Knowledge that Transforms

PROFESSOR HOTELLING has not questioned my reasoning, I am glad to note, but only the realism of the hypotheses employed, especially the hypothesis of rapid expansion. It is true, of course, that few such cases of rapid expansion as Henry Ford's could be cited. Nevertheless I believe Professor Hotelling's point is not well taken and for two chief reasons. (1) Though such high rates of expansion are relatively rare, they play a major role in the development of the country. (2) No such high rates are necessary to substantiate the realism of my contentions. As to (1) I will quote from my book' on the subject (which has come out since Professor Hotelling's manuscript was written).

To STUDY THE CAUSES of income distribution and its changes, it is necessary to formulate and test hypothetical random differential equationis describing the individual income growth in probability terms. Assume that everyone has started in 1910 with the same income, and that each subsequent year one-half of the families, chosen each year at random, has an income raise, the other half an income cut, of $1.00 a year. With a constant population, the mean income (a measure of prosperity) would remain the same. But the standard deviation would grow as the square root of time, because the variance of the sum of t independent random variables is equal to the sum of their t variances, which, in our case, are all equal. Hence, if the coefficient of variation (standard deviation divided by the mean) is used as the inequality measure, the inequality would turn out 10 per cent higher in 1933 than in 1929 (because /23: /19 =1.1) although, under the above assumptions there would have been no boom in the latter and no depression in the former year. Other inequality measures yield analogous results. As the second, more general example, denote by rt the income of a given family at time t, and by X a normally distributed variate with zero mean and with standard deviation (the same for all families and independent of time) o. If the individual income changes according to the random differential equation