Conjugate Paretian inefficiency measures

We study efficiency measurement using a partial ordering for the S-dimensional reals that generalizes the canonical less than or equal to partial ordering. We seek measures that judge outcomes as favorably as possible using a dual normalization strategy that generalizes those used in the minimum-norm and efficiency-measurement literatures.

We characterize the efficient frontier using dual methods and use that representation to identify a dual Nerlovian inefficiency measure. The Paretian inefficiency measure is defined as the minimal Nerlovian measure while constraining dual variates to fall in a predetermined closed convex set. We show that the Paretian inefficiency measure forms a dual conjugate pair with a restricted Nerlovian efficiency measure. We use those results to develop conditions that ensure that the Paretian inefficiency measure is an exhaustive function (cardinal) representation of the feasible set. We present a series of composition rules for different restrictions on the feasible set and dual-variate normalization that include generalizations of existing inefficiency measures. An empirical illustration of the concepts developed that is based on Catalan farming data closes the substantive part of the paper.