Lawrence Summers has an op-ed on a policy recommendation for improving equality of opportunity I largely agree with, but he makes a mathematical error that I see commonly in discussions of economic mobility:
The number of children not born into the top 1 percent who move into it must equal the number of children born into the top 1 percent who move out of it over their lifetimes.
The suppressed assumption here is that society is demographically static, with no entry and no exit and equal numbers of children born to households all across the income spectrum. This does not hold as a general matter. The great UC Davis economic historian Gregory Clark, for example, has shown that England in the centuries before the Industrial Revolution was characterized by a strong undercurrent of downward social mobility. The reason is simple: Richer households had more surviving children than poorer households and there was little international migration. The result was that most sons grew up to occupy a lower relative economic position than their father.
The demographics of the modern-day United States are probably not the same as those of medieval England. A high share of the positions at the bottom of the U.S. economic spectrum are occupied by people whose parents weren't in the United States at all. Consequently if we look at babies born in the 10th to 20th percentiles in the 1970s and ask where they are today, I'd imagine you'd see more above the 20th percentile than below the 10th percentile because all these new people have flooded in. I have no idea whether households in the top 1 percent of the population have more or fewer than 1 percent of the children, but knowing the answer to that question is important to determining whether there are net inflows or outflows. And ultimately the directionality of relative economic mobility is going to be driven much more by demographic structure than by anything related to tax policy or education or anything else.
This is just one of several reasons why I think discussions about opportunity, which are important, should not get too mixed up with discussions about inequality, which are also important. We ought to aspire to have an educational system in which kids pretty consistently end up with more and better schooling than their parents had, but we shouldn't think this guarantees an economically egalitarian outcome or consistently upward social and economic mobility.