Mathematical economics

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Economic mathematics is the application of mathematical methods to represent theory and analyze problems in economics. By convention, the applied method refers to things outside of simple geometry, such as differential and integral calculus, differential and differential equations, algebraic matrices, mathematical programming, and other computational methods. The advantages claimed for this approach are formulations that enable theoretical relationships with rigidity, generality, and simplicity.

Mathematics allows economists to form meaningful and testable propositions on a broad and complex subject that is less informal. Furthermore, the language of mathematics allows economists to make specific and positive claims about controversial or controversial subjects that are impossible without mathematics. Many economic theories today are presented in the form of mathematical economics models, a set of stylized and simplified mathematical relationships that assert to clarify assumptions and implications.

Extensive applications include:

• optimization issues such as goal balance, whether from home, business, or policy maker
A static (or equilibrium) analysis in which an economic unit (such as a household) or an economic system (such as a market or economy) is modeled as unchanged
• Comparative statics for change from one equilibrium to another caused by a change in one or more factors
• dynamic analysis, tracking changes in the economic system over time, for example from economic growth.

Formal economic modeling began in the 19th century with the use of differential calculus to represent and explain economic behavior, such as utility maximization, the initial economic application of mathematical optimization. Economics became more mathematical as discipline throughout the first half of the 20th century, but the introduction of new and common techniques in the period around the Second World War, as in game theory, would greatly expand the use of mathematical formulations in economics.

This rapid economic systematics worries the critics of discipline as well as some of the noted economists. John Maynard Keynes, Robert Heilbroner, Friedrich Hayek and others have criticized the widespread use of mathematical models for human behavior, arguing that some human choices can not be reduced to mathematics.

Video Mathematical economics

Histori

The use of mathematics in the service of social and economic analysis dates back to the 17th century. Later, especially in German universities, an instructional style emerged that specifically addressed the presentation of data in detail as it relates to public administration. Gottfried Achenwall speaks in this way, combining term statistics. At the same time, a small group of professors in England set the method of "reasoning by numbers on matters relating to government" and refers to this practice as Political Arithmetic . Sir William Petty writes at length about issues that will be of concern to economists, such as taxation, Speed ​​of money and national income, but while the analysis is numerical, it rejects the abstract mathematical methodology. The use of detailed numerical data by Petty (along with John Graunt) will affect statisticians and economists for some time, although the work of Petty is largely ignored by British scholars.

Economic mathematization began in the 19th century. Most of the current economic analysis was what came to be called classical economics. Subjects are discussed and shared by algebra, but calculus is not used. More importantly, until Johann Heinrich von ThÃÆ'¼nen The Isolated State in 1826, economists did not develop abstract and explicit models for behavior in order to apply mathematical tools. The model of ThÃÆ'¼nen agricultural land use is the first example of marginal analysis. The ThÃÆ'¼nen work is largely theoretical, but it also mines the empirical data to try to support its generalization. Compared with his contemporaries, ThÃÆ'¼nen built models and economic tools, rather than applying previous tools to new problems.

Meanwhile, a group of new intellectuals trained in the mathematical methods of the physical sciences are interested in economics, advocating and applying those methods to their subject, and described today as moving from geometry to mechanics. This includes W.S. Jevons who presented a paper on "general mathematical theory of political economy" in 1862, outlines the use of marginal utility theory in political economy. In 1871, he published The Principles of Political Economy, stating that the subject as a science "must be mathematical only because it relates to quantity." Jevons hopes the only collection of statistics for price and quantity will allow the subject presented to be an exact science. Others preceded and followed in broadening the mathematical representation of economic problems.

Marjinalists and neoclassical economic roots

Augustin Cournot and LÃÆ' Â © on Walras build axiomatic tools of discipline around the utility, arguing that individuals seek to maximize their utility across choices in ways that can be mathematically explained. At that time, there was the assumption that utilities could be quantified, in units known as utils. Cournot, Walras, and Francis Ysidro Edgeworth are regarded as the precursors of modern mathematical economics.

Augustin Cournot

Cournot, a professor of mathematics, developed mathematical medicine in 1838 for duopoly - a market condition determined by competition between two sellers. The treatment of this competition, first published in Research on the Principles of Maths of Wealth, is referred to as Cournot duopoly. It is assumed that both sellers have equal access to the market and can produce their goods at no cost. Furthermore, it is assumed that both goods are homogeneous. Each seller will vary its output based on output from another and the market price will be determined by the total amount given. The profits for each company will be determined by multiplying their output and market price per unit. Distinguish the function of profit with respect to the quantity provided for each firm leaving the system of linear equations, simultaneous solutions that give quantity, price and equilibrium return. Cournot's contribution to mathematization of economics will be neglected for decades, but ultimately affects many marginalists. Cournot's model of duopoly and oligopoly is also one of the first formulations of non-cooperative games. Today the solution can be given as the Nash equilibrium but Cournot's work preceded the modern game theory for more than 100 years.

LÃÆ' Â © in Walras

While Cournot provides a solution to what is then called partial equilibrium, LÃÆ'  © on Walra seeks to formalize the discussion of the economy as a whole through a general competitive equilibrium theory. The behavior of every economic actor will be considered on the side of production and consumption. Walras initially presented four separate exchange models, each recursively included in the next. The solution of the resulting system of equations (both linear and non-linear) is the general equilibrium. At the time, there was no general solution that could be expressed for many arbitrary equations systems, but Walras's efforts produced two well-known economic results. The first is Walras's law and the second is the principle of tà © ¢ tonnement. Walras's method was thought to be very mathematical for the time and Edgeworth commented at length on this fact in his review of the constellation politics pure (Elements of Pure Economy) Æ' m © ments d'ÃÆ'  © conomie politique pure.

Walra's law was introduced as a theoretical answer to the problem of determining solutions in general equilibrium. Notations differ from modern notations but can be constructed using a more modern notation of summation. Walras assumes that in equilibrium, all money will be spent on all goods: every item will be sold at market price for that item and each buyer will spend their last dollar for a basket of goods. Starting from this assumption, Walras can then show that if there is n market and n-1 market cleared (achieving equilibrium condition) that the market n will be clear as well. It's easiest to visualize with two markets (considered in most texts as a market for goods and the market for money). If one of the two markets has reached a state of equilibrium, no additional goods (or otherwise, money) can enter or exit the second market, so it should be in a balanced state as well. Walras uses this statement to move in the direction of evidence of a solution to general equilibrium but is commonly used today to illustrate the opening of the market in the money market at the undergraduate level.

TÃÆ' Â ¢ tonnement (roughly, French for groping toward ) is intended to serve as a practical expression of the general equilibrium of Walrasian. Walras abstracts the market as an auction of goods where the auctioneer will call the price and market participants will wait until they can meet their personal reservation price for the desired quantity (remember here that this is an auction in all goods, so everyone has a booking price for a basket of goods they want).

Only when all buyers are satisfied with the given market price will be a transaction. The market will be "clear" at that price - no surplus or deficiency will exist. The word tonnement is used to describe the direction the market takes in tapping into the balance, assigning a high or low price to different goods until the price is agreed for all goods. While the process looks dynamic, Walras presents only a static model, since no transaction will occur until all markets are in equilibrium. In practice very few markets operate in this way.

Francis Ysidro Edgeworth

Edgeworth introduced the mathematical elements to Economics explicitly in Psychology Mathematics: An Essay on Mathematical Applications for Moral Sciences, published in 1881. He adopted Jeremy Bentham's felic calculus for economic behavior, enabling the outcome of any decision converted into utility changes. Using this assumption, Edgeworth constructs an exchange model on three assumptions: individuals are attracted to themselves, individuals act to maximize utility, and individuals are "free to contract with others independently... third parties."

Given two individuals, a set of solutions in which both individuals can maximize the utility is described by the contract curve on what is now known as the Edgeworth Box. Technically, the construction of a two-person solution to the Edgeworth problem was not developed graphically until 1924 by Arthur Lyon Bowley. The Edgeworth box contract curve (or more commonly in any set of solutions for the Edgeworth problem for more actors) is referred to as the core of the economy.

Edgeworth devoted much effort to emphasize that mathematical evidence is appropriate for all schools of thought in economics. While at the helm of The Economic Journal he published articles criticizing mathematical rigor of rival researchers, including Edwin Robert Anderson Seligman, a skeptic who noted the mathematical economy. Articles are focused on back and forth over tax incidents and responses by producers. Edgeworth noticed that a monopoly produces goods that have a supply alliance but not a combined demand (such as first class and economy on an airplane, if the plane, both sets of chairs flying with it) may actually lower the price seen by the aircraft. the consumer for one of two commodities if the tax is applied. Common sense and more traditional, numerical analysis seems to suggest that this does not make sense. Seligman insisted that Edgeworth's result was the equation of his mathematical formulation. He suggests that the assumption of a persistent demand function and very small changes in taxes yields a paradoxical prediction. Harold Hotelling then pointed out that Edgeworth is correct and that the same result ("price reduction as a result of taxes") can occur with the function of disconnected demand and major changes in tax rates.

Maps Mathematical economics

Modern economic mathematics

From the 1930s, an array of new mathematical tools of differential calculus and differential equations, convex sets, and graph theory were deployed to advance economic theory in a manner similar to the new mathematical methods previously applied to physics. This process is then described as moving from mechanical to axiom.

Differential calculus

Vilfredo Pareto analyzes microeconomics by treating decisions by economic actors in an attempt to convert the designation of certain goods to another preferred part. The allocation set can then be considered as an efficient Pareto (optimal Pareto is an equivalent term) when no exchange can occur between the offender who can make at least one individual better without making the other individual worse. Pareto evidence is usually combined with Walrassian equilibrium or informally ascribed to the invisible Adam Smith hand hypothesis. In contrast, the Pareto statement is the first formal statement of what will be known as the first fundamental theorem of the welfare economy. These models have no inequalities in the next generation of mathematical economics.

In the important treatise of the Foundation for Economic Analysis (1947), Paul Samuelson identifies the general paradigm and mathematical structure in various fields in the subject, building on previous work by Alfred Marshall. Foundation takes the mathematical concept of physics and applies it to economic issues. This broad view (eg comparing the Le Chatelier principle with tà © tà © tonnement) encourages the fundamental premise of mathematical economics: the system of economic actors can be modeled and their behavior depicted as any other system. This extension followed the work of the marginalists in the previous century and expanded significantly. Samuelson approached the problem of applying individual utility maximization over aggregate groups with comparative statics, which compares two different equilibrium states after exogenous changes in the variables. These and other methods in this book provide a foundation for mathematical economics in the 20th century.

Linear model

The restricted general equilibrium model was formulated by John von Neumann in 1937. Unlike previous versions, the von Neumann model has inequality constraints. For his growing economic model, von Neumann proves the existence and uniqueness of balance using the generalization of Brouwer's fixed point theorem. Von Neumann's model of a developing economy is considered a pencil matrix A -? B with a non-negative matrix A and B ; von Neumann looks for the probability vector p and q and the positive number ? which will solve the complementarity equation

^T (( A -? B ) Ã, q = 0 ,

along with two inequal systems that state economic efficiency. In this model, the transposed vector p represents the price of the goods while the probability vector q represents the "intensity" in which the production process will proceed. Unique solution ? represents the rate of economic growth, which is equal to the interest rate. Proving a positive growth rate and proving that growth rates equal to the interest rate is a remarkable achievement, even for von Neumann. Von Neumann's results have been seen as a special case of linear programming, in which the von Neumann model uses only a non-negative matrix. The study of von Neumann's model of a widespread economy continues to attract the interest of mathematical economists with an interest in computing economics.

Economic input-output

In 1936, the Russian-born economist Wassily Leontief built his input-output analysis model from the 'material balance' table built by Soviet economists, who they themselves worked previously with the physiocrats. With its model, which describes the production and demand process system, Leontief describes how changes in demand in one economic sector will affect production in other sectors. In practice, Leontief estimates the simple model coefficients, to answer economically interesting questions. In a production economy, "Leontief technologies" produce output using a constant proportion of input, regardless of input prices, reducing the Leontief model's value for understanding the economy but allowing their parameters to be estimated relatively easily. In contrast, the von Neumann model of a developing economy allows for choice of techniques, but coefficients must be estimated for each technology.

Mathematical optimization

In mathematics, mathematical optimization (or optimization or mathematical programming) refers to the selection of the best elements of some available alternate series. In the simplest case, the optimization problem involves maximizing or minimizing the real function by selecting the input value of the function and calculating the corresponding values ​​of the function. The solution process includes meeting the general requirements required and sufficient for optimality. For optimization problems, custom notation can be used for functions and inputs. In general, optimization involves finding the best available elements of some of the functions given given domains and can use a variety of different computing optimization techniques.

Economics is quite closely related to the optimization by agents in economies whose related related definitions describe the economics of qua as "the study of human behavior as the relationship between ends and rarities means" with alternative uses. The optimization problem goes through the modern economy, many with explicit economic or technical constraints. In microeconomics, the problem of utility maximization and multiple problems, the problem of minimizing expenditure for certain utility levels, is a matter of economic optimization. Theories argue that consumers maximize their utility, subject to their budget constraints and that companies maximize their profits, subject to their production functions, input costs, and market demand.

Economic equilibrium is studied in optimization theory as the main ingredient of economic theorems which in principle can be tested against empirical data. New developments have taken place in dynamic programming and optimization modeling with risk and uncertainty, including applications for portfolio theory, information economics, and search theory.

The nature of the optimality for the whole market system can be expressed in mathematical terms, as in the formulation of two fundamental theories of economic well-being and in the Arrow-Debreu model of the general equilibrium (also discussed below). More concretely, many problems are suitable for analytical solutions (formulas). Many others may be complex enough to require a numerical method of solution, aided by software. Others are complex but quite easy to work with to enable a computable solution method, in particular a computable general equilibrium model for the whole economy.

Linear and nonlinear programming has greatly affected the microeconomics, which previously only considered equality constraints. Many mathematical economists who received the Nobel Prize in Economics have done important research using linear programming: Leonid Kantorovich, Leonid Hurwicz, Tjalling Koopmans, Kenneth J. Arrow, and Robert Dorfman, Paul Samuelson, and Robert Solow. Both Kantorovich and Koopmans recognize that George B. Dantzig deserves to distribute their Nobel Prize for linear programming. Economists conducting research in nonlinear programming have also won the Nobel Prize, especially Ragnar Frisch in addition to Kantorovich, Hurwicz, Koopmans, Arrow, and Samuelson.

Linear optimization

Linear programming was developed to help allocate resources in companies and in industries during the 1930s in Russia and during the 1940s in the United States. During the air transport of Berlin (1948), linear programming was used to plan delivery of supplies to prevent Berlin from starvation after the Soviet siege.

Nonlinear programming

In allowing inequality constraints, Kuhn-Tucker's approach generalizes the classical method of Lagrange multiplier, which (until then) permits only the boundaries of equality. Kuhn-Tucker's approach inspires further research on Lagrangian duality, including the handling of inequality constraints. The nonlinear programming duality theory is very satisfactory when applied to the problem of convex minimization, which enjoys the duality theory of Fenchel and Rockafellar analytes; this convexity is very strong for the polyhedral convex function, as arising in linear programming. Lagrangian duality and convex analysis are used daily in operations research, in scheduling power plants, planning production schedules for factories, and flight routes (routes, flights, planes, crew).

Variational calculus and optimal control

Economic dynamics allows for changes in economic variables over time, including in dynamic systems. The problem of finding the optimal function for such changes is studied in the variational calculus and in optimal control theory. Before the Second World War, Frank Ramsey and Harold Hotelling used the calculus of variations for that purpose.

Follow Richard Bellman's work on dynamic programming and English translation by L. Pontryagin et al . In 1962 previously, optimal control theory was used more widely in economics in addressing dynamic problems, especially for economic growth. balance and stability of the economic system, where textbook examples are optimal consumption and savings. The important difference is between the deterministic and stochastic control models. Other applications of optimum control theory include those in finance, inventory, and production for example.

Functional analysis

In order to prove the existence of an optimal balance in his 1937 model of economic growth, John von Neumann introduced a functional analytical method to incorporate topologies in economic theory, in particular, fixed-point theory through his generalization of Brouwer's fix- theorem point. Following the von Neumann program, Kenneth Arrow and GÃÆ' Â © rard Debreu formulated an abstract model of economic balance using convex devices and fixed-point theory. In introducing the Arrow-Debreu model in 1954, they prove the existence (but not uniqueness) of equilibrium and also prove that every Walras balance is efficient Pareto; In general, equilibrium does not have to be unique. In their model, the vector space ("primal") represents quantitites while the "double" vector space represents the price .

In Russia, mathematician Leonid Kantorovich developed an economic model in a partially ordered vector space, which emphasized the duality between quantity and price. Kantorovich renames price as "objectively determined valuation" abbreviated in Russian as "o.Ã, o.Ã, o.", Alluding to difficulties discussing prices in the Soviet Union.

Even in a finite dimension, the concept of functional analysis has illuminated economic theory, particularly in explaining the role of price as a normal vector to a hyperplane that supports a convex set, which represents the possibility of production or consumption. However, the problem of describing optimization over time or under uncertainty requires the use of dimensionless function space, since agents choose between stochastic functions or processes.

Decrease and increase of differential

John von Neumann's work on functional analysis and topology in new ground is divided in mathematical and economic theory. He also left the advanced mathematics economy with fewer applications of differential calculus. In particular, general balance theorists use general topology, convex geometry, and more optimization theory of differential calculus, since the differential approach of calculus has failed to establish the existence of equilibrium.

However, differential calculus calculus should not be overstated, since differential calculus is always used in postgraduate and in-app training. In addition, the differential calculus has returned to the highest level of economic mathematics, general equilibrium theory (GET), as practiced by "GET-set" (humor designation because Jacques H. DrÃÆ'¨ze). In the 1960s and 1970s, however, GÃÆ' Â © rard Debreu and Stephen Smale led the revival of the use of differential calculus in mathematical economics. In particular, they were able to prove the existence of a general equilibrium, in which previous writers had failed, because of their novel mathematics: Baire's category of general topology and Sard's lemma of differential topology. Other economists associated with the use of differential analysis include Egbert Dierker, Andreu Mas-Colell, and Yves Balasko. These advances have changed the traditional narrative of the economic history of mathematics, following von Neumann, which celebrates the neglect of differential calculus.

Game theory

John von Neumann, working with Oskar Morgenstern on game theory, solved a new mathematical foundation in 1944 by extending the functional analytical methods associated with convex devices and fixed-point topology theory to economic analysis. Their work thus avoids traditional differential calculus, where the maximum operator does not apply to undifferentiated functions. Continuing the work of von Neumann in cooperative game theory, game theorists Lloyd S. Shapley, Martin Shubik, HervÃÆ' © Moulin, Nimrod Megiddo, Bezalel Peleg affect economic research in politics and economics. For example, research on fair pricing in cooperative games and fair value for voting games led to changes in the rules to vote in the legislature and to calculate costs in public works projects. For example, cooperative game theory is used in designing water distribution systems in southern Sweden and to set tariffs for dedicated telephone lines in the US.

Previous neoclassical theory limits the bargaining range and in special cases, such as bilateral monopolies or along the Edgeworth box contract curve. The results of Von Neumann and Morgenstern are equally weak. Following the von Neumann program, however, John Nash uses fixed-point theory to prove the conditions under which non-cooperative negotiating and game problems can produce 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 led to the subject of designing mechanisms (sometimes called reversed game theories), which have public and private policy applications as a means of increasing economic efficiency through incentives for information sharing.

In 1994, Nash, John Harsanyi, and Reinhard Selten received the Nobel Prize Memorial in Economics, working for non-cooperative games. Harsanyi and Selten were awarded for their work on repeated games. Then work extends their results to the modeling computing method.

Agent-based computing economics

The agent-based computing economy (ACE) as a relatively new named field, originated from around the 1990s as published work. It studies the economic process, including the overall economy, as the dynamic systems of agents interact over time. Thus, it falls within the paradigm of a complex adaptive system. In the corresponding agent-based model, agents are not real people, but "computational objects are modeled as interacting according to the"... "rules whose micro-level interactions create patterns that appear" in time and space. The rules are formulated to predict behavior and social interactions based on incentives and information. The theoretical assumption of mathematical optimization by the agent market is replaced by a less restrictive postulate of agents with bounded to market forces.

The ACE model applies numerical analysis methods for computer-based simulations of complex dynamic problems that more conventional methods, such as theorem formulation, may not find ready use. Starting from the initial conditions specified, the computational economic system is modeled as evolving over time as its constituent agents repeatedly interact with each other. In this regard, ACE has been characterized as a bottom-up cultural food approach to studying economics. Unlike other standard modeling methods, ACE events are driven solely by the initial conditions, whether or not there is a balance or computationally tractability. ACE modeling, however, includes agent adaptation, autonomy, and learning. It has similarities to, and overlaps with, game theory as an agent-based method for modeling social interactions. Other dimensions of this approach include standard economic subjects such as competition and collaboration, market structure and industrial organization, transaction costs, welfare economics and mechanism design, information and uncertainty, and macroeconomics.

This method is said to be beneficial from continuous improvement in computer science modeling techniques and computer upgrading. Issues include those common to experimental economics in general and with comparison and development of a common framework for empirical validation and resolving open-ended questions in agent-based modeling. The main scientific goal of this method has been described as "testing theoretical findings of real-world data in a way that allows empirically supported theories to accumulate over time, with each researcher working to build precisely on the work that has happened before. "

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Economic mathematization

During the 20th century, articles in the "core journals" in economics were almost exclusively written by economists in the academic world. As a result, much of the material transmitted in these journals relates to economic theory, and "economic theory itself is continuously more abstract and mathematical." The subjective assessment of mathematical techniques used in these core journals shows a decrease in articles that use geometric representation as well as mathematical notation from 95% in 1892 to 5.3% in 1990. A 2007 survey of the top ten economic journals found that only 5, 8% of articles published in 2003 and 2004 both have no statistical data analysis and do not have mathematically displayed expressions that are indexed with numbers on page margins.

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Econometrics

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

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

The roots of modern econometrics can be traced to American economist Henry L. Moore. Moore studied agricultural productivity and attempted to adjust the changing productivity values ​​for corn and other plant plots to curves using different elasticity values. Moore made several mistakes in his work, some of his model choices and some of the limitations in the use of mathematics. The accuracy of Moore's models was also limited by bad data for national accounts in the United States at the time. While his first model of production was static, in 1925 he published a dynamic "equilibrium" model designed to explain the business cycle - the periodic variation of overcorrection in the supply and demand curves now known as the spider web model. The more formal derivation of this model was made later by Nicholas Kaldor, who was largely credited for his exposition.

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Apps

Many classical economics can be presented in simple geometric terms or basic mathematical notations. Economic mathematics, however, conventionally uses calculus and algebraic matrices in economic analysis to make strong claims that would be more difficult without such mathematical tools. These tools are a prerequisite for formal study, not only in mathematical economics but in contemporary economic theory in general. Economic problems often involve many variables so that math is the only practical way to attack and solve them. Alfred Marshall argues that any quantifiable economic problem, expressed analytically and solved, must be treated in a mathematical way.

Economics has become increasingly dependent on mathematical methods and the mathematical tools it uses to become more sophisticated. As a result, mathematics becomes more important for professionals in economics and finance. Graduate programs in both economics and finance require strong undergraduate preparation in mathematics to enter and, for this reason, attract a large number of mathematicians. Applied mathematics experts apply mathematical principles to practical problems, such as economic analysis and other related economic issues, and many economic problems are often defined as being integrated into the scope of applied mathematics.

This integration results from the formulation of economic problems as a stylish model with clear assumptions and false predictions. This modeling may be informal or prosaic, such as those in Adam Smith The Wealth of Nations , or may be formal, rigorous and mathematical.

Broadly speaking, formal economic models can be classified as stochastic or deterministic and as discrete or sustainable. At the practical level, quantitative modeling is applied to many areas of economics and some methodologies have evolved more or less independently of one another.

• Stochastic models are formulated using stochastic processes. They model the value that can be observed economically from time to time. Most econometrics are based on statistics to formulate and test hypotheses about these processes or to estimate parameters for those processes. Between World War, Herman Wold developed a representation of stationary stochastic processes in terms of autoregressive models and deterministic trends. Wold and Jan Tinbergen applied time series analysis to economic data. Contemporary research on time series statistics takes into account additional formulations of stationary processes, such as autoregressive moving average models. More general models include the autoregressive conditional heteroskedasticity (ARCH) model and the general ARCH (GARCH) model.
• Non-stochastic mathematical models may be purely qualitative (eg, models involved in some aspects of the theory of social choice) or quantitative (involving rationalization of financial variables, eg with hyperbolic coordinates, and/or specific forms of functional relationships between variables). In some cases, the economic prediction of a model merely confirms the direction of the movement of economic variables, and functional relationships are used only in a qualitative sense: for example, if the price of goods increases, the demand for the goods will decrease.. For such models, economists often use two-dimensional graphs rather than functions.
• Qualitative models are sometimes used. One example is the planning of qualitative scenarios in which the possibility of future events is played. Another example is non-numeric decision tree analysis. Qualitative models are often less precise.

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Classification

According to Mathematical Subject Classification (MSC), mathematical economics falls into Applied mathematics/other classifications of category 91:

Game theory, economics, social sciences and behavior

with the MSC2010 classification for 'Game Theory' in 91Axx code and for 'Economic Mathematics' in code 91Bxx.

Series of Handbook of Mathematical Economics (Elsevier), currently 4 volumes, distinguishes between mathematical methods in economics , v. 1, Part I, and economics in another volume where math is used.

Other sources with similar differences are The New Palgrave: Economic Dictionary (1987, 4 vols, 1,300 subject entries). In it, the "Subject Index" includes mathematical entries under 2 titles (volume IV, pp. 982-3):

Economic Mathematics (24 listed, such as "acyclicity", "aggregation problem", "comparative static", "lexicographic sequence", "linear model", "sequence", and "qualitative economy")

Methods of Math (42 listed, such as "calculus variation", "theory of disasters", "combinatorics," "general equilibrium calculations", "convexity", "convex programming", and "stochastic optimal control").

The most widely used system in economics that includes mathematical methods on the subject is the JEL classification code. It comes from the Journal of Economic Literature to classify new books and articles. The relevant categories are listed below (simplified below to remove the "Other" and "Other" JEL) codes, as reproduced from the JEL classification code # The mathematical and quantitative methods of JEL: C Subcategories. The New Palgrave Economic Dictionary (2008, second edition) also uses JEL codes to classify the entries. The related footnote below has a link to the abstract of The New Palgrave Online for each category of JEL (10 or fewer per page, similar to Google search).

JEL: C02 - Mathematical Method (following JEL: C00 - General and JEL: C01 - Econometrics)

JEL: C6 - Math Method; Programming Models; Mathematical Modeling and Simulation

JEL: C60 - General

JEL: C61 - Optimization techniques; Programming model; Dynamic analysis

JEL: C62 - Existence conditions and balance stability

JEL: C63 - Computational techniques; Simulation modeling

JEL: C67 - Model input-output

JEL: C68 - Computable General Equilibrium models

JEL: C7 - Game theory and bargaining theory

JEL: C70 - General

JEL: C71 - Co-operative game

JEL: C72 - Uncooperative Game

JEL: C73 - Stochastic and Dynamic games; Evolutionary game; Repeat Games

JEL: C78 - The bargaining theory; A suitable theory

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Criticism and defense

Mathematical adequacy for qualitative and elaborate economics

Friedrich Hayek argues that the use of formal techniques projects a scientific precision that does not precisely explain the limitations of information faced by real economic agents.

In an interview, economic historian Robert Heilbroner stated:

I think the scientific approach began to penetrate and soon dominated this profession in the last twenty or thirty years. This happens in part because of the "discovery" of mathematical analysis of various types and, indeed, considerable improvements in them. This is the age at which we not only have more data but also more sophisticated data usage. So there is a strong feeling that this is a science full of data and data-laden work, which, based on numbers, mere equations, and page views of journals, has a certain resemblance to science... One main activity that looks scientific. I understand that. I think it's original. This is close to becoming a universal law. But it resembles a different science from science.

Heilbroner states that "most/many economies are not naturally quantitative and therefore do not allow mathematical expositions."

Test the prediction of mathematical economy

The philosopher Karl Popper discussed the economic scientific position of the 1940s and 1950s. He argues that the mathematical economy suffers from being tautological. In other words, as far as economics is mathematical theory, mathematical economics ceases to rely on empirical rebuttal but rely more on mathematical evidence and incompetence. According to Popper, falsifiable assumptions can be tested by experiments and observations while unwarranted assumptions can be mathematically explored for the consequences and for consistency with other assumptions.

Sharing Popper's concerns about assumptions in economics in general, and not just mathematical economics, Milton Friedman states that "all assumptions are unrealistic". Friedman proposes to assess economic models with their predictive performance, not by a match between their assumptions and reality.

Economic mathematics as a pure mathematical form

Mempertimbangkan ekonomi matematika, J.M. Keynes menulis dalam The General Theory :

It is a big mistake of the symbolic pseudo-mathematical method to formalize the system of economic analysis... that they expressly assume strict freedom between the factors involved and the loss of efficacy and their authority if this hypothesis is not allowed; whereas, in ordinary discourse, where we do not blindly manipulate and know all the time what we do and what words mean, we can keep 'behind our head' the necessary reserves and qualifications and adjustments we will have to make later, in a way in which we can not keep a complicated partial differential 'behind' some algebraic pages that assume they are all gone. Too much of the recent proportion of 'mathematical' economics is merely an ingredient, just as unreliable as their original assumptions, allowing authors to forget the complexity and interdependence of the real world in the maze of symbols and unhelpful.

The economic defense of mathematics

In response to these criticisms, Paul Samuelson argues that mathematics is a language, repeating Josiah Willard Gibbs's thesis. In economics, the language of mathematics is sometimes necessary to represent substantive problems. In addition, mathematical economics has led to conceptual advances in economics. Specifically, Samuelson provides a microeconomic example, writing that "some people are clever enough to understand the more complicated parts... without switching to the math language, while most ordinary people can do so quite easily > with math help. "

Some economists argue that mathematical economics deserves support like other forms of mathematics, especially its neighbors in mathematical optimization and mathematical statistics and increasingly in theoretical computer science. Economic mathematics and other mathematical sciences have a history where theoretical advancements regularly contribute to more applied branches of the economy. In particular, following John von Neumann's program, game theory now provides a foundation for describing many applied economies, from statistical decision theory (as "game against nature") and econometrics to the theory of general equilibrium and industrial organization. In the last decade, with the advent of the internet, mathematical economists and optimization experts and computer scientists have worked on the issue of pricing for on-line services --- their contributions use the mathematics of cooperative game theory, indistinguishable optimization, and combinatorial games.

Robert M. Solow concludes that the mathematical economy is the core "infrastructure" of the contemporary economy:

Economics is no longer a piece of conversation that fits both women and men. This has become a technical issue. As with any technical subject, it attracts some people who are more interested in the technique than the subject. It's too bad, but it may be inevitable. However, do not cheat yourself: the technical core of the economy is an indispensable infrastructure for political economy. That is why, if you consult [references in contemporary economics] seeking enlightenment about the world today, you will be brought to a technical, or historical, or not at all.

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Mathematical economist

A prominent mathematical economist includes, but is not limited to, the following (by centuries of birth).

19th century

20th century

21st century

• Ejaz Gul

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See also

• Econophysics
• Mathematical finance

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References

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Further reading

• Alpha C. Chiang and Kevin Wainwright, [1967] 2005. Basic Methods of Mathematical Economics , McGraw-Hill Irwin. Contents.
• E. Roy Weintraub, 1982. Mathematics for Economists , Cambridge. Contents.
• Stephen Glaister, 1984. Mathematical Methods for Economists , 3rd ed., Blackwell. Contents.
• Akira Takayama, 1985. Economic Mathematics , 2nd ed. Cambridge. Contents.
• Nancy L. Stokey and Robert E. Lucas with Edward Prescott, 1989. Recursive Methods in Economic Dynamics , Harvard University Press. Preview and preview links of chapters.
• A. K. Dixit, [1976] 1990. Optimization in Economic Theory , 2nd ed., Oxford. Description and preview content.
• Kenneth L. Judd, 1998. Numerical Methods in Economics , MIT Press. Preview and preview links.
• Michael Carter, 2001. Yayasan Ekonomi Matematika , MIT Press. Contents.
• Ferenc Szidarovszky and SÃÆ'¨ndor MolnÃÆ'¡r, 2002. Introduction to the Matrix Theory: With Applications for Business and Economics , World Scientific Publishing. Description and preview.
• D. Wade Hands, 2004. Math Economics Introduction , second edition. Oxford. Contents.
• Giancarlo Gandolfo, [1997] 2009. Economic Dynamics , 4th ed., Springer. Description and preview.
• John Stachurski, 2009. Economic Dynamics: Theory and Computation , MIT Press. Description and preview.

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External links

• Journal of Mathematical Economics Purpose & amp; Coverage
• Mathematical Economics and Financial Mathematics at Curlie (based on DMOZ)
• Erasmus Mundus Master QEM - Quantitative Economic Models and Methods, Quantitative Economic Models and Methods - QEM

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