You're assuming a private supply-demand model. I lean towards a government controlled sector (through large crown corporations) whose mandate is not profit/bottom line. In fact, I'd argue that it should be subsidized and ran at a loss, if anything. Sort of like how we run public transit, the postal service, sewage, waste disposal, healthcare; you know, essential services. A two tier system where these crown corporations, operating at a loss, at least fulfill a large segment of the demand, and private corporations, operating at profit, cover the luxury segment or other niche segments, is ideal.
In an era where wealth aggregation is becoming a serious concern, and housing is the overwhelming largest pot of an average individual's spending, I'd say that mass subsidized housing is the best way to combat wealth inequality.