Consider a principal-agent model in which the principal is the risk- neutral owner of a firm, while the agent is the firm's manager who has pref- erences defined on the mean and the variance of his income w and on his effort level e as follows:
Expected utility = E(w) - r V ar(w) - a2/2, 2
where r > 0 and effort can be any nonnegative number. The output y is given by
y = a + ε,
where ε is a random variable with mean zero and variance σ2. (Thus, if the manager chooses effort level a, output will have mean a and variance σ2.) Assume that the utility value of the manager's outside option is 0.
(a) Suppose that a were observable so that a contract in which payments to the manager are made if and only if he chooses a contractually specified effort level a∗. What level of a∗ is optimal for the owner to specify in the contract? How should he structure compensation? (i.e. should it depend on y? how should it depend on a?).
(b) Suppose that effort is not observable and that the compensation scheme must assume the following linear form: w = s + by (thus it con- sists of a fixed salary s plus a share of output as a bonus). Show that under this compensation scheme the manager's expected utility as a function of a will be equal to s+ba- rb2σ2 -a2/2.
(c) Derive the optimal linear compensation scheme (i.e., the values of s, b and a - from the owner's point of view) when effort is not observable. (Hint: first derive the value of a the manager will choose, given a compensa- tion scheme, and plug this value into the firm's problem, which now entails choosing only s and b.)
(d) Compare the expected output in part (a) and part (c). What are the effects on b and a of changes in r and σ2? Give some intuition for your answer. Would an owner prefer a more or less risk averse manager (i.e. higher or lower value of r)?