A Note on the Speed of Convergence of Prices in Random Exchange Economies

This paper is a continuation of work by Hildenbrand (1971) and Bhattacharya and Majumdar (1973) (henceforth B - M). They consider pure exchange economies in which both preferences and endowments are random. Hildenbrand examines the convergence behaviour of prioce vectors for which total expected excess demand is zero. He shows that as an economy increases in size, if agents are stochastically independent, then the limit of such a sequence of price vectors is an equilibrium price vector in a suitablt defined limit economy. B - M considers the case where prices guarantee equilibrium in almost all states of the world. In other words, market-clearing prices are treated as rendom vectors. In particular, B - M show that under suitable assumptions there will exist a sequence of such random price vectors displaying almost sure convergence to any equilibrium price vecror in a deterministic limit economy. As in the case of any convergence result, the speed of convergence is a natural question to investigate and we will be concerned in this note with establishing a result on the speed of convergence of the random price vectors as the economy increases in size. In order to provide a characterization of the result in a simple case, let us suppose that the deterministic limit economy has a unique equilibrium Po. Assume also that the random economy consists of individuals with the same independently distributed random preferences and endowments. Then our result states that the probability that the random equilibrium price vector is further from Po than some distance which converges to zero slower than N-½ (where N is the number of agents in the random economy) is less than a term which converges to zero faster than N-½. In the more general case we consider both distance and probability will depend upon the proportions of different types of agent in the econopmy. In addition, the distance will also depend upon the rate at which the proportions of agents of different types approach their limiting values.