## Mathematical Economics

## Mathematical Economics

# Mathematical Economics

In a narrow sense, mathematical economics, as represented by the *Journal of Mathematical Economics* or the North-Holland *Handbook of Mathematical Economics*, is the specialization within advanced economic theory using sophisticated mathematical techniques such as functional analysis and topology. In a wider, more general sense, mathematical economics refers to the application of mathematical methods within economics (other than the application of probability theory and mathematical statistics in economics, which has acquired separate disciplinary status as econometrics). The use of mathematics, together with emphasis on formal economic theory, has become increasingly pervasive in economics since World War II (1939–1945), as general equilibrium analysis, game theory, and operations research became prominent. Training in at least calculus, matrix algebra, and constrained optimization (as well as econometrics) is now indispensable for comprehension of articles in general economics journals, not just journals specializing in economic theory.

There were numerous early attempts to use mathematics to analyze questions in political economy before a receptive audience existed for such works (see Baumol and Goldfeld 1968 for selections from this literature). A few of these early efforts, notably Antoine Augustin Cournot’s 1838 analysis of oligopoly, Francis Ysidro Edgeworth’s 1881 study of bilateral exchange, and Louis Bachelier’s 1900 dissertation on financial speculation and stochastic processes, have been rediscovered and reinterpreted in light of solution concepts developed much later (Cournot oligopoly being viewed by some as a special case of Nash equilibrium, Edgeworth’s analysis being related to the game-theoretic concept of the core), but their contemporary readership and influence was slight. Alfred Marshall was educated as a mathematician (Second Wrangler at Cambridge, ranked behind only the physicist Lord Rayleigh) but, unlike his contemporary Edgeworth, he relegated mathematics to a mathematical appendix in his *Principles of Economics* that, in eight editions from 1890 to 1920, dominated the teaching of economics in English. Although he advocated mathematics as an aid to thought, Marshall distrusted mathematical economics that could not be translated into plain English, and worried about restricting attention only to those considerations that can be quantified. Similarly, as late as 1939, J. R. Hicks restricted formal mathematics to an appendix in his *Value and Capital*, which brought the continental European tradition of general equilibrium analysis associated with Léon Walras and Vilfredo Pareto in Lausanne to the attention of English-language economists. Along with Roy Harrod, James Meade, and others, Hicks (1937) took a leading role in translating John Maynard Keynes’s theory of employment into a small system of simultaneous equations, which Hicks’s accompanying diagrams (later modified by Alvin Hansen) turned into the trained intuition of two generations of economists. Keynes himself had made such a translation into a four-equation system in his Cambridge lectures in December 1933 (and the first two published versions of such a model were by David Champernowne and Brian Reddaway, who had both attended that series of lectures), but following Marshall’s precedent, Keynes did not include such a formalized model in his *General Theory of Employment, Interest, and Money* in 1936.

The American economist Irving Fisher had proposed in 1912 the formation of an international society to promote the formal use of mathematics and statistics in economics, but it was only at the end of 1930 that the Econometric Society was founded with Fisher as president (succeeded by the French index-number theorist François Divisia). The Econometric Society’s U.S. and European conferences, and its journal *Econometrica* (founded 1933), provided a forum for mathematical economics and econometrics, as did the closely associated Cowles Commission for Research in Economics (in Colorado from 1933 to 1939, and then in Chicago from 1939 to 1955). The publication of Paul Samuelson’s *Foundations of Economic Analysis* in 1947 (based on his 1941 Harvard dissertation) was a turning point in general acceptance of formal, mathematical economic theory. Samuelson’s restatement and extension of neoclassical comparative static and dynamic analysis, introducing a generation of economists to difference and differential equations, emphasized Walrasian general equilibrium analysis, in which all markets are linked through the budget constraints of individual agents. Following from work by John von Neumann and Abraham Wald in Karl Menger’s mathematical colloquium in Vienna in the 1930s (see Baumol and Goldfeld 1968, Weintraub 2002), Kenneth Arrow, Gerard Debreu, and Lionel McKenzie provided formal proofs of the existence of general competitive equilibrium in the 1950s, but such works as the 1959 Cowles monograph *The Theory of Value* by Debreu could be read and understood by very few economists. Gradually, general equilibrium theory, requiring formal mathematics for its statement and application, came to dominate graduate courses in microeconomics (until challenged by game theory since the 1980s), with Marshallian partial equilibrium, susceptible to diagrammatic treatment, prominent only in undergraduate courses.

Military applications encouraged operations research in the United States (and operational research in Britain) during World War II, and postwar research in related fields was supported by the Office of Naval Research, the U.S. Air Force (initially the sole client of the RAND Corporation), and the U.S. Army through the Stanford Research Institute, later SRI (see Mirowski 2002). Discrete optimization (linear and nonlinear programming) was widely used in economics to implement the neoclassical paradigm of agents maximizing an objective function (expected utility) subject to budget and other constraints, as by Robert Dorfman, Paul Samuelson, and Robert Solow in their 1958 RAND study *Linear Programming and Economic Analysis*. Linear programming and related techniques were entrenched in business schools as operations research, management science, or decision science, and in the Soviet Union and other centrally planned economies as planometrics. Within economics, however, attention shifted from activity analysis (including linear programming) to game theory from the 1970s onward.

Game theory, the formal modeling of conflict and cooperation, emerged as a distinct field of study in applied mathematics and in economics and other social sciences with John von Neumann and Oskar Morgenstern’s *Theory of Games and Economic Behavior* in 1944. This work built on the minimax solution for two-player, zero-sum games, whose existence von Neumann had proved in 1928 (with a much simpler, non-topological proof provided by Jean Ville in 1938), and with John F. Nash Jr.’s equilibrium concept for n-player, general-sum games. Game theory stressed strategic interaction: the payoffs for each player depend not only on the strategy (a rule for selecting an action in response to a particular information set) chosen by that player, but also on the strategies chosen by all the other players (see “Strategic Games”). Nash equilibrium for a noncooperative game (one in which binding contracts, externally enforced, are not possible) requires that no player can gain from changing his or her strategy under the assumption that no other player changes strategy. Game theory, with its emphasis on strategic interaction among players, contrasts with general competitive equilibrium analysis, which assumes a single agent to be too small to affect market outcomes. In the 1950s and 1960s, game theoretic research, often published in the *Journal of Conflict Resolution* and some funded by the U.S. Arms Control and Disarmament Agency, focused on the understanding and control of arms races. Game theory is now pervasive in economics from industrial organization to trade policy and the credibility of macroeconomic policy, and is influential in other fields from political science and law to evolutionary biology.

## BIBLIOGRAPHY

Baumol, William J., and Stephen M. Goldfeld, eds. 1968. *Precursors in Mathematical Economics: An Anthology*. London: London School of Economics and Political Science.

Blaug, Mark. 2003. The Formalist Revolution of the 1950s. In *A Companion to the History of Economic Thought*, ed. Warren J. Samuels, Jeff E. Biddle, and John B. Davis, 395–410. Malden, MA, and Oxford: Blackwell.

Mirowski, Philip. 2002. *Machine Dreams: Economics becomes a Cyborg Science*. Cambridge, U.K.: Cambridge University Press.

Samuelson, Paul A. 1947. *Foundations of Economic Analysis*. Cambridge, MA: Harvard University Press (enlarged edition, 1983).

Weintraub, E. Roy. 2002. *How Economics Became a Mathematical Science*. Durham, NC: Duke University Press.

*Hichem Ben-El-Mechaiekh*

*Robert W. Dimand*