General equilibrium

General equilibrium theory is a branch of theoretical microeconomics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many markets. It is often assumed that agents are price takers and in that setting two common notions of equilibrium exist: Walrasian (or competitive) equilibrium and its generalization, a price equilibrium with transfers.

Broadly speaking, general equilibrium tries to give an understanding of the whole economy using a "bottom-up" approach, starting with individual markets and agents. Macroeconomics, as developed by the Keynesian economists, uses a "top-down" approach, where the analysis starts with larger aggregates, the "big picture". The distinction is not as clear as once thought, however.

This distinction has gradually become even less sharp, since much of modern macroeconomics has emphasized microeconomic foundations. However, many macroeconomic models simply have a "goods market" and study its interaction with, for instance, the financial market. General equilibrium models typically involve a multitude of different goods markets. Modern general equilibrium models are usually complex and require computers to help with numerical solutions.

In a market system, the prices and production of all goods, including the price of money and interest, are interrelated. A change in the price of one good -- say, bread -- may affect another price, such as bakers' wages. If bakers differ in tastes from others, the demand for bread might be affected by a change in bakers' wages, with a consequent effect on the price of bread. Calculating the equilibrium price of just one good, in theory, requires an analysis that accounts for all of the millions of different goods that are available.

History of general equilibrium modeling

The first attempt in neoclassical economics to model prices for a whole economy was made by Léon Walras. Walras' "Elements of Pure Economics" provides a succession of models, each taking into account more aspects of a real economy (two commodities, many commodities, production, growth, money). Some (for example, Eatwell (1989), see also Jaffe (1953)) think Walras was unsuccessful and the later models in this series are inconsistent. In particular, Walras's model was a long period model in which prices of capital goods are the same whether they appear as inputs or outputs and in which the same rate of profits is earned in all lines of industry. The cost price of each capital good must be equal in equilibrium, in this model, to the demand price. This is inconsistent with the quantities of capital goods being taken as data. But when Walras introduced capital goods in his later models, he took their quantities as given, in arbitrary ratios.(Kenneth Arrow and Gerard Debreu continued to take the initial quantities of capital goods as givens, but adopted a short run model in which the prices of capital goods vary with time and the own rate of interest varies across capital goods.)

Walras was the first to lay down a research program much followed by 20th century economists. In particular, Walras' agenda included the investigation of when equilibria are unique and stable.

Walras also proposed a dynamic process by which general equilibrium might be reached, that of the tâtonnement or groping process.

The tatonnement process is a model for investigating stability of equilibria. Prices are announced (perhaps by an "auctioneer"), and agents state how much of each good they would like to offer (supply) or purchase (demand). No transactions and no production take place at disequilibrium prices. Instead, prices are lowered for goods with positive prices and excess supply. Prices are raised for goods with excess demand. The question for the mathematician is under what conditions such a process will terminate in equilibrium in which demand equates to supply for goods with positive prices and demand does not exceed supply for goods with a price of zero. Walras was not able to provide a definitive answer to this question (see Unresolved Problems in General Equilibrium below).

In partial equilibrium analysis, the determination of the price of a good is simplified by just looking at the price of one good, and assuming that the prices of all other goods remain constant. The Marshallian theory of supply and demand is an example of partial equilibrium analysis. Partial equilibrium analysis is adequate when the first-order effects of a shift in, say, the demand curve do not shift the supply curve. Anglo-American economists became more interested in general equilibrium in the late 1920s and 1930s after Piero Sraffa's demonstration that Marshallian economists cannot account for theforces thought to account for the upward-slope of the supply curve for a consumer good.

If an industry uses little of a factor of production, a small increase in the output of that industry will not bid the price of that factor up. To a first order approximation, firms in the industry will not experience decreasing costs and the industry supply curves will not slope up. If an industry uses an appreciable amount of that factor of production, an increase in the output of that industry will exhibit decreasing costs. But such a factor is likely to be used in substitutes for the industry's product,and an increased price of that factor will have effects on the supply of those substitutes. Consequently, Sraffa argued, the first order effects of a shift in the demand curve of the original industry under these assumptions includes a shift in the supply curve of substitutes for that industry's product and consequent shifts in the original industry's supply curve. General equilibrium is designed to investigate such interactions between markets.

Continental European economists made important advances in the 1930s. Walras' proofs of the existence of general equilibrium often were based on the counting of equations and variables. Such arguments are inadequate for non-linear systems of equations and do not imply that equilibrium prices and quantities cannot be negative, a meaningless solution for his models. The replacement of certain equations by inequalities and the use of more rigorous mathematics improved general equilibrium modeling.

Modern concept of general equilibrium in economics

The modern conception of general equilibrium is provided by a model developed jointly by Kenneth Arrow, Gerard Debreu and Lionel W. McKenzie in the 1950s. Gerard Debreu presents this model in "Theory of Value" (1959) as an axiomatic model, following thestyle of mathematics promoted by Bourbaki. In such an approach, the interpretation of the terms in the theory (e.g., goods, prices) are not fixed by the axioms.

Three important interpretations of the terms of the theory have been often cited. First, suppose commodities are distinguished by the location where they are delivered. Then the Arrow-Debreu model is a spatial model of, for example, international trade.

Second, suppose commodities are distinguished by when they are delivered. That is, suppose all markets equilibrate at some initial instant of time. Agents in the model purchase and sell contracts, where a contract specifies, for example, a good to be delivered and the date at which it is to be delivered. The Arrow-Debreu model of intertemporal equilibrium contains forwardmarkets for all goods at all dates. No markets exist at any future dates.

Third, suppose contracts specify states of nature which affect whether a commodity is to be delivered: "A contract for the transfer of a commodity now specifies, in addition to its physical properties, its location and its date, an event on the occurrence of which the transfer is conditional. This new definition of a commodity allows one to obtain a theory of [risk] free from any probability concept..." (Debreu, 1959)

These interpretations can be combined. So the complete Arrow-Debreu model can be said to apply when goods are identified by when they are to be delivered, where they are to be delivered, and under what circumstances they are to be delivered, as well as their intrinsic nature. So there would be a complete set of prices for contracts such as "1 ton of Winter red wheat, delivered on 3rd of January in Minneapolis, if there is a hurricane in Florida during December". A general equilibrium model with complete markets of this sort seems to be a long way from describing the workings of real economies, however its proponents argue that it is still useful as a simplified guide as to how a real economies function.

Some of the recent work in general equilibrium has in fact explored the implications of incomplete markets, which is to say an intertemporal economy with uncertainty, where there do not exist sufficiently detailed contracts that would allow agents to fully allocate their consumption and resources through time. While it has been shown that such economies will generally still have an equilibrium, the outcome may no longer be Pareto optimal. The basic intuition for this result is that if consumers lack adequate means to transfer their wealth from one time period to another and the future is risky, there is nothing to necessarily tie any price ratio down to the relevant marginal rate of substitution, which is the standard requirement for Pareto optimality. However, under some conditions the economy may still be constrained Pareto optimal, meaning that a central authority limited to the same type and number of contracts as the individual agents may not be able to improve upon the outcome - what is needed is the introduction of a full set of possible contracts. Hence, one implication of the theory of incomplete markets is that inefficiency may be a result of underdeveloped financial institutions or credit constraints faced by some members of the public. Research still continues in this area.

Properties and characterization of general equilibrium

Basic questions in general equilibrium analysis are concerned with the conditions under which an equilibrium will be efficient, which efficient equilibria can be achieved, when an equilibrium is guaranteed to exist and when the equilibrium will be unique and stable.

First Fundamental Theorem of Welfare Economics

The first fundamental welfare theorem asserts that market equilibria are Pareto efficient. In a pure exchange economy, a sufficient condition for the first welfare theorem to hold is that preferences be locally nonsatiated. The first welfare theorem also holds for economies with production regardless of the properties of the production function. Implicitly, the theorem assumes complete markets and perfect information. In an economy with externalities, for example, it is possible for equilibria to arise that are not efficient.

The first welfare theorem is informative in the sense that it points to the sources of inefficiency in markets. Under the assumptions above, any market equilibrium is tautologically efficient. Therefore, when equilibria arise that are not efficient, the market system itself is not to blame, but rather some sort of market failure.

econd Fundamental Theorem of Welfare Economics

While every equilibrium is efficient, it is clearly not true that every efficient allocation of resources will be an equilibrium. However, the Second Theorem states that every efficient allocation can be supported by some set of prices. In other words all that is required to reach a particular outcome is a redistribution of initial endowments of the agents after which the market can be left alone to do its work. This suggests that the issues of efficiency and equity can be separated and need not involve a trade off. However, the conditions for the Second Theorem are stronger than those for the First, as now we need consumers' preferences to be convex (convexity roughly corresponds to the idea of diminishing marginal utility, or to preferences where "averages are better than extrema").

Existence

Even though every equilibrium is efficient, neither of the above two theorems say anything about the equilibrium existing in the first place. To guarantee that an equilibrium exists we once again need consumer preferences to be convex (although with enough consumers this assumption can be relaxed both for existence and the Second Welfare Theorem). Similarly, but less plausibly, feasible production sets must be convex, excluding the possibility of economies of scale.

Proofs of the existence of equilibrium generally rely on fixed point theorems such as Brouwer fixed point theorem or its generalization, the Kakutani fixed point theorem. In fact, one can quickly pass from a general theorem on the existence of equilibrium to Brouwer’s fixed point theorem. For this reason many mathematical economists consider proving existence a deeper result than proving the two Fundamental Theorems.

Uniqueness

Although generally (assuming convexity) an equilibrium will exist and will be efficient the conditions under which it will be unique are much stronger. While the issues are fairly technical the basic intuition is that the presence of wealth effects (which is the feature that most clearly delineates general equilibrium analysis from partial equilibrium) generates the possibility of multiple equilibria. When a price of a particular good changes there are two effects. First, the relative attractiveness of various commodities changes, and second, the wealth distribution of individual agents is altered. These two effects can offset or reinforce each other in ways that make it possible for more than one set of prices to constitute an equilibrium.

A result known as the Sonnenschein-Mantel-Debreu Theorem states that the aggregate (excess) demand function inherits only certain properties of individual's demand functions, and that these (Continuity, Homogeneity of degree zero, Walras' law, and boundary behavior when prices are near zero) are not sufficient to restrict the admissible aggregate excess demand function in a way which would ensure uniqueness of equilibrium.

There has been much research on conditions when the equilibrium will be unique, or which at least will limit the number of equilibria. One result states that under mild assumptions the number of equilibria will be finite (see Regular economy) and odd (see Index Theorem). Furthermore if an economy as a whole, as characterized by an aggregate excess demand function, has the revealed preference property (which is a much stronger condition than revealed preferences for a single individual) or the gross substitute property then likewise the equilibrium will be unique. All methods of establishing uniqueness can be thought of as establishing that each equilibrium has the same positive local index, in which case by the index theorem there can be but one such equilibrium.

Determinacy

Given that equilibria may not be unique, it is of some interest to ask whether any particular equilibrium is at least locally unique. If so, then comparative statics can be applied as long as the shocks to the system are not too large. As stated above, in a Regular economy equilibria will be finite, hence locally unique. One reassuring result, due to Debreu, is that "most" economies are regular. However recent work by Michael Mandler (1999) has challenged this claim. The Arrow-Debreu-McKenzie model is neutral between models of production functions as continuously differentiable and as formed from (linear combinations of) fixed coefficient processes. Mandler accepts that, under either model of production, the initial endowments will not be consistent with a continuum of equilibria, except for a set of Lebesgue measure zero. However, endowments change with time in the model and this evolution of endowments is determined by the decisions of agents (e.g., firms) in the model. Agents in the model have an interest in equilibria being indeterminate:

"Indeterminacy, moreover, is not just a technical nuisance; it undermines the price-taking assumption of competitive models. Since arbitrary small manipulations of factor supplies can dramatically increase a factor's price, factor owners will not take prices to be parametric."(Mandler 1999, p. 17)

When technology is modeled by (linear combinations) of fixed coefficient processes, optimizing agents will drive endowments to be such that a continuum of equilibria exist:

"The endowments where indeterminacy occurs systematically arise through time and therefore cannot be dismissed; the Arrow-Debreu-McKenzie model is thus fully subject to the dilemmas of factor price theory."(Mandler 1999, p. 19)

Critics of the general equilibrium approach have questioned its practical applicability based on the possibility of non-uniqueness of equilibria. Supporters have pointed out that this aspect is in fact a reflection of the complexity of the real world and hence an attractive realistic feature of the model.

tability

In a typical general equilibrium model the prices that prevail "when the dust settles" are simply those that coordinate the demands of various consumers for various goods. But this raises the question of how these prices and allocations have been arrived at and whether any (temporary) shock to the economy will cause it to converge back to the same outcome that prevailed before the shock. This is the question of stability of the equilibrium, and it can be readily seen that it is related to the question of uniqueness. If there are multiple equilibria, then some of them will be unstable. Then, if an equilibrium is unstable and there is a shock, the economy will wind up at a different set of allocations and prices once the convergence process terminates. However stability depends not only on the number of equilibria but also on the type of the process that guides price changes (for a specific type of price adjustment process see Tatonnement). Consequently some researchers have focused on plausible adjustment processes that guarantee system stability, i.e., that guarantee convergence of prices and allocations to some equilibrium. However, when more than one equilibrium exists, where one ends up will depend on where one begins.

Unresolved problems in general equilibrium

Research building on the Arrow-Debreu-McKenzie model has revealed some problems with the model. The Sonnenschein-Mantel-Debreu results show that, essentially, any restrictions on the shape of excess demand functions are stringent. Some think this implies that the Arrow-Debreu model lacks empirical content. At any rate, Arrow-Debreu-McKenzie equilibria cannot be expected to be unique, or stable.

A model organized around the tatonnement process has been said to be a model of a centrally planned economy, not a decentralized market economy. Some research has tried to develop general equilibrium models with other processes. In particular, some economists have developed models in which agents can trade at out-of-equilibrium prices and such trades can affect the equilibria to which the economy tends. Particularly noteworthy are the Hahn process, the Edgeworth process, and the Fisher process.

The data determining Arrow-Debreu equilibria include initial endowments of capital goods. If production and trade occur out of equilibrium, these endowments will bechanged further complicating the picture.

In a real economy, however, trading, as well as production and consumption, goeson out of equilibrium. It follows that, in the course of convergence to equilibrium (assuming that occurs), endowments change. In turn this changes the set of equilibria. Put more succinctly, the set of equilibria is path dependent... [This path dependence] makes the calculation of equilibria corresponding to the initial state of the system essentially irrelevant. What matters is the equilibrium that the economy will reach from given initial endowments, not the equilibrium that it would have been in, giveninitial endowments, had prices happened to be just right(Franklin Fisher, as quoted by Petri (2004)).

The Arrow-Debreu model in which all trade occurs in futures contracts at time zero requires a very large number of markets to exist. It is equivalent under complete markets to a sequential equilibrium concept in which spot markets for goods and assets open at each date-state event (they are not equivalent under incomplete markets); market clearing then requires that the entire sequence of prices clears all markets at all times. A generalization of the sequential market arrangement is the temporary equilibrium structure, where market clearing at a point in time is conditional on expectations of future prices which need not be market clearing ones.

Although the Arrow-Debreu-McKenzie model is set out in terms of some arbitrary numeraire, the model does not encompass money. Frank Hahn, for example, has investigated whether general equilibrium models can be developed in which money enters in some essential way. The goal is to find models in which existence of money can alter the equilibrium solutions, perhaps because the initial position of agents depends on monetary prices.

Some critics of general equilibrium modeling contend that much research in these models constitutes exercises in pure mathematics with no connection to actual economies. "There are endeavors that now pass for the most desirable kind of economic contributions although they are just plain mathematical exercises, not only without any economic substance but also without any mathematical value" (Nicholas Georgescu-Roegen 1979). Georgescu-Roegen cites as an example a paper that assumes more traders in existence than there are points in the set of real numbers.

Although modern models in general equilibrium theory demonstrate that under certain circumstances prices will indeed converge to equilibria, critics hold that the assumptions necessary for these results are extremely strong. As well as stringent restrictions on excess demand functions, the necessary assumptions include perfect rationality of individual complete information about all prices both now and in the future; and the conditions necessary for perfect competition. However some results from experimental economics suggest that even in circumstances where there are few, imperfectly informed agents, the resulting prices and allocations often wind up resembling those of a perfectly competitive market.

Frank Hahn defends general equilibrium modeling on the grounds that it provides a negative function. General equilibrium models show what the economy would have to be like for an unregulated economy to be Pareto efficient.

Computing general equilibrium

Until the 1970s, general equilibrium analysis remained theoretical. However, with advances in computing power, and the development of input-output tables, it became possible to model national economies, or even the world economy, and attempts were made to solve for general equilibrium prices and quantities empirically.

Applied general equilibrium (AGE) models were pioneered by Herbert Scarf in 1967, and offered a method for solving the Arrow-Debreu General Equilibrium system in a numerical fashion. This was first implemented by John Shoven and John Whalley (students of Scarf at Yale) in 1972 and 1973, and were a popular method up through the 1970's. In the 1980's however, AGE models faded from popularity due to their inability to provide a precise solution and its high cost of computation. Also, Scarf's method was proven non-computable to a precise solution by Velupillai (2006). (See AGE model article for the full references)

Computable general equilibrium (CGE) models surpassed and replaced AGE models in the mid 1980s, as the CGE model was able to provide relatively quick and large computable models for a whole economy, and was the preferred method of governments and the World Bank. CGE models are heavily used today, and while 'AGE' and 'CGE' is used inter-changeably in the literature, Scarf type AGE models have not been constructed since the mid 1980's, and the CGE literature at current is "not" based on Arrow-Debreu and General Equilibrium Theory as discussed in this article. CGE models, and what is today referred to as AGE models, are based on static, simultaneously solved, macro balancing equations (from the standard Keynesian macro model), giving a precise and explicitly computable result (Mitra-Kahn 2008).

References

* Arrow, K. J., and Hahn, F. H. (1971). "General Competitive Analysis", San Francisco: Holden-Day.
* Arrow K. J. and G. Debreu (1954). "The Existence of an Equilibrium for a Competitive Economy" "Econometrica", vol. XXII, 265-90
*Cite book
publisher = MIT Press
isbn = 0262023822
pages = 318
last = Black
first = Fischer
title = Exploring General Equilibrium
location = Cambridge Mass
date = 1995
* Debreu, G. (1959). "Theory of Value", New York: Wiley.
* Eaton, Eaton and Allen, "Intermediate Microeconomics" Chapters 13 and 18.
* Eatwell, John (1987). "Walras's Theory of Capital", "The " (Edited by Eatwell, J., Milgate, M., and Newman, P.), London: Macmillan.
*Geanakoplis, John (1987). "Arrow-Debreu model of general equilibrium," "The New Palgrave: A Dictionary of Economics", v. 1, pp. 116-24.
* Georgescu-Roegen, Nicholas (1979). "Methods in Economic Science", "Journal of Economic Issues", V. 13, N. 2 (June): 317-328.
* Grandmont, J. M. (1977). "Temporary General Equilibrium Theory", "Econometrica", V. 45, N. 3 (Apr.): 535-572.
* Hicks, John R. (1939, 2nd ed. 1946). "Value and Capital". Oxford: Clarendon Press.
* Jaffe, William (1953). "Walras's Theory of Capital Formation in the Framework of his Theory of General Equilibrium", "Economie Appliquee", V. 6 (Apr.-Sep.): 289-317.
* Kubler, Felix (2008). "computation of general equilibria (new developments)," "The New Palgrave Dictionary of Economics". 2nd Edition [http://www.dictionaryofeconomics.com/article?id=pde2008_C000564&q=computational%20economics&topicid=&result_number=2 Abstract.]
* Mandler, Michael (1999). "Dilemmas in Economic Theory: Persisting Foundational Problems of Microeconomics", Oxford: Oxford University Press.
* Mas-Colell, A., Whinston, M. and Green, J. (1995). "Microeconomic Theory", Oxford University Press
*McKenzie, Lionel W. (1981). "The Classical Theorem on Existence of Competitive Equilibrium", Econometrica.
* _____ (1983). "Turnpike Theory, Discounted Utility, and the von Neumann Facet", Journal of Economic Theory, 1983.
* _____ (1987). "general equilibrium", "The New Palgrave; : A Dictionary of Economics", 1987, v. 2, pp. 498-512.
* _____ (1987). turnpike theory," "The New Palgrave: A Dictionary of Economics", 1987, v. 4, pp. 712-20
* _____ (1999). "Equilibrium, Trade, and Capital Accumulation", Japanese Economic Review.
*Mitra-Kahn, Benjamin H., 2008, " [http://www.newschool.edu/cepa/publications/workingpapers/SCEPA%20Working%20Paper%202008-1%20Kahn.pdf Debunking the Myths of Computable General Equilibrium Models] ", "SCEPA Working Paper" 01-2008
* Petri, Fabio (2004). "General Equilibrium, Capital, and Macroeconomics: A Key to Recent Controversies in Equilibrium Theory", Edward Elgar.
* Samuelson, Paul A. (1947, Enlarged ed. 1983). "Foundations of Economic Analysis", Harvard University Press. ISBN 0-674-31301-1
* Scarf, Herbert E. (2008). "computation of general equilibria," "The New Palgrave Dictionary of Economics", 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000573&q=computational%20economics&topicid=&result_number=3 Abstract.]
* Walras, Léon (1877, printed 1954). "Elements of Pure Economics", Harvard University Press, ISBN 0678060282

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*Applied general equilibrium or AGE models
*Cobweb model
*convex preferences
*Computable general equilibrium or CGE models
*Dynamic stochastic general equilibrium or DSGE
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