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