Fat Right Tails


A further thought on income distribution. Itís occurred to me that most people who think about the subject at all probably have one of two wrong images in mind.

 One image thinks of class positions as stationary Ė poor people earn so much, middle-class people something more, rich people yet more; whatís happening is just that more formerly middle-class people become rich, and whatís wrong with that? This misses the point that itís not mainly a story of more people becoming rich; itís mainly about the rich getting far richer.

 The other wrong image is of income distribution as a bell-shaped, normal distribution.

 In fact, income distribution is no bell. (Does this mean that whoever figures out whatís going on gets a no bell prize? Sorry.) The reason is that it has a much fatter right tail Ė that is, there are many more people with really high incomes than you would expect from a normal distribution. Even a lognormal distribution doesnít capture it; the right tail is a lot fatter than that.

 None of this is news to people who have worked on the subject; since the 19th century work of Vilfredo Pareto it has been known that the right or upper tail of the distribution is well described by a power law. There are various ways to write this; one is to say that the share of total income above a given percentile P takes the form S = APb, where b<1 is an inverse index of income inequality. The smaller is b, the more income is concentrated in the hands of a really small minority.

 This relationship can be written in logs:

 ln(S) = ln(A) + b ln(P)

 In the figure that follows I have plotted data from Piketty and Saez. I used Table A-2 from the NBER draft of their paper, which contains data without capital gains Ė those data miss some important income at the top, but theyíre less noisy. The data points are taken at the 0.01th, 0.1th, 1st, and 10th percentiles; Iíve drawn schedules for 1929, 1970, and 1998.

 The first thing you learn is that Pareto was right: the power law works incredibly well. These really do look like straight lines. Iím sure that the fact that a power law works so well tells us something important, just as in the case of the city size distribution. In each case, however, itís not clear what that important lesson is.

 You can also see in dramatic form the shift toward greater equality and back again: the 1970 curve sits well below the 1929 curve, but by 1998 itís almost back where it was.

 Whatís really striking, however, is how the typical focus on broad measures, like the income of the top decile, misses the real action. The vertical distance between the data points on those schedules measures the (logarithmic) percentage change in income shares. So the shifts up and down at the right end show what was happening to the share of the top decile as a whole. It fell from 43.8 percent to 31.5 percent, then rose back to 41.4 percent Ė not trivial, but that gives you no sense of the extraordinary fall and rise of the income share of the truly rich. The point is that the schedule hasnít just moved back up; it has also flattened, so that the gains at the top have been enormous compared with the average gains of the top decile.

 At the very least, this is an extraordinary social change. Obviously I think it has important, and malign, political consequences too.

Originally published on the Official Paul Krugman Page, 10.20.02