We study the welfare effects of social security in an overlapping
generations general equilibrium model with aggregate and idiosyncratic
risk. Prior research on social security has only considered either
aggregate or idiosyncratic risk. We show analytically that the
aggregate and idiosyncratic risks interact due to the life-cycle structure
of the economy. This interaction increases the welfare gains of a
marginal introduction of an unfunded social security system. Adding
a second interaction by making the variance of the idiosyncratic risk
countercyclical further increases the welfare gains. In our quantitative
experiment, raising the contribution rate from zero to two percent
leads to long-run welfare gains of 3.5% of life-time consumption on average,
even though the economy experiences substantial crowding out
of capital. Approximately one third of these gains can be attributed
to the interactions between idiosyncratic and aggregate risk.