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

Quizplus · Accepted Answer

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 = 0.25

If 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 = 1

There 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 + 0.2

To 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 + 0.2

0.05 = 0.6/x

x = 0.6/0.05 = 12,000

Therefore, 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 + $6,000

To 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 = 0.25

We 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 = $90,000

Therefore, 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.

Buck Columbus Is Thinking of Starting a Pinball Palace near a Large