Demographic-economic paradox - Wikipedia, the free encyclopedia is an interesting graph showing number of offspring per woman against gross domestic product.

And on the reproductive magic front, a sufficiently advanced magic would most likely allow for selection of healthy offspring characteristics, number of offspring per birth, and so on. It would also potentially allow for the development of external non-human uterine replicators to allow for potentially unlimited numbers of offspring for a sufficiently adept or rich individual. Even application of magic to the two basic limiters in a population (food and hygiene) would make a huge difference in how the world develops (the longer unrestrained breeding can occur, the longer the population rides that J-curve).

In my opinion, the critical long-term drivers for the success of a culture are:
1) an increasing population (the number of geniuses per capita seems to be very roughly constant),
2) sufficient surplus in the culture to allow for specialization (if everyone is a subsistence farmer, you're not going to develop too much knowledge on any one subject),
3) a way and will to broadcast persistent information (writing + printing press to let everyone know about that neat stuff that your specialists have discovered).