Full text: Size Distributions in Economics

The size distributions of certain economic and 
socioeconomic variables—incomes, wealth, firms, 
plants, cities, etc.—display remarkably regular pat 
terns. These patterns, or distribution laws, are usu 
ally skew, the most important being the Pareto law 
(see Allais 1968) and the log-normal, or Gibrat, 
law (below). Some disagreement about the pat 
terns actually observed still exists. The empirical 
distributions often approximate the Gibrat law in 
the middle ranges of the variables and the Pareto 
law in the upper ranges. The study of size distribu 
tions is concerned with explaining why the ob 
served patterns exist and persist. The answer may 
be found in the conception of the distribution laws 
as the steady state equilibria of stochastic processes 
that describe the underlying economic or demo 
graphic forces. A steady state equilibrium is a 
macroscopic condition that results from the bal 
ance of a great number of random microscopic 
movements proceeding in opposite directions. Thus, 
in a stationary population a constant age struc 
ture is maintained by the annual occurrence of 
approximately constant numbers of births and 
deaths—the random events par excellence of hu 
man life. 
The steady state explanation is evidently inspired 
by the example of statistical mechanics in which 
the macroscopic conditions are heat and pressure 
and the microscopic random movements are per 
formed by the molecules. Characteristically, the 
steady state is independent of initial conditions, 
i.e., the initial size distribution. In economic appli 
cations this is important because it means that the 
pattern determined by certain structural constants 
tends to be re-established after a disturbance is 
imposed on the process. This will only be the case, 
however, if the process leading to the steady state 
is really ergodic, that is, if the influence of initial 
conditions on the state of the system becomes 
negligible after a certain time; and it will be rele 
vant in practice only if this time interval is suffi 
ciently short. 
The idea that the stable pattern of a distribution 
might be explained by the interplay of a multitude 
of small random events was first demonstrated in 
the case of the normal distribution. The central 
limit theorem shows that the addition of a great 
number of small independent random variables 
yields a variable that is normally distributed, when 
properly centered and scaled. A stochastic process 
that leads to a normal distribution is the random 
walk on a straight line with, for example, a 50 per 
cent probability each of a step in one direction and 
a step in the opposite direction. It is only natural 
that attempts to explain other distribution patterns 
should have started from this idea. The first exten 
sion was to allow the random walk to proceed on 
a logarithmic scale. The resulting distribution is 
log-normal on the natural scale and is known as 
the log-normal or Gibrat distribution. The basic as 
sumption, in economic terms, is that the chance 
of a certain proportionate growth or shrinkage is 
independent of the size already reached—the law 
of proportionate effect. This law was proposed by 
J. C. Kapteyn, by Francis Galton, and, later, by 
Gibrat (1931). 
O Let size (of towns, firms, incomes) at time t be 
denoted by Y(t), and let e(t) represent a random 
variable with a certain distribution. We have 
Y(t) = (1 + e(t))Y(t — 1) 
= Y(0)(l+e(l))---(l+e(t)), 
where Y(0) is size at time 0, the initial period. For 
small time intervals the logarithm of size can be 
represented as the sum of independent random 
variables and an initial size which will become 
negligible as t grows: 
log Y(t) = log Y(0) +6(1)+ 6(2) • • •+€(*). 
If the random variables e are identically distributed 
with mean m and variance cr 2 , the distribution of 
log Y(t) will be approximately normal with mean 
mt and variance a 2 t. 
This random walk corresponds to the process of 
diffusion in physics which is illustrated by the so- 
called Brownian movement of particles of dust 
put into a drop of liquid. Since it implies an ever 
growing variance, the idea of Gibrat is not itself 
enough to provide an explanation for a stable dis

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