Finding Equilibrium: Arrow, Debreu, McKenzie and the Problem of Scientific Credit

Finding Equilibrium: Arrow, Debreu, McKenzie and the Problem of Scientific Credit

by Till Düppe, E. Roy Weintraub

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ISBN-13: 9781400850129
Publisher: Princeton University Press
Publication date: 07/21/2014
Sold by: Barnes & Noble
Format: NOOK Book
Pages: 304
File size: 3 MB

About the Author

Till Düppe is assistant professor of economics at the Université du Québec à Montréal. He is the author of The Making of the Economy. E. Roy Weintraub is professor of economics at Duke University. He is the author of How Economics Became a Mathematical Science.

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Finding Equilibrium

Arrow, Debreu, Mckenzie and the Problem of Scientific Credit


By Till Düppe, E. Roy Weintraub

PRINCETON UNIVERSITY PRESS

Copyright © 2014 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-5012-9



CHAPTER 1

ARROW'S AMBITIONS


Kenneth Arrow was never shy about engaging his past. In contrast to our other two protagonists, he gave a large number of interviews and on various occasions written sketches of different portions of his life and the development of his interests. Likely his openness to interviewers and biographers is the result of his ebullience and his lifelong interest in thinking about how ideas develop and how individuals' natures are formed. At the same time Arrow was hesitant about claiming the last word about his past. When asked to write about his "life philosophy" he began, "it is part of my life philosophy that no life can ever be examined fully and that attempts to do so are never free of self-deception.... Like the state in which I live, we plan and build on ground that may open beneath us" (1992, 42). Accordingly, the historian who undertakes to add another account to Arrow's self-accounts will wish neither to repeat them nor to reconstruct a self that is hidden by these self-deceptions. Thus we ask: Where did Arrow's intellectual ambitions come from?


A Precocious Boyhood in New York (1921–40)

The first ground that opened beneath Arrow during his childhood was the Great Depression. He was born on August 23, 1921. Both of his parents had been born overseas and came to the United States as infants. They were both successful academically, as his mother graduated from high school and his father from college, not a usual event for immigrants in the 1900–1914 period. His father Harry's "family was very poor, [his] mother's hardworking and moderately successful shopkeepers" (Arrow, in Breit and Spencer 1995, 44). Arrow recalled that his father had some business successes fairly young and earned a law degree; he worked for a bank and as a result the family was fairly prosperous through the 1920s. With the Depression, his father lost his regular job and the family often had to sell household belongings in order to have money for food and rent and clothes. His father managed to do contract work for various legal firms from time to time in those years, but it wasn't until the end of the 1930s that the family began to reestablish itself economically.

Arrow was precocious. It was not simply that his school academic record was very strong, but he read extensively and deeply outside the school curriculum. He recalled that he read Bertrand Russell's Introduction to Mathematical Philosophy (1919) and other demanding books in philosophy, literature, and the sciences unrelated to his high school programs. On graduation he applied to Columbia University even though he was quite young (fifteen) compared to those who might have been his classmates. In an admission interview he asked the counselor about meeting the deadlines for financial aid decisions since he needed a scholarship in order to attend. The interviewer replied by telling him that he needn't bother about financial aid since he was not going to be admitted. In fact he was admitted, but the interviewer's comment had the effect of delaying the family's completing the scholarship application until after they heard about admission. By the time they realized what had happened it was too late for Arrow to apply for a scholarship. Many decades later Arrow discovered that his interviewer had been described by an historian as one of the most egregious anti-Semites in all of Ivy League education. Columbia, while not nearly as exclusionary with respect to Jews as were Harvard, Yale, and Princeton, had a numerus clausus (i.e., quota) arrangement to limit admissions of children of immigrant Jews living in the New York metropolitan region.

Without financial aid and with his family's own finances limited, Arrow applied to, and was accepted into, the City College of New York (CCNY). At that time, City College had free tuition for residents of the city, the result of earlier agreements and understandings that the prosperity of the city depended upon the education of its youth independent of their financial means. Admission was strictly by merit and Arrow was certainly meritorious. Moreover, CCNY was a commuter school so Arrow could live at home. CCNY was also, for the students especially, a particularly political cauldron. As the late Irving Kristol, the neoconservative editor and writer, recalled his student years at the City College:

Every alcove [of the City College lunchroom] had its own identity, there was the jock alcove ... there were alcoves for ROTC people—I don't think I ever met one—and then there was the Catholic alcove, the Newman Club. There was even, I am told, a Young Republican Club, but I don't think I ever met anyone who belonged to that club and maybe they didn't exist. But pretty much our life in City College was concentrated between alcove one and alcove two, the anti-Stalinist left and the Stalinist left. And that was our world, at least our intellectual universe. (Kristol, in Dorman 1998, 46–47)


Arrow was not so politically engaged as a student but was interested in many different subjects, and early on he decided that he would major in mathematics with the long-term objective of becoming a high school mathematics teacher: "I was concerned about getting a job. I didn't look beyond college very much at that point. All I wanted was security" (Arrow, in Horn 2009, 63). In that Depression period, secure civil service employment in the New York City public schools seemed reasonable to both him and his family. As a result, in his undergraduate program he took not only a lot of mathematics courses but also courses in education, and he did student intern teaching as well. In Arrow's case, that teaching consisted of conducting preparation classes for high school students who wished to overcome their initial failure on the New York State Regents exam through a retest process. He recalled that his students were the most motivated he had ever come across: "It was the biggest teaching success of my entire life" (ibid., 64). He loved teaching but a difficulty emerged: in 1932 the education administrators had constructed a list of qualified teachers from whom future recruits to the teaching profession would be drawn. The idea was that as teachers left the schools, those at the top of the list would be hired. But during the Depression teachers were not resigning to take other jobs. No new names, like that of Kenneth Arrow, could be added to the list until that preexisting pool of candidates had been drawn down. At least a year before his graduation, Arrow realized that he would not be able to find a job teaching high school mathematics. What was he to do?

In the summer after his junior year Arrow found work as an actuarial intern, even though he realized that if he wished to pursue this professional course after graduation, he would need to learn more statistics. Arrow thus took a course in statistics in the mathematics department, but the instructor apparently knew no statistics. However, one of the recommended books on the syllabus contained a large number of references to recent first-rate work in the field and so Arrow embarked on an independent reading program to become conversant with it all. He also did some other college work that would become important for his later success. In an interview with Jerry Kelly, Arrow spoke of how he encountered ideas in mathematical logic even though it wasn't part of any of his courses: he was "fascinated and used to aggravate my professors by writing out proofs in a very strictly logical form, avoiding words as much as possible and things of that kind" (Kelly 1987, 44). In his last term at CCNY he took a course titled "Logical Relations" with Alfred Tarski. As a result of his performance in that class, Tarski asked him to read and make necessary changes in the galley proofs of the English translation (from the German) of his textbook, Introduction to Logic (1941).

Arrow graduated magna cum laude in June 1940, having won the Gold Pell Medal in his junior year for having the highest average in all his studies and the Praeger Memorial Medal for having the highest average in his senior year, as well as the Ward Medal for Logic. He was also elected to Phi Beta Kappa. He was not yet nineteen years old.


Columbia, 1940–42

"By the time I graduated, in 1940, the job situation was not very good. So when I asked myself in my senior year, what do I do next, I thought—well, why not go to graduate school?" (Arrow, in Horn 2009, 64). The only place to learn advanced statistics in the New York City area was at Columbia University. There it was taught by Harold Hotelling, who had succeeded Henry Ludwell Moore. But Hotelling, as Moore before him, was not in the mathematics department but in the economics department. Since Arrow had no interest in economics, he decided that he could take Hotelling's courses as electives if he were at Columbia studying mathematics. Arrow was able, with $400 given him by his father (who had successfully borrowed it since he "knew somebody who knew somebody who was rather well off " [ibid.]), to accept Columbia's offer of admission to the graduate program in mathematics for fall 1940: "I went to Columbia because ... well there were several problems. One was that we were extremely poor and the question of going anywhere depended on resources. Columbia had the great advantage, of course, that I could live at home, which wasn't true anywhere else. I didn't get any financial support for my first year, none at all" (Arrow, in Kelly 1987, 45). To earn some money to help the family, Arrow got a summer job after his college graduation "as an actuarial clerk. That meant doing some elementary computations, calculating premiums ... I was very fast, I picked it up immediately. I got paid 20 dollars a week" (Arrow, in Horn 2009, 68). Beginning his graduate studies with the plan to become an actuary, he soon was taken by, and "bought" by, mathematical economics:

I had no interest in economics. I was in the mathematics department, taking courses like functions of a real variable, and I was going to take courses from Hotelling. In the first term he happened to give a course in mathematical economics. So out of curiosity I took this and got completely transformed. The course to an extent revolved around Hotelling's own papers.... Anyway, then I switched to economics from mathematics. I had gone to Hotelling asking for a letter of recommendation for a fellowship in the mathematics department [for my second year] and he said that, "Well, I'm sure I don't have any influence in the mathematics department, but if you should enroll in economics, I've found in the past that they are willing to give one of my students a fellowship." I was bought. Incidentally, I impressed him on about the second day of the class ... he said he was puzzled by the fact that he had never been able to produce an example of Edgeworth's paradox with linear demand functions. So I sat down and wrote out the conditions for linear demand functions to yield the paradox; these conditions were certain inequalities on the coefficients and the inequalities were inconsistent. So I came to him the next day and showed it to him. Really it was just a few lines, but from that point on he was really impressed with me.... Anyway, I enrolled in economics. (Arrow, in Kelly 1987, 45–47)


Hotelling's assistant at that time was Abraham Wald, who had ended up in that position through a grant from the Carnegie Foundation after a year at the Cowles Commission. We will later see the deep connection between Arrow's work with Debreu and work Wald had done in the early 1930s. It is thus worth pausing here to introduce this remarkable figure.

Wald had been a mathematics graduate student of Karl Menger's in Vienna in the late 1920s, but since he was a Romanian Ostjude he was precluded from finding a university faculty position in Austria. As a result Menger sent him to Karl Schlesinger, an economist-banker in Vienna who wished to be tutored in mathematics. It was as a result of that relationship, and Wald's participation in Menger's mathematical colloquium, that he wrote two papers on the existence of a competitive equilibrium, one using a model based on a system of exchange and the other based on a model of production and exchange; he published them both in the proceedings of Menger's colloquium (Wald 1934, 1935). Wald solved the equilibrium problem by cumbersome brute-force techniques. His approach employed a very strong assumption about household behavior, an assumption that assumed that there was only one consumer, which simplified the argument immensely. It would be one of the major accomplishments of our three protagonists to construct a proof of the existence of equilibrium that followed a more natural economic logic than did Wald's proof. This point is important, and we will return to it in a subsequent chapter.

Oskar Morgenstern, a member of the colloquium, subsequently hired Wald as a researcher at the Business Cycle Research Institute he directed in Vienna (one of the several Rockefeller institutes in Europe organized on this subject). There Wald became a mathematical statistician and wrote an important monograph on seasonal variation in time series. Following the Anschluss, and Schlesinger's suicide, and in the face of Morgenstern's earlier unwillingness to nominate Wald rather than others for a Rockefeller Fellowship to come to the United States, Wald had to escape from Vienna. As a Romanian citizen, however, he needed to get travel documents there, so he traveled first to Romania and then by boat to Cuba before he was able to enter the United States with some support from the Cowles Commission then in Colorado Springs.

It was during his time at Columbia from 1940 to 1942 that Arrow first met Wald while taking Hotelling's class in mathematical economics: "[As] I began to know a little more economics, I was hit by the number of extremely original papers that Hotelling had written. ... What [Hotelling] taught was essentially the theory of the firm and the theory of the consumer.... I was a complete master at bordered Hessians" (Arrow, in Horn 2009, 67), a matrix of partial derivatives that was the main tool used in solving the kinds of optimization problems that Hotelling's class addressed. Arrow still hoped to secure a full-time actuarial position, and so he applied in the spring of his first graduate year for a summer student position. But even as a mathematics graduate student he was found to be unqualified:

April 14, 1941. Dear Mr. Arrow: We've now reviewed the papers in connection with all applicants for actuarial student positions this year, and find that we have 27 candidates who have passed the mathematical test. As we propose to employ about 6 students this year, I'm afraid a lot of good men will be disappointed. I regret to state that you were not selected to fill one of the vacancies open at the present time. Mr. [XXX], Associate Actuary, The Prudential Company of America. (KJAP, Accession 2008–0037, Early Career)


In spring and summer 1941 he wrote his mathematics master's essay, titled "Stochastic Processes," a copy of which is preserved in the Arrow Papers at Duke University (KJAP, 28, Master's Thesis). Fully engaged with statistical work while finishing his mathematics master's degree in fall 1941, Arrow sailed through his courses in economics and reached the dissertation stage very quickly. He passed his oral Ph.D. examinations by December 1941 in economic theory, public finance, statistics, and business cycles, while being certified in economic history and mathematical economics. He won a University Fellowship for 1941–42 and a Lydig Fellowship for 1942 (which he would not take up until 1946). It was in this period that he read John Hicks's new (1939) book Value and Capital and realized that there was a way to think about economics in a systematic fashion: "You know, after reading through the mish mash like Marshall and things like that, suddenly there was this clear, well-organized view, you knew exactly what was happening. Just the sort of thing to appeal to me" (Arrow, in Kelly 1987, 47). This would be the entry point for Arrow's first attempt at writing a doctoral dissertation, which he hoped would take a Hicksian approach to some Marshallian production conundrums.

Even though Hotelling was Arrow's primary mentor, the dominant presence in the Columbia department was Wesley Clair Mitchell, who spent his time downtown at the National Bureau of Economic Research and so was generally unavailable in the department. His course on business cycles was data based, and Arrow recalled that he appreciated the statistical care with which matters were treated. He also had a course from Arthur Burns, who replaced Mitchell as a teacher for a period of time, and from A. G. Hart, who would eventually (postwar) serve as his thesis advisor: "The place was a little bit weird, even by the standards of the time, in the sense that it was very anti-neoclassical. One of the results of this mood was that there was not a course in price theory, at any level.... [Mitchell] said it was our duty to collect a lot of data. When you have collected enough data, then things will be [clear]" (Arrow, in Horn 2009, 69).


(Continues...)

Excerpted from Finding Equilibrium by Till Düppe, E. Roy Weintraub. Copyright © 2014 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

Preface ix
Chronology xxiii
Part I People 1
Chapter 1 Arrow's Ambitions 3
Chapter 2 McKenzie's Frustrations 24
Chapter 3 Debreu's Silence 47
Part II Context 65
Chapter 4 Sites 67
Chapter 5 Community 98
Part III Credit 129
Chapter 6 Three Proofs 131
Chapter 7 Aftermath 172
Chapter 8 The Proofs Become History 204
Conclusion 231
Coda 245
Acknowledgments 249
References 251
Index of Names 267
Index of Subjects 273

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