SIZE DISTRIBUTIONS IN ECONOMICS 997 
processes. One is a birth-and-death process of the 
population of cities or firms; the other is a sto 
chastic process of the growth of the city or firm 
itself. 
The way in which the interplay of these two 
processes brings about the Pareto law can be ex 
plained in elementary terms. We start with the size 
distribution of cities. The number of cities can be 
explained by a birth process, if we assume that 
cities do not die. Let us assume that new cities are 
appearing at a constant rate, e, the birth rate of 
cities. The number of cities increases exponentially, 
and the age distribution of cities at a given moment 
of time is 
(1) R(t) =R(0)er«*, 6 > 0, 
where t is age and R(t) is the number of cities 
with age in excess of t; in other words, R(t) is the 
rank of the town aged t 4- dt, and R( 0) is the total 
number of towns existing at the moment of time 
considered. The size of the city—its number of 
inhabitants—increases, on the average, with age. 
If the rate of births plus immigration, X, and of 
deaths plus emigration, fi, are constant, we obtain 
an exponential growth function for the size of the 
city: 
(2) n(t) = X > /i. 
Eliminating t between eqs. (1) and (2), we get 
(3) In R = — -r———In n 4- In R(0). 
A — fJL 
This is the Pareto law (ife/(X — /u,) > 1), and the 
Pareto coefficient is seen to be the ratio of the 
growth rate of the number of cities to the growth 
rate of a city. 
This demonstration, which on the face of it is 
deterministic in character, can be supplemented 
by a graphical illustration in which the stochastic 
features are included. In Figure 1 the distribution 
of cities according to age is plotted in the vertical 
(In R, t) plane. The abscissa shows the age of the 
city, and the ordinate shows the log of the rank 
of the city. Each city is thus represented by a dot, 
and the regression line fitted to these points repre 
sents relation (1). In the horizontal (t, In n) plane, 
we show the exponential growth of cities with age, 
as in relation (2). Again each city may be repre 
sented by a dot showing age and size. The scatter 
diagram in the horizontal plane may be regarded 
as a stochastic transformation of the time variable 
into the size variable. If the size of each city has 
been found on the scatter diagram, the cities can 
be reordered according to size; we then obtain, 
in the third (In R, Inn) plane, the transformed 
In R 
relation (3) between the number of cities (rank) 
and the size of a city. 
If firms are studied, we must take into account 
the death of firms. We might assume that a firm 
dies when it ceases to have customers. We can 
imagine that the age distribution in plane 1 of 
Figure 1 includes the dead firms; they are auto 
matically eliminated in the transformation to size, 
being transferred to the size class below one. In 
the exponential relation (1), e must now represent 
the net rate of growth of the number of firms if 
the birth of firms is assumed to be a constant ratio 
of the population. 
Figure 1 illustrates how the evolution in time 
of the number of firms (cities) is mapped onto 
the cross section of sizes. This process may be 
compared to sedimentation in geology, where a 
historical development is revealed in a cross sec 
tion of the layers. We can also see how irregulari 
ties in the evolution over time will affect the size 
distribution. If an exceptional spurt of births of 
nqw firms occurs at one point of time (after a war, 
for example), the regression line in plane 1 will 
be broken and its upper part shifted upward in a 
parallel fashion. The same thing will happen to the 
transformed distribution in plane 3. 
The complete model for firms may be described 
as follows. The size of a firm is measured by the 
number of customers attached to it. This is gov 
erned by a birth-and-death process. Let us denote 
by o(At) a magnitude that is small in comparison 
with At. There is a chance X At 4- o(At) of a cus 
tomer’s being acquired and a chance /jl At 4- o(At) 
of a customer’s being lost in a short period of time, 
At; multiple births and deaths have a chance of 
o(At). The probability that a firm has n or more 
than n customers is given by 
P(n) - [°°P(n, t)r(t) dt, 
Jo