Jake likes parties and he likes to drink beer. He knows that if he drinks too much beer he will not feel well the next day and won’t be able to get any work done. When Jake is at home, soberly thinking about the day-after effects of drinking, his preferences for drinking x glasses of beer at a party are represented by the utility function U0(x) = 10x − x2. Jake is invited to a party on Saturday night, and he knows that there will be free beer. His alternative is to spend a quiet evening with a tee-totaling friend. Spending the quiet evening with the friend would give him a utility of 20.
(a) If he goes to the party and drinks the amount of beer that maximizes U0(x), how many glasses will he drink? _____________ What will his utility be? _________ Is this higher or lower than the utility he would have by staying home? ___________
(b) Jake has noticed that beer has a strange effect on him. It changes his utility function. When he drinks more beer he seems to get thirstier and forget the morning-after costs. In fact, for any number of beers t, after he has drunk t beers, his utility function for drinking a total of x beers becomes Ut(x) = (10 + t)x − x2. For example, after he has drunk 5 beers, his utility for drinking a total of x beers will be 15x − x2, and his marginal utility for drinking more beer will be ____________ Since this marginal utility is positive when x = 5, he will choose to drink more than 5 beers. How many beers must he drink so that his marginal utility __________ of drinking more beer will be zero? __________ Suppose that before going to the party, Jake knows that the number of beers he would drink is not the number that his sober self tells him is optimal, but that he would drink until his beer-altered preferences tell him to quit. Using his sober preferences, what utility does he expect to get if he goes to the party? ___________ Would he be better off spending the quiet evening at home? ___________