Editing Arrow–Debreu model (section)

===Non-convexity in large economies===
{{See also|Shapley–Folkman lemma|Market failure}}

The assumption of convexity precluded many applications, which were discussed in the ''[[Journal of Political Economy]]'' from&nbsp;1959 to&nbsp;1961 by Francis&nbsp;M.&nbsp;Bator, [[Michael James Farrell|M.&nbsp;J.&nbsp;Farrell]], [[Tjalling Koopmans|Tjalling&nbsp;Koopmans]], and Thomas&nbsp;J.&nbsp;Rothenberg<!-- Starr's citation mis-spells "T."&nbsp;as&nbsp;"J." -->.<ref name="s69">{{citation
 | last = Starr | first = Ross&nbsp;M. | author-link = Ross Starr
 | issue = 1
 | journal = Econometrica
 | pages = 25–38
 | title = Quasi–equilibria in markets with non–convex preferences (Appendix&nbsp;2: The Shapley–Folkman theorem, pp.&nbsp;35–37)
 | volume = 37
 | year = 1969
 | jstor=1909201
 | doi=10.2307/1909201
| citeseerx = 10.1.1.297.8498 }}.</ref> {{harvs|first=Ross&nbsp;M.|last=Starr|authorlink=Ross Starr|year=1969|txt}} proved the existence of [[general equilibrium theory#Nonconvexities in large economies|economic equilibria]] when some [[convex preferences|consumer preferences]] need not be [[convex set|convex]].<ref name="s69"/> In his paper, Starr proved that a "convexified" economy has general equilibria that are closely approximated by "quasi-equilbria" of the original economy; Starr's proof used the [[Shapley–Folkman lemma|Shapley–Folkman theorem]].<ref name="Starr08" >{{cite book|last=Starr|first= Ross&nbsp;M.|author-link=Ross Starr|chapter=Shapley–Folkman theorem|title=The New Palgrave Dictionary of Economics|editor1-first=Steven&nbsp;N.|editor1-last=Durlauf|editor2-first=Lawrence&nbsp;E.|editor2-last=Blume|publisher=Palgrave Macmillan|year=2008|edition=Second|volume=4|pages=317–318 |chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_S000107|doi=10.1057/9780230226203.1518|isbn= 978-0-333-78676-5}}</ref>