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