I'm not sure why you're being downvoted. This is the theoretical prediction based on simple microeconomics. By example, I believe NY rent controls were ineffective to the point where rents outside of controlled areas went higher than they otherwise would have and within rent control areas there was (and is) scarcity of housing.

It also exacerbates other problems that go hand in hand with urban areas, such as traffic. Most people would rather stay in their rent controlled apartment and drive an hour to work instead of having to move and pay the current market value