Efficient Risk Sharing in the Presence of a Public Good

This paper studies the provision of a public good between two agents under lack of commitment and applies it to the problem of children's consumption in separated couples, where children are considered to be public goods. The custodial mother controls the child's consumption, whereas the father can contribute indirectly by making monetary transfers to the mother, but has no control over how the mother spends them. Using minmax punishments, I look for the Pareto frontier of the Subgame Perfect Equilibrium payoffs, and characterize the equilibrium and long term implications of the model. As in the previous literature, agents' consumptions and continuation values covary positively with their income levels. In the case where the constraint for the public good provision binds, both agents' private consumptions increase relative to the public good provision. In the long run, if some first best allocation is sustainable, the long-term equilibrium will converge to a first best allocation. Otherwise, agents' utilities oscillate over a finite set of values. I then study the theoretical implications of one-sided enforcement when the public good provider has the authority to enforce transfers from the second agent. This is motivated by the wave of US policy reforms to enforce child support payments from fathers. The model predicts an increase in the ratio of the mother's consumption to the child's.