Although I am by no means an expert in population, there are a number of interesting studies which show a deep statistical correlation between increased income and decreased birth rates over time. Higher income families and high income societies have significantly lower population replacement rates.

That being said it also seems that there is an immediate correlation between suddenly higher income and a individual family unit also increasing (i.e. when a family gets a sudden, large increase in income they will generally also increase their family size at that time).

So, with all of that being said, since in the words of the Gates this is a "bet", we could also potentially bet that by increasing the number of middle class individuals within a given population we will also see an overall increase in birthrate among those populations.