Mathematical economics refers to the application of mathematical Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions methods to represent economic theories and analyze problems posed in economics Economics is the social science that is concerned with the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek οἰκονομία from οἶκος (oikos, "house") + νόμος (nomos, "custom" or "law"), hence "rules of the house(hold)". Current. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity.[1] 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 In the humanities and social sciences, the term positive is used in a number of ways claims about controversial or contentious subjects that would be impossible without mathematics.[2] Much of economic theory is currently presented in terms of mathematical economic models 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 mathematical techniques. Frequently, economic models use, a set of stylized and simplified mathematical relationships that clarify assumptions and implications.

Formal economic modeling began in the late 19th century with the use of differential calculus In mathematics, differential calculus is a subfield of calculus concerned with the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential. The derivative of a function at a chosen input value describes the behavior of the function to help describe and predict economic behavior. Economics became more mathematical as a discipline throughout the first half of the 20th century, but it was not until the Second World War Albania · Australia · Austria · Azerbaijan · Belarus · Belgium · Brazil · Bulgaria · Burma · Cambodia · Canada · Ceylon (Sri Lanka) · Channel Islands · China · Czechoslovakia · Denmark · Dutch East Indies · Egypt · Estonia · Finland · France · Germany · Gibraltar · Greece · Greenland · Hong Kong · Hungary · Iceland · that new techniques would allow the use of mathematical formulations in almost all of economics. This rapid systematizing of economics alarmed critics of the discipline as well as some esteemed economists. John Maynard Keynes John Maynard Keynes, 1st Baron Keynes, CB was a British economist whose ideas have profoundly affected modern macroeconomics and social liberalism, both in theory and practice. He advocated interventionist economic policy, by which governments would use fiscal and monetary measures to mitigate the adverse effects of business cycles, economic, Robert Heilbroner, Friedrich Hayek Friedrich August von Hayek CH , was an Austrian-born economist and philosopher known for his defence of classical liberalism and free-market capitalism against socialist and collectivist thought. He is considered by some to be one of the most important economists and political philosophers of the twentieth century. Hayek's account of how changing and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to arbitrary quantities or probabilities.

Contents

History

Main article: History of economic thought The history of economic thought deals with different thinkers and theories in the subject that became political economy and economics from the ancient world to the present day. It encompasses many disparate schools of economic thought. Greek writers such as the philosopher Aristotle examined ideas about the "art" of wealth acquisition

The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly in German The Holy Roman Empire (HRE; German: Heiliges Römisches Reich , Latin: Imperium Romanum Sacrum (IRS), Italian: Sacro Romano Impero (SRI)) was for about a millennium a realm in Central Europe under a Holy Roman Emperor. Its character changed during the Middle Ages and the Early Modern period, when the power of the emperor gradually weakened in universities, a style of instruction emerged which dealt specifically with detailed presentation of data as it related to public administration. Gottfried Achenwall lectured in this fashion, coining the term statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, 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.[3] Sir William Petty Sir William Petty 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. He also managed to remain prominent under King Charles II and King James II, as did many wrote at length on issues that would later concern economists, such as taxation, Velocity of money The velocity of money is the average frequency with which a unit of money is spent in a specific period of time. Velocity associates the amount of economic activity associated with a given money supply. When the period is understood, the velocity may be present as a pure number; otherwise it should be given as a pure number over time. In the and national income 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 (GNP), and net national income (NNI). All are specially concerned with counting the total amount of goods and services produced within some "boundary&, but while his analysis was numerical, he rejected abstract mathematical methodology. Petty's use of detailed numerical data (along with John Graunt John Graunt was one of the first demographers, though by profession he was a haberdasher. Born in London, Graunt, along with William Petty, developed early human statistical and census methods that later provided a framework for modern demography. He is credited with producing the first life table, giving probabilities of survival to each age) would influence statisticians and economists for some time, even though Petty's works were largely ignored by English scholars.[4]

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 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 algebraic 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. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure means, but calculus was not used. More importantly, until Johann Heinrich von Thünen Johann Heinrich von Thünen was a prominent nineteenth century economist . Von Thünen was a Mecklenburg (north German) landowner, who in the first volume of his treatise, The Isolated State (1826), developed the first serious treatment of spatial economics, connecting it with the theory of rent. The importance lies less in the pattern of land use'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.[5] 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 apply previous tools to new problems.[6]

As the physical sciences Physical Science is an encompassing term for the branches of natural science and science that study non-living systems, in contrast to the biological sciences. However, the term "physical" creates an unintended, somewhat arbitrary distinction, since many branches of physical science also study biological phenomena became more systematized, economists pushed for a more formal methodology in economics. W.S. Jevons William Stanley Jevons was an English economist and logician. His book The Theory of Political Economy (1871) expounded upon the "final" (marginal) utility theory of value. Jevons' work, along with similar discoveries made by Carl Menger in Vienna (1871) and by Léon Walras in Switzerland (1874), marked the opening of a new period in the wrote the General Mathematical Theory of Political Economy in 1862, providing a rough outline for use of the theory of marginal utility In economics, the marginal utility of a good or service is the utility gained from an increase (or decrease) in the consumption of that good or service. In general, preferences display diminishing marginal utility. That is, the first unit of consumption of a good or service yields more utility than the second and subsequent units. The concept of in political economy. In 1874, he published The Principles of Science, declaring that "our science must be mathematical simply because it deals with quantities".[7] Jevons expected the practice of statistics would become sufficiently sophisticated as to permit analysis of all decisions under the marginal utility framework.[8] Independently, Carl Menger Carl Menger was the founder of the Austrian School of economics, famous for contributing to the development of the theory of marginal utility, which contested the cost-of-production theories of value, developed by the classical economists such as Adam Smith and David Ricardo presented the theory in Grundsätze der Volkswirtschaftslehre (Principles of Economics) in 1871. Menger's work found a significant and appreciative audience. This movement did not succeed in generating a complete mathematical system under which all economic theory would operate, but it did allow for powerful new interpretive tools. Economists continue to regard mathematical rigor Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism. A religion, too, may be worn lightly, or applied with rigour as a desirable objective.

Marginalists and the roots of neoclassical economics

Main article: 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 of a service, as opposed to some notion of the over-all significance of that class of good or service, or of some total quantity thereof Equilibrium quantities as a solution to two reaction functions in Cournot duopoly. Each reaction function is expressed as a linear equation dependent upon quantity demanded.

Augustin Cournot Antoine Augustin Cournot was a French economist, philosopher and mathematician and Léon Walras Marie-Esprit-Léon Walras was a French mathematical economist associated with the creation of the general equilibrium theory 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.[9] At the time, it was thought that utility was quantifiable, in units known as utils.[10] Cournot, Walras and Francis Ysidro Edgeworth Francis Ysidro Edgeworth was an Irish philosopher/politician/economist who made significant contributions to the methods of statistics during the 1880s. From 1891 onward he was the editor of a leading academic journal in economics and his own writings in economics were influential are considered the precursors to modern mathematical economics.[11]

Augustin Cournot

Cournot, a professor of Mathematics, developed a mathematical treatment in 1838 for duopoly A true duopoly (from Greek dyo / δυο + polein / πωλειν (to sell)) 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. In the field of industrial organization, it is the most commonly studied form of oligopoly—a market condition defined by competition between two sellers.[11] This treatment of competition, first published in Researches into the Mathematical Principles of Wealth,[12] is referred to as Cournot duopoly Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot after he observed competition in a spring water duopoly. It has the following. 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 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. Market value and market price are equal only under conditions of market efficiency, equilibrium, and rational expectations. 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.[13] Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced many of the marginalists 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 of a service, as opposed to some notion of the over-all significance of that class of good or service, or of some total quantity thereof.[13][14] Cournot's models of duopoly and Oligopoly In Economics, 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 (oligoi) "few" + πωλειν (polein) "to sell". Because there are few sellers, each oligopolist is likely to be aware of also represent one of the first formulations of non-cooperative games. Today the solution can be given as a 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 or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing but Cournot's work preceded modern Game theory Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology , engineering, political science, international relations, computer science, and philosophy. Game theory attempts to mathematically capture behavior in strategic situations, in which an individual's success in making by over 100 years.[15]

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 theory is a branch of theoretical neoclassical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many markets, by seeking to prove that equilibrium prices for goods exist and that all prices are at equilibrium, hence general equilibrium, in contrast to partial. 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.[16] 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 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 must sum to zero. That is, ΣXD = ΣXS = 0. Walras' and the second is the principle of tâtonnement 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. Thus, a Walrasian auction perfectly matches the supply. 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).[17]

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.[18]

An 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 famous book, Mathematical Psychics: An essay on the application of mathematics to the moral sciences, 1881. Edgeworth's original two axis depiction was developed into the now displaying the contract curve an economy with two participants. Referred to as the "core" of the economy in modern parlance, there are infinitely many solutions along the curve for economies with two participants[19]

Tâtonnement (roughly, French for groping toward) was meant to serve as the practical expression of Walrassian 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).[20] 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 Dynamics may refer to:, 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.[21]

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.[22] He adopted 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. He is best known for his advocacy of utilitarianism and animal rights, and the idea of the panopticon's 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 amount of pleasure or pain that it produced. The to economic behavior, allowing the outcome of each decision to be converted into a change in utility.[23] 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."[24] 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 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 famous book, Mathematical Psychics: An essay on the application of mathematics to the moral sciences, 1881. Edgeworth's original two axis depiction was developed into the now. Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until 1924 by Arthur Lyon Bowley.[25] The contract curve of the Edgeworth box (or more generally on any set of solutions to Edgeworth's problem for more actors) is referred to as the core of an economy.[26]

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 The Economic Journal is one of the leading scholarly journals of economics. It is published on behalf of the Royal Economic Society by Wiley-Blackwell Publishing. It is one of the oldest journals of economics. The journal was first published in 1891 "with a view of promoting the advancement of economic knowledge". John Maynard Keynes, he published several articles criticizing the mathematical rigor of rival researchers, including Edwin Robert Anderson Seligman, a noted skeptic of mathematical economics.[27] The articles focused on a back and forth over tax incidence and responses by producers. Edgeworth noticed that a monopoly producing a good that had jointness of supply but not jointness of demand (such as first class and economy on an airplane, if the plane flies, both sets of seats fly with it) might actually lower the price seen by the consumer for one of the two commodities if a tax were applied. Common sense and more traditional, numerical analysis seemed to indicate that this was preposterous. Seligman insisted that the results Edgeworth achieved were a quirk of his mathematical formulation. He suggested that the assumption of a continuous demand function and an infinitesimal change in the tax resulted in the paradoxical predictions. Harold Hotelling Harold Hotelling was a mathematical statistician and an influential economic theorist. His name is known to all statisticians because of Hotelling's T-square distribution and its use in statistical hypothesis testing and confidence regions. He also introduced canonical correlation analysis, and is the eponym of Hotelling's law, Hotelling's lemma, 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).[28]

Emergence of modern mathematical economics

Vilfredo Pareto Vilfredo Federico Damaso Pareto , born Wilfried Fritz Pareto, was an Italian industrialist, sociologist, economist, and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices. "His legacy as an economist was profound. Partly because of him, analyzed microeconomics Microeconomics is a branch of economics that studies how the individual parts of the economy, the household and the firms, make decisions to allocate limited resources, typically in markets where goods or services are being bought and sold. Microeconomics examines how these decisions and behaviours affect the supply and demand for goods and 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, or Pareto optimality, is a concept in economics with applications in all areas of the discipline as well as engineering and other 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. Informally, Pareto efficient (Pareto optimal is an equivalent term) when no exchanges could occur between actors that could make at least one individual better off without making any other individual worse off.[29] Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed to Adam Smith Adam Smith was a Scottish moral philosopher and a pioneer of political economics. 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. The latter, usually abbreviated as The Wealth of Nations, is considered his magnum opus and's Invisible hand In economics, the invisible hand, also known as the invisible hand of the market, the term economists use to describe the self-regulating nature of the marketplace, is a metaphor first coined by the economist Adam Smith in The Theory of Moral Sentiments. For Smith, the invisible hand was created by the conjunction of the forces of self-interest, hypothesis.[30][31] Rather, Pareto's statement was the first formal assertion of what would be known as the first fundamental theorem of welfare economics There are two fundamental theorems of welfare economics. The first states that any competitive equilibrium or Walrasian equilibrium leads to a Pareto efficient allocation of resources. The second states the converse, that any efficient allocation can be sustainable by a competitive equilibrium. Despite the apparent symmetry of the two theorems, in.[32] While it is known today that every Walras equilibrium is Pareto efficient, this was not known until more complex proofs were devised in 1936 by Abraham Wald and John von Neumann in 1938. A complete proof of both existence of a general equilibrium and uniqueness of the equilibrium would not come until Kenneth Arrow and Gérard Debreu introduced the Arrow-Debreu model in 1954[33][34]

In the late 1930s, economists saw the wider use of a broad array of mathematical tools, including convex sets and graph theory. Applied mathematicians and topologists began to discuss economic problems as a means to advance the state of pure mathematics in the same sense that solutions to problems in physics led to advancement in the underlying mathematics.[35] At roughly the same time, the Russian economist Wassily Leontief built his model of input-output analysis from 'material balance' tables constructed by Soviet economists. This model, which described a system of production and demand processes, could explain how variation in demand in one economic sector could influence production in another. Leontief published his first paper on the subject in 1936, but his work on both the theoretical foundations of his model and massive empirical studies of national economies in order to test it would continue through the 1960s.[36]

The exposure of (mostly American and British) economists to engineering problems and problems in large bureaucratic systems would bring about huge changes in the discipline and the nature of university research in general.[37] Linear programming, developed during the war, and generalized to nonlinear programming in 1951, would impact the study and practice of microeconomics heavily.[38] The War cemented the use of applied mathematics in many disciplines, including economics. Operations research, a newly formed discipline which influenced and was influenced by mathematical economics, would drive much new research and draw considerable government funding over the next few decades. Mathematical economics expanded in scope and use considerably during the immediate post-war period.[39] Optimal control theory began to be used in addressing dynamic problems in economics, especially the economic growth models, soon after the publication of the English translation of the book by Pontryagin et al.[40] For applications of optimal control theory to economic problems, see[41][42][43], for example.

In the landmark treatise Foundations of Economic Analysis (1947), Paul Samuelson identified a common paradigm and mathematical structure across multiple fields in the subject, building on previous work by Alfred Marshall. Foundations took mathematical concepts from physics and applied them to economic problems. This broad view (for example, comparing Le Chatelier's principle to tâtonnement) drives the fundamental premise of mathematical economics: systems of economic actors may be modeled and their behavior described much like any other system. This extension followed on the work of the marginalists in the previous century and extended it significantly. Samuelson approached the problems of applying individual utility maximization over aggregate groups with comparative statics, which compares two different equilibrium states after an exogenous change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.[44][45]

The surface of the Volatility smile is a 3-D surface whereby the current market implied volatility (Z-axis) for all options on the underlier is plotted against strike price and time to maturity (X & Y-axes).[46]

Over the course of the 20th century, articles in "core journals"[47] in economics have been almost exclusively written by economists in Academia. As a result, much of the material transmitted in those journals relates to economic theory, and "economic theory itself has been continuously more abstract and mathematical."[48] A subjective assessment of mathematical techniques[49] 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.[50] A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 are free of any numerical equations or regression tables.[51]

Econometrics

Main article: Econometrics

Between the world wars, advances in mathematical statistics and a cadre of mathematically trained economists led to econometrics, which was the name proposed for the discipline of advancing economics by using mathematics and statistics. Within economics, "econometrics" has often been used for statistical methods in economics, rather than mathematical economics. Statistical econometrics features the application of linear regression and time series analysis to economic data.

Ragnar Frisch coined the word "econometrics" and helped to found both the Econometric Society in 1930 and the journal Econometrica in 1933.[52][53] A student of Frisch's, Trygve Haavelmo published The Probability Approach in Econometrics in 1944, where he asserted that precise statistical analysis could be used as a tool to validate mathematical theories about economic actors with data from complex sources.[54] This linking of statistical analysis of systems to economic theory was also promulgated by the Cowles Commission (now the Cowles Foundation) throughout the 1930s and 1940s.[55]

Earlier work in econometrics

The roots of modern econometrics can be traced to the American economist Henry L. Moore. Moore studied agricultural productivity and attempted to fit changing values of productivity for plots of corn and other crops to a curve using different values of elasticity. Moore made several errors in his work, some from his choice of models and some from limitations in his use of mathematics. The accuracy of Moore's models also was limited by the poor data for national accounts in the United States at the time. While his first models of production were static, in 1925 he published a dynamic "moving equilibrium" model designed to explain business cycles—this periodic variation from overcorrection in supply and demand curves is now known as the cobweb model. A more formal derivation of this model was made later by Nicholas Kaldor, who is largely credited for its exposition.[56]

Application

The IS/LM model is a Keynesian macroeconomic model designed to make predictions about the intersection of "real" economic activity (e.g. spending, income, savings rates) and decisions made in the financial markets (Money supply and Liquidity preference). The model is no longer widely taught at the graduate level but is common in undergraduate macroeconomics courses.[57]

Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use of calculus and matrix algebra in economic analysis in order to make powerful claims that would be more difficult without such mathematical tools. These tools are prerequisites for formal study, not only in mathematical economics but in contemporary economic theory in general. Economic problems often involve so many variables that mathematics is the only practical way of attacking and solving them. Alfred Marshall argued that every economic problem which can be quantified, analytically expressed and solved, should be treated by means of mathematical work.[58] Economics has become increasingly dependent on 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 mathematicians. Applied mathematicians apply mathematical principles to practical problems, such as economic analysis and other economics-related issues, and many economic problems are often defined as integrated into the scope of applied mathematics.[9] This integration results from the formulation of economic problems as stylized models with clear assumptions and falsifiable predictions. This modeling may be informal or prosaic, as it was in Adam Smith's The Wealth of Nations, or it may be formal, rigorous and mathematical.

Broadly speaking, formal economic models may be classified as stochastic or deterministic and as discrete or continuous. At a practical level, quantitative modeling is applied to many areas of economics and several methodologies have evolved more or less independently of each other. [59]

Criticism of mathematical economics

The methods of mathematical economics are widely, though far from exclusively, used in professional publications. While Friedrich Hayek contended that the use of formal techniques projects a scientific exactness that does not appropriately account for informational limitations in the real world, this did not extend to a general critique of mathematical tools in economics.[60] Philosopher Karl Popper offered considerable criticism in the 1940s and 1950s. He argued that the fundamental problem with mathematical economics was that it was tautological. In other words, once economics became a mathematical discipline, it would cease to rely on empirical truth and instead rely on axiomatic proof.[61] Popper asserted that an economic model could either have verifiable assumptions and produce no new information or have unverifiable assumptions and sacrifice formalism for scope.[62] Milton Friedman responded to this by announcing that "all assumptions are unrealistic", charging that economic models should be judged on how well the theory predicts reality, not how well the assumptions accord with reality.[63] Samuelson argued a different tack. He proposed that economic theories should be refutable in principle; if they were refutable in principle, they could not be tautological.[64]

Another criticism of mathematical economics was popularized by Robert Heilbroner in the afterword to his popular book, The Worldly Philosophers. He elaborated on his feelings later in an interview:[65]

I guess the scientific approach began to penetrate and soon dominate the profession in the past twenty to thirty years. This came about in part because of the "invention" of mathematical analysis of various kinds and, indeed, considerable improvements in it. This is the age in which we have not only more data but more sophisticated use of data. So there is a strong feeling that this is a data-laden science and a data-laden undertaking, which, by virtue of the sheer numerics, the sheer equations, and the sheer look of a journal page, bears a certain resemblance to science...That one central activity looks scientific. I understand that. I think that is genuine. It approaches being a universal law. But resembling a science is different from being a science.

Heilbroner addresses one of the core critiques of economics in general here, that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition."[66] Some call this a mathematical romance that tends to eliminate the distinctively human elements from the economic equation.[67] This critique has been advanced in various forms by economists and other scientists, including Keynes and Paul Joskow. Joskow advanced a particularly harsh critique, observing that a good portion of economic insight came from outside formal models and that those formal, mathematical models were added "ex post" in order to provide a justification for the insight.[68][69]

J.M. Keynes considered excessively mathematical economics very limiting. In The General Theory he wrote:[70]

It is a great fault of symbolic pseudo-mathematical methods of formalising a system of economic analysis … that they expressly assume strict independence between the factors involved and lose their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are not blindly manipulating and know all the time what we are doing and what the words mean, we can keep ‘at the back of our heads’ the necessary reserves and qualifications and the adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials ‘at the back’ of several pages of algebra which assume they all vanish. Too large a proportion of recent ‘mathematical’ economics are merely concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.

Even noted practitioners have agreed in part with critics. Paul Samuelson (slightly) debunked use of mathematics in economics in affirming the virtues of understatement and lack of pretension. Still, he argued for its convenience or necessity in representing substantive problems and the success of its practitioners in arriving at truths.[71] Expressed more pointedly as to price theory, "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."[72]

Mathematical economists

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

19th century

20th century

Notes

  1. ^ Chiang, Alpha C.; Kevin Wainwright (2005). Fundamental Methods of Mathematical Economics. McGraw-Hill Irwin. pp. 1,2. ISBN 0-07-010910-9. http://books.google.com/books?id=KdDAAAAACAAJ.
  2. ^ Varian, Hal (29-30 October, 1992). "What use is Economic Theory?" (PDF). Is Economics Becoming a Hard Science?. Paris, France. pp. 1–11. http://www.sims.berkeley.edu/~hal/Papers/theory.pdf. Retrieved 2008-04-01.
  3. ^ Schumpeter, J.A. (1954). Elizabeth B. Schumpeter. ed. History of Economic Analysis. New York, NY: Oxford University Press. pp. 209–212. ISBN 9780043300862. OCLC 13498913. http://books.google.com/books?id=xjWiAAAACAAJ.
  4. ^ Schumpeter (1954) p. 212-215
  5. ^ Schnieder, Erich (1934). "Johann Heinrich von Thünen". Econometrica (The Econometric Society) 2 (1): 1–12. doi:10.2307/1907947. ISSN 00129682. OCLC 35705710. http://www.jstor.org/stable/1907947. Retrieved 2009-09-02.
  6. ^ Schumpeter (1954) p. 465-468
  7. ^ Jevons, W. S. (1874). The Principles of Science: A Treatise on Logic and Scientific Method. Macmillan. pp. 78. http://books.google.com/books?id=Qsmn7bAvzxoC.
  8. ^ Jevons, W. S. (1871). The Theory of Political Economy (Quoted in Dow ed.). London: Macmillan. http://www.econlib.org/library/YPDBooks/Jevons/jvnPE.html.
  9. ^ a b Sheila C., Dow (1999-05-21). "The Use of Mathematics in Economics". ESRC Public Understanding of Mathematics Seminar. Birmingham: Economic and Social Research Council. http://www.ioe.ac.uk/esrcmaths/sheila1.html. Retrieved 2008-07-06.
  10. ^ While the concept of cardinality has fallen out of favor in neoclassical economics, the differences between cardinal utility and ordinal utility are minor for most applications.
  11. ^ a b Nicola, PierCarlo (2000). Mainstream Mathermatical Economics in the 20th Century. Springer. pp. 4. ISBN 9783540670841. http://books.google.com/books?id=KR0Rbi8o4QQC. Retrieved 2008-08-21.
  12. ^ Augustin Cournot (1838, tr. 1897) Researches nto the Mathematical Principles of Wealth. Links to description and chapters.
  13. ^ a b Hotelling, Harold (1990). "Stability in Competition". in Darnell, Adrian C.. The Collected Economics Articles of Harold Hotelling. Springer. pp. 51, 52. ISBN 3540970118. OCLC 20217006. http://books.google.com/books?id=dYbbHQAACAAJ. Retrieved 2008-08-21.
  14. ^ "Antoine Augustin Cournot, 1801-1877". History of Economic Thought. The New School for Social Research. http://cepa.newschool.edu/het/profiles/cournot.htm. Retrieved 2008-08-21.
  15. ^ Gibbons, Robert (1992). Game Theory for Applied Economists. Princeton, New Jersey: Princeton University Press. pp. 14, 15. ISBN 0691003955. http://books.google.com/books?id=_6qgHgAACAAJ.
  16. ^ Nicola, p. 9-12
  17. ^ Edgeworth, Francis Ysidro (September 5, 1889). "The Mathematical Theory of Political Economy: Review of Léon Walras, Éléments d'économie politique pure" (PDF). Nature 40 (1036): 434–436. ISSN 0028-0836. http://cepa.newschool.edu/het/texts/edgeworth/edgewalras89.pdf. Retrieved 2008-08-21.
  18. ^ Nicholson, Walter; Snyder, Christopher, p. 350-353
  19. ^ Nicola, p. 14, 15, 258-261
  20. ^ Dixon, Robert. "Walras Law and Macroeconomics". Walras Law Guide. Department of Economics, University of Melbourne. http://www.economics.unimelb.edu.au/rdixon/wlaw.html. Retrieved 2008-09-28.
  21. ^ Dixon, Robert. "A Formal Proof of Walras Law". Walras Law Guide. Department of Economics, University of Melbourne. http://www.economics.unimelb.edu.au/rdixon/walproof.html. Retrieved 2008-09-28.
  22. ^ Rima, Ingrid H. (1977). "Neoclassicism and Dissent 1890-1930". in Weintraub, Sidney. Modern Economic Thought. University of Pennsylvania Press. pp. 10, 11. ISBN 0812277120. http://books.google.com/books?id=JDqAAAAAIAAJ&pgis=1.
  23. ^ Heilbroner, Robert L. (1953 [1999]). The Worldly Philosophers (Seventh ed.). New York, NY: Simon and Schuster. pp. 172–175, 313. ISBN 9780684862149. http://books.google.com/books?id=N_3cj4urgJcC&pgis=1.
  24. ^ Edgeworth, Francis Ysidro (1881 [1961]). Mathematical Physics. London: Kegan Paul [A. M. Kelley]. pp. 15–19. http://books.google.com/books?id=Q4WCGAAACAAJ.
  25. ^ Bowley, Arthur Lyon (1924 [1960]). The Mathematical Groundwork of Economics: an Introductory Treatise. Oxford: Clarendon Press [Kelly]. http://books.google.com/books?id=_cgkAAAAMAAJ&pgis=1.
  26. ^ Gillies, D. B. (1969). "Solutions to general non-zero-sum games". in Tucker, A. W. & Luce, R. D.. Contributions to the Theory of Games. Annals of Mathematics. 40. Princeton, New Jersey: Princeton University Press. pp. 47–85. ISBN 9780691079370. http://books.google.com/books?id=9lSVFzsTGWsC.
  27. ^ Moss, Lawrence S. (2003). "The Seligman-Edgeworth Debate about the Analysis of Tax Incidence: The Advent of Mathematical Economics, 1892–1910". History of Political Economy (Duke University Press) 35 (2): 207, 212, 219, 234–237. doi:10.1215/00182702-35-2-205. ISSN 0018-2702.
  28. ^ Hotelling, Harold (1990). "Note on Edgeworth's Taxation Phenomenon and Professor Garver's Additional Condition on Demand Functions". in Darnell, Adrian C.. The Collected Economics Articles of Harold Hotelling. Springer. pp. 94–122. ISBN 3540970118. OCLC 20217006. http://books.google.com/books?id=dYbbHQAACAAJ. Retrieved 2008-08-26.
  29. ^ Nicholson, Walter; Snyder, Christopher (2007). "General Equilibrium and Welfare". Intermediate Microeconomics and Its Applications (10th ed.). Thompson. pp. 364, 365. ISBN 0324319681.
  30. ^ Jolink, Albert (2006). "What Went Wrong with Walras?". in Backhaus, Juergen G.; Maks, J.A. Hans. From Walras to Pareto. The European Heritage in Economics and the Social Sciences. IV. Springer. doi:10.1007/978-0-387-33757-9_6. ISBN 978-0-387-33756-2.
  31. ^ Blaug, Mark (2007). "The Fundamental Theorems of Modern Welfare Economics, Historically Contemplated". History of Political Economy (Duke University Press) 39 (2): 186–188. doi:10.1215/00182702-2007-001. ISSN 0018-2702.
  32. ^ Blaug (2007), p. 185, 187
  33. ^ Weintraub, E. Roy (1977). "General Equilibrium Theory". in Weintraub, Sidney. Modern Economic Thought. University of Pennsylvania Press. pp. 107–109. ISBN 0812277120. http://books.google.com/books?id=JDqAAAAAIAAJ&pgis=1.
  34. ^ Arrow, Kenneth J.; Debreu, Gérard (1954). "Existence of an Equilibrium for a Competitive Economy". Econometrica (the Econometric Society) 22: 265–290. doi:10.2307/1907353. ISSN 0012-9682.
  35. ^ Herstein, I.N. (October 1953). "Some Mathematical Methods and Techniques in Economics". Quarterly of Applied Mathematics (American Mathematical Society) 11 (3): 249, 252, 260. ISSN 1552-4485.
  36. ^ Screpanti, Ernesto; Zamagni, Stefano (1993). An Outline of the History of Economic Thought. New York, NY: Oxford University Press. pp. 288–290. ISBN 0198283709. OCLC 57281275.
  37. ^ Turner, Fred (2006). From counterculture to cyberculture : Stewart Brand, the Whole Earth Network, and the rise of digital utopianism. Chicago: University Of Chicago Press. ISBN 0226817415. http://books.google.com/books?id=2SNFpgX_WigC.
  38. ^ Nicola, p. 133
  39. ^ Weintraub, E. Roy (2008). "Mathematics and economics". in Durlauf, Steven N.; Blume, Lawrence E.. The New Palgrave Dictionary of Economics (2nd Edition ed.). Macmillan. doi:10.1057/9780230226203.1063. http://www.dictionaryofeconomics.com/article?id=pde2008_M000372. Retrieved 2008-07-07.
  40. ^ Pontryagin, L. S.; Boltyanski, V. G., Gamkrelidze, R. V., Mischenko, E. F. (1962). The Mathematical Theory of Optimal Processes. New York, NY: Wiley. ISBN 68981.
  41. ^ Shell, K., Ed. > (1967). Essays on the Optimal Economic Growth. Cambridge, MA: The MIT Press.
  42. ^ Arrow, K. J.; Kurz, M. (1970). Public Investment, the Rate of Return, and Optimal Fiscal Policy. Baltimore, MD: The Johns Hopkins Press. ISBN 0801811244. Abstract.
  43. ^ Sethi, S. P.; Thompson, G. L.. (2000). Optimal Control Theory: Applications to Management Science and Economics, Second Edition. New York, NY: Springer. ISBN 0792386086.
  44. ^ Metzler, Lloyd (1948). "Review of Foundations of Economic Analysis". American Economic Review 38 (5): 905–910. ISSN 0002-8282. http://www.jstor.org/stable/1811704. Retrieved 2008-05-10.
  45. ^ Samuelson, Paul ((1947) [1983]). Foundations of Economic Analysis. Harvard University Press. ISBN 0-674-31301-1. http://books.google.com/books?id=461gAAAACAAJ.
  46. ^ Brockhaus, Oliver; Farkas, Michael; Ferraris, Andrew; Long, Douglas; Overhaus, Marcus (2000). Equity Derivatives and Market Risk Models. Risk Books. pp. 13–17. ISBN 9781899332878. http://books.google.com/books?id=JGZPAAAAMAAJ&pgis=1. Retrieved 2008-08-17.
  47. ^ Liner, Gaines H. (2002). "Core Journals in Economics". Economic Inquiry (Oxford University Press) 40 (1): 140. doi:10.1093/ei/40.1.138.
  48. ^ Stigler, George J.; Stigler, Steven J.; Friedland, Claire (April 1995). "The Journals of Economics". The Journal of Political Economy (The University of Chicago Press) 103 (2): 339. doi:10.1086/261986. ISSN 0022-3808. http://www.jstor.org/stable/2138643. Retrieved 2008-08-17.
  49. ^ Stigler et al. reviewed journal articles in core economic journals (as defined by the authors but meaning generally non-specialist journals) throughout the 20th century. Journal articles which at any point used geometric representation or mathematical notation were noted as using that level of mathematics as its "highest level of mathematical technique". The authors refer to "verbal techniques" as those which conveyed the subject of the piece without notation from geometry, algebra or calculus.
  50. ^ Stigler et al., p. 342
  51. ^ Sutter, Daniel and Rex Pjesky. "Where Would Adam Smith Publish Today?: The Near Absence of Math-free Research in Top Journals" (May 2007). [1]
  52. ^ Arrow, Kenneth J. (April 1960). "The Work of Ragnar Frisch, Econometrician". Econometrica (Blackwell Publishing) 28 (2): 175, 180. doi:10.2307/1907716. ISSN 0012-9682. http://www.jstor.org/stable/1907716. Retrieved 2008-08-17.
  53. ^ Bjerkholt, Olav (July 1995). "Ragnar Frisch, Editor of Econometrica 1933-1954". Econometrica (Blackwell Publishing) 63 (4): 755. doi:10.2307/2171799. ISSN 0012-9682. http://www.jstor.org/stable/1906940. Retrieved 2008-08-17.
  54. ^ Lange, Oskar (1945). "The Scope and Method of Economics". Review of Economic Studies (The Review of Economic Studies Ltd.) 13 (1): 21. doi:10.2307/2296113. ISSN 0034-6527. http://www.jstor.org/stable/2296113. Retrieved 2008-08-17.
  55. ^ Aldrich, John (January 1989). "Autonomy". Oxford Economic Papers (Oxford University Press) 41 (1, History and Methodology of Econometrics): 26, 27. ISSN 0030-7653. http://www.jstor.org/stable/2663180. Retrieved 2008-09-11.
  56. ^ Epstein, Roy J. (1987). A History of Econometrics. Contributions to Economic Analysis. North-Holland. pp. 13–19. ISBN 9780444702678. OCLC 230844893.
  57. ^ Colander, David C. (2004). "The Strange Persistence of the IS-LM Model". History of Political Economy (Duke University Press) 36 (Annual Supplement): 305–322. doi:10.1215/00182702-36-Suppl_1-305. ISSN 0018-2702.
  58. ^ Brems, Hans (Oct., 1975). "Marshall on Mathematics". Journal of Law and Economics (University of Chicago Press) 18 (2): 583–585. doi:10.1086/466825. ISSN 0022-2186. http://www.jstor.org/pss/725308.
  59. ^ Frigg, R.; Hartman, S. (February 27, 2006). Edward N. Zalta. ed. Models in Science. Stanford Encyclopedia of Philosophy. Stanford, CA: The Metaphysics Research Lab. http://plato.stanford.edu/entries/models-science/#OntWhaMod. Retrieved 2008-08-16.
  60. ^ Hayek, Friedrich (September 1945). "The Use of Knowledge in Society". American Economic Review 35 (4): 519–530. http://www.jstor.org/pss/1809376. Retrieved 2008-04-19.
  61. ^ Boland, L. A. (2007). "Seven Decades of Economic Methodology". in I. C. Jarvie, K. Milford, D.W. Miller. Karl Popper:A Centenary Assessment. London: Ashgate Publishing. pp. 219. ISBN 9780754653752. http://books.google.com/books?id=w-BEoTj0axoC. Retrieved 2008-06-10.
  62. ^ Beed, Clive; Kane, Owen (1991). "What Is the Critique of the Mathematization of Economics?". Kyklos 44 (4): 581–612. doi:10.1111/j.1467-6435.1991.tb01798.x.
  63. ^ Friedman, Milton (1953). Essays in Positive Economics. Chicago: University of Chicago Press. pp. 30, 33, 41. ISBN 9780226264035. http://books.google.com/books?id=rSGekjfpf4cC.
  64. ^ Boland, 220
  65. ^ Heilbroner, Robert (May-June 1999). "The end of the Dismal Science?". Challenge Magazine. http://findarticles.com/p/articles/mi_m1093/is_3_42/ai_54682627/print.
  66. ^ Beed & Owen, 584
  67. ^ Gibson, Warren C. "The Mathematical Romance: An Engineer's View of Mathematical Economics" (April 2005). [2]
  68. ^ Joskow, Paul (May 1975). "Firm Decision-making Policy and Oligopoly Theory". The American Economic Review 65 (2, Papers and Proceedings of the Eighty-seventh Annual Meeting of the American Economic Association): 270–279, Particularly 271. http://www.jstor.org/stable/1818864. Retrieved 2008-04-19.
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  70. ^ Keynes, John Maynard (1936). The General Theory of Employment, Interest and Money. Cambridge: Macmillan. pp. 297. ISBN 0333107292. http://www.marxists.org/reference/subject/economics/keynes/general-theory/ch21.htm.
  71. ^ Paul A. Samuelson (1952). "Economic Theory and Mathematics — An Appraisal," American Economic Review, 42(2), pp. 56, 64-65 (press +).
  72. ^ D.W. Bushaw and R.W. Clower (1957). Introduction to Mathematical Economics, p. vii.

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