DefensiveMarketingStrategieshauser/Hauser Articles 5.3.12/Hauser...HauserandShugan: Defensive...

Vol. 27, No. 1, January–February 2008, pp. 88–110issn 0732-2399 �eissn 1526-548X �08 �2701 �0088

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doi 10.1287/mksc.1070.0334©2008 INFORMS

Defensive Marketing StrategiesJohn R. Hauser

Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected]

Steven M. ShuganGraduate School of Business, University of Chicago, Chicago, Illinois 60637, [email protected]

This paper analyzes how a firm should adjust its marketing expenditures and its price to defend its positionin an existing market from attack by a competitive new product. Our focus is to provide usable managerial

recommendations on the strategy of response. In particular we show that if products can be represented by theirposition in a multiattribute space, consumers are heterogeneous and maximize utility, and awareness advertisingand distribution can be summarized by response functions, then for the profit maximizing firm:• it is optimal to decrease awareness advertising,• it is optimal to decrease the distribution budget unless the new product can be kept out of the market,• a price increase may be optimal, and• even under the optimal strategy, profits decrease as a result of the competitive new product.Furthermore, if the consumer tastes are uniformly distributed across the spectrum• a price decrease increases defensive profits,• it is optimal (at the margin) to improve product quality in the direction of the defending product’s strength

and• it is optimal (at the margin) to reposition by advertising in the same direction.In addition we provide practical procedures to estimate (1) the distribution of consumer tastes and (2) the

position of the new product in perceptual space from sales data and knowledge of the percent of consumerswho are aware of the new product and find it available. Competitive diagnostics, such as the angle of attack,are introduced to help the defending manager.This article was originally published in Marketing Science, Volume 2, Issue 4, pages 319–360, in 1983.

Key words : competition; pricing; product entry; defensive marketingHistory : This paper was received June 1981 and was has been with the authors for 3 revisions.

1. PerspectiveMany new products are launched each year. Whilesome of these products pioneer new markets, most arenew brands launched into a market with an existingcompetitive structure. For every new brand launchedthere are often four to five firms (or brands) whomust actively defend their share of the market. Thispaper investigates how the firms marketing the exist-ing brands should react to the launch of a competitivenew brand. We will refer to this topic as defensivemarketing strategy.The offensive launch of a new brand has been

well studied. See, for example, reviews by Pessemier(1982), Shocker and Srinivasan (1979), Urban andHauser (l980), and Wind (1982). Good strategies existfor brand positioning, the selection of brand featuresand price, the design of initial advertising strategies,and the selection of couponing, dealing, and samplingcampaigns. But the analysis to support these deci-sions is often expensive and time consuming. (A typ-ical positioning study takes 6 months and $50,000 to$100,000 (Urban and Hauser 1980, Chapter 3).) Whilesuch defensive expenditures are justified for some

markets and firms, most defensive actions demand amore rapid response with lower expenditures on mar-ket research. Indeed, defending firms often routinelymust determine if any response is even necessary. Lit-tle or no practical analytic models exist to supportsuch defensive reactions.The competitive equilibrium of markets has

also been well studied. See, for example, reviewsby Lancaster (1980), Lane (1980), Scherer (1980),Schmalensee (1981, 1982), and Stigler (1964). This lit-erature provides useful insights on how markets reachequilibrium, how market mechanisms lead to entrybarriers, and how product differentiation affects mar-ket equilibrium. This body of research attempts todescribe how markets behave and assess the socialwelfare implications of such behavior. This economicsresearch does not attempt to prescribe how an existingfirm—faced with a competitive new entrant—shouldreadjust its price, advertising expenditures, channelexpenditures and product quality to optimally defendits profit.We are fortunate to have a breadth of related con-

cepts in marketing and economics to draw upon.However, we find that the special nature of defensive

88

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Hauser and Shugan: Defensive Marketing StrategiesMarketing Science 27(1), pp. 88–110, © 2008 INFORMS 89

strategy requires the development of some additionalnew theory. Ultimately, researchers will develop aportfolio of methods to address the full complexity ofdefensive strategies. We choose a more selective focusby addressing an important subclass of problems thatare at the heart of many defensive strategies. We limitour research to the general strategic and qualitativeissues. Future research can then address other issues.

Problem Definition

Prior Information. We assume that the defendingfirm knows the positions of existing products (in amultiattribute space) prior to the launch of the com-petitive new product. We further assume that thedefending firm either is (1) willing to commissiona positioning study to determine the positioning ofthe new product or (2) has or can obtain data onsales, awareness, and availability for both the newand existing products. In the second case, we providea procedure to estimate the new product’s positionfrom such data.

Defensive Actions. Our focus is on price, adver-tising expenditures (broken down by spending onawareness and on repositioning), distribution expen-ditures, and product improvement expenditures. Wedo not address detailed allocation decisions such asthe advertising media decisions or timing decisions;we assume that once the level of an advertising ordistribution budget is set that standard normativemarketing techniques are used to allocate resourceswithin this budget. See Aaker and Myers (1975),Blattberg et al. (1981), Stern and El-Ansary (1982), andKotler (1980). This paper does not explicitly addressthe counter-launch of a “me-too” or “second-but-better” new product as a defensive strategy. Our anal-yses enable the firm to evaluate non-counter-launchstrategies. Once such strategies are identified they canbe compared to the potential profit stream from acounter-launch strategy.

Direction of First Response. By its very nature,defensive strategy often calls for a rapid response.If firms had full and perfect market research infor-mation they might be able to optimize strategy,but often such information is not available. How-ever, we can make statements about the directionof response (e.g., increase or decrease distributionspending) which are surprisingly robust over a widerange of circumstances. As Little (1966) found, suchdirectional responses can be almost as valuable asoptimal responses, especially with noisy data. We feelit is important to the development of a defensive mar-keting theory to know such directional forces andfocus our attention on such results.

Equilibrium Issues. Economists such as Lancaster(1971, 1980) and Lane (1980) have identified equilib-ria to which multiattributed markets evolve. But to

do so they have had to simplify greatly the complex-ity of such markets. For example, Lane (1980, p. 239)assumes that (1) market size is fixed, (2) consumershave uniformly distributed tastes, (3) consumers haveperfect information and all products are availableto them, (4) all firms face identical cost structures,(5) once firms choose their respective positions theycannot change them, (6) firms choose their prices asif all other firms will not change in response, and(7) various technical conditions. Even so, he mustresort to simulation for specific results beyond a proofof the existence of Nash equilibria.We wish to analyze a more complex problem. We

allow the firm to set advertising and distribution aswell as pricing, and we allow the firm to reposi-tion. Consumers may vary in awareness of productsand products may vary in availability. Consumers’tastes need not be uniform and firms’ cost structuresmay vary.For such marketing richness we pay a price; we

limit our analyses to how the target firm should reactto a new product. We assume all products, exceptthe new product and the product of the target firm,remain passive. We feel that this is an important steptoward a multiproduct equilibrium. For example, oneresearch direction might be to explore the equilib-rium that results if all firms iteratively followed ourtheories.There is some evidence that special cases of our

analyses, e.g., price response for uniformly dis-tributed tastes, are consistent with special cases ofequilibrium analyses found in the economic litera-ture. (Our theorem and Lane’s simulation both predictprice decreases (Lane 1980, Tables 1 and 2).) Thus,as a working hypothesis, subject to test by futureresearchers, we posit the directional results of our the-orems for a two-product equilibrium will not changefor a multiproduct equilibrium. By recognizing thislimitation we hope to encourage research into fullmarket equilibria with models that are rich in market-ing phenomena.

Deductive Analyses. This paper is deductive froma simple model of consumer response. We attemptto make our assumptions clear. Insofar as our modelaccurately abstracts reality, the results are true. Butwe caution the manager to examine his situation andcompare it to our model before using our results.We expect future research to examine and overcomethe limitations of our analysis both empirically andtheoretically.

Overview of the PaperSection 2 presents the consumer model and intro-duces notation and definitions that prove useful inour analyses. Section 3 derives a set of theorems thatindicate normatively how a defending firm should

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Hauser and Shugan: Defensive Marketing Strategies90 Marketing Science 27(1), pp. 88–110, © 2008 INFORMS

adjust its price, advertising budget, distribution bud-get, and product positioning in response to a newproduct. Section 4 indicates how to estimate the dis-tribution of consumer tastes, §5 indicates how to esti-mate the new product’s position, §6 suggests somecompetitive diagnostics, §7 discusses the issue of val-idating deductive models, and §8 discusses limita-tions of our analyses and suggests future research. Allproofs are placed in the appendices. In addition, weprovide a glossary of technical definitions.We begin with the consumer model.

2. Consumer ModelOur managerial interest is at the level of marketresponse. Thus, our primary concern in modelingconsumers is to predict how many consumers willpurchase our product and our competitors’ prod-ucts, under alternative defensive marketing strate-gies. However, to promote the evolution of defensiveanalysis we build up market response as resultingfrom the response of individual consumers to mar-ket forces. Although defensive models may ultimatelyincorporate complex micro-models such as thosereviewed by Bettman (1979), and decision-makingcost (e.g., Shugan 1980), we begin with several sim-plifying assumptions such as utility maximization. Inthis way, we can deal directly with aggregation acrossconsumers.A further requirement of a defensive model is that

it be sensitive to how a new product differentiallyimpacts each product. Finally, we require a modelthat incorporates important components of consumerbehavior, but is not too complex to be inestimablefrom available data. The following assumptions statethe tradeoffs we have made.

AssumptionsWe assume: (1) Existing brands can be representedby their position in a multiattribute space such asthe one shown in Figure 1. The position of the brandrepresents the amount of attributes the consumer canobtain by spending one dollar on that brand. (2) Eachconsumer chooses the brand that maximizes his util-ity, and (3) The utility of the product category is aconcave function of a summary measure linear in theproduct attributes (or some linear transformation ofthe product attributes). This assumption allows usto represent the brand choice decision with a lin-ear utility function; however, we do not require theactual utility function to be linear. (4) The effect of“awareness” and “availability” can be modeled byadvertising and distribution response functions asdefined below. This fourth assumption allows us tomodel consumer response when consumers do nothave full information about all products, when theyare interested in only a subset of the products, or

Figure 1 Hypothetical Perceptual Map of Four Liquid Dish WashingDetergents

PalmoliveIvory

Joy

Ajax

Efficacy/$

Mild

ness

/$when products are not available to all consumers.This assumption is particularly important when mar-kets are in turmoil as they are after a new productlaunch.Assumption 1, representation by a product posi-

tion, is commonly accepted in marketing. See Greenand Wind (1975), Johnson (1971), Kotler (1980),Pessemier (1982), and Urban (1974). Scaling by pricecomes from a budget constraint applied to the bun-dle of consumer purchases and from separabilityconditions that allow us to model behavior in onemarket (say liquid dish washing detergents) as inde-pendent of another market (say deodorants). (That is,the brand of deodorant used does not influencethe brand of liquid dishwashing detergent selected.)See Blackorby et al. (1975) and Strotz (1957). Thisimplicit assumption is discussed in Hauser andSimmie (1981), Hauser and Urban (1982), Keon (1980),Lancaster (1971, Chapter 8), Ratchford (1975, 1979),and Srinivasan (1982).Dividing by price implicitly assumes that the

dimensions of the multiattributed space are ratioscaled. This is clearly true for most physical attributes,say miles per gallon, but may require special mea-surement techniques for more qualitative attributessuch as “efficacy.” We require only that such scalesexist, not that they are easy to measure. Practicalmeasures remain an important empirical question;however some progress has been made. For exam-ple, Hauser and Simmie (1981) present methods toobtain appropriate ratio scales for perceptual dimen-sions and Hauser and Shugan (1980) derive statisticaltests for the necessary ratio-scaling properties.

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Assumption 2, utility maximization, is reasonablefor a market level model. At the individual levelstochasticity (Bass 1974, Massy et al. 1970), situationalvariation, and measurement error make it nearlyimpossible to predict behavior with certainty. At themarket level, utility maximization by a group of het-erogeneous consumers appears to describe and pre-dict sufficiently well to identify dominant marketforces. For example, see Givon and Horsky (1978),Green and Rao (1972), Green and Srinivasan (1978),Jain et al. (1979), Pekelman and Sen (1979), Shockerand Srinivasan (1979), Wind and Spitz (1976), andWittink and Montgomery (1979). Utility maximizationis particularly reasonable when we define utility on aperceptual space because perceptions are influencedby other product characteristics and psychosocial cuessuch as advertising image, and hence already incorpo-rate some of the effects due to information processing.Finally, any individual level stochasticity that resultsfrom randomness in consumers’ tastes can be readilyincorporated in our model by including such random-ness in the market distribution of consumers’ tastes.Such stochasticity when independent across individ-uals in no way affects our results.Assumption 3, linearity, is a simplification to obtain

analytic results. With linearity we sacrifice generalitybut obtain a manageable model of market behavior. Inmany cases the linearity assumption can be viewed asan approximation to the tangent of each consumer’sutility function in the neighborhood of his chosenproduct. Furthermore, Green and Devita (1975) showthat linear preference functions are good approxima-tions, and Einhorn (1970) and Einhorn et al. (1979)present evidence based on human information pro-cessing that justifies a linear approximation. In ourtheorems we treat the utility function as linear in theproduct attributes; however it is a simple matter tomodify these theorems for any utility function that islinear in the “taste” parameters that vary across thepopulation. For example, we can handle the standardideal point models since utility is linear in the squareddistance, along each attribute, from the ideal point.However, for the pricing theorem we must be carefulin modeling the price scaling involved in any trans-formation. In general, our analysis can incorporatemuch of the same class of nonlinear utility functionsdiscussed in McFadden (1973).Assumption 4, response function analysis, is a stan-

dard approximation in marketing that is based onmuch empirical experience (e.g., see Blattberg andGolanty 1978, Hauser and Urban 1982, Kotler 1980,Little 1979, Stern and El-Ansary 1982, Urban 1974).A response function relates firm expenditures to out-comes. For example, if kd dollars are spent optimallyon distribution (transportation, inventory, channelincentives, sales force, etc.), then we assume that sales

Figure 2 An Example Distribution Response Function ExhibitingDecreasing Marginal Returns over the Relevant Range

Dis

trib

utio

n in

dex,

D

1.0

Channel expenditure, kd

are proportional to D�kd� where D�·� is the responsefunction. Such functions always exist. Our assump-tion concerns their properties, i.e., that beyond a cer-tain point the functions are marginally decreasing(�2D/�k2d < 0) as in Figure 2 and proportionality holdsover the range of analysis.Assumption 4 is not as restrictive as it might appear.

Our theorems require only that the firm is operatingon the concave portion of the response function priorto the entry of the new product. This is reasonable,since, even if the response function were S-shaped,most optimization analyses would suggest operat-ing on either the concave portion of the responsefunction or at zero expenditure. Assumption 4 alsoimplies multiplicative interactions among marketingexpenditures but rules out more complex interactions.Based on the evidence summarized by Little (1979) webelieve Assumption 4 is a valid beginning.For ease of exposition we call the response analysis

effect of advertising expenditures, “awareness,” andthe response analysis effect of distribution expendi-tures, “availability.” We caution the reader that adver-tising and distribution affect, but are not limited to,consumer awareness of a product and the availabil-ity of the product in the store. Response functions,and our theorems, refer to the more general effects.The labels are for convenience of exposition. Finally,in addition to response analysis, we consider adver-tising expenditures directed at specific attributes.

Sensitivity to AssumptionsThe set of four assumptions about consumer responseis sufficient to prove a set of normative theoremswhich indicate qualitatively how a defending firmshould adjust its marketing strategy. However, assummarized in Table 1, each assumption only affectssome of the theorems. The others are independent ofthat particular assumption.We have selected theorems which we feel are rea-

sonably robust with respect to our consumer model.We have already indicated that the results hold forstochasticity of tastes and some nonlinearities in util-ity. It is possible that the theorems may hold for otherrelaxations of the assumptions. We leave such relax-ations for future research.

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Table 1 Assumptions of Theorems

Assumptions

Price Utility Linear ResponseTheorems scaling maximization tastes functions

1. Profit2. Price (uniform tastes) x x x

3. Price (dominance) x x x

4. Price (monotonic tastes) x x x

5. Price (segmented tastes) x x x

6. Price decoupling x

7. Distribution x

8. Pre-emption x

9. Repositioning x x

(product improvement)10. Advertising x

11. Repositioning (advertising) x x

12. Price (irregular markets) x x x

13. Price (general) x x x

14. Evoked sets

Note. “x” indicates our proof of the theorem uses of this assumption.

We make a final assumption for ease of expositionin the pricing and repositioning theorems. We limitour analyses to two product attributes. Two attributesmake possible pictorial representation and providevaluable intuition into the marketing forces whichdrive the theorems. As Schmalensee (1982) indicates,there is some evidence that results derived for one ortwo price-scaled attributes hold for three attributes.However, he also suggests that little is known aboutextensions to an arbitrary number of attributes.

NotationBy assumption, products are represented by theirattributes. Let x1j be the amount of attribute 1 obtain-able from one unit of brand j . Define x2j to be theamount of attribute 2 obtainable from one unit ofbrand j . (Note: �> xij ≥ 0 for i = 1 2 and xij is ratio-scaled.) Let pj be the price per unit of brand j . Letuj be the utility that a randomly selected consumerplaces on purchasing brand j . Note that uj is a ran-dom variable due to consumer heterogeneity. If everyconsumer is aware of each brand and finds it avail-able, all brands will be in everyone’s choice set, oth-erwise choice sets will vary. Let A be the set of allbrands, let Al, a subset of A, be the set of brandscalled choice set l, and let Sl be the probability thata randomly chosen consumer will select from choiceset l for l = 1 2 � � � L. See Silk and Urban (1978) forempirical evidence on the variation of choice sets.

Mathematical DerivationWe first compute the probability, mj , that a randomlychosen consumer purchases brand j . In marketingterms, mj is the market share of product j if all con-sumers purchase the same number of products per

time period.1 Applying assumption 2, utility maxi-mization, we obtain Equation (1) by definition.

mj = Prob�uj > ui for all i �= j�

where Prob[·] is a probability function. (1)

In addition, define mj � l = Prob�uj > ui for all i �= jin Al�. In marketing terminology, mj � l is the mar-ket share of product j among those customers whoevoke Al. Now, Equation (1) and assumption 3, lin-earity, imply

mj � l = Prob[� �w1x1j + �w2x2j �/pj

> � �w1x1i + �w2x2i�/pi for all i ∈Al

] (2)

where i ∈Al denotes all products contained in choiceset l. Here �w1 and �w2 are relative “weights” a ran-domly selected consumer places on attributes 1 and 2,respectively. Suitable algebraic manipulation of Equa-tion (2) yields

mj � l = Prob[�x1j /pj − x1i/pi� > � �w2/ �w1�

· �x2i/pi − x2j /pj � for all i ∈Al

]� (3)

Moreover, Equation (3) is equivalent to

mj � l = Prob[ �w2/ �w1 < rij for all i ∈A′

l and

�w2/ �w1 > rij for all i ∈A′′l

] (4)

where

rij = �x1j /pj − x1i/pi�/�x2i/pi − x2j /pj �

A′l =

{i � i ∈Al and x2i/pi > x2j /pj

}

A′′l =

{i � i ∈Al and x2i/pi < x2j /pj

}�

Note that Equation (4) illustrates that �w2/ �w1 is a suf-ficient parameter for computing choice probabilitiesand that rij = rji. Now consider Figure 3(a). Pictorially,“Joy” will be chosen if the angle of the indifferencecurve defined by �w2 w1� lies between the angle of theline connecting “Ivory” and “Joy” and the line con-necting “Joy” and “Ajax!” Note that consumers willonly choose those brands, called efficient brands, thatare not dominated on both dimensions by anotherproduct in the evoked set.2 Hence, we consider onlyefficient brands in evoked sets. (See the glossary for averbal summary of our definitions.)

1 We show later that mj is still the market share if consumers varyin the quantity that they purchase. This will become clear when weintroduce consumer heterogeneity with a taste distribution, f ��,that can incorporate multiple purchases by counting consumers inproportion to their purchase frequency. Our empirical procedureautomatically adjusts for this phenomenon. See also Footnote 3.2 Our definition of efficiency assumes utility is monotonic in x1jand x2j . Ideal points can be handled by redefining the scale to bemonotonic in distance from the ideal point.

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Figure 3 Geometric Illustration of the Relationship in Equation (3) fora Three Brand Market

Mildness/$

Efficacy/$

Ivory (X13/P3, X23/P3)

X23

Joy (X12/P2, X22/P2)

Ajax (X11/P1, X21/P1)

α23

α12

α23

90º

90º

90º

P3

X22

P2–

X22

P2

X21

P1–

X12

P2

X13

P3

–X11

P1

X12

P2

–

Mild

ness

/$

Efficacy/$

Ivoryα

90º

~

Indifferencecurve

Joy

Ajax

tan–1 w2w1

Mild

ness

/$

Efficacy/$

Ivory

90º

90º

Joypreferred

Ivorypreferred

Ajaypreferred

Joy

Ajax

(a)

(b)

(c)

Notes. Where �23 = tan−1�X12/P2 −X13/P3�/�X23/P3 −X22/P2�

�12 = tan−1�X11/P1 −X12/P2�/�X22/P2 −X21/P1�.

We simplify Equation (4) and make it easier tovisualize by introducing new notation. Let � =tan−1� �w2/ �w1� and let �ij = tan−1 rij . The angle, �, rep-resents a measure of consumer preference that variesbetween 0 and 90 . As indicated in Figure 3(b), � is

the angle the utility indifference curve makes withthe vertical axis. Figures 3(a), 3(b), and 3(c) graphi-cally illustrate the mathematical formulation. As canbe seen from Figure 3(a), three brands of liquiddishwashing detergent are efficiently positioned. Theangles �23 and �12 are tan−1 r23 and tan−1 r12 respec-tively. Moreover, these angles represent the relativepositions of Ivory, Joy, and Ajax. Figure 3(b) showshow consumer preferences can be represented. Con-sumers with w1 and w2 as shown in the figure (w1 andw2 are different for different consumers) would preferIvory because Ivory touches those consumers’ highestindifference curve. Note that whenever tan−1�w2/w1�is greater than �23 then Ivory is preferred. Figure 3(c)illustrates the regions where each product is pre-ferred. If tan−1�w2/w1� exactly equalled �23, then thoseconsumers would be indifferent between Joy andIvory. For continuous distributions, these ties occuronly on a set of measure zero.Moreover, we introduce the convention of index

numbering brands such that x2j /pj > x2i/pi if j > i.This will assume that index numbers (and �ij ’s)increase counter-clockwise for efficient brands in Al

for any l. For boundary conditions let �01 = 0 and�j� = 90 .Finally, we introduce the definition of adjacency. In

words, a lower (upper) adjacent brand is simply thenext efficient brand in Al as we proceed clockwise(counter-clockwise) around the boundary. Mathemat-ically, a brand, j−, is lower adjacent to brand j if(1) j−< j , (2) j−≥ k for all k < j where both k and jare contained in A′

l and (3) both brands are efficient.Define the upper adjacent brand, j+, similarly. Notethat j− and j+ depend on the choice set Al. Withthese definitions mj � l is simply computed by

mj � l = Prob[�jj− ≤ � < �jj+ for �jj−

�jj+ defined by Al

]� (5)

We now introduce consumer heterogeneity in theform of a variation in tastes across the consumerpopulation. Since each consumer’s utility function isdefined by the angle, �, we introduce a distribution,3

f ��, on the angles. (f �� may depend on l.) Onehypothetical distribution is shown in Figure 4. Asshown, small angles, �→ 0, imply that consumers aremore concerned with attribute 1, e.g., efficacy, thanwith attribute 2, e.g., mildness; large angles, �→ 90 ,imply a greater emphasis on attribute 2. This methodof introducing heterogeneity is similar to models pro-posed by Blin and Dodson (1980), Pekelman and

3 To assure that mj � l is the market share, we assume that consumersare counted in proportion to purchase frequency. This will be trueautomatically when f �� is calibrated on sales data.

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Figure 4 Hypothetical Distribution of Tastes with Respect to Efficacyand Mildness

90º

f (α)

Efficiency Midness

0 αjj – αjj +

α

Note. The shaded region represents those consumers who will choose“Joy.”

Sen (1975), and Srinivasan (1975). Equation (5) nowbecomes

mj � l =∫ �jj+

�jj−f ��d�� (6)

Thus, mj � l is the shaded area in Figure 4. Finally, if thechoice sets vary, the total market share, mj , of brandj is given by

mj =L∑l=1

mj � lSl where l= 1 2 � � � L� (7)

An interesting property of Equation (7) is that a brandcan be inefficient on A, but have nonzero total mar-ket share if it is efficient on some subset with nonzeroselection probability, Sl. This property, which is con-sistent with empirically observed consumer behavior,distinguishes our marketing model from the modelsassumed by Lancaster (1980) and Lane (1980).As an illustration for

A= {Ajax �j = 1� Ivory �j = 3� Palmolive �j = 4�

}

suppose that the positions of the three brandsare given by �x1j /pj � = �5 1� �2 4�6�, and (1 5) forj = 1 3 4, respectively. We first compute rij , findingr13 = 3/3�6 = 0�83 and r34 = 1/�0�4� = 2�5. Finding theangle whose tangent is rij gives us �01 = 0 , �13 = 40 ,�34 = 68 , and �4� = 90 . For this example “Ajax” islower adjacent to “Ivory” and “Palmolive” is upperadjacent to “Ivory.” See Figure 5.If tastes are uniformly distributed over �, then

f �� = �1/90 � and mj � l = ��jj+ − �jj−�/90 . For ourexample, if everyone evokes choice set A, marketshares are 0.44 for “Ajax,” 0.32 for “Ivory,” and 0.24for “Palmolive.”Suppose a new brand, “Attack,” is introduced at

(3 4) and suppose it is in everyone’s choice set. Since“Attack” is positioned between “Ajax” and “Ivory”we number it j = 2. See Figure 5. We compute r12 =

Figure 5 Hypothetical Market After the Entry of the New Product“Attack”

Palmolive (0.24)

Ivory (0.10)

Ajax (0.38)

Efficacy/$

Mild

ness

/$

Attack (0.28)

1

1

2

2

3

3

4

4

5

5

Note. Numbers in parentheses indicate new market shares.

0�67, r23 = 1�67, �12 = 34 , and �23 = 59 . The new mar-ket shares are 0.38 for “Ajax,” 0.28 for “Attack,” 0.10for “Ivory” and 0.24 for “Palmolive.” Thus “Attack”draws its share dominantly from “Ivory,” somewhatfrom “Ajax,” and not at all from “Palmolive.” “Ivory”will definitely need a good defensive strategy!Clearly the threat posed by the new brand depends

upon how it is positioned with respect to the defend-ing firm’s brand. We formalize this notion in §6.Section 4 derives estimates for f �� and §5 derivesestimates of the new product’s position.We turn now to the analysis of general strategies

for defensive pricing, advertising, distribution incen-tives, and product improvement which depend onlyon general properties of f ��, the new product’s posi-tion, and response functions. The theorems in thefollowing section are based on the consumer modelderived above. For analytic simplicity, unless other-wise noted, we base our derivations on the full prod-uct set, A. Extension to variation in evoked sets isdiscussed in §5.

3. Strategic ImplicationsWe sequentially examine pricing, distribution, prod-uct improvement, and advertising.

Pricing StrategyFor simplicity we assume that advertising and distri-bution strategies are fixed. We relax this assumptionin the following sections.Based on the consumer model, Equation (3) shows

profit for the defending brand, j , before entry of the

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new brand. We simplify the exposition by suppress-ing fixed costs which, in no way, affect our analysis.4

�b�p�= �p− c�Nb

∫ �jj+

�jj−f ��d�� (8)

Here Nb is the number of consumers before any com-petitive entry, c is our per unit costs,5 p is our price(we are brand j) and �b�p� is our before entry profitgiven price p. Note �jj+ and �jj− are functions of price.The new brand can enter the market in a number

of ways. (1) If it is inefficient, it does not affect ourprofits. (2) If it exactly matches one of our competi-tors, our profit is unchanged. (3) If the new brandis not adjacent in the new market, our profits areunchanged. Conditions (1)–(3) are based on the recog-nition that none of our consumers is lost to the newbrand.6 Defensive strategy only becomes importantunder condition (4) j+ ≥ n ≥ j− where n denotes theindex number of the new brand and, again, j denotesthe defending brand. Since the problem is symmetricwith respect to j+ ≥ n > j and j > n≥ j−, we analyzethe former case where the new product is efficient andupper adjacent. The latter case follows from transpos-ing the axes.In case (4), our profits (i.e., the defending brand)

after the launch of the competitive new brand, givenour price p, are given by

�a�p�= �p− c�Na

∫ �jn

�jj−f ��d�� (9)

Here, Na is the market volume after the competitiveentry.If we lose no customers, the new brand is no threat,

thus we are only concerned with the case of f �� notidentically zero in the range �jn to �jj+. Call this casefor an adjacent product, competitive entry.We begin with the intuitive result that we cannot

be better off after the new product attacks our marketthan we could have been by acting optimally beforethe attack. We state the result formally because it isused to derive later results. (The proofs of Theorem 1,and all subsequent theorems are in the appendix.)

4 Fixed costs, as usual in economic analysis, do not come into playuntil one considers barriers to entry and exit. See Schmalensee(1981, 1982), Scherer (1980), and Stigler (1964). Note also that cate-gory demand is modeled by Nb . Thus, our explicit model is primar-ily a market share model. Category demand impacts profit throughNb and (in Equation (9)) through Na. We allow the new product toaffect category demand but we do not derive the effect explicitlyfrom our consumer model.5 Note the implicit assumption of constant production costs to scale.Most of our theorems can be relaxed for nonconstant costs, butthe statements of the theorems become more mathematical and lessintuitive.6 In a full equilibrium analysis, conditions (1)–(3) are only approx-imate since nonadjacent brands can affect our profits by causingadjacent brands to reposition or change their price.

Theorem 1. If total market size does not increase, opti-mal defensive profits must decrease if the new product iscompetitive, regardless of the defensive price.

Theorem 1 illustrates the limits of defensive strat-egy. Unless the market is increasing at a rate that israpid enough to offset the loss due to competitiveentry, the competitive entry will lower our profit evenwith the best defensive price. Theorem 1 is true evenif the market is growing when growth is from con-sumers exclusively buying the new brand. Moreover,even if general growth is occurring, provided that ourgrowth is not due to the new brand, it is easy to showthat the competitor decreases potential profits.A key assumption in Theorem 1 is that the defend-

ing firm is acting optimally before the new brandenters. However, there exist cases where a newbrand awakens a “sleepy” market, the defending firmresponds with an active defensive (heavy advertis-ing and lower price), and finds itself with more salesand greater profits. A well-known case of this phe-nomenon is the reaction by Tylenol to a competitivethreat by Datril. Before Datril entered the marketactively, Tylenol was a little known, highly pricedalternative to aspirin. Tylenol is now the marketleader in analgesics. Theorem 1 implies that Tylenolcould have done at least as well had it moved opti-mally before Datril entered.Theorem 1 states that no pricing strategy can regain

the before-entry profit. Nonetheless, optimal defen-sive pricing is important. Defensive profits with anoptimal defensive price may still be significantlygreater than defensive profits with the wrong defen-sive price.We begin our study of price response by exam-

ining markets which are not highly segmented. Inparticular, we investigate the market forces presentwhen preferences are uniformly distributed. This lineof reasoning is not unlike that of Lancaster (1980)who assumes a different form of uniformity to studycompetition and product variety.7 As Lancaster (1980,p. 283) states: Uniformity provides “a background ofregularity against which variations in other featuresof the system can be studied.”For our first pricing theorem we introduce an

empirically testable market condition which we callregularity. Let "j be the angle of a ray connecting prod-uct j to the origin, "j = tan−1�x2j /x1j �. Then a market issaid to be regular if product j’s angle, "j , lies betweenthe angles, �jj− and �jj+, which define the limits ofthe tastes of product j’s consumers. Regularity is areasonable condition which we expect many markets

7 Lancaster’s uniformity assumption is more complex than anassumption of uniform tastes. However, to derive his results healso assumes a form of uniform tastes. See also analyses by Lane(1980), who assumes uniformly distributed tastes.

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to satisfy, however, as intuitive as regularity seems itis not guaranteed. For example, in the market withproducts positioned at (2�1 0), (2 2), and (1 10), wefind that "2 > �23, i.e., tan "2 = 1 and tan�23 = 1/8.We present the following theorem for regular marketsand then discuss its extension to irregular markets.

Theorem 2 (Defensive Pricing). If consumer tastesare uniformly distributed and the market is regular, thendefensive profits can be increased by decreasing price.

Theorem 2 provides us with useful insight ondefensive pricing; insight that with experience canevolve into very usable “rules of thumb.” For exam-ple, when some managers find the market shareof their brand decreasing they quickly increase theprice in the hopes of increasing lost revenue. Histori-cally, transit managers fall into this class (witness the1980–1981 fare increases in both Boston and Chicago).Other managers believe an aggressive price decreaseis necessary to regain lost share. This strategy is com-mon in package goods. Theorem 2 shows that if con-sumer tastes are uniform, the price decrease is likelyto be optimal in terms of profit. The result itself maynot surprise the aggressive package goods managers,but the general applicability of the result to regularmarkets is interesting. Note that Theorem 2 is trueeven if the market size is greater after the new entrant.We now examine what happens if we relax the

assumptions of regularity and uniformly distributedtastes.

Irregular Markets. Regularity is sufficient to for-mally prove Theorem 2, however regularity is notnecessary. As the appendix indicates we use only�jj+ ≥ "j to prove that profits are decreasing in price.Even these conditions are not necessary. For exam-ple, the following theorem is true in both regular andirregular markets. For further results see Theorem 12in the appendix.

Theorem 3 (Defensive Pricing). If the new brand’sangle with our brand (�jn) plus the upper adjacent brand’sangle with our brand (�jj+) exceeds 90 and if the distri-bution of consumer tastes is uniform, then the defensiveprofits will be increased by decreasing price.

Because we have not been able to develop a generalproof of Theorem 2 for irregular markets, we simu-lated 1,700,000 randomly generated irregular markets.In all cases, profits were found to be decreasing inprice. Since the functions in question are reasonablywell behaved, such a simulation on a compact set sug-gests that the theorem also holds for irregular mar-kets. At the very least, violation is extremely rare.

Nonuniform Taste Distributions. We, as the eco-nomic literature that has preceded us, have obtainedformal results for markets in which tastes are uni-formly distributed. Such markets are expected to bereasonably representative of unsegmented marketsand thus are a useful beginning to understand priceresponse. We now investigate whether Theorem 2 isrobust with respect to nonuniform distributions. Ourfirst result shows that a uniform distribution maybe sufficient but not necessary. (We state the resultfor regular markets, but it is easy to prove that ifTheorem 2 is true in irregular markets then so isTheorem 4.)

Theorem 4. For an upper adjacent attack in a regularmarket, if f ′��≤ 0, then defensive profits can be increasedby decreasing price from the before-entry optimal. For alower adjacent attack, the result holds if f ′��≥ 0.

Theorem 4 suggests that Theorem 2 may hold forconsumer tastes, f ��, represented by common prob-ability distributions that are monotonically decreas-ing (increasing) over their ranges. Such distributionsinclude the triangle distribution and some cases of theexponential and Beta distributions. See Drake (1967)for examples of such distributions. However, all mar-kets which satisfy Theorem 4 are unsegmented bydefinition.Consider the following example of highly seg-

mented market. Suppose f �� is discrete takingon values only at f �14 � = 1/27, f �18�5 � = 15/27,f �33�7 � = 10/27, and f �84 � = 1/27. Suppose we arepositioned at �x1j /p x2j /p�= �11/p 50/p� and our twoadjacent competitors are positioned at (1 60) and(21 20). If c = $�9/unit and Nb = 270 000 our profitat various price levels is given in Table 2. By inspec-tion, the optimal price is the highest price, $1.00, thatcaptures both segments 2 and 3. Suppose now thatthe new brand enters at (16 40) and that Na =Nb. Byinspection of Table 2, the optimal price is the high-est price, $1.03, that captures segment 3. Since theoptimal price after entry exceeds that before entry,we have generated an example where it is optimal toincrease price.We have shown this result to many marketing man-

agers of both consumer and industrial goods whohave found it surprising. Thus, we state the result asa theorem for emphasis.

Theorem 5. There exist distributions of consumertastes for which the optimal defensive price requires a priceincrease.

With careful inspection, the example used to illus-trate Theorem 5 is quite intuitive. The key idea in theexample is that there are two dominant segments inthe market. Thirty-seven percent (10/27) have a slight

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Table 2 Example to Demonstrate a Case of the Optimal DefensivePrice Requiring a Price Increase

Before entry After entry

Price Volume Profit Volume Profit

Under0.90 <0 <00.90 260,000 0 260,000 00.91 250,000 2,500 250,000 2,5000.92 250,000 5,000 250,000 5,0000.93 250,000 7,500 250,000 7,5000.94 250,000 10,000 100,000 4,0000.95 250,000 12,500 100,000 5,0000.96 250,000 15,000 100,000 6,0000.97 250,000 17,500 100,000 7,0000.98 250,000 20,000 100,000 8,0000.99 250,000 22,500 100,000 9,0001.000 250,0000 25,0000 100,000 10,0001.01 100,000 11,000 100,000 11,0001.02 100,000 12,000 100,000 12,0001.03∗ 100,000 13,000 100,000∗ 13,000∗

1.04 100,000 14,000 0 01.05 100,000 15,000 0 01.06 100,000 16,000 0 01.07 100,000 17,000 0 01.08 100,000 18,000 0 01.09 0 0 0 0

Over1.09 0 0 0 0

0 Indicates optimal price before competitive entry.∗ Indicates optimal price after entry.

preference for attribute 1, w2/w1 = tan�= 2/3; 56 per-cent (15/27) have a stronger preference for attribute 1,w2/w1 = 1/3. Before the competitive entry, we wereselling to both markets. Segment 3 clearly preferredour brand. However, we were forced to lower ourprice to compete for segment 2 for whom our productis less preferred. It was profitable to do so becauseof its large size. When the new brand entered, tar-geted directly at segment 2, we could no longer com-pete profitably for segment 2. However, since we arestill well positioned for segment 3 and they are will-ing to pay more for our brand, we raise our price toits new optimal profit level. This situation may oftenoccur near the end of a product’s life cycle when dif-ferentiated products have attracted all but the mostsatisfied customers. In the social welfare sense, con-sumers must pay a price for variety. (In Table 2 theprice increase is only 3¢, but it is possible to generateexamples where the price increase is very large.)We can, of course, generate necessary and sufficient

conditions which identify when a price decrease isoptimal (see Theorem 13 in the appendix for neces-sary conditions), but such conditions are excessivelyalgebraic and do not appear to provide any usefulintuition for the manager. On the other hand, we findTheorems 2, 3, 4, and 5 to be quite usable because theyprovide results for interpretable taste distributions.

Theorems 2, 3, and 4 suggest that a price decreaseis best in unsegmented markets. Theorem 5 suggeststhat in a highly segmented market, a price increasemay be optimal.

Summary. Together Theorems 2, 3, 4, and 5 pro-vide the manager with guidelines to consider in mak-ing defensive pricing decisions. Our proven resultsare specific, but we can generalize with the followingpropositions which are stated in the form of manage-rial guidelines:• If the market is highly segmented and the com-

petitor is attacking one of your segments, examine thesituation carefully for a price increase may be optimal.• If the market is not segmented and consumer

tastes are near uniform, a price decrease usually leadsto optimal profits. Finally, it is possible to show that ifa competitive brand leaves the market and consumertastes are near uniform, a price increase usually leadsto optimal profits.

Distribution StrategyOne strategy to combat a new entrant is for thedefending firm to exercise its power in the chan-nel of distribution. Channel power is a complexphenomenon, but to understand defensive distribu-tion strategy we can begin by summarizing the phe-nomenon with response functions as described in §2.In response analysis we assume that sales are pro-

portional to a distribution index, D. The index is inturn a function of the effort in dollars, kd, that weallocate to distribution assuming we spend kd dollarsoptimally in the channel.Under the conditions of response analysis, profit

after competitive entry is given by

�a�p kd�= �p− c�NaMa�p�D− kd where

Ma�p�=∫ �nj

�jj−f ��d�� (10)

That is, Ma�p� is the potential market share of prod-uct j as a function of price. We begin by examiningthe interrelationship between defensive pricing anddistribution strategies.

Theorem 6 (Price Decoupling). The optimal defen-sive pricing strategy is independent of the optimal defensivedistribution strategy, but the optimal distribution strategydepends on the optimal defensive price.

Theorem 7 (Defensive Distribution Strategy). Ifmarket size does not increase, the optimal defensive distri-bution strategy is to decrease spending on distribution.

Theorem 8 (Pre-emptive Distribution). If marketsize does not increase, optimal defensive profits mustdecrease if a new brand enters competitively, regardlessof the defensive price and distribution strategy. However,

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profit might be maintained (less prevention costs), undercertain conditions, if the new brand is prevented fromentering the market.

Together Theorems 6, 7, and 8 provide the defen-sive manager with valuable insights on how much ofhis budget to allocate to distribution. If he can preventcompetitive entry by dominating the channel and it islegal to do so, this may be his best strategy. However,in cases when it is illegal to prevent entry, the optimalprofit strategy is to decrease spending on distribution.Although decreasing distribution expenditures

appears counterintuitive to an active defense, it doesmake good economic sense. Marketing managersshould look at profits as well as sales. The market’spotential profitability and rate of return decreasesas the new product brings additional competition.Because the market is less profitable, the mathematicstells us that we should spend less of our resources inthis market. Instead, we should divert our resourcesto more profitable ventures. If demand shrinks fastenough, the reward for investment in the matureproduct decreases until new product developmentbecomes a more attractive alternative.Just the opposite would be true if a competitor

dropped out of the market; we would fight to geta share of its customers. Witness the active competi-tion between the Chicago Tribune and Chicago SunTimes when the Chicago Daily News folded and theactive competition by the Boston newspapers whenthe Record-American folded.Theorem 6 tells us that price strategy is indepen-

dent of distribution strategy but not vice versa. Thusthe results of Theorems 2, 3, 4, and 5 still hold. If pref-erences are uniformly distributed the defensive man-ager should decrease price and decrease distributionspending. In addition, if our response analysis is rea-sonable, these results imply that a defensive managershould first set price before distribution strategy. Thisimplication suggests that the pricing decision shouldbe made at a higher level in the organization than thedistribution level decision. In some sense, the pricingdecision is more important because the other deci-sions depend on it.Theorem 8 notes that there is no magic in distribu-

tion. The market is more competitive and there areless potential profits. Thus the manager should not bemisled by sales, but should focus on profits.Finally, we note that the tactics to implement a dis-

tribution strategy include details not addressed byresponse analysis. Nonetheless, although the detailsmay vary, the basic strategy, as summarized by totalspending, is to decrease the distribution budget whendefending against a new competitive brand. More-over, if the brand is sold in several markets, the resultis true for each market.

Product ImprovementWe now investigate whether or not the defendingmanager should invest money to improve his physi-cal product. We consider the case where the managerhas the option to improve consumer perceptions ofhis brand by modifying his brand to improve its posi-tion. That is, we assume that x1j (or x2j ) is an increas-ing function of c. Let cb be the optimal productioncost before competitive entry and let ca be the optimalproduction cost after entry. We further assume that x1j(or x2j ) is marginally decreasing (concave) in c.Profit is then given by

�a�p kd c�= �p− c�NaMa�p c�D�kd�− kd (11)

where x1j , and henceMa�p c�, is now an explicit func-tion of the per unit production cost c.To select the optimal physical product we examine

the first order condition implied by Equation (11). Itis given by

�Ma�p∗ c∗�/�c=Ma�p

∗ c∗�/�p∗ − c∗�� (12)

While Equation (12) appears to be relatively simple,production cost affects Ma�p c� in a complex way.We could repeat analyses similar to Theorems 2, 3, 4,and 5 to obtain results for production cost, but suchanalogous theorems do not provide the same usableintuition. Instead, we gain insight into the solution toEquation (12) by examining the marginal forces affect-ing the optimal production cost, cb, at which the firmwas operating prior to the competitive entry. Whilethis may not guarantee an optimum, it does suggesta directionality for improving profit.Define a conditional defensive profit function,

�a�c �0�, given by

�a�c �0�= �p0 − c�NbMa�p0 c�D�k0d�− k0d (13)

where p0 and k0d are the optimal price and distributionexpenditures prior to the competitive entry.The conditional defensive profit function can be

interpreted as the profit we obtain after competitiveentry if we are only allowed to improve (degrade)our physical product. We now examine how �a�c � 0�changes as we change c. (Our result is stated foruniformly distributed tastes. We leave it to futureresearch to extend these results as Theorems 3, 4,and 5 extended Theorem 2.)

Theorem 9 (Defensive Product Improvement).Suppose that consumer tastes are uniformly distributedand the competitive new brand attacks along attribute 2(i.e., an upper adjacent attack), then at the margin, if prod-uct improvement is possible,(a) Profits are increasing in improvements in attribute 1

(i.e., away from the attack);

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(b) Profits are increasing in improvements in attribute 2(i.e., toward the attack) if

�x1j /p− x1n/pn��1+ r−2nj �≤ �x1j /p− x1j+/p+��1+ r−2jj+�

where pn and p+ are the prices of the new and upper adja-cent brands, respectively.

Symmetric conditions hold for an attack alongattribute 1.

The results of Theorem 9 can be stated more simply.Part (a) says reposition by improving product qual-ity to your strength. I.e., the competitive new brandis attacking you on one flank, attribute 2, but youare still positioned better along your strength underthis attack, attribute 1. Theorem 9 says that if you canmove along attribute 1, do so.Part (b) says that improving product quality in the

direction of your competitor’s strength, attribute 2, isnot automatic. The testable condition given in The-orem 9 must be checked. (Section 5 of this paperprovides a practical procedure to estimate the newbrand’s position which is the data necessary tocheck the condition of Theorem 9.) Movement alongattribute 2 is more complex because there are twoconflicting effects. First, sales potential may be low inthe direction of the new brand’s competitive strengthand, hence, we seek to move away from our com-petitor. However, a second effect is also present. Ifthe competitor can be easily dominated on attribute 2,then we will seek to regain lost sales by attacking ourcompetitor’s strength. The first effect will dominatewhen our competitor is very strong, i.e., rjj+ � rjn. Inthis case, Theorem 9(b) directs us not to counterat-tack on the competitor’s strength. The second effectwill dominate when the competitor attacks stronglyon x1j as well as on x2j , i.e., when x1j /p � x1n/pn. Inthis case, Theorem 9(b) directs us to counterattack onour competitor’s strength in addition to moving toour strength. The opposite results will hold for thedeletion of an adjacent product.

Advertising StrategyThere are two components to advertising strategy.One goal of advertising is to entice consumers intointroducing our brand into their evoked sets, or ina dynamic framework, to keep consumers from for-getting our brand. We refer to this form of adver-tising, as awareness advertising, and handle it withresponse analysis much like we handled distribution.(Sales are proportional to the number of consumersaware of our product.) In some categories, anothergoal of advertising is to reposition our brand.8 For

8 While repositioning without changing the physical product is com-mon in image sensitive product categories such as softdrinks, per-fumes, and some beers, it may not be possible in many categories.

example, if we are under attack by a new liquid dish-washing detergent stressing mildness, we may wishto advertise to increase the perceived mildness (orefficacy) of our brand. If this form of advertising ispossible, we invest advertising dollars to increase x1jor x2j . Since repositioning advertising affects �nj and�jj−, it is more complex than awareness advertising.In the following analysis we consider expenditures onawareness advertising, ka, and repositioning, kr , sep-arately. In other words, our results enable the defen-sive manager to decide both on the overall advertisingbudget and on how to allocate it among awarenessand repositioning.We begin by considering awareness advertising. Let

A, a function of ka, be the probability a consumer isaware of our brand.9 Brands must continue spendingin order to maintain awareness levels. Equation (14)expresses our profit after the new brand enters:

�a�p kd ka�= �p− c�NaMa�p c�AD− ka− kd� (14)

Ma�p c� was defined in Equations (10) and (11).

Theorem 10 (Defensive Strategy for AwarenessAdvertising). If market size does not increase, the opti-mal defensive strategy for advertising includes decreasingthe budget for awareness advertising.

Theorem 10 is not surprising because awarenessadvertising is modeled with a response function thathas properties similar to the response function fordistribution. Thus all of our comments (and an anal-ogy to the preemptive distribution theorem) apply toawareness advertising.Now consider repositioning advertising. Follow-

ing arguments similar to those for physical productimprovement, we handle repositioning by a form ofresponse analysis where x1j (or x2j ) is an increasingbut marginally decreasing (concave) function of theinvestment, kr , in repositioning. Profit is given by

�a�p kd ka kr �

= �p− c�NaMa�p kr c�AD− kd − ka− kr � (15)

Following our previous analysis of product im-provement, we define a conditional profit function torepresent the profit attainable if we are only allowedto adjust kr . Based on such a definition (analogous toEquation (13)) we obtain the following theorem.

In which case, we combine repositioning with physical productimprovement. As is clear in Theorems 9 and 11, the results are inthe same direction.9 Our use of the letter “A” in the following discussion should notbe confused with our use of the letter “A” to denote choice sets.

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Theorem 11 (Repositioning by Advertising). Sup-pose consumer tastes are uniformly distributed and thenew competitive brand attacks along attribute 2 (i.e., anupper adjacent attack), then at the margin if repositioningis possible,(a) profits are increasing in repositioning spending

along attribute 1 (i.e., away from the attack);(b) profits are increasing in repositioning spending

along attribute 2 (i.e., toward the attack) if and only if

�x1j /p− x1n/pn��1+ r−2nj � < �x1j /p− x1j+/p+��1+ r−2jj+�

where pn and p+ are the prices of the new and upper adja-cent brands, respectively.

Symmetric conditions hold for an attack alongattribute 1.

Theorem 11 is a conditional result at the old opti-mal. But coupled with Theorem 10 it provides veryusable insight into defensive advertising strategy.Theorem 10 says decrease awareness advertising. Theo-rem 11 says that at least in the case of uniform con-sumer tastes, marginal gains are possible if we increaserepositioning advertising along our strength. Togetherthese results suggest that the defensive manager (fac-ing uniformly distributed tastes) should reallocateadvertising from an awareness function to a reposi-tioning function. For example, he might want to selectcopy that stresses “more effective in cleaning hard-to-clean dishes” over copy that simply gets attention forhis brand name.Finally, we caution the reader that in many cate-

gories repositioning by advertising may not be pos-sible without corresponding improvements in thephysical product. Since Theorems 9 and 11 suggestthe same directional adjustment of advertising andproduct improvement, we feel such changes shouldbe made together.

SummaryOur goal in this section was to provide insight ondefensive strategy, i.e., rules of thumb that rely onbelievable abstractions that model major componentsof market response. Toward this end we searched fordirectional guidelines that help the manager under-stand qualitatively how to modify his marketingexpenditures in response to a competitive new prod-uct. In many situations, especially where data arehard to obtain or extremely noisy, such qualitativeresults may be more usable than specific quantita-tive results. If good data are available the qualitativeresults may help the manager understand and acceptmore specific optimization results.In particular, our theorems show in general that:• distribution expenditures should be decreased,

unless the new brand can be prevented from enteringthe market,

• awareness advertising should be decreased, and• profit is always decreased by a competitive new

brand. More specifically, if consumer tastes are uni-formly distributed,• profits will increase if price is decreased,• (at the margin) the brand should be improved in

the direction of the defending brand’s strength,• (at the margin) advertising for repositioning

should be increased in the direction of the defendingbrand’s strength.We have also shown that there exist highly seg-

mented taste distributions for which a price increasemay be optimal.We caution the manager that like any mathemat-

ical scientific theory, the above results are based onassumptions. Our results are true to the extent thatthe market which the defensive manager faces canbe approximated by our model of the market. Sinceour model is based on previously tested assumptionsabout consumer behavior, we believe there will bemany situations that can be approximated by ourmodel. At the very least we believe that Theorems 1through 13 provide the foundations of a theory ofdefensive strategy that can be subjected to empiricaltesting and theoretical modification. (As referencedearlier, Theorems 12 and 13 are generalizations of thepricing theorems. See appendix.) In the long run itwill be the interplay of empirics and theory that willprovide greater understanding of defensive strategy.See §7 for a discussion of validation issues and §8 fora discussion of future research.

4. Estimation of the ConsumerTaste Distribution

The defensive pricing and repositioning theoremsdepend upon the distribution of consumer tastes.In some markets this information will be collectedthrough standard techniques. See the review by Greenand Srinivasan (1978). In other cases, the defensivemanager may only have sales and positioning dataavailable. We now describe a technique that providesan estimate of the taste distribution from such data.We begin with a technique to estimate consumer

tastes when everyone has the same evoked set. Wethen extend this technique to the case where evokedsets vary.

Homogeneous Evoked SetsWe return to the notation of §2. Let mj � l = Prob��jj− <� ≤ �jj+ for j ∈ Al] and let Mj � l be observed mar-ket share of product j for consumers who evoke Al.For this subsection we are concerned only with con-sumers who evoke Al, thus for notational simplicity,drop the argument l. Our problem is then to selectf �� such that mj =Mj for all j ∈Al.

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Figure 6 Approximating the Consumer Taste Distribution with a Seriesof Uniform Distributions

f (α)

0º a

b

d

c

90º

α

One solution is to select a parameterized fam-ily of functions, say a Beta distribution, f$�� % &�,and select % and & with maximum likelihood tech-niques. We feel most common distributions (Beta,Gamma, Normal, Lognormal, Exponential, Laplace,etc.) are not flexible enough to analyze defensivestrategy because they restrict the possible shape ofthe taste distribution. This is particularly importantbecause the directionality of defensive pricing strat-egy can be dependent on the detailed shape of thetaste distribution.An alternate solution is illustrated in Figure 6. We

approximate f �� with a series of uniform distribu-tions. This procedure can approximate any f �� and,in the limit, as the number of line segments gets large,the procedure converges to the true f ��. The prob-lem now becomes how to select the endpoints and theheights of the uniform distributions. If our approx-imation is a good one, the area under the approx-imate curve should be close to the area under theactual curve. In other words, if we calculate the areaunder the actual curve between any two angles, thatarea should roughly equal the area under the curveformed by the uniform approximation. Now, if wetake the area between the lower and upper adja-cent angles for any brand, that area should equal thebrand’s market share. For example, in Figure 6, if “a”and “d” are the lower and upper adjacent angles forthe brand, the area of the rectangle a−b−c−d shouldequal the market share for the brand. Moreover, ifwe choose the end points to be the adjacent angles,the brand market shares become the estimates of therespective areas. Then the area of the jth rectangle ismj and the height, hj , of the jth uniform distributionbecomes

mj/��jj+ −�jj−��

Given this approximation, we can exactly satisfy theobserved shares if mj = Mj . Figure 7 illustrates thisapproximation for a six-brand choice set. This pro-cedure provides one approximation to f �� that we

Figure 7 Estimation of Consumer Tastes for a HypotheticalSix-Product Choice Set

f (α)

0 α12 α23 α34 α45 α56 90ºα

ˆ

h2

h1

h5

h3

h4

h6

Area= M1

believe is sufficient to distinguish among alternativedefensive pricing strategies.Summarizing, f �� estimates f �� and is given

f ��=Mj/��jj+ −�jj−� where �jj+ ≥ �>�jj−� (16)

Heterogeneous Evoked SetsThe extension to heterogeneous evoked sets is deceiv-ingly simple. Since �jj− and �jj+ are dependent onthe evoked set, Al, we can obtain an aggregated f ��

by a weighted sum of f �� given Al, denoted f �� Al�.That is,

f ��=L∑l=1

Slf �� Al� (17)

where f �� estimates f ��, f �� Al� estimates f �� Al�and Sl is the proportion of consumers who evoke Al.Estimates for Sl are discussed in the next section. Forprediction we must work with the set of disaggre-gated f �� Al�’s, unless we can insure that evoked setswill change appropriately when we change positionsaccording to the aggregated f ��.

SummaryIn general, we posit that (17) will provide a goodapproximation if the number of brands, J , is somewhatlarger than the number of perceptual dimensions.

5. Estimation of the New Product’sPosition

Most of the theorems of §3 are independent ofthe new product’s position (as long as it’s compet-itive), but the repositioning theorems (Theorems 9and 11) depend explicitly upon the new product’sspecific position. In some cases, a defending man-ager will commission a positioning study to identifythat position. But such studies are expensive ($50–100thousand).In this section we provide alternative procedures

that depend upon only (1) the positioning map prior

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to competitive entry and (2) sales after competitiveentry. In this way a manager can select the techniqueappropriate to his situation. We close this section witha technique to estimate the probability that the newproduct enters each evoked set.

Maximum Likelihood EstimatesSuppose the new brand, n, enters the evoked set A∗

l

with the corresponding probability, S∗l . �A

∗l = Al ∪

)n*�� In other words, S∗l is the probability that a ran-

dom consumer evokes set l and is aware of brand n.Then the market share of the new brand as wellas after-entry market shares for previous brands canbe computed with the consumer model in §2 if thenew brand’s position, �x1n/pn x2n/pn�, is known. Thus,given the new brand’s position, we can obtain

m∗j =

L∑l=1

mj � l�Sl − S∗l �+

L∑l=1

m∗j � lS

∗l (18)

m∗n =

L∑l=1

m∗n � lS

∗l (19)

where

m∗j = the post-entry market share for brand j given

the new brand enters at xn.m∗

j � l = the post-entry market share for brand j amongthose with evoked set Al given the new brandis positioned at xn.

m∗n � l = the market share of the new brand given xn

and A∗l .

If we assume consumers are drawn at random toform an estimation sample then the log-likelihoodfunction, L�xn�, is given by

L�xn�=J∑

j=1Mj logm

∗j �xn�+Mn logm

∗n�xn� (20)

where

L�xn�= the log-likelihood function evaluated at xn =�x1n x2n�, the position of the new brand.

m∗j �xn�= m∗

j evaluated at xn = �x1n x2n�,m∗

n�xn�= m∗n evaluated at xn = �x1n x2n�,

Mj = post-entry market share of brand j .

The maximum likelihood estimator of the new brand’sposition is then the value of xn that maximizes L�xn�.Equation (20) is easy to derive but difficult to use.

The key terms, m∗j �xn� and m∗

n�xn�, are highly nonlin-ear in xn even for a uniform distribution. Thus theoptimization implied by (20) is solvable in theory,but extremely difficult in practice. Fortunately, there

is a more practical technique for obtaining estimatesof xn.

Bayes EstimatesIn discussing defensive strategies with product man-agers in both consumer and industrial products, wediscovered that most defending managers have rea-sonable hypotheses about how the competitive brandis positioned. For example, when Colgate-Palmolivelaunched Dermassage liquid dishwashing detergentwith the advertising message: “Dermassage actuallyimproves dry, irritated detergent hands and cuts eventhe toughest grease,” the experienced brand managersat Procter and Gamble, Lever Bros., and Purex couldbe expected to make informed estimates of Dermas-sage’s position in perceptual space. The existence ofsuch prior estimates suggests a Bayesian solution.Suppose that the defending manager provides a

prior estimate, fx�xn�, of the distribution of the newproduct’s position, xn. (Note that xn denotes xnexpressed as a random variable.) For tractability wediscretize the prior distribution. We then use the con-sumer model in §2 to derive m∗

j �xn�$�� and m∗n�xn�$��

where xn�$� for $= 1 2 � � � T are the possible posi-tions for the new product for each potential newproduct position, where $ indexes the discretized newbrand positions. See (18) and (19). If consumers aredrawn at random for the estimation sample, then theBayesian posterior distribution,fx�xn �M�, is given by

fx�xn�$� �M�

= fx�xn�$��K�M �m$�/∑

-

fx�xn�- ��K�M �m- � (21)

where

K�M �m$�=mn�xn�$��Mn. ·

J∏j=1)mj�xn�$��*

Mj.

M is the vector of post-entry market shares, K�M �m$�is the kernel of the sampling distribution for . con-sumers drawn at random from a population definedby the multinomial probabilities, and m$ = )mn�xn�$��and mj�xn�$�� for all j*. The use of (21) to update amanagerial prior is shown in Figure 8.Equation (21) looks complicated but it is relatively

easy to use. First, the manager specifies his priorbeliefs about the new brand’s position in the formof a discrete set of points xn�$�, and the probabil-ities, fx�xn�$��, that each point is the new position.For example, he may believe that points, (3�5 4�0),(4�0 3�5), (4�0 4�0), are the potential, equally likely,positions of the new brand, Dermassage.10 As shown

10 This hypothetical example does not necessarily reflect true mar-ket shares and market positions.

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Figure 8 Bayesian Updating Procedure for a Hypothetical Market

(a) Bayesian prior (b) Updated posterior

x2n

x1n x1n

f x(x

n)

f x(x

n|m

)

x2n

Notes. The Bayesian prior (a) is based on managerial judgment. The Bayesian posterior (b) is the result of using observed sales data to update managerialjudgment (see Equation 21).

(a) The manager believes the two separate areas on the map are not equally likely, but the manager’s priors are diffuse.(b) Bayesian procedure clearly identifies one area as the most likely position for the new product.

in Figure 9, the manager has a general idea of whereDermassage is positioned but does not know the exactposition. We first use (18) and (19) to compute thepredicted market shares, mj�xn�$��, for each of thethree potential new brand positions. For this example,assume that tastes are uniform and everyone evokesall brands. This case is shown in Table 3. Now sup-pose that we sample 50 consumers and find that,in our sample, the observed market shares are Ajax(20%), Dermassage (30%), Joy (20%), and Ivory (30%).We use (21) to compute the kernel of the samplingdistribution, e.g.,

K�M �m$=1�= �0�30�10�0�10�15�0�30�10�0�30�15 for $=1�

We can then compute the posterior probabilities asgiven in Table 3.

Figure 9 Example Managerial Priors for the New Product’s Position(Three Equally Likely Points for Dermassage)

Dermassage

Ajax

Efficacy/$

Ivory

Palmolive

Mild

ness

/$

As we have chosen the data, the market shares from50 consumers clearly identify (4�0 4�0� as the mostlikely position for Dermassage. The point, (4�0 3�5),still has an 11% chance of being the actual posi-tion; the point, (3�5 4�0), is all but ruled out. Not allapplications will be so dramatic in identifying thenew brand’s position with such small sample sizes,but all applications will follow the same conceptualframework.

Evoked Set ProbabilitiesEquations (17) through (21) depend on the evokedset probabilities, Sl and S∗

l . In §2 we assume that Slis known prior to the new product’s launch. If themanager collects data on the evoked sets after thelaunch of the new brand, (17) through (21) can beused directly.In many cases evoked set information will be too

expensive to collect after the new brand is launched.Equations (17) through (21) can still be used if weassume either that (1) the new brand enters evokedsets randomly or (2) that the new brand enters sets inproportion to the probability that it would be selectedif it were in that evoked set.Case 1 is based on the assumption that aware-

ness and availability are independent of preference.

Table 3 Calculations for Bayesian Updating Procedure

= 1 = 2 = 3 ObservedPosition of dermassage �35�40� �40�35� �40�40� share

Market sharesAjax 030 0.24 0.20 0.20Dermassage 010 0.19 0.30 0.30Joy 030 0.27 0.20 0.20Ivory 030 0.30 0.30 0.30

Prior probability 033 0.33 0.33 —Posterior probability < 001 0.11 0.88 —

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In case 1, (18) and (19) reduce to

m∗j �xn�=

∑l

��1−W�mj � l +Wm∗j � l�Sl (22)

m∗n�xn�=W

∑l

m∗n � lSl (23)

respectively, where W is the aggregate percent of con-sumers who are aware of the new brand and find itavailable.Case 2 is based on the assumption that evoking is

functionally dependent on preference. In other words,case 2 assumes that consumers are more likely tobecome aware of brands that closely match their pref-erences. In case 2, Equations (18) and (19) reduce to

m∗j �xn�

=∑l

mj �lSl−[W

∑l

�mj � l−m∗j � l�m

∗n � lSl

]/[∑l

m∗n � lSl

] (24)

m∗n�xn�=W

∑l

�m∗n � l�

2Sl

/∑l

m∗n � lSl (25)

respectively. The derivations of (24) and (25) are givenas Theorem 14 in the appendix.Equations (22) through (25) are useful computa-

tional results since together they provide flexibility inmodeling how a new brand enters evoked sets. Moreimportantly each can be applied if we know only theaggregate percent of consumers who evoke the newbrand.

SummaryThis section has provided a practical means to esti-mate the new brand’s position and its impact on sales.The only data on the new market that are requiredare (1) sales of each brand and (2) aggregate evok-ing of the new brand. (We assume, of course, that theold product positions, the old evoked set probabili-ties, and the taste distribution are known.)

6. Diagnostics on CompetitiveResponse

Defensive strategy is how to react to a competitivenew product. However, the concepts developed in §§2and 3 are useful for representing competitive threatsand, in particular, for representing how our competi-tors might react to the new brand. In this section webriefly outline these heuristic concepts.

Angle of AttackThe general equilibrium for a market depends on thecost structures faced by each firm as well as the distri-bution of consumer tastes. However, in talking withmanagers we have found it useful to represent thecompetitive threats posed by each firm as that portion

of the taste spectrum that that firm is capturing. Inparticular we define the “angle of attack,” 0j , to rep-resent the average tastes of product j’s consumers, i.e.:

0j = ��jj+ +�jj−�/2� (26)

With this definition we should be most concernedwith products that have angles of attack that are closeto ours. In defensive strategy formulation we believeit is useful to superimpose the angle of attack of thenew product on the preference distribution to deter-mine how competitive the new product is and todetermine which of our competitors is most likely toengage in an active defense. The angle of attack doesnot capture all of the richness of the analytic model,but does provide a useful visual interpretation of ourmodel that serves as a stimulus to discussion.

Price DiagnosticsTheorems 2, 3, 4, 5, 12, and 13 tell us how to changeour price in order to react to a competitive new prod-uct, but they can just as easily be applied to predicthow our competitors should react to the new prod-uct if they too are maximizing profit according to ourmodel.We can also gain some insight on their response

by examining the multiattributed space of productattributes. See Figure 10. Such a map tells us the max-imum price that our competitor (in this case Ajax) cancharge and still remain efficient. This point is marked“upper bound” in Figure 10. The map also indicatesthe point at which Ajax’s price forces the new prod-uct to lower its price to remain efficient. Beyond thispoint, labeled “lower bound” in Figure 10, Ajax andthe new product enter a (possibly escalating) pricewar. We cannot be more specific in our predictionsof Ajax’s behavior without knowing the detailed coststructure of the market, but Figure 10 does serve as auseful first cut at visualizing Ajax’s first response tothe new product.We leave the exact prediction of the competitor’s

price to future equilibrium analyses.

7. Validation IssuesThere are at least two validation issues relevant todeductive theory. The first is a test of the assumptionsand the sensitivity of the theorems to the assump-tions. The second is a test of whether or not managersalready appear to intuitively follow the predictions ofthe theorems. We discuss each issue in turn.

AssumptionsOur theorems analyze aggregate response, thus, anytest of our assumptions should test their validityas a model of aggregate market behavior. We choseassumptions that are well documented in the market-ing literature. See the detailed discussion in §2. As an

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aggregate market model, response analysis and util-ity maximization in a multiattributed space appear tobe reasonable. Linearity is a useful first order anal-ysis which encompasses a surprisingly broad rangeof commonly used utility functions including theideal point models, the Cobb-Douglas product form,and multiplicative conjoint models. Our price-scalingassumption, i.e., the assumption that the consumerchooses the product that maximizes u/p where uis the linearized utility function and p is price, fol-lows directly from economic theory (e.g., see Debreu1959). The assumption has been used in marketing(see Hauser and Simmie 1981, Ratchford 1975). Onlyour pricing theorems use this assumption.An alternative approach to testing our assumptions

is to test the robustness of our theorems to relaxationsin the assumptions. For example, some form of thepricing theorems may also hold for utility that is sim-ply decreasing in price.

Behavior of FirmsOur theorems suggest how managers should behave.We can provide a weak test of our results by observ-ing how managers do behave. If they behave as pre-dicted, then our model escapes falsification, but it isnot yet validated. If they do not behave as predicted,then either (1) the assumptions are wrong, (2) themodel is not sufficiently rich, or (3) they are actingsuboptimally. Unfortunately we cannot distinguishamong these three options without further study.Because defensive marketing strategy is a new

subject of inquiry, there is little empirical evidenceon defensive response. One empirical study of whichwe are aware is a study by Beshel (1982) of the pre-scription analgesics market. Before November 1980,the market was dominated by Tylenol with Codeine

Figure 10 Some Insights on Competitive Price Response to the NewProduct

Mild

ness

/$

Efficacy/$

Ivory

Upperbound

Lowerbound

Joy

Ajax

New product

()

(McNeil Laboratories), Empirin with Codeine(Burroughs Wellcome), and Darvocet N-100 (EliLilly). Tylenol w/Codeine and Empirin w/Codeinewere positioned as more effective; Darvocet N-100was positioned as gentler (fewer side effects). InNovember 1980, McNeil introduced Zomax whichwas positioned as gentler than Tylenol w/Codeineand Empirin w/Codeine and more effective thanDarvocet N-100. Since Zomax was attacking all thesebrands, Theorems 7 and 10 suggest that all thesebrands should reduce their distribution and promo-tion expenditures. Beshel (1982, pp. 91–93) presentsevidence that, that is probably what happened.Beshel’s study is indicative of the types of descriptiveanalyses that are necessary to further develop a bodyof defensive marketing theory.Another empirical study is also consistent with our

results. We would suspect that near the end of a prod-uct’s life cycle in highly segmented markets, the ulti-mate degree of product differentiation has occurred.Hence, brands would have been launched targeted ateach segment. At this point, conditions for Theorem 5would frequently be met within each segment. Hiseand McGinnis (1975) found that near the end of aproduct’s life cycle, price increases were more com-mon than price decreases. Of course, numerous branddeletions occur during this period which complicatethe findings.Both studies are indicative of the types of descrip-

tive analyses that are necessary to further develop abody of defensive marketing theory.

8. Limitations and Future ResearchTheorems 1 through 14 represent a beginning, butthey can be improved through further research.

Competitive IssuesAn explicit assumption of all of our analyses is thatof a two-product equilibrium. We react as if no otherproduct besides us reacts to the new product.Consider pricing under uniformly distributed

tastes. If we assume the product enters with per-fect foresight and an equilibrium exists, then in atwo-product market our results are the full equilib-rium. In a three-product market with the new productattacking both existing products, then, if our competi-tor reacts by reducing his price, his effect on us, ifhe has an effect, is to cause us to reduce our price.This response is in the same direction as the responsecaused by the new product. Thus, intuitively, it wouldseem that our pricing results would hold for a fullequilibrium in a three-product market. We can createsimilar intuitive arguments for multiproduct markets,for nonadjacent attacks, and for other marketing vari-ables, but such arguments are not rigorous and do notprove our results for a full equilibrium.

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Full equilibrium analyses are difficult but poten-tially fruitful directions for future study. One canbegin to study this issue by simulation or by exam-ining the equilibrium that results when each exist-ing firm follows a two-product equilibrium strategy.After such results are obtained, improved foresightcan be incorporated in the model and the model canbe extended to product line decisions.

Pricing TheoremsOur pricing theorems assume price scaling. It wouldbe interesting to examine their sensitivity to alterna-tive models of consumer price response. Furthermore,one can examine pricing strategy under different dis-tributional assumptions about consumer tastes. Forexample, is it always optimal to decrease price iftastes are approximately normally distributed (trun-cated at 0 and 90 )? Theorem 13 provides a startingpoint for such analyses.

9. SummaryAmong the key results of this paper are the theoremswhich investigate defensive marketing strategy. Thesetheorems are the logical consequences of a consumermodel based on the assumptions that consumers areheterogeneous and choose within a product categoryby maximizing a weighted sum of perceived prod-uct attributes. Since this consumer model is based onempirical marketing research we feel that it is a goodstarting point with which to analyze defensive mar-keting strategies.We feel that the theorems provide usable manage-

rial guidelines. When the appropriate data are avail-able normative optimization models may be the bestway to proceed, but, by their very nature, defensivemarketing strategies are often made quickly and with-out extensive data collection. The theorems tell themanager that as the result of a competitive entrant(1) the defender’s profit will decrease, (2) if entry can-not be prevented, budgets for distribution and aware-ness advertising should be decreased, and (3) thedefender should carefully compare the competitor’sangle of attack to his position and the distribution ofconsumer tastes. If tastes are segmented and if thecompetitors clearly out-position his product in oneof his consumer segments, a price increase may beoptimal. If consumer tastes are uniformly distributedacross the spectrum, the defender should decreaseprice, improve product quality to his strength, andadvertise to support changes. Sections 4, 5, and 6 pro-vide practical procedures to identify the consumertaste distribution, the new brand’s position and itsentry into evoked sets. These data allow us to decidewhen each theorem is appropriate. Because theseresults are robust with respect to many details of the

model they are good managerial rules of thumb evenwhen the data are extremely noisy.We feel that the theorems are a good beginning to

guide the development of empirical models and toencourage the development of a generalizable man-agerial theory of how to respond to competitive newproducts.

Glossary

Adjacent A lower (upper) adjacent brand is the next ef-ficient brand in the choice set as we proceedclockwise (counter-clockwise) around the effi-cient frontier.

Competitive A new product represents a competitiveattack if it is adjacent and there are at leastsome consumers whose tastes cause them toprefer the new product, i.e., f �� = 0 for all �in the range �jn ≤ �≤ ajj+.

Efficient A product is efficient in a multiattribute spaceif there is no other product or linear combina-tion of products in the space that dominatesthe efficient product on all attributes.

Regular A market is regular if, for every product inthat market, the angle of the ray connect-ing the product’s position to the origin liesbetween the angle which defines the limitsof the tastes of that product’s consumers, i.e.,�jj− ≤ "j ≤ �jj+.

Appendix

Lemma 1. Sales under uniformly distributed tastes are de-creasing in price.

Proof. Sales under the condition of uniform tastes isgiven by Na��nj − �jj−�. The result holds if �′

nj − �′jj− < 0

where the derivative is with respect to price, pj . By directcomputation,

�′nj = pn�x2j x1n − x1j x2n�

/[�pjx2n − pnx2j �

2 + �pnx1j − pjx1n�2]�

Since the denominator is positive we have �′nj < 0 if x2j x1n <

x1j x2n.Since n is now upper adjacent to j , x2j ≤ x2n and x1n ≤ x1j

where at least one inequality is strict. Thus �′nj < 0. By sym-

metry �′jj− > 0. By the same arguments �′

jj+ < 0, thus theresult holds both before and after entry. �

Lemma 2. Mb�p� is a nonincreasing function of price. Mb�p�is a decreasing function of price for f ��jj+� or f ��jj−� > 0, whereMb�p�=

∫ �jj+�jj− f ��d�. Similarly, let

Ma�p�=∫ �jn

�jj−f ��d��

Proof. Let (’) denote the derivative with respect to price,pj .M ′

b�p�= f ��jj+��′jj+−f ��jj−��′

jj−. Following Lemma 1, wesee that �′

jj+ < 0 and �′jj− > 0. Since f ��≥ 0, it follows that

M ′b�p�≤ 0. If f ��jj+� or f ��jj−� > 0, M ′

b�p� < 0. �

Lemma 3. �′jn < �′

jj+ implies �′a�p0� < 0 whenever f ′��≤ 0.

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Proof. By assumption and Lemma 1, �′jn < �′

jj+ < 0.Because �jn < �jj+ by the definition of competitive,f ��jn��

′jn < f ��jj+��′

jj+ < 0 whenever f ′�� ≤ 0. Subtractinga positive constant from each side yields

f ��jn��′jn − f ��jj−��

′jj− < f ��jj+��

′jj+ − f ��jj−��

′jj−�

Hence we find, by definition M ′a < M ′

b < 0. So becausep0 − c > 0, we know

�p0 − c�M ′a < �p0 − c�M ′

b < 0

but�′

b�p0�=Nb�Mb + �p0 − c�M ′

b�= 0

henceMb + �p0 − c�M ′

b = 0

so

Ma + �p0 − c�M ′a < 0 and Na�Ma + �p0 − c�M ′

a� < 0

for any Na > 0. Finally, by the definition of �′a�p

0�, we haveshown �′

a�p0� < 0. �

Theorem 1. If total market size does not increase, optimaldefensive profits must decrease if the new product is competitive,regardless of the defensive price.

Proof. Let p0 be our optimal price before entry and letp∗ be our optimal price after entry. First,

�b�p0�= �p0 − c�NbMb�p

0�≥ �p∗ − c�NbMb�p∗�

by the definition of optimality at p0. Nb ≥ Na by assump-tion. Mb�p� >Ma�p� since �jj+ >�jn and f �� not identicallyzero in the range, ��jj+ �jn�, by the definition of competi-tive. Thus,

�p∗ − c�NbMb�p∗� > �p∗ − c�NaMa�p

∗�=�a�p∗��

We have proven �b�p0� >�a�p

∗� which is the result. �

Theorem 2 (Defensive Pricing). If consumer tastes areuniformly distributed and the market is regular, then defensiveprofits can be increased by decreasing price.

Proof. To show that a price decrease increases profit wemust show that �′

a�p0� < 0. From regularity we know "j <

�jj+ hence "j < �jj+ + �nj given �nj > 0. If �jj+ + �nj < 90 ,then

tan "j < tan��jj+ +�nj �

tan "j < �tan�jj+ + tan�nj �/�1− tan�jj+ tan�nj �

and by definition

x2j /x1j < �rjj+ + rjn�/�1− rjj+rjn�

where 1 − rjj+rjn > 0, which implies x2j �1 − rjj+rjn� <x1j �rjj+ + rjn�. Moreover, if �jj+ + �nj > 90 , then tan "j >tan��jj+ +�nj � because tan��jj+ +�nj � < 0. However, tan "j >tan��jj+ +�nj � implies

x2j /x1j > �rjj+ + rjn�/�1− rjj+rjn�

where 1 − rjj+rjn < 0. Therefore, in both cases, we findx2j �1− rjj+rjn� < x1j �rjj+ + rjn�. Multiplying by �rjj+ − rjn� andrearranging terms we find that

�x1j + x2j rjn��1+ r2jj+� > �x1j + x2j rjj+��1+ r2jn�

which is algebraically equivalent to (note that both sides ofthe inequality are positive):

−�x2j+p−1j+ −x2j p−1�r ′jj+�1+r2jn�<−�x2np−1n −x2j p

−1�r ′jn�1+r2jj+�

but �x2j+p−1j+ − x2j p−1� > �x2np

−1n − x2j p

−1� > 0, hence it mustbe the case that

−r ′jj+�1+ r2jn� <−r ′jn/�1+ r2jj+�

dividing through

−r ′jj+/�1+ r2jj+� <−r ′jn/�1+ r2jn�

therefore by definition −�′jj+ < −�′

jn or �′jj+ > �′

jn andLemma 3 implies that �′

a�p0� < 0 since f ′��= 0. �

Theorem 3 (Defensive Pricing). If the new brand’s angle(�jn) plus the upper adjacent brand’s angle (�jj+) exceeds 90

and if the distribution of consumer tastes is uniformly distributedthen defensive profits can be increased by decreasing price fromthe before-entry optimal.

Proof. If �jj+ +�nj > 90 then

tan "j > tan��jj+ +�nj �

because tan��jj+ +�nj � < 0. Hence

x2j /x1j > �rjj+ + rjn�/�1− rjj+rjn�

and the proof follows from that point in the proof forTheorem 2. �

Theorem 4. For an upper adjacent attack in a regular market,if f ′�� ≤ 0 the defensive profits can be increased by decreasingprice from the before-entry optimal. For a lower adjacent attack,the result holds if f ′��≥ 0.

Proof. The proof to Theorem 2 shows that �′jn < �′

jj+.Lemma 3 then proves the result for f ′��≤ 0. The result fora lower adjacent attack follows by symmetry. �

Theorem 5. There exist distributions of consumer tastes forwhich the optimal defensive price requires a price increase.

Proof (By Construction). See example in text.

Theorem 6 (Price Decoupling). The optimal defensivepricing strategy is independent of the optimal defensive distribu-tion strategy, but the optimal distribution strategy depends on theoptimal defensive price.

Proof. Differentiating �a�p kd� with respect to p andsetting the derivative to zero yields �Ma�p� + �p − c�dMa�p�/dp�NaD = 0. Since NaD > 0 for the optimal price,the solution, p∗, of the first order conditions is independentof kd . Differentiating �a�p kd� with respect to kd and settingthe derivative to zero yields �p − c�NaMa�p��dD/dkd� = 1.The solution to this equation, k∗d , is clearly a function of p.Since the first order conditions for p are independent of

kd , so are the second order conditions. For k∗d dependent onp it is sufficient to examine first order conditions. �

Theorem 7 (Defensive Distribution Strategy). If themarket size does not increase then the optimal defensive distri-bution strategy to a competitive entry is to decrease spending ondistribution.

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Proof. Differentiating ��p kd�with respect to kd and rec-ognizing that �a�p� = �p − c�NaMa�p� as given by (9) and(10) yields the first order condition for the optimal k∗d as�a�p

∗��dD�k∗d�/dkd�= 1. Similarly we can show that the firstorder condition for the optimal before entry spending, k0d ,is �b�p

0��dD�k0d�/dkd� = 1. Since �b�p0� > �a�p

∗� by Theo-rem 1, we have dD�k∗d�/dkd > dD�k0d�/dkd . But dD�kd�/dkd isdecreasing in kd , hence k∗d < k0d . The second order conditionsare satisfied if �p∗ − c� > 0 since d2D�k∗d�/dk

∗d < 0 by assump-

tion. If �p∗ − c�≤ 0 then the optimal kd must be zero, i.e., noproduction. �

Theorem 8 (Preemptive Distribution). If market size doesnot increase, optimal defense profits must decrease if a new brandenters competitively, regardless of the defense price and distribu-tion strategy. However, profit might be maintained (less preven-tion costs), under certain conditions, if the new brand is preventedfrom entering the market.

Proof. Define �b�p kd� analogously to Equation (10).Note that �a�p kd�=�a�p�D�kd�−kd where �a�p� is definedby (8). Then

�b�p0 k0d� ≥ �b�p

0 k∗d�=�b�p0�D�k∗d�− k∗d

> �a�p∗�D�k∗d�− k∗d =�a�p

∗ kd��

The first step is by the definition of optimality; the secondstep is by definition, the third step is true by Theorem 1,and the fourth step is by definition. The last statement of thetheorem follows from the recognition that it is potentiallypossible to construct situations where the cost of preventingentry is smaller than �b�p

0 k0d�−�a�p∗ k∗d�. �

Theorem 9 (Defensive Product Improvement). Supposethat consumer tastes are uniformly distributed and the competi-tive new brand attacks along attribute 2 (i.e., an upper adjacentattack), then at the margin if product improvement is possible,(a) Profits are increasing in improvements in attribute 1 (i.e.,

away from the attack);(b) Profits are increasing in improvements in attribute 2 (i.e.,

toward the attack) if

�x1j /p− x1n/pn��1+ r−2nj �≤ �x1j /p− x1j+/p+��1+ r−2jj+��

Symmetric conditions hold for an attack along attribute 1.

Proof. We first recognize that for uniformly distributedtastes, before-entry profits, �b�c�, are given by

�b�c�=�a�c � 0�+H�p0 − c��jj+ −�jn�

where H = NbD�k0d�/90

. H is a positive constant indepen-dent of c. Let zj = x1j /p and let yj = x2j /p. Differentiating�b�c� with respect to c yields

�′b�c�=�′

a�c � 0�+H[�p0 − c��′

jj+ −�′jn�− ��jj+ −�jn�

]

where �′� denotes the derivative with respect to c. By thedefinition of cb being optimal, �′

b�c�= 0 at cb . Since H > 0,�′

a�c � 0� > 0 if the term in brackets is negative. Since�jj+ − �jn > 0 by the definition of competitive and since�p0 − cb�≥ 0 at the before-entry optimal, the term in bracketsis negative if �′

jj+ −�′jn ≤ 0.

Part (a). Consider spending with respect to attribute 1.Clearly dx1j /dc > 0, thus dzj/dc > 0. Since

d��jj+ −�jn�/dc= �d��jj+ −�jn�/dzj ��dzj/dc�

for spending along attribute 1, we need only show d��jj+ −�jn�/dzj ≤ 0.Redefine �′� to denote differentiation with respect to zj .

Then we must show �′jj+ < �′

jn. Since rjj+ = �zj − zj+�/�yj+−yj � r

′jj+ = �yj+−yj �

−1. Similarly, r ′jn = �yn−yj �−1. Hence

0< r ′jj+ < r ′jn. Now rjj+ > rjn, hence

0< �1+ r2jj+�−1 < �1+ r2jn�

−1�

Putting these results together yields

r ′jj+�1+ r2jj+�−1 < r ′jn�1+ r2jn�

−1�

Finally, �= tan−1 r , �′ = r ′�1+ r2�−1. Thus �′jj+ <�′

jn and theresult follows.Part (b). Analogously, we must show d��jj+ − �jn�/

dyj ≤ 0 to establish the result in part (b). Redefine �′� todenote differentiation with respect to yj . We derive condi-tions for �′

jj+ ≤ �′jn. As shown above this is equivalent to

r ′jj+�1+ r2jj+�−1 ≤ r ′jn�1+ r2jn�

−1�

Since rjj+ = �zj − zj+�/�yj+ − yj �,

r ′jj+ = �zj − zj+�/�yj+ − yj �2 = r2jj+/�zj − zj+��

Similarly r ′jn = r2jn/�zj −zn�. Substituting yields the conditionin part (b). Finally, the last statement of the theorem followsby symmetry. �

Theorem 10 (Defensive Strategy for AwarenessAdvertising). If market size does not increase, then the optimaldefensive strategy for advertising includes decreasing the budgetfor awareness advertising.

Proof. Differentiating �a�p∗ k∗d ka� with respect to ka

yields the equation ��a�p∗ k∗d� + k∗d��dA�k

∗a�/dka� = 1. Sim-

ilarly, we get ��b�p0 k0d� + k0d��dA�k

0a�/dka� = 1. The result

follows analogously to the proof of Theorem 7 since�b�p

0 k0d� > �a�p∗ k∗d� k

0d > k∗d , and A is marginally

decreasing in ka. Second order conditions follow fromd2A�k∗a�/dk

2a < 0. �

Theorem 11 (Repositioning by Advertising). Supposeconsumer tastes are uniformly distributed and the new com-petitive brand attacks along attribute 2 (i.e., an upper adjacentattack), then at the margin if repositioning is possible,(a) profits are increasing in repositioning spending along

attribute 1 (i.e., adjacent from the attack);(b) profits are increasing in repositioning spending along

attribute 2 (i.e., toward the attack) if and only if

�x1j /p− x1n/pn��1+ r−2nj � < �x1j /p− x1j+/p+��1+ r−2jj+��

Symmetric conditions hold for an attack along attribute 1.

Proof. Define a conditional profit function,

�a�kr � 0�= �p0 − cb�NbMb�p0 kr cb�D�k

0a�A− k0d − k0a − kr

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and recognize that under uniformly distributed tastes, thebefore-entry profit, �b�kr �, is given by

�b�kr �=�a�kr � 0�+G��jj+ −�jn�

where G = �p0 − cb�NbD�k0d�A�k

0a�/90 is a positive constant

independent of repositioning spending, kr .Let zj = x1j /p and let yj = x2j /p. Let kri be repositioning

spending along xij , for i= 1 2. Differentiating with respectto kri yields

�′b�k

0r �=�′

a�k0r � 0�+ �G/90��′

jj+ −�′jn�= 0

where �′� denotes the derivative with respect to kri. BecauseG> 0, �′

a�k0r � 0� > 0 if �′

jj+ <�′jn. Since zj is proportional to

x1j and x′1j > 0, part (a) follows if �d�jj+/dzj � < �d�jn/dzj �.Similarly, part (b) follows if �d�jj+/dyj � < �d�jn/dyj �. Theseare exactly the conditions we examined in the proof of The-orem 9. (Note x′ij > 0 is equivalent to the statement, “if repo-sitioning is possible.”) �

Note that in Theorem 9, condition (b) is sufficient butnot necessary but equality is allowed. In Theorem 11, con-dition (b) is necessary and sufficient but equality is notallowed.

Theorem 12 (Defensive Pricing in Irregular Mar-kets). If consumer tastes are uniformly distributed and �jj+ +�jn > "j , which is implied by regularity, then the optimal defen-sive price strategy is to decrease price.

Proof. The proof follows from line 2 of the proof to The-orem 2. Note that we could also prove the first order con-ditions for f ′��≤ 0 by Lemma 3. �

Theorem 13. If the size of the market remains constant,defensive profits are decreasing in price at the before-entry optimalif and only if the distribution of consumer tastes, f ��, satisfiesthe following equation:∫ �jj+

�jn

f ��d�+�p−c�{f ��jj+�pj+�x2j x1j+x1j x2j+�

÷[�pjx2j+−pj+x2j �2+�pj+x1j−pjx1j+�2

]−f ��jn�pn�x2j x1n−x1j x2n�÷[

�pjx2n−pnx2j �2+�pnx1j−pjx1n�2]}>0�

Proof. �b�p0� = �a�p

0� + Z�p0� where Z�p� = Na�p − c� ·∫ �jj+�jn

f ��d�. Since �′b�p

0� = 0 we need only show thatZ′�p0� > 0 to establish that �′

a�p0� < 0. �′� denotes the deriva-

tive with respect to price at p. Taking derivatives gives us

Z′�p�=Na

∫ �jj+

�jn

f ��d�+Na�p−c��f ��jj+��′jj+−f ��jn��

′jn�>0�

Finally substituting the explicit terms for �′jj+ and �′

jn yieldsthe result. �

Theorem 14. If evoking is proportional to the probability thatthe new product will be chosen if it is in the evoked set, then theforecasted market shares are given by (24) (25) in the text.

Proof. By (19),

m∗n�xn� =

∑l

m∗n � lS

∗l =

(∑l

m∗n � lS

∗l

)(∑k

m∗n �kSk

)/(∑k

m∗n �Sk

)

= ∑l

∑k

m∗n � lm

∗n �kS

∗l Sk�S

∗k S

∗−1k �

/(∑k

m∗n �kSk

)�

By assumption, S∗k is proportional to Skm∗n �k. Substituting

and rearranging terms yields

m∗n�xn�=

∑l

∑k

m∗n � lm

∗n �kSlm

∗n � lSkS

∗k S

−1k m∗−1

n �k/(∑

k

m∗n �kSk

)�

Recognizing W = ∑k S

∗k yields the equation for m∗

n�xn�in the text. Using this result and similar substitutionsyields (25). �

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DefensiveMarketingStrategieshauser/Hauser Articles 5.3.12/Hauser...HauserandShugan: Defensive...

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