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Most of the research related to this is on rent control rather than rent stabilization like Biden’s proposal, but the effects are likely similar. As with much economic research, rent control policy outcomes can be hard to assess. Economics 101 tells us that in a free market, rent control’s price cap decreases profitability, inevitably leading to less rental housing being available.
However, some studies disagree. This article by Rutgers professor Mark Paul goes over many studies that find positive effects of rent control, mainly for those people who live in rent-controlled units whose costs are frozen. The research is clear that some people do benefit from rent control.