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 preferences defined on the mean and the variance of his income w and on his effort level e as follows: Expected utility = E(w) r /2 Var(w) (a^2 /2), where r > 0 and effort can be any nonnegative number. The output y is given by y = a + , 1 where is a random variable with mean zero and variance ^2 . 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. Proceed similarly to how we did in class to 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 compensation scheme, and plug this value into the firm's problem, which now entails choosing only s and b.) (c) Given some intuition for how the bonus rate derived in part (b) depends on r and 2 ? What about the effort level a? (d) Compute the manager's expected compensation Es + by using the expressions in part (b). How does it depend on the riskiness of the job? How does this square with the idea of "compensating differentials," wherein that riskier jobs ought to be compensated with higher wages? Explain.