Housing authorities just need to lower housing prices. The model already exists there's just a lot of money fighting it:
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Build large 5-6 story buildings with dozens or hundreds of units in areas where prices are too high. These buildings should be mixed use on the first floor with no car parking.
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Sell or rent them at well below market rates to people that currently do not own homes anywhere under the condition they only sell under those conditions as well
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Watch the market prices fall.
There's currently enough housing in most places there just needs to be a needle applying downward pressure.