Narrow Banking Meets the Diamond Dybvig Model

A version of the Diamond-Dybvig model of banking is used to evaluate the narrow banking proposal, the idea that banks should be required to back demand deposits entirely by safe short-term assets. It is shown that the mere existence of an amount of safe short-term assets outside the banking system that exceeds banking system liabilities does not make the proposal either innocuous or desirable. In fact, despite such existence, using narrow banking to cope with banking system illiquidity eliminates the role of the banking system. The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. The current version of the 100 percent–reserve banking proposal, called the narrow banking proposal, begins with an observation: The magnitude of safe short-term assets outside the banking system exceeds the magnitude of banks’ demand deposit liabilities. Therefore, say the proponents of narrow banking, why not avoid the problems of an illiquid banking system portfolio—such as the threat of bank runs and the accompanying need for deposit insurance, regulation, and bailouts—by forcing a rearrangement of asset holdings in the economy? Why not require that demand deposits be backed entirely by safe short-term assets? This is the narrow banking proposal. However, this proposal both begins and ends with the same observation. That is, there is no theory or model of banking from which the proposal emerges. In particular, no model is offered which is consistent with the pervasiveness of illiquid banking systems and which also implies that the narrow banking proposal is desirable. This is a serious omission, for two reasons: the supposed problem that the narrow banking proposal is intended to solve would not exist if the banking system were not illiquid, and an explanation in the form of a theory or model of illiquid banking systems is likely to suggest that benefits accompany such systems. Models do exist, built on Diamond and Dybvig 1983, that are consistent with illiquid banking. These models offer a plausible explanation of the role played by an illiquid banking system; the explanation suggests that some benefits accompany banking system illiquidity. Although the original version of the model seems ill-suited to address the narrow banking proposal because, in that version, the banking system holds all the assets in the economy, simple extensions of the model may be made that are consistent with assets being held outside the banking system. It is presumably such extensions that Diamond and Dybvig have in mind in their critical discussions of the narrow banking proposal. (See Diamond and Dybvig 1986 and Dybvig 1993.) Diamond and Dybvig say that the proposal makes sense only if the safe short-term assets outside the banking system in the actual economy represent excess liquidity, a fact which they doubt. Therefore, Diamond and Dybvig suggest, implementation of the narrow banking proposal would have undesirable consequences. My purpose here is simply to make Diamond and Dybvig’s argument explicit. I set out a version of the Diamond-Dybvig model and point out what the model implies about the narrow banking proposal. My version of the model supports their position: there can be large amounts of safe short-term assets outside the banking system, but narrow banking is undesirable. It is undesirable relative to something that bears some resemblance to our current banking system and undesirable relative to something resembling, if anything, a banking system with a large amount of liabilities subordinate to its demand deposit liabilities. Of course, for a variety of reasons, advocates of the narrow banking proposal may be skeptical of this version of the Diamond-Dybvig model and, therefore, skeptical of its implications for their proposal. As with any banking model, this one does not capture some features of actual banking systems. Such skepticism, though, is hardly a persuasive argument for the narrow banking proposal. Any proposal for a major change in policy should be supported by a coherent view of the phenomenon under consideration. In this case, the phenomenon is illiquid banking. Such a view should imply that the policy is desirable, and such a proposal should argue that its view of the phenomenon should be accepted. The advocates of narrow banking have not even begun this process of argumentation. Not only have advocates of narrow banking not made a case, but some of them seem unaware that a plausible model of illiquid banking systems does not lend support to their proposal. Preliminaries Before examining the model, I will review a few basic concepts. The term illiquid banking system refers to a property of a consolidated balance sheet. The consolidation is over banks and may also include the central bank, the government, and even those in debt to banks. The illiquidity property of this balance sheet means that not all banking system obligations can be met if all holders of those obligations simultaneously claim what they have been promised. More generally, if the obligations are deposits that give the owners of the deposits the right to decide when to withdraw, then the banking system is illiquid if there is some possible pattern of withdrawals that cannot be accommodated. The most typical example of such a system is a fractional reserve banking system under a commodity standard such as the gold standard. Such a banking system has demand liabilities that exceed its reserves (in the form of the commodity standard). Banking system illiquidity seems to open the door to potential difficulties, and history is rife with instances of what are variously called bank panics or bank runs, which are generally viewed as realizations of the potential difficulties (Friedman and Schwartz 1963). Are these realizations inevitable? Can they be avoided, and at what cost? The narrow banking proposal says, Let’s eliminate the potential difficulties by eliminating illiquid banks. That may be a good idea, or it may be silly—as silly as a proposal to reduce automobile accidents by limiting automobile speeds to zero. Here I will appraise the narrow banking proposal using a model. Why use a model? The alternatives are to look at history or to try an experiment using the narrow banking proposal. As far as history is concerned, even if narrow banking had been in effect in the past, without a model we would not even know what to look for to judge narrow banking’s success or failure. The same difficulty arises when we consider an experiment. Moreover, experimenting on the actual economy may be very costly, particularly if narrow banking is not, in fact, a good idea. Using a model amounts to experimenting on an analog of the actual economy. Such experimentation is much cheaper, mainly because it avoids the risks of experimenting on the actual economy. A Version of the Diamond-Dybvig Model We want a plausible model that explains illiquid banking and lets us judge how people will be affected by various rules imposed on the banking system. Such a model is set out here. The model has three main ingredients, each of which, as will be explained below, is plausible: • Individual uncertainty about desired time profiles of consumption, including the assumption (referred to as private information uncertainty) that realizations of this uncertainty are known to the person, but not to others. • Investment technologies that offer a trade-off between those with good short-term returns and those with good long-term returns. • Isolation of people from each other in a way that, among other things, forces the banking system to deal with depositors on a first-come, first-served basis. The first ingredient, individual uncertainty, gives rise to a role for assets that can be cashed in at the request of the holder—something like actual demand deposit and savings deposit accounts. This ingredient is plausible in that such uncertainty has long been used to explain why people do not plan the pattern of their expenditures so as to avoid holding low-return assets such as checking accounts and traveler’s checks. The second ingredient, investment technologies that offer a trade-off between short-term and long-term technologies, makes it easy to have both banking system illiquidity and safe short-term assets outside the banking system. This ingredient seems eminently plausible as a feature of technologies in actual economies. The third ingredient, isolation that forces first-come, first-served treatment of depositors, is also plausible—if only because something like it is necessary to account for the dominant role of the first-come, first-served principle in almost all retail trade. When the above ingredients are combined in a way that produces a model of a complete economy, I am able to describe what is feasible in that economy as well as which feasible things are desirable. Desirability will be judged by the extent to which people’s preferences are satisfied—preferences that take into account the uncertainty people face about desired time profiles of consumption. According to the model, a best feasible outcome has features that resemble these: • Demand deposits. • An illiquid banking system where the consolidation is best viewed as over banks and over those indebted to banks. • Safe assets outside the banking system. This is the sense in which the model explains an illiquid banking system that has safe assets outside the banking system. The model is essentially the same as the one in Diamond-Dybvig 1983. There are three dates indexed by t, where t = 0, 1, 2, and there is one good per date. The economy is endowed with only date 0 good, an amount that is normalized, per person, to unity. There are two linear constant returns-to-scale technologies. There is a shortterm technology with gross return R1; output at t + 1 per unit input into this technology at t is R1. There is a longterm technology with gross two-period return R2; output at t + 2 per unit input into this technology at t is R2. There is also a return for liquidation of investment in the longterm technology after one period; this gross return is denoted r1. I assume R2 > (R1) 2 > (r1) 2 > 0, so that, according to the technology, it is best to provide for date 2 consumption by investment in the long-term technology and to provide for date 1 consumption by investment in the short-term technology. At date 0 there are a large number of identical people, and each person is uncertain about what his or her preferences over consumption at dates 1 and 2 will be. Those preferences may be of an impatient type, labeled type 1, or of a patient type, labeled type 2. At the beginning of date 1, each person learns his or her type. This is private information; a person’s type is known to that person, but not to anyone else. The preferences of each type 1 (impatient) person after learning his or her type are given by the utility function u(x,y), while the preferences of each type 2 (patient) person are given by the utility function u(x,y), where in both functions x is date 1 consumption and y is date 2 consumption. The accompanying chart depicts the relationship between the utility functions of the two types using indifference curves. The chart indicates that, at any consumption pair, each impatient person is willing to sacrifice less of date 1 consumption, per unit of additional date 2 consumption, than is a patient person; that is, at any consumption pair, the impatient indifference curve through that pair is steeper than that of the patient indifference curve through that pair. We let p be both the fraction of people who will turn out to be impatient and the subjective probability at date 0 that each person will be impatient. Welfare is judged at date 0 by the magnitude of expected utility, pu(x,y) + (1–p)u(x,y). Finally, I assume that people are isolated from each other at date 1, so that at date 1 they cannot get together and coordinate what they do, and so that, if a banking system exists, then it must accommodate withdrawal demands at date 1 sequentially, one person at a time. Implications of the Model Before I use the model to appraise narrow banking, I describe some of its other relevant implications. The Best Feasible Symmetric Outcome I now describe the best feasible symmetric outcome in the model. Let (c 1,c i 2) for i = 1, 2 be a symmetric allocation—symmetric in the sense that everyone at date 0 is given the same type-contingent consumption pairs, where the subscript represents the date and the superscript i represents the type. In other words, a symmetric allocation is a consumption pair for each type. The solution to the following problem is the best feasible symmetric outcome. Upper Bound Problem. Choose (c 1,c i 2) for i = 1, 2, and choose xs, xl, and xl1 (all nonnegative) to maximize pu(c1,c 1 2) + (1–p)u (c1,c 2 2), subject to