Consider an income distribution in which 1/3 of the population earns X dollars, 1/3 of the population earns Y> X dollars, and 1/3 of the population earns Z> Y> X dollars. (a) Derive a formula for GDP per capita. (b) Derive a formula for the income share of the top 10%. (c) Derive a formula i for the Gini index. (d) Assume that X changes to X', Y changes to Y' and Z changes to Z'. Do the two measures of inequality (the income share of the top 10% and the Gini index) necessarily change in the same direction, or is it possible that one increases while the other decreases?