Assume that 2 roommates share an apartment. Each roommate can spend his/her time by either studying or cleaning the apartment. Both roommates value higher grades and a cleaner apartment. A roommate's grade is increasing in how much he/she spends time on studying. The cleanliness of the apartment depends on the total time that the two roommates spend on studying. Hence the preferences of roommate 1 can be depicted by a utility function U(G1 + G2, X1) where G1 is the time that roommate 1 spends on cleaning the apartment, G2 is the time roommate 2 spends on cleaning the apartment, and X1 denotes the time that roommate 1 spends on studying. Similarly, the preferences of roommate 2 can be depicted by a utility function U(G1 + G2, X2); where G1 + G2 again denotes the total time that the two roommates spend on cleaning the apartment and X2 denotes the time that roommate 2 spends on studying.
Assume now that the utility function of roommate 1 is given by U (G1 + G2, X1) = (G1 + G2) X1 + ½ x X1,
and that the utility function of roommate 2 is given by
U (G1 + G2, X2) = (G1 + G2) X2 + 0 x X2,
Furthermore, assume that the total time that each roommate has at his/her disposal is 300 hours:
Solve for the Nash Equilibrium.