Universitat Pompeu Fabra, December 18, 1998

**1.** Consider a model with two possible actions (effort levels),
E={e_{H},e_{L}}. There are two possible outcomes
x_{H}=4 and x_{L}=1. The conditional probabilities
of the outcomes on the efforts are
p_{H}(e_{H})=2/3, p_{H}(e_{L})=1/3.
The von-Neumann-Morgenstern utility function of the principal is
B(x-w)=(x-w)^{1/2}.
The von-Neumann-Morgenstern utility from salary of the agent is
u(w)=w^{1/2}. The agent's cost from effort is
v(e_{H})=1/10, v(e_{L})=0. The reservation utility of the agent
is __U__=1/10. The effort is **not verifiable**.

- Is the principal risk loving, risk averse or risk neutral?
- Find the optimal contract that implements the low effort level. (It would be simpler to assume that the incentive compatibility constraint is not binding, and after calculating the contract verifying it).
- Find the optimal contract that implements the high effort level
- What effort level the principal would prefer?

**2.** Assume that there are two types of consumers
A and B, for a firm's product. The utility of a consumer of
buying the quantity x and paying a total amount of T is:

U_{A}(x,T)=z_{A}[1-(1-x)^{2}]/2 - T;
U_{B}(x,T)=z_{B}[1-(1-x)^{2}]/2 - T;
where z_{B}<z_{A}.

The firm is a monopoly and the cost per unit is c>0.

Answer the questions below. For each question, you can use the results of the previous one, even if you didn't answer it.

- Write the maximization problem of the firm for finding the optimal contracts.
- Show that the participation constraint of type A is redundant
- Use the first order conditions to show that the incentive compatibility constraint of A, and the participation constraint of B are binding.
- Assume that the incentive compatibility constraint of B is not binding and find the optimal contracts given the other constraints and first order conditions. Then, solve the incentive compatibility constraint of of B ans show that it is not binding.

**3.** Consider the following signalling game.
Felix considers to sue Pílez for selling him
bad cat food. Felix knows that he will win the case if
he brings it to court. (This information is his type)
Pílez (the principal) knows that Felix knows whether
he will win or not, and his a priori probabilities are 1/3 that
Felix is the type that wins in court. If Felix wins his payoff is 3
and of Pílez is -4. If Felix looses, his payoff is
-1 and of Pílez is zero.

Felix has two possible actions (signals), to offer not to bring the case to court for a low compensation (receiving m=1 from Pílez), or for a high compensation (receiving m=2 from Pílez). Pílez has two possible responses to accept the deal or not. If he doesn't the case goes to court. If he does accept the deal his payoff is -m and Felix's is m.

- Does the game have separating equilibria? If so, describe some, if not show that it doesn't have any.
- Does the game have pooling equilibria? If so, describe some, if not show that it doesn't have any.

(It may be helpful to draw a decision or game tree like in the game beer or quiche).