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 *p _{x}(e1)=2/3, p_{x}(e2)=1/2, p_{x}(e3)=1/3*.
The cost function of the effort is:

- 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(U _{A})=E(w)-(1/2)Var(w)-(1/2)e^{2}* and has
a reservation utility level

- What would the contract be under symmetric information?
- 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?
- 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.

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