The decomposition of productivity growth has been explored and measured extensively to include efficiency changes over time in addition to scale effect and technical change components (see Fried et al (2008) for a recent overview). This partitioning of the different contributions is important as it implies different incentives or remedies to influence different components. For example, expansionary investment involves impacting the scale effect of the growth decomposition, while replacement investment acts on the technical change effect. Decisions and incentives to learn how to extract the full potential of implemented technologies are acting on the efficiency change component of promoting growth.
The core theoretical concept for building these measures is the production technology, where one can define formally the notions of technical efficiency (operating on the boundary of the feasible technology set), technical progress (shifting the boundaries of this set) and scale effects (moving along the boundary of an existing set). The abundant economic literature on the estimation of stochastic production frontier functions and the subsequent measurement of technical inefficiency has assumed, in general, that the underlying production technology displays some degree of substitutability between factors of production. This is not unusual as a production technology with zero input elasticity of substitution would imply that the cost-minimizing inputs are independent of their prices, which is a restrictive assumption in many real world applications.
Although this is true for the agricultural sector, certain types of production activities may exhibit a zero elasticity of substitution among inputs. Some examples are given by Komiya (1962) who investigated the technological progress in the US steam power industry, Lau and Tamura (1972) who propose the use of a non-homothetic Leontief production function to analyze the Japanese petrochemical industry, Nakamura (1990) who utilized a non-homothetic generalized Leontief technological structure for empirically analyzing the Japanese iron and steel industry, Buccola and Sil (1996) who measured productivity in the agricultural marketing sector, Holvad et al., (2004) who maintain that the transport industry might be characterized by Leontief-type technologies when analyzing cost efficiency in the Norwegian bus industry. Furthermore, a stream of literature in agricultural economics in modeling crop response to different fertilizer’s nutrients levels, has maintained zero substitution among crop nutrients using a linear plateau specification motivated by the von Liebig farm technology.
Sorting out the components of productivity growth initially involves identifying the relationship between the input combination and the boundary of the production set. Measuring technical inefficiency in the case of Leontief type technologies is of interest in itself given Farrell’s (1957) radial measures are the basis for most applied work on the measurement of efficiency. However, the radial measures can be inadequate in that they may classify inefficient input combinations as being efficient, while input- and output- oriented measures might not coincide even under constant returns-to-scale.4 Once the technology is governed by a Leontief-type structure, it is plausible to have inefficiency displayed by none, all or a subset of the inputs, rendering radial measures unsatisfactory. In addition, output-oriented measures may fail to recognize inefficiencies when they affect a subset of the inputs only. If we accept that some specific production activities exhibit a zero elasticity of substitution among factors of production, then alternative ways are needed to define and empirically measure technical efficiency and to analyze factor productivity growth.
The purpose of this paper is to develop a framework for modeling productivity growth under factor non-substitution that accounts for technical inefficiency and technical progress. The econometric modeling framework accommodates the absence of substitution possibilities among inputs where inefficiency between factors can be correlated. Our theoretical model is based on the non-homothetic Leontief production function suggested by Lau and Tamura (1972) which is the most general function with zero elasticities of substitution between all pairs of inputs allowing at the same time differential returns-to-scale and technical progress (regress) to inputs. This Leontief frontier model adapts the copula approach to modeling the joint distributions between the one-sided error terms that capture factor-specific technical inefficiencies. Factor-specific technical efficiencies are specified and measured using Kopp’s (1981) orthogonal indices of technical efficiency, combined into an overall technical efficiency measure using Russell’s (1985; 1987) non-radial index of productive efficiency. Then, we proceed to the developing a tractable approach for the analysis of partial factor productivity growth.
The model is applied to a panel data set of 72 fossil-fuel fired steam electric power generation utilities in the US observed during the 1986-96 period. When analyzing the economies of scale and technical progress in the generation of steam-electric power also in the US, Komiya (1962) found that the Leontief factor limitation model provided a better representation of the data compared with the traditional Cobb-Douglas unitary substitution model. Hence, in this study we maintain a priori that the US steam electric power utilities offer a good case for applying the suggested theoretical framework for measuring technical efficiency and decomposing partial factor productivity in factor limitation models. Further, we assume that errors associated with factor demands and factor-specific technical inefficiencies can be correlated across factors.