That puts the horse before the cart, Michael. Poor people have more children, because children is their insurance policy against poverty, old age etc. So the proper counterfactual to poor having many children is not well off having fewer children - but poor having fewer children. With fewer or no children, the poor's economic condition would be even worse.
For example, the urban anthropologist patricia fernandez kelly found that teenage pregnancy in inner cities is often not a result of "irrational behavior" (i.e. teens not knowing how to prevent unwanted pregnancies) but a conscious decision to gain social status (mother having higher status than single women).
methinks, the causal connection between number of children and poverty is typical example of neo-classical mythology that put causality on its head. While economic well-being has a negative effect on the number of children, it doe not follow that the number of children has an effect on your socio-economic status. It is like saying: "wealthy people tend to go to the opera more often than poor people; so I better start going to the opera to increase my chances of becoming rich."
wojtek