# Convexity in economics

Last updated

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]

## Preliminaries

The economics depends upon the following definitions and results from convex geometry.

### Real vector spaces

A convex set covers the line segment connecting any two of its points.
A non‑convex set fails to cover a point in some line segment joining two of its points.

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 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

(1  λ) v0 + λv1

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 {v0, v1, . . . , vD} of a vector space is any weighted average  λ0v0 + λ1v1 + . . . + λDvD, for some indexed set of non‑negative real numbers {λd} satisfying the equation λ0 + 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 cover Q. 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 ${\displaystyle S}$ in the real n-space ${\displaystyle \mathbb {R} ^{n}}$ if it meets both of the following:

• ${\displaystyle S}$ is entirely contained in one of the two closed half-spaces determined by the hyperplane
• ${\displaystyle S}$ 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 ${\displaystyle S}$ is a closed convex set in ${\displaystyle \mathbb {R} ^{n},}$ and ${\displaystyle x}$ is a point on the boundary of ${\displaystyle S,}$ then there exists a supporting hyperplane containing ${\displaystyle x.}$

The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set ${\displaystyle S}$ is not convex, the statement of the theorem is not true at all points on the boundary of ${\displaystyle S,}$ 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.)

## Non-convexity

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.

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 graduate-level 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 supply-side 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]

### Nonsmooth analysis

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]

## Notes

1. Khan, M. Ali (2008). "Perfect competition". In Durlauf, Steven N.; Blume, Lawrence E., ed. (eds.). The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 354–365. doi:10.1057/9780230226203.1267. ISBN   978-0-333-78676-5.
2. Pages 392–399 and page 188: Arrow, Kenneth J.; Hahn, Frank H. (1971). . General competitive analysis. Mathematical economics texts [Advanced textbooks in economics]. San Francisco: Holden-Day, Inc. [North-Holland]. pp.  375–401. ISBN   978-0-444-85497-1. MR   0439057.
Pages 52–55 with applications on pages 145–146, 152–153, and 274–275: Mas-Colell, Andreu (1985). "1.L Averages of sets". The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs. Cambridge University Press. ISBN   978-0-521-26514-0. 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   978-0-691-04189-6. MR   0389160.
3. 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. ISBN   978-0-262-19443-3.
4. Pages 63–65: Laffont, Jean-Jacques (1988). "3 Nonconvexities". . MIT. ISBN   978-0-262-12127-9.
5. Varian, Hal R. (1992). . Microeconomic Analysis (3rd ed.). W. W. Norton & Company. pp.  393–394. ISBN   978-0-393-95735-8. MR   1036734.
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   978-0-19-507340-9.
6. 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. doi:10.1017/CBO9781139174749. ISBN   978-0-521-53386-7. MR   1462618.
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   978-0-521-31988-1.
7. 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. ISBN   978-0-12-370180-0. MR   0700688.
8. Pages 127 and 33–34: Cassels, J. W. S. (1981). "Appendix A Convex sets". Economics for mathematicians. London Mathematical Society lecture note series. 62. Cambridge, New York: Cambridge University Press. pp. xi+145. ISBN   978-0-521-28614-5. MR   0657578.
9. 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. ISBN   978-0-262-53192-4. MR   1865841.
Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory. 9. Berlin: Springer-Verlag. pp. xii+414. doi:10.1007/978-3-662-08544-8. ISBN   978-3-540-66235-8. 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: Springer-Verlag. pp. xii+154. doi:10.1007/978-3-642-56522-9. ISBN   978-3-540-41516-9. MR   1878374. S2CID   117240618.
10. Economists have studied non‑convex sets using advanced mathematics, particularly differential geometry and topology, Baire category, measure and integration theory, and ergodic theory: Trockel, Walter (1984). Market demand: An analysis of large economies with nonconvex preferences. Lecture Notes in Economics and Mathematical Systems. 223. Berlin: Springer-Verlag. pp. viii+205. doi:10.1007/978-3-642-46488-1. ISBN   978-3-540-12881-6. MR   0737006.
11. Page 1: Guesnerie, Roger (1975). "Pareto optimality in non‑convex economies". Econometrica. 43 (1): 1–29. doi:10.2307/1913410. JSTOR   1913410. MR   0443877. (Guesnerie, Roger (1975). "Errata". Econometrica. 43 (5–6): 1010. doi:10.2307/1911353. JSTOR   1911353. MR   0443878.)
12. Sraffa, Piero (1926). "The Laws of returns under competitive conditions". Economic Journal. 36 (144): 535–550. doi:10.2307/2959866. JSTOR   2959866. S2CID   6458099.
13. Hotelling, Harold (July 1938). "The General welfare in relation to problems of taxation and of railway and utility rates". Econometrica. 6 (3): 242–269. doi:10.2307/1907054. JSTOR   1907054.
14. 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. ISBN   978-0-19-506553-4.
15. 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. doi:10.2277/0521311128. ISBN   978-0-521-31112-0.
16. Starrett, David A. (1972). "Fundamental nonconvexities in the theory of externalities". Journal of Economic Theory. 4 (2): 180–199. doi:10.1016/0022-0531(72)90148-2. MR   0449575.
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). . Cambridge economic handbooks. Cambridge: Cambridge University Press. ISBN   9780521348010. nonconvex OR nonconvexities.
17. Radner, Roy (1968). "Competitive equilibrium under uncertainty". Econometrica. 36 (1): 31–53. doi:10.2307/1909602. JSTOR   1909602.
18. 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. doi:10.1017/CBO9780511559464. ISBN   978-0-521-26484-6. MR   0926685. (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.)
19. 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.
20. Mas-Colell, A. (1987). "Non‑convexity" (PDF). In Eatwell, John; Milgate, Murray; Newman, Peter (eds.). The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. pp. 653–661. doi:10.1057/9780230226203.3173. ISBN   9780333786765.
21. Rockafellar, R. Tyrrell; Wets, Roger J-B (1998). Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 317. Berlin: Springer-Verlag. pp. xiv+733. doi:10.1007/978-3-642-02431-3. ISBN   978-3-540-62772-2. MR   1491362. S2CID   198120391.
22. 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). 331. Springer. pp. i–xxii and 1–610. MR   2191745.
23. Brown, Donald J. (1991). "36 Equilibrium analysis with non‑convex technologies". In Hildenbrand, Werner; Sonnenschein, Hugo (eds.). Handbook of mathematical economics, Volume IV. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. pp. 1963–1995 [1966]. doi:10.1016/S1573-4382(05)80011-6. ISBN   0-444-87461-5. MR   1207195.
24. Chichilnisky, G. (1993). "Intersecting families of sets and the topology of cones in economics" (PDF). Bulletin of the American Mathematical Society. New Series. 29 (2): 189–207. arXiv:. Bibcode:1993math.....10228C. CiteSeerX  . doi:10.1090/S0273-0979-1993-00439-7. MR   1218037.

## Related Research Articles

In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.

Economics is a social science that studies the production, distribution, and consumption of goods and services.

In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium. General equilibrium theory contrasts to the theory of partial equilibrium, which analyzes a specific part of an economy while its other factors are held constant. In general equilibrium, constant influences are considered to be noneconomic, therefore, resulting beyond the natural scope of economic analysis.

Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by Hermann Minkowski (1910).

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

In geometry, the Minkowski sum of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set

In decision theory, subjective expected utility is the attractiveness of an economic opportunity as perceived by a decision-maker in the presence of risk. Characterizing the behavior of decision-makers as using subjective expected utility was promoted and axiomatized by L. J. Savage in 1954 following previous work by Ramsey and von Neumann. The theory of subjective expected utility combines two subjective concepts: first, a personal utility function, and second a personal probability distribution.

Harold Hotelling was an American mathematical statistician and an influential economic theorist, known for Hotelling's law, Hotelling's lemma, and Hotelling's rule in economics, as well as Hotelling's T-squared distribution in statistics. He also developed and named the principal component analysis method widely used in finance, statistics and computer science.

In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.

In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave.

In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.

Lawrence E. Blume is the Distinguished Arts and Sciences Professor of Economics and Professor of Information Science at Cornell University, US.

Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.

Demographic economics or population economics is the application of economic analysis to demography, the study of human populations, including size, growth, density, distribution, and vital statistics.

The Shapley–Folkman lemma is a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero, one, and two:

Roger Guesnerie is an economist born in France in 1943. He is currently the Chaired Professor of Economic Theory and Social Organization of the Collège de France, Director of Studies at the École des hautes études en sciences sociales, and the chairman of the board of directors of the Paris School of Economics.

Ross Marc Starr is an American economist who specializes in microeconomic theory, monetary economics and mathematical economics. He is a Professor at the University of California, San Diego.

In economics, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failures, where supply and demand differ or where market equilibria can be inefficient. Non-convex economies are studied with nonsmooth analysis, which is a generalization of convex analysis.

Ivar I. Ekeland is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well as popular books on mathematics, which have been published in French, English, and other languages. Ekeland is known as the author of Ekeland's variational principle and for his use of the Shapley–Folkman lemma in optimization theory. He has contributed to the periodic solutions of Hamiltonian systems and particularly to the theory of Kreĭn indices for linear systems. Ekeland helped to inspire the discussion of chaos theory in Michael Crichton's 1990 novel Jurassic Park.