"You're not allowed to build an apartment complex out of paper" regulations are not the main thing driving costs today. "You're not allowed to build an apartment at all" regulations are. End height and density restrictions.
I'd go with relax over end, because surrounding infrastructure still needs to scale with the amount of people living there, and having very dense skyscrapers next to low height buildings is not great from an urbanist perspective; but there's nothing stopping most American cities from upzoning to 8-10 floors, and upgrading transit and city infrastructure to match the increased density (you're going to need more internet, water, sewer, electricity but nothing fundamentally unfixable).
IMO, when we reach those limits if rents still don't tick downwards because the demand exceeds the market's ability to respond to it we're just going to go through another even more insufferable version of the argument I originally posted about except people will dig their heels in _even deeper_: "density didn't solve anything!"
Well, in 1920 to live in Midtown you would have had to be able to drop $3m ($55m inflation adjusted) for a ground-level mansion and now you can get an apartment there for under $1m, so I would say the cost has dropped precipitously over the past hundred years.
Demand is only induced when you give the thing away for free.
I don't know where to find rental pricing info for 1920s NYC so can't really check. But I'm fairly confident that $55M (inflation adjusted) was not the cheapest apartment available at the time so it's not an apples to apples comparison.
I'd go with relax over end, because surrounding infrastructure still needs to scale with the amount of people living there, and having very dense skyscrapers next to low height buildings is not great from an urbanist perspective; but there's nothing stopping most American cities from upzoning to 8-10 floors, and upgrading transit and city infrastructure to match the increased density (you're going to need more internet, water, sewer, electricity but nothing fundamentally unfixable).