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Final Exam
Microeconomic Theory II
ECON 522
May 27, 2008
4:3 0 pm C 7:30 pm

Problem 1 (Total 15 points)
In an Edgeworth Box economy, the total amount of the two goods are X and Y. Individual 1 and 2s utility functions are xay1-a and xby1-b respectively for a, b(0, 1). If player 1 consumes x1 and y1 in some equilibrium, find out the relation between x1 and y1.

Problem 2 (17 points)
Consider the interaction between a seller and a buyer, where the quality of the product, q {0, 1}, and the price, p, can take one of only three values, p {.5, 1.5, 2.5}. The value of the product to the buyer is v(q), where v(0) = 1, and v(1) = 3. The cost of production to the seller is c(q) = q. If a trade occurs, the sellers payoff is p C c(q), and the buyers payoff is v(q) C p; each receives zero if no activity takes place.

Quality is privately chosen by the seller, and it is not observable at the time of trade. The buyer observes the quality after purchasing the good. Note that, even under the restricted set of prices, both qualities could be traded if quality were observable. We will consider the following trading rule: the seller chooses both q and p first; observing only p the buyer decides whether to buy.

a) State the (unique) subgame perfect Nash equilibrium (SPNE), the associated outcome, and the equilibrium payoffs for each player. (7 points)

b) Now suppose the interaction is repeated infinitely many times. Hence, the buyer knows the quality of period t C 1 at the beginning of period t2. The common discount factor is (0, 1). Restrict attention to SPNE based on trigger-type strategies only. Is there an SPNE where p = 2.5 and q = 1 can occur in each period? If yes, state this equilibrium showing it is indeed an equilibrium and present the required condition on ; if no, explain why not. (10 points)

Problem 3 (18 points)
For the Spence signaling model with two types of job applicants L and H, suppose the population share of the two types is 50-50. For some e* > 0, all the employers hold the following belief:
. If a job applicant has education e < e*, he is of type L for certain.

. If e [e*, 2e*), he is type L with probability 0.5.

. If e 2e*, he is of type H for certain.

Let productivities be L = 1 and H = 2 and cost functions be cL(e) = e, cH(e) = e/2.
a) Given the belief, find the wage contract in a competitive labor market. (8 points)

b) Find the necessary and sufficient conditions on e*, by which a separating equilibrium with eL = 0 and eH = 2e* exists. Is the equilibrium outcome consistent with the belief? (10 points)

Problem 4 (Total 20 points)
Consider the following principal-agent model. The agent can choose a level of effort e[0 ,1] . Profits are random and relate to e according to the function:
= e +
where is a random variable uniformly distributed on [0, 1]. The principal is risk neutral and cares only a