A generalized distance function and the analysis of production efficiency.

1. Introduction

The measurement of productivity and efficiency has been a topic of considerable interest in economics. Much research has focused on the analysis of technical, allocative and scale efficiency of production activities (e.g., Debreu 1951; Farrell 1957; Farrell and Fieldhouse 1962;

Afriat 1972; Forsund, Lovell, and Schmidt 1980; Fare, Grosskopf, and Lovell 1985, 1994; Russell 1985; Banker and Maindiratta 1988; Bauer 1990; Seiford and Thrall 1990) and the analysis of technological progress and productivity (e.g., Diewert 1976; Caves, Christensen, and Diewert 1982a, b). Farrell's efficiency indexes commonly have been used in empirical research on production efficiency for two reasons: they can be combined easily into an overall efficiency index, and they have an intuitive interpretation in terms of cost ratios or average cost ratios (Farrell 1957; Farrell and Fieldhouse 1962).

Shephard's distance functions have guided much of the development in productivity analysis and efficiency analysis. For example, Caves, Christensen, and Diewert (1982b) have investigated productivity indexes derived from Shephard's distance functions. Fare, Grosskopf, and Lovell (1985, 1994) have shown how the Farrell efficiency indexes are closely related to Shephard's distance functions. In a multi-input multioutput framework, Shephard defines two distance functions: an input distance function that rescales all inputs toward the frontier technology and an output distance function that rescales all outputs toward the frontier. Unfortunately, unless the technology exhibits constant return to scale, these two distance functions differ and provide different measures of productivity and efficiency (Caves, Christensen, and Diewert 1982b; Fare, Grosskopf, and Lovell 1985, 1994). This appears rather undesirable. Also, to be empirically meaningful, Shephard's distance functions rely on an "attainability assumption." This assumption states that all output vectors can be obtained from the rescaling of any nonzero input vector or that all input vectors are feasible in the production of any rescaled nonzero output vector (see Shephard 1970, Chapter 9). However, in some situations, this attainability assumption may not be satisfied, especially if some inputs or outputs are not essential (see Shephard 1970; Fare and Mitchell 1987). This can greatly limit the empirical usefulness of the methodology. To illustrate, consider Ray and Desli's recent investigation of productivity growth and efficiency in industrialized countries. Using Shephard's output distance function, Ray and Desli (1997) were unable to report empirical estimates of technical change and scale efficiency for Ireland because the associated data did not satisfy the attainability assumption (Ray and Desli 1997, p. 1037). This suggests a need to extend Shephard's distance functions.