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The Nobel Series: The mathematical economist

Gerard Debreu’s axiomatization approach established coherent proof of general equilibrium theory, paving way for further research but it also underwent criticism for being unrealistic

The Nobel Series: The mathematical economist
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The Nobel Prize in Economics in 1983 was awarded to Gerard Debreu for applying innovative mathematical models and an axiom-based approach to general equilibrium theory. Debreu's paper in 1954, which he co-authored with Kenneth Arrow (the Nobel Prize winner in 1972), is his best-known work. It may be recalled that in this paper, he had proved that under unrestrictive assumptions, there was a set of prices at which the market of goods and services and that of factors of production would clear (i.e. the supply would equal demand) and there would be all-round equilibrium or general equilibrium. As Debreu stated in an article in the American Economic Review in 1984:

The benefits of the axiomatization of economic theory have been numerous. Making the assumptions of a theory entirely explicit permits a sounder judgement about the extent to which it applies to a particular situation.

When Debreu was a young student of mathematics in 1939 in Calais, France, World War II had begun. Despite the Nazi occupation, he continued his studies in Paris. In 1941, he joined the Ecole Normale Superieure and was there until 1944. He served briefly in the French occupation forces in Germany until the end of July 1945 but continued his studies in mathematics till 1946. Around this time, he was introduced to economics by Maurice Allais. He was most attracted by the general equilibrium model outlined by Leon Walras. In 1948, he was a student of Wassily Leontief (Nobel Prize winner in 1973) at Salzburg after which he got the Rockefeller Fellowship to the US, which enabled him to visit the great centres of learning in the US: Harvard, Chicago, Berkeley and Columbia Universities. In 1950, he joined the Cowles Commission at the University of Chicago. Here he honed his skills in mathematical economics and general equilibrium. He moved to Yale University when the Cowles Commission shifted there in 1955. In 1962, he became a faculty at the University of California at Berkeley, where he stayed till his retirement.

In this article, we will look at the major works of Debreu and see how they are relevant in various policy areas.

Main Works of Debreu

As noted above, Debreu's best-known work was the paper he co-authored with Kenneth Arrow in 1954 titled 'Existence of an Equilibrium for a Competitive Economy'. Professor Lionel McKenzie also provided proof of the existence of a general equilibrium independently of Arrow-Debreu.

The Arrow-Debreu model simply says that under a set of unrestrictive and reasonable assumptions (convex preferences, perfect competition and demand independence), aggregate demand of every good and service and factors of production will equal aggregate supply. In other words, there will be an economy-wide equilibrium in the market of all goods and services. Arrow-Debreu did not limit themselves to the existence of such an equilibrium but also its stability and normative aspects of the equilibrium. Later, Debreu also provided proof of the existence of equilibrium under less stringent assumptions.

Debreu, was, of course, influenced by Leon Walras's work on general equilibrium. Even before Walras, Adam Smith had outlined aspects of general equilibrium, when he had proposed that the invisible hand of the market drives the economy to a state where the resources are allocated efficiently and optimally. This happens by individuals maximizing their own self-interest and in doing so, ensure the best outcome for the economy as a whole, through the invisible hands of the market. In essence, Adam Smith said that the myriad decisions of economic agents which seem to take place without any coordination are coordinated through the price system, resulting in general equilibrium. Later, Léon Walras presented Smith's work in mathematical terms as a system of equations to represent consumers' demand for goods and services, producers' supply of these same goods and services and their demand for factors of production, and equality between supply and demand, i.e., equilibrium in each market. Debreu took forward this work and using mathematical tools, hitherto unused in economics, provided a logical and internally consistent proof. As he mentioned in an article in the American Economic Review in 1991:

Being denied a sufficiently secure experimental base, economic theory has to adhere to the rules of logical discourse and must renounce the facility of internal inconsistency. A deductive structure that tolerates a contradiction does so under the penalty of being useless since any statement can be derived flawlessly and immediately from that contradiction. In its mathematical form, economic theory is open to an efficient scrutiny for logical errors.

Debreu's other significant work was his book, 'Theory of Value', which was published in 1959. In this book, Debreu took forward his work on general equilibrium and expanded the definition of 'goods' to mean anything. The theory could therefore be applied in various areas. As the Nobel website tells us:

The theory developed in this study lends itself to many far-reaching interpretations and applications. The concept of "goods", for instance, is defined so broadly that the theory may be used in pure static equilibrium analysis, the analysis of the spatial distribution of production and consumption activities, intertemporal analysis and the analysis of an uncertainty. Thus, within the same model, Debreu's general equilibrium theory integrates the theory of location, the theory of capital, and the theory of economic behaviour under uncertainty.

Applications of Debreu's work

Debreu was primarily a theoretician and was well aware of this. He knew that his models may not work in the real world and that his formal theory was only a first approximation to understand the economic activity. The market equilibrium could only be reached theoretically, as prices for goods and services were constantly changing. Debreu left it to others to apply his theories to the real world. This is the reason he refused to be a consultant to many projects which required the application of theory. Though interested in politics, Debreu did not comment too often. After he won the Nobel Prize, he realized the greater influence he could have. "Time has always been scarce," Debreu said. "But I try not to waste it. I try to have purpose."

Debreu's work has been applied to the analyses of uncertainty, collective goods and taxes related to the public sector, environmental problems etc. In addition, his theory has also proved useful in understanding economic cost-benefits analysis. An even broader spectrum of applications has been forwarded by the creation of so-called computable general equilibrium models. These empirical models are built directly on Gerard Debreu's formulation of general equilibrium theory and may be used for problems requiring analysis of entire economic systems, such as those pursued by international agencies such as UN, IMF, World Bank and WTO. Debreu's work has also been applied by Kenneth Arrow, Campbell and Hal Varian in financial areas such as the Asset Pricing model, risk and uncertainty.

Debreu also opened many research areas for later researchers: he introduced the concept of a "quasi-equilibrium" (1962), the idea that a bundle of goods and their prices may be represented as a linear space and its dual (1954), integrated the concept of uncertainty into general equilibrium and defined the concept of "smooth preferences" which permitted the re-introduction of differential calculus into mathematical economics (1972), as well as the first venture into infinite-dimensional commodity spaces (1954).

One of his other important works is on market demand functions (1974), which yielded the famous Debreu-Sonnenschein-Mantel theorem. It may be recalled that the theorem theorized the shape of the excess demand function (excess demand is important in general equilibrium theory since it is a signal to the market to adjust prices). The theorem basically states that under the rationality assumption, the excess demand curve can take the shape of any function that is continuous and satisfies Walras' law. In other words, there may not be a unique and stable equilibrium.

Conclusion

Debreu's name is synonymous with the use of complex mathematics in economics. Along with Arrow, he was one of the earliest economists to do so. Debreu was also one of the first to attempt to provide a model of Adam Smith's invisible hand. Debreu had himself remarked in 1985, two years after winning the prize: "Among all the Nobel Laureates in Economics, I have done the most abstract work".

The Arrow-Debreu model has been criticised for being too abstract and far removed from reality. Some even say that the term 'general equilibrium' is a misnomer since most situations in the real world are hardly in 'equilibrium' and certainly not 'stable'. Further assumptions such as lumping together all goods as a single homogenous good and all consumers as a representative agent, used in the Arrow-Debreu model have been criticised as unrealistic and having no use in policy. Notwithstanding the criticism, Debreu's work was indeed pathbreaking and Debreu himself was acutely aware of the shortcomings of his models.

Views expressed are personal

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