Optimal insurance with time-inconsistent agents and hidden storage

Abstract

We examine the provision of insurance against non observable liquidity shocks for a time-inconsistent (hyperbolic) agent. This issue can be phrased as an optimal control problem, with mixed constraints and monotonicity condition on the control, and can be solved either when lack of self control is strong enough or when there is no time-inconsistency. In the latter case, the optimal contract for a time consistent agent resembles to an insurance contract that is ruled out when the agent can privately store resources. By contrast, in the former case, hidden storage is not binding and the optimal contract is similar to a credit contract. We study martingale property of the process in the presence of repeated shocks, and show that divergence between those two polar cases becomes even wider: the optimal contract leads time-consistent consumer to almost surely accumulate wealth over time, whereas optimal allocation for hyperbolic consumers leads almost surely to impoverishment. We discuss applications to social security design and consumer over-indebtedness.