We construct a Blanchard-style overlapping generations model consisting of long-lived individuals who have uninsurable idiosyncratic risk resulting from uncertain retirement periods and medical costs in retirement. Without social insurance, such individuals must save for these eventualities. We examine the impact of pay-as-you-go social insurance policies (public pensions and medicare coverage) on individual and aggregate consumption, saving, and wealth levels as well as wealth distribution. We also derive expressions for optimal (Pareto improving) social insurance policies.