Title:
Risk-sharing with Two-Sided Commitment: a Continuous Time Version
Abstract:
We consider two agents receiving exogenous endowments in a non-storablegood. On they own, each of the agents derives utility by consuming theirindividual endowment, which they receive continuously over an infinitetime horizon. The two agents may decide to pull their endowments andagree on a consumption allocation with a view to risk-sharing. We assumethat both agents have limited commitment, which gives rise to the constraintsthat the utilities they derive from their share of the allocation should be greaterthan or equal to their autarky utilities at all times. We address this problemusing a duality approach. The dual formulation, which involves Lagrangemultipliers to enforce the participation constraints, gives rise to a singularstochastic control problem. As an application, we consider a symmetricenvironment where endowment shares are driven by a mean-reverting diffusionprocess, allowing for aggregate uncertainty. The relevant co-state acts as atime-varying Pareto weight that determines the consumption allocation.We analyse the HJB equation associated with this problem and solve for thefree-boundaries that delineate regions of the state space in which participationconstraints become binding. In some configurations, perfect risk-sharing issustainable.