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Designing Performance Guarantee Contracts

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Title:
Designing Performance Guarantee Contracts
Creator:
Koschnick, Clay M
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (134 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Hartman, Joseph C
Committee Members:
Richard, Jean-Philippe
Geunes, Joseph P
Alba, Joseph W
Graduation Date:
8/11/2012

Subjects

Subjects / Keywords:
Capital costs ( jstor )
Commodities ( jstor )
Consumer prices ( jstor )
Cost allocation ( jstor )
Cost estimates ( jstor )
Guarantees ( jstor )
Manufactured products ( jstor )
Operating costs ( jstor )
Product performance ( jstor )
Warranties ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
dynamic -- guarantee -- performance -- programming -- warranty
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
This dissertation studies the use of contracts that can be used by a manufacturer to affect consumer behavior by providing the consumer some protection against poor product performance. We first define the concept of a performance based warranty (PBW) that reimburses a consumer if a product performs below a certain threshold and then show how these warranties can be used to entice consumers to make purchases sooner than planned. We show in a finite horizon equipment replacement problem with a single challenger that a manufacturer can increase their net revenue by offering PBWs at certain times. Next, we consider another finite horizon problem where there are multiple challengers. In this scenario, we model performance money back guarantees (MBG) that influence a consumer's expectation of future product performance. Specifically, we show how an incumbent manufacturer can use performance MBGs to entice the consumer to return for a follow-on purchase instead of taking their business to another company. Finally, we study the practicality of using extended performance based warranties (EPBWs) when the product's performance is based on the consumption of a given commodity. We model a renewable, deferrable EPBW that compensates a consumer according to the market price of the underlying commodity. In a semi-empirical example, we show the actual effect on revenue of offering EPBWs and the potential effect on revenue if the manufacturer can exploit the consumer's full willingness-to-pay. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Hartman, Joseph C.
Statement of Responsibility:
by Clay M Koschnick.

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UFRGP
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Copyright Koschnick, Clay M. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
858441661 ( OCLC )
Classification:
LD1780 2012 ( lcc )

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DESIGNINGPERFORMANCEGUARANTEECONTRACTSByCLAYM.KOSCHNICKADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012ClayM.Koschnick 2

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Idedicatethisdissertationtomyfamilyandfriends. 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,Dr.JosephHartman,fortheconstantsupportandoptimismthatthisprogramcouldbecompletedinthreeyears.Ourweeklymeetingsnotonlyprovidedmeguidanceonthisresearchbutalsoprovidedinvaluableinsightsintoacademia.IwouldalsoliketothankmycommitteemembersDr.JosephGeunes,Dr.JeanPhilippeRichard,andDr.JosephAlba.Yourresponsivenessandexibilitytosupportmycompressedtimelinewascriticaltocompletingthisdissertation.Next,tomyclassmatesandfriends.Withoutyoursupportandencouragement,Ineverwouldhavemadeitthroughthelastthreeyears.Fromgamenightstocurrenteventdiscussions,timewithyouwasalwaysawelcomebreakfromtheintenseschoolschedule.Withoutourgeneralexamstudygroup,Iwouldhavenevermadeitpasttherstyear.Forallyouhavedone,Iamgratefulandlookforwardtocontinuingtheselifelongfriendships.Andnally,tomyfamily.Thankyoutomyparentswhoprovidedconstantwordsofencouragement.Tomybrother,althoughmilesapart,wecouldcommiserateaswebothpursuedadvanceddegrees.Andtomywifeandson.Youarethedailyreminderofwhatlifeisallabout.Throughyourunlimitedsupportandsacrice,wewereabletoaccomplishthisgoal.WordscannotexpresshowblessedandthankfulIamtohaveyou.Withoutyou,nothingispossible,andwithyou,everythingisperfect. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 2USINGPERFORMANCEBASEDWARRANTIESTOINFLUENCECONSUMERPURCHASEDECISIONS .............................. 14 2.1MotivationandLiteratureReview ....................... 14 2.2PerformanceBasedWarrantyDenition ................... 18 2.3ConsumerDifferentiation ........................... 22 2.4MotivationThroughStationaryAnalysis ................... 24 2.4.1ConstantPerformanceWarranty ................... 24 2.4.2ConstantCostWarranty ........................ 29 2.5FiniteHorizonwithTechnologyUpgrade ................... 32 2.5.1ConstantPerformanceWarranty ................... 35 2.5.2EffectofMeanandVariance ...................... 44 2.5.3ConstantCostWarranty ........................ 46 2.6ImplementingPerformanceBasedWarranties ................ 51 2.7Summary .................................... 52 3PERFORMANCEGUARANTEESUNDERCOMPETITION ........... 54 3.1MotivationandLiteratureReview ....................... 54 3.2BayesianUpdatingMethodology ....................... 58 3.2.1Beta-BernoulliModelwithTwoCompetitors ............. 61 3.3DynamicProgramFormulation ........................ 64 3.3.1ScenarioAnalysis ........................... 67 3.3.2GuaranteeEffect ............................ 69 3.4Results ..................................... 82 3.5Summary .................................... 90 4EXTENDEDPERFORMANCEBASEDWARRANTIESWITHEXOGENOUSCOSTINPUTS .................................... 92 4.1MotivationandLiteratureReview ....................... 92 4.2ModelFormulationwithUpdatedCostEstimates .............. 98 4.2.1UncertaintyintheModel ........................ 98 5

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4.2.2CostEstimates ............................. 102 4.2.3DynamicProgram ........................... 104 4.2.4NumericalExample ........................... 107 4.3MarkovianPerformance ............................ 111 4.3.1ModelFormulationwithNon-IncreasingPerformance ........ 111 4.3.2DynamicProgram ........................... 115 4.3.3NumericalExample ........................... 118 4.4CombinedModel ................................ 122 4.5Summary .................................... 124 5CONCLUSIONANDFUTURERESEARCH .................... 126 REFERENCES ....................................... 130 BIOGRAPHICALSKETCH ................................ 134 6

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LISTOFTABLES Table page 2-1ParametersforPerformanceBasedWarranty(PBW)Example ......... 28 2-2ConsumerBeliefVectors .............................. 28 2-3ConsumerPoliciesforDifferentLevelsofTechnologyImprovement ....... 35 2-4OptimalZero-CostConstantPerformanceWarranties .............. 36 2-5OptimalUnrestricted-CostConstantPerformanceWarranties .......... 43 2-6EffectofVarianceonManufacturerRevenue ................... 45 2-7EffectofMeanonManufacturerRevenue ..................... 46 2-8OptimalZero-CostConstantCostWarranties ................... 46 2-9OptimalUnrestricted-CostConstantCostWarranties ............... 51 3-1ParametersforBayesianUpdatingExample .................... 83 3-2ManufacturerRevenuewithOptimalPerformanceGuarantee .......... 85 3-3OptimalPerformanceGuaranteeDesigns ..................... 88 3-4AnalysisofOptimalFront-EndGuarantees .................... 89 4-1ParametersforExtendedPerformanceBasedWarranty(EPBW)Example ... 108 4-2WarrantyEffectonManufacturerRevenueforBasicModel ............ 109 4-3PessimisticConsumerBeliefProles ........................ 118 4-4Single-PeriodWarrantyEffectonRevenueforMarkovianModel ......... 119 4-5SensitivityAnalysisforMarkovianModel ...................... 121 4-6WarrantyEffectonManufacturerRevenueforCombinedModel ......... 123 7

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LISTOFFIGURES Figure page 2-1ContinuousOperatingCostDistribution ...................... 18 2-2ConstantPerformanceWarrantyFrontiers ..................... 28 2-3ConstantPerformanceWarrantyChangeinRevenueCurves .......... 29 2-4ConstantCostWarrantyFrontiers .......................... 31 2-5ConstantCostWarrantyChangeinRevenueCurves ............... 32 2-6FeasibleConstantPerformanceWarrantiesforPessimisticConsumers ..... 37 2-7MaximumRevenueCurvesforConstantPerformanceWarrantiesforDifferentLevelsofTechnologyImprovement ......................... 38 2-8FeasibleConstantCostWarrantiesforPessimisticConsumers ......... 47 2-9MaximumRevenueCurvesforConstantCostWarrantiesforDifferentLevelsofTechnologyImprovement ............................. 48 3-1NetworkPathsWithoutPerformanceGuarantee ................. 70 3-2NetworkPathsAfterBack-EndPerformanceGuarantee ............. 80 3-3GuaranteeEffectonManufacturerRevenue .................... 84 3-4ComparisonofGuaranteeandNoGuaranteeEffectsonRevenue(HighProductPerformance) ..................................... 87 3-5ComparisonofGuaranteeandNoGuaranteeEffectsonRevenue(LowProductPerformance) ..................................... 87 4-1TimeVariantEffect-MechanicWagesvs.GasolinePrices ........... 93 4-2NetworkwithNon-IncreasingProductPerformance ................ 117 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDESIGNINGPERFORMANCEGUARANTEECONTRACTSByClayM.KoschnickAugust2012Chair:JosephC.HartmanMajor:IndustrialandSystemsEngineeringThisdissertationstudiestheuseofcontractsthatcanbeusedbyamanufacturertoaffectconsumerbehaviorbyprovidingprotectionagainstpoorproductperformance.Werstdenetheconceptofaperformancebasedwarranty(PBW)thatreimbursesaconsumerifaproductperformsbelowacertainthresholdandthenshowhowthesewarrantiescanbeusedtoenticeconsumerstomakepurchasessoonerthanplanned.WeshowinanitehorizonequipmentreplacementproblemwithasinglechallengerthatamanufacturercanincreasetheirnetrevenuebyofferingPBWsatcertaintimes.Next,weconsideranothernitehorizonproblemwheretherearemultiplechallengers.Inthisscenario,wemodelperformancemoneybackguarantees(MBG)thatinuenceaconsumer'sexpectationoffutureproductperformance.Specically,weshowhowanincumbentmanufacturercanuseperformanceMBGstoenticetheconsumertoreturnforafollow-onpurchaseinsteadoftakingtheirbusinesstoanothercompany.Finally,westudythepracticalityofusingextendedperformancebasedwarranties(EPBWs)whentheproduct'sperformanceisbasedontheconsumptionofagivencommodity.Wemodelarenewable,deferrableEPBWthatcompensatesaconsumeraccordingtothemarketpriceoftheunderlyingcommodity.Inasemi-empiricalexample,weshowtheactualeffectonrevenueofofferingEPBWsandthepotentialeffectonrevenueifthemanufacturercanexploittheconsumer'sfullwillingness-to-pay. 9

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CHAPTER1INTRODUCTIONAwarranty(orguarantee)isacontractualmechanismtoprovideaconsumeralevelofcompensationfromthemanufacturerwhenaproductdoesnotoperateaspromised.Fortheconsumer,thisprotectionmaylower,oratleastlimit,certainperiodiccostswhiletheproductisunderwarranty.Forthemanufacturer,thewarrantymaybeasignalofqualitytotheconsumer,generateasourceofadditionalrevenue,orincreaseconsumerloyalty.Thedesignsofawarrantycanvarysignicantlyastherearemanydifferentfactors,suchasprice,length,andcompensationrate,thatmustbespeciedforeachwarranty.MurthyandBlischke[ 1 2 ]provideanextensiveanalysisofwarrantiesthatarepredicatedonthefailurecharacteristicsoftheproduct,whichwerefertoastraditionalwarranties.Thesewarrantiesaremodeledanddesignedtoaddresstheconditionofwhetheranitemisabletoperformitsintendedfunctionproperlyornot,andthustheyimpacttheconsumer'smaintenancecosts.Anothermajortotallifecostcomponentthatisnotconsideredbytraditionalwarrantiesisoperatingcosts.Aperformancebasedwarranty(PBW)guaranteesthataproductoperatestoapre-denedlevelofperformanceoveracertainlengthoftime.APBWallowsacustomertocontroltheperiodicoperatingcostsassociatedwiththeproductjustasatraditionalwarrantyisusedtocontrolperiodicmaintenancecosts.Similarly,thedesignofaPBWhasawiderangeoffactorsthatmustbespeciedandtheconditionsofthewarrantymustbeadjustedtotheuniquenatureofoperatingcosts.Thereisextensiveresearchontraditionalwarranties.Conversely,thestudyofperformancebasedwarrantiesisrelativelysparsealthoughthereareexamplesinthecommercialsectorwheretheyhavebeenapplied: 10

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BoeingandAirbusincludeperformanceguaranteesintheircontractswithairlinecompanies.Anexampleperformanceguaranteeisonthemaximumleveloffuelconsumption. Photovoltaic(PV)cellmanufacturers,likeSchottorSolon,provideperformanceguaranteesforthepoweroutputoftheirPVmodules. LGenactedapostfactoguaranteeforsomerefrigeratormodelstobringtheenergyefciencyleveluptothatadvertisedonitsEnergyGuideLabel. Energyproducer,Raser,sueditspowergeneratorprovider,UTCPower,whenthegeneratorsdidnotproducethepoweroutputadvertised.TheseexamplesareinstructiveregardingthetypeofproductandbusinessenvironmentswherePBWswouldbemostsuitable.Thecentralcomponentoftheoperatingcostsintheseinstancesisenergy,whetherintheformofelectricityorfuel.Becausetherecanbeotherinputstooperatingcostssuchaslabor,caseswhereaproduct'soperatingcostsareheavilydependentonenergyconsumptionareanaturaltforPBWs.Asforthebusinessenvironments,therearetheretailsidewithindividualconsumersandtheindustrialsidewithbusinessesandgovernmentagencies.Ontheretailside,productsthatareminimallyaffectedbyaconsumer'soperatinghabits,suchasarefrigerator,signicantlyincreasethefeasibilityofusingaPBW.Asfortheindustrialside,thereisamuchgreaterlatitudetocraftoperatingguidancethatcanbemonitoredtoensuretheproductisusedwithinacertainoperatingenvelope.Clearly,therearemanyfactorsthataffectfuelconsumptionforanaircraftthatareindependentoftheenginesfuelefciency,butthestructurednatureofoperatinganaircraftmakescompliancetoaPBWpossible.Additionally,thesignicantinvolvementofthecustomerinthemanufacturer'sdevelopmentprocesscanhelpdeneanenforceablePBW.Besidesthetypeofproductsandbusinessenvironments,thesepracticalexamplesexistinanequipmentreplacementenvironment.Theservicesoftheproductsare 11

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requiredonacontinuousbasisandthereforethecustomersmustplanthereplacementscheduleoftheproductstoensuretheserviceisalwaysavailable.Becauseofthepossibilityofafollow-oncustomerpurchase,itislesslikelythatthemanufacturerwillexploitaPBWintheshort-runattheriskofalienatingacustomerwithregardstomakingafuturepurchase.Inthisenvironment,warrantiescanplayastrategicrolethataffectnotonlytheitemthatiscurrentlyunderwarrantybutthefutureproductreplacement.Withinwarrantyanalysis,thereisoftenanassociatedequipmentreplacementproblem.HartmanandLaksana[ 3 ]analyzetheuseofwarrantieswhenthereisonlyonechallengeratatimebutsubsequentchallengershavelowermaintenancecosts.Mamer[ 4 ]considersthetotalexpecteddiscountedandaverageperunitcostsofawarrantyusingrenewaltheorywhenthereisonestationarychallenger.HartmanandLaksana[ 5 ]alsoconsideronestationarychallengeranduseadynamicprogrammodeltoallowfortheadditionaldecisiontopurchaseanextendedwarranty.Additionally,thereareequipmentreplacementproblemswhicharewellsuitedfortheanalysisofaPBWaswellasothertypesofperformanceguaranteecontracts.NairandHopp[ 6 ]exploretheoptimalreplacementpolicywhenthecurrentlyowneditemisobsoleteandthesinglechallengerisatechnologicalimprovement.Oakfordet.al[ 7 ]considerequipmentreplacementpolicieswithmultiple,non-stationarychallengers.Inthisdissertation,westudytheuseofvariousperformanceguaranteecontractsinavarietyofequipmentreplacementproblems.InChapter2,ourrstmodelusesaPBWtoenticeaconsumertoreplacetheirobsoleteproductsoonerthantheyhadoriginallyplanned.TheninChapter3,wemodeltheuseofaperformanceguaranteetomotivateaconsumertomakeafollow-onpurchasefromtheincumbentmanufacturerinthepresenceofcompetition.Finally,inChapter4,weintroducetheconceptofanextendedperformancebasedwarranty(EPBW)thatallowsaconsumertoobtainawarrantyatanytimetheproductisowned,notjustatthetimeofpurchase.Ineachofthesechapters, 12

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weusedynamicprogramming(DP)tomodeltheconsumer'sdecisionmakingprocessanddeterminetheiroptimalpurchasepolicyoveranitehorizon.DPiswellsuitedforthesetypeofsequentialdecisionproblems,andusingtheconsumer'spurchasepolicies,thebestcontractdesignandassociatedeffectonamanufacturer'srevenuecanbedetermined. 13

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CHAPTER2USINGPERFORMANCEBASEDWARRANTIESTOINFLUENCECONSUMERPURCHASEDECISIONS 2.1MotivationandLiteratureReviewTraditionalwarrantiesaregenerallystructuredaroundtheoperationalstatusofaproduct(i.e.,thebinarystatesofworkingornotworking)andare,therefore,predicatedonthereliabilityoftheitem.Whiletraditionalwarrantiesaddressfailedproducts,thereislittleliteratureonwarrantiesthataddresstheoperationsofaproduct.Aperformancebasedwarranty(PBW)providestheconsumerinsuranceagainstaproductnotperformingtoitsadvertisedspecications.Similartootherwarranties,aPBWprovidesasignalofcondencethattheitemwillperformasadvertised.Thisaspectmaybeimportantwhenaconsumerisdecidingbetweenabasemodeloramoreexpensive,technologicallyadvancedmodelorbetweenmodelsfromcompetingmanufacturers.APBWalsoprovidesthepotentialtogeneraterevenueandbuildconsumerloyaltyforthemanufacturer,muchlikeatraditionalwarranty.AgoodexampletoillustratetheuseofaPBWwouldbeinthesaleofstandardhouseholdappliances.Forinstance,ifacompanyispromotinganew,highefciencyrefrigerator,theycanofferaperformancewarrantythatwillreimbursetheconsumerforenergyusageaboveaspeciedlevelorenergyefciencybelowsomegivenlevel.In2008,LGessentiallyenactedanexpostfactoPBWinresponsetoadowngradeintheenergyefciencyratingbytheDepartmentofEnergy(ConsumerReports.organdlg.com,2008).LGcompensatedconsumerswithaninitialpaymenttocoverthesubstandardperformancethroughthepointintimeofthedowngradeandagreedtopayannualinstallmentsthereaftertobringtheconsumer'scostdowntotheadvertisedperformancelevel.Similarexamplesexistforindustrialequipmentsuchasfarmcombinesandaircraft.Commercialaircraftmanufacturersoftenprovideguaranteesonengineperformance,measuredinfuelconsumptionorfuelburn,toairlinesthatpurchasetheirproduct[ 8 ]. 14

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Foraconsumer,aPBWlimitstheriskthataproductwillnotperformatitspromisedlevel.NotethataPBWimpliesthattheproductisfunctioningsufcientlytoperformthetaskforwhichitwaspurchased;protectionagainstalackoffunctionalityisaddressedbyatraditionalwarranty.APBWhasthepotentialtolowerthetotalexpectedcostsforaconsumerorminimizetheprobabilitythatcostswillexceedacertainthreshold.Also,sincebetterperformancegenerallycomesatahigherpurchasecost,aPBWcanlowerthepaybackperiodforthehigherinvestmentbyloweringtheexpectedoperatingcosts.Aswithtraditionalwarranties,themanufacturerrisksincurringadditionalcoststopayforanyperformancewarrantyclaims.Theadditionalriskcanprovideadditionalrevenueorgeneraterevenueearlier.APBWmayalsohelpchangeaconsumer'spurchasinghabits,whichmightsupportothercompanygoals.Forexample,acompanymayneedtomeetanon-revenuedrivengoal,suchasadherencetogovernmentregulation(CorporateAverageFuelEconomy(CAFE)standardsforexample)andthusneedtoincreasesalesofcertainproductsatcertaintimes.Or,acompanymayattempttoenteramarketandincreaseitscredibilitywithaPBW,suchasasolarpanelcompanyenteringtheresidentialenergymarket[ 9 ].Thereisextensiveliteratureonwarrantyanalysisandinparticular,onthestructureanddesignofwarrantiesthataddresswhetheraproductperformsthefunctionforwhichitwasintended.Althoughthetermfunctioninherentlyincludesareasonablelevelofperformance,itisnormallyreservedforthebinarystatesoffailedoroperating,andthereforefocusesonthefailurecharacteristicsoftheproduct.Thewarrantyliteratureprovidesawiderangeofwarrantydesignsthatprovidealevelofprotectionagainstthefailureofanitem.MurthyandBlischke[ 1 2 ]denenumerouswarrantypoliciesthatuseoneormoreofthreeprimarytypesofwarranties:1)Free-ReplacementWarranty(FRW),2)Pro-RataWarranty(PRW),and3)ReliabilityImprovementWarranty(RIW).TheFRWrepairsorreplacestheitematnochargetothecustomerwhileunderwarranty,whereasthePRWprovidesapartialreimbursementfortheunusedportionofthewarranty.A 15

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RIWisprimarilyusedwithproductswherethecustomerhasasignicantroleinproductdevelopment,suchasmilitaryprocurement,andthemanufacturer'scontractincentiveisbasedonimprovingthereliabilityoftheproduct.Severalvariationsofthesepoliciesexistbasedonthedimensionalityofthewarranty,numberofitemscovered(singleitemorlot),andwhetherthewarrantyisrenewableornot.Additionally,variousmodelswithawiderangeoffailurecharacteristics,repairrecticationactions,andrepaircostsareprovided.Anextendedwarranty(EW)isawarrantythatcanbepurchasedoncethebasewarrantythatcomeswiththeproductpurchaseexpires.Variousdesignsforextendedwarrantieshavebeenproposedandstudied.JackandMurthy[ 10 ]developanEWwheretheconsumerhastheexibilitytodeterminethelengthandstarttimeofthewarrantyandthemanufacturerthenpricesitaccordingly.HartmanandLaksana[ 5 ]showthatamenuofdifferentEWdesignscanincreaseamanufacturer'sprotversusastandalonewarrantybycapitalizingonthevariousriskpreferencesofcustomers.Gallegoetal.[ 11 ]studyavariationofaPRWwarrantycalledaResidualValueWarranty(RVW)wherethereimbursementisbasedonthenumberofclaimsoverthewarrantyperiod.Besidesvaryingthedesignofthewarranties,differentobjectivefunctionsandconsumerdifferentiationmethodsareaddressedthroughouttheliterature.Mamer[ 4 ]analyzesFRWandPRWwarrantiesusingtotaldiscountedcosts/protsandperunitcosts/prots.Theoptimalpoliciesforanextendedwarrantywhentheconsumerhasdifferentcostcriteria,totalexpectedcostsversusaverageperperiodcosts,areexploredbyLamandLam[ 12 ].ChunandTang[ 13 ]studytheeffectsofconsumerriskpreferencesonwarrantypricing.Infact,theyalsolookattheeffectofproducerriskpreferenceswhichsuggeststherearealternativemanufacturerobjectivesfromwhichtostudywarrantystructures.Padmanabhan[ 14 ]differentiatesbetweenconsumersbased 16

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onproductusage,andLutzandPadmanabhan[ 15 ]segmentthemarketbasedontheconsumer'svaluationoftheproduct.Wecontributetothewarrantyanalysisliteraturebyrstdevelopinganddesigningperformancebasedwarrantiesinthecontextofequipmentreplacementdecisions.SimilartoBellman[ 16 ],weconsiderequipmentreplacementintwocases:1)notechnologicalimprovementsoanyreplacementoccurswithasimilarmodel,and2)technologicalimprovementwhereareplacementmustbewithanupgradedproduct.WhileweareunawareofanypreviousresearchofPBWs,ourmodelsbuildonsomedesigncharacteristicsoftraditionalwarrantiessuchaslimitingwhenandhowoftenausermaypurchaseawarranty.Second,weshowhowprovidingaPBWwiththepurchaseofaproduct,atnocost,canincreaseamanufacturer'srevenuebyincentivizingaconsumertopurchaseareplacementearlierthanexpected,regardlessofwhethertheconsumerisoptimisticorpessimisticabouttheexpectedperformanceoftheitem.Third,wedevelopsufcientconditionsfordeterminingwhentochargeforaPBWandpresentapolynomial-timealgorithmtodeterminetheoptimalwarrantydesign(denedbyaprice,length,andperformancethreshold).Basedontheseconditions,wendthatamanufacturercanchargeapessimisticconsumerforaPBWandincreasetotalrevenuebycreatinganadditionalrevenuesource.Thischapterhasthefollowingstructure.Insection 2.2 ,wedeneageneralPBWanddiscussthetwospecictypesthatareanalyzedintheremainderofthechapter.Then,insection 2.3 ,wedeneourmethodofconsumerdifferentiationbasedontheconsumer'sbeliefaboutaproduct'sperformance.Section 2.4 identiestheoptimalwarrantytoincreasethemanufacturer'srevenueforaninnitehorizonequipmentreplacementproblemwithnotechnologicalchange.Insection 2.5 ,wepresentanalgorithmtosolvefortheoptimalwarrantyinanitehorizonequipmentreplacementproblemwheretheexistingproductbecomesobsoleteandmustbereplacedbyatechnologicallyadvancedproduct. 17

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2.2PerformanceBasedWarrantyDenitionBeforedeningaPBW,weintroducenotationwithregardstoanequipmentreplacementproblem.Asthisisasequentialdecisionproblem,wedenethetimehorizonbyTwhichrepresentsthetotalnumberofequallengthtimeperiodsoverwhichtheproblemissolved.AnassetcanbepurchasedatpricePtattimet2f1,...,Tg.Ineachperiodoftheitem'susefullifeN,theequipmenthasoperatingcostsCt(a)wherea2f1,...,Ngdenotestheageoftheassetatthebeginningofperiodt.Itisexpectedthattheoperatingcostsoftheassetareuncertainandthusfollowadistribution.Toprotecttheconsumeragainsthighoperatingcosts,themanufacturermayoffertocaptheoperatingcoststhattheconsumerwillpayeachperiodforacertainamountoftimeatapredeterminedprice.Thisistheessenceofaperformancebasedwarranty(PBW).First,considerthecontinuouscase.Letf(x;a,t)representtheprobabilitydensityfunctionofoperatingcostsofaproductofageaattimet.Alsoletla,tandua,tbethelowestandhighestpossibleoperatingcosts,respectively.Weassumeallcostsarepositiveandnitesothesupportoff(x;a,t)mustbeanitesegmentofthepositiverealnumberline(seeFigure 2-1 ). Figure2-1. ContinuousOperatingCostDistribution Theexpectedoperatingcostforanassetofageaattimetis: 18

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a,t=Zua,tla,txf(x;a,t)dx.NowassumeamanufactureroffersaPBWsuchthatoperatingcostsarecappedatba,tua,tforana-periodoldassetinperiodt.Thisisknownasthecostcaporcoveragelevel.TheexpectedoperatingcostforaconsumerwithaPBWforaproductofageaattimetwithacostcapofba,tis:a,t=Zba,tla,txf(x;a,t)dx+ba,t(1)]TJ /F5 11.955 Tf 11.95 0 Td[(F(ba,t;a,t)),suchthat,bydenition,a,ta,t.LetWbethenumberofperiodsthatoperatingcostsarecapped.Also,letbetheperiodicdiscountfactorinordertoaccountforthetimevalueofmoney.Wecannowexpressanupfrontfeeorwarrantycost,Pwt,thattheconsumeriswillingtopayattimetforlimitingperiodicoperatingcostsas: Pwt=WXi=1i(a+i,t+i)]TJ /F7 11.955 Tf 12.66 0 Td[(a+i,t+i)=WXi=1i(Zua+i,t+ila+i,t+ixf(x;a+i,t+i)dx)]TJ /F10 11.955 Tf 11.96 16.27 Td[(Zba+i,t+ila+i,t+ixf(x;a+i,t+i)dx)]TJ /F5 11.955 Tf 9.3 0 Td[(ba+i,t+i(1)]TJ /F5 11.955 Tf 11.96 0 Td[(F(ba+i,t+i;a+i,t+i)))=WXi=1i(Zua+i,t+iba+i,t+ixf(x;a+i,t+i)dx)]TJ /F5 11.955 Tf 11.96 0 Td[(ba+i,t+i(1)]TJ /F5 11.955 Tf 11.96 0 Td[(F(ba+i,t+i;a+i,t+i))). (2) Notethatthisassumesthatthecustomerisriskneutral(minimizesexpecteddiscountedcosts)andhasfullknowledgeofallcostsandparameters.Inthediscretecase,letLadenotethenumberofcostrealizationsforaproductofagea.Similarly,thiscanbedenedasLaoperationallevels,eachwithanassociatedoperatingcost.Thenca,tj:j2f1,...,Lagrepresentsthepossiblecostrealizationsattimetwithprobabilitypa,tj.Theexpectedoperatingcostforanassetofageainperiodtis: 19

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a,t=LaXj=1pa,tjca,tj.TheexpectedoperatingcostunderaPBWforanassetofageainperiodtis:a,t=LaXj=1pa,tjmin(ba,t,ca,tj).Nowassume,withoutlossofgenerality,thatforeacha2Wandt2T,thecostsareindecreasingordersuchthatca,tj>ca,tj+1forj2f1,...,La)]TJ /F7 11.955 Tf 12.79 0 Td[(1g.ThenwecandeneLaasthenumberofcostrealizationswhereba,t
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warranties(denedbyawarrantyprice,awarrantycoveragelevelforeachperiodofthewarranty,andthelengthofthewarranty)forwhichtheconsumerisindifferentdenestheconsumer'swarrantyfrontier.Thewarrantyfrontierfurtherdenesifanarbitrarywarrantymakesaconsumerbetterorworseoffbyassessingthewarranty'srelationshiptothewarrantyfrontier.Thewarrantydesign(Pw,b1,...,bW,W)isgeneral.Here,weexaminethecasewhereoperatingcostsfortheassetareexpectedtogrowfrom(la,ua)to(la+1,ua+1)atrategperperiod(astheproductages).WecannowdenespecictypesofPBWsbyprovidingastructuretothetruncationlevels,ba.Ifweconsidertruncationlevelsthatincreaseataconstantrate,r,overthelengthofthewarranty(i.e.,ba+1=(1+r)ba,8a),thenwecandenealltruncationlevelswithrespecttotheinitialtruncationlevelb1.Inordertosimplifynotation,wedropthesubscriptandletbrefertotheinitialtruncationlevel.Althoughtherearenorestrictionsonr,wewillcloselyexaminetwocases,r=gandr=0.LetusformallydenetwotypesofPBWs:(1)aconstantperformance(CP)warranty(r=g),and(2)aconstantcost(CC)warranty(r=0).ACPwarrantyallowsthecostcaptoincreaseatthesamegrowthrateastheexpectedperiodicoperatingcostsandthedesignofthewarrantyisdenedasthetriple(Pw,b,W)CPwherePwisthepurchasepriceofthewarranty,bistheinitialcaplevelintherstperiodofowningaproduct,andWisthelengthofthewarranty.Assumingcertainrestrictionsontheparametersofthedistributionovertime(suchasconstantshapeparametersinafourparameterbetadistribution),whenthecostcapincreasesatthesamegrowthrateastheperiodiccosts,theCPwarrantyalwaysprotectstheconsumerfromthesamepercentageofcostrealizationsinanyperiodthewarrantyisineffect.SimilartotheCPwarranty,theCCdesignisdenedasthetriple(Pw,b,W)CC,butthecostcapremainsconstantastheperiodiccostsgrowasaproductages.Thus, 21

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theCCwarrantyprotectstheconsumerfromthesameorahigherpercentageofcostrealizationsineachsubsequentperiodthewarrantyisineffect. 2.3ConsumerDifferentiationAsdiscussedintheliteraturereview,therearealternativewaystodifferentiateconsumers,butultimately,differentiationisanattempttocharacterizehowdifferentindividualsmakedecisions.Weareinterestedincapturingaconsumer'spreconceptionsaboutproductperformanceasitisexpectedthesepreconceptionswouldinuenceaconsumer'sdecisionregardingthepurchaseofaPBW.Thus,weintroducetheconceptofconsumerbeliefswheretheconsumermaymodifythemanufacturer'soperatingcostdistribution.Effectively,thesemodicationsareanindicationoftheconsumer'swillingnesstobelieveortrustthemanufacturerundertheassumptionofsymmetricinformation.Themanufacturer'sadvertisedinitialoperatingcostsdistributionisconsideredsymmetricinformationandtheconsumer'sbeliefismodeledbymodifyingtheprobabilitiesofthisdistribution.Forexample,anautomobilemanufacturermaydisplayanexpectedgasmileageandrange.Differentconsumersmaybelievethesenumbersdifferently.Usingtheconceptofawarrantyfrontier,wecanspecifydifferenttypesofwarrantyfrontiersbasedonconsumerbeliefs.Ifaconsumerisneutral,sheacceptsthemanufacturer'soperatingcostdistributionexactlyandwillhavearesultingwarrantyfrontierHnCPforaconstantperformancewarranty.Weusethesuperscriptntodenoteaconsumerwithneutralbeliefs.Iftheconsumerispessimistic(denotedwithasuperscriptp),theprobabilitiesaremodiedsuchthathiswarrantyfrontierisgreaterthanorequaltotheneutralwarrantyfrontier,i.e,HpCPHnCP8(Pw,b,W)andHpCP>HnCPforsome(Pw,b,W).Iftheconsumerisoptimistic(denotedwithasuperscripto),theprobabilitiesaremodiedsuchthattheconsumer'swarrantyfrontierisalwayslessthanorequaltotheneutralfrontier,i.e.,HoCPHnCP8(Pw,b,W)andHoCP
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EquivalentdenitionsexistforaconstantcostwarrantyandareindicatedwithaCCsubscript.Asalludedtointheprevioussection,theconsumer'sobjectiveistominimizetotalexpecteddiscountedcosts.Similarly,themanufacturerseekstomaximizetotalexpecteddiscountedrevenue.Thus,theanalysisinthischapterimplicityaddressesonlyriskneutrality,bututilityfunctionscouldbeincorporatedintothemethodologytoanalyzeotherriskpreferences.Additionally,ourcharacterizationofconsumerbeliefhasaninterestingrelationshiptoriskpreference.Baker[ 17 ]usesamethodofinatingmaintenancecoststoindicateriskaversion.Asshownlater,apessimisticconsumerhashigherexpectedoperatingcosts,anoptimisticconsumerhaslowerexpectedoperatingcosts,andaneutralconsumerhasthesameexpectedoperatingcosts.Therefore,theconsumerbeliefasameasureofconsumerriskiscompatiblewithBaker'smethodofcharacterizingriskpreference.Infact,usingtheproposeddenitionofconsumerbeliefasamodelofriskallowsforanevenhigherlevelofdiscriminationbetweendifferentriskpreferencesbecauseexpectationalonedoesnotfullydescribeaconsumer'sriskpreference.Theuseofadistributionfortheoperatingcostswouldalsoincorporateriskthroughthevarianceofthedistribution.Conceptually,thisissimilartoMarkowitz's[ 18 ]mean-varianceapproachaswellasmoremodernriskcharacterizations,suchasconditionalvalueatrisk(CVaR),wherevarianceisameasureofrisk.Ultimately,thepurposeofattemptingtodeneaconsumer'sriskistodifferentiatebetweenconsumersbasedontheunknown,inherentbeliefsaconsumerholdsthatinuencetheirdecisionmaking.Therefore,thepessimistic,optimistic,andneutralconsumerbeliefscouldequivalentlydescriberiskaverse,risktaking,andriskneutral,respectively.Inthisresearch,wetreatconsumerbeliefsasameasureoftheconsumer'sbeliefinthecredibilityofthemanufacturerandleaveopenthepossibilitytoincorporateriskthroughtheapplicationofutilitytheory[ 19 ]. 23

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2.4MotivationThroughStationaryAnalysisLetusconsiderastationary,innitehorizonequipmentreplacementproblemwithouttechnologicalchangeandassumetheproducthasthreelevelsofperformance:poor,average,andgoodwithinitialcostsc1>c2>c3andprobabilitiespm1,pm2,andpm3respectively.Themsuperscriptdenotesthemanufacturer'sprobabilitiesfortheproduct'sperformance.Theinitialcostsincreasewiththeageoftheproductattherategperperiod.Theconsumer'sbeliefsoftheperformancecostsarepc1,pc2,andpc3.Inthiscontext,theconsumerreplacestheproductatitseconomiclife. 2.4.1ConstantPerformanceWarrantyThemanufacturerisinterestedinofferingaCPPBWtotheconsumer.Intheabsenceofthewarranty,theconsumer'sinitialexpectedoperatingcostsareCc=pc1c1+pc2c2+pc3c3.Recallthesubscripttcanbedroppedduetostationarity.Usingthegrowthrateg,theproduct'smaximumusefullifeN,anddiscountfactor,theasset'seconomiclifeN*(agewhichminimizesdiscountedannualizedlife-cyclecosts)canbedetermined.Theconsumer'soptimalpolicywithoutaPBWistoreplacetheitemafterN*periods.UnderaPBW,theconsumer'sexpectedperiodiccostsofaoneperiodoldproductisCc=pc1min(c1,b)+pc2min(c2,b)+pc3min(c3,b).Settingtheannualequivalentcosts(AEC)withaPBWequaltotheAECwithoutaPBW,resultsin(assumingdiscrete,end-of-periodcashows): P+Pw+CcWXj=1j(1+g)j)]TJ /F8 7.97 Tf 6.59 0 Td[(1+CcNXj=W+1j(1+g)j)]TJ /F8 7.97 Tf 6.59 0 Td[(1=P+CcWXj=1j(1+g)j)]TJ /F8 7.97 Tf 6.59 0 Td[(1+CcNXj=W+1j(1+g)j)]TJ /F8 7.97 Tf 6.58 0 Td[(1 (2) Pw+CcWXj=1j(1+g)j)]TJ /F8 7.97 Tf 6.58 0 Td[(1=CcWXj=1j(1+g)j)]TJ /F8 7.97 Tf 6.59 0 Td[(1 24

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Cc=Cc)]TJ /F5 11.955 Tf 51.1 8.09 Td[(Pw PWj=1j(1+g)j)]TJ /F8 7.97 Tf 6.59 0 Td[(1pc1min(c1,b)+pc2min(c2,b)+pc3min(c3,b)=pc1c1+pc2c2+pc3c3)]TJ /F5 11.955 Tf 22.1 8.09 Td[(Pw G(W). (2) DeneG(v)=Pvj=1j(1+g)j)]TJ /F8 7.97 Tf 6.59 0 Td[(1tosimplifythenotation.TosolveEquation 2 ,weneedtoconsiderthreelevelsofperformance: ifc1bc2,thenEquation 2 is:pc1b+pc2c2+pc3c3=pc1c1+pc2c2+pc3c3)]TJ /F5 11.955 Tf 22.1 8.09 Td[(Pw G(W)b=c1)]TJ /F5 11.955 Tf 27.96 8.08 Td[(Pw pc1G(W), (2) ifc2bc3,thenEquation 2 is:pc1b+pc2b+pc3c3=pc1c1+pc2c2+pc3c3)]TJ /F5 11.955 Tf 22.1 8.09 Td[(Pw G(W)b=pc1c1+pc2c2 pc1+pc2)]TJ /F5 11.955 Tf 46 8.09 Td[(Pw (pc1+pc2)G(W), (2) ifc3b,thenEquation 2 is:pc1b+pc2b+pc3b=pc1c1+pc2c2+pc3c3)]TJ /F5 11.955 Tf 22.1 8.09 Td[(Pw G(W)b=C)]TJ /F5 11.955 Tf 22.1 8.09 Td[(Pw G(W). (2) Equations 2 2 ,and 2 formtheCPwarrantyfrontierforaconsumerwithbeliefs(pc1,pc2,pc3).Thefrontierispiecewiselinearandthesegmentscanbegeneralizedtotheform: b=Pkj=1pcjcj Pkj=1pcj)]TJ /F5 11.955 Tf 43.47 8.09 Td[(Pw G(W)Pkj=1pcjforckbck+1. (2) Wepresentanumericalexamplelater,andtheCPwarrantyfrontiersforthatexamplecanbeseeninFigure 2-2 .NotethatforagivenCPwarranty,(Pw,b,W),theeffectonthemanufacturer'stotaldiscountedrevenue,R,dependsontheconsumer'swarrantyfrontierandcanbefound 25

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bysubtractingthediscountedexpectedliabilityofprovidingthewarrantyfromtheextrarevenueprovidedbythesaleofthewarranty. RCP=Pw)]TJ /F5 11.955 Tf 11.96 0 Td[(G(W)kXj=1pmj(cj)]TJ /F5 11.955 Tf 11.95 0 Td[(b) (2) Bysubstitutingthegeneralizedwarrantyfrontierformula(Equation 2 )intoEquation 2 ,wecanthengeneralizeRas: RCP=Pw(1)]TJ /F10 11.955 Tf 13.15 18.44 Td[(Pkj=1pmj Pkj=1pcj))]TJ /F5 11.955 Tf 11.96 0 Td[(G(W)(kXj=1pmjcj)]TJ /F6 7.97 Tf 18.18 14.94 Td[(kXj=1pcjcjPkj=1pmj Pkj=1pcj). (2) Themanufacturer'schangeinrevenueisalsoapiecewiselinearfunction.Thetransitionpointsinthewarrantyfrontieroccurattheinitialperiodicoperatingcostsc1,c2,...,cL,thereforethetransitionpointsonthechangeinrevenuecurvealsocorrespondtoc1,c2,...,cL.Thus,aPBW,ifrevenueincreasing,willalwaysbeofferedsuchthatb2fc1,c2,...,cLg. Theorem2.1. Ifoneofthethreefollowingconditionshold,thenacustomerisindifferent(orbetteroff)byacceptingaCPwarrantyandamanufacturercanincreaseitsrevenue: i.) pc1>pm1 ii.) HpCPHnCP8(Pw,b,W)andHpCP>HnCPforsome(Pw,b,W) iii.) Cc(1)>Cm(1) Proof. i.) FromEquation 2 ,weseethateveryHCPbeginswiththewarranty(0,c1,W)8W.EvaluatingEquation 2 atthispointequatestoachangeofrevenueofzerosoeverychangeinrevenuecurvegoesthrough(0,0).FromEquation 2 ,theslopeoftherevenuecurveatthispointis(1)]TJ /F6 7.97 Tf 11.77 6.25 Td[(pm1 pc1).Sincepc1>pm1,then(1)]TJ /F6 7.97 Tf 11.77 6.25 Td[(pm1 pc1)>0 26

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whenc1bc2.ThereforeRCP>0andthemanufacturercanofferavalidwarrantyonthissegmentoftheHCPandincreaseitsrevenue. ii.) Ifacustomerispessimistic,thenhiswarrantyfrontier(HpCP)ispoint-wiselargerthanorequaltothewarrantyfrontieroftheneutralconsumer(HnCP),i.e.,HpCPHnCP8b2[0,c1]andHpCPHnCPforsomeb2[0,c1].Foraneutralconsumer,pcj=pmj8j.Therefore,fromEquation 2 ,RnCP=0foranywarrantyofferedontheneutralfrontier.Letb+representthesetofcaplevelswherePpw>Pnwandsincethereexistsatleastonepointwherethepessimisticwarrantyfrontierisstrictlylargerthantheneutralfrontier,thenb+isnon-empty.Foranyb2b+,Pnw)]TJ /F5 11.955 Tf 10.28 0 Td[(G(W)Pkj=1pmj(cj)]TJ /F5 11.955 Tf 10.27 0 Td[(b)=0andforthatsameb,Ppw)]TJ /F5 11.955 Tf 10.28 0 Td[(G(W)Pkj=1pmj(cj)]TJ /F5 11.955 Tf 10.27 0 Td[(b)>0sincePpw>Pnwandthusthemanufacturer'srevenueisincreased. iii.) Theprobabilitiesforboththeconsumerandmanufacturersumtoone.Then,Equation 2 simpliestoRCP=)]TJ /F5 11.955 Tf 9.3 0 Td[(G(W)(PLj=1pmjcj)]TJ /F10 11.955 Tf 13.45 8.97 Td[(PLj=1pcjcj)=G(W)(Cc(1))]TJ /F5 11.955 Tf 13.11 0 Td[(Cm(1))whenofferingavalidwarrantywithacostcapatcL.Therefore,whenCc(1)>Cm(1),wemusthaveRCP>0.Thus,themanufacturercanincreaseitsrevenue. Example:Toillustratetheeffectofaconsumer'sbeliefsonwarrantyfrontiersandthemanufacturer'schangeinrevenue,weuseanexamplewherea3-periodCPwarrantyisofferedforaproductwithvedifferentperformancelevels(c1>c2>c3>c4>c5).Table 2-1 providestheparametersfortheconsumerandmanufacturer,andTable 2-2 liststheperformancebeliefsforthreedifferenttypesofconsumers:neutral,pessimistic,andoptimistic.Notethatneutralbeliefscoincidewiththemanufacturer'sprobabilities. 27

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Table2-1. ParametersforPerformanceBasedWarranty(PBW)Example ParameterValue PurchasePrice(P)10000WarrantyLength(W)3ProductUsefulLife(N)10DiscountRate().9CostIncreaseRate(g).15InitialCostforLowPerformance(c1)1500InitialCostforMedium-LowPerformance(c2)1350InitialCostforMediumPerformance(c3)1200InitialCostforMedium-HighPerformance(c4)1100InitialCostforHighPerformance(c5)1000 Table2-2. ConsumerBeliefVectors ProbabilityNeutral/Manufacturer(N)Pessimistic(P)Optimistic(O) p1.2.3.1p2.2.25.15p3.2.2.2p4.2.15.25p5.2.1.3ExpectedInitialCost(C)12301292.51167.5 Figure 2-2 illustratestheCPwarrantyfrontiersforconsumerswitheachbeliefcharacterization.Thepessimisticconsumerwarrantyfrontierliesabovetheneutralfrontierandtheoptimisticconsumerfrontierfallsbelowtheneutralcurveasexpected.Thesquaresindicatewheretheslopeschange(atinitialoperatingcostsc1,c2,...,cL). Figure2-2. ConstantPerformanceWarrantyFrontiers 28

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Figure 2-3 showsthemanufacturer'sincreaseinrevenuefromofferingCPwarrantiesateachpointontheconsumerwarrantyfrontiersinFigure 2-2 .Thereisnowarrantythatcanbeofferedtoaneutraloroptimisticconsumerthatwillincreaserevenue.Sinceanywarrantyonaneutralconsumer'swarrantyfrontierwillhavenoeffectonthemanufacturer'srevenue,themanufacturercanonlygenerateadditionalrevenuebyofferingwarrantieswhenaconsumer'sfrontierisabovetheneutralwarrantyfrontier,e.g.,apessimisticconsumer.Forthepessimisticconsumerinthisexample,themanufacturerwillmaximizerevenuebyofferingthewarrantywithacostcapatcL. Figure2-3. ConstantPerformanceWarrantyChangeinRevenueCurves ThesquaresonarevenuecurveinFigure 2-3 againdenethelinearsegmentsofthegraphandcorrespondtothesquaresontheassociatedwarrantyfrontier.Thehorizontalportionsoftherevenuecurvesillustratethatrevenuescannotbeincreasedfurtherbyrelaxingtherestrictionthatthewarrantycoveragelevelbegreaterthanorequaltothelowestoperatingcost. 2.4.2ConstantCostWarrantyWearealsointerestedinevaluatingthewarrantyfrontierforaCCwarrantysuchthatr=0.Fromapracticalperspective,thismaybeattractivetoaconsumer 29

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becauseofitssimplicityinthatthecapdoesnotchangefromperiodtoperiod.Again,PwrepresentsthepriceofaCCwarrantywithcaplevelb,and(c1,c2,...,cL)representstheinitialperiodiccostsofthedifferentperformancelevels.Wemustalsointroducetheparameters(q1,q2,...,qL)whereqirepresentstherstperiodthatacostcapwillcovertheperformancelevelsassociatedwithc1,c2,...,cL,i.e.,qi(b)=argmini(1+g)i)]TJ /F8 7.97 Tf 6.58 0 Td[(1cj=fi:(1+g)i)]TJ /F8 7.97 Tf 6.59 0 Td[(1cjbg.Notethatqi(b)qi+1(b)8i.Undertheconditionc1bcL(i.e.,theinitialcaplevelisintherangeoftheinitialcosts),thenq1(b)=1.Thereforeqi(b)isafunctionofthecaplevelb,butfornotationalconvenience,weuseqiwiththeunderstandingitisdependentonthechoiceofthecaplevel.BeginningwithEquation 2 ,wedeterminetheconstantcapwarrantyfrontierbydiscountingthetermswherethewarrantycapisineffect:(pc1c1+pc2c2+pc3c3)WXi=1i(1+g)i)]TJ /F8 7.97 Tf 6.59 0 Td[(1=Pw+pc1(bWXi=1i)+pc2(c2q2)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xi=1i(1+g)i)]TJ /F8 7.97 Tf 6.58 0 Td[(1+bq2)]TJ /F8 7.97 Tf 6.58 0 Td[(1W)]TJ /F6 7.97 Tf 6.58 0 Td[(q2+1Xi=1i)+pc3(c3q3)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xi=1i(1+g)i)]TJ /F8 7.97 Tf 6.58 0 Td[(1+bq3)]TJ /F8 7.97 Tf 6.58 0 Td[(1W)]TJ /F6 7.97 Tf 6.58 0 Td[(q3+1Xi=1i).Solvingforthewarrantycoveragelevelb,weobtain: b=CcG(W))]TJ /F5 11.955 Tf 11.96 0 Td[(pc2c2G(q2)]TJ /F7 11.955 Tf 11.96 0 Td[(1))]TJ /F5 11.955 Tf 11.96 0 Td[(pc3c3G(q3)]TJ /F7 11.955 Tf 11.95 0 Td[(1))]TJ /F5 11.955 Tf 11.95 0 Td[(Pw pc1q1)]TJ /F8 7.97 Tf 6.58 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.96 0 Td[(q1+1)+pc2q2)]TJ /F8 7.97 Tf 6.59 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.95 0 Td[(q2+1)+pc3q3)]TJ /F8 7.97 Tf 6.58 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.96 0 Td[(q3+1).DeneU(v)=Pvj=1jtosimplifythenotation.Ingeneralizedformforanynumberofperformancelevels,thewarrantyfrontieris: b=CcG(W))]TJ /F10 11.955 Tf 11.96 8.97 Td[(PLj=2pcjcjG(qj)]TJ /F7 11.955 Tf 11.96 0 Td[(1) PLj=1pcjqj)]TJ /F8 7.97 Tf 6.58 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.96 0 Td[(qj+1))]TJ /F5 11.955 Tf 76.41 8.08 Td[(Pw PLj=1pcjqj)]TJ /F12 5.978 Tf 5.76 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.95 0 Td[(qj+1). (2) ForagivenCCwarranty(Pw,b,W),theeffectonthemanufacturer'srevenueis: RCC=Pw)]TJ /F6 7.97 Tf 18.18 14.94 Td[(LXj=1pmj(cj((1+g))qj)]TJ /F8 7.97 Tf 6.59 0 Td[(1G(W)]TJ /F5 11.955 Tf 11.95 0 Td[(qj+1))]TJ /F5 11.955 Tf 11.95 0 Td[(bqj)]TJ /F8 7.97 Tf 6.59 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.95 0 Td[(qj+1)). (2) 30

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Substitutingtheconsumer'swarrantyfrontier(Equation 2 )intoEquation 2 yieldsthefollowingchangeinrevenuecurveforthemanufacturer: RCC=Pw(1)]TJ /F10 11.955 Tf 13.15 17.12 Td[(PLk=1pmkqk)]TJ /F8 7.97 Tf 6.59 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.96 0 Td[(qk+1) PLj=1pcjqj)]TJ /F8 7.97 Tf 6.59 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.96 0 Td[(qj+1))+LXk=1pmkqk)]TJ /F8 7.97 Tf 6.58 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.96 0 Td[(qk+1)(CcG(W))]TJ /F10 11.955 Tf 11.96 8.96 Td[(PLj=2pcjcjG(qj)]TJ /F7 11.955 Tf 11.96 0 Td[(1) PLj=1pcjqj)]TJ /F12 5.978 Tf 5.75 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.96 0 Td[(qj+1)))]TJ /F6 7.97 Tf 17.51 14.94 Td[(LXk=1pmkck((1+g))qk)]TJ /F8 7.97 Tf 6.59 0 Td[(1G(W)]TJ /F5 11.955 Tf 11.95 0 Td[(qk+1). (2) UsingEquation 2 ,wecandeterminetheCCwarrantyfrontiersforthesameparametersandconsumersusedfortheCPexample.Figure 2-4 showstheCCwarrantyfrontiers,andtheassociatedchangesinrevenueareshowninFigure 2-5 Figure2-4. ConstantCostWarrantyFrontiers Thesquaresonthegraphsrepresenttheminimumandmaximumwarrantypricesnecessarytoensurethewarrantyleveliscontainedintherangeoftheinitialperiodicoperatingcosts(c1bcL).EachCCwarrantyfrontierandrevenuecurveincludewarrantypricesthatcorrespondtowarrantylevelsgreaterthanthehighestpossibleinitialoperatingcost;thisistoshowtheconvergenceofthecurvestothepoint(0,0).TheCCwarrantyfrontiersandrevenuecurvearealsopiecewiselinear.Thenumber 31

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ofsegmentsdeningthecurvesislargerincomparisontotheCPwarranty.Forthisexample,therearetwiceasmanysegmentssothegraphsappeartobesmooth. Figure2-5. ConstantCostWarrantyChangeinRevenueCurves 2.5FiniteHorizonwithTechnologyUpgradeNowassumeaconsumerownsanassetandwillrequiretheservicesofthisasset(orsimilarasset)foraniteperiodoftimeT.Atthebeginningofthehorizon,atechnologicallyadvancedassetisintroducedandtheexistingversionisobsoleteandnolongeravailableforpurchase.Theconsumermaykeeptheoriginalitem,uptoitsusefullife,thenpurchaseareplacement.Weareinterestedinstudyinghowawarrantymightbestructuredtopersuadetheconsumertoupgradeearlier.Therefore,thecustomeronlyreceivesthewarrantyiftheyreplacetheiroriginalproductsoonerthanplanned.Thisprovidesaneconomicincentivetothemanufacturertoofferthewarranty.Wemodeltheproblemasadynamicprogramwiththefollowingparameters:neistheageoftheexistingassetownedattimezero,Neisthemaximumusefullifeofthecurrentitem,N0eistheplannedreplacementagewhenawarrantyisnotoffered(i.e.,N0eNe),Nuisthemaximumusefullifeoftheupgradeditem,andWislengthofthewarranty. 32

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Thestateoftheprocessistheduple(n,w)wherenistheageoftheassettheconsumerownsandwistheremainingnumberofperiodsonthewarranty.Letgbethegrowthrateoftheperiodicoperatingcostsandristhegrowthrateofthewarrantylevelasnincreases.Pisthecostoftheupgradedasset(whichweassumeisconstantovertime)andc0j:j0=1,...,LrepresentsthecostsassociatedwiththeLperformancelevelsofthenewitem.Again,wemaintainc0j>c0j+1:j0=1,...,L.Foraconsumerwithbeliefspcj:j2f1,...,Lg,wedeneCe=PLi=1pcjcjtobetheexpectedinitialcostoftheexistingasset,Cu=PLi=1pcjc0jtobetheexpectedinitialcostoftheupgradedasset,andCu(n)=PLi=1pcjmin(b(1+r)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1,c0j(1+g)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1)tobetheexpectedinitialcostoftheupgradedassetunderaPBW.Note,theterminitialexpectedcostreferstotheoperatingcostofaproductinitsrstyearofoperation.Theperiodicdiscountrateisusedtoaccountforthetimevalueofmoney.Theconsumer'sdecisioneachperiodistokeeporreplacetheircurrentlyownedproduct,modeledas: vt(n,0)=min8><>:Keep:(Ce(1+g)n+vt+1(n+1,0))Replace:P+(Cu(1)+vt+1(1,W)]TJ /F7 11.955 Tf 11.96 0 Td[(1))9>=>;,nent(2) vt(n,0)=min8><>:Keep:(Ce(1+g)n+vt+1(n+1,0))Replace:P+(Cu+vt+1(1,0))9>=>;,N0ent(2) vt(n,w)=min8><>:Keep:(Cu(n)+vt+1(n+1,w)]TJ /F7 11.955 Tf 11.96 0 Td[(1))Replace:P+(Cu+vt+1(1,0))9>=>;,nt,1wW(2) vt(n,0)=min8><>:Keep:(Cu(1+g)n+vt+1(n+1,0))Replace:P+(Cu+vt+1(1,0))9>=>;,nt(2) vt(Ne,0)=minReplace:P+(Cu+vt+1(1,0)),Ne>t(2) vt(Nu,w)=minReplace:P+(Cu+vt+1(1,0)),Nut,8w.(2) vT+1(n,w)=0,8n,w.(2) 33

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Withntindicatesthatthecurrentitemistheobsoletetechnology.Equation 2 denestheconsumer'schoicetokeeptheoriginalproductthroughtheoriginallyplannedreplacementagesuchthatnowarrantyisoffered.Equation 2 showstheconsumer'schoicewhentheproductisunderwarranty.Notethatthewarrantyisonlyofferedwiththerstearlyupgrade.Equation 2 denestheconsumer'schoicewhentheupgradeisnotunderwarranty.Equations 2 2 ,and 2 representthereplacementoftheobsoleteitematitsmaximumusefullife,thereplacementoftheimproveditematitsmaximumusefullife,andtheboundaryconditionfortheDP,respectively.Notethatweassumetheproducthasnoresale/salvagevalue,butthisiseasilyincorporated.Tofurtherillustratetheprocess,letusconsiderthefollowingexample.SupposetheproductinTable 2-1 istheexistingitem,ne=3,andT=20.Atthestartofthetimehorizon(whichcorrespondstotheexistingitembeingthreeperiodsold),anewproductisintroducedwithapurchasepriceofP=10500.UsingEquations 2 2 andlettingCu(n)=Cu(1+g)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1,wecandetermineaconsumer'soptimalreplacementpolicywhennowarrantyisoffered.Table 2-3 showstheoptimalpolicyfortheconsumersfromTable 2-2 whenthereplacementitemisoneofsixdifferenttechnologies.Thetechnologiesarenumbered0-5intherstcolumn,andthesecondcolumnrepresentsthepercentreductionininitialperiodicoperatingcostscomparedtotheinitialoperatingcostsoftheoldtechnology(listedinTable 2-1 ).Theactualcostsarelistedincolumn3.Thetriple(x,y,z)incolumns4-6representstheconsumerspurchasepolicy:xisthenumberofperiodstheobsoleteitemiskept,yisthenumberofperiodstherstupgradeiskept,andzisthenumberofperiodsthenalupgradeiskept.Note,thevalueofxforagiventechnologydenesthewindowforwhichawarrantywouldbeoffered,i.e.N0e=xintheDPformulation.Sincexrepresentstheplannedreplacementwithnowarranty, 34

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thenx)]TJ /F7 11.955 Tf 12.18 0 Td[(1representsthenumberofperiodsthatthewarrantywouldbemadeavailabletotheconsumer. Table2-3. ConsumerPoliciesforVariousLevelsofTechnologyImprovement(ne=3) UpgradeVersion%ReductioninciCosts(c01,c02,c03,c04,c05)NeutralPessimisticOptimistic 00(1500,1350,1200,1100,1000)(6,8,6)(5,8,7)(6,8,6)15(1425,1282.5,1140,1045,950)(5,8,7)(5,8,7)(6,8,6)210(1350,1215,1080,990,900)(5,8,7)(5,8,7)(5,8,7)315(1275,1147.5,1020,935,850)(5,8,7)(5,8,7)(5,9,6)420(1200,1080,960,880,800)(5,9,6)(5,8,7)(5,9,6)525(1125,1012.5,900,825,750)(4,9,7)(4,9,7)(5,9,6) Forclarity,letuslookatupgradeversion3(thirdrow)inTable 2-3 wherethenewproducthas15%lowerperiodicoperatingcoststhantheexistingasset.Theneutralorpessimisticconsumerwillkeeptheoriginalproductthroughthefthperiod(theageoftheitemiseight),replaceitatthebeginningofperiodsix,andthenreplaceitonceagainatthebeginningofperiod14.Forthesametechnology,theoptimisticconsumerwillreplacetheexistingitematthesametimebutwillholdtherstupgradeoneperiodlonger,thusmakingthesecondupgradeaperiodlater.Tomotivateanearlierreplacementoftheoriginalitemandgenerateadditionalrevenue,themanufacturerconsidersofferingtheconsumereitheraCPorCCPBWwhentheobsoleteproductisreplacedearlierthantheplannedreplacementwhennoPBWisoffered. 2.5.1ConstantPerformanceWarrantyForaconstantperformancePBW,theexpectedgrowthrateofthewarrantylevelissetequaltothegrowthrateoftheoperatingcosts,i.e.,r=g.IntheDPformulationfortheCPwarranty,theexpectedperiodicoperatingcostscanbesimpliedtoCu(n)=(1+g)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1PLi=1pjmin(b,c0j).Table 2-4 showstheeffectsofofferingano-costCPPBWontheconsumer'spurchasepolicyandthemanufacturer'srevenueforthevedifferenttechnologyimprovementsdenedinTable 2-3 (recallthatthegoalistochangetheconsumer'spurchasepolicy).Thewarrantylevelslistedincolumn2andtheassociatedwarrantylengthincolumn3denetheno-costwarrantywhichgeneratesthegreatest 35

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revenueincreasefromthebaseline(nowarrantyoffered)solution.Column4representsthenewconsumerpurchasepolicywhenthewarrantyisoffered;thenotationisthesameasTable 2-3 .Column5indicatesthemanufacturer'stotaldiscountedrevenueovertheentireproblemhorizon,T,andcolumn6showsthepercentagechangeinrevenuerelativetothetotaldiscountedrevenuewhennowarrantyisoffered. Table2-4. OptimalZero-CostCPWarranties(ne=3) UpgradeVersionbWNewPolicyRevenuewithPBWRevenue Neutral 11131.455(4,8,8)9567.587.88%21197.019(4,9,7)9348.645.41%31128.086(4,9,7)9429.336.32%41134.806(4,9,7)9507.5710.52%5779.383(3,9,8)10316.357.93% Pessimistic 11274.688(4,8,8)9685.519.21%21217.017(4,8,8)9732.389.73%31212.508(4,9,7)9491.207.01%41112.951(4,9,7)9547.737.65%51003.009(3,9,8)10441.119.24% Optimistic 11422.524(5,8,7)8867.9911.10%2933.313(4,9,7)9227.214.04%3866.332(4,9,7)9338.798.56%4876.772(4,9,7)9410.719.40%5853.751(4,9,7)9501.7710.46% UnderthewarrantiespresentedinTable 2-4 ,theconsumerchangestheirbehaviorwhenpresentedwithawarrantythatisatleastasgoodaswhatislisted.Forexample,whenthetechnologyimprovementisversion3,apessimisticconsumerwillacceptanyCPwarrantythatiseightperiodsorlongerwithawarrantycoveragelevellessthanorequalto1212.50.Notethattheoptimalvalueswerefoundundertherestrictionthattheinitialwarrantylevelwaswithintherangeofinitialoperatingcosts(c01bc05).Withinthisrange,therearesomewarrantylengthsthatarenotfeasibleandthuscannotchangetheconsumer'sbehavior.Afeasiblewarrantyisdenedasawarrantythatdoesnotincreasetheconsumer'stotaldiscountedexpectedcosts.Inthesecases,theincreaseinperformanceisnotlargeenoughtooffsettheconsumer'sincreasedcostofpurchasinganupgradeearlier,ortheperformanceincreaseissolargethatthe 36

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availabilityoftheupgradealonecausedtheconsumertoupgradeearlyenoughthatawarrantywouldhavenoeffect.Theoptimalwarrantylengthcorrespondstotheminimumwarrantylengthaconsumerwouldacceptatagivencostcapbecauseifaconsumeriswillingtoacceptwarranty(0,b,W),hewouldsurelyaccept(0,b,W+1).TheeffectonrevenueofofferingaPBWsuchthattheconsumermakestherstupgradeearlierthanplannedisfoundforanyfeasiblewarranty.Figure 2-6 showsallfeasiblewarrantiesforapessimisticconsumerwhentheupgradeoffersa15%reductioninoperatingcosts.Theredhorizontallinerepresentsthemanufacturer'srevenuewhennowarrantyisofferedsoanywarrantyabovethelinewouldincreasesthemanufacturer'srevenue. Figure2-6. FeasibleConstantPerformanceWarrantiesforPessimisticConsumers Theeffectoftheleveloftechnologyimprovementcanhaveasignicanteffectonthemanufacturer'srevenue.Figure 2-7 showsthemaximumrevenueforapessimisticconsumerforthevedifferenttechnologicalimprovementslistedinTable 2-4 .Noticethecurvefortechnologyversion3(whichcorrespondstoa15%reductioninexpectedperiodicoperatingcosts)canbeobtainedbyndingthemaximumenvelopofthecurves 37

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inFigure 2-6 ,i.e.,theshortestlengthwarrantyforagivencostcapwherethewarrantywouldstillchangetheconsumer'sbehavior.Thehorizontalportionsofthesegraphsrepresentthemanufacturer'srevenuewhennowarrantyisofferedorequivalently,whenawarrantyisofferedthatdoesnotchangetheconsumersbehavior.Notethattheeffectoftechnologyonthemanufacturer'srevenueisnotmonotonic.Inthisexample,themanufacturermaximizesrevenuewiththemosttechnologicallyimprovedproduct,buttheproductwitha10%reductioninoperatingcostsprovidesalargerincreaseinrevenuethaneitherthe15%or20%technology. Figure2-7. MaximumRevenueCurvesforConstantPerformanceWarrantiesforDifferentLevelsofTechnologyImprovement Toobtaintheseoptimalno-costwarranties,theDPwassolvedNutimeswithb=c05andW2f1,...,Nug.ThisestablishedaninitialpointoneachoftherevenuecurvescorrespondingtoaxedvalueforW.Then,basedontheconsumer'sspendinglevelunderthewarrantyrelativetospendingwhennowarrantyisoffered,theremainderofthewarrantycurvecanbederived.Iftheconsumer'sspendingatthisinitialpointisgreaterthantheirspendingwhennowarrantyisoffered,thennofeasiblewarrantywith 38

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lengthWwillbeacceptabletotheconsumersinceincreasingthewarrantylevelonlyincreasesspending.Ifthespendinglevelunderthewarrantyislessthanorequaltothespendingwithnowarranty,thenthewarrantycoveragelevelcanbeincreaseduntilthespendinglevelunderthewarrantyequalsthespendingwithnowarranty.Notethatthechangefromtheinitialwarrantypointtowherethespendinglevelsareequivalentispiecewiselinearduetothefactthatchangesinthewarrantycoveragelevelaffectwhichlevelsofperformancearecoveredbythewarranty.ThemaximumenvelopoftheseNucurvesformthemaximumrevenuecurve.Wecanseethereisawiderangeofwarrantiesthatcanbeofferedtoincreasethemanufacturer'srevenue.Indeterminingtheoptimalwarranty,wefoundthelargestrevenuevalueofthelocalmaximaonthemaximumrevenuecurve(Figure 2-7 ).Foraneutralconsumer,allthelocalmaximaarethesamesincetheconsumerandmanufacturersharethesameperformancebeliefs.Foranon-neutralconsumer,thisisnotthecaseandeachlocalmaximumneedstobeevaluatedseparately.Notethegraphsareactuallydiscontinuouspiecewiselinearfunctions(theverticalsegmentsarebyproductsofconnectingthepointsofdiscontinuity).EachsegmentcorrespondstoawarrantylengthasshowninFigure 2-6 .Therefore,evaluatingthelocalmaximumwouldberequiredforatmostWpoints.IntheDPformulation,Equations 2 2 ,thecostofthewarrantywaszerounderthepremisethatthepurposeofthewarrantywastochangeconsumerbehavior.UsingtheresultsfromtheinnitehorizonmodelandtherevenuecurvesforeachW2f1,...,Nag,wecandetermineifthemanufacturercanchargeforthewarrantyandthusincreaserevenuefurtherwhilestillmaintainingtheconsumer'sbehaviorchange.Webegintoanswerthisquestionbylookingatthemaximumrevenuecurve.FromtheresultsinTable 2-4 ,weseethatthemaximumrevenueforapessimisticconsumerwhentheimprovedtechnologyreducesinitialoperatingcostsby15%isachievedbyofferinga(0,1212.5,8)CPwarranty(thismaximumisactuallyachievedateverypeakfor 39

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thispessimisticconsumerwhenthecostcapassociatedwiththemaximumrevenueisbetweenc1andc5).Forthiswarranty,themanufacturer'srevenueisR=9491.20.Ifwedenetheconsumer'sspendingforagivenwarranty(Pw,b,W)asS(Pw,b,W),thentheconsumer'stotalspendingassociatedwiththemanufacturer'srevenueisS(0,1212.5,8).Thisspendingmustnecessarilyequaltheconsumer'sspendingwhennowarrantyisoffered(S0).Otherwise,themanufacturercouldraisethecostcapleveluntilS(0,1212.5,8)=S0whichwoulddecreasethemanufacturer'sliabilityandincreasehisrevenue.SinceS(0,1212.5,8)=S0,theconsumerstillacceptsthisoffer.Onagivensegmentofthemaximumrevenuecurve,WisxedandS0isthedecisionmakingcriteria(insteadofannualequivalentcostwithnowarrantythatwasusedintheinnitehorizonmodel).UsingtheCPwarrantyfrontierequationfromtheinnitehorizonmodel,wecandeterminetherelationshipbetweenthewarrantypriceandcostcapleveltoensureawarrantyremainsacceptabletotheconsumerasthepriceincreasesfromzero.DifferentiatingEquation 2 withrespecttowarrantyprice,wehave: @Pw @b=)]TJ /F5 11.955 Tf 9.3 0 Td[(G(W)kXj=1pcj,forckbck+1. (2) Therefore,foreveryunitincreaseinthewarrantyprice,thecostcapmustdecreasebythemagnitudeofEquation 2 .Forthemanufacturer,totaldifferentiationandthelinearrelationshipamongthevariablesinEquation 2 allowsustodeterminetherelationshipbetweenwarrantypriceandcostcapthatkeepstherevenueunchangedforthemanufacturer:@R @b @R @Pw=@Pw @b=)]TJ /F5 11.955 Tf 9.3 0 Td[(G(W)kXj=1pmj,forckbck+1. (2)Forthemanufacturer,bmustdecreasebythemagnitudeofEquation 2 foreveryunitincreaseinwarrantypricefortherevenuetoremainunchanged.IfthechangeinbislessthanEquation 2 ,thenthemanufacturer'srevenueincreases.Bytakingthe 40

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ratiobetweenEquations 2 and 2 ,wecanstatetheconditionsforchargingforthewarrantyonaparticularsegmentofawarrantyrevenuecurve:Pkj=1pcj Pkj=1pmj=1:Revenuedoesnotchangebycharging,Pkj=1pcj Pkj=1pmj>1:Chargingforwarrantyincreasesrevenue,Pkj=1pcj Pkj=1pmj<1:Chargingforwarrantydecreasesrevenue.Thechargingconditionratiosrepresentthechangeinwarrantylevelneededfortheconsumertoremainatthesamespendinglevelaswhenthewarrantywaszerocostcomparedtothechangeinwarrantylevelneededforthemanufacturertoremainrevenueneutralgivenaunitchangeinthepriceofthewarranty.Furthermore,thedifferencebetweenthepartialderivativesfromEquations 2 and 2 ,adjustedbythediscountfactor,representsthechangeindiscountedrevenueforthemanufacturerforeveryunitchangeinthewarrantycostcapwhilecontrollingforthecorrespondingchangeinwarrantyprice,i.e.,@R @bjPw=dhG(W)(Pkj=1pcj)]TJ /F10 11.955 Tf 11.63 8.97 Td[(Pkj=1pmj)wherehrepresentsthenumberofperiodstheexistingitemisheld.NotethattherangeoftherulesdependsnotonlyonthevaluesofthecostcapthatdenethesegmentforaparticularvalueofWbutalsoontherangeckbck+1.Ifwedene^bwandbwasthelowerandupperbounds,respectively,ofthemaximumwarrantycurveforagivenW,thentherangeoftheconditionsaboveis[max(^bw,ck),bw].SincethecashowfactorG(W)isanelementinthechangeofrevenuerateandisincreasinginW,therevenuechangesatafasterrateonlowerrevenuecurves.Thisinsightprovidesanalgorithmtosolvefortheunrestricted(relaxingPw=0),optimalwarranty(Pw,b,W).First,determinetherestricted(Pw=0)revenuecurvesforeachpossiblewarrantylengthasshowninFigure 2-6 .Thiscanbedoneinpolynomial 41

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timeintheinput,O(T2N2u+NuL).Thestepstobuildtherevenuecurvesareprovidedbelow: 1) SolvetheDPformulation(Equations 2 2 )forthe(0,b,W)warranty,withb=cL,foreachW2f1,...,Nug.ThesolutionofoneDPcanbecompletedinO(T2Nu).Initializej=L. 2) Foreach(0,b,W)point,increasebfromcjuntilb=cj)]TJ /F8 7.97 Tf 6.58 0 Td[(1orS(0,b,W)=S0,whicheveroccursrst,throughlinearlyextrapolation.ThisisO(1). 3) Ifb=c1orS(0,b,W)=S0,therevenuecurveiscompleteforthatparticularW.Otherwise,ifb=cj)]TJ /F8 7.97 Tf 6.58 0 Td[(16=c1isthestoppingconditionforincreasingbinstep2,letj=j-1andreturntostep2.Notethattheextrapolationnowusedinstep2willbedifferentfromthepreviousextrapolationbecausethewarrantylevelisinadifferentrangeofperformancecosts.Buildingthefullrevenuecurvesforeachw2f1,...,NugfromtheNupointsinstep1isO(NuL).Toillustratethisprocess,letusconsiderpointAinFigure 2-6 .Step1determinestherevenueatthispointandrequirestheDPtobesolvedonceforthewarranty(0,b,4)whereb=cL.SinceS(0,cL,4)
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thethelargestperformancelevelcost(cj)thatislessthanthecostcapassociatedwiththemaximumrevenuecostcap(bmax).Repeatthisprocessmovingfromcjtocj+1untilcLhasbeenreached.Notethattherstiterationmovesfrombmaxtotheclosest,smallercjandeverysubsequentmoveisfromcjtocj+1.ImplementingthealgorithmoverallpossiblevaluesofWcanbecompletedinO(NuL).ThereforethetotaltimetondtheoptimalwarrantyisO(T2N2u+NuL).Table 2-5 showstheeffectonthewarrantydesignandmanufacturer'srevenueforaCPwarrantyifthemanufactureriswillingtochargeforthewarranty.Thepercentagechangeincolumn6iscalculatedwithrespecttothemaximumrevenuegeneratedwhentheoptimalno-costCPPBWisoffered. Table2-5. OptimalUnrestricted-CostCPWarranties(ne=3) UpgradeVersionPwbWRevenueRevenue Neutral 101131.4559567.580201197.0199348.640301128.0869429.330401134.8069507.57050779.38310316.350 Pessimistic 11880.4095089920.432.43%21865.2490089971.942.46%32452.23850109821.423.48%42444.04800109888.283.57%51904.737501010704.352.52% Optimistic 101422.5248867.99020933.3139227.21030866.3329338.79040876.7729410.71050853.7519501.770 Thepessimisticconsumerwe'veusedinthepreviousexampleisaspecialcase.Ingeneral,whentheconsumermodiestheirbeliefs,anewperformancecostdistributionisdenedsuchthatithasthesamerangeasthemanufacturer'scostdistribution.Afterthedistributionmodication,iftheconsumer'sdistributionofperformancecostsisrst 43

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orderstochasticdominanttothemanufacturer'sdistribution,thenthefollowingistrue: kXj=1pcjkXj=1pmj8k2f1,...,Lg. (2) Inthiscase,theoptimalcostcapoccursatcLforsomeW(seethepessimisticconsumersectionofTable 2-5 ).ThiscanbeseenfromthefactthatthechargingconditionismetalongeverysegmentofarevenuecurvewhichdrivesthewarrantyleveltocL.ThelengthofthewarrantywilltendtobeaslargeaspossiblebecausetherateofchangeinrevenueisgreaterforlargervaluesofW.Thiswon'talwaysdriveWtoNu(orevenensurethatthewarrantylengthislongerthantheoptimalsolutionforano-costCPwarranty)becausealargerwarrantylengthmightinduceadifferentconsumerpolicythanW)]TJ /F7 11.955 Tf 12.19 0 Td[(1,orthemaximumrevenueforano-costwarrantyoflengthWmaybesignicantlylessthanano-costwarrantyoflengthW)]TJ /F7 11.955 Tf 11.96 0 Td[(1.Similarly,aspecialcaseoftheoptimisticconsumerisiftheconsumer'sdistributionisrst-orderstochasticdominatedbythemanufacturer'sdistribution.Thisleadstothecondition: kXj=1pmjkXj=1pcj8k2f1,...,Lg. (2) Underthiscondition,theoptimalwarrantywillbetheoptimalno-costwarranty.Toseethis,comparetheoptimisticconsumersectionofTable 2-5 totheoptimisticconsumersectionofTable 2-4 .WhenEquation 2 holds,thechargingconditionwillneverbemetandthusthereisnorangeofwarrantylevelsoverwhichchargingforthewarrantywillincreaserevenue.Thisisnotsurprisingsincewewouldexpectanoptimisticconsumertohavefaithintheproductandtherefore,beunwillingtopayforthewarranty. 2.5.2EffectofMeanandVarianceUsingthepreviousalgorithm,wecaneasilyshowtheeffectofmodelingoperatingcostswithadistributionasopposedtoexpectedvalues.Table 2-6 showstheeffect 44

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onrevenueifthemanufactureroffersaCPwarrantyunderthesameconditionsastheexampleinsection 2.5.1 ,exceptthatthemanufacturerandconsumerbeliefsaredifferent.Inthesevencombinationsofbeliefsinthetable,theexpectedperiodicoperatingcostsforbothpartiesarethesame,butthevariancesofthecostsdiffer.Therstentryshowsthemanufacturer'srevenuewhenbothpartieshavethesamedeterministicbeliefofoperatingcosts.Thenextthreeentriesshowhowthemanufacturer'srevenueincreasesasthevarianceoftheconsumer'sbeliefsincreases.Thenextthreeentriesshowhowthemanufacturer'srevenuedecreasesasthevarianceoftheoperatingcostsincreases. Table2-6. EffectofVarianceonRevenuewith15%TechnologyImprovement(ne=3) ManufacturerBeliefConsumerBeliefRevenuePwbWVariance (0,0,1,0,0)(0,0,1,0,0)9415.670962.814 (0,0,1,0,0)(0,.24,.4,.36,0)9593.3759.88102089000(0,0,1,0,0)(0,.4,0,.6,0)9691.50226.071020815000(0,0,1,0,0)(.4,0,0,0,.6)9963.83641.541020860000 (0,.24,.4,.36,0)(0,0,1,0,0)9415.670901.5429000(0,.4,0,.6,0)(0,0,1,0,0)9415.670901.54215000(.4,0,0,0,.6)(0,0,1,0,0)9394.9446.93850160000 Additionally,wecandemonstratetheeffectofdifferentexpectationsonmanufacturerrevenue.Table 2-7 showshowthemanufacturer'srevenuevarieswhentheexpectedoperatingcostsofthetwopartiesaredifferent.Inthevebeliefcombinations,thevarianceofboththemanufacturer'sandconsumer'sbeliefsiszero.Therstentryprovidestherevenuewhenthemanufacturerandconsumerhavethesameexpectedoperatingcosts.Thesecondandthirdentriesshowhowthemanufacturer'srevenueincreasesastheconsumer'sexpectationofperformancecostsincreases.Thefourthandfthentriesshowthatthemanufacturer'srevenuedecreasesastheconsumer'sexpectationofhighperformance(i.e.,lowercosts)increases.Noticethatwhentheconsumer'sexpectationisatthelowestpossiblecost,themanufacturercan'tevenofferawarrantythattheconsumerwillaccept. 45

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Table2-7. EffectofMeanonRevenuewith15%TechnologyImprovement(ne=3) ManufacturerBeliefConsumerBeliefRevenuePwbW (0,0,1,0,0)(0,0,1,0,0)9415.670962.814 (0,0,1,0,0)(0,1,0,0,0)10193.941076.9610209(0,0,1,0,0)(1,0,0,0,0)11709.701660.8710208 (0,0,1,0,0)(0,0,0,1,0)9142.950853.234(0,0,1,0,0)(0,0,0,0,1)7741.99 2.5.3ConstantCostWarrantyAsdemonstratedintheinnitehorizonproblem,CCwarrantiesaffectconsumersdifferentlythanCPwarranties,andthus,theycanaffectthemanufacturer'srevenuedifferently.Foraconstantcostwarranty,thewarrantyleveldoesnotgrowsor=0.ThusCu(n)=PLi=1pjmin(b,(1+g)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1c0j)intheDPformulation.Table 2-8 belowshowsthemaximumrevenuegeneratedfromano-costCCwarrantyforagivenconsumerforthetechnologyimprovementsinTable 2-3 .NotethattheCPwarrantygeneratesmorerevenuefortheoptimisticconsumerbuttheCCwarrantygeneratesmorerevenueforthepessimisticconsumer.Aswouldbeexpected,thereisnodifferencebetweentherevenuesforaneutralconsumeralthoughthewarrantydesignsaredifferent. Table2-8. OptimalZero-CostCCWarrantiesonManufacturerRevenue(ne=3) UpgradeVersionbWNewPolicyRevenuewithPBWRevenue Neutral 11202.123(4,8,8)9567.587.88%21013.552(4,9,7)9348.645.41%31047.102(4,9,7)9429.336.32%4937.641(4,9,7)9507.5710.52%5908.533(3,9,8)10316.357.93% Pessimistic 11306.033(4,8,8)9667.459.00%21298.343(4,8,8)9720.899.60%31145.882(4,9,7)9487.106.97%41112.951(4,9,7)9547.737.65%5994.103(3,9,8)10419.559.01% Optimistic 11414.561(5,8,7)8867.9911.10%2920.312(4,9,7)9257.904.38%3936.732(4,9,7)9342.068.60%4952.882(4,9,7)9414.449.44%5853.751(4,9,7)9501.7710.46% 46

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Figure 2-8 showsthefeasibleCCwarrantiesforapessimisticconsumerwhentheupgradeoffersa15%reductioninoperatingcosts.UnliketheCPwarrantyinFigure 2-6 ,theCCwarrantycannotincreaserevenuewithawarrantyofanylength;onlyshorterlengthwarrantieswillincreaserevenue. Figure2-8. FeasibleConstantCostWarrantiesforPessimisticConsumers Figure 2-9 showsthemaximumrevenuecurveswhenapessimisticconsumerisofferedazero-costCCPBWforthevedifferenttechnologiesinTable 2-8 .TherelativerelationshipofthedifferenttechnologyversionsarethesameasfortheCPwarrantyasshowninFigure 2-7 47

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Figure2-9. MaximumRevenueCurvesforCCWarrantiesforDifferentLevelsofTechnologyImprovement WecannowdeterminetheoptimalunrestrictedCCwarrantybymodifyingthealgorithmfromsection 2.5.1 .FortheCCwarranty,therelationshipbetweenthewarrantypriceandcostcaplevelisfoundbydifferentiatingEquation 2 withrespecttowarrantyprice.Recallthatqisafunctionofthewarrantycap,b,thereforethechangeinwarrantypricewithrespecttothewarrantycoveragelevelisonlyvalidovertherangeofbsuchthattheqfunctionisconstant.Denebkandbk+1astherangeoverwhichthegivenqfunctiondoesnotchange. @Pw @b=)]TJ /F6 7.97 Tf 17.51 14.94 Td[(LXj=1pcjdqj)]TJ /F8 7.97 Tf 6.59 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.95 0 Td[(qj+1)forbkbbk+1 (2) Therefore,foreveryunitincreaseinthewarrantyprice,thecostcapmustdecreasebythemagnitudeofEquation 2 .SimilartotheCPwarranty,Equation 2 allowsustodeterminetherelationshipbetweenwarrantypriceandcostcapthatkeepstherevenueunchangedforthemanufacturer: 48

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@R @b @R @Pw=@Pw @b=)]TJ /F6 7.97 Tf 17.52 14.95 Td[(LXj=1pmjdqj)]TJ /F8 7.97 Tf 6.59 0 Td[(1U(W)]TJ /F5 11.955 Tf 11.95 0 Td[(qj+1)forbkbbk+1. (2)Forthemanufacturer,bmustdecreasebythemagnitudeofEquation 2 foreveryunitincreaseinwarrantypricefortherevenuetoremainunchanged.Clearly,bcannotdecreaseoutoftherangedenedbytheperformancecostsandtheqfunction.IfthechangeinbislessthanEquation 2 ,thenthemanufacturer'srevenueincreases.Again,theratiobetweenEquations 2 and 2 providestheconditionsforchargingforthewarrantyonaparticularsegmentofawarrantyrevenuecurve:PLj=1pcjdqj)]TJ /F12 5.978 Tf 5.75 0 Td[(1U(W)]TJ /F6 7.97 Tf 6.58 0 Td[(qj+1) Pkj=1pmjdqj)]TJ /F12 5.978 Tf 5.76 0 Td[(1U(W)]TJ /F6 7.97 Tf 6.59 0 Td[(qj+1)=1:Revenuedoesnotchangebycharging,PLj=1pcjdqj)]TJ /F12 5.978 Tf 5.75 0 Td[(1U(W)]TJ /F6 7.97 Tf 6.58 0 Td[(qj+1) Pkj=1pmjdqj)]TJ /F12 5.978 Tf 5.76 0 Td[(1U(W)]TJ /F6 7.97 Tf 6.59 0 Td[(qj+1)>1:Chargingforwarrantyincreasesrevenue,PLj=1pcjdqj)]TJ /F12 5.978 Tf 5.75 0 Td[(1U(W)]TJ /F6 7.97 Tf 6.58 0 Td[(qj+1) Pkj=1pmjdqj)]TJ /F12 5.978 Tf 5.76 0 Td[(1U(W)]TJ /F6 7.97 Tf 6.59 0 Td[(qj+1)<1:Chargingforwarrantydecreasesrevenue.Conceptually,thechargingconditionratiosarethesameasfortheCPwarranty.Sincethewarrantycancoverdifferentperformancelevelsoverthelifeofthewarranty,theprobabilitiesalonedonotcharacterizethechargingconditionsasbefore.WiththeCCwarranty,theprobabilitiesaremultipliedbythediscountfactorswhicharenotconstantforallb.ThedifferencebetweenthepartialderivativesfromEquations 2 and 2 ,adjustedbythediscountfactor,representsthechangeinrevenueforthemanufacturerforeveryunitchangeinthewarrantycostcapwhilecontrollingforthecorrespondingchangeinwarrantyprice,i.e.,@R @bjPw=dhPkj=1dqj)]TJ /F8 7.97 Tf 6.58 0 Td[(1U(W)]TJ /F5 11.955 Tf 12.05 0 Td[(qj+1)(pcj)]TJ /F5 11.955 Tf -455.56 -23.91 Td[(pmj).ThealgorithmtosolvefortheoptimalCCwarrantywithnorestrictiononchargingisthesameasfortheCPwarrantywithonemajorchange.Insteadofmovingbetween 49

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initialperformancelevelcosts,cjandcj+1,todetermineifthepriceofthewarrantycanbeincreased,thealgorithmmovesbetweenperformancelevelgrowthcostsbkandbk+1.Formally,wedenetheperformancelevelgrowthcostsastheorderedsetKsuchthatb2Kwhenfb=cj(1+g)iforsomej2f1,...,Lg,i2N+:c1bkcLg.ThenumberofperformancelevelgrowthcostsareO(L+Lln(c1 cL) ln(1+g)).NotethattheinitialperformancelevelcostsareasubsetofK.Table 2-9 showstheeffectonthewarrantydesignandmanufacturer'srevenueforaCCwarrantywhenthewarrantypriceisunrestricted.Thepercentagechangeinrevenueiswithrespecttothemaximumrevenuewhenano-costCCPBWisoffered.Fortheneutralandoptimisticconsumer,thereisnovaluethatcanbechargedforthewarrantythatwillincreasethemanufacturer'srevenuemorethanofferinganoptimalno-costwarranty.Note,feasiblewarrantieswithapricegreaterthanzerodoexistthatareacceptabletotheneutralconsumer,buttherevenueremainsthesameasthezero-costwarranty.Asforthepessimisticconsumer,revenuescanbeincreasedabovetheleveloftheoptimalzero-costwarrantybychargingforthewarranty.Theserevenueincreasesarenotaslargeastherevenueincreasesachievedbymotivatingtheconsumertopurchasetheassetearlier.Additionally,theoptimalwarrantypricesareextremelylargeasapercentageofthepurchaseprice.Whileacceptabletotheconsumerfromaminimumdiscountexpectedcostperspective,thepracticalityofsuchalargeadditionalcostisquestionable.Themanufacturer,therefore,couldofferasub-optimalCCwarrantywithalowercostthatstillincreasesrevenueovertheno-costwarranty,butnottothelevellistedincolumn6ofTable 2-9 50

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Table2-9. OptimalUnrestricted-CostCCWarranties(ne=3) UpgradeVersionPwbWRevenueRevenue Neutral 101202.1239567.580201013.5529348.640301047.1029429.33040937.6419507.57050908.53310316.350 Pessimistic 14750.0995089920.432.62%24583.8990089971.942.58%35413.0185099783.053.12%44747.0180099848.713.15%54547.27750910673.762.44% Optimistic 101414.5618867.99020920.3129257.90030936.7329342.06040952.8829414.44050853.7519501.770 2.6ImplementingPerformanceBasedWarrantiesClearly,therearechallengeswithregardstoimplementingPBWsbecauseoperatingcostscanbedependentoninputsoutsidethecontroloftheassetbeingcovered.Asstructuredinthischapter,theperformancelevelsaremeasuredinperiodiccosts.SuchaPBWcouldbeimplementedbypredeningtheusageoftheproductsuchasmilesdrivenorhoursusedperperiod.Deningproductusageinabusinessenvironmentmightbemuchmorerealisticthaninapersonaluseenvironment.Forinstance,aircraftoperators(whetherairlinecompaniesorthegovernment)cancraftcontractsthatreferenceusageamountsandconditions,butafamily'sabilitytocommittopre-denedcarusagecouldbemoretenuous.Thetrackingofactualusageovertheproblemhorizoncouldbeimplementedrelativelyeasilythroughusagetrackers,suchasodometersinautomobiles,enginehourgaugesinaircraft,orsmartgridtechnologyforhomeappliances.ProgressiveInsuranceactuallyusesadevicetotrackusage,includingtypeofusage;technologieslikethiscanimproveimplementationofPBWs.Next,performancemetrics,suchasmilespergallonorkilowattsperhours,wouldbeappliedtotheusagetogetthetotaloperatingcommodityused(e.g.,gasolineor 51

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electricity).Thenusingmutuallyagreedtocommodityrates(e.g.,pricepergallonorpriceperkilowatthour),theperiodicoperatingcostscouldbeobtained.Thisisimplicitlythemethodusedinthischapterwheretherangeofthedistributionistheperiodiccost.Byusingcostinsteadoftheperformancemetric,therelationshipbetweentheperformancemetricandunderlyingcommoditycostisconfounded.Weuseaninationrateonthecoststomodeldecreasingperformanceovertime,butthecostincreasecouldbefromeitheraperformancedecreaseorcommoditypriceincrease.Evenwithagreedtousageandcommoditycosts,implementingaprocesstotracktheperiodiccostswouldbedifcult.Theconsumercouldcollectreceiptsorbeissuedaspecialpaymentcardbutthepossiblyoffraud,especiallywiththeretailconsumer,mightbedifculttocontrol.Fortheretailconsumer,itwouldbepossiblethoughwithmoderntechnologies,liketheabilitytorequirenearreal-timeupdatesofpurchasesonawebsite,whichwouldallowthemanufacturertoidentifyfraudulentactivity.Inabusiness-to-businessenvironment,thisissuewouldnotbeasdifcultbecauseoftheaccountingandauditingstandardsthatrequireaccuratetrackingofcosts.Overall,theimplementationstrategyofreconcilingaPBWwouldneedtobeconsideredcarefully. 2.7SummaryWhencustomerspurchasedurablegoodssuchasacarorappliance,theynotonlyincurtheinitialpurchasecostbutalsoperiodicoperatingandmaintenancecosts.Thevasteldofwarrantyanalysishasfocusedonwarrantydesignsthataddressthefailurecharacteristicsoftheproductandthushelptheconsumercontrolmaintenancecosts.Likewise,consumer'shaveaninterestincontrollingoperatingcostsandarethusconcernedaboutproductperformancelevels.Performancebasedwarrantiesaddressthisconcernandofferamechanismforconsumerstocontroltheirperiodicoperatingcosts.ThebenetstoamanufacturerofferingPBWsincludeadditionalrevenueorincreasedconsumerloyalty. 52

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Weintroducetheconceptofaperformancebasedwarranty(PBW)thatcompensatesaconsumerifaproductdoesnotperformatacertainlevel.Specically,twotypesofPBWsareanalyzed:aconstantperformance(CP)warrantythatincreasesthecostcapatthesameexpectedgrowthrateastheoperatingcostsandaconstantcost(CC)warrantythatkeepstheoperatingcostcapconstantoverthelifeofthewarranty.Usingdiscountedcashowstomodelinnitehorizonequipmentreplacementproblemswithnotechnologicalchange,itisshownthatPBWscanincreaseamanufacturer'stotaldiscountedrevenueforconsumerswithapessimisticbeliefaboutthecredibilityofaproduct'sadvertisedperformance.Warrantyfrontiers,whichshowthecombinationsofwarrantypricesandperiodiccostcapsforwhichaconsumerisindifferent,andtheassociatedmanufacturerchangeinrevenuecurvesfromofferingwarrantiesonthewarrantyfrontiersaredeterminedforconsumerswithdifferentbeliefcharacterizationsforeachtypeofwarranty.WealsoanalyzetheuseofPBWsforanitehorizonequipmentreplacementproblemwithtechnologicalimprovementusingdynamicprogramming.Itisshownthatno-costPBWscanbeusedtoinuencetheconsumer'soptimalpurchasepolicyandincreasethemanufacturer'stotalrevenue.Additionally,ifthemanufactureriswillingtochargeforthewarranty,revenuescanbeincreasedfurtherforapessimisticconsumer.Thespecicdesignofthewarrantydependsonthetypeofwarranty,thelevelofproductimprovement,andtheconsumer'sbeliefoftheproduct'sexpectedperformance. 53

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CHAPTER3PERFORMANCEGUARANTEESUNDERCOMPETITION 3.1MotivationandLiteratureReviewConsumersareprovidedinformationabouttheproductsthatareavailabletopurchasewhenconsideringthereplacementofequipment.Forexample,whenpurchasingacar,consumersareinformedoftheestimatedmilespergallonratingbeforethepurchase.Thisinformationhelpsdeneaconsumer'sexpectationoftheproduct'sperformance.Afteraproducthasbeenpurchased,theconsumergainsadditionalinformationbyobservingtheproduct'sactualperformance.Astherearemanywaysinwhichtheconsumermaybeinuenced,theconsumer'sexpectationsmaychangeovertime.Furthermore,theperformanceofaproductwillsurelyinuenceaconsumer'sperceptionandexpectationoffutureproductperformancefromthesamemanufacturer,whichpresumablywillinuencetheirfutureproductpurchasedecisions.Inacompetitivemarket,theevolutionofaconsumer'sexpectationofamanufacturer'sproductlinecanhaveasignicantimpactontheprobabilitythattheconsumerreturnstothesamemanufacturerforafollow-onpurchase.Therefore,itisintheinterestofamanufacturertoevaluatewaysinwhichtoensureaconsumerissatisedwithaproduct.Whilethereareawiderangeofproductattributesthatcouldconstitutemeetingexpectedperformance,SwanandCombs[ 20 ]statethattheactualattributesthatdetermineaconsumer'ssatisfactionwithperformancearelimitedtothosethatareaffectedbytheconsumer'sperceptions.Forinstance,aroutineconsumermaybeunawarethatfront-wheeldriveandrear-wheeldrivevehiclesbehavedifferentlyinwetconditionswhileanautomobileexpertunderstandsthedifference.Fortheroutineconsumer,thisattributeisirrelevantwhileitdoesmatterforthecarexpert,thereforeeachisaffectedbytheirperception.Additionally,MyersandAlpert[ 21 ]andAlpert[ 22 ]discussthatonlyalimitednumberofattributesareimportantwithregardstopurchasingaproductandpost-purchaseperformancesatisfaction. 54

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Inourproblem,weconsiderthecaseofasingleperformanceattributethatdeterminesconsumersatisfaction.Specicallyinthecontextofanequipmentreplacementproblemwithmultiplepotentialreplacementoptions(challengers),theattributeofinterestisoperatingcost,thatformanyproductsishighlydependentontheeconomicaluseofanunderlyingcommodity.Forexample,thecommoditymaybegasforacarorelectricityforarefrigerator.Itisassumedherethattheonlydifferentiatorinproductsisoperatingcost.Oneexamplewouldbetheconsiderationofatraditionalsedananditshybridversionsuchthattheonly(assumed)differentiationistheoperatingcostofeachvehicle.Inthischapter,weexaminetheuseofaperformanceguarantee(withrespecttothesingleattributeofoperatingcosts)toincreaseamanufacturer'srevenueinaniteequipmentreplacementproblem.Sequentialdecisionproblems,suchasequipmentreplacementproblems,areoftensolvedwithdynamicprogramming.Inthecaseofadjustingtheconsumer'sperformanceexpectationovertime,aDPiswellsuitedsinceitprovidesanopportunitytoupdateexpectedperformanceeachperiodbasedonobservedperformance.Theprocessofupdatingaconsumer'sinformationusingaBayesianframeworkiswellstudied.CananandUlu[ 23 ]describeageneralBayesianupdatingstrategywheretheconsumerreceivesanewpieceofinformationeachperiodaboutthepotentialbenetofanewtechnologythathelpsdetermineiftheywanttopurchasetheproductorwaitformoreinformation.ErdemandKeane[ 24 ]useprevioususageandcurrentadvertisingtoupdateaconsumer'sproductexpectationandanalyzetheeffectontheprobabilityofbuyingfromagivencompany.Mehtaetal.[ 25 ]presentamodelwheretheconsumerupdatestheirpricebeliefsbasedondiverseinformationandthendevelopstheconsiderationsetofpossiblecompetitors.TheuseofBayesianupdatinginwarrantyanalysishasbeenusedtoshowhowamanufacturerupdatesproductfailurecharacteristicsindesigningawarrantytooptimizeprots.HuangandZhuo[ 26 ]updatethetimebetweenfailuresinaBayesiannaturetodeterminetheoptimalwarranty.Fang 55

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andHuang[ 27 ]updatethelevelofproductdeteriorationbasedonexpertopinionandlimiteddatainordertomakeoptimaldecisionswithrespecttoprice,production,andwarrantypolicy.Whileconsideringaconsumer'sequipmentreplacementpolicy,weareultimatelyinterestedindesigningcontractsthatamanufacturercanuseinordertoimproveacustomer'ssatisfactionwithaproductandthusimprovethelikelihoodofafuturepurchase.Anaturalcontracttoconsiderisaperformancewarrantyorguarantee.Theuseofwarrantiesandguaranteestoinuenceconsumerbehaviorisdiverseandhasbeenstudiedextensively.Generally,warrantiescanbedifferentiatedfromguaranteesinthatawarrantycoversalongerperiodandprovidesprotectionagainstcertainsub-standardproductattributesthatcannotbequicklydetermined,suchasreliability.Guarantees,specicallymoney-backguarantees(MBG),aretraditionallymuchshortertermandessentiallycoverunlimitedundesiredattributesascanbeseenwithlabelssuchasnoquestionsasked.Bothwarrantiesandguaranteescanbeusedtosignalproductqualitytotheconsumer.MoorthyandSrinivasan[ 28 ]andShieh[ 29 ]showhowaMBGisusedtosignalquality.TheuseofawarrantyasasignalofproductqualityisstudiedbyLutz[ 30 ]whenthereisapotentialmoralhazardbytheconsumer'schoiceofproductmaintenance.Gal-Or[ 31 ]examinestheeffectsofawarrantyasasignalincompetitivemarkets.Anothercommonpurposeforguaranteesandwarrantiesistoscreencustomersinordertooptimizeprotsbasedonaheterogenouspopulation.Kubo[ 32 ]demonstratestheuseofamenuofproductprices,withandwithoutaguarantee,toscreenconsumersbasedonincome.ThemenuconceptisfurtherdevelopedbyPadmanaban[ 14 ]andHartmanandLaksana[ 5 ]forextendedwarrantieswhereconsumersaredifferentiatedbasedonproductusageandriskpreference,respectively.TheconceptofawarrantytobothscreenandsignalsimultaneouslyisstudiedbySoberman[ 33 ]. 56

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Inthischapter,wepresenttheconceptofaperformanceguaranteethatdifferentiatesconsumersbasedontheirbeliefoftheproduct'sperformance.Weusethetermguaranteeoverwarrantyforseveralsemanticreasons.First,warrantiesoftencoverexplicitproductcharacteristics,mostoftenreliability.Inthecaseofusingoperatingcostsastheperformancemetric,weassumethattrueperformanceisnotobservedandperformanceisonlyindicatedbytherealizedcosts.Therefore,theremaybeacomponenttotheguaranteethatinsurestheconsumeragainstanoperatingcostfactoroutsideofthemanufacturer'scontrol.Inourparticularmodel,theunderlyingcommoditycostisthiscomponent,e.g.electricitypricesarenotcontrolledbythemanufacturerforaguaranteeontheenergyefciencyofahouseholdappliance.Second,awarranty,inmanycases,involvesaconsciousdecisionbytheconsumertopurchasethewarranty.Theguaranteewedesignisinherentintheproductprice,asMBGsoftenare,suchthatthereisnoconsumeractiontoacquireit.Thisisequivalenttothecasewherethereisasinglepurchasepricethatincludesabasewarranty.Ourcontributiontotheliteratureincludesaproposedmethodformodelingconsumerbeliefsaboutexpectedproductperformanceandhowthebeliefsareupdatedastheconsumerobservesperiodicoperatingcosts.Weshowunderthismodelthat:1)anyfeasibleguarantee,wherefeasibilityisdenedastheconsumerhavinglower(orequal)totalexpecteddiscountedcoststhanwithouttheguarantee,willbeacceptedbytheconsumer,andtheprobabilityofthemanufacturergettingafollow-onpurchaseincreasesinthelengthoftheguarantee;2)amanufacturercanincreaserevenuebyincreasingconsumersatisfactionsuchthattheprobabilityofafollow-onpurchaseincreases;3)theoptimalguaranteeisdependentnotonlyonthevalueoftheconsumer'sbeliefbutthecondenceofthatbelief.Thechapterhasthefollowingstructure.Insection 3.2 ,wepresentamodelforupdatingconsumerexpectationofproductperformancewithrespecttoexpectedperiodicoperatingcosts.Wealsopresentastructureforperformanceguaranteesthat 57

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canbeofferedtoconsumers.Theninsection 3.3 ,wepresentadynamicprogramthatsolvesfortheconsumer'soptimalpolicyandminimumtotaldiscountedexpectedcostswhenaspecicperformanceguaranteeisoffered.Weconductascenarioanalysis[ 34 ]todecreasethenumberofpotentialguaranteesthatmustbeevaluated.Section 3.4 presentstheresultsfromanexampleproblemandprovidessensitivityanalysis. 3.2BayesianUpdatingMethodologyOurdesignandanalysisofperformanceguaranteesareconsideredinanitehorizonequipmentreplacementproblemwithmultiplechallengersandtechnologicalchange.LetTrepresentthetotaltimehorizonwheretheconsumerhasthechoiceateachtimet2f0,1,...,T)]TJ /F7 11.955 Tf 12.5 0 Td[(1gtopurchaseaproductfromanymanufacturerorkeeptheproducttheycurrentlyownforanotherperiod.LetPi,jrepresentthepurchasepriceoftheithmanufacturer'sjthtechnologyversion.EachoftheseproductshasamaximumusefullifeNi,jandaperiodicoperatingcostdistributionfi,j,nwherenistheageoftheproduct.WeusethesameconsumerdifferentiationmethodfromChapter 2 wherebytheconsumermodiesthemanufacturer'scostdistributionsfi,j,n,whichareassumedtobeaccurate(representsthetrueperformancedistribution)andsymmetric(themanufacturerprovidesthedistributiontotheconsumer).Weannotatetheconsumer'soperatingcostdistribution,aftermodication,asfi,j,n.Correspondingly,Fi,j,nandFi,j,nrepresentthecumulativedistributionsforthemanufacturerandconsumer,respectively.Thepracticaljusticationforthemodicationstemsfrommanydifferentsources,suchastheconsumer'spastexperiencewiththemanufacturer,themanufacturer'sgeneralreputation,ortheinherentnatureoftheconsumertobeoptimisticorpessimistic(orneutral)intheirdecisionmaking.Theonlylimitationimposedontheconsumermodicationsisthatthemodieddistributionmusthavethesamesupportasthetruedistribution.Note,therequirementofidenticalsupportsdoesnotimposetherequirementofthesamedistribution.Considertheexampleofthemanufacturer 58

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claiminganexponentialdistributionandtheconsumermodifyingthistoachi-squareddistribution.Thesedistributionsaredenedoverthepositiverealnumberlinebutaredifferentdistributions.Thesamesupportassumptionessentiallytranslatestoagreementbetweenconsumerandmanufacturerontherangeofpossibleoperatingcosts.Ourchoiceofconsumerdifferentiationtswellintotheconceptofhavingtheconsumerupdatetheirbeliefsonproductperformanceaftereachtimeperiod.Thepreconceptionsthataconsumerhaspriortopurchasinganitemcanchangeafterobservingaproduct'sperformance.WeuseBayesianupdatingasthebasisfortheprogressionoftheconsumer'smodicationofthemanufacturer'sadvertisedoperatingcosts.Althoughthereareotherwaystoupdatethebeliefs(suchasexponentialsmoothing),thereisanintuitiveconnectionbetweenourproblemandtheBayesianframework.RecallinBayesianstatisticsthatwehaveaninitialdistribution()(calledtheprior)foragivenparameter.ThengivenadatasetX,weusethelikelihoodfunctionL(Xj)andBayesformulatodevelopanewprobabilitydistributionusingtherelationship: (jX)/L(Xj)().(3)Note,theproportionalrelationshipexistssinceL(Xj)isafunctionofandthereforeL(Xj)()maynotintegratetoone.Thusaconstantmaybeneededtonormalizetheright-handsidesuchthat(jX)isavaliddistributionfunction.Intheupdatingprocess,(jX)becomesthenewpriorthatisupdatedwhenthenextsetofdataisobserved(see[ 35 ]or[ 36 ]forfurtherdetailsonBayesianstatistics).Forexample,supposeadatasetYisobservedafterdetermining(jX).Thenwecanupdatethisprobabilityinthefollowingway:(jY)/L(Yj)(jX).Intermsofourproblem,issomevectorofparametersthatdescribestheperiodicoperatingcostsforaproductofagivenagen.Theconsumerhassomeinitialbelief 59

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beforeobservinganyperformance,fi,j,n,whichistheinitialpriorinEquation 3 .Afterowningaproductforaperiod,theconsumercanupdateanysubsetofpossibleoperatingcostsdistributions(fi,j,n,8i,j,n)dependingontheconsumersspecicstrategy.Forexample,iftheobservedperformanceisfromthejthtechnologyofmanufactureri,theconsumermaychoosetoupdateonlythedistributionsforthisfi,jgproduct,forallofmanufactureri'sproducts,orforallproductsintheequipmentreplacementproblem.Thestrategyforupdatingthecostdistributionsisdeterminedbytheconsumer'sparticularapplicationofobservationstotheeldofproductsaswellasthestructureofthelikelihoodfunctionL(Xj).Atthispoint,thenatureofthedata,X,hasnotbeenexplicitlystated.Sincethegoalistodesignaperformanceguaranteeandweareusingperiodicoperatingcostsasameasureofperformance,thenweletXrepresentameasurementofperformancecostsfortheproductthatiscurrentlyowned.Astherearemultiplecompetitors,theobservedperformanceandsubsequentupdatingofperformancedistributionscouldhaveaneffectonaconsumer'sfuturedecisiontokeeporreplaceaproduct.Theuseofaperformanceguaranteemighthelpamanufacturerretainthebusinessofaconsumerintheeventofsubstandardperformancewhichinturncouldincreasethemanufacturer'stotalrevenue.Thereforewedeneaperformanceguaranteebytheduple(W,L)whereWistheguaranteelengthandListhelevelofperformancethatisguaranteed.NotethatWandLmaynotbescalars.ForexampleWcouldbeabinomialcoefcient(i.e.,)]TJ /F8 7.97 Tf 5.48 -4.38 Td[(53)whichindicatesthat3outof5periodsareguaranteedorLcouldbeavectorrepresentingdifferentguaranteedlevelsofperformancefordifferentagesofaproduct.Astheproposedupdatingprocesssuggests,anaturalmethodforcharacterizingperiodiccostsgivenacostdistributionistousetheexpectedvalueofoperatingcostsforagivenproduct.Duetothesequentialdecisionprocessofthisproblem,minimizingtotalexpecteddiscountedcostsisareasonableoptimizationcriteriafortheconsumerfortheentireproblem,anddynamicprogrammingiswellsuitedtosolvethisproblem. 60

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Sincethereisuncertaintyintheperformanceoftheproductsandthedistributionsthatcharacterizethisuncertaintyareconditionalonpreviousperformance,astochasticdynamicprogramisformulated. 3.2.1Beta-BernoulliModelwithTwoCompetitorsThegeneralmodelaboveallowsforanunlimitedrangeofcompetitors,technologies,updatingrelationships,andguaranteedesigns.Thereforewewillstudythedesignandeffectofaperformanceguaranteeforaspecic,tractablemodel.Considerthecaseoftwocompetitorswhereeachoffersonetechnologyatatime.Withonlytwocompetitors,wecanuseasingleintegerparameter,p,todesignateboththemanufacturer(i)andtechnology(j).Whenpisodd,theproductisfrommanufacturer1,andwhenpiseventheproductisfrommanufacturer2.Thisrelationshipismathematicallydescribedbytherelationship:i=[(p+1)mod2]+1forp2f1,2,...,Rg.Additionally,thetechnologyversionisdescribedby:j=dp 2eforp2f1,2,...,Rg.IfweassumethateachmanufactureroffersaproducteachperiodthenR=2T.Thisdoesnotprecludeamanufacturerfromofferingthesameproductinconsecutiveperiods,i.e.parametersandfunctionssubscriptedwithpandp+2arenotrequiredtobedifferent.Thisnotationimplies(givenpisoddandthusfrommanufacturer1)thefollowing:p+1representsmanufacturer2'sproductwhenproductpwasoffered,p+2representsmanufacturer1'sproducttheperiodafterofferingproductp,andp+3representsmanufacturer2'sproducttheperiodafterofferingproductp.Asimilarrelationshipcanbemadewhenpisassumedtobeeven.Theparametersf,P,andNforthegeneralmodelcannowbesubscriptedwithpinsteadoffi,jg.Next,wedeneourupdatingrelationship.WeuseaBeta-BernoulliprocesssimilartothatusedbyMcCardle[ 37 ]todescribethegatheringofinformationaboutthebenetofaproduct.Thebetadistributioniswellsuitedtomodelarandomvariablewhere 61

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theminimum,maximum,andmostlikelyvaluesareknown.Inourcase,wetreatallminimumandmaximumvaluesasuniversallyknownandacceptedparameters,andthemostlikelyvalueisdeterminedbyeachindividualmanufacturerandconsumer.Weactuallyuseafour-parameterbetamodelthatincludesthestandardp,nandp,nshapeparametersforproductpatagenaswellasap,nandbp,nwhichdenetheminimumandmaximumpotentialoperatingcosts,respectively.Werepresentperformancedegradationbyusingauniversalperiodicrateofincreasegfortheoperatingcostssothattheminimumandmaximumvaluesforthecostdistributionofproductpcanbedenedwithrespecttotheinitialmaximumandminimumvaluesandtheageoftheproduct,i.e.ap,n=(1+g)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1ap,1andbp,n=(1+g)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1bp,1.Notetheagesubscriptmaybedroppedinwhichcaseaporbpareassumedtobeforaproductthatisoneperiodold.Themanufacturer'scostdistributionforproductpatagenis:fp,n=Beta(p,1,p,1,(1+g)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1ap,(1+g)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1bp)Whereasthemanufacturerhasadenedcostdistributionforaproductofeachage,theconsumeronlydenesinitialcostdistributionsforeachproduct.TheBayesianupdatingprocessdenestheconsumer'scostdistributionsforaproductwhenn>1.Thereforetheconsumer'scostdistributionforproductpbeforebeingpurchasedis:fp,1=Beta(p,1,p,1,ap,bp)Besidesthesuitabilityofthebetadistributiontodescribeperformancecosts,italsoprovidesaconjugatepriorwhenthelikelihoodfunctionisaBernoullidistribution.Thisguaranteesthattheposteriordistributions,andthusconsumer'scostdistributionsforn>1,arealsobetadistributions.ForthegeneralBeta-BernoulliBayesianupdate,givenapriordistributionBeta(,)andlikelihoodfunctionwhereXisdistributedasaBernoullirandomvariable,theposteriordistributionisBeta(+x,+1)]TJ /F5 11.955 Tf 11.55 0 Td[(x)wherexisasinglerealizationoftherandomvariableX.Typically,thisBayesianmodelisusedtodeterminethedistributionofanunknownprobability.Thestandardbetadistributionhassupport[0,1]soaftereachsetof 62

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observations,whicharedrawnfromaBernoullidistribution,thenewbetadistributionisalsooverthesupport[0,1].Inourmodelthough,thebetadistributionsareonsupportsoftheform[ap,n,bp,n].Additionally,weareinterestedinthedistributionofcosts,notofanunknownprobability.Morespecicallyweareinterestedintheexpectedvalueofoperatingcosts.Underthefour-parameterbetadistributiontheexpectedvalueis:a+ +(b)]TJ /F5 11.955 Tf 11.96 0 Td[(a)Theterm +isalwaysbetween0and1andcanbeinterpretedasthepercentageofthedistancebetweenthepointsaandbwheretheexpectedvaluelies.Saidanotherway, +isalevelofbeliefforwheretheexpectedcostsoccur.Forourproblem,p,n p,n+p,nistheconsumer'scurrentbelief.Notethatasimilarargumentcouldbemadewherethemodeisameasureoftheconsumer'sbeliefforproductpatagen.Sinceaconsumerdenestheshapeparametersforeachproductinitially,thechoiceoftheseparametersrevealstheconsumer'sinitialbeliefoftheperformanceofagivenproductbeforeobservinganyrelevantperformanceinformation.Ifp,n p,n+p,nisclosetoone,theconsumerexpectscoststobenearthemaximumlevel.Conversely,ifp,n p,n+p,nisnearzero,theconsumerexpectsperformancetobeclosetotheminimum.Basedontheperformanceobservedafterowningproduct,theconsumer'sbelieffortheexpectedcostswillchangedependingontheperformancecomparedtotheirpriorbeliefs.Therefore,wedeneourlikelihoodfunctionintermsofaBernoullidistributionwhereasuccessisdenedasperformancecostsgreaterthanwhattheconsumerexpectsandafailureasperformancecostslowerthanexpected.Usingthemanufacturer'sdistributionasthetruedistributionforoperatingcosts,theunderlyingBernoullidistributionis: Xp,n=8><>:0withprobabilityFp,n(p,n p,n+p,n)1withprobability(1)]TJ /F5 11.955 Tf 11.96 0 Td[(Fp,n)(p,n p,n+p,n)9>=>;(3)Whilethereisnoexplicitlimitationontheperformanceguaranteedesigngiventheproposedupdatingrelationship,thereisanaturalstructuretotheguaranteed 63

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performancelevelthatcomestomindfromthedenitionofXp,n.Recallwedescribedaperformanceguaranteebytheduple(W,L)whereLisavectorforperformanceguaranteelevels.BasedonEquation 3 ,wedesignguaranteelevelsbasedontheconsumer'sinitialbeliefandsubsequentproductperformance. 3.3DynamicProgramFormulationUsingthespecicmodelfromtheprevioussection,wecannowformulateastochasticdynamicprogram(SDP)thatallowsustodeterminetheequipmentreplacementpolicywhennoguaranteeisoffered.WerstmustdeneoneadditionalcomponentoftheupdatingprocessinordertogeneratetheSDP.Recallthatinthegeneralmodelthecurrentlyownedproduct'sperformancecouldbeusedtoupdatedifferentconsumercostdistributions(e.g.,justthecurrentlyownedproduct,allproductsfromthecurrentmanufacturer,orallproductsregardlessofthemanufacturer).Here,weassumethatproductperformanceonlyaffectsthebeliefsofproductsfromthatsamemanufacturer.Whileitisclearthatpastperformanceaffectsthebeliefsofthecurrentitem,thefactthatpastperformanceaffectsfutureproductsfromthesamemanufacturerisrealisticsinceconsumersdevelopageneralperceptionofamanufacturerbasedonpersonalexperience.WecannowdenethestatespaceoftheSDPby(p,n,m,s1,s2)wherepistheproductcurrentlyowned,nistheageofthecurrentlyownedproductatthebeginningoftheperiod,misthenumberofperiodsthatanyproductfrommanufacturer1hasbeenowned(mmaynotequaln),s1isthenumberofperiodsthataproductfrommanufacturer1hasbeenownedandperformedworsethanexpected,ands2isthenumberofperiodsthataproductfrommanufacturer2hasbeenownedandperformedworsethanexpected.Welettrepresenttimewheret2f0,1,...,Tg,andvt(p,n,m,s1,s2)isthecost-to-gofunctionattimetfromthestate(p,n,m,s1,s2).Thecost-to-gofunctionisfromtheconsumer'sperspectiveandthusrepresentstheexpectedminimumdiscountedcostofmakingoptimaldecisionsthroughtheremainingtimehorizon. 64

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Weaccountforthetimevalueofmoneyusingtheperiodicdiscountrate.Giventhataconsumer'sexpectationofperformanceofaproductpisbasedupontheinitialbeliefoftheproduct'sperformanceaswellasthepreviousperformanceofallproductsfromtheithmanufacturer,wecanannotatetheconsumer'sexpectationofperformanceasthefunctionEp(m,s1)fortherstmanufacturerandEp(t)]TJ /F5 11.955 Tf 12.09 0 Td[(m,s2)forthesecondmanufacturer.Similarly,werepresenttheexpectedperiodicoperatingcostofproductpatagenwiththefunctionCp(n,m,s1)foraproductfrommanufacturer1andCp(n,t)]TJ /F5 11.955 Tf 12.31 0 Td[(m,s2)foraproductfrommanufacturer2.NotethatthetimesubscriptisusedintheargumentsofEpandCpwhenpiseveninordertotrackthenumberofperiodsthataproductfrommanufacturer2isowned.Thereforeitisnotreallytimedependentaswecouldhavejustaddedanadditionalstatetothestatespace.Also,recallfromtheprevioussectionthatFp,nisthetruedistributionofproductpatagen.TheinclusionofuncertaintyincostsaddsastochasticcomponenttotherewardsbutdoesnotnecessarilyrequireanSDPinandofitself.Sincethemodelhastheconsumerupdatetheirbeliefsbasedonpreviousperformance,thetransitionsareuncertainandthusastochasticdynamicprogramformulationfollows.Theconsumer'sdecisionseachperiodaretokeeptheitemtheyhave(K),replaceitwiththecurrentproductofferedbymanufacturer1(R1),orreplaceitwiththecurrentproductofferedbymanufacturer2(R2).Clearlythedecisiontokeepisnotavailableiftheproducthasreacheditsmaximumusefullife.vt(p,n,m,s1,s2)=min8>>>>>>><>>>>>>>:R1:Pp0+(Cp0(1,m,s1)+Fp0,n(Ep0(m,s1))vt+1(p0,1,m+1,s1,s2)+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(Fp0,n(Ep0(m,s1)))vt+1(p0,1,m,s1+1,s2))R2:Pp0+1+(Cp0+1(1,t)]TJ /F5 11.955 Tf 11.96 0 Td[(m,s2)+Fp0+1,n(Ep0+1(t)]TJ /F5 11.955 Tf 11.96 0 Td[(m,s2))vt+1(p0+1,1,m,s1,s2)+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(Fp0+1,n(Ep0+1(t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2)))vt+1(p0+1,1,m,s1,s2+1))9>>>>>>>=>>>>>>>;,forn=0,Np (3) 65

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vt(p,n,m,s1,s2)=min8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:K:(Cp(n,m,s1)+Fp,n(Ep(m,s1))vt+1(p,n+1,m+1,s1,s2)+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(Fp,n(Ep(m,s1))vt+1(p,n+1,m+1,s1+1,s2))R1:Pp0+(Cp0(n,m,s1)+Fp0,n(Ep0(m,s1))vt+1(p0,1,m+1,s1,s2)+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(Fp0,n(Ep0(m,s1)))vt+1(p0,1,m+1,s1+1,s2))R2:Pp0+1+(Cp0+1(n,t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2)+Fp0+1,n(Ep0+1(t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2))vt+1(p0+1,1,m,s1,s2)+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(Fp0+1,n(Ep0+1(t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2)))vt+1(p0+1,1,m,s1,s2+1))9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;,whenpisodd:forp>0,Np>n>0,0s1+s2t,s1mt (3)vt(p,n,m,s1,s2)=min8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:K:(Cp(n,t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2)+Fp,n(Ep(t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2))vt+1(p,n+1,m,s1,s2)+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(Fp,n(Ep(t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2)))vt+1(p,n+1,m,s1,s2+1))R1:Pp0+(Cp0(n,m,s1)+Fp0,n(Ep0(m,s1))vt+1(p0,1,m+1,s1,s2)+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(Fp0,n(Ep0(m,s1)))vt+1(p0,1,m+1,s1+1,s2))R2:Pp0+1+(Cp0+1(n,t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2)+Fp0+1,n(Ep0+1(t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2))vt+1(p0+1,n+1,m,s1,s2)+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(Fp0+1,n(Ep0+1(t)]TJ /F5 11.955 Tf 11.95 0 Td[(m,s2)))vt+1(p0+1,n+1,m,s1,s2+1))9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;,whenpiseven:forp>0,Np>n>0,0s1+s2t,s1mt (3)vT(p,n,m,s1,s2)=0,8p,n,m,s1,s2 (3)Notethatweusep0torepresenttheproductofferedattimetfrommanufacturer1sop0>p.Theformulationaboveiswithoutamanufacturerguarantee.Undernoguarantee,weannotatetheinitialstatevaluev0(0,0,0,0,0)asvNG0.Ifthemanufacturerintroducesaguarantee,thenvG0vNG0(wherevG0istheconsumer'stotalexpecteddiscountedcostwiththeguarantee)mustholdfortheguaranteetobebenecialtotheconsumer.Whenaguaranteeisoffered,theconsumer'speriodicexpectedoperatingcostsarereduced,andwedenethesecostsasCp(n,m,s1)andCp(n,t)]TJ /F5 11.955 Tf 13.09 0 Td[(m,s2) 66

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forproductpatstates(p,n,m,s1,)]TJ /F7 11.955 Tf 9.3 0 Td[()and(p,n,m,)]TJ /F7 11.955 Tf 9.3 0 Td[(,s2)whenpisoddandeven,respectively.Equations 3 3 areupdatedwiththesenewcostsaccordingly.SincewealreadydenedtheguaranteecoverageLwithrespecttotheconsumer'sexpectedperformance,weareinterestedindeterminingthewarrantylength(W)thatmaximizesthemanufacturer'srevenue. 3.3.1ScenarioAnalysisTheSDPintheprevioussectionisforageneralizedequipmentreplacementmodel,thereforesolvingforanoptimalguaranteelength,W,couldbeprohibitivebasedonthelargenumberofpossibledesigns.Forexample,supposetherstmanufacturerisseekingtoofferaperformanceguaranteethersttimetheirproductispurchasedinordertoimprovethepossibilitytheconsumerreturnstothemforthefollow-onpurchase.ThemaximumusefullifeofthisrstproductisN1,thereforethereare2N1possibleguaranteelengthdesignsiftherearenolimitationsonwhichperiodstheproductperformanceisguaranteed.Withthenumberofguaranteedesignsbeingexponential,letusconsidertheimpactofdifferentassumptionsonthepossiblenumberofguaranteedperiods.Firstifweonlyconsiderguaranteesoverconsecutiveperiods,thenumberofguaranteedesignsbecomespolynomialwithN1(N1+1) 2possibledesigns.Whilethisrequirementseemsnaturalinaretailconsumerenvironment,wewillquicklyaddressanareawheretheconsumermayhaveaninterestinnon-consecutiveguarantees.Ingovernmentprocurement,minimumcostiscertainlyadesiredoutcome.Inthecontextofarestrictivebudgetingprocessthough,thetrade-offofbeingwithinbudgetincertainyearsatthecostofincreasedtotalprogramcostsisoftenmade.Inthisenvironment,thedesireforanon-consecutiveguaranteeforthegovernmentisrealistic.Regardless,wemaketheconsecutiveperiodassumptionhere.Next,considerthefurtherassumptionthattheconsecutiveperiodseitherstartatthebeginningofthepurchaseorendatthemaximumusefullife.Theguaranteethat 67

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startswiththepurchaseiseasilymarketedasguaranteedtomeetperformanceforatleastWnumberofperiods.Conversely,andonlyslightlymorecumbersometomarket,aWperiodguaranteethatendsattheusefullifeismarketedascoveringallunexpectedcostsfortheremainderoftheproduct'slife.Onecouldarguethatitismoredifcultforamanufacturertomarketandaretailconsumertoacceptaconsecutiveguaranteethatliesinthemiddleoftheproduct'slifecycle.Thisassumptionreducesthepossiblenumberofguaranteedesignsto2N1)]TJ /F7 11.955 Tf 11.96 0 Td[(1.Withthepossiblenumberofguaranteedesignsnowlinear,letusconsiderproblemswithaspecialcharacteristic:thesubsequentavailabletechnologies(challengers)aresignicanttechnologyupgrades.Wedenesignicantashavingalargeenoughpurchasepricesuchthattheitem'spredecessorwillbeheldtoitsmaximumusefullifebeforepurchasingthenewtechnology.Thereareseveralmathematicalwaystoensurethatthetechnologyimprovementissignicantwhicharesufcient,butnotnecessary,conditions.Considertwoproductspandp0wherep0isoneofthenextavailabletechnologiesafterhavingpurchasedp.Wesaythatp0isasignicanttechnologicalchangeoverpifoneofthefollowingconditionsholds: 1. Themaximumannualequivalentcost(AEC)ofpislessthantheminimumAECforp0,thenp0isasignicanttechnologychange. 2. TheAECforp0overanylengthoftimebeginningwiththepurchaseofp0isgreaterthantheAECforpoveranyequivalentlengthoftime(notnecessarilystartingwiththepurchase).Whiletheimpactofthesignicanttechnologychangeassumptiondoesnotproduceareductioninthenumberofguaranteedesigns,thesolutionmethodologyandanalysisissimpliedunderthischaracteristic.Withoutasignicanttechnologychange,replacementsmayhappenatdifferenttimeperiodsbasedonthepreviousperformance.Theprocessandinsightswegainfromanalyzingthesignicantchangeproblemthoughcanbeextendedtomoregeneralcases.Therearemanyexampleswhereasignicant 68

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technologychangeiscommon.ConsiderthemigrationfromacompactcartoalargeSUVormovingfromanobsoletetubestyledTVtoanewLEDatscreentelevision.Themilitaryisoftenfacedwiththescenarioinmovingfromonegenerationofaircrafttothenextgeneration. 3.3.2GuaranteeEffectWiththeassumptionsfromtheprevioussection,wecananalyzetheeffectaperformanceguaranteehasonamanufacturer'srevenueaswellaspresentingthemethodologyforsolvingfortheoptimalguaranteelength.Withthesignicantchangeassumption,thereplacementofaproductwilloccurattheendofthecurrentproduct'smaximumusefullife.Figure 3-1 showsthetransitionnetworkforaproductwithNp=4andaconsumer'sinitialbeliefdenedbyp,1=3andp,1=2,whichequatestoaconsumerbelievingcostswillbeskewedtowardsthehighend.Weannotatetheconsumer'sinitialbeliefwithE0.NoteFisthetruecostdistributionofthecurrentlyownedproduct.Basedontheperformanceoftheitem,theconsumerwillbeatoneofthevenodesinthefarrightcolumn(whichwecallterminalnodes)afterexperiencingthevariablecostsoftheproductoverfourperiods.Thenumberofpathstoeachterminalnodeisannotatedonthegraph.Thetotalprobabilityofbeingataterminalnodeisthesumoftheprobabilitiesofallpossiblepaths,andthenodesfromtoptobottomareinincreasingorderbasedonthenumberofperiodsofbelowexpectedperformance,i.e.s1inincreasing.Thedecisionatanygiventerminalnodeisnotaffectedbythepathtogettothenodesincethedecisionatthatstateisonlydependentonthenumberofsuccesses,andthenumberofsuccessesatanygiventerminalnodeisthesameregardlessofwhenthesuccessoccurred.Wecanndarelationshipbetweentheterminalnodes.Specially,wecanshowthattheremainingcosts-to-goforaconsumeratanytimetarenon-increasinginthenumberofperiodsofsubstandardperformance. 69

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Figure3-1. NetworkPathsWithoutPerformanceGuarantee(Np=4) Theorem3.1. vt(p,n,m,s1,s2)vt(p,n,m,s1+1,s2)whenaBeta-Bernoulliprocessisfollowed. Proof. Giventhestate(p,n,m,s1,s2),thereisacorrespondingoptimalpolicyattimetthatisannotatedbyDt(p,n,m,s1,s2)where:Dt(p,n,m,s1,s2)=8>>>><>>>>:1ifconsumerreplaceswithmanufacturer1'sproduct2ifconsumerreplaceswithmanufacturer2'sproduct3ifconsumerkeepsmanufacturer1'sproduct9>>>>=>>>>; IndeningDt(),weassumetheincumbentproductisfrommanufacturer1buttheproofcaneasilybeappliedtothecasewheretheincumbentproductisfrommanufacturer2.Notethatthetimesubscriptonthedecisionfunctiontracksthe 70

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numberofperiodsthataproductfrommanufacturer2isowned,thereforeitisnotreallytimedependentaswecouldhavejustaddedanadditionalstatetothestatespace.Thenon-discountedexpectedperiodicoperatingcostforthenextperiodifthecurrentitem(frommanufacturer1)iskeptorreplacedbyanotherproductfrommanufacturer1isCp(n,m,s1)=qp(an0+1,p+s1 1,p+1,p+m(bn0)]TJ /F5 11.955 Tf 12.11 0 Td[(an0))wheren0representstheageofitem,qpisthemultiplicativefactorforthechangeinoperatingcostsfromproduct1toproductp,and(an,bn)forn2f1,2,...,Ngrepresenttheminimumandmaximumoperatingcostsforaninitialproductofagenfrommanufacturer1.Sincetheageatthecurrenttimetisn,thenn02f1,ng.Similarly,giventhestate(p,n,m,s1+1,s2)attimet,Cp(n,m,s1+1)=qp(an0+1,p+s1+1 1,p+1,p+m(bn0)]TJ /F5 11.955 Tf 11.95 0 Td[(an0)). Therefore, Cp(n,m,s1+1)>Cp(n,m,s1)(3) Bythesamerationale,Cp(n,m,s1)>Cp(n,m+1,s1).LetCp(n,m,s2)representtheexpectedoperatingcostoftheitemfrommanufacturer2iftheproductisreplacedwhereCp=(n,m,s2)=qp(an0+2,p+s2 2,p+2,p+(t)]TJ /F6 7.97 Tf 6.58 0 Td[(m)(bn0)]TJ /F5 11.955 Tf 12.08 0 Td[(an0)).NotethatCp()doesnotincludethepurchasepriceoftheitem. RecallthatvT(p,n,m,s1,s2)=08p,n,m,s1,s2istheboundarycondition.NowconsiderthepossibledecisionsattimeT-1forthestate(p,n,m,s1+1,s2). Case1:Replacewithnewproductfrommanufacturer1 IfDT)]TJ /F8 7.97 Tf 6.58 0 Td[(1(p,n,m,s1+1,s2)=1,thenvT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p,n,m,s1+1,s2)=Pp0+Cp0(1,m,s1+1) IfDT)]TJ /F8 7.97 Tf 6.58 0 Td[(1(p,n,m,s1,s2)=1,thenvT)]TJ /F8 7.97 Tf 6.58 0 Td[(1(p,n,m,s1,s2)=Pp0+Cp0(1,m,s1+1) 71

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ByEquation 3 ,vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p,n,m,s1,s2)
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Therefore, vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p,n,m,s1,s2)vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p,n,m,s1+1,s2)(3) NowletusconsiderthedecisionsattimeT-2.Sincethereisafuturedecisiontomake,theanalysisofpossibledecisionsnowincludesuncertaintransitionsandnotjustuncertainrewards.Againwemustconsider3cases. Case1:Replacewithnewproductfrommanufacturer1 IfDT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1+1,s2)=1,thenvT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1+1,s2)=Pp0+(Cp0(1,m,s1+1)+Fp0,n(p0+s1+1 p0+p0+m)vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p0,1,m+1,s1+1,s2)+(1)]TJ /F16 10.909 Tf 10.91 0 Td[(Fp0,n(p0+s1+1 p0+p0+m))vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p0,1,m+1,s1+2,s2)) IfDT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1,s2)=1,thenvT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1,s2)=Pp0+(Cp0(1,m,s1)+Fp0,n(p0+s1 p0+p0+m)vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p0,1,m+1,s1,s2)+(1)]TJ /F16 10.909 Tf 10.91 0 Td[(Fp0,n(p0+s1 p0+p0+m))vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p0,1,m+1,s1+1,s2)) TodeterminetherelationshipbetweenvT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1+1,s2)andvT)]TJ /F8 7.97 Tf 6.59 0 Td[(2(p,n,m,s1,s2)let:r=Fp0,n(p0+s1+1 p0+p0+m)s=Fp0,n(p0+s1 p0+p0+m)x=vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p0,1,m+1,s1,s2)y=vT)]TJ /F8 7.97 Tf 6.58 0 Td[(1(p0,1,m+1,s1+1,s2)z=vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p0,1,m+1,s1+2,s2) Weknowthat0
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SincethepurchasepricesPp0cancelandCp0(1,m,s1+1)>Cp0(1,m,s1),wehaveshownthatvT)]TJ /F8 7.97 Tf 6.59 0 Td[(2(p,n,m,s1,s2)
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IfDT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1+1,s2)=3,thenvT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1+1,s2)=(Cp(n,m,s1+1)+Fp,n(p+s1+1 p+p+m)vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p,n+1,m+1,s1+1,s2)+(1)]TJ /F16 10.909 Tf 10.91 0 Td[(Fp,n(p+s1+1 p+p+m))vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p,n+1,m+1,s1+2,s2)) IfDT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1,s2)=3,thenvT)]TJ /F8 7.97 Tf 6.58 0 Td[(2(p,n,m,s1,s2)=(Cp(n,m,s1)+Fp,n(p+s1 p+p+m)vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p,n+1,m+1,s1,s2)+(1)]TJ /F16 10.909 Tf 10.91 0 Td[(Fp,n(p+s1 p+p+m))vT)]TJ /F8 7.97 Tf 6.59 0 Td[(1(p,n+1,m+1,s1+1,s2)) UsingEquations 3 and 3 ,wehavevT)]TJ /F8 7.97 Tf 6.59 0 Td[(2(p,n,m,s1,s2)
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theinitialDPthatstorestheconsumer'sdecisionatanygivenstate)describestheconsumer'sdecisionatallpossiblestates.Foragivenproductp,wedeterminethemaximumvalueofs1suchthattheconsumerdoesnotchangemanufacturerswhichisfoundbysearchingthestates(Np,p,Np,Np,s1,0):s12f0,1,...,Ng.Ifiisthemaximumnumberofsuccessesallowedinordertoremainwiththecurrentmanufacturer,thenitwillnotbenecessarytoofferaguaranteewithmoreperiodsthanNp)]TJ /F5 11.955 Tf 12.71 0 Td[(i.ThereforethenumberofadditionalDPstosolveafterthebaselineis2(Np)]TJ /F5 11.955 Tf 12.47 0 Td[(i))]TJ /F7 11.955 Tf 12.47 0 Td[(1.Note,inFigure 3-1 ,i=2wasusedasanexample.Theorem3.1providesthefollowingcorollaryfortheexistenceofathreshold. Corollary1. Atanytimetthatareplacementmustbemade,thereexistsathreshold,i,suchthatifs1ithereplacementismadefrommanufacturer1andfors1>ithereplacementismadefrommanufacturer2. Proof. TheprooffollowsdirectlyfromTheorem3.1.Forthelargestvalueofs1(whichweannotateasi)suchthatareplacementismadefrommanufacturer1,allvaluesofs1
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Theorem3.2. Theprobabilityofreplacingwiththeincumbentmanufacturer'sproductisincreasingasthelengthofguaranteeincreases. Proof. Givenaconsumerbeliefontheopeninterval(0,1)attimezeroandsinceanygivenproductisheldtoit'smaximumusefullifebyassumption,thenanetworkwithN+1terminalnodescanbeconstructedtorepresenttheprobabilitiesofreachingthestates(t,p,n,m,s1,s2)fors12f0,1,...,Ngatwhichareplacementwilloccur.Ateachnon-terminalnodeinthenetworktherearetwopossibleoutcomes:success(itemperformsworsethanexpected)andfailure(itemperformsbetterthanexpected),bothwithpositiveprobability.Thereforethetotalnumberofpathstotheterminalnodesis2Np. The2NppathstotheNp+1terminalnodescanbefurthercharacterizedbythebinomialcoefcientssuchthat2Np=PNpj=0)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(Npj.Thecombinationsinthissummationrepresentthenumberofpathsassociatedwithjsuccesses.LetRjbethesumoftheprobabilitiesofallpossiblepathstotheterminalnoderepresentedbyjsuccesseswhichwewillrefertoasthejthterminalnode.Furthermoresincethereare)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(Npjpathstothejthterminalnode,wecanrepresentthesumoftheprobabilitiesasRj=P(Npj)k=1rjkwhererj,kisthekthpathtoterminalnodej.Sinceaterminalnodemustbereached,P(Npj)j=0Rj=1. Recallthatirepresentsthemaximumnumberofsuccessestoremainwiththecurrentmanufacturer.ThereforetheRj'scanbepartitionedintotwosets:Rjforj2f0,...,igforthepathsthatleadtostayingwiththesamemanufacturersuchthatB1=Pij=0RjandRjforj2fi+1,...,NpgforthepathsthatleadtochangingmanufacturerssuchthatB2=PNpj=i+1Rj.B1andB2representthetotal 77

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probabilitiesofbuyingfrommanufacturer1andmanufacturer2,respectively.Nowletusconsidertheeffectofofferingasingleperiodback-endguarantee. Byofferingasingleperiodback-endguarantee,themanufacturereliminatesallNpsuccessarcsintheNthpperiod.Eliminationmeansdeletingthepathonanarcandmovingtheprobabilitytothefailurearccomingfromthesamesourcenode.Beforeelimination,wehave:R0=r0,1R1=r1,1+...+r1,(Np1)...RNp)]TJ /F8 7.97 Tf 6.59 0 Td[(2=rNp)]TJ /F8 7.97 Tf 6.59 0 Td[(2,1+...+rNp)]TJ /F12 5.978 Tf 5.76 0 Td[(2,(NpNp)]TJ /F12 5.978 Tf 5.76 0 Td[(2)RNp)]TJ /F8 7.97 Tf 6.59 0 Td[(1=rNp)]TJ /F8 7.97 Tf 6.59 0 Td[(1,1+...+rNp)]TJ /F12 5.978 Tf 5.76 0 Td[(1,(NpNp)]TJ /F12 5.978 Tf 5.76 0 Td[(1)RNp=rNp,1 Considereliminatingthesuccessarcsoneatatimebeginningatthebottomofthenetwork.Deletingthebottomarc(NthparcintheNthpperiod)resultsindecreasingthenumberofpathsby)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(Np)]TJ /F8 7.97 Tf 6.59 0 Td[(1Np)]TJ /F8 7.97 Tf 6.59 0 Td[(1=1butincreasesthechancesofreachingterminalnodeNp)]TJ /F7 11.955 Tf 12.4 0 Td[(1byrNp+1,1.Thereforethenewprobabilitiestotheremainingterminalnodesare:R0=r0,1R1=r1,1+...+r1,(Np1)...RNp)]TJ /F8 7.97 Tf 6.59 0 Td[(2=rNp)]TJ /F8 7.97 Tf 6.59 0 Td[(2,1+...+rNp)]TJ /F12 5.978 Tf 5.76 0 Td[(2,(NpNp)]TJ /F12 5.978 Tf 5.76 0 Td[(2)RNp)]TJ /F8 7.97 Tf 6.59 0 Td[(1=rNp)]TJ /F8 7.97 Tf 6.59 0 Td[(1,1+...+r1,(NpNp)]TJ /F12 5.978 Tf 5.76 0 Td[(1)+rNp,1 78

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Noteinthecaseofeliminatingthebottomarc,aterminalnodeisalsonolongerfeasible.Nowbyeliminatingthenextsuccessarc(Nthp)]TJ /F8 7.97 Tf 6.59 0 Td[(1arcintheNthpperiod),the)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(Np)]TJ /F8 7.97 Tf 6.59 0 Td[(1Np)]TJ /F8 7.97 Tf 6.59 0 Td[(2remainingpathstoterminalnodeNp)]TJ /F8 7.97 Tf 6.58 0 Td[(1areremovedandabsorbedbyterminalnodeNp)]TJ /F8 7.97 Tf 6.59 0 Td[(2thereforethenewprobabilitiestotheterminalnodesare:R0=r0,1R1=r1,1+...+r1,(Np1)...RNp)]TJ /F8 7.97 Tf 6.58 0 Td[(2=rNp)]TJ /F8 7.97 Tf 6.58 0 Td[(2,1+...+rNp)]TJ /F12 5.978 Tf 5.76 0 Td[(2,(NpNp)]TJ /F12 5.978 Tf 5.75 0 Td[(2)+rNp)]TJ /F8 7.97 Tf 6.59 0 Td[(1,1+...+rNp)]TJ /F8 7.97 Tf 6.59 0 Td[(1,(Np)]TJ /F12 5.978 Tf 5.76 0 Td[(1Np)]TJ /F12 5.978 Tf 5.76 0 Td[(2)RNp)]TJ /F8 7.97 Tf 6.58 0 Td[(1=rNp)]TJ /F8 7.97 Tf 6.58 0 Td[(1,(Np)]TJ /F12 5.978 Tf 5.75 0 Td[(1Np)]TJ /F12 5.978 Tf 5.75 0 Td[(2)+1+rNp,1 Continuingthisprocess,anysinglepathprobability,rjk,remainsintheterminalnodeprobability,RjthatitwasoriginallyinoritmovestotheRj)]TJ /F8 7.97 Tf 6.59 0 Td[(1terminalnode.Therefore,B1andB2onlychangevalueswhentheithsuccessarciseliminated.Afterthei+1,i+2,...,Npsuccessarcshavebeeneliminated,weknow:Ri=ri,1+...+ri,(Npi)Ri+1=ri+1,1+...+ri+1,(Npi+1)+ri+2,1+...+ri+2,(Np)]TJ /F12 5.978 Tf 5.76 0 Td[(1i) Ifwenoweliminatethei+1successarc,theaffectedterminalprobabilitiesbecome:Ri=ri,1+...+ri,(Npi)+ri+1,1+...+ri+1,(Np)]TJ /F12 5.978 Tf 5.76 0 Td[(1i) (3)Ri+1=ri+1,(Np)]TJ /F12 5.978 Tf 5.75 0 Td[(1i)+1+...+ri+1,(Npi+1)+ri+2,1+...+ri+2,(Np)]TJ /F12 5.978 Tf 5.75 0 Td[(1i) (3) Let^B1and^B2representthenewprobabilitiesofbuyingthereplacementfromtherstandsecondmanufacturer,respectively.UsingEquations 3 and 3 ,weknow^B1=B1+ri+1,1+...+ri+1,(Np)]TJ /F12 5.978 Tf 5.76 0 Td[(1i+1)therefore^B1>B1andtheprobabilityofremainingwithmanufacturer1hasincreasedsinceallrj,k>0byassumption. 79

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Note,aftereliminatingallsuccessarcsintheNthpperiod,thenewnetworkwillhaveNpterminalnodeswith2Np)]TJ /F8 7.97 Tf 6.59 0 Td[(1probabilitypaths.Thesameprocessusedaboveisappliediterativelytoaccountforamulti-periodback-endguarantee.Forexample,thetwo-periodguaranteewouldusethesameprocedureonthenewnetworkwheretheNp)]TJ /F7 11.955 Tf 12.85 0 Td[(1periodsuccessarcsareeliminated.Figure 3-2 showswhatthenewnetworkwouldlooklikeafterapplyingaoneperiodback-endguaranteewhenNp=4.Thehorizontalarcswithprobability1aredisplayedtoshowthenetworkstillaccountsforNptimeperiods;theycouldbedeletedsothenetworkisstructuredasbefore.Figure 3-2 illustratesthatthenumberofperformancepathsdecreases(comparedtothenetworkinFigure 3-1 )whenguaranteeingtheperformanceinthenalperiod.Notethatinthisexamplethenumberofpathsleadingtochangingmanufacturersisreducedfromvetoone(thetotalnumberofpathsisreducedfrom16toeight). Figure3-2. NetworkPathsAfterBack-EndPerformanceGuarantee(Np=4) 80

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Nowletusconsiderafront-endwarranty.ForagivenperformancenetworkwithNptimeperiods,withoutawarranty,thereareNp+1terminalnodes.Additionally,atanytime0ti.Noticethenthattheterminalnodesarenon-increasinginy.Wenowclaimthatthenodesinanygivencolumnarenon-increasingintherowindex,y.Usinginductionandthisinitialcase,wecanshowthattheclaimistrueforallcolumns. Foranyarbitrarynode(y,z),letqy,zrepresenttheprobabilityofmovingtonode(y,z+1)and(1)]TJ /F5 11.955 Tf 13.02 0 Td[(qy,z)representmovingtonode(y+1,z+1).Similarly,considernode(y+1,z)whereqy+1,zrepresentsmovingtonode(y+1,z+1)and(1)]TJ /F5 11.955 Tf 12.47 0 Td[(qy+1,z)representsmovingtonode(y+2,z+1).Byassumption,weknowP(y,z+1)P(y+1,z+1)P(y+2,z+1). ThereforeP(y,z)P(y+1,z)sinceanyconvexcombinationofP(y,z+1)andP(y+1,z+1)isatleastaslargeasanyconvexcombinationofP(y+1,z+1)andP(y+2,z+1).Thustheprobabilitiesofreachingaterminalnodeabovethethresholdarenon-increasingforanycolumn. 81

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Now,ifweconsidertherstnodeinthenetworkbeforeawarrantyisoffered,thenP(1,1)=q1,1P(1,2)+(1)]TJ /F5 11.955 Tf 12.46 0 Td[(q1,1)P(2,2).LetP(1,1)representtheprobabilityofreachingaterminalabovethethresholdafterofferingasingleperiodfront-endwarranty.Inthiscaseq1,1=1,soP(1,1)P(1,1)sinceP(1,2)P(2,2).Thus,asingleperiodfront-endwarrantyincreasestheprobabilityofreturningtotheincumbentmanufacturerforafollow-onpurchase.Thisproofiseasilyextendedtoafront-endwarrantyofanylength. Theorems3.1and3.2showthatanyguaranteeofferedwillmaketheconsumeratleastaswellasoffwithtorespecttotaldiscountedexpectedcost.Therefore,ndingtheoptimalguaranteeisatradeoffbetweenthemanufacturer'scostincreaseintheperiodswhenaguaranteeisofferedversustheincreasedprobabilitythattheconsumermakestheirnextpurchasefromthesamemanufacturer.Whenaguaranteeisprovided,themanufacturerincursaliabilityonlyintheperiodsthattheguaranteeisactive.Offeringtheguaranteeincreasestheprobabilitythattheconsumerwillpurchasefromthesamemanufacturer.Thusthegoalistondtheoptimallengthguaranteesuchthatthechangeinprobabilityofpurchasingfromthesamemanufacturer(multipliedbythepurchaseprice)islargerthanthesumoftheliabilitiesincurredovereachperiodoftheguarantee. 3.4ResultsWenowillustratetheeffectsofofferingaperformanceguaranteethroughanexample.Letusconsiderthescenariowherebothmanufacturershaveidenticalparameters.Inthiscasethetotalexpecteddiscountedcostfortheconsumerwouldbethesameforeachmanufacturer,sowearbitrarilychoosemanufacturer1tobethepreferredchoicefortheinitialpurchase.Table 3-1 liststheparameters.Notethattheconsumerinthiscaseisslightlypessimisticwithaninitialexpectationof.6foreachproductcomparedtoanexpectationof.5forthemanufacturers.Additionally,theperiodicoperatingcostsofthesignicanttechnologychangearethesameasthe 82

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precedingtechnology.Thiscapturestheconceptthatimprovedtechnologiesareoftenmoreexpensivetopurchasewhilereducingoperatingcosts(suchaspurchasingahybridorelectricvehicle);eventhoughtheoperatingcostsarethesame,thetimevalueofmoneyineffectmakestheoperatingcostsofthelatertechnologyanimprovement. Table3-1. ParametersforBayesianUpdatingExample ParameterValueRange TimeHorizon(T)15PurchasePrice(Pp)20000p=1,2PurchasePrice(Pp)30000p=3,4,...,2TProductUsefulLife(Np)10p=1,...,2TCostIncreaseRate(g).15InitialCostofLowestPerformance(ap)1500p=1,2,...,2TInitialCostofHighestPerformance(bp)1700p=1,2,...,2TConsumer'sInitialPerformanceExpectation(p,p)2.4,1.6p=1,2,3,4,...,2TManufacturer'sPerformanceDistribution(p,p)2,2p=1,2,3,4,...,2TDiscountRate().9 Evaluatinganexamplewiththesameparametersforeachmanufacturerattemptstoisolatetheeffectoftheguaranteeontheperformanceupdatingprocessoftheconsumerfromanydifferenceinparameters.InsolvingthebaselineDPwithnoguarantee,manufacturer1'srevenueis28664whilemanufacturer2'sis1796.Additionally,i=5soiftherstproducthassixormoreperiodsofbelowexpectedperformance,thentheconsumerwillswitchmanufacturers.Recall,thatweassumedthataguaranteeisnotonlyconsecutivebutstartsatpurchase(referredtoasafront-endguarantee)orendswithreplacement(referredtoasaback-endguarantee).Figure 3-3 showstheeffectontherstmanufacturer'srevenueforeachpossibleguaranteelength. 83

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Figure3-3. GuaranteeEffectonManufacturerRevenue Noticethattheeffectonrevenueisnotmonotonicinthelengthoftheguarantee.Themaximumrevenueisachievedwithaback-endthreeperiodperformanceguaranteeandincreasesrevenueto30367,approximatelya6%increase.Also,notethatneithertypeofguarantee,front-endorback-end,dominatestheotherforeachguaranteelength.Whilethisexampleshowstheuncertaintyinmakinggeneralizedstatementsabouttheoptimalguaranteelengthortype,wecanusesensitivityanalysistogainfurtherinsightintotheroleoftherelationshipbetweentheconsumer'sperformanceexpectationsandthetrueproductperformance.Undertheconditionthataproductwillbeheldtoitsmaximumusefullife,themaximumavailablerevenuebetweenthetwomanufacturerscanbecalculated.Inthepreviousexample,theproductisreplacedaftertenperiodssothemaximumavailablerevenueafterdiscountingis30460.Thisprovidesanupperboundontheeffectofaguarantee.Anyguaranteecanincreaserevenuetowardthisupperbound,butbecauseoftheliability,itcannotbeobtained.Ifwenowvarytheconsumer'sinitialexpectationandthemanufacturer'sadvertisedcostdistribution,weseetheimpact 84

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onthemanufacturer'srevenueandthestructureoftheoptimalguarantee.AllotherparametersarethesameaslistedinTable 3-1 .Table 3-2 showstheeffectonthemanufacturer'srevenue,intermsofpercentagechange,whentheoptimalguaranteeisoffered.Notethemaximumavailablerevenueisthesameupperboundaspreviouslystatedsincetheperformanceexpectationlevelsdonotaffectthisbound. Table3-2. ManufacturerRevenuewithOptimalPerformanceGuarantee ConsumerInitialBelief(E0) +0.1.2.3.4.5.6.7.8.91 015.224.440.030.000.000.000.000.000.000.000.00.148.2741.0231.464.280.070.010.000.000.000.000.00.248.1943.1335.3826.523.521.650.000.000.000.000.00.350.8250.4835.4827.0618.841.820.720.000.000.000.00.450.3550.1048.5845.3418.2311.530.640.180.000.000.00.549.6750.4850.4844.3638.6810.635.942.880.000.000.00.648.9150.0150.1650.1249.0630.1822.122.430.890.260.00.748.0949.4250.3250.4349.0346.6321.5014.037.950.130.00.847.2448.7549.8650.0450.6947.2743.4414.518.213.741.32.946.3748.0249.3149.5450.4651.0250.9142.3811.075.091.50151.6552.7153.4353.6954.0654.1754.0454.1553.6543.7927.46 ReviewingTable 3-2 ,therearesometrendsthatmayseemsurprising.First,foragivenleveloftrueperformance,theabilitytoaffectmanufacturerrevenuewithaguaranteeincreasesastheconsumerbecomesmoreoptimistic.Wemightexpecttheoppositesincepessimisticconsumers,whichareakintoriskaverseconsumersinthattheyhaveinatedexpectedvalues,areoftenmoreresponsivetowarrantyandguaranteetypecontracts.Thesecondobservationisthattherevenuechangesdecreasefasterastheconsumergetsmorepessimisticwhentheproductofferedbythemanufacturerhasbetterperformance.Thismaystemfromthefactthatwhenthetruedistributionisskewedtothelowercosts,theprobabilityofasuccessissmallwhenconsumerpessimismishighsincethethresholdforasuccesswouldbeinthetailofthebetadistributionwheneitheroftheshapeparametersislargerthan1.Thisleadstoasmallprobabilityofmakingafollow-onpurchasefromthesecondmanufacturer.Additionally, 85

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whentheconsumerispessimistic,themanufacturerhastoguaranteealargerrangeofcosts.Therefore,thereisalimitedbenetofusingaguaranteetochangetheconsumer'ssatisfactionsincethereisasmallamountofprobabilitytomovefromafollow-onpurchasefromthesecondmanufacturertotherstmanufacturerwhilesimultaneouslyhavingthepotentialforalargeliabilitywhentheguaranteeisactive.Ofimportancetonote,theexaminationofTable 3-2 doesnotimplyrevenueismaximizedwhentheconsumerhashighexpectationsortheperformanceoftheproductisverygood.Itonlyshowstheareaswheretheguaranteeismosteffective.Asamatteroffact,ifweconsiderthetotalmaximumdiscountedexpectedrevenue,wendthatforagivenproductcostdistribution,themanufacturer'srevenueincreaseswhenofferingtheoptimalguaranteeasthetheconsumerbecomesmorepessimistic.Figures 3-4 and 3-5 showthemanufacturer'stotalrevenuewiththeoptimalguaranteeandwithnoguarantee.Theconsumer'sexpectationisshownonthehorizontalaxisandthemanufacturer'sexpectationislistedinparenthesesinthelegend.Thegraphsshowthattherevenuecurvesareincreasingintheconsumer'sexpectationandthattheoptimalguaranteecurvesaresignicantlygreaterthanthenoguaranteecurvesformoreoptimisticconsumers.Astheconsumerbecomesmorepessimisticthough,thenoguaranteecurvesapproachtherevenueupperbound.Alsonoticethattherevenuewithnoguaranteeapproachestherevenuewiththeoptimalguaranteequickerwhenaproducthasgoodperformance.Inthiscase,quickermeansforamorepessimisticconsumer.NotethatFigure 3-4 showsrevenuecurvesforaproductwithbetterperformancethanFigure 3-5 86

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Figure3-4. ComparisonofGuaranteeandNoGuaranteeEffectsonRevenue(HighProductPerformance) Figure3-5. ComparisonofGuaranteeandNoGuaranteeEffectsonRevenue(LowProductPerformance) Besidestryingtounderstandtrendsinmanufacturerrevenuebasedontheconsumer'sandmanufacturer'sexpectation,wecanalsoanalyzetrendsinthestructureoftheoptimalguarantee.Table 3-3 showstheoptimalguaranteeforthesamecombinationsofconsumerandmanufacturerexpectationsasbefore. 87

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Table3-3. OptimalPerformanceGuaranteeDesigns ConsumerInitialBelief(E0) +0.1.2.3.4.5.6.7.8.91 0(10,B)(10,B)(9,F)(-,-)(-,-)(-,-)(-,-)(-,-)(-,-)(-,-)(-,-).1(9,B)(9,B)(9,F)(6,B)(2,B)(1,B)(-,-)(-,-)(-,-)(-,-)(-,-).2(8,B)(8,B)(8,B)(7,B)(4,B)(4,B)(-,-)(-,-)(-,-)(-,-)(-,-).3(8,B)(8,B)(7,B)(6,B)(6,B)(3,B)(2,B)(-,-)(-,-)(-,-)(-,-).4(8,B)(7,B)(7,B)(7,B)(5,B)(4,B)(2,B)(1,B)(-,-)(-,-)(-,-).5(8,B)(7,B)(7,B)(6,B)(6,B)(4,B)(3,B)(3,B)(1,F)(-,-)(-,-).6(8,B)(7,B)(7,B)(6,B)(6,B)(4,B)(4,B)(2,B)(2,B)(1,B)(1,F).7(8,B)(7,B)(6,B)(6,B)(5,B)(5,B)(3,B)(3,B)(3,B)(1,B)(1,F).8(8,B)(7,B)(6,B)(6,B)(5,B)(4,B)(4,B)(3,B)(2,B)(3,F)(1,B).9(8,B)(7,B)(6,B)(6,B)(5,B)(4,B)(4,B)(3,B)(2,F)(2,F)(2,F)1(8,B)(7,B)(6,B)(6,B)(5,B)(4,B)(3,B)(3,B)(2,B)(1,B)(1,B) Thestructureoftheguaranteeisrepresentedbytheduple(x,y)wherexrepresentsthelengthoftheguaranteeandyrepresentswhethertheguaranteeisfront-end(F)orback-end(B).Forsomecombinationofexpectations,offeringnoguaranteeproducesthemaximumrevenue.Whentheoptimalguaranteeisoffered,theback-endguarantee,inthisexample,isusuallypreferredtothefront-endguarantee.Note,94ofthe121scenariosshowthatthemanufacturer'srevenuecanbeincreasedbyofferingtheoptimalguarantee.Ofthese94,theoptimalguaranteeisafront-endguaranteeinonly9ofthem.Althoughthismightsuggestafront-endguaranteeplaysameaningfulroleinthisproblem(almost10%ofoptimalguaranteesarefront-end),furtheranalysismightsuggestotherwise.Ifwelookatthe9scenarioswherethefront-endguaranteeisoptimalandcompareittothebestback-endguarantee,we'llndthatthedifferenceintheeffectonmanufacturerrevenueisnegligible.Table 3-4 showsthepercentagedifferenceinthechangeinmanufacturerrevenuebyofferingtheoptimal(front-end)guaranteeasopposedtoofferingthebestback-endguarantee.Theabsolutedifferenceinmanufacturerrevenueisalsoprovided. 88

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Table3-4. AnalysisofOptimalFront-EndGuarantees (E0, +)%DifferenceAbsoluteDifference (.2,0).0005.1406(.2,.1).00912.0907(.8,.5).01895.7457(.8,.9).0012.3286(.9,.8).00521.5259(.9,.9).085524.7560(1,.6).082425.1040(1,.7).044813.6310(1,.9).02126.3582 TheresultsinTable 3-4 suggestforthisexamplethatwhentheoptimalguaranteeisonthefront-end,offeringthebestback-endguaranteedoesnotmakethemanufacturersignicantlyworseoff.Theabsolutedifferencessuggestthatthedifferencesarenotduetoroundingorotherpossiblenumericalanomalies.Thepurposeofthisanalysisistoprovideacounter-exampleagainstthehypothesisthatonetypeofguaranteemightdominatetheother.Theoutliers,inthisexample,occurnearthemarginwhereofferingaguaranteedoesnotincreaserevenueandnorwheretheconsumer'sbeliefisnotthatdifferentfromthetruedistribution.Theseresultssuggestthatforagivenproblemwithpredenedparameters,onetypeofguaranteemaybepreferredtotheotherinageneralsense.Additionally,foragivenconsumerexpectation,thelengthoftheguaranteetendstoincreaseasthetrueoperatingcostsincrease.Usingthisexample,aninterestingcharacteristicregardingtheconsumer'sexpectationforfractionalequivalentscanbeobserved.Considerthecasewherethetruedistributionisdenedbytheshapeparametersp,1=2andp,1=2andtheconsumerhasaninitialbeliefof.6.Underthisexpectation,theconsumer'sparametersarep,1=2.4andp,1=1.6.Aftertenperiodsofobservedcosts,thereisthepossibilitythattheconsumer'sexpectationremainsunchanged.Iftheconsumerrealizessixperiodsofworsethanexpectedperformanceandfourperiodsofbetterthanexpected 89

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performance,theconsumer'sexpectationofmanufacturer1'snextproduct(p0=p+20)isdenedbytheparametersp0,1=8.4andp0,1=5.6.Sincenoinformationhasbeenobservedontheperformanceofaproductfrommanufacturer2,theconsumermaintainstheiroriginalexpectationof.6denedbyp0+1,1=2.4andp0+1,1=1.6.Inthisinstance,theconsumerhasthesameexpectationforeachofthemanufacturer'sproducts.Theconsumer'sdecisionthoughistopurchasemanufacturer2'sproduct.Althoughtheexpectationforeachproductisthesame,thelargervalueoftheshapeparametersformanufacturer1'sproductmeansthatasuccessorfailurehaslessimpactontheconsumer'sfutureexpectation.Thisequatestotheconsumerbeingmorecondentintheproduct'speriodicoperatingcosts.Formanufacturer1,thiscondencehasanegativeimpactontheconsumer'schoiceatthemarginwhereproductexpectationsarethesameforeachmanufacturer.Thisresultcanbeinterpretedasthepessimismoftheconsumerhasbeenreinforcedbytherealizedcostswhereasthepessimismtowardsmanufacturer2'sproductisunprovenandthusismorelikelytobeupdated. 3.5SummaryTherearemanytypesofproductsthatconsumerscontinuallyneedandtechnologyimprovementsfortheseproductswillcommandalargerpurchaseprice.Automobilesandtelevisionsetsaretwocommonexamples.Thereforeitisinthebestinterestofmanufacturerstoconsiderwaystoenhanceconsumersatisfactionwiththeirproductinordertoimprovethechancesaconsumerreturnsforafollow-onpurchase.Inthischapter,wespecicallyconsiderthecasewhereconsumersarestronglyinuencedbyproductperformanceasmeasuredbyperiodicoperatingcosts.Whencostsarethedominantattributesuchthattheothervariableattributesareconsideredinsignicantorthesetofproductsunderconsiderationsharethesamequalitiesonallattributesotherthancosts,aconsumer'ssatisfactionwithaproduct'sperformancecanbemodeledbasedontheirinitialbeliefofaproduct'scostalongwiththeobservedproductperformanceasindicatedbytheobservedperiodicoperatingcosts.Wepresent 90

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aBayesianupdatingstrategywheretheconsumerhasaninitialbeliefaboutaproduct'scostbasedonanyrangeofinputssuchaspreviousexperiencewiththemanufacturer,andtheyupdatetheirexpectationeveryperiodbasedonthatperiod'srealizedcosts.Underthismodelandassumingtwoidenticalmanufacturers,weshowhowaperformanceguaranteecanbestructuredtoincreasethetotalexpecteddiscountedrevenueofamanufactureroveraxedhorizon.Witharenewedemphasisonproductperformancewithrespecttocommodityusage,thetypeofcontractualinstrumentspresentedinthischaptercangoalongwaytowardsimprovingtheprobabilityofaconsumerreturningforafuturepurchase.Ifproperlystructured,theseguaranteescanalsoincreasemanufacturerrevenue. 91

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CHAPTER4EXTENDEDPERFORMANCEBASEDWARRANTIESWITHEXOGENOUSCOSTINPUTS 4.1MotivationandLiteratureReviewAperformancebasedwarranty(PBW)guaranteesthataproductwillperformatsomeminimumlevel(i.e.thewarrantycoveragelevel)orelsethewarrantyprovidingagentmustcompensatetheconsumer.Whentheperformancelevelofaproductisrelatedtotheuseofacommodity,suchasfuelforaPBWongasmileageforacar,thevalueofaPBWisnon-stationaryovertimesincethepriceofthecommoditychanges,oftendramatically,overtime.Forexample,consideraproduct,suchasahouseholdappliance,thathasanadvertisedlevelofenergyusage.Ifamanufacturerguaranteesaminimumlevelofenergyefciency,thevalueoftheguaranteetotheconsumercanvarygreatlydependingontheunderlyingcostofenergy.Ifelectricityorgasratesarelow,thevalueofthewarrantymaybeminimaltotheconsumerwhereasiftheenergycostsareinated,thevaluetotheconsumermightbesignicant.Similarly,whenaPBWreimbursesaconsumerforcostsincurredfromsubstandardproductperformancerelativetotheuseofacommodity,thepriceamanufacturerchargesforaPBWcanalsobetime-dependent.Toillustratetheimportanceofnon-stationaritywhenmodelingPBWs,letuscompareatraditionalwarrantyandperformancebasedwarranty.Supposeweevaluatethepricechangesofkeyinputstomaintenanceandoperatingcostsintheautomobileindustry.Figure 4-1 comparestheaveragewageofanautomechanicandtheaveragegasolinepricefortheyears1999-2011.Theaveragemechanicwageisrelativelyatwhereasthepriceofgasolineishighlyvolatile.Althoughbothtypesofwarrantiescouldclaimnon-stationarityinthiscase,itisclearthatgascostsaremuchmoredependentontimethanmaintenancewages,andthuspricingaPBWbasedongascostswouldbemoresensitivetotime.NotethatthedataisfromtheBureauofLaborStatistics(BLS) 92

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andtheDepartmentofEnergy(DOE)withdatapointsforeachyearbasedonMaymeasurementsand1999usedasthebaselineyearinordertogeneratetheindex. Figure4-1. MechanicWagesvs.GasolinePrices Thenon-stationarityofthevalueofaxedlengthwarrantytotheconsumerandthemanufacturer'spriceofaPBWofthesamelengtharethereforedependentonestimatesoftheassociatedcommoditycostsfromboththeconsumerandmanufacturerperspectives.Itisnotonlylikelythattheconsumerandmanufacturerwillhavedifferentestimatesbutalsoreasonablethattheyhavenosystematicrelationship.InChapters 2 and 3 ,performancewasmodeledasasinglerandomvariablerepresentingoperatingcosts.Whilemathematicallysimple,thesemodelsconfoundtheeffectsofvariousfactorsonoperatingcosts.Additionally,themanufacturermaylimittheconsumer'srealizedoperatingcostsduetofactorsoutsidetheircontrolinsteadofasaresultoftheproduct'ssubstandardperformance.Undersimplifyingassumptionsaboutoperations,inordertomakethemodeltractable(e.g.constantmanpowerrequirementstooperateanyproduct),wecandecomposetheoperatingcostsintoanitenumberoffactorsbymodelingthecostsasafunctionofmultiplerandomvariablesorparameters.ThismethodallowsustoaccountfortheuniquecharacteristicsofaPBW,suchasnon-stationaryinputs.Usingthisdecompositionofoperatingcosts,weconsiderthe 93

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problemofhowamanufacturercanuseaPBWtoincreaserevenueinthepresenceofuncertaintyintheunderlyingcommoditycosts.Specically,westudyanequipmentreplacementproblemwithasinglechallengerwherethemanufactureroffersawarrantyeachperiodoveranitetimehorizon.TheconsumermaypurchasethePBWinanyperiodregardlessoftheirhistoryofpreviousPBWpurchases.Sinceawarrantyisofferedeveryperiodandtherearenolimitationsontheconsumer'schoicetopurchaseit,thePBWisbothrenewableanddeferrable.Additionally,thelengthandleveloftheperformanceguaranteearexedfortheentiretimehorizon.Conversely,thewarrantypricemaybedynamicovertimesincethewarrantycanbepricedaccordingtothemanufacturer'sforecastoffuturecommoditycosts(aswellasproductperformance).Also,theconsumer'sexpectedoperatingcostsarenotstationarysincetheydependontheconsumer'sforecastoffuturecommoditycosts.Boththemanufacturerandconsumerupdatetheirforecastseveryperiodafterobservingactualcommoditycosts.Inthischapter,weusedynamicprogramming(DP)toanalyzetheeffectofamanufacturerofferingPBWsineachperiodoveranitetimehorizonwhentheconsumerandmanufacturerviewperiodicoperatingcostsasaproductofmultiplerandomvariables.Differentestimationtechniquesareusedbythemanufacturerandconsumerforthecommoditycostparameters.Weassumethemanufacturerusesadvancedtime-seriesforecastingtechniquestomakesophisticatedpredictions.Theconsumerontheotherhandisassumedtousemorerudimentaryestimationtechniques.Whilethemodelsandcomputationalresultsthatfollowaredirectlyrelatedtotheestimation`techniqueschosen,theprimaryintentistodifferentiatebetweenthelevelofsophisticationoftheestimationtechniqueusedbyeachagent.InusingDPstomodelthissequentialdecisionproblem,itisrelativelystraightforwardtoallowthemanufacturerandconsumertoperiodicallyupdatetheircommoditycostestimatesbyiterativelysolvingasequenceofDPs.OurproposedstrategyofiterativelysolvingDPs 94

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alsosimultaneouslyallowsfortheinclusionofuncertaintransitioncharacteristics,suchasfutureperformancedependenceoncurrentperformance(Markovianperformance),seeninstochasticdynamicprograms.Aspreviouslymentioned,thepriceofthecommodity,suchasgasoline,usedbyaproduct,suchasacar,canhaveasignicantimpactonthevalueaconsumerassignstoaPBW,suchasaguaranteedminimumlevelofgasmileage.Therefore,theconsumer'scostestimationmethodisvitallyimportant.Wemightexpectthatconsumersoftenlackthestatisticalknowledgeandtoolstobuildasophisticatedforecastmodelorconductin-depthanalysis.TurrentineandKurani[ 38 ]discussconsumerbehaviorasitpertainstoestimatinggasolinepriceswhenconsideringfueleconomyandcarpurchases.Theyndthatconsumersoftenuserudimentaryestimatessuchasthecurrentpriceofgastoestimatefutureprices.InaNationalBureauofEconomicResearch(NBER)report,Kelloggetal.[ 39 ]alsofoundthatconsumers,onaverage,usethecurrentpriceofgasolineasanestimateoffutureprices.Theyfurthernotethatthereismuchheterogeneityamongconsumers,andthetechniqueofusingcurrentpricestoestimatethefuturepricesisnotobservedduringlargenancialshocks.Therefore,evenwithsimpleestimationtechniques,theconsumer'sforecastrelativetoactualpricescanvarydramatically.Greene[ 40 ]conductsasurveyofovertwodozenstatisticalstudiestodeterminehowconsumer'svaluefueleconomy.Hestatesthatinsomemodelsconsumersunderestimatethevalueoffuelsavingsandinothersoverestimatefuelsavings,butacommonassumptioninallthemodelsisconsumersusethecurrentpriceasanestimateforfutureprices.Forthemanufacturer,itwouldbeexpectedthattheycouldusetime-seriestechniquesandstatisticalsoftwaretoforecastcommodityprices.Althoughtheuseofamoresophisticatedtechniquedoesnotensurethatthemanufacturer'sestimateisbetterthantheconsumer's,wehypothesizethatthemanufacturercancapitalizeonthisasymmetry.Whiletheestimationtechniquesareimportant,ourgoalisnottoprovidean 95

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in-depthanalysisofmethodstopredictcommodityprices.WeonlyattempttoprovideafewreasonableexamplesthatclearlydifferentiateasophisticatedmanufacturerestimatefromarelativelyunsophisticatedconsumerestimateanddetermineifthemanufacturercanincreaserevenuebyofferingPBWsbasedontheirestimates.Regardlessoftheestimationtechniqueforeitherthemanufacturerorconsumer,wewouldexpectbothpartiestoupdatetheirestimatesbasedonnewinformation.Thiscouldbeaccomplishedbyusingaparametricdistributionforfuturecostsormerelyupdatingthecurrentpriceofthecommodityandtreatingitasadeterministicestimateforfuturecosts.Therefore,weproposesolvingasequenceofdynamicprogramsthatallowboththemanufacturerandconsumertoupdatetheirestimateseachperiod.PowellandSpivey[ 41 ]discusssolvingadynamicassignmentproblembysolvingasequenceofstaticassignmentproblemswhichallowsdecisionstobebemadeusingthemostup-to-dateinformation.ApplyingthissamemethodtoasequenceofdynamicprogramsallowstheconsumerandmanufacturertoincorporatethemostcurrentcostdatainmakingpricingandpurchasedecisionsforPBWs.Fortraditionalwarranties,anon-homogeneouspoissonprocessisoftenusedtomodelthenumberoffailuresovertime[ 42 44 ].Withthepropertyofindependentincrements,thenumberoffailuresinthecurrentperioddoesnotaffectthenumberoffailuresinthenextperiod.Conversely,whenmodelingproductperformanceintermsofaperformancemetricsuchasenergyusage,wemightexpectperformancetobenon-increasingovertheageoftheproduct,i.e.apropertyanalogoustoindependentincrementswouldnothold.Forexample,ahouseholdappliancelikearefrigeratorwouldnotbeexpectedtousealotofenergyearlyonandthenbecomemoreefcientasitages.Rather,theoppositemaybetrue.Tomodelthisdependency,astochasticdynamicprogram(SDP)isusedtomodeltheuncertaintransitionsbasedonthecurrentperformancelevel.Undercertainassumptionsabouthowtheconsumermodels 96

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non-increasingperformancewithage,theSDPcanbereplacedwithasequenceofDPsthatcanbesolvediteratively.OurcontributionstotheliteratureincludeproposingaframeworktomodelPBWsthataretiedtomarketcommoditycostsandadynamicprogrammingformulationtodeterminetheconsumer'soptimalpurchasepolicyforasetofdynamicallypricedwarranties.UsingactualtimeseriesdataforgasolinepricesfromtheDepartmentofEnergyandnominalproductparametersthatrepresentaneconomyclassautomobile,weshowthatthemagnitudeofamanufacturer'srevenuelossfromofferingperformancewarrantiesisrelativelysmall.Additionally,wendthatifthemanufacturercanfullyexploittheconsumer'swillingness-to-payforaPBW,thenthemanufacturercanincreasetotalrevenuebyofferingthePBWs.Finally,weincorporatenon-increasingperformanceintothemodelandshowthatwhenanaiveconsumerdoesnotaccountforthispracticalcharacteristic,themanufacturer'srevenuedecreases,onaverage,bylessthanonepercentforourdataset.Thischapterhasthefollowingstructure.Insection 4.2 ,wepresentamethodologyforpricingPBWsanddeterminetheeffectonmanufacturerrevenuebysolvingadynamicprogramiterativelyforasemi-empiricalexample.Wecomparetheeffectofdifferentcommoditycostestimates,forboththemanufacturerandconsumer,onmanufacturerrevenue.Next,insection 4.3 wepresentastochasticdynamicprogram(SDP)thatmodelsperformanceasnon-increasing.ThisSDPisalsosolvediterativelytodeterminethechangeinrevenuefromofferingPBWs.Becauseeachsolutionisonlyforaspecicperformancehistoryoverthetimehorizon,weconductaMonteCarlosimulationtodeterminetheaverageimpactonrevenuewhentheconsumerandmanufacturerusethesamecommoditycostestimates.Then,insection 4.4 wecombinethedisparatecostestimatesfromsection 4.2 withthenon-increasingperformanceassumptionfromsection 4.3 todeterminetheeffectonmanufacturerrevenue. 97

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4.2ModelFormulationwithUpdatedCostEstimatesOperatingcostsareasignicantinputwhenmakingequipmentreplacementdecisionsbasedonminimizingcosts.Whenperformancebasedwarrantiesareofferedthatpotentiallylimittheconsumer'soperatingcosts,itisimportantforboththemanufacturerandconsumertoaccountforthefactorsthatmostinuencethesecosts,andclearly,therearemanyfactorthataffectthem.Whileitisnotreasonabletoaccountforeverypossiblevariable,itisimportanttocapturetheeffectofthecriticalfactors.Byusingafunctionofthemostrelevantfactors,wecanprovideamorerealisticmodelofoperatingcostsandincorporatespecicconsumerbehaviortendenciesintothemodel. 4.2.1UncertaintyintheModelWeconsiderthecasewherethevariationinoperatingcostsisprimarilyduetotheconsumptionofaspeciccommodity.Withtotalproductusage,aunitcommodityprice,andaperformancemetricfortheamountofusageperunitofthecommodity,wemodeltheoperatingcostsasaproductofmultiplerandomvariables.DenearandomvariableMnwithdistributionfntorepresentthetrueperformancelevelofaproductofagen.Theconsumermaymodifythetrueperformancedistributionbasedontheirownbeliefstoderivetheirownperformancedistribution,whichweannotateasMcn.TheonlylimitationonthismodicationisthesupportforMnandMcnmustbethesame.Weassumetheperformancemetricispositivesuchthatthecommonsupportissomesubsetofthepositiverealnumberline.ThisissimilartoourmethodofaconsumermodifyingoperatingcostdistributionsinChapters 2 and 3 .Forthecommoditycosts,dene^ctasthemanufacturer'sestimateoftheunitcommoditycostattimet.Similarly,let^cctrepresenttheconsumer'sestimatedunitcommoditycostattimet.Theconsumer'sandmanufacturer'scostestimatesarenotnecessarilyrandomvariablesdependingontheirestimationtechnique(whichweaddresslater).Noticethat^ctand^cctaretimevariantrandomvariables(orparameters)whereasMnandMcnaretimeinvariant.LetU 98

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representthetotalperiodicusageofaproductwhichweassumetobeconstant.Whilethemodelcouldbeextendedtoallowforusagetovary,wearelimitingouranalysistothecasewhereUisxed.Weassumethatbothtandnrepresentdiscrete,equallyspacedperiodswhichmeanstheycanbeindexedoverasubsetofnon-negativeintegers.Specically,letTbethemaximumnumberoftimeperiodsandNbethemaximumusefullifeoftheproductsuchthatt2f0,...,Tgandn2f1,...,Ng.Theconsumer'sexpectedperiodicoperatingcosts,O,thenattimetforaproductofagenare:Ot(n)=E[U^cct Mcn]=UE[^cct]E[1 Mcn]. (4)Equation 4 describesthetotalperiodiccostastheproductofperiodicusage(e.g.milesperyear),unitcommoditycost(e.g.pricepergallonofgas),andaperformancemetric(e.g.milespergallon).Now,weareinterestedindeterminingifamanufacturercanincreaserevenuebyofferingperformancebasedwarranties.Specically,weintroducetheideaofanextendedperformancebasedwarranty(EPBW)denedbyalengthW,minimumperformancelevelX,andapriceP.ThetermEPBWisusedtodenotethatthesecontractscanbeobtainedapartfromthepurchaseofaproduct,similartothedenitionofanextendedwarrantyorservicecontract[ 45 ].ForanEPBWthatguaranteesaminimumperformancelevelX,theeffectiveperformance,forthemanufacturer,foraproductofagenunderawarrantyisdenedbythefollowingrandomvariable:Mn=min(Mn,X). (4)Themanufacturer'sliabilityisderivedfromthedifferencebetweeneffectiveperformance,Mn,andactualperformance,Mn.Similarly,wedeneMcnastherandomvariablecapturingtheconsumer'santicipatedeffectiveperformanceofaproductofagenwhenunderwarranty: 99

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Mcn=max(Mcn,X). (4)WithEquation 4 ,wecannowdenetheconsumer'sexpectedoperatingcostsunderanEPBWattimetandagenas:Ot(n)=E[U^cct Mcn]=UE[^cct]E[1 Mcn]. (4)InEquations 4 and 4 ,^cctandMcnareindependent.Wewillusethemoregeneralassumptionthroughoutthischapterthatstatesthatallcommoditypriceestimatesandperformancerandomvariablesareindependentforboththeconsumerandmanufacturer.Also,sincearandomvariableisinthedenominator,therearesomeimpliedrestrictionsonthedistributionalformofMnandMcn.Theserandomvariablescannotbefromadistributionsuchthattheexpectedvalueofthereciprocalisundened.Forexample,ifeitheroftheseareanexponentialdistribution,thenthereciprocalisaninverse-gammadistributionwithanundenedmean.WhilethewarrantyparametersWandXaresimilartoourpreviousperformancewarrantiesandguarantees,thereisasignicantdifferenceunderthemultiplerandomvariablemodel.Thewarrantylevelisdenedonaperformancemetricinsteadofoperatingcoststhereforethecompensationtotheconsumerisnotfullydenedbythewarranty.Underthismodel,theremustbesomefunctiontotranslatesubstandardperformancetoconsumerreimbursement.Additionally,wehavenotyetstatedanypricingconditionsontheEPBW.Supposethemanufacturerusestheactual,currentcommoditypricetodetermineanyreimbursementtotheconsumer.Whilethisexposesthemanufacturertotheriskofanexogenousfactorindeterminingtheirpotentialliability,thereexiststhepossibilitythatthemanufacturercouldstructureEPBWsthatcapitalizeonthedifferencebetween 100

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boththeconsumer'sperformancedistributionandcommoditycostestimateandthemanufacturer's(true)performancedistributionandcommoditycostestimate.Withthereimbursementstructuredened,thepriceoftheEPBWcannowbedetermined.Inmanywarrantyanalysisproblems,thepriceofthewarrantyisoftenastationaryparameter.Withourchosenreimbursementmethodology,thewarrantypriceisdynamic.Usingadiscountfactor,,toaccountforthetimevalueofmoney,thewarrantypricecanbecalculatedbysettingthepriceequaltothediscountedexpectedliabilityoverthelengthofthewarranty:Pt,n=WXi=1iE[U^ct+i Mn]=UWXi=1iE[^ct+i]E[1 Mn]. (4)OverthisWperiodsegmentstartingattimetandwiththecurrentlyownedproductofagen,themanufactureriswillingtoofferanEPBWtoaconsumeratpricePt,nwiththeunderstandingtherealizedliabilityisdependentontheactualcommoditypricesattheendofeachtimeperiod.Letctrepresenttheactualcommodityunitcostattimet.Underthiswarranty,theconsumerwouldpurchasethewarrantyunderthecondition:Pt,n+WXi=1iOt+i(n+i)WXi=1iOt+i(n+i).Conversely,themanufacturerrealizesanetincreaseinrevenueforthiswarrantyunderthecondition:Pt,nUWXi=1iE[ct+i]E[1 Mn]. (4)Interestingly,ifweconsidertheentireperiodofthewarrantywhenEquation 4 issatised,thechangeinmanufacturerrevenueisnotfullycharacterizedbythedifferenceintheleft-handsideandright-handsideoftheinequality.Theoretically,itispossiblefor 101

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theactualandestimatedcommoditycoststovaryoverthelifeofthewarrantysuchthatawarrantyisreplacedbeforeithasexpired. 4.2.2CostEstimatesFromtheprevioussection,therolesofconsumerandmanufacturercostestimatesindeterminingwhetherawarrantyispurchasedareextremelyimportant.Therearemanymodelsthatcouldbeusedtoestimateatimeserieslikegasolineprices.Onesuchmodelisamultiplicativedecompositionthatcanisolatelong-termtrendeffectsfromshorter-termbusinesscycleeffects[ 46 ].Whilemanyothertimeseriesmodelsarereasonable,suchasanautoregressivemodel,themultiplicativedecompositionhasaniceintuitivesimplicitythatisbenecialwhencomparingittotheconsumer'sestimate.Letusrstconsideraclassicaltimeseriesdecompositionmodelforthemanufacturer'sestimateofthecommoditycost.Weassumeamultiplicativemodeloftheform:ct=TtStBtIRtwhere:ct=commoditycostTt=trendcomponentSt=seasonalcomponentBt=businesscycliccomponentIRt=irregularcomponentInconductingthedecomposition,weassumeTt,St,Bt,andIRtareindependentsuchthatE(ct)=E(Tt)E(St)E(Bt)E(IRt).Forthetrendcomponent,anordinaryleastsquares(OLS)methodisusedwiththefollowinglinearform:T0t=0+1t0+,N(0,). 102

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Aquadratictcouldalsobeappliedtothetrendcomponentinwhichcase:T0t=a2(t0)2+a1t0+a0+,N(0,).Notet0isafunctionofthetimeperiodtsincethedatausedfortheestimatebeginsbeforetheproblemtimeperiodstarts.Toforecastthetrend,thepointestimatecouldbeusedorapredictionintervalcouldbedeterminedandtheupperlimitused.Usingtheupperlimitofthepredictionintervalindicatesamanufacturerwhowishestotakeonlessriskofunderestimatingthetrendeffect.Weannotatethesizeofthepredictionintervalbytheparameter,i.e..a100(1)]TJ /F9 11.955 Tf 11.95 0 Td[()predictioninterval.Fortheseasonalcomponent,St,wemakethesimplifyingassumptionthatthetimeperiodsareyearly;thusSt=1becauseeachperiodcoversthefullrangeofseasonaleffects,i.e.,aperiodisayearandthuscoverstheseasonaleffectofall12months.Thissimplicationisreasonablefortworeasons.First,anannualwarrantyreconciliation(determiningthereimbursementtotheconsumer)seemspracticalforboththemanufacturerandconsumer.Second,thepredictabilityandrepetitivenatureoftheseasonalimpactsoncertaincommoditiesmaybewell-knownbybothparties.TherandomcomponentBtrepresentsthebusinesscyclevariationandweassumethatBtG()whereGissomegeneraldistribution.Likewise,weassumeIRtG()whereGissomegeneraldistributionwithmeanone.Underthesesetofassumptionsthemanufacturermightusetheestimate:^ct=(^0+^1t0)^B. (4)InEquation 4 ,^Bisanestimateofthebusinesscycleeffect,andweconsiderthreedifferentestimatorsforthiscomponent.Therstisthesamplemeanofthecycliceffectscalculatedfromtheobservedcommodityvalues.Wecallthismethodtheaveragebusinesscycle(ABC).Thesecondisaquadraticttothelastthreebusinesscyclevalues.ThejusticationforthisestimateistheJuglarxedinvestmentcyclewhich 103

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hypothesizesthatbusinesscyclesarebetweensevenandelevenyearslong[ 47 ],andthuswecallthismethodtheJuglarbusinesscycle(JBC).Lastweconsiderthatthebusinesscyclehasnopositiveornegativeeffect(NBC)so^B=1.Whilethepreviousmanufacturerestimatesarenotsupposedtobeanauthoritativemodelforpredictingcommodityprices,wedoconsiderthemtobesophisticatedincomparisontothefollowingconsumermodels.Weconsiderthreedifferentmethodsfortheconsumerestimates.Iftisthecurrenttime,thenwedenethemethodsmathematicallyasfollows: i. Constant:^cct+i=ct,8i2f1,...,T)]TJ /F5 11.955 Tf 11.95 0 Td[(tg ii. Linear:^cct+i=i(ct)]TJ /F5 11.955 Tf 11.95 0 Td[(ct)]TJ /F8 7.97 Tf 6.58 0 Td[(1),8i2f1,...,T)]TJ /F5 11.955 Tf 11.95 0 Td[(tg iii. Geometric:^cct+i=(1+!)ict,8i2f1,...,T)]TJ /F5 11.955 Tf 11.95 0 Td[(tg,!2(0,1)Thethreeconsumerestimatemethodsarebasedontheconceptsthattheyaresimpleanddonotincorporatemuchmorethanthecurrentdata.Infact,onlythelinearmodelusesthepreviousperiod'scostalongwiththecurrentperiod.TheconstantmethodisbasedonTurrentineandKurani's[ 38 ]ndingwhilethelinearandgeometricmethodsallowfortheconsumertoaccountforcostinationinabasicway.Notethat!istheinationfactorforthegeometricestimate.Themodelpresentedthusfarisasubsetofalargerequipmentreplacementproblem.WeareinterestedintheuseofwarrantiesovertheentiretimehorizonTwheretheconsumerhastheoptionstoreplacetheproductandnorpurchaseanEPBWduringanyperiodinthehorizon. 4.2.3DynamicProgramBasedonthevariablesintheprevioussection,wedeneadynamicprogram(DP)todeterminetheconsumer'sreplacementpolicyandtheassociatedeffectonthemanufacturer'srevenue.LetP,withoutanysubscripts,bethepurchasepriceoftheproductwhichisstationaryoverthenitetimehorizon,T.ThestatespacefortheDP 104

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hastwodimensions:nistheageofthecurrentitemandwistheamountoftimeleftonthewarranty.Pt,nisthewarrantypricethatthemanufactureroffersforaproductofagenattimet.Foreachfeasiblestate(n,w)where0nN,0wW,wedenevt(n,w)astheconsumer'scost-to-gofunction.Equations 4 4 deneaDPwheretheconsumer'sexpectedoperatingcostsandthemanufacturer'swarrantypricesaredeterminedattimezero.Equation 4 representstheinitialpurchaseatthebeginningofthetimehorizonoramandatoryreplacementbecausetheproducthasreacheditsmaximumusefullife.Inthiscasetheconsumercanreplacetheitemwithoutawarranty(R)orreplaceitwithawarranty(R+PBW).Whentheconsumerdoesnothaveawarranty,Equation 4 allowsfortheadditionalchoicesofkeepingtheitem(K)orbuyingawarranty(K+PBW).Equation 4 representsthecasewheretheconsumerhasawarrantyandthushasthesamechoicesasinEquation 4 .Equation 4 istheterminalcondition. vt(n,w)=min8><>:R:P+(Ot(1)+vt+1(2,0))R+PBW:P+Pt,n+(O(1)+vt+1(2,W)]TJ /F7 11.955 Tf 11.96 0 Td[(1))9>=>;,(4)n=0,N,0wW vt(n,0)=min8>>>>>>><>>>>>>>:K:(Ot(n)+vt+1(n+1,0))K+PBW:Pt,n+(O(n)+vt+1(n+1,W)]TJ /F7 11.955 Tf 11.95 0 Td[(1))R:P+(Ot(1)+vt+1(2,0))R+PBW:P+Pt,n+(O(1)+vt+1(2,W)]TJ /F7 11.955 Tf 11.96 0 Td[(1))9>>>>>>>=>>>>>>>;,(4)1n>>>>>><>>>>>>>:K:(Ot(n)+vt+1(n+1,0))K+PBW:Pt,n+(O(n)+vt+1(n+1,w)]TJ /F7 11.955 Tf 11.96 0 Td[(1))R:P+(Ot(1)+vt+1(2,0))R+PBW:P+Pt,n+(O(1)+vt+1(2,w)]TJ /F7 11.955 Tf 11.96 0 Td[(1))9>>>>>>>=>>>>>>>;,(4)1n
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vT(n,w)=0,8n,w(4)Thisformulationrepresentsthescenariowherethecostestimatesaredeterminedupfrontandneverupdated.OnewaytomodifytheDPtoallowtheconsumertoupdatetheircostestimatesandaccountforwarrantypricechangesistoaddanotherdimensiontothestatespace.Thisthirddimensionwouldcapturethecurrentcommodityprice,butsincethetransitiontothenextstateisalsouncertain,theDPwouldbestochastic.Additionally,therangeforthecommoditypricewouldbecontinuousandthustherewouldbeaninnitenumberofstatessothetransitionswouldneedtobediscretized,whichcouldstilldramaticallyincreasethestatespace.Also,thepolicyofastochasticdynamicprogram(SDP)isdependentonknowingthestatesoverthehorizonoftheproblem.Finally,usingthismethodrequirestheconsumertodevelopadistributionwhichisamuchmorecomplicatedestimationprocessthanasimpleextrapolationofthecurrentcommoditycost.UsinganSDPispossibletomodelthisproblem,butwewanttouseamethodthatbettercapturestheconsumerbehaviorwithregardstocommoditycostestimation.Therefore,wesolvethedeterministicDPiterativelyand,ateachiteration,updatetheconsumer'sandmanufacturer'scostestimates.ThiswillrequireatotalofTDPstobesolvedwherethetimehorizonofthesubsequentiterationsdecreasesbyone.UsingthismethodmaintainsthesamelimitationastheSDPinthatanypolicyisdependentonknowingthestatesoverthetimehorizon,butitallowsfortheconsumerandmanufacturertoupdatetheirparametersandthuseliminatetheissueofhavingalargenumberofstateswhenthecommoditycostispartofthestatespace.Italsobetterreectstherealityofhowweexpecttheentitiestoactinthefactthattheywillincorporatethemostrecentinformation.Ultimately,weareinterestedindeterminingifitispossibletocapitalizeonthedifferenceincostestimates,andthisiterativemethodissufcientforthatpurpose.Weuseanumericalexamplewithanempiricalcomponent 106

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forthecommoditycostsandthecostestimationtechniquesfromtheprevioussectiontoseeifpricingandofferingEPBWs,asdescribed,caninfactimprovemanufacturerrevenueinarealisticenvironment. 4.2.4NumericalExampleTheconjecturethatamanufacturercanincreaserevenuethroughEPBWsbycapitalizingonitsmoresophisticatedcommoditycostestimatesisdifculttotest.Theestimationofthecommoditycostsaloneisanextremelycomplexendeavorbyitself.Theimportanceoftheproposedestimationtechniqueisnotthespecicestimateitself,butthedisparityinthesophisticationoftheestimationmethodversustheconsumer'sestimate.Additionally,duetotheuncertaintyandnatureofcommoditycosts,itisnearimpossibletomakeanygeneralizedcommentontheabilitytoaffectrevenuewhenthereimbursementistiedtoactualcosts.Therefore,ourbesteffortistopresentanumericalexamplethatincorporatesactualcommoditycostdatatodeterminethefeasibilityofusingEPBWstopositivelyaffectrevenue.TheempiricaldatasetusedforthecommoditycostscomesfromtheU.S.DepartmentofEnergy(DOE)andentailstheweeklygasolinepricesfortheU.S.fromJanuary1992toDecember2011.Theproductdataontheotherhandishypotheticalbutroughlymodeledonaneconomyclassautomobile.Table 4-1 belowliststheproblemparameters.Forthisexample,weuseadiscreteuniformdistributionforproductperformancesothevedifferentperformancelevelsoccurwithequalprobability.Additionally,weaccountforperformancedegradationbydeningaperiodicrate,,bywhichtheperformancechangeswitheachperiodincreaseinage,i.e.Mn+1=Mn.Sincewearetryingtodeterminetheeffectoftheestimationtechniquesinthepresenceofactualcosts,weassumetheconsumerhasneutralbeliefsaboutproductperformanceandacceptsthemanufacturer'sdistribution.Thewarrantycoveragelevelisdetermined 107

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aprioribythemanufacturer.Whilethisassumptioncouldberelaxedandtheoptimalcoveragelevelcouldbedetermined,wedonotstudythisaspect. Table4-1. ParametersforEPBWExample ParameterValue TimeHorizon(T)12MaximumUsefulLife(N)12PurchasePrice(P)20000LowPerformanceLevel26MediumLowPerformanceLevel29MediumPerformanceLevel32MediumHighPerformanceLevel35HighPerformanceLevel38PerformanceDegradation().99DiscountRate().9WarrantyCoverageLevel(X)32 Notethatthelengthoftheproblemhorizonis12periodsbutthelengthoftheempiricaltimeseriesis20periods.TheDPtimehorizonstartswithyear2000sothepreviouseightyearsofdataprovidetheinitialbasisforthecommoditycostestimates.FortheiterativeDPmethod,thenextperioddataisincorporatedinthenextsetofestimates.Also,theweeklydataisaveragedtoprovideannualvalues.Table 4-2 showstheresultsforwarrantylengthsuptosixperiodsandthedifferentcombinationsofconsumerandmanufacturerestimationmethods.TheestimatesthatusetheupperboundofthepredictionintervalareannotatedwithPr.TheentriesinTable 4-2 providethepercentagechangesinrevenuefromalwaysofferingEPBWsofagivenlengthforthedifferentcombinationsofconsumerandmanufacturercommoditycostestimates.Therstentryofthedupleisthechangeinactualrevenue.ItrepresentstherealizedrevenuewhenthemanufacturerpricesthewarrantiesinaccordancewithEquation 4 andreimbursestheconsumerbasedontheactualcommoditycostswhentheconsumerownsawarranty.Thesecondtermrepresentsthemaximumpossiblerevenueifthemanufacturerhadpricedthewarrantiesatthemaximumwillingness-to-payoftheconsumer.Theresultscanbeanalyzedinmanydifferentways,andwelimitourcommentstotwoparticularpoints. 108

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Table4-2. EPBWEffectonManufacturerRevenue(=.2and!=.2) W=1W=2 ConstantLinearGeometricConstantLinearGeometric LinearABC(-0.21,0.10)(-0.28,0.36)(-0.29,0.55)(-0.24,0.01)(-0.32,0.26)(-0.33,0.60)LinearABC-Pr(-0.18,0.08)(-0.25,0.32)(-0.25,0.50)(-0.22,-0.01)(-0.29,0.23)(-0.30,0.56)LinearJBC(-0.67,0.29)(-0.65,0.40)(-0.65,0.73)(-0.65,0.05)(-0.56,0.32)(-0.54,0.67)LinearJBC-Pr(-0.66,0.29)(-0.70,0.32)(-0.63,0.68)(-0.65,0.04)(-0.67,0.17)(-0.65,0.54)LinearNBC(-0.18,0.08)(-0.25,0.33)(-0.25,0.51)(-0.21,-0.01)(-0.29,0.23)(-0.30,0.56)LinearNBC-Pr(-0.15,0.05)(-0.22,0.29)(-0.22,0.47)(-0.19,-0.03)(-0.27,0.20)(-0.27,0.53)QuadraticABC(-0.04,-0.03)(-0.05,0.14)(-0.04,0.30)(-0.08,-0.07)(-0.07,0.19)(-0.10,0.36)QuadraticABC-Pr(0.00,0.00)(-0.03,0.03)(0.05,0.16)(0.00,0.00)(-0.10,-0.04)(-0.03,0.21)QuadraticJBC(-0.64,0.26)(-0.64,0.27)(-0.65,0.51)(-0.62,0.00)(-0.61,0.16)(-0.63,0.34)QuadraticJBC-Pr(-0.62,0.25)(-0.62,0.25)(-0.62,0.50)(-0.60,-0.04)(-0.59,0.15)(-0.60,0.30)QuadraticNBC(-0.05,-0.04)(-0.02,0.12)(-0.01,0.26)(-0.08,-0.07)(-0.04,0.15)(-0.07,0.33)QuadraticNBC-Pr(0.00,0.00)(-0.02,0.02)(-0.01,0.06)(0.00,0.00)(-0.09,-0.05)(-0.08,0.08) W=3W=4 ConstantLinearGeometricConstantLinearGeometric LinearABC(-0.28,-0.08)(-0.36,0.21)(-0.36,1.04)(-0.32,-0.02)(-0.42,0.26)(-0.42,1.02)LinearABC-Pr(-0.26,-0.09)(-0.34,0.18)(-0.33,1.00)(-0.30,-0.06)(-0.40,0.24)(-0.40,0.99)LinearJBC(-0.86,-0.02)(-0.58,0.19)(-0.58,0.75)(-1.18,-0.04)(-0.87,0.35)(-0.91,0.93)LinearJBC-Pr(-0.85,-0.03)(-0.84,0.20)(-0.55,0.70)(-1.17,-0.05)(-1.14,0.36)(-0.88,0.90)LinearNBC(-0.25,-0.10)(-0.33,0.19)(-0.33,1.00)(-0.29,-0.05)(-0.39,0.24)(-0.39,0.99)LinearNBC-Pr(-0.20,-0.06)(-0.31,0.16)(-0.30,0.97)(-0.26,-0.08)(-0.37,0.21)(-0.37,0.96)QuadraticABC(-0.11,-0.09)(-0.19,0.02)(-0.10,0.69)(-0.16,-0.12)(-0.18,0.15)(-0.21,0.72)QuadraticABC-Pr(0.00,0.00)(-0.14,-0.02)(-0.08,0.32)(0.00,0.00)(-0.21,-0.02)(-0.06,0.63)QuadraticJBC(-0.82,-0.07)(-0.81,0.09)(-0.82,0.43)(-1.16,-0.09)(-0.87,0.12)(-1.16,0.69)QuadraticJBC-Pr(-0.79,-0.11)(-0.79,0.08)(-0.79,0.40)(-1.12,-0.13)(-0.84,0.10)(-1.12,0.66)QuadraticNBC(-0.11,-0.09)(-0.16,0.00)(-0.06,0.64)(-0.16,-0.12)(-0.15,0.12)(-0.17,0.68)QuadraticNBC-Pr(0.00,0.00)(-0.13,-0.03)(-0.05,0.29)(0.00,0.00)(-0.19,-0.03)(-0.21,0.22) W=5W=6 ConstantLinearGeometricConstantLinearGeometric LinearABC(-0.34,-0.10)(-0.46,0.19)(-0.47,0.70)(-0.37,-0.12)(-0.50,0.62)(-0.50,1.33)LinearABC-Pr(-0.32,-0.11)(-0.45,0.17)(-0.45,0.68)(-0.34,-0.14)(-0.49,0.61)(-0.49,1.32)LinearJBC(-0.73,-0.06)(-0.79,0.28)(-0.83,0.83)(-0.84,-0.14)(-0.59,0.30)(-0.59,1.02)LinearJBC-Pr(-0.72,-0.07)(-0.79,0.27)(-0.82,0.81)(-0.84,-0.15)(-0.57,0.29)(-0.57,1.00)LinearNBC(-0.31,-0.12)(-0.44,0.17)(-0.44,0.68)(-0.33,-0.14)(-0.48,0.60)(-0.48,1.31)LinearNBC-Pr(-0.29,-0.13)(-0.43,0.16)(-0.43,0.66)(-0.31,-0.16)(-0.46,0.59)(-0.46,1.30)QuadraticABC(-0.23,-0.17)(-0.36,0.07)(-0.36,0.56)(-0.33,-0.25)(-0.33,0.38)(-0.33,1.09)QuadraticABC-Pr(0.00,0.00)(-0.25,0.03)(-0.25,0.52)(0.00,0.00)(-0.17,0.32)(-0.17,1.03)QuadraticJBC(-0.72,-0.11)(-0.71,0.17)(-0.64,0.64)(-0.85,-0.19)(-0.83,0.14)(-0.85,0.52)QuadraticJBC-Pr(-0.69,-0.14)(-0.68,0.17)(-0.69,0.41)(-0.81,-0.22)(-0.80,0.13)(-0.81,0.49)QuadraticNBC(-0.23,-0.17)(-0.35,0.06)(-0.35,0.55)(-0.33,-0.25)(-0.29,0.35)(-0.29,1.06)QuadraticNBC-Pr(0.00,0.00)(-0.22,0.02)(-0.22,0.51)(0.00,0.00)(-0.13,0.28)(-0.13,1.00) First,insummary,theaveragechangeinactualrevenueofallinstancesis)]TJ /F7 11.955 Tf 9.3 0 Td[(.41witharangeof[)]TJ /F7 11.955 Tf 9.3 0 Td[(1.18,.05]whenaveragesaregroupedbywarrantylength.Note,thebaselinerevenuewhenwarrantiesarenotofferedis20000sincetheeconomiclifeoftheproductis12periodsregardlessofwhichperformancelevelisrealized.Thechangeinmaximumpossiblerevenuehasanoverallaverageof.73witharangeof[)]TJ /F7 11.955 Tf 9.3 0 Td[(.25,1.33]forthedifferentwarrantylengths.Thesesummarystatisticssuggestthatthedownsideriskoftyingawarrantytoactualcostsmaybeacceptableasthedecreaseinrevenueislessthanahalf-percentonaverage.Thisslightdecreaseinrevenuehasthe 109

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potentialtoofferotherbenetssuchastheconsumerreturningtothepurchaselocationannuallytoreconcilethewarrantywhichcouldbetiedtoamarketingstrategytosellperiodicmaintenancefunctions(suchasanoilchangeinthecaseofanautomobile).Additionally,thewarrantypricesusingour12estimationmodelsleaverevenueonthetablesothepricesingeneralarenotcapturingthefullwillingness-to-payoftheconsumer.Whilethemanufacturer'sestimationtechniquesaremoresophisticatedthantheconsumer,ourimplementationsareformulaicanddonotreectthetrueabilityofamanufacturertomakemoreexibleadjustments,suchasvaryingthenumberofperiodsusedtodeterminethebusinessfactor,^B.Theconsumer'sestimates,especiallyforshorterlengthwarranties,doafairlygoodjobofcapturingthebusinesscycleeffectsinceitisbasedonthemostcurrentknownprice.TheABCandNBCtechniquesforestimatingthebusinesscyclemaynotactuallydoaswellastheconsumerinthiscase.TheJBC,orasimilartypeestimate,hasthepotentialtoexplainthebusinesscycleeffectbetterwhenitisexible(suchasallowingthenumberofdatapointsusedinthecalculationtovary).Ourformulationalsoassumesthemanufacturerwillalwaysofferawarrantywhereasinreality,basedonthemodelestimates,themanufacturermaychoosenottoofferanEPBWatcertaintimes.Second,thereisnosystematicrelationshipbetweentheresultswhengroupedbytheconsumer'scostestimationtechnique.Wemightexpecttheactualrevenueandmaximumpossiblerevenuetoincreaseastheconsumer'sestimationtechniquemovesfromconstanttolinearandfromlineartogeometric.Thisisnotalwaysthecase.Forexample,aconstantestimatemightleadtoalowerwillingness-to-pay(WTP)foraconsumerwhichmayinhibittheconsumerfrompurchasingawarrantyatacertainpointintime.Ifthesubsequentcommoditypricesrisesignicantly,thenthemanufacturercouldhaveincurredaliabilitylargerthanthewarrantypurchasepriceiftheconstantconsumerwouldhavepurchasedthewarranty.Aconsumerwhousesageometricestimatethoughhasahigherwillingness-to-pay,andmayhavepurchasedthesame 110

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warrantythattheconstantconsumerdidnot.Withthesharpriseincommodityprices,themanufacturermightrealizethelargeliabilityandthusthereisanegativeneteffectonthemanufacturer'srevenue.Therefore,nogeneralizationcanbemaderegardingtheabilityofamanufacturertocapitalizeonaconsumeroverestimatingfuturecommodityprices. 4.3MarkovianPerformanceInChapters 2 and 3 ,themethodusedtomodelperformanceuncertaintycontainedtheimpliedassumptionthatfutureperformanceisnotdependentonpreviousperformance.Fortraditionalwarranties,thisassumptionisreasonable,especiallywhenfailuresfollowapoissonprocess.However,wewouldexpectperformancetobenon-increasingandthusfutureperformancedependsonthecurrentlevelofperformance.TomodelperformanceinthismannerweproposeaMarkovchaintodescribethetransitionofproductperformancefromoneperiodtothenext. 4.3.1ModelFormulationwithNon-IncreasingPerformanceUptothispoint,distributionsdescribingproductperformancewerenotlimitedtobeingdiscrete.Withtheinclusionofnon-increasingperformance,welimittheanalysistodiscreteproductdistributionsinordertoutilizediscretespaceMarkovchains.NotethemodelingandanalysiscouldbeexpandedtocontinuousstateMarkovchainsoracontinuousdistributioncouldbediscretizedtoanacceptablenumberofperformancelevels.ThereforewebeginbydeningaproductwithLperformancelevels.Forl2f1,...,Lg,weassumethatperformancelevelsareindexedinincreasingordersuchthattheperformanceatlevell+1isbetterthantheperformanceatl.WedenetheinitialperformancedistributionbytheLx1vectorsuchthattheprobabilityofaproductperformingatleveliinit'srstperiodofoperationisdenedby(i).Nextwedenethesubsequenttransitionsbythefollowingone-steptransitionmatrix: 111

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PT=2666666666666664100...00p2,1p2,20...00p3,1p3,2p3,3...00..................pL)]TJ /F8 7.97 Tf 6.59 0 Td[(1,1pL)]TJ /F8 7.97 Tf 6.58 0 Td[(1,2pL)]TJ /F8 7.97 Tf 6.59 0 Td[(1,3...pL,L)]TJ /F8 7.97 Tf 6.59 0 Td[(10pL,1pL,2pL,3...pL,L)]TJ /F8 7.97 Tf 6.59 0 Td[(1pL,L3777777777777775ThediagonalstructureofPTisaby-productofnon-increasingperformanceandtherstdiagonalelementisalwaysone.Usingthisnewprobabilitystructure,wemustincludeanadditionalindexontheperformancelevels,theconsumer'speriodicoperatingcosts,andthemanufacturer'swarrantypricesinordertoaccountforthecurrentperformancelevel.LetMn,kandMcn,kbetherandomvariablesrepresentingthemanufacturer'sandconsumer'sperformancedistribution,respectively,wherenistheageandkisthecurrentoperatingleveloftheproduct.Asbefore,theconsumermodiesandPTtoconstructMcn,k.Similarly,weredeneperformanceunderwarrantyasMn,kandMcn,kaswellastheconsumer'scostwithandwithoutawarrantyasOt(n,k)andOt(n,k).Usingthenewdenitions,thewarrantypriceundernon-increasingperformancecanbecalculatedas:Pt,n,k=WXi=1kXj=1iE[U^ct+i Mn,j]=UWXi=1kXj=1iE[^ct+i]E[1 Mn,j].Underthisperformancestructure,theconsumerwouldpurchasethewarrantyunderthefollowingcondition:Pt,n,k+WXi=1kXj=1iOt+i(n+i,j)WXi=1kXj=1iOt+i(n+i,j). 112

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Conversely,themanufacturerrealizesanetincreaseinrevenueforthiswarrantyunderthefollowingcondition:Pt,n,kUWXi=1kXj=1iE[ct+i]E[1 Mn,j].Aspeciccaseunderthenon-increasingperformancemodelthatisinterestingtohighlightiswhentheconsumerbelievescurrentperformancewillcontinueindenitely.Inthiscase,theconsumerisprovidedtheinitialproductperformancedistribution,.AsinChapters 2 and 3 ,theconsumermaymodifyinsuchawaythattheyarecharacterizedasoptimistic,neutral,orpessimistic.Thetransitionmatrixisnotrevealed,though,becausetheconsumerisassumedtoexcludethisperformancecharacteristicfromtheirdecisionmaking.Whenthisisthecase,theconsumerusestheidentitymatrixastheirone-steptransitionmatrix.Whateverperformancelevelaconsumerobservesattimet,theyineffectexpecttheproducttoperformatthisleveluntilreplacement.Note,thissituation,althoughmathematicallyequivalent,isnotthesameasacustomerwhoisprovidedthetransitionmatrixandchoosestomodifyittobetheidentitymatrix.Weconsiderthisspecialcasebecauseitisareasonableassumptionregardinghowconsumersmakedecisions.Justasweevaluatedunsophisticatedcostestimatesfortheconsumerintheprevioussection,thisscenariorepresentsanunsophisticatedoutlookonperformance.Forthisspecialcase,ifweassumethemanufacturerandconsumerusethesamecostestimatesandinitialdistribution,thenaconsumerwillneverpurchaseawarrantyundernon-increasingperformance. Theorem4.1. Aneutraloroptimisticconsumerwhodoesnotaccountfornon-increasingperformancewillneverpurchaseawarrantyif^ct^cct8tandPT6=I. Proof. 113

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Giventhestate(n,w,k)attimet,thepriceofawarrantyis:Pt,n,k=WXi=1iUct+i E[Mn+i,k)]TJ /F5 11.955 Tf 11.96 0 Td[(Mn+i,k]=WXi=1iUct+i1 E[Mn+i,k)]TJ /F5 11.955 Tf 11.95 0 Td[(Mn+i,k]. (4) Similarly,thepricetheconsumeriswillingtopay,denotedbyPct,n,kis:Pct,n,k=WXi=1iUct+i E[Mn+i,k)]TJ /F5 11.955 Tf 11.96 0 Td[(Mn+i,k]=WXi=1iUct+i1 E[Mn+i,k)]TJ /F5 11.955 Tf 11.95 0 Td[(Mn+i,k]. (4) Sinceweareonlyconsideringdiscretedistributions,letpi:i2f1,...,Lgrepresentthemutuallyacceptedperformancelevelsfortheproductinincreasingorder,i.e.pi+1>pi.ThusforthefractionalcomponentofEquation 4 wehave:1 E[Mn+i,k)]TJ /F5 11.955 Tf 11.96 0 Td[(Mn+i,k]=1 Pkj=1pjPTk,j. Fortheconsumer,thefractionalcomponentofEquation 4 is:1 E[Mcn+i,k)]TJ /F5 11.955 Tf 11.96 0 Td[(Mcn+i,k]=1 Pkj=1pjIk,j. SincePTisalowerdiagonalMarkovtransitionmatrix(MTM),thenPkj=1pjPTk,jPct,n,kandthereforetheconsumerwillnotpurchaseawarranty. Notetheresultaboveonlyconsiderswarrantiesforproductsthatarealreadyowned.Whenawarrantyispurchasedatthesametimeaproductispurchased,theresultstill 114

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holdsbecausethefractionalcomponentundertherstperiodofoperationisthesameforthemanufacturerandconsumerwhentheconsumerisneutral.Whentheconsumerisoptimistic,theconsumer'sfractionalcomponentfortherstperiodislessthanthemanufacturer'scomponentintherstperiod.Foraneutraloroptimisticconsumer,thefollowingconditionholds:1 E[M1,k)]TJ /F5 11.955 Tf 11.95 0 Td[(Mn+i,k]1 E[Mc1,k)]TJ /F5 11.955 Tf 11.95 0 Td[(Mcn+i,k].Therefore,whetherthewarrantyisconcurrentwithaproductpurchaseornot,thetheoremisvalid. 4.3.2DynamicProgramWiththeinclusionoftheassumptionofnon-increasingperformance,therecursioninEquations 4 4 mustbeupdated.Besidesincreasingthestatespacefromtwotothreedimensionstoaccountforthecurrentperformancelevel,theuncertaintransitionsalsorequiretheuseofastochasticdynamicprogram(SDP).Equations 4 4 denetheSDPundertheassumptionofnon-increasingproductperformance.Thisrecursionisgeneralizedtoallowfortheconsumertoincorporatenon-increasingperformancesuchthattheconsumer'sMTMusedtodenetransitionsisdenotedbyPTc.Conceptually,PTccouldbesettothemanufacturer'sMTMorotherwisemodiedliketheconsumerdoeswiththeirinitialbelief.WhenPTc=I,theconsumerignoresnon-increasingperformance.AlthoughtheformulationisdifferentfromEquations 4 4 ,thedecisionspaceisthesameasthedeterministicmodelwhenfutureproductperformancewasnotinuencedbythecurrentperformancelevel.vt(n,w,k)=min8><>:R:P+(Ot(1,k)+PLi=1c(i)vt+1(2,0,i))R+PBW:P+Pt,n,k+(O(1,k)+PLi=1c(i)vt+1(2,W)]TJ /F7 11.955 Tf 11.95 0 Td[(1,i))9>=>;,n=0,N,0wW,1kL (4) 115

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vt(n,0,k)=min8>>>>>>><>>>>>>>:K:(Ot(n,k)+Pki=1PTck,ivt+1(n+1,0,i))K+PBW:Pt,n,k+(O(n,k)+Pki=1PTck,ivt+1(n+1,W)]TJ /F7 11.955 Tf 11.96 0 Td[(1,i))R:P+(Ot(1,k)+PLi=1c(i)vt+1(2,0,i))R+PBW:P+Pt,n,k+(O(1,k)+PLi=1c(i)vt+1(2,W)]TJ /F7 11.955 Tf 11.95 0 Td[(1,i))9>>>>>>>=>>>>>>>;,1n>>>>>><>>>>>>>:K:(Ot(n,k)+Pki=1PTck,ivt+1(n+1,0,i))K+PBW:Pt,n,k+(O(n,k)+Pki=1PTck,ivt+1(n+1,w)]TJ /F7 11.955 Tf 11.95 0 Td[(1,i))R:P+(Ot(1,k)+PLi=1c(i)vt+1(2,0,i))R+PBW:P+Pt,n,k+(O(1,k)+PLi=1c(i)vt+1(2,w)]TJ /F7 11.955 Tf 11.96 0 Td[(1,i))9>>>>>>>=>>>>>>>;,1n
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Figure4-2. NetworkwithNon-IncreasingProductPerformance Assumingageneralnon-increasingMarkoviannetworkwithLperformancelevelsandTtimeperiods,wedeneH(L,T)asthetotalnumberofpossibleperformancepathssuchthat:H(L,T)=1+(1+(T)]TJ /F7 11.955 Tf 11.96 0 Td[(1))+LXi=3[1+(T)]TJ /F7 11.955 Tf 11.96 0 Td[(1)+TXj=2TXk=j+1hi)]TJ /F8 7.97 Tf 6.59 0 Td[(1,k]=1+(L)]TJ /F7 11.955 Tf 11.95 0 Td[(1)T+LXi=3TXj=2TXk=j+1hi)]TJ /F8 7.97 Tf 6.58 0 Td[(1,k. (4)ForEquation 4 above,hi0,j0representsthetotalnumberofperformancepathsfromnode(i0,j0)wherei0representsthecurrentperformancelevelandj0representsthetimeperiod.Notei0andj0areusedasgenericindexing.Thehparametersaredeterminedbythefollowingrecursion:h1,j0=1forj0=2,...,T, (4)h2,j0=1+T)]TJ /F5 11.955 Tf 11.95 0 Td[(j0forj0=2,...,T, (4)hi0,j0=1+T)]TJ /F5 11.955 Tf 11.95 0 Td[(j0+i0Xq=1hq,j0+1fori0=3,...,L;j0=2,...,T. (4)Equation 4 representsthesingleperformancepathoncetheproducthasperformedatthelowestlevel.Forthesecond-to-lowestperformancelevel,Equation 4 representsthepathsofconstantperformancefortheremainderofthetimehorizonandprogressingtothelowestperformancelevelatanyfuturetimeperiod.InEquation 4 ,thersttermrepresentsconstantperformanceandmovingdirectlytothelowest 117

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performanceinafuturetimeperiod(similartoEquation 4 );thesummationtermrepresentsallotherpossiblepathsfromthegivennode.ApplyingthisrecursionandEquation 4 totheexampleinTable 4-1 withT=12andL=5,thetotalnumberofperformancepathsis1720. 4.3.3NumericalExampleWithTheorem4.1,itwasshownthataneutraloroptimisticconsumerwouldnotpurchaseawarrantywhentheydonotaccountfornon-increasingperformanceandcostestimatesareassumedtobethesameforboththemanufacturerandconsumer.Underthisassumption,itispossiblethoughthatnon-increasingperformance,ifnotaccountedforbytheconsumer,couldoffsetthemanufacturer'spricingadvantagethatcomesfromapessimisticconsumer.Inordertostudythecountereffectsofapessimisticconsumerthatdoesnotaccountfornon-increasingperformance,weassumetheconsumerandmanufacturerusethesamecommoditycostestimate.ConsiderthesameproblemparametersgiveninTable 4-1 butinsteadoftheconsumersharingthesamediscreteuniformdistributionasthemanufacturer,itismodiedtoapessimisticoutlook.Table 4-3 showsfourdifferentpessimisticconsumerbeliefs(inadditiontotheneutralconsumer)tobeanalyzed. Table4-3. PessimisticConsumerBeliefProles Consumer1Consumer2Consumer3Consumer4Consumer5 Probabilityk=1.2.3.33.51Probabilityk=2.2.25.33.50Probabilityk=3.2.2.3300Probabilityk=4.2.15000Probabilityk=5.2.1000 Furthermore,denethemanufacturer'sMTMas: 118

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PT=26666666666410000.1.9000.025.075.900.015.025.06.90.01.02.03.04.9377777777775Obviously,thechoiceofPTaffectstheresultsasmuchastheselectionofanyparameters,buttheparticularMTMchosenherehaspracticalcharacteristics.First,thediagonalelementsarelargerthan.5sothemostlikelytransitionistothesameperformancelevel.Next,theoffdiagonalelementsinagivenrowdecreaseinmagnitudeasthecolumnindexdecreases.Thisrepresentsthecharacteristicthatlargeperformancechangesarelesslikely.Usingthistransitionmatrix,Table 4-4 showstheaverageactualeffectandaveragemaximumpossibleeffect,inpercentages,onthemanufacturer'srevenueforthecustomerbeliefslistedinTable 4-3 forone-periodwarranties. Table4-4. Single-PeriodWarrantyEffectonRevenueforMarkovianModel W=1 Consumer1Consumer2Consumer3Consumer4Consumer5 LinearABC(-0.05,0.05)(-0.64,0.55)(-0.64,0.47)(-0.64,0.35)(-0.63,0.23)LinearABC-P(-0.07,0.07)(-0.65,0.56)(-0.62,0.44)(-0.65,0.37)(-0.64,0.24)LinearJBC(-0.07,0.07)(-2.42,2.41)(-2.42,2.40)(-2.40,2.38)(-2.42,2.39)LinearJBC-P(0.15,-0.15)(-2.19,2.18)(-2.19,2.18)(-2.18,2.15)(-2.18,2.14)LinearNBC(-0.03,0.03)(-0.60,0.50)(-0.59,0.41)(-0.59,0.31)(-0.60,0.19)LinearNBC-P(-0.05,0.05)(-0.60,0.51)(-0.60,0.42)(-0.60,0.31)(-0.60,0.19)QuadraticABC(-0.70,0.70)(-0.71,0.62)(-0.70,0.54)(-0.70,0.44)(-0.71,0.32)QuadraticABC-P(0.00,0.00)(-2.36,2.35)(-2.36,2.34)(-2.36,2.34)(-2.36,2.33)QuadraticJBC(0.20,-0.20)(-0.27,0.17)(-0.26,0.07)(-0.26,-0.03)(-0.27,-0.15)QuadraticJBC-P(0.00,0.00)(0.00,0.00)(0.00,0.00)(0.00,0.00)(0.00,0.00)QuadraticNBC(0.00,0.00)(0.00,0.00)(0.00,0.00)(0.00,0.00)(0.00,0.00)QuadraticNBC-P(0.00,0.00)(0.00,0.00)(0.00,0.00)(0.00,0.00)(0.00,0.00) Averages(-0.05,-0.05)(-0.87,-0.82)(-0.87,-0.77)(-0.87,-0.72)(-0.87,-0.66) IfweevaluatetheaveragesatthebottomofTable 4-4 ,weseethatastheconsumerbecomesmorepessimistic,themaximumpossiblerevenueincreaseswhichistobeexpectedsincethepessimisticconsumervaluesawarrantymorethananeutralor 119

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optimisticconsumerandiswillingtopaymore.Note,thischaracteristicmaynotalwaysbetrueeventhoughitisintuitiveandwhatweexpecttoobserve.Aconsumerwithahigherwillingness-to-paydoesnotnecessarilygeneratemorerevenuesincethecommoditypricescouldrisesufcientlysuchthatthereisanetnegativeeffectonrevenueoverthelifeofthewarranty.Asfortheactualchangesinrevenue,theystayconsistent.Thismaybeduetothelargevaluesonthediagonalofthetransitionmatrixorduetotheshortlengthofthewarranty.Therefore,weshouldcomparetheaveragesovervariouslengthwarrantiesandwithdifferenttransitionmatrices.Table 4-5 showstheaverageeffects(overallmanufacturerestimationtechniques)onactualrevenueandmaximumpossiblerevenueforwarrantylengthsuptosixperiodsandforthreedifferenttransitionmatrices.TherstMTM(labeled1inthetable)isthetransitionmatrixgivenin 4.3.3 ;thesecondMTM(labeled2inthetable)doublesalltheoff-diagonalelementsandreducesallbuttherstdiagonalelementto.8;andthethirdMTMtriplestheoff-diagonalelementsandreducesthediagonalelements(excepttherst)to.7.ReviewingtheentriesinTable 4-5 ,thereareseveraltrendsthatappear.FirstasthestructureoftheMTMchangessuchthatthelikelihoodofmovingtoalowerperformancelevelinthefutureincreases(i.e.,movingfromMTM1toMTM3),themanufactureexperiencesagreaterdecreaseinrevenue.Ineffect,thepriceofthewarrantyisincreasingrelativetotheconsumer'swillingness-to-pay(WTP)sotherearefewerchancesthattheconsumerwillpurchaseawarrantyandwhentheydo,itismoreexpensive.Notethatalltheaveragesarenegative(withtheexceptionofone0.00entry).Sinceweassumedtheconsumerusedthesamecostestimationtechniqueasthemanufacturer,weexpectthemtobenegativebecausethenaivetyoftheconsumernotaccountingfornon-increasingperformancegenerallykeepstheirWTPbelowthemanufacturer'swarrantyprice.Itispossible,though,thatfuturecommoditycostscouldmoveinsuchawaytoovercomethiseffect,butwedonotseethisresultonaverage. 120

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Table4-5. SensitivityAnalysisforMarkovianModel MTMWConsumer1Consumer2Consumer3Consumer4Consumer5 11(-0.05,-0.05)(-0.87,-0.82)(-0.87,-0.77)(-0.87,-0.72)(-0.87,-0.66)21(-0.05,-0.05)(-0.86,-0.82)(-0.86,-0.77)(-0.87,-0.72)(-0.86,-0.65)31(-0.05,-0.05)(-0.86,-0.81)(-0.86,-0.77)(-0.86,-0.71)(-0.86,-0.65) 12(-0.08,-0.08)(-0.08,-0.08)(-0.07,-0.07)(-0.85,-0.70)(-0.86,-0.64)22(-0.11,-0.11)(-0.08,-0.08)(-0.09,-0.09)(-0.88,-0.73)(-0.89,-0.68)32(-0.12,-0.12)(-0.12,-0.12)(-0.13,-0.13)(-0.92,-0.78)(-0.93,-0.71) 13(-0.04,-0.04)(-0.04,-0.04)(-0.04,-0.04)(-0.10,0.04)(-0.74,-0.52)23(-0.04,-0.04)(-0.04,-0.04)(-0.05,-0.05)(-0.14,0.00)(-0.77,-0.56)33(-0.05,-0.05)(-0.05,-0.05)(-0.05,-0.05)(-0.18,-0.04)(-0.83,-0.62) 14(-0.05,-0.05)(-0.05,-0.05)(-0.06,-0.06)(-0.09,-0.06)(-0.63,-0.42)24(-0.06,-0.06)(-0.05,-0.05)(-0.06,-0.06)(-0.07,-0.05)(-0.67,-0.46)34(-0.06,-0.06)(-0.06,-0.06)(-0.06,-0.06)(-0.11,-0.08)(-0.76,-0.55) 15(-0.02,-0.02)(-0.03,-0.03)(-0.03,-0.03)(-0.03,-0.03)(-0.61,-0.40)25(-0.02,-0.02)(-0.03,-0.03)(-0.03,-0.03)(-0.04,-0.04)(-0.70,-0.49)35(-0.04,-0.04)(-0.04,-0.04)(-0.05,-0.05)(-0.05,-0.05)(-0.82,-0.61) 16(-0.43,-0.42)(-0.44,-0.43)(-0.42,-0.42)(-0.45,-0.44)(-0.65,-0.43)26(-0.59,-0.58)(-0.56,-0.56)(-0.57,-0.56)(-0.56,-0.55)(-0.83,-0.61)36(-0.66,-0.66)(-0.70,-0.69)(-0.67,-0.66)(-0.67,-0.67)(-0.96,-0.74) Anothertrendweseeisthatthelargerdecreasesinrevenueoccurwithalongwarrantylengthoranextremelypessimisticconsumer.Theeffectofthelongerwarrantyperioddecreasingrevenuemorethanshorterlengthsmightbeexpectedsincethemanufacturercannotadjustforchangesincommoditycostsasoftenaswithshorterlengthwarranties.Asforthelargerdecreasesinrevenuefortheextremelypessimisticconsumer,thismaybeabyproductofahighWTPthatcausesthisconsumertopurchaseawarrantyintheinitialperiodwhenotherconsumerswouldnot.Dependingonthemovementoffuturecommodityprices,thisinitialpurchasemaynothaveapositiveneteffectonrevenue.Theseexplanationshighlightthedifcultyinmakinggeneralizationsforthesemodels.Resultsvarydependingontheactualrealizationsofcommoditypricesandwhenthenitehorizonstarts.Inourexample,thesixperiodwarrantydecreasesrevenuemorethanotherwarranties,butitiseasytoforeseeascenariowhenthismaynothold.Consideraconsumerwhopurchasesawarrantywhencommoditycostsareelevatedandthentheysubsequentlydropandremainrelativelyloweroverthelifeofthewarranty.Inthiscase,thelongerwarrantywouldbe 121

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advantageoustothemanufacturerbecausetheconsumerwaslockedinwhenpriceswerehigh. 4.4CombinedModelWenowconsideramodelthatcombinesthecharacteristicsofdistinctcommoditycostestimatesandnon-increasingperformance.Notetheconsumer'schoiceofcommoditycostestimatecouldcountertheirlackofacknowledgementofnon-increasingperformance.Inageneralsense,whenaconsumer'scommoditycostestimateishigh,theyessentiallyaretakingapessimisticoutlookoncostsbecausetheyprojecthighercostsrelativetoobservedcosts.Conversely,whenaconsumerdoesnotaccountfornon-increasingperformance,theyareineffecttakinganoptimisticperspectivebecausetheywillhave,allotherfactorsbeingequal,lowerexpectedoperatingcoststhanunderthetrueperformanceprocessdescribedbyPT.UsingtheparametersinTable 4-1 wheretheconsumerandmanufacturershareauniformdiscretedistributionandtheMTMfrom 4.3.3 ,wecandeterminetheeffectonthemanufacturer'srevenueunderdifferentcombinationsofthewarrantylengthandtheconsumer'scostestimate.Table 4-6 showtheresultsforwarrantylengthsuptosixperiodsforeachofthethreeconsumercostestimates.Similartotheprevioustables,thevaluesrepresenttheaverage(overallmanufacturerestimates)percentchangeinactualrevenueandmaximumpossiblerevenue.Table 4-6 showstwosignicantresultsfromourmodel.Firsttheaverageactualeffectsonrevenueforallthecombinationsofwarrantylength,consumerinitialbelief,andconsumercommodityestimateareintherangeof[)]TJ /F7 11.955 Tf 9.3 0 Td[(2.43,)]TJ /F7 11.955 Tf 9.3 0 Td[(1.98].Thisisafairlysmallrangegivenallthevariablecomponentsinthemodel.Toexplainthissurprisingresultwouldrequireamuchmorein-depthanalysisofthedatasoweofferonepossibleexplanation.NotethatinTable 4-5 whentheconsumerusedthesameestimatesasthemanufacturer,themagnitudeofthedecreasesweresmaller.Therefore,theconsumer'suseofthemostcurrentcommoditypriceasabasisofestimatesmaybeoutperforming 122

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themanufacturer'sestimates.Also,sincebothestimatesmaybetrackingtheactualpricesreasonablywell,thereisatightrangefortheeffectonactualrevenue. Table4-6. WarrantyEffectonManufacturerRevenueforCombinedModel WConsumerEstimateConsumer1Consumer2Consumer3Consumer4Consumer5 1Constant(-2.16,0.35)(-2.34,0.24)(-2.37,0.26)(-2.39,0.38)(-2.35,0.47)1Linear(-2.36,0.45)(-2.43,0.61)(-2.42,0.70)(-2.39,0.79)(-2.34,0.87)1Geometric(-2.37,1.06)(-2.36,1.15)(-2.44,1.34)(-2.38,1.39)(-2.43,1.60)2Constant(-2.08,0.25)(-2.10,0.30)(-2.13,0.32)(-2.27,0.34)(-2.25,0.46)2Linear(-2.13,0.68)(-2.17,0.64)(-2.25,0.75)(-2.26,0.88)(-2.27,1.03)2Geometric(-2.27,1.43)(-2.28,1.52)(-2.28,1.61)(-2.27,1.75)(-2.28,1.89)3Constant(-2.16,0.30)(-2.11,0.34)(-2.13,0.37)(-2.24,0.49)(-2.22,0.63)3Linear(-2.18,0.99)(-2.12,0.99)(-2.15,1.06)(-2.21,1.28)(-2.25,1.41)3Geometric(-2.15,1.80)(-2.16,2.18)(-2.18,2.35)(-2.19,2.45)(-2.15,2.58)4Constant(-2.26,0.27)(-2.26,0.30)(-2.25,0.34)(-2.27,0.42)(-2.26,0.62)4Linear(-2.21,1.17)(-2.16,1.19)(-2.21,1.26)(-2.20,1.48)(-2.23,1.59)4Geometric(-2.22,2.15)(-2.22,2.32)(-2.24,2.76)(-2.17,2.92)(-2.23,3.10)5Constant(-2.10,0.19)(-2.12,0.26)(-2.13,0.28)(-2.11,0.37)(-2.06,0.64)5Linear(-2.14,1.08)(-2.13,1.14)(-2.15,1.22)(-2.11,1.43)(-2.09,1.56)5Geometric(-2.12,2.24)(-2.13,2.28)(-2.13,2.44)(-2.10,3.09)(-2.05,3.23)6Constant(-2.15,0.17)(-2.17,0.20)(-2.20,0.23)(-2.21,0.33)(-2.13,0.61)6Linear(-2.18,1.30)(-2.18,1.39)(-2.17,1.44)(-2.05,2.13)(-1.98,2.24)6Geometric(-2.16,2.67)(-2.09,2.61)(-2.19,2.96)(-2.06,4.09)(-1.98,4.19) Thesecondresultthatdeservesattentionisthepositivevaluesforthemaximumpossiblerevenue.Therange([.17,4.19])iswiderandthemagnitudeofmaximumchangeislargerthantheactualeffectonrevenue.Therefore,ifthemanufacturercouldtakebetteradvantageoftheconsumer'swillingness-to-pay,thewarrantiescouldhaveapositiveeffectonrevenue.Again,themanufacturerestimatespresented,whilemoresophisticated,arerigidintheirapplicationofaformulaforidentifyingthebusinessandtrendeffects.Inreality,amanufacturercoulduseamuchmorerobustprocesstoanalyzethedataandchangethemodelasneeded.Whiletheconsumer'sestimatesarealsorigid,theliteraturesuggeststhatthisisindeedhowconsumersbehave,andthuswewouldnotexpectthemtoapplythesameexibilityinadjustingtheirestimates.Additionally,ifthemanufacturerwasnotlimitedtoalwaysofferingwarrantiesofthesamelength,theymaybeabletoincreaserevenueevenabovethelevelsindicatedbythemaximumpossiblerevenueaverages. 123

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4.5SummaryThevalueofaperformancebasedwarrantytoaconsumerwillvarybasedontheunderlyingcostofthecommoditythatisconsumedbyoperatingtheproduct.Sincethecostofthecommodityisnotknownapriori,thevalueofthePBWwillbedependentontheconsumer'sestimateoffuturecommoditycosts.Undertheassumptionthattheconsumerupdatesfuturecostestimateseachperiod,thewarrantyvalueisnon-stationary.Additionally,whenamanufacturerreimbursestheconsumerbasedontheactualcommoditycosts,thewarrantypricewillbedynamicallypricedaccordingtothemanufacturer'scurrentprojectionoffuturecommoditycostsaswellasexpectedproductperformance.Sinceitisreasonabletoassumethereisnosystematicrelationshipbetweenthemanufacturer'sandconsumer'sestimates,itisdifculttomakegeneralizationsabouttheeffectonrevenuewhenamanufactureroffersPBWs.Theresultsareultimatelydependentonthetruecommoditycostswhichareunknowntoeachagent.Inthischapter,weusedasemi-empiricalmodelwithanactualtimeseriestostudytheeffectsonmanufacturerrevenueofofferingextendedperformancebasedwarranties(EPBWs).Assumingthatconsumersuseunsophisticatedcommoditycostestimates,wehypothesizedthatthemanufacturercouldusemoresophisticatedestimatetechniquestobetterestimatefuturecommoditycostsandthusincreaserevenuebyofferingEPBWs.Usingvariousmanufacturercostestimates,wefoundthatamanufactureronaveragewoulddecreaserevenuebyofferingEPBWsbutthatthemagnitudeofthedecreasewaslessthanonepercent.ThisresultsuggeststhattherisktoamanufacturerofofferingEPBWswheretheliabilityisdependentonanexogenousfactor,likegasolineprices,maybeacceptableinapracticalexample.Withthemodestdecreaseinrevenue,theEPBWgivesthemanufactureratooltoensureareturnvisitfromtheconsumerperiodicallytoreconcilethewarranty,andthusitcouldbeastrongmarketingtooltosellperiodicmaintenanceservicesfortheproduct. 124

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Wealsodeterminedthemaximumpossiblerevenuethemanufacturercouldachieveifthewarrantieswerepricedatthewillingness-to-payleveloftheconsumerandfoundthatofferingEPBWsatthislevelincreasedthemanufacturer'srevenue.ThisshowsthepotentialforanEPBWtobeaadditionalrevenuesourceinadditiontobeingamarketingtooltoenticereturnvisits.Finally,weincorporatedthecharacteristicofnon-increasingperformanceintothemodel.Oftenmodelsforreliabilityusestochasticprocessesthatpossessindependentincrementssothecurrentlevelsoffailuresdonotinuencefuturefailures.Ontheotherhandwewouldexpectaproduct'sperformance,suchasenergyconsumption,toalwaysdecline(orremainconstant)withage.Westudiedthenon-increasingperformancemodelundertheassumptionthattheconsumerdoesnotaccountforitandusesthenaiveperspectivethatcurrentperformancewillcontinueindenitely.Theeffectofmodelingperformanceinthiswaygeneratesalowerwillingness-to-payfromtheconsumerwhichmaynotalwaysequatetoadecreaseinrevenue.Ingeneral,theresultsaredifculttogeneralizebecausetheyaredependentnotonlyondifferentestimationtechniquesbutmostimportantlyonanunknownandhardtopredicttimeseries.Themostmeaningfulresultcomesfromobservingthemagnitudeoftheeffectsonrevenue.Thefactthattherevenuedecreasesweresmallandthatthereisuntappedwillingness-topayprovidesmotivationforfurtherresearch. 125

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CHAPTER5CONCLUSIONANDFUTURERESEARCHTraditionalwarrantieshavebeenstudiedextensively,butthereisrelativelylittleresearchregardingperformancebasedwarrantiesandguarantees.Inthisdissertation,weprovideaframeworkforstudyingperformanceguaranteecontracts.Weproposedifferentmodelsforvariousequipmentreplacementscenariosandshowthepotentialforamanufacturertoaffectconsumerbehaviorinawaythatbenetsthemanufacturer.InChapter 2 ,weintroducetheconceptofaperformancebasedwarranty(PBW)thatcompensatesaconsumerintheeventaproductperformsbelowthelevelstatedinthewarranty.Withinthisframework,wealsoproposeamethodofconsumerdifferentiation(whichisusedthroughoutthedissertation)thatdescribesaconsumer'sexpectationofproductperformance.WhileourmethodofmodelingPBWsisgeneral,wespecicallyevaluatetheuseoftwotypesofPBWs:constantperformance(CP)andconstantcost(CC)warranties.WeconsidertheuseofCPandCCPBWsinthecontextofanitehorizonequipmentreplacementproblemwithasinglechallenger.Weshowthatwhenthesewarrantiesareproperlydesigned,theycanenticeaconsumertoupgradetoamoretechnologicallyadvancedproductearlierandthusincreasethemanufacturer'srevenue.Additionally,forconsumersthatbelieveaproductwillperformatalevelbelowthemanufacturer'sadvertisedperformance,themanufacturercanuseaPBWtogenerateanadditionalrevenuesource.Next,westudytheuseofaperformanceguaranteeinacompetitiveenvironmentinChapter 3 .Weextendtheconceptofaconsumer'spre-existingbeliefaboutaproduct'sperformancetoincludeperiodicupdatesofthisbeliefbasedontheproduct'spreviousperformance.WemodelthisextensionusingaBayesianupdatingmethodology.Assumingacompetitiveenvironmentwithtwochallengers,weshowthataperformanceguaranteecanbeusedtoincreasetheprobabilitythataconsumerreturnstotheincumbentmanufacturerforafollow-onpurchaseandthusincreasethemanufacturer's 126

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revenue.Whenthepurchasepriceislargecomparedtoperiodicoperatingcosts,amanufacturerhasastrongmotivationtosatisfyconsumerexpectationsaboutproductperformanceinordertoretaintheconsumer'sbusiness.InChapter 4 ,weintroducetheconceptofaspecictypeofPBWcalledanextendedperformancebasedwarranty(EPBW)inwhichthewarrantyisofferedatanytimetheproductisowned,notjustwiththepurchaseoftheproduct.Weuseanexampletodeterminetheactualrisktorevenuethatamanufacturermayexperiencewhenwarrantiesareofferedandproductperformanceislinkedtotheuseofanunderlyingcommodity.Althoughourresultscannotbegeneralizedduetotherelianceontheobservedcommodityprices,thismodelhighlightstheuniquenatureofperformancebasedwarrantiesincomparisontotraditionalwarranties,i.e.thehighdependencyonthecostoftheunderlyingcommodity.Thisdependencyinuencestheconsumer'swillingness-to-pay(WTP)andmakestheWTPnon-stationarywithrespecttotimeduetoarelianceontheconsumer'scommoditycostestimate.Whenthemanufacturertiesreimbursementtotheobservedcommodityprices,theyabsorbtheriskinvolvedwiththevolatilityofthecommodity.Weshowthatthisriskisreasonableandthatwithaneffectivecommodityestimateandwarrantystrategy,revenuecanbeincreasedbyofferingEPBWs.Thisresearchcanbeextendedinthefutureinseveralimportantways.First,therigidnatureofthemanufacturer'sestimationtechniquesinChapter 4 donotfullycapturetheabilityofamanufacturertoestimatefuturecommoditycosts.Whiletheestimationtechniqueswererelativelymoresophisticatedthanthoseoftheconsumer,theyneverthelesswerenotoftheextentandlevelthatwouldbeavailablefromaforecastingexpert.Additionally,weassumedthewarrantieswereofxedlengthandthattheywouldbeofferedeveryperiod.Inreality,thewarrantylength,andeventheperiodsinwhichtheyareoffered,couldvary.Oursimplisticapproachdidnotaccountforamanufacturer'sabilitytoevaluatecurrentmarketconditionstodecideifofferinga 127

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warrantyatagiventimeisintheirbestinterest.Also,wexedthewarrantylevelwhichcouldalsobestudiedtodeterminewhichminimumperformanceprovidesthegreatestincreaseinrevenue.Asthewarrantylengthcanbetreatedasvariable,thewarrantylevelcouldalsobevariable,especiallyastheproductrevealsitstrueperformance.ThemodelspresentedinChapters 2 and 3 usedthesimpliedmethodofmodelingperformancewithasinglerandomvariablerepresentingoperatingcosts.Whilethismadetheanalysismoretractable,iteliminatedacriticalaspectofperformancebasedwarrantiesandguarantees.Decomposingoperatingcostsintoitscomponentsforthesemodels,asinChapter 4 ,wouldallowforamorepracticalanalysis.Also,sincethetruebenetofaPBWcanonlybeseenwhenaccountingforcommoditycosts,extendingtheanalysisofthosemodelstoincludeanempiricalstudywouldbebenecial.Additionally,thecontractsstudiedinChapters 2 and 3 wereakintobasicwarrantieswarrantieswherethecoveragecannotbeextendedbeyondtheinitialterms.ItwouldbeinterestingtoseetheeffectonrevenueofofferingthesewarrantiesundertheoperatingcoststructureofChapter 4 becausethereisonlyonepricethatmustbedetermined.Thismayallowthemanufacturertobettercapitalizeontheconsumer'sunsophisticatedestimationtechniques.InChapter 2 ,thewarrantydesignsfocusedonthestructurallysimple,single-termconstantperformance(CP)andconstantcost(CC)PBWs.FutureresearchcouldinvestigatemorecomplexdesignssuchasallowingrenewalsifaninitialPBWispurchasedoravariablewarrantythatchangesastheobsoleteproductapproachesitseconomiclife.TheliteratureontraditionalwarrantiesoffermanywarrantydesignsthatcouldbeappliedtoPBWs.Additionally,thedecisiontoofferaCPorCCwarrantyisonlyconsideredinthecontextofaconsumerthatseekstominimizecosts.Thereareconsumerimplicationsfromthisdecisionthatwerenotmodeled,suchastheconsumer'spreferenceforthesimplicityofaCCwarrantyorthewillingnessoftheconsumertospendextramoneyonawarrantyonthesamedayasmakingsuchalargepurchase. 128

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Accountingforconsumerpsychologyissueslikethese(outsideofminimizingtotalcost)wouldbeaninterestingadditiontothemodel.WealsodiscussedtheimplementationofPBWsinChapter 2 .Furtherstudyofwaystoeffectivelyimplementthesenewtypesofcontractswouldincreasethelikelihoodofthembeingappliedinpractice.Therearesimplicitiestothesetypesofwarranties,suchasalackofrepairinfrastructure,buttherearealsocomplexities,suchasconsumerbehaviorthataffectsproductperformance,thatmakeimplementingPBWsachallenge.Exploringwaystoefcientlyaddressthesecomplexitieswouldbeanimportantextensionofthisresearch.Finally,theinclusionofbudgetconstraintswithinanyofourmodelswouldbeinsightfultoseehowtheseeffectconsumerbehaviorandmanufacturerrevenue.Businessesandgovernmentagenciesoftenhavebudgetandplanningprocessesthatputapremiumonlowcostvariability.Forcustomersthatseeksomemethodofsettingamaximumperiodiccostthreshold,aPBWcouldbeadesirablecontracttype.Thiswouldprovideahigherlevelofpossiblerevenueforthemanufacturerinexchangefortakingonmorerisk.Themanufacturercouldthenattempttomitigatethisriskwithhedgesonthecommoditycosts. 129

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BIOGRAPHICALSKETCH ClayKoschnickcompletedhisBachelorofScienceinOperationsResearchin1998andaMasterofScienceinOperationsResearchin2007.In2009,heenteredtheDoctorofPhilosophyprogramintheIndustrialSystemsandEngineeringDepartmentattheUniversityofFlorida.Inthesummerof2012,hegraduatedwithhisPhDinindustrialandsystemengineering.Hisresearchinterestsincludesolvingpracticalbusinessproblemsusingdynamicprogrammingandintegratingeconometricestimatesandforecastsintodynamicoptimizationmodels. 134