Lorenz curve is an important tool for the analysis of inequality in a variety of situations – inequality in the distribution of income within a population, inequality in the productivity of scientists in a give population, inequality in the allocation of research grants to different institutions by a funding agency, etc.
Lorenz curve is defined as follows.
Consider a sample of n individuals. Let xi denote the income of individual i (i= 1, 2, ... , n), such that x1 £ x2 £ x3 £ . . . £ xn. The sample Lorenz curve is the polygon joining the points (h/n, Lh/Ln), where L0 = 0 and is the total income of the poorest h individuals.
The Lorenz curve q = L(y) has as its abscissa the cumulative proportion of income receivers, ordered by increasing size of their income received. The Lorenz curve can be mathematically represented as:
where Y is a nonnegative income variable for which the mathematical expectation m = E (Y) exists and p = F(y) is the cumulative distribution function of the population of income receivers. When all the members of the population receive the same income, the Lorenz curve is the equidistribution or identity function F = L. The Lorenz curve bends downward to the right, as the distribution of income becomes more unequal.
Gini coefficient is related to the Lorenz Curve. Gini coefficient is twice the area between the Lorenz curve and the equidistribution function. Gini coefficient ranges between 0 and 1.
0 Þ perfect equality
1 Þ perfect inequality