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Additional resources for Advanced Robust and Nonparametric Methods in Efficiency Analysis: Methodology and Applications (Studies in Productivity and Efficiency)
This means of measuring the distance to the frontier yields an interpretation of performance or efficiency as maximal-minimal proportionate feasible changes in an activity given technology. This explanation is consistent with Debreu’s (1951) coefficient of resource utilization and with Farrell’s (1957) efficiency measures. However, neither Debreu nor Farrell formulated the efficiency measurement problem as a linear programming problem, even though Farrell and Fieldhouse (1962) envisaged the role of linear programming.
Their application leads to measures of technical efficiency from the potential for increasing outputs while reducing inputs at the same time. In order to provide a measure of “directional” efficiency, a direction, along which the observed DMU is projected onto the efficient frontier of the production set, has to be chosen. This choice is arbitrary and of course affects the resulting efficiency measures. In addition, those measures are no more scale-invariant. See F¨are and Grosskopf (2004) for more details on these “new directions” in efficiency analysis.
We follow here the presentation of Simar and Wilson (2002, 2006b). t. x ∈ ∂C(y) then αx ∈ ∂C(αy), ∀α > 0 or equivalently6 , ∀α > 0, C(αy) = αC(y). 6 Analogous expressions hold in terms of P (x): ∀α > 0, P (αx) = αP (x). t. x ∈ ∂C(y) are characterized by C(αy) = αC(y) for some α > 0. t. x ∈ ∂C(y) implies that (αx, αy) ∈ Ψ for α < 1. t. x ∈ ∂C(y) implies that (αx, αy) ∈ Ψ for α > 1. A frontier that exhibits increasing, constant and decreasing returns to scale in different regions is a Variable Returns to Scale (VRS) frontier.