The Measurement of Efficiency of Production

The subject matter of this inquiry is efficiency in production. We hold the view that efficiency is an important characteristic of producer performance, one that has suffered considerable and unfortunate neglect in the economic literature. The purpose of this inquiry, then, is to redress that neglect by constructing a model of the producer in which inefficiency is allowed to play a meaningful role. This construction enables us to systematically explore the various ways in which a producer might depart from overall efficiency, and to explore the structural relationships among the component types of inefficiency. It also enables us to derive indexes, or measures, of the degree of producer efficiency, both overall and by component, and to examine the properties each of these efficiency indexes satisfies. Finally, this approach suggests ways in which the incorporation of inefficiency into a model of the producer enriches the set of testable hypotheses concerning producer behavior.

Any investigation of efficiency in production must begin with a description of the structure of the technology that constrains production activities. The purpose of this chapter is to provide such a description.

Consider some feasible production plan {(u, x) : x ∈ L(u)}. It is natural to inquire as to whether there exists some smaller input vector 0 ≤ y ≤ x that remains feasible for output vector u, i.e., y ∈ L(u). A different, but related, question involves the existence of a less expensive, though not necessarily smaller, input vector z : pz < px, given input prices p ∈ R ₊₊ ⁿ , that remains feasible for output vector u, i.e.,z ∈ L(u). If either y or z exists, then x is clearly inefficient for u, and the efficiency of x can be calculated relative to y or z respectively.

In chapter 3 we modeled the technology of a production unit with an input correspondence uL(u) ⫅ R ₊ ⁿ , and we developed various measures of the efficiency with which inputs are used to produce a certain output vector ∈R ₊ ^m . Three of these measures are technical, and so are input price-independent, while one is allocative and input price-dependent. Thus the overall measure of input efficiency is also input price-dependent, having the cost minimizing set CM(u, p) as its reference set, and so the behavioral assumption underlying the construction of the overall measure of input efficiency is one of minimizing input cost in the production of a certain output vector.

In chapter 3 we developed a series of measures of the efficiency with which a production unit uses variable inputs to produce a given output vector. These measures are appropriate under a behavioral assumption of constrained cost minimization. In chapter 4 we developed an analogous series of measures of the efficiency with which a production unit produces variable outputs from a given input vector. These measures are appropriate under a behavioral assumption of constrained revenue maximization. In this chapter we drop the assumption that either outputs or inputs are given, and develop a similar series of measures of the efficiency with which a production unit uses variable inputs to produce variable outputs. That is, the production unit is assumed to be able to freely adjust all inputs and all outputs, subject only to the constraints imposed by the production technology. These measures are appropriate under a behavioral assumption of profit maximization. Since all inputs and all outputs are freely variable, we model technology with the graph rather than with the input correspondence or the output correspondence.

In the three previous chapters we established some relationships among various radial input measures of efficiency (chapter 3), among various radial output measures of efficiency (chapter 4), and among various hyperbolic graph measures of efficiency (chapter 5). In this brief chapter we establish some relationships among input, output, and graph measures of various types of efficiency. Our motivation for so doing is that we seek answers to questions such as (1) Under what conditions, if any, do input, output, and graph measures of a certain type of efficiency attach the same efficiency value to a given input-output vector? (2) Under what conditions, if any, can the three measures of a certain type of efficiency be ordered? And, (3) If an input-output vector is labeled efficient by one measure, when, if ever, is it labeled efficient by the other two measures?

In chapters 3 and 4 we introduced radial input and output efficiency measures. The hyperbolic graph efficiency measure introduced in chapter 5, though not a radial measure in the strict sense, nonetheless possesses many of the characteristics of the radial input and output efficiency measures. Among the virtues of this family of efficiency measures are their consistency with the original formulations of Farrell (1957), their ease of computation, their straightforward cost or revenue interpretation, and their consequent decomposability. That is, measures of overall (input, output, and graph) efficiency each have a multiplicative decomposition into technical, congestion, and allocative components.

From a private perspective, a firm facing a given technology and fixed output and input prices can do no better than to select an input-output vector that yields maximum profit. Indeed in section 5.4, an input-output vector (x, u) ∈ ПM(r, p) is termed overall graph efficient, i.e., the input-output vector is uncongested, technically efficient and allocatively efficient. However the structure of technology and the distribution of output and input prices may combine to yield the result that an overall graph efficient firm earns negative, zero, or positive profit.

In this brief concluding chapter we offer a summary of where we have been, where we are going, and how we might get there. In Section 9.1 we summarize the essential elements of efficiency measurement gleaned from the core of the book, chapters 3–8. In Section 9.2 we suggest two ways in which our work might be extended, to the development of dual efficiency measures, and to the measurement of dynamic efficiency. In Section 9.3 we provide a brief reader’s guide to the various extant approaches to empirical implementation. This guide condenses and updates earlier surveys of Førsund, Lovell and Schmidt (1980) and Lovell and Schmidt (1983).