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INCOME DISTRIBUTION AND INEQUALITY :
Lorenz Curve, Gini Coefficient and the Pietra Ratio

The LORENZ CURVE

Lorenz curves are an effective way of showing inequality of income within and between countries (see Figure 1). The cumulative percentage of population is plotted along the horizontal axis whilst the cumulative percentage of income is plotted along the vertical axis. The curve shows the actual relationship between the percentage of income recipients and the percentage of income that they did in fact actually receive.

The 45 degree line shows the situation when there is an even distribution of income i.e. 20% of the population earns 20% of the income and 50% of the households earn 50% of the income and so on. This is called the line of absolute equality (egalitarian line).

The closer the Lorenz curve of a region is to the 45-degree line the more equal the distribution of income is. In the case of the Lorenz curve in the diagram above 20% of the population earns 5% of the income and 50% of the population earns 20% of the income. The more the Lorenz curve bends away from the 45-degree line of absolute equality, the less equal is the distribution of income. In reality no country exhibits a totally equitable distribution of income.

The GINI COEFFICIENT

The Lorenz Curve construction also gives us a rough measure of the amount of inequality in the income distribution. The measure is called the Gini Coefficient. Computation of the Gini Coefficient is illustrated by Figure 2 below.

To compute the Gini Coefficient, we first measure the area between the Lorenz Curve and the 45 degree equality line. This area is divided by the entire area below the 45 degree line (which is always exactly one half). The quotient is the Gini coefficient, a measure of inequality. In other words, the Gini coefficient is the area shaded in pink divided by the total of the areas shaded in pink and light blue-green.

For a perfectly equal distribution, there would be no area between the 45 degree line and the Lorenz curve -- a Gini coefficient of zero. For complete inequality, in which only one person has any income (if that were possible) the Lorenz curve would coincide with the straight lines at the lower and right boundaries of the curve, so the Gini coefficient would be one. Real economies have some, but not complete inequality, so the Gini coefficients for real economic systems are between zero and one.

The PIETRA RATIO

The Pietra ratio (or Robin Hood Index) is a measure of inequality in a resource distribution in a population. It indicates the amount of the resource that needs to be taken from more affluent areas and given to the less affluent areas in order to achieve an equal distribution (in effect to rob from the rich to give to the poor).

The Robin Hood Index (See Figure 3) is equivalent to the maximum vertical distance between the Lorenz curve and the line of equal incomes. The value of the index approximates the share of total income that has to be transferred from households above the mean to those below the mean to achieve equality in the distribution of incomes.

MEASURES

The most common definition of the Gini Coefficient is in terms of the Lorenz Diagram-as the ratio between the Lorenz Curve and the line of equality, to the area below this line (see Figure 2). Various definitions have also been discussed in the literature and are useful for different purposes. The Geometric Definition is the one used to estimate the Gini Coefficient for the Philippines.

Suppose there are n individuals (or housholds) who are labeled in nondescending order of income as : y1 < y2 < … < Yn. Denote this (ordered) income distribution by the vector y = (y1, y2, … yn), and let µ be its mean. Let Fi be the cumulative population share and the cumulative income share corresponding to the individual i(i = 1,2,…n).

The Pietra Ratio

PR = SUM (percentages for deciles > 10%) – (10%*(number of deciles summed))