بانک مرکزی و مجموعه های اسکناس هایش : بهره وری هزینه تجارت کردن

One of the most important results of theoretical research on currency systems is that spacing denominations apart by a factor of two is better than a factor of three as this lowers the average number of notes and coins exchanged in transactions. These theoretical studies also claim that an efficient denominational mix has the additional benefit of keeping down the production costs incurred by the central bank. This paper challenges this claim and demonstrates that more efficient currency systems can also be more costly. Central banks therefore face an efficiency–cost trade-off and have to weigh the benefits for transactors against those for the central bank itself.

In the theoretical research on optimal denominations for coins and banknotes, efficiency has always been the predominant concern. This is reflected in the popularity of the “principle of least effort”, which holds that a denominational structure should make it possible for transactors to economize on the number of tokens exchanged. For the case of exact payment, Caianiello et al. (1982) demonstrated early on that modular currency systems – systems in which each denomination is X times the one below it (with X an integer) – are the most efficient, and that the number of tokens exchanged is a growing function of the spacing factor. For the case where overpayment and the return of change is allowed, Van Hove and Heyndels, 1996 and Van Hove, 2001 eventually, after many a controversy, showed that the optimal spacing factor is not three – as claimed by Sumner, 1993 and Telser, 1995 – but rather two, as in the case of exact payment. These theoretical studies supposedly also have important practical implications beyond efficiency. In particular, a reduction in the number of tokens exchanged is assumed not only to be more convenient for transactors, but also to keep down the number of coins and banknotes in circulation, and thus the handling and production costs incurred by the central bank.3 In short, the most efficient denominational mix would at the same time also be the most cost efficient, at least for the central bank. This matters because the currency expenses of central banks are quite high. In the U.S, for example, the cost for the Federal Reserve System of new currency alone is budgeted at $703 million for 2010, which is equivalent to 16% of the total budget.4 The objective of this paper is to show that reducing the spacing factor can actually increase the production costs incurred by the central bank. To that end, we build on Cramer's model of efficient payments ( Cramer, 1983) by incorporating the production costs of the tokens. The resulting model makes explicit how an increase in the “density” of a currency system – that is, an increase in the number of denominations over a given interval – on the one hand improves its efficiency and thus lowers the variable production costs, but on the other hand increases the fixed production costs. Using simulations in which we compare powers-of-two and powers-of-three currency systems, we demonstrate that, under certain conditions, the lower average frequency of use that comes with the powers-of-two system is not sufficient to offset the second effect. In other words, while the powers-of-two system is clearly more efficient, it can be more costly for central banks than a powers-of-three system, and this even with identical cost structures. Obviously, pure powers-of-two or powers-of-three systems – the latter having denominations of 1, 3, 9, 27, 81, etc. – are not really viable in practice. 5 We have therefore also included two real-life currency systems in our simulations, namely the series used by the European Central Bank (ECB) and the Federal Reserve. Again we find that the central bank faces an efficiency–cost trade-off and thus has to weigh the benefits for transactors against those for the bank itself. The paper contributes to the literature on optimal denominations for coins and banknotes on two crucial points. For one, it is the first to fully consider all production costs. Indeed, while Bouhdaoui and Bounie (2010) also show that the principle of least effort fails to minimize the production costs of the central bank and that it should thus be reconciled with cost considerations, they use a basic cost function of the central bank that ignores the fixed costs of production. As a result, they do not take into account economies of scale, which are critical in understanding the design of a denominational mix. Second, the present paper looks into the issue of what constitutes a (theoretically) optimal denominational mix. Bouhdaoui and Bounie (2010), for their part, only compare the cost of efficient and inefficient payments for a single real-life currency system, namely the U.S. series as it exists today. The remainder of the paper is structured as follows. In Section 2, we present an original framework that models the currency production costs of the central bank and in particular indicates how these are affected by the frequency of use of the tokens in circulation. In Section 3, we use this framework to study – still on a purely theoretical level – the conditions under which one currency system can be more costly than another. In Section 4, we then illustrate our analysis by performing simulations for selected currency systems. Section 5 discusses the implications of our results.

The main objective of this paper was to challenge one of the most important results of the theoretical research on currency systems. This result holds that, for modular currency systems, lowering the spacing factor between denominations not only improves the efficiency of the system but also reduces the production costs of the central bank. In order to debunk this finding, we compare the two most popular hypothetical currency systems, namely the powers-of-two and powers-of-three systems. We first develop a general model of the currency production costs incurred by the central bank and, using numerical simulations, we then show that while a powers-of-two system has a lower average number of tokens exchanged in transactions (and is thus indeed more efficient), it can be more costly for a central bank than a powers-of-three system. The intuition behind this result is that, for modular currency systems, a lower spacing factor goes hand in hand with a higher number of denominations, which tends to increase the production costs because of the origination costs attached to additional denominations and because the economies of scale are smaller. We also show that the validity of this result is not limited to hypothetical currency systems. Indeed, the choice between an ECB– or Fed– style series proves to be governed by the same efficiency-cost trade-off. This finding raises the question how central banks across the world have responded to the trade-off. Interestingly, two earlier studies have computed average spacing factors across a large number of countries. Wynne (1997) analyzes data on 156 countries and finds that the arithmetic mean of the spacing factors is… exactly 3. For the subset of OECD countries, the mean equals 2.8. Tschoegl (1997) performs a similar exercise for 50 countries, and comes up with slightly lower values, namely 2.60 for coins and 2.62 for notes. Still, both studies find that the mean average spacing factor is closer to 3 than to 2. Prima facie, this would seem to indicate that central banks opt not to maximize user convenience at all costs (pun intended), and do economize on the number of denominations. However, several qualifications are in order here. First, to ease mental arithmetic, currency systems have to be compatible with the decimal system. As a result, even in the case of central banks that opt for complete binary-decimal triplets, such as the ECB, the average spacing factor will be higher than 2 in casu. Second, Wynne in his paper finds that the distribution of the average multiples is not concentrated around the mean, but rather appears to be bi-modal, with peaks at 2.2 and 2.7. This would seem to indicate that some central banks handle the trade-off differently than others. Third, the average spacing factor says nothing about the number of denominations30 nor about where the “gaps” in the denominational structure are, if any. Fourth, and related to this, in deciding on their denominational structure, central banks also take into account other criteria besides the principle of least effort. There is, for example, the issue of surveyability: the larger the variety of denominations, the harder it becomes for the public to recognize the different coins and notes, to sort and store them, etc. Also, in reality the distribution of cash payments is not uniform; small payments occur more frequently. As a result, a low density in the upper regions of a denominational structure – where the store-of-value function of currency overwhelms its function as a medium of exchange – need not greatly affect the public's ability to handle cash payments efficiently, even though it raises the average multiple. In short, as acknowledged by the existing literature, multiple criteria play a role in the determination of the optimal number and spacing of currency denominations. Our contribution highlights that it is not only a multi-criteria, but also a multi-actor problem. We show that central banks face an efficiency–cost trade-off and thus have to weigh the benefits for transactors against those for the central bank itself.31 Extending our approach so that it can incorporate additional stakeholders and their activities – such as commercial banks and their ATM operations – would seem a fruitful avenue for further research.