# Exercise 3 of Microeconomics I

Universitat Pompeu Fabra, 1998/1999

Due October 21, 1998

1. Consider a model with three possible effort levels E={e1,e2,e3}. There are two possible outcomes: x=10 and y=0. The conditional probabilities of the outcomes given the effort are px(e1)=2/3, px(e2)=1/2, px(e3)=1/3. The cost function of the effort is: v(e1)=5/3, v(e2)=8/5, v(e3)=4/3. And u(w)=w1/2 and B(x-w)=x-w.

1. What is the optimal contract when the effort is not observable by the principal?

2. Consider the problem of moral hazard when the agent is risk averse and has mean-variance preferences, that is, E(UA)=E(w)-(1/2)Var(w)-(1/2)e2 and has a reservation utility level U=0. The principal is risk neutral. The firm's sales are proportional to the agent's effort: x=e×H; where the mean of the random variable H is E(H)=m>0 and the variance V(H)=s2. The principal offers a wage contract of the form w(x)=A+Bx. Recall that for any random variable z and non- random variables C,D, E(Cz+D)=CE(z)+D, V(Cz+D)=C2V(z)=C2E((z-E(z))2).

1. What would the contract be under symmetric information?
2. What is the effort level as a function of the parameters of the contract when the information is not symmetric? Can you apply the approach of first order conditions?
3. What is the contract under asymmetric information? (It is sufficient to indicate the system of equations that has to be solved).

3. In some universities the grades of the students are determined by by relative system. For example, the top 10% receive the highest grade, the next 30% receive the second highest grade, and so on.

1. Could you explain the possible effects of this system from the point of view of the theory of incentives and allocation of risk?
2. Could you give some reasons why Universitat Pompeu Fabra does not implement this grading system?