Question

Microeconomics:

1. There are two kinds of goods, food F and shelter S, which are only available in non-negative quantities. Ann's utility from consuming bundle (F, S) is given by the function U(F, S) = F0.690.2. Bob's utility from consuming bundle (F, S) is given by the function V(F, S) = F0.75 0.25 PF (a) (Time: 1 minute) What is Ann's utility level from bundle (F1, Si) = (1.1)? What about Bob's utility level? What is Ann's utility level from bundle (F2, S2) = (2, 2)? What about Bob's utility level? Based on your answers, would you agree that Ann's and Bob's utility functions are different? The consumer choice model focuses on analyzing the bundles chosen by a consumer from various budget sets. In the case of Ann and Bob, this would be equivalent to studying the demand of each of them from budget sets B defined by prices and income, where it is assumed they maximize utility subject to their budget constraint. Denote prices by PF and ps, and income by M. Then the budget constraint is pe F + psS = M. Ann's demand functions of F and S, obtained from maximizing her utility function subject to the budget constraint, are F Ann (PF, Ps, M) = 0,75M and SAnn(pf,ps, M) = 0.25M By solving Bob's utility maximization problem, one could show that his demand functions for F and S are F Bob(pF, PS, M) = 0.75M and SBob (PF, PS, M) = 0.25M . But these are the same! PS (b) (Time: 4 minutes) To understand what's going on, find the Ann's marginal utility for food and for shelter (as functions of F and S). So what is Ann's marginal rate of substitution (as a function of F and S)? (c) (Time: 4 minutes) Now, do the same for Bob. Find his marginal utility for food and for shelter, and then use this to find his marginal rate of substitution. (d) (Time: 1 minute) Are Ann's and Bob's marginal utilities given by the same function? (e) (Time: 1 minute) But what do you notice about the marginal rates of substitution of Ann and of Bob? So, even though Ann and Bob have different utility functions, they have the same demand functions. Notice that Bob's utility function is just Ann's utility function raised to a power, i.e. V(F, S) = (U(F, S))1.25. (f) (Time: 3 minutes) Why is the consumer choice model usually based on a theory of preference, rather than a theory of utility? Relate your answer to the observation on marginal rates of substitution of Ann and Bob above.

Answer 1

question (1). Annis utility function U(F, S) = 0.6 2002 Bobls utility Function VCFis) 0.25 70-75 S (a) Ann's Otility from C FI, S) = (1,1) UCF, SD = Foogoo 0.6 0.2 Bobls utility from (F1, S1) = (1,1) V(FIS) = 78.75,0.25 - (19.7619.25 = 1 Now from Bundle (F2, Sz) = (2,2) Ann's UCFS) = FOo6 Oo 2 0.6 0.2 S. O. 8 (2) Bob's VCFS) F 0.75 0.25 S 0075 0.25 = (2) (2) 0.75 to 25 (2) For Bundle (F2, S2 ) = (212) VLES) = 2 > UCF2, 92.)-(2) *** For Same Bundle Bob and Ann's has different Utility devels which implius they have dissesent utility Functions.

(b) MRTS = margina) rate of substitution = MOF = dolor Jults Mus U (FIS) = 0.6 Foo2 -0.4 JU (FIS) 0.67 soon (marginal utility of food) IF Jo (ES) 006 -0.8 002 F S (marginal utility of Shelter) ds - Ooh MRS = JUHF 0.6 002 S 0.2 0.8 65. S dults 0.2 F 006 -0.8 S 2 Foopool MRS 6 S Ann's 11 35 2 F mRs of Ann's = 35 F ) 0.75 0.25 V(FIS)= F 30:25 -0.25 0.25 marginal utility for food = do 6. 75 F S OF 0.75 marginal utility for Shelter = du - 0.75 0.25 F S ds -0.95 0.25 0.75 F Julde Jults MRS S 35 F 0.25 0.75 0.75

MRs of Bobo 35 F (d) marginal Utility for marginal utility for Food Sºo2 Shelter Ann's -0.4 0.6 F 0.6 -0.8 002 F S Bobls -0.25 0.25 0.75 F S 0075 -0.75 0.25 l S So it is clear that Ann's and bobls marginal utilities are no The same functions (e) Mrs of Bob Tolo mRs of Ann 35 F The Their mRs are same that is The rate at which they are giving up the shelter for Food are The same.while maintainging The Same devel of utility. 1025 Since VFS) = (UCFS)) That impies brining mono tonic Transformation of Ann's utility Function is equals To The Bob's

are The Utility function. Preyer cence cence are ordening That a person gives To different Available alternatives, and some Times it is listed use for To gives some numbers To different aiteratives (or goods J. These humbers are called utilities ana utility function tendue The utility assosiated with each good or alternatives For Ann's and bobls they are giving different numbers Food and Shelters Pellent (there by different utility Functions) But The rate they want to Change good for another good is a same which implice They have the preference odening for food and Shelters (from BRS That is why Consumer Choice model usually based a Theory of Preference, rather than a Theory of utility To One MRS Bob's ). Ann's
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