Remodeling your home (30 points): You want to hire a contractor to remodel your house. The value of the renovation to you depends on the contractorís e§ort e and is equal to = 10 + 4(e + X) where the random variable X has mean zero and variance equal to 2 = 3. You are risk neutral and care only about the Önal value of the house minus payment to the contractor w. That is your expected utility is uY = E[ w]. The contractor has the coe¢ cient of risk aversion r = 1 and his cost of e§ort is c(e) = e2=2. So his expected utility from payment w and e§ort e is uC = E[w] 1 rV ar(w) c(e). The contractor has an outside option of working as 2 a handyman for someone else instead of managing your renovation. The work as a handyman pays a Öxed salary of U = 10 and it requires zero e§ort. (a) First assume that you can observe e§ort e. Solve for the optimal linear contract w = + e by Örst formulating and solving the contractorís problem of choosing e§ort e to maximize uC and then formulating and solving your problem of choos- ing 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. Requirements: Show work (b) Solve for the optimal linear contract w = + (e + X) by Örst formulating and solving the agentís problem of choosing e§ort 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 V ar(Y ) = 2 and V ar(Z) = 1 and they are independent. Suppose Y is observable while Z is not observable. What is the optimal linear contract w = + (e+X Y) (i.e. what values of and maximize your expected utility)? [Hint: Start with . What maximizes the total value of the contract?] (b) Solve for the optimal linear contract w = + (e + X) by Örst formulating and solving the agentís problem of choosing e§ort 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 V ar(Y ) = 2 and V ar(Z) = 1 and they are independent. Suppose Y is observable while Z is not observable. What is the optimal linear contract w = + (e+X Y) (i.e. what values of and maximize your expected utility)? [Hint: Start with . What maximizes the total value of the contract?] ‘

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