Convexity in economics is included in the JEL classification codes as JEL: C65 
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Convexity is an important topic in economics.^{ [1] } In the Arrow–Debreu model of general economic equilibrium, agents have convex budget sets and convex preferences: At equilibrium prices, the budget hyperplane supports the best attainable indifference curve.^{ [2] } The profit function is the convex conjugate of the cost function.^{ [1] }^{ [2] } Convex analysis is the standard tool for analyzing textbook economics.^{ [1] } Non‑convex phenomena in economics have been studied with nonsmooth analysis, which generalizes convex analysis.^{ [3] }
This section may stray from the topic of the article into the topic of another article, convex analysis . (August 2013) 
The economics depends upon the following definitions and results from convex geometry.
A real vector space of two dimensions 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 added coordinatewise
further, a point can be multiplied by each real number λ coordinatewise
More generally, any real vector space of (finite) dimension D can be viewed as the set of all possible lists of D real numbers { (v_{1}, v_{2}, . . . , v_{D}) } together with two operations: vector addition and multiplication by a real number. For finitedimensional vector spaces, the operations of vector addition and realnumber multiplication can each be defined coordinatewise, following the example of the Cartesian plane.
In a real vector space, a set is defined to be convex if, for each pair of its points, every point on the line segment that joins them is covered by the set. For example, a solid cube is convex; however, anything that is hollow or dented, for example, a crescent shape, is non‑convex. Trivially, the empty set is convex.
More formally, a set Q is convex if, for all points v_{0} and v_{1} in Q and for every real number λ in the unit interval [0,1], the point
is a member of 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 {v_{0}, v_{1}, . . . , v_{D}} of a vector space is any weighted average λ_{0}v_{0} + λ_{1}v_{1} + . . . + λ_{D}v_{D}, for some indexed set of non‑negative real numbers {λ_{d}} satisfying the equation λ_{0} + λ_{1} + . . . + λ_{D} = 1.
The definition of a convex set implies that the intersection of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set.
For every subset Q of a real vector space, its convex hull Conv(Q) is the minimal convex set that contains Q. Thus Conv(Q) is the intersection of all the convex sets that cover Q. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in Q.
Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two halfspaces. A hyperplane is said to support a set in the real nspace if it meets both of the following:
Here, a closed halfspace is the halfspace that includes the hyperplane.
This theorem states that if is a closed convex set in and is a point on the boundary of then there exists a supporting hyperplane containing
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set is not convex, the statement of the theorem is not true at all points on the boundary of as illustrated in the third picture on the right.
An optimal basket of goods occurs where the consumer's convex preference set is supported 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 preference sets are closed. (The meanings of "closed set" is explained below, in the subsection on optimization applications.)
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 zookeeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zookeeper does not want to purchase a half an eagle and a half a lion (or a griffin)! Thus, the contemporary zookeeper's preferences are non‑convex: The zookeeper 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,^{ [4] } of market failures,^{ [5] } and of public economics.^{ [6] } These results are described in graduatelevel textbooks in microeconomics,^{ [7] } general equilibrium theory,^{ [8] } game theory,^{ [9] } mathematical economics,^{ [10] } and applied mathematics (for economists).^{ [11] } 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 economies with many small firms.^{ [12] }
In "oligopolies" (markets dominated by a few producers), especially in "monopolies" (markets dominated by one producer), non‑convexities remain important.^{ [13] } 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,^{ [14] } after which Harold Hotelling wrote about marginal cost pricing in 1938.^{ [15] } Both Sraffa and Hotelling illuminated the market power of producers without competitors, clearly stimulating a literature on the supplyside of the economy.^{ [16] } Non‑convex sets arise also with environmental goods (and other externalities),^{ [17] }^{ [18] } with information economics,^{ [19] } and with stock markets ^{ [13] } (and other incomplete markets).^{ [20] }^{ [21] } Such applications continued to motivate economists to study non‑convex sets.^{ [22] }
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Economists have increasingly studied non‑convex sets with 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 locally Lipschitz calculus), as described by Rockafellar & Wets (1998) ^{ [23] } and Mordukhovich (2006),^{ [24] } according to Khan (2008).^{ [3] } 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‑smooth analysis, as a [synthesis] of global analysis (differential topology) and [of] convex analysis." According to 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..^{ [25] } Economists have also used algebraic topology.^{ [26] }
Pages 52–55 with applications on pages 145–146, 152–153, and 274–275: MasColell, Andreu (1985). "1.L Averages of sets". The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs. Cambridge University Press. ISBN 9780521265140. MR 1113262.
Theorem C(6) on page 37 and applications on pages 115–116, 122, and 168: Hildenbrand, Werner (1974). Core and equilibria of a large economy. Princeton studies in mathematical economics. Princeton University Press. pp. viii+251. ISBN 9780691041896. MR 0389160.
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(help)Page 628: Mas–Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). "17.1 Large economies and nonconvexities". Microeconomic theory. Oxford University Press. pp. 627–630. ISBN 9780195073409.
In Ellickson, page xviii, and especially Chapter 7 "Walras meets Nash" (especially section 7.4 "Nonconvexity" pages 306–310 and 312, and also 328–329) and Chapter 8 "What is Competition?" (pages 347 and 352): Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. p. 420. doi:10.2277/0521319889. ISBN 9780521319881.
Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory. 9. Berlin: SpringerVerlag. pp. xii+414. doi:10.1007/9783662085448. ISBN 9783540662358. MR 1727000.
Pages 47–48: Florenzano, Monique; Le Van, Cuong (2001). Finite dimensional convexity and optimization. Studies in economic theory. 13. in cooperation with Pascal Gourdel. Berlin: SpringerVerlag. pp. xii+154. doi:10.1007/9783642565229. ISBN 9783540415169. MR 1878374.
Starrett discusses non‑convexities in his textbook on public economics (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): Starrett, David A. (1988). Foundations of public economics . Cambridge economic handbooks. Cambridge: Cambridge University Press. ISBN 9780521348010. nonconvex OR nonconvexities.
Mordukhovich, Boris S. (2006). Variational analysis and generalized differentiation II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences). 331. Springer. pp. i–xxii and , 1–610. MR 2191745.
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