This article analytically characterizes optimal social insurance in an economy with both ex ante heterogeneity and ex post risk, decomposing the benefits of social insurance into a redistributive and insurance benefit. Agents exert effort to increase the likelihood of high outcome events and are additionally supplied actuarially fair private insurance for this earnings risk. This article is novel in its joint consideration of two sources of heterogeneity, two potential sources of insurance, and an endogenous ex post distribution of outcomes. The introduction of optimal private insurance eliminates the insurance benefit of social insurance, but leaves the redistributive benefit intact. An income effect induced by the crowding out of private insurance generates an additional benefit to social insurance when it takes the form of a linear income tax. Finally, numerical simulations illustrate how the relative contributions of ex ante and ex post risk differentially impact the welfare loss associated with setting optimal social insurance without recognizing the presence of private insurance.
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