A lot of great discussion in this thread in the last few pages.
I think a big part of birth rate decline is that raising children relative to expectations has become exponentially more expensive/difficult than in the past. It may not actually be more of a sacrifice to raise kids in the world today than in the 70s, but I think people certainly feel like it is. Anecdotally, my experience as a parent thus far (currently have a 7 yo and 4 yo) has been that the sacrifice in freedom and the intensity of pressure to be a good parent has been a lot worse than expected; I am relatively well-off, so the financial burden hasn't stung much, although shelling out $3000/mo for daycare didn't feel good. Anyway, I don't think I'm actually having a harder time raising my kids than my parents did, but it certainly feels like I am; that influenced our decision to stop at 2, and it certainly influences how I talk about the experience of having children to people who have not yet had children. I think that summarizes the cause of the drop in birth rate in the 1st world over the last 30 years. To change that, you have to change the perception of the experience of having children.
My solution to address that would be similar to ideas that have been cited many times in this thread: paid parental leave (2 years per guardian per kid), free daycare (free starting at age 2), universal Pre-k, universal healthcare, and significant tax incentives for people having 3+ kids. I don't think that alone would get 1st world birth rates above 2.1, but it would make a difference.
Many first world countries (primarily the US, Canada, western Europe, and Australia) can likely bridge the gap in the short term with immigration and by extending the working age. Longer term, increases in worker productivity is the answer. If robotics and software can make a 2040 worker as productive as 3 2020 workers, then the economy and standard of living will be better off. But obviously if there is a net decline in population, eventually the economy and standard of living won't matter (unless we cure aging/death).
I personally am very optimistic on this issue. I think things will be done to address the problem. I do think Russia and China are proper fucked and that is risky for the world, but I think that risk will have either turned into reality or been mitigated within the next 10 years. For now I would predict the best years for humanity lie ahead and would guess world population stabilizes around 12 billion, accompanied by massive increases in standards of living worldwide. I for one really like the idea that we might finally be forced to treat every young person as the tremendously precious resource they are.