Optimal Social Insurance for Heterogeneous Agents with Private Insurance

ABSTRACT 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. INTRODUCTION Social insurance provides the government with the ability to insure individuals against risk and also to redistribute across agents in the economy. To account for each of these effects in an economic model, we must allow for an ex ante heterogenous population of individuals, each of whom face idiosyncratic risk. Moreover, it is reasonable to assume that in a market economy the social planner cannot directly control or exclude the provision of actuarially fair private insurance, which is of value to each individual. Thus, it is important for the government to internalize the endogenous response of private insurance markets and individual behavior to social policy. (1) Failure to recognize the presence of privately provided insurance can lead to incorrect prescriptions for optimal policy that may result in substantial welfare losses. This article therefore takes up the task of characterizing, analytically and numerically, the impact of introducing both ex ante heterogeneity across agents and ex post risk on optimal social insurance formulas in the presence of endogenously supplied private insurance. Specifically, this article builds a model of the economy in which a heterogeneous population of workers who differ with respect to productivity each face an ex post shock to earnings. Workers exert effort to increase the likelihood of a high shock. This effort is unobservable to insurers, both private and public, generating moral hazard. The important assumption, however, is that private insurers have better information than the government and in particular can observe the productivity of each agent. Thus, private insurance contracts are type contingent and vary across the population, while a single social insurance contract applies to everyone. I consider two cases of social insurance, depending on whether or not the social insurance contract can be state dependent for an individual. For example, social insurance for unemployment is state dependent since unemployment is verifiable, whereas in the case of productivity shocks, social insurance in the form of an income tax is state independent. In addition, I conclude with numerical simulations for the income tax as social insurance case. I establish four main results in this article: two analytically and two via numerical simulations. First, without private insurance, expressions are derived decomposing the social insurance benefit into separable insurance and redistributive components. In the state-dependent social insurance setting, the redistributive benefit, which depends on the covariance between risk and income, is new relative to standard models of social insurance (e.g., Baily, 1978; Chetty, 2006) that assume a representative agent. With ex ante heterogeneity, this redistributive benefit shows that the case for social insurance is greater when risk and income are negatively correlated across the population. …