We show that market discipline can be effective in resolving the moral hazard problem which arises when depositors do not know whether bankers are monitoring or not the projects they finance. Demandable debt, by allowing the possibility of bank runs, can induce bankers to monitor. However, market discipline comes at a cost. When depositors are not equally informed about the future value of bank assets, withdrawals caused by a liquidity shock may be confused with future insolvency and cause uninformed depositors to precipitate a run. Likewise, withdrawals due to upcoming insolvency may be confused with a liquidity shock and dissuade depositors from running. Bank runs are, therefore, costly and imperfect disciplinary devices for bankers. Our results offer a new perspective on the debate on market versus regulatory discipline of banks.