Re-examining the case for government deposit insurance: reply.

Dowd, Kevin

In a recent article in this journal, Kevin Dowd [2] presents a modified version of Diamond and Dybvig's [1] banking model. Dowd claims that for some parameters of his model, a bank capital holder can earn a profit by guaranteeing the optimal Diamond-Dybvig deposit withdrawals. According to Dowd, the capitalist willingly puts up his own resources as a guarantee on deposits, because of the profits he will earn from doing so. These profits, according to Dowd, make government-provided deposit insurance unnecessary. This comment shows that no such profits exist. By definition, the optimal Diamond-Dybvig deposit withdrawals require that all of the returns from deposited resources be paid out to depositors. There is no surplus for the bank capital holder to claim as profits.

Dowd's mistake came from forgetting that those Diamond-Dybvig depositors who ask for an early withdrawal receive a payout that is greater than the return provided by the underlying technology. The Diamond-Dybvig bank thus shares with early withdrawers some of the high return from long-term investment in the technology. Depositors insure themselves against the risk of being an impatient consumer by joining the Diamond-Dybvig bank, with its promise of a higher-than-autarkic return to those withdrawing early, and a lower-than-autarkic return for those withdrawing later. Let [r.sub.1] and [r.sub.2] be the optimal Diamond-Dybvig deposit payouts to the early and late withdrawers, respectively.(1) The underlying technology provides an early return of 1 or a later return of R on a unit deposited in it. The depositors are of measure one, so that if the fraction t of depositors asks for the early withdrawal, then (1 - t[r.sub.1]) is left in the technology to grow at the rate R. Dowd claims that the resources left in the bank, after all deposit payouts have been made and assuming that each depositor asks for the withdrawal intended for his type, would be KR + [t + (1 - t)R] - [t[r.sub.1] + (1 - t)[r.sub.2]], where K is the capital holder's contribution of resources, the first expression in brackets is supposedly the value of deposits invested in the technology, and the second term in brackets is the value of the bank's liabilities. In his expression for the value of deposits invested in the technology, Dowd has (1 - t) resources left in the technology to grow at rate R, rather than the smaller amount (1 - t[r.sub.1]) that are truly left in. Apparently he is forgetting that the bank promises early withdrawers [r.sub.1], which is greater than the underlying technological return of 1. This mistake provides the source of the profits Dowd describes the capitalist enjoying in exchange for the role of guaranteeing the deposits with his capital resources. When depositors ask for the withdrawals intended for their type, the true value of the invested deposits minus the value of the liabilities is [t[r.sub.1] + (1 - t[r.sub.1])R] - [t[r.sub.1] + (1 - t)[r.sub.2]]. This difference is equal to zero by virtue of the fact that, as optimal payouts, [r.sub.1] and [r.sub.2] satisfy a resource constraint [2, 364] requiring that all of the returns on deposited resources be paid out to depositors. Thus the capitalist earns KR, which is what he would earn in autarky.

Without deposit guarantees, this model has bank run equilibria. Suppose every depositor attempted to take the bank up on its promise of an early withdrawal of [r.sub.1]. Then the bank, which must deal with withdrawal requests sequentially, would run out of assets before everyone received a payout. So a patient depositor believing other patient types will try to withdraw early would ask for the early withdrawal too, before the payouts of [r.sub.1] per person completely deplete the bank's assets. If we introduce a capitalist with sufficiently large resources, then he can guarantee the optimal Diamond-Dybvig deposit payouts, and be reasonably certain that a run will not force him to actually spend any of his capital on the guarantees. However, since he will not earn any profits in doing so, the capitalist's willingness to provide a private deposit guarantee is much less likely than Dowd has claimed.

Denise Hazlett Whitman College Walla Walla, Washington

I would like to thank Neil Wallace for his comments.

1. A depositor's type is not known at date 0, when investments in the technology are made. Type, which is private information, is realized at date 1, at which point the impatient chose to withdraw. Preferences for the patient are such that they would rather have resources left in the technology until date 2, so that they can take advantage of the returns from long-term investment. The optimal Diamond-Dybvig payouts give the ex ante (as of period 0) optimal sharing of resources between patient and impatient, assuming that depositors ask for the withdrawal intended for their type.

References

1. Diamond, Douglas and Philip Dybvig, "Bank Runs, Deposit Insurance and Liquidity." Journal of Political Economy, June 1983, 401-19.

2. Dowd, Kevin, "Re-examining the Case for Government Deposit Insurance." Southern Economic Journal, January 1993, 363-70.