We study the impact of a fully-funded social security system in an economy with heterogeneous consumers. The unobservability of individual health conditions leads to adverse selection in the private annuity market. Introducing social security—which is immune to adverse selection—affects capital accumulation and individual welfare depending on its size and on the pension benefit rule that is adopted. If this rule incorporates some implicit or explicit redistribution from healthy to unhealthy individuals then the latter types are better off as a result of the pension system. In the absence of redistribution the public pension system makes everybody worse off in the long run. Though attractive to distant generations, privatization of social security is not generally Pareto improving to all generations.