Risk Sharing Under Limited Commitment and Private Information

What are the limits that private information and limited commitment impose on risk sharing? Previous literature considered both problems separately, or modeled the lack of commitment only as participation constraints. However, with private information, lack of commitment does not collapse to participation constraints and requires an extended notion. We argue that this narrow understanding of limited commitment is responsible for the difficulties to find a decentralization for the constrained efficient allocation, which rely on commitment of some parties or unreasonable off-the equilibrium beliefs. We propose a notion of limited commitment involving renegotiation-proofness of the contracts, which provides a more general notion of the lack of commitment in the presence of private information. We show that there can be risk sharing with ex-post efficient contracts, and we decentralize the constrained efficient allocation a la Alvarez & Jermann (2000), but with borrowing constraints that depend on the whole portfolio in all states. Finally, we derive implications for the optimal design of state-contingent sovereign debt, like GDP-linked government bonds.