General equilibrium

General equilibrium

Overview
General equilibrium theory is a branch of theoretical economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general equilibrium, in contrast to partial equilibrium
Partial equilibrium
Partial equilibrium is a condition of economic equilibrium which takes into consideration only a part of the market, ceteris paribus, to attain equilibrium....

, which only analyzes single markets. As with all models, this is an abstraction from a real economy; it is proposed as being a useful model, both by considering equilibrium prices as long-term prices and by considering actual prices as deviations from equilibrium.

General equilibrium theory both studies economies using the model of equilibrium pricing and seeks to determine in which circumstances the assumptions of general equilibrium will hold.
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Encyclopedia
General equilibrium theory is a branch of theoretical economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general equilibrium, in contrast to partial equilibrium
Partial equilibrium
Partial equilibrium is a condition of economic equilibrium which takes into consideration only a part of the market, ceteris paribus, to attain equilibrium....

, which only analyzes single markets. As with all models, this is an abstraction from a real economy; it is proposed as being a useful model, both by considering equilibrium prices as long-term prices and by considering actual prices as deviations from equilibrium.

General equilibrium theory both studies economies using the model of equilibrium pricing and seeks to determine in which circumstances the assumptions of general equilibrium will hold. The theory dates to the 1870s, particularly the work of French economist Léon Walras
Léon Walras
Marie-Esprit-Léon Walras was a French mathematical economist associated with the creation of the general equilibrium theory.-Life and career:...

.

Overview


It is often assumed that agents
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....

 are price takers
Market power
In economics, market power is the ability of a firm to alter the market price of a good or service. In perfectly competitive markets, market participants have no market power. A firm with market power can raise prices without losing its customers to competitors...

, and under that assumption two common notions of equilibrium exist: Walrasian (or competitive
Competitive equilibrium
Competitive market equilibrium is the traditional concept of economic equilibrium, appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis...

) 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
Macroeconomics
Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...

, as developed by the Keynesian economists
Keynesian economics
Keynesian economics is a school of macroeconomic thought based on the ideas of 20th-century English economist John Maynard Keynes.Keynesian economics argues that private sector decisions sometimes lead to inefficient macroeconomic outcomes and, therefore, advocates active policy responses by the...

, focused on a "top-down" approach, where the analysis starts with larger aggregates, the "big picture". Therefore, general equilibrium theory has traditionally been classified as part of microeconomics
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...

.

The difference is not as clear as it used to be, since much of modern macroeconomics has emphasized microeconomic foundations
Microfoundations
In economics, the term microfoundations refers to the microeconomic analysis of the behavior of individual agents such as households or firms that underpins a macroeconomic theory....

, and has constructed general equilibrium models of macroeconomic fluctuations
Dynamic stochastic general equilibrium
Dynamic stochastic general equilibrium modeling is a branch of applied general equilibrium theory that is influential in contemporary macroeconomics...

. General equilibrium macroeconomic models usually have a simplified structure that only incorporates a few markets, like a "goods market" and a "financial market". In contrast, general equilibrium models in the microeconomic tradition typically involve a multitude of different goods markets. They 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
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

 to model prices for a whole economy was made by Léon Walras
Léon Walras
Marie-Esprit-Léon Walras was a French mathematical economist associated with the creation of the general equilibrium theory.-Life and career:...

. 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 that the later models in this series are inconsistent.

In particular, Walras's model was a long-run 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. 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. (In contrast, Kenneth Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....

 and Gerard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...

 continued to take the initial quantities of capital goods as given, 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, the Walrasian agenda included the investigation of when equilibria are unique and stable.(Walras himself: Lesson 7 shows neither Uniqueness, nor Stability, nor even Existence of an agreement is guaranteed. Immediate after closing the deal, e.g.)

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 where 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
Partial equilibrium
Partial equilibrium is a condition of economic equilibrium which takes into consideration only a part of the market, ceteris paribus, to attain 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
Supply and demand
Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...

 is an example of partial equilibrium analysis. Partial equilibrium analysis is adequate when the first-order effects of a shift in 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
Piero Sraffa
Piero Sraffa was an influential Italian economist whose book Production of Commodities by Means of Commodities is taken as founding the Neo-Ricardian school of Economics.- Early life :...

'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
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....

, Gerard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...

 and Lionel W. McKenzie
Lionel W. McKenzie
Lionel Wilfred McKenzie was the Wilson Professor Emeritus of Economics at the University of Rochester. He was born in Montezuma, Georgia. He completed undergraduate studies at Duke University in 1939 and subsequently moved to Oxford that year as a Rhodes Scholar...

 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 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
Arrow-Debreu model
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.The model is central to the theory of...

 of intertemporal equilibrium
Intertemporal equilibrium
Intertemporal equilibrium is a notion of economic equilibrium conceived over many periods of time. The term has a different meaning in contemporary macroeconomics from its earlier meaning in Austrian economics....

 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 [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
Incomplete markets
In economics, incomplete markets refers to markets in which the number of Arrow–Debreu securities is less than the number of states of nature...

, 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
Marginal rate of substitution
In economics, the marginal rate of substitution is the rate at which a consumer is ready to give up one good in exchange for another good while maintaining the same level of utility.-Marginal rate of substitution as the slope of indifference curve:...

, which is the standard requirement for Pareto optimality. 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
Incomplete markets
In economics, incomplete markets refers to markets in which the number of Arrow–Debreu securities is less than the number of states of nature...

 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
Market failure
Market failure is a concept within economic theory wherein the allocation of goods and services by a free market is not efficient. That is, there exists another conceivable outcome where a market participant may be made better-off without making someone else worse-off...

.

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. The conditions for the second theorem are stronger than those for the first, as consumers' preferences now need to be convex (convexity roughly corresponds to the idea of diminishing rates of marginal substitution, 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, it suffices that consumer preferences be convex (although with enough consumers this assumption can be relaxed both for existence and the second welfare theorem). Similarly, but less plausibly, convex feasible production sets suffice for existence; convexity excludes economies of scale
Economies of scale
Economies of scale, in microeconomics, refers to the cost advantages that an enterprise obtains due to expansion. There are factors that cause a producer’s average cost per unit to fall as the scale of output is increased. "Economies of scale" is a long run concept and refers to reductions in unit...

.

Proofs of the existence of equilibrium traditionally rely on fixed-point theorems such as Brouwer fixed point theorem
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...

 for functions (or, more generally, the Kakutani fixed point theorem
Kakutani fixed point theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing...

 for set-valued functions
Multivalued function
In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...

). In fact, the converse holds, according to Uzawa's derivation of Brouwer’s fixed point theorem from Walras's law. Following Uzawa's theorem, many mathematical economists consider proving existence a deeper result than proving the two Fundamental Theorems.

Another method of proof of existence, global analysis, uses Sard's lemma
Sard's lemma
Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis which asserts that the image of the set of critical points of a smooth function f from one Euclidean space or manifold to another has Lebesgue measure 0 – they form a null set...

 and Baire category; this method was pioneered by Gerard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...

 and Stephen Smale
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...

.

Nonconvexities in large economies



applied the Shapley–Folkman–Starr theorem
Shapley–Folkman lemma
In geometry and economics, the Shapley–Folkman lemma describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of...

 to prove that even without convex preferences
Convex preferences
In economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes"...

 there exists an approximate equilibrium. The Shapley–Folkman–Starr results bound the distance from an "approximate" economic equilibrium
Economic equilibrium
In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...

 to an equilibrium of a "convexified" economy, when the number of agents exceeds the dimension of the goods. Following Starr's paper, the Shapley–Folkman–Starr results were "much exploited in the theoretical literature", according to Guesnerie (page 112), who wrote the following:

some key results obtained under the convexity assumption remain (approximately) relevant in circumstances where convexity fails. For example, in economies with a large consumption side, nonconvexities in preferences do not destroy the standard results of, say Debreu's theory of value. In the same way, if indivisibilities in the production sector are small with respect to the size of the economy, [ . . . ] then standard results are affected in only a minor way. (page 99)

To this text, Guesnerie appended the following footnote:

The derivation of these results in general form has been one of the major achievements of postwar economic theory.

In particular, the Shapley-Folkman-Starr results were incorporated in the theory of general economic equilibria and in the theory
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...

 of market failure
Market failure
Market failure is a concept within economic theory wherein the allocation of goods and services by a free market is not efficient. That is, there exists another conceivable outcome where a market participant may be made better-off without making someone else worse-off...

s and of public economics.

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 effect
Wealth effect
The wealth effect is an economic term, referring to an increase in spending that accompanies an increase in perceived wealth.-Effect on individuals:...

s (which is the feature that most clearly delineates general equilibrium analysis from partial equilibrium
Partial equilibrium
Partial equilibrium is a condition of economic equilibrium which takes into consideration only a part of the market, ceteris paribus, to attain 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
Sonnenschein-Mantel-Debreu Theorem
The Sonnenschein–Mantel–Debreu theorem is a result in general equilibrium economics. It states that the excess demand function for an economy is not restricted by the usual rationality restrictions on individual demands in the economy...

 states that the aggregate (excess) demand function inherits only certain properties of individual's demand functions, and that these (Continuity
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

, Homogeneity of degree zero
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

, Walras' law
Walras' law
Walras’ Law is a principle in general equilibrium theory asserting that when considering any particular market, if all other markets in an economy are in equilibrium, then that specific market must also be in equilibrium. Walras’ Law hinges on the mathematical notion that excess market demands ...

 and boundary behavior when prices are near zero) are the only real restriction one can expect from an aggregate excess demand function: any such function can be rationalized as the excess demand of an economy. In particular uniqueness of equilibrium should not be expected.

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
Regular economy
A regular economy is an economy characterized by an excess demand function which has the property that its slope at any equilibrium price vector is non-zero. In other words, if we graph the excess demand function against prices, then the excess demand function "cuts" the x-axis assuring that each...

) 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 preference
Revealed preference
Revealed preference theory, pioneered by American economist Paul Samuelson, is a method by which it is possible to discern the best possible option on the basis of consumer behavior. Essentially, this means that the preferences of consumers can be revealed by their purchasing habits...

s for a single individual) or the gross substitute property
Substitute good
In economics, one way we classify goods is by examining the relationship of the demand schedules when the price of one good changes. This relationship between demand schedules leads economists to classify goods as either substitutes or complements. Substitute goods are goods which, as a result...

 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
Comparative statics
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....

 can be applied as long as the shocks to the system are not too large. As stated above, in a regular economy
Regular economy
A regular economy is an economy characterized by an excess demand function which has the property that its slope at any equilibrium price vector is non-zero. In other words, if we graph the excess demand function against prices, then the excess demand function "cuts" the x-axis assuring that each...

 equilibria will be finite, hence locally unique. One reassuring result, due to Debreu, is that "most" economies are regular.

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
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

 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.

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 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. When more than one stable 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
Planned economy
A planned economy is an economic system in which decisions regarding production and investment are embodied in a plan formulated by a central authority, usually by a government agency...

, 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 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
Path dependence explains how the set of decisions one faces for any given circumstance is limited by the decisions one has made in the past, even though past circumstances may no longer be relevant....

... [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, given
initial 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
Market clearing
In economics, market clearing refers to either# a simplifying assumption made by the new classical school that markets always go to where the quantity supplied equals the quantity demanded; or# the process of getting there via price adjustment....

 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
Temporary equilibrium method
The temporary equilibrium method has been devised by Alfred Marshall for analyzing economic systems that comprise interdependent variables of different speed....

 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
Numéraire
Numéraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...

, the model does not encompass money. Frank Hahn
Frank Hahn
Frank Horace Hahn is a British economist whose work has focused on general equilibrium theory, monetary theory, Keynesian economics and monetarism...

, for example, has investigated whether general equilibrium models can be developed in which money enters in some essential way. One of the essential questions he introduces, often referred to as the Hahn's Problem
Hahn's Problem
The Hahn Problem refers to the theoretical challenge of building general equilibrium models where money does not enter preferences, but yet has a positive value...

 is : "Can one construct an equilibrium where money has value?" 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
Nicholas Georgescu-Roegen
Nicholas Georgescu-Roegen, born Nicolae Georgescu was a Romanian mathematician, statistician and economist, best known for his 1971 magnum opus The Entropy Law and the Economic Process, which situated the view that the second law of thermodynamics, i.e., that usable "free energy" tends to disperse...

 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
Rationality
In philosophy, rationality is the exercise of reason. It is the manner in which people derive conclusions when considering things deliberately. It also refers to the conformity of one's beliefs with one's reasons for belief, or with one's actions with one's reasons for action...

 of individual complete information
Complete information
Complete information is a term used in economics and game theory to describe an economic situation or game in which knowledge about other market participants or players is available to all participants. Every player knows the payoffs and strategies available to other players.Complete information...

 about all prices both now and in the future; and the conditions necessary for perfect competition
Perfect competition
In economic theory, perfect competition describes markets such that no participants are large enough to have the market power to set the price of a homogeneous product. Because the conditions for perfect competition are strict, there are few if any perfectly competitive markets...

. However some results from experimental economics
Experimental economics
Experimental economics is the application of experimental methods to study economic questions. Data collected in experiments are used to estimate effect size, test the validity of economic theories, and illuminate market mechanisms. Economic experiments usually use cash to motivate subjects, in...

 suggest that even in circumstances where there are few, imperfectly informed agents, the resulting prices and allocations may wind up resembling those of a perfectly competitive market(although certainly not a stable general equilibrium in all markets).

Frank Hahn
Frank Hahn
Frank Horace Hahn is a British economist whose work has focused on general equilibrium theory, monetary theory, Keynesian economics and monetarism...

 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. 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
Applied general equilibrium
In mathematical economics, applied general equilibrium models were pioneered by Herbert Scarf at Yale University in 1967, in two papers, and a follow up book with Terje Hansen in 1973, with the aim of empirically estimating the Arrow–Debreu general equilibrium model with empirical data, to provide...

 (AGE) models were pioneered by Herbert Scarf
Herbert Scarf
Herbert Eli "Herb" Scarf is an American mathematical economist and Sterling Professor of Economics at Yale University. He is a member of the American Academy of Arts and Sciences...

 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 1970s. In the 1980s 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
Computable general equilibrium
Computable general equilibrium models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors...

 (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
World Bank
The World Bank is an international financial institution that provides loans to developing countries for capital programmes.The World Bank's official goal is the reduction of poverty...

. 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 1980s, 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).

Other schools


General equilibrium theory is a central point of contention and influence between the neoclassical school and other schools of economic thought, and different schools have varied views on general equilibrium theory. Some, such as the Keynesian and Post-Keynesian schools, strongly reject general equilibrium theory as "misleading" and "useless"; others, such as the Austrian school, show more influence and acceptance of general equilibrium thinking, though the extent is debated. Other schools, such as new classical macroeconomics
New classical macroeconomics
New classical macroeconomics, sometimes simply called new classical economics, is a school of thought in macroeconomics that builds its analysis entirely on a neoclassical framework. Specifically, it emphasizes the importance of rigorous foundations based on microeconomics...

, developed from general equilibrium theory.

Keynesian and Post-Keynesian


Keynesian
Keynesian economics
Keynesian economics is a school of macroeconomic thought based on the ideas of 20th-century English economist John Maynard Keynes.Keynesian economics argues that private sector decisions sometimes lead to inefficient macroeconomic outcomes and, therefore, advocates active policy responses by the...

 and Post-Keynesian
Post-Keynesian economics
Post Keynesian economics is a school of economic thought with its origins in The General Theory of John Maynard Keynes, although its subsequent development was influenced to a large degree by Michał Kalecki, Joan Robinson, Nicholas Kaldor and Paul Davidson...

 economists, and their Underconsumption
Underconsumption
In underconsumption theory in economics, recessions and stagnation arise due to inadequate consumer demand relative to the amount produced. The theory has been replaced since the 1930s by Keynesian economics and the theory of aggregate demand, both of which were influenced by...

ist predecessors criticize general equilibrium theory specifically, and as part of criticisms of neoclassical economics
Criticisms of neoclassical economics
Neo-classical economics has come under critique on the basis of its core ideologies, assumptions and other matters.-Normative bias:Neoclassical economics is sometimes criticized for having a normative bias. In this view, it does not focus on explaining actual economies but instead on describing a...

 generally. Specifically, they argue that general equilibrium theory is neither accurate nor useful, that economies are not in equilibrium, that equilibrium may be slow and painful to achieve, and that modeling by equilibrium is "misleading", and that the resulting theory is not a useful guide, particularly for understanding of economic crises.
Robert Clower and others have argued for a reformulation of theory toward disequilibrium analysis to incorporate how monetary exchange fundamentally alters the representation of an economy as though a barter
Barter
Barter is a method of exchange by which goods or services are directly exchanged for other goods or services without using a medium of exchange, such as money. It is usually bilateral, but may be multilateral, and usually exists parallel to monetary systems in most developed countries, though to a...

 system.

More methodologically, it is argued that general equilibrium is a fundamentally static
Statics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...

analysis, rather than a dynamic
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

analysis, and thus is misleading and inapplicable. The theory of dynamic stochastic general equilibrium
Dynamic stochastic general equilibrium
Dynamic stochastic general equilibrium modeling is a branch of applied general equilibrium theory that is influential in contemporary macroeconomics...

 seeks to address this criticism.

Austrian economics


Whether Austrian economics supports or rejects general equilibrium theory and the precise relationship is unclear. Different Austrian economists have advocated differing positions, which have changed as Austrian economics developed over time. Some new classical economists argue that the work of Friedrich Hayek
Friedrich Hayek
Friedrich August Hayek CH , born in Austria-Hungary as Friedrich August von Hayek, was an economist and philosopher best known for his defense of classical liberalism and free-market capitalism against socialist and collectivist thought...

 in the 1920s and 1930s was in the general equilibrium tradition and was a precursor to business cycle equilibrium theory. Others argue that while there are clear influences of general equilibrium on Hayek's thought, and that he used it in his early work, he came to substantially reject it in his later work, post 1937. It is also argued by some that Friedrich von Wieser
Friedrich von Wieser
Friedrich Freiherr von Wieser was an early member of the Austrian School of economics. Born in Vienna, the son of Privy Councillor Leopold von Wieser, a high official in the war ministry he first trained in sociology and law...

, along with Hayek, worked in the general equilibrium tradition,
while others reject this, finding influences of general equilibrium on the Austrian economists superficial.

New classical macroeconomics



While general equilibrium theory and neoclassical economics generally were originally microeconomic theories, New classical macroeconomics
New classical macroeconomics
New classical macroeconomics, sometimes simply called new classical economics, is a school of thought in macroeconomics that builds its analysis entirely on a neoclassical framework. Specifically, it emphasizes the importance of rigorous foundations based on microeconomics...

 builds a macroeconomic theory on these bases. In new classical models, the macroeconomy is assumed to be at its unique equilibrium, with full employment and potential output, and that this equilibrium is assumed to always have been achieved via price and wage adjustment (market clearing). The best-known such model is Real Business Cycle Theory
Real business cycle theory
Real business cycle theory are a class of macroeconomic models in which business cycle fluctuations to a large extent can be accounted for by real shocks. Unlike other leading theories of the business cycle, RBC theory sees recessions and periods of economic growth as the efficient response to...

, in which business cycles are considered to be largely due to changes in the real economy, unemployment is not due to the failure of the market to achieve potential output, but due to equilibrium potential output having fallen and equilibrium unemployment having risen.

Socialist economics


Within socialist economics
Socialist economics
Socialist economics are the economic theories and practices of hypothetical and existing socialist economic systems.A socialist economy is based on public ownership or independent cooperative ownership of the means of production, wherein production is carried out to directly produce use-value,...

, a sustained critique of general equilibrium theory (and neoclassical economics generally) is given in Anti-Equilibrium , based on the experiences of János Kornai
János Kornai
János Kornai , is an economist noted for his analysis and criticism of the command economies of Eastern European communist states.- Biography :...

 with the failures of Communist central planning.

See also

  • Applied general equilibrium
    Applied general equilibrium
    In mathematical economics, applied general equilibrium models were pioneered by Herbert Scarf at Yale University in 1967, in two papers, and a follow up book with Terje Hansen in 1973, with the aim of empirically estimating the Arrow–Debreu general equilibrium model with empirical data, to provide...

     or AGE models
  • Cobweb model
    Cobweb model
    The cobweb model or cobweb theory is an economic model that explains why prices might be subject to periodic fluctuations in certain types of markets. It describes cyclical supply and demand in a market where the amount produced must be chosen before prices are observed. Producers' expectations...

  • Convex preferences
    Convex preferences
    In economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes"...

  • Computable general equilibrium
    Computable general equilibrium
    Computable general equilibrium models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors...

     or CGE models
    Computable general equilibrium
    Computable general equilibrium models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors...

  • Decision theory
    Decision theory
    Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...

  • Dynamic stochastic general equilibrium
    Dynamic stochastic general equilibrium
    Dynamic stochastic general equilibrium modeling is a branch of applied general equilibrium theory that is influential in contemporary macroeconomics...

     or DSGE
  • Game theory
    Game theory
    Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

  • Mechanism design theory
  • Partial equilibrium
    Partial equilibrium
    Partial equilibrium is a condition of economic equilibrium which takes into consideration only a part of the market, ceteris paribus, to attain equilibrium....