Mathematical Economics
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Mathematical economics is the application of
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
methods to represent theories and analyze problems in
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...
. Often, these applied methods are beyond simple geometry, and may include differential and integral
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, difference and
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
, matrix algebra, mathematical programming, or other computational methods.TOC.
/ref> Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. 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, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications. Broad applications include: * optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker * static (or equilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or the
economy An economy is an area of the production, distribution and trade, as well as consumption of goods and services. In general, it is defined as a social domain that emphasize the practices, discourses, and material expressions associated with t ...
) 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 * dynamic analysis, tracing changes in an economic system over time, for example from economic growth. 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 Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
. 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
Second World War World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...
, as in game theory, would greatly broaden the use of mathematical formulations in economics. * Debreu, Gérard (
987 Year 987 ( CMLXXXVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. Events By place Byzantine Empire * February 7 – Bardas Phokas (the Younger) and Bardas Skleros, two membe ...
2008). "mathematical economics", ''The New Palgrave Dictionary of Economics'', 2nd Edition
Abstract.
Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content", ''Econometrica'', 54(6), pp
1259
1270. * von Neumann, John, and Oskar Morgenstern (1944). ''
Theory of Games and Economic Behavior ''Theory of Games and Economic Behavior'', published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinar ...
''. Princeton University Press.
This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists.
John Maynard Keynes John Maynard Keynes, 1st Baron Keynes, ( ; 5 June 1883 – 21 April 1946), was an English economist whose ideas fundamentally changed the theory and practice of macroeconomics and the economic policies of governments. Originally trained in ...
, 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

The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly in German universities, a style of instruction emerged which dealt specifically with detailed presentation of data as it related to public administration.
Gottfried Achenwall Gottfried Achenwall (20 October 1719 – 1 May 1772) was a German philosopher, historian, economist, jurist and statistician. He is counted among the inventors of statistics. Biography Achenwall was born in Elbing (Elbląg) in the Polish provi ...
lectured in this fashion, coining the term statistics. At the same time, a small group of professors in England established a method of "reasoning by figures upon things relating to government" and referred to this practice as ''Political Arithmetick''.
Sir William Petty Sir William Petty FRS (26 May 1623 – 16 December 1687) was an English economist, physician, scientist and philosopher. He first became prominent serving Oliver Cromwell and the Commonwealth in Ireland. He developed efficient methods to su ...
wrote at length on issues that would later concern economists, such as taxation, Velocity of money and national income, but while his analysis was numerical, he rejected abstract mathematical methodology. Petty's use of detailed numerical data (along with John Graunt) would influence statisticians and economists for some time, even though Petty's works were largely ignored by English scholars. The mathematization of economics began in earnest in the 19th century. Most of the economic analysis of the time was what would later be called classical economics. Subjects were discussed and dispensed with through
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
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. 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. Meanwhile, a new cohort of scholars trained in the mathematical methods of the physical sciences gravitated to economics, advocating and applying those methods to their subject, and described today as moving from geometry to
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...
. These included W.S. Jevons who presented 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. 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. Others preceded and followed in expanding mathematical representations of economic
problem Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
s.


Marginalists and the roots of neoclassical economics

Augustin Cournot and Léon Walras built the tools of the discipline axiomatically around utility, arguing that individuals sought to maximize their utility across choices in a way that could be described mathematically. At the time, it was thought that utility was quantifiable, in units known as utils. Cournot, Walras and Francis Ysidro Edgeworth are considered the precursors to modern mathematical economics.


Augustin Cournot

Cournot, a professor of mathematics, developed a mathematical treatment in 1838 for duopoly—a market condition defined by competition between two sellers. This treatment of competition, first published in '' Researches into the Mathematical Principles of Wealth'', is referred to as
Cournot duopoly Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Au ...
. It is assumed that both sellers had equal access to the market and could produce their goods without cost. Further, it assumed that both goods were homogeneous. Each seller would vary her output based on the output of the other and the market price would be determined by the total quantity supplied. The profit for each firm would be determined by multiplying their output and the per unit Market price. Differentiating the profit function with respect to quantity supplied for each firm left a system of linear equations, the simultaneous solution of which gave the equilibrium quantity, price and profits. Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced many of the marginalists. Cournot's models of duopoly and Oligopoly also represent one of the first formulations of non-cooperative games. Today the solution can be given as a Nash equilibrium but Cournot's work preceded modern game theory by over 100 years.


Léon Walras

While Cournot provided a solution for what would later be called partial equilibrium, Léon Walras attempted to formalize discussion of the economy as a whole through a theory of general competitive equilibrium. The behavior of every economic actor would be considered on both the production and consumption side. Walras originally presented four separate models of exchange, each recursively included in the next. The solution of the resulting system of equations (both linear and non-linear) is the general equilibrium. At the time, no general solution could be expressed for a system of arbitrarily many equations, but Walras's attempts produced two famous results in economics. The first is
Walras' law Walras's law is a principle in general equilibrium theory asserting that budget constraints imply that the ''values'' of excess demand (or, conversely, excess market supplies) must sum to zero regardless of whether the prices are general equilibri ...
and the second is the principle of
tâtonnement A Walrasian auction, introduced by Léon Walras, is a type of simultaneous auction where each agent calculates its demand for the good at every possible price and submits this to an auctioneer. The price is then set so that the total demand across ...
. Walras' method was considered highly mathematical for the time and Edgeworth commented at length about this fact in his review of ''Éléments d'économie politique pure'' (Elements of Pure Economics). Walras' law was introduced as a theoretical answer to the problem of determining the solutions in general equilibrium. His notation is different from modern notation but can be constructed using more modern summation notation. Walras assumed that in equilibrium, all money would be spent on all goods: every good would be sold at the market price for that good and every buyer would expend their last dollar on a basket of goods. Starting from this assumption, Walras could then show that if there were n markets and n-1 markets cleared (reached equilibrium conditions) that the nth market would clear as well. This is easiest to visualize with two markets (considered in most texts as a market for goods and a market for money). If one of two markets has reached an equilibrium state, no additional goods (or conversely, money) can enter or exit the second market, so it must be in a state of equilibrium as well. Walras used this statement to move toward a proof of existence of solutions to general equilibrium but it is commonly used today to illustrate market clearing in money markets at the undergraduate level. Tâtonnement (roughly, French for ''groping toward'') was meant to serve as the practical expression of 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). Only when all buyers are satisfied with the given market price would transactions occur. The market would "clear" at that price—no surplus or shortage would exist. The word ''tâtonnement'' is used to describe the directions the market takes in ''groping toward'' equilibrium, settling high or low prices on different goods until a price is agreed upon for all goods. While the process appears dynamic, Walras only presented a static model, as no transactions would occur until all markets were in equilibrium. In practice, very few markets operate in this manner.


Francis Ysidro Edgeworth

Edgeworth introduced mathematical elements to Economics explicitly in '' Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences'', published in 1881. He adopted
Jeremy Bentham Jeremy Bentham (; 15 February 1748 O.S. 4 February 1747">Old_Style_and_New_Style_dates.html" ;"title="nowiki/>Old Style and New Style dates">O.S. 4 February 1747ref name="Johnson2012" /> – 6 June 1832) was an English philosopher, jurist, an ...
's felicific calculus to economic behavior, allowing the outcome of each decision to be converted into a change in utility. Using this assumption, Edgeworth built a model of exchange on three assumptions: individuals are self-interested, individuals act to maximize utility, and individuals are "free to recontract with another independently of...any third party". Given two individuals, the set of solutions where both individuals can maximize utility is described by the ''contract curve'' on what is now known as an Edgeworth Box. Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until 1924 by Arthur Lyon Bowley. The contract curve of the Edgeworth box (or more generally on any set of solutions to Edgeworth's problem for more actors) is referred to as the core of an economy. Edgeworth devoted considerable effort to insisting that mathematical proofs were appropriate for all schools of thought in economics. While at the helm of '' The Economic Journal'', he published several articles criticizing the mathematical rigor of rival researchers, including Edwin Robert Anderson Seligman, a noted skeptic of mathematical economics. The articles focused on a back and forth over tax incidence and responses by producers. Edgeworth noticed that a monopoly producing a good that had jointness of supply but not jointness of demand (such as first class and economy on an airplane, if the plane flies, both sets of seats fly with it) might actually lower the price seen by the consumer for one of the two commodities if a tax were applied. Common sense and more traditional, numerical analysis seemed to indicate that this was preposterous. Seligman insisted that the results Edgeworth achieved were a quirk of his mathematical formulation. He suggested that the assumption of a continuous demand function and an infinitesimal change in the tax resulted in the paradoxical predictions. Harold Hotelling later showed that Edgeworth was correct and that the same result (a "diminution of price as a result of the tax") could occur with a discontinuous demand function and large changes in the tax rate.


Modern mathematical economics

From the later-1930s, an array of new mathematical tools from the differential calculus and differential equations, convex sets, and
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
were deployed to advance economic theory in a way similar to new mathematical methods earlier applied to physics. The process was later described as moving from
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...
to axiomatics.


Differential calculus

Vilfredo Pareto analyzed microeconomics by treating decisions by economic actors as attempts to change a given allotment of goods to another, more preferred allotment. Sets of allocations could then be treated as Pareto efficient (Pareto optimal is an equivalent term) when no exchanges could occur between actors that could make at least one individual better off without making any other individual worse off. Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed to Adam Smith's Invisible hand hypothesis. Rather, Pareto's statement was the first formal assertion of what would be known as the first fundamental theorem of welfare economics. 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
tâtonnement A Walrasian auction, introduced by Léon Walras, is a type of simultaneous auction where each agent calculates its demand for the good at every possible price and submits this to an auctioneer. The price is then set so that the total demand across ...
) drives the fundamental premise of mathematical economics: systems of economic actors may be modeled and their behavior described much like any other system. This extension followed on the work of the marginalists in the previous century and extended it significantly. Samuelson approached the problems of applying individual utility maximization over aggregate groups with comparative statics, which compares two different equilibrium states after an exogenous change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.


Linear models

Restricted models of general equilibrium were formulated by John von Neumann in 1937.Neumann, J. von (1937). "Über ein ökonomisches Gleichungssystem und ein Verallgemeinerung des Brouwerschen Fixpunktsatzes", ''Ergebnisse eines Mathematischen Kolloquiums'', 8, pp. 73–83, translated and published in 1945-46, as "A Model of General Equilibrium", ''Review of Economic Studies'', 13, pp. 1–9. 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 Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compactness, compact convex set to itself there is a po ...
. Von Neumann's model of an expanding economy considered the matrix pencil '' A - λ B '' with nonnegative matrices A and B; von Neumann sought
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
vectors ''p'' and ''q'' and a positive number ''λ'' that would solve the complementarity equation :'' pT'' (''A'' − ''λ B'') ''q'' = 0, along with two inequality systems expressing economic efficiency. In this model, the ( transposed) 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 solution ''λ'' represents the rate of growth of the economy, which equals the
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...
. 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. Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.


Input-output economics

In 1936, the Russian–born economist
Wassily Leontief Wassily Wassilyevich Leontief (russian: Васи́лий Васи́льевич Лео́нтьев; August 5, 1905 – February 5, 1999), was a Soviet-American economist known for his research on input–output analysis and how changes in one ...
built his model of input-output analysis from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by the physiocrats. 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. In practice, Leontief estimated the coefficients of his simple models, to address economically interesting questions. In 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.


Mathematical optimization

In mathematics,
mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
(or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem involves maximizing or minimizing a real function by selecting input values of the function and computing the corresponding values of the function. The solution process includes satisfying general necessary and sufficient conditions for optimality. For optimization problems, specialized notation may be used as to the function and its input(s). More generally, optimization includes finding the best available element of some function given a defined domain and may use a variety of different computational optimization techniques.Schmedders, Karl (2008). "numerical optimization methods in economics", ''The New Palgrave Dictionary of Economics'', 2nd Edition, v. 6, pp. 138–57.
Abstract.
/ref> Economics is closely enough linked to optimization by agents in an
economy An economy is an area of the production, distribution and trade, as well as consumption of goods and services. In general, it is defined as a social domain that emphasize the practices, discourses, and material expressions associated with t ...
that an influential definition relatedly describes economics ''qua'' science as the "study of human behavior as a relationship between ends and
scarce In economics, scarcity "refers to the basic fact of life that there exists only a finite amount of human and nonhuman resources which the best technical knowledge is capable of using to produce only limited maximum amounts of each economic good ...
means" with alternative uses. 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. Theory posits that consumers maximize their utility, subject to their budget constraints and that firms maximize their profits, subject to their production functions, input costs, and market
demand In economics, demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given time. The relationship between price and quantity demand is also called the demand curve. Demand for a specific item ...
. Dixit, A. K. ( 9761990). ''Optimization in Economic Theory'', 2nd ed., Oxford
Description
and content
preview
Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that in principle could be tested against empirical data. Newer developments have occurred in dynamic programming and modeling optimization with
risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...
and
uncertainty Uncertainty refers to Epistemology, epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially ...
, including applications to
portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificatio ...
, the
economics of information Information economics or the economics of information is the branch of microeconomics that studies how information and information systems affect an economy and economic decisions. One application considers information embodied in certain types o ...
, and search theory. Optimality properties for an entire market system may be stated in mathematical terms, as in formulation of the two fundamental theorems of welfare economics and in the Arrow–Debreu model of general equilibrium (also discussed
below Below may refer to: *Earth * Ground (disambiguation) * Soil * Floor * Bottom (disambiguation) * Less than *Temperatures below freezing * Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fr ...
). More concretely, many problems are amenable to analytical (formulaic) solution. Many others may be sufficiently complex to require numerical methods of solution, aided by software. Still others are complex but tractable enough to allow computable methods of solution, in particular computable general equilibrium models for the entire economy. Linear and nonlinear programming have profoundly affected microeconomics, which had earlier considered only equality constraints. 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 Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972. In economics ...
,
Robert Dorfman Robert Dorfman (27 October 1916 – 24 June 2002) was professor of political economy at Harvard University. Dorfman made great contributions to the fields of economics, statistics, group testing and in the process of coding theory. His pape ...
, Paul Samuelson and Robert Solow. 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

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 (1948), linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet blockade.


Nonlinear programming

Extensions to nonlinear optimization with inequality constraints were achieved in 1951 by Albert W. Tucker and
Harold Kuhn Harold William Kuhn (July 29, 1925 – July 2, 2014) was an American mathematician who studied game theory. He won the 1980 John von Neumann Theory Prize along with David Gale and Albert W. Tucker. A former Professor Emeritus of Mathemati ...
, who considered the nonlinear optimization problem: :Minimize f(x) subject to g_i(x) \leq 0 and h_j(x) = 0 where :f(\cdot) is the function to be minimized :g_i(\cdot) are the functions of the m ''inequality constraints'' where i = 1, \dots, m :h_j(\cdot) are the functions of the l equality constraints where j = 1, \dots, l. In allowing inequality constraints, the Kuhn–Tucker approach generalized the classic method of Lagrange multipliers, which (until then) had allowed only equality constraints. The Kuhn–Tucker approach inspired further research on Lagrangian duality, including the treatment of inequality constraints. The duality theory of nonlinear programming is particularly satisfactory when applied to convex minimization problems, which enjoy the convex-analytic duality theory of Fenchel and Rockafellar; this convex duality is particularly strong for polyhedral convex functions, such as those arising in linear programming. Lagrangian duality and convex analysis are used daily in
operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve dec ...
, in the scheduling of power plants, the planning of production schedules for factories, and the routing of airlines (routes, flights, planes, crews).


Variational calculus and optimal control

''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 Ramsey and Harold Hotelling used the calculus of variations to that end. Following
Richard Bellman Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founde ...
's work on dynamic programming and the 1962 English translation of L. Pontryagin ''et al''.'s earlier work, optimal control theory was used more extensively in economics in addressing dynamic problems, especially as to economic growth equilibrium and stability of economic systems, of which a textbook example is optimal consumption and saving. A crucial distinction is between deterministic and stochastic control models. Other applications of optimal control theory include those in finance, inventories, and production for example.


Functional analysis

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 analytic methods to include
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
in economic theory, in particular, fixed-point theory through his generalization of Brouwer's fixed-point theorem. Following von Neumann's program, Kenneth Arrow and Gérard Debreu formulated abstract models of economic equilibria using convex sets 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. In their models, the ("primal") vector space represented ''quantities'' while the "dual" vector space represented ''prices''.Kantorovich, Leonid, and Victor Polterovich (2008). "Functional analysis", in S. Durlauf and L. Blume, ed., ''The New Palgrave Dictionary of Economics'', 2nd Edition.
Abstract.
ed., Palgrave Macmillan.
In Russia, the mathematician Leonid Kantorovich developed economic models in partially ordered vector spaces, that emphasized the duality between quantities and prices. Kantorovich renamed ''prices'' as "objectively determined valuations" which were abbreviated in Russian as "o. o. o.", alluding to the difficulty of discussing prices in the Soviet Union. Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly in clarifying the role of prices as normal vectors to a 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 processes.


Differential decline and rise

John von Neumann's work on functional analysis and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
broke new ground in mathematics and economic theory.Neumann, John von, and Oskar Morgenstern (1944) ''
Theory of Games and Economic Behavior ''Theory of Games and Economic Behavior'', published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinar ...
'', Princeton.
It also left advanced mathematical economics with fewer applications of differential calculus. In particular, general equilibrium theorists used general topology, convex geometry, and 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 " GET-set" (the humorous designation due to
Jacques H. Drèze Ancient and noble French family names, Jacques, Jacq, or James are believed to originate from the Middle Ages in the historic northwest Brittany region in France, and have since spread around the world over the centuries. To date, there are over ...
). 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 In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ex ...
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. 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

John von Neumann, working with Oskar Morgenstern on the theory of games, broke new mathematical ground in 1944 by extending functional analytic methods related to convex sets and topological fixed-point theory to economic analysis. 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 Hervé Moulin (born 1950 in Paris) is a French mathematician who is the Donald J. Robertson Chair of Economics at the Adam Smith Business School at the University of Glasgow. He is known for his research contributions in mathematical economics ...
, Nimrod Megiddo,
Bezalel Peleg In Book of Exodus, Exodus 31:1-6 and chapters 36 to 39, Bezalel, Bezaleel, or Betzalel ( he, בְּצַלְאֵל, ''Bəṣalʼēl''), was the chief artisan of the Tabernacle and was in charge of building the Ark of the Covenant, assisted by Oh ...
influenced economic research in politics and economics. For example, research on the fair prices in cooperative games and fair values for voting games 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 USA. Earlier neoclassical theory had bounded only the ''range'' of bargaining outcomes and in special cases, for example bilateral monopoly or along the
contract curve In microeconomics, the contract curve or Pareto set is the set of points representing final allocations of two goods between two people that could occur as a result of mutually beneficial trading between those people given their initial allocatio ...
of the Edgeworth box. Von Neumann and Morgenstern's results were similarly weak. Following von Neumann's program, however, John Nash used fixed–point theory to prove conditions under which the bargaining problem and noncooperative games can generate a unique equilibrium solution. Noncooperative game theory has been adopted as a fundamental aspect of experimental economics, behavioral economics, information economics, industrial organization, and
political economy Political economy is the study of how economic systems (e.g. markets and national economies) and political systems (e.g. law, institutions, government) are linked. Widely studied phenomena within the discipline are systems such as labour ...
. It has also given rise to the subject of mechanism design (sometimes called reverse game theory), which has private and public-policy applications as to ways of improving economic efficiency through incentives for information sharing. * ''The New Palgrave Dictionary of Economics'' (2008), 2nd Edition:
     Myerson, Roger B. "mechanism design.
Abstract.

     _____. "revelation principle.
Abstract.
br/>     Sandholm, Tuomas. "computing in mechanism design.
Abstract.
* Nisan, Noam, and Amir Ronen (2001). "Algorithmic Mechanism Design", ''Games and Economic Behavior'', 35(1-2), pp
166–196
* Nisan, Noam, ''et al''., ed. (2007). ''Algorithmic Game Theory'', Cambridge University Press
Description
.
In 1994, Nash, John Harsanyi, and Reinhard Selten received the
Nobel Memorial Prize in Economic Sciences The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...
their work on non–cooperative games. Harsanyi and Selten were awarded for their work on repeated games. Later work extended their results to computational methods of modeling. * Halpern, Joseph Y. (2008). "computer science and game theory", ''The New Palgrave Dictionary of Economics'', 2nd Edition.
Abstract

         * Shoham, Yoav (2008). "Computer Science and Game Theory", ''Communications of the ACM'', 51(8), pp
75-79
.
         * Roth, Alvin E. (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics", ''Econometrica'', 70(4), pp
1341–1378


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 economies, as dynamic systems of interacting agents over time. As such, it falls in the paradigm of complex adaptive systems. In corresponding agent-based models, 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. The rules are formulated to predict behavior and social interactions based on incentives and information. The theoretical assumption of mathematical ''optimization'' by agents markets is replaced by the less restrictive postulate of agents with ''bounded'' rationality ''adapting'' to market forces. ACE models apply numerical methods of analysis to computer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use. 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. 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. It has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions. Other dimensions of the approach include such standard economic subjects as
competition Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can arise between entities such as organisms, ind ...
and collaboration, market structure and industrial organization, transaction costs, welfare economics and mechanism design, information and uncertainty, and macroeconomics. The method is said to benefit from continuing improvements in modeling techniques of
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
and increased computer capabilities. Issues include those common to experimental economics in general and by comparison and to development of a common framework for empirical validation and resolving open questions in agent-based modeling. The ultimate scientific objective of the method has been described as "test ngtheoretical 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".


Mathematicization of economics

Over the course of the 20th century, articles in "core journals" in economics have been almost exclusively written by economists in
academia An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosophy ...
. As a result, much of the material transmitted in those journals relates to economic theory, and "economic theory itself has been continuously more abstract and mathematical." A subjective assessment of mathematical techniques employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1892 to 5.3% in 1990. A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 both lacked statistical analysis of data and lacked displayed mathematical expressions that were indexed with numbers at the margin of the page.


Econometrics

Between the world wars, advances in mathematical statistics and a cadre of mathematically trained economists 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 Frisch coined the word "econometrics" and helped to found both the Econometric Society in 1930 and the journal ''
Econometrica ''Econometrica'' is a peer-reviewed academic journal of economics, publishing articles in many areas of economics, especially econometrics. It is published by Wiley-Blackwell on behalf of the Econometric Society. The current editor-in-chief A ...
'' in 1933. A student of Frisch's, Trygve Haavelmo published ''The Probability Approach in Econometrics'' in 1944, where he asserted that precise statistical analysis could be used as a tool to validate mathematical theories about economic actors with data from complex sources. This linking of statistical analysis of systems to economic theory was also promulgated by the Cowles Commission (now the Cowles Foundation) throughout the 1930s and 1940s. The roots of modern econometrics can be traced to the American economist
Henry L. Moore Henry Ludwell Moore (November 21, 1869 – April 28, 1958) was an American economist known for his pioneering work in econometrics. Paul Samuelson named Moore (along with Harry Gunnison Brown, Allyn Abbott Young, Wesley Clair Mitchell, Frank Knig ...
. 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.


Application

Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
and 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. 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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s. Applied mathematicians apply mathematical principles to practical problems, such as economic analysis and other economics-related issues, and many economic problems are often defined as integrated into the scope of applied mathematics. This integration results from the formulation of economic problems as stylized models with clear assumptions and falsifiable predictions. This modeling may be informal or prosaic, as it was in Adam Smith's '' The Wealth of Nations'', or it may be formal, rigorous and mathematical. Broadly speaking, formal economic models may be classified as 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. * Stochastic models are formulated using stochastic processes. 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 Herman Ole Andreas Wold (25 December 1908 – 16 February 1992) was a Norwegian-born econometrician and statistician who had a long career in Sweden. Wold was known for his work in mathematical economics, in time series analysis, and in econometric ...
developed a
representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
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 In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. E ...
statistics consider additional formulations of stationary processes, such as autoregressive moving average models. More general models include autoregressive conditional heteroskedasticity (ARCH) models and generalized ARCH ( GARCH) models. * Non-stochastic mathematical models may be purely qualitative (for example, models involved in some aspect of
social choice theory Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense. Amartya Sen (2008). "So ...
) or quantitative (involving rationalization of financial variables, for example with hyperbolic coordinates, and/or specific forms of functional relationships between variables). In some cases economic predictions of a model merely assert the direction of movement of economic variables, and so the functional relationships are used only in a qualitative sense: for example, if the
price A price is the (usually not negative) quantity of payment or compensation given by one party to another in return for goods or services. In some situations, the price of production has a different name. If the product is a "good" in t ...
of an item increases, then the
demand In economics, demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given time. The relationship between price and quantity demand is also called the demand curve. Demand for a specific item ...
for that item will decrease. For such models, economists often use two-dimensional graphs instead of functions. * Qualitative models are occasionally used. One example is qualitative scenario planning in which possible future events are played out. Another example is non-numerical decision tree analysis. Qualitative models often suffer from lack of precision.


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 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 Prof. Greg Mankiw of
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
:
''An open economy has the production function y = f(k), where y is output per worker and k is capital per worker. The capital stock adjusts so that the after-tax marginal product of capital equals the exogenously given world interest rate r...How much will the tax cut increase wages?''
To answer this question, we follow
John H. Cochrane John Howland Cochrane ( ; born 26 November 1957) is an American economist specializing in financial economics and macroeconomics. Formerly a professor of economics and finance at the University of Chicago, Cochrane serves full-time as the Rose-Ma ...
of the Hoover Institution. Suppose an open economy has the production function:Y=F(K,L) = f(k)L, \quad k = K/LWhere the variables in this equation are: * Y is the total output * F(K,L) is the production function * K is the total capital stock * L is the total labor stock The standard choice for the production function is the Cobb-Douglas production function:Y = AK^L^ = A k^ L, \quad \alpha \in ,1/math>where A is the 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:J = \max_ \; (1-\tau)\left (K,L) - wL\right- rK \equiv \max_ \; (1-\tau)\left (k) - w\right - rKwhere \tau is the capital tax rate, w is wages per worker, and r is the exogenous interest rate. Then the first-order optimality conditions become:\begin \frac &= (1-\tau)f'(k) - r \\ \frac &= (1-\tau)\left (k) - f'(k)k - w\right\endTherefore, the optimality conditions imply that:r = (1-\tau)f'(k), \quad w = f(k)-f'(k)kDefine total taxes X = \tau (K,L)-wL/math>. This implies that taxes per worker x are:x = \tau (k)-w= \tau f'(k)kThen the change in taxes per worker, given the tax rate, is: = \underbrace_ + \underbrace_To find the change in wages, we differentiate the second optimality condition for the per worker wages to obtain:\frac = \left '(k)-f'(k)-f''(k)k \rightfrac = -f''(k)k \fracAssuming that the interest rate is fixed at r, so that dr/d\tau = 0, we may differentiate the first optimality condition for the interest rate to find: = For the moment, let's focus only on the static effect of a capital tax cut, so that dx/d\tau = f'(k)k. If we substitute this equation into equation for wage changes with respect to the tax rate, then we find that: = -\frac = - \fracTherefore, the static effect of a capital tax cut on wages is: = -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:\begin &= f'(k)k + \tau\left '(k) + f''(k)k \right \\ &= f'(k)k + \\ &= + f'(k)k \\ &= \left \tau + k \right \endRecalling that dw/d\tau = -f'(k)k/(1-\tau), we have that:\frac = - = -\fracUsing the Cobb-Douglas production function, we have that: = -Therefore, the dynamic effect of a capital tax cut on wages is: = -If we take \alpha = \tau = 1/3, 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:f(k) = A\left alpha k^ + (1-\alpha) \rightwhere \rho = 1-\sigma^; \sigma is the elasticity of substitution between capital and labor. The relevant quantity we want to calculate is f'/kf'''','' which may be derived as: = -Therefore, we may use this to find that:\begin 1+\tau &= 1- \\ pt &= \endTherefore, under a general CES model, the dynamic effect of a capital tax cut on wages is: = -We recover the Cobb-Douglas solution when \rho = 0. When \rho = 1, which is the case when perfect substitutes exist, we find that dw/dx = 0 - there is no effect of changes in capital taxes on wages. And when \rho = -\infty, which is the case when perfect complements exist, we find that dw/dx = -1 - a cut in capital taxes increases wages by exactly one dollar.


Criticisms and defences


Adequacy of mathematics for qualitative and complicated economics

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. In an interview in 1999, the economic historian Robert Heilbroner stated: Heilbroner stated that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition."


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 proofs and disproof. 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 consistency with other assumptions. 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.


Mathematical economics as a form of pure mathematics

Considering mathematical economics, J.M. Keynes wrote in ''The General Theory'':


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. In particular, Samuelson gave the example of microeconomics, writing that "few people are ingenious enough to grasp tsmore complex parts... ''without'' resorting to the language of mathematics, while most ordinary individuals can do so fairly easily ''with'' the aid of mathematics." Some economists state that mathematical economics deserves support just like other forms of mathematics, particularly its neighbors in
mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
and mathematical statistics and increasingly in
theoretical computer science Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumsc ...
. 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:
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 reference in contemporary economics looking for enlightenment about the world today, you will be led to technical economics, or history, or nothing at all.


Mathematical economists

Prominent mathematical economists include the following.


19th century

*
Enrico Barone Enrico Barone (; 22 December 1859, Naples, Kingdom of the Two Sicilies – 14 May 1924, Rome, Italy) was a soldier, military historian, and an economist. Biography Barone studied the classics and mathematics before becoming an army officer. He ta ...
* Antoine Augustin Cournot * Francis Ysidro Edgeworth * Irving Fisher * William Stanley Jevons * Vilfredo Pareto * Léon Walras


20th century

*
Charalambos D. Aliprantis Charalambos Dionisios Aliprantis ( el, Χαράλαμπος Διονύσιος Αλιπράντης; May 12, 1946 – February 27, 2009) was a Greek-American economist and mathematician who introduced Banach space and Riesz space methods in eco ...
* R. G. D. Allen * Maurice Allais *
Kenneth J. Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972. In economics ...
* Robert J. Aumann * Yves Balasko * David Blackwell * Lawrence E. Blume *
Graciela Chichilnisky Graciela Chichilnisky (born 1944) is an Argentine American mathematical economist. She is a professor of economics at Columbia University and has expertise in climate change.
* George B. Dantzig * Gérard Debreu * Mario Draghi *
Jacques H. Drèze Ancient and noble French family names, Jacques, Jacq, or James are believed to originate from the Middle Ages in the historic northwest Brittany region in France, and have since spread around the world over the centuries. To date, there are over ...
* David Gale * Nicholas Georgescu-Roegen * Roger Guesnerie * Frank Hahn * John C. Harsanyi * John R. Hicks *
Werner Hildenbrand Werner Hildenbrand (born 25 May 1936 in Göttingen) is a German economist and mathematician. He was educated at the University of Heidelberg, where he received his Diplom in mathematics, applied mathematics and physics in 1961. He continued his ...
* Harold Hotelling * Leonid Hurwicz * Leonid Kantorovich * Tjalling Koopmans * David M. Kreps * Harold W. Kuhn * Edmond Malinvaud * Andreu Mas-Colell * Eric Maskin * Nimrod Megiddo * Jean-François Mertens * James Mirrlees * Roger Myerson * John Forbes Nash, Jr. * John von Neumann *
Edward C. Prescott Edward Christian Prescott (December 26, 1940 – November 6, 2022) was an American economist. He received the Nobel Memorial Prize in Economics in 2004, sharing the award with Finn E. Kydland, "for their contributions to dynamic macroeconomics: ...
* Roy Radner * Frank Ramsey *
Donald John Roberts Donald John Roberts (born February 11, 1945) is a Canadian-American economist, and John H. and Irene S. Scully Professor of Economics, Strategic Management and International Business at the Stanford Graduate School of Business. Biography Born ...
* Paul Samuelson * Thomas Sargent * Leonard J. Savage * Herbert Scarf * Reinhard Selten * Amartya Sen * Lloyd S. Shapley * Stephen Smale * Robert Solow *
Hugo F. Sonnenschein Hugo Freund Sonnenschein (November 14, 1940 – July 15, 2021) was an American economist and educational administrator. He served as president of the University of Chicago from 1993 to 2000. Early life Sonnenschein was born in New York City on ...
* Nancy L. Stokey * Albert W. Tucker *
Hirofumi Uzawa was a Japanese economist. Biography Uzawa was born on July 21, 1928 in Yonago, Tottori to a farming family. He attended the Tokyo First Middle School (currently the Hibiya High School ) and the First Higher School, Japan (now the University o ...
*
Robert B. Wilson } Robert Butler Wilson, Jr. (born May 16, 1937) is an American economist and the Adams Distinguished Professor of Management, Emeritus at Stanford University. He was jointly awarded the 2020 Nobel Memorial Prize in Economic Sciences, together wit ...
* Abraham Wald * Hermann Wold *
Nicholas C. Yannelis Nicholas C. Yannelis ( el, Νικόλαoς Γιανvέλης; born 1953) is the Henry B. Tippie Research Professor of Economics and Applied Mathematics and Computation at the University of Iowa. He is an emeritus Commerce Distinguished Alumni Pr ...
* Yuliy Sannikov


See also

* Econophysics * Mathematical finance


References


Further reading

* Alpha C. Chiang and Kevin Wainwright,
967 Year 967 ( CMLXVII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. Events By place Europe * Spring – Emperor Otto I (the Great) calls for a council at Rome, to present the ne ...
2005. ''Fundamental Methods of Mathematical Economics'', McGraw-Hill Irwin
Contents.
* E. Roy Weintraub, 1982. ''Mathematics for Economists'', Cambridge.
Contents
* Stephen Glaister, 1984. ''Mathematical Methods for Economists'', 3rd ed., Blackwell
Contents.
* Akira Takayama, 1985. ''Mathematical Economics'', 2nd ed. Cambridge
Contents
* Nancy L. Stokey and Robert E. Lucas with
Edward Prescott Edward Christian Prescott (December 26, 1940 – November 6, 2022) was an American economist. He received the Nobel Memorial Prize in Economics in 2004, sharing the award with Finn E. Kydland, "for their contributions to dynamic macroeconomics: ...
, 1989. ''Recursive Methods in Economic Dynamics'', Harvard University Press
Desecription
and chapter-previe
links
* A. K. Dixit, 9761990. ''Optimization in Economic Theory'', 2nd ed., Oxford
Description
and content
preview
*
Kenneth L. Judd Kenneth Lewis Judd (born March 24, 1953) is a computational economist at Stanford University, where he is the Paul H. Bauer Senior Fellow at the Hoover Institution. He received his PhD in economics from the University of Wisconsin in 1980. ...
, 1998. ''Numerical Methods in Economics'', MIT Press.
Description
and chapter-previe
links
* Michael Carter, 2001. ''Foundations of Mathematical Economics'', MIT Press
Contents
* Ferenc Szidarovszky and Sándor Molnár, 2002. ''Introduction to Matrix Theory: With Applications to Business and Economics'', World Scientific Publishing
Description
an
preview
* D. Wade Hands, 2004. ''Introductory Mathematical Economics'', 2nd ed. Oxford
Contents
* Giancarlo Gandolfo, 9972009. ''Economic Dynamics'', 4th ed., Springer.
Description
an
preview
* John Stachurski, 2009. ''Economic Dynamics: Theory and Computation'', MIT Press

an
preview


External links

* ''Journal of Mathematical Economics'
Aims & Scope
*
Erasmus Mundus Master QEM - Models and Methods of Quantitative Economics
The
Models and Methods of Quantitative Economics - QEM A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models ...
{{DEFAULTSORT:Mathematical Economics Mathematical and quantitative methods (economics)