A question for Lant: Do your data and argument mean that the degree of income inequality is the same in each country? Is it not a counter intuitive conclusion?
Lant: No, it does not imply it is the same. Three points: First, for the standard head count poverty calculations the calculation is (thinking in continuous terms in a mathematical sense) the integral of the income distribution up to the poverty line. Therefore, the only inequality that matters is the inequality below the poverty line and many measures of inequality are actually very sensitive to what happens in the upper tail. So, if the 99th percentile pulls away from the median (50th) but the poverty line is at the 40th percentile, then this kind of inequality measure doesn’t feature in poverty calculations.
Second, I cheat, in an open way. That is, my work connects the median to the extent of headcount poverty. The mean is the measure of central tendency that is sensitive to inequality. In fact, in a log-normal distribution of income (and most are close to this) the difference between the median and mean is a summary measure of inequality. So I hope I was careful and never said “mean” or “average” but said “typical” or “median.” If I do the same regressions with the mean then the explanatory power is a bit (but not much) lower because then the mean is an inequality sensitive measure.
Third, though inequality is not constant across countries, it just varies a lot less than does the level of income (or the long run change over time). So (and Martin Ravallion and others have papers that work this out in gory detail) if inequality gets worse for a given income (in a way that affects the lower tail) then yes, poverty is higher for higher inequality, so your intuition that inequality can matter is right. But there is an empirical question of whether differences in (below poverty line) inequality are large relative to differences in median income and the answer is no, not enough to account for anything but a very small fraction of the differences in poverty. Similarly, the Gini (or other measures of inequality) tend to have large differences across countries but usually quite modest changes over time within countries. So again, empirically, over long spells, it is growth that explains changes in headcount poverty not inequality changes, not as a theorem or necessary fact but as an empirical finding.