document: Regarding the implications of the results of this paper, we note two points. From a practical point of view, we have endowed the weighted additive model with a distance function structure, which takes negative values for points located outside the technology and non-negative values for points into the production possibility set. In this respect, the weighted additive distance function methodologically supports the branch of the literature that resorts to the weighted additive model or some related approach to measure productivity over time (see, for example, Mahlberg & Sahoo, 2011 or Chang et al., 2012). From a theoretical point of view, we have provided a new distance function with some interesting properties in contrast to the usual ones, mainly (1) when technical inefficiency has to be estimated, the weighted additive distance function coincides with the weighted additive model, which means that technical inefficiency is measured following the Pareto-Koopmans notion of efficiency; and (2) when productivity has to be determined and decomposed over time the weighted additive distance function emerges as an attractive tool to be used for cross-period evaluation of returns to scale changes, since this distance function is always feasible, even under Variable Returns to Scale.
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extractive_keyphrases: a new distance function determined and decomposed over time distance function implications of the results measure productivity over time some related approach weighted additive model
abstractive_keyphrases: means that technical inefficiency is measured following the pareto-koopmans notion of efficiency takes negative values for points located outside the technology and non-negative values for points into the production possibility set the weighted additive distance function the weighted additive model with a distance function structure weighted additive distance function