Neighborhood Systems for Production Sets with Indivisibilities

Abstract

A production set with indivisibilities is described by an activity analysis matrix with activity levels which can assume arbitrary integral values. A neighborhood system is an association with each integral vector of activity levels of a finite set of neighboring vectors. The neighborhood relation is assumed to be symmetric and translation invariant. Each such neighborhood system can be used to define a local maximum for the associated integer programs obtained by selecting a single commodity whose level is to be maximized subject to specified factor endowments of the remaining commodities. It is shown that each technology matrix (subject to mild regularity assumptions) has a unique, minimal neighborhood system for which a local maximum is global. The complexity of such minimal neighborhood systems is examined for several examples.

This paper, which is based on the Presidential Address of the Econometric Society delivered at Northwestern University and at Pisa in 1983, was supported by a grant from the National Science Foundation. I am very much indebted to Andrew Caplin, Philip White, and Ludo Van der Heyden for many stimulating conversations on the subject of this paper, and to one of the referees of the paper for his insightful comments.