20¢ = 25¢

Hailing from Australia, I've often been puzzled by American currency mores. Why "nickel" and "dime"? (I still can't remember which is which after a decade in the U.S.) Why is a dime (10¢ if you're wondering) smaller than a nickel (5¢)? What do Americans have against their 50¢ and $1 coins? Why do all banknotes look the same? (A practice that has been ruled to be in violation of the Disabilities Act.) Why don't Americans use $50 and $100 bills?

Besides all of these questions, I've wondered whether quarters (25¢) or 20¢ coins (as we have in Australia) are more efficient. I define efficiency for a system of coinage as requiring the fewest coins for an arbitrary amount of change between 1¢ and 99¢. Although Australia withdrew 1¢ and 2¢ coins from circulation in 1992, I made the comparison as close as possible by looking at a system with 1¢, 5¢, 10¢, 20¢/25¢, and 50¢ coins. The result, as the title of this piece suggests, is that 20¢ and 25¢ coins are equally efficient--the average number of coins required for amounts of change between 1¢ and 99¢ is 4.24 in both cases (SD is 1.71 in both cases).

As I mentioned earlier, though, Americans have a strange aversion to 50¢ coins, while pre-1992 Australia had 2¢ coins. If we take this into account, the average number of coins required for change in the U.S. rises to 4.75 while the Australian average falls to 3.43, meaning that you get on average more than a coin extra in change in the U.S.

The next step, of course, is to ask whether it is possible to do better than 5¢, 10¢, 20¢/25¢, and 50¢ coins altogether. Might we in fact better be served by 4¢, 12¢, 37¢, and 74¢ coins with respect to minimizing change? (Actually, yes. This would require 4.02 coins on average.) I've cleared most hurdles to specifying this problem in AMPL (a mathematical programming language for optimization problems) and will return with my "rational" scheme for change in the near future.

Of course, not all amounts of change are created equal. In currency regimes with 1¢ coins, prices tend to end in nines (Basu 1992, 2006; Demery and Duck 2007), while in Australia prices end in fives. Ideally, we would weight the number of coins by the probability of receiving change of that amount in a transaction. However, a cursory search has not revealed a useful dataset for these calculations.

References

Basu, Kaushik. 1997. "Why are so many goods priced to end in nine? And why this practice hurts the producers." Economics Letters 54:41-44.

-----. 2006 "Consumer cognition and pricing in the nines in oligopolistic markets." Journal of Economics & Management Strategy 15:125-141.

Demery, David and Nigel W. Duck. 2007. "Two plus two equals six: an alternative explanation of why so many goods prices end in nine." Discussion Paper No. 07/598, Dept. of Economics, University of Bristol, Bristol, UK.)