|Non-convexity (economics) is included in the JEL classification codes as JEL: C65|
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.
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.
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.
The difficulties of studying non-convex preferences were emphasized by Herman Wold darkness ...", according to Diewert.and again by Paul Samuelson, who wrote that non-convexities are "shrouded in eternal
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 Michael Farrell, Francis Bator, Tjalling Koopmans, and Jerome 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.
Non-convex sets have been incorporated in the theories of general economic equilibria,.These results are described in graduate-level textbooks in microeconomics, general equilibrium theory, game theory, mathematical economics, and applied mathematics (for economists). 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.
Non-convexity is important under oligopolies and especially monopolies.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, after which Harold Hotelling wrote about marginal cost pricing in 1938. Both Sraffa and Hotelling illuminated the market power of producers without competitors, clearly stimulating a literature on the supply-side of the economy.
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.Non-convex sets arise also with environmental goods (and other externalities), and with market failures, and public economics. Non-convexities occur also with information economics, and with stock markets (and other incomplete markets). Such applications continued to motivate economists to study non-convex sets. In some cases, non-linear pricing or bargaining may overcome the failures of markets with competitive pricing; in other cases, regulation may be justified.
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.In economics, dynamic programing was used by Martin Beckmann and Richard F. Muth for work on inventory theory and consumption theory. Robert C. Merton used dynamic programming in his 1973 article on the intertemporal capital asset pricing model. (See also Merton's portfolio problem). In Merton's model, investors chose between income today and future income or capital gains, and their solution is found via dynamic programming. Stokey, Lucas & Prescott use dynamic programming to solve problems in economic theory, problems involving stochastic processes. 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. Dixit & Pindyck used dynamic programming for capital budgeting. For dynamic problems, non-convexities also are associated with market failures, just as they are for fixed-time problems.
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. & Wets (1998) and Mordukhovich (2006), according to Khan (2008). Brown (1995 , pp. 1967–1968) harvtxt error: no target: CITEREFBrown1995 (help) 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) 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."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
Exercise 45, page 146: Wold, Herman; Juréen, Lars (in association with Wold) (1953). "8 Some further applications of preference fields (pp. 129–148)". Demand analysis: A study in econometrics. Wiley publications in statistics. New York: John Wiley and Sons, Inc. Stockholm: Almqvist and Wiksell. 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.For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr (1969) harvtxt error: no target: CITEREFStarr1969 (help), 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 (1961a , pp. 482–483), Rothenberg (1960 , p. 438), and Starr (1969 , p. 26) harvtxt error: no target: CITEREFStarr1969 (help)—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]):
Koopmans, Tjalling C. (1957). "Allocation of resources and the price system". In Koopmans, Tjalling C (ed.). Three essays on the state of economic science. New York: McGraw–Hill Book Company. pp. 1–126. ISBN 0-07-035337-9.
Pages 52–55 with applications on pages 145–146, 152–153, and 274–275: Mas-Colell, Andreu (1985). "1.L Averages of sets". The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs. Cambridge University Press. ISBN 0-521-26514-2. MR 1113262.
Theorem C(6) on page 37 and applications on pages 115-116, 122, and 168: Hildenbrand, Werner (1974). Core and equilibria of a large economy. Princeton studies in mathematical economics. Princeton, NJ: Princeton University Press. ISBN 978-0-691-04189-6. MR 0389160.
Page 628: Mas–Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). "17.1 Large economies and nonconvexities". Microeconomic theory. Oxford University Press. pp. 627–630. ISBN 978-0-19-507340-9.
Ellickson (1994 , p. 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. doi:10.2277/0521319889. ISBN 978-0-521-31988-1.
Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory. 9. Berlin: Springer-Verlag. doi:10.1007/978-3-662-08544-8. ISBN 3-540-66235-9. MR 1727000.
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. doi:10.1007/978-3-642-56522-9. ISBN 3-540-41516-5. MR 1878374. S2CID 117240618.
Economics is a social science that studies the production, distribution, and consumption of goods and services.
Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics focuses on the study of individual markets, sectors, or industries as opposed to the national economy as whole, which is studied in macroeconomics.
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium. General equilibrium theory contrasts to the theory of partial equilibrium, which analyzes a specific part of an economy while its other factors are held constant. In general equilibrium, constant influences are considered to be noneconomic, therefore, resulting beyond the natural scope of economic analysis.
Kenneth Joseph Arrow was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972.
Gérard Debreu was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economic Sciences.
Harold Hotelling was an American mathematical statistician and an influential economic theorist, known for Hotelling's law, Hotelling's lemma, and Hotelling's rule in economics, as well as Hotelling's T-squared distribution in statistics. He also developed and named the principal component analysis method widely used in finance, statistics and computer science.
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.
William Jack Baumol was an American economist. He was a professor of economics at New York University, Academic Director of the Berkley Center for Entrepreneurship and Innovation, and Professor Emeritus at Princeton University. He was a prolific author of more than eighty books and several hundred journal articles.
Hirofumi Uzawa was a Japanese economist.
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave.
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.
Lawrence E. Blume is the Distinguished Arts and Sciences Professor of Economics and Professor of Information Science at Cornell University, US.
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
Demographic economics or population economics is the application of economic analysis to demography, the study of human populations, including size, growth, density, distribution, and vital statistics.
The Shapley–Folkman lemma is a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero, one, and two:
Roger Guesnerie is an economist born in France in 1943. He is currently the Chaired Professor of Economic Theory and Social Organization of the Collège de France, Director of Studies at the École des hautes études en sciences sociales, and the chairman of the board of directors of the Paris School of Economics.
Andreu Mas-Colell is an economist, an expert in microeconomics and a prominent mathematical economist. He is the founder of the Barcelona Graduate School of Economics and a professor in the department of economics at Pompeu Fabra University in Barcelona, Catalonia, Spain. He has also served several times in the cabinet of the Catalan government. Summarizing his and others' research in general equilibrium theory, his monograph gave a thorough exposition of research using differential topology. His textbook on microeconomics, co-authored with Michael Whinston and Jerry Green, is the most used graduate microeconomics textbook in the world.
Ross Marc Starr is an American economist who specializes in microeconomic theory, monetary economics and mathematical economics. He is a Professor at the University of California, San Diego.
Convexity is an important topic in economics. In the Arrow–Debreu model of general economic equilibrium, agents have convex budget sets and convex preferences: At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. The profit function is the convex conjugate of the cost function. Convex analysis is the standard tool for analyzing textbook economics. Non‑convex phenomena in economics have been studied with nonsmooth analysis, which generalizes convex analysis.
Ivar I. Ekeland is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well as popular books on mathematics, which have been published in French, English, and other languages. Ekeland is known as the author of Ekeland's variational principle and for his use of the Shapley–Folkman lemma in optimization theory. He has contributed to the periodic solutions of Hamiltonian systems and particularly to the theory of Kreĭn indices for linear systems. Ekeland helped to inspire the discussion of chaos theory in Michael Crichton's 1990 novel Jurassic Park.
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.