# Exercise 7 of Microeconomics I

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: Ui(w,e)=w-kiv(e), where ki is a parameter that satisfies k1<k2, v(0)=0, v'(e)>0, v"(e)>0. The value odf the output is independent of the education level and is: x1, x2, for agents of types 1 and 2 respectively. We assume that x1>x2.

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).

1. 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.
2. 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.
1. Show that in any separating (perfect Bayesian) equilibrium we have w1=x1, w2=x2.
2. Show that in any separating (perfect Bayesian) equilibrium e2 = 0.
3. Describe some separating equilibria.
4. Describe some pooling equilibria.

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.