesizeoftheirbetsorfromtheirfacialexpressions(ofcourse,abigbetmaybeabluffandafacialexpressionmaybefaked).Probabilitytheoryprovidesaformula,calledBayes’rule,formakinginferencesabouthiddeninformation.WewillencounterBayes’ruleinalatersection.TherelevanceofBayes’ruleingamesofincompleteinformationhasledthemtobecalledBayesiangames.Tolimitthecomplexityoftheanalysis,wewillfocusonthesimplestpossiblesettingthroughout.Wewillfocusontwo-playergamesinwhichoneoftheplayers(player1)hasprivateinformationandtheother(player2)doesnot.Theanalysisofgamesofincompleteinformationisdividedintotwosections.Thenextsectionbeginswiththesimplecaseinwhichtheplayersmovesimultaneously.Thesubsequentsectionthenanalyzesgamesinwhichtheinformedplayer1movesﬁrst.Suchgames,calledsignalinggames,aremorecom-plicatedthansimultaneousgamesbecauseplayer1’sactionmaysignalsomethingabouthisprivateinformationtotheuninformedplayer2.WewillintroduceBayes’ruleatthatpointtohelpanalyzeplayer2’sinferenceaboutplayer1’shiddeninformationbasedonobservationsofplayer1’saction.SIMULTANEOUSBAYESIANGAMESInthissectionwestudyatwo-player,simultaneous-movegameinwhichplayer1hasprivateinformationbutplayer2doesnot.(Wewilluse“he”forplayer1and“she”forplayer2inordertofacilitatetheexposition.)Webeginbystudyinghowtomodelprivateinformation.subtledifferencethatinChapter17theannuitypaymentscomeattheendofeachperiodratherthanatthebeginningasassumedhere).268Part2ChoiceandDemand PlayertypesandbeliefsJohnHarsanyi,whoreceivedtheNobelPrizeineconomicsforhisworkongameswithincompleteinformation,providedasimplewaytomodelprivateinformationbyintroducingplayercharacteristicsortypes.13Player1canbeoneofanumberofpossiblesuchtypes,denotedt:Player1knowshisowntype.Player2isuncertainabouttandmustdecideonherstrategybasedonbeliefsaboutt:Formally,thegamebeginsataninitialnode,calledachancenode,atwhichaparticularvaluetkisrandomlydrawnforplayer1’stypetfromasetofpossibletypesT¼ft1,…,tk,…,tKg:LetPrðtkÞbetheprobabilityofdrawingtheparticulartypetk:Player1seeswhichtypeisdrawn.Player2doesnotseethedrawandonlyknowstheprobabilities,usingthemtoformherbeliefsaboutplayer1’stype.Thustheprobabilitythatplayer2placesonplayer1’sbeingoftypetkisPrðtkÞ:Sinceplayer1observeshistypetbeforemoving,hisstrategycanbeconditionedont:Conditioningonthisinformationmaybeabigbeneﬁttoaplayer.Inpoker,forexample,thestrongeraplayer’shand,themorelikelytheplayeristowinthepotandthemoreaggressivelytheplayermaywanttobid.Lets1ðtÞbe1’sstrategycontingentonhistype.Sinceplayer2doesnotobservet,herstrategyissimplytheunconditionalone,s2:Aswithgamesofcompleteinformation,players’payoffsdependonstrategies.InBayesiangames,payoffsmayalsodependontypes.Wethereforewriteplayer1’spayoffasu1ðs1ðtÞ,s2,tÞand2’sasu2ðs2,s1ðtÞ,tÞ:Notethattappearsintwoplacesin2’spayofffunction.Player1’stypemayhaveadirecteffecton2’spayoffs.Player1’stypealsohasanindirecteffectthroughitseffecton1’sstrategys1ðtÞ,whichinturnaffects2’spayoffs.Since2’spayoffsdependontinthesetwoways,herbeliefsabouttwillbecrucialinthecalculationofheroptimalstrategy.Table8.11providesasimpleexampleofasimultaneousBayesiangame.Eachplayerchoosesoneoftwoactions.Allpayoffsareknownexceptfor1’spayoffwhen1choosesUand2choosesL:Player1’spayoffinoutcomeðU,LÞisidentiﬁedashistype,t:Therearetwopossiblevaluesforplayer1’stype,t¼6andt¼0;eachoccurringwithequalprobability.Player1knowshistypebeforemoving.Player2’sbeliefsarethateachtypehasprobability1=2.TheextensiveformisdrawninFigure8.12.Bayesian-NashequilibriumExtendingNashequilibriumtoBayesiangamesrequirestwosmallmattersofinterpretation.First,recallthatplayer1mayplayadifferentactionforeachofhistypes.Equilibriumrequiresthat1’sstrategybeabestresponseforeachandeveryoneofhistypes.Second,recallthatplayer2isuncertainaboutplayer1’stype.Equilibriumrequiresthat2’sstrategymaximizeanexpectedpayoff,wheretheexpectationistakenwithrespecttoherbeliefsabout1’stype.Weencounteredexpectedpayoffsinourdiscussionofmixedstrategies.ThecalculationsinvolvedincomputingthebestresponsetothepurestrategiesofdifferenttypesofrivalsinagameofTABLE8.11SimpleGameofIncompleteInformationPlayer2LRPlayer1Ut,20,0D2,02,4Note:t¼6withprobability1=2andt¼0withprobability1=2.13J.Harsanyi,“GameswithIncompleteInformationPlayedbyBayesianPlayers,”ManagementScience14(1967∕68):159–82,320–34,486–502.Chapter8StrategyandGameTheory269 incompleteinformationaresimilartothecalculationsinvolvedincomputingthebestre-sponsetoarival’smixedstrategyinagameofcompleteinformation.Interpretedinthisway,NashequilibriuminthesettingofaBayesiangameiscalledBayesian-Nashequilibrium.DEFINITIONBayesian-Nashequilibrium.Inatwo-player,simultaneous-movegameinwhichplayer1hasprivateinformation,aBayesian-Nashequilibriumisastrategyproﬁleðs1ðtÞ,s2Þsuchthats1ðtÞisabestresponsetos2foreachtypet2Tofplayer1,U1ðs1ðtÞ,s2,tÞU1ðs01,s2,tÞforalls012S1,(8.23)andsuchthats2isabestresponsetos1ðtÞgivenplayer2’sbeliefsPrðtkÞaboutplayer1’stypes:Xtk2TPrðtkÞU2ðs2,s1ðtkÞ,tkÞXtk2TPrðtkÞU2ðs02,s1ðtkÞ,tkÞforalls022S2.(8.24)SincethedifferencebetweenNashequilibriumandBayesian-Nashequilibriumisonlyamatterofinterpretation,allourpreviousresultsforNashequilibrium(includingtheexistenceproof)applytoBayesian-Nashequilibriumaswell.FIGURE8.12ExtensiveFormforSimpleGameofIncompleteInformationThisﬁguretranslatesTable8.11intoanextensive-formgame.Theinitialchancenodeisindicatedbyanopencircle.Player2’sdecisionnodesareinthesameinformationsetbecauseshedoesnotobserve1’stypeoractionpriortomoving.DDULLLLRRRRU1226, 20, 00, 20, 0Pr=1/2t=6Pr=1/2t=02, 02, 42, 42, 0221270Part2ChoiceandDemand EXAMPLE8.8Bayesian-NashEquilibriumofGameinFigure8.12TosolvefortheBayesian-NashequilibriumofthegameinFigure8.12,ﬁrstsolvefortheinformedplayer’s(player1’s)bestresponsesforeachofhistypes.Ifplayer1isoftypet¼0thenhewouldchooseDratherthanUbecauseheearns0byplayingUand2byplayingDregardlessofwhat2does.Ifplayer1isoftypet¼6,thenhisbestresponseisUto2’splayingLandDtoherplayingR.Thisleavesonlytwopossiblecandidatesforanequilibriuminpurestrategies:1playsðUjt¼6,Djt¼0Þand2playsL;1playsðDjt¼6,Djt¼0Þand2playsR.Theﬁrstcandidatecannotbeanequilibriumbecause,giventhat1playsðUjt¼6,Djt¼0Þ,2earnsanexpectedpayoffof1fromplayingL:Player2wouldgainbydeviatingtoR,earninganexpectedpayoffof2.ThesecondcandidateisaBayesian-Nashequilibrium.Giventhat2playsR,1’sbestresponseistoplayD,providingapayoffof2ratherthan0regardlessofhistype.Giventhatbothtypesofplayer1playD,player2’sbestresponseistoplayR,providingapayoffof4ratherthan0.QUERY:Iftheprobabilitythatplayer1isoftypet¼6ishighenough,cantheﬁrstcandidatebeaBayesian-Nashequilibrium?Ifso,computethethresholdprobability.EXAMPLE8.9TragedyoftheCommonsasaBayesianGameForanexampleofaBayesiangamewithcontinuousactions,considertheTragedyoftheCommonsinExample8.6butnowsupposethatherder1hasprivateinformationregardinghisvalueofgrazingpersheep:v1ðq1,q2,tÞ¼tðq1þq2Þ,(8.25)where1’stypeist¼130(the“high”type)withprobability2=3andt¼100(the“low”type)withprobability1=3.Herder2’svalueremainsthesameasinEquation8.11.TosolvefortheBayesian-Nashequilibrium,weﬁrstsolvefortheinformedplayer’s(herder1’s)bestresponsesforeachofhistypes.Foranytypetandrival’sstrategyq2,herder1’svalue-maximizationproblemismaxq1fq1v1ðq1,q2,tÞg¼maxq1fq1ðtq1q2Þg.(8.26)Theﬁrst-orderconditionforamaximumist2q1q2¼0.(8.27)Rearrangingandthensubstitutingthevaluest¼130andt¼100,weobtainq1H¼65q22andq1L¼50q22,(8.28)whereq1Histhequantityforthe“high”typeofherder1(thatis,thet¼130type)andq1Lforthe“low”type(thet¼130type).Nextwesolvefor2’sbestresponse.Herder2’sexpectedpayoffis23½q2ð120q1Hq2Þþ13½q2ð120q1Lq2Þ¼q2ð120_q1q2Þ,(8.29)(continued)Chapter8StrategyandGameTheory271 EXAMPLE8.9CONTINUEDwhere_q1¼23q1Hþ13q1L.(8.30)Rearrangingtheﬁrst-orderconditionfromthemaximizationofEquation8.29withrespecttoq2givesq2¼60_q12.(8.31)Substitutingforq1Handq1LfromEquation8.28intoEquation8.30andthensubstitutingtheresultingexpressionfor_q1intoEquation8.31yieldsq2¼30þq24,(8.32)implyingthatq2¼40:Substitutingq2¼40backintoEquation8.28impliesq1H¼45andq1L¼30:Figure8.13depictstheBayesian-Nashequilibriumgraphically.Herder2imaginesplayingagainstanaveragetypeofherder1,whoseaveragebestresponseisgivenbythethickdashedline.Theintersectionofthisbestresponseandherder2’satpointBdetermines2’sequilibriumquantity,q2¼40:Thebestresponseofthelow(resp.high)typeofherder1toq2¼40isgivenbypointA(resp.pointC).Forcomparison,thefull-informationNashequilibriaaredrawnwhenherder1isknowntobethelowtype(pointA0)orthehightype(pointC0).QUERY:Supposeherder1isthehightype.Howdoesthenumberofsheepeachherdergrazeschangeasthegamemovesfromincompletetofullinformation(movingfrompointC0FIGURE8.13EquilibriumoftheBayesianTragedyoftheCommonsBestresponsesforherder2andbothtypesofherder1aredrawnasthicksolidlines;theexpectedbestresponseasperceivedby2isdrawnasthethickdashedline.TheBayesian-Nashequilibriumoftheincomplete-informationgameisgivenbypointsAandC;Nashequilibriaofthecorrespondingfull-informationgamesaregivenbypointsA0andC0.High type’s best responseq1q2Low type’s best response2’s best responseC′CBA′A030454040272Part2ChoiceandDemand toC)?Whatifherder1isthelowtype?Whichtypeprefersfullinformationandthuswouldliketosignalitstype?Whichtypeprefersincompleteinformationandthuswouldliketohideitstype?Wewillstudythepossibilityplayer1cansignalhistypeinthenextsection.SIGNALINGGAMESInthissectionwemovefromsimultaneous-movegamesofprivateinformationtosequentialgamesinwhichtheinformedplayer,1,takesanactionthatisobservableto2before2moves.Player1’sactionprovidesinformation,asignal,that2canusetoupdateherbeliefsabout1’stype,perhapsalteringtheway2wouldplayintheabsenceofsuchinformation.Inpoker,forinstance,player2maytakeabigraisebyplayer1asasignalthathehasagoodhand,perhapsleading2tofold.Aﬁrmconsideringwhethertoenteramarketmaytaketheincumbentﬁrm’slowpriceasasignalthattheincumbentisalow-costproducerandthusatoughcompetitor,perhapskeepingtheentrantoutofthemarket.Aprestigiouscollegedegreemaysignalthatajobapplicantishighlyskilled.Theanalysisofsignalinggamesismorecomplicatedthansimultaneousgamesbecauseweneedtomodelhowplayer2processestheinformationin1’ssignalandthenupdatesherbeliefsabout1’stype.Toﬁxideas,wewillfocusonaconcreteapplication:aversionofMichaelSpence’smodelofjob-marketsignaling,forwhichhewonthe2001NobelPrizeineconomics.14Job-marketsignalingPlayer1isaworkerwhocanbeoneoftwotypes,high-skilledðt¼HÞorlow-skilledðt¼LÞ:Player2isaﬁrmthatconsidershiringtheapplicant.Alow-skilledworkeriscompletelyunproductiveandgeneratesnorevenuefortheﬁrm;ahigh-skilledworkergeneratesrevenueπ:Iftheapplicantishired,theﬁrmmustpaytheworkerw(thinkofthiswageasbeingﬁxedbygovernmentregulation).Assumeπ>w>0:Therefore,theﬁrmwishestohiretheapplicantifandonlyifheorsheishigh-skilled.Buttheﬁrmcannotobservetheapplicant’sskill;itcanobserveonlytheapplicant’sprioreducation.LetcHbethehightype’scostofobtaininganeducationandcLthelowtype’s.AssumecH0:Ifplayer2observesthatplayer1doesnotobtainaneduationthen2knowsitmustbeatnoden4ratherthann3,anditsbestresponseisnottoofferajob(NJ)because0>w:Thelaststepistogobackandcheckthatplayer1wouldnotwanttodeviatefromtheseparatingstrategyðEjH,NEjLÞgiventhat2playsðJjE,NJjNEÞ:TypeHofplayer1earnswcHbyobtaininganeducationinequilibrium.IftypeHdeviatesanddoesnotobtainaneducation,thenheorsheearns0becauseplayer2believesthat1istypeLanddoesnotofferajob.FortypeHnottoprefertodeviate,itmustbethatwcH0:NextturntotypeLofplayer1.TypeLearns0bynotobtaininganeducationinequilibrium.IftypeLdeviatesandobtainsaneducation,thenheorsheearnswcLbecauseplayer2believesthat1istypeHandoffersajob.FortypeLnottoprefertodeviate,wemusthavewcL0:Puttingtheseconditionstogether,thereisseparatingequilibriuminwhichtheworkerobtainsaneducationifandonlyifheorsheishigh-skilledandinwhichtheﬁrmoffersajobonlytoapplicantswithaneducationifandonlyifcHwcL:Anotherpossibleseparatingequilibriumisforplayer1toobtainaneducationifandonlyifheorsheislow-skilled.Thisisabizarreoutcome—sinceweexpecteducationtobeasignalofhighratherthanlowskill—andfortunatelywecanruleitoutasaperfectBayesianequilib-rium.Player2’sbestresponsewouldbetoofferajobifandonlyif1didnotobtainaneducation.TypeLwouldearncLfromplayingEandwfromplayingNE,soitwoulddeviatetoNE:QUERY:Whydoestheworkersometimesobtainaneducationeventhoughitdoesnotraisehisorherskilllevel?Wouldtheseparatingequilibriumexistifalow-skilledworkercouldobtainaneducationmoreeasilythanahigh-skilledone?EXAMPLE8.11PoolingEquilibriaintheJob-MarketSignalingGameLet’sinvestigateapossiblepoolingequilibriuminwhichbothtypesofplayer1chooseE:Forplayer1nottodeviatefromchoosingE,player2’sstrategymustbetoofferajobifandonlyiftheworkeriseducated—thatis,ðJjE,NJjNEÞ:If2doesn’tofferjobstoeducatedworkers,then1mightaswellsavethecostofobtaininganeducationandchooseNE:If2offersjobstouneducatedworkers,then1willagainchooseNEbecauseheorshesavesthecostofobtaininganeducationandstillearnsthewagefromthejoboffer.Next,weinvestigatewhenðJjE,NJjNEÞisabestresponsefor2.Player2’sposteriorbeliefsafterseeingEarethesameasitspriorbeliefsinthispoolingequilibrium.Player2’sexpectedpayofffromchoosingJisPrðHjEÞðπwÞþPrðLjEÞðwÞ¼PrðHÞðπwÞþPrðLÞðwÞ¼PrðHÞπw.(8.39)ForJtobeabestresponsetoE,Equation8.39mustexceed2’szeropayofffromchoosingNJ,whichuponrearrangingimpliesthatPrðHÞw=π:Player2’sposteriorbeliefsatnodesn3andn4arenotpinneddownbyBayes’rule,becauseNEisneverplayedin(continued)Chapter8StrategyandGameTheory277
EXAMPLE8.11CONTINUEDequilibriumandsoseeing1playNEisacompletelyunexpectedevent.PerfectBayesianequilibriumallowsustospecifyanyprobabilitydistributionwelikefortheposteriorbeliefsPrðHjNEÞatnoden3andPrðLjNEÞatnoden4:Player2’spayofffromchoosingNJis0.ForNJtobeabestresponsetoNE,0mustexceed2’sexpectedpayofffromplayingJ:0>PrðHjNEÞðπwÞþPrðLjNEÞðwÞ¼PrðHjNEÞπw,(8.40)wheretheright-handsidefollowsbecausePrðHjNEÞþPrðLjNEÞ¼1:RearrangingyieldsPrðHjNEÞw=π:Insum,inorderfortheretobeapoolingequilibriuminwhichbothtypesofplayer1obtainaneducation,weneedPrðHjNEÞw=πPrðHÞ:Theﬁrmhastobeoptimisticabouttheproportionofskilledworkersinthepopulation—PrðHÞmustbesufﬁcientlyhigh—andpessimisticabouttheskilllevelofuneducatedworkers—PrðHjNEÞmustbesufﬁcientlylow.Inthisequilibrium,typeLpoolswithtypeHinordertopreventplayer2fromlearninganythingabouttheworker’sskillfromtheeducationsignal.Theotherpossibilityforapoolingequilibriumisforbothtypesofplayer1tochooseNE:Thereareanumberofsuchequilibriadependingonwhatisassumedaboutplayer2’sposteriorbeliefsoutofequilibrium(thatis,2’sbeliefsafteritobserves1choosingE).PerfectBayesianequilibriumdoesnotplaceanyrestrictionsontheseposteriorbeliefs.Problem8.12asksyoutosearchforvariousoftheseequilibriaandintroducesafurtherreﬁnementofper-fectBayesianequilibrium(theintuitivecriterion)thathelpsruleoutunreasonableout-of-equilibriumbeliefsandthusimplausibleequilibria.QUERY:Returntothepoolingoutcomeinwhichbothtypesofplayer1obtainaneducation.Consider2’sposteriorbeliefsfollowingtheunexpectedeventthataworkershowsupwithnoeducation.PerfectBayesianequilibriumleavesusfreetoassumeanythingwewantabouttheseposteriorbeliefs.Supposeweassumethattheﬁrmobtainsnoinformationfromthe“noeducation”signalandsomaintainsitspriorbeliefs.Istheproposedpoolingoutcomeanequilibrium?Whatifweassumethattheﬁrmtakes“noeducation”asabadsignalofskill,believingthat1’stypeisLforcertain?EXAMPLE8.12HybridEquilibriaintheJob-MarketSignalingGameOnepossiblehybridequilibriumisfortypeHalwaystoobtainaneducationandfortypeLtorandomize,sometimespretendingtobeahightypebyobtaininganeducation.TypeLrandomizesbetweenplayingEandNEwithprobabilitieseand1e:Player2’sstrategyistoofferajobtoaneducatedapplicantwithprobabilityjandnottoofferajobtoanuneducatedapplicant.WeneedtosolvefortheequilibriumvaluesofthemixedstrategieseandjandtheposteriorbeliefsPrðHjEÞandPrðHjNEÞthatareconsistentwithperfectBayesianequilib-rium.TheposteriorbeliefsarecomputedusingBayes’rule:PrðHjEÞ¼PrðHÞPrðHÞþePrðLÞ¼PrðHÞPrðHÞþe½1PrðHÞ(8.41)andPrðHjNEÞ¼0:FortypeLofplayer1tobewillingtoplayastrictlymixedstrategy,heorshemustgetthesameexpectedpayofffromplayingE—whichequalsjwcL,given2’smixedstrategy—asfromplayingNE—whichequals0giventhatplayer2doesnotofferajobtouneducatedapplicants.HencejwcL¼0or,solvingforj,j¼cL=w:278Part2ChoiceandDemand
Player2willplayastrictlymixedstrategy(conditionalonobservingE)onlyifitgetsthesameexpectedpayofffromplayingJ,whichequalsPrðHjEÞðπwÞþPrðLjEÞðwÞ¼PrðHjEÞπw,(8.42)asfromplayingNJ,whichequals0.SettingEquation8.42equalto0,substitutingforPrðHjEÞfromEquation8.41,andthensolvingforegivese¼ðπwÞPrðHÞw½1PrðHÞ.(8.43)QUERY:Tocompleteouranalysis:inthisequilibrium,typeHofplayer1cannotprefertodeviatefromE:Isthistrue?Ifso,canyoushowit?HowdoestheprobabilityoftypeLtryingto“pool”withthehightypebyobtaininganeducationvarywithplayer2’spriorbeliefthatplayer1isthehightype?CheapTalkEducationisnothingmorethanacostlydisplayinthejob-marketsignalinggame.Thedisplaymustbecostly—indeed,itmustbemorecostlytothelow-skilledworker—orelsetheskilllevelscouldnotbeseparatedinequilibrium.Whilewedoseesomeinformationcommuni-catedthroughcostlydisplaysintherealworld,mostinformationiscommunicatedsimplybyhavingonepartytalktoanotheratlowornocost(“cheaptalk”).Gametheorycanhelpexplainwhycheaptalkisprevalentbutalsowhycheaptalksometimesfails,forcingpartiestoresorttocostlydisplays.Wewillmodelcheaptalkasatwo-playersignalinggameinwhichplayer1’sstrategyspaceconsistsofmessagessentcostlesslytoplayer2.Thetimingisotherwisethesameasbefore:player1ﬁrstlearnshistype(“stateoftheworld”mightbeabetterlabelthan“type”herebecauseplayer1’sprivateinformationwillenterbothplayers’payofffunctionsdirectly),player1communicatesto2,and2takessomeactionaffectingbothplayers’payoffs.Thespaceofmessagesispotentiallylimitless:player1canuseamoreorlesssophisticatedvocabulary,canwriteamoreorlessdetailedmessage,canspeakinanyofthethousandsoflanguagesintheworld,andsoforth.Sothesetofequilibriaisevenlargerthanwouldnormallybethecaseinsignalinggames.WewillanalyzetherangeofpossibleequilibriafromtheleasttothemostinformativeperfectBayesianequilibrium.Themaximumamountofinformationthatcanbecontainedinplayer1’smessagewilldependonhowwell-alignedtheplayers’payofffunctionsare.Player2wouldliketoknowthestateoftheworldbecauseshemighthavedifferentactionsthataresuitableindifferentsituations.Ifplayer1hasthesamepreferencesas2overwhichof2’sactionsarebestineachstateoftheworld,then1haseveryincentivetotell2preciselywhatthestateoftheworldis,and2haseveryreasontobelieve1’sreport.Ontheotherhand,iftheirpreferencesdiverge,then1wouldhaveanincentivetolieaboutthestateoftheworldtoinduce2totaketheactionthat1prefers.Ofcourse,2wouldanticipate1’slyingandwouldrefusetobelievethereport.Aspreferencesdiverge,messagesbecomelessandlessinformative.Inthelimit,1’smessagesarecompletelyuninformative(“babble”);tocommunicaterealinformation,player1wouldhavetoresorttocostlydisplays.Inthejob-marketsignalinggame,forexample,thepreferencesoftheworkerandﬁrmdivergewhentheworkerislow-skilled.Theworkerwouldliketobehiredandtheﬁrmwouldlikenottohiretheworker.Thehigh-skilledworkermustresorttothecostlydisplay(education)inordertosignalhisorhertype.Thereasonweseerelativelymorecheaptalkthancostlydisplaysintherealworldisprobablybecausepeopletrytoassociatewithotherswithwhomtheysharecommoninterestsandavoidthosewithwhomtheydon’t.Membersofafamily,playersonateam,orco-workerswithinaﬁrmtendtohavethesamegoalsandusuallyhavelittlereasontolietoeachChapter8StrategyandGameTheory279
other.Evenintheseexamples,players’interestsmaynotbecompletelyalignedandsocheaptalkmaynotbecompletelyinformative(thinkaboutteenagerstalkingtoparents).EXAMPLE8.13SimpleCheapTalkGameConsideragamewiththreestatesoftheworld:A,B,andC:Firstplayer1privatelyobservesthestate,then1sendsamessagetoplayer2,andthen2choosesanaction,LorR:Theinterestsofplayers1and2arealignedinstatesAandB:bothagreethat2shouldplayLinstateAandRinstateB:TheirinterestsdivergeinstateC:1prefers2toplayLand2preferstoplayR:AssumethatstatesAandBareequallylikely.LetdbetheprobabilityofstateC:Here,dmeasuresthedivergencebetweenplayers’preferences.Insteadoftheextensiveform,whichiscomplicatedbyhavingthreestatesandanill-deﬁnedmessagespaceforplayer1,thegameisrepresentedschematicallybythematricesinTable8.12.Ifd¼0thenplayers’incentivesarecompletelyaligned.Themostinformativeequilibriumresultsinperfectcommunication:1announcesthestatetruthfully;2playsLif1announces“A”andRif1announces“B”.16Ford>0;therecannotbeperfectcommunication.Ifcommunicationwereperfect,thenwhatevermessage1sendswhenthestateisAperfectlyrevealsthestateandsoleads2toplayL:Butthen1wouldhaveanincentivetoliewhenthetruestateisCandwouldthussendthesamemessageaswhenthestateisA:Player1’smessagescanbenomorereﬁnedthanissuingoneofthetwomessages“thestateiseitherAorC”or“thestateisB”;anyattempttodistinguishbetweenAandCwouldnotbebelieved.Ifthereisnottoomuchdivergencebetweenplayers’interests—inparticular,ifd1=3—thenthereisanequilibriumwithimperfectbutstillinformativecommunication.Inthisequilibrium,player1sendsoneoftwotruthfulmessages:“AorC”or“B.”Thenplayer2playsLconditionalonthemessage“AorC”andRconditionalon“B.”Player2’sexpectedpayofffromplayingLfollowingthemessage“AorC”equalsPrðAj“AorC”Þð1ÞþPrðCj“AorC”Þð0Þ¼PrðAj“AorC”Þ.(8.44)ByBayes’rule,PrðAj“AorC”Þ¼Prð“AorC”jAÞPrðAÞPrð“AorC”jAÞPrðAÞþPrð“AorC”jCÞPrðCÞ¼1d1þd.(8.45)TABLE8.12SimpleCheapTalkGamePlayer2Player1LRStateA1,10,0PrðAÞ¼ð1dÞ=2Player2Player1LRStateB0,01,1PrðBÞ¼ð1dÞ=2Player2Player1LRStateC0,11,0PrðCÞ¼d16Attheotherextreme,ford¼0andindeedforallparameters,thereisalwaysanuninformative“babbling”equilibriuminwhich1’smessagescontainnoinformationand2paysnoattentiontowhat1says.280Part2ChoiceandDemand
ThesecondequalityinEquation8.45holdsuponsubstitutingPrð“AorC”jAÞ¼Prð“AorC”jCÞ¼1(ifthestateisAorC,player1’sstrategyistoannounce“AorC”withcertainty)andsubstitutingthevaluesofPrðAÞandPrðCÞintermsofdfromTable8.12.Player2’sexpectedpayofffromdeviatingtoUcanbeshown(usingcalculationssimilartoEquations8.44and8.45)toequalPrðCj“AorC”Þ¼2d1þd.(8.46)Inequilibrium,Equation8.45mustexceedEquation8.46,implyingthatd1=3:Ifplayers’interestsareyetmoredivergent—inparticular,ifd>1=3—thenthereareonlyuninformative“babbling”equilibria.QUERY:Areplayersbetter-offinmoreinformativeequilibria?Whatdifferencewoulditmakeifplayer1announced“purple”insteadof“AorC”and“yellow”insteadof“B”?Whatfeaturesofalanguagewouldmakeitmoreorlessefﬁcientinacheap-talksetting?EXPERIMENTALGAMESExperimentaleconomicsisarecentbranchofresearchthatexploreshowwelleconomictheorymatchesthebehaviorofexperimentalsubjectsinlaboratorysettings.Themethodsaresimilartothoseusedinexperimentalpsychology—oftenconductedoncampususingunder-graduatesassubjects—althoughexperimentsineconomicstendtoinvolveincentivesintheformofexplicitmonetarypaymentspaidtosubjects.Theimportanceofexperimentaleco-nomicswashighlightedin2002,whenVernonSmithreceivedtheNobelPrizeineconomicsforhispioneeringworkintheﬁeld.Animportantareainthisﬁeldistheuseofexperimentalmethodstotestgametheory.ExperimentswiththePrisoners’DilemmaTherehavebeenhundredsoftestsofwhetherplayersﬁnkinthePrisoners’DilemmaaspredictedbyNashequilibriumorwhethertheyplaythecooperativeoutcomeofSilent.Inoneexperiment,subjectsplayedthegame20timeswitheachplayerbeingmatchedwithadifferent,anonymousopponenttoavoidrepeated-gameeffects.PlayconvergedtotheNashequilibriumassubjectsgainedexperiencewiththegame.Playersplayedthecooperativeaction43percentofthetimeintheﬁrstﬁverounds,fallingtoonly20percentofthetimeinthelastﬁverounds.17Asistypicalwithexperiments,subjects’behaviortendedtobenoisy.Although80percentofthedecisionswereconsistentwithNash-equilibriumplaybytheendoftheexperiment,still20percentofthemwereanomalous.Evenwhenexperimentalplayisroughlyconsistentwiththepredictionsoftheory,itisrarelyentirelyconsistent.ExperimentswiththeUltimatumGameExperimentaleconomicshasalsotestedtoseewhethersubgame-perfectequilibriumisagoodpredictorofbehaviorinsequentialgames.Inonewidelystudiedsequentialgame,theUltimatumGame,theexperimenterprovidesapotofmoneytotwoplayers.Theﬁrstmover(Proposer)proposesasplitofthispottothesecondmover.Thesecondmover(Responder)thendecideswhethertoaccepttheoffer,inwhichcaseplayersaregiventheamountofmoneyindicated,orrejecttheoffer,inwhichcasebothplayersgetnothing.Inthesubgame-perfect17R.Cooper,D.V.DeJong,R.Forsythe,andT.W.Ross,“CooperationWithoutReputation:ExperimentalEvidencefromPrisoner’sDilemmaGames,”GamesandEconomicBehavior(February1996):187–218.Chapter8StrategyandGameTheory281
equilibrium,theProposeroffersaminimalshareofthepotandthisisacceptedbytheResponder.Onecanseethisbyapplyingbackwardinduction:theRespondershouldacceptanypositivedivisionnomatterhowsmall;knowingthis,theProposershouldoffertheResponderonlyaminimalshare.Inexperiments,thedivisiontendstobemuchmoreeventhaninthesubgame-perfectequilibrium.18Themostcommonofferisa50–50split.Responderstendtorejectoffersgivingthemlessthan30percentofthepot.Thisresultisobservedevenwhenthepotisashighas$100,sothatrejectinga30percentoffermeansturningdown$30.Someeconomistshavesuggestedthatthemoneyplayersreceivemaynotbeatruemeasureoftheirpayoffs.Theymaycareaboutotherfactorssuchasfairnessandsoobtainabeneﬁtfromamoreequaldivisionofthepot.EvenifaProposerdoesnotcaredirectlyaboutfairness,thefearthattheRespondermaycareaboutfairnessandthusmightrejectanunevenofferoutofspitemayleadtheProposertoproposeanevensplit.ThedepartureofexperimentalbehaviorfromthepredictionsofgametheorywastoosystematicintheUltimatumGametobeattributedtonoisyplay,leadingsomegametheoriststorethinkthetheoryandaddanexplicitconsiderationforfairness.19ExperimentswiththeDictatorGameTotestwhetherplayerscaredirectlyaboutfairnessoractoutoffearoftheotherplayer’sspite,researchersexperimentedwitharelatedgame,theDictatorGame.IntheDictatorGame,theProposerchoosesasplitofthepot,andthissplitisimplementedwithoutinputfromtheResponder.ProposerstendtoofferalessevensplitthanintheUltimatumGamebutstilloffertheRespondersomeofthepot,suggestingthatRespondershavesomeresidualconcernforfairness.Thedetailsoftheexperimentaldesignarecrucial,however,asoneingeniousexperimentshowed.20TheexperimentwasdesignedsothattheexperimenterwouldneverlearnwhichProposershadmadewhichoffers.Withthiselementofanonymity,ProposersalmostnevergaveanequalsplittoRespondersandindeedtookthewholepotforthemselvestwothirdsofthetime.Proposersseemtocaremoreaboutappearingfairtotheexperimenterthantrulybeingfair.EVOLUTIONARYGAMESANDLEARNINGThefrontierofgame-theoryresearchregardswhetherandhowplayerscometoplayaNashequilibrium.Hyperrationalplayersmaydeduceeachothers’strategiesandinstantlysettleupontheNashequilibrium.HowcantheyinstantlycoordinateonasingleoutcomewhentherearemultipleNashequilibria?Whatoutcomewouldreal-worldplayers,forwhomhyper-rationaldeductionsmaybetoocomplex,settleon?Gametheoristshavetriedtomodelthedynamicprocessbywhichanequilibriumemergesoverthelongrunfromtheplayofalargepopulationofagentswhomeetothersatrandomandplayapairwisegame.GametheoristsanalyzewhetherplayconvergestoNashequilib-riumorsomeotheroutcome,whichNashequilibrium(ifany)isconvergedtoiftherearemultipleequilibria,andhowlongsuchconvergencetakes.Twomodels,whichmakevaryingassumptionsaboutthelevelofplayers’rationality,havebeenmostwidelystudied:anevolutionarymodelandalearningmodel.18ForareviewofUltimatumGameexperimentsandatextbooktreatmentofexperimentaleconomicsmoregenerally,seeD.D.DavisandC.A.Holt,ExperimentalEconomics(Princeton,NJ:PrincetonUniversityPress,1993).19See,forexample,M.Rabin,“IncorporatingFairnessintoGameTheoryandEconomics,”AmericanEconomicReview(December1993):1281–1302.20E.Hoffman,K.McCabe,K.Shachat,andV.Smith,“Preferences,PropertyRights,andAnonymityinBargainingGames,”GamesandEconomicBehavior(November1994):346–80.282Part2ChoiceandDemand
Intheevolutionarymodel,playersdonotmakerationaldecisions;instead,theyplaythewaytheyaregeneticallyprogrammed.Themoresuccessfulaplayer’sstrategyinthepopu-lation,themoreﬁtistheplayerandthemorelikelywilltheplayersurvivetopassitsgenesontofuturegenerationsandsothemorelikelythestrategyspreadsinthepopulation.EvolutionarymodelswereinitiallydevelopedbyJohnMaynardSmithandotherbiolo-giststoexplaintheevolutionofsuchanimalbehaviorashowhardalionﬁghtstowinamateoranantﬁghtstodefenditscolony.Whileitmaybemoreofastretchtoapplyevolutionarymodelstohumans,evolutionarymodelsprovideaconvenientwayofanalyzingpopulationdynamicsandmayhavesomedirectbearingonhowsocialconventionsarepasseddown,perhapsthroughculture.Inalearningmodel,playersareagainmatchedatrandomwithothersfromalargepopulation.Playersusetheirexperiencesofpayoffsfrompastplaytoteachthemhowothersareplayingandhowtheythemselvescanbestrespond.Playersusuallyareassumedtohaveadegreeofrationalityinthattheycanchooseastaticbestresponsegiventheirbeliefs,maydosomeexperimenting,andwillupdatetheirbeliefsaccordingtosomereasonablerule.Playersarenotfullyrationalinthattheydonotdistorttheirstrategiesinordertoaffectothers’learningandthusfutureplay.Gametheoristshaveinvestigatedwhethermore-orless-sophisticatedlearningstrategiesconvergemoreorlessquicklytoaNashequilibrium.Currentresearchseekstointegratetheorywithexperimentalstudy,tryingtoidentifythespeciﬁcalgorithmsthatreal-worldsubjectsusewhentheylearntoplaygames.SUMMARYThischapterprovidedastructuredwaytothinkaboutstra-tegicsituations.Wefocusedonthemostimportantsolutionconceptusedingametheory,Nashequilibrium.Wethenprogressedtoseveralmore-reﬁnedsolutionconceptsthatareinstandarduseingametheoryinmorecomplicatedsettings(withsequentialmovesandincompleteinformation).Someoftheprincipalresultsareasfollows.•Allgameshavethesamebasiccomponents:players,strat-egies,payoffs,andaninformationstructure.•Gamescanbewrittendowninnormalform(providingapayoffmatrixorpayofffunctions)orextensiveform(providingagametree).•Strategiescanbesimpleactions,morecomplicatedplanscontingentonothers’actions,orevenprobabilitydis-tributionsoversimpleactions(mixedstrategies).•ANashequilibriumisasetofstrategies,oneforeachplayer,thataremutualbestresponses.Inotherwords,aplayer’sstrategyinaNashequilibriumisoptimalgiventhatallothersplaytheirequilibriumstrategies.•ANashequilibriumalwaysexistsinﬁnitegames(inmixedifnotpurestrategies).•Subgame-perfectequilibriumisareﬁnementofNashequilibriumthathelpstoruleoutequilibriainsequentialgamesinvolvingnoncrediblethreats.•Repeatingastagegamealargenumberoftimesintro-ducesthepossibilityofusingpunishmentstrategiestoattainhigherpayoffsthanifthestagegameisplayedonce.Ifaﬁnitegamewithmultiplestagesisrepeatedoftenenoughorifplayersaresufﬁcientlypatientinaninﬁnitelyrepeatedgame,thenafolktheoremholdsimplyingthatessentiallyanypayoffsarepossibleintherepeatedgame.•Ingamesofprivateinformation,oneplayerknowsmoreabouthisorher“type”thananother.Playersmaximizetheirexpectedpayoffsgivenknowledgeoftheirowntypeandbeliefsabouttheothers’.•InaperfectBayesianequilibriumofasignalinggame,thesecondmoverusesBayes’ruletoupdatehisorherbeliefsabouttheﬁrstmover’stypeafterobservingtheﬁrstmover’saction.•Thefrontierofgame-theoryresearchcombinestheorywithexperimentstodeterminewhetherplayerswhomaynotbehyperrationalcometoplayaNashequilibrium,whichparticularequilibrium(iftherearemorethanone),andwhatpathleadstotheequilibrium.Chapter8StrategyandGameTheory283
PROBLEMS8.1Considerthefollowinggame:a.Findthepure-strategyNashequilibria(ifany).b.Findthemixed-strategyNashequilibriuminwhicheachplayerrandomizesoverjusttheﬁrsttwoactions.c.Computeplayers’expectedpayoffsintheequilibriafoundinparts(a)and(b).d.Drawtheextensiveformforthisgame.8.2Themixed-strategyNashequilibriumintheBattleoftheSexesinTable8.3maydependonthenumericalvaluesforthepayoffs.Togeneralizethissolution,assumethatthepayoffmatrixforthegameisgivenbywhereK1:Showhowthemixed-strategyNashequilibriumdependsonthevalueofK:8.3ThegameofChickenisplayedbytwomachoteenswhospeedtowardeachotheronasingle-laneroad.Theﬁrsttoveeroffisbrandedthechicken,whereastheonewhodoesn’tveergainspeer-groupesteem.Ofcourse,ifneitherveers,bothdieintheresultingcrash.PayoffstotheChickengameareprovidedinthefollowingtable.Player2DEFPlayer1A7,65,80,0B5,87,61,1C0,01,14,4Player2(Husband)BalletBoxingPlayer1ðWifeÞBalletK,10,0Boxing0,01,KTeen2VeerDon’tveerTeen1Veer2,21,3Don’tveer3,10,0284Part2ChoiceandDemand
a.Drawtheextensiveform.b.Findthepure-strategyNashequilibriumorequilibria.c.Computethemixed-strategyNashequilibrium.Aspartofyouranswer,drawthebest-responsefunctiondiagramforthemixedstrategies.d.Supposethegameisplayedsequentially,withteenAmovingﬁrstandcommittingtothisactionbythrowingawaythesteeringwheel.WhatareteenB’scontingentstrategies?Writedownthenormalandextensiveformsforthesequentialversionofthegame.e.Usingthenormalformforthesequentialversionofthegame,solvefortheNashequilibria.f.Identifythepropersubgamesintheextensiveformforthesequentialversionofthegame.Usebackwardinductiontosolveforthesubgame-perfectequilibrium.ExplainwhytheotherNashequilibriaofthesequentialgameare“unreasonable.”8.4Twoneighboringhomeowners,i¼1,2,simultaneouslychoosehowmanyhourslitospendmain-tainingabeautifullawn.Theaveragebeneﬁtperhouris10liþlj2,andthe(opportunity)costperhourforeachis4.Homeowneri’saveragebeneﬁtisincreasinginthehoursneighborjspendsonhisownlawn,sincetheappearanceofone’spropertydependsinpartonthebeautyofthesurroundingneighborhood.a.ComputetheNashequilibrium.b.Graphthebest-responsefunctionsandindicatetheNashequilibriumonthegraph.c.Onthegraph,showhowtheequilibriumwouldchangeiftheinterceptofoneoftheneighbor’saveragebeneﬁtfunctionsfellfrom6tosomesmallernumber.8.5TheAcademyAward–winningmovieABeautifulMindaboutthelifeofJohnNashdramatizesNash’sscholarlycontributioninasinglescene:hisequilibriumconceptdawnsonhimwhileinabarbanteringwithhisfellowmalegraduatestudents.Theynoticeseveralwomen,oneblondandtherestbrunette,andagreethattheblondismoredesirablethanthebrunettes.TheNashcharacterviewsthesituationasagameamongthemalegraduatestudents,alongthefollowinglines.Supposetherearenmaleswhosimultaneouslyapproacheithertheblondoroneofthebrunettes.Ifmaleialoneapproachestheblond,thenheissuccessfulingettingadatewithherandearnspayoffa:Ifoneormoreothermalesapproachtheblondalongwithi,thecompetitioncausesthemalltoloseher,andi(aswellastheotherswhoapproachedher)earnsapayoffofzero.Ontheotherhand,maleiearnsapayoffofb>0fromapproachingabrunette,sincetherearemorebrunettesthanmales,soiiscertaintogetadatewithabrunette.Thedesirabilityoftheblondimpliesa>b:a.Arguethatthisgamedoesnothaveasymmetricpure-strategyNashequilibrium.b.Solveforthesymmetricmixed-strategyequilibrium.Thatis,lettingpbetheprobabilitythatamaleapproachestheblond,ﬁndp.c.Showthatthemoremalesthereare,thelesslikelyitisintheequilibriumfrompart(b)thattheblondisapproachedbyatleastoneofthem.Note:ThisparadoxicalresultwasnotedbyS.AndersonandM.Engersin“ParticipationGames:MarketEntry,Coordination,andtheBeautifulBlond,”JournalofEconomicBehavior&Organization63(2007):120–37.Chapter8StrategyandGameTheory285
8.6Considerthefollowingstagegame.a.Computeaplayer’sminmaxvalueiftherivalisrestrictedtopurestrategies.Isthisminmaxvaluedifferentthaniftherivalisallowedtousemixedstrategies?b.Supposethestagegameisplayedtwice.Characterizethesubgame-perfectequilibriumprovid-ingthehighesttotalpayoffs.c.Drawagraphofthesetoffeasibleper-periodpayoffsinthelimitinaﬁnitelyrepeatedgameaccordingtothefolktheorem.8.7ReturntothegamewithtwoneighborsinProblem8.5.Continuetosupposethatplayeri’saveragebeneﬁtperhourofworkonlandscapingis10liþlj2.Continuetosupposethatplayer2’sopportunitycostofanhouroflandscapingworkis4.Supposethat1’sopportunitycostiseither3or5withequalprobabilityandthatthiscostis1’sprivateinformation.a.SolvefortheBayesian-Nashequilibrium.b.IndicatetheBayesian-Nashequilibriumonabest-responsefunctiondiagram.c.Whichtypeofplayer1wouldliketosendatruthfulsignalto2ifitcould?Whichtypewouldliketohideitsprivateinformation?8.8InBlindTexanPoker,player2drawsacardfromastandarddeckandplacesitagainstherforeheadwithoutlookingatitbutsoplayer1canseeit.Player1movesﬁrst,decidingwhethertostayorfold.Ifplayer1folds,hemustpayplayer2$50.Ifplayer1stays,theactiongoestoplayer2.Player2canfoldorcall.Ifplayer2folds,shemustpayplayer1$50.If2calls,thecardisexamined.Ifitisalowcard(2through8),player2paysplayer1$100.Ifitisahighcard(9,10,jack,queen,king,orace),player1paysplayer2$100.a.Drawtheextensiveformforthegame.b.Solveforthehybridequilibrium.c.Computetheplayers’expectedpayoffs.Player2ABCPlayer1A10,101,151,12B15,10,01,1C12,11,18,8286Part2ChoiceandDemand
AnalyticalProblems8.9DominantstrategiesProvethatanequilibriumindominantstrategiesistheuniqueNashequilibrium.8.10RottenKidTheoremInATreatiseontheFamily(Cambridge,MA:HarvardUniversityPress,1981),NobellaureateGaryBeckerproposeshisfamousRottenKidTheoremasasequentialgamebetweenthepotentiallyrottenchild(player1)andthechild’sparent(player2).Thechildmovesﬁrst,choosinganactionrthataffectshisownincomeY1ðrÞ½Y01ðrÞ>0andtheincomeoftheparentY2ðrÞ½Y02ðrÞ<0:Later,theparentmoves,leavingamonetarybequestLtothechild.Thechildcaresonlyforhisownutility,U1ðY1þLÞ,buttheparentmaximizesU2ðY2LÞþαU1,whereα>0reﬂectstheparent’saltruismtowardthechild.Provethat,inasubgame-perfectequilibrium,thechildwilloptforthevalueofrthatmaximizesY1þY2eventhoughhehasnoaltruisticintentions.Hint:Applybackwardinductiontotheparent’sproblemﬁrst,whichwillgiveaﬁrst-orderconditionthatimplicitlydeterminesL;althoughanexplicitsolutionforLcannotbefound,thederivativeofLwithrespecttor—requiredinthechild’sﬁrst-stageoptimizationproblem—canbefoundusingtheimplicitfunctionrule.8.11AlternativestoGrimStrategySupposethatthePrisoners’Dilemmastagegame(seeTable8.1)isrepeatedforinﬁnitelymanyperiods.a.Canplayerssupportthecooperativeoutcomebyusingtit-for-tatstrategies,punishingdeviationinapastperiodbyrevertingtothestage-gameNashequilibriumforjustoneperiodandthenreturningtocooperation?Aretwoperiodsofpunishmentenough?b.Supposeplayersusestrategiesthatpunishdeviationfromcooperationbyrevertingtothestage-gameNashequilibriumfortenperiodsbeforereturningtocooperation.Computethethresholddiscountfactorabovewhichcooperationispossibleontheoutcomethatmaximizesthejointpayoffs.8.12RefinementsofperfectBayesianequilibriumRecallthejob-marketsignalinggameinExample8.11.a.Findtheconditionsunderwhichthereisapoolingequilibriumwherebothtypesofworkerchoosenottoobtainaneducation(NE)andwheretheﬁrmoffersanuneducatedworkerajob.Besuretospecifybeliefsaswellasstrategies.b.Findtheconditionsunderwhichthereisapoolingequilibriumwherebothtypesofworkerchoosenottoobtainaneducation(NE)andwheretheﬁrmdoesnotofferanuneducatedworkerajob.Whatisthelowestposteriorbeliefthattheworkerislow-skilledconditionalonobtaininganeducationconsistentwiththispoolingequilibrium?Whyisitmorenaturaltothinkthatalow-skilledworkerwouldneverdeviatetoEandsoaneducatedworkermustbehigh-skilled?ChoandKreps’sintuitivecriterionisoneofaseriesofcomplicatedreﬁnementsofperfectBayesianequilibriumthatruleoutequilibriabasedonunreasonableposteriorbeliefsasidentiﬁedinthispart;seeI.K.ChoandD.M.Kreps,“SignallingGamesandStableEquilibria,”QuarterlyJournalofEconomics102(1987):179–221.SUGGESTIONSFORFURTHERREADINGFudenberg,D.,andJ.Tirole.GameTheory.Cambridge,MA:MITPress,1991.Acomprehensivesurveyofgametheoryatthegraduate-studentlevel,thoughselectedsectionsareaccessibletoadvancedundergraduates.Holt,C.A.Markets,Games,&StrategicBehavior.Boston:Pearson,2007.Anundergraduatetextwithemphasisonexperimentalgames.Rasmusen,E.GamesandInformation,4thed.Malden,MA:Blackwell,2007.Anadvancedundergraduatetextwithmanyreal-worldapplications.Watson,Joel.Strategy:AnIntroductiontoGameTheory.NewYork:Norton,2002.Anundergraduatetextthatbalancesrigorwithsimpleexamples(often22games).Emphasisonbargainingandcontractingexamples.Chapter8StrategyandGameTheory287
EXTENSIONSExistenceofNashEquilibriumThissectionwillsketchJohnNash’soriginalproofthatallﬁnitegameshaveatleastoneNashequilibrium(inmixedifnotinpurestrategies).Wewillprovidesomeofthedetailsoftheproofhere;theoriginalproofisinNash(1950),andacleartextbookpresentationofthefullproofisprovidedinFudenbergandTirole(1991).Thesectionconcludesbymentioningarelatedexis-tencetheoremforgameswithcontinuousactions.Nash’sproofissimilartotheproofoftheexistenceofageneralcompetitiveequilibriuminChapter13.Bothproofsrelyonaﬁxedpointtheorem.TheproofoftheexistenceofNashequilibriumrequiresaslightlymorepowerfultheorem.InsteadofBrouwer’sﬁxedpointtheorem,whichappliestofunctions,Nash’sproofreliesonKakutani’sﬁxedpointtheorem,whichappliestocorrespondences—moregeneralmappingsthanfunctions.E8.1CorrespondencesversusfunctionsAfunctionmapseachpointinaﬁrstsettoasinglepointinasecondset.Acorrespondencemapsasinglepointintheﬁrstsettopossiblymanypointsinthesecondset.FigureE8.1illustratesthedifference.FIGUREE8.1ComparisionofFunctionsandCorrespondencesThefunctiongraphedin(a)lookslikeafamiliarcurve.Eachvalueofxismappedintoasinglevalueofy.Withthecorrespondencegraphedin(b),eachvalueofxmaybemappedintomanyvaluesofy.Correspondencescanthushavebulgesasshownbythegrayregionsin(b).(a) Function(b) Correspondenceyxyx288Part2ChoiceandDemand
Anexampleofacorrespondencethatwehavealreadyseenisthebestresponse,BRiðsiÞ:Thebestresponseneednotmapotherplayers’strategiessiintoasinglestrategythatisabestresponseforplayeri:Theremaybetiesamongseveralbestresponses.AsshowninFigure8.3,intheBattleoftheSexes,thehusband’sbestresponsetothewife’splayingthemixedstrategyofgoingtoballetwithprobability2=3andboxingwithprobability1=3(orjustw¼2=3forshort)isnotjustasinglepointbutthewholeintervalofpossiblemixedstrategies.Boththehusband’sandthewife’sbestresponsesinthisﬁgurearecorrespon-dences,notfunctions.ThereasonNashneededaﬁxedpointtheoremin-volvingcorrespondencesratherthanjustfunctionsispreciselybecausehisproofworkswithplayers’bestresponsestoproveexistence.E8.2Kakutani’sfixedpointtheoremHereisthestatementofKakutani’sﬁxedpointtheorem:Anyconvex,upper-semicontinuouscorrrespondence½fðxÞfromaclosed,bounded,convexsetintoitselfhasatleastoneﬁxedpointðxÞsuchthatx2fðxÞ:ComparingthestatementofKakutani’sﬁxedpointtheoremwithBrouwer’sinChapter13,theyaresimilarexceptforthesubstitutionof“correspondence”for“function”andfortheconditionsonthecorrespon-dence.Brouwer’stheoremrequiresthefunctiontobecontinuous;Kakutani’stheoremrequiresthecorre-spondencetobeconvexanduppersemicontinuous.Theseproperties,whicharerelatedtocontinuity,arelessfamiliarandworthspendingamomenttounderstand.FigureE8.2providesexamplesofcorre-spondencesviolating(a)convexityand(b)uppersemi-continuity.Theﬁgureshowswhythetwopropertiesareneededtoguaranteeaﬁxedpoint.Withoutbothproperties,thecorrespondencecan“jump”acrossthe45°lineandsofailtohaveaﬁxedpoint—thatis,apointforwhichx¼fðxÞ:E8.3Nash’sproofWeuseRðsÞtodenotethecorrespondencethatunder-liesNash’sexistenceproof.Thiscorrespondencetakesanyproﬁleofplayers’strategiess¼ðs1,s2,…,snÞ(pos-siblymixed)andmapsitintoanothermixedstrategyproﬁle,theproﬁleofbestresponses:RðsÞ¼ðBR1ðs1Þ,BR2ðs2Þ,…,BRnðsnÞÞ.(i)Aﬁxedpointofthecorrespondenceisastrategyforwhichs2RðsÞ;thisisaNashequilibriumbecauseeachplayer’sstrategyisabestresponsetoothers’strategies.TheproofchecksthatalltheconditionsinvolvedinKakutani’sﬁxedpointtheoremaresatisﬁedbythebest-responsecorrespondenceRðsÞ:First,weneedtoshowthatthesetofmixed-strategyproﬁlesisclosed,bounded,andconvex.Sinceastrategyproﬁleisjustalistofindividualstrategies,thesetofstrategyproﬁleswillbeclosed,bounded,andconvexifeachplayer’sstrategysetSihasthesepropertiesindividually.AsFigureE8.3showsforthecaseoftwoandthreeactions,thesetofmixedstrategiesoveractionshasasimpleshape.1Thesetisclosed(containsitsbound-ary),bounded(doesnotgoofftoinﬁnityinanydirec-tion),andconvex(thesegmentbetweenanytwopointsinthesetisalsointheset).Wethenneedtocheckthatthebest-responsecorrespondenceRðsÞisconvex.Individualbestre-sponsescannotlooklike(a)inFigureE8.2,becauseifanytwomixedstrategiessuchasAandBarebestresponsestoothers’strategiesthenmixedstrategiesbetweenthemmustalsobebestresponses.Forexam-ple,intheBattleoftheSexes,if(1=3,2=3)and(2=3,1=3)arebestresponsesforthehusbandagainsthiswife’splaying(2=3,1=3)(where,ineachpair,theﬁrstnumberistheprobabilityofplayingballetandthesecondofplayingboxing),thenmixedstrategiesbe-tweenthetwosuchas(1=2,1=2)mustalsobebestresponsesforhim.Figure8.3showedthatinfactallpossiblemixedstrategiesforthehusbandarebestresponsestothewife’splaying(2=3,1=3).Finally,weneedtocheckthatRðsÞisuppersemi-continuous.Individualbestresponsescannotlooklike(b)inFigureE8.2.TheycannothaveholeslikepointDpunchedoutofthembecausepayofffunc-tionsuiðsi,siÞarecontinuous.Recallthatpayoffs,whenwrittenasfunctionsofmixedstrategies,areactuallyexpectedvalueswithprobabilitiesgivenbythestrategiessiandsi:AsEquation2.176showed,expectedvaluesarelinearfunctionsoftheunderlyingprobabilities.Linearfunctionsareofcoursecontinuous.1Mathematiciansstudythemsofrequentlythattheyhaveaspecialnameforsuchaset:asimplex.Chapter8StrategyandGameTheory289
FIGUREE8.2Kakutani’sConditionsonCorrespondencesThecorrespondencein(a)isnotconvexbecausethedashedverticalsegmentbetweenAandBisnotinsidethecorrespondence.Thecorrespondencein(b)isnotuppersemicontinuousbecausethereisapath(C)insidethecorrespondenceleadingtoapoint(D)that,asindicatedbytheopencircle,isnotinsidethecorrespondence.Both(a)and(b)failtohaveﬁxedpoints.(b) Correspondence that is not upper semicontinuous1D45°(a) Correspondence that is not convexf(x)11xf(x)xBA45°C290Part2ChoiceandDemand
E8.4GameswithcontinuousactionsNash’sexistencetheoremappliestoﬁnitegames—thatis,gameswithaﬁnitenumberofplayersandactionsperplayer.Nash’stheoremdoesnotapplytogames,suchastheTragedyoftheCommonsinEx-ample8.6,thatfeaturecontinuousactions.IsaNashequilibriumguaranteedtoexistforthesegames,too?Glicksberg(1952)provedthattheansweris“yes”aslongaspayofffunctionsarecontinuous.ReferencesFudenberg,D.,andJ.Tirole.GameTheory.Cambridge,MA:MITPress,1991,sec.1.3.Glicksberg,I.L.“AFurtherGeneralizationoftheKakutaniFixedPointTheoremwithApplicationtoNashEquilib-riumPoints.”ProceedingsoftheNationalAcademyofSciences38(1952):170–74.Nash,John.“EquilibriumPointsinn-PersonGames.”ProceedingsoftheNationalAcademyofSciences36(1950):48–49.FIGUREE8.3SetofMixedStrategiesforanIndividualPlayer1’ssetofpossiblemixedstrategiesovertwoactionsisgivenbythediagonallinesegmentin(a).Thesetforthreeactionsisgivenbytheshadedtriangleonthethree-dimensionalgraphin(b).(a) Two actions(b) Three actions101111p11p11p12p12p130Chapter8StrategyandGameTheory291
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PART3ProductionandSupplyCHAPTER9ProductionFunctionsCHAPTER10CostFunctionsCHAPTER11ProﬁtMaximizationInthispartweexaminetheproductionandsupplyofeconomicgoods.Institutionsthatcoordinatethetransformationofinputsintooutputsarecalledﬁrms.Theymaybelargeinstitutions(suchasMicrosoft,Sony,ortheU.S.DepartmentofDefense)orsmallones(suchas“MomandPop”storesorself-employedindividuals).Althoughtheymaypursuedifferentgoals(Microsoftmayseekmaximumproﬁts,whereasanIsraelikibbutzmaytrytomakemembersofthekibbutzaswelloffaspossible),allﬁrmsmustmakecertainbasicchoicesintheproductionprocess.ThepurposeofPart3istodevelopsometoolsforanalyzingthosechoices.InChapter9weexaminewaysofmodelingthephysicalrelationshipbetweeninputsandoutputs.Weintroducetheconceptofaproductionfunction,ausefulabstractionfromthecomplexitiesofreal-worldproductionprocesses.Twomeasurableaspectsoftheproductionfunctionarestressed:itsreturnstoscale(thatis,howoutputexpandswhenallinputsareincreased)anditselasticityofsubstitution(thatis,howeasilyoneinputmaybereplacedbyanotherwhilemaintainingthesamelevelofoutput).Wealsobrieﬂydescribehowtechnicalimprovementsarereﬂectedinproductionfunctions.TheproductionfunctionconceptisthenusedinChapter10todiscusscostsofproduction.Weassumethatallﬁrmsseektoproducetheiroutputatthelowestpossiblecost,anassumptionthatpermitsthedevel-opmentofcostfunctionsfortheﬁrm.Chapter10alsofocusesonhowcostsmaydifferbetweentheshortrunandthelongrun.InChapter11weinvestigatetheﬁrm’ssupplydecision.Todoso,weassumethattheﬁrm’smanagerwillmakeinputandoutputchoicessoastomaximizeproﬁts.Thechapterconcludeswiththefundamentalmodelofsupplybehaviorbyproﬁt-maximizingﬁrmsthatwewilluseinmanysubsequentchapters.293
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CHAPTER9ProductionFunctionsTheprincipalactivityofanyﬁrmistoturninputsintooutputs.Becauseeconomistsareinterestedinthechoicestheﬁrmmakesinaccomplishingthisgoal,butwishtoavoiddiscussingmanyoftheengineeringintricaciesinvolved,theyhavechosentoconstructanabstractmodelofproduction.Inthismodeltherelationshipbetweeninputsandoutputsisformalizedbyaproductionfunctionoftheformq¼fðk,l,m,…Þ,(9.1)whereqrepresentstheﬁrm’soutputofaparticulargoodduringaperiod,1krepresentsthemachine(thatis,capital)usageduringtheperiod,lrepresentshoursoflaborinput,mrepresentsrawmaterialsused,2andthenotationindicatesthepossibilityofothervariablesaﬀectingtheproductionprocess.Equation9.1isassumedtoprovide,foranyconceivablesetofinputs,theengineer’ssolutiontotheproblemofhowbesttocombinethoseinputstogetoutput.MARGINALPRODUCTIVITYInthissectionwelookatthechangeinoutputbroughtaboutbyachangeinoneoftheproductiveinputs.Forthepurposesofthisexamination(andindeedformostofthepurposesofthisbook),itwillbemoreconvenienttouseasimpliﬁedproductionfunctiondeﬁnedasfollows.DEFINITIONProductionfunction.Theﬁrm’sproductionfunctionforaparticulargood,q,q¼fðk,lÞ,(9.2)showsthemaximumamountofthegoodthatcanbeproducedusingalternativecom-binationsofcapitalðkÞandlaborðlÞ.Ofcourse,mostofouranalysiswillholdforanytwoinputstotheproductionprocesswemightwishtoexamine.Thetermscapitalandlaborareusedonlyforconvenience.Similarly,itwouldbeasimplemattertogeneralizeourdiscussiontocasesinvolvingmorethantwoinputs;occasionally,wewilldoso.Forthemostpart,however,limitingthediscussiontotwoinputswillbequitehelpfulbecausewecanshowtheseinputsontwo-dimensionalgraphs.MarginalphysicalproductTostudyvariationinasingleinput,wedeﬁnemarginalphysicalproductasfollows.1Hereweusealowercaseqtorepresentoneﬁrm’soutput.WereservetheuppercaseQtorepresenttotaloutputinamarket.Generally,weassumethataﬁrmproducesonlyoneoutput.Issuesthatariseinmultiproductﬁrmsarediscussedinafewfootnotesandproblems.2Inempiricalworkrawmaterialinputsoftenaredisregardedandoutput,q,ismeasuredintermsof“valueadded.”295
DEFINITIONMarginalphysicalproduct.Themarginalphysicalproductofaninputistheadditionaloutputthatcanbeproducedbyemployingonemoreunitofthatinputwhileholdingallotherinputsconstant.Mathematically,marginalphysicalproductofcapital¼MPk¼∂q∂k¼fk,marginalphysicalproductoflabor¼MPl¼∂q∂l¼fl.(9.3)Noticethatthemathematicaldeﬁnitionsofmarginalproductusepartialderivatives,therebyproperlyreﬂectingthefactthatallotherinputusageisheldconstantwhiletheinputofinterestisbeingvaried.Foranexample,considerafarmerhiringonemorelaborertoharvestthecropbutholdingallotherinputsconstant.Theextraoutputthislaborerproducesisthatfarmhand’smarginalphysicalproduct,measuredinphysicalquantities,suchasbushelsofwheat,cratesoforanges,orheadsoflettuce.Wemightobserve,forexample,that50workersonafarmareabletoproduce100bushelsofwheatperyear,whereas51workers,withthesamelandandequipment,canproduce102bushels.Themarginalphysicalproductofthe51stworkeristhen2bushelsperyear.DiminishingmarginalproductivityWemightexpectthatthemarginalphysicalproductofaninputdependsonhowmuchofthatinputisused.Labor,forexample,cannotbeaddedindeﬁnitelytoagivenﬁeld(whilekeepingtheamountofequipment,fertilizer,andsoforthﬁxed)withouteventuallyexhibitingsomedeteriorationinitsproductivity.Mathematically,theassumptionofdiminishingmar-ginalphysicalproductivityisanassumptionaboutthesecond-orderpartialderivativesoftheproductionfunction:∂MPk∂k¼∂2f∂k2¼fkk¼f11<0,∂MPl∂l¼∂2f∂l2¼fll¼f22<0:(9.4)Theassumptionofdiminishingmarginalproductivitywasoriginallyproposedbythenineteenth-centuryeconomistThomasMalthus,whoworriedthatrapidincreasesinpopulationwouldresultinlowerlaborproductivity.Hisgloomypredictionsforthefutureofhumanityledeconomicstobecalledthe“dismalscience.”Butthemathematicsoftheproductionfunctionsuggeststhatsuchgloommaybemisplaced.Changesinthemarginalproductivityoflaborovertimedependnotonlyonhowlaborinputisgrowing,butalsoonchangesinotherinputs,suchascapital.Thatis,wemustalsobeconcernedwith∂MPl=∂k¼flk.Inmostcases,flk>0,sodeclininglaborproductivityasbothlandkincreaseisnotaforegoneconclusion.Indeed,itappearsthatlaborproductivityhasrisensigniﬁcantlysinceMalthus’time,primarilybecauseincreasesincapitalinputs(alongwithtechnicalimprovements)haveoffsettheimpactofdiminishingmarginalproductivityalone.AveragephysicalproductivityIncommonusage,thetermlaborproductivityoftenmeansaverageproductivity.Whenitissaidthatacertainindustryhasexperiencedproductivityincreases,thisistakentomeanthatoutputperunitoflaborinputhasincreased.Althoughtheconceptofaverageproductivityisnotnearlyasimportantintheoreticaleconomicdiscussionsasmarginalproductivityis,itreceivesagreatdealofattentioninempiricaldiscussions.Becauseaverageproductivityis296Part3ProductionandSupply
easilymeasured(say,assomanybushelsofwheatperlabor-hourinput),itisoftenusedasameasureofefﬁciency.Wedeﬁnetheaverageproductoflabor(APl)tobeAPl¼outputlaborinput¼ql¼fðk,lÞl.(9.5)NoticethatAPlalsodependsonthelevelofcapitalemployed.Thisobservationwillprovetobequiteimportantwhenweexaminethemeasurementoftechnicalchangeattheendofthischapter.EXAMPLE9.1ATwo-InputProductionFunctionSupposetheproductionfunctionforﬂyswattersduringaparticularperiodcanberepresentedbyq¼fðk,lÞ¼600k2l2k3l3.(9.6)Toconstructthemarginalandaverageproductivityfunctionsoflabor(l)forthisfunction,wemustassumeaparticularvaluefortheotherinput,capital(k).Supposek¼10.Thentheproductionfunctionisgivenbyq¼60,000l21,000l3.(9.7)Marginalproduct.Themarginalproductivityfunction(whenk¼10)isgivenbyMPl¼∂q∂l¼120,000l3,000l2,(9.8)whichdiminishesaslincreases,eventuallybecomingnegative.Thisimpliesthatqreachesamaximumvalue.SettingMPlequalto0,120,000l3,000l2¼0(9.9)yields40l¼l2(9.10)orl¼40(9.11)asthepointatwhichqreachesitsmaximumvalue.Laborinputbeyond40unitsperperiodactuallyreducestotaloutput.Forexample,whenl¼40,Equation9.7showsthatq¼32millionﬂyswatters,whereaswhenl¼50,productionofﬂyswattersamountstoonly25million.Averageproduct.Toﬁndtheaverageproductivityoflaborinﬂyswatterproduction,wedivideqbyl,stillholdingk¼10:APl¼ql¼60,000l1,000l2.(9.12)Again,thisisaninvertedparabolathatreachesitsmaximumvaluewhen∂APl∂l¼60,0002,000l¼0,(9.13)whichoccurswhenl¼30.Atthisvalueforlaborinput,Equation9.12showsthatAPl¼900,000,andEquation9.8showsthatMPlisalso900,000.WhenAPlisatamaximum,averageandmarginalproductivitiesoflaborareequal.3(continued)3Thisresultisquitegeneral.Because∂APl∂l¼l⋅MPlql2,atamaximumlMPl¼qorMPl¼APl.Chapter9ProductionFunctions297
EXAMPLE9.1CONTINUEDNoticetherelationshipbetweentotaloutputandaverageproductivitythatisillustratedbythisexample.Eventhoughtotalproductionofﬂyswattersisgreaterwith40workers(32million)thanwith30workers(27million),outputperworkerishigherinthesecondcase.With40workers,eachworkerproduces800,000ﬂyswattersperperiod,whereaswith30workerseachworkerproduces900,000.Becausecapitalinput(ﬂyswatterpresses)isheldconstantinthisdeﬁnitionofproductivity,thediminishingmarginalproductivityoflaboreventuallyresultsinadeclininglevelofoutputperworker.QUERY:Howwouldanincreaseinkfrom10to11affecttheMPlandAPlfunctionshere?Explainyourresultsintuitively.ISOQUANTMAPSANDTHERATEOFTECHNICALSUBSTITUTIONToillustratepossiblesubstitutionofoneinputforanotherinaproductionfunction,weuseitsisoquantmap.Again,westudyaproductionfunctionoftheformq¼fðk,lÞ,withtheunderstandingthat“capital”and“labor”aresimplyconvenientexamplesofanytwoinputsthatmighthappentobeofinterest.Anisoquant(fromiso,meaning“equal”)recordsthosecombinationsofkandlthatareabletoproduceagivenquantityofoutput.Forexample,allthosecombinationsofkandlthatfallonthecurvelabeled“q¼10”inFigure9.1arecapableofproducing10unitsofoutputperperiod.Thisisoquantthenrecordsthefactthattherearemanyalternativewaysofproducing10unitsofoutput.OnewaymightberepresentedbypointA:WewoulduselAandkAtoproduce10unitsofoutput.Alternatively,wemightpreferFIGURE9.1AnIsoquantMapIsoquantsrecordthealternativecombinationsofinputsthatcanbeusedtoproduceagivenlevelofoutput.Theslopeofthesecurvesshowstherateatwhichlcanbesubstitutedforkwhilekeepingoutputconstant.Thenegativeofthisslopeiscalledthe(marginal)rateoftechnicalsubstitution(RTS).Intheﬁgure,theRTSispositiveanddiminishingforincreasinginputsoflabor.k per periodl per periodkAlAlBkBABq=30q=20q=10298Part3ProductionandSupply
touserelativelylesscapitalandmorelaborandthereforewouldchooseapointsuchasB.Hence,wemaydeﬁneanisoquantasfollows.DEFINITIONIsoquant.Anisoquantshowsthosecombinationsofkandlthatcanproduceagivenlevelofoutput(say,q0).Mathematically,anisoquantrecordsthesetofkandlthatsatisﬁesfðk,lÞ¼q0.(9.14)Aswasthecaseforindifferencecurves,thereareinﬁnitelymanyisoquantsinthek–lplane.Eachisoquantrepresentsadifferentlevelofoutput.Isoquantsrecordsuccessivelyhigherlevelsofoutputaswemoveinanortheasterlydirection.Presumably,usingmoreofeachoftheinputswillpermitoutputtoincrease.Twootherisoquants(forq¼20andq¼30)areshowninFigure9.1.Youwillnoticethesimilaritybetweenanisoquantmapandtheindividual’sindifferencecurvemapdiscussedinPart2.Theyareindeedsimilarconcepts,becausebothrepresent“contour”mapsofaparticularfunction.Forisoquants,however,thelabelingofthecurvesismeasurable—anoutputof10unitsperperiodhasaquantiﬁablemeaning.Economistsarethereforemoreinterestedinstudyingtheshapeofproductionfunctionsthaninexaminingtheexactshapeofutilityfunctions.Themarginalrateoftechnicalsubstitution(RTS)Theslopeofanisoquantshowshowoneinputcanbetradedforanotherwhileholdingoutputconstant.Examiningtheslopeprovidesinformationaboutthetechnicalpossibilityofsubstitutinglaborforcapital.Aformaldeﬁnitionfollows.DEFINITIONMarginalrateoftechnicalsubstitution.Themarginalrateoftechnicalsubstitution(RTS)showstherateatwhichlaborcanbesubstitutedforcapitalwhileholdingoutputconstantalonganisoquant.Inmathematicalterms,RTSðlforkÞ¼dkdlq¼q0.(9.15)Inthisdeﬁnition,thenotationisintendedasareminderthatoutputistobeheldconstantaslissubstitutedfork.Theparticularvalueofthistrade-offratewilldependnotonlyonthelevelofoutputbutalsoonthequantitiesofcapitalandlaborbeingused.Itsvaluedependsonthepointontheisoquantmapatwhichtheslopeistobemeasured.RTSandmarginalproductivitiesToexaminetheshapeofproductionfunctionisoquants,itisusefultoprovethefollowingresult:theRTS(oflfork)isequaltotheratioofthemarginalphysicalproductivityoflabor(MPl)tothemarginalphysicalproductivityofcapital(MPk).Webeginbysettingupthetotaldifferentialoftheproductionfunction:dq¼∂f∂l⋅dlþ∂f∂k⋅dk¼MPl⋅dlþMPk⋅dk,(9.16)whichrecordshowsmallchangesinlandkaffectoutput.Alonganisoquant,dq¼0(outputisconstant),soMPl⋅dl¼MPk⋅dk.(9.17)Thissaysthatalonganisoquant,thegaininoutputfromincreasinglslightlyisexactlybalancedbythelossinoutputfromsuitablydecreasingk.Rearrangingtermsabitgivesdkdlq¼q0¼RTSðlforkÞ¼MPlMPk.(9.18)HencetheRTSisgivenbytheratiooftheinputs’marginalproductivities.Chapter9ProductionFunctions299
Equation9.18showsthatthoseisoquantsthatweactuallyobservemustbenegativelysloped.BecausebothMPlandMPkwillbenonnegative(noﬁrmwouldchoosetouseacostlyinputthatreducedoutput),theRTSalsowillbepositive(orperhapszero).BecausetheslopeofanisoquantisthenegativeoftheRTS,anyﬁrmweobservewillnotbeoperatingonthepositivelyslopedportionofanisoquant.Althoughitismathematicallypossibletodeviseproductionfunctionswhoseisoquantshavepositiveslopesatsomepoints,itwouldnotmakeeconomicsenseforaﬁrmtooptforsuchinputchoices.ReasonsforadiminishingRTSTheisoquantsinFigure9.1aredrawnnotonlywithanegativeslope(astheyshouldbe)butalsoasconvexcurves.Alonganyoneofthecurves,theRTSisdiminishing.Forhighratiosofktol,theRTSisalargepositivenumber,indicatingthatagreatdealofcapitalcanbegivenupifonemoreunitoflaborbecomesavailable.Ontheotherhand,whenalotoflaborisalreadybeingused,theRTSislow,signifyingthatonlyasmallamountofcapitalcanbetradedforanadditionalunitoflaborifoutputistobeheldconstant.Thisassumptionwouldseemtohavesomerelationshiptotheassumptionofdiminishingmarginalproductivity.AhastyuseofEquation9.18mightleadonetoconcludethatariseinlaccompaniedbyafallinkwouldresultinafallinMPl,ariseinMPk,and,therefore,afallintheRTS.Theproblemwiththisquick“proof”isthatthemarginalproductivityofaninputdependsonthelevelofbothinputs—changesinlaffectMPkandviceversa.ItisnotpossibletoderiveadiminishingRTSfromtheassumptionofdiminishingmarginalproductivityalone.Toseewhythisissomathematically,assumethatq¼fðk,lÞandthatfkandflarepositive(thatis,themarginalproductivitiesarepositive).Assumealsothatfkk<0andfll<0(thatthemarginalproductivitiesarediminishing).Toshowthatisoquantsareconvex,wewouldliketoshowthatdðRTSÞ=dl<0.SinceRTS¼fl=fk,wehavedRTSdl¼dðfl=fkÞdl.(9.19)Becauseflandfkarefunctionsofbothkandl,wemustbecarefulintakingthederivativeofthisexpression:dRTSdl¼fkðfllþflk⋅dk=dlÞflðfklþfkk⋅dk=dlÞðfkÞ2.(9.20)Usingthefactthatdk=dl¼fl=fkalonganisoquantandYoung’stheorem(fkl¼flk),wehavedRTSdl¼f2kfll2fkflfklþf2lfkkðfkÞ3.(9.21)Becausewehaveassumedfk>0,thedenominatorofthisfunctionispositive.Hencethewholefractionwillbenegativeifthenumeratorisnegative.Becausefllandfkkarebothassumedtobenegative,thenumeratordeﬁnitelywillbenegativeiffklispositive.Ifwecanassumethis,wehaveshownthatdRTS=dl<0(thattheisoquantsareconvex)4.Importanceofcross-productivityeﬀectsIntuitively,itseemsreasonablethatthecross-partialderivativefkl¼flkshouldbepositive.Ifworkershadmorecapital,theywouldhavehighermarginalproductivities.But,althoughthisisprobablythemostprevalentcase,itdoesnotnecessarilyhavetobeso.Someproductionfunctionshavefkl<0,atleastforarangeofinputvalues.Whenweassumeadiminishing4AswepointedoutinChapter2,functionsforwhichthenumeratorinEquation9.21isnegativearecalled(strictly)quasi-concavefunctions.300Part3ProductionandSupply RTS(aswewillthroughoutmostofourdiscussion),wearethereforemakingastrongerassumptionthansimplydiminishingmarginalproductivitiesforeachinput—speciﬁcally,weareassumingthatmarginalproductivitiesdiminish“rapidlyenough”tocompensateforanypossiblenegativecross-productivityeffects.Ofcourse,asweshallseelater,withthreeormoreinputs,thingsbecomeevenmorecomplicated.EXAMPLE9.2ADiminishingRTSInExample9.1,theproductionfunctionforﬂyswatterswasgivenbyq¼fðk,lÞ¼600k2l2k3l3.(9.22)GeneralmarginalproductivityfunctionsforthisproductionfunctionareMPl¼fl¼∂q∂l¼1,200k2l3k3l2,MPk¼fk¼∂q∂k¼1,200kl23k2l3.(9.23)Noticethateachofthesedependsonthevaluesofbothinputs.Simplefactoringshowsthatthesemarginalproductivitieswillbepositiveforvaluesofkandlforwhichkl<400.Becausefll¼1,200k26k3landfkk¼1,200l26kl3,(9.24)itisclearthatthisfunctionexhibitsdiminishingmarginalproductivitiesforsufﬁcientlylargevaluesofkandl.Indeed,againbyfactoringeachexpression,itiseasytoshowthatfll,fkk<0ifkl>200.However,evenwithintherange2001),thenweclassifythereturnstoscaleoftheproductionfunctionbyInintuitiveterms,ifaproportionateincreaseininputsincreasesoutputbythesamepropor-tion,theproductionfunctionexhibitsconstantreturnstoscale.Ifoutputincreaseslessthanproportionately,thefunctionexhibitsdiminishingreturnstoscale.Andifoutputincreasesmorethanproportionately,thereareincreasingreturnstoscale.Asweshallsee,itistheo-reticallypossibleforafunctiontoexhibitconstantreturnstoscaleforsomelevelsofinputusageandincreasingordecreasingreturnsforotherlevels.5Often,however,economistsrefertothedegreeofreturnstoscaleofaproductionfunctionwiththeimplicitnotionthatonlyafairlynarrowrangeofvariationininputusageandtherelatedlevelofoutputisbeingconsidered.ConstantreturnstoscaleThereareeconomicreasonswhyaﬁrm’sproductionfunctionmightexhibitconstantreturnstoscale.Iftheﬁrmoperatesmanyidenticalplants,itmayincreaseordecreaseproductionsimplybyvaryingthenumberofthemincurrentoperation.Thatis,theﬁrmcandoubleoutputbydoublingthenumberofplantsitoperates,andthatwillrequireittoemploypreciselytwiceasmanyinputs.Alternatively,ifoneweremodelingthebehaviorofanentireindustrycomposedofmanyﬁrms,theconstantreturns-to-scaleassumptionmightEffectonOutputReturnstoScaleI.fðtk,tlÞ¼tfðk,lÞ¼tqConstantII.fðtk,tlÞtfðk,lÞ¼tqIncreasing5Alocalmeasureofreturnstoscaleisprovidedbythescaleelasticity,deﬁnedaseq,t¼∂fðtk,tlÞ∂t⋅tfðtk,tlÞ,wherethisexpressionistobeevaluatedatt¼1.Thisparametercan,inprinciple,takeondifferentvaluesdependingonthelevelofinputusage.Forsomeexamplesusingthisconcept,seeProblem9.9.302Part3ProductionandSupply
makesensebecausetheindustrycanexpandorcontractbyaddingordroppinganarbitrarynumberofidenticalﬁrms(seeChapter12).Finally,studiesoftheentireU.S.economyhavefoundthatconstantreturnstoscaleisareasonablygoodapproximationtouseforan“aggregate”productionfunction.Forallofthesereasons,then,theconstantreturns-to-scalecaseseemsworthexamininginsomewhatmoredetail.Whenaproductionfunctionexhibitsconstantreturnstoscale,itmeetsthedeﬁnitionof“homogeneity”thatweintroducedinChapter2.Thatis,theproductionishomogeneousofdegree1initsinputsbecausefðtk,tlÞ¼t1fðk,lÞ¼tq.(9.26)InChapter2weshowedthat,ifafunctionishomogeneousofdegreek,itsderivativesarehomogeneousofdegreek1.Inthiscontextthisimpliesthatthemarginalproductivityfunctionsderivedfromaconstantreturns-to-scaleproductionfunctionarehomogeneousofdegree0.Thatis,MPk¼∂fðk,lÞ∂k¼∂fðtk,tlÞ∂k,MPl¼∂fðk,lÞ∂l¼∂fðtk,tlÞ∂l(9.27)foranyt>0.Inparticular,wecanlett¼1=linEquations9.27andgetMPk¼∂fðk=l,1Þ∂k,MPl¼∂fðk=l,1Þ∂l.(9.28)Thatis,themarginalproductivityofanyinputdependsonlyontheratioofcapitaltolaborinput,notontheabsolutelevelsoftheseinputs.Thisfactisespeciallyimportant,forexample,inexplainingdifferencesinproductivityamongindustriesoracrosscountries.HomotheticproductionfunctionsOneconsequenceofEquations9.28isthattheRTSð¼MPl=MPkÞforanyconstantreturns-to-scaleproductionfunctionwilldependonlyontheratiooftheinputs,notontheirabsolutelevels.Thatis,suchafunctionwillbehomothetic(seeChapter2)—itsisoquantswillberadialexpansionsofoneanother.ThissituationisshowninFigure9.2.Alonganyraythroughtheorigin(wheretheratiok=ldoesnotchange),theslopesofsuccessivelyhigherisoquantsareidentical.Thispropertyoftheisoquantmapwillbeveryusefultousonseveraloccasions.Asimplenumericalexamplemayprovidesomeintuitionaboutthisresult.Supposearoofcanbeinstalledinonedaybythreeworkerswithonehammereachorbytwoworkerswithtwohammerseach(theseworkersareambidextrous).TheRTSofhammersforworkersisthereforeoneforone—oneextrahammercanbesubstitutedforoneworker.Ifthisproduc-tionprocessexhibitsconstantreturnstoscale,tworoofscanbeinstalledinonedayeitherbysixworkerswithsixhammersorbyfourworkerswitheighthammers.Inthelattercase,twohammersaresubstitutedfortwoworkers,soagaintheRTSisoneforone.Inconstantreturns-to-scalecases,expandingthelevelofproductiondoesnotaltertrade-offsamonginputs,soproductionfunctionsarehomothetic.Aproductionfunctioncanhaveahomotheticindifferencecurvemapevenifitdoesnotexhibitconstantreturnstoscale.AsweshowedinChapter2,thispropertyofhomotheticityisretainedbyanymonotonictransformationofahomogeneousfunction.Hence,increasingordecreasingreturnstoscalecanbeincorporatedintoaconstantreturns-to-scalefunctionChapter9ProductionFunctions303
throughanappropriatetransformation.Perhapsthemostcommonsuchtransformationisexponential.So,iffðk,lÞisaconstantreturns-to-scaleproductonfunction,wecanletFðk,lÞ¼½fðk,lÞγ,(9.29)whereγisanypositiveexponent.Ifγ>1thenFðtk,tlÞ¼½fðtk,tlÞγ¼½tfðk,lÞγ¼tγ½fðk,lÞγ¼tγFðk,lÞ>tFðk,lÞ(9.30)foranyt>1.Hence,thistransformedproductionfunctionexhibitsincreasingreturnstoscale.AnidenticalproofshowsthatthefunctionFexhibitsdecreasingreturnstoscaleforγ<1.Becausethisfunctionremainshomotheticthroughallsuchtransformations,wehaveshownthatthereareimportantcaseswheretheissueofreturnstoscalecanbeseparatedfromissuesinvolvingtheshapeofanisoquant.Inthenextsection,wewilllookathowshapesofisoquantscanbedescribed.Then-inputcaseThedeﬁnitionofreturnstoscalecanbeeasilygeneralizedtoaproductionfunctionwithninputs.Ifthatproductionfunctionisgivenbyq¼fðx1,x2,…,xnÞ(9.31)andifallinputsaremultipliedbyt>1,wehavefðtx1,tx2,…,txnÞ¼tkfðx1,x2,…,xnÞ¼tkq(9.32)forsomeconstantk.Ifk¼1,theproductionfunctionexhibitsconstantreturnstoscale.Dimin-ishingandincreasingreturnstoscalecorrespondtothecasesk<1andk>1,respectively.Thecrucialpartofthismathematicaldeﬁnitionistherequirementthatallinputsbeincreasedbythesameproportion,t.Inmanyreal-worldproductionprocesses,thisprovisionmaymakelittleeconomicsense.Forexample,aﬁrmmayhaveonlyone“boss,”andthatFIGURE9.2IsoquantMapforaConstantReturns-to-ScaleProductionFunctionForaconstantreturns-to-scaleproductionfunction,theRTSdependsonlyontheratioofktol,notonthescaleofproduction.Consequently,eachisoquantwillbearadialblowupoftheunitisoquant.Alonganyraythroughtheorigin(arayofconstantk=l),theRTSwillbethesameonallisoquants.l per periodk per periodq=3q=2q=1304Part3ProductionandSupply
numberwouldnotnecessarilybedoubledevenifallotherinputswere.Ortheoutputofafarmmaydependonthefertilityofthesoil.Itmaynotbeliterallypossibletodoubletheacresplantedwhilemaintainingfertility,becausethenewlandmaynotbeasgoodasthatalreadyundercultivation.Hence,someinputsmayhavetobeﬁxed(oratleastimperfectlyvariable)formostpracticalpurposes.Insuchcases,somedegreeofdiminishingproductivity(aresultofincreasingemploymentofvariableinputs)seemslikely,althoughthiscannotproperlybecalled“diminishingreturnstoscale”becauseofthepresenceofinputsthatareheldﬁxed.THEELASTICITYOFSUBSTITUTIONAnotherimportantcharacteristicoftheproductionfunctionishow“easy”itistosubstituteoneinputforanother.Thisisaquestionabouttheshapeofasingleisoquantratherthanaboutthewholeisoquantmap.Alongoneisoquant,therateoftechnicalsubstitutionwilldecreaseasthecapital-laborratiodecreases(thatis,ask=ldecreases);nowwewishtodeﬁnesomeparameterthatmeasuresthisdegreeofresponsiveness.IftheRTSdoesnotchangeatallforchangesink=l,wemightsaythatsubstitutioniseasybecausetheratioofthemarginalproductivitiesofthetwoinputsdoesnotchangeastheinputmixchanges.Alternatively,iftheRTSchangesrapidlyforsmallchangesink=l,wewouldsaythatsubstitutionisdifﬁcultbecauseminorvariationsintheinputmixwillhaveasubstantialeffectontheinputs’relativeproductivities.Ascale-freemeasureofthisresponsivenessisprovidedbytheelasticityofsubstitution,aconceptweencounteredinPart2.Nowwecanprovideaformaldeﬁnition.DEFINITIONElasticityofsubstitution.Fortheproductionfunctionq¼fðk,lÞ,theelasticityofsubsti-tutionðσÞmeasurestheproportionatechangeink=lrelativetotheproportionatechangeintheRTSalonganisoquant.Thatis,σ¼percent∆ðk=lÞpercent∆RTS¼dðk=lÞdRTS⋅RTSk=l¼∂lnk=l∂lnRTS¼∂lnk=l∂lnfl=fk.(9.33)Becausealonganisoquant,k=landRTSmoveinthesamedirection,thevalueofσisalwayspositive.Graphically,thisconceptisillustratedinFigure9.3asamovementfrompointAtopointBonanisoquant.Inthismovement,boththeRTSandtheratiok=lwillchange;weareinterestedintherelativemagnitudeofthesechanges.Ifσishigh,thentheRTSwillnotchangemuchrelativetok=landtheisoquantwillberelativelyﬂat.Ontheotherhand,alowvalueofσimpliesarathersharplycurvedisoquant;theRTSwillchangebyasubstantialamountask=lchanges.Ingeneral,itispossiblethattheelasticityofsubstitutionwillvaryasonemovesalonganisoquantandasthescaleofproductionchanges.Often,however,itisconvenienttoassumethatσisconstantalonganisoquant.Iftheproductionfunctionisalsohomothetic,then—becausealltheisoquantsaremerelyradialblowups—σwillbethesamealongallisoquants.Wewillencountersuchfunctionslaterinthischapterandinmanyofitsproblems.6Then-inputcaseGeneralizingtheelasticityofsubstitutiontothemany-inputcaseraisesseveralcomplications.OneapproachistoadoptadeﬁnitionanalogoustoEquation9.33;thatis,todeﬁnetheelasticityofsubstitutionbetweentwoinputstobetheproportionatechangeintheratioof6Theelasticityofsubstitutioncanbephraseddirectlyintermsoftheproductionfunctionanditsderivativesintheconstantreturns-to-scalecaseasσ¼fk⋅flf⋅fk,l.Butthisformisquitecumbersome.HenceusuallythelogarithmicdeﬁnitioninEquation9.33iseasiesttoapply.Foracompactsummary,seeP.BerckandK.Sydsaeter,Economist’sMathematicalManual(Berlin:Springer-Verlag,1999),chap.5.Chapter9ProductionFunctions305
thetwoinputstotheproportionatechangeintheRTSbetweenthemwhileholdingoutputconstant.7Tomakethisdeﬁnitioncomplete,itisnecessarytorequirethatallinputsotherthanthetwobeingexaminedbeheldconstant.However,thislatterrequirement(whichisnotrelevantwhenthereareonlytwoinputs)restrictsthevalueofthispotentialdeﬁnition.Inreal-worldproductionprocesses,itislikelythatanychangeintheratiooftwoinputswillalsobeaccompaniedbychangesinthelevelsofotherinputs.Someoftheseotherinputsmaybecomplementarywiththeonesbeingchanged,whereasothersmaybesubstitutes,andtoholdthemconstantcreatesaratherartiﬁcialrestriction.Forthisreason,analternativedeﬁnitionoftheelasticityofsubstitutionthatpermitssuchcomplementarityandsubstitutabilityintheﬁrm’scostfunctionisgenerallyusedinthen-goodcase.Becausethisconceptisusuallymeasuredusingcostfunctions,wewilldescribeitinthenextchapter.FOURSIMPLEPRODUCTIONFUNCTIONSInthissectionweillustratefoursimpleproductionfunctions,eachcharacterizedbyadifferentelasticityofsubstitution.Theseareshownonlyforthecaseoftwoinputs,butgeneralizationtomanyinputsiseasilyaccomplished(seetheExtensionsforthischapter).FIGURE9.3GraphicDescriptionoftheElasticityofSubstitutionInmovingfrompointAtopointBontheq¼q0isoquant,boththecapital-laborratio(k=l)andtheRTSwillchange.Theelasticityofsubstitution(σ)isdeﬁnedtobetheratiooftheseproportionalchanges;itisameasureofhowcurvedtheisoquantis.k perperiodl per periodq=q0AB(k/l)A(k/l)BRTSARTSB7Thatis,theelasticityofsubstitutionbetweeninputiandinputjmightbedeﬁnedasσij¼∂lnðxi=xjÞ∂lnðfj=fiÞformovementsalongfðx1,x2,…,xnÞ¼c.Noticethattheuseofpartialderivativesinthisdeﬁnitioneffectivelyrequiresthatallinputsotherthaniandjbeheldconstantwhenconsideringmovementsalongthecisoquant.306Part3ProductionandSupply
Case1:Linear(σ¼∞)Supposethattheproductionfunctionisgivenbyq¼fðk,lÞ¼akþbl.(9.34)Itiseasytoshowthatthisproductionfunctionexhibitsconstantreturnstoscale:Foranyt>1,fðtk,tlÞ¼atkþbtl¼tðakþblÞ¼tfðk,lÞ.(9.35)Allisoquantsforthisproductionfunctionareparallelstraightlineswithslopeb=a.Suchanisoquantmapispicturedinpanel(a)ofFigure9.4.BecausetheRTSisconstantalonganystraight-lineisoquant,thedenominatorinthedeﬁnitionofσ(Equation9.33)isequalto0andhenceσisinﬁnite.Althoughthislinearproductionfunctionisausefulexample,itisFIGURE9.4IsoquantMapsforSimpleProductionFunctionswithVariousValuesforσThreepossiblevaluesfortheelasticityofsubstitutionareillustratedintheseﬁgures.In(a),capitalandlaborareperfectsubstitutes.Inthiscase,theRTSwillnotchangeasthecapital-laborratiochanges.In(b),theﬁxed-proportionscase,nosubstitutionispossible.Thecapital-laborratioisﬁxedatb=a.Acaseoflimitedsubstitutabilityisillustratedin(c).k perperiodk perperiodk perperiodl per periodl per periodl per periodq3q2q1q3q2q1q3q2q1σ=∞σ=0σ=1(a)(b)(c)Slope = __–ba__q3a__q3bChapter9ProductionFunctions307
rarelyencounteredinpracticebecausefewproductionprocessesarecharacterizedbysucheaseofsubstitution.Indeed,inthiscase,capitalandlaborcanbethoughtofasperfectsubstitutesforeachother.Anindustrycharacterizedbysuchaproductionfunctioncoulduseonlycapitaloronlylabor,dependingontheseinputs’prices.Itishardtoenvisionsuchaproductionprocess:Everymachineneedssomeonetopressitsbuttons,andeverylaborerrequiressomecapitalequipment,howevermodest.Case2:Fixedproportions(σ¼0)Theproductionfunctioncharacterizedbyσ¼0istheimportantcaseofaﬁxed-proportionsproductionfunction.Capitalandlabormustalwaysbeusedinaﬁxedratio.TheisoquantsforthisproductionfunctionareL-shapedandarepicturedinpanel(b)ofFigure9.4.Aﬁrmcharacterizedbythisproductionfunctionwillalwaysoperatealongtheraywheretheratiok=lisconstant.Tooperateatsomepointotherthanatthevertexoftheisoquantswouldbeinefﬁcient,becausethesameoutputcouldbeproducedwithfewerinputsbymovingalongtheisoquanttowardthevertex.Becausek=lisaconstant,itiseasytoseefromthedeﬁnitionoftheelasticityofsubstitutionthatσmustequal0.Themathematicalformoftheﬁxed-proportionsproductionfunctionisgivenbyq¼minðak,blÞ,a,b>0,(9.36)wheretheoperator“min”meansthatqisgivenbythesmallerofthetwovaluesinparentheses.Forexample,supposethatakbl,thenlaboristhebindingconstraintonoutputandadditionalcapitalissuperﬂuous.Whenak¼bl,bothinputsarefullyutilized.Whenthishappens,k=l¼b=a,andproductiontakesplaceatavertexontheisoquantmap.Ifbothinputsarecostly,thisistheonlycost-minimizingplacetooperate.Thelocusofallsuchverticesisastraightlinethroughtheoriginwithaslopegivenbyb=a.Theﬁxed-proportionsproductionfunctionhasawiderangeofapplications.8Manymachines,forexample,requireacertainnumberofpeopletorunthem,butanyexcesslaborissuperﬂuous.Considercombiningcapital(alawnmower)andlabortomowalawn.Itwillalwaystakeonepersontorunthemower,andeitherinputwithouttheotherisnotabletoproduceanyoutputatall.Itmaybethatmanymachinesareofthistypeandrequireaﬁxedcomplementofworkerspermachine.9Case3:Cobb-Douglas(σ¼1)Theproductionfunctionforwhichσ¼1,calledaCobb-Douglasproductionfunction,10providesamiddlegroundbetweenthetwopolarcasespreviouslydiscussed.Isoquantsfor8WiththeformreﬂectedbyEquation9.35,theﬁxed-proportionsproductionfunctionexhibitsconstantreturnstoscalebecausefðtk,tlÞ¼minðatk,btlÞ¼tminðak,blÞ¼tfðk,lÞforanyt>1.Asbefore,increasingordecreasingreturnscanbeeasilyincorporatedintothefunctionsbyusinganonlineartransformationofthisfunctionalform—suchas½fðk,lÞγ,whereγmaybegreaterthanorlessthan1.9Thelawnmowerexamplepointsupanotherpossibility,however.Presumablythereissomeleewayinchoosingwhatsizeoflawnmowertobuy.Hence,priortotheactualpurchase,thecapital-laborratioinlawnmowingcanbeconsideredvariable:Anydevice,fromapairofclipperstoagangmower,mightbechosen.Oncethemowerispurchased,however,thecapital-laborratiobecomesﬁxed.10NamedafterC.W.CobbandP.H.Douglas.SeeP.H.Douglas,TheTheoryofWages(NewYork:MacmillanCo.,1934),pp.132–f35.308Part3ProductionandSupply
theCobb-Douglascasehavethe“normal”convexshapeandareshowninpanel(c)ofFigure9.4.ThemathematicalformoftheCobb-Douglasproductionfunctionisgivenbyq¼fðk,lÞ¼Akalb,(9.37)whereA,a,andbareallpositiveconstants.TheCobb-Douglasfunctioncanexhibitanydegreeofreturnstoscale,dependingonthevaluesofaandb.Supposeallinputswereincreasedbyafactoroft.Thenfðtk,tlÞ¼AðtkÞaðtlÞb¼Ataþbkalb¼taþbfðk,lÞ.(9.38)Hence,ifaþb¼1,theCobb-Douglasfunctionexhibitsconstantreturnstoscalebecauseoutputalsoincreasesbyafactoroft.Ifaþb>1thenthefunctionexhibitsincreasingreturnstoscale,whereasaþb<1correspondstothedecreasingreturns-to-scalecase.Itisasimplemattertoshowthattheelasticityofsubstitutionis1fortheCobb-Douglasfunction.11Thisfacthasledresearcherstousetheconstantreturns-to-scaleversionofthefunctionforageneraldescriptionofaggregateproductionrelationshipsinmanycountries.TheCobb-Douglasfunctionhasalsoprovedtobequiteusefulinmanyapplicationsbecauseitislinearinlogarithms:lnq¼lnAþalnkþblnl.(9.39)Theconstantaisthentheelasticityofoutputwithrespecttocapitalinput,andbistheelasticityofoutputwithrespecttolaborinput.12Theseconstantscansometimesbeestimatedfromactualdata,andsuchestimatesmaybeusedtomeasurereturnstoscale(byexaminingthesumaþb)andforotherpurposes.Case4:CESproductionfunctionAfunctionalformthatincorporatesallofthethreepreviouscasesandallowsσtotakeonothervaluesaswellistheconstantelasticityofsubstitution(CES)productionfunctionﬁrstintroducedbyArrowetal.in1961.13Thisfunctionisgivenbyq¼fðk,lÞ¼½kρþlργ=ρ(9.40)forρ1,ρ6¼0,andγ>0.ThisfunctioncloselyresemblestheCESutilityfunctiondiscussedinChapter3,thoughnowwehaveaddedtheexponentγ=ρtopermitexplicitintroductionofreturns-to-scalefactors.Forγ>1thefunctionexhibitsincreasingreturnstoscale,whereasforγ<1itexhibitsdiminishingreturns.11FortheCobb-Douglasfunction,RTS¼flfk¼bAkalb1aAka1lb¼baklorlnRTS¼lnðb=aÞþlnðk=lÞ.Henceσ¼∂lnk=l∂lnRTS¼1.12SeeProblem9.5.13K.J.Arrow,H.B.Chenery,B.S.Minhas,andR.M.Solow,“Capital-LaborSubstitutionandEconomicEfﬁciency,”ReviewofEconomicsandStatistics(August1961):225–50.Chapter9ProductionFunctions309 Directapplicationofthedeﬁnitionofσtothisfunction14givestheimportantresultthatσ¼11ρ.(9.41)Hencethelinear,ﬁxed-proportions,andCobb-Douglascasescorrespondtoρ¼1,ρ¼∞,andρ¼0,respectively.ProofofthisresultfortheﬁxedproportionsandCobb-Douglascasesrequiresalimitargument.OftentheCESfunctionisusedwithadistributionalweight,βð0β1Þ,toindicatetherelativesigniﬁcanceoftheinputs:q¼fðk,lÞ¼½βkρþð1βÞlργ=ρ:(9.42)Withconstantreturnstoscaleandρ¼0,thisfunctionconvergestotheCobb-Douglasformq¼fðk,lÞ¼kβl1β.(9.43)EXAMPLE9.3AGeneralizedLeontiefProductionFunctionSupposethattheproductionfunctionforagoodisgivenbyq¼fðk,lÞ¼kþlþ2ﬃﬃﬃﬃﬃﬃﬃﬃk⋅lp.(9.44)ThisfunctionisaspecialcaseofaclassoffunctionsnamedfortheRussian-AmericaneconomistWassilyLeontief.15Thefunctionclearlyexhibitsconstantreturnstoscalebecausefðtk,tlÞ¼tkþtlþ2tﬃﬃﬃﬃﬃklp¼tfðk,lÞ.(9.45)MarginalproductivitiesfortheLeontieffunctionarefk¼1þðk=lÞ0:5,fl¼1þðk=lÞ0:5.(9.46)Hence,marginalproductivitiesarepositiveanddiminishing.Aswouldbeexpected(becausethisfunctionexhibitsconstantreturnstoscale),theRTSheredependsonlyontheratioofthetwoinputsRTS¼flfk¼1þðk=lÞ0:51þðk=lÞ0:5.(9.47)ThisRTSdiminishesask=lfalls,sotheisoquantshavetheusualconvexshape.14FortheCESfunctionwehaveRTS¼flfk¼ðγ=ρÞ⋅qðγρÞ=γ⋅ρlρ1ðγ=ρÞ⋅qðγρÞ=γ⋅ρkρ1¼lkρ1¼kl1ρ.Applyingthedeﬁnitionoftheelasticityofsubstitutionthenyieldsσ¼∂lnðk=lÞ∂lnRTS¼11ρ.Noticeinthiscomputationthatthefactorρcancelsoutofthemarginalproductivityfunctions,therebyensuringthatthesemarginalproductivitiesarepositiveevenwhenρisnegative(asitisinmanycases).ThisexplainswhyρappearsintwodifferentplacesinthedeﬁnitionoftheCESfunction.15Lenotiefwasapioneerinthedevelopmentofinput-outputanalysis.Ininput-outputanalysis,productionisassumedtotakeplacewithaﬁxed-proportionstechnology.TheLeontiefproductionfunctiongeneralizestheﬁxed-proportionscase.FormoredetailsseethediscussionofLeontiefproductionfunctionsintheExtensionstothischapter.310Part3ProductionandSupply Therearetwowaysyoumightcalculatetheelasticityofsubstitutionforthisproductionfunction.First,youmightnoticethatinthisspecialcasethefunctioncanbefactoredasq¼kþlþ2ﬃﬃﬃﬃﬃklp¼ðﬃﬃﬃkpþﬃﬃlpÞ2¼ðk0:5þl0:5Þ2,(9.48)whichmakesclearthatthisfunctionhasaCESformwithρ¼0:5andγ¼1.Hencetheelasticityofsubstitutionhereisσ¼1=ð1ρÞ¼2.Ofcourse,inmostcasesitisnotpossibletodosuchasimplefactorization.Amoreex-haustiveapproachistoapplythedeﬁnitionoftheelasticityofsubstitutiongiveninfootnote6ofthischapter:σ¼fkflf⋅fkl¼½1þðk=lÞ0:5½1þðk=lÞ0:5q⋅ð0:5=ﬃﬃﬃﬃﬃklpÞ¼2þðk=lÞ0:5þðk=lÞ0:51þ0:5ðk=lÞ0:5þ0:5ðk=lÞ0:5¼2:(9.49)Noticethatinthiscalculationtheinputratioðk=lÞdropsout,leavingaverysimpleresult.Inotherapplications,onemightdoubtthatsuchafortuitousresultwouldoccurandhencedoubtthattheelasticityofsubstitutionisconstantalonganisoquant(seeProblem9.7).Butheretheresultthatσ¼2isintuitivelyreasonable,becausethatvaluerepresentsacompromisebetweentheelasticityofsubstitutionforthisproductionfunction’slinearpartðq¼kþl,σ¼∞ÞanditsCobb-Douglaspartðq¼2k0:5l0:5,σ¼1Þ.QUERY:Whatcanyoulearnaboutthisproductionfunctionbygraphingtheq¼4isoquant?Whydoesthisfunctiongeneralizetheﬁxedproportionscase?TECHNICALPROGRESSMethodsofproductionimproveovertime,anditisimportanttobeabletocapturetheseimprovementswiththeproductionfunctionconcept.AsimpliﬁedviewofsuchprogressisprovidedbyFigure9.5.Initially,isoquantq0recordsthosecombinationsofcapitalandlaborthatcanbeusedtoproduceanoutputlevelofq0.Followingthedevelopmentofsuperiorproductiontechniques,thisisoquantshiftstoq00.Nowthesamelevelofoutputcanbeproducedwithfewerinputs.Onewaytomeasurethisimprovementisbynotingthatwithalevelofcapitalinputof,say,k1,itpreviouslytookl2unitsoflabortoproduceq0,whereasnowittakesonlyl1.Outputperworkerhasrisenfromq0=l2toq0=l1.Butonemustbecarefulinthistypeofcalculation.Anincreaseincapitalinputtok2wouldalsohavepermittedareductioninlaborinputtol1alongtheoriginalq0isoquant.Inthiscase,outputperworkerwouldalsorise,althoughtherewouldhavebeennotruetechnicalprogress.Useoftheproductionfunctionconceptcanhelptodifferentiatebetweenthesetwoconceptsandthereforealloweconomiststoobtainanaccurateestimateoftherateoftechnicalchange.MeasuringtechnicalprogressTheﬁrstobservationtobemadeabouttechnicalprogressisthathistoricallytherateofgrowthofoutputovertimehasexceededthegrowthratethatcanbeattributedtothegrowthinconventionallydeﬁnedinputs.Supposethatweletq¼AðtÞfðk,lÞ(9.50)betheproductionfunctionforsomegood(orperhapsforsociety’soutputasawhole).ThetermAðtÞinthefunctionrepresentsalltheinﬂuencesthatgointodeterminingqotherthank(machine-hours)andl(labor-hours).ChangesinAovertimerepresenttechnicalprogress.Chapter9ProductionFunctions311 Forthisreason,Aisshownasafunctionoftime.PresumablydA=dt>0;particularlevelsofinputoflaborandcapitalbecomemoreproductiveovertime.DifferentiatingEquation9.50withrespecttotimegivesdqdt¼dAdt⋅fk,lðÞþA⋅dfðk,lÞdt¼dAdt⋅qAþqfðk,lÞ∂f∂k⋅dkdtþ∂f∂l⋅dldt.(9.51)Dividingbyqgivesdq=dtq¼dA=dtAþ∂f=∂kfðk,lÞ⋅dkdtþ∂f=∂lfðk,lÞ⋅dldt(9.52)ordq=dtq¼dA=dtAþ∂f∂k⋅kfðk,lÞ⋅dk=dtkþ∂f∂l⋅lfðk,lÞ⋅dl=dtl.(9.53)Now,foranyvariablex,(dx=dt)/xistheproportionalrateofgrowthofxperunitoftime.WeshalldenotethisbyGx.16Hence,Equation9.53canbewrittenintermsofgrowthratesasFIGURE9.5TechnicalProgressTechnicalprogressshiftstheq0isoquanttowardtheorigin.Thenewq0isoquant,q00,showsthatagivenlevelofoutputcannowbeproducedwithlessinput.Forexample,withk1unitsofcapitalitnowonlytakesl1unitsoflabortoproduceq0,whereasbeforethetechnicaladvanceittookl2unitsoflabor.k perperiodl per periodk1k2l2l1q0q′016Twousefulfeaturesofthisdeﬁnitionare:(1)Gx⋅y¼GxþGy—thatis,thegrowthrateofaproductoftwovariablesisthesumofeachone’sgrowthrate;and(2)Gx=y¼GxGy.312Part3ProductionandSupply
Gq¼GAþ∂f∂k⋅kfðk,lÞ⋅Gkþ∂f∂l⋅lfðk,lÞ⋅Gl,(9.54)but∂f∂k⋅kfðk,lÞ¼∂q∂k⋅kq¼elasticityofoutputwithrespecttocapitalinput¼eq,kand∂f∂l⋅lfðk,lÞ¼∂q∂l⋅lq¼elasticityofoutputwithrespecttolaborinput¼eq;l.GrowthaccountingTherefore,ourgrowthequationﬁnallybecomesGq¼GAþeq,kGkþeq,lGl.(9.55)Thisshowsthattherateofgrowthinoutputcanbebrokendownintothesumoftwocomponents:growthattributedtochangesininputs(kandl)andother“residual”growth(thatis,changesinA)thatrepresentstechnicalprogress.Equation9.55providesawayofestimatingtherelativeimportanceoftechnicalprogress(GA)indeterminingthegrowthofoutput.Forexample,inapioneeringstudyoftheentireU.S.economybetweentheyears1909and1949,R.M.Solowrecordedthefollowingvaluesforthetermsintheequation:17Gq¼2:75percentperyear,Gl¼1:00percentperyear,Gk¼1:75percentperyear,eq,l¼0:65,eq,k¼0:35.Consequently,GA¼Gqeq,lGleq,kGk¼2:750:65ð1:00Þ0:35ð1:75Þ¼2:750:650:60¼1:50.(9.56)TheconclusionSolowreached,then,wasthattechnologyadvancedatarateof1.5percentperyearfrom1909to1949.Morethanhalfofthegrowthinrealoutputcouldbeattributedtotechnicalchangeratherthantogrowthinthephysicalquantitiesofthefactorsofproduc-tion.MorerecentevidencehastendedtoconﬁrmSolow’sconclusionsabouttherelativeimportanceoftechnicalchange.Considerableuncertaintyremains,however,abouttheprecisecausesofsuchchange.17R.M.Solow,“TechnicalProgressandtheAggregateProductionFunction,”ReviewofEconomicsandStatistics39(August1957):312–f20.Chapter9ProductionFunctions313
EXAMPLE9.4TechnicalProgressintheCobb-DouglasProductionFunctionTheCobb-Douglasproductionfunctionprovidesanespeciallyeasyavenueforillustratingtechnicalprogress.Assumingconstantreturnstoscale,suchaproductionfunctionwithtechnicalprogressmightberepresentedbyq¼AðtÞfðk,lÞ¼AðtÞkαl1α.(9.57)Ifwealsoassumethattechnicalprogressoccursataconstantexponential(θ),thenwecanwriteAðtÞ¼Aeθtandtheproductionfunctionbecomesq¼Aeθtkαl1α.(9.58)Aparticularlyeasywaytostudythepropertiesofthistypeoffunctionovertimeistouse“logarithmicdifferentiation”:∂lnq∂t¼∂lnq∂q⋅∂q∂t¼∂q=∂tq¼Gq¼∂½lnAþθtþαlnkþð1αÞlnl∂t¼θþα⋅∂lnk∂tþð1αÞ⋅∂lnl∂t¼θþαGkþð1−αÞGl.(9.59)SothisderivationjustrepeatsEquation9.55fortheCobb-Douglascase.Herethetechnicalchangefactorisexplicitlymodeled,andtheoutputelasticitiesaregivenbythevaluesoftheexponentsintheCobb-Douglas.Theimportanceoftechnicalprogresscanbeillustratednumericallywiththisfunction.SupposeA¼10,θ¼0:03,α¼0:5andthataﬁrmusesaninputmixofk¼l¼4.Then,att¼0,outputis40ð¼10⋅40:5⋅40:5Þ.After20yearsðt¼20Þ,theproductionfunctionbecomesq¼10e0:03⋅20k0:5l0:5¼10⋅ð1:82Þk0:5l0:5¼18:2k0:5l0:5.(9.60)Inyear20theoriginalinputmixnowyieldsq¼72:8.Ofcourse,onecouldalsohaveproducedq¼72:8inyear0,butitwouldhavetakenalotmoreinputs.Forexample,withk¼13:25andl¼4,outputisindeed72.8butmuchmorecapitalisused.Outputperunitoflaborinputwouldrisefrom10(q=l¼40=4)to18:2ð¼72:8=4)ineithercircumstance,butonlytheﬁrstcasewouldhavebeentruetechnicalprogress.Input-augmentingtechnicalprogress.Itistemptingtoattributetheincreaseintheaverageproductivityoflaborinthisexampleto,say,improvedworkerskills,butthatwouldbemisleadingintheCobb-Douglascase.Onemightjustaswellhavesaidthatoutputperunitofcapitalrosefrom10to18.2overthe20yearsandattributethisrisetoimprovedmachinery.Aplausibleapproachtomodelingimprovementsinlaborandcapitalseparatelyistoassumethattheproductionfunctionisq¼AðeφtkÞαðeεtlÞ1α,(9.61)whereφrepresentstheannualrateofimprovementincapitalinputandεrepresentstheannualrateofimprovementinlaborinput.But,becauseoftheexponentialnatureoftheCobb-Douglasfunction,thiswouldbeindistinguishablefromouroriginalexample:q¼Ae½αφþð1αÞεtkαl1α¼Aeθtkαl1α,(9.62)whereθ¼αφþð1αÞε.Hence,tostudytechnicalprogressinindividualinputs,itisnecessaryeithertoadoptamorecomplexwayofmeasuringinputsthatallowsforimprovingqualityor(whatamountstothesamething)touseamulti-inputproductionfunction.314Part3ProductionandSupply
QUERY:ActualstudiesofproductionusingtheCobb-Douglastendtoﬁndα0.3.UsethisﬁndingtogetherwithEquation9.62todiscusstherelativeimportanceofimprovingcapitalandlaborqualitytotheoverallrateoftechnicalprogress.PROBLEMS9.1PowerGoatLawnCompanyusestwosizesofmowerstocutlawns.Thesmallermowershavea24-inchbladeandareusedonlawnswithmanytreesandobstacles.Thelargermowersareexactlytwiceasbigasthesmallermowersandareusedonopenlawnswheremaneuverabilityisnotsodifﬁcult.ThetwoproductionfunctionsavailabletoPowerGoatare:a.Graphtheq¼40,000squarefeetisoquantfortheﬁrstproductionfunction.Howmuchkandlwouldbeusedifthesefactorswerecombinedwithoutwaste?OutputperHour(squarefeet)CapitalInput(#of2400mowers)LaborInputLargemowers800021Smallmowers500011SUMMARYInthischapterweillustratedthewaysinwhicheconomistsconceptualizetheproductionprocessofturninginputsintooutputs.Thefundamentaltoolistheproductionfunction,which—initssimplestform—assumesthatoutputperperiod(q)isasimplefunctionofcapitalandlaborinputsduringthatperiod,q¼fðk,lÞ.Usingthisstartingpoint,wedevelopedseveralbasicresultsforthetheoryofproduction.•Ifallbutoneoftheinputsareheldconstant,arelation-shipbetweenthesingle-variableinputandoutputcanbederived.Fromthisrelationship,onecanderivethemar-ginalphysicalproductivity(MP)oftheinputasthechangeinoutputresultingfromaone-unitincreaseintheuseoftheinput.Themarginalphysicalproductivityofaninputisassumedtodeclineasuseoftheinputincreases.•Theentireproductionfunctioncanbeillustratedbyitsisoquantmap.The(negativeofthe)slopeofanisoquantistermedthemarginalrateoftechnicalsubstitution(RTS),becauseitshowshowoneinputcanbesubsti-tutedforanotherwhileholdingoutputconstant.TheRTSistheratioofthemarginalphysicalproductivitiesofthetwoinputs.•Isoquantsareusuallyassumedtobeconvex—theyobeytheassumptionofadiminishingRTS.Thisassumptioncannotbederivedexclusivelyfromtheassumptionofdiminishingmarginalphysicalproductivities.Onemustalsobeconcernedwiththeeffectofchangesinoneinputonthemarginalproductivityofotherinputs.•Thereturnstoscaleexhibitedbyaproductionfunctionrecordhowoutputrespondstoproportionateincreasesinallinputs.Ifoutputincreasesproportionatelywithinputuse,thereareconstantreturnstoscale.Iftherearegreaterthanproportionateincreasesinoutput,thereareincreasingreturnstoscale,whereasiftherearelessthanproportionateincreasesinoutput,therearede-creasingreturnstoscale.•TheelasticityofsubstitutionðσÞprovidesameasureofhoweasyitistosubstituteoneinputforanotherinpro-duction.Ahighσimpliesnearlylinearisoquants,whereasalowσimpliesthatisoquantsarenearlyL-shaped.•Technicalprogressshiftstheentireproductionfunctionanditsrelatedisoquantmap.Technicalimprovementsmayarisefromtheuseofimproved,more-productiveinputsorfrombettermethodsofeconomicorganization.Chapter9ProductionFunctions315
b.Answerpart(a)forthesecondfunction.c.Howmuchkandlwouldbeusedwithoutwasteifhalfofthe40,000-square-footlawnwerecutbythemethodoftheﬁrstproductionfunctionandhalfbythemethodofthesecond?Howmuchkandlwouldbeusedifthreefourthsofthelawnwerecutbytheﬁrstmethodandonefourthbythesecond?Whatdoesitmeantospeakoffractionsofkandl?d.Onthebasisofyourobservationsinpart(c),drawaq¼40,000isoquantforthecombinedproductionfunctions.9.2Supposetheproductionfunctionforwidgetsisgivenbyq¼kl0:8k20:2l2,whereqrepresentstheannualquantityofwidgetsproduced,krepresentsannualcapitalinput,andlrepresentsannuallaborinput.a.Supposek¼10;graphthetotalandaverageproductivityoflaborcurves.Atwhatleveloflaborinputdoesthisaverageproductivityreachamaximum?Howmanywidgetsareproducedatthatpoint?b.Againassumingthatk¼10,graphtheMPlcurve.AtwhatleveloflaborinputdoesMPl¼0?c.Supposecapitalinputswereincreasedtok¼20.Howwouldyouranswerstoparts(a)and(b)change?d.Doesthewidgetproductionfunctionexhibitconstant,increasing,ordecreasingreturnstoscale?9.3SamMaloneisconsideringrenovatingthebarstoolsatCheers.Theproductionfunctionfornewbarstoolsisgivenbyq¼0:1k0:2l0:8,whereqisthenumberofbarstoolsproducedduringtherenovationweek,krepresentsthenumberofhoursofbarstoollathesusedduringtheweek,andlrepresentsthenumberofworkerhoursemployedduringtheperiod.Samwouldliketoprovide10newbarstools,andhehasallocatedabudgetof$10,000fortheproject.a.Samreasonsthatbecausebarstoollathesandskilledbarstoolworkersbothcostthesameamount($50perhour),hemightaswellhirethesetwoinputsinequalamounts.IfSamproceedsinthisway,howmuchofeachinputwillhehireandhowmuchwilltherenovationprojectcost?b.Norm(whoknowssomethingaboutbarstools)arguesthatonceagainSamhasforgottenhismicroeconomics.HeassertsthatSamshouldchoosequantitiesofinputssothattheirmarginal(notaverage)productivitiesareequal.IfSamoptsforthisplaninstead,howmuchofeachinputwillhehireandhowmuchwilltherenovationprojectcost?c.UponhearingthatNorm’splanwillsavemoney,CliffarguesthatSamshouldputthesavingsintomorebarstoolsinordertoprovideseatingtomoreofhisUSPScolleagues.HowmanymorebarstoolscanSamgetforhisbudgetifhefollowsCliff’splan?d.CarlaworriesthatCliff’ssuggestionwilljustmeanmoreworkforherindeliveringfoodtobarpatrons.HowmightsheconvinceSamtosticktohisoriginal10–barstoolplan?316Part3ProductionandSupply
9.4SupposethattheproductionofcrayonsðqÞisconductedattwolocationsandusesonlylaborasaninput.Theproductionfunctioninlocation1isgivenbyq1¼10l0.51andinlocation2byq2¼50l0.52:a.Ifasingleﬁrmproducescrayonsinbothlocations,thenitwillobviouslywanttogetaslargeanoutputaspossiblegiventhelaborinputituses.Howshoulditallocatelaborbetweenthelocationsinordertodoso?Explainpreciselytherelationshipbetweenl1andl2:b.Assumingthattheﬁrmoperatesintheefﬁcientmannerdescribedinpart(a),howdoestotaloutputðqÞdependonthetotalamountoflaborhiredðlÞ?9.5Aswehaveseeninmanyplaces,thegeneralCobb-Douglasproductionfunctionfortwoinputsisgivenbyq¼fðk,lÞ¼Akαlβ,where0<α<1and0<β<1:Forthisproductionfunction:a.Showthatfk>0,fl>0,fkk<0,fll<0,andfkl¼flk>0.b.Showthateq,k¼αandee,l¼β:c.Infootnote5,wedeﬁnedthescaleelasticityaseq,t¼∂fðtk,tlÞ∂t⋅tfðtk,tlÞ,wheretheexpressionistobeevaluatedatt¼1:Showthat,forthisCobb-Douglasfunction,eq,t¼αþβ:Hence,inthiscasethescaleelasticityandthereturnstoscaleoftheproductionfunctionagree(formoreonthisconceptseeProblem9.9).d.Showthatthisfunctionisquasi-concave.e.Showthatthefunctionisconcaveforαþβ1butnotconcaveforαþβ>1:9.6Supposewearegiventheconstantreturns-to-scaleCESproductionfunctionq¼½kρþlρ1=ρ.a.ShowthatMPk¼ðq=kÞ1ρandMPl¼ðq=lÞ1ρ:b.ShowthatRTS¼ðl=kÞ1ρ;usethistoshowthatσ¼1=ð1ρÞ:c.Determinetheoutputelasticitiesforkandl,andshowthattheirsumequals1.d.Provethatql¼∂q∂lσandhencethatlnql¼σln∂q∂l.Note:Thelatterequalityisusefulinempiricalwork,becausewemayapproximate∂q=∂lbythecompetitivelydeterminedwagerate.Hence,σcanbeestimatedfromaregressionoflnðq=lÞonlnw:Chapter9ProductionFunctions317
9.7ConsiderageneralizationoftheproductionfunctioninExample9.3:q¼β0þβ1ﬃﬃﬃﬃﬃklpþβ2kþβ3l,where0βi1,i¼0,…,3.a.Ifthisfunctionistoexhibitconstantreturnstoscale,whatrestrictionsshouldbeplacedontheparametersβ0,…,β3?b.Showthat,intheconstantreturns-to-scalecase,thisfunctionexhibitsdiminishingmarginalproductivitiesandthatthemarginalproductivityfunctionsarehomogeneousofdegree0.c.Calculateσinthiscase.Althoughσisnotingeneralconstant,forwhatvaluesoftheβ’sdoesσ¼0,1,or∞?9.8ShowthatEuler’stheoremimpliesthat,foraconstantreturns-to-scaleproductionfunction½q¼fðk,lÞ,q¼fk⋅kþfl⋅l:Usethisresulttoshowthat,forsuchaproductionfunction,ifMPl>APlthenMPkmustbenegative.Whatdoesthisimplyaboutwhereproductionmusttakeplace?CanaﬁrmeverproduceatapointwhereAPlisincreasing?AnalyticalProblems9.9LocalreturnstoscaleAlocalmeasureofthereturnstoscaleincorporatedinaproductionfunctionisgivenbythescaleelasticityeq,t¼∂fðtk,tlÞ=∂t⋅t=qevaluatedatt¼l:a.Showthatiftheproductionfunctionexhibitsconstantreturnstoscaletheneq,t¼1:b.Wecandeﬁnetheoutputelasticitiesoftheinputskandlaseq,k¼∂fðk,lÞ∂k⋅kq,eq,l¼∂fðk,lÞ∂l⋅lq.Showthateq,t¼eq,kþeq,l:c.Afunctionthatexhibitsvariablescaleelasticityisq¼ð1þk1l1Þ1:Showthat,forthisfunction,eq,t>1forq<0.5andthateq,t<1forq>0.5:d.Explainyourresultsfrompart(c)intuitively.Hint:Doesqhaveanupperboundforthisproductionfunction?318Part3ProductionandSupply
9.10ReturnstoscaleandsubstitutionAlthoughmuchofourdiscussionofmeasuringtheelasticityofsubstitutionforvariousproductionfunctionshasassumedconstantreturnstoscale,oftenthatassumptionisnotnecessary.Thisproblemillustratessomeofthesecases.a.Infootnote6weshowedthat,intheconstantreturns-to-scalecase,theelasticityofsubstitutionforatwo-inputproductionfunctionisgivenbyσ¼fkflf⋅fkl.SupposenowthatwedeﬁnethehomotheticproductionfunctionFasFðk,lÞ¼½fðk,lÞγ,wherefðk,lÞisaconstantreturns-to-scaleproductionfunctionandγisapositiveexponent.Showthattheelasticityofsubstitutionforthisproductionfunctionisthesameastheelasticityofsubstitutionforthefunctionf:b.ShowhowthisresultcanbeappliedtoboththeCobb-DouglasandCESproductionfunctions.9.11MoreonEuler’stheoremSupposethataproductionfunctionfðx1,x2,…,xnÞishomogeneousofdegreek:Euler’stheoremshowsthatXixifi¼kf,andthisfactcanbeusedtoshowthatthepartialderivativesoffarehomogeneousofdegreek1:a.ProvethatXni¼1Xnj¼1xixjfij¼kðk1Þf:b.Inthecaseofn¼2andk¼1,whatkindofrestrictionsdoestheresultofpart(a)imposeonthesecond-orderpartialderivativef12?Howdoyourconclusionschangewhenk>1ork<1?c.Howwouldtheresultsofpart(b)begeneralizedtoaproductionfunctionwithanynumberofinputs?d.WhataretheimplicationsofthisproblemfortheparametersofthemultivariableCobb-Douglasproductionfunctionfðx1,x2,…,xnÞ¼∏ni¼1xαiiforαi0?SUGGESTIONSFORFURTHERREADINGClark,J.M.“DiminishingReturns.”InEncyclopaediaoftheSocialSciences,vol.5.NewYork:Crowell-CollierandMacmillan,1931,pp.144–f46.Luciddiscussionofthehistoricaldevelopmentofthediminishingreturnsconcept.Douglas,P.H.“AreThereLawsofProduction?”AmericanEconomicReview38(March1948):1–f41.Anicemethodologicalanalysisoftheusesandmisusesofproductionfunctions.Ferguson,C.E.TheNeoclassicalTheoryofProductionandDistribution.NewYork:CambridgeUniversityPress,1969.Athoroughdiscussionofproductionfunctiontheory(asof1970).Gooduseofthree-dimensionalgraphs.Fuss,M.,andMcFadden,D.ProductionEconomics:ADualApproachtoTheoryandApplication.Amsterdam:North-Holland,1980.Anapproachwithaheavyemphasisontheuseofduality.Mas-Collell,A.,M.D.Whinston,andJ.R.Green.Micro-economicTheory.NewYork:OxfordUniversityPress,1995.Chapter5providesasophisticated,ifsomewhatspare,reviewofproductiontheory.Theuseoftheproﬁtfunction(seeChapter11)isquitesophisticatedandilluminating.Shephard,R.W.TheoryofCostandProductionFunctions.Princeton,NJ:PrincetonUniversityPress,1978.Extendedanalysisofthedualrelationshipbetweenproductionandcostfunctions.Silberberg,E.,andW.Suen.TheStructureofEconomics:AMathematicalAnalysis,3rded.Boston:Irwin/McGraw-Hill,2001.Thoroughanalysisofthedualitybetweenproductionfunctionsandcostcurves.Providesaproofthattheelasticityofsubstitutioncanbederivedasshowninfootnote6ofthischapter.Stigler,G.J.“TheDivisionofLaborIsLimitedbytheExtentoftheMarket.”JournalofPoliticalEconomy59(June1951):185–f93.CarefultracingoftheevolutionofSmith’sideasabouteconomiesofscale.Chapter9ProductionFunctions319 EXTENSIONSMany-InputProductionFunctionsMostoftheproductionfunctionsillustratedinChap-ter9canbeeasilygeneralizedtomany-inputcases.HereweshowthisfortheCobb-DouglasandCEScasesandthenexaminetwoquiteﬂexibleformsthatsuchproductionfunctionsmighttake.Inalloftheseexamples,theβ’sarenonnegativeparametersandtheninputsarerepresentedbyx1,…,xn:E9.1Cobb-DouglasThemany-inputCobb-Douglasproductionfunctionisgivenbyq¼Yni¼1xβii.(i)a.ThisfunctionexhibitsconstantreturnstoscaleifXni¼1βi¼1.(ii)b.Intheconstant-returns-to-scaleCobb-Douglasfunction,βiistheelasticityofqwithrespecttoinputxi:Because0β<1;eachinputex-hibitsdiminishingmarginalproductivity.c.Anydegreeofincreasingreturnstoscalecanbeincorporatedintothisfunction,dependingonε¼Xni¼1βi.(iii)d.Theelasticityofsubstitutionbetweenanytwoinputsinthisproductionfunctionis1.Thiscanbeshownbyusingthedeﬁnitiongiveninfoot-note7ofthischapter:σij¼∂lnðxi=xjÞ∂lnðfj=fiÞ.Herefjfi¼βjxβj1j∏i6¼jxβiiβixβi1i∏j6¼ixβjj¼βjβi⋅xixj.Hence,lnfjfi¼lnβjβiþlnxixj !andσij¼1:Becausethisparameterissocon-strainedintheCobb-Douglasfunction,thefunctionisgenerallynotusedineconometricanalysesofmicroeconomicdataonﬁrms.How-ever,thefunctionhasavarietyofgeneralusesinmacroeconomics,asthenextexampleillus-trates.TheSolowgrowthmodelThemany-inputCobb-Douglasproductionfunctionisaprimaryfeatureofmanymodelsofeconomicgrowth.Forexample,Solow’s(1956)pioneeringmodelofequilibriumgrowthcanbemosteasilyderivedusingatwo-inputconstant-returns-to-scaleCobb-DouglasfunctionoftheformY¼AKαL1α,(iv)whereAisatechnicalchangefactorthatcanberepre-sentedbyexponentialgrowthoftheformA¼eat.(v)DividingbothsidesofEquationivbyLyieldsy¼eatkα,(vi)wherey¼Y=Landk¼K=L.Solowshowsthateconomieswillevolvetowardanequilibriumvalueofk(thecapital-laborratio).Hencecross-countrydifferencesingrowthratescanbeaccountedforonlybydifferencesinthetechnicalchangefactor,a:TwofeaturesofEquationviargueforincludingmoreinputsintheSolowmodel.First,theequationasitstandsisincapableofexplainingthelargedifferencesinpercapitaoutputðyÞthatareobservedaroundtheworld.Assumingα¼0:3,say(aﬁgureconsistentwithmanyempiricalstudies),itwouldtakecross-countrydifferencesinK=Lofasmuchas4,000,000-to-1toexplainthe100-to-1differencesinpercapitaincomeobserved—aclearlyunreasonablemagnitude.Byin-troducingadditionalinputs,suchashumancapital,thesedifferencesbecomemoreexplainable.AsecondshortcomingofthesimpleCobb-DouglasformulationoftheSolowmodelisthatitoffersnoexplanationofthetechnicalchangeparameter,a—itsvalueisdetermined“exogenously.”Byaddingaddi-tionalfactors,itbecomeseasiertounderstandhowtheparameteramayrespondtoeconomicincentives.320Part3ProductionandSupply Thisisthekeyinsightofliteratureon“endogenous”growththeory(forasummary,seeRomer,1996).E9.2CESThemany-inputconstantelasticityofsubstitution(CES)productionfunctionisgivenbyq¼Xβixρihiε=ρ,ρ1.(vii)a.Bysubstitutingmxiforeachoutput,itiseasytoshowthatthisfunctionexhibitsconstantreturnstoscaleforε¼1:Forε>1,thefunc-tionexhibitsincreasingreturnstoscale.b.Theproductionfunctionexhibitsdiminishingmarginalproductivitiesforeachinputbecauseρ1:c.Asinthetwo-inputcase,theelasticityofsub-stitutionhereisgivenbyσ¼11ρ,(viii)andthiselasticityappliestosubstitutionbetweenanytwooftheinputs.CheckingtheCobb-DouglasintheSovietUnionOnewayinwhichthemulti-inputCESfunctionisusedistodeterminewhethertheestimatedsubstitutionparameterðρÞisconsistentwiththevalueimpliedbytheCobb-Douglasðρ¼0,σ¼1Þ:Forexample,inastudyofﬁvemajorindustriesintheformerSovietUnion,E.Bairam(1991)ﬁndsthattheCobb-Douglasprovidesarelativelygoodexplanationofchangesinoutputinmostmajormanufacturingsectors.Onlyforfoodprocessingdoesalowervalueforσseemappropriate.Thenextthreeexamplesillustrateﬂexible-formproductionfunctionsthatmayapproximateanygen-eralfunctionofninputs.IntheChapter10exten-sions,weexaminethecostfunctionanaloguestosomeofthesefunctions,whicharemorewidelyusedthantheproductionfunctionsthemselves.E9.3NestedproductionfunctionsInsomeapplications,Cobb-DouglasandCESpro-ductionfunctionsarecombinedintoa“nested”singlefunction.Toaccomplishthis,theoriginalnprimaryinputsarecategorizedinto,say,mgeneralclassesofinputs.Thespeciﬁcinputsineachofthesecategoriesarethenaggregatedintoasinglecompositeinput,andtheﬁnalproductionfunctionisafunctionofthesemcomposites.Forexample,assumetherearethreepri-maryinputs,x1,x2,x3:Suppose,however,thatx1andx2arerelativelycloselyrelatedintheirusebyﬁrms(forexample,capitalandenergy)whereasthethirdinput(labor)isrelativelydistinct.ThenonemightwanttouseaCESaggregatorfunctiontoconstructacompos-iteinputforcapitalservicesoftheformx4¼½γxρ1þð1γÞxρ21=ρ:(ix)ThentheﬁnalproductionfunctionmighttakeaCobb-Douglasform:q¼xα3xβ4:(x)Thisstructureallowstheelasticityofsubstitutionbe-tweenx1andx2totakeonanyvalue½σ¼1=ð1ρÞbutconstrainstheelasticityofsubstitutionbetweenx3andx4tobeone.Avarietyofotheroptionsareavail-abledependingonhowpreciselytheembeddedfunc-tionsarespeciﬁed.Thedynamicsofcapital/energysubstitutabilityNestedproductionfunctionshavebeenwidelyusedinstudiesthatseektomeasuretheprecisenatureofthesubstitutabilitybetweencapitalandenergyinputs.Forexample,AtkesonandKehoe(1999)useamodelratherclosetotheonespeciﬁedinEquationsixandxtotrytoreconciletwofactsaboutthewayinwhichenergypricesaffecttheeconomy:(1)Overtime,useofenergyinproductionseemsratherunresponsivetoprice(atleastintheshort-run);and(2)acrosscoun-tries,energypricesseemtohavealargeinﬂuenceoverhowmuchenergyisused.ByusingacapitalserviceequationoftheformgiveninEquationixwithalowdegreeofsubstitutabilityðρ¼2:3Þ—alongwithaCobb-Douglasproductionfunctionthatcombineslaborwithcapitalservices—theyareabletoreplicatethefactsaboutenergypricesfairlywell.Theyconclude,however,thatthismodelimpliesamuchmorenegativeeffectofhigherenergypricesoneconomicgrowththanseemsactuallytohavebeenthecase.Hencetheyultimatelyoptforamorecomplexwayofmodelingproductionthatstressesdifferencesinenergyuseamongcapitalinvestmentsmadeatdifferentdates.E9.4GeneralizedLeontiefq¼Xni¼1Xnj¼1βijﬃﬃﬃﬃﬃﬃﬃﬃﬃxixjp,whereβij¼βji.Chapter9ProductionFunctions321
a.ThefunctionconsideredinProblem9.7isasimplecaseofthisfunctionforthecasen¼2:Forn¼3,thefunctionwouldhavelineartermsinthethreeinputsalongwiththreeradi-caltermsrepresentingallpossiblecross-pro-ductsoftheinputs.b.Thefunctionexhibitsconstantreturnstoscale,ascanbeshownbyusingmxi.Increasingreturnstoscalecanbeincorporatedintothefunctionbyusingthetransformationq0¼qε,ε>1.c.Becauseeachinputappearsbothlinearlyandundertheradical,thefunctionexhibitsdimin-ishingmarginalproductivitiestoallinputs.d.Therestrictionβij¼βjiisusedtoensuresym-metryofthesecond-orderpartialderivatives.E9.5Transloglnq¼β0þXni¼1βilnxiþ0:5Xni¼1Xnj¼1βijlnxilnxj;βij¼βji:a.NotethattheCobb-Douglasfunctionisaspe-cialcaseofthisfunctionwhereβ0¼βij¼0foralli,j:b.AsfortheCobb-Douglas,thisfunctionmayassumeanydegreeofreturnstoscale.IfXni¼1βi¼1andXnj¼1βij¼0foralli,thenthisfunctionexhibitsconstantreturnstoscale.Theproofrequiressomecareindealingwiththedoublesummation.c.Again,theconditionβij¼βjiisrequiredtoensureequalityofthecross-partialderivatives.ImmigrationBecausethetranslogproductionfunctionincorporatesalargenumberofsubstitutionpossibilitiesamongvar-iousinputs,ithasbeenwidelyusedtostudythewaysinwhichnewlyarrivedworkersmaysubstituteforexist-ingworkers.Ofparticularinterestisthewayinwhichtheskilllevelofimmigrantsmayleadtodifferingreac-tionsinthedemandforskilledandunskilledworkersinthedomesticeconomy.StudiesoftheUnitedStatesandmanyothercountries(Canada,Germany,France,andsoforth)havesuggestedthattheoverallsizeofsucheffectsismodest,especiallygivenrelativelysmallimmigrationﬂows.Butthereissomeevidencethatunskilledimmigrantworkersmayactassubstitutesforunskilleddomesticworkersbutascomplementstoskilleddomesticworkers.Henceincreasedimmigra-tionﬂowsmayexacerbatetrendstowardrisingwagedifferentials.Forasummary,seeBorjas(1994).ReferencesAtkeson,Andrew,andPatrickJ.Kehoe.“ModelsofEnergyUse:Putty-PuttyversusPutty-Clay.”AmericanEconomicReview(September1999):1028–43.Bairam,Erkin.“ElasticityofSubstitution,TechnicalProg-ressandReturnstoScaleinBranchesofSovietIndustry:ANewCESProductionFunctionApproach.”JournalofAppliedEconomics(January–March1991):91–f96.Borjas,G.J.“TheEconomicsofImmigration.”JournalofEconomicLiterature(December1994):1667–f1717.Romer,David.AdvancedMacroeconomics.NewYork:McGraw-Hill,1996.Solow,R.M.“AContributiontotheTheoryofEconomicGrowth.”QuarterlyJournalofEconomics(February1956):65–f94.322Part3ProductionandSupply
CHAPTER10CostFunctionsInthischapterweillustratethecoststhataﬁrmincurswhenitproducesoutput.InChapter11,wewillpursuethistopicfurtherbyshowinghowﬁrmsmakeproﬁt-maximizinginputandoutputdecisions.DEFINITIONSOFCOSTSBeforewecandiscussthetheoryofcosts,somedifﬁcultiesabouttheproperdeﬁnitionof“costs”mustbeclearedup.Speciﬁcally,wemustdifferentiatebetween(1)accountingcostand(2)economiccost.Theaccountant’sviewofcoststressesout-of-pocketexpenses,historicalcosts,depreciation,andotherbookkeepingentries.Theeconomist’sdeﬁnitionofcost(whichinobviouswaysdrawsonthefundamentalopportunity-costnotion)isthatthecostofanyinputisgivenbythesizeofthepaymentnecessarytokeeptheresourceinitspresentemployment.Alternatively,theeconomiccostofusinganinputiswhatthatinputwouldbepaidinitsnextbestuse.Onewaytodistinguishbetweenthesetwoviewsistoconsiderhowthecostsofvariousinputs(labor,capital,andentrepreneurialservices)aredeﬁnedundereachsystem.LaborcostsEconomistsandaccountantsregardlaborcostsinmuchthesameway.Toaccountants,expendituresonlaborarecurrentexpensesandhencecostsofproduction.Foreconomists,laborisanexplicitcost.Laborservices(labor-hours)arecontractedatsomehourlywagerateðwÞ,anditisusuallyassumedthatthisisalsowhatthelaborserviceswouldearnintheirbestalternativeemployment.Thehourlywage,ofcourse,includescostsoffringebeneﬁtsprovidedtoemployees.CapitalcostsInthecaseofcapitalservices(machine-hours),thetwoconceptsofcostdiffer.Incalculatingcapitalcosts,accountantsusethehistoricalpriceoftheparticularmachineunderinvestigationandapplysomemore-or-lessarbitrarydepreciationruletodeterminehowmuchofthatmachine’soriginalpricetochargetocurrentcosts.Economistsregardthehistoricalpriceofamachineasa“sunkcost,”whichisirrelevanttooutputdecisions.Theyinsteadregardtheimplicitcostofthemachinetobewhatsomeoneelsewouldbewillingtopayforitsuse.Thusthecostofonemachine-houristherentalrateforthatmachineinitsbestalternativeuse.Bycontinuingtousethemachineitself,theﬁrmisimplicitlyforgoingwhatsomeoneelsewouldbewillingtopaytouseit.Thisrentalrateforonemachine-hourwillbedenotedbyv.11Sometimesthesymbolrischosentorepresenttherentalrateoncapital.Becausethisvariableisoftenconfusedwiththerelatedbutdistinctconceptofthemarketinterestrate,analternativesymbolwaschosenhere.TheexactrelationshipbetweenvandtheinterestrateisexaminedinChapter17.323
CostsofentrepreneurialservicesTheownerofaﬁrmisaresidualclaimantwhoisentitledtowhateverextrarevenuesorlossesareleftafterpayingotherinputcosts.Toanaccountant,thesewouldbecalledproﬁts(whichmightbeeitherpositiveornegative).Economists,however,askwhetherowners(orentre-preneurs)alsoencounteropportunitycostsbyworkingataparticularﬁrmordevotingsomeoftheirfundstoitsoperation.Ifso,theseservicesshouldbeconsideredaninput,andsomecostshouldbeimputedtothem.Forexample,supposeahighlyskilledcomputerprogrammerstartsasoftwareﬁrmwiththeideaofkeepingany(accounting)proﬁtsthatmightbegener-ated.Theprogrammer’stimeisclearlyaninputtotheﬁrm,andacostshouldbeinputtedforit.Perhapsthewagethattheprogrammermightcommandifheorsheworkedforsomeoneelsecouldbeusedforthatpurpose.Hencesomepartoftheaccountingproﬁtsgeneratedbytheﬁrmwouldbecategorizedasentrepreneurialcostsbyeconomists.Economicproﬁtswouldbesmallerthanaccountingproﬁtsandmightbenegativeiftheprogrammer’sopportunitycostsexceededtheaccountingproﬁtsbeingearnedbythebusiness.Similarargumentsapplytothecapitalthatanentrepreneurprovidestotheﬁrm.EconomiccostsInthisbook,notsurprisingly,weuseeconomists’deﬁnitionofcost.DEFINITIONEconomiccost.Theeconomiccostofanyinputisthepaymentrequiredtokeepthatinputinitspresentemployment.Equivalently,theeconomiccostofaninputistheremunerationtheinputwouldreceiveinitsbestalternativeemployment.Useofthisdeﬁnitionisnotmeanttoimplythataccountants’conceptsareirrelevanttoeconomicbehavior.Indeed,accountingproceduresareintegrallyimportanttoanymanager’sdecision-makingprocessbecausetheycangreatlyaffecttherateoftaxationtobeappliedagainstproﬁts.Accountingdataarealsoreadilyavailable,whereasdataoneconomiccostsmustoftenbedevelopedseparately.Economists’deﬁnitions,however,dohavethedesirablefeaturesofbeingbroadlyapplicabletoallﬁrmsandofformingaconceptuallyconsistentsystem.Theythereforearebestsuitedforageneraltheoreticalanalysis.TwosimplifyingassumptionsAsastart,wewillmaketwosimpliﬁcationsabouttheinputsaﬁrmuses.First,weassumethatthereareonlytwoinputs:homogeneouslabor(l,measuredinlabor-hours)andhomoge-neouscapital(k,measuredinmachine-hours).Entrepreneurialcostsareincludedincapitalcosts.Thatis,weassumethattheprimaryopportunitycostsfacedbyaﬁrm’sownerarethoseassociatedwiththecapitalthattheownerprovides.Second,weassumethatinputsarehiredinperfectlycompetitivemarkets.Firmscanbuy(orsell)allthelabororcapitalservicestheywantattheprevailingrentalrates(wandv).Ingraphicterms,thesupplycurvefortheseresourcesishorizontalattheprevailingfactorprices.Bothwandvaretreatedas“parameters”intheﬁrm’sdecisions;thereisnothingtheﬁrmcandotoaffectthem.Theseconditionswillberelaxedinlaterchapters(notablyChapter16),butforthemomenttheprice-takerassumptionisaconvenientandusefulonetomake.EconomicproﬁtsandcostminimizationTotalcostsfortheﬁrmduringaperiodarethereforegivenbytotalcosts¼C¼wlþvk,(10.1)where,asbefore,landkrepresentinputusageduringtheperiod.Assumingtheﬁrmproducesonlyoneoutput,itstotalrevenuesaregivenbythepriceofitsproductðpÞtimesits324Part3ProductionandSupply
totaloutput[q¼fðk,lÞ,wherefðk,lÞistheﬁrm’sproductionfunction].EconomicproﬁtsðπÞarethenthedifferencebetweentotalrevenuesandtotaleconomiccosts.DEFINITIONEconomicprofits.EconomicproﬁtsðπÞarethedifferencebetweenaﬁrm’stotalrevenuesanditstotalcosts:π¼totalrevenuetotalcost¼pqwlvk¼pfðk,lÞwlvk.(10.2)Equation10.2showsthattheeconomicproﬁtsobtainedbyaﬁrmareafunctionoftheamountofcapitalandlaboremployed.If,aswewillassumeinmanyplacesinthisbook,theﬁrmseeksmaximumproﬁts,thenwemightstudyitsbehaviorbyexamininghowkandlarechosensoastomaximizeEquation10.2.Thiswould,inturn,leadtoatheoryofsupplyandtoatheoryofthe“deriveddemand”forcapitalandlaborinputs.Inthenextchapterwewilltakeupthosesubjectsindetail.Here,however,wewishtodevelopatheoryofcoststhatissomewhatmoregeneralandmightapplytoﬁrmsthatarenotnecessarilyproﬁtmaximizers.Hence,webeginthestudyofcostsbyﬁnessing,forthemoment,adiscussionofoutputchoice.Thatis,weassumethatforsomereasontheﬁrmhasdecidedtoproduceaparticularoutputlevel(say,q0).Theﬁrm’srevenuesarethereforeﬁxedatpq0.Nowwewishtoexaminehowtheﬁrmcanproduceq0atminimalcosts.COST-MINIMIZINGINPUTCHOICESMathematically,thisisaconstrainedminimizationproblem.Butbeforeproceedingwitharigoroussolution,itisusefultostatetheresulttobederivedwithanintuitiveargument.Tominimizethecostofproducingagivenlevelofoutput,aﬁrmshouldchoosethatpointontheq0isoquantatwhichtherateoftechnicalsubstitutionoflforkisequaltotheratiow=v:Itshouldequatetherateatwhichkcanbetradedforlinproductiontotherateatwhichtheycanbetradedinthemarketplace.Supposethatthiswerenottrue.Inparticular,supposethattheﬁrmwereproducingoutputlevelq0usingk¼10,l¼10,andassumethattheRTSwere2atthispoint.Assumealsothatw¼$1,v¼$1,andhencethatw=v¼1(whichisunequalto2).Atthisinputcombination,thecostofproducingq0is$20.Itiseasytoshowthisisnottheminimalinputcost.Forexample,q0canalsobeproducedusingk¼8andl¼11;wecangiveuptwounitsofkandkeepoutputconstantatq0byaddingoneunitofl.Butatthisinputcombination,thecostofproducingq0is$19andhencetheinitialinputcombinationwasnotoptimal.AcontradictionsimilartothisonecanbedemonstratedwhenevertheRTSandtheratiooftheinputcostsdiffer.MathematicalanalysisMathematically,weseektominimizetotalcostsgivenq¼fðk,lÞ¼q0.SettinguptheLagrangianexpressionℒ¼wlþvkþλ½q0fðk,lÞ,(10.3)theﬁrst-orderconditionsforaconstrainedminimumare∂ℒ∂l¼wλ∂f∂l¼0,∂ℒ∂k¼vλ∂f∂k¼0,∂ℒ∂λ¼q0fðk,lÞ¼0,(10.4)Chapter10CostFunctions325
or,dividingtheﬁrsttwoequations,wv¼∂f=∂l∂f=∂k¼RTSðlforkÞ.(10.5)Thissaysthatthecost-minimizingﬁrmshouldequatetheRTSforthetwoinputstotheratiooftheirprices.FurtherinterpretationsTheseﬁrst-orderconditionsforminimalcostscanbemanipulatedinseveraldifferentwaystoyieldinterestingresults.Forexample,cross-multiplyingEquation10.5givesfkv¼flw.(10.6)Thatis:forcoststobeminimized,themarginalproductivityperdollarspentshouldbethesameforallinputs.Ifincreasingoneinputpromisedtoincreaseoutputbyagreateramountperdollarspentthandidanotherinput,costswouldnotbeminimal—theﬁrmshouldhiremoreoftheinputthatpromisesabigger“bangperbuck”andlessofthemorecostly(intermsofproductivity)input.Anyinputthatcannotmeetthecommonbeneﬁt-costratiodeﬁnedinEquation10.6shouldnotbehiredatall.Equation10.6can,ofcourse,alsobederivedfromEquation10.4,butitismoreinstructivetoderiveitsinverse:wfl¼vfk¼λ.(10.7)Thisequationreportstheextracostofobtaininganextraunitofoutputbyhiringeitheraddedlabororaddedcapitalinput.Becauseofcostminimization,thismarginalcostisthesamenomatterwhichinputishired.ThiscommonmarginalcostisalsomeasuredbytheLagrangianmultiplierfromthecost-minimizationproblem.Asisthecaseforallconstrainedoptimizationproblems,heretheLagrangianmultipliershowshowmuchinextracostswouldbeincurredbyincreasingtheoutputconstraintslightly.Becausemarginalcostplaysanimportantroleinaﬁrm’ssupplydecisions,wewillreturntothisfeatureofcostminimizationfrequently.GraphicalanalysisCostminimizationisshowngraphicallyinFigure10.1.Giventheoutputisoquantq0,wewishtoﬁndtheleastcostlypointontheisoquant.Linesshowingequalcostareparallelstraightlineswithslopesw=v.ThreelinesofequaltotalcostareshowninFigure10.1;C11Þ,changesintheratioofwagestorentalratescauseagreaterthanproportionalincreaseinthecost-minimizingcapital-laborratio.Withlesssubstitutabilityðσ<1Þ,theresponseisproportionallysmaller.QUERY:IntheCobb-Douglasnumericalexamplewithw=v¼4,wefoundthatthecost-minimizinginputratioforproducing40unitsofoutputwask=l¼80=20¼4.Howwouldthisvaluechangeforσ¼2orσ¼0.5?Whatactualinputcombinationswouldbeused?Whatwouldtotalcostsbe?COSTFUNCTIONSWearenowinapositiontoexaminetheﬁrm’soverallcoststructure.Todoso,itwillbeconvenienttousetheexpansionpathsolutionstoderivethetotalcostfunction.DEFINITIONTotalcostfunction.Thetotalcostfunctionshowsthat,foranysetofinputcostsandforanyoutputlevel,theminimumtotalcostincurredbytheﬁrmisC¼Cðv,w,qÞ.(10.16)Figure10.2makesclearthattotalcostsincreaseasoutput,q,increases.Wewillbeginbyanalyzingthisrelationshipbetweentotalcostandoutputwhileholdinginputpricesﬁxed.Thenwewillconsiderhowachangeinaninputpriceshiftstheexpansionpathanditsrelatedcostfunctions.330Part3ProductionandSupply AverageandmarginalcostfunctionsAlthoughthetotalcostfunctionprovidescompleteinformationabouttheoutput-costrelationship,itisoftenconvenienttoanalyzecostsonaper-unit-of-outputbasisbecausethatapproachcorrespondsmorecloselytotheanalysisofdemand,whichfocusedonthepriceperunitofacommodity.Twodifferentunitcostmeasuresarewidelyusedineconomics:(1)averagecost,whichisthecostperunitofoutput;and(2)marginalcost,whichisthecostofonemoreunitofoutput.Therelationshipoftheseconceptstothetotalcostfunctionisdescribedinthefollowingdeﬁnitions.DEFINITIONAverageandmarginalcostfunctions.Theaveragecostfunction(AC)isfoundbycom-putingtotalcostsperunitofoutput:averagecost¼ACv,w,qðÞ¼Cðv,w,qÞq.(10.17)Themarginalcostfunction(MC)isfoundbycomputingthechangeintotalcostsforachangeinoutputproduced:marginalcost¼MCðv,w,qÞ¼∂Cðv,w,qÞ∂q:(10.18)Noticethatinthesedeﬁnitions,averageandmarginalcostsdependbothonthelevelofoutputbeingproducedandonthepricesofinputs.Inmanyplacesthroughoutthisbook,wewillgraphsimpletwo-dimensionalrelationshipsbetweencostsandoutput.Asthedeﬁnitionsmakeclear,allsuchgraphsaredrawnontheassumptionthatthepricesofinputsremainconstantandthattechnologydoesnotchange.Ifinputpriceschangeoriftechnologyad-vances,costcurvesgenerallywillshifttonewpositions.Laterinthischapter,wewillexplorethelikelydirectionandsizeofsuchshiftswhenwestudytheentirecostfunctionindetail.GraphicalanalysisoftotalcostsFigures10.4aand10.5aillustratetwopossibleshapesfortherelationshipbetweentotalcostandtheleveloftheﬁrm’soutput.InFigure10.4a,totalcostissimplyproportionaltooutput.Suchasituationwouldariseiftheunderlyingproductionfunctionexhibitsconstantreturnstoscale.Inthatcase,supposek1unitsofcapitalinputandl1unitsoflaborinputarerequiredtoproduceoneunitofoutput.ThenCðq¼1Þ¼vk1þwl1.(10.19)Toproducemunitsofoutput,then,requiresmk1unitsofcapitalandml1unitsoflabor,becauseoftheconstantreturns-to-scaleassumption.2HenceCðq¼mÞ¼vmk1þwml1¼mðvk1þwl1Þ¼m⋅Cðq¼1Þ,(10.20)andtheproportionalitybetweenoutputandcostisestablished.ThesituationinFigure10.5aismorecomplicated.Thereitisassumedthatinitiallythetotalcostcurveisconcave;althoughinitiallycostsriserapidlyforincreasesinoutput,thatrateofincreaseslowsasoutputexpandsintothemidrangeofoutput.Beyondthismiddlerange,however,thetotalcostcurvebecomesconvex,andcostsbegintoriseprogressivelymorerapidly.Onepossiblereasonforsuchashapeforthetotalcostcurveisthatthereissomethirdfactorofproduction(say,theservicesofanentrepreneur)thatisﬁxedascapitalandlaborusageexpands.Inthiscase,theinitialconcavesectionofthecurvemightbeexplainedbythe2Theinputcombinationml1,mk1minimizesthecostofproducingmunitsofoutputbecausetheratiooftheinputsisstillk1=l1andtheRTSforaconstantreturns-to-scaleproductionfunctiondependsonlyonthatratio.Chapter10CostFunctions331 increasinglyoptimalusageoftheentrepreneur’sservices—heorsheneedsamoderatelevelofproductiontoutilizehisorherskillsfully.Beyondthepointofinﬂection,however,theentrepreneurbecomesoverworkedinattemptingtocoordinateproduction,anddiminishingreturnssetin.Hence,totalcostsriserapidly.Avarietyofotherexplanationshavebeenofferedforthecubic-typetotalcostcurveinFigure10.5a,butwewillnotexaminethemhere.Ultimately,theshapeofthetotalcostcurveisanempiricalquestionthatcanbedeterminedonlybyexaminingreal-worlddata.IntheExtensionstothischapter,weillustratesomeoftheliteratureoncostfunctions.GraphicalanalysisofaverageandmarginalcostsInformationfromthetotalcostcurvescanbeusedtoconstructtheaverageandmarginalcostcurvesshowninFigures10.4band10.5b.Fortheconstantreturns-to-scalecase(Figure10.4),thisisquitesimple.Becausetotalcostsareproportionaltooutput,averageandmarginalcostsFIGURE10.4Total,Average,andMarginalCostCurvesfortheConstantReturns-to-ScaleCaseIn(a)totalcostsareproportionaltooutputlevel.Averageandmarginalcosts,asshownin(b),areequalandconstantforalloutputlevels.TotalcostsAverage andmarginal costsOutput perperiodOutput perperiodCAC=MC(a)(b)332Part3ProductionandSupply areconstantandequalforalllevelsofoutput.3ThesecostsareshownbythehorizontallineAC¼MCinFigure10.4b.Forthecubictotalcostcurvecase(Figure10.5),computationoftheaverageandmarginalcostcurvesrequiressomegeometricintuition.AsthedeﬁnitioninEquation10.18makesclear,marginalcostissimplytheslopeofthetotalcostcurve.Hence,becauseoftheassumedshapeofthecurve,theMCcurveisU-shaped,withMCfallingovertheconcaveportionofthetotalcostcurveandrisingbeyondthepointofinﬂection.Becausetheslopeisalwayspositive,however,MCisalwaysgreaterthan0.Averagecosts(AC)startoutbeingequaltoFIGURE10.5Total,Average,andMarginalCostCurvesfortheCubicTotalCostCurveCaseIfthetotalcostcurvehasthecubicshapeshownin(a),averageandmarginalcostcurveswillbeU-shaped.In(b)themarginalcostcurvepassesthroughthelowpointoftheaveragecostcurveatoutputlevelq.TotalcostsAverage andmarginal costsOutput perperiodOutput perperiodCACMC(a)(b)q*3Mathematically,becauseC¼aq(whereaisthecostofoneunitofoutput),AC¼Cq¼a¼∂C∂q¼MC.Chapter10CostFunctions333 marginalcostforthe“ﬁrst”unitofoutput.4Asoutputexpands,however,ACexceedsMC,becauseACreﬂectsboththemarginalcostofthelastunitproducedandthehighermarginalcostsofthepreviouslyproducedunits.SolongasAC>MC,averagecostsmustbefalling.Becausethelowercostsofthenewlyproducedunitsarebelowaveragecost,theycontinuetopullaveragecostsdownward.Marginalcostsrise,however,andeventually(atq)equalaveragecost.Beyondthispoint,MC>AC,andaveragecostswillberisingbecausetheyarebeingpulledupwardbyincreasinglyhighermarginalcosts.Consequently,wehaveshownthattheACcurvealsohasaU-shapeandthatitreachesalowpointatq,whereACandMCintersect.5Inempiricalstudiesofcostfunctions,thereisconsiderableinterestinthispointofminimumaveragecost.Itreﬂectsthe“minimumefﬁcientscale”(MES)fortheparticularproductionprocessbeingexamined.Thepointisalsotheoreticallyimportantbecauseoftheroleitplaysinperfectlycompetitivepricedeterminationinthelongrun(seeChapter12).COSTFUNCTIONSANDSHIFTSINCOSTCURVESThecostcurvesillustratedinFigures10.4and10.5showtherelationshipbetweencostsandquantityproducedontheassumptionthatallotherfactorsareheldconstant.Speciﬁcally,constructionofthecurvesassumesthatinputpricesandtheleveloftechnologydonotchange.6Ifthesefactorsdochange,thecostcurveswillshift.Inthissection,wedelvefurtherintothemathematicsofcostfunctionsasawayofstudyingtheseshifts.Webeginwithsomeexamples.EXAMPLE10.2SomeIllustrativeCostFunctionsInthisexamplewecalculatethecostfunctionsassociatedwiththreedifferentproductionfunctions.Laterwewillusetheseexamplestoillustratesomeofthegeneralpropertiesofcostfunctions.1.FixedProportions:q¼fðk,lÞ¼minðak,blÞ.Thecalculationofcostfunctionsfromtheirunderlyingproductionfunctionsisoneofthemorefrustratingtasksforeconomicsstudents.4Technically,AC¼MCatq¼0.ThiscanbeshownbyL’Hôpital’srule,whichstatesthatiffðaÞ¼gðaÞ¼0thenlimx!afðxÞgðxÞ¼limx!af0ðxÞg0ðxÞ.Inthiscase,C¼0atq¼0,andsolimq!0AC¼limq!0Cq¼limq!0∂C=∂q1¼limq!0MCorAC¼MCatq¼0,whichwastobeshown.5Mathematically,wecanﬁndtheminimumACbysettingitsderivativeequalto0:∂AC∂q¼∂ðC=qÞ∂q¼q⋅ð∂C=∂qÞC⋅1q2¼q⋅MCCq2¼0,orq⋅MCC¼0orMC¼C=q¼AC.6Formultiproductﬁrms,anadditionalcomplicationmustbeconsidered.Forsuchﬁrmsitispossiblethatthecostsassociatedwithproducingoneoutput(say,q1)arealsoaffectedbytheamountofsomeotheroutputbeingproducedðq2Þ.Inthiscasetheﬁrmissaidtoexhibit“economiesofscope,”andthetotalcostfunctionwillbeoftheformCðq1,q2,w,vÞ.Hence,q2mustalsobeheldconstantinconstructingtheq1costcurves.Presumablyincreasesinq2shifttheq1costcurvesdownward.334Part3ProductionandSupply
So,let’sstartwithasimpleexample.Whatwewishtodoisshowhowtotalcostsdependoninputcostsandonquantityproduced.Intheﬁxed-proportionscase,weknowthatproduc-tionwilloccuratavertexoftheL-shapedisoquantswhereq¼ak¼bl.Hence,totalcostsaretotalcosts¼Cv,w,qðÞ¼vkþwl¼vqaþwqb¼qvaþwb.(10.21)Thisisindeedthesortoffunctionwewantbecauseitstatestotalcostsasafunctionofv,w,andqonlytogetherwithsomeparametersoftheunderlyingproductionfunction.Becauseoftheconstantreturns-to-scalenatureofthisproductionfunction,ittakesthespecialformCðv,w,qÞ¼qCðv,w,1Þ.(10.22)Thatis,totalcostsaregivenbyoutputtimesthecostofproducingoneunit.Increasesininputpricesclearlyincreasetotalcostswiththisfunction,andtechnicalimprovementsthattaketheformofincreasingtheparametersaandbreducecosts.2.Cobb-Douglas:q¼fðk,lÞ¼kαlβ.Thisisourﬁrstexampleofburdensomecomputation,butwecanclarifytheprocessbyrecognizingthattheﬁnalgoalistousetheresultsofcostminimizationtoreplacetheinputsintheproductionfunctionwithcosts.FromExample10.1weknowthatcostminimizationrequiresthatwv¼βα⋅klandsok¼αβ⋅wv⋅l.Substitutionintotheproductionfunctionpermitsasolutionforlaborinputintermsofq,v,andwasq¼kαlβ¼αβ⋅wvαlαþβorl¼q1=ðαþβÞβαα=ðαþβÞwα=ðαþβÞvα=ðαþβÞ.(10.23)Asimilarsetofmanipulationsgivesk¼q1=ðαþβÞαββ=ðαþβÞwβ=ðαþβÞvβ=ðαþβÞ.(10.24)NowwearereadytoderivetotalcostsasCðv,w,qÞ¼vkþwl¼q1=ðαþβÞBvα=ðαþβÞwβ=ðαþβÞ,(10.25)whereB¼ðαþβÞαα=ðαþβÞββ=ðαþβÞ—aconstantthatinvolvesonlytheparametersαandβ.Althoughthisderivationwasabitmessy,severalinterestingaspectsofthisCobb-Douglascostfunctionarereadilyapparent.First,whetherthefunctionisaconvex,linear,orconcavefunctionofoutputdependsonwhethertherearedecreasingreturnstoscaleðαþβ<1Þ,constantreturnstoscaleðαþβ¼1Þ,orincreasingreturnstoscaleðαþβ>1Þ.Second,anincreaseinanyinputpriceincreasescosts,withtheextentoftheincreasebeingdeterminedbytherelativeimportanceoftheinputasreﬂectedbythesizeofitsexponentintheproductionfunction.Finally,thecostfunctionishomogeneousofdegree1intheinputprices—ageneralfeatureofallcostfunctions,asweshallshowshortly.3.CES:q¼fðk,lÞ¼ðkρþlρÞγ=ρ.Forthiscase,yourauthorwillmercifullyspareyouthealgebra.Toderivethetotalcostfunction,weusethecost-minimizationconditionspeciﬁedinEquation10.15,solveforeachinputindividually,andeventuallygetCðv,w,qÞ¼vkþwl¼q1=γðvρ=ðρ1Þþwρ=ðρ1ÞÞðρ1Þ=ρ¼q1=γðv1σþw1σÞ1=ð1σÞ,(10.26)wheretheelasticityofsubstitutionisgivenbyσ¼1=ð1ρÞ.OnceagaintheshapeofthetotalcostisdeterminedbythescaleparameterðγÞforthisproductionfunction,andthecostfunctionisincreasinginbothoftheinputprices.Thefunctionisalsohomogeneousofdegree1inthoseprices.OnelimitingfeatureofthisformoftheCESfunctionisthatthe(continued)Chapter10CostFunctions335
EXAMPLE10.2CONTINUEDinputsaregivenequalweights—hencetheirpricesareequallyimportantinthecostfunc-tion.ThisfeatureoftheCESiseasilygeneralized,however(seeProblem10.7).QUERY:HowarethevarioussubstitutionpossibilitiesinherentintheCESfunctionreﬂectedintheCEScostfunctioninEquation10.26?PropertiesofcostfunctionsTheseexamplesillustratesomepropertiesoftotalcostfunctionsthatarequitegeneral.1.Homogeneity.ThetotalcostfunctionsinExample10.3areallhomogeneousofdegree1intheinputprices.Thatis,adoublingofinputpriceswillpreciselydoublethecostofproducinganygivenoutputlevel(youmightcheckthisoutforyourself).Thisisapropertyofallpropercostfunctions.Whenallinputpricesdouble(orareincreasedbyanyuniformproportion),theratioofanytwoinputpriceswillnotchange.BecausecostminimizationrequiresthattheratioofinputpricesbesetequaltotheRTSalongagivenisoquant,thecost-minimizinginputcombinationalsowillnotchange.Hence,theﬁrmwillbuyexactlythesamesetofinputsandpaypreciselytwiceasmuchforthem.Oneimplicationofthisresultisthatapure,uniforminﬂationinallinputcostswillnotchangeaﬁrm’sinputdecisions.Itscostcurveswillshiftupwardinprecisecorrespon-dencetotherateofinﬂation.2.Totalcostfunctionsarenondecreasinginq,v,andw.Thispropertyseemsobvious,butitisworthdwellingonitabit.Becausecostfunctionsarederivedfromacost-minimizationprocess,anydeclineincostsfromanincreaseinoneofthefunction’sargumentswouldleadtoacontradiction.Forexample,ifanincreaseinoutputfromq1toq2causedtotalcoststodecline,itmustbethecasethattheﬁrmwasnotminimizingcostsintheﬁrstplace.Itshouldhaveproducedq2andthrownawayanoutputofq2q1,therebyproducingq1atalowercost.Similarly,ifanincreaseinthepriceofaninputeverreducedtotalcost,theﬁrmcouldnothavebeenminimizingitscostsintheﬁrstplace.Toseethis,supposetheﬁrmwasusingtheinputcombinationk1,l1andthatwincreases.Clearlythatwillincreasethecostoftheinitialinputcombination.Butifchangesininputchoicesactuallycausedtotalcoststodecline,thatmustimplythattherewasalower-costinputmixthank1,l1initially.Hencewehaveacontradiction,andthispropertyofcostfunctionsisestablished.73.Totalcostfunctionsareconcaveininputprices.Itisprobablyeasiesttoillustratethispropertywithagraph.Figure10.6showstotalcostsforvariousvaluesofaninputprice,say,w,holdingqandvconstant.Supposethatinitiallyawagerateofw1prevails7Aformalproofcouldalsobebasedontheenvelopetheoremasappliedtoconstrainedminimizationproblems.ConsidertheLagrangianexpressioninEquation10.3.AswaspointedoutinChapter2,wecancalculatethechangeintheobjectiveinsuchanexpression(here,totalcost)withrespecttoachangeinavariablebydifferentiatingtheLagrangianexpression.Performingthisdifferentiationyields∂C∂q¼∂ℒ∂q¼λð¼MCÞ0,∂C∂v¼∂ℒ∂v¼k0,∂C∂w¼∂ℒ∂w¼l0.Notonlydotheseenveloperesultsprovethispropertyofcostfunctions,theyalsoarequiteusefulintheirownright,aswewillshowlaterinthischapter.336Part3ProductionandSupply
andthatthetotalcostsassociatedwithproducingq1aregivenbyCðv,w1,q1Þ.Iftheﬁrmdidnotchangeitsinputmixinresponsetochangesinwages,thenitstotalcostcurvewouldbelinearasreﬂectedbythelineCPSEUDOð_v,w,q1Þ¼_v_k1þw_l1intheﬁgure.Butacost-minimizingﬁrmprobablywouldchangetheinputmixitusestoproduceq1whenwageschange,andtheseactualcosts½Cðv,w,q1Þwouldfallbelowthe“pseudo”costs.Hence,thetotalcostfunctionmusthavetheconcaveshapeshowninFigure10.6.Oneimplicationofthisﬁndingisthatcostswillbelowerwhenaﬁrmfacesinputpricesthatﬂuctuatearoundagivenlevelthanwhentheyremainconstantatthatlevel.Withﬂuctuatinginputprices,theﬁrmcanadaptitsinputmixtotakeadvantageofsuchﬂuctuationsbyusingalotof,say,laborwhenitspriceislowandeconomizingonthatinputwhenitspriceishigh.4.Averageandmarginalcosts.Some,butnotall,ofthesepropertiesoftotalcostfunctionscarryovertotheirrelatedaverageandmarginalcostfunctions.Homogeneityisonepropertythatcarriesoverdirectly.BecauseCðtv,tw,qÞ¼tCðv,w,qÞ,wehaveACtv,tw,qðÞ¼Cðtv,tw,qÞq¼tCðv,w,qÞq¼tACðv,w,qÞ(10.27)and8MCtv,tw,qðÞ¼∂Cðtv,tw,qÞ∂q¼t∂Cðv,w,qÞ∂q¼tMCðv,w,qÞ.(10.28)FIGURE10.6CostFunctionsAreConcaveinInputPricesWithawagerateofw1,totalcostsofproducingq1areCðv,w1,q1Þ.Iftheﬁrmdoesnotchangeitsinputmix,costsofproducingq1wouldfollowthestraightlineCPSEUDO.Withinputsubstitution,actualcostsCðv,w,q1Þwillfallbelowthisline,andhencethecostfunctionisconcaveinw.CostsC(v,w,q1)C(v,w1,q1)ww1CPSEUDO8Thisresultdoesnotviolatethetheoremthatthederivativeofafunctionthatishomogeneousofdegreekishomogeneousofdegreek−1,becausewearedifferentiatingwithrespecttoqandtotalcostsarehomogeneouswithrespecttoinputpricesonly.Chapter10CostFunctions337
Theeffectsofchangesinq,v,andwonaverageandmarginalcostsaresometimesambiguous,however.Wehavealreadyshownthataverageandmarginalcostcurvesmayhavenegativelyslopedsegments,soneitherACnorMCisnondecreasinginq.Becausetotalcostsmustnotdecreasewhenaninputpricerises,itisclearthataveragecostisincreasinginwandv.Butthecaseofmarginalcostismorecomplex.Themaincomplicationarisesbecauseofthepossibilityofinputinferiority.Inthat(admittedlyrare)case,anincreaseinaninferiorinput’spricewillactuallycausemarginalcosttodecline.Althoughtheproofofthisisrelativelystraightforward,9anintuitiveexpla-nationforitiselusive.Still,inmostcases,itseemsclearthattheincreaseinthepriceofaninputwillincreasemarginalcostaswell.InputsubstitutionAchangeinthepriceofaninputwillcausetheﬁrmtoalteritsinputmix.Hence,afullstudyofhowcostcurvesshiftwheninputpriceschangemustalsoincludeanexaminationofsubstitutionamonginputs.Tostudythisprocess,economistshavedevelopedasomewhatdifferentmeasureoftheelasticityofsubstitutionthantheoneweencounteredinthetheoryofproduction.Speciﬁcally,wewishtoexaminehowtheratioofinputusage(k=l)changesinresponsetoachangeinw=v,whileholdingqconstant.Thatis,wewishtoexaminethederivative∂ðk=lÞ∂ðw=vÞ(10.29)alonganisoquant.Puttingthisinproportionaltermsass¼∂ðk=lÞ∂ðw=vÞ⋅w=vk=l¼∂lnk=l∂lnw=v(10.30)givesanalternativeandmoreintuitivedeﬁnitionoftheelasticityofsubstitution.10Inthetwo-inputcase,smustbenonnegative;anincreaseinw=vwillbemetbyanincreaseink=l(or,inthelimitingﬁxed-proportionscase,k=lwillstayconstant).Largevaluesofsindicatethatﬁrmschangetheirinputproportionssigniﬁcantlyinresponsetochangesinrelativeinputprices,whereaslowvaluesindicatethatchangesininputpriceshaverelativelylittleeffect.SubstitutionwithmanyinputsWhenthereareonlytwoinputs,theelasticityofsubstitutiondeﬁnedinEquation10.30isidenticaltotheconceptwedeﬁnedinChapter9(seeEquation9.32).Thiscanbeshownbyrememberingthatcostminimization11requiresthattheﬁrmequateitsRTS(oflfork)totheinputpriceratiow=v.ThemajoradvantageofthedeﬁnitionoftheelasticityofsubstitutioninEquation10.30isthatitiseasiertogeneralizetomanyinputsthanisthedeﬁnitionbasedontheproductionfunction.Speciﬁcally,supposetherearemanyinputstotheproductionprocessðx1,x2,…,xnÞthatcanbehiredatcompetitiverentalratesðw1,w2,…,wnÞ.ThentheelasticityofsubstitutionbetweenanytwoinputsðsijÞisdeﬁnedasfollows.9Theprooffollowstheenvelopetheoremresultspresentedinfootnote7.BecausetheMCfunctioncanbederivedbydifferentiationfromtheLagrangianforcostminimization,wecanuseYoung’stheoremtoshow∂MC∂v¼∂ð∂ℒ=∂qÞ∂v¼∂2ℒ∂v∂q¼∂2ℒ∂q∂v¼∂k∂q.Hence,ifcapitalisanormalinput,anincreaseinvwillraiseMCwhereas,ifcapitalisinferior,anincreaseinvwillactuallyreduceMC.10ThisdeﬁnitionisusuallyattributedtoR.G.D.Allen,whodevelopeditinanalternativeforminhisMathematicalAnalysisforEconomists(NewYork:St.Martin’sPress,1938),pp.504–9.11InExample10.1wefoundthat,fortheCESproductionfunction,costminimizationrequiresthatk=l¼ðw=vÞσ,solnðk=lÞ¼σlnðw=vÞandthereforesk,l¼∂lnðk=lÞ=∂lnðw=vÞ¼σ.338Part3ProductionandSupply
DEFINITIONElasticityofsubstitution.Theelasticityofsubstitution12betweeninputsxiandxjisgivenbysi,j¼∂ðxi=xjÞ∂ðwj=wiÞ⋅wj=wixi=xj¼∂lnðxi=xjÞ∂lnðwj=wiÞ,(10.31)whereoutputandallotherinputpricesareheldconstant.Themajoradvantageofthisdeﬁnitioninamulti-inputcontextisthatitprovidestheﬁrmwiththeﬂexibilitytoadjustinputsotherthanxiandxj(whileholdingoutputconstant)wheninputpriceschange.Forexample,amajortopicinthetheoryofﬁrms’inputchoicesistodescribetherelationshipbetweencapitalandenergyinputs.ThedeﬁnitioninEquation10.31wouldpermitaresearchertostudyhowtheratioofenergytocapitalinputchangeswhenrelativeenergypricesrisewhilepermittingtheﬁrmtomakeanyadjustmentstolaborinput(whosepricehasnotchanged)thatwouldberequiredforcostminimization.Hencethiswouldgivearealisticpictureofhowﬁrmsactuallybehavewithregardtowhetherenergyandcapitalaremorelikesubstitutesorcomplements.Laterinthischapterwewilllookatthisdeﬁnitioninabitmoredetail,becauseitiswidelyusedinempiricalstudiesofproduction.QuantitativesizeofshiftsincostcurvesWehavealreadyshownthatincreasesinaninputpricewillraisetotal,average,and(exceptintheinferiorinputcase)marginalcosts.Wearenowinapositiontojudgetheextentofsuchincreases.First,andmostobviously,theincreaseincostswillbeinﬂuencedimportantlybytherelativesigniﬁcanceoftheinputintheproductionprocess.Ifaninputconstitutesalargefractionoftotalcosts,anincreaseinitspricewillraisecostssigniﬁcantly.Ariseinthewageratewouldsharplyincreasehome-builders’costs,becauselaborisamajorinputinconstruc-tion.Ontheotherhand,apriceriseforarelativelyminorinputwillhaveasmallcostimpact.Anincreaseinnailpriceswillnotraisehomecostsverymuch.Alessobviousdeterminantoftheextentofcostincreasesisinputsubstitutability.Ifﬁrmscaneasilysubstituteanotherinputfortheonethathasriseninprice,theremaybelittleincreaseincosts.Increasesincopperpricesinthelate1960s,forexample,hadlittleimpactonelectricutilities’costsofdistributingelectricity,becausetheyfoundtheycouldeasilysubsti-tutealuminumforcoppercables.Alternatively,iftheﬁrmﬁndsitdifﬁcultorimpossibletosubstitutefortheinputthathasbecomemorecostly,thencostsmayriserapidly.Thecostofgoldjewelry,alongwiththepriceofgold,roserapidlyduringtheearly1970s,becausetherewassimplynosubstitutefortherawinput.Itispossibletogiveaprecisemathematicalstatementofthequantitativesizesofalloftheseeffectsbyusingtheelasticityofsubstitution.Todoso,however,wouldriskfurtherclutteringthebookwithsymbols.13Forourpurposes,itissufﬁcienttorelyonthepreviousintuitivediscussion.Thisshouldserveasareminderthatchangesinthepriceofaninputwillhavetheeffectofshiftingﬁrms’costcurves,withthesizeoftheshiftdependingontherelativeimportanceoftheinputandonthesubstitutionpossibilitiesthatareavailable.TechnicalchangeTechnicalimprovementsallowtheﬁrmtoproduceagivenoutputwithfewerinputs.Suchimprovementsobviouslyshifttotalcostsdownward(ifinputpricesstayconstant).Although12ThisdeﬁnitionisattributedtotheJapaneseeconomistM.Morishima,andtheseelasticitiesaresometimesreferredtoas“Morishimaelasticities.”Inthisversion,theelasticityofsubstitutionforsubstituteinputsispositive.SomeauthorsreversetheorderofsubscriptsinthedenominatorofEquation10.31,andinthisusagetheelasticityofsubstitutionforsubstituteinputsisnegative.13ForacompletestatementseeFerguson,NeoclassicalTheoryofProductionandDistribution(Cambridge:CambridgeUniversityPress,1969),pp.154–60.Chapter10CostFunctions339
theactualwayinwhichtechnicalchangeaffectsthemathematicalformofthetotalcostcurvecanbecomplex,therearecaseswhereonemaydrawsimpleconclusions.Suppose,forexample,thattheproductionfunctionexhibitsconstantreturnstoscaleandthattechnicalchangeentersthatfunctionasdescribedinChapter9(thatis,q¼AðtÞfðk,lÞwhereAð0Þ¼1Þ.Inthiscase,totalcostsintheinitialperiodaregivenbyC0¼C0ðv,w,qÞ¼qC0ðv,w,1Þ.(10.32)Becausethesameinputsthatproducedoneunitofoutputinperiod0willproduceAðtÞunitsofoutputinperiodt,weknowthatCtðv,w,AðtÞÞ¼AðtÞCtðv,w,1Þ¼C0ðv,w,1Þ;(10.33)therefore,wecancomputethetotalcostfunctioninperiodtasCtv,w,qðÞ¼qCtv,w,1ðÞ¼qC0ðv,w,1ÞAðtÞ¼C0ðv,w,qÞAðtÞ.(10.34)Hence,totalcostsfallovertimeattherateoftechnicalchange.Notethatinthiscasetechnicalchangeis“neutral”inthatitdoesnotaffecttheﬁrm’sinputchoices(solongasinputpricesstayconstant).Thisneutralityresultmightnotholdincaseswheretechnicalprogresstakesamorecomplexformorwheretherearevariablereturnstoscale.Eveninthesemorecomplexcases,however,technicalimprovementswillcausetotalcoststofall.EXAMPLE10.3ShiftingtheCobb-DouglasCostFunctionInExample10.2wecomputedtheCobb-DouglascostfunctionasCðv,w,qÞ¼q1=ðαþβÞBvα=ðαþβÞwβ=ðαþβÞ,(10.35)whereB¼ðαþβÞαα=ðαþβÞββ=ðαþβÞ.AsinthenumericalillustrationinExample10.1,let’sassumethatα¼β¼0.5,inwhichcasethetotalcostfunctionisgreatlysimpliﬁed:Cðv,w,qÞ¼2qv0.5w0.5.(10.36)Thisfunctionwillyieldatotalcostcurverelatingtotalcostsandoutputifwespecifyparticularvaluesfortheinputprices.If,asbefore,weassumev¼3andw¼12,thentherelationshipisCð3,12,qÞ¼2qﬃﬃﬃﬃﬃﬃ36p¼12q,(10.37)and,asinExample10.1,itcosts480toproduce40unitsofoutput.HereaverageandmarginalcostsareeasilycomputedasAC¼Cq¼12,MC¼∂C∂q¼12.(10.38)Asexpected,averageandmarginalcostsareconstantandequaltoeachotherforthisconstantreturns-to-scaleproductionfunction.Changesininputprices.Ifeitherinputpriceweretochange,allofthesecostswouldchangealso.Forexample,ifwagesweretoincreaseto27(aneasynumberwithwhichtowork),costswouldbecomeCð3,27,qÞ¼2qﬃﬃﬃﬃﬃﬃ81p¼18q,AC¼18,MC¼18.(10.39)Noticethatanincreaseinwagesof125percentraisedcostsbyonly50percenthere,bothbecauselaborrepresentsonly50percentofallcostsandbecausethechangeininputpricesencouragedtheﬁrmtosubstitutecapitalforlabor.Thetotalcostfunction,becauseitis340Part3ProductionandSupply
derivedfromthecost-minimizationassumption,accomplishesthissubstitution“behindthescenes”—reportingonlytheﬁnalimpactontotalcosts.Technicalprogress.Let’slooknowattheimpactthattechnicalprogresscanhaveoncosts.Speciﬁcally,assumethattheCobb-Douglasproductionfunctionisq¼AðtÞk0.5l0.5¼e.03tk0.5l0.5.(10.40)Thatis,weassumethattechnicalchangetakesanexponentialformandthattherateoftech-nicalchangeis3percentperyear.Usingtheresultsoftheprevioussection(Equation10.34)yieldsCtv,w,qðÞ¼C0ðv,w,qÞAðtÞ¼2qv0.5w0.5e.03t:(10.41)So,ifinputpricesremainthesamethentotalcostsfallattherateoftechnicalimprovement—thatis,at3percentperyear.After,say,20years,costswillbe(withv¼3,w¼12)C20ð3,12,qÞ¼2qﬃﬃﬃﬃﬃﬃ36p⋅e.60¼12q⋅ð0.55Þ¼6.6q,AC20¼6.6,MC20¼6.6.(10.42)Consequently,costswillhavefallenbynearly50percentasaresultofthetechnicalchange.Thiswould,forexample,morethanhaveoffsetthewageriseillustratedpreviously.QUERY:Inthisexample,whataretheelasticitiesoftotalcostswithrespecttochangesininputcosts?Isthesizeoftheseelasticitiesaffectedbytechnicalchange?ContingentdemandforinputsandShephard’slemmaAswedescribedearlier,theprocessofcostminimizationcreatesanimplicitdemandforinputs.Becausethatprocessholdsquantityproducedconstant,thisdemandforinputswillalsobe“contingent”onthequantitybeingproduced.Thisrelationshipisfullyreﬂectedintheﬁrm’stotalcostfunctionand,perhapssurprisingly,contingentdemandfunctionsforalloftheﬁrm’sinputscanbeeasilyderivedfromthatfunction.TheprocessinvolveswhathascometobecalledShephard’slemma,14whichstatesthatthecontingentdemandfunctionforanyinputisgivenbythepartialderivativeofthetotalcostfunctionwithrespecttothatinput’sprice.BecauseShephard’slemmaiswidelyusedinmanyareasofeconomicresearch,wewillprovidearelativelydetailedexaminationofit.TheintuitionbehindShephard’slemmaisstraightforward.Supposethatthepriceoflabor(w)weretoincreaseslightly.Howwouldthisaffecttotalcosts?Ifnothingelsechanged,itseemsthatcostswouldrisebyapproximatelytheamountoflaborðlÞthattheﬁrmwascurrentlyhiring.Roughlyspeaking,then,∂C=∂w¼l,andthatiswhatShephard’slemmaclaims.Figure10.6makesroughlythesamepointgraphically.Alongthe“pseudo”costfunctionallinputsareheldconstant,soanincreaseinthewageincreasescostsindirectproportiontotheamountoflaborused.Becausethetruecostfunctionistangenttothepseudo-functionatthecurrentwage,itsslope(thatis,itspartialderivative)alsowillshowthecurrentamountoflaborinputdemanded.Technically,Shephard’slemmaisoneresultoftheenvelopetheoremthatwasﬁrstdiscussedinChapter2.Thereweshowedthatthechangeintheoptimalvalueinaconstrainedoptimizationproblemwithrespecttooneoftheparametersoftheproblemcanbefoundby14NamedforR.W.Shephard,whohighlightedtheimportantrelationshipbetweencostfunctionsandinputdemandfunctionsinhisCostandProductionFunctions(Princeton,NJ:PrincetonUniversityPress,1970).Chapter10CostFunctions341
differentiatingtheLagrangianexpressionforthatoptimizationproblemwithrespecttothischangingparameter.Inthecost-minimizationcase,theLagrangianexpressionisℒ¼vkþwlþλ½_qfðk,lÞ(10.43)andtheenvelopetheoremappliedtoeitherinputis∂Cðv,w,qÞ∂v¼∂ℒðv,w,q,λÞ∂v¼kcðv,w,qÞ,∂Cðv,w,qÞ∂w¼∂ℒðv,w,q,λÞ∂w¼lcðv,w,qÞ,(10.44)wherethenotationisintendedtomakeclearthattheresultingdemandfunctionsforcapitalandlaborinputdependonv,w,andq.Becausequantityproducedentersthesefunctions,inputdemandisindeedcontingentonthatvariable.Thisfeatureofthedemandfunctionsisalsoreﬂectedbythe“c”inthenotation.15Hence,thedemandrelationsinEquation10.44donotrepresentacompletepictureofinputdemandbecausetheystilldependonavariablethatisundertheﬁrm’scontrol.Inthenextchapter,wewillcompletethestudyofinputdemandbyshowinghowtheassumptionofproﬁtmaximizationallowsustoeffectivelyreplaceqintheinputdemandrelationshipswiththemarketpriceoftheﬁrm’soutput,p.EXAMPLE10.4ContingentInputDemandFunctionsInthisexample,wewillshowhowthetotalcostfunctionsderivedinExample10.2canbeusedtoderivecontingentdemandfunctionsfortheinputscapitalandlabor.1.FixedProportions:Cðv,w,qÞ¼qðv=aþw=bÞ.Forthiscostfunction,contingentde-mandfunctionsarequitesimple:kcðv,w,qÞ¼∂Cðv,w,qÞ∂v¼qa,lcðv,w,qÞ¼∂Cðv,w,qÞ∂w¼qb.(10.45)Inordertoproduceanyparticularoutputwithaﬁxedproportionsproductionfunctionatminimalcost,theﬁrmmustproduceatthevertexofitsisoquantsnomatterwhattheinputs’pricesare.Hence,thedemandforinputsdependsonlyonthelevelofoutput,andvandwdonotenterthecontingentinputdemandfunctions.Inputpricesmay,however,affecttotalinputdemandsintheﬁxedproportionscasebecausetheymayaffecthowmuchtheﬁrmcansell.2.Cobb-Douglas:Cðv,w,qÞ¼q1=ðαþβÞBvα=ðαþβÞwβ=ðαþβÞ.Inthiscase,thederivationismessierbutalsomoreinstructive:kcðv,w,qÞ¼∂C∂v¼ααþβ⋅q1=ðαþβÞBvβ=ðαþβÞwβ=ðαþβÞ¼ααþβ⋅q1=ðαþβÞBwvβ=ðαþβÞ,lcðv,w,qÞ¼∂C∂w¼βαþβ⋅q1=ðαþβÞBvα=ðαþβÞwα=ðαþβÞ¼βαþβ⋅q1=ðαþβÞBwvα=ðαþβÞ:(10.46)15ThenotationmirrorsthatusedforcompensateddemandcurvesinChapter5(whichwerederivedfromtheexpenditurefunction).Inthatcase,suchdemandfunctionswerecontingentontheutilitytargetassumed.342Part3ProductionandSupply
Consequently,thecontingentdemandsforinputsdependonbothinputs’prices.Ifweassumeα¼β¼0.5(soB¼2),thesereducetokcv,w,qðÞ¼0.5⋅q⋅2⋅wv0.5¼qwv0.5,lcv,w,qðÞ¼0.5⋅q⋅2⋅wv0.5¼qwv0.5.(10.47)Withv¼3,w¼12,andq¼40,Equations10.47yieldtheresultweobtainedpreviously:thattheﬁrmshouldchoosetheinputcombinationk¼80,l¼20tominimizethecostofproducing40unitsofoutput.Ifthewageweretoriseto,say,27,theﬁrmwouldchoosetheinputcombinationk¼120,l¼40=3toproduce40unitsofoutput.Totalcostswouldrisefrom480to520,buttheabilityoftheﬁrmtosubstitutecapitalforthenowmoreexpensivelabordoessaveconsiderably.Forexample,theinitialinputcombinationwouldnowcost780.3.CES:Cðv,w,qÞ¼q1=γðv1σþw1σÞ1=ð1σÞ.TheimportanceofinputsubstitutionisshownevenmoreclearlywiththecontingentdemandfunctionsderivedfromtheCESfunction.Forthatfunction,kcðv,w,qÞ¼∂C∂v¼11σ⋅q1=γðv1σþw1σÞσ=ð1σÞð1σÞvσ¼q1=γðv1σþw1σÞσ=ð1σÞvσ,lcðv,w,qÞ¼∂C∂w¼11σ⋅q1=γðv1σþw1σÞσ=ð1σÞð1σÞwσ¼q1=γðv1σþw1σÞσ=ð1σÞwσ.(10.48)Thesefunctionscollapsewhenσ¼1(theCobb-Douglascase),butwecanstudyexampleswitheithermoreðσ¼2Þorlessðσ¼0.5ÞsubstitutabilityanduseCobb-Douglasasthemiddleground.Ifweassumeconstantreturnstoscaleðγ¼1Þandv¼3,w¼12,andq¼40,thencontingentdemandsfortheinputswhenσ¼2arekcð3,12,40Þ¼40ð31þ121Þ2⋅32¼25:6,lcð3,12,40Þ¼40ð31þ121Þ2⋅122¼1:6:(10.49)Thatis,thelevelofcapitalinputis16timestheamountoflaborinput.Withlesssub-stitutabilityðσ¼0.5Þ,contingentinputdemandsarekcð3,12,40Þ¼40ð30:5þ120:5Þ1⋅30:5¼120,lcð3,12,40Þ¼40ð30:5þ120:5Þ1⋅120:5¼60.(10.50)So,inthiscase,capitalinputisonlytwiceaslargeaslaborinput.Althoughthesevariouscasescannotbecompareddirectlybecausedifferentvaluesforσscaleoutputdifferently,wecan,asanexample,lookattheconsequenceofariseinwto27inthelow-substitutabilitycase.Withw¼27,theﬁrmwillchoosek¼160,l¼53.3.Inthiscase,thecostsavingsfromsubstitutioncanbecalculatedbycomparingtotalcostswhenusingtheinitialinputcombination(¼ð3Þ120þ27ð60Þ¼1980)tototalcostswiththeoptimalcombination(¼ð3Þ160þ27ð53:3Þ¼1919).Hence,movingtotheoptimalinputcombinationreducestotalcostsbyonlyabout3percent.IntheCobb-Douglascase,costsavingsareover20percent.QUERY:Howwouldtotalcostschangeifwincreasedfrom12to27andtheproductionfunctiontookthesimplelinearformq¼kþ4l?Whatlightdoesthisresultshedontheothercasesinthisexample?Chapter10CostFunctions343
SHEPHARD’SLEMMAANDTHEELASTICITYOFSUBSTITUTIONOneespeciallynicefeatureofShephard’slemmaisthatitcanbeusedtoshowhowtoderiveinformationaboutinputsubstitutiondirectlyfromthetotalcostfunctionthroughdifferenti-ation.UsingthedeﬁnitioninEquation10.31yieldssi,j¼∂lnðxi=xjÞ∂lnðwj=wiÞ¼∂lnðCi=CjÞ∂lnðwj=wiÞ,(10.51)whereCiandCjarethepartialderivativesofthetotalcostfunctionwithrespecttotheinputprices.Oncethetotalcostfunctionisknown(perhapsthrougheconometricestimation),informationaboutsubstitutabilityamonginputscanthusbereadilyobtainedfromit.IntheExtensionstothischapter,wedescribesomeoftheresultsthathavebeenobtainedinthisway.Problems10.11and10.12providesomeadditionaldetailsaboutwaysinwhichsub-stitutabilityamonginputscanbemeasured.SHORT-RUN,LONG-RUNDISTINCTIONItistraditionalineconomicstomakeadistinctionbetweenthe“shortrun”andthe“longrun.”Althoughnoveryprecisetemporaldeﬁnitioncanbeprovidedfortheseterms,thegeneralpurposeofthedistinctionistodifferentiatebetweenashortperiodduringwhicheconomicactorshaveonlylimitedﬂexibilityintheiractionsandalongerperiodthatprovidesgreaterfreedom.Oneareaofstudyinwhichthisdistinctionisquiteimportantisinthetheoryoftheﬁrmanditscosts,becauseeconomistsareinterestedinexaminingsupplyreactionsoverdifferingtimeintervals.Intheremainderofthischapter,wewillexaminetheimplicationsofsuchdifferentialresponse.Toillustratewhyshort-runandlong-runreactionsmightdiffer,assumethatcapitalinputisheldﬁxedatalevelofk1andthat(intheshortrun)theﬁrmisfreetovaryonlyitslaborinput.16Implicitly,weareassumingthatalterationsinthelevelofcapitalinputareinﬁnitelycostlyintheshortrun.Asaresultofthisassumption,theshort-runproductionfunctionisq¼fðk1,lÞ,(10.52)wherethisnotationexplicitlyshowsthatcapitalinputsmaynotvary.Ofcourse,thelevelofoutputstillmaybechangediftheﬁrmaltersitsuseoflabor.Short-runtotalcostsTotalcostfortheﬁrmcontinuestobedeﬁnedasC¼vkþwl(10.53)forourshort-runanalysis,butnowcapitalinputisﬁxedatk1.Todenotethisfact,wewillwriteSC¼vk1þwl,(10.54)wheretheSindicatesthatweareanalyzingshort-runcostswiththelevelofcapitalinputﬁxed.Throughoutouranalysis,wewillusethismethodtoindicateshort-runcosts,whereaslong-runcostswillbedenotedbyC,AC,andMC.Usuallywewillnotdenotethelevelofcapitalinputexplicitly,butitisunderstoodthatthisinputisﬁxed.16Ofcourse,thisapproachisforillustrativepurposesonly.Inmanyactualsituations,laborinputmaybelessﬂexibleintheshortrunthaniscapitalinput.344Part3ProductionandSupply
FixedandvariablecostsThetwotypesofinputcostsinEquation8.53aregivenspecialnames.Thetermvk1isreferredtoas(short-run)ﬁxedcosts;becausek1isconstant,thesecostswillnotchangeintheshortrun.Thetermwlisreferredtoas(short-run)variablecosts—laborinputcanindeedbevariedintheshortrun.Hencewehavethefollowingdeﬁnitions.DEFINITIONShort-runfixedandvariablecosts.Short-runﬁxedcostsarecostsassociatedwithinputsthatcannotbevariedintheshortrun.Short-runvariablecostsarecostsofthoseinputsthatcanbevariedsoastochangetheﬁrm’soutputlevel.Theimportanceofthisdistinctionistodifferentiatebetweenvariablecoststhattheﬁrmcanavoidbyproducingnothingintheshortrunandcoststhatareﬁxedandmustbepaidregardlessoftheoutputlevelchosen(evenzero).Nonoptimalityofshort-runcostsItisimportanttounderstandthattotalshort-runcostsarenottheminimalcostsforproducingthevariousoutputlevels.Becauseweareholdingcapitalﬁxedintheshortrun,theﬁrmdoesnothavetheﬂexibilityofinputchoicethatweassumedwhenwediscussedcostminimizationearlierinthischapter.Rather,tovaryitsoutputlevelintheshortrun,theﬁrmwillbeforcedtouse“nonoptimal”inputcombinations:TheRTSwillnotbeequaltotheratiooftheinputprices.ThisisshowninFigure10.7.Intheshortrun,theﬁrmisconstrainedtousek1unitsofcapital.Toproduceoutputlevelq0,itthereforewillusel0unitsoflabor.Similarly,itwillusel1unitsoflabortoproduceq1andl2unitstoproduceq2.ThetotalcostsoftheseinputcombinationsaregivenbySC0,SC1,andSC2,respectively.Onlyfortheinputcombinationk1,l1isoutputbeingproducedatminimalcost.OnlyatthatpointistheRTSequaltotheratiooftheinputprices.FromFigure10.7,itisclearthatq0isbeingproducedwith“toomuch”capitalinthisshort-runsituation.Costminimizationshouldsuggestasoutheasterlymovementalongtheq0isoquant,indicatingasubstitutionoflaborforcapitalinproduction.Similarly,q2isbeingproducedwith“toolittle”capital,andcostscouldbereducedbysubstitutingcapitalforlabor.Neitherofthesesubstitutionsispossibleintheshortrun.Overalongerperiod,however,theﬁrmwillbeabletochangeitslevelofcapitalinputandwilladjustitsinputusagetothecost-minimizingcombina-tions.Wehavealreadydiscussedthisﬂexiblecaseearlierinthischapterandshallreturntoittoillustratetheconnectionbetweenlong-runandshort-runcostcurves.Short-runmarginalandaveragecostsFrequently,itismoreusefultoanalyzeshort-runcostsonaper-unit-of-outputbasisratherthanonatotalbasis.Thetwomostimportantper-unitconceptsthatcanbederivedfromtheshort-runtotalcostfunctionaretheshort-runaveragetotalcostfunction(SAC)andtheshort-runmarginalcostfunction(SMC).TheseconceptsaredeﬁnedasSAC¼totalcoststotaloutput¼SCq,SMC¼changeintotalcostschangeinoutput¼∂SC∂q,(10.55)whereagainthesearedeﬁnedforaspeciﬁedlevelofcapitalinput.Thesedeﬁnitionsforaverageandmarginalcostsareidenticaltothosedevelopedpreviouslyforthelong-run,fullyﬂexiblecase,andthederivationofcostcurvesfromthetotalcostfunctionproceedsinexactlythesameway.Becausetheshort-runtotalcostcurvehasthesamegeneraltypeofcubicshapeasdidthetotalcostcurveinFigure10.5,theseshort-runaverageandmarginalcostcurveswillalsobeU-shaped.Chapter10CostFunctions345
Relationshipbetweenshort-runandlong-runcostcurvesItiseasytodemonstratetherelationshipbetweentheshort-runcostsandthefullyﬂexiblelong-runcoststhatwerederivedpreviouslyinthischapter.Figure10.8showsthisrelation-shipforboththeconstantreturns-to-scaleandcubictotalcostcurvecases.Short-runtotalcostsforthreelevelsofcapitalinputareshown,althoughofcourseitwouldbepossibletoshowmanymoresuchshort-runcurves.Theﬁguresshowthatlong-runtotalcostsðCÞarealwayslessthanshort-runtotalcosts,exceptatthatoutputlevelforwhichtheassumedﬁxedcapitalinputisappropriatetolong-runcostminimization.Forexample,asinFigure10.7,withcapitalinputofk1theﬁrmcanobtainfullcostminimizationwhenq1isproduced.Hence,short-runandlong-runtotalcostsareequalatthispoint.Foroutputlevelsotherthanq1,however,SC>C,aswasthecaseinFigure10.7.Technically,thelong-runtotalcostcurvesinFigure10.8aresaidtobean“envelope”oftheirrespectiveshort-runcurves.Theseshort-runtotalcostcurvescanberepresentedpara-metricallybyshort-runtotalcost¼SCðv,w,q,kÞ,(10.56)andthefamilyofshort-runtotalcostcurvesisgeneratedbyallowingktovarywhileholdingvandwconstant.Thelong-runtotalcostcurveCmustobeytheshort-runrelationshipinEquation10.56andthefurtherconditionthatkbecostminimizingforanylevelofoutput.Aﬁrst-orderconditionforthisminimizationisthatFIGURE10.7“Nonoptimal”InputChoicesMustBeMadeintheShortRunBecausecapitalinputisﬁxedatk,intheshortruntheﬁrmcannotbringitsRTSintoequalitywiththeratioofinputprices.Giventheinputprices,q0shouldbeproducedwithmorelaborandlesscapitalthanitwillbeintheshortrun,whereasq2shouldbeproducedwithmorecapitalandlesslaborthanitwillbe.SC0l2l1k1l0q2q1q0SC1= CSC2k perperiodl per period346Part3ProductionandSupply
∂SCðv,w,q,kÞ∂k¼0.(10.57)SolvingEquations10.56and10.57simultaneouslythengeneratesthelong-runtotalcostfunction.Althoughthisisadifferentapproachtoderivingthetotalcostfunction,itshouldgivepreciselythesameresultsderivedearlierinthischapter—asthenextexampleillustrates.FIGURE10.8TwoPossibleShapesforLong-RunTotalCostCurvesByconsideringallpossiblelevelsofcapitalinput,thelong-runtotalcostcurve(C)canbetraced.In(a),theunderlyingproductionfunctionexhibitsconstantreturnstoscale:inthelongrun,thoughnotintheshortrun,totalcostsareproportionaltooutput.In(b),thelong-runtotalcostcurvehasacubicshape,asdotheshort-runcurves.Diminishingreturnssetinmoresharplyfortheshort-runcurves,however,becauseoftheassumedﬁxedlevelofcapitalinput.TotalcostsTotalcosts(a) Constant returns to scale(b) Cubic total cost curve caseOutput per periodOutputper periodSC (k0)SC (k0)q0q1q2q0q1q2CCSC (k1)SC (k1)SC (k2)SC (k2)
EXAMPLE10.5EnvelopeRelationsandCobb-DouglasCostFunctionsAgainwestartwiththeCobb-Douglasproductionfunctionq¼kαlβ,butnowweholdcapitalinputconstantatk1.So,intheshortrun,q¼kα1lβorl¼q1=βkα=β1,(10.58)andtotalcostsaregivenbySCðv,w,q,k1Þ¼vk1þwl¼vk1þwq1=βkα=β1.(10.59)Noticethattheﬁxedlevelofcapitalentersintothisshort-runtotalcostfunctionintwoways:(1)k1determinesﬁxedcosts;and(2)k1alsoinpartdeterminesvariablecostsbecauseitdetermineshowmuchofthevariableinput(labor)isrequiredtoproducevariouslevelsofoutput.Toderivelong-runcosts,werequirethatkbechosentominimizetotalcosts:∂SCðv,w,q,kÞ∂k¼vþαβ⋅wq1=βkðαþβÞ=β¼0.(10.60)Althoughthealgebraismessy,thisequationcanbesolvedforkandsubstitutedintoEquation10.59toreturnustotheCobb-Douglascostfunction:Cðv,w,qÞ¼Bq1=ðαþβÞvα=ðαþβÞwβ=ðαþβÞ.(10.61)Numericalexample.Ifweagainletα¼β¼0.5,v¼3,andw¼12,thentheshort-runcostfunctionisSCð3,12,q,kÞ¼3k1þ12q2k11.(10.62)InExample10.1wefoundthatthecost-minimizinglevelofcapitalinputforq¼40wask¼80.Equation10.62showsthatshort-runtotalcostsforproducing40unitsofoutputwithk¼80isSCð3,12,q,80Þ¼3.80þ12⋅q2⋅180¼240þ3q220¼240þ240¼480,(10.63)whichisjustwhatwefoundbefore.WecanalsouseEquation10.62toshowhowcostsdifferintheshortandlongrun.Table10.1showsthat,foroutputlevelsotherthanq¼40,short-runcostsarelargerthanlong-runcostsandthatthisdifferenceisproportionallylargerthefartheronegetsfromtheoutputlevelforwhichk¼80isoptimal.TABLE10.1DiﬀerencebetweenShort-RunandLong-RunTotalCost,k¼80qC¼12qSC¼240þ3q2=2010120255202403003036037540480480506006156072078070840975809601200348Part3ProductionandSupply
Itisalsoinstructivetostudydifferencesbetweenthelong-runandshort-runper-unitcostsinthissituation.HereAC¼MC¼12.Wecancomputetheshort-runequivalents(whenk¼80)asSAC¼SCq¼240qþ3q20,SMC¼∂SC∂q¼6q20.(10.64)Bothoftheseshort-rununitcostsareequalto12whenq¼40.However,asTable10.2shows,short-rununitcostscandiffersigniﬁcantlyfromthisﬁgure,dependingontheoutputlevelthattheﬁrmproduces.Noticeinparticularthatshort-runmarginalcostincreasesrapidlyasoutputexpandsbeyondq¼40becauseofdiminishingreturnstothevariableinput(labor).Thisconclusionplaysanimportantroleinthetheoryofshort-runpricedetermination.QUERY:Explainwhyanincreaseinwwillincreasebothshort-runaveragecostandshort-runmarginalcostinthisillustration,butanincreaseinvaffectsonlyshort-runaveragecost.Graphsofper-unitcostcurvesTheenvelopetotalcostcurverelationshipsexhibitedinFigure10.8canbeusedtoshowgeometricconnectionsbetweenshort-runandlong-runaverageandmarginalcostcurves.ThesearepresentedinFigure10.9forthecubictotalcostcurvecase.Intheﬁgure,short-runandlong-runaveragecostsareequalatthatoutputforwhichthe(ﬁxed)capitalinputisappropriate.Atq1,forexample,SACðk1Þ¼ACbecausek1isusedinproducingq1atminimalcosts.Formovementsawayfromq1,short-runaveragecostsexceedlong-runav-eragecosts,thusreﬂectingthecost-minimizingnatureofthelong-runtotalcostcurve.Becausetheminimumpointofthelong-runaveragecostcurve(AC)playsamajorroleinthetheoryoflong-runpricedetermination,itisimportanttonotethevariouscurvesthatpassthroughthispointinFigure10.9.First,asisalwaystrueforaverageandmarginalcostcurves,theMCcurvepassesthroughthelowpointoftheACcurve.Atq1,long-runaverageandmarginalcostsareequal.Associatedwithq1isacertainlevelofcapitalinput(say,k1);theshort-runaveragecostcurveforthislevelofcapitalinputistangenttotheACcurveatitsminimumpoint.TheSACcurvealsoreachesitsminimumatoutputlevelq1.Formovementsawayfromq1,theACcurveismuchﬂatterthantheSACcurve,andthisreﬂectsthegreaterﬂexibilityopentoﬁrmsinthelongrun.Short-runcostsriserapidlybecausecapitalinputsareﬁxed.Inthelongrun,suchinputsarenotﬁxed,anddiminishingmarginalproductivitiesdoTABLE10.2UnitCostsintheLongRunandtheShortRun,k¼80qACMCSACSMC10121225.5320121215.0630121212.5940121212.01250121212.31560121213.01870121213.92180121215.024Chapter10CostFunctions349
notoccursoabruptly.Finally,becausetheSACcurvereachesitsminimumatq1,theshort-runmarginalcostcurve(SMC)alsopassesthroughthispoint.TheminimumpointoftheACcurvethereforebringstogetherthefourmostimportantper-unitcosts:atthispoint,AC¼MC¼SAC¼SMC.(10.65)Forthisreason,asweshallshowinChapter12,theoutputlevelq1isanimportantequi-libriumpointforacompetitiveﬁrminthelongrun.SUMMARYInthischapterweexaminedtherelationshipbetweenthelevelofoutputaﬁrmproducesandtheinputcostsassociatedwiththatlevelofproduction.Theresultingcostcurvesshouldgenerallybefamiliartoyoubecausetheyarewidelyusedinmostcoursesinintroductoryeconomics.Herewehaveshownhowsuchcurvesreﬂecttheﬁrm’sunderlyingproductionfunctionandtheﬁrm’sdesiretominimizecosts.Bydevelopingcostcurvesfromthesebasicfoundations,wewereabletoillustrateanumberofimportantﬁndings.•Aﬁrmthatwishestominimizetheeconomiccostsofpro-ducingaparticularlevelofoutputshouldchoosethatinputcombinationforwhichtherateoftechnicalsubsti-tution(RTS)isequaltotheratiooftheinputs’rentalprices.•Repeatedapplicationofthisminimizationprocedureyieldstheﬁrm’sexpansionpath.Becausetheexpansionpathshowshowinputusageexpandswiththelevelofoutput,italsoshowstherelationshipbetweenoutputlevelandtotalcost.Thatrelationshipissummarizedbythetotalcostfunction,Cðq,v,wÞ,whichshowsproduc-tioncostsasafunctionofoutputlevelsandinputprices.•Theﬁrm’saveragecostðAC¼C=qÞandmarginalcostðMC¼∂C=∂qÞfunctionscanbederiveddirectlyfromthetotalcostfunction.IfthetotalcostcurvehasageneralcubicshapethentheACandMCcurveswillbeU-shaped.•Allcostcurvesaredrawnontheassumptionthattheinputpricesareheldconstant.Wheninputpriceschange,FIGURE10.9AverageandMarginalCostCurvesfortheCubicCostCurveCaseThissetofcurvesisderivedfromthetotalcostcurvesshowninFigure10.8.TheACandMCcurveshavetheusualU-shapes,asdotheshort-runcurves.Atq1,long-runaveragecostsareminimized.Theconﬁgurationofcurvesatthisminimumpointisquiteimportant.CostsOutput perperiodq0q1q2SMC (k0)SMC (k1)SMC (k2)MCSAC (k0)SAC (k1)ACSAC (k2)350Part3ProductionandSupply
PROBLEMS10.1Inafamousarticle[J.Viner,“CostCurvesandSupplyCurves,”ZeitschriftfurNationalokonomie3(September1931):23–46],VinercriticizedhisdraftsmanwhocouldnotdrawafamilyofSACcurveswhosepointsoftangencywiththeU-shapedACcurvewerealsotheminimumpointsoneachSACcurve.Thedraftsmanprotestedthatsuchadrawingwasimpossibletoconstruct.Whomwouldyousupportinthisdebate?10.2Supposethataﬁrmproducestwodifferentoutputs,thequantitiesofwhicharerepresentedbyq1andq2.Ingeneral,theﬁrm’stotalcostscanberepresentedbyCðq1,q2Þ.ThisfunctionexhibitseconomiesofscopeifCðq1,0ÞþCð0,q2Þ>Cðq1,q2Þforalloutputlevelsofeithergood.a.Explaininwordswhythismathematicalformulationimpliesthatcostswillbelowerinthismultiproductﬁrmthanintwosingle-productﬁrmsproducingeachgoodseparately.b.Ifthetwooutputsareactuallythesamegood,wecandeﬁnetotaloutputasq¼q1þq2.Supposethatinthiscaseaveragecostð¼C=qÞfallsasqincreases.Showthatthisﬁrmalsoenjoyseconomiesofscopeunderthedeﬁnitionprovidedhere.10.3ProfessorSmithandProfessorJonesaregoingtoproduceanewintroductorytextbook.Astruescientists,theyhavelaidouttheproductionfunctionforthebookasq¼S1=2J1=2,whereq¼thenumberofpagesintheﬁnishedbook,S¼thenumberofworkinghoursspentbySmith,andJ¼thenumberofhoursspentworkingbyJones.Smithvalueshislaboras$3perworkinghour.Hehasspent900hourspreparingtheﬁrstdraft.Jones,whoselaborisvaluedat$12perworkinghour,willreviseSmith’sdrafttocompletethebook.a.HowmanyhourswillJoneshavetospendtoproduceaﬁnishedbookof150pages?Of300pages?Of450pages?b.Whatisthemarginalcostofthe150thpageoftheﬁnishedbook?Ofthe300thpage?Ofthe450thpage?10.4Supposethataﬁrm’sﬁxedproportionproductionfunctionisgivenbyq¼minð5k,10lÞ.a.Calculatetheﬁrm’slong-runtotal,average,andmarginalcostfunctions.b.Supposethatkisﬁxedat10intheshortrun.Calculatetheﬁrm’sshort-runtotal,average,andmarginalcostfunctions.costcurveswillshifttonewpositions.Theextentoftheshiftswillbedeterminedbytheoverallimportanceoftheinputwhosepricehaschangedandbytheeasewithwhichtheﬁrmmaysubstituteoneinputforanother.Technicalprogresswillalsoshiftcostcurves.•Inputdemandfunctionscanbederivedfromtheﬁrm’stotalcostfunctionthroughpartialdifferentiation.Theseinputdemandfunctionswilldependonthequantityofoutputthattheﬁrmchoosestoproduceandarethere-forecalled“contingent”demandfunctions.•Intheshortrun,theﬁrmmaynotbeabletovarysomeinputs.Itcanthenalteritslevelofproductiononlybychangingitsemploymentofvariableinputs.Insodoing,itmayhavetousenonoptimal,higher-costinputcom-binationsthanitwouldchooseifitwerepossibletovaryallinputs.Chapter10CostFunctions351
c.Supposev¼1andw¼3.Calculatethisﬁrm’slong-runandshort-runaverageandmarginalcostcurves.10.5Aﬁrmproducinghockeystickshasaproductionfunctiongivenbyq¼2ﬃﬃﬃﬃﬃﬃﬃﬃk⋅lp.Intheshortrun,theﬁrm’samountofcapitalequipmentisﬁxedatk¼100.Therentalrateforkisv¼$1,andthewagerateforlisw¼$4.a.Calculatetheﬁrm’sshort-runtotalcostcurve.Calculatetheshort-runaveragecostcurve.b.Whatistheﬁrm’sshort-runmarginalcostfunction?WhataretheSC,SAC,andSMCfortheﬁrmifitproduces25hockeysticks?Fiftyhockeysticks?Onehundredhockeysticks?Twohundredhockeysticks?c.GraphtheSACandtheSMCcurvesfortheﬁrm.Indicatethepointsfoundinpart(b).d.WheredoestheSMCcurveintersecttheSACcurve?ExplainwhytheSMCcurvewillalwaysintersecttheSACcurveatitslowestpoint.Supposenowthatcapitalusedforproducinghockeysticksisﬁxedat_kintheshortrun.e.Calculatetheﬁrm’stotalcostsasafunctionofq,w,v,and_k.f.Givenq,w,andv,howshouldthecapitalstockbechosentominimizetotalcost?g.Useyourresultsfrompart(f)tocalculatethelong-runtotalcostofhockeystickproduction.h.Forw¼$4,v¼$1,graphthelong-runtotalcostcurveforhockeystickproduction.Showthatthisisanenvelopefortheshort-runcurvescomputedinpart(a)byexaminingvaluesof_kof100,200,and400.10.6Anenterprisingentrepreneurpurchasestwoﬁrmstoproducewidgets.Eachﬁrmproducesidenticalproducts,andeachhasaproductionfunctiongivenbyq¼ﬃﬃﬃﬃﬃﬃﬃﬃkilip,i¼1,2.Theﬁrmsdiffer,however,intheamountofcapitalequipmenteachhas.Inparticular,ﬁrm1hask1¼25whereasﬁrm2hask2¼100.Rentalratesforkandlaregivenbyw¼v¼$1.a.Iftheentrepreneurwishestominimizeshort-runtotalcostsofwidgetproduction,howshouldoutputbeallocatedbetweenthetwoﬁrms?b.Giventhatoutputisoptimallyallocatedbetweenthetwoﬁrms,calculatetheshort-runtotal,average,andmarginalcostcurves.Whatisthemarginalcostofthe100thwidget?The125thwidget?The200thwidget?c.Howshouldtheentrepreneurallocatewidgetproductionbetweenthetwoﬁrmsinthelongrun?Calculatethelong-runtotal,average,andmarginalcostcurvesforwidgetproduction.d.Howwouldyouranswertopart(c)changeifbothﬁrmsexhibiteddiminishingreturnstoscale?10.7Supposethetotal-costfunctionforaﬁrmisgivenbyC¼qw2=3v1=3.a.UseShephard’slemmatocomputetheconstantoutputdemandfunctionsforinputslandk.b.Useyourresultsfrompart(a)tocalculatetheunderlyingproductionfunctionforq.352Part3ProductionandSupply
10.8Supposethetotal-costfunctionforaﬁrmisgivenbyC¼qðvþ2ﬃﬃﬃﬃﬃﬃvwpþwÞ.a.UseShephard’slemmatocomputetheconstantoutputdemandfunctionforeachinput,kandl.b.Usetheresultsfrompart(a)tocomputetheunderlyingproductionfunctionforq.c.YoucanchecktheresultbyusingresultsfromExample10.2toshowthattheCEScostfunctionwithσ¼0:5,ρ¼1generatesthistotal-costfunction.AnalyticalProblems10.9GeneralizingtheCEScostfunctionTheCESproductionfunctioncanbegeneralizedtopermitweightingoftheinputs.Inthetwo-inputcase,thisfunctionisq¼fðk,lÞ¼½ðakÞρþðblÞργ=ρ.a.Whatisthetotal-costfunctionforaﬁrmwiththisproductionfunction?Hint:Youcan,ofcourse,workthisoutfromscratch;easierperhapsistousetheresultsfromExample10.2andreasonthatthepriceforaunitofcapitalinputinthisproductionfunctionisv=aandforaunitoflaborinputisw=b.b.Ifγ¼1andaþb¼1,itcanbeshownthatthisproductionfunctionconvergestotheCobb-Douglasformq¼kalbasρ!0.WhatisthetotalcostfunctionforthisparticularversionoftheCESfunction?c.Therelativelaborcostshareforatwo-inputproductionfunctionisgivenbywl=vk.ShowthatthisshareisconstantfortheCobb-Douglasfunctioninpart(b).Howistherelativelaborshareaffectedbytheparametersaandb?d.CalculatetherelativelaborcostshareforthegeneralCESfunctionintroducedabove.Howisthatshareaffectedbychangesinw=v?Howisthedirectionofthiseffectdeterminedbytheelasticityofsubstitution,σ?Howisitaffectedbythesizesoftheparametersaandb?10.10InputdemandelasticitiesTheown-priceelasticitiesofcontingentinputdemandforlaborandcapitalaredeﬁnedaselc,w¼∂lc∂w⋅wlc,ekc,v¼∂kc∂v⋅vkc.a.Calculateelc,wandekc,vforeachofthecostfunctionsshowninExample10.2.b.Showthat,ingeneral,elc,wþelc,v¼0.c.Showthatthecross-pricederivativesofcontingentdemandfunctionsareequal—thatis,showthat∂lc=∂v¼∂kc=∂w.Usethisfacttoshowthatslelc,v¼skekc,wwheresl,skare,respectively,theshareoflaborintotalcostðwl=CÞandofcapitalintotalcostðvk=CÞ.d.Usetheresultsfromparts(b)and(c)toshowthatslelc,wþskekc,w¼0.e.Interpretthesevariouselasticityrelationshipsinwordsanddiscusstheiroverallrelevancetoageneraltheoryofinputdemand.10.11TheelasticityofsubstitutionandinputdemandelasticitiesThedeﬁnitionofthe(Morishima)elasticityofsubstitution(Equation10.51)canalsobedescribedintermsofinputdemandelasticities.Thisillustratesthebasicasymmetryinthedeﬁnition.Chapter10CostFunctions353
a.Showthatifonlywjchanges,si,j¼exci,wjexcj,wj.b.Showthatifonlywichanges,sj,i¼excj,wiexci,wi.c.ShowthatiftheproductionfunctiontakesthegeneralCESformq¼½Pnxρi1=ρforρ6¼0,thenalloftheMorishimaelasticitiesarethesame:si,j¼1=ð1ρÞ¼σ.ThisistheonlycaseinwhichtheMorishimadeﬁnitionissymmetric.10.12TheAllenelasticityofsubstitutionManyempiricalstudiesofcostsreportanalternativedeﬁnitionoftheelasticityofsubstitutionbetweeninputs.ThisalternativedeﬁnitionwasﬁrstproposedbyR.G.D.Alleninthe1930sandfurtherclariﬁedbyH.Uzawainthe1960s.Thisdeﬁnitionbuildsdirectlyontheproductionfunction–basedelasticityofsubstitutiondeﬁnedinfootnote6ofChapter9:Ai,j¼CijC=CiCj,wherethesubscriptsindicatepartialdifferentiationwithrespecttovariousinputprices.Clearly,theAllendeﬁnitionissymmetric.a.ShowthatAi,j¼exci,wj=sj,wheresjistheshareofinputjintotalcost.b.ShowthattheelasticityofsiwithrespecttothepriceofinputjisrelatedtotheAllenelasticitybyesi,pj¼sjðAi,j1Þ.c.Showthat,withonlytwoinputs,Ak,l¼1fortheCobb-DouglascaseandAk,l¼σfortheCEScase.d.ReadBlackorbyandRussell(1989:“WilltheRealElasticityofSubstitutionPleaseStandUp?”)toseewhytheMorishimadeﬁnitionispreferredformostpurposes.SUGGESTIONSFORFURTHERREADINGAllen,R.G.D.MathematicalAnalysisforEconomists.NewYork:St.Martin’sPress,1938,variouspages—seeindex.Complete(thoughdated)mathematicalanalysisofsubstitutionpos-sibilitiesandcostfunctions.Notationsomewhatdifﬁcult.Blackorby,C.,andR.R.Russell.“WilltheRealElasticityofSubstitutionPleaseStandUp?(AComparisonoftheAllen/UzawaandMorishimaElasticities).”AmericanEconomicReview(September1989):882–88.Aniceclariﬁcationoftheproperwaytomeasuresubstitutabilityamongmanyinputsinproduction.ArguesthattheAllen/UzawadeﬁnitionislargelyuselessandthattheMorishimadeﬁnitionisbyfarthebest.Ferguson,C.E.TheNeoclassicalTheoryofProductionandDistribution.Cambridge:CambridgeUniversityPress,1969,Chap.6.Nicedevelopmentofcostcurves;especiallystrongongraphicanalysis.Fuss,M.,andD.McFadden.ProductionEconomics:ADualApproachtoTheoryandApplications.Amsterdam:North-Holland,1978.Difﬁcultandquitecompletetreatmentofthedualrelationshipbe-tweenproductionandcostfunctions.Somediscussionofempiricalissues.Knight,H.H.“CostofProductionandPriceoverLongandShortPeriods.”JournalofPoliticalEconomics29(April1921):304–35.Classictreatmentoftheshort-run,long-rundistinction.Silberberg,E.,andW.Suen.TheStructureofEconomics:AMathematicalAnalysis,3rded.Boston:Irwin/McGraw-Hill,2001.Chapters7–9haveagreatdealofmaterialoncostfunctions.Espe-ciallyrecommendedaretheauthors’discussionsof“reciprocityeffects”andtheirtreatmentoftheshort-run–long-rundistinctionasanapplicationoftheLeChatelierprinciplefromphysics.Sydsaeter,K.,A.Strom,andP.Berck.Economists’Mathe-maticalManual,3rded.Berlin:Springer-Verlag,2000.Chapter25providesasuccinctsummaryofthemathematicalcon-ceptsinthischapter.Anicesummaryofmanyinputcostfunctions,butbewareoftypos.354Part3ProductionandSupply
EXTENSIONSTheTranslogCostFunctionThetwocostfunctionsstudiedinChapter10(theCobb-DouglasandtheCES)areveryrestrictiveinthesubstitutionpossibilitiestheypermit.TheCobb-Douglasimplicitlyassumesthatσ¼1betweenanytwoinputs.TheCESpermitsσtotakeanyvalue,butitrequiresthattheelasticityofsubstitutionbethesamebetweenanytwoinputs.Becauseempiricaleconomistswouldprefertoletthedatashowwhattheactualsubstitutionpossibilitiesamonginputsare,theyhavetriedtoﬁndmoreﬂexiblefunctionalforms.Oneespe-ciallypopularsuchformisthetranslogcostfunction,ﬁrstmadepopularbyFussandMcFadden(1978).Inthisextensionwewilllookatthisfunction.E10.1ThetranslogwithtwoinputsInExample10.2,wecalculatedtheCobb-Douglascostfunctioninthetwo-inputcaseasCðq,v,wÞ¼Bq1=ðαþβÞvα=ðαþβÞwβ=ðαþβÞ.Ifwetakethenaturalloga-rithmofthiswehavelnCðq,v,wÞ¼lnBþ½1=ðαþβÞlnqþ½α=ðαþβÞlnvþ½β=ðαþβÞlnw.(i)Thatis,thelogoftotalcostsislinearinthelogsofoutputandtheinputprices.Thetranslogfunctiongeneralizesthisbypermittingsecond-ordertermsininputprices:lnCðq,v,wÞ¼lnqþβ0þβ1lnvþβ2lnwþβ3ðlnvÞ2þβ4ðlnwÞ2þβ5lnvlnw,(ii)wherethisfunctionimplicitlyassumesconstantreturnstoscale(becausethecoefﬁcientoflnqis1.0)—althoughthatneednotbethecase.Someofthepropertiesofthisfunctionare:•Forthefunctiontobehomogeneousofdegree1ininputprices,itmustbethecasethatβ1þβ2¼1andβ3þβ4þβ5¼0.•ThisfunctionincludestheCobb-Douglasasthespecialcaseβ3¼β4¼β5¼0.Hence,thefunc-tioncanbeusedtoteststatisticallywhethertheCobb-Douglasisappropriate.•Inputsharesforthetranslogfunctionareespe-ciallyeasytocomputeusingtheresultthatsi¼ð∂lnCÞ=ð∂lnwiÞ.Inthetwo-inputcase,thisyieldssk¼∂lnC∂lnv¼β1þ2β3lnvþβ5lnw,sl¼∂lnC∂lnw¼β2þ2β4lnwþβ5lnv.(iii)IntheCobb-Douglascaseðβ3¼β4¼β5¼0Þthesesharesareconstant,butwiththegeneraltranslogfunctiontheyarenot.•CalculatingtheelasticityofsubstitutioninthetranslogcaseproceedsbyusingtheresultgiveninProblem10.11thatsk,l¼ekc,welc,w.Mak-ingthiscalculationisstraightforward(providedonekeepstrackofhowtouselogarithms):ekc,w¼∂lnCv∂lnw¼∂lnCv⋅∂lnC∂lnv∂lnw¼∂lnClnvþln∂lnC∂lnv∂lnw¼sl0þ∂lnsk∂sk⋅∂2lnC∂v∂w¼slþβ5sk.(iv)Observethat,intheCobb-Douglascaseðβ5¼0Þ,thecontingentpriceelasticityofde-mandforkwithrespecttothewagehasasimpleform:ekc,w¼sl.Asimilarsetofmanipulationsyieldselc,w¼skþ2β4=sland,intheCobb-Douglascase,elc,w¼sk.Bringingthesetwoelasticitiestogetheryieldssk,l¼ekc,welc,w¼slþskþβ5sk2β4sl¼1þslβ52skβ4sksl.(v)Again,intheCobb-Douglascasewehavesk,l¼1,asshouldhavebeenexpected.•TheAllenelasticityofsubstitution(seeProblem10.12)forthetranslogfunctionisAk,l¼1þβ5=sksl.Thisfunctioncanalsobeusedtocalculatethatthe(contingent)cross-priceelas-ticityofdemandisekc,w¼slAk,l¼slþβ5=sk,aswasshownpreviously.Hereagain,Ak,l¼1intheCobb-Douglascase.Ingeneral,however,theAllenandMorishimadeﬁnitionswilldifferevenwithjusttwoinputs.Chapter10CostFunctions355
E10.2Themany-inputtranslogcostfunctionMostempiricalstudiesincludemorethantwoinputs.Thetranslogcostfunctionisespeciallyeasytogeneral-izetothesesituations.Ifweassumethereareninputs,eachwithapriceofwiði¼1,nÞ,thenthisfunctionisCðq,w1,…,wnÞ¼lnqþβ0þXni¼1βilnwiþ0:5Xni¼1Xnj¼1βijlnwilnwj,(vi)wherewehaveonceagainassumedconstantreturnstoscale.Thisfunctionrequiresβij¼βji,soeachtermforwhichi6¼jappearstwiceintheﬁnaldoublesum(whichexplainsthepresenceofthe0.5intheex-pression).Forthisfunctiontobehomogeneousofdegree1intheinputprices,itmustbethecasethatXni¼1βi¼1andXni¼1βij¼0.Twousefulpropertiesofthisfunctionare:•Inputsharestakethelinearformsi¼βiþXnj¼1βijlnwj.(vii)Again,thisshowswhythetranslogisusuallyestimatedinashareform.Sometimesaterminlnqisalsoaddedtotheshareequationstoallowforscaleeffectsontheshares(seeSydsæter,Strøm,andBerck,2000).•Theelasticityofsubstitutionbetweenanytwoinputsinthetranslogfunctionisgivenbysi,j¼1þsjβijsiβjjsisj.(viii)Hence,substitutabilitycanagainbejudgeddi-rectlyfromtheparametersestimatedforthetranslogfunction.E10.3SomeapplicationsThetranslogcostfunctionhasbecomethemainchoiceforempiricalstudiesofproduction.Twofactorsac-countforthispopularity.First,thefunctionallowsafairlycompletecharacterizationofsubstitutionpatternsamonginputs—itdoesnotrequirethatthedataﬁtanyprespeciﬁedpattern.Second,thefunction’sformatincorporatesinputpricesinaﬂexiblewaysothatonecanbereasonablysurethatheorshehascontrolledforsuchpricesinregressionanalysis.Whensuchcontrolisassured,measuresofotheraspectsofthecostfunction(suchasitsreturnstoscale)willbemorereliable.OneexampleofusingthetranslogfunctiontostudyinputsubstitutionisthestudybyWestbrookandBuckley(1990)oftheresponsesthatshippersmadetochangingrelativepricesofmovinggoodsthatresultedfromderegulationoftherailroadandtruckingindustriesintheUnitedStates.TheauthorslookspeciﬁcallyattheshippingoffruitsandvegetablesfromthewesternstatestoChicagoandNewYork.Theyﬁndrelativelyhighsubstitutionelasticitiesamongshippingoptionsandsoconcludethatderegu-lationhadsigniﬁcantwelfarebeneﬁts.DoucouliagosandHone(2000)provideasimilaranalysisofderegu-lationofdairypricesinAustralia.Theyshowthatchangesinthepriceofrawmilkcauseddairyprocessingﬁrmstoundertakesigniﬁcantchangesininputusage.Theyalsoshowthattheindustryadoptedsigniﬁcantnewtechnologiesinresponsetothepricechange.AninterestingstudythatusesthetranslogprimarilytojudgereturnstoscaleisLatzko’s(1999)analysisoftheU.S.mutualfundindustry.Heﬁndsthattheelastic-ityoftotalcostswithrespecttothetotalassetsmanagedbythefundislessthan1forallbutthelargestfunds(thosewithmorethan$4billioninassets).Hence,theauthorconcludesthatmoneymanagementexhibitssubstantialreturnstoscale.Anumberofotherstudiesthatusethetranslogtoestimateeconomiesofscalefocusonmunicipalservices.Forexample,GarciaandThomas(2001)lookatwatersupplysystemsinlocalFrenchcommunities.Theyconcludethattherearesigniﬁcantoperatingeconomiesofscaleinsuchsystemsandthatsomemergingofsystemswouldmakesense.Yatchew(2000)reachesasimilarconclusionaboutelectricitydistributioninsmallcommunitiesinOntario,Canada.Heﬁndsthatthereareeconomiesofscaleforelectricitydistributionsystemsservinguptoabout20,000custo-mers.Again,someefﬁcienciesmightbeobtainedfrommergingsystemsthataremuchsmallerthanthissize.ReferencesDoucouliagos,H.,andP.Hone.“DeregulationandSub-equilibriumintheAustralianDairyProcessingIndus-try.”EconomicRecord(June2000):152–62.Fuss,M.,andD.McFadden,Eds.ProductionEconomics:ADualApproachtoTheoryandApplications.Amsterdam:NorthHolland,1978.Garcia,S.,andA.Thomas.“TheStructureofMunicipalWaterSupplyCosts:ApplicationtoaPanelofFrench356Part3ProductionandSupply
LocalCommunities.”JournalofProductivityAnalysis(July2001):5–29.Latzko,D.“EconomiesofScaleinMutualFundAdministra-tion.”JournalofFinancialResearch(Fall1999):331–39.Sydsæter,K.,A.Strøm,andP.Berck.Economists’Mathe-maticalManual,3rded.Berlin:Springer-Verlag,2000.Westbrook,M.D.,andP.A.Buckley.“FlexibleFunctionalFormsandRegularity:AssessingtheCompetitiveRelationshipbetweenTruckandRailTransportation.”ReviewofEconomicsandStatistics(November1990):623–30.Yatchew,A.“ScaleEconomiesinElectricityDistribution:ASemiparametricAnalysis.”JournalofAppliedEconomet-rics(March/April2000):187–210.Chapter10CostFunctions357
CHAPTER11ProﬁtMaximizationInChapter10weexaminedthewayinwhichﬁrmsminimizecostsforanylevelofoutputtheychoose.Inthischapterwefocusonhowthelevelofoutputischosenbyproﬁt-maximizingﬁrms.Beforeinvestigatingthatdecision,however,itisappropriatetodiscussbrieﬂythenatureofﬁrmsandthewaysinwhichtheirchoicesshouldbeanalyzed.THENATUREANDBEHAVIOROFFIRMSAswepointedoutatthebeginningofouranalysisofproduction,aﬁrmisanassociationofindividualswhohaveorganizedthemselvesforthepurposeofturninginputsintooutputs.Differentindividualswillprovidedifferenttypesofinputs,suchasworkers’skillsandvarietiesofcapitalequipment,withtheexpectationofreceivingsomesortofrewardfordoingso.ContractualrelationshipswithinﬁrmsThenatureofthecontractualrelationshipbetweentheprovidersofinputstoaﬁrmmaybequitecomplicated.Eachprovideragreestodevotehisorherinputtoproductionactivitiesunderasetofunderstandingsabouthowitistobeusedandwhatbeneﬁtistobeexpectedfromthatuse.Insomecasesthesecontractsareexplicit.Workersoftennegotiatecontractsthatspecifyinconsiderabledetailwhathoursaretobeworked,whatrulesofworkaretobefollowed,andwhatrateofpayistobeexpected.Similarly,capitalownersinvestinaﬁrmunderasetofexplicitlegalprinciplesaboutthewaysinwhichthatcapitalmaybeused,thecompensationtheownercanexpecttoreceive,andwhethertheownerretainsanyproﬁtsorlossesafteralleconomiccostshavebeenpaid.Despitetheseformalarrangements,itisclearthatmanyoftheunderstandingsbetweentheprovidersofinputstoaﬁrmareimplicit;relationshipsbetweenmanagersandworkersfollowcertainproceduresaboutwhohastheauthoritytodowhatinmakingproductiondecisions.Amongworkers,numerousimplicitunderstandingsexistabouthowworktasksaretobeshared;andcapitalownersmaydelegatemuchoftheirauthoritytomanagersandworkerstomakedecisionsontheirbehalf(GeneralMotors’shareholders,forexample,areneverinvolvedinhowassembly-lineequipmentwillbeused,thoughtechnicallytheyownit).Alloftheseexplicitandimplicitrelationshipschangeinresponsetoexperiencesandeventsexternaltotheﬁrm.Muchasabasketballteamwilltryoutnewplaysanddefensivestrategies,sotooﬁrmswillalterthenatureoftheirinternalorganizationstoachievebetterlong-termresults.11TheinitialdevelopmentofthetheoryoftheﬁrmfromthenotionofthecontractualrelationshipsinvolvedcanbefoundinR.H.Coase,“TheNatureoftheFirm,”Economica(November1937):386–405.358
Modelingﬁrms’behaviorAlthoughsomeeconomistshaveadopteda“behavioral”approachtostudyingﬁrms’deci-sions,mosthavefoundthatapproachtoocumbersomeforgeneralpurposes.Rather,theyhaveadopteda“holistic”approachthattreatstheﬁrmasasingledecision-makingunitandsweepsawayallthecomplicatedbehavioralissuesaboutrelationshipsamonginputproviders.Underthisapproach,itisoftenconvenienttoassumethataﬁrm’sdecisionsaremadebyasingledictatorialmanagerwhorationallypursuessomegoal,usuallyproﬁtmaximization.Thatistheapproachwetakehere.InChapter18welookatsomeoftheinformationalissuesthatariseinintraﬁrmcontracts.PROFITMAXIMIZATIONMostmodelsofsupplyassumethattheﬁrmanditsmanagerpursuethegoalofachievingthelargesteconomicproﬁtspossible.Hencewewillusethefollowingdeﬁnition.DEFINITIONProfit-maximizingfirm.Aproﬁt-maximizingﬁrmchoosesbothitsinputsanditsoutputswiththesolegoalofachievingmaximumeconomicproﬁts.Thatis,theﬁrmseekstomakethedifferencebetweenitstotalrevenuesanditstotaleconomiccostsaslargeaspossible.Thisassumption—thatﬁrmsseekmaximumeconomicproﬁts—hasalonghistoryineco-nomicliterature.Ithasmuchtorecommendit.Itisplausiblebecauseﬁrmownersmayindeedseektomaketheirassetasvaluableaspossibleandbecausecompetitivemarketsmaypunishﬁrmsthatdonotmaximizeproﬁts.Theassumptionalsoyieldsinterestingtheoreticalresultsthatcanexplainactualﬁrms’decisions.ProﬁtmaximizationandmarginalismIfﬁrmsarestrictproﬁtmaximizers,theywillmakedecisionsina“marginal”way.Theentrepreneurwillperformtheconceptualexperimentofadjustingthosevariablesthatcanbecontrolleduntilitisimpossibletoincreaseproﬁtsfurther.Thisinvolves,say,lookingattheincremental,or“marginal,”proﬁtobtainablefromproducingonemoreunitofoutput,orattheadditionalproﬁtavailablefromhiringonemorelaborer.Aslongasthisincrementalproﬁtispositive,theextraoutputwillbeproducedortheextralaborerwillbehired.Whentheincrementalproﬁtofanactivitybecomeszero,theentrepreneurhaspushedthatactivityfarenough,anditwouldnotbeproﬁtabletogofurther.Inthischapter,wewillexploretheconsequencesofthisassumptionbyusingincreasinglysophisticatedmathematics.OutputchoiceFirstweexamineatopicthatshouldbeveryfamiliar:whatoutputlevelaﬁrmwillproduceinordertoobtainmaximumproﬁts.Aﬁrmsellssomelevelofoutput,q,atamarketpriceofpperunit.TotalrevenuesðRÞaregivenbyRðqÞ¼pðqÞ⋅q,(11.1)wherewehaveallowedforthepossibilitythatthesellingpricetheﬁrmreceivesmightbeaffectedbyhowmuchitsells.Intheproductionofq,certaineconomiccostsareincurredand,asinChapter10,wewilldenotethesebyCðqÞ.ThedifferencebetweenrevenuesandcostsiscalledeconomicproﬁtsðπÞ.Becausebothrevenuesandcostsdependonthequantityproduced,economicproﬁtswillalso.Thatis,πðqÞ¼pðqÞ⋅qCðqÞ¼RðqÞCðqÞ.(11.2)Chapter11ProﬁtMaximization359
ThenecessaryconditionforchoosingthevalueofqthatmaximizesproﬁtsisfoundbysettingthederivativeofEquation11.2withrespecttoqequalto0:2dπdq¼π0ðqÞ¼dRdqdCdq¼0,(11.3)sotheﬁrst-orderconditionforamaximumisthatdRdq¼dCdq.(11.4)Thisisamathematicalstatementofthe“marginalrevenueequalsmarginalcost”ruleusuallystudiedinintroductoryeconomicscourses.Hencewehavethefollowing.OPTIMIZATIONPRINCIPLEProfitmaximization.Tomaximizeeconomicproﬁts,theﬁrmshouldchoosethatoutputforwhichmarginalrevenueisequaltomarginalcost.Thatis,MR¼dRdq¼dCdq¼MC.(11.5)Second-orderconditionsEquation11.4or11.5isonlyanecessaryconditionforaproﬁtmaximum.Forsufﬁciency,itisalsorequiredthatd2πdq2q¼q¼dπ0ðqÞdqq¼q<0,(11.6)orthat“marginal”proﬁtmustbedecreasingattheoptimallevelofq.Forqlessthanq(theoptimallevelofoutput),proﬁtmustbeincreasing½π0ðqÞ>0;andforqgreaterthanq,proﬁtmustbedecreasing½π0ðqÞ<0.Onlyifthisconditionholdshasatruemaximumbeenachieved.Clearlytheconditionholdsifmarginalrevenueisdecreasing(orconstant)inqandmarginalcostisincreasinginq.GraphicalanalysisTheserelationshipsareillustratedinFigure11.1,wherethetoppaneldepictstypicalcostandrevenuefunctions.Forlowlevelsofoutput,costsexceedrevenuesandsoeconomicproﬁtsarenegative.Inthemiddlerangesofoutput,revenuesexceedcosts;thismeansthatproﬁtsarepositive.Finally,athighlevelsofoutput,costsrisesharplyandagainexceedrevenues.Theverticaldistancebetweentherevenueandcostcurves(thatis,proﬁts)isshowninFigure11.1b.Hereproﬁtsreachamaximumatq.Atthislevelofoutputitisalsotruethattheslopeoftherevenuecurve(marginalrevenue)isequaltotheslopeofthecostcurve(marginalcost).Itisclearfromtheﬁgurethatthesufﬁcientconditionsforamaximumarealsosatisﬁedatthispoint,becauseproﬁtsareincreasingtotheleftofqanddecreasingtotherightofq.Outputlevelqisthereforeatrueproﬁtmaximum.Thisisnotsoforoutputlevelq.Althoughmarginalrevenueisequaltomarginalcostatthisoutput,proﬁtsareinfactataminimumthere.2Noticethatthisisanunconstrainedmaximizationproblem;theconstraintsintheproblemareimplicitintherevenueandcostfunctions.Speciﬁcally,thedemandcurvefacingtheﬁrmdeterminestherevenuefunction,andtheﬁrm’sproductionfunction(togetherwithinputprices)determinesitscosts.360Part3ProductionandSupply MARGINALREVENUEItistherevenueobtainedfromsellingonemoreunitofoutputthatisrelevanttotheproﬁt-maximizingﬁrm’soutputdecision.Iftheﬁrmcansellallitwisheswithouthavinganyeffectonmarketprice,themarketpricewillindeedbetheextrarevenueobtainedfromsellingonemoreunit.Phrasedinanotherway:ifaﬁrm’soutputdecisionswillnotaffectmarketprice,thenmarginalrevenueisequaltothepriceatwhichaunitsells.FIGURE11.1MarginalRevenueMustEqualMarginalCostforProﬁtMaximizationBecauseproﬁtsaredeﬁnedtoberevenuesðRÞminuscostsðCÞ,itisclearthatproﬁtsreachamaximumwhentheslopeoftherevenuefunction(marginalrevenue)isequaltotheslopeofthecostfunction(marginalcost).Thisequalityisonlyanecessaryconditionforamaximum,asmaybeseenbycomparingpointsq(atruemaximum)andq(atrueminimum),pointsatwhichmarginalrevenueequalsmarginalcost.Revenues,costsProfitsLossesOutput per periodOutput per period(a)(b)q**q*q*CR0Chapter11ProﬁtMaximization361 Aﬁrmmaynotalwaysbeabletosellallitwantsattheprevailingmarketprice,however.Ifitfacesadownward-slopingdemandcurveforitsproduct,thenmoreoutputcanbesoldonlybyreducingthegood’sprice.Inthiscasetherevenueobtainedfromsellingonemoreunitwillbelessthanthepriceofthatunitbecause,inordertogetconsumerstotaketheextraunit,thepriceofallotherunitsmustbelowered.Thisresultcanbeeasilydemonstrated.Asbefore,totalrevenueðRÞistheproductofthequantitysoldðqÞtimesthepriceatwhichitissoldðpÞ,whichmayalsodependonq.MarginalrevenueðMRÞisthendeﬁnedtobethechangeinRresultingfromachangeinq.DEFINITIONMarginalrevenue.Wedeﬁnemarginalrevenue¼MRðqÞ¼dRdq¼d½pðqÞ⋅qdq¼pþq⋅dpdq:(11.7)Noticethatthemarginalrevenueisafunctionofoutput.Ingeneral,MRwillbedifferentfordifferentlevelsofq.FromEquation11.7itiseasytoseethat,ifpricedoesnotchangeasquantityincreasesðdp=dq¼0Þ,marginalrevenuewillbeequaltoprice.Inthiscasewesaythattheﬁrmisapricetakerbecauseitsoutputdecisionsdonotinﬂuencethepriceitreceives.Ontheotherhand,ifpricefallsasquantityincreasesðdp=dq<0Þ,marginalrevenuewillbelessthanprice.Aproﬁt-maximizingmanagermustknowhowincreasesinoutputwillaffectthepricereceivedbeforemakinganoptimaloutputdecision.Ifincreasesinqcausemarketpricetofall,thismustbetakenintoaccount.EXAMPLE11.1MarginalRevenuefromaLinearDemandFunctionSupposeashopsellingsubsandwichs(alsocalledgrinders,torpedoes,or,inPhiladelphia,hoagies)facesalineardemandcurveforitsdailyoutputoverperiodðqÞoftheformq¼10010p.(11.8)Solvingforthepricetheshopreceives,wehavep¼q10þ10,(11.9)andtotalrevenues(asafunctionofq)aregivenbyR¼pq¼q210þ10q.(11.10)Thesubﬁrm’smarginalrevenuefunctionisMR¼dRdq¼q5þ10,(11.11)andinthiscaseMR1Þ,marginalrevenuewillbenegative.Increasesinqcanbeobtainedonlythrough“large”declinesinmarketprice,andthesedeclineswillactuallycausetotalrevenuetodecrease.TherelationshipbetweenmarginalrevenueandelasticityissummarizedbyTable11.1.Price–marginalcostmarkupIfweassumetheﬁrmwishestomaximizeproﬁts,thisanalysiscanbeextendedtoillustratetheconnectionbetweenpriceandmarginalcost.SettingMR¼MCyieldsMC¼p1þ1eq,p !orpMC¼eq,p1þeq,p.(11.13)Thatis,the“markup”ofpriceovermarginalcostdependsinaveryspeciﬁcwayontheelastic-ityofdemandfacingtheﬁrm.First,noticethatthisdemandmustbeelasticðeq,p<1Þ.TABLE11.1RelationshipbetweenElasticityandMarginalRevenueeq,p<1MR>0eq,p¼1MR¼0eq,p>1MR<0Chapter11ProﬁtMaximization363 Ifdemandwereinelastic,theratioinEquation11.13wouldbenegativeandtheequationwouldbenonsensical.Thissimplyreﬂectsthat,whendemandisinelastic,marginalrevenueisnegativeandcannotbeequatedtoapositivemarginalcost.Itisimportanttostressthatitisthedemandfacingtheﬁrmthatmustbeelastic.Thismaybeconsistentwithanoverallinelasticdemandfortheproductinquestioniftheﬁrmfacescompetitionfromotherﬁrmsproducingthesamegood.Equation11.13impliesthatthemarkupovermarginalcostwillbehighertheclosereq,pisto1.Ifthedemandfacingtheﬁrmisinﬁnitelyelastic(perhapsbecausetherearemanyotherﬁrmsproducingthesamegood),theneq,p¼∞andthereisnomarkupðp=MC¼1Þ.Ontheotherhand,withanelasticityofdemandof(say)eq,p¼2,themarkupovermarginalcostwillbe100percent(i.e.,p=MC¼2).MarginalrevenuecurveAnydemandcurvehasamarginalrevenuecurveassociatedwithit.If,aswesometimesassume,theﬁrmmustsellallitsoutputatoneprice,itisconvenienttothinkofthedemandcurvefacingtheﬁrmasanaveragerevenuecurve.Thatis,thedemandcurveshowstherevenueperunit(inotherwords,theprice)yieldedbyalternativeoutputchoices.Themarginalrevenuecurve,ontheotherhand,showstheextrarevenueprovidedbythelastunitsold.Intheusualcaseofadownward-slopingdemandcurve,themarginalrevenuecurvewillliebelowthedemandcurvebecause,accordingtoEquation11.7,MR0;marginalcostmustbeincreasing.366Part3ProductionandSupply
loweroutputlevelwillresultinlowermarginalcoststomeetthislowerprice.Byconsideringallpossiblepricestheﬁrmmightface,wecanseebythemarginalcostcurvehowmuchoutputtheﬁrmshouldsupplyateachprice.Theshutdowndecision.Forverylowpriceswemustbecarefulaboutthisconclusion.ShouldmarketpricefallbelowP1,theproﬁt-maximizingdecisionwouldbetoproducenothing.AsFigure11.3shows,priceslessthanP1donotcoveraveragevariablecosts.Therewillbealossoneachunitproducedinadditiontothelossofallﬁxedcosts.Byshuttingdownproduction,theﬁrmmuststillpayﬁxedcostsbutavoidsthelossesincurredoneachunitproduced.Because,intheshortrun,theﬁrmcannotclosedownandavoidallcosts,itsbestdecisionistoproducenooutput.Ontheotherhand,apriceonlyslightlyaboveP1meanstheﬁrmshouldproducesomeoutput.Althoughproﬁtsmaybenegative(whichtheywillbeifpricefallsbelowshort-runaveragetotalcosts,thecaseatP),theproﬁt-maximizingdecisionistocontinueproductionaslongasvariablecostsarecovered.Fixedcostsmustbepaidinanycase,andanypricethatcoversvariablecostswillproviderevenueasanoffsettotheﬁxedcosts.5Hencewehaveacompletedescriptionofthisﬁrm’ssupplydecisionsinresponsetoalternativepricesforitsoutput.Thesearesummarizedinthefollowingdeﬁnition.DEFINITIONShort-runsupplycurve.Theﬁrm’sshort-runsupplycurveshowshowmuchitwillproduceatvariouspossibleoutputprices.Foraproﬁt-maximizingﬁrmthattakesthepriceofitsoutputasgiven,thiscurveconsistsofthepositivelyslopedsegmentoftheﬁrm’sshort-runmarginalcostabovethepointofminimumaveragevariablecost.Forpricesbelowthislevel,theﬁrm’sproﬁt-maximizingdecisionistoshutdownandproducenooutput.Ofcourse,anyfactorthatshiftstheﬁrm’sshort-runmarginalcostcurve(suchaschangesininputpricesorchangesinthelevelofﬁxedinputsemployed)willalsoshifttheshort-runsupplycurve.InChapter12wewillmakeextensiveuseofthistypeofanalysistostudytheoperationsofperfectlycompetitivemarkets.EXAMPLE11.3Short-RunSupplyInExample10.5wecalculatedtheshort-runtotal-costfunctionfortheCobb-DouglasproductionfunctionasSCðv,w,q,kÞ¼vk1þwq1=βkα=β1,(11.17)(continued)5Somealgebramayclarifymatters.Weknowthattotalcostsequalthesumofﬁxedandvariablecosts,SC¼SFCþSVC,andthatproﬁtsaregivenbyπ¼RSC¼P⋅qSFCSVC.Ifq¼0,thenvariablecostsandrevenuesare0andsoπ¼SFC.Theﬁrmwillproducesomethingonlyifπ>SFC.ButthatmeansthatP⋅q>SVCorP>SVC=q.Chapter11ProﬁtMaximization367
EXAMPLE11.3CONTINUEDwherek1isthelevelofcapitalinputthatisheldconstantintheshortrun.6Short-runmarginalcostiseasilycomputedasSMCðv,w,q,k1Þ¼∂SC∂q¼wβqð1βÞ=βkα=β1.(11.18)Noticethatshort-runmarginalcostisincreasinginoutputforallvaluesofq.Short-runproﬁtmaximizationforaprice-takingﬁrmrequiresthatoutputbechosensothatmarketpriceðPÞisequaltoshort-runmarginalcost:SMC¼wβqð1βÞ=βkα=β1¼P,(11.19)andwecansolveforquantitysuppliedasq¼wββ=ð1βÞkα=ð1βÞ1Pβ=ð1βÞ.(11.20)Thissupplyfunctionprovidesanumberofinsightsthatshouldbefamiliarfromearliereconomicscourses:(1)thesupplycurveispositivelysloped—increasesinPcausetheﬁrmtoproducemorebecauseitiswillingtoincurahighermarginalcost;7(2)thesupplycurveisshiftedtotheleftbyincreasesinthewagerate,w—thatis,foranygivenoutputprice,lessissuppliedwithahigherwage;(3)thesupplycurveisshiftedoutwardbyincreasesincapitalinput,k—withmorecapitalintheshortrun,theﬁrmincursagivenlevelofshort-runmarginalcostatahigheroutputlevel;and(4)therentalrateofcapital,v,isirrelevanttoshort-runsupplydecisionsbecauseitisonlyacomponentofﬁxedcosts.Numericalexample.WecanpursueoncemorethenumericalexamplefromExample10.5,whereα¼β¼0.5,v¼3,w¼12,andk1¼80.Forthesespeciﬁcparameters,thesupplyfunctionisq¼w0.51⋅ðk1Þ1⋅P1¼40⋅Pw¼40P12¼10P3.(11.21)Thatthiscomputationiscorrectcanbecheckedbycomparingthequantitysuppliedatvariouspriceswiththecomputationofshort-runmarginalcostinTable10.2.Forexample,ifP¼12thenthesupplyfunctionpredictsthatq¼40willbesupplied,andTable10.2showsthatthiswillagreewiththeP¼SMCrule.IfpriceweretodoubletoP¼24,anoutputlevelof80wouldbesuppliedand,again,Table10.2showsthatwhenq¼80,SMC¼24.Alowerprice(sayP¼6)wouldcauselesstobeproducedðq¼20Þ.BeforeadoptingEquation11.21asthesupplycurveinthissituation,weshouldalsochecktheﬁrm’sshutdowndecision.Isthereapricewhereitwouldbemoreproﬁtabletoproduceq¼0thantofollowtheP¼SMCrule?FromEquation11.17weknowthatshort-runvariablecostsaregivenbySVC¼wq1=βkα=β1(11.22)andsoSVCq¼wqð1βÞ=βkα=β1.(11.23)6Becausecapitalinputisheldconstant,theshort-runcostfunctionexhibitsincreasingmarginalcostandwillthereforeyieldauniqueproﬁt-maximizingoutputlevel.Ifwehadusedaconstantreturns-to-scaleproductionfunctioninthelongrun,therewouldhavebeennosuchuniqueoutputlevel.WediscussthispointlaterinthischapterandinChapter12.7Infact,theshort-runelasticityofsupplycanbereaddirectlyfromEquation11.20asβ=ð1βÞ.368Part3ProductionandSupply
AcomparisonofEquation11.23withEquation11.18showsthatSVC=qP1thenΠðP2,…ÞΠðP1,…Þ,anditwouldbenaturaltomeasurethewelfaregaintotheﬁrmfromthepricechangeaswelfaregain¼ΠðP2,…ÞΠðP1,…Þ.(11.29)Figure11.4showshowthisvaluecanbemeasuredgraphicallyastheareaboundedbythetwopricesandabovetheshort-runsupplycurve.Intuitively,thesupplycurveshowstheminimumpricethattheﬁrmwillacceptforproducingitsoutput.Hence,whenmarketpricerisesfromP1toP2,theﬁrmisabletosellitsprioroutputlevelðq1Þatahigherpriceandalsooptstoselladditionaloutputðq2q1Þforwhich,atthemargin,itlikewiseearnsaddedproﬁtsonallbuttheﬁnalunit.Hence,thetotalgainintheﬁrm’sproﬁtsisgivenbyareaP2ABP1.Mathematically,wecanmakeuseoftheenveloperesultsfromtheprevioussectiontoderive10Thisrelationshipissometimesreferredtoas“Hotelling’slemma”—aftertheeconomistHaroldHotelling,whodiscovereditinthe1930s.11UnliketheinputdemandfunctionsderivedinChapter10,theseinputdemandfunctionsarenotconditionalonoutputlevels.Rather,theﬁrm’sproﬁt-maximizingoutputdecisionhasalreadybeentakenintoaccountinthefunctions.ThisdemandconceptisthereforemoregeneralthantheoneweintroducedinChapter10,andwewillhavemuchmoretosayaboutitinthenextsection.Chapter11ProﬁtMaximization371
welfaregain¼∫P2P1qðPÞdP¼∫P2P1∂Π∂PdP¼ΠðP2,…ÞΠðP1,…Þ.(11.30)Thus,thegeometricandmathematicalmeasuresofthewelfarechangeagree.Usingthisapproach,wecanalsomeasurehowmuchtheﬁrmvaluestherighttoproduceattheprevailingmarketpricerelativetoasituationwhereitwouldproducenooutput.Ifwedenotetheshort-runshutdownpriceasP0(whichmayormaynotbeapriceofzero),thentheextraproﬁtsavailablefromfacingapriceofP1aredeﬁnedtobeproducersurplus:producersurplus¼ΠðP1,…ÞΠðP0,…Þ¼∫P1P0qðPÞdP.(11.31)ThisisshownasareaP1BP0inFigure11.4.Hencewehavethefollowingformaldeﬁnition.DEFINITIONProducersurplus.Producersurplusistheextrareturnthatproducersearnbymakingtransactionsatthemarketpriceoverandabovewhattheywouldearnifnothingwereproduced.Itisillustratedbythesizeoftheareabelowthemarketpriceandabovethesupplycurve.Inthisdeﬁnitionwehavemadenodistinctionbetweentheshortrunandthelongrun,thoughourdevelopmentsofarhasinvolvedonlyshort-runanalysis.InthenextchapterwewillseethatthesamedeﬁnitioncanservedualdutybydescribingproducersurplusinthelongFIGURE11.4ChangesinShort-RunProducerSurplusMeasureFirmProﬁtsIfpricerisesfromP1toP2thentheincreaseintheﬁrm’sproﬁtsisgivenbyareaP2ABP1.AtapriceofP1,theﬁrmearnsshort-runproducersurplusgivenbyareaP0BP1.Thismeasurestheincreaseinshort-runproﬁtsfortheﬁrmwhenitproducesq1ratherthanshuttingdownwhenpriceisP0orbelow.MarketpriceP2qq1SMCq2P1P0AB372Part3ProductionandSupply
run,sousingthisgenericdeﬁnitionworksforbothconcepts.Ofcourse,aswewillshow,themeaningoflong-runproducersurplusisquitedifferentfromwhatwehavestudiedhere.Onemoreaspectofshort-runproducersurplusshouldbepointedout.Becausetheﬁrmproducesnooutputatitsshutdownprice,weknowthatðP0,…Þ¼vk1;thatis,proﬁtsattheshutdownpricearesolelymadeupoflossesofallﬁxedcosts.Therefore,producersurplus¼ΠðP1,…ÞΠðP0,…Þ¼ΠðP1,…Þðvk1Þ¼ΠðP1,…Þþvk1.(11.32)Thatis,producersurplusisgivenbycurrentproﬁtsbeingearnedplusshort-runﬁxedcosts.Furthermanipulationshowsthatmagnitudecanalsobeexpressedasproducersurplus¼ΠðP1,…ÞΠðP0,…Þ¼P1q1vk1wl1þvk1¼P1q1wl1.(11.33)Inwords,aﬁrm’sshort-runproducersurplusisgivenbytheextenttowhichitsrevenuesexceeditsvariablecosts—thisis,indeed,whattheﬁrmgainsbyproducingintheshortrunratherthanshuttingdownandproducingnothing.EXAMPLE11.4AShort-RunProﬁtFunctionThesevarioususesoftheproﬁtfunctioncanbeillustratedwiththeCobb-Douglasproduc-tionfunctionwehavebeenusing.Sinceq¼kαlβandsincewetreatcapitalasﬁxedatk1intheshortrun,itfollowsthatproﬁtsareπ¼Pkα1lβvk1wl.(11.34)Toﬁndtheproﬁtfunctionweusetheﬁrst-orderconditionsforamaximumtoeliminatelfromthisexpression:∂π∂l¼βPkα1lβ1w¼0sol¼wβPkα11=ðβ1Þ.(11.35)WecansimplifytheprocessofsubstitutingthisbackintotheproﬁtequationbylettingA¼ðw=βPkα1Þ.Makinguseofthisshortcut,wehaveΠðP,v,w,k1Þ¼Pkα1Aβ=ðβ1Þvk1wA1=ðβ1Þ¼wA1=ðβ1ÞPkα1Aw1vk1¼1βββ=ðβ1Þwβ=ðβ1ÞP1=ð1βÞkα=ð1βÞ1vk1.(11.36)Thoughadmittedlymessy,thissolutioniswhatwaspromised—theﬁrm’smaximalproﬁtsareexpressedasafunctionofonlythepricesitfacesanditstechnology.Noticethattheﬁrm’sﬁxedcostsðvk1Þenterthisexpressioninasimplelinearway.Thepricestheﬁrmfacesdeterminetheextenttowhichrevenuesexceedvariablecosts;thenﬁxedcostsaresubtractedtoobtaintheﬁnalproﬁtnumber.Becauseitisalwayswisetocheckthatone’salgebraiscorrect,let’stryoutthenumericalexamplewehavebeenusing.Withα¼β¼0.5,v¼3,w¼12,andk1¼80,weknowthatatapriceofP¼12theﬁrmwillproduce40unitsofoutputanduselaborinputofl¼20.Henceproﬁtswillbeπ¼RC¼12⋅403⋅8012⋅20¼0.TheﬁrmwilljustbreakevenatapriceofP¼12.UsingtheproﬁtfunctionyieldsΠðP,v,w,k1Þ¼Πð12,3,12,80Þ¼0:25⋅121⋅122⋅803⋅80¼0.(11.37)Thus,atapriceof12,theﬁrmearns240inproﬁtsonitsvariablecosts,andthesearepreciselyoffsetbyﬁxedcostsinarrivingattheﬁnaltotal.Withahigherpriceforitsoutput,(continued)Chapter11ProﬁtMaximization373
EXAMPLE11.4CONTINUEDtheﬁrmearnspositiveproﬁts.Ifthepricefallsbelow12,however,theﬁrmincursshort-runlosses.12Hotelling’slemma.WecanusetheproﬁtfunctioninEquation11.36togetherwiththeenvelopetheoremtoderivethisﬁrm’sshort-runsupplyfunction:qðP,v,w,k1Þ¼∂Π∂P¼wββ=ðβ1Þkα=ð1βÞ1Pβ=ð1βÞ,(11.38)whichispreciselytheshort-runsupplyfunctionthatwecalculatedinExample11.3(seeEquation11.20).Producersurplus.Wecanalsousethesupplyfunctiontocalculatetheﬁrm’sshort-runproducersurplus.Todoso,weagainreturntoournumericalexample:α¼β¼0.5,v¼3,w¼12,andk1¼80.Withtheseparameters,theshort-runsupplyrelationshipisq¼10P=3andtheshutdownpriceiszero.Hence,atapriceofP¼12,producersurplusisproducersurplus¼∫12010P3dP¼10P26120¼240.(11.39)Thispreciselyequalsshort-runproﬁtsatapriceof12ðπ¼0Þplusshort-runﬁxedcostsð¼vk1¼3⋅80¼240Þ.Ifpriceweretoriseto(say)15thenproducersurpluswouldincreaseto375,whichwouldstillconsistof240inﬁxedcostsplustotalproﬁtsatthehigherpriceð¼135Þ.QUERY:Howistheamountofshort-runproducersurplushereaffectedbychangesintherentalrateforcapital,v?Howisitaffectedbychangesinthewage,w?PROFITMAXIMIZATIONANDINPUTDEMANDThusfar,wehavetreatedtheﬁrm’sdecisionproblemasoneofchoosingaproﬁt-maximizinglevelofoutput.Butourdiscussionthroughouthasmadeclearthattheﬁrm’soutputis,infact,determinedbytheinputsitchoosestoemploy,arelationshipthatissummarizedbytheproductionfunctionq¼fðk,lÞ.Consequently,theﬁrm’seconomicproﬁtscanalsobeexpressedasafunctionofonlytheinputsitemploys:πðk,lÞ¼PqCðqÞ¼Pfðk,lÞðvkþwlÞ.(11.40)Viewedinthisway,theproﬁt-maximizingﬁrm’sdecisionproblembecomesoneofchoos-ingtheappropriatelevelsofcapitalandlaborinput.13Theﬁrst-orderconditionsforamaximumare∂π∂k¼P∂f∂kv¼0,∂π∂l¼P∂f∂lw¼0.(11.41)12InTable10.2weshowedthatifq¼40thenSAC¼12.HencezeroproﬁtsarealsoindicatedbyP¼12¼SAC.13Throughoutourdiscussioninthissection,weassumethattheﬁrmisapricetakersothepricesofitsoutputanditsinputscanbetreatedasﬁxedparameters.Resultscanbegeneralizedfairlyeasilyinthecasewherepricesdependonquantity.374Part3ProductionandSupply
Theseconditionsmaketheintuitivelyappealingpointthataproﬁt-maximizingﬁrmshouldhireanyinputuptothepointatwhichtheinput’smarginalcontributiontorevenueisequaltothemarginalcostofhiringtheinput.Becausetheﬁrmisassumedtobeapricetakerinitshiring,themarginalcostofhiringanyinputisequaltoitsmarketprice.Theinput’smarginalcontributiontorevenueisgivenbytheextraoutputitproduces(themarginalproduct)timesthatgood’smarketprice.Thisdemandconceptisgivenaspecialnameasfollows.DEFINITIONMarginalrevenueproduct.Themarginalrevenueproductistheextrarevenueaﬁrmreceiveswhenitemploysonemoreunitofaninput.Intheprice-taking14case,MRPl¼PflandMRPk¼Pfk.Hence,proﬁtmaximizationrequiresthattheﬁrmhireeachinputuptothepointatwhichitsmarginalrevenueproductisequaltoitsmarketprice.Noticealsothattheproﬁt-maximizingEquations11.41alsoimplycostminimizationbecauseRTS¼fl=fk¼w=v.Second-orderconditionsBecausetheproﬁtfunctioninEquation11.40dependsontwovariables,kandl,thesecond-orderconditionsforaproﬁtmaximumaresomewhatmorecomplexthaninthesingle-variablecaseweexaminedearlier.InChapter2weshowedthat,toensureatruemaximum,theproﬁtfunctionmustbeconcave.Thatis,πkk¼fkk<0,πll¼fll<0,(11.42)andπkkπllπ2kl¼fkkfllf2kl>0.Therefore,concavityoftheproﬁtrelationshipamountstorequiringthattheproductionfunctionitselfbeconcave.Noticethatdiminishingmarginalproductivityforeachinputisnotsufﬁcienttoensureincreasingmarginalcosts.Expandingoutputusuallyrequirestheﬁrmtousemorecapitalandmorelabor.Thuswemustalsoensurethatincreasesincapitalinputdonotraisethemarginalproductivityoflabor(andtherebyreducemarginalcost)byalargeenoughamounttoreversetheeffectofdiminishingmarginalproductivityoflaboritself.ThesecondpartofEquation11.42thereforerequiresthatsuchcross-productivityeffectsberelativelysmall—thattheybedominatedbydiminishingmarginalproductivitiesoftheinputs.Iftheseconditionsaresatisﬁed,thenmarginalcostswillbeincreasingattheproﬁt-maximizingchoicesforkandl,andtheﬁrst-orderconditionswillrepresentalocalmaximum.InputdemandfunctionsInprinciple,theﬁrst-orderconditionsforhiringinputsinaproﬁt-maximizingwaycanbemanipulatedtoyieldinputdemandfunctionsthatshowhowhiringdependsonthepricesthattheﬁrmfaces.Wewilldenotethesedemandfunctionsbycapitaldemand¼kðP,v,wÞ,labordemand¼lðP,v,wÞ.(11.43)Noticethat,contrarytotheinputdemandconceptsdiscussedinChapter10,thesedemandfunctionsare“unconditional”—thatis,theyimplicitlypermittheﬁrmtoadjustitsoutputtochangingprices.Hence,thesedemandfunctionsprovideamorecompletepictureofhowpricesaffectinputdemandthandidthecontingentdemandfunctionsintroducedinChapter10.Wehavealreadyshownthattheseinputdemandfunctionscanalsobederivedfromtheproﬁtfunctionthroughdifferentiation;inExample11.5,weshowthatprocess14Iftheﬁrmisnotapricetakerintheoutputmarket,thenthisdeﬁnitionisgeneralizedbyusingmarginalrevenueinplaceofprice.Thatis,MRPl¼∂R=∂l¼∂R=∂q⋅∂q=∂l¼MR⋅MPl.Asimiliarderivationholdsforcapitalinput.Chapter11ProﬁtMaximization375
explicitly.First,however,wewillexplorehowchangesinthepriceofaninputmightbeexpectedtoaffectthedemandforit.Tosimplifymatterswelookonlyatlabordemand,buttheanalysisofthedemandforanyotherinputwouldbethesame.Ingeneral,weconcludethatthedirectionofthiseffectisunambiguousinallcases—thatis,∂l=∂w0nomatterhowmanyinputsthereare.Todevelopsomeintuitionforthisresult,webeginwithsomesimplecases.Single-inputcaseOnereasonforexpecting∂l=∂wtobenegativeisbasedonthepresumptionthatthemarginalphysicalproductoflabordeclinesasthequantityoflaboremployedincreases.Adecreaseinwmeansthatmorelabormustbehiredtobringabouttheequalityw¼P⋅MPl:AfallinwmustbemetbyafallinMPl(becausePisﬁxedasrequiredbytheceterisparibusassumption),andthiscanbebroughtaboutbyincreasingl.Thatthisargumentisstrictlycorrectforthecaseofoneinputcanbeshownasfollows.Writethetotaldifferentialoftheproﬁt-maximizingEquation11.41asdw¼P⋅∂fl∂l⋅∂l∂w⋅dwor∂l∂w¼1P⋅fll0,(11.44)wheretheﬁnalinequalityholdsbecausethemarginalproductivityoflaborisassumedtobediminishingðfll0Þ.Hencewehaveshownthat,atleastinthesingle-inputcase,aceterisparibusincreaseinthewagewillcauselesslabortobehired.Two-inputcaseForthecaseoftwo(ormore)inputs,thestoryismorecomplex.Theassumptionofadiminishingmarginalphysicalproductoflaborcanbemisleadinghere.Ifwfalls,therewillnotonlybeachangeinlbutalsoachangeinkasanewcost-minimizingcombinationofinputsischosen.Whenkchanges,theentireflfunctionchanges(labornowhasadifferentamountofcapitaltoworkwith),andthesimpleargumentusedpreviouslycannotbemade.Firstwewilluseagraphicapproachtosuggestwhy,eveninthetwo-inputcase,∂l=∂wmustbenegative.Amoreprecise,mathematicalanalysisispresentedinthenextsection.SubstitutioneﬀectInsomeways,analyzingthetwo-inputcaseissimilartotheanalysisoftheindividual’sresponsetoachangeinthepriceofagoodthatwaspresentedinChapter5.Whenwfalls,wecandecomposethetotaleffectonthequantityoflhiredintotwocomponents.Theﬁrstofthesecomponentsiscalledthesubstitutioneffect.Ifqisheldconstantatq1,thentherewillbeatendencytosubstitutelforkintheproductionprocess.ThiseffectisillustratedinFigure11.5a.Becausetheconditionforminimizingthecostofproducingq1requiresthatRTS¼w=v,afallinwwillnecessitateamovementfrominputcombinationAtocombina-tionB.AndbecausetheisoquantsexhibitadiminishingRTS,itisclearfromthediagramthatthissubstitutioneffectmustbenegative.Adecreaseinwwillcauseanincreaseinlaborhiredifoutputisheldconstant.OutputeﬀectItisnotcorrect,however,toholdoutputconstant.Itiswhenweconsiderachangeinq(theoutputeffect)thattheanalogytotheindividual’sutility-maximizationproblembreaksdown.376Part3ProductionandSupply
Consumershavebudgetconstraints,butﬁrmsdonot.Firmsproduceasmuchastheavailabledemandallows.Toinvestigatewhathappenstothequantityofoutputproduced,wemustinvestigatetheﬁrm’sproﬁt-maximizingoutputdecision.Achangeinw,becauseitchangesrelativeinputcosts,willshifttheﬁrm’sexpansionpath.Consequently,alltheﬁrm’scostcurveswillbeshifted,andprobablysomeoutputlevelotherthanq1willbechosen.Figure11.5bshowswhatmightbeconsideredthe“normal”case.TherethefallinwcausesMCtoshiftdownwardtoMC0.Consequently,theproﬁt-maximizinglevelofoutputrisesfromq1toq2.Theproﬁt-maximizingcondition(P¼MC)isnowsatisﬁedatahigherlevelofoutput.ReturningtoFigure11.5a,thisincreaseinoutputwillcauseevenmoreltobedemandedaslongaslisnotaninferiorinput(seebelow).TheresultofboththesubstitutionandoutputeffectswillbetomovetheinputchoicetopointContheﬁrm’sisoquantmap.Botheffectsworktoincreasethequantityoflaborhiredinresponsetoadecreaseintherealwage.TheanalysisprovidedinFigure11.5assumedthatthemarketprice(ormarginalrevenue,ifthisdoesnotequalprice)ofthegoodbeingproducedremainedconstant.Thiswouldbeanappropriateassumptionifonlyoneﬁrminanindustryexperiencedafallinunitlaborcosts.However,ifthedeclinewereindustrywidethenaslightlydifferentanalysiswouldbere-quired.Inthatcaseallﬁrms’marginalcostcurveswouldshiftoutward,andhencetheindustrysupplycurvewouldshiftalso.Assumingthatoutputdemandisdownwardsloping,thiswillleadtoadeclineinproductprice.Outputfortheindustryandforthetypicalﬁrmwillstillincreaseand(asbefore)morelaborwillbehired,buttheprecisecauseoftheoutputeffectisdifferent(seeProblem11.11).Cross-priceeﬀectsWehaveshownthat,atleastinsimplecases,∂l=∂wisunambiguouslynegative;substitutionandoutputeffectscausemorelabortobehiredwhenthewageratefalls.FromFigure11.5itshouldbeclearthatnodeﬁnitestatementcanbemadeabouthowcapitalusagerespondstoFIGURE11.5TheSubstitutionandOutputEﬀectsofaDecreaseinthePriceofaFactorWhenthepriceoflaborfalls,twoanalyticallydifferenteffectscomeintoplay.Oneofthese,thesubstitutioneffect,wouldcausemorelabortobepurchasedifoutputwereheldconstant.ThisisshownasamovementfrompointAtopointBin(a).AtpointB,thecost-minimizingconditionðRTS¼w=vÞissatisﬁedforthenew,lowerw.Thischangeinw=vwillalsoshifttheﬁrm’sexpansionpathanditsmarginalcostcurve.AnormalsituationmightbefortheMCcurvetoshiftdownwardinresponsetoadecreaseinwasshownin(b).WiththisnewcurveðMC0Þahigherlevelofoutputðq2Þwillbechosen.Consequently,thehiringoflaborwillincrease(tol2),alsofromthisoutputeffect.Pricek1k2Pq1q2ABCl1l2lper periodkper periodq1q2Outputper period(a) The isoquant map(b) The output decisionMCMC′Chapter11ProﬁtMaximization377
thewagechange.Thatis,thesignof∂k=∂wisindeterminate.Inthesimpletwo-inputcase,afallinthewagewillcauseasubstitutionawayfromcapital;thatis,lesscapitalwillbeusedtoproduceagivenoutputlevel.However,theoutputeffectwillcausemorecapitaltobedemandedaspartoftheﬁrm’sincreasedproductionplan.Thussubstitutionandoutputeffectsinthiscaseworkinoppositedirections,andnodeﬁniteconclusionaboutthesignof∂k=∂wispossible.AsummaryofsubstitutionandoutputeﬀectsTheresultsofthisdiscussioncanbesummarizedbythefollowingprinciple.OPTIMIZATIONPRINCIPLESubstitutionandoutputeffectsininputdemand.Whenthepriceofaninputfalls,twoeffectscausethequantitydemandedofthatinputtorise:1.thesubstitutioneffectcausesanygivenoutputleveltobeproducedusingmoreoftheinput;and2.thefallincostscausesmoreofthegoodtobesold,therebycreatinganadditionaloutputeffectthatincreasesdemandfortheinput.Forariseininputprice,bothsubstitutionandoutputeffectscausethequantityofaninputdemandedtodecline.Wenowprovideamoreprecisedevelopmentoftheseconceptsusingamathematicalapproachtotheanalysis.AmathematicaldevelopmentOurmathematicaldevelopmentofthesubstitutionandoutputeffectsthatarisefromthechangeinaninputpricefollowsthemethodweusedtostudytheeffectofpricechangesinconsumertheory.TheﬁnalresultisaSlutsky-styleequationthatresemblestheonewederivedinChapter5.However,theambiguitystemmingfromGiffen’sparadoxinthetheoryofconsumptiondemanddoesnotoccurhere.Westartwithareminderthatwehavetwoconceptsofdemandforanyinput(say,labor):(1)theconditionaldemandforlabor,denotedbylcðv,w,qÞ;and(2)theunconditionaldemandforlabor,whichisdenotedbylðP,v,wÞ.Attheproﬁt-maximizingchoiceforlaborinput,thesetwoconceptsagreeabouttheamountoflaborhired.Thetwoconceptsalsoagreeonthelevelofoutputproduced(whichisafunctionofalltheprices):lðP,v,wÞ¼lcðv,w,qÞ¼lcðv,w,qðP,v,wÞÞ.(11.45)Differentiationofthisexpressionwithrespecttothewage(andholdingtheotherpricesconstant)yields∂lðP,v,wÞ∂w¼∂lcðv,w,qÞ∂wþ∂lcðv,w,qÞ∂q⋅∂qðP,v,wÞ∂w.(11.46)So,theeffectofachangeinthewageonthedemandforlaboristhesumoftwocomponents:asubstitutioneffectinwhichoutputisheldconstant;andanoutputeffectinwhichthewagechangehasitseffectthroughchangingthequantityofoutputthattheﬁrmoptstoproduce.Theﬁrstoftheseeffectsisclearlynegative—becausetheproductionfunctionisquasi-concave(i.e.,ithasconvexisoquants),theoutput-contingentdemandforlabormustbenegativelysloped.Figure11.5bprovidesanintuitiveillustrationofwhytheoutputeffectinEqua-tion11.46isnegative,butitcanhardlybecalledaproof.Theparticularcomplicatingfactoristhepossibilitythattheinputunderconsideration(here,labor)maybeinferior.Perhapsoddly,inferiorinputsalsohavenegativeoutputeffects,butforratherarcanereasonsthatarebest378Part3ProductionandSupply
relegatedtoafootnote.15Thebottomline,however,isthatGiffen’sparadoxcannotoccurinthetheoryoftheﬁrm’sdemandforinputs:inputdemandfunctionsareunambiguouslydownwardsloping.Inthiscasethetheoryofproﬁtmaximizationimposesmorerestrictionsonwhatmighthappenthandoesthetheoryofutilitymaximization.InExample11.5weshowhowdecomposinginputdemandintoitssubstitutionandoutputcomponentscanyieldusefulinsightsintohowchangesininputpricesactuallyaffectﬁrms.EXAMPLE11.5DecomposingInputDemandintoSubstitutionandOutputComponentsTostudyinputdemandweneedtostartwithaproductionfunctionthathastwofeatures:(1)thefunctionmustpermitcapital-laborsubstitution(becausesubstitutionisanimportantpartofthestory);and(2)theproductionfunctionmustexhibitincreasingmarginalcosts(sothatthesecond-orderconditionsforproﬁtmaximizationaresatisﬁed).Onefunctionthatsatisﬁestheseconditionsisathree-inputCobb-Douglasfunctionwhenoneoftheinputsisheldﬁxed.So,letq¼fðk,l,gÞ¼k0.25l0.25g0.5,wherekandlarethefamiliarcapitalandlaborinputsandgisathirdinput(sizeofthefactory)thatisheldﬁxedatg¼16(squaremeters?)forallofouranalysis.Theshort-runproductionfunctionisthereforeq¼4k0.25l0.25.Weassumethatthefactorycanberentedatacostofrpersquaremeterperperiod.Tostudythedemandfor(say)laborinput,weneedboththetotalcostfunctionandtheproﬁtfunctionimpliedbythisproductionfunction.Mercifully,yourauthorhascomputedthesefunctionsforyouasCv,w,r,qðÞ¼q2v0.5w0.58þ16r(11.47)andΠðP,v,w,rÞ¼2P2v0.5w0.516r.(11.48)Asexpected,thecostsoftheﬁxedinputðgÞenterasaconstantintheseequations,andthesecostswillplayverylittleroleinouranalysis.EnvelopeResultsLabor-demandrelationshipscanbederivedfrombothofthesefunctionsthroughdifferentiation:lcðv,w,r,qÞ¼∂C∂w¼q2v0.5w0.516(11.49)andlðP,v,w,rÞ¼∂Π∂w¼P2v0.5w1.5.(11.50)Thesefunctionsalreadysuggestthatachangeinthewagehasalargereffectontotallabordemandthanitdoesoncontingentlabordemandbecausetheexponentofwismorenegativeinthetotaldemandequation.Thatis,theoutputeffectmustalsobeplayingarolehere.Toseethatdirectly,weturntosomenumbers.(continued)15Inwords,anincreaseinthepriceofaninferiorreducesmarginalcostandtherebyincreasesoutput.Butwhenoutputincreases,lessoftheinferiorinputishired.Hencetheendresultisadecreaseinquantitydemandedinresponsetoanincreaseinprice.Aformalproofmakesextensiveuseofenveloperelationships:outputeffect¼∂lc∂q⋅∂q∂w¼∂lc∂q⋅∂l∂P¼∂lc∂q2⋅∂q∂P.Becausethesecond-orderconditionsforproﬁtmaximizationrequirethat∂q=∂P>0,theoutputeffectisclearlynegative.Chapter11ProﬁtMaximization379
EXAMPLE11.5CONTINUEDNumericalexample.Let’sstartagainwiththeassumedvaluesthatwehavebeenusinginseveralpreviousexamples:v¼3,w¼12,andP¼60.Let’sﬁrstcalculatewhatoutputtheﬁrmwillchooseinthissituation.Todoso,weneeditssupplyfunction:qðP,v,w,rÞ¼∂Π∂P¼4Pv0.5w0.5.(11.51)Withthisfunctionandthepriceswehavechosen,theﬁrm’sproﬁt-maximizingoutputlevelis(surprise)q¼40.Withthesepricesandanoutputlevelof40,bothofthedemandfunctionspredictthattheﬁrmwillhirel¼50.BecausetheRTShereisgivenbyk=l,wealsoknowthatk=l¼w=v,soatthesepricesk¼200.Supposenowthatthewageraterisestow¼27butthattheotherpricesremainun-changed.Theﬁrm’ssupplyfunction(Equation11.51)showsthatitwillnowproduceq¼26.67.Theriseinthewageshiftstheﬁrm’smarginalcostcurveupwardand,withaconstantoutputprice,thiscausestheﬁrmtoproduceless.Toproducethisoutput,eitherofthelabor-demandfunctionscanbeusedtoshowthattheﬁrmwillhirel¼14.8.Hiringofcapitalwillalsofalltok¼133.3becauseofthelargereductioninoutput.Wecandecomposethefallinlaborhiringfroml¼50tol¼14.8intosubstitutionandoutputeffectsbyusingthecontingentdemandfunction.Iftheﬁrmhadcontinuedtoproduceq¼40eventhoughthewagerose,Equation11.49showsthatitwouldhaveusedl¼33.33.Capitalinputwouldhaveincreasedtok¼300.Becauseweareholdingoutputconstantatitsinitiallevelofq¼40,thesechangesrepresenttheﬁrm’ssubstitutioneffectsinresponsetothehigherwage.Thedeclineinoutputneededtorestoreproﬁtmaximizationcausestheﬁrmtocutbackonitsoutput.Indoingsoitsubstantiallyreducesitsuseofbothinputs.Noticeinparticularthat,inthisexample,theriseinthewagenotonlycausedlaborusagetodeclinesharplybutalsocausedcapitalusagetofallbecauseofthelargeoutputeffect.QUERY:Howwouldthecalculationsinthisproblembeaffectedifallﬁrmshadexperiencedtheriseinwages?Wouldthedeclineinlabor(andcapital)demandbegreaterorsmallerthanfoundhere?SUMMARYInthischapterwestudiedthesupplydecisionofaproﬁt-maximizingﬁrm.Ourgeneralgoalwastoshowhowsuchaﬁrmrespondstopricesignalsfromthemarketplace.Inad-dressingthatquestion,wedevelopedanumberofanalyticalresults.•Inordertomaximizeproﬁts,theﬁrmshouldchoosetoproducethatoutputlevelforwhichmarginalrevenue(therevenuefromsellingonemoreunit)isequaltomarginalcost(thecostofproducingonemoreunit).•Ifaﬁrmisapricetakerthenitsoutputdecisionsdonotaffectthepriceofitsoutput,somarginalrevenueisgivenbythisprice.Iftheﬁrmfacesadownward-slopingde-mandforitsoutput,however,thenitcansellmoreonlyatalowerprice.Inthiscasemarginalrevenuewillbelessthanpriceandmayevenbenegative.•MarginalrevenueandthepriceelasticityofdemandarerelatedbytheformulaMR¼P1þ1eq,p !,wherePisthemarketpriceoftheﬁrm’soutputandeq,pisthepriceelasticityofdemandforitsproduct.•Thesupplycurveforaprice-taking,proﬁt-maximizingﬁrmisgivenbythepositivelyslopedportionofitsmar-ginalcostcurveabovethepointofminimumaveragevariablecost(AVC).IfpricefallsbelowminimumAVC,theﬁrm’sproﬁt-maximizingchoiceistoshutdownandproducenothing.•Theﬁrm’sreactionstochangesinthevariouspricesitfacescanbestudiedthroughuseofitsproﬁtfunction,380Part3ProductionandSupply
PROBLEMS11.1John’sLawnMovingServiceisasmallbusinessthatactsasapricetaker(i.e.,MR¼P).Theprevailingmarketpriceoflawnmowingis$20peracre.John’scostsaregivenbytotalcost¼0.1q2þ10qþ50,whereq¼thenumberofacresJohnchoosestocutaday.a.HowmanyacresshouldJohnchoosetocutinordertomaximizeproﬁt?b.CalculateJohn’smaximumdailyproﬁt.c.GraphtheseresultsandlabelJohn’ssupplycurve.11.2Wouldalump-sumproﬁtstaxaffecttheproﬁt-maximizingquantityofoutput?Howaboutapropor-tionaltaxonproﬁts?Howaboutataxassessedoneachunitofoutput?Howaboutataxonlaborinput?11.3Thisproblemconcernstherelationshipbetweendemandandmarginalrevenuecurvesforafewfunctionalforms.a.Showthat,foralineardemandcurve,themarginalrevenuecurvebisectsthedistancebetweentheverticalaxisandthedemandcurveforanyprice.b.Showthat,foranylineardemandcurve,theverticaldistancebetweenthedemandandmarginalrevenuecurvesis1=b⋅q,wherebð<0Þistheslopeofthedemandcurve.c.Showthat,foraconstantelasticitydemandcurveoftheformq¼aPb,theverticaldistancebetweenthedemandandmarginalrevenuecurvesisaconstantratiooftheheightofthedemandcurve,withthisconstantdependingonthepriceelasticityofdemand.d.Showthat,foranydownward-slopingdemandcurve,theverticaldistancebetweenthedemandandmarginalrevenuecurvesatanypointcanbefoundbyusingalinearapproximationtothedemandcurveatthatpointandapplyingtheproceduredescribedinpart(b).e.Graphtheresultsofparts(a)–(d)ofthisproblem.ðP,v,wÞ.Thatfunctionshowsthemaximumproﬁtsthattheﬁrmcanachievegiventhepriceforitsoutput,thepricesofitsinput,anditsproductiontechnology.Theproﬁtfunctionyieldsparticularlyusefulenveloperesults.Differentiationwithrespecttomarketpriceyieldsthesupplyfunctionwhiledifferentiationwithre-specttoanyinputpriceyields(thenegativeof)thedemandfunctionforthatinput.•Short-runchangesinmarketpriceresultinchangestotheﬁrm’sshort-runproﬁtability.Thesecanbemeasuredgraphicallybychangesinthesizeofproducersurplus.Theproﬁtfunctioncanalsobeusedtocalculatechangesinproducersurplus.•Proﬁtmaximizationprovidesatheoryoftheﬁrm’sde-riveddemandforinputs.Theﬁrmwillhireanyinputuptothepointatwhichitsmarginalrevenueproductisjustequaltoitsper-unitmarketprice.Increasesinthepriceofaninputwillinducesubstitutionandoutputeffectsthatcausetheﬁrmtoreducehiringofthatinput.Chapter11ProﬁtMaximization381 11.4UniversalWidgetproduceshigh-qualitywidgetsatitsplantinGulch,Nevada,forsalethroughouttheworld.ThecostfunctionfortotalwidgetproductionðqÞisgivenbytotalcost¼0.25q2.WidgetsaredemandedonlyinAustralia(wherethedemandcurveisgivenbyq¼1002P)andLapland(wherethedemandcurveisgivenbyq¼1004P).IfUniversalWidgetcancontrolthequantitiessuppliedtoeachmarket,howmanyshoulditsellineachlocationinordertomaximizetotalproﬁts?Whatpricewillbechargedineachlocation?11.5Theproductionfunctionforaﬁrminthebusinessofcalculatorassemblyisgivenbyq¼2ﬃﬃlp,whereqdenotesﬁnishedcalculatoroutputandldenoteshoursoflaborinput.Theﬁrmisapricetakerbothforcalculators(whichsellforP)andforworkers(whichcanbehiredatawagerateofwperhour).a.Whatisthetotalcostfunctionforthisﬁrm?b.Whatistheproﬁtfunctionforthisﬁrm?c.Whatisthesupplyfunctionforassembledcalculators½qðP,wÞ?d.Whatisthisﬁrm’sdemandforlaborfunction½lðP,wÞ?e.Describeintuitivelywhythesefunctionshavetheformtheydo.11.6Themarketforhigh-qualitycaviarisdependentontheweather.Iftheweatherisgood,therearemanyfancypartiesandcaviarsellsfor$30perpound.Inbadweatheritsellsforonly$20perpound.Caviarproducedoneweekwillnotkeepuntilthenextweek.AsmallcaviarproducerhasacostfunctiongivenbyC¼0:5q2þ5qþ100,whereqisweeklycaviarproduction.Productiondecisionsmustbemadebeforetheweather(andthepriceofcaviar)isknown,butitisknownthatgoodweatherandbadweathereachoccurwithaprobabilityof0.5.a.Howmuchcaviarshouldthisﬁrmproduceifitwishestomaximizetheexpectedvalueofitsproﬁts?b.Supposetheownerofthisﬁrmhasautilityfunctionoftheformutility¼ﬃﬃﬃﬃπp,whereπisweeklyproﬁts.Whatistheexpectedutilityassociatedwiththeoutputstrategydeﬁnedinpart(a)?c.Canthisﬁrmownerobtainahigherutilityofproﬁtsbyproducingsomeoutputotherthanthatspeciﬁedinparts(a)and(b)?Explain.d.Supposethisﬁrmcouldpredictnextweek’spricebutcouldnotinﬂuencethatprice.Whatstrategywouldmaximizeexpectedproﬁtsinthiscase?Whatwouldexpectedproﬁtsbe?382Part3ProductionandSupply 11.7TheAcmeHeavyEquipmentSchoolteachesstudentshowtodriveconstructionmachinery.Thenumberofstudentsthattheschoolcaneducateperweekisgivenbyq¼10minðk,lÞr,wherekisthenumberofbackhoestheﬁrmrentsperweek,listhenumberofinstructorshiredeachweek,andγisaparameterindicatingthereturnstoscaleinthisproductionfunction.a.Explainwhydevelopmentofaproﬁt-maximizingmodelhererequires0<γ<1.b.Suppposingγ¼0.5,calculatetheﬁrm’stotalcostfunctionandproﬁtfunction.c.Ifv¼1000,w¼500,andP¼600,howmanystudentswillAcmeserveandwhatareitsproﬁts?d.IfthepricestudentsarewillingtopayrisestoP¼900,howmuchwillproﬁtschange?e.GraphAcme’ssupplycurveforstudentslots,andshowthattheincreaseinproﬁtscalculatedinpart(d)canbeplottedonthatgraph.11.8Howwouldyouexpectanincreaseinoutputprice,P,toaffectthedemandforcapitalandlaborinputs?a.Explaingraphicallywhy,ifneitherinputisinferior,itseemsclearthatariseinPmustnotreducethedemandforeitherfactor.b.Showthatthegraphicalpresumptionfrompart(a)isdemonstratedbytheinputdemandfunctionsthatcanbederivedintheCobb-Douglascase.c.UsetheproﬁtfunctiontoshowhowthepresenceofinferiorinputswouldleadtoambiguityintheeffectofPoninputdemand.AnalyticalProblems11.9ACESproﬁtfunctionWithaCESproductionfunctionoftheformq¼ðkρþlρÞγ=ρawholelotofalgebraisneededtocomputetheproﬁtfunctionasðP,v,wÞ¼KP1=ð1γÞðv1σþw1σÞγ=ð1σÞðγ1Þ,whereσ¼1=ð1ρÞandKisaconstant.a.Ifyouareagluttonforpunishment(orifyourinstructoris),provethattheproﬁtfunctiontakesthisform.PerhapstheeasiestwaytodosoistostartfromtheCEScostfunctioninExample10.2.b.Explainwhythisproﬁtfunctionprovidesareasonablerepresentationofaﬁrm’sbehavioronlyfor0<γ<1.c.ExplaintheroleoftheelasticityofsubstitutionðσÞinthisproﬁtfunction.d.Whatisthesupplyfunctioninthiscase?Howdoesσdeterminetheextenttowhichthatfunctionshiftswheninputpriceschange?e.Derivetheinputdemandfunctionsinthiscase.Howarethesefunctionsaffectedbythesizeofσ?11.10SomeenveloperesultsYoung’stheoremcanbeusedincombinationwiththeenveloperesultsinthischaptertoderivesomeusefulresults.a.Showthat∂lðP,v,wÞ=∂v¼∂kðP,v,wÞ=∂w.Interpretthisresultusingsubtitutionandoutputeffects.b.Usetheresultfrompart(a)toshowhowaunittaxonlaborwouldbeexpectedtoaffectcapitalinput.c.Showthat∂q=∂w¼∂l=∂P.Interpretthisresult.d.Usetheresultfrompart(c)todiscusshowaunittaxonlaborinputwouldaffectquantitysupplied.Chapter11ProﬁtMaximization383 11.11MoreonthederiveddemandwithtwoinputsThedemandforanyinputdependsultimatelyonthedemandforthegoodsthatinputproduces.Thiscanbeshownmostexplicitlybyderivinganentireindustry’sdemandforinputs.Todoso,weassumethatanindustryproducesahomogeneousgood,Q,underconstantreturnstoscaleusingonlycapitalandlabor.ThedemandfunctionforQisgivenbyQ¼DðPÞ,wherePisthemarketpriceofthegoodbeingproduced.Becauseoftheconstantreturns-to-scaleassumption,P¼MC¼AC.ThroughoutthisproblemletCðv,w,1Þbetheﬁrm’sunitcostfunction.a.ExplainwhythetotalindustrydemandsforcapitalandlaboraregivenbyK¼QCvandL¼QCw.b.Showthat∂K∂v¼QCvvþD0C2vand∂L∂w¼QCwwþD0C2w.c.ProvethatCvv¼wvCvwandCww¼vwCvw.d.Usetheresultsfromparts(b)and(c)togetherwiththeelasticityofsubstitutiondeﬁnedasσ¼CCvw=CvCwtoshowthat∂K∂v¼wLQ⋅σKvCþD0K2Q2and∂L∂w¼vKQ⋅σLwCþD0L2Q2.e.Convertthederivativesinpart(d)intoelasticitiestoshowthateK,v¼sLσþsKeQ,PandeL,w¼sKσþsLeQ,P,whereeQ,Pisthepriceelasticityofdemandfortheproductbeingproduced.f.Discusstheimportanceoftheresultsinpart(e)usingthenotionsofsubstitutionandoutputeffectsfromChapter11.Note:ThenotionthattheelasticityofthederiveddemandforaninputdependsonthepriceelasticityofdemandfortheoutputbeingproducedwasﬁrstsuggestedbyAlfredMarshall.TheproofgivenherefollowsthatinD.Hamermesh,LaborDemand(Princeton,NJ:PrincetonUniversityPress,1993).11.12Cross-priceeﬀectsininputdemandWithtwoinputs,cross-priceeffectsoninputdemandcanbeeasilycalculatedusingtheprocedureoutlinedinProblem11.11.a.Usesteps(b),(d),and(e)fromProblem11.11toshowthateK,w¼sLðσþeQ,PÞandeL,v¼sKðσþeQ,PÞ.b.Describeintuitivelywhyinputsharesappearsomewhatdifferentlyinthedemandelasticitiesinpart(e)ofProblem11.11thantheydoinpart(a)ofthisproblem.c.Theexpressioncomputedinpart(a)canbeeasilygeneralizedtothemany-inputcaseasexi,wj¼sjðAi,jþeQ,PÞ,whereAi,jistheAllenelasticityofsubstitutiondeﬁnedinProblem10.12.ForreasonsdescribedinProblems10.11and10.12,thisapproachtoinputdemandinthemulti-inputcaseisgenerallyinferiortousingMorishimaelasticities.Oneodditymightbementioned,however.Forthecasei¼jthisexpressionseemstosaythateL,w¼sLðAL,LþeQ,PÞ,andifwejumpedtotheconclusionthatAL,L¼σinthetwo-inputcasethenthiswouldcontradicttheresultfromProblem11.11.YoucanresolvethisparadoxbyusingthedeﬁnitionsfromProblem10.12toshowthat,withtwoinputs,AL,L¼ðsK=sLÞ⋅AK,L¼ðsK=sLÞ⋅σandsothereisnodisagreement.384Part3ProductionandSupply SUGGESTIONSFORFURTHERREADINGFerguson,C.E.TheNeoclassicalTheoryofProductionandDistribution.Cambridge,UK:CambridgeUniversityPress,1969.Providesacompleteanalysisoftheoutputeffectinfactordemand.Alsoshowshowthedegreeofsubstitutabilityaffectsmanyoftheresultsinthischapter.Hicks,J.R.ValueandCapital,2nded.Oxford:OxfordUniversityPress,1947.TheAppendixlooksindetailatthenotionoffactorcomplementarity.Mas-Colell,A.,M.D.Whinston,andJ.R.Green.Microeco-nomicTheory.NewYork:OxfordUniversityPress,1995.Providesanelegantintroductiontothetheoryofproductionusingvectorandmatrixnotation.Thisallowsforanarbitrarynumberofinputsandoutputs.Samuelson,P.A.FoundationsofEconomicAnalysis.Cam-bridge,MA:HarvardUniversityPress,1947.Earlydevelopmentoftheproﬁtfunctionideatogetherwithanicediscussionoftheconsequencesofconstantreturnstoscaleformarketequilibrium.Sydsaeter,K.,A.Strom,andP.Berck.Economists’Mathe-maticalManual,3rded.Berlin:Springer-Verlag,2000.Chapter25offersformulasforanumberofproﬁtandfactordemandfunctions.Varian,H.R.MicroeconomicAnalysis,3rded.NewYork:W.W.Norton,1992.Includesanentirechapterontheproﬁtfunction.Varianoffersanovelapproachforcomparingshort-andlong-runresponsesusingtheLeChatelierprinciple.Chapter11ProﬁtMaximization385 EXTENSIONSApplicationsoftheProﬁtFunctionInChapter11weintroducedtheproﬁtfunction.Thatfunctionsummarizestheﬁrm’s“bottomline”asitdependsonthepricesitfacesforitsoutputsandinputs.Intheseextensionsweshowhowsomeofthepropertiesoftheproﬁtfunctionhavebeenusedtoassessimportantempiricalandtheoreticalquestions.E11.1ConvexityandpricestabilizationConvexityoftheproﬁtfunctionimpliesthataﬁrmwillgenerallypreferaﬂuctuatingoutputpricetoonethatisstabilized(say,throughgovernmentintervention)atitsmeanvalue.Theresultrunscontrarytothedirectionofeconomicpolicyinmanylessdevelopedcountries,whichtendstostressthedesirabilityofstabilizationofcommodityprices.Severalfactorsmayaccountforthisseemingparadox.First,manyplansto“stabilize”com-moditypricesareinrealityplanstoraisetheaverageleveloftheseprices.Cartelsofproducersoftenhavethisastheirprimarygoal,forexample.Second,theconvexityresultappliesforasingleprice-takingﬁrm.Fromtheperspectiveoftheentiremarket,totalreven-uesfromstabilizedorﬂuctuatingpriceswilldependonthenatureofthedemandfortheproduct.1Athirdcomplicationthatmustbeaddressedinassessingpricestabilizationschemesisﬁrms’expectationsoffutureprices.Whencommoditiescanbestored,optimalpro-ductiondecisionsinthepresenceofpricestabilizationschemescanbequitecomplex.Finally,thepurposeofpricestabilizationschemesmayinsomesituationsbefocusedmoreonreducingrisksfortheconsumersofbasiccommodities(suchasfood)thanonthewelfareofproducers.Still,thisfundamentalpropertyoftheproﬁtfunctionsuggestscautionindevisingpricestabi-lizationschemesthathavedesirablelong-runeffectsonproducers.Foranextendedtheoreticalanalysisoftheseissues,seeNewburyandStiglitz(1981).E11.2Producersurplusandtheshort-runcostsofdiseaseDiseaseepisodescanseverelydisruptmarkets,leadingtoshort-runlossesinproducerandconsumersurplus.Forﬁrms,theselossescanbecomputedastheshort-runlossesofproﬁtsfromtemporarilylowerpricesfortheiroutputorfromthetemporarilyhigherinputpricestheymustpay.AparticularextensivesetofsuchcalculationsisprovidedbyHarrington,Krupnick,andSpofford(1991)intheirdetailedstudyofagiardi-asisoutbreakinPennsylvaniain1983.Althoughcon-sumerssufferedmostofthelossesassociatedwiththisoutbreak,theauthorsalsocalculatesubstantiallossesforrestaurantsandbarsintheimmediatearea.Suchlossesarosebothfromreducedbusinessfortheseﬁrmsandfromthetemporaryneedtousebottledwaterandotherhigh-costinputsintheiroperations.Quantitativecalculationsoftheselossesarebasedonproﬁtfunctionsdescribedbytheauthors.E11.3ProﬁtfunctionsandproductivitymeasurementInChapter9weshowedthattotalfactorproductivitygrowthisusuallymeasuredasGA¼GqskGkslGl,whereGx¼dx=dtx¼dlnxdtandwhereskandslarethesharesofcapitalandlaborintotalcosts,respectively.Onedifﬁcultywithmakingthiscalculationisthatitrequiresmeasuringchangesininputusageovertime—ameasurementthatcanbeespeciallydifﬁcultforcapital.Theproﬁtfunctionpro-videsanalternativewayofmeasuringthesamephe-nomenonwithoutestimatinginputusagedirectly.Tounderstandthelogicofthisapproach,considertheproductionfunctionwewishtoexamine,q¼fðk,l,tÞ.Wewanttoknowhowoutputwouldchangeovertimeifinputlevelswereheldconstant.Thatis,wewishtomeasure∂ðlnqÞ=∂t¼ft=f.Noticetheuseofpartialdifferentiationinthisexpression—inwords,wewanttoknowtheproportionatechangeinfovertimewhenotherinputsareheldconstant.Iftheproductionfunc-tionexhibitsconstantreturnstoscaleandiftheﬁrmisapricetakerforbothinputsanditsoutput,itisfairlyeasy2toshowthatthispartialderivativeisthemeasureof1Speciﬁcally,foraconstantelasticitydemandfunction,totalrevenuewillbeaconcavefunctionofpriceifdemandisinelasticbutconvexifdemandiselastic.Hence,intheelasticcase,producerswillobtainhighertotalrevenuesfromaﬂuctuatingpricethanfromapricestabilizedatitsmeanvalue.2Theproofproceedsbydifferentiatingtheproductionfunctionlogarith-micallywithrespecttotimeasGq¼dðlnqÞ=dt¼eq,kGkþeq,lGlþft=fandthenrecognizingthat,withconstantreturnstoscaleandprice-takingbehavior,eq,k¼skandeq,l¼sl.386Part3ProductionandSupply changingtotalfactorproductivitywewant—thatis,GA¼ft=f.Nowconsidertheproﬁtfunction,ðP,v,w,tÞ.Bydeﬁnition,proﬁtsaregivenbyπ¼Pqvkwl¼Pfvkwl,so∂lnΠ∂t¼PftΠandthusGA¼ftf¼ΠPf⋅∂lnΠ∂t¼ΠPq⋅∂lnΠ∂t.(i)So,inthisspecialcase,changesintotalfactorpro-ductivitycanbeinferredfromtheshareofproﬁtsintotalrevenueandthetimederivativeofthelogoftheproﬁtfunction.Butthisconclusioncanbereadilygeneralizedtocasesofnonconstantreturnstoscaleandeventoﬁrmsthatproducemultipleoutputs(seee.g.Kumbhakar,2002).Hence,forsituationswhereinputandoutputpricesaremorereadilyavailablethaninputquantities,usingtheproﬁtfunctionisanattractivewaytoproceed.Threeexamplesofthisusefortheproﬁtfunctionmightbementioned.KaragiannisandMergos(2000)reassessthemajorincreasesintotalfactorproductivitythathavebeenexperiencedbyU.S.agricultureduringthepast50yearsusingtheproﬁtfunctionapproach.Theyﬁndresultsthatarebroadlyconsistentwiththoseusingmoreconventionalmeasures.Huang(2000)adoptsthesameapproachinastudyofTaiwanesebankingandﬁndssigniﬁcantincreasesinproductivitythatcouldnotbedetectedusingothermethods.Fi-nally,CoelliandPerelman(2000)useamodiﬁedproﬁtfunctionapproachtomeasuretherelativeefﬁ-ciencyofEuropeanrailroads.Perhapsnotsurprisingly,theyﬁndthatDutchrailroadsarethemostefﬁcientinEuropewhereasthoseinItalyaretheleastefﬁcient.ReferencesCoelli,T.,andS.Perelman.“TechnicalEfﬁciencyofEuropeanRailways:ADistanceFunctionApproach.”AppliedEconomics(December2000):1967–76.Harrington,W.A.,J.Krupnick,andW.O.Spofford.EconomicsandEpisodicDisease:TheBeneﬁtsofPrevent-ingaGiardiasisOutbreak.Baltimore:JohnsHopkinsUniversityPress,1991.Huang,T.“EstimatingX-EfﬁciencyinTaiwaneseBankingUsingaTranslogShadowProﬁtFunction.”JournalofProductivityAnalysis(November2000):225–45.Karagiannis,G.,andG.J.Mergos.“TotalFactorProduc-tivityGrowthandTechnicalChangeinaProﬁtFunctionFramework.”JournalofProductivityAnalysis(July2000):31–51.Kumbhakar,S.“ProductivityMeasurement:AProﬁtFunctionApproach.”AppliedEconomicsLetters(April2002):331–34.Newbury,D.M.G.,andJ.E.Stiglitz.TheTheoryofCommodityPriceStabilization.Oxford:OxfordUniver-sityPress,1981.Chapter11ProﬁtMaximization387 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PART4CompetitiveMarketsCHAPTER12ThePartialEquilibriumCompetitiveModelCHAPTER13GeneralEquilibriumandWelfareInParts2and3wedevelopedmodelstoexplainthedemandforgoodsbyutility-maximizingindividualsandthesupplyofgoodsbyproﬁt-maximizingﬁrms.Inthispartwewillbringthesetwostrandsofanalysistogethertodescribetheprocessbywhichpricesaredetermined.Wewillfocusononlyonespeciﬁcmodelofpricedetermination,theperfectlycompetitivemodel.Thatmodelassumesalargeenoughnumberofdemandersandsuppliersofeachgoodsothateachmustbeapricetaker.InPart5wewillillustratesomeofthemodelsthatresultfromrelaxingthestrictprice-takingassumptionsofthecompetitivecase,butinthispartweassumeprice-takingbehaviorthroughout.Chapter12developsthefamiliarpartialequilibriummodelofpricedeterminationincompetitivemarkets.TheprincipalresultistheMarshallian“cross”diagramofsupplyanddemandthatweﬁrstdiscussedinChapter1.Thismodelillustratesa“partial”equilibriumviewofpricedeterminationbecauseitfocusesononlyasinglemarket.Intheconcludingsectionsofthechapterweshowsomeofthewaysinwhichsuchmodelsareapplied.Aspeciﬁcfocusisonillustratinghowthecompetitivemodelcanbeusedtojudgethewelfareconsequencesformarketparticipantsofchangesinmarketequilibria.Althoughthepartialequilibriumcompetitivemodelisusefulforstudyingasinglemarketindetail,itisinappropriateforexaminingrelationshipsamongmarkets.Tocapturesuchcross-marketeffectsrequiresthedevelopmentof“general”equilibriummodels—atopicwetakeupinChapter13.Thereweshowhowanentireeconomycanbeviewedasasystemofinterconnectedcompetitivemarketsthatdetermineallpricessimultaneously.Wealsoexaminehowwelfareconsequencesofvariouseconomicquestionscanbestudiedinthismodel. 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CHAPTER12ThePartialEquilibriumCompetitiveModelInthischapterwedescribethefamiliarmodelofpricedeterminationunderperfectcompetitionthatwasoriginallydevelopedbyAlfredMarshallinthelatenineteenthcentury.Thatis,weprovideafairlycompleteanalysisofthesupply-demandmechanismasitappliestoasinglemarket.Thisisperhapsthemostwidelyusedmodelforthestudyofprices.MARKETDEMANDInPart2weshowedhowtoconstructindividualdemandfunctionsthatillustratechangesinthequantityofagoodthatautility-maximizingindividualchoosesasthemarketpriceandotherfactorschange.Withonlytwogoods(xandy)weconcludedthatanindividual’s(Marshallian)demandfunctioncanbesummarizedasquantityofxdemanded¼xðpx,py,IÞ.(12.1)Nowwewishtoshowhowthesedemandfunctionscanbeaddeduptoreflectthedemandofallindividualsinamarketplace.Usingasubscriptiði¼1,nÞtorepresenteachperson’sdemandfunctionforgoodx,wecandefinethetotaldemandinthemarketasmarketdemandforX¼Xni¼1xiðpx,py,IiÞ.(12.2)Noticethreethingsaboutthissummation.First,weassumethateveryoneinthismarketplacefacesthesamepricesforbothgoods.Thatis,pxandpyenterEquation12.2withoutperson-specificsubscripts.Ontheotherhand,eachperson’sincomeentersintohisorherownspecificdemandfunction.Marketdemanddependsnotonlyonthetotalincomeofallmarketparticipantsbutalsoonhowthatincomeisdistributedamongconsumers.Finally,observethatwehaveusedanuppercaseXtorefertomarketdemand—anotationwewillsoonmodify.ThemarketdemandcurveEquation12.2makesclearthatthetotalquantityofagooddemandeddependsnotonlyonitsownpricebutalsoonthepricesofothergoodsandontheincomeofeachperson.ToconstructthemarketdemandcurveforgoodX,weallowpxtovarywhileholdingpyandtheincomeofeachpersonconstant.Figure12.1showsthisconstructionforthecasewherethereareonlytwoconsumersinthemarket.Foreachpotentialpriceofx,thepointonthemarketdemandcurveforXisfoundbyaddingupthequantitiesdemandedbyeachperson.Forexample,atapriceofpxperson1demandsx1andperson2demandsx2.Thetotalquantitydemandedinthistwo-personmarketisthesumofthesetwoamountsðX¼x1þx2Þ.Thepointpx,XisthereforeonepointonthemarketdemandcurveforX.Otherpointsonthe391 curvearederivedinasimilarway.Themarketdemandcurveisthusa“horizontalsum”ofeachindividual’sdemandcurve.1ShiftsinthemarketdemandcurveThemarketdemandcurve,then,summarizestheceterisparibusrelationshipbetweenXandpx.Itisimportanttokeepinmindthatthecurveisinrealityatwo-dimensionalrepresenta-tionofamany-variablefunction.Changesinpxresultinmovementsalongthiscurve,butchangesinanyoftheotherdeterminantsofthedemandforXcausethecurvetoshifttoanewposition.Ageneralriseinincomeswould,forexample,causethedemandcurvetoshiftoutward(assumingXisanormalgood)becauseeachindividualwouldchoosetobuymoreXateveryprice.Similarly,ariseinpywouldshiftthedemandcurvetoXoutwardifindividualsregardedXandYassubstitutes,butitwouldshiftthedemandcurveforXinwardifthegoodswereregardedascomplements.Accountingforallsuchshiftsmaysometimesrequirereturningtoexaminetheindividualdemandfunctionsthatconstitutethemarketrelationship,especiallywhenexaminingsituationsinwhichthedistributionofincomechangesandtherebyraisessomeincomeswhilereducingothers.Tokeepmattersstraight,economistsusuallyreservethetermchangeinquantitydemandedforamovementalongaﬁxeddemandcurveinresponsetoachangeinpx.Alternatively,anyshiftinthepositionofthedemandcurveisreferredtoasachangeindemand.EXAMPLE12.1ShiftsinMarketDemandTheseideascanbeillustratedwithasimplesetoflineardemandfunctions.Supposeindividual1’sdemandfororanges(x,measuredindozensperyear)isgivenby2x1¼102pxþ0:1I1þ0:5py,(12.3)FIGURE12.1ConstructionofaMarketDemandCurvefromIndividualDemandCurvesAmarketdemandcurveisthe“horizontalsum”ofeachindividual’sdemandcurve.Ateachpricethequantitydemandedinthemarketisthesumoftheamountseachindividualdemands.Forexample,atpxthedemandinthemarketisx1þx2¼x.pxpxpxpx*x1Xx1*x2x2*Xx*(a) Individual 1(b) Individual 2(c) Market demandx1x21Compensatedmarketdemandcurvescanbeconstructedinexactlythesamewaybysummingeachindividual’scom-pensateddemand.Suchacompensatedmarketdemandcurvewouldholdeachperson’sutilityconstant.2Thislinearformisusedtoillustratesomeissuesinaggregation.Itisdifficulttodefendthisformtheoretically,however.Forexample,itisnothomogeneousofdegree0inallpricesandincome.392Part4CompetitiveMarkets wherepx¼priceoforangesðdollarsperdozenÞ,I1¼individual1’sincomeðinthousandsofdollarsÞ,py¼priceofgrapefruitðagrosssubstitutefororanges—dollarsperdozenÞ.Individual2’sdemandfororangesisgivenbyx2¼17pxþ0:05I2þ0:5py.(12.4)HencethemarketdemandfunctionisXðpx,py,I1,I2Þ¼x1þx2¼273pxþ0:1I1þ0:05I2þpy.(12.5)Herethecoefficientforthepriceoforangesrepresentsthesumofthetwoindividuals’coefficients,asdoesthecoefficientforgrapefruitprices.Thisreflectstheassumptionthatorangeandgrapefruitmarketsarecharacterizedbythelawofoneprice.Becausethein-dividualshavedifferingcoefficientsforincome,however,thedemandfunctiondependsoneachperson’sincome.TographEquation12.5asamarketdemandcurve,wemustassumevaluesforI1,I2,andpy(becausethedemandcurvereﬂectsonlythetwo-dimensionalrelationshipbetweenxandpx).IfI1¼40,I2¼20,andpy¼4,thenthemarketdemandcurveisgivenbyX¼273pxþ4þ1þ4¼363px,(12.6)whichisasimplelineardemandcurve.Ifthepriceofgrapefruitweretorisetopy¼6thenthecurvewould,assumingincomesremainunchanged,shiftoutwardtoX¼273pxþ4þ1þ6¼383px,(12.7)whereasanincometaxthattook10(thousanddollars)fromindividual1andtransferredittoindividual2wouldshiftthedemandcurveinwardtoX¼273pxþ3þ1:5þ4¼35:53px(12.8)becauseindividual1hasalargermarginaleffectofincomechangesonorangepurchases.Allofthesechangesshiftthedemandcurveinaparallelwaybecause,inthislinearcase,noneofthemaffectseitherindividual’scoefficientforpx.Inallcases,ariseinpxof0.10(tencents)wouldcauseXtofallby0.30(dozenperyear).QUERY:Forthislinearcase,whenwoulditbepossibletoexpressmarketdemandasalinearfunctionoftotalincomeðI1þI2Þ?Alternatively,supposetheindividualshaddifferingco-efﬁcientsforpy.Wouldthatchangetheanalysisinanyfundamentalway?GeneralizationsAlthoughourconstructionconcernsonlytwogoodsandtwoindividuals,itiseasilygeneral-ized.Supposetherearengoods(denotedbyxi,i¼1,n)withpricespi,i¼1,n.Assumealsothattherearemindividualsinsociety.Thenthejthindividual’sdemandfortheithgoodwilldependonallpricesandonIj,theincomeofthisperson.Thiscanbedenotedbyxi,j¼xi,jðp1,…,pn,IjÞ,(12.9)wherei¼1,nandj¼1,m.Chapter12ThePartialEquilibriumCompetitiveModel393 Usingtheseindividualdemandfunctions,marketdemandconceptsareprovidedbythefollowingdeﬁnition.DEFINITIONMarketdemand.ThemarketdemandfunctionforaparticulargoodðXiÞisthesumofeachindividual’sdemandforthatgood:Xi¼Xmj¼1xi,jðp1,…,pn,IjÞ.(12.10)ThemarketdemandcurveforXiisconstructedfromthedemandfunctionbyvaryingpi,whileholdingallotherdeterminantsofXiconstant.Assumingthateachindividual’sdemandcurveisdownwardsloping,thismarketdemandcurvewillalsobedownwardsloping.Ofcourse,thisdeﬁnitionisjustageneralizationofourpriordiscussion,butthreefeatureswarrantrepetition.First,thefunctionalrepresentationofEquation12.10makesclearthatthedemandforXidependsnotonlyonpibutalsoonthepricesofallothergoods.Achangeinoneofthoseotherpriceswouldthereforebeexpectedtoshiftthedemandcurvetoanewposition.Second,thefunctionalnotationindicatesthatthedemandforXidependsontheentiredistributionofindividuals’incomes.Althoughinmanyeconomicdiscussionsitiscustomarytorefertotheeffectofchangesinaggregatetotalpurchasingpoweronthedemandforagood,thisapproachmaybeamisleadingsimpliﬁcationbecausetheactualeffectofsuchachangeontotaldemandwilldependonpreciselyhowtheincomechangesaredistributedamongindividuals.Finally,althoughtheyareobscuredsomewhatbythenotationwehavebeenusing,theroleofchangesinpreferencesshouldbementioned.Wehaveconstructedindividuals’demandfunctionswiththeassumptionthatpreferences(asrepre-sentedbyindifferencecurvemaps)remainﬁxed.Ifpreferencesweretochange,sowouldindividualandmarketdemandfunctions.Hence,marketdemandcurvescanclearlybeshiftedbychangesinpreferences.Inmanyeconomicanalyses,however,itisassumedthatthesechangesoccursoslowlythattheymaybeimplicitlyheldconstantwithoutmisrepre-sentingthesituation.AsimplifiednotationOfteninthisbookweshallbelookingatonlyonemarket.Inordertosimplifythenotation,inthesecasesweshalluseQDtorefertothequantityoftheparticulargooddemandedinthismarketandPtodenoteitsmarketprice.Asalways,whenwedrawademandcurveintheQ–Pplane,theceterisparibusassumptionisineffect.Ifanyofthefactorsmentionedintheprevioussection(otherprices,individuals’incomes,orpreferences)shouldchange,theQ–Pdemandcurvewillshift,andweshouldkeepthatpossibilityinmind.Whenweturntoconsiderrelationshipsamongtwoormoregoods,however,wewillreturntothenotationwehavebeenusingupuntilnow(thatis,denotinggoodsbyxandyorbyxi).ElasticityofmarketdemandWhenweusethisnotationformarketdemand,wewillalsouseacompactnotationforthepriceelasticityofthemarketdemandfunction:priceelasticityofmarketdemand¼eQ,P¼∂QDðP,P0,IÞ∂P⋅PQD,(12.11)wherethenotationisintendedasareminderthatthedemandforQdependsonmanyfactorsotherthanitsownprice,suchasthepricesofothergoodsðP0Þandtheincomesofallpotentialdemanders(I).Theseotherfactorsareheldconstantwhencomputingthe394Part4CompetitiveMarkets own-priceelasticityofmarketdemand.AsinChapter5,thiselasticitymeasurestheproportionateresponseinquantitydemandedtoa1percentchangeinagood’sprice.MarketdemandisalsocharacterizedbywhetherdemandiselasticðeQ,P<1Þorinelas

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