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Continuity properties of the private core

Authors:

Institution:

(1) Mathematical Institute, University of Utrecht, P.O. Box 80.010, 3508, TA, Utrecht, Netherlands;(2) Department of Economics, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA

Abstract:

Let $({\mathcal E}_k)_{k \in {\mathbb N}}$ be a sequence of differential information economies, converging to a limit differential information economy ${\mathcal E}_{\infty}$ (written as ${\mathcal E}_k \Rightarrow {\mathcal E}_{\infty}$ ). Denote by $C_{\epsilon}({\mathcal E}_k)$ the set of all ε-private core allocations, ε ≥ 0 (for ε=0 we get the private core of Yannelis (1991), denoted by $C({\mathcal E}_k)$ ). Under appropriate conditions, we prove the following stability results

(1)	(upper semicontinuity): if ${\mathcal E}_k \Rightarrow {\mathcal E}_{\infty}$ , $f_k \in C({\mathcal E}_k)$ , and if f _k → f _∞ L ¹-weakly, then $f_{\infty} \in C({\mathcal E}_{\infty})$ .
(2)	(lower semicontinuity): if ${\mathcal E}_k \Rightarrow {\mathcal E}_{\infty}$ , $f_{\infty} \in C({\mathcal E}_{\infty})$ , ε > 0, then there exist $f_k \in C_{\epsilon}({\mathcal E}_k)$ , with f _k → f _∞ L ¹-weakly.

JEL Classification Numbers D82, D50, D83, C62, C71, D46, D61Most of this work was done in Spring 2001, when Balder held a visiting professorship at the University of Illinois. Presentations based on this paper were given by Balder at the Midwestern Theory Conference in Madison, Wisconsin (May, 2001) and at the SAET Conference in Ischia, Italy (June, 2001).

Keywords:

Private core Upper semicontinuity Lower semicontinuity Weak L ¹-convergence Martingales

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