Right, but what if one is not the cause of the other, but rather they both have the same root cause. e.g. Well-off people have a personality that both makes them less likely to procreate and also likely to be a big earner.

Then, you give people that don't have that personality trait a bunch of money, you can't expect that persons behavior to be the same.

As I understood, there is a causal relationship. Not needing children to support you when you get old, and being able to get your first couple of kids past the infant mortality barrier, tend to result in fewer children overall.