# A new and useful measure of inequality

Ask most economists about the level of inequality in a country and chances are they will first point to its Gini coefficient. The statistic gives a single number that shows the level of inequality in the distribution of any kind of good. It could be income, wealth, or consumption. The “Gini” is widely cited and very useful, but a new inequality measure, the Palma index, can help complement the Gini coefficient and other measures of inequality.

To understand the “Palma,” though, one first needs to grasp the intricacies of the Gini. The Gini coefficient ranges from 0 to 1, with a coefficient of 0 signifying a perfectly equal distribution and 1 a perfectly unequal distribution. In the first case, every individual or household has an equal amount of whatever it is that’s being measured—let’s say income. And in the second case, only one individual or household has all the income.

The calculation of the coefficient isn’t straightforward. Graphically, imagine a coordinated plane. The x-axis is the cumulative share of the population. If you’re at 0.9 on the axis then you’re at the point representing the first 90 percent of the population. The y-axis is the cumulative share of (for our purposes here) income.

Then there are two lines. The first is the line of equality. It has a slope of exactly 1, so that for each cumulative share of the population it has the same share of the cumulative income. For example, the first 50 percent of the population would have 50 percent of total income. The second line is the so called Lorenz curve, (named after the early 20^{th} century U.S. economist Max O. Lorenz), which traces out the actual distribution of income in a given population, say, a country. So, for example, the first 50 percent of the population may have only 30 percent of income.

The coefficient is the ratio between the area under the line of equality to the area between the Lorenz curve and the line of equality. To give a concrete example, the Gini coefficient in the United States for post-tax and transfer income in 2010 was 0.434, a 21 percent increase since 1979.

Clearly, one flaw of the coefficient is that it’s not easy to explain. Furthermore, changes in the Gini coefficient don’t tell us where the changes in the distribution occurred. The Lorenz curve could shift in a number of ways and end up having the same area between it and the line of equality.

Yet another issue with the Gini is that it’s overly responsive to the changes in the middle of the distribution. If there are changes in the so called tails of the distribution—the low and high ends—then the Gini might not pick it up.

Enter the Palma index. The index was created by economists Alex Cobham of the Center for Global Development and Andy Sumner of King’s College London. Based on work of University of Cambridge economist Gabriel Palma, they point out that there isn’t much change across time or across countries in income going to the middle class after adjusting by income per capita for each country. In other words, middle classes don’t shrink or expand all that much overall so a measure of inequality that focuses on the tails may be more useful.

The Palma is simply the ratio of the share of income held by the top 10 percent of the population compared to the share of income held by the bottom 40 percent. Cobham and Sumner find that the Palma tracks the Gini well, so it captures the overall level of inequality well. But in some cases it can show increases in inequality that the Gini misses. Case in point: the Palma for post-tax income in the United States for 2010 was 1.98, a 43 percent increase since 1979. The larger increase compared to the Gini (21 percent) shows how much of the rising inequality was due to changes at the tails.

The likelihood that the Palma replaces the Gini coefficient as the singularly important inequality statistic is unlikely. But that isn’t the best way to use the statistic or any statistic for that matter. The Palma can serve as a complement to, not a substitute for, the Gini. And the two statistics should be used in concert with data on the shares possessed by certain segments of the population as well as other measures. When it comes to statistical measures, at least in this case, more is better.