Thelma and Louise have a big exam tomorrow. Each is trying to decide how much time to spend socializing, and how much to spend studying. Let's use T to stand for the amount of time Thelma spends socializing, and L to stand for the amount of time Louise spends socializing. Thelma's utility benefit from socializing is
12 T + T ∙ L
And Louise's utility benefit is
12 L + L ∙ T
Notice that Thelma enjoys socializing more when she does it with Louise, and Louise feels likewise. Unfortunately, socializing will adversely affect their exam scores. Thelma's utility cost of socializing is 2T2, and Louise's is 2L2. Suppose they choose T and L simultaneously. Find the Nash equilibrium. How many hours do they spend socializing? Can they do better? Would they be happier if they both studied more than in the Nash equilibrium or less?
a. At a Nash equilibrium, how many hours will they spend socializing?
b. What are Thelma and Louise's utility levels at the Nash equilibrium?
c. What are Thelma and Louise's utility levels if they socialize four hours each?
d. True or false? Both Thelma and Louise would be better off if they socialize slightly more than their Nash equilibrium levels of socialization.