

Department of Social Systems and Management
Discussion Paper Series
No.1322

Are manufacturers’ efforts to improve their brands’
reputation really rewarded?
The case of Japanese yogurt market
byTomohito KAMAI and Yuichiro KANAZAWA
Oct 2014

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPAN

Are manufacturers’ efforts to improve theirbrands’ reputation really rewarded? The caseof Japanese yogurt market.
Tomohito KAMAI and Yuichiro KANAZAWA ∗October 22, 2014

Abstract
More than 65 years have passed since yogurt was first introduced inJapan, and the yogurt market is still growing there. The recent market growth is said to be stimulated by a group of products with newly
found lactic-acid bacilli which are claimed to have features including
protection from virus-infection, allergies and so forth. However, it is∗

Tomohito KAMAI is a doctoral student, Graduate School of Systems and Informa-tion, University of Tsukuba (E-mail: tkamai@sk.tsukuba.ac.jp). Yuichiro KANAZAWA
is Professor of Statistics, Department of Social Systems and Management, University ofTsukuba (Email: kanazawa@sk.tsukuba.ac.jp). This work is supported by Grant-in-Aid
for Scientific Research (B)20310081 and Challenging Exploratory Research 25590051. The
R
CCL-CAFÉ"dataemployed in this manuscript for illustration purpose are provided by

CUSTOMER COMMUNICATIONS, Ltd.,through Prof. Tadahiko Sato of the Graduate School of Business Sciences of the University of Tsukuba. His work is supported
by Grant-in-Aid for Scientific Research(A)21243030. We thank very much CUSTOMERCOMMUNICATIONS, Ltd. and Prof. Tadahiko Sato for generous offer.

1

empirically unknown if manufacturers are really rewarded with higher
margins from brands with these features as the average retail price of
yogurt kept decreasing over the last decade. To uncover factors responsible for such a phenomenon, we employ Che et al. (2007) because
they incorporate important facets of enterprises such as the strategicinteraction among manufacturers and retailers, consumer state dependence, and forward-looking behavior of firms. With Japanese yogurt
panel data from January 2007 to December 2008, we find that (1) manufacturers producing brands with special features successfully charge
more margins as expected; (2) a retailer also charges higher margins for these brands; and (3) a retailer has slightly higher amount ofmargins than manufacturers, reflecting Bertrand competition among
manufacturers and vertical Nash game between manufacturers and aretailer.

1

Introduction

Since yogurt was first introduced in Japan in 1950’s, the market kept growing.
The recent market growth is said to be stimulated by a group of productswith newly found lactic-acid bacilli, which are claimed to enhance immune
strength and prevent consumers from virus-infection, allergies and so forth.
The traditional marketing theory would predict that these manufacturers’
efforts are rewarded with high margins. The average price of yogurt, however, kept decreasing over the last decade and the temporal price reduction(“TPR” henceforth) is prevalent practice in this category, with 66.7% of supermarket engaged in TPR in a sampled week according to the retail survey2

in 2007.1
In this paper, we try to uncover factors responsible for such a phenomenon. The possible explanations include market pressure for lower price,
bargaining power imbalance between manufacturers and retailers where oneof channel members squeezes the margin of the other, or increasing pricecompetition among manufacturers. To identify which of these factors are actually in effect, we need to decompose prices into margins of manufacturers,
those of retailers, and marginal costs and assess their relative magnitude.
This requires to model both demand and supply-side behavior as supply-side
behavior would be affected by the market demand condition. Moreover, sincesome consumers switch brands in this market, inferred from prevalent practices of TPR and the nature of product, consumer state dependence mustalso be incorporated in demand model for plausible investigation.
In addition to strategic interaction among manufacturers and retailers andconsumer state dependence, a model needs to accommodate forward-looking
behavior of firms as pricing behavior of firms could be drastically altered ifthey engage in such behavior as found in empirical paper of Che et al. (2007).
In their research, both manufacturers and retailers in U.S. cereal market
are shown to set prices accounting for the effect of current prices on future
profit while incorporating strategic interaction among firms and consumer
state dependence. The fundamental marketing issues of unobserved demand
1Compared to 2003, the average retail price of boxed yogurt in stores in Tokyo areafell by 7.5% in 2008, and it further fell by 14.9% in 2013 according to “Retail Survey”
conducted by Statistics Bureau, Ministry of Internal Affairs and Communications, Japan.
Data regarding TPR are obtained from “National Survey of Prices” conducted by Statistics
Bureau, Ministry of Internal Affairs and Communications and calculated from data of“Distribution of Regular Prices and Sale Prices by Sales Floor Space, Type of Outlets Japan, City Groups, Prefectures.”3

characteristics, heterogeneity across household, and price endogeneity arealso accounted for in their model. Therefore in this research, we employ thecomprehensive model of Che et al. (2007) to the yogurt data in Japanesemarket to uncover the structure of this market.
The rest of paper is organized as follows. The next section describes themodel. In section 3, we present our estimation procedure. We briefly explain
our data in section 4. In section 5, we will present and discuss results for
empirical analysis. Section 6 concludes.

2

The Model

In this section, we specify both demand- and supply-side models. As we implied, there are three major dimensions in the modeling framework, which are
strategic interaction among manufacturers and retailers, consumer state dependence, and forward-looking behavior of firms. Out of them, consumer
state dependence is modeled in demand-side and the rest is modeled insupply-side. This approach of structural market equilibrium model enables
the analysis of supply-side behavior by observing only the demand-side data,
which is an advantage of the model as supply-side information is rarely available to researchers. Examples of papers in this line include Besanko et al.
(1998), Sudhir (2001), Yang et al. (2003), Villas-Boas and Zhao (2005) and
Che et al. (2007) to name a few. Because supply-side behavior is estimatedconditional on the estimation results of demand-side model, we start withdemand-side model.

4

2.1

Demand−Side Specification

The brand choice model Let us suppose there are j = 1, . . . , J brands
in the market and each household i = 1, . . . , I has ti = 1, . . . , Ti purchasingoccasions. We employ the multinomial logit model for household brand choice
behavior. Specifically, the probability of household i choosing brand j at its
ti -th purchasing occasion is defined to be Prijti and is written as
Prijti =exp(vijti )!J
1 + k=1 exp(vikti )(2.1)

where vijti = xjti · βs + simkj · SDs + ξjti is the deterministic part of theutility function.2 The addition of 1 in the denominator stands for the outside
option which results from the specification vi0ti = 0. The set of explanatory
variables indexed by vector xjti include brand dummy variables and price ofbrand j a household i faces on purchasing occasion ti , simkj is the attribute
similarity index for brand j with respect to the previously purchased brand k,and ξjti is the unobserved demand characteristics which can be observed byfirms and households but not by a researcher. The examples of unobserved
demand characteristics are national advertisement, coupon availability, shelf
space allocations and so forth. As prevalent in this study field, we assume
it commonly affects all households (Besanko et al., 1998; Villas-Boas andWiner, 1999; Villas-Boas and Zhao, 2005). Parameters to be estimated are
βs and SDs , where a subscript s = 1, . . . , S corresponds to segment (i.e.,a subset to which households belong to, where those in the same segment
are assumed to be the same in terms of responsiveness to marketing mix
2

The term ξjti is a subset of ξjt where the latter is defined for all calendar dates andbrands in the panel, and the former is retrieved from the latter according to ti . On theother hand, the values of xjti may be different depending on households even when twohouseholds shop at the same time as temporal price reduction such as coupon may only
be available to a specific household.

5

variables), as we will employ the latent class model (Kamakura and Russell,1989).
The attribute similarity index We use the attribute similarity index toexpress the state dependence in household brand choice behavior, following
Che et al. (2007).3 In their specification, each brand is allocated with a setof attributes by a researcher. Each attribute has different levels, and brands
are assumed to be similar if they share the same level of attributes. The
degree of similarity between brands increases with the number of attributelevels shared by these brands.
Employing the attribute similarity index enables a researcher to examine how brand attributes contribute to the perception of similarity between
brands among consumers. Apparently, this approach would yield richer insight on consumer brand choice behavior and on brand positioning comparedto the prevalent approach such as employing the lagged brand indicator variable. Specifically, the similarity between the brand purchased on the previous
occasion (brand k) and the brand a household faces on the current purchaseoccasion (brand j) is specified as!
Ikj + Pp=1 Ikjp · rp
simkj =,!
1 + Pp=1 rp(2.2)

where Ikj is an indicator variable taking unity if k = j, Ikjp is an indicator variable taking unity if two brands share the same level of attribute3

The idea of the attribute similarity index can be found in previous papers (e.g., Lattin(1987)), but the specification in previous literature requires questionnaire which explicitly
asks subjects for the perceived similarities between listed brands. The advantage of thespecification of Che et al. (2007) is that it does not require such information and similarities
between brands can be calibrated from the data, although the level of attributes sharedby brands must be arbitrarily set by researchers.

6p = 1, · · · , P , and rp > 0 is importance weight associated with attribute p
to be estimated. As (2.2) implies, the similarity index is designed to takevalue between 0 (brands are totally dissimilar) and 1 (brands are identical).
The parameter of the attribute similarity index, SDs , can either be positiveor negative which corresponds to inertia (i.e., a previous brand consumption
experience raises the probability of repurchasing a brand) and variety-seeking(i.e., a previous brand consumption experience lowers the probability of repurchasing a brand) respectively. Following Che et al. (2007), we specify
SDs to be the function of demographic variables as
SDs = γs0 + Di · γs
where Di is vector of demographic characteristics of household i, γs0 is anintercept term, and γs is vector of parameters for Di .4
The price endogeneity It is empirically well known that ignoring unobserved product characteristics leads to a biased estimate of price effect asthey could be correlated with prices (Berry, 1994; Villas-Boas and Winer,
1999; Besanko et al., 1998, 2003; Nevo, 2001; Villas-Boas and Zhao, 2005).
This price endogeneity problem arises because if the desirable unobserved
product characteristic is not modeled, its hidden impact on utility will bepicked up by price, mitigating its negative effect on the utility (Train, 2009).
To avoid this problem, we employ an idea of two-stage least squares. The
details are explained in the next section.

4

In estimating γs0 and γs , we write out each component of the term simkj · SDs andestimate them by least squares. The detailed estimation algorithm can be provided upon
the request to the author.

7

2.2

Supply−Side Specification

Following the preceding research, we assume that the retailer is a local monopolist which maximizes its joint category profit.5 The assumption of a
local monopolist is often justified by empirical reports which find that there
is little evidence of among store competitions (Besanko et al., 1998; Sudhir,
2001; Villas-Boas and Zhao, 2005; Che et al., 2007). We further assume that
there are multiple manufacturers which sell their brands through a common
retailer. Manufacturers are allowed to produce multiple brands.
To examine if firms forward-look in setting prices, we will test three models − namely static, myopic, and forward-looking model. The static model
is a standard multinomial logit model without state dependence. The myopic model assumes that firms account for state dependence in demand (i.e.,
firms consider the effect of a household previous brand choice via the attribute similarity index) but do not account for the current price impact on
future profit, while those in forward-looking model are assumed to accountfor such an effect (Che et al., 2007). In all models, we will estimate the margins of manufacturers and a retailer under four different games, which arise
from the combination of two games in horizontal strategic interaction among
manufacturers and two games in vertical strategic interaction between manufacturers and a retailer. Two games in horizontal strategic interaction are
Bertrand competition and tacit collusion, where Bertrand competition refers
to own-brands profit maximizing behavior of each manufacturer and tacit
collusion refers to the behavior of manufacturers which collectively maximize
5

A retailer could use the other pricing rules such as brand profit maximization where it

sets up a profit function for each brand. However, Sudhir (2001) empirically shows that aretailer attains a maximum profit when it engages in category profit maximization, which
supports the assumption widely adopted in the literature.

8

total profit from all brands in the market. Two games in vertical strategicinteraction are manufacturer Stackelberg and vertical Nash. In the manufacturer Stackelberg game, manufacturers act as Stackelberg leaders withrespect to a retailer and choose their wholesale prices anticipating a reaction
from a retailer and wholesale prices of competing brands. In this case, theretailer chooses retail prices to maximize its profit taking wholesale prices asgiven. In the vertical Nash game, manufacturers and a retailer move simultaneously; they choose prices anticipating the profit maximizing behavior of
the others (Choi, 1991; Sudhir, 2001). We reserve the derivation of margins
in Appendix. Our derivation much follows Villas-Boas and Zhao (2005) and
Che et al. (2007).
After calculating margins of manufacturers and a retailer, we will estimate
marginal cost of each brand using variables such as prices of ingredients.
Finally, we will calculate likelihood for each model and game, and compare
the results by Vuong test statistics.

3

Estimation

3.1

Demand−Side Estimation

Pricing equation As prices may be correlated with unobserved demand
characteristics, we first set up the pricing equationpjt = κ0 + zjt · κ1 + ηjt(3.1)

where zjt is an instrument which is correlated with pjt but not with ξjt , κ0and κ1 are parameters to be estimated, and ηjt is a random error term. Notethat this equation is defined for calendar date t = 1, . . . , T . We estimate p"
jt
and η"
jt by ordinary least squares.

9

Next, ξjt is obtained as residual in the following equation:ln S̃jt − ln S̃0t = xjt · β + simkj · SD + ξjt(3.2)

where ln S̃jt and ln S̃0t are the log of observed market shares of brand j andoutside good at time t respectively.
If price endogeneity exists, the terms ξjt and ηjt will be correlated.6 This
correlation should arise as ηjt can represent both demand and cost shock(i.e., if the unobserved demand characteristic is desirable, it is reasonable toassume it incurs cost). In order to check the existence of price endogeneity,
we assume that ξjt and ηjt jointly follow the bivariate normal distribution ascorrelation in that distribution equates dependence between them. We also
assume that their means are both zero, and their moments exist up to thesecond order.
Likelihood function The likelihood of purchase history of household i iswritten as
Li =

Ti $
#

ti =1

% J
#
j=0

&

[Prijti ]yijti × f (ξjti |ηjti ) × f (ηjti ) dξjti(3.3)

where yijti is an indicator function taking unity if household i chooses brandj at time t and 0 otherwise, f (ξjt |ηjt ) is the conditional density of ξjt , andf (ηjt ) is the density function of ηjt . Similarly to ξjti , the term ηjti is a subsetof ηjt , which is defined for all calendar dates in the panel. In this paper, we
employ the latent class model under which the likelihood function as in (3.3)
for household i is replaced with Li (Si = s), the likelihood of household ibelonging to the segment s or Si = s. Then we have the likelihood for whole6

As κ0 + zjt · κ1 is uncorrelated with ξjt by construction, ηjt represents a correlated(with ξjt ) part of pjt .

10panel data as
L=

% S
I
#
#
i=1

s=1

Li (Si = s) × Pri (s)

&

(3.4)

where S is the number of segments and Pri (s) is the membership probability
to segment s of household i. Parameters βs and SDs are estimated by
maximizing this likelihood function.

3.2

Supply−Side Estimation

Marginal cost We specify the marginal cost equation as
mcjt = wj0 + inputjt · wr

(3.5)

where wj0 is a brand-specific intercept term, inputjt is vector of observable
cost shifters, and wr is corresponding vector of parameters. For the notational convenience, let w ≡ (wj0 , wr ). Now to estimate w, we utilize thefollowing equation!
!
pjt − CMM jt − CM
Rjt = mcjt + εjt

(3.6)

!
!
where CMM jt and CM
Rjt are computed margin of manufacturers and aretailer for brand j at time t respectively, and εjt is a random error term.
Assuming the error term εjt follows a normal distribution with mean zero andfinite variance (which is to be estimated), the right-hand side of the equation!
!
εjt = pjt − CM
M jt − CM
Rjt − wj0 − inputjt · wr

(3.7)

also follows the normal distribution. Then we have the likelihood functionof the supply-side as
T #J#g(εjt )

(3.8)

t=1 j=1

where g(·) is the marginal density of εjt , to estimate w and to calculate
Vuong test statistics.
11

4

Data

We use scanner-panel data of yogurt purchases from anonymous retail chain
in the western Tokyo starting January 2007 to December 2008. There aremainly two type yogurts; box type and snack type. We chose the latter typefor empirical analysis as the former type may also be used for cooking. There
were 157 brands during the period of study, but only 16 brands were on salethroughout the period. Out of these brands, we chose 7 brands which had
enough purchasing records across stores, as we would like to use the average
yogurt prices in these stores as instruments for prices of yogurt in particular
store we would analyze.7 Next, we choose the store carrying these brands
with the least missing values. Then we chose households who only purchased
the selected 7 brands at least twice as we were to incorporate the effect ofstate dependence. This lefts 183 households who made 15,194 shopping trips
and 2,550 yogurt purchases.8 In the data, 76.5% of purchases were made bya female member of household. The average age of consumers in the panel
is 59.4 with standard deviations of 19.6. The minimum and maximum agesof consumers in the panel are 14 and 94 respectively.
The information on chosen brands is summarized in Table 4.1. The attributes we used for the attribute similarity index were “Raw milk usage”(the proportion of raw milk in yogurt, 3 levels), “Fat level” (the fat amountcontained, 3 levels) and “Ager usage” (whether yogurt contains ager or not,2 levels). Ager is used to produce so called “hard-type” yogurt, which has
7

The other stores had at least 20 dates without a single sale of any brands during

two years. We chose to exclude them from our analysis, as brand switch could have been
attributed to the fact that some of them were out-of-stock in these stores. In this paper,
we are not focusing on this kind of forced brand switching behavior.
8
We explicitly counted the shopping trips which did not lead to yogurt purchase forthose households to measure the share of outside option.

12

Table 4.1: Summary of brands.
Average price

Manufacturer Market Raw milk

Fat level

usage

Ager

Fat content Sugar content

usage(g/100g)(g/100g)(yen per gram)

ID

share

Brand 1

0.459

1

1.14%

No

Middle

Yes

2.47

7.77

Brand 2

0.486

2

2.95%

PartialMiddle

Yes

2.05

14.6

Brand 3

0.488

3

0.86%

All

High

Yes

4.10

14.9

Brand 4

0.483

4

1.08%

Partial

Low

No

1.76

15.2

Brand 5

1.113

4

3.22%

PartialMiddle

No

3.04

9.73

Brand 6

1.113

4

1.35%

Partial

Low

No

1.43

9.20

Brand 7

0.834

5

2.31%

Partial

Low

No

1.88

13.4

texture like pudding unlike plain-type yogurt. Out of these brands, we areespecially interested in brand 3, 5, and 6; brand 3 is differentiated in termsof taste (it is the only brand using only raw milk), brand 5 is the yogurt withspecial lactic-acid bacilli, and brand 6 is a low fat version of brand 5. To
compare the margins of these brands with those of the others would answer
the question we addressed − whether these brands bring higher margins to
manufacturers. Relatively small numbers in market share column in Table
4.1 are because of outside option as consumers did not buy any of these 7
brands 87.0% of their shopping trips. Brand 7 is a brand containing a fruit,
which is thought to justify its higher retail price.
As for marginal cost shifters, we collected data of raw milk price, labor
wage in four prefectures where 7 brands of yogurt are produced, internationalsugar price, cream price index, and international oil price.9 Because all data
were only available in monthly basis, we transformed them into weekly data
9

The information sources are as follows: Raw milk price and cream price index are ob-tained from the database of Jmilk (2014); labor wage in four prefectures are obtained fromstatistical departments of corresponding prefectures; international sugar price is obtained
from the database of Agriculture & Livestock Industries Corporation (2014); international
oil price is obtained from U.S. Energy Information Administration (2014).

13

by the linear filtering process employed by Slade (1995):
Wt = 0.25Wt−1 + 0.50Wt + 0.25Wt+1
where Wt in week t is the input price in the corresponding month (Besanko
et al., 1998). As for international sugar price, we multiplied it to the sugar
amount each brand contains. Also, since cream is mixed in yogurt to increase
fat content, we multiplied cream price index to the fat amount each brandcontains. We used raw milk price as they were, and we took log for labor wage
and for international oil price because their scales were of different ordersof magnitude. In addition, we employed manufacturer dummy variables toincorporate firm-specific cost structure with manufacturer 1 as baseline.

5

Empirical Results

5.1

Demand-Side Results

We estimate the latent class model by increasing the number of segments until there is no improvement in AIC. We find that the model with six segments
maximizes AIC. Additionally, we construct and estimate a multinomial logit
model without state dependence and the model with lagged brand choicedummy variable with the same number of segments to compare the fits. The
model fit is presented in Table 5.1. As we see from Table 5.1, there is large
improvement in log-likelihood by employing lagged brand choice dummy variable model relative to a multinomial logit model as log-likelihood improves
from -9,056.5 to -6,725.2. However, the additional improvement by employing the model with the attribute similarity index instead of the lagged brandchoice dummy variable is minimal, as log-likelihood only improves by 0.88%.
Nonetheless, we retain the result of the model with the attribute similarity
14

Table 5.1: Model Fit.
Model specification

Multinomial logit
model

Number of parameters

The model with the lagged

The model with thebrand choice dummy variable attribute similarity index

54

60

72

Log-likelihood-9,056.5-6,725.2-6,666.2

AIC

18,221

13,57013,476index (henceforth “optimal model”) since it is still the best fitting model.
The parameter estimates of the optimal model with standard errors arepresented in Table 5.2. All variables are significant at 1% level. “Brand”
in Table 5.2 represent brand-specific intercept terms relative to outside options, presented below “Demographics” entry are parameters for calculating
SDs , which is a parameter associated with the similarity index in (2.2), andpresented below “The attribute similarity index” entry are the estimates ofimportance weight for two attributes in calculating the attribute similarity
index.10 Because we find that using all three attributes results in anomalies
in estimation, we choose to remove “Fat level” attribute. We see that “Agerusage” has almost as six times greater effect as that of “Raw milk usage”
with importance weights of 0.358 and 0.060 respectively. This suggests thatperceived similarity between brands largely depends on the type of yogurt(i.e., whether yogurt is hard-type or plain-type). Estimated segment sizes
are reported below price coefficients in Table 5.2.
10

We only have importance weight estimates for segment 1 in Table 5.2. This is because

we estimated them with the model without segment and used these estimates for themodels with the greater number of segments. In other words, we assumed perceptionsof similarities between brands were common across segments as in Che et al. (2007). In
fact, estimating the model without this assumption would have increased the number ofparameters by 66, and this could have made the estimation unstable.

15

Table 5.2: Parameter estimates of the optimal model.

Brand 1

Brand 2

Brand 3

Brand 4

Brand 5

Brand 6

Brand 7

Price Coefficient

Segment sizes

Segment 1

Segment 2

Segment 3

Segment 4

Segment 5

Segment 6

1.803-2.733-3.2991.635-1.071-4.097(0.0002)(0.0003)(0.0000)(0.0000)(0.0001)(0.0002)

3.067-10.24

0.119

3.973

2.577

-20.62(0.0001)(0.0000)(0.0002)(0.0000)(0.0021)(0.0000)

2.358

0.829

-4.435

1.887

1.629

-2.142(0.0002)(0.0028)(0.0000)(0.0000)(0.0005)(0.0004)

2.245-1.8273.938-0.4031.054-2.943(0.0000)(0.0001)(0.0037)(0.0000)(0.0004)(0.0002)

14.00

3.580

4.542

8.090

3.525

5.026

(0.0002)(0.0093)(0.0000)(0.0000)(0.0000)(0.003)12.88-0.598-2.733

14.50

4.882

4.045

(0.0001)(0.0001)(0.0000)(0.0000)(0.0000)(0.0002)

7.178-1.093

2.702

7.478

-2.304-1.478(0.0001)(0.0001)(0.0000)(0.0002)(0.0000)(0.0000)-17.10-1.754-14.30-21.12-10.90-5.812(0.0025)(0.0048)(0.0042)(0.0006)(0.0037)(0.0077)

41.3%

2.7%

8.9%

30.4%

6.9%

9.8%

0.518

4.119

1.151

-7.100-1.5250.099(0.0131)(0.0134)(0.0585)(0.008)(0.0236)(0.0062)

0.696

4.138

-1.953-1.7860.517-1.419(0.0015)(0.0003)(0.0001)(0.0001)(0.0002)(0.0001)

Demographics
Intercept

Male dummyAge (logged)

0.143-2.644

1.641

2.283

0.962

-0.152(0.0471)(0.0536)(0.2405)(0.0414)(0.0943)(0.0231)

The attribute similarity index
Raw milk usage

0.060(0.0016)

Ager usage

0.358(0.0012)

Number of parameters

72

Number of observations

15,194Log-likelihood-6,666.2

16

State dependence tendencies across households are derived from the attribute similarity index calculated based on the estimated importance weights
of attributes and SDs calculated from demographic variables of households
and parameter estimates for them. They are presented in Table 5.3. The
Table 5.3: State dependence tendencies across segments.
Segment 1

Segment 2

Segment 3

Segment 4

Segment 5

Segment 6

1.464

3.646

2.058

-4.9050.670-1.584

Female0.768-0.491

4.012-3.1190.152-0.165Weighted Average

0.931

0.481

3.553

-3.5390.274-0.498

Malevalues in rows “Male” and “Female” are corresponding values for each group,
calculated with mean age. We see that households in segment 4 and 6 arevariety-seekers from Table 5.3. The rest is almost all inertial. With thisalong with the results in Table 5.2, about 40.2% of households under study
are assumed to be variety-seekers. However, also from Table 5.2, we do not
see the consistent relationship between state dependence tendencies and demographic variables. Being male affects the utility of the similar brand topreviously purchased one either positively or negatively, and the same is true
for age.
Now we briefly summarize the results presented in Table 5.2. First ofall, two largest segments are segment 1 and 4, consisting of 71.7% of themarket as “Segment sizes” indicate. Though they are similar in terms ofthe magnitude of price sensitivities and signs for brand dummies, Table 5.3
indicates that they differ substantially in their state dependence tendencies;
while segment 1 shows moderate inertia, segment 4 exhibits the greatest
variety-seeking tendency among all segments. Segment 1 and 4 being large
and price sensitive should induce rigorous price competition in the market,

17despite the fact that state dependence exists within the targeted segment.
As we see from Table 5.2, all segments except for segment 1 have varying
number of negative coefficients for brand dummies. For example, segment 2
and 6 endow positive values only to two brands (brand 3 and 5 for segment
2 and brand 5 and 6 for segment 6). This combined with the fact thatsegment 6 is variety-seeking implies that households in segment 6 are likely
to often switch between brand 5 and 6, which turn out to be the same
product with brand 6 being a low fat version of brand 5. This segment
is thought to be health-conscious as brand 5 and 6 are the yogurt withimmune-enhanced feature. Segment 6 consists 9.8% of the market according
to Table 5.2. Households in segment 3 are the least price sensitive withstrong inertia with size of 8.9%. Segment 5 is modest in all aspects withsize of 6.9%. The description of such segment can be difficult, but what-if
analysis would be helpful to understand the behavior of the segment, which
is another advantage of the structural equilibrium model (Kadiyali et al.,2001).

5.2

Supply-Side Results

In this subsection, we will present the results of margins, marginal cost andmodel comparison. Though the actual calculations proceed in this order, wefirst present the result of model comparison as it helps the interpretation of
the results of margins.
Log-likelihood for supply-side and Vuong test statistics After calculating margins, we calculated the log-likelihood for supply-side in (3.8) and
Vuong test statistics to compare the fits of three models and games in thesemodels as presented in Table 5.4. As we see, the vertical Nash−Bertrand
18

Table 5.4: The log-likelihood and Vuong test statistics under each game andmodel.
Log-likelihood(Vuong test statistics)

Vertical Interaction

Horizontal Interaction

Static model

Myopic model

Forward-looking model

manufacturer Stackelberg

Bertrand competition-209.76-242.45-118.10(22.64)(6.11)(44.05)

Tacit collusion-216.13-214.74-129.53(38.24)(53.30)(45.64)-203.48-134.75-37.99

(−)

(−)

(−)

-209.56-148.80-49.84(58.94)(34.53)(28.82)

vertical Nash

Bertrand competition

Tacit collusioncompetition game dominates across all models in terms of log-likelihood.
Thus Vuong test statistics in each column are with respect to this game.
They indicate that the vertical Nash−Bertrand competition game is statistically better than the other games. Moreover, log-likelihood shows thatforward-looking model fits the data most with value -37.99. We also compared the Vuong test statistics across models, to find that the best-fitting
model (the vertical Nash−Bertrand competition game in forward-looking
model) is statistically better than any other models and games.
Margins The margins (in yen per gram) under three models are presented
in Table 5.5. Presented in the entries of “Average delta” in the last row
are the mean value of ∆j in (B.3) in Appendix B for each brand, which is!
defined to be θj2|j1 − Jl=1,l"=j θj2|l1 , where θj2|j1 is the probability of purchasing
brand j in period 2 given the purchase of the brand in period 1 and θj2|l1 isdefined likewise for brand l (please see Appendix for detail). In a nutshell,∆j evaluates how the market share of brand j changes from the previous time19

Table 5.5: Margins (unit: yen per gram) under each model and game.

Average Prices

Brand 1

Brand 2

Brand 3

Brand 4

Brand 5

Brand 6

Brand 7

0.451

0.504

0.513

0.480

1.127

1.128

0.859

0.112

0.139

0.167

0.157

0.145

0.119

0.167

(0.0006) (0.0002)(0.0013)(0.0005)(0.0015)(0.0006)(0.0016)

Static model
Retail margin

manufacturer Stackelberg
Bertrand competition

Tacit collusion

0.137

0.124

0.141

0.107

0.108

0.133

(0.0006) (0.0005)

0.093(0.0016)(0.0007)(0.0007)(0.0006)(0.0020)

0.101

0.144

0.131

0.156

0.112

0.112

0.143

(0.0011) (0.0006)(0.0015)(0.0010)(0.0008)(0.0007)(0.0021)

vertical Nash
Bertrand competition

0.132

0.159

0.142

0.139

0.115

0.156

(0.0006) (0.0001)

0.102(0.0015)(0.0003)(0.0015)(0.0007)(0.0015)Myopic model
Retail margin

0.105

0.167

0.121

0.198

0.135

0.083

(0.0006) (0.0013)

0.088(0.0043)(0.0024)(0.0042)(0.0023)(0.0019)

0.067

0.097

0.011

0.145

0.025

0.083

0.068

(0.0005) (0.0028)(0.0057)(0.0016)(0.0028)(0.0010)(0.0009)

manufacturer Stackelberg
Bertrand competition

Tacit collusion

0.127

0.029

0.172

0.028

0.086

0.086

(0.0010) (0.0023)

0.083(0.0060)(0.0017)(0.0030)(0.0010)(0.0019)

0.072

0.082

0.140

0.098

0.192

0.131

0.066

(0.0004) (0.0002)(0.0037)(0.0016)(0.0041)(0.0023)(0.0011)

0.051

0.109

0.105

0.104

0.313

0.187

0.103

(0.0006) (0.0013)(0.0031)(0.0024)(0.0051)(0.0026)(0.0019)

0.039

0.100

0.007

0.125

0.039

0.115

0.085

(0.0006) (0.0028)(0.0044)(0.0015)(0.0033)(0.0014)(0.0010)

0.048

0.131

0.018

0.147

0.045

0.119

0.107

(0.0010) (0.0023)(0.0046)(0.0015)(0.0035)(0.0013)(0.0021)

vertical Nash
Bertrand competition

Forward-looking model
Retail margin

manufacturer Stackelberg
Bertrand competition

Tacit collusion

vertical Nash
Bertrand competition

Average delta

0.084

0.088

0.084

0.304

0.182

0.083

(0.0005) (0.0002)

0.042(0.0026)(0.0016)(0.0050)(0.0027)(0.0011)

0.444

0.396

0.151

-0.621-0.410-0.264-0.03520

period in response to a minute change of market share of the same brand.
We omit the margins of the vertical Nash−Tacit collusion game as they are
identical to retail margins. It should be noted that, in view of Table 5.4,
margins in the vertical Nash−Bertrand competition game in forward-looking
entries are the most accurate ones within the framework in Table 5.5, andthose in the other entries are counter-factual in the sense that, had these sorts
of games and perspectives were in play, these margins would have resulted.
Now we briefly overview the results in Table 5.5.
First of all, manufacturers’ margins under tacit collusion always exceed
those under Bertrand competition as expected. However, for brand 1, 3, 5
and 6, the margins under manufacturer Stackelberg are lower than verticalNash counterparts in both myopic and forward-looking models regardless ofwhich game in horizontal interaction is assumed. This is one piece of evidence
that manufacturer Stackelberg game between manufacturers and a retailer
cannot be justified with data.
Comparing margins under myopic and forward-looking models, we seethat both manufacturers and a retailer charge more margins for brands withnegative ∆j and charge less for those with positive ∆j in forward-looking
model relative to myopic model. In other words, forward-looking firms would
price lower if brands are likely to be repurchased and higher if they would
be switched to. This is consistent with economical rationale. In comparing
static and myopic models, however, we do not see such relationship for either
manufacturers nor a retailer.
Remember that brand 3 has a distinct taste advantage due to the fact thatit uses only raw milk, while brand 5 and 6 are the yogurt with special lacticacid bacilli. Therefore we expect that these brands to command higher margins. As expected, brand 3, 5 and 6 command three largest margins under the21

vertical Nash−Bertrand competition game in forward-looking model (0.088,
0.304, and 0.182 respectively), which we estimate to reflect Japanese yogurt
market.11 Meanwhile, brand 3 and 5 have the second least and the least margins respectively under the manufacturer Stackelberg−Bertrand competition
counter-factual (0.007 and 0.039), which is another evidence that manufacturer Stackelberg game cannot be justified with data.
These facts and the market being characterized by the vertical Nash−Bertrand
competition game jointly imply that differentiating brands 3,5, and 6 enable
manufacturers to charge the three highest margins among the seven brands.
In this sense, manufacturers’ efforts to differentiate their brands are reasonably rewarded.
However, we must note that a retailer also charges the largest and thesecond largest margins for brand 5 and 6 and charges the fourth largest
margin for brand 3. In fact, the amount of retailer’s margins in (A.3) arehigher than manufacturers’ margins in (A.17) for all brands in the verticalNash−Bertrand competition game. These facts lead us to the conclusionthat a retailer has more power than manufacturers. The decreasing priceof yogurt over the last decade is at least partially due to decreasing powerof manufacturers relative to the retailer in addition to competition among
manufacturers as indicated by our result. The existence of fierce competition among manufacturers makes sense, as 157 yogurt brands existed in themarket in January 2007 to December 2008.

11

The margins of Brand 1 and 5 under the manufacturer Stackelberg−Bertrand com-

petition game in forward-looking model are the same in Table 5.5 but this is because ofrounding. The margin of brand 1 is slightly larger than that of brand 5, even though thedifference is minimal.

22

Marginal costs The estimation result for marginal cost in forward-looking
model is presented in Table 5.6.12 We find that after including manufacturer dummies, labor wage and international sugar price (multiplied by sugar
amount) result in negative coefficients in all models and games, thus we exclude them.13 We find that remaining variables − cream price index (multiplied by fat amount), raw milk price, and international oil price − havepositive coefficients in manufacturer Stackelberg game, even though t-valuesof raw milk price and international oil price are not necessarily significant.
However, only international oil price has a positive coefficient in forwardlooking model under vertical Nash game regardless of horizontal interaction.
The high values for manufacturers 4 and 5 are consistent with the fact that
manufacturer 4 produces brand 5 and 6 and manufacturer 5 produces brand
7.
The price endogeneity After estimating ξ""
jt and η
jt , we tested the correlation between them using one of Pearson’s product moment correlation

coefficient test. The test reveals that they are significantly correlated andthus prices are proven to be endogenously determined, which is consistent
with the general finding in literature.

6

Conclusion

In this paper, we empirically analyzed Japanese yogurt market incorporating consumer heterogeneity, consumer state dependence, forward-looking behavior of manufacturers and a retailer, and price endogeneity arises from
12
13

Results for the other models can be provided upon the request to the author.
If we use only labor wage, their coefficients are positive. The effect of labor wage

seems to be absorbed by manufacturer dummies.

23

Table 5.6: Marginal cost estimation in forward-looking model.
manufacturer Stackelberg

Bertrandcompetition

Tacit

collusion

Estimate

Std.Errt-value p-value

Estimate

Std.Errt-value p-value

Intercept-0.7420.233-3.1820.002-0.8160.240-3.4080.001

Manufacturer 2-0.0410.026-1.5980.110-0.0610.027-2.3190.021

Manufacturer 3-0.0260.030-0.8800.379-0.0310.031-1.004

0.316

Manufacturer 4

0.287

0.022

13.22

0.000

0.287

0.022

12.87

0.000

Manufacturer 5

0.346

0.027

13.04

0.000

0.335

0.027

12.28

0.000

Cream price index

0.133

0.032

4.137

0.000

0.139

0.033

4.192

0.000

International oil price

0.038

0.024

1.546

0.123

0.037

0.025

1.480

0.139

Raw milk price

0.003

0.002

1.104

0.270

0.003

0.002

1.278

0.202

vertical Nash

Bertrandcompetition

Tacit

collusion

Estimate

Std.Err

Estimate

Std.Err

Intercept

0.196

0.082

2.401

0.017

0.187

0.084

2.225

0.026

Manufacturer 2-0.0480.021-2.2800.023-0.0630.021-2.9180.004

Manufacturer 3-0.0370.021-1.7780.076-0.0440.021-2.075

0.038

Manufacturer 4

0.163

0.017

9.563

0.000

0.161

0.018

9.193

0.000

Manufacturer 5

0.315

0.021

15.11

0.000

0.304

0.021

14.17

0.000

International oil price

0.037

0.018

2.015

0.044

0.037

0.019

1.957

0.051

t-value p-value

24

t-value p-value

the interaction between unobserved demand characteristics and prices. Ourdemand-side findings are consistent with those of previous literature; consumers are heterogeneous in their responsiveness to marketing variables anddegrees of state dependence. On supply-side, we find prices are endogenously
determined, manufacturers engage in Bertrand competition game, manufacturers and a retailer play vertical Nash game, and they set prices considering
their impact on future profit.
We find that brands with differentiating features (brand 3, 5 and 6) docommand higher margins, proving that manufacturers’ efforts are rewarded.
However, a retailer also charges higher margins for these brands and obtains
larger split of the profit. We also find that there are rigorous competitions
among manufacturers in this market which is consistent with the findings in
the other papers such as Nevo (2001) and Che et al. (2007), where Bertrand
competition was the case in the U.S. cereal market with large number ofbrands. Finally, our work adds another evidence to the body of literature inthis field of intersection between marketing and neo empirical industrial organization, as lack of empirical study is general concern in this area (Kadiyali
et al., 2001).
One major limitation of this research is the assumption of a monopolistic
retailer as retailers are likely to compete in reality. In fact, “National Surveyof Prices” conducted by Statistics Bureau, Ministry of Internal Affairs andCommunications in Japan indicates that the average retail prices of yogurt
are higher in stores with no competitors around. Incorporating retail competition in the framework employed in this study would be an interestingsource of future research. The other possible direction of future research isinclusion the effect of store brand. This topic is common in the literature,
and widely investigated in the context such as its effect on power balance25between manufacturers, store loyalty and so forth. As state dependence isoften neglected in these analysis, investigating the effect of store brand in
the presented framework may provide new insight to the literature.

26

Appendix
In section A, we derive margins in myopic model. In section B, we derive
margins in forward-looking model.

A

Margins in Myopic Model

We start with margins of a retailer as it will be used in calculating margins
of manufacturers.

A.1

Margins of a Retailer

The profit function for the retailer is defined asJ'πR =(pjt − wjt )Sjt M

(A.1)

j=1

where wjt is the wholesale price for brand j at time t, Sjt is market share,
and M is the market size. The retail margin for brand j is pjt − wjt .
Now by partially differentiating (A.1) with respect to each retail price pjt ,setting them zero, and algebraic manipulations, we have



−1 

∂SJt
∂S1t
p − w1t, · · · , ∂p1t
S
 1t

 ∂p1t  1t 
..
..



  ..  = −
  . .
.
.



 

∂SJt
∂S1t
pJt − wJt
, · · · , ∂pJt
SJt
∂pJt

(A.2)

Using the notation of Che et al. (2007), we have(pt − wt ) = Φ−1
t St

(A.3)

where (pt − wt ) ≡ (p1t − w1t , · · · , pJt − wJt )T is J × 1 vector of retail margins,
Φt is J × J matrix with elements∂SktΦjkt = −∂pjt
for brand j, k = 1, · · · , J, and St is J × 1 vector St = (S1t , . . . , SJt )T .
27

A.2

Margins of Manufacturers

Now we derive margins of manufacturers under different games. Unlike inthe retailer’s case, the profit function of manufacturers differs depending on
which game in horizontal strategic interaction is assumed. The profit function
πf of manufacturer f under Bertrand competition is given by'πf =(wjt − mcjt )Sjt M,

(A.4)

j∈Jf

where Jf is a subset of brands produced by manufacturer f and mcjt is themarginal cost of producing brand j at time t. The manufacturer’s margin
from brand j is wjt − mcjt . On the other hand, the total profit function π∀fof collusive manufacturers is given by
π∀f =

J
'
j=1

(wjt − mcjt )Sjt M.

The first order condition of the profit function in tacit collusion game is
4
4
55
J
J
'
'
∂π∀f
∂Sjt ∂pkt= M Slt +(wjt − mcjt )
·
=0
(A.5)
∂wlt
∂p
∂w
kt
lt
j=1
k=1

for l = 1, . . . , J. By algebraic manipulation, we have



 
−1 

∂p1t
∂pJt
∂SJt
∂S1t
w1t − mc1t
,
·
·
·
,
,
·
·
·
,
S
∂w1t   ∂p1t
∂p1t  ∂w1t
 1t 
..
..
..



 
  ..  = − ·  .  ,
.
.
.



 
 

∂p1t
∂pJt
∂SJt
∂S1t
wJt − mcJt
, · · · , ∂wJt
, · · · , ∂pJt
SJt
∂wJt
∂pJt

(A.6)

where the left hand side of equation (A.6) is J × 1 vector of manufacturers’ margins. The first order condition of profit function of Bertrand
competition can be derived similarly.

In equation (A.6), the terms Sjt

and ∂Sjt /∂pkt can be directly obtained from the estimated demand parameters but ∂pkt /∂wlt cannot be. Thus we must infer these terms indirectly, and the difference between manufacturer Stackelberg and vertical28Nash stems from how these terms are specified. We start with the manufacturer Stackelberg−Tacit collusion game because margins under the manufacturer Stackelberg−Bertrand competition game is a special case of thoseunder the manufacturer Stackelberg−Tacit collusion game.
A.2.1

Margins under the manufacturer Stackelberg−Tacit collusion gameTo infer ∂pkt /∂wlt , we exploit the first order condition of the retail profit
function defined in (A.1);
7
J 6
'
∂πR
∂Skt
= Sgt +
(pkt − wkt )
=0
∂pgt
∂pgt
k=1

(A.7)

for g = 1, . . . , J with the market size M removed. Since a retailer is assumed
to maximize the category profit, the change in wholesale price of one brandwould affect all retail prices in the category. Thus we totally differentiate(A.7) with respect to prices pjt , j = 1, . . . , J, and wholesale price wlt forbrand l, to obtain, for some g,
4
5
J
J
'
∂Sgt ∂Sjt '∂ 2 Skt∂Slt++(pkt − wkt )dpjt −· dwlt = 0.
∂pjt
∂pgt k=1∂pjt ∂pgt
∂pgt
j=1

(A.8)

Denoting the terms inside the bracket on the left hand side of equation(A.8) as ν(g, j), we have the set of J equations for some l as

∂Slt


ν(1, 1)dp1t + ν(1, 2)dp2t + · · · + ν(1, J)dpJt = ∂p
· dwlt ,

1t

..
.



 ν(J, 1)dp + ν(J, 2)dp + · · · + ν(J, J)dp = ∂Slt · dw .
1t
2t
Jt
lt
∂pJt

(A.9)

Defining Gg ≡ (ν(g, 1), . . . , ν(g, J)), we rewrite the expression in (A.9) in

29

matrix form and rearrange it as

 
−1 

∂Slt
∂p1t /∂wlt
G1

 

 ∂p1t ..
  ..  .  =  .  ·  ..  ,
.

 



∂Slt
∂pJt /∂wlt
GJ∂pJt(A.10)

assuming the inverse of the J × J matrix (G1 , . . . , GJ )T exists. Transposing
both sides of equation (A.10) and stacking them vertically for l = 1, · · · , J,
we have






∂p1t
∂pJt
, · · · , ∂w
∂w1t
1t

..
.

∂p1t
∂pJt
, · · · , ∂w
∂wJt
Jt





 
 
=
 

∂S1t
∂S1t
, · · · , ∂p
∂p1t
Jt

..
.

∂SJt
Jt
, · · · , ∂S
∂p1t
∂pJt



 <
=−1

.
 · GT1 , · · · , GTJ
(A.11)

Substituting (A.11) into (A.6), we have the manufacturers’ margins underthe manufacturer Stackelberg−Tacit collusion game as(wt − mct ) = −(ΦTt G−1 Φt )−1 St(A.12)

where (wt − mct ) = (w1t − mc1t , · · · , wJt − mcJt )T and G = (GT1 , · · · , GTJ ).
A.2.2

Margins under the manufacturer Stackelberg−Bertrand competition game

In Bertrand competition, each manufacturer maximizes a profit from its own
brands. Thus in Bertrand competition, (A.6) applies only to the brands aparticular manufacturer produces. This requires replacement of the third
term Φt in matrix (ΦTt G−1 Φt )−1 in (A.12) with Φt · ∗Ω, where ·∗ denotes
element-by-element multiplication, and Ω is J × J matrix whose (j, k) elements are indicator functions taking unity if brands j and k are made by
the same manufacturer and zero otherwise. Then we have the manufacturers’
margins under the manufacturer Stackelberg−Bertrand competition game as(wt − mct ) = −(ΦTt G−1 Φt · ∗Ω)−1 St .
30

(A.13)

A.2.3

Margins under the vertical Nash−Tacit collusion gameIn the vertical Nash game, manufacturers and a retailer move simultaneously.
More specifically, manufacturers set wholesale price expecting a certain levelof retail margin for the brand; a retailer sets its retail margin for each brandbased on its profit maximizing behavior. Now by assumption, we have therelationship∂(pjt − wjt )=0∂wjt
or equivalently
∂pjt
=1
∂wjt

(A.14)

for all j = 1, . . . , J since the retail margin of brand j, pjt − wjt , is notaffected by the wholesale price of the brand as manufacturers and a retailer
move simultaneously. Similarly, since the retail margin of brand, pjt − wjt ,is not affected by the wholesale price of the other brands, we have∂(pjt − wjt )=0∂wkt
or equivalently
∂pjt
=0
∂wkt

(A.15)

for j = 1, . . . , J, j &= k.14 Finally, from (A.14) and (A.15), the matrix
with elements ∂pjt /∂wkt on the right-hand side of equation (A.6) becomes14

We note that this behavioral principle of retailer is consistent with its profit maxi-

mizing behavior, as the predetermined retail margins are determined from the first ordercondition of its profit function
7
J 6
'
∂πR
∂Skt
= Sgt +
(pkt − wkt )
=0
∂pgt
∂pgt
k=1

even in the vertical Nash game.

31

an identity matrix and equation (A.6) becomes



−1 

∂SJt
∂S1t
w − mc1t, · · · , ∂p1t
S
 1t

 ∂p1t  1t 
..
..



  ..  = −
  . .
.
.



 

∂SJt
∂S1t
wJt − mcJt
, · · · , ∂pJt
SJt
∂pJt

Thus we have manufacturers’ margins under the vertical Nash−Tacit collusion game as(wt − mct ) = Φ−1t St(A.16)

which is identical to margin of the retailer. This makes sense as the verticalNash game assumes approximately equal power between manufacturers anda retailer (Choi, 1991).
A.2.4

Margins under the vertical Nash−Bertrand competition game

Since the retailer behaves the same independent of whether manufacturers
compete or tacitly collude, the conditions (A.14) and (A.15) still hold in
the vertical Nash−Bertrand competition game. And by the same reasoningof the manufacturer Stackelberg−Bertrand competition game, we have the
manufacturers’ margins under the vertical Nash−Bertrand competition gameas(wt − mct ) = (Φt · ∗Ω)−1 St .

(A.17)

Table A.1 summarizes margins under each game.
Table A.1: The manufacturers’ margin under myopic model.
Bertrand competition

Tacit collusion

manufacturer Stackelberg −(ΦTt G−1 Φt · ∗Ω)−1 St−(ΦTt G−1 Φt )−1 Stvertical Nash(Φt · ∗Ω)−1 St

32

Φ−1
t St

B

Margins in Forward-Looking Model

Here we derive the margins in forward-looking model. We start with themargin of a retailer.

B.1

Margins of a Retailer (Forward-Looking Model)

The objective function of one-period forward-looking retailer is VR = πR1 +δπR2 , where πRt is a profit function defined in (A.1) for period t = 1, 2, andthe term δ is some exogenously given discount rate. Then the first orderconditions are ∂πR1 + δ !J ∂πR2 · ∂Sj2 · ∂Sj1 = 0
j=1 ∂Sj2∂pk1∂Sj1 ∂pk1
∂πR2

= 0
∂pk2

(B.1)

for k = 1, . . . , J. In (B.1), the first equation corresponds to the first ordercondition of the first period profit function and the second equation corresponds to that of the second period profit. As the first order conditionin the second period is already known, we only concern for the first equation in (B.1) in the following derivation. Furthermore, in that equation, theunknown terms are ∂πR2 /∂Sj2 and ∂Sj2 /∂Sj1 .
Clearly, ∂πR2 /∂Sj2 is (pj2 − wj2 ). To calculate ∂Sj2 /∂Sj1 , we exploit thefollowing relationship:
Sj2 = θj2|j1 × Sj1 +J'

l=1,l"=jθj2|l1 × Sl1(B.2)

where θj2|j1 is the probability of purchasing brand j in period 2 given thepurchase of the brand in period 1, and θj2|l1 is defined likewise. Since themarket share sums up to one, the term Sl1 is rewritten as Sl1 = (1 − S11 −· · · − Sl−1,1 − Sl+1,1 − · · · − SJ1 ) for all l = 1, . . . , J, l &= j, which includes the33term −Sj1 . Thus, the partial derivative of the second term on the right-handside of equation (B.2) with respect to Sj1 is
>!
?
J
J
∂
θ
×
S
'
l1
l=1,l"=j j2|l1=−
θj2|l1∂Sj1
l=1,l"=jas ∂Sl1 /∂Sj1 = −1 for l = 1, . . . , J, l &= j. Thus taking partial derivative ofboth sides of (B.2) with respect to Sj1 , we haveJ'∂Sj2
= θj2|j1 −θj2|l1 .
∂Sj1
l=1,l"=j(B.3)

We define the right-hand side of equation (B.3) as ∆j .
In the same manner as in the derivation of vector (pt − wt ), the second
term on the left-hand side of the first equation in (B.1) can be expressed bymatrix form as

 
 

∂SJ1
∂S11
, · · · , ∂p11
∆ ,··· ,0
p − w12 ∂p11
  1
  12

..
..
..

 
 

δ
·
·

.
.
.

 
 

∂SJ1
∂S11
, · · · , ∂pJ1
0, · · · , ∆J
pJ2 − wJ2
∂pJ1

where the second matrix is diagonal matrix with diagonal elements ∆j , which
we will express as ∆. Thus we have the margin in the first period as



−1 


 

∂SJ1
∂S11
p − w11, · · · , ∂p11
S
∆ ,··· ,0
p − w12 11 ∂p11 11 
 1
  12

..
..
..
..




 .. 

 

=
−
·
−
δ
·










.
.
.
.
.







 

∂SJ1
∂S11
pJ1 − wJ1
, · · · , ∂pJ1
SJ1
0, · · · , ∆J
pJ2 − wJ2
∂pJ1
or (p1 − w1 ) = {ΦT }−1 S1 − δ∆(p2 − w2 ), assuming the inverse of ΦT exists.

To derive margins in forward-looking model, we first calculate the margins in
the myopic case from week 2, and use these margins in calculating margins
in forward-looking model starting from week 1.

34

B.2

Margins of Manufacturers (Forward-Looking Model)

The derivation of margins of manufacturers in one-period forward-looking
model much follows the case of the retailer. Here we consider the margin in
the manufacturer Stackelberg−Tacit collusion game as those under the othergames are special case of this game. The objective function is VM = πf 1 +δπf 2
and the first order conditions are ∂πf 1 + δ !J ∂πf 2 · ∂Sj2 · ∂Sj1 = 0
j=1 ∂Sj2 ∂Sj1 ∂wk1
∂wk1
∂πf 2

= 0.
∂wk2

(B.4)

As was the case in (B.1), the first equation of (B.4) corresponds to the firstorder condition of the first period profit function and the second equation
corresponds to that of the second period profit. Clearly, ∂πf 2 /∂Sj2 = (wj2 −mcj2 ). Then the product of this term and ∂Sj1 /∂wk1 turns out to be thesecond term of the first order condition of the profit function of manufacturersin (A.5) except for the subscript being 2 instead of t in wholesale price wj2
and marginal cost mcj2 . Then this product term can be written as

 
 

∂p11
∂pJ1
∂SJ1
∂S11
,
·
·
·
,
,
·
·
·
,
w
−
mc
12
12
∂w11   ∂p11
∂p11   ∂w11

..
..
..

 
 

·
·






.
.
.

 
 

∂p11
∂pJ1
∂SJ1
∂S11
, · · · , ∂wJ1
, · · · , ∂pJ1
wJ2 − mcJ2
∂wJ1
∂pJ1

or simply ΦTt G−1 Φt (w2 −mc2 ). Thus the second term on the left-hand side of

the first equation of (B.4) becomes δ(ΦTt G−1 Φt )∆(w2 − mc2 ). Then we haveS1 + ΦTt G−1 Φt (w1 − mc1 ) + δ(ΦTt G−1 Φt )∆(w2 − mc2 ) = 0 or (w1 − mc1 ) =−(ΦTt G−1 Φt )−1 S1 −δ·∆(w2 −mc2 ), assuming the inverse of ΦTt G−1 Φt exists.
The margins in the other games are derived similarly as we presented in the
myopic case. They are summarized in Table B1.

35

Table B.1: The manufacturers’ margins under forward-looking model in thefirst period.
Bertrand competition
manufacturer Stackelberg −(ΦTt G−1 Φt · ∗Ω)−1 S1−δ · ∆(w2 − mc2 )vertical Nash(Φt · ∗Ω)−1 St−δ · ∆(w2 − mc2 )36

Tacit collusion−(ΦTt G−1 Φt )−1 S1−δ · ∆(w2 − mc2 )

Φ−1
t St

−δ · ∆(w2 − mc2 )References
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