Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical optimization, or other computational methods.Elaborated at the JEL classification codes, Mathematical and quantitative methods JEL: C Subcategories. TOC. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.Debreu, Gérard ( 2008). "mathematical economics", section II, The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content", Econometrica, 54(6), pp. 1259 -1270.
Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects that would be less easily expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial subjects that would be impossible without it.Varian, Hal (1997). "What Use Is Economic Theory?" in A. D'Autume and J. Cartelier, ed., Is Economics Becoming a Hard Science?, Edward Elgar. Pre-publication PDF. Retrieved 2008-04-01. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.* As in Handbook of Mathematical Economics, 1st-page chapter links: Arrow, Kenneth J., and Michael D. Intriligator, ed., (1981), v. 1 _____ (1982). v. 2 _____ (1986). v. 3 Hildenbrand, Werner, and Hugo Sonnenschein, ed. (1991). v. 4. Debreu, Gérard (1983). Mathematical Economics: Twenty Papers of Gérard Debreu, Contents . Glaister, Stephen (1984). Mathematical Methods for Economists, 3rd ed., Blackwell. Contents. Takayama, Akira (1985). Mathematical Economics, 2nd ed. Cambridge. Description and Contents . Michael Carter (2001). Foundations of Mathematical Economics, MIT Press. Description and Contents .
Broad applications include: optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker static (or equilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or the economy) is modeled as not changing comparative statics as to a change from one equilibrium to another induced by a change in one or more factors dynamic analysis, tracing changes in an economic system over time, for example, from economic growth.Chiang, Alpha C. (1992). Elements of Dynamic Optimization, Waveland. TOC & Amazon.com link to inside, first pp.
Formal economic modeling began in the 19th century with the use of differential calculus to represent and explain economic behavior, such as utility maximization, an early economic application of mathematical optimization. Economics became more mathematical as a discipline throughout the first half of the 20th century, but the introduction of new and generalized techniques in the period around the Second World War, as in game theory, would greatly broaden the use of mathematical formulations in economics.* Debreu, Gérard ( 2008). "mathematical economics", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content", Econometrica, 54(6), pp. 1259 -1270. von Neumann, John, and Oskar Morgenstern (1944). Theory of Games and Economic Behavior. Princeton University Press.
This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists. John Maynard Keynes, Robert Heilbroner, Friedrich Hayek, and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics.
History
The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly in German universities, a style of instruction emerged that focused on the detailed presentation of data as it related to public administration. Gottfried Achenwall lectured in this fashion, coining the term statistics. At the same time, a small group of professors in England established a method of "reasoning by figures upon things relating to government" and referred to this practice as Political Arithmetick. Sir William Petty wrote at length on issues that would later concern economists, such as taxation, Velocity of money and national income, but while his analysis was numerical, he rejected abstract mathematical methodology. Petty's use of detailed numerical data (along with John Graunt) would influence statisticians and economists for some time, even though English scholars largely ignored Petty's works.Schumpeter (1954) p. 212-215
The mathematization of economics began in earnest in the 19th century. Most of the economic analysis of the time was what would later be called classical economics. Subjects were discussed and dispensed with through algebraic means, but calculus was not used. More importantly, until Johann Heinrich von Thünen's The Isolated State in 1826, economists did not develop explicit, abstract models of behavior to which mathematical tools could be applied. Thünen's model of farmland use represents the first example of marginal analysis. Thünen's work was largely theoretical, but he also mined empirical data to attempt to support his generalizations. Compared with his contemporaries, Thünen developed economic models and tools rather than applying existing ones to new problems.Schumpeter (1954) p. 465-468
Meanwhile, a new cohort of scholars trained in the mathematical methods of the physical sciences gravitated toward economics, advocating and applying those methods to their subject,Philip Mirowski, 1991. "The When, the How and the Why of Mathematical Expression in the History of Economics Analysis", Journal of Economic Perspectives, 5(1) pp. 145-157. and described today as moving from geometry to mechanics.Weintraub, E. Roy (2008). "mathematics and economics", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract . These included W.S. Jevons, who presented a paper on a "general mathematical theory of political economy" in 1862, providing an outline for applying the theory of marginal utility to political economy.Jevons, W.S. (1866). "Brief Account of a General Mathematical Theory of Political Economy", Journal of the Royal Statistical Society, XXIX (June) pp. 282–87. Read in Section F of the British Association, 1862. PDF. In 1871, he published The Principles of Political Economy, declaring that the subject as science "must be mathematical simply because it deals with quantities". Jevons expected that only the collection of statistics on prices and quantities would permit the subject, as presented, to become an exact science. Others preceded and followed in expanding mathematical representations of economic problems. See the preface to Irving Fisher's 1897 work, A brief introduction to the infinitesimal calculus: designed especially to aid in reading mathematical economics and statistics.
Marginalists and the roots of neoclassical economics
Augustin Cournot and Léon Walras built the tools of the discipline axiomatically around utility, arguing that individuals sought to maximize their utility across choices in a way that could be described mathematically. At the time, it was thought that utility was quantifiable, in units known as utils.While the concept of cardinality has fallen out of favor in neoclassical economics, the differences between cardinal utility and ordinal utility are minor for most applications. Cournot, Walras and Francis Ysidro Edgeworth are considered the precursors to modern mathematical economics.
Augustin Cournot Cournot, a professor of mathematics, developed a mathematical treatment in 1838 for duopoly—a market condition defined by competition between two sellers. This treatment of competition, first published in Researches into the Mathematical Principles of Wealth,Augustin Cournot (1838, tr. 1897) Researches into the Mathematical Principles of Wealth. Links to description and chapters. is referred to as Cournot duopoly. It is assumed that both sellers had equal access to the market and could produce their goods without cost. Further, it assumed that both goods were homogeneous. Each seller would vary their output based on the other's output, and the market price would be determined by the total quantity supplied. The profit for each firm would be determined by multiplying its output by the per unit market price. Differentiating the profit function with respect to quantity supplied for each firm left a system of linear equations, the simultaneous solution of which gave the equilibrium quantity, price, and profits. Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced many of the marginalists. Cournot's models of duopoly and oligopoly also represent one of the first formulations of non-cooperative games. Today, the solution can be given as a Nash equilibrium, but Cournot's work preceded modern game theory by over 100 years.
Léon Walras While Cournot provided a solution for what would later be called partial equilibrium, Léon Walras attempted to formalize the discussion of the economy as a whole through a theory of general competitive equilibrium. The behavior of every economic actor would be considered on both the production and consumption sides. Walras originally presented four separate models of exchange, each recursively included in the next. The solution of the resulting system of equations (both linear and non-linear) is the general equilibrium.Nicola, p. 9-12 At the time, no general solution could be expressed for a system of arbitrarily many equations, but Walras's attempts produced two famous results in economics. The first is Walras' law and the second is the principle of tâtonnement. Walras' method was considered highly mathematical for the time and Edgeworth commented at length about this fact in his review of Éléments d'économie politique pure (Elements of Pure Economics).
Walras' law was introduced as a theoretical answer to the problem of determining the solutions in general equilibrium. His notation is different from modern notation, but it can be expressed using modern summation notation. Walras assumed that in equilibrium, all money would be spent on all goods: every good would be sold at the market price for that good, and every buyer would expend their last dollar on a basket of goods. Starting from this assumption, Walras could then show that if there were n markets and n-1 markets cleared (i.e., reached equilibrium), the nth market would clear as well. This is easiest to visualize with two markets (as considered in most texts: a goods market and a money market). If one of two markets has reached an equilibrium state, no additional goods (or conversely, money) can enter or exit the second market, so it must be in a state of equilibrium as well. Walras used this statement to advance a proof of the existence of general equilibrium solutions, but it is commonly used today at the undergraduate level to illustrate market clearing in money markets.Nicholson, Walter; Snyder, Christopher, p. 350-353.
Tâtonnement (roughly, French for groping toward) was meant to serve as the practical expression of Walrasian general equilibrium. Walras abstracted the marketplace as an auction of goods where the auctioneer would call out prices and market participants would wait until they could each satisfy their personal reservation prices for the quantity desired (remembering here that this is an auction on all goods, so everyone has a reservation price for their desired basket of goods).