Summary of the 62th Conference, JSHET@@
(24-3-2)
The Study of the Existence of General Equilibrium: A History As Viewed From Japan by Aiko Ikeo, Kokugakuin University
1. Introduction
This paper aims to investigate how the Japanese mathematical economists studied the questions relating to the existence of a general equilibrium from the late 1920s till the early 1960s. We will trace the research line including Kazuo Midutani, Shizuo Kakutani, Hukukane Nikaido, Takashi Negishi and Hirofumi Uzawa. (See Ikeo (1996) on Kei Shibata's contribution.) We will shed light on the complicated history of the study of the existence question, by focusing on Japan's direct connection with Karl Menger, John von Neumann, Oskar Morgenstern and Kenneth J. Arrow. The Japanese scholars who began to study mathematics before 1960 mastered the mathematics which had been developed in the German-speaking world. In this respect, the Japanese did mathematics in a tradition different from those who had studied mathematics in other areas such as France or North America. Within a few years of the end of the war in 1945, the Japanese were working on the similar subjects as were American and European economists thanks to the prompt circulation of scientific, refereed journals. We discuss the rejection of Nikaido's existence paper at Econometrica in detail. The paper was later published in Metroeconomica.
2. Proof of the Existence of a General Equilibrium until 1956
The proof of existence, stability and uniqueness are important topics for the study of general equilibrium theory. Set theory and the convex set method were used for the proof of existence in the 1950s, and these were mathematical tools different from those used for the proof of stability. Moreover, the study of stability analysis was promoted by a group of scholars prior to the study of the existence question carried out by another group. It has been already discussed that in the 1940s several Japanese economists made important contributions to stability analysis, which were comparable to the studies which were developed in North America and Europe in the 1950s (Ikeo 1994).
The research line of the existence question was especially blurred by the controversy over the foundation of mathematics, which culminated in the clash between the formalist David Hilbert and the intuitionist L.E.J. Brouwer in 1927. This controversy did not matter, at least for the study of the so-called existence question, in the sense that Brouwer's fixed point theorem has been formalized by Hilbert's students and become available for economists as well as the economists who used the traditional language in mathematics. Later fixed point theorems became familiar to economists by von Neumann (1937) and Kakutani (1941). The historical development of the study of existence of general equilibrium was further complicated by the development of relevant mathematical tools and game theory, and the interactions and communications among migrating and traveling scholars in the 1930s and 1940s.
In retrospect, T.C. Koopmans (Koopmans ed. 1951: 1) mentioned in English the importance of the intellectual legacy of general equilibrium analysis from Europe in the 1930s. A little later, one pair of economists and three individual economists independently proved the existence of a competitive economy with the use of set theory and convex set method including a fixed point theorem.
3. Karl Menger's Mathematical Colloquium
It is well known that Karl Menger informally organized the mathematical colloquium for mathematicians and economists in Vienna from 1928 until 1936 (Menger 1973: 47). He published their reports and proceedings as Ergebnisse eines Mathematischen Kolloquiums from 1931 until 1937. However, it is relatively unknown that three Japanese scholars, mathematician Yukio Mimura (Osaka University), economists Kazuo Midutani (Kobe University) and Yuzo Yamada (Tokyo University of Commerce, Hitotsubashi University since 1949), attended Menger's mathematical colloquium in Vienna. Mimura made a report on the colloquium and Midutani (1939) discussed Abraham Wald's study of the existence problem. Mimura advised young Shizuo Kakutani (Osaka University) to study the works of von Neumann.
Kakutani was naturally fascinated by von Neumann's works as were many other mathematicians of the day. Kakutani followed Mimura's advice and started with von Neumann's 'Uber ein okonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes' ('On a system of economic equilibrium and the generalization of Brouwer's fixed point theorem' in German, 1928; 'A model of general equilibrium' in English, 1945-6). He found the Ergebnisse eines mathematische Kolloquiums in the library and 'Zur Theorie der Gesellschaftsspiele' (in German, 1937; 'On the theory of games and strategy' in 1959). Kakutani recalled and said, "Von Neumann's papers were rather difficult" (Personal communication with Kakutani). Kakutani also met Midutani in the mathematical seminars held in Osaka-Kobe area.
4. John von Neumann -- Shizuo Kakutani
Kakutani received a chance to study at the Institute for Advanced Study in Princeton University. In October 1940, Kakutani started to attend the seminar run by von Neumann and gave a talk on the extension of Brouwer's fixed point theorem. Brouwer's fixed point theorem is related to point-to-point mapping, while Kakutani's related to multi-valued or point-to-set mapping. Kakutani's idea was published as 'A generalization of Brouwer's fixed point theorem' in the Duke Mathematical Journal of 1941. In October 1941, von Neumann and Morgenstern started to run the seminar on the theory of games. Kakutani and A.W. Tucker were the only participants. Although the Institute for Advanced Study allowed Kakutani to stay there and continue his research after December 1941, he left the United States by exchange ship in May. At that time, von Neumann and Morgenstern's manuscript for their The Theory of Games and Economic Behavior (1944) was quite incomplete. Kakutani confirmed that he did not help them produce a clean manuscript. After the conclusion of World War II, Kakutani came back to Princeton in 1948, and moved to Yale University. He found that the mathematics relating to economics had been developing during the war.
5. Japanese Scholars -- K. J. Arrow
Around 1950 in Japan, a mathematics student, Hukukane Nikaido (b. 1923), realized that John von Neumann's and Kenneth J. Arrow's works of mathematical economics were different from those of Hicks and Samuelson's, which were based on calculus. In the new approach, the abstract economy was modeled based on the knowledge of set theory and convex set methods to establish the existence of general equilibrium and to clarify the welfare aspects of the competitive economy.
The proceedings of the first conference on mathematical programming entitled Activity Analysis of Production and Allocation were published as a Cowles Commission monograph in 1951 and soon copies arrived in Japan. Also in 1951, K. J. Arrow's 'An extension of the basic theorems of classical welfare economics' appeared in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability edited by Jerzy Neyman. Arrow reviewed Pareto optimality from the viewpoint of convex set theory. Gerald Debreu in his 'the coefficient of resource utilization' (1951), independently of Arrow, embarked on the set-theoretic and convex-set method in the study of the optimality of competitive equilibrium.
In the winter of 1952 at the Chicago meeting of the Econometric Society, Arrow and Debreu presented their 'Existence of an equilibrium for a competitive economy' and L. W. McKenzie his 'On equilibrium in Graham's model of world trade and other competitive systems'. Takuma Yasui, who had studied the conditions for the stability of a competitive equilibrium with the use of a system of ordinary differential equations in Japan in the 1940s, attended the sessions. Yasui for the first time learned the fixed point theorem, which was the key to the proof of the existence of a competitive economy. Yet Yasui did not report the heated argument between McKenzie, and Arrow and Debreu. McKenzie proved the existence and uniqueness of equilibrium in Frank D. Graham's model for world trade by using Kakutani's fixed point theorem. The production aspect of the model was represented by a linear activity model in which the primary goods are the labor supplies of the several countries. Arrow and Debreu used set-theoretical techniques to specify the precise assumptions of a competitive economy as the basic starting point. They confined themselves to proving the existence of competitive equilibrium by Eilenberg-Montgomery fixed point theorem and extended J. F. Nash's notion of an equilibrium point for a game to their abstract economy, which was first discussed in Debreu (1952). They were discussing the question of the existence of a competitive equilibrium through a generalization of the concept of a game. Their papers were both published in Econometrica in 1954.
Hukukane Nikaido attended Shokichi Iyanaga's seminar for graduate students, when he was a undergraduate student at the mathematics department of the University of Tokyo. Tsuneyoshi Seki (b. 1924) began to attend Iyanaga's seminar to become a mathematical economist in 1948 after he graduated from the economics department of Hitotsubashi University. Seki was interested in the question of the existence of general equilibrium which was discussed not only in Watanabe and Hisatake's Application of Mathematics to Economics (in Japanese, 1933) but also in K. Menger's Ergebnisse eines mathematische Kolloquiums. Seki delivered a talk on von Neumann's 1937 paper, which stimulated Nikaido to read the paper and von Neumann and Morgenstern's 1944 book.
Nikaido was in Japan working out the existence problem of competitive equilibrium along with the minimax theorem in game theory, the theorem of Nash's equilibrium in non-cooperative games, the von Neumann growth model, and, the Brouwer and Kakutani's fixed point theorems. He did not know McKenzie or Arrow and Debreu's presentations on the same problem at the 1952 Chicago meeting. When he came across McKenzie's 'On equilibrium in Graham's model of world trade and other competitive systems' in the April issue of Econometrica in June or July 1954 in Japan, Nikaido immediately submitted his (first) existence paper to Econometrica. Then Arrow and G. Debreu's 'Existence of an equilibrium for a competitive economy' was published in the next July issue of Econometrica. Nikaido received the rejection letter in October 1954. He admitted that he had had no opportunity to read Arrow and Debreu's article before having submitted the manuscript to the editor of Econometrica (Nikaido's letter to Strotz, 7 October 1954). With minor modifications, such as adding Arrow and Debreu (1954) to the list of references and 'more economic merit', he submitted his (second) existence paper entitled 'On the classical multilateral exchange problem' to Econometrica. This time, unexpectedly, Arrow wrote to Nikaido and suggested that he submit the paper to Metroeconomica, a journal which he had never heard of. Fortunately the paper was published in Metroeconomica in 1956. Nikaido formulated the basic propositions of the existence of general equilibrium as a theorem relating to the excess demand function in the case of multilateral exchange of many commodities, and proved this with the direct use of Kakutani's fixed point theorem.
There are several ways of proving the existence of competitive equilibrium with the use of a fixed point theorem. In any case, it was proved that fixed point theorems imply Walras's existence theorem. Hirofumi Uzawa in his 'Equilibrium and stability' (1962) proved that Walras' theorem implies Brouwer's fixed point theorem. This means that Walras's existence theorem and Brouwer's fixed point theorem are equivalent.
Moreover, Takashi Negishi in his 'Monopolistic competition and general equilibrium' (1961) initiated the study of imperfect competition in general equilibrium analysis. He assumed that consumers were price takers while firms were monopolistically competitive. His firms had subjective inverse demand (supply) functions for their outputs (inputs), being consistent with the given information of the present state of the market. He further assumed the convexity of possible production sets of firms. Then he proved the existence of equilibrium in an imperfect market.
From 1950 to 1960, Nikaido, Negishi, and Uzawa all joined Arrow's project on the Efficiency of Decision Making in Economic Systems at Stanford, which was backed by the Office of Naval Research. Other Japanese mathematical economists such as Ken-ichi Inada and Hajime Oniki also joined Arrow's project. They played active roles in the study of the existence and stability of a general equilibrium in a competitive economy, two sector growth models and welfare economics.
Personal Communications
Shizuo Kakutani at Yale University in New Haven on 5 January and 3-4 April 1995. Hukukane Nikaido on the phone on 7 July 1993, at Tokyo International University on 6 May 1994, and correspondence etc. during September 1996 and January 1997.
Selected References
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