A generalized distance function and the analysis of production efficiency.

• Author: Chavas, Jean-Paul

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.

   Shephard's distance functions have been generalized in a number of ways. Graph measures of production efficiency have been developed by Fare, Grosskopf, and Lovell (1985, Chapters 5-7; 1994, Chapter 8). For example, Fare, Grosskopf, and Lovell (1985, p. 110; 1994, p. 198) defined a "Farrell graph technical efficiency index" that rescales both inputs and outputs equiproportionally. Other extensions of Farrell technical efficiency include a "generalized Farrell graph" measure proposed by Fare et al. (1985, p. 125), nonradial efficiency measures discussed by Russell (1985) and Fare, Grosskopf, and Lovell (1985, Chapter 7), a "Farrell proportional distance" measure defined by Briec (1997), and the shortage and benefit functions developed by Luenberger (1992a, b, 1995). Briec's "Farrell proportional distance" function and Luenberger's shortage function are the same: they both allow the rescaling of inputs and outputs in any particular direction. As such, they provide a broad generalization to Shephard's distance functions (Chambers, Chung, and Fare 1996a, b). They include as special cases most measures of technical efficiency found in the literature (Briec 1997).

   Thus it appears desirable to rescale inputs and outputs in a more flexible way than done in Shephard's distance functions. The Luenberger-Briec approach provides a general framework for doing so. However, it does not provide clear guidance for choosing the rescaling direction for inputs and outputs in efficiency analysis. Also, although the Farrell efficiency measures can be easily interpreted in terms of average cost, such an interpretation is not straightforward in the Luenberger-Briec approach. This is somewhat unfortunate since average cost is a basic concept found in all production economic textbooks and is commonly used in empirical economic analysis. This suggests considering a rescaling scheme for inputs and outputs that extends Shephard's distance functions while retaining the intuitive average cost interpretation of the Farrell indexes.

   The objective of this paper is to propose a generalized Shephard's distance function with the following characteristics. First, it includes as special cases both Shephard's input and output distance functions while relaxing Shephard's attainability assumptions. Second, it generates efficiency indexes that have an intuitive interpretation in terms of average cost (or ray-average cost in a multioutput framework; see Baumol, Panzar, and Willig 1982). Third, these efficiency indexes can be combined easily into an overall efficiency index.

   Our analysis is presented in a multi-input/multioutput framework. Our generalized Shephard's distance function considers the simultaneous rescaling of both inputs and outputs. The direction of rescaling depends on a single parameter [Alpha] that can vary between zero and one. As special cases, the parameter a taking the value 0 (1) implies only input (output) rescaling. Thus, our generalized distance function nests, as special cases, both Shephard's input and output distance functions. It applies without Shephard's attainability assumption, thus widening the range of applications of distance functions in economic analysis. Also, our proposed approach resolves the current dilemma concerning which Shephard's distance function (i.e., the input distance function or the output distance function) to use when the technology departs from constant return to scale.(1)

   The paper is organized as follows. Section 2 defines a generalized distance function and investigates its properties. Our generalized distance function provides a basis for investigating production efficiency and productivity growth. We propose new indexes of productivity, technical efficiency, allocative efficiency, and scale efficiency. Indexes of technical efficiency and productivity are presented in section 3. Our proposed technical efficiency index nests as special cases both the traditional input-based and output-based technical efficiency indexes commonly found in the literature (e.g., Fare, Grosskopf, and Lovell 1985, 1994). Similarly, our productivity index nests as special cases both the input-based and output-based Malmquist (1953) productivity indexes discussed by Caves, Christensen, and Diewert (1982b). Our allocative efficiency indexes are presented in section 4. They are motivated from the cost function, from the revenue function, and from the profit function. Section 5 discusses our proposed scale efficiency indexes, based again on the cost function, the revenue function, and the profit function. In section 6, we show how all our proposed indexes can be conveniently interpreted in terms of the properties of ray-average cost, ray-average revenue, and cost-to-revenue ratios. This simple economic interpretation provides intuitive appeal to our proposed approach.

2. A Generalized Distance Function

   Consider a production process involving an (n x 1) input vector [Mathematical Expression Omitted] used in the production of an (m x 1) output vector [Mathematical Expression Omitted].(2) The underlying technology is represented by the feasible set T, [Mathematical Expression Omitted], or equivalently by the associated input requirement set [Mathematical Expression Omitted]. The following assumptions will be made throughout the paper:(3)

   A1. The feasible set T is nonempty, and there exists an x [greater than or equal to] 0 and y [greater than or equal to] 0 such that (x, y) [element of] T.

   A2. Nested: If x [element of] V(y, T) and y [greater than or equal to] y[prime], then x [element of] V (y[prime], T), [for every] y, [Mathematical Expression Omitted].

   A3. Monotonic: If x [element of] V(y, T) and x[prime] [greater than or equal to] x, then x[prime] [element of] V(y, T), [Mathematical Expression Omitted].

   A4. The feasible set T is closed.

   Assumption A1 states that it is possible to produce some positive output from some positive input. Assumptions A2 and A3 have been called "strong disposability" of outputs and inputs, respectively. Assumption A4 implies the existence of isoquants at the boundary of the feasible set.

   In general, we will assume that T represents a variable return to scale (VRTS) technology. The nature of returns to scale can be characterized globally as follows.(4)

   DEFINITION 1. For all y [greater than or equal to] 0,

   V([Lambda]y, T = [Lambda]V(y, T), for [Lambda] [greater than] 0, under constant return to scale (CRTS); (1a)

   V([Lambda]y, T) [subset or equal to] ([contains or equal to]) [Lambda],V(y, T), for 0 [less than] [Lambda] [less than or equal to] 1 ([Lambda] [greater than or equal to] 1), under increasing return to scale (IRTS); (1b)

   V([Lambda]y, T) [contains or equal to]([subset or equal to]) [Lambda] V(y, T), for 0 [less than] [Lambda] [less than or equal to] 1 ([Lambda] [greater than or equal to] 1), under decreasing return to scale (DRTS). (1c)

   From Equation la under CRTS, a proportional change in all outputs is associated with the same proportional change in all inputs. In this case, the underlying production frontier is linearly homogeneous. From Expressions 1b and 1c under...