Scale economies, elasticities of substitution

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by Pablo Coto Millan
Departamento dc Economía
Universidad de Cantabria
Avda. de los Castros s/n
39005 – Santander


In this paper the behaviour of railroad transport costs in Spain is analyzed. Generalized cost functions are estimated. From estimatedequations, the Allen and Morishima elasticities of substitution are calculated. This paper indicates that the great number of Allen elasticities of substitution that have appeared in the literature should be interpreted with caution. In addition, we show that the industry is characterized by constant returns to scale.


En esta nota se analiza el comportamiento de los costes deltransporte de ferrocarril en España. Se estiman funciones de costes generalizados. Con las ecuaciones estimadas se calculan las elasticidades de sustitución Allen y Morishima. Este articulo muestra que gran numero de elasticidades de sustitución de Allen que han aparecido en los diferentes trabajos deben ser interpretadas con precaución. Adicionalmente se muestra que esta industria posee una tecnologíacaracterizada por rendimientos constantes de escala.
1.- Introduction

The wish to know the business reaction to the relative prices of production factors, has led to studies of cost and production functions. The elasticity of substitution, formulated by Hicks in 1932 (Ferguson, 1979), is the key concept provided by such a function. In 1938 Allen extended the concept of elasticity ofsubstitution to multiproduct technologies, generating a new concept which carries his name in the economics literature. Subsequently, according to this, there have been some contributions on constant elasticity of substitution production functions, such as the labourby Arrow et al. (1961). However, more recent studies question the accuracy of the Allen elasticity of substitution, since it is aone-factor one-price elasticity and is a derived demand elasticity divided by a share proportion. Thus, other authors state that the Morishima elasticity (Blackorby and Russel, 1981) provides some information about economics which is more relevant than Allen's. Blackcorby and Russell (1989) point out the need to assume both of these elasticities of substitution, and warn seriously about the use of theAllen elasticities of substitution when multifactor technologies are being studied.

Railway transport is managed in Spain by a state company, RENFE, which has the monopoly of passengers and goods transport services. Transport prices are fixed by the government which not only finances the differences between the operating costs of the services and the revenues produced by them, but it alsoprovides the infrastructure.

In the present paper, I estimate cost functions for the passengers railway transport by the state company RENFE, from 1964 to 1992. The functional form used is the translog drawn by Christensen, Jorgensen and Lau (1973). I also estimate the Allen and Morishima elasticities of substitution, and I compare and interpret the results as well.

2.- The Model

Thepresent model is based upon Keeler's (1974) research , and Caves et al. (1981).

It is assumed that the production function has only one output and three inputs. The railway activity is multiproduct as regards passengers and goods services, departures and arrivals, timetables, and so on. For this reason, the production function must strictly have a vector of products. There are also fourfactors at least. However, the availability of data forces us to consider only three.

It also assumed that production input prices are exogenous.

Due to these two assumptions about product homogeneity and exogeneity of inputs prices, it is possible to represent a transformation surface of combinations of the three production factors in order to obtain the output implicitly:

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