File Name: applications of calculus in business and economics .zip
Without choice, there is nothing to study.
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus , difference and differential equations , matrix algebra , mathematical programming , and other computational methods.
Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics.
Formal economic modeling began in the 19th century with the use of differential calculus to represent 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 introduction of new and generalized techniques in the period around the Second World War , as in game theory , would greatly broaden the use of mathematical formulations in economics.
This rapid systematizing of economics alarmed critics of the discipline as well as some noted 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. The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly in German universities, a style of instruction emerged which dealt specifically with detailed presentation 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 established a method of "reasoning by figures upon things relating to government" and referred to this practice as Political Arithmetick.
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. Most 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. 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.
Jevons who presented paper on a "general mathematical theory of political economy" in , providing an outline for use of the theory of marginal utility in political economy. Jevons expected that only collection of statistics for price and quantities would permit the subject as presented to become an exact science. Cournot, a professor of mathematics, developed a mathematical treatment in for duopoly —a market condition defined by competition between two sellers.
It is assumed that both sellers had equal access to the market and could produce their goods without cost. Further, it assumed that both goods were homogeneous.
Each seller would vary her output based on the output of the other and the market price would be determined by the total quantity supplied. The profit for each firm would be determined by multiplying their output and the per unit Market price. Differentiating the profit function with respect to quantity supplied for each firm left a system of linear equations, the simultaneous solution of which gave the equilibrium quantity, price and profits.
Today the solution can be given as a Nash equilibrium but Cournot's work preceded modern game theory by over years. The behavior of every economic actor would be considered on both the production and consumption side.
Walras originally presented four separate models of exchange, each recursively included in the next. The solution of the resulting system of equations both linear and non-linear is the general equilibrium. Walras' law was introduced as a theoretical answer to 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 if 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 most texts as a market for goods and a market for money. If one of two markets has reached an equilibrium state, no additional goods or conversely, money can enter or exit the second market, so it must be in a state of equilibrium as well.
Walras used this statement to move toward a proof of existence of solutions to general equilibrium but it is commonly used today to illustrate market clearing in money markets at the undergraduate level. Walras abstracted the marketplace as an auction of goods where the auctioneer would call 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 everyone has a reservation price for their desired basket of goods.
Only when all buyers are satisfied with the given market price would transactions occur. The market would "clear" at that price—no surplus or shortage would exist.
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 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. 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  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. Pontryagin et al. It was in the course of proving of the existence of an optimal equilibrium in his model of economic growth that John von Neumann introduced functional analytic methods to include topology in economic theory, in particular, fixed-point theory through his generalization of Brouwer's fixed-point theorem.
In introducing the Arrow—Debreu model in , they proved the existence but not the uniqueness of an equilibrium and also proved that every Walras equilibrium is Pareto efficient ; in general, equilibria need not be unique. In Russia, the mathematician Leonid Kantorovich developed economic models in partially ordered vector spaces , that emphasized the duality between quantities and prices. Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly in clarifying the role of prices as normal vectors to a hyperplane supporting a convex set, representing production or consumption possibilities.
However, problems of describing optimization over time or under uncertainty require the use of infinite—dimensional function spaces, because agents are choosing among functions or stochastic processes. John von Neumann 's work on functional analysis and topology broke new ground in mathematics and economic theory. In particular, general equilibrium theorists used general topology , convex geometry , and optimization theory more than differential calculus, because the approach of differential calculus had failed to establish the existence of an equilibrium.
However, the decline of differential calculus should not be exaggerated, because differential calculus has always been used in graduate training and in applications. Moreover, differential calculus has returned to the highest levels of mathematical economics, general equilibrium theory GET , as practiced by the " GET-set " the humorous designation due to Jacques H.
In particular, they were able to prove the existence of a general equilibrium, where earlier writers had failed, because of their novel mathematics: Baire category from general topology and Sard's lemma from differential topology.
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus , difference and differential equations , matrix algebra , mathematical programming , and other computational methods. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Formal economic modeling began in the 19th century with the use of differential calculus to represent 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 introduction of new and generalized techniques in the period around the Second World War , as in game theory , would greatly broaden the use of mathematical formulations in economics. This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists.
functions; applications to business and economics, such as maximum- S–8. MATH + Calculus with Business and Economic Applications. NOTES.
Gordon, Walter O. Wang, and April Allen Materowski. Published by Pearson Learning Solutions.
Gordon W. Pearson Custom Publishing, This single text may be used to cover the content of an applied calculus course for non-science majors.
The first stage of the Memory Revolution in economics is associated with the works published in and by Clive W. Bailey for sharing his outstanding [CrossRef] 2. Meng, it is considered a fractional generalization of business cycle model with memory and time delay, Further, this collection continues with works, nio M. Lopes, the fractional calculus and concept of pseudo-phase space are used for modeling the. Moreover, we compare the results for the fractional model with the integer order model. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. The use of the memory function instead of the indicator of nervousness allows us to take into account the memory effects in the Cagan model.
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In this paper, we discuss the properties of kernels of fractional operators to distinguish the following types of phenomena: fading memory forgetting and power-law frequency dispersion, spatial non-locality and power-law spatial dispersion, distributed lag time delay , distributed scaling dilation , depreciation, and aging.