Hildenbrand Distribution Economies as Limiting Empirical Distributions of Random Economies
The present note shows that the concept of a distribution economy (Hildenbrand (1974)) is closely related to a framework of an exchange economy in which the agents’ individual characteristics (i.e. preferences and endowments) are random (Hildenbrand (1971), Bhattacharya and Majumdar (1973), Föllmer (1974)). A random exchange economy is fully specified by the distribution of the family of random variables representing the agents’ individual characteristics. This distribution is a probability measure µ on (SA, B(S)A) with S denoting the space of individual characteristics, B(S) the Borel o-algebra generated by an appropriate topology on S and A a denumerable set of agents. The linkage between a Hildenbrand distribution economy and an ergodic random exchange economy with a countably infinite set of agents endowed with the graph topology of the integer lattice Zd is established in this paper by a convergence result for the empirical distribution of the latter. For any increasing sequence of finite subsets of A exhausting A, the associated sequence of empirical distributions converges almost everywhere on the underlying probability space to some distribution v on (S, B(S)). As far as aggregate variables of the economy, such as the mean demand or the equilibrium price system are concerned, any infinite random exchange economy with converging limiting empirical distribution v is equivalent to a Hildenbrand distribution economy characterized by the same distribution v. This relationship suggests an approach to endogenous modelling of distributions of individual characteristics in General Equilibrium Theory. Thereby, specific distributions of characteristics can be obtained from a specific stochastic microstructure of local interaction between agents affecting their individual characteristics.
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