Actually, statistics show that wealthy peoples (as measured by per capita GNP) tend also to have the lowest fertility rates.
And wealth strongly correlates with health (as measured by any number of parameters, such as health spending, life expectancy, etc.) - think "typical Nigerian" vs. "typical Scandinavian" or "typical Japanese."
Thus, these billionaires are on the right track: Raise the overall prosperity and health of a society, and you will likely also reduce its fertility.
Regards,
Population will also drop if there is a significant fall in prosperity. Fertility rates may rise, but fatality rates will rise much faster. Populations do not tend to rise when people are starving to death.
On the other hand, a sudden increase in prosperity will reduce mortality much faster than it changes reproductive habits. Posterity under those conditions will descend from those who lack prosperity. The more successful economically will be less successful in the Darwinian sense.