A Unified Approach to Measuring Poverty and Inequality
Cumulative distribution (%)
Figure 2.14: Poverty Incidence Curve and Headcount Ratio
100 Fx ′ PH(x ′; z ″)
Fx
PH(x ′; z ′) PH(x; z ″) PH(x; z ′) PH(x ′; z) PH(x; z) z
z′ z″
xN
Income
Suppose there is another distribution x'. One can see in figure 2.14 that the headcount ratios corresponding to poverty lines z, z', and z" lie above the respective headcount ratios for distribution x. Is there any other poverty line that reflects a higher headcount ratio in x than in x'? The answer is no. The cdf of x lies to the right of the cdf of x', which means that the headcount ratio for x' for no poverty line can be lower than the headcount ratio for x. When a cdf lies to the right of another cdf, first-order stochastic dominance (introduced earlier) occurs. When such dominance relation holds between two cdfs, not only do the headcount ratios agree for all poverty lines, but the poverty gap measure, the squared gap measure, the mean gap measure, the Watts index, and the CHUC indices also agree for all poverty lines. This approach also answers the second question, which asks when all poverty measures agree. Therefore, if the first-order stochastic dominance holds, then there is no need to compare any two distributions by any poverty measure introduced earlier with respect to varying the poverty line. The choice of poverty measure and the choice of poverty line simply do not matter when the first-order dominance condition holds. The cdf in the context of poverty measurement is also known as the poverty incidence curve. Poverty Deficit Curve What if two poverty incidence curves cross? Then a unanimous relationship in terms of the headcount ratio does not hold. However, there are two other
136