In a recent AMA, Bill Gates was asked:
Mr Gates, Do you feel that we are facing an overpopulation problem on this planet? If so, what do you think needs to be done about it?
He gave a short answer, stating that as people become more healthy they choose to have fewer children. To back this up he referred to the work of Hans Rosling from Gapminder. At times I found his style a bit odd. That didn’t detract from how amazing it was at tying together so many topics that seem big and hard to fix.
Over population, extreme poverty, birth rate, infant mortality, climate change, education, access to healthcare, and ignorance of the west are all interconnected in this video. The basic gist is that since birth rates are falling the population will begin growing much more slowly in the near future. He outlines a number of reasons why birth rates are falling like education, lower infant mortality rates and a general cultural shift in poor countries.
I am not sure about everything being as cut and dry as he makes it. You really should watch the whole video and judge for yourself.