Universitat Pompeu Fabra, 1998/1999

**Due November 25, 1998**

**1.** Consider a market with two types of agents
whose utility functions depend on their salary and their education
level, in the following way:
*U _{i}(w,e)=w-k_{i}v(e)*,
where

The firms (principals) are risk neutral. The worker choose their education level and the firms offer them contracts after the education decisions has been made and observed by the firms. The wages may be conditional on the education levels but not on the output (which cannot be verified in a court of law).

- Assume that the worker's type is known both to the worker and to the firms. (And everybody knows that before education decisions were made). What would be the equilibrium in this situation.
- Assume now that the firms do not know the type of the worker, but the worker does.
The firms know that with probability
*q*the worker is of type 1.- Show that in any separating (perfect Bayesian) equilibrium
we have
*w*._{1}=x_{1}, w_{2}=x_{2} - Show that in any separating (perfect Bayesian) equilibrium
*e*._{2}= 0 - Describe some separating equilibria.
- Describe some pooling equilibria.

- Show that in any separating (perfect Bayesian) equilibrium
we have

**2.** A researcher in the Pompeu
Fabra has shown that the grades of students in the university are
highly correlated with grades of their earlier studies (high school
and the *selectividad*). How this fact may be used to
falsify the theory that university studies only constitute a signaling
mechanism and have no effect on the productivity of the students.