Abstract
The directional distance function encompasses Shephard’s input and output distance functions and also allows nonradial projections of the assessed firm onto the frontier of the technology in a preassigned direction. However, the criteria underlying the choice of its associated directional vector are numerous. When market prices are observed and firms have a profit maximizing behavior, it seems natural to choose as the directional vector that projecting inefficient firms towards profit maximizing benchmarks. Based on that choice of directional vector, we introduce the directional profit efficiency measure and show that, in this general setting, profit inefficiency can be categorized as either technical, for firms situated within the interior of the technology, or allocative, for firms lying on the frontier. We implement and illustrate the analytical model by way of Data Envelopment Analysis techniques, and introduce the necessary optimization programs for profit inefficiency measurement.
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Notes
Also, the approach could be extended to a free disposal hull setting.
We note that we could render (8) more flexible by letting it solve for the directional vector \( ( {g_{x}^{T} ,g_{y}^{T} } ) \in R_{ + + }^{N} \times R_{ + + }^{M} \), including pg T y + wg T x = 1 and g T x , g T y ≥ 0 as additional constraints, thereby projecting the evaluated firm in the direction leading to the reference benchmark with the highest possible profit compatible with the non-negativity constraint, i.e., the firm is forced to move northwesterly.
Moreover, the case of multiple profit maximizing benchmarks could also arise when input and output prices are not the same across firms (see, for example, Portela and Thanassoulis 2007).
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Acknowledgments
We are grateful to the participants and discussants at the International Workshop on Efficiency and Productivity in honor of Prof. Knox Lovell (October 4–5, 2010, Elche, Spain). We acknowledge financial support from the Ministerio de Ciencia e Innovacion, Spain, and the Conselleria de Educacion, Generalitat Valenciana, for supporting this research with grants MTM2009-10479 and ACOMP/2011/115, respectively. Finally, we are also indebted to two referees for their comments and suggestions that have contributed to improving this paper.
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A previous version was published in Economics Working Papers Series, Universidad Autónoma de Madrid, Department of Economics, No. 2010/09.
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Zofio, J.L., Pastor, J.T. & Aparicio, J. The directional profit efficiency measure: on why profit inefficiency is either technical or allocative. J Prod Anal 40, 257–266 (2013). https://doi.org/10.1007/s11123-012-0292-0
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DOI: https://doi.org/10.1007/s11123-012-0292-0