## 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.