Direct estimation of marginal characteristics of nonparametric production frontiers in the presence of undesirable outputs

There is extensive literature on the estimation of marginal characteristics of nonparametric production frontiers, including various marginal rates and elasticity measures. It has recently been shown that all such characteristics can be evaluated by a unifying linear programming approach applicable to any polyhedral production technology. In this paper we show how this approach can be applied to polyhedral technologies incorporating undesirable outputs. In particular, we derive a linear programming method for the direct assessment of the marginal rate of transformation between a bad and a good output often used for the estimation of the unobserved price of the bad output. In contrast with the existing methods based on a conventionally specified directional distance function, the new approach does not require the assessment of two shadow prices of the good and bad outputs. It also correctly estimates one-sided marginal rates in all cases in which the shadow prices on nonsmooth production frontiers are not unique.