复旦大学：《保险学原理 Insurance》课程教学资源（学习资料）The Utility Analysis of Choices Involving Risk

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THE JOURNAL OF POLITICAL EC○NOMY Volume LVI AUGUST 1948 Number 4 THE UTILITY ANALYSIS OF CHOICES INVOLVING RISKI MILTON FRIEDMAN AND L.J. SAVAGE I. THE PROBLEM AND This choice among different degrees of ITS BACKGROUND sk so prominent in insurance and gam purpose of this paper is to sug- bling, is clearly present and important in a gest that an important class of much broader range of economic choices reactions of individuals to risk can Occupations differ greatly in the variabl- be rationalized by a rather simple exten- ity of the income they promise: in some, sion of orthodox utility analysis. for example, civil serviceemployment,the Individuals frequently must, or can, prospective income is rather clearly de hoose among alternatives that differ, fined and is almost certain to be within among other things, in the degree of risk rather narrow limits; in others, for ex- to which the individual will be subject. ample, salaried employment as an ac re insurance and gambling. An individual ability yet almost no chance of either an who buys fire insurance on a house he extremely high or an extremely low in owns is accepting the certain loss of a come; in still others, for example, mo- small sum (the insurance premium) in tion-picture acting, there is extreme vari- preference to the combination of a small ability, with a small chance of an ex chance of a much larger loss(the value of treme ly high income and a larger chance the house)and a large chance of no loss. of an extremely low income. Securities That is, he is choosing certainty in vary similarly, from government bonds preference to uncertainty An individual and industrial "blue chips"to"blue- who buys a lottery ticket is subjecting sky" common stocks; and so do business himself to a large chance of losing a enterprises or lines of business activity small amount( the price of the lottery Whether or not they realize it and wheth ticket)plus a small chance of winning a er or not they take explicit account of the large amount (a prize)in preference to varying degree of risk involved, indi avoiding both risks. He is choosing un- viduals certainty in preference to certainty. securities, or lines of business activity are making choices analogous to those ideas of this paper were that they make when they decide wheth was written primarily by the senior author er to buy insurance or to gamble. Is the

MILTON FRIEDMAN AND L. J. SAVAGE any consistency among the choices of ginal utility. If the marginal utility of this kind that individuals make? Do money diminishes, an individual seeking they neglect the element of risk? Or does to maximize utility will never participate it play a central role? If so, what is that in a" fair"game of chance, for example, a role? game in which he has an equal chance of These problems have, of course, been winning or losing a dollar. The gain in considered by economic theorists, par- utility from winning a dollar will be less ticularly in their discussions of earnings than the loss in utility from losing a dol- in different occupations and of profits lar, so that the expected utility from par in different lines of business. Their treat- ticipation in the game is negative. D ment of these problems has, however, minishing marginal utility plus maxim never been integrated with their ex- zation of expected utility would thus im- planation of choices among riskless alter- ply that individuals would always have atives. Choices among riskless alterna- to be paid to induce them to bear risk. 4 tives are explained in terms of maximiza- But this implication is clearly contra- tion of utility individuals are supposed dicted by actual behavior. People not to choose as they would if they at- only engage in fair games of chance,they tributed some common quantitative engage freely and often eagerly in such haracteristic designated utility-to unfair games as lotteries. Not only do various goods and then selected the risky occupations and risky investments combination of goods that yielded the not always yield a higher average return largest total amount of this common than relatively safe occupations or in characteristic. Choices among alterna- vestments, they frequently yield a much tives involving different degrees of risk, lower average return r example, among different occupa- Marshall resolved this contradiction tions, are explained in utterly different by rejecting utility maximization as an by ignorance of the odds or by explanation of choices involving risk. He ct that young men of an ad- need not have done so, since he did not venturous disposition are more attracted need diminishing marginal utility or by the prospects of a great success than indeed, any quantitative concept of util- hey are deterred by the fear of failure, ity--for the analysis of riskless choices by "the overweening conceit which the greater part of men have of their 4 See Marshall, op. cit, p. I35 n; Mathematical abilities, by"their absurd presump Appendix, n. ix(p. 843)."Gambling involves an their own good fortune, or by fair and even terms. .. a theoretically fair insu s The rejection of utility maximization 135). The argument that fair gambling is an eco- similar deus ex machina. 3 an explanation of choices among dif- than that, firstly the pla es no further assumpti ferent degrees of risk was a direct conse- values of 1, where g(a)is the pleasure derived from quence of the belief in diminishing mar- wealth equal to x.. It is true that this loss of a E. g, see Adam Smith, The Wealth of pleasure derived from the excitement of gambling ed. ) pp. 106-11; Alfred Marshall, Prin that pleasures of gambling are in Bentham's phrase s (8th ed i London: Macmillan 'impure '; since experience shows that they are likely Ltd,rg20),pp.398-4o,554-55,613 3 Marshall, op. cil,, p. 554(first quotation); for steady work as well as for the higher and more Smith, op. cit p. Io7(last two quotation solid pleasures of life"(p. 843)

THE UTILITY ANALYSIS OF CHOICES INVOLVING RISK 28I The shift from the kind of utility ginal utility is unnecessary to explain sis employed by Marshall to the riskless choices, writers have continued ference-curve analysis of F. Y to reject maximization of expected utility worth, Irving Fisher, and Vilfredo Pa- as"unrealistic. This rejection of maxi- reto revealed that to rationalize riskless mization of expected utility has been choices, it is sufficient to suppose that in- challenged by John von Neumann and dividuals can rank baskets of goods by Oskar Morgenstern in their recent book, total utility. It is unnecessary to suppose Theory of Games and Economic behavior that they can compare differences be- They argue that "under the conditions tween utilities. But diminishing, or in- on which the indifference curve analysis creasing, marginal utility implies a com- is based very little extra effort is needed parison of differences between utilities to reach a numerical utility, ' the ex and hence is an entirely gratuitous as- pected value of which is maximized in sumption in interpreting riskless choices. choosing among alternatives involv The idea that choices among alterna- risk. 8 The present paper is based on their ives involving risk can be explained by treatment but has been made self-con the maximization of expected utility is tained by the paraphrasing of essentia ancient, dating back at least to D. Ber- parts of their argument noulli's celebrated analysis of the St. If an individual shows by his market Petersburg paradox. It has been re peatealy referred to since then but almost 6"It has been the assumption in the classical invariably rejected as the correct expla- question will always try to maximize the mathe- ing belief in diminishing marginal utility may appear plausible, but it is certainly not an his nation--commonly because the prevail- matical made it appear that the existence of surmpilointed out tust hold true in all cases. It has t the individual may also be gambling could not be so explained. interested in, and influenced by, the range or the Even since the widespread recognition standard deviation of the different derived or some other measure of that the assumption of diminishing mar- pears pretty evident from the behavior of e In aniel Bernoulli, Versuch einer neuen influenced by the skewness of the probability dis- Theori Wertbestimmmung won Gliicksfaillen (L tribute hard Tintner, "A Contribution to zig, I896), translated by A Pringsheim from"Speci- the Non-Static Theory of Choice, "Quarterly Journe politanae, Vol. V, for the years I730 and I731, ourselves to the mathematical expectation In an interesting note appended to his Marschak, "Money and the Cramer[1704-52 ), a famous mathemat Econometrica, VI [1938], 320 letter in French by Cramer contain what, to us, is consistent with maximization of expected utility is the truly essential point in Bernoullis paper, name- erroneous(see secs. 3 and 4 below). He is led to ly, the idea of using the mathematical expectation consider a formally more general solution because of his failure to appreciate the real generality of the the mathematical on of income to com- kinds of behavior explicable by the maximization of pare alternatives involving risk. Cramer has not expected utility. in general been attributed this much credit apparently because the essential point in Ber- Princeton University Press, Ist ed, 194 ed 1947: pp. I5-3I(both eds ) pp, that the logarithm of income is an appropriate ed, only); succeeding references are to 2d ed s Ibid., P. I7

MILTON FRIEDMAN AND L J. SAVAGE behavior that he prefers A to B and b to ing rationally to maximize expected util- C, it is traditional to rationalize this be- ity in accordance with a given utility havior by supposing that he attaches function. Or it may be that some types more utility to A than to B and more of reactions to risk can be rationalized in utility to B than to C. All utility func- this way while others cannot. Whether tions that give the same ranking to pos- a numerical utility function will in fact sible alternatives will provide equally serve to rationalize any particular class good rationalizations of such choices, and of reactions to risk is an empirical ques it will make no difference which particu- tion to be tested; there is no obvious con- lar one is used. If, in addition, the indi- tradiction such as was once thought to vidual should show by his market be- exist havior that he prefers a 5o-5o chance of This paper attempts to provide a A orC to the certainty of B, it seems nat- crude empirical test by bringing to- ural to rationalize this behavior by sup gether a few broad observations about posing that the diference between the the behavior of individuals in choosing utilities he attaches to A and B is greater among alternatives involving risk(sec than the difference between the utilities and investigating whether these observa he attaches to B and C, so that the ex- tions are consistent with the hypothesis pected utility of the preferred combina- revived by von Neumann and Morgen tion is greater than the utility of B. The stern(secs. 3 and 4). It turns out that class of utility functions, if there be any, these empirical observations are entirely hat can provide the same ranking of al- consistent with the hypothesis if a rather ternatives that involve risk is much more special shape is given to the total utility restricted than the class that can provide curve of money(sec. 4). This special he same ranking of alternatives that are shape, which can be given a tolerably certain. It consists of utility functions satisfactory interpretation(sec. 5),not hat differ only in origin and unit of only brings under the aegis of rational measure (i.e, the utility functions in the utility maximization much behavior that class are linear functions of one an- is ordinarily explained in other terms but other). Thus, in effect, the ordinal prop. also has implications about observable erties of utility functions can be used to behavior not used in deriving it(sec. 6) rationalize riskless choices, the numerical Further empirical work should make it properties to rationalize choices involv- possible to determine whether or not ons conform h t does not, of course, follow that there It is a testimony to the strength of the ill exist a utility function that will ra- belief in diminishing marginal utility tionalize in this way the reactions of indi- that it has taken so long for the possibil viduals to risk. It may be that individ- ity of interpreting gambling and similar uals behave inconsistently--sometimes phenomena as a contradiction of univer choosing a so-so chance of A or C instead sal diminishing marginal utility, rather of B and sometimes the reverse; or some- than of utility maximization, to be recog times choosing A instead of B, B instead nized. The initial mistake must have of C, and c instead of A-or that in some been at least partly a product of a strong other way their behavior is different introspective belief in diminishing mar- from what it would be if they were seek- ginal utility: a dollar must mean less to a Ibid., pp. 15-31, esp. p. 25 rich man than to a poor man; see how

THE UTILITY ANALYSIS OF CHOICES INVOLVING RISK much more a man will spend when he is sumption is contradicted by experience rich than when he is poor to avoid any can easily be shown by hundreds of ex given amount of pain or discomfort o amples [including] the participation of Some of the comments that have been individuals in lotteries in which their the utility analysis of von Neumann and ity)is negative, Tation of gain(util- published by competent economists on mathematical expe Morgenstern are even more remarkable testimony to the hold that diminishing 2. OBSERVABLE BEHA TO BE RATIONALIZED rey remarks: "There is abundant evi- The economic phenomena to which the dence that individual decisions in situ- hypothesis revived by von Neumann and ations involving risk are not always made Morgenstern is relevant can be divided in ways that are compatible with the nto, first, the phenomena ordinarily sumption that the decisions are made ra- garded as gambling tionally with a view to maximizing the ond, other economic phenomena involv mathematical expectation of a utility ing risk. The latter are clearly the more function. The purchase of tickets in lot- important, and the ultimate significance teries, sweepstakes, and'numbers'pools of the hypothesis will depend primarily would imply, on such a basis, that the on the contribution it makes to an under marginal utility of money is an increasing standing of them. At the same time, the rather than a decreasing function of in- influence of risk is revealed most mark come. Such a conclusion is obviously edly in gambling and insurance, so that unacceptable as a guide to social pol- these phenomena have a significance for icy. Kaysen remarks, " Unfortunately, testing and elaborating the hypothesis these postulates [underlying the von out of proportion to their importance in Neumann and Morgenstern discussion of actual economic behavior utility measurement] involve an assump- At the outset it should be confessed tion about economic behavior which is that we have conducted no extensive em- contrary to experience.. That this as- pirical investigation of either class of This elemental argument seems so clearly to phenomena. For the present, we are con justify diminishing marginal desirable even now to sta how this phe- the literature, or obvious from casual ob assumption of increasing marginal utility of money. servation, to provide a first test of the It is only necessary to suppose that the avoidance of hypothesis and to impose significant sub Lin and the other goods that can be bought wi money are related goods and that, while the margin- stantive restrictions on it. al utility of money increases as the amount The major economic isions of an money increases, the marginal utility of avoiding dividual in which risk plays an important pain increases even faster. role concern the employment of the re William Vickrey, ""Measuring M XIII (1945), sources he controls: what occupation to by Reactions to Risk, "econometr The quotation is from pp 327 and 328."The follow, what entrepreneurial activity to f tickets in lotteries, sweepstakes oIs"does not imply that marginal engage in, how to invest (nonhuman) ncreases with income everywhere capital. Alternative possible uses of re- sources can be classified into three broad the quantity that individuals are as maximizing with a quantity C. Kaysen, " A Revolutic en s

MILTON FRIEDMAN AND L J. SAVAGE volved:(a)those involving little or no which does not appeal to great ambition groups according to the degree of risk in- ing only C200. Uncertainty, therefor risk about the money return to be re- and lofty aspirations, has special attrac ceived-occupations like schoolteaching, tions for very few; while it acts as a de other civil service employment, clerical terrent to many of those who are making work; business undertakings of a stand- their choice of a career. And as a rule the ard, predictable type like many pub- certainty of moderate success attra lic utilities: securities like ke government more than an expectation of an uncertain bonds, high-grade industrial bo ss that has an equal actuarial OD rticularly owner-o "But on the other hand. if mely high prizes, its rate degree of risk but unlikely to lead attractiveness is increased out of all pro- to either extreme gains or extreme losses portion to their aggregate value. 3 -occupations like dentistry, account- Adam Smith comments similarly about ancy, some kinds of managerial work; occupational choices and, in addition business undertakings of fairly standard says of entrepreneurial undertakings kinds in which, however, there is suf- "The ordinary rate of profits always ficient competition to make the outcome rises more or less with the risk. It does fairly uncertain; securities like lower- not, however, seem to rise in proportion grade bonds, preferred stocks, higher- to it, or so as to compensate it com grade common stocks; (c) those involving pletely. he presumptuous hope much risk, with some possibility of ex- success seems to act here as upon al tremely large gains and some of extreme- other occasions, and to entice so many ly large losses occupations involving adventurers into those hazardous trades, risks, like piloting aircraft,auto- tha competition re bile racing, or professions like medi- below what is sufficient to compensate cine and law business undertakings in the risk. 1 untried fields; securities like highly Edwin Cannan, in discussing the rate speculative stocks some types of real of return on investments, concludes that 'the probability is that the classes of in- The most significant generalization in vestments which on the average return the literature about choices among these most to the investor are neither the very three uses of resources is that, other safest of all nor the very riskiest, but the things the same, uses a or c tend in gen- intermediate classes which do not appeal eral to be preferred to use b; that is, either to timidity or to the gambling people must in general be paid a premium instinct. I5 to induce them to undertake moderate This asserted preference for extremely risks instead of subjecting themselves to safe or extremely risky investments over either small or large risks. Thus Marshall says:"There are many people of a sober 0p cil, PP.554-55 steady-going temper, who like to know 140pcil,pIII what is before them, and who would far s Article on"Profit, "in Dictionary of Political rather have an appointment which of-He f6oo,but had an equal chance of afford- and Political Science, 1933), pp. 362-oz. economi H. Knight, Risk, Uncertainty, and Profit(Ner than one which was not unlikely to yield York, 1921; reprint London School of

THE UTILITY ANALYSIS OF CHOICES INVOLVING RISK 5 investments with an intermediate degree ance companies have costs of operation of risk has its direct counterpart in the that are covered by their premium re- willingness of persons to buy insurance ceipts, the purchaser is obviously paying nd also to buy lottery tickets or engage a larger premium than the average com- in other forms of gambling involving a pensation he can expect to receive for the small chance of a large gain. The exten- sive market for highly speculative stocks listed and varies from 75 per cent in the income class the kind of stocks that"blue-sky"laws 5oo-$749 to over os per cent in the upper-income case that could equally well be desig- but ranger wan t oP highet for teve, other clas clhs nated as investment or gambling ry is repeated for city after city, the The empirical evidence for the willing- entries in the table for the percentage of families ness of persons of all income classes to These figures cannot be regarded as direct esti buy insurance is extensive, 16 S mates of the percentage of families willing to pay E.g, see U.S. Bureau of Labor Statistics, value-in order to escap e risk, the technical meaning Bulletin 648: Family Expenditures in Selected Cities, of the purchase of insurance that is relevant for our 035-36, Vol. I: Family Expenditures for Housing, purpose. (I)The purchase of automobile and hous- 935-36, Vol, VI: Family Expenditures for Trans- ing insurance may not be a matter of choice. Most Table 6 of the Tabular Summary of Vol. I gives carried. The relevant figure for mortgaged homcs the percentage of home-owning families reportin ould be the fraction of owners carrying a larger come class in each of a number of cities or groups of that insurance be carried on automobiles purchased cities. Since premiums are often paid less frequently on the instalment plan and not fully paid for, and than once a year, the percentages given definitely the purchase of automobile insurance is compuls surance. Yet the bulk of the percentages are well and liability insurance(but not collision insurance Table s of the Tabular Summary of Vol. VI gives pany may not be the same, particularly to persons the percentage of fa the lower-income classes. The loss to the uninsured and cities or groups of cities)reporting expenditures operator is limited by his wealth and bo ce. These fig show a very power, and the maximum amount that he apid increase in the percentage of automobile opera- may be well below the face value of th tors that had insurance(this figure is derived by that he would purchase. The excess of the premium obileinsuranceby the percentage of families operat- for a person with more wealth or borrowing c,gcars)as income increases. In the bottom inc The rise in the percentage of persons ca arvin a minority of those who operate cars carry insurance. reflect not an increased willingness to carry insur- n the upper income classes, where most families ance but a reduction in the effective price that must perate cars, the majority of operators carry insur- be paid for insurance. (3)This tendency may be re percentages versed for the relatively high-income classes for both for selected income classes in cities,given automobile and housing insurance by the operation n text Table Io(p. 26), has for of income tax under the United States federa able 3 of the Tabular Summary of Vol. VIII come tax, while insurance premiums are not. This gives the percentage of families in each income class tends to make the net expected in various cities or groups of cities reporting the individual than for the insurance yment of life, endowment, or ar insurance effect is almost certainly negligible for the figures remiums. The percentages are uniformly high. For cited above, both because they do not effectively example, for New York City the percentage of white cover very high incomes and because the families reporting the payment of insurance pre- come tax was relatively low in 1935-36 iums is 75 per cent or higher for every income class insurance at times comes closer to gaml

MILTON FRIEDMAN AND L J. SAVAGE losses against which he carries insurance. It seems highly unlikely that there is a That is, he is paying something to escape sharp dichotomy between the individuals who purchase insurance and those who The empirical evidence for the willing- gamble. It seems much more likely that ness of individuals to purchase lottery. willing to. We can cite no direct evidence many do both or, at any rate, would be tickets, or engage in similar forms of gambling, is also extensive. Many gov. for this asserted fact, though indirect evi- ernments find, and more governments dence and casual observation give us con have found, lotteries an effective means siderable confidence that it is correct. It of raising revenue I7 Though illegal, the validity is suggested by the extensiveness numbers"game and similar forms of of both gambling and the purchase of in gambling are reported to flourish in the surance. It is also suggested by some of ed States, 8 particularly among the the available evidence on how people in- lower income class vest their funds. The widespread legisla- tion against "bucket shops certain alternative with a higher expected value that relatively poor people must have For example, special life-insurance policies pur- stocks of a"blue-sky"variety. Yet the hased to cover a single railroad or airplane trip are bulk of the property income of the lower han a means of achieving certainty. (5)Even aside income classes consists of interest and from these qualifications, actual purchase of insur- rents and relatively little of dividends ince would give at best a lower limit to the number whereas the reverse is true for the upper These qualifications ofi et one another thigh. income classes. 9 Rents and interest are some types of receipts that tend to be derived from investments with relatively little ed by the evidence cited that a large fraction of risk, and so correspond to the purchase ole in all income classes are willing to buy insur- of insurance, whereas investment in spe examples, currently conduct lotteries he but three chase of lottery ticke ds to the pur- ulative stocks corresp ublic. Great Britain conducted lotteries from 1694 same person both to buy insurance and thereafter, both directly by state governments and mium, in the one case to avoid risk, in under state charters granted to further specific the other. to bear risk. And indeed it tory of lotteries in Great Britain see C. L'Estrange wou ld be inconsistent for a person to be New York State, A. F. Ross, "History of Lotteries in much value. Since most legal wagering is 1907). There seem to be no direct estimates of the track, gambling is available only to those who go to fraction of the people who purchase tickets in state watch the races and is combined with participation or other legal lotteries, and it is clear that such in the mechanics of the game of chance figures would be difficult to get from data obtained 19 Delaware I ncome Statistics, Vol. I (Bureau of ith running the Economic and B from legal lotteries, and casual impressions of ob- Delaware, 1941, Table r; Minnesota Incomes, 1938- relevant units (families or income recipients) purchase tickets ely, individual Table 27; FAHanna,J.APechman,SM.Lerner Analysis of wisconsin Income(Studies in Income Evidence from wagering on horse races, where and Wealth, "Vol. IX (National Bureau of Eco- this has been legalized, is too ambiguous to be of nomic Research, 1948)), Part Il, Table I

THE UTILITY ANALYSIS OF CHOICES INVOLVING RISK willing to pay something(no matter how lotteries would carry over to other ttle)in excess of actuarial value to choices involving risk. There would,how avoid every possible risk and also some- ever, seem to be some presumption that ning in excess of actuarial value to as- they would do so, though of course the sume every possible risk. One must dis- validity of this presumption would have tinguish among different kinds of insur- to be tested ance and different kinds of gambling, The one general feature of lotteries ince a willingness to pay something for that is worth noting in this preliminary only some kinds of insurance would not survey, in addition to the general willing necessarily be inconsistent with a willing- ness of people to participate in them, is ness to engage in only some kinds of the structure of prizes that seems to have gambling. Unfortunately, very little em- developed. Lotteries rarely have just pirical evidence is readily available on single prize equal to the total sum to be willing to buy and the kinds ofe ble are paid out as prizes. Instead, they tend to that they are willing to engage in. about prize is ordinarily not very much larger the only clear indication is that people than the next largest, and often there is are willing to enter into gambles that not one largest prize but several of the offer a small chance of a large gain-as in same size. This tendency is so general lotteries and‘"blue-sky” securities. that one would expect it to reflect some otteries seem to be an extremely consistent feature of individual reac fruitful, and much neglected, source of tions, and any hypothesis designed to information about reactions of individ- explain reactions to uncertainty should uals to risk. They present risk in rela- explain it tively pure form, with little admixture of other factors; they have been conducted 3. THE FORMAL HYPOTHESIS in many countries and for many cen- that is turies, so that a great deal of evidence is rationalizing the behavior just summar available about them; there has been ex- ized can be stated compactly as follows tensive experimentation with the terms In choosing among alternatives open to and conditions that would make them it, whether or not these alternatives in- attractive, and much competition in con- volve risk, a consumer unit(generally a ducting them, so that any regularities family, sometimes an individual) be they may show would have to be inter- haves as if (a) it had a consistent set of preted as reflecting corresponding regu- larities in human behavior. 0 It is. of preferences; (6)these preferences could be completely described by a function course, not certain that inferences from attaching a numerical value-to be des Aside from their value in providinginformation ignatedutility'to alternatives each about reactions to risk, data from lotteries may be of of which is regarded as certain;(c its broader interest in providing evidence about the objective were to make its expected util stability of tastes and preferences over time and their similarity in different parts of the world. Her s a "commodity) which has remained unchanged at See Smith, op, cit, P, 1o8, f which is the same all over the globe, a See Ewen, op cit, passim and which has been dealt in widely for the entire of state lotteries in chap. vii, pp period and over much of the globe. It is hard to con- the large numbers of bills adve eive of any other commodity for which this is John Ashton, A History of Engl 't e eop d4 imag ies(Lon true don: Leadenhall Press, 1893)

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