CES and VES Production Functions in a Cross-Section Context

Abstract

Recent years have witnessed a growing interest in variable-elasticity-ofsubstitution (VES) production functions. Alternative parametric forms have been derived, analyzed, and tested on various bodies of data by Lovell (1968), Lu and Fletcher (1968), and Revankar (1971), among others. Several studies have found the elasticity of substitution between capital and labor to vary, usually inversely and often significantly, as capital deepening occurs. Such evidence naturally casts some measure of doubt on the empirical usefulness of any constant-elasticity-of-substitution (CES) specification as a descriptive tool.' The purpose of this study is to shed some theoretical and empirical light on the subject. It is well known that time-series evidence can be misleading if consideration is not accorded the manner in which technical change affects a production function. If technical change is biased in Hicks's sense, the net relationship between the elasticity of substitution and factor proportions along an isoquant is obscured by a nonradial displacement of the isoquant map as technical change occurs. Cross-section evidence is subject to analogous, but less widely recognized, reservations. Nonneutral geographical variation in production technology causes the same estimation problems as biased technical changes does in a time-series context. In either case discrimination between alternative functional forms becomes a difficult proposition. After elaborating on this point in Section II, we introduce two specifications of an aggregate production function that are amenable to estimation in a cross-section context. One function is CES, the other VES, and both encompass the Cobb-Douglas (CD) form as a special case. Both functions contain three factors of production (capital, production labor, and non-