Full text: 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 
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)], 
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 
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

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