SIZE DISTRIBUTIONS IN ECONOMICS 99 5
tribution. There must be a stabilizing influence to
offset the tendency of the variance to increase;
indeed, a distinguishing feature of the various
theories presently to be reviewed lies in the kind
of stabilizer they introduce to offset the diffusion.
Two interesting cases may be noted here. One
possibility is to modify the law of proportionate
effect and assume that the chances of growth de
cline as size increases. This approach has been
taken by Kalecki (1945), who assumes a negative
correlation between the size and the jump and
obtains a Gibrat law with constant variance. An
other possibility is to combine the diffusion process
of the random walk with a steady inflow of new,
small units (firms, cities, incomes). Some units
may continue indefinitely to increase in size, but
their weight will be offset by that of a continuous
stream of many new, small entrants, so that both
the mean and the variance of the distribution will
remain constant. This approach, which leads to
the Pareto law, has been taken by Simon (1955).
Review of various models. Descriptions of var
ious models will illustrate the methods employed.
Models differ with regard to the distribution law
explained, the field of application (towns, incomes,
etc.), and the type of stochastic process used.
Champernoivne’s model. Champernowne (1953 )
presents a model that explains the Pareto law for
the size distribution of incomes. The stochastic
process employed is the so-called Markov chain [see
Markov chains]. The model is based on a matrix
of probabilities of transition from one income class
to another in a certain interval of time, say a year.
The rows are the income classes of one year, the
columns the income classes of the next year. The
income classes are chosen in such a way that they
are equal on the logarithmic scale (for example,
incomes from 1 to 10, from 10 to 100, etc.). The
probability of a jump from one income class to
the next income class in the course of a year is
assumed to be independent of the income from
which the jump is made (the law of proportionate
effect). The number of income earners is constant.
The number of income earners in income class s
is then determined as follows. The number of in
comes in class s at time £ + 1 is
f(s, t+l) = Sf(s- W„t)p(ti),
u--n
where s, it, and t take on integer values, p(u) is
the probability of a jump over u intervals (i.e.,
the transition probability), and the size of the jump
is constrained to the range +1, —n. In the steady
state equilibrium reached after a sufficiently long
time has passed, the action of the transition matrix
leaves the distribution unchanged. We then have
f(s) = Y,f(s-u)p(u), s >°.
u = -n
as This difference equation is solved by
putting f(s) = z\ The characteristic equation
g(z) = Z z 1 U p(u) -z = o
w=-n
has two positive real roots, one of which is unity.
To assure that the other root will be between 0
and 1, Champernowne introduces the following
stability condition:
g'(i) = - S up(u) > o.
M = -n
The relevant solution is f(s) = b*, 0 < b < 1, which
gives the number of incomes in income class s. If
the lower bound of this class is the log of the
income Y a , then the probability of an income ex
ceeding Y s is given by
log P(Y f ) = s log b.
Since s is determined by
log Y, = sh + log Y mjn)
where h is the class interval and Y min is the lower
boundary of the lowest income class, it follows that
log P(Y*) = y-arlogY,,
where the parameters y and a are determined by
b, h, and Y min . This is the Pareto law with Pareto
coefficient a.
Champernowne’s stability condition implies that
the mathematical expectation of a change in in
come is negative. This counteracts the diffusion.
How can the stability condition be justified on
economic grounds? It may be connected with the
fact that in this model every income earner who
drops out is replaced by a new income earner.
Since, in practice, young people have on the aver
age lower and more uniform incomes than old
people, the replacement of old income earners by
young ones usually means a drop in income. Thus,
Champernowne’s stability condition, as far as its
economic basis is concerned, is very similar to the
entry of new, small units that act as a stabilizer
in Simon’s model.
Rutherford's model. Rutherford’s model (1955)
leads, in his opinion, to the Gibrat law for the size
distribution of incomes. Newly entering income
earners, assumed to be log-normally distributed at
the start, are subject to a random walk and thus