How private equity is changing housing

I recently learned about Singapore’s seemingly excellent public housing system that is used by over 75% of its population. Singapore being so capitalist about everything else while carving out housing I think provides evidence that capitalism can be made stronger by keeping certain things away from strict market forces.

https://en.wikipedia.org/wiki/Public_housing_in_Singapore

We don't have a free market in housing at all. It's a cartel with restricted supply.

You can't have a free market if supply (or demand) is constrained.

You either go the liberal+regulated free market way, which will get lobbied, then overregulated, or you go the socialized way, where you allow local government and associations/unions to compete in the market... with the exact same rights as the companies, and no weird rules that prevent them to truly compete.

And if you have a natural monopoly (energy distribution is the worst offender), you have to go the central planning way I guess, until we find something better.

Who is “we”? Even within the USA that statement certainly isn’t true everywhere.