A Unified Approach to Measuring Poverty and Inequality
All partial means agree that distribution y has higher welfare than distribution x. Similarly, if there is another distribution x' whose generalized Lorenz curve, GLx', lies completely below GLx (also shown in figure 2.9), then all partial means agree that distribution x has higher welfare than distribution x'. The heights of the generalized Lorenz curves for distributions y, x, and x' at the 50th percentile are GLy(50), GLx(50), and GLy'(50), respectively. The generalized Lorenz curve represents second-order stochastic dominance, which signals higher welfare according to every utilitarian welfare function with identical and increasing utility function exhibiting diminishing marginal utility. Example 2.4 provides a practical illustration of generalized Lorenz calculations. The generalized Lorenz curve is also closely related to the Sen mean. For distribution x, the Sen mean, WS(x), is twice the area underneath GLx. Example 2.4: Suppose per capita income in India is Rs 25,000. If only 3 percent of this mean income is received by the poorest 20 percent of the population, then GLInd(20) = Rs 750. Suppose incomes in India were redistributed, thereby keeping the average income unaltered so that everyone in India has identical income. Let us denote this income distribution by y. Then the cumulative average income received by the poorest 20 percent of the population is 20 percent and GLy(20) = Rs 5,000. Thus, GLy(20) – GLInd(20) = Rs 5,000 – Rs 750 = Rs 4,250. The loss of welfare because of unequal distribution of income for the poorest 20 percent of the population is Rs 4,250. In relative terms, the loss of welfare is 4,250/5,000 = 85 percent. However, note that the loss presented in terms of the height of the generalized Lorenz curve is not the potential loss in the mean income of the poorest 20 percent of the population. The mean income of the poorest 20 percent of the population is GLInd(20)/0.2 = Rs 3,750. Had income been equally distributed, the mean income of the poorest 20 percent would have been Rs 25,000. In that scenario, the potential loss of mean income is Rs 21,250. But in a relative sense, the percentage loss in mean income is 25,205/25,000 = 85 percent, which is the same as the percentage loss in terms of the height of the generalized Lorenz curve. In fact, the percentage loss of welfare using the height of the generalized Lorenz curve is always the same as the percentage loss of mean income of that percentile.
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