General equilibrium

General equilibrium theory is a branch of theoretical microeconomics. It seeks to explain production, consumption and prices in a whole economy.

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. Since modern macroeconomics has emphasized microeconomic foundations, this distinction has been slightly blurred. However, many macroeconomic models simply have a 'goods market' and study its interaction with for instance the financial market. General equilibrium models typically model a multitude of different goods markets. Modern general equilibrium models are typically 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, for example, the wages of bakers. 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 modelling

The first attempt in Neoclassical economics to model prices for a whole economy was made by Leon 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 programme much followed by 20th century economists. In particular, Walras' agenda included the investigation of when equilibria are unique and stable.

Walras also first introduced a restriction into general equilibrium theory that some think has never been overcome, that of the tâtonnement or groping process.

The tatonnement process is a tool for investigating stability of equilibria. Prices are cried, and agents register 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 the forces 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 and Gerard Debreu in the 1950s. Gerard Debreu presents this model in Theory of Value (1959) as an axiomatic model, following the style 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 equilibriate 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 forward markets 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 ^* 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 Theorem states that every equilibrium is Pareto efficient. The technical condition for the result to hold is the fairly weak one that consumer preferences are locally nonsatiated, which rules out a situation where all commodities are "bads". Additional implicit assumptions are that consumers are rational, markets are complete, there are no externalities and information is perfect. While these assumptions are certainly unrealistic, what the theorem basically tells us is that the sources of inefficiency found in the real world are not due to the decentralized nature of the market system, but rather have their sources elsewhere.

Second 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 enough to guarantee uniqueness.

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 whether any particular equilibrium is at least locally unique. This means that 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 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 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.

Stability

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 converging process completes. However stability depends not only on the number 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 will guarantee system stability, that is, prices and allocations always converging to some equilibrium, though when there exists more than one, which equilibrium it is will depend on where one begins.

Unresolved problems in general equilibrium

Research building on the Arrow-Debreu 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 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, not very successfully, 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 be changed further complicating the picture.

In a real economy, however, trading, as well as production and consumption, goes on 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... 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, given initial endowments, had prices happened to be just right (Franklin Fisher, as quoted by Petri (2004)).

The Arrow-Debreu model of intertemporal equilibrium, in which forward markets exist at the initial instant for goods to be delivered at each future point in time, can be transformed into a model of sequences of temporary equilibrium. Sequences of temporary equilibrium contain spot markets at each point in time. Roy Radner found that in order for equilibria to exist in such models, agents (e.g., firms and consumers) must have unlimited computational capabilities.

Although the Arrow-Debreu 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 individuals; 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.

Computable general equilibrium

(main article Computable general equilibrium)

Until the 1970s, general equilibrium analysis remained theoretical. However, with advances in computing power, and the development of input-output tables, it become possible to model national economies, or even the world economy, and solve for general equilibrium prices and quantities under a range of assumptions.

References

Arrow, K., 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
Debreu, G. (1956). "Theory of Value", New York: Wiley.
Eatwell, John (1989). "Walras's Theory of Capital", The New Palgrave: A Dictionary of Economics (Edited by Eatwell, J., Milgate, M., and Newman, P.), London: Macmillan.
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.
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.
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
Petri, Fabio (2004). General Equilibrium, Capital, and Macroeconomics: A Key to Recent Controversies in Equilibrium Theory, Edward Elgar.

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