Mathematical economics

Modern mathematical economics

From the late 1930s, an array of new mathematical tools, including differential calculus and differential equations, convex sets, and graph theory, were deployed to advance economic theory in a way similar to the new mathematical methods earlier applied to physics. The process was later described as moving from mechanics to axiomatics.

Differential calculus

Vilfredo Pareto analyzed microeconomics by treating decisions by economic actors as attempts to change a given allotment of goods to another, more preferred allotment. Sets of allocations could then be treated as Pareto efficient (Pareto optimal is an equivalent term) when no exchanges could occur between actors that could make at least one individual better off without making any other individual worse off. Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed to Adam Smith's Invisible hand hypothesis. Rather, Pareto's statement was the first formal assertion of what would be known as the first fundamental theorem of welfare economics.

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 to aggregate groups using comparative statics, which compares two equilibrium states after an exogenous change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.

Linear models

Restricted models of general equilibrium were formulated by John von Neumann in 1937. Unlike earlier versions, the models of von Neumann had inequality constraints. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of Brouwer's fixed point theorem. Von Neumann's model of an expanding economy considered the matrix pencil ${\displaystyle \mathbf{A}-\lambda \mathbf{B}}$ with nonnegative matrices ${\displaystyle \mathbf{A}}$ and ${\displaystyle \mathbf{B}}$; von Neumann sought probability vectors ${\displaystyle \overrightarrow{p}}$ and ${\displaystyle \overrightarrow{q}}$, and a positive number ${\displaystyle \lambda }$ that would solve the complementarity equation $${\displaystyle p^{\mathrm{T}}(\mathbf{A}-\lambda \mathbf{B})q=0,}$$ along with two systems of inequalities expressing economic efficiency. In this model, the (transposed) probability vector ${\displaystyle \overrightarrow{p}}$ represents the prices of the goods, while the probability vector ${\displaystyle \overrightarrow{q}}$ represents the "intensity" at which the production process would run. The unique solution ${\displaystyle \lambda }$ represents the rate of growth of the economy, which equals the interest 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. Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.

Input-output economics

In 1936, the Russian–born economist Wassily Leontief built his model of input-output analysis from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by the physiocrats. With his model, which described a system of production and demand processes, Leontief described how changes in demand in one economic sector would influence production in another. In practice, Leontief estimated the coefficients of his simple models, to address economically interesting questions. In production economics, "Leontief technologies" produce outputs using constant proportions of inputs, regardless of the price of inputs, reducing the value of Leontief models for understanding economies but allowing their parameters to be estimated relatively easily. In contrast, the von Neumann model of an expanding economy allows for choice of techniques, but the coefficients must be estimated for each technology.

Mathematical optimization

In mathematics, mathematical optimization (or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem involves maximizing or minimizing a real function by selecting input values of the function and computing the corresponding values of the function. The solution process includes satisfying general necessary and sufficient conditions for optimality. For optimization problems, specialized notation may be used for 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. Optimization problems run through modern economics, many with explicit economic or technical constraints. In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem for a given level of utility, are economic optimization problems. Theory posits that consumers maximize their utility, subject to their budget constraints and that firms maximize their profits, subject to their production functions, input costs, and market demand.

Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that, in principle, could be tested against empirical data. Newer developments have occurred in dynamic programming and modeling optimization with risk and uncertainty, including applications to portfolio theory, the economics of information, and search theory.

Optimality properties for an entire market system may be stated in mathematical terms, as in the formulation of the two fundamental theorems of welfare economics and in the Arrow–Debreu model of general equilibrium (also discussed below). More concretely, many problems are amenable to analytical (formulaic) solution. Many others may be sufficiently complex to require numerical methods of solution, aided by software. Still others are complex but tractable enough to allow computable methods of solution, in particular computable general equilibrium models for the entire economy.

Linear and nonlinear programming have profoundly affected microeconomics, which had previously been considered only in terms of equality constraints. Many of the mathematical economists who received Nobel Prizes in Economics had conducted notable research using linear programming: Leonid Kantorovich, Leonid Hurwicz, Tjalling Koopmans, Kenneth J. Arrow, Robert Dorfman, Paul Samuelson and Robert Solow.

Linear optimization

Linear programming was developed to aid the allocation of resources in firms and in industries during the 1930s in Russia and during the 1940s in the United States. During the Berlin airlift (1948), linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet blockade.

Nonlinear programming

Extensions to nonlinear optimization with inequality constraints were achieved in 1951 by Albert W. Tucker and Harold Kuhn, who considered the nonlinear optimization problem:

Minimize ${\displaystyle f(x)}$ subject to ${\displaystyle g_i(x)\le 0}$ and ${\displaystyle h_j(x)=0}$ where

${\displaystyle f(\cdot )}$ is the function to be minimized

${\displaystyle g_i(\cdot )}$ are the functions of the ${\displaystyle m}$ inequality constraints where ${\displaystyle i=1,\dots ,m}$

${\displaystyle h_j(\cdot )}$ are the functions of the ${\displaystyle l}$ equality constraints where ${\displaystyle j=1,\dots ,l}$.

In allowing inequality constraints, the Kuhn–Tucker approach generalized the classic method of Lagrange multipliers, which (until then) had allowed only equality constraints. The Kuhn–Tucker approach inspired further research on Lagrangian duality, including the treatment of inequality constraints. The duality theory of nonlinear programming is particularly satisfactory when applied to convex minimization problems, which enjoy the convex-analytic duality theory of Fenchel and Rockafellar; this convex duality is particularly strong for polyhedral convex functions, such as those arising in linear programming. Lagrangian duality and convex analysis are used daily in operations research, in the scheduling of power plants, the planning of production schedules for factories, and the routing of airlines (routes, flights, planes, crews).

Variational calculus and optimal control

Economic dynamics refers to changes in economic variables over time, including within dynamic systems. The problem of finding optimal functions for such changes is studied in variational calculus and in optimal control theory. Before the Second World War, Frank Ramsey and Harold Hotelling used the calculus of variations to that end. Following Richard Bellman's work on dynamic programming and the 1962 English translation of L. Pontryagin et al.'s earlier work, optimal control theory was used more extensively in economics in addressing dynamic problems, especially as to economic growth equilibrium and stability of economic systems, of which a textbook example is optimal consumption and saving. A crucial distinction is between deterministic and stochastic control models. Other applications of optimal control theory include those in finance, inventories, and production, for example.

Functional analysis

It was in the course of proving the existence of an optimal equilibrium in his 1937 model of economic growth that John von Neumann introduced functional analytic methods to include topology in economic theory, in particular, fixed-point theory through his generalization of Brouwer's fixed-point theorem. Following von Neumann's program, Kenneth Arrow and Gérard Debreu formulated abstract models of economic equilibria using convex sets and fixed-point theory. In introducing the Arrow–Debreu model in 1954, they proved the existence (but not the uniqueness) of an equilibrium and also proved that every Walras equilibrium is Pareto efficient; in general, equilibria need not be unique. In their models, the ("primal") vector space represented quantities while the "dual" vector space represented prices.

In Russia, the mathematician Leonid Kantorovich developed economic models in partially ordered vector spaces, emphasizing the duality between quantities and prices. Kantorovich renamed prices as "objectively determined valuations" which were abbreviated in Russian as "o. o. o.", alluding to the difficulty of discussing prices in the Soviet Union.

Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly by clarifying the role of prices as normal vectors to a supporting hyperplane of 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.

Game theory

John von Neumann, working with Oskar Morgenstern on the theory of games, broke new mathematical ground in 1944 by extending functional analytic methods related to convex sets and topological fixed-point theory to economic analysis.

Earlier neoclassical theory had bounded only the range of bargaining outcomes and, in special cases, for example, in a bilateral monopoly or along the contract curve of the Edgeworth box. Von Neumann and Morgenstern's results were similarly weak. Following von Neumann's program, however, John Nash used fixed–point theory to prove conditions under which the bargaining problem and noncooperative games can generate a unique equilibrium solution. Noncooperative game theory has been adopted as a fundamental aspect of experimental economics, behavioral economics, information economics, industrial organization, and political economy. It has also given rise to the subject of mechanism design (sometimes called reverse game theory), which has private and public-policy applications as to ways of improving economic efficiency through incentives for information sharing.

In 1994, Nash, John Harsanyi, and Reinhard Selten received the Nobel Memorial Prize in Economic Sciences for their work on non–cooperative games. Harsanyi and Selten were awarded for their work on repeated games. Later work extended their results to computational methods of modeling.

Agent-based computational economics

Agent-based computational economics (ACE), as a named field, is relatively recent, dating to about the 1990s in published work. It studies economic processes, including whole economies, as dynamic systems of interacting agents over time. As such, it falls in the paradigm of complex adaptive systems. In corresponding agent-based models, agents are not real people but "computational objects modeled as interacting according to rules" ... "whose micro-level interactions create emergent patterns" in space and time. The rules are formulated to predict behavior and social interactions based on incentives and information. The theoretical assumption of mathematical optimization by agent 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 be readily applicable. Starting from specified initial conditions, the computational economic system is modeled as evolving as its constituent agents repeatedly interact with each other. In these respects, ACE has been characterized as a bottom-up culture-dish approach to studying the economy. In contrast to other standard modeling methods, ACE events are driven solely by initial conditions, whether or not equilibria exist or are computationally tractable. ACE modeling, however, includes agent adaptation, autonomy, and learning. It has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions. Other dimensions of the approach include such standard economic subjects as competition and collaboration, market structure and industrial organization, transaction costs, welfare economics and mechanism design, information and uncertainty, and macroeconomics.

The method is said to benefit from ongoing improvements in modeling techniques in computer science and from increased computer capabilities. Issues include those common to experimental economics in general and by comparison and to the development of a common framework for empirical validation and resolving open questions in agent-based modeling. The ultimate scientific objective of the method has been described as "test[ing] theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher's work building appropriately on the work that has gone before".

Mathematicization of economics

Over the course of the 20th century, articles in "core journals" in economics have been almost exclusively written by economists in academia. As a result, much of the material published in those journals concerns economic theory, and "economic theory itself has been continuously more abstract and mathematical." A subjective assessment of mathematical techniques employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1892 to 5.3% in 1990. A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 lacked both statistical analysis of data and mathematical expressions that were indexed with numbers at the margin of the page.

Econometrics

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

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

Application

Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use of calculus and matrix algebra in economic analysis 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 on mathematical methods, and the tools it employs have grown 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.

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 across many areas of economics, and several methodologies have evolved more or less independently of one another.

Discussions of validity

The Austrian school — while making many of the same normative economic arguments as mainstream economists from marginalist traditions, such as the Chicago school — differed methodologically from mainstream neoclassical schools of economics, in particular in their sharp critiques of the mathematization of economics. In an interview in 1999, the economic historian Robert 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. He added that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition."

Philosopher Karl Popper argued that mathematical economics was tautological, meaning 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 on mathematical proofs and disproof. According to Popper, falsifiable assumptions can be tested by experiment and observation while unfalsifiable assumptions can be explored mathematically for their consequences and for their consistency with other assumptions. 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.

J.M. Keynes wrote in The General Theory that the assumption of strict independence among factors was problematic and unrealistic, given their interrelatedness in the real world; this undermined much research in mathematical economics.

In response to these criticisms, Paul Samuelson argued that mathematics is a language, repeating a thesis of Josiah Willard Gibbs. In economics, the language of mathematics is sometimes necessary to represent substantive problems. Moreover, mathematical economics has led to conceptual advances in economics. In particular, Samuelson gave the example of microeconomics, writing that "few people are ingenious enough to grasp [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." Robert M. Solow wrote that mathematical economics was the core "infrastructure" of contemporary economics, and a technical subject in its own right.