基于需求扩散和随机技术进步的连续产品引入过程课件

TimingSuccessiveProductIntroductionswithDemandDiffusionandStochastic

TechnologyImprovement

基于需求扩散和随机技术进步的连续产品引入过程

R.MarkKrankelDepartmentofIndustrialandOperationsEngineering,UniversityofMichigan,IzakDuenyas,RomanKapuscinskiRossSchoolofBusiness,UniversityofMichigan,AnnArbor,MichiganPresentbyLiWeiTimingSuccessiveProductIntr1CONTENTSIntroductionLiteratureModelOptimalPolicyComputationalStudyandInsightsExtensionsCONTENTSIntroduction2IntroductionConsideraninnovativefirmthatmanagesthedevelopmentandproductionofasingle,durableproduct.Overtime,thefirm’sresearchanddevelopment(R&D)departmentgeneratesastochasticstreamofnewproducttechnology,features,andenhancementsfordesignintosuccessiveproductgenerations.IntroductionConsideraninnova3IntroductionThefirmcapturesthebenefitsofsuchadvancesbyintroducinganewproductgeneration.Duetofixedproduct-introductioncosts,itmaybeunreasonabletoimmediatelyreleaseanewproductgenerationaftereachtechnologydiscovery.Rather,thefirmmayprefertodelayanintroductionuntilsufficientincrementalnewproducttechnologyhasaccumulatedinR&D.Theobjectiveofthispaperistocharacterizethefirm’soptimalproduct-introductionpolicyIntroductionThefirmcaptures4IntroductionThetotalnumberofproductgenerationsisnotpre-specified;rather,itisdeterminedbythepaceoftechnologyimprovementalongwiththefirm’sdynamicdecisionsonwhentointroduce.Analysisiscenteredupontwokeyinfluencesaffectingtheintroductiontimingdecisions:(1)demanddiffusiondynamics,wherefutureproductdemandisafunctionofpastsales(2)technologyimprovementprocess,specificallytheconceptthatdelayingintroductiontoalaterdatemayleadtothecaptureoffurtherimprovementsinproducttechnology.IntroductionThetotalnumbero5IntroductionPreviousliteratureexaminingincrementaltechnologyintroductionhasfocusedoneither(1)or(2),butnonehaveconsideredbothfactorssimultaneously.Asaresult,thepresentanalysisprovidesnewinsightintothestructureoftheoptimalintroductiontimingpolicyforaninnovativefirm.Usingaproposeddecisionmodelthatincorporatesbothkeyinfluences,weprovetheoptimalityofathresholdpolicy:itisoptimalforthefirmtointroducethenextproductgenerationwhenthetechnologyofthecurrentgenerationisbelowastate-dependentthreshold,inwhichthestateisdefinedbythefirm’scumulativesalesandthetechnologylevelinR&D.IntroductionPreviousliteratur6IntroductionRelativepapersWilsonandNorton(1989)&MahajanandMuller(1996)Thesetwopapersproceedunderademanddiffusionframework,butdonotmodeltheprogressionofproducttechnology.Rather,theyassumethatthenextgenerationproducttobeintroducedisavailableatalltimesstartingfromTime0.Asaresult,theyrespectivelyconcludetheoptimalityof“nowornever”(thenewgenerationproductisintroducedimmediatelyornever)and“noworatmaturity”(thenewgenerationproductisintroducedimmediatelyorwhenthepresentgenerationproducthasreachedsufficientsales)rulesgoverningproductintroductions.IntroductionRelativepapers7LiteratureTwomainresearchareasaredirectlyrelevanttothecurrentwork.Thefirstcentersonmodelsofdemand.Papersinthisareadescribethepatternsofdemandexhibitedbysingleormultipleproductgenerations,specificallyinrelationtonewinnovations.Thesepapersconcentrateonsystemdynamicsand/ormodelfitwithempiricaldata.Thesecondresearchareaexaminesdecisionmodelsfortechnologyadoptiontiming.Asubsetofthisgroupincludespapersthatmodeltheintroductionofnewproductssubjecttodemanddiffusion.LiteratureTwomainresearchar8Literature—modelsofdemandBass(1969)initiatesthestreamthatexaminesdemanddiffusionmodelsbyformulatingamodelforasingle(innovative)product.TheBassmodelspecifiesapotentialadopterpopulationoffixedsizeandidentifiestwotypesofconsumerswithinthatpopulation:innovatorsandimitators.Innovatorsactindependently,whereastherateofadoptionduetoimitatorsdependsonthenumberofthosewhohavealreadyadopted.Theresultingdifferentialequationforsalesrateasafunctionoftimedescribestheempiricallyobserveds-shapedpatternofcumulativesales:exponentialgrowthtoapeakfollowedbyexponentialdecay.Literature—modelsofdemandBa9Literature—modelsofdemandBass,F.M.1969.Anewproductgrowthmodelforconsumerdurables.ManagementSci.15215–227.ProfDr.FrankM.Bass(1926-2006)wasaleadingacademicinthefieldofmarketingresearch,andisconsideredtobeamongthefoundersofMarketingscience.HebecamefamousasthecreatoroftheBassdiffusionmodelthatdescribestheadoptionofnewproductsandtechnologiesbyfirst-timebuyers.HediedonDecember1,2006.Literature—modelsofdemandBa10Literature—modelsofdemandNortonandBass(1987)extendtheoriginalBassmodelbyincorporatingsubstitutioneffectstodescribethegrowthanddeclineofsalesforsuccessivegenerationsofafrequentlypurchasedproduct.JunandPark(1999)examinemultiple-generationdemanddiffusioncharacteristicsbycombiningdiffusiontheorywithelementsofchoicetheory.WilsonandNorton(1989)proposeamultiple-generationdemanddiffusionmodelbasedoninformationflow.KumarandSwaminathan(2003)modifytheBassmodelforthecaseinwhichafirm’scapacityconstraintsmaylimitthefirm’sabilitytomeetalldemand.Usingtheirreviseddemanddiffusionmodel,theydetermineconditionsunderwhichacapacitatedfirm’soptimalproduction/salesplanisa“build-uppolicy,”inwhichthefirmbuildsupaninitialinventorylevelbeforethestartofproductsalesandalldemandismetthereafter.Literature—modelsofdemandNo11Literature—technologyadoptiontimingGjerdeetal.(2002)modelafirm’sdecisionsonthelevelofinnovationtoincorporateintosuccessiveproductgenerations.Thepaperdoesnotconsiderthediffusiondynamicsoftheexistingproductsinthemarket(productsalesratesdonotdependoncumulativesales).Cohenetal.(1996)assumethatproductcanonlybesoldduringafixedwindowoftime.Therefore,delayingtheproductintroductionforfurtherdevelopmentwillleadtoabetterproductandhigherrevenuesbutoverashortertime.Cohenetal.furtherassumethattheproductcurrentlyinthemarketorthenewlyintroducedproductbothhavesalesataconstantrate.Thus,theydonotconsiderthediffusiondynamics.TheyalsodonotconsiderthestochasticnatureoftheR&DProcess.Literature—technologyadoption12Literature—technologyadoptiontimingBalcerandLippman(1984)concludethatafirmwilladoptthecurrentbesttechnologyifitslaginprocesstechnologyexceedsacertainthreshold.Thethresholdiseithernonincreasingornondecreasingintime,dependentonexpectationswithrespecttopotentialfortechnologydiscovery.Farzinetal.(1998)considersasimilarproblemunderadynamicprogrammingframework.Thepaperexplicitlyaddressestheoptionvalueofdelayingadoptionandcomparesresultstothoseusingtraditionalnetpresentvaluemethods,inwhichtechnologyadoptiontakesplaceiftheresultingdiscountednetcashflowsarepositive.Ineachoftheseworks,thetechnologyadoptiondecisiondoesnotexplicitlyconsidertheeffectsofadoptiontimingonproduct-demanddynamics.Literature—technologyadoption13Literature—technologyadoptiontimingWilsonandNorton(1989)considertheone-timeintroductiondecisionforanewproductgeneration.Intheirmodel,productintroductionhasfixedpositiveeffectsonmarketpotentialalongwithnegativeeffectsduetocannibalization.Theyconcludethattheoptimalpolicyforthefirmisgivenbya“nowornever”rule.Thatis,itwilleitherbeoptimaltointroducetheimprovedproductassoonasitisavailableorneveratall.MahajanandMuller(1996)concludethatitwillbeoptimaltoeitherintroducetheimprovedproductassoonasitisavailableorwhenenoughsaleshavebeenaccumulatedforthepreviousproductgeneration.(“noworatmaturity”rule)BothWilsonandNorton(1989)andMahajanandMuller(1996)implicitlyassumethatthenextproductgenerationisavailableandremainsunchangedregardlessofwhenitisintroduced.Incontrast,weassumethatafirmthatdelaysintroductionofthenextproductgenerationexpectstocapturegreatertechnologicaladvancesatalaterdate.Literature—technologyadoption14ModelUnderadiscrete-time,infinite-horizonscenario,considerasinglebaseproductthatprogressesthroughaseriesofproductgenerationsovertime.ThebenefitsofimprovedtechnologyarerealizedonlythroughintroductionofanewproductgenerationthatincorporatesthelatesttechnologyavailableinR&D.Animprovementintheincumbentproducttechnologyleadstoahighersalespotentialforthenewproductgeneration.However,eachnewgenerationrequiresafixedintroductioncost.Thefirmseeksanintroductionpolicythatmaximizesnetprofits.ModelUnderadiscrete-time,in15ModelIneachperiod,thefirmhastheoptiontoeitherintroducethelatesttechnologyorcontinuesellingatthecurrentincumbenttechnologylevel(wait).WemodeltheleveloftechnologyinR&Dusingasingleindex,andassumethatthislevelimprovesstochasticallyduringeachperiod.Ourobjectiveistocharacterizethefirm’soptimalintroductionpolicygiventhisstochasticR&Dprocess.ModelIneachperiod,thefirm16Model—NotationandAssumptionsWebeginwiththefollowingdefinitionsunderadynamic-programmingframework:Model—NotationandAssumptions17Model—NotationandAssumptionsModel—NotationandAssumptions18Model—NotationandAssumptionsWeconsideradurablebaseproductforwhichproducttechnologyisadditiveandintroductionofanewproductgenerationresultsincompleteobsolescenceofthepreviousgeneration;i.e.,onceanewgenerationisintroduced,salesofthepreviousgenerationimmediatelydroptoandremainatzero.Thispropertyisreferredtolaterasthe“completereplacement”condition.Itisassumedthat(1)availableproducttechnologyimprovesineachperiodaccordingtoastochasticprocess,and(2)salesforanygivengenerationfollowademanddiffusionprocess.Model—NotationandAssumptions19Model—NotationandAssumptionsBoththetechnologylevelandthepriceofanewproductareexpectedtoinfluencetheproduct’smarketpotentialandassociateddemanddiffusiondynamics.Tounderstandtheeffectsofprogressingtechnologyindependentofothercompoundingfactors,weassumeaveryspecificbutrealisticpricingstrategythatmaintainsconstantunitprofitmargins.Model—NotationandAssumptions20Model—NotationandAssumptionsAsmentionedabove,salespotentialisassumedtobeanincreasingfunctionofproducttechnologylevel.Moreover,wedonotmodelcapacityconstraintsandassumethatalldemandcanbemetsothatsalesequalsdemand.Model—NotationandAssumptions21ModelFormulationthefollowingassumptionismadeonthesalesratecurves:ModelFormulationthefollowing22ModelFormulation(i)ensuresthat,allelseequal,productsalesrateisnondecreasinginproducttechnology.Part(ii)accommodatesrealisticdurable-goodmarketscenariosinwhichthepotentialmarketsizeisfiniteandcurrentperiodsalesdonotexceedtotalremainingmarketpotential.Condition(iii)limitstherateatwhichsalesdecreaseandinadiscrete-timeframeworkguaranteesthatthesalesratefromoneperiodtothenextdoesnotdecreaseatafasterpacethansalesaccumulatedwithintheperiod.ModelFormulation(i)ensurest23ModelFormulationModelFormulation24ModelFormulationTheoptimumintroductionpolicyiscomputedfromtheoptimalityequation:ModelFormulationTheoptimumi25Model—RelationshiptoDemandDiffusionForthescenarioconsideredinthispaper,thereisanaturallinkbetweenthissalesmodelandthatofatypical(continuous-time)diffusionmodel.ConsidertheBassdiffusionmodelforasingleinnovativeproduct:Model—RelationshiptoDemandD26Model—RelationshiptoDemandDiffusionMahajanandMuller(1996)presentanextensionoftheBassmodelforthecaseofmultipleproductgenerations.Model—RelationshiptoDemandD27Model—RelationshiptoDemandDiffusionModel—RelationshiptoDemandD28Model—RelationshiptoDemandDiffusionwhereaandbarecoefficientsofinnovationandimitation,respectively.Becausecumulativesalesistrackedasastatevariable,thedecisionmodel(1)–(3)clearlycapturestheinteractionbetweenproductgenerationswhensalescurvesareofthedemanddiffusionform(6).Moreover,anexaminationof(6)showsthatthedemanddiffusionformsatisfiesAssumption1subjecttoamildrestrictiononproblemparameters.Model—RelationshiptoDemandD29OptimalPolicyOptimalPolicy30OptimalPolicyOptimalPolicy31OptimalPolicyThefirstresultstatesthatasthetwosystemsprogressovertime,thecumulativesaleslevelofthefirmwithlowerinitialcumulativesaleswillneversurpassthefirmwithhigherinitialcumulativesales.OptimalPolicyThefirstresult32OptimalPolicyOptimalPolicy33OptimalPolicyTheresultstatesthatallelseequal,thediscountedoptimalprofit-to-goforafirmwithlowercumulativesaleswillnotexceedthatofafirmwithhighercumulativesalesbymorethanthenetvalueoftheircumulativesalesdifference.Thatis,futurebenefitscannotmakeupforthecurrentsalesdeficit.OptimalPolicyTheresultstate34OptimalPolicyOptimalPolicy35OptimalPolicyOptimalPolicy36OptimalPolicyOptimalPolicy37OptimalPolicyOptimalPolicy38OptimalPolicyOptimalPolicy39OptimalPolicyOptimalPolicy40OptimalPolicyOptimalPolicy41OptimalPolicyOptimalPolicy42OptimalPolicyOptimalPolicy43ComputationalStudyandInsightsThenumericalstudyfocusesontheinfluencesofasimpletechnologydiscoveryrate,fixedproduct-introductioncosts,andmarketparametersincludingthediffusioncoefficientsandaparameterdescribingthesensitivityofproductmarketpotentialtochangesinproducttechnology.ComputationalStudyandInsigh44ComputationalStudyandInsightsforpurposesofnumericalinvestigationwebeginwithasimplifiedbaselinescenario.Salesratecurvesforthebaselinescenarioaregeneratedwithinadiscrete-timeframeworktoapproximateademanddiffusionprocessaccordingtotheformgivenin(6).

TechnologyimprovementisassumedtofollowasimplifiedstochasticprocessinwhichavailabletechnologyinR&Dincreasesbyoneineachperiodwithprobabilityp.ComputationalStudyandInsigh45ComputationalStudyandInsightsComputationalStudyandInsigh46ComputationalStudyandInsightsComputationalStudyandInsigh47ComputationalStudyandInsightsThebaselineoptimalpolicyiscomputedbysolvingthedynamicprogram(3).Insolving(3)numerically,weuselinearinterpolationtohandlecasesinwhichthecurrentperiodsalesgszisanonintegermultipleoftheindexingunitusedforcumulativesales.ComputationalStudyandInsigh48ComputationalStudyandInsightsComputationalStudyandInsigh49ComputationalStudyandInsightsNumericalapproximationgeneratesthebaselinesetoftechnologyswitchingcurvesillustratedinFigure6.ComputationalStudyandInsigh50ComputationalStudyandInsightsTheswitchingcurvesinFigure6suggestthatoptimalintroductionofthenextproductgenerationmaybetriggeredinoneoftwoways:(1)throughsufficientproductsalesatthecurrenttechnologylevels,(2)throughsignificantadvancesinavailableproducttechnology.(1)impliesthat,evenwithoutfurthergainsinR&D,afirmthatcontinuestosellthecurrentgenerationlongenoughmayeventuallyfinditoptimaltointroducethetechnologyonhandeventhoughintroducingatthesametechnologylevelwasnotprofitableinthepast.(2)impliesthat,regardlessofthecurrentgeneration’spositionalongitssalescurve,itmaybeoptimaltointroduceanewgenerationwithlargeenoughgainsinR&Dtechnology.ComputationalStudyandInsigh51ComputationalStudyandInsightsConsidertheexpectedrateoftechnologydiscoveryasmeasured(forthebaselinescenario)bythetechnologydiscoveryprobabilityp.ComputationalStudyandInsigh52ComputationalStudyandInsightsItisnaturalthatunderfixedintroductioncosts,afirmwithahighertechnologydiscoveryratewillintroducenewproductgenerationswhenthegainintechnologyoverthepreviousgenerationislarger.Hence,foragiventechnologylagbetweentheproductgenerationinthemarketandthatavailableinR&D,anincreaseinthetechnologydiscoveryprobabilityshouldincreasetheattractivenessofthedecisiontowaitversusintroduce.ComputationalStudyandInsigh53ComputationalStudyandInsightsComputationalStudyandInsigh54ComputationalStudyandInsightsItisevidentthatalthoughthetechnologyintroductionthresholdsareincreasinginthediscoveryprobability(asshowninFigure7),theexpectedrate,bothintermsoftimeandsales,atwhichthefirmwillintroducenewgenerationsisalsoincreasing;i.e.,firmswithahigherexpectedtechnologydiscoveryrateareexpectedtointroducenewproductgenerationsmorefrequentlyandwithlargertechnologygainsbetweengenerations.ComputationalStudyandInsigh55ComputationalStudyandInsightsNext,weexaminetheinfluenceofthefirm’scoststructureontheoptimalpolicy.AsillustratedinFigure8,adecreaseinthefixedintroductioncostKdecreasestheoptimalintroductionthresholdatanygivencumulativesaleslevel.ComputationalStudyandInsigh56ComputationalStudyandInsightsLetusturntotheparametersthatdescribetheproductmarket.Themodeledproductsalesdynamicswillbeaffectedbythediffusioncoefficientsaandbin(6)aswellasthemarketpotentialparameterm,thatdeterminestheproductmarketpotentialforaspecifictechnologylevel.ComputationalStudyandInsigh57ComputationalStudyandInsightsMarketPotentialParameterAnincreaseinmtranslatestolargergainsinmarketpotentialperunitgainintechnology.Inturn,thesalesrateatagivenlevelofcumulativesalesismoresensitivetoincreasesinproducttechnologywhenmishigher.ComputationalStudyandInsigh58ComputationalStudyandInsightsDemandDiffusionCoefficientsAproductwithahighercoefficientofinnovationa

wouldexhibitasalesratecurvethatstartswithhigherone-periodsales,peaksearlier,andliescompletelyabovethatofaproductwithlowera.Figure10illustrateshowthecoefficientofinnovationinfluencestheproductsalesratecurves.ComputationalStudyandInsigh59ComputationalStudyandInsightsDemandDiffusionCoefficientsWefindthataproductwithhighercoefficientofinnovationisassociatedwithmore-frequentproductintroductions.ComputationalStudyandInsigh60ComputationalStudyandInsightsDemandDiffusionCoefficientsSimilartotheeffectofa,ahighercoefficientofimitationb

translatestoasalesratecurvethatliescompletelyabovethatforlowerb.ComputationalStudyandInsigh61ComputationalStudyandInsightsDemandDiffusionCoefficientsAswitha,theintroductionthresholdsaredecreasingintheproduct’scoefficientofimitationb.Thus,afirmshouldintroducenewproductgenerationsmorefrequentlygivenabasetechnologythatdiffusesthroughitspotentialadopterpopulationfaster.ComputationalStudyandInsigh62ComputationalStudyandInsightsComputationalStudyandInsigh63ComputationalStudyandInsightsUncertainDemandHere,wedescribetwopossiblescenariosforuncertaindemandalongwiththereviseddecision-modelformulations.ComputationalStudyandInsigh64ComputationalStudyandInsightsComputationalStudyandInsigh65ComputationalStudyandInsightsThedecisionmodel(1)–(3)isreformulatedasfollowstoaccommodatethissingle-perioddemanduncertainty:ComputationalStudyandInsigh66ComputationalStudyandInsightsComputationalStudyandInsigh67ComputationalStudyandInsightsAfirmmayalsowishtocapturetheuncertaintyinoverallmarketacceptanceofanewproductgenerationwhiletakingintoaccountpotentialcorrelationbetweenthemarketsuccessoftwosequentialgenerations.Forthedemanddiffusioncase,theestimationofanewgeneration’smarketpotentialN(z)isakeysourceofsuchuncertainty.ComputationalStudyandInsigh68ComputationalStudyandInsightsThepresentmodelframeworkcancapturethemarketsuccessuncertaintyforanewgenerationusingaMarkovmodulateddemandformulation.Apossibleimplementationispresentedhereforillustration.ComputationalStudyandInsigh69ComputationalStudyandInsightsComputationalStudyandInsigh70ComputationalStudyandInsightsThedecisionmodel(12)–(13)implementsthisMarkovmodulateddemandframeworkforaccommodatinguncertaintyinproductsuccess.ComputationalStudyandInsigh71ExtensionsAlimitationofouranalysisisthatthemodeldoesnotconsidertheeffectsofintroductiontimingonconsumers’purchasestrategiesandresultingdemandpatterns.Significantincreaseinthefirm’spaceofproductintroductionsmaycauseconsumerstopostponepurchasedecisionsinanticipationofforthcomingproductimprovements(e.g.,Dhebar1994,Kornish2001).ExtensionsAlimitationofour72ExtensionsThemodeldoesnothowevercapturethetime-to-marketconcernsthatmayariseinacompetitivemarketsetting.AsdescribedinHendricksandSinghal(1997)thecostofdelay(andhencevalueofearlierintroduction)insuchenvironmentscanbesignificant.Onepossibledirectionforfutureresearchistoconsideragametheoreticmodelwithmultiplefirms.AnalternativeapproachmaybetofollowCohenetal.(1996)andimposeafixedwindowoftimefornewproductintroduction.Suchaconstraintimplicitlycapturestime-to-marketconcerns.ExtensionsThemodeldoesnoth73ExtensionsGeneralizetheframeworktoincorporateupgradepurchasesortopermitsimultaneoussaleofmultipleproductgenerationswithsubstitutioneffects.Allowproductpricetofluctuateovertimeanddifferbetweenproductgenerations.Incorporateadecisionvariablefortheexpectedpaceofproductinnovation(e.g.,asmeasuredbyR&Dinvestment)wouldenableamore-completeanalysisoffirmpolicies.ExtensionsGeneralizetheframe74THANKS!THANKS!75演讲完毕，谢谢观看！演讲完毕，谢谢观看！76TimingSuccessiveProductIntroductionswithDemandDiffusionandStochastic

TechnologyImprovement

基于需求扩散和随机技术进步的连续产品引入过程

R.MarkKrankelDepartmentofIndustrialandOperationsEngineering,UniversityofMichigan,IzakDuenyas,RomanKapuscinskiRossSchoolofBusiness,UniversityofMichigan,AnnArbor,MichiganPresentbyLiWeiTimingSuccessiveProductIntr77CONTENTSIntroductionLiteratureModelOptimalPolicyComputationalStudyandInsightsExtensionsCONTENTSIntroduction78IntroductionConsideraninnovativefirmthatmanagesthedevelopmentandproductionofasingle,durableproduct.Overtime,thefirm’sresearchanddevelopment(R&D)departmentgeneratesastochasticstreamofnewproducttechnology,features,andenhancementsfordesignintosuccessiveproductgenerations.IntroductionConsideraninnova79IntroductionThefirmcapturesthebenefitsofsuchadvancesbyintroducinganewproductgeneration.Duetofixedproduct-introductioncosts,itmaybeunreasonabletoimmediatelyreleaseanewproductgenerationaftereachtechnologydiscovery.Rather,thefirmmayprefertodelayanintroductionuntilsufficientincrementalnewproducttechnologyhasaccumulatedinR&D.Theobjectiveofthispaperistocharacterizethefirm’soptimalproduct-introductionpolicyIntroductionThefirmcaptures80IntroductionThetotalnumberofproductgenerationsisnotpre-specified;rather,itisdeterminedbythepaceoftechnologyimprovementalongwiththefirm’sdynamicdecisionsonwhentointroduce.Analysisiscenteredupontwokeyinfluencesaffectingtheintroductiontimingdecisions:(1)demanddiffusiondynamics,wherefutureproductdemandisafunctionofpastsales(2)technologyimprovementprocess,specificallytheconceptthatdelayingintroductiontoalaterdatemayleadtothecaptureoffurtherimprovementsinproducttechnology.IntroductionThetotalnumbero81IntroductionPreviousliteratureexaminingincrementaltechnologyintroductionhasfocusedoneither(1)or(2),butnonehaveconsideredbothfactorssimultaneously.Asaresult,thepresentanalysisprovidesnewinsightintothestructureoftheoptimalintroductiontimingpolicyforaninnovativefirm.Usingaproposeddecisionmodelthatincorporatesbothkeyinfluences,weprovetheoptimalityofathresholdpolicy:itisoptimalforthefirmtointroducethenextproductgenerationwhenthetechnologyofthecurrentgenerationisbelowastate-dependentthreshold,inwhichthestateisdefinedbythefirm’scumulativesalesandthetechnologylevelinR&D.IntroductionPreviousliteratur82IntroductionRelativepapersWilsonandNorton(1989)&MahajanandMuller(1996)Thesetwopapersproceedunderademanddiffusionframework,butdonotmodeltheprogressionofproducttechnology.Rather,theyassumethatthenextgenerationproducttobeintroducedisavailableatalltimesstartingfromTime0.Asaresult,theyrespectivelyconcludetheoptimalityof“nowornever”(thenewgenerationproductisintroducedimmediatelyornever)and“noworatmaturity”(thenewgenerationproductisintroducedimmediatelyorwhenthepresentgenerationproducthasreachedsufficientsales)rulesgoverningproductintroductions.IntroductionRelativepapers83LiteratureTwomainresearchareasaredirectlyrelevanttothecurrentwork.Thefirstcentersonmodelsofdemand.Papersinthisareadescribethepatternsofdemandexhibitedbysingleormultipleproductgenerations,specificallyinrelationtonewinnovations.Thesepapersconcentrateonsystemdynamicsand/ormodelfitwithempiricaldata.Thesecondresearchareaexaminesdecisionmodelsfortechnologyadoptiontiming.Asubsetofthi