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SIZE DISTRIBUTIONS IN ECONOMICS
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