Diamond-Dybvig Model: Unrealistic Assumptions and No Historical Support

Douglas Diamond and Philip Dybvig (DD) recently received their nobel prize in economy, along with Ben Bernanke. This has to do with their influential paper “Bank Runs, Deposit Insurance, and Liquidity” (1983) in which they demonstrate that free banks are inherently subject to bank runs and panics that are, according to them, stochastic (i.e., random). The only remedy is for the government to provide tax-based deposit insurance.

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Description of DD model

Their model considers 3 time periods with 2 types of agents: T=0 when everyone deposits their goods inside the bank, T=1 when Type 1 (impatient) agents are willing to consume immediately and withdraw, T=2 when Type 2 (patient) agents consume by withdrawing. Everyone has access to a production technology which converts each unit of input invested in T=0 to one unit of output in T=1 but greater than one in T=2. Each agent knows his type at T=1, this is meant to capture the idea that agents have unpredictable consumption shocks.

The problem arises when, due to the combination of information asymmetry and self-fulfilling processes, both Type 1 and 2 agents decide to run a bank during period 1, ultimately resulting in a panic. Because the model assumes the sequential (first-come, first-served) service constraint, which is meant to capture the flavor of continuous time, the depositors who recover their initial deposits plus interest in period 1 leave the rest with less than their initial deposits. Therefore, if some type 2 agents believe the other type 2 agents will run a bank, everyone would be better off running as well. Banks become insolvent as runs spread and the production process is unnecessarily interrupted by type 2 agents who would yield greater returns had they waited until period 2.

While DD considers suspension of payment, they believe it interrupts type 1 agents consumption and therefore is not pareto-optimal. They opt for a tax-based insurance that they consider costless to implement.

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Avalanche of Criticism

Dowd (1993, 1996) first describes several assumptions of the DD intermediary (i.e., bank) which don’t make sense for the real-world intermediation. The DD intermediary issues only one class of liability, a hybrid debt-equity that looks like debt to type 1 agents and equity to type 2 agents, whereas real-world intermediaries issue both claims. The DD intermediary must also be subject to bank run due to having no capital because it ignores bank shareholders who would typically pledge their own capital as reassurance.

Although DD themselves acknowledged that runs can trigger due to bad earnings reports, a failure of another bank, government report, their model cannot explain why these events could exist. So, DD ends up relying on extraneous uncertainty (i.e., sunspots) to generate expectations but yet again they do not explain why rational agents form their expectations in this way.

Now, Dowd reveals what is probably the most illogical aspect of the model. In order to provide this deposit guarantee to type 2 agents, the government must get the resources from type 1 agents who have already withdrawn in period 1. But this implies it must be able to overcome the sequential service constraint, which in turn implies it can overcome their isolation. Banks cannot get back to agents after they have withdrawn due to this isolation. If agents are isolated, then the government must possess some exclusive “information-producing” technology that the market does not possess in order to get back to these agents. Because if the market possesses this technology that produces the “information”, banks would have no reason to exist in the real world since, without isolation, agents would directly invest in the production process instead. Indeed there would be no market frictions for the intermediary to overcome.

While Diamond and Dybvig argued that private insurers must hold reserves to deliver insurance, a suboptimal outcome, the government doesn't need to because it has the power to tax. Dowd believes that argument doesn’t make sense because a tax comes from the private sector, and if the private sector doesn’t have the resources to provide the insurance, there is nothing to tax. Moreover, it is not reserves that private insurers need to offer credible insurance but adequate wealth (such as collateral).

Dowd now deals a final blow to the DD solution of public insurance. To understand its self-destructive nature, let me quote:

The state intervention might also create moral hazard problems that can only be dealt with by further intervention. Since the expected value of the bailout to any individual banker rises with the probability of default, and the probability of default rises with the promised return to withdrawals in T = 1, the latter would rise as each individual banker attempted to maximise the value of his bailout‑subsidy, so the bailout facility might encourage competitive intermediaries to bid more aggressively for deposits by raising deposit rates paid to those who withdraw in T = 1. T = 1 deposit rates would then rise indefinitely, and a point would come where the anticipated returns to T = 2 withdrawals had fallen so low that there was no point keeping deposits beyond T = 1.

Then, Dowd reviews several models attempting to provide an equilibrium in which there is no run, unlike DD. While most of these models do have some shortcomings similar to DD, notably why financial intermediation had any reason to exist in their model, one overlooked paper is worth mentioning. Williamson’s (1988) model assumes risk-neutral agents with random preferences to generate a demand for assets that are liquid in the short term. Risk-neutrality implies that there is no risk‑sharing benefit from financial intermediation at first glance. But because agents cannot distinguish claims to good assets and claims to bad ones when investing directly in the primitive assets (either short‑term or long‑term assets), they might then find themselves unable to meet unexpected liquidity needs without having to liquidate what would be good assets. Dowd explains that “A financial intermediary could exploit the fact that individual liquidity shocks tended to cancel out in large numbers and issue claims that enabled individuals to invest in the underlying production process while simultaneously holding assets that could be liquidated at little loss in the short run.” and then recognizes that “In Williamson’s model, banks generate superior outcomes to unintermediated markets in some states of the world, and the same outcome in others, so it always makes sense ex ante for agents to invest in a bank rather than invest directly themselves in the primitive assets.”

Another model worth considering is that of Gorton (1985a). It assumes that the depositors are imperfectly informed, causing them to either overestimate or underestimate the bank’s assets. None of which is optimal. To prevent runs, Gorton proposed a suspension clause which will operate on condition that the bank submits to a verification about its true state, cost of which would be borne by shareholders. Dowd commented however that these shareholders could simply augment bank capital instead, a less expensive solution.

Dowd (2000) later proposed a free market alternative to the governmental deposit insurance. By considering Type 3 agents (e.g., investors) who want to consume at period 2, provided sufficient capital investment (e.g., purchasing shares in a bank), to charge an insurance premium that is high enough to induce those investors to sell insurance but low enough to make it worth for other agents to buy that insurance. And, as capital rises, Type 3 agents become less risk averse. The overall situation satisfies the condition that all agents are better off with having financial intermediaries (banks) than under autarky.

Similarly, Selgin & White (1996) pointed out that capital adequacy is achieved by having bank shareholders accept unlimited liability, as this was the case in Sweden and Scotland. Or double liability in Canada. They also argued that competition in insurance would generate risk-based premiums and avoid, by doing so, the moral hazard usually associated with the flat premium structure of the public insurance (e.g., FDIC) who lacks profit motive. And one should not even be obsessed with insurance, especially the regulators. Selgin argued recently that international crises were primarily currency crises involving speculations on pegged foreign exchange rates, which means deposit insurance cannot protect against such withdrawals.

Selgin (1993) identifies one weakness of the DD model. Its inability to distinguish disturbances to consumption due to media of exchange shortages from disturbances to consumption due to consumption goods shortages. This is curious because the DD panic assumed shortages in consumption goods (due to the DD bank being the only source of consumable commodities) whereas banking panics historically have typically involved media of exchange shortages but not consumption goods shortages. A remedy to this problem is to simply add in the model gold and a substitute medium of exchange (e.g., bank deposits convertible into paper bank notes). This means the bank now distributes gold and deposits as means to purchase consumables from merchants. The modified model now is more coherent with the real world. Selgin then distinguishes one glaring element of the DD model: in that suspension is necessarily harmful to depositors because it prevents consumption of type 1 agents. He finally considers three suspension options:

1. Full suspension or bank holiday. Deposits cease to be convertible in notes or gold. Bank operations are shut down.

2. Partial external suspension. Deposits are convertible into bank notes but not in gold. Depositors can still purchase and consume. Banks are open for other activities.

3. Internal suspension. Temporary suspension of interbank clearings, assuming interbank consent. Banks are now supplied with additional gold to satisfy type 1 agents.

Notice that only bank holiday (which in practice can only happen if enforced by decree) would cause a panic due to the consequence of shutting down money payments. Unlike the other two, this type of suspension is closer to the DD suspension. Selgin finally recounts the episode of the U.S. prior to the Civil War when banks were allowed to suspend in response to emergencies. Not only did they engage in agreements to continue accepting each other’s notes at par value during these hard times but depositors could receive payments of checks in notes but not in specie, with merchants formally agreeing to accept notes at par. Of course, such success was hampered by the US laws imposing unit banking and banning branch banking. Yet in areas where cooperation was possible welfare losses were avoided. Historically the costs of suspension have been greatly increased by regulations which reduce the substitutability of bank money for outside money. Suspension therefore is better appreciated assuming unregulated banking. Also worth noting, Rockoff (1986), following the observation of Friedman and Schwartz’s (1963, pp. 163-168) on the 1907 panic, argued that suspension prevents the failure of sound banks which were temporarily illiquid, therefore avoiding contagion effects.

Earlier, Selgin (1989, 1996) explained that the contagion effect underlying the DD model is not supported by strong evidence when looking at the US panics and crises of 1873, 1884, 1890, 1893, 1907 and 1930-33. And bank runs were actually triggered by news indicating probable insolvency (see also Gorton 1988). Only during 1932 a panic did happen but only because the government officials declared a bank holiday despite protests from the bankers that it was harmful to their reputation. A less known feature of unregulated banking is the existence of a secondary bank note market portrayed as involving professional brokers and bank note reporters using weekly publications with information on note discounts. If brokers do not request any risk-related discount, it gives the signal to note holders that the bank is safe (Selgin, 1988, ch. 2 and 9). Another source of information is provided by a clearinghouse through which banks share information about member banks’ solvency. Clearinghouses may even conduct audits to ensure others are worthy clearing partners. Notes still being traded at par by rival banks again suggest to note holders that their bank is solvent. As to the utility of the lender of last resort, not only free banking episodes were free of runs and panics without it, but private banks would provide even better last-resort assistance than central banks because doing so is consistent with profit-maximization. While private banks would lend only to solvent institutions at penalty rates, central banks lend at subsidy rates and often to insolvent banks. This means, instead of signaling the public that an institution is safe as would be the case with the private alternative, the aid provided by central banks does not tell the public which institution is indeed safe. This explains why sometimes the Fed was unable to prevent runs.

Wallace (1988) faulted DD for not applying the sequential service to the government insurance because otherwise the model falls apart. Wallace illustrates this problem by using a camping trip analogy with the options of late-night snacks (period 1) versus breakfasts (period 2). During the night, isolated people may wake up but at different times, and some of them will find out they are hungry (i.e., they discover that they are type 1 agents). If so, they can decide to consume immediately (snack) or later (breakfast). If they consume, then, the optimal consumption for each agent would be different since they wake up and realize they were hungry at different points in time. This result contradicts the DD model in which the optimal consumption of type 1 agents must be equal among them. But Wallace model is obscured by having imposed the sequential constraint since, as noted by McCulloch and Yu (1998), a sequential constraint can be imposed on the economy without assuming that individuals are isolated.

For McCulloch and Yu (1998) the tax-based insurance is not efficient as it implies that the tax must be collected before the deposits are withdrawn or otherwise it would cause an inefficient interruption of the production process. Furthermore, the assumption of DD for the resources being all deposited in the banks during period 0 implies there is no nondeposit wealth to tax and that such a tax merely redistributes between period 1 withdrawers and non-withdrawers, which is something a private bank itself can do. Yet without that assumption, the DD insurance scheme falls apart. The solution as for the private insurance is a base payment provided to type 1 agents who withdraw with the remainder coming later but still in period 1, as a contingent bonus to be determined after the withdrawal volume is known. It should be noted here that this outcome is made possible by the authors applying the DD system in which the banks are subject to sequential constraint but not the government insurance. Likewise here, the private insurance isn’t constrained.

White (1999, pp. 128-129) points out that the DD bank is run-prone despite sharing some features of money-market mutual funds which are run-proof because they only issue equity shares. Yet mutual funds pursue two strategies to insure solvency: 1) maintaining adequate capital in the form of a high equity ratio, 2) holding a diversified portfolio containing assets of high grade and short maturity. Because the DD bank has no equity, it has no cushion. Büttler (1999) actually simulated such a condition of a high equity-to-debt ratio and found that the bank stays both solvent and liquid, as opposed to a low ratio.

White (1995, 1999) goes on to say that a contractual option to delay redemption for a stipulated period would prevent runs and panics. Scottish banks during their free banking era issued such “option-clause” bank notes and they circulated at par. In principle, option clauses will be invoked only if it reduces the risk of the bank becoming insolvent. This outcome is beneficial to both parties. On one hand banks can delay redemption for long enough to liquidate assets in an orderly manner, on the other hand customers will be compensated by earning higher interests for the delay. Bédard (2016) however showed that option clause is beneficial only if we assume that the DD model, in which runs and panics are triggered by self-fulfilling prophecies, is correct.

According to White (1999, ch. 4), clearinghouse associations can “offer loans at an interest rate below the market rate prevailing in a panic, thus providing the burden-sharing of insurance, but above the normal market rate, thus reducing the potential moral hazard of a bank responding to the insurance by taking insufficient care to keep adequate reserves.” (p. 76). Successful experiment of such policy against panics was apparent in the US between 1857 and 1907. Gorton (1985b) and Gorton & Mullineaux (1987) made the same point. They provide a detailed account of the New York clearinghouse’s tasks and functions during panics prior to the creation of the Fed. Any member bank had every reason not to behave recklessly as the expulsion would destroy the bank’s reputation. When needed, the clearinghouse would hide the identity of the banks who borrowed the loan certificate. These certificates were also accepted by depositors because they insured against individual bank failure. Therefore, the information asymmetry was solved.

The complaints voiced by Dowd (1996) and White (1999) about DD model not issuing both demand deposits and equity have been considered by Gangopadhyay and Singh (2000). They modify the DD model so as to incorporate debt and equity. They incorporate risk-neutral equityholders (or shareholders) and risk-averse agents who purchase deposits. This condition allows part of the depositors’ risk to be transferred to shareholders who are residual earners. Their model reveals that when capital is large enough, depositors can be fully insured whereas when the amount of capital is less, runs are also prevented but depositors are only partially insured.

Goldstein & Pauzner (2005) studied the relation between the short-term payment offered by demand deposit contracts (that are observed in practice) and purely panic-based bank runs, a topic that is understudied. Their simulation shows that the probability of run increases when banks offer more risk sharing. However, one key element was the inclusion of fundamentals, which serve as a coordination device for the expectations of agents. Their model finally reveals a unique equilibrium in which, conditioning on the range of fundamental news not being too large, banks are viable and increase welfare.

Bédard & Gentier (2013) analyzed the differential impact of the 1837 US crisis on New York State (NY) and Massachusetts (MA). The interesting fact being that NY did have a public deposit insurance (which defaulted during the panic of 1837) and experienced greater depositor losses compared to MA. They also observed that NY financed its credit expansion through deposits whereas MA financed it through savings. They also argue that the marginal credit expansion of MA is due to competition preventing inflation because adverse clearings cause banks to become illiquid if they overexpand (see Selgin 1988). They rightly remind us again that deposit insurance increases the banker’s risk-taking behaviour and reduces the incentive of the depositors/investors to select the bank that is best managed.

A less discussed topic is the cost of running a government insurance. Hogan and Luther (2014) examined the operating expenses of the FDIC over time. It is anything but costless. This is worth mentioning because the DD model assumed that a tax-based insurance was costless.

To make things worse, several reviews concluded that the literature on the impact of public insurance has produced mixed results so far (Anginer & Demirgüç-Kunt, 2018).

If one is still unconvinced, one should ask why historical experiences of free banking were so successful when regulations were light and non-binding. Weren’t contagion effects supposed to be stochastic (i.e., random) after all? Because if bank runs really should emerge as an equilibrium, i.e., it is a “known” feature of a free banking system, then no one would ever put his money in any bank knowing that a panic is likely coming.

References.

1. Anginer, D., & Demirgüç-Kunt, A. (2018). Bank runs and moral hazard: A review of deposit insurance. World Bank Policy Research Working Paper, (8589).

2. Bédard, M., & Gentier, A. (2013). Deposit Insurance and Coordination Failure: The Case of the 1837 Crisis in New York State and Massachusetts. Working paper CAE CERGAM no. Search in.

3. Bédard, M. (2016). In Which Context is the Option Clause Desirable?. Journal of business ethics, 139(2), 287-297.

4. Büttler, H. J. (1999). The optimal capital structure of a liquidity‐insuring bank. The Econometrics Journal, 2(2), 268-291.

5. Diamond, D. W., & Dybvig, P. H. (1983). Bank runs, deposit insurance, and liquidity. Journal of political economy, 91(3), 401-419.

6. Dowd, K. (1993). Laissez Faire Banking. Routledge.

7. Dowd, K. (1996). Recent Models of Banking Instability. In Competition and Finance (pp. 222-240). Palgrave Macmillan, London.

8. Dowd, K. (2000). Bank capital adequacy versus deposit insurance. Journal of Financial Services Research, 17(1), 7-15.

9. Gangopadhyay, S., & Singh, G. (2000). Avoiding bank runs in transition economies: The role of risk neutral capital. Journal of Banking & Finance, 24(4), 625-642.

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12. Gorton, G. (1985b). Clearinghouses and the origin of central banking in the United States. The Journal of Economic History, 45(2), 277-283.

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14. Gorton, G. (1988). Banking panics and business cycles. Oxford economic papers, 40(4), 751-781.

15. Hogan, T. L., & Luther, W. J. (2014). The explicit costs of government deposit insurance. Cato J., 34, 145.

16. McCulloch, J. H., & Yu, M. T. (1998). Government deposit insurance and the Diamond-Dybvig model. The Geneva Papers on Risk and Insurance Theory, 23(2), 139-149.

17. Rockoff, H. (1986). Institutional requirements for stable free banking. Cato J., 6, 617.

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