Abstract
This paper proposes two new Luenberger-type indicators, one for measuring productivity change of decision making units in the full input–output space, and the other for determining profit inefficiency change over time when information on market prices is also available. Both approaches are based upon the recently introduced weighted additive distance function, which permits the well-known weighted additive model in data envelopment analysis to be endowed with a distance function structure. We also show how the two indicators may be decomposed into their drivers.
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Notes
Luenberger (1992, 1995) introduced the concepts of benefit function and shortage function. In particular, the shortage function measures the distance in the direction of a vector g of a production plan from the boundary of the production possibility set, i.e., the shortage function measures the amount by which a specific plan is short of reaching the frontier of the technology. A few years later, Chambers et al. (1998) redefined the shortage function as a technical inefficiency measure, introducing the directional distance function.
In addition, Briec and Kerstens (2009b) notice that the computation of mixed-period Directional Distance Functions can lead to projections with a negative output, which in general have little meaning in standard economic production applications. In order to avoid such a problem, one needs to add an additional constraint into program (3): the output translated by the Directional Distance Function into the direction of the directional vector must be positive (that is, \(y_{r0} +\beta g_r^O \ge 0)\). It is worth noticing that imposing this constraint may lead to additional infeasibilities.
Note that \(\Pi \left( {kc,kp} \right) =k\Pi \left( {c,p} \right) \) because the profit function is homogeneous of degree one in prices (see Färe and Primont 1995).
Note that the denominator suggested by Chambers et al. (1998) is homogeneous of degree one in prices and, therefore, \(\frac{\Pi \left( {kc,kp} \right) -\left( {\sum \limits _{r=1}^s {kp_r y_{r0} } -\sum \limits _{i=1}^m {kc_i x_{i0} } } \right) }{\left( {\sum \limits _{i=1}^m {kc_i g_i^I } +\sum \limits _{r=1}^s {kp_r g_r^O } } \right) }=\frac{k\Pi \left( {c,p} \right) -k\left( {\sum \limits _{r=1}^s {p_r y_{r0} } -\sum \limits _{i=1}^m {c_i x_{i0} } } \right) }{k\left( {\sum \limits _{i=1}^m {c_i g_i^I } +\sum \limits _{r=1}^s {p_r g_r^O } } \right) }= \quad \frac{\Pi \left( {c,p} \right) -\Pi _0 }{\left( {\sum \limits _{i=1}^m {c_i g_i^I } +\sum \limits _{r=1}^s {p_r g_r^O } } \right) }\).
For instance, Lozano et al. (2004) resorted to the MIP in order to estimate the performance of a set of municipalities in the context of glass recycling. It is an example of a cross-sectional study (one period). However, this same measure could not have been used if a panel data were available and determining productivity change was the focus. Now it is possible thanks to the new methodology introduced in this paper.
- $$\begin{aligned}&{\left( {c,p} \right) }\big /{\max \left\{ {\frac{c_1 }{w_1^- },\ldots ,\frac{c_m }{w_m^- },\frac{p_1 }{w_1^+ },\ldots ,\frac{p_s }{w_s^+ }} \right\} }\\&\quad =\left( {\frac{c_1 }{\max \left\{ {\frac{c_1 }{w_1^- },\ldots ,\frac{c_m }{w_m^- },\frac{p_1 }{w_1^+ },\ldots ,\frac{p_s }{w_s^+ }} \right\} },\ldots ,\frac{p_s }{\max \left\{ {\frac{c_1 }{w_1^- },\ldots ,\frac{c_m }{w_m^- },\frac{p_1 }{w_1^+ },\ldots ,\frac{p_s }{w_s^+ }} \right\} }} \right) . \end{aligned}$$
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Acknowledgements
We thank two anonymous referees for providing constructive comments and help. Also, the authors would like to express their gratitude to the Spanish Ministry for Economy and Competitiveness for supporting this research through Grant MTM2013-43903-P.
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Aparicio, J., Borras, F., Ortiz, L. et al. Luenberger-type indicators based on the weighted additive distance function. Ann Oper Res 278, 195–213 (2019). https://doi.org/10.1007/s10479-017-2620-2
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DOI: https://doi.org/10.1007/s10479-017-2620-2