That's not how it turned out in the UK. As this graph illustrates, when the local government authorities were prohibited from building more housing, from the 1980s onwards, the private sector didn't pick up any of the slack: https://i.stack.imgur.com/MmJ5N.png

The capitalists instead decided to optimize for steadily increasing the price of housing by restricting supply, so these are now primarily seen as an investment with nearly-guaranteed returns, rather than places for people to live.