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Mathematical economics
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Mathematical economics refers to the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could not be adequately expressed informally.
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Mathematical economics refers to the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could not be adequately expressed informally. Further, the language of mathematics allows economists to make clear, specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships that clarify assumptions and implications.
Formal economic modeling began in the late 19th century with the use of differential calculus to describe and predict economic behavior. Economics became more mathematical as a discipline throughout the first half of the 20th century, but it was not until the Second World War that new techniques would allow the use of mathematical formulations in almost all of economics. This rapid systematizing of economics alarmed critics of the discipline as well as some esteemed 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 arbitrary quantities or probabilities.
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 which dealt specifically with 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 Petty's works were largely ignored by English scholars.
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 and abstract models for behavior in order to apply the tools of mathematics. 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 in order to attempt to support his generalizations. In comparison to his contemporaries, Thünen built economic models and tools, rather than apply previous tools to new problems.
As the physical sciences became more systematized, economists pushed for a more formal methodology in economics. W.S. Jevons wrote the General Mathematical Theory of Political Economy in 1862, providing a rough outline for use of the theory of marginal utility in political economy. In 1874, he published The Principles of Science, declaring that "our science must be mathematical simply because it deals with quantities". Jevons expected the practice of statistics would become sufficiently sophisticated as to permit analysis of all decisions under the marginal utility framework. Independently, Carl Menger presented the theory in Grundsätze der Volkswirtschaftslehre (Principles of Economics) in 1871. Menger's work found a significant and appreciative audience. This movement did not succeed in generating a complete mathematical system under which all economic theory would operate, but it did allow for powerful new interpretive tools.
Around the end of the 19th century, the effort to place mathematics on secure logical foundations reached its most ambitious heights with the publication of Principia Mathematica by Russell and Whitehead. In 1918, David Hilbert continued the call for a complete axiomatic system in mathematics.. Although the ultimate goal of a complete, consistent and self-contained mathematical system was ultimately shown by Godel to be unachievable, economists continue to regard mathematical rigor as a desirable objective.
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. 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, 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 her output based on the output of the other and the market price would be determined by the total quantity supplied. The profit for each firm would be determined by multiplying their output and 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 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 side. 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. 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 can be constructed using more 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 (reached equilibrium conditions) that the nth market would clear as well. This is easiest to visualize with two markets (considered in most texts as a market for goods and a market for money). 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 move toward a proof of existence of solutions to general equilibrium but it is commonly used today to illustrate market clearing in money markets at the undergraduate level.
Tâtonnement (roughly, French for groping toward) was meant to serve as the practical expression of Walrassian 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). Only when all buyers are satisfied with the given market price would transactions occur. The market would "clear" at that price—no surplus or shortage would exist. The word tâtonnement is used to describe the directions the market takes in groping toward equilibrium, settling high or low prices on different goods until a price is agreed upon for all goods. While the process appears dynamic, Walras only presented a static model, as no transactions would occur until all markets were in equilibrium. In practice very few markets operate in this manner.
Francis Ysidro Edgeworth
Edgeworth introduced mathematical elements to Economics explicitly in Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences, published in 1881. He adopted Jeremy Bentham's Felicific calculus to economic behavior, allowing the outcome of each decision to be converted into a change in utility. Using this assumption, Edgeworth built a model of exchange on three assumptions: individuals are self interested, individuals act to maximize utility, and individuals are "free to recontract with another independently of...any third party." Given two individuals, the set of solutions where the both individuals can maximize utility is described by the contract curve on what is now known as an Edgeworth Box. Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until 1924 by Arthur Lyon Bowley. The contract curve of the Edgeworth box (or more generally on any set of solutions to Edgeworth's problem for more actors) is referred to as the core of an economy.
Edgeworth devoted considerable effort to insisting that mathematical proofs were appropriate for all schools of thought in economics. While at the helm of The Economic Journal, he published several articles criticizing the mathematical rigor of rival researchers, including Edwin Robert Anderson Seligman, a noted skeptic of mathematical economics. The articles focused on a back and forth over tax incidence and responses by producers. Edgeworth noticed that a monopoly producing a good that had jointness of supply but not jointness of demand (such as first class and economy on an airplane, if the plane flies, both sets of seats fly with it) might actually lower the price seen by the consumer for one of the two commodities if a tax were applied. Common sense and more traditional, numerical analysis seemed to indicate that this was preposterous. Seligman insisted that the results Edgeworth achieved were a quirk of his mathematical formulation. He suggested that the assumption of a continuous demand function and an infinitesimal change in the tax resulted in the paradoxical predictions. Harold Hotelling later showed that Edgeworth was correct and that the same result (a "diminution of price as a result of the tax") could occur with a discontinuous demand function and large changes in the tax rate).
Emergence of modern mathematical economics
Vilfredo Pareto analyzed microeconomics by treating decisions by economic actors as attempts to change a given allotment of goods to another, more preferred allotment. Sets of allocations could then be treated as Pareto efficient (Pareto optimal is an equivalent term) when no exchanges could occur between actors that could make at least one individual better off without making any other individual worse off. Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed to Adam Smith's Invisible hand hypothesis. Rather, Pareto's statement was the first formal assertion of what would be known as the first fundamental theorem of welfare economics. While it is known today that every Walras equilibrium is Pareto efficient, this was not known until more complex proofs were devised in 1936 by Abraham Wald and John von Neumann in 1938. A complete proof of both existence of a general equilibrium and uniqueness of the equilibrium would not come until Kenneth Arrow and Gérard Debreu introduced the Arrow-Debreu model in 1954
In the late 1930s, economists saw the wider use of a broad array of mathematical tools, including convex sets and graph theory. Applied mathematicians and topologists began to discuss economic problems as a means to advance the state of pure mathematics in the same sense that solutions to problems in physics led to advancement in the underlying mathematics. At roughly the same time, the Russian economist Wassily Leontief built his model of input-output analysis from 'material balance' tables constructed by Soviet economists. This model, which described a system of production and demand processes, could explain how variation in demand in one economic sector could influence production in another. Leontief published his first paper on the subject in 1936, but his work on both the theoretical foundations of his model and massive empirical studies of national economies in order to test it would continue through the 1960s.
The exposure of (mostly American and British) economists to engineering problems and problems in large bureaucratic systems would bring about huge changes in the discipline and the nature of university research in general. Linear programming, developed during the war, and generalized to nonlinear programming in 1951, would impact the study and practice of microeconomics heavily. The War cemented the use of applied mathematics in many disciplines, including economics. Operations research, a newly formed discipline which influenced and was influenced by mathematical economics, would drive much new research and draw considerable government funding over the next few decades. Mathematical economics expanded in scope and use considerably during the immediate post-war period.
In the landmark text Foundations of Economic Analysis (1947), Paul Samuelson identified a common paradigm and mathematical structure across multiple fields in the subject, building on previous work by Alfred Marshall. Foundations took mathematical concepts from chemistry and physics and applied them to economic problems. This broad view (for example, comparing Le Chatelier's principle to tâtonnement) drives the fundamental premise of mathematical economics: systems of economic actors may be modeled and their behavior described much like any other system. This extension followed on the work of the marginalists in the previous century and extended it significantly. Samuelson approached the problems of applying individual utility maximization over aggregate groups with comparative statics, which compares two different equilibrium states after an exogenous change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.
Over the course of the 20th century, articles in "core journals" in economics have been almost exclusively written by economists in Academia. As a result, much of the material transmitted in those journals relates to economic theory, and "economic theory itself has been continuously more abstract and mathematical." A subjective assessment of mathematical techniques employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1892 to 5.3% in 1990. A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 are free of any numerical equations or regression tables.
Econometrics
Between the world wars, advances in Probability theory resulted in the application of linear regression and time series analysis to economic data, a new method referred to as econometrics. The roots of modern econometrics can be traced to the American economist Henry L. Moore. Moore studied agricultural productivity and attempted to fit changing values of productivity for plots of corn and other crops to a curve using different values of elasticity. Moore made several errors in his work, some from his choice of models and some from limitations in his use of mathematics. The accuracy of Moore's models also was limited by the poor data for national accounts in the United States at the time. While his first models of production were static, in 1925 he published a dynamic "moving equilibrium" model designed to explain business cycles—this periodic variation from overcorrection in supply and demand curves is now known as the cobweb model. A more formal derivation of this model was made later by Nicholas Kaldor, who is largely credited for its exposition.
Ragnar Frisch coined the word "econometrics" and helped to found both the Econometric Society in 1930 and the journal Econometrica in 1933. A student of Frisch's, Trygve Haavelmo published The Probability Approach in Econometrics in 1944, where he asserted that precise statistical analysis could be used as a tool to validate mathematical theories about economic actors with data from complex sources. This linking of statistical analysis of systems to economic theory was also promulgated by the Cowles Commission (now the Cowles Foundation) throughout the 1930s and 1940s.
Application
Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use of calculus and matrix algebra in economic analysis in order to make powerful claims that would be more difficult without these mathematical tools. These are prerequisites for formal study, not only in mathematical economics but in contemporary economic theory generally. Economic problems often involve so many variables that mathematics is the only practical way of attacking and solving them. Alfred Marshall argued that every economic problem which can be quantified, analytically expressed and solved should be treated by means of mathematical work. Economics has become increasingly dependent on mathematical methods and the mathematical tools it employs have become more sophisticated. As a result, mathematics has become considerably more important to professionals in economics and finance. Graduate programs in economics and finance programs in graduate schools of management require strong undergraduate preparation in mathematics for admission and attract an increasingly high number of mathematicians. Applied mathematicians apply mathematical principles to practical problems, such as economic analysis and other economics-related issues, and many economic problems are often defined as integrated into the scope of applied mathematics. This integration results from the formulation of economic problems as stylized models with clear assumptions and falsifiable predictions. This modeling may be informal or prosaic, as it was in Adam Smith's The Wealth of Nations, or it may be formal, rigorous and mathematical.
Broadly speaking, formal economic models are stochastic or non-stochastic and discrete or continuous. At a practical level, quantitative modeling is applied to many areas of economics and several methodologies have evolved more or less independently of each other.
- Non-stochastic mathematical models may be purely qualitative (for example, models involved in some aspect of social choice theory) or quantitative (involving rationalization of financial variables, for example with hyperbolic coordinates, and/or specific forms of functional relationships between variables). In some cases economic predictions of a model merely assert the direction of movement of economic variables, and so the functional relationships are used only in a qualitative sense: for example, if the price of an item increases, then the demand for that item will decrease. For such models, economists often use two-dimensional graphs instead of functions.
- Qualitative models are occasionally used. One example is qualitative scenario planning in which possible future events are played out. Another example is non-numerical decision tree analysis. Qualitative models often suffer from lack of precision.
Criticism of mathematical economics
The methods of mathematical economics are widely, though far from exclusively, used in professional publications. While Friedrich Hayek contended that the use of formal techniques projects a scientific exactness that does not appropriately account for informational limitations in the real world, this did not extend to a general critique of mathematical tools in economics.
Philosopher Karl Popper offered considerable criticism in the 1940s and 1950s. He argued that the fundamental problem with mathematical economics was that it was tautological. In other words, once economics became a mathematical discipline, it would cease to rely on empirical truth and instead rely on axiomatic proof. Popper asserted that an economic model could either have verifiable assumptions and produce no new information or have unverifiable assumptions and sacrifice formalism for scope. Milton Friedman responded to this by announcing that "all assumptions are unrealistic", charging that economic models should be judged on how well the theory predicts reality, not how well the assumptions accord with reality. Samuelson argued a different tack. He proposed that economic theories should be refutable in principle; if they were refutable in principle, they could not be tautological.
Another criticism of mathematical economics was popularized by Robert Heilbroner in the afterword to his popular book, The Worldly Philosophers. He elaborated on his feelings later in an interview:
Heilbroner addresses one of the core critiques of economics in general here, that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition." Some call this a mathematical romance that tends to eliminate the distinctively human elements from the economic equation. This critique has been advanced in various forms by economists and other scientists, including Keynes and Paul Joskow. Joskow advanced a particularly harsh critique, observing that a good portion of economic insight came from outside formal models and that those formal, mathematical models were added "ex post" in order to provide a justification for the insight.
See also
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