A month or so ago, it was announced that Ireland was planning to abolish its 1 and 2 euro cent coins. In this respect, Ireland follows the likes of Belgium, Canada, and the Netherlands abolishing some of their low denomination coins in recent years (see here for more details and examples).

There have been various arguments regarding whether abolishing such low denomination, some in favour and others against doing so. Indeed, such arguments regarding “tradition” and “rounding” have been well covered elsewhere. However, one issue that does not seem to have been mentioned is the optimality (in terms of efficiency) of currency denominations. Although the research regarding this issue is a few years old now (the most recent paper I could find was from about 2001), it is still relevant for this discussion.

The design of a set of optimal currency denominations can be based on two possible approaches. The first follows “the principle of least effort”. This approach is espoused by the likes of van Hove & Heyndels (1996) and states that the number of denominations should be designed so as to allow a cash payment to be made with the fewest number of coins/notes handed over. At the extreme this would imply having one denomination for each possible value of a transaction, but this is impractical.

More practical is a set of denominations that have some “spacing” between them (i.e. the multiple applied to the smallest denomination to get the next largest denomination, and so on), but needing to account for the fact that the number of notes and coins required for a transaction increases with the spacing. (To see this last point, note that if the spacing were 2, such that denominations were 1, 2, 4, 8 etc, in order to pay for an item that cost 18 would require the use of 2 coins – one 16 and one 2. If, on the other hand, a spacing of 10 were used – implying denominations of 1, 10, 100 etc – this would require as many as 9 coins: one 10 and eight 1s.)

Caianello et al (1982) found that a spacing of 2 would be optimal, but due to the need to fit denominations around a decimal system, using a 1-2-5 structure would be best in terms of minimising effort. Indeed, this structure corresponds to the 1p, 2p, 5p, 10p, 20p, 50p set-up present in the UK. Moreover, this set-up is supported by more complicated models that allow for overpayment and change (these were not included in the model used by Caianello et al).

Under this approach it is easy to see why the penny comes in handy – it allows the smallest exchange of coins to occur, while still allowing producers maximum flexibility to set prices at the level of the nearest penny. (Note that if, for example, coins below the 10p denomination were removed, producers would no longer have as much flexibility to set prices – they would need to set prices to the nearest 10p as opposed to the nearest 1p.)

The second, alternative, approach to determining the optimum denominations of a currency is one of minimising the number of denominations. This approach has been suggested by, for example, Wynne (1997) and Tschoegl (1997) and views the problem as one that should be solved by minimising the number of denominations in circulation subject to the constraint that all possible transaction values can still be achieved. In other words, the main aim should be to set the number of denominations at the smallest level that still allows for all transaction values to be paid.

In the “real world” scenario in which overpayments and change occur, it transpires that the optimal system under this approach is one that uses a spacing of 3 – i.e. that denominations should be 1p, 3p, 9p and so on. However, under this approach, (and although the denominations conventionally start at 1p) it is easy to see that getting rid of the 1p could be optimal. Specifically, it reduces the number of denominations. On the other hand, given that the denomination would then start at 2p (or 5p), the optimal form of subsequent denominations would increase by a factor of 3 each time – i.e. 2p, 6p, 18p and so on.

It is unlikely that this change in denomination spacing will actually happen if the penny is abolished, such that even if the minimisation of denominations were used as the justification for getting rid of the penny, the resulting denominations are likely to be sub-optimal. Hence, it seems that getting rid of the penny is unlikely to be optimal from the perspective of designing currency denominations.