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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.〔 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.〔〔〔 ==Demand with many consumers== 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:
The difficulties of studying non-convex preferences were emphasized by Herman Wold〔Pages 231 and 239 (Figure 10 a–b: Illustration of lemma 5 (240 )): Exercise 45, page 146: 〕 and again by Paul Samuelson, who wrote that non-convexities are "shrouded in eternal 〔: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 , 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〕 according to Diewert.〔: 〕 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 preferences were illuminated from 1959 to 1961 by a sequence of papers in ''The Journal of Political Economy'' (''JPE''). The main contributors were Farrell,〔 〕 Bator,〔 〕 Koopmans,〔 and others—for example, and , , , and —commented on : 〕 and Rothenberg.〔: () 〕 In particular, Rothenberg's paper discussed the approximate convexity of sums of non-convex sets. 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".〔: 〕 The ''JPE''-papers and the Shapley–Shubik paper influenced another notion of "quasi-equilibria", due to Robert Aumann.〔: builds on two papers: 〕〔Taking the convex hull of non-convex preferences had been discussed earlier by and by , according to .〕 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: 〕 These results are described in graduate-level textbooks in microeconomics,〔Page 628: 〕 general equilibrium theory,〔Page 169 in the first edition: 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): 〕 game theory,〔Theorem 1.6.5 on pages 24–25: 〕 mathematical economics,〔Pages 127 and 33–34: 〕 and applied mathematics (for economists).〔Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Page 309: Pages 47–48: 〕 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.〔Economists have studied non-convex sets using advanced mathematics, particularly differential geometry and topology, Baire category, measure and integration theory, and ergodic theory: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Non-convexity (economics)」の詳細全文を読む スポンサード リンク
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