Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific,positive claims about controversial or contentious subjects that would be impossible without mathematics.[4] Much of economic theory is currently presented in terms of mathematicaleconomic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.[5]
Broad applications include:
optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker
static (orequilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or theeconomy) is modeled as not changing
comparative statics as to a change from one equilibrium to another induced by a change in one or more factors
Formal economic modeling began in the 19th century with the use ofdifferential calculus to represent and explain economic behavior, such asutility maximization, an early economic application ofmathematical 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 theSecond World War, as ingame theory, would greatly broaden the use of mathematical formulations in economics.[8][7]
This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists.John Maynard Keynes,Robert Heilbroner,Friedrich Hayek and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics.
The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly inGerman 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 termstatistics. 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 asPolitical Arithmetick.[9]Sir William Petty wrote at length on issues that would later concern economists, such as taxation,Velocity of money andnational income, but while his analysis was numerical, he rejected abstract mathematical methodology. Petty's use of detailed numerical data (along withJohn Graunt) would influence statisticians and economists for some time, even though Petty's works were largely ignored by English scholars.[10]
The mathematization of economics began in earnest in the 19th century. Most of the economic analysis of the time was what would later be calledclassical economics. Subjects were discussed and dispensed with throughalgebraic means, but calculus was not used. More importantly, untilJohann Heinrich von Thünen'sThe 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.[11] 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.[12]
Meanwhile, a new cohort of scholars trained in the mathematical methods of thephysical sciences gravitated to economics, advocating and applying those methods to their subject,[13] and described today as moving from geometry tomechanics.[14]These includedW.S. Jevons who presented a paper on a "general mathematical theory of political economy" in 1862, providing an outline for use of the theory ofmarginal utility in political economy.[15] In 1871, he publishedThe Principles of Political Economy, declaring that the subject as science "must be mathematical simply because it deals with quantities". Jevons expected that only collection of statistics for price and quantities would permit the subject as presented to become an exact science.[16] Others preceded and followed in expanding mathematical representations of economicproblems.[17]
Marginalists and the roots of neoclassical economics
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 andLéon Walras built the tools of the discipline axiomatically around utility, arguing that individuals sought to maximize their utility across choices in a way that could be described mathematically.[18] At the time, it was thought that utility was quantifiable, in units known asutils.[19] Cournot, Walras andFrancis Ysidro Edgeworth are considered the precursors to modern mathematical economics.[20]
Cournot, a professor of mathematics, developed a mathematical treatment in 1838 forduopoly—a market condition defined by competition between two sellers.[20] This treatment of competition, first published inResearches into the Mathematical Principles of Wealth,[21] is referred to asCournot duopoly. It is assumed that both sellers had equal access to the market and could produce their goods without cost. Further, it assumed that both goods werehomogeneous. Each seller would vary her output based on the output of the other and the market price would be determined by the total quantity supplied. The profit for each firm would be determined by multiplying their output by the per unitmarket price. Differentiating the profit function with respect to quantity supplied for each firm left a system of linear equations, the simultaneous solution of which gave the equilibrium quantity, price and profits.[22] Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced many of themarginalists.[22][23] Cournot's models of duopoly andoligopoly also represent one of the first formulations ofnon-cooperative games. Today the solution can be given as aNash equilibrium but Cournot's work preceded moderngame theory by over 100 years.[24]
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 ofgeneral competitive equilibrium. The behavior of every economic actor would be considered on both the production and consumption side. Walras originally presented four separate models of exchange, each recursively included in the next. The solution of the resulting system of equations (both linear and non-linear) is the general equilibrium.[25] 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 isWalras' law and the second is the principle oftâtonnement. Walras' method was considered highly mathematical for the time and Edgeworth commented at length about this fact in his review ofÉléments d'économie politique pure (Elements of Pure Economics).[26]
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.[27]
Tâtonnement (roughly, French forgroping 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 onall goods, so everyone has a reservation price for their desired basket of goods).[28]
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 wordtâtonnement is used to describe the directions the market takes ingroping 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.[29]
Edgeworth introduced mathematical elements to Economics explicitly inMathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences, published in 1881.[30] He adoptedJeremy Bentham'sfelicific calculus to economic behavior, allowing the outcome of each decision to be converted into a change in utility.[31] 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".[32]
AnEdgeworth box displaying the contract curve on an economy with two participants. Referred to as the "core" of the economy in modern parlance, there are infinitely many solutions along the curve for economies with two participants[33]
Given two individuals, the set of solutions where both individuals can maximize utility is described by thecontract curve on what is now known as anEdgeworth Box. Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until 1924 byArthur Lyon Bowley.[34] 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 thecore of an economy.[35]
Edgeworth devoted considerable effort to insisting that mathematical proofs were appropriate for all schools of thought in economics. While at the helm ofThe Economic Journal, he published several articles criticizing the mathematical rigor of rival researchers, includingEdwin Robert Anderson Seligman, a noted skeptic of mathematical economics.[36] The articles focused on a back and forth overtax incidence and responses by producers. Edgeworth noticed that a monopoly producing a good that had jointness of supply but not jointness of demand (such as first class and economy on an airplane, if the plane flies, both sets of seats fly with it) might actually lower the price seen by the consumer for one of the two commodities if a tax were applied. Common sense and more traditional, numerical analysis seemed to indicate that this was preposterous. Seligman insisted that the results Edgeworth achieved were a quirk of his mathematical formulation. He suggested that the assumption of a continuous demand function and an infinitesimal change in the tax resulted in the paradoxical predictions.Harold Hotelling later showed that Edgeworth was correct and that the same result (a "diminution of price as a result of the tax") could occur with a discontinuous demand function and large changes in the tax rate.[37]
From the later-1930s, an array of new mathematical tools from differential calculus and differential equations,convex sets, andgraph theory were deployed to advance economic theory in a way similar to new mathematical methods earlier applied to physics.[8][38] The process was later described as moving frommechanics toaxiomatics.[39]
Vilfredo Pareto analyzedmicroeconomics 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 asPareto 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.[40] Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed toAdam Smith'sInvisible hand hypothesis.[41] Rather, Pareto's statement was the first formal assertion of what would be known as thefirst fundamental theorem of welfare economics.[42]
In the landmark treatiseFoundations of Economic Analysis (1947),Paul Samuelson identified a common paradigm and mathematical structure across multiple fields in the subject, building on previous work byAlfred Marshall.Foundations took mathematical concepts from physics and applied them to economic problems. This broad view (for example, comparingLe Chatelier's principle totâtonnement) 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 withcomparative statics, which compares two differentequilibrium states after anexogenous change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.[7][43]
Restricted models of general equilibrium were formulated byJohn von Neumann in 1937.[44] 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 ofBrouwer's fixed point theorem. Von Neumann's model of an expanding economy considered thematrix pencil with nonnegative matrices and; von Neumann soughtprobabilityvectors and, and a positive number that would solve thecomplementarity equationalong with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector represents the prices of the goods, while the probability vector represents the "intensity" at which the production process would run. The uniquesolution represents therate of growth of the economy, which equals theinterest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.[45][46][47] Von Neumann's results have been viewed as a special case oflinear programming, where von Neumann's model uses only nonnegative matrices.[48] The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.[49][50][51]
In 1936, the Russian–born economistWassily Leontief built his model ofinput-output analysis from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by thephysiocrats. With his model, which described a system of production and demand processes, Leontief described how changes in demand in oneeconomic sector would influence production in another.[52] In practice, Leontief estimated the coefficients of his simple models, to address economically interesting questions. Inproduction economics, "Leontief technologies" produce outputs using constant proportions of inputs, regardless of the price of inputs, reducing the value of Leontief models for understanding economies but allowing their parameters to be estimated relatively easily. In contrast, the von Neumann model of an expanding economy allows forchoice of techniques, but the coefficients must be estimated for each technology.[53][54]
Economics is closely enough linked to optimization byagents in aneconomy that an influential definition relatedly describes economicsqua science as the "study of human behavior as a relationship between ends andscarce means" with alternative uses.[57] Optimization problems run through modern economics, many with explicit economic or technical constraints. In microeconomics, theutility maximization problem and itsdual problem, theexpenditure minimization problem for a given level of utility, are economic optimization problems.[58] Theory posits thatconsumers maximize theirutility, subject to theirbudget constraints and thatfirms maximize theirprofits, subject to theirproduction functions,input costs, and marketdemand.[59]
Linear programming was developed to aid the allocation of resources in firms and in industries during the 1930s in Russia and during the 1940s in the United States. During theBerlin airlift (1948), linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet blockade.[66][67]
are the functions of theinequalityconstraints where
are the functions of the equality constraints where.
In allowing inequality constraints, theKuhn–Tucker approach generalized the classic method ofLagrange multipliers, which (until then) had allowed only equality constraints.[68] The Kuhn–Tucker approach inspired further research on Lagrangian duality, including the treatment of inequality constraints.[69][70] The duality theory of nonlinear programming is particularly satisfactory when applied toconvex minimization problems, which enjoy theconvex-analyticduality theory ofFenchel andRockafellar; this convex duality is particularly strong forpolyhedral convex functions, such as those arising inlinear programming. Lagrangian duality and convex analysis are used daily inoperations research, in the scheduling of power plants, the planning of production schedules for factories, and the routing of airlines (routes, flights, planes, crews).[70]
Economic dynamics allows for changes in economic variables over time, including indynamic systems. The problem of finding optimal functions for such changes is studied invariational calculus and inoptimal control theory. Before the Second World War,Frank Ramsey andHarold Hotelling used the calculus of variations to that end. FollowingRichard Bellman's work on dynamic programming and the 1962 English translation of L.Pontryaginet al.'s earlier work,[71] optimal control theory was used more extensively in economics in addressing dynamic problems, especially as toeconomic growth equilibrium and stability of economic systems,[72] of which a textbook example isoptimal consumption and saving.[73] A crucial distinction is between deterministic and stochastic control models.[74] Other applications of optimal control theory include those in finance, inventories, and production for example.[75]
In Russia, the mathematicianLeonid Kantorovich developed economic models inpartially ordered vector spaces, that emphasized the duality between quantities and prices.[79] Kantorovich renamedprices as "objectively determined valuations" which were abbreviated in Russian as "o. o. o.", alluding to the difficulty of discussing prices in the Soviet Union.[78][80][81]
Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly in clarifying the role of prices asnormal vectors to ahyperplane supporting a convex set, representing production or consumption possibilities. However, problems of describing optimization over time or under uncertainty require the use of infinite–dimensional function spaces, because agents are choosing among functions orstochastic processes.[78][82][83][84]
Agent-based computational economics (ACE) as a named field is relatively recent, dating from about the 1990s as to published work. It studies economic processes, including wholeeconomies, asdynamic systems of interactingagents over time. As such, it falls in theparadigm ofcomplex adaptive systems.[95] In correspondingagent-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.[96] The rules are formulated to predict behavior and social interactions based on incentives and information. The theoretical assumption ofmathematicaloptimization by agents markets is replaced by the less restrictive postulate of agents withbounded rationalityadapting to market forces.[97]
ACE models applynumerical methods of analysis tocomputer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.[98] Starting from specified initial conditions, the computationaleconomic system is modeled as evolving over time as its constituent agents repeatedly interact with each other. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of the economy.[99] In contrast to other standard modeling methods, ACE events are driven solely by initial conditions, whether or not equilibria exist or are computationally tractable. ACE modeling, however, includes agent adaptation, autonomy, and learning.[100] It has a similarity to, and overlap with,game theory as an agent-based method for modeling social interactions.[94] Other dimensions of the approach include such standard economic subjects ascompetition andcollaboration,[101]market structure andindustrial organization,[102]transaction costs,[103]welfare economics[104] andmechanism design,[93]information and uncertainty,[105] andmacroeconomics.[106][107]
The method is said to benefit from continuing improvements in modeling techniques ofcomputer science and increased computer capabilities. Issues include those common toexperimental economics in general[108] and by comparison[109] and to development of a common framework for empirical validation and resolving open questions in agent-based modeling.[110] 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".[111]
The surface of theVolatility 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).[112]
Over the course of the 20th century, articles in "core journals"[113] in economics have been almost exclusively written by economists inacademia. 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."[114] A subjective assessment of mathematical techniques[115] 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.[116] A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 lacked both statistical analysis of data and mathematical expressions that were indexed with numbers at the margin of the page.[117]
Ragnar Frisch coined the word "econometrics" and helped to found both theEconometric Society in 1930 and the journalEconometrica in 1933.[118][119] A student of Frisch's,Trygve Haavelmo publishedThe 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.[120] This linking of statistical analysis of systems to economic theory was also promulgated by the Cowles Commission (now theCowles Foundation) throughout the 1930s and 1940s.[121]
The roots of modern econometrics can be traced to the American economistHenry L. Moore. Moore studied agricultural productivity and attempted to fit changing values of productivity for plots of corn and other crops to a curve using different values of elasticity. Moore made several errors in his work, some from his choice of models and some from limitations in his use of mathematics. The accuracy of Moore's models also was limited by the poor data for national accounts in the United States at the time. While his first models of production were static, in 1925 he published a dynamic "moving equilibrium" model designed to explain business cycles—this periodic variation from over-correction in supply and demand curves is now known as thecobweb model. A more formal derivation of this model was made later byNicholas Kaldor, who is largely credited for its exposition.[122]
TheIS/LM model is aKeynesianmacroeconomic 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 andLiquidity preference). The model is no longer widely taught at the graduate level but is common in undergraduate macroeconomics courses.[123]
Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use ofcalculus andmatrix 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 thatmathematics 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.[124]
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 ofmathematicians.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.[18]
Broadly speaking, formal economic models may be classified asstochastic ordeterministic 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.[125]
TheAustrian school — while making many of the samenormative economic arguments as mainstream economists from marginalist traditions, such as theChicago school — differedmethodologically from mainstream neoclassical schools of economics, in particular in their sharp critiques of the mathematization of economics.[126][127] In an interview in 1999, the economic historianRobert Heilbroner stated that the use of mathematical analysis in economics had brought the feeling that it was a "data-laden science", which did not mean that it actually was a science.[128] He added that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition."[129]
PhilosopherKarl Popper argued that mathematical economics suffered from being tautological, meaning that it consisted merely of mathematics without connection to the real world. In other words, insofar as economics became a mathematical theory, mathematical economics ceased to rely on empirical refutation but rather relied onmathematical proofs and disproof.[130] According to Popper, falsifiable assumptions can be tested by experiment and observation while unfalsifiable assumptions can be explored mathematically for their consequences and for theirconsistency with other assumptions.[131]Milton Friedman declared that "all assumptions are unrealistic". Friedman proposed judging economic models by their predictive performance rather than by the match between their assumptions and reality.[132]
J.M. Keynes wrote inThe General Theory that the assumption that factors were strictly independent was problematic and unrealistic given the interrelatedness of factors in the real world; this undermined much research in mathematical economics.[133]
In response to these criticisms, Paul Samuelson argued that mathematics is a language, repeating a thesis ofJosiah Willard Gibbs. In economics, the language of mathematics is sometimes necessary to represent substantive problems. Moreover, mathematical economics has led to conceptual advances in economics.[134] In particular, Samuelson gave the example ofmicroeconomics, 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 easilywith the aid of mathematics."[135]Robert M. Solow wrote that mathematical economics was the core "infrastructure" of contemporary economics, and a technical subject in its own right.[136]
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