The Economist argues that the main thing holding back nations in sub-Saharan Africa is the high birth rate, and that the primary reason for this is the lack of contraception. I think they are mistaken. I have linked to two dynamic graphs. The relationship between fertility rates and contraception is uneven. Far, far stronger is the relationship between fertility rates and infant mortality. The main single factor determining how many children a woman has is how likely it seems that those children will survive.
The position is of course more complex. Fertility rates fall wherever women have economic opportunities, education and better health; the reason is partly because they then have alternatives, and partly because they are more likely to delay childbearing. Contraception contributes to this process, but it is not decisive. Economic development is far more important.
This, by the way, is the 400th entry I have made on this blog.