We introduce two desirable properties concerning the allocative efficiency term in the decomposition of economic efficiency. Since Farrell, economic efficiency, defined in terms of the cost, revenue, profitability, or profit functions, is decomposed into a technical efficiency measure and allocative efficiency. Resorting to duality theory, allocative efficiency is calculated as a residual. In this framework, we show that this residual is numerically inconsistent for several economic efficiency decompositions: those based on the Russell Efficiency Measures, the Slack-Based Measure and the Weighted Additive Measures. It should be expected that a technical inefficient firm, if projected to the optimal economic benchmark, e.g., that maximizing profit, should be allocative efficient, yet we show that the above decompositions may signal that it is allocative inefficient. Our first property, called ‘essential,’ demands that economic efficiency decompositions satisfy the above criterion. We also extend this property by requiring that the allocative efficiency of a technically inefficient firm, evaluated at the projected benchmark on the production frontier, coincides with the allocative efficiency of the benchmark itself. Regarding this extension of the property, we show that, besides the measures mentioned above, the Directional Distance Function and the Hölder Distance Function fail in general to comply with it. However, we also prove that, thanks to the flexibility of these two approaches in choosing a common directional vector for all the assessed firms or a specific Hölder norm, these measures may satisfy the extended essential property. We conclude that unless both properties are fulfilled, the (residual) allocative component of many previously published decompositions cannot be correctly interpreted as price inefficiency.
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We thank two anonymous reviewers for providing constructive comments and help in improving the contents and presentation of this paper. The authors also thank the grant PID2019-105952GBI00 funded by Ministerio de Ciencia e Innovación/Agencia Estatal de Investigación/ https://doi.org/10.13039/501100011033 . José L. Zofío also acknowledges financial support from the Grant EIN2020-112260/AEI/ https://doi.org/10.13039/501100011033 and from the Comunidad de Madrid Government under Grant H2019/HUM-5761 (INNJOBMAD-CM). Additionally, J. Aparicio thanks the Grant PROMETEO/2021/063 funded by the Valencian Community (Spain).
© 2023, The Author(s).