It's not so much that the birth rates go down when the level of poverty goes down, but birth rates go down when the level of education go up and then the standard of living goes up. This seems like a semantic argument, I know but it isn't just birth rates that goes down that goes down but age of conception goes up as well, allowing for more independent and more qualified choices. Not to mention a more egalitarian society for the sexes.

If we focus on education and not poverty, that would seem an easier problem to solve. Unfortunately, there are more societal and cultural boundaries to overcome to educating certain segments of specific populations (women in some societies, homosexuals in others, untouchables in some).