Editing General equilibrium theory (section)

===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>