5
towns grows at the rate of and the number of inhabitants
of the town grows at the rate of ^ then after a
sufficiently long time there will be a steady state
distribution with Pareto coefficient yp •
Mandelbrot (1960,1961) deals with the problem from the
point of view of a mathematician and therefore on a
very general level. He starts from the concept of stable
a
laws (compare Feller Vol II ch.VI ). If thre sum of
independent identically distributed random variables
is distributed in the same way afjits components, except
for a scale factor and possibly of a location factor,
then this distribution is stable. The best known example
is the normal distribution. It has been shown by P.Levy
that there is a class of distributions with infinite
variance which are stable and which converge to the law
of Pareto when the variable in question ( say income )
tends to infinity. The Pareto law in this context is
confined to the range 1 2. Mandelbrot surmises,
owing to the reproductive quality, in the above sense,
of the Pareto law, that its importance empirically must
be very great. He also considers that this must have
implications for some statistical meljtods which depend
on the assumption of normalcy.
As to income Mandelbrot suggests that it can be regarded
as composed of a number of independent elements which
are identically distributed. We can easily imagine
f
decomposition into a few parts such as earned income,
property income and transfer income. Mandelbrot requires^,
however, in order to assure convergence, a large number of
components, and these, as he admits, have hardly any