Bank runs and the optimality of limited banking

We extend the Diamond-Dybvig model of bank runs to include a specification of how much to deposit. When the propensity to run is zero, we prove an equivalence result, that the efficient allocation (satisfying resource, IC, and sequential service constraints) can be achieved in equilibrium as long as the deposit level is above a threshold. Within this range, the lower the deposit level, the more tempted patient depositors are to withdraw early. When the propensity to run is positive and certain conditions are met, the optimal banking system entails less than full deposits and runs on the equilibrium path. We extend the analysis to consider a propensity to run that depends on the risk factor of the run equilibrium.