I'm a bit less skeptical after reading more, but only up to a point. When examining surnames in England, Clark looks at more than just medicine and law (rather specific professions). He looks at estates and probate cases in English courts, enrollment in Cambridge and Oxford, and even elections to the House of Commons. Perhaps most interesting is that he doesn't just look at common elite surnames (where people might benefit from everyone around them knowing that, say, Montgomery is an elite name), but also rare elite surnames. This is an important way to rule out the possibility that people benefit from name recognition. Interestingly, his patterns hold up through the 20th century, even though the past few centuries saw substantial changes in social policy, educational admissions standards, professional standards, etc.
In the US, he is looking at shorter time periods (his analysis of England goes back to the Middle Ages), and some of his patterns are less convincing, but plausible. Perhaps most interesting is that he looks at both downward mobility (e.g. decline in common surnames of doctors and lawyers) but also upward mobility (rises in surnames from disadvantaged groups). When looking at disadvantaged groups he includes both the obvious ones (African Americans, Native Americans, Jews) and also a less obvious one (French people in the northeastern US). The less obvious one is interesting because totally different mechanisms should be in play. While I have no doubt that families lower on the socioeconomic scale will be known locally, and face stigma accordingly, white people of French descent have some clearly plausible ways to blend in if they move a little ways away. Nonetheless, multi-generational persistence in comparatively lower status shows patterns similar to other situations in both the US and elsewhere.
In the case of the US, the nice Markovian upward regressions to the mean are a bit less obvious. The 20th century saw some pretty clear and significant changes to laws that barred advancement for certain groups, so one would expect upticks, and some of the data is messy. If we take away any message, it's that some of the regressions to the mean have not involved changes as sharp as one would have hoped. We have an obvious explanation: Ongoing discrimination. And no doubt that is part of it. But Clark invites us to consider the possibility that the patterns of adjustment are close enough to other societies that maybe slow progress is sadly inevitable.
There are obvious things that a 21st century college professor must say at that point. Some of those mandatory responses even have some validity! As much as I'm a skeptic of rapid progress, even I hold out hope that we can change more rapidly than the English class system. I mean, there's a ton of room between "We'll totally transform things in one generation" (which some people seem to think is the only goal we can allow ourselves to have) and "Well, we'll improve as slowly as a country that still has a House of Lords."
Clark proposes in chapters 6-7 that families pass on knowledge and skills, which, yeah. It's pretty much what we advanced primates do. He gives much less consideration to genetic explanations than I feared, acknowledging that anything passed from parents to children will show similar statistical patterns whether it involves nature or nurture. That's encouraging. Also, as much as the slow progress is dispiriting, the Markovian nature of the patterns means that each generation is, in some sense, a new start. A start with loaded dice, but at least a fresh dice roll.
There's one other encouraging thing in these chapters (and I note that encouragement about social problems is a rarity from the Dismal Science of economics): He argues that whatever parents are doing to make their kids successful, it isn't (for the most part) about their money. If it were about money, families with more kids would have less successful kids than families with similar financial resources but fewer kids. The per-child investment would differ. But the data he cites doesn't support such a model.
Now, at this point one might say "What kind of idiot thinks rich kids don't have it better?" Of course they do. His point is that it's not JUST the money spent on the kids. It's also the fact that people who have money know how to get money and they know how to teach kids to do the same. That includes connections.
I can't find the reference now, but I vaguely recall reading that after the Communist revolutions in Russia and China, the bureaucracies remains in the hands of more or less the same families. Yes, yes, there were some high-profile punishments of selected upper-class people, but you can't run a complex country without keeping the people who are experienced at running it. You can hand over a few things to the revolutionaries (e.g. punishing ideological crimes), but if you want the public infrastructure to run you keep more or less the same people at the Department of Water and Sanitation. An extreme example is Egypt, where my understanding is that even after successive conquests by Greeks, Romans, and Arabs, the public records continued to be kept in Coptic (the language that evolved from Ancient Egyptian) until at least a century after Arab conquest. Yes, eventually they switched to Arabic, but when the Arabs first arrived the bureaucracy was run by Copts. I imagine that the Copts simply explained that this place only functions if they properly manage all of the water infrastructure projects associated with the annual rise and fall of the Nile, and if the new rulers want to avoid bread riots they'll let the pros continue to handle all of the civil engineering projects.
Finally, if the role of families is as great as Clark suggests, it makes more sense that revolutionaries frequently contemplate (but never succeed at) major changes to family structure. It also makes sense that Rawls would call for massive projects to ensure truly equal lifetime chances (from the earliest stages) for children. You don't really have equal opportunity unless you massively engineer the most influential period of life. Except you can't actually do it, at least not without horrors.
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