# 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.

#### MikeM

Surely the ease of making change counts for something! How quickly can you add up coins that equal 83 cents in your head? Our system isn't that bad.

#### steve staudaher

how interesting

I find this measure of efficiency somewhat suspect. It is pretty hard to give out 4.7 coins in a transaction. None of the solutions gets to 4 coins or less, so I think this discussion is somewhat moot.

The most reasonable thing I have heard on currency simplification is to get rid of the penny, nickle, and the dime! The main reason is because the Lincoln penny when first issued was worth about 25 cents in today's currency.

The paper dollar lasts about 18 months in circulation, the dollar coin would be good for about 20 years. About half the currency printed is the one dollar bill.

#### Carl Zetie

You guys are lucky you didn't grow up in the UK pre-1971. As I recall, the coins were:

Farthing (one quarter of a penny!)
Halfpenny (pronounced "hayp-nee")
Penny
Two penny (pronounced "tuppeny")
Three penny (pronounced "thruppeny bit")
Sixpence (also called a "joey")
Shilling (equal to 12 pence, also called a "bob")
Half a crown (two shillings and sixpence; also pronounced "two and six")
[For some reason, there was no crown (five shillings), or at least not in common circulation.]

After that there were bills (notes):
Ten shillings (known as a "ten bob note", and equal to half a pound)
One pound (equal to 20 shillings, or 240 pence).

The British equivalent of the phrase "\$3 bill" was "nine bob note".

Decimalization was a great relief, I can tell you.

#### Ryan Huebner

Interesting. However this is all assuming that \$1 is the ideal note to achieve. Wouldn't it be more economical to find the average small purchase amount and change that to be our dollar note?

#### Joe D

1, 5, 25 also approximates the exponential (1x5 = 5, 5x5 = 25, 25x4 = 100), which is why dropping the dime and going to a three-coin system is nearly as efficient.

I do like the Euro 1, 2, 5, 10, 20, 50, 1€, 2€ coinage.

An improvement on Patrick's question of how many coins are required to *pay* is to ask how many coins I need to *carry* so that I can pay with the most efficient combination. With our current system, if I carry 3x25, 2x10, 1x5, 4x1, I can always pay with the fewest coins, and I only have to carry ten. With the Euro system (thinking only of the six coins under 1€), I have to carry 1x50, 2x20, 1x10, 1x5, 2x2, and 2x1, so I only have to carry nine coins. With {1, 5, 25} that I mentioned above, I have to carry 3x25, 4x5, and 4x1, or eleven coins. I wonder if the ratio of nearly two (how many I have to carry/average efficiency) is close to fixed. There's also the question of how many transactions I expect to make between trips to the coin dish at home.

#### Brent Edwards

I recommend introducing the "deficent": a coin with value -1 cent.

A purchase of \$4.99 could be made with a five-dollar bill and one deficent. A purchase of \$9.99 could be made with a ten-dollar bill and one deficent.

And best of all, I can save up millions of deficents, send them to someone, and bankrupt them!

#### Michael

I think we'd be surprised how quickly we'd be able to learn the value of various combinations: two 37-cent pieces and a penny make 75 cents, four 37-cent pieces make \$1.11, etc.

#### alex

Can anyone explain to me where the 152,096 number comes from. That seems pretty low to me. My first inclination is that there would be 99*98*97=941,094 possible combinations.

#### conor

Yes, unfortunately patrick is trying to minimize coins needed, whereas the coins in circulation provide many more functions which are incompatible with this objective. Chief amongst these is that each coin must divide evenly into \$1.

#### Steve

I'm surprised 1,3,10,30 wasn't highlighted, it's the most efficient system 4 coin system with a regular pattern from decade to decade. And why stop with cents? \$1, \$3, \$10, and \$30 would be more efficient than \$1, \$5, \$10, \$20

#### BMF

Building on earlier comments, not only does our current system provide multiple methods to add up to a dollar, but also each smaller denominated coin, or relatively easy combination of smaller denominated coins can be added to the next highest denomination (i.e. 5 pennies = 1 nickel, 2 nickels = 1 dime, 2 dimes and a nickel - 1 quarter, 4 quarters = 1 dollar, etc.).

Secondly, wouldn't the elimination of a penny effectively cause inflation on a micro scale? Even with accurate rounding, if an item that is now valued at 3 cents would now cost 5 cents, why wouldn't that same premium be added to items previously valued at 5 cents (i.e price increase to the next highest level, or 10 cents.

Finally, what the analysis, although interesting, doesn't take into consideration, is what I would guess is an aversion to paying with correct change. A transaction that would utilize 5 coins (a quarter, a dime, a nickel and two pennies, for instance) would easily be replaced by two quarters and receipt of a nickel and 3 pennies.

#### KarenS

By the time we finally settle on a better system, all monetary transactions will be electronic.

#### Nick

It would be intriguing for a respected young economist to get data from some of the leading chain businesses with face-to-face transactions under \$10 to give up a sample of their data - stripped down to take out the dollar value, just the number of cents. Getting a sample of say 100,000 transactions each from a few dozen mini-marts (both in gas stations and in airports) and quick-service restaurants would produce a serious sample from which to challenge the assumptions in the posted analysis.

But for me, the question has become moot over the last 3-4 years. A preponderance of my sub-\$10 transactions have migrated to rewards-based credit cards. I only do cash transactions now at places like my dry cleaners, and just enough to have \$1 bills for tipping the guy at the parking lot.

#### jacob

The US once had a 3 cent piece. the public hated it.

I'll also point out that the US currently has a 50cent piece. the public hates it.

The smallest Euro coin is 5 cent piece. i used a Euro 5 cent at the dutry free store and was given Euro 2 cents off my bill for it, i didnt bother asking the cashier if it was a typo or actual value.

#### jacob

correction - there is a euro one cent and euro 2 cent, but i only learned that from reading this blog and not from my 2 week trip to europe last month

#### Patrick DeJarnette

Charles: Thanks!

Publius3: I don't feel strongly about the penny, although I wasn't aware of the half-cent's existence.

Ty: I like to think of it as exercising my curiousity. :) For no pennies, round to the nearest 0.05, there are still 18 more efficient coin systems.

Greg: Actually, a portion of the original email to Steve did focus on bills - the (1,5,10,20) system for bills is analagous to (1,5,10,25) for coins. I wasn't so focused on bills vs. coins though.

Jch: Found that pretty clever.

Chris Manly: That's a good suggestion. The only factors of 100 are 1,2,4,5,10,20,25,50, and 100. The most efficient combination of coins in that case is (1,4,10,25) - which decreases coins per transaction to 4.26.

SP: I did branch out into changing the number of coins. One issue is that increasing the number of coins allowed will always decrease the number of coins per transaction (if you are optimizing properly).
In other words, we could have 99 coins, the 1 cent, 2 cent, 3 cent, etc. which would push the number of coins per transaction down to 1. I mostly held fixed the number of coins because it seemed fair to compare the US currency system to other coin systems, holding the number of coins fixed.

IfIdontmind: That's an interesting observation. At first I attempted a general solution but quickly opted for a simulated one. You are probably onto something, however.

James: Yes, that is still the most efficient set at 2.4 coins per transaction as compared to 2.7 if we kept (5,10,25).

qingl: I'm using requirement 2 as an assumption. I agree that the actual distribution is likely not to be uniform. Please let me know if I've misunderstood.

LG: On comment 1, the only factors of 100 are 1,2,4,5,10,20,25,50, and 100. Given your requirements (a) and (b), I believe the set of possible systems is pretty slim.
On comment 2, I agree completly. I don't think enacting these systems would be efficient for those very reasons. This was more of a hypothetical "if humans could compute costlessly".

Jason: If you exchanged \$1 bills for \$1 coins, then you would have an increase in the number of coins, surely, but there would be no effect on the total number of bills+coins that you would carry, unless I'm missing something.

Keith: My original analysis (not included above) did look at the cost of printing money, but I hadn't considered the weight! I think that's a pretty interesting problem...

Brad: Yeah, the entire thing is predicated on the uniform assumption. I agree that in many situations, the assumption of uniform is violated. That is just one of the many reasons I considered this a flight of fancy rather than a serious proposal. For example, companies that have few goods will probably tend to price appropriately to minimize change, and as you mention, the \$x.99 effect.

Ka Keng, Lee: I did originally. There are 18 more efficient penny-less systems (using "round to the nearest 0.05") than if we removed the penny today. See my reply to James above.

Mike Scott: We'll take it under consideration. :)

ScotterOtter: I agree with both of your points. The latter in particular helps demonstrate why the uniform assumption is not valid.

MikeM: I agree completely, but I also consider a system in which all coins are multiples of 5 in the article. Furthermore, see my reply to Chris Manly above - (1,4,10,25) does rather well compared to (1,5,10,25) and all are factors of 100.

Steve Staudaher: Each individual transaction has an integer number of coins, but the transactions have a distribution. 4.7 coins referred to the number of coins when averaging across all different transactions.

Carl Zetie: Thank you for the rundown on UK currency. I hadn't heard of these terms before (my favorite is "thrumppeny bit").

Ryan Huebner: You are correct that this is assuming the \$1 is the target. I suspect you may be right in your suggestion.

Joe D: I hadn't considered that. I was mostly thinking about getting change back, but I like your question as well. I haven't thought about it much though.

#### Gary

How about we mandate a change to base 12 while we're at it?

#### Bobby G

Good read. I'll have to think about other non-mathematical factors that may play into our current system.

Not to mention the cost of changing out all existing coins would be rather expensive, possibly reducing any value gained from the switch from the get-go.

Could be one of those cases where the best outcome is to the west, but we are so far to the east that going west would be too expensive.

Why use coins?

#### spencer

I like the comment above about fractional coins in making a transaction.

I would like to see the math recalculated so that you are only dealing in whole coins.

Second, the number of coins is one form of efficiency, but there are other forms.

How about the efficiency of the overwhelming number of transactions being in multiples of five that are very easy to do in your head. Once you start trying to do calculations in units of \$0.37 it gets extremely difficult to do it quickly in your head.
Consequently the number of incorrect math operations at the point of exchange will expand exponentially. I suspect the the inefficiency of this would massively offset any efficiency gains proposed in the article.