Concepts of Mathematical Modeling (Dover Books on Mathematics)

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- Concepts of Mathematical Modeling (Dover Books on Mathematics) by Walter J. Meyer

Youll need a lot of mathematical of backgrounds to understand and tackle this book. In practice very few markets operate in this manner. Edgeworth introduced mathematical elements to Economics explicitly in Mathematical Psychics: Given two individuals, the set of solutions where the both individuals can maximize utility is described by the contract curve on what is now known as an Edgeworth Box.

Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until by Arthur Lyon Bowley.

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Edgeworth devoted considerable effort to insisting that mathematical proofs were appropriate for all schools of thought in economics. While at the helm of The Economic Journal , he published several articles criticizing the mathematical rigor of rival researchers, including Edwin Robert Anderson Seligman , a noted skeptic of mathematical economics.

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. From the laters, an array of new mathematical tools from the differential calculus and differential equations, convex sets , and graph theory were deployed to advance economic theory in a way similar to new mathematical methods earlier applied to physics.

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. In the landmark treatise Foundations of Economic Analysis , 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 extension followed on the work of the marginalists in the previous century and extended it significantly. Samuelson approached the problems of applying individual utility maximization over aggregate groups with comparative statics , which compares two different equilibrium states after an exogenous change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.

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. 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. 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 [60] 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. 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. 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. 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.

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.

9780486435152 - Concepts of Mathematical Modeling (Dover Books on Mathematics) by Walter J. Meyer

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. 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. This integration results from the formulation of economic problems as stylized models with clear assumptions and falsifiable predictions. This modeling may be informal or prosaic, as it was in Adam Smith 's The Wealth of Nations , or it may be formal, rigorous and mathematical. Broadly speaking, formal economic models may be classified as stochastic or deterministic and as discrete or continuous.

At a practical level, quantitative modeling is applied to many areas of economics and several methodologies have evolved more or less independently of each other. The Handbook of Mathematical Economics series Elsevier , currently 4 volumes, distinguishes between mathematical methods in economics , v.

Another source with a similar distinction is The New Palgrave: A Dictionary of Economics , 4 vols. In it, a "Subject Index" includes mathematical entries under 2 headings vol. A widely used system in economics that includes mathematical methods on the subject is the JEL classification codes. It originated in the Journal of Economic Literature for classifying new books and articles. The New Palgrave Dictionary of Economics , 2nd ed.

The corresponding footnotes below have links to abstracts of The New Palgrave Online for each JEL category 10 or fewer per page, similar to Google searches. Friedrich Hayek contended that the use of formal techniques projects a scientific exactness that does not appropriately account for informational limitations faced by real economic agents. In an interview in , the economic historian Robert Heilbroner stated: I guess the scientific approach began to penetrate and soon dominate the profession in the past twenty to thirty years.

Retrieved June 16, The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis. Mathematical Logic and the Foundations of Mathematics: An Informal Approach to the Mathematics of Change. Calculus is the study of change—how things change, and how quickly they change. The mathematical study of change, motion, growth or decay is calculus.

An Invitation to Mathematics: From Competitions to Research. The Science of Patterns Science , The Science of Patterns: Communications on Pure and Applied Mathematics. Archived from the original on February 28, A History of Greek Mathematics: From Thales to Euclid.

Archived from the original on March 7, Lion Hunting and Other Mathematical Pursuits: Oxford English Dictionary, on-line version Educational Studies in Mathematics. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form. The quote is Einstein's answer to the question: A History of Mathematics.

American Mathematical Society reprint. Logicism, Intuitionism, and Formalism". Archived from the original on September 6, The Principles of Mathematics. Outlines of a Formalist Philosophy of Mathematics. A Brief History of Mathematics. Event occurs at min. Archived from the original on December 16, Retrieved October 26, Out of Their Minds: Archived PDF from the original on May 14, Retrieved November 9, Proof and Other Dilemmas: Proofs from The Book. Archived from the original on February 20, Retrieved September 14, This may be because humans haven't evolved over the millennia to manipulate mathematical ideas, which are frequently more abstractly encrypted than those of conventional language.

You can point to a real live cow chewing its cud in a pasture and equate it with the letters c—o—w on the page. Archived June 20, , at the Wayback Machine. Putting Chance to Work , World Scientific. Mathematical programming in statistics.

Wiley Series in Probability and Mathematical Statistics. Probability, statistics and optimisation: A Tribute to Peter Whittle previously "A realised path: The Cambridge Statistical Laboratory upto revised " ed.

Special order items

Archived from the original on December 19, A History of Mathematics 2nd ed. Courant, Richard ; Robbins, Herbert An Elementary Approach to Ideas and Methods 2nd ed. Retrieved 26 October Geometry and experience translated by G.

An Introduction to the History of Mathematics 6th ed. Retrieved July 28, A Mind For Numbers: Pappas, Theoni June The Joy Of Mathematics Revised ed. American Journal of Mathematics Corrected, expanded, and annotated revision with an paper by B. Peirce and annotations by his son, C. Peirce, of the lithograph ed. Corrected, expanded, and annotated revision with an paper by B. Google Eprint and as an extract, D. Van Nostrand, , Google Eprint.

In Search of a Better World: Lectures and Essays from Thirty Years. Riehm, Carl August Notices of the AMS. Bulletin of the American Mathematical Society. Retrieved June 24, Waltershausen, Wolfgang Sartorius von [first published ]. The Moment of Proof: The Mathematical Experience Reprint ed. From the Birth of Numbers 1st ed.

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The World of Mathematics. At the Threshold of Exact Science: University of Pennsylvania Press. This article's use of external links may not follow Wikipedia's policies or guidelines. Please improve this article by removing excessive or inappropriate external links, and converting useful links where appropriate into footnote references. October Learn how and when to remove this template message. Mathematics at Wikipedia's sister projects. At Wikiversity , you can learn more and teach others about Mathematics at the School of Mathematics. Linear Multilinear Abstract Elementary. Philosophy of mathematics Mathematical logic Set theory Category theory.

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