Knowledge that Transforms

A model is presented for the derivation and implementation of optimal linear decision rule for a firm producing and dealing in a number of interacting products, and possessing partial influence on their prices. The behavior of a multi-item production-inventory complex is represented as the dynamics of suitably defined state variables under the influence of decision rules that are stable and linear in the state variables, but otherwise unspecified. The dynamical equations are stochastic owing to the presence of stochastic processes in the forcing terms. The statistical properties of these processes, together with the decision rules, determine the statistics of the outcome or the criterion functional. The optimum inventory decision is then derived as the "best" linear transformation on the past of the state variables such that the mean value of the criterion functional is optimized subject to the system constraints. [Likely published between 1961-1966.]

The sampling properties of the usual estimator of marginal product from a CobbDouglas production function are studied under the assumption of normally distributed errors. The asymptotic normality of these estimators is demonstrated and certain special cases are considered for which the asymptotic distribution is particularly simple. An alternative estimator of marginal product is suggested which has both a smaller bias and greater precision than the usual estimator. Estimators of the variance-covariance matrix of all marginal product estimators mentioned are given. These are accurate to order n-1. A GREAT deal of attention is given nowadays to the statistical properties of estimators of parameters in simultaneous equations models. By contrast, little attention is given to the statistical properties of estimators of parameters in simpler economic models. One such model that has been virtually ignored by statisticians is the Cobb-Douglas production function. The usefulness of this function in the cross section analysis of an industry should not be underrated although, as with all models, care must be taken to ensure that the data to which the function is fitted correspond reasonably well with the assumptions, both economic and statistical, implied by the function. The purpose of the present paper is to consider some of the statistical properties of estimators of marginal product obtained from a CobbDouglas function. Section 2 will discuss the estimator most frequently used (see, for example, Tintner and Brownlee [8] and Heady [4]). The third section will discuss alternative estimators which reduce bias to order n-2 where n is the size of sample. These alternative estimators may be useful when the sample size is small and the variability of observed production about the fitted Cobb-Douglas function is large. The fourth section considers the problem of estimating the variance matrices of the estimators discussed in Sections 2 and 3. A numerical illustration is given in Section 5.

Abstract : The paper reformulates Houthakker's capacity method for quadratic programming in the framework of the Simplex and dual methods for quadratic programming, thereby greatly reducing the conceptual and computational complexities of the method. It is shown that the method is applicable for all convex quadratic programming problems, including the case of a semi-definite matrix of the quadratic form and that of constraints in equality form. The method reduces in the linear programming case to a parametric version of the dual method. (Author)