Graph Efficiency Measurement

Abstract

Introduction Chapters 3 and 4 of this monograph introduce and decompose cost and revenue efficiency, respectively. In this chapter we are interested in profit efficiency, which requires that we simultaneously adjust input and output quantities, given input and output prices. Since we simultaneously wish to increase output quantities and associated revenues and decrease input quantities and their associated costs, we require a different reference technology and perhaps a different means of attaining the frontier of that technology than in Chapters 3 and 4. In order to judge adjustments of both input and output quantities simultaneously, we model technology with the graph. Specifically, we introduce a series of efficiency measures which judge performance relative to technology as described by the graph. In contrast to most of the efficiency measures introduced in earlier chapters, those introduced here are not radial contractions or expansions of observed data, but rather follow a hyperbolic path to the frontier of technology. More specifically, both inputs and outputs are allowed to vary by the same proportion, but inputs are proportionately decreased while outputs are simultaneously increased at the same proportion. The fact that we require increases in output quantities and decreases in input quantities to occur at the same proportion or rate, yields a hyperbolic path to the frontier of the graph. One can imagine alternatives: a Russell type measure which allows variation in the rate of increase of individual outputs and in the rate of decrease of individual inputs, or a measure which increases outputs at a proportionate rate which may differ, however, from the proportionate rate of decrease of inputs. These variations are left to the interested reader.