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,
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