# Technological inefficiency indexes: a binary taxonomy and a generic theorem

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## Abstract

Over the years following Debreu’s (1951) seminal formulation of a “coefficient of resource utilization”, 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 particular, 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 additive 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 contraction/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 specified 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.

## Keywords

Efficiency indexes Inefficiency indexes Specification## JEL classification

C6 D2x## Notes

### Acknowledgements

We thank Walter Briec, Rolf Färe, Antonio Peyrache, and Valentin Zelenyuk, as well as the referees of the JPA, for comments and suggestions. The paper has also benefited from discussions during seminars at Lecce University and the University of Queensland.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no competing interests.

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