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. Restricted models of general equilibrium were formulated by John von Neumann in 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.
In this model, the transposed probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run.
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. In , the Russian—born economist Wassily Leontief built his model of input-output analysis from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by the physiocrats.
With his model, which described a system of production and demand processes, Leontief described how changes in demand in one economic sector would influence production in another.
In production economics , "Leontief technologies" produce outputs using constant proportions of inputs, regardless of the price of inputs, reducing the value of Leontief models for understanding economies but allowing their parameters to be estimated relatively easily. In contrast, the von Neumann model of an expanding economy allows for choice of techniques , but the coefficients must be estimated for each technology. In mathematics, mathematical optimization or optimization or mathematical programming refers to the selection of a best element from some set of available alternatives.
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The solution process includes satisfying general necessary and sufficient conditions for optimality. For optimization problems, specialized notation may be used as to the function and its input s. More generally, optimization includes finding the best available element of some function given a defined domain and may use a variety of different computational optimization techniques. Economics is closely enough linked to optimization by agents in an economy that an influential definition relatedly describes economics qua science as the "study of human behavior as a relationship between ends and scarce means" with alternative uses.
In microeconomics, the utility maximization problem and its dual problem , the expenditure minimization problem for a given level of utility, are economic optimization problems.
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Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that in principle could be tested against empirical data. Optimality properties for an entire market system may be stated in mathematical terms, as in formulation of the two fundamental theorems of welfare economics  and in the Arrow—Debreu model of general equilibrium also discussed below.
Many others may be sufficiently complex to require numerical methods of solution, aided by software.
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Linear and nonlinear programming have profoundly affected microeconomics, which had earlier considered only equality constraints. Dantzig deserved to share their Nobel Prize for linear programming.
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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 programming was developed to aid the allocation of resources in firms and in industries during the s in Russia and during the s in the United States. During the Berlin airlift , linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet blockade.
Extensions to nonlinear optimization with inequality constraints were achieved in by Albert W.
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Tucker and Harold Kuhn , who considered the nonlinear optimization problem :. In allowing inequality constraints, the Kuhn—Tucker approach generalized the classic method of Lagrange multipliers , which until then had allowed only equality constraints.
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. Economic dynamics allows for changes in economic variables over time, including in dynamic systems. The problem of finding optimal functions for such changes is studied in variational calculus and in optimal control theory.
Following Richard Bellman 's work on dynamic programming and the English translation of L. Pontryagin et al. It was in the course of proving of the existence of an optimal equilibrium in his model of economic growth that John von Neumann introduced functional analytic methods to include topology in economic theory, in particular, fixed-point theory through his generalization of Brouwer's fixed-point theorem.
In introducing the Arrow—Debreu model in , 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 Russia, the mathematician Leonid Kantorovich developed economic models in partially ordered vector spaces , that emphasized the duality between quantities and prices. Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly in clarifying the role of prices as normal vectors to a hyperplane supporting a convex set, representing production or consumption possibilities.
However, problems of describing optimization over time or under uncertainty require the use of infinite—dimensional function spaces, because agents are choosing among functions or stochastic processes. John von Neumann 's work on functional analysis and topology broke new ground in mathematics and economic theory.
In particular, general equilibrium theorists used general topology , convex geometry , and optimization theory more than differential calculus, because the approach of differential calculus had failed to establish the existence of an equilibrium. However, the decline of differential calculus should not be exaggerated, because differential calculus has always been used in graduate training and in applications. Moreover, differential calculus has returned to the highest levels of mathematical economics, general equilibrium theory GET , as practiced by the " GET-set " the humorous designation due to Jacques H.
In particular, they were able to prove the existence of a general equilibrium, where earlier writers had failed, because of their novel mathematics: Baire category from general topology and Sard's lemma from differential topology. John von Neumann, working with Oskar Morgenstern on the theory of games , broke new mathematical ground in by extending functional analytic methods related to convex sets and topological fixed-point theory to economic analysis.
Continuing von Neumann's work in cooperative game theory , game theorists Lloyd S. For example, research on the fair prices in cooperative games and fair values for voting games led to changed rules for voting in legislatures and for accounting for the costs in public—works projects. For example, cooperative game theory was used in designing the water distribution system of Southern Sweden and for setting rates for dedicated telephone lines in the USA.
Earlier neoclassical theory had bounded only the range of bargaining outcomes and in special cases, for example bilateral monopoly or along the contract curve of the Edgeworth box. 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. Harsanyi and Selten were awarded for their work on repeated games. Later work extended their results to computational methods of modeling.
Agent-based computational economics ACE as a named field is relatively recent, dating from about the s as to published work. It studies economic processes, including whole economies , as dynamic systems of interacting agents over time. As such, it falls in the paradigm of complex adaptive systems. The theoretical assumption of mathematical optimization by agents markets is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces.
ACE models apply numerical methods of analysis to computer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.
In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of the economy. ACE modeling, however, includes agent adaptation, autonomy, and learning. The method is said to benefit from continuing improvements in modeling techniques of computer science and increased computer capabilities. Issues include those common to experimental economics in general  and by comparison  and to development of a common framework for empirical validation and resolving open questions in agent-based modeling.
Over the course of the 20th century, articles in "core journals"  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. 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 and the journal Econometrica in The roots of modern econometrics can be traced to the American economist Henry L.
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 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.
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. 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.