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The logic of currency denominations. And the possible irrelevance of it all

As a student entering the eighth grade, I was appalled to discover on the classroom door a sign with ‘VIII B’ on it. Not that I had anything in particular against being in section B. Rather, I had not, until then, realised that the number eight in Roman numerals was expressed that way. Given the odd structure of the Roman system of numerals, I had always assumed that the accepted way of displaying all numbers involved using the fewest possible letters. The sign on the door should have read ‘IIX B’ as far as I was concerned. My one pillar of sense and order in that anarchic system had just been destroyed. Why not just write ‘IVIV B’ or even ‘ IIIIIIII B’, I wondered.

Some years earlier, in an attempt to introduce the concept of money to us, our second grade teacher had asked us all to bring things we did not use anymore to class. We were then asked to design our own money, and ‘buy’ and ‘sell’ our ‘goods’. Most of us just copied the currency notes we had seen in circulation. There were one or two, however, who incurred the ridicule of our teacher because they had designed money in non-standard denominations. One of them created notes corresponding to two, four and eight units. We made fun of him. That he had a valid point, we did not quite realise at the time.

The problem of deciding the denominations of currency notes and coins is surprisingly subtle and difficult. One method is based on the principle of least effort. The ‘effort’ in this case is the number of notes (or coins, which follow the same logic and therefore need not be explicitly discussed) required to be able to produce a certain amount of money (say, rupees). What is the easiest way (involving the fewest notes) of attaining every amount between one and three? That’s simple—a one-rupee and two-rupee note. What about four rupees? You could use two notes of two-rupees each, but it’s more efficient to introduce a four-rupee note. This means we can also furnish every amount up to seven rupees. The next note to be added would be an eight-rupee note. A pattern thus emerges: each new note we introduce is twice the value of the previous new note. This is a ‘power of two’ system, like the one devised by my second grade classmate.

A competing approach was proposed by the anagrammatically endowed Professor of Economics at University of Chicago, Lester Telser. It is based on a famous problem involving a farmer and weights, known as the Problem of Bachet. Originally, the problem was to figure out the least number of different pound weights to be used on a scale pan to be able to measure all whole number weights between one and 40. Weights can be placed in either scale. It turns out that the answer is four weights—one each weighing one pound, three pounds, nine pounds and 27 pounds. Here, the pattern is clear: each new weight is three times the previously introduced weight; a ‘power of three’ system.

The astute reader would have noticed that neither method described above includes the ubiquitous five-rupee note. This is because all arithmetic that we are familiar and comfortable with is based on the decimal system (which means that we have different symbols for ten numbers, which are then reused in various combinations to express all other numbers). With our natural grasp of decimal logic, a potentially useful system would involve the prime factors of ten, namely two and five. This has fewer different denominations than the 1-2-4-8 system. The 1-2-5 system is closer to the power-of-two system, since the average ratio of two successive denominations is 2.2.

Historically, many currencies did not use the decimal structure. In the pre-decimal period, the Indian rupee followed a strict power of two system. One rupee consisted of 16 annas, and annas were produced in denominations of a quarter, half, one, two, four, eight and 16. The Indian rupee switched over to the decimal system in 1957, fourteen years before the similarly archaic British system.

When countries in Europe abandoned their individual currencies in favour of the euro, most made the switch easily because euro denominations matched their own currency denominations. The Dutch guilder, however, posed a bit of problem. The guilder coin range started at 0.05 and moved through 0.1, 0.25,1, 2.5 and 5. The presence of the two-and-a-half guilder coin meant that the average ratio of this system was 2.6, bringing it closer to Telser’s power-of-three system. The US dollar has an average ratio of 3 because of the absence of a 50 cent coin or two dollar bill.

So, which is the better way? Opinion is divided. Empirical evidence from over 150 countries published by Mark Wynne in 1997 shows that the average ratio is 3. In an article published on Allbusiness.

com, Leo van Hove argues in favour of the superiority of systems closer to the power-of-two system.

Arguments arise because efficiency for spenders of money is not the same as efficiency as seen by the mint. The 1-2-5 system represents a compromise between the spending efficiency of the power-of-two system and printing efficiency of the power-of-three system.

All of which could become irrelevant in a few years because of the proliferation of online and other cash-free transactions, where denominations of currency make no difference. The concept of Roman numerals, annas, shillings and guineas seems outdated today. Perhaps the argument over efficient denominations of currency has a similar fate in store by the year MMXLVIII.