Social Insurance, Information Revelation, and Lack of Commitment

We study the optimal provision of insurance against unobservable idiosyncratic shocks in a setting in which a benevolent government cannot commit. A continuum of agents and the government play an infinitely repeated game. Actions of the government are constrained only by the threat of reverting to the worst perfect Bayesian equilibrium (PBE). We construct a recursive problem that characterizes the resource allocation and information revelation on the Pareto frontier of the set of PBE. We prove a version of the Revelation Principle and find an upper bound on the maximum number of messages that are needed to achieve the optimal allocation. Agents play mixed strategies over that message set to limit the amount of information transmitted to the government. The central feature of the optimal contract is that agents who enter the period with low implicitly-promised lifetime utilities reveal no information to the government and receive no insurance against current period shock, while agents with high promised utilities reveal precise information about their current shock and receive insurance as in economies with full commitment by the government.