In
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall
general equilibrium
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
. General equilibrium theory contrasts to the theory of
''partial'' equilibrium, which analyzes a specific part of an economy while its other factors are held constant. In general equilibrium, constant influences are considered to be noneconomic, therefore, resulting beyond the natural scope of economic analysis. The noneconomic influences is possible to be non-constant when the economic variables change, and the prediction accuracy may depend on the independence of the economic factors.
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
Marie-Esprit-Léon Walras (; 16 December 1834 – 5 January 1910) was a French mathematical economist and Georgist. He formulated the marginal theory of value (independently of William Stanley Jevons and Carl Menger) and pioneered the developmen ...
in his pioneering 1874 work ''Elements of Pure Economics''. The theory reached its modern form with the work of
Lionel W. McKenzie
Lionel Wilfred McKenzie (January 26, 1919 – October 12, 2010) was an American economist. He was the Wilson Professor Emeritus of Economics at the University of Rochester. He was born in Montezuma, Georgia. He completed undergraduate studies at ...
(Walrasian theory),
Kenneth Arrow
Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972.
In economics ...
and
Gérard Debreu
Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. 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 ...
(Hicksian theory) in the 1950s.
Overview
Broadly speaking, general equilibrium tries to give an understanding of the whole economy using a "bottom-up" approach, starting with individual markets and agents. Therefore, general equilibrium theory has traditionally been classified as part of
microeconomics
Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...
. The difference is not as clear as it used to be, since much of modern macroeconomics has emphasized
microeconomic foundations, and has constructed
general equilibrium models of macroeconomic fluctuations. 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 calculate
numerical solutions.
In a market system the prices and production of all goods, including the price of money and
interest
In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distinct ...
, are interrelated. A change in the price of one good, say bread, may affect another price, such as bakers' wages. If bakers don't 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. It is often assumed that
agents are
price takers
In economics, market power refers to the ability of a firm to influence the price at which it sells a product or service by manipulating either the supply or demand of the product or service to increase economic profit. In other words, market powe ...
, and under that assumption two common notions of equilibrium exist: Walrasian, or
competitive equilibrium
Competitive equilibrium (also called: Walrasian equilibrium) is a concept of economic equilibrium introduced by Kenneth Arrow and Gérard Debreu in 1951 appropriate for the analysis of commodity markets with flexible prices and many traders, and s ...
, and its generalization: a price equilibrium with transfers.
Walrasian equilibrium
The first attempt in
neoclassical economics
Neoclassical economics is an approach to economics in which the production, consumption and valuation (pricing) of goods and services are observed as driven by the supply and demand model. According to this line of thought, the value of a good ...
to model prices for a whole economy was made by
Léon Walras
Marie-Esprit-Léon Walras (; 16 December 1834 – 5 January 1910) was a French mathematical economist and Georgist. He formulated the marginal theory of value (independently of William Stanley Jevons and Carl Menger) and pioneered the developmen ...
. 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 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 Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972.
In economics ...
and
Gérard Debreu
Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. 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 ...
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 widely followed by 20th-century economists. In particular, the Walrasian agenda included the investigation of when equilibria are unique and stable— Walras' Lesson 7 shows neither uniqueness, nor stability, nor even existence of an equilibrium is guaranteed. Walras also proposed a dynamic process by which general equilibrium might be reached, that of the
tâtonnement or groping process.
The tâtonnement 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
In economics, an excess supply, economic surplus market surplus or briefly surply is a situation in which the quantity of a good or service supplied is more than the quantity demanded, and the price is above the equilibrium level determined by sup ...
. 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).
Marshall and Sraffa
In
partial equilibrium
In economics, partial equilibrium is a condition of economic equilibrium which analyzes only a single market, ''ceteris paribus'' (everything else remaining constant) except for the one change at a time being analyzed. In general equilibrium ana ...
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
In microeconomics, supply and demand is an economic model of price determination in a Market (economics), market. It postulates that, Ceteris paribus, holding all else equal, in a perfect competition, competitive market, the unit price for a ...
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 (5 August 1898 – 3 September 1983) was an influential Italian economist who served as lecturer of economics at the University of Cambridge. His book ''Production of Commodities by Means of Commodities'' is taken as founding the neo- ...
'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 experience constant 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 increasing 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' arguments for the existence of general equilibrium often were based on the counting of equations and variables. Such arguments are inadequate for
non-linear systems
Non-Linear Systems is an electronics manufacturing company based in San Diego, California. Non-Linear Systems was founded in 1952, by Andrew Kay, the inventor of the digital voltmeter in 1954.Kenneth Arrow
Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972.
In economics ...
,
Gérard Debreu
Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. 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 ...
, and
Lionel W. McKenzie
Lionel Wilfred McKenzie (January 26, 1919 – October 12, 2010) was an American economist. He was the Wilson Professor Emeritus of Economics at the University of Rochester. He was born in Montezuma, Georgia. He completed undergraduate studies at ...
in the 1950s. Debreu presents this model in ''Theory of Value'' (1959) as an axiomatic model, following the style of mathematics promoted by
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...
. 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
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggreg ...
of
intertemporal equilibrium
Intertemporal equilibrium is a notion of economic equilibrium conceived over many periods of time.
In modern economic theory, most models explicitly take into account the fact that the economy evolves over time, and that its equilibrium cannot be ...
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
iskfree from any probability concept..."
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 real economies function.
Some of the recent work in general equilibrium has in fact explored the implications of
incomplete markets In economics, incomplete markets are markets in which there does not exist an Arrow–Debreu security for every possible state of nature. In contrast with complete markets, this shortage of securities will likely restrict individuals from transfer ...
, 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
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engine ...
. 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
In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no exte ...
, which is the standard requirement for Pareto optimality. Under some conditions the economy may still be
constrained Pareto optimal
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engine ...
, 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 In economics, incomplete markets are markets in which there does not exist an Arrow–Debreu security for every possible state of nature. In contrast with complete markets, this shortage of securities will likely restrict individuals from transfer ...
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
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...
. In other words, the allocation of goods in the equilibria is such that there is no reallocation which would leave a consumer better off without leaving another consumer worse off. 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
In economics, an externality or external cost is an indirect cost or benefit to an uninvolved third party that arises as an effect of another party's (or parties') activity. Externalities can be considered as unpriced goods involved in either co ...
, 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
In neoclassical economics, market failure is a situation in which the allocation of goods and services by a free market is not Pareto efficient, often leading to a net loss of economic value. Market failures can be viewed as scenarios where indiv ...
.
Second Fundamental Theorem of Welfare Economics
Even if every equilibrium is efficient, it may not be that every efficient allocation of resources can be part of an equilibrium. However, the second theorem states that every Pareto efficient allocation can be supported as an equilibrium by some set of prices. In other words, all that is required to reach a particular Pareto efficient 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 and production sets now need to be convex (convexity roughly corresponds to the idea of diminishing marginal rates of substitution i.e. "the average of two equally good bundles is better than either of the two bundles").
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 strictly convex. With enough consumers, the convexity 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
In microeconomics, economies of scale are the cost advantages that enterprises obtain due to their scale of operation, and are typically measured by the amount of output produced per unit of time. A decrease in cost per unit of output enables ...
.
Proofs of the existence of equilibrium traditionally rely on fixed-point theorems such as
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest ...
for functions (or, more generally, the
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 set, convex, compact set, compact subset of a Euclidean sp ...
for
set-valued functions). See
Competitive equilibrium#Existence of a competitive equilibrium. The proof was first due to
Lionel McKenzie
Lionel Wilfred McKenzie (January 26, 1919 – October 12, 2010) was an American economist. He was the Wilson Professor Emeritus of Economics at the University of Rochester. He was born in Montezuma, Georgia. He completed undergraduate studies at ...
, and
Kenneth Arrow
Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972.
In economics ...
and
Gérard Debreu
Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. 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 fact, the converse also 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 In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite-dimensional manifold th ...
, uses
Sard's lemma
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth functi ...
and the
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
; this method was pioneered by
Gérard Debreu
Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. 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 ...
and
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
.
Nonconvexities in large economies
Starr (1969) applied the
Shapley–Folkman–Starr theorem to prove that even without
convex preferences In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly ...
there exists an approximate equilibrium. The Shapley–Folkman–Starr results bound the distance from an "approximate"
economic equilibrium
In economics, economic equilibrium is a situation in which economic forces such as supply and demand are balanced and in the absence of external influences the ( equilibrium) values of economic variables will not change. For example, in the st ...
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,
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, . . .
The ellipsis (, also known informally as dot dot dot) is a series of dots that indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning. The plural is ellipses. The term origin ...
then standard results are affected in only a minor way.
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
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
of
market failure
In neoclassical economics, market failure is a situation in which the allocation of goods and services by a free market is not Pareto efficient, often leading to a net loss of economic value. Market failures can be viewed as scenarios where indiv ...
s and of
public economics
Public economics ''(or economics of the public sector)'' is the study of government policy through the lens of economic efficiency and equity. Public economics builds on the theory of welfare economics and is ultimately used as a tool to improve ...
.
Uniqueness
Although generally (assuming convexity) an equilibrium will exist and will be efficient, the conditions under which it will be unique are much stronger. The
Sonnenschein–Mantel–Debreu theorem
The Sonnenschein–Mantel–Debreu theorem is an important result in general equilibrium economics, proved by Gérard Debreu, , and Hugo F. Sonnenschein in the 1970s. It states that the excess demand curve for an exchange economy populated with ...
, proven in the 1970s, 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
Walras's law is a principle in general equilibrium theory asserting that budget constraints imply that the ''values'' of excess demand (or, conversely, excess market supplies) must sum to zero regardless of whether the prices are general equilib ...
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 represent the excess demand of an economy populated with rational utility-maximizing individuals.
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
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
). 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 theory, pioneered by economist Paul Anthony Samuelson in 1938, is a method of analyzing choices made by individuals, mostly used for comparing the influence of policies on consumer behavior. Revealed preference models assume t ...
s 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
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous variable, exogenous parameter.
As a type of ''static analysis'' it compares two different economic equ ...
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.
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
In measure theory, a branch of mathematics, 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 wit ...
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.
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.
Some have questioned the practical applicability of the general equilibrium approach based on the possibility of non-uniqueness of equilibria.
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
Walrasian auction
A Walrasian auction, introduced by Léon Walras, is a type of simultaneous auction where each agent calculates its demand for the good at every possible price and submits this to an auctioneer. The price is then set so that the total demand across ...
). 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. The theorems that have been mostly conclusive when related to the stability of a typical general equilibrium model are closed related to that of the most local stability.
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 tâtonnement process has been said to be a model of a centrally
planned economy
A planned economy is a type of economic system where investment, production and the allocation of capital goods takes place according to economy-wide economic plans and production plans. A planned economy may use centralized, decentralized, part ...
, 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... his path dependencemakes 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
Franklin Marvin Fisher (December 13, 1934 – April 29, 2019) was an American economist. He taught economics at the Massachusetts Institute of Technology from 1960 to 2004.
Biography
Fisher attended Harvard University, where he was inducted into ...
).
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
In economics, market clearing is the process by which, in an economic market, the supply of whatever is traded is equated to the demand so that there is no excess supply or demand. The new classical economics assumes that in any given market, assum ...
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
numéraire
The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. In a monetary economy, actin ...
, the model does not encompass money.
Frank Hahn
Frank Horace Hahn FBA (26 April 1925 – 29 January 2013) was a British economist whose work focused on general equilibrium theory, monetary theory, Keynesian economics and critique of monetarism. A famous problem of economic theory, the condi ...
, 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 (or Hahn's question) refers to the theoretical challenge of building general equilibrium models where money does not enter preferences, yet has a positive equilibrium value. Money, since it is intrinsically worthless and is not deman ...
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. In a 1979 article,
Nicholas Georgescu-Roegen
Nicholas Georgescu-Roegen (born Nicolae Georgescu, 4 February 1906 – 30 October 1994) was a Romanian mathematician, statistician and economist. He is best known today for his 1971 ''The Entropy Law and the Economic Process'', in which he argu ...
complains: "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." He 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 is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abil ...
of individuals;
complete information
In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...
about all prices both now and in the future; and the conditions necessary for
perfect competition
In economics, specifically general equilibrium theory, a perfect market, also known as an atomistic market, is defined by several idealizing conditions, collectively called perfect competition, or atomistic competition. In Economic model, theoret ...
. However, some results from
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 expe ...
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 Horace Hahn FBA (26 April 1925 – 29 January 2013) was a British economist whose work focused on general equilibrium theory, monetary theory, Keynesian economics and critique of monetarism. A famous problem of economic theory, the condi ...
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
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...
.
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 In mathematical economics, applied general equilibrium (AGE) 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 ...
(AGE) models were pioneered by
Herbert Scarf
Herbert Eli "Herb" Scarf (July 25, 1930 – November 15, 2015) was an American mathematical economist and Sterling Professor of Economics at Yale University.
Education and career
Scarf was born in Philadelphia, the son of Jewish emigrants from ...
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.
Computable general equilibrium
Computable general equilibrium (CGE) 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 are also referred to as AGE (appl ...
(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
The World Bank is an international financial institution that provides loans and grants to the governments of low- and middle-income countries for the purpose of pursuing capital projects. The World Bank is the collective name for the Interna ...
. 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.
Other schools
General equilibrium theory is a central point of contention and influence between the neoclassical school and other
schools of economic thought
In the history of economic thought, a school of economic thought is a group of economics, economic thinkers who share or shared a common perspective on the way economy, economies work. While economists do not always fit into particular schools, pa ...
, 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". Other schools, such as
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 economics, neoclassical framework. Specifically, it emphasizes the importa ...
, developed from general equilibrium theory.
Keynesian and Post-Keynesian
Keynesian
Keynesian economics ( ; sometimes Keynesianism, named after British economist John Maynard Keynes) are the various macroeconomic theories and models of how aggregate demand (total spending in the economy) strongly influences economic output and ...
and
Post-Keynesian
Post-Keynesian economics is a school of economic thought with its origins in ''The General Theory'' of John Maynard Keynes, with subsequent development influenced to a large degree by Michał Kalecki, Joan Robinson, Nicholas Kaldor, Sidney Wei ...
economists, and their
underconsumption Underconsumption is a theory in economics that recessions and stagnation arise from an inadequate consumer demand, relative to the amount produced. In other words, there is a problem of overproduction and overinvestment during a demand crisis. The ...
ist predecessors criticize general equilibrium theory specifically, and as part of criticisms of neoclassical economics 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
A financial crisis is any of a broad variety of situations in which some financial assets suddenly lose a large part of their nominal value. In the 19th and early 20th centuries, many financial crises were associated with banking panics, and man ...
.
Robert Clower
Robert Wayne Clower (February 13, 1926 – May 2, 2011) was an American economist. He is credited with having largely created the field of stock-flow analysis in economics and with seminal works on the microfoundations of monetary theory and macr ...
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
In trade, barter (derived from ''baretor'') is a system of exchange in which participants in a transaction directly exchange goods or services for other goods or services without using a medium of exchange, such as money. Economists distingu ...
system.
New classical macroeconomics
While general equilibrium theory and neoclassical economics generally were originally microeconomic theories,
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 economics, neoclassical framework. Specifically, it emphasizes the importa ...
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 (RBC theory) is a class of new classical macroeconomics models in which business-cycle fluctuations are accounted for by real (in contrast to nominal) shocks. Unlike other leading theories of the business cycle, RBC th ...
, in which
business cycle
Business cycles are intervals of Economic expansion, expansion followed by recession in economic activity. These changes have implications for the welfare of the broad population as well as for private institutions. Typically business cycles are ...
s 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 comprises the economic theories, practices and norms of hypothetical and existing socialist economic systems. A socialist economic system is characterized by social ownership and operation of the means of production that may ...
, 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 (21 January 1928 – 18 October 2021) was a Hungarian economist noted for his analysis and criticism of the command economies of Eastern European communist states. He also covered macroeconomic aspects in countries undergoing pos ...
with the failures of Communist
central planning
A planned economy is a type of economic system where investment, production and the allocation of capital goods takes place according to economy-wide economic plans and production plans. A planned economy may use centralized, decentralized, pa ...
, although
Michael Albert
Michael Albert (born April 8, 1947) is an American economist, speaker, writer, and political critic. Since the late 1970s, he has published books, articles, and other contributions on a wide array of subjects. He has also set up his own media ...
and
Robin Hahnel
Robin Eric Hahnel (born March 25, 1946) is an American economist and professor emeritus of economics at American University. He was a professor at American University for many years and traveled extensively advising on economic matters all over t ...
later based their
Parecon
Participatory economics, often abbreviated Parecon, is an economic system based on participatory decision making as the primary economic mechanism for allocation in society. In the system, the say in decision-making is proportional to the impa ...
model on the same theory.
See also
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General equilibrium theorists
A general officer is an officer of high rank in the armies, and in some nations' air forces, space forces, and marines or naval infantry.
In some usages the term "general officer" refers to a rank above colonel."general, adj. and n.". OED O ...
(category)
*
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 bef ...
*
Decision theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
*
Game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
*
Mechanism design
Mechanism design is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts a ...
Notes
Further reading
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{{Authority control
Microeconomic theories
Welfare economics
Game theory