Monte Carlo simulation of macroeconomic risk with a continuum of agents: the symmetric case

Authors

Peter J. Hammond

Department of Economics, Stanford University, Stanford, CA 94305-6072, USA (e-mail: peter.hammond@stanford.edu)

Yeneng Sun

Institute for Mathematical Sciences, National University of Singapore, 3 Prince George's Park, Singapore 118402, REPUBLIC OF SINGAPORE, and Department of Mathematics and the Centre for Financial Engineering, National University of Singapore, Singapore, REPUBLIC OF SINGAPORE (e-mail: matsuny@nus.edu.sg)

Symposium Article

DOI:
10.1007/s00199-002-0302-y

Cite this article as:

Hammond, P. & Sun, Y. Econ Theory (2003) 21: 743. doi:10.1007/s00199-002-0302-y

Suppose a large economy with individual risk is modeled by a continuum of pairwise exchangeable random variables (i.i.d., in particular). Then the relevant stochastic process is jointly measurable only in degenerate cases. Yet in Monte Carlo simulation, the average of a large finite draw of the random variables converges almost surely. Several necessary and sufficient conditions for such “Monte Carlo convergence” are given. Also, conditioned on the associated Monte Carlo \( \sigma \)-algebra, which represents macroeconomic risk, individual agents' random shocks are independent. Furthermore, a converse to one version of the classical law of large numbers is proved.

Keywords and Phrases: Large economy, Continuum of agents, Law of large numbers, Exchangeability, Joint measurability problem, de Finetti's theorem, Monte Carlo convergence, Monte Carlo $\sigma$-algebra.