Assessing Slacks through a Nested Radial Approach in an FDH Reference Technology

After having reiterated the point that the traditional Dehreu-Farrell efficiency measure is not appropriate within an FDH technology, we show in this paper that for such a technology it is possible to obtain through a nested radial procedure various technical efficiency measures comprehensive of the existence of slacks. Considering for simplicity the case of input-oriented efficiency only,1 we show that the input vector (of order m) of an inefficient observation can be decomposed in m radial projections on m subspaces. This decomposition procedure follows a hierarchically ordered (or nested) sequence from the space of dimension m (which allows the greatest possible radial reduction associated with the disappearance of slack in at least one input) to a space of dimension one. At every step of the procedure, Dehreu-Farrell measures are obtained which correspond to the amount of slack in a given input that is cancelled by a radial contraction in the relevant subspace, or in other words to the ratio between the input quantity utilised by the efficient plan to that utilised by the inefficient plan. For all inputs we compute the product of these Debreu-Farrell measures, obtaining the total input contraction required to bring an inefficient observation in the efficient subset. This approach allows (a) to generate some nonradial measures proposed in the literature (asymmetric Färe, Färe-Lovell)2 as the products of nested radial contractions; and (b) to show that the correct measure of technical efficiency always falls in an interval bounded by the asymmetric Färe and the radial Debreu-Farrell measures.