The owner of a firm hires a manager. There are 2 possible effort levels that the manager can

exert -high effortehand low effortel, whereeh> el. The manager is risk averse and his

utility is given byu=phsh+plslce, whereceis his cost for exerting efforte=h,l.hhlle e

High effort increases the probability of success, but is more costly to the manager:ch= 4 andcl= 0. Letshbe the wage in the high output state andslis the wage in the low output state.u = 10 is the manager's outside utility from some other job. There are 2 possible output levels:yh= 2000,yl= 1000.

The probabilities of high output state and low output state are given as follows.

	y=yh	y=yl
e=eh	0.8	0.2
e=el	0.6	0.4

(a) Write down the maximization problems for the Principal when effort is observable under high effort and low effort. Find the wage ratessh,sl.

(b) Write down the maximization problem for the Principal when effort is unobservable. Find the optimal wage ratessh,slfor implementing high effort, as well as the Lagrange multipliers. What is the interpretation of Lagrange multipliers in this context?

(c) Will the Principal ask for a high effort from the worker when the effort level is unobserv- able? Why or why not? Show your work.

(d) What is the answer to the previous questions if, instead, the high output level is$2500? Does the manager have a preference between the case of observability and the case when effort is not observable?

(e) Suppose now that, instead of increasing the probability of a high output state, effort has no effect on it (i.e., it remains at 0.6 even if the manager exerts high effort). However, with low effort, the output in the low state is 700, while with high effort, it remains at 1000. How does this change the optimal contract under unobservability?