Chapter 18: Risky Choice and Risk

Transcription

1 Chapter 18: Risky Choice and Risk Risky Choice Probability States of Nature Expected Utility Function Interval Measure Violations Risk Preference State Dependent Utility Risk-Aversion Coefficient Actuarially Fair Prices Actuarially Unfair Prices Risk Aversion Risk Seeking Risk Neutral Insurance Actuarially Fair Actuarially Unfair Outline and Conceptual Inquiries Understanding Probability Subjective Probability: Good Guess What is the probability you will read this chapter? Objective Probability: Frequency of Occurrence What is the probability you will take a bath tonight? Mutually Exclusive Probabilities States of Nature Can you be at two places at once? Deriving the Expected Utility Function Application: Expected Tornado Hit and Demand for Shelters Risk Preference Are you a risk taker, or do you purchase insurance? Risk Aversion: School Principals Risk Seeking: Self-Employed Risk Neutrality: Government Are zombies risk-neutral? Certainty Equivalent

2 Why have Insurance? Optimal Level of Insurance How much insurance should you purchase? Application: Elderly Willingness to Pay for Medicare Actuarially Favorable Outcomes Calculating the Risk-Aversion Coefficient What is your number? Application: Precautionary Principle Should it be policy first, then science? Or science first, then policy? State-Dependent Utility Actuarially Fair Prices: Fully Insure Actuarially Unfair Prices: Accept some Risk Risk Seeking Appendix to Chapter 18 Interval Measure of Utility Linear Transformations Violations of Expected Utility Allais s Paradox Machina s Paradox If you cannot go to the football game, would you watch it on television? St Petersburg Paradox Should you always play the lottery? Summary 1. Probability is a measure of the likelihood an outcome will occur and can be classified as subjective and objective. Subjective probability is where a household has a perception that an event may occur; objective probability is the frequency with which an outcome will occur. 2. In making future consumption plans, it is assumed that households must choose among alternatives with outcomes of known probabilities. These outcomes result in contingent commodities whose consumption levels depend on which state of nature occurs. 3. The ranges of risk preferences for households are classified as risk averse, risk seeking, and risk neutral. Risk-averse preferences are associated with concave utility functions, which imply diminishing marginal utility of wealth. Risk-seeking preferences are associated with convex utility functions, and linear utility functions result from risk-neutral preferences. 4. Certainty equivalent is the amount of return a household would receive from a certain outcome so that it is indifferent between a risky outcome and this certain outcome. An implication of certainty equivalence is that risk-averse households are willing to trade some expected return for certainty.

3 5. If insurance is actuarially fair, a risk-averse household attempting to maximize expected utility will fully insure against all losses. In contrast, with actuarially unfair insurance, a household will not fully insure and thus will to accept some risk. 6. For actuarially favorable outcomes, a risk-averse household will accept some risk. The degree of risk a household is willing to accept is generally greater the less risk-averse a household is. 7. A measure of the degree of risk aversion is the Arrow Pratt risk-aversion coefficient, which is based on the curvature of the expected utility function. The larger the risk-aversion coefficient, the more risk-averse is a household. 8. A household s expected utility function is used to determine the utility-maximizing state of nature for a given level of wealth. A risk-averse household facing actuarially fair markets will be willing to pay for facing a state with certainty where the final level of wealth is the same regardless of which state occurs. 9. If the market is actuarially unfair, a risk-averse household may choose to accept some risk. A risk-seeking household will also accept risk and seek out the state of nature with the greatest risk. 10. (Appendix) An expected utility function is the weighted average of utility obtained from alternative states of nature. It is an interval measure of utility, so only a linear, not a monotonic, transformation is possible without changing the preference ordering. Key Concepts actuarially fair games actuarially favorable Allais s Paradox Arrow Pratt risk-aversion coefficient asset certainty equivalent contingent commodity expected utility function golden parachutes increasing linear transformations Independence Axiom lotteries Machina s Paradox outcome positive affine transformations probability risk risk-averse preferences risk-loving preferences risk-neutral preferences risk-seeking preferences St Petersburg Paradox state dependent utility states of nature von Neuman Morgenstern utility function

4 Key Equations An expected utility function is the weighted sum of the utility from consumption in the states of nature. Risk aversion is where the level of utility associated with the expected wealth is greater than the expected utility of wealth. Risk seeking is where the level of utility associated with the expected wealth is less than the expected utility of wealth. Certainty equivalent is the amount of return a household would receive from a certain outcome so that it is indifferent between a risky outcome and this certain outcome. pa < W C For a household to be willing to purchase insurance, the insurance premium must be less than the difference in initial risky wealth and the certainty equivalent. υ = U /U The risk-aversion coefficient is based on the curvature of the expected utility function. Risk neutrality is where the level of utility associated with the expected wealth is equal to the expected utility of wealth.

5 TESTING YOURSELF Multiple Choice 1. Which statement about probability is true? I: The probability of an impossible outcome is 0. II: The probability of a certain outcome is 1. III: The sum of the probabilities of all possible outcomes must equal 1. a. I b. I and II c. I and III d. I, II and III. 2. Donald runs a gas station. He estimates that he can earn a $50,000 profit this year if he can raise the price of gasoline to $4.50 and convince his competitor across the street to match his price increase. However, if his competitor only raises his price to $4.25, he will suffer a loss of $25,000. Donald estimates the probability that his competitor will match his price is 0.7. What is the expected monetary value of raising his price to $4.50? a. $42,500 b. $25,000 c. $27,500 d. $37, Suppose you must determine the degree of confidence that an event will occur. This is known as a. Subjective probability b. Expected value c. Objective probability d. The Independence Axiom. 4. Pearl flips a coin. This state of nature can be described as a. (0, 1) b. (⅓, ⅔) c. (½, ½) d. (¼, ¾). 5. Alberta s plans for the evening will either involve going to the movies or going to a concert. She is offered two states of nature, with the outcomes (¼, ¾) and (⅔, ⅓), and prefers the first state over the second. If she is offered a third state of nature with probabilities (½, ½), she will a. Now prefer the second state over the first state b. Not change her preferences between the first and second states c. Be indifferent between all three states d. None of the above.

6 6. Eugene will win either commodity A or commodity B. Suppose A provides Eugene with a utility of 10 and B provides him with a utility of 15. If he has a 40 percent chance of receiving A, his expected utility is a. 4 b. 10 c. 12 d Suppose Karen s utility function is U = 10W 2W². This implies that Karen is a. Risk-neutral b. Always interested in increasing wealth c. Risk-loving d. Risk-averse. 8. Suppose Tim is playing a game where he flips a nickel. If it is heads, his payoff is $8. For this game to be actuarially fair, the payout for tails is a. $0 b. $4 c. $8 d. $ A risk-seeking individual will have a utility function that is a. Convex b. Downward-sloping c. Linear d. Concave. 10. Gale is an organic-apple grower. She is willing to receive $10,000 today for her newly harvested apples. However, if she puts the apples in cold storage and sells them in three months, there is a 40 percent probability that the crop will be valued at $5000 and a 60 percent probability that the crop will be valued at $20,000. Is Gale a. Risk-neutral b. Risk-seeking c. Irrational d. Risk-averse.

7 11. Consider the following graph: U E U(W) B D C A 0 W 1 D 1 W 1 + D 2 W 2 W 2 W What type of risk preferences does this graph represent? a. Risk-neutral b. Prefers W1 over W2 c. Risk-averse d. Prefers the gamble over wealth of ρ1w1 + ρ2w2 with certainty. 12. Refer to Question 11. To avoid the gamble, an individual with these risk preferences is willing to give up some wealth. The amount the individual is willing to pay is a. E C b. C A c. B A d. C D. 13. Erick is purchasing actuarially fair insurance if a. The insurance premium is equal to the expected cost of the company s claims b. He pays a premium that is lower than his expected loss c. His premium is equal to the difference between his initial wealth and certainty equivalence d. None of the above. 14. Which of the following is correct? a. A risk seeker will purchase insurance b. Risk aversion implies that the marginal utility of wealth rises with wealth c. A risk averter will totally insurance, if the premium is actuarially fair d. The expected value of a gamble can never be negative. 15. Suppose p is the cost of a per-unit decrease in loss that is covered by insurance and ρ is the probability that a loss will occur. As p ρ rises, a. The optimal level of insurance rises b. The individual will become more risk-averse c. The optimal level of insurance declines d. The insurance premium for a given level of expected loss falls.

8 16. Let r C be the rate of return for a certain asset. A risky asset you are considering purchasing may have a return of r 1 with probability ρ or r 2 with probability of 1 ρ. If you are risk averse and are considering the purchase of the risky asset, it must be true that a. r 1 + r 2 > r C b. r 1 + r 2 < r C c. ρr 1 + (1 + ρ)r 2 > r C d. ρr 1 + (1 + ρ)r 2 < r C. 17. A risk-neutral individual has a risk-aversion coefficient a. Greater than zero. b. Negative for low and positive for high levels of wealth c. Equal to zero d. Less than zero. 18. Suppose Bernie s utility function is U = e 3W. What is her risk-aversion coefficient? a. 3 b. 3 c. ⅓ d. ⅓. 19. Suppose you are examining an individual s preferences for contingent commodities. If his indifference curves are concave, a. He will purchase actuarially fair insurance b. He is risk-averse. c. His marginal utility of wealth falls with wealth d. He is a risk seeker. 20. (Appendix) Which of the following does not violate expected utility theory? a. The Independence Axiom b. The St Petersburg Paradox c. Allais s Paradox d. None of the above.

9 Short Answer 1. Explain the difference between subjective and objective probability. 2. Paul s choices for dinner are either spaghetti or spaghetti in meat sauce. He is offered two states of nature: (½, ½) and (⅓, ⅔). Being a meat lover, he prefers the second state to the first. How will his preferences between the first and second states be affected by a third state of nature with probabilities (¼, ¾)? Explain. 3. Explain a household s certainty equivalence. How is this measure related to insurance premiums? 4. Tony is offered two different gambles that he can play for free: I. A 50 percent probability of winning $10 and a 50 percent probability of winning nothing. II. A 50 percent probability of winning $15 and a 50 percent probability of losing $5. If he is risk-averse, will he be indifferent to these two gambles? Explain. 5. If a household purchases actuarially fair insurance, is it risk-averse? 6. Illustrate graphically an expected utility function representing risk-neutral preferences. Why does it have this shape? 7. Illustrate graphically an expected utility function representing risk-averse preferences. Assume that the household has a ρ 1 probability of wealth W 1 and a 1 ρ probability of wealth W 2.

10 Problems 1. Brain s utility function can be represented by U = 10W W 2 and he has the opportunity to invest $20,000 in a new bar and grill. He believes that the probability he will lose his entire investment is ⅔ and that there is a ⅓ chance he will gain $30,000. If he makes the investment, what is his expected utility? Should he make this investment? 2. Assume a Ski Resort faces a 2 percent risk of no snow resulting in a $200 million loss. The owners utility function is U = W 1/2. If the resort s initial value of wealth W is $225 million, what is the maximum premium the owners will be willing to pay for insurance? 3. Suppose Sara s utility function is U = 25W 5W 2. Is she risk-averse, risk-neutral, or riskseeking? Explain. Calculate her risk-aversion coefficient. How is it affected by increases in wealth? 4. Suppose Brian s utility function is U = 2W 2 4W. Is he risk-averse, risk-neutral or riskseeking? Explain. Calculate his risk-aversion coefficient. How is it affected by increases in wealth? 5. Consider Katherine s utility function U = 0.002W². Assume she has a 50 percent probability of W 1 = 100 and a 50 percent probability of W 2 = 500. What is her certainty-equivalent value of wealth? 6. Suppose Sean s initial wealth is $250,000 and faces a 10 percent probability of incurring a $50,000 loss. Assume his insurance company charges an actuarially unfavorable premium of p = 12 percent. If Sean is risk-averse with a logarithmic expected utility function, what is the optimal level of insurance he should purchase? How does your answer change if his insurance company lowers the price to p = 11 percent?