Factor Demand and Substitution under Decreasing Returns to Scale. An Application to the Rybczynski Theorem

Abstract

In the first part of this paper, the elasticity of substitution and factor demand are determined under decreasing returns to scale. By applying the properties of homogeneous functions, elasticities of marginal products with respect to factor uses are shown to be useful when determining the elasticity of factor substitution and factor demand elasticities. Second, the results are applied to the theorem developed by Rybczynski (1955), according to which an increase in the endowment of one factor, while the endowment of the other factor and commodity prices remain unchanged, will increase production of the good intensive in this factor, and will reduce production of the good intensive in the other factor. This theorem has been developed within the context of a two-good, two-factor model under constant returns to scale. In the more general case, under variable returns to scale, the conditions for the Rybczynski theorem have been derived by Jones (1968), Panagariya (1980) and Herberg, Kemp and Tawada (1982). The case of variable returns to scale includes both decreasing and increasing returns to scale. In this paper, however, we assume the simple model under decreasing returns to scale formulated by Hansson and Lundahl (1983). We interpret the necessary and sufficient conditions for the Rybczynski theorem using the elasticities of substitution and the factor demand elasticities and derive the condition under which expansion can occur in both sectors.