# PDF Mathematical Economics: Twenty Papers of Gerard Debreu

Although there had been quantum leaps of mathematical sophistica- tion before in the history of economics, there had never been anything like this. The way that this happened might go some distance in explaining the otherwise totally unprecedented character of this kind of mathematical economics. Gerard Debreu was born on 4 July in Calais, France.

He experienced a successful early school career, preparing for the baccalaureate by studying physics and mathematics. His plans to study at a lycee for entrance into one of the Grandes Ecoles were disrupted by the beginning of the war, but he did manage further preparation in mathematics at Grenoble; he won the Concours General in physics in , and later admission into the Ecole Normale Superieure.

The group entering the Ecole Normale Superieure was divided roughly in half, with around fifty students each in the humanities and sciences. Around twenty were thus mathematics students. The sciences were divided basically between mathematics on one hand and physics and chemistry on the other the two went together and there was a third possibility but very few students went that way , that was biology.

And I imagine that in our group maybe only one or two went the way of biology whereas the division between mathematics and physics and chemistry was about even. All science students took the same examination [to] enter the school, and then we decided which way to go.

## Gérard Debreu

In mathematics it was normally a three year course and in physics I think it was four. And at one point I thought I wanted to take my distance from mathematics because it was very abstract, and as I wrote somewhere else I was interested in several other directions. One of them was economics as you well know, but one was astrophysics, though I did not go very far. The problem in astrophysics was that first of all, the faculty at the University of Paris was depleted during the Second World War. I think some of them were Jewish and it was unwise for them to stay in Paris. And others were communists and some were both!

Debreu b The mathematical training that Debreu received at the Ecole Normale Supe- rieure was very different from that which he had had earlier. Instruction was carried out, in mathematics, in a complicated fashion.

It's very strange. Again, it is unique. If you take another Grande Ecole like the Ecole Polytechnique, they have all their teaching within the school only for students there. Not at the Ecole Normale Superieure. It is close to the Sorbonne, geographically close, and we were supposed to take the standard courses at the Sorbonne. And what we had at the Ecole Normale Superieure was very small seminars; that is where we were taught by Cartan. There was no fixed curriculum, and it was attended by about 10 people whereas in the fundamental courses at the Sorbonne the attendance was at first in the hundreds.

I do remember a course taught by the physicist What was lacking in them then was the enthusiasm that Cartan generated. The instructor Debreu remembers best is Henri Cartan, one of the Bourbaki "founders. In any case I was aware of what Bourbaki volumes had already appeared, which in fact by was very little, I think it only two volumes. And even then, one was a summary.

- Equilibrium with Arbitrary Market Structure.
- Gerard Debreu | IDEAS/RePEc.
- Mathematical Economics: Twenty Papers of Gerard Debreu - Gerard Debreu - Google книги.
- SearchWorks Catalog.
- Selected Works.
- On the Existence of Price Equilibrium in Economies with Excess Demand Functions.

For Debreu the mathematical work was interesting, but he already had some idea that he was perhaps going to be more involved with mathematics in another discipline. Perhaps this was because of his earlier success in physics, perhaps it is because he reached a limit as it were in his ability to sustain interest in pure mathematics under the conditions of wartime Paris. At any rate, Debreu seems to have understood, fairly early in his career at the Ecole Normale Superieure, that his own path was to be a bit different from those of the mathematicians.

The objective at the Ecole Normale Superieure was basically to produce teachers of mathematics; and that was understood in the days when I was there, to mean teachers of mathematics at the Mathematiques Speciale Preparatoire and Mathematiques Sp6ciale level. Students had to make decisions then whether they wanted to become teachers or research workers, and some of them went one way and some went the other. I do not know whether the decision was made as we entered, or whether we discovered two years later we might want to do research.

I had to decide whether I wanted to spend my entire life doing research in a very abstract subject. You must also remember that during that last year of high school when I was influenced by my physics teacher I had thought that physics was going to be my field. His training at the Ecole Normale Superieure was at the highest university level, and in fact can better be compared to the work done at the graduate level at most other universities, because the students had to do the standard university mathe- matics curriculum on their own, as it were.

At the Sorbonne it was lectures, fairly polished lectures. I faithfully attended the lectures by Garnier, and Gamier taught differential geometry.

## Mathematical economics : twenty papers of Gerard Debreu in SearchWorks catalog

I'm sure I took other lectures as well. And then in the seminar [at ENS] it was a mixed bag; we occasionally had a lecture by Elie Cartan, the father of Henri, who was of course already at that time a very revered mathematician. We had a lecture by De Broglie, the physicist, Nobel Prize winner.

So the seminar was a little of different things by different people. Henri Cartan was still young, and did great things later, and the seminar was simply supposed to review mathematics, and it did that; it was also to give us a taste of a variety of mathematical researches, and no text was used. On my own, I read most of Goursat.

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The point of the story, of course, is that Debreu was as well trained in mathemat- ics as was possible for any student to have been at that time. He had the remarkable good fortune to be at the place, at the time, when mathematics itself was being re-represented by Bourbaki as a discipline defined by its pursuit of the implications of, and the investigation and exposition of, the idea of mathematical structure.

In this mathematical hothouse, isolated because of the war and the dislocations it produced, Debreu pursued mathematics but did not want to have it define his intellectual life. But there were no real alternatives; he was a mathematics student first, and other possibilities would have to be deferred to the end of the war, since the only applied alternative that he might have considered at ENS, astrophysics, seemed to be ruled out by the absence of any real instructional program — the professor was not present. Stuck in mathematics, around he looked at possibilities for later work: When I became interested in economics as a possibility as before I had become interested in astrophysics I got hold of the standard text studied by students of economics at the university.

I don't remember who the author was but it was very non-theoretical somebody I never met. Debreu has recounted, in a couple of places where autobiographical material is available, the happenstance of his receiving a copy of Allais' book, A la recherche d'une discipline economique. In conversation he noted that his move to economics was a feature of his own intellectual changes as well as a circumstance of the times: To a large extent it was pure chance because Allais had sent his book to a friend of mine, who was a humanist, who was the president of his class at ENS — he was actually not in my own class but one year after, but we were friends and he gave me his copy.

I suppose otherwise if I had persevered in my interest in economics that I might not have been aware of the Allais book for months, and maybe by then it would have been too late. But one part of my interest in economics — although it was not too elegant a field — was simply that the war economy in France was special. We believed, though we found it a long time coming, that Germany was going to be defeated. It was clear that there would be a lot of reconstruction in particular in France. There would be a lot of reconstruction work to do after the war and it proved to be the case. And that may be why I came to know people like Pierre Massey, who was at one time president of Electricite de France, but who also wrote a book on stock management.

I came to know him very well and I saw him regularly until his death. He was succeeded as President of Electricite de France by a friend of mine, Marcel Boiteaux, who also had his career disturbed by the war. He was an officer in Italy somewhere and later on after the Aggregation cast his lot with Electricite de France. We shared the extraordinary story of the coin-tossing for the Rockefeller Fellowship.

## On the Existence of Price Equilibrium in Economies with Excess Demand Functions

So a number of random events were surely very important [in my move to economics]. The book by Allais arrived in Debreu's hands at a very crucial moment, for Debreu was searching for meaningful work, as many young people search at that age. The Allais book had been more or less just sent around to individuals; not many copies had been printed. It was very much outside the established French economics channels.

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In retrospect, it is not only remarkable that it was able to be printed under those wartime conditions, but also that it received any attention at all, given the number of unusual books that merely drop out of sight. Even had it gone to all those interested in mathematical economics at that time, there would have been problems. It was very primitive mathematics from a Bourbaki point of view, though it was more sophisticated than most neoclassical texts. Nevertheless in Debreu it found a friendly reader. First of all I saw that mathematics could be used in economics in a rigorous way, even though it was not the kind of mathematics I was most fond of.

My interest in economics wasn't ready made. I became interested in economics in or maybe a year before. And the circumstances were such that Bonpere gave me that book around April of But I think that the circumstances were not sort of ideal for me to read it just then — recall that D-Day was June 6. It was only in September I suppose when it was clear first of all that I would not start another academic year normally. Things were too chaotic in France and it was then I think that I became serious about economics; I may have looked at it when Bonpere gave it to me but I did not study it.

Since a few people got hold of copies, my guess would be that most of them must have been repulsed by all the characteristics of the book, it was lengthy, technical, et cetera. But remember that I was in a very bizarre situation because it had become clear to me that I would not be a professional mathematician. It was late when I decided that, and my career was disturbed by events of the war.