More support for the theory that fertility has fallen because status now puts more weight on having well educated kids:
Discussions … typically assume that the poor have more children than the rich. Micro-data from [62,146 women in] 48 developing countries suggest that this assumption was false until recently. …
In earlier birth cohorts (mostly of the 1940s and 1950s), both the number of ever-born siblings and the number of surviving siblings are positively associated with years of education in 25 countries and negatively associated in two. In contrast, in later birth cohorts (mostly of the 1980s), 20 countries exhibit negative associations between both measures of sibship size and education, while seven show the opposite. …
Changes in women’s labor force participation, sectoral composition, GDP per capita, [urbanization,] and child mortality do not predict changes in the education-sibship size association. Instead, one variable stands out …: the average educational attainment of the parent generation. …
Two explanations … stand out. … The first involves subsistence consumption constraints. … This explanation is somewhat difficult to square with the finding that economic growth is uncorrelated with changes in the education-sibship size association. … The second explanation is … that mass education induces widespread change in fertility norms … [and so] increases the importance of child quality relative to quantity in the utility function, [with] the preferences of the most educated couples [being] most sensitive. (more)
"is "education" one of those intrinsically limited status goods [?]"
Yes, the quality of public education is roughly related to the living standard of society so it takes wealth to send your children to a better-than-average school. In the far future this may of course change because there are limits to how much a person can learn.
"Bill Gates could clearly afford to send any number of his children to Harvard."
Yes, there is clearly more going on than just the quality over quantity factor in education: modern people like their spare time, they do not feel the religious pressure to multiply excessively anymore and there exist reliable constraceptives now. Additionally it may be that the ultra rich with their humungous egos feel that they are themselves the best teachers: that their children each need to spend a lot of time with them, so even if you can buy all the teachers in the world the time YOU can spend with each child is still limited.
Does this imply that, if economic growth enables everyone to have three or four well-educated kids, population growth will pick up again? Or is "education" one of those intrinsically limited status goods, so that the scenario I outline is impossible? Even if it is, why don't the ultra-wealthy have more children than the middle class? Bill Gates could clearly afford to send any number of his children to Harvard.