View source for Convexity in economics

{{short description|Significant topic in economics}}
{{JEL code|[[JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories|C65]]}}
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'''Convexity''' is a geometric property with a variety of applications in '''economics'''.<ref name="Newman1987c" >{{harvtxt|Newman|1987c|}}</ref> Informally, an economic phenomenon is convex when "intermediates (or combinations) are better than extremes".  For example, an economic agent with [[convex preferences]] prefers ''combinations'' of goods over having a lot of any ''one'' sort of good; this represents a kind of [[Marginal utility#Law of diminishing marginal utility|diminishing marginal utility]] of having more of the same good. 

Convexity is a key simplifying assumption in many economic models, as it leads to market behavior that is easy to understand and which has desirable properties. For example, the [[Arrow–Debreu model]] of [[general equilibrium|general economic equilibrium]] posits that if preferences are convex and there is perfect competition, then [[Aggregate supply|aggregate supplies]] will equal [[Aggregate demand|aggregate demands]] for every commodity in the economy.

In contrast, [[Non-convexity (economics)|non-convexity]] is associated with [[Market failure|market failures]], where [[supply and demand]] differ or where [[Market equilibrium|market equilibria]] can be [[Pareto efficiency|inefficient]]. 

The branch of mathematics which supplies the tools for convex functions and their properties is called [[convex analysis]]; non-convex phenomena are studied under [[subgradient|nonsmooth analysis]].

==Preliminaries==
{{off topic|convex analysis|date=August 2013}}
The economics depends upon the following definitions and results from [[convex geometry]].

===Real vector spaces===
{{multiple image
   | width     = 107
   | footer    = [[Line segment]]s test [[convex set|convexity]].
   | image1    = Convex polygon illustration1.svg
   | 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.svg
   | 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 non‑convex set fails to [[cover (topology)|cover]] a point in some [[line segment]] joining two of its points.
}}

A [[real number|real]] [[vector space]] of two [[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 ''x'' and ''y''. Two points in the Cartesian plane can be ''[[Euclidean vector#Addition and subtraction|added]]'' coordinate-wise
: (''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>);
further, a point can be ''[[scalar multiplication|multiplied]]'' by each real number ''λ'' coordinate-wise 
: ''λ'' (''x'', ''y'') = (''λx'', ''λy'').

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

===Convex sets===
[[File:Extreme points.svg|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 set, each blue point is a [[convex combination]] of some red points.]]
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 ''Q'' is convex if, for all points ''v''<sub>0</sub> and ''v''<sub>1</sub> in ''Q'' and for every real number ''λ'' in the [[unit interval]] {{closed-closed|0,1}}, the point 
: (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 ''Q'' is convex if and only if every [[convex combination]] of members of ''Q'' also belongs to ''Q''. By definition, a ''convex combination'' of an indexed 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 set of non‑negative real numbers {''λ''<sub>d</sub>} satisfying the [[affine combination|equation]] {{nowrap|''λ''<sub>0</sub> + ''λ''<sub>1</sub> + . . . + ''λ''<sub>D</sub>}}&nbsp;=&nbsp;1.

The definition of a convex set implies that the ''[[intersection (set theory)|intersection]]'' of two convex 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. -->

===Convex hull===
For every subset ''Q'' of a real vector space, its {{nowrap|''[[convex hull]]'' Conv(''Q'')}} is the [[minimal element|minimal]] convex set that contains ''Q''. Thus Conv(''Q'') is the intersection of all the convex sets that [[cover (mathematics)|cover]] ''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==

[[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 that contains <math>S</math> (in light blue). ]]
''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 the [[real coordinate space|real ''n''-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===
[[File:Supporting hyperplane2.svg|right|thumb|A convex set can have more than one supporting hyperplane at a given point on its boundary.]]
This [[theorem]] states that if <math>S</math> is a closed [[convex set]] in <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>

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.

[[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.]]

===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 an eagle and a {{nowrap|half a 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. |author-link1=Kenneth Arrow |last2=Hahn |first2=Frank&nbsp;H. |author-link2=Frank Hahn |year=1971 |chapter=Appendix&nbsp;B: Convex and related sets |title=General competitive analysis |chapter-url=https://archive.org/details/generalcompetiti0000arro |chapter-url-access=registration |publisher=Holden-Day, Inc. [North-Holland] |pages=[https://archive.org/details/generalcompetiti0000arro/page/375 375–401] |mr=439057 |series=Mathematical economics texts [Advanced textbooks in economics] |number=6 [12] |location=San Francisco |isbn=978-0-444-85497-1}}<br> 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 |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 |number=9 |publisher=Cambridge University Press |isbn=978-0-521-26514-0 |mr=1113262}}<br> 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 |author-link=Werner Hildenbrand |title=Core and equilibria of a large economy |series=Princeton studies in mathematical economics |number=5 |publisher=Princeton University Press |year=1974 |pages=viii+251 |isbn=978-0-691-04189-6 |mr=389160}} </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 Press |pages=107–125 |isbn=978-0-262-19443-3}}</ref> and of [[public economics]].<ref>Pages&nbsp;63–65: {{cite book |last=Laffont |first=Jean-Jacques |author-link=Jean-Jacques Laffont |year=1988 |chapter=3 Nonconvexities <!-- Not "Non–convexities" --> |title=Fundamentals of public economics |chapter-url=https://books.google.com/books?id=O5MnAQAAIAAJ&q=editions:ISBN%200-262-12127-1 |publisher=MIT |isbn=978-0-262-12127-9 |url-access=registration |url=https://archive.org/details/fundamentalsofpu0000laff }}</ref> These results are described in graduate-level textbooks in [[microeconomics]],<ref>{{cite book |author-link=Hal Varian |last=Varian |first=Hal&nbsp;R. |chapter=21.2&nbsp;Convexity and size |pages=[https://archive.org/details/microeconomicana00vari_0/page/393 393–394] |title=Microeconomic Analysis |publisher=W.&nbsp;W. Norton & Company |edition=3rd |year=1992 |isbn=978-0-393-95735-8 |mr=1036734 |chapter-url-access=registration |chapter-url=https://archive.org/details/microeconomicana00vari_0 |url=https://archive.org/details/microeconomicana00vari_0/page/393 }} <br> Page 628: {{cite book |last1=Mas–Colell |first1=Andreu |author-link=Andreu Mas-Colell |last2=Whinston |first2=Michael&nbsp;D. |first3=Jerry&nbsp;R. |last3=Green |chapter=17.1 Large economies and nonconvexities |title=Microeconomic theory |publisher=Oxford 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 equilibrium theory: An introduction |edition=Second |publisher=Cambridge University Press |location=Cambridge |year=2011 |isbn=978-0-521-53386-7 |mr=1462618|doi=10.1017/CBO9781139174749}} <br> 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 |year=1994 |pages=420}} 
</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, Inc. [Harcourt Brace Jovanovich, Publishers] |location=New York |year=1983 |pages=x+164 |isbn=978-0-12-370180-0 |mr=700688}}</ref> [[mathematical economics]],<ref>Pages 127 and 33–34: {{cite book |last=Cassels |first=J.&nbsp;W.&nbsp;S. |author-link=J. W. S. Cassels |chapter=Appendix&nbsp;A Convex&nbsp;sets |title=Economics for mathematicians |series=London Mathematical Society lecture note series |volume=62 |publisher=Cambridge University Press |location=Cambridge, New York |year=1981 |pages=xi+145 |isbn=978-0-521-28614-5 |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 Press |year=2001 |pages=xx+649 |isbn=978-0-262-53192-4 |mr=1865841}} <br> 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=978-3-540-66235-8 |mr=1727000|doi=10.1007/978-3-662-08544-8 }} <br> Pages 47–48: {{cite book |mr=1878374 |last1=Florenzano |first1=Monique |last2=Le&nbsp;Van |first2=Cuong |title=Finite dimensional convexity and optimization |others=in cooperation with Pascal Gourdel |s2cid=117240618 |series=Studies in economic theory |volume=13 |publisher=Springer-Verlag |location=Berlin |year=2001 |pages=xii+154 |isbn=978-3-540-41516-9|doi=10.1007/978-3-642-56522-9}}</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 [[differential topology|topology]], [[Baire category]], [[measure (mathematics)|measure]] and [[integral|integration theory]], and [[ergodic theory]]: {{cite book |last=Trockel |first=Walter |title=Market demand: An analysis of large economies with nonconvex preferences |series=Lecture Notes in Economics and Mathematical Systems |volume=223 |publisher=Springer-Verlag |location=Berlin |year=1984 |pages=viii+205 |isbn=978-3-540-12881-6 |mr=737006|doi=10.1007/978-3-642-46488-1 }}</ref>

In "[[oligopoly|oligopolies]]" (markets dominated by a few producers<!-- with increasing [[returns to scale]] -->), especially in "[[monopoly|monopolies]]" (markets dominated by one 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 journal |last=Guesnerie |first=Roger |author-link=Roger Guesnerie |title=Pareto optimality in non‑convex economies |journal=Econometrica |volume=43 |issue=1 |year=1975 |pages=1–29 |jstor=1913410 |doi=10.2307/1913410 |mr=443877}} ({{cite journal <!-- |last=Guesnerie |first=Roger |author-link=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|last1=Guesnerie |first1=Roger}})</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 1926,<ref>{{cite journal |last=Sraffa |first=Piero |s2cid=6458099 |author-link=Piero Sraffa |year=1926 |title=The Laws of returns under competitive conditions |journal=Economic Journal |volume=36 |number=144 |pages=535–550 |jstor=2959866 |doi=10.2307/2959866}}</ref> after which [[Harold Hotelling]] wrote about [[marginal cost pricing]] in 1938.<ref>{{cite journal |
first=Harold |last=Hotelling |author-link=Harold Hotelling |title=The General welfare in relation to problems of taxation and of railway and utility rates |journal=Econometrica |volume=6 |number=3 |date=July 1938 |pages=242–269 |jstor=1907054 |doi=10.2307/1907054}}</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|author-link= Martine Quinzii |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=978-0-19-506553-4}}</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 |author-link1=William Baumol |last2=Oates |first2=Wallace&nbsp;E. |isbn=978-0-521-31112-0 |year=1988 |pages=x+299 |publisher=Cambridge University Press |location=Cambridge |others=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><ref>{{cite journal |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 |doi=10.1016/0022-0531(72)90148-2 }}<br>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 |year=1988 |publisher=Cambridge University Press |location=Cambridge |url=https://archive.org/details/foundationsofpub0000star |url-access=registration |quote=nonconvex OR nonconvexities. |isbn=9780521348010 }}</ref> with [[information economics]],<ref>{{cite journal |first=Roy |last=Radner |author-link=Roy Radner |title=Competitive equilibrium under uncertainty |journal=Econometrica |volume=36 |issue=1 |year=1968 |pages=31–53 |doi=10.2307/1909602|jstor=1909602 }} <!-- 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 |author-link=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. |author-link=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=978-0-521-26484-6 |chapter=14 Investment under private ownership: Optimality, equilibrium and stability |doi=10.1017/CBO9780511559464 }} (Originally published as {{cite book |last=Drèze |first=Jacques&nbsp;H. |author-link=Jacques H. Drèze |year=1974 |chapter=Investment under private ownership: Optimality, equilibrium and stability |editor-last=Drèze |editor-first=J.&nbsp;H.|title=Allocation under Uncertainty: Equilibrium and Optimality |publisher=Wiley |location=New York |pages=129–165 }})</ref><ref>Page 371: {{cite book |last1=Magill |first1=Michael |last2=Quinzii |first2=Martine |author2-link= Martine Quinzii |year=1996 |chapter=6 Production in a finance economy, Section 31 Partnerships|pages=329–425 |title=The Theory of incomplete markets |publisher=MIT Press |location=Cambridge, Massachusetts}}</ref> Such applications continued to motivate economists to study non‑convex sets.<ref name="MCNC">{{cite book |last=Mas-Colell |first=A. |author-link=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|doi=10.1057/9780230226203.3173 |pages=653–661 |chapter-url=http://www.econ.upf.edu/~mcolell/research/art_083b.pdf |isbn=9780333786765}}</ref>

===Nonsmooth analysis===
{{cleanup|section|date=August 2013|reason=Relationship between subderivatives and non‑convexity remains cryptic}}
Economists have increasingly studied non‑convex sets with [[subderivative|nonsmooth analysis]], which generalizes [[convex 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 |author-link1=R. Tyrrell Rockafellar |last2=Wets |first2=Roger&nbsp;<!-- NO PERIODS -->J-B |s2cid=198120391 |author-link2=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=978-3-540-62772-2 |mr=1491362 |doi=10.1007/978-3-642-02431-3 }}</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: <br>{{cite book |author-link=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> according to {{harvtxt|Khan|2008}}.<ref name="Khan" >{{cite book |last=Khan |first=M.&nbsp;Ali |chapter=Perfect competition |title=The New Palgrave Dictionary of Economics |editor-first=Steven&nbsp;N. |editor-last=Durlauf |editor2-first=Lawrence E. |editor2-last=Blume |publisher=Palgrave Macmillan |year=2008 |edition=Second |pages= 354–365|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_P000056 |doi=10.1057/9780230226203.1267 |isbn=978-0-333-78676-5}}</ref> {{harvtxt|Brown|1991|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‑smooth analysis, as a [synthesis] of global analysis (differential topology) and [of] convex analysis." According to {{harvtxt|Brown|1991|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 |last=Brown |first=Donald&nbsp;J. |chapter=36 Equilibrium analysis with non‑convex technologies |doi=10.1016/S1573-4382(05)80011-6 |title=Handbook of mathematical economics, Volume&nbsp;'''IV''' |pages=1963–1995 [1966] |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> Economists have also used [[<!-- reduced -->singular homology|algebraic topology]].<ref>{{cite journal |mr=1218037 |last=Chichilnisky |first=G. |author-link=Graciela Chichilnisky |title=Intersecting families of sets and the topology of cones in economics |journal=Bulletin of the American Mathematical Society |series=New Series |volume=29 |year=1993 |number=2 |pages=189–207 |doi=10.1090/S0273-0979-1993-00439-7 |url=https://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00439-7/S0273-0979-1993-00439-7.pdf |arxiv=math/9310228 |citeseerx=10.1.1.234.3909 |bibcode=1993math.....10228C}}</ref>

==See also==
* [[Convex duality]]
{{clear}}

==Notes==
{{reflist}}

==References==
* {{cite book |last=Blume |first=Lawrence&nbsp;E. |author-link=Lawrence E. Blume |chapter=Convexity |year=2008a |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= 225–226|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000508 |doi=10.1057/9780230226203.0315 |isbn=978-0-333-78676-5}}
* {{cite book |last=Blume |first=Lawrence&nbsp;E. |chapter=Convex programming |year=2008b |title=The New 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= 220–225|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000348 |doi=10.1057/9780230226203.0314 |isbn=978-0-333-78676-5}}
* {{cite book |last=Blume |first=Lawrence&nbsp;E. |chapter=Duality |year=2008c |title=The New 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= 551–555|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626 |doi=10.1057/9780230226203.0411 |isbn=978-0-333-78676-5}}
* {{cite book |last=Crouzeix |first=J.-P. |chapter=Quasi-concavity |title=The New Palgrave Dictionary of Economics |editor-first=Steven&nbsp;N. |editor-last=Durlauf |editor2-first=Lawrence&nbsp;E<!-- . --> |editor2-last=Blume |publisher=Palgrave Macmillan |year=2008 |edition=Second |pages= 815–816|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008 |doi=10.1057/9780230226203.1375 |isbn=978-0-333-78676-5}}
* {{cite book |first=W.&nbsp;E. |last=Diewert |chapter=12 Duality approaches to microeconomic theory|pages=535–599|doi=10.1016/S1573-4382(82)02007-4|title=Handbook of mathematical 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}}
* {{cite book |first1=Jerry |last1=Green |first2=Walter&nbsp;P. |last2=Heller |chapter=1 Mathematical analysis and convexity with applications to economics |pages=15–52 |doi=10.1016/S1573-4382(81)01005-9 |title=Handbook of mathematical 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=978-0-444-86126-9 |mr=634800}}
* Luenberger, David G. ''Microeconomic Theory'', McGraw-Hill, Inc., New York, 1995.
* {{cite book |last=Mas-Colell |first=A. |author-link=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|doi=10.1057/9780230226203.3173 |pages=653–661 |chapter-url=http://www.econ.upf.edu/~mcolell/research/art_083b.pdf |isbn=9780333786765}}
* {{cite book |last=Newman |first=Peter |author-link=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|doi=10.1057/9780230226203.2282<!-- SNAFU at NP? 30 Jan 2011--> |pages= 1|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000453 |isbn=9780333786765}}
* {{cite book |last=Newman |first=Peter |author-link=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|doi=10.1057/9780230226203.2412<!-- SNAFU at NP? 30 Jan 2011--> |pages= 1|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626 |isbn=9780333786765}}
* {{cite book |last=Rockafellar |first=R.&nbsp;Tyrrell |author-link=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|isbn=978-0-691-01586-6|mr=274683}}.
* {{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 |isbn=978-0-521-35220-8 |mr=1216521 |doi=10.1017/CBO9780511526282 |url=https://archive.org/details/convexbodiesbrun0000schn}}

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{{Microeconomics|state=collapsed}}
{{Convex analysis and variational analysis}}

{{DEFAULTSORT:Convexity In Economics}}
[[Category:Convex hulls]]
[[Category:Convex geometry]]
[[Category:General equilibrium theory]]
[[Category:Convex optimization|*]]