Medicaid finances U.S. nursing home care through federal matching grants that reward verifiable volume, not quality. We show that states exploit this through creative financing, diverting funds earmarked for nursing homes. This turns the federal match into a pure volume subsidy and generates a novel allocative distortion we term public moral hazard. We develop the mechanism theoretically and test its predictions using 24 federal audits and administrative microdata from Indiana. Event studies show Medicaid dementia volume rises 12\%; structural quality estimates show the expansion concentrates in the lowest-quality facilities, reallocating patients toward worse providers and reducing one-year survival.