Factor Substitution and the Stability of Two-Sector Growth Models: Generalizations and a Synthesis

Abstract

The many possible applications of twosector models of economic growth to other fields in Economics indicate a need for generalizing the stability conditions of the models to less restrictive terms. The purpose of this paper is to generalize the existing conditions imposed on the elasticities of factor substitution for causality and stability.' The work in this area has been done mainly by Drandakis [6], Amano [1], and also by Sato [10]. Using aggregate elasticities of factor substitution and the properties of its components, we are able to extend the causality region (Causality Theorem II(c)) and prove the existing results in a very simple and systematic way. Similarly, all the stability conditions which hitherto required that the elasticity of substitution be no less than unity have been extended to conditions that it merely be no less than certain fraction of unity. Such extension thus conforms with the recent empirical studies [9] on the estimation of the elasticities of substitution. Other conditions are also given in this paper. These conditions are derived for the general proportional saving function (Stability Theorem II), the general classical saving function (SIII), and the general Keynesian saving functions (SIV). They are again proved by using simple arguments and synthesized with the existing conditions through the use of aggregate elasticities of factor substitution and its components. Five causality conditions and fifteen stability conditions are then classified in a table in accordance with technical conditions (elasticities of substitution), behavioral conditions (saving and capital intensity assumptions), and mixtures of these conditions. We then derive the necessary and sufficient conditions for causality and stability of the two-sector growth model. By these conditions, the effectiveness of the causality and stability theorems, all of which are sufficient conditions, can be illustrated. Lastly, it is also shown that all the stability conditions imposed directly on the aggregate elasticity of factor substitution can be resolved into at least one of the more specific existing conditions. Instead of the usual approaches, the equations in this paper are expressed in terms of distributive shares of factors of production in each industry. Since the values of these shares are restricted to the unit interval, the range of the value of the variables expressed in this way is more easily determined. The approach generally yields more familiar economic interpretations and lends itself to further manipulations as well as symmetric expressions.