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.