1. Consider a simple adverse selection model where a firm wants to       delegate to an agent the production of s units of a good, whose       value to the firm is s(s) and s\ > 0, s\\ < 0. The agent can be of       two possible types, θ with probability u and θ¯ with probability 1 _ u .       The agent’s cost function is C(s, θ) = θs, where θ e {θ , θ¯}. The       firm offers the agent a menu of contracts, which are tuples (s, u),       where s is the desired level of production and u is the transfer to the       agent. Draw in a graph the indifference curves of the firm and each       agent type. Using a graph, describe the complete information, first       best contracts for both types. State the incentive compatibility and       participation constraints in an environment where the firm does not       learn the agent’s type.  In a different graph, show the second best       contracts.          [25]

2. Suppose a firm is negotiating over a wage-employment package with a union representing L members. The production function is f (g), where g is the number of workers. Assume that f is strictly concave, f (0) = 0 and f (g) > gw0 for some g, where w0  is the wage a worker can get outside of the firm. Normalise the price of the output to be one. Agreements are restricted to pairs (w, g) such that profits are positive and the wage is at least w0 . Hence define d = (0, Lw0 ).

(b) Draw in a graph the bargaining problem.                                  [10]

(c) Applying the Kalai-Smorodinsky solution, find the optimal wage w* .                         [10]

3. In the axiomatic bargaining problem, describe a solution which sat-       isfies Invariance, Independence of Irrelevant Alternatives, Symmetry       but not Pareto. Prove your claims for each of these four Axioms.     [25]

4. Consider the dictatorial solution of the bargaining problem (s, d) , where agent 1 is the dictator, s is the set of all allocations and d is the disagreement point.  In particular, for every problem (s, d) , agent 2 gets d2 , whereas agent 1 gets the remaining:

f (s, d) = {(t1(*), d2 ) e s : (t1(*), d2 ) > (t1 , d2 ) for all (t1 , d2 ) e s}．

Show which axioms this solution satisfies.                                      [25]

5. Consider a standard moral hazard problem. An agent exerts costly effort e e {0, 1}, with cost ①(e), where ①(0) = 0 and ①(1) = ① . He receives a transfer u from the principal, who cannot observe his effort.   The agent’s utility is U = o(u) _ ①(e), with o(.) being strictly increasing and strictly concave.  There are two production levels, s¯ > s . If e = 1, then the probability of high production, s¯, is π 1 , whereas if e = 0, the probability of high production is π0 , where π 1  > π0 .  Let  be the principal’s profit from high production and

s be his profit from low production.  State the principal’s problem       in the case where the agent is risk neutral.  What is the expected       payment of the principal? What is his expected gain from inducing       positive effort? Is delegation costly for the principal?                       [25]