Skip to main content

A law of large numbers for large economies

Summary

LetX(i),iε[0; 1] be a collection of identically distributed and pairwise uncorrelated random variables with common finite meanμ and variance σ2. This paper shows the law of large numbers, i.e. the fact that ∝ 10 X(i)di=μ. It does so by interpreting the integral as a Pettis-integral. Studying Riemann sums, the paper first provides a simple proof involving no more than the calculation of variances, and demonstrates, that the measurability problem pointed out by Judd (1985) is avoided by requiring convergence in mean square rather than convergence almost everywhere. We raise the issue of when a random continuum economy is a good abstraction for a large finite economy and give an example in which it is not.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Al-Najjar, N. I.: Decomposition and characterization of risk with a continuum of random variables. Econometrica63, 1195–1224 (1995)

    Google Scholar 

  2. 2.

    Bewley, T.: Appendix to Stationary monetary equilibrium with a continuum of independently fluctuating consumers. In: Hildenbrand, Mas-Colell (eds.) Contributions to mathematical economics in Honor of G. Debreu, Amsterdam: North-Holland, 1986

    Google Scholar 

  3. 3.

    Diestel, J., Uhl, J. J. Jr.: Vector measures. Mathematical surveys # 15. Providence, R. I.: American Mathematical Society, 1977

    Google Scholar 

  4. 4.

    Diamond, D. W., Dybvig, P. H.: Bandk runs, deposit insurance and liquidity. Journal of Political Economy91, 401–419 (1983)

    Google Scholar 

  5. 5.

    Green, E.: Lending and the smoothing of uninsurable income. In: E. C. Prescott, and N. Wallace (eds.) Contractual arrangements for intertemporal trade. Minnesota studies in macroeconomis, Vol. 1, pp. 3–25. Minneapolis: University of Minnesota Press, 1987

    Google Scholar 

  6. 6.

    Hildenbrand, W.: Core and equilibria of a large economy. Princeton: Princeton University Press, 1974

    Google Scholar 

  7. 7.

    Judd, K. L.: The law of large numbers with a continuum of IID random variables. Journal of Economic Theory35, 19–25 (1985)

    Google Scholar 

  8. 8.

    Lucas, R. E.: Equilibrium in a pure currency economy. Economic Inquiry18, 203–220 (1980)

    Google Scholar 

  9. 9.

    Mas-Colell, A.: The theory of general equilibrium: A differentiable approach. Cambridge: Cambridge University Press, 1985

    Google Scholar 

  10. 10.

    Mas-Colell, A., Vives, X.: Implementation in economies with a continuum of agents. Review of Economic Studies60, 613–629 (1993)

    Google Scholar 

  11. 11.

    Prescott, E. C., Townsend, R. M.: Pareto optima and competitive equilibria with adverse selection and moral hazard. Econometrica52, 21–45 (1984)

    Google Scholar 

  12. 12.

    Pettis, B. J.: On integration in vector spaces. Trans. Amer. Math. Soc.44, 277–304 (1938)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

I am indebted to Hugo Hopenhayn. Furthermore I would like to thank Dilip Abreu, Glenn Donaldson, Ed Green, Ramon Marimon, Nabil Al-Najjar, Victor Rios-Rull, Timothy van Zandt and the editor for useful comments. The first version of this paper was written in 1987.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Uhlig, H. A law of large numbers for large economies. Econ Theory 8, 41–50 (1996). https://doi.org/10.1007/BF01212011

Download citation

Keywords

  • Economic Theory
  • Simple Proof
  • Measurability Problem
  • Large Economy
  • Uncorrelated Random Variable