Non-commitment and savings in dynamic risk-sharing contracts

We characterize the solution to a dynamic model of risk sharing under non-commitment when saving is possible. Savings can play two important roles. First savings can be used to smooth aggregate consumption across different periods. Second, when savings are observable, they can act as a collateral that can be seized in the case of default. This relaxes the non-commitment constraint. When the aggregate income is fixed or when one of the agent is risk neutral, the allocation tends to first-best consumption. When one of the agent is risk neutral, this convergence occurs in an expected finite number of periods.