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{{Short description|Branch of applied mathematics}}
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'''Mathematical economics''' is the application of [[Mathematics|mathematical]] methods to represent theories and analyze problems in [[economics]].  Often, these [[Applied mathematics#Economics|applied methods]] are beyond simple geometry, and may include differential and integral [[calculus]], [[Recurrence relation|difference]] and [[differential equations]], [[Matrix (mathematics)|matrix algebra]], [[mathematical programming]], or other [[Computational economics|computational methods]].<ref>Elaborated at the [[JEL classification codes]], [[JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories|Mathematical and quantitative methods JEL: C Subcategories]].</ref><ref name = "Chiang">{{cite book | last1 = Chiang | first1 = Alpha C. |author-link1= Alpha C. Chiang | author2= Kevin Wainwright | title = Fundamental Methods of Mathematical Economics | publisher = McGraw-Hill Irwin | year = 2005 | pages = 3–4 | isbn = 978-0-07-010910-0}} [http://www.mhprofessional.com/product.php?isbn=0070109109 TOC.] {{Webarchive|url=https://web.archive.org/web/20120308171226/http://www.mhprofessional.com/product.php?isbn=0070109109 |date=2012-03-08 }}</ref> Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.<ref>[[Gérard Debreu|Debreu, Gérard]] ([1987] 2008). "mathematical economics", section II, ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_M000107&edition=current&q=Mathematical%20economics&topicid=&result_number=1 Abstract.] {{Webarchive|url=https://web.archive.org/web/20130516144807/http://www.dictionaryofeconomics.com/article?id=pde2008_M000107&edition=current&q=Mathematical%20economics&topicid=&result_number=1 |date=2013-05-16 }} Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content", ''Econometrica'', 54(6), pp. [https://www.jstor.org/pss/1914299 1259] {{Webarchive|url=https://web.archive.org/web/20170805220327/https://www.jstor.org/pss/1914299 |date=2017-08-05 }}-1270.</ref>

Mathematics allows economists to form meaningful, testable propositions about  wide-ranging and complex subjects which could less easily  be expressed informally. Further, the language of mathematics allows economists to make specific, [[positivism|positive]] claims about controversial or contentious subjects that would be impossible without mathematics.<ref>[[Hal Varian|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 [http://www.sims.berkeley.edu/~hal/Papers/theory.pdf PDF.] {{Webarchive|url=https://web.archive.org/web/20060625062619/http://www.sims.berkeley.edu/~hal/Papers/theory.pdf |date=2006-06-25 }} Retrieved 2008-04-01.</ref> Much of economic theory is currently presented in terms of mathematical [[Model (economics)|economic models]], a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.<ref>* As in ''Handbook of Mathematical Economics'', 1st-page chapter links:<br/>&nbsp;&nbsp;&nbsp;&nbsp; Arrow, Kenneth J., and Michael D. Intriligator, ed., (1981), v. [http://www.sciencedirect.com/science?_ob=PublicationURL&_tockey=%23TOC%2324615%231981%23999989999%23565707%23FLP%23&_cdi=24615&_pubType=HS&_auth=y&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=01881ea3fe7d7990fed1c5b78d9f7be6 1]<br/>&nbsp;&nbsp;&nbsp;&nbsp; _____ (1982). v. [http://www.sciencedirect.com/science?_ob=PublicationURL&_tockey=%23TOC%2324615%231982%23999979999%23565708%23FLP%23&_cdi=24615&_pubType=HS&_auth=y&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=80dbd8f22c229a3640dc02e59ff80fe4 2]<br/>&nbsp;&nbsp;&nbsp;&nbsp; _____ (1986). v. [http://www.sciencedirect.com/science?_ob=PublicationURL&_tockey=%23TOC%2324615%231986%23999969999%23565709%23FLP%23&_cdi=24615&_pubType=HS&_auth=y&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=56f86ec2f0a1d2881e15bd1d0a45accd  3]<br/>&nbsp;&nbsp;&nbsp;&nbsp; [[Werner Hildenbrand|Hildenbrand, Werner]], and [[Hugo Sonnenschein]], ed. (1991). v. [http://www.sciencedirect.com/science/handbooks/15734382 4.] {{Webarchive|url=https://web.archive.org/web/20130415064616/http://www.sciencedirect.com/science/handbooks/15734382 |date=2013-04-15 }}
* [[Gérard Debreu|Debreu, Gérard]] (1983).   ''Mathematical Economics: Twenty Papers of Gérard Debreu'',  [https://books.google.com/books?id=wKJp6DepYncC&pg=PR7 Contents] {{Webarchive|url=https://web.archive.org/web/20230701102231/https://books.google.com/books?id=wKJp6DepYncC&pg=PR7 |date=2023-07-01 }}.
* [[Stephen Glaister|Glaister, Stephen]] (1984). ''Mathematical Methods for Economists'', 3rd ed., Blackwell. [https://books.google.com/books?id=Ct2nrJSHxsQC&pg=PR5=onepage Contents.] {{Webarchive|url=https://web.archive.org/web/20230701102234/https://books.google.com/books?id=Ct2nrJSHxsQC&pg=PR5=onepage |date=2023-07-01 }}
* Takayama, Akira (1985). ''Mathematical Economics'', 2nd ed. Cambridge.
[https://books.google.com/books?id=685iPEaLAEcC Description] {{Webarchive|url=https://web.archive.org/web/20230701102232/https://books.google.com/books?id=685iPEaLAEcC |date=2023-07-01 }} and [https://books.google.com/books?id=685iPEaLAEcC&pg=PR9=onepage Contents] {{Webarchive|url=https://web.archive.org/web/20230701102741/https://books.google.com/books?id=685iPEaLAEcC&pg=PR9=onepage |date=2023-07-01 }}.
* Michael Carter (2001). ''Foundations of Mathematical Economics'', MIT Press. [https://web.archive.org/web/20060915230536/http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=8630 Description] and [https://books.google.com/books?id=KysvrGGfzq0C&pg=PR7=onepage Contents] {{Webarchive|url=https://web.archive.org/web/20230701102740/https://books.google.com/books?id=KysvrGGfzq0C&pg=PR7=onepage |date=2023-07-01 }}.</ref>

Broad applications include:
* [[mathematical optimization|optimization]] problems as to goal equilibrium, whether of a household, business firm, or policy maker
* static (or [[Economic equilibrium|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
*  [[Comparative dynamics|dynamic]] analysis, tracing changes in an economic system over time, for example from [[economic growth]].<ref name = "Chiang"/><ref>Chiang, Alpha C. (1992). ''Elements of Dynamic Optimization'', Waveland. [https://web.archive.org/web/20050211210424/http://waveland.com/Titles/Chiang.htm TOC] & Amazon.com [https://www.amazon.com/Elements-Dynamic-Optimization-Alpha-Chiang/dp/157766096X link] {{Webarchive|url=https://web.archive.org/web/20160303201037/https://www.amazon.com/Elements-Dynamic-Optimization-Alpha-Chiang/dp/157766096X |date=2016-03-03 }} to inside, first pp.</ref><ref name="Samuelson">{{cite book|last=Samuelson|first=Paul| author-link =Paul Samuelson|title=Foundations of Economic Analysis|publisher=Harvard University Press|date=1947|orig-year=1983|isbn=978-0-674-31301-9|title-link=Foundations of Economic Analysis}}</ref>

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 introduction of new and generalized techniques in the period around the [[World War II|Second World War]], as in [[game theory]], would greatly broaden the use of mathematical formulations in economics.<ref name="DebreuNeumann">* [[Gérard Debreu|Debreu, Gérard]] ([1987] 2008). "mathematical economics", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_M000107&edition=current&q=Mathematical%20economics&topicid=&result_number=1 Abstract.] {{Webarchive|url=https://web.archive.org/web/20130516144807/http://www.dictionaryofeconomics.com/article?id=pde2008_M000107&edition=current&q=Mathematical%20economics&topicid=&result_number=1 |date=2013-05-16 }} Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content", ''Econometrica'', 54(6), pp. [https://www.jstor.org/pss/1914299 1259] {{Webarchive|url=https://web.archive.org/web/20170805220327/https://www.jstor.org/pss/1914299 |date=2017-08-05 }}-1270.
* [[John von Neumann|von Neumann, John]], and [[Oskar Morgenstern]] (1944). ''[[Theory of Games and Economic Behavior]]''. Princeton University Press.</ref><ref name="Samuelson"/>

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==
{{main|History of economic thought}}
The use of mathematics in the service of social and economic analysis dates back to the 17th century.  Then, mainly in [[Holy Roman Empire|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''.<ref>{{cite book|last=Schumpeter|first=J.A.|title=History of Economic Analysis|editor=Elizabeth B. Schumpeter|publisher=Oxford University Press|location=New York|year=1954|pages=209–212|isbn=978-0-04-330086-2|oclc=13498913|url=https://books.google.com/books?id=xjWiAAAACAAJ|access-date=2020-05-28|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701102741/https://books.google.com/books?id=xjWiAAAACAAJ|url-status=live}}</ref>  [[William Petty|Sir William Petty]] wrote at length on issues that would later concern economists, such as taxation, [[Velocity of money]] and [[Measures of national income and output|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.<ref>Schumpeter (1954) p. 212-215</ref>

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 [[algebra]]ic 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.<ref>{{cite journal|last= Schnieder|first=Erich|title=Johann Heinrich von Thünen|journal=[[Econometrica]]|volume=2|issue=1|pages=1–12|issn=0012-9682|oclc=35705710|jstor=1907947|doi= 10.2307/1907947|year= 1934}}</ref>  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 applying previous tools to new problems.<ref>Schumpeter (1954) p. 465-468</ref>

Meanwhile, a new cohort of scholars trained in the mathematical methods of the [[physical science]]s  gravitated to economics, advocating and applying those methods to their subject,<ref>[[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.</ref> and described today as moving from geometry to [[mechanics]].<ref>[[E. Roy Weintraub|Weintraub, E. Roy]] (2008). "mathematics and economics", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_M000372&edition=current&q=&topicid=&result_number=1 Abstract] {{Webarchive|url=https://web.archive.org/web/20130516145005/http://www.dictionaryofeconomics.com/article?id=pde2008_M000372&edition=current&q=&topicid=&result_number=1 |date=2013-05-16 }}.</ref>
These included [[William Stanley Jevons|W.S. Jevons]] who presented a paper on a "general mathematical theory of political economy" in 1862, providing an outline for use of the theory of [[marginal utility]] in political economy.<ref>Jevons, W.S. (1866). "Brief Account of a General Mathematical Theory of Political Economy", ''Journal of the Royal Statistical Society'',  XXIX (June) pp.&nbsp;282–87. Read in Section F of the British Association, 1862. [https://web.archive.org/web/20220622033317/http://www.adelinotorres.com/economia/STANLEY%20JEVONS_Teoria%20Matem%EF%BF%BDtica%20e%20Economia%20Politica.pdf PDF.]</ref>  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 collection of statistics for price and quantities would permit the subject as presented to become an exact science.<ref>{{cite book|last=Jevons|first=W. Stanley|year= 1871|title=The Principles of Political Economy, pp. 4, 25.|publisher=Macmillan|url=https://archive.org/details/bub_gb_Sw8ZAAAAYAAJ|quote=The Theory of Political Economy, jevons 1871.}}</ref>  Others preceded and followed in expanding mathematical representations of economic [[mathematical problem|problem]]s.
<ref>See [https://babel.hathitrust.org/cgi/pt?id=uc2.ark:/13960/t4zg6mw37&view=1up&seq=9 the preface] {{Webarchive|url=https://web.archive.org/web/20230701102745/https://babel.hathitrust.org/cgi/pt?id=uc2.ark:/13960/t4zg6mw37&view=1up&seq=9 |date=2023-07-01 }} to Irving Fisher's 1897 work, ''A brief introduction to the infinitesimal calculus: designed especially to aid in reading mathematical economics and statistics''.</ref>

===Marginalists and the roots of neoclassical economics===
{{main|Marginalism}}
[[Image:Economics cournot diag4 svg.svg|thumb|275 px|Equilibrium quantities as a solution to two reaction functions in Cournot duopoly. Each reaction function is expressed as a linear equation dependent upon quantity demanded.]]

[[Antoine Augustin Cournot|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.<ref name="Dow">
{{cite conference
| first = Dow
| last = Sheila C.
| title = The Use of Mathematics in Economics
| book-title = ESRC Public Understanding of Mathematics Seminar
| publisher = Economic and Social Research Council
| date = 1999-05-21
| location = [[Birmingham]]
| url = http://www.ioe.ac.uk/esrcmaths/sheila1.html
| access-date = 2008-07-06}}
</ref>  At the time, it was thought that utility was quantifiable, in units known as [[util]]s.<ref>While the concept of [[Cardinal utility|cardinality]] has fallen out of favor in [[neoclassical economics]], the differences between cardinal utility and [[ordinal utility]] are minor for most applications.</ref>  Cournot, Walras and [[Francis Ysidro Edgeworth]] are considered the precursors to modern mathematical economics.<ref name="CWE">{{cite book|last=Nicola|first=PierCarlo|title=Mainstream Mathermatical Economics in the 20th Century|publisher=Springer|year=2000|page=4|isbn=978-3-540-67084-1|url=https://books.google.com/books?id=KR0Rbi8o4QQC|access-date=2008-08-21|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701102743/https://books.google.com/books?id=KR0Rbi8o4QQC|url-status=live}}</ref>

====Augustin Cournot====
Cournot, a professor of mathematics, developed a mathematical treatment in 1838 for [[duopoly]]—a market condition defined by competition between two sellers.<ref name="CWE"/>  This treatment of competition, first published in ''Researches into the Mathematical Principles of Wealth'',<ref>Augustin Cournot (1838, tr. 1897) '' Researches into the Mathematical Principles of Wealth''. Links to [https://books.google.com/books?id=AGoGpyJY_SAC description] {{Webarchive|url=https://web.archive.org/web/20230701103744/https://books.google.com/books?id=AGoGpyJY_SAC |date=2023-07-01 }} and  [https://books.google.com/books?id=AGoGpyJY_SAC chapters.] {{Webarchive|url=https://web.archive.org/web/20230701103744/https://books.google.com/books?id=AGoGpyJY_SAC |date=2023-07-01 }}</ref> is referred to as [[Cournot competition|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 [[Homogeneity and heterogeneity|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 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.<ref name="Stability">{{cite book|last=Hotelling|first=Harold|author-link=Harold Hotelling|title=The Collected Economics Articles of Harold Hotelling|editor=Darnell, Adrian C.|publisher=Springer|year=1990|pages=51, 52|chapter=Stability in Competition|isbn=978-3-540-97011-8|oclc=20217006|chapter-url=https://books.google.com/books?id=dYbbHQAACAAJ|access-date=2008-08-21|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701103745/https://books.google.com/books?id=dYbbHQAACAAJ|url-status=live}}</ref> Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced many of the [[Marginalism|marginalists]].<ref name="Stability"/><ref>{{cite web|url=http://cepa.newschool.edu/het/profiles/cournot.htm |archive-url=https://web.archive.org/web/20000709103727/http://cepa.newschool.edu/het/profiles/cournot.htm |url-status=dead |archive-date=2000-07-09 |title=Antoine Augustin Cournot, 1801-1877 |work=The History of Economic Thought Website |publisher=The New School for Social Research |access-date=2008-08-21 }}</ref>  Cournot's models of duopoly and [[oligopoly]] also represent one of the first formulations of [[non-cooperative game]]s.  Today the solution can be given as a [[Nash equilibrium]] but Cournot's work preceded modern [[game theory]] by over 100 years.<ref>{{cite book|last=Gibbons|first=Robert|title=Game Theory for Applied Economists|publisher=Princeton University Press|location=Princeton, New Jersey|year=1992|pages=14, 15|isbn=978-0-691-00395-5}}</ref>

====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 equilibrium|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.<ref>Nicola, p. 9-12</ref>  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 [[Walrasian auction|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).<ref>{{cite journal|last=Edgeworth |first=Francis Ysidro |author-link=Francis Ysidro Edgeworth |date=September 5, 1889 |title=The Mathematical Theory of Political Economy: Review of Léon Walras, Éléments d'économie politique pure |journal=[[Nature (journal)|Nature]] |volume=40 |issue=1036 |pages=434–436 |issn=0028-0836 |doi=10.1038/040434a0 |s2cid=21004543 |url=http://cepa.newschool.edu/het/texts/edgeworth/edgewalras89.pdf |archive-url=https://web.archive.org/web/20030411130712/http://cepa.newschool.edu/het/texts/edgeworth/edgewalras89.pdf |url-status=dead |archive-date=April 11, 2003 |access-date=2008-08-21 }}</ref>

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.<ref>Nicholson, Walter; Snyder, Christopher, p. 350-353.</ref>

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).<ref>{{cite web|url=http://www.economics.unimelb.edu.au/rdixon/wlaw.html|title=Walras Law and Macroeconomics|last=Dixon|first=Robert|work=Walras Law Guide|publisher=Department of Economics, University of Melbourne|access-date=2008-09-28 |archive-url = https://web.archive.org/web/20080417102559/http://www.economics.unimelb.edu.au/rdixon/wlaw.html |archive-date = April 17, 2008}}</ref>

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.<ref>{{cite web|url=http://www.economics.unimelb.edu.au/rdixon/walproof.html|title=A Formal Proof of Walras Law|last=Dixon|first=Robert|work=Walras Law Guide|publisher=Department of Economics, University of Melbourne|access-date=2008-09-28 |archive-url = https://web.archive.org/web/20080430033548/http://www.economics.unimelb.edu.au/rdixon/walproof.html |archive-date = April 30, 2008}}</ref>

====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.<ref>{{cite book|last=Rima|first=Ingrid H.|title=Modern Economic Thought|editor=Weintraub, Sidney|publisher=University of Pennsylvania Press|year=1977|pages=10, 11|chapter=Neoclassicism and Dissent 1890-1930|isbn=978-0-8122-7712-8|chapter-url=https://books.google.com/books?id=s7cJAAAAIAAJ&pg=PR5|access-date=2021-05-31|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701103744/https://books.google.com/books?id=s7cJAAAAIAAJ&pg=PR5|url-status=live}}</ref>  He adopted [[Jeremy Bentham]]'s [[felicific calculus]] to economic behavior, allowing the outcome of each decision to be converted into a change in utility.<ref>{{cite book|last=Heilbroner|first=Robert L.|author-link=Robert Heilbroner|title=The Worldly Philosophers|publisher=Simon and Schuster|location=New York|orig-year=1953|date=1999|edition=Seventh|pages=172–175, 313|isbn=978-0-684-86214-9|url=https://books.google.com/books?id=N_3cj4urgJcC|access-date=2020-05-28|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701103745/https://books.google.com/books?id=N_3cj4urgJcC|url-status=live}}</ref>  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".<ref>{{cite book|last=Edgeworth|first=Francis Ysidro|title=Mathematical Psychics|publisher=Kegan Paul [A. M. Kelley]|location=London|orig-year=1881|date=1961|pages=15–19|url=https://books.google.com/books?id=Q4WCGAAACAAJ|access-date=2020-05-28|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701103745/https://books.google.com/books?id=Q4WCGAAACAAJ|url-status=live}}</ref>

[[File:Contract-curve-on-edgeworth-box.svg|right|thumb|350px|An [[Edgeworth box]] displaying the contract curve on an economy with two participants.  Referred to as the "core" of the economy in modern parlance, there are infinitely many solutions along the curve for economies with two participants<ref>Nicola, p. 14, 15, 258-261</ref>]]

Given two individuals, the set of solutions where 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]].<ref name="Bowley">{{cite book|last=Bowley|first=Arthur Lyon|title=The Mathematical Groundwork of Economics: an Introductory Treatise|publisher=Clarendon Press [Kelly]|location=Oxford|orig-year=1924|date=1960|url=https://books.google.com/books?id=_cgkAAAAMAAJ|access-date=2020-05-28|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701104747/https://books.google.com/books?id=_cgkAAAAMAAJ|url-status=live}}</ref>  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 (economics)|core]] of an economy.<ref>{{cite book|last=Gillies|first=D. B.|title=Contributions to the Theory of Games|editor=Tucker, A. W.|editor2=Luce, R. D.|publisher=[[Princeton University Press]]|location=Princeton, New Jersey|series=[[Annals of Mathematics]]|volume=40|pages=47–85|chapter=Solutions to general non-zero-sum games|isbn=978-0-691-07937-0|chapter-url=https://books.google.com/books?id=9lSVFzsTGWsC|year=1969|access-date=2020-05-28|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701104747/https://books.google.com/books?id=9lSVFzsTGWsC|url-status=live}}</ref>

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.<ref name="Moss">{{cite journal|last=Moss|first=Lawrence S.|year=2003|title=The Seligman-Edgeworth Debate about the Analysis of Tax Incidence: The Advent of Mathematical Economics, 1892–1910 |journal=History of Political Economy|volume=35|issue=2|pages=207, 212, 219, 234–237|issn=0018-2702|doi=10.1215/00182702-35-2-205}}</ref>  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.<ref>{{cite book|last=Hotelling|first=Harold|author-link=Harold Hotelling|title=The Collected Economics Articles of Harold Hotelling|editor=Darnell, Adrian C.|publisher=Springer|year=1990|pages=94–122|chapter=Note on Edgeworth's Taxation Phenomenon and Professor Garver's Additional Condition on Demand Functions|isbn=978-3-540-97011-8|oclc=20217006|chapter-url=https://books.google.com/books?id=dYbbHQAACAAJ|access-date=2008-08-26|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701103745/https://books.google.com/books?id=dYbbHQAACAAJ|url-status=live}}</ref>

==Modern mathematical economics==
From the later-1930s, an array of new mathematical tools from differential calculus and differential equations, [[convex set]]s, and [[graph theory]] were deployed to advance  economic theory in a way similar to new mathematical methods earlier applied to physics.<ref name="DebreuNeumann"/><ref>{{cite journal|last=Herstein|first=I.N.|date=October 1953|title=Some Mathematical Methods and Techniques in Economics|journal=Quarterly of Applied Mathematics|volume=11|issue=3|pages=249–262|issn=1552-4485|doi=10.1090/qam/60205|doi-access=free}} [Pp. [https://web.archive.org/web/20000914192808/http://cowles.econ.yale.edu/P/cp/p00b/p0078.pdf 249-62].</ref> The process was later described as moving from [[mechanics]] to [[axiomatic system|axiomatics]].<ref>* Weintraub, E. Roy (2008). "mathematics and economics", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_M000372&edition=current&q=&topicid=&result_number=1 Abstract] {{Webarchive|url=https://web.archive.org/web/20130516145005/http://www.dictionaryofeconomics.com/article?id=pde2008_M000372&edition=current&q=&topicid=&result_number=1 |date=2013-05-16 }}.
* _____ (2002). ''How Economics Became a Mathematical Science''. Duke University Press. [https://web.archive.org/web/20100729022635/http://www.dukeupress.edu/Catalog/ViewProduct.php?productid=6807&viewby=series&categoryid=47&sort=newest Description] and [https://books.google.com/books?id=ntc_Fs36AoQC&pg=PR7 preview] {{Webarchive|url=https://web.archive.org/web/20230604194155/https://books.google.com/books?id=ntc_Fs36AoQC&pg=PR7 |date=2023-06-04 }}.</ref>

===Differential calculus===
{{Main|Foundations of Economic Analysis|Differential calculus}}
{{See also|Pareto efficiency|Walrasian auction}}

[[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 efficiency|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.<ref>{{cite book|last1=Nicholson|first1=Walter|author2=Snyder, Christopher|title=Intermediate Microeconomics and Its Applications|publisher=Thompson|year=2007|edition=10th|pages=364, 365|chapter=General Equilibrium and Welfare|isbn=978-0-324-31968-2}}</ref>  Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed to [[Adam Smith]]'s [[Invisible hand]] hypothesis.<ref>* {{cite book|author1 =Jolink, Albert|chapter=What Went Wrong with Walras?|title=From Walras to Pareto|editor1=Backhaus, Juergen G. |editor2=Maks, J.A. Hans |publisher=Springer|year=2006|series=The European Heritage in Economics and the Social Sciences |volume=IV|pages=69–80 | isbn=978-0-387-33756-2 |doi=10.1007/978-0-387-33757-9_6}}
* {{cite journal|last=Blaug|first=Mark|year=2007|title=The Fundamental Theorems of Modern Welfare Economics, Historically Contemplated|journal=History of Political Economy|volume=39|issue=2|pages=186–188|doi=10.1215/00182702-2007-001|s2cid=154074343|issn=0018-2702}}</ref> Rather, Pareto's statement was the first formal assertion of what would be known as the [[Fundamental theorems of welfare economics#Proof of the first fundamental theorem|first fundamental theorem of welfare economics]].<ref>Blaug (2007), p. 185, 187</ref>  These models lacked the inequalities of the next generation of mathematical economics.

In the landmark treatise ''[[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  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 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 [[economic equilibrium|equilibrium]] states after an [[exogenous variable|exogenous]] change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.<ref name="Samuelson"/><ref>{{cite journal|last=Metzler|first=Lloyd |author-link=Lloyd Metzler |year=1948|title=Review of ''Foundations of Economic Analysis''|journal=American Economic Review|volume=38|issue=5|pages=905–910|issn=0002-8282|jstor=1811704}}</ref>

===Linear models===

{{See also|Linear algebra|Linear programming|Perron–Frobenius theorem}}

Restricted models of general equilibrium were formulated  by [[John von Neumann]] in 1937.<ref name="Neumann1937">Neumann, J. von (1937). "Über ein ökonomisches Gleichungssystem und ein Verallgemeinerung des Brouwerschen Fixpunktsatzes", ''Ergebnisse eines Mathematischen Kolloquiums'', 8, pp.&nbsp;73–83,  translated and published in 1945-46, as "A Model of General Equilibrium", ''Review of Economic Studies'', 13, pp. 1–9.</ref> Unlike earlier versions, the models of von Neumann had inequality constraints. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of [[Brouwer's fixed point theorem]]. Von Neumann's model of an expanding economy considered the [[generalized eigenvalue problem|matrix pencil]]&nbsp;'' '''A''' - λ '''B''' '' with nonnegative matrices&nbsp;'''A''' and '''B'''; von Neumann sought [[probability vector|probability]] [[generalized eigenvector|vector]]s&nbsp;''p'' and&nbsp;''q'' and a positive number&nbsp;''λ'' that would solve the [[complementarity theory|complementarity]] equation
:'' p<sup>T</sup>''&nbsp;('''''A''''' − ''λ&nbsp;'''B''''')&nbsp;''q'' = 0,
along with two inequality systems expressing economic efficiency. In this model, the ([[transpose]]d) probability vector ''p'' represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run.  The unique [[generalized eigenvalue problem|solution]] ''λ'' represents the [[economic growth|rate of growth]] of the economy, which equals the [[interest rate]]. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.<ref>For this problem to have a unique solution, it suffices that the nonnegative matrices&nbsp;'''A''' and&nbsp;'''B''' satisfy an [[Perron–Frobenius theorem|irreducibility condition]], generalizing that of the [[Perron–Frobenius theorem]] of nonnegative matrices, which considers the (simplified) [[eigenvalue|eigenvalue problem]]
:'' '''A''''' − ''λ '''I''' q'' = 0,
where the nonnegative matrix&nbsp;'''''A''''' must be square and where the [[diagonal matrix]]&nbsp;'''''I''''' is the [[identity matrix]]. Von Neumann's irreducibility condition was called the "whales and [[wrangler (profession)|wrangler]]s" hypothesis by David Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every economic good. Weaker "irreducibility" conditions were given by [[David Gale]] and by [[John George Kemeny|John Kemeny]], [[Oskar Morgenstern]], and [[Gerald L. Thompson]] in the 1950s and then by [[Stephen M. Robinson]] in the&nbsp;1970s.</ref><ref>David Gale. ''The theory of linear economic models''. McGraw-Hill, New York, 1960.</ref><ref>{{cite book | last1=Morgenstern | first1=Oskar | author-link1=Oskar Morgenstern |  last2=Thompson| first2=Gerald L. | author-link2=Gerald L. Thompson | title=Mathematical theory of expanding and contracting economies | series=Lexington Books| publisher=D. C. Heath and Company | year=1976 | location=Lexington, Massachusetts | pages=xviii+277 }}</ref> Von Neumann's results have been viewed as a special case of [[linear programming]], where von Neumann's model uses only nonnegative matrices.<ref>[[Alexander Schrijver]], ''Theory of Linear and Integer Programming''. John Wiley & sons, 1998, {{isbn|0-471-98232-6}}.</ref> The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.<ref>
*{{cite book|last=Rockafellar|first=R. Tyrrell|title=Monotone processes of convex and concave type|series=Memoirs of the American Mathematical Society|publisher=American Mathematical Society|location=Providence, R.I.|pages=i+74|year=1967|issue=77}}
* {{cite book|last=Rockafellar|first=R. T.|author-link=R. Tyrrell Rockafellar|chapter=Convex algebra and duality in dynamic models of production|title=Mathematical models in economics (Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972)|editor=Josef Loz |editor2=Maria Loz|pages=351–378|publisher=North-Holland and Polish Academy of Sciences (PAN)|location=Amsterdam|year=1974}}
*{{cite book|author-link=R. Tyrrell Rockafellar|first=R. T.|last=Rockafellar|title=Convex analysis|publisher=Princeton University Press|location=Princeton, New Jersey|orig-year=1970|date=1997}}
</ref><ref>{{cite book|title=John Von Neumann and modern economics|editor1=Mohammed Dore |editor2=Sukhamoy Chakravarty |editor3= Richard Goodwin |editor3-link=Richard Murphy Goodwin | publisher=Oxford:Clarendon|year=1989|page=261|author-link1=Kenneth Arrow |author-link2=Paul Samuelson |author-link3=John Harsanyi |author-link4=Sidney Afriat |author-link5=Gerald L. Thompson |author-link6=Nicholas Kaldor |first1=Kenneth |last1=Arrow |first2=Paul |last2= Samuelson |first3=John |last3=Harsanyi |first4=Sidney |last4=Afriat |first5=Gerald L. |last5=Thompson |first6=Nicholas |last6=Kaldor}}</ref><ref>Chapter 9.1 "The von Neumann growth model" (pages 277–299): [[Yinyu Ye]]. ''Interior point algorithms: Theory and analysis''. Wiley. 1997.</ref>

====Input-output economics====
{{Main|Input-output model}}

In 1936, the Russian–born economist [[Wassily Leontief]] built his model of [[Input-output model|input-output analysis]] from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by the [[physiocrat]]s.  With his model, which described a system of production and demand processes, Leontief described how changes in demand in one [[economic sector]] would influence production in another.<ref>{{cite book|last1=Screpanti|first1=Ernesto|author2=Zamagni, Stefano|title=An Outline of the History of Economic Thought|publisher=[[Oxford University Press]]|location=New York|year=1993|pages=288–290|isbn=978-0-19-828370-6|oclc=57281275}}</ref> In practice, Leontief  estimated the coefficients of his simple models, to address economically interesting questions. <!-- Leontief's work made him a hero to [[George B. Dantzig]], the inventor of the [[simplex algorithm]] of [[linear programming]]. --> In [[production set|production economics]], "Leontief technologies" produce outputs using constant proportions of inputs, regardless of the price of inputs, reducing the value of Leontief models for understanding economies but allowing their parameters to be estimated relatively easily. In contrast, the von Neumann model of an expanding economy allows for [[choice of techniques]], but the coefficients must be estimated for each technology.<ref>[[David Gale]]. ''The theory of linear economic models''. McGraw-Hill, New York, 1960.</ref><ref>{{cite book|last1=Morgenstern |first1=Oskar|author-link1=Oskar Morgenstern|last2=Thompson|first2=Gerald&nbsp;L.|author-link2=Gerald L. Thompson|title=Mathematical theory of expanding and contracting economies|series=Lexington Books|publisher=D.&nbsp;C.&nbsp;Heath and Company|year=1976| location=Lexington, Massachusetts|pages=xviii+277}}</ref>

===Mathematical optimization===
[[File:MaximumParaboloid.png|right|thumb|Red dot in z direction as [[maximum (mathematics)|maximum]] for [[paraboloid]] function of (x, y) inputs]]{{Main|Mathematical optimization|Dual problem}}
{{See also|Convexity in economics|Non-convexity (economics)}}
In mathematics, [[mathematical optimization]] (or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives.<ref>"[https://web.archive.org/web/20060624034203/http://glossary.computing.society.informs.org/index.php?page=nature.html The Nature of Mathematical Programming]", ''Mathematical Programming Glossary'', INFORMS Computing Society.</ref> In the simplest case, an [[Energy function|optimization problem]]  involves [[maxima and minima|maximizing or minimizing]] a [[Function of a real variable|real function]] by selecting [[Argument of a function|input]] values of the function and computing the corresponding [[Value (mathematics)|value]]s of the function. The solution process includes satisfying general [[Mathematical optimization#Classification of critical points and extrema|necessary and sufficient conditions for optimality]]. For optimization problems, [[Mathematical optimization#Notation|specialized notation]] may be used as to the function and its input(s). More generally, optimization includes finding the best available [[Element (mathematics)|element]] of some function given a defined [[domain of a function|domain]] and may use a variety of different  [[Optimization algorithm|computational optimization techniques]].<ref name="Schmedders">Schmedders, Karl (2008). "numerical optimization methods in economics", ''The New Palgrave Dictionary of Economics'', 2nd Edition, v. 6, pp.&nbsp;138–57.  [http://www.dictionaryofeconomics.com/article?id=pde2008_N000148&edition=current&q=optimization&topicid=&result_number=1 Abstract.] {{Webarchive|url=https://web.archive.org/web/20170811183849/http://www.dictionaryofeconomics.com/article?id=pde2008_N000148&edition=current&q=optimization&topicid=&result_number=1 |date=2017-08-11 }}</ref>

Economics is closely enough linked to optimization by [[agent (economics)|agents]] in an [[economy]] that an influential definition relatedly describes economics ''qua'' science as the "study of human behavior as a relationship between ends and [[scarce]] means" with alternative uses.<ref>[[Lionel Robbins|Robbins, Lionel]] (1935, 2nd ed.). ''[[An Essay on the Nature and Significance of Economic Science#Major propositions|An Essay on the Nature and Significance of Economic Science]]'', Macmillan, p. 16.</ref> Optimization problems run through modern economics, many with explicit economic or technical constraints. In microeconomics, the [[utility maximization problem]] and its [[dual problem]], the [[expenditure minimization problem]] for a given level of utility, are economic optimization problems.<ref>[[Lawrence E. Blume|Blume, Lawrence E.]] (2008). "duality", ''The New Palgrave Dictionary of Economics'', 2nd Edition.
[http://www.dictionaryofeconomics.com/article?id=pde2008_D000196 Abstract.] {{Webarchive|url=https://web.archive.org/web/20170202015620/http://www.dictionaryofeconomics.com/article?id=pde2008_D000196 |date=2017-02-02 }}</ref> Theory posits that [[consumer]]s  maximize their [[utility]], subject to their [[budget constraint]]s and that [[firm]]s maximize their [[Profit (economics)|profit]]s, subject to their [[production function]]s, [[Factors of production|input]] costs, and market [[demand]].<ref name="Dixit1990">[[Avinash Dixit|Dixit, A. K.]] ([1976] 1990).  ''Optimization in Economic Theory'', 2nd ed., Oxford. [https://books.google.com/books?id=dHrsHz0VocUC&pg=PA194=false Description] {{Webarchive|url=https://web.archive.org/web/20230701104748/https://books.google.com/books?id=dHrsHz0VocUC&pg=PA194=false |date=2023-07-01 }} and contents [https://books.google.com/books?id=dHrsHz0VocUC&dq=false&pg=PR7 preview] {{Webarchive|url=https://web.archive.org/web/20230701104748/https://books.google.com/books?id=dHrsHz0VocUC&dq=false&pg=PR7 |date=2023-07-01 }}.</ref>

[[Economic equilibrium]] is studied in optimization theory as a key ingredient of economic theorems that in principle could be tested against empirical data.<ref name="Samuelson"/><ref>* Samuelson, Paul A., 1998. "How ''Foundations'' Came to Be", ''Journal of Economic Literature'', 36(3), pp. [https://www.jstor.org/pss/2564803 1375]–1386.
* _____ (1970).[https://www.nobelprize.org/nobel_prizes/economics/laureates/1970/samuelson-lecture.pdf "Maximum Principles in Analytical Economics"] {{Webarchive|url=https://web.archive.org/web/20121011025043/http://www.nobelprize.org/nobel_prizes/economics/laureates/1970/samuelson-lecture.pdf |date=2012-10-11 }},  Nobel Prize lecture.</ref>  Newer developments have occurred in [[dynamic programming]] and modeling optimization with [[risk]] and [[uncertainty]], including applications to [[portfolio theory]], the [[economics of information]], and [[search theory]].<ref name="Dixit1990"/>

Optimality properties for an entire [[market system]] may be stated in mathematical terms, as in formulation of the two [[fundamental theorems of welfare economics]]<ref>* Allan M. Feldman (3008). "welfare economics", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_W000050&edition=current&q=fundamental%20theorems%20&topicid=&result_number=3 Abstract] {{Webarchive|url=https://web.archive.org/web/20170811183948/http://www.dictionaryofeconomics.com/article?id=pde2008_W000050&edition=current&q=fundamental%20theorems%20&topicid=&result_number=3 |date=2017-08-11 }}.  
* [[Andreu Mas-Colell|Mas-Colell, Andreu]], Michael D. Whinston, and Jerry R. Green (1995), ''Microeconomic Theory'', Chapter 16. Oxford University Press, {{isbn|0-19-510268-1}}. [http://www.oup.com/us/catalog/general/subject/Economics/MicroeconomicTheory/?view=usa&ci=9780195073409 Description] {{webarchive|url=https://web.archive.org/web/20120126115547/http://www.oup.com/us/catalog/general/subject/Economics/MicroeconomicTheory/?view=usa |date=2012-01-26 }} and  [http://www.oup.com/us/catalog/general/subject/Economics/MicroeconomicTheory/?view=usa&sf=toc&ci=9780195073409 contents] {{Webarchive|url=https://web.archive.org/web/20120126115547/http://www.oup.com/us/catalog/general/subject/Economics/MicroeconomicTheory/?view=usa&sf=toc&ci=9780195073409 |date=2012-01-26 }} .</ref> and in the [[Arrow–Debreu model]] of [[general equilibrium]] (also discussed [[#Functional analysis|below]]).<ref>* [[John Geanakoplos|Geanakoplos, John]] ([1987] 2008). "Arrow–Debreu model of general equilibrium", ''The New Palgrave Dictionary of Economics'',  2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_A000133&edition=current&q=optimization&topicid=&result_number=20 Abstract] {{Webarchive|url=https://web.archive.org/web/20170811183914/http://www.dictionaryofeconomics.com/article?id=pde2008_A000133&edition=current&q=optimization&topicid=&result_number=20 |date=2017-08-11 }}.
* Arrow, Kenneth J., and Gérard Debreu (1954). "Existence of an Equilibrium for a Competitive Economy", ''Econometrica'' 22(3), pp. [https://www.jstor.org/pss/1907353 265]-290.</ref> More concretely, many problems are amenable to [[Mathematical analysis|analytical]] (formulaic) solution. Many others may be sufficiently complex to require [[numerical methods]] of solution, aided by software.<ref name="Schmedders"/> Still others are complex but tractable enough to allow [[computational economics|computable methods]] of solution, in particular [[computable general equilibrium]] models for the entire economy.<ref>* [[Herbert Scarf|Scarf, Herbert E.]] (2008). "computation of general equilibria", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000573&q=computational%20economics&topicid=&result_number=3 Abstract.] {{Webarchive|url=https://web.archive.org/web/20090523073724/http://www.dictionaryofeconomics.com/article?id=pde2008_C000573&q=computational%20economics&topicid=&result_number=3 |date=2009-05-23 }}
<br/>&nbsp;
* Kubler, Felix (2008). "computation of general equilibria (new developments)", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000564&q=computational%20economics&topicid=&result_number=2 Abstract.] {{Webarchive|url=https://web.archive.org/web/20170811183202/http://www.dictionaryofeconomics.com/article?id=pde2008_C000573&q=computational%20economics&topicid=&result_number=3 |date=2017-08-11 }}</ref>

Linear and nonlinear programming have profoundly affected microeconomics, which had earlier considered only equality constraints.<ref>Nicola, p. 133</ref> Many of the mathematical economists who received Nobel Prizes in Economics had conducted notable research using linear programming: [[Leonid Kantorovich]], [[Leonid Hurwicz]], [[Tjalling Koopmans]], [[Kenneth J. Arrow]], [[Robert Dorfman]], [[Paul Samuelson]] and [[Robert Solow]].<ref>Dorfman, Robert, Paul A. Samuelson, and Robert M. Solow (1958). ''Linear Programming and Economic Analysis''. McGraw–Hill. Chapter-preview [https://books.google.com/books?id=k5_vzaCNQP4C&pg=PP11 links.] {{Webarchive|url=https://web.archive.org/web/20230701105252/https://books.google.com/books?id=k5_vzaCNQP4C&pg=PP11 |date=2023-07-01 }}</ref> Both Kantorovich and Koopmans acknowledged that [[George B. Dantzig]] deserved to share their Nobel Prize for linear programming. Economists who conducted research in nonlinear programming also have won the  Nobel prize, notably [[Ragnar Frisch]] in addition to Kantorovich, Hurwicz, Koopmans, Arrow, and Samuelson.

====Linear optimization====
{{Main|Linear programming|Simplex algorithm}}

[[Linear programming]] was developed to aid the allocation of resources in firms and in industries during the 1930s in Russia and during the 1940s in the United States. During the [[Berlin airlift|Berlin airlift (1948)]], linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet  blockade.<ref>M. Padberg, ''Linear Optimization and Extensions'', Second Edition, Springer-Verlag, 1999.</ref><ref>Dantzig, George B. ([1987] 2008). "linear programming", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_L000106&edition=current&q=Linear%20programming&topicid=&result_number=1 Abstract] {{Webarchive|url=https://web.archive.org/web/20170811222114/http://www.dictionaryofeconomics.com/article?id=pde2008_L000106&edition=current&q=Linear%20programming&topicid=&result_number=1 |date=2017-08-11 }}.</ref>

====Nonlinear programming====
{{See also|Nonlinear programming|Lagrangian multiplier|Karush–Kuhn–Tucker conditions|Shadow price}}

Extensions to [[nonlinear programming|nonlinear optimization with inequality constraints]] were achieved in 1951 by [[Albert W. Tucker]] and [[Harold Kuhn]], who considered the nonlinear [[optimization problem]]:
:Minimize <math>f(x)</math> subject to <math>g_i(x) \leq 0</math> and <math>h_j(x) = 0</math> where
:<math>f(\cdot)</math> is the [[function (mathematics)|function]] to be minimized
:<math>g_i(\cdot)</math> are the functions of the <math>m</math> ''inequality [[Constraint (mathematics)|constraints]]'' where <math>i = 1, \dots, m</math>
:<math>h_j(\cdot)</math> are the functions of the <math>l</math> equality constraints where <math>j = 1, \dots, l</math>.

In allowing inequality constraints, the [[Karush–Kuhn–Tucker conditions|Kuhn–Tucker approach]] generalized the classic method of [[Lagrange multipliers]], which (until then) had  allowed only equality constraints.<ref>* Intriligator, Michael D. (2008). "nonlinear programming", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_N000083&edition=current&q=non-linear%20programming&topicid=&result_number=3 TOC] {{Webarchive|url=https://web.archive.org/web/20160304052812/http://www.dictionaryofeconomics.com/article?id=pde2008_N000083&edition=current&q=non-linear%20programming&topicid=&result_number=3 |date=2016-03-04 }}.
* Blume, Lawrence E. (2008). "convex programming", ''The New Palgrave Dictionary of Economics'', 2nd Edition.
[http://www.dictionaryofeconomics.com/article?id=pde2008_C000348&edition=current&q=optimization&topicid=&result_number=4 Abstract] {{Webarchive|url=https://web.archive.org/web/20171018182532/http://www.dictionaryofeconomics.com/article?id=pde2008_C000348&edition=current&q=optimization&topicid=&result_number=4 |date=2017-10-18 }}.
* {{cite conference| first1 = H. W.| last1 = Kuhn|author-link1=Harold W. Kuhn| first2 = A. W.| last2 = Tucker|author-link2=Albert W. Tucker| book-title = Proceedings of 2nd Berkeley Symposium| pages = 481–492| title=Nonlinear programming| publisher = University of California Press| year = 1951| location = Berkeley}}</ref>  The Kuhn–Tucker approach inspired further research on Lagrangian duality, including the treatment of inequality constraints.<ref>
* {{cite book| last = Bertsekas| first = Dimitri P.| author-link = Dimitri P. Bertsekas| title = Nonlinear Programming| edition = Second| publisher = Athena Scientific| year = 1999| location = Cambridge, Massachusetts.| isbn = 978-1-886529-00-7}}
* {{springer | id=L/l057190 | title=Lagrange multipliers| first=I.B. | last=Vapnyarskii }}.
* {{cite book|last=Lasdon|first=Leon&nbsp;S.|title=Optimization theory for large systems|publisher=The Macmillan Company|series=Macmillan series in operations research|location=New York|year=1970|pages=xi+523|mr=337317}}
* {{cite book|last=Lasdon|first=Leon&nbsp;S.|title=Optimization theory for large systems|publisher=Dover Publications, Inc.|location=Mineola, New York|year=2002|edition=reprint of the 1970 Macmillan|pages=xiii+523|mr=1888251}}
* {{cite book|last1=Hiriart-Urruty|first1=Jean-Baptiste|last2=Lemaréchal|first2=Claude|chapter=XII&nbsp;Abstract duality for practitioners|title=Convex analysis and minimization algorithms, Volume&nbsp;II: Advanced theory and bundle methods | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | volume=306 | publisher=Springer-Verlag | location=Berlin|year=1993|pages=136–193 (and Bibliographical comments on pp.&nbsp;334–335) | isbn=978-3-540-56852-0|mr=1295240|author-link2=Claude Lemaréchal}}</ref><ref name="Lemaréchal 2001 112–156">{{cite book | last=Lemaréchal | first=Claude| chapter=Lagrangian relaxation | pages=112–156 | title=Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May&nbsp;15–19,&nbsp;2000|editor=Michael Jünger |editor2=Denis Naddef | series=Lecture Notes in Computer Science|volume=2241|publisher=Springer-Verlag | location=Berlin | year=2001 | isbn=978-3-540-42877-0 | mr=1900016 | doi=10.1007/3-540-45586-8_4 | s2cid=9048698| author-link=Claude Lemaréchal}}</ref> The duality theory of nonlinear programming is particularly satisfactory when applied to [[convex minimization]] problems, which enjoy the [[convex analysis|convex-analytic]] [[Legendre transformation|duality theory]] of [[Werner Fenchel|Fenchel]] and [[R. Tyrrell Rockafellar|Rockafellar]]; this convex duality is particularly strong for [[polyhedral convex function]]s, such as those arising in [[linear programming]]. Lagrangian duality and convex analysis are used daily in [[operations research]], in the scheduling of power plants, the planning of production schedules for factories, and the routing of airlines (routes, flights, planes, crews).<ref name="Lemaréchal 2001 112–156"/>

====Variational calculus and optimal control====
{{See also|Calculus of variations|Optimal control|Dynamic programming}}

''Economic dynamics'' allows for changes in economic variables over time, including in [[dynamic systems]].  The problem of finding optimal functions for such changes is studied in [[variational calculus]] and in [[optimal control theory]].  Before the Second World War, [[Frank P. Ramsey|Frank Ramsey]] and [[Harold Hotelling]] used the calculus of variations to that end.

Following [[Richard Bellman]]'s work on dynamic programming and the 1962 English translation of L. [[Pontryagin]] ''et al''.'s earlier work,<ref>{{cite book|last1=Pontryagin|first1=L. S.|author2=Boltyanski, V. G., Gamkrelidze, R. V., Mischenko, E. F.|url=https://books.google.com/books?id=kwzq0F4cBVAC|title=The Mathematical Theory of Optimal Processes|publisher=Wiley|location=New York|year=1962|isbn=9782881240775|access-date=2015-06-27|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701105250/https://books.google.com/books?id=kwzq0F4cBVAC|url-status=live}}</ref> optimal control theory was used more extensively in economics in addressing dynamic problems, especially as to [[economic growth]] equilibrium and stability of economic systems,<ref>* [[Mikhail Zelikin|Zelikin, M. I.]] ([1987] 2008). "Pontryagin's principle of optimality",  ''The New Palgrave Dictionary of Economics'', 2nd Edition. Preview [http://www.dictionaryofeconomics.com/search_results?q=%22Pontryagin%27s+principle+of+optimality%22&edition=current&button_search=GO link] {{Webarchive|url=https://web.archive.org/web/20170811183438/http://www.dictionaryofeconomics.com/search_results?q=%22Pontryagin%27s+principle+of+optimality%22&edition=current&button_search=GO |date=2017-08-11 }}.
* Martos, Béla (1987). "control and coordination of economic activity",  ''The New Palgrave: A Dictionary of Economics''.  Description [http://www.dictionaryofeconomics.com/search_results?q=control+coordination+Martos+&field=content&edition=all&topicid= link] {{Webarchive|url=https://web.archive.org/web/20160306035519/http://www.dictionaryofeconomics.com/search_results?q=control+coordination+Martos+&field=content&edition=all&topicid= |date=2016-03-06 }}.
* Brock, W. A. (1987). "optimal control and economic dynamics", ''The New Palgrave: A Dictionary of Economics''. [http://www.dictionaryofeconomics.com/article?id=pde1987_X001613 Outline] {{Webarchive|url=https://web.archive.org/web/20170811183836/http://www.dictionaryofeconomics.com/article?id=pde1987_X001613 |date=2017-08-11 }}.
* {{cite book|editor-last=Shell|editor-first=K.|title=Essays on the Theory of Optimal Economic Growth|publisher=The MIT Press|location=Cambridge, Massachusetts|year=1967|isbn=978-0-262-19036-7|url=https://archive.org/details/essaysontheoryof00karl}}]</ref> of which a textbook example is [[Dynamic programming#Optimal consumption and saving|optimal consumption and saving]].<ref>[[Nancy Stokey|Stokey, Nancy L.]] and [[Robert E. Lucas]] with [[Edward Prescott]] (1989). ''Recursive Methods in Economic Dynamics'', Harvard University Press, chapter 5. [http://www.hup.harvard.edu/catalog.php?isbn=9780674750968 Desecription] {{Webarchive|url=https://web.archive.org/web/20170811222354/http://www.hup.harvard.edu/catalog.php?isbn=9780674750968 |date=2017-08-11 }} and chapter-preview [https://books.google.com/books?id=tWYo0QolyLAC&q=f%3Dfalse&pg=PR11 links] {{Webarchive|url=https://web.archive.org/web/20230701105309/https://books.google.com/books?id=tWYo0QolyLAC&q=f%3Dfalse&pg=PR11 |date=2023-07-01 }}.</ref> A crucial distinction is between deterministic and stochastic control models.<ref>Malliaris, A.G. (2008). "stochastic optimal control", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_S000269&edition=&field=keyword&q=Taylor's%20th&topicid=&result_number=1 Abstract] {{Webarchive|url=https://web.archive.org/web/20171018182459/http://www.dictionaryofeconomics.com/article?id=pde2008_S000269&edition=&field=keyword&q=Taylor%27s%20th&topicid=&result_number=1 |date=2017-10-18 }}.</ref>  Other applications of optimal control theory include those in finance, inventories, and production for example.<ref>* {{cite book|last1=Arrow|first1=K. J.|author2=Kurz, M.|title=Public Investment, the Rate of Return, and Optimal Fiscal Policy|url=https://archive.org/details/publicinvestment0000arro|url-access=registration|publisher=The Johns Hopkins Press|location=Baltimore, Maryland|year=1970|isbn=978-0-8018-1124-1}} [http://md1.csa.com/partners/viewrecord.php?requester=gs&collection=ENV&recid=7107596&q=&uid=788819967&setcookie=yes Abstract.] {{webarchive|url=https://web.archive.org/web/20130309003925/http://md1.csa.com/partners/viewrecord.php?requester=gs&collection=ENV&recid=7107596&q=&uid=788819967&setcookie=yes |date=2013-03-09 }}
* {{cite book|last1=Sethi|first1=S. P.|last2=Thompson|first2=G. L.|author-link2=Gerald L. Thompson|title=Optimal Control Theory: Applications to Management Science and Economics, Second Edition|publisher=Springer|location=New York|year=2000|isbn=978-0-7923-8608-7 }} Scroll to chapter-preview [https://books.google.com/books?id=gLMmaLYRy4QC links.] {{Webarchive|url=https://web.archive.org/web/20230701105253/https://books.google.com/books?id=gLMmaLYRy4QC |date=2023-07-01 }}</ref>

====Functional analysis====
{{See also|Functional analysis|Convex set|Supporting hyperplane|Hahn–Banach theorem|Fixed point theorem|Dual space|}}

It was in the course of proving of the existence of an optimal equilibrium in his 1937 model of [[economic growth]] that [[John von Neumann]] introduced [[functional analysis|functional analytic]] methods to include [[topology]] in economic theory, in particular,  [[fixed point theorem|fixed-point theory]] through his generalization of [[Brouwer's fixed-point theorem]].<ref name="DebreuNeumann"/><ref name="Neumann1937"/><ref>Andrew McLennan, 2008. "fixed point theorems", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_F000135&edition=current&q= Abstract] {{Webarchive|url=https://web.archive.org/web/20160306100148/http://www.dictionaryofeconomics.com/article?id=pde2008_F000135&edition=current&q= |date=2016-03-06 }}.</ref> Following von Neumann's program, [[Kenneth Arrow]] and [[Gérard Debreu]] formulated abstract models of economic equilibria using [[convex set]]s  and fixed–point theory.  In introducing the [[Arrow–Debreu model]] in 1954, they proved the existence (but not the uniqueness) of an equilibrium and also proved that every Walras equilibrium is [[Pareto efficient]]; in general, equilibria need not be unique.<ref>{{cite book|last=Weintraub|first=E. Roy|author-link=E. Roy Weintraub|pages=107–109|chapter=General Equilibrium Theory|title=Modern Economic Thought|editor=Weintraub, Sidney|publisher=University of Pennsylvania Press|year=1977|isbn=978-0-8122-7712-8|chapter-url=https://books.google.com/books?id=JDqAAAAAIAAJ|access-date=2020-05-28|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701105752/https://books.google.com/books?id=JDqAAAAAIAAJ|url-status=live}}
* {{cite journal|last1=Arrow|first1=Kenneth J.|author-link1=Kenneth Arrow|last2=Debreu|first2=Gérard|author-link2=Gérard Debreu|year=1954|title=Existence of an equilibrium for a competitive economy|journal=[[Econometrica]]|volume=22|pages=265–290|issn=0012-9682|doi=10.2307/1907353|jstor=1907353|issue=3}}</ref> In their models, the ("primal") vector space represented ''quantities'' while the [[Dual space|"dual" vector space]] represented ''prices''.<ref name="LK08" >Kantorovich, Leonid, and Victor Polterovich (2008).  "Functional analysis", in S. Durlauf and L. Blume, ed., ''The New Palgrave Dictionary of Economics'', 2nd Edition.  [http://www.dictionaryofeconomics.com/article?id=pde2008_F000236 Abstract.] {{Webarchive|url=https://web.archive.org/web/20160303171917/http://www.dictionaryofeconomics.com/article?id=pde2008_F000236 |date=2016-03-03 }}, ed., Palgrave Macmillan.</ref>

In Russia, the mathematician [[Leonid Kantorovich]] developed economic models in [[Riesz space|partially ordered vector space]]s, that emphasized the duality between quantities and prices.<ref>{{cite book|first=L.&nbsp;V|last=Kantorovich|author-link=Leonid Kantorovich|chapter="My journey in science (supposed report to the Moscow Mathematical Society)" [expanding ''Russian Math. Surveys''&nbsp;42 (1987), no.&nbsp;2, pp.&nbsp;233–270]|pages=8–45|mr=898626|editor=Lev J. Leifman |title=Functional analysis, optimization, and mathematical economics: A collection of papers dedicated to the memory of Leonid Vitalʹevich Kantorovich|publisher=The Clarendon Press, Oxford University Press|location=New York|year=1990|isbn=978-0-19-505729-4}}
</ref> Kantorovich renamed ''prices'' as "objectively determined valuations" <!-- "objective–conditioned estimates", Polyak --> which were abbreviated in Russian as "o.&nbsp;o.&nbsp;o.", alluding to the difficulty of discussing prices in the Soviet Union.<ref name="LK08"/><ref>Page&nbsp;406: {{Cite journal|last=Polyak|first=B.&nbsp;T.|author-link=Boris T. Polyak|title=History of mathematical programming in the USSR: Analyzing the phenomenon (Chapter&nbsp;3 The pioneer: L.&nbsp;V.&nbsp;Kantorovich, 1912–1986, pp.&nbsp;405–407)|issue=ISMP&nbsp;2000, Part&nbsp;1 (Atlanta,&nbsp;GA), number 3 |journal=Mathematical Programming |series=Series&nbsp;'''B'''|volume=91|year=2002|pages=401–416|doi=10.1007/s101070100258|mr=1888984|s2cid=13089965 }}</ref><ref>{{cite web|title=Leonid Vitaliyevich Kantorovich&nbsp;— Prize Lecture ("Mathematics in economics: Achievements, difficulties, perspectives")|work=Nobelprize.org|access-date=12 Dec 2010|url=http://nobelprize.org/nobel_prizes/economics/laureates/1975/kantorovich-lecture.html|archive-date=14 December 2010|archive-url=https://web.archive.org/web/20101214230226/http://nobelprize.org/nobel_prizes/economics/laureates/1975/kantorovich-lecture.html|url-status=live}}</ref>

Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly in clarifying the role of prices as [[normal vector]]s to a [[supporting hyperplane|hyperplane supporting]] a convex set, representing production or consumption possibilities. However, problems of describing optimization over time or under uncertainty require the use of infinite–dimensional function spaces, because agents are choosing among functions or [[stochastic process]]es.<ref name="LK08"/><ref>{{cite book|last1=Aliprantis |first1=Charalambos&nbsp;D.|author-link1=Charalambos D. Aliprantis|last2=Brown|first2=Donald&nbsp;J. |last3=Burkinshaw|first3=Owen|title=Existence and optimality of competitive equilibria|publisher=Springer–Verlag|location=Berlin |year=1990|pages=xii+284|isbn=978-3-540-52866-1|mr=1075992}}</ref><ref>[[R. Tyrrell Rockafellar|Rockafellar, R. Tyrrell]]. ''Conjugate duality and optimization''. Lectures given at the Johns Hopkins University, Baltimore, Maryland, June, 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. vi+74 pp.</ref><ref>[[Lester G. Telser]] and [[Robert L. Graves]] ''Functional Analysis in Mathematical Economics: Optimization Over Infinite Horizons'' 1972. University of Chicago Press, 1972, {{isbn|978-0-226-79190-6}}.</ref>

===Differential decline and rise===
{{See also|Global analysis|Baire category|Sard's lemma}}

[[John von Neumann]]'s work on [[functional analysis]] and [[topology]] broke new ground in mathematics and economic theory.<ref name="Neumann1937"/><ref name="NeumannMorgenstern">Neumann, John von, and Oskar Morgenstern (1944) ''[[Theory of Games and Economic Behavior]]'', Princeton.</ref> It also left advanced mathematical economics with fewer applications of differential calculus. In particular, general equilibrium theorists used [[general topology]], [[convex geometry]], and [[mathematical optimization|optimization theory]] more than differential calculus, because the approach of differential calculus had failed to establish the existence of an equilibrium.

However, the decline of differential calculus should not be exaggerated, because differential calculus has always been used in graduate training and in applications. Moreover, differential calculus has returned to the highest levels of mathematical economics, [[general equilibrium theory]] (GET), as practiced by the "[[jet set|GET-set]]" (the humorous designation due to [[Jacques H. Drèze]]). In the 1960s and 1970s, however, [[Gérard Debreu]] and [[Stephen Smale]] led a revival of the use of differential calculus in mathematical economics. In particular, they were able to prove the existence of a general equilibrium, where earlier writers had failed, because of their novel mathematics: [[Baire category]] from [[general topology]] and [[Sard's lemma]] from [[differential topology]]. Other economists associated with the use of differential analysis include Egbert Dierker, [[Andreu Mas-Colell]], and [[Yves Balasko]].<ref>{{cite book|last=Mas-Colell|first=Andreu|author-link=Andreu Mas-Colell|year=1985|title=The Theory of general economic equilibrium: A ''differentiable'' approach|series=Econometric Society monographs|publisher=Cambridge&nbsp;UP|isbn=978-0-521-26514-0|mr=1113262|issue=9}}</ref><ref>[[Yves Balasko]]. ''Foundations of the Theory of General Equilibrium'', 1988, {{isbn|0-12-076975-1}}.</ref> These advances have changed the traditional narrative of the history of mathematical economics, following von Neumann, which celebrated the abandonment of differential calculus.

===Game theory===
{{Main|Game theory}}
{{See also|Cooperative game theory|Non-cooperative game theory|John von Neumann|Theory of Games and Economic Behavior|John Forbes Nash, Jr.}}

John von Neumann, working with [[Oskar Morgenstern]] on the [[Theory of Games and Economic Behavior|theory of games]], broke new mathematical ground in 1944 by extending [[functional analysis|functional analytic]] methods related to [[convex set]]s and [[topology|topological]] [[fixed point theorem (economics)|fixed-point theory]] to economic analysis.<ref name="DebreuNeumann"/><ref name="NeumannMorgenstern"/> Their work thereby avoided the traditional [[differential calculus]], for which the [[maximum]]–operator did not apply to non-differentiable functions. Continuing von Neumann's work in [[cooperative game theory]], game theorists [[Lloyd S. Shapley]], [[Martin Shubik]], [[Hervé Moulin]], [[Nimrod Megiddo]], [[Bezalel Peleg]] influenced economic research in politics and economics. For example, research on the [[Shapley value|fair prices]] in cooperative games and [[Shapley–Shubik power index|fair values]] for [[voting theory|voting game]]s led to changed rules for voting in legislatures and for accounting for the costs in public–works projects. For example, cooperative game theory was used in designing the water distribution system of Southern Sweden and for setting rates for dedicated telephone lines in the US.

Earlier [[neoclassical economics|neoclassical]] theory had bounded only the ''range'' of bargaining outcomes and in special cases, for  example [[bilateral monopoly]] or along the [[contract curve]] of the [[Edgeworth box]].<ref>Creedy, John (2008). "Francis Ysidro (1845–1926)", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_E000041&edition=current&q=edgeworth&topicid=&result_number=3 Abstract] {{Webarchive|url=https://web.archive.org/web/20170811182927/http://www.dictionaryofeconomics.com/article?id=pde2008_E000041&edition=current&q=edgeworth&topicid=&result_number=3 |date=2017-08-11 }}.</ref>  Von Neumann and Morgenstern's results were similarly weak. Following von Neumann's program, however, [[John Forbes Nash Jr.|John Nash]] used fixed–point theory to prove conditions under which the [[Bargaining#Game theory|bargaining problem]] and [[noncooperative game]]s can generate a unique [[Nash equilibria|equilibrium]] solution.<ref>* Nash, John F., Jr. (1950). "The Bargaining Problem", ''Econometrica'', 18(2),  pp. [http://www.stern.nyu.edu/networks/Nash_The_Bargaining_Problem.pdf 155-162] {{Webarchive|url=https://web.archive.org/web/20160304131554/http://www.stern.nyu.edu/networks/Nash_The_Bargaining_Problem.pdf |date=2016-03-04 }}.
* Serrano, Roberto (2008). "bargaining", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_B000073&edition=current&q=bargaining&topicid=&result_number=1 Abstract] {{Webarchive|url=https://web.archive.org/web/20170811183937/http://www.dictionaryofeconomics.com/article?id=pde2008_B000073&edition=current&q=bargaining&topicid=&result_number=1 |date=2017-08-11 }}.</ref> Noncooperative game theory has been adopted as a fundamental aspect of  [[experimental economics]],<ref>* [[Vernon L. Smith|Smith, Vernon L.]] (1992).  "Game Theory and Experimental Economics: Beginnings and Early Influences", in E. R. Weintraub, ed., ''Towards a History of Game Theory'', pp. [https://books.google.com/books?id=9CHY2Gozh1MC&pg=PA241 241-] {{Webarchive|url=https://web.archive.org/web/20230701105752/https://books.google.com/books?id=9CHY2Gozh1MC&pg=PA241 |date=2023-07-01 }} 282.
* _____ (2001). "Experimental Economics", ''[[International Encyclopedia of the Social & Behavioral Sciences]]'', pp.&nbsp;5100–5108. [http://www.sciencedirect.com/science/article/pii/B0080430767022324 Abstract] {{Webarchive|url=https://web.archive.org/web/20181014005510/http://www.sciencedirect.com/science/article/pii/B0080430767022324 |date=2018-10-14 }} per sect. 1.1 & 2.1.
* [[Charles R. Plott|Plott, Charles R.]], and Vernon L. Smith, ed. (2008). ''Handbook of Experimental Economics Results'', v. 1, Elsevier,  Part 4, Games,  ch. 45-66 preview [https://archive.today/20120910050854/http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574072207X00015&_cid=277334&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=49f8b6d5e3024eac39ed5fad351fe568 links].
* Shubik, Martin (2002).  "Game Theory and Experimental Gaming", in R. Aumann and S. Hart, ed., ''Handbook of Game Theory with Economic Applications'', Elsevier, v. 3, pp.&nbsp;2327–2351. [http://www.sciencedirect.com/science/article/pii/S1574000502030254 Abstract] {{Webarchive|url=https://web.archive.org/web/20181107105923/http://www.sciencedirect.com/science/article/pii/S1574000502030254 |date=2018-11-07 }}.</ref> [[behavioral#Behavioral game theory|behavioral economics]],<ref>From ''The New Palgrave Dictionary of Economics'' (2008), 2nd Edition: 
* [[Faruk Gul|Gul, Faruk]].  "behavioural economics and game theory." [http://www.dictionaryofeconomics.com/article?id=pde2008_G000210&q=Behavioral%20economics%20&topicid=&result_number=2 Abstract.] {{Webarchive|url=https://web.archive.org/web/20170807112808/http://www.dictionaryofeconomics.com/article?id=pde2008_G000210&q=Behavioral%20economics%20&topicid=&result_number=2 |date=2017-08-07 }}
* [[Colin F. Camerer|Camerer, Colin F.]] "behavioral game theory." [http://www.dictionaryofeconomics.com/article?id=pde2008_B000302&q=Behavioral%20economics%20&topicid=&result_number=13 Abstract.] {{Webarchive|url=https://web.archive.org/web/20111123034346/http://www.dictionaryofeconomics.com/article?id=pde2008_B000302&q=Behavioral%20economics%20&topicid=&result_number=13 |date=2011-11-23 }}</ref> [[information economics]],<ref>* Rasmusen, Eric (2007). ''Games and Information'', 4th ed. [http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP001009.html Description] {{Webarchive|url=https://web.archive.org/web/20170624054901/http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP001009.html |date=2017-06-24 }} and chapter-preview [https://books.google.com/books?id=5XEMuJwnBmUC&dq=gbs_atb&pg=PR5 links.] {{Webarchive|url=https://web.archive.org/web/20230701105816/https://books.google.com/books?id=5XEMuJwnBmUC&dq=gbs_atb&pg=PR5 |date=2023-07-01 }}
* Aumann,  R., and S. Hart, ed. (1992, 2002). ''Handbook of Game Theory with Economic Applications'' v. 1, links at [http://www.sciencedirect.com/science/handbooks/15740005/1 ch. 3-6] {{Webarchive|url=https://web.archive.org/web/20170816004452/http://www.sciencedirect.com/science/handbooks/15740005/1 |date=2017-08-16 }} and v. 3, [http://www.sciencedirect.com/science/article/pii/S1574000502030060 ch. 43] {{Webarchive|url=https://web.archive.org/web/20181014004759/http://www.sciencedirect.com/science/article/pii/S1574000502030060 |date=2018-10-14 }}.</ref> [[industrial organization]],<ref>* [[Jean Tirole|Tirole, Jean]] (1988).  ''The Theory of Industrial Organization'', MIT Press. [https://web.archive.org/web/20011122194726/http://mitpress.mit.edu/catalog/item/default.asp?tid=8224&ttype=2 Description] and chapter-preview links, pp. [https://archive.org/details/theoryofindustri00jean vii-ix],  "General Organization", pp. [https://archive.org/details/theoryofindustri00jean/page/5 <!-- quote=find. --> 5-6], and "Non-Cooperative Game Theory: A User's Guide Manual,' " ch. 11, pp. [https://archive.org/details/theoryofindustri00jean/page/423 <!-- quote=find. --> 423-59].
* Bagwell, Kyle, and Asher Wolinsky (2002). "Game theory and Industrial Organization", ch. 49, ''Handbook of Game Theory with Economic Applications'',
v. 3, pp. [http://www.sciencedirect.com/science/article/pii/S1574000502030126 1851] {{Webarchive|url=https://web.archive.org/web/20160102040935/http://www.sciencedirect.com/science/article/pii/S1574000502030126 |date=2016-01-02 }}-1895.</ref> and  [[political economy]].<ref>
* Shubik, Martin (1981). "Game Theory Models and  Methods in Political Economy", in ''Handbook of Mathematical Economics'', v. 1, pp. 285–330. {{doi|10.1016/S1573-4382(81)01011-4}}.</ref> It has also given rise to the subject of [[mechanism design]] (sometimes called reverse game theory), which has private and [[public policy|public-policy]] applications as to ways of improving [[economic efficiency]] through incentives for information sharing.<ref name="Abstract">* ''The New Palgrave Dictionary of Economics'' (2008), 2nd Edition: <br/>&nbsp;&nbsp;&nbsp;&nbsp; [[Roger B. Myerson|Myerson, Roger B.]] "mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 Abstract.] {{Webarchive|url=https://web.archive.org/web/20111123042038/http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 |date=2011-11-23 }}   <br/>&nbsp;&nbsp;&nbsp;&nbsp; _____. "revelation principle." [http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 Abstract.] {{Webarchive|url=https://web.archive.org/web/20130516143359/http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 |date=2013-05-16 }}<br/>&nbsp;&nbsp;&nbsp;&nbsp; Sandholm, Tuomas. "computing in mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 Abstract.] {{Webarchive|url=https://web.archive.org/web/20111123042038/http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 |date=2011-11-23 }}
* Nisan, Noam, and Amir Ronen (2001). "Algorithmic Mechanism Design", ''Games and Economic Behavior'',  35(1-2), pp. [https://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf 166–196] {{Webarchive|url=https://web.archive.org/web/20181014005314/http://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf |date=2018-10-14 }}.
* Nisan, Noam, ''et al''., ed. (2007). ''Algorithmic Game Theory'', Cambridge University Press. [http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 Description] {{webarchive|url=https://web.archive.org/web/20120505140924/http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 |date=2012-05-05 }}.</ref>

In 1994, Nash, [[John Harsanyi]], and [[Reinhard Selten]] received the [[Nobel Memorial Prize in Economic Sciences]] their work on non–cooperative games. Harsanyi and Selten were awarded for their work on [[repeated game]]s. Later work extended their results to [[computational economics|computational methods]] of modeling.<ref name="COMP&GT">* [[Joseph Y. Halpern|Halpern, Joseph Y.]] (2008). "computer science and game theory", ''The New Palgrave Dictionary of Economics'', 2nd Edition.  [http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&q=&topicid=&result_number=1 Abstract] {{Webarchive|url=https://web.archive.org/web/20171105200352/http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&q=&topicid=&result_number=1 |date=2017-11-05 }}.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
* Shoham, Yoav (2008). "Computer Science and Game Theory", ''Communications of the ACM'',  51(8), pp.
[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] {{Webarchive|url=https://web.archive.org/web/20120426005917/http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf |date=2012-04-26 }}.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
* [[Alvin E. Roth|Roth, Alvin E.]] (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics", ''Econometrica'', 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref>

===Agent-based computational economics===
{{main|Agent-based computational economics}}
Agent-based computational economics (ACE) as a named field is relatively recent, dating from about the 1990s as to published work. It studies economic processes, including whole [[economy|economies]], as [[dynamic systems]] of interacting [[Agent (economics)|agents]] over time. As such, it falls in the [[paradigm]] of [[complex adaptive system]]s.<ref>* Kirman, Alan (2008).  "economy as a complex system", ''The New Palgrave Dictionary of Economics '', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_E000246&edition=current&q=Computational%20economics&topicid=&result_number=20 Abstract] {{Webarchive|url=https://web.archive.org/web/20170811150018/http://www.dictionaryofeconomics.com/article?id=pde2008_E000246&edition=current&q=Computational%20economics&topicid=&result_number=20 |date=2017-08-11 }}.
* [[Leigh Tesfatsion|Tesfatsion, Leigh]] (2003). "Agent-based Computational Economics: Modeling Economies as Complex Adaptive Systems", ''Information Sciences'', 149(4), pp. [https://web.archive.org/web/20120426000037/http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf 262-268].</ref> In corresponding [[agent-based model]]s, agents are not real people but "computational objects modeled as interacting according to rules" ... "whose micro-level interactions create emergent patterns"   in space and time.<ref>Scott E. Page (2008), "agent-based models", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_A000218&edition=current&q=agent-based%20computational%20modeling&topicid=&result_number=1 Abstract] {{Webarchive|url=https://web.archive.org/web/20180210023520/https://www.springernature.com/gp/ |date=2018-02-10 }}.</ref> The rules are formulated to predict behavior and social interactions based on incentives and information. The theoretical assumption of [[mathematical optimization|mathematical ''optimization'']] by agents markets is replaced by the less restrictive postulate of agents with [[bounded rationality|''bounded'' rationality]] ''adapting'' to market forces.<ref>* [[John H. Holland|Holland, John H.]], and John H. Miller (1991). "Artificial Adaptive Agents in Economic Theory", ''American Economic Review'', 81(2),  pp. [http://www.santafe.edu/media/workingpapers/91-05-025.pdf  365-370] {{Webarchive|url=https://web.archive.org/web/20110105015853/http://www.santafe.edu/media/workingpapers/91-05-025.pdf |date=2011-01-05 }} p. 366.
*  [[W. Brian Arthur|Arthur, W. Brian]], 1994. "Inductive Reasoning and Bounded Rationality", ''American Economic Review'', 84(2), pp. [https://web.archive.org/web/20130521145936/http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf 406-411].
* [[Thomas C. Schelling|Schelling, Thomas C.]] (1978 [2006]).  ''Micromotives and Macrobehavior'', Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] {{Webarchive|url=https://web.archive.org/web/20171102093240/http://books.wwnorton.com/books/978-0-393-32946-9/ |date=2017-11-02 }},  [https://books.google.com/books?id=DenWKRgqzWMC&pg=PA1= preview] {{Webarchive|url=https://web.archive.org/web/20230701105752/https://books.google.com/books?id=DenWKRgqzWMC&pg=PA1= |date=2023-07-01 }}.
* [[Thomas J. Sargent|Sargent, Thomas J.]] (1994). ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links] {{Webarchive|url=https://web.archive.org/web/20230701105755/https://www.gale.com/databases/questia |date=2023-07-01 }}.</ref>

ACE models apply [[numerical methods]] of analysis to [[Computer simulation|computer-based simulations]] of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>*  Judd, Kenneth L. (2006). "Computationally Intensive Analyses in Economics", ''Handbook of Computational Economics'', v. 2, ch. 17, Introduction, p. 883. Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [https://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20220121012922/https://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf |date=2022-01-21 }}.<br/>&nbsp;&nbsp;•  _____ (1998). ''Numerical Methods in Economics'', MIT Press. Links to [https://web.archive.org/web/20020923144108/http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from specified initial conditions, the computational [[economic system]] is modeled as evolving over time as its constituent agents repeatedly interact with each other. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of the economy.<ref>* Tesfatsion, Leigh (2002). "Agent-Based Computational Economics: Growing Economies from the Bottom Up", ''Artificial Life'', 8(1), pp.55-82. [http://www.mitpressjournals.org/doi/abs/10.1162/106454602753694765 Abstract] {{Webarchive|url=https://web.archive.org/web/20200306001153/http://www.mitpressjournals.org/doi/abs/10.1162/106454602753694765 |date=2020-03-06 }} and pre-pub [https://web.archive.org/web/20130514143904/http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf PDF].<br/>&nbsp;&nbsp;• _____ (1997). "How Economists Can Get Alife", in W. B. Arthur, S. Durlauf, and D. Lane, eds., ''The Economy as an Evolving Complex System, II'',  pp.&nbsp;533–564. Addison-Wesley. Pre-pub [http://ageconsearch.umn.edu/bitstream/18196/1/er37.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20120415135342/http://ageconsearch.umn.edu/bitstream/18196/1/er37.pdf |date=2012-04-15 }}.</ref> In contrast to other standard modeling methods, ACE events are driven solely by initial conditions, whether or not equilibria exist or are computationally tractable.   ACE modeling, however, includes agent adaptation, autonomy, and learning.<ref>Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] {{Webarchive|url=https://web.archive.org/web/20180809120427/https://www.sciencedirect.com/science/article/pii/S1574002105020162 |date=2018-08-09 }} and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20170811162709/http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf |date=2017-08-11 }}.</ref> It has a similarity to, and overlap with, [[game theory]] as an agent-based method for modeling social interactions.<ref name="COMP&GT"/> Other dimensions of the approach include such standard economic subjects as [[competition]] and [[collaboration]],<ref>[[Robert Axelrod (political scientist)|Axelrod, Robert]] (1997). ''The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration'', Princeton. [http://press.princeton.edu/titles/6144.html Description] {{Webarchive|url=https://web.archive.org/web/20180102004530/https://press.princeton.edu/titles/6144.html |date=2018-01-02 }}, [http://press.princeton.edu/titles/6144.html#TOC contents] {{Webarchive|url=https://web.archive.org/web/20180102004530/https://press.princeton.edu/titles/6144.html#TOC |date=2018-01-02 }}, and [https://books.google.com/books?id=J0dgRGMdjmQC&pg=PR11 preview] {{Webarchive|url=https://web.archive.org/web/20230701110255/https://books.google.com/books?id=J0dgRGMdjmQC&pg=PR11 |date=2023-07-01 }}.</ref> [[market structure]] and [[industrial organization]],<ref>* Leombruni, Roberto, and Matteo Richiardi, ed. (2004), ''Industry  and Labor Dynamics: The Agent-Based Computational Economics Approach.'' World Scientific Publishing {{isbn|981-256-100-5}}. [http://www.worldscibooks.com/economics/5706.html Description] {{Webarchive|url=https://web.archive.org/web/20100727221149/http://www.worldscibooks.com/economics/5706.html |date=2010-07-27 }} and chapter-preview [https://books.google.com/books?id=P5O7A5D55nQC&pg=PR5 links] {{Webarchive|url=https://web.archive.org/web/20230701110255/https://books.google.com/books?id=P5O7A5D55nQC&pg=PR5 |date=2023-07-01 }}.
* [[Joshua M. Epstein|Epstein, Joshua M.]] (2006). "Growing Adaptive Organizations: An Agent-Based Computational Approach", in ''Generative Social Science: Studies in Agent-Based Computational Modeling'', pp. 309 - [https://books.google.com/books?id=543OS3qdxBYC&dq=false&pg=PA326] {{Webarchive|url=https://web.archive.org/web/20230701112303/https://books.google.com/books?id=543OS3qdxBYC&dq=false&pg=PA326 |date=2023-07-01 }} 344. [http://press.princeton.edu/titles/8277.html Description] {{Webarchive|url=https://web.archive.org/web/20120126180655/http://press.princeton.edu/titles/8277.html |date=2012-01-26 }} and [http://www.santafe.edu/research/working-papers/abstract/99895b6465e8b87656612f8e3570b34c/ abstract] {{Webarchive|url=https://web.archive.org/web/20161019055158/http://www.santafe.edu/research/working-papers/abstract/99895b6465e8b87656612f8e3570b34c/ |date=2016-10-19 }}.</ref> [[transaction cost]]s,<ref>Klosa, Tomas B., and [[Bart Nooteboom]], 2001. "Agent-based Computational Transaction Cost Economics",  ''Journal of Economic Dynamics and Control'' 25(3–4), pp. 503–52. [http://www.sciencedirect.com/science/article/pii/S0165188900000348 Abstract.] {{Webarchive|url=https://web.archive.org/web/20200622114131/https://www.sciencedirect.com/science/article/pii/S0165188900000348 |date=2020-06-22 }}</ref> [[welfare economics]]<ref>[[Robert Axtell|Axtell, Robert]] (2005). "The Complexity of Exchange",  ''Economic Journal'', 115(504, Features), pp. [http://econfaculty.gmu.edu/pboettke/workshop/archives/f05/Axtell.pdf F193-F210] {{Webarchive|url=https://web.archive.org/web/20170808183932/http://econfaculty.gmu.edu/pboettke/workshop/archives/f05/Axtell.pdf |date=2017-08-08 }}.</ref> and [[mechanism design]],<ref name="Abstract"/>  [[Information economics|information and uncertainty]],<ref>Sandholm, Tuomas W., and Victor R. Lesser (2001)."Leveled Commitment Contracts and Strategic Breach", ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/afs/.cs.cmu.edu/Web/People/sandholm/leveled.geb.pdf 212-270] {{Webarchive|url=https://web.archive.org/web/20201204120103/https://www.cs.cmu.edu/afs/.cs.cmu.edu/Web/People/sandholm/leveled.geb.pdf |date=2020-12-04 }}.</ref> and [[macroeconomics]].<ref>* [[David Colander|Colander, David]], [[Peter Howitt]], Alan Kirman, [[Axel Leijonhufvud]], and [[Perry Mehrling]] (2008). "Beyond DSGE Models: Toward an Empirically Based Macroeconomics", ''American Economic Review'', 98(2), pp. [https://www.jstor.org/pss/29730026 236]-240. Pre-pub [https://web.archive.org/web/20100622214223/http://www.econ.brown.edu/fac/Peter_Howitt/publication/complex%20macro6.pdf PDF].
* [[Thomas J. Sargent|Sargent, Thomas J.]] (1994). ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links] {{Webarchive|url=https://web.archive.org/web/20230701105755/https://www.gale.com/databases/questia |date=2023-07-01 }}.</ref><ref>Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16, ''Handbook of Computational Economics'', v. 2, pp.&nbsp;832–865. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] {{Webarchive|url=https://web.archive.org/web/20180809120427/https://www.sciencedirect.com/science/article/pii/S1574002105020162 |date=2018-08-09 }} and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20170811162709/http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf |date=2017-08-11 }}.</ref>

The method is said to benefit from continuing improvements in modeling techniques of [[computer science]] and increased computer capabilities.  Issues include those common to [[experimental economics]] in general<ref>[[Vernon L. Smith|Smith, Vernon L.]] (2008). "experimental economics", ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_E000277&q=experimental%20&topicid=&result_number=2 Abstract] {{Webarchive|url=https://web.archive.org/web/20120119071348/http://www.dictionaryofeconomics.com/article?id=pde2008_E000277&q=experimental%20&topicid=&result_number=2 |date=2012-01-19 }}.</ref> and by comparison<ref>[[John Duffy (economist)|Duffy, John]] (2006). "Agent-Based Models and Human Subject Experiments",  ch. 19,  ''Handbook of Computational Economics'', v.2, pp. 949–101. [http://www.sciencedirect.com/science/article/pii/S1574002105020198 Abstract] {{Webarchive|url=https://web.archive.org/web/20150924191026/http://www.sciencedirect.com/science/article/pii/S1574002105020198 |date=2015-09-24 }}.</ref> and to development of a common framework for empirical validation and resolving open questions in agent-based modeling.<ref>* Namatame, Akira, and Takao Terano (2002). "The Hare and the Tortoise: Cumulative Progress in Agent-based Simulation", in ''Agent-based Approaches in Economic and Social Complex Systems''. pp. [https://books.google.com/books?id=nTkwxVUYvfIC&pg=PA3=onepage 3-] 14, IOS Press. [http://www.iospress.nl/book/agent-based-approaches-in-economic-and-social-complex-systems-2/ Description] {{Webarchive|url=https://web.archive.org/web/20120405084745/http://www.iospress.nl/book/agent-based-approaches-in-economic-and-social-complex-systems-2/ |date=2012-04-05 }}.
* Fagiolo, Giorgio, Alessio Moneta, and Paul Windrum (2007). "A Critical Guide to Empirical Validation of Agent-Based Models in Economics: Methodologies, Procedures, and Open Problems", ''Computational Economics'', 30, pp. [https://doi.org/10.1007%2Fs10614-007-9104-4 195] {{Webarchive|url=https://web.archive.org/web/20230701112420/https://link.springer.com/article/10.1007/s10614-007-9104-4 |date=2023-07-01 }}–226.</ref> The ultimate scientific objective of the method has been described as "test[ing] theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher's work building appropriately on the work that has gone before".<ref>* Tesfatsion, Leigh (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16, ''Handbook of Computational Economics'', v. 2, [pp.&nbsp;831–880] sect. 5. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] {{Webarchive|url=https://web.archive.org/web/20180809120427/https://www.sciencedirect.com/science/article/pii/S1574002105020162 |date=2018-08-09 }} and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20170811162709/http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf |date=2017-08-11 }}.
* [[Kenneth L. Judd|Judd, Kenneth L.]] (2006). "Computationally Intensive Analyses in Economics", ''Handbook of Computational Economics'', v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [https://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20220121012922/https://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf |date=2022-01-21 }}.
* Tesfatsion, Leigh, and Kenneth L. Judd, ed. (2006). ''Handbook of Computational Economics'', v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] {{Webarchive|url=https://web.archive.org/web/20120306100156/http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description |date=2012-03-06 }} & and chapter-preview
[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref>

==Mathematicization of economics==
[[Image:Ivsrf.gif|thumb|275px|right|The surface of the [[Volatility smile]] is a 3-D surface whereby the current market implied volatility (Z-axis) for all options on the underlier is plotted against strike price and time to maturity (X & Y-axes).<ref name="Black Scholes">{{cite book|last1=Brockhaus|first1=Oliver|author2=Farkas, Michael; Ferraris, Andrew; Long, Douglas; Overhaus, Marcus|title=Equity Derivatives and Market Risk Models|publisher=Risk Books|year=2000|pages=13–17|isbn=978-1-899332-87-8|url=https://books.google.com/books?id=JGZPAAAAMAAJ|access-date=2008-08-17|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701113301/https://books.google.com/books?id=JGZPAAAAMAAJ|url-status=live}}</ref>]]

Over the course of the 20th century, articles in "core journals"<ref>{{cite journal| last = Liner| first = Gaines H.| title = Core Journals in Economics| journal = Economic Inquiry| volume = 40| issue = 1| page =140| year = 2002| doi =10.1093/ei/40.1.138}}</ref> 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."<ref>{{cite journal|last1=Stigler|first1=George J.|author-link=George Stigler|author2=Stigler, Steven J.|author3=Friedland, Claire|date=April 1995|title=The Journals of Economics|journal=[[The Journal of Political Economy]]|volume=103|issue=2|pages=331–359|issn=0022-3808|jstor=2138643|doi=10.1086/261986|s2cid=154780520}}</ref> A subjective assessment of mathematical techniques<ref>Stigler et al. reviewed journal articles in core economic journals (as defined by the authors but meaning generally non-specialist journals) throughout the 20th century.  Journal articles which at any point used geometric representation or mathematical notation were noted as using that level of mathematics as its "highest level of mathematical technique".  The authors refer to "verbal techniques" as those which conveyed the subject of the piece without notation from [[geometry]], [[algebra]] or [[calculus]].</ref> 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.<ref>Stigler et al., p. 342</ref>  A 2007 survey of ten of the top economic journals finds that only&nbsp;5.8% of the articles published in 2003 and 2004 both lacked statistical analysis of data and  lacked displayed mathematical expressions that were indexed with numbers at the margin of the page.<ref>Sutter, Daniel and Rex Pjesky. "Where Would Adam Smith Publish Today?: The Near Absence of Math-free Research in Top Journals" (May 2007). [http://econjwatch.org/issues/volume-4-number-1-may-2007] {{Webarchive|url=https://web.archive.org/web/20171010015443/https://econjwatch.org/issues/volume-4-number-1-may-2007 |date=2017-10-10 }}</ref>

==Econometrics==
{{main|Econometrics}}
Between the world wars, advances in [[mathematical statistics]] and a cadre of mathematically trained economists <!-- and  [[probability theory]] --> led to [[econometrics]], which was the name proposed for the discipline of advancing economics by using mathematics and statistics. Within economics, "econometrics" has often been used for statistical methods in economics, rather than mathematical economics. Statistical econometrics features the application of linear regression and time series analysis to economic data.

[[Ragnar Anton Kittil Frisch|Ragnar Frisch]] coined the word "econometrics" and helped to found both the [[Econometric Society]] in 1930 and the journal ''[[Econometrica]]'' in 1933.<ref name="Arrow bio of Frisch">{{cite journal|last=Arrow|first=Kenneth J.|author-link=Kenneth Arrow|date=April 1960|title=The Work of Ragnar Frisch, Econometrician|journal=[[Econometrica]]|volume=28|issue=2|pages=175–192|jstor=1907716|issn=0012-9682|doi=10.2307/1907716}}</ref><ref>{{cite journal|last=Bjerkholt|first=Olav|date=July 1995|title=Ragnar Frisch, Editor of Econometrica 1933-1954|journal=[[Econometrica]]|volume=63|issue=4|pages=755–765|issn=0012-9682|jstor=1906940|doi=10.2307/2171799}}</ref>  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.<ref>{{cite journal|last=Lange|first=Oskar|year=1945|title=The Scope and Method of Economics|journal=Review of Economic Studies|volume=13|issue=1|pages=19–32|issn=0034-6527|jstor=2296113|doi=10.2307/2296113|s2cid=4140287 }}</ref>  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.<ref>{{cite journal|last=Aldrich|first=John|date=January 1989|title=Autonomy|journal=Oxford Economic Papers|volume=41|issue=1, History and Methodology of Econometrics |pages=15–34|issn=0030-7653 |jstor=2663180|doi=10.1093/oxfordjournals.oep.a041889}}</ref>

The roots of modern econometrics can be traced to the American economist [[Henry Ludwell Moore|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 over-correction 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.<ref>{{cite book|last=Epstein|first=Roy J.|title=A History of Econometrics|publisher=North-Holland|series=Contributions to Economic Analysis|pages=13–19|isbn=978-0-444-70267-8|oclc=230844893|year=1987}}</ref>

==Application==
[[Image:Islm.svg|thumb|300 px|The [[IS/LM model]] is a [[Keynesian economics|Keynesian]] [[macroeconomics|macroeconomic]] model designed to make predictions about the intersection of "real" economic activity (e.g. spending, [[Measures of national income and output|income]], savings rates) and decisions made in the financial markets ([[Money supply]] and [[Liquidity preference]]).  The model is no longer widely taught at the graduate level but is common in undergraduate macroeconomics courses.<ref>{{cite journal|last=Colander|first=David C.|author-link=David Colander|year=2004|title=The Strange Persistence of the IS-LM Model|journal=History of Political Economy|volume=36|issue=Annual Supplement|doi=10.1215/00182702-36-Suppl_1-305|pages=305–322|issn=0018-2702 |citeseerx=10.1.1.692.6446|s2cid=6705939}}</ref>]]
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 (mathematics)|matrix algebra]] in economic analysis in order to make powerful claims that would be more difficult without such mathematical tools. These tools are prerequisites for formal study, not only in mathematical economics but in contemporary economic theory in general. 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.<ref>{{cite journal|last=Brems|first=Hans|date=October 1975|title=Marshall on Mathematics|journal=Journal of Law and Economics|volume=18|issue=2|pages=583–585|issn=0022-2186|jstor=725308|doi=10.1086/466825|s2cid=154881432 }}</ref>

Economics has become increasingly dependent upon 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 both economics and finance require strong undergraduate preparation in mathematics for admission and, for this reason, attract an increasingly high number of [[mathematician]]s. [[applied mathematics|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.<ref name="Dow"/>

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 may be classified as [[stochastic process|stochastic]] or deterministic and as 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.<ref>{{cite book|last1=Frigg|first1=R.|author2=Hartman, S.|title=Models in Science|editor=Edward N. Zalta|publisher=The Metaphysics Research Lab|location=Stanford, California|date=February 27, 2006|issn=1095-5054|series=Stanford Encyclopedia of Philosophy|url=http://plato.stanford.edu/entries/models-science/#OntWhaMod|access-date=2008-08-16|archive-date=2007-06-09|archive-url=https://web.archive.org/web/20070609102029/http://plato.stanford.edu/entries/models-science/#OntWhaMod|url-status=live}}</ref>

* [[Stochastic|Stochastic models]] are formulated using [[stochastic process]]es. They model economically observable values over time. Most of [[econometrics]] is based on [[statistics]] to formulate and test [[hypotheses]] about these processes or estimate parameters for them. Between the World Wars, [[Herman Wold]] developed a [[Wold decomposition|representation]] of stationary stochastic processes in terms of [[autoregressive]] models and a determinist trend. Wold and [[Jan Tinbergen]] applied time-series analysis to economic data. Contemporary research on [[time series]] [[statistics]] consider additional formulations of stationary processes, such as  [[autoregressive moving average model]]s. More general models include  [[autoregressive conditional heteroskedasticity]] (ARCH) models and generalized ARCH ([[GARCH]]) models.
* [[Deterministic system (mathematics)|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 [[Function (mathematics)|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 economics|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.
<!--maybe later
===Early Models of competition===
====Perfect competition====
====Monopoly====
====Duopoly====
===Models of equilibrium===
====Walrassian equilibrium====
====IS/LM====
===Models of choice===
====Game theory and oligopoly====
====Public choice====
===The theory of the firm===
-->
<!--More sections commented out
==Relationship to the discipline as a whole==
===Mathematical economics as methodology===
===Mathematical finance===
===Views from inside and outside the profession===
-->

== Example: The effect of a corporate tax cut on wages ==
The great appeal of mathematical economics is that it brings a degree of rigor to economic thinking, particularly around charged political topics. For example, during the discussion of the efficacy of a [[Tax Cuts and Jobs Act of 2017|corporate tax cut]] for increasing the wages of workers, a simple mathematical model proved beneficial to understanding the issues at hand.

As an intellectual exercise, the following problem was posed by [[Greg Mankiw|Prof. Greg Mankiw]] of [[Harvard University]]:<ref>{{Cite web|url=http://gregmankiw.blogspot.com/2017/10/an-exercise-for-my-readers.html|title=Greg Mankiw's Blog: An Exercise for My Readers|language=en|access-date=2019-08-07|archive-date=2019-08-07|archive-url=https://web.archive.org/web/20190807055309/http://gregmankiw.blogspot.com/2017/10/an-exercise-for-my-readers.html|url-status=live}}</ref><blockquote>'''''An open economy has the production function <math display="inline">y = f(k)</math>, where <math display="inline">y</math> is output per worker and <math display="inline">k</math> is capital per worker. The capital stock adjusts so that the after-tax marginal product of capital equals the exogenously given world interest rate <math display="inline">r</math>...How much will the tax cut increase wages?'''''</blockquote>To answer this question, we follow [[John H. Cochrane]] of the [[Hoover Institution]].<ref>{{Cite web|url=https://johnhcochrane.blogspot.com/2017/10/gregs-algebra.html|title=The Grumpy Economist: Greg's algebra|last=Cochrane|first=John H.|date=2017-10-21|website=The Grumpy Economist|access-date=2019-08-07|archive-date=2023-07-01|archive-url=https://web.archive.org/web/20230701114303/https://johnhcochrane.blogspot.com/2017/10/gregs-algebra.html|url-status=live}}</ref> Suppose an open economy has the [[production function]]:<math display="block">Y=F(K,L) = f(k)L, \quad k = K/L</math>Where the variables in this equation are:

* <math display="inline">Y</math> is the total output
* <math display="inline">F(K,L)</math> is the production function
* <math display="inline">K</math> is the total capital stock
* <math display="inline">L</math> is the total labor stock

The standard choice for the production function is the [[Cobb–Douglas production function|Cobb-Douglas production function]]:<math display="block">Y = AK^{\alpha}L^{1-\alpha} = A k^{\alpha} L, \quad \alpha \in [0,1]</math>where <math display="inline">A</math> is the [[Total factor productivity|factor of productivity]] - assumed to be a constant. A corporate tax cut in this model is equivalent to a tax on capital. With taxes, firms look to maximize:<math display="block">J
= \max_{K,L} \; (1-\tau)\left[F(K,L) - wL\right] - rK
\equiv \max_{K,L} \; (1-\tau)\left[f(k) - w\right]L - rK</math>where <math display="inline">\tau</math> is the capital tax rate, <math display="inline">w</math> is wages per worker, and <math display="inline">r</math> is the exogenous interest rate. Then the [[Karush–Kuhn–Tucker conditions|first-order optimality conditions]] become:<math display="block">\begin{aligned}
\frac{\partial J}{\partial K} &= (1-\tau)f'(k) - r \\
\frac{\partial J}{\partial L} &= (1-\tau)\left[f(k) - f'(k)k - w\right]
\end{aligned}</math>Therefore, the optimality conditions imply that:<math display="block">r = (1-\tau)f'(k), \quad w = f(k)-f'(k)k</math>Define total taxes <math display="inline">X = \tau[F(K,L)-wL]</math>. This implies that taxes per worker <math display="inline">x</math> are:<math display="block">x = \tau[f(k)-w] = \tau f'(k)k</math>Then the change in taxes per worker, given the tax rate, is:<math display="block">{dx\over{d\tau}} = \underbrace{f'(k)k}_{\text{Static}} + \underbrace{\tau \left[ f'(k) + f''(k)k \right]{dk\over{d\tau}} }_{\text{Dynamic}}</math>To find the change in wages, we differentiate the second optimality condition for the per worker wages to obtain:<math display="block">\frac{dw}{d\tau}
= \left[f'(k)-f'(k)-f''(k)k \right]\frac{dk}{d\tau}
= -f''(k)k \frac{dk}{d\tau}</math>Assuming that the interest rate is fixed at <math display="inline">r</math>, so that <math display="inline">dr/d\tau = 0</math>, we may differentiate the first optimality condition for the interest rate to find:<math display="block">{dk\over{d\tau}} = {f'(k)\over{(1-\tau)f''(k)}}</math>For the moment, let's focus only on the '''static''' effect of a capital tax cut, so that <math display="inline">dx/d\tau = f'(k)k</math>. If we substitute this equation into equation for wage changes with respect to the tax rate, then we find that:<math display="block">{dw\over{d\tau}}
= -\frac{f'(k)k}{1-\tau}
= -{1\over{1-\tau}} \frac{dx}{d\tau}</math>Therefore, the '''static''' effect of a capital tax cut on wages is:<math display="block">{dw\over{dx}} = -{1\over{1-\tau}}</math>Based on the model, it seems possible that we may achieve a rise in the wage of a worker greater than the amount of the tax cut. But that only considers the static effect, and we know that the dynamic effect must be accounted for. In the dynamic model, we may rewrite the equation for changes in taxes per worker with respect to the tax rate as:<math display="block">\begin{aligned} {dx\over{d\tau}} &= f'(k)k + \tau\left[f'(k) + f''(k)k \right ]{dk\over{d\tau}} \\ &= f'(k)k + {\tau\over{1-\tau}}{[f'(k)]^{2} + f'(k)f''(k)k\over{f''(k)}} \\ &= {\tau\over{1-\tau}}{f'(k)^{2}\over{f''(k)}} + {1\over{1-\tau}}f'(k)k \\ &= {f'(k)\over{1-\tau}}\left[ \tau{ f'(k)\over{f''(k)}} + k \right ] \end{aligned}</math>Recalling that <math display="inline">dw/d\tau = -f'(k)k/(1-\tau)</math>, we have that:<math display="block">\frac{dw}{dx}
= -{\frac{f'(k)k}{1-\tau} \over {\frac{f'(k)}{1-\tau} \left[ \tau \frac{ f'(k)}{f''(k)} + k \right ]}}
= -\frac{1}{\tau \frac{ f'(k)}{kf''(k)} + 1}</math>Using the Cobb-Douglas production function, we have that:<math display="block">{f'(k)\over{kf''(k)}} = -{1\over{1-\alpha}}</math>Therefore, the '''dynamic''' effect of a capital tax cut on wages is:<math display="block">{dw\over{dx}} = -{1-\alpha\over{1-\tau-\alpha}}</math>If we take <math display="inline">\alpha = \tau = 1/3</math>, then the dynamic effect of lowering capital taxes on wages will be even larger than the static effect. Moreover, if there are positive externalities to [[capital accumulation]], the effect of the tax cut on wages would be larger than in the model we just derived. It is important to note that the result is a combination of:

# The standard result that in a small open economy labor bears 100% of a small capital income tax
# The fact that, starting at a positive tax rate, the burden of a tax increase exceeds revenue collection due to the first-order deadweight loss

This result showing that, under certain assumptions, a corporate tax cut can boost the wages of workers by more than the lost revenue does not imply that the magnitude is correct.  Rather, it suggests a basis for policy analysis that is not grounded in handwaving.  If the assumptions are reasonable, then the model is an acceptable approximation of reality; if they are not, then better models should be developed.

=== CES production function ===
Now let's assume that instead of the Cobb-Douglas production function we have a more general [[Constant elasticity of substitution#CES production function|constant elasticity of substitution (CES) production function]]:<math display="block">f(k) = A\left[\alpha k^{\rho} + (1-\alpha) \right]^{1/\rho}</math>where <math display="inline">\rho = 1-\sigma^{-1}</math>; <math display="inline">\sigma</math> is the elasticity of substitution between capital and labor. The relevant quantity we want to calculate is <math display="inline">f'/kf''</math>'','' which may be derived as:<math display="block">{f'\over{kf''}} = -{1\over{1-\rho -{\alpha(1-\rho)\over{\alpha + (1-\alpha)k^{-\rho}}}}}</math>Therefore, we may use this to find that:<math display="block">\begin{aligned}
1+\tau {f'\over{kf''}}
&= 1-{\tau[\alpha + (1-\alpha)k^{-\rho}]\over{(1-\rho)[\alpha+(1-\alpha)k^{-\rho}] -\alpha(1-\rho)}} \\[6pt]
 &= {(1-\rho-\tau)[\alpha+(1-\alpha)k^{-\rho}] -\alpha(1-\rho)\over{(1-\rho)[\alpha+(1-\alpha)k^{-\rho}] -\alpha(1-\rho)}}
\end{aligned}</math>Therefore, under a general CES model, the dynamic effect of a capital tax cut on wages is:<math display="block">{dw\over{dx}} = -{(1-\rho)[\alpha+(1-\alpha)k^{-\rho}] -\alpha(1-\rho)\over{(1-\rho-\tau)[\alpha+(1-\alpha)k^{-\rho}] -\alpha(1-\rho)}}</math>We recover the Cobb-Douglas solution when <math display="inline">\rho = 0</math>.  When <math display="inline">\rho = 1</math>, which is the case when perfect substitutes exist, we find that <math display="inline">dw/dx = 0</math> - there is no effect of changes in capital taxes on wages.  And when <math display="inline">\rho = -\infty</math>, which is the case when perfect complements exist, we find that <math display="inline">dw/dx = -1</math> - a cut in capital taxes increases wages by exactly one dollar.

==Criticisms and defences==

===Adequacy of mathematics for qualitative and complicated economics===
The [[Austrian school of economics|Austrian school]] — while making many of the same ''normative'' economic arguments as mainstream economists from marginalist traditions, such as the [[Chicago school of economics|Chicago school]] — differs ''methodologically'' from mainstream neoclassical schools of economics, in particular in their sharp critiques of the mathematization of economics.<ref>{{cite book |last1=Ekelund |first1=Robert |last2=Hébert |first2=Robert |title=A History of Economic Theory & Method |date=2014 |publisher=Waveland Press |location=Long Grove, IL |pages=574–575 |edition=6th}}</ref> Friedrich Hayek contended that the use of formal techniques projects a scientific exactness that does not appropriately account for informational limitations faced by real economic agents. <!-- A REFERENCE IS NEEDED for this: Hayek's concern did not extend to a general critique of mathematical tools in economics. --><ref>
{{cite journal
| last = Hayek
| first = Friedrich
| author-link = Friedrich Hayek
| title = The Use of Knowledge in Society
| journal = American Economic Review
| volume = 35
| issue = 4
| pages = 519–530
|date=September 1945
| jstor =1809376
}}
</ref>

In an interview in 1999, the economic historian  [[Robert Heilbroner]] stated:<ref>
{{Cite news
| last = Heilbroner
| first = Robert
| author-link = Robert Heilbroner
| title = The end of the Dismal Science?
| newspaper = Challenge Magazine
| date = May–June 1999
| url =http://findarticles.com/p/articles/mi_m1093/is_3_42/ai_54682627/print | archive-url =https://web.archive.org/web/20081210091203/http://findarticles.com/p/articles/mi_m1093/is_3_42/ai_54682627/print | url-status =dead | archive-date =2008-12-10 }}
</ref>
{{blockquote|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 stated that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition."<ref>Beed & Owen, 584</ref>

===Testing predictions of mathematical economics===
Philosopher [[Karl Popper]] discussed the scientific standing of economics in the 1940s and 1950s. He argued that mathematical economics suffered from being tautological. In other words, insofar as economics became a mathematical theory, mathematical economics ceased to rely on empirical refutation  but rather relied on [[mathematical proof]]s and disproof.<ref name="Boland">{{cite book|last=Boland|first=L. A.|title=Karl Popper:A Centenary Assessment|editor=I. C. Jarvie |editor2=K. Milford |editor3=D.W. Miller |publisher=Ashgate Publishing|location=London|year=2007|page=219|chapter=Seven Decades of Economic Methodology|isbn=978-0-7546-5375-2|chapter-url=https://books.google.com/books?id=w-BEoTj0axoC|access-date=2008-06-10}}</ref> According to Popper,  falsifiable assumptions can be tested by experiment and observation while unfalsifiable assumptions can be explored mathematically for their consequences and for their [[logical consistency|consistency]] with other assumptions.<ref>
{{cite journal
| last1= Beed
| first1= Clive
| author2= Kane, Owen
| title = What Is the Critique of the Mathematization of Economics?
| journal = Kyklos
| volume = 44
| issue = 4
| pages = 581–612
| year = 1991
| doi =10.1111/j.1467-6435.1991.tb01798.x
}}
</ref>

Sharing Popper's concerns about assumptions in economics generally, and not just mathematical economics, [[Milton Friedman]] declared that "all assumptions are unrealistic".  Friedman proposed judging economic models by their predictive performance rather than by the match between their assumptions and reality.<ref>{{cite book
| last = Friedman
| first = Milton
| author-link = Milton Friedman
| title = Essays in Positive Economics
| publisher = University of Chicago Press
| year = 1953
| location = Chicago
| pages = [https://archive.org/details/essaysinpositive00milt/page/30 30], 33, 41
| url =https://archive.org/details/essaysinpositive00milt
| url-access = registration
| isbn = 978-0-226-26403-5}}
</ref>

===Mathematical economics as a form of pure mathematics===
{{See also|Pure mathematics|Applied mathematics|Engineering}}

Considering mathematical economics, [[J.M. Keynes]] wrote in ''The General Theory'':<ref>{{cite book
| last = Keynes
| first = John Maynard
| author-link = John Keynes
| title = The General Theory of Employment, Interest and Money
| publisher = Macmillan
| year = 1936
| location = Cambridge
| page = 297
| url = http://www.marxists.org/reference/subject/economics/keynes/general-theory/ch21.htm
| isbn = 978-0-333-10729-4
| access-date = 2009-04-30
| archive-date = 2019-05-28
| archive-url = https://web.archive.org/web/20190528142836/https://www.marxists.org/reference/subject/economics/keynes/general-theory/ch21.htm
| url-status = live
}}</ref>
{{blockquote|It is a great fault of symbolic pseudo-mathematical methods of formalising a system of economic analysis ... that they expressly assume strict independence between the factors involved and lose their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are not blindly manipulating and know all the time what we are doing and what the words mean, we can keep ‘at the back of our heads’ the necessary reserves and qualifications and the adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials ‘at the back’ of several pages of algebra which assume they all vanish. Too large a proportion of recent ‘mathematical’ economics are merely concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.}}

===Defense of mathematical economics===
In response to these criticisms, Paul Samuelson argued that mathematics is a language, repeating a thesis of [[Josiah Willard Gibbs]]. In economics, the language of mathematics is sometimes necessary for representing substantive problems. Moreover, mathematical economics has led to conceptual advances in economics.<ref>Paul A. Samuelson (1952). "Economic Theory and Mathematics&nbsp;— An Appraisal", ''American Economic Review'', 42(2), pp. [https://web.archive.org/web/20000914192525/http://cowles.econ.yale.edu/P/cp/p00b/p0061.pdf 56, 64-65] (press '''+''').</ref>  In particular, Samuelson gave the example of [[microeconomics]], writing that "few people are ingenious enough to grasp [its] more complex parts... ''without'' resorting to the language of mathematics, while most ordinary individuals can do so fairly easily ''with'' the aid of mathematics."<ref>D.W. Bushaw and [[Robert W. Clower|R.W. Clower]] (1957). ''Introduction to Mathematical Economics'', p. [http://babel.hathitrust.org/cgi/pt?id=uc1.b3513586;page=root;view=image;size=100;seq=11;num=vii vii.] {{Webarchive|url=https://web.archive.org/web/20220318043822/http://babel.hathitrust.org/cgi/pt?id=uc1.b3513586;page=root;view=image;size=100;seq=11;num=vii |date=2022-03-18 }}</ref>

Some economists state that mathematical economics deserves support just like other forms of mathematics, particularly its neighbors in [[mathematical optimization]] and [[mathematical statistics]] and increasingly in [[theoretical computer science]]. Mathematical economics and other mathematical sciences have a history in which theoretical advances have regularly contributed to the reform of the more applied branches of economics. In particular, following the program of [[John von Neumann]], game theory now provides the foundations for describing much of applied economics, from  statistical decision theory (as "games against nature") and econometrics to general equilibrium theory and industrial organization. In the last decade, with the rise of the internet, mathematical economists and optimization experts and computer scientists have worked on problems of pricing for on-line services --- their contributions using mathematics from cooperative game theory, nondifferentiable optimization, and combinatorial games.

[[Robert M. Solow]] concluded that mathematical economics was the core "[[infrastructure]]" of contemporary economics:
<blockquote>
Economics is no longer a fit conversation piece for ladies and gentlemen. It has become a technical subject. Like any technical subject it attracts some people who are more interested in the technique than the subject. That is too bad, but it may be inevitable. In any case, do not kid yourself: the technical core of economics is indispensable infrastructure for the political economy. That is why, if you consult [a reference in contemporary economics]  looking for enlightenment about the world today, you will be led to technical economics, or history, or nothing at all.<ref>{{Cite news|title=The Wide, Wide World Of Wealth (''The New Palgrave: A Dictionary of Economics''. Edited by John Eatwell, Murray Milgate and Peter Newman. Four volumes. 4,103 pp. New York: Stockton Press. $650)|last=Solow|first=Robert M.|author-link=Robert M. Solow|date=20 March 1988|journal=New York Times|url=https://www.nytimes.com/1988/03/20/books/the-wide-wide-world-of-wealth.html?scp=1|access-date=11 February 2017|archive-date=1 August 2017|archive-url=https://web.archive.org/web/20170801204606/http://www.nytimes.com/1988/03/20/books/the-wide-wide-world-of-wealth.html?scp=1|url-status=live}}</ref>
</blockquote>

==Mathematical economists==
Prominent mathematical economists include the following.

===19th century===
{{Col-begin}}
{{Col-break}}
* [[Enrico Barone]]
* [[Antoine Augustin Cournot]]
* [[Francis Ysidro Edgeworth]] <!-- Core, etc.  -->
{{Col-break}}
* [[Irving Fisher]]
* [[William Stanley Jevons]]
{{Col-break}}
* [[Vilfredo Pareto]]
* [[Léon Walras]]

{{col-end}}

===20th century===
{{Col-begin}}
{{Col-break}}
* [[Charalambos D. Aliprantis]]
* [[R. G. D. Allen]]
* [[Maurice Allais]]
* [[Kenneth J. Arrow]]
* [[Robert J. Aumann]]
* [[Yves Balasko]]
* [[David Blackwell]]
* [[Lawrence E. Blume]]
* [[Graciela Chichilnisky]]
<!-- * [[Bernard Cornet]] -->
* [[George B. Dantzig]]
* [[Gérard Debreu]]
* [[Mario Draghi]]
* [[Jacques H. Drèze]]
* [[David Gale]]
* [[Nicholas Georgescu-Roegen]]
{{Col-break}}
* [[Roger Guesnerie]]
* [[Frank Hahn]]
* [[John C. Harsanyi]]
* [[John Hicks|John R. Hicks]]
* [[Werner Hildenbrand]]
* [[Harold Hotelling]]
* [[Leonid Hurwicz]]
* [[Leonid Kantorovich]]
* [[Tjalling Koopmans]]
* [[David M. Kreps]]
* [[Harold W. Kuhn]]
* [[Edmond Malinvaud]]
* [[Andreu Mas-Colell]]
* [[Eric Maskin]]
<!-- * [[David Luenberger]] notable for optimization and control-->
* [[Nimrod Megiddo]]
{{Col-break}}
*[[Jean-François Mertens]]
* [[James Mirrlees]]
* [[Roger Myerson]]
* [[John Forbes Nash, Jr.]]
* [[John von Neumann]]
* [[Vladimir Pokrovskii]]
* [[Edward C. Prescott]]
* [[Roy Radner]]
* [[Frank P. Ramsey|Frank Ramsey]]
* [[Donald John Roberts]]
* [[Paul Samuelson]]
* [[Yuliy Sannikov]]
* [[Thomas Sargent]]
* [[Leonard J. Savage]]
* [[Herbert Scarf]]
{{Col-break}}
* [[Reinhard Selten]]
* [[Amartya Sen]]
* [[Lloyd S. Shapley]]
* [[Stephen Smale]]
* [[Robert Solow]]
* [[Hugo F. Sonnenschein]]
* [[Nancy L. Stokey]]
* [[Albert W. Tucker]]
* [[Hirofumi Uzawa]]
* [[Robert B. Wilson]]
* [[Abraham Wald]] <!-- proof that Walras’s equilibrium equations had a solution in suitable conditions -->
* [[Hermann Wold]]
* [[Nicholas C. Yannelis]]
{{col-end}}

==See also==
{{Portal|Business and economics|Mathematics}}
* [[Econophysics]]
*[[Mathematical finance]]

==References==
{{Reflist|30em}}

==Further reading==
* Alpha C. Chiang and Kevin Wainwright, [1967] 2005. ''Fundamental Methods of Mathematical Economics'', McGraw-Hill Irwin. [http://www.mhprofessional.com/product.php?isbn=0070109109 Contents.]
* [[E. Roy Weintraub]], 1982. ''Mathematics for Economists'',  Cambridge.  [https://archive.org/details/mathematicsforec0000wein Contents].
* [[Stephen Glaister]], 1984. ''Mathematical Methods for Economists'', 3rd ed., Blackwell. [https://books.google.com/books?id=Ct2nrJSHxsQC&pg=PR5=onepage Contents.]
* Akira Takayama, 1985. ''Mathematical Economics'', 2nd ed. Cambridge. [https://books.google.com/books?id=685iPEaLAEcC&pg=PR9=onepage Contents].
* [[Nancy Stokey|Nancy L. Stokey]] and [[Robert E. Lucas]] with [[Edward Prescott]], 1989. ''Recursive Methods in Economic Dynamics'', Harvard University Press. [http://www.hup.harvard.edu/catalog.php?isbn=9780674750968 Desecription] and chapter-preview [https://books.google.com/books?id=tWYo0QolyLAC&q=f%3Dfalse&pg=PR11 links].
* [[Avinash Dixit|A. K. Dixit]], [1976] 1990.  ''Optimization in Economic Theory'', 2nd ed., Oxford. [https://books.google.com/books?id=dHrsHz0VocUC&pg=PA194=false Description] and contents [https://books.google.com/books?id=dHrsHz0VocUC&dq=false&pg=PR7 preview].
* [[Kenneth L. Judd]], 1998. ''Numerical Methods in Economics'', MIT Press.  [https://web.archive.org/web/20020923144108/http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 Description] and chapter-preview [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 links].
* Michael Carter, 2001. ''Foundations of Mathematical Economics'', MIT Press. [https://books.google.com/books?id=KysvrGGfzq0C&pg=PR7=onepage Contents].
* Ferenc Szidarovszky and Sándor Molnár, 2002. ''Introduction to Matrix Theory: With Applications to Business and Economics'', World Scientific Publishing. [http://www.worldscientific.com/worldscibooks/10.1142/4595 Description] and [http://www.worldscientific.com/doi/suppl/10.1142/4595/suppl_file/4595_chap01.pdf preview].
* D. Wade Hands, 2004. ''Introductory Mathematical Economics'', 2nd ed. Oxford. [http://www.oup.com/us/catalog/he/subject/Economics/OtherCourses/MathematicalEconomics/?view=usa&sf=toc&ci=9780195133783 Contents].
* [[Vladimir Pokrovskii]], 2018. ''Econodynamics. The Theory of Social Production'', 3th ed., Springer.   
* Giancarlo Gandolfo, [1997] 2009. ''Economic Dynamics'', 4th ed., Springer.   [https://www.springer.com/economics/economic+theory/book/978-3-642-03862-4 Description] and [https://books.google.com/books?id=ZMwXi67nhHQC&pg=PR9 preview].
* John Stachurski, 2009. ''Economic Dynamics: Theory and Computation'', MIT Press. [https://web.archive.org/web/20120916032346/http://johnstachurski.net/book/book.html  Description] and [https://books.google.com/books?id=IzW5rSzlluIC&pg=PR7 preview].

==External links==
{{wikiquote}}
{{Wiktionary}}
* ''Journal of Mathematical Economics'' [http://www.elsevier.com/wps/find/journaldescription.cws_home/505577/description#description Aims & Scope]
* [http://erasmusmundus-qem.univ-paris1.fr/ Erasmus Mundus Master QEM - Models and Methods of Quantitative Economics], The [[Models and Methods of Quantitative Economics - QEM]]
{{good article}}

{{Areas of mathematics}}
{{Economics}}
{{Social sciences}}
{{Authority control}}

{{DEFAULTSORT:Mathematical Economics}}
[[Category:Mathematical economics| ]]
[[Category:Mathematical and quantitative methods (economics)]]