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{{JEL code|[[JEL_classification_codes#Mathematical_and_quantitative_methods_JEL:_C_Subcategories|C65]]}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
{{Economics sidebar}}
'''Convexity''' is an important topic in '''economics'''.<ref name="Newman1987c" >{{harvtxt|Newman|1987c|}}</ref> In the [[Arrow-Debreu model]] of [[general equilibrium|general economic equilibrium]], agents have convex [[budget set]]s and [[convex preferences]]: At equilibrium prices, the budget [[supporting hyperplane|hyperplane supports]] the best attainable [[indifference curve]].<ref name="Newman1987d" >{{harvtxt|Newman|1987d|}}</ref> The [[profit (economics)|profit function]] is the [[convex conjugate]] of the [[cost function]].<ref name="Newman1987c"/><ref name="Newman1987d" /> [[Convex analysis]] is the standard tool for analyzing textbook economics.<ref name="Newman1987c"/> Non-convex phenomena in economics have been studied with [[subgradient|nonsmooth analysis]], which  generalizes [[convex analysis]].<ref name="Khan"/>


==Preliminaries==
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The economics depends upon the following definitions and results from [[convex geometry]].
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===Real vector spaces===
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{{multiple image
  | width    = 107
  | footer    = [[Line segment]]s test [[convex set|convexity]].
  | image1    = Convex polygon illustration1.png
  | alt1      = Illustration of a convex set, which looks somewhat like a disk: A (green) convex set contains the (black) line-segment joining the points x and y. The entire line segment lies in the interior of the convex set.
  | caption1  = A [[convex set]] [[cover (topology)|covers]] the [[line segment]] connecting any two of its points.
  | image2    = Convex polygon illustration2.png
  | alt2      = Illustration of a non-convex set, which looks somewhat like a boomerang or cashew nut. A (green) non-convex set contains the (black) line-segment joining the points ''x'' and ''y''. Part of the line segment lies outside of the (green) non-convex set.
  | caption2  = A [[convex set|non-convex set]] fails to [[cover (topology)|cover]] a point in some [[line segment|line-segment]] joining two of its points.
}}


A [[real number|real]] [[vector space]] of two&nbsp;[[dimension (vector space)|dimension]]s may be given a [[Cartesian coordinate system]] in which every point is identified by a list of two real numbers, called "coordinates", which are conventionally denoted by&nbsp;''x'' and&nbsp;''y''. Two points in the Cartesian&nbsp;plane can be ''[[Euclidean_vector#Addition_and_subtraction|added]]'' coordinate-wise
'''MathML'''
: (''x''<sub>1</sub>,&nbsp;''y''<sub>1</sub>) + (''x''<sub>2</sub>,&nbsp;''y''<sub>2</sub>) = (''x''<sub>1</sub>+''x''<sub>2</sub>, ''y''<sub>1</sub>+''y''<sub>2</sub>);
:<math forcemathmode="mathml">E=mc^2</math>
further, a point can be ''[[scalar multiplication|multiplied]]'' by each real&nbsp;number&nbsp;''λ'' coordinate-wise
: ''λ''&nbsp;(''x'',&nbsp;''y'') = (''λx'', ''λy'').


More generally, any real vector space of (finite) dimension&nbsp;''D'' can be viewed as the [[set (mathematics)|set]] of all possible lists of&nbsp;''D'' real&nbsp;numbers {{nowrap|{&nbsp;(''v''<sub>1</sub>, ''v''<sub>2</sub>, .&nbsp;.&nbsp;.&nbsp;, ''v''<sub>D</sub>)}}&nbsp;} together with two&nbsp;[[operation (mathematics)|operation]]s: [[Euclidean_vector#Addition_and_subtraction|vector&nbsp;addition]] and [[scalar multiplication|multiplication by a real&nbsp;number]]. For finite-dimensional vector spaces, the operations of vector&nbsp;addition and real-number&nbsp;multiplication can each be defined coordinate-wise, following the example of the Cartesian plane.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


===Convex sets===
'''source'''
[[File:Extreme points illustration.png|thumb|right|alt=A picture of a smoothed triangle, like a triangular tortilla-chip or a triangular road-sign. Each of the three rounded corners is drawn with a red curve. The remaining interior points of the triangular shape are shaded with blue.|In the [[convex hull]] of the red&nbsp;set, each blue&nbsp;point is a [[convex combination]] of some red&nbsp;points.]]
:<math forcemathmode="source">E=mc^2</math> -->
In a real vector space, a set is defined to be ''[[convex set|convex]]'' if, for each pair of its points, every point on the [[line segment]] that joins them is [[cover (mathematics)|covered]] by the set. For example, a solid [[cube (geometry)|cube]] is convex; however, anything that is hollow or dented, for example, a [[crescent]] shape, is non-convex. [[Vacuous truth|Trivially]], the [[empty set]] is convex.


More formally, a set&nbsp;''Q'' is convex if, for all points&nbsp;''v''<sub>0</sub> and&nbsp;''v''<sub>1</sub> in&nbsp;''Q'' and for every real number&nbsp;''λ'' in the [[unit interval]]&nbsp;[0,1], the point
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
: (1&nbsp;&minus;&nbsp;''λ'')&nbsp;''v''<sub>0</sub> + ''λv''<sub>1</sub>
is a [[element (mathematics)|member]] of&nbsp;''Q''.  


By [[mathematical induction]], a set&nbsp;''Q'' is convex if and only&nbsp;if every [[convex combination]] of members of&nbsp;''Q'' also belongs to&nbsp;''Q''. By definition, a ''convex combination'' of an indexed&nbsp;subset&nbsp;{''v''<sub>0</sub>,&nbsp;''v''<sub>1</sub>,&nbsp;.&nbsp;.&nbsp;.&nbsp;,&nbsp;''v''<sub>D</sub>} of a vector space is any [[weighted mean|weighted average]]&nbsp;{{nowrap|''λ''<sub>0</sub>''v''<sub>0</sub> + ''λ''<sub>1</sub>''v''<sub>1</sub> + .&nbsp;.&nbsp;. +  ''λ''<sub>D</sub>''v''<sub>D</sub>,}} for some indexed&nbsp;set of non-negative real numbers&nbsp;{''λ''<sub>d</sub>} satisfying the [[affine combination|equation]]&nbsp;{{nowrap|''λ''<sub>0</sub> + ''λ''<sub>1</sub> + .&nbsp;.&nbsp;.&nbsp; +  ''λ''<sub>D</sub>}}&nbsp;=&nbsp;1.
==Demos==


The definition of a convex&nbsp;set implies that the ''[[intersection (set theory)|intersection]]'' of two convex&nbsp;sets is a convex set. More generally, the intersection of a family of convex sets is a convex set. <!-- In this proposition, the family can be empty, finite, countably infinite, or uncountably infinite. -->
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


===Convex hull===
For every subset&nbsp;''Q'' of a real vector&nbsp;space, its {{nowrap|''[[convex hull]]''&nbsp;Conv(''Q'')}} is the [[minimal element|minimal]] convex set that contains&nbsp;''Q''. Thus&nbsp;Conv(''Q'') is the intersection of all the convex sets that [[cover (mathematics)|cover]]&nbsp;''Q''. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in&nbsp;''Q''.


==Duality:Intersecting half-spaces==
* accessibility:
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** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


[[File:Supporting hyperplane1.svg|right|thumb|A [[convex set]] <math>S</math> (in pink), a supporting hyperplane of <math>S</math> (the dashed line), and the half-space delimited by the hyperplane which contains <math>S</math> (in light blue). ]]
==Test pages ==
''Supporting hyperplane'' is a concept in [[geometry]]. A [[hyperplane]] divides a space into two [[Half-space (geometry)|half-space]]s. A hyperplane is said to '''support''' a [[Set (mathematics)|set]] <math>S</math> in [[Euclidean space]] <math>\mathbb R^n</math> if it meets both of the following:
* <math>S</math> is entirely contained in one of the two [[closed set|closed]] half-spaces determined by the hyperplane
* <math>S</math> has at least one point on the hyperplane.
Here, a closed half-space is the half-space that includes the hyperplane.


===Supporting hyperplane theorem===
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
[[File:Supporting hyperplane2.svg|right|thumb|A convex set can have more than one supporting hyperplane at a given point on its boundary.]]
*[[Displaystyle]]
This [[theorem]] states that if <math>S</math> is a closed [[convex set]] in [[Euclidean space]] <math>\mathbb R^n,</math> and <math>x</math> is a point on the [[boundary (topology)|boundary]] of <math>S,</math> then there exists a supporting hyperplane containing <math>x.</math>
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set <math>S</math> is not convex, the statement of the theorem is not true at all points on the boundary of <math>S,</math> as illustrated in the third picture on the right.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
[[File:Supporting hyperplane3.svg|right|thumb|A supporting hyperplane containing a given point on the boundary of <math>S</math> may not exist if <math>S</math> is not convex.]]
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
===Economics===
[[File:Indifference curves showing budget line.svg|thumb|right|The consumer prefers the vector of goods&nbsp;(''Q''<sub>''x''</sub>,&nbsp;''Q''<sub>''y''</sub>) over other affordable vectors. At this optimal vector, the budget line supports the indifference curve&nbsp;''I''<sub>2</sub>.]]
An optimal basket of goods occurs where the consumer's convex [[convex preferences|preference set]]  is [[supporting hyperplane|supported]]<!-- "tangent" is wrong, unless differentiability is needlessly supposed, and ambiguous even then, if foliation happens --> by the budget constraint, as shown in the diagram. If the preference set is convex, then the consumer's set of optimal decisions is a convex set, for example, a unique optimal basket (or even a line-segment of optimal baskets).
 
For simplicity, we shall assume that the preferences of a consumer can be described by a [[utility function]] that is a [[continuous function]], which implies that the [[convex preferences|preference set]]s are [[closed set|closed]]. (The meanings of "closed set" is explained below, in the subsection on optimization applications.)
 
==Non-convexity==
{{Main|Non-convexity (economics)}}
{{See also|Shapley–Folkman lemma}}
[[File:NonConvex.gif|thumb|right|When consumer preferences have concavities, then the linear budgets need not support equilibria: Consumers can jump between allocations.]]
If a preference set is non-convex, then some prices produce a budget supporting two different optimal consumption decisions. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeper does not want to purchase a half&nbsp;an&nbsp;eagle and a {{nowrap|half&nbsp;a&nbsp;lion}} (or a [[griffin]])! Thus, the contemporary zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.
 
Non-convex sets have been incorporated in the theories of general economic equilibria,<ref>Pages&nbsp;392–399 and  page&nbsp;188: {{cite book |last1=Arrow |first1=Kenneth&nbsp;J. |authorlink1=Kenneth Arrow |last2=Hahn |first2=Frank&nbsp;H. |authorlink2=Frank Hahn |year=1971 |chapter=Appendix&nbsp;B: Convex and related sets |title=General competitive analysis |publisher=Holden-Day, Inc. [North-Holland] |pages=375–401 |mr=439057 |series=Mathematical&nbsp;economics texts [Advanced textbooks in economics] |number=6 [12] |location=San&nbsp;Francisco,&nbsp;CA |isbn=0-444-85497-5}}</p>  <p> Pages&nbsp;52–55 with applications on pages&nbsp;145–146,&nbsp;152–153, and&nbsp;274–275: {{cite book |last=Mas-Colell |first=Andreu |authorlink=Andreu Mas-Colell |year=1985 |chapter=1.L Averages of sets |title=The Theory of General Economic Equilibrium: A ''Differentiable'' Approach |series=Econometric&nbsp;Society Monographs |number=9 |publisher=Cambridge&nbsp;UP |isbn=0-521-26514-2 |mr=1113262}}</p>  <p> Theorem C(6) on page&nbsp;37 and applications on pages&nbsp;115-116,&nbsp;122, and&nbsp;168: {{cite book |last=Hildenbrand |first=Werner |authorlink=Werner Hildenbrand |title=Core and equilibria of a large economy |series=Princeton studies in mathematical economics |number=5 |publisher=Princeton&nbsp;University Press |location=Princeton,&nbsp;N.J. |year=1974 |pages=viii+251 |isbn=978-0-691-04189-6 |mr=389160}} </p></ref> of [[market failure]]s,<ref>Pages 112–113 in Section&nbsp;7.2 "Convexification by numbers" (and more generally pp.&nbsp;107–115): {{cite book |last=Salanié |first=Bernard |chapter=7 Nonconvexities <!-- Not "Non–convexities" --> |title=Microeconomics of market failures |edition=English translation of the (1998) French ''Microéconomie: Les défaillances du&nbsp;marché'' (Economica, Paris) |year=2000 |publisher=MIT&nbsp;Press |location=Cambridge,&nbsp;MA |pages=107–125 |isbn=0-262-19443-0, 978-0-262-19443-3}}</ref> and of [[public economics]].<ref>Pages&nbsp;63–65: {{cite book |last=Laffont |first=Jean-Jacques |authorlink=Jean-Jacques Laffont |year=1988 |chapter=3 Nonconvexities <!-- Not "Non–convexities" --> |title=Fundamentals of public economics |url=http://books.google.com/books?q=editions:ISBN 0-262-12127-1&id=O5MnAQAAIAAJ |publisher=[http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=7534 MIT]
|isbn=0-262-12127-1, 978-0-262-12127-9}}</ref> These results are described in graduate-level textbooks in [[microeconomics]],<ref>{{cite book |authorlink=Hal Varian |last=Varian |first=Hal&nbsp;R. |chapter=21.2&nbsp;Convexity and size |pages=393–394 |title=Microeconomic Analysis |publisher=W.&nbsp;W.&nbsp;Norton&nbsp;& Company |edition=3rd |year=1992 |isbn=978-0-393-95735-8 |mr=1036734}} <p> Page 628: {{cite book |last1=Mas–Colell |first1=Andreu |authorlink=Andreu Mas-Colell |last2=Whinston |first2=Michael&nbsp;D. |first3=Jerry&nbsp;R. |last3=Green |chapter=17.1 Large&nbsp;economies and nonconvexities |title=Microeconomic theory |publisher=Oxford&nbsp;University Press |year=1995 |pages=627–630 |isbn=978-0-19-507340-9}}</ref> general equilibrium theory,<ref>Page&nbsp;169 in the first edition: {{cite book |last=Starr |first=Ross&nbsp;M. |chapter=8 Convex sets, separation theorems, and non-convex sets in&nbsp;'''R'''<sup>N</sup> |title=General&nbsp;equilibrium theory: An introduction |edition=Second |publisher=Cambridge University Press |location=Cambridge |year=2011 |pages= |isbn=978-0-521-53386-7 |mr=1462618}} <p> In Ellickson, page&nbsp;xviii, and especially Chapter&nbsp;7 "Walras meets Nash" (especially section&nbsp;7.4 "Nonconvexity" pages&nbsp;306–310 and&nbsp;312, and also&nbsp;328–329) and Chapter&nbsp;8 "What is Competition?" (pages&nbsp;347 and&nbsp;352): {{cite book |title=Competitive equilibrium: Theory and applications |first=Bryan |last=Ellickson |publisher=Cambridge University Press |isbn=978-0-521-31988-1 |doi=10.2277/0521319889 |year=1994 |pages=420}} </p>
</ref> [[game theory]],<ref>Theorem&nbsp;1.6.5 on pages&nbsp;24–25: {{cite book |last=Ichiishi |first=Tatsuro |title=Game theory for economic analysis |series=Economic theory, econometrics, and mathematical economics |publisher=Academic Press,&nbsp;Inc. [Harcourt Brace Jovanovich, Publishers] |location=New&nbsp;York |year=1983 |pages=x+164 |isbn=0-12-370180-5 |mr=700688}}</ref> [[mathematical economics]],<ref>Pages 127 and 33–34: {{cite book |last=Cassels |first=J.&nbsp;W.&nbsp;S. |authorlink=J. W. S. Cassels |chapter=Appendix&nbsp;A Convex&nbsp;sets |title=Economics for mathematicians |series=London&nbsp;Mathematical&nbsp;Society lecture&nbsp;note series |volume=62 |publisher=Cambridge&nbsp;University Press |location=Cambridge, New&nbsp;York |year=1981 |pages=xi+145 |isbn=0-521-28614-X |mr=657578}}</ref>
and applied mathematics (for economists).<ref>Pages&nbsp;93–94 (especially example&nbsp;1.92),&nbsp;143,&nbsp;318–319,&nbsp;375–377, and&nbsp;416: {{cite book |last=Carter |first=Michael |title=Foundations of mathematical economics |publisher=MIT&nbsp;Press |location=Cambridge,&nbsp;MA |year=2001 |pages=xx+649 |isbn=0-262-53192-5 |mr=1865841}} <p> Page 309: {{cite book |last=Moore |first=James&nbsp;C. |title=Mathematical methods for economic theory: Volume&nbsp;'''I'''
|series=Studies in economic theory |volume=9 |publisher=Springer-Verlag |location=Berlin |year=1999 |pages=xii+414 |isbn=3-540-66235-9 |mr=1727000}} <p> Pages 47–48: {{cite book |mr=1878374 |last1=Florenzano |first1=Monique |last2=Le&nbsp;Van |first2=Cuong |Finite dimensional convexity and optimization |author3<!-- not an author -->=in cooperation with Pascal Gourdel |series=Studies in economic theory |volume=13 |publisher=Springer-Verlag |location=Berlin |year=2001 |pages=xii+154 |isbn=3-540-41516-5}}</ref> The [[Shapley–Folkman lemma]] results establish that non-convexities are compatible with approximate equilibria in markets with many consumers; these results also apply to [[production (economics)|production economies]] with many small [[business|firm]]s.<ref>Economists have studied non-convex sets using advanced mathematics, particularly [[differential geometry]] and&nbsp;[[differential topology|topology]], [[Baire&nbsp;category]], [[measure (mathematics)|measure]]&nbsp;and&nbsp;[[integral|integration theory]], and [[ergodic&nbsp;theory]]: {{cite book |last=Trockel |first=Walter |title=Market&nbsp;demand: An analysis of large&nbsp;economies with nonconvex&nbsp;preferences |series=Lecture Notes in Economics and Mathematical Systems |volume=223 |publisher=Springer-Verlag |location=Berlin |year=1984 |pages=viii+205 |isbn=3-540-12881-6 |mr=737006}}</ref>
 
In "[[oligopoly|oligopolies]]" (markets dominated by a few producers<!-- with increasing [[returns to&nbsp;scale]] -->), especially in "[[monopoly|monopolies]]" (markets dominated by one&nbsp;producer), non-convexities remain important.<ref name="GuesnerieNonConvex" ><!-- Guenserie discusses public economies a bit later, but he introduces "indivisibilities" on page 1, and economists would recognize that this synopsis is faithful and accessible to the public, imho -->Page 1: {{cite article |last=Guesnerie |first=Roger |authorlink=Roger Guesnerie |title=Pareto optimality in non-convex economies |journal=Econometrica |volume=43 |year=1975 |page=1–29 |jstor=1913410 |doi=10.2307/1913410 |mr=443877.{{jstor|1913410}}|ref=harv}} ({{cite article |<!-- last=Guesnerie |first=Roger |authorlink=Roger Guesnerie --> |title=Errata<!-- : "Pareto optimality in non-convex economies" (''Econometrica'' 43 (1975), 1–29) --> |journal=Econometrica |volume=43 |year=1975 |number=5–6 |page=1010 |jstor=1911353 |doi=10.2307/1911353 |mr=443878.{{jstor|1911353}}|}})</ref> Concerns with large producers exploiting market power in fact initiated the literature on non-convex sets, when [[Piero Sraffa]] wrote about on firms with increasing [[returns to scale]] in&nbsp;1926,<ref>{{cite article |last=Sraffa |first=Piero |authorlink=Piero Sraffa |year=1926 |title=The Laws of returns under competitive conditions |journal=Economic Journal |volume=36 |number=144 |pages=535-550 |jstor=2959866 |jstor=2959866 |ref=harv}}</ref> after which [[Harold Hotelling]] wrote about [[marginal&nbsp;cost pricing]] in&nbsp;1938.<ref>{{cite article |
first=Harold |last=Hotelling |authorlink=Harold Hotelling |title=The General welfare in relation to problems of taxation and of railway and utility rates |title=Econometrica |volume=6 |number=3 |month=July |year=1938 |pages=242–269 | jstor=1907054 |jstor=1907054}}</ref> Both Sraffa and Hotelling illuminated the [[market power]] of producers without competitors, clearly stimulating a literature on the supply-side of the economy.<ref>Pages 5–7: {{cite book |last=Quinzii |first=Martine |title=Increasing returns and efficiency |location=New York |publisher=Oxford University Press |year=1992 |edition=Revised translation of (1988) ''Rendements croissants et efficacité economique''. Paris: Editions du Centre National de la Recherche Scientifique |pages=viii+165 |isbn=0-19-506553-0}}</ref>
Non-convex sets arise also with [[environmental economics|environmental goods]] (and other [[externality|externalities]]),<ref>Pages 106, 110–137, 172, and 248: {{cite book |title=The Theory of environmental policy |edition=Second |first1=William&nbsp;J. |last1=Baumol |authorlink1=William Baumol |last2=Oates |first2=Wallace&nbsp;E. |isbn=978-0-521-31112-0 |doi=10.2277/0521311128 |year=1988 |pages=x+299 |publisher=Cambridge University Press |location=Cambridge |author3=with contributions by V.&nbsp;S.&nbsp;Bawa and David&nbsp;F.&nbsp;Bradford |chapter=8 Detrimental externalities and nonconvexities in the production set |ref=harv}}
</ref><ref>{{cite article |mr=449575 |last=Starrett |first=David&nbsp;A. |title=Fundamental nonconvexities in the theory of externalities |journal=Journal of Economic Theory |volume=4 |year=1972 |number=2 |pages=180–199 |url=http://www.sciencedirect.com/science/article/B6WJ3-4CYGBWD-NX/2/0f7447ebad01895b6e454dfee4ac481b |doi=10.1016/0022-0531(72)90148-2 |ref=harv}}<p>Starrett discusses non-convexities in his textbook on [[public economics]] (pages&nbsp;33,&nbsp;43,&nbsp;48,&nbsp;56,&nbsp;70–72,&nbsp;82,&nbsp;147, and&nbsp;234–236): {{cite book |last=Starrett |first=David&nbsp;A. |title=Foundations of public economics |series=Cambridge economic handbooks |volume= |year=1988 |number= |pages= |publisher=Cambridge University Press |location=Cambridge |url=http://books.google.com/books?id=R35yljdyyIkC&pg=PR11&dq=David+A.+Starrett,+public+economics#v=onepage&q=nonconvex%20OR%20nonconvexities&f=false |ref=harv}}<p/></ref> with [[information economics]],<ref>{{cite article |first=Roy |last=Radner |authorlink=Roy Radner |title=Competitive equilibrium under uncertainty |journal=Econometrica |volume=36 |year=1968 |pages=31–53 |ref=harv}} <!-- Apparently not [[Radner Equilibrium]]: {{cite journal |authorlink=Roy Radner |last=Radner |first=R. |year=1967 |title=Equilibre des marchés à terme et au comptant en cas d’incertitude [Equilibrium of temporal sequences of markets under uncertainty] |language=French |journal=Cahiers du Séminaire d’Econométrie |volume=17 |pages=35-52 |}} and {{cite article |authorlink=Roy Radner |first=Roy |last=Radner |mr=381655 |title=Existence of equilibrium of plans, prices, and price expectations in a sequence of markets |journal=Econometrica |volume=40 |year=1972 |pages=289–304 |}} -->
</ref> and with [[stock market]]s<ref name="GuesnerieNonConvex"/> (and other  [[incomplete markets]]).<ref>Page 270: {{cite book |mr=926685 |last=Drèze |first=Jacques&nbsp;H. |authorlink=Jacques H. Drèze |title=Essays on economic decisions under uncertainty |publisher=Cambridge University Press |editor-last=Drèze |editor-first=J.&nbsp;H. |<!-- editor-link=Jacques H. Drèze -->location=Cambridge |year=1987 | pages=261–297<!-- xxviii+424 --> |isbn=0-521-26484-7 |chapter=14 Investment under private ownership: Optimality, equilibrium and stability |ref=harv}} (Originally published as {{cite book |last=Drèze |first=Jacques&nbsp;H. |authorlink=Jacques H. Drèze |year=1974 |chapter=Investment under private ownership: Optimality, equilibrium and stability |editor-last=Drèze |editor-first=J.&nbsp;H. |<!-- editor-link=Jacques H. Drèze --> |title=Allocation under Uncertainty: Equilibrium and Optimality |publisher=Wiley |location=New York |pages=129–165 |ref=harv}})</ref><ref>Page 371: {{cite book |last1=Magill |first1=Michael |last2=Quinzii |first2=Martine |year=1996 |chapter=6 Production in a finance economy |pages=329–425 |section=31 Partnerships |title=The Theory of incomplete markets |publisher=MIT Press |location=Cambridge, Massachusetts |ref=harv}}</ref> Such applications continued to motivate economists to study non-convex sets.<ref name="MCNC">{{cite book |last=Mas-Colell |first=A. |authorlink=Andreu Mas-Colell |chapter=Non-convexity |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |publisher=Palgrave Macmillan |year=1987 |edition=first |newedition=The New Palgrave Dictionary of Economics Online |doi=10.1057/9780230226203.3173 |pages=653–661 |url=http://www.econ.upf.edu/~mcolell/research/art_083b.pdf |ref=harv}}
</ref>
 
===Nonsmooth analysis===
Economists have increasingly studied non-convex sets with [[subderivative|nonsmooth analysis]], which generalizes [[convex&nbsp;analysis]]. "Non-convexities in [both] production and consumption ... required mathematical tools that went beyond convexity, and further development had to await the invention of non-smooth calculus" (for example, Francis Clarke's [[Rademacher's theorem|locally Lipschitz]] calculus), as described by {{harvtxt|Rockafellar|Wets|1998}}<ref>{{cite book |last1=Rockafellar |first1=R.&nbsp;Tyrrell |authorlink1=R. Tyrrell Rockafellar |last2=Wets |first2=Roger&nbsp;<!-- NO PERIODS -->J-B |authorlink2=Roger J-B Wets |title=Variational analysis |series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] |volume=317 |publisher=Springer-Verlag |location=Berlin |year=1998 |pages=xiv+733 |isbn=3-540-62772-3 |mr=1491362 |ref=harv}}</ref> and {{harvtxt|Mordukhovich|2006}},<ref name=Mordukhovich2>Chapter&nbsp;8 "Applications to economics", especially Section&nbsp;8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page&nbsp;495: <p>{{cite book |authorlink=Boris Mordukhovich |first=Boris S. |last=Mordukhovich |title=Variational analysis and generalized differentiation&nbsp;'''II''': Applications |series=Grundlehren Series (Fundamental Principles of Mathematical Sciences) |volume=331 |publisher=Springer |year=2006 |pages=i–xxii and&nbsp;1–610 |mr=2191745 |ref=harv}}<p/></ref> according to {{harvtxt|Khan|2008}}.<ref name="Khan" >{{cite book |last=Khan |first=M.&nbsp;Ali |chapter=Perfect competition |title=The New&nbsp;Palgrave Dictionary of Economics |editor-first=Steven&nbsp;N. |editor-last=Durlauf |editor2-first=Lawrence&nbsp;E.,&nbsp;ed. |editor2-last=Blume |publisher=Palgrave Macmillan |year=2008 |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde2008_P000056 |doi=10.1057/9780230226203.1267 |ref=harv}}</ref> {{harvtxt|Brown|1995|pp=1967–1968}} wrote that the "major methodological innovation in the general equilibrium analysis of firms with pricing rules" was "the introduction of the methods of non-<!-- and end of line breaks "non-smooth" at the hyphen --> analysis, as a [synthesis] of global analysis (differential topology) and [of] convex analysis." According to {{harvtxt|Brown|1995|p=1966}}, "Non-smooth analysis extends the local approximation of manifolds by tangent planes<!-- , --> [and extends] the analogous approximation of convex sets by tangent cones to sets" that can be non-smooth or non-convex.<!-- "which [deprecated] are neither smooth nor convex [false: Ioffe notes all subdifferentials specialize to the convex-analysis subdifferential for convex functions] -->.<ref>{{cite book |page=1966 |last=Brown |first=Donald&nbsp;J. |chapter=36 Equilibrium analysis with non-convex technologies |doi=10.1016/S1573-4382(05)80011-6 |url=http://www.sciencedirect.com/science/article/B7P5Y-4FKY4C6-C/2/0bd1a73374b0b690f702691e1f7fe671 |title=Handbook of mathematical economics, Volume&nbsp;'''IV''' |pages=1963–1995 |mr=1207195 |editor1-first=Werner |editor1-last=Hildenbrand |editor1-link=Werner Hildenbrand |editor2-first=Hugo |editor2-last=Sonnenschein |editor2-link=Hugo Sonnenschein |series=Handbooks in Economics |volume=1 |publisher=North-Holland Publishing&nbsp;Co |location=Amsterdam |year=1991 |isbn=0-444-87461-5<!-- The terminal "5" is correct, while Elsevier's on-line "0" is for volume 2 --> |ref=harv}}</ref> Economists have also used [[<!-- reduced -->singular homology|algebraic topology]].<ref>{{cite article |mr=1218037 |last=Chichilnisky |first=G. |authorlink=Graciela Chichilnisky |title=Intersecting families of sets and the topology of cones in economics |journal=Bulletin of the American Mathematical Society (New Series) |volume=29 |year=1993 |number=2 |pages=189–207 |doi=10.1090/S0273-0979-1993-00439-7 |url=http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00439-7/S0273-0979-1993-00439-7.pdf |ref=harv}}
</ref>
 
==Notes==
<references/>
 
==References==
* {{cite book |last=Blume |first=Lawrence&nbsp;E. |authorlink=Lawrence E. Blume |editor2-link=Lawrence E. Blume |chapter=Convexity |year=2008c |title=The New&nbsp;Palgrave Dictionary of Economics |editor-first=Steven&nbsp;N. |editor-last=Durlauf |editor2-first=Lawrence&nbsp;E<!-- . --> |editor2-last=Blume |publisher=Palgrave Macmillan |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000508 |doi=10.1057/9780230226203.0315 |ref=harv}}
* {{cite book |last=Blume |first=Lawrence&nbsp;E. |chapter=Convex programming |year=2008cp |authorlink=Lawrence E. Blume |editor2-link=Lawrence E. Blume |title=The New&nbsp;Palgrave Dictionary of Economics |editor-first=Steven&nbsp;N. |editor-last=Durlauf |editor2-first=Lawrence&nbsp;E<!-- . --> |editor2-last=Blume |publisher=Palgrave Macmillan |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000348 |doi=10.1057/9780230226203.0314 |ref=harv}}
* {{cite book |last=Blume |first=Lawrence&nbsp;E. |authorlink=Lawrence E. Blume |editor2-link=Lawrence E. Blume |chapter=Duality |year=2008d |title=The New&nbsp;Palgrave Dictionary of Economics |editor-first=Steven&nbsp;N. |editor-last=Durlauf |editor2-first=Lawrence&nbsp;E<!-- . --> |editor2-last=Blume |publisher=Palgrave Macmillan |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626 |doi=10.1057/9780230226203.0411 |ref=harv}}
* {{cite book |last=Crouzeix |first=J.-P. |chapter=Quasi-concavity |title=The New&nbsp;Palgrave Dictionary of Economics |editor-first=Steven&nbsp;N. |editor-last=Durlauf |editor2-first=Lawrence&nbsp;E<!-- . --> |editor2-last=Blume |editor2-link=Lawrence E. Blume |publisher=Palgrave Macmillan |year=2008 |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008 |doi=10.1057/9780230226203.1375 |ref=harv}}
* {{cite book |first=W.&nbsp;E. |last=Diewert |chapter=12 Duality approaches to microeconomic theory
|pages=535–599
|url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-R/2/dcc0f8c9352eb054c96b3ff481976ce7
|doi=10.1016/S1573-4382(82)02007-4
|title=Handbook of mathematical&nbsp;economics, Volume&nbsp;'''II''' |editor1-link=Kenneth Arrow |editor1-first=Kenneth&nbsp;Joseph |editor1-last=Arrow |editor2-first=Michael&nbsp;D<!-- . --> |editor2-last=Intriligator |series=Handbooks in economics |volume=1 |publisher=North-Holland Publishing&nbsp;Co. |location=Amsterdam |year=1982 |isbn=978-0-444-86127-6 |mr=648778 |ref=harv}}
* {{cite book |first1=Jerry |last1=Green |first2=Walter&nbsp;P. |last2=Heller |chapter=1 Mathematical&nbsp;analysis and&nbsp;convexity with applications to economics |pages=15–52 |url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-5/2/613440787037f7f62d65a05172503737 |doi=10.1016/S1573-4382(81)01005-9 |title=Handbook of mathematical&nbsp;economics, Volume&nbsp;'''I''' |editor1-link=Kenneth Arrow |editor1-first=Kenneth&nbsp;Joseph |editor1-last=Arrow |editor2-first=Michael&nbsp;D<!-- . --> |editor2-last=Intriligator |series=Handbooks in economics |volume=1 |publisher=North-Holland Publishing&nbsp;Co. |location=Amsterdam |year=1981 |isbn=0-444-86126-2 |mr=634800 |ref=harv}}
* Luenberger, David G. ''Microeconomic Theory'', McGraw-Hill, Inc., New York, 1995.
* {{cite book |last=Mas-Colell |first=A. |authorlink=Andreu Mas-Colell |chapter=Non-convexity |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |editor3-link=Peter Kenneth Newman |publisher=Palgrave Macmillan |year=1987 |edition=first |newedition=The New Palgrave Dictionary of Economics Online |doi=10.1057/9780230226203.3173 |pages=653–661 |url=http://www.econ.upf.edu/~mcolell/research/art_083b.pdf |ref=harv}}
* {{cite book |last=Newman |first=Peter |authorlink=Peter Kenneth Newman |chapter=Convexity |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |editor3-link=Peter Kenneth Newman |publisher=Palgrave Macmillan |year=1987c |edition=first |newedition=The New Palgrave Dictionary of Economics Online |doi=10.1057/9780230226203.2282<!-- SNAFU at NP? 30 Jan 2011--> |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000453 |ref=harv}}
* {{cite book |last=Newman |first=Peter |authorlink=Peter Kenneth Newman |chapter=Duality |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |editor3-link=Peter Kenneth Newman |publisher=Palgrave Macmillan |year=1987d |edition=first |newedition=The New Palgrave Dictionary of Economics Online |doi=10.1057/9780230226203.2412<!-- SNAFU at NP? 30 Jan 2011--> |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626 |ref=harv}}
* {{cite book |last=Rockafellar |first=R.&nbsp;Tyrrell |<!-- authorlink=R. Tyrrell Rockafellar --> |title=Convex analysis |edition=Reprint of the 1979 Princeton mathematical series&nbsp;'''28''' |series=Princeton landmarks in mathematics |publisher=Princeton University Press |location=Princeton, NJ |year=1997 |pages=xviii+451 |isbn=0-691-01586-4 |id={{MR|1451876}}, {{MR|274683}} |ref=harv |name="RockCA" }}
* {{cite book |last=Schneider |first=Rolf |title=Convex&nbsp;bodies: The Brunn–Minkowski theory |series=Encyclopedia of mathematics and its applications |volume=44 |publisher=Cambridge University Press |location=Cambridge |year=1993 |pages=xiv+490 |ref=harv |isbn=0-521-35220-7 |mr=1216521}}
 
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{{DEFAULTSORT:Convexity In Economics}}
[[Category:Convex hulls]]
[[Category:Convex geometry]]
[[Category:Mathematical and quantitative methods (economics)]]
[[Category:Mathematical economics]]
[[Category:General equilibrium and disequilibrium]]
[[Category:Convexity in economics| ]]
 
[[es:Convexidad (economía)]]
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