# Debreu, Gerard

# Debreu, Gerard *1921-2005*

Gerard Debreu, a French-born American mathematical economist and 1983 Nobel laureate, was a major force in the advancement of the theory of general equilibrium. His early influence came from the 1988 French Nobel laureate, Maurice Allais, who introduced him to the writings of Leon Walras (1834-1910), the founder of the mathematical theory of general equilibrium analysis.

In 1949 Debreu visited several premier universities, including Harvard and the University of California, Berkeley, on a Rockefeller fellowship. Following the fellowship, he spent a decade with the Cowles Commission, then attached to the University of Chicago, working on Pareto optima, the existence of a general economics equilibrium, and utility theory. Subsequently, after one year at Stanford, he moved to Berkeley in 1962 where he remained until his retirement in 1991. Besides mathematics, Debreu was so taken by the U.S. rule of law during the Watergate scandal that he became an American citizen in 1975.

Debreu’s research interest was general equilibrium analysis, in line with Walras, whose method was to equate equations and unknowns. The essential idea is to show how prices gravitate from disequilibrium to their natural or equilibrium levels. Debreu liberated the analysis by opting for fixed-point and convex tools for general equilibrium analysis.

Convexity requires that the production and consumption sets be bowl-shaped. Debreu would “slip a hyperplane,” as he would say, through the tangent of those convex sets. He first revealed a proof at the 1950 meeting of the Econometric Society at Harvard that did not use the convexity solution. Many optimal solutions of the Pareto type and some possible competitive solutions resulted where prices are given and agents maximize their affairs. In 1954, in collaboration with Kenneth Arrow and using topological methods, which Debreu had used previously, he proved the existence of general equilibrium in an “epoch-making” paper, “Existence of an Equilibrium for a Competitive Economy.”

Debreu’s early book, *The Theory of Value* (1959), produced the hard-core ideas of his lifetime research program.

He expressed the economy as , where each consumer, *i =* 1.… *m*, has an initial endowment, ω_{i}, a share of j^{th} producer’s profit, θ_{i, j}, and an underlying utility function for his or her preferences. Each producer, *j =* 1.… *n*, takes the prices that are announced, and maximizes profits, π_{j }. The inner-bracketed terms represent the consumer budget from the initial endowment and shares of profits. Consumers will maximize their utility subject to their budget constraints, yielding demand functions. Excess demand functions are now possible by summing all the demand functions less endowment. Thus we can derive the equilibrium prices from the excess demand functions, which in two dimensions are usually the zeros or solutions of a quadratic equation.

Debreu built his theories on axioms, where a primitive such as a commodity becomes a mathematical object with spatiotemporal and physical characteristics. He urged his students to be concise, mentioning that John Nash demonstrated the equilibrium for finite games on only one page. During his lectures his eyes would light up when he demonstrated the superiority of mathematical reasoning. He used to say that mathematical economists have the best of two worlds—mathematical discoveries when they are young and economic discoveries when they are old.

**SEE ALSO** *Arrow-Debreu Model; General Equilibrium; Pareto Optimum; Tatonnement*

## BIBLIOGRAPHY

Arrow, Kenneth J., and Gerard Debreu. 1954. Existence of an Equilibrium for a Competitive Economy. *Econometrica* 22: 265–290.

Arrow, Kenneth J., and Michael D. Intriligator. 1982. Existence of Competitive Equilibrium. In *Handbook of Mathematical* *Economics*, eds. Kenneth J. Arrow and Michael D. Intriligator, Vol. 2: *Mathematical Approaches to Microeconomic Theory*, 677–743. Amsterdam, NY: Elsevier North-Holland.

Debreu, Gerard. 1959. Separation Theorem for Convex Set. *SIAM Review* 1: 95–98.

Debreu, Gerard. 1959. *The Theory of Value: An Axiomatic Analysis of Economic Equilibrium*. New York: Wiley.

Debreu, Gerard. 1983. *Mathematical Economics: Twenty Papers of Gerard Debreu*. New York and Cambridge, U.K.: Cambridge University Press.

Debreu, Gerard. 1993. Random Walk and Life Philosophy. In *Eminent Economists, Their Life Philosophies*, ed. Michael Szenberg, 107–114. New York and Cambridge, U.K.: Cambridge University Press.

Ramrattan, Lall, and Michael Szenberg. 2005. Gerard Debreu: The General Equilibrium Model (1921-2005). *The American Economist* 49 (1): 3–14.

*Lall Ramrattan*

*Michael Szenberg*

# Debreu, Gerard

# Debreu, Gerard

(*b*. 4 July 1921 in Calais, France; *d*. 31 December 2004 in Paris, France), mathematician and economist of the highest rank whose fundamental contributions in many areas of economics, especially the general equilibrium theory, earned him the 1983 Nobel Memorial Prize in Economic Sciences.

Debreu was born to Camille and Fernande (Decharne) Debreu. His father and maternal grandfather were partners in a local lace manufacturing business. He attended the College of the City of Calais until 1939, when France’s entry into World War II began disrupting the school system. Debreu studied the Mathématiques Spéciales Préparatoires curriculum, first in Ambert (1939–1940) and then at the Grenoble Lycée (1940–1941). He then studied mathematics and physics at the École Normale Supérieure in Paris until 1944, before becoming interested in bringing mathematical rigor to economics after reading Maurice Allais’s *À la recherche d’une discipline économique* (1943). Following the Allied invasion of Normandy, France, in June 1944, Debreu joined the French Army for a year. He married Françoise Bled on 14 June 1945, and they later had two daughters. Debreu worked as a research associate at the Centre National de la Recherche Scientifique in Paris from 1946 to 1948 and also attended the University of Paris in 1946, later earning a DSc from the university in 1956.

As a Rockefeller Fellow from 1948 to 1950, Debreu traveled in the United States, Sweden, and Norway. During this journey he came into contact with the Cowles Commission for Research in Economics, which was then affiliated with the University of Chicago. The Cowles Commission, which was the institutional home and patron of the Econometric Society, was the leading center for mathematical economics in the United States at the time. Debreu’s visit to Cowles became an extended stay. He worked as a research associate in Chicago from 1950 to 1955 and then followed Cowles to Yale University in New Haven, Connecticut, where he was an associate professor of economics until 1961. In 1962 Debreu accepted an appointment as a professor of economics at the University of California, Berkeley, where he taught until his retirement in 1991. In 1975 he became a professor of mathematics, and in 1985 he was named a university professor. During his time at Berkeley, Debreu continued his world travels as a visiting academic. He became a naturalized U.S. citizen in 1975.

Thanks to his training in France, Debreu embraced an approach to mathematics that emphasized the need to ground the analysis of all things in systems of axioms. This approach followed the ideas of Nicolas Bourbaki, the collective name for a group of French twentieth-century mathematicians. Indeed, Debreu was perhaps the most prominent apostle of the new faith in formal mathematical rigor in postwar economics. To axiomatize a theory, in Debreu’s view, was to create “a consistent set of definitions, hypotheses, and theorems that can be used in turn to provide a formalized representation of different concepts and problems in economic theory,” according to Bruna Ingrao and Giorgio Israel in *The Invisible Hand* (1990). Thus, as Debreu wrote, “Allegiance to rigor dictates... the axiomatic form of the analysis where the theory, in the strict sense, is logically entirely disconnected from its interpretations.” To Debreu, the economic part of mathematical economics was analogous to the introduction and conclusion of a book, with the mathematical part being the main text in between.

The great virtue of this approach, to Debreu, was that “an axiomatized theory substitutes for an ambiguous economic concept a mathematical object that is subject to entirely definite rules of reasoning,” bringing clarity and precision to economic analysis by forcing analysts to make their assumptions perfectly explicit. The great weakness of this approach, to its critics, is that this divorce of the “theory” from its economic “interpretations” also separated it from empirical observation. As the joke goes, a mathematical economist is someone who lies awake at night wondering if reality is theoretically possible.

When Debreu began his career, the use of formal, sophisticated mathematics in economics was rare, finding strong advocates primarily at the Cowles Commission and at Harvard, Princeton, and Columbia universities. For example, the University of Chicago’s department of economics, one of the leaders in the mathematization of economics in the 1930s and 1940s because of its intimate connections to the Cowles Commission, at that time required only one quarter-long course in mathematics for its undergraduates and only one additional course in statistics for its graduate students. By the end of Debreu’s career, however, high-powered (and highly abstract) mathematics had become the lingua franca of economic theory. Even the type of mathematics used by economists had changed: those who applied mathematics to economics before World War II usually employed the calculus or fairly elementary statistical analysis. Debreu and his generation, however, added to their arsenal an array of mathematical weapons drawn from modern set theory, number theory (especially as related to ordinal numbers), topology, and functional analysis. A classic example of Debreu’s use of these new tools was his application of theories of convex sets to analyzing economic indifference curves, which led to a series of influential papers in the early 1950s.

Debreu applied this axiomatic approach and these powerful mathematical tools to many of the central questions in postwar economic theory. Of these, the most important were questions related to the problem of equilibrium in a market economy. A basic assumption of much of twentieth-century economics was that market economies are governed by forces that tend to produce an equilibrium. A further common assumption was that the equilibrating nature of a market system is a good thing, ensuring maximum efficiency and an optimal allocation of resources. For the professional economist interested in bringing mathematical rigor to economics, the equilibrium-system nature of a market economy was extremely important as well, since many of the most valuable mathematical tools only can be applied to economic analysis if the economy in question is an equilibrium system. Some of the central questions both for economic policy and philosophy are: Is a market economy really an equilibrium system? If so, what are the forces that keep it in equilibrium? How do they operate? Is there one point of equilibrium or many? Are these equilibria stable, or do they require the “visible hand” of human management? Would a state of equilibrium really produce the desirable outcomes (maximum efficiency and optimal resource allocation) long associated with equilibrium systems?

Among Debreu’s greatest achievements were creating the first formal proofs of answers to several of these questions. Most notably, in 1954 he and Kenneth J. Arrow collaborated to prove the existence of competitive equilibrium in a market economy after having simultaneously, but independently, proven that a market economy that is an equilibrium system would produce an optimal allocation of resources. (The latter is known as the proof of the First Welfare Theorem.) In 1970 Debreu proved that there could be multiple points of equilibrium for an economy, but that the set of possible equilibria is finite and thus such equilibria are locally unique. Debreu also is renowned for his collaboration with Herbert Scarf on a pair of landmark articles in the early 1960s that used game theory to define the “core” of a market economy and prove that this core converges on an equilibrium position in a competitive market. Debreu’s *Theory of Value: An Axiomatic Analysis of Economic Equilibrium* (1959) remains a classic statement of neo-Walrasian general equilibrium theory, presented in precise, rigorous, axiomatic form.

According to the economic historian E. Roy Weintraub in *How Economics Became a Mathematical Science* (2002), “When the place of mathematics in twentieth-century economics is broached, it is Debreu who is always mentioned with awe, and not a little apprehension.” Even more than his specific theoretical contributions to economics, which were remarkable, perhaps Debreu’s greatest single contribution was as a proof by example of the power and utility of mathematics in economic theory. Debreu’s work was honored in both the United States and France. He was inducted into the U.S. National Academy of Sciences in 1977 and was named as a member of the French Legion of Honor in 1976 and a commander of the French National Order of Merit in 1984. He also received the Nobel Memorial Prize in Economic Sciences in 1983. Debreu died of natural causes and is buried in Père Lachaise Cemetery in Paris.

Late in his career, Debreu published two autobiographical recollections linking his work directly to the role of mathematics in postwar economics, “Economic Theory in the Mathematical Mode,” *American Economic Review* 74, no. 3 (1984): 267–278, and “The Mathematization of Economic Theory,” *American Economic Review* 81, no. 1 (1991): 1–7. Valuable secondary sources include Bruna Ingrao and Giorgio Israel, *The Invisible Hand: Economic Equilibrium in the History of Science* (1990), translated by Ian McGilvray; and E. Roy Weintraub, *How Economics Became a Mathematical Science* (2002). An obituary is in the *New York Times* (6 Jan. 2005).

*Hunter Crowther-Heyck*

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