Aggregation and the Law of Large Numbers in Large Economies

This paper introduces a new model of environments with a large number of agents and stochastic characteristics. We consider sequences of finite but increasingly large economies that `discretize' the continuum. In the limit we obtain a model that is continuum-like in important respects, yet it has a countable set of agents with a finitely additive, `uniform' distribution. In this model, the law of large numbers is meaningful and holds on all subintervals. This framework provides, among other things, a new interpretation of the measurability problem and the failure of the law of large numbers in the continuum. It is also shown that the Pettis integral in the continuum coincides with the empirical frequencies in the discrete model almost surely. Finally, the model is used to study a mechanism design problem in a large economy with private information.