turnip,
US's current birthrate is right at replacement level (or close to it). Plus we have a large amount of immigration (not even counting illegal immigration, which the effects on economy I know very little about). And our economy is growing. So we're in OK shape economically speaking.
The reason a low birth rate hurts economically is very basic from a high level. In Europe, it's basically a bunch of cradle->grave welfare states. Unemployment rates are large. Work weeks in many countries are 30 hours a week, with 4-6 weeks guaranteed vacation and early retirement that bring with it expensive, lavish retirement packages provided by the gov't. Who pays? The workers, and the tax burdens are enormous. When your workforce is dwindling, it's clear that the burden per worker increases. So some will leave to find greater reward for their efforts. Some will simply say "screw it" and join the welfare crowd, seeing little benefit in working hard. Eventually it all collapses.
Yes, immigration helps curtail this, but it's not safe to rely on. I believe in places like France and Germany where unemployment is around 10%, around half of the people on welfare are immigrants.
As for the places with high birthrates and awful economies (Africa, Middle East), there are a bunch of factors that I'm not entirely educated on. Part of it might have to do with agricultural based societies (kids = workers, like early US), part of it's an ideology (devout Muslims want kids, much like devout Christians), and I'm sure there are some other reasons.
For more on the subject, I suggest Mark Steyn's
America Alone. Yes he's a hardcore conservative and you may not agree with it all. I didn't. But it's a pretty basic look at the importance of demography and ideology and there's a lot of good data in it. Worth a read one way or the other, IMO.
I'm currently looking for other reads about demography and economic impact, so if anyone has some books to suggest please provide.