Andy and Beth are neighbors in a small duplex. In the evenings after work, Andy enjoys practicing the tuba, while Beth likes to relax and read novels. Unfortunately, Andy is not very good at his instrument, and noise from his playing penetrates the walls and annoys Beth. The daily utility Andy derives from playing the tuba for m minutes and spending xA dollars on other consumption is given by

UA = xA + 32 log(m):

Andy would be happy to play his horn all day, except that he gets tired from blowing and he needs to drink Red Bull (which is costly) to keep up his energy. (For simplicity, assume Andy gets no direct utility benet from drinking Red Bull.) In fact, because there are diminishing returns to the eectiveness of energy drinks, Andy has to increase his rate of Red Bull consumption the longer he plays the tuba. Thus, Andy incurs c(m) dollars of Red Bull expense from playing the tuba m minutes in a day, where c(m) =m2/36. Beth's happiness in a day is simply a function of how many dollars xB she spends on consumption and how many minutes m of Andy's tuba playing she must endure. She becomes increasingly irritated by the tuba the longer the playing goes on. Her utility is given by

UB = xB -m2/12:

Assume that Beth and Andy have $150 of income to spend each day, and that they cannot save or borrow any extra (they either use it or lose it).

1. From the perspective of a social planner with a utilitarian social welfare function, what is the socially optimal amount of tuba playing each day?

2. Suppose there is no law stipulating whether Andy has a right to play his horn, or whether Beth has a right to peace and quiet (it is hard to measure noise levels and sources, and to give rights to this).

(a) Describe intuitively whether a market failure exists in this context.

(b) Calculate how many minutes m Andy chooses to play each day, and the resulting utilities of Andy and Beth.

(c) Is there any deadweight loss from Andy's choice (if so, calculate it)?

3. Beth complains to her Landlord about the tuba noise, and in response the Landlord installs noise meters that precisely record the level and source of noise in the apartments. The Landlord is considering a policy where residents would be charged a fee of per minute of noise above a certain threshold (the tuba would exceed this threshold). The Landlord wants to set to maximize total welfare, as in part 1.

(a) In one concise sentence, describe intuitively how the optimal should be set.

(b) Calculate the optimal .

(c) What is the most Beth would be willing to pay the Landlord to induce him to implement the policy in (b) (vs. the status quo described in part 2)?

(d) The Landlord does not want to make Andy upset. How much must the Landlord pay Andy before he would agree to the policy in (b)?

4. Suppose the Landlord considers two alternative policies of \noise rights:"

(a) The Landlord gives Beth the rights to peace and quiet.

(b) The Landlord gives Andy the right to make noise.

These rights would be written in the leases and would be enforceable because of the noise meters.

Assume that Andy and Beth can costlessly bargain. Separately for both policies (a) and (b), please describe (and quantify when appropriate)

(i) the amount of any side payments between Andy and Beth,

(ii) how much tuba Andy chooses to play (also compare to the optimal tax policy in 3(b)), and (iii) Andy and Beth's utilities.

Are there any dierences between these two \noise rights" policies (a) and (b)? Also compare and contrast these results with the optimal tax policy in 3(b).