It's not very clear what the connection is between your two paragraphs. In fact, they seem contradictory; why does natural selection lead to fewer smart people having children?
It is just a consequence of the human environment changing. Birth control and the modern education system disproportionally effect the number of children smart people have. The end result is the smarter you are the less children you will have all other factors being equal.
Another major factor contributing is children are very expensive and smart children are much more expensive to raise than the average child.
Smart children are not more expensive then average child. Competitive, ambitious parents and parents who want their children to be smart have additional expenses.
Yes they are (on average anyway). Their parents sacrifice greater income to have them and even their education is more expensive in that living in areas with good schools cost more.
Of course you are right that provided identical environments smart and average children cost the same, but since smart parents tend to earn more, smart children cost more.
Parents who choose good schools choose them regardless of whether kids are smart. High income parents don't want the children in bad schools even if kids are not smart (whether it means stupid or just normal).
Non-smart child of smart parent costs even more - on tutors and such to raise the kid.
That is not smart children costs more. That is ambitious/smart parents spend more on children. The cost of smart child here is merely function of correlation with parent.
You are assuming that smart kids are randomly distributed through the population. High income parents have disproportionally more smart kids than average income parents and disproportionally live in above average cost locations.
The reason for the correlation between child raising cost and intelligence is there is a strong correlation between intelligence and income and between the intelligence of parents and their children. Smart kids tend to have smart parents who tend to have high incomes. Of course there are many exceptions, but we are talking about group averages not individuals.
It started with "Another major factor contributing is children are very expensive and smart children are much more expensive to raise than the average child."
In that context, my response that "The cost of smart child here is merely function of correlation with parent." is valid.
Another major factor contributing is children are very expensive and smart children are much more expensive to raise than the average child.