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Case Study 5Modeling ConsumerDemand for Variety

Consumers are often observed to purchase more than one variety of a product ona given shopping trip. The simultaneous demand for varieties is observed not onlyfor packaged goods such as yogurt or soft drinks but also in many other productcategories such as movies, music CDs, and apparel. Linear utility models used to justify arandom utility logit or probit choice specification cannot accommodate interior solutions

with more than one variety (alternative) chosen, and standard demand models in theeconomics literature exhibit only interior solutions. The reason is that models withmixtures of interior and corner solutions are computationally demanding, particularly

when heterogeneity is incorporated into the model specification. Modern Bayesianmethods are effective at reducing this computational burden, facilitating the estimationof models with nonstandard utility specifications.

In this case study, we examine the translated additive utility structure of Kim et al.(2002) that nests the linear utility structure, while allowing for the possibility of interiorsolutions where more than one variety is selected. The model can be used, for example,to undertake calculations to value assortments by asking how much prices must belowered to compensate households (in utility terms) for the removal of flavors or todirectly compute the monetary value stemming from the contribution of a given flavorto the assortment. This sort of computation gets at the fundamental question of the

value of variety.

BACKGROUND

Standard choice models are based on a linear utility structure in which one and onlyone variety will be selected at each purchase occasion. This is clearly at variance with

data on the purchase of multiple varieties of the same product. We develop a new

Bayes ian Statis tics and M arketing P. E. Rossi, G. M. Allenby and R. McCulloch

2005 John Wiley & Sons, Ltd. ISBN: 0-470-86367-6

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270 CASE STUDY 5 MODELING CONSUMER DEMAND FOR VARIETY

utility-based model and estimation procedure that can accommodate a mixture of cornerand interior solutions.

In order to allow for simultaneous demand for multiple varieties, each variety can

be thought of as an imperfect substitute for the other varieties. The proposed demandmodel is based on a translated, nonlinear, but additive utility structure. This modelprovides a parsimonious specification that allows both interior and corner solutions as

well as diminishing marginal utility, while nesting the standard linear utility structure.The likelihood function for this model is derived from normal random errors in marginalutility. Evaluation of the likelihood involves high-dimensional integrals of normaldistributions over rectangular regions and, for this reason, has not been used in eitherthe economics or marketing literature. Coupled with a simulation approach to evaluatethe likelihood, a Bayesian hierarchical model of household heterogeneity is used.

The model is applied to data on purchases of varieties of yogurt. In this data set,

all purchases involve corner solutions and households frequently purchase more thanone variety on the same shopping trip. Estimates from the hierarchical model revealdifferences between varieties in base preference as well as different rates of diminishingmarginal utility. Households are found to differ greatly in their preferences for varieties,

with some households showing extreme preferences for particular flavors. Finally, webriefly discuss use of the model to investigate the value households place on particular

varieties by computing compensating values at the household level the monetaryequivalent of the households loss in utility.

MODEL

As introduced in Section 4.4.3, we define utility over the i= 1, . . . , mvarieties as

U(x) =

i

i(xi+ i)i, (CS5.1)

where x is the vector of quantity demanded with elements xi, and i, i, and i areparameters of the utility function. The utility function in (CS5.1) is an additive butnonlinear utility function. Equation (CS5.1) defines a valid utility function under the

restrictions that i > 0 and 0 < i < 1. An additive utility structure is used because theproducts we consider are not jointly consumed. For example, there is no interaction inutility from consuming two flavors together.

The utility in (CS5.1) is a family of translated utility functions where i controls thetranslation and i influences the rate of diminishing marginal returns. If i is zero, thenthere will be no corner solutions as the indifference curves are tangent to the axes.However, for positive , the indifference curves will have finite nonzero slope at the axes,creating the possibility of a corner solution. As pointed out in Chapter 4, the likelihoodfunction for the variety model is based on the KuhnTucker conditions which involvemarginal utility. Marginal utility for the variety model is given by

Ui = ii(xi+ i)i1, (CS5.2)

which suggests a more natural parameterization for estimation: i = ln(ii) and i =i 1. In this parameterization, i is estimated unrestricted and i is constrained to lie

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CASE STUDY 5 MODELING CONSUMER DEMAND FOR VARIETY 271

in the interval (1, 0). A uniform prior distribution over (1, 0) is assumed for thei parameters.

Heterogeneity is introduced into the model by specifying a random effects distribution

for the parameters:h = ln(hh) N(, V), (CS5.3)

where h= 1, . . . , H indexes the households. The multivariate normal distribution ofheterogeneity for h is flexible as it does not restrict individual parameters to a specificsupport and allows elements to covary across the population. The model in (CS5.3)could easily be modified to include a set of covariates in the mean function, = zh, asdiscussed in Chapter 5. Marketing mix variables (other than price) such as display andfeature are also candidates for inclusion. Display and feature activity are the same for all

varieties of the same brand, which implies these covariates will drop out of the resulting

demand functions used in our empirical analysis.A random effects distribution is not specified for the other model parameters because

there does not exist sufficient information in the data. For example, allowing householdto exhibit different curvature parameters (i) would require long purchase histories that

we do not observe. However, there are no conceptual problems specifying a randomeffects distribution for all model parameters.

We specify the standard conditionally conjugate priors for , V:

|V N(, V A1),

V IW(, V).

(CS5.4)

In this case, A is a scalar. Recognizing that the {h} are conditionally independent, wecan define a hybrid MCMC algorithm for this model which is suggested by the followingconditional distributions:

h|, Xh, Ph, , V, (CS5.5)

|{h}, X, P, (CS5.6)

, V|{h}. (CS5.7)

X is a matrix of the demand for all products, P is the matrix of observed prices. Xh, Phare the submatrices corresponding to demand and price for household h. (CS5.5) can beachieved via H RW Metropolis draws, one for each household. Data for all householdscan be pooled and another RW Metropolis draw can be used for the common parameters. Given {h} draws, (CS5.7) can be accomplished by a standard one-for-onedraw, as illustrated for hierarchical models in Chapters 3 and 5. The likelihood functionrequired to implement the Metropolis steps in (CS5.5) and (CS5.6) is developed inChapter 4.

DATAThe model is estimated using scanner-panel data from the yogurt category. Five flavorsof Dannon yogurt (blueberry, mixed berry, pina colada, plain, and strawberry) in thepopular 8 oz size are examined. In these data, there is virtually no brand-switching

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between the Dannon and Yoplait brands of yogurt. Less than 2 % of the observationsinvolve purchase of both brands. Therefore, attention is restricted to the dominantmarket share Dannon brand varieties. However, there is nothing conceptually different

about applying our utility specification to multiple varieties from multiple brands. Thedata are from 332 households and include 2380 purchase occasions.

Table CS5.1 presents information on the frequency of corner and interior solutions. Acorner solution is defined as a purchase occasion on which only one variety is purchased.Table CS5.1 shows the problem with application of standard choice models since morethan 20 % of the purchase occasions involve the simultaneous purchase of more thanone variety, but never all varieties. We can eliminate the possibility that householdportfolio effects are the only possible explanation for the simultaneous purchase ofmultiple varieties. The 69 single-person households in our data set have 464 purchaseoccasions with the same frequency of interior solutions (26 %).

Tables CS5.2 and CS5.3 provide information on the quantity of yogurt purchased.Table CS5.2 reports the distribution of purchase quantity. More than 50 % of thepurchase occasions involve quantities of at least two units, indicating that the data

Table CS5.1 Frequency of corner and interior solutions in the yogurt data

Purchase incidence Corner solution Interior solution

Strawberry 989 571 418Blueberry 545 309 236

Pina colada 570 281 289Plain 361 338 23Mixed berry 570 325 245Total 2380 1824 (76.6 %) 556 (23.4 %)

Table CS5.2 Distribution of purchase quantity

Purchase quantity Frequency %

1 1140 47.902 745 31.303 228 9.584 133 5.595 42 1.766 60 2.527 8 0.348 17 0.719 1 0.04

10 3 0.1311 1 0.04

12 1 0.0421 1 0.04

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Table CS5.3 Frequency of corner and interior solutions for multiple unitpurchases

Observations Corner Solution Interior Solution

Strawberry 336 208 (0.62) 128 (0.38)Blueberry 207 129 (0.62) 78 (0.38)Pina colada 183 99 (0.54) 84 (0.46)Plain 117 108 (0.92) 9 (0.08)Mixed berry 198 140 (0.71) 58 (0.29)

is capable of providing information about flavor satiation. Table CS5.3 displays the

incidence of corner and interior solutions for purchase occasions in which at least twounits are demanded. This table shows that plain and mixed berry yogurt are more oftenpurchased in isolation, while the other flavors are typically not. One possible explanationfor the differing behavior of plain and mixed berry varieties is that they are stronglypreferred by some households. Alternatively, consumers may not tire of these varietiesas rapidly as the others. The model discussed above has both features and will allow usto distinguish between these effects.

RESULTS

Table CS5.4 presents parameter estimates. As expected, there is considerable evidence ofnonlinear utility and different rates of satiation for different flavors. The flavor satiationparameters = 1 range from a maximum of 0.02 for plain, indicating littlecurvature in the utility function, to a minimum of0.57 for strawberry that is associated

with greater curvature. Estimates of the mean of the random effects distribution for areall negative, indicating that strawberry has the highest marginal utility when x= 0, andplain has the lowest marginal utility, on average, when x= 0. Estimates of the elementsof the covariance matrix, V, indicate substantial heterogeneity in the population, withthe most diverse preferences for plain.

DISCUSSION

Demand systems derived from valid utility functions can be used to make policyrecommendations in pricing and assortment. Without a valid utility structure, reduced-form models must be used in which the demand for yogurt is linked in an ad hoc mannerto the assortment of yogurts available. Even if agreement could be reached on the natureof the link between yogurt expenditure and variety as well as on a measure of variety,

estimation of such a reduced form would require time series variation in variety offered,which is not present in our data. A utility-based model allows us to analyze counterfactualexperiments such as what demand would be in the absence of a particular variety. Directutility calculations can be used to determine the utility loss and compensating value ofdeleting a flavor from the assortment.

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Table CS5.4 Parameter estimates (posterior standard deviations)

Common parameters

= 1

Strawberry 0.00a 0.57 (0.07)Blueberry 0.85 (0.13) 0.40 (0.06)Pina colada 0.70 (0.13) 0.46 (0.07)Plain 2.66 (0.33) 0.02 (0.21)Mixed berry 0.67 (0.12) 0.33 (0.06)

Covariance/Correlationmatrix (V)Blueberry 2.89 (0.44) 0.21 0.18 0.31Pina colada 0.53 (0.29) 2.15 (0.33) 0.30 0.46 Plain 1.12 (0.74) 1.59 (0.62) 13.33 (2.55) 0.28Mixed berry 0.70 (0.26) 0.90 (0.23) 1.36 (0.69) 1.76 (0.25)

aFixed for identification.

Note: : i = ln(ii).

Compensating Values

The value of an assortment can be determined by computing the compensating valueof adding or removing an offering from the product line. As the assortment is altered,the utility attainable for a fixed level of expenditure changes. The compensating valueis the amount that the budgetary allotment would have to increase or decrease to yieldthe same level of utility as that attained prior to any change in the assortment. As utilityis measured on an arbitrary scale, it is converted to the monetary scale of compensating

value for interpretability purposes.The removal of a flavor from a product line will result in a decrease in the attainable

utility of all consumers, and is affected by factors such as whether other offerings areconsidered good substitutes and the marginal utility of consumption. These factorsinvolve more than one of the model parameters and depend on the current level ofexpenditure. The compensating value depends on the set of substitutes available.

Compensating value is computed by numerically evaluating the indirect utility functionand computing the increase in expenditures required to attain the level of utility derivedfrom the full assortment. Since prices and total grocery expenditure vary across purchaseoccasions, computations are undertaken observation by observation and then summedfor each household.

Define for each observation the indirect utility function

Vht(pht, Eht) = maxx

Eh,|data[U(x|h, )]

such that phtx= Eht,(CS5.8)

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where U is defined over all five flavors. Eht is the total yogurt expenditure for householdh at purchase occasion t. Note that we integrate or average utility with respect to theposterior distribution of(h, ).

It is also possible to include an outside good in addition to the yogurt varieties (asin Kim et al. 2002). Some would argue that it is important to include an outside goodto allow for the substitution possibility of consuming more of the outside good whena flavor is removed. Failure to include the outside good may overstate the value of anassortment. However, Kim et al. (2002) had difficulty estimating the parameters in amodel with the outside good. There is also some ambiguity as to the exact definition ofthe outside good. Kim et al. use the total expenditure on grocery products as a measureof the outside good. Given the large expenditures on this definition of the outsidegood relative to yogurt expenditures, the outside good will be viewed as exhibiting littlesatiation and the yogurt flavors will have high satiation parameters. This will tend to

push the parameters to the boundaries of their admissible region. For this reason, wewill confine attention here to the model with only inside goods.

To find the compensating value (CV) for deletion of flavor i, we use the indirect utilityfunction defined in (CS5.8). CV is the solution to

Vht(pht, Eht) = V(i)

ht (pht, Eht + CV(i)ht ), (CS5.9)

where V(i)ht is defined by

V

(i)

ht(pht

,Eht

) =maxx Eh,|data[U

(x|

h, )

]

such that phtx= Eht and xi = 0.(CS5.10)

The indirect utility functions can be evaluated by numerical optimization methods (e.g.,R routine constrOptim). These utility calculations are made conditional on estimatesRof household-level parameters that are not available in non-Bayesian models.

CV(i)ht is defined as the amount by which expenditure must be increased to compensatehousehold hon purchase occasion t so that utility will remain unaffected by the deletionof flavor i. If a flavor has unique value with poor substitutes, then the compensating

value will be high. In addition, some households may have an extreme preference for a

given flavor which may also cause the CV to be large. We can sum up the CV(i)ht over

purchase occasions to the household level, CV(i)h =Th

t=1 CV(i)ht .

As discussed in Kim et al. (2002), compensating values for the deletion of particularflavors can be large because of the presence of heterogeneous preferences. Somehouseholds have strong preference for one flavor. These households require largecompensating values when their favorite flavor is deleted. Calculations based on pluggingin the average household preference parameter, , underestimate the compensating valueof flavors.

Implications for Pricing Policy

As a policy matter, retailers face space constraints and do not offer a complete assortmentof products. If a retailer were to delete a flavor from a store inventory, utility losses

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might drive customers away from the yogurt category and away from the store. Theretailer would need to compensate customers for reduced variety by lowering priceseither store-wide or in the yogurt category. This strategy has been adopted by warehouse

format competitors such as Wal-Mart; less variety is offered with lower prices. Theindirect utility function offers a means of computing the price reductions necessary tocompensate for loss in variety.

To compute the price reductions necessary to compensate for loss in variety, we againuse the indirect utility functions defined above. To calculate utility-compensating pricereductions, we solve for the percentage reduction in prices required to restore aggregateutility to the level present prior to its deletion from the product line. In performingthis computation, we integrate over the observed purchases and posterior distributionof model parameters, including the distribution of heterogeneity. We can perform twotypes of calculations: we will compute the amount by which we must reduce the price

of the remaining yogurts to compensate for loss of a flavor; and we will calculate theamount by which all store prices (outside good as well as yogurt) must be reduced tocompensate for loss of variety. Here we assume that overall grocery expenditure will notbe affected by deletion of a yogurt flavor. We will only comment on the first calculation;the reader is referred to Kim et al. for discussion of the second.

We compute price reductions necessary to compensate for the loss of a flavor, usingonly yogurt prices. That is, the retailer lowers prices of yogurts alone to compensate forloss of variety without changing the prices of other goods in the store (as captured bythe outside good). We solve the following problem to determine the level by which wemust reduce yogurt prices, ri,yogurt, to compensate for deletion of flavor i:

find ri,yogurt such thatV(i)

(ri,yogurt) =V,

where

V =

H

h=1

Th

t=1

Vht,

Vht = max Eh,|data[U(x|h, )] such that xpht = Eht

(CS5.11)

and

V(i)

(ri,yogurt) =H

h=1

Th

t=1

V(i)

ht (ri,yogurt),

V(i)

ht (ri,yogurt) = max Eh,|data[U(x|h, )]

such that (xpht)ri = Eht and xi = 0.

(CS5.12)

Our computations take into account the distribution of flavor preferences by inte-grating over the distribution of household parameters. These results show that deletionof all flavors (except for mixed berry) result in losses in utility that require substantial

price reductions (15 21 %). Plain yogurt requires the highest price cut even though,on average, households do not like plain yogurt. The reason is that there is a subset ofhouseholds that place a high value on plain yogurt. In a traditional retailing environment,the only way the retailer can compensate for this is to lower prices on the remaining

yogurts substantially.

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These price reduction results all hinge on the distribution of household heterogeneity.On average, households regard the yogurt flavors as quite substitutable, but thisstatement is very misleading. Many households have a decided preference for one flavor

which drives these utility and price reduction results. To illustrate this point, we considerthe price reduction experiment conditional on the average value of the utility parameter,. Here we see that, conditional on , there are only negligible reductions in utility,

which means that we do not have to lower prices at all to compensate for the lossof a flavor. In models with nonlinear utility functions, any sort of welfare calculationsuch as the compensating value computation undertaken here is very sensitive to howhousehold heterogeneity is handled. We cannot simply plug in the mean value withoutobtaining very misleading results. This illustrates the importance of integrating over thedistribution of parameters in a full decision-theoretic approach.

If it were possible to customize the assortment to each household, the utility loss

incurred by reduction in the size of the assortment would be reduced. In web retailing,this is a very real possibility. A web retailer can have the full assortment of varieties,but displaying this information to the buyer may be costly in terms of navigation of the

website and purchase decisions. For example, a web retailer such as Peapod could offerits customers the full array of 25 or more Dannon yogurt flavors. The danger here is thatthe customers incur a larger search and ordering cost than a customer facing a muchmore limited variety at the standard bricks-and-mortar retailer. One way of avoidingthese information-processing mental costs is to customize the assortment based on pastpurchase behavior.

R IMPLEMENTATION

We have implemented the MCMC algorithm in the R function rhierVarietyRw,Rwhich is available on the book web site. At the core of this function is the routine tocompute the log-likelihood for the variety model. As discussed in Chapter 4, this requiresintegration of a multivariate normal density over a half-plane whenever there is a zeroin the vector of demands. In addition, if there is more than one nonzero demand, wemust compute a Jacobian term. This function implements a partial vectorization of thecomputation of the variety likelihood for observations with one or two nonzero demands.

Care should be taken to ensure that there has been an adequate burn-in for thisprocedure. Our experience suggests than at least 5000 draws are required. Also, someexperimentation with scaling of the RW increment matrix (the parameter s) may berequired for optimal performance. We recommend making trial runs of shorter duration,followed by relative numerical efficiency calculations (using numEff) on the resultingRoutput to optimize the choice of the RW increment size.

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