Production Functions with Variable Elasticity of Factor Substitution: Some Analysis and Testing

Abstract

A POWERFUL index of important properties of a production function is its elasticity of factor substitution function. Under certain assumptions such as first degree homogeneity (H.D. 1), the form of the production function is uniquely determined by the form and value of the elasticity of factor substitution function. Production functions generated under the assumption of constant elasticity of factor substitution (CES) [1] have been extensively studied both in regard to their theoretical properties and their empirical implications. Once one drops the assumption of constancy and admits a variable elasticity of factor substitution (VES), the resultant production function depends on the assumptions involved in the elasticity of factor substitution function. This paper proposes to show the extent to which one can generalize beyond the assumption of CES and still derive an explicit form of the production function. A pair of approaches to particular explicit VES production functions, each derived from a different hypothesis regarding the (variable) elasticity of factor substitution function will be discussed. One approach relates the elasticity of factor substitution to capital intensity.' It is shown that if the elasticity of factor substitution is a linear function of the capital labor ratio, then a unicue explicit production function exists [12]. This result is extended to cases where the shares of capital and labor are linear functions of the capital labor ratio. The second approach considers the elasticity of factor substitution as a linear function of time. The result is a production function which may be characterized as a shifting CES. The basic assumptions of our analysis appear in section II. In section III we begin with a variable elasticity of factor substitution function o-(k) and show its implications for VES production functions. Section IV deals with the special case of a VES production function developed from the hypothesis that capital's share (a) is a linear function of k. Empirical tests of this case using both United States and Japanese data are reported. In section V we develop a VES production function where cr is a linear function of k. Some estimates of the VES production function where o= 1 + bk are reported in section VI. Section VII deals with the VES production function which is derived for the case where cr is a linear function of time.