Private Money and Bank Runs

This paper studies bank runs in a model with coexistence of fiat money and private money. When fiat money is the only medium of exchange, there exist a bank run equilibrium and an equilibrium that achieves the optimal risk sharing. In contrast, when private money is also a medium of exchange, there exists a unique equilibrium where no one demands early withdrawals of fiat money and agents in need of liquidity only use private money to finance consumption. The unique equilibrium achieves the first-best outcome and eliminates bank runs without having resort to any government intervention.