Pareto Distribution
Using certain data on personal income V.Pareto (1897 )
plotted income on the abscissa and the number of people
who received more than that on the ordinate of logarithmic
paper and found a roughly linear relation. This
Pareto distribution or Pareto law may be written as
—* / . .
x = a y or log x = a — log y (1)
where oi ( the negative slope of the straight line )
is called the Pareto coefficient. The density of the
distribution is
dx = a c< y dy
irv
The Pareto coefficient is occasionally used as a measure
of inequality: The larger 0( the less unequal is the
distribution. According to Champernowne 0( is useful
as a measure of inequality for the high income range
whereas for medium and low incomes other measures are
preferable (Champernowne 1952 ).
(£ takes only positive values. If o( ^ 2 the distribution
has no variance; if oC ^ 1 it has no mean either.
In practice the Pareto law applies only to the tail of
the empirical distributions i.e. to incomes above a certain
size. Thus the law (1) is valid assymptotically as y —^ oo
The range in which the empirical distributions conform
i
to thejlaw is different in different cases. It seems to
be larger for wealth than for income ( perhaps because
we have only data for large wealth ) and even larger for
tuwns. In the case of firm sizes only the very large firms
are covered by the law.