Buck Columbus is thinking of starting a pinball palace near a large Midwestern university.Buck is an expected utility maximizer with a von Neuman-Morgenstern utility function, U(W)  1  (6,000/W) , where W is his wealth.Buck's total wealth is $24,000.With probability .2 the palace will be a failure and he'll lose $18,000, so that his wealth will be just $6,000.With probability .8 it will succeed and his wealth will grow to $x.What is the smallest value of x that would be sufficient to make Buck want to invest in the pinball palace rather than have a wealth of $24,000 with certainty?

A) $28,500
B) $150,000
C) $96,000
D) $72,000
E) $30,000

Question

Traci Kozakiewicz · Accepted Answer

Final Answer : C, Explanation :  To solve the problem, we need to find the expected utility of both options and compare them. Firstly, if Buck doesn't invest in the pinball palace, he will have a certain wealth of $24,000 with a utility of:U(24,000) = 6,000/24,000 = 0.25If he invests in the pinball palace, there is a 0.2 probability that it will fail, and he will end up with a wealth of $6,000 with a utility of:U(6,000) = 6,000/6,000 = 1There is also a 0.8 probability that it will succeed, and his wealth will grow to $x. In this case, his expected utility is:0.8U(x) + 0.2U(6,000)Using Buck's utility function, we can substitute U(x) and simplify:0.8(6,000/x) + 0.2(1) = 0.6/x + 0.2To find the smallest value of x that would make Buck indifferent between the two options, we need to set these expected utilities equal to each other and solve for x:0.25 = 0.6/x + 0.20.05 = 0.6/xx = 0.6/0.05 = 12,000Therefore, if the pinball palace can generate at least $12,000 in profits, Buck should invest in it. However, we also need to consider his total wealth, which is $24,000. If he invests in the pinball palace and it succeeds, his new wealth will be:$24,000 + $x - $18,000 = $x + $6,000To find the smallest value of x that would make him prefer this option, we need to compare the utility of $x + $6,000 with the utility of $24,000. Using Buck's utility function, we get:U($x + $6,000) = 6,000/($x + $6,000)U($24,000) = 6,000/$24,000 = 0.25We want to find the smallest value of x that would make Buck prefer the pinball palace, so we set the two utilities equal to each other and solve for x:6,000/($x + $6,000) = 0.25$x + $6,000 = 24,000/0.25$x + $6,000 = 96,000$x = $90,000Therefore, if the pinball palace can generate at least $90,000 in profits, Buck should invest in it. However, the question asked for the smallest value of x, which is $12,000. This means that the answer choice is C.

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