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
*U _{i}=k_{i}q_{i}-t_{i}*
where

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
*U _{P}=t-C(q)*.

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

**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.

- What is the maximum amount IF would pay for PIG in order to have a non-negative expected profit?
- 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:
*U _{i}(w,e)=w-k_{i}v(e)*,
where

- Assume that the worker's type in known to the firms, what would be the equilibrium contract?
- 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.- The expected profits of the firms is zero.
- There are no pooling equilibria.
- In all separating equilibria the profit of each contract is zero,
that is
*w*._{1}=x_{1}and w_{2}= x_{2} - In all separating equilibria the contract of type 2 is the same as under symmetric information.
- In all separating equilibria the contract of type 1
satisfies
*U*that is, type 2 is indifferent between his contract and the one offered to to type 1._{2}(w_{1},e_{1})=U_{2}(w_{2},e_{2}) - There are cases where an equilibrium exists, and there are where it does not.