Observations on the Great Gatsby Curve
My own view is that this correlation is not particularly surprising. Let me give you an analogy to explain why.
Suppose we collected data on various chess clubs (nations). In every club, we have data on each member's win-loss record over the year (income). We can then measure the variance of individuals' win-loss records (inequality). We can also measure how a person's win-loss record in one year predicts his win-loss record in the subsequent year (mobility).
Some clubs have a bunch of players with similar levels of skill at chess. In this case, everyone would have a win-lose record that is close to each other, and a person's club ranking one year would not have a lot of predictive value for his ranking the next. That is, we would have small inequality and substantial mobility.
Other clubs are more heterogeneous. They have some masters and some novices. The masters have much better records than the novices, and their better records tend to persist year to year. That is, we would have substantial inequality and little mobility.
If we put all these clubs together in a scatterplot, we would get something close to the Great Gatsby curve.
Notice a corollary: Suppose we combined two clubs, one that with mostly masters and one with mostly novices. The new combined club would be more heterogeneous and, therefore, would exhibit more inequality and less mobility than either of the clubs separately.
The application of this corollary to the Great Gatsby curve is that if we looked at Europe as a whole, rather than each nation separately, we would find that Europe as a whole has more inequality and less mobility than the individual countries. That is, Germans are richer on average than Greeks, and that difference in income tends to persist from generation to generation. When people look at the Great Gatsby curve, they omit this fact, because the nation is the unit of analysis. But it is not obvious that the political divisions that divide people are the right ones for economic analysis. We combine the persistently rich Connecticut with the persistently poor Mississippi, so why not combine Germany with Greece?
The bottom line for me that the Great Gatsby curve is a bit interesting, but neither particularly surprising nor suggestive of any specific conclusions or policy recommendations.