Diversified Consumption Characteristics and Conditionally Dispersed Endowment Distribution: Regularizing Effect and Existence of Equilibria

Abstract

The paper introduces the concept of a conditionally endowment distribution (given the consumption set) in the study of regularizing effect of aggregation and existence of equilibria in an exchange economy with an atomless measure space of consumers. It is shown that there is a regularizing effect of aggregation whenever the endowment distribution of an economy is conditionally dispersed. On the basis of this regularizing effect, the existence of a competitive equilibrium is proved in a very general setting where some of the commodities are indivisible and where different consumers may have different consumption sets. GENERAL EQUILIBRIUM THEORY saw, in recent years, explicit formulation of models with indivisible commodities in which existence of competitive equilibria was discussed. (References are Henry [12], E. Dierker [10], Broome [7], and Mas-Colell [19].) A pathbreaking paper by Mas-Colell [19] clearly reconfirmed the power of a continuum of traders model introduced by Aumann [1], and it emphasized, for the first time in the literature of the equilibrium existence problem, the importance of a condition relating to the form of distribution of agents' characteristics. More specifically, a competitive equilibrium was shown to exist in an economy with indivisible commodities where a single perfectly divisible commodity exists and the distribution of its initial holdings among agents is dispersed in the sense that it is absolutely continuous with respect to the Lebesgue measure. (See [19, Theorem 1, p. 446].) This development in the existence literature is of much interest to equilibrium theory and coincides with the way in which the analysis of a certain aggregation problem is being pursued in equilibrium theory. (See, e.g., Sonderman [33], Mas-Colell [20], and Hildenbrand [18].) In both of these problems, elaboration of the concept of dispersedness in the distribution of agents' characteristics is required.3 The work of Mas-Colell [19] was followed by [35] in which the concept of a endowment distribution was introduced in the study of existence of equilibria in an exchange economy with an atomless measure space of consumers where no convexity assumptions on preferences and consumption sets were