5. MODELINGINFLUENCE IN

THE MARICETING-MIXNEV/-PRODUCT DIFFUSION

FRANK M. BASS

&hool of Management, Uniuerity of Texas at Dallu

DIPAI( JAIN

Kellog Gruduate School oJ Management, Northwestem l)niversity.

TRICI{Y KRISHNAN

Jesse H. Jones Graduate School of Managemen!, Rie lJniuersity

Absftact. Following the publication of the Bass model in 1969 in ManagementScience, the earliest attempt to modify this model to include decision variables wasan often-cited 1975 paper by Robinson and Lakhani thar was also published inManagement Scierce. In the ensuing years numerous modifications and extensions ofthe Bass model have been proposed to study the effects of decision variables on thediffusion process for new products.'we review here the various published papers thatinclude decision variables in diffusion models. In evaluating these publicarions weprovide benchmarks of desirable properties of diffirsion models with markering-mixvariables. In further discussion and exploration of desirable properries for suchmodels we present and evaluate proportional-hazard models, originally proposed byCox (1972) for application to surrrival data, that include decision variables. Weprovide a comparison of these models with the generalized Bass model developedby Bass, Krishnan, andJain (1994).Wb conclude with a discussion of the limitationsof the generalized Bass model and suggest possible areas for future research.

5.1 INTRODUCTION

The importance of the role of marketing-mix variables in understanding diffusiondynamics has been repeatedly emphasized by diffrsion scholars (Mahajan and Muller1979; Kalish and Sen 1986; Mahajan and Wind 1986; Mahajan, Mulleq and Bass1990). A major limitation of the early models of innovation diffirsion such as theBass model (1969) is that they do not explicitly incorporare marketing-mix vari-

V Mohajan, E MuUq andYWnd (eds.) NEW-PRODUCT DIFFUSION MODEIS Coryight @ 2000 Kluw Acade-mic Publishen. Boston All ights rcwil

100 IlI. Diffirsion Models

ables that are under the conrrol of managers. These models therefore become lessattractive for strategic planning with respect to policies for pricing and other vari-ables under managerial conffol. In addition, because of the rapid growth of succes-sive generations of high-technology products with short life cycles (Norton andBass, 1987, 1992),the importance of control variables to management has increased.

Because of the growing importance in real-world applications of difiimion modelswith decision variables, it is desirable to develop an understanding of the weaknessesand strengths of the available models. We undertake here to provide evaluarions ofthe existing models in the literature and to suggest possible fruitful areas for furtherresearch.

We organize our discussion along the following lines. In Section 5.2, we startwith a description of the Bass model (1969) and then discuss the desired proper-ties of diffusion models in Section 5.3. In Section 5.4 we present those models thathave analyzed only the effects of price on new product diftrsion, and in Section5.5 models with only advertising effects. In Section 5.6 we introduce models thathave incorporated both price and advertising effects. In Section 5.7 we present theempirical analysis and results, and in Section 5.8 we conclude the chapter givingdirections for future research.

5.2 O\rERVIEV OF THE BASS MODEL

5.2.1 The Bass Model (1969)

The Bass (1969) model has proved to be a seminal model in marketing in analyz-ing the first-purchase sales of new durables. lt postulates that the likelihood that anindividual would adopt the new product at rime l, given that he has not yet adopted,is represented by the following equation:

1@lg- nO)= p+ qF(t), (s.1)

where the parameters p and q represent the coefficients of innovation and imita-rion, respectively, F(r) rs the cumulative distribution function (probabiliry of adop-tion by time r), and flt) is the probability densiry function of the random variable/, the time to adoption for the new product.

Let S(t) and Y(r) denote the sales and the cumulative sdes of the new product,respectively, at time t, and m be the total market potential. Assuming S(t) = *11t1,we can rewrite equation (5.1) as

s(t) = p + (q - p)mr Q) - qn[r Q)f" = o + bv (t) + cfv (t)]'z,

where a - pm, b = (q - p), c = -(q/*),and Y(l) = mF(t).Assuming F(0) = 0, the solution to the differential equation in (5.1) is

F(,) = (1 - e-o.'v ) | (t + (ql p1r-t"'0'7.

Thus,

.tQ)=lb+ q)'lrf, *" f 11t*(qlp), t")l'-

(5.2)

5. Modeling the Markedng-Mix Influence in New-Product Diftrsion 101

To find the time at which sales will peak, differentiate nflt) and ser rhe result equdro 0 to find

f =(lb+q))rn(qld. (s.s)

Assuming 4 is larger than p, which is the case for successful new products, an inte-rior maximum will exist and at the maximum

.( '-)= G-dlzq. (s.6)

(5.7)t('.)= rrt-) = ̂ G+ q\' l+q.

Because p is normally a small number compared to 4, cumulative sales (zF) will beapproximately lzm when sales reaches its peak.

5.2.2 A Special Case: The Logistic Model

A model that has long been used to fit and project cumulative output or sales ofvarious produccs is the logistic model. Mansfield (1961) used the logistic model insrudies of diffirsion of locomorives and other industrial products. This model maybe written as

Y(t) = a,f 11* r-(a2+a3')\.

The "sales rate" for the logistic will then be

dYf dt = arare-Q""t') f (7+ e-Q2+ar'|t)z

= a,(y(t)e-Gr*.i'r)f (t+ e4"2.dn)

= (a,l a,)lv(t)1z ,-toz,ot,\

=(a,f a,)v(t)la, -v(r)l =o,Y!)-Q,lo,)[v(r)]'. (s.8)

Let a, = b and (a3/a) = t ot d1 = at/c. In that case, the logistic is a special case ofthe Bass model in equation (5.2) when a = 0-{hat is, p = Q.

Note that for the logisric discribution, F(t*) = 72, which is approximately thesame as for the Bass model but slighdy to right, or later, than the peak for the Bassmodel. For the logistic, S(r*) is mq/4 as compared to m(1t + q)2/4q for the Bassmodel.

If 4 is the same for both models, the Bass model will peak slighdy earlier thanthe logisric. Empirical estimates of m wtlJ be slighdy smaller and of q's slighdy largerfor the fitted logisric than for the corresponding values of m and 4 for the fittedBass model.

5.2.3 A Special Case: The Negative Exponential

If 4 = g, 11t.tt

1@l(-r@)=p,

1O2 III. Diffusion Models

and solving this differential equation, we get

F(t) = 1- s-t' .

Therefore, the sales rate will be

S(t) = ar-o'.

Hence, when 4 = 0, the Bass model becomes the negative exponential, and the sales

rate will continuously decline. Unlike the logistic or the Bass model, the percent-

agechangeinsales (=(dS/dt)/S) wi l lbeaconstant(d.FourtandWoodlock(1960)used the negative exponential in the study of timing of first purchase of new grocery

products.

5.3 DESIRABLE PROPERIIES OF DIFFUSIONMODELS WITH DECISION VARIABLES

To provide a basis for evaluation of di{fusion models with decision variables, we

develop and discuss desirable properties of such models (for another discussion of

desirable properties, see Bass, Krishnan, and Jain 1994):

. Because the Bass model is an empirical generalization that almost always describes

the diffusion and adoption timing for new products (see "Inrroduction" 1995), it

is also desirable for extended diffusion models with decision variables to also

describe the empirical diffusion process. This implies that when decision variables

follow frequendy observed patterns such as exponentially falling prices associated

with the experience curve, diffusions models that include these variables should

reduce to or be approximately equiualent to the Bass moilel curue.. An essential properry of the Bass model stems from the behavioral rationale under-

lying the model. The model indicates that the timing of adoption (probabiliry of

adoption) of those who have not yet adopted at any time I is influenced by the

number of previous adopters (because of learning or imitation).This implies that

the greater the number of adopters today the greater will be the influence on the

remaining potential adopters to adopt at each future time period. One can think

of this property as the "carry-through" property in that events today carry-through

and have impact in all future rime periods. In the context of models with deci-

sion variables this means that a price reduction today will not only increase adop-

rions today but have a positive impact in all future time periods due to diffusion

force.. Diffsion models with decision variables should have empitical support.This implies

that such models should be capable of empirical estimation and that the models

should yield plausible parameter estimates. In addition, the credibility of models

is enhanced as the number of data sets for which empirical estirnates of parame-

ters rncreases.. In examining diffusion models with decision variables it is important to recogaize

that goodness of ft il,one is an inadequate criterion for evaluating model validiry.

5. Modeling the Marketing-Mix Infuence in New-Product Diffusion 103

' Moilels with closed-fortn solutions have nice properries. Although closed-form solu-tions do not guarantee model validity and are, at bimes, special cases of moregeneral models, they are appealing in their simpliciry and aid in interprering andunderstanding relationships among the variables and the time pattern of adoption.

' It is desirable for diffusion models with decision variables to be constructed in sucha way that they are managerially useful.This implies that parameters of the model canbe expressed in a way that allows management interpretation either by intuition orwith reference to comparisons with other products. Parameter interpretation mayalso be enhanced by conversion to equivalent expressions that allow interpretation.For example, although p and q in the Bass model may be difiicult to interpret bythemselves, they may be converted to "time of peak," a number that may be judgedeither by intuition or by comparison with other products.

' Related to the previous property is the desirable property that diffirsion modelswith decision variables should be imgtlementable for new proilucts without ilata. Afterall, the period of greatest scrategic importance is the period before the productis launched when no data are available. The basic Bass model has been appliednumerous rimes under this condition by "guessing by andogy." Managementdetermines m by judgment modified by information about the market size,but pand q arc determined by analogy with previously introduced products where thep's and qt are known. In determining p's and q's for new-product models withdecision variables it is desirable, then, to have benchmark values for determiningp's, q's, and other parameters, such as elasticities, so as to enhance the ease of"guessing by analogy." For several examples of applications of the Bass model andthe generalized Bass model before product launch using the "guessing by analogymethod" see http:,//www.utdallas.edu/-mzjbl (click on Bass Model Presentarionto download a Power Point presentation), and see also http:,/,/wwws-M-A-R-T. com/Exp_penetration.

5.4 MODELS THAT INCORPORATE PRICE ALONE

5.4.1 Robinson and Lakhani (1975) and Others

Robinson and Lakhani (1975) were the first to introduce decision variables intodiffusion models. Their purpose was to examine the effects on profits of differencesbetween optimal pricing policies that take into account the dynamic effects of prices(that is, optimal price sequence over the planning horizon) in conrrast to an myopi-cally optimal policy. They constructed their results through simulations. The modelthey proposed was the following modification of the Bass model:

Sales = (1z -Y)(p+ qV)s-hh('t.

The first two terms on the right-hand side of the equation represent the differen-tial equation of the Bass model. In this model price effects are represented by theexpression e rq'), where Pr(t) is price at time t and k is a price coefficient.The modeldoes, on the suface, seem ro be plausible, but it does nor reduce to the Bass modelunless price is constant and it does not have a closed-form expression. However, it

104 lII. DifFusion Models

does have the desirable carry-through properry. For the Robinson-Lakhani model

the problem is empirical estimabion because the estimation becomes "confused" in

that it is unable to distinguish between parameters unless the coefiicient k is near

0 or price is constant, in which case the model reduces to the Bass model.The fact that the Robinson-Lakhani model cannot be esrimated has obvious

implications for the normative conclusions that are derived ftom this model. For a

detailed discussion of this see, Krishnan, Bass, and Jain (1999). Dolan and Jeuland(1981) and Jeuland and Dolan (1982) also used the Robinson-Lakhani model in

explorarions of normative implicarions of the model.Other studies that used models similar but not identical to the Robinson-Lakhani

model are Horsky and Simon (1983) and Dockner and Jorgenson (1988). Teng

and Thomson (1983) analyzed an elegant and sophisticated optimal control

theory problem using the framework of Dockner and Jorgenson. \Jnfortunately,

these near-Robinson-Lakhani models are also not generally viable for empiricalesumanon.

5.4.2 Bass (1980)

The Bass (1980) model starts with the simple constant-elasticiry demand function:

q(t)= f(t)d1(t)la,

where Q(r) = quanrity demanded at time t, P,(t) = price of the product at time

t, JU) = Bass model specification given in equation (5.1), c = cost function, and

\ - price elasticiry parameter. ff) serves to shift the demand funcrion in time. In

this model it is assumed that firms are choosing prices to maximize profits myopi-

cally and therefore equaring marginal revenue to marginal cost in each period. If

marginal costs follow the experience curve, as they normally do for technologyproducts, then prices will be proportional to accumulated output raised to the power

l, where i, is the learning rate parameter. Substitution of the optimal price, Q(/)into the equation above results in the following closed-form solution for demand

at trme ,:

q(t) =[nl( - h1t)]J(t)e(t)^n/t'-^nt .

For esrimation purposes.;(r) must be specified. Bass used the basic Bass model for

this specificarion so that the demand function for the new durables changes in time

as a result of diffirsion forces, prices, and saturalion effecs. The model was estimated

on data for six consumer durable products and generally provided good descriptionsof the empirical sales and price series.

In the last equation above demand is a function of time only, but the price effects

are embedded in the model under the assumption of myopic profit optimization byfirms. lJnder myopic behavior price today affects the demand today but does not

have effects that carry through in the future. The parameter i, is estimated from the

historical relationship of price and the experience curve-that is, 4(t) = [y(0]^-

5. Modeling the Marketing-Mix Influence in New-Product Diffirsion 105

and then substituted into the equation allowing empirical estimation of m, l|, andthe parameters offlr) and F(r), p and q.

Applying the seven desirable properties previously suggested it is possible to con-clude that this model has the desirable properties of being capable of empiricd esti-mation and it has a closed-form expression. However, in all other respects it hasundesirable properties. It does not reduce to the Bass model when prices are chang-ing. It is a "current-effects" model and does not have the desirable carry-throughProperry and, as a result of these two undesirable properries, its managerial usefrrl-ness and implementarion Gatures are limited.

5.4.3 Kalish (1985)

Kalish (1985) builds his model on the interrelationship berween how a new productis perceived in the market with respect to rhe uncertainty surrounding is perfor-mance evaluation, its utility ro the potential adopter and the price offered in themarket. He argues that information about a new product, and not the new productper se, is the one that diffirses in the society through aduertising and word ofmouth. He further claims that those who become aware of the product may notadopt immediately because of the uncertainty he has with respect to its perfor-mance (which is modeled as the perceived value rninus the price) and that theuncertainty reduces as more and more consumen adopt the product. Kalishmodels this rwo-stage process through the following set of simultaneous differentialequauons:

Awareness: 4 =l- rll f4(t)l+ tr +a'f44).l. ""0d t - 1 " " "

\ m ) l '

Adoption: dY.(t) -["f ,1t) .)r-v(tt]p.' dt L'\u(v@ln)) - -'J-'

where Y(l) is the cumulative sales up to r, P(r) ir the price ar rime r, ,,4(t) is theadvertising at time l, f is the information or awareness level, m is the initial marketpotentid, g,J and I are function operators, and b, b', and k are parameters.

In the empirical stage, Kalish reduces this model by assuming away f to be 1 (thatis, the first equation ceases to exist) and proposes for g an exponential and for r aquadraric specification. Thus advertising is not present in the empirically testifiablemodel.t The final reduced equation is

where B and y are new paramerers. Kalish tested this reduced model on a product(whose data description was reported confidenrial) and found the fit to be betterthan Robinson and Lakhani model (1975).

Kalish (1985) model includes price in such a way that that it has the desirablecarry-through properry in that an impact on sales at time t created by a price change

106 III. Diffirsion Models

at time t is carried forward to future periods through the diffusion effect. Although

the two-state process of Kalish (1985) looks interesting, it is not very appealing fiom

a managerial perspective because it contains constructs that are usually diffrcult to

measure in a real market. Moreover, there is no closed-from solution for the Kalish

model in the sense that sales is not expressed as a function of time, price, and adver-

tising alone.

5.4.4 Kamakuta and Balasubrarnanian (1988)

Unlike Kalish (1985), the Kamakura and Balasubramanian (1988) model was not

developed from any utiliry maximization principle but from a simple extension of

the Bass (1969) model. The main purpose of their model was to use an extended

data set (that is, the set that contains sales and price data extending well beyond the

usually employed first few years data) in a modified Bass model and test for the

impact of price on diffirsion speed or market potential. The model used is

s(t) = l p + qy (t)lnQ)e' feu Q)p,(,)o' - y0)1,

where p1 represents the impact of price on the adoprion speed while p2 representsthe impact of price on the market potential, M(t) is the number of households withelectricity at rime t, and 0 is the ultimate penetration level.

Testing the model on six data sets (that is, extended data sets)2 the authors showthat price has no effect at all in three product categories (toasters, blenders, andmixers), which is understandable considering that these categories are relarively inex-pensive compared to air conditioners, refrigerators, and vacuum cleaners, for whichprice was found to significandy alfect the adoption rate but not the marketpotential.

Although Kamakura and Balasubramanian (1988) incorporate price explicitly inthe Bass model (1969), th.y use a discrete-rime formularion for estimating a con-tinuous model. Therefore, there may be a time-interval bias present in the parame-ter estimates (Schmitdein and Mahajan 1982). This model does have the desirablecarry-through properry but it does not reduce to the Bass model unless price isconstant and it does not have a closed-form solution.

5.4.5 Jain and Rao (1990)

To overcome the methodological shortcomings of the Kamakura and Balasubra-manian (1988) model,Jain and Rao (1990) used the continuous-time formulationof the Bass model (Schmittlein and Mahajan 1982) and proposed two model spec-ifications that include price. These are

s(') = (n r+(')-" - v1r - 9) r(4:-F(H),

s(t) = (n - v(t- 0)a0)-' F(t) - F(t - 1)

.1 -F( r -1 )

5. Modeling the Marketing-Mix Influence in New-Product Diffi:sion lO7

where F(.) is the Bass (1969) model given by equation (5.3). Note that the termF G ) - F - ( r - 1 ) .- -* is thediscret izat ionof thehazardratef l t ) / |_F( t ) .Thef i rs tmodel

1- F(r - 1)above suggests that price affects the market potential, and the second rnodel sug-gests that the price affects the effective market potential (that is, the remaining salespotential).

Jain and Rao (1990) tested both models with four product categories-namely,room air conditioners, clothes dryers, color televisions, and can openers..With canopeners they found the pdce to be insignificant, which was not a surprise. Withthe other products, they found the second model to be the best in terms of fit andhence concluded that the price affected the effective market potential.

Although the Jain and Rao (1990) model provides sales as an explicit functionof price and time, the price variable is not modeled to affect the basic diffirsionprocess given by the differential equarion (5.1) but modeled to act on the sales func-tion (equation (5.3)) that had been independently derived by solving differential equa-tion (5.1).A closed-form solution exists for part of the model bur only for that partthat does not include the price variable. Note that the Jain and Rao model doesnot have the carry-through properry. The model formulation provides for a priceinfluence on "effective market potential;' and this is identical to price affecting therate of adoption.

5.4.6 Horsky (1990)

Horsky (1990) built a model from an economic principle rhat a new product isadopted because it saves time (such as, dishwashers) or enhances utility (stch as tele-visions). Such benefim are evaluated by a consumer using reseruation pricc (that is,how much maximum he is willing to pay for those benefits),his wage rate,andthepriee of the new product. Thus, the adoption of the new product depends on howit maximizes the net utility for the household. Here, both the wage-rare distribu-tion (across population) and the price distribution change over time. Adding the dif-fusion effect, Horsky arrives at the final model as follows:

where u(r) and P(t) are the average wage rare of the population and the averagemarket price of the product, respectively, d represents the dispersion of both the dis-tributions, K and ft represenr the rime-saving and utiliry-enhancing attribute ofthe new product, Y(l) is the cumulative sales up to t, M(t) is the total number ofhouseholds in the U.S. population, 0 is the fraction who are potential buyers, andp and q are the diffusion parameters. The first part of the model is the income-price-market saturation force, and the second part is the diffusion force.

In the empirical analysis Horsky used data sets with long rime intervals so that

108 III. Diffusion Models

a large component of the demand in later years was made up of replacement buyersrather than adopters. To take replacement into account Horsky assumed that 70percent of the demand in later years was made up of replacement purchases. Thereis no empirical support for this assumption, and the empirical estimates depend inan important way on it. It seems likely that his result relating income to sales isbecause replacement demand is more income sensitive than initial demand. Thus,Horskyt empirical results can be quescioned with respect to their applicabiliry toadoption.

The principle behind the Horsky model is very appealing. It is perhaps the onlymodel that attempts to explain why an individual buys a new product contrary toother diftrsion models that focus on when an individual would adopt a new product.Further, the Horsky model possesses the desirable carry-through Property dthoughit does not have a closed-form solution.

5.5 MODELS TIIAT INCORPORATE ADVERTISING ALONE

Only rwo articles analyze advertising effects empirically. These are the Horsky and

Simon (1983) and Simon and Sebastian (1987) models.

5.5.1 Horsky and Simon (1983)

Horsky and Simon (1983) proposed that advertising affects the innovative charac-

teristics of the adoption population, and hence they formulated the model as follows:

s(t) =lu + BLn(aQ\ + qv(t -r)lln -v(t -r)1,

where .,4(t) is the advertising at rime l, and p is its effectiveness.The model was tested with one product, telephone banking, a new type of

banking introduced in 1970s. Out of the five cities where the new system was intro-

duced, the model provided good fit and significant estimates for advertising effec-

tiveness on the spread of the system in four cities.

5.5.2 Simon and Sebastian (1987)

Simon and Sebastian (1987) argued that advertising could affect either innovation

or imitation component of the diffusion, and hence they set to empirically test the

following models:

s(t) =fp+ p1@(r))+ av(r- t)[n-v(t-t)],s(r) = [p + {a + qf (A@)ly(t - r)]ln - v(t - r)1,

where,4(r) is the advertising time r.The funcrion;fis assumed to have one of threeforms: Ln(A(t)), or 2t6Ln(A(tD, or Ln(W A(t)), where W A(t) is the weightedaverage of the advertising from 0 to l. The third formulation is Nerlove-Arrow

model (1962).Thus, in effect, Simon and Sebastian tested six models.

The product that they used for empirical testing was telephone adoption in

Germany. The results showed that the Nerlove-Arrow form of advertising that was

5. Modeling the Marketing-Mix Influence in New-Product Diffrsion 109

modeled to affect either the innovation or the imitation component of the diffi.r-sion performed better than the other four models.

Note that both the advertising-diffirsion models possess the desirable carry-through property but do not have closed-form solutions. However, compared to theprice-diffusion models we saw in the previous section, these advertising-diffirsionmodels are different in the following ways. First, there are only very few models.Second, these advertising models concenrrate on whether advertising affects the inno-vation or the imitation component of the diffusion. The two articles we discussedreached different conclusions on this, however: one article concluded that advertisingaffected the innovarion component, and the other article concluded that adverrisingaffected the imitation component of diffusion. Third, both the Horsky and Simon(1'983) and the Simon and Sebastian (1987) models have advertisingincluded in thediftrsion process and use the ordinary-least-squares procedure to estimate the modelparameters, which has some limitations as discussed in Schmitdein and Mahajan(1,982). Further, these models do not seem ro have srrong empirical support (forexternal validiry) because they were tested with just one product each.

5.6 MODELS THAT INCORPORATE BOTH PRICE AND ADVERTISING

5.6.1 The Generalized Bass Model

Bass, Krishnan, and Jain (199\ build a generalized Bass model (GBM) presentedbelow:

f ( t \

ffiiDj =[p+qr(t)]x(t),

where r(r) is termed as the current marketing effort that can be modeled to includethe impact of all marketing mix variables. Specifically, they use

x(r) = r+ mh'!') * BzA'l',) .' Pr(t) '- ,q(t) '

where Pr(r) and A(t) are the price and advertising at dme t, and Pr'(t) and A'(t) arethe rate of change in price and advertising respectively ar time t.

GBM has the appealing property that both price and advertising are incorporatedin the diffirsion process. At the same rime, it is solvable to yield sales as an explicitfunction of time, price, and advertising as follows:

S(r) =, e-b+d0+fib(hl+P2b(A)l

The GBM was tested on three product categories (room air conditioners, clothesdryers, and color televisions) and found to provide significant estimares for priceand advertising (for room air conditioners and clothes dryers) and for price (forcolor televisions).

(p* q)'p

l r* q

,-t *r,u*^t,or*u"ur^rr]t '

L p l

110 III. Diffirsion Models

An interesting property of the GBM is that if the percentage changes in price

and advertising remain the same from one period to the next, then r(t) would

reduce to a constant, thus giving us the Bass (1969) model. This provides a reason

as to why the Bass (1969) model provides a good fit to most of the new-product

sales-growth data even though it does not have the marketing-mix variables. Prob-

ably because of this reason, the GBM estimates of the diffusion parameters-namely,

p, q, and tn-are very close to the estimates produced by the Bas (1969) model.

This is an attractive properry for forecasting purposes.

GBM is perhaps the only model in the literature that has both price and

advertising included in a single framework and, at the same time, is solvable to yield

sales as an explicit function of independent variables alone (that is, time, price,

and advertising). It has been empirically tested with multiple product categories.

This model carries an additional property in that it generalizes the Bass (1969)

model by including price and advertising as additional variables to the time variable

in such a way that these marketing-mix effects could be captured by the time vari-

able under the plausible scenario of exponential changes in those variables along

the time path.

The generalized Bass model has all of the desirable properties that we have sug-

gested for evaluating diffusion models that include decision variables. It (1) reduces

to the Bass model when prices and other decision variables are changing by a con-

stant percentage, (2) has the carry-through properry and (3) has a closed-form solu-

tion. Because decision variables are incorporated in the model as percentage changes,

,thus eliminating strong trends in the level of the variables, the danger of spurious

regression is substantially mitigated for this model. The model is managerially usefirl

and implementable because the pt and q's are close to the pi and 4's for the Bass

model estimates without decision variables. Therefore, "guessing by analogy" with

the generalized model may proceed in the same fashion as it would for the basic

Bass model for p's and q's. Decision variable coeflicients may be easily guessed on

the basis of guesses of elasticities (see Bass, Krishnan, and Jain 1994).

The generalized Bass model has strong theoretical and empirical support, and it

provides for a generalized explanation of the existence of the empirical generaliza-

tion of the Bass curve almost always being observed in di{Frsion phenomena. From

a practical viewpoint, however, GBM is limited by the fact that marketing-mix vari-

ables act only through changes in the variables and not in the leuels of variables. In

other words, it implies that irrespective of what price a new product starts with, its

sales path is controlled only by the changes happening along the price path. This is

a serious limitacion in developing optimal policies from the model. This problem is

partially handled by an assumption that the initial price acts as a reference price so

that subsequent changes in price may be used to convert to price levels (Krishnan,

Bass, and Jain, forthcoming).

Proportional-hazard models that include decision variables in levels have been

employed in several applied contexts. It therefore would seem to be desirable to

study such a model in the diffusion context. In the next section we develop and

use the proporrional-hazard framework (Cox 1972) introduced in marketing by Jain

5. Modeling the Marketing-Mix Influence in New-Product Diffirsion 111

and Vilcassim (1991), Jain (1992), and Helsen and Schmitdein (1992). Mahajan,Muller, and Bass (1990, pp. 1,9-20) in a survey piece on diffusion models in mar-keting suggested that a "hazard modeling framework may provide a unifying themefor understanding of covariate,/marketing-mix effects in diffusion models." Althoughproportional-hazard models have the rather severe limitation of being "current-

effects" models, they do appear to be worth exploring in comparison with the gen-

eralized Bass model.

We model the rate of adoption of a new innovation as a function of a baselinehazard specification that depends only on time and a function that captures theeffect of the marketing-mix variables. For the baseline hazard function we use thespecification proposed by Bass (1969).A nonnegative function is specified to measurethe impact of marketing-mix variables on the rate of adoption.

An appealing feature of the proposed modeling framework in this chapter is thatthe underlying model parameters can be easily estimated by the nonlinear least-squares procedure (Srinivasan and Mason 1986;Jain and Rao 1990).This procedurehas been found to be superior to the ordinary-least-squares and the maximum-likelihood-estimation procedures for the data on consumer durables and educationaland medical innovations (Mahajan, Mason, and Srinivasan 1986).

In the empirical analysis, we compare the parameter esbimates and the predicriveperformance of the proposed model that incorporates price and advertising vari-ables with the basic Bass diffusion model and the GBM.'We use the data sets onconsumer durables that have commonly been analyzed in the diffirsion literature.For model validation, we obtain the one-step-ahead forecasts and compare them tothe actual values in the holdout samples.

.We present below our continuous time hazard-modeling framework.

5.6.2 Proportional-flazard Models

Let the random variable l, the time to adoption, have the distribucion function Fand the density function J Let L (t, Z) denote the conditional probabiliry rhat anindividual will adopt the product in the time interval (t, t + dt) given that he hasnot adopted by time f and given that he observes the set of marketing-mix vari-ables Z in the marketplace at rime t. In the literature of survival analysis, )" (t, Z)is referred to as t}re hazard function, or the failure-rate function for an individual attime I with covariate Z. It represents the instantaneous probabiliry of adoprion attime / given that the individual has not adopted before t. Using the definirion of ,1,(t , Z),we can write

),(t,z)=#dr4-; (s.e)

Integrating (5.9) using F(0, Z) = 0 as rhe initial condition, we obtain the distribu-tion function

r(t. z) =1 - e*p[-f .t(,, 4du]1. (s.10)

ll2 lll. Diftrsion Models

Equation (5.10) shows the one-to-one correspondence benveen the disrributionfunction and the hazard function of the random variable T. Following Cox (1972),the conditiond, hazard rute h (t, Z) can be expressed as

x.(t, z)= x"(t)Nz(t)\. (s .1 1 )

Equation (5.11) is referred to as Cox's proportional-hazard modelLo 0 is referredto as a base-line hazard function, and Q {Z(t)\ is a function of marketing-mix vari-ables Z(t), which act multiplicarively on the baseline hazard rate. Customarily, it isassumed that

Nz(r)\ = exp{z(t)Fl, (s.r2)

where p = (Ft, k, . . . , Br) is the set of unknown coefficients (to be estimated),associated with Z(t).

We propose that fu(t) follows the specification postulated by Bass (1969)-thatls ,

which implies that the hazard function resulting from the Bass model follows a logis-tlc CUrVe.

Substituting (5.12) and (5.14) in (5.11) and defining a= p I qand b= q/p,weget

)n(t) = JQ) | t - r(t) = p + qF(t).

Evaluating fu(t) using (5.3) and (5.4), we get

h( t )= o*u ,t+ L.*p[-Qt+ q)t]

p

).( t ,z)= * ! ^exp{2iL)\.r+ aexp\-dt)

Now subsdtuting (5.15) in (5.10), we can write

F (t, z) =t - .*[- j { *rfu }.*v{ z {,) F}n)

F (t, z) =' - .*[->;=, { i,{ffi6} *r{'aw},"}f

(s.14)

(s.1s)

(s.16)

Using the properties of definite integrals, we can write the above expression as

='-"*[-X =,{.*,{'{drrtj{dCA}r,}] (s,7)

since Z(t) is constant in the interval r - 7 to t. After evaluating the integral in(5.17), we get

5. Modeling the Marketing-Mix In-fluence in New-Pmduct Diftrsion 113

F (t, z) =t - *n[I"=, { exp{z(t) B} to-{#m}]

= t-.-d->, =,sG)1, (s.1 8)

where

eG) =,*p{ z ("tBr .4#?O] (s.1 e)

Equation (5.18) represents the distribution function of adoption rime Tin the pres-

ence of marketing-mix variables.

In real-world applications, the individud adoption times are not known, and

instead only the number of individuals who have adopted the product in each timeinterval is known.Therefore, to estimate the diffusion parameters, we need to present

the discrete proportional-hazard model that is appropriate for grouped data(Kalbfeisch and Prentice 1980).

Disnete hoportional-Hazdril Model

In discrete form, equation (5.18) can be writcen as

F(t,, z) = r -.*p[-)"-, s1r;]

or, equivalendy,

r- F(t,, z) = r*pl-t"=, g(")] (s.20)

Using (5.20), we can write the following:

I - F(r,-,, z1 = .*p[-);=',e(r)} (s.2r)

Dividing (5.20) W (5.27),we obtatn

f f i= .*p[-r l ' g1z1 + )i=] r(z)]=.*p[_s(r,)].

Equation (5.22) can also be expressed as

7- F(t,, Z) = r*p{-eQ,)}l- r(t,-,, z)l=

"*p[-g(r, )] - expl- s(t,)\r(t,-,, z).

Equation (5.23) implies

FQ,, z) = t - exp{-g(r,)} +."p[-g(r,)]F(t,-,, z).

Subnacting F(tA, Z) from both sides of (5.24) and simplifying, we get

(5.22)

(5.23)

(s.24)

ll1 IIl. Diffirsion Models

F(t,, z)- r(t,-,, z) =fr- n(t,-,, z)l[1- exp{-sQ,)}].

Therefore,

F(t,, Z) - F(t,-,, Z)--_------- - r -!^Pt-g\ri,fl

1- F(t t - t , Z)

(5.2s)

(5.26)

= r -.*p[-.*ptrt,,lBr.*{#i}]

= r -[.d-,(r, -,,, r{ffifrj}]"*'""'' (s.27)

after substituting (5.19) for g(t) in equation (5.26). Equation (5.27) is the appropri-ate model for grouped data where Z(t) affects the rate of adoption mulriplicatively.

Time to Peak

To derive the expression for t*, the time at which the sales rate reaches its peak,we have to differenriate the probabiliry densiry function J(t,.), set it to zero, andsolve for t*. Differentiating F(r, z) in (5.10) we get;(t, z) as follows:

( , . \f(t, z) = 7(t, z)expl-J L(u, z)au l.

\ 0 , /

Differentiating flt, z) arrd setring the derivative to zero, we find that the time topeak sales t* is the time at which the following equaliry holds:

/ " \ t

t[f)= b+q)t* - p,z,(r)- F,t^(o),

wherc Zo(*) and ZaQ*) denote the price and adverrising variables, respectively, and

81, fudenote the coefiicients associated with these variables. Note that in the Bassmodel, Fr= g, 0z= 0. Substituring this in the above equaliry we get

r"(l)=tr*o'.

or

1 / - \

r * = ' h l q l ,p + q \ p )

which is identical to the expression given in equation

Estimation Proceilure'We

now discuss the esrimation procedure for the proportional-hazard model. Con-sider a popularion of m potential adopters. Let X(r-1) denote the cumulative numberof adopters prior to time /-1.Then the potential number of adopters likely to adopt

5. Modeling the Markenng-Mix Influence in New-Prcduct Diffusion 115

the product in the time intervd (r-1, r) is (m - X(t;_)). Consequently, E[S(r)], theexpected number of adopters in the rime interval (r-,, t) or the expected sales ofthe product in the interval (t,_,, t) is given by

E[s(,,)] = (n - x(t,-,)) n{r{t,_,, t,ll r> t,_}. (s.28)

Using the terminology of survival analysis (Aalen 1976) , m - X(t,-1) can be referredto as the size of the "risk set" at time t;4 and p{T e ft_r, tl l T > 4_1} as the failurerate for an individual. Using (5.27), the sales process s(/) can be expressed

s(r, ) = [, - x(t,-)f1 Q,_,, t,) + u(t,),

where

(5.2e)

r Q _,, t,)= r _ [.*p(_,){##}]"*'"",, (5.30)

assuming the intervals (t,-t, t) ro be equally spaced and of unit length. Note thatl(l) represents an additive random error component similar to the Srinivasan andMason (1986) formulation.'we propose that nonlinear-least-squares (NLS) proce-dure can be used to estirnate the parameters rn,p,q,and B.The details of the NLSprocedure for estimating the basic Bass model are given in Srinivasan and Mason(1986), Mahajan, Mason, and Srinivasan (1986), and Jain and Rao (1990).

5.7 EMPIRICAL COMPARISON OF PROPORTIONAL-HAZARDMODELS VTITH THE BASS MODEL AND THE GBM

5.7.1 Two Proportional-Hazatd Models: PHM-L and pHM-D

The decision variables in the generalized Bass model are measured as percenragechanges of the variables from one rime period to rhe nexr. with the proporrional-hazard model decision variables may be measured as either levels of the variablesor, as wich GBM, as changes or percentage changes.'we have chosen to estimatemodels with both measures in comparing estimates and other statistics with the gen-eralized Bass model. we denote the proportional-hazard model with decision vari-ables measured as levels by PHM-L, and for percentage changes we denote themodel as PHM-D.

5.7.2 Price Advertising Measures

The price variable is measured as the average retail dollar value of unit shipment.Prices are deflated using the durable commodities consumer price index. Adver-tising is measured by advertising expenditures with Simon's (1982) rationale sothat

Advertising = max{0, A4},

and thus if advertising is positive it is measured as the change in advertising forPHM-L and also for the model we esrimared for PHM-D. For GBM the price and

116 III. Diffirsion Models

Table 5.1. Pameter Estimates for Four Models

B2pl

Room airconditionen

Clothes dryers

Colortelevisions

18,321 (1,223)19,503 (1,129)26,802 (r,611)18,965 (929\

16,239 (1,011)16,756 (910)22,708 (2,24r)17,066 (1,090)

3e,s24 (1,708)3e,754 (e96)48,798 (s,r42)3e,072 (8e2)

-1 4 (0.6s)-{.73 (0 07)-1.s (0.s4)

-0.81 (0.s7)-{.74 (0.1s)-0.86 (0 s8)

-4.8 (1 87)-{.28 (0.09)-5.11 0.4)

0 9400 62 (O.26',) 0.9750 02 (0 01) 0ee'l0 05 (o o1s) 0.978

0.9280 66 (O 27) 0.9680.08 (0.02) 0.9840.08 (0.03) 0.e68

0 9850.00 0.9940.00 0 9890 00 0.996

Bos 0.01 (0 002) 0 38 (0.04)cBM o.o1 (0.002) 0.33 (0 03)PHM-L 0 03 (0 01) 0 79 (0.1s)PHM-D o 01 (o 001) 0.37 (0.03)

Bas 0 01 (0.002) 0.33 (0.04)cBM 0 01 (0.002) 0.30 (0.03)PHM-L o.o4 (0.01) o.4s (0 06)PHM-D 0 014 (0 002) 0.29 (0.08)

Bass 0.01 (0.002) 0 64 (0.06)cBM 0004 (0001) 060 (0.30)PHM-L 0.02 (0.00s) 0.e6 (0.19)PHM-D 0.005 (0.001) o 64 (0.03)

advertising variables are measured as percentage changes. For the estimated PHM-

D model the price variable is measured as a percentage change.

5.7.3 Products Studied and Data'We

use yearly data on the following consumer durables for the emPirical estima-

tion: room air conditioners (1949 to 1,961), clothes dryers (1949 to 1961), and color

television (1961 to 1970). These products have been extensively analyzed in the

diffusion literature (Mahajan, Mason, and Srinivasan 1986; Horsky 1990; Bass,

Krishnan, andJain 1.994).The data on prices are the same as used inJain and Rao

(1990). The annual advertising expenditures for these products are obtained form

the Leading Nationd Advertisers (LNA) publication.

5.7.4 Panrneter Estimate for Four Models: BM, GBM, PHM-L' and PHM-D

Table 5.1 shows the parameter esrimates, standard errors, and R3t for all four models

for each of the three products. It is no surprise that the estimates fot p, q, and m

are similar for BM and GBM as that result had been previously reported by Bass,

Krishnan, and Jain (7994), but we were surprised to find that the pararneter

estimates for p, Q, and m for the PHM-D model are also quite close to those of

BM and GBM. On reflection, however, considering that the price-input variable

(percentage change) is the same for PHM-D as for GBM as well as the fact that

both models are based on hazard functions, perhaps this result should not have

been surprising. The price coefficient B1 for PHM-D is also very close to the esti-

mate for GBM, and, again, the price input variable is the same in both cases.

The advertising coefiicient ft is quite different for both room air conditioners and

clothes dryers, but the advertising variable is measured differendy for PHM-D (dif-

ferences if differences are positive) than for GBM (percentage changes if changes

are positive). The fits as measured by R2 are also extremely close for GBM and

PHM-D.

For the PHM-L model the estimates of p, q, and ffi as well as of the price and

advertising coefiicients are very different from the estimates of GBM and PHM-D.

5. Modeling the Marketing-Mix Influence in New-Product Diffirsion 117

The PHM-L model produces higher R2s and generally higher r-rarios than eitherGBM or PHM-D. The differences in the parameter estimares for pHM-L and theother rwo models can be explained by the difference in the input variable for theprice variable (level versus percentage change), and the higher R2 for pHM-L isalso very likely the result of this di{Ierence. This issue of empirical validity associ-ated with the use of levels of decision variables in diffusion models should be care-fully examined in future research. On the other hand, because the decision variablesin the other two models are expressed as differences or percentage d.ifferences,the effecs of regressions with nonstationary variables have been mitigated if noteliminated.

5.7.5 Step-Ahead Forecasts

Table 5.2 shows the one-step-ahead forecasts compared with actual sales and mea-sures of forecaseing error for the four models for each of the three product cate-gories'These forecasts were constructed by estimating parameters with data throughyear t - 1 and using the estimates to predict sales for the following year. whethermeasured by mean absolute deviacion, mean squared error, or mean absolute per-cenage deviation, the PHM-L model outperforms the other models in forecastingfor one-step-ahead. This, however, is simply another way of measuring the fit of themodel.

5.8 SUMMARY AND CONCLUSIONS

5.8.1 GBM and the Seven Desirable properties'We

compared GBM with the seven desinble properries for diffusion models thatinclude decision variables:

' Reiluce to the Bass moilel GBM does reduce to BM when the decision variablesare changing by constant percencages. prices for new technologies tend to fallapproximately exponentially (with constant percentage decline in time). tlndernormally observed patterns for the behavior of decision variables GBM willreduce to BM thus explaining why a Bass-like curve is almost always observed.

' The carry-through property GBM has the carry-through property that character-izes diffusion models with imitation or learning effects.

' Empirical support GBM has produced plausible parameter estimates for severalproduct categories in both published and unpublished studies.

' Casual ualidity Because the decision variables in GBM are expressed as percent-age differences, the effects of nonstationarity have in all likelihood been removed,and thus the input variable has a property that is prescribed by modern rime-series econometricians for inclusion in regression models to avoid spuriousinferences.

' Closed-form expression The fact that GBM has a closed-form expression is usefulin interpreting and understanding the relationships among the variables and indemonstrating how differenc policies can shift the diffusion curve.

@

FI

3o

Table 5.2. One-Step-Ahead Forecast

Room Air Conditionen Clothes Drvers Color Televisiors

Actual Bass GBM PHM-L PHM-D Actual Bass GBM PHM-L PHM-D Actud Bass PHM-L PHM-D

79s9 1,800

1960 1,580

7967 1,500

MAD

MSE

MAPD _

1,258 7,326 1,706

7,346 1,355 1,504

7,207 7,477 1,567

356 247 79

744,790 91,943 6,367

0 27 0.14 0.05

7,288 1,425

7,382 7,260

7,527 7,236

228

700,692

0.13

1,011 1,170 1,388 1,429

1,069 1,015 995 967

986 954 1,154 942

285 267 728 794

90,126 68,791 26,706 57,434

0.27 0.20 0.10 0.16

5,982 4,744

s,962 4,277

4,637 3,976

1,675,744

U . Z r

4,824 -

4,617 5,188 4,754

4,419 5,577 4,402

90s 850 778

t,o64,978 728,276 755,624

0.16 0 .165 0 .13

Note: MAD = Mean absolute dwietion, MSE = Mean sqwd errcr, MAPD = Mem absolute Percent dryiation

5. Modeling the Marketing-Mix Influence in New-Product Diftrsion ll9

Managerially useJul The parameters of GBM may be expressed in ways thatpermit interpretation. The p's and q's may be converted to time of peak sales.Market potential n has intuitive interpreration and can be buttressed by data. Thedecision variable coefficients can be easily related to elasticities. All of this is use{irlin "gtressing without data."Ease oJ implementation The period when model implementation is most impor-tant (and most difticult) is the period before product launch when sales data areavailable for information. Forrunarely, the method that has been applied nruner-ous times for BM of "guessing by analogy" is easily transferred ro the use of GBMbecause the p's and 4's are very similar for the rwo models. (For examples of appli-carions of BM and GBM see the websites indicated in Secrion 5.3.)Thep's and4's have been estimated for a large number of producs and are readily available.Guesses of the coefficients of decision variables are also feasible because of theirclose connection to elasricities. There have been managerial applications of GBMin comparing and evaluating the effects of different pricing strategies in shiftingthe diffusion curve.

5.8.2 GBM and Other Models

It is clear the GBM dominates each of the existing models in the literarure whenevaluated on the basis of desirable properries. In comparison with the pHM models:

' PHM-L produces better fim and is easier to use for opttm)zation purposes becausedecision variables are expressed as levels, but further investigation is needed todetermine the validiry of the PHM-L model.

. PHM-L is a "current-effects" model and thus does not have the carry-throughproperry of a diffusion model.

. PHM-D on the other hand, yields parameter estimates similar to those of GBM,but it too is a "current-effects" model.

Models of the PHM class, then, may not be appropriate for application to diffusionprocesses that involve learning and imitation.

5.8-3 Veaknesses of GBM and Areas for Further Research

Absence of Ituels

The absence of levels in the decision variables makes the application of optimiza-tion methods more difiicult for GBM. There are ways of dealing with the problemsuch as treating the initial levels of variables as "reference" points as suggested byKrishnan, Bass, and Jain (1999), but other methods for handling the problem needto be explored.

Estimation Issues

If the decision variables have litde variation over the period for which parametersare estimated or if the effects of these variables are weak, estimation of GBM mav

120 III. Diffusion Models

not be able to pick up these effects.Also, experience indicates that it is often helpfirl

to have rwo or more decision variables in the set of variables to be included in the

model. Efiicient methods for estimation need to be explored as well as studies of

the appropriate data intervals to be included in the analysis.

Iags of the Decision Vaiables

Studies of the inclusion of lags of the decision variables in the model might be

&uitfirl. Because the model has carry-through properties there are carry-over eflects

of changes in the decision variable even without lags, but additional dynamics in

the model might be useful.

Aililitional Cenerality

Because of the way the model is formulated the ratio of q to p is the same in the

GBM as it is in the Bass model. This properry allows a closed-form solution. Addi-

tional research is needed to find more general specifications of the model that have

such interesting relationships among diffusion parameters.

Caus al Connections Among Variables

The causal connecrions berween the output variable (sales) and input variables, such

as price and adverbising, needs further study. In the existing formulation with a single

equation causality runs in one direction only, and there may be a simultaneous equa-

tions bias in that, for example, sales (or cumulative sales) is a cause of price as well

as the other way around.

In conclusion. the causal connection between decision variables in diffusion

models as between levels versus differences and one-way versus simultaneous rela-

tionships should be carefully studied in future research.

ACKNOVLEDGMENTS

The authors thank Subrata Sen for his valuable comments on this article. In addi-

tion, the article has benefited from the suggesrions of the participants at the New-

Product Diffusion Models Conference at the Wharton School held on September

18-19. 1998.

NOTES

1. For this reason, we decided to include Kalish (1985) model in the grcup of models that incorpo-mte price alone.

2. The authors use a procedure to extmct the fint purchase data ftom the sales.

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