We develop a mixed duopoly model with quality-differentiated products. The public firm chooses its product quality and offers it for free to eligible individuals as a form of social assistance to maximize consumer welfare, while the private firm chooses both the product quality and price to maximize profit. We first characterize the pattern of market segmentation for given product offerings, highlighting the non-monotonic relationship between market participation and individual income. We then calibrate the model to health insurance for the U.S. working-age population, with Medicaid acting as the public firm. We examine the distributional implications of policy changes, both actual and hypothetical, that lead to various degrees of public program expansion. Despite potentially significant inefficiency of the public firm, the overall effect of its expansion is welfare improving. Central to these findings is the significant market power enjoyed by the private firm that results in high profit margins if left unchecked. As more individuals become eligible for the public program, the resulting increase in competitive pressure disciplines the private firm’s ability to exercise market power.