The Mathematization of Economic Theory

Gerard Debreu

The American Economic Review, Vol. 81, No. 1. (Mar., 1991), pp. 1-7.

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http://www.jstor.org Thu Aug 23 23:26:40 2007 Number 92 of a series of photographs of past presidents of the Association

The Mathematization of Economic Theoryt

Morgenstern published the Theory of Games and Economic Behavior. As the Second World War was drawing While the professional journals in the field near its resolution, economic theory entered of mathematical economics grew at an un- a phase of intensive mathematization that sustainably rapid rate, the American Eco- profoundly transformed our profession. In nomic Review underwent a radical change several of its main features that phase had in identity. In 1940, less than 3 percent of no precedent, and it will have no successor. the refereed pages of its 30th volume ven- Assessing it requires a multidimensional tured to include rudimentary mathematical analysis acknowledging the contributions to expressions. Fifty years later, nearly 40 per- economics that were made, as well as the cent of the refereed pages of the 80th vol- tensions among economists that were ume display mathematics of a more elabo- heightened. rate type. The development of mathematical eco- At the same time, the mathematization of nomics during the past half-century can be economists proceeded at an even faster pace read in the total number of pages published in the 13 American de~artmentsof eco- each year by the leading periodicals in the nomics labeled by a recent assessment of field, an index that I will follow at first. research-doctorate programs in the United From 1933, the date when they both started States (Lyle V. Jones et al., 1982) as "dis- publication, to 1959, those periodicals were tinguished" or "strong" according to the Econometrica and the Review of Economic scholarly quality of their faculties. Every Studies, and the index tells of the decline year the Fellows of the Econometric Society from a high point, above 700 pages in 1935 (ES) certify new members by election into to the lowest point, below 400 pages in their international guild, which increased in 1943-1944. But 1944 marked the beginning size from 46 in 1940 to 422 in 1990. For of a period of explosive growth in which those 13 departments together, the propor- Econometrica and the Review of Economic tion of ES Fellows among professors was Studies were joined in 1960 by the Interna- less than 1 percent in 1940; it is now close tional Economic Review, in 1969 by the to 50 percent. It equals or exceeds 50 per- Journal of Economic Theory, and in 1974 by cent for six of them, which were among the Journal of Mathematical Economics. In those assessed as the eight strongest. So 1977, these five periodicals together pub- mathematized a faculty expects its students lished over 5,000 pages. During the period to have what it considers to be minimal 1944-1977, the index more than doubled mathematical proficiency, and knowledge of every nine years. By that measure, 1944 was calculus and linear algebra is required, or a sharp turning point in the history of math- forcefully recommended, for admission to ematical economics. It was also the year in all 13 graduate programs. which John von Neumann and Oskar Several scholarly recognitions lay addi- tional emphasis on the role that mathemati- cal culture is now playing in our profession. Of the 152 members of the economics sec- 'presidential address delivered at the one-hundred tion of the American Academy of Arts and third meeting of the American Economic Association, December 29, 1990, Washington, DC. Sciences, 87 are Fellows of the Econometric *Department of Economics, University of California Society; and of the 40 members of the eco- at Berkeley, Berkeley, CA 94720. nomics section of the National Academy of 2 THE AMERICAN ECONOMIC REWEW MARCH 1991

Sciences of the United States, 34 are ES been an inaccessible ideal toward which Fellows. From 1969 to 1990, 30 economics economic theory sometimes strove. During Nobel awards were made, and 25 of the that period, this striving became a powerful laureates are, or were, ES Fellows. Since it stimulus in the mathematization of eco- was first presented to Paul Samuelson in nomic theory. 1947, the John Bates Clark medal of the The great theories of physics cover an American Economic Association has been immense range of phenomena with a given to 21 economists, of whom 20 are ES supreme economy of expression. Of this, Fellows; and of the 26 living past presidents James Clerk Maxwell (1865) had given a of our Association, 13 are ES Fellows. notable example, as he described the elec- One may wish that those counts had not tromagnetic field by means of eight equa- been made. One may argue about points of tions at the time when mathematical eco- their interpretation. But they belong in our nomics was born and came of age in the common knowledge, and their thrust is un- middle of the 19th century. This extreme equivocal. They indicate how extensive the conciseness is made possible by the privi- mathematization of economics and how leged relationship that developed over sev- deep the accompanying change of our field eral centuries between physics and mathe- were over the past five decades. matics. In turn, the former presented the The perception of the depth of that latter with open problems, or found to ques- change is reinforced by a comparison of the tions raised by physical theory ready-made levels of mathematics required in 1940 and answers discovered by mathematicians in in 1990 to follow the development of eco- their abstract universe. Sometimes the nomic theory in every direction it was tak- causal linkage of research done in each one ing. Fifty years ago, basic undergraduate of the two fields could not easily be unrav- preparation in mathematics was almost al- eled; and, on occasion, the same scientist ways sufficient. Today, graduate training in made inextricably intertwined contributions mathematics is necessary. If, instead of be- to both disciplines. ing a follower, one wishes to be an active The benefits of that special relationship participant in that development along its were large for both fields; but physics did most technical avenues, a high degree of not completely surrender to the embrace of mathematical professionalism is called for. mathematics and to its inherent compulsion Several faculty members of the 13 depart- toward logical rigor. The experimental re- ments of economics mentioned previously sults and the factual observations that are at were actually identified as mathematicians the basis of physics, and which provide a by their doctorates; four of them served as constant check on its theoretical construc- chairmen of those departments during the tions, occasionally led its bold reasonings to past 25 years. If still sharper focus brings violate knowingly the canons of mathemati- out the intellectual leaders of that develop- cal deduction. ment, prominent among them is John von In these directions, economic theory could Neumann, one of the foremost mathemati- not follow the role model offered by physi- cians of his generation. cal theory. Next to the most sumptuous In that development process, mathemati- scientific tool of physics, the Superconduct- cal economics was continuously redefined as ing Super Collider whose construction cost new territories were included within its out- is estimated to be on the order of $10"' ward-moving frontier and as topics that were (David P. Hamilton, 1990; see also Science, once at that frontier became standard parts 5 October 19901, the experiments of eco- of the graduate, if not of the undergradu- nomics look excessively frugal. Being denied ate, economic-theory curriculum. a sufficiently secure experimental base, eco- nomic theory has to adhere to the rules of logical discourse and must renounce the facility of internal inconsistency. A deduc- Before the contemporary period of the tive structure that tolerates a contradiction past five decades, theoretical physics had does so under the penalty of being useless, VOL. 81 NO. I DEBREU: MA THEM TIZATION OF ECONOMIC THEORY 3 since any statement can be derived flawlessly commodity (opposite signs differentiating and immediately from that contradiction. inputs from outputs). That list can be treated In its mathematical form, economic the- as the list of the coordinates of a point in ory is open to an efficient scrutiny for logi- the linear commodity space. Similarly, the cal errors. The rigor that has been reached price system of an economy can be treated as a consequence is in sharp contrast to the as a point in the linear price space, dual of standards of reasoning that were accepted the commodity space, whose dimension is in the late 1930's. Few of the articles pub- also the number of commodities. lished then by Econometrica or by the Re- In those two linear spaces, the stage was view of Economic Studies would pass the set for sometimes dazzling mathematical acid test of removing all their economic developments that began with the elements interpretations and letting their mathemati- of differential calculus and linear algebra cal infrastructure stand on its own. The and that gradually called on an ever broader greater logical solidity of more recent analy- array of powerful techniques and funda- ses has contributed to the rapid contempo- mental results offered by mathematics. Thus, rary construction of economic theory. It has the three roles of prices given earlier as enabled researchers to build on the work of instances were illuminated by basic mathe- their predecessors and to accelerate the cu- matical theorems: the first, the achievement mulative process in which they are partici- of an efficient use of resources, by results of pating. convex analysis; the second, the equaliza- But a Grand Unified Theory will remain tion of supply and demand for commodities, out of the reach of economics. which will by results of fixed point theory; the third, keep appealing to a large collection of indi- the prevention of the formation of destabi- vidual theories. Each one of them deals lizing coalitions, by results of the theory of with a certain range of phenomena that it integration and of nonstandard analysis. In attempts to understand and to explain. those three cases, the lag between the date When it acquires an axiomatic form, its of a mathematical discovery and the date of explicit assumptions delimit its domain of its application to economic theory de- applicability and make illegitimate overstep- creased over time. It was notably short for ping of its boundary flagrant. Some of those nonstandard analysis, founded at the begin- theories take a comprehensive view of an ning of the 1960's by Abraham Robinson' economic system and bring insights into the and applied to economics by Donald Brown solutions of several global problems. For and Abraham Robinson (1972). instance, prices contribute to achieving an The last, and most recently developed, of efficient use of resources, to equalizing sup- those three instances can be chosen. as can ply and demand for commodities, and to either of the other two, for a more detailed preventing the formation of destabilizing illustration. Competition is perfect when ev- coalitions. In every case, a theoretical expla- ery agent's influence on the outcome of nation must be provided. The assumptions, economic activity is insignificant. The in- which cannot be satisfied by all economic fluence of their totality on that outcome is, observations, are the present outcome of a however, significant. It is to solve the prob- continuing weakening process. lem of aggregating negligible quantities so A global view of an economy that wants as to obtain a nonnegligible sum that inte- to take into account the large number of its gration was invented. In this perspective, commodities, the equally large number of the application of integration theory to the its prices, the multitude of its agents, and study of economic competition is entirely their interactions requires a mathematical natural. That application requires the set of model. Economists have successfully con- agents to be large-larger than the set of structed such a model because the central integers. Treating the set of the agents of an concept of the quantity of a commodity has economy as the rich collection of the points a natural linear structure. The action of an agent can then be described by listing the quantity of its input or output for each 'see the preface in Robinson (1966). 4 THE AMERICAN ECONOMIC REVIEW MARCH 1991 of an interval of real numbers has long been bibliographical key (John Eatwell et al., familiar in descriptions of economic data. It 1987-1989). Walras's consumers and pro- became familiar in economic theory as well ducers have been freed from many of their after Robert J. Aumann (1964) showed that, constraining characteristics; Edgeworth's in a pure exchange economy composed of universe of two consumers and two com- insignificant agents, the formation of desta- modities has been vastly expanded. bilizing coalitions is prevented if and only if Mathematics also dictates the imperative all those agents base their decisions on a of simplicity. It relentlessly searches for price system. short transparent proofs and for the theo- The concept of a convex set (i.e., a set retical frameworks in which they will be containing the segment connecting any two inserted. Participating in that pursuit, eco- of its points) had repeatedly been placed at nomic theory was sometimes drawn by drives the center of economic theory before 1964. toward greater generality and toward greater It appeared in a new light with the intro- simplicity in the same direction, rather than duction of integration theory in the study of in opposite directions. Cohort after cohort, economic competition: if one associates with students of consumer theory have learned every agent of an economy an arbitrary set about the concept of decreasing marginal in the commodity space and if one averages rate of substitution for two commodities on those individual sets over a collection of an indifference curve and about its exten- insignificant agents, then the resulting set is sion to the multicommodity case. Notably necessarily c~nvex.~But explanations of the more general, and notably simpler, is the three functions of prices taken as examples concept of convexity of the set of points can be made to rest on the convexity of sets preferred to a given point in the commodity derived by that averaging process. Convexity space. Welfare economics presents another in the commodity space obtained by aggre- instance. One of its main theorems formu- gation over a collection of insignificant lates precisely the principle enunciated by agents is an insight that economic theory Adam Smith (1776). If all the agents of an owes in its revealing clarity to integration economy are in equilibrium relative to a theory. price system, then they utilize their collec- An economist who experiences such an tive resources optimally. The proof of that insight belongs to the group of applied theorem (Kenneth J. Arrow, 1951) has be- mathematicians, whose values he espouses. come so simple that it can be given without Mathematics provides him with a language mathematical symbols. It is, at the same and a method that permit an effective study time, of utmost generality; in relating two of economic systems of forbidding complex- basic concepts of economic theory to each ity; but it is a demanding master. It cease- other, it uses no assumption. lessly asks for weaker assumptions, for In its attempts to attain its many objec- stronger conclusions, for greater generality. tives, economic theory was helped by greater In taking a mathematical form, economic abstraction. Preference theory supplies an theory is driven to submit to those demands. example again. Significant research efforts The gains in generality that it has achieved were expended on solutions of the integra- as a result, in little more than a century, bility problem. That problem can be by- stand out when the first formulations of passed altogether, and greater simplicity can the theories of general equilibrium (Lton be achieved by moving from the commodity Walras, 1874-1877) and of the core of an space to the more abstract space of the economy (Francis Y. Edgeworth, 1881 pp. pairs of its points. In this space, whose 34-8) are placed side by side with the re- dimension is twice the number of commodi- cent treatments of those subjects to which ties, the pairs of commodity points indif- The New Palgrave is an introduction and a ferent to each other are now assumed to form a smooth (hyperlsurface. As another 'on this direct consequence of a theorem of A. A. instance of the generality permitted by ab- Lyapunov, see Karl Vind (1964). straction, consider the notion of a commod- VOL. 81 NO. I DEBREU: MA THEMTI24 7'ION OF ECONOMIC THEORY 5 ity, which can be treated as a primitive were speaking, and in spite of the wide concept, with an unspecified interpretation, diffusion of their critiques, neither Leontief in an axiomatic economic theory. A newly nor Gordon altered the course of the devel- discovered interpretation can then increase opment they were assessing. In the past two considerably the range of applicability of decades, economic theory has been carried the theory without requiring any change in away further by a seemingly irresistible cur- its structure. Thus, by making the transfer rent that can be explained only partly by the of a good or service between two agents intellectual successes of its mathematiza- contingent on the state of the world that tion. will obtain, Arrow (1953) made possible the Essential to an attempt at a fuller expla- immediate extension of the economic theory nation are the values imprinted on an of certainty to an economic theory of uncer- economist by his study of mathematics. tainty by a simple reinterpretation of the When a theorist who has been so typed concept of a commodity. The theory of fi- judges his scholarly work, those values do nancial markets has been influenced by that not play a silent role; they may play a deci- view of uncertainty, and their practice has sive role. The very choice of the questions not been unaffected. Finally, take the prob- to which he tries to find answers is influ- lem of existence of a general equilibrium, enced by his mathematical background. once considered to be one of the most ab- Thus, the danger is ever present that the stract questions of economic theory. The part of economics will become secondary, if solutions that were proposed in the early not marginal, in that judgment. 1950's paved the way for the algorithms for The reward system of our profession rein- the computation of equilibria of Herbert E. forces the effects of that autocriticism. De- Scarf (1973) and for several of the develop- cisions that shape the career of an economic ments of applied general equilibrium analy- theorist are made by his peers. Whether sis (Scarf and John B. Shoven, 1984). In this they are referees of a journal or of a re- case, abstraction in economic theory led to search organization, members of an ap- the study of fundamental problems of great pointment or of a promotion committee, generality, but also to a broad range of when they sit as judges in any capacity, their applications. verdicts will not be independent of their own values. An economist who appears in their court rarely ignores his perception of those values. If he believes that they rate The list of advances that the mathemati- mathematical sophistication highly, and if zation of economic theory helped or permit- he can prove that he is one of the sophisti- ted is already long; and in one aspect it may cates, the applause that he expects to re- appear lengthy. Ceteris paribus, one cannot ceive will condition his performance. prefer less to more rigor, lesser to greater The same effects are also amplified by the generality, or complexity to simplicity; but relentless pressure to publish exerted by his other things are not equal, and in the esti- environment. There are indeed instances of mate of many members of our Association extreme restraint in scientific publication, the cost of that mathematization sometimes and some of them have become legend. The outweighs its benefit. Two of its presidential mathematical papers of Bernhard Riemann addresses notably confronted that difficult (1826-1866) take 506 pages in the volume analysis and stressed the price that eco- that collected them (Riemann, 1876). The nomics paid for its increased use of mathe- molecular structure of DNA was announced matics. Wassily Leontief's (1971) observa- by James Watson and Francis Crick (1953) tions were factual, and Robert A. Gordon's in a one-page article. But it is easier to (1976) comments relevant when they were explain those examples away than to follow made in 1970 and in 1975. They still a.re them. The environment of a scholar de- today, for, in spite of their authorities, en- mands papers, and the temptation to supply hanced by the platform from which they them without restraint may become over- 6 THE AMERICAN ECONOMIC REVIEW MARCH 1991 powering to an economic theorist who has ments of the phase that economic theory developed proficiency in his research style. underwent and the effectiveness of attempts The precocious development of that profi- to alter the course of its evolution will gain ciency is a comparative advantage that a from a detailed analysis of the processes mathematical approach bestows on him. that led to its present state. The spread of mathematized economic theory was helped even by its esoteric char- REFERENCES acter. 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2 Edgeworth-Allocations in an Exchange Economy with Many Traders Karl Vind International Economic Review, Vol. 5, No. 2. (May, 1964), pp. 165-177. Stable URL: http://links.jstor.org/sici?sici=0020-6598%28196405%295%3A2%3C165%3AEIAEEW%3E2.0.CO%3B2-4

References

Markets with a Continuum of Traders Robert J. Aumann Econometrica, Vol. 32, No. 1/2. (Jan. - Apr., 1964), pp. 39-50. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28196401%2F04%2932%3A1%2F2%3C39%3AMWACOT%3E2.0.CO%3B2-6

A Limit Theorem on the Cores of Large Standard Exchange Economies Donald J. Brown; Abraham Robinson Proceedings of the National Academy of Sciences of the United States of America, Vol. 69, No. 5. (May, 1972), pp. 1258-1260. Stable URL: http://links.jstor.org/sici?sici=0027-8424%28197205%2969%3A5%3C1258%3AALTOTC%3E2.0.CO%3B2-W

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Rigor and Relevance in a Changing Institutional Setting Robert Aaron Gordon The American Economic Review, Vol. 66, No. 1. (Mar., 1976), pp. 1-14. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197603%2966%3A1%3C1%3ARARIAC%3E2.0.CO%3B2-W

The SSC Takes On a Life of Its Own David P. Hamilton Science, New Series, Vol. 249, No. 4970. (Aug. 17, 1990), pp. 731-732. Stable URL: http://links.jstor.org/sici?sici=0036-8075%2819900817%293%3A249%3A4970%3C731%3ATSTOAL%3E2.0.CO%3B2-0

Theoretical Assumptions and Nonobserved Facts Wassily Leontief The American Economic Review, Vol. 61, No. 1. (Mar., 1971), pp. 1-7. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197103%2961%3A1%3C1%3ATAANF%3E2.0.CO%3B2-7

A Dynamical Theory of the Electromagnetic Field J. Clerk Maxwell Philosophical Transactions of the Royal Society of London, Vol. 155. (1865), pp. 459-512. Stable URL: http://links.jstor.org/sici?sici=0261-0523%281865%29155%3C459%3AADTOTE%3E2.0.CO%3B2-A

Edgeworth-Allocations in an Exchange Economy with Many Traders Karl Vind International Economic Review, Vol. 5, No. 2. (May, 1964), pp. 165-177. Stable URL: http://links.jstor.org/sici?sici=0020-6598%28196405%295%3A2%3C165%3AEIAEEW%3E2.0.CO%3B2-4

NOTE: The reference numbering from the original has been maintained in this citation list.