A law of large numbers for large economies
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 ∝ 0 1 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.
KeywordsEconomic Theory Simple Proof Measurability Problem Large Economy Uncorrelated Random Variable
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- 1.Al-Najjar, N. I.: Decomposition and characterization of risk with a continuum of random variables. Econometrica63, 1195–1224 (1995)Google Scholar
- 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, 1986Google Scholar
- 3.Diestel, J., Uhl, J. J. Jr.: Vector measures. Mathematical surveys # 15. Providence, R. I.: American Mathematical Society, 1977Google Scholar
- 4.Diamond, D. W., Dybvig, P. H.: Bandk runs, deposit insurance and liquidity. Journal of Political Economy91, 401–419 (1983)Google Scholar
- 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, 1987Google Scholar
- 6.Hildenbrand, W.: Core and equilibria of a large economy. Princeton: Princeton University Press, 1974Google Scholar
- 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.Lucas, R. E.: Equilibrium in a pure currency economy. Economic Inquiry18, 203–220 (1980)Google Scholar
- 9.Mas-Colell, A.: The theory of general equilibrium: A differentiable approach. Cambridge: Cambridge University Press, 1985Google Scholar
- 10.Mas-Colell, A., Vives, X.: Implementation in economies with a continuum of agents. Review of Economic Studies60, 613–629 (1993)Google Scholar
- 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.Pettis, B. J.: On integration in vector spaces. Trans. Amer. Math. Soc.44, 277–304 (1938)Google Scholar