Not necessarily. Demographics indicates that developed nations seem to have near-replacement birth rates. Anything less than 1 and you get exponential decay. So a small nudge will eventually lead to large reductions in population, but the rate of decline need not be severe.
There are of course issues that a lot of the economy is predicated on growth, but again, I'm actually pretty optimistic that markets and humans are pretty resilient and adaptable given enough warning and time.
I don't know, Japan seems to be doing OK. There are all sorts of problems that are going to eventually arise simply due to the fact that our economies are built on endless growth. But IF they embraced it and were able to advance automation fast enough, most of those would be overcome. Mainly: who will take care of the elderly.
HA part of that could be easily solved if suicide weren't so stigmatized. If everyone could just agree that once you're past the age of 80, once you get to a point of being infirm you just take a magic pill and peace out on your own terms, we'd be far better off as a species.
From the article:
"In a 1974 United Nations population conference in Bucharest, Karan Singh, a former minister of population in India, illustrated this trend by stating 'Development is the best contraceptive.'"