# 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.

## charles

I love it.

## Publius3

Getting rid of the penny now will be no worse than getting rid of the half-cent in 1857 - there were howls of protest, but eventually people learned to live without it, and then wondered why it was still being manufactured. Yes, prices were rounded up to the nearest cent, instead of the nearest half-cent. The U.S. economy did not fall apart.

## Ty

SO academic. How about no pennies, round to the nearest .05? It works in Monopoly...

## Greg

How can you tell that this young economist is American? From the fact that he does not question the assumption that dollar bills should be retained in place of dollar coins (or perhaps 123-cent coins...)!

## jch

The 37 cent coin would give us a golden opportunity to finally put Nixon on a coin (our 37th president).

## Chris Manly

I think you're missing one thing that, at a practical level, would be a requirement (or at least strongly desirable): you should be able to come up with an even dollar with any single coin denomination. (4 quarters, 10 dimes, etc.) Of all the alternate denominations you listed as possible, only the 4 cent piece would fit that description.

I think any coin that didn't fit that criteria would and should be shot down.

## SP

there must be a reason why the Euro has 1, 2, 5, 10, 20, and 50 cents coins. The fallacy of this approach is that it starts with the inherent limitation of 4 coins in circulation.

One should instead solve for the optimal coins per transaction (the absolute minimum, as shown, might not be practical to implement, e.g., 37c coins etc).

## ifidontmind

Its interesting that these combinations -- particularly the most efficient ones -- approximate an exponential curve. It's just a hunch, but I'd bet that in general, the optimal solution for any problem like this is an exponential (allow, say, a different interval than [0,99], and/or a different number of coins).

## James

What is the efficiency gain of ditching the penny entirely and only having to deal with multiples of five? Is the most efficient set in that case still (5, 15, 40)?

## qingl

I don't think that for requirement 2 the distribution is uniform. I could be wrong but I'd like to see more rigorous proof of that.

Good first order approximation though and lots to think about. Well done.

## LG

Am I the only one whose first reaction was that it's amazing and great that our current system of coins that (a) are super easy to count and play well to our 5-finger-centric brains, (b) all can add to a dollar evenly (whereas 3 or 11 cent pieces cannot), (c) were chosen without an exhaustive computer search of all coin combinations, still managed to be in the 98th percentile of all possible choices and well within a full coin of the optimum? Wow, good work, guys!

## LG

This wouldn't be complete without an economic analysis of the enormous cost to our society (in time, effort, and errors) of EVERY freaking transaction involving arithmetic of quantities that are harder to add and subtract in your head.

## ifidontmind

@SP -- thanks for the reminder about the Euro. The "1, 2, 5, 10, 20, 50... " is a common approximation to an exponential series, since it roughly doubles with each step but also sticks to relatively round numbers (I think the Euro also continues with 1 and 2 euro coins, 5 and 10 euro notes, etc.). This would be consistent with my earlier hypothesis the exponential is probably optimal.

## Jason

There seems to be no way to get rid of the penny, as much as many people think we should. Can we consider the effect on efficiency of transactions using both coins and paper currency when replacing paper dollar bills for $1 coins? Or even adding $2 coins? Does adding these coins while eliminating their respective bills increase or decrease efficiency?

## KB

I'd add one more stipulation. A type of coin must divide evenly into a dollar.

## keith

Why is the measure of efficiency quanta and not mass? I know from hauling coins to the bank to be counted that pennies are not so efficient and dimes are very efficient! The dime, quarter, Kennedy 50 cent, and Eisenhower dollar all have the same mass-to-value ratio (0.2268 grams per cent), probably as a remnant from when many of these coins were cast in silver alloy. Sacajawea and Presidential dollars are even more efficient, at 0.081 g/cent.

Pennies and nickles are heavier (2.5 g/cent and 1 g/cent).

These are more efficient both from the point of view of the mint, that has to buy raw materials, as well as users, who have to carry around the coins.

## Brad

So is it really a good assumption that all change amount from 1 to 99 cents are equally likely? I don't know about you all, but it seems like everywhere I look things cost $x.99, where x is some positive integer, so it seems like a non-uniform distribution should be considered for different cent values in your pocket.

Sort of reminds me of when I went to the diner the other day and the guy in front me was amazed that we had the same total, even though we had order two different things (for two people), but if you looked at the menu almost everything is priced 9.99, so it really should not be that big of a shock...

## Ka Keng, Lee

Perhaps Patrick could compute the efficiency of a penny-less system as well and compare it to the ones above?

## Mike Scott

You should just forget about this independence business, and go back to pounds, shillings and pence, with 240 pence to the pound. It makes it much easier to find sets of coins that are both sensible-looking and efficient.

## ScotterOtter

I would prefer the European model of mandatory listing of prices that includes tax. Also, I noticed on my recent trips that most prices are rounded to the nearest .10 Euro or, if not, to .05 Euro. I rarely saw .99 like we do here so often in the US

You always see this at places where it's difficult to make exact change and the need for speedy service ( ie. sporting event concessions, taxes are included and rounded to the nearest $.25)