# Uzawa, Hirofumi

# Uzawa, Hirofumi *1928–*

Hirofumi Uzawa was a native of Yonago, northwest Tottori Prefecture, on Honshu. His father was a schoolteacher. At the age of four, Uzawa and his brother moved with their parents to Tokyo, where they grew up. In 1948 he enrolled in the Department of Mathematics at the University of Tokyo, where he was chosen as a Special Research Fellow in the department. He received the BS degree in mathematics at the age of twenty-three in 1951, having majored in algebraic number theory. After receiving the BS Uzawa entered graduate school and taught mathematics at the University of Tokyo for five years.

To fully appreciate Uzawa’s formative years, one must keep in mind that when he was born the ruling power in Japan was the militaristic Meiji dynasty. This dynasty engineered the Japanese defeat of Russia in 1904, Japanese control of South Korea and Taiwan through League of Nations mandates after World War I (1914–1918), and Japan’s attack on Pearl Harbor in 1941. After Japan’s defeat, the U.S. military occupied the country from 1945 until 1952, and Japan was ruled by a military governor during that period. Japan’s September 1945 surrender resulted in its loss of control of South Korea and Taiwan, both of which then were occupied by U.S. troops. During its occupation of Japan, the United States engaged in a war in Korea, across the Sea of Japan from Uzawa’s birthplace. These events are implied in Uzawa’s autobiographical sketch, especially in his discussion of poverty, starvation, and underdevelopment in Japan.

While studying undergraduate mathematics and seeking to become a professional mathematician, Uzawa was led by the postwar poverty of Japan to study economics. He and several others began a systematic reading of Marxian economics. He contemplated joining the Japanese Communist Party but was advised by a friend who was already a member that he could not pass the entrance examination given to prospective members. He therefore decided to quit mathematics and study economics so that he could learn enough to pass the examination. But he did continue studying mathematics until he earned a degree.

After Uzawa graduated in 1951, he secured a job with the Institute of Statistical Mathematics at the Ministry of Education, and subsequently as a statistician with a life insurance company. During 1955 and 1956, he published four articles in the institute’s quarterly publication. The first two were purely statistical theory. The third was on Leontieff input-output models. The last was “A Note on Preferences and Axioms of Choice.” Three years later, he published “Preference and Rational Choice in the Theory of Consumption.”

During the early 1950s, Uzawa remained actively affiliated with the University of Tokyo. He joined a small group of economists in the Faculty of Economics and read Keynes’s *General Theory of Employment, Interest and Money* and Kenneth J. Arrow’s *Social Choice and Individual Values*. He worked for six months in 1954 as an assistant to Everett E. Hagen (1906–1993) of MIT, who was in Japan with a mission of the World Bank. Hagen was in charge of macroeconomic analysis, and on this project, Uzawa first learned about Keynesian economic policy administration. In the summer of 1954, he also attended the annual joint University of Tokyo–Stanford University seminar conducted by Dutch economist Hendrik Houtakker (b. 1924) of Stanford on demand analysis. In this seminar, Uzawa was reintroduced to the work of Arrow of Stanford, and this time read everything he could by Arrow, including especially his work with Leonid Hurwicz on the feasibility and stability of the “allocative mechanism” of a socialist economy.

In economics, Uzawa studied under Hyoe Ouch (1888–1980), who led the fight to resuscitate the Ohara Institute for Social Research after World War II (1939–1945). This institute studied labor economics, Marxian economics, and social issues and published the *Journal of the Ohara Institute for Social Research*, the *Labor Yearbook*, pamphlets, and a publications series. Keynesian economics in Japan began with Keynes’s *The Economic Consequences of the Peace*, and *A Treatise on Money*.

In 1955, through Houthakker, Uzawa reviewed the unpublished manuscript of Kenneth J. Arrow and Leonid Hurwicz’s article on local stability. Expanding on this article on his own, Uzawa wrote “Gradient Method for Concave Programming II: Global Stability in the Strictly Convex Case.” His manuscript led to the receipt in 1955 of an invitation from Arrow to work with him at Stanford. Having become interested in pursuing a career in economics rather than mathematics, in 1956 he applied for and received a Fulbright Fellowship to finance the Arrow enterprise. He was a research assistant at Stanford from 1956 to 1964. Here, he was exposed to a rigorous mathematical treatment of neoclassical economics, Keynesian theory, and general equilibrium theory rather than Marxist economics. Uzawa published at least three papers in the Technical Reports series of the Stanford economics department between 1956 and 1958. Two of these papers contributed to the Houthakker research program in consumer economics, specifically his interest in preference functions.

In 1958 Uzawa, Arrow, and Hurwicz published *Studies in Linear and Non-Linear Programming* with the support of the U.S. Office of Naval Research. In this book, Uzawa presented a general mathematical theory. This book was concerned with deriving existence proofs for solutions to programs in linear topological spaces. These proofs involved the theory of convex polyhedral cones (CPC) in point-set topology, which are treated algebraically by means of analytical geometry. Using set theory and linear algebra, Uzawa presented the theory of topology on which the editors based the application to programming in the remainder of the book. Uzawa defined CPC conventionally as “the intersection of a finite number of half-spaces,” that is, affine spaces bounded by hyperplanes. In the linear programming problem, the intersection of half-spaces creates a pyramid. The linear constraints of the problem constitute the edges of the polyhedron. The intersections of these edges constitute the vertices of the polyhedron. The solution algorithm evaluates each vertex in turn to find the one constituting the maximum or minimum. In the case of the pyramid, or three-dimensional polyhedron, four or five vertices must be evaluated, depending upon whether the base is a triangle or a square.

The gradient [slope] method is defined as the solution set to a system of differential or difference equations. The gradient is the ratio of the change in the slope of the plane triangle constituting one side of the pyramid. Further, the gradient method is applied to several particular problems, including economic development and growth. It is found that it is slower than the simplex method because it calculates all surface vectors to a linear programming problem, while the simplex method calculates only the optimum vectors, that is, only vertices. A simplex is the simplest form that can be constructed between points in a given space.

Uzawa remained away from Japan for thirteen years, serving on the faculties of the University of California at Berkeley (1960–1961), Stanford (1961–1964), Cambridge (1964–1965), and the University of Chicago (1965–1969). He was thoroughly immersed in the project to mathematize neoclassical economic theory. He left Stanford to attain intellectual independence. By 1960, he was married to his wife, Hiroko, and had children.

In 1968 Uzawa returned to the University of Tokyo as professor of economics until retiring to emeritus status in 1993. In 1973 and 1976, his interest shifted to the shortrun fluctuations of a capitalist economy, or business cycles, and he published papers on Keynesian theory. In 1976, he served as president of the Econometric Society. Beginning around 1970, and continuing to the present, he has been senior advisor to the Research Institute of Capital Formation of the Development Bank of Japan. He also taught at Niigata University, Chuo University, United Nations University, and Doshisha University. From about 2003 to 2005, he was director of Doshisha University’s Research Center of Social Overhead Capital.

Uzawa’s general equilibrium analysis concluded that Walrasian *tâtonnement* mechanism is globally stable. This again was part of the project to study the nature of equilibrium mathematically but also of the project to renew general equilibrium analysis. Uzawa is most renowned, though, for the development of two-sector neoclassical endogenous growth models, and for a theory of economic growth. His method was taken from Marx’s two-department model of simple reproduction in volume 1 of *Das Kapital*. The language and categories employed, however, were those of neoclassical economics. In these growth articles, he was disaggregating the one-sector models of Robert M. Solow (b. 1924) and T. W. Swan (1918–1989). The two sectors were a consumption sector and an investment sector, each using labor and capital to produce an output. The investment sector produced a capital good, and the consumption sector produced a consumption good. The production functions for each sector exhibited diminishing returns to scale. Technological change was not included in the Solow-Swan model, and so was exogenous. Uzawa included terms for technology in his model, thus endogenizing technological change. His articles stimulated an explosion of research into growth models in the 1960s, but this interest subsided thereafter. The practical motivation driving his interest in growth was the underdeveloped state of the Japanese economy as he perceived it.

**SEE ALSO** *Economic Growth; Economics; Economics, Keynesian; Industrialization; Lucas, Robert E., Jr.; Mathematical Economics; McFadden, Daniel L.; Solow, Robert M.; Underdevelopment; World War II*

## BIBLIOGRAPHY

Solow, Robert M. 1961. Note on Uzawa’s Two-Sector Model of Economic Growth. *Review of Economic Studies* 29: 48–50.

Uzawa, Hirofumi. 1958. Gradient Method for Concave Programming II: Global Stability in the Strictly Convex Case. In *Studies in Linear and Non-Linear Programming,* ed. Kenneth J. Arrow, Leonid Hurwicz, and Hirofumi Uzawa, 127–132. Stanford, CA: Stanford University Press.

Uzawa, Hirofumi. 1961. On a Two-Sector Model of Economic Growth, I. *Review of Economic Studies* 29: 40–47.

Uzawa, Hirofumi. 1963. On a Two-Sector Model of Economic Growth, II. *Review of Economic Studies* 30: 105–118.

Uzawa, Hirofumi. 1964. Duality Principles in the Theory of Cost and Production. *International Economic Papers* 5: 216–220.

Uzawa, Hirofumi. 1964. Optimal Growth in a Two-Sector Model of Capital Accumulation. *Review of Economic Studies* 31: 1–24.

Uzawa, Hirofumi. 1969. Optimum Fiscal Policy in an Aggregative Model of Economic Growth. In *The Theory and Design of Economic Development*, eds. Irma Adelman and Erik Thorbecke, 113–139. Baltimore, MD: Johns Hopkins Press.

Uzawa, Hirofumi, and Kenneth J. Arrow. 1988. *Preference, Production, and Capital: Selected Papers of Hirofumi Uzawa*. Cambridge, U.K.: Cambridge University Press.

*Julian Ellison*

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