A
realvector 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 coordinate-wise
(x1, y1) + (x2, y2) = (x1+x2, y1+y2);
further, a point can be 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 of all possible lists of D real numbers { (v1, v2, . . . , vD) } together with two
operations:
vector addition and
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
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 v0 and v1 in Q and for every real number λ in the
unit interval[0,1], the point
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 {v0, v1, . . . , vD} of a vector space is any
weighted averageλ0v0 + λ1v1 + . . . + λDvD, for some indexed set of non‑negative real numbers {λd} satisfying the
equationλ0 + λ1 + . . . + λD = 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.
Convex hull
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
coverQ. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in Q.
Duality: Intersecting half-spaces
Supporting hyperplane is a concept in
geometry. A
hyperplane divides a space into two
half-spaces. A hyperplane is said to support a
set in the
real n-space if it meets both of the following:
is entirely contained in one of the two
closed half-spaces determined by the hyperplane
has at least one point on the hyperplane.
Here, a closed half-space is the half-space that includes the hyperplane.
Supporting hyperplane theorem
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.
Economics
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 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 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.
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 (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
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.[25] Economists have also used
algebraic topology.[26]
^Pages 392–399 and page 188:
Arrow, Kenneth J.;
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MR0439057. Pages 52–55 with applications on pages 145–146, 152–153, and 274–275:
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MR1113262. 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.
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MR0389160.
^Pages 112–113 in Section 7.2 "Convexification by numbers" (and more generally pp. 107–115): Salanié, Bernard (2000). "7 Nonconvexities". Microeconomics of market failures (English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris) ed.). MIT Press. pp. 107–125.
ISBN978-0-262-19443-3.
^Page 169 in the first edition: Starr, Ross M. (2011). "8 Convex sets, separation theorems, and non‑convex sets in RN". General equilibrium theory: An introduction (Second ed.). Cambridge: Cambridge University Press.
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MR1462618. 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.
ISBN978-0-521-31988-1.
^Theorem 1.6.5 on pages 24–25: Ichiishi, Tatsuro (1983). Game theory for economic analysis. Economic theory, econometrics, and mathematical economics. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. x+164.
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MR0700688.
^Pages 127 and 33–34:
Cassels, J. W. S. (1981). "Appendix A Convex sets". Economics for mathematicians. London Mathematical Society lecture note series. Vol. 62. Cambridge, New York: Cambridge University Press. pp. xi+145.
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MR0657578.
^Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Carter, Michael (2001). Foundations of mathematical economics. MIT Press. pp. xx+649.
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MR1865841. Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory. Vol. 9. Berlin: Springer-Verlag. pp. xii+414.
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ISBN978-3-540-66235-8.
MR1727000. Pages 47–48: Florenzano, Monique; Le Van, Cuong (2001). Finite dimensional convexity and optimization. Studies in economic theory. Vol. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. pp. xii+154.
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^Pages 5–7:
Quinzii, Martine (1992). Increasing returns and efficiency (Revised translation of (1988) Rendements croissants et efficacité economique. Paris: Editions du Centre National de la Recherche Scientifique ed.). New York: Oxford University Press. pp. viii+165.
ISBN978-0-19-506553-4.
^Pages 106, 110–137, 172, and 248:
Baumol, William J.; Oates, Wallace E. (1988). "8 Detrimental externalities and nonconvexities in the production set". The Theory of environmental policy. with contributions by V. S. Bawa and David F. Bradford (Second ed.). Cambridge: Cambridge University Press. pp. x+299.
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^Starrett, David A. (1972). "Fundamental nonconvexities in the theory of externalities". Journal of Economic Theory. 4 (2): 180–199.
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MR0449575. 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.
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^Page 270:
Drèze, Jacques H. (1987). "14 Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. (ed.). Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press. pp. 261–297.
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MR0926685. (Originally published as
Drèze, Jacques H. (1974). "Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. (ed.). Allocation under Uncertainty: Equilibrium and Optimality. New York: Wiley. pp. 129–165.)
^Page 371: Magill, Michael;
Quinzii, Martine (1996). "6 Production in a finance economy, Section 31 Partnerships". The Theory of incomplete markets. Cambridge, Massachusetts: MIT Press. pp. 329–425.
^Chapter 8 "Applications to economics", especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495: Mordukhovich, Boris S. (2006). Variational analysis and generalized differentiation II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences). Vol. 331. Springer. pp. i–xxii and 1–610.
MR2191745.
Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In
Arrow, Kenneth Joseph; Intriligator, Michael D (eds.). Handbook of mathematical economics, Volume I. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 15–52.
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MR0634800.
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