Chapter 4 UTILITY MAXIMIZATION AND CHOICE. Copyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.

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1 Chater 4 UTILITY MAXIMIZATION AND CHOICE Coyright 2005 by South-Western, a division of Thomson Learning. All rights reserved. 1

2 Comlaints about the Economic Aroach No real individuals make the kinds of lightning calculations ( 闪电计算 ) required for utility maimization But the utility-maimization model redicts many asects of behavior Thus, economists assume that eole behave as if they made such calculations 2

3 Comlaints about the Economic Aroach The economic model of choice is etremely selfish because no one has solely self-centered goals But nothing in the utility-maimization model revents individuals from deriving satisfaction from doing good 3

4 Otimization Princile To maimize utility, given a fied amount of income to send, an individual will buy the goods and services: that ehaust his or her total income for which the sychic rate of trade-off between any goods (the MRS) is equal to the rate at which goods can be traded for one another in the marketlace 4

5 A Numerical Illustration Assume that the individual s MRS 1 willing to trade one unit of for one unit of y Suose the rice of $2 and the rice of y $1 The individual can be made better off trade 1 unit of for 2 units of y in the marketlace 5

6 The Budget Constraint Assume that an individual has I dollars to allocate between good and good y + y y I Quantity of y I y If all income is sent on y, this is the amount of y that can be urchased The individual can afford to choose only combinations of and y in the shaded triangle If all income is sent on, this is the amount of that can be urchased I Quantity of 6

7 First-Order Conditions for a Maimum We can add the individual s utility ma to show the utility-maimization rocess Quantity of y A B C The individual can do better than oint A by reallocating his budget The individual cannot have oint C because income is not large enough U 1 U 2 U 3 Point B is the oint of utility maimization Quantity of 7

8 First-Order Conditions for a Maimum Utility is maimized where the indifference curve is tangent to the budget constraint Quantity of y sloe of budget constraint y B sloe of indifference curve dy d U constant U 2 dy - d y U constant MRS Quantity of 8

9 Second-Order Conditions for a Maimum The tangency rule is only necessary but not sufficient unless we assume that MRS is diminishing if MRS is diminishing, then indifference curves are strictly conve If MRS is not diminishing, then we must check second-order conditions to ensure that we are at a maimum 9

10 Second-Order Conditions for a Maimum The tangency rule is only a necessary condition we need MRS to be diminishing Quantity of y B There is a tangency at oint A, but the individual can reach a higher level of utility at oint B A U 2 U 1 Quantity of 10

11 Corner Solutions In some situations, individuals references may be such that they can maimize utility by choosing to consume only one of the goods Quantity of y U 1 U 2 U 3 At oint A, the indifference curve is not tangent to the budget constraint Utility is maimized at oint A A Quantity of 11

12 The (general) n-good Case The individual s objective is to maimize utility U( 1, 2,, n ) subject to the budget constraint I n n Set u the Lagrangian: L U( 1, 2,, n ) + λ(i n n ) 12

13 The n-good Case First-order conditions for an interior maimum: L/ 1 U/ 1 - λ 1 0 L/ 2 U/ 2 - λ 2 0 L/ n U/ n - λ n 0 L/ λ I n n 0 13

14 Imlications of First-Order Conditions For any two goods, U U / / i j i j This imlies that at the otimal allocation of income MRS ( for ) i j i j 14

15 15 Interreting the Lagrangian Multilier λ is the marginal utility of an etra dollar of consumtion eenditure the marginal utility of income n n U U U λ /... / / n MU MU MU n λ

16 Interreting the Lagrangian Multilier At the margin, the rice of a good reresents the consumer s evaluation of the utility of the last unit consumed how much the consumer is willing to ay for the last unit i MU i λ 16

17 Corner Solutions When corner solutions are involved, the firstorder conditions must be modified: L/ i U/ i - λ i 0 (i 1,,n) If for some i, L/ i U/ i - λ i < 0, then i 0 This means that U / MU i i i > λ λ any good whose rice eceeds its marginal value to the consumer will not be urchased 17

18 Cobb-Douglas Demand Functions Cobb-Douglas utility function: U(,y) α y β Setting u the Lagrangian: L α y β + λ(i - - y y) First-order conditions: L/ α α-1 y β - λ 0 L/ y β α y β-1 - λ y 0 L/ λ I - - y y 0 18

19 Cobb-Douglas Demand Functions First-order conditions imly: Since α + β 1: αy/β / y y y (β/α) [(1- α)/α] Substituting into the budget constraint: I + [(1- α)/α] (1/α) 19

20 Cobb-Douglas Demand Functions Solving for yields * Solving for y yields y* αi The individual will allocate α ercent of his income to good and β ercent of his income to good y βi y 20

21 Cobb-Douglas Demand Functions The Cobb-Douglas utility function is limited in its ability to elain actual consumtion behavior the share of income devoted to articular goods often changes in resonse to changing economic conditions A more general functional form might be more useful in elaining consumtion decisions 21

22 CES Demand Assume that δ 0.5 U(,y) y 0.5 Setting u the Lagrangian: L y λ(i - - y y) First-order conditions: L/ λ 0 L/ y 0.5y λ y 0 L/ λ I - - y y 0 22

23 This means that CES Demand (y/) 0.5 / y Substituting into the budget constraint, we can solve for the demand functions * I [1+ y ] y* y I [1+ y ] 23

24 CES Demand In these demand functions, the share of income sent on either or y is not a constant deends on the ratio of the two rices The higher is the relative rice of (or y), the smaller will be the share of income sent on (or y) 24

25 25 CES Demand If δ -1, U(,y) y -1 First-order conditions imly that y/ ( / y ) 0.5 The demand functions are * y I * y y y I

26 If δ -, CES Demand U(,y) Min(,4y) The erson will choose only combinations for which 4y This means that I + y y + y (/4) I ( y ) 26

27 CES Demand Hence, the demand functions are * I y y* I 4 + y 27

28 Indirect Utility Function It is often ossible to maniulate firstorder conditions to solve for otimal values of 1, 2,, n These otimal values will deend on the rices of all goods and income * 1 1 ( 1, 2,, n,i) * 2 2 ( 1, 2,, n,i) * n n ( 1, 2,, n,i) 28

29 Indirect Utility Function We can use the otimal values of the s to find the indirect utility function maimum utility U(* 1,* 2,,* n ) Substituting for each * i, we get maimum utility V( 1, 2,, n,i) The otimal level of utility will deend indirectly on rices and income if either rices or income were to change, the maimum ossible utility will change 29

30 Lum Sum Princile(ski) Taes on an individual s general urchasing ower are suerior to taes on a secific good an income ta allows the individual to decide freely how to allocate remaining income a ta on a secific good will reduce an individual s urchasing ower and distort his choices 30

31 The Lum Sum Princile A ta on good would shift the utilitymaimizing choice from oint A to oint B Quantity of y B A U 1 U 2 Quantity of 31

32 An income ta that collected the same amount would shift the budget constraint to I Quantity of y The Lum Sum Princile I Utility is maimized now at oint C on U 3 B C A U 3 U 1 U 2 Quantity of 32

33 Indirect Utility and the Lum Sum Princile If the utility function is Cobb-Douglas with α β 0.5, we know that * I 2 y* I 2 y So the indirect utility function is V(, y, I ) (*) (y*) 2 I y 33

34 Indirect Utility and the Lum Sum Princile If a ta of $1 was imosed on good the individual will urchase *2 indirect utility will fall from 2 to 1.41 An equal-revenue ta will reduce income to $6 indirect utility will fall from 2 to

35 Indirect Utility and the Lum Sum Princile If the utility function is fied roortions with U Min(,4y), we know that * I y y* I 4 + y So the indirect utility function is V(, y, I) Min( *,4y *) 4y* y I * I y y 35

36 Indirect Utility and the Lum Sum Princile If a ta of $1 was imosed on good indirect utility will fall from 4 to 8/3 An equal-revenue ta will reduce income to $16/3 indirect utility will fall from 4 to 8/3 Since references are rigid, the ta on does not distort choices 36

37 Eenditure Minimization Dual minimization roblem for utility maimization allocating income in such a way as to achieve a given level of utility with the minimal eenditure this means that the goal and the constraint have been reversed 37

38 Eenditure Minimization Point A is the solution to the dual roblem Quantity of y Eenditure level E 2 rovides just enough to reach U 1 Eenditure level E 3 will allow the individual to reach U 1 but is not the minimal eenditure required to do so A Eenditure level E 1 is too small to achieve U 1 U 1 Quantity of 38

39 Eenditure Minimization The individual s roblem is to choose 1, 2,, n to minimize total eenditures E n n subject to the constraint utility U 1 U( 1, 2,, n ) The otimal amounts of 1, 2,, n will deend on the rices of the goods and the required utility level 39

40 Eenditure Function ( 支出函数 ) The eenditure function shows the minimal eenditures necessary to achieve a given utility level for a articular set of rices minimal eenditures E( 1, 2,, n,u) The eenditure function and the indirect utility function are inversely related both deend on market rices but involve different constraints 40

41 Two Eenditure Functions The indirect utility function in the two-good, Cobb-Douglas case is V(, y, I ) 2 I If we interchange the role of utility and income (eenditure), we will have the eenditure function y E(, y,u) y 0.5 U 41

42 Two Eenditure Functions For the fied-roortions case ( δ - ), the indirect utility function is V(, y, I) + I 0.25 y If we again switch the role of utility and eenditures, we will have the eenditure function E(, y,u) ( y )U 42

43 Proerties of Eenditure Functions Homogeneity a doubling of all rices will recisely double the value of required eenditures homogeneous of degree one Nondecreasing in rices E/ i 0 for every good, i Concave in rices 43

44 Concavity of Eenditure Function At * 1, the erson sends E(* 1, ) E( 1, ) E seudo E( 1, ) E(* 1, ) If he continues to buy the same set of goods as * 1 changes, his eenditure function would be E seudo Since his consumtion attern will likely change, actual eenditures will be less than E seudo such as E( 1, ) *

45 Imortant Points to Note: To reach a constrained maimum, an individual should: send all available income choose a commodity bundle such that the MRS between any two goods is equal to the ratio of the goods rices the individual will equate the ratios of the marginal utility to rice for every good that is actually consumed 45

46 Imortant Points to Note: Tangency conditions are only firstorder conditions the individual s indifference ma must ehibit diminishing MRS the utility function must be strictly quasiconcave 46

47 Imortant Points to Note: Tangency conditions must also be modified to allow for corner solutions the ratio of marginal utility to rice will be below the common marginal benefitmarginal cost ratio for goods actually bought 47

48 Imortant Points to Note: The individual s otimal choices imlicitly deend on the arameters of his budget constraint choices observed will be imlicit functions of rices and income utility will also be an indirect function of rices and income 48

49 Imortant Points to Note: The dual roblem to the constrained utility-maimization roblem is to minimize the eenditure required to reach a given utility target yields the same otimal solution as the rimary roblem leads to eenditure functions in which sending is a function of the utility target and rices 49