Reference no: EM132461423

Remodeling your home:

You want to hire a contractor to remodel your house. The value of the renovation to you depends on the contractor's effort e, and is equal to P = 15 + 6(e + X), where the random variable X has mean zero and variance equal to V = 2.

You are risk-neutral and care only about the expectation final value of the house minus payment to the contractor, w. That is, your expected utility is uY = E[p - w].

The contractor has the coefficient of risk aversion r = 1 and his cost of effort is c(e) = e²/2. So, his expected utility from payment w and effort e is uC = E[w] - 1/2 * r * Var(w) - c(e).

The contractor has an outside option of working as a handyman for someone else instead of managing your renovation. The work as a handyman pays a fixed salary of U = 20 and it requires zero effort.

Questions

(a) First, assume that you can observe effort e. Solve for the optimal linear contract w = + e by first formulating and solving the contractor's problem of choosing effort e to maximize uC, and then formulating and solving your problem of choosing and to maximize uY. What is your expected utility from this optimal contract?

For the rest of the question, assume that e is not observable to you, but you can observe (and contract upon) e + X.

(b) Solve for the optimal linear contract w = α + β (e + X) by first formulating and solving the agent's problem of choosing effort e to maximize uC, and then formulating and solving your problem of choosing and to maximize uY. What is your expected utility from this optimal contract?

(c) Now, assume that the noise term equals X = Y + Z, where Y is the availability of parts and Z is the residual noise. Y and Z are random variables with mean zero and variance equal to Var(Y ) = 1 and Var(Z) = 1, and they are independent. Suppose Y is observable while Z is not observable. What is the optimal linear contract w = α + β(e + X - γY ) (i.e., what values of α,β,γ and maximize your expected utility)? [Hint: Start with γ. What γ maximizes the total value of the contract?]