The Optimal Denomination of Currency

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The Powers-of-three Principle

Recently, Telser (1995) and Sumner (1993) have argued that the optimal system of denominations of currency (coins and notes) would minimize the number of denominations while it increased the probability of proffering exact change. In their optimal system, the denominations differ from each other by a factor of three (that is, denominations of 1, 3, 9, and so forth). One can then describe the whole system in terms of powers of three (that is [3.sup.0], [3.sup.1], [3.sup.2] and so forth).

Telser shows that the problem is analogous to Bachet's problem of finding the smallest number of weights capable of weighing any unknown integer quantity between one and a finite upper bound, using a bilaterally symmetric balance (like the scales of justice). When the person doing the weighing puts weights in both pans, in the optimal set of weights each weight is three times as heavy as the next smaller weight.(1) Sumner and Telser then point out that the average of the ratio of adjacent denominations for the U.S. dollar and the British pound is "close to three. "

Even though for many countries the average is "close to three," the average is not three. Table 1 shows the average of the ratio of adjacent denominations for coins, the small change of daily life, of some fifty countries. Across the fifty countries, the average of the averages is 2.60.(2) For notes it is 2.62.(3)


Today no currency follows the powers-of-three principle; to the best of my knowledge, all currency systems are decimal. Even in the past we can observe only a few examples of the use of the principle. However, pure decimal systems (with denominations of 0.01, 0.10, 1, 10, 100, etc.) would provide too few denominations; most systems augment the decimal denominations by denominations that are two times or 1/2 the decimal (for example, 0.02, 0.05, 0.20, 0.50, 2, 5, 20, 50, etc.). The systems thus modify the decimal principle with a powers-of-two principle ([2.sup.1] or [2.sup.1]).

The failure of countries to achieve an average ratio "close to three" is not due to the rigidities of an augmented decimal system. As the example of Singapore shows, one can achieve an average ratio that is close to three (that is, 3.08) with denominations of 1, 5, 10, 20, and 50 cents and 1 and 5 dollars.

Moreover, the ratios within a country display substantial variation (Table 1; column labeled Stnd. Dev.) where optimality would argue for no variability.(4) For both coins and notes there is a strong correlation between the average ratios and their variability (0.87 and 0.91); countries with high average ratios tend to be countries with highly variable ratios. However, among coins, countries with many coins tend to have them closely grouped; among notes, there is no correlation between the number of notes and the grouping.

Existing currency systems are the result of an evolution in which historical and social influences have modified what is optimal in the Telser-Sumner sense. Most especially, currency systems embody the past as well as the present counting systems that the society uses. The evolution of our current pattern of denominations recalls David's (1986) story about the origins and durability of the QWERTY typewriter keyboard and the importance of path-dependence as a factor explaining the logic (or illogic) or the world around US.(5)

The Two Most Common Principles

The two most common organizing principles are (1) powers of two of the standard unit, and (2) a combination of this principle with the decimal principle.(6) The powers commonly range from -3 (that is, from one-eighth of the standard unit) to two (that is, to four times the standard unit). This principle also shows up in financial markets. In many older, Anglo-Saxon financial markets, tick (minimum price increment) sizes are in negative powers of two to at least the third power and often less. …