applications to marketing

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1、Chapter7ApplicationstoMarketingOvertheyears,anumberofapplicationsofoptimalcontroltheoryhavebeenmadetothefieldofmarketing.Manyoftheseapplicationsdealwiththeproblemoffindingorcharacterizinganoptimaladvertisingpriceovertime.Othersdealwiththeproblemofdetermini

2、ngoptimalpriceandqualityovertime,inadditiontoorwithoutadvertising.ThereaderisreferredtoSethi(1977a)andFeichtinger,Hartl,andSethi(1994)forcomprehensivereviewsondynamicoptimalcontrolproblemsinadvertisingandrelatedproblems.Inthischapterwediscussoptimaladverti

3、singpoliciesfortwoofthewell-knownmodelscalledtheNerlove-ArrowmodelandtheVidale-Wolfemodel.Todescribethespecificproblemsimderconsideration,letusassimiethatafirmhassomewayofknowingorestimatingthedynamicsofsalesandadvertising.Suchknowledgeisexpressedintermsof

4、adifferentialequationwithgoodwillorrateofsalesasthestatevariableandtherateofadvertisingexpendituresasthecontrolvariable.Weassumethatthefirmwishestomaximizeanobjectivefunction(thecriterionfunction)whichrefiectsitsprofitmotivesexpressedintermsofsalesandadver

5、tisingrates.Theoptimalcontrolproblemistofindanadvertisingpolicywhichmaximizesthefirm'sobjectivefimction.Theplanofthischapterisasfollows.Section7.1wiUcovertheNerlove-Arrowmodelaswellasanonlinearextensionofit.Section7.2dealswiththeVidale-Wolfeadvertisingmode

6、landitsdetailedanaly•sisusingGreen'stheoreminconjunctionwiththemaximimaprinciple.Theswitching-pointanalysisforthisproblemisagoodexampleofthereverse-timeconstructiontechniqueusedearlierinChapters4and5.Ex•tensionsofthesemodelstomulti-stateproblemsaretreatedi

7、nTurner1867.ApplicationstoMarketingandNeuman(1976)andSrinivasan(1976).7.1TheNerlove-ArrowAdvertisingModelThebeliefthatadvertisingexpendituresbyafirmaffectsitspresentandfuturesales,andhenceitspresentandfuturenetrevenues,hasledanumberofeconomistsincludingNer

8、loveandArrow(1962)totreatad•vertisingasaninvestmentinbuildingupsomesortofadvertisingcapitalusuallycalledgoodwillFurthermore,thestockofgoodwilldepreciatesovertime.VidaleandWolfe(1957),Palda(1964),andotherspres