Gerard Debreu was an austere man who brought that austerity to economics.

He, almost single-handedly, introduced the axiomatic approach to economic theory. He never speculated about what might be true and confined himself to saying what he knew to be true.

His approach was essentially that of a mathematician and he used that to protect himself from getting involved in debates about economic matters. His reserved attitude to the profession and to the world can, at least in part, be attributed to the sad events that marked his early years (see Duppe 2012).

Life

Born in Calais in 1921, he was orphaned at a very young age and spent most of his youth in boarding schools, until, with the outbreak of the war, he was evacuated to the unoccupied part of France where he prepared for the competition to enter a “Grande Ecole”. He succeeded and was admitted to the Ecole Normale Superieure in Paris, then occupied by the Germans, to study mathematics. When he had finished he was going to take the aggregation to become a mathematics teacher but went into the army until the end of the war when he finally took the examination and was also admitted as a researcher in the Centre national de la recherche scientifique (CNRS). He had been heavily influenced by the Bourbaki School of mathematics and his research reflects that school’s approach.

His interest, nevertheless, turned to economics and followed the courses given in Paris by Maurice Allais, who was later also to obtain the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel. He was offered, through the good offices of Maurice Allais, a Rockefeller fellowship to go to the United States. Allais could propose two candidates, but one would have to wait a year. Marcel Boiteux was the other potential candidate and the two decided on the toss of a coin. Boiteux went on to become the Chief Executive Officer of Electricite de France, and never took up the scholarship.

Debreu spent six months at Harvard, then a summer at Berkeley before going to the Cowles Commission in Chicago. He visited Ragnar Frisch before accepting a permanent position at the Cowles Commission. When the Commission moved to Yale, Debreu was appointed as an associate professor. In 1954 he was elected as a Fellow of the Econometric Society and defended his PhD thesis at the Sorbonne in 1956. This thesis was the first version of his seminal work, Theory of Value published in 1959. He spent a sabbatical year at Stanford in 1960-61 before becoming full professor of economics at Berkeley and later, in 1975, also professor of mathematics. He was elected president of the Econometric Society in 1972 and was named Chevalier de la Legion d’Honneur in 1976. He received the Sveriges Riksbank Prize in Economic Sciences in 1983. Gerard Debreu returned to France at the end of his career and died in Paris on the last day of 2004.

Debreu’s Contributions to Economics

In addition to Debreu’s own interest in economics from an abstract and mathematical point of view, we should not forget the strong influence that the Second World War and

its aftermath had on all those involved in economics at the time. There was a general sentiment that the wartime experience had, to an important extent, vindicated economic planning. Thus there was a certain confidence in the government as a resource manager. Furthermore, rightly or wrongly, the Marshall Plan was held up as a shining example of successful government intervention. It is not surprising then that Debreu’s first published contribution was entitled “The coefficient of resource utilization” (1951) and involved a discussion of the nature of Pareto optimality and measuring the extent to which an economy approached that situation. At the same time, the major thrust in pure economic theory to which Debreu devoted his whole career was general equilibrium theory and the notion of social optimality derived from the earlier work of Walras and Pareto.

This tradition had been extended and developed by Maurice Allais in his book published in 1943 and entitled A la recherche d’une discipline economique which was revised and then republished in 1953 with the title Traite d’economiepure. Debreu’s interest in the efficient allocation of scarce resources had then two origins, one being the impact of the wartime experience in France and the other being that of the intellectual tradition of refining and perfecting the basic mathematical structure of economic theory.

From his earliest encounters with economics, Debreu already saw the major task in economic theory as being that of showing that the general equilibrium model, which originated with Walras and Pareto, was internally consistent. He was later to say, in his Nobel lecture that “the highest prominence will be given in this lecture to Leon Walras (1834-1910), the founder of the mathematical theory of general economic equilibrium, to Francis Y. Edgeworth (1845-1926), and to Vilfredo Pareto (1848-1923)” (Debreu 1983 [1992]: 87).

For Debreu, the basic problem boiled down to showing that there exists a vector of prices which will clear all markets simultaneously. This amounts to finding a solution to a set of equations and had already, in large part, been solved by Wald (1935). Framing the problem in these terms was very much in the spirit of the Bourbaki tradition and it is not surprising that it should have been Debreu who pushed Walrasian theory in this direction. He was following on from Maurice Allais, his mentor, who was clearly oriented in the direction of mathematical purity rather than realism. “The fundamental Anglo-Saxon quality is satisfaction with the accumulation of facts. The need for clarity, for logical coherence and for synthesis is, for an Anglo-Saxon, only a minor need, if it is a need at all. For a Latin, and particularly a Frenchman, it is exactly the opposite” (Allais (1953: 58).

However, in thinking in this way Allais was encouraging the distancing of economics from reality.

This was exactly in the Bourbaki tradition for as Bourbaki (1949: 2) wrote:

Why do applications [of mathematics] ever succeed? Why is a certain amount of logical reasoning occasionally helpful in practical life? Why have some of the most intricate theories in mathematics become an indispensable tool to the modern physicist, to the engineer, and to the manufacturer of atom-bombs? Fortunately for us, the mathematician does not feel called upon to answer such questions.

Already, when Debreu received the Sveriges Riksbank Prize the message was clear, the way forward was to use the mathematical approach that he introduced. As the Sveriges Riksbank Prize committee said at the time,

Gerard Debreu symbolizes the use of a new mathematical apparatus, an apparatus comprehended by most economists only abstractly. Nevertheless, his work has given us an improved intuitive understanding of the underlying economic relevance. His clarity and analytical rigor, as well as the distinction drawn by him between an economic theory and its interpretation, have given his work important bearing on the choice of methods and analytical techniques within economic theory on a par with any other living economist. (Sveriges Riksbank 1983)

Undoubtedly Debreu’s contribution that will mark the history of economic thought the most is that with Arrow (Arrow and Debreu 1954), where they gave a proof of the existence of an equilibrium for an economy under rather general conditions. But, more importantly, what were the specific contributions made by Gerard Debreu which convinced an important part of the economics profession to accept his methodology in the 1960s and 1970s?

The Theory of Value

The most symbolic of these was his Theory of Value (Debreu 1959). This slim book was, during that period, a sort of bible for all mathematically inclined economists. This for two reasons: the axiomatic approach seemed to provide the rigorous foundations that put economics on a par with the “hard sciences”, and secondly, it gave the impression of limiting the assumptions to a minimum.

The first of these was, in large measure, an illusion. Physics, for example, which had provided the theoretical structure for Walras and Pareto, had moved in quite a different direction, and what Debreu wanted to achieve was to place economics firmly in the field of abstract mathematics. Nowhere, in Debreu’s work, is there any reference to an empirical phenomenon and all the emphasis is on building an internally consistent model without any concern for its use in explaining economic data.

The second reason for the admiration for Debreu’s approach, that it minimized the assumptions used to arrive at his basic results, is also, in retrospect erroneous. Debreu made great play, in the Theory of Value, with the absence of any assumption of differentiability, “the differential calculus and its compromises with logic”. This seemed like a major and even revolutionary step. The whole marginal revolution had been formalized in terms of marginal products, or marginal utility, which were formally interpreted as derivatives of the relevant production, cost, or utility functions. Thus, to dispense with this assumption gave the impression of a significant advance. Yet as Mas-Colell (1990), in many ways a disciple of Debreu, showed later, given the other assumptions that Debreu made, it is always possible to approximate any of the functions in question by a differentiable function. Thus, the period that a generation of young theoretical economists spent wrestling with the mathematical structure that Debreu had imposed was, in large part, wasted.

However, what was the structure of the economy that Debreu worked with and what were its basic properties? The first pillar was the definition of goods, of which there were a finite number, and were characterized by their physical characteristics, and the time and place at which they were available. Of course, if time is continuous the assumption that there is only a finite number of good is contradictory; or, again, even if time is discrete, if no limit is put on the lifespan of the individuals in the economy, the fact that goods are dated will make them infinite in number.

The second pillar of the economy is provided by the characteristics of the consumers who were assumed to have preferences over the goods which satisfied certain axioms. These axioms were not based on observing people’s choices but on the introspection of economists who had preceded Debreu. Although each of them, such as continuity, convexity and monotonicity can be given an intuitive explanation, they are essentially imposed for mathematical convenience. Given these assumptions, one can derive the individual and aggregate demand of the consumers, that is, what they would like to consume at any given prices. Although Debreu did not use utility functions in the Theory of Value, one question of some importance to theoretical economists in this connection is, under what conditions can one represent the ordinal preferences that he used, by a continuous utility function? The idea, that it was enough to know how individuals order bundles of goods, rather than attribute quantities of utility to those bundles, was developed by Pareto. To analyse choices one has only to know whether bundle x is preferred or indifferent to bundle y, written x * y. Then the question becomes, under what conditions can one find a function u(x) such that u (x) $ u (y) 3 x * y? Debreu had already given the conditions for this earlier, in Debreu (1954), but chose not to work with this notion even if it is more familiar to economists, because it entails some restrictions that can be dispensed with if one works with preferences alone.

The third pillar is given by the producers, who are present in a fixed number and who are characterized by their production possibilities. The latter specify all the possible production plans, that is, all the possible combinations of inputs and outputs. Given the prices of the inputs and outputs the producer simply chooses the production plan that generates the maximum profit. These choices define the individual and aggregate supplies of the economy. Debreu made a number of assumptions about the set of production possibilities, or technologies. Perhaps the most important of these is that this set is convex. This rules out the inconvenient possibility of increasing returns to scale. However, it does not rule out the possibility that firms will make positive profits and then, to complete the model, what happens to those profits has to be specified. Debreu’s answer was to assume that every consumer, at the outset, is somehow endowed with a share in each firm’s profits and that they include this in their income. There is an obvious simultaneity problem here and, if Debreu’s general equilibrium model is thought of as a model of how individuals make their choices, it is not clear how consumers are supposed to know what the profits of the firms will be at equilibrium. In fact, Debreu ignored this problem by simply viewing an equilibrium as a set of prices and choices such that aggregate demand is equal to aggregate supply, or, put another way, the excess demands for all goods are zero. No question is asked as to how such an equilibrium is established; the statement is simply, if these were the prices and these were all the choices made by the individual consumers and producers in the economy at those prices then the economy would be in equilibrium. The examination of the properties of an equilibrium state was Debreu’s fundamental contribution in the Theory of Value. He showed that a competitive equilibrium allocation of resources is efficient by the Pareto criterion and, what is more, that for any such efficient allocation, one can always redistribute initial resources in the economy so that, for the new resources, the efficient allocation in question is the competitive allocation. These are the basic fundamental theorems of welfare economics, which were to lead the Figaro newspaper (Le Figaro 1984), to claim, when Debreu received the Sveriges Riksbank Prize, that he had shown that free markets were socially optimal.

However, all of this raises an important question, for much of the justification for the market economy which the general equilibrium model is supposed to epitomize rests on the idea that the economy will somehow organize itself into an equilibrium state. Given this, one is then interested in what the properties of such states are. Yet, Debreu, unlike many of his contemporaries, such as Arrow and Fisher, did not concern himself with the “stability” problem, that of how an economy would reach equilibrium, but instead just assumed that it would do so. Indeed, he asserted in private conversations that the problem was fundamentally intractable. He also considered that global uniqueness of equilibrium was out of reach with the standard assumptions. The idea that the equilibrium of an economy is unique is not just an abstract consideration. Macroeconomists, in seeking to determine the effects of a policy measure, rely on the comparative statics approach. This means considering “the equilibrium” of an economy and then changing some parameter, and analysing the new equilibrium that results from the change. However, this makes little sense if there are several equilibria since it is not known from where one starts and where one would end up. Yet, the use or misuse of economic theory was never something which interested Debreu; he simply ignored work which he thought made unjustified claims.

The core of an economy

One of the major figures cited by Debreu was Edgeworth. The latter argued that one should not be interested, as was Walras, in showing how a vector of prices, one for each good, converged to an equilibrium but one should consider haggling between individuals as reflecting the process involved. He then went on to develop a solution to this negotiation process which later was known as the core of an economy. When formalized, this amounts to considering allocations of goods to which no group, or “coalition” of agents could object. Such a group could object if they could take their own resources and distribute them among themselves, and, in so doing, make all the members of their group better off. What Edgeworth showed, in a very limited case, was that allocations with this property were competitive equilibria if the number of agents was large enough. This has led some to suggest, erroneously, that a bargaining process consisting of proposed allocations and objections to them would lead to an equilibrium. All that it, in fact, says is that an allocation to which no coalition has an objection is a competitive equilibrium when the number of agents in the economy becomes arbitrarily large. Nothing is said about how the economy gets there. Nevertheless, the result is a curiosity since, whether or not there is an objection to an allocation, it in no way involves prices, yet this criterion gives the same result as the more familiar price mechanism. Debreu, and Herbert Scarf (1963), showed that Edgeworth’s result holds for a reasonably general class of economies. Later, Debreu (1975) discussed how fast this would happen, in the sense of asking how close this result was to holding as the number of agents increased. Although Lloyd Shapley (1975) had shown that the convergence could be arbitrarily slow, Debreu showed that for a certain class of economies the “core converges to the competitive equilibrium” at the rate of 1/N where N is the number of agents in the economy. This has sometimes been misinterpreted as saying how fast a bargaining process would lead to competitive equilibrium and this is, of course, not true.

This work shows again clearly how fascinated Debreu was with the beauty of proving results for very abstract economies rather than being concerned with analysing real economic phenomena. Yet it has also to be said that the refined puzzles on which he worked, usually had their origins in the work of leading figures in the history of economic theory.

Uniqueness of equilibrium

Debreu (1970) showed in a paper, again reverting to differentiability, that economies, in general, only have a finite set of equilibria. This means that even if there are many of them they will each be isolated and, in that sense, locally unique. This result is not particularly comforting for macroeconomists but does introduce an interesting notion, the idea that something may be true in general for economies but does not always hold. Somewhat inaccurately it can be said that the probability of finding an economy with an infinite number of equilibria is zero. Such a property is referred to as being “generic”. This idea is not as innocent as it might seem and such results should be treated with caution (see Grandmont et al. 1974). Nevertheless, this was an innovation in a discipline in which researchers had been concerned with proving results that held without exception, albeit for a very limited class of economies.

Undermining the foundations

The general equilibrium framework which Debreu was, in large measure, responsible for perfecting, turned out to be extremely fragile. A reflection of this is seen in his 1974 paper, where he contributed to the results obtained by Sonnenschein (1972) and Mantel (1974) by showing that, even with the restrictive standard assumptions on the individual participants in the economy, neither stability nor uniqueness could be guaranteed. Ironically, this paper was heavily dependent on a differentiability assumption, something which he had disdained earlier. Here Debreu’s major contribution was to refine in an elegant way the insight provided by Sonnenschein’s contribution. He essentially showed that the only properties that could be deduced from the standard assumptions on individuals in an economy were the continuity of the aggregate excess demand function, Walras’s Law, and the homogeneity of degree 0 of the economies aggregate excess demand function (increasing all prices by a multiple has no effect on excess demand). If not confined to prices that are bounded away from zero, it has to be added that if the price of any good goes to zero then the average excess demand for all commodities goes to infinity. Since it is easy to give examples of functions which satisfy these conditions but which have neither unique nor stable equilibria, then the usefulness of general equilibrium models for macroeconomists, for example, is obviously limited. It took some time before the devastating implications of this result for general equilibrium theory were realized, and there are still many economists who refer to their models as general equilibrium models but who, in reality, simplify away the difficulties by making some special assumption, such as assuming the existence of a “representative agent”. Debreu never approved of this.

Conclusion

Debreu provides a very special and very important example of the relationship between economics and mathematics in what might be thought of as the golden age of high theory in economics. He felt that he had provided a method which, even if destructive in some ways, enforced clear and rigorous thinking in economics and even had an influence on mathematics. As he said in his Nobel lecture:

[T]he axiomatization of economic theory has helped its practitioners by making available to them the superbly efficient language of mathematics. It has permitted them to communicate with each other, and to think, with a great economy of means. At the same time, the dialogue between economists and mathematicians has become more intense. The example of a mathematician of the first magnitude like John von Neumann devoting a significant fraction of his research to economic problems has not been unique. Simultaneously, economic theory has begun to influence mathematics... (Debreu 1983 [1992]: 99)

Yet, looking at the recent evolution of the world’s economy, it is perhaps legitimate to enquire whether the edifice that Debreu contributed so much to building has produced much insight into the working of that economy. Hildenbrand (1981) speaking of Debreu’s contribution to general equilibrium theory said that while Walras was the architect, Debreu was the master builder. Yet more recently it has been said (Kirman 1998), that this magnificent building, once full of fervent believers is now only visited by curious tourists. This was said in a volume in honour of Debreu’s seventy-fifth birthday, and his response was characteristic; he was happy that the weaknesses of the structure, to which he had contributed so much, were revealed by those who had built it.

Alan Kirman

Gerard Debreu was an austere man who brought that austerity to economics.

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