Gerard Debreu Wins the Nobel Prize Steve Smale

I was at the office early Monday, October 17, when my wife Clara called with the news that our friend Gerard Debreu had won the Nobel Prize in economics. Although I had anticipated this event, it was exciting and a pleasure to hear that it had actually happened.

Gerard and I have been close friends since we met fifteen years ago. I still remember our first encounter. He came to my office to ask about some mathematics needed for his work (he was trying to show that a general econorrry could have only a finite number of equilibria). I found Debreu friendly; we could com- municate easily about mathematics and economics. I was especially impressed with his ability as a mathe- matician, his clarity and rigor.

His questions were the beginning of my own work in economic theory. In the following years we spent long hours in discussions. (Our day-hikes at Point Reyes have been especially memorable.) We ex- changed questions, his more mathematical and mine about economics. Eventually, Gerard joined the math- ematics department; I joined the economics depart- ment.

Debreu's great contribution is his profound use of mathematics in the central theme of economic theory, consolidating an insight of Adam Smith more than 200 years ago. Debreu has given the foundations of gen- eral equilibrium theory in his classic work "Theory of Value." The award of the Nobel prize to Debreu gives a valuable impetus to basic research in mathematical economics.

In 1776, "The Wealth of Nations," Adam Smith sug- gested an "invisible h a n d " promoted economic ac- tivity. He wrote:

Every individual endeavors to employ his capital so that its produce may be of greatest value. He generally neither intends to promote the public interest, nor knows how much he is promoting it. He intends only his own secu- rity, only his own gain. And he is in this led by an IN- VISIBLE HAND to promote an end which was no part of his intention. By pursuing his own interest he frequently promotes that of society more effectually than when he really intends to promote it.

However, Smith did not give any mechanism as to how this theory could actually work. The invisible

hand was a mysterious force. To see the problem, one can consider, as in Paul Samuelson 's famous text "ECONOMICS", the city of New York:

Without a constant flow of goods in and out of the city, it would be on the verge of starvation within a week. A variety of the right kinds and amounts of food is required. From the surrounding counties, from 50 states, and from the far comers of the world, goods have been traveling for days and months with New York as their destination.

How is it that that 12 million people are able to sleep easily at night, without living in mortal terror of a break- down in the elaborate economic processes upon which the city's existence depends? For all this is undertaken without coercion or centralized direction by any conscious body!

This is the operation of the "invisible hand". The price system somehow gives order to a highly decen- tralized economic process. The key to understanding this mechanism lies in the equations of supply and demand. If there were only a single isolated market, i.e., one good, it is not hard to see that the equilibrium price of that good would be such that supply and de- mand would balance.

There are many markets, however , in an actual economy and those markets are highly interrelated (for example gasoline and cars). The theory of a balanced supply and demand in this complex situation is called "General Equilibrium". It is quite subtle even to for- mulate the problem sensibly.

About one hundred years after "The Wealth of Na- tions", Leon Walras made an important initial contri- bution by putting forth a mathematical framework in which the equations of supply and demand were gen- erated by individual agents, producers and con- sumers. Moreover, Walras plausibly asserted that a price equilibrium solution to his equations existed.

It wasn' t until the 1930s, however, that Abraham Wald made the Walras theory correct and this was only in a limited economic context.

In the early 1950s Kenneth Arrow (Nobel Prize in 1972) and Gerard Debreu gave a general mathematical economic setting for the ideas of Walras and in this setting proved the existence of an economic equilib- rium. Debreu laid all this forth in his monograph, "Theory of Value".

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The work on general equilibrium theory does not imply that it should be the model for society. For one thing, in order that the conclusions be valid, it is hy- pothesized that there is no monopoly. But monopoly does arise in a decentralized system. Putting aside mo- nopoly, there is the problem of inequity. Arrow and Debreu showed that at an equilibrium allocation, no one can be made better off without hurting someone else. However , the theory allows grossly unequal shares of the goods of society. Thus, substantial gov- ernment mediation of the decentralized price system is required.

There are serious problems of a different kind---lim- itations of the theory itself. In the Arrow-Debreu

economy the passage of time is not satisfactorily taken into account. This lack of dynamics keeps the theory from explaining why prices adjust to equilibrium or why they stay there. Another, possibly related, weak- ness is that the model makes unrealistic demands on the rationality of its agents. Even equipped with m o d e r n computers , the consumers and pro- ducers could not make the decisions required for the theory.

Although fascinating challenges remain, there is now a framework in which we can try to meet them. That is the basis established by economists of two cen- turies and, in particular, Smith, Walras, Wald, Arrow and Debreu.

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