This paper characterizes the welfare gains from redistributive taxation and social insurance in an environment where the private sector provides partial insurance. We analyze stylized models in which adverse selection, pre-existing information, or imperfect optimization in private insurance markets create a role for government intervention. We derive simple formulas that map reduced-form empirical estimates into quantitative predictions for optimal tax and social insurance policy. Applications to unemployment and health insurance show that taking private market insurance into account matters significantly for optimal benefit levels given existing empirical estimates of the key parameters.