Technological Inefficiency Indexes: A Binary Taxonomy and a Generic Theorem

Over the years, a large number of indexes of technological inefficiency have been specified, and a spate of papers has examined the properties satisfied by these indexes. This paper approaches the subject more synthetically, presenting generic results on classes of indexes and their properties. In part- icular, we consider a broad class of indexes containing almost all known indexes and a partition of this class into two subsets, slacks-based indexes and path-based indexes. Slacks-based indexes are expressed in terms of addi- tive or multiplicative slacks for all inputs and outputs, and particular indexes are generated by specifying the form of aggregation over the coordinate-wise slacks. Path-based indexes are expressed in terms of a common contrac- tion/expansion factor, and particular indexes are generated by specifying the form of the path to the frontier of the technology. Owing to an impossibility result in one of our earlier papers, we know that the set of all inefficiency indexes can be partitioned into three subsets: those that satisfy continuity (in quantities and technologies) and violate indication (equal to some spec- ified value if and only if the quantity vector is efficient), those that satisfy indication and violate continuity, and those that satisfy neither. We prove two generic theorems establishing the equivalence of these two partitions: all slacks-based indexes satisfy indication and hence violate continuity, and all path-based indexes satisfy continuity and hence violate indication. We also discuss the few indexes that do not belong to either of these two sets. Our hope is that these results will help guide decisions about specification of the form of efficiency indexes used in empirical analysis.