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: =$10003(6)2; Bob: =$1500+(2)2 a) 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 this Pareto optimal? b) What curfew time, , maximizes the sum of their utilities, + ? What is the sum of their utilities at the optimal curfew? Does this maximization determine the payment y? Explain. c) 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) What is the payment that would equalize their utilities, = ? What would their payoffs be with this arrangement? e) What payment would equalize their gains from the deal, 1000 = 1500? What would their utilities be with this arrangement? f) What are the ethical arguments for each of the three payment deals described in parts c), d), and e)?