A Crises-Bailouts Game

This paper studies the optimal design of a liability-sharing arrangement as an infinitely repeated game. We construct a noncooperative model with two agents: one active and one passive. The active agent can take a costly and unobservable action to reduce the incidence of crisis, but a crisis is costly for both agents. When a crisis occurs, each agent decides unilaterally how much to contribute mitigating it. For the one-shot game, when the avoidance cost is too high relative to the expected loss of crisis for the active agent, the first-best is not achievable. That is, the active agent cannot be induced to put in effort to minimize the incidence of crisis in a static game. We show that with the same stage-game environment, the first-best cannot be implemented as a perfect public equilibrium (PPE) of the infinitely repeated game either. Instead, at any constrained efficient PPE, the active agent "shirks" infinitely often, and when crisis happens, the active agent is "bailed out" infinitely often. The frequencies of crisis and bailout are endogenously determined in equilibrium. The welfare optimal equilibrium being characterized by recurrent crises and bailouts is consistent with historical episodes of financial crises, which features varying frequency and varied external responses for troubled institutions and countries in the real world. We explore some comparative statics of the PPEs of the repeated game numerically.