Distance Functions and Generalized Means: Duality and Taxonomy

TL;DR

This paper explores the relationships between utility functions, distance functions, and efficiency measures in production economics, introducing a generalized mean distance function inspired by the Atkinson index and establishing duality theorems.

Contribution

It introduces a new generalized mean distance function and establishes duality theorems linking it to profit functions, broadening the understanding of efficiency measures.

Findings

Established duality theorems linking distance and profit functions.

Introduced a generalized mean distance function inspired by the Atkinson index.

Identified classes of measures with duality results without convexity assumptions.

Abstract

This article demonstrates how a large number of efficiency measures known in the literature in production economics can be interpreted through the notion of utility function, based on the concept of Stone-Geary utility. Several relationships between these utility functions and distance functions, a commonly used tool in production theory, are established. To achieve these objectives, a generalized mean distance function is introduced, inspired by the Atkinson inequality index, itself derived from the notion of the Aczel mean. It measures the maximum sum of netput expansions required to reach an efficient point. Several duality theorems are established, linking the new distance functions to the profit function. For all feasible production vectors, the results include as special cases most of the dual correspondences previously established in the literature. Finally, a large class of measures is identified for which these duality results can be obtained without requiring convexity. A numerical example is provided.