Two
neighbors, Alice and Bob, live in a building with poor sound insulation. Alice
likes to play music loud, and Bob likes silence. The building coop board will
enforce a curfew forbidding loud music playing after a certain time, x, measured as hours after 8 pm. Thus if x = 3 the curfew will be 11 pm. There is also the
possibility that Alice can pay Bob a certain amount per week, y, to extend the curfew hours. If y < 0 the payment would go from Bob to Alice. Their
utility functions are:

Alice: ;

Bob:

A. [10points] What are Alice and Bob's own preferred curfew times,
ignoring the presence of the other? If the coop board sets x = 3, and there is no payment
between them, what would their utilities be? Is setting x =3 lead to a Pareto optimal
outcome?

B. [6 points] What curfew
time, , maximizes the sum of their
utilities, ? What is the sum of their utilities at ? Does this maximization determine the payment
y? Explain.

C. [8 points] What is the minimum payment Alice could offer Bob on
a take-it-or-leave-it basis to shift the curfew to 1 am (x = 5)? What would their
payoffs be if they made this deal? Is this a Pareto-improvement over the board
curfew?

D. [6 points] What is the payment that would equalize their
utilities, ? What would their payoffs be with this arrangement?

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