>Logic would seem to indicate that a more plentiful environment (a richer society) would lead to more children due to a surplus of life's necessities.
Based on seeing my wife’s experience of pregnancy, childbirth, and breastfeeding, without any complications, I would not agree the above statement.
Based on speaking to my women members of my family, friends, and the data, it is evident that childbirth/child rearing is a costly undertaking, not measurable in any “normal” amount of money.
So logically, I would expect women to have a couple kids (due to wanting to be mothers, or whatever), but each additional kid is at a very high marginal cost (on their body).
I would posit poorer regions have higher birthrates simply because women lack the knowledge and power to obtain and use contraception. Even in poorer regions, giving women freedom (economic freedom and security such that they are deciding when to get pregnant), and access to contraception will make the birthrate drop.