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In economics, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences (that do not prefer extremes to in-between values) and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood.^{[1]}^{[2]} 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,^{[3]}^{[4]} where supply and demand differ or where market equilibria can be inefficient.^{[1]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]} Non-convex economies are studied with nonsmooth analysis, which is a generalization of convex analysis.^{[8]}^{[9]}^{[10]}^{[11]}
If a preference set is non-convex, then some prices determine a budget-line that supports two separate optimal-baskets. 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 half of an eagle and half of a lion (or a griffin)! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.
When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connected; A disconnected demand implies some discontinuous behavior by the consumer, as discussed by Harold Hotelling:
If indifference curves for purchases be thought of as possessing a wavy character, convex to the origin in some regions and concave in others, we are forced to the conclusion that it is only the portions convex to the origin that can be regarded as possessing any importance, since the others are essentially unobservable. They can be detected only by the discontinuities that may occur in demand with variation in price-ratios, leading to an abrupt jumping of a point of tangency across a chasm when the straight line is rotated. But, while such discontinuities may reveal the existence of chasms, they can never measure their depth. The concave portions of the indifference curves and their many-dimensional generalizations, if they exist, must forever remain in unmeasurable obscurity.^{[12]}
The difficulties of studying non-convex preferences were emphasized by Herman Wold^{[13]} and again by Paul Samuelson, who wrote that non-convexities are "shrouded in eternal darkness ...",^{[14]} according to Diewert.^{[15]}
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.^{[1]} Non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in The Journal of Political Economy (JPE). The main contributors were Farrell,^{[16]} Bator,^{[17]} Koopmans,^{[18]} and Rothenberg.^{[19]} In particular, Rothenberg's paper discussed the approximate convexity of sums of non-convex sets.^{[20]} These JPE-papers stimulated a paper by Lloyd Shapley and Martin Shubik, which considered convexified consumer-preferences and introduced the concept of an "approximate equilibrium".^{[21]} The JPE-papers and the Shapley–Shubik paper influenced another notion of "quasi-equilibria", due to Robert Aumann.^{[22]}^{[23]}
Non-convex sets have been incorporated in the theories of general economic equilibria,.^{[24]} These results are described in graduate-level textbooks in microeconomics,^{[25]} general equilibrium theory,^{[26]} game theory,^{[27]} mathematical economics,^{[28]} and applied mathematics (for economists).^{[29]} The Shapley–Folkman lemma establishes that non-convexities are compatible with approximate equilibria in markets with many consumers; these results also apply to production economies with many small firms.^{[30]}
Non-convexity is important under oligopolies and especially monopolies.^{[8]} Concerns with large producers exploiting market power initiated the literature on non-convex sets, when Piero Sraffa wrote about on firms with increasing returns to scale in 1926,^{[31]} after which Harold Hotelling wrote about marginal cost pricing in 1938.^{[32]} Both Sraffa and Hotelling illuminated the market power of producers without competitors, clearly stimulating a literature on the supply-side of the economy.^{[33]}
Recent research in economics has recognized non-convexity in new areas of economics. In these areas, non-convexity is associated with market failures, where equilibria need not be efficient or where no equilibrium exists because supply and demand differ.^{[1]}^{[4]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]} Non-convex sets arise also with environmental goods (and other externalities),^{[6]}^{[7]} and with market failures,^{[3]} and public economics.^{[5]}^{[34]} Non-convexities occur also with information economics,^{[35]} and with stock markets^{[8]} (and other incomplete markets).^{[36]}^{[37]} Such applications continued to motivate economists to study non-convex sets.^{[1]}
The previously mentioned applications concern non-convexities in finite-dimensional vector spaces, where points represent commodity bundles. However, economists also consider dynamic problems of optimization over time, using the theories of differential equations, dynamic systems, stochastic processes, and functional analysis: Economists use the following optimization methods:
In these theories, regular problems involve convex functions defined on convex domains, and this convexity allows simplifications of techniques and economic meaningful interpretations of the results.^{[43]}^{[44]}^{[45]} In economics, dynamic programing was used by Martin Beckmann and Richard F. Muth for work on monetary policy, fiscal policy, taxation, economic growth, search theory, and labor economics.^{[49]} Dixit & Pindyck used dynamic programming for capital budgeting.^{[50]} For dynamic problems, non-convexities also are associated with market failures,^{[51]} just as they are for fixed-time problems.^{[52]}
Economists have increasingly studied non-convex sets with nonsmooth analysis, which generalizes convex analysis. Convex analysis centers on convex sets and convex functions, for which it provides powerful ideas and clear results, but it is not adequate for the analysis of non-convexities, such as increasing returns to scale.^{[53]} "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, Clarke's differential calculus for Lipschitz continuous functions, which uses Rademacher's theorem and which is described by Rockafellar & Wets (1998)^{[54]} and Mordukhovich (2006),^{[9]} according to Khan (2008).^{[10]} 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.^{[11]}^{[55]}
Exercise 45, page 146:
It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.
For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr (1969), Arrow & Hahn (1971, p. 169) quote John Milton's description of the (non-convex) Serbonian Bog in Paradise Lost (Book II, lines 592–594):
A gulf profound as that Serbonian Bog Betwixt Damiata and Mount Casius old, Where Armies whole have sunk.
Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator (1961, pp. 482–483), Rothenberg (1960, p. 438), and Starr (1969, p. 26)—commented on Koopmans (1957, pp. 1–126, especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality], and 35–37 [1.7 The role of convexity assumptions in the analysis]):
Pages 52–55 with applications on pages 145–146, 152–153, and 274–275:
Theorem C(6) on page 37 and applications on pages 115-116, 122, and 168:
Page 628:
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.
Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory 9. Berlin: Springer-Verlag. pp. xii+414.
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.
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