A General Formula for the Optimal Level of Social Insurance

In an influential paper, Baily (1978) showed that the optimal level of unemployment insurance (UI) in a stylized static model depends on only three parameters: risk aversion, the consumption-smoothing benefit of UI, and the elasticity of unemployment durations with respect to the benefit rate. This paper examines the key economic assumptions under which these parameters determine the optimal level of social insurance. A Baily-type expression, with an adjustment for precautionary saving motives, holds in a very general class of dynamic models subject to weak regularity conditions. For example, the simple reduced-form formula derived here applies with arbitrary borrowing constraints, endogenous insurance markets, and search and leisure benefits of unemployment. A counterintuitive aspect of this result is that the optimal benefit rate appears not to depend on (1) any benefit of UI besides consumption-smoothing or (2) the relative magnitudes of income and substitution effects in the link between UI benefits and durations. However, these parameters enter implicitly in the optimal benefit calculation, and estimating them can be useful in testing whether the values of the primary inputs are consistent with observed behavior.