I assume you mean less children, right? More wealth means less children but only up to a certain point, after that the trend reverses and even more wealth means more children again. [1] I did not check the numbers but I will give the author the benefit of doubt. It certainly makes intuitive sense that more wealth gives you the opportunity to have the luxury of more children.

It is actually harder to understand the initial decline because it is not obvious that having more children when you are less wealthy is a good thing. Maybe it is just a indirect effect because of less education and human development in general but I discussed this before and am not sure of the best way to think about it, especially because of arguments like having many children is a substitute for other social security system in less wealthy societies which implies that it is not really an indirect effect.

But setting this aside and assuming that the observed pattern does not radically change in the future, then more wealth will not stop population growth but at best slow it down somewhat and the extend of that depends on where the countries finally end up. But it gets worse, not only does increasing wealth not reduce population growth in the long term, it also increases energy consumption per capita. [2]

So increasing wealth gives us more people, which is not good, and higher energy consumption, which is also not good unless it is clean energy. And there is another aspect - India adds 5 times as many people to the world population per year as the USA. But the energy consumption per capita in the USA is so large that the USA nonetheless adds more than twice the additional energy consumption of India per year. [3]

While population growth is a serious problem - a constant 1 % population growth per year will definitely kill every human on Earth within 1400 years - it just won't happen. If there are to many people, they will just die, either starve or get killed in resource wars. The problem will solve itself although I would much prefer that we proactively ensured that population growth will not be stopped by some disaster.

But the more pressing issue is the climate change due to our energy consumption and the way we produce it. We probably can not solve that problem by simply killing enough people. Therefore population growth is at least currently only an issue insofar as it contributes to the increase in energy consumption and trying to reduce population growth by increasing wealth and energy consumption is pretty likely counterproductive for both issues.

But admittedly the exact outcome is not so easy to predict. If countries would stay at the point where the trend in [1] reverses for a long time, then we could get reduced population growth or even a population decline without further increase in energy consumption per capita and the former might offset the later depending on the pace of the increase in wealth. But I do not see what kind of policy could keep countries there, one would essentially have to stop the further development of countries when the trend reverses.

And here is the blog [4] I took the graphs from.

[1] http://physics.ucsd.edu/do-the-math/wp-content/uploads/2013/...

[2] http://physics.ucsd.edu/do-the-math/wp-content/uploads/2013/...

[3] http://physics.ucsd.edu/do-the-math/wp-content/uploads/2013/...

[4] http://physics.ucsd.edu/do-the-math/2013/09/the-real-populat...