My index is failing me here but I recall a paper that had looked into falling birth rates as standards of living increased and found that birth rate correlated strongly with the "cost" of children. The supposition (it really wasn't a theory or even a solid hypothesis) was that as parents became more educated and society's measure of success became more material, the parental effort to provide the "best" for their kids went up.
It worked kind of like this, you have kids, you want them to have as good or better life than you have. If you've defined "better" here as wealth and social status, then you want them to have more opportunities.
Getting them those opportunities (which you may have missed out on) costs money (wealth). Examples, good schools, good nutrition, good after school programs, Etc.
Also, the developing society is putting more social status points on "successful" people (at least outwardly so by their display of wealth) which incentivizes couples to both continue their careers and "success trajectories." That path costs still more because now you have to add day care and possibly a nanny or two to the mix.
Here in the U.S. there is actually an interesting empirical example that supports this, its called foster care fraud. When the state offers to pay money to someone as a foster parent, a small number of people will exploit this by becoming foster parents for many many children to accrue the state benefit for "caring" for them. As an example, it suggests to me that if the state provided support to people with larger families that offset their costs significantly, the result would be larger families.
I would not argue that this "proves" the supposition that child care costs drive down the size of families, but to me it argues that it is likely a large factor in the decision.