Mathematical economics 50676 224229968 2008-07-07T22:16:32Z SmackBot 433328 Date the maintenance tags or general fixes {{Expand|date=July 2008}} [[Image:Yield curves.gif|frame|10 Year Treasury Yields in a Recession.<ref>Powell, Nathan. ''What the Yield Curve does (and doesn't) tell us.'' http://www.fdic.gov/bank/analytical/fyi/2006/022206fyi.html</ref>]] '''Mathematical economics''' refers to the application of [[mathematical]] methods to represent economic theory or analyze [[Mathematical problem|problems]] posed in [[economics]]. It allows formulation and derivation of key relationships in the [[economic model|theory]] with clarity, generality, rigor, and simplicity.<ref> {{cite book | last = Chiang | first = Alpha C. | authorlink = | coauthors = Kevin Wainwright | title = Fundamental Methods of Mathematical Economics | publisher = McGraw-Hill Irwin | date = 2005 | location = | pages = 1,2 | url = | doi = | id = | isbn = 0-07-010910-9}} </ref> 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 science|positive]] claims about controversial or contentious subjects that would be impossible without mathematics.<ref> {{cite conference | first = Varian | last = Hal | authorlink = Hal_Varian | coauthors = | title = What use is Economic Theory? | booktitle = | pages = 1-11 | publisher = | date = 29-30 October, 1992 | location = ``Is Economics Becoming a Hard Science?'', Paris, France | url = http://www.sims.berkeley.edu/~hal/Papers/theory.pdf | doi = | id = | accessdate = 2008-04-01}} </ref> == History == The mathematization of economics began in the 19th century. As the [[physical science]]s became more systematized, economists pushed for a quantification in the discipline around the notion of '[[marginal utility]]'. [[Antoine Augustin Cournot|Augustin Cournot]] and [[Léon Walras]] built the tools of the discipline axiomatically about utility, arguing that individuals sought to maximize their utility across choices, a condition that "lends itself to mathematical treatment."<ref> {{cite conference | first = Dow | last = Sheila C. | authorlink = | coauthors = | title = The Use of Mathematics in Economics | booktitle = ESRC Public Understanding of Mathematics Seminar | pages = | publisher = Economic and Social Research Council | date = 1999-05-21 | location = [[Birmingham]] | url = http://www.ioe.ac.uk/esrcmaths/sheila1.html | doi = | id = | accessdate = 2008-07-06}} </ref> At the time, it was thought that utility was expressible in quantities, known as '[[util]]s'.<ref>While the concept of [[Cardinal utility|cardinality]] has fallen out of favor in [[neoclassical economics]], the differences in application between cardinal utility and [[ordinal utility]] are minor for most applications</ref> The expectation at the time was that advances in commercial statistics would allow for sufficient information to formulate a proper theory of marginal utility.<ref>Jevons, W.S. (1871) ''The Theory of Political Economy''. London: Macmillan. Quoted in Dow</ref> 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]].<ref>Hilbert, D. (1918). ''"Axiomatisches Denken."'' Mathematische Annalen 78: 405-415</ref>. Change would not come until the end of the [[World War II|Second World War]]. The exposure of (mostly American and British) economists to problems in large bureaucratic systems and engineering problems would bring about huge changes in the discipline and the nature of university research in general.<ref> Turner, Fred (2006). ''From counterculture to cyberculture : Stewart Brand, the Whole Earth Network, and the rise of digital utopianism''. University Of Chicago Press</ref> The War cemented the use of applied mathematics in many disciplines, including economics. [[Operations research]], a newly formed discipline from this marriage of mathematics and 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.<ref> {{cite encyclopedia | last = Weintraub | first = E. Roy | authorlink = E._Roy_Weintraub | editor = Durlauf, Steven N.; Blume, Lawrence E. | encyclopedia =The New Palgrave Dictionary of Economics | title = Mathematics and economics | url = http://www.dictionaryofeconomics.com/article?id=pde2008_M000372 | accessdate = 2008-07-07 | edition = 2nd Edition | date = 2008 | publisher = Macmillan | isbn = | doi =10.1057/9780230226203.1063 }}</ref> [[Paul Samuelson]], in the landmark text, ''[[Foundations of Economic Analysis]]'' (1947) identified a common mathematical structure across multiple fields in the subject. This assertion of structure was both a connection to previous work by [[Alfred Marshall]] and an attempt to turn economics into a [[normal science]]. ''[[Foundations of Economic Analysis|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 [[Walrasian auction|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 last century and extended it significantly. Samuelson approached the problems of applying individual utility maximization over aggregate groups with [[comparative statics]]. Comparative statics compares two different [[economic equilibrium|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.<ref>Metzler, Lloyd (1948), Review of ''Foundations of Economic Analysis'', in ''American Economic Review'' 38 (5), pp. 905-10.</ref><ref>Samuelson, Paul A. (1947, Enlarged ed. 1983). ''Foundations of Economic Analysis'', Harvard University Press. ISBN 0-674-31301-1</ref> == Application == Much of modern economics can be presented in 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 mathematical tools. These are prerequisites for formal study, not only in mathematical economics but in contemporary economic theory generally. Both ''[[Journal of Economic Theory]]'' and ''[[Econometrica]]'' are core journals in economics,<ref> {{cite journal | last = Liner | first = Gaines H. | authorlink = | coauthors = | title = Core Journals in Economics | journal = Economic Inquiry | volume = 40 | issue = 1 | pages = | publisher = Oxford University Press | location = | date = 2002 | url = http://econpapers.repec.org | doi = | id = | accessdate = }} </ref> focusing on mathematical economics.<ref>[http://www.nyu.edu/jet/index.html Journal of Economic Theory<!-- Bot generated title -->]</ref><ref>[http://www.econometricsociety.org/aims.asp Welcome to the website of The Econometric Society An International Society for the Advancement of Economic Theory in its Relation to Statistics and Mathematics<!-- Bot generated title -->]</ref> According to the Editors, each is a ''non-specialist'' journal. For the diverse theoretical topics in economics examined in them, most articles use a mathematical representation of the theory presented. 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.<ref>McGinty, Matthew. ''"International Environmental Agreements among Asymmetric Nations,"'' Oxford Economic Papers, '''59'''(1), January 2007: 45-62.</ref><ref>Wolfers, Justin and Stevenson, Betsy. ''"Marriage and Divorce: Changes and their Driving Factors."'' Journal of Economic Perspectives, '''21'''(2) 27-52, Spring 2007.</ref><ref>[http://www.econbrowser.com/archives/2008/04/commodity_arbit.html Econbrowser: Commodity arbitrage<!-- Bot generated title -->]</ref>. === Applied mathematics === {{main|Applied mathematics}} An economic problem often involves so many variables that [[mathematics]] is the only practical way of handling it - "handling" in the sense of solving it. For many academics interested on the subject, like [[Alfred Marshall]], every economic problem which can be quantified, analytically expressed and solved, should be treated by means of mathematical work.<ref>[http://links.jstor.org/sici?sici=0022-2186(197510)18%3A2%3C583%3AMOM%3E2.0.CO%3B2-M JSTOR: Marshall on Mathematics], in [[JSTOR]]</ref> Economic analysis relies more and more on mathematical foundations. Economics has become increasingly dependent on mathematical methods, and the mathematical tools it employs have become more sophisticated. As a result, mathematically competent professionals are needed in industry and government for dealing with the subject. Graduate programs in economics and finance programs in graduate schools of management require strong undergraduate preparation in mathematics for admission and are attractive to an increasingly high number of [[mathematician]]s. [[applied mathematics|Applied mathematician]]s apply mathematical principles to practical problems, such as economic analysis and other economics-related issues, in such a way that many economic problems are often defined as being integrated into the scope of applied mathematics.<ref>Sheila C Dow - [http://www.ioe.ac.uk/esrcmaths/sheila1.html THE USE OF MATHEMATICS IN ECONOMICS], For presentation to the ESRC Public Understanding of Mathematics Seminar, Birmingham, 21-2 May 1999</ref> === Econometrics === {{main|Econometrics}} Another tool of mathematical economics is [[econometrics]], the use of statistical techniques to interpret quantitative empirical data.<ref> Geweke, John, Joel Horowitz and Hashem Pesaran. "econometrics." The New Palgrave Dictionary of Economics. Second Edition. Eds. Steven N. Durlauf and Lawrence E. Blume. Palgrave Macmillan, 2008. The New Palgrave Dictionary of Economics Online. Palgrave Macmillan. 22 April 2008 <http://www.dictionaryofeconomics.com/article?id=pde2008_E000007> doi:10.1057/9780230226203.0425</ref> Strictly speaking, econometrics refers to empirical interpretation of data while mathematical economics refers to the formulation of models. A recent study of economic articles submitted for publication shows an increasing focus on empirical data and analysis.<ref> {{Citation | first = Oswald | last = Andrew J. | author-link = | first2 = Ralsmark | last2 = Hilda | author2-link = | editor-last = | editor-first = | editor2-last = | editor2-first = | contribution = Warwick Economic Research Papers | contribution-url = | title = Some Evidence on the Future of Economics | year = 2008 | pages = 7 | place = | publisher = Warwick Department of Economics | url = | doi = | id = }} </ref> This technique focuses strongly on statistical methods. Some university programs cross-list courses in the statistics department and require students studying for their PhD enroll in supporting courses offered by the statistics department.<ref>[http://www.ssc.wisc.edu/econ/grad/fields.html Department of Economics | University of Wisconsin - Madison<!-- Bot generated title -->]</ref> The [[Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel|Nobel prize]] has been awarded to econometricians, most recently in 2003<ref>[http://nobelprize.org/nobel_prizes/economics/laureates/2003/index.html 2003 Economics Nobel Laureates <!-- Bot generated title -->] from the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel website. Accessed on 2008-06-23</ref> for methods of estimating data with [[Autoregressive conditional heteroskedasticity|time series volatility]] and methods in [[cointegration]]. Econometrics was developed partially as a tool to validate mathematical theories about economic actors with data from complex sources. Mathematical theory provides the form and econometrics is used to test that form to a given degree of [[Statistical significance|significance]]. This approach is widespread in the profession today but the original intent by [[Richard Lipsey]] and Chris Achibald at the [[London School of Economics]] to formalize [[Falsifiability|falsifiable]] statements about mathematical theory through the use of econometrics did not produce fruitful results.<ref name="Boland">Boland, L. A. (2006) Seven Decades of Economic Methodology. pp. 219-223 '''In''': I. C. Jarvie, K. Milford, D.W. Miller Eds ''Karl Popper:A Centenary Assessment''. London:Ashgate Publishing p. 219</ref> ==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.<ref> {{cite journal | last = Hayek | first = Friedrich | authorlink = Friedrich_Hayek | coauthors = | title = The Use of Knowledge in Society | journal = American Economic Review | volume = 35 | issue = 4 | pages = 519–530 | publisher = | location = | date = September, 1945 | url = | doi = | id = | accessdate = 2008-04-19}} </ref> A considerable source of criticism was written in the 1940s and 1950's by philosopher [[Karl Popper]]. 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.<ref name="Boland">Boland, L. A. (2006) Seven Decades of Economic Methodology. pp. 219-223 '''In''': I. C. Jarvie, K. Milford, D.W. Miller Eds ''Karl Popper:A Centenary Assessment''. London:Ashgate Publishing p. 219</ref> 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.<ref> {{cite journal | last = Beed | first = Clive | authorlink = | coauthors = Kane, Owen | title = Critique of the Mathematization of Economics | journal = Kyklos | volume = 44 | issue = 4 | pages = 581–612 | publisher = | location = | date = 1991 | url = | doi = | id = | accessdate = 2008-04-19}} </ref> [[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<ref> {{cite book | last = Friedman | first = Milton | authorlink = Milton_Friedman | coauthors = | title = Essays in Positive Economics | publisher = University of Chicago Press | date = 1953 | location = Chicago | pages = | url = | doi = | id = | isbn = }} </ref>. Samuelson argued a different tack. He proposed that economic theories should be [[Falsifiability|refutable in principle]]--if they were refutable in principle, they could not be tautological.<ref>§ Boland, L. A. (2006) Seven Decades of Economic Methodology. pp. 219-223 '''In''': I. C. Jarvie, K. Milford, D.W. Miller Eds ''Karl Popper:A Centenary Assessment''. London:Ashgate Publishing p. 220</ref> 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 in an interview later.<ref> {{Citation | last = Heilbroner | first = Robert | author-link = | last2 = | first2 = | author2-link = | title = The end of the Dismal Science? | newspaper = Challenge Magazine | pages = | year = 1999 | date = May-June 1999 | url =http://findarticles.com/p/articles/mi_m1093/is_3_42/ai_54682627/print }} </ref> {{quote|I guess the scientific approach began to penetrate and soon dominate the profession in the past twenty to thirty years. This came about in part because of the "invention" of mathematical analysis of various kinds and, indeed, considerable improvements in it. This is the age in which we have not only more data but more sophisticated use of data. So there is a strong feeling that this is a data-laden science and a data-laden undertaking, which, by virtue of the sheer numerics, the sheer equations, and the sheer look of a journal page, bears a certain resemblance to science...That one central activity looks scientific. I understand that. I think that is genuine. It approaches being a universal law. But resembling a science is different from being a science.}} 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."<ref> {{cite journal | last = Beed | first = Clive | authorlink = | coauthors = Kane, Owen | title = Critique of the Mathematization of Economics | journal = Kyklos | volume = 44 | issue = 4 | pages = 584 | publisher = | location = | date = 1991 | url = | doi = | id = | accessdate = 2008-04-19}} </ref> This critique has been advanced in various forms by economists and other scientists, including [[J.M. Keynes|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.<ref> {{cite journal | last = Joskow | first = Paul | authorlink =Paul_Joskow | coauthors = | title = Firm Decision-making Policy and Oligopoly Theory | journal = The American Economic Review | volume = 65 | issue = 2, Papers and Proceedings of the Eighty-seventh Annual Meeting of the American Economic Association | pages = 270–279, Particularly 271 | publisher = | location = | date = May, 1975 | url = | doi = | id = | accessdate = 2008-04-19}} </ref><ref> {{cite journal | last = Keynes | first = J. M. | authorlink = J.M._Keynes | coauthors = | title = Alfred Marshall 1842-1924 | journal = The Economic Journal | volume = 34 | issue = 135 | pages = 333,356 | publisher = | location = | date = September, 1924 | url = | doi = | id = | accessdate = 2008-04-19}} </ref> ==Mathematical economics applications== * [[Arbitrage]] * [[Black-Scholes equation]] * [[Econometrics]] * [[Game theory]] * [[General equilibrium theory]] * [[Growth theory]] * [[Information theory]] * [[Qualitative economics]] * [[Wealth condensation]] ==Mathematical economists== Famous mathematical economists include, but are not limited to, the following list (by century of birth). ===19th century=== {{MultiCol}} * [[Antoine Augustin Cournot]] * [[Enrico Barone]] {{ColBreak}} * [[Francis Ysidro Edgeworth]] * [[Irving Fisher]] {{ColBreak}} * [[Vilfredo Pareto]] * [[Léon Walras]] {{EndMultiCol}} ===20th century=== {{MultiCol}} * [[R. G. D. Allen]] * [[Maurice Allais]] * [[Kenneth Arrow]] * [[Robert J. Aumann]] * [[Gerard Debreu]] * [[John Hicks|John R. Hicks]] * [[Leonid Hurwicz]] * [[Leonid Kantorovich]] {{ColBreak}} * [[Tjalling Koopmans]] * [[Finn E. Kydland]] * [[James Mirrlees|Sir James Mirrlees]] * [[John Forbes Nash Jr. (mathematician)|John Nash]] * [[Edmond Malinvaud]] * [[Eric Maskin]] * [[Roger Myerson]] * [[Edward C. Prescott]] {{ColBreak}} * [[Frank P. Ramsey|Frank Ramsey]] * [[Reinhard Selten]] * [[Paul Samuelson]] * [[Amartya Sen]] * [[Lloyd Shapley]] * [[Herbert Simon|Herbert A. Simon]] * [[Robert Solow]] * [[John von Neumann#economics|John von Neumann]] {{EndMultiCol}} ==See also== <div style="-moz-column-count:2; column-count:2;"> * [[Mathematical finance]] * Mathematics of [[random variable]]s * [[Pareto distribution]] * [[Probability theory]] * [[Zipf's law]] * [[Extreme value theory]] * [[Fractal]] * [[Optimization (mathematics)]] * [[Systems theory]] * [[Self-organization]] * [[Self-similarity]] * [[Randomness]] </div> ==Notes== {{reflist|2}} ==References== *[[Gerard Debreu]] (1987). "Mathematical Economics," ''[[The New Palgrave: A Dictionary of Economics]]'', v. 3, pp. 399-404. * [[F.Y. Edgeworth]] ([1925] 1987). "Mathematical Method in Political Economy," ''The New Palgrave: A Dictionary of Economics'', v. 3, pp. 404-05. * Eugene Silberberg and Wing Suen (2000). ''The Structure of Economics: A Mathematical Analysis'', 3rd ed. McGraw-Hill. * ''[[Econometrica]], leading journal in mathermatical economics and [[econometrics]] ==External links== {{Wiktionary}} *[http://www.elsevier.com/cgi-bin/cas/tree/store/mateco/cas_free/browse/browse.cgi?key=mkwa Keyword list from Journal of Mathematical Economics] *{{dmoz|Science/Math/Applications/Mathematical_Economics_and_Financial_Mathematics/|Mathematical Economics and Financial Mathematics}} *[http://www.ma.hw.ac.uk/~des/HWM00-26.pdf Wealth Condensation in Pareto Macro-Economies] [[Category:Applied mathematics]] [[Category:Mathematical economics| ]] [[Category:Mathematical and quantitative methods (economics)]] [[de:Wirtschaftsmathematik]] [[el:Μαθηματική Οικονομική]] [[eo:Vikipedio:Projekto matematiko/Matematika ekonomio]] [[hu:Matematikai közgazdaságtan]] [[ja:数理経済学]] [[pl:Ekonomia matematyczna]] [[pt:Economia matemática]] [[ru:Математическая экономика]] [[vi:Toán kinh tế]] [[tr:Matematiksel ekonomi]] [[uk:Математична економіка]] [[zh:数理经济学]]