# Properties of inefficiency indexes on 〈input, output〉 space

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## Abstract

We analyze efficiency measurement in the full \(\langle\)input, output\(\rangle\) space. We posit four types of axioms: indication (of efficient production bundles), monotonicity, independence of units of measurement, and continuity (in technologies as well as input and output quantities). Impossibility results establish a tension between indication and continuity. We focus on seven well-known inefficiency indexes from the operations-research and economics literature, establishing the properties they satisfy—and do not satisfy—on a general class of technologies satisfying minimal regularity conditions and on the subset of these technologies satisfying convexity. We also discuss several other indexes that are dominated by or very similar to these seven indexes. The set of properties satisfied by these indexes elucidates the trade-offs faced in selecting among the indexes.

## Keywords

Technical efficiency indexes Technical efficiency axioms## JEL classification

C43 C61 D24## Notes

### Acknowledgments

We thank Erwin Diewert, Rolf Färe, Daniel Primont and the referees for suggestions that substantially improved the paper. The paper has also benefitted from discussions with conference participants at the 2007 European Workshop on Efficiency and Productivity Analysis in Lille, the 2009 Economic Measurement Group Workshop at the University of New South Wales, and the 2010 Efficiency and Productivity Workshop at the Auckland University of Technology.

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