In this chapter, issues of different coins and notes in terms of denomination are determined seemingly by the Ministry of Finance. But in fact, they are determined by the market mechanism in a broad sense. This is our basic framework as to how the choice of optimal currency denomination can be made.
If money is a general good, differences in the use-value would be adjusted both in terms of price and quantity by the market mechanism. The price of money (legal tender) is fixed as its face value, thus there is no room for price adjustment. If the use-value or money demand among coins and notes differs, it would be a quantity that is adjustable. We will investigate how coins and notes with different denomination would circulate in the market.
2.1 Introduction
The innovation in currency such as digital money has been discussed worldwide. Digital money may replace coins and notes in a substantial way. Under such a circumstance as the coexistence of coins, notes and electronic money, a small distortion of coins and notes in terms of use-value would change the flow of money circulation to a great extent.
In this chapter, issues of different coins and notes in terms of denomination are determined seemingly by the Ministry of Finance.1 But in fact, they are determined by the market mechanism in a broad sense. This is our basic framework as to how the choice of optimal currency denomination can be made.
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In a textbook of monetary economics, money has three functions: medium of exchange, medium of account, and store of value. Actual coins and notes indeed have these three functions. At the same time, the relative use-value would differ among different coins and notes, thus the demand for coins and notes would differ.2
If money is a general good, differences in the use-value would be adjusted both in terms of price and quantity by the market mechanism. The price of money (legal tender) is fixed as its face value, thus there is no room for price adjustment. If the use-value or money demand among coins and notes differs, it would be a quantity that is adjustable.
Is currency denomination in Japan optimal? To begin with, can we define optimal distribution of currency denomination? If we can obtain the optimal distribution of currency denomination, how is the actual distribution distorted from the optimal one? Can we measure it?
To answer this question a priori, the optimal currency denomination is defined as the relative use-values of each coins and notes are set as equal, and thus relative demand for each coin and note is indifferent. The high use-value means that the buyer can settle his/her payment optimally by means of a minimum exchange of coins and notes (including changes). A distortion from the optimal distribution can be measured as the relative difference in actual circulation of coins and notes.
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In the following section, we will discuss how to define the optimal currency denomination, how to measure a distortion from the optimal, and the policy implication for the actual circulation of coins and notes.
2.2 Optimal Distribution of Currency Denominations
How can we obtain the optimal currency denomination structure, based on the assumption that the relative use-values of all coins and notes are set equal and thus relative demand for each coin and note is irrelevant? The basic idea is to minimize the number of transactions (exchanges) of coins and notes given any payment amount. To do so, we need to identify the denomination structure (e.g., 1 yen, 5 yen, 10 yen, 50 yen, 100 yen, 500 yen, 1,000 yen, 5,000 yen,10,000 yen) to minimize the cash carrying cost in daily life. Telser (1995) discusses this problem. He argues that the problem of Bâchet, a famous problem in number theory, can help solve this problem. The problem of Bâchet seeks the smallest number of weights capable of weighting any unknown integer quantity between one and 41 on a two-pan balance. Telser restates the problem, assuming each weight costs the same, finds the least expensive set of weights that can weight any integer quantity between one and a finite upper bound. Hardy and Wright (1979) define the Bâchet problem in two cases: (a) when weights may be put into one pan only and (b) when weights may be put into either pan. The version pertinent to the currency denomination issue is applicable to case (b) because we usually allow changes to be made. Hardy and Wright (1979) obtained the solutions for both (a) and (b) cases. In case of (a), the weights should follow the powers of two (i.e., 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,..). In case of (b), the weights follow the powers of three (i.e., 1, 3, 9, 27, 81, 243…) up to the sum of the weights 1 + 3 + 9 + …. + 3 k = (3 k+1–1)/2, the upper bound.
It is clear that the unknown amount to be weighted corresponds to a retail transaction to be paid in cash. Allowing weights to go in either pan corresponds to making change. The weights that can weight any quantity between one and the upper bound correspond to the assumption that a retail transaction is equally likely to be anywhere between one and some finite upper bound. So the Bâchet problem itself corresponds the choice of optimal currency denominations as we see above.
In case of retail transactions, if change can be made, case (b) applies and if change can not be made (e.g., old vending machines or bus payments), case (a) applies. In the following, we will mainly discuss case (b). The concrete problem is to determine the optimal denominations of coins and notes by way of minimizing cash holdings of consumers or cashiers in retail shops (e.g., supermarkets or convenience stores). In the Appendix, we will give a mathematical proof of the optimal currency denomination problem, as a natural extension of the Bâchet problem. Here we assume the distribution of transaction sizes are uniform.3
Figure 2.1 illustrates the progressivity of currency denomination in the US, the UK, and Japan and the optimal distribution (i.e., 3 k).
A line graph represents progressivity of currency denominations, with optimal distribution higher than U S, ad U K, and lower than Japan.
Fig. 2.1
Progressivity of currency denominations
×
As Telser (1995) reports, distribution of currency denominations in the US is fairly close to the optimal distribution. Numerically, the arithmetic mean of progressivity of optimal distribution is 3.0 (power of 3) while it is 3.34 in the US. It was 3.52 for Japan and 2.53 for the UK. After k = 6, the progressivity accelerates in Japan and decelerates in the UK.4
Does the degree of progressivity exceeding 3 (the optimal level) in Japan create any distortions in circulations of coins and notes? See Table 2.1. The gap between the optimal and the actual denominations becomes clear when k = 3, 5, 7. The denominations will be approximately 25 yen, 250 yen and 2,000 yen, while the actual denominations take 50 yen, 500 yen and 5,000 yen. These denominations make progressivity higher than 3. Of course, if these denominations of coins and notes are used indifferently, there is no problem.
Table 2.1
Distribution of currency denominations
k
0
1
2
3
4
5
6
7
8
Degree of progressivity
Optimal distribution
1
3
9
27
81
243
729
2187
6561
3.00
Japan
1
5
10
50
100
500
1000
5000
10,000
3.52
US
1
5
10
25
100
500
1000
2000
5000
3.34
UK
1
3
6
12
30
60
240
1200
2400
2.53
Note (1) Figures are based on the minimum unit (yen in Japan, cent in US, penny in UK)
(2) In case of US and UK, old denomination coins and notes are omitted because they are rarely used
(3) In Japan, the distribution was before introduction of 2000 yen note in July 2000
(4) In UK, the distribution was before the decimal reform of February 1971
In order to identify the actual circulation of coins and notes, see Fig. 2.2. Surprisingly, circulation of 5 yen, 50 yen, and 5,000 yen are rather low, while circulation of the 500 yen coin is steadily increasing over time. In particular, that of the 5,000 yen note is distinguishably lower than those of the 1,000 yen and 10,000 yen notes. Does this imply the use-value of the 5,000 yen note is very low?
A line graph represents a mild increasing trend of the quantities of notes and coins in circulation. 1 Yen coin is above all, and 2000 Yen note lies flat at the bottom.
Fig. 2.2
Quantities of notes and coins in circulation. Notes (1) 500-yen note is omitted because of low circulation quantity. (2) The unit of notes is billion and that of coins is 10 billion
×
By the way, how can we measure the use-value of coins and notes? It can be measured by the relative space among coins and notes. That is, in case of 1 yen and 5 yen, 1 yen is used when the payment is less than 3 yen and 5 yen is used when that is more than 3 yen. In other words, the use-value of the 1 yen coin is 3 yen because it is best used in a payment between 0 and 3 yen. That of the 5 yen coin is 4.5yen because the borderline between 5 and 10 yen is 7.5 yen.
Similarly, that of the 10 yen coin is 22.5 yen, the 50 yen coin is 45 yen, the 100 yen coin is 225 yen, the 500 yen coin is 450 yen, the 1,000 yen note is 2,250 yen, the 5,000 yen note is 4,500 yen, and the 10,000 yen note has no upper bound. The greater the denomination (face value), the larger the use-value becomes. We can standardize the use-value by dividing the denomination (face value). The standardized use-value for a 1 yen coin is 3; those of a 10 yen coin, 100 yen coin, and 1,000 yen note are equally 2.25. That of a 10,000 yen note is infinity. Those of a 5 yen coin, 50 yen coin, 500 yen coin, and a 5,000 yen note are equally 0.9. That is, the use-value for a 10 yen coin, 100 yen coin, and a1,000 yen note is 2.5 times larger than that of a 5 yen coin, 50 yen coin, 500 yen coin, and a 5,000 yen note. This is a theoretical explanation for the low circulations of the 5 yen coin, 50 yen coin, and 5,000 yen note.5 The relative use-value of the optimal denominations (i.e., the power of three, 3 k) is equally spaced 1.33 except for the minimum 1. This equality of the relative use-value is the fundamental reason for equal demand for the optimal denominations of coins and notes.
Let us turn to compare the relative ratio of actual coins and notes in circulation with the theoretical relative ratio of corresponding currency (i.e., 2.5). In so doing, we can understand how distortions from the optimal denomination structure can explain the actual circulations of coins and notes. We use the Chi-square goodness of fit test between the actual value and the theoretical value. This is one of the standard goodness of fit tests. The results are given in Table 2.2 for annual data 1984–1998.
A 5 yen coin vis-à-vis a 1 yen coin and a 5 yen coin vis-à-vis a 10a yen coin are both theoretically consistent. A 50 yen coin vis-à-vis 100 yen coin is consistent with the theoretical value. A 50 yen coin vis-à-vis a 10 yen coin is not consistent with the theoretical value because a 10 yen coin in circulation exceeds the theoretical value. A 500 yen coin vis-à-vis a 100 yen coin is consistent with the 10% significant level. A 500 yen coin vis-à-vis a 1,000 yen note is not consistent with the theoretical value. A 5,000 yen note is not theoretically consistent with either a 1,000 yen note or a 1,0000 yen note—i.e., a 5,000 yen note in circulation is by far lower or both 1,000 yen and 10,000 yen notes in circulation exceed the theoretical value vis-à-vis a 5,000 yen note. In summary, while 5 yen, 50 yen, and 500 yen coins, with some exceptions, are broadly consistent with the theoretical ratio, s 5,000 yen note in circulation is exceedingly low in value.
In the real world, coins and notes in circulation are determined not only by the relative use-values, but also by the average price levels and the average payment amounts per shopper. It may not be appropriate to conclude that 1,000 yen and 10,000 yen notes are widely circulated because of distorted use-values vis-à-vis a 5,000 yen note. On the contrary, low circulation of the 5,000 yen note, compared with all other coins and notes is clearly shown in Fig. 2.2. This is due to inappropriate choice of denomination spaces.
I raised the following question when I originally wrote in 1996: How about adding the 2,000 yen note for the current denomination structure? I thought that probably the 1,000 yen note, 5,000 yen note, and 10,000 yen notes would be substituted for by the 2,000 yen note to some extent and thus overall circulation of these notes would drop. In particular, substitution for the 1,000 yen note would be large and thus the relative distortion between the 500 yen coin and the 1,000 yen note would be corrected.
Lastly, excess circulation of 10,000 yen notes vis-à-vis 5,000 yen notes is not only due to a distortion between the two denominations, but is also due to no upper boundary above the 10,000 yen note. Due to the expansionary monetary policy in recent years, the circulation of 10 000 yen notes has steadily increased. We may think about the next denomination value above the 10,000 yen note. The Bâchet problem provides a clear answer for this. That is, the next denomination above the 10,000 yen note theoretically would be, 3 k = 39 = 19,683 which is a 20,000 yen note in practice.
2.3 Conclusion
In recent years, policymakers in the government seem to adopt a market-based policymaking framework. They pay attention to economic deregulations and liberalization of the markets on the surface, but they rarely pay attention to such institutional distortions as discussed in this chapter. As is discussed in the framework of comparative institutional analysis, an institution affects other institutions (this is called institutional complements). Deregulation of a part of market may not be so effective because other parts of markets complement the deregulated part. Non-optimality of currency denominations may cause unnecessary cash hoarding costs, and price level concentration around certain digits (e.g., 3,990 yen for UNIQLO jeans) may also cause distortions in cash hoarding. Adjusting currency denominations to the optimal spacing is certainly a part of market-based policy making.
The Bâchet problem falls in the field of number theory in mathematics, which was described as the queen of pure mathematics by Carl Friedrich Gauss. Application of this problem almost directly to an economic problem is surprising, because in social science, rigorously controlled experiments are basically impossible. Cryptocurrency or cryptoassets use, by definition, cryptographic techniques which are also based on the field of number theory. Given that the market economy is fundamentally based on the world of natural numbers (i.e., 1, 2, 3,…), the analysis of the market economy may adopt more results from number theory.6
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2.4 Appendix: Application of the Bâschet Problem to the Choice of Optimal Currency Denominations
The Bâschet problem can be interpreted as equivalent to the choice of optimal currency denomination, setting different denominations to minimize transactions of coins and notes between buyers and sellers. We can consider either the case of allowing changes or that of allowing no change. We will focus on the case with change, which is more general. The solution is divided by two sections: (1) We can pay any amount by combining different denominations of coins and notes (Lemma 1) and (2) There exist the optimal denominations of coins and notes to minimize transactions (Theorem1). Proof of theorem 1 is based on Hardy and Wright (pp.116–117, theorem 141).
Lemma 2.1
Given denominations, k0, k1, k2, ….,kn, any positive integer (payment) a can be expressed uniquely as a combination of denominations, allowing changes.
Equation (2.7) implies that en-1 minimizes en, and in turn, en-2 minimizes en-1 and so on. Given denominations, k0, k1, k2, ….,kn,, e0, e1, e2, …..en are the minimum number of integers to make payment a. (Q.E.D.)
The most efficient (non-redundant) way of using currency denominations n + 1 is the case where ei takes 0, –1, or 1. This case satisfies the condition of denominations following the powers of three. Proof is given in the following.
Theorem 1
Allowing changes, currency denominations 1, 3, 32, 33, 34, …0.3n can pay any amount (integers) up to (3n+1 – 1)/2 and transactions of coins and notes for this payment becomes minimum.
Proof
From Lemma 1, any positive integer a (\(\mathrm{up to }\,\,{3}^{n+1}-1)\) can be expressed uniquely as a weighted sum of the ternary scale.
$$a=\sum_{i=0}^{n}{f}_{i}{3}^{i}$$
(2.8)
where every \({f}_{i}\) is 0, 1, or 2. And
$$b={\sum }_{i=0}^{n}{3}^{i}=({3}^{n+1}-1)/2$$
(2.9)
Subtracting (2.9) from (2.8), c = a–b, we see that every positive or negative integer between –(\({3}^{n+1}\)–1)/2 and (\({3}^{n+1}\)–1)/2 inclusive can be expressed uniquely in the form
$${\sum }_{i=0}^{n}{g}_{i}{3}^{i}$$
(2.10)
where every \({g}_{i}\) is –1, 0 or 1. Hence our currency denominations (n + 1 different coins and notes), allowing changes, will pay any amount up to \(({3}^{n+1}-1)/2\) by using only one (maximum) denomination each.
Now we turn to proving that no other combination of n + 1 denominations can pay the same amount as efficiently (small transactions) as a weighted sum of the ternary scale.
Currency denominations ki differ and are put in order such as
$${k}_{0}<{k}_{1}<{k}_{2}<\dots <{k}_{n}$$
(2.11)
The two largest amounts using one denomination each are plainly
Leaving the denominations \({k}_{s}, {k}_{s+1}, \dots .,{k}_{n}\) undisturbed, and removing some of the smaller denominations, or transferring them to changes, we can pay up to the minimum amount,
In Japan as of 1996, in which this research was conducted, the Japanese government issued coins (1 yen 5 yen, 10 yen, 50 yen, 100 yen, 500 yen) and the Bank of Japan notes (500 yen; 1,000 yen; 5,000 yen; 10,000 yen).
It is rare to find a person who is indifferent about paying 10,000 yen in 1-yen coins or in a 10,000 yen note. By the same token, it is rare to find a person who pays 1 yen with 10,000 yen note. The use-value of each coin and note would differ according to the size of payment, even though every coin and note can be used equally as a medium of exchange.
If we look at individual prices, there is a tendency to quote prices ending in eight, nine or zero because it looks a bit cheaper (compare 102 yen vs 99 yen; people prefer 99 to 102. In fact, 3 yen difference would not matter from the retailer’s point of view, so they set 99 instead of 102). Many goods are purchased all together and with value-added tax on consumption, but the distribution of final payments can still be assumed to be uniform.
