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Random walk and life philosophy.


As a particle performs a random walk in a high-dimensional space, an observer may discover a subspace in which the projection of its path approximates a straight line. The observer may then be tempted to anthropomorphize the particle, and to believe that it has "a system which a person forms for the conduct of life."(1) In an inversion of roles, a scientist or a humanist who is asked to expound his life philosophy must feel inclined to identity with that particle if he is aware of the many chance events that shaped his career, and of the inchoate system that he formed for its conduct as it began.

That very beginning I owe to a high school teacher who gave me my first glimpse of the austere beauty of mathematics and my first mathematical experiences. For several years, Jules Dermie challenged the ingenuity of my class with the geometric puzzles that were then at the core of the French curriculum. A seductive subject presented by a dedicated master at a critical time in my intellectual development pointed me, unaware, toward a career of scientific research. Chance intervened again when, in my last year at the high school I had been attending in Calais, an uninspiring teacher of mathematics and an inspiring teacher of physics, Albert Javelle, shifted my allegiance from the first subject to the second and when, at the end of that year, my school sent me to participate, in the field of physics, in one of the national competitions of which the French are fond. In 1939, the Concurs General was a venerable institution dating back to the first half of the 18th century, but a year before, Louis Mandel, after becoming minister of colonies, had added munificent awards to the symbolic value of its prizes. Thus in late August 1939, in Bordeaux, I boarded a liner heading for Dakar. The travel program was to include a rail trip from Dakar to Bamako, a journey by more primitive means to Abidjan, and a return sea voyage to Bordeaux. So exotic an experience and the sudden perception of the unimagined possibilities that the future held would have elated an adolescent brought up in a provincial town if his joy had not been tempered by the storm that was about to break over Europe. By the time the liner reached Dakar, World War II had started for France. Passenger ships were commandeered as troop transports, and the return voyage was made on a mixed freighter slowly steaming for Marseilles.

The next step in my walk, as I began to study for admission to one of the Grandes Ecoles, was not random. But the conditions for that first year of study would have been unpredictable a few weeks earlier. An improvised curriculum had been set up in the small town of Ambert in the Massif Central. There, Joseph Coissard, a retired teacher called back to active duty, communicated his enthusiasm to his students as he initiated them into college mathematics. The isolation of the Ambert novitiate often made it possible to forget that France was at war. It also made the defeat that brought the school year to an end stunning.

Entering the Ecole Normale Superieure in the fall of 1941 meant another initiation, this time into living science. The three years during which I studied and lived at the Ecole Normale were rich in revelations. Nicolas Bourbaki was beginning to publish his Elements de Mathematique, and his grandiose plan to reconstruct the entire edifice of mathematics commanded instant and total adhesion. Henri Cartan, who represented him at the Ecole Normale, influenced me then as no other faculty member did.

The student body of some one hundred and fifty men formed a microcosm that the pressure of the dark outside world of Paris under German occupation contained in a state of permanent implosion. Daily interaction and the diversify of their fields of study in the humanities and the sciences helped to create a charged intellectual atmosphere that I did not experience again elsewhere with the same intensity. Another experience remained unique, that of living in a totalitarian state that does not concede any right to its subjects. It made walking through Paris in 1943-44, with papers that were not in perfect order, a game of evasion with a high payoff.

The new levels of abstraction and of purity to which the work of Bourbaki was raising mathematics had won a respect that was not to be withdrawn. But, by the end of 1942, I began to question whether I was ready for a total commitment to an activity so detached from the real world, and during the following year I explored several alternatives. Economics was one of them. In 1943-44 the teaching of the subject in French universities paid little attention to theory, and the first textbook that I undertook to read reflected this neglect. The distance between the pedestrian approach I was invited to follow, and the ever higher flight I had been riding for several years looked immense, perhaps irreducible. Reason counseled retreat to a safe course. What kept me on an unreasonable heading? The formless feeling that the intellectual gap could be bridged? The wishful thought that the end of the war was near, and the perception that economists had a contribution to make to the task of reconstruction that would follow? An improbable event brought my search to a close. Maurice Allais, whose A la Recherche d'une Discipline Economique had appeared in 1943, sent copies of his book to several class presidents at the Ecole Normale. One of them, Jacques Bompaire, knew of my interest in the application of mathematics to economics, an interest that, as a Hellenist, he did not share. In the copy that he gave me in the spring of 1944, I discovered the theory of general economic equilibrium, and found a scientific vacation.

But several years would elapse before an observer of my wandering course could reasonably believe that it would not take yet another direction. As the war was nearing its conclusion, the French army gave me the opportunity, unrepeated to this day, of experiencing life outside the academic cocoon. Then in 1945-46, I had to go through the scholastic exercise of the agregation de mathematiques, rendered specially pointless by my new heading. After that, the way was clear for the conversion from mathematics to economics which the Centre National de la Recherche Scientific made possible by its tolerance of the absence of results of one of its research associates. In 1948, at a critical bifurcation point, chance occurred in its purest form. The Rockefeller Foundation had earmarked a fellowship for a young French economist. Maurice Allais, whose recommendation was to be decisive, brought together in his office the two candidates, Marcel Boiteux and me, and suggested that a coin choose between us. The winner would spend the year 1949 in the U.S. and the loser the year 1950. The coin rotated on a table for a long time, and eventually decided that I would leave Paris at the end of 1948. But before that, my visit to the United States was prepared by a summer at the Salzburg Seminar in American Studies. From Wassily Leontief and Robert Solow I learned about developments in economics from which France had been cut off, and in the Schloss Leopoldskron library I started reading Theory of Games and Economic Behavior.

The Rockefeller Fellowship I had won in a game of heads or tails took me on a discovery tour of American universities in 1949, and to Uppsala and Oslo in the first four months of 1950. It led my path to another pivotal point in Chicago in the fall of 1949 when Tjalling Koopmans, then director of research of the Cowles Commission, invited me to join his group.


Shortly after my association with Cowles began on June 1, 1950, the methodology that became explicit a few years later started to delineate itself. A central tenet was respect for mathematical rigor in economic theory.

In the early fifties, the tolerance of mathematical economists for approximate rigor in their field strongly tempted one to overlook bothersome, apparently insignificant technical points, and Econometrica published articles then that would not go past its referees now. In that climate, it was easy to stray from the narrow path that I had followed until my formal study of mathematics ended in 1946. In contrast, the Cowles Commission, by welcoming strict intellectual discipline, prompted one to look at economic theory from the viewpoint of a mathematician. An even more compelling motivation came from the nature of two problems on which I began to work. The first was the existence of a utility function representing a preference relation. The second was the existence of a general economic equilibrium. In either case the propose solution would be pointless if an irremediable defect invalidated the argument claiming to prove that its assumptions ensured existence.

The rewards of allegiance to rigor were many. It helped one to choose the mathematical tools most appropriate for a particular question of economic theory. Taking the uncompromising stance of a mathematician, one also shared in his insights into the behavior of mathematical objects, in his drive for ever weaker assumptions and ever stronger conclusions, and in his compulsive quest for simplicity.

The studies of welfare economics by means of convex analysis that Kenneth Arrow and I separately published in 1951 had already left a lasting personal imprint on these points. By 1950, the characterization of Pareto optima had been the subject of an extensive literature whose reliance on the differential calculus had several drawbacks. The differentiability assumptions on which it rested did not allow, for instance, for consumers who do not consume some commodities. Nor did they allow for producers whose technologies are generated by a finite number of elementary processes. The alternative analysis by means of convexity showed that those disturbing assumptions were superfluous, in a proof of the second theorem of welfare economics that can be summarized in two steps. Given a Pareto-optimal allocation of the resource-vector e of an economy, define the set E in the commodity space by adding the preferred sets of all the consumers, and by subtracting the production sets of all the producers. A hyperplane H through the point e supporting the set E yields an intrinsic price vector orthogonal to H. The derivation so obtained was at the same time simpler and more insightful, more general, and more rigorous.

Powerful reinforcement was provided by the two existences problems, whose solutions did not tolerate fudging of rigor. The former gave rise to the first of a series of articles on preference, utility, and demand theory. The aim of that initial paper was to give topological conditions on the space of actions of an agent implying that a closed preference relation can be represented by a continuous utility function. The latter eventually led to the article on the existence of a general equilibrium that Kenneth Arrow and I published jointly in 1954, after having worked independently on that question until Tjalling Koopmans brought us together. The paper is based on the presentation of a general equilibrium as a Kakutani fixed point. But first the concepts of an economy and of a general equilibrium had to be defined with the precision that an existence proof requires. The reexamination of the theory of general equilibrium that was entitled must be credited to the pursuit of rigor.

The developing mathematization of the theory of general equilibrium was giving it an axiomatic form in which the structure of the theory could be entirely divorced from its interpretations. The value of that axiomatization was illustrated when Kenneth Arrow conceived the idea of a contingent commodity whose delivery is conditional on a specified state of the world. In a paper I wrote at Electricite de France in 1953 and elaborated as the last chapter of my Theory of Value, a theory of general equilibrium under uncertainty, formally identical to the theory under certainty, was obtained by the reinterpretation of the concept of a commodity along this line.

The Cowles Commission provided an ideal environment for research on each one of those problems. The work of any member was of interest to all. Opportunities for interaction arose every week in staff meetings and every fortnight in seminars. The compact cluster of offices, some of which were shared, created many other occasions for exchange of ideas. But the state of the economics profession in that period also helped. The small number of working papers to read, and of colloquia to attend, the general lack of interest in exacting mathematical approaches gave one freedom in choosing problems, and time to work on them.


As the fifties drew to a close, Edgeworth's contract curve was retrieved from a neglect of nearly eighty years by an article of Martin Shubik which set off the contemporary theory of the core of an economy. After Herbert Scarf had proved a first extension of Edgeworth's limit theorem, we were given a chance to collaborate on another generalization in the winter of 1962 when I moved from New Haven to Berkeley and before he left Stanford for Yale. From this brief overlap resulted the article that we published in 1963. The study of the core did more than elucidate the function of prices in bringing about allocations of the resources of an economy that are stable relative to coalition formation. It led to the introduction of powerful mathematical techniques into economic theory when the concept of a large set of small agents was formalized by means of measure theory by Robert Aumann in 1964 and by means of non-standard analysis by Donald Brown and Abraham Robinson in 1972. It also led to the definition of a distance between two preference relations giving a precise meaning to the notion of their similarity by Yakar Kannai in 1970. The idea sown by Edgeworth in 1881, and long dormant, had suddenly entered a phase of explosive growth, and the solutions of several problems of economic theory progressed rapidly. An insight of fundamental importance was made clear by the measure-theoretical framework when integration of a family of arbitrary sets over a space of insignificant agents was seen to yield an aggregate convex set. In several results central to economic theory, convexity was needed only for aggregate sets. In large economies made up of small agents, convexity assumptions on their individual characteristics could therefore be dispensed with. Karl Vind, also in 1964, presented that aggregation property as a direct consequence of Lyapunov's theorem on the range of an atomless vector measure.

The question of determinacy of general equilibrium had received partial answers in the form of various existence theorems. A more complete solution would have been provided by additional conditions implying that there is exactly one equilibrium. But the requirement of global uniqueness revealed itself to be excessively demanding. Requiring only that every equilibrium be locally unique was not a sufficient weakening since one can display in Edgeworth's box an economy having a continuum of equilibrium, even though each one of its agents is mathematically exceeding well behaved. Taking a generic viewpoint, however, one could show that almost every differentiable economy has a set of locally unique equilibria. The basis of the proof was Sard's theorem on the set of critical values of a differentiable function, of which I learned in the late sixties in a first encounter with Steve Smale, whose path had remained separate from mine during the period of campus turbulence that began in September 1964. Differentiability assumptions that had been expurgated earlier from economic theory when they were irrelevant were essential now for the generic discreteness of the set of equilibria. One of the by-products of the scrutiny of those assumptions was a reconsideration of the problem of representation of a preference relation by a differentiable utility function. An alternative to the usual approach via the Frobenius integrability conditions was proposed in an article of 1972 in which I defined a differentiable preference relation by the requirement that the indifferent pairs of commodity vectors from a differentiable manifold.

Basic to the theory of general equilibrium are the concept of the aggregate excess demand function of an economy and its properties. In an exchange economy, the individual excess demand of a consumer is a function f of prices obtained by subtracting his endowment-vector from the commodity vector that he demands. Under customary assumptions on his preferences, (1) f is continuous, and (2) for every price vector p, the value p * f(p) of f(p) relative to p vanishes. Since these two properties are preserved by summation over a set of consumers, the aggregate excess demand function F of the economy also has properties (1) and (2). In 1972 and 1973, Hugo Sonnenschein conjectured that, conversely, any function F satisfying (1) and (2) can be written as a sum of individual excess demand functions generated by traditional preferences, and made the first attack on the problem he had posed. The proof given by Rolf Mantel in 1974 was based on differentiability assumptions. It was followed, in the same year, by another proof in which I dispensed with differentiability and characterized the aggregate excess demand function of an exchange economy by means of an additive decomposition with the minimal number of consumers. The paucity of properties enjoyed by the aggregate excess demand function, expressed by this result, threw a hard light on several aspects of general equilibrium models. But it also suggested endowing them with a riche structure by imposing restrictions on the distribution of the characteristics of agents. A significant step in this direction was taken by Werner Hildebrand in 1983 in his study of the Law of Demand.

Leaving the Cowles Foundation for the University of California at the end of 1961, I had made a move to a new environment that proved to be more than another powerful research stimulant. It also expanded a teaching experience that had been limited to specialized graduate courses. Essential dimensions were now added by an exceptional succession of doctoral students and of younger colleagues, and I discovered that there was pleasure in lecturing to undergraduates.


In the phase of exuberant growth that mathematical economics entered in 1944, close scrutiny of its detail sometimes blurred its dominant features. Bringing them into focus in the analysis of general equilibrium stresses, as one of its central themes, the study of the multiple function of prices -- notably their roles in the efficiency of resource allocation, in the equality of demand and supply, and in the stability of allocations with respect to coalition formation. Explaining these functions raises major scientific questions. Any attempt at answering them must take into account the large number of agents, the large number of commodities, and prices, and the interdependence of the many variables involved. Only a mathematical model can deal with these characteristics. Once such a model is formulated, appeal to deductive reasoning requires an explicit set of assumptions as its foundation. Those assumptions delimit the domain of validity of the theory that is based on them. At a given moment in the development of economic analysis, they are the outcome of a continuing cumulative weakening process.

From that perspective, engaging in controversy over the merits and demerits of the theory of general economic equilibrium has low priority in contrast with participating in its construction. In this task, mathematical form powerfully contributes to defining a philosophy of economic analysis whose major tenets include rigor, generality and simplicity. It commands the long search for the most direct secure routes from assumptions to conclusions. It dictates its aesthetic code, and it imposes its terse language. Another tenet of that philosophy is recognition, and acceptance, of the limits of economic theory, which cannot achieve a grand unified explanation of economic phenomena. Instead, it adds insights to the perception of the areas to which it turns its search. When they are gained by accepting mathematical challenges, those insights are the highest prizes sought by a mathematical economist.


(1)A New English Dictionary on Historical Principles, vol. vii, Part II, James A. H. Murray, ed., "Philosophy, 9.a," Oxford; Clarendon Press, 1909. Gerard Debreu is a University Professor, and Class of 1958 Professor of Economics and of Mathematics, University of California at Berkeley. He is also the 1983 winner of the Nobel Memorial Prize in Economic Science.
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Author:Debreu, Gerard
Publication:American Economist
Date:Sep 22, 1991
Next Article:Life and philosophy.

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