Poor people everywhere have more kids than richer people in the same places because various welfare programs make the net cost of each kid less. There's also a massive cost cliff once you get below the threshold at which parents have enough income to justify saving for their kids college.
That doesn't mean family sizes aren't smaller in high cost areas. This is a well known trend.
Are we talking locally rich or globally rich?
People in industrialized (rich) nations don't have tons of kids because that's not how you get ahead in that kind of economy. Subsistence farmers have tons of kids because that's how they get ahead economically.
Within richer nations richer people have fewer kids because the way social safety nets and social expectations of how one should raise one's kids make kids more expensive the higher up you go.
I'm skeptical of local explanations because the relationship is so universal.
Because below a certain agricultural productivity level having more kids is actually beneficial for your economic success. Once you get over a certain productivity level it makes sense to have fewer kids and invest in them moving up the economic ladder.
>Builders in Arkensas have more kids than tech workers in San Francisco?
Because the cost of every warm body in a household in SF is completely asinine so people have fewer kids than they would like to.