Do We Need a 37-Cent Coin?

Dubner thinks we should do away with the penny.
A young economist I know, Patrick DeJarnette, believes a much more radical change in currency is warranted. Here is what Patrick writes:

Late one night I was curious how efficient the “penny, nickel, dime, quarter” system was, so I wrote a little script to compare all possible 4-coin systems, with the following stipulations:
1. Some combination of coins must reach every integer value in [0,99].
2. Probability of a transaction resulting in value v is uniform from [0,99].
In other words, you start with $10 and no coins. You buy something at the store. Afterward, the chance you have 43 cents in your pocket is equal to the probability that you have 29 or 99 cents in your pocket (in addition to any bills).
Requirement (1) implies the penny is necessary, as you must have a combination of coins that reach value = 1 cent.
With this in mind, the current combination of coins (penny, nickel, dime, quarter) results in an average of 4.70 coins per transaction. What’s a little surprising is how inefficient our current setup is! It’s only the 2,952-nd most efficient combination. There are effectively 152,096 different combinations of penny + three coins. In other words, it’s only in the 98th percentile for efficiency.

How can you tell that Patrick is a young economist from the preceding discussion? Because he finds that the current government solution for the coins we use is 98 percent efficient and thinks this is inefficient. The other day I was walking through the halls of the University of Chicago economics department and heard a faculty member say that the right rule of thumb for government spending is that it is worth only 10 cents on the dollar because of inefficiency.
Anyway, Patrick then tackles the question of which combinations of coins would be most efficient:

The most efficient systems?
The penny, 3-cent piece, 11-cent piece, 37-cent piece, and (1,3,11,38) are tied at 4.10 coins per transaction.
But no one wants an 11-cent piece! There are other ways to look at efficiency; and given human limitations, this would result in a lot of errors and transactions would take more time.

• (1,4,15,40) is the first “reasonable looking” combination, with 4.14 coins per transaction.
• (1,3,10,35) also does well, with 4.16 coins per transaction.

But what if we restrict ourselves to “all coins (except pennies) are multiples of 5”? There are 18 different combinations that are more efficient than our current setup, (1,5,15,40) being the most efficient at 4.40 coins per transaction. Some other examples:

• (1,5,15,35) at 4.50 coins.
• (1,5,10,30) at 4.60 coins.

If we were to change just one of our current coins, what would be the most efficient?

• Changing the nickel to a 3-cent piece increases efficiency to 4.22 coins per transaction.
• Changing the dime to an 11-cent piece increases efficiency to 4.46 coins per transaction. (Although the 11-cent piece is unreasonable).
• Changing the quarter to a 30-cent piece increases efficiency to 4.60 coins per transaction. (Changing it to a 28-cent piece increases efficiency to 4.50, but that seems unreasonable.)

Therefore, changing the nickel is the most efficient thing. Not surprisingly, losing the dime entirely only costs us ~0.8 coins per transaction in efficiency; it does the least good of the existing coins.