A Unified Approach to Measuring Poverty and Inequality
Figure 2.13: Lorenz Curve 100
Lx(100)
Income share
Ly
Lx ′ Lx
A
Ly(20) Lx ′(20) Lx(20)
B C 0
20
Population share
100
20 14 4 0
respectively. For income distribution x, Lx represents its Lorenz curve, denoted by the dotted curve. Following the example of Nigeria, Lx(20) = 4 percent, which is the height of the curve Lx at point C. If distribution y is obtained from distribution x by distributing income equally across the population, then the Lorenz curve becomes a 45-degree straight line, Ly (the solid line in figure 2.13). In this case, the share of the population’s bottom 20 percent in distribution y is Ly(20) = 20 percent. This is obtained at point A on Lorenz curve Ly. Now, suppose the income distribution in Nigeria improves over time and the new income distribution is denoted by x'. The Lorenz curve for x' is denoted by the dashed curve Lx' in figure 2.13. The share of the bottom 20 percent in the total income increases from 4 percent to 14 percent. This is shown at point B on the Lorenz curve Lx'. Notice that every portion of Lorenz curve Lx' lies above that of Lorenz curve Lx. This is what we mean by Lorenz dominance: the income share of every cumulative population share in x' is higher than that in x. Thus, distribution x' Lorenz dominates distribution x'. Similarly, distribution x Lorenz dominates both distributions x and x'. Any inequality measure satisfying the four basic properties—symmetry, population invariance, scale invariance, and the weak transfer principle— would evaluate distribution y as more equal than distributions x and x' and distribution x' as more equal than distribution x. Thus, before comparing distributions using different inequality measures, the distributions’ Lorenz
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