GuadagniandLittle:ALogitModelofBrandChoiceCalibratedonScannerData

44

Table2

AlthoughCustomers,theAttributeValuesResponseModelCoef?cientsAretheSameforAll

atTimeCanofBePurchaseQuiteDifferentDependingonHypotheticaltwoproductmarket:AandB

LoyaltyPricecoef?cient:7.0

Noothercoef?cient:attributes.

?30oz/dolPriceLoyaltyresponsePricetoA=ofloyaltycustomertoBwithequalloyaltiesProbabilityofA=ofpricechoosingB=0??A15=0??5=dol/oz

e??3??5?4??5??/??e?1+e?110%??=0??5ProbabilitypricereductionPriceelasticityofchoosingofAforA=?Ato2=??2

00.135??61dol/ozPriceLoyaltyresponsePricetoA=of0customer??8,loyaltywithtoBunequal=0??loyaltiesProbabilityofA=priceofB=0??15dol/oz210%ProbabilitypricereductionofchoosingofAA=to00.135??985

dol/ozPriceelasticityofchoosingforA=?A0=??06

0??991theuct.productshasanelasticityof?0.06tothatprod-action.ThisourAnalogously,isanexamplealthoughofthetheS-shapeofFigure2inforularalleight-productbrand-sizes,marketthemarkethavethecontrolsamevariablescoef?cientsinbrand-sizesproductwilldependontheresponseactionsforoftheapartic-othercustomers.

andthedistributionofloyaltiesacrosstheresponseAnalytically,determinationofaggregatemarketresponsetomerfunctioncallsforoveranintegrationajointdistributionofthecustomerofbrand.sionalOneloyalties,standardprices,approachandpromotionstosuchaforeverycus-andcustomers,easierintegrationmethodistoMonteimplementCarlosampling.highistouseAthesimplerdimen-actualdata.timeperiodsandattributevariablesofthesizeingbyBy1%changing,overthesay,wholetheregularpriceofabrand-aggregatethecorrespondingtimeperiodandobserv-sharechangeinshare,weobtainanSomeMany(2)examplesdifferentresponseareresponsetoregularprice.

(1)calculationsarepossible.another’sthecrosselasticityregularofonepricebrand-size’selasticityofshareshare,to(4)promotionthecannibalizationprice,(3)shareofoneresponsesizeofatopromotion,duringapromotion.

ofanother,and(5)theelasticitybrandofbypricetheeffect,Eachpurchasei.e.,ofastheseanalterationcanbeevaluatedasashort-termorfutureasaoccasiononwhichofchoicethevariableprobabilitiesischanged,atthehereimpactlong-termofchangedeffect,takingloyalty.alsoWerestrictintoaccountthepricetoelasticity,threeexamplespromotionofshort-termourselvesresponse,response:andpromotional

regularMarketingScience27(1),pp.29–48,?2008INFORMS

Table3

Short-TermVariables

MarketResponsetoThreeDifferentControlRegularfromShareprice(depromoted)promotionincreasePromotionalpricesharePurchase(%)

ofelasticityshare

pricewithmediancut(%)

elasticitycutofshareSmallSmallC28?LargeB17???1??9173SmallC15??4?2334?LargeA14??110??50?2????3273?1?1????17LargeA?23362SmallB9??6?2????57363?1??7SmallDE

3??81????06

?2??6502?1?32????44

409?1????88568

?2?3??2??0??24

priceberfollows:

ofcutimportantelasticity.points.TheresultsThecalculationswillillustratearemadeanum-assample(1)RegularularofthePrice32-weekcalibrationElasticity.Usingperiod,thecalibrationandchoosingotherpricesof,say,SmallAareincreasedallthebyreg-1%producesSmallvariablesAisheldcalculatedconstant.forallThepurchases.probabilityofcalibrationanewshareperiod.purchaseThepercentshareforSmallAoverThistheasdividedbythepresentchangechangeinpriceinpurchaseistakenConsider(2)theResponseshort-termtoaregularPromotionpricewithelasticity.

aMedianPricesionpromotion.calculateagaintheSmallprobabilityA.Forofeverypurchaseocca-Cut.RepeatpurchasetheCalculateprocessbuttheaveragepurchaseshareforwithoutSmallA.amedianwhenSmallAcomputehasapromotiontheprobabilitywithofastanttionthroughout.pricecut.OurAllmeasureotherattributesofshort-termareheldcon-purchaseresponseshareisoverthethepercentincreaseofthepromo-secondchase(3)PromotionalaoccasioncalculatePriceCut?rst.

probabilityElasticity.ofForpurchaseeverypur-1%SmallpurchaseofregularApromotionpriceaddedhavingbackin.amedianCalculatepricecutwithSmallwithAtheshare.Takethepercentagedifferencefromdividesharepriceby1%.withThepromotionresultistheandshort-termmedianpricepromotionalcutandThecutresultselasticity.

areinTable3.7.2.ADiscussion

widely?rstallbyobservationbrand-size.isThisthatwasresponseexpectedvariesevenquitetrolbrand-sizestionsvariables,becauseusethesamethemarketcoef?cientsforthethoughcon-includingtakeThecompetitiveintoaccountconditions,thecompleteresponsecalcula-brandenvironment,loyalties,icantvariationimplicationsinresponseforretailersbybrand-sizeetc.andmanufacturers.

hassignif-

GuadagniandLittle:ALogitModelofBrandChoiceCalibratedonScannerData

MarketingScience27(1),pp.29–48,?2008INFORMS

Differencesdifferentinpriceelasticity,forexample,suggestaThenextpricingpointpoliciesisthattheacrossresponsebrand-sizes.

measuresshowthede?niteofbrand-sizespattern.areTablelisted3inhasorderbeenarrangedsothatmagnitudespurchases.withofAglancedownthetableofdecreasingshowsthatsharetheaveragedecreasingtheshare.responseForexample,numbersintendeachtoincreasecasethelutetionshipmagnitudeofthefourthanlowerthatbrand-sizesofislargerinabso-forofsitivetheirbrandisnotperfect,therebythehappilyhigherfour.leavingTheroomrela-ownindividuality.share,highNevertheless,sharebrand-sizesasaarepercentagelesssen-brand-sizes.totheirthreeOnownthecontrolotherhand,variableswethanlowshareingchangessharesresponsetomeasuresofTable3bycanthemultiplycorrespond-thediscoveredincontrolobtainvariables.absoluteshareIfthisincrementsisdone,itforthelargerthesharethatincrementthehigh-sharethanbrandstendtoobtainwillbeaityWesamearguemarketingthelow-sharebrandsforthattheseaction.

relationshipsbetweensensitiv-model.andpatternsDetailsshareareappearforcedinbythestructureofthelogitlevelofTable3existAppendixattheindividual2.We?ndpurchasethatthesurprising.andsolarWetheiralsomanifestation?ndthattheinaggregateisnotpurchasepriceelasticitychaseprobabilitieswillmagnitudeofregu-arebelargestwhenallindividualthenlowersshare.identicalandequaltopur-theHeterogeneitymagnitudeofpurchaseprobabilitiesillustratedThenaturepurchaseinFigureofthe8,heterogeneitiesofelasticity.

whichinpracticeisesteachandlowestprobabilitiessharebrand-sizes.acrosspurchasesshowshistogramsforthehigh-ofchasers,productingabroadhasrangeagroupofintermediateofhighAsprobabilitymaybeseen,valuesindicat-pur-Theswitchers,andaclusterofnear-zeroprobabilities.Aahighlatterpurchaseisverylargeprobabilityfortheforlowasharebrand-size.ofhighthehighutilityinthelogitmodel,oftenproductasaimpliesresultorS-shapedloyalty.curveAsofmentionedFigure3,incustomersthediscussionwithofrespondinglowutilitiesinsensitivepurchaseforagivenhighprobabilitiesbrand-sizethatwillhavecor-overalltomarketingactions,thusarerelativelytoInresponsivenessoftheproduct.

reducingtheables,producesummary,short-termthestructuresensitivitiesofthetologitmarketingmodeltendsvari-decreasewhich,whenexpressedinelasticityform,share.inabsolutevaluewithincreasingpurchaseprobabilitiesAbroaddistributionofloyalties(andthereforetomentsasetofpurchase)reducessensitivityrelativeci?cisofdependentidenticalcustomers.onthecoffeeNeithermarketofthesestate-ofthebrandsmodelinvolved,form.

butinsteadisonlyorafunctionthespe-45

Figure8

TheforBimodalitytheDistributionHighestwithandofPurchaseHigherLowestProbabilitiesAcrossPurchasesExtremesShareBrand-SizesandaBroadShowsMiddle

aSlightDistribution of purchase probabilities for Small C

es

sahrcup fo egatnecrePs

eashcrup fo egatnecreP05050505

50

50

5

50

5050

01122334455667788990....................00000000000000000001Probability of purchase

weAsgivesremarkona?nalobservationthecompletenessaboutmarketresponsebutinformationnotjustaboutaofsinglethebrand-sizemodel.ItTableaboutsures3withthecross-elasticitieswholemarket.andWeothercouldresponsesupplementtheerationstudyasdescribedalsoofvariousbyofacompanystrategicearlier.issues,Thecompletenesspermitsmea-possiblenotcompetitiveonlyforofexample,itsreactions.

owntheactionsconsid-but8.ThelentmultinomialConclusions

logitmodelketCoef?cientsatrepresentationtheindividualoftheregularhasgroundprovidedcoffeeanexcel-mar-manyofthemodelcustomerarestatisticallyandretailstorelevel.dictssatisfyinglytheofbehaviorthemhighlyofso.Thecalibratedmodelsigni?cant,pre-iswell.Aremarkableahold-outfeaturesampleofofthecustomersmodelforitsarebrandparsimony.andsizeTheloyaltymajorcoef?cients,andnamely,thoseandthesameacrossallbrand-sizestheandcontrolallcustomers,variables,widelyyetdifferentthemodelshares,ablyfollowspredictsdifferentbrand-sizestrendswithand

GuadagniandLittle:ALogitModelofBrandChoiceCalibratedonScannerData

46

turnsferentoverreasonablytypestimeofstores.andtracksTheresultsbrandperformancearenotperfectinand,dif-areandmissingso,sincevariousmarketingphenomenaiesmediaadvertising)(forexample,asdisplayweknowquality,fromcouponing,otherstud-less,thatthetheseresultsactionshereseemin?uenceverypromising.

purchases.Neverthe-dueThenesstosuccessthemicroofthedetailmodelingeffortappearstoberespondoftheturers.toscannerpanelanddata.competitivePeople,notcomplete-markets,ablesThegreatertheactionsvariabilityoftheofretailerstheexplanatoryandmanufac-fororcalibratingattheindividualresponselevelthanoffersvari-thecorrespondingricheropportunitiesresponseManipulationmarketdata.

storehavetoseveralofthecalibratedmodelyieldssharethat,onlyscratchedthemarketingsurface.variables.However,itHereisclearwesizes,becausestronglytheanswersweareproducedmodelingbytheactionsofallbrand-answersonthequestionsitisasked.themodelInparticularwilldependthecompetitivewilldependonthecustomerloyaltiesandfeatureMuchworkremainsactionsassumed.

tobeOuristhemodelingofthepurchasedone.Amajoroccasionmissingitself.tainworkhaspurchasesmarketactions,focusednotablyentirelypromotion,onshare,whereastendtocer-expandintimeandthereforeatshiftdescribesThemodelthemarket.

leasttemporarilyneedsnotwhathappensotherinsideextensions.theretailstore.Presently,Thisisitcernedadequatehapsaboutforcompetitiontheretailer,amongwhoisbrands,relativelyexceptuncon-competitionforhouseShapirofrombrands,otherbutretailers.isveryconcernedaboutper-turerstion,are1980.)stronglyCorrespondingly,interestedininterbrandalthough(SeeLittlemanufac-andcompeti-theufacturer.retailerthecontrolresultsVariousandonlyvariablesofourmodelarethoseofstepsindirectlyshouldin?uencedbetakentobyextendtheman-theencouraging,Agoodcloserrepresentationtothedecisionofmakers’thecoffeeneeds.

marketuctstrongcategories.butWecoffeedoisjustoneofthemanyprod-isupassumptionsofthenotmultinomialyetknowlogitwhetherwillholdtheple,iningproductmorecomplicateddifferentiationsituationsisgreaterwhere,orvarietyforexam-ofemergeproducts.iscommonWeorexpectthecustomermaintainsaportfolioseek-tionNevertheless,andrequirethatnewmodelingissueswillintheingenuitycoffeetoresolve.

brandofanewunderstandingofcasetheweinterplayseethebetweensugges-model,franchiseandfrugalinandparameters,marketingthatactions.?tsHereisachasesrevealstionshipandadistinctpatternbetweenthesharedataofwellpur-normarkettrackingresponse.isperfect,Sincethereneitherappearthetorela-be

MarketingScience27(1),pp.29–48,?2008INFORMS

idiosyncraticpurchaseeffectivenessvariablesbybrandvariablesatawaitoccasioninclusionand,ofincourse,themodel.othermarketingandthethehaspictureleveloftivelyasetofemergesgeneralunderstandingoftheseNeverthelessmarkets,loyalcustomersthatawell-entrenchedwhomakeitsbrand-sizeshareincontainstheinsensitiveshortsizeapoolrun.tocertainmarketingactions,atrela-leastofAtswitchersthesamelesstimeloyalthemarketalsoingamongvariables.whorespondfairlyreadilytochangestoanyinmarket-brand-strategyattributesReliableknowledgeabouttheinterplayouremergingnowseemandtheirrami?cationsforbrandmarketingtobetechnology.withinstrikingdistanceofAcknowledgments

TheauthorsthankSellingAreasMarketing,Inc.(SAMI)forprovidingthescannerpaneldataonwhichthisworkisbased.TheauthorsareespeciallyappreciativeofthehelpofKennethSilvers,FrankSmith,ValerieGager,andJackMof?y.

AppendixTheIrrelevant1.multinomialAlternatives

TestforIndependencefromlogitmodelimplicitlyassumes“indepen-dencefromirrelevantalternatives”(IIA).McFaddenetal.(1977)presentaresidualstestthatweshallapplytoeval-uatewhethertheassumptionreasonablyholdsinoursetofdata.ThemotivatingideaofthetestisthatviolationoftheIIApropertywillcausesystematicerrorsinthechoiceprobabilities.Theprocedure?rstcalculatestheprobabilitiesbythecalibratedmodelfor,say,alternativejforeachofthe1021observations.Theprobabilitiesarethenrankedandsortedintosomenumberofcells,roughlythesamenumberofobservationsineachcell.Foreachcellwecalculateanexpectednumberofchoicesofjfromtheprobabilitiesandcompareitwiththeactualnumberinagoodnessof?ttest.Thestatistic

??2

=

??

M??SN??m?mP??jm??2NmP

??jwhere

m=1

m=indexofcell,

M=totalnumberofcells,

Sm=numberofactualchoicesofjincell,Ntotalnumberofobservationsincell,

P??m=jmaverageprobabilityforalternativejincellm,andP

??=j=averagehasanasymptomaticprobabilityfordistributionalternativeboundedjintotalbysample,??2dis-tributionswithM?1andM?K?1degreesoffree-domwhereKisthenumberofestimatedparameters.Theteststatisticsarenotindependentacrossalterna-tives.

Thetestwasrunforeachbrand-size,dividingtheobser-vationsinto50cellseachtimeorabout20observationsper

GuadagniandLittle:ALogitModelofBrandChoiceCalibratedonScannerData

MarketingScience27(1),pp.29–48,?2008INFORMS

cell.Theresulting??2were

??2

??2SmallLargeA24Small17SmallA18??6LargeBB

33??5LargeC18??917????00

SmallCSmallDE

309????6??52

The49degreesoffreedomfortheupperbound??2

statisticisandwithwell49.7.criticalAs(0.05)level66.1andforthelowerbound35IIAisbelowdetected.

bothmaycriticalbeseenlevels.theTherefore??2foreachnobrand-sizedeparturefromfallsAppendixTable2.EffectofShareonPriceresponse,3showsshare,whenthatexpressedthemagnitudeResponse

asofshort-termmarket

share,increasesphenomenaresponsewithdecreasingashare.percentageAftermultiplyingofpurchasebymialregularlogitmodel.aredecreasestheOurresultwithanalysisoftheshare.willstructureWeclaimthatthesebedoneofforthethemultino-caseofTheConsiderprice.

theprobabilityexpected?rstshareasingleofpurchase.ofpurchasescustomerandpurchasedecision.Let

foroneoccasionissimply??i

kk

??n??=theforcustomerbrand-sizeregularpricei??konelasticitypurchaseofpurchaseoccasionnshareof=xi??n????pi??n??

(A1)kkwhereabilityxiofk??n??i

brand-sizeistheregularpriceandpk??n??thepurchaseprob-mialPurchaselogitprobabilitieskonpurchasearecalculatedoccasionfromnofthecustomermultino-i.pik??n??=e

vi??k

??n????evi

j??n??(A2)j

with

vik??n??=i=aiutilityk??n??+ofbxbbrand-size??n??yk

kforcustomerionnthai

=chaseeffectsoccasion,

pur-k??n??fortheofsameallotherpurchaseconstantsoccasion,andvariablesonutility

yb=coef?cientofregularprice,

k=aofparameterintroducedinorderNominalbrand-sizevaluesimultaneouslytochangeallprices

=1??0.

andproportionally.ofFor(Al)immediateinto(A2)andpurposesimplifyingyk=1.Substitutingyields

thederivative??ikk??n??=bxik??n????1?pik??n??????

(A3)

Therefore,ofonanindividualbasis,theabsolutemagnitude

explainsregular??ithepriceaggregateelasticityeffectincreasesfoundasshareindecreases.Thisinpk

??n????iprice,??n??decreases??,theincrementalwithdecreasingshareforTable3.Furtherkkshareapercentageonanindividual

change47

basispreponderanceovertherangeof0≤pi

k??n??≤1/2.Providedthatthebesecondthecasetypeforofbrand-sizesoftheprobabilitiespatterninTablehavingfalls3cansharesinthisbeexpected.

lessrange,than1/2,aswilltheinsightAlthoughcanbe(A3)gainedexplainsaboutaggregatethepatternselasticity.found,Let

furthermk=expectedpurchaseshareofbrand-sizek

=

1????Ni

pi

k??n??(A4)

in

where??

N.FurtherNi=thelet

numberofpurchasesbycustomeriandN=i??kj=cross-elasticitywithrespecttoofpricepurchaseofjshareofk

=yj??mkkj

??(A5)

From(A2)

?

??pi??n??

?pik??n???pik??n??2??j=k??ij

=bxj??n??

??pi

k??n??pji??n????

j=k??

(A6)

Using(A5),(A4),and(A6),andsettingyj=1,weobtain

??????kk=

b1iii

kxk??n????pk??n???pk??n??2????

j=k??(A7)in

??kj=

b1????kxij??n??pi

k??n??pji??n????

j=k??

(A8)

in

learnAlthough(A7)looksalittlecomplicated,i

weto?rsttheaskhoweffectofheterogeneityofpk

??n??onwe??canuseitto

kk.Todothis,i

??n??sharemaximize????wewouldkk??subject????topickthethei

constraintdistributionthatpurchaseofpk

studyis?xed,i.e.,i

??1/N??pk

??n??=mk.(Forsimplicity,weMathematically,thecasexkmaximize??1??uthe??n??=problemxk,aconstantisequivalentpriceforallpurchases.)i?u2

??to?nding??ui??toi??subjecttouiject????????byto??N=Nmk,0≤u??i≤1,i=the.Insameturn,constraints.thisisequivalentItcantobeminimizingu2isub-??Thisdynamicmeansprogramming,thatthethatthesolutionshown,isforuexamplei=mk.

have??belargestformagnitudeasetofcustomerofregularpurchasespriceelasticitykk??willwhichgate??purchaseidenticalpurchaseshareofprobabilitiesthegroup.Heterogeneityallequaltotheaggre-i

would??kk??fromreducesgivethere.to

??In0.fact,Forthepushingidenticalthepprobabilityk

??n??toallcase,0sreduces

and(A7)1skk=??kk=bxk??1?mk????Usingfee??pricetheincalibrationthemarket,value0.165b=?29??9andtheaveragecof-decreasing=?4??93??1?mdollars/ounce,weobtainkkk??.Thisshows??thiserogeneityformulashare,withas??kk??tobeincreasingwiththoseexpected.inComparisonofvaluesfrom????=3??5byonthis????Table3showstheeffectofhet-kkformula??.Asanbutexample,is1.9bySmallTableC3.

wouldhavekk??Tosummarize,thelogitstructureinherentlyproducesthepatternsofresponseobtained.Further,thegreaterthedis-persionoftheindividualpurchasepi

probabilitiesawayfrom??n??=mktowardall0sand1s,thelowerthemagnitudeofelasticity.

k