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Logically Consistent Market Share ModelsPHILIPPE A. NAERT and ALAIN BULTEZ*

INTRODUCTIONA logically consistent m arket share model should pre-dict market shares that are between zero and one, andsum to one. Few authors have worried about this typeof problem in empirical studies, mainly because of theusual interest in a particular brand. It is then implicitlyassumed that if predicted market share for that brand isMSt, the other firms combined will get (1 MSt).This point m ay not be as obvious as it seems. If wewereto estimate the market share response functions for theother brands as well, we would often find that the sumof the market shares is not one. The problem becomesapparent particularly when the response functions forail brands or for a group of brands are estimated simul-taneously. An article by Neil E. Beckwith in a recentissue of JMR [2] reports an application of Zellner'sjoint generalized least squares method (joint GLS) [13]to the estimation of linear market share response func-tions of various competing brands. Beckwith's article

illustrates an interesting way of obtaining more efficientestimators, but at the same time, raises the issue oflogical consistency more clearly than in other applica-tions. This article will address the problems which arisewhen market response models are sum-constrained.Reference will be made to Beckwith's study because itis one of the few examples where response functions ofvarious brands have been estimated simultaneously.We will first derive restrictions on the explanatoryvariables and on the parameters which are implied by asum constraint on the dependent variable. Beckwith'sstudy will serve as an illustration. The constraint in hiswork relates to the sum of the individual firms' marketshares. Market share models are such that few distribu-tions can describe the behavior of the disturbances.This is discussed in the second section. In the third sec-tion we reach the conclusion tha t for m arket share func-tions to be logically consistent their functional formshould almost invariably be intrinsically nonlinear.' It

* Philippe A. Naert is Professor of M anagem ent, Eu ropeanInstitute for Advanced Studies in Management, and AssociateProfessor, St. Ignatius Faculty, University of Antwerp, Bel-gium. Alain Bultez is Assistant Professor, FUCAM, Mons,Belgium.

would seem intuitively reasonable to require logicalconsistency as a criterion for judging a model's ap-propriateness. However, logical consistency leads tomore complicated market share functions, and necessi-tates more sophisticated estimation techniqu es. So therewill be a trade-off between requiring logical consistencyand model simplicity. This point will be examined in thefinal section.

IMPLICATIONS OF THE ADDITIVITYCONSTRAINTZellner's method will usually lead to considerablegains in efficiency of the estimators provided that dis-turbances of the different equations are contemporane-ously highly correlated [13, pp. 353-4]. This conditionis satisfied in the case of market share response func-tions for various brands competing in an oligopolisticmarket since, for example, a positive disturbance for onebrand in a particular time period implies that at least

one other brand will have a negative disturbance in tha tsame period. This will be true especially in cases wherethe total market consists of just a few bran ds. Thus itwould appear that Zellner's method may be profitablyapplied for joint estimation of a set of m arket share re-sponse functions.However, Zellner's method does not guarantee thatthe sum of the market shares predicted from the esti-mated functions will add up to one, or to the sum of themarket shares of the brands considered. Yet such aconstraint is necessary if a logically consistent model isdesired. For the time being we will assume that logicalconsistency is indeed desirable.Dependent variables may be sum-constrained in thefollowing sense: Suppose the dependent variable ismarket share. If all brands are considered, predictedmarket shares should add up to one. If one considers alimited number of brands which make up a well-definedsubmarket, then predicted market shares should add upto whatever share they actually sum to. For example, if' By intrinsically nonlinear forms we mean equations whichcannot be linearized by transformations (e.g., logarithmic trans-formation), except by such approximating procedures as a firstorder Taylor expansion. In the sequel "linearization" will thusrefer to transformation without approximation.

334Journat of M arketing ResearchVol. X (August 1973), 334-40

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L O G I C A L L Y C O N S I S T E N T M A R K E T S H A R E M O D E L S 335in period t these brands have rt = .951 (95.1 %) marketshare, and in t -\- 1, rt+i = .953, then predicted sharesshould sum to 95.1 and 95.3 in period t and / 4- 1respectively. Another example of a sum constraintwhich is time-dependent occurs when the dependentvariable is brand sales. If ?,,< is sales of brand / inperiod t, then the sum of brand sales over all brandsshould be equal to Qt, industry sales in period /.First we will examine the implications of imposing asum constraint on the dependent variable in a set oflinear func tions, and then apply the results to the specificmodel proposed by Beckwith. Consider the followinglinear function (e.g., market share function for brand

(1)Let

y '< =

4 - 0ipXipt, f o r / = 1 , , .

fory = 1 , ,p.

) , for ; = I,Let J = 1 correspond to the constant term in the re-gression, that is, Z.K = 1 for / = 1, , n and t = I,, T. Finally let u be a nxl column vector of ones,

u ' = ( 1 , 1 , , 1 ) .Theorem:^ The necessary and sufficient conditions fora linear model, ;;,( = J3?=i ffuXnt, to predict sum-constra ined dependent variables, i.e., u'yi = / for all /, andfoTj = 2, ,p follows from the constraint on the de-pendent variables, and that no such restriction isneeded for the constant term. Since Beckwith's formula-

tion does not include a constant term, the procedureproposed by McGuire et al. is directly applicable. Theprocedure suggested by McGuire et al. as applied toBeckwith's case is briefly outlined below:(a) Pool the data on the various brands into one bigregression equation,MSi.t = \MSi,t-i + (r, - X rt-i)ASu. + u.t,

in whic h the c on stra int s X,- = X, 7,- = 7 for all /, an d7 = r( X7-,_i are em bod ied. T liat is, we have to esti-mate the constrained form.

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336 JOURNAL 0F MARKETING RESEARCH, AUGUST 1973

0} = \{MSi.t-i - rt-iASi.t) + e,-,,.Alternatively, we could have substituted

X = ( ' / - y)/rt-i,in (5). Tiie equation to be estimated would then be,

,,,_i) +(8)So if r, is not constant, either X or 7 will be time de-pendent. This results froni the peculiar assiiniptionthat 100% advertising leads to r < 1.0 share in a modelwhere advertising is the only decision variable. It wouldbe better in this case to regard thie market ca!ptured bythe 5, brands as the total m arket, i.e., express the marketshares of the individual brands as a percentage pf /,.Industry sales should then also be defined as r, tiniestrue industry sales. This would seem quite appropriatesince, as Beckwith states, the 2 other brands' appeal tovery specialized niarket segments.In fact,' 'we have no more information than that totalshare for the 5 brands is approximately 98% in eachperiod; r, is then, /-, = rt-i'= r = .98, for all /, and (7)becomes,MSi.t - .9,8 ASit = X{MSi.t-i - .98 ASi.i) ^- ei.t,

I 1 , while (8) reduces to,Si.t - MSi.t-i = 7(^5.-,, - ii/,9i)MSi,t-i) -I- u.i.(b) The second step is the ordinary least squares(OLS) estimation pf the equation obtained in the first

step. This will provide uswith anestimate of the matrixof conteniporaneous covariances of the various bran ds'disturbances: Si;(cj then, we can estimate our constrained equationby iGLS using * = il I. Note, howeyer, that a issingular since if the market shares suin to r,,Z . , . = P

(see [7, p. 1203]); as a result one of the brands observa-tions ought to be deleted from the set, so that the n iethodshould be applied to n 1 brands.RESTRICTIONS ON THE DISTURBANCE TERMS

Market share is a quantity obviously confiped to theinterval from zero to one. This natui-al restriction limitsi;iie set of distributions capable of describing the be-havior of the disturbances (see [7, p. 1205]) sinpe0 g MSi.t ^ 1 iniplies;- 1 g ^i\,MSi,t-i -\- y-ASi^t) g e.-., g

1 - :ASi.t) ^

As pointed out by Theil [11, pp. 629 ff.], this aspectis awkward in" regression-type situations and thereforea monotonic transformation is usually applied to thisparticular kind of dependent variable so ' that the newdependent variable constructed is then defined over the[00, +00] range. In fact, niany transformations havethis property. The logit is' ohe of them, and in p^r casethe firs't step a mo unts to defining a new variable:

mi.t = MSi.t/il - MSj.t},a quantity with range [0, ] since yvhen MSi.t = 0,mi.t = 0 and when MSi.t = 1, nti.t - 1/0 = .The log of mi.t is then defined over the [ 00, + 00 ]-interval since (lp,g 0) = - and (log -|- ) = + .Adjusting the priginal niodel accordingly we obtain theequation to estiniiate. In Beckwith's case we come upwith:

\ -M5. , , - i + y. , ,_i -f-

-ASi.t 1yASi.t)j

orlog ^ ^ 1 ' ' 1 = log i^-MSi.t + yASi,t) MSi.tj

- lpg (1 - \ MSi.t-l -which has now an intrinsically nonlinear form.

INTRINSICALLY NONHNPAR FOR MS: ANECESSITY?Npw one might argye that the requirements Xi = Xand 7< = 7 for all / are not very ap pealing. Indeed theydo not allow for diffe.rentiation between brands, in the

way market shares respond to advertising decisions. Wethen come to the conclusion that, if we want a logicallyconsistent model which allows for differences in theresponse parameters between the various brands, themodel striicture should simply be nonlinear.We n ay wonder at this point whether such a struc-ture may easily be defined, Usually when market sharefunetions are nonlinear, they are of such a form as tobecome linear upon transformation. For example,multiplicative functions are widely used in empiricalwork. The multiplicative equivalent of the expectedvalue of Beckwith's market share function is:(9) MSi.t = MS^i't^i AS^ffFor a multiplicative response function such as (9), it isnot possible to determine meaningful restrictions on theexplanatory variables and on the p arame ters, which willguarantee that market shares add up to one. Morecomplex functions are generally needed.For exapiple, with advertising and lagged market

that thenprmality assufpption is irrele-vant.

*For example in (9) assuming rt = 1 for all t, we have2 ? - , MSi.t-i = 1, and Ti'-iASi., = 1.With X. = 0, and 7. = 1for all I, the sum of the expected market shares is 1. However,these restrictions are, at least a priori, not very meaningful.

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LOGICALLY CONSISTENT MARKET SHARE MODELSsha r e as the qn ly e xp la na to r y v a r i ab le s , let:(10) MSi.t = \MSi.t-x + i\ -\)

T Al'* - l

where /4,,, is b ran d /'s adver t i s ing expendi tures . It isreadi ly seen that the sum of th'e niarket sha'res as definedin (10) is one, a'hd this without specific constraints on the7 , parameters . '' Unfor tunate ly , such formulat ions havesome problem's of the i r own! Determin ing ' the values oft h e p a rame t e r s is less s t raightforward thari in the l inearcase , a l thougfi many non l inear p rogram ming proceduresare now avai lab le which make nonl ihear es t imat ionmuch less of' a probleni' i Also , the stat is t ical propert iesof nonl inear ' est imators ' are weaker . Thi s expla ins whyempi r ica l work has usual ly avoided intrinsical ly non-l inear funct ional forms. Only a fewappl ica t ions ofnonl inear es t i tna t ion t echniques were repor ted in themarket ing l i t e ra ture [4, 12].So me r ead e r s may now ask themselves whether it is

possible to"design any logically consistent m arke t sharemodel which may be liiie'arized. The answer is yes , butr a t h e r " h e ro i c" a s s u mp t i o n s h av e to bem a d e a n d / o rnont r iv ia l t ransformat ions have to be devised. To il-lus t ra te , suppose theconsumpt ion pat tern on a two-b ran d mark e t may be descr ibed by a M ark o v - t y p emat r i x :/ + 1

where X, is the p ro p o r t i o n of consumers loyal tob r a n d/; ffc is the p ro p o r t i o n of consumers switching fromb ran d c to b r a n d /; Xc and tr, are similarly defined butrefer to b ran d c.

W e can define the i mp ac t of the competi t ive forceson the co n s u mp t i o n h ab i t s ' b y mak i n g X,, (7,, X , and(Tc ex pl ici t furictio n's of the 2 b ran d s ' mark e t i n g mix.Cons ider ing on ly the effect of advert is ing (A), we couldp o s t u l a t e :

(11)X , , ( = 1 exp {oiiA'i.t),ac t = exp i-acAc.t),

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338 JOURNAL OF MARKETING RESEARCH, AUGUST 1973He applied a direct search technique to estimate theparameters. Nakanishi [9] has shown that (15) can bemade linear after applying the following transformation:Let

MSt =a = I Z log Oi)/

\i=i /

The linearized form of (15) is then,(log AfS,-,, - M S, ) = (logo.- - a)

(16) i-p+ i:

This example illustrates that sometimes it is possibleto linearize models by means of nontrivial transforma-tions. In many cases, however, one will find that im-posing logical consistency leads to response functionsthat are intrinsically non linear. In such cases, nonlinearprocedures are to be applied to estimate the parameters.LOGICAL CONSISTENCY AS A CRITERIONFOR JUDGING MODEL STRUCTUREWithout additivity constraint, predicted market sharesummed over all brands might be more or less than o ne.Logical consistency, therefore, would seem intuitivelyappealing as one criterion for judging a particularmarket share model structure. In terms of Little'sdecision calculus [6] one would say that without addi-

tivity constraint a market share model lacks robustness.Yet, the advertising budgeting model that illustratesthe concept of a decision calculus is itself not robustwith regard to that constraint. Market share in Little'smodel is:(17)where:

',-,, = Min -\- {Max - Min) adv:,,/{S +

i.t is market share of company /, at time /,advi.t is i's advertising expenditure at t, Min, Max,y, 5 are parameters with 0 g Min g Max ^ 1,and 5, y positive.

If we were to define a similar market share functionfor competitors, no meaningful constraints on theparameters could be found to guarantee that the sumof predicted market shares be 1. The easy way out thenis to assume (implicitly) that competitors' marketshare is 1 MSi.t. However, that implies:(18) Com petitors' Share = 1 - Min- {Max - Min)adv]_J{& +

not depend on their own advertising. On the other h and,returning to (17), we see that for any value of /'s ad-vertising, its market share will be between 0 and 1, andwill be nondecreasing with advertising. So in many re-spects this model is robust. Suppose now for a momentthat competitive advertising data are not available.Without such data, (17) perhaps represents the mostrobust possible functional form. Competitive activityis then implicitly reflected in the response parameters,and the value of additional information on competitiveactivity may not even be worth the cost. So, robustnessis not som ething absolute, but has to be looked at froma cost-benefit point of view. And thus, robustnessshould also be related to what the model will be usedfor. For example, if the problem is to find out howcompetitors will react to changes in z's advertising, (17)would be of little help. On the other hand if we wantto know how much share to expect for a given adver-tising budget, (17) will probably perform quite well,assuming that the brand in question is well-established.

Returning now to the comparison between linear andnonlinear models, we do not argue then that most(intrinsically) linear models should no longer be used.We should recognize the fact that linear (or linearizable)models are easier and less expensive to estimate, thestatistical results are believed to be more straightforwardto interpret, and when used as aids in decision making,optimal allocation rules are simpler to derive and toapply. In a cost-benefit sense, predicted market sharevalues from an intrinsically linear model may oftenprovide sufficiently close approximations.*Furthermore, estimated parameters are often used asprior estimates, which are subsequently adjusted bymanagerial judgment. Defining an approximate modelmay then perhaps be even more acceptable. For a dis-cussion of some of these issues, see :Lambin [5, pp .120-1].CONCLUSION

In this article we derived restrictions on explanatoryvariables, parameters, and disturbances implied by anadditivity constraint on the dependent variable, e.g.,market share. These restrictions are such that linearmarket share structures do not allow for differences inthe response parameters for various brands. We arguedthat to be logically consistent, market share modelswill often be intrinsically nonlinear. We examined logi-cal consistency as a criterion for judging model struc-ture and argued that this should be looked at from acost-benefit point of view.

APPENDIXTheorem: The necessary and sufficient conditions foa linear model.

that is, competitors' market share is a decreasing func-tion of company ;'s advertising expenditures, but does'The authors are currently investigating the cost-benefiaspects in more depth based on empirical studies as well as bymeans of simulation.

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L O G I C A L L Y C O N S I S T E N T M A R K E T S H A R E M O D E L S 339

to predict sum-constrained dependent variables , i.e.,a'yt = rt, are:(a ) u'X,, = Cjt, for ally and ;,(b ) Pij = ;9', for all / and al ly 5 1,(c) u'3i -h h^'cu = rt, f o r a l W .

Sufficient Condition: If u'Xy< = c,, for all j and /,and 0ij = /3 ' for all i and for; = 2, , ; ? , and if u'^i -|-2ZIL2 0'Cjt = /(, it follows that u'y< = rt.Proof: First we take the sum of the dependent vari-ables(19) u'v, = y x'-,Bi

Xit is a vector of ones, and thus X(,/3i = u'^i. Foral ly not equal to one 3j is a vector of constants /8, i.e.,3 ,' = (|3 ' , |3 ' , , 0'), and therefore we can write:With u'Xjt = Cjt, we obtain after substitution in (19):(20) u'y< ^ u'^i -f ; = 2With the right-hand side of (20) equal to r,, the proofof sufficiency is complete .Necessary Condition: Suppose now tha t the depend-ent variables are sum-cons tra ined, i.e., u'y< = /

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340 JbURNAL OF MARKETING RESEARCH, AUGUST 197 3REFERENCES

1. Beckwith, Neil E. "The Rfcsponse of Competing Brands toAdvertising: A Multivariate Regression test," unpublisheddoctoral dissertation, Pufdiie University, 1970.2. . "Multivariate Analysis of Sales, Responses ofCompetihg Bj^nds to Advertising," Journal of MarketingResearch, 9 (Nlay 1972), l6S-iiS.3. Hartung, Philip H. and James L. Fisher, "riband Switchingand Mathematical Programmihg in Market Expansion,"Man agemen t Science: Application, 2 (August 1965), 23 1-4 3 .4. Kuehn, Alfred A., Timothy W. McGuire, and Doyle L.Weiss. "Measuring the Eflfectlvehess of Advertising," inScience Technology and Marketing, American MarketingAssociation, 1966, 185-94; also in John J. Wheatley, ed..Measuring Advertising Effectiveness. Homewood, 111.:Richard D. Irwin, 1969, 30-9.5. Lambin, Jean-Jacques. "A Computer On-Line MarketingMix Model," Journal of Marketing Research, 9 (May 1972),119-26.6. Little, John D. C. "Models and Manager^: The Concept ofa Decision Calculus," Management Science, 16 (April 1970),B466-85.7. Mcduire, Timothy W.) John U. FaHey, Robert E. Lucas,

Jr., and W ihston L . Rihg. "EstiiilbtioH, and Infereflde forLinear Models in Which Sllbsets of the Dependent Vii'ldbleare Constrained," Journal vf the AnieAcdn Statistical Aiio-ciation, 63 (December 1968), 1201-13 .8. Naert, Philippe A. and Alain Bultez. "An Aggregate kfetailOutlet Location Model," unpublished working paper, AlfredP. Sloan School of Managemeht, Massachusetts Institute bfTechnology, 1972.9. Naka nishi, Masao. "Measurem ent of Sales Promo tion Ef-fect at the Retail Level: A NeW Approach," unpublishedworking paper. Graduate School of Management, Univei'-sity of California, Los Angeles, 1972.10. Schmalensee, Richard Lee. "On the Economics of Adver-tising," unpublished doctoral dissertation, MassachusettsInstitute of Technology, 1970.11. Theil, Henri. Principles of Econorrietrics. New York: JohnWiley & Sons, 1971.12. Urban, Glen. "A Mathematical Modeling Approach toProduct Line Decisions," Joi4rnal of Marketing Research, 6(February 1969), 40 -7.

13 . Zellner, Arnold. "An Efficient Method of Estimating Seem-ingly Unrelated Regressions and Tests for AggregationBias," Journal of the American Statistical Association, 57(June 1962), 348-68.

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