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Optimal Advertising Policies for Diffusion Models of New Product Innovation in Monopolistic

SituationsAuthor(s): Engelbert Dockner and Steffen JrgensenSource: Management Science, Vol. 34, No. 1 (Jan., 1988), pp. 119-130Published by: INFORMSStable URL: http://www.jstor.org/stable/2632189

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MANAGEMENT SCIENCEVol. 34, No. 1, January 1988Printed in U.S.A.

OPTIMAL ADVERTISING POLICIES FOR DIFFUSIONMODELS OF NEW PRODUCT INNOVATION INMONOPOLISTIC SITUATIONS*ENGELBERTDOCKNERAND STEFFENJ0RGENSENInstitute of Economic Theoryand Policy, Universityof Economics,Augasse2-6, A-1090 Vienna,AustriaDepartmentof Economics,Universityof Saskatchewan,Saskatoon, Saskatchewan,S7N OWO,CanadaInstitute of TheoreticalStatistics, CopenhagenSchool of Jconomics and BusinessAdministration, ulius ThomsensPlads 10, DK-1925 FrederiksbergC, Denmark

This paper deals with the determination of optimal advertising strategies for new productdiffusion models. We consider the introduction of a new consumer durable in a monopolisticmarket and the evolution of sales is modelled by a flexible diffusion model. Repeat sales andpossible entry of rivals are disregarded but we allow for discounting of future revenue streamsand cost learning curve. Using standard methods of optimal control theory we characterizequalitatively the structure of an optimal advertising strategy for different versions of the diffu-sion model.(MARKETING; MARKETING-ADVERTISING; MARKETING-NEW PRODUCTS)

1. IntroductionIn the last 15-20 yearstherehas been a growing nterestin applyingmethods ofdynamicoptimization o marketingproblems,castin the framework f salesresponsemodels or diffusionmodelsof newproduct nnovation.In the areasof optimalpricingoradvertisinghere s nowa considerable odyof literature;he reader s referredo thesurveyarticlesby Sethi (1977), Little (1979) who deal with the advertisingarea,andRao (1984) who is concernedwith pricing.Survey papersthat particularly tressthecompetitiveaspectsareEliashbergndChatterjee1985),Dolanet al.(1986), J0rgensen(1986). Diffusion models of new product innovation are treatedby MahajanandMuller(1979), Mahajanand Wind(1986).Innovationdiffusionmodels arethe main focusof thispaperand we wish to designoptimal advertising trategies or such models. Briefly stated,a diffusion model at-

temptsto describe he spreadof a new product nnovationamong a set of potentialadopters.Typically, he model considers he evolutionof the numberof adoptersovertime,andthe diffusioncomes aboutthroughwordof mouth and is dueto advertising.Thepotentialvalue of such models iesin explanationof frequentlyobservedphenom-ena associatedwith new product nnovation(e.g. the S-shapeddiffusioncurve).Fur-thermore,diffusionmodelscan be used in forecasting.A thirdapplication,whichweshallconsider n this paper, s for normativepurposes, hat is, to resolveproblemsofoptimalmarketing trategies.In ?2 we give a shortreview of the literatureon diffusionmodelsthat incorporatemarketingdecisionvariables. n ?3, a flexiblediffusionmodelwithadvertisings ana-lyzedby optimalcontrol methods.A main resultof this sectionis Theorem1 wherequalitativestatementsabout an optimaladvertisingpolicy are made underminimalhypotheses.We includecostexperiencebutdisregard iscounting. t can be shownthatin a diffusionmodel of newproduct nnovation t paysto increase decrease) dvertis-* Accepted by John R. Hauser; received November 20, 1985. This paper has been with the authors 9months for 3 revisions.

119 0025-1909/88/3401/01 19$01.25Copyright ?) 1988, The Institute of Management Sciences


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120 ENGELBERT DOCKNER AND STEFFEN J0RGENSEN

ing over time if salesincrease decrease)as penetrationncreases. n ?4 which,in ouropinion, containsthe main resultsof the paper,we look at more specificcasesandobtainstronger esults Corollary1 and Theorem2).A model of this sectiongeneralizestheHorskyandSimon(1983)model and hasrelationshipso a seriesof otheradvertis-ing models.?5 containssome policyrecommendations nd suggests utureavenuesofresearch.

2. Literature ReviewAny innovationdiffusionmodelrests on a numberof simplifyingassumptions hatmust be recognized,discussedand,wheneverpossible,empiricallyested.Mostly, heseare assumptionsmade to facilitateanalytical nsightsinto the problemand we en-counter he problemof obtaininga reasonablebalancebetweencomplexityand tracta-bility.The major assumptionsunderlyinga diffusionmodel, and variousattempts o

relax these assumptions,are well summarized n Mahajanand Wind (1986, Tables1-1, 1-2).Diffusionmodelshavingconstantparameters re oftencriticized or their failure otake into account the effects of the marketingvariables.To relax this assumption,variousproposalshavebeen madeto let the parameters ependon relevantmarketingvariables.But no unifiedtheoreticalrameworkexists to provideguidelines or whatmarketing ariableso include,andwhere.Different uggestions anbe found n Robin-son and Lakhani(1975), Dodson and Muller (1978), Mahajanand Muller (1979),Monahan(1984), Horskyand Simon (1983), Kalish(1983), Lilien, Rao and Kalish(1981), ThompsonandTeng (1984),Kalish(1985),Kalish and Sen (1986).A unified heory,supportedbyconclusiveevidence,dealingwiththequestionhow toincorporatehemarketingdecisionvariables nto diffusionmodels, s stillnotapparent.Theoreticalmodelshave been builtwith the purposeof remedying omeshortcomingsof a certainmodel,or makingsome kind of generalization.Onlyin veryfew casesthemodelling effortshave been empiricallyvalidated.Neitherempirical nor theoreticalworkhaveprovided irmguidelines orthe choice of a particular unctional ormto beusedfor a specificdiffusionscenario,and the modellingof the impactsof e.g.advertis-ing in innovationdiffusionprocesseshasprogressed atherad hoc.In this paperwe suggest hat a feasibleavenueof researchmay be to assume moreflexible, .e. moregeneraldiffusionmodels.'Then we avoid,at leastto a certainextent,theproblem hatoptimalmarketing trategiesmaybe highlysensitive o the particularfunctional orm chosenforanalysis.Butthis benefit s obtainedat a cost.Ourapproachinvolves a risk that the conclusionsdrawnfrom such 'general'models may be lessspecific.However,we cannothardlyexpectmore thanqualitative ecommendationsoemergefrom generalmodels;this is, of course,a drawbackcomparedto the moreclearcutprescriptions btained romhighly specializedmodels.

3. Model FormulationWebeginthe analysisby statinga model withas fewassumptionsas possible giventhe overall ramework).Thismodelwillyield general,but alsolessspecificconclusions.In ?4we considera morespecializedmodel whichwillyieldmorepreciseconclusions.Consider irstpurchasesof a new consumerdurable n a monopolisticmarket.Werestrict he analysis o the case of a monopolistfirmwhichonly manipulatests adver-tising expenditureovera fixedplanningperiod.Lett denote time such that0 < t < T. The lengthof the planningperiod, T, is fixed.

I This approach resembles the one used by Sasieni (1971) and Kalish (1983).


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OPTIMAL ADVERTISING FOR NEW PRODUCT DIFFUSION 121Definex = x(t): cumulativesalesvolume (demand)by time t,A = A(t):the firm'srate of advertising xpenditureat time t.A flexiblediffusionmodelis given by

x= (t) = g(x(t), A(t)), x(O) = xO? 0 and fixed, (1)where x represents he current sales rate. By (1), the currentsales rate is relatedtoaccumulatedpast sales and the currentrate of advertising xpenditure.Functiong istwice differentiablend

gA>O; gAA< 0; g 0 for all (x, A) (2)where a subscripton a variabledenotes partialdifferentiationwith respectto thatvariable.The inequalitiesn (2) implythat salesarenonnegativeand concave ncreasingas a function of advertising.23We introducea cost learningcurveby assuming hat marginalcosts,denotedby c,dependon cumulativesales (production) uch that marginalcosts decreasewith in-creasing umulativeoutput(experience):

c = c(x); dc(x)/dx = c'(x) ? 0. (3)Note thatmarginalcosts could be constant(c' = 0). The learningcurvephenomenonhas been applied in a number of innovation diffusion models: see Robinson andLakhani 1975),Bass(1980),Kalish andLilien(1983),Kalish(1983, 1985).Assume that the firmchargesa constantprice, p, over the planningperiod.4Thisassumptions primarilymade for mathematical onvenience. deally, he optimizationproblem hould ncludepriceas well asadvertising andothermarketing ariables). eeThompsonandTeng (1984), Kalish(1985)wherebothpriceandadvertising re deci-sion variablesof the firm. It shouldbe noted, however,that the interactionbetweenoptimal price and optimal advertisingmay be sensitive to the particular unctionalformof the diffusionmodel.Inthe model(1), pricereductions ouldpossiblysubstituteincreasedadvertisingn order o increase nstantaneous ales.It is, however,our con-jecturethat in a model includingpriceas well as advertising,he qualitative tructureof an optimal advertising policy will be the same as the ones that will be estab-lishedbelow.5The instantaneous profit stream to the firm is given by P(x, A) = (p - c(x))g(x, A)- A. A convexadvertising xpenditureerm in P(x, A) was suggestedby Gould (1970)but was questioned by Schmalensee 1972). Instead of the term -A, Gould (1970)introduced term:-h(A) where unctionh is convex,but assumedg,s = 0, that s, salesare inearly ncreasingwithadvertising.NoticethatGould'sproblem convexadvertis-ing cost and linearsales)will yield a solutionwhich is structurally quivalent o thesolution of ourproblemwheresales areconcaveandadvertising ost is linear.)The intertemporal ptimizationproblem or a profitmaximizing irmcanbe stated

2In diffusion models with advertising, a concave function has been employed by e.g. Horsky and Simon(1983), Kalish (1985).3The concavity assumption guarantees that it is never optimal to increase advertising without bound.Hence we avoid nonsensical conclusions. The assumption in (2) means that the firmis operating on a concaveportion of the advertising-sales response function. See Little (1979), Hauser and Shugan (1983), Lilien andKotler ( 1983) for a further discussion of possible shapes of the response function.4 In Theorem 2 below (where marginal costs are constant) we also need to assume that the (constant) term p

- c (i.e. the unit margin) is strictly positive. This is not an unusual assumption because otherwise the firmwould possibly not produce the product at all.5 This conjecture has support in calculations done for a general price-advertising model in a preliminarypaper by Dockner et al. (1985).


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as follows.Determinean advertisingpolicyover the fixed time interval rom t = 0 tot = T such that the total discounted profitTJ(x, A) = exp(-rt)[(p - c(x))g(x, A) - A]dt (4)

is maximized,subject o thedynamicconstraint 1). In (4), r is a constantnonnegativediscountrate.Asadmissible ontrolsA = A(t)we take those that aretwice differentiablein t and satisfyA(t) 2 0 for all relevant .6To solvethe optimalcontrolproblem 4) subject o (1) andA(t) admissible,we applythe maximumprinciple ArrowandKurz1970).Definethecurrent-valueHamiltonianH=H(x,A, X) =g(x,A)(p-c(x)+ X)-A (5)

where X= X(t) s the current-value ostatevariable shadowpriceof x) whichsatisfiesthe differential quation X = rX gx[p - c + X]+ c'g (6)with the transversalitycondition at t = T

X(T) =O. (7)In (6),andin the sequel,we omit the arguments x, A) and x of functionsg andc (andtheirderivatives)whereno confusioncan arise.For an optimal nonnegativeA we have the firstordernecessarycondition that HA< 0. Confiningourinterest o interiorsolutionswe have

H4 = 0 == gA[p-c + X] = 1 (8)which admitsa unique solution because of the assumptions n (2). A second orderconditionfor H-maximization s

HAA


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OPTIMAL ADVERTISING FOR NEW PRODUCT DIFFUSION 123tion X = uJ*/ax,where J* is the optimal value of the profitfunctional.Hence thecostatevariable epresentshe dollarvalue(attime t) of a marginalncrease n cumula-tive sales. (Equation 10) showshow this dollarvaluechangesovertime.)Next noticethat gx > 0 (gx < 0) means that additionalsales now will increase(decrease) uturedemand.Put in anotherway, gx > 0 (gx < 0) reflectspositive (negative) diffusion'effectson the demand side. Finallynotice that the costate variabledepends on threeimportant dynamic factors:diffusion effect on the demand side, cost learninganddiscounting.From(1 1)it is easilyseen that if(a) gxis uniformlypositive,then from (10) the costate is positive forall t. Further-more,if r = 0, thenthe costatevariable s monotonicallydecreasing.This seems to beconsistentwith intuition,since in this case the firmhas no preferenceor earlyprofitsandgx positivemeans thatadditional alesnow stimulate ales ateron. In this case wewould expect an optimal advertisingpolicy to be one of initially heavyadvertising oreap the benefits romrapidly ncreasing umulativesales which will stimulate uturedemand. Then advertisingcan be graduallydecreasedas the diffusionprocessgainsmomentum.(b) gxis uniformlynegativeand cost learning s absent(i.e. c' = 0), then the costatevariable s negativebut increasing or all t. Here, increasedpenetrationhas an adverseeffecton futuredemandandwewould expectthatoptimaladvertising houldbe low inthe beginningof the planningperiod. Also, concerningcosts, there is no incentive toincreasecumulativesalesin order o benefitfrom lower costsin the future.We are readyto state a result for the case where sales are given by the generalspecification 1), i.e. x = g(x, A). Therearecost experienceeffectsbut a zerodiscountrate.(The case wherethe discountrate s positive s apparently ntractable.)

THEOREM 1. Withdemandgiven by (1), a cost learningcurvebutno discounting,then an optimaladvertising olicyis characterized y:A is decreasingovertime if gx> 0 andgAx < 0 for all t,A is increasingovertime if gx < 0 andgAx> 0 for all t.8PROOF. Seethe appendix.The results n Theorem1 canbe given the following nterpretations:1. Thedecrease increase)ofA is global, .e.A decreases increases)or all t, and the

advertising athis monotonic.2. An optimal advertising olicyis independentof the cost learningcurve since theassumption = 0 removes he effectof the costatevariable.Seealsoequation 0) in theappendix.)Hence the shapeof an optimal advertisingpathis only influencedby thediffusioneffects,embodied n the signsof the partials x andgAx, respectively.3. The conditiongx > 0 and gAx < 0 means that demand(sales)will increasewithpenetrationor, put in anotherway, additional sales now increase future demand.Moreover, he increase n current ales s weaker orhigher evels of advertising,.e. wehave a kindof 'diminishing eturns'phenomenon.The policy implication s to adver-tise heavilyin the beginningof the planning periodin orderto stimulateadoptionwhichin turnstimulates uturesales.The conditiongx < 0 andgAx> 0 is related o asituation wheredemand decreaseswith penetrationand the decrease s weaker forhigherlevels of advertising.The policy recommendation s to increaseadvertising

8 Noticethatthecases gx > 0, gAx > 0) and(g, < 0, gAx < 0), respectively,renot consideredn Theorem1.There are two reasons for this omission. First, the cases treated in the theorem are those we found analyticallytractable (cf. the appendix, Equations (0), (i)). Second, a situation where, for instance, g, > 0 and gAx > 0would imply that sales are increasing with penetration, and the increase in (current) sales being higher forhigher levels of (current) advertising, i.e. a 'increasing returns'phenomenon. We conjecture that the oppositecase is the more realistic one, but this has to be decided in each particular application.


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graduallyover the horizon in orderto counterbalance he decrease n currentsalescaused by increasingpenetration. See also the remarksconcerning(11).) A typicalsituationfor a durablegood is as follows:at introduction,a positiveword of mouthstimulatesdemand(i.e.g, is positive),while lateron, saturation ffectsdominateandfuturedemanddecreaseswith additionalsales now (i.e.g, is negative).In this caseanoptimaladvertising olicy maynot be monotonicasthe onesin Theorem1,but ratherU-shaped.4. Noticethatalthougha zerodiscountratemayseem somewhatunrealistic,he casecan serveas an approximationo situationswhere the discount rate is relatively ow.Also noticethat a zerodiscountrate means that the firmis indifferentof havingonedollar odayor one dollarat some instant n the future.4. Specific Functional Forms

In this section,the demandfunction g(x, A) is specifiedas a Bass (1969) diffusionmodelwithparameters ependingon theadvertisingate.Consider he following peci-fication9 x = g(x, A) = [a + Of (A) + yx + af (A)x](M-x) (12)where all parameters re nonnegativeand constant.For = 6 = 0, a and 'ycan beinterpreted s in the Bass (1969) model;a is the innovation coefficientand 'y is theimitation coefficient.The parameter B s relatedto the effectivenessof advertisingvis-'a-vishe innovators,whereas6 is related o the effectiveness f advertising owardthe imitators.The advertising fficiency unction, (A)is twice differentiable nd

f '(A)> 0; f "(A)< 0 (13)implying decreasingreturnsfrom advertising. See also assumption (2).) Teng andThompson(1983), Thompsonand Teng (1984)considereda modelof the same struc-ture as (12) but withf(A) linear. (Hence A = 0, i.e. there are constant returnstoadvertisingand (13) is not satisfied.)Due to the linearity,they obtainedadvertisingpoliciesofthebang-bang ypesuchthatexpenditures repiecewiseconstant; ometimesadvertisings zero,andsometimes t is at the maximal evel.

Let us considersome specialcases of (12). Putting6 = 0 yieldsx = g(x, A) = [a + Of (A) + yx](M-x). (14)In (14)it is assumed hatadvertisingnforms he innovators;mitatorsare nformedbyword of mouth. The model (14) could describea situation where the product s aninnovationat an early stageof the life cycle. Advertisings targetedat the innovatorsandemphasizesnewness,uniquenessandpersonalvalue for the potentialadopter.Thediffusionmodelin (14)has beenanalyzedby Horskyand Simon(1983)whoemployeda specificadvertising fficiency unction: (A) = ln A. The model(14) is alsorelated othe modelsof Stigler 1961),Fourtand Woodlock 1960), namelyfor B= y = 0. Takingthe innovation coefficientas a linearfunctionof advertising nd introducinga decaytermon theright-handideof (14),say, -dx, turns he model into theVidaleand Wolfe(1957) model. Optimal advertising trategies or this model have been designed byGould(1970),Sethi(1973).Putting ,B= 0 in (12) yields

x = g(x, A) = [a + yx +af (A)x](M-x). (15)9 See also Simon and Sebastian ( 1987)where an empirical study of some variants of ( 12)is carried out, usingdata from an advertising campaign of the German telephone company. The purpose of the analysis was to test

for goodness-of-fit various subcases of (12); hence, no optimization was involved.


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OPTIMAL ADVERTISING FOR NEW PRODUCT DIFFUSION 125In (15) it is supposed that advertising informs the imitators. The model could describe asituation where an innovation is at an intermediate stage of the life cycle. The product isnow an object of social communication which can be stressed by advertising thatemphasizes buyer evaluation, demonstration effects and social pressure. The diffusionmodel in (1 5) is related to the models of Ozga (1960), Mansfield (1961), namely for a= 6 = 0. Taking the imitation coefficient as a linear function of advertising and ap-pending a decay term yields the optimal advertising model treated by Gould (1970),Sethi (1979).In the model (12) we distinguish four components of the diffusion process:a(M - x): adoption without interaction with previous adopters and without influ-ence from advertising,3f (A)(M - x): adoption without interaction with previous adopters but stimulatedby advertising,

'yx(M - x): adoption by interaction with previous adopters but without influencefrom advertising,af (A)x(M - x): adoption by interaction with previous adopters and stimulated byadvertising.In the two subsections to follow we analyze the optimal advertising problem (4),subject to the evolution equation (12). First we assume a zero discount rate; next wederive some results for the more realistic case of positive discounting. In the latter wespecify the advertising efficiency function as logarithmic (cf. Sethi 1975, Horsky andSimon 1983).4.1. The UndiscountedCaseThroughout this subsection, the discount rate, r, is equal to zero. The following is animplication of Theorem 1.

COROLLARY1. If the demandfunction s given by (12), thenoptimaladvertisingexpenditures re increasing f ab - 0,y > 0, anddecreasingor ab - fry< 0.PROOF. See the appendix.The result of Corollary 1 can be seen as a nonlinear counterpart to the result inTheorem 2 of Teng and Thompson (1983). However, these authors could prove their

result for a positive discount rate. The inequalities that must hold for parameters togenerate the two different types of policies correspond to those employed by Teng andThompson (1983). Notice that advertising should, for example, be increased if ab - fiY> 0 O a/y > A/6, i.e. if the independent innovative effect is strongerthan the imitativeeffect.For the two special cases of (12), that is (14) and (15), respectively, we can obtainmore definite answers.(a) If the demand function is given by (14), i.e. advertising informs the innovators,then optimal advertising expenditures are decreasing. (This is easily seen from thecorollary by putting 6 = 0.) As noted, equation (14) is a generalization of the Horskyand Simon (1983) model to the case of a general, concave advertising efficiency func-tion. Hence we have demonstrated that the result of Horsky and Simon (1983) holds ina more general set-up. Moreover, we have proved that a policy of decreasing advertisingexpenditures is also optimal in a model incorporating cost learning. Horsky and Simon(1983) proved their results for a model with a logarithmic advertising efficiency func-tion and a constant marginal cost.(b) If the demand function is given by (15), i.e. advertising informs the imitators,then optimal advertising expenditures are increasing. (This is easily seen from thecorollary by putting A = 0.) The result is the opposite of the above but when advertisinginforms the imitators it is beneficial to postpone expenditures. (In contrast hereto,


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126 ENGELBERT DOCKNER AND STEFFEN J0RGENSENexpenditureshouldbe decreasedwhenadvertisingnforms heinnovators incein thatcaseit paysto use initiallya considerable mountof advertisingo informthe innova-tors and get the diffusionprocessstarted.)4.2. The DiscountedCase

Throughout his subsection he discountrate is strictlypositive. FollowingHorskyand Simon (1983) we specify the advertising fficiency unctionas logarithmic:(A)= ln A. Furthermorewe assume that there is no cost learning.This impliesconstantmarginal osts.We state a resultfor the demandfunction(12), recalling hat(14) and(15), respectively, an be obtainedas specialcasesof (12).THEOREM 2. For a constantmarginalcost,c, such thatp - c > 0, and a demand

functiongiven by (12) with (A)= ln A, then the ollowingholds:(i) If ab - 0ry> 0 and x(T) ? (M/2) - (f/26), then theoptimaladvertising xpendi-turesare increasing for all t).TABLE 1

Summary of Results> ~~Discounting/ lll

Demand No Discounting DiscountingSpecification Cost Learning No Cost Learning

g(x, A)= (a + ff(A) + yx +6xf(A))(M- x)

g(x,~ ~ ~).=g(a) ,Bf(A)+ yx+ 6xf(A))(M- x) |tt**ab - 1!


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OPTIMAL ADVERTISING FOR NEW PRODUCT DIFFUSION 127(ii) If ab - 0,By 0 and x(O)2 (M/2) - (f/26), then the optimaladvertising xpendi-turesare decreasing for all t).PROOF. See the appendix.Theorem2 tends to confirm he results hatwerederived or the undiscounted ase.The theoremstatesthat(i) optimaladvertisings increasingunder he same conditionasin the undiscountedcase, providedthat penetrationat the horizondate is not too high. This additionalconditioncanbe satisfiedf, forinstance, he marketpotential s sufficiently arge.(ii) optimal advertising s decreasingunder the same conditions as in the undis-countedcase, provided hat the initialpenetrationexceeds a certain(minimal)level.Thisextraconditioncan be satisfiedf, forexample, nitialpenetrations relativelyhighcompared o the maximalpenetrationevel.As in the undiscounted ase we can obtainstronger esults or the two models(14)

and (15).(a) Underthe samehypotheses sin Theorem2, except hat demand sgivenby (14),then the optimal advertisingexpendituresare decreasing. This is easily seen fromTheorem2 with6 = 0.) This resultconfirmsa conjectureof Horskyand Simon(1983)that the structureof an optimal policy will be the same in the discountedas in theundiscounted ase. Asalreadymentioned,Horskyand Simon(1983) only proved heirresults or the case of a zero discountrate.(b) Underthe same hypothesesas in Theorem2, exceptthat demandis given by(15), then the optimal advertising xpendituresare increasingwhenever t holds thatx(T) ? M/2. (This s easilyseen fromTheorem2 with,B= 0.) We obtain hepolicythatadvertising xpenditureshouldbe increasedas longas saturation ffectsare not domi-nating, in the sense that the market is not 'too much saturated'at the end of theplanningperiod.In Table 1 our resultshave been summarizedn the form of diagramsdepicting hequalitative tructure f the time pathof optimal advertising xpenditures.The focus ison the results romthe three diffusionmodelsin ?4.

5. ConcludingRemarksThispaperhasdealtwiththeproblemof determining ptimaladvertising oliciesformonopolisticnew product diffusion models. In particular,we have tried to designmodels of some flexibility cf. (1) and (12)). Results of Horskyand Simon (1983) ondecreasingadvertising trategieshave been extended to cases with a generalconcaveadvertising fficiency unction,cost learningor a positivediscount rate.A nonlinearcounterparto the modelof TengandThompson(1983) was analyzedbut only with azero discountrate.The followingmanagerial mplicationsareemphasized.* Ourapproach o modellingdiffusionof a newproductexplicitlyrecognizes hreedynamicfactors hataremanageriallyelevant.First,therearedynamic effectson thedemandside,reflectedn the diffusiondifferential quation.Sucheffectsarisebecauseofphenomena uchaswordof mouth and saturation.Second, herearedynamiceffectson the costside,due to thepresenceof a costlearning urve.Third,dynamicconsidera-tions are involvedvia the discount rate whichreveals he manager's ime preferenceformoney.* In all cases we obtainedmonotonicadvertising trategies ver the entireplanningperiod.Hence, optimal advertising houldeitherbe initially ow, and then increasing,or initially heavy and then graduallydecreased. t should be noticed, however,thatthesetypesof strategies oulddependon the factthat the profitfunctional 4) did not

includea salvagevalueterm. If the firmputs a value on the finalamount of cumulative


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128 ENGELBERT DOCKNER AND STEFFEN J0RGENSENsales, then nonmonotonic strategies may be optimal. Moreover, such strategies couldemerge in cases where the demand diffusion effects are not uniform over the interval[0, T]. Finally, it should be stressed that in a number of cases monotonicity wasdependent on the length of the planning period.

* Strategies of decreasing advertising typically occur when there are positive diffu-sion effects throughout the planning period, or advertisingis targeted at the innovators.This type of policy has been reported in the majority of cases treated in the literature;apparently it is quite robust against the choice of model assumptions.* Strategies of increasing advertising occur when negative diffusion effects arepresent and advertising is aimed at the imitators. This case may, however, be the lessinteresting case since the product suffers from negative word of mouth. Moreover,advertising toward imitators has only indirect effects.The diffusion models under study in this paper are restricted to the case of a mo-nopolist firm which only manipulates its advertising. Evident extensions are multi-player models, incorporating several marketingdecision variables.For such models, seeJ0rgensen (1982), Eliashberg and Chatteree (1985), Dolan et al. (1986), J0rgensen(1986), Kalish and Sen (1986). Recently, also studies have begun of marketing policiesfor diffusion in stochastic environments. See Eliashberget al. (1987), Raman (1987).10

'? The research of the first author was supported by a grant from the Osterreichischen Bundeswirtschafts-kammer (Exportakademie).

AppendixPROOF OF THEOREM 1. Differentiation with respect to time in (8), and using (10) yields

AA X g' g-"g-A(g) = SgA+r )2 (0)For r = 0 we obtain in (0):

gAA gx gxA (i)(g -)2 gA (g)

and the results follow immediately. Q.E.D.PROOF OF COROLLARY 1. Forr = 0 we obtain i) from(0),and evaluation f

-gxgA + gAxg = F(x, A) (ii)for demand specification (12) shows that

F(x, A) = f'(A)(M - x)2 [ab - ly].Using (i)-(ii) completes the proof. Q.E.D.

PROOF OF THEOREM2. Withdemandgiven by (12)andf(A) = ln A, we obtainfrom(O)A(ca - 03y)1(ii= (O + bx)(M -x) Xr + (o + 6X)2 ]J* (iii)

From (8) follows A _ + c(iv)and substitution of (iv) into (iii) yields

A=A[r a-yM+ - J)]-(d + x)(M-x)(p-c)r. (v)Differentiation in (v) with respect to time and evaluation for A = 0 yields

A| -A(ab-y) + A(a - y)(M - x) + r(p-c)[w(M-x)- - x]- * (vi)AO X x (fl+ 6X)2


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OPTIMAL ADVERTISING FOR NEW PRODUCT DIFFUSION 129Whenever A is constant, it must hold that (cf. (iii) and (iv))

Ar+ - r(p - c)6(M - x) = _ A(a - y)(M -x) (vii)fi+ 5x (i+ 6X)2(v)Substitution of (vii) into (vi) yields

AIA=o XI-d+A [a - y - r] - r(p - c)[26(M- x) - -x]} (viii)First consider the case where

ab - de > 0. (ix)From (iii) and (ix) it is clear that A(T) > 0. If, in (vi), it holds that x c (M/2) - (fl/26) then A < 0 along A = 0.Hence A > 0 for all t. If, in (viii), it holds that x c (2M/3) - (fl/36), and the stronger condition (than (ix)) ab- ,ly - br 2 0 is satisfied, then A < 0 along A = 0. Hence A > 0 for all t. This shows the first part of thetheorem. Next consider the case

ab - fy < 0. (x)From (iii) and (x) we haveA(T) < 0. If, in (vi), it holds that x0 2 (M/2) - (j3/26) hen A > 0 forA = 0 and henceA < 0 for all t. This completes the proof. Q.E.D.

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