Gerard Debreu

Gerard Debreu( 1921– 2004) was a French-born economist and mathematician who became a naturalized citizen of the United States)in 1975. Best known as a professor of economics, and, from 1985, mathematics, at the University of California, Berkeley where he began work in 1962.

In 1983 he won the Nobel Memorial Prize in Economics, “…… because his work was of ‘vital importance for the understanding of the market’ ……”(press release). Even if not exactly celebrated in certain quarters, economists profited from Debreu since he pushed the believe in and fostered the reputation of economics as a science and, roughly in other words, for "…….having proven the invisible hand of Smith…..".

Biography

He was born in Calais on July 4, 1921. His father was the business partner of his maternal grandfather in lace manufacturing, a traditional industry in Calais.

Just prior to the start of World War II he received his baccalauréat, and went to Ambert to begin preparing for the exam for entering a grande école. Later on he moved from Ambert to Grenoble to complete his preparation, both being in the so-called "Free Zone" during World War II.

In 1941 he was admitted to the École Normale Supérieure in Paris, along with Marcel Boiteux. He was totally, and forever, influenced by his mathematics teacher Henri Cartan and the Bourbaki circle.

Debreu’s Bourbaki roots

The encounter with Henri Cartan alias Bourbaki, as his mathematics teacher, shaped Debreu’s image of mathematics and impressed his intellectual ethos for all years to come, and in particular insofar Bourbaki ‘commanded instant and total adhesion’. The influence goes so far that everything one could say about Bourbaki, “…..applies with equal force to Gerard Debreu……” ( Weintraub 2002, p. 113 ).

“…..The three years during which I studied and lived at the Ecole Normale were rich in revelations. Nicolas Bourbaki was beginning to publish his Eléments de Mathématique, 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 as no other faculty member did. 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of abstraction and of purity to which the work of Bourbaki was raising mathematics had won a respect that was not to be withdrawn….” (Debreu 1991, p. 3. )

What is Bourbaki? The seven founding members all caming from Ecole Normale: Jean Dieudonne , Claude Chevalley, Szolem Mandelbroijt, Rene de Possel, Jean Delsarte, Andre Weil, and Henri Cartan who brough in Debreu. At their beginnings, keeping their names secret, “Bourbaki” ---as new philosophical and methodological apparatus of finite mathematics became known, and which has, till today, provided the very roots of applied mathematics ---soon was surrounded by mysticism.

In Bourbaki’s words, mathematic forms appear without being “freighted with special intuitive references” (Bourbaki 1950, p. 227) or, in Debreu’s words, without being “…….marred by a substantial margin of ambiguity……” (Debreu 1986, p. 1266).

Within this hierarchy the first three “mother-structures”, as they called them, are topology, order, and algebra. Bourbaki wanted to bring order into mathematics that was diffused in various fields out of which mathematic problems arouse.

They wanted to avoid mathematics “…..becoming a tower of babel……” (Bourbaki 1950, pp. 221, 227), so that mathematics could speak with one voice. The so called “axiomatic method” was one of the tools for it.

As Mandelbrot explained Bourbaki’s “top-down” approach in opposition to more “bottom-up” approaches to mathematics: “……the former tend to be build around one key principle or structure (…) the latter tend to organize themselves around a class of problems……” (Mandelbrot 1989, p. 11).

In this effect, Bourbaki can justly be called ‘the ideology of rigour’ --- and/or proponent of “axiomatic method” that eventually gave rise to Operations Research and other quantitative applications in economics --- Mandelbrot confirmed the political agenda behind that ideology:

“…..Bourbaki showed extraordinarily wide-reaching concern with political influence across the age groups and across the disciplines. Power to school the children ( of which Debreu was one ) , to educate the young to have the ‘correct’ taste (…). And ‘export’ of their standards of rigour and taste they do not belong to has done untold harm……”( ibid. 1989, p.12).

Debreu’s "economic transformation"

Later, under the tutelage of economist Maurice Allais, who was not Bourbakian but a scientist of many-layered interests, Debreu had become interested in economics. That “interest” was slowly transformed into a lifetime dedication when he was introduced into the mathematical theory of general economic equilibrium.

The general economic equilibrium was founded by Léon Walras in 1874-77, and young Debreu came to it via the formulation given by Maurice Allais in his book, A la Recherche d'une Discipline Économique, 1943. In his biography, Debreu writes: “….The two and a half years following the Agrégation were devoted to my conversion from mathematics to economics…..”( Debreu, 1984.)

Still, under the equilibrium spell, he published his first economic article in French ( Debreu 1949 ). This article, which reads like a survey of Hicksean general equilibrium theory, is his least formal article he ever wrote, and he shows considerable respect to the interpretive sensitivity of economic claims. At the end of the article we find one of the rare cases that Debreu made some effort to interpret economic terms. He even made the reader aware of a “certain danger” regarding the welfare interpretation Pareto-optimum, which, at that time, was a center of major discussion between left-leaning economists and the “free-market” ones .

Here, as in many crossroad-points later, he could not simply digest the mathematics because dynamic equations implied computational methods, which as a Bourbakian he cannot consider rigorous. We clearly hear the echo of the dilemma he will be faced in all his years. Again, he is torn between his mathematic values and his interest in making an economic claim.

However, as Bourbaki believed to be good for the working mathematician, Debreu believed that he could help the economist to be more explicit, precise, concise, and simple, to see contradictions more easily, and unravel unnecessary assumptions. He wanted to give economists a feeling for consistency, a sense of rigor and let them participate in the fascination he experienced with Bourbaki. He may have never believed that this sense of consistency could meet the full need of scientification in economics, yet, it certainly should have added something valuable to it. Debreu never conceived the anti-scientific attitude of Bourbaki.

Nevertheless, after Bourbaki’s fascination of an aesthetic void --- than a magic spell for the young student Debreu --- in his real-life work, he was soon faced with a dilemma between pure and mundane reason; the separation of structure and meaning now results in a discreet attitude of Debreu as an economist. Being a mathematical economist for Debreu meant to be discreet.

Debreu’s American career

In 1948, two years after he married Françoise Bled in 1946 and had two daughters --- Chantal and Florence, born in 1946 and 1950 respectively --- Debreu came to the USA on a Rockefeller Fellowship which allowed him to visit several American universities, as well as those in Uppsala and Oslo in 1949-50.

At a seminar in Salzburg he acquired a first taste of the New World when ---after meeting Leontief and Solow ---- started reading the Theory of Games ( encountering thus the use of Bourbaki-proof fix points ). Thereafter, in the summer of 1950, he was approached by Tjalling Koopmans, who just became the Director of Research of the Cowles commission. Debreu was welcome because he would help Koopmans to push “Cowles Mark II”.

Cowles, at that time, was advancing mathematical economics of the rather “theoretical” type; the Bourbakian label, with which Debreu now appeared, was therefore eminently useful, thoroughly discussed and, even emulated.

Debreu thus joined the Cowles Commission at the University of Chicago and became a Cowles research associate in June 1950 for an eleven-year term.

The Bourbakian axiomatic method, serving him very well, certainly represents the peak of abstraction in the history of mathematical economics. It is commonly ascribed to a particular school of thought, “neo-Walrasian” economics. How little, however, Walras and Debreu had in common is obvious if we remember Debreu’s polemic not only against the Walras-Cassel representation of the economy, but mainly against the differential analysis of Pareto and Hicks. The main connector of Debreu with the Walrasian tradition was the rejection of the mathematics that has been applied leaving the economic intuitions rather untouched. We certainly could point to Abraham Wald, one of the members of the Menger colloquium in Vienna, as the mediator between Walrasian and neo-Walrasian economics.

Debreu remained with Cowles for five years, returning to Paris periodically. In 1954 he published a breakthrough paper titled Existence of an Equilibrium for a Competitive Economy (together with Kenneth Arrow), dealing with basic question of whether a preference preorder on a topological space can be represented by a real-valued utility function. In this paper Debreu provided a definitive mathematical proof of the existence of general equilibrium, using topological rather than calculus methods.

“…….The paper of 1954 was indeed closer to von Neumann than to Wald in that it presented an equilibrium as a generalization of a game. It was a paper that could not reach the common interest of economists as for example Hicks and was rather an internal success at Cowles. How difficult it was in 1954 to get the paper accepted, shows the objection of the first referee, who rejected the paper because it was not rigorous!.........” ( Weintraub 2002, ch. 6.)

In 1955 Debreu moved to Yale Universityand in 1959 he published his classical monograph --- henceforth The Monograph --- Theory of Value: An Axiomatic Analysis of Economic Equilibrium, (Cowles Foundation Monographs Series), which is one of the most important works in mathematical economics. He also studied several problems in the theory of cardinal utility, the additive decomposition of a utility function defined on a Cartesian product of sets.

Overview of The Monograph

In The Monograph, Debreu sets up an axiomatic foundation for competitive markets. He establishes the existence of equilibrium using a novel approach. The main idea is to show that there exists a price system for which the aggregate excess demand correspondence vanishes. He does so by proving a type of fixed point theorem based on the Kakutani fixed point theorem. In Chapter 7 of the book Debreu introduces uncertainty and shows how it can be incorporated into the deterministic model. Here he introduces the notion of a contingent commodity, which is a promise to deliver a good should a state of nature realize. This concept is very frequently used in financial economics as a so called Arrow Debreu security.

However, two major questions had to be answered in The Monograph first:

• Uniqueness, that is, the logical determinability ( “is there one equilibrium or could all states be an equilibrium?” ), and, then

• Stability (“does an equilibrium hold more than one moment or are we every moment in another equilibrium?”). For an economist these two questions are essential regarding the intuition of general equilibrium theory.

The issue of stability was so pressing because only then the market “brings about” something, makes a difference, and can be played out in a context where there are also other alternatives to the market. Only then the market matters.

And here Debreu showed his utmost theoretical discreetness; he did not engage in the question of stability ( Ingrao and Israel 1990.) Consider the following reply he gave when asked about “dynamic analysis”. His ambiguous position in economics and the history of economic thought is fully present in these lines. Debreu had reservations about “dynamic analysis”.

“……I had my own reservations about dynamics in spite of the fact that I had studied classical mechanics (…) I thought that the whole question was very facile, and that in economics one did not specify, then test, the dynamic equations that we so easily taken up because of the analogy to classical mechanics. So I was very, always very, suspicious of dynamics and that is a view I have held very consistently (…) I thought about those questions of course as every economist must, but it seemed to me that the contributions made were not important……” ( Weintraub 2002, p. 146 e.a. )

Certainly, Debreu never promoted or even referred to a particular school of mathematics. He never was an outspoken Bourbakist in economics, which was crucial for both, Debreu’s self-image in economics and the silent and creepy influence he had on the discipline. For Debreu Bourbaki is mathematics.

In The Monograph, the Bourbakian document in economics, he refers to the method he applies merely with “…….the contemporary formalist school of mathematics….” ( Bourbaki, 1959). And so the very name of Bourbaki did not enter economists’ consciousness as the word “axiomatic” did, although economists were in full impact of Bourbakian values for at least 20 years. Economists, like Debreu, hardly discuss what kind of mathematics they use. With Debreu economics became mathematized, not bourbakized.

Concluding remarks to The Monograph

Here we come back to the void of Bourbaki’s platonic vision of mathematics, which translates to Debreu’s methodological discreetness:

“……..(W)hen you are out of equilibrium, in economics you cannot assume that every commodity has a unique price because that is already an equilibrium determination…….” (  Weintraub 2002, p. 146)   

Also this statement is as unique as striking: the concept of disequilibrium for Debreu is a contradiction in itself ! If we talk about markets we necessarily always already talk about an equilibrium, since in disequilibria prices have no conceivable identity whatsoever. Equilibrium is tantamount to consistent.

Debreu does thus not avoid speaking about disequilibria over and against the possible fact that we are most of the times in such a state (as the canon goes today), but because it is beyond a consistent economic theory.

The existence of an equilibrium is the condition of the possibility of economic science. This is why all economic models have to proof first the possibility of an equilibrium solution. This is the point where mathematics and “scientification” of economics fall together. As long as economics cannot conceive of economic theory without any reference to an equilibrium, it is Debreuan-Bourbakian.

Debreu in 60s and thereafter

In 1960-61, he worked at the Center for Advanced Study in the Behavioral Sciences at Stanford and devoted mostly to the complex proof that appeared in 1962 of a general theorem on the existence of an economic equilibrium.

In January of 1962, he started worked at the University of California, Berkeley where he held the title University Professor and Class of 1958 Professor of Economics and Mathematics Emeritus. During his leaves in late sixties and seventies he visited universities in Leiden, Cambridge, Bonn, and Paris.

His later studies centred mainly on the theory of differentiable economies where he showed that in general aggregate excess demand functions vanish at a finite number of points. Basically, showing that economies have a finite number of price equilibria.

In 1976 he received the French Legion of Honor. He was awarded the 1983 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel for having incorporated new analytical methods into economic theory and for his rigorous reformulation of general equilibrium theory.

In 1990, Debreu served as President of the American Economic Association. He died in Paris at age 83 of natural causes on December 31, New Year's Eve, 2004 and was interred in the Père Lachaise Cemetery.

Conclusion

First, the ( Nobel ) committee reasoned, along the economic history, with the following Debreu-Smith link that could be presented in the following rough statement; Debreu and Smith had the same in mind, but Debreu had the scientific articulation:

“…….In a decentralized market system, individual consumers and firms make decisions on the purchase and sale of goods and services solely on the basis of self-interest. Adam Smith had already raised the question of how these decisions apparently independent of one another, are coordinated (…) Smith’s answer was that (…) prize systems automatically bring about the desired coordination of individual plans. Toward the end of the 19th century, Leon Walras formulated this idea in mathematical terms as a system of equations (…). But it was not until long afterwards that this system of equations was scrutinized to ascertain whether it had an economically meaningful solution, i.e. whether this theoretical structure of vital importance for understanding the market system was logically consistent (…). Arrow and Debreu managed to prove the existence of equilibrium prizes, i.e., they confirmed the internal logical consistency of Smith’s and Walras’ model of the market economy…….” ( Duppe 2007).

The argument says roughly that “….Debreu proved Smith, i.e., that markets work automatically, by showing that equilibrium has an economic meaning, i.e., that it is logically consistent…..” The hierarchy it establishes is exclusive: however it was Debreu who made it a genuine economic science because only since then one can speak of a generic market of inner-consistency and thus a proper object to study; only since Debreu markets have an “economic meaning” and as such can be analyzed.

In the Nobel Lecture, Debreu said:

“…….The axiomatisation may also give ready answers to new questions when a novel interpretation of primitive concepts is discovered…….” ( Debreu 1983 )

Press release after his Memorial Lecture : “…His clarity, analytical stringency, and insistence on always making a clear cut distinction between a theory and its interpretation have had a profound and unsurpassed effect on the choice of methods and analytical techniques in economics….”

Secondly, if Debreu had an effect, it was on the ethos of economists. Theoretic experience matters less for economic theory, or, to evoke once more Husserl’s words, in economics experience does not function as experience.

Economic science today can be called “Debreuan” insofar its theories are indifferent to the intuition and interpretation economists invest in their theoretical engagement.

As Heilbronner and Milberg (1995) correctly observed, our affective history of Debreu’s entrance into economics is its genetic code. There are identifiable affective “Debreu-symptoms” that clearly separate the time before and after Debreu. In order to quote only one recent example of this symptom, here the game-theorist Ariel Rubinstein: “……my greatest dilemma is between my attraction to economic theory, on the one hand, and my doubts about its relevance, on the other……” (Rubinstein 2006, p. 866).

Major publications and References

• Bourbaki, N., “The Architecture of Mathematics”, The American Mathematical Monthly, 57, 4, 1950, pp. 221-232

• Bourbaki, N., “Foundations of Mathematics for the Working Mathematician”, The Journal of Symbolic Logic 14 (1), 1949, pp. 1-8

• Bourbaki, N., Elements of Mathematics: Theory of Sets. Reading et al., Addison-Wesley, 1968

• Debreu, G., “Les Fins du système économique: un essai de définition objective”, in: Revue d’économie politique, 1949, pp. 600-615

• Debreu, G., "Existence of an Equilibrium for a Competitive Economy,"( with K.J.Arrow) , Econometrica, 1954

• Debreu, G., Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Yale University Press , New Haven, London, 1959

• Debreu,G.,. “Mathematical Economics at Cowles”. In Klevorick (ed.), Cowles Fiftieth Anniversary. Yale U., Cowles Foundation, 1983

• Debreu, G., “Economic Theory in the Mathematical Model”, Nobel Memorial Lecture, Royal Swedish Academy of Sciences, www.nobelprize.org. 1983a, p. 98

• Debreu, G., “Autobiography”, in Odelberg (ed.), The Nobel Prizes 1983, Stockholm: Nobel Foundation, 1984

• Debreu, G., “Theoretic Models: Mathematical Form and Economic Content”, Econometrica 54, 6, 1986, pp. 1259-1270 (Frisch Memorial Lecture of the Econometric Society, August 17-24, 1985).]

• Debreu, G, “The Mathematization of Economic Theory”, The American Economic Review 81,1, 1991 pp. 1-7 (presidential address of the American Economic Association, December 29, 1990).

• Duppe, T., “Gerard Debreu from Nicolas Bourbaki to Adam Smith”, Paper presented at European Conference on theHistory of Economics, 2007, University of Siena, Italy
• Heilbroner, R. and W. Milberg, The Crisis of Vision in Modern Economic Thought. Cambridge University Press, New York, 1995

• Ingrao, B., and G.Israel, The Invisible Hand: Economic Equilibrium in the History of Science, MIT Press, Cambridge, MA, 1990

• Mandelbrot, B., “Chaos, Bourbaki, and Poincaré”, The Mathematical Intelligencer, 11, 3, 1989, pp. 10-12

• Rubinstein, A., “Dilemmas of an economic theorist”, Econometrica, Vol. 74, 4, 2006,pp.865–883

• Weintraub,E.R., How Economics became a Mathematical Science, Duke University Press, Durham and London, 2002

External links

• The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 1983
• Existence of an Equilibrium for a Competitive Economy
• Obituary for Debreu

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