What problem is public housing supposed to solve? The current housing crisis is a supply problem. We aren't building enough housing in places with high demand,

What problem is public housing supposed to solve? The current housing crisis is a supply problem. We aren't building enough housing in places with high demand, so prices are going up. Even if public housing was subsidized to be cheaper than the market rate, it doesn't matter because the public housing will just fill up right away and then we're back to square one. Is there something about public housing that would allow them to build more than the private sector?