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