Bank Runs, Deposit Insurance, and Liquidity

This is a three-period model in which households face a privately observed, uninsurable risk of needing to consume early, and in which a demand deposit contract lets banks share that risk better than any market can — at the price of admitting a second equilibrium in which everyone withdraws at once. The technology is the source of everything: one unit invested at T = 0 yields R > 1 at T = 2, but interrupting production at T = 1 returns only the initial investment. Agents are identical at T = 0; at T = 1 each privately learns whether they are a type 1 who cares only about T = 1 consumption or a type 2 who cares only about T = 2, with a fraction t of type 1s. Because the type is never publicly verifiable, “simple competitive markets cannot provide this liquidity insurance,” and holding assets directly delivers only consumption of 1 to a type 1 and R to a type 2. Full-information optimal risk-sharing instead equates the type 1’s marginal utility to ρR times the type 2’s, which under the paper’s assumptions that ρR > 1 and relative risk aversion exceeds one everywhere implies early consumption above 1 and late consumption below R. The demand deposit contract, promising a fixed r_1 per unit withdrawn at T = 1 under a sequential service constraint, attains exactly this allocation as one Nash equilibrium when r_1 equals the optimal early consumption. It also admits a bank run, and the paper states the trade-off in its starkest form: runs are an equilibrium for every r_1 > 1, while r_1 = 1 eliminates them but makes the bank “no improvement on simple competitive-claims markets” — “A demand deposit contract which is not subject to runs provides no liquidity services.” Runs here are costly in a specific way: they destroy risk-sharing and interrupt production, so that “everyone receives a risky return that has a mean of one” where direct holding was riskless and at least one. What triggers them is a change in beliefs that can attach to anything — “a bad earnings report, a commonly observed run at some other bank, a negative government forecast, or even sunspots” — and “the observed variable need not convey anything fundamental about the bank’s condition.” Suspension of convertibility at a threshold between t and (R − r_1)/[r_1(R − 1)] turns the good allocation into a dominant-strategy equilibrium, but “works perfectly only in the case where the normal volume of withdrawals, t, is known and not stochastic.” Once withdrawals are random, Proposition 1 establishes that no bank contract obeying sequential service can achieve optimal risk-sharing at all, and Proposition 2 that demand deposits with government deposit insurance can, as a unique dominant-strategy equilibrium, provided the government imposes the optimal tax. The mechanism is precise: “What is crucial is that deposit insurance frees the asset liquidation policy from strict dependence on the volume of withdrawals.” The policy implication the authors draw is that “the real damage from bank runs is primarily from the direct damage occurring when production is interrupted by the recalling of loans,” so “much of the economic damage in the Great Depression was caused directly by bank runs,” citing Bernanke’s finding that run counts predict economic distress better than the money supply. Their own caveats are extensive: Proposition 2 “may be too strong, since it allows the government to follow an unconstrained tax policy,” and “if a sufficiently perverse tax provided the revenues for insurance, social welfare could be higher without the insurance”; the model has one bank, no currency, no risky technology and hence no moral hazard in portfolio choice; and the equivalence between deposit insurance and the discount window holds only “because the technology is riskless.”

Philip H. Dybvig | Macro Paper Warehouse

Bank Runs, Deposit Insurance, and Liquidity