Closed cone extensions of convex and nonconvex production technologies

Cone extensions of production technologies and their lower and upper cone variants are often referred to as the constant, nonincreasing and nondecreasing returns-to-scale technologies and are used for the evaluation of different scale characteristics of production frontiers. These include the scale efficiency, returns to scale and most productive scale size of decision making units. It may not be widely known that such cone extensions of production technologies, including the cone extension of the standard variable returns-to-scale technology, are usually not closed sets. Theoretically, for the correct evaluation of scale characteristics, we need to perform computations in cone extensions of the technology that are generally not closed and whose operational statements are not known. Instead, it is common to use the closures of cone technologies without acknowledging that this is essentially a heuristic approach which is assumed valid without a proof. In this paper, we address two goals. First, for a very large class of production technologies, which includes all polyhedral technologies and their finite unions, i.e., most of technologies of data envelopment analysis, we establish equivalence between the evaluation of scale characteristics using cone extensions and their closures, therefore theoretically validating the existing heuristic approaches. Second, we establish new relationships between the production technology and its closed cone extensions which lead to a simplification of the known methods of assessing returns to scale.