##### Value of Remodeling a Home Question

### Question Description

I’m working on a management multi-part question and need a sample draft to help me understand better.

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.

- (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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