Distance Functions and Generalized Means : Duality and Taxonomy

This paper defines a new distance function in production theory that generalizes several existing efficiency measures. The new distance function is inspired from the Atkinson inequality index and maximizes the sum of the netput expansions required to reach an efficient point. This article also establishes a relation between this distance function and Stone-Geary utility functions and shows, more generally, that a large class of efficiency measures can be derived from the notion of utility function. A generalized duality theorem is proved and a duality result linking the new distance function and the profit function is obtained. For all feasible production vectors, it includes as special cases most of the dual correspondences previously established in the literature. Finally, we identify a large class of measures for which these duality results can be obtained without convexity