# Utility

# Utility

The principle of decreasing marginal utility

Revealed preference; multiple choice

In a broad perspective the history of economics emerges as a struggle with the problem of value. The importance of the concept known in modern economics by the name of utility derives precisely from the great light it shed upon this problem from the outset.

The term “utility” appeared quite early in the economic literature, but its meaning has shifted continually. Initially it had the same meaning as its common synonym “usefulness.” Thus, Galiani (1750) defined *utilità* as “the capacity of a thing to procure us felicity.” For Bentham, too, “utility” meant “that property in any object, whereby it tends to produce benefit, advantage, pleasure, good, or happiness … to the party whose interest is considered” *(The Works of Jeremy Bentham,* vol. 1, pp. Iff.).

It is with Jevons (1871) that utility no longer refers to “an intrinsic quality” of a thing but means “the sum of pleasure and the pain prevented” by its use. Actually, this new meaning slowly began creeping into the economic literature long before Jevons. The switch first occurred in Bentham’s works: he used the term both in its old sense *and* in the sense of the “principle of utility,” i.e., the principle of organizing society so as to achieve “the greatest happiness of the greatest number.” Bentham was, nevertheless, disturbed by his own license, and late in life the architect of modern utilitarianism lamented, *“Utility* was an unfortunately chosen word.”

It is hard to say whether Jevons, who was quite familiar with Bentham’s main writings, knew of Bentham’s verdict. But by choosing an old term— which even nowadays circulates with its old meaning—for a new and highly subtle concept, Jevons created a permanent source of confusion. Various writers have proposed a change in terminology, but without success. Even “ophelimity,” the word expressly coined for this concept by Pareto (1896–1897), did not prevail over “utility” and is now found only in esoteric works, where it has a different meaning.

For Bentham, and to a large extent for Jevons too, “utility” was strongly associated with the pleasure felt during the act of consumption. Gossen (1854) and Edgeworth (1881) reduced everything to this pleasure alone. Actually, with Edgeworth’s *Mathematical Psychics* the hedonistic school of economic behavior reached its climax. Gradually, however, economists came to realize that economic behavior cannot be compared with that of an individual who decides whether or not to have more coffee *while drinking coffee.* The modern concept of utility was worked out by Jennings (1855), who, curiously, arrived at it after an excursion in psycho–physical parallelism far more extensive than that undertaken by Edgeworth. Jennings pointed out that in choice the individual is guided by *expected pleasure,* which in turn is based on the remembrance of the pleasures actually experienced in the past; but he called this expected pleasure “value.”

In truth, what we now call utility theory need not require any shift from the old meaning of “utility.” As early as 1833 Lloyd stated its basic principles, with perfect clarity, in the old terminology:

The utility [usefulness] of corn is the same after an abundant harvest as in time of famine…. The term value [utility], therefore, does not express a quality inherent in a commodity, [but] a feeling of the mind, and is variable with the variations of the external circumstances which can influence that feeling, without any variation of the intrinsic qualities of the commodity which is the object of it. ([1833] 1927, pp. 174, 181)

Even Jevons wrote, “value in use = total utility,” and Pareto, too, admitted that “utility” is another word for “value in use.” No doubt “value in use” fits better the modern concept of utility.

It is curious, then, that after Aristotle set forth the concept of value in use (see below), it took more than twenty centuries for utility to become the basis of a revolutionary theory in economics. The quantitative scaffold, which now constitutes both the pride and the burden of modern economics, was erected, however, not on the utility concept alone but on the “principle of decreasing marginal utility.” Simply stated, this principle is that each additional unit of the same commodity increases utility by a decreasing magnitude. It obviously implies that pleasure, nay, *expected* pleasure, can be measured just as length or weight can. The numberless preutilitarian economists must not be blamed for being as incapable of thinking up this idea as we are now reluctant to part with it.

**Historical background.** Long and tortuous though the intellectual struggle with the problem of value has been, a preliminary survey reveals only a few truly important landmarks. The first, laid by the philosophers of ancient Greece, is the idea that two distinct elements are involved in the problem of value: an intrinsic quality of the object of value and a subjective evaluation by the user. But the common belief that only monistic explanation befits genuine science prompted one student after another to seek a single cause for value. Thus, the problem took the form of the famous epistemo–logical puzzle: the pain must be either in the needle or in the sentient being, not in both. And since the thought that pleasure and pain can be measured seemed extravagant, economic thought was dominated for centuries by commodity fetishism: “the value of a thing lies in the thing itself,” as we find it frankly stated by J. B. Say. This orientation led to the second landmark: value is determined by the amount of labor crystallized in the commodity.

Some writers still played with the idea that there is, nonetheless, a subjective element in value, but any hint in this direction crumbled against the “paradox of value,” known even to the poet Pindar. The paradox is that many very useful commodities, such as water, have a low exchange value or none at all, whereas much less useful ones, such as diamonds, have a high exchange value. Perhaps nothing illustrates the complexity of the problem of value better than the fact that although the paradox once constituted the greatest obstacle on the road to utility theory, the only light on it comes from this very theory.

The characteristic ferment of the Enlightenment produced a reaction against commodity fetishism, which—as happens with most reactions—swung to the opposite extreme: “A thing does not have value because, as is assumed, it costs; but it costs because it has value [in use],” as Condillac ([1776] 1948, p. 246) was to summarize the position representing the third landmark.

The fourth landmark, the most important of all, is utility theory itself. Upon it the great synthesis of general economic equilibrium is now based: in value, utility and scarcity (both in their modern meanings) are linked together through the whole economic process. But the prevalent belief to the contrary, not all issues pertaining to value or to consumer behavior are thereby elucidated (see the last section of this article, “A critique of utility”).

Even though the philosophers of ancient Greece never undertook a systematic study of the economic problems beyond *oikonomia* ("housekeeping"), Plato’s *Dialogues* contain a remarkable analysis of pleasure and pain (e.g., *Protagoras, Gorgias,* and especially *Philebus*) and are studded with other ideas on a par with or even superior to many modern works on utility. Plato argued that life is a “juxtaposition” of pleasure and pain: these alone form the object of man’s choice. Centuries later Bentham was to open his *Principles of Morals and Legislation* with exactly the same thought: “Nature has placed mankind under the governance of two sovereign masters, *pain* and *pleasure” (The Works of Jeremy Bentham,* vol. 1, p. 1). In an equally modern vein, Plato *(Philebus* 21) proclaimed that if you were unable “to calculate on future pleasure, your life would be the life, not of a man, but of an oyster or *pulmo marinus.”* And again we find Bentham insisting that “all men [even madmen] calculate” with pleasures and pains.

The legacy of Aristotle also is not confined to the basic distinction between value in use and value in exchange *(Politica* 1257a6–14). Regrettably, Aristotle’s most important thoughts on the relation between value and prices are crowded into a few pages of *Ethica Nicomachea*(1132a–1133b). There we find the fundamental idea that underlies both the labor and the utility theories of value in modern times: since commodities must be compared in order to be exchanged for each other, every commodity must possess in some definite degree a measurable quality common to all. There is, then, a unit by which this common quality, value, can be measured. Most interesting of all is that Aristotle goes on to say, first, that it is “[social] need” that renders all commodities commensurable and, second, that the exchange value of a commodity is proportional to the work necessary to produce it. This led him to conclude—an argument repeated by Marx—that exchange cannot increase value: after a “just exchange,” everyone must come out without gain or loss. Coming from one of the intellectual giants of all time, this fallacy maintained an unrelenting grip on economic thought for centuries.

Fifteen centuries after the golden period of the Athenian Academy came to an end, interest in the analysis of the economic life of society was revived by the Scholastics, who found Aristotle’s just exchange and just remuneration perfectly congenial to their Christian ideals. In the end, however, the Scholastics broke away from the normative viewpoint to ask, not what value should be, but what it is. The final position is explicitly crystallized by St. Antoninus. The value of an article rests upon (*a*) its comparative quality with other similar articles, *(b)* its scarcity, and (*c*) its *complacibilitas*—a concept equivalent to that introduced later by Galiani and Bentham.

The publication in 1750 of Galiani’s treatise on money marks the beginning of the subjectivist reaction already mentioned. Galiani’s psychological approach led him to the important observation that value is “an idea of the balance between the possession of one thing and that of another in the mind of an individual” ([1750] 1915, p. 27), a thought which adumbrates the basis of the modern theory of choice. But his apologists to the contrary, Galiani did not come close to perceiving the principle of decreasing marginal utility. Instead, Galiani touched upon some ideas that made history in the hands of later writers. He anticipated the classical school by arguing that the only invariable standard of value is man himself. Equally interesting is Galiani’s thesis, which bears upon a recent idea of Friedman and Savage ([1948] 1952, pp. 87 ff.), that the desire for social distinction—rank, titles, honors, nobility, authority—is stronger than that for luxuries and that the desire for luxuries is stronger than the desire of the hungry for food.

Another valuable contribution along the same line is a little–known essay written by Turgot in 1768 but not published until 1808. Turgot was the first economist to relate explicitly the consumer’s behavior to choice and also to see that in a barter each party values what he gets more than he values what he gives. But Turgot still could not free himself completely from the Aristotelian tradition, for he went on to argue that in a free barter no party can gain more than the other.

The pragmatic reason why utility theory constitutes an important chapter of modern economics is that it greatly simplifies demand theory. Unfortunately, in their groping for a solution to the problem of value the early economists were handicapped by an entirely inadequate conception of demand as some invariant to price. The idea that demand is an invariant quantity goes back to Montanari, who in the 1680s argued that a commodity is abundant, not when there is in fact a large quantity of it in an absolute sense, but when there is plenty of it relative to the need, esteem, and desire people have for it (see 1687). Later on we find demand conceived as an invariant expenditure. This is most clearly expressed by Cantillon: “A pound of Beef will be in value to a piece of money pretty nearly as all the Beef offered for sale in the Market is to all the money brought thither to buy Beef” ([1755] 1931, p. 118), a fallacy that for a long time passed as an economic truism. J. B. Say built upon it his famous theorem that a rise in the price of a commodity is in direct proportion to the demand and in inverse proportion to the supply of the commodity.

The post–Scholastic writers were partly justified in relating demand to an invariant. On the whole, economic life in Europe remained both stagnant and at a low level until late in the eighteenth century; for the overwhelming majority of people, expenditure covered only basic needs and consequently had an invariable pattern. It was only normal, therefore, for these writers to generalize from what applied to most consumers in their own society. But the view that demand consists of an invariant to price survived even after historical conditions no longer justified it, for example, in the writings of Adam Smith and Karl Marx. The reason for the survival is, no doubt, that any invariant element simplifies matters immensely. The usual tendency to steer away from complicating ideas may also explain why the law that the smaller the crop, the greater its money value, formulated by Gregory King in 1696, made no history—which lack of impact was rightly deplored by Jevons.

No excuse can be offered, however, for the failure of the early writers to discriminate between value in use and value in exchange. Adam Smith therefore gets the glory of rediscovering Aristotle’s dichotomy. But no sooner was this dichotomy recognized than classical economists discarded altogether the concept of value in use, on the ground that it was unscientific. Smith and, more skillfully, Ricardo argued that only the labor necessary to produce a commodity constitutes as objective and invariable a standard for value as the yardstick for length. Ricardo scorned the idea that the propensities of the individual may have anything to do with value. To be sure, classical economists admitted that value in use is “absolutely essential” to exchangeable value; but by barring from economics the subjective side of value, they closed the door to any progress in the analysis of demand. For them, the problem of why consumers do not spend their incomes entirely on bread or entirely on pearls simply did not arise. This is why J. S. Mill could say, “Happily, there is nothing in the laws of value which remains for the present or for any future writer to clear up; the theory of the subject is complete” ([1848] 1961, p. 436).

Although ideas similar to Gregory King’s appeared now and then in the English literature, the notion of “demand at a price” as a general schedule was not reached until 1870, by Fleeming Jenkin. In France, where the classical viewpoint failed to monopolize economic thought, Cournot had already come out with the general law of demand in 1838, and six years later Dupuit (1844), an engineer, not only formulated the principle of decreasing marginal utility for a commodity but also derived from it a synthesis of utility and demand.

That the three authors just mentioned were in varying degrees familiar with the mathematical tool is not mere accident. But one of the most convincing proofs of the power of mathematics in clarifying or solving economic problems is the fact that the first major breakthrough in the problem of value was achieved a hundred years before Cournot’s theory of demand, by two mathematicians, Gabriel Cramer and Daniel Bernoulli, and a natural scientist versed in mathematics, Leclerc de Buffon. The traditional economists’ lack of familiarity with mathematics does not suffice, however, to explain why they did not learn of the discovery. Even nonmathematical formulations of the principle, subsequently made by its rediscoverers—Bentham, Lloyd, and Jennings—failed to attract much attention from British economists. No doubt the shadows of Smith and Ricardo were still strong enough to cover that cast by Bentham. The fate of Gossen’s *Entwicklung* (1854) provides an even stronger proof that in social science new ideas must also suit the temper of the age in order to be accepted. This splendid work—which, as Jevons admitted, completely anticipated his own —was wholly ignored by the German economists, who in those times were all of the historical school.

## The principle of decreasing marginal utility

Simple and transparent though the usual formulation of the principle of decreasing marginal utility seems, only an axiomatic analysis can reveal all that this formulation leaves unsaid or unsolved. The elements involved in the axioms, to be presented and discussed one by one, require little elaboration. Following Jevons, we may define a commodity as an object whose use yields utility (produces pleasure or prevents pain) to at least some individual. The axioms are as follows:

*Axiom A*(Bentham–Gossen axiom): Every commodity is a quantum, i.e., any instance of it is a definite multiple of some arbitrarily chosen unit.

Another way of saying the same thing is that every commodity is cardinally measurable. It was Bentham who first insisted upon this condition explicitly. The same idea appeared in Gossen’s writings and, later, in Menger’s (1871). Most economists, however, have followed Jevons in ignoring the issue of the cardinality of commodities. Yet the issue is vital for any quantitative principle in economics, not only for the one discussed here.

*Axiom B*(Bentham’s axiom): For any given individual, the utility of any commodity is a cardinally measurable variable.

This is certainly the most critical assumption ever adopted in economics. Actually, Bentham assumed much more, namely, that there is a standard of utility *common to all individuals.* Without this common standard he could not possibly have formulated his famous utility principle: “It is the greatest happiness of the greatest number that is the measure of right and wrong.” In an unpublished manuscript, Bentham’s skepticism did show itself in the admission that “one man’s happiness will never be another man’s happiness: a gain to a man is no gain to another: you might as well pretend to add twenty apples to twenty pears.” In one place he even denounced the axiom of cardinally measurable utility, but like countless others, he argued that this is the voice of “indifference or incapacity,” explaining later that even though the “addibility of the happiness of different individuals” is groundless, without it “all political reasoning is at a stand.”

Gossen, too, recognized that no method of measuring pleasure is available and expressed the hope that one might be discovered some day. But at the same time, he remarked that all the new theory needs is the comparability of utilities, a thought found again only in later writings. Jevons quoted Bentham extensively, but except for a categorical denial of interpersonal comparison of utility, he added nothing new. Walras (1874–1877), on the other hand, set a lasting pattern in the way he belittled the issue raised by the utility measure.

Edgeworth, however, defended the cardinal measurability of utility with unsurpassed fervor. Profoundly influenced by the discoveries of Fechner and Weber in experimental psychology—of which Jevons and Walras seem to have had no knowledge—Edgeworth argued that pleasure is measurable in terms of its atoms. Bentham had already alluded to a “moral thermometer,” but Edgeworth is the first economist to believe earnestly in the possibility of measuring utility by a “hedonimeter.” This extreme position failed to impress his contemporaries, for by then economists were ready for the idea that utility is not identical with pleasure.

Unwittingly, for reasons of simplicity, Gossen and Jevons worked with another axiom, stated explicitly only by Walras:

*Axiom C*: The utility of a commodity to an individual depends solely upon the amount of that commodity in his possession.

This axiom proclaims the independence of utilities.

From the above axioms, it follows that the utility of a *given* commodity for a *given* individual is a function, *U(x),* of the amount *x.* Jevons called *U(x)* total utility. The choice was rather unfortunate because the term evokes the utility,

(1) *U(x _{1}, x_{2}, ..., x_{n}) = U_{1}(x_{t}) + U_{2}(x_{2}) + ... + U_{n}(x_{n})*,

derived from all commodities in the possession of the individual. But Jevons wanted to emphasize the distinction between *U(x)* and the utility increment, *ΔU — U(x + Δx)*— *U(x),* produced by an additional dose *Ax* of the given commodity. Jevons used “degree of utility” for the ratio *ΔU/Ax.* Quite reasonably, he assumed that for a continuous commodity this ratio tends toward a limit, *U’(x),* if *Δx* tends toward zero, or, as is said in mathematics, that *U(x)* has a derivative function. Jevons referred to *U’(x)* as “the final degree of utility” of the amount *x;* Walras used *rareté* Pareto, “elementary ophelimity.” The equivalent German word now in use, *Grenznutzen,* was introduced by Wieser (1889); initially, however, the term meant exactly what it says: use (not utility) at the margin.

Not happy with Jevons’ terminology, Wicksteed, in his *Alphabet of Economic Science,* proposed “marginal effect” for ΔU, and “marginal effectiveness” or “marginal utility” for *U’x).*“Marginal” has replaced “final” ever since Marshall, too, adopted it, but Marshall used the term to denote indifferently any of the concepts mentioned above. To avoid the ambiguity now current, we shall denote *ΔU* by “marginal utility increment” and *ΔU/Ax* by “average marginal utility,” and reserve “marginal utility” solely for *U’(x).*

The principle of decreasing marginal utility can now be stated as follows: Given a sequence of equal doses of the same commodity, the marginal utility increment decreases with each successive dose; ultimately it becomes negative. The usual argument by which most discoverers of this principle justified it will appear in a better light in the discussion below. Here we need only note that the principle does not work in all cases. The second cocktail or the second sip of coffee may yield greater satisfaction than the first. Moreover, some commodities may yield no satisfaction until their amount exceeds a certain level. All commodities indispensable to life belong to this category, but the example of fishing for pleasure proves that there are other cases in point.

From the assumption of a decreasing marginal utility increment, it follows that the average marginal utility, *ΔU/Δx,* of a dose also decreases if *x* increases. And since the amount of a dose is arbitrary, for a continuous commodity we can make Δx tend to zero. We then obtain the following axiom:

*Axiom D:* The marginal utility of a commodity, *U’(x),* decreases as *x* increases; ultimately it becomes negative.

The first part of this axiom is currently the most used formulation of the principle of decreasing marginal utility.

The logical content of the preceding axioms can be represented directly by some elementary diagrams. The case of a “regular” commodity, for which U(0) = 0 but *U(x)* is positive and finite in the neighborhood of *x*= 0, will be considered first. In figures la and *lb* the amounts of the given commodity in, say, pounds are measured on the axis *Ox.* In Figure 1*a* the ordinate represents (total) utility, measured in units usually referred to as “utils” in Figure 1*b* it represents marginal utility, measured in utils per pound. In accord with the last part of the principle of decreasing marginal utility, beyond S marginal utility is negative. At *S* utility is maximum; *OS* represents the saturation amount. With Jevons and Menger, we may also think of a point of “suffocation,” *R,* beyond which utility itself becomes negative.

In Figure 1a the average marginal utility of an increment Δx = *AB* is represented by *ΔU/Δx = B’L/A’L,* that is, by the slope of *A’B’.* And since by definition *U’* is the limit of ΔU/Δx, the marginal utility of the amount *OA* is the slope of the tangent *A’T.* Turning to Figure 1*b,* we let *AB* be divided into a certain number of very small equal doses, Δ*x* = *Δa.* Because *ΔU/Δx* approaches *U’* as *Δx* tends to zero, *ΔU//Δx* is approximately equal to *AA” =* U’. In the first approximation, therefore, the area of the rectangle *AA"a"a*= *AA"* x *Aa = U’x Δx = ΔU*. Hence, the area under the step-line *A"a"a’b"b’c"B"* approximates the increase in utility corresponding to AB. If *Δx* tends to zero, the step-line tends to the curve *A"a’b’B”* hence, the area under this curve represents the utility of the increment *AB.* In the special case of a regular commodity—considered in figures 1*a* and 1*b*—the (total) utility of *OB* is the area under the curve *KA"B".*

The principle of decreasing marginal utility is expressed in Figure 1*b* by the downward slope of the curve *KA˝S* and in Figure 1*a* by the decreasing slope of the tangent *A’T* as x increases. Intuitively. the last property is equivalent to the upward convexity of OA’B’R. In mathematical terms, the principle is tantamount to saying that *U˝*(the derivative of *V,* i.e., the second derivative of *U)* is negative: *U”* 0 is the simplest condition for the upward convexity of *U.* By a faulty argument Pareto concluded that, in addition, *U’”>0* this means that the marginal utility curve *U’* is convex downward. But aside from the curious tendency of almost everyone to draw this curve so, no ground seems to exist for accepting Pareto’s inequality.

The two representations just described are not always wholly equivalent: one can always derive Figure 1*b* from Figure 1*a,* but not the converse. The total utility curve can be lowered or raised by any amount whatever; the slopes of its tangents and, hence, the marginal utility curve are not affected thereby; to put it mathematically, the derivative *U’(x)* determines the function *U(x)* except for an arbitrary additive constant. Only for a regular commodity is this constant determined, because we already know that the utility curve must pass through the origin. Actually, mathematics tells us further that the function *U* corresponding to a given *U’* may be infinite. The point is illustrated by the case of *U’(x) = a/x,* the famous formula used by Bernoulli to represent the marginal utility of wealth (see below). The area under this curve (i.e., the total utility) for any amount of the commodity is infinite. Imagine that the axis Ox in Figure 1*a* is moved down to infinity, and you have a simpler illustration of infinite utility. In all these cases, the level of utility can be visualized as a bottomless ocean; the wave on top can, nevertheless, be seen by a navigator and described by the curve of marginal utility.

The idea of an infinite utility is not a mere analytical nicety. Up to the minimum of subsistence, say, *x _{0},* the utility of a basic food item is that of life itself and, hence, must be infinite; the utility of any amount greater than that minimum also must be infinite. To circumvent the ensuing analytical difficulties, it has been suggested that we “take life for granted” and measure utility above the level of subsistence. Even so, the area under the marginal utility curve,

*KS,*in Figure 1

*b*may be infinite; only if it is not, can the truncated utility be represented by a curve like

*A0A’R*in Figure

*2a.*

The foregoing observations show that the principle of decreasing marginal utility does not require that utility should be cardinally measurable (Bentham’s axiom); only its differences must be so. The Bentham-Gossen axiom can also be made

weaker in this respect. However, neither axiom can be weakened further.

In step with the conceptual framework of this section, any stamp collection must have a definite utility. Yet no basis exists for saying that the utility of, say, the fifth stamp is smaller than that of the fourth. A stamp collection is not a multiple of some unit stamp—it can be arranged in any order we please. This proves that the principle of decreasing marginal utility loses all meaning without some cardinality. To prove the same thing for Bentham’s axiom, it suffices to suppose that even though the individual can *feel* that utility increases or decreases, he is unable to compare the utility increments of different doses.

Curiously, the first quantitative law ever proposed for utility was not the principle of decreasing marginal utility but the principle of the decreasing *average* utility—set forth by Gabriel Cramer around 1730. The two principles, it should be noted, are not equivalent. If we bear in mind that the average utility of *OA* is the slope of *OA’,* Figure 2a immediately shows that the marginal principle does not entail the average principle. The converse is proved by the case in which the total utility for the first three doses is 10, 17, and 25 utils.

Even for a modern student, the idea that a man does not become ten times happier if he becomes ten times richer is far more transparent than the idea expressed by the principle of decreasing marginal utility. True, Cramer’s principle cannot work at all if utility is infinite. But since such a consideration was foreign to the early writers, it cannot explain why, in spite of its simplicity, only very few came to think of Cramer’s principle, and especially why the marginal prevailed over the average principle from the outset. The explanation is that economists felt, however faintly, that man’s economic life is like that of a navigator on high seas, for whom the depth of the ocean, whether infinite or finite, is immaterial. In other words, they somehow intuited the characteristic orientation of modern economics—utility theory must be a theory of marginal utility, not of average utility nor of total utility itself. Recent extensions have not denied the justice of this view, and in the end it seems not only that “utility” was an unfortunately chosen term but also that the concept was not needed at all.

Officially, the edifice erected by Gossen, Jevons, and Walras for the theory of value was abandoned quite early. Yet everyone continued to speak the language of utility and to resort now and then to the old framework, for—as we shall see in the following sections—many notions and issues are set in stronger light within that simple framework than through the technically complex apparatus of modern extensions.

### Utility and value

Which utility coordinate truly represents “value” has been the object of considerable disagreement, not only between the classical and the utility schools but also between the members of the latter. Clearly, if the sense in which value is used is not specified, the problem has no determinate answer. But if it is value in use that one has in mind, there is no doubt that the value of an amount *x* is its utility, *U(x).* The case of exchange value is more intricate. The difficulty stems from the fact—first pointed out by Jevons—that the concept of value in exchange as understood by Aristotle and the classical economists has no *direct* relation to either total or marginal utility. In that sense the exchange value of a unit of *C _{1}* is the amount of some other commodity, C

_{1}, for which it can be exchanged on the market at the prevailing exchange rate. Although in the ultimate analysis this exchange rate

is related to the utilities that all commodities separately have for each trader, it is a market datum, not one pertaining to the individual. Jevons (1871) therefore was justified in referring to this market datum as purchasing power, instead of exchange value.

Obviously, the purchasing power of C_{1} has a different measure according to the particular commodity against which it is exchanged. However, as Marx (1867-1879) first insisted, with a wealth of detail, on the same competitive market the only exchange ratios that can prevail are those that render the purchasing power of any commodity equivalent in any exchange. The consequence is that the purchasing power of any commodity can be expressed either in terms of some arbitrarily chosen commodity (which need not be gold or silver) or in terms of some money of account representing *general* purchasing power.

The fact that most economists have held fast to the Aristotelian terminology may be due to a common fallacy, particularly conspicuous in the work of Walras. The fallacy is the argument that marginal utility “is the cause of [the Aristotelian] value in exchange” because price ratios are proportional to the marginal utilities of the respective commodities for *all* traders. This is to ignore that it is the way a competitive market functions that accounts for the proportionality, not vice versa. Besides, even in a pure exchange market the relationship between marginal utilities and exchange values is not as strict as we have it from Walras. The exchange value of a commodity for which some people have absolutely no use—a not uncommon situation—should be nil if the relationship held strictly.

Marginal utility is connected with valuation in exchange, not in the impersonal meaning “exchange value” had for Aristotle, but in a purely subjective one. For the individual who contemplates exchanging one commodity against another, only the balance of the utilities lost and gained counts. In exchange, therefore, the individual’s “esteem” of the last dose △x is *U’*△x. Per unit of C_{1,} this esteem is *U’(x),* but “exchange value of a unit of C_{1},” describes the concept better than “esteem.” And since all available doses of a commodity stand undifferentiated vis-a-vis the satisfaction they yield together, we reach the conclusion that xU’(x) measures the value in exchange of the entire amount x.

This manner of bringing to the surface the relation between marginal utility and value is the hallmark of Jevons’ teaching, as well as that of Menger. But the champion of the thesis that *economic* value can have no other measure than *xU’(x)* is Wieser. Perhaps he wanted to teach not only that “final degree of utility determines value”—as Jevons did —but also that utility (value in use) plays no direct role in this determination. In the end Wieser struck a more acceptable chord by saying that “value in use measures utility; exchange value measures a combination of [marginal] utility and purchasing power” ([1889] 1956, p. 57).

### Utility, money, and exchange

There are two pure types of money used by analytical economics to epitomize two main attributes of actual money. The first type, Walras’ *numeraire,* corresponds to the situation where one ordinary commodity serves also as money. The second type, now generally known as Marshall’s money (1890), typifies the opposite extreme, where money consists solely of pecuniary tokens of no *direct* use whatever. To the holder such money represents only some general purchasing power.

Whichever is the case, from any amount of money, *m,* spent on some commodity, C, at the price p, the individual derives an amount of utility, *V(m,p).* The corresponding formula is almost a tautology: *V(m,p)*= *U(x),* where *U* is the utility of C and *x* is equal to *m/p.* It becomes a full tautology if C is the *numeraire: V(m)*= *U(m).* But all this is of no use if *U* is infinite. However, regardless of whether *U* is infinite or not, we are on solid ground if, instead of *V(m,p),* we consider *V’(m,p),*“the marginal utility of money through C.” Since Am = pAx, the average marginal utility of △m is *△U/(p△x),* or *(△U/△x)/p.* At the limit, this yields

With the aid of this formula, we can easily see how to construct the curve representing *V’(m,p)* if *U’(x)* is given and also how to show the effect of a change in p on the same curve.

**Equilibrium of the consumer** . The problem of the consumer’s equilibrium consists of how an individual described by our theory is going to distribute his budget money, *m _{0},* among the various commodities, C

_{1}, available at fixed prices, p

*if it is also assumed that he economizes, i.e., strives to get the highest possible satisfaction from his budget money. A very elementary solution is obtained with the aid of two graphical devices imagined by Gossen. The first device applies to the case of two commodities. In Figure 3,*

_{i},*V’*short for V’

_{1}(m),_{1}(m,p

_{1}), is represented by

*A*The curve

_{1}S_{1}.*A’*represents

_{2}S’_{2}*V’*when

_{2}(m)*m*is measured in reverse from M toward O, and

*OM*= m0 is the money pos-

sessed by the individual. Let *A _{1}S_{1}* and intersect at B’. The utility corresponding to budget B, where m1, = OB is spent on C

_{i}and

*m*, =

*MB*is spent on C

_{2}, is greater than that of any other budget. Clearly, for budget

*b*the utility is smaller by the amount measured by the area

*B’a’b’.*Hence, B is the

*optimal*budget. (Similar conclusions apply if

*A*and

_{1}S_{1}*A’*do not intersect.)

_{2}S’_{2}Figure 3 also shows, in a simple and direct fashion, what Turgot first pointed out: even though exchange does not create objects of utility, un-coerced exchange *always* increases the trader’s utility. For if our individual possesses OM of Ct and is offered budget *b* as the *sole* alternative to not trading at all, he will freely accept *b,* provided that the area under *b’A’ _{2}* is greater than that under

*a’A.*If the latter is finite and the former infinite, he will accept any

*b.*Whatever the case, the difference between the two areas represents the trader’s gain. The other trader, too, must gain, for otherwise he would not offer to the first the choice of

*b.*The just exchange of Aristotle corresponds to the case where the trader’s gain is nil, which can never happen if trade is uncoerced. Also, if the

prices are p_{1} and p_{2}, nothing can prevent a consumer in a free market from choosing B. The gain he derives from being able to buy C_{2}*in addition to C _{1}* is shown by the area

*B’A’*This is all the gain if C

_{2}A._{1}is the

*numeraire.*But in the case of Marshall’s money, the whole gain is the aggregate utility of B, that is, the sum of the areas under

*A*and

_{1}B’*A’*

_{2}B’.In Gossen’s second graphical device, money is again measured on the horizontal axis (Figure 4). For every level of the ordinate, we add the corresponding abscissas of the curves representing and obtaining the curve *A _{1}A_{2}S".* On this diagram we can read directly any optimal budget. If

*m*for instance, the optimal budget is m

_{0}= OM,_{1}=

*OB, m*for which the marginal utilities of money,

_{2}= BM,*V’*and

_{1}(m)*V’*are equal to

_{2}(m),*OP*in both uses. If

*C*also is available, we can add horizontally

_{3}*A*and the curve representing

_{1}A_{2}S"*V’*The resulting curve,

_{3}(m).*A*has the same convenient property: for

_{1}A_{2}A_{3}“,*m*we can read directly the optimal budget,

_{0}“= Oc*m*for which the indirect marginal utilities of money are equal to

_{1}, = Oa, m_{2}— ab, m_{3}, = bc,*Ow.*We can extend the curve A

_{1}A

_{2}A

_{3}A

_{4}... for any number of commodities.

It is with the aid of simple diagrams such as these that Gossen proved the fundamental proposition of consumer’s equilibrium, in a form found elsewhere only in much later works: “Man maximizes his total life pleasure if he distributes his entire money income, E, among the various enjoyments ... so that the last atom of money spent on each single pleasure yields the same amount of pleasure.”

The curve *A _{1}A_{2}A_{3}A_{4} ...* also has another important property. Take the case of m

_{0}=

*OM*as an illustration. A comparison of Figure 4 with Figure 3 shows immediately that the area under the curve

*A*represents the aggregate utility of the optimal allocation of m” =

_{1}A_{2}Q*OM.*We may denote this aggregate utility for any

*m*by

*W(m),*or by W(m,p

_{1},p

_{2}, ...p

_{n}) if we wish to remind ourselves that this utility depends on prices as well. And since the area under the curve

*A*represents W(w), the ordinate of the curve itself represents

_{1}A_{2}A_{3}A_{4}...*W’(m),*i.e., the marginal utility of

*m*if spent optimally and prices remain constant.

The idea that W(m) measures the utility of money goes back to Walras; as a hint, it is also found in Jevons. It is, however, with Marshall that it acquired the important significance it now has for our topic. Within any analytical framework that assumes cardinal measurability of utility and ignores uncertainty, the idea is not a mere analytical convenience but is intrinsically legitimate.

In his early writings Pareto argued that since Marshall’s money does not satisfy *directly* any human need, it cannot have a utility in the same sense milk has. But if we were to confine the concept of utility to commodities that satisfy some need *directly,* a great number of consumer goods —beef, fish, eggs, wool, etc.—would no longer have a utility. Besides, the utility of milk through its use in, say, an eggnog would also have to be ignored. Another objection to equating *W(m)* with utility is that its measure is based upon the assumption that the individual always chooses the optimal budget, that he always economizes *rationally.* Yet, without assuming that the individual somehow calculates on all types of pleasures, we cannot attribute any utility to potatoes either. The utility of potatoes, too, corresponds to an optimal “potato budget,” i.e., to an optimal distribution among various uses—boiling, baking, etc.—determined by given transformation rates. The case of Marshall’s money is completely similar. Milk, which has both direct and indirect uses, illustrates the case of Walras’ *numeraire.* There is finally the well-known objection against any tool of partial analysis: since *W(m)* depends upon prices, the notion of money utility has “meaning only after the [general] equilibrium has been attained” (Schultz 1938, p. 34). Yet, for consumer theory *W(m)* is as indispensable a tool as total cost is for the theory of production.

The real difficulty of the concept of the utility of money comes from an entirely different direction: actual money is legal tender for future purchases, the prices and the utilities of which are subject to pure uncertainty. Replacing the utilities of all such purchases by a single utility—the utility of money hoarded, or *encaisse desiree,* as Walras called it—is a tempting thought. But such a utility is beset with all the difficulties inherent in the elusive nature of uncertainty itself. This is the main reason why we need a separate theory of money proper.

The diagram of Figure 4 shows neatly why *W’(m)* should not be confused with the marginal utility of the *numeraire* C1,, which is represented by *A _{1}S_{2}.* Yet, in the early works this difference is not sharply marked, perhaps not even suspected. The Walrasian tradition of considering only the case of a

*numeraire*accounts for other omissions as well. In the case of Marshall’s money, a proportional variation of

*m*and all prices obviously affects neither the optimal budget nor the utility of money. Hence, the optimal purchases (the demand functions) and the money utility depend only on the ratios

*p*and consequently they can be written respectively as follows:

_{1}/m,*x*and

_{i}(p_{1}/m,p_{2}/m,^{...})*W(p*

_{1}/m,p_{2}/m,^{...}). From equation (2) above, it also follows that the marginal utility of money can be written as

*W ’ w{p*As great a simplification as these forms represent, economists did not think of them until very recently. Naturally, if one commodity proper is used as money, there can be no proportional variation of money and

_{1}/m, p_{2}/m,^{...})/m.*all*prices. Besides, in the case of a

*numeraire,*the form W(1/m,

*p*) represents no simplification over

_{2}/m,^{...}*W(m, p*).

_{2},p_{3},^{...}Utility and demand. From Figure 4 it can be seen that the optimal budget includes no commodity for which V’_{i}(0) is smaller than *W’(m _{0})* and that the indirect marginal utility of a commodity included in the budget is equal to

*W’(m*Consequently, the complete system of the optimal distribution of budget

_{0}).*m*among the expenditures

_{0}*m*is

_{1},m_{2}^{...}m_{n}*m _{l} + m_{2} + ... + m_{n} =* m

_{0};

With the aid of (2), this yields the correct Walrasian form of the system for the demands

*x _{1},x_{2},^{...},x_{n}:*

*p _{1}x_{1} + p_{2}x_{2} + ... + p_{n}x_{n} = m_{0};*

Employing elaborate mathematics, Pareto (1906) and Slutsky (1915) have derived from (4) a series of propositions regarding the change in the demand *X;* following a change either in one of the prices or in money. All these propositions can be established directly, with the aid of the elementary diagrams used above. For example, Figure 4 shows immediately that an increase in money increases the expenditure on each commodity.

The price effect (i.e.,the change in the demand for a commodity due to a change in the price of the same or another commodity) can be determined by using an idea implicitly used by Marshall and explicitly stated by Pareto: when an individual contemplates spending a dollar on any of the commodities C_{1},C_{2},..., C_{i}, *i ∠ n,* he compares the resulting utility with the utility which the dollar would have if spent on all other commodities, *C _{i+1},* C

_{i+2},...,

*C*

_{n}. The money spent only on these last commodities appears, therefore, as a commodity with its own utility. Let be the marginal utility of money when spent on all commodities but

*C*(assumed not to be the

_{1}*numeraire).*In Figure 5 let

*AS*represent V’

_{1}(m) and

*CD*represent W’

_{1}(m), the latter curve being drawn with the abscissas reversed. If

*m*=

_{0}*OM*can be spent on

*all*

commodities, the optimal budget is *B: OB = m _{1}* is spent on C

_{1}and

*BM*is spent optimally on the other commodities. The marginal utility of money is

*W’(m) =bB.*Now, if the price of

*C*increases, the new V’

_{1}alone_{1}(m) is represented by

*A’S’.*Hence, in the situation of the diagram the marginal utility of money, as well as the expenditure on C

_{1}, increases. But since

*c’d/cd*measures the relative increase in price, this expenditure increases proportionally less than the price. Therefore, the demand for C

_{1}decreases. If

*CD*happens to pass through

*K,*then everything is left unchanged except that the demand for C

_{1}decreases. If

*CD*cuts

*AS*above

*K,*the marginal utility of money and the expenditure on C

_{1}decrease.

Table 1, in which *i*≠ 1, summarizes the various effects of an increase in p_{1}. The fourth column establishes the Cournot-Marshall law of demand for the case of independent utilities—that an increase in the price of a commodity leads to a reduction in the consumer’s demand for the same commodity. On the other hand, the last column shows that even in this most simple case the frequent assertion that an increase in the price of C_{1} increases the demand for C_{i} is incorrect.

Table 1 - Effects of an increase in P_{i} | |||||
---|---|---|---|---|---|

W’(m) | m_{1} | m_{i} | X_{1} | X_{i} | |

CD cuts AS below K | + | + | - | - | - |

CD cuts at K | 0 | 0 | 0 | - | 0 |

CD cuts above K | - | - | + | - | + |

Two singularities have a special importance for the next topic. The first is the case where *AS* and A’S’ coincide for any price. In this case, U’_{1}(x_{1}) must have unit elasticity, and V’_{1}(m) = *a/m,* where a is a constant. As is easily seen from Figure 5, the demand for C_{i} is also of unit elasticity. Moreover, the marginal utility of money is constant with respect to p_{i}. The second singularity arises if *CD* becomes a horizontal straight line before it intersects *AS.* In this case, an increase in *m _{0}* leaves the expenditure m

_{1}( as well as the marginal utility of money, unchanged. The marginal utility of money also remains constant for moderate variations of P

_{i}. What this singularity, which bears on the notion of the constant marginal utility of money, implies is simple yet important. From Figure 4 it is immediately clear that the marginal utility of one commodity (say,

*C*

_{n})

*included in the budget*must be constant. This condition exposes two commonly ignored points: first, that

*U’*constant does not necessarily imply

_{n}=*W’(m) =*constant; and second, that normally W’(m) = constant can be true only if

*m*exceeds a certain value, say,

*m*.*

### Money as a measure of utility

In retrospect it seems quite natural that the idea of measuring utility by money should have occurred to an engineer, Dupuit, and also to an economist, such as Marshall, whose highest ambition was to keep theory as close as possible to “the practice of everyday life.” Although the idea is mentioned also by Jevons, who may have been Marshall’s source of inspiration, only Dupuit and Marshall developed it into a practical procedure. Their points of departure are identical. For Dupuit, “…the measure of the utility of an object [is] the maximum sacrifice [in money] which each consumer would be willing to make in order to acquire the object” ([1844] 1952, p. 89). Similarly, Marshall asserts, “…[the money] which a person would be just willing to pay for any satisfaction rather than go without it, is … the ‘economic measure’ of the satisfaction to him” ([1879] 1930, p. 20). In the entire economic science there is perhaps no idea more transparent to the uninitiated than the Dupuit-Marshall principle; but, also, none illustrates the contradictions often inherent in such transparency better than this very principle when interpreted with full rigor.

Let the maximum amounts of money a given individual is willing to sacrifice for each additional dose △x be *p _{1}*△x,

*p*△x,

_{2}^{...}. On the ground that each additional dose serves a less important need, Dupuit asserted that

*P*In other words, he took it for granted that utility is measured by money. Marshall, however, strove to present this decreasing sequence as a consequence of Jevons’ marginal utility principle. But both authors argued that if the price is, say, p

_{1}>p>^{...},>_{2}, the individual would buy exactly two doses: to buy only one dose would be not to take full advantage of the offer; to buy three doses would be to pay more for the third than its maximum worth to him. Knowing, then, that

at the price *p*_{n} the individual will buy *n* doses, we can represent the individual’s demand schedule by a sequence of points. A continuous demand curve (Figure 6) is obtained by making *△x* tend toward zero. In this case the utility of, say, the second dose is the area under the arc *b _{1}b_{2},* not that of the rectangle

*a*but the difference (the area

_{1}c_{2}b_{2}a_{2},*b*is of the second order of magnitude.

_{1}C_{2}b_{2})Areas in Figure 6 represent money. Thus, *Op _{1}b_{4}a_{4} =* p

_{4}x Oa

_{4}represents the total money the individual would freely spend on the amount

*Oa*; the area under the curve Pb

_{4}_{4}represents the maximum sacrifice this amount is worth in money to him. The picture is similar to that of Figure lb, and the distinction between the meanings of the two areas recalls the earlier discussion of value. Also the “relative utility” of Dupuit or the “consumer’s surplus” of Marshall, measured by the triangular area

*Pp*recalls the trader’s gain. Yet the difference between the Dupuit-Marshall scheme and the utility representation is fundamental.

_{4}b_{4},In the case of utility there is absolutely no difference between one util and the next, whereas in the Dupuit-Marshall scheme the utility of the money that the individual has to pay for each additional dose must always increase, for it is drawn away from increasingly important uses. Fully aware of this difficulty, Marshall tried to circumvent it by arguing that the marginal utility of money is constant if the elements that “generally belong to the second order of small quantities” are neglected. Unfortunately, apart from another assertion that this neglect is justified by the fact that the “expenditure on any one thing ... is only a small part” of the budget, Marshall never explained his position fully. While for some Marshall’s reasoning became an article of faith, others challenged it repeatedly. A *cause célèbre* thus emerged in the economic annals, with Pareto opening the debates.

Pareto argued (1896-1897) that Marshall was correct only if the marginal utility of “the commodity serving as money” is constant and, curiously, maintained that in the first approximation one may take it to be so. If *V’ _{n}(m) =k =* constant, then

*W*’

_{1}

*(m)*=

*k*for

*m > m** regardless of whether

*C*

_{n}is

*numeraire*or not. Hence, the optimal distribution of

*m*between C

_{0}_{1}and all other commodities is given by the system

The first equation shows that, save for the scale,the demand curve for C,is identical to the curve representing the marginal utility of the same commodity. Consequently, the areas under the demand curve measure *both* money and utility. But neither Pareto nor other mathematical economists seem to have realized that the above model vindicates Marshall only if *m′* > *m**, i.e., only if 0 <: *P*_{1}*x* < *m*_{0} - *m**. Also, Marshall’s supposition that *p*_{1}*x*/*m*_{0} is small would not do if *m*_{0} < *m**.

Pareto then set out (1906) to determine under what conditions *W’(m)* is constant, not with respect to *m*, but with respect to some price, *p*_{1}. The shift only attests to the confusion Marshall created by not specifying whether he understood the marginal utility of money to be constant with respect to *m* or to *p.* Pareto reached two conclusions. The first is that *W′(m)* is constant with respect to *p*′ if U′_{1} has unit elasticity, a case which corresponds to the first singularity discussed earlier: *U′*_{1} = *a/x,* and the expenditure for C_{1}, *p*_{1}*x* = *a/W′*_{1} (*m*′), does not vary with *p*_{1}. Perhaps the average Englishman used to set aside a fixed sum to buy tea at any price, but to assume the same behavior for every other commodity is, realistically, absurd. Besides, nothing suggests that Marshall had in mind a demand of unit elasticity.

Pareto’s second result is that *W’(m)* is constant with respect to any price if the absolute value of the analytical expression

is infinite. Obviously, *T* is infinite if some *U*′_{1} = constant, and hence, *U*”_{1} = 0, a case discussed earlier. Pareto goes on to say, “*T* can be large, even very large, if the number of commodities is great,” but he dismisses the thought as totally uninteresting.

More recently, Samuelson (1942) proved that even within the most general utility structure Marshall is right only for absurd cases. However, Samuelson’s proof, too, is based on a strict interpretation of “constancy” and on the assumption that *all* commodities are included in every budget. It seems, therefore, that in none of these mathematical works did the quasi constancy of the marginal utility of money get a fair hearing. Surprisingly enough, it is far more difficult to probe the correctness of the quasi-constancy assumption than that of the strict-constancy assumption. The problem has been tackled only in part and only in comparatively recent times (Friedman 1935; Georgescu-Roegen 1936*a*). An example of the sort of difficulties encountered in the case of quasi constancy is the assertion that *T* tends toward infinity if the number of commodities increases without limit. Since *T* is a sum of terms having no connection with each other, there is no basis on which to predict its limit in the general case. Besides, note that *T* is a dimensional variable with dimension given by (money)^{2}/(utility). Therefore, a change in the units of the *p*_{i} from dollars to cents increases the numerical value of *T* by 10,000. Thus, there is no scale for ascertaining whether *T* is sufficiently large to warrant the quasi constancy of *W′(m).* Actually, 1/*T* represents the slope of the tangent to *W′(m).* And it was Marshall himself who taught us to use elasticity, never slope. Moreover, examples have been constructed to prove that even the elasticity of *W′(m)* does not necessarily tend toward zero when *n* increases.

One can nonetheless approach the problem differently. A substantial part of the budget of any middle-class consumer is in fact spent on mere conveniences, such as magazines, greeting cards, flowers, movie tickets, and the like. These constitute *marginal* expenditures, in the sense that a small variation in the budget would cause one of them either to disappear completely from the budget or to become a new budget item. And since such conveniences usually are numerous, the marginal utility of money remains quasi constant for the income range in point—as can be shown with the aid of a diagram analogous to Figure 4. For this income range, Marshall’s reasoning can be applied, with the appropriate approximation, to any commodity for which the expenditure is not a large fraction of the budget.

### Utility and disutility

In the picture that Plato and every utilitarian after him drew of man’s motives and actions, opposite the utility procured by commodities there stands the disutility caused by labor. However, the law governing disutility is not a mirror image of the principle of marginal utility. The general description of the sensations caused by toiling was formulated in 1855 by Jennings: following a brief period of discommodity at the outset, a laborer experiences for a while some pleasure, but soon irksomeness sets in progressively, and “the amount of toilsome sensation attending each succeeding increment [of time is] greater than that of the increment preceding.” The last part of this description constitutes the “principle of increasing marginal disutility.” The entire law is represented by the curve of marginal disutility, *ABCK* in Figure 7*a*. Naturally, if *t*_{0} = *OT* represents the maximum length of time the individual can work *daily,* the marginal disutility, *D′(t),* is infinite for *t*_{0}.

This seems simple enough. Yet disutility raises

some fundamental issues with greater force than does utility. One such issue pertains to the time-factor. What has happened to the pleasure experienced during the interval *BC* by the time the worker feels exhausted? In other words, can utilities or disutilities be added over time? Some sport enthusiasts incur the discommodity of rowing a boat all Sunday long for “sheer pleasure.” The point, as Plato argued, is that nothing is either pure pleasure or pure pain; everything is a “juxtaposition” of both. Even the activity ordinarily called labor is not pure discommodity; otherwise we would not hear so many people say, “I truly enjoy my work.” Consequently, whether or not the disutility of labor can be treated as a simple quantum depends upon whether we accept the Benthamite thesis that pleasure and pain cancel each other or we believe, with Plato, that even though “both admit of more and less,” pain is not the negation of pleasure. Modern economic theory goes along with Bentham, taking as its first article the idea that pleasure is the positive, and pain the negative, face of the same essence. Even the elemental fact that the discommodity of labor is not simply the absence of leisure is generally ignored; leisure is equated with freedom from the burden of work; and the satisfaction it yields is reduced to the enjoyment of not working. As a result, in the current analyses of the supply of labor, the commodity of leisure never appears side by side with the discommodity of labor. We find either the model originated by Gossen and Jevons, involving only the disutility of labor, or that of Walras, involving only the utility of leisure.

Let *r*_{1}, *r*_{2},… represent the number of work hours necessary to produce one unit of *C*_{1}, *C*_{2},… respectively. The “marginal utility of labor-time through *C*_{i},”

has exactly the same general properties as *V′ _{i}(m).* By the same artifice of horizontal addition, we can construct the curve

*P*in Figure 7

_{1},P_{2}…Q*a,*representing the marginal utility of labor-time,

*P′(t).*The difference

*N′(t) = P′(t) - D′(t)*is the

*net*marginal utility of labor-time; it is positive for

*t > OT*. and negative for

_{g}*t < OT*Therefore,

_{g}.*T*is the saturation point of the net utility of labor-time. If one ignores the utility of leisure, as Gossen does, the optimal labor-time is

_{0}*OT*In the optimal distribution,

_{g}.*t*

_{1}, =

*aa*

_{1},

*t*

_{2}=

*a*

_{1},a_{2}, t_{3}=

*a*are the number of hours spent on producing

_{2}a_{3}, t_{4}= a_{3}E_{g},*C*

_{1},

*C*

_{2},

*C*

_{3},

*C*

_{4}, respectively. This is another general theorem first formulated by Gossen: “in order to maximize his life pleasure, man should distribute his time and labor power among the satisfaction of various enjoyments in such a way that for every enjoyment the value of the atom produced last should be equal to the labor hardship that he would have experienced if he had to produce that atom at the last moment of his labor exertion.”

The above solution can be immediately adapted to the case of a wage earner simply by choosing the money unit so that the wage rate is unity and replacing the labor coefficients, *r _{i},* by the market prices,

*p*. The scale of the abscissa then measures both money and labor-time, and

_{i}*P*represents the marginal utility of earned money,

_{1}Q*W′(t).*The Gossenian supply of labor-time is

*t*=

_{0}*OT*

_{g}.

The Walrasian approach ignores labor disutility: the individual merely maximizes the satisfaction derived from two “commodities,” earned money and leisure. In Figure 7*b, P _{1}Q* represents the marginal utility of earned money and

*LM*the marginal utility of leisure

*L′(t),*the latter being drawn with the abscissa reversed. Let

*E*be the intersection of

_{w}*P*and

_{1}Q*LM.*The Walrasian supply of labor is

*t*

_{w}= OT_{w}Let*′Q′* represent the net marginal utility of labor-time *N′(t),* and let *E _{c}* be the intersection of

*R′Q′*and

*LM.*Clearly, the time distribution that maximizes the individual’s total utility—i.e., the net utility of labor-time and the utility of leisure— corresponds to

*T*. This correct solution, according to which the supply of labor is

_{c}*t*=

_{c}*OT*was given by S. N. Patten in 1892 but now seems to have been forgotten. Interestingly, Patten argued that

_{c},*t*characterizes an advanced economy.

_{c}< t_{g}### Utility and the time-factor

The time-factor, an issue to which the early writers paid considerable attention, has gradually become a rather neglected topic in utility theory. For Gossen, the economic problem consisted precisely of how man should distribute his *time* so as to obtain the greatest pleasure out of life. His two fundamental laws refer only to time:

(1) As the same enjoyment continues in time, the magnitude [intensity] of pleasure continuously decreases until ultimately satiety is reached.

(2) A similar diminution of pleasure occurs if we repeat a previously satisfied pleasure: both the initial magnitude [intensity] and the maximum duration of the enjoyment decrease with each repetition and, moreover, the sooner the repetition the greater is the decrease.

The second law brings up one important aspect of the time-factor: in essence it says that enjoyments at different periods of time are not independent. But as stated, it negates the familiar fact that many pleasures are not worn out by their daily repetition. Through some questionable logic, Gossen sought to prove that for each particular pleasure there is an optimum rhythm of repetition. This is tantamount to saying that every pleasure— whether that of eating bread or listening to a symphony concert—is rejuvenated after the lapse of some definite time interval. By this amendment— rediscovered later by others and now known as the “law of the periodic recurrence of wants”—Gossen implicitly abandoned his initial formulation of the second law. Also, like many after him, he considered only the case in which pleasures are not affected by age and, moreover, assumed that everyone knows his own life span and plans his life from its first moment. Obviously, then, everyone should distribute each commodity equally among the successive and periodic enjoyments covered by his life span. Needless to say, this oversimplified scheme completely evicts the time-factor from economic behavior.

One aspect of the time-factor, however, burst through this static scheme from the outset. In an argument so clumsy that it could not possibly fare well, Jevons contended that *x* in *U′(x)* must always represent a flow rate, “*so much commodity producing a certain amount of pleasurable effect per unit of time.*” Now, if *x* is to represent a rate per unit of time, this rate can only be the rate at which, say, milk is available to the consumer daily, not the rate at which he drinks milk from a glass.

Many contemporary writers, however, are satisfied with the ambiguous formula “*x* is the amount of tea.” Others specify that *x* represents a *uniform* flow rate; still others insist that for a realistic representation, *x* must be a quantity (Fisher 1892). True, apart from a few exceptional cases, the consumer chooses between quantities of commodities, not between flow rates for the duration of his life. To be more realistic, one must add that the utility of any quantity depends also upon how long the consumer expects this quantity to last him—as Menger insisted. The time-factor thus brings into the picture the time horizon. And it is notorious that the time horizon has an extremely blurred limit, a consequence of the pure uncertainty that envelops the future.

There are still other important problems raised by the time-factor. There is, first, the hysteresis effect resulting from the individual’s continuous adaptation, to which Marshall ([1890] 1948, p. 807) called our attention. (See also Georgescu-Roegen 1950 and the works of J. Duesenberry and F. Modi-gliani quoted there.) Actually, the hysteresis effect is a main pillar of the modern concept of utility.

Another complication created by the time-factor was first pointed out by Jevons. On a hint from Bentham, Jevons observed that the more remote from the present a pleasure is, the dimmer its importance appears to the individual. The resulting time preference, or impatience, later became “the nub and kernel” of Bohm-Bawerk’s theory of interest (1884-1912). Curiously, according to Jevons’ law, if an individual were given his life income in one lump sum, he would not distribute it over time in such a way as to obtain the maximum *actual* pleasure out of it. Further interesting inconsistencies resulting from the “myopia” produced by a discount factor of future utility have been pointed out by Strotz (1956).

It must be admitted, however, that we have no satisfactory answer for many problems raised by the time-factor. The reason is that any simile embracing all its aspects, or even some of them, exceeds the present limits of analytical manageability. The mathematical analysis of the utility of infinite sequences of income over time—initiated by Koopmans (1960)—affords a glimpse of the difficulty. There is little doubt that by far the greatest amount of work still to be done in utility theory concerns the time-factor.

### The marginal principle of Carl Menger

Although Menger is listed along with Jevons and Walras as a founder of utility theory, his approach is fundamentally different. His theory is rather akin to the ordinalist doctrine of later days and also to Edgeworth’s general scheme of interdependent utilities. Instead of axioms B, C, and D, Menger uses a far more transparent basis, which, recast in axiomatic form, is as follows:

*Axiom E:* The individual has various general needs, each consisting of a sequence of concrete needs that can be satisfied only in succession.

*Axiom F:* All concrete needs of an individual are ordered on a scale of importance *(Bedeutung),* with the successive concrete needs of the same general need in decreasing order of importance.

*Axiom G:* The concrete needs are such that each is satisfied over a period of time by one definite dose *(Teil-quantitdt)* of one or several commodities.

Table 2 - Menger’s lexicographic orderingneeds | ||||||
---|---|---|---|---|---|---|

CONCRETE | GENERAL NEED | |||||

NEED | I | II | III | IV | V | |

First | 10 | 8 | 7 | 5 | 4 | … |

Second | 9 | 4 | 2 | 3 | 3 | … |

Third | 6 | 2 | 2 | 2 | … | |

Fourth | 5 | 1 | 1 | 1 | … | |

Fifth | 3 | … | ||||

Sixth | 1 | … |

Menger’s position is illustrated by Table 2. The Roman numerals denote the general needs, in the order of their relative importance. The Arabic numerals show the relative importance of the successive concrete needs of each general need. The idea is that even though the second concrete need of II has the same importance rating as the first concrete need of v, the former is more urgent than the latter. In other words, Menger’s table is a *lexicographic* ordering of concrete needs. We can then assign a unique rank to each concrete need and obtain a “need scale.” (This formulation is carefully explained in Böhm-Bawerk [1884-1912] 1959, vol. 2, pp. 140 ff.)

Menger, however, is far from being explicit on the nature of the need scale. He begins by explaining that the Arabic numerals of the table “are not intended to express numerically the *absolute* but merely the *relative* magnitudes of importance of the satisfactions in question.” But he immediately adds that if the importance of two satisfactions are represented by 40 and 20, then “the first of the two satisfactions has twice the importance of the second,” which makes sense only if importance is a cardinal variable. Nevertheless, the general tone of his *Principles* leaves little doubt that Menger actually had in mind an ordinal, not a cardinal, scale.

To observe that any set of concrete needs can be ordered according to their importance and to proclaim that the value of a commodity is determined by the least important need the commodity satisfies does not suffice for a theory of value. Menger added the proposition that *because the individual economizes,* an additional dose lowers the value of a commodity to that of the next most important need. This constitutes Menger’s “principle of decreasing marginal importance.” Menger thought that with its aid the problem of passing from the scale of needs to the scale of value had been solved. The only case considered by Menger— that of a single commodity satisfying all needs—raises no difficulty. The situation is entirely different for the case of several commodities, and this case became the problem child of the Austrian school, involving all Menger’s followers in endless controversies.

As Menger was first to note, the same need may be satisfied alternatively by several commodities. Consequently, which need is satisfied by a particular dose of a commodity depends upon the available doses of *all* commodities. The need fulfilled by an additional dose of a commodity may very well have the same importance rating as the need fulfilled by the preceding dose in the initial allocation. Böhm-Bawerk tried to solve the impasse by distinguishing between the *direct* and the *indirect* importance of a marginal dose—the basis for the current opinion that the Austrian school explained values as opportunities forgone indirectly. However, even with Bohm-Bawerk’s amendment, Menger’s theory cannot explain prices. If Menger’s illustration leaves a different impression, it is because of a series of numerical coincidences. Mending this gap in the theory without adulterating its characteristic rationale would require that Menger’s scale be extended to include ratings of all *sets* of concrete needs. Menger’s followers, however, moved in an entirely different, easier direction. Both Wieser and Bohm-Bawerk, by a verbal legerdemain, equated *Grenznutzen* with Jevons’ marginal utility, and Menger’s ordinal importance rating with Jevons’ cardinal utility. Actually, Bohm-Bawerk used more words than any other economist in arguing that satisfaction has a cardinal measure. Thus, Menger’s followers abandoned the most salient idea of his theory: they all continued to use his jargon, but they reasoned like Jevons or, rather, like Edgeworth, without admitting it overtly.

### The general utility function; interdependent utilities

A new vista was opened by Edgeworth, who disposed of the unrealistic assumption of independent utilities (Axiom *C* above). His idea, which in retrospect seems ultrasimple, is to conceive of total utility as a general function, *U = U(x _{l}x_{2},…x_{n}*), of the amounts of all commodities, instead of the sum of separate utilities, as in (1). Figure 8,

devised by Edgeworth to represent the new utility function, is one of the most popular in the economic literature. The cardinal measures, x and *y,* of the commodities *C*_{1} and *C*_{2} are represented on the axes *Ox* and *Oy.* A curve such as (*u*_{0}) represents all combinations of *x* and *y* that have the same utility, *U(x, y) = u _{0},* for the given individual. Edgeworth called such a curve an “indifference curve,” to express the fact that since the combinations A and A′ have the same utility, the individual must be indifferent as to whether he has one or the other. For reasons to become apparent later, the term “utility curve”—used by Johnson (1913)— seems more appropriate. In the case of more than two commodities, the term “utility varieties” will be used.

In a three-dimensional space the utility function *U = U(x,y)* can be represented as a surface, which Pareto wittily referred to as “the hill of pleasure.” The utility curves corresponding to various amounts of utility constitute the two-dimensional map of this hill. The point S corresponds to the hill’s top, i.e., to the absolute saturation combination, for which *U(x, y)* is maximum. A point such as S′ is one of relative saturation: at S′, *U(x, y)* is maximum if *x* alone is allowed to vary. The area *ONSM* covers the normal situations. For any combination inside this area the marginal utilities of both *C*_{1}, and *C*_{2} are positive: *U _{x} ≡ ∂U/∂x* > 0,

*U*> 0.

_{y}≡ ∂U/≡yEdgeworth retained the principle of decreasing marginal utility. By and large, there seems to be no immediate reason against continuing to assume that the utility increment of a second pound of beef is smaller than that of the first pound, regardless of how much potatoes the individual has. The principle is expressed now by the negative sign of the second partial derivatives: *U _{xx}* ≡

*∂*< 0,

^{2}U/∂x^{2}*U*

_{yy}≡ ∂

^{2}∂U/∂y

^{2}< 0. On a grid of utility curves separated by the same utility increment Δ

*u*(Figure 8), the same property is expressed by

*AB*< BD,

*AC < CE.*

Edgeworth and everyone after him drew the utility curves convex toward the origin. It should be emphasized, however, that this property is not a necessary consequence of the principle of decreasing marginal utility. Conversely, the convexity of the utility curves does not imply the principle of decreasing marginal utility.

At first Edgeworth justified this convexity on the additional assumption that the marginal utility *U _{x}* increases with,

*y*i.e.,

But later on he contended that the contrary assumption,

which means that the marginal utility of C_{1}, decreases with the amount of *C _{2},* is more in the spirit of the principle of decreasing marginal utility. In the end, he admitted that both situations may occur: he called the goods satisfying (8) complementary, those satisfying (9) rival or competitive. The intermediate case, where

*U*= 0, corresponds to independent utilities. An equivalent but more transparent definition is due to Pareto: two commodities are complementary, independent, or competitive according to whether they yield together a greater, equal, or smaller utility than they yield separately. If the two commodities are complementary or independent, the utility curves must be convex. The same applies to mildly competitive commodities. But cases of strongly competitive commodities can be conceived which, although satisfying the principle of decreasing marginal utility, yield utility curves concave toward the origin (Georgescu-Roegen 1966, pp. 60 ff.).

_{ry}**The optimal budget.** The optimal budget, B, is represented in Figure 9 by that point of the budget line *AM* which lies on the highest possible utility curve. If B contains both commodities, it is the point of tangency of this utility curve and *AM.* Now, if Δ*x* is sufficiently small and if we substitute Δ*y* for Δ*x* so as to keep the individual on the same utility curve, we have *U*_{x} Δ*x* + *U _{y} Δ_{x}* = 0. Hence, the “marginal rate of substitution,” which corresponds to the slope of the tangent to the utility curve, is

*dy/dx = -(U*On the other hand, the slope of the budget line

_{x}/U_{y}).*p*is — (

_{x}x + p_{y}y = m*p*). From the equation of the budget line and the equality of the two slopes, we obtain a system of equations that determines the optimal budget—a system formally identical to (4).

_{x}/p_{y}For the analysis of the changes in the optimal budget caused by changes in m or in the prices, it is again convenient to group all commodities whose prices remain constant into a single coordinate. Let the prices of *C*_{i+1}, *C*_{i+2}…,*C*_{n}, 1 < *i < n,* be kept constant, and let *m* be the amount of money that the individual can spend (optimally) on these commodities. It can easily be shown that his utility is now represented by *V(x _{1},x_{2},…,x_{i},m*) and that this utility function yields convex utility varieties.

Let us now assume that the utility curves of Figure 9 are those of *V(x, m).* Let *m = OM* be the money at the disposal of the individual, and let *MA* be the budget line corresponding to some price, *p,* of the commodity C, measured on *Ox*. The individual’s demand for C is *Oa,* and *Ob* is his reserve demand for money to be spent on other commodities. An increase in *m,* i.e., a parallel shift of *MA* to *M*_{1}*A*_{1}, normally increases the demand for C. But, as shown in Figure 9, this demand may decrease. In this case, *C* is said to be an “inferior” commodity. If *m* remains constant and *p* decreases, the new budget line is *MA′.* Again, the demand for *C* normally increases, but in some exceptional cases it may decrease. These cases, first brought to light by R. Giffen, were considered paradoxical because they contradicted the Cournot-Marshall law of demand. Clearly, they can occur only if *C* is an inferior commodity, but this is not a sufficient condition.

Two general relations between the changes in demand of the sort just considered were first established by Slutsky (1915). In a different form, they were rediscovered by Hicks and Allen (1934), who also completed the interpretation supplied by Slutsky. Moreover, Hicks (1939) offered a very simple graphical proof of Slutsky’s first relation. In Figure 9, the change in the demand for C when money increases is the “income-effect,” (Δ*x*)_{m} = *aa”;* the change in the demand for *C* when *p* decreases is the “price-effect,” (Δ*x*)_{p} = *aa”.* The price-effect can be decomposed into two movements: from B to B’ and from B’ to *B".* The second movement, which takes place on the same utility curve, is a substitution of C for money. Hence, the corresponding change in *x* is called the “substitution-effect,” (Δ*x*)_{s} = *a′a”.* The elementary form of the first Slutsky relation is immediately apparent from the graph:

Noting that because of the convexity of the utility curves Δ*x*)_{s} is always positive if *p* decreases, from (10) we obtain the general law of demand for a decrease in p: If *C* is a normal commodity, i.e., if (Δ*x*)_{m} < 0, then (Δ*x*)_{p} > 0. If *C* is an inferior commodity, i.e., if (Δ*x*)_{m} < 0, then (Δ*x*)_{p} may be positive, negative, or zero, depending on the relative magnitudes of (Δ*x*)_{m} and (Δ*x*)_{s}. Demand, therefore, may increase with price if the commodity is a strongly inferior good—as bread is for the poor. Of course, if one neglects the income-effect—as Marshall implicitly did in his analysis of demand—then (Δ*x*)_{p}, = (Δ*x*)_{s} is always positive.

To determine the effect of a decrease in *p* on the demand of a commodity C′, other than *C,* all we need do is visualize the same argument applied to a three-dimensional map representing the utility varieties of *V(x, y, m).* We thus obtain a relation analogous to (10):

which is the elementary form of the second Slutsky relation. However, in contrast with (Δ*x*)_{s}, the cross-substitution-effect (Δ*y*)′_{s} may have any sign. Using a contour map of the utility variety *V(x,y,m)* = constant for money levels *m*, one can show, however, that the two cross-substitution-effects, (Δ*x*)′_{s}, and (Δ*y*)′_{s});, have the same sign. On this property Hicks and Allen based a new definition: C and *C’* are complementary or competitive according to whether this sign is positive or negative.

It can be shown that the Hicks-Allen definition is *formally* identical to the Edgeworth-Pareto definition. Both definitions say in essence that two goods are complementary or competitive according to whether the individual values two increments taken together more or less than he values them taken separately. The only difference is that in the Edgeworth-Pareto definition “values” refers to *utility,* whereas in the Hicks-Allen definition it refers to the amount of *money* that the individual is just willing to pay for an additional increment (Georgescu-Roegen 1952). It should be emphasized, however, that the two definitions are not equivalent. They are so only if money measures utility, i.e., only if the marginal utility of money is constant.

### Ordinal utility and binary choice

The small volume published in 1886 by an Italian engineer, Antonelli, in which he presented a theory of consumer behavior without resorting to any measure of utility, must be considered one of the most important milestones in utility theory. Fate willed that Antonelli’s work should remain largely ignored in spite of the high praise it received from a few writers, and until recently, the glory of being the first to hint at the nonnecessity of cardinal utility has belonged to Fisher (1892). Fisher’s hint and Antonelli’s work inspired Pareto to develop two distinct theories, each capable of replacing Edgeworth’s. One of these is now known as the theory of choice.

Pareto argued that it is unnecessary, as well as unwarranted, to assume that the individual is capable of attributing a definite utility quantum to commodity combinations. All we need to (and can) say is that if the individual is asked to choose between two bundles of commodities, *M*_{1} and *M*_{2}, he always either chooses the same bundle or is indifferent as to which one he gets. This “postulate of binary choice” obviously does not imply the cardinality of utility. But Pareto, as was often the case with him, summarily concluded that if we have enough answers from the individual, we can represent his preferences by a map of indifference curves (varieties) having the same basic properties as the utility curves of Edgeworth: the individual is indifferent as to whether he has M, or Ma if and only if *M*_{1} and *M*_{2} are on the same indifference variety; he prefers *M*_{1} to *M*_{2} if and only if *M*_{1} is on a higher indifference variety than *M*_{2}. Granting, however, the existence of the indifference varieties, one can rank them in the order of the individual’s preference. Pareto proposed referring to any ranking of this sort as an “ophelimity index.” Once such an index, *ϕ(M),* is chosen, then

depending on whether *M*_{1} is preferred, considered indifferent, or nonpreferred to *M*_{2}—and conversely. Like any index, ϕ is to a large extent arbitrary. If ϕ*(M)* is an ophelimity index, so is any *arbitrary* increasing function *F(ϕ).* For example, if then F = ϕ^{2} + 2 = *xy* + 2 is another ophelimity index: in both cases all the points on the curve *xy = c* have the same index, which is lower than that of the higher indifference curve *xy* = c′ > c.

As we now say, *cardinal* utility is replaced by *ordinal* utility, i.e., by ophelimity. Or, to put it differently, utility can be ordered, but it cannot be measured in the same sense in which length or mass can. The nature of an ordinal variable is illustrated by the Mohs scale for hardness. On that scale diamond and talc have the indexes 10 and 1, respectively; however, this does not mean that diamond is ten times harder than talc in any conceivable sense. Obviously, the Mohs scale can be replaced, if we so wish, by any other arbitrary but increasing sequence of ten numbers.

Let us write *M*_{1}*eM*_{2}, to indicate that Af, is chosen from the pair (*M*_{1}, *M*_{2}). If *M*_{2}*eM*_{1} also applies, we have the indifference case, for which we can write either *M*_{1}, *lM*_{2} or *M*_{2}.*lM*_{1}. If only *M*_{1}*eM*_{2} applies, then *M*_{1}, is preferred to *M*_{2} and we write *M*_{1}*pM*_{2}. Assuming now that for three different combinations we had *A*_{1},*PA*_{3}, *A*_{2},*PA*_{3}, *A*_{3},*PA*_{1}, we could not possibly represent them by an ophelimity index, for no numbers can satisfy the inequalities ϕ(*A*_{1}) > ϕ(*A*_{2}), ϕ(*A*_{2})> (*A*_{3}), ϕ(*A*_{3}) > (*A*_{1}). Pareto’s theory, therefore, should also include the postulate that preference is always transitive, i.e., if *A*_{1}*PA*_{2} and *A*_{2}*PA*_{3} then *A*_{1}*PA*_{3} The usual justification for the transitivity postulate is that it represents the “rationality” of the individual’s choice.

But even the transitivity condition is not sufficient for establishing an ophelimity index. One needs in addition the “indifference postulate,” which asserts the existence of indifferent combinations or, what comes to the same thing, *perfect* substitutability in choice. For a most intuitive introduction of this new idea, we may refer first to another elementary (and necessary) proposition: if *M*_{1} ≠ *M*_{2} are *regular* combinations (see Figure 1 and the discussion adjoining it) and if no commodity is contained in *M*_{2} in a smaller amount than in *M*_{1}*PM*_{2} then *M*_{1}*PM*_{2}. A simple form of the indifference postulate states that given a combination M such that *MPM*_{2} and *M*_{1}*PM* there exists on the segment *M*_{1}*M*_{2} a combination *N* indifferent to *M* (Georgescu-Roegen 1936*b,* pp. 136 ff.).

For a long time this postulate has been (and it often still is) treated as a logical necessity, on the grounds that one cannot pass from nonpreference to preference without passing through indifference. But by the same token, there should be an intermediary state between preference and indifference, and so on. The indifference postulate, therefore, is not a tautology: it is conceivable that it may not fit the facts.

The three postulates mentioned above suffice to prove the following: (1) the loci of combinations indifferent to a given *M* is a curve (surface) passing through M, (2) all combinations on such a curve are indifferent to each other (transitivity of indifference), (3) the indifference curves do not intersect. It is customary to make the additional assumption that in the normal region the indifference curves are convex toward the origin, an assumption which Hicks and Allen (1934) translated as the “principle of decreasing marginal rate of substitution.” However, as Hicks (1939) observed, no transparent property of choice seems to support it (but see Georgescu-Roegen 1966, pp. 60 ff., 188 ff.).

Thus completed, the theory of binary choice provides us with a map of indifference curves structurally identical with that of utility curves (Figure 8). Moreover, all propositions concerning the consumer’s demand (or supply) remain true because even in the case of measurable utility the various effects are movements in the commodity space, not on the utility hill. This is true, in particular, of relations (10) and (11). Naturally, we can no longer speak of utility, including the utility of money, but the only important analytical loss is that we can no longer have a grid of equally distant indifference curves and, hence, the Edgeworth-Pareto definition of related goods becomes useless. The merit of the Hicks-Allen definition is that it works for ordinal utility as well.

It seems hard to accept that in his choices the individual is like a perfectly accurate instrument. Choice must be affected by a psychological threshold. Confronted, at various but small intervals of time, with *M*_{1}, and *M*_{2}, the individual will on some occasions prefer *M*_{1} on others *M*_{2}. The structure of the individual’s choice is then described by a binary function, ω(*M*_{1},*M*_{2}), representing the probability that he will choose *M*_{1}. Clearly, ω(*M*_{1},*M*_{2}) + ω(*M*_{1},*M*_{2}) = 1. But it should be emphasized that since no individual cares which of two *identical* combinations he gets, ω(*M, M*) is completely indeterminate. Indifference and preference can now be *defined* with relation to the value of ω(*M*_{1},*M*_{2}). For example, indifference may be defined as either ω(*M*_{1},*M*_{2}) = ½ or ω(*M*_{1},*M*_{2} ≠ 0,1. Various postulates have been suggested for this structure (e.g., Georgescu-Roegen 1966, pp. 228-240), but the structure is apt to become very complicated as we pass to multiple choice. For example, *M*_{3} may be the most frequently chosen among (*M*_{1}, *M*_{2},*M*_{3}), even though *M*_{1} and *M*_{2} are more frequently chosen than *M*_{3} in the binary choices (*M*_{1}, *M*_{3}), (*M*_{2}, *M*_{3}).

### Directional choice

The second method suggested by Pareto for constructing an ophelimity index from behavioral data is unmistakably inspired by Antonelli’s idea. The data are the prices and the quantities demanded, as observed in the market. Pareto’s argument again rides on extraneous positions, yet his idea contains the seeds of two entirely new approaches to consumer behavior—the theory of directional choice (Georgescu-Roegen 1936*b*; 1954*a*) and the theory of revealed preference (see below). In the theory of binary or multiple choice, man’s choice is analogous to that of a bird which, after surveying from above a large piece of ground, dives directly at the most preferred spot. In the theory of directional choice, man’s choice is rather like that of a worm which, from any position, chooses some direction and then moves along it. There are reasons to believe that the latter approach may be the more realistic.

From market data we can find out the optimal allocation, *M,* of each budget *AB* (Figure 10). If the budget does not cover a point of absolute saturation, it is clear that any direction below the budget line is a *nonpreference* direction and, if “kinky” structures are barred, that any direction ' above the budget line is a *preference* direction. There remain the *limiting* directions and If the extreme case of perfect substitutability is left out, they, too, must be nonpreference directions— a clear-cut case in which no indifference state exists between nonpreference and preference.

To allow for a saturation point, we should also consider the case of a point such as M’. But if the individual’s budget allows him to buy AT, he will choose the saturation point S and throw away part of his money. Therefore, in order to obtain the same information for M′ as for M, we have to oblige the individual to spend all his money on commodities.

The ground is thus prepared for introducing the axioms of directional choice:

(1) To any regular point, i.e., a point that is not a point of saturation, there corresponds a linear (in general, a planar) element such that any direction in one of the closed half-planes determined by it is one of nonpreference, all others being preference directions.

(2) All directions away from a saturation point are nonpreference directions.

(3) If is a limiting direction for *M* and if *K* is any point on the half-line *MB,* the same direction is a nonpreference direction for *K.*

Let Σ,L_{i}(*X*_{i}— *x*_{i}) =0’ be the equation of the planar element corresponding to *M*(*x*_{1}, *x*_{2},…, *x*_{n}), where the *L*_{i} are chosen so that the direction ', *M*′ = *M* + ΔM, is a preference, limiting, or nonpreference direction, depending on whether

Assuming that the *L*_{i} are differentiable, from Axiom 3 we obtain

This is known as the convexity condition of the planar elements. It further yields

which means that if is a nonpreference direction for M, the same direction is a nonpreference direction for any point *N* on the half-line *MN.* This is the “principle of persisting nonpreference.” From it and Axiom 2 (in the strong form stated above), it follows that, S being a saturation point, a preference direction for any *M*. The corollary is that at a finite distance there can be only one saturation point.

For the case of two commodities, by a mathematical operation called the integration of the linear elements we can construct a family of “integral curves,” such that the tangent to any curve at any of its points is the linear element of that point. Through each ordinary point passes at least one integral curve. From (13) it follows that the integral curves are everywhere convex; but as to shape, they may be nonintersecting, closed curves around S, as in Figure 8, spirals converging asymptotically toward S, half-curves originating in *S,* as in Figure 10, or curves that meet in the pattern of Figure 13 (see below).

For a bridge between directional and binary choice, we may adopt the following reasonable convention: if the individual can move from *A*_{2} to *A*_{1} on a path going always in a preference direction, we shall say that *A*_{1} is directionally preferred to *A*_{2} and write *A*_{1}*P _{D} A*

_{2}. In the case of Figure 8, if

*A*

_{1}is on a “higher” integral curve than

*A*

_{2}, then

*A*

_{1}

*P*and conversely. We can then attribute to every integral curve a ranking number increasing toward S. But this ranking,

_{D}A_{2}*ψ*(M), is only a

*pseudo*index of ophelimity, because the preceding axioms and convention imply nothing about the ranking of two combinations on the same integral line, i.e., about the case where ψ(

*M*

_{1}= ψ((

*M*

_{2}). Integral curves are not necessarily indifference curves. The point is that the indifference postulate does not fit into the present scheme; all that demand data can reveal is preference.

Turning to Figure 10, we see that *M*_{1}*P _{D} M*

_{3}and also

*M*

_{3}

*P*

_{D}M_{1}. This contradiction proves that our convention may provide no basis even for ordering all combinations into a “preference” chain; we cannot, then, speak of a pseudo index of ophelimity, because there is no way to decide which of two integral curves is “higher” than another. The whole truth is that we cannot even say that two combinations are comparable.

In the Italian edition of the *Manuel*(1906), Pareto asserted that one can always determine the indifference varieties by integrating the planar elements. In his review the Italian mathematician Vito Volterra reminded Pareto that for more than two commodities the planar elements may not be integrable and, hence, the integral varieties do not always exist. The “nonintegrability problem” was thus born. With it, a paradox emerged: why can an ophelimity index be analytically constructed from demand data for two commodities but not for more than two? All this goes back to Volterra’s criticism, which overlooked the fact that integrabil-ity does not suffice to establish an index of ophelimity. The necessary and sufficient condition is that the integral varieties should be closed and nonintersecting (as in Figure 8) or, in mathematical terms, should constitute a potential. As illustrated by figures 10 and 13, a potential may not exist even in the case of two commodities. And since only in the case of a potential is the directional preference transitive, the integrability issue is “without any meaning outside the transitivity condition” (Georgescu-Roegen 1936*b*).

Pareto subsequently tried to explain the nonintegrability case by arguing that the ophelimity index depends on the order of consumption, i.e., on the path followed from one position to another. By bringing in the order of consumption, he missed an excellent occasion to solve the puzzle: as he himself admitted, ophelimity is a function of quantities possessed, not consumed. For some time thereafter, opinions about the nonintegrability case were strongly divided: some treated it with concern, others regarded it as a “will-o’-the-wisp.” More recently, models of demand theory without an ophelimity ordering have formed the subject of numerous studies, some of which—as pointed out by P. K. Newman and R. C. Read (1958)—contain the ordering assumption implicitly. The simple diagram of Figure 10 suffices to show that demand theory does not require even binary comparability. To see the generality of this statement, let us take a budget plane in a space of three commodities. In this plane there is a system of linear elements. Their integration yields a picture identical to that for two commodities: convex integral curves and a saturation point. The only difference is that now this point is one of *relative* saturation and coincides with the optimal allocation of the budget. And if the planar elements in the three-commodity space are not integrable, the integral curves in the budget plane are, in general, not closed and hence there is no basis for comparability.

## Revealed preference; multiple choice

Searching for a theory of consumer’s behavior freed from any “vestigial traces of the utility concept,” Samuelson (1938a) developed a new approach, known as “revealed preference.” The basic idea is essentially the same as that of Antonelli and Pareto, namely, to use the objective data of demand as a foundation for the ophelimity concept (cf. Samuelson 1947, pp. Ill, 145-154). However, the distinctive merit of Samuelson’s approach is that his point of departure consists of a single and highly transparent proposition involving only finite terms.

Let be an optimal allocation of a certain budget, . Let *M*_{2} ≠ *M*_{1} be a combination that does not cost more than *m*_{1}, i.e., a combination such that

The fact that the individual chooses *M*_{1} even though he can also buy *M*_{2}*reveals* that he prefers *M*_{1} to *M*_{2},. We may then write *M*_{1}*P _{R} M*

_{2}. The postulate introduced by Samuelson is now known as the “weak axiom of revealed preference": If

*M*

_{2}, is revealed to be preferred to

*M*

_{2}, then

*M*

_{2}cannot be revealed to be preferred to

*M*

_{1}.

Using *f*:*A* to denote that proposition *A* is false, we can express this axiom as follows:

In algebra this means that if (15) is true, then

where p_{1}^{(2)}, p_{2}^{(2)}, ..., p_{n}^{(2)} are the prices for which *M*_{2} is the optimal allocation of the budget . With this postulate alone, Samuelson proved that demand is uniquely determined for any budget data and that demand depends only on the ratios *P*_{i}/*m*.

It is immediately clear that the budget plane for which *M*_{1} is the optimal allocation is the planar element of *M*_{1} and that the relation *M*_{1}*P _{R} M*

_{2}is a special case of

*M*

_{1}

*P*

_{D}M_{2}. Revealed preference, therefore, is a

*finite*formulation of the theory of directional choice. The only difference is that Samuel-son’s postulate implies that the optimum allocation always exhausts the budget; obviously, this eliminates any point of absolute saturation at a

*finite*distance. Without this special condition, his proof (1948) that in the case of two commodities the integral curves separate the plane into “better off” and “worse off” regions, just as the indifference curves do, would not be valid.

From the weak axiom, Samuelson derived the same convexity condition as that established for directional choice. The important point is that he expressed this geometrical condition in an economically more transparent form. Applying (15) and (17) to *M P _{R} M*′,

*M*′ =

*M*+ Δ

*M,*Samuelson straight-forwardly derived

and the simpler corollary

both relations being valid for any *finite* displacement Δ*M*. In case the *p*_{i} are differentiable, (19) yields

The preceding relations, it should be emphasized, are valid whether or not integral varieties exist. But if they exist, (20) states that they are everywhere convex. Conversely—a point to remember— if they are convex, the weak axiom is true. Another interpretation of (20) is that for any infinitesimal displacement on an integral variety, we must have Σ*d P*_{i}*dx*_{j} < 0. Limiting himself to the case in which the indifference varieties exist, Hicks (1939) saw in this inequality the general law of demand. Subsequently, Samuelson proved that this law is valid even for finite displacements on an indifference variety:

This result should not be confused with (19), which applies to displacements in the budget plane of *M*.

The concept of revealed preference presents a few slippery points, about which Samuelson tried to warn us. The first is that revealed preference, like any theory based on preference alone, can never arrive at a criterion of indifference. As he put it, integral lines, if they exist, can only “by courtesy” be given the title of indifference curves. The second point concerns the negation of revealed preference. In the theory of choice the negation of *M*_{1}*M*_{2}always entails *M*_{2}*M*_{1}. But as can be seen on a diagram of budgets, *M*_{1} and *M*_{2} may be such that neither *M*_{1}*M*_{2}nor hence, *M _{2}_{R} M_{1}* the negation of one relation does not entail the other. This case proves that revealed preference, too, does not lead to general comparability.

To illustrate some of the above points, Samuel-son considered the situation where preferences are already ordered by an ophelimity index. It is obvious that in this case (a pseudo index would also do) revealed preference reveals ordinary preference:

However, the implication is not reversible: ϕ(*M*_{1}) > ϕ(*M*_{2}) does not entail *M*_{1}_{R}*M*_{2}; all it entails is that *M*_{2}_{R}*M*_{1} is not true. Samuelson further observed that *M1*_{1}_{R}*M*_{2}, *M*_{2}_{R}*M*_{3} yields ϕ(*M*_{3}) < ϕ(*M*_{1}), which we may generalize by writing

He also came to talk about “pseudo transitivity,” i.e.,

which within the same framework can immediately be extended to

However, Samuelson stops short of saying anything about the status of (23) or (25) in a *pure* theory of revealed preference. He speaks of *M*_{3} as being “revealed to be better” than Mt (1948, pp. 247 ff.), but he is not explicit on whether he has in mind (23) or a new definition of “indirect revealed preference,” based on the relation

That (25) is not a consequence of (16) can be immediately shown if we recall that for three commodities the optimal allocation is a point of *relative* saturation in relation to the integral curves in the budget plane and take the case where these curves are those of Figure 10. All that is required by the weak axiom is that the integral curves be convex. And since the budget planes for *M*_{1}, *M*_{2}, *M*_{3} must be tangent to the corresponding integral curves, we see immediately that *M*_{1}_{R}*M*_{2}, *M*_{1}_{R}*M*_{3}, and yet, *M*_{3}_{R}*M*_{1}. True, one can nonetheless combine, as Uzawa (1960) did, the weak axiom with some other axioms so as to imply (25).

The French mathematician Ville (1946), moving on the same path as Samuelson, was first to formulate (25) as an axiom and to show how it leads to a preference map equivalent to that of the theory of binary choice. In the Anglo-American literature the same feat was achieved, independently, by Houthakker (1950). Obviously, condition (25), now known as the “strong axiom of revealed preference,” generalizes the weak axiom.

Some points concerning the weak and the strong axioms deserve emphasis. First, since the weak axiom is equivalent only to the convexity condition, it cannot imply the strong axiom. It does not even deny the existence of a point of absolute saturation, unless the exhaustion of the budget is implicitly assumed. Moreover, this denial would be insufficient for the existence of a preference ordering for more than two commodities. Second, the strong axiom entails more than integrability but less than the existence of an ophelimity index. And since it says, in essence, that one cannot start from a point and return to it moving always in a revealed preference direction, it appears that, after all, Pareto was not a fool in trying to relate the problem of integrability to a shift in the path between two combinations.

Revealed preference theory is a special case of what we may call a theory of multiple choice: its basic postulate is that confronted with the set B of all combinations that can be bought with a given budget, Σ*P*_{i}*x*_{i} ≤ *m,* the individual chooses consistently. But his choice need not be unique. Consistency means only that the combinations that he may choose form a *nonnull* subset, *C*(*B*), of *B*. Samuelson’s theory is a *special* theory of multiple choice, because it considers only a special class of sets, those determined by a budget. It is also a *pure* theory of multiple choice, because it does not include binary choice (at least, as long as commodities are assumed continuous).

In 1954 Arrow suggested a general formalization of a theory of multiple choice (see Arrow 1959 and references given there). Let *X, Y,* … denote the sets belonging to a certain family, *J*. The first axiom is that the combinations the individual may choose from set *X* form a nonnull subset, *C*(*X*). Any *X* is thus divided into two separate subsets, the choice set, *C*(*X*), and the set *X* - *C*(*X*). We can then define revealed preference, *M _{1}_{R}M2,* to mean that for

*some X*we have

*M*

_{1}∈

*C*(

*X*),

*M*

_{2}∈

*X*-

*C*(

*X*). The equivalent form of the weak axiom is that if

*M*

_{1}

_{R}

*M*

_{2}, then

Arrow considered the case in which *J* consists of all finite sets. In contrast with Samuelson’s, this approach is not a pure theory of multiple choice; it includes the binary choice between any *M*_{1}, *M*_{2}. The point bears upon Arrow’s proof that the weak axiom suffices for ordering all combinations. Indeed, this may no longer be true if *J* includes only multiple choices, as is shown by the following analytical example, which satisfies even the strong axiom:

C{M,N,L,K} = {M,N},C{M,N,L} = {M,N},C{M, N, K} = {M, N},C{M,L,K} = {M},C{N,L,K} = {N}.

The individual always chooses, but the preferences revealed by his choices do not lead to a comparison between *M* and *N* or between *L* and *K.*

## Utility and its measure

In spite of the repeated blows suffered by the Bentham-Edgeworth position, economists kept hoping that utility might after all be measurable in some sense other than mere ordering. Some endeavored to derive a utility measure from the objective data of consumption; others searched for reasons other than Edgeworth’s for the existence of a cardinal scale. The pioneering works of Frisch (1926; 1932) and Fisher (1927) practically exhaust the first category. Both authors arrived, independently, at the idea of measuring the marginal utility of money with the aid of consumption data of an “auxiliary” commodity, *C*_{1}. Their first assumption is that marginal utility, not necessarily utility itself, is cardinally measurable; the second, that the marginal utility of *C*_{1}, is a function of the amount of *C*_{1} alone; and the third, that the marginal utility of money is of the form

where *P* is a cost-of-living index. The budget system (4) then yields

where *p*_{1}, is the price of the auxiliary commodity, α = *P*/*p*_{1}, the inverted deflated price, and r = *m/P* the “real income.” In the space of coordinates r, α, and *x,* relation (29) represents a surface, which Frisch called the consumption surface. In Figure 11a the intersection, *AB,* of this surface with a plane parallel to *xOα* at the distance r_{0} represents the inverted demand for Ct corresponding to the real income *r*_{0}, i.e., it represents the relation between *x* and the reciprocal of the deflated price of . α = *P*/*p*_{1}. The intersection, *AC,* with a plane parallel to *xOr* at the distance *α*_{0} represents the

income demand for C_{1} for the corresponding deflated price, *C _{1}, a=P/p_{1}*. These last intersections are, therefore, Engel curves (Schultz 1938). Finally, there are the intersections, such as

*DE*and

*D’E’,*of the same surface with horizontal planes at various values of

*x.*Projected onto the

*αOr*plane, they yield the curves of Figure 11

*b*. Because of the relation

*ac*/

*ab U*—(

*x*

_{1})/

*U*—(

*x*

_{2}), all these curves can be derived from any one of them by uniform expansion of the vertical scale of the figure. In particular, if

*x*is chosen so that

*U’(x)*= 1, the curve is

*α = w(r).*With enough observations, one can determine the consumption surface by interpolation and thus obtain

*w*(

*r*) and, from (28), the marginal utility of money. Both Fisher and Frisch used aggregate consumption data on

*C*

_{1}, instead of data for the

*same*individual as required by the theoretical scheme. The procedure is based on the fact that the preferences of people living in the same environment and having comparable incomes do not vary much. The whole scheme met with strong criticism (Allen 1933; Bergson 1936) which, curiously, concentrated on relation (29), not on the assumption of measurability. Although it was a most ingenious econometric piece, the Fisher-Frisch contribution had a short-lived vogue and produced no imitators.

Frisch endeavored to justify the measurability of marginal utility by an idea with which Pareto first played. Pareto observed that if the individual is able not only to compare two different combinations but to compare, as well, the utility interval between *M*_{1}, and *M*_{2}, with that between *M*_{2} and *M*_{3}, and if, also, he can always say for what *M*_{3} the two differences are exactly equal—a possibility in which Pareto did not believe—then, among all possible systems of ophelimity indexes, we should retain only those that display the same differential order as that felt by the individual. Pareto’s remarks on this point indicate that he had a better understanding of what measurability of utility means than many subsequent writers on the topic.

Much of the controversy of the last forty years has fed on the ambiguous use of the term “measurability.” A most enlightened discussion was, however, presented by E. H. Phelps Brown and H. Bernardelli as early as 1934. These authors reminded us of the well-known epistemological distinction between extensive magnitudes, which have homogeneous parts, and intensive magnitudes, which have no parts at all. “Difference,” therefore, cannot have the same meaning in both cases. Even though words can be ordered in a dictionary, “the difference between two words is not itself a word,” the implication being that the difference between two utilities is not a utility. If ophelimity indexes are replaced by ophelimity words—"cow” for M,, “energy” for *M*_{2}, “grate” for *M*_{3}—for “differences” we must use another class of indexes, say, French words. Given that the “difference” between *M*_{2} and *M*_{1}, is “equal” to that between *M*_{3} and *M*_{2}, we may assign to each the index *maison.* To the interval from *M*_{1}, to *M*_{3} there also corresponds a definite word, say, *melon.* But without some strong additional assumption we cannot conclude that *maison* plus *maison* is *melon.*

Most illuminating though this argument is, it may leave the impression that no intensive magnitude can have a measure that is not purely ordinal. The familiar counterexample is the measure of the feeling of temperature, but a more instructive one is provided by the chronological time. Certainly, time is not a cardinal variable. No meaning can be attached to subsuming chronological dates or multiplying them by a number. Like the Paretian ophelimity, they can only be ordered; yet there is the feeling of the passage of time between two dates. The question is whether a person can say that the amount of time that has passed between breakfast and lunch is more than, less than, or equal to the amount of time between tea and bedtime. It is similar to the question of whether the individual can compare the feeling of passing from *M*_{1} to *M*_{2}, to that of passing from *M*_{3} to *M*_{4}. The difference is that for time intervals we do have an instrumental measure—that provided by the special mechanisms called clocks. However, there is no basis whatever for believing, as Newton did, that one clock-hour contains in some absolute sense the same amount of “time” as another clock-hour. Consequently, the fact that time intervals have an instrumental measure implies neither that the individual can compare such intervals nor that, if he can do so, he feels that a two-hour interval is twice as long as a one-hour interval. [Scales *of measurement are discussed further in* Psychometrics*and* Statistics, descriptive.]

The above points bear upon Pareto’s argument, further elaborated by Lange (1934), that the simple comparability of ophelimity intervals suffices to construct an ophelimity scale similar to that of the thermometer. The argument seems to enjoy great popularity in spite of Samuelson’s proof of its insufficiency (1938c). Samuelson’s argument focuses on the point that without some sort of a *metric* axiom there can be no question of a stronger measure than the ordinal one. If some appropriate metric axioms are added to the assumption that an ophelimity index exists, we can prove the following:

(*a*) There exists an ophelimity index *ϕ ^{*}_{0}* such that

implies that the utility interval [*M’, N’*] is not greater than [*M, N*].

*(b)* If ϕ*is another ophelimity index satisfying this same condition, then, *a* and *b* being some parameters, we have

The scale determined by ϕ*, it should be emphasized, is not a cardinal scale: such a scale requires that . It is, however, analogous to that for temperature and for time: it depends on the unit of measurement *and* on the origin of the scale. We may refer to it as a “weak cardinal scale,” to reflect the fact that a cardinal scale exists for the *intervals,* for according to (31),

One should, however, be on guard against one unwarranted conclusion. The fact that we can compute

the ratio, *r,* between *ϕ*^{(M) — ϕ(N) and ϕ(M′) — ϕ(N′) and then write symbolically [M, N] = r[M′, N′]does not prove at all that this paper-and-pencil operation has any meaning for the individual concerned. Should he feel that two successive intervals represented by maison are equivalent to the interval melon, this equivalence ought to be stated as an axiom from the outset. But then, all other axioms would become superfluous.}

Another idea leading to a measure of utility was the root of the first formulation of the principle of decreasing marginal utility. The earliest discoverers of this principle were confronted with a puzzle known as the St. Petersburg paradox. In gambling, a game is “fair” if the stake is equal to the gain multiplied by the probability of winning, i.e., in general, equal to the mathematical expectation of the gambler, which is Σ*p _{i}g_{i},* where

*g*are the mutually exclusive gains, p; is the probability of

_{1}, g_{2},…,g_{n}*g*and Σ

_{i},*p*= 1. Let us take the simple game in which a fair coin is tossed twice and the rules are as follows: if tails shows on the first toss, the gain is 1 and the game terminates; if tails shows on the second toss but not on the first toss, the gain is 2; if both tosses show heads, the gain is 2

_{i}^{2}. The mathematical expectation is l × (½) + 2 × (¼) + 2

^{2}× (¼) = 2, a result which seems within reason. But let the game be generalized so that it ends when tails occurs for the first time; if this happens on the nth toss, the gain is 2

^{n-1}. In this case the mathematical expectation is

which is infinite. The fact that no gambler would be willing to bet even an enormous sum at this game constitutes the St. Petersburg paradox. Seeking a solution of the paradox, Buffon, Cramer, and Bernoulli were led to observe that the “moral fortune” (the utility of wealth) is not measured by the number of “dollars.” For the utility of wealth, Bernoulli proposed his famous formula *U*(*x*) = *b* log(*x/a*), where *a* is the wealth initially possessed by the gambler and *b* a characterizing parameter. The “moral expectation,” i.e., the mathematical expectation in *utility units,* of the St. Petersburg game then is

A simple mathematical proposition shows that *E _{n},* in contrast to

*E*has a finite value (provided that a ≠ 0). The value of the game in

_{m},*money terms, m*is given by the equation log [(a +

_{0},*m*)/

_{0}*a*] =

*E*; hence,

_{n}*m*is finite. This result may be restated as follows: an individual who neither likes nor abhors gambling must be indifferent as to whether he receives

_{0}*m*dollars or a ticket in a St. Petersburg lottery.

_{0}Bernoulli’s basic idea can be analyzed into two general postulates of increasing strength. Let us denote a *risk* combination by

where M_{i}M_{i+1}, Σ*P _{i}* = 1, and

*P*(> 0) is the probability that the individual should get the sure commodity combination

_{i}*M*. The postulate is that for any risk combination, there is an

_{i}*M*such that M

_{1}M, M M

_{n}, and

A stronger form of this postulate, which presupposes the existence of a utility measure for all sure combinations, replaces (36) by the formula of “moral expectation,"

Bernoulli used this relation to determine *M* in a problem where *ϕ* was already given. Ramsey (1923-1928) reversed the problem: he used (37) as a basis for constructing a weak cardinal measure for utility, an idea also used, more recently, by von Neumann and Morgenstern (1944). It is obvious that (37) is the *metric* relation necessary to restrict the arbitrariness of the ophelimity index; for example, (37) cannot be true for both and *ϕ* and *ϕ ^{2}*

An elementary illustration of the procedure for constructing an ophelimity index satisfying (37) is offered by the case of a single commodity, say, money (Friedman & Savage 1948). Let *m′* < *m″,* and let us put *W*(*m′*) = 0, *W*(*m″*)= 1. For any *m _{1}, m′* <

*m*<

_{1}*m″,*there is a

*p,*<

*p*< 1, such that (1 -

*p*)

*W*(

*m′*) +

*pW*(

*m″*) =

*W*(

*m*), and conversely; hence,

_{1}*W*(

*m*) =

_{1}*p*. Similarly, for

*m*<

_{2}*m″*there is a

*q,*0 <

*q*< 1, such that (1 -

*q*)

*W*(

*m′*) +

*qW*(

*m*) =

_{2}*W*(

*m″*); hence,

*W*(

*m*) = l/q. Clearly, the ophelimity index thus constructed depends upon the chosen origin,

_{2}*W*(

*m′*), and the chosen unit,

*W*(

*m″*) -

*W*(

*m′*). It can be further shown that if ϕ

^{*}

_{0}satisfies (37), then any other such index

*ϕ*satisfies , and conversely— a result identical to (31). It is thus seen that this second approach is essentially equivalent to that based on the comparability of ophelimity intervals. Actually, Ramsey arrived at a weak cardinal measure of utility by first establishing the comparability of ophelimity intervals. However, the great advantage of the approach based on risk is that its axiomatic foundation is far more obvious and also more convenient experimentally than that based on the comparability of intervals. Perhaps the best illustration of this advantage is the system of axioms proposed by Marschak (1950).

^{}The analogy between the measure of temperature and that of utility does not seem as close as von Neumann and Morgenstern, among others, want us to believe. The thermometer and the clock dispense man from measuring temperature and time intervals by *his own scale.* For utility there is as yet no hedonimeter and, hence, no measure outside what man can compare and gauge. Even an economist such as Samuelson, to whom modern utility theory owes an immense debt, confesses that he is unable to construct a weak cardinal scale for his own utility. And when we find ourselves incapable of doing what the cardinal theory claims that we can do, certainly something must be wrong with that theory.

Various authors came to believe that this paradox of the cardinalist doctrine has its roots in one of the assumptions which are required, in addition to (37), for the complete proof of the existence of a weak cardinal scale and which pertain to the ordering of *all* commodities. Samuelson, for example, felt that the culprit is the axiom known as the “strong independence axiom”: If , and *p ^{'}_{i}* =

*p*for

_{i}*i ≠ l, k,*then

In view of the high transparency of this axiom, however, Samuelson’s conclusion should be suspect. There are good reasons for regarding, instead, the postulate (36) itself as the real culprit: this postulate extends the indifference postulate to *all* combinations and hence denies that risk adds an essentially different dimension to the object of choice.

The fallacy of the ordinalist resides precisely in not seeing that once he has sworn by the principle that between preference and nonpreference one must necessarily pass through indifference, he must accept (36) and therefore can no longer maintain that utility is a purely ordinal variable. In fact, for *both* theories, the indifference postulate is the only one that is really critical. It is also the only one that can be neither refuted nor confirmed by behavioral experiments, any more than the irrationality of V^ can be proved on the workbench. And if the indifference postulate is dropped, there is no longer any basis for ordering all commodity combinations by an ophelimity index.

### Utility and wants

The prominent position utility theory occupies in modern economics does not mean that before utility entered economics there was no theory of consumer behavior. The earlier economists analyzed this behavior in terms of needs or wants, an approach which even Marshall found worthy of attention. Actually, it was the early theorizing about wants that constituted the major source of inspiration for those who laid the foundation of utility theory in economics. The classic parable used by Menger for explaining his marginal principle refers to the *needs* of an isolated farmer: with a miserable harvest, such a farmer would satisfy only the most commanding need, to keep himself and his family alive; should the harvest exceed this minimum of subsistence, he would allocate the excess to filling his other needs in the order of their decreasing importance—seeds for the next season, fodder for his animals, mash for beverages, food for his pet parrot, and so on down the line. And no subsequent author, it seems, has been able to justify the principle of decreasing marginal utility without invoking the diminishing importance of the needs satisfied by the *same* commodity.

There is no denying that the structure of wants is not amenable to ordinary analysis. Wants are dialectical concepts, with blurred, not sharply drawn, boundaries. But this is no reason for refusing to describe and study them. After all, their structure is not completely amorphous. The first general and the most obvious feature of the structure of wants is their hierarchy. The idea goes back to Plato (*Republic* II, 369). Even Pareto implicitly recognized the hierarchy of wants by endeavoring to establish a hierarchy of commodities.

The hierarchy of wants can be further analyzed into a number of principles of a more intuitive character (Georgescu-Roegen 1954*b*). The “principle of the subordination of wants,” emphasized by T. C. Banfield and reformulated by Jevons, says that “the satisfaction of a lower want …permits the higher want to manifest itself.” It obviously implies Gossen’s law of satiable wants. Moreover, not only does a want have to reach satiety before the next want manifests itself, but also it appears that there is always a *next* want. The “principle of the growth of wants” states that the number of wants has no end.

The terms being interpreted with the flexibility appropriate to the case, we can distinguish three broad classes of wants. The first class consists of necessities, i.e., of those wants that pertain to the immediate requirements of maintaining life—water, food, rest, and shelter, in that order. These wants are common to all men. Above this class there are the conveniences, i.e., the wants which have the same hierarchy for all members of the same culture. Finally, in every society, people who can afford the satisfaction of all wants of the two classes already mentioned manifest wants for luxuries. Purely personal in character, these wants no longer have the same hierarchy for all individuals: one person may prefer a movie camera to a fishing rod; another may prefer a rod to a camera. The luxury wants are also less stable, even for the same individual.

Admittedly, this description of the structure of wants ignores many details which may become very important as the analysis is pushed further. The suggestion that wants form a tree structure instead of a linear chain (Strotz 1957) opens up a more realistic vista.

One principle, the “principle of the irreducibility of wants,” although the most critical of all, has received little attention. It is beyond question that no amount of food can save someone from dying of thirst and that no fancy jewelry can be a substitute for food. Curiously, even Pareto, the author of the indifference theory, mentions this irreducibility. Jevons, too, admits in one place that “motives and feelings are certainly of the same kind to the extent that we are able to weigh them against each other; but they are, nevertheless, almost incomparable in power and authority.” In fact, this very thought was expressed by Aristotle (*Ethica Nico-machea* 1133).

The upshot is that it is the principle of irreducibility, not the postulate of indifference, that should be a part of a realistic theory of choice. The consequences can be illustrated by a very simple scheme. Take an imaginary individual whose hierarchy of wants is food, taste, and social companionship and who lives by two commodities only, margarine and butter. Let us also assume that he prefers the taste of butter and that his saturation for food is *K* calories. Let both margarine and butter be measured in calories, *x*_{1}, *x*_{2}, respectively. For the choice between two combinations for which *x*_{1} + *x*_{2} ≤ *K,* i.e., which belong to the triangle *OAB* in Figure 12, only the most important want, that for food, counts. Any combination above *A′B′* is preferred to any combination, M, on *A′B′.* However, the want for food alone cannot decide which of the two combinations M and N—containing the same amount of food—should be chosen. In this case it is the next want in the hierarchy that decides: *N* because *N* contains more butter than M. Now, for the choice between two combinations that satisfy the want for food completely, taste becomes the primary criterion: any combination above *Bab* is preferred to *G* or *H.* However, *HPG* because the individual can entertain more if he chooses *H.* The same hierarchic choice can be extended to the case in which wants are not related linearly to commod

ities. In Figure 13, all combinations above *A′B′* are preferred to M, and *N P M;* also all combinations above *Bab* are preferred to *G,* and *H P G*.

The preceding schemes tell several stories. Because they satisfy all axioms of choice except the indifference postulate, they prove that this postulate, upon which the observations of the preceding sections cast substantial doubts, is completely unnecessary for a theory of choice. The curves, such as *A′B′* or *Bab,* separate the commodity plane into a preferred and a nonpreferred domain in relation to any combination lying on one of them. Borrowing a term from Little, we may call them “behavior curves.” But as is seen from figures 12 and 13, they may meet each other, a most important difference from the traditional indifference curves. Figure 13, which has been drawn so that *ba* is tangent at *a* to *BA,* proves another point: the fact that every combination is an optimal combination for only one

budget does not entail the nonintersection of behavior (integral) curves.

Last but not least, the hierarchy of wants throws overboard not only utility—which it originally bred —but also any ophelimity index. No such index can be constructed for maps like those in figures 12 and 13. To be sure, the hierarchy of wants completely orders all commodity combinations, but the order is a lexicographic ordering that cannot be reduced to a single ranking index.

### A critique of utility

The endeavors to explain economic value by a single “cause” have followed two trails. In the chronological order, it was Marx, in his labor theory of value, who first claimed to have discovered the same thing that Aristotle asserted exists in every exchangeable good. Marx did see that for such a claim it is necessary, first of all, to bring *all forms* of labor to a common denominator. And he went to great pains to convince us that any concrete labor power is only a particular form of the same “jelly”—general abstract labor. The founders of the utility theory, on the other hand, steamrollered over the parallel issue of whether every concrete want is only a particular form of a general abstract want—utility. For, in essence, this is the meaning of utility.

The fact that at one time economists held that utility is measurable, although no one could devise a hedonimeter, led them to defend their position by arguing that a cardinal scale exists only for the utility of an individual person, not for the utilities of all persons. The dogma of the interpersonal non-comparability of utility has ever since been strongly advocated. But if this dogma is accepted, economics must reconcile itself to being a science (perhaps the only one) unable to recognize at least a modicum of standards in the phenomenal domain which it purports to study. Fortunately, the dogma flies in the face of two irresistible forces: the faculty of man called empathy, without which “there is really no game we can play at all, whether in philosophy, literature, science, or family” and the hierarchy of wants. To be sure, the interpersonal comparison of wants does not always work. At the same time, it can hardly be denied that it makes objective *economic* sense to help starving people by taxing those who spend their summers at luxurious resorts. There is economic sense even in taxing the latter more heavily than those who cannot afford any luxuries. However, in view of the absence of any order among the luxury wants, there is no objective justification for taxing those who have motorboats and using the money to help others buy hunting equipment. The fact that the advocates of the interpersonal noncomparability of utility had in front of their eyes only a society of relatively high incomes is certainly responsible for their view.

The hierarchy of wants can also reveal and explain some phenomena that disappear under the colorless blanket of utility. The technical problems of a general economic plan are simplest in the case of an economy where the incomes of the overwhelming majority of people are so low that they barely cover the basic human wants. Since these wants are irreducible and rather inflexible, it is a simple problem to estimate the demand for consumer goods in this case. Even for an economy where a large stratum can enjoy many social but not individual wants, the same problem presents little difficulty once these wants and their hierarchy are known with sufficient clarity. It is only when the bulk of incomes can also satisfy some personal wants that the prediction of demand for “luxury” commodities becomes a real thorn for central planning. This is the deep-seated reason why such planning would work poorly in any advanced economy and also why the continuous rise in the average personal income in the Soviet Union has called for increasing decentralization in the production of “luxuries.”

Finally, another drawback of the utility theory accounts for the despair of most economists who have been called on to make policy recommendations for underdeveloped countries. The general complaint has been that the local people are “irrational,” because their choices do not conform to the basic principles of utility theory. But since these people conform to the Fisher-Pareto principle that “each individual acts as he desires,” how can they be irrational? The explanation is that utility theory passes, quite furtively, from the above principle to one which is no longer innocuous: each individual desires only commodities. In the theory of choice, therefore, only commodity vectors, ** X** = (

*x*

_{1},,

*x*

_{2}, …,

*x*), count. However, this is not generally true, even for a modern urban society. In making a choice, man generally takes into consideration, in addition to the commodity vector, the action by which he can obtain this vector. In other words, man always chooses between complexes of two coordinates, [

_{n}**X**,A], where

**X**is a commodity vector and

*A*the action by which

**X**may be obtained. The former has a utility on the personal “utility scale,” the latter a value determined by the cultural matrix of the society in which the individual lives (Georgescu-Roegen 1966, pp. 124-126). A concrete act of choice is, in general, not a

*culturally*

*free*choice. True, in the so-called traditional societies the cultural coordinate often counts more heavily than the utility coordinate; but this constitutes no irrationality at all. One can only say that a theory that would describe adequately the economic behavior of the individuals belonging to a traditional society raises far more complex analytical issues than those of the modern theory of utility.

Nicholas Georgescu-Roegen

[*See also* Decision making,*article on*economic aspects; Welfare economics.*Other relevant material may be found in the biographies of* Bentham; Böhm-Bawerk; Edgeworth; Gossen; Jevons; Marshall; Marx; Menger; Pareto; Ricardo; Smith, Adam; Turgot; Walras.]

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# utility

u·til·i·ty / yoōˈtilətē/ • n. (pl. -ties) 1. the state of being useful, profitable, or beneficial: *he had a poor opinion of the utility of book learning.* ∎ (in game theory or economics) a measure of that which is sought to be maximized in any situation involving a choice. 2. a public utility. ∎ stocks and bonds in public utilities. 3. Comput. a utility program. • adj. 1. useful, esp. through being able to perform several functions: *a utility truck.* ∎ denoting a player capable of playing in several different positions in a sport. 2. functional rather than attractive: *utility clothing.* 3. of or relating to the lowest U.S. government grade of beef.4. (of domestic animals) raised for potential profit and not for show or as pets.

# utility

# utility

**utility** In economic theory, utility is defined as the benefit or satisfaction which is derived from the consumption of a commodity. In eighteenth-century moral philosophy, it meant ‘the greatest happiness principle’: actions are right if they tend to promote happiness. Early sociologists, for example Émile Durkheim, criticized the utility principle, because it provided an inadequate explanation of social order.

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