Consider the following information about Bruce Wayne. His current wealth is 100.

Question

Consider the following information about Bruce Wayne. His current wealth is 100. He will meet with Miss Kyle tomorrow. If she is in the good mood, she will help him to fight crime, increasing his wealth to 121. However, she can also decrease his wealth to 64 if she decides to join Bane. You can consider both events equally likely. You can assume that Mr. Wayne’s utility function is u(w) = w.

(a) What is Mr. Wayne’s attitude towards the risk? Explain and show your work!

(b) Suppose that in order to hedge himself from Miss Kyle, Mr. Wayne can purchase the insurance: hire Robin. What is the maximum amount Bruce is willing to pay Robin to eliminate potential losses?

(c) How will your answer to the previous question change, if Mr. Wayne has the utility function: u(w) = w? u(w) = w2 ?

Explanation / Answer

a) The initial wealth of Bruce is 100 and there is 50% chance of losing 36 and 50% chance of gaining 21. His marginal utility is (1/2)W-1/2 where the second order derivative is (-1/4)W-3/2 which is negative. This suggests that Bruce is risk averse.

b) Expected amount of gain/loss is

0.5*(21) - 0.5*(36) = -7.5

The expected wealth is:

0.5*(121) + 0.5*(64) = $92.5

Bruce’s expected utility is:

= 0.5 U(121) + 0.5 U(64)

= 0.5(11) + 0.5(8)

= 9.5

His certainty equivalent wealth is the certain wealth that gives him the same expected utility. His certainty equivalent for an expected utility of 9.5 is:

U(Wce) = 9.5.

(Wce)1/2 = 9.5

Wce = 90.25

With this wealth, the maximum amount he is willing to pay for the insurance is his initial wealth minus expected certain wealth, that is 100 - 90.25 or $9.75.

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