Mathematical economics

# Mathematical economics

Overview
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity. By convention, the methods refer to those beyond simple geometry, such as differential and integral calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, difference and differential equations, matrix algebra
Matrix algebra
Matrix algebra may refer to:*Matrix theory, is the branch of mathematics that studies matrices*Matrix ring, thought of as an algebra over a field or a commutative ring...

, and mathematical programming
Mathematical Programming
Mathematical Programming, established in 1971, and published by Springer Science+Business Media, is the official scientific journal of the Mathematical Optimization Society. It currently consists of two series: A and B. The "A" series contains general publications. The "B" series focuses on topical...

and other computational methods
Computational economics
Computational economics is a research discipline at the interface between computer science and economic and management science. Areas encompassed include agent-based computational modeling, computational modeling of dynamic macroeconomic systems and transaction costs, other applications in...

.

Mathematics allows economists to form meaningful, testable propositions about many wide-ranging and complex subjects which could not be adequately expressed informally.
Discussion

Encyclopedia
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity. By convention, the methods refer to those beyond simple geometry, such as differential and integral calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, difference and differential equations, matrix algebra
Matrix algebra
Matrix algebra may refer to:*Matrix theory, is the branch of mathematics that studies matrices*Matrix ring, thought of as an algebra over a field or a commutative ring...

, and mathematical programming
Mathematical Programming
Mathematical Programming, established in 1971, and published by Springer Science+Business Media, is the official scientific journal of the Mathematical Optimization Society. It currently consists of two series: A and B. The "A" series contains general publications. The "B" series focuses on topical...

and other computational methods
Computational economics
Computational economics is a research discipline at the interface between computer science and economic and management science. Areas encompassed include agent-based computational modeling, computational modeling of dynamic macroeconomic systems and transaction costs, other applications in...

.

Mathematics allows economists to form meaningful, testable propositions about many wide-ranging and complex subjects which could not be adequately expressed informally. Further, the language of mathematics allows economists to make clear, specific, positive
Positive science
In the humanities and social sciences, the term positive is used in at least two ways.The most common usage refers to analysis or theories which only attempt to describe how things 'are', as opposed to how they 'should' be. Positive means also 'value free'. In this sense, the opposite of 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
Model (economics)
In economics, a model is a theoretical construct that represents economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified framework designed to illustrate complex processes, often but not always using...

, a set of stylized and simplified mathematical relationships that clarify assumptions and implications.

• optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker

• static (or equilibrium
Economic equilibrium
In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...

) analysis in which the economic unit (such as a household) or economic system (such as a market or the economy
Economy
An economy consists of the economic system of a country or other area; the labor, capital and land resources; and the manufacturing, trade, distribution, and consumption of goods and services of that area...

) is modeled as not changing

comparative statics
Comparative statics
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....

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
Economic growth
In economics, economic growth is defined as the increasing capacity of the economy to satisfy the wants of goods and services of the members of society. Economic growth is enabled by increases in productivity, which lowers the inputs for a given amount of output. Lowered costs increase demand...

.

Formal economic modeling began in the 19th century with the use of differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

to represent and explain economic behavior, such as utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

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 Second World War
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...

, as in game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

, would greatly broaden the use of mathematical formulations in economics.

This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists. John Maynard Keynes
John Maynard Keynes
John Maynard Keynes, Baron Keynes of Tilton, CB FBA , was a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments...

, Robert Heilbroner
Robert Heilbroner
Robert L. Heilbroner was an American economist and historian of economic thought. The author of some twenty books, Heilbroner was best known for The Worldly Philosophers , a survey of the lives and contributions of famous economists, notably Adam Smith, Karl Marx, and John Maynard...

, Friedrich Hayek
Friedrich Hayek
Friedrich August Hayek CH , born in Austria-Hungary as Friedrich August von Hayek, was an economist and philosopher best known for his defense of classical liberalism and free-market capitalism against socialist and collectivist thought...

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
Holy Roman Empire
The Holy Roman Empire was a realm that existed from 962 to 1806 in Central Europe.It was ruled by the Holy Roman Emperor. Its character changed during the Middle Ages and the Early Modern period, when the power of the emperor gradually weakened in favour of the princes...

universities, a style of instruction emerged which dealt specifically with detailed presentation of data as it related to public administration. Gottfried Achenwall
Gottfried Achenwall
Gottfried Achenwall was a German philosopher, historian, economist, jurist and statistician. He is counted among the inventors of statistics.-Biography:...

lectured in this fashion, coining the term statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

. 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
William Petty
Sir William Petty FRS was an English economist, scientist and philosopher. He first became prominent serving Oliver Cromwell and Commonwealth in Ireland. He developed efficient methods to survey the land that was to be confiscated and given to Cromwell's soldiers...

wrote at length on issues that would later concern economists, such as taxation, Velocity of money
Velocity of money
300px|thumb|Similar chart showing the velocity of a broader measure of money that covers M2 plus large institutional deposits, M3. The US no longer publishes official M3 measures, so the chart only runs through 2005....

and national income
Measures of national income and output
A variety of measures of national income and output are used in economics to estimate total economic activity in a country or region, including gross domestic product , gross national product , and net national income . All are specially concerned with counting the total amount of goods and...

, but while his analysis was numerical, he rejected abstract mathematical methodology. Petty's use of detailed numerical data (along with John Graunt
John Graunt
John Graunt was one of the first demographers, though by profession he was a haberdasher. Born in London, the eldest of seven or eight children of Henry and Mary Graunt. His father was a draper who had moved to London from Hampshire...

) 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
Classical economics
Classical economics is widely regarded as the first modern school of economic thought. Its major developers include Adam Smith, Jean-Baptiste Say, David Ricardo, Thomas Malthus and John Stuart Mill....

. Subjects were discussed and dispensed with through algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

ic means, but calculus was not used. More importantly, until Johann Heinrich von Thünen
Johann Heinrich von Thünen
Johann Heinrich von Thünen was a prominent nineteenth century economist. Von Thünen was a Mecklenburg landowner, who in the first volume of his treatise, The Isolated State , developed the first serious treatment of spatial economics, connecting it with the theory of rent...

'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 science
Physical science
Physical science is an encompassing term for the branches of natural science and science that study non-living systems, in contrast to the life sciences...

s gravitated to economics, advocating and applying those methods to their subject.
These included W.S. Jevons
William Stanley Jevons
William Stanley Jevons was a British economist and logician.Irving Fisher described his book The Theory of Political Economy as beginning the mathematical method in economics. It made the case that economics as a science concerned with quantities is necessarily mathematical...

who presented paper on a "general mathematical theory of political economy" in 1862, providing an outline for use of the theory of marginal utility
Marginal utility
In economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...

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 the 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
Mathematical problem
A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's...

s.

### Marginalists and the roots of neoclassical economics

Augustin Cournot
Antoine Augustin Cournot
Antoine Augustin Cournot was a French philosopher and mathematician.Antoine Augustin Cournot was born at Gray, Haute-Saone. In 1821 he entered one of the most prestigious Grande École, the École Normale Supérieure, and in 1829 he had earned a doctoral degree in mathematics, with mechanics as his...

and Léon Walras
Léon Walras
Marie-Esprit-Léon Walras was a French mathematical economist associated with the creation of the general equilibrium theory.-Life and career:...

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
Francis Ysidro Edgeworth
Francis Ysidro Edgeworth FBA was an Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s...

are considered the precursors to modern mathematical economics.

#### Augustin Cournot

Cournot, a professor of Mathematics, developed a mathematical treatment in 1838 for duopoly
Duopoly
A true duopoly is a specific type of oligopoly where only two producers exist in one market. In reality, this definition is generally used where only two firms have dominant control over a market...

—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
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 Augustin Cournot who was inspired by observing...

. 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
Market price
In economics, market price is the economic price for which a good or service is offered in the marketplace. It is of interest mainly in the study of microeconomics...

. 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
Marginalism
Marginalism refers to the use of marginal concepts in economic theory. Marginalism is associated with arguments concerning changes in the quantity used of a good or service, as opposed to some notion of the over-all significance of that class of good or service, or of some total quantity...

. Cournot's models of duopoly and Oligopoly
Oligopoly
An oligopoly is a market form in which a market or industry is dominated by a small number of sellers . The word is derived, by analogy with "monopoly", from the Greek ὀλίγοι "few" + πόλειν "to sell". Because there are few sellers, each oligopolist is likely to be aware of the actions of the others...

also represent one of the first formulations of non-cooperative game
Non-cooperative game
In game theory, a non-cooperative game is one in which players make decisions independently. Thus, while they may be able to cooperate, any cooperation must be self-enforcing....

s. Today the solution can be given as a Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

but Cournot's work preceded modern Game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

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
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...

. 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' law
Walras’ Law is a principle in general equilibrium theory asserting that when considering any particular market, if all other markets in an economy are in equilibrium, then that specific market must also be in equilibrium. Walras’ Law hinges on the mathematical notion that excess market demands ...

and the second is the principle of tâtonnement
Walrasian auction
A Walrasian auction, introduced by Leon 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 all agents equals the total amount of the good...

. 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
Jeremy Bentham was an English jurist, philosopher, and legal and social reformer. He became a leading theorist in Anglo-American philosophy of law, and a political radical whose ideas influenced the development of welfarism...

's felicific calculus
Felicific calculus
The felicific calculus is an algorithm formulated by utilitarian philosopher Jeremy Bentham for calculating the degree or amount of pleasure that a specific action is likely to cause. Bentham, an ethical hedonist, believed the moral rightness or wrongness of an action to be a function of the...

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 the both individuals can maximize utility is described by the contract curve on what is now known as an Edgeworth Box
Edgeworth box
In economics, an Edgeworth box, named after Francis Ysidro Edgeworth, is a way of representing various distributions of resources. Edgeworth made his presentation in his book Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences, 1881...

. Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until 1924 by Arthur Lyon Bowley
Arthur Lyon Bowley
Sir Arthur Lyon Bowley was an English statistician and economist who worked on economic statistics and pioneered the use of sampling techniques in social surveys....

. 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
Core (economics)
The core is the set of feasible allocations that cannot be improved upon by a subset of the economy's consumers. A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first...

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
Edwin Robert Anderson Seligman
Edwin Robert Anderson Seligman , was an American economist.-Biography:He was born in New York City, a son of Joseph Seligman, a banker. He was educated at Columbia University, where he graduated in 1879...

, a noted skeptic of mathematical economics. The articles focused on a back and forth over tax incidence
Tax incidence
In economics, tax incidence is the analysis of the effect of a particular tax on the distribution of economic welfare. Tax incidence is said to "fall" upon the group that, at the end of the day, bears the burden of the tax...

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
Harold Hotelling
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...

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

In the late 1930s, economists saw the wider use of a broad array of mathematical tools, including convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

s and graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. Mathematicians began to discuss economic problems as a means to advance the state of pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

in the same sense that solutions to problems in physics led to advancement in the underlying mathematics.

### Differential calculus

Vilfredo Pareto
Vilfredo Pareto
Vilfredo Federico Damaso Pareto , born Wilfried Fritz Pareto, was an Italian engineer, sociologist, economist, political scientist and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices....

analyzed microeconomics
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...

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 efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...

(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
Adam Smith was a Scottish social philosopher and a pioneer of political economy. One of the key figures of the Scottish Enlightenment, Smith is the author of The Theory of Moral Sentiments and An Inquiry into the Nature and Causes of the Wealth of Nations...

's Invisible hand
Invisible hand
In economics, invisible hand or invisible hand of the market is the term economists use to describe the self-regulating nature of the marketplace. This is a metaphor first coined by the economist Adam Smith...

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
Foundations of Economic Analysis
Foundations of Economic Analysis is a book by Paul A. Samuelson published in 1947 by Harvard University Press. It sought to demonstrate a common mathematical structure underlying multiple branches of economics from two basic principles: maximizing behavior of agents and stability of equilibrium...

(1947), Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...

identified a common paradigm and mathematical structure across multiple fields in the subject, building on previous work by Alfred Marshall
Alfred Marshall
Alfred Marshall was an Englishman and one of the most influential economists of his time. His book, Principles of Economics , was the dominant economic textbook in England for many years...

. Foundations took mathematical concepts from physics and applied them to economic problems. This broad view (for example, comparing Le Chatelier's principle
Le Châtelier's principle
In chemistry, Le Chatelier's principle, also called the Chatelier's principle, can be used to predict the effect of a change in conditions on a chemical equilibrium. The principle is named after Henry Louis Le Chatelier and sometimes Karl Ferdinand Braun who discovered it independently...

to tâtonnement
Walrasian auction
A Walrasian auction, introduced by Leon 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 all agents equals the total amount of the good...

) 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
Comparative statics
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....

, which compares two different equilibrium
Economic equilibrium
In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...

states after an exogenous
Exogenous
Exogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....

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
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

in 1938: 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 matrix pencil  A - λ B with nonnegative matrices A and B; von Neumann sought probability
Probability vector
Stochastic vector redirects here. For the concept of a random vector, see Multivariate random variable.In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one....

vector
Generalized eigenvector
In linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis – a matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue λ is greater than its geometric multiplicity...

p and q and a positive number λ that would solve the complementarity
Complementarity theory
A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing a function of two vector variables subject to certain requirements which include: that the inner product of the two variables must equal zero, i.e.  = 0...

equation
pT (A - λ B) q = 0,

along with two inequality systems expressing economic efficiency. In this model, the (transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

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 solution λ represents the rate of growth
Economic growth
In economics, economic growth is defined as the increasing capacity of the economy to satisfy the wants of goods and services of the members of society. Economic growth is enabled by increases in productivity, which lowers the inputs for a given amount of output. Lowered costs increase demand...

of the economy, which equals the interest rate
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...

. 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
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...

, 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 Leontief
Wassily Wassilyovich Leontief , was a Russian-American economist notable for his research on how changes in one economic sector may have an effect on other sectors. Leontief won the Nobel Committee's Nobel Memorial Prize in Economic Sciences in 1973, and three of his doctoral students have also...

built his model of input-output analysis
Input-output model
In economics, an input-output model is a quantitative economic technique that represents the interdependencies between different branches of national economy or between branches of different, even competing economies. Wassily Leontief developed this type of analysis and took the Nobel Memorial...

from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by Austrian economists and the physiocrats. With his model, which described a system of production and demand processes, Leontief described how changes in demand in one economic sector
Economic sector
An economy may include several sectors , that evolved in successive phases.* The ancient economy was mainly based on subsistence farming....

would influence production in another. In practice, Leontief estimated the coefficients of his simple models, to address economically interesting questions. In production economics
Production set
A production set is the set of all possible output bundles that a firm can produce given its vector of inputs. Used as part of profit maximization calculations....

, "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

Economics is closely enough linked to optimization by agents
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....

in an economy
Economy
An economy consists of the economic system of a country or other area; the labor, capital and land resources; and the manufacturing, trade, distribution, and consumption of goods and services of that area...

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. Optimization problems run through modern economics, many with explicit economic or technical constraints. In microeconomics, the utility maximization problem
Utility maximization problem
In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem.-Basic setup:...

and its dual problem
Dual problem
In constrained optimization, it is often possible to convert the primal problem to a dual form, which is termed a dual problem. Usually dual problem refers to the Lagrangian dual problem but other dual problems are used, for example, the Wolfe dual problem and the Fenchel dual problem...

, the expenditure minimization problem
Expenditure minimization problem
In microeconomics, the expenditure minimization problem is another perspective on the utility maximization problem: "how much money do I need to reach a certain level of happiness?". This question comes in two parts...

for a given level of utility, are economic optimization problems. Theory posits that consumer
Consumer
Consumer is a broad label for any individuals or households that use goods generated within the economy. The concept of a consumer occurs in different contexts, so that the usage and significance of the term may vary.-Economics and marketing:...

s maximize their utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

, subject to their budget constraint
Budget constraint
A budget constraint represents the combinations of goods and services that a consumer can purchase given current prices with his or her income. Consumer theory uses the concepts of a budget constraint and a preference map to analyze consumer choices...

s and that firm
Firm
A firm is a business.Firm or The Firm may also refer to:-Organizations:* Hooligan firm, a group of unruly football fans* The Firm, Inc., a talent management company* Fair Immigration Reform Movement...

s maximize their profit
Profit (economics)
In economics, the term profit has two related but distinct meanings. Normal profit represents the total opportunity costs of a venture to an entrepreneur or investor, whilst economic profit In economics, the term profit has two related but distinct meanings. Normal profit represents the total...

s, subject to their production function
Production function
In microeconomics and macroeconomics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...

s, input
Factors of production
In economics, factors of production means inputs and finished goods means output. Input determines the quantity of output i.e. output depends upon input. Input is the starting point and output is the end point of production process and such input-output relationship is called a production function...

costs, and market demand
Demand
- Economics :*Demand , the desire to own something and the ability to pay for it*Demand curve, a graphic representation of a demand schedule*Demand deposit, the money in checking accounts...

.

Optimality properties for an entire market system
Market system
A market system is any systematic process enabling many market players to bid and ask: helping bidders and sellers interact and make deals. It is not just the price mechanism but the entire system of regulation, qualification, credentials, reputations and clearing that surrounds that mechanism and...

may be stated in mathematical terms, such as for the Arrow–Debreu model of general equilibrium
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...

(also discussed below). More concretely, many problems are amenable to analytical
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

(formulaic) solution. Many others may be sufficiently complex to require numerical methods of solution, aided by softwares. Still others are complex but tractable enough to allow computable methods
Computational economics
Computational economics is a research discipline at the interface between computer science and economic and management science. Areas encompassed include agent-based computational modeling, computational modeling of dynamic macroeconomic systems and transaction costs, other applications in...

of solution, in particular computable general equilibrium
Computable general equilibrium
Computable general equilibrium models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors...

models for the entire economy.

Linear and nonlinear programming profoundly enriched 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 Kantorovich
Leonid Vitaliyevich Kantorovich was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources...

, Leonid Hurwicz
Leonid Hurwicz
Leonid "Leo" Hurwicz was a Russian-born American economist and mathematician. His nationality of origin was Polish. He was Jewish. He originated incentive compatibility and mechanism design, which show how desired outcomes are achieved in economics, social science and political science...

, Tjalling Koopmans
Tjalling Koopmans
Tjalling Charles Koopmans was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences....

, Kenneth J. Arrow, and Robert Dorfman
Robert Dorfman
Robert Dorfman was emeritus professor of political economy at Harvard University. Dorfman made great contributions to the fields of economics, group testing and in the process of coding theory....

, Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...

, and Robert Solow
Robert Solow
Robert Merton Solow is an American economist particularly known for his work on the theory of economic growth that culminated in the exogenous growth model named after him...

. 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
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...

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
Nonlinear programming
In mathematics, nonlinear programming is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are...

were achieved in 1951 by Albert W. Tucker
Albert W. Tucker
Albert William Tucker was a Canadian-born American mathematician who made important contributions in topology, game theory, and non-linear programming....

and Harold Kuhn, who considered the nonlinear optimization problem
Optimization problem
In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete...

:
Minimize () subject to i() ≤ 0 and j() = 0 where(.) is the function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

to be minimizedi(.) ( = 1, ..., ) are the functions of the inequality constraints
Constraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...

j(.) ( = 1, ..., ) are the functions of the equality constraints.

In allowing inequality constraints, the Kuhn–Tucker approach generalized the classic method of Lagrange multipliers
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...

, 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
Convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....

duality theory
Legendre transformation
In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another...

of Fenchel
Werner Fenchel
Moritz Werner Fenchel was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory. Fenchel's monographs and lecture-notes were very influential also...

and Rockafellar
R. Tyrrell Rockafellar
* for the George Dantzig Prize in 1994 in Optima, Issue 44 page 5.- Books :* Rockafellar, R. Tyrrell. Conjugate duality and optimization. Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied...

; this convex duality is particularly strong for polyhedral convex functions, such as those arising in linear programming
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...

. 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).

### Game theory

Working with Oskar Morgenstern
Oskar Morgenstern
Oskar Morgenstern was a German-born Austrian-School economist. He, along with John von Neumann, helped found the mathematical field of game theory ....

on the theory of games
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 interdisciplinary research field of game theory...

, von Neumann declared that economic theory needed to use functional analytic
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

methods, especially convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

s and topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

fixed point theorem, rather than the traditional differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

, because the maximum–operator did not preserve differentiable functions. Continuing von Neumann's work in cooperative game theory
Cooperative game
In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...

, game theorists Lloyd S. Shapley, Martin Shubik
Martin Shubik
Martin Shubik is an American economist, who is Professor Emeritus of Mathematical Institutional Economics at Yale University. He was educated at the University of Toronto and Princeton University...

, Hervé Moulin, Nimrod Megiddo
Nimrod Megiddo
Nimrod Megiddo is a mathematician and computer scientist. He is research scientist at the IBM Almaden Research Center.His interests include optimization, algorithm design and analysis, game theory, and machine learning....

, Bezalel Peleg influenced economic research in politics and economics. For example, research on the fair prices
Shapley value
In game theory, the Shapley value, named in honour of Lloyd Shapley, who introduced it in 1953, is a solution concept in cooperative game theory. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players...

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
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

theory had bounded only the range of bargaining outcomes and in special cases, for example bilateral monopoly
Bilateral monopoly
In a bilateral monopoly there is both a monopoly and monopsony in the same market.In such, market price and output will be determined by forces like bargaining power of both buyer and seller...

or along the contract curve
Contract curve
In microeconomics, the contract curve is the set of points, representing final allocations of two goods between two people, that could occur as a result of voluntary trading between those people given their initial allocations of the goods...

of the Edgeworth box
Edgeworth box
In economics, an Edgeworth box, named after Francis Ysidro Edgeworth, is a way of representing various distributions of resources. Edgeworth made his presentation in his book Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences, 1881...

. 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 industrial organization
Industrial organization
Industrial organization is the field of economics that builds on the theory of the firm in examining the structure of, and boundaries between, firms and markets....

, political economy
Political economy
Political economy originally was the term for studying production, buying, and selling, and their relations with law, custom, and government, as well as with the distribution of national income and wealth, including through the budget process. Political economy originated in moral philosophy...

, experimental economics
Experimental economics
Experimental economics is the application of experimental methods to study economic questions. Data collected in experiments are used to estimate effect size, test the validity of economic theories, and illuminate market mechanisms. Economic experiments usually use cash to motivate subjects, in...

, behavioral economics, and information economics. It has also given rise to the subject of mechanism design
Mechanism design
Mechanism design is a field in game theory studying solution concepts for a class of private information games...

(sometimes called reverse game theory), which has private and public-policy
Public policy
Public policy as government action is generally the principled guide to action taken by the administrative or executive branches of the state with regard to a class of issues in a manner consistent with law and institutional customs. In general, the foundation is the pertinent national and...

applications as to ways of improving economic efficiency through incentives for information sharing.

In 1994, Nash, John Harsanyi
John Harsanyi
John Charles Harsanyi was a Hungarian-Australian-American economist and Nobel Memorial Prize in Economic Sciences winner....

, and Reinhard Selten
Reinhard Selten
-Life and career:Selten was born in Breslau in Lower Silesia, now in Poland, to a Jewish father, Adolf Selten, and Protestant mother, Käthe Luther. For his work in game theory, Selten won the 1994 Nobel Memorial Prize in Economic Sciences...

received the Nobel Memorial Prize in Economic Sciences
Nobel Memorial Prize in Economic Sciences
The Nobel Memorial Prize in Economic Sciences, commonly referred to as the Nobel Prize in Economics, but officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel , is an award for outstanding contributions to the field of economics, generally regarded as one of the...

their work on non–cooperative games. Harsanyi and Selten were awarded for their work on repeated game
Repeated game
In game theory, a repeated game is an extensive form game which consists in some number of repetitions of some base game . The stage game is usually one of the well-studied 2-person games...

s. Later work extended their results to computational methods
Computational economics
Computational economics is a research discipline at the interface between computer science and economic and management science. Areas encompassed include agent-based computational modeling, computational modeling of dynamic macroeconomic systems and transaction costs, other applications in...

of modeling.         • Shoham, Yoav (2008). "Computer Science and Game Theory," Communications of the ACM, 51(8), pp.
75-79.
• Roth,Alvin E.
Alvin E. Roth
Alvin E. "Al" Roth is an American economist currently serving as the George Gund Professor of Economics and Business Administration at Harvard Business School...

(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
Agent-Based Computational Economics
Agent-based computational economics is the major aspect of computational economics that studies economic processes, including whole economies, as dynamic systems of interacting agents. As such, it falls in paradigm of complex adaptive systems...

(ACE) as a named field is relatively recent, dating from about the 1990s as to published work. It studies economic processes, including whole economies
Economy
An economy consists of the economic system of a country or other area; the labor, capital and land resources; and the manufacturing, trade, distribution, and consumption of goods and services of that area...

, as dynamic systems of interacting agents
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....

. As such, it falls in the paradigm
The word paradigm has been used in science to describe distinct concepts. It comes from Greek "παράδειγμα" , "pattern, example, sample" from the verb "παραδείκνυμι" , "exhibit, represent, expose" and that from "παρά" , "beside, beyond" + "δείκνυμι" , "to show, to point out".The original Greek...

Complex adaptive systems are special cases of complex systems. They are complex in that they are dynamic networks of interactions and relationships not aggregations of static entities...

s. 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 is replaced by the less restrictive postulate of agents adapting to market forces. Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other and learn from their interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of economic systems. ACE has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions. But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.

The method is said to benefit from continuing improvements in modeling techniques of computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

and increased computer capabilities. Issues include those common to experimental economics
Experimental economics
Experimental economics is the application of experimental methods to study economic questions. Data collected in experiments are used to estimate effect size, test the validity of economic theories, and illuminate market mechanisms. Economic experiments usually use cash to motivate subjects, in...

in general and 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[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."

### Functional analysis

Following von Neumann's program, Kenneth Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....

and Gérard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...

formulated abstract models of economic equilibria using convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

s and fixed–point theory. Introduced the Arrow-Debreu model
Arrow-Debreu model
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.The model is central to the theory of...

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 quantitites while the "dual" vector space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

represented prices.

In Russia, the mathematician Leonid Kantorovich
Leonid Kantorovich
Leonid Vitaliyevich Kantorovich was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources...

developed economic models in partially ordered vector space
Riesz space
In mathematics a Riesz space, lattice-ordered vector space or vector lattice is an ordered vector space where the order structure is a lattice....

s, that emphasized the duality between quantities and prices. Oppressed by communism, 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
Supporting hyperplane
Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set S in Euclidean space \mathbb R^n if it meets both of the following:...

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 chosing among functions or stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es.

### Variational calculus and optimal control

Economic dynamics allows for changes economic variables over time. 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
Frank Ramsey
Frank Ramsey may refer to:*Frank P. Ramsey, mathematician, philosopher, and economist*Frank Ramsey , basketball Hall of Famer...

and Harold Hotelling
Harold Hotelling
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...

used the calculus of variations to that end.

Following Richard Bellman
Richard Bellman
Richard Ernest Bellman was an American applied mathematician, celebrated for his invention of dynamic programming in 1953, and important contributions in other fields of mathematics.-Biography:...

'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
Economic growth
In economics, economic growth is defined as the increasing capacity of the economy to satisfy the wants of goods and services of the members of society. Economic growth is enabled by increases in productivity, which lowers the inputs for a given amount of output. Lowered costs increase demand...

equilibrium and stability of economic systems. 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.

### Differential renaissance

As discussed below, following John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

's break-throughs in economics, and particularly after his introduction of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

in economic theory, advanced mathematical economics had fewer applications of differential calculus. In particular, general equilibrium theorists used general topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...

, convex geometry
Convex geometry
Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...

, 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
Jet set
"Jet set" is a journalistic term that was used to describe an international social group of wealthy people, organizing and participating all around the world in social activities that are unreachable to ordinary people...

" (the humorous designation due to Jacques H. Drèze). In the 1960s and 1970s, however, Gérard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...

and Stephen Smale
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...

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
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...

and Sard's lemma
Sard's lemma
Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis which asserts that the image of the set of critical points of a smooth function f from one Euclidean space or manifold to another has Lebesgue measure 0 – they form a null set...

from differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

. Other economists asssociated with the use of differential analysis include Egbert Dierker, Andreu Mas-Colell
Andreu Mas-Colell
Andreu Mas-Colell is a Spanish economist, an expert in microeconomics and one of the world's leading mathematical economists. He is the founder of the Barcelona Graduate School of Economics and a professor in the department of economics at Pompeu Fabra University in Barcelona, Catalonia, Spain...

Yves Balasko was born in Paris on August 9, 1945 from a Hungarian father and a French mother. After studying mathematics at Ecole Normale Superieure, Paris, his interests switched to economic theory. He then went for six years at Electricite de France where he was involved in the application of the...

. These advances have changed the traditional narrative of the history of mathematical economics, following von Neumann, which celebrated the abandonment of differential calculus.

## 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
Academia is the community of students and scholars engaged in higher education and research.-Etymology:The word comes from the akademeia in ancient Greece. Outside the city walls of Athens, the gymnasium was made famous by Plato as a center of learning...

. 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
Mathematical statistics
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...

and a cadre of mathematically trained economists led to econometrics
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

, 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
Ragnar Anton Kittil Frisch
Ragnar Anton Kittil Frisch was a Norwegian economist and the co-winner with Jan Tinbergen of the first Nobel Memorial Prize in Economic Sciences in 1969...

coined the word "econometrics" and helped to found both the Econometric Society
Econometric Society
The Econometric Society is an international society for the advancement of economic theory in its relation with statistics and mathematics. It was founded on December 29, 1930 at the Stalton Hotel in Cleveland, Ohio....

in 1930 and the journal Econometrica
Econometrica
Econometrica is a peer-reviewed academic journal of economics, publishing articles not only in econometrics but in many areas of economics. It is published by the Econometric Society and distributed by Wiley-Blackwell. Econometrica is one of the most highly ranked economics journals in the world...

in 1933. A student of Frisch's, Trygve Haavelmo
Trygve Haavelmo
Trygve Magnus Haavelmo , born in Skedsmo, Norway, was an influential economist with main research interests centered on the fields of econometrics and economics theory. During World War II he worked with Nortraship in the Statistical Department in New York City. He received his Ph.D...

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
Cowles Foundation
The Cowles Commission for Research in Economics is an economic research institute, founded in Colorado Springs in 1932 by Alfred Cowles, a businessman and economist. In 1939, the Cowles Commission moved to the University of Chicago under the directorship of Theodore O. Yntema. Jacob Marschak took...

) throughout the 1930s and 1940s.

### Earlier work in econometrics

The roots of modern econometrics can be traced to the American economist Henry L. Moore
Henry Ludwell Moore
Henry Ludwell Moore was an American economist known for his pioneering work in econometrics.Moore was born in Charles County, Maryland, the first of 15 children. He received a B.A. from Randolph-Macon College in 1892 and a Ph.D. from Johns Hopkins University in 1896. His thesis was on von Thünen's...

. 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 overcorrection in supply and demand curves is now known as the cobweb model
Cobweb model
The cobweb model or cobweb theory is an economic model that explains why prices might be subject to periodic fluctuations in certain types of markets. It describes cyclical supply and demand in a market where the amount produced must be chosen before prices are observed. Producers' expectations...

. A more formal derivation of this model was made later by Nicholas Kaldor
Nicholas Kaldor
Nicholas Kaldor, Baron Kaldor was one of the foremost Cambridge economists in the post-war period...

, 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
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

and matrix algebra
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

is the only practical way of attacking and solving them. Alfred Marshall
Alfred Marshall
Alfred Marshall was an Englishman and one of the most influential economists of his time. His book, Principles of Economics , was the dominant economic textbook in England for many years...

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
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s. Applied mathematicians
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

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
Adam Smith was a Scottish social philosopher and a pioneer of political economy. One of the key figures of the Scottish Enlightenment, Smith is the author of The Theory of Moral Sentiments and An Inquiry into the Nature and Causes of the Wealth of Nations...

's The Wealth of Nations
The Wealth of Nations
An Inquiry into the Nature and Causes of the Wealth of Nations, generally referred to by its shortened title The Wealth of Nations, is the magnum opus of the Scottish economist and moral philosopher Adam Smith...

, or it may be formal, rigorous and mathematical.

Broadly speaking, formal economic models may be classified as stochastic
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

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
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

are formulated using stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es. They model economically observable values over time. Most of econometrics
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

is based on statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

to formulate and test hypotheses about these processes or estimate parameters for them. Between the World Wars, Herman Wold
Herman Wold
Herman Ole Andreas Wold was a Norwegian-born econometrician and statistician who had a long career in Sweden...

developed a representation
Wold decomposition
In operator theory, the Wold decomposition, named after Herman Wold, or Wold-von Neumann decomposition, after Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space...

of stationary stochastic processes in terms of autoregressive models and a determinist trend. Wold and Jan Tinbergen
Jan Tinbergen
Jan Tinbergen , was a Dutch economist. He was awarded the first Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1969, which he shared with Ragnar Frisch for having developed and applied dynamic models for the analysis of economic processes...

applied time-series analysis to economic data. Contemporary research on time series
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

consider additional formulations of stationary processes, such as autoregressive moving average model
Autoregressive moving average model
In statistics and signal processing, autoregressive–moving-average models, sometimes called Box–Jenkins models after the iterative Box–Jenkins methodology usually used to estimate them, are typically applied to autocorrelated time series data.Given a time series of data Xt, the ARMA model is a...

s. More general models include autoregressive conditional heteroskedasticity
Autoregressive conditional heteroskedasticity
In econometrics, AutoRegressive Conditional Heteroskedasticity models are used to characterize and model observed time series. They are used whenever there is reason to believe that, at any point in a series, the terms will have a characteristic size, or variance...

(ARCH) models and generalized ARCH (GARCH) models.

• Non-stochastic mathematical models
Deterministic system (mathematics)
In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:...

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 functional relationships
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

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
Price
-Definition:In ordinary usage, price is the quantity of payment or compensation given by one party to another in return for goods or services.In modern economies, prices are generally expressed in units of some form of currency...

of an item increases, then the demand
Demand
- Economics :*Demand , the desire to own something and the ability to pay for it*Demand curve, a graphic representation of a demand schedule*Demand deposit, the money in checking accounts...

for that item will decrease. For such models, economists often use two-dimensional graphs instead of functions.

• Qualitative models
Qualitative economics
Qualitative economics refers to representation and analysis of information about the direction of change in some economic variable as related to change of some other economic variable...

are occasionally used. One example is qualitative scenario planning
Scenario planning
Scenario planning, also called scenario thinking or scenario analysis, is a strategic planning method that some organizations use to make flexible long-term plans. It is in large part an adaptation and generalization of classic methods used by military intelligence.The original method was that a...

in which possible future events are played out. Another example is non-numerical decision tree analysis. Qualitative models often suffer from lack of precision.

### 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, the economic historian Robert Heilbroner
Robert Heilbroner
Robert L. Heilbroner was an American economist and historian of economic thought. The author of some twenty books, Heilbroner was best known for The Worldly Philosophers , a survey of the lives and contributions of famous economists, notably Adam Smith, Karl Marx, and John Maynard...

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
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...

discussed the scientific standing of economics in the 1940s and 1950s. He argued that mathematical economics suffered from being tautological. In other words, insofar that economics became a mathematical theory, mathematical economics ceased to rely on empirical refutation but rather relied on mathematical proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

s 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
Milton Friedman
Milton Friedman was an American economist, statistician, academic, and author who taught at the University of Chicago for more than three decades...

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
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...

. 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
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...

, 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."

Some economists state that mathematical economics deserves support just like other forms of mathematics, particularly its neighbors in mathematical optimization and mathematical statistics
Mathematical statistics
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...

and increasingly in theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

. 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
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

, 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 economicists 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
Infrastructure
Infrastructure is basic physical and organizational structures needed for the operation of a society or enterprise, or the services and facilities necessary for an economy to function...

" 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 [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.

## Mathematical economists

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

### 19th century

• Enrico Barone
Enrico Barone
Enrico Barone was a soldier, military historian, and economist.Barone studied the classics and mathematics before becoming an army officer. He taught military history for eight years from 1894 at the Officers' Training School. There he wrote a series of influential historical military works...

• Antoine Augustin Cournot
Antoine Augustin Cournot
Antoine Augustin Cournot was a French philosopher and mathematician.Antoine Augustin Cournot was born at Gray, Haute-Saone. In 1821 he entered one of the most prestigious Grande École, the École Normale Supérieure, and in 1829 he had earned a doctoral degree in mathematics, with mechanics as his...

• Francis Ysidro Edgeworth
Francis Ysidro Edgeworth
Francis Ysidro Edgeworth FBA was an Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s...

• Irving Fisher
Irving Fisher
Irving Fisher was an American economist, inventor, and health campaigner, and one of the earliest American neoclassical economists, though his later work on debt deflation often regarded as belonging instead to the Post-Keynesian school.Fisher made important contributions to utility theory and...

• William Stanley Jevons
William Stanley Jevons
William Stanley Jevons was a British economist and logician.Irving Fisher described his book The Theory of Political Economy as beginning the mathematical method in economics. It made the case that economics as a science concerned with quantities is necessarily mathematical...

### 20th century

• Charalambos D. Aliprantis
Charalambos D. Aliprantis
Charalambos Dionisios Aliprantis was a Greek-American economist who introduced Banach space and Riesz space methods in economic theory. He was born in Cefalonia, Greece in 1946 and came to the US in 1969, where he obtained his PhD in Mathematics from Caltech in June 1973.He was a distinguished...

• R. G. D. Allen
R. G. D. Allen
Sir Roy George Douglas Allen, CBE, FBA was an English economist, mathematician and statistician.Allen was born in Worcester and educated at the Royal Grammar School Worcester, from which he won a scholarship to Sidney Sussex College, Cambridge...

• Maurice Allais
Maurice Allais
Maurice Félix Charles Allais was a French economist, and was the 1988 winner of the Nobel Memorial Prize in Economics "for his pioneering contributions to the theory of markets and efficient utilization of resources."...

• Kenneth J. Arrow
• Robert J. Aumann
Yves Balasko was born in Paris on August 9, 1945 from a Hungarian father and a French mother. After studying mathematics at Ecole Normale Superieure, Paris, his interests switched to economic theory. He then went for six years at Electricite de France where he was involved in the application of the...

• David Blackwell
David Blackwell
-Honors and awards:*President, Institute of Mathematical Statistics, 1956*National Academy of Sciences, 1965*American Academy of Arts and Sciences, 1968*Honorary Fellow, Royal Statistical Society, 1976*Vice President, American Statistical Association, 1978...

• Lawrence E. Blume
Lawrence E. Blume
Lawrence E. Blume is a Goldwin Smith Professor of Economics at Cornell University, USA.He is a Visiting Research Professor at IHS and a member of the external faculty at the Santa Fe Institute, where he has served as Co-Director of the Economics Program and on the Institute's steering committee...

• Graciela Chichilnisky
Graciela Chichilnisky
Graciela Chichilnisky is an Argentine American mathematical economist and an expert on climate change. She is a professor of economics at Columbia University....

• George B. Dantzig
• Gérard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...

• Jacques H. Drèze
• David Gale
David Gale
David Gale was a distinguished American mathematician and economist. He was a Professor Emeritus at University of California, Berkeley, affiliated with departments of Mathematics, Economics, and Industrial Engineering and Operations Research...

• Nicholas Georgescu-Roegen
Nicholas Georgescu-Roegen
Nicholas Georgescu-Roegen, born Nicolae Georgescu was a Romanian mathematician, statistician and economist, best known for his 1971 magnum opus The Entropy Law and the Economic Process, which situated the view that the second law of thermodynamics, i.e., that usable "free energy" tends to disperse...

• Roger Guesnerie
Roger Guesnerie
Roger Guesnerie is an economist born in France in 1943. He is currently the Chaired Professor of Economic Theory and Social Organization of the Collège de France, Director of Studies at the École des hautes études en sciences sociales, and the Chairman of the Board of Directors of the Paris School...

• Frank Hahn
Frank Hahn
Frank Horace Hahn is a British economist whose work has focused on general equilibrium theory, monetary theory, Keynesian economics and monetarism...

• John C. Harsanyi
• John R. Hicks
John Hicks
Sir John Richard Hicks was a British economist and one of the most important and influential economists of the twentieth century. The most familiar of his many contributions in the field of economics were his statement of consumer demand theory in microeconomics, and the IS/LM model , which...

• Werner Hildenbrand
Werner Hildenbrand
Werner Hildenbrand is a German economist and mathematician. He was educated at the University of Heidelberg, were he received his Diplom in mathematics, applied mathematics and physics in 1961. He continued his education at the University of Heidelberg and received his Ph.D...

• Harold Hotelling
Harold Hotelling
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...

• Leonid Hurwicz
Leonid Hurwicz
Leonid "Leo" Hurwicz was a Russian-born American economist and mathematician. His nationality of origin was Polish. He was Jewish. He originated incentive compatibility and mechanism design, which show how desired outcomes are achieved in economics, social science and political science...

• Leonid Kantorovich
Leonid Kantorovich
Leonid Vitaliyevich Kantorovich was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources...

• Tjalling Koopmans
Tjalling Koopmans
Tjalling Charles Koopmans was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences....

• David M. Kreps
David M. Kreps
David Marc "Dave" Kreps is a game theorist and economist and professor at the Graduate School of Business at Stanford University. He is known for his analysis of dynamic choice models and non-cooperative game theory, particularly the idea of sequential equilibrium, which he developed with Stanford...

• Harold W. Kuhn
Harold W. Kuhn
Harold William Kuhn is an American mathematician who studied game theory. He won the 1980 John von Neumann Theory Prize along with David Gale and Albert W. Tucker...

• Edmond Malinvaud
Edmond Malinvaud
Edmond Malinvaud is a French economist. He was the first president of the Pontifical Academy of Social Sciences....

• Andreu Mas-Colell
Andreu Mas-Colell
Andreu Mas-Colell is a Spanish economist, an expert in microeconomics and one of the world's leading mathematical economists. He is the founder of the Barcelona Graduate School of Economics and a professor in the department of economics at Pompeu Fabra University in Barcelona, Catalonia, Spain...

Eric Stark Maskin is an American economist and Nobel laureate recognized with Leonid Hurwicz and Roger Myerson "for having laid the foundations of mechanism design theory." He is the Albert O...

• Nimrod Megiddo
Nimrod Megiddo
Nimrod Megiddo is a mathematician and computer scientist. He is research scientist at the IBM Almaden Research Center.His interests include optimization, algorithm design and analysis, game theory, and machine learning....

• James Mirrlees
James Mirrlees
Sir James Alexander Mirrlees is a Scottish economist and winner of the 1996 Nobel Memorial Prize in Economic Sciences. He was knighted in 1998....

• Roger Myerson
Roger Myerson
Roger Bruce Myerson is an American economist and Nobel laureate recognized with Leonid Hurwicz and Eric Maskin for "having laid the foundations of mechanism design theory." A professor at the University of Chicago, he has made contributions as an economist, as an applied mathematician, and as a...

• John Forbes Nash, Jr.
• John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

• Edward C. Prescott
Edward C. Prescott
Edward Christian Prescott is 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: the time consistency of economic policy and the driving forces behind business cycles"...

Roy Radner is currently Leonard N. Stern School Professor of Business at New York University. He is a micro-economic theorist with varied interests...

• Frank Ramsey
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...

• Donald John Roberts
Donald John Roberts
Donald John Roberts is the John H. and Irene S. Scully Professor of Economics, Strategic Management and International Business at the Stanford Graduate School of Business . He has been a member of the Stanford faculty since 1980...

• Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...

• Thomas Sargent

• Leonard J. Savage
• Herbert Scarf
Herbert Scarf
Herbert Eli "Herb" Scarf is an American mathematical economist and Sterling Professor of Economics at Yale University. He is a member of the American Academy of Arts and Sciences...

• Reinhard Selten
Reinhard Selten
-Life and career:Selten was born in Breslau in Lower Silesia, now in Poland, to a Jewish father, Adolf Selten, and Protestant mother, Käthe Luther. For his work in game theory, Selten won the 1994 Nobel Memorial Prize in Economic Sciences...

• Amartya Sen
Amartya Sen
Amartya Sen, CH is an Indian economist who was awarded the 1998 Nobel Prize in Economic Sciences for his contributions to welfare economics and social choice theory, and for his interest in the problems of society's poorest members...

• Lloyd S. Shapley
• Stephen Smale
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...

• Robert Solow
Robert Solow
Robert Merton Solow is an American economist particularly known for his work on the theory of economic growth that culminated in the exogenous growth model named after him...

• Hugo F. Sonnenschein
Hugo F. Sonnenschein
Hugo Freund Sonnenschein is a prominent American economist and educational administrator. Currently the Adam Smith Distinguished Service Professor in Economics at the University of Chicago, his specialty is microeconomic theory; with a particular interest in game theory...

• Albert W. Tucker
Albert W. Tucker
Albert William Tucker was a Canadian-born American mathematician who made important contributions in topology, game theory, and non-linear programming....

• Hirofumi Uzawa
• Robert B. Wilson
Robert B. Wilson
Robert Butler "Bob" Wilson, Jr. is an American economist and the Adams Distinguished Professor of Management, Emeritus at Stanford University. He is known for his contributions to management science and business economics. His doctoral thesis introduced sequential quadratic programming, which...

• Hermann Wold
• Nicholas C. Yannelis
Nicholas C. Yannelis
Nicholas C. Yannelis is the current editor of Economic Theory. He obtained his PhD at the University of Rochester under the direction of Lionel McKenzie. He is also the Commerce Distinguished Alumni Professor of Economics at the University of Illinois at Urbana-Champaign and the Sir John Hicks...