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Previous works related to optimal denominations for coins and banknotes consider that the “principle of least effort” that defines an efficient payment is the most important criterion for two main reasons. Firstly, it is more convenient for transactors and, secondly, it limits the production costs of denominations incurred by the central bank. Exploiting production cost data for the U.S. currency system in 2010, we show using simulations that efficient payments actually increase the annual production costs of the Federal Reserve by $156 million. As a consequence, we raise a larger issue for central banks which consists in issuing an efficient denominational mix that is more convenient for transactors and that reduces the production costs of denominations.

In recent years, abundant research has been devoted to the study of denominational structures of currency systems (Bouhdaoui et al., 2011, Caianiello et al., 1982, Franses and Kippers, 2007, Lee et al., 2005, Sumner, 1993, Telser, 1995, Tschoegl, 1997, Van Hove, 2001, Van Hove and Heyndels, 1996 and Wynne, 1997). Among the multiple properties of a currency system, the principle of least effort (PLE) is considered the most important.2 This principle that defines an efficient payment states that the settlement of cash transactions should involve as few coins and notes as possible. The preeminence of this principle, supported by many economists such as Boeschoten and Fase (1989), Eriksson and Kokkola (1993), Abrams (1995), Pedersen and Wagener (1996), Van Hove and Heyndels (1996) and Van Hove (2001), is justified by two main arguments. Firstly, the PLE states that it is more convenient for transactors given that it reduces the bulk and weight carried around by the cash-using public in turn limiting handling costs. Secondly, it keeps down the number of coins and notes in circulation and thus, so the reasoning goes, the production costs incurred by the central bank. Following this argument, it is therefore preferable for the central bank to opt for a currency system that limits the number of coins and notes used in transactions. In this article, we demonstrate that the second argument is biased and that efficient payments increase the production costs incurred by the central bank. Our results tend therefore to support the idea that the private benefits of transactors emphasized in the economic literature can be undermined by the private costs of central banks. To prove this, we proceed in three stages. Firstly, we propose a general framework that links the costs of cash transactions to the production costs of the central bank. Secondly, we compare the costs of cash transactions using the PLE and a hypothetical cost-minimizing payment behavior named the “principle of least cost” (PLC). This latter minimizes the costs of cash transactions without considering the number of tokens exchanged in transactions; this model is only used to identify inefficient payments from the viewpoint of the PLE. Thirdly, we perform simulations on a set of cash transactions using production cost data for the U.S. currency system for the year 2010. The simulation results show that while the number of notes and coins used in transactions is certainly efficient (minimum) with the PLE, the costs of cash transactions are on average 24.2% greater than those obtained with the principle of least cost. Hence, while the PLE keeps down the total number of coins and notes in circulation it can also contribute to an increase in the costs of cash transactions and thus in the production costs of denominations incurred by the central bank. We precisely estimate the increase in the annual production cost to $156 million. The remainder of the paper is structured as follows. In Section 2, we present the general framework and the cash payment behavior models. In Section 3, we describe the data used to perform simulations and comment on the results obtained. Finally, in Section 4, we discuss the implications of our results.

The issue of the cost of cash has become today an important topic for economists, banks and monetary authorities. The different central banks all over the world have indeed tried on numerous occasions to reduce the cost of banknote production either by withdrawing a denomination (Chen, 1976) or by introducing coins in place of banknotes (Lotz and Rocheteau, 2004) or, finally, by introducing a new technology such as the polymer one (Bouhdaoui et al., 2012). The commercial banks have also encouraged the public to use electronic payment instruments supposedly less costly for the society. These strategies seem to be supported by empirical studies that conclude that debit card payments are socially less costly than paper-based payments (Guibourg and Seggendorff, 2007). Our paper contributes to this topic in two different ways. First, our results show in the case of the U.S. currency system that efficient payments can increase the costs of cash transactions and then the annual production costs of denominations incurred by the Federal Reserve. The main argument is that the usage costs of denominations significantly differ and that it can be efficient from a cost perspective to use more cheaper coins and notes in transactions. Now, by definition, the principle of least effort can fail to reach this objective. Second, our paper provides not only a simple method to assess the cost efficiency of a denomination in a currency system but also some ways to restore the overall efficiency. On the one hand, at the level of each denomination, our model provides an original approach to measure the usage cost of a denomination (Eq. (10)). This measure is more useful for a central bank than the mere knowledge of the unit production cost as a denomination can certainly have a high unit production cost but be widely used in transactions; as a result, its usage cost is low. This is the case for example for the 25 cents denomination (comparison of Table 1 and Table 2). Now, by measuring and comparing usage costs of denominations, a central bank can determine the denominations that quickly deteriorate with regard to their production cost and come up with solutions to reduce their usage cost. These concerns are lively in a large number of countries. For instance, in Europe, some countries have decided to stop issuing the lowest denominations (1 and 2 cents) as they are supposed to have a limited utility in transactions. Our model can then accurately measure the effects of such decision on the use of related denominations and on the induced changes in production costs for the central bank. On the other hand, at the level of the overall cost efficiency of a currency system, assuming that the public behaves according to the principle of least effort, which is not a restrictive assumption (Franses and Kippers, 2007), central banks can be interested in designing a currency system so as to reduce the production costs of denominations according to this payment behavior. In this perspective, our paper introduces a hypothetical cash payment behavior, the principle of least cost that can be used as a benchmark by central banks to measure the “extra cost” of their currency system. As a consequence, our paper raises a larger issue for monetary authority: making a denominational mix more convenient for transactors while reducing the resulting production costs. This issue of the cost efficiency of a denominational structure has gone largely unresearched (Massoud, 2005) even though recent monetary contributions attempt to tackle this issue (Bouhdaoui et al., 2011 and Lee et al., 2005).