The majority of U.S. households that qualify for federal rental housing assistance do not receive it. In the absence of an entitlement to housing assistance, an underexplored cause of the shortfall is that higher rents in some areas driven by supply-constraining local regulations increase program costs, leaving fewer funds available to serve additional families. In this paper, we simulate the effect of increasing housing supply on the cost of Section 8 housing assistance programs in Los Angeles, as well as all 11 metropolitan areas most constrained by local regulations. If Los Angeles (all 11 metropolitan areas) produced new housing units at the same rate as the 90th percentile metropolitan area for a decade, market rents would fall by 18.1 percent (2.0 to 24.0 percent), and federal cost savings would equal $353 million ($1.8 billion), enough to increase the number of assisted families by 23.8 percent (18.6 percent). For comparison, doubling the number of units placed in service through the Low Income Housing Tax Credit for a decade—an alternative method for increasing housing supply—would lead to much lower cost savings of $18 million in Los Angeles, and $231 million across all 11 metropolitan areas.