Recently, in a Data Envelopment Analysis (DEA) framework, Färe and Grosskopf [15] argued that the input directional distance function is invariant to affine data transformations under variable returns to scale (VRS), which includes, as a particular case, the property of translation invariance. In this paper we show that, depending on the directional vector used, the translation invariance may fail. In order to identify the directional distance functions (DDFs) that are translation invariant under VRS, we establish a necessary and sufficient condition that the directional vector must fulfill. As a consequence, we identify the characteristics that the DDFs should verify to be translation invariant. We additionally show some distinguished members that satisfy the aforementioned condition. We finally give several examples of DDFs, including input and output DDFs, which are not translation invariant

Overall inefficiency measurement and decomposition are important for firms facing a world of changing prices since the resultant loss has implications on managers׳ decision making. In this paper, we draw attention to some problems within existing approaches to decompose overall inefficiency into its sources and propose new revenue and cost inefficiency measures based on the well-known Russell measures. Specifically, the technical inefficiency component is calculated by the Russell output (input) measure, which is able to incorporate all sources of inefficiency corresponding to the output (input) side, specifically output (input) slacks, whereas allocative inefficiency is retrieved residually. All our results are derived from a new Fenchel–Mahler inequality using the theory of convex conjugates. This paper has several implications in theory and practice. From a theoretical point of view, we establish a natural dual relationship between the revenue (cost) function and the Russell output (input) measure; despite the previous unsuccessful attempts in the literature to provide such duality result. From a practical point of view, we provide a way of decomposing revenue (cost) inefficiency into allocative inefficiency and a component that measures technical inefficiency in the sense of Pareto, contrasting with the usual approaches for decomposing revenue (cost) inefficiency, such as those based on Shephard׳s output (input) distance function and the directional output (input) distance function

Distance functions in production theory are mathematical structures that characterize the belonging to the reference technology through a numerical value, behave as technical efficiency measures when the focus is analyzing an observed input–output vector within its production possibility set and present a dual relationship with some support function (profit, revenue, cost function). In this paper, we endow the well-known weighted additive models in Data Envelopment Analysis with a distance function structure, introducing the Weighted Additive Distance Function and showing its main properties

Efficiency analysis of the designations of origin in the Spanish wine sector