α-Returns to scale and multi-output production technologies

Abstract

This contribution proposes a specification of strictly increasing and decreasing returns to scale in multi-output technologies. Along this line a notion of α-returns to scale is derived from that of homogeneous multi-output technology. For a large class of technologies we establish necessary and sufficient conditions characterizing strictly increasing and strictly decreasing returns to scale to scale. Furthermore, a relationship between input, output and graph distance functions is established. These connections lead naturally to a link between the various Malmquist indexes and the Chavas–Cox productivity index. Finally, we show that these concepts can be implemented in a DEA context using a piecewise homogeneous constant elasticity substitution–transformation model due to [Färe, R., Grosskopf, S., Njinkeu, D., 1988b. On piecewise reference technologies. Management Science 34, 1507–1511].

Introduction

Productivity growth – defined as output variations that are not explained by input variations – is made up of technical efficiency gains and technological change. Macroeconomic productivity growth measures usually neglect inefficiencies in input usage or output production and identify-via the Solow residual – total factor productivity growth as a shift in technology. More complete measures, based upon the Malmquist (1953) index, separate productivity growth into its two components. Adopting the method of distance functions, the interest of this approach is reinforced since (1) it imposes no a priori functional form – thus no restrictive assumptions regarding input remuneration-, (2) it allows for productive inefficiencies in production and (3) it offers the “benchmarking” of an economy’s performance with respect to both its past experience and the best practices established by other countries.

Internal economies of scale, or increasing returns to scale in more modern language, occur if output expansion leads to a less than proportional increase in the use of inputs. This idea was suggested first by Smith (1776) and refined further by Marshall (1898) but the literature on productivity measurement seems to have much neglected the case where production technologies satisfy strictly increasing returns to scale. This might be due to the fact that strictly increasing returns to scale imply a nonconvexity assumption of the production technology. At least to our knowledge, there do not exist multi-output empirical models measuring productivity and postulating strictly increasing returns to scale. However, there exist in industrial organization theory and in growth theory a tremendous number of models based on these assumptions. Increasing returns to scale play a crucial role in oligopolistic competition and network economics. Especially, the existence of fixed costs and the possible exploitation of economies of scale have some strong implications on the market structure. The focus on increasing returns also arises from the fact that it provides a simple and plausible explanation of intra-industry specialization and of non-traditional comparative advantages. In turn, the introduction of increasing returns into the analysis has left no room for the assumption of perfect competition, thus opening the door for an increasing integration of international trade theory and industrial organization studies. In growth theory, Romer (1986) went to great lengths to disqualify the restriction of a constant returns to scale assumption in growth models. Romer claimed that the rate of growth of the capital alone may yield increasing returns. Hence, if growth is to be fostered by greater productivity, such is enhanced, by implication, in capital-intensive production processes.

Since the early eighties, some models for estimating nonparametric technologies have been developed to characterize nondecreasing and nonincreasing returns to scale. The seminal model was introduced by Charnes et al. (1978) and it postulated a constant returns to scale assumption. Banker et al. (1984) have introduced a model implying a variable returns to scale assumption and Färe et al. (1994a), proposed a taxonomy of such nonparametric models. However, all these nonparametric models are also based upon a convexity assumption and a dual representation of the production technology using the profit function.

In this paper, we propose to model returns to scale in almost homogeneous technologies. This terminology comes from Lau (1978) who has analyzed the structure of profit functions associated with homogeneous multiple output technologies. To simplify the terminologies, we refer to “homogeneous technology” throughout the paper. More recently Färe and Mitchell (1993), have established necessary and sufficient conditions to characterize a technology that is homogeneous of degree α. Among other things they established some link between input and output distance functions under an assumption of almost homogeneous technology. In this paper, we investigate this concept and its relationship to returns to scale. In particular, defining strictly increasing and strictly decreasing returns to scale, we justify a notion of α-returns to scale. It should be pointed out that the case of strictly decreasing returns to scale is probably less relevant. It is mainly added as a matter of treating the alternative case symmetrically. Following traditional literature on productivity analysis, we analyze some implications of this notion on efficiency measures and distance functions. These measures evaluate the proximity between a firm and a frontier of technical efficiency of the technology to make comparisons of performances. In particular, we establish several connections between input and output Farrell technical efficiency measures (see Farrell, 1957), the hyperbolic measure (see Färe et al., 1985) and the recent generalized distance function introduced by Chavas and Cox (1999). In any case the relationship we establish depends on the degree of homogeneity of the technology.

Since technical efficiency measures are interesting tools to evaluate productivity growth, we study the impact of returns to scale on productivity measures. Along this line we propose a Chavas–Cox productivity index we relate to the traditional measures of productivity.

Distance functions and productivity indexes are especially useful to measure technical efficiency and productivity changes when the technology is modelled using nonparametric methods. A taxonomy to model variable, constant, nondecreasing and nonincreasing returns to scale was proposed among others by Färe et al. (1985). Finally, we show that these concepts can be implemented using a piecewise homogeneous constant elasticity substitution–transformation model due to Färe et al. (1988b). Along this line, a nonparametric model involving α-returns to scale is proposed. Hence, given a set of firms operating on the same sector of the economy, one can model a production technology involving either strictly increasing or strictly decreasing returns to scale. More importantly, these distance functions and productivity indexes can be calculated using linear programming.

The paper unfolds as follows: Section 2 lays down the ground-work presenting distance functions and traditional productivity indexes. In addition homogeneous multi-output technologies are introduced and a formal definition of strictly increasing and decreasing returns to scale is proposed. Section 3 investigates the implications of α-returns to scale on input, output and directional distance functions. Furthermore, the impact of an α-returns to scale assumption on productivity measurement is analyzed. In particular we focus on the Malmquist productivity index. Finally, Section 4 proposes a nonparametric CES-CET production model satisfying α-returns to scale.