The marginalist revolution and the development of the neoclassical paradigm: models and methods

"The conclusion to which I am ever more clearly coming is that the only hope of attaining a true system of Economics is to fling aside, once and for ever, the mazy and preposterous assumptions of the Ricardian School. Our English Economists have been living in a fool's paradise. The truth is with the French School, and the sooner we recognize this fact, the better it will be for the world."

(William Stanley Jevons, Theory of Political Economy, 1871: p.xliv-xlv).

"If, instead of welcoming inquiry and criticism, the admirers of a great author accept his writings as authoritative, both in their excellences and in their defects, the most serious injury is done to truth. In matters of philosophy and science, authority has ever been the great opponent of truth. A despotic calm is usually the triumph of error. In the republic of the sciences, sedition and even anarchy are beneficial in the long run to the greatest happiness of the greatest number."

(William Stanley Jevons, Theory of Political Economy, 1871: p.275-6)
The crux of the Neoclassical theory of value is the notion of subjective scarcity. The Neoclassical answer to the famous "water-diamond" paradox is that diamonds are naturally more valuable than water not because diamonds are costlier to produce (the Classical answer), but rather because diamonds are more scarce than water. Some may object to this distinction: if diamonds are very costly to produce, then one should expect to see somewhat less of them around, thus the cost-of-production and rarity arguments seem to boil down to the same thing. Adam Smith seems to imply this when he writes:

"[T]he value of [precious] metals has, in all ages and nations, arisen chiefly from their scarcity, and that their scarcity has arisen from the very small quantities of them which nature has any where deposited in one place, from the hard and intractable substance with which she has almost every where surrounded those small quantities, and consequently from the labour and expence [sic] which are every where necessary in order to penetrate and get at them." (A. Smith, 1776: p.563).

But this is not quite true for Neoclassicals. The Neoclassical notion of scarcity is not merely that something is "rare", but rather that it is perceived as rare by consumers. To take Lionel Robbins's (1932: p.46) famous example, bad eggs may be "rare", but if people do not desire bad eggs, then even one bad egg is already "too many" in their eyes and thus will not have much value. In contrast, if people's desire for diamonds is very great indeed, then in their perception, even a large number of diamonds may be "too few" in their eyes, thus they will have a high price. Consequently, the Neoclassical concept of scarcity is quite distinct from the Classical notion: the subjective element of desire is an integral part of the story.

There are thus two essential ingredients of Neoclassical value theory: (1) that the relative values of things arise from their relative scarcity and (2) that subjective desires are an integral part in determining the relative scarcity. Both of these notions have an old history predating 1871-4, but they were not always wedded together. Some economists believed that rarity gave rise to value without thinking too hard about whether rarity was a subjective or objective thing; in contrast, others have thought that subjective notions such as utility and demand were important in determining price, but did not really connect it to scarcity.

The Classicals -- Adam Smith, David Ricardo, John Stuart Mill, Karl Marx, etc. -- believed in neither of these ideas.  Following the pattern set by Richard Cantillon (1755), they argued that subjective desires and scarcity may be important factors in determining market (or temporary or short-run) prices, but they insisted that the natural (or equilibrium or long-run) prices were determined solely by relative costs of production (usually, relative labor costs).

The Classicals perceived rarity to be an aberration: if goods can be produced -- i.e. created -- then there is no inherent scarcity of them. Consequently, scarcity prices were what  Ricardo called "monopoly prices" -- i.e. the prices which arose only "when by no possible device their quantity can be augmented; and where, therefore, the competition is wholly on one side -- amongst the buyers." (Ricardo, 1817: p.165) and thus "their price is limited only by the extent of the power and will of purchasers" (ibid.) But this is not the natural, long-run price. "The exchangeable value...of a commodity which is at a monopoly price is nowhere regulated by the cost of production." (Ricardo, ibid.) Thus, scarcity may play a role in the short-run (when quantities are fixed), but not in the long-run.

They had granted that rarity might be a determinant of value in a few cases, "rare statues and pictures, scarce book and coins, wines of a peculiar quality" (Ricardo, 1817: p.6) -- goods which cannot be produced and thus whose value is regulated by "monopoly prices". But these cases were so exceptional that they could be safely ignored. At best, as our earlier quotation from Smith indicates, they were willing to discuss scarcity as a foundation of value only insofar as it arose from high costs of production. Certainly, whatever lip service they paid to scarcity, they did not incorporate it into their central theoretical schema.

Utility was a slightly different story. The Classicals confused utility of a good with its usefulness. They agreed that a good must have usefulness if it is to be produced. The mercantilist Nicholas Barbon was perhaps the first to explicitly claim that price was influenced by utility: "the Value of all Wares arise from their Use; Things of no Use, have no Value, as the English phrase is, They are good for nothing." (Barbon, 1690: p.13). In this, he was followed up by John Locke (1692) and John Law (1705).

Richard Cantillon  (1755) -- like all the Classicals thereafter -- acknowledged that a good must have utility in order to be produced. But utility itself did not determine the relative prices of the goods. It is relative costs of production that will determine the natural prices of goods. Utility merely determines that a good will be produced, period. That's where its role both begins and ends. Utility is of no further use beyond that.

Why did the Classicals cut utility's role so short? The reason is that utility seemed to run into trouble when confronted with the old water-diamond paradox set forth by John Law (1704: p.4) and made famous by Adam Smith (1776: p.44-5). As Smith noted, water is useful to humans, diamonds are useless to humans, thus water should have a higher "use-value" or "utility" than diamonds. But clearly, water commands a lower "exchange-value" than diamonds. Thus, like Aristotle before them, the Classicals gave up on the utility-value connection: it seems as if utility simply could not be incorporated successfully into a theory of natural price.

Of course, Smith's error was to confuse "utility" with "use-value". The concept of "utility" handed down by the Scholastics to the modern 18th Century economics by writers such as Samuel von Pufendorf  (1675) was to connect utility to desiredness and not to usefulness. Diamonds may be "useless", as Smith asserted, but they could still have utility in the sense that they are desired. With the notable exceptions of Jean-Baptiste Say and Nassau Senior, the misleading argument by Adam Smith was accepted by the rest of the Classical School.

At best, utility (like scarcity) will have a prominent role to play in Classical  theory only for the temporary case of short-run market prices. Indeed,  Cantillon (1755) was the first to suggest a clear supply and demand mechanism for the determination of market prices which includes both utility and scarcity (inexplicably, a lot of Anglo-Saxon literature tends to credit Sir James Steuart (1767) for this). But for long-run natural prices, neither utility nor rarity have a role in the Classical schema.

(B) The Franco-Italian Tradition: Subjective Scarcity

Some writers during the Classical period refused to relegate the utility explanation to a temporary or minor phenomenon and disputed the cost-of-production solution to the water-diamond paradox. The most notable of the disputants was Jean-Baptiste Say (1803, 1815, 1828). Although a follower of Smith in many other respects, he rejected Smith's labor theory of value. Or rather, he argued that utility and thus demand must play a part in the determination of natural price. At times, he went quite far in this pursuit. James Maitland Earl Lauderdale (1804) also rejected Smith's theory and proposed a long-run demand-and-supply mechanism.

The Classicals were not amused: David Ricardo (1817: Ch. 20) and John Stuart Mill (1845, 1945) took both Say and Lauderdale to task for their heresy. For instance, Ricardo writes:

"M. Say acknowledges that the cost of production is the foundation of price, and yet in various parts of his book he maintains that price is regulated by the proportion which demand bears to supply. The real and ultimate regulator of the relative value of any two commodities is the cost of their production, and not the respective quantities which may be produced, nor the competition amongst the purchasers." (Ricardo, 1817: p.231)

However, we should note that the resistance of the Classicals was not mere pig-headedness or simply a reiteration of Smith's "use-value" confusion. As particularly expressed by J.S. Mill (1845), if one was to acknowledge the role of both demand and supply in long-run price-determination, one is effectively mixing together mathematically heterogeneous things which cannot be juxtaposed upon each other.

"It seems to me necessary, when we mean to speak of the ratio between the demand for a commodity & the supply of it, that the two quantities should be, in the mathematical sense, homogeneous -- that both of them should be estimated in numbers of the same unit." (J.S. Mill, 1945: p.143)

Although insisting on the importance of subjective utility in price determination, Law, Say and Lauderdale were less clear about the role of rarity in all of this. This is understandable given the traditional difficulty of distinguishing a costly item from a rare item. Objectively-determined rarity had been of central importance in the work of Bernardo Davanzati (1588), Juan de Lugo (1642) and Pierre de Boisguilbert (1695), but the connection with utility was not immediately and clearly made.

The first explicit recognition of scarcity, i.e. subjectively-determined rarity, as the source of value is contained in the remarkable work of Ferdinando Galiani (1751). Galiani's brilliant performance was followed up by the anti-Physiocrat philosopher, Abbé Condillac (1776). Condillac explicitly employed both utility and rarity in determining scarcity and value and was willing to confront the Classical solution directly. As he wrote, "a thing does not have value because of its cost, as some suppose; but it costs because it has value." Condillac's argument was reiterated in a relatively obscure note by the ambiguous Physiocrat, Jacques Turgot (1769).

It is evident, then, that Say's groping for a subjectivist theory of price was not isolated. There was already a somewhat long history in France and Italy. Under Say's own influence, this Franco-Italian tradition sustained itself in these countries throughout the 19th Century. The ground-breaking work of French proto-marginalist economists such as Louis Auguste Say (1822) (J.B. Say's brother), Auguste Walras (1831) (L. Walras's father), Augustin Cournot (1838) and Jules Dupuit (1844) can thus be seen as natural outgrowths of a long tradition and not merely a series of brilliant isolated sparks of insight. It was upon this tradition that Léon Walras  was to draw in composing his 1874 masterpiece..

In Germany, another ambivalent follower of Smith and popular textbook writer, Johann Friedrich Rau (1827) did not discard the role of demand entirely -- indeed he showed how demand-and-supply diagrams can be used to determine price explicitly!  In addition, the weight of the German Historical School ensured that the Classical Ricardian theory never penetrated very deeply in Germany either. Together, this can perhaps explain the German-language contributions to the utility-cum-scarcity tradition, such as F.B.W. Hermann (1832), Hans von Mangoldt  (1863) and, above everything, Hermann Heinrich Gossen (1854). Carl Menger was thoroughly soaked in Rau and Hermann before trying his hand in 1871.

In Great Britain, where the Ricardians reigned supreme, the subjective scarcity notion had more trouble catching on. Nonetheless, the idea had been hatched by Nassau Senior  (1836) and his associates at Oxford and Dublin -- Richard Whately  (1832), William F. Lloyd (1837) and Mountiford Longfield (1834). These fledgling Neoclassicals did not mince their words when confronting the Classicals. As Whately asserts heretically, "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." (Whately, 1832: p.253).

These tentative efforts in Britain, however, were smashed by John Stuart Mill's Principles of Political Economy (1848), a weighty restatement of the Classical Ricardian doctrine. It was with Lloyd, Whately and company in mind that Mill went on to assert that "Happily, there is nothing in the laws of Value which remains for the present or any future writer to clear up; the theory of the subject is complete" (Mill, 1848: Ch. III.1).

Despite Mill's abrupt interception, the grumbling continued. By the 1860s, the Classical Ricardian doctrine had came under siege, not only from the usual suspects (e.g. Carlyle, Ruskin, Cliffe Leslie) but, more importantly, from their own. The gutting of the wages fund doctrine by Thornton, Sidgwick and Walker, and the wide-ranging assault on the "vulgar economists" by Karl Marx, had dented the confidence of Classical theory. As a consequence, a window of opportunity opened in Britain during this time for outcasts such as  Richard Jennings (1855), William E. Hearn (1864), Fleeming Jenkin (1870) and Henry Dunning Macleod (1857, 1881), to pursue subjective scarcity and/or supply-and-demand mechanisms in their work. Thus the claim that William Stanley Jevons was working in a vaccum in 1871 with little more than Bentham to draw upon, is not strictly correct.

[Note: Emil Kauder (1957) has argued that the reason for the retarded acceptance of subjective scarcity theory in English economics and the predominance of labor-cost theories was due to differing philosophical and religious traditions. Protestant Britain was wary of the hedonistic conception of utility, and the labor-cost theories seemed quite more compatible with its work-oriented Puritanical traditions. Thus it is in Catholic countries, like France and Italy, where sensualism is not altogether dead, that we find the great expounders of subjective scarcity theory. This is an interesting hypothesis, but it does not perfectly fit with the facts and certainly overlooks more straightforward explanations.]

(C) Marginal Utility

Discussions of utility, scarcity and the mechanism of demand and supply, however suggestive, were not well-integrated in the efforts of the early proto-Neoclassical economists. The great missing ingredient was the connection between utility and demand.  Auguste Walras (1831) and  Mountiford Longfield (1834) attempted an explicit connection, but their theories ended tied up in knots. As was to be discerned later, the key to successful integration was marginal utility -- specifically, diminishing marginal utility.

The concept of diminishing marginal utility -- i.e. that equal increments of a good yield diminishing increments of utility -- was already widely known. Daniel Bernoulli (1738) had employed this concept to solve the  St. Petersburg Paradox.  The utilitarian Jeremy Bentham  (1789, 1802) had certainly stated the idea. Lloyd (1833), Senior (1836), Jennings (1855) and  Hearn  (1864) were well aware of diminishing marginal utility as well. The question was one of connecting it to demand, which these writers failed to do clearly.

(i) Auguste Cournot

The idea of a demand function itself was proposed by Charles D'Avenant  (1699), who even attempted to estimate one for wheat (on the basis of data allegedly provided by Gregory King (1696)). The first concrete expression of a demand function was accomplished by Pietro Verri (1760). Thereafter silence reigned until the enormous leap of Augustin Cournot (1838).  Cournot did not bother with the niceties of utility; his concern was focused on demand functions directly which he considered to be deducible from empirical fact. He was the first to express the demand function in algebraic form as D = F(p) and the first to draw demand-and-supply functions in price-quantity space (Cournot, 1838: p.92, Fig. 6). This, of course, was not all: in addition to demand functions, Cournot introduced the concepts of marginal revenue, marginal cost, the concept of the profit-maximizing firm, monopoly, duopoly, perfect competition and, of course, his famous "reaction functions". But marginal utility was nowhere in sight. As he argued, the "accessory ideas of utility, scarcity, and suitability to the needs and enjoyments of mankind...are variable and by nature indeterminate, and consequently ill suited for the foundation of a scientific theory" (Cournot, 1838: p.10).

The first successful connection between marginal utility and demand was accomplished by the French engineer Jules Dupuit (1844). His remarkable effort at developing a cost-benefit analysis of public works led him to draw the demand curve in price-quantity space. Unlike Cournot, Dupuit did not rest his demand curve on empirical intuition but rather identified the demand curve as the marginal utility curve itself. Dupuit's basic idea was this: as quantity rises, the marginal utility of the good declines. Consequently, one should also say that as the quantity rises, the willingness of a person to pay for that good declines. Thus, the concept of diminishing marginal utility should translate itself into a downward-sloping demand function.

Of course, Dupuit's logic was suspect in at least one place: marginal utility is particular to an individual, while market demand is an aggregate, so something must be said about the interpersonal comparability of utility in order to proceed with the connection. Dupuit skimped on this. Nonetheless, the important point was that the connection was made between demand and utility. Dupuit, however, did not draw a supply curve and thus did not get price-determination into his story.

[Dupuit went on to define "relative utility" (what later became known as Marshall's "consumer surplus") as the area under the demand/marginal utility curve above the price and used it as a measure of the welfare effects of different prices -- yielding his famous conclusion that public welfare is maximized when the price (in his case, the toll rate on a bridge) is zero.]

The final step came from Hermann Heinrich Gossen (1854). Unlike Dupuit, Gossen clearly distinguished the marginal utility curve from the demand curve. Gossen posited that demand is derived from the utility-maximizing choices of the consumer. Gossen's "Three Laws" can be stated as follows:

(1) the amount of utility derived from the consumption of a good declines with each additional unit of that commodity (i.e. diminishing marginal utility, or, to use Gossen's term, "diminishing worth of the last atom".)

(2) a person maximizes his utility when he distributes his income among various goods so that he obtains the same amount of satisfaction from the last unit of each good or, if money is being used, he obtains the same amount of satisfaction from the last unit of money spent upon each commodity (i.e. equality of the ratio of marginal utilities to the ratio of prices, i.e. MU_i/p_i = MU_j/p_j for any two goods i, j).

(3) a good has value only when the demand for it exceeds supply (i.e. subjective scarcity is source of value).

Of Gossen's three laws, the second is perhaps the most remarkable. The idea that, at the margin, the consumer substitutes between goods so that he obtains the same marginal utility (in terms of money) across goods yields the downward-sloping demand curve for each of the goods. To see this, merely note that when the price of a good rises, the marginal utility in terms of money (MU_i/p_i) declines and thus, by Gossen's first law (diminshing marginal utility), less of that good will be bought. The foundations of the Marginalist Revolution were thus in place.

H.H. Gossen's (1854) work had already anticipated much of the Marginalist Revolution. However, this "ingenius idiot", as Schmoller called him, was an unknown man -- a retired Prussian civil servant -- whose work was entirely neglected. His work was only accidentally discovered in 1878 during a search by Jevons for fellow travellers. As such, the works of  Jevons, Menger and Walras came forth without the benefit of Gossen's insights.

William Stanley Jevons developed his results on marginal utility (which Jevons called "final degree of utility") independently and first announced them in an abstract of a 1862 lecture (published in 1866). The publication of Fleeming Jenkin's (1870) diagrammatic representation of the demand-and-supply mechanism led Jevons to quickly write and publish his own 1871 treatise, Theory of Political Economy in order to establish priority. Jevons couched his construction in the context of pure exchange. Specifically, assuming two goods (call them x₁ and x₂) and two agents (call them A and B), then in equilibrium, Jevons proposed that:

MU₁^A/MU₂^A = -dx₂/dx₁ = MU₁^B/MU₂^B

where MU_i^h is the marginal utility of good x_i to household h. The term dx₁ is the amount of good x₁ given by agent A to agent B and dx₂ is the amount of good x₂ given by agent B to agent A. Thus, for agent A, the marginal utility of good x₁ after amount dx₁ has been surrendered divided by the marginal utility of good x₂ after amound dx₂ has been gained, is inversely related to the exchange ratio dx₂/dx₁. The analogous reasoning applies for agent B. Thus: "The ratio of exchange of any two commodities will be the reciprocal of the ratio of the final degrees of utility of the quantities of commodity available for consumption after the exchange is completed." (Jevons, 1871: p.95).

Through the medium of a market, the exchange ratio can be expressed as prices, i.e. -dx₂/dx₁ = p₁/p₂, so that this can be rewritten as:

MU₁^A/p₁ = MU₂^A/p₂ = MU₁^B/p₁ = MU₂^B/p₂

so not only does Gossen's Second Law hold for every single consumer, but it also holds across consumers.

However, Jevons got somewhat tangled up in this derivation. In particular, he was not quite sure how to get from dx₁/dx₂ to p₁/p₂ or back again. His difficulty stems in part from his wariness of the difference between bilateral exchange between two people and multi-lateral exchange via the medium of the market. However, he confused himself a bit in this respect. Apparently, he believed it was easier to determine exchange ratios in bilateral exchange than in competitive situations - thus he tried to reduce his "market" situation into one of simple bilateral exchange. He did so with the help of two pieces of scaffolding: (1) by constructing "trading bodies" ( Jevons, 1871: p.88-90), so that the enormous mass of heterogeneous traders in markets could be reduced to a pair of what would today be called "representative agents" that would exchange with each other; (2) then via arbitrage-theoretic reasoning -- in his famous "Law of Indifference"  (Jevons, 1871: p.90-5) -- he goes on to argue for price-taking agents so that, by definition, dx₁/dx₂ = p₁/p₂.

Ten years later, Francis Ysidro Edgeworth (1881) was to show that Jevons's instincts should be reversed: determining the exchange ratio between two bilateral trading bodies, Edgeworth argued, is in fact more difficult than determining the exchange ratio when there are numerous, heterogeneous agents in a competitive market situation. Edgeworth's demonstration of the indeterminacy of exchange ratios in bilateral exchange was captured in his famous notion of the "contract curve" and the "core". His famous conjecture, that indeterminacy is eliminated when the number of traders increases ("perfect competition"), however, was not picked up immediately.

 Jevons established equilibrium in pure exchange, but, not having a good theory of production, was unable to construct the familiar "supply-and-demand" theory with variable output levels. His resolution was considerably unsatisfying. Jevons argued, as Gossen had before him, that labor supply was governed by disutility of labor: the greater the amount of work, the greater the marginal disutility of labor. Consequently, he went on to argue, by Gossen's Second Law, that the marginal utility of consuming a good must be equal to the marginal disutility of producing it. In other words, the quantity of a good produced is determined by the intersection of a downward-sloping marginal utility of consumption curve and an upward sloping marginal disutility of labor curve.

With the quantity of the good thus determinate,  Jevons's next step was quickly and sloppily reasoned: once the supply of the good is given, then we can apply the pure exchange scenario we had before to determine the price of the good. This entire story is summarized by Jevons in a famous "catena":

"Cost of production determines supply;

(W.S. Jevons, 1871: p.165)

The first line seeks to explain the output-determination process we have just summarized: namely, how higher or lower wages shift around the the marginal utility of income/disutility of labor curves so that output level, i.e. supply, changes. The second line merely states that once we have supplies of goods determined, we know what marginal utility of those goods will be; finally, the third line summarizes the pure exchange process.

This is confusing. The immediate temptation is to remove all the intermediate steps and reduce the catena to the simple claim that "cost of production determines value", a complete restatement of the Classical theory of  Ricardo and Mill! Obviously, Jevons did not quite want to put it this way. Wages ought not to be hanging in the air by themselves. Unlike Classical theory, Jevons argued, the value of labour "is determined by the value of the produce, not the value of the produce by that of the labour." (Jevons, 1871: p.166). Thus, years later, Alfred Marshall would invert Jevons's catena into its proper order:

"Utility determines the amount that has to be supplied,
The amount that has to be supplied determines cost of production
Cost of production determines value."

(A. Marshall, 1890: p.674)

Without deriving demand and supply functions from his marginal utility/disutility schedules, Jevons discussion is quite confusing. It was Alfred Marshall (1890: Ch. 3; Math. App.), that got Jevons' out of this knot by deriving it formally. Now, as stated, MU_i/p_i is the marginal utility of dollar spend on good x_i. By Gossen's Second Law, MU_i/p_i = MU_j/p_j for all goods i, j. Consequently, we can define the marginal utility of income (what Marshall called the marginal utility of money) as:

MU_Y = MU_i/p_i = MU_j/p_j = ....

Jevons, as we know, was clearly aware of this and even offered the identity MU_i = p_iMU_Y in his work (Jevons, 1871: p.146). But how is one to construct a demand function? Marshall's process is to follow Jevons in assuming that the marginal utility of income MU_Y is a constant. Doing so, we immediately recognize that differentiating with respect to quantity demanded x_i:

dMU_i/dx_i = (dp_i/dx_i)·MU_Y

so that:

dx_i/dp_i = MU_Y/( MU_i/ x_i)

as, by the rule of diminishing marginal utility, dMU_i/dx_i < 0, then dx_i/dp_i < 0, so a rise in the price of good x_i leads to a decrease in demand for it. Thus, the demand curve is downward-sloping.

[Note: the constancy of the marginal utility of income is a dubious assumption, discussed more fully elsewhere. Effectively, what it does to eliminate the famous "income effect" of a change in price (note that this is not the same as saying that all other prices are held constant! rather, intercommodity effects are neutralized). However, this also implies that as p_i rises and x_i falls, total expenditure on this good, p_ix_i, is unchanged, i.e. the demand function is unit elastic. Marshall was uncomfortable with this assumption, thus he cautioned that this was only approximately true, provided that the expenditure by a consumer on any particular good is only "a small part of his total resources" (Marshall, 1890: p.279). The derivation of demand from utility without the constant marginal utlity of income assumption had to wait until Vilfredo Pareto (1892)]