Umbralla Branding

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Switching Costs and Market Power Under Umbrella Branding

Polykarpos Pavlidis Paul B. Ellickson

September 14, 2012

Abstract

This paper extends recent research on the importance of switching cost for dynamic pricingto the area of umbrella branding and multi-product firms. We examine household level scanner

data from the category of yogurt, focusing on parent brands that offer multiple sub-brands (e.g.

Yoplait offers Yoplait Original, Yoplait Light and Yoplait Thick and Creamy). Our demand

framework allows us to estimate state dependence arising from both the parent brand and

the sub-brand level and in turn, to set up a dynamic pricing game in which we evaluate the

impact of parent brand state dependence on forward looking prices and profits. Additionally,

we use the dynamic pricing model to study two strategic issues pertaining to pricing: i) the

benefit of centralized decision making (multi-product pricing) and ii) the potential loss associated

with setting uniform prices across sub-brands of the same parent brand. Findings for the

market leader in this category, Yoplait, can be summarized as follows: parent brand state

dependence effects increase its profits by about 8%, centralized decision making generates only

0.1% incremental profits and this number is mediated by cross-sub-brand dynamics, and finally,

uniform pricing across three sub-brands would cost only a small fraction of profits for the parent

firm.

Keywords: state dependence, multiproduct firms, umbrella branding, forward looking

prices

We would like to thank Guy Arie, Dan Horsky, and Sanjog Misra for helpful comments and suggestions. Allremaining errors are our own.

Nielsen Marketing Analytics, [email protected] of Rochester, [email protected].

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1 Introduction

Consumer packaged goods are usually marketed under a brand name that encompasses a wide

variety of both physical (size, package type) and taste attributes (low fat, enriched with fruit,

etc.). The practice of umbrella branding is observed across many consumer products including

durables (e.g. automobiles, electronics, computers) and services (e.g. TV shows, hotels). The usual

interpretation of a brand name is one of product identity: brands are identifiable in consumers mind

and they are associated with unique images that are both comparable and subject to preference

rankings. Through exposures to a brands performance, consumers form beliefs about that brand

and employ these beliefs in their contingent purchase decision within a given product category.

Several studies have established that a consumers most recent experience with a brand is an

important factor in their next purchase decision (Seetharaman, Ainslie and Chintagunta 1999,

Seetharaman 2004, Anand & Shachar 2004, Horsky and Pavlidis 2011). This type of persistence

in demand (often described as state dependence or switching costs) has strong implications for the

pricing behavior of firms. While demand state dependence has been studied in academic research

extensively over the last two decades, the pricing side of the problem has received considerably

less attention. In general, state dependence in the demand for frequently purchased goods has

been found to create two countervailing forces to the equilibrium prices that forward looking firms

may charge (Farrell and Klemperer 2007, Dube, Hitsch and Rossi 2009). On one hand, it poses

an upward pressure on prices because consumers past choices partially lock them in and make

them less likely to switch. On the other hand, this lock-in creates a threat of lost future sales

that will realize if todays prices are relatively higher, pushing prices downwards.The goal of this paper is to study the effect of demand state dependence on multiproduct

firms forward looking pricing behavior and resulting profitability. In particular, we focus on the

implications of parent brand state dependence under umbrella branding. Given that the existence

of switching costs is very often motivated by brand loyalty (e.g. Klemperer 1995) this is a natural

and important question that has not been addressed so far. There are two basic differentiating

factors between the analysis of forward looking pricing for single and multiproduct firms. First, it

is important to examine whether current demand for a specific product depends on past experience

with the specific product only or if it also depends on past experience with other products from the

same firm, effectively making them dynamic complements. Second, when there is state dependence

to the parent firm, the two countervailing forces described above become more involved because

a firm that maximizes joint profits of different product offerings should internalize the impact of

dynamic demand from one product to the pricing decisions of its others.

It is well known that optimal multi-product pricing increases market power since it allows a

firm to internalize part of the competition between its different products, thereby leveraging local

monopoly power (Nevo 2001). The existence of firm state dependence should have non-trivial im-

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plications on the magnitude of the benefit that joint pricing offers because, for a multiproduct firm,

two additional countervailing forces come into play. In addition to the harvest-invest dilemma

that single product price managers face, there is also a cross-product harvest-invest dilemma.Consider the simple case of two variants, under the same brand name, where the respective higher

level manager maximizes joint profits and consumer demand is state dependent at both the sub-

brand and the parent brand level. Deciding on the price of each sub-brand, a forward looking

manager has to consider the following two intuitive dimensions: a) charge a little more for each

sub-brand since part of the demand losses inter-temporally will go to the other sub-brand through

state dependence to the parent brand, b) lower the price of each sub-brand because this will create

greater future demand for the other sub-brand as well (the dynamic complement effect). The total

impact of any parent brand loyalty that past experiences create is naturally determined by the

balance of these two forces.

To study the impact of firm state dependence on forward looking prices of multiproduct firms,

we examine household level scanner data from the category of yogurt. Our approach draws heavily

on the framework developed by Dube, Hitsch and Rossi (DHR 2009) which we extend to the case

of multiproduct firms. Analyzing the demand of parent brands that offer a portfolio of sub-brands,

we estimate state dependence both to the parent brand (PBSD) and to sub-brands. Using our

demand estimates and model of forward looking pricing behavior, we first recover estimates of

the relevant marginal costs. This is one of the contributions of this study and it is a necessary

step because we do not observe any information on the firms costs. With estimates of consumer

demand dynamics and costs in hand, we solve for the best response pricing policies of the firms

in our sample, within the context of a dynamic game. The computed policies are then used to

calculate steady state equilibrium prices under different market structure scenarios. The design of

the simulation experiments allows us to explore three specific questions.

First, we explore the impact of parent brand state dependence on equilibrium prices and profits

by contrasting the base case of estimated demand parameters with one where we eliminate the

parent brand level dynamics. We find that similarly to results reported in the literature for sub-

brand state dependence (e.g. DHR (2009)) for single-product decision makers, the relation between

firm level choice dynamics and equilibrium prices is negative. Absence of parent brand choice

dynamics is associated with higher prices. This effect is a direct consequence of the elimination of

the investing motive for firms when consumers future demand is not enhanced by their currentpurchases. We also find that firm profits increase with parent brand state dependence because the

associated lower prices lead to higher per period demand in a profit enhancing way (note that we

are considering the impact of state dependence on a single firm in isolation, rather than for the full

set of firms together).

Second, we quantify the benefit of centralized price setting. With the use of the forward looking

pricing model, we evaluate the incremental profits accrued from coordination in sub-brands price

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setting. A basic question that is also addressed is whether the cross-line invest-harvest motivation

increases or decreases the benefit of multi-product pricing. In particular, we ask whether the

internalization of competition between sub-brands pays off more or less under parent brand statedependence and if so, how important is the difference? Our results suggest that the market power

benefits arising from joint price-setting are affected by parent brand state dependence. Centralized

price decision making pays off more under no parent firm state dependence effects.

Third, we quantify the losses associated with uniform pricing, a form of constrained profit

maximization. That is, we compute equilibrium prices for a firm that institutionally decides on a

single price for all its sub-brands and compare the resulting situation with the one that obtains

when the focal firm sets sub-brand specific prices (and thus fully internalizes the externality). The

results provide a lower bound on menu costs that is required for uniform pricing to be optimal.

If setting and maintaining separate sub-brand prices (i.e. menu costs), is more expensive than

the loss associated with charging uniform prices, then the latter is preferable. We find that the

aforementioned losses are also mediated by parent brand state dependence.

2 Literature Review

2.1 State Dependence and Pricing Implications

The implications of state dependence for firm behavior, market structure, and consumer welfare

have been analyzed extensively in both the economics and marketing literatures. Farrell and Klem-

perer (2007) provide a comprehensive survey, highlighting both empirical and theoretical results.Of particular relevance to our study is the impact on product market competition. Klemperer

(1995) expresses the traditional view that switching costs make markets less competitive, as the

lock-in effect is likely to dominate the desire to harvest. Viard (2007) empirically tests the impact

of portability switching costs on prices for toll-free services and finds a positive relation; prices

decreased after the introduction of portability, which reduced customer switching costs.

DHR (2009) challenge the conventional wisdom and show that switching costs can result in

pricing equilibria which are more competitive; characterized by lower manufacturer prices and

profits. The main intuition behind this result is that if switching costs due to state dependence are

relatively low, the strategic effect of attracting and retaining customers in a competitive market

can outweigh the harvest incentive. They demonstrate that switching costs for the categories of

refrigerated orange juice and margarine, estimated with transactional data, are well within the

range that increases competition. In a related work, Dube, Hitsch, Rossi and Vitorino (2008)

establish a similar result for category managers setting a portfolio of prices in a product category:

the prices of higher quality products decline relative to lower quality substitutes in the presence of

loyalty.

These new empirical findings have sparked additional theoretical research examining the impact

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of switching costs on equilibrium pricing. Arie and Grieco (2010) demonstrate theoretically the

existence of an additional compensating effect that pushes prices downwards when switching costs

increase from zero to moderate levels. Based on this effect, single product firms may reduce pricesto compensate marginal consumers who are loyal to rivals. In their analysis too, the investing effect

of attracting future loyal customers decreases prices when switching costs increase, provided that

there is no very dominant firm in the market. Similar theoretical results, with prices decreasing

in the presence of relatively low switching costs, are reported by Doganoglou (2010) who further

proves the existence of a MPE with switching at equilibrium.

2.2 Multiproduct Firms / Umbrella Branding

Erdem (1998) shows the existence of cross-category brand spillover effects. Consumers choices

in the categories of toothpaste and toothbrush proved to be consistent with the predictions of signal-ing theory. A positive experience with a brand in one category reduces the perceptive uncertainty

of the brands quality in a different but related category. Erdem and Sun (2002) extend the above

framework to include advertising. For the same product categories, they show that advertising as

well as experience, reduce uncertainty about quality and also affect preferences. Under umbrella

branding firms enjoy marketing mix synergies that go beyond advertising. Draganska & Jain (2005)

examine empirically the category of yogurt and find that consumers perceive product lines to be

different and they are willing to pay analogously to quality but do not perceive different flavors of

the same line to deserve different prices. Miklos-Thal (2012) shows that forward looking firms have

incentives to employ umbrella branding for new products only if their existing products are of high

quality. Anand and Shachar (2004) show that consumers are loyal to multiproduct firms because

of information related benefits. Anderson and De Palma (2006) show that firms tend to restrict

their product ranges in order to relax price competition but that in turn generates relatively high

profits which attract more entrants in the market.

3 Model Setting

3.1 Demand Utility Function

Demand is modeled using a discrete choice framework. We assume consumers are in the marketfor yogurt every week they visit a grocery store or supermarket, denoting the purchase choice set

by J and reserving the subscript 0 for the outside option of not buying yogurt. An individual

consumers conditional indirect utility function is composed of a linear deterministic component,

U, and additive Type I Extreme Value demand shocks, , yielding the well known logit probabilities

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for purchase of each sub-brand.

uhjt

= hj

h1P

hjt+

h3P BL

hjt+

h4SBL

hjt+

hjt

= Uhjt + hjt , j {1,...,J} (1)

uh0t = h0t (2)

P r(Chjt = 1) = P r (uhjt > uhkt k J, k = j, & maxuhkt > uh0t, k J) (3)

P r(Chjt = 1) =eUhjt

1 +

kJ eUhkt

(4)

In the utility functions (1), hj denotes the intrinsic preference for each choice alternative j

while Phjt stands for the alternatives net price at week t. The choice set is defined over sub-brands

of popular parent brands that may cover one or more different alternatives. The different levels

of state dependence, for the Parent Brand and/or the specific Sub-brand, are captured with the

variables P BLhjt and SBLhjt respectively. These are both indicator variables, the former taking

the value one if the households last purchase was for the parent brand of the choice alternative

j and the latter being equal to one if last purchase was for the same sub-brand. For a given

alternative j there are three possibilities with respect to the loyalty variables: a) both P BLhjt

and SBLhjt could be equal to one if the same product line was bought last time, b) both P BLhjt

and SBLhjt could be zero if an alternative of a different brand than j was bought and c) P BLhjt

could be equal to one and SBLhjt could be equal to zero if another product line of the same

parent brand was chosen last. Coefficients h3 and h4 thus capture the effect of last purchaseto the current periods utility. Significant positive values for both would imply the existence of

demand state dependence associated with both the specific sub-brand as well as the parent brand.

If consumers tend to repeat their choices only with respect to specific alternatives but there is no

purchase reinforcement from the parent brand, then h3 would not be statistically different from

zero. Finally, we note that concerns regarding the potential endogeneity of price are mitigated by

the inclusion of brand intercepts. As is the case in many scanner data settings, the bulk of the

variation in prices occurs across brands, presumably due to differences in (permanent) unobserved

quality which will be captured through these intercepts.

Dube, Hitsch and Rossi (2010) highlight an important confound between state dependence andunobserved preference heterogeneity. To address this concern, we follow their approach which

controls for heterogeneity by approximating the distribution of household coefficients across the

population with a very flexible mixture of Normal distributions (Rossi, Allenby and McCullough

2006).

h N(lh, lh) (5)

lh multinomial () (6)

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3.2 Demand Estimation

Following Dube, Hitsch and Rossi (2010), we estimate the demand model using a Bayesian

MCMC framework. Given diffuse standard priors for the parameters, we take draws through a

mixed Gibbs sampler with a Metropolis-Hastings step. To assess the convergence of the posterior

parameters we inspect visually the series of draws and experiment with more draws to ensure

that the estimates are stable. We compare different model specifications based on the Decision

Information Criterion (DIC). DIC is a criterion of model selection based on the deviance of

each specification (Gelman 2004, Gamerman & Lopes 2006). The deviance reflects the discrepancy

between data and the model; the specification with the lowest discrepancy, or deviance, is considered

to fit the data better.

3.3 Total Demand & Evolution of States

Similarly to DHR (2009), the market is approximated by N types of consumers, with each type

n having its own parameters and comprising a portion n of the overall market. At any period t, a

fraction snkt of each types total consumers will have chosen a particular alternative k in their last

purchase. The fractions of consumers loyal to each sub-brand comprise the state space that is the

foundation of the firms dynamic pricing problems. For each type n the probability of choice for j

conditional on k being chosen last is denoted by P r(Cnjt = 1|k). Since each product is associated

with one parent brand only, tracking past sub-brand choices is sufficient to characterize the vector

of state variables at both the sub-brand and the brand level. That is, both P BLhjt and SBLhjt

are uniquely defined by the last product line chosen.1

Dnjt =J

k=1

snkt P r(Cnjt = 1|k) (7)

Jk=1

snkt = 1 (8)

Djt =N

n=1n D

njt (9)

The state vector of each type snt = (sn1t,...,s

nJ t)

evolves deterministically over time based on

the choices made by consumers the previous period. This evolution is operationalized with a type-

specific Markovian transition matrix. The element in the jth row and kth column of the transition

1The dimensionality of the state space depends on the number of choice alternatives J and on the number ofconsumer types since each type has its own taste, price sensitivity and state dependence parameters. For example,in a market of one type and twelve sub-brands belonging to less than twelve parent brands, the state space hasdimension of eleven ((12-1)x1). If one wants to add a consumer type, the dimensionality increases to twenty-two((12-1)x2).

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specific allocation of consumer loyalties across sub-brands, are described by the Bellman equation

Vf

(s) = maxpj

0, jf

{f

(s, P) + Vf

[g(P, s)]} s S (12)

in which g() is the transition kernel described above. The value function Vf in (12) represents

all payoffs given a specific strategy profile for firm f, f. We use the notion of Markov Perfect

Equilibrium to compute equilibrium prices in the form of pure strategies. At equilibrium each firm

has an optimal strategy that prescribes the best response to all rivals and for all possible states.

Denoting the strategy profiles of competitors by f, the optimal strategy for f, f, satisfies the

following Bellman equation.

Vf(s) = maxpj0, jff s,P,f(s) + Vf g(P,s,

f(s)) s S, f K (13)

The specification of the pricing game is flexible enough to accommodate both a single price for all

products of a firm and separate prices for each sub-brand. Under full firm-level profit maximization,

the value function is firm-specific in both cases. The difference is that in the latter case the state-

specific strategy profile of each firm contains as many prices as there are sub-brands under the firms

parent brand name. The base case of the pricing model assumes that multiproduct firms decide

jointly on sub-brand specific prices. Results from the base specification are then compared with

results from specifications where the games structure is modified in a controlled way that allows

the examination of various potential strategic factors (e.g. separate sub-brand pricing, centralized

price setting, or uniform pricing).

3.4.2 Sub-brand Profit Maximization

With minor adjustments, the game can be adapted to allow the computation of optimal prices

independently for each sub-brand. This situation is equivalent to one where product-line rather than

firm-level managers set prices. In this case, the value function is sub-brand specific and strategy

profiles include a single price for each point in the state space. There are two major implications

from such an adjustment to the game: i) sub-brand prices do not internalize the current demand

for other products of the same parent brand, a condition that would enhance their market power

and ii) they also do not account for the inter-temporal demand synergies that the parent brand

state dependence creates between lines of the same brand (i.e. dynamic complementarity).

Vj(s) = maxpj0 {j(s, P) + Vj [g(P, s)]} s S (14)

Vj(s) = maxpj0

j

s,P,j(s)

+ Vj

g(P,s,j(s))

s S, j J (15)

Comparing the optimal prices in two different scenarios, one where firms maximize sub-brand

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profits jointly and one where each sub-brand has its own separate profit function, we evaluate the

impact of full firm-level profit maximization on profitability.

3.4.3 Price Equilibrium

The existence and uniqueness of a Bertrand Nash equilibrium between multiproduct firms under

logit demand is currently a challenging theoretical problem even in the static case where firms

compete by simply maximizing current period profits. In general, there are a few existence results

for various utility specifications that are limited to standard logit models (without heterogeneity).2

Following DHR (2009) we use a numerical approach in computing pure strategy equilibrium for

the full game. Accordingly, existence and uniqueness of equilibrium are evaluated numerically by

means of convergence criteria and using a variety of different starting conditions for the algorithm.

3.5 Equilibrium Computation

To study the implications of parent brand state dependence on firm pricing we solve explicitly

for equilibrium steady state prices under different scenarios regarding the role of parent brand state

dependence and the level at which prices are set. We will assume for now that costs are observed

to the researcher and then describe below our method for recovering estimates of these objects.

Before obtaining steady state prices, we compute the optimal competitive policies for each firm for

each unique point in the state space. Given the formulation of the game, the optimal price for each

firm (or sub-brand, depending on the scenario) depends on the state of the market, namely how

many consumers from each type are loyal to the sub-brands of the respective firm at each point intime. The policy functions, which condition on the measure of each consumer type, are infinite-

dimensional functions. The value function in (12) is also of infinite dimension since it is defined

uniquely for each point in the state space. To work around the insurmountable problem of solving

for infinite-dimensional functions we approximate the solution to the dynamic game specified by

equations (12) and (13) by discretizing the state space. This general approach is described in Judd

(1998) and has been implemented successfully by DHR (2009) and others. Details regarding our

implementation of the algorithm can be found in an appendix.

2Morrow & Skerlos (2010) prove and characterize the existence of Bertrand Nash equilibrium between multiproductfirms for very general logit based demand. Their framework imposes relatively week restrictions on the specification of

utility functions and price effects. To overcome the problem that the logit-based profit functions of multiproduct firmsare not quasi-concave, they base their approach on fixed-point equations. Although they generalize their numericalfixed-point approach for equilibrium price computations to the Mixed logit context, they do not prove existence there.

In forward looking settings, there are proofs for existence of equilibrium between single product firms understandard logit demand. A relatively common approach is to prove the quasi-concavity of the profit function (i.e.current period profits plus the value function for a given state vector) which in turn guarantees the existence ofa unique profit maximizing price. Besanko, Doraszelski, Kryukov and Satterthwaite (2010) and DHR (2009) fora simple case of their model, prove the existence of equilibrium through quasi-concavity. Unfortunately, the sameapproach cannot be followed in the case of multiproduct firms as the profit function is no longer quasi-concave.

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3.5.1 Cost Estimates

To define the profit objective function for each firm we need measures of product level marginal

costs. Since we do not observe costs (or even wholesale prices), we rely on the pricing policy implied

by our framework (together with the estimated demand system) to back out consistent estimates.

Our approach draws heavily on methods developed by Bajari, Benkard and Levin (2007), adapted

to match the dynamic pricing model discussed in subsection 3.4. Specifically, we estimate costs

that rationalize observed prices for the observed structure in which firms decide jointly on separate

prices for their sub-brands taking into account both sub-brand and parent brand state dependence.

The estimated costs are then used to evaluate alternative (counterfactual) pricing strategies, such

as decentralization or uniform pricing, by re-solving the full pricing game.

The approach of Bajari, Benkard and Levin (2007) works in two steps. In the first step,

one recovers the policy function from the data and then uses it to forward simulate the valuefunction of each firm. Along with the value functions corresponding to the firms policies, one also

computes value functions corresponding to counterfactual or perturbed policies. In the second

step, a minimum distance estimator is used to recover the structural parameters that rationalize

the observed behavior. Neither step requires solving for equilibria.

In practice, we forward simulate value functions for the steady state that corresponds to the

average sub-brand prices observed in the data. For the same state variable vector we simulate value

functions based on prices that deviate from the observed prices by varying amounts (e.g. +/- 0.01

and +/- 0.025) separately for each sub-brand. Denoting the correct and perturbed value functions

with Vf

(s; j

, j

; ) and Vf

(s; j

, j

; ) respectively, the second stage estimator minimizes the

following objective function.

Q() =1

H

Hh=1

min{Vf(s; j, j; ) Vf(s;

j, j; ), 0}

2

Applying the methodology is facilitated by the fact that the value functions for the pricing model

of this study can be written as a linear function of the structural parameters which, in this case,

are costs. This allows for significant computational economies in that the forward simulation needs

only be done once, before the estimation, and not for every trial parameter vector of the estimation

routine. For demonstration purposes, we describe the linear form of the model below.

Vf(s; ; ) =

t=0

t

jf

(Pjt cj) Djt M

=

t=0

t

jf

Pjt Djt M

jf

t=0

t(Djt M)

cj

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The linearity of the value function allows us to construct empirical analogs of both the true

and perturbed value functions using forward simulated estimates of Djt , pricing policies, market

size, discount factor and any trial values for the cost parameter vector. For the particular pricingmodel considered here, the existence of a steady state after a finite number of time periods gives

an additional, albeit minor, advantage. In computing the discounted future flows, one can forward

simulate until the steady state and then add a perpetuity term (discounted for the periods until

the steady state obtains) which by definition has a finite value3. This term is equal to per period

profit flows at the steady state. As a result of this advantage, one avoids any simulation error in

value function estimates induced by the practice of terminating forward simulating at some finite

period T when the discount factor makes profit flows in the far future very close to zero.

4 Empirical Application4.1 Data

To study the implications of parent brand loyalty on multiproduct firms market power we

use scanner data from the yogurt category. The source of the data is the IRI Marketing Dataset

(Bronnenberg, Kruger and Mela 2008). The sample used for the analysis includes household level

shopping trips for groceries, purchases of well known yogurt products of the most popular size in

the category (6 oz) and the respective prices for each sub-brand. The time period covered is years

2002 and 2003 for a total length of 104 weeks. Purchase histories from year 2001 are also available

and are used to provide the initial conditions of brand and product loyalty for each household. Toensure proper tracking of choices over time, we exclude from the sample households who do not

satisfy the IRI criteria for regular reporting of information. The choice set used for the estimation

covers five parent brands with twelve sub-brands lines in total. Table 1 reports average price and

market share by sub-brand.

Sub-brands of the same brand name in this category are moderately differentiated in packaging,

labeling, shelf space and ingredients. In all cases, the parent brand name is prominent and clearly

visible on the package of each sub-brand. Hence, consumers are able to identify or remember both

the parent brand and the specific sub-brand name when they shop. In some cases the product

lines of the same brand have very similar prices, like Yoplait Light and Yoplait Original. On the

other hand, some brands use price as a differentiating factor between their various sub-brands; for

example Kemps Classic is priced 10 cents lower than Kemps Free per pound.

3The present value of a stream of cash payments (A) that continue for ever is given by: PV = Ar

. Here r denotes

the appropriate discounting rate which in our case is: r = 1

.

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Table 1: Descriptive Statistics

Parent Brand Sub-Brand Average Price/lb Share

Dannon Dannon Creamy Fruit Blends 1.581 0.01Dannon Dannon Fruit on the Bottom 1.712 0.04Dannon Dannon Light N Fit 1.790 0.07Dannon Dannon Light N Fit Creamy 1.674 0.03Kemps Kemps Classic 1.261 0.04Kemps Kemps Free 1.361 0.05Old Home Old Home 1.383 0.01Old Home Old Home 100 Calories 1.341 0.04Wells Blue Bunny Wells Blue Bunny Lite 85 1.403 0.14Yoplait Yoplain Light 1.671 0.23Yoplait Yoplait Original 1.671 0.24

Yoplait Yoplait Thick and Creamy 1.676 0.09

4.2 Identification

The identification of demand state dependence is a challenging empirical task due to the classic

confound between state dependence and heterogeneity in brand preferences (Heckman 1981). We

follow Dube, Hitsch and Rossi (2010) in allowing for a flexible distribution of unobserved hetero-

geneity in the demand model 4. Data on observed heterogeneity, for example initial stated brand

preferences of the panelists whose purchase histories form the estimation sample, can help even

further to disentangle state dependence from differences in intrinsic preferences. This has been

shown by Horsky, Misra & Nelson (2006) for state dependence and Shin, Misra & Horsky (2010)for Bayesian learning in particular. In the absence of stated preference data one should make ev-

ery effort to assess the existence of revealed preference data patterns that imply state dependence

effects and provide a source of identification in estimation. We report and discuss such model-free

evidence below.

Since the new element in this study is the existence and analysis of parent brand state depen-

dence, the discussion is focused on the identification of the corresponding parameter, h3. Given

the utility functions and purchase probabilities in sub-section 3.1, for h3 to be positive it must be

that for any given purchase of a specific variant, the probability of switching to a different variant

of the same parent brand is higher than the probability of switching to a variant that belongs to adifferent brand (i.e. expression 16).

P r(j Jf, j = k | k Jf) > P r(l / Jf | k Jf) (16)

Table 2 shows that this is indeed the case for the sample of household purchases in this product

4Computational considerations with respect to the curse of dimensionality may limit the dimension of theheterogeneity used in the pricing model. It is important that a flexible heterogeneity is allowed for in estimation.

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Table 2: Choice Summary Statistics

1st Quart. Median Mean 3rd Quart

Number of sub-brands bought 3 5 5.2 7Number of brands bought 2 3 3.2 4Switchings within brand 3 6 7.9 11Switchings to other brand 1 4 6 9

category. The first two rows of the table report summary statistics for the number of different

variants and different parent brands bought by each household in the sample; the average number of

sub-brands a household purchased is 5.2 while the average number of brands is 3.2. This is evidence

that there is switching observed in the data. Rows three and four show a more specific pattern,

namely that the number of sub-brand switchings to a variant of the same parent name ( Switchings

within brand) is higher than the number of switchings to a variant of a different parent brand name

(Switchings to other brand). This pattern holds across the range of the household distribution (i.e.

1st quartile, median, 3rd quartile) with the respective averages being 7.9 against 6. These average

figures correspond to the sample frequency probabilities in (16) which are conditional on the same

event, the purchase of any sub-brand from the choice set.

A second issue that can potentially hamper identification of state dependence effects relates to

initial conditions. If the initial loyalty states of sample households are unobserved, the required

assumptions made for the states of the first period observations can affect the estimates of state

dependence in non-trivial ways. To address this concern, we use the sample households purchase

histories from the previous year to observe and incorporate the initial brand and sub-brand loyaltystates in our data. Given that the demand model dynamics in this work are limited to only one

purchase back, the identity of the last variant bought is sufficient to completely characterize the

initial state conditions for each household.

5 Results & Discussion

5.1 Demand Estimation

We start the discussion of estimation results by reporting DIC values for model specification

selection. In the first three rows of Table 3 we evaluate the plausibility of including state depen-

dence to both the sub-brand and the parent brand. Based on the fit measures, adding sub-brand

state dependence improves the fit of the demand model to the data. This is evident by DIC val-

ues decreasing from the first row of Table 3 to the second. Moreover, parent brand level state

dependence adds even more explanatory power and is supported by the data in this category; DIC

values decrease further in the third row of the table. This is evidence that: i) consumers do tend to

repeat their sub-brand choices over time controlling for price effects and intrinsic preferences and

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Table 3: Model Selection ResultsMixtures State Dependence DIC

1 Comp. None 68117.61 Component No Brand SD (only sub-brand) 67290.81 Component Full State Dependence 67224.92 Components No SD at all 68025.32 Components No Brand SD (only sub-brand) 67236.92 Components Full State Dependence 67194.83 Components Full State Dependence 67196.94 Components Full State Dependence 67234.0

ii) they also tend to become loyal to parent brand names in cases when they switch their product

line choices. In the next three rows of Table 3 we report similar results from model specifications

where consumer heterogeneity is a mixture of two Normal components. The two component mix-

ture attains lower DIC values, for each comparable specification, and provides a better fit to the

data. Also in this case, incorporating sub-brand and parent brand state dependence improves the

fit of the model. In the last two rows we explore whether adding more components to the het-

erogeneity mixture provides even better fit. We see that DIC values increase when we use three

or four components, showing that the additional explanatory power of the respective flexibility is

not as high relative to the added complexity in the model. Based on these results, we use average

household estimates from the two component mixture specification to compute market equilibrium

and counterfactual scenarios.

The demand estimates used for the pricing model are reported in Table 4. These results refer to

mean posterior estimates across the households in the estimation sample 5. The sub-brand specific

preferences reflect the popularity of each alternative in the market place with Yoplait Original and

Yoplait Light being the market share leaders in this category. Preference estimates are negative

because they are estimated against the normalized outside option that has the greatest share in

the sample as most consumers do not purchase yogurt in every shopping trip they make to one

of the grocery stores. The estimates of state dependence show that the effect of lagged purchase

on the utility of the same sub-brand is about two times stronger than the respective effect on the

utilities of sub-brands of the same parent brand name; the corresponding estimates are 0.79 and

0.34 respectively. The quantification of this estimate for parent brand state dependence follows inthe next subsection with results from counterfactual scenarios.

5To keep the dimension of the state space manageable we limit our pricing analysis to one consumer type. Wehave computed our results using two consumer types and a subset (two) of the parent brands in our sample; theconclusions of that analysis were similar to the ones reported in the paper.

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Table 5: Cost EstimatesSub-Brand Static Game Dynamic Game

Dannon Creamy Fruit blends 1.114 1.152Dannon Fruit on the Bottom 1.256 1.292Dannon Light N Fit 1.323 1.394Dannon Light N Fit Creamy 1.208 1.259Kemps Classic 0.795 0.830Kemps Free 0.902 0.943Old Home 0.931 0.951Old Home 100 Calories 0.882 0.927Wells Blue Bunny Lite 85 0.935 1.015Yoplait Light 1.187 1.283Yoplait Original 1.185 1.283Yoplait Thick and Creamy 1.193 1.239

5.2.2 Effect of Parent Brand State Dependence on Firm Profits

To evaluate the impact of parent brand state dependence on firms profits, we perform two

counterfactual experiments. In the first experiment, we eliminate the dynamic effects from Yoplait

Thick and Creamy to Yoplait Original and Yoplait Light and vice versa. This scenario is equivalent

to a situation in which the Thick and Creamy sub-brand is not branded under the Yoplait umbrella

name but under a different name that is not associated directly with Yoplait and therefore does not

create any dynamic inter-dependence in demand. Equilibrium prices and per period profits in this

counterfactual scenario (C2) are compared with prices and profits in a base scenario (C1) wherethe branding structure of Yoplait, or any other brand, does not change. The difference between

the two scenarios is that, in the counterfactual, demand for Yoplait Thick and Creamy does not

increase with loyalty to the other Yoplait sub-brands and demand for the other Yoplait sub-brands

does not increase with loyalty to Yoplait Thick and Creamy. The results are reported in Table 6.

The reader should focus their attention on the bottom three rows, as the equilibrium impact on

rival firms is negligible here. There are two basic conclusions that emerge from Table 6: i) the per

pound price of Yoplait Thick and Creamy is higher by about 5 cents when this sub-brand does not

have any parent brand state dependence effects and ii) the firm Yoplait loses about 2.1% of its

per period profits as a result. The first conclusion indicates that, in this case, the effect of parent

brand state dependence is to increase the firms motivation to invest in future demand for its other

sub-brands by lowering the price of Yoplait Thick and Creamy. The second conclusion reveals that

this lower price is profitable when there is parent brand state dependence because it increases total

demand for the brands sub-brands.

Results from a second counterfactual are also reported in Table 6. In this case, for the perturbed

scenario (C3), we eliminate the parent brand state dependence effects for all three sub-brands of

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Table 6: Effect of Yoplait PBSD

Base Case (C1) Yoplait T&C Yoplait All

PBSD = 0 (C2) PBSD = 0 (C3)Brand Sub-Brand Prices Profit Prices Profit Prices Profit

Dannon Creamy Fruit Blends 1.589 1.588 1.589Dannon Fruit on the Bottom 1.730 1.730 1.730Dannon Light N Fit 1.791 1.791 1.790Dannon Light N Fit Creamy 1.683 737.9 1.683 739.3 1.683 741.9Kemps Classic 1.264 1.263 1.263Kemps Free 1.368 242.4 1.368 242.8 1.367 243.5Old Home Classic 1.398 1.398 1.398Old Home 100 Calories 1.347 208.6 1.347 209 1.347 209.6Wells B.B. Lite 85 1.408 442.2 1.408 443.1 1.408 444.7

Yoplait Light 1.691 1.690 1.696Yoplait Original 1.688 1.688 1.693Yoplait Thick and Creamy 1.716 1891.2 1.765 1851.7 1.750 1757.9

Yoplait. This is equivalent to all three sub-brands being marketed under different, unrelated names,

without the common factor Yoplait. Similar to the first counterfactual, the own state dependence

effects remain the same and only cross-sub-brand effects are eliminated. The comparison of this

counterfactual with the base scenario (C1) indicates a price increase for all three sub-brands of

Yoplait and a total reduction of per period profits by 7.6%, when the demand for Yoplait sub-

brands is conditionally independent. This can be seen as the total impact of parent brand state

dependence on the profitability of the firm Yoplait, holding other things constant.

5.2.3 Firm-Level Profit Maximization and the Impact of Parent Brand State Depen-

dence

Next, we examine a multiproduct firms ability to benefit from dynamic loyalty to the firms

portfolio of products. The first step in this part of the analysis is to uncover the impact of full

firm-level profit maximization on equilibrium prices and per period profits. We do so by comparing

two different scenarios, the base one where each firm decides on the prices of its sub-brands via a

unified objective function and one where the profit function and pricing policies of a major firm,

Yoplait, are sub-brand specific (i.e. sub-brand profit maximization). Any differences, in prices and

per period profits, between the two scenarios can be attributed to the centralization of the pricing

decision for Yoplait alone. In other words, this experiment allows us to quantify the benefit that

a major parent brand like Yoplait can derive by unifying the profit objective functions of all its

sub-brands and accounting for respective externalities.

The first two numerical columns in Table 7, the base case (C4) in the sub-brand profit max-

imization scenarios, reports results similar to the base case (C1) in Table 6 with the difference

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Table 7: Sub-brand Profit Maximization for Yoplait

Base Case (C4) Yoplait All

PBSD = 0 (C5)Brand Sub-Brand Prices Profit Prices Profit

Dannon Creamy Fruit Blends 1.588 1.589Dannon Fruit on the Bottom 1.730 1.730Dannon Light N Fit 1.798 1.790Dannon Light N Fit Creamy 1.682 733.5 1.682 733.9Kemps Classic 1.263 1.264Kemps Free 1.367 241.4 1.368 241.8Old Home Classic 1.398 1.398Old Home 100 Calories 1.347 207.7 1.348 208.0Wells B.B. Lite 85 1.408 439.2 1.408 439.3

Yoplait Light 1.673 844.0 1.663 786.6Yoplait Original 1.672 881.7 1.662 824.1Yoplait Thick and Creamy 1.673 163.5 1.665 140.9

that now all sub-brands of Yoplait maximize sub-brand profits and do not internalize each others

demand in their respective profit objective functions. To see the impact of setting prices jointly for

Yoplait, one should compare the base case in Table 7, (C4), with the base case numbers in Table 6

(C1). Again, focusing on Yoplait alone is sufficient as the impact on rivals is small. With respect

to prices, we see that full multi-product pricing leads Yoplait to charge higher prices, reflecting an

increase in market power. The prices of the three sub-brands are 1.691, 1.688 and 1.716 under full

firm-level profit maximization versus 1.673, 1.672, and 1.673 under sub-brand profit maximization.

However, the impact of the increased market power on profits is quite modest; per period profits

only decrease by 0.1%6 when Yoplait decentralizes its pricing strategy completely. This number

can be seen as the incremental benefit of fully internalizing the dynamic multi-product pricing

externality for Yoplait and can be managerially useful for a possible organizational decision with

respect to price setting. If, for example, the coordination of profit maximizing prices across the

three sub-brands is costly (e.g. because of managers workloads) and this cost is higher than 0.1%

of per period profits, our experiment shows that decentralized pricing might be preferable.

While centralized pricing should clearly be more profitable in all cases, it is interesting to

examine how its benefits relate to parent brand state dependence. Given the inter-temporal com-plementarities that parent brand state dependence creates, it is possible that it reduces the benefits

of multi-product pricing for firms. This can be so because when a firm internalizes demand de-

pendencies across different sub-brands, under parent brand state dependence it has an incentive

to lower prices and by doing so increase its profitability. On the other hand, when there are no

6This number is computed as the difference between total Yoplait profits with sub-brand profit maximizationminus total profits with full firm-level profit maximization, expressed as a percentage of the latter.

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inter-temporal dependencies between sub-brands, joint pricing tends to increase prices because it

only internalizes competition. The far right columns in Table 7 reports results from a scenario in

which Yoplait employs decentralized pricing (i.e. sub-brand profit maximization) and there are nodynamic effects from one sub-brand to the others. This scenario is comparable with the results in

Table 6 where again cross-sub-brand effects were eliminated but Yoplait decided on prices jointly

for all three sub-brands. Comparing the two sets of results, we see that without the cross-sub-brand

effects, decentralized pricing still leads to lower prices (1.663, 1.662 and 1.665 versus 1.696, 1.693,

and 1.750) but in this case the associated profit loss is 0.4% of per period profits. This means

that the lack of coordination in pricing between the three sub-brands of Yoplait would be more

costly if there were no inter-dependencies in their respective demand. This result is reasonably

expected to have implications in all cases where demand is likely characterized by parent brand

state dependence and one seeks to evaluate centralized pricing.

5.2.4 Uniform Pricing

It is not uncommon for sub-brands of yogurt belonging to the same parent brand to be marketed

under a single common price (i.e. uniform pricing). Our forward looking pricing model can be used

to quantify the loss associated with this type of constrained pricing7. This loss can be benchmarked

against possible menu costs that lead managers to impose uniform prices. Such costs may exist

because of labeling, data storing and processing reasons for example. If menu costs are higher than

the loss that sub-optimal prices generate, then uniform pricing is the rational decision. To examine

this issue, we compare the base case scenario with counterfactual scenarios where brands charge

the same price for all sub-brands under their parent name. The results in Table 8 refer to Yoplait

and show that even though the prices of the three sub-brands of Yoplait change noticeably, the

associated loss in per period profits is only 0.02%. The reason behind this effect being very small

probably has to do with the fact that the biggest change in prices is for Yoplait Thick and Creamy

which has a lower price under uniform pricing. The results imply that any losses in revenues from

this price decrease are likely counterbalanced from increases in demand and revenue for the other

two Yoplait sub-brands; something implied by parent brand state dependence.

Table 8 also reports the comparison results for the case of Dannon. Notably, a uniform pricing

strategy for Dannon is much more costly the associated loss is about 1.1% of per period profits.

This increase in the cost of imposing a single price can be attributed to Dannon offering sub-brands

that have very different prices in the base case (i.e. 1.589, 1.730, 1.791 and 1.683). Therefore the

sub-optimality of imposing a single price (1.749 for Dannon) is a more significant change. The

different results between the two major parent brands arguably demonstrate the usefulness of the

counterfactual experiment. Managers must evaluate the cost of uniform pricing on a case by case

7Holding everything else constant, constraining the price of each sub-brand to be the same, inevitably produces asub-optimal decision compared to the unconstrained profit maximization problem.

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Table 8: Uniform Pricing

Base Case (C1) Yoplait (C6) Dannon (C7)

Brand Sub-Brand Prices Profit Prices Profit Prices ProfitDannon Creamy Fruit Blends 1.589 1.589 1.749Dannon Fruit on the Bottom 1.730 1.730 1.749Dannon Light N Fit 1.791 1.791 1.749Dannon Light N Fit Creamy 1.683 737.9 1.683 737.9 1.749 729.7Kemps Classic 1.264 1.263 1.263Kemps Free 1.368 242.4 1.367 242.4 1.368 242.5Old Home Classic 1.398 1.399 1.398Old Home 100 Calories 1.347 208.6 1.348 208.6 1.347 208.7Wells B.B. Lite 85 1.408 442.2 1.408 442.2 1.408 442.7Yoplait Light 1.691 1.692 1.691

Yoplait Original 1.688 1.692 1.688Yoplait Thick and Creamy 1.716 1891.2 1.692 1890.9 1.716 1894.3

basis. Looking at the price data, indeed the sub-brands of Yoplait, especially Original and Light,

are usually offered at the same regular price. The prices of Dannon sub-brands on the other hand

are almost always different from each other.

It is telling that in both counterfactual experiments discussed above, the prices of the brands

that are not being evaluated do not change very much when either Yoplait or Dannon change their

pricing strategies with respect to uniform prices. This result suggests that competitive reactions are

not very strong and only become visible when a major brand changes its behavior more dramatically.

Of course it is important to remember that we use steady state prices to summarize our results and

these are by definition relatively stable. Pricing policies corresponding to particular state variable

values can show greater responsiveness. The long-term equilibrium prices are much less sensitive

to relatively small changes in rivals prices, like the ones we evaluated here proved to be.

6 Conclusion

There is increasing academic interest in the long term impact of marketing actions and the

perils of myopic behavior on the part of managers and decision makers. In this work, we take a

forward looking perspective on the pricing decision of multiproduct firms and examine the effectof demand choice dynamics that reflect consumers tendency to stay loyal to a firm when choosing

between multiple sub-brands of a broader category. We examine a product category, refrigerated

yogurt, where consumers choices depend not only on past choices of specific product alternatives

but also the parent brands. Such cross-sub-brand state dependence for repeated purchase goods

generates interesting dynamics with non-trivial implications on equilibrium prices and profitability.

In order to specify a tractable model and explore some of the dynamic implications we had

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to abstract from important factors like frequent price discounts and retailer intervention for the

formation of shelf price. Nevertheless, studying equilibrium steady state prices in a dynamic game

context reveals novel aspects of price dynamics for multiproduct firms. A more realistic game withthe inclusion of the retailer/manufacturer game and/or the exploration of equilibrium pricing with

frequent discounts is an interesting avenue for future research.

Our findings for the yogurt category-leading brand, Yoplait, can be summarized in three key

points. First, the existence of parent brand state dependence of demand pushes equilibrium prices

downwards, because it incentivizes the firm to invest in future demand, and increases profitability,

because the lower prices increase per period demand for the same firm. Second, the incremental

benefit of centralized pricing for Yoplait is not very high, around 0.1% of per period profits, and this

is in part due to the existence of parent brand dynamics in demand. Third, profit losses associated

with uniform pricing in the forward looking model are relatively low for Yoplait and they are also

mediated by parent brand state dependence. The equivalent losses are significantly larger for the

second higher share brand in the category, Dannon. Overall, the novel findings reported in this work

underline the intricate role of demand choice dynamics for prices of forward looking multi-product

firms and highlight new dimensions of it. It seems reasonable to expect that the methodology and

findings in this work are relevant for several other product categories where demand is likely to

characterized by parent brand dynamics.

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References

[1] Anand, B., Shachar, R., 2004. Brands as Beacons: A New Source of Loyalty to Multiproduct

Firms, Journal of Marketing Research. Vol. 16, p135.

[2] Anderson, S., De Palma, A., 2006. Market Performance with Multiproduct Firms. The Journal

of Industrial Economics. Vol. 54, No 1, p 95.

[3] Arie, G., Grieco, P., 2010. Who Pays for Switching Costs?. Working Paper.

[4] Bajari, P., Benkard, L., Levin, J., 2007. Estimating Dynamic Models of Imperfect Competition.

Econometrica. Vol. 75, No 5, p1331.

[5] Besanko, D., Doraszelski, U., Kryukov, Y., Satterthwaite, M., 2010. Learning-by-doing, Orga-

nizational Forgetting, and Industry Dynamics. Econometrica. Vol. 78, No. 2, p453.

[6] Bronnenberg, B., Kruger, M., Mela, C., 2008. Database Paper: The IRI Marketing Data Set.

Marketing Science. Vol. 27, No. 4, p745.

[7] Doganoglou, T., 2010. Switching Costs, Experience Goods and Dynamic Price Competition.

Quantitative Marketing and Economics. No. 8, p167.

[8] Doraszelski, U., Satterthwaite, M., 2010, Computable Markov-perfect Industry dynamics.

RAND Journal of Economics. Vol. 41, No. 2, p215.

[9] Draganska, M., Jain, D.C., 2005. Consumer Preferences and Product-Line Pricing Strategies:An Empirical Analysis. Marketing Science. Vol. 25, No. 2, p164.

[10] Dube, JP., Hitsch, G., Rossi, P., Vitorino, MA., 2008. Category Pricing with State Dependent

Utility. Marketing Science. Vol. 27, No. 3, p417.

[11] Dube, JP., Hitsch, G., Rossi P., 2009. Do Switching Costs Make Markets Less Competitive?

Journal of Marketing Research. Vol. 46, No. 4, p435.

[12] Dube, JP., Hitsch, G., Rossi, P., 2010. State Dependence and Alternative Explanations for

Consumer Inertia. RAND Journal of Economics. Vol. 41, No. 3, p417.

[13] Erdem, T., 1998. An Empirical Analysis of Umbrella Branding. Journal of Marketing Research.

Vol. 35, p339.

[14] Farrell, J., Klemperer, P., 2007. Coordination and Lock-In: Competition with Switching Costs

and Network Effects. In Handbook of Industrial Organization. Vol. 3. Elsevier.

[15] Erdem, T., Sun, B., 2002. An Empirical Investigation of the Spillover Effects of Advertising

and Sales Promotions in Umbrella Branding. Journal of Marketing Research. Vol. 39, p408.

23


24/28

[16] Gamerman, D., Lopes, H., 2006. Markov Chain Monte Carlo: Stochastic Simulation for

Bayesian Inference. Second Edition, Chapman & Hall/CRC.

[17] Gelman, A., Carlin, J., Stern, H., Rubin, D., 2004. Bayesian Data Analysis. Second Edition,

Chapman & Hall/CRC.

[18] Heckman, J.J., 1981. Statistical Models for Discrete Panel Data. In Structural Analysis of

Discrete Data with Econometric Applications. Cambridge, Mass.: MIT Press.

[19] Horsky, D., Misra, S., Nelson, P. 2006. Observed and Unobserved Preference Heterogeneity in

Brand-Choice Models. Marketing Science. Vol. 25, No. 4, p322-335.

[20] Horsky, D., Pavlidis, P., 2011. Brand Loyalty Induced Price Promotions: An Empirical Inves-

tigation. Working paper, University of Rochester.

[21] Judd, K., 1998. Numerical Methods in Economics. The MIT Press, Cambridge, Massachusetts.

[22] Klemperer, Paul, 1995. Competition When Consumers Have Switching Costs: An Overview

With Applications To Industrial Organization, Macroeconomics And International Trade. The

Review Of Economic Studies. Vol. 62, No. 4, p515.

[23] Miklos-Thal, J., 2012. Linking Reputations through Umbrella Branding. Quantitative Market-

ing and Economics. Vol. 10, No. 3, p335.

[24] Morrow, R., Skerlos, S., 2010. On the Existence of Bertrand-Nash Equilibrium Prices Under

Logit Demand. Working Paper. University of Michigan.

[25] Nevo, A., 2001. Measuring Market Power in the Ready-to-Eat Cereal Industry. Econometrica.

Vol. 69, No. 2, p307.

[26] Rossi, P., Allenby, G., McCulloch, R., 2005. Bayesian Statistics and Marketing. New York:

John Wiley and Sons.

[27] Seetharaman, P.B., 2004. Modelling Multiple Sources of State Dependence in Random Utility

Models: A Distributed Lag Approach. Marketing Science. Vol. 23, No. 2, p263.

[28] Seetharaman, P.B., Ainslie, A., Chintagunta, P., 1999. Investigating Household State Depen-

dence Effects Across Categories. Journal of Marketing Research. Vol. 36, No. 4, p488.

[29] Shin, S., Misra, S., Horsky D., 2012. Disentangling Preferences, Inertia, and Learning in Brand

Choice Models. Marketing Science. Vol. 31, No. 1, p115-137.

[30] Shin, J., Sudhir, K., Cabral, L., Dube, JP., Hitsch, G., Rossi, P., 2009. Commentaries and Re-

joinder to Do Switching Costs Make Markets Less Competitive? Journal of Marketing Research.

Vol. 46, No. 4, p446.

24


25/28

[31] Viard, B., 2007. Do Switching Costs Make Markets More or Less Competitive? The Case of

800-Number Portability. RAND Journal of Economics. Vol. 38, No. 1, p146.

25


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A Computational Details

A.1 State Space Discretization

The approximation starts with discretizing the state space on a multidimensional grid. Each axis

of the grid corresponds to one dimension of the state space, namely the fraction of a particular type

consumers loyal to a specific sub-brand. We discretize each axis of the grid with a finite number of

points g, such that: 0 = gn1jt < ... < gnL

jt = 1, j, n. The grid is formed by the Cartesian product

of all finite sets of points for each axis, such thatJ

k=1 gnlkt 1, l = 1,...,L, n. Intuitively, this

condition says that, for any consumer type n, the fraction of the market loyal to each sub-brand

in the choice set should sum up to one across all sub-brands. The main computational challenge

in solving a discretized version of a dynamic programming problem with continuous state space is

caused by high dimensionality. This is because the number of grid points at which one must solve

for the value and policy functions increase exponentially with the dimensionality of the state space.

For the game described in section 3 the state space has dimension equal to G = (J 1) N, that is

the number of choice alternatives minus one, times the number of consumer types. Because of the

condition that loyalty states for each type sum up to one across sub-brands, only J-1 loyalty states

need to be tracked for each type. For a regular grid with L points in each axis, the total number of

points would be LG. The grid for this work in particular is not rectangular but rather triangular

(if we think about it in three dimensions) because of the condition that the states of loyalty of

a consumer type to all sub-brands must sum up to one. Even though this condition reduces the

number of grid points a little, as the dimensionality of the state space increases, the computational

requirements of the grid become exorbitant. It is relatively easy to see that the dimensionality

of the space increases faster with the number of brands when the number of consumer types is

higher. This creates a trade-off between using more consumer types or more choice alternatives.

By settling with one consumer type, we were able to include all twelve sub-brands of the sample

in the pricing model. To the best of our knowledge, a forward looking pricing model with such a

large number of competing alternatives appears for the first time in the literature.

A.2 Interpolation

During the computation, in all cases where we need to compute the value function or the policy

action on state space points outside the grid we use polynomial based interpolation. Our polynomial

approximation function has the general form given in expression 17. It includes all the first, second

and third order terms of the state variables, their square root, and several sets of (two way, three

way, four way etc) interactions between states of different brands for the same consumer type.

In the implementation of the algorithm, the correlation between predicted polynomial values and

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actual values is about 0.9. This suggests that the approximation works well in practice.

yj (s) =

a0 +N

n=1

J1j=1

m{0.5,1,2,3} anjm

snj

m(17)

+K

k=1 ak

lIksnl

A.3 Policy Function Iteration

The dynamic game analysis proceeds in two steps. In a first phase we compute optimal pricing

policies for each point in the state space and in a next step we compute steady state prices and

shares for all sub-brands of the game. The steps for the policy function iteration are as follows:

1. Start with initial guesses of value functions and price strategies. For all reported resultswe use zero to initiate the value functions and optimal prices for static period profits to

initiate the policy functions, for each point in the state space; V0f (s) = 0 an d 0f(s) =

maxPjf

s,P,0f(s)

s S, f. The initial policies are iterated so that they are best

responses for the static case.

We experiment with different initial guesses, for the value and policy functions, to exam-

ine whether the equilibrium policies are the same or change depending on the starting

values; the latter would imply the existence of multiple equilibria. It is noteworthy that

all trials generated the same equilibrium policies.

2. For each firm, given Viter1f and iter1, compute Viterf ; here superscripts refer to iterations

of the algorithm. The computation of value functions involves the purchase probabilities that

determine both static and future discounted profits.

(a) Iterate the Bellman equation until convergence,max|Viter

fViter1

f|

max|Viterf

| v, f. Then move

to the next step.

3. For each firm, given Viterf and iter1, compute iter. This involves finding the optimal price

for f, for each point on the grid, given the right hand side of the Bellman equation and the

rivals policy. Optimal prices satisfy simultaneously the J first order conditions of the current

and discounted future profits objective functions. Conditional on the value functions, which

are functions of prices and demand parameters, the first order conditions are well defined.

(a) If the computed policies converge for each firm ,max|iter

fiter1

f|

max|iterf

| , f, stop the

algorithm.

(b) If not, update the policies and return to step 2.

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Doraszelski and Satterthwaite (2010) give the following definition for a Markov-perfect equilibrium:

An equilibrium involves value and policy functions V and such that i) given f, V solves the

Bellman equation for all f and ii) given f(s) and Vf, f(s) solves the maximization problemon the right hand side of the Bellman equation for all s and all f. Markov-perfect equilibria are

by definition sub-game perfect, that is firms have optimal strategies for each possible state of the

game. Upon convergence, the algorithm described above satisfies these general conditions and thus

computes a Markov-perfect equilibrium.

A.4 Steady State Computation

Once the policy functions are computed, a potential way of summarizing the market outcome

is to compute a steady state for the market. This is equivalent to forward simulating demand until

the prices and states of loyalty, to each brand and sub-brand, stabilize. In this study, we treatthe steady state as a succinct summary of the price equilibrium that reflects general price levels,

loyalty shares and per period profits.

To compute the steady state of a pricing game specification, we start from some initial state

vector and given the best response price policies we first find the optimal price of each firm for

the given period. Then we compute the states that prevail in the market under the prices of the

last iteration and use them as the next periods state vector. We repeat the process until both

the state variables and the prices of each specific product alternative converge. To verify if the

obtained steady state is unique we repeat the computations several times starting from different

initial states. Throughout all trials the algorithm always converges to the same unique steady state.

A.5 Game Parametrization

To complete the game specification we also need to make assumptions regarding parameters

that are part of the model but are unobserved, namely the discount factor and the total size of the

market. The parametrization used in the algorithm is: = 0.998, MS = 100000.

In order to keep the dimensionality of the problem tractable we limit the number of consumer

types to one. Increasing the number of types, even by one, is literally infeasible unless we limit the

number of brands to much less than twelve (e.g. four). Using one consumer type is probably more

realistic with regards to tracking the loyalty shares of each brands as it doesnt require that firmstrack loyalties for consumer groups based on the groups different demand parameter vectors.

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