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An important function of banks is to issue liabilities, like demand deposits, that are relatively safe and liquid. We introduce a risk of theft and a safe-keeping role for banks into modern monetary theory. This provides a general equilibrium framework for analyzing banking in historical and contemporary contexts. The model can generate the concurrent circulation of cash and bank liabilities as media of exchange, or inside and outside money. It also yields novel policy implications. For example, negative nominal interest rates are feasible, and for some parameters optimal; for other parameters, strictly positive nominal rates are optimal.

Banks perform many functions in modern economies, but one very important function is to issue liabilities, like demand deposits, that are relatively safe and also liquid. Putting money in the bank obviously reduces the risk that it will get lost or stolen without excessively hindering its use as a means of payment. Moreover, using something other than cash reduces other risks, since one may be able to stop payment with a check or credit card, for example, if a purchase turns out to be flawed or fraudulent. While these points may be obvious, this does not mean they are uninteresting or unimportant for our understanding of money and banking. Yet they have been all but ignored in the literature.1 In a previous attempt to rectify this situation (He et al., 2005), we introduced a risk of theft into a micro-founded model of monetary exchange based on search theory. This allowed us to study the role of banks as institutions that provide safekeeping plus liquidity in a setting where there is an endogenous role for a means of payment in the first place. A drawback with that analysis, however, is that we used a rather crude physical environment. As in all simple first-generation search models, we adopted the assumption that money is indivisible and agents can only hold at most 1 unit. While this is unsatisfying for a number of reasons, perhaps the main limitation is that it is impossible to discuss many aspects of monetary policy, especially the effect of inflation or nominal interest rates on the use of currency and bank liabilities in payments. The goal of this project is to continue the integration of banking and monetary theory by reconsidering these ideas in a more recent generation of search models where agents can hold any amount of money. This allows us to go well beyond the earlier work, especially concerning policy and the effects of inflation or nominal interest rates, and provides a general equilibrium framework in which to formalize venerable ideas about how banks evolved historically. Although we do not dwell on history here, it may be helpful to review the story told in standard reference books: “The direct ancestors of modern banks were … the goldsmiths. At first the goldsmiths accepted deposits merely for safe keeping; but early in the 17th century their deposit receipts were circulating in place of money.” (Encyclopedia Britannica 1954, vol. 3, p. 41, emphasis added). 2 This is precisely how banks operate and how consumers use them in the model. While this history is fascinating, there are also contemporary issues for which our analysis is relevant. In terms of policy, an improved understanding of the design and implementation of modern payment systems may arise when we better understand simple situations like the one studied here. In terms of empirical work, research surveyed by Boyd and Champ (2003), for example, describes many findings concerning relations between inflation or interest rates and financial markets, including the banking sector. We do not attempt to address these observations directly, but we think our framework provides a step in the right direction, in that if one is to make sense of such empirical results, especially those concerning monetary policy, it might be useful to have a framework that better integrates banks and other financial institutions into monetary theory. The rest of the paper is summarized as follows. In Section 2 we present basic assumptions. In Section 3 we study the case with exogenous risk of theft and no banks to show how the value of money depends on this risk. We show that it is possible in equilibrium to have negative nominal interest rates, although there is a lower bound. In fact, in this model it is optimal to go to the lower bound, which means deflation in excess of the Friedman Rule i=0. In Section 4 we endogenize the risk associated with cash, still with no banks. In this version of the model, depending on parameters, it may or may not be possible to have negative nominal rates, but it will never be optimal: the optimal interest rate is either i=0 or i>0. The reason that some inflation in excess of the Friedman Rule may be optimal is that in equilibrium it reduces the risk associated with cash. In Section 5 we introduce banks with exogenous theft. We show that generically agents either put all or none of their money in the bank, so we cannot get the concurrent circulation of multiple means of payment: bank liabilities drive cash out of circulation (or vice versa) when their operating costs are small (big). The optimal policy is i<0 with banks and exogenous theft. In Section 6 we endogenize both theft and banking. Now we can generate concurrent circulation of multiple means of payment. We find in this version of the model the optimal policy is either i<0 or i>0. This is interesting because usually the Friedman Rule is extremely robust: i=0 is optimal in a wide variety of models. In Section 7 we conclude. Before proceeding, we comment further on the applicability of these ideas. The fact that mainstream banking theory mainly ignores payments and banks’ role in the provision of convenient, efficient, and safe instruments that facilitate this process might mean people who work in this tradition will not recognize many of the issues or the tools here, but this is no reason to dismiss the approach. In any monetary economy, or payment system, generally, safety is a real concern and significant resources are devoted to this end. It should be clear this does not just mean petty theft. Theft here is a way to capture formally fraud, embezzlement, counterfeiting and many other kinds of opportunistic behavior. It is also obvious that, as economists, we can only study tradeoffs between alternative payment instruments when they have different properties. We focus narrowly on the fact that some instruments, like cash, are low cost but risky; hopefully, the ideas and results are applicable to anyone interested generally in alternative payment instruments and the role of banks in the exchange system.3

We studied models where as a medium of exchange agents may use cash, bank liabilities, or both. Basically, the advantage of currency is that it can be exchanged at low cost in situations where agents have little knowledge of each other. This leads to a disadvantage—currency can be stolen. Alternative means of payment, modeled here as bank liabilities, mitigate the theft problem, but systems with these alternatives are costly to operate. Our theory differs from the mainstream banking literature by emphasizing the role of banks and their liabilities in payments. The model generates novel policy predictions. It is feasible to have i<0, and for some parameters this is optimal. For other parameters it is optimal to have i>0. Our setup is simplistic, but one could combine it with other banking models, and alternative environments could be considered. One could, for example, try to replace theft by private information, as in the monetary models of Williamson and Wright (1994), Trejos (1997) or Berentsen and Rocheteau (2004). Other extensions include reducing the reserve ratio below ρ=1 and deriving a money multiplier, as in our earlier paper. One can also think of ρ as a policy tool. With ρ<1 we obtain ϕ<a, since banks earn revenue from loans as well as fees, and we obtain ϕ<0 (interest on checking account) if a and ρ are low. In this case b=1, since deposits are equally liquid, have a higher yield, and are safer than money. While this captures a “cashless economy” one could add features to make deposits less liquid—some agents do not accept checks, say. All this is left for future research.