A Unified Approach to Measuring Poverty and Inequality
As in figure 2.3, the horizontal axis in figure 2.4 denotes the population share in percentage, which lies between 0 and 100. The left-hand vertical axis denotes the corresponding value of a quantile function Qx and the right-hand vertical axis reports the quantile incomes. By definition, the quantile income for a certain percentile is the value of the quantile function at that percentile, so WQI (x; p) = Qx(p). In the figure, WQI (x; 50) = bM is the median of distribution x. Likewise, WQI (x; 25) and WQI (x; 75) are the first and the third quartiles of distribution x. The wellknown 10th and 90th percentiles of distribution x are WQI (x; 10) = Qx(10) and WQI (x; 90) = Qx(90), respectively. Given that a cdf is an inverse of a quantile function, quantile incomes can also be graphically portrayed and calculated using a cdf. What properties do quantile incomes satisfy? It is straightforward to verify that any quantile income satisfies symmetry, normalization, population invariance, linear homogeneity, and weak monotonicity. However, no quantile income satisfies the other dominance properties: monotonicity, transfer principle, and subgroup consistency. Quantile incomes do not satisfy monotonicity because a personâ&#x20AC;&#x2122;s income may increase, but as long as it does not surpass a certain quantile, that quantile income remains unaltered. Similarly, quantile incomes do not satisfy the transfer principle because they do not change to a transfer that takes place at a nonrelevant part of the distribution. The income standards are not subgroup consistent because the quantile incomes of the subregions may increase, but the overall quantile income may fall. Consider the following example. Suppose the income vector of society X is x = ($10k, $20k, $30k, $50k, $60k, $80k) and the income vector of two subgroups is x' = ($10k, $20k, $30k) and x" = ($50k, $60k, $80k). The 67th quantile of the three distributions is WQI (x'; 67) = $20k, WQI (x"; 67) = $60k, and WQI (x; 67) = $50k. Now, suppose the subgroup income vectors over time become y' = ($10k, $20k, $30k) and y" = ($45k, $65k, $80k). Apparently, the quantile income at the 67th percentile of the first group does not change, but that of the second does. In fact, WQI (x'; 67) = WQI (y'; 67) but WQI (y"; 67) > WQI (x"; 67). What happens to the quantile income at the 67th percentile of the overall distribution? It turns out that WQI (y; 67) = 45 < WQI (x; 67). Partial Mean The next set of commonly used means is the partial means. There are two types of partial means: lower partial means and upper partial means. A lower
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