On Estimation of the CES Production Function

Abstract

THE ORIGINAL SPECIFICATION of the constant-elasticity-of-substitution (CES) production function by Arrow, Chenery, Minhas, and Solow [1] was restricted to the case of constant returns to scale.With this restriction it is possible to estimate the elasticity of substitution from the marginal productivity condition by regressing the value of production per worker on wage rate (both variables measured in logarithms).If, however, the CES production function is generalized to allow for the possibility of non-constant returns to scale, this method of estimation is no longer feasible.The purpose of this paper is to consider estimation procedures applicable to the generalized version of the CES function under various circumstances.An obvious starting point is to consider estimates obtained by fitting the production function to observations on output and inputs alone.These estimates are consistent if the input variables are non-stochastic or, if stochastic, independent of the disturbance in the production function.The CES function can be written in the form (1.1) log Xi log r--log [JK -+ (1 -)Lt ] + ui .The subscript i refers to the i-th firm, and ut is the stochastic error term assumed to be independently and normally distributed with, zero mean and constant variance.This specification is analogous to that used in the context of the Cobb-Douglas function.The parameters of (1.1) could be estimated by nonlinear least squares methods for which computer programs are now available.An alternative method, based on simple least squares estimation, is possible if we replace (1.1) by its approximation which is linear in p.This can be derived by using Taylor's formula for expansion around p = 0.2 After disregarding the terms of third and higher orders, the expansion leads to3 log Xi log 7 + >d log K& + v(1 -3) log Li (1.2) 1 pl1 -3)[log Ki -log L&]2 + us *The approximation to the CES function given by (1.2) can then be con-*