996 SIZE DISTRIBUTIONS IN ECONOMICS
to increasing variance during their lifetimes. The
process of birth and death of income earners, which
is explicitly introduced into the model, acts as the
stabilizer.
The distribution of total income is obtained by
summing the distributions for all age cohorts that
contribute survivors. Rutherford's method is to de
rive the moments of the distribution by integration
over time of the moments for the entrance groups.
The distribution is built up "synthetically” from the
moments, as it were. In the absence of an analyti
cal solution with a definite distribution law, some
disagreement remains about the result.
Simons model. In Simon's model (1955), which
leads to what he calls the Yule distribution, the
aggregate growth of firms, cities, or incomes is
given a priori. The stochastic process apportions
this given increment to various units according to
certain rules, which are weakened forms of the law
of proportionate effect and rules of new entry. As
a consequence of this procedure, there is no possi
bility of shrinkage of individual units. The given
aggregate emphasizes the interdependence of for
tunes of different firms (the gain of one is the loss
of another)—a point that is neglected in other
models, such as that of Steindl (1965). On the
other hand, the aggregate is, in reality, not given;
it is not independent of the action of the firms,
which may increase their total market by adver
tising, product innovation, and so on.
The process of apportionment may be described
as follows. We may conveniently think of popula
tions of cities, so that f(rz, N) is the frequency of
cities with n inhabitants in a total urban popula
tion of N; to be realistic, we shall assume that a
city exceeds a certain minimum number of inhabi
tants; n will measure the excess over this mini
mum, and N will correspondingly be the sum of
these excess populations. An additional urban in
habitant is allocated to a new city with a proba
bility a and to an existing city, of any size class,
with a probability proportionate to the number of
(excess) inhabitants in that size class. Then,
f(n, JV -h 1) — f(n, N)
= [(« - 1 )f(n - 1, N) - nf(n, N)],
f(l,N+l)-f(l,N)
We assume that there is a steady state solution
in which the frequencies of all classes of cities
change in the same proportion, that is, in which
f(tt, N + 1) N i 1
“ f(n, NJ~ ' N ""
for all n.
Using this relation and defining a relative frequency
of cities as fCn ) ~ f(n, N)/(aN), we obtain from
the above equations
// \ /v 1 n (l “ oc)(n — l)
or, setting l/( 1 - «) = p, d f )
(n- 1 )(n — 2) • • • 2 • 1
f(n) =
(n + p)(n + p - 1) • • • (2 + p)
f( 1)
r(n)IYp + 2)
F (n + p + 1)
f( 1).
This expression is the Yule distribution. Using a
property of the T-function, it can be shown that the
Yule distribution asymptotically approaches the
Pareto law for large values of rz, that is, f(n)~*
n p A f( l)T(p + 2) as rz -> °o.
This model is applicable to cases in which size
is measured by a stock, for example, number of
employees of a firm. Simon provides an alternative
interpretation of it that applies to flows, such as
income and turnover of firms. For example, the
total flow of income is given, and each dollar is
apportioned to existing and new income earners
according to the rules given above.
Using simulation techniques, Ijiri and Simon
(1964) show that the pattern of the Yule distribu
tion persists if serial correlation of the growth of
individual firms in different periods is assumed.
This finding is important because, in reality, growth
is often affected by “constitutional'' factors, such
as financial resources and research done in the
past.
The 7iiodel of Wold and Whittle. Wold and
Whittle (1957) present a model of the size distri
bution of wealth in which stability is provided by
the turnover of generations, as in Rutherford’s
model. On the death of a wealth owner, his fortune
is divided among his heirs (in equal parts, as a
simplification). The diffusion effect is provided by
the growth of wealth of living proprietors, which
proceeds deterministically at compound interest.
The model is shown to lead to a Pareto distribution,
the Pareto coefficient depending on the number of
heirs to an estate and the ratio of the growth rate
of capital to the mortality rate of the wealth owners.
Steindis models. Steindl’s models (1965, chap
ters 2, 3) are designed to explain the size distri
bution of firms, but they can equally well be applied
to the size distribution of cities. The distribution
laws obtained are. for large sizes, identical with
the Pareto law. Like Rutherford’s model, Steindl’s
models rest on a combination of two stochastic