Convexity in economics

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Convexity in economics is included in the JEL classification codes as JEL: C65
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Convexity is an important topic in economics.

nonsmooth analysis, which generalizes convex analysis.[3]

Preliminaries

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

Real vector spaces

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

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

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

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

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

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

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

A convex set (in pink), a supporting hyperplane of (the dashed line), and the half-space delimited by the hyperplane that contains (in light blue).

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

A convex set can have more than one supporting hyperplane at a given point on its boundary.

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.

A supporting hyperplane containing a given point on the boundary of may not exist if is not convex.

Economics

The consumer prefers the vector of goods (Qx, Qy) over other affordable vectors. At this optimal vector, the budget line supports the indifference curve I2.

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

Main article: Non-convexity (economics)
See also: Shapley–Folkman lemma
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 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 "

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 (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]

See also

  • Convex duality

Notes

  1. ^ a b c Newman (1987c)
  2. ^ a b Newman (1987d)
  3. ^
    ISBN 978-0-333-78676-5
    .
  4. MR 0389160
    .
  5. ISBN 978-0-262-19443-3
    .
  6. ISBN 978-0-262-12127-9
    .
  7. ISBN 978-0-19-507340-9
    .
  8. ISBN 978-0-521-31988-1
    .
  9. MR 0700688
    .
  10. MR 0657578
    .
  11. S2CID 117240618
    .
  12. MR 0737006
    .
  13. ^
    MR 0443878
    .)
  14. S2CID 6458099
    .
  15. JSTOR 1907054
    .
  16. ISBN 978-0-19-506553-4
    .
  17. ISBN 978-0-521-31112-0
    .
  18. ISBN 9780521348010
    . nonconvex OR nonconvexities.
  19. JSTOR 1909602
    .
  20. 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.)
  21. ^ 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.
  22. ISBN 9780333786765
    .
  23. S2CID 198120391
    .
  24. MR 2191745
    .
  25. MR 1207195
    .
  26. MR 1218037
    .

References
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