2 The Econometric Approach to Efficiency Analysis William H. Greene 2.1 Introduction Chapter 1 describes two broad paradigms for measuring economic efficiency, one based on an essentially nonparametric, programming approach to anal- ysis of observed outcomes, and one based on an econometric approach to estimation of theory-based models of production, cost, or profit. This chapter presents an overview of techniques for econometric analysis of technical (pro- duction) and economic (cost) efficiency. The stochastic frontier model of Aigner, Lovell, and Schmidt (1977) is now the standard econometric platform for this type of analysis. I survey the underlying models and econometric tech- niques that have been used in studying technical inefficiency in the stochastic frontier framework and present some of the recent developments in econo- metric methodology. Applications that illustrate some of the computations are presented in the final section. 2.1.1 Modeling production The empirical estimation of production and cost functions is a standard exer- cise in econometrics. The frontier production function or production frontier is an extension of the familiar regression model based on the theoretical premise that a production function, or its dual, the cost function, or the convex conjugate of the two, the profit function, represents an ideal, the maximum output attain- able given a set of inputs, the minimum cost of producing that output given the prices of the inputs, or the maximum profit attainable given the inputs, 92 FRIED: “CHAP02” — 2007/8/24 — 19:02 — PAGE 92 — #1 The Econometric Approach to Efficiency Analysis 93 outputs, and prices of the inputs. The estimation of frontier functions is the econometric exercise of making the empirical implementation consistent with the underlying theoretical proposition that no observed agent can exceed the ideal. In practice,the frontier function model is (essentially) a regression model that is fit with the recognition of the theoretical constraint that all observations lie within the theoretical extreme. Measurement of (in)efficiency is, then, the empirical estimation of the extent to which observed agents (fail to) achieve the theoretical ideal. My interest in this chapter is in this latter function. The estimated model of production, cost, or profit is the means to the objective of measuring inefficiency. As intuition might suggest at this point, the exercise here is a formal analysis of the “residuals” from the production or cost model. The theory of optimization, production, and/or cost provides a description of the ultimate source of deviations from this theoretical ideal. 2.1.2 History of thought The literature on frontier production and cost functions and the calculation of efficiency measures begins with Debreu (1951) and Farrell (1957) [though there are intellectual antecedents, e.g., Hicks’s (1935) suggestion that monop- olists would enjoy their position through the attainment of a quiet life rather than through the pursuit of economic profits, a conjecture formalized some- what more by Leibenstein (1966, 1975)]. Farrell suggested that one could usefully analyze technical efficiency in terms of realized deviations from an idealized frontier isoquant. This approach falls naturally into an economet- ric approach in which the inefficiency is identified with disturbances in a regression model. The empirical estimation of production functions had begun long before Farrell’s work, arguably with Cobb and Douglas (1928). However, until the 1950s, production functions were largely used as devices for studying the func- tional distribution of income between capital and labor at the macroeconomic level. The celebrated contribution of Arrow et al. (1961) marks a milestone in this literature. The origins of empirical analysis of microeconomic production structures can be more reasonably identified with the work of Dean (1951, a leather belt shop), Johnston (1959, electricity generation), and, in his seminal work on electric power generation, Nerlove (1963). It is noteworthy that all three of these focus on costs rather than production, though Nerlove, follow- ing Samuelson (1938) and Shephard (1953), highlighted the dual relationship between cost and production.1 Empirical attention to production functions at a disaggregated level is a literature that began to emerge in earnest in the 1960s (see, e.g., Hildebrand and Liu, 1965; Zellner and Revankar, 1969). 2.1.3 Empirical antecedents The empirical literature on production and cost developed largely indepen- dently of the discourse on frontier modeling. Least squares or some variant FRIED: “CHAP02” — 2007/8/24 — 19:02 — PAGE 93 — #2 94 The Measurement of Productive Efficiency and Productivity Growth was generally used to pass a function through the middle of a cloud of points, and residuals of both signs were, as in other areas of study, not singled out for special treatment. The focal points of the studies in this literature were the esti- mated parameters of the production structure, not the individual deviations from the estimated function. An argument was made that these “averaging” estimators were estimating the average,rather than the“best-practice”technol- ogy. Farrell’s arguments provided an intellectual basis for redirecting attention from the production function specifically to the deviations from that function, and respecifying the model and the techniques accordingly. A series of papers includingAigner and Chu (1968) and Timmer (1971) proposed specific econo- metric models that were consistent with the frontier notions of Debreu (1951) and Farrell (1957). The contemporary line of research on econometric models begins with the nearly simultaneous appearance of the canonical papers of Aigner, Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977), who proposed the stochastic frontier models that applied researchers now use to combine the underlying theoretical propositions with a practical economet- ric framework. The current literature on production frontiers and efficiency estimation combines these two lines of research. 2.1.4 Organization of the survey This survey presents an overview of this literature and proceeds as follows: Section 2.2 presents the microeconomic theoretical underpinnings of the empirical models. As in the other parts of our presentation, this section gives only a cursory survey because of the very large literature on which it is based. The interested reader can find considerable additional detail in chapter 1 of this book and in a gateway to the larger literature, chapter 2 of Kumbhakar and Lovell (2000). Section 2.3 constructs the basic econometric framework for the econo- metric analysis of efficiency. This section also presents some intermediate results on “deterministic” (orthodox) frontier models that adhere strictly to the microeconomic theory. This part is brief. It is of some historical inter- est and contains some useful perspective for the more recent work. However, with little exception, current research on the deterministic approach to effi- ciency analysis takes place in the environment of “data envelopment analysis” (DEA), which is the subject of chapter 3 of this book.2 This section provides a bridge between the formulation of orthodox frontier models and the modern stochastic frontier models. Section 2.4 introduces the stochastic production frontier model and presents results on formulation and estimation of this model. Section 2.5 extends the stochastic frontier model to the analysis of cost and profits and describes the important extension of the frontier concept to multiple-output technologies. Section 2.6 turns to a major econometric issue, that of accommodating heterogeneity in the production model. The assumptions made in sections 2.4 FRIED: “CHAP02” — 2007/8/24 — 19:02 — PAGE 94 — #3 The Econometric Approach to Efficiency Analysis 95 and 2.5 regarding the stochastic nature of technical inefficiency are narrow and arguably unrealistic. Inefficiency is viewed as simply a random shock distributed homogeneously across firms. These assumptions are relaxed at the end of section 2.5 and in section 2.6. Here, I examine proposed models that allow the mean and variance of inefficiency to vary across firms, thus producing a richer, albeit considerably more complex, formulation. This part of the econometric model extends the theory to the practical consideration of observed and unobserved influences that are absent from the pure theory but are a crucial aspect of the real-world application. The econometric analysis continues in section 2.7 with the development of models for panel data. Once again, this is a modeling issue that provides a means to stretch the theory to producer behavior as it evolves through time. The analysis pursued here goes beyond the econometric issue of how to exploit the useful features of longitudinal data. The literature on panel data estimation of frontier models also addresses the fundamental question of how and whether inefficiency varies over time, and how econometric models can be made to accommodate the theoretical propositions. The formal measurement of inefficiency is considered in sections 2.8 and 2.9. The use of the frontier function model for estimating firm-level inefficiency that was suggested in sections 2.3 and 2.4 is formalized in the stochastic frontier model in section 2.8. Section 2.9 considers the separate issue of allocative inefficiency. In this area of study, the distinction between errors in optimization and the consequences of those errors for the goals or objectives of