Two roommates, 1 and 2, have preferences over leisure (x) and the cleanliness of their room, Suppose c = y1 + y2, where yi is the amount of time spent by roommate i in cleanup activities. Each roommate has a total of 1 unit of time available (per week): thus x1 + y1 =1 and similarly for roommate 2. Let 1's preferences be given by

U(x1 , c) = x1 + 1/2 log(c)
and let 2's preferences be given by
V(x2 , c) = x2 + 1/2 log(c)

a. Suppose 1 and 2 take each other's clean up effort as given. Derive the optimal effort of 1 as a function of 2's effort. Find the symmetric Cournot equilibrium. Are there other Cournot equilibria?

b. Suppose 1 and 2 agree on a schedule of clean up that maximizes the sum of their utilities. What is the optimal level of cleanliness, c*?

c. Suppose that 1 thinks 2 is following the jointly optimal policy agreed in part b. What is 1's optimal choice?

d. Suppose that for the entire year the utility a roommate gets is just the undiscounted sum of the weekly utilities. The roommates agree to hire a friend to inspect the room each Sunday. If the room is dirtier than c* the roomates will "do pennance" by volunteering some fraction of time (possibly over more than 1 week) to a local charity. Can you come up with a penalty that will effectively enforce the jointly optimal cleanliness level?