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| Mathematical economics | |
|---|---|
| Name | Mathematical economics |
| Caption | Mathematical modeling in economics |
| Discipline | Economics, Mathematics |
| Introduced | 20th century |
| Notable figures | Leon Walras; Vilfredo Pareto; Kenneth Arrow; Gerard Debreu; John von Neumann; Paul Samuelson; John Nash; Wassily Leontief; Ragnar Frisch; Trygve Haavelmo; Herbert A. Simon |
Mathematical economics Mathematical economics applies mathematics to analyze, formalize, and derive implications for economic phenomena using models, proofs, and quantitative methods. The field synthesizes contributions from figures associated with Walras, Pareto, Arrow, Debreu, Samuelson, and Nash to inform theoretical and empirical work across many institutions, including Cowles Commission, RAND Corporation, Massachusetts Institute of Technology, and London School of Economics. It relies on tools developed in conjunction with advances linked to Hilbert, Lebesgue, von Neumann, Kolmogorov, and Tarski.
Mathematical economics formalizes questions historically posed by scholars such as Adam Smith, David Ricardo, John Stuart Mill, Alfred Marshall, and Karl Marx into systems amenable to analysis via methods originating with Isaac Newton, Pierre-Simon Laplace, Carl Friedrich Gauss, Bernhard Riemann, and Augustin-Louis Cauchy. The approach became institutionalized through departments at Princeton University, University of Chicago, University of Cambridge, Harvard University, and University of California, Berkeley and through journals like Econometrica, Journal of Political Economy, Review of Economic Studies, and American Economic Review. Key prizewinners tied to the field include recipients of the Nobel Memorial Prize in Economic Sciences such as Arrow, Debreu, Samuelson, Nash, Frisch, Leontief, Haavelmo, and Simon.
Early formalization traces to exchange theories by Walras and Pareto efficiency by Pareto. The marginal revolution involving Jevons, Menger, and Edgeworth introduced calculus-based comparative statics akin to work by Lagrange and Euler. The 20th century saw synthesis at the Cowles Commission involving Samuelson and Nicholas Kaldor and connections to game theory via von Neumann and Morgenstern and later Nash equilibrium. The development of econometrics tied to Haavelmo, Theil, Goldberger, and Tinbergen integrated statistical inference influenced by Fisher, Pearson, and Wald. Formal general equilibrium theory advanced through Arrow and Debreu while input–output analysis advanced by Leontief linked to industrial planners in Soviet Union and practice at United Nations. The mathematization intersected with computational shifts at Bell Labs, IBM, Microsoft Research, and Los Alamos National Laboratory.
Techniques derive from calculus traditions traced to Newton and Leibniz, measure theory from Lebesgue, and topology from Brouwer and Schauder used in fixed-point theorems like Brouwer fixed-point theorem and Kakutani fixed-point theorem essential for equilibrium existence proofs by Debreu and Arrow. Linear algebra and matrix theory from Cayley and Sylvester underpin input–output models by Leontief and principal component methods linked to Karhunen–Loève theorem. Optimization methods reference works by Lagrange, Kuhn–Tucker, and Pontryagin with duality theory informed by Fenchel and Rockafellar. Probability theory from Kolmogorov and stochastic calculus from Itô inform dynamic stochastic general equilibrium methods developed in curricula at MIT and Stanford University. Game-theoretic equilibrium refinements rely on concepts by Nash, Harsanyi, and Selten.
Canonical models include general equilibrium theory by Walras, formalized by Arrow and Debreu; the Solow growth model associated with Robert Solow and growth literature linked to Paul Romer and Robert Lucas Jr.; overlapping generations models by Paul Samuelson and Peter Diamond; input–output analysis by Leontief; consumption theory by Engel and Friedman; portfolio theory by Harry Markowitz and asset pricing models like CAPM by William Sharpe and Black–Scholes option pricing by Fischer Black and Myron Scholes. Auction theory connected to William Vickrey and Paul Milgrom integrates mechanism design pioneered by Leonid Hurwicz and Eric Maskin. Matching theory owes to Gale and Shapley and spurred applications by Lloyd Shapley and Alvin Roth.
Applications span macroeconomic policy models used at Federal Reserve System, European Central Bank, and International Monetary Fund, welfare analysis in legal cases before U.S. Supreme Court and regulatory agencies like Federal Trade Commission, and industrial organization models tested using techniques by Bresnahan and Berry. Empirical methods include econometric identification strategies advanced by Angrist and Imbens, time-series methods by Box and Jenkins, structural estimation influenced by Rust and Heckman, and experimental methods developed by Vernon Smith and Daniel Kahneman. Computable general equilibrium implementations are used by organizations such as World Bank and OECD and calibrated macro models draw from work at NBER and CEPR.
Critiques originate from thinkers like Keynes and Friedman who questioned realism of assumptions and predictive power, and from methodological critics including Herbert Simon regarding bounded rationality and complex systems research inspired by Ilya Prigogine and John Holland. Debates involve representativeness highlighted by Thomas Kuhn-style paradigm critiques and robustness concerns raised after financial crises such as the Great Recession and 2008 financial crisis. Philosophers of science like Karl Popper and Paul Feyerabend have debated falsifiability of highly abstract models. Ethical considerations have been raised in contexts involving policy failures examined by Sen and Amartya Sen-related welfare critiques.
Mathematical methods influenced policy through central bankers at Federal Reserve Bank of New York, fiscal institutions at International Monetary Fund, and trade negotiators at World Trade Organization. Interdisciplinary crossovers include behavioral economics linking to Daniel Kahneman and Amos Tversky; computational economics engaging with Alan Turing-inspired computation theory and John McCarthy's artificial intelligence; network economics connecting to Erdős–Rényi graph theory and Mark Granovetter; and environmental economics drawing on models by Nicholas Stern and institutions like Intergovernmental Panel on Climate Change. The legacy extends into finance at Goldman Sachs and JP Morgan Chase and into public choice theory associated with James Buchanan.