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I. The steady course on which mathematical economics has held for the past four decades sharply contrasts with its progress during the preceding century, which was marked by several major scientific accidents. One of them occurred in 1838, at the beginning of that period, with the publication of Augustin Cournot’s Recherches sur les principes mathématiques de la théorie des richesses. By its mathematical form and by its economic content, his book stands in splendid isolation in time; and in explaining its data historians of economic analysis in the first half of the 19th century must use a wide confidence interval.

The University of Lausanne was responsible for two other of those accidents. When Léon Walras delivered his first professorial lecture there on 16 December 1870, he had held no previous academic appointment; he had published a novel and a short story but he had not contributed to economic theory before 1870; and he was exactly 36. The risk that his university took was vindicated by the appearance of the Eléments d’économie politique pure in 1874–7. For Vilfredo Pareto, who succeeded Walras in his chair in 1893, it was also a first academic appointment; he had not contributed to economic theory before 1892; and he was 45. This second gamble of the University of Lausanne paid off when Pareto’s Cours d’économie politique appeared in 1896–97, followed by his Manuel d’économie politique in 1909, and by the article ‘Economie mathématique’ in 1911.

In the contemporary period of development of mathematical economics, profoundly influenced by John von Neumann, his article of 1928 on games and his paper of 1937 on economic growth also stand out as major accidents, even in a career with so many facets.

The preceding local views would yield a distorted historical perception, however, if they were not complemented by a global view which sees in the development of mathematical economics a powerful, irresistible current of thought. Deductive reasoning about social phenomena invited the use of mathematics from the first. Among the social sciences, economics was in a privileged position to respond to that invitation, for two of its central concepts, commodity and price, are quantified in a unique manner, as soon as units of measurement are chosen. Thus for an economy with a finite number of commodities, the action of an economic agent is described by listing his input, or his output, of each commodity. Once a sign convention distinguishing inputs from outputs is made, the action of an agent is represented by a point in the commodity space, a finite-dimensional real vector space. Similarly the prices in the economy are represented by a point in the price space, the real vector space dual of the commodity space. The rich mathematical structure of those two spaces provides an ideal basis for the development of a large part of economic theory.

Finite dimensional commodity and price spaces can be, and usually are, identified and treated as a Euclidean space. The stage is thus set for geometric intuition to take a lead role in economic analysis. That role is manifest in the figures that abound in the economics literature, and some of the great theorists have substituted virtuosity in reasoning on diagrams for the use of mathematical form. As for mathematical economists, geometric insight into the commodity-price space has often provided the key to the solution of problems in economic theory.

The differential calculus and linear algebra were applied to that space at first as a matter of course. By the time John Hicks’s Value and Capital appeared in 1939, Maurice Allais’ A la recherche d’une discipline économique in 1943, and Paul Samuelson’s Foundations of Economic Analysis in 1947, they had both served economic theory well. They would serve it well again, but the publication of the Theory of Games and Economic Behavior in 1944 signalled that action was also going to take new directions. In mathematical form, the book of von Neumann and Oskar Morgenstern set a new level of logical rigour for economic reasoning, and it introduced convex analysis in economic theory by its elementary proof of the MiniMax theorem. In the next few years convexity became one of the central mathematical concepts, first in activity analysis and in linear programming, as the Activity Analysis of Production and Allocation edited by Tjalling Koopmans attested in 1951, and then in the mainstream of economic theory. In consumption theory as in production theory, in welfare economics as in efficiency analysis, in theory of general economic equilibrium and in the theory of the core, the picture of a convex set supported by a hyperplane kept reappearing, and the supporting hyperplane theorem supplied a standard technique for obtaining implicit prices. The applications of that theorem to economics were a ready consequence of the real vector space structure of the commodity space; yet they were made more than thirty years after Minkowski proved it in 1911.

Algebraic topology entered economic theory in 1937, when von Neumann generalized Brouwer’s fixed point theorem in a lemma devised to prove the existence of an optimal growth path in his model. The lag from Brouwer’s result of 1911 to its first economic application was shorter than for Minkowski’s result. It should, however, have been significantly longer, for von Neumann’s lemma was far too powerful a tool for his proof of existence. Several authors later obtained more elementary demonstrations, and David Gale in particular based his in 1956 on the supporting hyperplane theorem. Thus von Neumann’s lemma, reformulated in 1941 as Kakutani’s fixed point theorem, was an accident within an accidental paper. But in a global historical view, the perfect fit between the mathematical concept of a fixed point and the social science concept of an equilibrium stands out. A state of a social system is described by listing an action for each one of its agents. Considering such a state, each agent reacts by selecting the action that is optimal for him given the actions of all the others. Listing those reactions yields a new state, and thereby a transformation of the set of states of the social system into itself is defined. A state of the system is an equilibrium if, and only if, it is a fixed point of that transformation. More generally, if the optimal reactions of the agents to a given state are not uniquely determined, one is led to associate a set of new states, instead of a single state, with every state of the system. A point-to-set transformation of the set of states of the social system into itself is thereby defined; and a state of the system is an equilibrium if, and only if, it is a fixed point of that transformation. In this view, fixed point theorems were slated for the prominent part they played in game theory and in the theory of general economic equilibrium after John Nash’s one-page note of 1950.

A perfect fit of mathematical form to economic content was also found when the traditional concept of a set of negligible agents was formulated exactly. In 1881, in Mathematical Psychics, Francis Edgeworth had studied in his box the asymptotic equality of the ‘contract curve’ of an economy and of its set of competitive allocations. Basic to his proof of convergence is the fact that in his limiting process every agent tends to become negligible. A long period of neglect of his contribution ended in 1959, when Martin Shubik brought out the connection between the contract curve and the game theoretic concept of the core. After the second impulse given in 1962 by Herbert Scarf’s first extension of Edgeworth’s result, a new phase of development of the economic theory of the core was under way; and in 1964 Robert Aumann formalized the concept of a set of negligible agents as the unit interval of the real line with its Lebesgue measure. The power of that formulation was demonstrated as Aumann proved that in an exchange economy with that set of agents, the core and the set of competitive allocations coincide. Karl Vind then gave, also in 1964, a different formulation of this remarkable result in the context of a measure space of agents without atoms, and showed that it is a direct consequence of Lyapunov’s theorem of 1940 on the range of an atomless vector measure. The convexity of that range explains the convexing effect of large economies. In the important case of a set of negligible agents, it justifies the convexity assumption on aggregate sets to which economic theory frequently appeals. A privileged place was clearly marked for measure theory in mathematical economics.

An alternative formulation of the concept of a set of negligible agents was proposed by Donald Brown and Abraham Robinson in 1972 in terms of Non-standard Analysis, created by Robinson in the early 1960s. Innovations in the mathematical tools of economic theory had not always been immediately and universally adopted in the past. In this case the lag from mathematical discovery to economic application was exceptionally short, and Non-standard Analysis had not been widely accepted by mathematicians themselves. Predictably the intrusion of this strange, sophisticated new tool in economic theory was greeted mostly with indifference or with scepticism. Yet it led to the form given by Robert Anderson to inequalities on the deviation of core allocations from competitive allocations, which are central to the theory of the core. In the article published by Anderson in 1978 those inequalities are stated and proved in an elementary manner, but their expression was found by means of Non-standard Analysis.

The differential calculus, which had been used earlier on too broad a spectrum of economic problems, turned out in the 1970s to supply the proper mathematical machinery for the study of the set of competitive equilibria of an economy. A partial explanation of the observed state of an economic system had been provided by proofs of existence of equilibrium based on fixed point theorems. A more complete explanation would have followed from persuasive assumptions on a mathematical model of the economy ensuring uniqueness of equilibrium. Unfortunately the assumptions proposed to that end were excessively stringent, and the requirement of global uniqueness had to be relaxed to that of local uniqueness. Even then an economy composed of agents on their best mathematical behaviour (for instance each having a concave utility function and a demand function both indefinitely differentiable) may be ill-behaved and fail to have locally unique equilibria. If one considers the question from the generic viewpoint, however, one sees that the set of those ill-behaved economies is negligible. This time the ideal mathematical tool for the proof of that assertion is Sard’s theorem of 1942 on the set of critical values of a differentiable function. By providing appropriate techniques for the study of the set of equilibria, differential topology and global analysis came to occupy in mathematical economics a place that seemed to have been long reserved for them.

As new fields of mathematics were introduced into economic theory and solved some of its fundamental problems, a growth-generating cycle operated. The mathematical interest of the questions raised by economic theory attracted mathematicians who in turn made the subject mathematically more interesting. The resulting expansion of mathematical economics was unexpectedly rapid. Attempting to quantify it, one can use as an index the total number of pages published yearly by the five main periodicals in the field: Econometrica and the Review of Economic Studies (which both started publishing in 1933), the International Economic Review (1960), the Journal of Economic Theory (1969), and the Journal of Mathematical Economics (1974). The graph of that index is eloquent. It shows a first phase of decline to 1943, followed by a 33-year period of exuberant, nearly exponential growth. The annual rate of increase that would carry the index exponentially from its 1944 level to its 1977 level is 8.2 per cent, a rate that implies doubling in slightly less than nine years and that cannot easily be sustained. The years 1977–84 have indeed marked a pause that will soon resemble a stagnation phase if it persists. Among its imperfections the index gives equal weights to Econometrica, the Review of Economic Studies, and the International Economic Review, all of which publish articles on econometrics as well as on mathematical economics, and to the Journal of Economic Theory and the Journal of Mathematical Economics, which do not. But given lower relative weights to the first three yields even higher annual rates of exponential growth of the index for the period 1944–77.

The sweeping movement that took place from 1944 to 1977 suggests an inevitable phase in the evolution of mathematical economics. The graph illustrating that phase hints at the deep transformation of departments of economics during those 33 years. It also hints at the proliferation of discussion papers and at the metamorphosis of professional journals like the American Economic Review, which was almost pure of mathematical symbols in 1933 but had lost its innocence by the late 1950s (Fig. 1).

Mathematical Economics, Fig. 1
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Number of pages published yearly by the leading journals in mathematical economics (Econometrica (abbr. Eta), Review of Economic Studies, (For the first 29 years the Review of Economic Studies was published on an academic rather than on a calendar year basis. As a result, only one issue appeared in 1933, compared with three in 1934; hence the spurious initial increase in the graph.) International Economic Review, Journal of Economic Theory, Journal of Mathematical Economics)

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II. As a formal model of an economy acquires a mathematical life of its own, it becomes the object of an inexorable process in which rigour, generality and simplicity are relentlessly pursued.

Before 1944, articles on economic theory only exceptionally met the standards of rigour common in mathematical periodicals. But several of the exceptions were outstanding, among them the two papers of von Neumann of 1928 and of 1937, and the three papers of Abraham Wald of 1935–6 on the existence of a general economic equilibrium. In 1944 the Theory of Games and Economic Behavior gained full rights for uncompromising rigour in economic theory and prepared the way for its axiomatization. An axiomatized theory first selects its primitive concepts and represents each one of them by a mathematical object. For instance the consumption of a consumer, his set of possible consumptions and his preferences are represented respectively by a point in the commodity space, a subset of the commodity space and a binary relation in that subset. Next, assumptions on the objects representing the primitive concepts are specified, and consequences are mathematically derived from them. The economic interpretation of the theorems so obtained is the last step of the analysis. According to this schema, an axiomatized theory has a mathematical form that is completely separated from its economic content. If one removes the economic interpretation of the primitive concepts, of the assumptions and of the conclusions of the model, its bare mathematical structure must still stand. This severe test is passed only by a small minority of the papers on economic theory published by Econometrica and by the Review of Economic Studies during their first decade.

The divorce of form and content immediately yields a new theory whenever a novel interpretation of a primitive concept is discovered. A textbook illustration of this application of the axiomatic method occurred in the economic theory of uncertainty. The traditional characteristics of a commodity were its physical description, its date, and its location when in 1953 Kenneth Arrow proposed adding the state of the world in which it will be available. This reinterpretation of the concept of a commodity led, without any formal change in the model developed for the case of certainty, to a theory of uncertainty which eventually gained broad acceptance, notably among finance theorists.

The pursuit of logical rigour also contributed powerfully to the rapid expansion of mathematical economics after World War II. It made it possible for research workers to use the precisely stated and flawlessly proved results that appeared in the literature without scrutinizing their statements and their proofs in every detail. Another cumulative process could thus gather great momentum.

The exact formulation of assumptions and of conclusions turned out, moreover, to be an effective safeguard against the ever-present temptation to apply an economic theory beyond its domain of validity. And by the exactness of that formulation, economic analysis was sometimes brought closer to its ideology-free ideal. The case of the two main theorems of welfare economics is symptomatic. They respectively give conditions under which an equilibrium relative to a price system is a Pareto optimum, and under which the converse holds. Foes of state intervention read in those two theorems a mathematical demonstration of the unqualified superiority of market economies, while advocates of state intervention welcome the same theorems because the explicitness of their assumptions emphasizes discrepancies between the theoretic model and the economies that they observe.

Still another consequence of the axiomatization of economic theory has been a greater clarity of expression, one of the most significant gains that it has achieved. To that effect, axiomatization does more than making assumptions and conclusions explicit and exposing the deductions linking them. The very definition of an economic concept is usually marred by a substantial margin of ambiguity. An axiomatized theory substitutes for that ambiguous concept a mathematical object that is subjected to definite rules of reasoning. Thus an axiomatic theorist succeeds in communicating the meaning he intends to give to a primitive concept because of the completely specified formal context in which he operates. The more developed this context is, the richer it is in theorems, and in other primitive concepts, the smaller will be the margin of ambiguity in the intended interpretation.

Although an axiomatic theory may flaunt the separation of its mathematical form and its economic content in print, their interaction is sometimes close in the discovery and elaboration phases. As an instance, consider the characterization of aggregate excess demand functions in an l-commodity exchange economy. Such a function maps a positive price vector into an aggregate excess demand vector, and Walras’ Law says that those two vectors are orthogonal in the Euclidean commodity-price space. That function is also homogeneous of degree zero. For a mathematician, these are compelling reasons for normalizing the price vector so that it belongs to the unit sphere. Then aggregate excess demand can be represented by a vector tangent to the sphere at the price vector with which it is associated. In other words, the aggregate excess demand function is a vector field on the positive unit sphere. Hugo Sonnenschein conjectured in 1973 that any continuous function satisfying Walras’ Law is the aggregate excess demand function of a finite exchange economy. A proof of that conjecture (Debreu 1974) was suggested by the preceding geometric interpretation since any vector field on the positive unit sphere can be written as a sum of l elementary vector fields, each one obtained by projecting a positive vector on one of the l coordinate axes into the tangent hyperplane. There only remains to note that every continuous elementary vector field is the excess demand function of a mathematically well-behaved consumer. Mathematical form and economic content alternately took the lead in the development of this proof.

The pursuit of generality in a formalized theory is no less imperative than the pursuit of rigour, and the mathematician’s compulsive search for ever weaker assumptions is reinforced by the economist’s awareness of the limitations of his postulates. It has, for example, expurgated superfluous differentiability assumptions from economic theory, and prompted its extension to general commodity spaces.

Akin in motivation, execution and consequences is the pursuit of simplicity. One of its expressions is the quest for the most direct link between the assumptions and the conclusions of a theorem. Strongly motivated by aesthetic appeal, this quest is responsible for more transparent proofs in which logical flaws cannot remain hidden, and which are more easily communicated. In extreme cases the proof of an economic proposition becomes so simple that it can dispense with mathematical symbols. The first main theorem of welfare economics, according to which an equilibrium relative to a price system is a Pareto optimum, is such a case.

In the demonstration, we study an economy consisting of a set of agents who have collectively at their disposal positive amounts of a certain number of commodities and who want to allocate these total resources among themselves. By the consumption of an agent, we mean a list of the amounts of each commodity that he consumes. And by an allocation, we mean a specification of the consumption of each agent such that the sum of all those individual consumptions equals the total resources. Following Pareto, we compare two allocations according to a unanimity principle. We say that the second allocation is collectively preferred to the first allocation if every agent prefers the consumption that he receives in the second to the consumption that he receives in the first. According to this definition, an allocation is optimal if no other allocation is collectively preferred to it. Now imagine that the agents use a price system, and consider a certain allocation. We say that each agent is in equilibrium relative to the given price system if he cannot satisfy his preferences better than he does with his allotted consumption unless he spends more than he does for that consumption. We claim that an allocation in which every agent is in equilibrium relative to a price system is optimal. Suppose, by contradiction, that there is a second allocation collectively preferred to the first. Then every agent prefers his consumption in the second allocation to his consumption in the first. Therefore the consumption of every agent in the second allocation is more expensive than his consumption in the first. Consequently the total consumption of all the agents in the second allocation is more expensive than their total consumption in the first. For both allocations, however, the total consumption equals the total resources at the disposal of the economy. Thus we asserted that the value of the total resources relative to the price system is greater than itself. A contradiction has been obtained, and the claim that the first allocation is optimal has been established.

This result, which provides an essential insight into the role of prices in an economy and which requires no assumption within the model, is remarkable in another way. The two concepts that it relates might have been isolated, and its symbol-free proof might have been given early in the history of economic theory and without any help from mathematics. In fact that demonstration is a late by-product of the development of the mathematical theory of welfare economics. But to economists who have even a casual acquaintance with mathematical symbols, the previous exercise is not more than an artificial tour de force that has lost the incisive conciseness of a proof imposing no bar against the use of mathematics. That conciseness is one of the most highly prized aspects of the simplicity of expression of a mathematized theory.

In close relationship with its axiomatization, economic theory became concerned with more fundamental questions and also more abstract. The problem of existence of a general economic equilibrium is representative of those trends. The model proposed by Walras in 1874–7 sought to explain the observed state of an economy as an equilibrium resulting from the interaction of a large number of small agents through markets for commodities. Over the century that followed its publication, that model came to be a common intellectual framework for many economists, theorists as well as practitioners. This eventually made it compelling for mathematical economists to specify assumptions that guarantee the existence of the central concept of Walrasian theory. Only through such a specification, in particular, could the explanatory power of the model be fully appraised. The early proofs of existence of Wald in 1935–6 were followed by a pause of nearly two decades, and then by the contemporary phase of development beginning in 1954 with the articles of Arrow and Debreu, and of Lionel McKenzie.

In the reformulation that the theory of general economic equilibrium underwent, it reached a higher level of abstraction. From that new viewpoint a deeper understanding both of the mathematical form and of the economic content of the model was gained. Its role as a benchmark was also perceived more clearly, a role which prompted extensions to incomplete markets for contingent commodities, externalities, indivisibilities, increasing returns, public goods, temporary equilibrium, … .

In an unanticipated, yet not unprecedented, way greater abstraction brought Walrasian theory closer to concrete applications. When different areas of the field of computable general equilibrium were opened to research at the University of Oslo, at the Cowles Foundation, and at the World Bank, the algorithms of Scarf included in their lineage proofs of existence of a general economic equilibrium by means of fixed point theorems. This article has credited the mathematical form of theoretic models with many assets. Their sum is so large as to turn occasionally into a liability, as the seductiveness of that form becomes almost irresistible. In its pursuit, research may be tempted to forget economic content and to shun economic problems that are not readily amenable to mathematization. No attempt will be made here, however, to draw a balance sheet, to the debit side of which justice would not be done. Economic theory is fated for a long mathematical future, and in other editions of Palgrave authors will have the opportunity, and possibly the inclination, to choose as a theme ‘Mathematical Form vs. Economic Content’.

First published in Econometrica, November 1986, with revisions.

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