Mathematical economics

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Mathematical economics refers to the application of mathematical methods to represent economic theory or analyze problems posed in economics. Expositors maintain that it allows formulation and derivation of key relationships in the theory with clarity, generality, rigor, and simplicity [1]. For example, Paul Samuelson's Foundations of Economic Analysis (1947) identifies a common mathematical structure across multiple fields in the subject.

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. 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[2], focusing on mathematical economics [3][4]. According to the Editors, each is a non-specialist journal. For the diverse theoretical topics in economics examined in them, most articles there use a mathematical representation of the theory.

Yield Curves in a Recession.
Yield Curves in a Recession[5].

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[6][7][8]. 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[9].



Contents

[edit] Applied mathematics

Main article: 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.[10] 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 mathematicians. Applied mathematicians 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.[11]

[edit] Econometrics

Main article: Econometrics

Another tool of mathematical economics is econometrics, the use of statistical techniques to interpret quantitative empirical data [12]. 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[13]. 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[14]. The Nobel prize has been awarded to econometricians, most recently in 2003[15] for methods of estimating data with time series volatility and methods in cointegration.

[edit] 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[16], this did not extend to a general critique of mathematical tools in economics.

A considerable source of criticism was written in the 1940's 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 (In Popper's view) to rely on empirical truth and instead rely on axiomatic proof[17]. This meant that an economic model could either have verifiable assumptions and produce no new information or have unverifiable assumptions and sacrifice formalism for scope [18]. Milton Friedman responded to this by announcing that "all assumptions are unrealistic"[19]. By this he meant 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[20].

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[21]:

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"[22]. This critique has been advanced in various forms by economists and other scientists, including (but not limited to) Keynes[23] 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[24].

[edit] Mathematical economics applications

[edit] Mathematical economists

Famous mathematical economists include, but are not limited to, the following list (by century of birth).

[edit] 19th century

[edit] 20th century

[edit] See also


[edit] References

  1. ^ Chiang, Alpha C.; Kevin Wainwright (2005). Fundamental Methods of Mathematical Economics. McGraw-Hill Irwin, 1,2. ISBN 0-07-010910-9. 
  2. ^ Liner, Gaines H. (2002). "Core Journals in Economics". Economic Inquiry 40 (1). Oxford University Press. 
  3. ^ Journal of Economic Theory
  4. ^ Welcome to the website of The Econometric Society An International Society for the Advancement of Economic Theory in its Relation to Statistics and Mathematics
  5. ^ Powell, Nathan. What the Yield Curve does (and doesn't) tell us. http://www.fdic.gov/bank/analytical/fyi/2006/022206fyi.html
  6. ^ McGinty, Matthew. "International Environmental Agreements among Asymmetric Nations," Oxford Economic Papers, 59(1), January 2007: 45-62.
  7. ^ Wolfers, Justin and Stevenson, Betsy. "Marriage and Divorce: Changes and their Driving Factors." Journal of Economic Perspectives, 21(2) 27-52, Spring 2007.
  8. ^ Econbrowser: Commodity arbitrage
  9. ^ Hal, Varian (29-30 October, 1992). "What use is Economic Theory?".: 1-11. Retrieved on April, 2008. 
  10. ^ JSTOR: Marshall on Mathematics, in JSTOR
  11. ^ Sheila C Dow - THE USE OF MATHEMATICS IN ECONOMICS, For presentation to the ESRC Public Understanding of Mathematics Seminar, Birmingham, 21-2 May 1999
  12. ^ 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
  13. ^ Andrew J., Oswald & Hilda, Ralsmark (2008), “Warwick Economic Research Papers”, Some Evidence on the Future of Economics, Warwick Department of Economics, pp. 7 
  14. ^ Department of Economics | University of Wisconsin - Madison
  15. ^ Economics 2003
  16. ^ Hayek, Friedrich (September, 1945). "The Use of Knowledge in Society". American Economic Review 35 (4): 519-530. 
  17. ^ 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
  18. ^ Beed, Clive; Kane, Owen (1991). "Critique of the Mathematization of Economics". Kyklos 44 (4): 581-612. 
  19. ^ Friedman, Milton (1953). Essays in Positive Economics. Chicago: University of Chicago Press. 
  20. ^ § 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
  21. ^ Heilbroner, Robert (May-June 1999), “The end of the Dismal Science?”, Challenge Magazine, <http://findarticles.com/p/articles/mi_m1093/is_3_42/ai_54682627/print> 
  22. ^ Beed, Clive; Kane, Owen (1991). "Critique of the Mathematization of Economics". Kyklos 44 (4): 584. 
  23. ^ Keynes, J. M. (September, 1924). "Alfred Marshall 1842-1924". The Economic Journal 34 (135): 333,356. 
  24. ^ Joskow, Paul (May, 1975). "Firm Decision-making Policy and Oligopoly Theory". The American Economic Review 65 (2, Papers and Proceedings of the Eighty-seventh Annual Meeting of the American Economic Association): 270-279, Particularly 271. 


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