Non-convexity (economics)

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In

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]

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. Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.

utility
).

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 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 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.[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]

Supply with few producers

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]

Contemporary economics

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 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.[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] In some cases, non-linear pricing or bargaining may overcome the failures of markets with competitive pricing; in other cases, regulation may be justified.

Optimization over time

The previously mentioned applications concern non-convexities in finite-dimensional

: 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.

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]

Nonsmooth analysis

Economists have increasingly studied non-convex sets with

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]

See also

Notes

  1. ^ .
  2. .
  3. ^ .
  4. ^ a b c Salanié (2000, p. 36)
  5. ^ .
  6. ^ .
  7. ^ .
  8. ^ .)
  9. ^ .

  10. ^ .
  11. ^ .
  12. ^ Hotelling (1935, p. 74):
    JSTOR 1907346
    .
  13. .

  14. ^ Samuelson (1950, pp. 359–360):

    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.

  15. ^ Diewert (1982, pp. 552–553).
  16. S2CID 153653926
    . Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5): 484–489. . Farrell, M. J. (October 1961b). "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5): 493. .
  17. .
  18. .

  19. .)
  20. ^ Arrow & Hahn (1980, p. 182)
  21. .
  22. .

    .

  23. ^ Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold & Juréen (1953, p. 146), according to Diewert (1982, p. 552).

  24. .

  25. .

  26. .

  27. .
  28. .
  29. .

    Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory. Vol. 9. Berlin: Springer-Verlag.

    .

  30. .
  31. .
  32. .
  33. .
  34. .
  35. .
  36. 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.)
  37. ^ Magille & Quinzii, Section 31 "Partnerships", p. 371): Magill, Michael; Quinzii, Martine (1996). "6 Production in a finance economy". The Theory of incomplete markets. Cambridge, Massachusetts: MIT Press. pp. 329–425.
  38. S2CID 154223797
    .
  39. .
  40. ^ Adda, Jerome; Cooper, Russell (2003), Dynamic Economics, MIT Press, archived from the original on 2008-12-05, retrieved 2011-03-01
  41. ^ Howard, Ronald A. (1960). Dynamic Programming and Markov Processes. The M.I.T. Press.
  42. .
  43. .
  44. .
  45. ^ Beckmann, Martin; Muth, Richard F. (1954). "On the solution to the fundamental equation of inventory theory". Cowles Commission Discussion Paper. 2116.
  46. S2CID 1504746
    .
  47. .
  48. .
  49. .
  50. ^ Dasgupta & Heal (1979, pp. 96–97, 285, 404, 420, 422, and 429)
  51. ^ Dasgupta & Heal (1979, pp. 51, 64–65, 87, and 91–92)
  52. . Retrieved 5 March 2011.
  53. .
  54. .

References

External links

Heal, G. M. (April 1998). The Economics of Increasing Returns (PDF). PaineWebber working paper series in money, economics, and finance. Columbia Business School. PW-97-20. Archived from the original (PDF) on 15 September 2015. Retrieved 5 March 2011.