Abstract
A general principle of obtaining equivalence of core and quasiWalrasian allocations in nonatomic markets with an infinite number of commodities is formulated through four ingredients: the set of arbitrage, the coalitional representation, the space of allocations and the (weak) Lyapunov Convexity Theorem.
Special thanks are due to T. Armstrong for correcting an error in definitions; to A. Khan for helping me in the literature; to N. Yannelis for showing me his unpublished work; and to W. Zame and J. Ostroy for fruitful discussions I had with them. Thanks are also due to the referee for correcting many errors. All remaining errors are my responsibility.
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
Aliprantis, C. D., Brown, D. J., and Burkinshaw, 0., 1985, “Edgeworth Equilibria,” Cowles Foundation Discussion paper.
Aliprantis, C. D., Brown, D. J. and Burkinshaw, 0., 1986, “Edgeworth Equilibria in Production Economies,” Cowles Foundation Discussion paper.
Anderson, R. 1978, “An Elementary Core Equivalence Theorem,” Econometrica 46 1483–1487.
Armstrong, T. and Richter, M. K., 1984, “The Core-Walras Equivalence,” J. Econ. Theory 33, 116–151.
Armstrong, T. and Richter, M. K., 1986, “Existence of Non-atomic Core-Walras Allocations,” J. Econ. Theory 38, 137–159.
Aumann, R., 1964, “Markets with a Continuum of Traders,” Econometrica 32, 39–50.
Bewley, T., 1972, “Existence of Equilibria in Economies with Infinitely Many Commodities,” J. Econ. Theory 4, 514–540.
Bewley, T., 1973, “The Equality of the Core and the Set of Equilibria in Economies with Infinitely Many Commodities and Continuum of Agents,” Int. Econ. Rev. 14, 383–393.
Brown, D. J. and Khan, M. A., 1980, “An Extension of the Brown-Robinson Equivalence Theorem,” Appl. Math. Comp. 6, 167–175.
Brown, D. J. and Robinson, A., 1975, “Nonstandard Exchange Economies,” Econometrica 43, 41–55.
Champsaur, P. and Laroque, G., 1971, “Notes Sur Les Mechanismes de Blocage,” unpublished manuscript, Paris.
Cheng, Harrison H. C., 1986, “The Coalitional Approach to the Core Theory,” J. Math. Econ.,forthcoming.
Cheng, Harrison H. C., 1987a, “Arbitrage Theory and Equilibrium: The Case of Unconstrained Consumption,” MRG Working Paper M8719, Department of Economics, University of Southern California.
Cheng, Harrison H. C., 1987b, “The Existence of Arbitrage-Free Equilibria in Banach Spaces,” MRG Working Paper M8712, Department of Economics, University of Southern California.
Cheng, Harrison H. C., 1987c, “Asset Market Equilibrium in Infinite Dimensional Economies,” J. Math. Econ., to appear.
Cheng, Harrison H. C., 1987d, “Dynamic Hedging, Arbitrage Pricing, and the Efficient Market Hypothesis: An Axiomatic Approach,” Department of Economics, University of Southern California.
Debreu, G., 1962, “New Concepts and Techniques for Equilibrium Analysis,” Int. Econ. Rev. 3, 257–273.
Debreu, G., 1967, “Preference Functions on Measure Spaces of Economic Agents,” Econometrica 35, 111–122.
Debreu, G., 1986, “Theoretic Models: Mathematical Form and Economic Content,” Econometrica 54, 1259–1270.
Diestel, J. and Uhl, J. J., 1977, Vector Measures, Amer. Math. Soc., Math. Surveys 15, Providence, Rhode Island.
Duffle, D. and Huang, C. F., 1985, “Implementing Arrow-Debreu Equilibria by Continuous Trading of Few Long-lived Securities,” Econometrica 53, 1337–1357.
Edgeworth, F., 881, Mathematical Psychics,Kegan Paul, London.
Gabszewicz, J., 1968, “Coeurs et Allocations Concurrentielles dans des Economies d’Echange avec un Continu de Biens,” Librarie Universitair, Louvain.
Gretsky, N. and Ostroy, J., 1985, “Thick and Thin Market Non-Atomic Exchange Economies” in Advances in Equilibrium Theory, C. D. Aliprantis, O. Burkinshaw and N. J. Rothman, eds., Springer-Verlag, New York.
Grodal, B., 1972, “A Second Remark on the Core of an Atomless Economy,” Econometrica 40, 581–583.
Hansen, T., 1969, “A Note on the Limit of the Core of an Exchange Economy,” Int. Econ. Rev. 10, 479–483.
Harrison, J. M. and Kreps, D., 1979, “Martingales and Arbitrage in Multi-period Securities Markets,” J. Econ. Theory 20, 381.-408.
Hart, O., 1974, “On the Existence of Equilibrium in a Securities Model,” J. Econ. Theory 9, 293–311.
Hildenbrand, W., 1968, “The Core of an Economy with a Measure Space of Economic Agents,” Rev. Econ. Stud. 35, 443–452.
Holmes, R., 1975, Geometric Functional Analysis and Its Applications, Springer-Verlag, New York.
Jones, L., 1983, “Existence of Equilibria with Infinitely Many Consumers and Infinitely Many Commodities,” J. Math. Econ. 12, 119–138.
Jones, L., 1984, “A Competitive Model of Product Differentiation,” Econometrica 52, 507–530.
Khan, M. Ali, 1974a, “Some Remarks on the Core of a ‘Large Economy’,” Econometrica 42, 633–642.
Khan, M. Ali, 1974b, “Some Equivalence Theorems,” Rev. Econ. Stud. 41, 549–565.
Kingman, J. and Robertson, A., 1968, “On a Theorem of Lyapunov,” J. London Math. Soc. 43 347–351.
Kluvânek, I. 1973, “The Range of a Vector Valued Measure,” Math. Sys. Theory 7 44–54.
Knowles, G., 1974, “Lyapunov Vector Measures,” SIAM J. Control 13, 294–303.
Kreps, D., 1981, “Arbitrage and Equilibrium in Economies with Infinitely Many Commodities,” J. Math. Econ. 8, 15–36.
Lindenstrauss, L., 1966, “A Short Proof of Lyapunov’s Convexity Theorem,” J. Math. Mech. 15, 971–972.
Lyapunov, A. A., 1940, “ Sur les Fonctions-Vecteurs Complèment Additives,” Izvestija Akademii Nauk SSSR, 465–478.
Mas-Colell, A., 1975, “A Model of Equilibrium with Differentiated Commodities,” J. Math. Econ. 2, 263–295.
Mas-Colell, A., 1978, “A Note on the Core Equivalence Theorem: How Many Blocking Coalitions Are There?” J. Math. Econ. 5, 207–215.
Mas-Colell, A., 1982, “Perfect Competition and the Core,” Rev. Econ. Stud. 49, 15–30.
Mas-Colell, A., 1986, “The Price Equilibrium Existence Problem in Topological Vector Lattices,” Econometrica 54, 1039–1054.
Ostroy, J., 1984, “The Existence of Walrasian Equilibrium in Large-Square Economies,” J. Math. Econ. 13, 143–163.
Peleg, B. and Yaari, M., 1970, “Markets with Countably Many Commodities,” Int. Econ. Rev. 11, 369–377.
Radner, R., 1972, “Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets,” Econometrica 40, 289–303.
Rashid, S., 1979, “The Relationship Between Measure-Theoretic and Non-Standard Exchange Economies,” J. Math. Econ. 6, 195–202.
Richter, M. K., 1971, “Coalitions, Core and Competition,” J. Econ. Theory 3, 323–334.
Rustichini, A. and Yannelis, N. C., 1991, “Edgeworth’s Conjecture in Economies with a Continuum of Agents and Commodities,” J. Math. Econ., to appear.
Scarf, H. 1962, “Analysis of Markets with a Large Number of Participants,” Recent Advances in Game Theory,Princeton University Press, Princeton, NJ.
Schaefer, H. 1980, Topological Vector Spaces,Springer-Verlag, New York. Schmeidler, D., 1972, “A Remark on the Core of an Atomless Economy,” Econometrica 40 579–580.
Uhl, J., 1969, “The Range of a Vector-Valued Measure,” Proc. Amer. Math. Soc. 23, 158–163.
Vind, K., 1964, “Edgeworth Allocations in an Exchange Economy with Many Traders,” Int. Econ. Rev. 5, 165–177.
Vind, K., 1973, “ A Third Remark on the Core of an Atomless Economy,” Econometrica 40, 585–586.
Yannelis, N. C. and Zame, W. R., 1986, “Equilibria in Banach Lattices without Ordered Preferences,” J. Math. Econ. 15, 85–110.
Zame, W., 1986, “Markets with a Continuum of Traders and Infinitely Many Commodities,” Department of Mathematics, SUNY at Buffalo.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Cheng, H.H.C. (1991). The Principle of Equivalence. 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_8
Download citation
DOI: https://doi.org/10.1007/978-3-662-07071-0_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08114-9
Online ISBN: 978-3-662-07071-0
eBook Packages: Springer Book Archive