Advances in Mathematical Economics Volume 20 by Shigeo Kusuoka & Toru Maruyama

Author:Shigeo Kusuoka & Toru Maruyama
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

Note that X is a complete separable metric space. From now on, we will denote (homogeneous goods) and for .

We claim that D is a closed subset of in the weak*-topology. To see this, suppose that , . Since is a metric space, we can discuss the (weak*) topology on D by the sequences rather than the nets. Since is a Cauchy sequence, for 0 < ε < 1 there exists an N such that

for all n ≥ 1, where for all t ∈ K. Hence for all n ≥ 1. Therefore for all n ≥ 1, where # S is the cardinality of a set S. Consequently we have , or has a finite support, . We can take pairwise disjoint neighborhood U j of t j and write . Since , it is then easy to show that for , m j n  = m j , k = k n for n large enough, and t j n  → t j , . Therefore .

Set . Then we can write and let , where means that for some ε > 0 and for all t ∈ K. In this paper, a price vector for a commodity bundle is a nonzero element of . Then for and , we denote .

As usual, a preference relation is a complete, transitive and reflexive binary relation on X, and we denote by . Mas-Colell [49] made the following assumptions on the preferences and we will keep them. Let , where 1 is at the i’th place. We denote the usual topology on by .

Assumption (PR)

(i) is closed in X × X in the (product of) -topology,

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