Decision Theory Problems and Solutions

Problem a

Nancy is indifferent between a gamble that pays $625 with a probability of 20% and $100 with a probability of 80%, versus a sure payment of $169. She is also indifferent between another gamble that pays $169 with a probability of 50% and $49 with a probability of 50%, versus a sure payment of $100. Lastly, she is indifferent between a sure payment of $49

versus a gamble of receiving $100 with probability 0.625 and $4 with probability 0.375. Let (p1, p2, p3, p4, p5: 625, 169, 100, 49, 4) represent a prospect (gamble) where p1, p2, p3, p4, and p5 represent the probabilities of the prizes $625, 169, 100, 49, and 4, respectively.

a. Determine whether Nancy is risk-averse, risk-loving, or risk-neutral, based on her preferences.

b.                Rank the following four prospects according to Nancy’s preferences. A: (0.2, 0.2, 0.2, 0.2, 0.2: 625, 169, 100, 49, 4)
B: (0.4, 0, 0, 0, 0.6: 625, 169, 100, 49, 4)
C: (0, 0.1, 0.8, 0, 0.1: 625, 169, 100, 49, 4) 
D: (0, 0, 1.0, 0, 0: 625, 169, 100, 49, 4)

A2-02 (20 marks)

After graduating, Sallie Handshake’s best job offer will either be with a Big-8 accounting firm for $160,000 a year or as a State Farm agent in Grand Rapids for $40,000 a year. She can increase the probability of the former outcome by studying more, but such studying has its costs. If S represents her amount of studying (where S = 0 is no study and S = 1 is all-out effort), her probability of getting the job with a Big-8 firm just equals S. Her utility depends on how hard she studies and her subsequent annual income Y. She tries to maximize the expected value of the von Neumann-Morgenstern utility functionU (S,Y ) = Y 0.5 - 400S 2 . If she chooses S to maximize

her expected utility, how much will she study?

A2-03 (20 marks)

Zoe Goggles is a deep sea underwater photographer. Her camera and lenses are valued at $4,000. There is a chance of 1/20 that she will lose her equipment on a dive over the course of the year. Her wealth, including the value of her equipment, is $30,000. Zoe’s utility function is
U (w) = ln w . Zoe wishes to purchase insurance against the risk of losing her equipment on a

dive. The price per dollar of coverage is γ.

Problem b

a. Write an equation to represent Zoe’s net wealth in the state of the world where she does not lose her equipment and she has purchased K dollars of coverage. Call this w1.

b.                Write an equation to represent Zoe’s net wealth in the state of the world where she does lose her equipment and she has purchased K dollars of coverage. Call this w2.

c. Combine your equations from parts (a) and (b) to write her budget constraint. What is the price of a claim on one dollar of wealth in state of the world 1? What is the price of a claim on one dollar of wealth in state of the world 2? What is the slope of her budget constraint, if w2 is on the vertical axis and w1 on the horizontal axis?

d.                Provide an expression for Zoe’s marginal rate of substitution, MRS = - dw2 / dw1 .

e. If γ = 0.10, how much insurance coverage K will Zoe buy? What is her total premium γK in that case?

f.  If, instead, the price of insurance is actuarially fair, show that Zoe will purchase full insurance: K = $4,000.

g.Draw a diagram depicting Zoe’s budget constraint and her optimal choice of w1 and w2.

A2-04 (10 marks)

Sara has a utility function u=500-100/c, where c is her consumption in thousands of dollars. If Sarah becomes a clerk, she will make $30,000 per year for certain. If she becomes a pediatrician, she will make $60,000 if there is a baby boom and $20,000 if there is a baby bust. The probability of a baby boom is 75% and of a baby bust, 25%. A consulting firm has prepared

demographic projections that indicate which event will occur. What is the most that she should be willing to pay for this information?

A2-05 (20 marks)

Jane owns a house worth $100,000. She cares only about her wealth, which consists entirely of the house. In any given year, there is a 20 percent chance that the house will burn down. If it does, its scrap value will be $30,000. Jane’s utility function is

U(w)= w.

a. Draw Jane’s utility function (plotting utility against wealth).

b.                Is Jane risk averse or risk loving?

c. What is the expected monetary value of Jane’s gamble?

d.                How much would Jane be willing to pay to insure her house against being 
destroyed by fire (leaving her with the scrap value only)?

e. Say that Homer is the president of an insurance company. He is risk neutral and 
has the utility function U(w) = w. Between what two prices (premiums) could a beneficial insurance contract be made by Jane and Homer?

A2-06 (20 marks)

Lucky Midas is a risk-averse gold prospector who has struck it rich. He has $W worth of gold— his only wealth—safely stashed away on his claim in the Yukon. He wants to get his gold from his claim to the big city where he hopes to spend it. His friend Pandora will transport the gold for him free of charge. With probability q, all the gold on any trip will be stolen; with probability 1- q, none of it will be stolen. Show that Lucky’s expected utility is larger if half the gold is transferred in each of two trips than if all the gold is transferred in a single trip.