Ž . Econometrica, Vol. 67, No. 3 May, 1999 , 527542 SOCIAL SECURITY AND DEMOGRAPHIC SHOCKS BY GABRIELLE DEMANGE AND GUY LAROQUE An overlapping generations model of social security with shocks to the productivity of labor and capital and demographic shocks is studied. We focus attention on stationary long run allocations. An allocation is interim optimal if there does not exist another feasible allocation that improves the expected welfare of all generations, computed conditionally on the state of the world when they are born. We characterize the set of interim optimal allocations and study the equilibria associated with various institutional forms of social security from the point of view of this optimality criterion. We obtain the analogs of the two traditional welfare theorems of microeconomic theory. Assume that Ž . there exists a financial asset in fixed quantity, which supports some non null intergenera- tional transfers. Then the rational expectations equilibrium allocation of this economy is interim optimal. Conversely, any stationary interim optimal allocation can be supported by such an equilibrium, with adequate lump sum transfers. KEYWORDS: Overlapping generations, demographic shocks, optimality, welfare theo- rems. 1. INTRODUCTION SOCIAL SECURITY INSTITUTIONS typically are means of implementing intergenera- tional transfers. In the presence of demographic shocks, it is natural to see how the various social security designs distribute those risks over time. This paper studies the sharing of risks between generations in the framework of a simple overlapping generations model with a single physical good where there is macroeconomic uncertainty: there are productivity shocks, which affect both the wage rate and the return on capital, and there are demographic shocks. The growth rate of the population is random, which is intended to allow for the uncertainties associated with an unfunded social security system. The shocks are Markov, and we limit our attention to stationary population growth rates and stationary allocations of resources. Therefore we are investigating how risks are shared between generations in an ideal world, leaving for further work the study of transition paths. We use the concept of interim optimality, which amounts to standard Pareto optimality once the state of the world in which the agents are born is known. It selects the golden rule path in an environment without uncertainty. The usual optimality condition, the equality of the interest rate to the population growth rate, is extended to this stochastic setting: a matrix made up of the marginal rates of substitution between the future and current states, weighted by the rate of growth of population, has its maximal eigenvalue equal to one. When this eigenvalue is smaller than 1, one may qualify the situation as one where the young agents do not consume enough, andor do not invest enough, given the rate of return on capital and the rate of growth of the population. However, contrary to the certainty case, a simple uniform transfer from the old to the 527