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