Daniel l. Rubinfeld

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158 Part 2 Producers, Consumers, and Competitive Markets risk premium Maximum amount of money that a riskaverse person will pay to avoid taking a risk Some criminologists might describe criminals as risk lovers, especially if they commit crimes despite a high prospect of apprehension and punishment Except for such special cases, huwever, few people are risk 10"ing, at least with respect to major purchases or large amounts of income or wealth. The risk premium is the maximum amount of money that a risk-averse person will pay to avoid taking a risk. In general, the magrutude of the risk premium depends on the risky alternatives that the person faces. To determine the risk premium, "lve have reproduced the utility function of Figure 5.3(a) in Figure 5.4 and extended it to an income of 540,000. Recall that an expected utility of 14 is achieved by a woman \\'ho is going to take a risky job with an expected income of 520,000. This outcome is shown graphically by drawing a horizontal line to the vertical axis from point F, v{hich bisects straight line AE (thus representing an average of $10,000 and 530,000) .. But the utility level of 14 can also be achieved if the woman has a certllin income of 516,000, as shown by dropping a vertical line from point C. Thus the risk premium of 54000, given by line segment CF, is the amount of expected income (520,000 minus 516,000) that she 'would give up in order to remain indifferent between the risky job and the safe one. The extent of an individual's risk aversion depends on the nature of the risk and on the person's income. Other things being equal, risk-averse people prefer a smaller variability of outcomes. We saw that vvhen there are h\'o outcomes-an income of $10,000 and an income of $30,000-the risk premium is $4000. Now consider a second risky job, involving a .5 probability of receiving an income of $40,000 and, as shown in Figure 5.4, with a utility le\"el of 20; and a .5 probability of getting an income of SO, vvith a utility level of o. The expected income is again S20,000, but the expected utility is onl;/10: Expected utility .511(50) + .511(540,000) ° + .5(20) 10 Because the utility of ha\"ing a certain income of 520,000 is 16, our consumer loses 6 units of utility if she is required to accept the job. The risk premium in this case is equal to S10,000 because the utility of a certain income of S10,000 is 10: She is willing to give up 510,000 of her 520,000 expected income to ensure a certain income of S10,000 with the same le,"el of expected utilitv. Thus the (Treater the variability, the more a person is willing to pay to avoid a l:isky situati~1. We can also describe the extent of a person's ris~ a,:e.rsion .in terms of indifference curves that relate expected income to the vanablhty of ll1come, 'where the latter is measured bv the standard deviation. Figure 5.5 shows such indifference cun-es for two individuals, one who is wry risk averse and another who is only slightly risk averse. Each indifference curve shows the combinations of expected income and standard deviation of income that give the individual the same amount of utility. Observe that all of the indifference curves are up,vard sloping: Because risk is L{ndesirable, the greater the amount of risk, the greater the expected income needed to make the individual equally \'>'ell off. Figure 5.5(a) describes an individual who is highly risk averse. Observe that an increa~e in the standard deviation of income requires a large increase in e~pected .1l1come to leave this person equally well off Figure S.S(b) applies to a slIghtly nsk-averse person. In this case, a large increase in the standard deviation at income requires only a small increase in expected income. 5 Choice Under Uncertainty 59 In §3.1, we define an indifference curve to be all market baskets that generate the same le\'eJ of sa tis faction for a consumer Utility 20 18 ------------------ G Expected Income Expected Income 14 _____________________ AW~~ Risk Premium 10 10 16 20 30 40 Income (S1000) Standard Deviation of Income (a) Standard Deviation of Income (b) The risk premium, CF, measures the amotmt of income that an individual would give up to leave her indifferent between a risky choice and a certain one. Here, the risk premium is $4000 because a certain income of $16,000 (a! point C) gives her the same expected utility (14) as the uncertain income (a .5 probability of being at point A and a .J probability of being at point E) that has an expected value of $20,000. "", Part (a) applies :0 a pers.on who is highly risk averse: An increase in this individual's standard deviation of income ~ larg~ illc:eas~ ill .expe~ted income if he is ~o ~emau: equally well off. Part (b) applies to a person who is only nsk a\ erse. An mClease ill the standard deViation of ll1come requires only a small increase in expected income
160 Part:2 Producers, Consumers, and Competitive Markets We will return to the use of indifference curves as a means of describing risk aversion in Section SA, where we discuss the dernand for risky assets. First, however , we will turn to the ways , in which an individual can reduce risk. A re business executi\'es more risk loving than most people \I\Then they are P resented 'with alternative strateaies, some risky, some safe, which do they LJ - , choose In one Shldv, -l6-l executives 'were asked to respond to a questionnaire describina riskv sitL~ations that an individual might face as vice president of a LJ , hypothetical company! Respondents ,>vere presented with, four risky events, each of which had a given probability of a favorable and untavorable outcome. The payoffs and probabilities were chosen so that each event had the same expected \'alue. In increasing order of the risk involved (as measured by the difference between the favorable and unfavorable outcomes), the four items were: 1. A lawsuit involving a patent violation 2. A customer tlU'eat concerning the supplying of a competitor 3. A union dispute 4. A joint venture with a competitor To aauae their Willlll1l11eSS to take or avoid risks, researchers asked respondents LJ LJ LJ , a series of questions, In different situations, they could opt to delay a cholCe, to collect information, to bargain, or to delegate a decision. Each option permitted respondents to a\'oid taking risks or to modify the risks that they would take lateL The study found that executives vary substantially in their preferences toward risk~ Roughly 20 percent indicat~d that they were relatively neutral toward risk; 40 percent opted for the more risky alternatives; and 20 percent were clearly risk averse (20 percent did not respond). More importantly, executi\'es (inch~ding those who chose risky alternatives) typically made efforts to reduce or eliminate risk, usually by delaying decisions and collecting more lllformation. In general, risk can arise when the expected gain is either positive (e.g., a chance for a larae reward versus a small one) or negative (e.g., a chance for a large loss or for ~lO loss). The Shldy fmmd that differing preferences toward risk depended on whether a given risk involved gains or losses. In general, those who liked risky situations did so when losses ,'>'ere involved. (Perhaps they ,'>'ere willina t; aamble aaainst a larae loss in the hope of breaking even.) LJ LJ LJ LJ However, ,yhen the risks llwolved gallls, the same executives were more consen'ative, for the less risky alternatives. s 5.3 As the recent gro'wth in state lotteries shows, people sometimes choose risky alternatives that suggest risk-loving rather than risk-averse beha\'ior. In the face of a broad variety of risky situations, however, people are generally risk a\'erse. In this section, we describe three ways by which consumers and managers commonly reduce risks: diversification, insurt1llce, and obtaining more ill/orll/ation about choices and payoffs. Diversification Recall the old saying, "Don't put all your eggs in one basket." Ignoring this advice is unnecessarily risky: If your basket turns out to be a bad bet, all will be lost. Instead, one can reduce risk through diversification: allocating one's resources to a variety of risky sihlations. Suppose, for example, that you plan to take a part-time job selllllg appliances on a commission basis. You can decide to sell only air conditioners or only heaters, or you can spend half your time selling each. Of course, you can't be sure how hot 01' cold the weather will be next year, How should you apportion your time in order to minimize the risk involved III the job , Risk can be minimized by diversification-by allocating your time so that you sell two or more products (whose sales are not closely related) rather than a single product. Suppose there is a 0.5 probability that it will be a relatively hot year, and a 0.5 probability that it will be cold. Table 5.5 gives the earnings that you can make selling air conditioners and heaters. If you sell only air conditioners or only heaters, your actual income will be either $12,000 or $30,000, but your expected income will be $21,000 (.5[$30,000] + .5[$12,000]). But suppose you diversify by dividing your time evenly between the tvvo products. In that case, your income will certainly be $21,000, regardless of the weather. If the weather is hot, you will earn $15,000 from air conditioner sales and $6000 from heater sales; if it is cold, you will earn $6000 from air conditioners and $15,000 from heaters. In this instance, diversification eliminates all risk. Of course, diversification is not always this easy. In our example, heater and air conditioner sales are negatively correlated-they tend to move in opposite directions. In other 'Nords, ,,\'henever sales of one are strong, sales of the other are weak. But the principle of diversification is a general one: As long as you can allocate your resources toward a variety of activities whose outcomes are not closely related, you can eliminate some risk. Diversification is especially important for people who invest in the stock market. On any given day, the price of an llldividual stock can go up or down by a large ammmt, but some stocks rise in price while others fall. 5 Choice Under Uncertainty diversification Reducing risk by alloca ting resources to a variety of activities whose outcomes are not closely related. . negatively correlated Having a tendency to move in opposite directions (said of two variables). 7 This example is based on Kenneth R. MacCrimmon and Donald A Wehrung, "The Risk In Basket," JOIII'IIlll of BIiSillcss 57 (1984): 367-87 s Interestingly, s~me people treat the risk of a small gain in income \'ery differently from the risk of a small loss. Prospcct theon/. de\'eloped by ps\'chologists <strong>Daniel</strong> Kahneman and ~;nos T \'ersky, 11elg: ~~ explain this phenomenon See "Rational CholCe and the Frammg of DeclSlons, JOllllwl oj BU,lI1l,:.. (1986): S251-78, and "Prospect Theory: An Analysis of Decision under Risk," EcollOlllciricll -17 (19/9): 263-92. HOT WEATHER COLD WEATHER Air conditioner sales 30,000 12,000 Heater sales 12,000 30,000
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Daniel l. Rubinfeld

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