What is Perfect Competition?

  • Aldo Rustichini
  • Nicholas C. Yannelis
Part of the Studies in Economic Theory book series (ECON.THEORY, volume 1)


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.


Measure Space Competitive Equilibrium Borel Subset Separable Banach Space Convexity Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aumann, R. J., 1964, “Markets with a Continuum of Traders,” Econometrica 32, 39–50.Google Scholar
  2. Aumann, R. J., 1965, “Integrals of Set-Valued Functions,” J. Math. Anal. Appl. 12, 1–12.CrossRefGoogle Scholar
  3. Aumann, R. J., 1966, “Existence of a Competitive Equilibrium in Markets with a Continuum of Traders,” Econometrica 34, 1–17.CrossRefGoogle Scholar
  4. Bewley, T., 1990, “A Very Weak Theorem on the Existence of Equilibria in Atomless Economies with Infinitely Many Commodities,” this volume.Google Scholar
  5. Castaing, C. and Valadier, M., 1977, “Convex Analysis and Measurable Mul-tifunctions,” Lect. Notes Math. 580, Springer-Verlag, New York.Google Scholar
  6. Debreu, G., 1967, “Integration of Correspondences,” Proc. Fifth Berkeley Symp. Math. Stat. Prob., University of California Press, Berkeley, Vol. II, Part I, 351–372.Google Scholar
  7. Diestel, J. and Uhl, J., 1977, Vector Measures,Mathematical Surveys, No. 15, American Mathematical Society, Providence, Rhode Island.Google Scholar
  8. Dunford, N. and Schwartz, J. T., 1958, Linear Operators, Part I, Interscience, New York.Google Scholar
  9. Gretsky, N. E. and Ostroy, J. 1985, “Thick and Thin Market Nonatomic Exchange Economies,” in Advances in Equilibrium Theory,C. D. Aliprantis, Springer-Verlag.Google Scholar
  10. Halmos, P. R. and Von Neumann, J. 1941, “Operator Methods in Classical Mechanics, II,” Ann. Math. 43:2 Google Scholar
  11. Himmelberg, C. J. 1975, “Measurable Relations,” Fund. Mali. 87 53–72.Google Scholar
  12. Ionescu-Tulcea, A. and C., 1969, Topics in the Theory of Lifting, Springer-Verlag, Berlin.CrossRefGoogle Scholar
  13. Khan, M. Ali, 1976, “On the Integration of Set-Valued Mappings in a Nonreflexive Banach Space II,” Simon Stevin 59, 257–267.Google Scholar
  14. Khan, M. Ali, 1986, “Equilibrium Points of Nonatomic Games over a Banach Space,” Trans. Amer. Math. Soc. 293:2, 737–749.Google Scholar
  15. Khan, M. Ali and Yanelis, N. C., 1990, “Existence of a Competitive Equilibrium in Markets with a Continuum of Agents and Commodities,” this volume.Google Scholar
  16. Kluvanek, I. and Knowles, G., 1975, “Vector Measures and Control System,” Math. Stud. 20, North Holland.Google Scholar
  17. Knowles, G., 1974, Liapunov Vector Measures, SIAM J. Control13, 294–303.Google Scholar
  18. Kuratowski, K. and Ryll-Nardzewski, C., 1962, “A General Theorem on Selectors,” Bull. Acad. Polon. Sci. Ser. Sci. Marsh. Astronom. Phys. 13, 397–403.Google Scholar
  19. Lewis, L., 1977, Ph.D. Thesis, Yale University.Google Scholar
  20. Loeb, P., 1971, “A Combinatorial Analog of Lyapinov’s Theorem for Infinitesimal Generated Atomic Vector Measures,” Proc. Amer. Math. Soc. 39, 585–586.Google Scholar
  21. Lindenstrauss, J., 1966, “A Short Proof of Lyapunov’s Convexity Theorem,” J. Math. Mech. 15, 971–972.Google Scholar
  22. Maharam, D., 1942, “On Homogeneous Measure Algebras,” Proc. Natl. Acad. Sci. 28, 108–111.CrossRefGoogle Scholar
  23. Masani, P., 1978, “Measurability and Pettis Integration in Hilbert Spaces,” J. Reine Angew. Math. 297, 92–135.Google Scholar
  24. Mas-Colell, A., 1975, “A Model of Equilibrium with Differentiated Commodities,” J. Math. Econ. 2, 263–295.CrossRefGoogle Scholar
  25. Mertens, J.-F., 1990, “An Equivalence Theorem for the Core of an Economy with Commodity Space L τ(L , L 1),” this volume.Google Scholar
  26. Ostroy, J. and Zame, W. R., 1988, “Non-Atomic Exchange Economies and the Boundaries of Perfect Competition,” mimeo.Google Scholar
  27. Robertson, A. P. and Kingman, J. F. C., 1968, “On a Theorem of Lyapunov,” J. London Math. Soc. 43, 347–351.CrossRefGoogle Scholar
  28. Rustichini, A., 1989, “A Counterexample and an Exact Version of Fatou’s Lemma in Infinite Dimensions,” Archiv der Mathematic, 52, 357–362.CrossRefGoogle Scholar
  29. Rustichini, A. and Yannelis, N. C., 1989, “Commodity Pair Desirability and the Core Equivalence Theorem,” mimeo.Google Scholar
  30. Rustichini, A. and Yannelis, N. C., 1991, “Edgeworth’s Conjecture in Economies with a Continuum of Agents and Commodities,” J. Math. Econ., to appear.Google Scholar
  31. Schmeidler, D., 1973, “Equilibrium Points of Non Atomic Games, J. Stat. Phys. 7:4 295–300.Google Scholar
  32. Von Neumann, J., 1950, “Functional Operators,” Ann. Math. Stud. 22, Princeton University Press.Google Scholar
  33. Yannelis, N. C., 1988, “Fatou’s Lemma In Infinite Dimensional Spaces,” Proc. Amer. Math. Soc. 102, 303–310.Google Scholar
  34. Yannelis, N. C., 1990, “Integration of Banach-Valued Correspondences,” this volume.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Aldo Rustichini
  • Nicholas C. Yannelis

There are no affiliations available

Personalised recommendations