Modeling the Distribution of Price Sensitivity and Implications for Optimal Retail Pricing
Author(s): Byung-Do Kim, Robert C. Blattberg, Peter E. Rossi
Source: Journal of Business & Economic Statistics, Vol. 13, No. 3, (Jul., 1995), pp. 291-303
Published by: American Statistical Association
Stable URL: http://www.jstor.org/stable/1392189
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@ 1995AmericanStatisticalAssociation
Modeling the
Sensitivity and
Optimal Retail
Statistics,July1995,Vol.13, No.3
Journalof Business&Economic
Distribution of
Price
for
Implications
Pricing
Byung-Do KIM
PA 15213
GraduateSchoolof Industrial
Administration,
Pittsburgh,
CarnegieMellonUniversity,
Robert C. BLATTBERG
Northwestern
Evanston,IL 60201
University,
KelloggGraduateSchoolof Management,
Peter E. Rossi
of Chicago,Chicago,IL 60637
GraduateSchoolof Business,University
of pricesensitivity
acrossconsumers.We employa
Thisarticlefocuseson the distribution
andprice-slope
coefficients
are
random-coefficient
intercepts
logitmodelinwhichbrand-specific
allowedto varyacrosshouseholds.Themodelis estimatedwithpaneldatafortwoproduct
of the estimatedmodelare deducedthroughan optimalretail
categories. The implications
costfigures.Wetest parametric
pricinganalysisthatcombinesthe paneldatawithchain-level
distributional
densityestimatesbasedon seriesexpansions.
assumptions
usingsemiparametric
Random-coefficient
KEYWORDS:
Optimal
pricing;
logit.
Heterogeneity;
Marketingresearchershave long recognized that differences among consumersplay an importantrole in the development of pricing policy and the positioning of consumer
products. Consumerpreferencesor perceivedqualityof different brandswithin a productcategory are criticalin determining the pricing of existing brandsas well as in planning
the introductionof new brands.In additionto qualityperceptions, the distributionof reservationpricesfor a given level of
perceivedquality,or theprice sensitivityof consumers,is also
fundamentalto the pricing decision. Traditionalcategorymanagement models do not incorporateconsumer heterogeneity [see BlattbergandNeslin (1990) for a briefdiscussion
of category-managementmodels]. The goal of this articleis
to demonstratethe importanceof proper modeling of consumerheterogeneityin the context of the pricingproblem.
The availabilityof detailedhouseholdpaneldatacombined
with the application of econometric methods for handling
unobservedheterogeneityhas fostereda rapidlygrowingliterature in marketing. Table 1 summarizesthe models for
heterogeneity and estimation methods used in the choice
literature.In the choice-model context, Guadagniand Little
(G&L) (1983) were among the first to recognize that traditional demographicvariableswere not sufficientto explain
the differentpatternsof productloyalty observed in household panel data. Their solution is to introduce a "loyalty"
variable, which has the effect of making the interceptsof a
logit model vary accordingto the pastpurchasehistoryof the
household.The G&L approachhas been adoptedby manyin
the choice field. Kamakuraand Russell (1989) used a finitemixture random-coefficientapproachto modeling heterogeneity thathas also attracteda large following in the choice
literature.Chintagunta,Jain, and Vilcassim (1991) reviewed
many parametricapproachesto interceptheterogeneityand
comparedthese to a nonparametricapproach. Recently, researchersareexperimentingwith models in which the slopes
of the price variable,as well as the intercepts,are household
specific. Allenby and Lenk (1994), McCulloch and Rossi
(1994), andGonulandSrinivasan(1993) implementedchoice
models in which both the interceptsand the slopes vary according to some joint distributionover households. Table 1
summarizes some of the key works in the heterogeneity
literature.
Althoughthe recent literaturefocuses on methods for estimationof random-coefficientchoice models, there has not
been a carefulassessmentof the importanceof heterogeneity
and, in particular,slope heterogeneity,for actual marketing
decisions. Allenby and Rossi (1991) were among the firstto
use a retailerpricingproblemas a meansof comparisonof alternativechoice models, but they did not considerthe impact
of incorporatingheterogeneity.Vilcassim and Chintagunta
(1992) discussed the problem of optimal retail pricing in a
model with interceptheterogeneitybut made no assessment
of the importanceof heterogeneityin terms of profitsor optimal prices. Gupta(1993) extendedthe model of Vilcassim
and Chintaguntato include slope heterogeneityand concentratedon derivingoptimaldynamicprice discountschedules.
Again, he did not evaluatethe substantiveimportanceof incorporatingheterogeneityin the model in termsof its effects
on the dynamic schedule of optimal prices. While he emphasized the dynamic problem of choosing a promotional
or discountingschedule over time, our work focused on the
problemof choosing the regularprice level against which a
schedule of discounts of the sort derived by Gupta can be
applied.
291
292
Journalof Business&Economic
Statistics,
July1995
Table1. Heterogeneity
in ChoiceModeling:
of theLiterature
Summary
brands.
analysisto groupsof similaror highlysubstitutable
Thereis animplicitassumption
thatgroupsof similarproductsareweaklyseparablein the householdutilityfunction;
Distributional
Heterogeneity
thisreducesthesize of thedemand-system
model
Study
type
parameterization.
In addition,thepaneldatacommonlyavailableto marketing
andLittle Intercept
None.Weighted
Guadagni
average
researchers
areonlyavailablefora few productcategories.
pastpurchases
(1983)
The
demand
for a categoryor groupof brandsof a given
Kamakura
andRussell Intercept/slopeDiscretemixture
different
sizes andbrandsof cannedtunafish)
product(e.g.,
(1989)
et
Discrete
mixture
al.
can
be
broken
into
two
beIntercept
Chintagunta
components.Thesubstitutability
(1991)
tweenbrandsin thisgivencategoryandotherproductswill
RossiandAllenby
Intercept/slopeBayesianfixedeffect
determinethe overallcategorydemand(sometimestermed
(1993)
"categoryexpansion"in the marketingliterature)and the
andLenk
Intercept/slopeBayesianrandom
Allenby
coefficient
substitutability
amongbrandsin the category.The substi(1994)
logistic
in the categoryis modeledby brandof
brands
regression
tutability
andRossi Intercept/slopeBayesianrandom
McCulloch
choiceormarket-share
models.Inthisarticle,we will focus
coefficient
multinominal on
(1994)
modelingheterogeneityin the brand-choice
portionof
probit
the
In
model.
category-demand
manycategories,suchas the
GonulandSrinivasan Intercept/slopeRandom-coefficient
logit
considered
decilater,the quantity-choice
ketchupcategory
(1993)
sion is not criticalbecausemost consumersbuy only one
andcategory-purchase
decisions
unit;it is the brand-choice
thatare mostimportant.Evenfor the categoriesin which
A pointof departure
forouranalysisis theuseof themodel
multipleunitsarepurchased,
priceplaysmuchmoreof a role
to solveanoptimal-retail-pricing
problem.Weusecostdata
in the brand-choice
decisionthanin the category-purchase
obtainedfroma largeChicagogrocerychainto solve for
decision(seeChiang1991).
regular(shelf)retailprices.In additionto
profit-maximizing
We follow the standardrandom-utility
frameworkintroprovidinginsightsinto the optimalityof the existingretail
ducedby McFadden(1973)to formulatea household-level
analysisprovidesauseful
pricingsystem,theoptimal-pricing
choice model.To fix the notationand clarifythe sources
tool.
model-evaluation
of randomness,we will brieflyreview this approach.If
contributions.In
We also makeseveralmethodological
we assumethatthe householdsubutilityfunctionover the
modelsin marketing,
the applicationof random-coefficient
brandsin the productcategoryis linearwithmarginalutilformfor the
it is commonto assumea specificparametric
of brandj exp(V0),then the choice model is derived
ity
distribution
of coefficientsacrosshouseholds.We employ
fromthe first-order
conditions,and we choosebrandj iff
a seminonparametric
densityestimatordue to Gallantand
exp(VPp)/pj
>_ exp('m.)/pm for m = 1,2,... ,J (J brandsin
of a lognormal
Nychka(1987)to checkourassumption
slope
thecategory);
pj is thepriceof brandj.
distribution
of thepricecoefficient.It is alsocommonto reTo developaneconometricspecification,
anerrortermis
strictanalysisto a smallsubsetof thetotalnumberof houseintroducedinto the marginalutilityof brandj. We write
holdsin the panel.Frequently,
the sampleof householdsis
the marginalutilityof consumeri (i = 1,... , I) for brand
tohouseholdswhohavemadeoveracertainnumber
restricted
j
(j = 1,... ,J) on purchaseoccasionk (k = 1,..., Ki) as
of purchasesin theproductcategory(particularly
forstudies
= exp(,ij)exp(eijk). The marginalutilityconstantVii
uijk
thatemploya G&Lloyaltymeasure).KimandRossi(1994)
variesacrossconsumersas well as brands,reflectingdifferdemonstrated
a strongbiasfromincludingonly households
ent levelsof intrinsicbrandpreferencefor differenthousewithhighvolumeor frequencyof purchase.In ourcontinuholds. It is important
to differentiate
betweenrandomness
itis notnecessarytorestrict
ousrandom-coefficient
approach,
induced
differences
between
households
thatareunobby
the sampleto householdswithlongpurchasehistories,and
servableto thedataanalystandrandomness
acrosspurchase
we use the full sample of over 3,000 households.
The organizationof the article is as follows: Section 1
introducesthe model and lays out the statistical specification, Section 2 discusses the data and parameterestimates,
Section 3 discusses optimal retail pricing, Section 4 discusses methodological issues, and Section 5 provides some
conclusions.
1.
MODEL AND STATISTICALSPECIFICATION
To formulatepricing and positioning strategies,we must
firstdevelop and estimatea demandsystem for the items under consideration. At the lowest level of UniversalProduct
Code (UPC) aggregation,the averagesupermarketcontains
some 25,000 to 40,000 items. Itis common,therefore,to limit
occasionsforthe samehousehold. The errorterm,eik, should
be viewed as representingfactorsaffectingpurchasebehavior
beyondthe includedprice variableconditionalon the values
of household-specificparameters.Later we will introducea
random-coefficientspecification that will capture variation
across householdsin 4'and otherkey parameters.
As is well known, the distributionof the errorterms will
determinethe functionalformof these probabilities.Withthe
errorsassumedto be iid as theTypeI extremevalue,we obtain
a standardlogit specificationwith slopes and interceptsthat
vary acrosshouseholds:
P(J)
=
exp(?fr,- 1/ai In pijk)
2,,m
exp(4,• - 1/ai In Pik)
andRossi:PriceSensitivity
andOptimal
RetailPricing
Kim,Blattberg,
Note that the Y'are normalizedintercepts,V5'= VPij/ai;ai
is the scale parameterfor the errorterm for household i; ai
representsthe relativesize of the unobservablecomponentof
the ith household's behaviorto thatdeterminedby the intercept parameters,V)',andprices. We can interpretthis termby
writingthe price coefficients as 3 = - 1/ai. Householdsthat
are influencedprimarilyby price and not by otherconsiderations will have a low value of ai and a very large(negative)
pricecoefficient,which will makethemverysensitiveto price
changes.
To summarize,we have now specified and interpretedthe
parametersof a logit model with,interceptsand a price coefficient that vary across households:
exp(fo, + /, In pij)
293
evidence from these individuallogit coefficient estimatesto
supportthefirstassumptionof independence.Frombothscatterplotsand correlationanalysis, we could detect no relationship at all between the slope and intercepts.(We constructed
a sample of households with 10 or more purchasesand for
which the individual-levelestimatesexist. This leaves a sample of 225 households.The correlationbetweenthe intercepts
and slope for these estimatesis .00091.) For a sample of 100
households,Allenby and Lenk (1992) found only weak evidence of correlation.[Allenby and Lenk (1994) allowed for
price, display,and featureeffects to be correlatedwith three
interceptterms.Of a total of nine covarianceterms,only one
has any appreciablemass away from0. Allenby andLenkdid
not, however,compute the posteriordistributionof the correlationcoefficient so that it is difficultto gauge the strength
(J)=exp(in + 3, Inp,,)k)
of theirevidence againstthe assumptionof zero correlation.]
istoallowfordifferent
Theroleof0' parameters
of
patterns
Allenby and Lenk (1994), McCulloch and Rossi (1994),
brand
across
whereas
thepricecoef- andGonulandSrinivasan(1993) assumedthatthe pricecoefconsumers,
preference
inpricesensitivity.
ficients
allowfordifferences
ficient is normallydistributed.Given the overwhelmingdeintheparameters
Tomodel
theheterogeneity
ofthehouse- mand theoretic argumentsand empirical evidence that the
arandom-coefficientprice coefficient must be negative, we decided instead to
holdlogitmodel
givenby(1),weadopt
framework (e.g., see Heckman 1982). In this approach,
employ a reflected lognormal distributionthat is only deeach household is viewed as obtainingits parametervector
fined over negativevalues. Our assumptionof lognormality
=
a
draw
from
some
as
is strongly supportedby nonparametricdensity estimation
1,...,
superpopulation
J,
(Oii,j
J3/)
distribution.The form of the heterogeneitydistributionis the
methodsappliedto ourdata as shown in Section 4.1.
Our third assumption concerning the nature of heterokey modelingdecision in random-coefficientmodels. Forany
reasonablenumberof brands,this J-dimensionaldistribution
geneity in the interceptor qualityperceptionterms deserves
furtherdiscussion. We exploit certain observed patternsof
(J - 1 interceptsand one slope coefficient)can be quitecomplex and highly parameterized.It is common, therefore,to
loyalty that characterizethe data. For example, in the tuna
restrictthe dimensionalityof the problemby eithereliminatcategory, there is strong loyalty to form [e.g., households
are loyal to the form in which canned tuna is packed (in
ing heterogeneity in some of the parametersor simplifying
the structureof the multivariatedistribution.
oil or water) not to brands].In the ketchup category, only
Our approach is to build a parsimonious randomone nationalbrandearns any appreciabledegree of loyalty.
coefficientmodel thatcapturestheessentialfeaturesof houseThus, there is only one majordimension in quality percephold behavior without the introductionof many potentially
tions along which households differ. We model this by inWe
will
identified
the
exact
choice
poorly
parameters.
justify
cluding a "type"-shiftervariableinto the logit model and by
of our model specification by examinationof the purchase
allowing this variableto be randomacross households. We
believe thatthis approachcapturesthe salient featuresof inpatterns in our data and by comparison to less restricted
models. Three key assumptions are made in the developtercept heterogeneitywithout the introductionof many pament of our random-coefficientmodel: (1) The slope and
rameters.In addition,the form of interceptheterogeneitywe
interceptsare assumedto be independent,(2) the negativeof
adoptis easily interpretablefrom the marketingperspective.
the price-sensitivityparameteris assumedto be lognormally
Section 4.2 provides a comparisonto an unrestrictedmodel
distributed[i.e., the price-sensitivityparameter,/, is parameof interceptheterogeneitythat supportsthese views.
terizedas / = - exp(y), 7 - N(ji, c)], and (3) heterogeneity
Under these assumptions,the likelihood for a sample of
in the interceptis restrictedto a one-dimensionaldiscreteranhouseholdsgiven in the following equationinvolves averagdom variable. Each of the intercepts•0 is parameterizedin
ing each householdlikelihoodover thejoint distributionof 7
terms of the loyalty-shiftervariable.We assume thatloyalty
and7y.In the equation,•' is the vector of the J - 1 identified
patternsare of two "types,"A andB; for example,A is oil and
intercepts:
B is water:
Id= qi +7"Dj ,
(2)
where Di is an indicatorvariablethatswitches on if brandjis
of the certainkey loyalty type and 7i is the randomdrawfor
household i from the loyalty heterogeneitydistribution- ,iidf.
Before making these assumptions,we experimentedextensively with fixed-effects or individuallogit models fit to
householdswith relativelylong purchasehistories.We found
L( ', qA,qB,I, 0")
i=1
i=1
oo
x (r Iq,
k=l j=l
qB)q(7
I,u,a)dlrd-j.
294
Journalof Business &EconomicStatistics,July1995
Here Pik is defined in (1), Yjk= 1 if brandj is purchasedon
occasion k, and 0( ) is the normaldensity function;ni is the
numberof purchaseoccasions for householdi.
In our application,f is a discretedistributionthatputs all
of its mass on the points qAandq, withp as the probabilityof
value qA.To identify the model, E(i) shouldbe 0, leaving 0j
as the mean of Oij.In otherwords,becausethe meanvalueof
the interceptfor each brand,0j, is separatelyestimated,we do
not requirean additionalmeanparameterfor the distribution
of ri. Since E(ri) = qAP + qB(1 - p) = 0, p = qB/(q - qA).
Thus,theprobabilityof'ri = qA(orp) canbe implicitlydefined
as a function of qAand qB. In addition,p is constrainedto be
greaterthanor equalto 0 andless thanor equalto 1 in estimation becausep is a probability.This constrainton the probability (p) can be achievedby restrictingqA> 0 and q, < 0.
RESULTS
2. DATAAND ESTIMATION
2.1 Data
To estimate the price-sensitivitydistributionand the perceived quality level of each brand,scannerpanel data from
A. C. Nielsen on cannedtunaandketchupwereused. Tofacilitatecomparisonsacrosscategoriesandto keep the dataanalysis manageable,we restrictattentionto one "everydaylow
price"chainin Springfield,Missouri. Althoughthepurchasehistory files for each household are complete and accurate,
there are problems in constructingcompetitive prices (see
Kim 1992 for details).
In both categories,we restrictattentionto householdswho
remainin the sample for at least 100 weeks. This was determined from the shopping-occasionfile, which lists all shopping tripsfor the householdregardlessof whetheror not the
householdmadepurchasesin the productcategory.Note that
this is a very differentsample-selectionrule from specifying
a minimumnumberof purchaseoccasions as is common in
the scannerliterature.Oursample-inclusionrule is free from
the choice-based sampling bias that would afflict samples
chosen on the basis of the numberof purchaseoccasions [see
Narasimhanand Renken (1991) for more discussion on this
point]. It is possible, however,that our sample suffers from
attritionbias, as discussed by Winer(1983). Comparisonof
measureddemographicvariablesof the entirepopulationof
householdswith our sample of householdswho remainedin
the panel shows negligible differences.
The tunafishmarketis complex in the sense thattunafishis
availablein variousforms (e.g., water vs. oil, light meat vs.
white,etc.) anddifferentsizes (e.g., 6.5 oz., 3.25 oz., 9.25 oz.,
etc.). Furthermore,for each form and size thereare national
brands,privatelabels, and generics. We restrictattentionto
"lightmeat 6.5 oz." brandsbecause this type dominateswith
more than 90% of the market. We will analyze the top four
nationalbrandsand one store-specificbrandfor each chain.
One reason for including a store brandin our analysis is to
obtaina wide dispersionof perceivedproductquality,which
will reducethe variancesof estimatesof interceptparameters.
Table2 shows summarystatisticsfor each UPC of canned
tunaanalyzed. Three"water"brands,Starkistwater(SKW),
Table2. SummaryStatisticsfor Tuna
ADPa
SKWd
COSW
PW
SKO
COSO
.75
.80
.63
.75
.78
WAPb
.69
.75
.63
.73
.70
MSc
44.2
16.3
7.7
17.8
14.0
NOTE: Thetotalnumberof purchasesis 13,705. The numberof householdsis 3,093. The
distribution
of numberof purchasesforthe householdhas a mean of 4.43 and a median
of 3; 0Q1= 1, 03 = 5.
aADP(averagedailyprice)representsthe dailyaverageshelf price.
bWAP(weightedaverageprice)representsthe dailyaverageshelfpriceweightedbydaily
sales.
CMSrepresentsthe marketshare.
dSKWis "Starkist
COSWis "Chicken-of-the-Sea
Labelwater,"
water,"
PWis "Private
water,"
SKOis "Starkist
oil"
oil:'COSOis "Chicken-of-the-Sea
Chicken-of-the-Seawater (COSW), a store-specificprivate
label, and two "oil"brands,Starkistoil (SKO) and Chickenof-the-Seaoil (COSO),areincludedin the analysis. Theprice
measures,averagedaily price (ADP) and weighted average
price (WAP),arecomputedfor each UPC. The averagedaily
price is simplythe daily averageshelf price, andthe weighted
average price is the daily average price weighted by daily
sales. The weighted averageprice is computedto determine
the percentageof sales for a given UPC thatare made during
a promotionalperiod. For example, compare the average
and weighted averageprices of SKO and COSO. The ADP
of SKO ($.75) is lower than that of COSO ($.78), but the
WAP of SKO ($.73) is higher than that of COSO ($.70).
This implies that much of the COSO sales are made during
promotionalperiods.
The total numberof panelists is over 3,000, and the total
numberof purchaserecords is over 13,000 in all chains for
canned tuna. The large numberof panelists makes it possible to more accuratelyestimate the price-sensitivitydistributionacross panelists. Many other studies of household
heterogeneityuse samples with a much smaller numberof
households[for example,Chintaguntaet al. (1992) used 135
panelists,GonulandSrinivasan(1993) used 152 households,
and Allenby and Lenk (1994) used 100 households]. Only
KamakuraandRussell (1989) with 585 householdsandRossi
and Allenby (1993) with 777 householdsused a large number of households. The mean numberof purchasesfor each
household is four during the observationperiod (e.g., 120
weeks from Februaryin 1985 to May in 1987). There are
many households with only one or two purchase records.
The total sample of panelists is used to estimate the model
because we are interestedin the behaviorof all panelists.
In the choice literatureand particularlyin those studies
that employ the G&L loyalty measure, it is common to
eliminate households with short purchase histories. Kim
and Rossi (1994) demonstratedthat there is a strong bias
from restrictingthe analysis to a sample of households with
long purchasehistories and/or large volume/frequency. In
a random-coefficientapproachto modeling heterogeneity,it
is not necessaryto restrictanalysis to households with long
purchasehistories because even households with only one
choice observationadd informationaboutthe distributionof
interceptsand slopes in the populationof households.
Kim,Blattberg,and Rossi: PriceSensitivityand OptimalRetailPricing
of WaterPurchasesinTuna
Proportion
295
of Heinz
Proportion
r
CD
0
c0
0.0
0.2
0.6
0.4
0.8
1.0
0.0
0.2
0.4
0.6
0.8
proportion
proportion
of SKamongWaterTunaPurchases
Proportion
of HuntsamongNonHeinzPurchases
Proportion
a)
1.0
a
0
o
0.0
0.2
0.4
C0
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
proportion
proportion
Figure 1. Formand BrandLoyaltyforTuna.
Figure2. BrandLoyaltyin Ketchup.
The tuna panelists display an interestingpatternof form
but not brandloyalty as shown in Figure 1. The top portion
of the figureis a histogramof the proportionof water-packed
tuna purchases by household. Most households purchase
only water-packedtuna, with a small minoritybuying tuna
in oil. The bottom portion of the figure is a histogramof
the proportionof purchasesof Starkistamong water-packed
tunapurchasesfor each household. Veryfew householdsare
loyal to Starkist. The price of Starkistand Chicken-of-theSea are very similar with frequentpromotionsthat cause a
great deal of brandswitching. Given this patternof loyalty,
it seems most importantto capturethe differences in tastes
for oil- versus water-packedtunaratherthanto try to model
subpopulationswith specific brandloyalty. Thisjustifies our
use of an oil/watershiftervariablein the specificationoutlined
previously.
Table 3 presents the summary statistics for the ketchup
data. Some 2,000 panelists made nearly 5,000 purchases
from among four majorbrandsof ketchup. Heinz is by far
the market-shareleader,with the storebrandandHuntsgrappling for second place. The differences between WAP and
ADP suggestthatHeinz is the most promotedbrand,whereas
the store brandis the least promoted. Due to the low purchase frequencyof ketchup,thereare very few purchasesper
householdin this category.
Figure 2 is designed to illustrate the key characteristic
of ketchuployalty. We see a significant fraction of households who purchaseonly Heinz brandketchup, but there is
a good deal of switching among other national brandsand
the private-labelbrand.Again, our strategyis to use a Heinz
shiftervariableto captureheterogeneityin the perceivedquality of Heinz. It appearsthat some fraction of households
perceivethatHeinz has a significantlyhigherqualitythanthe
otherbrands.
Table 3. SummaryStatisticsforKetchup
2.2 EstimationResults
Brand
ADP
WAP
MS
Heinz
Hunts
Del Monte
Store brand
1.32
1.36
1.43
.92
1.25
1.34
1.42
.92
51.0
20.6
5.2
23.3
NOTE: Allbrandsare 32 ounces. The totalnumberof purchasesis 4,956. The number
of the numberof purchasesfora householdhas a
of householdsis 1,956. The distribution
mean of 2.53 and a medianof 2; 0Q1= 1, 03 = 3.
In this section, we discuss the resultsof fittingourrandomcoefficient specification to each of the product categories.
First, it is useful to documentthe evidence in the data that
supports the assumptions of heterogeneity.As many have
noted (see, in particular,Chintaguntaet al. 1991), incorporatinginterceptheterogeneityis very importantin improving
model fitandexplanatorypower. It is interestingto ask: What
296
Journalof Business &EconomicStatistics,July1995
Table 4. ModelSelection Criteria
Model
Log-likelihoodParameters AIC
Homogeneouslogit
BIC
-16,264
5
-16,267 -16,288
-15,201
7
-15,205 -15,234
-11,553
8
-11,557 -11,591
Intercept
heterogeneityonly
Slope and intercept
heterogeneity
Table 5. ParameterEstimatesforthe TunaCategory
Parameterestimates*
Intercept
SK water
1.016
(.019)
COSwater
Storewater
.000
-1.608
(.021)
-.154
(.043)
-1.125
(.049)
SK oil
is the marginalcontributionfrom accommodatingslope heterogeneityas well as interceptheterogeneity?
To address this issue, we fit successive variantsof our
model to the entire sample of 13,705 tuna purchases. We
startwith a highly restrictedmodel in which all heterogeneity has been eliminated by restrictingall coefficients to be
constant across households. We then free up the intercept
coefficient by allowing form-loyaltyheterogeneity.Finally,
we allow both interceptand slope heterogeneityby allowing
the varianceterm in the slope-coefficientdistributionto be
freely determinedby the data. The results are summarized
in Table4. AIC is the Akaike informationcriterion,AIC =
LL - q/2, whereLL is log-likelihoodandq is the numberof
parameters.BIC is the Bayesian informationcriterionintroducedby Schwarz(1978), BIC = LL - 1/2q In(v),wherev is
the degrees of freedom. Unlike the AIC, the BIC is a consistentmodel-selectioncriterion.Both the AIC andBIC figures
dramaticallyemphasizethe importanceof slope heterogeneity. By adding only one parameterto the model, the loglikelihood increases by over 20%. Furthermore,it appears
thatslope heterogeneityis relativelymore importantthaninterceptheterogeneityin this dataset becauseimprovementin
fit fromintroducingthe slope heterogeneityis approximately
four times the improvementfrom introducinginterceptheterogeneity(this findingis robustto the orderof introduction
of intercept/slopeheterogeneity).
Theparameterestimatesforthe tunacategoryarepresented
in Table5. In the tunaspecification,the shiftervariableis an
oil/waterindicatorvariablethattakes on two values, qoi'and
qwaer. The interceptsare relatedto this variableas Vhj= hj+
rDj, whereDj = 1 if the brandis oil packed. The oil constant
is estimatedat 2.218, and the waterconstantis set to -.854.
Thus,the "oil-loyal"householdsact as if the interceptsfor the
oil brandsare equal to the estimates+2.218, but the "waterloyal" households add -.854 to the interceptsof oil brands.
As mentioned previously, identificationrestrictionsrequire
thatthe mean of 7 be set to 0. This allows us to computethe
proportionof oil-loyal households from the values qoil and
qwater;
P =
qwater/(qwater
-
qoil).
We can insert the estimates
of the oil and water constantsinto this expressionto obtain
the maximum likelihood estimate (MLE) of the fractionof
oil-loyal households of .28 with a standarderrorof .013.
The mean and standarddeviation of the 7 distribution
are not particularlyinterpretableparametersin and of themselves. Forthis reason,we will use these parameterestimates
to compute the implied moments for the price-sensitivity
coefficient distributions. The price-sensitivity coefficient
f = -exp(7). The mean and standarddeviation of the
COS oil
Gammadistribution
1.523
(.026)
a
.794
(.027)
Shifterconstants
qoil
2.218
qwater
-.854
(.054)
(.043)
NOTE: The log-likelihood
is -11,326.03, the numberof purchases is 11,427, and the
numberof householdsis 2,593.
* Standarderrorsare in
parentheses.
distributionof 3 implied by the estimatesof p1and a are
f, (pricesensitivity)
Mean
-6.29 (.12)
Std. dev.
5.89 (.14).
The estimatedstandarddeviationof 5.89 shows the dramatic
variationfrom householdto householdin price sensitivity.
Results for the ketchupcategory are given in Table 6. In
the ketchup specification, we introduce a dummy variable
that is a contrastbetweenHeinz and all otherbrands,DA= 1
if not Heinz, 0 if Heinz. Householdswho are loyal to Heinz
Table 6. ParameterEstimatesforthe KetchupCategory
Parameterestimates*
Intercept
Heinz
.782
(.058)
.000
Hunts
Del Monte
Private
Gammadistribution
/
a
Shifterconstants
qothers
qHeinz
-1.139
(.066)
-2.265
(.069)
1.715
(.039)
.618
(.041)
.946
(.075)
-1.920
(.146)
NOTE: Thelog-likelihood
is -2.696.29, the numberof purchasesis 4,956, andthe number
of householdsis 1,956.
* Standarderrorsare in
parentheses.
Kim,Blattberg,and Rossi: PriceSensitivityand OptimalRetailPricing
Tuna
0
0
-25
-20
-15
-10
-5
0
pricecoefficient
Ketchup
-25
-20
-15
-10
-5
0
pricecoefficient
Distributions.
Figure3. PriceCoefficient
subtractan estimated 1.92 from the interceptsof all other
brands.The proportionof householdswho areloyal to Heinz
can be inferredfrom the shifter constantsto be .33 with a
standarderrorof .012. Again, we find very substantialpricesensitivity differences across households. The moments of
the price-sensitivitydistributionsare
p3(price sensitivity)
-6.73 (.12)
Mean
4.60 (.22).
Std. dev.
Figure 3 shows that the price-sensitivitydistributionsfor
the tuna and ketchup categories are quite similar.The high
varianceof the price-sensitivitydistributionsis striking,butit
remainsto be seen if this high degreeof heterogeneityaffects
the outcome of key marketingdecisions such as productcategorypricing.In the next section, we explorethe implications of the heterogeneitydistributionforthepricingproblem.
2.3
Price Elasticities
One importantway of summarizingthe effect of price is
to compute the price-elasticity matrix. This matrix shows
all own- and cross-price elasticities between the set of
brands. Note that we define elasticity as the derivativeto
choice probability with respect to the logarithm of price
with all other prices set to their sample averages. Matrices for the homogeneous andheterogeneouslogit models are
297
Homogeneouslogit
skw cosw
.57
skw
-2.0
1.8 -3.2
cosw
.57
1.8
pw
1.8
.57
sko
1.8
.57
coso
pw
.38
.38
-3.43
.38
.38
sko
.74
.74
.74
-3.07
.74
Heterogeneouslogit
skw cosw
-3.6
kw
.45
1.7 -4.2
cosw
.46
2.8
pw
sko
1.7
.38
1.5
coso
.36
pw
.87
.55
-6.6
.79
.58
sko
coso
.50
1.8
.45
1.54
.60
2.7
.60
-3.46
1.96 -4.44.
coso
.30
.30
.30
.30
-3.5
Of course, the homogeneous logit elasticities exhibit the
well-known proportional-drawproperty,which implies that
the elasticities are proportionalto marketshares. The heterogeneous logit model displays a richer patternof crosselasticities and generally larger own-price elasticities. The
implicationsof these elasticities for the optimal pricing decision are not straightforward.For example, one cannot use
a simple elasticity-basedmarkuprule. In Section 3, we pose
a stylized version of the category-pricingproblemand show
how heterogeneityaffects the optimal-pricingproblem.
3. OPTIMAL
RETAIL
PRICING
3.1 The Optimal-Pricing
ProblemUnder
Heterogeneity
The growing literatureon household heterogeneity has
focused mainlyon the importantmethodologicalissues of the
form of the heterogeneitydistributionand estimationmethods. In this section, we demonstrateboth the importanceof
heterogeneityand the usefulness of our random-coefficient
model by applying the model to a version of the retailer's
optimalpricingproblem.
Recently, some researchershave started to examine the
retailer problem using models fitted to panel data. In a
modelwithoutslope heterogeneity,Allenby andRossi (1991)
posed a highly stylized retailer problem as a method of
evaluating a nonhomotheticchoice model. Vilcassim and
Chintagunta(1992) consideredpricingproblemswith household heterogeneity in "intrinsicbrand preferences"(intercepts) and in the household consumption rates but not in
price sensitivity. Furthermore,the main emphasis of their
paperwas on promotionalissues such as durationand depth
of deals ratherthan on regularshelf pricing, which is our
focus. None of the preceding works used actual cost data
but, instead, made assumptionsabout the size of nationalbrandand private-labelmargins.
The general problem of determiningan optimal retailer
strategymust involve many possible policy tools including
choice of regularorlong-runaverageprices,choice of promotionaldepthandfrequency,optimalpass-throughof manufacturerpromotions,featureand display policy, and reactionto
policies of competingchains. A full analysis of this problem
298
Journalof Business&Economic
Statistics,
July1995
wouldrequirea complex demandmodel of consumerbehavior that would take into account consumer expectationsof
futurepromotions,couponing, and inventorydecisions coupled with a complex model of the supply side that would
includeexit and entryof retailersandstrategicdetermination
of pricingpolicies. A fully articulatedand reliablemodel of
these complex featuresof the retailerproblemsdoes not exist and would have very formidabledata requirementsonce
developed.
To keep the problem of optimal pricing manageable,we
have made some simplifying assumptions. We focus on the
optimal choice of regular or shelf prices conditional on a
given promotional strategy. We assume that promotional
policies such as the depth and frequency of deals, as well
as feature/displayuse, remainin place while the level of the
variousprice series is variedto maximize retailerprofits. In
discussion with majorgroceryretailers,the retailerconveys
some degree of confidence in his promotionalstrategybut
often has little idea of how to set shelf prices for each item
within categories. One of the reasonsrelativepricingwithin
a categoryhas been extremelyproblematicis thatretailersdo
not know how willing consumersare to pay for brandswith
higherperceivedquality and how to price theirprivatelabel
relativeto nationalbrands.
We did not include promotionalvariablessuch as display
and feature-addummies because the principalaim of the article was to addressthe issue of the substantiveimportanceof
heterogeneityfor regularor long-runpricingissues. Thatis,
we think of the optimal-regular-pricing
problemas holding
the promotionaltiminganddiscountschedulefixed andvarying the long-runprice. As such, ourestimateof the marginal
distributionof the pricecoefficientis perfectlyvalidfor use in
the analysis. In our opinion,thereis a good deal of confusion
regardingthe problemof omitted-variablebias in marketing
applications. For example, consider the world with no heterogeneity. There are those who would say that the price
coefficientin a model withoutdisplay/featureis inconsistent
because of the omitted and correlatedvariables. The coefficient consistentlyestimatesthe marginalimpactof a change
in pricegiven thejoint distributionof priceandthe othervariables. Thus, when the promotionalpolicy is unchanged,this
is the appropriatecoefficient to use to predict the response
to price. If, on the otherhand, we were derivingan optimal
promotionalpolicy, we would need to add these variables.
The retailer's optimal-pricing problem is posed as a
category-managementproblemin which we focus on determining the prices within only one category at a time. For a
given productcategory,the retailer'sproblemof findingthe
optimalset of prices for each brandcan be writtenas
maximize ir = pi - cj) D (p,,s = 1,...,J),
0 ,... j}
(3)
wherepi is the unit price of brandj(j = 1,..., J), cj is the retailer'sunitbuyingcost of brandj,andDi (ps, s = 1,..., J) is
the numberof units demandedfor brandj, which is a function
of the price of all brandsin the category.
The solution to the retailer's problem requires cost
estimates. Unfortunately,ERIM scanner-paneldata does
not provide information on the wholesale cost of each
brand. Therefore, we have calculated the cost of each
brandusing the margindata supplied by the University of
Chicago/Dominick'sFiner Food project. We restrictour attentionto the tunacategorybecause cost dataare morereadily availablefor that category. All 85 stores of Dominick's
carry all four national brands (e.g., Starkistoil and water,
Chicken-of-the-Seaoil and water) we are interestedin and
a private-labelbrand. The average (percentage)marginfor
each brandis computedacross 85 stores and the 115 weeks
of availabledata. These averagepercentagemarginsfor five
brandsare used to computethe cost of each brandin Springfield (Chain1). Inotherwords,we assumethatthepercentage
marginof each brandin Springfield(Chain 1) is the same as
the averagepercentagemarginof each brandin Dominick's.
To completely specify the retailer profit function, we
couple the choice-model system with a simple log-linear
category-volumemodel to specify the demand system facing the retailer. We assume that the unit demand of brand
j is equal to the category demand, which is a function of
prices of all brandstimes the marketshare of brandj. That
is, Dj(p) = CD(p) MSj(p), whereCD is the categoryunitdemandandMSjrepresentsthe expectedmarketshareof brand
j. Both theCD andMS arefunctionsof the price of all brands.
As an alternative to the aggregate category demand
function approachjust taken, we could have adopted the
Chiang (1991) "outside"good model as a starting micromodel andthenaggregate.We choose to use an approachthat
couples the sort of model thatcan be fit by the retailerusing
store-levelscannerdatawith ourpanel-calibratedlogit. This
is simple to implementand easily interpretable.Moreover,a
simple implementationof the Chiangapproachassumes that
the reasonhouseholdsdo not purchasein the categoryin one
week versusthenextis thatthepricesforall brandsin thecategory exceed theirreservationpricefortheproduct.It seems to
us thatthismisses importantaspectsof theproblem,including
consumerstockpilingandspeculationaboutthe futurecourse
of prices. For these reasons, we choose to keep the analysis
simple anduse an aggregatecategory-demandmodel.
To estimatethe categorydemandfunction,CD, we assume
that the unit categorydemandat week t is a function of the
pricesof all five brandsat week t. Then, we computethe total
weekly unit sales of all five brandsand the weekly average
price of each brandusing the purchasesby all panelists in
Springfield(Chain 1).
The expectedmarketshares,MSr(P), arecomputedby aggregatingour random-coefficientlogit model. We integrate
the choice system over the heterogeneitydistributionconditional on our estimates of the heterogeneity-distribution
parameters:MS(p) = fMSj(p I 9)f(9 I •) dO,where 9 is
the vectorof both the interceptand slope parametersandijis
the vectorof estimatedhyperparameters
of the heterogeneity
distribution.
We can also solve the retailer'sproblem using a homogeneous or constant-coefficientlogit model to compute the
andRossi:PriceSensitivity
andOptimal
RetailPricing
Kim,Blattberg,
householdsis given underthe threesets of pricesandprovides
a measureof the importanceof heterogeneity.
The optimal prices computed under the assumptionof a
heterogeneousmodel differmarkedlyfromthe optimalprices
from a homogeneous logit specification. The homogeneous
optimalprices are much more extreme and have the retailer
dramaticallyincreasingthe price of the waterbrandwith the
highest intercept(SK) and lowering the COS oil brandprice
to a level at which there is almost no margin. On the other
hand,theheterogeneous-modeloptimalpricesaremuchmore
reasonable. We see a lowering of the oil brandprices and a
moremoderateincreasein the SK waterprice. The difference
betweenthe homogeneousandheterogeneousoptimalprices
is accountedfor by underestimationof the price-sensitivity
coefficient in models that do not properly account for heterogeneity. The homogeneous logit model-pricecoefficient
estimate is -3.8, which is much lower than the mean of the
price-sensitivitydistributionin the heterogeneousand model
(-6.3). Retailerswho fail to take into accountheterogeneity
will underestimatethe extent of switching behaviorinduced
by price changes.
Based on the profitmetric, the currentprices are far from
optimal, primarilybecause of the insistence on pricing the
oil- andwater-packedversionsof the same brandequally.The
optimal prices underthe heterogeneousmodel specification
produce a 15% higher level of profits. The prices derived
under the misspecified homogeneous model are associated
with a 5% lower level of profits than is available from the
heterogeneousspecification.In the intensely competitiveretail environment, a change in profitabilityof even a few
per cent is very valuable.Ourresults suggest, however,that
gross pricing errorsthat are based on incorrect inferences
aboutthe "average"or representativeconsumercan be more
importantthan a fine tuning based on proper modeling of
heterogeneity.
expected marketshares in the profitfunction. This provides
an importantsubstantivemetric with which we can measure
the importanceof heterogeneity.
3.2 OptimalPricingin the TunaCategory
We first consider the solution to the optimalpricing exercise for the tuna category. The estimatedcategory-demand
(CD) model is given by
InCD, = 3.95 - 2.07 InpSK,t - 1.70 In Pcos,:
(.31)
(.30)
(.30)
- .35 In ppw,,,
R2 = .43;
(.55)
T= 110,
(standarderrorsin parentheses). Notice that in the estimation of the preceding CD function, the price of each of the
five brandsis not used but instead the share-weightedaverage prices are used for SK and COS because the correlation
between the price of SK oil and SK water(and COS oil and
COSwater)is veryhigh.
In the profit-maximizationproblem,optimalprices aredeterminedby the trade-offamongthe category-demandeffect,
the own-demandeffect, and the cross-demandeffect. The
price of brandj will influence the category unit demandby
the CD function, while it influencesthe sales of otherbrands
by the function MS(j), which is involved in the logit function. The profitfunctionof the retailer(3) is highly nonlinear
and the maximizationproblemdoes not have a closed-form
solution.
Table 7 shows the results of the optimal-pricingexercise.
The left panel of the table shows the averagepriceandthe assumedmargin(fromthe Dominick'sdata). The middlepanel
shows the optimizationresultsfora homogeneouslogit model
in which all parametersare constantacross households. The
column marked"InitialMS" shows the marketshareof each
brandcomputed by evaluatingthe choice system at the average prices in the ERIM data. The column labeled "Opt
MS" presents the marketshare computedby evaluatingthe
choice probabilitiesat the optimalset of prices. The last column gives the new margins assuming that the costs do not
change as a result of the pricing exercise. The right panel
of the table shows the results of an optimal-pricingsolution
thatassumesthatthe market-demandsystem is an aggregated
heterogeneous logit model. At the bottom of the table, the
expected profitreportedin dollarsper week for our panel of
3.3 OptimalPricingin the KetchupCategory
The estimated CD function for the ketchup category is
given by
In CD, = 4.60 - 2.77 In PHeinz,t - 1.15 In PHunts,t
(.25)
(.37)
(.35)
- .02 In pDelM,t- .79 In privat,,,,
(.54)
(.37)
R2= .38;
Table 7. OptimalRetailPricing:ERIMTunaData
Homogeneouslogit
Brands
SKW
COSW
PW
COSO
SKO
Profits
Current
Price
Margin
.75
.80
.63
.78
.75
.27
.27
.26
.27
.27
$31.71
299
Heterogeneouslogit
Initial
MS
Opt.
MS
Opt.
price
Margin
Initial
MS
Opt.
MS
Opt.
price
Margin
47.4
15.0
10.0
8.1
19.5
35.4
19.2
9.0
14.3
22.2
.81
.73
.65
.61
.70
.32
.21
.28
.06
.21
47.3
12.7
13.3
6.4
20.4
35.7
18.2
10.1
12.7
23.2
.79
.77
.64
.71
.70
.30
.26
.27
.23
.21
$34.50
$36.37
T = 110.
300
Journalof Business & EconomicStatistics,July1995
Table 8. OptimalRetailPricing:ERIMKetchupData
Homogeneouslogit
Current
Brands
Price
Margin
Heinz
Hunts
DelM
Private
1.32
1.36
1.43
.92
.27
.27
.27
.27
Profits
Initial Opt.
MS
MS
Opt.
price
47.0
20.0
5.0
28.0
1.20
1.24
1.56
.91
54.0
23.0
2.0
21.0
$11.34
Heterogeneouslogit
Initial Opt.
MS
Margin MS
.20
.20
.33
.26
45.0
20.0
5.0
30.0
47.0
26.0
3.0
24.0
$12.32
Table8 presentsthe resultsof the ketchupoptimal-pricing
exercise in the same format as Table 7 for the cannedtuna category.Again, we see large differencesbetween the
optimal prices computed under a homogeneous versus a
heterogeneous logit specification. The heterogeneousoptimal price for Heinz is much closer to the actual retail
price thanthe optimalprice derivedunderthe assumptionof
homogeneity.
3.4 HeterogeneityParameter-Sensitivity
Analysis
As discussedpreviously,the role of the I anda parameters
in determiningthe shape of the price-sensitivitydistribution
is difficult to determine without careful analysis. Furthermore, the translationfrom changes in the price-sensitivity
distributionto changes in the optimal-pricingexperimentis
complicated. To develop an intuitionfor the role of shape
parameters,we perturb/I and a away from the estimatesfor
the tunadata,plot the resultingprice- and quality-sensitivity
distributions,and resolve the optimal-pricingproblem for
each of the new sets of parametervalues. Table 9 presents
the results of experimentsin which the a parameteris held
fixed at the estimated value and /t is changed. The column
of the table labeled "A"has a value of kIthat is only 50% of
the estimatedvalue, whereas the column labeled "C" has a
value of kI that is 50% largerthan the estimatedvalue. As
ILincreases from .766 to 2.288, the price-sensitivitydistribution shifts to the left with many more households showing a high price sensitivity as shown in the top panel of
Figure 4. As the populationdistributionof price sensitivity shifts towardhouseholdswith high sensitivity,we should
expect that the retailer will not be able to support large
price differentials between low- and high-quality brands.
Opt.
price
Margin
1.30
1.25
1.68
.92
.30
.24
.37
.27
$12.59
This intuition is supportedby the optimal prices shown in
Table 9. With large numbersof price-sensitivehouseholds
(col. C), the difference between national and private label
water-packedprices is markedly smaller than the spread
for a situation with many more households that are price
insensitive.
The bottompanel of Figure4 shows the results of experiments in which iz is held fixed while a is varied. Changes
in o affect primarilythe dispersionof the price/qualitysensitivity distributionswith a small effect on the mean level
of sensitivity. Increasesin the dispersion of price sensitivity that leave the mean relatively unchangedhave a much
SigmaFixed
0
cJ
0
0A
.
C.
S
-
-20
-
-
-
---------
-
~
-10
-15
-5
0
pricecoefficient
MuFixed
Table 9. Sensitivityto Changes in the Shape of the
Case I: Fixed= .782
Price-SensitivityDistribution:
Optimal
prices
A
=
.51j .766
B
1.Op= 1.525
C
1.54 = 2.288
PSKW
.87
.79
.75
Pcosw
PPW
.75
.58
.77
.64
.75
.66
PSKO
.67
.70
.71
Pcoso
.67
.71
.72
-"""---"-" --"-"-"---"'-""(•,
"----"-=-
-20
-15
--"-
--
-"=
-10
-5
0
pricecoefficient
Figure4. Effectsof Shape Parameterson Price-SensitivityDistributions.
Kim,Blattberg,and Rossi: PriceSensitivityand OptimalRetailPricing
Table 10. Sensitivityto Changes in the Shape of the
Case II:u Fixed= 1.525
Price-SensitivityDistribution:
PSKW
Pcosw
PPw
PSKO
Pcoso
301
4. METHODOLOGICAL
ISSUESAND
DIAGNOSTIC
CHECKS
A
.5a = .391
B
1.0a= .782
C
1.5a= 1.173
4.1 Nonparametric
Checks on the Lognormality
Assumption
.79
.77
.79
.77
.78
.77
.63
.64
.65
.70
.70
.70
.71
.70
.72
In the analysis reportedup to this point, we have assumed
that the negative of the price-sensitivityparameteris lognormallydistributedacross households. We use the lognormal distributionbecause of its simplicity and flexibility. In
addition,the lognormaldistributionrestrictsthe price coefficient to be negative. It can be argued, however, that the
assumptionof normalityof y = In(-/3) is arbitrary. It is
importantto rememberthat misspecificationof the randommixturedistributionis a fundamentalproblem that can lead
to inconsistentparameterestimates and incorrectdecisions.
In this section, we check the lognormalityassumptionusing the seminonparametricapproachadvanced by Gallant
and Nychka (1987). They used a series expansion-basedestimatorto approximatedensityof y. They provedthat,under
very mild regularityconditions, a particularclass of series
expansionestimatorscan consistentlyestimatethe unknown
density and many functionsof the density such as moments,
derivatives,and so forth. The difference between the seminonparametricapproach(SNP) followed here and the discrete approximationsused in the marketingliterature(see
Chintaguntaet al. 1991; Kamakuraand Russell 1989) is that
we explicitly assume that y is a continuousrandomvariable
with unknowndensity.
The basic idea of Gallantand Nychka (1987) is to approximate the unknowndensity by a normaldensity x a polynomial, p(7) oc 0(7yI A, o) Pk(y)2, where Pk(7) is a kth-order
polynomial in y. The polynomial terms in Pk act to modify the shapeof the normaldistribution,providingthe option
of skewness and excess kurtosis. In addition,the SNP density can easily be multimodel,allowing, for example, for the
possibility of two groups-one price sensitive and the other
price insensitive. The polynomialis squaredto enforce positivity of the density. The strategyadvocatedby Gallantand
Nychka(1987) is to keep addingtermsto the polynomialpart
as the sample size increases. The importanceof the nonconstantterms in the polynomialpartprovides a naturalmethod
for evaluatingthe extent of nonnormalityin the data.
We implement the following SNP density estimate (We
also considered higher-orderquarticpolynomials in the Pk
term of the SNP density estimate and found no evidence of
nonnormality.):
p(Y I ,u, o, 60,61) = kf (y I,, o)
smallereffect on the optimalprices thanchanges in the mean
as shown in Table 10. Thus, it appearsthat the centraltendency or location of the price-sensitivitydistributionis the
key parameterin determiningoptimalprices. This does not
mean that household heterogeneityis not important.As we
have shown, models that restrictheterogeneityto the intercepts alone will produce strongly biased estimates of price
sensitivity.
Because the parameterestimates used in computing the
optimalprices are subjectto samplingerror,it is very important to assess the role of sampling errorin the analysis. If
changes in the parametersat the magnitudeexpected from
sampling variationaffect the optimalprices, then the results
of the optimal-pricingexercise are of little practicaluse. It
is not enough to simply observe that the standarderrorsare
small relativeto the parameterestimates. Forthis reason,we
approximatedthe samplingdistributionof the optimalprices
by using an approximatesimulationmethod. The vector of
optimal prices, p*, can be viewed as a vector-valuedfunction of the parametersof the choice model, conditionalon
a given level of prices: p* = g(O; price). g( ) is a function
that is only implicitly defined by the optimizationproblem
that produces the optimal prices. Standardasymptoticdistributiontheory for the MLE allows us to approximatethe
sampling distributionof the MLE as 0 r N(O,I"'), where
10 is Fisher's informationmatrix. We draw from this normal distributionand solve the optimizationproblemfor each
drawof 0, therebybuildingup the samplingdistributionofp*.
Table 11 summarizesthis samplingdistribution.We observe
that the optimal prices have very small sampling variation
and are very insensitive to parametervariationdue to sampling error.This insensitivityis undoubtedlydue to the high
degree of precision of estimation of the random-coefficient
distributionparametersthat our very large sample of households affords.
of OptimalPrices
Table 11. SamplingDistribution
Price
Mean
Std. error
SKW
COSW
PW
.785
.767
.641
.0017
.0012
.0012
SKO
COSO
.698
.713
.0021
.0026
x [1 + 6o((y- i)/l) + 61((Y- ))22
where k is the integratingconstantthat is a function of o, 60,
and 61. This specificationallows for skewness, excess kurtosis, and bimodality. We insert this new SNP density in
the random-coefficientspecification outlined in Section 2.
That is, we integratethe household likelihood over p(7y
w,o, 60,61) instead of over the normal density 1(7 I , o).
One of the advantagesof the SNP approximationmethod
is that it lends itself readily to the use of Gauss-Hermite
302
Journalof Business &EconomicStatistics,July 1995
Table 12. Comparisonto InterceptFinite-Mixture
Models
Model:
"type"-shifter
Interceptmixture
2 mass pts.
3
4
5
6
Log-likelihood:Parameters: AIC:
8
-11,553
-11,557
-11,524
-11,510
-11,501
-11,472
-11,468
11
16
21
26
31
BIC:
-11,591
-11,530 -11.576
-11,518 -11,586
-11,511 -11,601
-11,485 -11,596
-11,484 -11,615
quadraturemethods to perform the integrals necessary to
evaluatethe likelihood [see Davidianand Gallant(1991) for
more on this point].
Using a randomsubsampleof 200 households,we fit both
our standardlognormalmodel and the SNP mixturemodel.
The log-likelihood increasesby less than .1%. Conventional
likelihood ratio tests for inclusion of the SNP nonnormal
terms fail to reject the null hypothesis of normality. Thus,
thereis no evidence in the data of nonnormality,and we can
have a high degree of confidencethatthe normalassumption
for -yis justified.
4.2 DiagnosticChecks on the Formof Intercept
Heterogeneity
As discussed in Section 2, we restrictthe form of intercept
heterogeneityacross householdsto only one key dimension,
loyalty to form in the case of tuna and loyalty to one key
brandin the case of ketchup. This restrictionis based on
the observed patternsof loyalty in the data, as depicted in
Figures 1-2. As a more formal test of our restrictedmodel
of interceptheterogeneity,we compare our model with an
unrestrictedinterceptmodel in which the distributionover
the interceptsis a finite mixture. This unrestrictedmodel is
developedfrom threekey assumptions:(1) We retainthe assumptionthatthe slope is independentof the intercept,(2) the
joint distributionof the J - 1 interceptsis approximatedby a
discretedistributionwith a specified numberof mass points,
and (3) we retain the assumption of lognormalityfor the
quality-sensitivitycoefficient. Using the whole sample of
13,705 tunapurchases,our restricted"type"-shiftermodel is
comparedto the unrestrictedmixture/lognormalmodel with
between two and six mass points for the mixtureon the intercepts in Table 12. The less restrictedinterceptmixture
models fit the data slightly better with an .8% higher likelihood value. The interceptmixturemodels, however,have
substantiallymore parameters.As measuredby the BIC criterion,only the two- andthree-segmentmixturemodels have
a slight edge over our model. It should be emphasizedthat
our"type"-shiftermodel is not nestedin the interceptmixture
specificationso that it is not possible to performa standard
likelihood ratio test. Ourconclusion is thatwe miss little of
importanceby using the more restrictedspecification.
5.
CONCLUSIONS
Modeling and measuringconsumerheterogeneityalone is
not sufficientto help managersprice and position theirprod-
ucts. We must take a furtherstep by making the estimated
models an input into an optimal decision process. In this
article, we documenta large degree of slope heterogeneity
in the panel data and find that this degree of heterogeneity
has a materialimpact on the categorypricing decision. We
demonstratehow a random-coefficientlogit model can be
used to derive optimal retailpricing for brandsin a product
category.
In bothproductcategoriesconsidered,thereis a greatdeal
of heterogeneityin the price-sensitivityparameter. Previous studies that concentrateon interceptheterogeneityare
subject to a large heterogeneitybias in the price-sensitivity
parameter. Retailers who fail to take into account heterogeneity in the price-sensitivityparameterwill underestimate
the extentof switchingbehaviorinducedby changes in price
and, consequently,obtainsuboptimalpricingstrategiesfrom
the category-profit-maximization
problem. We find that the
optimal-pricingstrategyis more sensitive to movements in
the mean of the price-sensitivitydistributionthan changes in
the dispersionof thatdistribution.
ACKNOWLEDGMENTS
We acknowledgethe helpful comments of Greg Allenby,
Pradeep Chintagunta,Kris Helsen, and Naufel Vilcassim.
We thank Steve Hoch of the GraduateSchool of Business,
Universityof Chicago, and Dan Nelson of Dominick's Finer
Foods for supplyingcost data.
[ReceivedApril1993. RevisedJanuary1995.]
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