Can number of children in rich country is small due to this simple economic factor? In rich countries, people are expensive.

Even taxi driver in Singapore earns $6k per month.

Now, the "market" tend to correct it self.

If something is expensive, and it's marginal productivity is low, then the "market" would produce less of that thing.

The market produces far more computer than abacus for obvious reason. Once the market produces more and more computer abacus become obsolete and too expensive.

The same way with humans. Humans are getting more and more expensive. So the market produce less of them.

And that's why rich countries produce less and less of them.

Now, I have many doubt about this theory. After all, the number of children you have is not up to the market fully.

It depends on how well you can attract mates to knock up. Yes. That would be market mechanism.

However, there are many government rules that sort make this not market mechanism. Polygamy tend to be illegal. Government tend to subsidize children of the poor with public schools and welfare. Minimum wage means poor people's salary are set by government. Child support laws are set proportional to man's wealth.

But still, market "distortion" may not be that "big". The market may still handle it. 

And overall, things still follow market mechanism. The market, decides that we have too many humans. The market stop producing it.

Poor men, for example, can't get a girlfriend. Or poor people think that cost of raising children is too high they decided not to produce children and so on. Those seem like the way the market handles things.

Am I right? Wrong? Anyone study this?
Let me fist answer that individual decisions on whether to try to have children or not is a decision that is not driven by economics alone. There are many other factors that influence these decisions such as individual experiences and preferences and social norms.

That being said there is a relatively large literature on the economics of families and children. Some of the arguments as to why people have fewer children in developed countries include:

The costs of children are higher and the direct economic benefits are lower in developed countries. In developing countries children constitute extra labour (although they also need to be fed) and they require little education and other investments. In developed countries children typically do not contribute towards family income, and need large investments in terms of education and raising. A related argument that was put forward by Becker and Tomes is that there is a trade-off between the "quality" and the quantity of children, and in developed countries there is a larger demand for quality as opposed to quantity of children.

Another argument is that the need for a large number of children is lower in developed countries. In developing countries children are your old-age insurance policy, and infant mortality is high. In developed countries we have pension systems, good (partly) public health care and elderly homes, all of which reduce the necessity of having many children. The good health care system reduces the infant mortality so you don't need more children to be sure that you'll have a few when you get old. The pension systems, good public health care system and elderly homes mean you have to rely less on your children for income and care when you get old.

A final argument that has been made is that the opportunity costs of children are larger in developed countries. Most people are highly educated, and receive a relatively high salary. If they have to reduce their working hours to provide care to children, the opportunity costs are higher (all else being equal).