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Elasticity:ComprehensiveCalculationsIngeneral,elasticityisthepercentagechangeinsomethingrelativetothepercentagechangein

somethingelse. Becausethechangeismeasuredintermsofpercent,unitsofmeasurearenolonger

relevant.

Thereareanumberofwaystocalculateelasticity. Forsimplicity,Iwillfocusontheprice

elasticityofdemandonthissheet. Theonethingtorememberaboutpriceelasticityofdemandisthat

becauseyouaremeasuringthesensitivityofanegativerelationship(recallthelawofdemand)the

resultingnumberisgoingtobenegative. Therefore,itiscustomarytoreporttheabsolutevalueofthe

priceelasticityofdemand. Iwillexplicitlyshowtheabsolutevaluefunction.

|%QD/%P|

StartingPointElasticity:Inmanypracticalapplicationsofelasticityitisofinteresttoknowwhatwillhappenifthereisa

changefromastatusquo. Forexample,howmuchwillsalesincreaseifyoudecreasepricebyten

percent? Therefore,itisappropriatetocalculateelasticityrelativetoacurrentstartingpoint.

ExamineFigure1below:

Figure1

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Price

Quantity

Elasticity

Elasticity

ImaginethatcurrentsalesareatQuantity=40andPrice=15(40,15). IfIdecreasepriceto10,sales

wouldincreaseto60. LetQ1=StartingQuantity=40andQ2=EndingQuantity=60. LetP1=Starting

Price=15andP2=EndingPrice=10.Thepercentagechangeinquantitydemandedismeasuredrelative

tothestartingquantityof40:%QD=(Q2Q1)/Q1= (6040)/40=20/40=0.5or50%.Thepercentage

changeinpriceismeasuredrelativetotheStartingPriceof15:%P=(P2P1)/P1= (1015)/15=5/15=

0.33or 331/3%. Thepriceelasticityofdemandstartingat(40,15)is|50%/331/3%|=1.5.

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Startingpointelasticityismostbeneficialtothosepeoplewhoareconsideringachangeina

currentsituationandwouldlikeawaytoanticipatetheresultsofaparticularchange. Theonereason

whysometextbookauthorsdonotlikestartingpointelasticityisthatitisdependentonwhatyou

chooseasyourstartingpoint. Noticeintheexampleaboveifwechoosethepoint(60,10)asour

startingpointandmoveto(40,15)themeasureofelasticitychanges:

|(4060)/60/(1510)/10)|=0.66.

Therefore,oneapproachtoavoidthispotentialconfusionistointroducemidpointelasticity.

MidpointElasticity:Someauthorsliketointroducethemidpointelasticityapproachtoavoidtheproblemofgetting

differentmeasuresofelasticityifstudentsusethewrongdenominatorwhencalculatingpercentage

changes. Furthermore,ifyouwantasimpledescriptionofconsumerbehavioroveraportionofa

demandcurveratherthanatasinglepointonthedemandcurve,midpointelasticityisappropriate. The

onlychangefromaboveistoredefinethepercentagechangestobeintermsoftheaveragepointbetweenthestartingpointandtheendpoint. Fromtheexampleabovewecanseethattheaverage

quantityis:(Q1+Q2)/2=(40+60)/2=50. Theaveragepriceis:(P1+P2)/2=(15+10)/2=12.5. Theformula

forthemidpointpriceelasticityofdemandis:

|%QD/%P|=|(Q2Q1)/((Q1+Q2)/2)/(P2P1)/((P1+P2)/2)|

Thetwoscanceltocleanuptheequationalittle:|(Q2Q1)/(Q1+Q2)/(P2P1)/(P1+P2)|

Thenicethingabouttheformulaisthatitdoesnotmatterifyouconfusethestartingpointandtheend

point. Themathematicalresultwillbethesame. Considerthepointsweusedabove,(40,15)and(60,

10):

|(6040)/50/(1015)/12.5|=1

|(4060)/50/(1510)/12.5|=1

Itisworthmentioningthatthenewmeasureofelasticityisdifferentfromthetwoprevious

measures. Noticethatitisnotsimplytheaverageofthetwo.

ArcElasticity:Arcelasticityissimplyavariantofthemidpointelasticitybutisconvenientifyouareworking

withanonlinearfunctionandyouhavenotyettakencalculus(gotakecalculusifyouhavethechance

itimpressesallofthepeoplewhoareafraidofmathclasses).

Initssimplestform,arcelasticityisexactlylikethemidpointmethod. Thedifferenceisthatthe

demandcurveofinterestisactuallycurved. ThedemandcurveinFigure2isaconstantelasticity

demandcurve. Itconvenientlygoesthroughthesametwopointswehavebeenexamining. Therefore,

youcanseefairlyquicklythatifyouweretoevaluatethepercentchangequantitydemandedrelativeto

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thepercentchangepriceoverthesametwopointsyouwillgetthesamemeasureofpriceelasticity. It

isworthnoting,however,thatasyouworkyourwaytowardeitheraxisthecurveflattensout. Thisis

whatallowsitselasticitytoremainconstant.

Figure2

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Linear

ExactPointElasticity:Oneofthewayseconomistsmakealivingisbycalculatingelasticities. Becauseunderstanding

elasticityissoimportant,ownersoffirmsbothlargeandsmallarewillingtopayhandsomelyfor

informationaboutthesensitivityofthedemandtheyface. Wheneconomistsconductempirical

researchtheyexamineaproblematthemargin. Thatis,theylookatthesensitivityofdemandtochangesinpricewithinaverysmallrangerelativetotheobservedpriceandquantity. Theexactpoint

elasticityformulaprovidesameasureofpriceelasticityofdemandthatrelatestoonepointona(linear)

demandcurve. Formymoney,itistheeasiestofallofthemeasuresofelasticitytorememberanditis

themostuseful. But,itdoesrequireaslightlydeeperunderstandingofdemand. ReferbacktoFigure1.

Noticethatataparticularprice(25)thequantitydemandedgoestozero. Refertothispointasa.

Someeconomistscallaachokepricethepriceatwhichalldemandischokedoff. Atanygivenprice,

P,thepointelasticityofdemandissimply:1

|P/(Pa)|

Quicklycheckthreepointsofinterest:

AtP=15:|15/(1525)|=11/3

AtP=10:|10/(1025)|=2/3

1Notethatthequantitydemandedisnotneededtocalculatetheexactpointelasticity. Thisisbecauseallofthe

informationthatisnecessaryiscapturedintherelativepositionofthecurrentpriceandthechokeprice.

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AtP=12.5:|12.5/(12.525)|=1

Noticethatwegetthesameanswersasthestartingpointmethodinthefirsttwocasesand,byusing

themidpointprice,wegetthesameanswerasthemidpointmethodinthelastcase. Therearesome

additionalnicethingsaboutusingtheexactpointelasticity. Forone,itremindsusthatasprice

increases,elasticityincreasesandaspricedecreases,elasticitydecreases. Anotherniceaspectabouttheexactpointelasticityisthatitisanalogoustousingmoreadvancedcalculusapproaches. Toseethis,

consideranonlineardemandfunctionsuchastheoneinFigure2. Now,examineanarccreatedby

squeezingtwopointseverclosertogether. Oncethosetwopointsbecomeonepoint,drawatangentto

thearcatthatpoint. Thetangentrepresentsthestraightlinedemandfunctionatthatpoint. Theexact

pointelasticitygivesyoutheelasticityatthatpoint.