Chapter 2: Income Standards, Inequality, and Poverty
and the geometric mean of x. The arithmetic mean represents the level of welfare obtained when the overall income is distributed equally across everyone in the society. This is an ideal situation when there is no inequality in the society. The geometric mean, in contrast, is the equally distributed equivalent (ede) income, which, if received by everyone in the society, would yield the same welfare level as in x for the degree of inequality aversion a = 0. So IA(x; 0) = 0.162 implies that the loss of welfare because of inequality in distribution x is 16.2 percent of what the welfare level would be if the overall income had been equally distributed. Suppose the society becomes more averse to inequality and a is reduced to −1. In this case, the equally distributed equivalent income is the harmonic mean of x. The loss of total welfare because of unequal distribution increases from 16.2 percent to 31.7 percent. Likewise, the percentage loss of welfare would increase to 42.7 percent if the society became even more averse to inequality and a fell to −2. What properties does any index in this family satisfy? Any measure in this family satisfies all four invariance properties: symmetry, population invariance, scale invariance, and normalization. In addition, unlike the quantile ratios and the partial mean ratios, measures in this class satisfy the transfer principle, transfer sensitivity, and subgroup consistency. If distribution x' is obtained from distribution x by at least one regressive transfer, then the level of inequality in x' is strictly higher than that in x. Furthermore, if transfers take place between poor people, then the inequality measure changes more than if the same amounts of transfers take place among rich people. Finally, because these measures satisfy subgroup consistency, they do not lead to any inconsistent results while decomposing across subgroups. If inequality in certain subgroups increases while inequality in the others does not fall, then overall inequality increases. However, measures in this class are not additively decomposable. Gini Coefficient The Gini coefficient, developed by Italian statistician Corrado Gini (1912), is the most commonly used inequality measure. It measures the average difference between pairs of incomes in a distribution, relative to the distribution’s mean. The most common formulation of the Gini coefficient for the distribution x is
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