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General Equilibrium

General Equilibrium

CONTEXT AND HISTORY

THE MODEL AND PROOF

PROBLEMS WITH THE ASSUMPTIONS AND DEVELOPMENTS IN GET

WHAT LIES BEYOND?

BIBLIOGRAPHY

General equilibrium theory deals with the existence of efficient competitive prices in an individual private enterprise economy. This discussion of general equilibrium theory (GET) is divided into four parts: (1) the context and history of GET, (2) comments on the appropriate mathematization and method of proof, (3) the problems with the parsimonious modeling and solution concept used, and (4) what lies beyond the theory.

CONTEXT AND HISTORY

The French economist Léon Walras (1834-1910) was the first to attempt a mathematization of the conditions necessary and sufficient to determine the prices of all goods and services produced and consumed by firms and individuals in a closed economy. He distinguished between services and durable production goods, and he developed the theory of competitive exchange, noting the possibility of multiple solutions to the existence of competitive prices. In particular he did not prove rigorously the existence of an efficient price system, but he did recognize that this would require the simultaneous solution to the set of equations he had utilized to describe the overall system of production and consumption. In writing down his production conditions, he utilized simple fixed coefficients for the structure of production.

Walras sketched the roles of both a rate of interest and a currency in his system, but these were not fully developed. These features, unlike his integration of production and consumption into a consistent whole, may be regarded as more or less undeveloped when compared with the market structure for the production and consumptions of goods and services. Although Walras discussed the important role of government, he left it out as being beyond the more strictly economic problems he was addressing.

The full modeling of an exchange and production economy presented a challenge. The construction of a formal closed mathematical model of the economy called for Draconian simplification, and it was thus a major achievement in abstraction. As is often the case, even in the purest of mathematical economics, the distinction between the verbal treatment and the mathematization offered is often considerable. Words permit a richness of discourse at the cost of precision, while mathematics offers precision and logical tightness at the cost of qualification, nuance, and the recognition of complexity. Mathematics thus facilitates the analysis at the cost of minimizing concern with context.

The work of Walras was accepted relatively slowly. The mathematical problems surrounding the solution of the simultaneous equation structure of the general equilibrium model were posed in a modern mathematical context by Abraham Wald, who considered production with inputs in fixed proportion, and with a single output. He also, however, assumed the existence of demand functions rather than deriving them from utility maximization.

THE MODEL AND PROOF

The key papers presenting the first rigorous proofs of GET were written by Kenneth Arrow and Gerard Debreu (1954) and Lionel McKenzie (1959). The sketch here primarily follows the treatment presented in Debreus Theory of Value (1959).

An economy is considered initially with l commodities (goods or services), m utility maximizing agents, and n profit maximizing multi-product firms. The economy exists over a specific finite period of time sectioned into T time periods. The existence of money is not considered. It is as though there exists some vast smoothly functioning central clearinghouse that balances all accounts at the end of the trading period after which the economy ceases. A commodity or service is characterized not merely by its physical aspects, but by both a time of availability and a location, providing a considerable simplification of the model structure. It is as though all trade takes place in a single time period, with a vast array of futures markets available that make further trade unnecessary.

All real persons or individuals are assumed to have well-defined preferences that can be represented by utility functions. It is further assumed that individual preferences are such that more goods are always of value.

Production is described by a convex production set, in which each individual producer selects a set of inputs to maximize the profits obtained from the sale of its outputs. In the process of production, free disposal is assumed. The profits are distributed to individuals who hold shares in the firms. It is proved (under reasonably plausible conditions on the utility functions and production sets) that a set of prices exist (not necessarily unique) such that supply equals demand in every market and that the resultant imputation of goods and services is efficient or Pareto optimal, meaning that there is no way that any individual can improve utility without another individual obtaining less utility. The method of proof is highly technical; it utilized fixed-point theory (Kakutani 1941), which has been of considerable use in both subsequent developments in general equilibrium theory and in the application of the theory of games in economics.

PROBLEMS WITH THE ASSUMPTIONS AND DEVELOPMENTS IN GET

Since the development of the original models, there have been many modifications of the original stringent assumptions. In particular, the original models deal with a finite sector of time, but in the real world there is a past and a future. There is thus a question as to how one can extend the general equilibrium analysis to infinite horizon models. Furthermore, given an infinite horizon model, an overlapping generations structure to the population appears to be more appropriate than viewing individuals as living forever. The seminal work of Maurice Allais (1946) and Paul Samuelson (1958) opened up a literature extending the investigation of competitive markets (see Geanakoplos 1987 for an extensive summary).

When dealing with a high level of abstraction, the linkage between the assumption and the underlying reality must be considered, for there are empirical exceptions to virtually every assumption made. But, on the whole, the question to be considered is whether the rigor, when confronted with the reality, provides a good enough fit to cast light on the function of a significant part of the economy. The considerable developments in the computation of applied general equilibrium models suggests that it does. The computational methods are based on the work of Herbert Scarf.

A basic assumption in the original proofs is that individual consumers and firms are price takers. In other words, they are so small that their actions have no influence on market prices. But, as presented, the proofs were based on there being a finite number of agents. This affects the stated basic assumption of price taking. The Debreu proof does not depend on whether there is one agent or a million in any market. If this finite number is taken into account, then the actions of individual agents may influence price. A precise proof is needed to show the conditions under which the individual influence can be ignored. Robert Aumann (1964), using technical results from the mathematics of measure theory, provided a solution reflecting the lack of power of an individual small agent. The strategic market game (SMG) model of Lloyd Shapley and Martin Shubik (1977) provided the basis for non-cooperative game models of a closed economy, while Pradeep Dubey and Shubik (1978) showed the basic inefficiency of the noncooperative equilibria. However, the equilibria approach a competitive price as the number of agents in the economy increases. The noncooperative game models and the general equilibrium models are mathematically distinct, but for a continuum of agents the solutions may coincide.

The original models avoided problems with uncertainty through an ingenious but unsatisfactory enlargement of the number of goods, including a myriad of markets with contingent goods. This is currently being avoided in the development of general equilibrium with incomplete (GEI) markets.

The SMG models of the economy are more institutional than those of GET. They require an explicit specification of the price formation mechanism in the markets and a description of what happens to the traders under all circumstances. The mathematical model is so complete that it could be played as an experimental game, and the role of money and markets appears naturally and explicitly as a way of simplifying trading activity. The game-theoretic formulation has permitted the handling of three important items left out by GET: the influence of oligopolistic markets (i.e., markets with powerful players whose individual action influences price), the role of possible default, and the role of different levels of information.

At a high level of abstraction, there is an important link between work in cooperative game theory and GET. There are two cooperative game theory solutions called the core and value that can be related to the competitive equilibrium. These solutions stress group power and individual equity. The core characterizes outcomes that cannot be effectively challenged by group behavior, while the value reflects the expected marginal productivity of an individual over all groups he or she could join. It can be shown that in an economy with many individuals the price system has the properties of both the core and the value.

WHAT LIES BEYOND?

One of the major concerns in economic theory is the reconciliation between general equilibrium microeconomic theory and macroeconomics. It has been suggested that a natural extension beyond basic general equilibrium will incorporate both the financial structure and government. The next series of basic micro-macro models will have a continuum of private agents plus government as a large player. Beyond this, the addition of taxation and public goods to the basic micro-macro models of the economy will be difficult but rewarding. Game-theoretic methods that permit the blending of price taking and oligopolistic elements, and that take into account nonsymmetries in information, are currently being developed in the study of macroeconomic control.

SEE ALSO Arrow, Kenneth J.; Arrow-Debreu Model; Debreu, Gerard; Economic Model; Economics; Equilibrium in Economics; Game Theory; Macroeconomics; Market Economy; Microeconomics; Stability in Economics

BIBLIOGRAPHY

Allais, Maurice. 1946 Economie et Interet. Imprimerie National: Paris.

Aumann, Robert. J. 1964. Markets with a Continuum of Traders. Econometrica 32: 3950.

Arrow, Kenneth J. and Gerard Debreu. 1954. Existence of Equilibrium for a Competitive Economy. Econometrica 22: 265290.

Debreu, Gerard. 1959. Theory of Value, an Axiomatic Analysis of Economic Equilibrium. New York: Wiley.

Dubey, Pradeep, and Martin Shubik. 1978. The Noncooperative Equilibria of a Closed Trading Economy with Market Supply and Bidding Strategies. Journal of Economic Theory 17 (1): 120.

Geanakoplos, John D. 1987. The Overlapping Generations Model of General Equilibrium. In The New Palgrave: A Dictionary of Economics, ed. J. Eatwell, M. Millgate and P. Newman, 767779. London: Macmillan.

Kakutani, Shizuo. 1941. A Generalization of Brouwers Fixed Point Theorem. Duke Mathematical Journal 8 (3): 437459.

McKenzie, Lionel W. 1959. On the Existence of General Equilibrium for a Competitive Market. Econometrica 27: 5471.

Samuelson, Paul. 1958. Aspects of Public Expenditure Theory. Review of Economics and Statistics 40: 335338.

Shapley, Lloyd S., and Martin Shubik. 1977. Trade Using One Commodity as a Means of Payment. Journal of Political Economy 85 (5): 937968.

Walras, Léon. Eléments déconomie politique pure. 4th ed. Lausanne: L. Corbaz.

Martin Shubik

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