閤 The Utility analysis of Choices Involving Risk OR。 Milton Friedman: L. J. Savage The Journal of political Economy, Vol. 56, No. 4.(Aug, 1948), pp. 279-304 Stable url: http://inks.jstororg/sici?sici0022-3808%28194808%02956%03a4%3c279%03atuaoc1%03e2.0.co%3b2-%23 The Journal of political Economy is currently published by The University of Chicago Press Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyouhaveobtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the jsTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that ap on the screen or printed page of such transmission STOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support @jstor. org http://www.jstor.org Sun mar1815:55:162007
The Utility Analysis of Choices Involving Risk Milton Friedman; L. J. Savage The Journal of Political Economy, Vol. 56, No. 4. (Aug., 1948), pp. 279-304. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28194808%2956%3A4%3C279%3ATUAOCI%3E2.0.CO%3B2-%23 The Journal of Political Economy is currently published by The University of Chicago Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ucpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Sun Mar 18 15:55:16 2007
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