What if all housing was owned by the government?

This question stems from a monologue I was having about how I would cure the homeless problem in America. I'm in no way saying this is what we should do, but it's just a radical idea.

Say that all housing was owned by the government. You don't have to pay for a mortgage, or pay rent. Instead it is paid for through taxes. That way even if you lose your job, you wouldn't have to worry about losing your house. When you turn 18, you can apply for your first home. When you get into said home, it's yours to live in forever, and when you die your kids will be first on the list to acquire said home if they want.

If you make enough money, then it would be possible to buy the home from the government at a standard mortgage rate. Once it's paid for, it's then your own house and you can do with what you whim. You can rent it out, make changes to it, or sell it to another private party, or even back to the government.

I understand, it's radical, Marxist, and not very thought out, but I'm just curious. Tell me why this would or wouldn't work, thanks.

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