The Pareto Distribution

Abstract

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

$$x = a\,{y^{ - \alpha }}\,{\text{or}}\,\log \,x = a' - \alpha \,\log \,y$$

(23.1)

where α (the negative slope of the straight line) is called the Pareto coefficient. The density of the distribution is

$$dx = a\,\alpha \,{y^{ - \alpha - 1}}dy$$

The Pareto coefficient is occasionally used as a measure of inequality: the larger α, the less unequal is the distribution. According to Champernowne α is useful as a measure of inequality for the high income range, whereas for medium and low incomes other measures are preferable (Champernowne, 1953).

First published in W.H. Kruskol and J. Tanur, International Encyclopedia of Statistics (New York: Macmillan, 1978) and J. Eatwell, M. Milgate and P. Newman, The New Palgrave (London: Macmillan, 1987).