A Monte Carlo Study of Alternative Estimates of the Cobb-Douglas Production Function

Abstract

A Monte Carlo experiment is carried out to examine the small sample properties of ordinary least squares, indirect least squares, Hoch's, and Klein's estimates of the parameters of the Cobb-Douglas production function. A perfectly competitive model of firms irn a single industry is considered in nine situations which differ in the behavior of the disturbances, the variability of inputs, and the position of the average firm. In each case 200 samples of size 20 anld 200 samples of size 100 were obtained to approximate the sampling distribution of the various estimators. THE CONDITIONS of profit maximization and the specification of the production function fully determine the equilibrium position of a firm that operates under conditions of perfect competition in the product market, obtains its inputs at fixed prices, and experiences decreasing returns to scale. If all the relationships hold exactly, all firms in the industry will be producing identical quantities of output and will be employing identical quantities of inputs, providing the inputs are freely variable and substitutable.2 Variations from firm to firm will exist if one or more of the inputs are fixed; in this case, the profit maximizing quantities of output and of inputs will depend on the amount of the fixed input or inputs in each firm. If, however, the production function as well as the profit-maximizing decision equations contain stochastic disturbances, differences in actual outputs and inputs of firms will appear even in the absence of fixed factors of production. In this case a solution of the system of equations for the quantities of variable inputs and of output of any firm shows that each quantity is a function of all disturbances in the system. Consequently, the inputs are not independent of the disturbance in the production function, and single-equation least-squares estimates of the production function parameters based on cross-sectional data will be, in general, biased and inconsistent. This was first noted in a classical article by Marschak and Andrews [5] in 1944. Alternative methods of estimation have since been proposed; these include Klein's, Hoch's, and the indirect leastsquares procedure. With the exception of the first, no small sample properties of the suggested estimators have been derived. This paper represents EDITORS' NOTE: We regret to record that Mr. M. E. Joseph is now deceased and his authorship of the final version of this article is posthumous.