Editing General equilibrium theory (section)

==Properties and characterization of general equilibrium==
{{See also|Fundamental theorems of welfare economics}}

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 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]], 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 [[convex preferences|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 function|set-valued functions]]). See [[Competitive equilibrium#Existence of a competitive equilibrium]]. The proof was first due to [[Lionel McKenzie]],<ref>{{cite journal |first=Lionel W. |last=McKenzie |title=On Equilibrium in Graham's Model of World Trade and Other Competitive Systems |journal=[[Econometrica]]|year=1954 |volume=22 |issue=2 |pages=147–161 |jstor=1907539 |doi=10.2307/1907539}}</ref> and [[Kenneth Arrow]] and [[Gérard Debreu]].<ref>{{cite journal | last1 = Arrow | first1 = K. J. | last2 = Debreu | first2 = G. | year = 1954 | title = Existence of an equilibrium for a competitive economy | journal = Econometrica | volume = 22 | issue =3 | pages = 265–290 | doi = 10.2307/1907353 | jstor = 1907353 }}</ref> In fact, the converse also holds, according to [[Hirofumi Uzawa|Uzawa]]'s derivation of Brouwer's fixed point theorem from Walras's law.<ref>{{cite journal |first=Hirofumi |last=Uzawa |title=Walras' Existence Theorem and Brouwer's Fixed-Point Theorem |journal=Economic Studies Quarterly |volume=13 |year=1962 |issue=1 |pages=59–62 |doi=10.11398/economics1950.13.1_59 }}</ref> 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====
{{Main|Shapley–Folkman lemma}}
Starr (1969) applied the [[Shapley–Folkman lemma|Shapley–Folkman–Starr theorem]]<!-- his corollary to the [[Shapley–Folkman lemma|Shapley–Folkman 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.<ref name="s69">{{Cite journal |doi=10.2307/1909201 |last=Starr |first=Ross M. |author-link=Ross Starr |issue=1 |journal=Econometrica |pages=25–38 |title=Quasi-equilibria in markets with non-convex preferences |jstor=1909201 |volume=37 |year=1969 |url=http://econ.ucsd.edu/~rstarr/Non-Convex%20Preferences.pdf |archive-url=https://web.archive.org/web/20170809014202/http://econ.ucsd.edu/~rstarr/Non-Convex%20Preferences.pdf |archive-date=2017-08-09 |url-status=live |citeseerx=10.1.1.297.8498 }}</ref> Following Starr's paper, the Shapley–Folkman–Starr results were "much exploited in the theoretical literature", according to Guesnerie,<ref name="Guesnerie1989">{{cite book |last=Guesnerie |first=Roger |year=1989 |chapter=First-best allocation of resources with nonconvexities in production|pages=99–143|editor=Bernard Cornet and Henry Tulkens |title= Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987) |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-03149-3 |mr=1104662}}</ref>{{rp|112}} who wrote the following:
<blockquote>
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.<ref name="Guesnerie1989"/>{{rp|99}}
</blockquote>
To this text, Guesnerie appended the following footnote:
<blockquote>
The derivation of these results in general form has been one of the major achievements of postwar economic theory.<ref name="Guesnerie1989" />{{rp|138}}
</blockquote>
In particular, the Shapley-Folkman-Starr results were incorporated in the theory of general economic equilibria<ref>See pages 392–399 for the Shapley-Folkman-Starr results and see p. 188 for applications in {{cite book|last1=Arrow|first1=Kenneth J.|author-link1=Kenneth J. Arrow|last2=Hahn|first2=Frank H.|author-link2=Frank Hahn|year=1971|chapter=Appendix B: Convex and related sets|title=General Competitive Analysis|url=https://archive.org/details/generalcompetiti0000arro|url-access=registration|publisher=Holden-Day [North-Holland]|pages=[https://archive.org/details/generalcompetiti0000arro/page/375 375–401]|mr=439057|series=Mathematical economics texts [Advanced textbooks in economics]|location=San Francisco|isbn=978-0-444-85497-1|issue=6 [12]}}</ref><ref>Pages 52–55 with applications on pages 145–146, 152–153, and 274–275 in {{cite book|last=Mas-Colell|first=Andreu|author-link=Andreu Mas-Colell|year=1985|chapter=1.L Averages of sets| title=The Theory of General Economic Equilibrium: A ''Differentiable'' Approach
| series=Econometric Society Monographs|publisher=Cambridge University Press|isbn=978-0-521-26514-0
|mr=1113262|issue=9}}</ref><ref>{{cite book|last=Hildenbrand|first=Werner |title=Core and Equilibria of a Large Economy|series=Princeton Studies in Mathematical Economics|publisher=Princeton University Press|location=Princeton, New Jersey|year=1974 |pages=viii+251|isbn=978-0-691-04189-6|mr=389160|issue=5}}</ref> and in the [[microeconomics|theory]] of [[market failure]]s<ref>See section 7.2 Convexification by numbers in Salanié: {{cite book|last=Salanié|first=Bernard |chapter=7 Nonconvexities|title=Microeconomics of market failures|edition=English translation of the (1998) French ''Microéconomie: Les défaillances du marché'' (Economica, Paris)|year=2000|publisher=MIT Press|location=Cambridge, Massachusetts|pages=107–125|isbn=978-0-262-19443-3}}</ref> and of [[public economics]].<ref>An "informal" presentation appears in pages 63–65 of Laffont: {{cite book|last=Laffont|first=Jean-Jacques|author-link=Jean-Jacques Laffont|year=1988|chapter=3 Nonconvexities|title=Fundamentals of Public Economics|url=https://archive.org/details/fundamentalsofpu0000laff|url-access=registration|publisher=MIT|isbn=978-0-585-13445-1}}</ref>

===Uniqueness===

{{See also|Sonnenschein–Mantel–Debreu theorem}}

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]], proven in the 1970s, states that the aggregate [[excess demand function]] inherits only certain properties of individual's demand functions, and that these ([[Continuous function|continuity]], [[Homogeneous function|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 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 [[Poincaré–Hopf theorem|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]]s for a single individual) or the [[substitute good|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.<ref name="Mandler 1999">{{cite book |last=Mandler |first=Michael |year=1999 |title=Dilemmas in Economic Theory: Persisting Foundational Problems of Microeconomics |location=Oxford |publisher=Oxford University Press |isbn=978-0-19-510087-7 |url-access=registration |url=https://archive.org/details/dilemmasineconom0000mand }}</ref> 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:

<blockquote>
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.<ref name="Mandler 1999" />{{rp|17}}
</blockquote>

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:

<blockquote>
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.<ref name="Mandler 1999" />{{rp|19}}
</blockquote>

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. 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.