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This paper develops an overlapping generations model with stochastic production and incomplete markets to assess whether the introduction of an unfunded social security system can lead to a Pareto improvement, even if the initial equilibrium is neither production-inefficient in the spirit of Diamond (1965) nor dynamically inefficient in the spirit of Samuelson (1957). When returns to capital and wages are imperfectly correlated and subject to aggregate shocks, then the consumption variance of all generations can be reduced if private markets or government policies enable them to pool their labor and capital incomes. A social security system that endows retired households with a claim to labor income may serve as an effective tool to share aggregate risk between generations, in the absence of financial securities that could serve a similar purpose. We construct numerical examples which demonstrate that the intergenerational risk sharing role of social security can be sufficient to warrant its introduction on welfare grounds. For a realistically calibrated economy, however, we find that this role is insufficiently strong quantitatively to offset the negative effects an unfunded system has on capital accumulation and thus does not constitute a Pareto improving policy.