Full text: Pareto Distribution

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 
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 
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.

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