Poor heath, large acute and long-term care medical expenses and spousal death are significant drivers of impoverishment among retirees. We document these facts and build a rich overlapping generations model that reproduces them. We use the model to assess the incentive and welfare effects of U.S. Social Security and means-tested social insurance programs, such as Medicaid and food stamp programs, for the aged. We find that U.S. means-tested social insurance programs for retirees provide significant welfare benefits for all newborn. Moreover, when means-tested social insurance benefits are of the scale in the U.S., all newborn prefer that Social Security be removed. Finally, we find that the current scale of means-tested social insurance in the U.S. is about right in the following sense. If we condition on the current Social Security program, the benefits of increasing means-tested social insurance are small or negative.