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 7

^{th}decile. Thus, unless something really radical happens, real income of the 7^{th}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 7^{th}decile, the two above (8^{th}and 9^{th}) and the two below (5^{th}and 6^{th}). 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 R

^{2}reaching 0.97. (I am not sure though what would be the value of such a relationship.)^{th}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.

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