When Should We Replace $5 Bills With $3 Bills?

Almost all currencies in the world have denominations that start with 1, 5, or 0 and the main difference among them is whether $2, $20, $200, etc. bills are used. But prevalence does not prove efficiency. In this post, I will compare the efficiency of different bill denomination systems.

An efficient currency minimizes the cost of transaction. To simplify, we assume that the cost of transaction is proportional to the time spent on it. The cost of transaction can be dichotomized into time spent by the customer (while the cashier waits) and time spent by the cashier (while the customer waits). Each component further divides into the following categories denoted by a, b, and c:

  • Fixed transaction cost (a): time spent on taking out the wallet, opening the cash register, thinking about which bills to use (while doing nothing else), and handing and receiving the money. Avoid double counting if these jobs overlap in time or if the other party is not waiting but doing some other necessary work, such as printing the receipt.
  • Fixed cost for using bills in each denomination (b): time spent on moving the hand to reach a specific slot in the cash register; time spent on finding the place for a specific denomination in the wallet and putting all bills of this denomination on the other hand or the counter before moving on to work on the next denomination.
  • Cost for counting a bill (c).

We acknowledge that cost a can be substantially lower if no change is required from the cashier, which saves the time of passing the changes back to the customer. However, most transactions involve taxes and multiple items that lead to the usage of coins (in the US). Most of the time the customer does not have the exact amount of coins or does not want to pay any coins. Secondly, for the same reason, charges are usually not psychologically convenient numbers (e.g. $5, $20). Nice numbers can make buying decisions easier but they don’t show up in the payment. Therefore we can assume that there are always changes involved and a is constant and can be entirely dropped from this analysis.

Analytical Solution

To derive the the most efficient currency denominations analytically, we set the largest possible payment to M. It’s easy to see that the ratios between successive denominations should be the same to be efficient. We observe that the largest number of denominations needed in a transaction is \log_{x}{M}  where x is the inter-denomination ratio we need to find. The largest number of bills needed in a single denomination is x – 1. Thus, to minimize the cost in the worst scenario, we have:

\min\limits_{x}(b\log_{x}{M}+c(x-1)+a)

The first order condition can be arranged as:

cx \log {x} =b \log_{x}{M}

This shows that increasing the denominational fixed cost b increases the ratio x, which decreases the number of denominations. Increasing the per bill cost c has the opposite effect. If we set b and c to 1 so that we are minimizing the sum of the number of denominations used and the number of bills, then the formula simplifies to x \log {x} = \log_{x}{M} . When M is about $15, we have x = log_{x}{M} = e , which equates the number of denominations with the inter-denomination ratio. When M is as high as $100, x is slightly higher at 3.27 (see graph below). To be practical, this suggests using a ratio of 3, but 3 is not a factor of 10, which is almost a required denomination given the decimal system. So we could use denominations of $1, $3, $10, $30, etc., which alternates the ratio between 3 and 3.333.  The United States dollar varies in x from 2 (between $5 and $10 bills) to 5 (between $1 and $5 bills) which is not extremely far from the ideal ratios.

Rplot02.png

If we set b = 3c, then x would be almost twice higher for each M. This would call for a system with $1, $5, $20, $100 bills since x is close to 5 for larger M’s. Some currencies have ratios as low as 2 to 2.5 by using $1, $2, $5, $10… bills, such as the Euro. If efficiency is maximized in Euro, it would suggest b < c or that most transactions are less than €10.

We have analyzed the optimization of the inter-denomination ratio in worst case transactions. Average case solution depends on the distribution of transaction sizes and is algebraically more complicated; x tends to be a little higher than before in most of the cases.

 

Simulation

In the first part of the simulation, we try to set the values of the parameters to be as realistic as possible and compare the cost of transaction of six different denomination systems on a real economic scale, which are:

  1. $1, $5, $20, $100 (inter-denomination ratio x ≈ 5)
  2. $1, $3, $10, $30, $100 (x ≈ 3)
  3. $1, $5, $10, $50, $100 (no denominations that start with 2)
  4. $1, $3, $10, $20, $50, $100 (changing the $5 bill to $3 in USD)
  5. $1, $5, $10, $20, $50, $100 (current USD system)
  6. $1, $2, $5, $10, $20, $50, $100 (with all denominations that start with 2. x ≈ 2)

All denominations are paper bills. Coins are not considered. We set c = 1 second and b = 1.5 seconds. Since customers are slower than cashiers due to lower dexterity and wallets have lower productivity compared to cash registers, we increase c and b by 30% when the customer gives bills. Note that the time attributable to the customer includes both finding cash in the wallet and putting the cash in the right places in the cash register by the cashier.

Since c and b are the same for all six denomination systems, we are assuming that using $3 bills does not add any bill counting time or additional time to think about what bills to use. This assumption might be too unrealistic outside of East Asia as the time to thinking about $8 = $3 × 2 + $1× 2 might be much longer than thinking about $8 = $5 + $1 × 3 (although in the $9 case, $3 bills give an advantage). Another fragile assumption is that b does not increase with the number of denominations in the system. When there are many slots in the cash register (or wallet), it could take longer to locate a specific slot. However, incorporating these nuances in the simulation requires more empirical data to be realistic and it would make interpretation harder.

We assume that the transaction amounts follow a Log-Normal(μ = 2.7, σ = 0.9) distribution which has a mean of $22.31 and a median of $14.88. Since coin counting times are not simulated, the transaction amounts are rounded to the nearest integer \in [1, 200] (amounts higher than $200 are ignored). The customers and cashiers both use the combination of bills that use the least time and both have an unlimited supply of bills. This implies that for each payment, the least number of bills is chosen (e.g. customers don’t pay $10 with ten $1 bills). The results are presented below.

Rplot03.png

This bar chart clearly shows that when the number of denomination is fixed, a relatively constant inter-denomination ratio is better. Secondly, changing the $5 bill to $3 bill does save 3.8% of the (variable) transaction time. To relieve the added mental burden of thinking in multiples of 3 and 20 & 50, the second system (x ≈ 3) can be considered, which reduces the transaction time by 0.6%.

We have also tried a discrete Uniform(1, 200) transaction amount distribution. The mean transaction times are about 30% longer but the relative rankings of the six systems stay the same. When b is increased to 3, systems with a low x are penalized and the last system (x ≈ 2) loses its lead only when b is greater than around 3. The reason that the last system performs better than what the analytical solution suggested is that changes greatly reduce the number of different denominations used and thus lowering the significance of b. For the 200 prices (from $1 to $200), only 58 of them suggest an exact payment as the fastest payment under the last system. In fact, in most of the 200 prices, the customer pays using only 3 to 4 of 7 different denominations. The following table shows how much the customer should pay to minimize transaction time. The cells are highlighted if changes are required.

2131.png

However, we realize that the customer does not always have enough bills of every denomination in the wallet. Some bills are use so much that they tend to deplete quickly; other bills accumulate in the wallet since the supply (from receiving changes) is greater than demand. The following table shows the expected changes of the number of bills in the wallet after 100 transactions under the same log-normal distribution of the transaction amounts.

微信截图_20171029000345.png

One dollar bills accumulate over time if the most time saving payments are utilized. Generally, when the inter-denomination ratio is constant, the demand is more even across different denominations. The $20 to $30 bills are in most shortage because our distribution of prices is most concentrated in the $5-$20 range. This supports the usage of $20 bills in ATMs.

Wallet simulation

Since it’s not sustainable to always use the most time saving combination of bills, we have to do a more realistic simulation by keeping track of which bills are in the customer’s wallet. For each transaction, we find all the exact amounts (greater than or equal to the charge) the wallet can give and for each amount we use the most time saving combination. We then make the customer pay the amount that uses the least time; changes he receive are added to his wallet. The wallet has an infinite reserve of the largest bills since the customer can always withdraw from the bank.

The customer pays the best combination of bills as we analyzed earlier in only a third of the situations due to the lack of certain bills in the wallet. The wallet ultimately does not accumulate small bills after 3000 transactions. Consider a situation when the wallet accumulates 30 $1 bills. He will make payments exclusively using his $1 bills (when he’s out of $3 or $5 bills) to pay for small charges less than around $10. His wallet will eventually get thinner.

The mean transaction times of each denomination system is shown below. The relative values of the systems are similar to that in the previous section. Error bars are 95% (individual, not simultaneous) confidence intervals. The only pair without statistically significantly difference at 5% alpha level in mean transaction time is x ≈ 5 vs USD after controlling for family wise error rate. This means removing the $20 bill would not adversely affect efficiency, which is different from our earlier conclusion. We see that the current USD system can benefit from changing $5 bills to $3 bills and additional merging $20 and $50 bills to form $30 bills is beneficial too. Specifically, the gain in efficiency is 4.3% with a 95% (FWER) margin of error of 0.8%. Furthermore, the x ≈ 2 system remains the best and is 6.8% ± 0.5% more efficient than USD; therefore, the best improvement for the USD is actually restoring the $2 bill.

Rplot06.png

Varying costs

So far we have assumed the per bill counting time c and the fixed per denomination counting time b to be 1 and 1.5 respectively. However, these two costs are affected by many factors and I have not justified the set values using empirical data. Since these values might not be realistic, we would like to compute the efficiency of each denomination system with other possible values of the costs. Since only the relative cost between c and b matters, we fix c to 1 again and vary b from 0 to 5, which should capture any plausible values of it. Since all the mean transaction time data points are evaluated on the same 15,000 transaction amounts, the margin of error should be relatively small. The results are shown below. To clearly view the difference among the denomination systems, the mean transaction time shown is standardized by dividing (1 + b).

Rplot04.png

Consistent with our previous conclusion, x ≈ 2 is the most efficient denomination system for b = 1.5. This graph further shows that x ≈ 2 is best as long as b < 2. For b > 2, the x ≈ 3 system takes over. This is expected since a larger b favors a denomination system with fewer number of denominations .

ATM bill choice

Additionally, we can compute the efficiency of various types of bills supplied from ATMs or enterprises. If the ATM only dispenses $20 bills, consumers who only gain cash from the ATM will never use any bills larger than $20; therefore it can be modeled as abolishing the $50 and $100 bills. The following graphs compares the mean transaction time for different choices of ATM bills and denomination systems (assuming everyone gains bills from ATMs).

Rplot08.png

For all denomination systems, the mean transaction times are substantially lowered by using a bill smaller than $100 in the ATMs which is optimized at $20 or $30. This shows that fewer choices of bills can easily be beneficial. It also supports the current usage of $20 in ATMs in the US. The x ≈ 2 system remains most efficient for all ATM bills but the difference between $5 to $3 and x ≈ 3 shrinks at $20. Note that the results are heavily influenced by the choice of the distribution of transaction amounts. In our model we have been using a right skewed log-normal distribution  and 62% of the purchases are less than $20. Different countries should take the distribution of prices into account when designing denomination systems.

 

Weaknesses

  • The model did not consider street vendors who don’t charge taxes or use any coins which violates our assumption of always receive changes. The charges are often related to the bill denominations (e.g. $5 chicken over rice) which adds another layer of consideration. This also violates the smooth probability function of the log-normal distribution of transaction amounts. It’s efficient to (1) match the price with bill denominations or (2) make the prices cognitively nice numbers (e.g. $10). It’s best if these two can be achieved simultaneously. This is the case for most of the tested denomination systems except the two systems involving $3 or $30, which are more difficult to process than $2 or $5.
  • The model did not consider vending machines or other machines that accept small bills. Removing $2 bills can increase the frequency of $1 bills which can make vending machines more accessible (if they only accept $1 bills or the cost of developing machines that accept $2 bills is high).
  • The model did not consider the occurrences of small private payments in which both sides are usually consumers. For example, when someone wants to pay his friend $3 for using his metrocard, the chances of having exact change is increased when $2 bills are not used (this might not always be the case for all amounts).
  • The model assumes that cashiers have an unlimited supply of all denominations of bills. Having a limited supply would favor systems with fewer denominations.
  • The wallet has an unlimited capacity. In the main simulation, the wallet only occasionally contains over 30 bills but in the “varying costs” and “ATM” simulations it happened much more often which could make the results less valid.
  • Some parameters were assumed to be true without empirical evidence, such as the 30% added transaction time for the customers and the parameters for the log-normal distribution of transaction amounts.

 

 

R code

 

 

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