Individual Risk and Lebesgue Extension without Aggregate Uncertainty

Sun, Yeneng and Zhang, Yongchao (2008): Individual Risk and Lebesgue Extension without Aggregate Uncertainty.

Abstract

Many economic models include random shocks imposed on a large number (continuum) of economic agents with individual risk. In this context, an exact law of large numbers and its converse is presented in Sun (2006) to characterize the cancelation of individual risk via aggregation. However, it is well known that the Lebesgue unit interval is not suitable for modeling a continuum of agents in the particular setting. The purpose of this note is to show that an extension of the Lebesgue unit interval does work well as an agent space with various desirable properties associated with individual risk.

• Item Type: MPRA Paper
• Language: English
• Keywords: No aggregate uncertainty; independence; exact law of large numbers; Fubini extension; Lebesgue measure
• Subjects: C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C43 - Index Numbers and Aggregation; E - Macroeconomics and Monetary Economics > E0 - General > E00 - General; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D80 - General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General

References

C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, Springer, Berlin, 1994.

R.M. Anderson, Non-standard analysis with applications to economics, in Handbook of Mathematical Economics, Volume IV (W. Hildenbrand and H. Sonnenschein Eds.), North-Holland, New York, 1991.

R.J. Aumann, Markets with a continuum of traders, Econometrica 32(1964), 39-50.

J.L. Doob, Stochastic processes depending on a continuous parameter, Transactions of the American Mathematical Society 42(1937), 107-140.

J.L. Doob, Stochastic Processes, Wiley, New York, 1953.

D. Duffie, N. Garleanu and L.H. Pedersen, Over-the-counter markets, Econometrica 73(2005), 1815-1847.

D. Duffie, N. Garleanu and L.H. Pedersen, Valuation in over-the-counter markets, Review of Financial Studies 20 (2007), 1865-1900.

R. Durrett, Probability: Theory and Examples, 3rd edition, Brooks/Cole, Belmont, 2005.

M. Feldman and C. Gilles, An expository note on individual risk without aggregate uncertainty, Journal of Economic Theory 35 (1985), 26-32.

D.H. Fremlin, Measure Theory, Volume 4: Topological Measure Spaces, Torres Fremlin, Colchester, 2003.

D.H. Fremlin, Measure Theory, Volume 5: Set-theoretic Measure Theory (version 8.5.03/29.6.06)}, 2005. Available at http://www.essex.ac.uk/maths/staff/fremlin/mt.htm

E.J. Green, Individual-level randomness in a nonatomic population, University of Minnesota, mimeo., 1994.

P.J. Hammond and Y.N. Sun, Joint measurability and the one-way fubini property for a continuum of independent random variables, Proceedings of American Mathematical Society 134 (2006), 737-747.

K.L. Judd, The law of large numbers with a continuum of i.i.d. random variables, Journal of Economic Theory 35 (1985), 19-25.

H.J. Keisler and Y.N. Sun, Loeb measures and Borel algebras, in Reuniting the Antipodes - Constructive and Nonstandard Views of the Continuum (Ulrich Berger, Horst Osswald and Peter Schuster eds.), Kluwer Academic Publishers, Dordrecht, 2001, pp. 111-117.

M.A. Khan, Perfect competition, to appear in The New Palgrave, mimeo., Johns Hopkins University, 2007.

R. Lagos and G. Rocheteau, Liquidity in asset markets with search frictions, NYU Working Paper, 2007.

P.A. Loeb and M. Wolff, eds. Nonstandard Analysis for the Working Mathematician, Kluwer Academic Publishers, Dordrecht, 2000.

R. McLean and A. Postlewaite, Informational size and incentive compatibility, Econometrica 70(2002), 2421-2453.

R. McLean and A. Postlewaite, Core convergence with asymmetric information, Games and Economic Bahavior 50 (2005), 58-78.

K. Podczeck, On the convexity and compactness of the integral of a Banach space valued correspondence, Journal of Mathematical Economics, published online.

H.L. Royden, Real Analysis, Macmillan, New York, 1968.

Y.N. Sun, The exact law of large numbers via Fubini extension and characterization of insurable risks, Journal of Economic Theory 126 (2006), 31-69.

Y.N. Sun and N.C. Yannelis, Core, equilibria and incentives in large asymmetric information economies, Games and Economic Behavior 61 (2007), 131-155.

P.-O. Weill, Leaning against the wind, Review of Economic Studies 74 (2007), 1329-1354.