On the Gibrat Distribution

Abstract

1. IT WAS a great achievement of Gibrat2 to show that the distribution of the logarithms of some economic variates (for instance, the distribution of factories according to the number of workers) is approximately normal. The explanation of this phenomenon by Gibrat may be presented in a rigorous form as follows: Let us denote the variate X (for instance the number of workers in a factory) at a certain date by XO. Let us further assume that subsequently it undergoes a series of random independent proportionate changes mi, M2, *, mn, (Gibrat's loi de l'effet proportionnel).3 Thus at the end of the period in which these changes have taken place the value of the variate will have become XO(1+ml)(1M+m2) * (1 ++m,) and its natural logarithm=log Xo+log (1+ml)+log (1+m2)+ + +log (1 +mn). If we denote the deviation from the mean of log XO by Yo and the deviation from the mean of log (l+mk) by yk, the deviation from the mean of this expression becomes YO+yl+Y2+ +Yn. The absolute value of mk may be assumed small as compared with 1. It follows that the absolute value of log (1 +mk) and consequently that of yk is also small as compared with 1. As the second moment of yl+y2+ * +y. is equal to the sum of the second moments of Y1, Y2, * .., yn, it may be assumed that if n is sufficiently large the standard deviation of Yl+y2+ * +yn is equal to or greater than 1 (provided the standard deviation of yn does not fall below a certain level as n increases.) Thus yk is small as compared with the standard deviation of Yl+Y2+ +Yn. With this condition fulfilled the distribution of Yl+Y2+ +y,, is approximately normal (according to the Laplace-Liapounoff theorem4). Further if n is so large that the standard deviation of yl+y2+ + yn is large as compared with the standard deviation of Yo also, the distribution of Yo+yl+y2+ * * * +yn will not differ much from normality. Whatever the distribution of Y at the initial date, with the lapse of time it approaches normality more and more. 2. This argument is formally correct but it may be shown that its

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