Econometrica: May 1986, Volume 54, Issue 3

Neighborhood Systems for Production Sets with Indivisibilities

DOI: 0012-9682(198605)54:3<507:NSFPSW>2.0.CO;2-K
p. 507-532

Herbert E. Scarf

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.

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