This paper shows that bank deposit contracts can provide allocations superior to those of exchange markets, offering an explanation of how banks subject to runs can attract deposits. Investors face privately observed risks which lead to a demand for liquidity. Traditional demand deposit contracts which provide liquidity have multiple equilibria, one of which is a bank run. Bank runs in the model
cause real economic damage, rather than simply reflecting other problems. Contracts which can prevent runs are studied, and the analysis shows that there are circumstances when government provision of deposit insurance can produce superior contracts.