THE PROBABILITY DISTRIBUTION OF THE AVERAGE MARGINAL PRODUCTS OF COBB-DOUGLAS FACTORS WITH APPLICATIONS

The comparative productivity of inputs is often the focus of applied studies of production. For example, one might wish to compare the marginal product of water in irrigation districts to provide guidelines for capital investment in new water projects or to assist in water allocation decisions. The purpose of this note is to illustrate a straightforward econometric method for making such comparisons. It is assumed that panel data are available for T periods on N cross-sectional units and that output is produced according to a stochastic Cobb-Douglas production function. The Cobb-Douglas function was chosen because it has a simple formula for the marginal product and has been found to perform well in studies of agricultural production. The statistical results derived in the paper are obtained assuming that the error term of the production function is independently and identically distributed. This assumption can be easily relaxed and the asymptotic results presented in the paper can be generalised for the case where the variance-covariance matrix of the error term is known or can be estimated consistently. An important generalisation is the Zellner (1962) seemingly unrelated regressions framework. Only the simplest case is developed here because of space limitations. To illustrate the wide applicability of these results a test for risk aversion is developed which can be applied when the production setting satisfies specified assumptions.