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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.

Abstract This paper studies bank runs in a model with private money. We show that allowing claims on demand deposits to circulate as a medium of exchange can help prevent bank runs. In our model, there exists a unique banking equilibrium where no one demands early withdrawals of real goods and agents in need of liquidity use private money to finance consumption. With private money, the unique equilibrium not only eliminates bank runs but also improves banking efficiency. The implications of our model are consistent with the evidence from the banking history of the United States.