Non-convexity (economics)
[8][9][10][11] If a preference set is non-convex, then some prices determine a budget-line that supports two separate optimal-baskets.
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.
The concave portions of the indifference curves and their many-dimensional generalizations, if they exist, must forever remain in unmeasurable obscurity.
[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 Michael Farrell,[16] Francis Bator,[17] Tjalling Koopmans,[18] and Jerome 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.
[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.
[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.
In these areas, non-convexity is associated with market failures, where equilibria need not be efficient or where no competitive equilibrium exists because supply and demand differ.
[1][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.
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.
[46] Robert C. Merton used dynamic programming in his 1973 article on the intertemporal capital asset pricing model.
[48] Dynamic programming has been used in optimal economic growth, resource extraction, principal–agent problems, public finance, business investment, asset pricing, factor supply, and industrial organization.
Ljungqvist & Sargent apply dynamic programming to study a variety of theoretical questions in monetary policy, fiscal policy, taxation, economic growth, search theory, and labor economics.
[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).
According to Brown (1995, p. 1966) harvtxt error: no target: CITEREFBrown1995 (help), "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.
MR 0064385.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.A gulf profound as that Serbonian Bog Betwixt Damiata and Mount Casius old,
