## Wednesday, June 10, 2015

### Why I am still not excited by the Palma index

Those who follow inequality studies and debates might remember that about a year ago there  was a fierce attack on “the tyranny of the Gini coefficient” from the proponents of a new index of inequality named after the Cambridge professor Gabriel Palma.  Alex Cobham and Andy Sumner (both my good friends and among the top researchers in development and inequality) argue that Gini lacks an intuitive meaning (which is true) and that the Palma index which shows the ratio between the share of income received by the top 10% of recipients and the share of income received by the bottom 40% is much more intuitive, more sensitive to the changes, and should be used in the UN’s Millennium Development Goals to monitor changes in inequality.

Let me define the logic of Palma. It has been observed throughout the years, and codified by Gabriel Palma, that the “middle five deciles” (which I will call “the central deciles”) in most countries get about the same share of total income. (I have noticed that regularity here; see Table 6 on page 29). Thus almost any inequality change, if we look at what happens in a single country over time, or any inequality difference, if we compare various countries, must be due to the high (low) share of the top decile vs. correspondingly low (high) share of the bottom deciles. Nothing will be changed among the central deciles. Now, which are the central  deciles?

Not what you might expect: the deciles around the median and slightly below and above it.  To find which are the central deciles, we have to recall the rationale of the Palma approach: find the parts of the income distribution such that that their  shares will be constant regardless of what happens elsewhere in the distribution. Once we have identified such “central deciles” we can just drop them from the picture and represent income inequality by some measure of what happens among other parts of the income distribution. The first part of the Palma approach consists in “gutting out” or ignoring one part of the distribution.

Now, income distributions are skewed to the right which means  that the right-end tail is longer than the left-end tail (there are many rich people but there are no people who can live on zero income or consumption). Skewness to the right implies  that the mean is greater than the median. Normally, the mean in empirical income distributions is  found in the 7th decile. Thus, unless something really radical happens, real income of the 7th decile will increase at the same rate as the mean income of a country—which precisely means that its share will be fixed. Similarly to it will also rise incomes of the deciles around the 7th decile, the two above (8th and 9th) and the two below (5th and 6th). Therefore, these five deciles together should, approximately, take one-half of total income, regardless of what happens elsewhere in the distribution, and we can just drop them from the rest of the investigation.

And indeed Palma and people following him find that, across countries, these five central deciles take about one-half of total income. The Palma measure of inequality is then defined simply as

Figure below shows the range of the Palma (calculated from my World Income distribution WYD-2008 data, the same as used by Alice Krozer; see below).  The mean Palma is just below 2, the maximum value (for South Africa) is 10, while 14 countries have vales below 1 (the minimum being Slovenia with a Palma of 0.8). Gini mean is 0.38; median 0.36.

The next figure shows the empirical relationship between Palma  and Gini. As can be seen, Palma is convex with respect to Gini. It is much more “elastic” at high values of the Gini, that is, the increase in Palma for a given change in Gini is much greater at high values of Gini. If one wishes, one can even establish a strong empirical relationship between the two with R2 reaching 0.97. (I am not sure though what would be the value of such a relationship.)

In  a recent paper, Alive Krozer notices on the same dataset that I used here that the stability in the income shares extends further and goes over to the 19th ventile, that is to the lower half of the top decile. Thus, she decides to drop these five percent of the population  too, and redefines Palma, calling it Palma v.2, as follows:

She then goes even further and defines Palma v. 3, by putting the share of the top percentile only in the numerator since we know that most of the recent increase in inequality in rich countries was driven by its rising share. Thus, not only are the five “central deciles” left out, plus the percentiles 91-95, but also the percentiles  96-99. At that point, Krozer seems to realize that the things have gone too far and rightly decides to ignores Palma version 3.

Her paper is valuable in a way which I think was not part of her original intention. The paper shows a rather thin basis on which the Palma is constructed.  Palma’s logic is, as we have seen, to find parts of the distribution that, in terms of their shares,  do not change regardless of the changes elsewhere, and to build a measure of inequality around these immovable parts.  But these immobile chunks are no more immobile than Pareto’s top shares were. What is immobile may change between the countries, or across time. We see this it in Krozer’s own paper:  there is no superior argument to assume that only the “central five deciles”  are constant than to assume that “the central 55 percentiles” are constant.

With Palma we are thus building a general measure of inequality on the quicksand of what seems today more or less an empirical regularity.  (Note that even when the regularity holds the five central deciles do not take exactly 50% of total income, but approximately 50%.)

But if  the distribution changes and the middle loses while the bottom gains, and it turns out, for examples, that the deciles’ 4-7 shares are suddenly fixed, should we change our measure of inequality to look at the ratio between the top three deciles and the bottom three? Or if growth of incomes is concentrated in the top 1% or the top 5%, should be again redefine the Palma formula as Krozer has done? An infinite number of such permutations is possible, and an endless dispute will open up regarding what deciles’ shares are fixed and what not. The virtue of Krozer’s paper, despite what I think was her original intention,  is to highlight the fragility of the empirical nature of the index and thus its basic arbitrariness.

This is, in my opinion, a big and insoluble problem, separate from other methodological problems that have already been raised with respect to Palma: insensitivity to transfers within each of its “chunks” (bottom 40%, central 50% and top 10%) which is indeed a big factor that should disqualify any inequality measure. Namely, if there are either progressive or regressive transfers within such “chunks”,  Palma does not change. (Another problem is that its decomposition properties—what is the Palma of two distributions whose Palmas and mean incomes are known—cannot, I think, be determined in general.)

In conclusion, using Palma, among other indexes  of inequality, may be useful. But believing, like Pareto, that there is some fundamental immutability that has been revealed by the measure, and that we can dispense with other measures, is plain wrong.