A Unified Approach to Measuring Poverty and Inequality
For income distribution x, let the quantile income at the pth percentile be denoted by WQI(x; p), and let the quantile income at the p'th percentile be denoted by WQI(x; p'), such that p > p'. A quantile ratio is commonly reported as a ratio of the larger quantile income to the smaller quantile income. However, this view leads the values of inequality measures to range from one to ∞. This range is not comparable to other inequality measures, which commonly range from zero to one. In this book, we formulate the quantile ratio in such a way that it ranges from zero to one. The p/p' quantile ratio is represented by the following formula: IQR (x; p / p ′) =
WQI (x; p) − WQI (x; p ′) WQI (x; p)
= 1−
WQI (x; p ′) WQI (x; p)
.
(2.16)
In this case, the quantile income at the pth percentile WQI(x; p) is the higher income standard, and the quantile income at the p'th percentile WQI(x; p') is the lower income standard. The higher the quantile ratio, the higher the level of inequality across two percentiles of the population in the society. A quantile ratio is zero when both the upper and the lower quantile incomes are equal. A quantile ratio reaches its maximum value of one when the lower quantile income WQI(x; p') is zero. This means that no one in the lower percentile earns any income and that the upper quantile income is positive. Note that if all people in the society have equal incomes, then any quantile ratio is zero. However, a quantile ratio of zero does not necessarily mean that incomes are equally distributed across everyone in the society. The quantile ratios used most often include the 90/10 ratio, 80/20 ratio, 50/10 ratio, and 90/50 ratio. The 90/10 ratio, for example, captures the distance between the quantile income at the 90th percentile and the quantile income at the pth percentile as a proportion of the quantile income at the 10th percentile. How should the number IQR(x; 90/10) = 0.9 be interpreted? There are, in fact, several ways to interpret the number: • The number may be directly read as the gap between the lowest income of the richest 10 percent and the highest income of the poorest 10 percent of the population, being 90 percent of the lowest income of the richest 10 percent of the population. • The number may be seen as the highest income of the poorest 10 percent of the population, being 10 percent (1 − 0.9 = 0.1) of the lowest income of the richest 10 percent of the population.
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