Right now you can guess that for many couples, 2 kids becomes close to unaffordable, and 3 becomes nearly unmanageable. Individually, you will see couples choose to do it and even to do it in ways that others both envy and chastise; but overall, it's not happening.
It requires many things to change not just in economy, but in society as a whole. It's not going to happen in a society devoted to growth, that's for sure.
As long as some people are still having kids, the birth rate will eventually reach replacement rate.
This won't save us. By the time civilization can't make birth control pills, it also means we've lost advanced medical care and now maternal/infant mortality kicks in. Nearly every baby born now survives into adulthood, barring rare misfortunes. But the idea that obstetrics will still be cruising along while we can't crank out simple pharmaceuticals is nonsense.
The trend is accelerating. We'll see 0.1 fertility rates in our lifetimes, you and me, and I'm old. We'll see central Africa hit sub-replacement fertility rates in 25 years. And even then we'll still have to listen to retarded jackasses tell us how it's no big deal, population will bounce back once things clear out. Buried deep in the human psyche is some truly superhuman level of obliviousness and denial of reality, and no logic or long term observation or plain facts are a match for it.
I'd take that bet. Except... the term of the bet is "in our lifetimes", which means that it ends when one of us dies, which means that collecting is going to be a problem.
For the record, I'm 64. "In my lifetime" is somewhere between 1 day and 40 years.
We'll still be getting the cycle of "politicians are trying tax incentives" and "someone needs to fix the economy so people feel safe having children" headlines in North America, which is apparently enough to lull everyone into thinking it's not a problem.