Pietro Poggi-Corradini on Inequality

My and Russ’s friend Pietro Poggi-Corradini, a mathematician with a deep appreciation of Hayek, sent the following e-mail to us a few weeks ago:

Research
on inequality usually keeps track of percentiles. So let’s look at the
following simple example. A society at the beginning consists of 10
individuals, 9 of which make 1 dollar and 1 who makes 10 dollars.
Social scientists decide to keep track of the top 20%. So the top 20%
makes an average of 5.5 dollars while the bottom 80% makes an average
of 1 dollar. Now suppose that after 1 year there are now 8 people
making 1 dollar and 2 people making 10 dollars. The top 20% now makes
an average of 10 dollars. Dividing 4.5 by 5.5 this represents an 82%
increase for the top quintile. The bottom 80%  on the other hand sees a
0% increase in income. One would like to conclude that "inequality has
risen".  But  if you were given a choice to live in a society like the
earlier one with 9 people making the same income of 1 dollar and one
very rich person making 10 dollars, or live in the latter society where
less people make 1 dollar and more people make 10 dollar, what would
you choose? A simple calculus of probability tells me that the latter
society might be more appealing to most people.