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On Optimal Denominations for Coins and Currency

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On Optimal Denominations for Coins and Currency More on Optimal Denominations for Coins and Currency Mark A. Wynne February 1997 Research Department Working Paper 97-02 Federal Reserve Bank of Dallas This publication was digitized and made available by the Federal Reserve Bank of Dallas' Historical Library ([email protected]) More on optimal denominationsfor coins and currerrcy" Mark A. Wynne ResearchDepartrnent FederalReserve Bank of Dallas 2200 North Pearl Street DallasTX 75201 October,1996 Revised:February 1997 Abstract: Telser (1995) has shownthat the problem of BAchethelps answerthe questionof the optimal denominationalstrucnrre of currency in the U.S. and U.K. This note provides further evidenceto supportthis claim using cross-countrydata. Keywords: Currency denominations;problem of Bichet. IEL classification:E42 .Correspondence to: Mark A. Wynne, ResearchDeparEnent, Federal ReserveBank of Dallas,2200 North PearlStreet, Dallas, TX 75201. Phone:214-V22-5159. Fax: 214-922- 5194. Email: [email protected] thankCarrie Kelleher for her assistancein preparing this note and a refereefor comments. The views in this paper are thoseof the author and do not necessarilyreflect the views of the FederalReserve Bank of Dallas or the FederalReserve System. 1. Introduction In a recentcommunication Telser (1995) arguedthat the problem of Bichet can shed light on the problem of choosingdenominations for coins and currency. The relevantversion of the problem of BAchetis the one that seeksthe smallestnumber of weights capableof weighing any unknown quantity up to someprespecified amount to a given degreeof accuracy,using a two pan balarrceand allowing the weights to be placedin either pan. The solution to the problem is weights that are powers of three, and a systemof t suchweights allowsone to weighany quantityup to (3t.r-1)/2. The analogywith the problemof choosing denominatiorsfor coins and currency is obvious. The unknown quantity to be weighedcan be viewed as the nominal value of a cashtransaction; allowing the weights to be placedin either pan correspondsto the ability to make change. Tetserargues that a "proclivity for the decimal system" meansthat the optimal denominationsrnay deviatefrom thoseobserved in reality, but then goeson to show that for the denominationsof U.S. coins and currency that circulate (i.e. excluding the 50 cent coin and the $2 bilD the face value of eachdenomination is on average three times the face value of the denominationbelow it. 2. Currency Denominations Around the World. This striking conformity betweenthe predictionsof the simple theory and what we observein the U.S. currency systemraises the questionof whether we seea similar conformity when we look acrossa larger group of countries. A usefrrl sourceof data for addressingthis questionis the Stalesman'sYearbook (Htnter (1994)), which includesamong other information about countriesdetails on the units and denominationsof eachcountry's currency. An important caveataccompanying this data is tlat we cannotidentify denominationsthat are issuedbut do not circulate. A cursory analysisof the data (which is too spaceconsuming to be presentedhere but is availableon requestfrom the author) revealsthat the so calledbinary-decimal system (consistingofthetriplets {0.01,0.02,0.05}, {0.10,0.20,0.50}, {1,2,5}, {10,20,50}, {100, 20O,500} etc.) is by far the most prevalentsystem of currency denominationsthat we observe. For 20 of the 156 countrieslisted, the binarydecimal systemcompletely characterizesthe currency system.r For another42, the binary-decimaltriplet appearsat least rwice in the cunency system. The binarydecimal triplet appearsa lot more frequently than thefractional4ecimal tiplet {1, 2.5, 5}, and none of the countriesin the samplehas a denominationalstructure based solely on the fractionaldecimal triplet. The counfiies that comeclosest are Lebanon(which issuescoins of 1,2.5, 5,10, 25, and50 piastres,and notes of 100,250, 500, 10,000,25,000 and 50,000 Lebanese pounds, as well asnotes of0.1, 0.5, 1,000,5,000 and 100,000l:banese pounds),Madagascar (which issues coins of 10, 25, 50, 100,and 250 Malagasyfrancs, and notes of500, 1,000,2,500, 5,000, 10,000,and 25,000 Malagasyfrancs, as well as coinsof 1, 2, 5, and20 Malagasyftancs), and the Netherlands (whichissues coins of l, 2.5, and5 guilders,and notes of 10, 25, 50, 100,and 250 guilders, as well as low denominationcoins of5, 10, and 25 centsand a high denominationnote of 1,000 guilders). For another10 of the countriesthe currency systemis basedexclusively on rThesecountries are Argentina, Australia, Bulgaria, Colombia, Fiji, France, Ghana, Gibraltar, Honduras,Italy, Mexico, Mongolia, New Zealand,Papua New Guinea,Solomon Islands,South Africa, Swaziland,Tonga, Uruguay, and Western Samoa. thedecimal pair {1, 5} andmultiples thereof (i.e. {0.01,0.05}, {0.10,0.05}, {1, 5}, {10, s0),{100, s00}, {1000, 50oo})., In light of Telser's argumentit is striking that only 5 countrieshave denominationsthat are either powers or integer multiples of three: Albania issues3 lek note, the Bahamasissue a 3 Bahamiandollar note, Cuba issues3 pesonote, Romaniaissues a 3 bani (fractional unit) coin, and Russiaissues a 3 rouble note. Burma issuesnotes at the 15 kyat, 45 kyat and 90 kyat denominations.Before proceeding,we might note that the comparativerarity of notesor coins at denominationsthat are powers or integer multiples of three doesnot seemto characterizecurrency systemsof the past. For example,tlrc ftactional cwrency issuedin the U.S. during and after the Civil War included notesat the 3c and 6a denominations. Likewise, almostevery issueof Continentalcurrency in the U.S. during the RevolutionaryWar irrcluded notesat the $3 and $6 denominations,most issuesincluded a note at the $30 denomination,the issueof February17 , 1776included notes at the $1i6, $l/3, and $2i3 denominations,while the last issue(of January 14, 1779)included notes at the $45 and $60 denoninations. There were also numerotlsissues of colonial currency at denomimtionsthat were integer multiples of three.3 2These countriesare Chile, Comoros, Iceland, Japal, Korea, North Korea, Norway, Paraguay,Taiwan, and Yugoslavia. 3SeeFriedberg (1995) for an htroduction to denominationsof U.S. papermoney. Theseobservations raise the interestingquestion of why the currency systemin the United Stat€sevolved away ftom thesedenominations to its current structue. A refereepoints out that a 3-mark coin was also issuedin Germanvn 1y24. So how well doesTelser's argumentwork when we look acrosscountries? Using the data on currency denominatiors,for eachcountry I calculatedthe ratio of the face value of eachdenomination to the one immediatelybetow it and then calculatedthe averagemultiple for eachcountry. Thus for a country with a denominationalstructure based exclusively on the binarydecimal triplet with, say, four completetriplets appearing,the sequenceof denominationswould be lc, 24, 5c, 104, 204, 50c, $1, $2, $5, $10, $20, and$50. Each denominationhas a face value equal to on average2.2 times the face value of the denominationbelow ir (i.e. Q+2.5+2+2+2.5+2+2+2s+2+2+2.5)l|l). Notethat the averagemultiple for a systembased on the fractionaldecimal triplet with four complete tripletsappearing would alsobe2.2, arf, either2.1 or2.3 ifthe systemincluded incomplete triplets.4 A currency systembased on the decimal pair {1, 5} would have an averagemultiple of 3.5 or 4 dependingon whetherthe systemincluded cornplete or incompletepairs. Figure I is a plot of the histogramsof the averagemultiple for all countriesand for the subsetof OECD countries. What is remarkableis that the arithmetic meanof the average multiples acrosscountries is exactly equal to three as predictedby Telser! However the histogramalso revealstlat the distribution is not concentratedaround the arithmetic mean, and if anything seemsto be bi-modal with peaksat 2.2 and2.7 . T-heformer value would be the meanof the distribution if all countrieshad denominationalstructures consisting of four binary-decimaltriplets, while the latter would be the meanif all countrieshad denominational structuresconsisting of four binarydecimal triplets, plus two more denominationsat 100 units 4Telser(1995) points out that denominationsthat are powers of two would be optimal if all transactionshad to be conductedwith exact chanee. and 1000units. While it is comparativelyrare for two denominatiorsto b€ separatedby a factor of 10, it is not uncommon:thus in Canadathe two highestdenominations are the C$100 and C$1,000 notes,while in Israel the two lowest denominationcoins are the 5 and 50 agorot. In no fewer than 17 countriesdo we find two denominationsseparated by a factor of l0 or more, and Vanuatuhas the distinction of a currency systemwherc the highestdenomination coin is the 1 vatu, while the lowest denominationnote is the 100 vatu! PanelB of Figure I showswhat happenswhen we excludecountries with large *gaps" in their denominationstrucfirre from consideration(defined as thosecountries with at least two denominationsseparated by a factor of ten or more), and panelsC and D presentthe sameinformation for the subsetof OECD countries. Excluding the outliers only marginally reducesthe various measuresof central tendency. 3. Conclusions The evidencepresented here lends supportto Telser's (1995) argumentthat the problem of Bechetprovides insights into the issueof optimal denominationsfor currency, but also suggeststhat the observeddenominational structures reflect other considerations. For example,the problem of Bichet ignoresthe fact that mentalcalculations seem to be easierwith
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