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 agent who can privately store resources. When lack of self control is strong enough, hidden storage does not constrain the allocation of resources. The optimal contract is strictly concave and similar to a credit contract: It allows the agent to borrow at an increasing cost, and save at a decreasing rate of return. Extending the model to an infinite horizon, we then show that, in the presence of repeated shocks, the optimal contract leads to impoverishment almost surely. By contrast, the optimal contract for a time consistent agent only allows him to borrow at the economy's marginal rate of transformation, and induces him to almost surely accumulate wealth indefinitely. We discuss applications to social security design, Pigovian regulation, sovereign debt crises and consumer over-indebtedness.