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 in his pioneering 1874 work Elements of Pure Economics.
Overview
It is often assumed that agents are price takers, and under that assumption two common notions of equilibrium exist: Walrasian, or competitive equilibrium, and its generalization: a price equilibrium with transfers.
Broadly speaking, general equilibrium tries to give an
understanding of the whole economy using a "bottom-up" approach,
starting with individual markets and agents. (Macroeconomics, as developed by the Keynesian economists,
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.
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 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 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.
The first attempt in neoclassical economics to model prices for a whole economy was made by Léon Walras. Walras' Elements of Pure Economics
provides a succession of models, each taking into account more aspects
of a real economy (two commodities, many commodities, production,
growth, money). Some 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 and Gérard Debreu
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' 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.
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.
In partial equilibrium
analysis, the determination of the price of a good is simplified by
just looking at the price of one good, and assuming that the prices of
all other goods remain constant. The Marshallian theory of supply and demand
is an example of partial equilibrium analysis. Partial equilibrium
analysis is adequate when the first-order effects of a shift in the
demand curve do not shift the supply curve. Anglo-American economists
became more interested in general equilibrium in the late 1920s and
1930s after Piero Sraffa's
demonstration that Marshallian economists cannot account for the forces
thought to account for the upward-slope of the supply curve for a
consumer good.
If an industry uses little of a factor of production, a small
increase in the output of that industry will not bid the price of that
factor up. To a first-order approximation, firms in the industry will
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' 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, Gérard Debreu, and Lionel W. McKenzie 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. In such an approach, the interpretation of the terms in the theory (e.g., goods, prices) are not fixed by the axioms.
Three important interpretations of the terms of the theory have
been often cited. First, suppose commodities are distinguished by the
location where they are delivered. Then the Arrow-Debreu model is a
spatial model of, for example, international trade.
Second, suppose commodities are distinguished by when they are
delivered. That is, suppose all markets equilibrate at some initial
instant of time. Agents in the model purchase and sell contracts, where a
contract specifies, for example, a good to be delivered and the date at
which it is to be delivered. The Arrow–Debreu model of intertemporal equilibrium contains 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..."
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,
which is to say an intertemporal economy with uncertainty, where there
do not exist sufficiently detailed contracts that would allow agents to
fully allocate their consumption and resources through time. While it
has been shown that such economies will generally still have an
equilibrium, the outcome may no longer be Pareto optimal.
The basic intuition for this result is that if consumers lack adequate
means to transfer their wealth from one time period to another and the
future is risky, there is nothing to necessarily tie any price ratio
down to the relevant marginal rate of substitution, which is the standard requirement for Pareto optimality. Under some conditions the economy may still be constrained Pareto optimal,
meaning that a central authority limited to the same type and number of
contracts as the individual agents may not be able to improve upon the
outcome, what is needed is the introduction of a full set of possible
contracts. Hence, one implication of the theory of incomplete markets
is that inefficiency may be a result of underdeveloped financial
institutions or credit constraints faced by some members of the public.
Research still continues in this area.
Properties and characterization of general equilibrium
Basic questions in general equilibrium analysis are concerned with
the conditions under which an equilibrium will be efficient, which
efficient equilibria can be achieved, when an equilibrium is guaranteed
to exist and when the equilibrium will be unique and stable.
First Fundamental Theorem of Welfare Economics
The First Fundamental Welfare Theorem asserts that market equilibria are Pareto efficient. In a pure exchange economy, a sufficient condition for the first welfare theorem to hold is that preferences be locally nonsatiated.
The first welfare theorem also holds for economies with production
regardless of the properties of the production function. Implicitly, the
theorem assumes complete markets and perfect information. In an economy
with externalities, for example, it is possible for equilibria to arise that are not efficient.
The first welfare theorem is informative in the sense that it
points to the sources of inefficiency in markets. Under the assumptions
above, any market equilibrium is tautologically efficient. Therefore,
when equilibria arise that are not efficient, the market system itself
is not to blame, but rather some sort of market failure.
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.
Proofs of the existence of equilibrium traditionally rely on fixed-point theorems such as Brouwer fixed-point theorem for functions (or, more generally, the Kakutani fixed-point theorem for set-valued functions). See Competitive equilibrium#Existence of a competitive equilibrium. The proof was first due to Lionel McKenzie, and Kenneth Arrow and Gérard Debreu. 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, uses Sard's lemma and the Baire category theorem; this method was pioneered by Gérard Debreu and Stephen Smale.
Nonconvexities in large economies
Starr (1969) applied the Shapley–Folkman–Starr theorem to prove that even without convex preferences there exists an approximate equilibrium. The Shapley–Folkman–Starr results bound the distance from an "approximate" economic equilibrium 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, [ . . . ] 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 of market failures 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 effects (which is the feature that most clearly delineates general equilibrium analysis from partial equilibrium)
generates the possibility of multiple equilibria. When a price of a
particular good changes there are two effects. First, the relative
attractiveness of various commodities changes; and second, the wealth
distribution of individual agents is altered. These two effects can
offset or reinforce each other in ways that make it possible for more
than one set of prices to constitute an equilibrium.
A result known as the Sonnenschein–Mantel–Debreu theorem states that the aggregate excess demand function inherits only certain properties of individual's demand functions, and that these (Continuity, Homogeneity of degree zero, Walras' law
and boundary behavior when prices are near zero) are 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 and odd.
Furthermore, if an economy as a whole, as characterized by an aggregate
excess demand function, has the revealed preference property (which is a
much stronger condition than revealed preferences for a single individual) or the gross substitute property
then likewise the equilibrium will be unique. All methods of
establishing uniqueness can be thought of as establishing that each
equilibrium has the same positive local index, in which case by the
index theorem there can be but one such equilibrium.
Determinacy
Given
that equilibria may not be unique, it is of some interest to ask
whether any particular equilibrium is at least locally unique. If so,
then comparative statics can be applied as long as the shocks to the system are not too large. As stated above, in a regular economy equilibria will be finite, hence locally unique. One reassuring result, due to Debreu, is that "most" economies are regular.
Work by Michael Mandler (1999) has challenged this claim.
The Arrow–Debreu–McKenzie model is neutral between models of production
functions as continuously differentiable and as formed from (linear
combinations of) fixed coefficient processes. Mandler accepts that,
under either model of production, the initial endowments will not be
consistent with a continuum of equilibria, except for a set of Lebesgue measure
zero. However, endowments change with time in the model and this
evolution of endowments is determined by the decisions of agents (e.g.,
firms) in the model. Agents in the model have an interest in equilibria
being indeterminate:
Indeterminacy, moreover, is not just a technical nuisance; it undermines the price-taking assumption of competitive models. Since arbitrary small manipulations of factor supplies can dramatically increase a factor's price, factor owners will not take prices to be parametric.
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).
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 tâtonnement process has been said to be a model of a centrally planned economy,
not a decentralized market economy. Some research has tried to develop
general equilibrium models with other processes. In particular, some
economists have developed models in which agents can trade at
out-of-equilibrium prices and such trades can affect the equilibria to
which the economy tends. Particularly noteworthy are the Hahn process, the Edgeworth process and the Fisher process.
The data determining Arrow-Debreu equilibria include initial
endowments of capital goods. If production and trade occur out of
equilibrium, these endowments will 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... [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).
The Arrow–Debreu model in which all trade occurs in futures contracts
at time zero requires a very large number of markets to exist. It is
equivalent under complete markets to a sequential equilibrium concept in
which spot markets for goods and assets open at each date-state event
(they are not equivalent under incomplete markets); market clearing
then requires that the entire sequence of prices clears all markets at
all times. A generalization of the sequential market arrangement is the temporary equilibrium
structure, where market clearing at a point in time is conditional on
expectations of future prices which need not be market clearing ones.
Although the Arrow–Debreu–McKenzie model is set out in terms of some arbitrary numéraire, the model does not encompass money. Frank Hahn,
for example, has investigated whether general equilibrium models can be
developed in which money enters in some essential way. One of the
essential questions he introduces, often referred to as the Hahn's problem
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 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 of individuals; complete information about all prices both now and in the future; and the conditions necessary for perfect competition. However some results from experimental economics
suggest that even in circumstances where there are few, imperfectly
informed agents, the resulting prices and allocations may wind up
resembling those of a perfectly competitive market (although certainly
not a stable general equilibrium in all markets).
Frank Hahn
defends general equilibrium modeling on the grounds that it provides a
negative function. General equilibrium models show what the economy
would have to be like for an unregulated economy to be Pareto efficient.
Computing general equilibrium
Until the 1970s general equilibrium analysis remained theoretical. With advances in computing power and the development of input–output
tables, it became possible to model national economies, or even the
world economy, and attempts were made to solve for general equilibrium
prices and quantities empirically.
Applied general equilibrium (AGE) models were pioneered by Herbert Scarf
in 1967, and offered a method for solving the Arrow–Debreu General
Equilibrium system in a numerical fashion. This was first implemented by
John Shoven and John Whalley (students of Scarf at Yale) in 1972 and
1973, and were a popular method up through the 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
(CGE) models surpassed and replaced AGE models in the mid-1980s, as the
CGE model was able to provide relatively quick and large computable
models for a whole economy, and was the preferred method of governments
and the World Bank.
CGE models are heavily used today, and while 'AGE' and 'CGE' is used
inter-changeably in the literature, Scarf-type AGE models have not been
constructed since the mid-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,
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, developed from general equilibrium theory.
Keynesian and Post-Keynesian
Keynesian and Post-Keynesian economists, and their underconsumptionist 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.
Let us beware of this dangerous theory of equilibrium which is supposed to be automatically established. A certain kind of equilibrium, it is true, is reestablished in the long run, but it is after a frightful amount of suffering.
— Simonde de Sismondi, New Principles of Political Economy, vol. 1, 1819, pp. 20-21.
The long run is a misleading guide to current affairs. In the long run we are all dead. Economists set themselves too easy, too useless a task if in tempestuous seasons they can only tell us that when the storm is past the ocean is flat again.
— John Maynard Keynes, A Tract on Monetary Reform, 1923, ch. 3
It is as absurd to assume that, for any long period of time, the variables in the economic organization, or any part of them, will "stay put," in perfect equilibrium, as to assume that the Atlantic Ocean can ever be without a wave.
— Irving Fisher, The Debt-Deflation Theory of Great Depressions, 1933, p. 339
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 system.
New classical macroeconomics
While general equilibrium theory and neoclassical economics generally were originally microeconomic theories, new classical macroeconomics
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, 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, a sustained critique of general equilibrium theory (and neoclassical economics generally) is given in Anti-Equilibrium, based on the experiences of János Kornai with the failures of Communist central planning, although Michael Albert and Robin Hahnel later based their Parecon model on the same theory.
