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.

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The Five Goals of Lawmaking and Three Approaches of Combining Them

Making and executing ideal laws with the right penalty is an optimization process with debatable objectives. But at least we can list the five components of the objectives and three components of the costs in an attempt to model lawmaking. Later we can look for the prevailing approaches to combine these goals in ideal lawmaking and determine the origins of these approaches.

Goals of lawmaking

1. Change future behavior of criminals

This is often thought of as the goal of imprisonment. We would like criminals to have limited opportunities to commit another crime while in prison and less likely to commit crimes after they are released, which we attempt to achieve through enrichment programs in prison.

2. Compensate the victims

Victims can be financially compensated through fine or psychologically compensated through apology or by knowing that the criminals will be punished.

3. Deter people from committing crimes

When the punishment is severe enough, it is no longer worth it to commit a crime even when the probability of getting caught is small.

4. Improve the perceived fairness of society

Most people would like to live in a just society where people who harm others without permission are punished.

5. Make the criminals better people

Criminals are people too and many believe that the society should be responsible to help them by pulling them out of the wrong path and teaching them what’s right.

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