a mathematical approach to (jmr 1969)

8/6/2019 A Mathematical Approach to (Jmr 1969)

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A Mathematical Modeling Approach to Product Line Decisions

Glen L. Urban

Journal of Marketing Research , Vol. 6, No. 1. (Feb., 1969), pp. 40-47.

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GLEN L. URBAN*

Interdependencies among brands in a firm's product line should be considered

when marketing strategies are formulated. This article develops a mathematical

model of the interaction among products for normative st rategy recommenda-

tions. An empirical exampl e apply ing the a priori model to a product line de -cision suggests that the model prespecification is relevant and useful.

A Mathematical Modeling Approach to ProductLine Decisions

Most firms market several somewhat similar productscalled a product line. Policies of product diversificationand new product introduction have been implementedby widening the product line. Depth in the product linehas emerged as firms attempt to meet competition andsatisfy the nee ds of the m arket's subsegments. Althoughthe multiproduct firm has grown in importance, therehas not been a corresponding growth in model buildingand research to help solve the marketing problems offirms with a product line.

Product line decisions are difficult because the prod-ucts in the line are not usually independent. Productscannot be optimized individually and then added to theline to produce optimum product line results. Themarketing mix established for one product may affectthe sales of another product; interdependency is thekey consideration in product line decision making. Thisarticle develops a mathematical product line modelthat analyzes the marketing strategy implications ofproduct interdependency. The model will be developedby a priori reasoning and will be subjected to a prelimi-nary test based on empirical market data.

A M AT H E M AT I C A L M O D E L OF T H E P R O D U C T L I N E D E C IS IO N

Model Development Cr i fer ionIn developing a model of product line effects many

approaches are available ranging from micro-analytic

* Glen L. Urban is assistant professor of management, Mas-sachusetts Institute of Technology. The author would like toacknowledge the Computation Center and Sloan School ofManagement at MIT for the computer resources necessary forthis project. Appreciation is also expressed for data manipulationassistance provided by Roy Dorrance and Richard Ch andler.

simulation and its potential for a highly disaggregatedconsideration of the consumer choice process1 to ahighly aggregated model that might be represented in asimple linear regression model. Between these extremesthere a re several other levels of aggregation such ascomplex single equation and multiple equation models.

In developing the model proposed in this article, twocriteria were established. The first was decision rele-vance, i.e., the model should encompass the major fac-tors and market phenomena affecting the problem offinding the best marketing mix for a product line. Thesecond criterion is reflected in the doc trine of parsimon yand requires that simple models be preferred wheneverpossible. These criteria suggest the development goalthat the product line model should be the simplestmodel that encom passes the relevant market phenom enaand is useful in decision mak ing.

To specify the relevant phenomena the basic con-sumer choice process should be e ~ a m i n e d . ~or ex-ample, consider the purchasing process for four classesof goods related to shaving: electric shavers, safetyrazors, aerosal shave cream, and after shave lotion. Theconsumer choice process originates with the develop-ment of an awareness of these classes of goods andthe particular brands in these classes. ~ w a r i n e s smaybe produced by advertising, personal selling, word ofmouth, o r post-buying experience.

The consumer also forms attitudes about each prod-uct class, the relationship between these classes, andthe brands in the classes. These attitudes may bedirected toward product characteristics or advertisingappeals. For example, consumers will form attitudes

See [ I ] .This brief description is consistent with Nicosia's basic

structure. See [9].

Journal of Marketing Research,Vol. VI (February 1969), 40-7


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- -

PRODUCT LINE DECISIONS

about electric shavers and their advertising appeals.They will also form attitudes about electric versussaftey razors, and electric razors versus after shavelotion. These consumer attitudes become prime factorsfor developing a perceived need for the products. Whenthe level of perceived need is sufficient, a search effort-a shopping trip in this example-is conducted and apurchase decision is made. In the store the consumeris infiuenced by point of purchase communication, andhe integrates this new information with existing atti-tudes. The consumer will choose the product with thegreatest perceived utility per dollar, assuming a satis-factory alternative is present. It can be expected thatthe consumer's willingness to buy a product at a givenprice will depend on his attitude toward the product'scharacteristics and appeals. This implies a marketingmix effect between price and advertising since priceresponse will depend on the level of advertising.

In examining the purchase decision the effects be-tween and within product classes should be con-sidered. Since the proneness to purchase from a groupis a function of the perceived need for that group, thecombined advertising of all brands in the group mayinfluence the attitudes and utility ascribed to the prod-uct class. For example, if all brands of after shavelotion increase their advertising, the attitudes and salesof the group may increase. Th e same product groupphenomena could occur for other product classes, suchas safety razors and electric razors. However, in thesecases the additional sales generated by a group may beobtained from a related product group. For example, ifall safety razor brands increase their advertising, con-sumers may develop attitudes that suggest substitutionof safety razors for electric shavers. Substitution is notthe only possible intergroup effect; some groups may becomplementary. If advertising of safety razors increases,the perceived need for shaving cream and perhaps aftershave lotion may increase.

An intragroup phenomenon, the competition ofbrands within the group, also exists. If consumerschoose a brand because of relative utility per dollar, theselection reflects a combination of attitudes and priceand implies that the relative marketing mix effects ofbrands are significant. For example, the in-store choiceof Schick over Gillette would probably reflect the com-bined advertising, promotion and price advantage ofSchick over Gillette as perceived in the customer's

utility assessment.From this brief and simplified consideration of theprocess, an a priori specification of the most importantphenomena can be derived. For the model three fac-tors were identified as having high behavioral anddecision relevance: (1) aggregate product class market-ing mix effects, (2) product class interdependencies,and (3) intragroup relative competitive brand effects.The goal was to design a simple model of these phe-nomena for aiding product line decisions.

Aggregate Prod uct C lass Marketing M ix EfJects

For a simple model of the combined effects of priceand nonprice marketing activity, an aggregate sales re-sponse function will be postulated. This function shouldinclude three basic marketing variables: advertising,

price, and distribution. Distribution may be measuredby the percent of outlets carrying the product, the num-ber of salesmen selling the product, the middleman'smargin on the product , or a combination of these. Thesimplest form would be a linear equation of these vari-ables, but this has two disadvantages. First, it does notallow marketing mix effects since the sales response toa variable is not affected by other variables. Second, thelinear form would imply a linear response to advertisingwhich can lead to unreasonable decision implicationssince it usually implies extremes of large o r small levelsof advertising. A linear form would also not allow fordecreasing effects on advertising ex pe nd it ~ re .~ he nextmost appropriate form for representing the mix effects

is a linear log function. In unlogged form the formula-tion would be:4

(1) Xi, = a Pyf' A;:' D;p'X jI is industry sales of Product j

a is scale constantP j I is average price level of all brands in product

group jAil is total advertising of all brands in product

group jD jI is total distribution level for all brands in

product group jEPI is industry price elasticity for Product jEA I is industry advertising elasticity for Product jED1 is industry distribution elasticity for Product

j.

This function captures marketing mix effects andallows nonlinearity in response to marketing variables.The nonlinearity is reflected in the parameters EPZ,EAZ, and EDZ. For example, if 0 < EAZ < 1, themarginal sales response to advertising would be con-stantly decreasing as advertising increases. If EAZ = 0,total group advertising does not affect the group's totalsales. In general, EAI and ED1 should be expected tofall between zero and plus one. The price parameter EPZshould be negative because as price increases, salesshould decrease. The parameters EAI, EDZ, and EPZare elasticities and reflect the proportionate changes

in the product group's sales resulting from a propor-tionate change in one variable.Equation 1 reflects marketing mix effects since the

sales response of one variable depends on other vari-ables as established, for example, by differentiating

For empirical consideration of decreasing returns to adver-tising, see [2 , 31.

This is similar in form to the Cobb-Douglas productionfunction used by many authors in marketing, see [4]. For theo-retical uses see [6 ,7] . Fo r empirical support see [8].


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Equat ion 1 with respect to price. The marginal responseto price changes ( d X j r / d P j I )depends on the level ofadvertising and distribution. Differentiating Equation3with respect to the other variables will yield similarresults.

Equat ion 1 is appropriate for this model since itsatisfies the twin criteri a of d ecision relevan ce and sim-plicity; that is, it includes nonlinear marketing mix ef-fects in a simple f0rm.j I n addition, Equation1 can beestimated by linear logarithmic regression.

I'roduct Class Interdependen ciesTwo basic kinds of interdependencies exist, comple-

mentarity and substitutability. Substitutability impliesan introspective con sume r attitude of substituting oneproduct for another under certain conditions; com-plementarity implies that one product will be purchasedalong with another product. Substitution or complemen-tary effects of one product group with other productgroups are essential to a product line model. To includenonlinearity and marketing mix effects in the consid-eration of interdependency, the general form of Equa-tion l seems applicable. I n considering intergroup effectsthe variables would relate to other groups, and a rea-sonable form w ould be:

b is scale constantP I , is average price of product groupMA I M is total a dvertising level for p roduct gr oup

MDIM is total distribution level for product group

MC P j M is cross price elasticity fo r Prod uctsj and MC A j M is cross advertising elasticity for Products

j and MC D j M is cross distribution elasticity for Products

j and M .

T he equation 's parameters-cross elasticities of price,advertising , and distribution-have theoretica l econo miccontent; they measure product interdependency. Thecross price elasticity between Prod ucts1 and 2 is:

xl is sales of Product 1

Pa is price of Product 2.In gene ral if th e cross price ela sticity is positive, the

products are substitutes and if ne gative, the productsare complement^.^ Since price is not the only appro-

' For a discussion of more complex response forms, see [ l l ] .If Equation 1 is not empirically viable, more complex formsshoul d be investigated.

This reasoning is not valid for a product that violates thelaw of demand (e.g., a Giffin good) because as price increasessales increase.

JOURN AL O F MAR KETING RESEARCH, FEBRUARY 1969

priate variable for monitoring interdependencies, pro-motion and distribution cross elasticities should beconsidered. The cross advertising elasticity is:

xl is sales of Product I

Aa is advertising for Product 2.

If this elasticity is positive, the goods demonstrate com-plementarity and if negative, substitutability. The sameimplications are true for distribution. Interdependenciesshould be monitored through several variables since aproduct may be a substitute with respect to one and acomplement with respect to another.

Notation 2 is a good model choice since it allows non-linear interdependency effects and con siders the mark et-ing mix effects between products as it retains a simpleform fo r log-linear regressions.

The group marketing mix and intergroup productinterdependencies can be combined to specify the totalsales of one product class as:

EP Z E.41 ED 1( 3 ) x,, = k P j I A , ~D~~ (nAW~:ZiM~rCiiM~rC:jM),where nx is product sum over M, M # j and k isscale constant.

Intragroup Competitive Brand EfJectsTh e market share a brand gets will reflect that brand's

relative marketing effectiveness compared with that ofother bran ds in the product group. Relative effectivenesswas prespecified as relevant for the theoretical presump-tion that the consumer-buying process entails compar-

ing the perceived utility of competing products. Asimple form for representing relative market shareeffects would be by a firm's advertising expenditurecompared with the total industry's advertising. How-ever, this does not include the marketing mix effects ofeach brand. Since the consumer judges each productby its overall utility, brand choice could be formulatedby representing each brand's mix effects and addingthe relative effectiveness.A form for representing mixeffects of a product was developed in Equation 1. Amarket share expression using this format and includ-ing relative m ix effects is:

pSp1~f4'L)' ;p1( 4 ) Market share for Product =

iD?Yi,j in Firm 1 c p $ j ' i ~ q f1

P ij is price of Productj by Firm i

A ij is advertising level for Productj by Firm i

D ij is distribution level for Productj by Firm i

S P i is competitive price sensitivity for Firm i and Product j


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SAi is competitive advertising sensitivity for Firmi and Product j

SDi is competitive distribution sensitivity for Firmi and Product j.

Equation 3 seems complicated, but it is the simplestequation that captures the prespecified relevant phe-nomena of marketing mix and the relative brand ~ h o i c e . ~The parameters of the equation reflect the market sharesensitivity of each brand's marketing variable^.^ These

are individualized to allow the considerationof product differentiation within each product group.

Given Equation 3, the effects of various strategiesand counterstrategies can be related to the market sharea firm will receive. For example, if Firm 1 is the priceleader for a homogeneous product market (EP1 = EPi),price lowering by that firm would be followed by pricelowering of other firms with no change in market share.However, industry effects described in the previoussection may be produced. Given a strategy and set ofcounterstrategies, this expression could be used toconsider intraproduct-group competitive brand effectsSg

Demand, Cost and Profit Models for a FirmThe submodels developed in the previous three sec-

tions can be combined into one equation to describethe sales of one brand of a product class. The sales ofFirm 1's brand in Product j class is:

E P I E A I C P l M C A j M C D j M( 5 ) xj = kPj r AiI AIM D1.u ]? ? I [ I I . + ~ P ~ ~

where lIx is product sum over M, M'

j, andother not ations as previously defined. Given constantdirect and cross elasticities and sensitivities, this equa-tion represents the demand for one product of a firm'sproduct line. This formulation could be extended toinclude more than three marketing variables by specify-ing the appropriate direct and cross elasticities and sen-sitivities.

A firm's total revenue is the sum of each product'sprice times its sales. To calculate profit, costs must bespecified. The costs may be in the simple form of fixedplus variable costs, but if the products share commonproduction resources, this is unlikely. If there are pro-duction interdependencies, a linear programming model

For a discussion of a more complex form including adver-tising interdependenc y in the consi deration of compe titive ef-fects, see [lo].

The sensitivities appear to be similar to elasticities, but theyare not elasticities. he; do not represent proportionate changesin market shar e as the result of proportionate changes in thevariable. However, they do represent the sensitivity of the mar-ket share to changes in the marketing mix for each firm. Equa-tion 4 is similar to Kotler (see [ 6 ] ) except the sensitivities aresubscripted to allow the possibility of differentiated productsand response.

'F or game theoretic considerations of this form, see [ 6 , 101.

designed to minimize the cost of producing specifiedquantities of the products could be used. Successiveruns of this model or cost records could provide thedata for estimating an interdependent cost function suchas:

( 6 ) TVC,=

A VC, (x;) n[ H (xH) C C i M ,TVCj is total variable cost of producing the firm's

brand of Product jAVC; is average variable cost function for the

firm's brand of Pioduct j, if producedindependently of other products

x j is quantity of brand of Product j producedXM is quantity of brand of Product M pro-

duced, (M # j )CCjMis cross cost elasticity of firm's brands

Products j and M, ( M # j ) .

Subtracting the variable cost and fixed production,advertising and distribution costs from the total revenue

will yield total profit.Combining the cost and demand equations in thecalculation of profit results in a simple model that in-cludes the phenomena that were considered a priori tohave high decision relevance.

Output of ModelAssuming the firm's problem in the short run is to

maximize the total profit subject to existing technical,managerial, financial, and production constraints, theoutput of the model should be the best marketingstrategy for each brand in the firm's product line. Thisrequires the optimization of the model's profit functionwhich is difficult since the model is not amenable to

mathematical programming or other analytical tech-niques. However, it may be solved by an iterativesearch routine.1

The feasibility of gaining the described output fromthe model rests on the ability to generate meaningfulinput and on the presence of a practical solution method.The direct and cross elasticities could be estimated ona subjective basis that reflects the decision maker's bestjudgment. This approach might be justified since thedecision must be made and if the model is not used, asimpler and perhaps less accurate decision procedurewould be used. However, subjective inputs should beused only after all empirical information relating to theproblem has been considered.

The model developed could be especially useful tofirms using brand managers since it can be a basis ofallocating resources to each brand in the product line.The brand manager concept artificially imposes in-dependence between specific products in the line bydelegating products to competing brand managers. Butif resources are allocated on the basis of product inter-dependencies at the top marketing management level,the motivational advantages of the brand manager con-

See 1141 and the application section of this article.


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Table 1 PRODU CT GR OU P ELASTICITIES A N D CRO SS ELASTICITIES

SEPARATED BY PRODUCER A N D COMP ETITOR Sa

Product I Product 2 Product 3Elasticity Class ( j = I ) ( j = 2 ) ( j = 3 )

Price elasticity of USProduct j ( EPI ) them

Facing elasticity of USProduct j (EFI) them

Cross price elastic- USity between Prod-ucts j & 1 (CPj1)

Cross facing elas- USticity betweenProducts j & 1(CFj1

Cross price elastic- USity between Prod- themucts j & 2 (CPj2)

Cross facing elas- U Sticity between themProducts j & 2(CFj2)

Cross urice elastic- us - .29 -1.55bity between Prod- them .207c .136'ucts j & 3 (CPj3)

Cross facing elastic- us . I 1 - .42dity between Prod- them - .301d .09ucts j & 3 (CFj3)

R2 .52' .61b .51b

a Significance is based on one tail test for direct elasticitiesand two tail tests on cross elasticities.

b Significant a t .O1 level. Significant a t .05 level.

d Significant a t .10 level. There was no competition in Product 1 ' s market.

NOTE: "us" is our brand in product group j; 'them" is allother brands in product group j

CPiN > 0 + substitutes CFiN > 0 =+ complementsCPiN < 0 ==+ complements CFiN < 0 =+ substitutes.

cept and the use of product line resources to maximizetotal line profits can be compatible.

A N E M P I R I C A L A P P L IC AT IO N

To test the descriptive adequacy and usefulness ofthe proposed model in a real product line problem, 100grocery store audits of a three product line of related,frequently purchased consumer goods were used.llThese product line data were used to estimate theparameters of the product line model, and an on-linecomputer search program derived the optimum market-ing mix for each product in the producer's line.

The audited product line contained three classes ofproducts that served the same food need but haddifferent product features. Product group 1 was a new

"Th e author thanks Samuel G. Barton and I. J. Abrams ofthe Market Research Corporation of America for use of thesedata. The product line is not identified to protect the interests ofthe producer and MRCA.

JOURN AL O F MARK ETING RESEARCH, FEBRUARY1969

product and only the firm to be examined offered abrand in this class. The competitors in product groups2 and 3 were aggregated into one competitor in eachmarket. The aggregation resulted in a firm with a threeproduct line and brands which faced no competitors inproduct group 1, one competitor in product group 2,and one competitor in product group 3 .

In each product class the brand, shelf price, numberof facings, deals, and special displays were recorded inthe audits. Over ninety-five percent of the data wererecorded with cents-off or bonus-size deals. Hence,dealing was not considered a separate variable since itcould be reflected in the price per unit. Special displaysoccurred so infrequently (less than one percent of thedata) that they were not considered in the analysis. Theaudits did not monitor national or local advertisingin the test area. It was assumed that none of the brandsreceived a disproportionate amount of local advertisingat any of the audited stores, and advertising was notconsidered as a variable in the testing.

Model Parameter Estimation

Product Class Marketing Effects and Produ ct Inter-dependency. The product group elasticities and crosselasticities for each product class were obtained bylinear logarithm regressions of Equation 3 where distri-bution is represented by the number of total facings ofthe product group and where advertising is omitted. Allthe direct price elasticities obtained from this regressionwere negative, and all facings' elasticities were positiveas expected and significant at least at the .05 level. Thesignificant cross elasticities of price and facings forthe products indicated that the three product groupswere basically complementary. This complementaritydid not agree with a priori feelings and past studies thatindicated these kinds of products could be competingfor the same buyers in the general product class.12 Thisfinding implies that the prespecification of the model'sproduct interdependency section was not satisfactory.

To explore alternative forms for updating the model,the interdependencies between our brand in a productgroup and other product groups were postulated to bedifferent from other brands in the group. To evaluatethis updated model structure, the industry price andfacing data for each product group were subdividedinto "us" and "them" classes. Us was our brand priceand facings and them was the average price and facingfor all other competitors. The elasticities and crosselasticities for the firm us and the competitors them ineach product are indicated in Table 1.

An examination of cross elasticities for Product 1indicates that Product 1 was complementary to bothour brand and other competitive brands of Product 2.Cross elasticities for Product 3 indicate that Product 1was complementary to our brand of Product 3 but showssubstitution effects with other brands in Product 3.

'%e [5].


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Product group 2 was complementary to our brand inthe Product 3 group but was a substitute for competi-tive brands in Product 3's market. Although the initialregression showed Products 1 and 2 to be complemen-tary to Product 3, the new regressions (Table 1) indicatethat complementarity was with our brands of Product3, and that Products 1 and 2 were competitive with otherbrands of Product 3. This is the interdependency pat-tern a marketing manager would desire.

Product group 3 showed no significant interdepend-encies with Product 1 but displayed some interestingrelationships with product group 2. CP32,, = 1.67indicates our brand of Product 2 was a substitute forProduct 3. But CP23,, = -1.55 indicates Product 2'ssales were complementary to our brand of Product 3.The asymmetry in the interdependency is significant atthe five percent level and the elasticities are large, i.e.,greater than one.13

That product group 3 felt substitution effects fromour brand of Product 2 is understandable. As the priceof our brand of Product 2 increased, our brand buyerssubstituted Product 3. That product group 2 felt com-plementary effects from our brand of Product 3 is moredifficult to explain. CP23Us = - 1.55 so that a 10 per-cent reduction in the price of our brand of Product 3caused a 15.5 percent increase in the sales of Product2. This could have occurred if buyers of our brand ofProduct 3 had also bought Product 2 when theyperceived low prices for this class of goods. If theperception is based on the price of Product 3, loweringour brand's prices of Product 3 could have caused aperception of low prices for these buyers. They mighthave bought more of Product 2 with little sensitivity toits price. The income effect caused by lower prices inBrand 3 also indicates complementarity. This increasein real income might have led to additional purchases ofProduct 2.

Except for the asymmetry, the new regressions in-dicate that the products within the firm's product lineare complementary but are substitutes for products inother firms' product lines. The split product groupregressions explained 30 percent more of the variancein the data than the earlier regressions so updatingthe model for split interactions is advisable.

Zntragroup Competitive Brand Sensitivities. Theestimation of the competitive sensitivities to be used indescribing market share effects (see Equation 4 withfacings representing D and omitting A) was carried out

l 3 This asymmetry was difficult to accept, so stepwise multipleregressions were run for our brand sales in markets for products2 and 3 . The asymmetry again app eared at the five percent sig-nificance level. Analysis of the correlation matrix showed littlemulticolinearity between the variables. Since the data was takenat one point in time, autocorrelation in the data was not sus-pected. If there had been multicolinearity in the autocorrelation,this might have caused the asymmetry. The elasticity of storesize averaged .O8 for the relevant brands. This appeared reason-able and supported the assumption that the stores represented asimple sample with respect to market responses to the variables.

by a computer program to minimize the total variationbetween the actual market shares and the market sharespredicted by Equation 4 given a set of sensitivities, ob-served prices, and observed facings. The estimationwas executed on the MIT computation center com-patible time-sharing system. The interactive ability ofan on-line system was used in a conversational programthat asks the researcher or manager to supply initialestimates of the price and facing sensitivities for thefirm and its competitors. These initial sensitivities areincremented by an amount prescribed by the manager.The number of incremental steps to be taken for eachsensitivity is also an on-line input supplied by themanager. All combinations of the initial and in-cremented sensitivities are evaluated; the set of sen-sitivities producing the minimum total variation for theaudit data points is recorded.

Then the manager is asked to supply a new incrementand number of steps for the search. The next evalua-tion uses the best past estimates as initial values. Bycontinuing this process the manager can guide thesearch until it has reached a prescribed level of accuracy.This procedure does not guarantee that the optimal fithas been achieved; rather, it is a heuristic procedurebased on the manager's best judgment and the computa-tional power of a high speed computer.14

The minimum variation estimates for the two com-petitive product markets are shown in the tabulation.The estimates explain 24 percent of the variance ofthe market shares in Product 2 and 54 percent ofProduct 3's market shares. This empirical success addsconfidence to the a priori specification of the competi-tive phenomena.15

Product 2 Product 3Producer

Price Facing Price Facing

Our firm - . 2 7 1.31 - . 8 5 5 1.13Competitor .00 .7 5 -1.24 1.20

The competitive sensitivities in Product 2 indicatethat the competitor has little or no effect on marketshare by his change in prices while changes in our pricehave a negative sensitivity. The facings sensitivities ofProduct 2 indicated our firm's facings were 1.5 times aseffective as the competitor's in producing market sharechanges. The Product 3 competitive estimates indicatedour firm's price and facing variables were less effective

in changing market share than the competitor's.The estimation of the competitive sensitivities com-pletes the parameter estimation for the model's demandequation (see Equation 5). The remaining demand in-put is the strategy competitors were expected to use;

l4 For details of the search technique, see [13].The prespecified structure was also reinforced since simple

log-linear regressions of the two firms' prices and facings againstsales produced low RZ values and unreasonable coefficients inthe two products. This suggests that the proposed form is a rele-vant level of detail.


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--

46 JOURN AL O F MARK ETING RESEARCH, FEBRUARY 1969

Table 2

--

SEARCH RESULTS

Variable Reference No in terac t ionProduct group

in terac t ionSpl it product group

in terac t ion

Price of Product 1" L C" 2

" "" 3

Facings of Product 1" "

" "

" 2" 3

Total Profit 382,600 594,500 585,180 908,700

Profit for Product 1" "

" 2" "

" 3

Sales for Product 1" "

" "

" 2" 3

Market share for Product 1" "

" 2" "

" 3

100%29. 64y074. 7970

100yc28. 76707o.O3Y0

100%31.01%72.03%

100%31.17080. 37y0

a These values are at the upper or lower limit of the data used for estimation.

they were expected to be nonadaptive with respect tothe number facings and price for Product 2 and befollowers with respect to Product 3. The final input tothe model is the cost function (see Equation 6). In thistest application the costs for each product were in-dependent, and the m arginal costs were considered con-stant over the meaningful range of production.

Optimization of Product Line ProfitThe maximization of the model's product line profit

was carried out by an on-line computer search routine.16The trial and error routine began by evaluating a refer-ence set of prices and facings for each of the three prod-ucts and then examining a range of values on eachside of the reference values in discrete steps. The rangeand steps were specified from the remote computerconsole. Given the increments and ranges all combina-tions of the trial values were run, and the best totalproduct line profit based on Equations1 to 6 with theestimated param eters was recorded. After th e first seriesof trials have been repo rted, the ma nager c an respecifythe step intervals and ranges. By continuing this processhe can achieve the desired level of accuracy. One con-strain t was placed on the search-the total facings hadto be less than eight for o ur pro duct line. Th is constraintforces the search rou tine to allocate the shelf spacebetween the produ cts in the line.

The search program results are shown in Table 2.The first column gives the price per unit, facingsweighted by package size, profit, sales, and market

'"See [13].

share for our brand of each product a t the referencelevel. The reference results are based on the averageprice and facings for each group as observed in theaudits.

The next question was whether the interdependenciesbetween products should be considered at the aggregateproduct group level or at the split-product group level.The empirical estimation indicated that the modelshould be updated for the split, butif the decision wereinsensitive to the add ed complexity of the sp lit, it mightbe omitted. A sensitivity analysis determ ined if theoptimum marketing strategy was different under varyinginteraction assumptions.

To establish a reference base in evaluating decisionsensitivity, the most profitable program with no interac-tions was found. It was considerably different from thereference program. The price of Product 1 was de-creased but the prices for Products 2 an d 3 were in-creased. The facing's allocation was also altered. Prod-ucts 1 an d 2 received fewer facings but Product 3received more. T he result of an improv emen t in profitof over 50 percent and an increase in the sales of eachprodu ct implies that o n the basis of the mod el theexisting strategy was nono ptimal.

The best program with aggregate product groupinteraction led to the same price structure as the nointeraction case. The facing allocation was changed,however (see Table 2). The facings were concentratedin Product 2 because of the complementarity betweenthe facings of Product 2 and sales of Products 1 an d 3in the regressions.


9/9


The next phase of the sensitivity analysis was todetermine the maximum profit for the most empiricallyvalid case-split-product-group interactions. In thiscase a different pricing strategy should be used. Theprice of Product 3 is lower than even the reference priceprimarily because of the complementarity of the hriceof our brand of Product 3 and the sales of Product 2(see Table 1). The product line profits in this analysiswere 50 percent greater than in-the no interaction orgroup interaction cases. Additional profits occurred inProduct 2, but the profits in Product 3 decreased almost70 percent because of the asymmetric product inter-dependencies between our brands of Products 2 and 3 .The decision output is sensitive to the splitting of theinteractions; since this is the most empirically viablemodel structure, the model should be updated to reflectthe differences in the product interdependencies be-tween our brands and other brands in other productgroups.

The output of the optimization and sensitivity testingcan be summarized in the recommendations that theprice of Product 1 be lowered, the price of Product 2be raised, and the facings allocation be more con-centrated on Product 2. The vrice of Product 3 shouldbe lowered if the asymmetricLinterdependency betweenProducts 2 and 3 is real as it appears to be. Additionalstudy is necessary to ascertain the underlying behaviorthat generates the asymmetric condition, but updatingthe model for split interactions is recommended sinceapplication of the model identified significant productinterdependencies and recommended changes in themarketing mix of the products in the line so that theinte rde~endencies could be exvloited for additionalprofit. he updated model posses'ses reasonable descrip-tive adequacy and decision relevance.

S U M M A R Y A N D E XT E NS IO N S

This article presented an a priori product line modelfor finding the best marketing mix for each product ina line. The model includes aggregate product groupmarketing mix, product interdependency, and com-petitive brand effects. The initial testing of the modelsuggests that the basic structure is appropriate and themodel deserves additional consideration, testing, anddevelopment.

The model analysis could be extended in several

ways. First, the model was a static one period model;the analysis could be extended to include carry-overeffects and the problem of multiperiod marketing mixdetermination. Second, the model test application ex-amined only three related products; it would be useful

to expand the test product definitions to include otherclasses of products or to narrow the product definitionto consider package sizes in each product. A hierarchyof interdependencies exists, and a sequential applica-tion of the model to increasingly more specific productdefinitions would be appropriate. Third, the multipleregression and iterative search routines used to esti-mate the model's parameters were applied to a limiteddata base of 100 store audits. Although reasonabledescriptive adequacy was found, it would be usefulto have an information system that builds a data bankon the products' performance to obtain more accurateinput estimates and to test more complex responseforms. And consideration of the effects of adding ordropping a product from the line and the effects of thisaction on the marketing mix would extend the analysis.17

REFERENCES

1. Arnold E. Amstutz, Computer Simulat ion of Compet i t ive

Market Response, Cambridge, Mass.: MIT Press, 1967.2. B. Benjamin and J. Maitland, "Operational Research a ndAdvertising: Some Experiments in the Use of Analogies,"Operational Research Quarterly, 9 (September 1958), 207-17.

3. B. Benjamin, W. P. Jolly, and J. Maitland, "OperationalResearch and Advertising: Theories of Response," Opera-tional Research Quarterly, 11 (December 1960), 205-18.

4. Paul H . Douglas, Theory o f W ages, New York: The Mac-millan Company, 1934.

5. Peter L. Henderson, James F. Hind, and Sidney E. Brown,"Sales Effects of Two Campaign Themes," Journal of Ad-vertising Research , 1 (December 1961), 2-1 1.

6. Philip Kotler, "Competitive Strategies for New ProductMarketing Over the Life Cycle," Management Science , 12(December 1965), 104-19.

7. Alfred A. Kuehn and Doyle L. Weiss, "Market AnalysisTraining Exercise," Behavioral Science, 10 (January 1965),5 1-62.

8. William F. Massy and Ronald E. Frank, "Short Term Priceand Dealing Effects in Selected Market Segments," Journalo f Marketing Research, 3 (May 1965), 171-85.

9. Francesco M. Nicosia, Consumer Decision Processes, En-glewood Cliffs, N. J.: Prentice-Hall, Inc., 1966.

10. Melvin Shakun, "A Dynamic Model for Competitive Mar-keting in Coupled Markets," Management Science , 12 (Au-gust 1966), 525-30.

11. Glen L. Urban, "Sprinter: A Tool for New Product Deci-sion Makers," Industrial Management Review, 8 (Spring1967), 43-55.

12.-

"A New Product Analysis and Decision Model,"Management Science, 14 (April 1968), 490-517.

13.-

"An On-Line Technique for Estimating a nd Ana-lyzing Complex Models," in Reed Moyer, ed., ChangingMarketing Systems,Winter Conference Proceedings, Ameri-can Marketing Association, 1967, 322-7.

14. Douglass Wilde and Charles S. Beightler, Foundations ofOpt imizat ion, Englewood Cliffs, N. J.: Prentice-Hal l, Inc.,1967.

a mathematical approach to (jmr 1969)

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