Marginal utility

Bold text:''“Marginal revolution” redirects here. For the economics weblog, see Marginal Revolution (blog).'' The marginal utility of a good or service is its utility in its least urgent use of the most-desired available uses, in other words, the use that is just in the margin. The same object may have different marginal utilities for different people. The concept grew out of attempts by economists to explain the determination of price. The term “marginal utility” arises from a translation of “Grenznutzen”, coined by the Austrian economist Friedrich von Wieser.

The conception of utility here is simply that of use. (Indeed, “Grenz·nutzen” could be translated “border-use”.) All that is essential to analysis is that uses be partially orderable by desirability. Nonethless, there has been a proclivity amongst most economists to conceptualize utility here as corresponding to a measure, that is to say, as being quantified. This proclivity has significantly affected the development and reception of theories of marginal utility.

A marginal change is as large as the smallest relevant division. For reasons of tractability, it is often assumed in neoclassical analysis that goods and services are continuously divisible. In such context, a marginal change may be an infinitesimal change or a limit. However, strictly speaking, the smallest relevant division may be quite large.

Placement of margins
The location of the margin for any individual corresponds to his or her endowment, broadly conceived to include opportunities. This endowment is determined by many things including physical laws (which constrain how forms of energy and matter may be transformed), accidents of nature (which determine the presence of natural resources), and the outcomes of past decisions made both by others and by the individual him- or herself.

The “law” of diminishing marginal utility
An individual will typically be able to partially order the potential uses of a good or service. For example, a ration of water might be used to sustain oneself, a dog, or a rose bush. Say that a given person gives her own sustenance highest priority, that of the dog next highest priority, and lowest priority to saving the roses. In that case, if the individual has two rations of water, then the marginal utility of either of those rations is that of watering the dog. The marginal utility of a third gallon would be that of watering the roses.

(The diminishing of utility should not necessarily be taken to be itself an arithmetic subtraction. It may be no more than a purely ordinal change. )

The notion that marginal utilities are diminishing across the ranges relevant to decision-making is called “the law of diminishing marginal utility” (and also known as a “Gossen's First Law”). However, it will not always hold. The case of the person, dog, and roses is one in which potential uses operate independently — there is no complementarity across the three uses. Sometimes an amount added brings things past a desired tipping point, or an amount subtracted causes them to fall short. In such cases, the marginal utility of a good or service might actually be increasing.

Independence of the “law” from presumptions of self-interested behavior
While the above example conforms to ordinary notions of self-interested behavior, the concept and logic of marginal utility are independent of the presumption that people pursue self-interest. For example, a different person might give highest priority to the rose bush, next highest to the dog, and last to himself. In that case, if the individual has three rations of water, then the marginal utility of any one of those rations is that watering the person. With just two rations, the person is left unwatered and the marginal utility of either ration is that of the dog. Likewise, a person could give highest priority to the needs of one of her neighbors, next to another, and so forth, placing her own welfare last; the concept of diminishing marginal utility would still apply.

Marginalist theory
Marginalism explains choice with the hypothesis that people decide whether to effect any given change based on the marginal utility of that change, with rival alternatives being chosen based upon which has the greatest marginal utility.

Market price and diminishing marginal utility
If an individual has a stock or flow of a good or service whose marginal utility is less than would be that of some other good or service for which he or she could trade, then it is in his or her interest to effect that trade. Of course, as one thing is traded-away and another is acquired, the respective marginal gains or losses from further trades are now changed. On the assumption that the marginal utility of one is diminishing, and the other is not increasing, all else being equal, an individual will demand an increasing ratio of that which is acquired to that which is sacrificed. If any trader can better his or her own marginal position by offering a trade more favorable to complementary traders, then he or she will do so.

In an economy with money, the marginal utility of a quantity is simply that of the best good or service that it could purchase. On the assumption that purchasable goods and services have diminishing marginal utility, the marginal utility of money will itself be diminishing.

Hence, the “law” of diminishing marginal utility provides an explanation for diminishing marginal rates of substitution and thus for the “laws” of supply and demand, as well as essential aspects of models of “imperfect” competition.

The paradox of water and diamonds
The “law” of diminishing marginal utility is said to explain the “paradox of water and diamonds”, most commonly associated with Adam Smith (though recognized by earlier thinkers). Human beings cannot even survive without water, whereas diamonds were in Smith's day mere ornamentation or engraving bits. Yet water had a very small price, and diamonds a very large price, by any normal measure. Marginalists explained that it is the marginal usefulness of any given quantity that matters, rather than the usefulness of a class or of a totality. For most people, water was sufficiently abundant that the loss or gain of a gallon would withdraw or add only some very minor use if any; whereas diamonds were in much more restricted supply, so that the lost or gained use were much greater.

That is not to say that the price of any good or service is simply a function of the marginal utility that it has for any one individual nor for some ostensibly typical individual. Rather, individuals are willing to trade based upon the respective marginal utilities of the the goods that they have or desire (with these marginal utilities being distinct for each potential trader), and prices thus develop constrained by these marginal utilities.

The “law” does not tell us such things as why diamonds are naturally less abundant on the earth than is water, but helps us to understand how this affects the value imputed to a given diamond and the price of diamonds in a market.

Criticism of the marginalist explanation of the paradox of water and diamonds
Many critics of marginalism would reply that the reason that diamonds are more expensive than water is not because of their relative natural abundance but because of their cost of production. The reason water is available abundantly and diamonds in relatively smaller quantities is because one is inexpensive to produce and one very expensive. Critics claim that thus the reason water is cheaper than diamonds is simply because it costs less to produce. If diamonds could be produced cheaply from carbon, as modern technology may make possible in the short term, then the price of diamonds will fall, even though the demand for their use has not altered. Therefore, as these critics would claim, it is the cost of production which determines price, not the marginal utility.

Marginalists simply respond that if this were true then, rather than our seeing some goods and services not produced because their costs exceeded their prices, consumers would make a practice of seeking expensive wares without regard to their use. (As proto-marginalist Richard Whately put it, “It is not that pearls fetch a high price because men have dived for them; but on the contrary, men dive for them because they fetch a high price.” ) Marginalists explain that costs of production may be what limit supply, but that these costs of production are themselves sacrificed marginal uses, and will not be borne when they are expected to exceed the marginal use of what is produced. In other words, the marginalist certainly does not explain price as a simple function of the marginal utility of a single good for one person or for some “average” person, but nonetheless insists that it results from the trade-offs that each participant would be willing to make for the various goods and services at stake, with those trade-offs being determined by marginal uses. The critics who believe that costs of production determine price, by assuming a demand that will bear the cost, have begged the essential question that the marginalists purport to answer.

Quantified marginal utility
Under the special case in which usefulness can be quantified, the change in utility of moving from state $$S_1$$ to state $$S_2$$ is
 * $$\Delta U=U(S_2)-U(S_1)\,$$

Moreover, if $$S_1$$ and $$S_2$$ are distinguishable by values just one variable $$g\,$$ which is itself quantified, then it becomes possible to speak of the ratio of the marginal utility of the change in $$g\,$$ to the size of that change:
 * $$\left.\frac{\Delta U}{\Delta g}\right|_{c.p.}$$

(where “c.p.” indicates that the only independent variable to change is $$g\,$$).

Mainstream neoclassical economics will typically assume that
 * $$\lim_{\Delta g\to 0}{\left.\frac{\Delta U}{\Delta g}\right|_{c.p.}}$$

is well defined, and use “marginal utility” to refer to a partial derivative
 * $$\frac{\partial U}{\partial g}\approx\left.\frac{\Delta U}{\Delta g}\right|_{c.p.}$$

and diminishing marginal utility is similarly taken to correspond to
 * $$\frac{\partial^2 U}{\partial g^2}<0$$

Proto-marginalist approaches
A great variety of economists concluded that there was some sort of inter-relationship between utility and rarity that effected economic decisions, and in turn informed the determination of prices.

Marginalists before the Revolution
The first published statement of a sort of theory of marginal utility was by Daniel Bernoulli, in “Specimen theoriae novae de mensura sortis”. This paper appeared in 1738, but a draft had been written in 1731 or in 1732. In 1728, Gabriel Cramer produced fundamentally the same theory in a private letter. Each had sought to resolve the St. Petersburg paradox, and had concluded that the marginal desirability of money decreased as it was accumulated, more specifically such that the desirability of a sum were the natural logarithm (Bernoulli) or square root (Cramer) thereof. However, the more general implications of this hypothesis were not explicated, and the work fell into obscurity.

In “A Lecture on the Notion of Value”, delivered in 1833 and included in Lectures on Population, Value, Poor Laws and Rent (1837), William Forster Lloyd explicitly offered a general marginal utility theory, but did not offer its derivation nor elaborate its implications. The importance of his statement seems to have been lost on everyone (including Lloyd) until the early 20th century, by which time others had independently developed and popularized the same insight.

In An Outline of the Science of Political Economy (1836), Nassau William Senior asserted that marginal utilities were the ultimate determinant of demand, yet apparently did not pursue implications, though some interpret his work as indeed doing just that.

In 1854, Hermann Heinrich Gossen published Die Entwicklung der Gesetze des menschlichen Verkehrs und der daraus fließenden Regeln für menschliches Handeln, which presented a marginal utility theory and to a very large extent worked-out its implications for the behavior of a market economy. However, Gossen's work was not well received in the Germany of his time, most copies were destroyed unsold, and he was virtually forgotten until rediscovered after the so-called Marginal Revolution.

The Marginal Revolution
Marginalism eventually found a foot-hold by way of the work of three economists, Jevons in England, Menger in Austria, and Walras in Switzerland.

William Stanley Jevons first proposed the theory in “A General Mathematical Theory of Political Economy” (PDF), a little-noticed paper delivered in 1862 and published in 1863. He later presented the theory in The Theory of Political Economy in 1871, which was fairly widely read but not much appreciated. Jevons' conception of utility was that in the hedonic tradition of Jeremy Bentham and of John Stuart Mill, and Jevons explained demand but not supply by reference to marginal utility.

Carl Menger presented the theory in Grundsätze der Volkswirtschaftslehre (translated as Principles of Economics) in 1871. Menger's presentation is peculiarly notable on two points. First, he took special pains to explain why individuals should be expected to rank possible uses and then to use marginal utility to decide amongst trade-offs. (For this reason, Menger and his followers are sometimes called “the Psychological School”, though they are more frequently known as “the Austrian School” or as “the Vienna School”.) Second, while his illustrative examples present utility as quantified, his essential assumptions do not. Menger's work found a significant and appreciative audience.

Marie-Esprit-Léon Walras introduced the theory in Éléments d'économie politique pure, the first part of which was published in 1874. Walras's work found relatively few readers.

(An American, John Bates Clark, is sometimes also mentioned. But, while Clark independently arrived at a marginal utility theory, he did little to advance it until it was clear that the followers of Jevons, Menger, and Walras were revolutionizing economics.  Nonetheless, his contributions thereafter were profound.)

The second generation
Although the Marginal Revolution flowed from the work of Jevons, Menger, and Walras, their work might have failed to enter the mainstream were it not for a second generation of economists. In England, the second generation were exemplified by Philip Henry Wicksteed, by William Smart, and by Alfred Marshall; in Austria by Eugen von Böhm-Bawerk and by Friedrich von Wieser; in Switzerland by Vilfredo Pareto; and in America by Herbert Joseph Davenport and by Frank A. Fetter.

There were significant, distinguishing features amongst the approaches of Jevons, Menger, and Walras, but the second generation did not maintain distinctions along national or linguistic lines. The work of von Wieser was heavily influenced by that of Walras. Wicksteed was heavily influenced by Menger. Fetter referred to himself and Davenport as part of “the American Psychological School”, named in imitation of the Austrian “Psychological School”. (And Clark's work from this period onward similarly shows heavy influence by Menger.) William Smart began as a conveyor of Austrian School theory to English-language readers, though he fell increasingly under the influence of Marshall.

Böhm-Bawerk was perhaps the most able expositor of Menger's conception. He was further noted for producing a theory of interest and of profit in equilibrium based upon the interaction of diminishing marginal utility with diminishing marginal productivity of time and with time preference. (This theory was adopted in full and then further developed by Knut Wicksell and, with modifications including formal disregard for time-preference, by Wicksell's American rival Irving Fisher. )

Marshall was the second-generation marginalist whose work on marginal utility came most to inform the mainstream of neoclassical economics, especially by way of his Principles of Economics, the first volume of which was published in 1890. Marshall constructed the demand curve with the aid of assumptions that utility was quantified, and that the marginal utility of money was constant (or nearly so). Like Jevons, Marshall did not see an explanation for supply in the theory of marginal utility, so he synthesized an explanation of demand thus explained with supply explained in a more classical manner, determined by costs which were taken to be objectively determined. (Marshall later actively mischaracterized the criticism that these costs were themselves ultimately determined by marginal utilities. )

The Marginal Revolution and Marxism
The doctrines of marginalism and the Marginal Revolution are often interpreted as somehow a response to Marxist economics. In fact, the first volume of Das Kapital was not published until 1867, after the works of Jevons, Menger, and Walras were written or well under way; and Marx was still a relatively obscure figure when these works were completed. It is unlikely that that any of them knew anything of him. (On the other hand, Hayek or Bartley has suggested that Marx, voraciously reading at the British Museum, may have come across the works of one or more of these figures, and that his inability to formulate a viable critique may account for his failure to complete any further volumes of Kapital before his death. )

Nonetheless, it is not unreasonable to suggest that part of what contributed to the success of the generation who followed the preceptors of the Revolution was their ability to formulate straight-forward responses to Marxist economic theory. The most famous of these was that of Böhm-Bawerk, “Zum Abschluss des Marxschen Systems” (1896), but the first was Wicksteed's “The Marxian Theory of Value. Das Kapital: a criticism” (1884, followed by “The Jevonian criticism of Marx: a rejoinder” in 1885 ). Only a few Marxist replies were made to marginalism, of which the most famous were Rudolf Hilferding's Böhm-Bawerks Marx-Kritik (1904) and Политической экономии рантье (1914) by Никола&#769;й Ива&#769;нович Буха&#769;рин (Nikolai Bukharin).

(It might also be noted that some followers of Henry George similarly consider marginalism and neoclassical economics a reaction to Progress and Poverty, which was published in 1879. )

Eclipse
In his 1881 work Mathematical Psychics, Francis Ysidro Edgeworth presented the indifference curve, deriving its properties from marginalist theory which assumed utility to be a differentiable function of quantified goods and services. But it came to be seen that indifference curves could be considered as somehow given, without bothering with notions of utility.

In 1915, Евгений Евгениевич Слуцкий (Eugen Slutsky) derived a theory of consumer choice solely from properties of indifference curves. Because of the World War, the Bolshevik Revolution, and his own subsequent loss of interest, Slutsky's work drew almost no notice, but similar work in 1934 by John Richard Hicks and R. G. D. Allen derived much the same results and found a significant audience. (Allen subsequently drew attention to Slutksy's earlier accomplishment.)

Although some of the third generation of Austrian school economists had by 1911 rejected the quantification of utility while continuing to think in terms of marginal utility, most economists presumed that utility must be a sort of quantity. Indifference curve analysis seemed to represent a way of dispensing with presumptions of quantification, albe&iuml;t that a seemingly arbitrary assumption (later admitted by Hicks to be a “rabbit out of a hat”) about decreasing marginal rates of substitution would then have to be introduced to have convexity of indifference curves.

For those who accepted that indifference curve analysis superseded marginal utility analysis, the former became at best somewhat analogous to the Bohr model of the atom — perhaps pedagogically useful, but “old fashioned” and ultimately incorrect.

Revival
When Cramer and Bernoulli introduced the notion of diminishing marginal utility, it had been to address a paradox of gambling, rather than the paradox of value. The marginalists of the revolution, however, had been formally concerned with problems in which there was neither risk nor uncertainty. So too with the indifference curve analysis of Slutsky, Hicks, and Allen.

The expected utility hypothesis of Bernoulli et alii was revived by various 20th century thinkers, perhaps most notably Ramsey (1926), v. Neumann and Morgenstern (1944), and Savage (1954). Although this hypothesis remains controversial, it brings not merely utility but a quantified conception thereof back into the mainstream of economic thought, and would dispatch the Ockhamistic argument. (It should perhaps be noted that, in expected utility analysis, the “law” of diminishing marginal utility corresponds to what is called “risk aversion”.)

Meanwhile, the Austrian School continues to develop its ordinalist notions of marginal utility analysis, formally demonstrating that from them proceed the decreasing marginal rates of substitution of indifference curves.