Chapter 2: Income Standards, Inequality, and Poverty
FSD ensures higher welfare according to every utilitarian welfare function with identical, increasing utility functions. The robustness implied by an unambiguous comparison of quantile functions extends to all income standards and all symmetric welfare functions for which “more is better.� However, if the resulting curves cross, the final judgment is contingent on which income standard is selected. Even in this case, the quantile function can be helpful in identifying the winning and losing portions of the distribution. Figure 2.7 depicts the situation where x FSD y. Panel a shows the FSD by quantile functions, and panel b shows the FSD by cdfs. In panel a, the quantile function of x lies completely above that of y, which means that every quantile income of distribution x is larger than the corresponding quantile income of distribution y, so x FSD y. The same argument applies to the cdfs in panel b, where the cdf of x lies to the right of y. Later, we will find the concept of FSD that is based on cdfs useful, especially in poverty analysis. The generalized Lorenz (GL) curve is a second curve that is useful for dominance analysis. The generalized Lorenz curve graphs the area under the quantile function up to each percent p of the population. Thus, any point on a generalized Lorenz curve is the cumulative mean income held by the bottom p percent of the population. We denote the generalized Lorenz function of distribution x by GLx, and that for the p percent of the population by GLx(p). By construction, for income distribution x, GLx(100) = WA(x) and GLx (0) = 0. Figure 2.7: First-Order Stochastic Dominance Using Quantile Functions and Cumulative Distribution Functions b. Cumulative distribution function
Cumulative distribution
Income
a. Quantile function
Qx Qy
0
Population share
100
100 Fy
0
Fx
Income
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