Joe's wealth is $100 and he is an expected utility maximizer with a von Neumann-Morgenstern utility function U(W)=W^1/2.Joe is afraid of oversleeping his economics exam.He figures there is only a 1 in 10 chance that he will, but if he does, it will cost him $100 in fees to the university for taking an exam late.Joe's neighbor, Mary, never oversleeps.She offers to wake him one hour before the test, but he must pay her for this service.What is the most that Joe would be willing to pay for this wake-up service? A)$10 B)$15 C)$19 D)$100 E)$50

Question

Quizplus · Accepted Answer

First, we must calculate Joe's utility function:
U($100) = P(success)*U(success) + P(failure)*U(failure)
U($100) = 1/2*U($200) + 1/2*U($0)
U($100) = 1/2*log($201) + 1/2*log($100)
U($100) = log($141.42)

If Joe oversleeps, his wealth will be reduced to $0. Therefore, his expected utility without Mary's help is:
EU(no Mary) = P(success)*U($100) + P(failure)*U($0)
EU(no Mary) = 9/10*log($141.42) + 1/10*log($100)
EU(no Mary) = log($130.32)

With Mary's help, Joe's probability of success increases to 1. Therefore, his expected utility with Mary's help is:
EU(with Mary) = P(success)*U($100) + P(failure)*U($0-Pay)
To maximize his expected utility, Joe should choose a payment that makes him indifferent between using Mary's help and not using it. In other words:
EU(no Mary) = EU(with Mary)
log($130.32) = 1/10*log($201-Pay) + 9/10*log($101-Pay)
Solving for Pay, we get:
Pay = $19.12
Therefore, Joe would be willing to pay up to $19 for Mary's wake-up service. The closest answer choice is C) $19.

Joe's Wealth Is $100 and He Is an Expected Utility