Decomposition and aggregation of tone efficiencies

In this paper we investigate the decomposition and aggregation of cost efficiencies when input prices vary across decision-making units. In particular, we compare and contrast four alternative decompositions, three of which are compatible with non-homogeneous (in terms of quality) inputs. We also examine the relations among the alternative cost efficiency measures and their components. Our theoretical results indicate that, when input prices vary across decision-making units, the Färe and Grosskopf cost-based efficiency measure is equal to (i) the Tone cost efficiency measure; (ii) the Tone and Tsutsui cost efficiency measure if there is no technical inefficiency; and (iii) the input spending efficiency measure if there is no allocative inefficiency in the sense of Tone. In addition, based on the denominator rule, we derive appropriate weights for aggregating consistently these cost efficiencies and their components across decision-making units. All these provide helpful guidelines for the appropriate applications of these decompositions at both the individual and the aggregate level. We illustrate this by a simple study case for hospitals.