That's a great question. It addresses the basis for many people's concerns over the viability of so many 'programs' that we might undertake.
Yes, I said "under" population, not overpopulation. You see, you hit the nail on the head later when you compared educated (sic) to undereducated reproduction. The problem is not that we are multiplying too fast and creating too many human bodies on the planet. That's a myth of statistics for people who don't understand trends. The real problem is that natural selection is no longer determining which category takes over the reproduction swings. As you might guess, that one's easy to understand so I won't bother explaining it. For more questions, watch the movie, "Idiocracy". :)
The reason we are heading for underpopulation is more complex. The fact is that you "can" force people to not have kids but you can't force them to HAVE kids. Once a social trend sets in that it's bad in some way to have too many kids, there is no social mechanism to reverse that. You simply have to wait it out. Why is this a problem? Another great question.
When you break the globe into various societies, each roughly falls into one of two categories. Either they are "wealthy" (blind label) and their replacement rate has dropped below 1.8ish or they are "struggling" (poor) and are still having lots of kids.
Since there are so many countries struggling, the majority are still having large families. There are two problems with that. First, many of those kids are not surviving to reproduce for themselves or they are coming up sterile.
Second, due to this large number, the 'news' (propaganda) reaching the wealthy countries is that we're overpopulated. So, despite the US, UK, Europe, etc. having 1.5-1.7 kids per woman (It takes 2.15 realistically to replace equally)... despite that it's too low, it's dropping further. This is setting the trend for all newcomers to the 'wealthy' group. As individual villages in Africa become wealthy (due to some tech entering or whatever) when they get online and healthy, their rate immediately drops from 3.1 to 2 or less. That's a huge drop very quickly.
What will happen when some energy technology makes electricity, potable water, cell phones, internet, refrigerators and vegetables instantly ubiquitous across that entire group of countries over a very short 5 year span? What will happen is the globe's overall replacement rate will drop from 2.4 down to 1.95. With that new rate, you will see the population clock drop by a full 10% per generation with no way to slow it.
Estimating this against real numbers, this means we are headed for a peak population of maybe 11 billion people in less than 35 years. After that, it will drop at a rate faster than it ever grew.
That's if we do nothing. If we go into this fearing OVERpopulation, it could get much worse. If we undergo a mass viral die-off, worse. If we ramp up the wars, worse. If everything goes against this in the worst way, we literally could be down to a 100 million global population by 2150. We don't have things designed for that few people but what about the first problem of who breeds? Do we really want only the dumbest and laziest 100 million people on this planet when our great grandkids are born? (Oh, wait. If only the dumb survive, then ours will be long gone, right?)
Long story short, population rates are diving with no fixes so overpopulation is not of a concern. And with the Earth having a carrying capacity of 12 Billion people, all we really need to do it to stop the waste to live sustainably. That's the easy part. It's getting this through to the law makers that's hard.