optimal insurance for time-inconsistent agents

We examine the provision of insurance against non-observable liquidity shocks for a time-inconsistent agent who can privately store resources. When the 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 consumers to impoverishment almost surely. By contrast, the optimal contract for a time-consistent agent only allows him to borrow at the economy’s interest rate, and induces him to almost surely accumulate wealth indefinitely. Those results bring out how lack of self-control changes the nature of optimal savings and borrowing instruments, and may lead individuals and states to impoverishment. We discuss applications to consumer over-indebtedness, social security design, sovereign debt crises and Pigovian regulation.