Switching Costs and Market Power Under Umbrella Branding∗
Polykarpos Pavlidis†
Paul B. Ellickson‡
August 18, 2015
Abstract
This paper extends recent research on the importance of switching costs to forward looking
pricing outcomes in the area of umbrella branding and multi-product firms. Focusing on the
Yogurt category, we consider parent brands with several sub-brands. We show through numerical analysis that state dependence to the parent brand can i) decrease equilibrium prices, ii)
mitigate or even reverse the benefits of joint profit maximization compared to sub-brand profit
maximization and iii) lead multi-product firms to price their various sub-brands more uniformly.
Using household level scanner data, we estimate the parameters that characterize consumer demand and firm cost. We then use counterfactual analysis to study the general effect of parent
brand state dependence on prices and profits, as well as two strategic issues pertaining to pricing
practice: 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 5%, centralized decision making generates only 0.2% incremental profits and this number is mediated by cross-sub-brand dynamics,
and finally, switching to uniform pricing across three sub-brands yields only a 0.2% decrease in
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. All
remaining errors are our own.
†
Nielsen Marketing Analytics, polykarpos.pavlidis@nielsen.com.
‡
University of Rochester, paul.ellickson@simon.rochester.edu.
1
1
Introduction
Consumer packaged goods are frequently 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. Past exposure to a brand’s performance should guide a consumer’s purchase decision the
next time they shop in 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, Dubé, Hitsch and Rossi 2009). On one hand, it places 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
occur if today’s prices are relatively higher, yielding a downward force.
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 often motivated by brand loyalty (e.g. Klemperer (1995)), this is an important
question that has not been fully addressed in the literature. 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 that product only or if it also depends on past experience with other products offered by 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 on the pricing decisions regarding its other products.
It is well known that optimal multi-product pricing increases market power since it allows a
firm to internalize part of the competition across its different products, thereby leveraging ”local”
monopoly power as in Nevo (2001). The existence of firm state dependence should have non-trivial
2
implications 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 product offerings, under the same brand name, where
the relevant 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 must carefully weigh two options: 1) charge a little more for each subbrand since part of the lost demand will inter-temporally accrue to the other sub-brand through
state dependence to the parent brand, or 2) lower the price of each sub-brand since 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 countervailing 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 Dubé, Hitsch and Rossi (DHR 2009), which we extend to the case
of multiproduct firms. Analyzing the demand for parent brands that offer a portfolio of sub-brands,
we estimate state dependence (inertia) both to the parent brand (PBSD) and to sub-brands. Our
data rejects the alternative explanation of learning and points to the existence of true structural
state dependence. Using our demand estimates and model of forward looking pricing behavior, we
first recover estimates of the relevant marginal costs - 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 is aimed at
exploring three specific questions, enumerated below.
First, we examine the impact of parent brand state dependence on equilibrium prices and profits
by contrasting the base case of the estimated demand parameters with one where we eliminate the
parent brand level dynamics. We run two sets of counterfactual scenarios, one where we vary
the magnitude of the parent brand state dependence coefficient and another where we simulate a
market structure where there are no state dependence effects across a subset of the sub-brands.
In both cases we find that, consistent with results reported in the literature for sub-brand state
dependence (e.g. DHR (2009)) for single-product decision makers, the relation between parent
brand state dependence 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 current
purchases. We also find that firm profits increase with parent brand state dependence because the
associated lower prices lead to higher per period demand. When consumers exhibit parent brand
3
state dependence, as prices decrease, firms can capitalize on the intertemporal complementarities
to their other sub-brands and this additional dimension drives profitability upwards. For the case
of Yoplait, we find that eliminating cross-sub-brand dynamics would decrease net discounted profits
by 5.4%.
Second, we quantify the benefit of centralized price setting. Employing the forward looking
pricing model, we evaluate the incremental profits accrued from coordination across sub-brands
prices. 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 state
dependence 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
pricing is more lucrative when parent firm state dependence effects are absent. For example,
decentralized pricing would cost Yoplait 0.2% of per period profits under the estimated levels of
state dependence but 0.5% absent parent brand loyalty.
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 the menu cost 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 (0.2% for Yoplait and 0.3% for Dannon), then the
latter is preferable.
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 Klemperer (2007) provide a comprehensive survey, covering 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 incentive to harvest. Viard (2007) empirically tests the impact
of portability switching costs on prices for toll-free services and finds a positive impact - 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 arising from state dependence
4
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 using consumer transaction data, are well within
the range that increases competition. In related work, Dubé, 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
greater loyalty.
These new empirical findings have sparked additional theoretical research examining the impact
of switching costs on equilibrium pricing. Arie and Grieco (2014) 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
prices to compensate marginal consumers who are loyal to rivals. In their setting, the investing
effect of attracting future loyal customers decreases prices when switching costs increase, provided
that no “very” dominant firm is present 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 an MPE that supports switching in equilibrium.
2.2
Multiproduct Firms / Umbrella Branding
In the context of multiproduct firms, Erdem (1998) shows the existence of cross-category brand
spillover effects. Consumers’ choices in the categories of toothpaste and toothbrush reveal behavior
consistent with the predictions of signaling theory. A positive experience with a brand in one
category reduces the perceived uncertainty of the brand’s 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, that they are willing to pay for quality,
but that they consider flavors belonging to the same product line to be of comparable quality.
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)
test and confirm that consumer loyalty to multiproduct firms is also driven by information-related
benefits. Anderson and De Palma (2006) outline how 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 to the market.
5
3
Model Setting
3.1
Demand Utility Function
We model demand using a discrete choice framework. We assume consumers are in the market
for 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 consumer
h’s conditional indirect utility from consuming product j is composed of a linear deterministic
component, U , and additive Type I Extreme Value demand shocks, ε, yielding the well known logit
probabilities for purchase of each sub-brand.
uhjt = βhj − βhp Phjt + βhP B P BLhjt + βhSB SBLhjt + βhT T Lhjt + εhjt
= Uhjt + εhjt , j ∈ {1, ..., J}
(1)
uh0t = εh0t
(2)
P r(Chjt = 1) = P r (uhjt > uhkt ∀k ∈ J, k 6= j, & max uhkt > uh0t , ∀k ∈ J)
(3)
P r(Chjt = 1) =
1+
eUhjt
P
k∈J
eUhkt
(4)
In the utility functions (1), βhj denotes the intrinsic preference for each choice alternative j,
while Phjt represents the alternative’s 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 household’s last purchase was for the parent brand of the choice alternative j
and the latter being equal to one if last purchase was of the same sub-brand. For a given alternative
j there are three possibilities with respect to the brand related 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 or 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 last chosen by the consumer. Coefficients βhP B and βhSB thus capture the effect of
last purchase on the current period’s 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 βhP B should not be statistically
different from zero. In addition to brand related state dependence, we allow for state dependence
with respect to the type of yogurt. The indicator variable T Lhjt takes the value one for sub-brands
that are of the same type (i.e. light or regular ) as the last sub-brand purchased by consumer h.
6
By including this additional state variable in our utility function, we ensure that the brand related
variables do not confound sub-brand/brand state dependence with state dependence to yogurt type.
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 (to the
researcher) ingredient quality or brand capital which will be captured through these intercepts.
Dubé, Hitsch and Rossi (2010) highlight an important confound between state dependence and
unobserved preference heterogeneity: absent a rich representation of consumer preferences, state
dependence could simply be an artifact of unobserved tastes. 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).
βh ∼ N (β̄lh , Σlh )
(5)
lh ∼ multinomial (π)
(6)
The idea is that including this flexible heterogeneity distribution accounts for any persistence in
choice that is driven by consumer tastes.
3.2
Demand Estimation
Following Dubé, 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).
h
i
D(y, θ̂|M ) = −2 × log l(θ̂|M )
(7)
R
1 X
D̂Avg (y|M ) =
D(y, θr |M )
R
(8)
DIC(y|M ) = 2 × D̂Avg (y|M ) − D(y, θ̂|M )
(9)
r=1
DIC (defined in equation 9) is based on the Deviance of the model (equation 7) which reflects
discrepancy between the model and data. DIC is a more complete model selection metric compared
to the simple Deviance because it reflects coefficient uncertainty and penalizes for model complexity
(Gelman 2004, Gamerman & Lopes 2006). Lower values of DIC indicate a preferred model.
7
3.3
Total Demand & Evolution of States
While we use a very flexible heterogeneity distribution in demand estimation, we use a single
consumer type to approximate total demand and evolution of states that lead to future discounted
profits. In theory it is straight forward to add more than one consumer type but in combination
with the number of choice alternatives the pricing game eventually yields a curse of dimensionality.1
To mitigate this we restrict attention to a single consumer type for purposes of the pricing model.
At any period t, a fraction skt of all consumers will have chosen a particular alternative k in their
last purchase. The fraction of consumers loyal to each sub-brand comprises the state space that is
the foundation of each firm’s dynamic pricing problem2 . The probability of choice for j conditional
on k being chosen last is denoted by P r(Cjt = 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.
Djt =
J
X
skt × P r(Cjt = 1|k)
(10)
k=1
J
X
skt = 1
(11)
k=1
The state vector st = (s,1t ..., sJt )0 evolves deterministically over time based on the choices made
by consumers the previous period. This evolution is operationalized with a Markovian transition
matrix. The element in the jth row and kth column of the transition matrix Q, is denoted by Qjk
and is equal to the conditional probability that a consumer will choose sub-brand j given that he or
she is loyal to sub-brand k. The transition matrix is naturally a function of the demand parameters
and prices of all sub-brands.
st+1 = g(Pt , st ) = Q(Pt ) × st

P r(C = 1|k) + P r(C = 1|k)
jt
0t
Qjk (Pt ) =
P r(C = 1|k)
jt
(12)
if j = k
(13)
if j 6= k
Given the choice probabilities in (4) and the transition function described in (12), the share at
time t of a specific alternative j belonging to brand B depends on its share at t-1 and also on the
1
For example, in a market of one type and twelve sub-brands belonging to less than twelve parent brands, the
state space has dimension of eleven ((12-1)x1). If one wants to add a consumer type, the dimensionality increases
to twenty-two ((12-1)x2). In previous iterations of this work we used two consumer types for a subset of sub-brands
and reached similar qualitative results.
2
An advantage of specifying the pricing game with a single consumer type is that it makes it easier for firms to
track the current state when applying our methodology in practice. It is significantly more costly to track the loyalty
of different consumer types in conjunction with each type’s preference and price sensitivity parameters.
8
share at t-1 of the other alternatives that belong to the same parent brand. The dependence is
operationalized through i) the state vector at t which determines the distribution of loyalty values
across consumers for the particular period, and ii) the three state dependence variables P BLhjt ,
SBLhjt , T Lhjt and their associated coefficients, in the utility function.
3.4
Pricing Behavior
In this section we specify a model of forward looking pricing based on the demand system
just described. There are K firms in the market and each one may carry several sub-brands
j ∈ f, j = 1, .., Jf , f = 1, ...K. Time is discrete and the profits of each firm f in any period t
P
are a function of the state vector and all market prices: πf t (st , Pt ) = j∈f Djt × M × (Pjt − cj )
where st is the complete state vector, cj is the marginal cost of production for sub-brand j, M is
the total market size, and D and P denote demand and prices respectively. The assumption of the
state vector entering the profit function is equivalent to assuming that firms are fully aware of the
state dependence dynamics and observe the fractions of the market loyal to each sub-brand, setting
prices accordingly. The wide availability of rich scanner data makes this assumption plausible,
especially for established brand manufacturers with long presence in their respective markets, as
is the case here. At equilibrium, firms have best response pricing strategies that maximize their
current and future profits conditional on payoff relevant information that is captured in the state
vector. The strategies are functions that map the state space into the continuum of possible prices,
σf : S → R. Given the transition function in (12), strategies are Markovian and depend only on
last period’s actions and outcomes, which in turn determine the current period’s states.
3.4.1
Firm-Level Profit Maximization
Firms maximize discounted profits for all their products jointly over an infinite horizon. The
current and future payoffs of each firm, for any point in the state space which corresponds to a
specific allocation of consumer loyalties across sub-brands, are described by the Bellman equation
Vf (s) = maxpj ≥0, j∈f {πf (s, P ) + βVf [g(P, s)]}
∀s ∈ S
(14)
in which g(·) is the transition kernel described above. The value function Vf in (14) represents
all payoffs given a specific strategy profile for firm f, σf . 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 σ−f , the optimal strategy for f, σf∗ , satisfies the
following Bellman equation.
∗
∗
Vf (s) = maxpj ≥0, j∈f πf s, P, σ−f
(s) + βVf g(P, s, σ−f
(s))
9
∀s ∈ S, ∀f ∈ K
(15)
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 statespecific strategy profile of each firm contains as many prices as there are sub-brands under the firm’s
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 game’s 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) = maxpj ≥0 {πj (s, P ) + βVj [g(P, s)]}
∀s ∈ S
∗
∗
Vj (s) = maxpj ≥0 πj s, P, σ−j
(s) + βVj g(P, s, σ−j
(s))
∀s ∈ S, ∀j ∈ J
(16)
(17)
Comparing the optimal prices in two different scenarios, one where firms maximize sub-brand
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).3
Following DHR (2009) we use a numerical approach in computing pure strategy equilibrium for the
3
Morrow & Skerlos (2010) prove and characterize the existence of Bertrand Nash equilibrium between multiproduct
firms 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 firms
are not quasi-concave, they base their approach on fixed-point equations. Although they generalize their numerical
fixed-point approach for equilibrium price computations to the Mixed logit context, they do not prove existence there.
10
full game. Since the game may admit more than one equilibrium, we are effectively assuming that
our algorithm finds the correct one. While we do employ a variety of different starting conditions
for the algorithm, there are no known methods for finding all equilibrium and thus no guarantee
that we find the correct one.
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 constructs.
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 are loyal to the sub-brands of the respective firm at each point in time. The policy
functions are infinite-dimensional functions. The value function in (14) is also of infinite dimension
since it is defined uniquely for each point in the state space. To circumvent the intractable problem
of solving for infinite-dimensional functions, we approximate the solution to the dynamic game
specified by equations (14) and (15) by discretizing the state space and interpolating the values for
points that do not fall on this finite grid. 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.
3.6
Theoretical Implications of PBSD
In this sub-section, we use numerical simulation to establish the theoretical implications of
parent brand state dependence on prices and firm profits. Our results are based on a simplified
model and help us flesh out implications for a range of state dependence parameter values. In our
streamlined model, there are just two firms and a single consumer. The first firm carries sub-brands
1 and 2 while the second firm carries sub-brands 3 and 4. The utility function is defined as in subsection 3.1, with the following symmetric parameter values: sub-brand intercepts βj = 1 ∀j, price
coefficient βp = -1 and costs cj = 0.5 ∀j ∈ 1, .., 4. The parameters for parent brand state dependence
(βP B ) and sub-brand state dependence (βSB ) vary across different simulation scenarios4 . Each
In forward looking settings, there are proofs for existence of equilibrium between single product firms under
standard 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 of
a unique profit maximizing price. Besanko, Doraszelski, Kryukov and Satterthwaite (2010) and DHR (2009) for
a simple case of their model, prove the existence of equilibrium through quasi-concavity. Unfortunately, the same
approach cannot be followed in the case of multiproduct firms as the profit function is no longer quasi-concave.
4
Similar to DHR 2009 we adjust βj as we increase the state dependence parameters so that the size of the outside
good remains constant. This allows us to isolate the effect of state dependence magnitude from market size expansion.
11
2.50
2.40
2.30
Price
2.20
2.10
Forward Looking
Myopic
2.00
1.90
1.80
1.70
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
PBSD
Figure 1: Equilibrium Policy: Forward Looking vs Myopic
period, brands (or sub-brands depending on the scenario) compete for the consumer’s business
while maximizing their discounted net future profits. State dependence in each period is based on
the previous period’s choice by the consumer (e.g. state 1 refers to sub-brand 1 being chosen the
previous period) and it induces imperfect lock-in as the consumer always has positive probability
of switching and choosing any of the remaining sub-brands.
In each of tables 1, 2 and 3 we report the average transaction price (p =
PJ
j=1
σj (s=1)×P r(j|s=1)
PJ
)
k=1 P r(k|s=1)
of the pricing game for state s=1.This is the average price that the consumer is expected to pay if
last period he/she chose sub-brand 1. Due to symmetry the average transaction price is the same
across all states. We also report the optimal pricing policy and the value function of sub-brand
1 for each possible state. Since sub-brands are symmetric in our numerical analysis, the policies
and value functions of all sub-brands are identical up to the value of the state variable. Table 1
reports results for various values of parent brand state dependence (βP B ) and no sub-brand state
dependence (βSB = 0) (Case 1). Tables 2 and 3 expand reported results to positive values of
sub-brand state dependence (Case 2 for βSB = 0.5 and Case 3 for βSB = 1 respectively). All three
tables include two sets of results, one corresponding to firms maximizing joint profits over their
sub-brands and one for sub-brand profit maximization. We note that the value function in both
cases includes the profit flow from two sub-brands, in other words we report profits for firm 1 that
owns sub-brands 1 and 2.
The first conclusion we draw from our numerical analysis is that parent brand state dependence
has a U-shaped relation to price policies in the forward looking multiproduct pricing model. Prices
tend to initially decrease as the magnitude of PBSD increases and then start increasing. This
result holds for all cases and is similar to the impact of sub-brand state dependence that DHR 2009
uncovered. At moderate levels of PBSD, prices are more aggressive as firms compete for future
12
PBSD
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
Net Discounted Profits (Value Function)
Joint Profit Maximization
Sub-brand Profit Maximization
V1 (s=1) V1 (s=2) V1 (s=3) V1 (s=4)
V1 (s=1) V1 (s=2) V1 (s=3) V1 (s=4)
219.4
219.4
219.4
219.4
199.5
199.5
199.5
199.5
205.9
205.9
205.6
205.6
191.7
191.7
191.4
191.4
188.9
188.9
188.3
188.3
184.1
184.1
183.4
183.4
171.2
171.2
170.3
170.3
177.2
177.2
176.0
176.0
153.7
153.7
152.4
152.4
171.1
171.1
169.4
169.4
138.0
138.0
136.3
136.3
166.0
166.0
163.8
163.8
153.9
153.9
151.5
151.5
147.3
147.3
144.8
144.8
193.7
193.7
187.8
187.8
179.7
179.7
173.5
173.5
216.6
216.6
201.4
201.4
199.2
199.2
183.3
183.3
250.0
1.00
0.90
200.0
0.80
0.70
150.0
0.60
0.50
100.0
0.40
Myopic profits per period
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
p
1.94
1.77
1.62
1.49
1.37
1.27
1.39
1.68
1.85
Net discounted profits (Value Function)
PBSD
Table 1: Theoretical Implications of PBSD - Case 1: SBSD = 0
Average Transaction Price (p) and Equilibrium Policies (σ1 )
Joint Profit Maximization
Sub-brand Profit Maximization
σ1 (s=1) σ1 (s=2) σ1 (s=3) σ1 (s=4)
p
σ1 (s=1) σ1 (s=2) σ1 (s=3) σ1 (s=4)
1.94
1.94
1.94
1.94
1.70
1.70
1.70
1.70
1.70
1.91
1.91
1.61
1.61
1.62
1.69
1.69
1.53
1.53
1.88
1.88
1.25
1.25
1.56
1.68
1.68
1.33
1.33
1.84
1.84
0.88
0.88
1.51
1.68
1.68
1.09
1.09
1.81
1.81
0.50
0.50
1.47
1.67
1.67
0.83
0.83
1.78
1.78
0.11
0.11
1.44
1.67
1.67
0.53
0.53
1.81
1.81
0
0
1.32
1.67
1.67
0
0
1.89
1.89
0
0
1.53
1.68
1.68
0
0
1.93
1.93
0
0
1.64
1.70
1.70
0
0
Forward Looking
Myopic
0.30
50.0
0.20
0.10
0.0
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
PBSD
Figure 2: Equilibrium Profits: Forward Looking vs Myopic
13
PBSD
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
PBSD
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
p
1.85
1.69
1.55
1.42
1.32
1.30
1.49
1.74
1.88
Table 2: Theoretical Implications of PBSD - Case 2: SBSD = 0.5
Average Transaction Price (p) and Equilibrium Policies (σ1 )
Joint Profit Maximization
Sub-brand Profit Maximization
σ1 (s=1) σ1 (s=2) σ1 (s=3) σ1 (s=4)
p
σ1 (s=1) σ1 (s=2) σ1 (s=3) σ1 (s=4)
1.92
1.92
1.76
1.76
1.58
1.69
1.53
1.53
1.53
1.89
1.89
1.41
1.41
1.50
1.68
1.50
1.34
1.34
1.86
1.86
1.05
1.05
1.44
1.67
1.47
1.12
1.12
1.82
1.82
0.67
0.67
1.39
1.66
1.45
0.87
0.87
1.79
1.79
0.28
0.28
1.35
1.66
1.43
0.59
0.59
1.78
1.78
0
0
1.32
1.65
1.42
0.29
0.29
1.84
1.84
0
0
1.31
1.65
1.41
0
0
1.90
1.90
0
0
1.46
1.67
1.41
0
0
1.94
1.94
0
0
1.53
1.69
1.41
0
0
Net Discounted Profits (Value Function)
Joint Profit Maximization
Sub-brand Profit Maximization
V1 (s=1) V1 (s=2) V1 (s=3) V1 (s=4)
V1 (s=1) V1 (s=2) V1 (s=3) V1 (s=4)
212.4
212.4
212.2
212.2
186.3
186.3
186.1
186.1
196.7
196.7
196.3
196.3
176.9
176.9
176.4
176.4
179.1
179.1
178.3
178.3
168.5
168.5
167.6
167.6
161.3
161.3
160.2
160.2
161.2
161.2
159.8
159.8
144.7
144.7
143.2
143.2
155.3
155.3
153.4
153.4
141.9
141.9
139.9
139.9
150.5
150.5
148.0
148.0
167.5
167.5
164.4
164.4
148.3
148.3
145.1
145.1
201.2
201.2
193.5
193.5
173.1
173.1
165.0
165.0
221.9
221.9
202.2
202.2
189.4
189.4
168.6
168.6
14
PBSD
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
PBSD
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
p
1.74
1.59
1.46
1.35
1.25
1.43
1.59
1.80
1.91
Table 3: Theoretical Implications of PBSD - Case 3: SBSD = 1
Average Transaction Price (p) and Equilibrium Policies (σ1 )
Joint Profit Maximization
Sub-brand Profit Maximization
σ1 (s=1) σ1 (s=2) σ1 (s=3) σ1 (s=4)
p
σ1 (s=1) σ1 (s=2) σ1 (s=3) σ1 (s=4)
1.90
1.90
1.52
1.52
1.45
1.67
1.32
1.32
1.32
1.87
1.87
1.17
1.17
1.38
1.66
1.26
1.11
1.11
1.83
1.83
0.79
0.79
1.33
1.65
1.22
0.87
0.87
1.80
1.80
0.41
0.41
1.28
1.65
1.19
0.61
0.61
1.77
1.77
0.02
0.02
1.25
1.64
1.16
0.33
0.33
1.82
1.82
0
0
1.21
1.64
1.14
0
0
1.86
1.86
0
0
1.29
1.65
1.12
0
0
1.92
1.92
0
0
1.38
1.66
1.10
0
0
1.96
1.96
0
0
1.42
1.68
1.10
0
0
Net Discounted Profits (Value Function)
Joint Profit Maximization
Sub-brand Profit Maximization
V1 (s=1) V1 (s=2) V1 (s=3) V1 (s=4)
V1 (s=1) V1 (s=2) V1 (s=3) V1 (s=4)
202.1
202.1
201.7
201.7
170.7
170.7
170.3
170.3
184.9
184.9
184.2
184.2
160.4
160.4
159.7
159.7
166.9
166.9
165.9
165.9
151.7
151.7
150.6
150.6
149.9
149.9
148.5
148.5
144.7
144.7
143
143
134.4
134.4
132.7
132.7
139.1
139.1
136.9
136.9
160.0
160.0
157.3
157.3
133.4
133.4
130.7
130.7
181.3
181.3
177.1
177.1
146.8
146.8
142.6
142.6
209.0
209.0
198.4
198.4
164.0
164.0
153.2
153.2
228.5
228.5
201.6
201.6
178.6
178.6
151.2
151.2
15
1.00
0.90
Price Difference between JPM and SPM
0.80
0.70
0.60
0.50
SBSD=0
SBSD=0.5
0.40
SBSD=1
0.30
0.20
0.10
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
PBSD
Figure 3: Difference in σ1 (s=2) between Joint Profit Maximization and Sub-brand Profit Maximization
business across their portfolio of sub-brands. As we show in figure 1 the U-shaped pattern we
observe for equilibrium policies is driven by forward looking behavior and does not manifest itself
in the equivalent myopic game. Figure 2 depicts the corresponding difference in profits between
the forward looking and myopic case. Similar to prices, profits also decrease for moderate levels of
PBSD. In the forward looking case, as the magnitude of PBSD increases brands charge relatively
higher prices to their loyal consumer (e.g. policy at state 1 in the top left panel of table 1) or offer
deep discounts to attract the non-loyal consumer (policy at states 3 or 4). For these high values of
PBSD the loyal consumer is relatively unlikely to switch away and firms price accordingly. So at a
basic level, PBSD has similar effects with SBSD as we would expect. However, there are differences
that arise as we dig deeper into aspects of pricing strategy that multiproduct firms face.
The second finding that emerges is that PBSD can actually decrease, and in some cases reverse,
the benefit of centralized pricing. Starting with Table 1 we see that in most cases the value
function for state 1 (V1 (s=1)) is greater under joint profit maximization. Figure 4 summarizes
the difference in net discounted profits under joint and sub-brand profit maximization for the
whole range of parameters we analyze. There is a sizeable range of PBSD values where forward
looking profits under joint profit maximization are lower than forward looking profits under subbrand profit maximization. This is a striking result where the internalization of competition with
centralized pricing leads to lower profits compared to de-centralized pricing. Our interpretation of
this outcome is that, for this range of parameter values, the future benefit of locking in consumers
becomes more important under joint profit maximization and leads firms to compete harder for
consumer’s business. Indeed, as we show in Figure 5 joint profit maximization always generates
higher profits when firms are not forward looking. But with a long term horizon, the increased
probability of future profit streams across the sub-brand portfolio drives firms to invest in their
16
60
50
Net Discounted Profits Difference
40
30
20
10
SBSD=0
SBSD=0.5
0
SBSD=1
(10)
(20)
(30)
(40)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
PBSD
Figure 4: Benefit from Joint Profit Maximization - Forward Looking
0.07
Net Discounted Profits Difference
0.06
0.05
0.04
SBSD=0
0.03
SBSD=0.5
SBSD=1
0.02
0.01
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
PBSD
Figure 5: Benefit from Joint Profit Maximization - Myopic
17
clientele via lower prices. It is important to note that this is an equilibrium effect. When only
one of the two firms applies sub-brand profit maximization while the other is maximizing joint
profits, sub-brand profit maximization yields a lower value function than joint profit maximization
for the whole range of parameters5 . It is competition between sub-brand portfolios that induces
lower prices and profits. The level of PBSD where we observe this behavior is also the level where
the average transaction price is at its lowest value for this game. We see a similar result in Table
2 where, at PBSD = 2.5, the value function for state 1 under joint profit maximization is 144.7
while the one under sub-brand profit maximization is 155.3. The difference in profits between subbrand profit maximization and joint profit maximization is smaller in this case but still positive;
centralized pricing leads to stronger competition and reduces profits because of parent brand state
dependence. Interestingly, the negative effect of PBSD on the benefits of joint profit maximization
is mitigated by sub-brand state dependence. As Figure 4 shows, when SBSD is higher, joint profit
maximization tends to yield superior profits for a wider range of PBSD values. In other words, when
sub-brand state dependence is significant, the internalization of competition between sub-brands
tends to surpass the competitive pressure between firms that PBSD induces. This is reflected
more clearly in the equilibrium policies of sub-brand 1 when the consumer is loyal to sub-brand 2
(which belongs to the same parent), σ1 (s=2). The difference in these policies between joint profit
maximization and sub-brand profit maximization, shown in Figure 3, tends to increase with SBSD.
As SBSD increases, sub-brand 1 prices more and more aggressively to win back the consumer to
maximize its own profits. Centralized pricing internalizes this increased competition and therefore
offers higher benefits compared to de-centralized pricing. In light of this intuitive result, it is very
important to measure existing levels of PBSD and SBSD when evaluating centralized pricing.
The numerical analysis of the simplified model also yields insights into the relative pricing of
sub-brands and how this is affected by PBSD and centralized pricing. Starting with Table 1, we
see that both under joint and sub-brand profit maximization, the pricing policy of sub-brand 1
prescribes a very similar price for states 1 and 2. With only parent brand state dependence, under
joint profit maximization, firm 1 considers loyalty to sub-brand 1 or sub-brand 2 inter-changeably
and prices in a way that mitigates within-brand competition. Optimal prices of sub-brand 1 for
states 1 and 2 are also very similar under sub-brand profit maximization. Since we assume that
firms have full information on the state variable and the demand system, sub-brand 1 knows that,
in this game, state 1 is as good as state 2 and prices accordingly. When sub-brand state dependence
is introduced in addition to PBSD we see a different picture (Tables 2 and 3). Under joint profit
maximization, firms still price their sub-brands quite similarly as long as they either have or not the
state in their favor. For example, firm 1 has almost exactly the same price for sub-brand in states
1 and 2. Its pricing is sophisticated enough to internalize sub-brand competition under the firm
umbrella. It also prices similarly when the consumer’s state favors the rival firm, irrespective of
5
These results are available from the authors upon request.
18
which rival sub-brand was purchased last by the consumer. To be clear, this result is partly driven
by symmetry, but nevertheless contrasts in an informative way with the corresponding pricing
policies under sub-brand profit maximization. The same symmetrical sub-brands have markedly
different pricing policies depending on which one is favored by the consumer’s state. Under state
2, where the consumer is loyal to sub-brand 2, sub-brand 1 charges a lower price compared to
state 1 albeit still higher than the rival firm’s sub-brands which do not benefit from PBSD. In this
sense, we see that PBSD tends to lead firms more towards uniform pricing compared to SBSD,
to the extent this is supported by sufficient symmetry in sub-brand preferences. Intuitively, when
consumers tend to repeat their choices within the brand portfolio and there is no state dependence
to the sub-brand, even self-interested sub-brand managers will not compete with sub-brands of the
same parent but only with the rival parent brand. From a forward looking perspective, consumers
in this case would be equally likely to buy sub-brand 1 next time irrespective of whether they
chose sub-brand 1 or 2 before. What is important therefore is a good understanding of the loyalty
dynamics across sub-brands, even if each sub-brand maximizes its own profits.
3.7
Cost Estimation
To solve for optimal counterfactual prices 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 implied
estimates. Our approach is based on methods developed by Bajari, Benkard and Levin (2007)
(BBL), adapted to match the dynamic pricing model discussed in subsection 3.4. Specifically, we
estimate costs that rationalize observed prices for the intuitive structure in which firms decide jointly
on separate prices for their sub-brands taking into account both sub-brand and parent brand state
dependence. Since joint profit maximization is a priori expected to generate superior profitability,
we assume this is the behavior firms actually implement in the market. 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 BBL works in two steps. In the first step, one recovers the policy function from
the data and then uses it to forward simulate the value function 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, but does rely on a maintained assumption that the same equilibrium is played
throughout the data frame.
Our implementation involves the following steps:
1. We estimate policies in a first stage using spline-based regression. Policies for each sub-brand
19
are estimated as very flexible functions of the loyalty shares of the various sub-brands (i.e.
observed state vectors)
2. We define a set of inequalities H based on deviations from the optimal estimated policies by
varying amounts (e.g. +/- 0.005, +/- 0.01, +/- 0.02) separately for each sub-brand.
3. Using our first stage policies, we forward simulate ”correct” and ”perturbed” value functions
for each inequality in H in the following way:
(a) Draw a random state vector s0 and use as a starting point
(b) Simulate current period profit flow πf t (st , σ̂(st , vt )) using the current state vector and
estimated policies. The estimated policies σ̂(st , vt ), also include i.i.d. random shocks
drawn from the residuals of the first stage estimation.
(c) Calculate next period’s state vector based on choice probabilities that correspond to
the demand model, the policies and the state vector from the previous step. This is
specified explicitly in our pricing game and is described in subsection 3.3 and equation
12, snt+1 = g(Pt , snt ) = Qn (Pt ) × snt .
(d) Continue forward simulation until discounted period profit flows are negligible
6
(e) Repeat for R different starting state vectors and average the results. This step averages
out both different starting state vectors and the random shocks of the policies.
" T
#
R
1 X X t
V̂f (s; σ; θ) =
β πf t (st , σ̂(st , vt ); θ)
R
r=1
(18)
t=0
4. We use the approach described above to obtain ”correct” and ”perturbed” value functions for a
set of inequalities that constitute the equilibrium conditions of our model. The final step of our
approach recovers cost parameters through a minimum distance estimator that penalizes the
violation of the equilibrium conditions. Denoting the correct and perturbed value functions
with Vf (s; σj , σ−j ; θ) and Vf (s; σj0 , σ−j ; θ) respectively, the second stage estimator minimizes
the following objective function.
H
2
1 X
Q(θ) =
min{Vf (s; σf , σ−f ; θ) − Vf (s; σf0 , σ−f ; θ), 0}
H
(19)
h=1
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
6
We use 48 years or 2500 weeks - we have also tested a horizon of 3000 and 3500 weeks and results do not change
substantially, evidence that any simulation error is already too small when using 2500 weeks.
20
need 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.
hP
i
P
t
Vf (s; σ; θ) = ∞
β
(P
−
c
)
×
D
×
M
jt
j
jt
t=0
j∈f
i P
P∞ t
P∞ t hP
= t=0 β
t=0 β (Djt × M ) × cj
j∈f Pjt × Djt × M −
j∈f
(20)
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
sizes and any trial values for the cost parameter vector.
The demand parameters we use when solving for the cost parameters are Bayesian posterior
estimates that correspond to the average household. We note that the dynamic parameter estimation requires consistency of the demand estimates. Our Bayesian estimates are asymptotically
equivalent to MLE estimates and therefore satisfy this requirement, so long as the demand model
is correctly specified.
4
Empirical Application
4.1
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 includes 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. To
ensure 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, as shown below.
Sub-brands of the same brand name in this category are moderately differentiated in packaging,
labelling, 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 recall both
the parent brand and the specific sub-brand name when shopping. 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. Table 4 reports
the number of different flavor varieties per sub-brand, average price, standard deviation of price
21
Parent Brand
Dannon
Dannon
Dannon
Dannon
Kemps
Kemps
Old Home
Old Home
Wells BB
Yoplait
Yoplait
Yoplait
Table 4: Marketplace Descriptive Statistics
Sub-Brand
Total
New
Price/lb
Flavors Flavors Average Std Dev.
Dannon Creamy Fruit Blends
6
6
1.581
0.18
Dannon Fruit on the Bottom
9
9
1.712
0.32
Dannon Light N Fit
12
12
1.790
0.24
Dannon Light N Fit Creamy
6
6
1.674
0.10
Kemps Classic
9
0
1.261
0.12
Kemps Free
11
0
1.361
0.20
Old Home
8
1
1.383
0.19
Old Home 100 Calories
9
1
1.341
0.18
Wells Blue Bunny Lite 85
16
2
1.403
0.11
Yoplait Light
15
1
1.671
0.16
Yoplait Original
24
1
1.671
0.14
Yoplait Thick and Creamy
15
2
1.676
0.13
Total
140
40
1.551
0.24
Share
0.01
0.04
0.07
0.03
0.04
0.05
0.01
0.04
0.14
0.23
0.24
0.09
1
and category market share. Yoplait and Wells blue Bunny carry the greatest number of flavors,
consistent with their share leadership positions which allow for ample shelf space and a large pool
of loyal customers. Prices vary across brands, sub-brands and across weeks/stores within any
given sub-brand. Kemps is the lowest priced brand of the sample and carries the fewest flavors
while Dannon has the two most expensive sub-brands with a moderate number of flavors but the
greatest number of sub-brands. The product category we are modeling is mature and relatively
stable. For example, if we exclude Dannon, the rest of the brands collectively only had eight new
flavor introductions in the course of two years covered by our sample. Dannon introduced several
new varieties at this time building its position with new SKUs and refreshed labels at the size class
we analyze. It is important to note that Dannon enjoyed wide distribution in the US for many
years before our analysis sample and is a well-established brand. The new varieties involved a
re-positioning of 0.5 pound SKUs to match the 0.375 pound size of the market leader Yoplait.
Table 5 reports quartile, median and mean statistics for several metrics across households. The
average number of shopping trips in our sample is 88.3, while the average number of trips with
purchase of Yogurt is 23.1. Only about 3.4 of those trips involve the purchase of multiple subbrands during a single trip, mitigating somewhat concerns regarding variety seeking. To simplify
estimation we treat these multiple sub-brand purchases as separate observations, a practice typical
in the scanner data literature. When a household is observed to buy more than one sub-brand on
a given week, we track the household’s state dependence to all purchased sub-brands (e.g loyal to
Yoplait Original and to Dannon Light N Fit). Rows four and five of the table report the number of
different variants and different parent brands bought by each household in the sample; the average
22
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, which is clearly needed to identify
switching costs. 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. As we discuss in the next sub-section,
these sample frequencies point to a source of identification for the parent-brand state dependence
parameter.
Table 5: Choice Summary Statistics
1st Quart. Median
Shopping Trips
82
91
Trips with Purchase
13
20
Trips with Multiple Sub-brand Purchase
1
2
Number of brands bought
2
3
Number of sub-brands bought
3
5
Switchings within brand
3
6
Switchings to other brand
1
4
4.2
Mean
88.3
23.1
3.4
3.2
5.2
7.9
6
3rd Quart.
97
29
4
4
7
11
9
Identification
Analyzing and evaluating the impact of state dependence empirically requires identifying the
state dependence parameters in the demand model. Recent and older literature in this area has
outlined the challenge of identifying choice specific state dependence separately from heterogeneity.
We follow this literature, primarily Dubé, Hitsch and Rossi (2009, 2010) and we specify a very
flexible heterogeneity distribution to ensure we control for household level differences in observed
preferences and price sensitivity.
Since the new element in this study is the existence and analysis of parent brand state dependence, we devote a short discussion to how the parent brand state dependence parameter is
identified separately from the typical sub-brand state dependence parameter (i.e. corresponding
to same purchase). The two mentioned parameters are identified by different pairs of conditional
probabilities.
P r(j ∈ Jf , | j) > P r(l 6= j | j)
(21)
For the sub-brand state dependence parameter to be positive, it needs to be that the probability
of buying a choice alternative conditional on the same alternative being chosen last time is greater
23
compared to the case when a different alternative was chosen last (i.e. expression 21).
P r(j ∈ Jf , j 6= k | k ∈ Jf ) > P r(l ∈
/ Jf | k ∈ Jf )
(22)
On the other hand, for the parent brand state dependence parameter to be positive it must be that
for any given purchase of a specific sub-brand, the probability of switching to a different sub-brand
of the same parent brand is higher than the probability of switching to a variant that belongs to a
different parent brand (i.e. expression 22).
Clearly, the two conditions do not imply each other, showing that the two different types of state
dependence can be separately identified if the choice data frequencies support the inequalities above.
Table 5 shows that this is indeed the case for the sample of household purchases in this product
category. The numbers in rows six and seven correspond to the sample frequency probabilities in
(22) which are conditional on the same event, the purchase of any sub-brand from the choice set.
Table 6:
Mixtures
3 Components
3 Components
3 Components
3 Components
Identification Testing on Shuffled Sample
State Dependence
DIC
No SD at all
68039.6
No PBSD (only sub-brand and type) 68269.2
Only PBSD (no sub-brand and type) 68193.8
Full State Dependence
68293.6
To provide additional support for identification, we ran empirical tests similar to those in Dubé,
Hitsch and Rossi (2010). We randomly shuffled the order of observations within each household of
our sample. We then estimated versions of our model without state dependence, with only subbrand state dependence and finally also adding parent-brand state dependence. The various model
specifications are compared based on Deviance Information Criterion (DIC) values.
If our estimated coefficients of inertia were spurious and only captured unobserved heterogeneity,
when we added them to the model and estimated on the shuffled observations we should still find
that they add predictive power to the model (e.g. DIC decreases). In contrast, we find that adding
sub-brand state dependence to the model when observations are shuffled does not add significant
information to the model. Similarly, parent-brand state dependence also does not improve the
model when the purchase order is shuffled. As we can see in Table 6, the preferred model in these
series of tests is the one without any state dependence. This is strong evidence that what we
estimate in our model with the correct order of purchases is structural inertia and not masked
heterogeneity.
24
5
Results & Discussion
5.1
Demand Estimation
We begin discussion of our empirical results by reporting DIC values for model specification
selection. In the first three rows of Table 7 we evaluate the plausibility of including state dependence
to both the sub-brand and the parent brand. Based on the fit measures, adding sub-brand and
type state dependence improves the fit of the demand model to the data. This is evident by DIC
values decreasing from the first row of Table 7 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 (or type) choices over time controlling for price effects and intrinsic
preferences and ii) they also tend to become loyal to parent brand names in cases when they switch
their product line choices. In the next rows of Table 7 we report results from model specifications
where consumer heterogeneity is a mixture of Normal components. The preferred heterogeneity
specification, based on DIC, is the one with three Normal components. Also in the case of three
components, incorporating sub-brand, type and parent brand state dependence improves the fit
of the model. In the last row we explore whether adding more components to the heterogeneity
mixture provides even better fit. Based on these results, we use average posterior mean household
estimates from the three component mixture specification to estimate costs and compute market
equilibrium and counterfactual scenarios.
Mixtures
1 Comp.
1 Component
1 Component
2 Components
3 Components
3 Components
3 Components
4 Components
Table 7: Model Selection Results
State Dependence
None
No Brand SD (only sub-brand and type)
Full State Dependence
Full State Dependence
No SD at all
No Brand SD (only sub-brand)
Full State Dependence
Full State Dependence
DIC
68117.6
67215.4
67114.3
67157.9
67954.3
67201.4
67113.7
67150.0
The demand estimates used for the cost and the pricing model are reported in Table 8. These
results refer to mean posterior estimates across the households in the estimation sample. The subbrand 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
25
Table 8: Demand Results
Sub-Brand
Household Estimates
Average
90% CI
Dannon Creamy Fruit blends
-2.896 [-7.99, 2.98]
Dannon Fruit on the Bottom
-2.690 [-8.17, 3.86]
Dannon Light N Fit
-1.083 [-6.42, 4.86]
Dannon Light N Fit Creamy
-2.126 [-7.29, 3.26]
Kemps Classic
-3.460 [-8.33, 2.03]
Kemps Free
-2.860 [-7.82, 2.51]
Old Home
-4.255 [-9.05, 1.01]
Old Home 100 Calories
-2.790 [-7.86, 2.47]
Wells Blue Bunny Lite 85
-1.781 [-6.10, 2.88]
Yoplait Light
-0.685 [-5.90, 4.19]
Yoplait Original
-0.628 [-6.04, 4.52]
Yoplait Thick and Creamy
-2.077 [-7.33, 3.26]
Price Coefficient
State Dependence
Parent Brand
Product Line
Reg./ Light Type
26
-2.186
[-5.20, 0.41]
0.286
0.729
0.313
[-0.56, 1.15]
[-0.27, 1.85]
[-0.45, 1.09]
purchase on the utility of the same sub-brand is a little more than two times stronger than the
respective effect on the utilities of sub-brands of the same parent brand name; the corresponding
estimates are 0.729 and 0.286 respectively. State dependence based on fat content has comparable
size to PBSD. The quantification of the estimate for parent brand state dependence follows in the
next subsection with the results from counterfactual scenarios.
Table 9: Household Coefficient Correlation Table: Price and State Dependence Parameters
Dannon CFB
Dannon FOB
Dannon LnF
Dannon LnFC
Kemps C
Kemps F
Old Home
Old Home 100
WBBL
Yoplait L
Yoplait O
Yoplait TnC
Price
PBSD
SBSD
Reg/L SD
Price
PBSD
SBSD
-0.73
-0.71
-0.76
-0.74
-0.73
-0.74
-0.69
-0.74
-0.69
-0.73
-0.73
-0.72
1
-0.18
-0.28
-0.22
-0.22
-0.17
-0.16
-0.18
-0.19
-0.15
-0.15
-0.15
-0.18
0.14
1
-0.32
-0.24
-0.3
-0.32
-0.29
-0.32
-0.28
-0.3
-0.31
-0.32
-0.31
-0.28
0.24
-0.04
1
Reg/L
SD
-0.08
-0.1
-0.06
-0.1
-0.11
-0.09
-0.1
-0.11
-0.08
-0.13
-0.13
-0.11
0.03
0.03
-0.05
1
As one can see in Table 9, the state dependence coefficients are negatively correlated with
preference intercepts (correlation around -0.2) and positively correlated with price sensitivity (correlation around 0.2). This shows that state dependence tends to be higher for households without
very strong preferences and for households who are not very price sensitive. At the same time, the
state dependence coefficients are not strongly correlated with each other. The correlation table of
sub-brand intercepts is reported in the Appendix for economy of space.
5.2
5.2.1
Pricing Implications
Cost Estimates
Table 10 reports cost estimates from two different pricing regimes. The first one is a single
period Bertrand pricing game evaluated at the average sub-brand prices observed in the data and
at the respective steady state that corresponds to these prices7 . The cost estimates of the forward
looking pricing game, obtained using the methodology detailed in subsection 3.5, are reported in
the rightmost column of Table 10. For most sub-brands, the cost estimates of the dynamic game
7
Note that given our state dependent model, we always need to condition the firm’s profit function to a state
vector. While the forward looking costs are estimated based on a random sample of observed states the myopic costs
are only calibrated based on one state.
27
Sub−brand SD
density
0.0
0.0
0.1
0.1
0.2
0.2
density
0.3
0.3
0.4
0.4
0.5
0.5
Parent Brand SD
−3
−2
−1
0
1
2
3
−2
−1
0
1
2
3
4
Figure 6: Marginal Densities of State Dependence Parameters
are higher than those of the static version and statistically different. The only two exceptions where
myopic costs are higher than the forward looking estimates are Dannon Fruit on the Bottom and Old
Home 100 Calories. The myopic cost estimate of Dannon Fruit on the Bottom is not statistically
different from the forward looking one. The case of Old Home 100 Calories is different and we
attribute its relatively low dynamic cost estimate to its strong preference estimate in the demand
model. For example, if we look at the average price of Old Home 100 Calories, it is only a little
lower from that of Old Home (1.34 vs 1.38) while it’s mean posterior preference parameter is much
higher (-2.8 vs -4.3). The model infers that, since Old Home 100 Calories is priced comparably to
Old Home despite having tastes so much in its favor, it must have a much lower cost.8 And while
the myopic estimate for the 100 Calories sub-brand is also lower, the difference is more pronounced
in the forward looking case. Except for these two exceptions, forward looking cost estimates are
higher than costs calibrated against a myopic model. The difference is greater for Kemps (more than
15%) and Yoplait (more than 7%) This result already implies that for our empirical application
the ’invest’ incentive outweighs the ’harvest’ incentive. This is so because of the duality in the
problems of obtaining costs given prices or obtaining prices given costs. If forward looking prices
under state dependent demand are lower conditional on costs, it follows that costs from the same
model will be higher conditional on prices. For the rest of the analysis we use the cost estimates
corresponding to the dynamic game, in accordance with the forward looking view we take for the
pricing problem.
8
This is also supported by light milk being less expensive than full fat milk.
28
Table 10: Cost Estimates
Static Game
Dynamic Game
Est.
Est.
90% CI
Dannon Creamy Fruit blends
1.117
1.175 [1.157, 1.204]
Dannon Fruit on the Bottom
1.256
1.243 [1.217, 1.267]
Dannon Light N Fit
1.325
1.338 [1.322, 1.385]
Dannon Light N Fit Creamy
1.210
1.322 [1.292, 1.347]
Kemps Classic
0.797
0.942 [0.879, 0.962]
Kemps Free
0.904
1.052 [1.022, 1.083]
Old Home
0.933
0.953 [0.924, 0.986]
Old Home 100 Calories
0.884
0.853 [0.839, 0.876]
Wells Blue Bunny Lite 85
0.937
0.995 [0.982, 1.022]
Yoplait Light
1.190
1.296 [1.275, 1.310]
Yoplait Original
1.186
1.274 [1.228, 1.293]
Yoplait Thick and Creamy
1.194
1.281 [1.235, 1.330]
Sub-Brand
5.2.2
Effect of Parent Brand State Dependence on Firm Profits through Umbrella
Branding
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 with Yoplait and therefore does not create
any dynamic inter-dependence in demand due to branding. Equilibrium prices and per period
profits in this counterfactual scenario (C2) are compared with prices and profits in a base scenario
(C1) where the 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 11. The reader should focus their attention on the bottom three rows, as the equilibrium
impact on rival firms is negligible in this example. There are two basic conclusions that emerge
from Table 11: i) the per pound price of Yoplait Thick and Creamy is higher by about 4 cents when
this sub-brand does not have any parent brand state dependence effects and ii) the firm Yoplait
loses about 1.9% 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 firm’s 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 parent brand’s sub-brands.
Results from a second counterfactual are also reported in Table 11. In this case, for the alterna29
Brand
Dannon
Dannon
Dannon
Dannon
Kemps
Kemps
Old Home
Old Home
Wells B.B.
Yoplait
Yoplait
Yoplait
Table 11: The Effect of Yoplait PBSD on Prices and Profits
Base Case (C1)
Yoplait T&C
Yoplait All
PBSD = 0 (C2) PBSD = 0 (C3)
Sub-Brand
Prices Profit
Prices Profit
Prices Profit
Creamy Fruit Blends 1.622
1.622
1.622
Fruit on the Bottom 1.690
1.690
1.690
Light N Fit
1.733
1.733
1.733
Light N Fit Creamy 1.751 754.1
1.751 755.6
1.751 757.4
Classic
1.380
1.380
1.379
Free
1.483 192.3
1.483 192.5
1.483 193
Classic
1.403
1.403
1.403
100 Calories
1.271 235.1
1.271 235.5
1.270 236
Lite 85
1.386 456.5
1.386 457.6
1.385 458.8
Light
1.713
1.712
1.717
Original
1.671
1.671
1.674
Thick and Creamy
1.754 1895
1.797 1859.1
1.785 1791.9
Table notes: Prices and per period profits are reported for forward simulated steady state in each scenario
(C1, C2, C3).
tive scenario (C3), we eliminate the parent brand state dependence effects for all three sub-brands
of Yoplait. This is equivalent to all three sub-brands being marketed under different, unrelated
names, without the common Yoplait banner. 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 5.4%, when the demand for
Yoplait sub-brands is conditionally independent. This can be viewed as the total impact of parent
brand state dependence on the profitability of Yoplait as a whole, holding other things constant.
5.2.3
Effect of Parent Brand State Dependence Level
In this subsection, we report results on the effects of PBSD on firm prices and profits based
on various counterfactual values of the PBSD coefficient. In contrast with the previous subsection
where we experimented with whether Yoplait is offered under umbrella branding, for the experiments of this section we keep the branding structure fixed at its observed state and vary the
magnitude of PBSD.
Table 12 has steady state prices and per period profits for various levels of PBSD, from zero
to five times the estimated levels. For most brands steady state prices decrease as the magnitude
of PBSD increases. A notable exception is Yoplait for which prices also decrease initially but
then reverse course and start increasing when PBSD is three times or higher the estimated level
depending on the sub-brand. Overall, for the range of PBSD we tried, we observe a U shaped
30
Table 12: Effect of PBSD Magnitude on Prices and Profits
PBSD Scale Factor
0
0.5
1
2
3
4
5
Steady State Prices
Dannon Creamy Fruit Blends
Dannon Fruit On The Bottom
Dannon Light N Fit
Dannon Light N Fit Creamy
Kemps Classic
Kemps Free
Old Home
Old Home 100 Calorie
Wells Blue Bunny Lite 85
Yoplait Light
Yoplait Original
Yoplait Thick And Creamy
1.642
1.710
1.755
1.770
1.389
1.494
1.413
1.284
1.404
1.731
1.691
1.771
1.632
1.700
1.744
1.761
1.385
1.489
1.408
1.277
1.395
1.722
1.680
1.761
1.622
1.69
1.733
1.751
1.380
1.483
1.403
1.271
1.386
1.713
1.671
1.754
1.603
1.671
1.714
1.734
1.369
1.472
1.393
1.257
1.369
1.702
1.659
1.749
1.585
1.655
1.697
1.721
1.358
1.461
1.383
1.243
1.354
1.696
1.652
1.758
1.571
1.643
1.684
1.714
1.348
1.450
1.373
1.230
1.342
1.697
1.651
1.787
1.561
1.638
1.677
1.714
1.339
1.440
1.366
1.218
1.333
1.705
1.654
1.838
Steady State per Period Profits
Dannon
Kemps
Old Home
Wells Blue Bunny
Yoplait
695.5
188.9
228.4
426.4
1610.3
722.9
190.5
231.5
440.5
1743.9
754.1
192.3
235.1
456.5
1895
830.9
196.7
243.9
496.3
2257.8
932.4
202.8
255.9
549.2
2716
1066.7
211.4
272.5
619.9
3289.4
1243.7
223.4
295.4
714.7
3998.7
31
relation between the magnitude of PBSD and prices for Yoplait and a decreasing relationship for
the other brands. Our interpretation of the different pattern for Yoplait is that having significantly
higher market share than all other brands leads the harvesting incentive to dominate investment
as PBSD increases for this brand. In terms of per period profits, these increase for all firms as the
magnitude of PBSD increases. Our interpretation of these results is that as PBSD strength increases
firms find it optimal to charge lower prices because they capitalize on increased demand through
PBSD reinforcement effects. As a result, their profits increase due to the associated increase in
market power. When consumers exhibit parent brand state dependence, as prices decrease, firms
can capitalize on the intertemporal complementarities to their other sub-brands and this additional
dimension drives profitability upwards.
5.2.4
Firm-Level Profit Maximization and the Impact of Parent Brand State Dependence
Next, we examine a multiproduct firm’s ability to exploit dynamic loyalty to the firm’s 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 the respective externalities.
The top panel in Table 13, shows steady state prices and per period profits for the base case (C1)
where Yoplait maximizes profits jointly across all its sub-brands and a counterfactual (C4) where
all sub-brands of Yoplait maximize sub-brand profits and do not internalize each other’s demand
in their respective pricing problems. With respect to prices, we see that full multi-product pricing
leads Yoplait to charge higher prices, reflecting the increase in market power. The prices of the three
sub-brands are 1.713, 1.671 and 1.754 under full firm-level profit maximization versus 1.682, 1.649,
and 1.708 under sub-brand profit maximization. However, the impact of the increased market power
on profits is quite modest; per period profits only decrease by 0.2% 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, full coordination of
prices across the three sub-brands is costly (e.g. because of managers’ workloads) and this cost is
higher than 0.2% of per period profits, our experiment shows that decentralized pricing might be
preferable.
32
Table 13: Sub-brand Profit Maximization for Yoplait: Effect on Prices and Profits
Estimated PBSD Levels
JPM (C1)
SPM (C4)
Brand
Sub-Brand
Prices Profit Prices Profit % Difference in Profit
Yoplait Light
1.713
1.682 823.3
Yoplait Original
1.671
1.649 900
Yoplait Thick and Creamy 1.754
1.708 167.8
Yoplait Total
1895
1891.1 -0.21
Brand
Yoplait
Yoplait
Yoplait
Yoplait
Sub-Brand
Light
Original
Thick and Creamy
Total
Yoplait All PBSD = 0
JPM (C3)
SPM (C5)
Prices Profit Prices Profit % Difference in Profit
1.717
1.675 780.8
1.674
1.641 855.3
1.785
1.700 147.2
1791.9
1783.3 -0.48
Table notes: Yoplait prices and per period profits are reported for forward simulated steady state in each
scenario.
JPM: Joint Profit Maximization
SPM: Sub-brand Profit Maximization
While centralized pricing should clearly be more profitable in all cases, abstracting away from
strategic effects, it is interesting to examine how its benefits relate to parent brand state dependence. Given the inter-temporal complementarities that parent brand state dependence creates, it
is possible that it reduces the benefits of multi-product pricing for firms. This can occur because,
when a firm internalizes demand dependencies 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 inter-temporal dependencies between sub-brands, joint pricing tends
to increase prices because it only internalizes competition. The bottom panel of Table 13 reports
results similar to those of the top panel, contrasting joint profit maximization with sub-brand profit
maximization for the case where there are no dynamic effects from one Yoplait sub-brand to the
others. This scenario is comparable to the results in Table 11, 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 (1.717, 1.674 and 1.785 versus 1.675, 1.641, and 1.700) but in this case the associated
profit loss is about 0.5% 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 for broader
situations in which demand is likely characterized by parent brand state dependence and one seeks
33
Brand
Dannon
Dannon
Dannon
Dannon
Kemps
Kemps
Old Home
Old Home
Wells B.B.
Yoplait
Yoplait
Yoplait
Table 14: Uniform Pricing
Base Case (C1)
Yoplait (C6)
Sub-Brand
Prices Profit
Prices Profit
Creamy Fruit Blends 1.622
1.622
Fruit on the Bottom 1.690
1.690
Light N Fit
1.733
1.734
Light N Fit Creamy 1.751 754.1
1.751 754.9
Classic
1.380
1.380
Free
1.483 192.3
1.483 192.4
Classic
1.403
1.403
100 Calories
1.271 235.1
1.271 235.4
Lite 85
1.386 456.5
1.386 457.4
Light
1.713
1.698
Original
1.671
1.698
Thick and Creamy
1.754 1895
1.698 1891.6
Dannon (C7)
Prices Profit
1.724
1.724
1.724
1.724 751.9
1.380
1.483 192.4
1.403
1.271 235.4
1.386 457.4
1.713
1.671
1.754 1895.8
to evaluate centralized pricing.
5.2.5
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 pricing9 . This loss can be benchmarked
against possible menu costs that lead managers to impose uniform prices. Such costs may exist
because of labeling, or 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 14 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 0.2%.
Table 14 also reports the comparison results for the case of Dannon. Notably, a uniform pricing
strategy for Dannon is more costly − the associated loss is about 0.3% 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. 1.622, 1.690, 1.733 and 1.751). Therefore, imposing
a single price (1.724 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 basis.
In both counterfactual experiments discussed above, the prices of the brands that are not being
9
Holding everything else constant and abstracting from strategic effects, constraining the price of each sub-brand to
be the same inevitably produces a sub-optimal decision compared to the unconstrained profit maximization problem.
34
evaluated do not change very much when either Yoplait or Dannon change their pricing strategies
with respect to uniform prices. We think this is not all that surprising, as our counterfactuals are
mostly changing the relative prices within a parent brand, rather than the overall level relative to
their rival firms. For example, when Yoplait is using uniform pricing, it is charging a little higher
price for Yoplait Original and Light and a little lower price for Yoplait Thick and Creamy. In the
long run, the rest of the brands do not face a big overall competitive price change from Yoplait,
they simply face different relative prices for Yoplait sub-brands.
5.3
Discussion
5.3.1
Learning
As we demonstrated in the previous sections, parent brand state dependence is an important
economic force. An alternative explanation of consumers’ tendency to repeat their choices over
time, controlling for their preferences and price effects, is if they are uncertain about a brand’s true
quality and they engage in learning. Overall, the category we are analyzing has well established
brands and does not involve large purchase quantities. This makes it less likely that consumers
engage in learning or are otherwise forward looking in their every day decision making process. The
sunk cost of a sub-brand choice that might not prove ideal is fairly small. To test whether our model
captures structural state dependence rather than learning, we re-estimate our model controlling for
consumers’ experience and how it interacts with state dependence. If our estimates reflect learning
rather that true state dependence, then we would find that the interaction of cumulative brand
purchases with sub-brand state dependence would add information to the model and increase its
explanatory power. As consumers learn more about all the sub-brands they are buying their state
dependence should decrease to zero. This test was first implemented in Dubé, Hitsch and Rossi
2010. Our results in Table 15 reveal that adding the interaction of brand experience with state
dependence does not add meaningful information to the model. The DIC value of the model with
the interaction (M2) is much higher than the one for the model without the interaction (M1)10 . This
shows that state dependence exists controlling for consumers’ experience accumulation, providing
additional justification for our focus on state dependence.
Mixtures
3 Components
3 Components
Table 15: Alternative Explanation Test - Model Selection
Test Specification
Add Brand Experience only
Add Brand Experience and interaction with state dependence
10
DIC
66508.8
66568.0
We note that similar to Dubé, Hitsch and Rossi 2010 we only test for the interaction term comparing to a model
specification which includes brand experience but not the interaction. Comparing to our full base model would not
be appropriate since the models evaluated for this sub-section contain additional information.
35
Table 16: Alternative Explanation Test - State Dependence
State Dependence Average Household Estimates
M1
M2
Parent Brand
0.330
0.301
Product Line
0.659
1.141
Reg./ Light Type 0.371
0.333
6
Conclusion
There is increasing academic interest in the long term impact of marketing policies 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 effect of demand
choice dynamics that reflect consumers’ tendency to stay loyal to a firm when choosing between
multiple sub-brands of a broader umbrella brand. 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
to abstract from important factors like frequent price discounts and retailer involvement in the
choice 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 complex game with
the 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 0.2% 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 relatively larger for the
second leading brand in the category, Dannon. Overall, the novel findings reported in this work
highlight the intricate role of demand choice dynamics for prices of forward looking multi-product
firms and suggest new dimensions to explore. It seems reasonable to expect that the methodology
and findings in this work are relevant for several other product categories where demand is likely
characterized by parent brand dynamics.
36
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39
7
Computational Details
7.1
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
n1 < ... < g nL = 1, ∀j, ∀n. The grid is formed by the Cartesian product
points g, such that: 0 = gjt
jt
P
nl ≤ 1, ∀l = 1, ..., L, ∀n. Intuitively, this
of all finite sets of points for each axis, such that Jk=1 gkt
condition says that, for any consumer type n, the fraction of the market loyal to each sub-brand
in the choice set should sum 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. Due to the
fact that loyalty states for each type sum 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 to one. Even though this condition reduces the number
of grid points significantly, as the dimensionality of the state space increases, the computational
requirements of the grid become exorbitant. In practice we are using a grid consisting of six points
in each dimension (0, 0.2, 0.4, 0.6, 0.8, 1). While the complete Cartesian product for such a grid
would have 362 million points (611 ), the condition that the state shares for all brands should sum up
to one limits our total number of grid points to 4368. To see this consider the case when sub-brand
A has a state vector of 1, then the only possible states for the other sub-brands is to have zero
state share. Similarly, when sub-brands A and B have state shares of 0.4 and 0.6 respectively, the
only possible states for the other sub-brands are zero state shares.
It is relatively easy to see that the dimensionality of the state space increases faster with the
number of brands when the number of consumer types is higher. This is true especially because
adding consumer types is not economizing as much from the condition that state shares sum up to
one (this condition allowed us to work with 4368 grid points instead of 363 million grid points). An
additional consumer type would imply another vector of sub-brand state shares (so another 4368
points to enumerate) and then the total state space would be the product of the two consumer
type grids (43682 ). 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 thus ensuring internal consistency.
40
7.2
Interpolation
During the computation, in all cases where we need to compute the value or policy functions
on state space points outside the grid we use polynomial based interpolation. Our polynomial
approximation function has the general form given in expression 23. 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
actual values is about 0.99 while the average percent error of the prediction (MAPE) is at most
0.1% . This suggests that the approximation works well in practice.
aˆ0 +
7.3
PN
n=1
yˆj (s) =
PJ−1 P
m
n
a
ˆ
j=1
m∈{0.5,1,2,3} njm sj
Q
P
n
+ K
l∈Ik sl
k=1 aˆk
(23)
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 results
we 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; Vf0 (s) = 0 and σf0 (s) =
i
h
0 (s)
∀s ∈ S, ∀f . The initial policies are iterated so that they are best
maxPj πf s, P, σ−f
responses for the static case.
• We experiment with different initial guesses, for the value and policy functions, to examine 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 Vfiter−1 and σ iter−1 , compute Vfiter ; 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,
to the next step. We set εv = 0.0000001.
41
max|Vfiter −Vfiter−1 |
max|Vfiter |
≤ εv , ∀f . Then move
3. For each firm, given Vfiter and σ iter−1 , compute σ 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
rival’s policy. In practice we iterate across firms, solving for optimal prices at all points of our
state space grid, given rival prices from the previous iteration. Upon convergence, no firm can
get a higher value function by changing its policy, policies are best responses for each state,
conditional on value functions. To speed our price optimization sub-routine and to avoid
local maxima issues we start with a simple global search to identify the region of the global
maximum for prices. That is we evaluate the right hand side of the Bellman equation in 20
cent intervals to find the neighborhood of the solution. We then bound our quasi-Newton
optimization algorithm in a 40 cent neighborhood of the solution identified with the initial
search.
(a) If the computed policies converge for each firm ,
max|σfiter −σfiter−1 |
max|σfiter |
≤ εσ , ∀f , stop the
algorithm. We set εσ = 0.00001.
(b) If not, update the policies and return to step 2.
Doraszelski and Satterthwaite (2010) give the following definition for a Markov-perfect equilibrium:
“An equilibrium involves value and policy functions V and σ such that i) given σ−f , V solves the
Bellman equation for all f and ii) given σ−f (s) and Vf , σf (s) solves the maximization problem
on 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.
To ensure that our solution is not a local minimum but a global maximum as well as best
response for all profit maximizing firms we complemented our algorithm with an additional step
that checks our equilibrium policies with a global optimizer that combines genetic algorithms with
derivative based search. Indeed, when we used this global optimizer for our final policy iteration,
equilibrium policies did not change, evidence that our original approach was suitably robust. Details
about this optimizer can be found in Mebane W. and Sekhon J., 2011, ”Genetic Optimization Using
Derivatives: The rgenoud package for R.”, Journal of Statistical Software, 42(11)
7.4
Steady State Computation
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 period’s state vector. We repeat the process until both
the state variables and the prices of each specific product alternative converge. To verify if the
42
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.
7.5
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: β = 0.998, M S = 100000.
7.6
Demand Coefficient Details
Table 17: Household Coefficient Correlation Table: Sub-brand Intercepts
Dannon CFB
Dannon FOB
Dannon LnF
Dannon LnFC
Kemps C
Kemps F
Old Home
Old Home 100
WBBL
Yoplait L
Yoplait O
Yoplait TnC
Dannon
CFB
1
Dannon
FOB
0.81
1
Dannon
LnF
0.82
0.8
1
Dannon
LnFC
0.82
0.78
0.87
1
Kemps
Classic
0.81
0.78
0.78
0.79
1
Kemps
Free
0.77
0.71
0.82
0.81
0.8
1
Old
Home
0.78
0.77
0.76
0.77
0.81
0.77
1
Old
H 100
0.79
0.78
0.83
0.82
0.81
0.84
0.82
1
WBBL
0.74
0.68
0.77
0.78
0.73
0.8
0.72
0.79
1
Yoplait
Light
0.78
0.68
0.79
0.8
0.74
0.78
0.69
0.76
0.74
1
Yoplait
Orig.
0.81
0.72
0.76
0.79
0.78
0.75
0.73
0.75
0.7
0.82
1
Table 18: Demand Results - Alternative Specifications
State Dependence
Average Household Estimates
6oz packs,
6oz packs,
Multiple sizes,
Single Person HH Loyalty from all sizes 24 alternatives
Parent Brand
0.625
0.333
0.261
Product Line
0.750
0.634
0.854
Reg./ Light Type
0.353
0.281
0.196
43
Yoplait
TnC
0.8
0.74
0.73
0.75
0.75
0.71
0.71
0.74
0.67
0.77
0.82
1