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998 SIZE DISTRIBUTIONS IN ECONOMICS where P(n, £) is the probability that a firm of age £ has n or more than n customers. The term r(£) is the density of the age distribution of firms, in cluding dead firms; for large £ it is the steady state of a renewal process and is given by r(t) = ce~ ft , where e is the net rate of growth of the firm popu lation and c is a constant. The number of firms with less than one customer, P(0, £) — P(l, £), equals the dead firms. The value of P(n, £) is ob tained as the solution of a birth-and-death process for the customers of a firm: j(l — a) n 1 J(1 — a' ) n where a = e a a' = (jjl/\)e~ a and X > /x. This expression can be expanded in series and fh- serted in the above integral; this yields, integrating term by term, w-i /^W-nVn-lN P(n) = CYY A fe / p + k + a) 00 =cr.cW n ) b(w, <o+p), v-o ' / where <o = e/(X — /x) > 0 and B(n, a) + v) is the Beta integral. Hence, P(«) = CyY-£Y—4—:B(n, oj + v). If /x/X < 1, we can neglect the terms with v above a certain value. Thus, if n °o and v has a mod erate value, we can use the approximations B(n, a) -f v) n~ <a ~ ,/ V(o) + v), B(n, r>) « n~ v T{v); therefore, as n -» °o, P(n) -» C'n - ". The following features of the solution may be remarked: Since the approximation depends on the value of /x/X, which is the mortality of firms of high age, the smaller the mortality of firms, the greater will be the proportion of the distribution that conforms to Pareto's law. The mean of the distribution will be finite if o > 1. This is important in connection with disequilibria, which can arise through changes in X, /a, and e. It can be shown that the Pareto solution applies to the growing firm (X > /x, the above case) and, in a modified form, to the shrinking firm (X</x); but it does not obtain for the stationary firm (X - /x). The above solution for the distribution according to customers can be shown to be valid also for the distribution according to sales, if firms grow mainly by acquiring more customers and not by getting bigger customers. This is often true in retail trade but not in manufacturing. An alternative model assumes the other extreme—that firms grow only by getting bigger orders. This model is based on the theory of collective risk. The capital of the firm, a continuous variable, is subject to sudden jumps at the instant when orders are executed and to a continuing drain of costs, which is represented deterministically by an exponential decline. The steady state solution obtained from this process is, for large values of capital, identical with the Pareto law; for moderate values, the distribution has a mode and represents, albeit with some complica tions, a modification of the “first law of Laplace," which was proposed by Frechet (1939) for income distributions. Size as a vector. It would be natural to measure the size of a firm by a vector, including employ ment, output, capital, etc., and apply the steady state concept to the joint distribution of several variables. Regression and correlation coefficients obtained in a cross section could then be regarded, like the Pareto coefficient, as characteristics of the steady state. It may be guessed that the growth of the number of firms will have an influence on these parameters as well. Practically no work has been done in this direc tion, but it is the only way to clear up the meaning of cross-section data and their relation to time series data and to the theoretical parameters of the underlying stochastic process. The situation in economics is totally unlike that in physics, where the processes are stationary and the ergodic law establishes the identity of time and phase averages. (Only the cosmogony of F. Hoyle, in which the continuous creation of matter offsets the expansion of the universe to establish a steady state of the cosmos, offers a parallel to the growth processes considered above.) The surprise expressed at one time at the difference in estimates of income elas ticities from cross-section data and from time series data appears naive in this light because we could only expect them to be equal if the processes gen erating households, incomes, and consumption were stationary. But the population of households or the popu lation of firms is not stationary. A cross section of firms shows the growth path of the firm through its different stages of evolution; but the number of firms of a given age depends on the past growth of the total number of firms, and this may influ ence the regression coefficient. Moreover, the growth path is not unique, because there are several proc esses superimposed upon one another (growth