Bank Runs, Deposit Insurance, and Liquidity

That demand deposits improve on competitive markets, that the contract doing so admits a run equilibrium, and that runs cause real economic damage rather than merely reflecting it. The authors list them: “First, banks issuing demand deposits can improve on a competitive market by providing better risk-sharing among people who need to consume at different random times. Second, the demand deposit contract providing this improvement has an undesirable equilibrium (a bank run) in which all depositors panic and withdraw immediately, including even those who would prefer to leave their deposits in if they were not concerned about the bank failing. Third, bank runs cause real economic problems because even healthy banks can fail, causing the recall of loans and the termination of productive investment.” A fourth claim is about the model’s use rather than its content: “our model provides a suitable framework for analysis of the devices traditionally used to stop or prevent bank runs, namely, suspension of convertibility and demand deposit insurance (which works similarly to a central bank serving as lender of last resort).”

Two: why bank contracts in particular are unstable, and what depositors are strategically doing. The paper credits Friedman and Schwartz with providing “substantial insight into the properties of past bank runs in the United States” through description and analysis, but says “existing theoretical analysis has neglected to explain why bank contracts are less stable than other types of financial contracts or to investigate the strategic decisions that depositors face.” Against Patinkin, Tobin and Niehans, who “provide insights into characterizing the liquidity of assets,” the claim is that “this article gives the first explicit analysis of the demand for liquidity and the transformation service provided by banks.” And the conceptual point the authors single out as new is the source of liquidity demand: “our simple model shows that asymmetric information lies at the root of liquidity demand, a point not explicitly noted in the previous literature.”

Illiquidity is a property of the production technology — interrupting it forfeits the return — and it survives the absence of any transaction cost. The technology gives R > 1 at T = 2 per unit invested at T = 0, with salvage value equal to the initial investment if production is interrupted at T = 1, and the choice is made in period 1 with constant returns so fractional continuation is possible. One reading is offered — “long-term capital investments are somewhat irreversible, which appears to be a reasonable characterization” — and an alternative is granted: “The results would be reinforced (or can be alternatively motivated) by any type of transaction cost associated with selling a bank’s assets before maturity.” The point the authors insist on is that the friction is not a market friction: “this illiquidity is a property of the financial assets in the economy in our model, even though they are traded in competitive markets with no transaction costs.” And illiquidity does double duty: “Illiquidity of assets provides the rationale both for the existence of banks and for their vulnerability to runs.”

State-dependent utility with private states, a discount factor ρ satisfying 1 > ρ > R⁻¹ so that ρR > 1, and relative risk aversion greater than one everywhere. Type 1 agents get u(c₁); type 2 agents get ρu(c₁ + c₂), where the sum reflects that a type 2 who receives goods at T = 1 can store them costlessly and privately until T = 2 — storage being unobservable, and never worth using between T = 0 and T = 1 “because the productive technology does at least as well.” u is twice continuously differentiable, increasing, strictly concave, and satisfies Inada conditions. The two quantitative assumptions are load-bearing: ρR > 1 and relative risk aversion above one together imply, via the optimality conditions, that optimal early consumption exceeds 1 and optimal late consumption falls below R, which is precisely “room for improvement on the competitive outcome,” and that late consumption still exceeds early consumption. A fraction t of the continuum is type 1, and “conditional on t, each agent has an equal and independent chance of being of type 1” — t is taken as a constant initially and made random later.

Because the payoff would have to be conditioned on information only the agent observes, and there is no mechanism to elicit it. In the competitive solution “agents can write only uncontingent contracts, since there is no public information on which to condition,” prices are pinned down by constant returns, “there is never any trade, and agents can do no better or worse than if they produced only for their own consumption.” With publicly observable types, optimal insurance contracts would be writeable; without, “the lack of observability of agents’ types rules out a complete market of Arrow-Debreu state-contingent claims, because this market would require claims that depend on the nonverifiable private information.” The authors state the underlying structural reason carefully: “Under optimal risk-sharing, this private risk implies that agents have different time patterns of return in different private information states and that agents want to allocate wealth unequally across private information states. Because only the agent ever observes the private information state, it is impossible to write insurance contracts in which the payoff depends directly on private information without an explicit mechanism for information flow.”

Because the optimum satisfies the self-selection constraints, which guarantees some contract implements it as a Nash equilibrium — but Nash implementation is not unique implementation. The self-selection argument is given in a footnote and is worth carrying in full because it is where the paper’s whole logical structure sits: since “c₁¹* > 1 and c₂¹* = 0, type 1 agents do not envy type 2 agents. Furthermore, because c₁²* + c₂²* = c₂²* > c₁¹* = c₁¹* + c₂¹*, type 2 agents do not envy type 1 agents. Because the optimal contract satisfies the self-selection constraints, there is necessarily a contract structure which implements it as a Nash equilibrium — the ordinary demand deposit is a contract which will work. However, the optimal allocation is not the unique Nash equilibrium under the ordinary demand deposit contract. Another inferior equilibrium is what we identify as a bank run. Our model gives a real-world example of a situation in which the distinction between implementation as a Nash equilibrium and implementation as a unique Nash equilibrium is crucial.” Setting r_1 to the full-information optimal early consumption, “it is an equilibrium for type 1 agents to withdraw at T = 1 and for type 2 agents to wait, provided this is what is anticipated.”

That a payout can depend only on a depositor’s place in the queue, not on anything learned later — imposed to capture in three periods what really happens in continuous time. “Withdrawal tenders are served sequentially in random order until the bank runs out of assets. This approach allows us to capture the flavor of continuous time (in which depositors deposit and withdraw at different random times) in a discrete model.” Formally, the constraint “specifies that a bank’s payoff to any agent can depend only on the agent’s place in line and not on future information about agents later in line.” The bank is assumed mutually owned and liquidated in period 2, so non-withdrawers get a pro rata share. The constraint is later described as standing in for unmodelled banking services — “the realistic sequential service constraint represents some services that a bank provides but which we do not explicitly model” — and it is the friction that Proposition 1 turns on and that government taxation escapes.

Every promise of more than the liquidation value creates a run equilibrium, and a promise equal to the liquidation value provides no insurance at all. “For all r₁ > 1, runs are an equilibrium. If r₁ = 1, a bank would not be susceptible to runs… but if r₁ = 1, the bank simply mimics direct holding of the assets, and the bank is therefore no improvement on simple competitive-claims markets. A demand deposit contract which is not subject to runs provides no liquidity services.” A footnote generalizes the threshold away from the specific numbers: “The value r₁ = 1 is the value which rules out runs and mimics the competitive market because that is the per unit T = 1 liquidating value of the technology. If that liquidating value were θ < 1, then r₁ = θ would have this property. The connection between runs and liquidity service has nothing directly to do with the zero rate of interest on deposits.” The cause of the run is stated in one line: “This is because the face value of deposits is larger than the liquidation value of the bank’s assets.”

Worse for everyone than never having had a bank, with losses on both the risk-sharing and the production margins. “The bank run equilibrium provides allocations that are worse for all agents than they would have obtained without the bank (trading in the competitive-claims market). In the bank run equilibrium, everyone receives a risky return that has a mean of one. Holding assets directly provides a riskless return that is at least one (and equal to R > 1 if an agent becomes a type 2). Bank runs ruin the risk-sharing between agents and take a toll on the efficiency of production because all production is interrupted at T = 1, when it is optimal for some to continue until T = 2.” The forced-liquidation logic is spelled out: “The bank must liquidate all its assets, even if not all depositors withdraw, because liquidated assets are sold at a loss.”

Because the good equilibrium dominates direct holding, so a small enough probability of a run still leaves deposit attractive — and the run probability is driven by an extrinsic signal. The apparent contradiction is raised by the authors themselves: “If we take the position that outcomes must match anticipations, the inferiority of bank runs seems to rule out observed runs, since no one would deposit anticipating a run. However, agents will choose to deposit at least some of their wealth in the bank even if they anticipate a positive probability of a run, provided that the probability is small enough, because the good equilibrium dominates holding assets directly. This could happen if the selection between the bank run equilibrium and the good equilibrium depended on some commonly observed random variable in the economy.” Hence the list of possible coordinating signals, and hence the conclusion: “banks with pure demand deposit contracts will be very concerned about maintaining confidence because they realize that the good equilibrium is very fragile.” The existence explanation is then complete: “The pure demand deposit contract is feasible, and we have seen that it can attract deposits even if the perceived probability of a run is positive. This explains why the contract has actually been used by banks in spite of the danger of runs.”

From Friedman and Schwartz on the channel, and from Fisher and Bryant on the cause. On the channel: the model’s costliness result “is consistent with the Friedman and Schwartz (1963) observation of large costs imposed on the U.S. economy by the bank runs in the 1930s, although Friedman and Schwartz assert that the real damage from bank runs occurred through the money supply.” On the cause, the alternative view is that “a run occurs because the bank’s assets, which are liquid but risky, no longer cover the nominally fixed liability (demand deposits), so depositors withdraw quickly to cut their losses. The real losses are indirect, through the loss of collateral caused by falling prices. In contrast, a bank run in our model is caused by a shift in expectations, which could depend on almost anything.” The empirical claim the authors enlist in their own support is Bernanke’s: it “shows that the number of bank runs is a better predictor of economic distress than the money supply.”

It removes type 2 agents’ incentive to withdraw early no matter what they expect others to do, delivering a dominant-strategy equilibrium. The contract is the demand deposit contract plus a clause that anyone attempting to withdraw at T = 1 after a fraction f-bar has already withdrawn receives nothing. Setting r_1 to the optimal early consumption and choosing f-bar in the interval between t and (R − r_1)/[r_1(R − 1)], “no type 2 agent will withdraw at T = 1 because no matter what the agent anticipates about others’ withdrawals, the agent receives higher proceeds by waiting until T = 2.” Type 1s withdraw because late consumption is worthless to them, so the equilibrium is unique at f = t, and it is stronger than Nash: “In fact, this is a dominant strategy equilibrium, because each agent will choose the equilibrium action even if it is anticipated that other agents will choose nonequilibrium or even irrational actions. This makes this contract very stable.” The reason it works: “A policy of suspension of convertibility at f-bar guarantees that it will never be profitable to participate in a bank run because the liquidation of the bank’s assets is terminated while type 2s still have an incentive not to withdraw.” The paper connects this to the historical record: its results “are consistent with the claim by Friedman and Schwartz (1963) that in the 1930s, the newly organized Federal Reserve Board may have made runs worse by preventing banks from suspending convertibility: the total week-long banking ‘holiday’ that followed was more severe than any of the previous suspensions.”

Because suspension then actually binds in equilibrium, cutting off genuinely early consumers — and more fundamentally because Proposition 1 rules out any sequential-service contract reaching the optimum. Proposition 1 states that “bank contracts (which must obey the sequential service constraint) cannot achieve optimal risk-sharing when t is stochastic and has a nondegenerate distribution.” The proof is a two-part contradiction. If the T = 1 payout varies with place in line, then “there is a positive probability of different consumption levels by two type 1 agents who will withdraw at T = 1, and this contradicts an unconstrained optimum.” If instead it is constant in place in line for every realization, then early consumption is independent of t while the resource constraint forces late consumption to vary with t, which “contradict[s] optimal risk-sharing.” Hence “optimal risk-sharing is inconsistent with sequential service.” Suspension nevertheless remains useful: it “can generally improve on the uninsured demand deposit contract by preventing runs,” and although binding suspension means “some type 1 agents cannot withdraw, which is inefficient ex post… This can be desirable ex ante, however, because the threat of suspension prevents runs and allows a relatively high value of r₁.” The historical verdict the authors align with is Friedman and Schwartz’s: suspensions were “regarded as anything but a satisfactory solution by those who experienced them, which is why they produced such strong pressure for monetary and banking reform.”

It breaks the link between how much the bank must pay out and how much it must liquidate, by taxing on the basis of realized total withdrawals rather than place in line. Proposition 2: “Demand deposit contracts with government deposit insurance achieve the unconstrained optimum as a unique Nash equilibrium (in fact, a dominant strategies equilibrium) if the government imposes an optimal tax to finance the deposit insurance.” The asymmetry that makes this possible is stated exactly: the government “can base its tax on f, the realized total value of T = 1 withdrawals. This is in marked contrast to a bank, which must provide sequential service and cannot reduce the amount of a withdrawal after it has been made. This asymmetry allows a potential benefit from government intervention.” The construction is a proportionate tax on all wealth held at the beginning of T = 1, payable in goods or deposits, with any excess collection “plowed back into the bank (to minimize the fraction of assets liquidated)”; the resulting after-tax payoffs make waiting weakly better for every type 2 and withdrawing strictly better for every type 1, so the unique dominant-strategy equilibrium reproduces the full-information optimum. The general statement of why: “This insurance allows the bank to follow a desirable asset liquidation policy, which can be separated from the cash-flow constraint imposed directly by withdrawals. Furthermore, deposit insurance prevents runs because, for all possible anticipated withdrawal policies of other agents, participating in a bank run never pays. As a result, no strategic issues of confidence arise.”

Nothing in equilibrium when withdrawals are non-stochastic, because the guarantee is never called; potentially something real when they are, and possibly enough to make insurance welfare-reducing. The authors flag the strength of their own result before developing it: “A very strong result (which may be too strong) about the optimality of deposit insurance will illuminate the more general reasons it is desirable,” and afterwards: “The proposition may be too strong, since it allows the government to follow an unconstrained tax policy. If a nonoptimal tax must be imposed, then when t is stochastic, there will be some tax distortions and resource costs associated with government deposit insurance. If a sufficiently perverse tax provided the revenues for insurance, social welfare could be higher without the insurance.” In the non-stochastic case, “as long as the government can impose some tax to finance the insurance, no matter how distortionary, there will be no runs and the distorting tax need never be imposed,” a property shared with the adoption-externalities model of Dybvig and Spatt: “In both models, the credible promise to provide the insurance means that the promise will not need to be fulfilled.” The framing of what policy is for follows from this: “The role of government policy in our model focuses on providing an institution to prevent a bad equilibrium rather than a policy to move an existing equilibrium. Generally, such a policy need not cause distortion.”

Because the guarantee’s credibility rests on taxation, and private insurers must instead hold reserves. “If this is a guarantee of a real value, the amount that can be guaranteed is constrained: the government must impose real taxes to honor a deposit guarantee. If the deposit guarantee is nominal, the tax is the (inflation) tax on nominal assets caused by money creation.” Since “a private insurance company is constrained by its reserves in the scale of unconditional guarantees which it can offer, we argue that deposit insurance probably ought to be governmental for this reason.” The argument is functional rather than institutional — “the deposit guarantee could be made by a private organization with some authority to tax or create money… although we would usually think of such an organization as being a branch of government” — and it leaves room for “a small competitive fringe of commercially insured deposits, limited by the amount of private collateral.” The paper also positions itself against the prior deposit insurance literature, which “assumes away any real service from deposit insurance, concentrating instead on the question of pricing the insurance, taking as given the likelihood of failure.”

Only because the technology is riskless; once assets are risky, a lender of last resort is strictly less credible and introduces perverse incentives. In the model, “the Fed would buy bank assets with (money creation) tax revenues at T = 1 for prices greater than the assets’ liquidating value. If the taxes and transfers were set to be identical to what is implicit in the optimal deposit insurance, the effect would be the same. The identity of deposit insurance and discount window services occurs because the technology is riskless.” With risky technology the equivalence breaks two ways. If bailouts are unconditional, “there would be perverse incentives for banks to take on risk, even if bailouts occurred only when many banks fail together. For instance, if a bailout is anticipated, all banks have an incentive to take on interest rate risk by mismatching maturities of assets and liabilities, because banks will all be bailed out together.” If bailouts are conditional, discretion reintroduces the run: “a bank run can occur in response to changes in depositor expectations about the bank’s creditworthiness. A run can even occur in response to expectations about the general willingness of the lender of last resort to rescue failing banks, as illustrated by the unfortunate experience of the 1930s when the Federal Reserve misused its discretion and did not allow much discounting.” The contrast drawn is institutional: “deposit insurance is a binding commitment which can be structured to retain punishment of the bank’s owners, board of directors, and officers in the case of a failure.”

Currency, risky assets, multiple banks and moral hazard — and the authors say what each omission costs. On currency and risk: “It is interesting that the problems of runs and the differing effects of suspension of convertibility and deposit insurance manifest themselves in a model which does not introduce currency or risky technology. This demonstrates that many of the important problems in banking are not necessarily related to those factors, although a general model will require their introduction.” On the single bank: it “represents the financial intermediary industry and… withdrawals represent net withdrawals from the system. If many banks were introduced into the model, then there would be a role for liquidity risk-sharing among banks, and phenomena such as the federal funds market or the impact of bank-specific risk on deposit insurance could be analyzed.” On moral hazard, the omission is the most consequential: if portfolio risk were chosen by a manager and imperfectly observed, “a moral hazard problem would exist. In this case there is a trade-off between optimal risk-sharing and proper incentives for portfolio choice, and introducing deposit insurance can influence the portfolio choice.” The authors note that the existing moral hazard literature sits in complete-market settings “where deposit insurance is redundant and can provide no social improvement… But of course in this case there is no trade-off.” Their conjecture about the extension is hedged: “It appears likely that some form of government deposit insurance could again be desirable but that it would be accompanied by some sort of bank regulation. Such bank regulation would serve a function similar to restrictive covenants in bond indentures. Interesting but hard to model are questions of regulator discretion which then arise.”

Yes — to any firm whose liabilities are more liquid than its assets — and the paper explains why it nonetheless focuses on intermediaries. “The potential for multiple equilibria when a firm’s liabilities are more liquid than its assets applies more generally, not simply to banks. Consider a firm with illiquid technology which issues very short-term bonds as a large part of its capital structure. Suppose one lender expects all other lenders to refuse to roll over their loans to the firm. Then it may be the lender’s best response to refuse to roll over its loans even if the firm would be solvent if all loans were rolled over. Such liquidity crises are similar to bank runs.” Bankruptcy law then plays suspension’s role: “The protection from creditors provided by the bankruptcy laws serves a function similar to the suspension of convertibility. The firm which is viable but illiquid is guaranteed survival.” The reason to keep banks central is institutional fact rather than theory: “Our focus on intermediaries is supported by the fact that banks directly hold a substantial fraction of the short-term debt of corporations. Also, there is frequently a requirement (or custom) that a firm issuing short-term commercial paper obtain a bank line of credit sufficient to pay off the issue if it cannot be rolled over.” The inference is stated with its own condition attached: “This suggests that most of the aggregate liquidity risk in the U.S. economy is channeled through its insured financial intermediaries, to the extent that lines of credit represent binding commitments.”

Deregulation and the savings and loan crisis of the early 1980s, with runs described as a live issue despite fifty years of quiet. “Institutions in place since the Great Depression have successfully prevented bank runs in the United States since the 1930s. Nonetheless, current deregulation and the dire financial condition of savings and loan associations make bank runs and institutions to prevent them a current policy issue, as shown by recent aborted runs.” The footnote names them: “the aborted runs on Hartford Federal Savings and Loan (Hartford, Conn., February 1982) and on Abilene National Bank (Abilene, Texas, July 1982) are two recent examples. The large amounts of uninsured deposits in the recently failed Penn Square Bank (Oklahoma City, July 1982) and that failure’s repercussions are another symptom of banks’ current problems.” The paper also notes uninsured exposure outside the safety net: “Internationally, Eurodollar deposits tend to be uninsured and are therefore subject to runs, and this is true in the United States as well for deposits above the insured amount.” The normative conclusion is put plainly: “It is good that deregulation will leave banking more competitive, but policymakers must ensure that banks will not be left vulnerable to runs.”