# Midterm exam of Microeconomics I

October 28, 1998, Universitat Pompeu Fabra.

1.Consider the model with two effort levels E={eH,eL}. There are two possible outcomes: xH and xL. The conditional probabilities (of the outcomes given the effort) are pH(eH)=P, pH(eL)=p. The utility function of the principal is B(x-w)=x-w, the agent's utility from the wage is u(w)=w1/2. The agent's cost of effort is v(eH)=1, v(eL)=0. The agent's reservation utility level is U. The effort is not observable by the principal

1. Is the principal risk averse, risk neutral or risk loving? and the agent?
2. Find the optimal contracts for inducing high and low efforts.
3. Write the equation that determines when the principal is indifferent between inducing the high and the low effort levels. (It may be a function of P, p, xH, xL and U).

2. Consider the problem of moral hazard, where both the principal and the agent have mean-variance preferences. The agent's preferences are: EUA=E(w)-½rAVar(w)-½e2 and his reservation utility level is U=0. The principal's preferences are: EUP=E(x-w)-½rPVar(x-w). The outcome is x=e+d has a mean of zero and variance s2.

1. Find the optimal effort level the will be chosen under symmetric information.
2. Assume that the contract has the form w(x)=A+Bx. Find the optimal effort of the agent e*. (Note that for any random variable x and non-random variables a,b, we have V(a+bx)=b2V(x)).
3. Find the optimal contract under asymmetric information.