# UCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory

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1 UCLA Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally. Answering fewer than 5 questions is not advisable, so do not spend too much time on any question. Do NOT answer all questions. Use a SEPARATE bluebook to answer each question.

2 1. Consumer Problem with Quota Consider a convenience store that sells two kind of (divisible) goods x 1 and x 2 : x 1 and x 2 are on sale, but there is a limit on the amount each customer can purchase; the total amount of x 1 and x 2 cannot exceed some quota Q > 0: Let p = (p 1 ; p 2 ) 0 be the price of x 1 and x 2 respectively. Alice is planning to spend W > 0 at this convenience store to purchase these goods. Let u : R 2 +! R be Alice s utility function, which is di erentiable. Answer the following questions. (a) Write down Alice s utility maximization problem and, assuming interior solutions, write down the Kuhn-Tucker conditions for the problem. Explain why the Kuhn-Tucker conditions are necessary for any optimal solution. (b) Suppose that u (x 1 ; x 2 ) = x 0:5 1 x0:5 2 ; W = 80; Q = 40; (p 1; p 2 ) = (1; 5) : Find Alice s optimal consumption (x 1 ; x 2 ) : (c) Let x (p; W; Q) be Alice s demand correspondence for this problem. De ne Alice s indirect utility function by v (p; W; Q) := u(x (p; W; Q)): Show that v is quasi-convex (i.e. the lower contour set is convex) in (p; W ) given Q and quasi-convex in Q given (p; W ): Is v (p; W; Q) also quasi-convex in (p; W; Q)? Answer yes or no and explain why. (d) Suppose that u is strictly quasi-concave. Let x 2 R 2 + be an optimal consumption vector for Alice given some (p 0 ; W 0 ; Q 0 ) 0: Given x ; consider the following cost minimization problem. min p 0 x s.t. u (x) u (x ) and x 1 + x 2 Q 0 : x2r 2 + Show that x is the unique solution for this problem. 2

3 2. Excess Demand Function and Existence of Competitive Equilibrium. Consider a standard pure exchange economy E pure = (fx i ; i ; e i g n i=1 ) where X i = R L +; i is consumer i s (rational and continuous) preference, and e i 2 R L + is consumer i s initial endowment. Suppose that i is locally nonsatiated, strictly convex and monotone for every nx i and r = e i 0: Let z : R L ++! R L be the excess demand function de ned by i=1 z (p) := nx [x i (p; p e i ) e i ] i=1 where x i (p; p e i ) is consumer i s (Walrasian) demand function. We know that there exists a competitive equilibrium when z (p) satis es the following ve properties in R L ++. (I) z is continuous: (II) z is homogeneous of degree 0: (III) p z(p) = 0 (Walras Law) (IV) z` is bounded below for every `: (V) kz(p n )k! 1 for any sequence of strictly positive prices in the price simplex fp n g 1 n=1 int4 such that p n converges to a boundary point of 4 (kk is Euclidean norm). Show that these ve properties are indeed satis ed by z(p) in this economy. explicitly which assumptions are used in each step of your proof. State 3

4 3. Equilibrium For the game shown below, nd: 1. all the sequential equilibria (actions and beliefs) in pure strategies 2. all the subgame perfect equilibria in pure strategies that are not sequential equilibria (if any) 3. all the Bayesian Nash equilibria in pure strategies that are not subgame perfect equilibria (if any) In each case, explain thoroughly. (The given labeling of nodes and information sets may be useful.) In particular, if you nd a subgame perfect equilibrium that is not a sequential equilibrium explain why not; if you nd a Bayesian Nash equilibrium that is not a subgame perfect equilibrium, explain why not. 4

5 4. Repeated Games Consider the stage game G below. C D C 12; 12 4; 8 D 0; 0 6; 6 1. Find the Nash equilibria in pure strategies and the minmax payo s in pure strategies for G. 2. Make a careful sketch of the set of vectors (x; y) 2 R 2 for which there exists some discount factor < 1 such that (x; y) can be achieved as time-average payo s of some not necessarily equilibrium play of the in nitely repeated game G 1 (). 3. Let W be the union of the point (6; 6) with the line segment from (8; 10) to (12; 12). For what discount factors < 1 is W self-generating (in the time-average sense)? 5

6 5. Auctions Buyers are risk neutral and values are independently and identically distributed with support [0; 1], c.d.f.f () and p.d.f. p (). A single item is for sale using an auction in which the buyer with the high value is the winner. In the case of a tie the winner is selected randomly from the tying high bidders. (a) Prove that the equilibrium expected payo depends only on the c.d.f. and is independent of the payment rules. (b) Hence prove revenue equivalence. Two sisters have decided not to divide up their parents estate. Instead they will allocate it using a sealed rst price auction. The sister who makes the high bid and receives all the items in the estate pays her bid to the sister who makes the losing bid. (There is no third party collecting revenue.) The c.d.f. of the estate s value is uniform on [0; 1]. (c) Write down an expression for the equilibrium expected payo. Di erentiate this expression and hence show that the equilibrium bid function satis es the following di erential equation. B 0 () + 2B () = 2 B 0 () + 2B () 2 = 0 (d) Solve for the equilibrium bid function. 6

7 6. A Chicken-Egg Problem Each individual has a chicken with a probability of laying either 0 or 2 eggs. (Probabilities are independent and eggs, once laid, are divisible.) Individual i s quasilinear preferences are v i (z i ) + m i = az i (b i =2)z 2 i + m i, and z i 0. There are two sources of private information: knowledge of one s own tastes, here b i, and knowledge of one s endowment whether or not one s chicken has laid eggs. For (a) and (b), below, there is private information with respect to tastes but endowments are public information once they are realized. An insurance market (mechanism) maximizes the sum of individuals expected utilities based on their reports. (a) There are two individuals. Assuming a = 2, b 1 = b 2 = 1, nd the e cient allocations of (z 1 ; z 2 ) for the 4 equally likely expost outcomes, (2; 2); (0; 0); (2; 0); (0; 2), (1 s eggs; 2 s eggs). Suppose the mechanism calculates a money payment for the individual from whom it takes eggs equal to what that individual would receive in the expost price-taking equilibrium, with the opposite payment for the buyer. Each individual s net money payment is the probability weighted average of these expost transfers. (b) If individual 1 were to report b 1 = 1=2 (when b 1 = 1) before endowments are realized, that would change the allocation and the net money transfers. Show that 1 s expected utility including the transfers would not increase. For the following, information about tastes is common knowledge: a = 2 and b i = 1. There is, however, private information about individual endowments (! 1 ;! 2 ), where! i 2 [0; 2]. If an individual s endowment is! i he can report 0 r i! i and keep (! i r i ) to add to his nal consumption. (c) When there are two individuals and! 1 >! 2, 1 will be the seller. Show that the scheme in (b) is not consistent with truthful reporting. Focus on individual 1 having! 1 = 2 when! 2 = 0 and individual 1 can report r 1 = 1 (after! 2 is realized). (d) Is there a scheme that is e cient in the sense of maximizing the sum of utilities with respect to the z-commodity that would encourage truthful reporting no matter what the reported endowment of the other individual? 7

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