THE UTILITY OF WEALTH', II. Friedman and Savage 3 have ex- in Figure I, We may assume it to be a con- plained the existence of insurance and lot- tinuous curve with at least first and second teries by the following joint hypothesis derivatives. 5 Let U be utility and W be (I)Each individual (or consumer unit) wealth. Below some point 4, (a U)/(awa)o; called utility)to every level of wealth and above B, (a'U)/(aw )<o To tell geometrically whether or not an individual would prefer Wo with certainty or a"fair chance of rising to w or falling to (b)acts in the face of known odds so as to aximize expected utility 2)The uti This paper will be reprinted as Cowles Com- W,,, draw a line from the point(Wi, U(W1)) ission Paper, New Series, No. 5 to the point(Wa, U(W2). If this line pass C Hildreth, E Malinvaud, L. ]. Sava above the point Wo, U(o), then the ex- the article of Friedman and Savage, quoted in n3, U(Wo); the bet is preferred to having W I take it as axiomatic that the Friedman-Savage with certainty. The opposite is true if the article has been one of the major contributions to the line(W, U(W),(Wa, U(Wa))passes below nt paper leads only to a small modification of the the point(Wo, U(Wo)). In Figure to Wior W。is Friedman-Savage analysis. This modification, how- preferred to a fair chance of rising ever, materially increases the extent to which com nly observed behavior is implied by the analy The existence of derivatives is not essential to the hypothesis. what ntial is that the M. Friedman and L nalysis of Choices In y be convex below A an Risk, Journal of A and B. The discussi 说 Political Economy, LVI (August, 1948), 279-304 affected if these more general assumptions were I wish to avoid delicate of whether made the relevant utility function is the " utility 6 A fair bet is defined as one with expected gain or the utility of income. I shall assume that in- or loss o Ith equal to zero. In particular if a 15 come is discounted by some interest rate, and I shall the probability of Wr and(I-a)is that of w speak of the"utility of wealth hen aWrt(I-aw,= we
HARRY MARKOWITZ falling to wa. The chance of rising to wI or "poor"; a person with wealth greater than D falling to W, is preferred to having Wa with is presumably well to do. Friedman and certainty. The first example may be thought Savage go so far as to suggest that these may of as an insurance situation. A person with represent different social classes. The wealth W, would prefer to be sure of W. amount(D-C)is the size of the optimal than to take a chance of falling to wa. The lottery prize (i.e, the size of prize which it is second example may be thought of as a lot most profitable for lottery managers to of tery situation. The person with wealth wo fer). Those poorer than C will never take a Wa) for a lottery ticket in the fair bet. Those richer than D will never take hope of winning(W'I-W1). Even if the a fair bet. Those with wealth between C and insurance and the lottery were slightly "un- D will take some fair bets. fair, "7 the insurance would have been taken We shall now look more closely at the and the lottery ticket bought pothesized behavior of persons with vari- Thus the Friedman-Savage hypothesis ous levels of wealth. We shall see that for explains both the buying of insurance and some points on the W axis the F-S hypothe- the buying of lottery tickets. sis implies behavior which not only is not observed but would generally be considered peculiar if it were. At other points on the curve the hypothesis implies less peculiar, but still questionable, behavior. At only one region of the curve does the F-S hypothesis imply behavior which is only ob his in itself analysis should be modified Consider two men with wealth equal to C+i(D-C)(i. e, two men who are mid between C and D). The which these men would prefer, in the way of a fair bet. rather than one in which the loser would fall to C and the winner would rise to FIC. 3 D. The amount bet would be(D-C)/2- half the size of the optimal lottery prize. At I.2. In this section I shall argue that the the flip of a coin the loser would become Friedman-Savage(F-S) hypothesis contra- poor; the winner, rich. Not only would such dicts common observation in important re- a fair bet be acceptable to them but spects. In the following section I shall pre- would please them more sent a hypothesis which explains what the We do not observe persons of middle in- F-S hypothesis explains, avoids the contra- come taking large symmetric bets, we dictions with common observation to which pect people to be repelled by such bets. If the F-S hypothesis is subject, and explains such a bet were made, it would certainly be still other phenomena concerning behavior considered unusual and probably irrational Inder uncertainty Consider a person with wealth slightly In Figure 3 a line l has been drawn tan- less than D. This person is"almost rich gent to the curve at two points .& A person The bet which this person would like most, with wealth less than C is presumably according to the F-S hypothesis, is one hich if we to D, W+ (aW.>Wo, ai+ would lower him to C. He would be willing (I-aoywi<W. For limits on the amount of un- to take a small chance of a large loss for a Friedman and Savage, opcil,p large chance of a small gain. He would not
UTILITY OF WEALTH I53 contrary he would be anxious to underwrite mally of many people and have typically re insurance. He would even be willing to ex- ceived the answers indicated. But these tend insurance at an expected loss to him-"surveys"have been too unsystematic to serve as evidence; I present these questions gain such behavior is not observed On and typical answers only as a heuristic in- the contrary we find that the insurance busi- troduction. After this hypothesis is intro ness is done by companies of such great duced, I shall compare its ability and that of wealth that they can diversify to the point the F-S hypothesis to explain well-estab of almost eliminating risk. In general, it lished phenomena. The hypothesis as a seems to me that circumstances in which a whole is presented on page I55 moderately wealthy person is willing to risk Suppose a stranger offered to give you a large fraction of his wealth at actuarially either io cents or else one chance in ten of unfair odds will arise very rarely. Yet such a getting SI(and nine chances in ten of get willingness is implied by a utility function ting nothing). If the situation were quite ke that of Figure 3 for a rather large range impersonal and you knew the odds were as stated, which would you prefer? Another implication of the utility func tion of Figure 3 is worth noting briefly. A (r) Io cents with certainty or one chance in person with wealth less than C or more than ten of getting Sr? D will never take any fair bet(and, a for- Similarly which would you prefer(why not tiori, never an unfair bet). This seems pe- circle your choice? culiar, since even poor people, apparently as (2)SI with certainty or one chance in ten of much as others, buy sweepstakes tickets, getting SI play the horses, and participate in other (3)Sio with certainty or one chance in ten of forms of gambling. Rich people play roulette getting SIoo? and the stock market. We might rationalize (4)Soo with certainty or one chance in ter this behavior by ascribing it to the "fun of participation,or to inside information. but (s)SI, ooo with certainty or one chance in ten people gamble even when there can be no of getting SIo, inside information; and, as to the joy of(6)S1,ooo, ooo with certainty or one chance in participation, if people like participation but en of getting $Io,ooo,ooo do not like taking chances, why do they not Suppose that you owed the stranger always play with stage money It is desir- cents, would you prefer to pay the able(at least according to Occam's razor )to (7)Io cents or take one chance in ten of have an alternative utility analysis which owing Sr? can help to explain chance- taking among the rich and the poor as well as to avoid the less Similarly would you prefer to owe defensible implications of the F-S hypothe-( 8)SI or take one chance in ten of owing SIo? (o)Sio or take one chance in ten of owing Another level of wealth of interest corre- SIoo? F-S curve. We shall find that the implica- St,0otoo or take one chance in ten of owing ons of the F-s hypothesis are quite plau (Ir)SI, ooo, ooo or take one chance in ten of sible for this level of wealth. I shall not dis cuss these implicatons at this point, for the The typical answers (of my middle-in alysis is essentially the hat of the come acquaintances)to these questions are modified hypothesis to be presented below. as follows: most prefer to take a chance on 2. I. I shall introduce this modified hy- I rather than get Io cents for sure; take a pothesis by means of a set of questions and chance on Sio rather than get SI for sure answers. I have asked these questions infor- Preferences begin to differ on the choice be-
HARRY MARKOWITZ tween SIo for sure or one chance in ten of ting the Sio rather than take SI for sure SIoo. Those who prefer the SIo for sure in take a chance on Soo rather than take SIo situation(3) also prefer SIoo for sure in situ- for sure; perhaps take a chance on Sr,ooo ation(4); while some who would take a rather than take SIoo for sure But the point chance in situation(3)prefer the SIoo for would come when he too would become cau sure in situation (4). By situation(6)every- tious. For example, he would prefer Sr,ooo one erer for sure rather ooo rather than one chance in ten of All this may be explained by assuming essentially the same, in situations(1)- act than one chance in ten of SIo, ooo, ooo SIo, ooo, oo. In other words, he would that the utility function for levels of wealth someone with more moderate wealth, except bove present wealth is first concave and that his third inflection point would be then convex(Fig. 4) farther from the origin. Similarly we hy pothesize that in situations (7)-(II)he would act as if his first inflection point also were farther from the origin Conversely, if the chooser were rather poor, I should expect him to act as if his first and third inflection points were closer 0· present wealth Let us continue our heuristic introduc ion. People have generally indicated a pref than one chance in ten of owing SI; owing SI for sure rather than taking one chance in ten of owing SIo; SIo for sure rather than one in ten of SIoo. There comes a point, however where the individual is willing to take a chance. In situation(II), for example, the FIG. 5 individual generally will prefer one chance in ten of owing SIo, oo, ooo rather than Generally people avoid symmetric bets. owing SI, ooo, ooo for sure. All this may be his suggests that the curve falls faster to plained by assuming that the utility func- the left of the origin than it rises to the right tion going from present wealth downward is of the origin. (I.e, U(X)>U(-X)I a curve as in Figure 5, with three inflection To avoid the famous St. Petersburg Pa resent wealth. The functi mediately above present wealth; convex, from above. For analogous reasons I assume mmediately below it to be bounded from below How would choices in situations(1)-(Il) So far I have assumed that the second in differ if the chooser were rather rich? My flection corresponds to present wealth guess is that he would take a chance on get- There are reasons for believing that this
UTILITY OF WEALTH I55 not always the case. For example, suppose curve which is consistent with our specifica that our hypothetical stranger, rather than tions is given in Figure 5 offering to give you SX or a chance of SY 2.2. An examination of had instead first given you the SX and then that the above hypothesis is consistent with ad offered you a fair bet which if lost would the existence of both "fair (or slightly"un cost you -SX and if won would net you fair")insurance and"fair"(or slightly"un- S(r-X). These two situations are essen- fair")lotteries. The same individual will buy ially the same, and it is plausible to expect insurance and lottery tickets. He will take the chooser to act in the same manner in large chances of a small loss for a small oth situations. But this will not always be chance for a large gain. the implication of our hypotheses if we in- The hypothesis implies that his behavior st that the second inflection point always will be essentially the same whether he is corresponds to present wealth. We can re- poor or rich--except the meaning of"large" solve this dilemma by assuming that in the and"small"will be different. In particular case of recent windfall gains or losses the there are no levels of wealth where people second inflection point may, temporarily, prefer large symmetric bets to any other wealth which corresponds to the second in- ance companies, even at an expected loss flection point will be called customar Thus we see that the hypothesis is con- wealth. Unless I specify otherwise, I shall sistent with both insurance and lotteries, as assume that there have been no recent wind- was the F-S hypothesis. We also see that the fall gains or losses, and that present wealth hypothesis avoids the contradictions with the two dife wealth"are equal. Where common observations to which the F-S hy wealth"(i.e, the second inflection point) 2.3. I shall now apply the modified hy remain at the origin of the graph. Later I pothesis to other phenomena. I shall only will present evidence to support my conten- consider situations wherein there are objec tions concerning the second inflection point tive odds. This is because we are concerned and justify the definition of"customary with a hypothesis about the utility function wealth and do not want to get involved in questions To summarize my hypothesis: the utility concerning subjective probability beliefs.It function has three inflection points. The may be hoped, however, that a utility func middle inflection point is defined to be at the tion which is successful in explaining be stomary"level of wealth. Except in havior in the face of known odds(risk) will cases of recent windfall gains and losses, cus- also prove useful in the explanation of be- tomary wealth equals present wealth. The havior under uncertainty. first inflection point is below, the third It is a common observation that in card flection games,dice games, and the like, people play The distance between the inflection p nore conservatively when losing moder a nondecreasing function of wealth ately, more liberally when winning moder ately. Anyone who wishes evidence of this is curve is monotonically increasing but referred to an experiment of mosteller and ounded;it is first concave, then convex, Nogee to Participants in the experiment were then concave, and finally convex. We may asked to write instructions as to how their also assume that U(-XI>U(X), X> money should be bet by others. The instruc- o(where X=o is customary wealth). A tions consisted of indicating what bets It may also be a function of other things, Thereto"An imental Measurement of utilit is reason to believe, for example, that the distance Journal of Political Economy, LIX (1951), 3 between inflection points is typically greater for The above evidence would be more conclusive if it bachelors than for married men represented a greater range of income levels
HARRY MARKOWITZ should be accepted when offered and"fur- plies the desirability of playing conserva ther(written) instructions. "The "further tively when losing moderately and playi instructionsare revealing; for example, liberally when winning moderately A-II--Play till you drop to 75 cents then This implication holds true whatever be you get low, play only the level of customary wealth of the individ very good odds;"C-I--If you are ahead, ual. In the F-S analysis a person with wealth you may play the four 4's for as low as $3; equal to D in Figure 3 would play liberally C-II-If player finds that he is winning, when losing, conservatively when winning, he shall go ahead and bet at his discretion; so as to attain negative skewness of the fre- C-IV--If his winnings exceed $2. 50, he may quency distribution. This, I should say, is play any and every hand as he so desires, another one of those peculiar implications but, if his amount should drop below 6o which flow from the F-S analys cents, he should use discretion in regard to Now let us consider the effect of wins or the odds and hands that come up. No one losses on the liberality of betting when we do gave instructions to play more liberally not have the strategic considerations which when losing than when winning. The tend- were central in the previous discussion. For ency to play liberally when winning, con- example, suppose that the"evening"is over servatively when losing, can be explained in The question arises as to whether or not the two different ways. These two explanations game should be continued into the morning apply to somewhat different situations whether or not a new series of games a bet which a person makes during a se- should be initiated ). There is also a question ries of games("plays"in the von Neumann of whether or not the stakes should be higher sense)cannot be explained without reference or lower. We abstract from fatigue or loss of to the gains and losses which have occurred interest in the game before and the possibilities open afterward. How do the evenings wins or losses af- What is important is the outcome for the fect the individuals preferences on these whole series of games: the winnings or los- questions? Since his gain or loss is a"wind ening a the evening consists of a series of inde- middle inflection point (presumably by the endent games(say matching pennies); sup- amount of the gain or loss) pose that the probability(frequency) dis- A person who broke even would, by hy tribution of wins and losses for a particular pothesis, have the same preferences as at the game is symmetric about zero. Suppose that beginning of the evening at each particular game the player has a A person who had won moderately would choice of betting liberally or conservatively (by definition of "moderate") be between (i.e,, he can influence the dispersion of the the second and third infecti on point. The wins and losses). If he bet with equal liberal- moderate winner would wish to continue the ity at each game, regardless of previous wins game and increase the stakes or losses, then the frequency distribution of a person who had won very much would final wins and losses(for the evening as a (by the definition of "very much") be to the laying conservatve metric. The effect of right of the third inflection point.He would when winning, is to make the frequency dis- at all. In the vernacular, the heavy winner tribution of final outcomes skewed to the would have made his <killing", and would right. Such skewness is implied as desirable wish to"quit while winning n a large neighborhood of customary in- The moderate loser, between the first and come) by our utility function. In sum, our second inflection points, would wish to play utility function implies the desirability of for lower stakes or not to play at all. some positive skewness of the final outcome A person who lost extremely heavily(to frequency distribution, which in turn im- the left of the first infection point) would
UTILITY OF WEALTH ish to continue the game(somewhat in a strategy leading to a positively skewed desperation) probability distribution of final outcome for We see above the use of the distinction the evening as a whole. Second, for very between customary and present wealth. In small symmetric bets the loss in utility from the explanation use was made of both the the bet is negligible and is compensated for ry income) is at present income and (6)im- inflection point at W= o; therefore, the mediately after such gains or losses custom- utility function is almost linear in the neigh ary income and present income are not the borhood of w=o therefore there is little loss of utility from sI To have an exact hypothesis-the sort bets one finds in physics-we should have to 3. I Above I used the concept of"fun of specify two things:(a) the conditions under participation. If we admit thisas we hich customary wealth is not equal to pres- must-as one of the determinants of be- ent wealth (i.e the conditions referred to as havior under uncertainty, then we must con recent windfall gains or losses)and (b)the value of customary wealth(i.e,, the position of the second inflection point) when cus- tomary wealth is not equal to present which atns w wealth. It would be very convenient if I had a rule which in every actual situation told whether or not there had been a recent wind fall gain or loss, It would be convenient if I had a formula from which customary wealth could be calculated when this was not equal to present wealth. But I do not have such a rule and formula. For some clear-cut cases I am willing to assert that there are or are not recent windfall gains or losses: the man who just won or lost at cards; the man who has tend with the following hypothesis: The experienced no change income for years. I utility function is leave it to the readers intuition to recognize( fair)chance-taking is due to the"fun of other clear-cut cases. I leave it to future re- participation. "This"classical" hypothesis search and reflection to classify the am- is simpler than mine and is probably rather iguous, border-line cases. We are even more popular. If it explained observable behavior ent wealth or how long it takes to catch up. pothesis woula be preferable to mine al hy gnorant of the way customary follows pres- as well as my hypothesis, this classic I have assumed that asymmetric bets are Before examining the hypothesis, we undesirable. This assumption could be must formulate it more exactly. It seems to dropped or relaxed without much change in nble is the ex the rest of the hypothesis; but I believe this pected utility of the outcomes plus the util assumption is correct and should be kept. ity of playing the game(the latter utility is Symmetric bets are avoided when moderate independent of the outcome of the game or large amounts are at stake Sometimes This can be presented graphically as in Fig small symmetric bets are observed. How can ure 6. One implication of this hypothesis is hese be explained? I offer three explana- that, for given( fair)odds, the smaller the tions, one or more of which may apply in amount bet, the higher the expected utility any particular situation. First, we saw pre- In particular, when millionaires play poker viously that a symmetric bet may be part of together, they play for pennies; and no one
HARRY MARKOWITZ will buy more than one lottery ticket. This persons desiring large symmetric bets and contradicts observation, It the implausibility of the one(moderately <6. One might hypothesize that the utility of rich)man insurance company.Perhaps the ame, to be added to the utility of the only evidence of mine which could, so to outcomes, is a function of the possible loss speak, "stand up in court""is the testimony W,or the difference between gain and of the Mosteller- Nogee experiment. But this loss(W,-W2) Neither of these hypotheses does not fully suit our needs, since only a explains why people prefer small chances of narrow range of wealth positions were sam- gains with large chances of small losses realize that i have not demon- rather than vice versa In "beyond a shadow of a doubt? the why people play more conservatively when of the hypothesis introduced. 2 I losing than when winning have tried to present, motivate, and, to a In short, the classical hypothesis may be certain extent, justify and make plausible a consistent with the existence of chance-tak- hypothesis which should be kept in mind ing, but it does not explain the particular when explaining phenomena or designing chances which are taken. To explain such experiments concerning behavior under risk choices, while maintaining simple hypothe- or uncertainty ses concerning " fun of participation, "we 1 Even now we are of one class of com- must postulate a utility function as in Fig- monly observed phenomena which seems to be in ure 5 consistent with the hypothesis introduced in this I. It may be objected that the argu- pas intended to supersede. The existence of multiple as well as the hypotheses which this one ments in this paper are based on flimsy evi- lottery prizes with various sized prizes may con dence. It is true that many arguments are tradict the theory presented. If we are forced to based on"a priori"evidence. Like most"a concede that the individual (lottery-ticket buyer) priori, evidence, these are presumptions of fair lotteries, then my hypothesis cannot explain this the writer which he presumes are also held fact. Nor can any other hypothesis the reader. Such a priori evidence in- this paper explain a preference for dife. idered in udes bility of middle-income lotter or can hypothesis which sumes that people maximize expected utility. Ever vG, The statement that millionaires"ought"to now we must seek hypotheses which explain what for pennies is irrelevant. We seek a hypothesis our present hypotheses explain, avoid the con- o explain behavior, not a moral principle by which tradictions with observation to which they are to judge beh subject, and perhaps explain still other phenomena
