That is about all demographers know for sure. The good news is that it does not seem to be nearly as difficult to reduce population growth as previously thought. As economies modernise, it seems, traditional attitudes change and limiting numbers of children becomes more attractive. Most educated women who are free to choose have few children. What we really need to do, it seems, is get rid of the barriers to people having as few children as the small number freely chosen by the prosperous and educated.
More good news is that limiting fertility is technologically feasible and cheap.
I think it is the only anti-poverty program cheap enough to have a chance of success in the poorest countries. Unless you can think of a way to keep GDP increasing forever, limiting population growth is the only feasible long run solution to the economic problem. Moreover, I think limiting population has positive effects on political freedoms and increasing the value of each individual. My kid will have more human capital, because I invested more in him, than Rosa's 8 kids. The missing bit is for economists to design and endorse institutions to speed up reduction in population growth.