This is a very interesting idea, but I’m not convinced it’s true. Why do you think that might be the outcome?

Because people are urbanizing despite the price hikes. Which means they prefer higher rents to staying where they are or moving. Probably because of other factors like commuting, other common distances being shorter, friends and family.

So if higher prices are indeed a problem, holding people back from urbanizing, you'd think they'd urbanize faster if the pressures against it decrease.

It's certainly fair to think that there's some backlog of people who would move in if prices reach some price threshold, which could be considered resistance against the price falling lower than that threshold.

But in the case that you dont build units, you're still going to get some of that urbanization by those who can afford it (or are willing to struggle to survive in the city), and this will drive up prices even further.