A Unified Approach to Measuring Poverty and Inequality
source of ambiguity. Thus, conducting a dominance analysis that is based on these curves should be the first step in welfare comparison. A second important motivation for dominance analysis might be focus, or an identified concern with different parts of the distribution. Has the rapid growth for the higher-income group been matched by growth of the middle-income group or the lower-income group? We spend some time in this subsection finding answers to these questions by plotting entire classes of income standards using the various curves to be defined next. If one curve always remains above another curve, then all income standards in that class agree in ranking—for example, two income distributions. However, if the curves cross, then situations may arise in which different income standards in the same class disagree with each other. A first such curve is the quantile function itself, which simultaneously depicts incomes from lowest to highest. When all income quantiles are the same, then one income distribution always lies above another income distribution. When two distributions never cross, the situation is known as first-order stochastic dominance (FSD). An income distribution x first order stochastically dominates another distribution y, denoted by x FSD y, if and only if (a) no portion of x’s quantile function lies below y’s quantile function and (b) at least some part of x’s quantile function lies above y’s quantile function. Let us denote quantile function using the notations introduced earlier. So x’s quantile function is denoted by Qx and that of y is denoted by Qy. Then, the definition of FSD is as follows: First-Order Stochastic Dominance: Distribution x first order stochastically dominates another distribution y if and only if Qx(p) ≥ Qy(p) for all p in the range [0,100] and Qx(p) > Qy(p) for some p. The concept of FSD may also be understood in terms of cumulative distribution functions. Recall that a quantile function is just an inverse of a cdf. Using the notations introduced earlier, we denote the cdf of x by Fx and that of y by Fy. The formal definition of FSD in terms of cdfs is as follows:
First-Order Stochastic Dominance: Distribution x first order stochastically dominates another distribution y if and only if Fx(b) ≤ Fy(b) for all b in the range [0, ∞] and Fx(b) < Fy(b) for some b. 70