Knowledge that Transforms

TH:E following note is the result of a study by the Engineering Experiment Station of Iowa State College. It represents an attempt to find the form of the functional relation between cost and output in an electric light and power plant. A broader knowledge of such relations should be useful in discussions of light-plant economics. For instance, the extent to which the rate schedule should encourage consumption increases depends partly on the shape of the cost curves.

The following article presents preliminary estimates of the economic importance of nuclear energy and is part of a study being conducted by the Cowles Commission* at the University of Chicago under the direction of Dr. J. Marschak, and Dr. Sam Schurr. Mr. Menke joined the Manhattan Project as an engineer in 1942, and is now on the staff of the Clinton Laboratories in Oak Ridge. This article has appeared in ECONOMETRIKA and is reprinted by permission of its editors. Minor revisions have been made by the author.

An estimate of y obtained by applying the method of maximum likelihood under the assumption that ut is normally distributed is consistent and asymptotically normally distributed. The asymptotic standard deviation is given in this note. Although Kendall considers many estimates of the period in his publication, he does not use the maximum-likelihood estimate although it has desirable properties in large samples that several of the other estimates do not have.4 It is interesting to compare the numerical results of using this estimate with those Kendall applies to four artificial series generated by (1), each series with a different pair of coefficients a and j3.5 If the ut (t , 2, . . . , T) are assumed to be normally distributed and if x-, and x0 are assumed to be fixed, the estimate defined by the method of maximum likelihood is obtained by substituting in (2) the estimates of a and ,B found by the method of maximum likelihood under these assumptions [see equations (8)]. H. B. Mann and A. Wald6 have

what are the conditions under which the regression of xi on x2 is linear, $, a, and ,B being independent variables and a and b being constants but unknown. The problem requires the condition for linearity of regression of xi on X2 whatever may be the values of a and b. A partial answer was given by H. V. Allen,' who has proved that if the first two moments of a and all the moments of t and ,B are finite then the necessary and sufficient condition for the linearity of regression of xi on X2 whatever a and b may be is that both t and ,B should be normally distributed. A fairly complete answer to this problem was given by the present author in a thesis submitted to the Calcutta University in 1943. The restrictions imposed on {, a, and / are that their first moments exist. The proof is now extended to a more general case which is discussed below together with a general problem that may be of interest to economists. Let xi and X2 be two variables with the joint probability density G(x1, X2) and marginal distributions Gi(x,) and G2(x2). If the regression of xi on X2 is linear and the mean values of xi and x2 exist, in which case they may be supposed to be zero without loss of generality, then the regression equation may be written as