Mathematical economics

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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.

3239:. 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 1925 he published a dynamic "moving equilibrium" model designed to explain business cycles—this periodic variation from over-correction in supply and demand curves is now known as the 3181:. 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." A subjective assessment of mathematical techniques employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1892 to 5.3% in 1990. A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 both lacked statistical analysis of data and lacked displayed mathematical expressions that were indexed with numbers at the margin of the page. 7586:

not blindly manipulating and know all the time what we are doing and what the words mean, we can keep ‘at the back of our heads’ the necessary reserves and qualifications and the adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials ‘at the back’ of several pages of algebra which assume they all vanish. Too large a proportion of recent ‘mathematical’ economics are merely concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.

14287: 8060: 1076: 7630:, game theory now provides the foundations for describing much of applied economics, from statistical decision theory (as "games against nature") and econometrics to general equilibrium theory and industrial organization. In the last decade, with the rise of the internet, mathematical economists and optimization experts and computer scientists have worked on problems of pricing for on-line services --- their contributions using mathematics from cooperative game theory, nondifferentiable optimization, and combinatorial games. 1500: 5837: 40: 1436:. 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. At the time, no general solution could be expressed for a system of arbitrarily many equations, but Walras's attempts produced two famous results in economics. The first is 1353: 16128: 7518:

more data but more sophisticated use of data. So there is a strong feeling that this is a data-laden science and a data-laden undertaking, which, by virtue of the sheer numerics, the sheer equations, and the sheer look of a journal page, bears a certain resemblance to science . . . That one central activity looks scientific. I understand that. I think that is genuine. It approaches being a universal law. But resembling a science is different from being a science.

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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.

5325: 1654:) drives the fundamental premise of mathematical economics: systems of economic actors may be modeled and their behavior described much like any other system. This extension followed on the work of the marginalists in the previous century and extended it significantly. Samuelson approached the problems of applying individual utility maximization over aggregate groups with 3256: 7101: 1459:) 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 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 3203:, 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. 1404:. 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. Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced many of the 1334:, 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 permit the subject as presented to become an exact science. Others preceded and followed in expanding mathematical representations of economic 11426: 3072:, agents are not real people but "computational objects modeled as interacting according to rules" ... "whose micro-level interactions create emergent patterns" in space and time. The rules are formulated to predict behavior and social interactions based on incentives and information. The theoretical assumption of 1831:, "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 6167: 8242: 8237: 8232: 5321:

Based on the model, it seems possible that we may achieve a rise in the wage of a worker greater than the amount of the tax cut. But that only considers the static effect, and we know that the dynamic effect must be accounted for. In the dynamic model, we may rewrite the equation for changes in taxes

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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

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in general and by comparison and to development of a common framework for empirical validation and resolving open questions in agent-based modeling. The ultimate scientific objective of the method has been described as "test theoretical findings against real-world data in ways that permit empirically

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It is a great fault of symbolic pseudo-mathematical methods of formalising a system of economic analysis ... that they expressly assume strict independence between the factors involved and lose their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are

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I guess the scientific approach began to penetrate and soon dominate the profession in the past twenty to thirty years. This came about in part because of the "invention" of mathematical analysis of various kinds and, indeed, considerable improvements in it. This is the age in which we have not only

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This result showing that, under certain assumptions, a corporate tax cut can boost the wages of workers by more than the lost revenue does not imply that the magnitude is correct. Rather, it suggests a basis for policy analysis that is not grounded in handwaving. If the assumptions are reasonable,

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and responses by producers. 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

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in 1826, economists did not develop explicit and abstract models for behavior in order to apply the tools of mathematics. Thünen's model of farmland use represents the first example of marginal analysis. Thünen's work was largely theoretical, but he also mined empirical data in order to attempt to

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Economics is no longer a fit conversation piece for ladies and gentlemen. It has become a technical subject. Like any technical subject it attracts some people who are more interested in the technique than the subject. That is too bad, but it may be inevitable. In any case, do not kid yourself: the

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from mainstream neoclassical schools of economics, in particular in their sharp critiques of the mathematization of economics. Friedrich Hayek contended that the use of formal techniques projects a scientific exactness that does not appropriately account for informational limitations faced by real

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to economic behavior, allowing the outcome of each decision to be converted into a change in utility. Using this assumption, Edgeworth built a model of exchange on three assumptions: individuals are self-interested, individuals act to maximize utility, and individuals are "free to recontract with

5832:{\displaystyle {\begin{aligned}{dx \over {d\tau }}&=f'(k)k+\tau \left{dk \over {d\tau }}\\&=f'(k)k+{\tau \over {1-\tau }}{^{2}+f'(k)f''(k)k \over {f''(k)}}\\&={\tau \over {1-\tau }}{f'(k)^{2} \over {f''(k)}}+{1 \over {1-\tau }}f'(k)k\\&={f'(k) \over {1-\tau }}\left\end{aligned}}} 1451:

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

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Stigler et al. reviewed journal articles in core economic journals (as defined by the authors but meaning generally non-specialist journals) throughout the 20th century. Journal articles which at any point used geometric representation or mathematical notation were noted as using that level of

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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

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in 1944, where he asserted that precise statistical analysis could be used as a tool to validate mathematical theories about economic actors with data from complex sources. This linking of statistical analysis of systems to economic theory was also promulgated by the Cowles Commission (now the

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is modeled as evolving over time as its constituent agents repeatedly interact with each other. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of the economy. In contrast to other standard modeling methods, ACE events are driven solely by initial

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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

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discussed the scientific standing of economics in the 1940s and 1950s. He argued that mathematical economics suffered from being tautological. In other words, insofar as economics became a mathematical theory, mathematical economics ceased to rely on empirical refutation but rather relied on

9713:" hypothesis by David Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every economic good. Weaker "irreducibility" conditions were given by 6731: 4736: 12592: 2939:

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 US.

2523:, which (until then) had allowed only equality constraints. The Kuhn–Tucker approach inspired further research on Lagrangian duality, including the treatment of inequality constraints. The duality theory of nonlinear programming is particularly satisfactory when applied to 1613:(Pareto optimal is an equivalent term) when no exchanges could occur between actors that could make at least one individual better off without making any other individual worse off. Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed to 11190:. Lectures given at the Johns Hopkins University, Baltimore, Maryland, June, 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. vi+74 pp. 6726: 1367:

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 known as

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Linear and nonlinear programming have profoundly affected microeconomics, which had earlier considered only equality constraints. Many of the mathematical economists who received Nobel Prizes in Economics had conducted notable research using linear programming:

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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,

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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

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technical core of economics is indispensable infrastructure for the political economy. That is why, if you consult looking for enlightenment about the world today, you will be led to technical economics, or history, or nothing at all.

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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,

1400:. 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 by the per unit 5922: 7599:. In economics, the language of mathematics is sometimes necessary for representing substantive problems. Moreover, mathematical economics has led to conceptual advances in economics. In particular, Samuelson gave the example of 4062: 11826: 10794: 10151: 7626:. Mathematical economics and other mathematical sciences have a history in which theoretical advances have regularly contributed to the reform of the more applied branches of economics. In particular, following the program of 2811:

led a revival of the use of differential calculus in mathematical economics. In particular, they were able to prove the existence of a general equilibrium, where earlier writers had failed, because of their novel mathematics:

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between variables). In some cases economic predictions of a model merely assert the direction of movement of economic variables, and so the functional relationships are used only in a qualitative sense: for example, if the

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in 1937. Unlike earlier versions, the models of von Neumann had inequality constraints. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of

6253: 1793:. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann. Von Neumann's results have been viewed as a special case of 13718: 11666: 3862: 10664: 10221: 10835: 10395: 10006: 3315:

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.

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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.

11358: 6332: 11316: 11488: 10682: 11678: 11472: 10281: 10262: 1797:, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics. 4413: 7539:

and disproof. According to Popper, falsifiable assumptions can be tested by experiment and observation while unfalsifiable assumptions can be explored mathematically for their consequences and for their

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displaying the contract curve on an economy with two participants. Referred to as the "core" of the economy in modern parlance, there are infinitely many solutions along the curve for economies with two

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The great appeal of mathematical economics is that it brings a degree of rigor to economic thinking, particularly around charged political topics. For example, during the discussion of the efficacy of a

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broke new ground in mathematics and economic theory. It also left advanced mathematical economics with fewer applications of differential calculus. In particular, general equilibrium theorists used

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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.

7096:{\displaystyle {\begin{aligned}1+\tau {f' \over {kf''}}&=1-{\tau \over {(1-\rho )-\alpha (1-\rho )}}\\&={(1-\rho -\tau )-\alpha (1-\rho ) \over {(1-\rho )-\alpha (1-\rho )}}\end{aligned}}} 6602: 12429: 11863: 3105:

conditions, whether or not equilibria exist or are computationally tractable. ACE modeling, however, includes agent adaptation, autonomy, and learning. It has a similarity to, and overlap with,

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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

2621:. A crucial distinction is between deterministic and stochastic control models. Other applications of optimal control theory include those in finance, inventories, and production for example. 8667: 5135: 10026: 7551:

declared that "all assumptions are unrealistic". Friedman proposed judging economic models by their predictive performance rather than by the match between their assumptions and reality.

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Fagiolo, Giorgio, Alessio Moneta, and Paul Windrum (2007). "A Critical Guide to Empirical Validation of Agent-Based Models in Economics: Methodologies, Procedures, and Open Problems",

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of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use. Starting from specified initial conditions, the computational

2836:. These advances have changed the traditional narrative of the history of mathematical economics, following von Neumann, which celebrated the abandonment of differential calculus. 10348: 6594: 3048:

Agent-based computational economics (ACE) as a named field is relatively recent, dating from about the 1990s as to published work. It studies economic processes, including whole

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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

5130: 4741: 11663: 6533: 1220:. 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 12739:

mathematics as its "highest level of mathematical technique". The authors refer to "verbal techniques" as those which conveyed the subject of the piece without notation from

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is a 3-D surface whereby the current market implied volatility (Z-axis) for all options on the underlier is plotted against strike price and time to maturity (X & Y-axes).

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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. The word

1269:. 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 10661: 1609:

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

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Equilibrium quantities as a solution to two reaction functions in Cournot duopoly. Each reaction function is expressed as a linear equation dependent upon quantity demanded.

11071:(2002). "History of mathematical programming in the USSR: Analyzing the phenomenon (Chapter 3 The pioneer: L. V. Kantorovich, 1912–1986, pp. 405–407)". 3891: 2451: 2350: 1622: 10218: 2509: 2414: 12484: 10832: 10392: 10003: 7431: 1992:

is studied in optimization theory as a key ingredient of economic theorems that in principle could be tested against empirical data. Newer developments have occurred in

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Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use of

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claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical

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as "objectively determined valuations" which were abbreviated in Russian as "o. o. o.", alluding to the difficulty of discussing prices in the Soviet Union.

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would influence production in another. In practice, Leontief estimated the coefficients of his simple models, to address economically interesting questions. In

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were deployed to advance economic theory in a way similar to new mathematical methods earlier applied to physics. The process was later described as moving from

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in which possible future events are played out. Another example is non-numerical decision tree analysis. Qualitative models often suffer from lack of precision.

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support his generalizations. In comparison to his contemporaries, Thünen built economic models and tools, rather than applying previous tools to new problems.

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deserved to share their Nobel Prize for linear programming. Economists who conducted research in nonlinear programming also have won the Nobel prize, notably

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means" with alternative uses. Optimization problems run through modern economics, many with explicit economic or technical constraints. In microeconomics, the

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was developed to aid the allocation of resources in firms and in industries during the 1930s in Russia and during the 1940s in the United States. During the

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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

11844: 10515:. Grundlehren der Mathematischen Wissenschaften . Vol. 306. Berlin: Springer-Verlag. pp. 136–193 (and Bibliographical comments on pp. 334–335). 6374:, then the dynamic effect of lowering capital taxes on wages will be even larger than the static effect. Moreover, if there are positive externalities to 6262: 4308:{\displaystyle {\begin{aligned}{\frac {\partial J}{\partial K}}&=(1-\tau )f'(k)-r\\{\frac {\partial J}{\partial L}}&=(1-\tau )\left\end{aligned}}} 1883:(or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an 12426: 12344: 4977: 12303: 11784: 10770: 10188: 11203: 2554:, in the scheduling of power plants, the planning of production schedules for factories, and the routing of airlines (routes, flights, planes, crews). 1396:. 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 12362: 12201: 11962: 11729: 11524: 9979: 11912: 11068: 6378:, the effect of the tax cut on wages would be larger than in the model we just derived. It is important to note that the result is a combination of: 13155: 10090: 6172: 3538:

is capital per worker. The capital stock adjusts so that the after-tax marginal product of capital equals the exogenously given world interest rate

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represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique

1524:. 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 1243:

and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics.

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Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly in clarifying the role of prices as

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universities, a style of instruction emerged which dealt specifically with detailed presentation of data as it related to public administration.

12528: 12391: 12253: 12078: 9857: 6399: 6162:{\displaystyle {\frac {dw}{dx}}=-{{\frac {f'(k)k}{1-\tau }} \over {{\frac {f'(k)}{1-\tau }}\left}}=-{\frac {1}{\tau {\frac {f'(k)}{kf''(k)}}+1}}} 2924: 1292:

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

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Moss, Lawrence S. (2003). "The Seligman-Edgeworth Debate about the Analysis of Tax Incidence: The Advent of Mathematical Economics, 1892–1910".

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argued that every economic problem which can be quantified, analytically expressed and solved, should be treated by means of mathematical work.

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Functional analysis, optimization, and mathematical economics: A collection of papers dedicated to the memory of Leonid Vitalʹevich Kantorovich

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The fact that, starting at a positive tax rate, the burden of a tax increase exceeds revenue collection due to the first-order deadweight loss

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Heilbroner stated that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition."

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Sutter, Daniel and Rex Pjesky. "Where Would Adam Smith Publish Today?: The Near Absence of Math-free Research in Top Journals" (May 2007).

1285:, but while his analysis was numerical, he rejected abstract mathematical methodology. Petty's use of detailed numerical data (along with 5263: 4318: 15434: 15390: 13130: 12502: 11388: 897: 829: 12138: 11940: 10315: 2787:

more than differential calculus, because the approach of differential calculus had failed to establish the existence of an equilibrium.

1155:. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. 12181: 11804: 11541: 10759: 9462: 9126: 8278: 8139: 7614:

Some economists state that mathematical economics deserves support just like other forms of mathematics, particularly its neighbors in

4731:{\displaystyle {dx \over {d\tau }}=\underbrace {f'(k)k} _{\text{Static}}+\underbrace {\tau \left{dk \over {d\tau }}} _{\text{Dynamic}}} 3109:

as an agent-based method for modeling social interactions. Other dimensions of the approach include such standard economic subjects as

12772: 12564: 11995: 11894: 9078: 15471: 8258: 3023: 2024: 1444:. Walras' method was considered highly mathematical for the time and Edgeworth commented at length about this fact in his review of 1326:

who presented a paper on a "general mathematical theory of political economy" in 1862, providing an outline for use of the theory of

12054: 3888:- assumed to be a constant. A corporate tax cut in this model is equivalent to a tax on capital. With taxes, firms look to maximize: 3177:

Over the course of the 20th century, articles in "core journals" in economics have been almost exclusively written by economists in

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require strong undergraduate preparation in mathematics for admission and, for this reason, attract an increasingly high number of

13454: 9048: 1289:) would influence statisticians and economists for some time, even though Petty's works were largely ignored by English scholars. 13705: 8209: 2613:.'s earlier work, optimal control theory was used more extensively in economics in addressing dynamic problems, especially as to 1666:

change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.

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supported theories to cumulate over time, with each researcher's work building appropriately on the work that has gone before".

1823:. With his model, which described a system of production and demand processes, Leontief described how changes in demand in one 1318:

gravitated to economics, advocating and applying those methods to their subject, and described today as moving from geometry to

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A brief introduction to the infinitesimal calculus: designed especially to aid in reading mathematical economics and statistics

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in 1954, they proved the existence (but not the uniqueness) of an equilibrium and also proved that every Walras equilibrium is

1520:. Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until 1924 by 1282: 13700: 12170: 12127: 13429: 13398: 13368: 13299: 12984: 12615: 11261: 11231: 11214: 11160: 11119:"Leonid Vitaliyevich Kantorovich — Prize Lecture ("Mathematics in economics: Achievements, difficulties, perspectives")" 11043: 10927: 10891: 10858: 10825: 10722: 10569: 10520: 10444: 9917: 9524: 9490: 9420: 9352: 9274: 9161: 9120: 8944: 8871: 8791: 8514: 8487: 8419: 8129: 4125: 2877: 2863: 2516: 2137: 12621: 12220: 10933: 10869: 10618: 9634:

Neumann, J. von (1937). "Über ein ökonomisches Gleichungssystem und ein Verallgemeinerung des Brouwerschen Fixpunktsatzes",

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identified a common paradigm and mathematical structure across multiple fields in the subject, building on previous work by

14110: 12579: 12472:* Namatame, Akira, and Takao Terano (2002). "The Hare and the Tortoise: Cumulative Progress in Agent-based Simulation", in 12105: 6405: 3765: 2120:, linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet blockade. 1039: 12116: 10554:

Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May 15–19, 2000

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for increasing the wages of workers, a simple mathematical model proved beneficial to understanding the issues at hand.

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took mathematical concepts from physics and applied them to economic problems. This broad view (for example, comparing

14255: 13740: 12520:* Tesfatsion, Leigh (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16, 11118: 3398: 1257:

The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly in

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Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16,

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Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16,

11296: 10612: 10173: 9789: 3282:). The model is no longer widely taught at the graduate level but is common in undergraduate macroeconomics courses. 1096: 13482:. Edited by John Eatwell, Murray Milgate and Peter Newman. Four volumes. 4,103 pp. New York: Stockton Press. $ 650)" 13477: 12365: 11965: 11704: 1558:

From the later-1930s, an array of new mathematical tools from the differential calculus and differential equations,

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then the model is an acceptable approximation of reality; if they are not, then better models should be developed.

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from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by the

1694: 1630: 1583: 1388:—a market condition defined by competition between two sellers. This treatment of competition, first published in 11799:(2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics", 11034:(1990). ""My journey in science (supposed report to the Moscow Mathematical Society)" ". In Lev J. Leifman (ed.). 8378: 1911:

may be used as to the function and its input(s). More generally, optimization includes finding the best available

16154: 16044: 14205: 4738:

To find the change in wages, we differentiate the second optimality condition for the per worker wages to obtain:

890: 16132: 11866: 5132:. If we substitute this equation into equation for wage changes with respect to the tax rate, then we find that: 3422:

for that item will decrease. For such models, economists often use two-dimensional graphs instead of functions.

1167:, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications. 16001: 15655: 15496: 14872: 14303: 9808:. Memoirs of the American Mathematical Society. Providence, R.I.: American Mathematical Society. pp. i+74. 8468: 8182: 6721:{\displaystyle {f' \over {kf''}}=-{1 \over {1-\rho -{\alpha (1-\rho ) \over {\alpha +(1-\alpha )k^{-\rho }}}}}} 2932: 2052: 1779: 1698: 1538: 1908: 15608: 15464: 14842: 14832: 14525: 13733: 13581: 12300: 11781: 11380:(1992). "Game Theory and Experimental Economics: Beginnings and Early Influences", in E. R. Weintraub, ed., 11006:

Kantorovich, Leonid, and Victor Polterovich (2008). "Functional analysis", in S. Durlauf and L. Blume, ed.,

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In response to these criticisms, Paul Samuelson argued that mathematics is a language, repeating a thesis of

5842: 3443: 1397: 228: 15053: 13665: 12046: • _____ (1997). "How Economists Can Get Alife", in W. B. Arthur, S. Durlauf, and D. Lane, eds., 1512:

Given two individuals, the set of solutions where both individuals can maximize utility is described by the

1301: 490: 15949: 15859: 14992: 14965: 14315: 12603: 12199: 11726: 11521: 9664: 9660: 7623: 7490: 6555:

is the elasticity of substitution between capital and labor. The relevant quantity we want to calculate is

1943: 1683: 1231:

This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists.

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Sharing Popper's concerns about assumptions in economics generally, and not just mathematical economics,

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2963:. Von Neumann and Morgenstern's results were similarly weak. Following von Neumann's program, however, 1184:) analysis in which the economic unit (such as a household) or economic system (such as a market or the 15394: 14970: 14660: 14650: 13801: 12276: 9534:

Blaug, Mark (2007). "The Fundamental Theorems of Modern Welfare Economics, Historically Contemplated".

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Sandholm, Tuomas W., and Victor R. Lesser (2001)."Leveled Commitment Contracts and Strategic Breach",

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model designed to make predictions about the intersection of "real" economic activity (e.g. spending,

2699:; in general, equilibria need not be unique. In their models, the ("primal") vector space represented 15763: 15491: 14540: 14215: 14187: 13824: 12525: 12453: 12388: 12250: 12075: 12030: 11605: 11549: 11510: 11437: 11405: 11148: 7801: 7731: 7615: 3885: 3073: 2784: 1880: 1874: 1860: 1217: 1174: 1054: 1044: 1034: 243: 13628: 13595: 13555: 13545: 13025: 12477: 11337: 10076: 8348: 8326: 8296: 2198: 1621:

hypothesis. Rather, Pareto's statement was the first formal assertion of what would be known as the

1537:, he published several articles criticizing the mathematical rigor of rival researchers, including 16096: 15944: 15934: 15731: 15457: 15424: 15370: 15113: 15068: 14907: 14778: 14655: 14260: 9968:. Lexington Books. Lexington, Massachusetts: D. C. Heath and Company. pp. xviii+277. 8969: 8097: 7678: 7673: 6337: 2420: 2374: 2319: 2009: 1373: 1360: 1148: 1128: 812: 550: 505: 358: 127: 71: 15128: 11011: 10697: 10047: 8460:

Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content",

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Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content",

7889: 7607:

resorting to the language of mathematics, while most ordinary individuals can do so fairly easily

5078: 2476: 2381: 565: 16127: 15906: 15794: 15689: 15078: 14912: 14902: 14892: 14882: 14620: 14610: 14570: 14560: 14433: 14354: 14145: 14135: 14105: 14039: 13774: 10600: 9909: 9014: 7619: 3196: 3122: 3065: 3031: 2992: 2904: 2851: 2642: 2531: 2048: 1723: 1152: 515: 323: 313: 273: 263: 170: 93: 13177: 10556:. Lecture Notes in Computer Science. Vol. 2241. Berlin: Springer-Verlag. pp. 112–156. 9638:, 8, pp. 73–83, translated and published in 1945-46, as "A Model of General Equilibrium", 7407: 2655:

It was in the course of proving of the existence of an optimal equilibrium in his 1937 model of

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Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods

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The standard result that in a small open economy labor bears 100% of a small capital income tax

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Brockhaus, Oliver; Farkas, Michael; Ferraris, Andrew; Long, Douglas; Overhaus, Marcus (2000).

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Shubik, Martin (2002). "Game Theory and Experimental Gaming", in R. Aumann and S. Hart, ed.,

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Beed, Clive; Kane, Owen (1991). "What Is the Critique of the Mathematization of Economics?".

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Bagwell, Kyle, and Asher Wolinsky (2002). "Game theory and Industrial Organization", ch. 49,

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Now let's assume that instead of the Cobb-Douglas production function we have a more general

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Jevons, W.S. (1866). "Brief Account of a General Mathematical Theory of Political Economy",

8275: 8136: 2828:. Other economists associated with the use of differential analysis include Egbert Dierker, 1541:, a noted skeptic of mathematical economics. The articles focused on a back and forth over 15886: 15726: 15699: 15694: 15591: 15429: 15133: 14877: 14847: 14800: 14763: 14689: 14640: 14605: 14545: 14510: 14445: 14428: 14250: 14130: 14125: 14049: 13950: 13121:. Stanford Encyclopedia of Philosophy. Stanford, California: The Metaphysics Research Lab. 12561: 11992: 11891: 11271: 11170: 11096: 11053: 10579: 10530: 10499: 10482: 10432: 9818:

Mathematical models in economics (Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972)

9651:

For this problem to have a unique solution, it suffices that the nonnegative matrices

9086: 8078: 8023: 7988: 7596: 6375: 4907:{\displaystyle {\frac {dw}{d\tau }}=\left{\frac {dk}{d\tau }}=-f''(k)k{\frac {dk}{d\tau }}} 3402: 3291: 3279: 2964: 2584: 2520: 2159: 1989: 1916: 1832: 1659: 1533: 1413: 1335: 1197: 1181: 1144: 1019: 978: 947: 570: 392: 343: 308: 248: 213: 117: 107: 54: 16:

Application of mathematical methods to represent theories and analyze problems in economics

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Therefore, under a general CES model, the dynamic effect of a capital tax cut on wages is:

2903:–operator did not apply to non-differentiable functions. Continuing von Neumann's work in 2169: 1625:. These models lacked the inequalities of the next generation of mathematical economics. 8: 16066: 16060: 15986: 15901: 15846: 15721: 15586: 15532: 15359: 15168: 14987: 14887: 14867: 14827: 14783: 14768: 14724: 14665: 14590: 14580: 14550: 14473: 14265: 14175: 14097: 13996: 13930: 13887: 13877: 13857: 12051: 11918: 9777: 9767:. Lexington Books. Lexington, Massachusetts: D. C. Heath and Company. pp. xviii+277. 8231: Arrow, Kenneth J., and Michael D. Intriligator, ed., (1981), v. 7929: 7564: 7541: 7367: 7341: 7315: 6558: 6538: 4418: 3573: 3312: 3264: 3236: 3210: 3097: 3080: 3008: 2972: 2881: 2768: 2672: 2664: 2646: 2630: 2571: 2551: 2524: 2032: 1993: 1975: 1900: 1655: 1433: 1393: 1293: 1262: 1232: 1194:

as to a change from one equilibrium to another induced by a change in one or more factors

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Optimal Control Theory: Applications to Management Science and Economics, Second Edition

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their work on non–cooperative games. Harsanyi and Selten were awarded for their work on

2800: 465: 16011: 15576: 15414: 15385: 15343: 15148: 14857: 14837: 14805: 14719: 14714: 14694: 14645: 14585: 14575: 14520: 14515: 14347: 14291: 14210: 14150: 14082: 14072: 14011: 13986: 13862: 13819: 13814: 13531: 13334: 13233: 13097: 13089: 13046: 12957: 12945: 12914: 12906: 12863: 12818: 12721: 12713: 12636: 11100: 11031: 10982: 10846: 10814: 10633: 10583: 9961: 9861: 9760: 9726: 9718: 9612: 9559: 9056: 9006: 8585: 8247: 8065: 7914: 7845: 7830: 7536: 4067: 3569: 3454: 3365: 3345: 3333: 2952: 2739: 2715: 2547: 2456: 2355: 2113: 2104: 2060: 2047:

of solution, aided by software. Still others are complex but tractable enough to allow

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Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences

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problems as to goal equilibrium, whether of a household, business firm, or policy maker

1079: 983: 824: 780: 585: 422: 417: 397: 348: 288: 278: 223: 218: 190: 185: 155: 23: 11635: 8206: 4057:{\displaystyle J=\max _{K,L}\;(1-\tau )\left-rK\equiv \max _{K,L}\;(1-\tau )\leftL-rK} 3405:) or quantitative (involving rationalization of financial variables, for example with 1770:

along with two inequality systems expressing economic efficiency. In this model, the (

15981: 15976: 15964: 15959: 15831: 15821: 15741: 15684: 15628: 15278: 15253: 15163: 15043: 14931: 14734: 14670: 14635: 14625: 14495: 14286: 14006: 13991: 13935: 13882: 13394: 13364: 13353: 13295: 13251: 13122: 13101: 13081: 13038: 12990: 12980: 12949: 12898: 12855: 12810: 12725: 12705: 12611: 12536: 12399: 12350: 12328: 12191: 12159: 12086: 11950: 11926: 11751: 11583: 11292: 11257: 11245: 11210: 11156: 11039: 10974: 10923: 10854: 10821: 10718: 10712: 10608: 10565: 10516: 10440: 10169: 10161: 10039: 9957: 9923: 9913: 9785: 9756: 9722: 9604: 9563: 9551: 9520: 9486: 9410: 9358: 9348: 9318: 9270: 9157: 9147: 9116: 8998: 8940: 8914: 8877: 8867: 8787: 8593: 8577: 8520: 8510: 8482: 8415: 8251: 8125: 8059: 7949: 7870: 7850: 7776: 7766: 7510: 7497:

economic arguments as mainstream economists from marginalist traditions, such as the

3429: 3229: 3148:

The method is said to benefit from continuing improvements in modeling techniques of

3130: 3093: 3069: 3057: 2996: 2927:

influenced economic research in politics and economics. For example, research on the

2873: 2829: 2108: 2088: 2068: 2044: 1963: 1954:

for a given level of utility, are economic optimization problems. Theory posits that

1927: 1888: 1651: 1610: 1597: 1593: 1441: 1278: 1258: 1236: 1164: 1075: 715: 690: 600: 480: 437: 373: 338: 328: 160: 122: 78: 15203: 13034: 12194:(2006). "Growing Adaptive Organizations: An Agent-Based Computational Approach", in 11104: 9010: 8666:, XXIX (June) pp. 282–87. Read in Section F of the British Association, 1862. 7484: 4974:, we may differentiate the first optimality condition for the interest rate to find: 1384:

Cournot, a professor of mathematics, developed a mathematical treatment in 1838 for

640: 15996: 15841: 15816: 15623: 15613: 15603: 15328: 15273: 15258: 15243: 15228: 15158: 15138: 15118: 15073: 14680: 14630: 14600: 14595: 14485: 14411: 14220: 14195: 14067: 13915: 13852: 13537: 13523: 13446: 13330: 13073: 13050: 13030: 12941: 12918: 12890: 12847: 12802: 12697: 12686:; Stigler, Steven J.; Friedland, Claire (April 1995). "The Journals of Economics". 12665: 12167: 11836: 11659: 11631: 11481: 11418: 11199: 11084: 10966: 10911: 10601:

Pontryagin, L. S.; Boltyanski, V. G., Gamkrelidze, R. V., Mischenko, E. F. (1962).

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The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration

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Shubik, Martin (1981). "Game Theory Models and Methods in Political Economy", in

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Nicholson, Walter; Snyder, Christopher (2007). "General Equilibrium and Welfare".

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solution. Noncooperative game theory has been adopted as a fundamental aspect of

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Paul A. Samuelson (1952). "Economic Theory and Mathematics — An Appraisal",

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wrote at length on issues that would later concern economists, such as taxation,

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Bjerkholt, Olav (July 1995). "Ragnar Frisch, Editor of Econometrica 1933-1954".

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Industry and Labor Dynamics: The Agent-Based Computational Economics Approach.

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about these processes or estimate parameters for them. Between the World Wars,

3267: 3142: 3079:

by agents markets is replaced by the less restrictive postulate of agents with

2956: 2920: 2813: 2755: 2535: 2117: 2080: 2076: 2064: 1828: 1675: 1635: 1618: 1606: 1488: 1463:

goods, so everyone has a reservation price for their desired basket of goods).

1437: 1314:

Meanwhile, a new cohort of scholars trained in the mathematical methods of the

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equilibrium and stability of economic systems, of which a textbook example is

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analysis, tracing changes in an economic system over time, for example from

39: 15871: 15811: 15559: 15517: 15348: 15293: 15188: 15178: 15173: 15098: 14943: 14468: 14396: 13968: 13963: 13867: 13714: 13561: 12838: 12793: 12324: 11284: 10957: 10740: 8560: 8073: 8018: 8013: 7983: 7786: 7756: 3857:{\displaystyle Y=AK^{\alpha }L^{1-\alpha }=Ak^{\alpha }L,\quad \alpha \in } 3349: 3275: 3240: 3215: 3200: 3190: 2833: 2583:. The problem of finding optimal functions for such changes is studied in 2141: 2084: 1947: 1938:

science as the "study of human behavior as a relationship between ends and

1864: 1563: 1401: 1228:, would greatly broaden the use of mathematical formulations in economics. 1221: 871: 785: 730: 625: 615: 610: 535: 83: 13725: 13636:

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may be purely qualitative (for example, models involved in some aspect of

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The roots of modern econometrics can be traced to the American economist

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applied time-series analysis to economic data. Contemporary research on

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Martos, Béla (1987). "control and coordination of economic activity",

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Edgeworth introduced mathematical elements to Economics explicitly in

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are beyond simple geometry, and may include differential and integral

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and increased computer capabilities. Issues include those common to

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in addition to Kantorovich, Hurwicz, Koopmans, Arrow, and Samuelson.

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Nisan, Noam, and Amir Ronen (2001). "Algorithmic Mechanism Design",

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Frigg, R.; Hartman, S. (February 27, 2006). Edward N. Zalta (ed.).

13077: 12748: 12740: 12701: 9887: 5064:{\displaystyle {dk \over {d\tau }}={f'(k) \over {(1-\tau )f''(k)}}} 3383:

consider additional formulations of stationary processes, such as

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Formal economic modeling began in the 19th century with the use of

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may be stated in mathematical terms, as in formulation of the two

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of the function. The solution process includes satisfying general

15598: 13844: 13210:(6th ed.). Long Grove, IL: Waveland Press. pp. 574–575. 12744: 12245:, 2001. "Agent-based Computational Transaction Cost Economics", 10955:(1954). "Existence of an equilibrium for a competitive economy". 10439:(Second ed.). Cambridge, Massachusetts.: Athena Scientific. 7485:

Adequacy of mathematics for qualitative and complicated economics

7433:, which is the case when perfect complements exist, we find that 7364:, which is the case when perfect substitutes exist, we find that 3049: 2900: 2895:

to economic analysis. Their work thereby avoided the traditional

2796: 2579:

allows for changes in economic variables over time, including in

1959: 1931: 1385: 1297: 1213: 1185: 964: 10816:

Public Investment, the Rate of Return, and Optimal Fiscal Policy

10307:

Dorfman, Robert, Paul A. Samuelson, and Robert M. Solow (1958).

7525: 7476:- a cut in capital taxes increases wages by exactly one dollar. 3449:

As an intellectual exercise, the following problem was posed by

1376:

are considered the precursors to modern mathematical economics.

13709: 8045: 3419: 3348:. They model economically observable values over time. Most of 1983: 1939: 854: 10692:

Brock, W. A. (1987). "optimal control and economic dynamics",

6248:{\displaystyle {f'(k) \over {kf''(k)}}=-{1 \over {1-\alpha }}} 3332:

Broadly speaking, formal economic models may be classified as

3274:, savings rates) and decisions made in the financial markets ( 3003:(sometimes called reverse game theory), which has private and 1835:, but the coefficients must be estimated for each technology. 12474:

Agent-based Approaches in Economic and Social Complex Systems

10417:. Berkeley: University of California Press. pp. 481–492. 6400:

constant elasticity of substitution (CES) production function

3415: 3392: 1934:

that an influential definition relatedly describes economics

1697:. Von Neumann's model of an expanding economy considered the 1688:

Restricted models of general equilibrium were formulated by

14339: 10367:* Intriligator, Michael D. (2008). "nonlinear programming", 9886:

Chapter 9.1 "The von Neumann growth model" (pages 277–299):

9844: 9480: 4570:

Then the change in taxes per worker, given the tax rate, is:

3243:. A more formal derivation of this model was made later by 3209:

coined the word "econometrics" and helped to found both the

2967:

used fixed–point theory to prove conditions under which the

12791:(April 1960). "The Work of Ragnar Frisch, Econometrician". 12552:(2006). "Computationally Intensive Analyses in Economics", 10027:

An Essay on the Nature and Significance of Economic Science

2557: 2550:. Lagrangian duality and convex analysis are used daily in 1997: 1967: 1369: 1300:

means, but calculus was not used. More importantly, until

12881:

Lange, Oskar (1945). "The Scope and Method of Economics".

10544: 10542: 9966:

Mathematical theory of expanding and contracting economies

9765:

Mathematical theory of expanding and contracting economies

9667:

of nonnegative matrices, which considers the (simplified)

8102:

Mathematical and quantitative methods JEL: C Subcategories

3299:

is the only practical way of attacking and solving them.

2951:

of bargaining outcomes and in special cases, for example

15640: 11774:

Shoham, Yoav (2008). "Computer Science and Game Theory",

11147: 9903: 9432:* Weintraub, E. Roy (2008). "mathematics and economics", 7654:

Prominent mathematical economists include the following.

6169:

Using the Cobb-Douglas production function, we have that:

3199:

and a cadre of mathematically trained economists led to

12154:* Leombruni, Roberto, and Matteo Richiardi, ed. (2004), 11674: _____. "revelation principle." 10506: 8503:

Schumpeter, J.A. (1954). Elizabeth B. Schumpeter (ed.).

3165: 2687:

formulated abstract models of economic equilibria using

13064:

Brems, Hans (October 1975). "Marshall on Mathematics".

12656:

Liner, Gaines H. (2002). "Core Journals in Economics".

11907:, 1994. "Inductive Reasoning and Bounded Rationality", 11818:* Kirman, Alan (2008). "economy as a complex system", 11332:* Nash, John F., Jr. (1950). "The Bargaining Problem", 10539: 10511:(1993). "XII Abstract duality for practitioners". 9389:"Some Mathematical Methods and Techniques in Economics" 8558:

Schnieder, Erich (1934). "Johann Heinrich von Thünen".

8509:. New York: Oxford University Press. pp. 209–212. 3764:

The standard choice for the production function is the

3255: 1899:

values of the function and computing the corresponding

1470:

is used to describe the directions the market takes in

13011:(2004). "The Strange Persistence of the IS-LM Model". 11083:(ISMP 2000, Part 1 (Atlanta, GA), number 3): 401–416. 10922:. University of Pennsylvania Press. pp. 107–109. 10786:

Malliaris, A.G. (2008). "stochastic optimal control",

10213:( 2008). "Arrow–Debreu model of general equilibrium", 8783:

Mainstream Mathermatical Economics in the 20th Century

8272:

Mathematical Economics: Twenty Papers of Gérard Debreu

8116: 7439: 7410: 7370: 7344: 7318: 6561: 6541: 6499: 6340: 5845: 5081: 4940: 4920: 4480: 4421: 4110: 4090: 4070: 3870: 3746: 3724: 3687: 3665: 3544: 3524: 3504: 3469: 3037: 1926:

Economics is closely enough linked to optimization by

1296:. Subjects were discussed and dispensed with through 1123:

methods to represent theories and analyze problems in

13224:(September 1945). "The Use of Knowledge in Society". 12940:(1, History and Methodology of Econometrics): 15–34. 11425:, v. 1, Elsevier, Part 4, Games, ch. 45-66 preview 11256:. Econometric Society monographs. Cambridge UP. 11225: 11223: 9115:. University of Pennsylvania Press. pp. 10, 11. 8819:

Researches into the Mathematical Principles of Wealth

7109: 6734: 6605: 6408: 6265: 6175: 5925: 5328: 5266: 5138: 4980: 4744: 4576: 4500: 4321: 4134: 3894: 3774: 3582: 2880:, broke new mathematical ground in 1944 by extending 2479: 2459: 2423: 2384: 2358: 2322: 2287: 2243: 2201: 2172: 1390:

Researches into the Mathematical Principles of Wealth

13390:

The General Theory of Employment, Interest and Money

12781: 12574:

Tesfatsion, Leigh, and Kenneth L. Judd, ed. (2006).

9832:. Princeton, New Jersey: Princeton University Press. 9345:

The Collected Economics Articles of Harold Hotelling

8913:. The New School for Social Research. Archived from 8864:

The Collected Economics Articles of Harold Hotelling

8041: 7636:

concluded that mathematical economics was the core "

7590: 7555:

Mathematical economics as a form of pure mathematics

5314:{\displaystyle {dw \over {dx}}=-{1 \over {1-\tau }}} 4408:{\displaystyle r=(1-\tau )f'(k),\quad w=f(k)-f'(k)k} 1342:

Marginalists and the roots of neoclassical economics

1212:

to represent and explain economic behavior, such as

13290:. In I. C. Jarvie; K. Milford; D.W. Miller (eds.). 11308:Creedy, John (2008). "Francis Ysidro (1845–1926)", 9956: 9755: 9037:

Nicholson, Walter; Snyder, Christopher, p. 350-353.

3437:

Example: The effect of a corporate tax cut on wages

13719:Models and Methods of Quantitative Economics - QEM 13352: 11599:Handbook of Game Theory with Economic Applications 11535:Handbook of Game Theory with Economic Applications 11434:Handbook of Game Theory with Economic Applications 11220: 11153:Existence and optimality of competitive equilibria 11151:; Brown, Donald J.; Burkinshaw, Owen (1990). 10813: 10656:( 2008). "Pontryagin's principle of optimality", 10343:Dantzig, George B. ( 2008). "linear programming", 10164:, Michael D. Whinston, and Jerry R. Green (1995), 7468: 7425: 7396: 7356: 7330: 7304: 7095: 6720: 6588: 6547: 6527: 6485: 6366: 6326: 6247: 6161: 5911: 5831: 5313: 5248: 5124: 5063: 4966: 4926: 4906: 4730: 4562: 4486: 4466: 4407: 4307: 4116: 4096: 4076: 4056: 3876: 3856: 3752: 3730: 3708: 3671: 3648: 3550: 3530: 3510: 3490: 3428:are occasionally used. One example is qualitative 3329:, or it may be formal, rigorous and mathematical. 2503: 2465: 2445: 2408: 2364: 2344: 2302: 2271: 2229: 2187: 2148:nonlinear optimization with inequality constraints 2087:. Both Kantorovich and Koopmans acknowledged that 2039:). More concretely, many problems are amenable to 1905:necessary and sufficient conditions for optimality 16160:Mathematical and quantitative methods (economics) 13359:. Chicago: University of Chicago Press. pp. 11793: 11771: 10385:Blume, Lawrence E. (2008). "convex programming", 7509:In an interview in 1999, the economic historian 3160: 2542:; this convex duality is particularly strong for 16146: 13706:Mathematical Economics and Financial Mathematics 13152:"Greg Mankiw's Blog: An Exercise for My Readers" 13114: 11289:Foundations of the Theory of General Equilibrium 11229:Neumann, John von, and Oskar Morgenstern (1944) 10820:. Baltimore, Maryland: The Johns Hopkins Press. 10143:* Allan M. Feldman (3008). "welfare economics", 9483:Intermediate Microeconomics and Its Applications 8732:ESRC Public Understanding of Mathematics Seminar 4315:Therefore, the optimality conditions imply that: 3983: 3902: 2745: 1553: 1412:also represent one of the first formulations of 10886:Andrew McLennan, 2008. "fixed point theorems", 10714:Essays on the Theory of Optimal Economic Growth 8763:, the differences between cardinal utility and 8437: 8435: 8433: 8431: 8157:( 2008). "mathematical economics", section II, 3368:of stationary stochastic processes in terms of 1216:maximization, an early economic application of 13320: 12597: 12074:, v. 2, part 2, ACE study of economic system. 11746: 11744: 11250:The Theory of general economic equilibrium: A 9892:Interior point algorithms: Theory and analysis 9630: 9628: 9626: 9506: 9485:(10th ed.). Thompson. pp. 364, 365. 8682:The Principles of Political Economy, pp. 4, 25 3247:, who is largely credited for its exposition. 1623:first fundamental theorem of welfare economics 15465: 14355: 13741: 13634:Ferenc Szidarovszky and Sándor Molnár, 2002. 13515:Fundamental Methods of Mathematical Economics 13513:Alpha C. Chiang and Kevin Wainwright, 2005. 13205: 12050:, pp. 533–564. Addison-Wesley. Pre-pub 12048:The Economy as an Evolving Complex System, II 11155:. Berlin: Springer–Verlag. pp. xii+284. 10844: 10811: 10254:(2008). "computation of general equilibria", 9994: 9992: 9906:An Outline of the History of Economic Thought 9904:Screpanti, Ernesto; Zamagni, Stefano (1993). 9868:(1989). Mohammed Dore; Sukhamoy Chakravarty; 9806:Monotone processes of convex and concave type 8687:The Theory of Political Economy, jevons 1871. 8122:Fundamental Methods of Mathematical Economics 7526:Testing predictions of mathematical economics 3389:autoregressive conditional heteroskedasticity 2599:used the calculus of variations to that end. 1496:another independently of...any third party". 1330:in political economy. In 1871, he published 1097: 891: 12641:: CS1 maint: multiple names: authors list ( 11858:Scott E. Page (2008), "agent-based models", 11754:(2008). "computer science and game theory", 11533:Aumann, R., and S. Hart, ed. (1992, 2002). 11468:. "behavioural economics and game theory." 10638:: CS1 maint: multiple names: authors list ( 10604:The Mathematical Theory of Optimal Processes 10123:"Maximum Principles in Analytical Economics" 10064: 10062: 8850: 8848: 8428: 4914:Assuming that the interest rate is fixed at 3558:...How much will the tax cut increase wages? 3463:An open economy has the production function 3011:through incentives for information sharing. 2691:and fixed–point theory. In introducing the 1838: 13755: 13480:The New Palgrave: A Dictionary of Economics 13294:. London: Ashgate Publishing. p. 219. 11741: 11648: 11646: 11030: 10947: 10717:. Cambridge, Massachusetts: The MIT Press. 10694:The New Palgrave: A Dictionary of Economics 10676:The New Palgrave: A Dictionary of Economics 10453: 9824: 9812: 9803: 9636:Ergebnisse eines Mathematischen Kolloquiums 9623: 9452:How Economics Became a Mathematical Science 8400: 8398: 8396: 8394: 5322:per worker with respect to the tax rate as: 2999:. It has also given rise to the subject of 1478: 1265:lectured in this fashion, coining the term 15472: 15458: 14362: 14348: 13748: 13734: 13250: 12932:Aldrich, John (January 1989). "Autonomy". 11987:, v. 2, ch. 17, Introduction, p. 883. Pp. 11423:Handbook of Experimental Economics Results 11244: 10548: 10405: 9989: 9146: 8502: 8241: _____ (1986). v. 8236: _____ (1982). v. 7649: 7479: 7312:We recover the Cobb-Douglas solution when 6393: 3998: 3917: 3656:Where the variables in this equation are: 3649:{\displaystyle Y=F(K,L)=f(k)L,\quad k=K/L} 1104: 1090: 898: 884: 13677:Economic Dynamics: Theory and Computation 13208:A History of Economic Theory & Method 13024: 12835: 12607:Equity Derivatives and Market Risk Models 11820:The New Palgrave Dictionary of Economics 11209:1972. University of Chicago Press, 1972, 10910: 10431: 10059: 9404: 9335: 9255:"Solutions to general non-zero-sum games" 9187: 8968: 8854: 8845: 8725: 8557: 8404: 4124:is the exogenous interest rate. Then the 3372:models and a determinist trend. Wold and 3024:Nobel Memorial Prize in Economic Sciences 2025:fundamental theorems of welfare economics 1800: 13347: 13206:Ekelund, Robert; Hébert, Robert (2014). 13175: 13007: 12423:The New Palgrave Dictionary of Economics 12247:Journal of Economic Dynamics and Control 11860:The New Palgrave Dictionary of Economics 11756:The New Palgrave Dictionary of Economics 11654:The New Palgrave Dictionary of Economics 11643: 11460:The New Palgrave Dictionary of Economics 11398:_____ (2001). "Experimental Economics", 11352:The New Palgrave Dictionary of Economics 11310:The New Palgrave Dictionary of Economics 11008:The New Palgrave Dictionary of Economics 10888:The New Palgrave Dictionary of Economics 10788:The New Palgrave Dictionary of Economics 10658:The New Palgrave Dictionary of Economics 10387:The New Palgrave Dictionary of Economics 10369:The New Palgrave Dictionary of Economics 10345:The New Palgrave Dictionary of Economics 10334:, Second Edition, Springer-Verlag, 1999. 10309:Linear Programming and Economic Analysis 10275:The New Palgrave Dictionary of Economics 10256:The New Palgrave Dictionary of Economics 10215:The New Palgrave Dictionary of Economics 10145:The New Palgrave Dictionary of Economics 10044:The New Palgrave Dictionary of Economics 10000:The New Palgrave Dictionary of Economics 9782:Theory of Linear and Integer Programming 9434:The New Palgrave Dictionary of Economics 9386: 9257:. In Tucker, A. W.; Luce, R. D. (eds.). 8664:Journal of the Royal Statistical Society 8640:The New Palgrave Dictionary of Economics 8447:The New Palgrave Dictionary of Economics 8391: 8160:The New Palgrave Dictionary of Economics 8112: 8110: 6728:Therefore, we may use this to find that: 6259:effect of a capital tax cut on wages is: 5912:{\textstyle dw/d\tau =-f'(k)k/(1-\tau )} 5260:effect of a capital tax cut on wages is: 5071:For the moment, let's focus only on the 3254: 3225:The Probability Approach in Econometrics 3164: 2558:Variational calculus and optimal control 2515:In allowing inequality constraints, the 2123: 2036: 1842: 1577: 1498: 1416:. Today the solution can be given as a 1351: 13288:"Seven Decades of Economic Methodology" 13279: 12974: 12931: 12271:(2005). "The Complexity of Exchange", 11350:Serrano, Roberto (2008). "bargaining", 11002: 11000: 10998: 10755:, Harvard University Press, chapter 5. 10168:, Chapter 16. Oxford University Press, 10002:, 2nd Edition, v. 6, pp. 138–57. 9587: 9265:. Vol. 40. Princeton, New Jersey: 9252: 9194:. London: Kegan Paul . pp. 15–19. 8934: 8911:The History of Economic Thought Website 3030:. Later work extended their results to 16147: 13574:Recursive Methods in Economic Dynamics 13451:Introduction to Mathematical Economics 13383: 13285: 13178:"The Grumpy Economist: Greg's algebra" 11436:, Elsevier, v. 3, pp. 2327–2351. 11067: 10753:Recursive Methods in Economic Dynamics 10489: 10472: 9980:The Nature of Mathematical Programming 9224: 9218: 8779: 8775: 8773: 8738:: Economic and Social Research Council 8721: 8719: 8678: 2624: 2098: 15453: 14343: 13729: 13625:Foundations of Mathematical Economics 13472: 13393:. Cambridge: Macmillan. p. 297. 13220: 13063: 12880: 12787: 12655: 12355:Bounded Rationality in Macroeconomics 11955:Bounded Rationality in Macroeconomics 11576:The Theory of Industrial Organization 11232:Theory of Games and Economic Behavior 11129:from the original on 14 December 2010 10710: 10492:Optimization theory for large systems 10475:Optimization theory for large systems 10415:Proceedings of 2nd Berkeley Symposium 9874:John Von Neumann and modern economics 9533: 9109:"Neoclassicism and Dissent 1890-1930" 8907:"Antoine Augustin Cournot, 1801-1877" 8728:"The Use of Mathematics in Economics" 8638:(2008). "mathematics and economics", 8488:Theory of Games and Economic Behavior 8341:Foundations of Mathematical Economics 8203:Is Economics Becoming a Hard Science? 8107: 6486:{\displaystyle f(k)=A\left^{1/\rho }} 5075:effect of a capital tax cut, so that 4474:. This implies that taxes per worker 3007:applications as to ways of improving 2864:Theory of Games and Economic Behavior 1921:computational optimization techniques 16103: 14310: 10995: 9945:The theory of linear economic models 9744:The theory of linear economic models 9300: 9294: 9259:Contributions to the Theory of Games 9106: 7725: 7662: 7573:Considering mathematical economics, 3391:(ARCH) models and generalized ARCH ( 3385:autoregressive moving average models 3195:Between the world wars, advances in 3060:over time. As such, it falls in the 2803:). In the 1960s and 1970s, however, 2671:in economic theory, in particular, 2019:Optimality properties for an entire 1919:and may use a variety of different 1811:In 1936, the Russian–born economist 15479: 14439:Agent-based computational economics 14322: 13651:Introductory Mathematical Economics 13542:Mathematical Methods for Economists 12576:Handbook of Computational Economics 12554:Handbook of Computational Economics 12522:Handbook of Computational Economics 12450:Handbook of Computational Economics 12385:Handbook of Computational Economics 12072:Handbook of Computational Economics 11985:Handbook of Computational Economics 11421:, and Vernon L. Smith, ed. (2008). 9188:Edgeworth, Francis Ysidro (1961) . 8770: 8716: 8445:( 2008). "mathematical economics", 8293:Mathematical Methods for Economists 8124:. McGraw-Hill Irwin. pp. 3–4. 4563:{\displaystyle x=\tau =\tau f'(k)k} 3564:To answer this question, we follow 3044:Agent-based computational economics 3038:Agent-based computational economics 2799:" (the humorous designation due to 2679:. Following von Neumann's program, 1440:and the second is the principle of 1420:but Cournot's work preceded modern 1408:. Cournot's models of duopoly and 1379: 1332:The Principles of Political Economy 13: 13507: 13492:from the original on 1 August 2017 13335:10.1111/j.1467-6435.1991.tb01798.x 13292:Karl Popper:A Centenary Assessment 12946:10.1093/oxfordjournals.oep.a041889 12421:(2008). "experimental economics", 11628:Handbook of Mathematical Economics 11188:Conjugate duality and optimization 10332:Linear Optimization and Extensions 9689:where the nonnegative matrix 8937:Game Theory for Applied Economists 8817:Augustin Cournot (1838, tr. 1897) 8227:Handbook of Mathematical Economics 7420: 4219: 4211: 4150: 4142: 3572:. Suppose an open economy has the 3399:Non-stochastic mathematical models 3232:) throughout the 1930s and 1940s. 2519:generalized the classic method of 1446:Éléments d'économie politique pure 14: 16171: 13698:Journal of Mathematical Economics 13691: 13478:"The Wide, Wide World Of Wealth ( 10413:(1951). "Nonlinear programming". 10106:* Samuelson, Paul A., 1998. "How 9984:Mathematical Programming Glossary 9076: 9046: 8205:, Edward Elgar. Pre-publication 7591:Defense of mathematical economics 6528:{\textstyle \rho =1-\sigma ^{-1}} 4126:first-order optimality conditions 3219:in 1933. A student of Frisch's, 1915:of some function given a defined 1789:of the economy, which equals the 1487:, published in 1881. He adopted 1455:Tâtonnement (roughly, French for 16126: 16115: 16102: 16091: 16090: 14897:neoclassical–Keynesian synthesis 14321: 14309: 14298: 14297: 14285: 13466: 13439: 13418: 13377: 13341: 13314: 13270: 13256:"The end of the Dismal Science?" 13244: 13214: 13199: 13176:Cochrane, John H. (2017-10-21). 13169: 13144: 13108: 13057: 13001: 12968: 12925: 12874: 12829: 12763: 12754: 12732: 12689:The Journal of Political Economy 12676: 12649: 12514: 12466: 12439: 12412: 12377: 12313: 12289: 12262: 12235: 12148: 12099: 12064: 12019: 11977: 11382:Towards a History of Game Theory 9876:. Oxford:Clarendon. p. 261. 9804:Rockafellar, R. Tyrrell (1967). 9593:Foundations of Economic Analysis 9393:Quarterly of Applied Mathematics 8767:are minor for most applications. 8624:Journal of Economic Perspectives 8411:Foundations of Economic Analysis 8371:Elements of Dynamic Optimization 8058: 8044: 7493:— while making many of the same 3766:Cobb-Douglas production function 2591:. Before the Second World War, 1952:expenditure minimization problem 1669: 1631:Foundations of Economic Analysis 1584:Foundations of Economic Analysis 1074: 865: 853: 38: 16045:List of social science journals 14206:Computational complexity theory 13638:, World Scientific Publishing. 13592:Optimization in Economic Theory 13407:from the original on 2019-05-28 13188:from the original on 2023-07-01 13158:from the original on 2019-08-07 13133:from the original on 2007-06-09 13035:10.1215/00182702-36-Suppl_1-305 12624:from the original on 2023-07-01 11876: 11852: 11812: 11618: 11586:, "General Organization", pp. 11582:and chapter-preview links, pp. 11564: 11499: 11452: 11370: 11326: 11302: 11278: 11238: 11193: 11177: 11149:Aliprantis, Charalambos D. 11141: 11111: 11060: 11024: 10936:from the original on 2023-07-01 10904: 10880: 10812:Arrow, K. J.; Kurz, M. (1970). 10804: 10780: 10734: 10646: 10621:from the original on 2023-07-01 10594: 10507:Hiriart-Urruty, Jean-Baptiste; 10423: 10361: 10337: 10324: 10311:. McGraw–Hill. Chapter-preview 10301: 10292: 10244: 10203: 10137: 10100: 10073:Optimization in Economic Theory 10033: 10015: 9972: 9950: 9934: 9897: 9880: 9838: 9795: 9784:. John Wiley & sons, 1998, 9771: 9749: 9736: 9645: 9581: 9572: 9499: 9474: 9426: 9387:Herstein, I.N. (October 1953). 9380: 9369:from the original on 2023-07-01 9343:. In Darnell, Adrian C. (ed.). 9329: 9283:from the original on 2023-07-01 9246: 9235:from the original on 2023-07-01 9209: 9198:from the original on 2023-07-01 9181: 9170:from the original on 2023-07-01 9140: 9129:from the original on 2023-07-01 9100: 9070: 9049:"Walras Law and Macroeconomics" 9040: 9031: 8962: 8953: 8928: 8899: 8888:from the original on 2023-07-01 8862:. In Darnell, Adrian C. (ed.). 8811: 8800:from the original on 2023-07-01 8749: 8692: 8672: 8656: 8629: 8613: 8604: 8551: 8542: 8531:from the original on 2023-07-01 7720: 7657: 4363: 3832: 3628: 3418:of an item increases, then the 3387:. More general models include 3184: 2872:John von Neumann, working with 2720:partially ordered vector spaces 2158:, who considered the nonlinear 2055:models for the entire economy. 1996:and modeling optimization with 1711:with nonnegative matrices 1658:, which compares two different 1434:general competitive equilibrium 140:Concepts, theory and techniques 16002:Science and technology studies 13610:Numerical Methods in Economics 13019:(Annual Supplement): 305–322. 12610:. Risk Books. pp. 13–17. 12006:Numerical Methods in Economics 11923:Micromotives and Macrobehavior 11725:, Cambridge University Press. 10918:. In Weintraub, Sidney (ed.). 10112:Journal of Economic Literature 9947:. McGraw-Hill, New York, 1960. 9746:. McGraw-Hill, New York, 1960. 9111:. In Weintraub, Sidney (ed.). 9079:"A Formal Proof of Walras Law" 8709:to Irving Fisher's 1897 work, 8496: 8363: 8219: 8192: 8148: 8090: 7296: 7284: 7275: 7259: 7247: 7238: 7235: 7217: 7212: 7200: 7191: 7175: 7163: 7154: 7151: 7139: 7083: 7071: 7062: 7046: 7034: 7025: 7022: 7010: 7005: 6993: 6984: 6968: 6956: 6947: 6944: 6926: 6907: 6895: 6886: 6870: 6858: 6849: 6846: 6834: 6829: 6813: 6801: 6792: 6696: 6684: 6673: 6661: 6461: 6449: 6418: 6412: 6367:{\textstyle \alpha =\tau =1/3} 6215: 6209: 6193: 6187: 6144: 6138: 6122: 6116: 6070: 6064: 6051: 6045: 6009: 6003: 5971: 5965: 5906: 5894: 5883: 5877: 5808: 5802: 5789: 5783: 5747: 5741: 5714: 5708: 5673: 5667: 5648: 5641: 5596: 5590: 5574: 5568: 5557: 5551: 5531: 5527: 5521: 5510: 5480: 5474: 5425: 5419: 5405: 5399: 5374: 5368: 5182: 5176: 5116: 5110: 5055: 5049: 5038: 5026: 5021: 5015: 4878: 4872: 4827: 4821: 4807: 4801: 4787: 4781: 4685: 4679: 4665: 4659: 4618: 4612: 4554: 4548: 4531: 4522: 4516: 4510: 4461: 4449: 4437: 4431: 4399: 4393: 4379: 4373: 4357: 4351: 4340: 4328: 4284: 4278: 4264: 4258: 4247: 4235: 4195: 4189: 4178: 4166: 4028: 4022: 4011: 3999: 3953: 3941: 3930: 3918: 3851: 3839: 3703: 3691: 3619: 3613: 3604: 3592: 3485: 3479: 3250: 3161:Mathematicization of economics 2839: 2675:through his generalization of 2619:optimal consumption and saving 2440: 2434: 2339: 2333: 2297: 2291: 2260: 2254: 2230:{\displaystyle g_{i}(x)\leq 0} 2218: 2212: 2182: 2176: 2053:computable general equilibrium 1539:Edwin Robert Anderson Seligman 1448:(Elements of Pure Economics). 1427: 1: 14833:Critique of political economy 14369: 12361:and chapter-preview 1st-page 12004: • _____ (1998). 11961:and chapter-preview 1st-page 11636:10.1016/S1573-4382(81)01011-4 10490:Lasdon, Leon S. (2002). 10473:Lasdon, Leon S. (1970). 9695:must be square and where the 9347:. Springer. pp. 94–122. 9309:(2): 207, 212, 219, 234–237. 9225:Bowley, Arthur Lyon (1960) . 8866:. Springer. pp. 51, 52. 8726:Sheila C., Dow (1999-05-21). 8491:. Princeton University Press. 8084: 8052:Business and economics portal 7640:" of contemporary economics: 5125:{\textstyle dx/d\tau =f'(k)k} 2746:Differential decline and rise 2718:developed economic models in 2714:In Russia, the mathematician 2677:Brouwer's fixed-point theorem 2446:{\displaystyle h_{j}(\cdot )} 2345:{\displaystyle g_{i}(\cdot )} 2138:Karush–Kuhn–Tucker conditions 1907:. For optimization problems, 1695:Brouwer's fixed point theorem 1554:Modern mathematical economics 13576:, Harvard University Press. 13355:Essays in Positive Economics 13066:Journal of Law and Economics 13013:History of Political Economy 12158:World Scientific Publishing 10916:"General Equilibrium Theory" 9986:, INFORMS Computing Society. 9536:History of Political Economy 9303:History of Political Economy 9231:. Oxford: Clarendon Press . 8610:Schumpeter (1954) p. 465-468 8548:Schumpeter (1954) p. 212-215 8506:History of Economic Analysis 8414:. Harvard University Press. 7624:theoretical computer science 2947:theory had bounded only the 2795:(GET), as practiced by the " 2504:{\displaystyle j=1,\dots ,l} 2409:{\displaystyle i=1,\dots ,m} 2004:, including applications to 1944:utility maximization problem 1188:) is modeled as not changing 1170:Broad applications include: 7: 13660:Giancarlo Gandolfo, 2009. 12297:Games and Economic Behavior 11701:Games and Economic Behavior 10461:Encyclopedia of Mathematics 10454:Vapnyarskii, I.B. (2001) , 9663:, generalizing that of the 9517:10.1007/978-0-387-33757-9_6 8759:has fallen out of favor in 8679:Jevons, W. Stanley (1871). 8120:; Kevin Wainwright (2005). 8037: 7426:{\textstyle \rho =-\infty } 3409:, and/or specific forms of 3139:information and uncertainty 2856:Non-cooperative game theory 2546:, such as those arising in 2544:polyhedral convex functions 2473:equality constraints where 2051:of solution, in particular 1734:and a positive number 1516:on what is now known as an 1253:History of economic thought 10: 16176: 14971:Real business-cycle theory 14256:Films about mathematicians 13528:Mathematics for Economists 12883:Review of Economic Studies 12387:, v. 2, pp. 832–865. 12275:, 115(504, Features), pp. 11484:"behavioral game theory." 11021:, ed., Palgrave Macmillan. 10865:Scroll to chapter-preview 9640:Review of Economic Studies 9267:Princeton University Press 9215:Nicola, p. 14, 15, 258-261 8860:"Stability in Competition" 8780:Nicola, PierCarlo (2000). 7558: 3738:is the total capital stock 3716:is the production function 3188: 3098:computer-based simulations 3041: 2849: 2843: 2793:general equilibrium theory 2749: 2628: 2561: 2527:problems, which enjoy the 2272:{\displaystyle h_{j}(x)=0} 2127: 2102: 1868: 1858: 1847:Red dot in z direction as 1804: 1673: 1591: 1581: 1345: 1302:Johann Heinrich von Thünen 1250: 1246: 1014:Relationship with sciences 16086: 16053: 16037: 15784: 15510: 15487: 15411: 15369: 15011: 14745: 14494: 14459: 14377: 14279: 14229: 14186: 14096: 14058: 14025: 13977: 13949: 13896: 13843: 13825:Philosophy of mathematics 13800: 13765: 12977:A History of Econometrics 12589:& and chapter-preview 11776:Communications of the ACM 11658: 11505:* Rasmusen, Eric (2007). 9729:in the 1950s and then by 9578:Blaug (2007), p. 185, 187 9548:10.1215/00182702-2007-001 9454:. Duke University Press. 9315:10.1215/00182702-35-2-205 8970:Edgeworth, Francis Ysidro 8369:Chiang, Alpha C. (1992). 8246: 8229:, 1st-page chapter links: 7802:Nicholas Georgescu-Roegen 7732:Charalambos D. Aliprantis 7616:mathematical optimization 7611:the aid of mathematics." 4104:is wages per worker, and 4084:is the capital tax rate, 3518:is output per worker and 2931:in cooperative games and 2453:are the functions of the 2352:are the functions of the 2303:{\displaystyle f(\cdot )} 2150:were achieved in 1951 by 1881:mathematical optimization 1875:Non-convexity (economics) 1861:Mathematical optimization 1855:function of (x, y) inputs 1839:Mathematical optimization 1628:In the landmark treatise 1218:mathematical optimization 14261:Recreational mathematics 13426:American Economic Review 13226:American Economic Review 12975:Epstein, Roy J. (1987). 12337:American Economic Review 11909:American Economic Review 11888:American Economic Review 11415:per sect. 1.1 & 2.1. 11073:Mathematical Programming 9962:Thompson, Gerald L. 9665:Perron–Frobenius theorem 9661:irreducibility condition 9597:American Economic Review 9153:The Worldly Philosophers 9107:Rima, Ingrid H. (1977). 8935:Gibbons, Robert (1992). 8308:Takayama, Akira (1985). 8098:JEL classification codes 7679:Francis Ysidro Edgeworth 7674:Antoine Augustin Cournot 7544:with other assumptions. 6599:which may be derived as: 4967:{\textstyle dr/d\tau =0} 3760:is the total labor stock 3411:functional relationships 3213:in 1930 and the journal 3066:complex adaptive systems 2010:economics of information 1889:maximizing or minimizing 1684:Perron–Frobenius theorem 1648:Le Chatelier's principle 1479:Francis Ysidro Edgeworth 1374:Francis Ysidro Edgeworth 1149:mathematical programming 128:JEL classification codes 15690:international relations 14611:Industrial organization 14434:Computational economics 14146:Mathematical statistics 14136:Mathematical psychology 14106:Engineering mathematics 14040:Algebraic number theory 13675:John Stachurski, 2009. 13664:, 4th ed., Springer. 12496:Computational Economics 11723:Algorithmic Game Theory 11184:Rockafellar, R. Tyrrell 10920:Modern Economic Thought 10711:Shell, K., ed. (1967). 10660:, 2nd Edition. Preview 10562:10.1007/3-540-45586-8_4 10132:, Nobel Prize lecture. 9910:Oxford University Press 9253:Gillies, D. B. (1969). 9113:Modern Economic Thought 8786:. Springer. p. 4. 8339:Michael Carter (2001). 7650:Mathematical economists 7620:mathematical statistics 7480:Criticisms and defences 6394:CES production function 3197:mathematical statistics 3123:industrial organization 2993:industrial organization 2905:cooperative game theory 2852:Cooperative game theory 1372:. Cournot, Walras and 314:Industrial organization 171:Computational economics 16155:Mathematical economics 16017:Quantum social science 14816:Modern monetary theory 14481:Experimental economics 14451:Pluralism in economics 14424:Mathematical economics 14292:Mathematics portal 14141:Mathematical sociology 14121:Mathematical economics 14116:Mathematical chemistry 14045:Analytic number theory 13926:Differential equations 13623:Michael Carter, 2001. 13552:Mathematical Economics 13550:Akira Takayama, 1985. 13544:, 3rd ed., Blackwell. 13286:Boland, L. A. (2007). 12934:Oxford Economic Papers 12760:Stigler et al., p. 342 12008:, MIT Press. Links to 11462:(2008), 2nd Edition: 11404:, pp. 5100–5108. 11032:Kantorovich, L. V 10853:. New York: Springer. 10456:"Lagrange multipliers" 8761:neoclassical economics 8310:Mathematical Economics 8295:, 3rd ed., Blackwell. 7693:William Stanley Jevons 7647: 7588: 7520: 7470: 7427: 7398: 7358: 7332: 7306: 7097: 6722: 6590: 6549: 6529: 6487: 6368: 6328: 6249: 6163: 5913: 5833: 5315: 5250: 5126: 5065: 4968: 4928: 4908: 4732: 4564: 4488: 4468: 4409: 4309: 4118: 4098: 4078: 4058: 3886:factor of productivity 3878: 3858: 3754: 3732: 3710: 3673: 3650: 3562: 3552: 3532: 3512: 3492: 3407:hyperbolic coordinates 3356:to formulate and test 3313:Applied mathematicians 3283: 3174: 3154:experimental economics 2981:experimental economics 2975:can generate a unique 2589:optimal control theory 2564:Calculus of variations 2505: 2467: 2447: 2410: 2366: 2346: 2304: 2273: 2231: 2189: 1871:Convexity in economics 1856: 1801:Input-output economics 1509: 1357: 1141:differential equations 1119:is the application of 1117:Mathematical economics 166:Experimental economics 16054:Other categorizations 15907:International studies 15892:History of technology 15827:Communication studies 15710:public administration 14271:Mathematics education 14201:Theory of computation 13921:Hypercomplex analysis 13649:D. Wade Hands, 2004. 13554:, 2nd ed. Cambridge. 13517:, McGraw-Hill Irwin. 12249:25(3–4), pp. 503–52. 12241:Klosa, Tomas B., and 11656:(2008), 2nd Edition: 11630:, v. 1, pp. 285–330. 11507:Games and Information 11089:10.1007/s101070100258 10437:Nonlinear Programming 10433:Bertsekas, Dimitri P. 9509:From Walras to Pareto 9263:Annals of Mathematics 9191:Mathematical Psychics 9148:Heilbroner, Robert L. 8972:(September 5, 1889). 8755:While the concept of 8216:Retrieved 2008-04-01. 7905:John Forbes Nash, Jr. 7890:Jean-François Mertens 7772:Graciela Chichilnisky 7642: 7583: 7515: 7471: 7469:{\textstyle dw/dx=-1} 7428: 7399: 7359: 7333: 7307: 7098: 6723: 6591: 6550: 6530: 6488: 6369: 6329: 6250: 6164: 5914: 5834: 5316: 5251: 5127: 5066: 4969: 4929: 4909: 4733: 4565: 4489: 4469: 4410: 4310: 4119: 4099: 4079: 4059: 3879: 3859: 3755: 3733: 3711: 3674: 3651: 3553: 3533: 3513: 3493: 3459: 3344:are formulated using 3326:The Wealth of Nations 3258: 3168: 3032:computational methods 2989:information economics 2897:differential calculus 2868:John Forbes Nash, Jr. 2826:differential topology 2735:hyperplane supporting 2639:Supporting hyperplane 2506: 2468: 2448: 2411: 2367: 2347: 2305: 2274: 2232: 2190: 2134:Lagrangian multiplier 2130:Nonlinear programming 2124:Nonlinear programming 2118:Berlin airlift (1948) 1846: 1817:input-output analysis 1774:) probability vector 1738:that would solve the 1719:; von Neumann sought 1588:Differential calculus 1578:Differential calculus 1502: 1414:non-cooperative games 1355: 1271:Political Arithmetick 1210:differential calculus 1153:computational methods 15887:Historical sociology 14690:Social choice theory 14446:Behavioral economics 14429:Complexity economics 14251:Informal mathematics 14131:Mathematical physics 14126:Mathematical finance 14111:Mathematical biology 14050:Diophantine geometry 13616:and chapter-preview 13580:and chapter-preview 13385:Keynes, John Maynard 13276:Beed & Owen, 584 13182:The Grumpy Economist 12556:, v. 2, ch. 17, pp. 12452:, v.2, pp. 949–101. 12177:and chapter-preview 11919:Schelling, Thomas C. 11841:Information Sciences 11662:"mechanism design." 11520:and chapter-preview 10766:and chapter-preview 10166:Microeconomic Theory 9912:. pp. 288–290. 8388:to inside, first pp. 8312:, 2nd ed. Cambridge. 8079:Mathematical finance 8024:Nicholas C. Yannelis 7989:Hugo F. Sonnenschein 7622:and increasingly in 7597:Josiah Willard Gibbs 7437: 7408: 7397:{\textstyle dw/dx=0} 7368: 7357:{\textstyle \rho =1} 7342: 7331:{\textstyle \rho =0} 7316: 7107: 6732: 6603: 6589:{\textstyle f'/kf''} 6559: 6548:{\textstyle \sigma } 6539: 6497: 6406: 6376:capital accumulation 6338: 6263: 6173: 5923: 5843: 5326: 5264: 5136: 5079: 4978: 4938: 4918: 4742: 4574: 4498: 4478: 4467:{\textstyle X=\tau } 4419: 4319: 4132: 4108: 4088: 4068: 3892: 3868: 3772: 3744: 3722: 3685: 3663: 3580: 3542: 3522: 3502: 3467: 3403:social choice theory 3346:stochastic processes 3280:Liquidity preference 2985:behavioral economics 2973:noncooperative games 2740:stochastic processes 2585:variational calculus 2521:Lagrange multipliers 2517:Kuhn–Tucker approach 2477: 2457: 2421: 2382: 2356: 2320: 2285: 2241: 2199: 2188:{\displaystyle f(x)} 2170: 2160:optimization problem 1990:Economic equilibrium 1976:production functions 1909:specialized notation 1885:optimization problem 1833:choice of techniques 1829:production economics 1534:The Economic Journal 1392:, is referred to as 979:Discrete mathematics 393:Social choice theory 16067:Geisteswissenschaft 16061:Behavioral sciences 15987:Political sociology 15902:Information science 15847:Development studies 14774:American (National) 14474:Economic statistics 14266:Mathematics and art 14176:Operations research 13931:Functional analysis 13594:, 2nd ed., Oxford. 12670:10.1093/ei/40.1.138 10607:. New York: Wiley. 10075:, 2nd ed., Oxford. 10042:(2008). "duality", 10030:, Macmillan, p. 16. 9862:Thompson, Gerald L. 9778:Alexander Schrijver 9761:Thompson, Gerald L. 9731:Stephen M. Robinson 9591:(1948). "Review of 8626:, 5(1) pp. 145-157. 8248:Hildenbrand, Werner 7930:Donald John Roberts 7565:Applied mathematics 7537:mathematical proofs 4415:Define total taxes 3709:{\textstyle F(K,L)} 3679:is the total output 3574:production function 3491:{\textstyle y=f(k)} 3211:Econometric Society 3169:The surface of the 3068:. In corresponding 3009:economic efficiency 2884:methods related to 2882:functional analytic 2785:optimization theory 2769:functional analysis 2705:"dual" vector space 2667:methods to include 2665:functional analytic 2647:Fixed point theorem 2643:Hahn–Banach theorem 2631:Functional analysis 2625:Functional analysis 2572:Dynamic programming 2552:operations research 2525:convex minimization 2099:Linear optimization 2033:general equilibrium 1994:dynamic programming 1974:, subject to their 1962:, subject to their 1815:built his model of 1656:comparative statics 1424:by over 100 years. 1294:classical economics 1263:Gottfried Achenwall 1233:John Maynard Keynes 1192:comparative statics 913:Part of a series on 860:Business portal 181:Operations research 161:National accounting 16122:Society portal 15609:auxiliary sciences 14211:Numerical analysis 13820:Mathematical logic 13815:Information theory 13653:, 2nd ed. Oxford. 13460:2022-03-18 at the 13260:Challenge Magazine 13252:Heilbroner, Robert 13009:Colander, David C. 12775:2017-10-10 at the 12684:Stigler, George J. 12585:2012-03-06 at the 12567:2022-01-21 at the 12542:2017-08-11 at the 12531:2018-08-09 at the 12524:, v. 2, sect. 5. 12505:2023-07-01 at the 12487:2012-04-05 at the 12459:2015-09-24 at the 12432:2012-01-19 at the 12405:2017-08-11 at the 12394:2018-08-09 at the 12368:2023-07-01 at the 12351:Sargent, Thomas J. 12306:2020-12-04 at the 12282:2017-08-08 at the 12256:2020-06-22 at the 12226:2016-10-19 at the 12215:2012-01-26 at the 12204:2023-07-01 at the 12192:Epstein, Joshua M. 12184:2023-07-01 at the 12173:2010-07-27 at the 12141:2023-07-01 at the 12130:2018-01-02 at the 12119:2018-01-02 at the 12092:2017-08-11 at the 12081:2018-08-09 at the 12057:2012-04-15 at the 12036:2020-03-06 at the 12029:, 8(1), pp.55-82. 11998:2022-01-21 at the 11968:2023-07-01 at the 11951:Sargent, Thomas J. 11943:2023-07-01 at the 11932:2017-11-02 at the 11897:2011-01-05 at the 11869:2018-02-10 at the 11829:2017-08-11 at the 11787:2012-04-26 at the 11765:2017-11-05 at the 11752:Halpern, Joseph Y. 11732:2012-05-05 at the 11710:2018-10-14 at the 11693:2011-11-23 at the 11681:2013-05-16 at the 11669:2011-11-23 at the 11611:2016-01-02 at the 11555:2018-10-14 at the 11544:2017-08-16 at the 11527:2023-07-01 at the 11516:2017-06-24 at the 11491:2011-11-23 at the 11475:2017-08-07 at the 11443:2018-11-07 at the 11411:2018-10-14 at the 11391:2023-07-01 at the 11361:2017-08-11 at the 11343:2016-03-04 at the 11319:2017-08-11 at the 11246:Mas-Colell, Andreu 11069:Polyak, B. T. 11017:2016-03-03 at the 10897:2016-03-06 at the 10872:2023-07-01 at the 10838:2013-03-09 at the 10797:2017-10-18 at the 10773:2023-07-01 at the 10762:2017-08-11 at the 10703:2017-08-11 at the 10685:2016-03-06 at the 10667:2017-08-11 at the 10550:Lemaréchal, Claude 10509:Lemaréchal, Claude 10398:2017-10-18 at the 10378:2016-03-04 at the 10354:2017-08-11 at the 10318:2023-07-01 at the 10284:2017-08-11 at the 10265:2009-05-23 at the 10224:2017-08-11 at the 10194:2012-01-26 at the 10183:2012-01-26 at the 10162:Mas-Colell, Andreu 10154:2017-08-11 at the 10128:2012-10-11 at the 10093:2023-07-01 at the 10082:2023-07-01 at the 10053:2017-02-02 at the 10040:Blume, Lawrence E. 10009:2017-08-11 at the 9958:Morgenstern, Oskar 9826:Rockafellar, R. T. 9814:Rockafellar, R. T. 9757:Morgenstern, Oskar 9733:in the 1970s. 9727:Gerald L. Thompson 9669:eigenvalue problem 9465:2023-06-04 at the 9443:2013-05-16 at the 9269:. pp. 47–85. 8839:2023-07-01 at the 8828:2023-07-01 at the 8705:2023-07-01 at the 8649:2013-05-16 at the 8471:2017-08-05 at the 8456:2013-05-16 at the 8384:2016-03-03 at the 8354:2023-07-01 at the 8332:2023-07-01 at the 8321:2023-07-01 at the 8302:2023-07-01 at the 8281:2023-07-01 at the 8261:2013-04-15 at the 8212:2006-06-25 at the 8185:2017-08-05 at the 8170:2013-05-16 at the 8142:2012-03-08 at the 8096:Elaborated at the 8066:Mathematics portal 7915:Edward C. Prescott 7846:Leonid Kantorovich 7831:Werner Hildenbrand 7579:The General Theory 7506:economic agents. 7466: 7423: 7394: 7354: 7328: 7302: 7093: 7091: 6718: 6586: 6545: 6525: 6483: 6364: 6324: 6245: 6159: 5909: 5829: 5827: 5311: 5246: 5122: 5061: 4964: 4924: 4904: 4728: 4727: 4720: 4635: 4628: 4560: 4484: 4464: 4405: 4305: 4303: 4114: 4094: 4077:{\textstyle \tau } 4074: 4054: 3997: 3916: 3874: 3854: 3750: 3728: 3706: 3669: 3646: 3570:Hoover Institution 3548: 3528: 3508: 3488: 3455:Harvard University 3426:Qualitative models 3284: 3175: 3089:to market forces. 3070:agent-based models 2969:bargaining problem 2953:bilateral monopoly 2893:fixed-point theory 2716:Leonid Kantorovich 2693:Arrow–Debreu model 2673:fixed-point theory 2548:linear programming 2501: 2463: 2443: 2406: 2362: 2342: 2300: 2269: 2227: 2185: 2114:Linear programming 2105:Linear programming 2061:Leonid Kantorovich 2049:computable methods 2029:Arrow–Debreu model 1982:costs, and market 1964:budget constraints 1857: 1807:Input-output model 1795:linear programming 1680:Linear programming 1522:Arthur Lyon Bowley 1510: 1493:felicific calculus 1358: 1307:The Isolated State 1080:Mathematics Portal 191:Industrial complex 186:Middle income trap 16142: 16141: 15982:Political economy 15977:Political ecology 15832:Community studies 15822:Cognitive science 15785:Interdisciplinary 15685:Political science 15447: 15446: 14978:New institutional 14337: 14336: 13936:Harmonic analysis 13662:Economic Dynamics 13476:(20 March 1988). 13400:978-0-333-10729-4 13370:978-0-226-26403-5 13301:978-0-7546-5375-2 13254:(May–June 1999). 13118:Models in Science 12986:978-0-444-70267-8 12789:Arrow, Kenneth J. 12617:978-1-899332-87-8 12329:Axel Leijonhufvud 11837:Tesfatsion, Leigh 11660:Myerson, Roger B. 11482:Camerer, Colin F. 11419:Plott, Charles R. 11263:978-0-521-26514-0 11215:978-0-226-79190-6 11162:978-3-540-52866-1 11045:978-0-19-505729-4 10949:Arrow, Kenneth J. 10929:978-0-8122-7712-8 10912:Weintraub, E. Roy 10860:978-0-7923-8608-7 10827:978-0-8018-1124-1 10724:978-0-262-19036-7 10571:978-3-540-42877-0 10522:978-3-540-56852-0 10446:978-1-886529-00-7 10252:Scarf, Herbert E. 10211:Geanakoplos, John 10024:(1935, 2nd ed.). 9919:978-0-19-828370-6 9723:Oskar Morgenstern 9526:978-0-387-33756-2 9492:978-0-324-31968-2 9406:10.1090/qam/60205 9354:978-3-540-97011-8 9337:Hotelling, Harold 9276:978-0-691-07937-0 9163:978-0-684-86214-9 9122:978-0-8122-7712-8 9089:on April 30, 2008 9059:on April 17, 2008 9020:on April 11, 2003 8989:(1036): 434–436. 8946:978-0-691-00395-5 8873:978-3-540-97011-8 8856:Hotelling, Harold 8793:978-3-540-67084-1 8636:Weintraub, E. Roy 8516:978-0-04-330086-2 8483:Oskar Morgenstern 8479:von Neumann, John 8421:978-0-674-31301-9 8377:& Amazon.com 8289:Glaister, Stephen 8254:, ed. (1991). v. 8252:Hugo Sonnenschein 8131:978-0-07-010910-0 8034: 8033: 7950:Leonard J. Savage 7871:Andreu Mas-Colell 7851:Tjalling Koopmans 7777:George B. Dantzig 7767:Lawrence E. Blume 7717: 7716: 7511:Robert Heilbroner 7300: 7128: 7087: 6911: 6771: 6716: 6713: 6629: 6322: 6284: 6243: 6219: 6157: 6148: 6088: 6074: 6024: 5989: 5944: 5812: 5762: 5698: 5677: 5628: 5600: 5505: 5454: 5351: 5309: 5285: 5244: 5224: 5200: 5157: 5059: 4999: 4902: 4856: 4763: 4725: 4714: 4641: 4639: 4633: 4602: 4600: 4595: 4226: 4157: 3982: 3901: 3451:Prof. Greg Mankiw 3444:corporate tax cut 3430:scenario planning 3342:Stochastic models 3230:Cowles Foundation 3131:welfare economics 3127:transaction costs 3094:numerical methods 3092:ACE models apply 2997:political economy 2907:, game theorists 2874:Oskar Morgenstern 2830:Andreu Mas-Colell 2577:Economic dynamics 2466:{\displaystyle l} 2365:{\displaystyle m} 2109:Simplex algorithm 2089:George B. Dantzig 2069:Tjalling Koopmans 2045:numerical methods 1598:Walrasian auction 1594:Pareto efficiency 1322:. These included 1316:physical sciences 1279:Velocity of money 1275:Sir William Petty 1237:Robert Heilbroner 1114: 1113: 1069: 1068: 908: 907: 16167: 16130: 16120: 16119: 16106: 16105: 16094: 16093: 15997:Regional science 15842:Cultural studies 15817:Business studies 15474: 15467: 15460: 15451: 15450: 14651:Natural resource 14486:Economic history 14412:Mechanism design 14364: 14357: 14350: 14341: 14340: 14325: 14324: 14313: 14312: 14301: 14300: 14290: 14289: 14221:Computer algebra 14196:Computer science 13916:Complex analysis 13750: 13743: 13736: 13727: 13726: 13701:Aims & Scope 13538:Stephen Glaister 13524:E. Roy Weintraub 13502: 13501: 13499: 13497: 13474:Solow, Robert M. 13470: 13464: 13445:D.W. Bushaw and 13443: 13437: 13422: 13416: 13415: 13413: 13412: 13381: 13375: 13374: 13358: 13349:Friedman, Milton 13345: 13339: 13338: 13318: 13312: 13311: 13309: 13308: 13283: 13277: 13274: 13268: 13267: 13262:. Archived from 13248: 13242: 13241: 13222:Hayek, Friedrich 13218: 13212: 13211: 13203: 13197: 13196: 13194: 13193: 13173: 13167: 13166: 13164: 13163: 13148: 13142: 13141: 13139: 13138: 13112: 13106: 13105: 13061: 13055: 13054: 13028: 13005: 12999: 12998: 12972: 12966: 12965: 12929: 12923: 12922: 12878: 12872: 12871: 12833: 12827: 12826: 12785: 12779: 12767: 12761: 12758: 12752: 12736: 12730: 12729: 12680: 12674: 12673: 12658:Economic Inquiry 12653: 12647: 12646: 12640: 12632: 12630: 12629: 12601: 12595: 12550:Judd, Kenneth L. 12518: 12512: 12470: 12464: 12443: 12437: 12419:Smith, Vernon L. 12416: 12410: 12381: 12375: 12317: 12311: 12293: 12287: 12273:Economic Journal 12266: 12260: 12239: 12233: 12152: 12146: 12103: 12097: 12068: 12062: 12023: 12017: 12014:chapter previews 11981: 11975: 11905:Arthur, W. Brian 11884:Holland, John H. 11880: 11874: 11856: 11850: 11816: 11810: 11758:, 2nd Edition. 11748: 11739: 11703:, 35(1-2), pp. 11650: 11641: 11622: 11616: 11568: 11562: 11503: 11497: 11456: 11450: 11378:Smith, Vernon L. 11374: 11368: 11330: 11324: 11306: 11300: 11282: 11276: 11275: 11242: 11236: 11227: 11218: 11204:Robert L. Graves 11200:Lester G. Telser 11197: 11191: 11181: 11175: 11174: 11145: 11139: 11138: 11136: 11134: 11115: 11109: 11108: 11064: 11058: 11057: 11028: 11022: 11010:, 2nd Edition. 11004: 10993: 10990: 10944: 10942: 10941: 10908: 10902: 10884: 10878: 10864: 10831: 10819: 10808: 10802: 10784: 10778: 10741:Stokey, Nancy L. 10738: 10732: 10728: 10650: 10644: 10643: 10637: 10629: 10627: 10626: 10598: 10592: 10591: 10546: 10537: 10534: 10503: 10486: 10468: 10450: 10427: 10421: 10418: 10365: 10359: 10341: 10335: 10328: 10322: 10305: 10299: 10296: 10290: 10248: 10242: 10217:, 2nd Edition. 10207: 10201: 10141: 10135: 10104: 10098: 10066: 10057: 10037: 10031: 10019: 10013: 9996: 9987: 9976: 9970: 9969: 9954: 9948: 9938: 9932: 9931: 9901: 9895: 9884: 9878: 9877: 9866:Kaldor, Nicholas 9842: 9836: 9833: 9821: 9809: 9799: 9793: 9775: 9769: 9768: 9753: 9747: 9740: 9734: 9649: 9643: 9632: 9621: 9620: 9585: 9579: 9576: 9570: 9567: 9530: 9503: 9497: 9496: 9478: 9472: 9430: 9424: 9418: 9408: 9384: 9378: 9377: 9375: 9374: 9333: 9327: 9326: 9298: 9292: 9291: 9289: 9288: 9250: 9244: 9243: 9241: 9240: 9222: 9216: 9213: 9207: 9206: 9204: 9203: 9185: 9179: 9178: 9176: 9175: 9144: 9138: 9137: 9135: 9134: 9104: 9098: 9097: 9095: 9094: 9083:Walras Law Guide 9074: 9068: 9067: 9065: 9064: 9053:Walras Law Guide 9044: 9038: 9035: 9029: 9028: 9026: 9025: 9019: 9013:. Archived from 8995:10.1038/040434a0 8978: 8966: 8960: 8957: 8951: 8950: 8932: 8926: 8925: 8923: 8922: 8903: 8897: 8896: 8894: 8893: 8852: 8843: 8815: 8809: 8808: 8806: 8805: 8777: 8768: 8753: 8747: 8746: 8744: 8743: 8723: 8714: 8696: 8690: 8689: 8676: 8670: 8660: 8654: 8633: 8627: 8617: 8611: 8608: 8602: 8601: 8555: 8549: 8546: 8540: 8539: 8537: 8536: 8500: 8494: 8439: 8426: 8425: 8402: 8389: 8367: 8361: 8223: 8217: 8196: 8190: 8152: 8146: 8135: 8118:Chiang, Alpha C. 8114: 8105: 8094: 8068: 8063: 8062: 8054: 8049: 8048: 8009:Robert B. Wilson 7999:Albert W. Tucker 7974:Lloyd S. Shapley 7910:John von Neumann 7866:Edmond Malinvaud 7836:Harold Hotelling 7821:John C. Harsanyi 7792:Jacques H. Drèze 7752:Robert J. Aumann 7747:Kenneth J. Arrow 7726: 7663: 7628:John von Neumann 7561:Pure mathematics 7503:methodologically 7475: 7473: 7472: 7467: 7450: 7432: 7430: 7429: 7424: 7403: 7401: 7400: 7395: 7381: 7363: 7361: 7360: 7355: 7337: 7335: 7334: 7329: 7311: 7309: 7308: 7303: 7301: 7299: 7274: 7273: 7215: 7190: 7189: 7137: 7129: 7127: 7119: 7111: 7102: 7100: 7099: 7094: 7092: 7088: 7086: 7061: 7060: 7008: 6983: 6982: 6924: 6916: 6912: 6910: 6885: 6884: 6832: 6828: 6827: 6787: 6772: 6770: 6769: 6757: 6749: 6727: 6725: 6724: 6719: 6717: 6715: 6714: 6712: 6711: 6710: 6676: 6656: 6638: 6630: 6628: 6627: 6615: 6607: 6595: 6593: 6592: 6587: 6585: 6574: 6569: 6554: 6552: 6551: 6546: 6534: 6532: 6531: 6526: 6524: 6523: 6492: 6490: 6489: 6484: 6482: 6481: 6477: 6468: 6464: 6445: 6444: 6373: 6371: 6370: 6365: 6360: 6333: 6331: 6330: 6325: 6323: 6321: 6304: 6293: 6285: 6283: 6275: 6267: 6254: 6252: 6251: 6246: 6244: 6242: 6228: 6220: 6218: 6208: 6196: 6186: 6177: 6168: 6166: 6165: 6160: 6158: 6156: 6149: 6147: 6137: 6125: 6115: 6106: 6097: 6089: 6087: 6086: 6082: 6075: 6073: 6063: 6054: 6044: 6035: 6025: 6023: 6012: 6002: 5993: 5990: 5988: 5977: 5964: 5955: 5953: 5945: 5943: 5935: 5927: 5918: 5916: 5915: 5910: 5893: 5876: 5856: 5838: 5836: 5835: 5830: 5828: 5824: 5820: 5813: 5811: 5801: 5792: 5782: 5773: 5763: 5761: 5750: 5740: 5731: 5723: 5707: 5699: 5697: 5683: 5678: 5676: 5666: 5657: 5656: 5655: 5640: 5631: 5629: 5627: 5613: 5605: 5601: 5599: 5589: 5580: 5567: 5550: 5539: 5538: 5520: 5508: 5506: 5504: 5490: 5473: 5459: 5455: 5453: 5445: 5437: 5435: 5431: 5418: 5398: 5367: 5352: 5350: 5342: 5334: 5320: 5318: 5317: 5312: 5310: 5308: 5294: 5286: 5284: 5276: 5268: 5255: 5253: 5252: 5247: 5245: 5243: 5235: 5227: 5225: 5223: 5209: 5201: 5199: 5188: 5175: 5166: 5158: 5156: 5148: 5140: 5131: 5129: 5128: 5123: 5109: 5092: 5070: 5068: 5067: 5062: 5060: 5058: 5048: 5024: 5014: 5005: 5000: 4998: 4990: 4982: 4973: 4971: 4970: 4965: 4951: 4933: 4931: 4930: 4925: 4913: 4911: 4910: 4905: 4903: 4901: 4893: 4885: 4871: 4857: 4855: 4847: 4839: 4837: 4833: 4820: 4800: 4780: 4764: 4762: 4754: 4746: 4737: 4735: 4734: 4729: 4726: 4723: 4721: 4716: 4715: 4713: 4705: 4697: 4695: 4691: 4678: 4658: 4634: 4631: 4629: 4624: 4611: 4596: 4594: 4586: 4578: 4569: 4567: 4566: 4561: 4547: 4493: 4491: 4490: 4485: 4473: 4471: 4470: 4465: 4414: 4412: 4411: 4406: 4392: 4350: 4314: 4312: 4311: 4306: 4304: 4300: 4296: 4277: 4227: 4225: 4217: 4209: 4188: 4158: 4156: 4148: 4140: 4123: 4121: 4120: 4115: 4103: 4101: 4100: 4095: 4083: 4081: 4080: 4075: 4063: 4061: 4060: 4055: 4041: 4037: 3996: 3969: 3965: 3915: 3883: 3881: 3880: 3875: 3863: 3861: 3860: 3855: 3825: 3824: 3809: 3808: 3793: 3792: 3759: 3757: 3756: 3751: 3737: 3735: 3734: 3729: 3715: 3713: 3712: 3707: 3678: 3676: 3675: 3670: 3655: 3653: 3652: 3647: 3642: 3566:John H. Cochrane 3557: 3555: 3554: 3549: 3537: 3535: 3534: 3529: 3517: 3515: 3514: 3509: 3497: 3495: 3494: 3489: 3171:Volatility smile 3150:computer science 3135:mechanism design 3119:market structure 3001:mechanism design 2909:Lloyd S. Shapley 2899:, for which the 2860:John von Neumann 2818:general topology 2801:Jacques H. Drèze 2777:general topology 2765:John von Neumann 2697:Pareto efficient 2661:John von Neumann 2597:Harold Hotelling 2510: 2508: 2507: 2502: 2472: 2470: 2469: 2464: 2452: 2450: 2449: 2444: 2433: 2432: 2415: 2413: 2412: 2407: 2371: 2369: 2368: 2363: 2351: 2349: 2348: 2343: 2332: 2331: 2309: 2307: 2306: 2301: 2278: 2276: 2275: 2270: 2253: 2252: 2236: 2234: 2233: 2228: 2211: 2210: 2194: 2192: 2191: 2186: 2152:Albert W. Tucker 2073:Kenneth J. Arrow 2035:(also discussed 2006:portfolio theory 1879:In mathematics, 1813:Wassily Leontief 1690:John von Neumann 1662:states after an 1611:Pareto efficient 1548:Harold Hotelling 1418:Nash equilibrium 1380:Augustin Cournot 1361:Augustin Cournot 1328:marginal utility 1222:Second World War 1127:. Often, these 1106: 1099: 1092: 1078: 942: 941: 910: 909: 900: 893: 886: 872:Money portal 870: 869: 868: 858: 857: 354:Natural resource 146:Economic systems 42: 19: 18: 16175: 16174: 16170: 16169: 16168: 16166: 16165: 16164: 16145: 16144: 16143: 16138: 16114: 16082: 16049: 16033: 16007:Science studies 15791:Administration 15780: 15506: 15483: 15481:Social sciences 15478: 15448: 15443: 15440:Business portal 15407: 15406: 15405: 15365: 15129:von Böhm-Bawerk 15017: 15016: 15007: 14779:Ancient thought 14757: 14756: 14750: 14741: 14740: 14739: 14490: 14455: 14407:Contract theory 14392:Decision theory 14373: 14368: 14338: 14333: 14284: 14275: 14225: 14182: 14161:Systems science 14092: 14088:Homotopy theory 14054: 14021: 13973: 13945: 13892: 13839: 13810:Category theory 13796: 13761: 13754: 13694: 13606:Kenneth L. Judd 13570:Edward Prescott 13566:Robert E. Lucas 13562:Nancy L. Stokey 13530:, Cambridge. 13510: 13508:Further reading 13505: 13495: 13493: 13471: 13467: 13462:Wayback Machine 13444: 13440: 13423: 13419: 13410: 13408: 13401: 13382: 13378: 13371: 13346: 13342: 13319: 13315: 13306: 13304: 13302: 13284: 13280: 13275: 13271: 13249: 13245: 13219: 13215: 13204: 13200: 13191: 13189: 13174: 13170: 13161: 13159: 13150: 13149: 13145: 13136: 13134: 13113: 13109: 13062: 13058: 13026:10.1.1.692.6446 13006: 13002: 12987: 12973: 12969: 12930: 12926: 12895:10.2307/2296113 12879: 12875: 12852:10.2307/2171799 12834: 12830: 12807:10.2307/1907716 12786: 12782: 12777:Wayback Machine 12768: 12764: 12759: 12755: 12737: 12733: 12681: 12677: 12654: 12650: 12634: 12633: 12627: 12625: 12618: 12602: 12598: 12587:Wayback Machine 12569:Wayback Machine 12544:Wayback Machine 12533:Wayback Machine 12519: 12515: 12507:Wayback Machine 12489:Wayback Machine 12480:14, IOS Press. 12471: 12467: 12461:Wayback Machine 12444: 12440: 12434:Wayback Machine 12425:, 2nd Edition. 12417: 12413: 12407:Wayback Machine 12396:Wayback Machine 12382: 12378: 12370:Wayback Machine 12327:, Alan Kirman, 12321:Colander, David 12318: 12314: 12308:Wayback Machine 12299:, 35(1-2), pp. 12294: 12290: 12284:Wayback Machine 12267: 12263: 12258:Wayback Machine 12240: 12236: 12228:Wayback Machine 12217:Wayback Machine 12206:Wayback Machine 12186:Wayback Machine 12175:Wayback Machine 12153: 12149: 12143:Wayback Machine 12132:Wayback Machine 12121:Wayback Machine 12106:Axelrod, Robert 12104: 12100: 12094:Wayback Machine 12083:Wayback Machine 12069: 12065: 12059:Wayback Machine 12045: 12038:Wayback Machine 12027:Artificial Life 12024: 12020: 12003: 12000:Wayback Machine 11982: 11978: 11970:Wayback Machine 11945:Wayback Machine 11934:Wayback Machine 11899:Wayback Machine 11881: 11877: 11871:Wayback Machine 11862:, 2nd Edition. 11857: 11853: 11831:Wayback Machine 11822:, 2nd Edition. 11817: 11813: 11792: 11789:Wayback Machine 11770: 11767:Wayback Machine 11749: 11742: 11734:Wayback Machine 11721:., ed. (2007). 11712:Wayback Machine 11695:Wayback Machine 11685: 11683:Wayback Machine 11673: 11671:Wayback Machine 11657: 11651: 11644: 11623: 11619: 11613:Wayback Machine 11569: 11565: 11557:Wayback Machine 11546:Wayback Machine 11537:v. 1, links at 11529:Wayback Machine 11518:Wayback Machine 11504: 11500: 11493:Wayback Machine 11477:Wayback Machine 11457: 11453: 11445:Wayback Machine 11413:Wayback Machine 11393:Wayback Machine 11375: 11371: 11363:Wayback Machine 11354:, 2nd Edition. 11345:Wayback Machine 11331: 11327: 11321:Wayback Machine 11312:, 2nd Edition. 11307: 11303: 11283: 11279: 11264: 11243: 11239: 11228: 11221: 11198: 11194: 11182: 11178: 11163: 11146: 11142: 11132: 11130: 11117: 11116: 11112: 11066:Page 406: 11065: 11061: 11046: 11029: 11025: 11019:Wayback Machine 11005: 10996: 10971:10.2307/1907353 10939: 10937: 10930: 10909: 10905: 10899:Wayback Machine 10890:, 2nd Edition. 10885: 10881: 10874:Wayback Machine 10861: 10847:Thompson, G. L. 10840:Wayback Machine 10828: 10809: 10805: 10799:Wayback Machine 10790:, 2nd Edition. 10785: 10781: 10775:Wayback Machine 10764:Wayback Machine 10749:Edward Prescott 10745:Robert E. Lucas 10739: 10735: 10725: 10705:Wayback Machine 10687:Wayback Machine 10678:. Description 10669:Wayback Machine 10651: 10647: 10631: 10630: 10624: 10622: 10615: 10599: 10595: 10572: 10547: 10540: 10523: 10447: 10428: 10424: 10400:Wayback Machine 10380:Wayback Machine 10371:, 2nd Edition. 10366: 10362: 10356:Wayback Machine 10347:, 2nd Edition. 10342: 10338: 10329: 10325: 10320:Wayback Machine 10306: 10302: 10297: 10293: 10286:Wayback Machine 10277:, 2nd Edition. 10269: 10267:Wayback Machine 10258:, 2nd Edition. 10249: 10245: 10226:Wayback Machine 10208: 10204: 10196:Wayback Machine 10185:Wayback Machine 10156:Wayback Machine 10147:, 2nd Edition. 10142: 10138: 10130:Wayback Machine 10105: 10101: 10095:Wayback Machine 10084:Wayback Machine 10067: 10060: 10055:Wayback Machine 10046:, 2nd Edition. 10038: 10034: 10022:Robbins, Lionel 10020: 10016: 10011:Wayback Machine 9997: 9990: 9977: 9973: 9955: 9951: 9939: 9935: 9920: 9902: 9898: 9885: 9881: 9870:Richard Goodwin 9850:Samuelson, Paul 9843: 9839: 9830:Convex analysis 9800: 9796: 9776: 9772: 9754: 9750: 9741: 9737: 9707:identity matrix 9697:diagonal matrix 9650: 9646: 9633: 9624: 9586: 9582: 9577: 9573: 9527: 9504: 9500: 9493: 9479: 9475: 9467:Wayback Machine 9445:Wayback Machine 9436:, 2nd Edition. 9431: 9427: 9385: 9381: 9372: 9370: 9355: 9334: 9330: 9299: 9295: 9286: 9284: 9277: 9251: 9247: 9238: 9236: 9223: 9219: 9214: 9210: 9201: 9199: 9186: 9182: 9173: 9171: 9164: 9145: 9141: 9132: 9130: 9123: 9105: 9101: 9092: 9090: 9077:Dixon, Robert. 9075: 9071: 9062: 9060: 9047:Dixon, Robert. 9045: 9041: 9036: 9032: 9023: 9021: 9017: 8976: 8967: 8963: 8959:Nicola, p. 9-12 8958: 8954: 8947: 8933: 8929: 8920: 8918: 8905: 8904: 8900: 8891: 8889: 8874: 8853: 8846: 8841:Wayback Machine 8830:Wayback Machine 8816: 8812: 8803: 8801: 8794: 8778: 8771: 8765:ordinal utility 8754: 8750: 8741: 8739: 8724: 8717: 8707:Wayback Machine 8697: 8693: 8677: 8673: 8661: 8657: 8651:Wayback Machine 8642:, 2nd Edition. 8634: 8630: 8620:Philip Mirowski 8618: 8614: 8609: 8605: 8574:10.2307/1907947 8556: 8552: 8547: 8543: 8534: 8532: 8517: 8501: 8497: 8473:Wayback Machine 8458:Wayback Machine 8449:, 2nd Edition. 8440: 8429: 8422: 8406:Samuelson, Paul 8403: 8392: 8386:Wayback Machine 8368: 8364: 8356:Wayback Machine 8334:Wayback Machine 8323:Wayback Machine 8304:Wayback Machine 8283:Wayback Machine 8263:Wayback Machine 8245: 8240: 8235: 8230: 8224: 8220: 8214:Wayback Machine 8197: 8193: 8187:Wayback Machine 8172:Wayback Machine 8163:, 2nd Edition. 8153: 8149: 8144:Wayback Machine 8132: 8115: 8108: 8095: 8091: 8087: 8064: 8057: 8050: 8043: 8040: 8035: 7994:Nancy L. Stokey 7964:Reinhard Selten 7811:Roger Guesnerie 7762:David Blackwell 7723: 7718: 7702:Vilfredo Pareto 7660: 7652: 7634:Robert M. Solow 7593: 7571: 7557: 7549:Milton Friedman 7528: 7491:Austrian school 7487: 7482: 7446: 7438: 7435: 7434: 7409: 7406: 7405: 7377: 7369: 7366: 7365: 7343: 7340: 7339: 7317: 7314: 7313: 7266: 7262: 7216: 7182: 7178: 7138: 7136: 7120: 7112: 7110: 7108: 7105: 7104: 7090: 7089: 7053: 7049: 7009: 6975: 6971: 6925: 6923: 6914: 6913: 6877: 6873: 6833: 6820: 6816: 6788: 6786: 6773: 6762: 6758: 6750: 6748: 6735: 6733: 6730: 6729: 6703: 6699: 6677: 6657: 6655: 6642: 6637: 6620: 6616: 6608: 6606: 6604: 6601: 6600: 6578: 6570: 6562: 6560: 6557: 6556: 6540: 6537: 6536: 6516: 6512: 6498: 6495: 6494: 6473: 6469: 6440: 6436: 6432: 6428: 6427: 6407: 6404: 6403: 6396: 6356: 6339: 6336: 6335: 6305: 6294: 6292: 6276: 6268: 6266: 6264: 6261: 6260: 6255:Therefore, the 6232: 6227: 6201: 6197: 6179: 6178: 6176: 6174: 6171: 6170: 6130: 6126: 6108: 6107: 6105: 6101: 6096: 6056: 6055: 6037: 6036: 6034: 6030: 6026: 6013: 5995: 5994: 5992: 5991: 5978: 5957: 5956: 5954: 5952: 5936: 5928: 5926: 5924: 5921: 5920: 5919:, we have that: 5889: 5869: 5852: 5844: 5841: 5840: 5839:Recalling that 5826: 5825: 5794: 5793: 5775: 5774: 5772: 5768: 5764: 5751: 5733: 5732: 5730: 5721: 5720: 5700: 5687: 5682: 5659: 5658: 5651: 5647: 5633: 5632: 5630: 5617: 5612: 5603: 5602: 5582: 5581: 5560: 5543: 5534: 5530: 5513: 5509: 5507: 5494: 5489: 5466: 5457: 5456: 5446: 5438: 5436: 5411: 5391: 5390: 5386: 5360: 5353: 5343: 5335: 5333: 5329: 5327: 5324: 5323: 5298: 5293: 5277: 5269: 5267: 5265: 5262: 5261: 5256:Therefore, the 5236: 5228: 5226: 5213: 5208: 5189: 5168: 5167: 5165: 5149: 5141: 5139: 5137: 5134: 5133: 5102: 5088: 5080: 5077: 5076: 5041: 5025: 5007: 5006: 5004: 4991: 4983: 4981: 4979: 4976: 4975: 4947: 4939: 4936: 4935: 4919: 4916: 4915: 4894: 4886: 4884: 4864: 4848: 4840: 4838: 4813: 4793: 4773: 4772: 4768: 4755: 4747: 4745: 4743: 4740: 4739: 4722: 4706: 4698: 4696: 4671: 4651: 4650: 4646: 4642: 4640: 4630: 4604: 4603: 4601: 4587: 4579: 4577: 4575: 4572: 4571: 4540: 4499: 4496: 4495: 4479: 4476: 4475: 4420: 4417: 4416: 4385: 4343: 4320: 4317: 4316: 4302: 4301: 4270: 4254: 4250: 4228: 4218: 4210: 4208: 4205: 4204: 4181: 4159: 4149: 4141: 4139: 4135: 4133: 4130: 4129: 4109: 4106: 4105: 4089: 4086: 4085: 4069: 4066: 4065: 4018: 4014: 3986: 3937: 3933: 3905: 3893: 3890: 3889: 3869: 3866: 3865: 3820: 3816: 3798: 3794: 3788: 3784: 3773: 3770: 3769: 3745: 3742: 3741: 3723: 3720: 3719: 3686: 3683: 3682: 3664: 3661: 3660: 3638: 3581: 3578: 3577: 3543: 3540: 3539: 3523: 3520: 3519: 3503: 3500: 3499: 3468: 3465: 3464: 3439: 3301:Alfred Marshall 3253: 3245:Nicholas Kaldor 3221:Trygve Haavelmo 3193: 3187: 3163: 3102:economic system 3096:of analysis to 3056:of interacting 3054:dynamic systems 3046: 3040: 3020:Reinhard Selten 3014:In 1994, Nash, 2878:theory of games 2870: 2848: 2842: 2781:convex geometry 2762: 2752:Global analysis 2748: 2657:economic growth 2653: 2627: 2615:economic growth 2604:Richard Bellman 2581:dynamic systems 2574: 2568:Optimal control 2560: 2529:convex-analytic 2478: 2475: 2474: 2458: 2455: 2454: 2428: 2424: 2422: 2419: 2418: 2383: 2380: 2379: 2357: 2354: 2353: 2327: 2323: 2321: 2318: 2317: 2314:to be minimized 2286: 2283: 2282: 2248: 2244: 2242: 2239: 2238: 2206: 2202: 2200: 2197: 2196: 2171: 2168: 2167: 2144: 2126: 2111: 2103:Main articles: 2101: 1970:maximize their 1958:maximize their 1877: 1867: 1859:Main articles: 1841: 1825:economic sector 1809: 1803: 1785:represents the 1740:complementarity 1686: 1672: 1640:Alfred Marshall 1603:Vilfredo Pareto 1600: 1590: 1582:Main articles: 1580: 1556: 1528:of an economy. 1481: 1430: 1394:Cournot duopoly 1382: 1350: 1344: 1283:national income 1255: 1249: 1241:Friedrich Hayek 1202:economic growth 1165:economic models 1129:applied methods 1110: 1065: 1064: 1015: 1007: 1006: 1002:Decision theory 950: 904: 866: 864: 852: 845: 844: 815: 805: 804: 803: 802: 566:von Böhm-Bawerk 454: 443: 442: 204: 196: 195: 151:Economic growth 141: 133: 132: 74: 72:classifications 17: 12: 11: 5: 16173: 16163: 16162: 16157: 16140: 16139: 16137: 16136: 16124: 16112: 16100: 16087: 16084: 16083: 16081: 16080: 16075: 16070: 16063: 16057: 16055: 16051: 16050: 16048: 16047: 16041: 16039: 16035: 16034: 16032: 16031: 16026: 16021: 16020: 16019: 16014: 16004: 15999: 15994: 15989: 15984: 15979: 15974: 15973: 15972: 15967: 15962: 15954: 15953: 15952: 15950:social science 15947: 15942: 15937: 15932: 15924: 15919: 15914: 15909: 15904: 15899: 15894: 15889: 15884: 15882:Global studies 15879: 15877:Gender studies 15874: 15869: 15868: 15867: 15862: 15860:social science 15856:Environmental 15854: 15849: 15844: 15839: 15834: 15829: 15824: 15819: 15814: 15809: 15804: 15803: 15802: 15797: 15788: 15786: 15782: 15781: 15779: 15778: 15777: 15776: 15771: 15766: 15761: 15756: 15746: 15745: 15744: 15739: 15734: 15729: 15724: 15714: 15713: 15712: 15707: 15702: 15697: 15692: 15682: 15681: 15680: 15670: 15669: 15668: 15663: 15658: 15653: 15648: 15638: 15637: 15636: 15631: 15626: 15621: 15616: 15611: 15606: 15596: 15595: 15594: 15589: 15584: 15579: 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14352: 14344: 14335: 14334: 14332: 14331: 14319: 14307: 14295: 14280: 14277: 14276: 14274: 14273: 14268: 14263: 14258: 14253: 14248: 14247: 14246: 14239:Mathematicians 14235: 14233: 14231:Related topics 14227: 14226: 14224: 14223: 14218: 14213: 14208: 14203: 14198: 14192: 14190: 14184: 14183: 14181: 14180: 14179: 14178: 14173: 14168: 14166:Control theory 14158: 14153: 14148: 14143: 14138: 14133: 14128: 14123: 14118: 14113: 14108: 14102: 14100: 14094: 14093: 14091: 14090: 14085: 14080: 14075: 14070: 14064: 14062: 14056: 14055: 14053: 14052: 14047: 14042: 14037: 14031: 14029: 14023: 14022: 14020: 14019: 14014: 14009: 14004: 13999: 13994: 13989: 13983: 13981: 13975: 13974: 13972: 13971: 13966: 13961: 13955: 13953: 13947: 13946: 13944: 13943: 13941:Measure theory 13938: 13933: 13928: 13923: 13918: 13913: 13908: 13902: 13900: 13894: 13893: 13891: 13890: 13885: 13880: 13875: 13870: 13865: 13860: 13855: 13849: 13847: 13841: 13840: 13838: 13837: 13832: 13827: 13822: 13817: 13812: 13806: 13804: 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Pre-pub 12333:Perry Mehrling 12312: 12288: 12269:Axtell, Robert 12261: 12243:Bart Nooteboom 12234: 12232: 12231: 12147: 12098: 12063: 12018: 11976: 11974: 11973: 11948: 11916: 11890:, 81(2), pp. 11875: 11851: 11849: 11848: 11843:, 149(4), pp. 11811: 11809: 11808: 11797:Roth, Alvin E. 11780: 11779: 11740: 11738: 11737: 11715: 11642: 11640: 11639: 11617: 11603: 11602: 11563: 11561: 11560: 11498: 11496: 11495: 11479: 11451: 11449: 11448: 11430: 11416: 11369: 11367: 11366: 11336:, 18(2), pp. 11325: 11301: 11277: 11262: 11252:differentiable 11237: 11219: 11192: 11176: 11161: 11140: 11123:Nobelprize.org 11110: 11075:. Series 11059: 11044: 11023: 10994: 10992: 10991: 10965:(3): 265–290. 10953:Debreu, Gérard 10928: 10903: 10879: 10877: 10876: 10859: 10845:Sethi, S. P.; 10826: 10803: 10779: 10733: 10731: 10730: 10723: 10708: 10690: 10654:Zelikin, M. I. 10645: 10613: 10593: 10570: 10538: 10536: 10535: 10521: 10504: 10487: 10470: 10451: 10445: 10422: 10420: 10419: 10391: 10390: 10389:, 2nd Edition. 10360: 10336: 10323: 10300: 10298:Nicola, p. 133 10291: 10289: 10288: 10243: 10241: 10240: 10202: 10200: 10199: 10136: 10134: 10133: 10099: 10058: 10032: 10014: 9988: 9971: 9949: 9933: 9918: 9896: 9894:. Wiley. 1997. 9879: 9858:Afriat, Sidney 9854:Harsanyi, John 9846:Arrow, Kenneth 9837: 9835: 9834: 9822: 9810: 9794: 9770: 9748: 9735: 9688: 9687: 9644: 9642:, 13, pp. 1–9. 9622: 9603:(5): 905–910. 9589:Metzler, Lloyd 9580: 9571: 9569: 9568: 9542:(2): 186–188. 9525: 9498: 9491: 9473: 9471: 9470: 9450:_____ (2002). 9425: 9399:(3): 249–262. 9379: 9353: 9328: 9293: 9275: 9245: 9217: 9208: 9180: 9162: 9139: 9121: 9099: 9069: 9039: 9030: 8961: 8952: 8945: 8927: 8898: 8872: 8844: 8810: 8792: 8769: 8748: 8715: 8691: 8671: 8655: 8628: 8612: 8603: 8550: 8541: 8515: 8495: 8493: 8492: 8443:Debreu, Gérard 8427: 8420: 8390: 8362: 8360: 8359: 8314: 8313: 8306: 8286: 8268:Debreu, Gérard 8218: 8191: 8155:Debreu, Gérard 8147: 8130: 8106: 8088: 8086: 8083: 8082: 8081: 8076: 8070: 8069: 8055: 8039: 8036: 8032: 8031: 8027: 8026: 8021: 8016: 8011: 8006: 8004:Hirofumi Uzawa 8001: 7996: 7991: 7986: 7981: 7976: 7971: 7966: 7959: 7958: 7957: 7952: 7947: 7945:Thomas Sargent 7942: 7940:Yuliy Sannikov 7937: 7935:Paul Samuelson 7932: 7927: 7922: 7917: 7912: 7907: 7902: 7897: 7895:James Mirrlees 7892: 7887: 7885:Nimrod Megiddo 7880: 7879: 7878: 7873: 7868: 7863: 7861:Harold W. Kuhn 7858: 7856:David M. Kreps 7853: 7848: 7843: 7841:Leonid Hurwicz 7838: 7833: 7828: 7823: 7818: 7813: 7806: 7805: 7804: 7799: 7794: 7789: 7784: 7779: 7774: 7769: 7764: 7759: 7754: 7749: 7744: 7742:Maurice Allais 7739: 7737:R. G. D. Allen 7734: 7724: 7722: 7719: 7715: 7714: 7710: 7709: 7704: 7697: 7696: 7695: 7690: 7683: 7682: 7681: 7676: 7671: 7661: 7659: 7656: 7651: 7648: 7638:infrastructure 7601:microeconomics 7592: 7589: 7556: 7553: 7527: 7524: 7499:Chicago school 7486: 7483: 7481: 7478: 7465: 7462: 7459: 7456: 7453: 7449: 7445: 7442: 7422: 7419: 7416: 7413: 7393: 7390: 7387: 7384: 7380: 7376: 7373: 7353: 7350: 7347: 7327: 7324: 7321: 7298: 7295: 7292: 7289: 7286: 7283: 7280: 7277: 7272: 7269: 7265: 7261: 7258: 7255: 7252: 7249: 7246: 7243: 7240: 7237: 7234: 7231: 7228: 7225: 7222: 7219: 7214: 7211: 7208: 7205: 7202: 7199: 7196: 7193: 7188: 7185: 7181: 7177: 7174: 7171: 7168: 7165: 7162: 7159: 7156: 7153: 7150: 7147: 7144: 7141: 7135: 7132: 7126: 7123: 7118: 7115: 7085: 7082: 7079: 7076: 7073: 7070: 7067: 7064: 7059: 7056: 7052: 7048: 7045: 7042: 7039: 7036: 7033: 7030: 7027: 7024: 7021: 7018: 7015: 7012: 7007: 7004: 7001: 6998: 6995: 6992: 6989: 6986: 6981: 6978: 6974: 6970: 6967: 6964: 6961: 6958: 6955: 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5482: 5479: 5476: 5472: 5469: 5465: 5462: 5460: 5458: 5452: 5449: 5444: 5441: 5434: 5430: 5427: 5424: 5421: 5417: 5414: 5410: 5407: 5404: 5401: 5397: 5394: 5389: 5385: 5382: 5379: 5376: 5373: 5370: 5366: 5363: 5359: 5356: 5354: 5349: 5346: 5341: 5338: 5332: 5331: 5307: 5304: 5301: 5297: 5292: 5289: 5283: 5280: 5275: 5272: 5242: 5239: 5234: 5231: 5222: 5219: 5216: 5212: 5207: 5204: 5198: 5195: 5192: 5187: 5184: 5181: 5178: 5174: 5171: 5164: 5161: 5155: 5152: 5147: 5144: 5121: 5118: 5115: 5112: 5108: 5105: 5101: 5098: 5095: 5091: 5087: 5084: 5057: 5054: 5051: 5047: 5044: 5040: 5037: 5034: 5031: 5028: 5023: 5020: 5017: 5013: 5010: 5003: 4997: 4994: 4989: 4986: 4963: 4960: 4957: 4954: 4950: 4946: 4943: 4927:{\textstyle r} 4923: 4900: 4897: 4892: 4889: 4883: 4880: 4877: 4874: 4870: 4867: 4863: 4860: 4854: 4851: 4846: 4843: 4836: 4832: 4829: 4826: 4823: 4819: 4816: 4812: 4809: 4806: 4803: 4799: 4796: 4792: 4789: 4786: 4783: 4779: 4776: 4771: 4767: 4761: 4758: 4753: 4750: 4719: 4712: 4709: 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4010: 4007: 4004: 4001: 3995: 3992: 3989: 3985: 3981: 3978: 3975: 3972: 3968: 3964: 3961: 3958: 3955: 3952: 3949: 3946: 3943: 3940: 3936: 3932: 3929: 3926: 3923: 3920: 3914: 3911: 3908: 3904: 3900: 3897: 3877:{\textstyle A} 3873: 3853: 3850: 3847: 3844: 3841: 3838: 3835: 3831: 3828: 3823: 3819: 3815: 3812: 3807: 3804: 3801: 3797: 3791: 3787: 3783: 3780: 3777: 3762: 3761: 3753:{\textstyle L} 3749: 3739: 3731:{\textstyle K} 3727: 3717: 3705: 3702: 3699: 3696: 3693: 3690: 3680: 3672:{\textstyle Y} 3668: 3645: 3641: 3637: 3634: 3631: 3627: 3624: 3621: 3618: 3615: 3612: 3609: 3606: 3603: 3600: 3597: 3594: 3591: 3588: 3585: 3551:{\textstyle r} 3547: 3531:{\textstyle k} 3527: 3511:{\textstyle y} 3507: 3487: 3484: 3481: 3478: 3475: 3472: 3438: 3435: 3434: 3433: 3423: 3396: 3370:autoregressive 3366:representation 3309:mathematicians 3292:matrix algebra 3252: 3249: 3237:Henry L. Moore 3189:Main article: 3186: 3183: 3162: 3159: 3143:macroeconomics 3042:Main article: 3039: 3036: 3028:repeated games 2957:contract curve 2921:Nimrod Megiddo 2844:Main article: 2841: 2838: 2814:Baire category 2756:Baire category 2747: 2744: 2731:normal vectors 2626: 2623: 2559: 2556: 2532:duality theory 2513: 2512: 2500: 2497: 2494: 2491: 2488: 2485: 2482: 2462: 2442: 2439: 2436: 2431: 2427: 2416: 2405: 2402: 2399: 2396: 2393: 2390: 2387: 2361: 2341: 2338: 2335: 2330: 2326: 2315: 2299: 2296: 2293: 2290: 2280: 2268: 2265: 2262: 2259: 2256: 2251: 2247: 2226: 2223: 2220: 2217: 2214: 2209: 2205: 2184: 2181: 2178: 2175: 2146:Extensions to 2125: 2122: 2100: 2097: 2081:Paul Samuelson 2077:Robert Dorfman 2065:Leonid Hurwicz 1840: 1837: 1805:Main article: 1802: 1799: 1787:rate of growth 1768: 1767: 1676:Linear algebra 1671: 1668: 1636:Paul Samuelson 1619:Invisible hand 1607:microeconomics 1579: 1576: 1555: 1552: 1514:contract curve 1489:Jeremy Bentham 1480: 1477: 1472:groping toward 1457:groping toward 1429: 1426: 1381: 1378: 1346:Main article: 1343: 1340: 1251:Main article: 1248: 1245: 1206: 1205: 1195: 1189: 1178: 1145:matrix algebra 1112: 1111: 1109: 1108: 1101: 1094: 1086: 1083: 1082: 1071: 1070: 1067: 1066: 1063: 1062: 1057: 1052: 1047: 1042: 1037: 1032: 1027: 1022: 1016: 1013: 1012: 1009: 1008: 1005: 1004: 995: 990: 981: 976: 967: 962: 957: 951: 946: 945: 938: 937: 936: 935: 930: 922: 921: 915: 914: 906: 905: 903: 902: 895: 888: 880: 877: 876: 875: 874: 862: 847: 846: 843: 842: 837: 827: 822: 816: 811: 810: 807: 806: 801: 800: 793: 788: 783: 778: 773: 768: 763: 758: 753: 748: 743: 738: 733: 728: 723: 718: 713: 708: 703: 698: 693: 688: 683: 678: 673: 668: 663: 658: 653: 648: 643: 638: 633: 628: 623: 618: 613: 608: 603: 598: 593: 588: 583: 578: 573: 568: 563: 558: 553: 548: 543: 538: 533: 528: 523: 518: 513: 508: 503: 498: 493: 488: 483: 478: 473: 468: 463: 457: 456: 455: 449: 448: 445: 444: 441: 440: 435: 430: 425: 420: 415: 410: 405: 400: 395: 386: 381: 376: 371: 366: 361: 359:Organizational 356: 351: 346: 341: 336: 331: 326: 321: 316: 311: 306: 301: 296: 291: 286: 281: 276: 271: 266: 261: 256: 251: 246: 241: 236: 231: 226: 221: 216: 211: 205: 203:By application 202: 201: 198: 197: 194: 193: 188: 183: 178: 173: 168: 163: 158: 153: 148: 142: 139: 138: 135: 134: 131: 130: 125: 120: 115: 110: 105: 96: 91: 86: 81: 75: 69: 68: 65: 64: 63: 62: 57: 52: 44: 43: 35: 34: 28: 27: 15: 9: 6: 4: 3: 2: 16172: 16161: 16158: 16156: 16153: 16152: 16150: 16135: 16134: 16129: 16125: 16123: 16118: 16113: 16111: 16110: 16101: 16099: 16098: 16089: 16088: 16085: 16079: 16076: 16074: 16073:Human science 16071: 16069: 16068: 16064: 16062: 16059: 16058: 16056: 16052: 16046: 16043: 16042: 16040: 16036: 16030: 16029:Vegan studies 16027: 16025: 16022: 16018: 16015: 16013: 16010: 16009: 16008: 16005: 16003: 16000: 15998: 15995: 15993: 15992:Public health 15990: 15988: 15985: 15983: 15980: 15978: 15975: 15971: 15968: 15966: 15963: 15961: 15958: 15957: 15955: 15951: 15948: 15946: 15943: 15941: 15938: 15936: 15933: 15931: 15928: 15927: 15926:Philosophies 15925: 15923: 15922:Media studies 15920: 15918: 15915: 15913: 15910: 15908: 15905: 15903: 15900: 15898: 15897:Human ecology 15895: 15893: 15890: 15888: 15885: 15883: 15880: 15878: 15875: 15873: 15870: 15866: 15863: 15861: 15858: 15857: 15855: 15853: 15850: 15848: 15845: 15843: 15840: 15838: 15835: 15833: 15830: 15828: 15825: 15823: 15820: 15818: 15815: 15813: 15810: 15808: 15807:Anthrozoology 15805: 15801: 15798: 15796: 15793: 15792: 15790: 15789: 15787: 15783: 15775: 15772: 15770: 15767: 15765: 15762: 15760: 15757: 15755: 15752: 15751: 15750: 15747: 15743: 15740: 15738: 15735: 15733: 15732:developmental 15730: 15728: 15725: 15723: 15720: 15719: 15718: 15715: 15711: 15708: 15706: 15705:public policy 15703: 15701: 15698: 15696: 15693: 15691: 15688: 15687: 15686: 15683: 15679: 15676: 15675: 15674: 15671: 15667: 15664: 15662: 15659: 15657: 15656:legal systems 15654: 15652: 15651:legal history 15649: 15647: 15646:jurisprudence 15644: 15643: 15642: 15639: 15635: 15632: 15630: 15627: 15625: 15622: 15620: 15617: 15615: 15612: 15610: 15607: 15605: 15602: 15601: 15600: 15597: 15593: 15590: 15588: 15585: 15583: 15580: 15578: 15575: 15574: 15573: 15570: 15566: 15563: 15561: 15558: 15556: 15553: 15551: 15548: 15547: 15546: 15543: 15539: 15536: 15534: 15531: 15529: 15526: 15524: 15521: 15520: 15519: 15516: 15515: 15513: 15509: 15503: 15500: 15498: 15495: 15493: 15490: 15489: 15486: 15482: 15475: 15470: 15468: 15463: 15461: 15456: 15455: 15452: 15441: 15438: 15436: 15433: 15431: 15428: 15426: 15423: 15421: 15418: 15416: 15413: 15410: 15402: 15399: 15396: 15392: 15389: 15387: 15384: 15382: 15379: 15378: 15374: 15372: 15368: 15362: 15361: 15357: 15355: 15352: 15350: 15347: 15345: 15342: 15340: 15337: 15335: 15332: 15330: 15327: 15325: 15322: 15320: 15317: 15315: 15312: 15310: 15307: 15305: 15302: 15300: 15297: 15295: 15292: 15290: 15287: 15285: 15282: 15280: 15277: 15275: 15272: 15270: 15267: 15265: 15262: 15260: 15257: 15255: 15252: 15250: 15247: 15245: 15242: 15240: 15237: 15235: 15232: 15230: 15227: 15225: 15222: 15220: 15217: 15215: 15212: 15210: 15207: 15205: 15202: 15200: 15197: 15195: 15192: 15190: 15187: 15185: 15182: 15180: 15177: 15175: 15172: 15170: 15167: 15165: 15162: 15160: 15157: 15155: 15152: 15150: 15147: 15145: 15142: 15140: 15137: 15135: 15132: 15130: 15127: 15125: 15122: 15120: 15117: 15115: 15112: 15110: 15107: 15105: 15102: 15100: 15097: 15095: 15092: 15090: 15087: 15085: 15082: 15080: 15077: 15075: 15072: 15070: 15067: 15065: 15062: 15060: 15057: 15055: 15052: 15050: 15047: 15045: 15042: 15040: 15037: 15035: 15032: 15030: 15027: 15025: 15024:de Mandeville 15022: 15021: 15019: 15015: 15010: 15004: 15001: 14999: 14996: 14994: 14991: 14989: 14986: 14984: 14981: 14979: 14976: 14972: 14969: 14968: 14967: 14966:New classical 14964: 14960: 14957: 14956: 14955: 14952: 14950: 14947: 14945: 14942: 14938: 14935: 14934: 14933: 14930: 14928: 14925: 14923: 14922:Malthusianism 14920: 14914: 14911: 14910: 14909: 14906: 14904: 14901: 14898: 14894: 14891: 14890: 14889: 14886: 14884: 14883:Institutional 14881: 14879: 14876: 14874: 14871: 14869: 14866: 14864: 14861: 14859: 14856: 14854: 14851: 14849: 14846: 14844: 14841: 14839: 14836: 14834: 14831: 14829: 14826: 14824: 14821: 14817: 14814: 14813: 14812: 14809: 14807: 14804: 14802: 14799: 14797: 14794: 14790: 14787: 14786: 14785: 14782: 14780: 14777: 14775: 14772: 14770: 14767: 14765: 14762: 14761: 14759: 14754: 14749: 14744: 14736: 14733: 14731: 14728: 14726: 14723: 14721: 14718: 14716: 14713: 14711: 14708: 14706: 14703: 14701: 14698: 14696: 14693: 14691: 14687: 14686:Public choice 14684: 14682: 14679: 14677: 14674: 14672: 14669: 14667: 14664: 14662: 14661:Participation 14659: 14657: 14654: 14652: 14649: 14647: 14644: 14642: 14639: 14637: 14634: 14632: 14629: 14627: 14624: 14622: 14621:Institutional 14619: 14617: 14614: 14612: 14609: 14607: 14604: 14602: 14599: 14597: 14594: 14592: 14589: 14587: 14584: 14582: 14579: 14577: 14574: 14572: 14571:Expeditionary 14569: 14567: 14564: 14562: 14561:Environmental 14559: 14557: 14554: 14552: 14549: 14547: 14544: 14542: 14539: 14537: 14534: 14532: 14529: 14527: 14524: 14522: 14519: 14517: 14514: 14512: 14509: 14507: 14504: 14503: 14499: 14497: 14493: 14487: 14484: 14482: 14479: 14475: 14472: 14471: 14470: 14467: 14466: 14464: 14462: 14458: 14452: 14449: 14447: 14444: 14440: 14437: 14436: 14435: 14432: 14430: 14427: 14425: 14422: 14420: 14417: 14413: 14410: 14408: 14405: 14403: 14400: 14398: 14395: 14393: 14390: 14389: 14388: 14385: 14384: 14382: 14380: 14376: 14372: 14365: 14360: 14358: 14353: 14351: 14346: 14345: 14342: 14330: 14329: 14320: 14318: 14317: 14308: 14306: 14305: 14296: 14294: 14293: 14288: 14282: 14281: 14278: 14272: 14269: 14267: 14264: 14262: 14259: 14257: 14254: 14252: 14249: 14245: 14242: 14241: 14240: 14237: 14236: 14234: 14232: 14228: 14222: 14219: 14217: 14214: 14212: 14209: 14207: 14204: 14202: 14199: 14197: 14194: 14193: 14191: 14189: 14188:Computational 14185: 14177: 14174: 14172: 14169: 14167: 14164: 14163: 14162: 14159: 14157: 14154: 14152: 14149: 14147: 14144: 14142: 14139: 14137: 14134: 14132: 14129: 14127: 14124: 14122: 14119: 14117: 14114: 14112: 14109: 14107: 14104: 14103: 14101: 14099: 14095: 14089: 14086: 14084: 14081: 14079: 14076: 14074: 14071: 14069: 14066: 14065: 14063: 14061: 14057: 14051: 14048: 14046: 14043: 14041: 14038: 14036: 14033: 14032: 14030: 14028: 14027:Number theory 14024: 14018: 14015: 14013: 14010: 14008: 14005: 14003: 14000: 13998: 13995: 13993: 13990: 13988: 13985: 13984: 13982: 13980: 13976: 13970: 13967: 13965: 13962: 13960: 13959:Combinatorics 13957: 13956: 13954: 13952: 13948: 13942: 13939: 13937: 13934: 13932: 13929: 13927: 13924: 13922: 13919: 13917: 13914: 13912: 13911:Real analysis 13909: 13907: 13904: 13903: 13901: 13899: 13895: 13889: 13886: 13884: 13881: 13879: 13876: 13874: 13871: 13869: 13866: 13864: 13861: 13859: 13856: 13854: 13851: 13850: 13848: 13846: 13842: 13836: 13833: 13831: 13828: 13826: 13823: 13821: 13818: 13816: 13813: 13811: 13808: 13807: 13805: 13803: 13799: 13793: 13790: 13788: 13785: 13781: 13778: 13776: 13773: 13772: 13771: 13768: 13767: 13764: 13759: 13751: 13746: 13744: 13739: 13737: 13732: 13731: 13728: 13724: 13720: 13716: 13713: 13711: 13707: 13704: 13702: 13699: 13696: 13695: 13686: 13682: 13679:, MIT Press. 13678: 13674: 13671: 13667: 13663: 13659: 13656: 13652: 13648: 13645: 13641: 13637: 13633: 13630: 13627:, MIT Press. 13626: 13622: 13619: 13615: 13611: 13607: 13604: 13601: 13598:and contents 13597: 13593: 13589: 13586: 13583: 13579: 13575: 13571: 13567: 13563: 13560: 13557: 13553: 13549: 13547: 13543: 13539: 13536: 13533: 13529: 13525: 13522: 13520: 13516: 13512: 13511: 13491: 13487: 13483: 13481: 13475: 13469: 13463: 13459: 13456: 13452: 13448: 13442: 13435: 13431: 13428:, 42(2), pp. 13427: 13421: 13406: 13402: 13396: 13392: 13391: 13386: 13380: 13372: 13366: 13362: 13357: 13356: 13350: 13344: 13336: 13332: 13328: 13324: 13317: 13303: 13297: 13293: 13289: 13282: 13273: 13265: 13261: 13257: 13253: 13247: 13239: 13235: 13231: 13227: 13223: 13217: 13209: 13202: 13187: 13183: 13179: 13172: 13157: 13153: 13147: 13132: 13128: 13124: 13120: 13119: 13111: 13103: 13099: 13095: 13091: 13087: 13083: 13079: 13075: 13071: 13067: 13060: 13052: 13048: 13044: 13040: 13036: 13032: 13027: 13022: 13018: 13014: 13010: 13004: 12996: 12992: 12988: 12982: 12978: 12971: 12963: 12959: 12955: 12951: 12947: 12943: 12939: 12935: 12928: 12920: 12916: 12912: 12908: 12904: 12900: 12896: 12892: 12888: 12884: 12877: 12869: 12865: 12861: 12857: 12853: 12849: 12845: 12841: 12840: 12832: 12824: 12820: 12816: 12812: 12808: 12804: 12800: 12796: 12795: 12790: 12784: 12778: 12774: 12771: 12766: 12757: 12750: 12746: 12742: 12735: 12727: 12723: 12719: 12715: 12711: 12707: 12703: 12699: 12695: 12691: 12690: 12685: 12679: 12671: 12667: 12663: 12659: 12652: 12644: 12638: 12623: 12619: 12613: 12609: 12608: 12600: 12594: 12588: 12584: 12581: 12577: 12573: 12570: 12566: 12563: 12560:893. Pre-pub 12559: 12555: 12551: 12548: 12547: 12545: 12541: 12538: 12534: 12530: 12527: 12523: 12517: 12508: 12504: 12501: 12497: 12493: 12492: 12490: 12486: 12483: 12479: 12475: 12469: 12462: 12458: 12455: 12451: 12447: 12442: 12435: 12431: 12428: 12424: 12420: 12415: 12408: 12404: 12401: 12397: 12393: 12390: 12386: 12380: 12371: 12367: 12364: 12360: 12356: 12352: 12349: 12348: 12346: 12342: 12339:, 98(2), pp. 12338: 12334: 12330: 12326: 12322: 12316: 12309: 12305: 12302: 12298: 12292: 12285: 12281: 12278: 12274: 12270: 12265: 12259: 12255: 12252: 12248: 12244: 12238: 12229: 12225: 12222: 12218: 12214: 12211: 12207: 12203: 12200: 12197: 12193: 12190: 12189: 12187: 12183: 12180: 12176: 12172: 12169: 12165: 12164:981-256-100-5 12161: 12157: 12151: 12144: 12140: 12137: 12133: 12129: 12126: 12122: 12118: 12115: 12112:, Princeton. 12111: 12107: 12102: 12095: 12091: 12088: 12084: 12080: 12077: 12073: 12067: 12060: 12056: 12053: 12049: 12043: 12039: 12035: 12032: 12028: 12022: 12015: 12011: 12007: 12001: 11997: 11994: 11991:893. Pre-pub 11990: 11986: 11980: 11971: 11967: 11964: 11960: 11956: 11952: 11949: 11946: 11942: 11939: 11935: 11931: 11928: 11924: 11920: 11917: 11914: 11911:, 84(2), pp. 11910: 11906: 11903: 11902: 11900: 11896: 11893: 11889: 11885: 11879: 11872: 11868: 11865: 11861: 11855: 11846: 11842: 11838: 11835: 11834: 11832: 11828: 11825: 11821: 11815: 11806: 11803:, 70(4), pp. 11802: 11798: 11795: 11794: 11790: 11786: 11783: 11778:, 51(8), pp. 11777: 11773: 11772: 11768: 11764: 11761: 11757: 11753: 11747: 11745: 11735: 11731: 11728: 11724: 11720: 11717:Nisan, Noam, 11716: 11713: 11709: 11706: 11702: 11698: 11697: 11696: 11692: 11689: 11684: 11680: 11677: 11672: 11668: 11665: 11661: 11655: 11649: 11647: 11637: 11633: 11629: 11625: 11624: 11621: 11614: 11610: 11607: 11600: 11596: 11595: 11593: 11589: 11585: 11581: 11578:, MIT Press. 11577: 11573: 11567: 11558: 11554: 11551: 11547: 11543: 11540: 11536: 11532: 11531: 11530: 11526: 11523: 11519: 11515: 11512: 11508: 11502: 11494: 11490: 11487: 11483: 11480: 11478: 11474: 11471: 11467: 11464: 11463: 11461: 11455: 11446: 11442: 11439: 11435: 11431: 11428: 11424: 11420: 11417: 11414: 11410: 11407: 11403: 11402: 11397: 11396: 11394: 11390: 11387: 11383: 11379: 11373: 11364: 11360: 11357: 11353: 11349: 11348: 11346: 11342: 11339: 11335: 11329: 11322: 11318: 11315: 11311: 11305: 11298: 11297:0-12-076975-1 11294: 11290: 11286: 11281: 11273: 11269: 11265: 11259: 11255: 11251: 11247: 11241: 11234: 11233: 11226: 11224: 11216: 11212: 11208: 11205: 11201: 11196: 11189: 11185: 11180: 11172: 11168: 11164: 11158: 11154: 11150: 11144: 11128: 11124: 11120: 11114: 11106: 11102: 11098: 11094: 11090: 11086: 11082: 11078: 11074: 11070: 11063: 11055: 11051: 11047: 11041: 11037: 11033: 11027: 11020: 11016: 11013: 11009: 11003: 11001: 10999: 10988: 10984: 10980: 10976: 10972: 10968: 10964: 10960: 10959: 10954: 10950: 10946: 10945: 10935: 10931: 10925: 10921: 10917: 10913: 10907: 10900: 10896: 10893: 10889: 10883: 10875: 10871: 10868: 10862: 10856: 10852: 10848: 10843: 10842: 10841: 10837: 10834: 10829: 10823: 10818: 10817: 10807: 10800: 10796: 10793: 10789: 10783: 10776: 10772: 10769: 10765: 10761: 10758: 10754: 10750: 10746: 10742: 10737: 10726: 10720: 10716: 10715: 10709: 10706: 10702: 10699: 10695: 10691: 10688: 10684: 10681: 10677: 10673: 10672: 10670: 10666: 10663: 10659: 10655: 10649: 10641: 10635: 10620: 10616: 10614:9782881240775 10610: 10606: 10605: 10597: 10589: 10585: 10581: 10577: 10573: 10567: 10563: 10559: 10555: 10551: 10545: 10543: 10532: 10528: 10524: 10518: 10514: 10510: 10505: 10501: 10497: 10493: 10488: 10484: 10480: 10476: 10471: 10467: 10463: 10462: 10457: 10452: 10448: 10442: 10438: 10434: 10430: 10429: 10426: 10416: 10412: 10411:Tucker, A. W. 10408: 10404: 10403: 10401: 10397: 10394: 10388: 10384: 10383: 10381: 10377: 10374: 10370: 10364: 10357: 10353: 10350: 10346: 10340: 10333: 10327: 10321: 10317: 10314: 10310: 10304: 10295: 10287: 10283: 10280: 10276: 10272: 10271: 10268: 10264: 10261: 10257: 10253: 10247: 10238: 10234: 10230: 10229: 10227: 10223: 10220: 10216: 10212: 10206: 10197: 10193: 10190: 10186: 10182: 10179: 10175: 10174:0-19-510268-1 10171: 10167: 10163: 10160: 10159: 10157: 10153: 10150: 10146: 10140: 10131: 10127: 10124: 10121:_____ (1970). 10120: 10119: 10117: 10114:, 36(3), pp. 10113: 10110:Came to Be", 10109: 10103: 10096: 10092: 10089: 10086:and contents 10085: 10081: 10078: 10074: 10070: 10065: 10063: 10056: 10052: 10049: 10045: 10041: 10036: 10029: 10028: 10023: 10018: 10012: 10008: 10005: 10001: 9995: 9993: 9985: 9981: 9975: 9967: 9963: 9959: 9953: 9946: 9942: 9937: 9929: 9925: 9921: 9915: 9911: 9907: 9900: 9893: 9889: 9883: 9875: 9871: 9867: 9863: 9859: 9855: 9851: 9847: 9841: 9831: 9827: 9823: 9819: 9815: 9811: 9807: 9802: 9801: 9798: 9791: 9790:0-471-98232-6 9787: 9783: 9779: 9774: 9766: 9762: 9758: 9752: 9745: 9739: 9732: 9728: 9724: 9720: 9716: 9712: 9708: 9704: 9703: 9698: 9694: 9693: 9685: 9683: 9677: 9676: 9672: 9671: 9670: 9666: 9662: 9658: 9654: 9648: 9641: 9637: 9631: 9629: 9627: 9618: 9614: 9610: 9606: 9602: 9598: 9594: 9590: 9584: 9575: 9565: 9561: 9557: 9553: 9549: 9545: 9541: 9537: 9532: 9531: 9528: 9522: 9518: 9514: 9510: 9502: 9494: 9488: 9484: 9477: 9468: 9464: 9461: 9457: 9453: 9449: 9448: 9446: 9442: 9439: 9435: 9429: 9422: 9416: 9412: 9407: 9402: 9398: 9394: 9390: 9383: 9368: 9364: 9360: 9356: 9350: 9346: 9342: 9338: 9332: 9324: 9320: 9316: 9312: 9308: 9304: 9297: 9282: 9278: 9272: 9268: 9264: 9260: 9256: 9249: 9234: 9230: 9229: 9221: 9212: 9197: 9193: 9192: 9184: 9169: 9165: 9159: 9155: 9154: 9149: 9143: 9128: 9124: 9118: 9114: 9110: 9103: 9088: 9084: 9080: 9073: 9058: 9054: 9050: 9043: 9034: 9016: 9012: 9008: 9004: 9000: 8996: 8992: 8988: 8984: 8983: 8975: 8971: 8965: 8956: 8948: 8942: 8938: 8931: 8917:on 2000-07-09 8916: 8912: 8908: 8902: 8887: 8883: 8879: 8875: 8869: 8865: 8861: 8857: 8851: 8849: 8842: 8838: 8835: 8831: 8827: 8824: 8820: 8814: 8799: 8795: 8789: 8785: 8784: 8776: 8774: 8766: 8762: 8758: 8752: 8737: 8733: 8729: 8722: 8720: 8712: 8708: 8704: 8701: 8695: 8688: 8685:. Macmillan. 8684: 8683: 8675: 8669: 8665: 8659: 8652: 8648: 8645: 8641: 8637: 8632: 8625: 8621: 8616: 8607: 8599: 8595: 8591: 8587: 8583: 8579: 8575: 8571: 8567: 8563: 8562: 8554: 8545: 8530: 8526: 8522: 8518: 8512: 8508: 8507: 8499: 8490: 8489: 8484: 8480: 8477: 8476: 8474: 8470: 8467: 8464:, 54(6), pp. 8463: 8459: 8455: 8452: 8448: 8444: 8438: 8436: 8434: 8432: 8423: 8417: 8413: 8412: 8407: 8401: 8399: 8397: 8395: 8387: 8383: 8380: 8376: 8372: 8366: 8357: 8353: 8350: 8346: 8343:, MIT Press. 8342: 8338: 8337: 8335: 8331: 8328: 8324: 8320: 8317: 8311: 8307: 8305: 8301: 8298: 8294: 8290: 8287: 8284: 8280: 8277: 8273: 8269: 8266: 8265: 8264: 8260: 8257: 8253: 8249: 8244: 8239: 8234: 8228: 8222: 8215: 8211: 8208: 8204: 8200: 8195: 8188: 8184: 8181: 8178:, 54(6), pp. 8177: 8173: 8169: 8166: 8162: 8161: 8156: 8151: 8145: 8141: 8138: 8133: 8127: 8123: 8119: 8113: 8111: 8103: 8099: 8093: 8089: 8080: 8077: 8075: 8072: 8071: 8067: 8061: 8056: 8053: 8047: 8042: 8030: 8025: 8022: 8020: 8017: 8015: 8012: 8010: 8007: 8005: 8002: 8000: 7997: 7995: 7992: 7990: 7987: 7985: 7982: 7980: 7979:Stephen Smale 7977: 7975: 7972: 7970: 7967: 7965: 7962: 7961: 7960: 7956: 7955:Herbert Scarf 7953: 7951: 7948: 7946: 7943: 7941: 7938: 7936: 7933: 7931: 7928: 7926: 7923: 7921: 7918: 7916: 7913: 7911: 7908: 7906: 7903: 7901: 7900:Roger Myerson 7898: 7896: 7893: 7891: 7888: 7886: 7883: 7882: 7881: 7877: 7874: 7872: 7869: 7867: 7864: 7862: 7859: 7857: 7854: 7852: 7849: 7847: 7844: 7842: 7839: 7837: 7834: 7832: 7829: 7827: 7826:John R. Hicks 7824: 7822: 7819: 7817: 7814: 7812: 7809: 7808: 7807: 7803: 7800: 7798: 7795: 7793: 7790: 7788: 7785: 7783: 7782:Gérard Debreu 7780: 7778: 7775: 7773: 7770: 7768: 7765: 7763: 7760: 7758: 7755: 7753: 7750: 7748: 7745: 7743: 7740: 7738: 7735: 7733: 7730: 7729: 7728: 7727: 7713: 7708: 7705: 7703: 7700: 7699: 7698: 7694: 7691: 7689: 7688:Irving Fisher 7686: 7685: 7684: 7680: 7677: 7675: 7672: 7670: 7669:Enrico Barone 7667: 7666: 7665: 7664: 7655: 7646: 7641: 7639: 7635: 7631: 7629: 7625: 7621: 7617: 7612: 7610: 7606: 7602: 7598: 7587: 7582: 7580: 7576: 7570: 7566: 7562: 7552: 7550: 7545: 7543: 7538: 7533: 7523: 7519: 7514: 7512: 7507: 7504: 7500: 7496: 7492: 7477: 7463: 7460: 7457: 7454: 7451: 7447: 7443: 7440: 7417: 7414: 7411: 7391: 7388: 7385: 7382: 7378: 7374: 7371: 7351: 7348: 7345: 7325: 7322: 7319: 7293: 7290: 7287: 7281: 7278: 7270: 7267: 7263: 7256: 7253: 7250: 7244: 7241: 7232: 7229: 7226: 7223: 7220: 7209: 7206: 7203: 7197: 7194: 7186: 7183: 7179: 7172: 7169: 7166: 7160: 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4325: 4322: 4297: 4293: 4290: 4287: 4281: 4274: 4271: 4267: 4261: 4255: 4251: 4244: 4241: 4238: 4232: 4230: 4222: 4214: 4201: 4198: 4192: 4185: 4182: 4175: 4172: 4169: 4163: 4161: 4153: 4145: 4127: 4111: 4091: 4071: 4051: 4048: 4045: 4042: 4038: 4034: 4031: 4025: 4019: 4015: 4008: 4005: 4002: 3993: 3990: 3987: 3979: 3976: 3973: 3970: 3966: 3962: 3959: 3956: 3950: 3947: 3944: 3938: 3934: 3927: 3924: 3921: 3912: 3909: 3906: 3898: 3895: 3887: 3871: 3848: 3845: 3842: 3836: 3833: 3829: 3826: 3821: 3817: 3813: 3810: 3805: 3802: 3799: 3795: 3789: 3785: 3781: 3778: 3775: 3767: 3747: 3740: 3725: 3718: 3700: 3697: 3694: 3688: 3681: 3666: 3659: 3658: 3657: 3643: 3639: 3635: 3632: 3629: 3625: 3622: 3616: 3610: 3607: 3601: 3598: 3595: 3589: 3586: 3583: 3575: 3571: 3567: 3561: 3560: 3559: 3545: 3525: 3505: 3482: 3476: 3473: 3470: 3458: 3456: 3452: 3447: 3445: 3431: 3427: 3424: 3421: 3417: 3412: 3408: 3404: 3400: 3397: 3394: 3390: 3386: 3382: 3379: 3375: 3374:Jan Tinbergen 3371: 3367: 3363: 3359: 3355: 3351: 3347: 3343: 3340: 3339: 3338: 3335: 3330: 3328: 3327: 3322: 3317: 3314: 3310: 3304: 3302: 3298: 3293: 3289: 3281: 3277: 3273: 3269: 3268:macroeconomic 3266: 3262: 3257: 3248: 3246: 3242: 3238: 3233: 3231: 3226: 3222: 3218: 3217: 3212: 3208: 3207:Ragnar Frisch 3204: 3202: 3198: 3192: 3182: 3180: 3172: 3167: 3158: 3155: 3151: 3146: 3144: 3140: 3136: 3132: 3128: 3124: 3120: 3116: 3115:collaboration 3112: 3108: 3103: 3099: 3095: 3090: 3088: 3085: 3083: 3078: 3077: 3074:mathematical 3071: 3067: 3063: 3059: 3055: 3051: 3045: 3035: 3034:of modeling. 3033: 3029: 3025: 3022:received the 3021: 3017: 3016:John Harsanyi 3012: 3010: 3006: 3005:public-policy 3002: 2998: 2994: 2990: 2986: 2982: 2978: 2974: 2970: 2966: 2962: 2961:Edgeworth box 2958: 2955:or along the 2954: 2950: 2946: 2941: 2938: 2934: 2930: 2926: 2925:Bezalel Peleg 2922: 2918: 2914: 2913:Martin Shubik 2910: 2906: 2902: 2898: 2894: 2891: 2887: 2883: 2879: 2875: 2869: 2865: 2861: 2857: 2853: 2847: 2837: 2835: 2831: 2827: 2823: 2819: 2815: 2810: 2809:Stephen Smale 2806: 2805:Gérard Debreu 2802: 2798: 2794: 2788: 2786: 2782: 2778: 2774: 2770: 2766: 2761: 2757: 2753: 2743: 2741: 2736: 2732: 2727: 2725: 2721: 2717: 2712: 2710: 2706: 2702: 2698: 2694: 2690: 2686: 2685:Gérard Debreu 2682: 2681:Kenneth Arrow 2678: 2674: 2670: 2666: 2662: 2658: 2652: 2648: 2644: 2640: 2636: 2632: 2622: 2620: 2616: 2612: 2609: 2605: 2600: 2598: 2594: 2590: 2586: 2582: 2578: 2573: 2569: 2565: 2555: 2553: 2549: 2545: 2541: 2537: 2533: 2530: 2526: 2522: 2518: 2498: 2495: 2492: 2489: 2486: 2483: 2480: 2460: 2437: 2429: 2425: 2417: 2403: 2400: 2397: 2394: 2391: 2388: 2385: 2377: 2376: 2359: 2336: 2328: 2324: 2316: 2313: 2294: 2288: 2281: 2266: 2263: 2257: 2249: 2245: 2224: 2221: 2215: 2207: 2203: 2179: 2173: 2165: 2164: 2163: 2161: 2157: 2153: 2149: 2143: 2139: 2135: 2131: 2121: 2119: 2115: 2110: 2106: 2096: 2094: 2093:Ragnar Frisch 2090: 2086: 2082: 2078: 2074: 2070: 2066: 2062: 2056: 2054: 2050: 2046: 2042: 2038: 2034: 2030: 2026: 2022: 2021:market system 2017: 2015: 2014:search theory 2011: 2007: 2003: 1999: 1995: 1991: 1987: 1985: 1981: 1977: 1973: 1969: 1965: 1961: 1957: 1953: 1949: 1945: 1941: 1937: 1933: 1929: 1924: 1922: 1918: 1914: 1910: 1906: 1902: 1898: 1895:by selecting 1894: 1893:real function 1890: 1886: 1882: 1876: 1872: 1866: 1862: 1854: 1850: 1845: 1836: 1834: 1830: 1826: 1822: 1818: 1814: 1808: 1798: 1796: 1792: 1791:interest rate 1788: 1784: 1781: 1777: 1773: 1765: 1761: 1760: 1754: 1753: 1748: 1745: 1744: 1743: 1741: 1737: 1733: 1729: 1725: 1722: 1718: 1714: 1710: 1709: 1705: 1700: 1699:matrix pencil 1696: 1691: 1685: 1681: 1677: 1670:Linear models 1667: 1665: 1661: 1657: 1653: 1649: 1645: 1641: 1637: 1633: 1632: 1626: 1624: 1620: 1616: 1612: 1608: 1604: 1599: 1595: 1589: 1585: 1575: 1573: 1569: 1565: 1561: 1551: 1549: 1544: 1543:tax incidence 1540: 1536: 1535: 1529: 1527: 1523: 1519: 1518:Edgeworth Box 1515: 1506: 1505:Edgeworth box 1501: 1497: 1494: 1490: 1486: 1476: 1473: 1469: 1464: 1462: 1458: 1453: 1449: 1447: 1443: 1439: 1435: 1425: 1423: 1419: 1415: 1411: 1407: 1403: 1399: 1395: 1391: 1387: 1377: 1375: 1371: 1366: 1362: 1354: 1349: 1339: 1337: 1333: 1329: 1325: 1321: 1317: 1312: 1309: 1308: 1303: 1299: 1295: 1290: 1288: 1284: 1280: 1276: 1272: 1268: 1264: 1260: 1254: 1244: 1242: 1238: 1234: 1229: 1227: 1223: 1219: 1215: 1211: 1203: 1199: 1196: 1193: 1190: 1187: 1183: 1179: 1176: 1173: 1172: 1171: 1168: 1166: 1162: 1156: 1154: 1150: 1146: 1142: 1138: 1134: 1130: 1126: 1122: 1118: 1107: 1102: 1100: 1095: 1093: 1088: 1087: 1085: 1084: 1081: 1077: 1073: 1072: 1061: 1058: 1056: 1053: 1051: 1048: 1046: 1043: 1041: 1038: 1036: 1033: 1031: 1028: 1026: 1023: 1021: 1018: 1017: 1011: 1010: 1003: 999: 996: 994: 991: 989: 985: 982: 980: 977: 975: 971: 968: 966: 963: 961: 958: 956: 955:Number theory 953: 952: 949: 944: 943: 940: 939: 934: 931: 929: 926: 925: 924: 923: 920: 917: 916: 912: 911: 901: 896: 894: 889: 887: 882: 881: 879: 878: 873: 863: 861: 856: 851: 850: 849: 848: 841: 838: 835: 831: 828: 826: 823: 821: 818: 817: 814: 809: 808: 799: 798: 794: 792: 789: 787: 784: 782: 779: 777: 774: 772: 769: 767: 764: 762: 759: 757: 754: 752: 749: 747: 744: 742: 739: 737: 734: 732: 729: 727: 724: 722: 719: 717: 714: 712: 709: 707: 704: 702: 699: 697: 694: 692: 689: 687: 684: 682: 679: 677: 674: 672: 669: 667: 664: 662: 659: 657: 654: 652: 649: 647: 644: 642: 639: 637: 634: 632: 629: 627: 624: 622: 619: 617: 614: 612: 609: 607: 604: 602: 599: 597: 594: 592: 589: 587: 584: 582: 579: 577: 574: 572: 569: 567: 564: 562: 559: 557: 554: 552: 549: 547: 544: 542: 539: 537: 534: 532: 529: 527: 524: 522: 519: 517: 514: 512: 509: 507: 504: 502: 499: 497: 494: 492: 489: 487: 484: 482: 479: 477: 474: 472: 469: 467: 464: 462: 461:de Mandeville 459: 458: 453: 447: 446: 439: 436: 434: 431: 429: 426: 424: 421: 419: 416: 414: 411: 409: 406: 404: 401: 399: 396: 394: 390: 389:Public choice 387: 385: 382: 380: 377: 375: 372: 370: 367: 365: 364:Participation 362: 360: 357: 355: 352: 350: 347: 345: 342: 340: 337: 335: 332: 330: 327: 325: 324:Institutional 322: 320: 317: 315: 312: 310: 307: 305: 302: 300: 297: 295: 292: 290: 287: 285: 282: 280: 277: 275: 274:Expeditionary 272: 270: 267: 265: 264:Environmental 262: 260: 257: 255: 252: 250: 247: 245: 242: 240: 237: 235: 232: 230: 227: 225: 222: 220: 217: 215: 212: 210: 207: 206: 200: 199: 192: 189: 187: 184: 182: 179: 177: 174: 172: 169: 167: 164: 162: 159: 157: 154: 152: 149: 147: 144: 143: 137: 136: 129: 126: 124: 121: 119: 116: 114: 111: 109: 106: 104: 100: 97: 95: 94:International 92: 90: 87: 85: 82: 80: 77: 76: 73: 70:Branches and 67: 66: 61: 58: 56: 53: 51: 48: 47: 46: 45: 41: 37: 36: 33: 30: 29: 25: 21: 20: 16131: 16107: 16095: 16065: 15872:Food studies 15812:Area studies 15565:mathematical 15564: 15560:econometrics 15518:Anthropology 15435:Publications 15391:Publications 15358: 14954:Neoclassical 14944:Mercantilism 14853:Evolutionary 14715:Sociological 14688: / 14586:Geographical 14566:Evolutionary 14541:Digitization 14506:Agricultural 14469:Econometrics 14423: 14397:Price theory 14326: 14314: 14302: 14283: 14216:Optimization 14120: 14078:Differential 14002:Differential 13969:Order theory 13964:Graph theory 13868:Group theory 13723: 13697: 13676: 13661: 13650: 13635: 13624: 13609: 13591: 13578:Desecription 13573: 13551: 13541: 13527: 13514: 13494:. 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Retrieved 12606: 12599: 12575: 12553: 12535:and pre-pub 12521: 12516: 12495: 12473: 12468: 12449: 12441: 12422: 12414: 12398:and pre-pub 12384: 12379: 12354: 12336: 12325:Peter Howitt 12315: 12296: 12291: 12272: 12264: 12246: 12237: 12198:, pp. 309 - 12195: 12155: 12150: 12109: 12101: 12085:and pre-pub 12071: 12066: 12047: 12040:and pre-pub 12026: 12021: 12005: 11984: 11979: 11954: 11922: 11908: 11887: 11878: 11859: 11854: 11840: 11819: 11814: 11801:Econometrica 11800: 11775: 11755: 11722: 11718: 11700: 11653: 11627: 11620: 11598: 11575: 11572:Tirole, Jean 11566: 11534: 11506: 11501: 11459: 11454: 11433: 11422: 11399: 11381: 11372: 11351: 11334:Econometrica 11333: 11328: 11309: 11304: 11288: 11285:Yves Balasko 11280: 11253: 11249: 11240: 11235:, Princeton. 11230: 11206: 11195: 11187: 11179: 11152: 11143: 11131:. Retrieved 11122: 11113: 11080: 11076: 11072: 11062: 11035: 11026: 11007: 10962: 10958:Econometrica 10956: 10938:. 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Retrieved 8505: 8498: 8486: 8462:Econometrica 8461: 8446: 8410: 8373:, Waveland. 8370: 8365: 8340: 8309: 8292: 8271: 8226: 8221: 8202: 8194: 8176:Econometrica 8175: 8158: 8150: 8121: 8092: 8074:Econophysics 8028: 8019:Hermann Wold 8014:Abraham Wald 7984:Robert Solow 7925:Frank Ramsey 7787:Mario Draghi 7757:Yves Balasko 7721:20th century 7711: 7658:19th century 7653: 7643: 7632: 7613: 7608: 7604: 7594: 7584: 7578: 7572: 7546: 7530:Philosopher 7529: 7521: 7516: 7508: 7502: 7494: 7488: 6596: 6397: 6388: 6256: 5257: 5072: 3763: 3563: 3462: 3461: 3460: 3448: 3440: 3364:developed a 3352:is based on 3350:econometrics 3331: 3324: 3318: 3305: 3285: 3276:Money supply 3241:cobweb model 3234: 3224: 3216:Econometrica 3214: 3205: 3201:econometrics 3194: 3191:Econometrics 3185:Econometrics 3176: 3147: 3091: 3086: 3081: 3076:optimization 3075: 3047: 3013: 2948: 2945:neoclassical 2942: 2937:voting games 2917:Hervé Moulin 2871: 2834:Yves Balasko 2822:Sard's lemma 2789: 2763: 2760:Sard's lemma 2728: 2723: 2713: 2708: 2707:represented 2700: 2654: 2610: 2601: 2593:Frank Ramsey 2576: 2575: 2514: 2372: 2145: 2142:Shadow price 2112: 2085:Robert Solow 2057: 2018: 1988: 1948:dual problem 1935: 1925: 1878: 1865:Dual problem 1810: 1782: 1775: 1769: 1763: 1758: 1756: 1751: 1750: 1746: 1735: 1731: 1727: 1716: 1712: 1707: 1703: 1702: 1687: 1643: 1629: 1627: 1601: 1564:graph theory 1557: 1532: 1530: 1513: 1511: 1508:participants 1484: 1482: 1471: 1467: 1465: 1460: 1456: 1454: 1450: 1445: 1431: 1406:marginalists 1402:market price 1389: 1383: 1359: 1331: 1313: 1305: 1291: 1270: 1256: 1230: 1207: 1175:optimization 1169: 1157: 1121:mathematical 1116: 1115: 1049: 830:Publications 795: 418:Sociological 391: / 289:Geographical 269:Evolutionary 244:Digitization 209:Agricultural 113:Mathematical 112: 84:Econometrics 16133:Wikiversity 16024:Social work 15912:Linguistics 15837:Criminology 15754:criminology 15737:personality 15695:comparative 15673:Linguistics 15666:private law 15523:archaeology 15229:von Neumann 14998:Supply-side 14983:Physiocracy 14927:Marginalism 14616:Information 14556:Engineering 14536:Development 14531:Demographic 14402:Game theory 14379:Theoretical 14328:WikiProject 14171:Game theory 14151:Probability 13888:Homological 13878:Multilinear 13858:Commutative 13835:Type theory 13802:Foundations 13758:mathematics 13681:Description 13666:Description 13640:Description 13614:Description 13596:Description 13588:A. K. Dixit 13496:11 February 13447:R.W. Clower 12580:Description 12482:Description 12446:Duffy, John 12359:Description 12210:Description 12168:Description 12114:Description 12010:description 11959:Description 11927:Description 11727:Description 11580:Description 11511:Description 10407:Kuhn, H. W. 10235:22(3), pp. 10178:Description 10108:Foundations 10077:Description 9719:John Kemeny 9659:satisfy an 9456:Description 8823:description 8821:. Links to 8757:cardinality 8700:the preface 8568:(1): 1–12. 8345:Description 8316:Description 8199:Varian, Hal 7969:Amartya Sen 7876:Eric Maskin 7707:Léon Walras 7575:J.M. Keynes 7569:Engineering 7542:consistency 7532:Karl Popper 6334:If we take 3378:time series 3362:Herman Wold 3297:mathematics 3261:IS/LM model 3251:Application 3111:competition 3107:game theory 3084:rationality 2977:equilibrium 2933:fair values 2929:fair prices 2890:topological 2886:convex sets 2846:Game Theory 2840:Game theory 2767:'s work on 2689:convex sets 2663:introduced 2540:Rockafellar 2375:constraints 2373:inequality 2195:subject to 2156:Harold Kuhn 2027:and in the 2002:uncertainty 1821:physiocrats 1721:probability 1660:equilibrium 1652:tâtonnement 1644:Foundations 1560:convex sets 1468:tâtonnement 1442:tâtonnement 1438:Walras' law 1428:Léon Walras 1422:game theory 1398:homogeneous 1365:Léon Walras 1348:Marginalism 1324:W.S. Jevons 1287:John Graunt 1226:game theory 1182:equilibrium 1180:static (or 1151:, or other 1045:Linguistics 1035:Computation 1030:Geosciences 993:Probability 919:Mathematics 666:von Neumann 319:Information 259:Engineering 239:Development 234:Demographic 176:Game theory 118:Methodology 16149:Categories 16078:Humanities 16012:historical 15945:psychology 15917:Management 15759:demography 15717:Psychology 15700:philosophy 15661:public law 15592:integrated 15386:Economists 15259:Schumacher 15164:Schumpeter 15134:von Wieser 15054:von Thünen 15014:Economists 14913:Circuitism 14878:Humanistic 14873:Historical 14848:Ecological 14838:Democratic 14811:Chartalism 14801:Behavioral 14764:Mainstream 14725:Statistics 14720:Solidarity 14641:Managerial 14606:Humanistic 14601:Historical 14546:Ecological 14511:Behavioral 14156:Statistics 14035:Arithmetic 13997:Arithmetic 13863:Elementary 13830:Set theory 13590:, 1990. 13411:2009-04-30 13363:, 33, 41. 13307:2008-06-10 13192:2019-08-07 13162:2019-08-07 13137:2008-08-16 12664:(1): 140. 12628:2008-08-17 12498:, 30, pp. 12357:, Oxford. 11957:, Oxford. 11925:, Norton. 11921:(1978 ). 11604:v. 3, pp. 11548:and v. 3, 11509:, 4th ed. 11466:Gul, Faruk 10940:2020-05-28 10625:2015-06-27 10071:( 1990). 9941:David Gale 9715:David Gale 9373:2008-08-26 9287:2020-05-28 9239:2020-05-28 9202:2020-05-28 9174:2020-05-28 9133:2021-05-31 9093:2008-09-28 9063:2008-09-28 9024:2008-08-21 8921:2008-08-21 8892:2008-08-21 8804:2008-08-21 8742:2008-07-06 8736:Birmingham 8535:2020-05-28 8270:(1983). 8085:References 7920:Roy Radner 7816:Frank Hahn 7797:David Gale 7559:See also: 7501:— differs 4934:, so that 3381:statistics 3358:hypotheses 3354:statistics 3334:stochastic 3321:Adam Smith 3223:published 2850:See also: 2750:See also: 2703:while the 2701:quantities 2651:Dual space 2635:Convex set 2629:See also: 2608:Pontryagin 2602:Following 2562:See also: 2128:See also: 2041:analytical 1869:See also: 1853:paraboloid 1772:transposed 1674:See also: 1615:Adam Smith 1592:See also: 1572:axiomatics 1267:statistics 1137:difference 1055:Philosophy 998:Statistics 988:Set theory 825:Economists 696:Schumacher 601:Schumpeter 571:von Wieser 491:von Thünen 452:economists 428:Statistics 423:Solidarity 344:Managerial 309:Humanistic 304:Historical 249:Ecological 214:Behavioral 108:Mainstream 15956:Planning 15935:economics 15852:Education 15749:Sociology 15727:cognitive 15678:semiotics 15629:political 15587:technical 15572:Geography 15545:Economics 15304:Greenspan 15269:Samuelson 15249:Galbraith 15219:Tinbergen 15159:von Mises 15154:Heckscher 15114:Edgeworth 14993:Stockholm 14988:Socialist 14888:Keynesian 14868:Happiness 14828:Classical 14789:Mutualism 14784:Anarchist 14769:Heterodox 14666:Personnel 14626:Knowledge 14591:Happiness 14581:Financial 14551:Education 14526:Democracy 14461:Empirical 14371:Economics 14083:Geometric 14073:Algebraic 14012:Euclidean 13987:Algebraic 13883:Universal 13546:Contents. 13519:Contents. 13430:56, 64-65 13127:1095-5054 13102:154881432 13086:0022-2186 13043:0018-2702 13021:CiteSeerX 12995:230844893 12954:0030-7653 12903:0034-6527 12860:0012-9682 12815:0012-9682 12726:154780520 12710:0022-3808 12637:cite book 12277:F193-F210 12251:Abstract. 11805:1341–1378 11688:Abstract. 11676:Abstract. 11664:Abstract. 11574:(1988). 11486:Abstract. 11470:Abstract. 11012:Abstract. 10979:0012-9682 10833:Abstract. 10634:cite book 10466:EMS Press 10279:Abstract. 10260:Abstract. 10048:Abstract. 10004:Abstract. 9828:(1997) . 9711:wranglers 9655:and 9609:0002-8282 9564:154074343 9556:0018-2702 9415:1552-4485 9323:0018-2702 9150:(1999) . 9003:0028-0836 8834:chapters. 8582:0012-9682 8451:Abstract. 8408:(1947) . 8297:Contents. 8165:Abstract. 7577:wrote in 7495:normative 7461:− 7421:∞ 7418:− 7412:ρ 7346:ρ 7320:ρ 7294:ρ 7291:− 7282:α 7279:− 7271:ρ 7268:− 7257:α 7254:− 7242:α 7233:τ 7230:− 7227:ρ 7224:− 7210:ρ 7207:− 7198:α 7195:− 7187:ρ 7184:− 7173:α 7170:− 7158:α 7149:ρ 7146:− 7134:− 7081:ρ 7078:− 7069:α 7066:− 7058:ρ 7055:− 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4072:τ 4046:− 4032:− 4009:τ 4006:− 3980:≡ 3971:− 3957:− 3928:τ 3925:− 3837:∈ 3834:α 3822:α 3806:α 3803:− 3790:α 3395:) models. 3265:Keynesian 3050:economies 2965:John Nash 2493:… 2438:⋅ 2398:… 2337:⋅ 2295:⋅ 2222:≤ 2166:Minimize 1966:and that 1956:consumers 1887:involves 1742:equation 1730:and 1664:exogenous 1605:analyzed 1568:mechanics 1410:oligopoly 1320:mechanics 1298:algebraic 1125:economics 1060:Education 1050:Economics 1025:Chemistry 741:Greenspan 706:Samuelson 686:Galbraith 656:Tinbergen 596:von Mises 591:Heckscher 551:Edgeworth 369:Personnel 329:Knowledge 294:Happiness 284:Financial 254:Education 229:Democracy 123:Political 89:Heterodox 32:Economics 16097:Category 15965:regional 15960:land use 15795:business 15764:internet 15722:abnormal 15624:military 15614:economic 15604:cultural 15577:physical 15538:physical 15528:cultural 15415:Category 15395:journals 15381:Glossary 15334:Stiglitz 15299:Rothbard 15279:Buchanan 15264:Friedman 15254:Koopmans 15244:Leontief 15224:Robinson 15109:Marshall 14959:Lausanne 14863:Georgism 14858:Feminist 14806:Buddhist 14796:Austrian 14695:Regional 14671:Planning 14646:Monetary 14576:Feminist 14521:Cultural 14516:Business 14304:Category 14060:Topology 14007:Discrete 13992:Analytic 13979:Geometry 13951:Discrete 13906:Calculus 13898:Analysis 13853:Abstract 13792:Glossary 13775:Timeline 13655:Contents 13629:Contents 13608:, 1998. 13572:, 1989. 13556:Contents 13540:, 1984. 13532:Contents 13526:, 1982. 13490:Archived 13458:Archived 13449:(1957). 13405:Archived 13387:(1936). 13351:(1953). 13186:Archived 13156:Archived 13131:Archived 12773:Archived 12749:calculus 12741:geometry 12622:Archived 12583:Archived 12578:, v. 2. 12565:Archived 12540:Archived 12529:Archived 12526:Abstract 12503:Archived 12485:Archived 12457:Archived 12454:Abstract 12430:Archived 12427:Abstract 12403:Archived 12392:Archived 12389:Abstract 12366:Archived 12353:(1994). 12304:Archived 12280:Archived 12254:Archived 12224:Archived 12221:abstract 12213:Archived 12202:Archived 12182:Archived 12171:Archived 12139:Archived 12128:Archived 12125:contents 12117:Archived 12108:(1997). 12090:Archived 12079:Archived 12076:Abstract 12055:Archived 12034:Archived 12031:Abstract 11996:Archived 11966:Archived 11953:(1994). 11941:Archived 11930:Archived 11901:p. 366. 11895:Archived 11867:Archived 11864:Abstract 11827:Archived 11824:Abstract 11785:Archived 11763:Archived 11760:Abstract 11730:Archived 11708:Archived 11691:Archived 11679:Archived 11667:Archived 11609:Archived 11553:Archived 11542:Archived 11525:Archived 11514:Archived 11489:Archived 11473:Archived 11441:Archived 11438:Abstract 11409:Archived 11406:Abstract 11389:Archived 11359:Archived 11356:Abstract 11341:Archived 11317:Archived 11314:Abstract 11291:, 1988, 11254:approach 11248:(1985). 11127:Archived 11105:13089965 11015:Archived 10934:Archived 10914:(1977). 10895:Archived 10892:Abstract 10870:Archived 10849:(2000). 10836:Archived 10795:Archived 10792:Abstract 10771:Archived 10760:Archived 10751:(1989). 10701:Archived 10683:Archived 10665:Archived 10619:Archived 10435:(1999). 10396:Archived 10393:Abstract 10376:Archived 10352:Archived 10349:Abstract 10316:Archived 10282:Archived 10263:Archived 10222:Archived 10219:Abstract 10192:Archived 10189:contents 10181:Archived 10152:Archived 10149:Abstract 10126:Archived 10091:Archived 10080:Archived 10051:Archived 10007:Archived 9964:(1976). 9928:57281275 9888:Yinyu Ye 9872:(eds.). 9763:(1976). 9463:Archived 9441:Archived 9438:Abstract 9367:Archived 9363:20217006 9339:(1990). 9281:Archived 9233:Archived 9196:Archived 9168:Archived 9127:Archived 9011:21004543 8886:Archived 8882:20217006 8858:(1990). 8837:Archived 8826:Archived 8798:Archived 8703:Archived 8647:Archived 8644:Abstract 8598:35705710 8529:Archived 8525:13498913 8485:(1944). 8469:Archived 8454:Archived 8382:Archived 8352:Archived 8349:Contents 8330:Archived 8327:Contents 8319:Archived 8300:Archived 8291:(1984). 8279:Archived 8276:Contents 8259:Archived 8225:* As in 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15349:Krugman 15214:Kuznets 15204:Kalecki 15179:Polanyi 15069:Cournot 15064:Bastiat 15049:Ricardo 15039:Malthus 15029:Quesnay 14932:Marxian 14823:Chicago 14753:history 14748:Schools 14735:Welfare 14705:Service 14496:Applied 14316:Commons 14098:Applied 14068:General 13845:Algebra 13770:History 13685:preview 13670:preview 13644:preview 13600:preview 13432:(press 13238:1809376 13051:6705939 12962:2663180 12919:4140287 12911:2296113 12868:1906940 12823:1907716 12745:algebra 12718:2138643 12301:212-270 12136:preview 11938:preview 11913:406-411 11892:365-370 11845:262-268 11705:166–196 11539:ch. 3-6 11338:155-162 11272:1113262 11171:1075992 11097:1888984 11054:0898626 10987:1907353 10698:Outline 10588:9048698 10580:1900016 10531:1295240 10500:1888251 10483:0337317 10270: 10118:–1386. 10088:preview 9717:and by 9705:is the 9617:1811704 9460:preview 8590:1907947 8475:-1270. 7605:without 6257:dynamic 4724:Dynamic 4128:become: 3884:is the 3568:of the 3082:bounded 2995:, and 2959:of the 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