I don't know that the cause of reduced fertility is prosperity. It might even be the other way around: as population growth decreases people have more because less people to split the wealth with. Our father's dug the gold from the ground and only had 1 child between them. It's probably a bit of both, when you're prosperous you have more to lose by feeding hungry babies.

But it most certainly doesn't balance out. Either a population grows or it shrinks, when a population grows it can produce more but there are more to feed, when it shrinks it ages, at first you have prosperity from that but eventually you wind up poor and old.