We study a monetary, general equilibrium economy in which banks exist because they provide inter-temporal insurance to risk-averse depositors. A "banking crisis" is defined as a case in which banks exhaust their reserve assets. This may (but need not) be associated with liquidation of a storage asset. When such liquidation does occur, the result is a real resource loss to the economy and we label this a "costly banking crisis." There is a monetary authority whose only policy choice is the long-run, constant rate of growth of the money supply, and thus the rate of inflation. Under different model specifications, the banking industry is either a monopoly bank or a competitive banking industry. It is shown that the probability of a banking crisis may be higher either under competition or under monopoly. This is shown to depend on the rate of inflation. In particular, if the nominal rate of interest (rate of inflation) is below (above) some threshold, a monopolistic banking system will always result in a higher (lower) crisis probability. Thus, the relative crisis probabilities under the two banking systems cannot be determined independently of the conduct of monetary policy. We further show that the probability of a costly banking crisis is always higher under competition than under monopoly. However, this apparent advantage of the monopoly banking is strictly due to the fact that it provides relatively less valuable inter-temporal insurance.