Chapter 1: Introduction
Dominance and Unanimity One alternative to numerical inequality measures for making inequality comparisons is the so-called Lorenz curve and its associated criterion of Lorenz dominance. The Lorenz curve graphs the share of income received by the lowest p percent of the population as p varies from 0 percent to 100 percent. A completely equal distribution yields a Lorenz curve where the lowest p percent receives p percent of the overall income, or the 45 degree line. Inequality results in a Lorenz curve that falls below this line in accordance with the extent and location of the inequality. When one compares two distributions, a higher Lorenz curve is associated with lower inequality. This is the case of Lorenz dominance in which one distribution is unambiguously less unequal than another. Alternatively, if the two Lorenz curves cross, no unambiguous determination can be made. The Lorenz curve is a useful tool for locating pockets of inequality along the distribution. For example, if a portion of the curve is straight, then there is no inequality over that slice of the population. If it is very curved, then there is significant inequality over the relevant population range. It also can help determine if a given inequality comparison is robust to the choice of inequality measure. Indeed, when the Lorenz curve of one distribution dominates the Lorenz curve of another distribution, it follows that every inequality measure satisfying the four basic properties (symmetry, replication invariance, scale invariance, and the weak transfer principle) will not go against this judgment, whereas the subsets of measures satisfying the transfer principle are in strict agreement with the Lorenz judgment (that the first has less inequality than the second). So these unambiguous judgments are also unanimous judgments across wide classes of inequality measures. The Lorenz curve is also the generalized Lorenz curve divided by the mean. At p = 0 percent, both curves have the value 0 percent; at p = 100 percent, the Lorenz curve has the value 100 percent, whereas the generalized Lorenz curve takes the mean as its value. At any percentage of the population p, the generalized Lorenz curve is p times the associated lower partial mean at p, and the Lorenz curve is p times the lower partial mean over the mean. If one recalls the link between second-order stochastic dominance and the generalized Lorenz curve, it follows that when the means of the two distributions under comparison are the same, a distribution has greater equality according to Lorenz dominance exactly when it has higher welfare for the
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