# Exercise 5-6 of Microeconomics I

Universitat Pompeu Fabra, 1998/1999

Due November 11, 1998

1. Consider a model with two types of consumers (agent), each of which would like to buy a piano. Each consumer has a utility function Ui=kiqi-ti where qi is the piano's quality and ti is the price payed. And ki is a parameter that satisfies k1<k2. If a consumer does not buy a piano his utility is zero.

The seller (principal) is a monopoly in the local piano market. He can choose any non-negative piano's quality. The production cost of a piano of quality q is C(q), the marginal cost at zero is zero, and infinity at infinity. The seller's utility is UP=t-C(q).

1. Under symmetric information, what will be the contracts offered to the two different types. In addition, show your answer graphically.
2. Assume now that the consumer knows his type but the principal does not. The probability that the agent is of type 1 is p.
1. What is the maximization problem of the principal?
2. Show that the participation constraint of type 2 is redundant.
3. Show that q2 is larger or equal q1.
4. Show that the participation constraint of type 1 is binding.
5. Show that the incentives compatibility constraint of type 2 is binding.
6. Show that q2>q1.
7. Show that the incentive compatibility of type 1 is not binding.
8. Give a four equation system with four variables whose solution is the optimal contracts.
9. Solve the above system for C(q)=½(q2).

2. The investment company IF (Investing Fantasy) is considering to buy the petrol company PIG (Promising Industrial Gasoline), whose present value is V, if it stays with its current management. The managers of PIG know the exact value of V, but the managers of IF only know that it is distributed uniformly between 0 and 100. If IF buys PIGS its value would be V+40 due to better management.

1. What is the maximum amount IF would pay for PIG in order to have a non-negative expected profit?
2. Assume that IF is the only company competing for buying PIG, what offer would maximize its expected profit?

3. Consider a market with two types of workers with utility functions of wages and effort: Ui(w,e)=w-kiv(e), where e is the effort of type i, w his salary and ki is a parameter with k1<k2, v(0)=0, v'(e)>0, v"(e)>0. The value of the output of the worker is independent of his effort and equal to: xi, i=1,2. We assume that x1>x2. There are two firms (principals) who are risk neutral.

1. Assume that the worker's type in known to the firms, what would be the equilibrium contract?
2. Assume now that the firms know that with probability q the worker is of type 1. The worker knows his type. Prove the following claims. You may use diagrams.
1. The expected profits of the firms is zero.
2. There are no pooling equilibria.
3. In all separating equilibria the profit of each contract is zero, that is w1=x1 and w2 = x2.
4. In all separating equilibria the contract of type 2 is the same as under symmetric information.
5. In all separating equilibria the contract of type 1 satisfies U2(w1,e1)=U2(w2,e2) that is, type 2 is indifferent between his contract and the one offered to to type 1.
6. There are cases where an equilibrium exists, and there are where it does not.