Mathematical economics

Mathematical economics is the application of applied methods are beyond simple geometry, and may increase differential together with integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it enable the formulation of theoretical relationships with rigor, generality, and simplicity.

Mathematics helps economists to draw meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the Linguistic communication of mathematics allows economists to gain specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic opinion is currently submission in terms of mathematical economic models, a race of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.

Broad application include:

Formal economic modeling began in the 19th century with the use of differential calculus to survive and explain economic behavior, such as utility maximization, an early economic application of mathematical optimization. Economics became more mathematical as a discipline throughout the first half of the 20th century, but first design of new and generalized techniques in the period around the Second World War, as in game theory, would greatly broaden the usage of mathematical formulations in economics.

This rapid systematizing of economics alarmed critics of the discipline as well as some covered economists. John Maynard Keynes, Robert Heilbroner, Friedrich Hayek and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics.

History

The use of mathematics in the utility of social and economic analysis dates back to the 17th century. Then, mainly in German universities, a race of instruction emerged which dealt specifically with detailed introduced of data as it related to public administration. Gottfried Achenwall lectured in this fashion, coining the term statistics. At the same time, a small group of professors in England determining a method of "reasoning by figures upon matters relating to government" and covered to this practice as Political Arithmetick. Sir William Petty wrote at length on issues that would later concern economists, such(a) as taxation, Velocity of money and national income, but while his analysis was numerical, he rejected summary mathematical methodology. Petty's use of detailed numerical data along with John Graunt would influence statisticians and economists for some time, even though Petty's works were largely ignored by English scholars.

The mathematization of economics began in earnest in the 19th century. almost of the economic analysis of the time was what would later be called classical economics. Subjects were discussed and dispensed with through algebraic means, but calculus was not used. More importantly, until Johann Heinrich von Thünen's The Isolated State in 1826, economists did not defining explicit and abstract models for behavior in an arrangement of parts or elements in a particular form figure or combination. to apply the tools of mathematics. Thünen's utility example of farmland use represents the number one example of marginal analysis. Thünen's work was largely theoretical, but he also mined empirical data in order to try to help his generalizations. In comparison to his contemporaries, Thünen built economic models and tools, rather than applying preceding tools to new problems.

Meanwhile, a new cohort of scholars trained in the mathematical methods of the physical sciences gravitated to economics, advocating and applying those methods to their subject, and described today as moving from geometry to mechanics. These included W.S. Jevons who presented paper on a "general mathematical opinion of political economy" in 1862, providing an outline for use of the theory of marginal utility in political economy. In 1871, he published The Principles of Political Economy, declaring that the subject as science "must be mathematical simply because it deals with quantities". Jevons expected that only collection of statistics for price and quantities would let the subject as presented to become an exact science. Others preceded and followed in expanding mathematical representations of economic problems.

Augustin Cournot and Léon Walras built the tools of the discipline axiomatically around utility, arguing that individuals sought to maximize their utility across choices in a way that could be described mathematically. At the time, it was thought that utility was quantifiable, in units invited as utils. Cournot, Walras and Francis Ysidro Edgeworth are considered the precursors to innovative mathematical economics.

Cournot, a professor of mathematics, developed a mathematical treatment in 1838 for Researches into the Mathematical Principles of Wealth,homogeneous. used to refer to every one of two or more people or matters seller would restyle her output based on the output of the other and the market price would be determined by the statement quantity supplied. The profit for regarded and identified separately. firm would be determined by multiplying their output and the per constituent Market price. Differentiating the profit function with respect to quantity supplied for used to refer to every one of two or more people or things firm left a system of linear equations, the simultaneous or situation. of which gave the equilibrium quantity, price and profits. Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced numerous of the marginalists. Cournot's models of duopoly and Oligopoly also represent one of the first formulations of non-cooperative games. Today the solution can be precondition as a Nash equilibrium but Cournot's work preceded modern game theory by over 100 years.

While Cournot provided a solution for what would later be called partial equilibrium, Léon Walras attempted to formalize discussion of the economy as a whole through a theory of Walras' law and theis the principle of tâtonnement. Walras' method was considered highly mathematical for the time and Edgeworth commented at length about this fact in his review of Éléments d'économie politique pure Elements of Pure Economics.

Walras' law was introduced as a theoreticalto the problem of determining the solutions in general equilibrium. His notation is different from modern notation but can be constructed using more modern summation notation. Walras assumed that in equilibrium, all money would be spent on all goods: every good would be sold at the market price for that good and every buyer would expend their last dollar on a basket of goods. Starting from this assumption, Walras could then show that whether there were n markets and n-1 markets cleared reached equilibrium conditions that the nth market would clear as well. This is easiest to visualize with two markets considered in nearly texts as a market for goods and a market for money. whether one of two markets has reached an equilibrium state, no extra goods or conversely, money can enter or exit themarket, so it must be in a state of equilibrium as well. Walras used this statement to proceed toward a proof of existence of solutions to general equilibrium but it is ordinarily used today to illustrate market clearing in money markets at the undergraduate level.

Tâtonnement roughly, French for groping toward was meant to serve as the practical expression of Walrasian general equilibrium. Walras abstracted the marketplace as an auction of goods where the auctioneer would required out prices and market participants would wait until they could each satisfy their personal reservation prices for the quantity desired remembering here that this is an auction on all goods, so programs has a reservation price for their desired basket of goods.

Only when all buyers arewith the assumption market price would transactions occur. The market would "clear" at that price—no surplus or shortage would exist. The word tâtonnement is used to describe the directions the market takes in groping toward equilibrium, settling high or low prices on different goods until a price is agreed upon for all goods. While the process appears dynamic, Walras only presented a static model, as no transactions would occur until all markets were in equilibrium. In practice, very few markets operate in this manner.

Edgeworth introduced mathematical elements to Economics explicitly in Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences, published in 1881. He adopted Jeremy Bentham's felicific calculus to economic behavior, allowing the outcome of each decision to be converted into a change in utility. Using this assumption, Edgeworth built a framework of exchange on three assumptions: individuals are self-interested, individuals act to maximize utility, and individuals are "free to recontract with another independently of...any third party".

Given two individuals, the set of solutions where 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 1924 by Arthur Lyon Bowley. The contract curve of the Edgeworth box or more generally on any set of solutions to Edgeworth's problem for more actors is referred to as the core of an economy.

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. The articles focused on a back and forth over tax incidence and responses by producers. Edgeworth noticed that a monopoly producing a good that had jointness of render but not jointness of demand such as first a collection of things sharing a common attaches and economy on an airplane, if the plane flies, both sets of seats cruise 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 modify in the tax resulted in the paradoxical predictions. Harold Hotelling later showed that Edgeworth was adjustment and that the same result a "diminution of price as a result of the tax" could arise with a discontinuous demand function and large reform in the tax rate.