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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.[1] 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.[2] Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships that make clear their assumptions and their 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.
[edit] History[edit] Marginalists and the roots of neoclassical economics
The mathematization of economics began in the 19th century. As the physical sciences became more systematized, economists pushed for a quantification in the discipline around the notion of marginal utility. 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.[3] At the time, it was thought that utility was quantifiable, in units known as utils.[4] The expectation was that advances in statistics would allow formulation of a proper theory of marginal utility. [5] 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. In 1918, David Hilbert continued the call for a complete axiomatic system in economics similar to physics and chemistry.[6] 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 or Pareto optimal when no exchanges could occur between actors that could make at least one individual better off without making any other individual worse off.[7] This frontier of optimal choices between two actors can be shown on the plane as an Edgeworth box while greater numbers of actors requires more dimensions. [edit] Emergence of modern mathematical economics
Change across the entire discipline would not come until the end of the Second World War. 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.[9] 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.[10] 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.[11][12] Over the course of the 20th century, articles in "core journals"[13] 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."[14] A subjective assessment of mathematical techniques[15] employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1982 to 5.3% in 1990.[16] [edit] EconometricsBetween 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. Ragnar Frisch coined the word "econometrics" and helped to found both the Econometric Society in 1930 and the journal Econometrica in 1933.[17][18] Econometrics was originally developed as a tool to validate mathematical theories about economic actors with data from complex sources.[19] While this approach is widespread in the profession today, Richard Lipsey and Chris Achibald at the London School of Economics adopted econometrics to formalize falsifiable statements about mathematical theory. This particular application of econometrics did not produce fruitful results.[20] Strictly speaking, econometrics refers to empirical interpretation of data while mathematical economics refers to the formulation of models.[21] A study in 2008 of economic articles submitted for publication shows an increasing focus on empirical data and analysis.[22] Some university economics programs cross-list courses in the statistics department and require doctoral students to enroll in supporting courses offered by the statistics department.[23] The Nobel prize has been awarded to econometricians, most recently in 2003 for methods of estimating data with time series volatility and methods in cointegration.[24] [edit] 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.[26] 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.[3] 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 modelling is applied to many areas of economics and several methodologies have evolved more or less independently of each other. [27]
Mathematical economics provides methods to model behavior in diverse, real world situations, including international climate agreements, reactions to changes in divorce laws, and pricing in the futures markets for commodities.[28] [29][30] [edit] Criticism of mathematical economicsThe 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.[31] 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.[20] 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.[32] 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.[33] 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.[34] 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:[35]
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."[36] 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.[37][38] [edit] Mathematical economistsFamous mathematical economists include, but are not limited to, the following list (by century of birth). [edit] 19th century[edit] 20th century[edit] See also
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