Efficiency evaluation in the presence of undesirable outputs: the most favorable shadow price approach

Undesirable outputs are generally by-products of producing desirable ones, and they can thus be reduced only with an accompanying reduction in the latter. To capture this notion, the shadow price of the undesirable output must be negative, as opposed to positive for the desirable output. Based on this condition, this paper proposes a data envelopment analysis model that allows the production units being evaluated to determine the shadow prices for both the desirable and undesirable outputs by themselves so that the measured efficiency score will achieve the highest possible level. The proposed model satisfies the assumption of weak disposability of outputs. It is also shown that it falls into one category of the directional distance function model that has been widely applied in modeling undesirable outputs. However, different from the conventional directional distance measures, the proposed model is able to provide efficiency measures in the range of zero and one for easy comparison among inefficient production units.