## Saturday, February 28, 2015

### What remains of Pareto?

My class on personal income distribution theories (within-nations) begins with Pareto. Pareto will indeed  be for ever part of such classes because he was the first economist to have been seriously interested in empirical analysis of inter-personal inequality.  Before him,  economics was about functional income distribution which, of course, makes sense if you assume that all workers are at subsistence, all capitalists rich, and all landlords even richer. Then, you do not need to bother with inter-personal inequality. It is just a transformation of factoral income distribution. (This is, I think, most obvious in Ricardo who cannot  even conceive of a possibility of wages ever going above subsistence—except temporarily to precipitate the Malthusian movement. The advantage for modeling—since Ricardo’s “Principles”  are in reality an exercise in verbal modelling—is that you have one fixed quantity and you can then let others vary).

It is just a minor simplification to say that Pareto thought that there was an iron law of income distribution, namely that inequality did not change whatever social system was in power.  It gave consistency to his theory of the circulation of the elites, because whatever elite be in power (land-owning, capitalist or bureaucratic), income distribution would be the same although the people who would be rich or poor would be different.  It was a serious critique of the idea that Marxist socialism would reduce  income inequalities.

What remains of Pareto’s claim?  Several things are clear now--more than a century after Pareto defined his power law and showed that the percentage of recipients of a given income decreases in proportion as that income threshold is raised.

"Pareto's law" does not apply to any entire income distribution. A typical empirical inverse cumulative function is shown on the vertical axis of Figure1. (Explanation: the highest number on the vertical axis is 1, that is 100% of the distribution, or in log terms log of 1=0. The corresponding number on the horizontal axis is the minimum income. Interpretation: 100% of people have an income higher than the minimum. As you increase income on the horizontal axis, the percentage of people with income higher than x goes down.)

If you try to fit a straight line (with both axis in logs), it gets you nowhere: the fit is like R2=0.01. So, we can forget about entire distributions following a power law. (Pareto was not aware of that because he used fiscal data from various European cities and regions which, of course, referred only to the incomes of the rich for which, as we shall see, the fit of the Pareto line is good.)

Figure 1. Pareto line fitted across the entire income distribution

And here is the thing. A straight line can almost  always be drawn, and in a meaningful fashion, across the top 1%, top 2% or top 5%, or even top 10%.  On the vertical axis in Figure 2 you have these top percentiles (in logs) with lines drawn at top 5% and top 1%. On the horizontal axis, you have the threshold incomes (such as:  top 2% begins with income level x, top 1% begins with income level x+y etc.). When  you link the two, you get a straight line.

But the problem is that the straight line, or rather its slope, changes depending on whether you draw it across top 10%, top 5%, top 2% or top 1%. In Figure 2, I show the Pareto line drawn over the top 5% from the same distribution that we just used in the previous graph. It does a relatively good job. But if I were to draw the line across the top 1% (rather than top 5%), its slope would have been very different. We would normally, I think, expect that the line would get steeper (the fat tails are less fat) as we move towards the very top of the distribution. But it does not always happen. In this example, it certainly does not: the line drawn over the top 1% would be flatter than the line drawn over the top 5%.  Sometimes, the very top is the fattest part of an income distribution.

Figure 2. Pareto line drawn over the top 5% of income distribution

So, neither (1) does Pareto constant exist across the entire distribution, (2) nor is it the same across different countries, nor (3) is it the same across different top percentiles of a given income distribution. You would think that Pareto’s contribution was almost nil.

But this is not the case. We use Pareto whenever we need to extend the line to the part of the distribution about which we have no information. For example, if I have percentile data (all 100 of them), but would like to estimate the share received by the top 0.1%, I would then draw a Pareto line over the top 5% (or 10%; this is my call) and extend it all the way to the top 0.1%. That would give me an estimate for the top 0.1% income share.

Second, Pareto is also useful for revealing data anomalies. Consider this example in Figure 3 using US micro data (CPS, year 2008). US Census Bureau practices top-coding, which means that top incomes are artificially reduced. Drawing a Pareto line reveals this. Instead of households continuing to “exist” in very high income areas (all the way to the right), their numbers abruptly drop and the richest households (whom we would expect to exist according to the Pareto fit) are just not there at all. Why are they not there? Because the Census Bureau decided to simply “truncate” their incomes.

Figure 3. Pareto line used to show anomalies in the data

Third, Pareto’s power law is indispensable for all non-symmetrical distributions, of which income or wealth are among the best examples. These are the distributions where the presence of high values does not decrease nearly as fast as implied by Gaussian distributions.  Pareto’s law and power laws in general play an important role in Nassim Taleb’s “Black Swan” (an impressive and absolutely essential book about which I might write in a different post). So, we owe a lot to Pareto.

Finally, what remains of Pareto’s sociology? He does not seem to be very much taught at universities these days, and his books are difficult to find. Many are out of print and extremely expensive (yet, you can get them on Amazon and elsewhere). What he said of Marxist socialism proved half true and half false. Socialist systems indeed created a “new class”, with incomes depending on bureaucratic position and no longer on ownership of assets. But socialism also succeeded in reducing the spread of income which Pareto believed impossible.

He was always an uncomfortable  writer. (Raymond Aron in his discussion of Pareto explains it extremely well.) Pareto’s writing style was atrocious (I just borrowed from the library his “The mind and society” and it is quasi unreadable). And he was a very unusual person: a mathematician, an engineer, steeped in Greek and Latin, and ancient history, for whom (as Schumpeter wrote) 4th century BC Athens was much more real than contemporary United States or Russia (let alone China).
He was also that very unusual breed: a conservative with anti-religious feelings. I think that he was basically a nihilist. But that would seem to be a good philosophy to have in our age of globalization: atomistic individuals, caring only about own gain and loss, not believing in any community or religious ties, and thinking (as Pareto did) that all religions, grand social theories and the like are simply fairy tales. Far from what Pareto called “logico-experimental” theories, religion peddled in “theories transcending experience”. But, he believed, chastened by reality and being of somber disposition, that every ruling class has to justify its power by resorting to such fairy tales. So we cannot have a society without fairly tales, and yet we know that all fairy tales are false.