Abstract
We provide a mathematical formulation of the idea of perfect competition for an economy with infinitely many agents and commodities. We conclude that in the presence of infinitely many commodities the Aumann (1964, 1966) measure space of agents, i.e., the interval [0,1] endowed with Lebesgue measure, is not appropriate to model the idea of perfect competition and we provide a characterization of the “appropriate” measure space of agents in an infinite dimensional commodity space setting. The latter is achieved by modeling precisely the idea of an economy with “many more” agents than commodities. For such an economy the existence of a competitive equilibrium is proved. The convexity assumption on preferences is not needed in the existence proof. We wish to thank Tom Armstrong for useful comments. As always we are responsible for any remaining errors.
We wish to thank Tom Armstrong for useful comments. As always we are responsible for any remaining errors.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aumann, R. J., 1964, “Markets with a Continuum of Traders,” Econometrica 32, 39–50.
Aumann, R. J., 1965, “Integrals of Set-Valued Functions,” J. Math. Anal. Appl. 12, 1–12.
Aumann, R. J., 1966, “Existence of a Competitive Equilibrium in Markets with a Continuum of Traders,” Econometrica 34, 1–17.
Bewley, T., 1990, “A Very Weak Theorem on the Existence of Equilibria in Atomless Economies with Infinitely Many Commodities,” this volume.
Castaing, C. and Valadier, M., 1977, “Convex Analysis and Measurable Mul-tifunctions,” Lect. Notes Math. 580, Springer-Verlag, New York.
Debreu, G., 1967, “Integration of Correspondences,” Proc. Fifth Berkeley Symp. Math. Stat. Prob., University of California Press, Berkeley, Vol. II, Part I, 351–372.
Diestel, J. and Uhl, J., 1977, Vector Measures,Mathematical Surveys, No. 15, American Mathematical Society, Providence, Rhode Island.
Dunford, N. and Schwartz, J. T., 1958, Linear Operators, Part I, Interscience, New York.
Gretsky, N. E. and Ostroy, J. 1985, “Thick and Thin Market Nonatomic Exchange Economies,” in Advances in Equilibrium Theory,C. D. Aliprantis, Springer-Verlag.
Halmos, P. R. and Von Neumann, J. 1941, “Operator Methods in Classical Mechanics, II,” Ann. Math. 43:2
Himmelberg, C. J. 1975, “Measurable Relations,” Fund. Mali. 87 53–72.
Ionescu-Tulcea, A. and C., 1969, Topics in the Theory of Lifting, Springer-Verlag, Berlin.
Khan, M. Ali, 1976, “On the Integration of Set-Valued Mappings in a Nonreflexive Banach Space II,” Simon Stevin 59, 257–267.
Khan, M. Ali, 1986, “Equilibrium Points of Nonatomic Games over a Banach Space,” Trans. Amer. Math. Soc. 293:2, 737–749.
Khan, M. Ali and Yanelis, N. C., 1990, “Existence of a Competitive Equilibrium in Markets with a Continuum of Agents and Commodities,” this volume.
Kluvanek, I. and Knowles, G., 1975, “Vector Measures and Control System,” Math. Stud. 20, North Holland.
Knowles, G., 1974, Liapunov Vector Measures, SIAM J. Control13, 294–303.
Kuratowski, K. and Ryll-Nardzewski, C., 1962, “A General Theorem on Selectors,” Bull. Acad. Polon. Sci. Ser. Sci. Marsh. Astronom. Phys. 13, 397–403.
Lewis, L., 1977, Ph.D. Thesis, Yale University.
Loeb, P., 1971, “A Combinatorial Analog of Lyapinov’s Theorem for Infinitesimal Generated Atomic Vector Measures,” Proc. Amer. Math. Soc. 39, 585–586.
Lindenstrauss, J., 1966, “A Short Proof of Lyapunov’s Convexity Theorem,” J. Math. Mech. 15, 971–972.
Maharam, D., 1942, “On Homogeneous Measure Algebras,” Proc. Natl. Acad. Sci. 28, 108–111.
Masani, P., 1978, “Measurability and Pettis Integration in Hilbert Spaces,” J. Reine Angew. Math. 297, 92–135.
Mas-Colell, A., 1975, “A Model of Equilibrium with Differentiated Commodities,” J. Math. Econ. 2, 263–295.
Mertens, J.-F., 1990, “An Equivalence Theorem for the Core of an Economy with Commodity Space L ∞ τ(L ∞, L 1),” this volume.
Ostroy, J. and Zame, W. R., 1988, “Non-Atomic Exchange Economies and the Boundaries of Perfect Competition,” mimeo.
Robertson, A. P. and Kingman, J. F. C., 1968, “On a Theorem of Lyapunov,” J. London Math. Soc. 43, 347–351.
Rustichini, A., 1989, “A Counterexample and an Exact Version of Fatou’s Lemma in Infinite Dimensions,” Archiv der Mathematic, 52, 357–362.
Rustichini, A. and Yannelis, N. C., 1989, “Commodity Pair Desirability and the Core Equivalence Theorem,” mimeo.
Rustichini, A. and Yannelis, N. C., 1991, “Edgeworth’s Conjecture in Economies with a Continuum of Agents and Commodities,” J. Math. Econ., to appear.
Schmeidler, D., 1973, “Equilibrium Points of Non Atomic Games, J. Stat. Phys. 7:4 295–300.
Von Neumann, J., 1950, “Functional Operators,” Ann. Math. Stud. 22, Princeton University Press.
Yannelis, N. C., 1988, “Fatou’s Lemma In Infinite Dimensional Spaces,” Proc. Amer. Math. Soc. 102, 303–310.
Yannelis, N. C., 1990, “Integration of Banach-Valued Correspondences,” this volume.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Rustichini, A., Yannelis, N.C. (1991). What is Perfect Competition?. In: Khan, M.A., Yannelis, N.C. (eds) Equilibrium Theory in Infinite Dimensional Spaces. Studies in Economic Theory, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07071-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-662-07071-0_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08114-9
Online ISBN: 978-3-662-07071-0
eBook Packages: Springer Book Archive