Chapter 2: Income Standards, Inequality, and Poverty
Dominance and Unanimity An inequality measure estimates, with a single number, the inequality level in a society. A question may naturally arise: Do all inequality measures compare two income distributions in the same way? In other words, if an inequality measure evaluates income distribution x to be more equal than distribution y, would another inequality measure evaluate distributions x and y in the same way? The answer depends on the two inequality measures we use for evaluation. Not all inequality measures evaluate various distributions in the same manner. We can clarify this concern with an example. Consider the two income vectors x = ($4k, $5k, $6k, $7k, $14k, $16k) and y = ($3.4k, $6.1k, $6k, $6.5k, $14k, $16k). These two vectors have the same mean. The Gini coefficient indicates that the inequality level in x is 0.282, which is higher than the inequality in y (0.280). However, the Atkinson’s measure that is based on the geometric mean shows that the inequality level in x is 0.127, which is lower than the level of inequality in y (0.132). Therefore, different inequality measures may disagree in different situations. Is there any condition in which different inequality measures agree with each other? The answer is yes. Inequality measures that satisfy the four basic properties—symmetry, population invariance, scale invariance, and the weak transfer principle—agree with each other when Lorenz dominance holds between two distributions. To understand Lorenz dominance, we need to understand the Lorenz curve. The Lorenz curve of an income distribution shows the proportion of total income held by the poorest p percent of the population.10 We denote the Lorenz curve of distribution x by Lx. Then Lx(p) is the share of total income held by the poorest p percent of the population. Indeed, Lx(100) = 100 percent and Lx(0) = 0 percent. Suppose the total income of Nigeria is N25 trillion and only N1 trillion is received by the poorest 20 percent of the population. Then LNig(20) = 4 percent. Suppose that income in Nigeria is redistributed, keeping the total income unaltered, so that everyone has identical income. Let us denote the equal income distribution by y. Then the percentage of total income enjoyed by the poorest 20 percent of the population is 20 percent, and Ly(20) = 20 percent. In figure 2.13, the horizontal axis denotes the cumulative share of the population (p), and the vertical axis shows the share of total income. Note that the lowest and the highest values for both axes are 0 and 100,
101