Estimating Consumer Demand for Music CD: A Bayesian Analysis
Yuji Nakayama, Tomonori Ishigaki, Nagateru Araki, Osaka Prefecture University
Abstract
In this study, we conduct an empirical analysis using sales data from a music CD store in
Japan. The distinguishing feature of our data is that it contains each customer’s ID number,
which enables us to examine each customer’s preference for music genre using his/her
purchase history. We use this scanner panel data to estimate each customer’s demand for
music CDs. For estimation, we adopt a Bayesian hierarchical model and use a Markov Chain
Monte Carlo (MCMC) method, following an earlier work by Kim, Allenby and Rossi (2002).
Utilizing estimated customers’ preference, we propose the retailer’s assortment c hange
combined with couponing both to improve its profitability and to maintain its reputation with
customers.
Introduction: Background and Motivation
Innovative Internet technology and broadband distribution of digital content has changed the
music industry. According to OECD (2005), in recent years the music industry experienced a
significant 16% decline of sales from USD 38 billion in 1999 to USD 32 billion in 2003 (see
OECD 2005 Annex Table 3.1). The Japanese market, which has the world’s second largest
sales, encountered a more severe sales decline of nearly 20% from USD 322 million in 1999
to USD 260 million in 2003 (see OECD 2005 Annex Table 3.3). Among various factors, the
decline of music sales is said to be caused by CD-burning technology and online piracy, both
of which is enabled by the spread of computer in the home and the Internet.
Apart from illegal file-sharing of recorded music, the advent of Internet-based music stores is
a remarkable phenomenon in the music market. They are noteworthy challengers to
traditional music stores. A vast number of products to sell and an easy-to-use technology to
search for a particular piece of music are the most remarkable features of Internet music stores,
whether they sell p h y s i c a l music CDs ( e.g. Amazon.com) or digital music contents
downloadable directly via the Internet ( e.g. the iTunes music store). Conventional music
stores on the ground have limited shelf space. Thus, in order to compete with Internet-based
music stores, conventional stores should examine their assortment of merchandise and plan
effective category management, particularly in categories of music products that do not have
enough sales. In addition, to maintain their reputation with customers, in particular those who
have purchased less popular genre CDs, they should consider how to compensate such
disadvantaged customers. Several papers deal with the music industry (Bradlow and Fader,
2001; Lee et al., 2003; Moe and Fader, 2001; Oberholzer-Gee and Strumpf, 2007; Yamada et
al., 2002). In this paper, we conduct an empirical analysis to tackle the above problem faced
by a conventional music store.
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Model
Consider the following utility maximization problem with a budget constraint. The utility
function of customer i i s given as:
( )
J
(
U i x i = åy ij x ij + g ij
)
aj
( )
exp e ij ,
j =1
where first x ij is the demand for category j , which is an element of vector x i , second y ij ,
g ij and a j are parameters of the utility function, and finally e ij is the random utility term
which is drawn from standard normal distribution N (0,1) . Customer i knows the value of
e ij when he/she chooses x i , while the researcher does not, which yields the likelihood
function for the parameter estimation.
Customers maximize their utility, taking account of the budget constraint, p ' x i £ E i where
p = ( p1 ,.., p J )' is the price vector and E i i s the budget that customer i c a n spend. The
Lagrangian for constrained maximization is given by L = U i (x i ) - l ( p ' x i - E i ) where l i s
the Lagran
ge multiplier. Differentiating the Lagrangian with respect to x ij yields the Kuhn-Tucker
(KT) first order conditions for optimization:
( )
when x > 0,
U (x ) - lp < 0, when x = 0,
where U (x ) = a y (x + g ) exp(e ) is the marginal utility of good j and x i s an
element of the vector of optimal demands: x = (x ,.., x )' . If y and/or e have a large
U ij x ij* - lp j = 0,
i
j
i
j
i*
j
j
i
j
i*
j
i a j -1
j
i*
j
i*
j
i*
j
j
i*
j
i
j
i*
i*
1
i*
J
i
j
i
j
value, customer i ’s optimal choice is x ij* > 0 and the first part of KT condition holds. On
the other hand, if not, its choice is x ij* = 0 and the second part of KT condition holds. In the
following, we use the above framework to examine each customer’s preference for music
genre.
Data
We use scanner panel data from one of the major music CD store chains in Japan. The store,
whose name is confidential, is located in Tokyo. Period of data is from November 1 in 2002
to December 21 in 2003. The distinguishing feature of our data is that i t contains the ID
number of each customer, thus distinguishing one customer’s purchase history from that of
another. The original data has 23 music genres. For our study, we merge some genres into one
to make the following 5 genres: (1) Japanese Pop (J-Pop), (2) (Overseas) Pop, Rock, Blues,
(3) Dance, Soul, Hip Hop, (4) Classical, and (5) Others (inc. Jazz). In addition, we convert
daily original data into monthly data and select customers who visit the store more than 5
times and purchase only album CDs. We remain with a database of 384 customers with 2827
purchase occasions. We regard multiple visits in a month as one purchase occasion. Note that
we need price data for each genre for each customer. We construct the data as follows: first,
when a customer bought one CD, we use the actual price that the customer paid, second,
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when he/she bought multiple CDs of one genre, we use the average price that the customer
paid, and finally, when he/she buys no CDs of a particular genre, we use the average price
that other customers paid for CDs of that genre.
Table 1 shows total quantities purchased in each genre by the 384 customers. We find that
J-Pop, Pop et al., and Dance et al. comprise an overwhelming share of nearly 90%. However,
it does not mean that each customer buys music CDs in the same proportion.
Table 1. Quantity Purchased in Each Genre
Genre
JapanesePop
Pop,Rock,Bl
ues
Dance,Soul
,Hi
pHop
Cl
assi
cal
Others(i
nc.Jazz)
Total
Quanti
ty Percentage
1,
295
23.
9%
2,
135
39.
4%
1,
389
25.
6%
93
1.
7%
505
9.
3%
5,
417
100.
0%
Table 2 shows the purchase histories of two customers chosen from the 384 customers. Table
2 (a) shows that customer no. 1 strongly prefers the Pop genre and only purchases CDs of this
genre, while Table 2 (b) reveals that customer no. 276 prefers J-Pop, Dance, and Others and
buys CDs of these three genres. These results indicate the diversity of consumers’ preference
for music.
Table 2. Examples of Purchase History for Specific Customers
(a)
Year Month J-Pop
2002
2002
2003
2003
2003
2003
2003
(b)
CustomerNo.1
11
12
1
5
6
7
8
Pop
0
0
0
0
0
0
0
Dance Cl
assi
cal Others
3
3
2
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
CustomerNo.276
Year Month J-Pop
2002
2003
2003
2003
2003
2003
2003
2003
2003
12
2
3
4
6
7
8
11
12
1
2
0
0
1
1
1
2
1
Pop
Dance Cl
assi
cal Others
0
0
0
0
0
0
0
0
0
11
3
2
2
1
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
1
0
Estimation and Results
Estimation
We adopt a Bayesian hierarchical approach to estimate our model. Following Case Study 5 in
Rossi, Allenby, and McCulloch (2005), which is a modified version of Kim, Allenby, and
Rossi (2002), we specify the prior distributions of the parameters for the utility function as
follows:
b i = b 2i ,..., b 5i ' ~ N (b ,Vb ), b ~ N 0, a -1Vb , Vb ~ IW (n ,V ), d j ~ U [- 1,0],
(
)
(
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)
where b ij and d j a r e the conversion of a j a n d y ij
(
such that b ij = ln a jy ij
)
and
d j = a j - 1 respectively. Note that N (×) , IW (×) U (×) denote normal, inverted Wishart and
uniform distributions, respectively. We set b1i = 0 (the parameter of J-Pop) for all i a n d
g ij = 1 for all i a n d j
for identification. A l s o , w e s e t a = 0.01 n = 7 a n d
V = 7 ´ diag (4 ) f o r the above prior distributions to be relatively diffuse. We conduct a
Markov Chain Monte Carlo (MCMC) method to draw samples of the parameters from
posterior distributions using the statistical software R (R Development Core Team, 2006) and
an associated package bayesm (Rossi, Allenby, and McCulloch, 2005). Specifically, we
discard the first 5,000 draws for burn- in and use every 10 draws of the next 10,000 draws (i.e.,
thin rate is equal to 10) for estimation. For details of the MCMC algorithm we use, see Rossi,
Allenby,
and
McCulloch
(2005) a n d
the
accompanying
website
(http://faculty.chicagogsb.edu/peter.rossi/rearch/bsm.html).
Results
Table 3 shows posterior mean and standard deviation of parameters, betabar ( b ), and delta
( d j ), which is common to all customers. We find the followings from this table.
Table 3. Posterior Mean and Standard Deviation of Common Parameters
Genre
JapanesePop
Pop,Rock,Bl
ues
Dance,Soul
,Hi
pHop
Cl
assi
cal
Others(i
nc.Jazz)
*Fi
xedfori
denti
fi
cati
on
Commonparameters
Betabar
Del
ta
Mean
SD
Mean
SD
0.
00*
-0.
38
0.
05
0.
18
0.
11
-0.
57
0.
05
-0.
56
0.
12
-0.
53
0.
06
-4.
18
0.
42
-0.
48
0.
14
-0.
86
0.
09
-0.
40
0.
06
First, the Pop et al. genre has the highest mean betabar, while the Classical genre has the
lowest, which means that the marginal utility of the former (the latter) when x i = 0 is the
largest (the smallest), on average. Most mean betabar figures are consistent with quantities
purchased, shown in Table 1. The mean betabar of the Dance et al. genre is lower than that of
the J-Pop genre, while the quantity purchased of the former genre is larger than that for the
latter genre. It indicates that the average customer prefers the J-Pop genre to the Dance et al.
genre when he/she buys only one CD, and that there are some customers who strongly prefer
the Dance et al. genre. Second, the Pop et al. genre has the lowest mean delta, which signifies
that the curvature in utility is largest of all genres, thus customers, on average, are satiated
with music of this genre and purchase other genres if they buy more than one CD. Finally,
standard deviations of both betabar and delta for the Classical genre are the largest. This result
suggests diversity of preference for the Classical genre; some customers like and buy classical
CDs, while others do not.
As noted before, customers have heterogeneous preferences for music. We can obtain each
customer’s posterior distribution for music genre, y ij , using formulae, y 1i = 1 a 1 a n d
{ }
{ }
( )
y ij = exp b ij a j f o r j ¹ 1 . Figures 1 and 2 display the posterior distribution of y ij for
customers i = 1 and 276, which is congruous with their purchase history in Table 2.
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{ }
Figure 1. Posterior Distribution of y ij for Customer No. 1
1.5
3.0
0
40
80
0
1
2
3
4
0.4
Density
0
0.0
Density
Others
2 4
Classical
0.0
Density
Dance
0.3
Pop
0.015
Density
0.0
0.000
0.0 2.0
Density
J-Pop
0.00
0.10
0.20
0.0
1.5
3.0
{ }
Figure 2. Posterior Distribution of y ij for Customer No. 276
1.5
3.0
0
40
80
0.0
0
0
1
2
3
4
0.00
1.0
Density
Others
10
Density
Density
0.0
0.0
0.0
Classical
0.3
Dance
1.5
Density
Pop
0.0 2.0
Density
J-Pop
0.10
0.20
0.0
1.5
3.0
Assortment Change Combined with Couponing for a Conventional Music Store
Finally, we examine the assortment change for a conventional music store. We know from
Table 1 that there are not enough classical CD sales. Thus, one plausible change is that the
store removes CDs of this genre in order to add those of other genres that have enough sales.
For example, if the store holds larger stocks of the latest hit CDs of Japanese pop music, it
will get new customers and its sales and profit will expand.
However, this assortment change lowers the utility attainable by customers who prefer
classical music, given their expenditure. Maintenance of the store’s reputation with customers
is indispensable for a conventional store which faces strong competition from Internet-based
stores. Therefore, in addition to the above assortment change of removing classical CDs, we
consider providing coupons which customers can use when buying CDs at the store.
Following Rossi, Allenby, and McCulloch (2005), we compute the value of a coupon by
calculating the increase in spending in order to attain the utility derived from the full
assortment. Specifically, we use a R routine constrOptim for solving the utility maximization
with a budget constraint. We set the upper limit on a coupon at 3,000 yen, with which one can
buy a standard album CD in Japan. The computations are made for each purchase occasion of
each customer. Then, we calculate his/her mean of coupon value for each customer. We
assume that the store gives a coupon with the mean value to each customer. It does not mean
that the store gives the same coupon to each customer, because there exists the diversity of
customers’ preference for music genre. In our calculation, the minimum coupon value is 11
yen, while the maximum value is 3000 yen. That is, based on data of their purchase history,
one who gets the coupon of the minimum value does not prefer classical music, while another
one who gets the coupon of the maximum value prefers this music genre very much. The total
coupon value that the store gives to the 384 customers is 182,690 yen.
The store needs to mail a different coupon to each customer. In Japan, an “electronic coupon”,
which a store sends to each customer’s mobile phone via e-mail, is well known to consumers.
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This is very convenient and labour-saving for customers. Thus, using an electronic coupon
customized for each client is a possible and plausible option to maintain the store’s reputation.
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