Modeling the Legend, or, the Trouble with Diamond and Dybvig: Part I

Bank Run

Bank RunHas any theoretical work on banking been more influential than Douglas Diamond and Phillip Dybvig's 1983 JPE article, "Bank Runs, Deposit Insurance, and Liquidity"? If so, I can't think of it. With well over 12,000 Google citations and counting, it's certainly among the most cited academic papers in economics, let alone in the sub-discipline of monetary economics.

Nor has that paper's influence been merely academic. Far from it: it is routinely cited by policymakers as supplying a rationale for government intervention in banking, and for explicit national deposit insurance schemes in particular. That the number of countries that adopted such schemes more than quadrupled during the two decades immediately following the article's appearance—from just 20 to 87—almost certainly owes something to Diamond and Dybvig's influential publication, which is bound to have informed the thinking of experts at the IMF and other international agencies who recommended deposit insurance as the best cure for banking crises.[1] Alas for Diamond and Dybvig and the citizens of the countries that followed it, the practical results of this expert advice can't be said to have been universally benign.

That the Diamond-Dybvig model should have become so popular in policy circles isn't hard to fathom. So far as many policymakers (and more than a few economists) are concerned, it shows, "rigorously," that ordinary (that is, fractional-reserve) banking systems are inherently unstable, and that deposit insurance or an alert and efficient lender of last resort or some other form of intervention, like narrow banking, is needed to stabilize them.  Ask an expert who advocates any of these solutions for proof that it's necessary, and chances are he or she will eventually get around to saying, in essence, "See Diamond and Dybvig (1983)." Q.E.D.

There's just one problem. It's that Diamond and Dybig (1983) demonstrate no such thing.

Although I plan to criticize rather than praise the Diamond-Dybvig model, I have no desire to bury it. I completely agree with  Ricardo Cavalcanti when he writes that it

was a significant conceptual and methodological advance in studying banking arrangements. Its methodological contribution was the use of mechanism-design theory rather than the old strategy, still prevalent in textbooks and some of macro, of tacking a banking sector onto a model of market exchange.

Indeed. Diamond and Dybvig's model is cited so often in part because it has been criticized and refined so often, with each criticism and refinement adding, if only a teeny bit, to our understanding of the true causes of systemic banking crises.[2] So my beef isn't with Diamond and Dybvig per se. It's with those who assume that their model supplies adequate grounds for government intervention in banking. As experts have long understood, it does nothing of the sort. Here I review some reasons why.

When the Legend Becomes Fact…

In "The Man Who Shot Liberty Valence," John Ford's 1962 Western, Jimmy Stewart plays distinguished U.S. Senator Ranse Stoddard, who has just returned to the frontier town he left 25 years earlier. Maxwell Scott, who edits the town newspaper, wants his story. Stoddard regales him with his many accomplishments since he left; but Scott, in the meantime, hears the rumor that young Stoddard killed the town villain in a shootout. The rumor turns out to be false (John Wayne did it). But that doesn't stop Scott from deciding that it makes for better copy than Stoddard's real accomplishments. "This is the West, sir," he tells the crestfallen Senator after tearing up his notes. "When the legend becomes fact, print the legend."

Banking, too, has its legends. And in the U.S., at least, those legends have more in common with John Ford's movies than one might expect. For they, too, are at least partly legends of the old West, which besides its gunslingers and regular shootouts had its fly-by-night "wildcat" banks and worthless shinplasters.

Or so we're told. But just as shootouts were actually rare, even in Dodge City, so (as numerous studies have now shown) was wildcat banking.  That's not to say that there were no bank runs or bank failures: there were plenty. But that wasn't because antebellum bank liabilities weren't insured, or because their issuers only held fractional reserves, or because, after 1836, there was no central bank, or (despite misleadingly named "free banking" statues) because banks weren't otherwise regulated. On the contrary: it was because most of the banks that failed were regulated, irresponsibly, especially by being prevented from branching and by being compelled to invest in doubtful securities as a condition for issuing currency.

Yet the myth that banks failed in droves before the Civil War because they weren't regulated enough persists. As I observed in a previous essay, this is partly because many naively assume that if government authorities call something "free banking" it must be so, and also because a story about banks running "wild"

makes for more titillating reading than ones about the mass of less colorful, if no less unfortunate, free-bank failures. Wildcat banking is to the history of banking what the O.K. Corral and Wild Bill Hickok are to the history of the far west.

If people still subscribe to the wildcat banking myth today, it wouldn't be at all surprising to learn that Diamond and Dybvig subscribed to it in 1983, when its exposure by economic historians was still not widely appreciated, even among economists. However, it's more likely that their assumption that banks are inherently failure-prone was informed by a different myth: that concerning the bank runs and failures of the early 1930s, and especially the systemic banking crisis that took place during late February and early March, 1933.[3]

As I've also addressed this "Great American Banking Myth" elsewhere on this site, I'll only observe here that, just as Hugh Rockoff, Arthur Rolnick, and Warren Weber began to demolish the myth of antebellum wildcat banking starting in the mid 1970s, others, including Barry Wigmore and Elmus Wicker, took a hard look at banking crises of the 1930s, and found that self-fulfilling fears of bank insolvency played a much more limited part in them than others had supposed. Wigmore, in particular, has argued quite persuasively that the 1933 run was "a run on the dollar," meaning that, instead of imagining that all the banks still standing were about to go belly-up, people worried that, once he became President, FDR would devalue the dollar. That made it perfectly reasonable for them to run on banks they considered perfectly sound, so they could acquire gold by staging a run on the Federal Reserve banks, and ultimately on the Federal Reserve Bank of New York, which held most of the system's gold. It was because the New York Fed was running short of gold that it ultimately urged the government to declare a nationwide bank holiday.

Alas, the theoretical literature on banking panics, including Diamond and Dybvig's seminal article, appears to owe more to tenacious myths about banking in the U.S. than to any well-informed survey of U.S. banking experience, let alone experience elsewhere. "It does not seem to be an exaggeration," Gary Gorton and Andrew Winton observe, in their chapter on "Financial Intermediation"  for Elsevier's Handbook on the Economics of Finance, "to say that most of the theoretical work on panics has been motivated by the U.S. experience, which has then been incorrectly generalized. Panics simply are not a feature of most economies that have banks." What Gorton and Andrew ought to have said is that the theoretical work in question has been exclusively motivated by misinterpretations of U.S. experience.

In short, when Diamond and Dybvig set out to write their famous paper, U.S. banking legends were quickly giving way to facts. But like Maxwell Scott, though unwittingly, Diamond and Dybvig printed the legend. That is, they modeled the legend, and the JPE printed it.

The Baseline Diamond-Dybvig Model

Not that modeling the legend was easy. On the contrary: coming up with a reasonably tractable model economy in which a "bank" performed some essential function, yet was almost certain to eventually fail, was anything but. (Not for nothing did their paper make it into the JPE!) Indeed, the real lesson the Diamond-Dybvig exercise teaches couldn't be further removed from the one many draw from it, particularly if they aren't familiar with the many follow-up studies it inspired.[3] The lesson is that, while it's possible to come up with a model bank that succumbs easily to panic, making one that also resembles real-world banks is well-nigh impossible.

To see why, we must first come to grips with the Diamond-Dybvig (or D-D, for short) model itself. Having done my best to explain that model elsewhere, I can't do much better than repeat that effort here. "The D-D economy," I wrote,

begins with N consumers endowed with equal quantities of the economy's single consumption good, e.g., … a bushel of corn. There are three periods, 0, 1, 2—a "planting" period, an "intermediate" period, and a "harvest" period. A bushel of corn planted in period 0 yields R > 1 bushels in period 2, but only one bushel in period 1. Because consumers may meet with emergencies in period 1, investment in corn production is risky. Unlucky "type 1" consumers will have to liquidate their corn investments prematurely, realizing a net return of zero. In contrast, lucky "type 2" consumers can afford to delay consumption until the harvest, enjoying a positive return. All consumers feel vulnerable as of period 0, however, because they do not learn their types until period 1.

According to Diamond and Dybvig, a bank is a device that allows optimal risk sharing by pooling investments and dividing anticipated returns among type 1 and type 2 consumers. Assuming that the fraction, t, of type 1 consumers is less than one, risk sharing takes the form of (nontransactable) deposit contracts entitling depositors to a preset payoff of r1 bushels of corn per bushel deposited in period 0 for period 1 withdrawals (Rr1 > 1) and to a residual payoff of r2 (< R) bushels of corn per bushel deposited in period 0 for period 2 withdrawals, where r2 represents a pro rata share of corn harvested in period 2.  A "good" banking equilibrium with optimal risk sharing occurs when deposit contracts are taken advantage of and type 2 consumers behave like type 2 consumers, delaying their withdrawals until the harvest.

Unfortunately, Diamond and Dybvig demonstrate that such a "good" equilibrium is only one of two possibilities. A "bad" equilibrium may also occur in which type 2 agents panic and join type 1s in withdrawing their deposits prematurely. Because the D-D model assumes a single bank, such panic-based withdrawals are the equivalent of systemwide panic in a multibank system. A panic spoils the risk-sharing arrangement because, with r1> 1, the value of the bank's assets in period 1 (= N bushels of com) falls short of the promised period 1 redemption value of bank deposits (= Nr1). Assuming… a "sequential [first-come, first-served] service constraint" is in effect, depositors who recover their initial deposits plus interest in period 1 leave the rest with less than their initial deposits. It follows that "anything that causes [type 2 consumers] to anticipate a run will lead to a run," including intrinsically irrelevant random events such as sunspots.

At first blush, the Diamond-Dybvig model seems to answer all the requirements of a rigorous model of banking panics. It starts with an economy in which uncertain consumption needs coexist with a production technology that calls for illiquid investment. In that environment, a "bank" can make everyone better off by allowing consumers to pool their investments to share the risk of having to consume early. But the insurance banks offer makes them vulnerable to panic-based runs. When panic strikes, the insurance arrangement breaks down, and instead of being better off, some depositors—those who take too long to cash-in their deposits—lose out, earning less than they would have by fending for themselves.

But there's a hitch. In the simple or "baseline" version just described, the D-D model suggests a straightforward, contractual remedy to the problem of bank runs: provided it knows the value of t—the fraction of impatient, type-1 depositors—instead of allowing unlimited deposit withdrawals, the bank can include clauses in its deposit agreements allowing it to "suspend" period one payments as soon as the share of withdrawn deposits reaches t . By assuring patient, type-2 depositors that they will always receive their promised return, the option of suspending payments eliminates their incentive to panic, ruling out the bad, bank run equilibrium. Because the option to suspend makes everyone better off, it constitutes a free-market solution to the problem of bank panics. So the baseline Diamond-Dybvig bank turns out not to be inherently unstable after all.

"Aggregate Uncertainty" and the Case for Intervention

But Diamond and Dybvig show that a minor and perfectly plausible tweak to their baseline model suffices to make the suspension solution less than optimal. If, instead of being known with certainty, the fraction of type-1 depositors is a random variable, the bank has to estimate t. If it underestimates it, the bank will suspend payments even when no type 2s are panicking, but before all of its type-1 depositors cash out. The left-out type 1s then suffer a welfare loss compared to the case of a "good" equilibrium without suspension. It follows that in a situation of "aggregate uncertainty" (that is, where the value of aggregate type-1 withdrawals is uncertain), deposit agreements that allow for suspension may not be attractive enough to make them a viable, free-market solution to the problem of panics.

The government, on the other hand, can do something to stop panics—or so Diamond and Dybvig claim. "Government deposit insurance," they write, "can improve on the best allocations that private markets provide" by guaranteeing "that the promised return will be paid to all who withdraw." By taking advantage of its power to tax by committing to tax bank customers who withdraw their deposits in period 1, at a rate based on the realized value of total withdrawals, it can reduce the post-tax return type 2s can look forward to if they panic. Insurance can thus achieve the same run-preventing structure of returns in a situation of aggregate uncertainty as suspension contracts are able to achieve only when the share of type 1 depositors is known.

This isn't to say that deposit insurance is the only possible solution. Diamond and Dybvig claim that appropriate central bank action can also rule-out runs. The Fed, for example, might "provide a service similar to deposit insurance" by using its last-resort lending facilities to "buy bank assets … for prices greater than their liquidating value. If the taxes and transfers were set to be identical to that of the optimal deposit insurance, it would have the same effect."

***

So much for Diamond and Dybvig's model and the claims they and others make with reference to it. In the second half of this essay, I'll explain why their model actually says very little about how real world banks work, and even less about how we can get them to work better.

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[1] A rising incidence of financial crises abroad undoubtedly played their part as well. However it's worth noting that the period's international crises were primarily currency crises, involving speculative attacks on pegged foreign exchange rates, rather than banking crises in the strict sense. While deposit insurance can rule out ordinary bank runs, it generally can't prevent runs on banks by persons seeking to acquire domestic currency for the sake of converting it into foreign exchange.

[2] Space and time prevent me from mentioning, let alone doing justice to, most of these Diamond-Dybvig inspired writings: while some of the strictures I make here against Diamond and Dybvig's original effort pertain to many of these other works, it's far from being the case that all of my criticisms apply to all of them. This is most obviously so for those articles that themselves underscore problems with Diamond and Dybvig's analysis, even as they retain and build upon many of its features.

[3] By way of motivating their article, Diamond and Dybvig themselves mention runs on Hartford Federal Savings and Loan (February 1982) and the Abilene National Bank of Texas (July 1982) along with the large losses suffered by uninsured depositors upon the failure of Oklahoma's Penn Square Bank (July 1982). That Penn Square failed because it made all sorts of bad loans is notorious. Abilene National Bank was one of Penn Square's major correspondents, which was run upon when word got out that it had suffered major loan losses. Hartford Federal, finally, endured a run after the Hartford Current reported, accurately, that it "lost a record $7.3 million in 1981 and is now taking drastic steps to stay afloat." Sheer panic played no obvious part in any of these runs, while deposit insurance failed to prevent the last of them. In short, while Diamond and Dybvig may well have been inspired by these episodes, their famous article can hardly be said to shed light on them.