Description
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modelling.
ABSTRACT
Title of Document: ADVANCES IN MATHEMATICAL MODELS
IN MARKETING.
Ashwin Aravindakshan, Doctor of Philosophy,
2007.
Directed By: Professor Roland T. Rust, Department of
Marketing
This dissertation comprises a series of three essays that relate advances made
to both theoretical and empirical issues in marketing.
The first essay discusses the issue of endogeneity of market share and price in
logit models and provides a theoretical procedure to solve this problem. The
inseparability of demand and price make the possibility of drawing definite
conclusions about either almost impossible. We employ a recently rediscovered
mathematical function called the ‘LambertW’ to solve this problem of endogeneity
and in turn yield logit models more conducive to theoretical study. We also employ
this methodology to the problem studied by Basuroy and Nguyen (1998).
The second essay deals with the issue of pricing implicit bundling. Implicit
bundles are products that are sold separately but provide an enhanced level of
satisfaction if purchased together. We develop a model that would account for the
possible relationships of the products across the different product lines. We show that
accounting for these relationships would decrease the amount of price competition in
the market and also allow the Firm to enjoy higher profits. We also account for the
endogeneity of price and market share when deriving the optimal solutions. We show
that optimal prices first increase as the relationship between the firm’s two products
become stronger and then decrease as the two products become more exclusive to
each other. Finally, we also find that a firm’s prices increase as the competitor’s
contingent valuations increase.
The third essay helps improve the efficacy of CRM interventions by analyzing
the latent psychological loyalty states of the customer. We use state space models to
predict these latent loyalty states using observed data. We then use the predicted
values of loyalty to derive the probability of repurchase of the customer. We also
identify the types of CRM interventions that play a role in improving the loyalty of
the customer to the firm and those interventions that have no effect. We compare our
model’s predictions to those derived from two other estimation methods. We find that
our predictions are better than those computed from the other methods discussed.
ADVANCES IN MATHEMATICAL MODELS IN MARKETING
By
Ashwin Aravindakshan
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2007
Advisory Committee:
Professor Roland T. Rust, Chair
Professor Brian T. Ratchford
Professor P. K. Kannan
Professor Wendy Moe
Professor Roger Betancourt
© Copyright by
Ashwin Aravindakshan
2007
ii
Acknowledgements
I am indebted to Dr. Roland Rust for his constant help and guidance towards
the completion of this dissertation. I am grateful to Dr. Brian Ratchford for his
unconditional support and guidance.
I would also like to thank Dr. P. K. Kannan, Dr. Wendy Moe and Dr. Roger
Betancourt for taking the time to help enrich this effort.
iii
Table of Contents
Acknowledgements....................................................................................................... ii
Table of Contents......................................................................................................... iii
List of Tables ................................................................................................................ v
List of Figures .............................................................................................................. vi
Overview....................................................................................................................... 1
Chapter 1: Essay 1 - The LambertW Transformation As An Approach To Solving
Share Equations In Logit Models ................................................................................. 5
Summary................................................................................................................... 5
1. Introduction........................................................................................................... 5
2. A Brief Note on the ‘LambertW’ Function........................................................... 6
3. The Logit Model ................................................................................................... 9
3.1 Consumer’s Demand Function ....................................................................... 9
3.2 Firm’s Profit Function................................................................................... 11
4. The Analytical Solution...................................................................................... 11
4.1 Solving the Firm Level Endogeneity Problem.............................................. 11
4.2 Estimation Procedures .................................................................................. 14
5. A More General Application Incorporating Marketing Expenditure ................. 15
6. Conclusion ......................................................................................................... 19
Chapter 2: Essay 2 - Pricing Related Products In A Competitive Environment – The
Role Of Contingent Valuations Between Products..................................................... 20
Summary................................................................................................................. 20
1. Introduction......................................................................................................... 20
2. Literature Review................................................................................................ 22
2.1 Product Relationships ................................................................................... 22
2.2 Complementary Pricing ................................................................................ 27
3. Model Formulation ............................................................................................. 29
3.1 Defining the Market Structure ...................................................................... 29
3.2 Consumer Demand Formulation................................................................... 31
3.3 Manufacturer’s Profit Function..................................................................... 37
3.4 Consumer Surplus......................................................................................... 46
4. Conclusion And Managerial Implications ........................................................ 48
5. Future Research .................................................................................................. 54
Chapter 3: Essay 3 - Modeling Loyalty For Better Customer Relationship
Management................................................................................................................ 55
Summary................................................................................................................. 55
1. Introduction......................................................................................................... 56
2. CRM Models....................................................................................................... 59
3. Model Development............................................................................................ 63
3.1 Modeling the Customer’s Probability of Purchase ....................................... 67
3.2 Modeling Loyalty Transitions....................................................................... 72
4. Data and Estimation Procedure........................................................................... 74
5. Results and Discussion ....................................................................................... 77
iv
5.1 The Loyalty Equation ................................................................................... 78
5.2 The Repurchase Equation ............................................................................. 81
5.3 Model Validation .......................................................................................... 83
6. Conclusion .......................................................................................................... 86
Appendices.................................................................................................................. 90
Appendix 1: Solving for price and market share in logit demand models.............. 90
References................................................................................................................... 92
v
List of Tables
Chapter 3: Essay 3 - Modeling Loyalty For Better Customer Relationship
Management
Table 1a: Descriptive Statistics – Mean and Standard Deviation 77
Table 1b: Descriptive Statistics – Frequency 78
Table 2: Model Parameter Estimates 82
Table 3: Model Comparison Table – Comparing the Predictive
Ability of the State Space Model with the Hidden
Markov Model and the Loyalty Model 86
vi
List of Figures
Chapter 1: Essay 1 - The LambertW Transformation As An Approach To Solving
Share Equations In Logit Models
Figure 1 – The LambertW Function 7
Figure 2 – Relationship between market share of firm ‘i’ and price of
firm ‘k’ 12
Figure 3 – Relationship between the price of firm ‘i’ and price of
firm ‘k’ 13
Figure 4 – The decrease in P
1
* with increase in P
2
18
Chapter 2: Essay 2 - Pricing Related Products In A Competitive Environment – The
Role Of Contingent Valuations Between Products
Figure 5 – The Market Structure 30
Figure 6 – Variation of PA
1
with respect to ?
1
44
Figure 7 – Variation of PA
1
with respect to ?
2
45
Figure 8 – Variation of PA
1
with respect to ?
1
and
?
2
46
Figure 9 – The Consumer Surplus 47
Chapter 3: Essay 3 - Modeling Loyalty For Better Customer Relationship
Management
Figure 10 – The standard State Space Model 67
Figure 11 – Population-wide distribution of the individual means
of the loyalty function 77
1
Overview
My dissertation consists of three essays. All three essays detail and develop
new methodologies that can be used by researchers to better understand the market.
In ‘Essay 1’ and ‘Essay 2’ I develop a new methodology to solve the issue of
endogeneity of price and market share in logit models. Endogeneity in logit models
prevent meaningful interpretation of marketing dynamics when data for the particular
phenomenon or situation is absent. Hence in Essay 1 we solve this problem by
developing a procedure that explicitly separates the price and demand variables by
utilizing a special class of functions called the LambertW. The LambertW function
helps solve several complex equations that involve either exponential or logarithmic
functions. Using this procedure we analytically solve the endogeneity problem
between price and demand in logit models. We do so without the need for
instrumental variables that would otherwise have been used. We also show the ease
with which these functions can be adapted to different simulation and estimation
procedures. Finally this procedure is validated by employing it to explicitly derive the
results obtained by the Basuroy and Ngyuen (1998).
Essay 2 adds to the dissertation by applying the procedure developed in Essay
1 to study a new problem. We study the market for implicit bundles. Implicit bundles
refer to the group of products that are sold separately (also referred to as pure
components) but could be perceived by the consumer as products that would provide
an enhanced level of satisfaction if purchased together. In many cases, the implicit
bundles are sold by the same firm, in the form of products sold across different
product lines. The aim of this study is to address the issue of pricing these implicit
2
bundles. We do so by developing a model that would account for the possible
relationships, also known as contingent valuations, of the products across the
different product lines. The degree of contingency determines the strength of the
relationship between the two products. If the contingency level is zero then the
products are independent. As the level increases the relationship between the products
increases, thereby increasing the exclusivity between the products. We show that
accounting for these relationships would decrease the amount of price competition in
the market and also allow the firm to enjoy higher profits, because it would be able to
charge the consumer for the additional surplus gained by purchasing both products
from the same Firm. We also account for the endogeneity of price and market share
when deriving the optimal solutions.
The analysis carried out in this essay helps us establish the following results:
1. The prices of the products of a firm (X) increase with increase in the
contingent valuation of the products manufactured by the same firm (X).
However if the contingent valuation is increased beyond a particular value,
then the prices of firm (X) decrease with increase in the relationship. Hence
the relationship between prices and the contingent valuation is initially
monotonically increasing and then monotonically decreasing.
2. The prices of the products of the firm (X) increase with increase in the
contingent valuation of the products manufactured by a competing firm (Y).
3
3. Firm X experiences a higher profit for the same market share, when its
products are priced considering the contingent valuations than when they are
priced independently of each other.
Consumers who purchase the products would have a lower level of consumer
surplus if the contingent valuation is recognized by a firm.
CRM (Customer Relationship Management) interventions like direct mailings
have long been used by firms to improve customer relationships. In the third essay we
develop a method that will allow the firm to understand the effect of these
interventions for customer loyalty. Loyalty is assumed to be unobserved and hence is
modeled as a latent variable. We use a generalization of the ‘Hidden Markov Model’
(HMM) called the ‘State Space Model’ (SSM) to better predict a customer’s loyalty
function towards a particular firm or product. The SSM models are structurally
different from HMM models and they offer three main advantages over HMM
models. First they are continuous and are described across all possible relationship
states of the customer, hence we avoid the problem of explicitly choosing the number
of states; second, they can be used to model an infinite number of relationship states;
and third they are better at modeling recursive behavior, which is necessary when
modeling customer behavior that involves the effect of experience. We also predict
the customer’s probability of purchase given certain marketing actions and the
predicted loyalty state using a hazard model. We combine the hazard model and the
SSM to predict the customer’s probability of purchase at a given loyalty state. We
apply this model to data from a retailer of health and beauty aids, in order to help
them better understand the effect of their CRM interventions on the customer’s
4
loyalty towards the firm and their repurchase intentions. We also point out the types
of CRM interventions that play a role in improving the loyalty of the customer to the
firm and those interventions that have no effect. This information can hence help the
firm better organize its menu of CRM interventions. We can also compute the
probability distributions across the loyalty states for each individual customer, thus
providing the researcher with knowledge of each customer’s loyalty state. Finally we
introduce a new methodology to the literature on modeling relationships in marketing.
The methodology improves upon existing methods by allowing for a more flexible
and efficient estimation procedure. We also compare our model’s predictions to those
derived from two other estimation methods. We find that the predictions derived from
our estimation procedure are better than those computed from the other methods
discussed.
5
Chapter 1: Essay 1 - The LambertW Transformation As An
Approach To Solving Share Equations In Logit Models
Summary
Logit models allow the expression of individual demand and supply
equations. However, closed-form solutions for equilibrium shares and prices are
highly nonlinear and cannot readily be derived. This hinders the employment of logit
models in theoretical studies, and also makes it difficult to develop reduced-form
expressions for share and price as a function of exogenous variables for use in
empirical studies. In this paper we propose that a recently rediscovered mathematical
function called the ‘LambertW’ be employed in solving logit models for equilibrium
shares and prices. We demonstrate this methodology on the problem studied by
Basuroy and Nguyen (1998).
1. Introduction
Discrete choice models have been extensively used in both the marketing and
economics literature to study various aspects of consumer behavior using data on
market share, price and other variables that affect demand. Logit models are widely
used in the empirical literature (Abramson, Andrews, Currim and Jones 2000;
Kamakura and Russell 1989; Guadagni and Little 1983; McFadden 1978). However,
these models have seldom have been used to derive theoretical relationships between
6
variables of interest. Some exceptions include Basuroy and Nguyen (1998), Carpenter
and Lehmann (1985), Lillien and Kotler (1983) and Lillien and Ruzdic (1982). The
complexity of the resultant expressions is often attributed to be the cause of this non-
usage (Gruca, Kumar and Sudharshan 1992; Gruca and Sudharsdhan 1991; Karnani
1985).
In this paper, we propose that a recently rediscovered mathematical function
(first studied by Euler 1779), termed ‘LambertW’, be employed in obtaining
equilibrium solutions for share and price in logit models. Traditionally, the
LambertW function has been used to solve several exponential equations (Corless, et
al. 1996). The rest of the paper is organized as follows— in §2, we provide a brief
overview of the ‘LambertW’ function. We develop the standard logit model in §3,
and in §4 we present the solution and simplify the model for estimation purposes. We
provide a theoretical application of this methodology in §5. We provide our final
conclusions in §6.
2. A Brief Note on the ‘LambertW’ Function
LambertW is the inverse function associated with the equation,
x
W
We = (1).
The LambertW function belongs to the family of exponential and logarithmic
functions. The function given in (1) resembles the exponential function and the
inverse of this function resembles the logarithmic function. Hence, the shape of the
LambertW function closely follows that of shape of the exponential function and the
logarithmic function. The LambertW function differs from the exponential to the left
of point x 0 = . The exponential is always positive, however the LambertW dips to a
7
minimum of -1 at
1
x
e
= ? . Similarly, the LambertW function differs from the
logarithmic function for values of x ? 0 as while the logarithmic function is not
defined for these values of x, the LambertW function continues to have a value
till
1
x
e
= ? . A special case of the LambertW is the case when x lies in the range
between
1
e
? and 0. In this case, W(x) has not just a single valued function but has two
values. In the case of LambertW, a single valued function W
0
(x) is defined for values
of
1
x
e
= ? and W(x) ? -1. W
0
(x) is also referred to as the principal branch of the
LambertW function. The other branch satisfying W(x) ? -1 is denoted by W
-1
(x). The
shape of the LambertW function is shown in Figure 1.
Figure 1: The LambertW function
LambertW(x)
8
At this point, let us examine and question the importance of the LambertW
function. Similar to the exponential, logarithmic and square root functions, the
LambertW is helpful in solving a series of previously unsolvable equations (readers
may refer Corless, et al. 1996 for an illustration). The LambertW function has already
seen widespread application in the fields of physics (Warburton and Wang 2004;
Valluri, Jeffrey and Corless 2000) and applied mathematics (Corless, Jeffrey and
Knuth 1997, Jeffrey, Hare and Corless 1996, Jeffrey, et al. 1995). Hence, there has
been a movement to include the LambertW function in the core set of elementary
functions that are used to solve equations (Hayes 2005, FOCUS 2000). Many
equations involving exponentials can be solved using the LambertW function.
In this paper, our primary interest is to employ the LambertW function to
solve a previously analytically unsolvable simultaneous equations problem. Before
we proceed with our study, we will briefly outline a few important properties of the
LambertW function. For a more detailed exposition on the properties of the
LambertW function, the reader is advised to refer Corless, et al. (1996).
The LambertW function has the series expansion,
n
n
n
x
n
n
x W
?
?
=
?
?
=
1
1
0
!
) (
) ( , (2)
where n! is a factorial. This series however oscillates for values of x ? 0.4 and hence
cannot always be used for practical numerical computation. However, there exists an
asymptotic formulation, that is convergent for all x, which yields reasonably accurate
real values. This is given by:
9
(
(
¸
(
¸
|
|
¹
|
\
|
+
+ ? + ?
+
+ ? + ?
+
+ ?
+
+ ?
+ + ? =
6
1
2
5
1
4
2
3
2
2
2 2 2
4
1
3
2
2
2 2 2
3
1
2
2 2 2
2
1
2 2
1
2
2 1
60
) 12 125 350 300 60 (
12
) 3 22 36 12 (
6
) 9 6 (
2
) 2 (
) (
L
L
O
L
L L L L L
L
L L L L
L
L L L
L
L L
L
L
L L x W
, (3a)
where L
1
= ln(x) and L
2
= ln(ln(x)).
A further simplification of this formulation, namely,
04 . 0 ) 1 ln( )] 1 ln( 0195 . 0 1 [ 665 . 0 ) ( + + + + ? x x x W , (3b)
provides an accurate approximation to the LambertW function and is used primarily
for numerical estimation purposes.
The derivative of LambertW is,
)] ( 1 [
) (
)] ( exp[ )] ( 1 [
1
) (
'
x W x
x W
x W x W
x W
+
=
+
= and (4)
the anti-derivative of W(x) is,
?
+ + ? = C
x W
x W x dx x W ]
) (
1
1 ) ( [ ) ( . (5)
Also W(0)=0 and W’(0) =1.
Finally the logarithm of W(x) is given by,
log(W(x)) = log(x)-W(x) (6)
3. The Logit Model
3.1 Consumer’s Demand Function
We consider the simplified scenario with two firms, namely firm i and firm k.
Each of these firms sells a product. The products sold by these firms compete for
market share. Let product i denote the product manufactured by firm i and product k
denote the product manufactured by firm k. We conduct our analysis with respect to
firm i. The scenario may be extended to include multiple firms.
10
As the aim of this paper is to develop techniques to analytically solve discrete choice
problems, we proceed to define the probability of a consumer choosing a particular
firm’s product, i.e. we model the consumer’s demand for each brand. If a consumer
purchases product i then the utility to the consumer from the purchase is:
ji i ji i i i ji
U P u ? ? ? ? + = + ? =
0
, (7)
where u
ji
is the utility obtained by consumer j from product i, ?
0i
is the brand or
product specific parameter, ?
i
is the price response parameter for product i, P
i
is the
price of i, U
i
is the deterministic component of utility that is assumed to be constant
across consumers, and
ji
? is the random error term.
We wish to compute the share of i relative to the competing brand k and an
outside good that represents non-purchase in the focal category. The utility of the
outside good is normalized to 0; i.e. we assume U
0
= 0. Given that error terms for i, k
and 0 follow an iid type 1 extreme value distribution, the choice share for i
with
respect to the outside product option, and with respect to the competitor k’s product
is,
k i
i
U U
U
i
e e
e
S
+ +
=
1
, (8)
where S
i
is the probability of choosing product i. The elasticities of demand for firm i
with respect to the prices charged by firm i (own price) and firm k (competitor’s
price) are:
( )
(
¸
(
¸
? ?
=
(
(
(
(
¸
(
¸
?
?
?
?
k i k
i i i
k
i
i
i
S S
S S
P
S
P
S
?
? 1
. (9)
11
3.2 Firm’s Profit Function
The firms in the market are price setters. Let C
i
be the marginal cost incurred
by the firm to provide product i. Hence, the firm’s profit from selling product i is:
( )
i i i i
S C P ? = ? , (10)
where
i
? is the profit firm i earns from selling the product at price P
i
. Assuming the
existence of a pure strategy interior equilibrium, the price vector satisfies the first
order conditions. Hence differentiating (10) with respect to P
i
, and setting the result
as equivalent to zero, we derive the optimal price that the firm must charge so as to
maximize profit as:
( )
( )
i i
i i i
i
S
S C
P
?
? +
=
1
1 1
*
?
?
, (11)
where
*
i
P is the optimal price charged by the firm. This optimal price is a function of
share, which, in turn, is a function of price.
4. The Analytical Solution
4.1 Solving the Firm Level Endogeneity Problem
In equilibrium, equations (8) and (11) must hold simultaneously. The solution
to these simultaneous equations is highly nonlinear, and does not readily admit to a
closed form solution. However, a closed-form solution can be provided in terms of
the LambertW transformation. As demonstrated in the appendix, the following
equations for price and share can be derived:
0i i
0k k k
1 C
i i P
*
i
i
e
1 LambertW C
1 e
P
? ?
? ?
?
?
? ?
?
| |
+ +
|
+
\ ¹
= , (12)
12
and
|
|
¹
|
\
|
+
+
|
|
¹
|
\
|
+
=
?
? ?
?
? ?
k k k
i i i
k k k
i i i
P
C
P
C
i
e
e
LambertW
e
e
LambertW
S
? ?
? ?
? ?
? ?
0
0
0
0
1
1
1
1
1
. (13)
This solution allows the price equation to be expressed in a form that allows
the effects of market share and price to be separated. Hence the LambertW
transformation allows us to rewrite the equation for market share and price in a
manner in which they are independent of each other. Initially the market share of firm
i was an integral part of the price equation (see equation 11) and hence we could not
make any predictions about the price that the firm charges. Hence, through the
LambertW transformation we can eliminate the econometric endogeneity problem
that exists between P
i
and S
i
. The solution also allows for reduced-form expressions
for price and share, which allows direct determination of changes in optimal price
resulting from changes are made to the cost, other independent variables, or the
competitor’s price. For example if we assume that the prices of the competitors are
given, then figures (2) and (3) show the variation of the market share of firm i (S
i
) and
the price charged by firm i (P
i
) in relation to the price of the competitor’s price (P
k
)
respectively.
Figure 2: Relationship between market share of firm i and price of firm ‘k’
Market Share
of firm i: S
i
Price of firm k: P
k
13
Figure 2 describes the market share increase for firm ‘i’ when firm ‘k’
increases its prices. Figure 3 shows the amount by which firm i can increase its price
when firm k increases its price, if firm i intends to retain the same level of market
share.
Figure 3: Relationship between the price of firm ‘i’ and price of firm ‘k’
While the above provides separate solutions for prices and shares of firm i by
eliminating the econometric endogeneity between them, the price charged by firm i is
still dependent on firm k’s price (see equation 12). Hence, the endogeneity arising
from the fact that the prices of the two firms need to be jointly determined still
remains, i.e. the problem of determining equilibrium market prices remains. Since P
i
and P
k
are symmetric, the price that firm k charges can be derived in the same way as
the price of firm i, and can be expressed as:
Price of
firm i: P
i
Price of firm k: P
k
14
0k k k
0i i i
1 C
k k P
*
k
k
e
1 LambertW C
1 e
P
? ?
? ?
?
?
? ?
?
| |
+ +
|
+
\ ¹
= (14)
While equations 12 and 14 do not appear to have a readily expressible closed-
form solution, Nash equilibrium prices can be determined as the point of intersection
of plots of P
i
given P
k
and of P
k
given P
i
.
4.2 Estimation Procedures
Equation (3b) provides a formulation that accurately approximates the
LambertW function for all x. Using the formulation in (3b) we can rewrite equation
(13) in the following way,
0 0
0 0
( 1 ) ( 1 )
* 2
1 1 1 ( ) ( )
(1.04) (.665) ln( 1) (.0129)[ln( 1)]
1 1
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ?
= + + + + +
+ +
i i i i i i
k k k k k k
C C
i i P P
e e
P C
e e
,
(15)
where ?
1
= 1/?
1
. As we are now interested in estimating equation (15), assume an
unobservable term for the price equation denoted by ?
i
, and that the price is linear in
the unobservable term ?
i
. Hence equation (15) can be rewritten as:
0 0
0 0
( 1 ) ( 1 )
* 2
1 1 1 ( ) ( )
(1.04) (.665) ln( 1) (.0129)[ln( 1)]
1 1
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
= + + + + + +
+ +
i i i i i i
k k k k k k
C C
i i i P P
e e
P C
e e
(16)
Equation (16) is now independent of the effect of the firm’s own market share
and hence can be estimated without the use of firm level instrumental variables. The
estimation procedure would follow a standard nonlinear regression where the
15
parameters to be estimated would include
1
? and all the ?’s. It is important to note
that if the market prices are set endogenously, the researcher would estimate (16) and
a similar equation for the prices of firm k simultaneously, using suitable market level
instrumental variables.
5. A More General Application Incorporating Marketing Expenditure
We apply this methodology to the multinomial logit problem studied by
Basuroy and Nguyen (1998). The authors discuss the appropriateness of Multinomial
Logit (MNL) market share models for equilibrium analysis. Their results show that a
linear price response in conjunction with the typical concavity assumed in a large
range of marketing response functions would yield an interior equilibrium solution.
The authors then consider the optimal pricing and marketing expenditure reactions to
entry and potential market expansion. In the context of the MNL models, they
demonstrate that the entry of a new brand evokes a defensive reaction through a
decrease in the equilibrium prices of the existing brands. They also note that while
new entry into a fixed market triggers the incumbents to lower marketing
expenditure, when faced with market expansion, firms tend to raise marketing
activities. Consequently, there exist distinct possibilities that marketing efforts for the
existing brands increase in view of entry in an expanding market.
In this section we will incorporate the characteristics outlined by the authors,
namely—the linearity of price response and the concavity of market expenditure
response functions into our MNL market share model. We then derive optimal
16
solutions for price and market expenditure independent of market share by employing
the methodology developed in the earlier part of this paper.
Consistent with Basuroy and Nguyen, let P
1
and m
1
be the price charged and
marketing expenditure of Firm 1 for its product. Let P
2
and m
2
the price charged and
marketing expenditure of Firm 2 for its product. We assume a linear price response
function and a concave marketing response function. The market share function for
Firm 1 is given by,
1
1 1 1 2 1
1 2
1 1 1 2 1 1 2 2 2 2
( 1 )
1
( 1 ) ( 1 )
m
m m
P e
P e P e
e
S
e e
? ?
? ? ? ?
?
? ?
+ ?
+ ? + ?
=
+
, (17)
where S
1
is the market share for Firm 1, ?
11
is the price response parameter that is
assumed to be negative and ?
21
is the market response parameter that is assumed to be
positive. We assume that Firm 2 has a market share function that is analogous to Firm
1. Given the market share function for Firm 1, the profit function for Firm 1 becomes,
1 1 1 1 1 1
( ) P c N S m FC ? = ? ? ? ? ? , (18)
where ?
1
is the profit that Firm 1 gets from selling its product, c
1
is the marginal cost
incurred by Firm 1 and FC
1
is the firm’s fixed cost and N is the market size.
Assuming that the firm is a price setter and assuming the existence of a pure
strategy interior equilibrium, the price vector satisfies the first order conditions. We
can obtain the optimal price that Firm 1 should charge for its product by solving the
first order conditions for P
1
. The first derivative with respect to P
1
is given by,
1
.
1 1 1 1 1 1
( ) (1 )
P
S N P c N S S ? ? = ? ? ? ? ? ? (19)
Similarly, the first order conditions for marketing expenditure m
1
can be obtained by
differentiating (2) with respect to m
1
. This is given by:
17
1
1
.
1 1 11 1 1
( ) (1 ) 1
m
m P c N e S S ? ?
?
= ? ? ? ? ? ? ? (20)
We can now use the LambertW functional form to solve equations (17), (19)
and (20) and obtain the optimal solutions for price P
1
and marketing expenditure m
1
,
independent of the firm’s market share. We solve the equations simultaneously and
obtain the following solutions for the optimal price and optimal marketing
expenditure:
11
21 11 1
22
12 2 22
2
1
1
*
1
11
1
1
m
c
N
P
e
e
LambertW
e
P c
?
? ?
?
? ?
?
| |
+ + ?
|
\ ¹
| |
+ ?
|
\ ¹
(
(
? ?
(
(
+
¸ ¸
= + , (21)
where
1
*
P is the optimal price charged by Firm 1 and,
1
* 11
21
ln m
N
?
?
| | ?
= ?
|
\ ¹
, (22)
where
1
*
m is the optimal marketing expenditure. As ?
11
is assumed to have a negative
value, both P
1
and m
1
from equations (21) and (22) will always be positive. Hence,
the methodology developed in this paper allows us to derive closed form solutions to
both the optimal price and the optimal marketing expenditure, where the optimal
price depends on cost and competitor price.
Comparing our results with Basuroy and Nguyen (1998) we find, as shown in
figure 4, that the optimal price charged by the Firm 1 would decrease upon the
introduction of a new product in the market. This is consistent with the results
obtained by the authors.
18
Figure 4: The decrease in P
1
* with increase in P
2
It is relatively harder to analyze the marketing expenditure function as the
functions only involve the price response parameter of Firm 1. Thus, we cannot make
predictions about the change in marketing expenditure with respect to the entry of a
competitor. However we can make predictions of the effect of an increasing market
size on the marketing expenditure. Taking the first derivative of marketing
expenditure function given in equation (22) we find that
*
1
1 m
N N
?
=
?
(23)
This implies that the marketing expenditure function shares a positive
relationship with the market size. Hence as market size increases, the marketing
expenditure of Firm 1 must also increase, albeit at a decreasing rate. This result is
also consistent with the results obtained by Basuroy and Nguyen (1998).
P
1
*
P
2
19
6. Conclusion
In this paper we analytically solved the problem of econometric endogeneity
in discrete choice models through the use of the LambertW function. The LambertW
function lends itself to analytically solving exponential equations, and thereby
facilitates the derivation of closed-form solutions for price and market share. The
LambertW function also lends itself to easy estimation through a simple yet accurate
approximation as explained in sections 2 and 4.2. This approximation can be used in
estimating expressions for price and share that depend only on competitor actions,
and that therefore do not require firm level instrumental variables. Market level
instrumental variables, which are easier to obtain, suffice. The LambertW function is
potentially useful whenever logit models are employed in theoretical or empirical
work, or more generally, whenever solutions to exponential functions are required.
20
Chapter 2: Essay 2 - Pricing Related Products In A Competitive
Environment – The Role Of Contingent Valuations Between
Products
Summary
This paper extends the research by Venkatesh and Kamakura (2003) to
consider the case of pure component pricing in a competitive setting. We develop an
optimal pricing scheme for a Firm that has two products, when its products are sold as
pure components. We show that the optimal price increases monotonically with
respect to the contingent valuation of both the Firm’s and its competitor’s products.
We also derive the consumer surplus when the products are priced independently and
when they are priced by taking into account the contingent valuation. We conclude
that consumer surplus is higher in the former case.
1. Introduction
Implicit bundles refer to the group of products that are sold separately (also
referred to as pure components) but could be perceived by the consumer as products
that would provide an enhanced level of satisfaction if purchased together. In many
cases, the implicit bundles are sold by the same Firms. An example would be
Microsoft selling both its Windows Operating System and Office Suite. It has been
shown that even though Microsoft sells these products separately, they still price
them as if they were selling them as a bundle, where Windows is priced lower than
21
Office, even though Windows costs almost twice as much to develop (Economides
and Viard 2003). Most Firms market products in many different product lines, so it is
very important for the Firms to develop optimal pricing strategies for their products in
the different product lines. Many issues go into establishing optimal pricing schemes.
The study of the effect of inter-category relationships between the products made by
the same manufacturer on the pricing scheme is very important, because this could
affect the consumption behavior of the consumer. This topic has been studied in some
detail in the retail setting (Manchanda et al. 1999). However, with the exception of a
handful of studies (Reibstein and Gatignon 1984; Urban 1969), the literature has
generally ignored how manufacturers must treat this problem when pricing their
products across the different product lines (Elrod et al. 2002).
The aim of this paper is to address the issue of pricing across different product
lines. We do so by developing a model that would account for the possible
relationships of the products across the different product lines. We hope to show that
accounting for these relationships would decrease the amount of price competition in
the market and also allow the Firm to enjoy higher profits, because it would be able to
charge the consumer for the additional surplus gained by purchasing both products
from the same Firm. We also account for the endogeneity of price and market share
when deriving the optimal price.
The rest of the paper is organized as follows. We review the literature in §2.
Model Development is carried out in §3. The conclusions and managerial
implications are presented in §4. Directions for future research are provided in §5.
22
2. Literature Review
2.1 Product Relationships
Traditionally inter-category relationships between products have been studied
in the form of substitutes or complements. However these relationships are far more
complex than just substitutes or complements. Product relationships can no longer be
viewed upon as a simple dichotomy (either complements or substitutes) but these
relationships should be viewed in terms of degree of complementarity and
substitutability, not absolute substitutability or complementarity.
Venkatesh and Kamakura (2003) explain these relationships in the form of
contingent valuations. Many recent studies have indicated that buyers evaluate the
components of a bundle by assigning certain valuations to the strength of the
relationships in the bundle. (Jedidi et al. 2003, Yadav 1994). Contingent valuations
measure this relationship between the components in the bundle. Contingent
valuations may be defined as the degree of complementarity (or substitutability)
between two products as perceived by the consumer, if the products were to be sold
as a bundle (Venkatesh and Kamakura 2003). Contingent valuations can be either
positive or negative. A positive relationship implies that the two products share a
complementary relationship and a negative relationship implies substitutability. In
this study we focus only on the positive values of the contingent valuations, i.e., we
confine our analysis to only complementary products as we are interested in
understanding how a firm can take advantage of a complementary relationship in
23
pricing its products. More recently, Wang, Venkatesh and Chatterjee (2006) develop
a new methodology to reformulate the the way reservation prices can be calculated.
Their procedure named ICERANGE draws on literature on buyers’ uncertainty in
preference and product knowledge. Their results demonstrate that the ICERANGE
method significantly outperforms previous models in terms of predictive validity.
For the purpose of this study we develop a pricing policy for two products that
can be perceived by the consumer as having a relationship with each other. We then
compare this pricing scheme to another pricing scheme where the products are priced
independently of each other. In this study, we only consider implicit bundles, as we
assume that it is not convenient for the manufacturer to explicitly bundle its products.
The main difference between an explicit and implicit bundle would be that products
in explicit bundles are sold together, while products in implicit bundles are sold
separately. An example of an explicit bundle is the Microsoft Windows and Internet
Explorer bundle. Here the consumer typically attains a higher level of satisfaction
using the two products, but they are not sold separately. An example of an implicit
bundle is the Windows-Office bundle, where the products are sold separately, and the
consumer enjoys a higher level of satisfaction by owning both the products. We will
explain both the economic and legal reasons for this later on in this section.
Implicit bundles can be composed of both complements and substitutes. For
the purpose of this study, we only consider complements. Hicks (1939) was one of
the first researchers to define complementary relationships. He argued that
complementarity could be determined through a modification to the cross price
elasticity test, using observable variables like price and demand. Hicks (1939),
24
Schultz (1938) and Hicks and Allen (1934) suggested that instead of using the sign of
the cross price elasticity term as a measure of complementarity, it would be better to
use a compensated price change to measure complementarity. The compensated price
change term considers the effect of not only the change in price, but also the effect of
income. A negative value of the compensated price change implied a complementary
relationship. (Deaton and Muellbauer 1980). Samuelson (1972) also developed a
measure for complementarity called the money metric. He proposed that if we
consider the von Neumann utility function (as a money metric, with diminishing
returns to the marginal utility of income), then two products are complements if
0
[.]
2
>
? ?
?
j i
q q
V
, where V[.] is the von Neumann utility function and q
i
is demand for
product i and q
j
is demand for product j.
In marketing, Guilitinan (1987) provides an excellent overview of how
complementarity can arise between any two products. Two products become
complements in the following situations:
Savings in Search Economies: A motorist prefers going to a dealer who does
both an engine tune up and an oil change as opposed to two dealers separately
offering only one of the services each. The savings in time and effort gained
by the consumer, by going to a single dealer creates a degree of
complementarity between the services offered by the dealer.
Enhance Customer Satisfaction: Guilitinan (1987) offers the example of a ski
lodge that provides both ski lessons and rentals. The combination of services
offered would enhance the satisfaction that a beginner gets from the ski lodge.
25
Improved Total Image: He provides the example of a Firm that offers both
lawn care as well as shrub care services, thereby enhancing the image of the
Firm.
Shocker et al. (2004) also attempt to define complementary products. They
identify three types of complementary relationships, perfect complements (e.g., video
cassette recorder and video cassette), augmenting complements (products that add
new benefits not present in an already existing one, e.g., washing machine and dryer)
and enhancing complements (new products that improves the sales of an already
existing product, e.g., clipart and presentation software).
Inspite of there being a lot of research on bundling of products (Hanson and
Martin 1990, Bakos and Brynjolfsson 1999, Venkatesh and Mahajan 1993, Chung
and Rao 2003), only Venkatesh and Kamakura (2003) have looked at the
relationships between complementary products using contingent valuations. They
derive optimal bundling and pricing strategies taking into account the contingent
valuations of the two products of a monopolist. They compare the derived pricing
schemes to the pricing scheme if these products were priced independently of each
other. They show that the price increases monotonically as a function of the
contingent valuation.
The aim of this study is to extend the paper by Venkatesh and Kamakura
(2003) to incorporate the effects that competition would have on the optimal pricing
strategy. This condition has not been explored in their study. We propose that the
price will depend on both the manufacturer’s and competitor’s contingent valuations.
26
We also propose that the price that accounts for these valuations, will always be
higher than if they were priced independently (when market share is the same),
provided the valuations always have a positive value (in the case of substitutes these
valuations could be negative).
We assume that the firm finds it is disadvantageous to offer an explicit bundle of its
products in the market. This situation can arise under two conditions:
1. Competition: Anderson and Leruth (1993), show that in a duopoly
environment, only pure components pricing may be offered in
equilibrium since Firms fear the extra degree of competition inherent
in offering the option of a bundle (in the case of complements). Our
study differs from Anderson and Leruth (1993) in that they do not
consider contingent valuations or try to formally develop a pricing
scheme. Our assumption is further supported by Matutes and Regibeau
(1992) who also show that in a competitive setting the pure
components strategy is dominant.
2. Conditions for Legality: Stremersch and Tellis (2002) raise the issue
of legality of introducing product bundles in a competitive market.
U.S. Law has two rules which determine the legality of any bundle, the
‘per se rule’ and ‘rule of reason’. The per se rule says that bundling is
illegal when it involves pure bundling of separate products by a Firm
with market power and when a substantial amount of commerce is at
stake. The rule of reason says that bundling is illegal when it involves
pure bundling of separate products by a Firm with market power,
27
involving a substantial amount of commerce, which poses a threat that
the bundling Firm will acquire additional market power over at least
one of the products that is bundled with the tying products and no
plausible consumer benefits offset the potential damage to
competition. Details of the above rules are provided in the Stremersch
and Tellis (2002) paper.
Hence, under such conditions, as explicit bundling is either disadvantageous
or restricted due to legal reasons, it is important for a Firm that manufactures two
different yet related complementary products to develop an optimal pricing scheme
that accounts for the relationship of the two products. We restrict the analysis in this
paper to this scenario.
2.2 Complementary Pricing
Liao et al. (2002a, 2002b) study the effect of pure component pricing on the
equilibrium of the market. They show that if Firms are restricted to pure component
pricing scenario, then the market attains a stable equilibrium only when the two
products made by the Firm are incompatible with the products made by the
competing Firms. However, if the products are compatible, then the market attains a
stable pure strategy equilibrium only if the bundle pricing of the complementary
products is allowed.
In the marketing literature, Reibstein and Gatignon (1984) develop a model for
optimally pricing related products in a product line by extending the mathematical
28
model developed by Urban (1969). They develop a model that includes cross-
elasticities of the various products made by the manufacturer, and use the method of
seemingly unrelated regression to estimate the model.
Complementarity for the most part has been largely ignored by the marketing
literature (Elrod et al. 2002), however there do exist some models of pricing
summarized by Tellis (1986), which are presented below:
1. Captive Pricing: Captive pricing occurs when the manufacturer
charges a low price for the base good and then charges a higher
monopoly price for the accessory product (the base good is assumed to
work only with the accessory). This is successful because the
consumers might not view the basic product they purchased as a sunk
cost and hence may try and recover their ‘money’ by purchasing the
accessories and using it (‘Sunk Cost Effect’, Thaler 1980, 1985). The
consumer may also use this product more than expected and hence buy
more of the accessory. This has led researchers to label this scheme as
the ‘captive pricing scheme’.
The main restraint on captive pricing is that, if the manufacturer
charges a high price for the accessory, it could lead to the entry of
competing Firms in the market for the accessory, thereby reducing the
Firm’s net profit.
2. Two-Part Pricing: The price here is broken into a fixed fee and a usage
fee. Two part pricing is a type of captive pricing scheme that is
adapted to services (Oi 1972). The usage fee helps Firms exploit the
29
heterogeneity in demand for the products. Hence the heavy user would
pay more to use the service than the light user. Oi (1972) illustrates
this by applying this model to the case of Disneyland where the
customer would pay an entry fee to enter the park and then a usage fee,
i.e., spend on rides and other items in the park based upon their
demands for these items.
Sales from both captive and two-part pricing are classified as types of tie-in
sales. Apart from the legal difficulties mentioned earlier, tie-in sales suffer from some
other legal difficulties also. For example the procedure of tying the buyer to purchase
the supplies from the same Firm that manufactured the base product may be
considered to be illegal under the Sherman Act of 1890 or the Clayton act of 1914
(Burstein 1960, Mathewson and Winer 1997). This paper thus aims to develop a
pricing scheme, when tie-in sales are not an option, by considering the contingent
valuations of the products.
3. Model Formulation
3.1 Defining the Market Structure
First we describe the market structure. We consider two multi-product Firms
(i = 1, 2). Each Firm has two products, one in category A and one in category B. The
Firms compete with each other for market share in both markets A and B. Firm i
30
manufactures products A
i
and B
i
and competes with the products made by the other
Firm. This structure is illustrated in figure 5.
Figure 5: The Market Structure
Both products A
i
and B
i
share a complementary relationship which is
parameterized by ?
i
. ?
i
measures the degree of complementarity between the two
products within the Firm i. Thus ?
i
is the contingent valuation of relationship between
the two products.
The notion of contingent valuations arises when a consumer perceives the two
products as if they were a bundle, complementing each other in some way. This
perception of complementarity could be because the products belong to categories
that could be consumed together. Schmalansee (1982) indicated that the consumer’s
reservation price for the bundle of A
i
and B
i
would be higher than the sum of the
FIRM 1 FIRM 2
M
A
R
K
E
T
A
1
B
1
A
2
B
2
?
1
?
2
Competing
Products/
Firms
31
stand alone reservation prices for the products when the products are complements.
The contingent valuation ?
i
is defined as:
) (
) (
Bi Ai
Bi Ai Bi Ai
i
RP RP
RP RP RP
+
+ ?
=
+
? …………..(1)
where:
RP
Ai+Bi
= Consumer’s reservation price for the perceived bundle of products A
i
and B
i
RP
Ai
= Consumer’s reservation price for the product A
i
RP
Bi
= Consumer’s reservation price for the product B
i
?
i
lies in the range (0,1). When ?
i
= 0, the products do not share a relationship
and thus are independent of each other. When ?
i
= 1, the products are perfect
complements and hence cannot be used independently. An example of this case
would be a video cassette player and a video cassette. When 0 < ?
i
< 1, the products
can be used independently, however the consumer obtains a higher utility buying both
the products (e.g., MS Windows and MS Office). When ?
i
< 0, the products are
substitutes. We however do not consider this case for the purpose of this study.
When ?
i
> 0, it is assumed that the consumer will prefer to buy both products
from Firm i.
3.2 Consumer Demand Formulation
3.2.1 Contingent Valuation Condition
The next step in the analysis would be to model the consumer’s demand for
the two products. For convenience we perform our entire analysis with respect to
32
Firm 1, i.e., we provide optimal pricing schemes for Firm 1 when it competes with
Firm 2.
Therefore we must first define the utility that each consumer would derive
from the each product marketed by Firm 1.
If a consumer buys only product A
1
, then the utility that the consumer derives
owning A
1
is:
……………(2a)
where
u
j,A1,t
= utility obtained by consumer j from product A
1
at time t
?
0
= brand / product - specific parameter
?
1
= price response parameter for product A
1
PA
1
= price of A
1
?
A1,t
= unobserved component of utility derived from product A
1
?
j,A1,t
= the random error term, which follows an iid type 1 extreme value distribution
Similarly the utility for product B
1
in period t for consumer j is given by:
……………(2b)
where
u
j,B1,t
= utility obtained by consumer j from product B
1
at time t
?
0
= brand / product specific parameter
?
1
= price response parameter for product B
1
PB
1
= price of B
1
?
B1,t
= unobserved component of utility derived from product B
1
?
j,B1,t
= the random error term, which follows an iid type 1 extreme value distribution
t A j t A t A j
PA u
, , , 1 1 0 , ,
1 1 1
? ? ? ? + + ? =
t B j t B t B j
PB u
, 1 , , 1 1 1 0 , ,
1
? ? ? ? + + ? =
33
We are interested in determining a pricing scheme for when the consumer
might be interested in purchasing both the products. For this we will modify the
Venkatesh and Kamakura (2003) definition slightly to make the value of one item
depend on whether or not the companion item is purchased.
Let V
A1
=
t A
PA
, 1 1 0
1
? ? ? + ? ………….(3a) and
V
B1
=
t B
PB
, 1 1 1 0
? ? ? + ? ……………….(3b),
where V
A
and V
B
are the values that the consumer attaches to the products A and B
respectively.
Therefore the value of A
1
if B
1
is purchased is:
V
A1|B1
= V
A1
+ ?V
B1
……..(4a)
where ? is the contingent value between the two products.
Conversely the value of A
1
if B
1
is not purchased becomes:
V
A1|NB1
= V
A1
……….(4b)
The expected value of product A
1
thus depends on whether or not the companion item
is purchased.
V
A1
*
= P
B1
V
A1|B1
+ (1-P
B1
)V
A1|NB1
= V
A1
+ P
B
?
1
V
B1
…………(5)
where P
B
is the probability that the companion item B
1
will be purchased. This
probability depends on the price of B
1
: PB
1
, as well as the value of B
1
. Hence the
expected value of owning A
1
can be expressed as:
V
A1
*
= V
A1
+ E [f (V
B1
, PB
1
)]…………….(6a),
where ‘E’ is the expectation and ‘f’ represents a function. For simplicity we will
write:
34
E [f (V
B1
, PB
1
)] = ?
0
+ ?
1
PB
1
+ ?
B1
…………..(6b)
Thus V
A1
*
= V
A1
+ ?
0
+ ?
1
PB
1
+ ?
B1
+ ?………….(7)
Thus the consumer’s utility function for A
1
would now depend on whether B
1
is purchased or not. Therefore the consumer’s utility for product A
1
in period t for
consumer j, given that consumer j expects to purchase B
1
also, is given by:
………….(8a)
Similarly utility of B
1
given A
1
is
…………..(8b)
The next step would be to model the net demand for the products A
1
and B
1
.
To model demand we consider the logit demand model. The logit model has been
used extensively in the marketing literature for modeling both household level data
(Guadagni and Little 1983) as well as aggregate market share data (Allenby 1989).
Here we model the aggregate market share using the logit formulation. The utility can
be rewritten as a function of the deterministic part and the random component:
u
j,A1,t
= U
A1,t
+ ?
j,A1,t
…………(9a)
u
j,B1,t
= U
B1,t
+ ?
j,B1,t
…………(9b)
U
A1,t
and U
B1,t
represents the deterministic part, which is the aggregate utility
obtained from A
1
and B
1
. ?
j,A1,t
and ?
j,B1,t
represent the heterogeneity in consumer
preferences for the two products, and they are assumed to be distributed iid type-1
extreme value.
t A j t A t B t A j
PB PA u
, 1 , , 1 , 1 1 1 0 1 1 1 0 , 1 ,
) ( ? ? ? ? ? ? ? ? + + + + + + =
t B j t B t A t B j
PA PB u
, 1 , , 1 , 1 1 1 0 1 1 1 0 , 1 ,
) ( ? ? ? ? ? ? ? ? + + + + + + =
35
To allow for the market share of the brand to expand and contract over the
different periods with the choice of the marketing mix we allow for non-purchases
(eg: an outside good, or choosing not to purchase), this is denoted as product 0. The
utility of the outside product is normalized to 0 across periods; i.e. we assume U
0,t
=
0.
We now derive the choice shares for products A
1
and B
1
with respect to the
option of the outside product 0 and the competitor’s products A
2
and B
2
respectively.
Thus the choice share for A
1
is:
S
A1,t
=
t A t A
t A
U U
U
e e
e
, 2 , 1
, 1
1 + +
…………..(10)
where S
A1,t
is the probability of the choice of product A
1
. Hence the choice share for
product A
1
is dependent on the utility of A
2
, which is a function of the relationship
between A
2
and B
2
, which is given by ?
2
. Thus the choice share for B
1
is hence
analogous to Equation (10) and S
B1,t
is the probability of the choice of product B
1
.
For the next step in our analysis we derive the elasticities of the market share
of the products A
1
and B
1
with respect to the prices of A
1
, B
1
, A
2
and B
2
.
Therefore the first derivatives of S
A1,t
and S
B1,t
with respect to prices of A
1
and B
1
are:
(
¸
(
¸
? ?
? ?
=
(
(
(
(
¸
(
¸
?
?
?
?
?
?
?
?
) 1 ( ) 1 (
) 1 ( ) 1 (
, 1 , 1 1 , 1 , 1 1 1
, 1 , 1 1 1 , 1 , 1 1
1
, 1
1
, 1
1
, 1
1
, 1
t B t B t A t A
t B t B t A t A
t B t A
t B t A
S S S S
S S S S
PB
S
PB
S
PA
S
PA
S
? ? ?
? ? ?
……………(11a)
The first derivatives of S
A1,t
and S
B1,t
with respect to A
2
and B
2
are:
36
(
¸
(
¸
? ?
? ?
=
(
(
(
(
¸
(
¸
?
?
?
?
?
?
?
?
t B t B t A t A
t B t B t A t A
t B t A
t B t A
S S S S
S S S S
PB
S
PB
S
PA
S
PA
S
, 2 , 1 2 , 2 , 1 2 2
, 2 , 2 2 2 , 2 , 1 2
2
, 1
2
, 1
2
, 1
2
, 1
? ? ?
? ? ?
……………(11b)
3.2.2 No Contingent Valuation Condition
We also derive the case when manufacturers are unaware of the existence of
the contingent valuations of the two products, and price the two products accordingly.
This no contingent valuation condition is labeled ‘nc’. This step is carried out to
facilitate comparison and to elucidate the importance of identifying the various
contingent effects.
Therefore in this case the Firm expects that the consumer’s utility function would be
given by:
……….(12a)
………..(12b)
Hence we again split the model into the deterministic component and random
component as follows:
u
i,A1,t,nc
= V
A1t
+ ?
iA1t,nc
………..(13a)
u
i,B1,t,nc
= V
B1t
+ ?
iB1t,nc
………..(13b)
where:
V
A1,t
= deterministic component of utility for product A
1
V
B1,t
= deterministic component of utility for product B
1
Thus market share of product A
1
at time t is given by:
nc t A j nc t A nc nc nc nc t A j
PA u
, , 1 , , , 1 , 1 , 1 , 0 , , 1 ,
? ? ? ? + + + =
nc t B j nc t B nc nc nc nc t B j
PB u
, , 1 , , , 1 , 1 , 1 , 0 , , 1 ,
? ? ? ? + + + =
37
S
A1,t,nc
=
t A t A
t A
V V
V
e e
e
2 1
1
1 + +
…………(14)
Thus market share of product B
1
at time t is analogous to Equation (14).
For the next step in our analysis we derive the elasticities of the market share of the
products A
1
and B
1
with respect to the prices of A
1
, B
1
, A
2
and B
2
.
Therefore the first derivatives of S
A1,t,nc
and S
B1,t,nc
with respect to A
1
and B
1
are:
(
¸
(
¸
?
?
=
(
(
(
(
¸
(
¸
?
?
?
?
?
?
?
?
) 1 ( 0
0 ) 1 (
, , 1 , , 1 , 1
, , 1 , , 1 , 1
, 1
, , 1
, 1
, , 1
, 1
, , 1
, 1
, , 1
nc t B nc t B nc
nc t A nc t A nc
nc
nc t B
nc
nc t A
nc
nc t B
nc
nc t A
S S
S S
PB
S
PB
S
PA
S
PA
S
?
?
…………(15a)
The first derivatives of S
A1,t,nc
and S
B1,t,nc
with respect to A
2
and B
2
are:
(
¸
(
¸
?
?
=
(
(
(
(
¸
(
¸
?
?
?
?
?
?
?
?
nc t B nc t B nc
nc t A nc t A nc
nc
nc t B
nc
nc t A
nc
nc t B
nc
nc t A
S S
S S
PB
S
PB
S
PA
S
PA
S
, , 2 , , 1 , 2
, , 2 , , 1 , 2
, 2
, , 1
, 2
, , 1
, 2
, , 1
, 2
, , 1
0
0
?
?
……………(15b)
In the above equations 15a and 15b, as we assume the ‘nc’ condition the off-
diagonal terms are zero in contrast to equations 11a and 11b. Specifically there is no
effect of the change of price of A
2
on the market share of B
1
or of the change of price
of B
2
on the market share of A
1
. It is also evident that the change in price of A
1
has to
effect on the market share of B
1
and vice versa.
3.3 Manufacturer’s Profit Function
We now proceed to derive the manufacturer’s profit function for the two
situations, namely, the contingent valuation condition and the no contingent valuation
condition.
38
3.3.1 Pricing with no contingent valuation
We first consider the case when the manufacturer prices not realizing the
presence of a relationship between the two products it manufactures.
The manufacturer profit function ?
t,nc
in the nc condition is given by:
nc t B nc nc t A nc nc t
S N CB PB S N CA PA
, , 1 1 , 1 , , 1 1 , 1 ,
) ( ) ( ? ? + ? ? = ? ……………..(16)
where:
CA
1
: marginal cost of product A
1
CB
1
: marginal cost of product B
1
N: Total number of consumers in the market for the products.
Maximizing the profit equation with respect to PA
1,nc
and PB
1,nc
we get:
PA*
1,nc
=
) 1 (
) 1 ( 1
, , 1 1
, , 1 1 1
nc t A
nc t A
S
S CA
?
? +
?
?
………………(17a)
PB*
1,nc
=
) 1 (
) 1 ( 1
, , 1 1
, , 1 1 1
nc t B
nc t B
S
S CB
?
? +
?
?
………………(17b)
Equations (17a) and (17b) represent the optimal prices charged in the ‘nc’ condition.
3.3.2 Pricing with Contingent Valuations
The next step in our formulation of the optimal prices is to define the
manufacturer’s profit function when contingent valuations are recognized by the
Firm. As we account for the market share of each product in each period, the
manufacturer’s profit function ?
t
can be formulated as:
t B t A t
S N CB PB S N CA PA
, 1 1 1 , 1 1 1
) ( ) ( ? ? ? + ? ? ? = ? …………..(18)
39
Firm 1’s aim is thus to maximize profit with respect to PA
1
and PB
1
in each period.
Therefore we differentiate the profit function with respect to PA
1
and PB
1
, and obtain
the following equations:
N S S CB PB S S CA PA S
PA
t B t B t A t A t A
t
? ? ? ? ? ? + ? ? ? ? + =
?
?
)) 1 ( ) ( ) 1 ( ) ( (
, 1 , 1 1 1 1 1 , 1 , 1 1 1 1 , 1
1
? ? ?
?
..(1
9a)
and
N S S CB PB S S CA PA S
PB
t B t B t A t A t B
t
? ? ? ? ? + ? ? ? ? ? + =
?
?
)) 1 ( ) ( ) 1 ( ) ( (
, 1 , 1 1 1 1 , 1 , 1 1 1 1 1 , 1
1
? ? ?
?
..(19b)
We then solve equations 19a and 19b to get the optimal prices PA
1
and PB
1
for Firm
1’s products A
1
and B
1
. Thus the optimal prices are:
1
, 1 , 1 1 1
, 1 1 , 1 1 1 *
1
) 1 (
CA
S S
S S
PA
t A t A
t A t B
+
?
?
=
? ?
? ? ?
………….(20a)
1
, 1 , 1 1 1
, 1 1 , 1 1 1 *
1
) 1 (
CB
S S
S S
PB
t B t B
t B t A
+
?
?
=
? ?
? ? ?
………….(20b)
This leads us to our first proposition.
40
Proposition 1
The price charged by the manufacturer when contingent valuations are
considered is always greater than the price charged by the manufacturer in the no
contingent valuation condition for all positive values of ?
1
.
Proof
The Equations (20a) and (20b) for the price of the products when contingent
valuations are considered are given below:
1
, 1 , 1 1 1
, 1 1 , 1 1 1 *
1
) 1 (
CA
S S
S S
PA
t A t A
t A t B
+
?
?
=
? ?
? ? ?
………….(20a)
1
, 1 , 1 1 1
, 1 1 , 1 1 1 *
1
) 1 (
CB
S S
S S
PB
t B t B
t B t A
+
?
?
=
? ?
? ? ?
………….(20b)
The equations 17a and 17b when the manufacturer prices in the ‘nc’ condition are
given below:
PA*
1,nc
=
1
, , 1 1
) 1 (
1
CA
S
nc t A
+
? ?
………………(17a)
PB*
1,nc
=
1
, , 1 1
) 1 (
1
CB
S
nc t B
+
? ?
………………(17b)
Comparing the equation (17a) with (20a) we can see that equation (20a) ?
(17a) for all ?
1
? 0, when the market shares are assumed to be equal. Also (20b) ?
(17b) for all ?
1
? 0. We can then rewrite equations (20a) and (20b) as functions of the
price charged in the ‘nc’ condition (17a and 17b) and the premium charged when
contingent valuations are considered to express this result:
41
) 1 (
, 1 , 1 1
, 1 1 *
, 1
*
1
t A t A
t B
nc
S S
S
PA PA
?
+ =
?
?
……….(21a)
) 1 (
, 1 , 1 1
, 1 1 *
, 1
*
1
t B t B
t A
nc
S S
S
PB PB
?
+ =
?
?
……….(21b)
Thus the premium charged when contingent valuations are considered is an
increasing function of the contingent valuations.
Result 1
For a given market share, net profit will always be greater when the
manufacturer prices the products utilizing the symbiotic relationship between them.
Proof
From the above analysis, we can conclude that, when the market shares are
the same, the profit obtained under the ‘nc’ condition will be less than the profit
obtained when the contingent valuations are considered. Thus ?
t
? ?
t,nc
.
One of the main issues of using the logit model in empirical analysis is the
endogeneity of price with market share (Berry 1994). For example, if we consider
product A1, its price PA
1
* and market share S
A1,t
will be correlated. It is thus difficult
to predict whether market share drives the price or vice versa. Berry (1994) suggests
that when estimating the equations with data, the researcher should make use of
appropriate instrumental variables so as to overcome this endogeneity problem.
However, as we are deriving an analytical model for the optimal prices for the Firm,
we follow the procedure employed by Aydin and Ryan (Working Paper). They
simultaneously solve the market share and price equation, to derive the expression for
42
price, which is independent of the market share drivers. Following this approach we
solve for PA
1
* and S
A1,t
, so as to make the price equation of product A
1
dependent on
only the market share of product B
1
and ?
1
. Similarly we solve for PB
1
* and S
B1,t
so
as to make the price equation of B
1
dependent on only the market share of A
1
and ?
1
.
Solving for PA
1
*
and PB
1
*
we get the following expressions:
1
( )
( )
1
1
1
1 , 1 1
1
) 1 (
1 , 1 1
*
1
1
1
) 1 (
PA
2 2 2
1 1 1
1
1 , 1 1
?
?
? ?
?
? ?
? ?
? ?
?
? ?
(
(
(
(
(
¸
(
¸
? +
(
¸
(
¸
?
(
(
(
(
¸
(
¸
+
?
=
|
|
¹
|
\
|
+ ? ?
CA
S
e
e S
LambertW
t B
PA
CA
S
t B
t B
….(22a)
2, 3
( )
( )
1
1
1
1 , 1 1
1
) 1 (
1 , 1 1
*
1
1
1
) 1 (
PB
2 2 2
1 1 1
1
1 , 1 1
?
?
? ?
?
? ?
? ?
? ?
?
? ?
(
(
(
(
(
¸
(
¸
? +
(
¸
(
¸
?
(
(
(
(
¸
(
¸
+
?
=
|
|
¹
|
\
|
+ ? ?
CB
S
e
e S
LambertW
t A
PB
CB
S
t A
t A
……(22b)
2,3
1
Solving 21a and 21b simultaneously eliminates the endogeneity between the firm’s own price and
own market share. However endogeneity between the competing firms’ prices still remains.
2
The Lambert W function is the inverse of the function given by f(x) = xe
x
, where W is the function
that satisfies W(x)e
W(x)
= x for all real values of x. The Lambert W function has a concave shape
(Chapeau-Blondeau, F. and Monir, A 2002; Corless et.al. 1996).
3
Equation (22a) and (22b) are now independent of the effects of the ‘own market share’ of the product.
Hence it is possible to isolate the effects of the contingent valuations on the price without the
possibility of price endogeneity bias due to the relation between price and market share, i.e., changes in
price due to changes in ?
1
will have no effect on the own market share of the product. In Equation
(20a) and (20b) it would not have been possible to isolate the effect of the contingent valuation on
price alone due to the presence of the market share of the product.
43
RESULT 2
Concluding from proposition 1 we know that PA
1
* and PB
1
* will always be
greater than PA*
1,nc
and PB*
1,nc
for all ?
1
> 0. Thus when ?
1
> 0, the premium
charged above the ‘nc’ price will increase monotonically with respect to the
contingent valuation ?
1
, for values between 0? ?
1
?0.45. For values between 0.45 < ?
1
? 1, the premiums decrease monotonically with respect to the contingent valuation ?
1
.
Hence premiums always increase for A
1
are B
1
when 0? ?
1
?0.45 but decrease
slightly when 0.45 < ?
1
? 1.
COMMENT
The variation of PA
1
* with respect to ?
1
is given in Figure 6. It is evident
from the graph that the price of A
1
varies with respect to the contingent valuation.
This result is also applicable to the relationship between the price of B
1
and ?
1
.
For values 0? ?
1
?0.45, valuation of a pair of complements (?
1
> 0) exceed
those for independently valued products (?
1
= 0), the seller thus gains more by
charging higher prices while stimulating the consumers to buy both the products.
Venkatesh and Kamakura (2003) also obtained the same results. However as seen in
our results (Figure 2a), when ?
1
lies between (0.45,1) the optimal price decreases
slightly. This result is different from the results obtained by Venkatesh and Kamakura
(2003). The main reason for this difference is that they did not consider
interdependent utilities and competitive effects.
When ?
1
increases from 0.45 to 1, the Firm’s products become tied to each
other and hence they cannot be used individually. Thus if ?
2
is low, and ?
1
increases
44
the consumers will be locked to Firm 1. When the consumer is locked to a Firm, all
three marketing strategies (pure components, pure bundling and mixed bundling) are
equivalent to pure bundling (Matutes and Regibeau 1992). Hence the price charged
for each individual item would be equivalent to price charged if the products were
sold as a bundle. As the price charged for a bundle is usually lower than when the two
products are sold separately, we would expect a decline in the premium that would be
charged.A very high degree of complementarity between the two products
manufactured by the same Firm implies that the products cannot be used individually
and hence the consumer gets no utility from purchasing just one of the products. This
might explain the reason why we see that the optimal price starts decreasing as ?
1
?
(0.45, 1). However it is important to note that even thought the premium decreases
slightly, it is still higher than if the products were priced in the ‘nc’ condition.
Figure 6: Variation of PA
1
with respect to ?
1
.
RESULT 3
PA
1
* and PB
1
* increase monotonically with respect to the contingent valuation ?
2
.
45
COMMENT
The variation of PA
1
* with respect to ?
2
is given in Figure 7. It is evident
from the graph that the price of A
1
increases monotonically with the contingent
valuation of its competitor’s products. This result is also applicable to the relationship
between the price of B
1
and ?
2
. As ?
2
increases the competitor can afford to increase
its products prices without having any effect on its market share. As the competitor
increases her prices, price competition between the two firms decreases, and hence
Firm 1 can also increase her price slightly.
As ?
2
increases the competitor’s products become increasingly dependent on
each other. Thus if ?
1
is low, Firm 1 can capitalize on the exclusivity of the
competitors products and charge a higher price for its own line of products. This
could be the reason why we do not see a decrease in the premium that Firm 1 charges,
even though the premium charged by Firm 2 decreases when ?
2
lies in the region
(0.45, 1).
Figure 7: Variation of PA
1
with respect to ?
2
.
46
Figure 8 shows the variation of profit for Firm 1 with respect to both ?
1
and
?
2
. We see that profit is maximum for high values of ?
2
and low values of ?
1
. This
result follows directly from result 2 and result 3.
Figure 8: Variation of Profit With Respect to ?
1
and ?
2
3.4 Consumer Surplus
Proposition 2
Consumer surplus is lower when the Firm prices in the contingent valuation condition
than when the Firm prices in the ‘nc’ condition.
Proof:
We derive the consumer surplus for consumer j. Assuming utility has a dollar
value, the consumer surplus, when the Firm recognizes the complementary effects
and charges prices accordingly, is given by,
CS
j,t
= u
j,A1,t
+ u
j,B1,t
– PA
1
* - PB
1
*…………(23a)
When the Firm prices in the ‘nc’ condition, the consumer surplus is,
47
CS
j,t,nc
= u
j,A1,t
+ u
j,B1,t
– PA
1,nc
* - PB
1,nc
*…………(23b)
In equation 23b, the Firm is unaware of the complementary relationships between its
two products, however the consumer enjoys the effects of complementarity between
the two products. Therefore s/he is charged the price derived in the ‘nc’ condition.
Hence (s)he would enjoy a higher utility with a lower price, thereby higher surplus. It
is straightforward to see CS
j,t
? CS
j,t,nc
as PA
1
* > PA
1,nc
* and PB
1
* > PB
1,nc
*. This
can also be observed pictorially in Figure 9. Consumer surplus when the
manufacturer prices the product independently is given by the areas numbered (5) +
(2) + (4). When the products are priced by considering the contingent value, the
consumer surplus is given by the area numbered (5). Thus consumer surplus is higher
when products are priced in the ‘nc’ condition. Thus, it is evident that consumer
surplus decreases when the consumer purchases both products from the same
manufacturer, who prices the products taking into account the contingent valuations.
X Axis: Quantity Purchased
Y Axis: Net Price Paid
A: u
i,A1,t,nc
+ u
i,B1,t,nc
B: u
i,A1,t
+ u
i,B1,t
Figure 9: Consumer Surplus
48
C: Marginal cost of producing A
1
and B
1
(CA
1
+CB
1
)
D: Net Price Paid when the products are charged independently (PA
1,nc
* + PB
1,nc
*)
E: Net Price Paid when the products are charged considering the contingent effects PA
1
* +
PB
1
*
1: Expected Profit at PA
1,nc
* + PB
1,nc
*
2: Expected Profit at PA
1
* + PB
1
*. It is also the lost profit when products are not
priced as if they are related.
1.+ 3.: Net profit achievable at PA
1,nc
* + PB
1,nc
*, given that the entire market is not yet
satisfied.
4: Deadweight loss when the product are increased in price from PA
1,nc
* + PB
1,nc
*
to PA
1
* + PB
1
*.
5: Consumer Surplus after pricing at PA
1
* + PB
1
*
5+2+4: Consumer Surplus when priced at PA
1,nc
* + PB
1,nc
*.
4. Conclusion And Managerial Implications
Main Contribution
This study contributes to the pricing literature by extending the paper by
Venkatesh and Kamakura (2003) study to the case of competition. The study adds to
the literature in the following ways:
1. Venkatesh and Kamakura (2003) show that the optimal price charged
increases linearly and monotonically with respect to contingent valuations
when products are sold as pure components. We show that this is not
necessarily the case, while the optimal price does increase with respect to the
contingent valuation the relationship is not linear or monotonically increasing.
49
The price first increases rapidly for small values of ? and then increases at a
slower rate till ? = 0.45. When 0.45< ? < 1, we see that the optimal price
actually decreases. Thus locking the consumer to the product reduce the
optimal price that the manufacturer can charge and hence would reduce the
optimal profit.
2. We show that the optimal price will also increase with respect to the
competitor’s contingent valuation, albeit at a slower rate.
3. We also derive the surplus that the consumer would enjoy in each of the cases
and show that the Firm can charge a higher price based on the contingent
valuation mainly because the creation of a contingent valuation causes an
increase in the overall surplus that the consumer could enjoy. The firm can
thus take advantage of this increase and charge a higher price to the consumer.
4. We isolate the effect of ? on the price of the product by accounting for the
endogeneity of market share and price. This helps us predict the variation of
price with respect to ?, without the price having any effect on its own market
share. This is significant because we can predict changes in price with respect
to ?, without the effect of market share of the product.
The analysis carried out in the previous sections have helped us establish the
following results:
4. The prices of the products of Firm 1 increase with increase in the contingent
valuation of the products manufactured by Firm 1.
50
5. The prices of the products of Firm 1 increase with increase in the contingent
valuation of the products manufactured by Firm 2.
6. Firm 1 experiences a higher profit for the same market share, when its
products are priced considering the contingent valuations than when they are
priced independently of each other.
7. Consumers who purchase the products would have a lower level of consumer
surplus if the contingent valuation is recognized by the Firm.
Managerial Implications
We have established that the Firm can charge a premium for the products. It is
important for a manager to decide how to split the net premium that can be charged
between the two products. Should the net premium be split equally between the two
products or applied solely to one of the products?
Economides and Viard (2003) answer this question on the basis of the
network externality of the products. They consider two products MS Windows and
MS Office, both made by the same manufacturer. They propose that an optimal
pricing scheme would be to charge high for Office and low for Windows (even
though it costs Microsoft twice as much to develop Windows than it does to develop
Office). They show that the network externality generated through the sale of
Windows would increase the value of Office, and hence the profit lost by Windows
can be recouped by Office. This is however only true of markets with strong network
externality effects.
51
For other markets, we could use the axioms of prospect theory (Kahneman
and Tversky 1979) and mental accounting (Thaler 1980,1985) to understand how to
distribute the premium among the products. For example, when the consumer has to
purchase a base product (e.g., printer) and then make repeated purchases of an
accessory (it is assumed that the accessory is not exclusive) (e.g., ink) to use the base
product, it might be better to charge a higher price for the base product. This is
because the consumer would mostly relegate the price paid for the base product to the
level of a sunk cost (decoupling the costs and benefits, Soman and Gourville 2001),
and look to only the future price of the accessory as the basis for his/her choice of the
accessory product.
Another important example is the razor and blade pricing strategy. This is a form of
captive pricing, since the consumer is locked to the Firm (? = 1) once they purchase
the razor.
Purchase behavior may also be affected by heterogeneity in the willingness to
pay for the base product. For a light user/low end segment it might be better to lower
the cost of the base product, so that the segment enters the market, and increase the
cost of the accessory as they would not use as much of the accessory. This is akin to
the captive pricing scenario talked about earlier. It suffers from an obvious weakness,
the entry of competitors in the accessory market would cause the low (light) end
(user) segment to choose the competitor. In such a situation, for the low end segment,
it is better to split the premium equally between the two products.
52
Implications for Marketing Management
The analysis in this paper provides the following managerial conclusions:
1. Building contingent valuations is important as it helps in limiting the role of
price competition in the market.
2. If the Firm accounts for the existence of contingent valuations when setting
the prices of the different products, the Firm would get a higher profit as it
would get some of the extra surplus that the consumer accrues from owning
the two products.
Therefore it is important for Firms to ensure that its new product lines have a
positive relationship to its existing range of products. To better understand how a
Firm could build contingent valuations into its product line we consider the example
of Apple Computer. In October 2001, Apple introduced its portable MP3 player iPod.
The iPod allowed for ease of portability of each customers MP3 music collection and
was considered a better option to the available MP3 players at that time due to the
fact that it could hold significantly more songs that the average MP3 player and also
because it had a more sophisticated design. Although MP3 music was easy to
download, the practice of downloading music suffered from a multitude of legal
difficulties. This did not permit iPod sales to reach their full potential.
In the beginning of 2003, several new competitors like Dell began entering the
market and selling their own MP3 players at significantly lower prices. Thus Apple
now faced two challenges, one from the legal problems of downloaded MP3 music,
and the second challenge was a price war with the new competitors.
53
As a means to overcome this problem, Apple introduced its iTunes online music
download service. This service allowed consumers to download music both legally
and at a very low price. However Apple ensured exclusivity of the downloaded
music, as it was not in the MP3 format (Kanellos 2004, CNET News). This ensured
that the downloaded music could be played only on the iPod (which was played both
the MP3 and the AAC format, the AAc format was copyrighted by Apple) and not on
competing MP3 players. As the iTunes service allowed for easy and legal downloads,
consumers increasingly turned to iTunes as a source of music. This simultaneously
increased demand for the iPod, as it allowed for portability of the consumer’s new
music collection. Similarly consumers who purchased the iPod found iTunes to be a
convenient source for music for their iPod player. Thus consumers derived a higher
level of satisfaction when they purchased both the iPod and the iTunes service as
opposed to purchasing only one of them. This allowed Apple to charge a higher price
for its iPod player. Apple also escaped a future price competition that would have
ensued with the introduction of new MP3 players by its competitors like Dell
(Dalrymple, Technology Business Research 2003). Hence the introduction of the
iTunes service created a contingent valuation between iTunes and iPod and this
allowed Apple to gain higher market share without sacrificing profits or bundling the
two products together. After introducing iTunes, iPod sales were up 235%
(Hasseldahl, Forbes 2004).
54
5. Future Research
The procedure employed in this paper accounts for and solves the econometric
endogeneity that is present between each firm’s own price and own marketshare.
However, there is an extra level of endogeneity, namely between the focal firm’s
price and the competitor’s price that needs to be considered when deriving
equilibrium prices.
To keep the model tractable we did not consider heterogeneity of the
contingent valuation. In reality, the contingent valuations could vary across
consumers, hence it is important to account for it. However there would be no closed
form solution when accounting for the heterogeneity and hence one must develop
simulation methods when trying to derive optimal pricing schemes accounting for
differences in ?
i
. We could also consider the case when ?
A,B
? ?
A,B
, however we do
not expect the results to be qualitatively different.
55
Chapter 3: Essay 3 - Modeling Loyalty For Better Customer
Relationship Management
Summary
CRM (Customer Relationship Management) interventions like direct mailings
have long been used by firms to improve customer relationships. In this study we
develop a method that will allow the firm to understand the effect of these
interventions on customer loyalty. Loyalty is assumed to be unobserved and hence is
modeled as a latent variable. We use an adaptation of a generalization of the ‘Hidden
Markov Model’ (HMM) called the ‘State Space Model’ (SSM) to better predict a
customer’s loyalty function towards a particular firm or product. The SSM models are
structurally different from HMM models and they offer three main advantages over
HMM models. First they are continuous and are described across all possible
relationship states of the customer, hence we avoid the problem of explicitly choosing
the number of states; second, they can be used to model an infinite number of
relationship states; and third they are better at modeling recursive behavior, which is
necessary when modeling customer behavior that involves the effect of experience.
We adapt the SSM model to our study by combining the generic model with a set of
covariates that we use to better understand customer loyalty. We call this model the
SSMC (State Space Model with Covariates). We also predict the customer’s
probability of purchase given certain marketing actions and the predicted loyalty state
using a hazard model. We combine the hazard model and the SSMC to predict the
56
customer’s probability of purchase at a given loyalty state. We apply this model to
data from a retailer of health and beauty aids, to help them better understand the
effect of their CRM interventions on the customer’s loyalty towards the firm and their
repurchase intentions. We also point out the types of CRM interventions that play a
role in improving the loyalty of the customer to the firm and those interventions that
have no effect. This information can hence help the firm better organize its menu of
CRM interventions. We can also compute the probability distributions across the
loyalty states for each individual customer, thus providing the researcher with
knowledge of each customer’s loyalty state. Finally we introduce a new methodology
to the literature on modeling relationships in marketing. The methodology improves
upon existing methods by allowing for a more flexible and efficient estimation
procedure. We also compare our model’s predictions to those derived from two other
estimation methods. We find that the predictions derived from our estimation
procedure are better than those computed from the other methods discussed.
1. Introduction
Customer relationship management (CRM) is a firm-wide approach to
understanding and influencing customer acquisition, customer retention and customer
value through interactive and relevant information exchange between the firm and the
customer. The past few years have seen a multitude of research in the field of CRM,
exploring several important facets such as selecting the right candidates from a
mailing list and then selecting the appropriate communication or marketing
57
intervention for each of the selected candidates (Elsner, Krafft and Huchzermeier
2004; Venkatesan and Kumar 2004; Rust and Verhoef 2005; Gonul and Shi 1998;
Bitran and Mondschein 1996; Bult and Wansbeck 1995; Banslaben 1992; Roberts
and Berger 1989; Kass 1976; Sonquist 1970), studying the link between satisfaction
and commercial success (Kamakura et al. 2002), the link between customer loyalty
and profitability (Rust, Lemon and Zeithaml 2004; Reinartz and Kumar 2000),
customer profitability heterogeneity (Niraj, Gupta and Narasimhan 2001), customer
loyalty programs (Verhoef 2003) and towards establishing a sound construct of the
CRM process (Reinartz, Krafft and Hoyer 2004). In this essay we develop a
methodology that actually estimates a customer’s loyalty state and allows one to
predict the effect of a customer’s loyalty on their intention to repurchase.
Previous marketing literature defines loyalty as being either psychological or
behavioral in nature. Psychological loyalty considers the underlying motivation of the
consumer to repurchase the same brand. It is based on the attitudes of the customer to
the brand in question (Jacoby and Chestnut 1978). Behavioral loyalty on the other
hand depends on the purchase patterns of the customer. It is hence defined by the
revealed behavior of the customer (Fader and Hardie 1996). Our model derives
estimates of loyalty that indicate a tendency to engage in behavior that results in a
purchase. This is similar to psychological loyalty as described in the literature. We
then predict the probability of repurchase, thereby linking psychological loyalty to the
revealed behavior of the customer, i.e. the customer’s behavioral loyalty. We carry
out the tests in the following way. First we test whether a firm’s CRM interventions
have an effect on the customer’s psychological loyalty towards the firm’s products
58
and estimate the customer’s loyalty states. We then check whether loyalty affects the
probability that a customer will repurchase a product from the firm.
We conduct the above tests using an adaptation of state space models. We
combine the state space model with a set of covariates that are used to predict loyalty.
These models, continuous generalizations of hidden Markov models, allow one to
develop a continuous loyalty function that can help the firm better understand the
customer and also allow the firm to understand the efficacy of their marketing
interventions. State space models supersede hidden Markov models on two counts:
In the hidden Markov model the researcher would have to make assumptions
about the possible number of loyalty states, while in the state space model no
such assumptions have to be made due to the continuous nature of the
function over all possible states
The second advantage lies in the fact that the hidden Markov model assumes
loyalty to be a discrete variable, which is a simplifying approximation. The
state space model allows us to relax this assumption.
Hence using state space models, the firm will learn which CRM
interventions actually have an effect on customer loyalty and hence can better
organize their interventions to take advantage of this new insight. The other
advantage of this methodology lies in the fact that one can even use discrete data to
obtain a continuous loyalty function.
The rest of the paper is organized as follows. We review the existing CRM
models in §2 and point out how our paper improves upon the current literature. Model
59
development is carried out in §3. We describe the data and estimation procedure in §4
and present results in §5. Finally in §6 we provide our conclusions.
2. CRM Models
CRM has become increasingly important in marketing. There are now many
different ways to manage customer relationships in marketing. Direct mail, which
was once synonymous with customer relationship management studies, is now
complemented by a host of other techniques like email marketing and other advanced
procedures for customer targeting etc., which are now increasingly used by retailers
and manufacturers to manage customer relationships. Hence the efficient
management of these customer interventions forms the essence of customer
relationship management. Towards the goal of efficient management, there are many
factors that play an important role, for example the timing of the marketing
intervention, the frequency of purchase, the monetary value of the purchase, the
customer characteristics etc. (Nash 1984). There are many models proposed in
marketing to help firms manage customer relationships by choosing both the right
customers and the right market intervention scheme.
One of the first CRM models was the RFM model (recency, frequency and
monetary value) (Bitran and Mondschein 1996; Roberts and Berger 1989). Recency
refers to the time since last purchase or the number of mailings since last purchase.
Frequency refers to the number of purchases in a given period of time and monetary
value refers to the monetary value of all purchases in a given period of time. Gonul
and Shi (1998) extend the RFM to include a dynamic component and analyze the key
60
determinants of an optimal mailing policy, while simultaneously maximizing both
customer utility and firm profit. The authors discuss that due to its dynamic nature the
model outperforms its single period counterparts. The model proposed in this paper
incorporates the RFM variables and also includes a dynamic component.
There are several other models that exist in the literature with an aim of
improving the CRM intervention policy. These include AID and CH-AID (Sonquist
1970, Kass 1976); the ‘Gains Chart Analysis’ (Banslaben 1992); finite mixture
models (Bult and Wansbeck 1995); the DMLM Procedure (Elsner, Krafft and
Huchzermeier 2004) which determines the optimal frequency size and customer
segmentation of direct marketing activities; customer lifetime value models (Dwyer
1989, Blattberg and Deighton 1996, Berger and Nasr 1998) which used the
customer’s lifetime value as a guide towards developing a better CRM intervention
schedule.
More recently Rust, Lemon and Zeithaml (2004) develop an approach that
allows firms to achieve financial accountability by considering the effect of strategic
marketing expenditures and by relating the improvements in customer equity to the
expenditure required to achieve it. This thus allows firms to distinguish between
customers and hence concentrate on its most profitable customers. Venkatesan and
Kumar (2004) also use a CLV metric for customer selection and marketing resource
allocation by developing a dynamic framework that enables managers to maintain and
improve customer relationships proactively through marketing contacts across various
channels and maximize CLV simultaneously. The authors show that customers who
61
are selected on the basis of their lifetime value provide higher profits in the future
than do customers selected on the basis of several other customer based metrics.
Apart from the methods discussed above there are several other procedures
that allow firms to better manage customer relationships. For example Ansari and
Mela (2003) develop a model to show that (in the case of an email manufacturer) it is
better to customize the email marketing interventions to each individual customer's
taste as this would increase the number of click throughs and hence increase revenue
for the firm. As previously discussed the customer-centric approach in marketing lays
great emphasis on the calculation of customer lifetime value (CLV) which is defined
as the value of future cash flows associated with a customer (Pfeifer, Haskins and
Conroy 2005). CLV measures focus on the future and not on the past. In contrast, the
direct marketing literature use measures of customer’s prior behavior to predict their
future behavior. This is best summarized through the RFM (Recency Frequency
Monetary value) models.
Fader, Hardie and Lee (2005a) by the means of a stochastic model integrate
the RFM paradigm with CLV. They use ‘iso-value’ curves to illustrate the
interactions between RFM measures and CLV and discuss the practical application of
the model. Their approach proposes using the RFM variables as sufficient variables
for an individual customer’s purchasing history and eliminating the need for
additional data to calculate CLV. Additionally Fader, Hardie and Lee (2005b)
develop the beta-geometric/NBD (BG/NBD) as an alternative model to the
Pareto/NBD model developed by Schmittlein et al. (1987). The BG/NBD model
allows for easier estimation of parameters as compared to the Pareto/NBD model.
62
The model is used in cases where predicting the future purchase of the customer is
important to managers as they are interested in estimating the customer’s lifetime
value to the firm.
In our study we are interested in learning about the psychological loyalty of
the customer to the firm. As we do not have any data on the measures of
psychological loyalty of the customer, we need to incorporate loyalty as a latent
variable. Additionally we also model the effect of the CRM interventions on the
transition of the customer across the different loyalty states. Past research (Rust,
Zeithaml and Lemon 2000) used Markov Chain Models to study customer
relationships, migration and retention scenarios. More recently Netzer, Lattin and
Srinivasan (2005) model the dynamics of customer relationships using transaction
data using Hidden Markov models (HMM). Their model of relationship dynamics
incorporates the idea that customer encounters may have an enduring impact by
shifting the customer from one unobservable relationship state to another. The hidden
Markov model (HMM) allows for transitions among latent relationship states and
effects on buying behavior. The dynamics of customer relationships with firms are
more or less continuous, owing not just to purchase encounters with the firm, but also
with competing firms, CRM interventions etc. Hence, we extend this methodology by
employing a state space model, a method that allows us to model relationships as a
continuous function, as a more efficient approach towards studying customer
relationships. We elaborate on the advantages of our method over the HMM in the
next section.
63
Finally, Rust and Verhoef (2005) propose a hierarchical model to manage
customer relationships that individualize rather than segmentize the population of
customers. Their results show that customers are highly heterogeneous in their
responses to marketing interventions. As we conduct our analysis at the level of the
individual customer, we use a hierarchical model to study the customer relationships
and interactions between the customer and the firm.
3. Model Development
We will provide a short description of the scenario that will be considered in
this paper. The case is of a multi-product customer goods company that wants to sell
products to its customers through its own line of stores. The company sends out a
variety of marketing interventions (email, snail mail, etc.) periodically. The aim of
our study is to help the firm understand which CRM interventions strengthen the
loyalty of the customer towards the brand and in turn study the impact of loyalty on a
customer’s intention to repurchase from the firm. We do so by predicting the
customer’s probability of purchase at a given point in time, their loyalty state at a
given time and the likelihood that a customer will make a purchase at a given loyalty
state.
The model described here is one of individual level buying behavior. We
consider a panel of customers and their repeated interactions with the firm. The data
that we use is the typical transaction data that is commonly used in various models of
customer choice. Hence the manufacturer observes not only each customer’s choice
64
history but also the marketing environment at that time. This information will help the
manufacturer better understand the relationship between the customer and the firm.
The relationship is comprised of a longitudinal sequence of encounters, each of which
contain information including, but not limited, to purchases made, whether mailings
were sent, the recency and frequency of purchases etc.
Additionally we define a series of hidden or latent loyalty states for each
customer. These loyalty states are indicative of a tendency of the customer to engage
in behavior that results in a purchase. The transitions between these states are
probabilistically determined and are affected by each relationship encounter. To date
hidden Markov models (HMM) have been used to identify these states and predict
transitions between these states. Hidden Markov models are discrete models. They
have a finite number of different internal states that produce different kinds of
outputs. Typically there are a couple of states for each encounter or a pair of
encounters. The whole dynamical process of producing a relationship function is thus
modeled by discrete transitions between the states corresponding to the different
encounters.
Prior research, as mentioned above, has looked at psychological loyalty as
comprising of discrete states that the customers transition between. While it is useful
to consider loyalty as a discrete variable, it is a simplifying approximation. The
approach we develop in this paper enables us to relax this approximation in a
computationally feasible way. Psychological loyalty is modeled in this paper as a
variable that is continuous across all possible loyalty states. We elaborate on this in
the next section.
65
Additionally, the dynamics of customer relationships with firms are more or
less continuous, owing to not just purchase encounters with the firm, but also with
competing firms, CRM interventions etc. This is of importance to a manager since
customer interactions with the firm are no longer confined to just purchases. They
now also include the firm’s own CRM interventions as well as their competitors’
interventions, which subsequently play a significant role in determining a customer’s
future purchases. For example, an individual who receives a CRM intervention after
making a purchase might be influenced positively by the mailing even if it does not
result in an immediate purchase. Therefore, it is important to understand how these
loyalty or relationship states evolve continuously over time with respect to the CRM
interventions. It would be more beneficial to model the data with a continuous model,
as the frequency with which CRM interventions are sent out is typically much higher
than the frequency of purchase. Loyalty to a firm is hence assumed to be a continuous
function and a good candidate for the task of modeling these loyalty transitions is a
state space model (SSM).
The SSM can be described as the continuous counterpart of the HMM. SSMs
are a general method for the probabilistic modeling of sequences and time-series.
They take the form of iterated maps on continuous state-spaces, and can have either
discrete or continuous valued output functions. They are basically generalizations of
the better known state-space models such as Hidden Markov models (HMMs). A
SSM is, however, more powerful than a HMM. For example, a SSM can represent
infinitely many distinct states as a consequence of their real-valued state-spaces
(Moore 1990, Olivier Bournez 1996). By contrast, HMMs have a finite number of
66
states and hence can be no more powerful than strictly finite-automata. Finite-
automata cannot model many of the recursive structures found in human behavior,
especially instances where learning or the effect of experience is involved. This
occurs because a researcher is forced into organizing such behavior into a
predetermined finite number of states, a step that is not necessary when using a SSM.
A SSM can be expressed across all possible real valued states and transitions between
these states are not confined to a predetermined number. Additionally the estimation
of a hidden Markov model becomes much more inefficient as the number of states
becomes very large (a situation encountered when modeling recursive behavior), a
problem that is avoided in state space models due to their continuous nature.
If we assume x(t) is the observed data, and s(t) is the collection of internal
hidden states of the dynamical system then a standard/generic SSM can be expressed
as follows:
( 1) ( ( )) ( )
( ) ( ( )) ( )
s t g s t m t
x t f s t n t
+ = +
= +
(1)
Both vectors m(t) and n(t) are the noise components of the two equations with
m(t) being the process noise and n(t) the observation noise. Functions f and g are the
linear or nonlinear mappings, with f being the observation mapping and g the process
mapping. Figure 10 provides a visual interpretation of the process involved in a
standard SSM.
67
Figure 10: A standard State Space Model
3.1 Modeling the Customer’s Probability of Purchase
To model customer loyalty and likelihood of purchase we construct a model to
predict the customer’s probability of re-purchase. We are interested in examining the
effect of mailings and other variables on the probability of repurchase. At the same
time we need to control for the time between two consecutive purchases made by the
same household. A model that incorporates both these features (i.e. intrinsic purchase
patterns over time and the effect of marketing variables) is the hazard model. The
hazard model allows us to model the survival of the customer through the subsequent
periods of interest, at the same time controlling for the effects of the marketing
variables that are important to manufacturers, like the effect of CRM interventions on
repurchase.(Gonul and Srinivasan 1993, Vilcassim & Jain 1991). The customer is
s(t-1)
s(t) s(t+1)
x(t-1)
x(t+1)
x(t)
f(.)
f(.)
f(.)
g(.) g(.) g(.)
68
assumed to survive as long as s/he does not make a purchase and the customer dies
when s/he repurchases from the firm in a future period. The hazard model captures
this intrinsic propensity of the customer to repurchase in a computationally feasible
way.
Before defining the probability of purchase we first model the loyalty states
of the customer. There are two approaches to loyalty considered in the literature. The
first being a psychological approach towards loyalty, as was proposed by Jacoby and
Chestnut (1978). The approach distinguishes itself from the behavioral approach (i.e.
repeat purchases of a particular product in a given period of time) by including the
attitudinal aspects of loyalty like the cognitive (the brand is preferable to competitive
offerings), affective (preferential attitude for the brand) and conative (higher intention
to buy the brand as compared to the alternatives) elements (Oliver 1999). For
example loyalty could be viewed as a favorable set of stated beliefs towards the brand
purchased. These attitudes can be gauged by asking how much people like the brand,
feel attached to it, will recommend it to others, and have positive beliefs and feelings
about it – relative to some other competing brands (Reichheld 1996, Dick and Basu,
1994). This then translates into behaviors that result in repurchase. The tendency to
repeatedly purchase the same brand leads to behavioral loyalty. Thus behavioral
loyalty is the propensity to repurchase from the same firm in future time periods and
is defined mainly on the pattern of past purchases. (Fader and Hardie 1996,
Ehrenberg and Scriven 1999).
The approach we develop in our study estimates the hidden loyalty states that
indicate a tendency to engage in behavior that results in a purchase. Hence the loyalty
69
states we derive are similar to the psychological loyalty as described in the literature.
Our study relates the psychological loyalty of the customer to the behavioral
outcomes explained above. We construct a model to estimate the psychological
loyalty of the customer to the firm and then relate this estimated psychological loyalty
of the customer to the behavioral loyalty that the customer exhibits in the form of a
repurchase. Thus the framework developed in this paper relates the two loyalties, one
observed (behavioral loyalty) and the other unobserved (psychological loyalty).
We assume that there is a set of latent psychological loyalty states that
influence behavior. The loyalty states can range from the customer being completely
disloyal to the brand to one in which the customer is completely loyal to the brand. At
a given point in time, each customer is assumed to occupy a particular loyalty state. A
customer’s loyalty state can be affected by several variables like satisfaction, the
purchase experience, affinity to a competitor’s product, marketing activities of the
focal firm, quantity of the brand purchased etc. If the impact is positive it can cause
the customer to transition to a loyalty state that is more loyal than the one s/he was
previously in. On the other hand a negative impact can cause the customer to
transition to a less loyal state.
We divide the time a customer is in each loyalty state into intervals of 3
months. Each quarter is represented by ?, where ? = 1 in quarter 1. At time ?, the
loyalty state for each individual ‘i’ is represented by l
i
(?), and the distribution of these
loyalty states within each individual, is represented by p
i
(l
i
(?)). As we are interested
in estimating p
i
(l
i
(?)) using a Bayesian approach, we assume a prior distribution
across each individual’s set of loyalty states at ? =1, denoted by p
i
(l
i
(1)). The
70
distribution for loyalty varies along the real number line from -? to +?. If the loyalty
state has a value of -? then the customer would be completely disloyal. At the value
of 0, the customer is neither loyal nor disloyal to the brand and is more likely to
engage in switching behavior, as (s)he is indifferent to the experience of using the
brand. At this stage the customer behaves similar to a switcher. At +? the customer
would be completely psychologically loyal to the brand.
We will now proceed to model the customer’s probability of purchase. The
inter-purchase times in the equation for probability of purchase is given by the
number of days since purchase. Here time is indicated by ‘t’ and is calibrated at the
daily level. The probability that customer ‘i’ chooses to purchase product j at time t is
given by
,
i
j t
h . Let ( ( ))
it i
l ? ? be the baseline hazard function for customer i at time t and
loyalty state l
i
(?).
ij
? is a vector of response parameters for customer i at time t for
product j. We assume a prior distribution for
ij
? given by p
i
(
i
? ). X
ijt
is a row vector
of covariates, which include customer attitudes and behaviors towards the brand that
could have an impact on the probability of purchase. Some examples of such
variables that affect the utility of a purchase and hence the probability of purchase
include satisfaction, quantity purchased, price sensitivity, number of repeat purchases,
distance to the store, interaction with frontline employees, trust and the variety of
product offerings (Agustin and Singh 2005; Sirdeshmukh, Singh and Sabol 2002;
Oliver 1999, 1997). The covariates also include marketing variables that are under the
researcher’s control.
Loyalty is also a covariate in the equation for the probability of purchase and
is represented by l
i
(?). Oliver (1999 & 1997) highlights the significant role of loyalty
71
in understanding purchase behavior. He states that the psychological elements, as
defined earlier that constitute our definition of loyalty can lead the customer to
repurchase the brand. At a higher loyalty state a customer’s intention to make a
purchase becomes stronger. Hence, it is important to include loyalty as a covariate in
the purchase equation.
ij
?
is a measure of how much loyalty affects the probability of
purchase. We assume a prior for
ij
?
given by p
i
(
i
? ).
The utility that the customer will derive from purchasing the particular brand
directly impacts the probability of purchase. Brands with a higher utility will have a
higher probability of purchase. As both X
ijt
and l
i
(?) impact the probability of
purchase and hence the utility, we include them as elements in the utility function
(Heilman, Bowman and Wright 2000, Krishnamurthi and Raj 1991). We assume that
the utility that a customer i obtains from purchasing product j is given by
( )
ijt ijt ij i ij ijt
U X l ? ? ? ? = + + (2)
ijt
? is assumed to follow an iid type-1 extreme value distribution.
Therefore, the probability that customer ‘i’ purchases the product ‘j’ at time
‘t’ in a particular loyalty state is given by:
( ( ) ) ( ( ) )
,
( ( ) ) ( ( ) )
( ( ) , , , )
1
? ? ? ? ?
? ? ? ? ?
? ? ?
+
+
?
=
+ ?
i j t i j i i j i t i
i j t i j i i j i t i
X l l
i
j t i i j t i j i j
X l l
e e
h l X
e e
(3)
We assume a general functional form for the baseline hazard as given by
(Vilcassim & Jain 1991):
72
0 1 2 2 3
( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ln
it i it i it i it i it i
l l l t l t l t ? ? ? ? ? ? ? ? ? ? = + × ? + × ? + × ? (4)
where ?t is the time elapsed since the product was purchased. The baseline hazard
function is a very general specification that nests most of the commonly used
probability distributions for inter-purchase times. Vilcassim & Jain (1991) show that
setting certain ?’s to zero allows the function to assume various distributions ranging
from the Weibull, Exponential and the second order power series approximation of an
Erlang-2 type distribution.
We now proceed to model the likelihood function for the hazard model, which
is given by:
1
, ,
( | , ( ), , ) ( ( , ( ), , )) (1 ( , ( ), , ))
? ?
? ? ? ? ? ? ? ? ?
?
= ?
?
v v
i i
iH ijt ijt i ij ij j t ijt i ij ij j t ijt i ij ij
t
L Y X l h X l h X l
(5)
where Y
ijt
is the choice made by the customer and 1
v
? = if a purchase
occurred at time t, and 0 otherwise.
3.2 Modeling Loyalty Transitions
The second step in our analysis is to model the transitions of customer loyalty.
The transitions are allowed to occur in either direction, i.e. in the direction of
increasing or decreasing loyalty. In the state space equation l
i
(?+1) is the loyalty state
in quarter ? +1, l
i
(?) is the loyalty state in quarter ?, and ( ) ? ?
i
is the error term,
assumed to be distributed N(0,?), where ? is the variance of the prior distribution.
We know from previous research (Johnson, Herrmann and Huber 2006, Yi and Jeon
2003 and Oliver 1997) that the loyalty function is also dependant on customer
73
attitudes and behaviors towards the brand as well as firm level marketing variables,
given by X
ij?
, which could have an impact on the probability of purchase. ?
i
measures the effect of X
ij?
on the transition of loyalty from a state at time t to a state
at time ? +1. We also assume a prior distribution for ?
i
given by p
i
(?
i
). To capture
the evolution of loyalty from one time period to another, we need to establish a link
between loyalties across adjacent time periods. Hence we use a state space equation
that allows us to model this link across adjacent time periods. We adapt the SSM
model to our study by combining the generic model with the above mentioned set of
covariates that we use to better understand customer loyalty. We call this model the
SSMC (State Space Model with Covariates).
The state space equation is then defined as follows:
( 1) ( ) ( )
?
? ? ? ? ? + = + +
i i ij i i
l l X (6),
The likelihood function can then be written in the following manner,
L
i
(l
i
(1), l
i
(2),.....,l
i
(T)| ,
?
?
i ij
X ) =
1
1
( (1)) ( ( 1) | ( ), , )
T
i i i i i i ij
P l P l l X
?
?
? ? ?
?
=
? + (7)
Hence the combined likelihood of the system of equations to predict both the
probability of purchase and the loyalty state is given by multiplying equations (5) and
(7):
, ( ),
( | , ( ), , ) ,
i
i l h iH ijt ijt i ij ij i i i i i ij
L L Y X l L (l (1), l (2),.....,l (T) | X )
? ?
? ? ? ? = ? (8)
The likelihood function given in equation (7) has three components, namely, the
initial state distribution, the transitions and the state dependent choice.
74
3.3 The Bayesian Framework
Combining the likelihood function given by equation (7) with the priors on
i
?
given by p
i
(
i
? ), the prior on
i
? given by p
i
(
i
? ), the prior on ?
i
given by p
i
(?
i
) and
the prior on l
i
(?), the loyalty state, given by p
i
(l
i
(?)), we obtain, by Bayes rule the joint
posterior distribution for the hazard model, which is given by:
i
i i i i i i i i i i i i i,l ( ),h
p ( , l ( ), , | data) p ( ) p ( )p ( ) p ( ) L
?
? ? ? ? ? ? ? ? ? ? ? ?
i
l ( ) (9)
The priors for
i
? ,
i
? , ?
i
and l
i
(?) vary across customers as follows:
p
i
(
i
? ) ~ MVN(µ, ?); (10)
p
i
(
i
? ) ~ N(?, ?); (11)
p
i
(?
i
) ~ MVN(?, ?); (12)
µ ~ MVN(0,1.0); ? ~ Gamma(1.0,1.0); (13)
? ~ N(0,1.0); ?~ Gamma(1.0,1.0); (14)
? ~ MVN(0,1.0); ? ~ Gamma(1.0,1.0); (15)
We draw the initial state prior for loyalty from a population wide distribution
of loyalty. We assume that the mean of this distribution is 0 and the variance is 100I
(where ‘I’ is the identity matrix) as we have no specific information about the
variance of the distribution, implying an uninformative prior (uninformative due to
the high numerical value of the variance).
p
i
(l
i
(1)) ~ N(0,100I) (16)
4. Data and Estimation Procedure
We obtained data from a leading manufacturer of beauty products. The
products manufactured by the firm are sold through its own retail outlets. The dataset
75
is divided into two parts: a calibration dataset (also referred to simply as the dataset)
and a validation dataset. The calibration dataset consists of purchases of 250,717
customers over a period of 18 months. The validation dataset consists of purchase
data on the same panel of customers who made purchases over the next 6 months.
The firm has provided data on several variables including the date the customer was
first included in the database, the dates the purchases were made, the amount spent
per purchase, the different types of mailings that were sent and whether the
customer’s email address and physical address was in the system. The company had
provided data on 15 different types of mailings that were sent to the customers in the
dataset. If a mailing was sent to the customer the value of the particular mailing
variable was 1 and if a mailing was not sent then the value of the mailing variable
was 0. We also calculate a cumulative mailings variable which is the sum total of all
the different types of mailings sent to the customer. The average number of mailings
sent out by the firm to a customer over the 18 months was 2.61. We then calculate the
recency of purchase (the time since last purchase), the frequency with which
purchases were made (the number of times the customer made a purchase at the store)
and the monetary value of each purchase for each customer (the total dollar value of
purchases made on each purchase occasion). Finally, we divide the data into three
month intervals at ?=0, ? =1, ? =2, ? =3, ? =4, ? =5, and ? =6.
Estimation is carried out using the MCMC procedure based on a Gibbs
sampling scheme (Geman and Geman 1984). We approximate the posterior
distribution as described in (9) by sampling from the full conditional distributions.
We use equations (10)-(16) to fully specify the model. We derive the estimates of the
76
loyalty states for each quarter and then we derive the estimates of the probability of
purchase at the daily level. We ran 50000 iterations using the WinBUGS software
package, where the first 40000 iterations were used for burn-in and the last 10000
were used for estimation. We controlled for autocorrelation by thinning the
observations-only every 4
th
observation was used for our estimation procedure.
Finally we also checked for convergence by running two chains simultaneously and
monitoring the Brooks-Gelman-Rubin (BGR) convergence diagnostic, where
convergence is said to have been achieved if the BGR statistic for each chain
approaches the value of unity.
We estimate the two equations (3) and (6) using the following covariates.
The repurchase equation (3) included the following covariates: monetary value of the
purchase, whether the customer’s email address is available, whether the customer’s
name and physical address are available, frequency of purchase, whether the purchase
was made on a national holiday, the total number of mailings sent mailings, the
recency of purchase, whether the visit was to return a previous purchase, and finally
the predicted loyalty of the customer. The equation for predicted loyalty (6) includes
the following covariates: the recency of purchase, monetary value of the purchase,
frequency of purchase, and 15 variables that indicate whether the particular type of
mailing was sent or not. These variables have a value of 1 if the mailing was sent and
0 if they were not. The mailing variables included are: product mail1, product mail2,
product mail3, product mail4, product mail5, product mail6, product mail6, product
mail7, relationship mailing1, relationship mailing2, relationship mailing3, action
mailing1, action mailing2, action mailing3 and action mailing4.
77
5. Results and Discussion
Table 1a and 1b present the descriptive statistics of the variables in the
dataset. Table 2 reports the posterior means and posterior standard deviations of the
estimation procedure outlined in the preceding section. In figure 11 we see the
distribution of the population wide loyalty function. The function is centered close to
a mean of zero on the x-axis. An implication of this result is that on average
customers are neither loyal nor disloyal to the firm’s products.
Figure 11: Population-wide distribution of the individual means of the
loyalty function
Variable Mean Std Dev
Monetary Value 11.0881 15.63892
Recency 543.9177 136.3432
Frequency 3.001363 1.184601
Mailings 2.612951 1.800003
Table 1a: Descriptive Statistics – Mean and Standard Deviation
ploy1 sample: 10001
-0.5 -0.25 0.0 0.25 0.5
0.0
2.0
4.0
6.0
78
Variable Value Frequency (%)
Customer Email Address 1 24.86
Customer Name & Address 1 79.87
Product mail 1 1 4.02
Product mail 2 1 43.78
Product mail 3 1 5.87
Product mail 4 1 11.13
Product mail 5 1 2.36
Product mail 6 1 7.05
Product mail 7 1 4.59
Product mail 8 1 4.41
Relationship mailings 1 1 2.62
Relationship mailings 2 1 4.87
Relationship mailings 3 1 5.74
Action mailings 1 1 7.63
Action mailings 2 1 0.91
Action mailings 3 1 3.53
Action mailings 4 1 3.11
Holiday 1 0.66
Table 1b: Descriptive Statistics – Frequency
5.1 The Loyalty Equation
The variables included in the analysis are presented in table 2. We see from
the results that frequency of visits has a significant effect on loyalty of the customer
towards the firm. We find that frequency has a positive effect on the customer’s
loyalty, indicating that customers who visit the company store more often tend to be
more loyal to the company’s products. Neither the recency of purchase nor the
monetary value of purchase has a significant effect on the customer’s loyalty towards
the firm. This result is of interest to firms because it implies that the amount a
customer spends at a store on a given purchase occasion does not necessarily relate to
how loyal the customer is to the firm’s products.
79
We now examine the effect of the firm’s CRM interventions on the predicted
loyalty of the customer. We segment the CRM interventions in the following way:
1. Interventions that provide information about a specific product type or
offering are classified as product mails. The variables in this category include:
Product mailings 1 through 8. From the results shown in table 2, we find that a
majority of the mailings that fall into this category have a significant effect on
the loyalty of the customer towards the firm. Product mailings 1, 2, 3, 4, and 7
have significant positive effects on the loyalty of the customer to the firm.
Mailings that are in this category provide information about the specific
product, thereby arming the customer with knowledge about the firm’s
product offerings. This helps reduce the information uncertainty that the
customer has with respect to the firm’s products. This reduction in uncertainty
also reduces the risk that the customer accepts when purchasing the firm’s
products. We hypothesize that this reduction in risk and increase in
information available to customer help them make a more informed decision
about the firm’s products and hence have a positive effect on the loyalty of the
customer to the firm. Hence such mailings help in improving the loyalty of
customer to the firm.
2. Interventions that provide information about the entire suite of products
offered by the firm in the form of a catalog can be classified as relationship
mails. The variables that fall under this classification are: Relationship
80
mailing 1 through 3. Relationship mailings provide more social benefits (i.e.,
information on the product lines and lifestyle information) and focus on both
relationship building and the creation of additional sales. These mailings tend
to be more involved as they provide more information than just simple
product mailings, and also include information on a much larger variety of
product offerings. As they also involve greater customer participation, these
mailings add towards strengthening the relationship between the customer and
the firm. Hence we find that all the relationship mailings, 1, 2 and 3 are
significant and have a positive effect on the loyalty of the customer to the
firm.
3. Finally, interventions sent at specific times of the year that are meant to
prompt the customer to visit the store are referred to as action oriented mails.
These mails are generally devoid of any information about a particular
product or other offerings. The main purpose of such mailings is to remind the
customer of the firm’s products and induce them to make a purchase. The
mailings that fall into this category are: action mailings 1 through 4. We find
that of these mailings only action mailing 2 plays a significant towards
improving the loyalty of the customer to the firm. These mailings are
generally less involved than the other mailings described above, and their
main purpose as mentioned before is to provide a quick increase in sales.
Hence to a large extent these mailings would inspire feelings of opportunism
in the customer rather than feelings of loyalty, as the amount of investment
81
that the customer makes in using these mailings is less than both relationship
mailings and product mails.
5.2 The Repurchase Equation
The next step of our analysis involved estimating the repurchase equation. The
parameter estimates are also given in table 2. We first analyze the effect of the three
RFM variables, namely recency, frequency and monetary value, on the customer’s
probability to repurchase. The estimates indicate that monetary value of purchase and
frequency or purchase visits are most likely to have a significant effect on the
probability that the customer comes back to the store to make a repeat purchase. The
monetary value of a purchase has a negative effect on the likelihood of repurchase,
i.e. the higher the monetary value of purchase, lower the probability of the customer
coming back to the store. We also find that the frequency with which a customer
makes a purchase visit has a positive effect on the likelihood of the customer
repurchasing from the firm.
Next we examine the effect of CRM interventions on the probability of
repurchase. We model this variable as the cumulative total of all the different types of
mailings received by the customer. We find that the total number of mailings received
by the customer does in fact have a very strong positive effect on the probability of
repurchase. Hence, the efforts undertaken by the firm to send out their CRM
interventions do in fact bear fruit. The analysis suggests that the firm, given the
82
appropriate profit constraints, might even find it in its interest to increase the total
number of CRM interventions sent out.
Repurchase Equation
node Mean sd mean/s.d
Monetary Value µ
1
-0.05 0.007243 -6.90322
Customer Email Address µ
2
-0.02047 1.023 -0.02001
Frequency µ
3
0.3999 0.007179 55.73365
Holiday µ
4
2.14E-04 0.00817 0.026144
Total Number of Mailings µ
5
0.5002 0.007173 69.70584
Recency µ
6
-6.15E-05 0.00829 -0.00741
Returns µ
7
-2.30E-05 0.008258 -0.00279
Customer Physical Address µ
8
8.39E-05 0.008263 0.010156
Predicted Loyalty µ
9
0.02084 0.00702 2.968661
Loyalty Equation
node Mean sd mean/s.d
Frequency ?
1
4.67E-04 1.59E-04 2.939547
Monetary Value ?
2
-9.05E-05 0.01098 -0.00824
Product mail 1 ?
3a
.990 0.1012 9.881423
Product mail 2 ?
3c
1.001 0.1084 9.234317
Product mail 3 ?
3f
1.09 0.09905 10.18677
Product mail 4 ?
3g
1.02 0.09441 10.82512
Product mail 5 ?
3h
0.007276 0.1048 0.069427
Product mail 6 ?
3i
0.008962 0.09926 0.090288
Product mail 7 ?
3k
1.006 0.09742 10.32642
Product mail 8 ?
3n
1.658 2.149 0.771522
Relationship mailings 1 ?
3j
1.112 0.1078 10.3154
Relationship mailings 2 ?
3l
0.998 0.1004 9.940239
Relationship mailings 3 ?
3m
1.007 0.1008 9.990079
Action mailings 1 ?
3b
-0.00332 0.09832 -0.0338
Action mailings 2 ?
3d
1.021 0.09019 11.09879
Action mailings 3 ?
3e
0.8881 0.5952 1.492103
Action mailings 4 ?
3o
1.632 2.142 0.761905
Recency ?
4
2.99E-05 0.00824 0.003625
Table 2: Model Parameter Estimates
We also find that predicted loyalty has a significant effect on the probability
of repurchase. This effect is positive, implying that an increase in loyalty does in fact
increase the probability of the customer coming back to repurchase from the firm.
83
Hence, it would bode well for the firm to work towards increasing the loyalty of the
customer towards its products. We have also shown in section 5.5.1 that certain CRM
interventions like product mails and relationship mailings help improve the
customer’s loyalty towards the firm. Thus the firm would do well to orient its CRM
intervention schedule to include more of these types of mailings as they have been
shown to improve loyalty and loyalty has in turn been shown to increase the
probability that the customer would repurchase from the firm.
Through our analysis we have shown that it is possible to infer a customer’s
latent loyalty towards the firm and examine the effects of a firm’s CRM schedule on
this derived loyalty function. We have also shown that the derived loyalty does in fact
have a strong positive effect on the probability of the customer repurchasing from the
same firm. Hence a firm can use these results to better design their CRM schedule.
5.3 Model Validation
We validated our model’s results by comparing the prediction ability of our
model to two additional models that are used in the literature to study relationship
dynamics. We used the validation dataset to compare our predictions to those of the
benchmark models that are described below. The first model estimated was the
hidden Markov model (Netzer, Lattin and Srinivasan 2005) and the second model
estimated was the loyalty model (Guadagni and Little 1983). Netzer, Lattin and
Srinivasan (2005) show that the HMM is useful at estimating the relationship states of
the customer and then predicting each customer’s probability of purchase. Therefore
we decided to compare our model to the HMM model as described by them.
84
Additionally, the authors also state that the Guadagni and Little (1983) model
was closest in terms of predictive ability to the results obtained from their model.
Hence we chose the loyalty model as described by Guadagni and Little (1983) as our
second benchmark model. The models are summarized below:
Model 1: The first model we compared our predictions to was the hidden Markov
model. We assumed that the customer transitions between three possible loyalty states
namely: loyal, switcher and disloyal. The probability of transition between these
loyalty states is modeled using a logit framework. Similar to the methodology
adopted in our model, we assume that the mailing variables have an impact on the
probability of transitioning from one state to another or remaining in the same state.
Hence the transition probabilities are defined using the following function, where s is
the state at time t, s’ is the state at time t+1 and ?
its
is the effect of the relationship
encounter on the probability of transition between loyalty states and ?
ss’
is the state
specific threshold. The threshold is the value that the cumulative impact of the
encounters has to pass in order for a transition to occur:
'
'
( )
( )
e
( , ')
1 e
? ?
? ?
+
+
=
+
ss its ijt
ss its ijt
X
it X
l s s (19)
Model 2: In the second model, loyalty is modeled using the same formulation as that
devised by Guadagni and Little (1983), where ? is the decay parameter and purchase
it-
1
= 1 if customer ‘i’ made a purchase occurred in the previous period t-1.
l
it
=?l
it-1
+ (1-?)(purchase)
it-1
(20)
The probability of choice is modeled similar to the model shown in (3).
85
Following the procedure employed by Netzer, Lattin and Srinivasan (2005)
we used the RMSPE (Root Mean Squared Predicted Error) and validation log-
likelihood procedures to compare the predictive powers of the three models. The
RMSPE (Root Mean Squared Predicted Error) measures the error between the
predicted purchase probabilities and the actual purchases across customers and time.
The validation log likelihood compares the predictive performance of the models. The
results are shown in table 3. We can see from the results that the state space model
results outperform those obtained from model 1 and 2.
We also compared the differences in means between the repurchase
predictions made by the three models. We checked for differences using a pairwise t-
test, the Bonferroni test and the Tukey test. The tests show that all the means are
significantly different from each other. The results of the analysis are also presented
in table 3.
The SSMC model was also superior to the HMM in terms of computational
efficiency. The HMM model took significantly longer time (almost twice the time) to
achieve convergence than the SSMC. Additionally the SSMC spared the researcher
from making any assumptions about the number of states that the customer could
have. The continuous nature of the SSMC hence provided the researcher with a
significant advantage over the HMM in terms of both the efficiency of the estimation
procedure as well as the ability to avoid making assumptions about the relationship
state of the customer.
86
Model SSMC HMM Loyalty Model
RMSPE 0.3571 0.3703 0.3827
Log Likelihood 80788 83434 85542
Means Comparison t-value Significance
Pairwise t-test of SSMC and HMM -79.181 <.0001
Pairwise t-test of SSMC and Loyalty Model -84.733 <.0001
Means Comparison HMM Loyalty Model
Bonferroni Test SSMC <.0001 <.0001
Tukey Test
SSMC <.0001 <.0001
Table 3: Model Comparison Table – Comparing the Predictive Ability of the State
Space Model with the Hidden Markov Model and the Loyalty Model
Legend: SSMC – State Space Model with Covariates; HMM – Hidden Markov Model
Model 2, while incorporating the importance of state dependence, only
includes the effects of lagged loyalty on the loyalty state of the customer in current
time period. Hence our model, which incorporates customer-firm interactions, allows
additional insights on the effects of different CRM interventions and other variables
of importance to the researcher on the current relationship state of the customer.
Hence from the validation tests carried out above we can see that the SSMC
provides advantages over previous methods in terms of efficiency, better predictive
ability and it also frees the researcher from making any assumptions about the
number of relationship states that the customer might be in.
6. Conclusion
In this paper we used standard transaction data on customer purchase behavior
to predict the relationship dynamics that exist between a customer and the firm.
87
Previous marketing literature solved problems of this type either by developing
models that incorporate only state dependence, or using hidden Markov models. We
develop a state space model to estimate the dynamics of relationships between the
customer and the firm. The SSMC was estimated using a hierarchical Bayes MCMC
procedure to account for both observed and unobserved heterogeneity.
The main contribution of this research is the introduction of a new
methodology to study the relationship dynamics between the customer and the firm,
which helps manufacturers infer the underlying structure of relationship states. The
researcher can dynamically classify customers into the relationship states, and assess
the dynamic effect of interactions between the customer and the firm on the
customer’s relationship state and consequent buying behavior. While the HMM can
also achieve similar results, we show in section 5.3.2 that our model outperforms the
HMM both in terms of its predictive ability as well as the efficiency with which
convergence can be achieved. The number of states of the HMM determines the
number of parameters that need to be estimated, as the researcher has to estimate a set
of parameters for each individual state. On the other hand, the SSMC provides a
major advantage in computational efficiency by allowing the researcher to derive a
continuous probability distribution for the parameters across the states of the
individual customer. Hence the number of parameters to be estimated no longer
increases due to the continuous nature of the state space distribution.
An additional advantage of the SSMC over the HMM is that the researcher no
longer needs to make an assumption about the number of states that the customer
might have. The flexibility of the SSMC lies in the fact that the researcher can predict
88
the most likely relationship state of the customer at a given point in time by simply
studying the probability distribution of the customer at that time across all the
possible states.
The empirical application of the SSMC model is demonstrated in sections 4
and 5. From the results we can see the usefulness of the model in studying dynamic
relationships. The results indicate that CRM interventions do play a role in shifting
the customer from a state of lower loyalty to one of a higher loyalty. Specifically, we
find that CRM interventions that belong to the product mail category and the
relationship mailings category tend to have more of an impact on increasing the
loyalty of the customer than interventions that are more action–oriented, i.e., those
that incentivize the customer into making a purchase immediately through means of a
coupon or sale.
We also find that loyalty plays a positive and significant role in affecting the
customer’s probability of repurchase. The higher the loyalty state of the customer, the
greater is the probability of the customer coming back to the store to make a
purchase. Hence using the SSMC we can test long term impact of the customer-firm
interactions on the relationship between the customer and the firm.
There are two main limitations of this study. Our dataset is comprised of only
those customers who made a purchase in the time period we consider. Customers in
the database who never purchased in the two year span were not included in the
dataset. The exclusion of customers who never make a purchase from the dataset
leads to a selection bias. Such biases can be controlled for by using an indicator
variable that equals 1 if the customer is a purchaser and 0 otherwise. However the
89
lack of data on non-purchasers prevents us from doing so. Hence, due to the
limitation of the dataset, the results derived must be weighted taking the selection bias
into consideration.
The second limitation arises because of the presence of endogeneity in the
model. The endogeneity exists due to the fact that relationship mailings were sent
only to customers who spent >$70 at the store. The problem therefore is that the
marginal distribution of the relationship mailing variable is not independent of the
conditional distribution of the loyalty variable given the relationship mailings. Hence
the relationship mailings variable is endogenous, and this might lead to a bias in the
estimates of the effect of relationship mailings on the loyalty of customer.
The main purpose of this paper is to demonstrate the use of a state space
model to study customer – firm relationship dynamics and illustrate the advantage of
using the state space model over methods prescribed in the previous literature. Hence
our effort was mainly concentrated on using this method to study the customer
relationship characteristics in order to deduce their loyalty states, taking into account
various customer-firm interactions. To keep the model parsimonious we made
simplifying assumptions with respect to the model parameters. A richer dataset,
which includes survey data on variables like customer satisfaction levels, would
provide a better insight into variables that drive loyalty and provide an additional
insight into the effect of these CRM interventions. The use of longitudinal survey data
would be extremely beneficial in determining and shaping a customer’s loyalty state.
Data on channels of purchase would also play an important role in shaping the loyalty
function of the customer.
90
Appendices
Appendix 1: Solving for price and market share in logit demand models.
Solving for price P
i
From (12) in the paper we have,
*
1 (1 )
(1 )
?
?
+ ?
=
?
i i it
i
i it
C S
P
S
. (a)
Simplifying and rewriting (12) we get,
*
1
(1 ) ?
= +
?
i i
i it
P C
S
(b)
Substituting (9) from the paper in (12) we get,
*
1
(1 )
1
?
= +
?
+ +
it
it kt
i i U
i U U
P C
e
e e
, (c)
which can be simplified as
0
*
1
(1 )
? ?
? ?
?
= + +
+
i i i
kt
P
i i U
i i
e
P C
e
. (d)
If we let
kt
U
(1 e ) ? + = , then substituting this in (d) we get,
0
*
1
? ?
? ? ?
?
= + +
i i i
P
i i
i i
e
P C .(e)
Multiplying (e) by
i
? and then subtracting
0
? from both sides, we get
0
*
0 0
1
? ?
? ? ? ?
?
?
? = + + ?
i i i
P
i i i i
e
P C . (f)
Rewriting (f) we have,
0
*
0 0
1
? ?
? ? ? ?
?
?
? + = ? ? +
i i i
P
i i i i
e
P C . (f1)
Taking exponentials on both sides of (f1) and then dividing both sides by ? we have,
* 0
0 0
1
? ?
? ? ? ?
?
? ?
?
? + ? ? +
=
P
i i i
i i i i
e P C
e e
e . (g)
Let
*
0
? ?
?
? +
=
i i
P
e
W , then rewriting (g) we have
0
1 ? ?
?
? ? +
=
i i
C
W
e
We . (h)
It is hence easy to see that (h) is similar to equation (1) in the paper, hence the
solution to (h) is given by
0
1
( )
? ?
?
? ? +
=
i i
C
e
W LambertW . (i.1)
Substituting for W, we have
*
0 0
1
( )
? ? ? ?
? ?
? + ? ? +
=
i i i i
P C
e e
LambertW . (i.2)
Taking the natural logarithms on both sides of (i1) and using the result from equation
(6) we have
0
1
*
0 0
ln( ) 1 ln( ) ( )
? ?
? ? ? ? ? ?
?
? ? +
? + ? = ? ? + ? ?
i i
C
i i i i
e
P C LambertW . (j)
Equation (j) further simplifies to
0
1
*
1 ( )
? ?
?
?
?
? ? +
+ +
=
i i
C
i i
i
i
e
C LambertW
P , (k)
91
which is the closed form solution of P
i
independent of the effect of its own market
share. Substituting for ?, we get the equation (13), which is,
0
0
( 1 )
( )
*
1 ( )
1
? ?
? ?
?
?
? ?
?
+ +
+
=
i i i
k k k
C
i i P
i
i
e
LambertW C
e
P . (k’)
Solving for market share S
it
To simplify notations, we replace 1 ? +
i i
C with ‘?’,
0
? with ‘?’ and
0
1 ? ? ? ? +
i i
C
e with
‘?’. As previously noted, we still maintain the notation(1 ) ? + =
kt
U
e . Thus
substituting this into equation (9) from the paper we have,
S
it
=
*
i i
*
i i
P
P
e
e
?
?
?
? ?
? ?
+
. (m)
Rewriting (m) we have, S
it
=
*
i i
P
e
e e
?
?
?
?
+
. (m1)
Substituting (k) in (m) we have, S
it
=
?
?
?
+e e e
e
LambertW ) (
?
?
?
.
Using the result in equation (1), we can rewrite as
S
it
=
?
?
+
?
?
e
LambertW
e
e
) (
?
?
?
?
. (o)
It is evident that ? = e
-?
e
?
; therefore the (o) becomes,
S
it
=
1
) (
1
1
+
?
?
LambertW
, which can be rewritten as S
it
=
) ( 1
) (
?
?
?
+
?
LambertW
LambertW
. (p)
The equation in (p) is analogous to the result obtained in equation (14). Simply
substituting for the values of ? and ? we get,
0
0
0
0
( 1 )
( )
( 1 )
( )
( )
1
1 ( )
1
? ?
? ?
? ?
? ?
? ?
?
? ?
?
+
=
+
+
i i i
k k k
i i i
k k k
C
P
it C
P
e
LambertW
e
S
e
Lambert W
e
. (p’)
92
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doc_482309460.pdf
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modelling.
ABSTRACT
Title of Document: ADVANCES IN MATHEMATICAL MODELS
IN MARKETING.
Ashwin Aravindakshan, Doctor of Philosophy,
2007.
Directed By: Professor Roland T. Rust, Department of
Marketing
This dissertation comprises a series of three essays that relate advances made
to both theoretical and empirical issues in marketing.
The first essay discusses the issue of endogeneity of market share and price in
logit models and provides a theoretical procedure to solve this problem. The
inseparability of demand and price make the possibility of drawing definite
conclusions about either almost impossible. We employ a recently rediscovered
mathematical function called the ‘LambertW’ to solve this problem of endogeneity
and in turn yield logit models more conducive to theoretical study. We also employ
this methodology to the problem studied by Basuroy and Nguyen (1998).
The second essay deals with the issue of pricing implicit bundling. Implicit
bundles are products that are sold separately but provide an enhanced level of
satisfaction if purchased together. We develop a model that would account for the
possible relationships of the products across the different product lines. We show that
accounting for these relationships would decrease the amount of price competition in
the market and also allow the Firm to enjoy higher profits. We also account for the
endogeneity of price and market share when deriving the optimal solutions. We show
that optimal prices first increase as the relationship between the firm’s two products
become stronger and then decrease as the two products become more exclusive to
each other. Finally, we also find that a firm’s prices increase as the competitor’s
contingent valuations increase.
The third essay helps improve the efficacy of CRM interventions by analyzing
the latent psychological loyalty states of the customer. We use state space models to
predict these latent loyalty states using observed data. We then use the predicted
values of loyalty to derive the probability of repurchase of the customer. We also
identify the types of CRM interventions that play a role in improving the loyalty of
the customer to the firm and those interventions that have no effect. We compare our
model’s predictions to those derived from two other estimation methods. We find that
our predictions are better than those computed from the other methods discussed.
ADVANCES IN MATHEMATICAL MODELS IN MARKETING
By
Ashwin Aravindakshan
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2007
Advisory Committee:
Professor Roland T. Rust, Chair
Professor Brian T. Ratchford
Professor P. K. Kannan
Professor Wendy Moe
Professor Roger Betancourt
© Copyright by
Ashwin Aravindakshan
2007
ii
Acknowledgements
I am indebted to Dr. Roland Rust for his constant help and guidance towards
the completion of this dissertation. I am grateful to Dr. Brian Ratchford for his
unconditional support and guidance.
I would also like to thank Dr. P. K. Kannan, Dr. Wendy Moe and Dr. Roger
Betancourt for taking the time to help enrich this effort.
iii
Table of Contents
Acknowledgements....................................................................................................... ii
Table of Contents......................................................................................................... iii
List of Tables ................................................................................................................ v
List of Figures .............................................................................................................. vi
Overview....................................................................................................................... 1
Chapter 1: Essay 1 - The LambertW Transformation As An Approach To Solving
Share Equations In Logit Models ................................................................................. 5
Summary................................................................................................................... 5
1. Introduction........................................................................................................... 5
2. A Brief Note on the ‘LambertW’ Function........................................................... 6
3. The Logit Model ................................................................................................... 9
3.1 Consumer’s Demand Function ....................................................................... 9
3.2 Firm’s Profit Function................................................................................... 11
4. The Analytical Solution...................................................................................... 11
4.1 Solving the Firm Level Endogeneity Problem.............................................. 11
4.2 Estimation Procedures .................................................................................. 14
5. A More General Application Incorporating Marketing Expenditure ................. 15
6. Conclusion ......................................................................................................... 19
Chapter 2: Essay 2 - Pricing Related Products In A Competitive Environment – The
Role Of Contingent Valuations Between Products..................................................... 20
Summary................................................................................................................. 20
1. Introduction......................................................................................................... 20
2. Literature Review................................................................................................ 22
2.1 Product Relationships ................................................................................... 22
2.2 Complementary Pricing ................................................................................ 27
3. Model Formulation ............................................................................................. 29
3.1 Defining the Market Structure ...................................................................... 29
3.2 Consumer Demand Formulation................................................................... 31
3.3 Manufacturer’s Profit Function..................................................................... 37
3.4 Consumer Surplus......................................................................................... 46
4. Conclusion And Managerial Implications ........................................................ 48
5. Future Research .................................................................................................. 54
Chapter 3: Essay 3 - Modeling Loyalty For Better Customer Relationship
Management................................................................................................................ 55
Summary................................................................................................................. 55
1. Introduction......................................................................................................... 56
2. CRM Models....................................................................................................... 59
3. Model Development............................................................................................ 63
3.1 Modeling the Customer’s Probability of Purchase ....................................... 67
3.2 Modeling Loyalty Transitions....................................................................... 72
4. Data and Estimation Procedure........................................................................... 74
5. Results and Discussion ....................................................................................... 77
iv
5.1 The Loyalty Equation ................................................................................... 78
5.2 The Repurchase Equation ............................................................................. 81
5.3 Model Validation .......................................................................................... 83
6. Conclusion .......................................................................................................... 86
Appendices.................................................................................................................. 90
Appendix 1: Solving for price and market share in logit demand models.............. 90
References................................................................................................................... 92
v
List of Tables
Chapter 3: Essay 3 - Modeling Loyalty For Better Customer Relationship
Management
Table 1a: Descriptive Statistics – Mean and Standard Deviation 77
Table 1b: Descriptive Statistics – Frequency 78
Table 2: Model Parameter Estimates 82
Table 3: Model Comparison Table – Comparing the Predictive
Ability of the State Space Model with the Hidden
Markov Model and the Loyalty Model 86
vi
List of Figures
Chapter 1: Essay 1 - The LambertW Transformation As An Approach To Solving
Share Equations In Logit Models
Figure 1 – The LambertW Function 7
Figure 2 – Relationship between market share of firm ‘i’ and price of
firm ‘k’ 12
Figure 3 – Relationship between the price of firm ‘i’ and price of
firm ‘k’ 13
Figure 4 – The decrease in P
1
* with increase in P
2
18
Chapter 2: Essay 2 - Pricing Related Products In A Competitive Environment – The
Role Of Contingent Valuations Between Products
Figure 5 – The Market Structure 30
Figure 6 – Variation of PA
1
with respect to ?
1
44
Figure 7 – Variation of PA
1
with respect to ?
2
45
Figure 8 – Variation of PA
1
with respect to ?
1
and
?
2
46
Figure 9 – The Consumer Surplus 47
Chapter 3: Essay 3 - Modeling Loyalty For Better Customer Relationship
Management
Figure 10 – The standard State Space Model 67
Figure 11 – Population-wide distribution of the individual means
of the loyalty function 77
1
Overview
My dissertation consists of three essays. All three essays detail and develop
new methodologies that can be used by researchers to better understand the market.
In ‘Essay 1’ and ‘Essay 2’ I develop a new methodology to solve the issue of
endogeneity of price and market share in logit models. Endogeneity in logit models
prevent meaningful interpretation of marketing dynamics when data for the particular
phenomenon or situation is absent. Hence in Essay 1 we solve this problem by
developing a procedure that explicitly separates the price and demand variables by
utilizing a special class of functions called the LambertW. The LambertW function
helps solve several complex equations that involve either exponential or logarithmic
functions. Using this procedure we analytically solve the endogeneity problem
between price and demand in logit models. We do so without the need for
instrumental variables that would otherwise have been used. We also show the ease
with which these functions can be adapted to different simulation and estimation
procedures. Finally this procedure is validated by employing it to explicitly derive the
results obtained by the Basuroy and Ngyuen (1998).
Essay 2 adds to the dissertation by applying the procedure developed in Essay
1 to study a new problem. We study the market for implicit bundles. Implicit bundles
refer to the group of products that are sold separately (also referred to as pure
components) but could be perceived by the consumer as products that would provide
an enhanced level of satisfaction if purchased together. In many cases, the implicit
bundles are sold by the same firm, in the form of products sold across different
product lines. The aim of this study is to address the issue of pricing these implicit
2
bundles. We do so by developing a model that would account for the possible
relationships, also known as contingent valuations, of the products across the
different product lines. The degree of contingency determines the strength of the
relationship between the two products. If the contingency level is zero then the
products are independent. As the level increases the relationship between the products
increases, thereby increasing the exclusivity between the products. We show that
accounting for these relationships would decrease the amount of price competition in
the market and also allow the firm to enjoy higher profits, because it would be able to
charge the consumer for the additional surplus gained by purchasing both products
from the same Firm. We also account for the endogeneity of price and market share
when deriving the optimal solutions.
The analysis carried out in this essay helps us establish the following results:
1. The prices of the products of a firm (X) increase with increase in the
contingent valuation of the products manufactured by the same firm (X).
However if the contingent valuation is increased beyond a particular value,
then the prices of firm (X) decrease with increase in the relationship. Hence
the relationship between prices and the contingent valuation is initially
monotonically increasing and then monotonically decreasing.
2. The prices of the products of the firm (X) increase with increase in the
contingent valuation of the products manufactured by a competing firm (Y).
3
3. Firm X experiences a higher profit for the same market share, when its
products are priced considering the contingent valuations than when they are
priced independently of each other.
Consumers who purchase the products would have a lower level of consumer
surplus if the contingent valuation is recognized by a firm.
CRM (Customer Relationship Management) interventions like direct mailings
have long been used by firms to improve customer relationships. In the third essay we
develop a method that will allow the firm to understand the effect of these
interventions for customer loyalty. Loyalty is assumed to be unobserved and hence is
modeled as a latent variable. We use a generalization of the ‘Hidden Markov Model’
(HMM) called the ‘State Space Model’ (SSM) to better predict a customer’s loyalty
function towards a particular firm or product. The SSM models are structurally
different from HMM models and they offer three main advantages over HMM
models. First they are continuous and are described across all possible relationship
states of the customer, hence we avoid the problem of explicitly choosing the number
of states; second, they can be used to model an infinite number of relationship states;
and third they are better at modeling recursive behavior, which is necessary when
modeling customer behavior that involves the effect of experience. We also predict
the customer’s probability of purchase given certain marketing actions and the
predicted loyalty state using a hazard model. We combine the hazard model and the
SSM to predict the customer’s probability of purchase at a given loyalty state. We
apply this model to data from a retailer of health and beauty aids, in order to help
them better understand the effect of their CRM interventions on the customer’s
4
loyalty towards the firm and their repurchase intentions. We also point out the types
of CRM interventions that play a role in improving the loyalty of the customer to the
firm and those interventions that have no effect. This information can hence help the
firm better organize its menu of CRM interventions. We can also compute the
probability distributions across the loyalty states for each individual customer, thus
providing the researcher with knowledge of each customer’s loyalty state. Finally we
introduce a new methodology to the literature on modeling relationships in marketing.
The methodology improves upon existing methods by allowing for a more flexible
and efficient estimation procedure. We also compare our model’s predictions to those
derived from two other estimation methods. We find that the predictions derived from
our estimation procedure are better than those computed from the other methods
discussed.
5
Chapter 1: Essay 1 - The LambertW Transformation As An
Approach To Solving Share Equations In Logit Models
Summary
Logit models allow the expression of individual demand and supply
equations. However, closed-form solutions for equilibrium shares and prices are
highly nonlinear and cannot readily be derived. This hinders the employment of logit
models in theoretical studies, and also makes it difficult to develop reduced-form
expressions for share and price as a function of exogenous variables for use in
empirical studies. In this paper we propose that a recently rediscovered mathematical
function called the ‘LambertW’ be employed in solving logit models for equilibrium
shares and prices. We demonstrate this methodology on the problem studied by
Basuroy and Nguyen (1998).
1. Introduction
Discrete choice models have been extensively used in both the marketing and
economics literature to study various aspects of consumer behavior using data on
market share, price and other variables that affect demand. Logit models are widely
used in the empirical literature (Abramson, Andrews, Currim and Jones 2000;
Kamakura and Russell 1989; Guadagni and Little 1983; McFadden 1978). However,
these models have seldom have been used to derive theoretical relationships between
6
variables of interest. Some exceptions include Basuroy and Nguyen (1998), Carpenter
and Lehmann (1985), Lillien and Kotler (1983) and Lillien and Ruzdic (1982). The
complexity of the resultant expressions is often attributed to be the cause of this non-
usage (Gruca, Kumar and Sudharshan 1992; Gruca and Sudharsdhan 1991; Karnani
1985).
In this paper, we propose that a recently rediscovered mathematical function
(first studied by Euler 1779), termed ‘LambertW’, be employed in obtaining
equilibrium solutions for share and price in logit models. Traditionally, the
LambertW function has been used to solve several exponential equations (Corless, et
al. 1996). The rest of the paper is organized as follows— in §2, we provide a brief
overview of the ‘LambertW’ function. We develop the standard logit model in §3,
and in §4 we present the solution and simplify the model for estimation purposes. We
provide a theoretical application of this methodology in §5. We provide our final
conclusions in §6.
2. A Brief Note on the ‘LambertW’ Function
LambertW is the inverse function associated with the equation,
x
W
We = (1).
The LambertW function belongs to the family of exponential and logarithmic
functions. The function given in (1) resembles the exponential function and the
inverse of this function resembles the logarithmic function. Hence, the shape of the
LambertW function closely follows that of shape of the exponential function and the
logarithmic function. The LambertW function differs from the exponential to the left
of point x 0 = . The exponential is always positive, however the LambertW dips to a
7
minimum of -1 at
1
x
e
= ? . Similarly, the LambertW function differs from the
logarithmic function for values of x ? 0 as while the logarithmic function is not
defined for these values of x, the LambertW function continues to have a value
till
1
x
e
= ? . A special case of the LambertW is the case when x lies in the range
between
1
e
? and 0. In this case, W(x) has not just a single valued function but has two
values. In the case of LambertW, a single valued function W
0
(x) is defined for values
of
1
x
e
= ? and W(x) ? -1. W
0
(x) is also referred to as the principal branch of the
LambertW function. The other branch satisfying W(x) ? -1 is denoted by W
-1
(x). The
shape of the LambertW function is shown in Figure 1.
Figure 1: The LambertW function
LambertW(x)
8
At this point, let us examine and question the importance of the LambertW
function. Similar to the exponential, logarithmic and square root functions, the
LambertW is helpful in solving a series of previously unsolvable equations (readers
may refer Corless, et al. 1996 for an illustration). The LambertW function has already
seen widespread application in the fields of physics (Warburton and Wang 2004;
Valluri, Jeffrey and Corless 2000) and applied mathematics (Corless, Jeffrey and
Knuth 1997, Jeffrey, Hare and Corless 1996, Jeffrey, et al. 1995). Hence, there has
been a movement to include the LambertW function in the core set of elementary
functions that are used to solve equations (Hayes 2005, FOCUS 2000). Many
equations involving exponentials can be solved using the LambertW function.
In this paper, our primary interest is to employ the LambertW function to
solve a previously analytically unsolvable simultaneous equations problem. Before
we proceed with our study, we will briefly outline a few important properties of the
LambertW function. For a more detailed exposition on the properties of the
LambertW function, the reader is advised to refer Corless, et al. (1996).
The LambertW function has the series expansion,
n
n
n
x
n
n
x W
?
?
=
?
?
=
1
1
0
!
) (
) ( , (2)
where n! is a factorial. This series however oscillates for values of x ? 0.4 and hence
cannot always be used for practical numerical computation. However, there exists an
asymptotic formulation, that is convergent for all x, which yields reasonably accurate
real values. This is given by:
9
(
(
¸
(
¸
|
|
¹
|
\
|
+
+ ? + ?
+
+ ? + ?
+
+ ?
+
+ ?
+ + ? =
6
1
2
5
1
4
2
3
2
2
2 2 2
4
1
3
2
2
2 2 2
3
1
2
2 2 2
2
1
2 2
1
2
2 1
60
) 12 125 350 300 60 (
12
) 3 22 36 12 (
6
) 9 6 (
2
) 2 (
) (
L
L
O
L
L L L L L
L
L L L L
L
L L L
L
L L
L
L
L L x W
, (3a)
where L
1
= ln(x) and L
2
= ln(ln(x)).
A further simplification of this formulation, namely,
04 . 0 ) 1 ln( )] 1 ln( 0195 . 0 1 [ 665 . 0 ) ( + + + + ? x x x W , (3b)
provides an accurate approximation to the LambertW function and is used primarily
for numerical estimation purposes.
The derivative of LambertW is,
)] ( 1 [
) (
)] ( exp[ )] ( 1 [
1
) (
'
x W x
x W
x W x W
x W
+
=
+
= and (4)
the anti-derivative of W(x) is,
?
+ + ? = C
x W
x W x dx x W ]
) (
1
1 ) ( [ ) ( . (5)
Also W(0)=0 and W’(0) =1.
Finally the logarithm of W(x) is given by,
log(W(x)) = log(x)-W(x) (6)
3. The Logit Model
3.1 Consumer’s Demand Function
We consider the simplified scenario with two firms, namely firm i and firm k.
Each of these firms sells a product. The products sold by these firms compete for
market share. Let product i denote the product manufactured by firm i and product k
denote the product manufactured by firm k. We conduct our analysis with respect to
firm i. The scenario may be extended to include multiple firms.
10
As the aim of this paper is to develop techniques to analytically solve discrete choice
problems, we proceed to define the probability of a consumer choosing a particular
firm’s product, i.e. we model the consumer’s demand for each brand. If a consumer
purchases product i then the utility to the consumer from the purchase is:
ji i ji i i i ji
U P u ? ? ? ? + = + ? =
0
, (7)
where u
ji
is the utility obtained by consumer j from product i, ?
0i
is the brand or
product specific parameter, ?
i
is the price response parameter for product i, P
i
is the
price of i, U
i
is the deterministic component of utility that is assumed to be constant
across consumers, and
ji
? is the random error term.
We wish to compute the share of i relative to the competing brand k and an
outside good that represents non-purchase in the focal category. The utility of the
outside good is normalized to 0; i.e. we assume U
0
= 0. Given that error terms for i, k
and 0 follow an iid type 1 extreme value distribution, the choice share for i
with
respect to the outside product option, and with respect to the competitor k’s product
is,
k i
i
U U
U
i
e e
e
S
+ +
=
1
, (8)
where S
i
is the probability of choosing product i. The elasticities of demand for firm i
with respect to the prices charged by firm i (own price) and firm k (competitor’s
price) are:
( )
(
¸
(
¸
? ?
=
(
(
(
(
¸
(
¸
?
?
?
?
k i k
i i i
k
i
i
i
S S
S S
P
S
P
S
?
? 1
. (9)
11
3.2 Firm’s Profit Function
The firms in the market are price setters. Let C
i
be the marginal cost incurred
by the firm to provide product i. Hence, the firm’s profit from selling product i is:
( )
i i i i
S C P ? = ? , (10)
where
i
? is the profit firm i earns from selling the product at price P
i
. Assuming the
existence of a pure strategy interior equilibrium, the price vector satisfies the first
order conditions. Hence differentiating (10) with respect to P
i
, and setting the result
as equivalent to zero, we derive the optimal price that the firm must charge so as to
maximize profit as:
( )
( )
i i
i i i
i
S
S C
P
?
? +
=
1
1 1
*
?
?
, (11)
where
*
i
P is the optimal price charged by the firm. This optimal price is a function of
share, which, in turn, is a function of price.
4. The Analytical Solution
4.1 Solving the Firm Level Endogeneity Problem
In equilibrium, equations (8) and (11) must hold simultaneously. The solution
to these simultaneous equations is highly nonlinear, and does not readily admit to a
closed form solution. However, a closed-form solution can be provided in terms of
the LambertW transformation. As demonstrated in the appendix, the following
equations for price and share can be derived:
0i i
0k k k
1 C
i i P
*
i
i
e
1 LambertW C
1 e
P
? ?
? ?
?
?
? ?
?
| |
+ +
|
+
\ ¹
= , (12)
12
and
|
|
¹
|
\
|
+
+
|
|
¹
|
\
|
+
=
?
? ?
?
? ?
k k k
i i i
k k k
i i i
P
C
P
C
i
e
e
LambertW
e
e
LambertW
S
? ?
? ?
? ?
? ?
0
0
0
0
1
1
1
1
1
. (13)
This solution allows the price equation to be expressed in a form that allows
the effects of market share and price to be separated. Hence the LambertW
transformation allows us to rewrite the equation for market share and price in a
manner in which they are independent of each other. Initially the market share of firm
i was an integral part of the price equation (see equation 11) and hence we could not
make any predictions about the price that the firm charges. Hence, through the
LambertW transformation we can eliminate the econometric endogeneity problem
that exists between P
i
and S
i
. The solution also allows for reduced-form expressions
for price and share, which allows direct determination of changes in optimal price
resulting from changes are made to the cost, other independent variables, or the
competitor’s price. For example if we assume that the prices of the competitors are
given, then figures (2) and (3) show the variation of the market share of firm i (S
i
) and
the price charged by firm i (P
i
) in relation to the price of the competitor’s price (P
k
)
respectively.
Figure 2: Relationship between market share of firm i and price of firm ‘k’
Market Share
of firm i: S
i
Price of firm k: P
k
13
Figure 2 describes the market share increase for firm ‘i’ when firm ‘k’
increases its prices. Figure 3 shows the amount by which firm i can increase its price
when firm k increases its price, if firm i intends to retain the same level of market
share.
Figure 3: Relationship between the price of firm ‘i’ and price of firm ‘k’
While the above provides separate solutions for prices and shares of firm i by
eliminating the econometric endogeneity between them, the price charged by firm i is
still dependent on firm k’s price (see equation 12). Hence, the endogeneity arising
from the fact that the prices of the two firms need to be jointly determined still
remains, i.e. the problem of determining equilibrium market prices remains. Since P
i
and P
k
are symmetric, the price that firm k charges can be derived in the same way as
the price of firm i, and can be expressed as:
Price of
firm i: P
i
Price of firm k: P
k
14
0k k k
0i i i
1 C
k k P
*
k
k
e
1 LambertW C
1 e
P
? ?
? ?
?
?
? ?
?
| |
+ +
|
+
\ ¹
= (14)
While equations 12 and 14 do not appear to have a readily expressible closed-
form solution, Nash equilibrium prices can be determined as the point of intersection
of plots of P
i
given P
k
and of P
k
given P
i
.
4.2 Estimation Procedures
Equation (3b) provides a formulation that accurately approximates the
LambertW function for all x. Using the formulation in (3b) we can rewrite equation
(13) in the following way,
0 0
0 0
( 1 ) ( 1 )
* 2
1 1 1 ( ) ( )
(1.04) (.665) ln( 1) (.0129)[ln( 1)]
1 1
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ?
= + + + + +
+ +
i i i i i i
k k k k k k
C C
i i P P
e e
P C
e e
,
(15)
where ?
1
= 1/?
1
. As we are now interested in estimating equation (15), assume an
unobservable term for the price equation denoted by ?
i
, and that the price is linear in
the unobservable term ?
i
. Hence equation (15) can be rewritten as:
0 0
0 0
( 1 ) ( 1 )
* 2
1 1 1 ( ) ( )
(1.04) (.665) ln( 1) (.0129)[ln( 1)]
1 1
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
= + + + + + +
+ +
i i i i i i
k k k k k k
C C
i i i P P
e e
P C
e e
(16)
Equation (16) is now independent of the effect of the firm’s own market share
and hence can be estimated without the use of firm level instrumental variables. The
estimation procedure would follow a standard nonlinear regression where the
15
parameters to be estimated would include
1
? and all the ?’s. It is important to note
that if the market prices are set endogenously, the researcher would estimate (16) and
a similar equation for the prices of firm k simultaneously, using suitable market level
instrumental variables.
5. A More General Application Incorporating Marketing Expenditure
We apply this methodology to the multinomial logit problem studied by
Basuroy and Nguyen (1998). The authors discuss the appropriateness of Multinomial
Logit (MNL) market share models for equilibrium analysis. Their results show that a
linear price response in conjunction with the typical concavity assumed in a large
range of marketing response functions would yield an interior equilibrium solution.
The authors then consider the optimal pricing and marketing expenditure reactions to
entry and potential market expansion. In the context of the MNL models, they
demonstrate that the entry of a new brand evokes a defensive reaction through a
decrease in the equilibrium prices of the existing brands. They also note that while
new entry into a fixed market triggers the incumbents to lower marketing
expenditure, when faced with market expansion, firms tend to raise marketing
activities. Consequently, there exist distinct possibilities that marketing efforts for the
existing brands increase in view of entry in an expanding market.
In this section we will incorporate the characteristics outlined by the authors,
namely—the linearity of price response and the concavity of market expenditure
response functions into our MNL market share model. We then derive optimal
16
solutions for price and market expenditure independent of market share by employing
the methodology developed in the earlier part of this paper.
Consistent with Basuroy and Nguyen, let P
1
and m
1
be the price charged and
marketing expenditure of Firm 1 for its product. Let P
2
and m
2
the price charged and
marketing expenditure of Firm 2 for its product. We assume a linear price response
function and a concave marketing response function. The market share function for
Firm 1 is given by,
1
1 1 1 2 1
1 2
1 1 1 2 1 1 2 2 2 2
( 1 )
1
( 1 ) ( 1 )
m
m m
P e
P e P e
e
S
e e
? ?
? ? ? ?
?
? ?
+ ?
+ ? + ?
=
+
, (17)
where S
1
is the market share for Firm 1, ?
11
is the price response parameter that is
assumed to be negative and ?
21
is the market response parameter that is assumed to be
positive. We assume that Firm 2 has a market share function that is analogous to Firm
1. Given the market share function for Firm 1, the profit function for Firm 1 becomes,
1 1 1 1 1 1
( ) P c N S m FC ? = ? ? ? ? ? , (18)
where ?
1
is the profit that Firm 1 gets from selling its product, c
1
is the marginal cost
incurred by Firm 1 and FC
1
is the firm’s fixed cost and N is the market size.
Assuming that the firm is a price setter and assuming the existence of a pure
strategy interior equilibrium, the price vector satisfies the first order conditions. We
can obtain the optimal price that Firm 1 should charge for its product by solving the
first order conditions for P
1
. The first derivative with respect to P
1
is given by,
1
.
1 1 1 1 1 1
( ) (1 )
P
S N P c N S S ? ? = ? ? ? ? ? ? (19)
Similarly, the first order conditions for marketing expenditure m
1
can be obtained by
differentiating (2) with respect to m
1
. This is given by:
17
1
1
.
1 1 11 1 1
( ) (1 ) 1
m
m P c N e S S ? ?
?
= ? ? ? ? ? ? ? (20)
We can now use the LambertW functional form to solve equations (17), (19)
and (20) and obtain the optimal solutions for price P
1
and marketing expenditure m
1
,
independent of the firm’s market share. We solve the equations simultaneously and
obtain the following solutions for the optimal price and optimal marketing
expenditure:
11
21 11 1
22
12 2 22
2
1
1
*
1
11
1
1
m
c
N
P
e
e
LambertW
e
P c
?
? ?
?
? ?
?
| |
+ + ?
|
\ ¹
| |
+ ?
|
\ ¹
(
(
? ?
(
(
+
¸ ¸
= + , (21)
where
1
*
P is the optimal price charged by Firm 1 and,
1
* 11
21
ln m
N
?
?
| | ?
= ?
|
\ ¹
, (22)
where
1
*
m is the optimal marketing expenditure. As ?
11
is assumed to have a negative
value, both P
1
and m
1
from equations (21) and (22) will always be positive. Hence,
the methodology developed in this paper allows us to derive closed form solutions to
both the optimal price and the optimal marketing expenditure, where the optimal
price depends on cost and competitor price.
Comparing our results with Basuroy and Nguyen (1998) we find, as shown in
figure 4, that the optimal price charged by the Firm 1 would decrease upon the
introduction of a new product in the market. This is consistent with the results
obtained by the authors.
18
Figure 4: The decrease in P
1
* with increase in P
2
It is relatively harder to analyze the marketing expenditure function as the
functions only involve the price response parameter of Firm 1. Thus, we cannot make
predictions about the change in marketing expenditure with respect to the entry of a
competitor. However we can make predictions of the effect of an increasing market
size on the marketing expenditure. Taking the first derivative of marketing
expenditure function given in equation (22) we find that
*
1
1 m
N N
?
=
?
(23)
This implies that the marketing expenditure function shares a positive
relationship with the market size. Hence as market size increases, the marketing
expenditure of Firm 1 must also increase, albeit at a decreasing rate. This result is
also consistent with the results obtained by Basuroy and Nguyen (1998).
P
1
*
P
2
19
6. Conclusion
In this paper we analytically solved the problem of econometric endogeneity
in discrete choice models through the use of the LambertW function. The LambertW
function lends itself to analytically solving exponential equations, and thereby
facilitates the derivation of closed-form solutions for price and market share. The
LambertW function also lends itself to easy estimation through a simple yet accurate
approximation as explained in sections 2 and 4.2. This approximation can be used in
estimating expressions for price and share that depend only on competitor actions,
and that therefore do not require firm level instrumental variables. Market level
instrumental variables, which are easier to obtain, suffice. The LambertW function is
potentially useful whenever logit models are employed in theoretical or empirical
work, or more generally, whenever solutions to exponential functions are required.
20
Chapter 2: Essay 2 - Pricing Related Products In A Competitive
Environment – The Role Of Contingent Valuations Between
Products
Summary
This paper extends the research by Venkatesh and Kamakura (2003) to
consider the case of pure component pricing in a competitive setting. We develop an
optimal pricing scheme for a Firm that has two products, when its products are sold as
pure components. We show that the optimal price increases monotonically with
respect to the contingent valuation of both the Firm’s and its competitor’s products.
We also derive the consumer surplus when the products are priced independently and
when they are priced by taking into account the contingent valuation. We conclude
that consumer surplus is higher in the former case.
1. Introduction
Implicit bundles refer to the group of products that are sold separately (also
referred to as pure components) but could be perceived by the consumer as products
that would provide an enhanced level of satisfaction if purchased together. In many
cases, the implicit bundles are sold by the same Firms. An example would be
Microsoft selling both its Windows Operating System and Office Suite. It has been
shown that even though Microsoft sells these products separately, they still price
them as if they were selling them as a bundle, where Windows is priced lower than
21
Office, even though Windows costs almost twice as much to develop (Economides
and Viard 2003). Most Firms market products in many different product lines, so it is
very important for the Firms to develop optimal pricing strategies for their products in
the different product lines. Many issues go into establishing optimal pricing schemes.
The study of the effect of inter-category relationships between the products made by
the same manufacturer on the pricing scheme is very important, because this could
affect the consumption behavior of the consumer. This topic has been studied in some
detail in the retail setting (Manchanda et al. 1999). However, with the exception of a
handful of studies (Reibstein and Gatignon 1984; Urban 1969), the literature has
generally ignored how manufacturers must treat this problem when pricing their
products across the different product lines (Elrod et al. 2002).
The aim of this paper is to address the issue of pricing across different product
lines. We do so by developing a model that would account for the possible
relationships of the products across the different product lines. We hope to show that
accounting for these relationships would decrease the amount of price competition in
the market and also allow the Firm to enjoy higher profits, because it would be able to
charge the consumer for the additional surplus gained by purchasing both products
from the same Firm. We also account for the endogeneity of price and market share
when deriving the optimal price.
The rest of the paper is organized as follows. We review the literature in §2.
Model Development is carried out in §3. The conclusions and managerial
implications are presented in §4. Directions for future research are provided in §5.
22
2. Literature Review
2.1 Product Relationships
Traditionally inter-category relationships between products have been studied
in the form of substitutes or complements. However these relationships are far more
complex than just substitutes or complements. Product relationships can no longer be
viewed upon as a simple dichotomy (either complements or substitutes) but these
relationships should be viewed in terms of degree of complementarity and
substitutability, not absolute substitutability or complementarity.
Venkatesh and Kamakura (2003) explain these relationships in the form of
contingent valuations. Many recent studies have indicated that buyers evaluate the
components of a bundle by assigning certain valuations to the strength of the
relationships in the bundle. (Jedidi et al. 2003, Yadav 1994). Contingent valuations
measure this relationship between the components in the bundle. Contingent
valuations may be defined as the degree of complementarity (or substitutability)
between two products as perceived by the consumer, if the products were to be sold
as a bundle (Venkatesh and Kamakura 2003). Contingent valuations can be either
positive or negative. A positive relationship implies that the two products share a
complementary relationship and a negative relationship implies substitutability. In
this study we focus only on the positive values of the contingent valuations, i.e., we
confine our analysis to only complementary products as we are interested in
understanding how a firm can take advantage of a complementary relationship in
23
pricing its products. More recently, Wang, Venkatesh and Chatterjee (2006) develop
a new methodology to reformulate the the way reservation prices can be calculated.
Their procedure named ICERANGE draws on literature on buyers’ uncertainty in
preference and product knowledge. Their results demonstrate that the ICERANGE
method significantly outperforms previous models in terms of predictive validity.
For the purpose of this study we develop a pricing policy for two products that
can be perceived by the consumer as having a relationship with each other. We then
compare this pricing scheme to another pricing scheme where the products are priced
independently of each other. In this study, we only consider implicit bundles, as we
assume that it is not convenient for the manufacturer to explicitly bundle its products.
The main difference between an explicit and implicit bundle would be that products
in explicit bundles are sold together, while products in implicit bundles are sold
separately. An example of an explicit bundle is the Microsoft Windows and Internet
Explorer bundle. Here the consumer typically attains a higher level of satisfaction
using the two products, but they are not sold separately. An example of an implicit
bundle is the Windows-Office bundle, where the products are sold separately, and the
consumer enjoys a higher level of satisfaction by owning both the products. We will
explain both the economic and legal reasons for this later on in this section.
Implicit bundles can be composed of both complements and substitutes. For
the purpose of this study, we only consider complements. Hicks (1939) was one of
the first researchers to define complementary relationships. He argued that
complementarity could be determined through a modification to the cross price
elasticity test, using observable variables like price and demand. Hicks (1939),
24
Schultz (1938) and Hicks and Allen (1934) suggested that instead of using the sign of
the cross price elasticity term as a measure of complementarity, it would be better to
use a compensated price change to measure complementarity. The compensated price
change term considers the effect of not only the change in price, but also the effect of
income. A negative value of the compensated price change implied a complementary
relationship. (Deaton and Muellbauer 1980). Samuelson (1972) also developed a
measure for complementarity called the money metric. He proposed that if we
consider the von Neumann utility function (as a money metric, with diminishing
returns to the marginal utility of income), then two products are complements if
0
[.]
2
>
? ?
?
j i
q q
V
, where V[.] is the von Neumann utility function and q
i
is demand for
product i and q
j
is demand for product j.
In marketing, Guilitinan (1987) provides an excellent overview of how
complementarity can arise between any two products. Two products become
complements in the following situations:
Savings in Search Economies: A motorist prefers going to a dealer who does
both an engine tune up and an oil change as opposed to two dealers separately
offering only one of the services each. The savings in time and effort gained
by the consumer, by going to a single dealer creates a degree of
complementarity between the services offered by the dealer.
Enhance Customer Satisfaction: Guilitinan (1987) offers the example of a ski
lodge that provides both ski lessons and rentals. The combination of services
offered would enhance the satisfaction that a beginner gets from the ski lodge.
25
Improved Total Image: He provides the example of a Firm that offers both
lawn care as well as shrub care services, thereby enhancing the image of the
Firm.
Shocker et al. (2004) also attempt to define complementary products. They
identify three types of complementary relationships, perfect complements (e.g., video
cassette recorder and video cassette), augmenting complements (products that add
new benefits not present in an already existing one, e.g., washing machine and dryer)
and enhancing complements (new products that improves the sales of an already
existing product, e.g., clipart and presentation software).
Inspite of there being a lot of research on bundling of products (Hanson and
Martin 1990, Bakos and Brynjolfsson 1999, Venkatesh and Mahajan 1993, Chung
and Rao 2003), only Venkatesh and Kamakura (2003) have looked at the
relationships between complementary products using contingent valuations. They
derive optimal bundling and pricing strategies taking into account the contingent
valuations of the two products of a monopolist. They compare the derived pricing
schemes to the pricing scheme if these products were priced independently of each
other. They show that the price increases monotonically as a function of the
contingent valuation.
The aim of this study is to extend the paper by Venkatesh and Kamakura
(2003) to incorporate the effects that competition would have on the optimal pricing
strategy. This condition has not been explored in their study. We propose that the
price will depend on both the manufacturer’s and competitor’s contingent valuations.
26
We also propose that the price that accounts for these valuations, will always be
higher than if they were priced independently (when market share is the same),
provided the valuations always have a positive value (in the case of substitutes these
valuations could be negative).
We assume that the firm finds it is disadvantageous to offer an explicit bundle of its
products in the market. This situation can arise under two conditions:
1. Competition: Anderson and Leruth (1993), show that in a duopoly
environment, only pure components pricing may be offered in
equilibrium since Firms fear the extra degree of competition inherent
in offering the option of a bundle (in the case of complements). Our
study differs from Anderson and Leruth (1993) in that they do not
consider contingent valuations or try to formally develop a pricing
scheme. Our assumption is further supported by Matutes and Regibeau
(1992) who also show that in a competitive setting the pure
components strategy is dominant.
2. Conditions for Legality: Stremersch and Tellis (2002) raise the issue
of legality of introducing product bundles in a competitive market.
U.S. Law has two rules which determine the legality of any bundle, the
‘per se rule’ and ‘rule of reason’. The per se rule says that bundling is
illegal when it involves pure bundling of separate products by a Firm
with market power and when a substantial amount of commerce is at
stake. The rule of reason says that bundling is illegal when it involves
pure bundling of separate products by a Firm with market power,
27
involving a substantial amount of commerce, which poses a threat that
the bundling Firm will acquire additional market power over at least
one of the products that is bundled with the tying products and no
plausible consumer benefits offset the potential damage to
competition. Details of the above rules are provided in the Stremersch
and Tellis (2002) paper.
Hence, under such conditions, as explicit bundling is either disadvantageous
or restricted due to legal reasons, it is important for a Firm that manufactures two
different yet related complementary products to develop an optimal pricing scheme
that accounts for the relationship of the two products. We restrict the analysis in this
paper to this scenario.
2.2 Complementary Pricing
Liao et al. (2002a, 2002b) study the effect of pure component pricing on the
equilibrium of the market. They show that if Firms are restricted to pure component
pricing scenario, then the market attains a stable equilibrium only when the two
products made by the Firm are incompatible with the products made by the
competing Firms. However, if the products are compatible, then the market attains a
stable pure strategy equilibrium only if the bundle pricing of the complementary
products is allowed.
In the marketing literature, Reibstein and Gatignon (1984) develop a model for
optimally pricing related products in a product line by extending the mathematical
28
model developed by Urban (1969). They develop a model that includes cross-
elasticities of the various products made by the manufacturer, and use the method of
seemingly unrelated regression to estimate the model.
Complementarity for the most part has been largely ignored by the marketing
literature (Elrod et al. 2002), however there do exist some models of pricing
summarized by Tellis (1986), which are presented below:
1. Captive Pricing: Captive pricing occurs when the manufacturer
charges a low price for the base good and then charges a higher
monopoly price for the accessory product (the base good is assumed to
work only with the accessory). This is successful because the
consumers might not view the basic product they purchased as a sunk
cost and hence may try and recover their ‘money’ by purchasing the
accessories and using it (‘Sunk Cost Effect’, Thaler 1980, 1985). The
consumer may also use this product more than expected and hence buy
more of the accessory. This has led researchers to label this scheme as
the ‘captive pricing scheme’.
The main restraint on captive pricing is that, if the manufacturer
charges a high price for the accessory, it could lead to the entry of
competing Firms in the market for the accessory, thereby reducing the
Firm’s net profit.
2. Two-Part Pricing: The price here is broken into a fixed fee and a usage
fee. Two part pricing is a type of captive pricing scheme that is
adapted to services (Oi 1972). The usage fee helps Firms exploit the
29
heterogeneity in demand for the products. Hence the heavy user would
pay more to use the service than the light user. Oi (1972) illustrates
this by applying this model to the case of Disneyland where the
customer would pay an entry fee to enter the park and then a usage fee,
i.e., spend on rides and other items in the park based upon their
demands for these items.
Sales from both captive and two-part pricing are classified as types of tie-in
sales. Apart from the legal difficulties mentioned earlier, tie-in sales suffer from some
other legal difficulties also. For example the procedure of tying the buyer to purchase
the supplies from the same Firm that manufactured the base product may be
considered to be illegal under the Sherman Act of 1890 or the Clayton act of 1914
(Burstein 1960, Mathewson and Winer 1997). This paper thus aims to develop a
pricing scheme, when tie-in sales are not an option, by considering the contingent
valuations of the products.
3. Model Formulation
3.1 Defining the Market Structure
First we describe the market structure. We consider two multi-product Firms
(i = 1, 2). Each Firm has two products, one in category A and one in category B. The
Firms compete with each other for market share in both markets A and B. Firm i
30
manufactures products A
i
and B
i
and competes with the products made by the other
Firm. This structure is illustrated in figure 5.
Figure 5: The Market Structure
Both products A
i
and B
i
share a complementary relationship which is
parameterized by ?
i
. ?
i
measures the degree of complementarity between the two
products within the Firm i. Thus ?
i
is the contingent valuation of relationship between
the two products.
The notion of contingent valuations arises when a consumer perceives the two
products as if they were a bundle, complementing each other in some way. This
perception of complementarity could be because the products belong to categories
that could be consumed together. Schmalansee (1982) indicated that the consumer’s
reservation price for the bundle of A
i
and B
i
would be higher than the sum of the
FIRM 1 FIRM 2
M
A
R
K
E
T
A
1
B
1
A
2
B
2
?
1
?
2
Competing
Products/
Firms
31
stand alone reservation prices for the products when the products are complements.
The contingent valuation ?
i
is defined as:
) (
) (
Bi Ai
Bi Ai Bi Ai
i
RP RP
RP RP RP
+
+ ?
=
+
? …………..(1)
where:
RP
Ai+Bi
= Consumer’s reservation price for the perceived bundle of products A
i
and B
i
RP
Ai
= Consumer’s reservation price for the product A
i
RP
Bi
= Consumer’s reservation price for the product B
i
?
i
lies in the range (0,1). When ?
i
= 0, the products do not share a relationship
and thus are independent of each other. When ?
i
= 1, the products are perfect
complements and hence cannot be used independently. An example of this case
would be a video cassette player and a video cassette. When 0 < ?
i
< 1, the products
can be used independently, however the consumer obtains a higher utility buying both
the products (e.g., MS Windows and MS Office). When ?
i
< 0, the products are
substitutes. We however do not consider this case for the purpose of this study.
When ?
i
> 0, it is assumed that the consumer will prefer to buy both products
from Firm i.
3.2 Consumer Demand Formulation
3.2.1 Contingent Valuation Condition
The next step in the analysis would be to model the consumer’s demand for
the two products. For convenience we perform our entire analysis with respect to
32
Firm 1, i.e., we provide optimal pricing schemes for Firm 1 when it competes with
Firm 2.
Therefore we must first define the utility that each consumer would derive
from the each product marketed by Firm 1.
If a consumer buys only product A
1
, then the utility that the consumer derives
owning A
1
is:
……………(2a)
where
u
j,A1,t
= utility obtained by consumer j from product A
1
at time t
?
0
= brand / product - specific parameter
?
1
= price response parameter for product A
1
PA
1
= price of A
1
?
A1,t
= unobserved component of utility derived from product A
1
?
j,A1,t
= the random error term, which follows an iid type 1 extreme value distribution
Similarly the utility for product B
1
in period t for consumer j is given by:
……………(2b)
where
u
j,B1,t
= utility obtained by consumer j from product B
1
at time t
?
0
= brand / product specific parameter
?
1
= price response parameter for product B
1
PB
1
= price of B
1
?
B1,t
= unobserved component of utility derived from product B
1
?
j,B1,t
= the random error term, which follows an iid type 1 extreme value distribution
t A j t A t A j
PA u
, , , 1 1 0 , ,
1 1 1
? ? ? ? + + ? =
t B j t B t B j
PB u
, 1 , , 1 1 1 0 , ,
1
? ? ? ? + + ? =
33
We are interested in determining a pricing scheme for when the consumer
might be interested in purchasing both the products. For this we will modify the
Venkatesh and Kamakura (2003) definition slightly to make the value of one item
depend on whether or not the companion item is purchased.
Let V
A1
=
t A
PA
, 1 1 0
1
? ? ? + ? ………….(3a) and
V
B1
=
t B
PB
, 1 1 1 0
? ? ? + ? ……………….(3b),
where V
A
and V
B
are the values that the consumer attaches to the products A and B
respectively.
Therefore the value of A
1
if B
1
is purchased is:
V
A1|B1
= V
A1
+ ?V
B1
……..(4a)
where ? is the contingent value between the two products.
Conversely the value of A
1
if B
1
is not purchased becomes:
V
A1|NB1
= V
A1
……….(4b)
The expected value of product A
1
thus depends on whether or not the companion item
is purchased.
V
A1
*
= P
B1
V
A1|B1
+ (1-P
B1
)V
A1|NB1
= V
A1
+ P
B
?
1
V
B1
…………(5)
where P
B
is the probability that the companion item B
1
will be purchased. This
probability depends on the price of B
1
: PB
1
, as well as the value of B
1
. Hence the
expected value of owning A
1
can be expressed as:
V
A1
*
= V
A1
+ E [f (V
B1
, PB
1
)]…………….(6a),
where ‘E’ is the expectation and ‘f’ represents a function. For simplicity we will
write:
34
E [f (V
B1
, PB
1
)] = ?
0
+ ?
1
PB
1
+ ?
B1
…………..(6b)
Thus V
A1
*
= V
A1
+ ?
0
+ ?
1
PB
1
+ ?
B1
+ ?………….(7)
Thus the consumer’s utility function for A
1
would now depend on whether B
1
is purchased or not. Therefore the consumer’s utility for product A
1
in period t for
consumer j, given that consumer j expects to purchase B
1
also, is given by:
………….(8a)
Similarly utility of B
1
given A
1
is
…………..(8b)
The next step would be to model the net demand for the products A
1
and B
1
.
To model demand we consider the logit demand model. The logit model has been
used extensively in the marketing literature for modeling both household level data
(Guadagni and Little 1983) as well as aggregate market share data (Allenby 1989).
Here we model the aggregate market share using the logit formulation. The utility can
be rewritten as a function of the deterministic part and the random component:
u
j,A1,t
= U
A1,t
+ ?
j,A1,t
…………(9a)
u
j,B1,t
= U
B1,t
+ ?
j,B1,t
…………(9b)
U
A1,t
and U
B1,t
represents the deterministic part, which is the aggregate utility
obtained from A
1
and B
1
. ?
j,A1,t
and ?
j,B1,t
represent the heterogeneity in consumer
preferences for the two products, and they are assumed to be distributed iid type-1
extreme value.
t A j t A t B t A j
PB PA u
, 1 , , 1 , 1 1 1 0 1 1 1 0 , 1 ,
) ( ? ? ? ? ? ? ? ? + + + + + + =
t B j t B t A t B j
PA PB u
, 1 , , 1 , 1 1 1 0 1 1 1 0 , 1 ,
) ( ? ? ? ? ? ? ? ? + + + + + + =
35
To allow for the market share of the brand to expand and contract over the
different periods with the choice of the marketing mix we allow for non-purchases
(eg: an outside good, or choosing not to purchase), this is denoted as product 0. The
utility of the outside product is normalized to 0 across periods; i.e. we assume U
0,t
=
0.
We now derive the choice shares for products A
1
and B
1
with respect to the
option of the outside product 0 and the competitor’s products A
2
and B
2
respectively.
Thus the choice share for A
1
is:
S
A1,t
=
t A t A
t A
U U
U
e e
e
, 2 , 1
, 1
1 + +
…………..(10)
where S
A1,t
is the probability of the choice of product A
1
. Hence the choice share for
product A
1
is dependent on the utility of A
2
, which is a function of the relationship
between A
2
and B
2
, which is given by ?
2
. Thus the choice share for B
1
is hence
analogous to Equation (10) and S
B1,t
is the probability of the choice of product B
1
.
For the next step in our analysis we derive the elasticities of the market share
of the products A
1
and B
1
with respect to the prices of A
1
, B
1
, A
2
and B
2
.
Therefore the first derivatives of S
A1,t
and S
B1,t
with respect to prices of A
1
and B
1
are:
(
¸
(
¸
? ?
? ?
=
(
(
(
(
¸
(
¸
?
?
?
?
?
?
?
?
) 1 ( ) 1 (
) 1 ( ) 1 (
, 1 , 1 1 , 1 , 1 1 1
, 1 , 1 1 1 , 1 , 1 1
1
, 1
1
, 1
1
, 1
1
, 1
t B t B t A t A
t B t B t A t A
t B t A
t B t A
S S S S
S S S S
PB
S
PB
S
PA
S
PA
S
? ? ?
? ? ?
……………(11a)
The first derivatives of S
A1,t
and S
B1,t
with respect to A
2
and B
2
are:
36
(
¸
(
¸
? ?
? ?
=
(
(
(
(
¸
(
¸
?
?
?
?
?
?
?
?
t B t B t A t A
t B t B t A t A
t B t A
t B t A
S S S S
S S S S
PB
S
PB
S
PA
S
PA
S
, 2 , 1 2 , 2 , 1 2 2
, 2 , 2 2 2 , 2 , 1 2
2
, 1
2
, 1
2
, 1
2
, 1
? ? ?
? ? ?
……………(11b)
3.2.2 No Contingent Valuation Condition
We also derive the case when manufacturers are unaware of the existence of
the contingent valuations of the two products, and price the two products accordingly.
This no contingent valuation condition is labeled ‘nc’. This step is carried out to
facilitate comparison and to elucidate the importance of identifying the various
contingent effects.
Therefore in this case the Firm expects that the consumer’s utility function would be
given by:
……….(12a)
………..(12b)
Hence we again split the model into the deterministic component and random
component as follows:
u
i,A1,t,nc
= V
A1t
+ ?
iA1t,nc
………..(13a)
u
i,B1,t,nc
= V
B1t
+ ?
iB1t,nc
………..(13b)
where:
V
A1,t
= deterministic component of utility for product A
1
V
B1,t
= deterministic component of utility for product B
1
Thus market share of product A
1
at time t is given by:
nc t A j nc t A nc nc nc nc t A j
PA u
, , 1 , , , 1 , 1 , 1 , 0 , , 1 ,
? ? ? ? + + + =
nc t B j nc t B nc nc nc nc t B j
PB u
, , 1 , , , 1 , 1 , 1 , 0 , , 1 ,
? ? ? ? + + + =
37
S
A1,t,nc
=
t A t A
t A
V V
V
e e
e
2 1
1
1 + +
…………(14)
Thus market share of product B
1
at time t is analogous to Equation (14).
For the next step in our analysis we derive the elasticities of the market share of the
products A
1
and B
1
with respect to the prices of A
1
, B
1
, A
2
and B
2
.
Therefore the first derivatives of S
A1,t,nc
and S
B1,t,nc
with respect to A
1
and B
1
are:
(
¸
(
¸
?
?
=
(
(
(
(
¸
(
¸
?
?
?
?
?
?
?
?
) 1 ( 0
0 ) 1 (
, , 1 , , 1 , 1
, , 1 , , 1 , 1
, 1
, , 1
, 1
, , 1
, 1
, , 1
, 1
, , 1
nc t B nc t B nc
nc t A nc t A nc
nc
nc t B
nc
nc t A
nc
nc t B
nc
nc t A
S S
S S
PB
S
PB
S
PA
S
PA
S
?
?
…………(15a)
The first derivatives of S
A1,t,nc
and S
B1,t,nc
with respect to A
2
and B
2
are:
(
¸
(
¸
?
?
=
(
(
(
(
¸
(
¸
?
?
?
?
?
?
?
?
nc t B nc t B nc
nc t A nc t A nc
nc
nc t B
nc
nc t A
nc
nc t B
nc
nc t A
S S
S S
PB
S
PB
S
PA
S
PA
S
, , 2 , , 1 , 2
, , 2 , , 1 , 2
, 2
, , 1
, 2
, , 1
, 2
, , 1
, 2
, , 1
0
0
?
?
……………(15b)
In the above equations 15a and 15b, as we assume the ‘nc’ condition the off-
diagonal terms are zero in contrast to equations 11a and 11b. Specifically there is no
effect of the change of price of A
2
on the market share of B
1
or of the change of price
of B
2
on the market share of A
1
. It is also evident that the change in price of A
1
has to
effect on the market share of B
1
and vice versa.
3.3 Manufacturer’s Profit Function
We now proceed to derive the manufacturer’s profit function for the two
situations, namely, the contingent valuation condition and the no contingent valuation
condition.
38
3.3.1 Pricing with no contingent valuation
We first consider the case when the manufacturer prices not realizing the
presence of a relationship between the two products it manufactures.
The manufacturer profit function ?
t,nc
in the nc condition is given by:
nc t B nc nc t A nc nc t
S N CB PB S N CA PA
, , 1 1 , 1 , , 1 1 , 1 ,
) ( ) ( ? ? + ? ? = ? ……………..(16)
where:
CA
1
: marginal cost of product A
1
CB
1
: marginal cost of product B
1
N: Total number of consumers in the market for the products.
Maximizing the profit equation with respect to PA
1,nc
and PB
1,nc
we get:
PA*
1,nc
=
) 1 (
) 1 ( 1
, , 1 1
, , 1 1 1
nc t A
nc t A
S
S CA
?
? +
?
?
………………(17a)
PB*
1,nc
=
) 1 (
) 1 ( 1
, , 1 1
, , 1 1 1
nc t B
nc t B
S
S CB
?
? +
?
?
………………(17b)
Equations (17a) and (17b) represent the optimal prices charged in the ‘nc’ condition.
3.3.2 Pricing with Contingent Valuations
The next step in our formulation of the optimal prices is to define the
manufacturer’s profit function when contingent valuations are recognized by the
Firm. As we account for the market share of each product in each period, the
manufacturer’s profit function ?
t
can be formulated as:
t B t A t
S N CB PB S N CA PA
, 1 1 1 , 1 1 1
) ( ) ( ? ? ? + ? ? ? = ? …………..(18)
39
Firm 1’s aim is thus to maximize profit with respect to PA
1
and PB
1
in each period.
Therefore we differentiate the profit function with respect to PA
1
and PB
1
, and obtain
the following equations:
N S S CB PB S S CA PA S
PA
t B t B t A t A t A
t
? ? ? ? ? ? + ? ? ? ? + =
?
?
)) 1 ( ) ( ) 1 ( ) ( (
, 1 , 1 1 1 1 1 , 1 , 1 1 1 1 , 1
1
? ? ?
?
..(1
9a)
and
N S S CB PB S S CA PA S
PB
t B t B t A t A t B
t
? ? ? ? ? + ? ? ? ? ? + =
?
?
)) 1 ( ) ( ) 1 ( ) ( (
, 1 , 1 1 1 1 , 1 , 1 1 1 1 1 , 1
1
? ? ?
?
..(19b)
We then solve equations 19a and 19b to get the optimal prices PA
1
and PB
1
for Firm
1’s products A
1
and B
1
. Thus the optimal prices are:
1
, 1 , 1 1 1
, 1 1 , 1 1 1 *
1
) 1 (
CA
S S
S S
PA
t A t A
t A t B
+
?
?
=
? ?
? ? ?
………….(20a)
1
, 1 , 1 1 1
, 1 1 , 1 1 1 *
1
) 1 (
CB
S S
S S
PB
t B t B
t B t A
+
?
?
=
? ?
? ? ?
………….(20b)
This leads us to our first proposition.
40
Proposition 1
The price charged by the manufacturer when contingent valuations are
considered is always greater than the price charged by the manufacturer in the no
contingent valuation condition for all positive values of ?
1
.
Proof
The Equations (20a) and (20b) for the price of the products when contingent
valuations are considered are given below:
1
, 1 , 1 1 1
, 1 1 , 1 1 1 *
1
) 1 (
CA
S S
S S
PA
t A t A
t A t B
+
?
?
=
? ?
? ? ?
………….(20a)
1
, 1 , 1 1 1
, 1 1 , 1 1 1 *
1
) 1 (
CB
S S
S S
PB
t B t B
t B t A
+
?
?
=
? ?
? ? ?
………….(20b)
The equations 17a and 17b when the manufacturer prices in the ‘nc’ condition are
given below:
PA*
1,nc
=
1
, , 1 1
) 1 (
1
CA
S
nc t A
+
? ?
………………(17a)
PB*
1,nc
=
1
, , 1 1
) 1 (
1
CB
S
nc t B
+
? ?
………………(17b)
Comparing the equation (17a) with (20a) we can see that equation (20a) ?
(17a) for all ?
1
? 0, when the market shares are assumed to be equal. Also (20b) ?
(17b) for all ?
1
? 0. We can then rewrite equations (20a) and (20b) as functions of the
price charged in the ‘nc’ condition (17a and 17b) and the premium charged when
contingent valuations are considered to express this result:
41
) 1 (
, 1 , 1 1
, 1 1 *
, 1
*
1
t A t A
t B
nc
S S
S
PA PA
?
+ =
?
?
……….(21a)
) 1 (
, 1 , 1 1
, 1 1 *
, 1
*
1
t B t B
t A
nc
S S
S
PB PB
?
+ =
?
?
……….(21b)
Thus the premium charged when contingent valuations are considered is an
increasing function of the contingent valuations.
Result 1
For a given market share, net profit will always be greater when the
manufacturer prices the products utilizing the symbiotic relationship between them.
Proof
From the above analysis, we can conclude that, when the market shares are
the same, the profit obtained under the ‘nc’ condition will be less than the profit
obtained when the contingent valuations are considered. Thus ?
t
? ?
t,nc
.
One of the main issues of using the logit model in empirical analysis is the
endogeneity of price with market share (Berry 1994). For example, if we consider
product A1, its price PA
1
* and market share S
A1,t
will be correlated. It is thus difficult
to predict whether market share drives the price or vice versa. Berry (1994) suggests
that when estimating the equations with data, the researcher should make use of
appropriate instrumental variables so as to overcome this endogeneity problem.
However, as we are deriving an analytical model for the optimal prices for the Firm,
we follow the procedure employed by Aydin and Ryan (Working Paper). They
simultaneously solve the market share and price equation, to derive the expression for
42
price, which is independent of the market share drivers. Following this approach we
solve for PA
1
* and S
A1,t
, so as to make the price equation of product A
1
dependent on
only the market share of product B
1
and ?
1
. Similarly we solve for PB
1
* and S
B1,t
so
as to make the price equation of B
1
dependent on only the market share of A
1
and ?
1
.
Solving for PA
1
*
and PB
1
*
we get the following expressions:
1
( )
( )
1
1
1
1 , 1 1
1
) 1 (
1 , 1 1
*
1
1
1
) 1 (
PA
2 2 2
1 1 1
1
1 , 1 1
?
?
? ?
?
? ?
? ?
? ?
?
? ?
(
(
(
(
(
¸
(
¸
? +
(
¸
(
¸
?
(
(
(
(
¸
(
¸
+
?
=
|
|
¹
|
\
|
+ ? ?
CA
S
e
e S
LambertW
t B
PA
CA
S
t B
t B
….(22a)
2, 3
( )
( )
1
1
1
1 , 1 1
1
) 1 (
1 , 1 1
*
1
1
1
) 1 (
PB
2 2 2
1 1 1
1
1 , 1 1
?
?
? ?
?
? ?
? ?
? ?
?
? ?
(
(
(
(
(
¸
(
¸
? +
(
¸
(
¸
?
(
(
(
(
¸
(
¸
+
?
=
|
|
¹
|
\
|
+ ? ?
CB
S
e
e S
LambertW
t A
PB
CB
S
t A
t A
……(22b)
2,3
1
Solving 21a and 21b simultaneously eliminates the endogeneity between the firm’s own price and
own market share. However endogeneity between the competing firms’ prices still remains.
2
The Lambert W function is the inverse of the function given by f(x) = xe
x
, where W is the function
that satisfies W(x)e
W(x)
= x for all real values of x. The Lambert W function has a concave shape
(Chapeau-Blondeau, F. and Monir, A 2002; Corless et.al. 1996).
3
Equation (22a) and (22b) are now independent of the effects of the ‘own market share’ of the product.
Hence it is possible to isolate the effects of the contingent valuations on the price without the
possibility of price endogeneity bias due to the relation between price and market share, i.e., changes in
price due to changes in ?
1
will have no effect on the own market share of the product. In Equation
(20a) and (20b) it would not have been possible to isolate the effect of the contingent valuation on
price alone due to the presence of the market share of the product.
43
RESULT 2
Concluding from proposition 1 we know that PA
1
* and PB
1
* will always be
greater than PA*
1,nc
and PB*
1,nc
for all ?
1
> 0. Thus when ?
1
> 0, the premium
charged above the ‘nc’ price will increase monotonically with respect to the
contingent valuation ?
1
, for values between 0? ?
1
?0.45. For values between 0.45 < ?
1
? 1, the premiums decrease monotonically with respect to the contingent valuation ?
1
.
Hence premiums always increase for A
1
are B
1
when 0? ?
1
?0.45 but decrease
slightly when 0.45 < ?
1
? 1.
COMMENT
The variation of PA
1
* with respect to ?
1
is given in Figure 6. It is evident
from the graph that the price of A
1
varies with respect to the contingent valuation.
This result is also applicable to the relationship between the price of B
1
and ?
1
.
For values 0? ?
1
?0.45, valuation of a pair of complements (?
1
> 0) exceed
those for independently valued products (?
1
= 0), the seller thus gains more by
charging higher prices while stimulating the consumers to buy both the products.
Venkatesh and Kamakura (2003) also obtained the same results. However as seen in
our results (Figure 2a), when ?
1
lies between (0.45,1) the optimal price decreases
slightly. This result is different from the results obtained by Venkatesh and Kamakura
(2003). The main reason for this difference is that they did not consider
interdependent utilities and competitive effects.
When ?
1
increases from 0.45 to 1, the Firm’s products become tied to each
other and hence they cannot be used individually. Thus if ?
2
is low, and ?
1
increases
44
the consumers will be locked to Firm 1. When the consumer is locked to a Firm, all
three marketing strategies (pure components, pure bundling and mixed bundling) are
equivalent to pure bundling (Matutes and Regibeau 1992). Hence the price charged
for each individual item would be equivalent to price charged if the products were
sold as a bundle. As the price charged for a bundle is usually lower than when the two
products are sold separately, we would expect a decline in the premium that would be
charged.A very high degree of complementarity between the two products
manufactured by the same Firm implies that the products cannot be used individually
and hence the consumer gets no utility from purchasing just one of the products. This
might explain the reason why we see that the optimal price starts decreasing as ?
1
?
(0.45, 1). However it is important to note that even thought the premium decreases
slightly, it is still higher than if the products were priced in the ‘nc’ condition.
Figure 6: Variation of PA
1
with respect to ?
1
.
RESULT 3
PA
1
* and PB
1
* increase monotonically with respect to the contingent valuation ?
2
.
45
COMMENT
The variation of PA
1
* with respect to ?
2
is given in Figure 7. It is evident
from the graph that the price of A
1
increases monotonically with the contingent
valuation of its competitor’s products. This result is also applicable to the relationship
between the price of B
1
and ?
2
. As ?
2
increases the competitor can afford to increase
its products prices without having any effect on its market share. As the competitor
increases her prices, price competition between the two firms decreases, and hence
Firm 1 can also increase her price slightly.
As ?
2
increases the competitor’s products become increasingly dependent on
each other. Thus if ?
1
is low, Firm 1 can capitalize on the exclusivity of the
competitors products and charge a higher price for its own line of products. This
could be the reason why we do not see a decrease in the premium that Firm 1 charges,
even though the premium charged by Firm 2 decreases when ?
2
lies in the region
(0.45, 1).
Figure 7: Variation of PA
1
with respect to ?
2
.
46
Figure 8 shows the variation of profit for Firm 1 with respect to both ?
1
and
?
2
. We see that profit is maximum for high values of ?
2
and low values of ?
1
. This
result follows directly from result 2 and result 3.
Figure 8: Variation of Profit With Respect to ?
1
and ?
2
3.4 Consumer Surplus
Proposition 2
Consumer surplus is lower when the Firm prices in the contingent valuation condition
than when the Firm prices in the ‘nc’ condition.
Proof:
We derive the consumer surplus for consumer j. Assuming utility has a dollar
value, the consumer surplus, when the Firm recognizes the complementary effects
and charges prices accordingly, is given by,
CS
j,t
= u
j,A1,t
+ u
j,B1,t
– PA
1
* - PB
1
*…………(23a)
When the Firm prices in the ‘nc’ condition, the consumer surplus is,
47
CS
j,t,nc
= u
j,A1,t
+ u
j,B1,t
– PA
1,nc
* - PB
1,nc
*…………(23b)
In equation 23b, the Firm is unaware of the complementary relationships between its
two products, however the consumer enjoys the effects of complementarity between
the two products. Therefore s/he is charged the price derived in the ‘nc’ condition.
Hence (s)he would enjoy a higher utility with a lower price, thereby higher surplus. It
is straightforward to see CS
j,t
? CS
j,t,nc
as PA
1
* > PA
1,nc
* and PB
1
* > PB
1,nc
*. This
can also be observed pictorially in Figure 9. Consumer surplus when the
manufacturer prices the product independently is given by the areas numbered (5) +
(2) + (4). When the products are priced by considering the contingent value, the
consumer surplus is given by the area numbered (5). Thus consumer surplus is higher
when products are priced in the ‘nc’ condition. Thus, it is evident that consumer
surplus decreases when the consumer purchases both products from the same
manufacturer, who prices the products taking into account the contingent valuations.
X Axis: Quantity Purchased
Y Axis: Net Price Paid
A: u
i,A1,t,nc
+ u
i,B1,t,nc
B: u
i,A1,t
+ u
i,B1,t
Figure 9: Consumer Surplus
48
C: Marginal cost of producing A
1
and B
1
(CA
1
+CB
1
)
D: Net Price Paid when the products are charged independently (PA
1,nc
* + PB
1,nc
*)
E: Net Price Paid when the products are charged considering the contingent effects PA
1
* +
PB
1
*
1: Expected Profit at PA
1,nc
* + PB
1,nc
*
2: Expected Profit at PA
1
* + PB
1
*. It is also the lost profit when products are not
priced as if they are related.
1.+ 3.: Net profit achievable at PA
1,nc
* + PB
1,nc
*, given that the entire market is not yet
satisfied.
4: Deadweight loss when the product are increased in price from PA
1,nc
* + PB
1,nc
*
to PA
1
* + PB
1
*.
5: Consumer Surplus after pricing at PA
1
* + PB
1
*
5+2+4: Consumer Surplus when priced at PA
1,nc
* + PB
1,nc
*.
4. Conclusion And Managerial Implications
Main Contribution
This study contributes to the pricing literature by extending the paper by
Venkatesh and Kamakura (2003) study to the case of competition. The study adds to
the literature in the following ways:
1. Venkatesh and Kamakura (2003) show that the optimal price charged
increases linearly and monotonically with respect to contingent valuations
when products are sold as pure components. We show that this is not
necessarily the case, while the optimal price does increase with respect to the
contingent valuation the relationship is not linear or monotonically increasing.
49
The price first increases rapidly for small values of ? and then increases at a
slower rate till ? = 0.45. When 0.45< ? < 1, we see that the optimal price
actually decreases. Thus locking the consumer to the product reduce the
optimal price that the manufacturer can charge and hence would reduce the
optimal profit.
2. We show that the optimal price will also increase with respect to the
competitor’s contingent valuation, albeit at a slower rate.
3. We also derive the surplus that the consumer would enjoy in each of the cases
and show that the Firm can charge a higher price based on the contingent
valuation mainly because the creation of a contingent valuation causes an
increase in the overall surplus that the consumer could enjoy. The firm can
thus take advantage of this increase and charge a higher price to the consumer.
4. We isolate the effect of ? on the price of the product by accounting for the
endogeneity of market share and price. This helps us predict the variation of
price with respect to ?, without the price having any effect on its own market
share. This is significant because we can predict changes in price with respect
to ?, without the effect of market share of the product.
The analysis carried out in the previous sections have helped us establish the
following results:
4. The prices of the products of Firm 1 increase with increase in the contingent
valuation of the products manufactured by Firm 1.
50
5. The prices of the products of Firm 1 increase with increase in the contingent
valuation of the products manufactured by Firm 2.
6. Firm 1 experiences a higher profit for the same market share, when its
products are priced considering the contingent valuations than when they are
priced independently of each other.
7. Consumers who purchase the products would have a lower level of consumer
surplus if the contingent valuation is recognized by the Firm.
Managerial Implications
We have established that the Firm can charge a premium for the products. It is
important for a manager to decide how to split the net premium that can be charged
between the two products. Should the net premium be split equally between the two
products or applied solely to one of the products?
Economides and Viard (2003) answer this question on the basis of the
network externality of the products. They consider two products MS Windows and
MS Office, both made by the same manufacturer. They propose that an optimal
pricing scheme would be to charge high for Office and low for Windows (even
though it costs Microsoft twice as much to develop Windows than it does to develop
Office). They show that the network externality generated through the sale of
Windows would increase the value of Office, and hence the profit lost by Windows
can be recouped by Office. This is however only true of markets with strong network
externality effects.
51
For other markets, we could use the axioms of prospect theory (Kahneman
and Tversky 1979) and mental accounting (Thaler 1980,1985) to understand how to
distribute the premium among the products. For example, when the consumer has to
purchase a base product (e.g., printer) and then make repeated purchases of an
accessory (it is assumed that the accessory is not exclusive) (e.g., ink) to use the base
product, it might be better to charge a higher price for the base product. This is
because the consumer would mostly relegate the price paid for the base product to the
level of a sunk cost (decoupling the costs and benefits, Soman and Gourville 2001),
and look to only the future price of the accessory as the basis for his/her choice of the
accessory product.
Another important example is the razor and blade pricing strategy. This is a form of
captive pricing, since the consumer is locked to the Firm (? = 1) once they purchase
the razor.
Purchase behavior may also be affected by heterogeneity in the willingness to
pay for the base product. For a light user/low end segment it might be better to lower
the cost of the base product, so that the segment enters the market, and increase the
cost of the accessory as they would not use as much of the accessory. This is akin to
the captive pricing scenario talked about earlier. It suffers from an obvious weakness,
the entry of competitors in the accessory market would cause the low (light) end
(user) segment to choose the competitor. In such a situation, for the low end segment,
it is better to split the premium equally between the two products.
52
Implications for Marketing Management
The analysis in this paper provides the following managerial conclusions:
1. Building contingent valuations is important as it helps in limiting the role of
price competition in the market.
2. If the Firm accounts for the existence of contingent valuations when setting
the prices of the different products, the Firm would get a higher profit as it
would get some of the extra surplus that the consumer accrues from owning
the two products.
Therefore it is important for Firms to ensure that its new product lines have a
positive relationship to its existing range of products. To better understand how a
Firm could build contingent valuations into its product line we consider the example
of Apple Computer. In October 2001, Apple introduced its portable MP3 player iPod.
The iPod allowed for ease of portability of each customers MP3 music collection and
was considered a better option to the available MP3 players at that time due to the
fact that it could hold significantly more songs that the average MP3 player and also
because it had a more sophisticated design. Although MP3 music was easy to
download, the practice of downloading music suffered from a multitude of legal
difficulties. This did not permit iPod sales to reach their full potential.
In the beginning of 2003, several new competitors like Dell began entering the
market and selling their own MP3 players at significantly lower prices. Thus Apple
now faced two challenges, one from the legal problems of downloaded MP3 music,
and the second challenge was a price war with the new competitors.
53
As a means to overcome this problem, Apple introduced its iTunes online music
download service. This service allowed consumers to download music both legally
and at a very low price. However Apple ensured exclusivity of the downloaded
music, as it was not in the MP3 format (Kanellos 2004, CNET News). This ensured
that the downloaded music could be played only on the iPod (which was played both
the MP3 and the AAC format, the AAc format was copyrighted by Apple) and not on
competing MP3 players. As the iTunes service allowed for easy and legal downloads,
consumers increasingly turned to iTunes as a source of music. This simultaneously
increased demand for the iPod, as it allowed for portability of the consumer’s new
music collection. Similarly consumers who purchased the iPod found iTunes to be a
convenient source for music for their iPod player. Thus consumers derived a higher
level of satisfaction when they purchased both the iPod and the iTunes service as
opposed to purchasing only one of them. This allowed Apple to charge a higher price
for its iPod player. Apple also escaped a future price competition that would have
ensued with the introduction of new MP3 players by its competitors like Dell
(Dalrymple, Technology Business Research 2003). Hence the introduction of the
iTunes service created a contingent valuation between iTunes and iPod and this
allowed Apple to gain higher market share without sacrificing profits or bundling the
two products together. After introducing iTunes, iPod sales were up 235%
(Hasseldahl, Forbes 2004).
54
5. Future Research
The procedure employed in this paper accounts for and solves the econometric
endogeneity that is present between each firm’s own price and own marketshare.
However, there is an extra level of endogeneity, namely between the focal firm’s
price and the competitor’s price that needs to be considered when deriving
equilibrium prices.
To keep the model tractable we did not consider heterogeneity of the
contingent valuation. In reality, the contingent valuations could vary across
consumers, hence it is important to account for it. However there would be no closed
form solution when accounting for the heterogeneity and hence one must develop
simulation methods when trying to derive optimal pricing schemes accounting for
differences in ?
i
. We could also consider the case when ?
A,B
? ?
A,B
, however we do
not expect the results to be qualitatively different.
55
Chapter 3: Essay 3 - Modeling Loyalty For Better Customer
Relationship Management
Summary
CRM (Customer Relationship Management) interventions like direct mailings
have long been used by firms to improve customer relationships. In this study we
develop a method that will allow the firm to understand the effect of these
interventions on customer loyalty. Loyalty is assumed to be unobserved and hence is
modeled as a latent variable. We use an adaptation of a generalization of the ‘Hidden
Markov Model’ (HMM) called the ‘State Space Model’ (SSM) to better predict a
customer’s loyalty function towards a particular firm or product. The SSM models are
structurally different from HMM models and they offer three main advantages over
HMM models. First they are continuous and are described across all possible
relationship states of the customer, hence we avoid the problem of explicitly choosing
the number of states; second, they can be used to model an infinite number of
relationship states; and third they are better at modeling recursive behavior, which is
necessary when modeling customer behavior that involves the effect of experience.
We adapt the SSM model to our study by combining the generic model with a set of
covariates that we use to better understand customer loyalty. We call this model the
SSMC (State Space Model with Covariates). We also predict the customer’s
probability of purchase given certain marketing actions and the predicted loyalty state
using a hazard model. We combine the hazard model and the SSMC to predict the
56
customer’s probability of purchase at a given loyalty state. We apply this model to
data from a retailer of health and beauty aids, to help them better understand the
effect of their CRM interventions on the customer’s loyalty towards the firm and their
repurchase intentions. We also point out the types of CRM interventions that play a
role in improving the loyalty of the customer to the firm and those interventions that
have no effect. This information can hence help the firm better organize its menu of
CRM interventions. We can also compute the probability distributions across the
loyalty states for each individual customer, thus providing the researcher with
knowledge of each customer’s loyalty state. Finally we introduce a new methodology
to the literature on modeling relationships in marketing. The methodology improves
upon existing methods by allowing for a more flexible and efficient estimation
procedure. We also compare our model’s predictions to those derived from two other
estimation methods. We find that the predictions derived from our estimation
procedure are better than those computed from the other methods discussed.
1. Introduction
Customer relationship management (CRM) is a firm-wide approach to
understanding and influencing customer acquisition, customer retention and customer
value through interactive and relevant information exchange between the firm and the
customer. The past few years have seen a multitude of research in the field of CRM,
exploring several important facets such as selecting the right candidates from a
mailing list and then selecting the appropriate communication or marketing
57
intervention for each of the selected candidates (Elsner, Krafft and Huchzermeier
2004; Venkatesan and Kumar 2004; Rust and Verhoef 2005; Gonul and Shi 1998;
Bitran and Mondschein 1996; Bult and Wansbeck 1995; Banslaben 1992; Roberts
and Berger 1989; Kass 1976; Sonquist 1970), studying the link between satisfaction
and commercial success (Kamakura et al. 2002), the link between customer loyalty
and profitability (Rust, Lemon and Zeithaml 2004; Reinartz and Kumar 2000),
customer profitability heterogeneity (Niraj, Gupta and Narasimhan 2001), customer
loyalty programs (Verhoef 2003) and towards establishing a sound construct of the
CRM process (Reinartz, Krafft and Hoyer 2004). In this essay we develop a
methodology that actually estimates a customer’s loyalty state and allows one to
predict the effect of a customer’s loyalty on their intention to repurchase.
Previous marketing literature defines loyalty as being either psychological or
behavioral in nature. Psychological loyalty considers the underlying motivation of the
consumer to repurchase the same brand. It is based on the attitudes of the customer to
the brand in question (Jacoby and Chestnut 1978). Behavioral loyalty on the other
hand depends on the purchase patterns of the customer. It is hence defined by the
revealed behavior of the customer (Fader and Hardie 1996). Our model derives
estimates of loyalty that indicate a tendency to engage in behavior that results in a
purchase. This is similar to psychological loyalty as described in the literature. We
then predict the probability of repurchase, thereby linking psychological loyalty to the
revealed behavior of the customer, i.e. the customer’s behavioral loyalty. We carry
out the tests in the following way. First we test whether a firm’s CRM interventions
have an effect on the customer’s psychological loyalty towards the firm’s products
58
and estimate the customer’s loyalty states. We then check whether loyalty affects the
probability that a customer will repurchase a product from the firm.
We conduct the above tests using an adaptation of state space models. We
combine the state space model with a set of covariates that are used to predict loyalty.
These models, continuous generalizations of hidden Markov models, allow one to
develop a continuous loyalty function that can help the firm better understand the
customer and also allow the firm to understand the efficacy of their marketing
interventions. State space models supersede hidden Markov models on two counts:
In the hidden Markov model the researcher would have to make assumptions
about the possible number of loyalty states, while in the state space model no
such assumptions have to be made due to the continuous nature of the
function over all possible states
The second advantage lies in the fact that the hidden Markov model assumes
loyalty to be a discrete variable, which is a simplifying approximation. The
state space model allows us to relax this assumption.
Hence using state space models, the firm will learn which CRM
interventions actually have an effect on customer loyalty and hence can better
organize their interventions to take advantage of this new insight. The other
advantage of this methodology lies in the fact that one can even use discrete data to
obtain a continuous loyalty function.
The rest of the paper is organized as follows. We review the existing CRM
models in §2 and point out how our paper improves upon the current literature. Model
59
development is carried out in §3. We describe the data and estimation procedure in §4
and present results in §5. Finally in §6 we provide our conclusions.
2. CRM Models
CRM has become increasingly important in marketing. There are now many
different ways to manage customer relationships in marketing. Direct mail, which
was once synonymous with customer relationship management studies, is now
complemented by a host of other techniques like email marketing and other advanced
procedures for customer targeting etc., which are now increasingly used by retailers
and manufacturers to manage customer relationships. Hence the efficient
management of these customer interventions forms the essence of customer
relationship management. Towards the goal of efficient management, there are many
factors that play an important role, for example the timing of the marketing
intervention, the frequency of purchase, the monetary value of the purchase, the
customer characteristics etc. (Nash 1984). There are many models proposed in
marketing to help firms manage customer relationships by choosing both the right
customers and the right market intervention scheme.
One of the first CRM models was the RFM model (recency, frequency and
monetary value) (Bitran and Mondschein 1996; Roberts and Berger 1989). Recency
refers to the time since last purchase or the number of mailings since last purchase.
Frequency refers to the number of purchases in a given period of time and monetary
value refers to the monetary value of all purchases in a given period of time. Gonul
and Shi (1998) extend the RFM to include a dynamic component and analyze the key
60
determinants of an optimal mailing policy, while simultaneously maximizing both
customer utility and firm profit. The authors discuss that due to its dynamic nature the
model outperforms its single period counterparts. The model proposed in this paper
incorporates the RFM variables and also includes a dynamic component.
There are several other models that exist in the literature with an aim of
improving the CRM intervention policy. These include AID and CH-AID (Sonquist
1970, Kass 1976); the ‘Gains Chart Analysis’ (Banslaben 1992); finite mixture
models (Bult and Wansbeck 1995); the DMLM Procedure (Elsner, Krafft and
Huchzermeier 2004) which determines the optimal frequency size and customer
segmentation of direct marketing activities; customer lifetime value models (Dwyer
1989, Blattberg and Deighton 1996, Berger and Nasr 1998) which used the
customer’s lifetime value as a guide towards developing a better CRM intervention
schedule.
More recently Rust, Lemon and Zeithaml (2004) develop an approach that
allows firms to achieve financial accountability by considering the effect of strategic
marketing expenditures and by relating the improvements in customer equity to the
expenditure required to achieve it. This thus allows firms to distinguish between
customers and hence concentrate on its most profitable customers. Venkatesan and
Kumar (2004) also use a CLV metric for customer selection and marketing resource
allocation by developing a dynamic framework that enables managers to maintain and
improve customer relationships proactively through marketing contacts across various
channels and maximize CLV simultaneously. The authors show that customers who
61
are selected on the basis of their lifetime value provide higher profits in the future
than do customers selected on the basis of several other customer based metrics.
Apart from the methods discussed above there are several other procedures
that allow firms to better manage customer relationships. For example Ansari and
Mela (2003) develop a model to show that (in the case of an email manufacturer) it is
better to customize the email marketing interventions to each individual customer's
taste as this would increase the number of click throughs and hence increase revenue
for the firm. As previously discussed the customer-centric approach in marketing lays
great emphasis on the calculation of customer lifetime value (CLV) which is defined
as the value of future cash flows associated with a customer (Pfeifer, Haskins and
Conroy 2005). CLV measures focus on the future and not on the past. In contrast, the
direct marketing literature use measures of customer’s prior behavior to predict their
future behavior. This is best summarized through the RFM (Recency Frequency
Monetary value) models.
Fader, Hardie and Lee (2005a) by the means of a stochastic model integrate
the RFM paradigm with CLV. They use ‘iso-value’ curves to illustrate the
interactions between RFM measures and CLV and discuss the practical application of
the model. Their approach proposes using the RFM variables as sufficient variables
for an individual customer’s purchasing history and eliminating the need for
additional data to calculate CLV. Additionally Fader, Hardie and Lee (2005b)
develop the beta-geometric/NBD (BG/NBD) as an alternative model to the
Pareto/NBD model developed by Schmittlein et al. (1987). The BG/NBD model
allows for easier estimation of parameters as compared to the Pareto/NBD model.
62
The model is used in cases where predicting the future purchase of the customer is
important to managers as they are interested in estimating the customer’s lifetime
value to the firm.
In our study we are interested in learning about the psychological loyalty of
the customer to the firm. As we do not have any data on the measures of
psychological loyalty of the customer, we need to incorporate loyalty as a latent
variable. Additionally we also model the effect of the CRM interventions on the
transition of the customer across the different loyalty states. Past research (Rust,
Zeithaml and Lemon 2000) used Markov Chain Models to study customer
relationships, migration and retention scenarios. More recently Netzer, Lattin and
Srinivasan (2005) model the dynamics of customer relationships using transaction
data using Hidden Markov models (HMM). Their model of relationship dynamics
incorporates the idea that customer encounters may have an enduring impact by
shifting the customer from one unobservable relationship state to another. The hidden
Markov model (HMM) allows for transitions among latent relationship states and
effects on buying behavior. The dynamics of customer relationships with firms are
more or less continuous, owing not just to purchase encounters with the firm, but also
with competing firms, CRM interventions etc. Hence, we extend this methodology by
employing a state space model, a method that allows us to model relationships as a
continuous function, as a more efficient approach towards studying customer
relationships. We elaborate on the advantages of our method over the HMM in the
next section.
63
Finally, Rust and Verhoef (2005) propose a hierarchical model to manage
customer relationships that individualize rather than segmentize the population of
customers. Their results show that customers are highly heterogeneous in their
responses to marketing interventions. As we conduct our analysis at the level of the
individual customer, we use a hierarchical model to study the customer relationships
and interactions between the customer and the firm.
3. Model Development
We will provide a short description of the scenario that will be considered in
this paper. The case is of a multi-product customer goods company that wants to sell
products to its customers through its own line of stores. The company sends out a
variety of marketing interventions (email, snail mail, etc.) periodically. The aim of
our study is to help the firm understand which CRM interventions strengthen the
loyalty of the customer towards the brand and in turn study the impact of loyalty on a
customer’s intention to repurchase from the firm. We do so by predicting the
customer’s probability of purchase at a given point in time, their loyalty state at a
given time and the likelihood that a customer will make a purchase at a given loyalty
state.
The model described here is one of individual level buying behavior. We
consider a panel of customers and their repeated interactions with the firm. The data
that we use is the typical transaction data that is commonly used in various models of
customer choice. Hence the manufacturer observes not only each customer’s choice
64
history but also the marketing environment at that time. This information will help the
manufacturer better understand the relationship between the customer and the firm.
The relationship is comprised of a longitudinal sequence of encounters, each of which
contain information including, but not limited, to purchases made, whether mailings
were sent, the recency and frequency of purchases etc.
Additionally we define a series of hidden or latent loyalty states for each
customer. These loyalty states are indicative of a tendency of the customer to engage
in behavior that results in a purchase. The transitions between these states are
probabilistically determined and are affected by each relationship encounter. To date
hidden Markov models (HMM) have been used to identify these states and predict
transitions between these states. Hidden Markov models are discrete models. They
have a finite number of different internal states that produce different kinds of
outputs. Typically there are a couple of states for each encounter or a pair of
encounters. The whole dynamical process of producing a relationship function is thus
modeled by discrete transitions between the states corresponding to the different
encounters.
Prior research, as mentioned above, has looked at psychological loyalty as
comprising of discrete states that the customers transition between. While it is useful
to consider loyalty as a discrete variable, it is a simplifying approximation. The
approach we develop in this paper enables us to relax this approximation in a
computationally feasible way. Psychological loyalty is modeled in this paper as a
variable that is continuous across all possible loyalty states. We elaborate on this in
the next section.
65
Additionally, the dynamics of customer relationships with firms are more or
less continuous, owing to not just purchase encounters with the firm, but also with
competing firms, CRM interventions etc. This is of importance to a manager since
customer interactions with the firm are no longer confined to just purchases. They
now also include the firm’s own CRM interventions as well as their competitors’
interventions, which subsequently play a significant role in determining a customer’s
future purchases. For example, an individual who receives a CRM intervention after
making a purchase might be influenced positively by the mailing even if it does not
result in an immediate purchase. Therefore, it is important to understand how these
loyalty or relationship states evolve continuously over time with respect to the CRM
interventions. It would be more beneficial to model the data with a continuous model,
as the frequency with which CRM interventions are sent out is typically much higher
than the frequency of purchase. Loyalty to a firm is hence assumed to be a continuous
function and a good candidate for the task of modeling these loyalty transitions is a
state space model (SSM).
The SSM can be described as the continuous counterpart of the HMM. SSMs
are a general method for the probabilistic modeling of sequences and time-series.
They take the form of iterated maps on continuous state-spaces, and can have either
discrete or continuous valued output functions. They are basically generalizations of
the better known state-space models such as Hidden Markov models (HMMs). A
SSM is, however, more powerful than a HMM. For example, a SSM can represent
infinitely many distinct states as a consequence of their real-valued state-spaces
(Moore 1990, Olivier Bournez 1996). By contrast, HMMs have a finite number of
66
states and hence can be no more powerful than strictly finite-automata. Finite-
automata cannot model many of the recursive structures found in human behavior,
especially instances where learning or the effect of experience is involved. This
occurs because a researcher is forced into organizing such behavior into a
predetermined finite number of states, a step that is not necessary when using a SSM.
A SSM can be expressed across all possible real valued states and transitions between
these states are not confined to a predetermined number. Additionally the estimation
of a hidden Markov model becomes much more inefficient as the number of states
becomes very large (a situation encountered when modeling recursive behavior), a
problem that is avoided in state space models due to their continuous nature.
If we assume x(t) is the observed data, and s(t) is the collection of internal
hidden states of the dynamical system then a standard/generic SSM can be expressed
as follows:
( 1) ( ( )) ( )
( ) ( ( )) ( )
s t g s t m t
x t f s t n t
+ = +
= +
(1)
Both vectors m(t) and n(t) are the noise components of the two equations with
m(t) being the process noise and n(t) the observation noise. Functions f and g are the
linear or nonlinear mappings, with f being the observation mapping and g the process
mapping. Figure 10 provides a visual interpretation of the process involved in a
standard SSM.
67
Figure 10: A standard State Space Model
3.1 Modeling the Customer’s Probability of Purchase
To model customer loyalty and likelihood of purchase we construct a model to
predict the customer’s probability of re-purchase. We are interested in examining the
effect of mailings and other variables on the probability of repurchase. At the same
time we need to control for the time between two consecutive purchases made by the
same household. A model that incorporates both these features (i.e. intrinsic purchase
patterns over time and the effect of marketing variables) is the hazard model. The
hazard model allows us to model the survival of the customer through the subsequent
periods of interest, at the same time controlling for the effects of the marketing
variables that are important to manufacturers, like the effect of CRM interventions on
repurchase.(Gonul and Srinivasan 1993, Vilcassim & Jain 1991). The customer is
s(t-1)
s(t) s(t+1)
x(t-1)
x(t+1)
x(t)
f(.)
f(.)
f(.)
g(.) g(.) g(.)
68
assumed to survive as long as s/he does not make a purchase and the customer dies
when s/he repurchases from the firm in a future period. The hazard model captures
this intrinsic propensity of the customer to repurchase in a computationally feasible
way.
Before defining the probability of purchase we first model the loyalty states
of the customer. There are two approaches to loyalty considered in the literature. The
first being a psychological approach towards loyalty, as was proposed by Jacoby and
Chestnut (1978). The approach distinguishes itself from the behavioral approach (i.e.
repeat purchases of a particular product in a given period of time) by including the
attitudinal aspects of loyalty like the cognitive (the brand is preferable to competitive
offerings), affective (preferential attitude for the brand) and conative (higher intention
to buy the brand as compared to the alternatives) elements (Oliver 1999). For
example loyalty could be viewed as a favorable set of stated beliefs towards the brand
purchased. These attitudes can be gauged by asking how much people like the brand,
feel attached to it, will recommend it to others, and have positive beliefs and feelings
about it – relative to some other competing brands (Reichheld 1996, Dick and Basu,
1994). This then translates into behaviors that result in repurchase. The tendency to
repeatedly purchase the same brand leads to behavioral loyalty. Thus behavioral
loyalty is the propensity to repurchase from the same firm in future time periods and
is defined mainly on the pattern of past purchases. (Fader and Hardie 1996,
Ehrenberg and Scriven 1999).
The approach we develop in our study estimates the hidden loyalty states that
indicate a tendency to engage in behavior that results in a purchase. Hence the loyalty
69
states we derive are similar to the psychological loyalty as described in the literature.
Our study relates the psychological loyalty of the customer to the behavioral
outcomes explained above. We construct a model to estimate the psychological
loyalty of the customer to the firm and then relate this estimated psychological loyalty
of the customer to the behavioral loyalty that the customer exhibits in the form of a
repurchase. Thus the framework developed in this paper relates the two loyalties, one
observed (behavioral loyalty) and the other unobserved (psychological loyalty).
We assume that there is a set of latent psychological loyalty states that
influence behavior. The loyalty states can range from the customer being completely
disloyal to the brand to one in which the customer is completely loyal to the brand. At
a given point in time, each customer is assumed to occupy a particular loyalty state. A
customer’s loyalty state can be affected by several variables like satisfaction, the
purchase experience, affinity to a competitor’s product, marketing activities of the
focal firm, quantity of the brand purchased etc. If the impact is positive it can cause
the customer to transition to a loyalty state that is more loyal than the one s/he was
previously in. On the other hand a negative impact can cause the customer to
transition to a less loyal state.
We divide the time a customer is in each loyalty state into intervals of 3
months. Each quarter is represented by ?, where ? = 1 in quarter 1. At time ?, the
loyalty state for each individual ‘i’ is represented by l
i
(?), and the distribution of these
loyalty states within each individual, is represented by p
i
(l
i
(?)). As we are interested
in estimating p
i
(l
i
(?)) using a Bayesian approach, we assume a prior distribution
across each individual’s set of loyalty states at ? =1, denoted by p
i
(l
i
(1)). The
70
distribution for loyalty varies along the real number line from -? to +?. If the loyalty
state has a value of -? then the customer would be completely disloyal. At the value
of 0, the customer is neither loyal nor disloyal to the brand and is more likely to
engage in switching behavior, as (s)he is indifferent to the experience of using the
brand. At this stage the customer behaves similar to a switcher. At +? the customer
would be completely psychologically loyal to the brand.
We will now proceed to model the customer’s probability of purchase. The
inter-purchase times in the equation for probability of purchase is given by the
number of days since purchase. Here time is indicated by ‘t’ and is calibrated at the
daily level. The probability that customer ‘i’ chooses to purchase product j at time t is
given by
,
i
j t
h . Let ( ( ))
it i
l ? ? be the baseline hazard function for customer i at time t and
loyalty state l
i
(?).
ij
? is a vector of response parameters for customer i at time t for
product j. We assume a prior distribution for
ij
? given by p
i
(
i
? ). X
ijt
is a row vector
of covariates, which include customer attitudes and behaviors towards the brand that
could have an impact on the probability of purchase. Some examples of such
variables that affect the utility of a purchase and hence the probability of purchase
include satisfaction, quantity purchased, price sensitivity, number of repeat purchases,
distance to the store, interaction with frontline employees, trust and the variety of
product offerings (Agustin and Singh 2005; Sirdeshmukh, Singh and Sabol 2002;
Oliver 1999, 1997). The covariates also include marketing variables that are under the
researcher’s control.
Loyalty is also a covariate in the equation for the probability of purchase and
is represented by l
i
(?). Oliver (1999 & 1997) highlights the significant role of loyalty
71
in understanding purchase behavior. He states that the psychological elements, as
defined earlier that constitute our definition of loyalty can lead the customer to
repurchase the brand. At a higher loyalty state a customer’s intention to make a
purchase becomes stronger. Hence, it is important to include loyalty as a covariate in
the purchase equation.
ij
?
is a measure of how much loyalty affects the probability of
purchase. We assume a prior for
ij
?
given by p
i
(
i
? ).
The utility that the customer will derive from purchasing the particular brand
directly impacts the probability of purchase. Brands with a higher utility will have a
higher probability of purchase. As both X
ijt
and l
i
(?) impact the probability of
purchase and hence the utility, we include them as elements in the utility function
(Heilman, Bowman and Wright 2000, Krishnamurthi and Raj 1991). We assume that
the utility that a customer i obtains from purchasing product j is given by
( )
ijt ijt ij i ij ijt
U X l ? ? ? ? = + + (2)
ijt
? is assumed to follow an iid type-1 extreme value distribution.
Therefore, the probability that customer ‘i’ purchases the product ‘j’ at time
‘t’ in a particular loyalty state is given by:
( ( ) ) ( ( ) )
,
( ( ) ) ( ( ) )
( ( ) , , , )
1
? ? ? ? ?
? ? ? ? ?
? ? ?
+
+
?
=
+ ?
i j t i j i i j i t i
i j t i j i i j i t i
X l l
i
j t i i j t i j i j
X l l
e e
h l X
e e
(3)
We assume a general functional form for the baseline hazard as given by
(Vilcassim & Jain 1991):
72
0 1 2 2 3
( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ln
it i it i it i it i it i
l l l t l t l t ? ? ? ? ? ? ? ? ? ? = + × ? + × ? + × ? (4)
where ?t is the time elapsed since the product was purchased. The baseline hazard
function is a very general specification that nests most of the commonly used
probability distributions for inter-purchase times. Vilcassim & Jain (1991) show that
setting certain ?’s to zero allows the function to assume various distributions ranging
from the Weibull, Exponential and the second order power series approximation of an
Erlang-2 type distribution.
We now proceed to model the likelihood function for the hazard model, which
is given by:
1
, ,
( | , ( ), , ) ( ( , ( ), , )) (1 ( , ( ), , ))
? ?
? ? ? ? ? ? ? ? ?
?
= ?
?
v v
i i
iH ijt ijt i ij ij j t ijt i ij ij j t ijt i ij ij
t
L Y X l h X l h X l
(5)
where Y
ijt
is the choice made by the customer and 1
v
? = if a purchase
occurred at time t, and 0 otherwise.
3.2 Modeling Loyalty Transitions
The second step in our analysis is to model the transitions of customer loyalty.
The transitions are allowed to occur in either direction, i.e. in the direction of
increasing or decreasing loyalty. In the state space equation l
i
(?+1) is the loyalty state
in quarter ? +1, l
i
(?) is the loyalty state in quarter ?, and ( ) ? ?
i
is the error term,
assumed to be distributed N(0,?), where ? is the variance of the prior distribution.
We know from previous research (Johnson, Herrmann and Huber 2006, Yi and Jeon
2003 and Oliver 1997) that the loyalty function is also dependant on customer
73
attitudes and behaviors towards the brand as well as firm level marketing variables,
given by X
ij?
, which could have an impact on the probability of purchase. ?
i
measures the effect of X
ij?
on the transition of loyalty from a state at time t to a state
at time ? +1. We also assume a prior distribution for ?
i
given by p
i
(?
i
). To capture
the evolution of loyalty from one time period to another, we need to establish a link
between loyalties across adjacent time periods. Hence we use a state space equation
that allows us to model this link across adjacent time periods. We adapt the SSM
model to our study by combining the generic model with the above mentioned set of
covariates that we use to better understand customer loyalty. We call this model the
SSMC (State Space Model with Covariates).
The state space equation is then defined as follows:
( 1) ( ) ( )
?
? ? ? ? ? + = + +
i i ij i i
l l X (6),
The likelihood function can then be written in the following manner,
L
i
(l
i
(1), l
i
(2),.....,l
i
(T)| ,
?
?
i ij
X ) =
1
1
( (1)) ( ( 1) | ( ), , )
T
i i i i i i ij
P l P l l X
?
?
? ? ?
?
=
? + (7)
Hence the combined likelihood of the system of equations to predict both the
probability of purchase and the loyalty state is given by multiplying equations (5) and
(7):
, ( ),
( | , ( ), , ) ,
i
i l h iH ijt ijt i ij ij i i i i i ij
L L Y X l L (l (1), l (2),.....,l (T) | X )
? ?
? ? ? ? = ? (8)
The likelihood function given in equation (7) has three components, namely, the
initial state distribution, the transitions and the state dependent choice.
74
3.3 The Bayesian Framework
Combining the likelihood function given by equation (7) with the priors on
i
?
given by p
i
(
i
? ), the prior on
i
? given by p
i
(
i
? ), the prior on ?
i
given by p
i
(?
i
) and
the prior on l
i
(?), the loyalty state, given by p
i
(l
i
(?)), we obtain, by Bayes rule the joint
posterior distribution for the hazard model, which is given by:
i
i i i i i i i i i i i i i,l ( ),h
p ( , l ( ), , | data) p ( ) p ( )p ( ) p ( ) L
?
? ? ? ? ? ? ? ? ? ? ? ?
i
l ( ) (9)
The priors for
i
? ,
i
? , ?
i
and l
i
(?) vary across customers as follows:
p
i
(
i
? ) ~ MVN(µ, ?); (10)
p
i
(
i
? ) ~ N(?, ?); (11)
p
i
(?
i
) ~ MVN(?, ?); (12)
µ ~ MVN(0,1.0); ? ~ Gamma(1.0,1.0); (13)
? ~ N(0,1.0); ?~ Gamma(1.0,1.0); (14)
? ~ MVN(0,1.0); ? ~ Gamma(1.0,1.0); (15)
We draw the initial state prior for loyalty from a population wide distribution
of loyalty. We assume that the mean of this distribution is 0 and the variance is 100I
(where ‘I’ is the identity matrix) as we have no specific information about the
variance of the distribution, implying an uninformative prior (uninformative due to
the high numerical value of the variance).
p
i
(l
i
(1)) ~ N(0,100I) (16)
4. Data and Estimation Procedure
We obtained data from a leading manufacturer of beauty products. The
products manufactured by the firm are sold through its own retail outlets. The dataset
75
is divided into two parts: a calibration dataset (also referred to simply as the dataset)
and a validation dataset. The calibration dataset consists of purchases of 250,717
customers over a period of 18 months. The validation dataset consists of purchase
data on the same panel of customers who made purchases over the next 6 months.
The firm has provided data on several variables including the date the customer was
first included in the database, the dates the purchases were made, the amount spent
per purchase, the different types of mailings that were sent and whether the
customer’s email address and physical address was in the system. The company had
provided data on 15 different types of mailings that were sent to the customers in the
dataset. If a mailing was sent to the customer the value of the particular mailing
variable was 1 and if a mailing was not sent then the value of the mailing variable
was 0. We also calculate a cumulative mailings variable which is the sum total of all
the different types of mailings sent to the customer. The average number of mailings
sent out by the firm to a customer over the 18 months was 2.61. We then calculate the
recency of purchase (the time since last purchase), the frequency with which
purchases were made (the number of times the customer made a purchase at the store)
and the monetary value of each purchase for each customer (the total dollar value of
purchases made on each purchase occasion). Finally, we divide the data into three
month intervals at ?=0, ? =1, ? =2, ? =3, ? =4, ? =5, and ? =6.
Estimation is carried out using the MCMC procedure based on a Gibbs
sampling scheme (Geman and Geman 1984). We approximate the posterior
distribution as described in (9) by sampling from the full conditional distributions.
We use equations (10)-(16) to fully specify the model. We derive the estimates of the
76
loyalty states for each quarter and then we derive the estimates of the probability of
purchase at the daily level. We ran 50000 iterations using the WinBUGS software
package, where the first 40000 iterations were used for burn-in and the last 10000
were used for estimation. We controlled for autocorrelation by thinning the
observations-only every 4
th
observation was used for our estimation procedure.
Finally we also checked for convergence by running two chains simultaneously and
monitoring the Brooks-Gelman-Rubin (BGR) convergence diagnostic, where
convergence is said to have been achieved if the BGR statistic for each chain
approaches the value of unity.
We estimate the two equations (3) and (6) using the following covariates.
The repurchase equation (3) included the following covariates: monetary value of the
purchase, whether the customer’s email address is available, whether the customer’s
name and physical address are available, frequency of purchase, whether the purchase
was made on a national holiday, the total number of mailings sent mailings, the
recency of purchase, whether the visit was to return a previous purchase, and finally
the predicted loyalty of the customer. The equation for predicted loyalty (6) includes
the following covariates: the recency of purchase, monetary value of the purchase,
frequency of purchase, and 15 variables that indicate whether the particular type of
mailing was sent or not. These variables have a value of 1 if the mailing was sent and
0 if they were not. The mailing variables included are: product mail1, product mail2,
product mail3, product mail4, product mail5, product mail6, product mail6, product
mail7, relationship mailing1, relationship mailing2, relationship mailing3, action
mailing1, action mailing2, action mailing3 and action mailing4.
77
5. Results and Discussion
Table 1a and 1b present the descriptive statistics of the variables in the
dataset. Table 2 reports the posterior means and posterior standard deviations of the
estimation procedure outlined in the preceding section. In figure 11 we see the
distribution of the population wide loyalty function. The function is centered close to
a mean of zero on the x-axis. An implication of this result is that on average
customers are neither loyal nor disloyal to the firm’s products.
Figure 11: Population-wide distribution of the individual means of the
loyalty function
Variable Mean Std Dev
Monetary Value 11.0881 15.63892
Recency 543.9177 136.3432
Frequency 3.001363 1.184601
Mailings 2.612951 1.800003
Table 1a: Descriptive Statistics – Mean and Standard Deviation
ploy1 sample: 10001
-0.5 -0.25 0.0 0.25 0.5
0.0
2.0
4.0
6.0
78
Variable Value Frequency (%)
Customer Email Address 1 24.86
Customer Name & Address 1 79.87
Product mail 1 1 4.02
Product mail 2 1 43.78
Product mail 3 1 5.87
Product mail 4 1 11.13
Product mail 5 1 2.36
Product mail 6 1 7.05
Product mail 7 1 4.59
Product mail 8 1 4.41
Relationship mailings 1 1 2.62
Relationship mailings 2 1 4.87
Relationship mailings 3 1 5.74
Action mailings 1 1 7.63
Action mailings 2 1 0.91
Action mailings 3 1 3.53
Action mailings 4 1 3.11
Holiday 1 0.66
Table 1b: Descriptive Statistics – Frequency
5.1 The Loyalty Equation
The variables included in the analysis are presented in table 2. We see from
the results that frequency of visits has a significant effect on loyalty of the customer
towards the firm. We find that frequency has a positive effect on the customer’s
loyalty, indicating that customers who visit the company store more often tend to be
more loyal to the company’s products. Neither the recency of purchase nor the
monetary value of purchase has a significant effect on the customer’s loyalty towards
the firm. This result is of interest to firms because it implies that the amount a
customer spends at a store on a given purchase occasion does not necessarily relate to
how loyal the customer is to the firm’s products.
79
We now examine the effect of the firm’s CRM interventions on the predicted
loyalty of the customer. We segment the CRM interventions in the following way:
1. Interventions that provide information about a specific product type or
offering are classified as product mails. The variables in this category include:
Product mailings 1 through 8. From the results shown in table 2, we find that a
majority of the mailings that fall into this category have a significant effect on
the loyalty of the customer towards the firm. Product mailings 1, 2, 3, 4, and 7
have significant positive effects on the loyalty of the customer to the firm.
Mailings that are in this category provide information about the specific
product, thereby arming the customer with knowledge about the firm’s
product offerings. This helps reduce the information uncertainty that the
customer has with respect to the firm’s products. This reduction in uncertainty
also reduces the risk that the customer accepts when purchasing the firm’s
products. We hypothesize that this reduction in risk and increase in
information available to customer help them make a more informed decision
about the firm’s products and hence have a positive effect on the loyalty of the
customer to the firm. Hence such mailings help in improving the loyalty of
customer to the firm.
2. Interventions that provide information about the entire suite of products
offered by the firm in the form of a catalog can be classified as relationship
mails. The variables that fall under this classification are: Relationship
80
mailing 1 through 3. Relationship mailings provide more social benefits (i.e.,
information on the product lines and lifestyle information) and focus on both
relationship building and the creation of additional sales. These mailings tend
to be more involved as they provide more information than just simple
product mailings, and also include information on a much larger variety of
product offerings. As they also involve greater customer participation, these
mailings add towards strengthening the relationship between the customer and
the firm. Hence we find that all the relationship mailings, 1, 2 and 3 are
significant and have a positive effect on the loyalty of the customer to the
firm.
3. Finally, interventions sent at specific times of the year that are meant to
prompt the customer to visit the store are referred to as action oriented mails.
These mails are generally devoid of any information about a particular
product or other offerings. The main purpose of such mailings is to remind the
customer of the firm’s products and induce them to make a purchase. The
mailings that fall into this category are: action mailings 1 through 4. We find
that of these mailings only action mailing 2 plays a significant towards
improving the loyalty of the customer to the firm. These mailings are
generally less involved than the other mailings described above, and their
main purpose as mentioned before is to provide a quick increase in sales.
Hence to a large extent these mailings would inspire feelings of opportunism
in the customer rather than feelings of loyalty, as the amount of investment
81
that the customer makes in using these mailings is less than both relationship
mailings and product mails.
5.2 The Repurchase Equation
The next step of our analysis involved estimating the repurchase equation. The
parameter estimates are also given in table 2. We first analyze the effect of the three
RFM variables, namely recency, frequency and monetary value, on the customer’s
probability to repurchase. The estimates indicate that monetary value of purchase and
frequency or purchase visits are most likely to have a significant effect on the
probability that the customer comes back to the store to make a repeat purchase. The
monetary value of a purchase has a negative effect on the likelihood of repurchase,
i.e. the higher the monetary value of purchase, lower the probability of the customer
coming back to the store. We also find that the frequency with which a customer
makes a purchase visit has a positive effect on the likelihood of the customer
repurchasing from the firm.
Next we examine the effect of CRM interventions on the probability of
repurchase. We model this variable as the cumulative total of all the different types of
mailings received by the customer. We find that the total number of mailings received
by the customer does in fact have a very strong positive effect on the probability of
repurchase. Hence, the efforts undertaken by the firm to send out their CRM
interventions do in fact bear fruit. The analysis suggests that the firm, given the
82
appropriate profit constraints, might even find it in its interest to increase the total
number of CRM interventions sent out.
Repurchase Equation
node Mean sd mean/s.d
Monetary Value µ
1
-0.05 0.007243 -6.90322
Customer Email Address µ
2
-0.02047 1.023 -0.02001
Frequency µ
3
0.3999 0.007179 55.73365
Holiday µ
4
2.14E-04 0.00817 0.026144
Total Number of Mailings µ
5
0.5002 0.007173 69.70584
Recency µ
6
-6.15E-05 0.00829 -0.00741
Returns µ
7
-2.30E-05 0.008258 -0.00279
Customer Physical Address µ
8
8.39E-05 0.008263 0.010156
Predicted Loyalty µ
9
0.02084 0.00702 2.968661
Loyalty Equation
node Mean sd mean/s.d
Frequency ?
1
4.67E-04 1.59E-04 2.939547
Monetary Value ?
2
-9.05E-05 0.01098 -0.00824
Product mail 1 ?
3a
.990 0.1012 9.881423
Product mail 2 ?
3c
1.001 0.1084 9.234317
Product mail 3 ?
3f
1.09 0.09905 10.18677
Product mail 4 ?
3g
1.02 0.09441 10.82512
Product mail 5 ?
3h
0.007276 0.1048 0.069427
Product mail 6 ?
3i
0.008962 0.09926 0.090288
Product mail 7 ?
3k
1.006 0.09742 10.32642
Product mail 8 ?
3n
1.658 2.149 0.771522
Relationship mailings 1 ?
3j
1.112 0.1078 10.3154
Relationship mailings 2 ?
3l
0.998 0.1004 9.940239
Relationship mailings 3 ?
3m
1.007 0.1008 9.990079
Action mailings 1 ?
3b
-0.00332 0.09832 -0.0338
Action mailings 2 ?
3d
1.021 0.09019 11.09879
Action mailings 3 ?
3e
0.8881 0.5952 1.492103
Action mailings 4 ?
3o
1.632 2.142 0.761905
Recency ?
4
2.99E-05 0.00824 0.003625
Table 2: Model Parameter Estimates
We also find that predicted loyalty has a significant effect on the probability
of repurchase. This effect is positive, implying that an increase in loyalty does in fact
increase the probability of the customer coming back to repurchase from the firm.
83
Hence, it would bode well for the firm to work towards increasing the loyalty of the
customer towards its products. We have also shown in section 5.5.1 that certain CRM
interventions like product mails and relationship mailings help improve the
customer’s loyalty towards the firm. Thus the firm would do well to orient its CRM
intervention schedule to include more of these types of mailings as they have been
shown to improve loyalty and loyalty has in turn been shown to increase the
probability that the customer would repurchase from the firm.
Through our analysis we have shown that it is possible to infer a customer’s
latent loyalty towards the firm and examine the effects of a firm’s CRM schedule on
this derived loyalty function. We have also shown that the derived loyalty does in fact
have a strong positive effect on the probability of the customer repurchasing from the
same firm. Hence a firm can use these results to better design their CRM schedule.
5.3 Model Validation
We validated our model’s results by comparing the prediction ability of our
model to two additional models that are used in the literature to study relationship
dynamics. We used the validation dataset to compare our predictions to those of the
benchmark models that are described below. The first model estimated was the
hidden Markov model (Netzer, Lattin and Srinivasan 2005) and the second model
estimated was the loyalty model (Guadagni and Little 1983). Netzer, Lattin and
Srinivasan (2005) show that the HMM is useful at estimating the relationship states of
the customer and then predicting each customer’s probability of purchase. Therefore
we decided to compare our model to the HMM model as described by them.
84
Additionally, the authors also state that the Guadagni and Little (1983) model
was closest in terms of predictive ability to the results obtained from their model.
Hence we chose the loyalty model as described by Guadagni and Little (1983) as our
second benchmark model. The models are summarized below:
Model 1: The first model we compared our predictions to was the hidden Markov
model. We assumed that the customer transitions between three possible loyalty states
namely: loyal, switcher and disloyal. The probability of transition between these
loyalty states is modeled using a logit framework. Similar to the methodology
adopted in our model, we assume that the mailing variables have an impact on the
probability of transitioning from one state to another or remaining in the same state.
Hence the transition probabilities are defined using the following function, where s is
the state at time t, s’ is the state at time t+1 and ?
its
is the effect of the relationship
encounter on the probability of transition between loyalty states and ?
ss’
is the state
specific threshold. The threshold is the value that the cumulative impact of the
encounters has to pass in order for a transition to occur:
'
'
( )
( )
e
( , ')
1 e
? ?
? ?
+
+
=
+
ss its ijt
ss its ijt
X
it X
l s s (19)
Model 2: In the second model, loyalty is modeled using the same formulation as that
devised by Guadagni and Little (1983), where ? is the decay parameter and purchase
it-
1
= 1 if customer ‘i’ made a purchase occurred in the previous period t-1.
l
it
=?l
it-1
+ (1-?)(purchase)
it-1
(20)
The probability of choice is modeled similar to the model shown in (3).
85
Following the procedure employed by Netzer, Lattin and Srinivasan (2005)
we used the RMSPE (Root Mean Squared Predicted Error) and validation log-
likelihood procedures to compare the predictive powers of the three models. The
RMSPE (Root Mean Squared Predicted Error) measures the error between the
predicted purchase probabilities and the actual purchases across customers and time.
The validation log likelihood compares the predictive performance of the models. The
results are shown in table 3. We can see from the results that the state space model
results outperform those obtained from model 1 and 2.
We also compared the differences in means between the repurchase
predictions made by the three models. We checked for differences using a pairwise t-
test, the Bonferroni test and the Tukey test. The tests show that all the means are
significantly different from each other. The results of the analysis are also presented
in table 3.
The SSMC model was also superior to the HMM in terms of computational
efficiency. The HMM model took significantly longer time (almost twice the time) to
achieve convergence than the SSMC. Additionally the SSMC spared the researcher
from making any assumptions about the number of states that the customer could
have. The continuous nature of the SSMC hence provided the researcher with a
significant advantage over the HMM in terms of both the efficiency of the estimation
procedure as well as the ability to avoid making assumptions about the relationship
state of the customer.
86
Model SSMC HMM Loyalty Model
RMSPE 0.3571 0.3703 0.3827
Log Likelihood 80788 83434 85542
Means Comparison t-value Significance
Pairwise t-test of SSMC and HMM -79.181 <.0001
Pairwise t-test of SSMC and Loyalty Model -84.733 <.0001
Means Comparison HMM Loyalty Model
Bonferroni Test SSMC <.0001 <.0001
Tukey Test
SSMC <.0001 <.0001
Table 3: Model Comparison Table – Comparing the Predictive Ability of the State
Space Model with the Hidden Markov Model and the Loyalty Model
Legend: SSMC – State Space Model with Covariates; HMM – Hidden Markov Model
Model 2, while incorporating the importance of state dependence, only
includes the effects of lagged loyalty on the loyalty state of the customer in current
time period. Hence our model, which incorporates customer-firm interactions, allows
additional insights on the effects of different CRM interventions and other variables
of importance to the researcher on the current relationship state of the customer.
Hence from the validation tests carried out above we can see that the SSMC
provides advantages over previous methods in terms of efficiency, better predictive
ability and it also frees the researcher from making any assumptions about the
number of relationship states that the customer might be in.
6. Conclusion
In this paper we used standard transaction data on customer purchase behavior
to predict the relationship dynamics that exist between a customer and the firm.
87
Previous marketing literature solved problems of this type either by developing
models that incorporate only state dependence, or using hidden Markov models. We
develop a state space model to estimate the dynamics of relationships between the
customer and the firm. The SSMC was estimated using a hierarchical Bayes MCMC
procedure to account for both observed and unobserved heterogeneity.
The main contribution of this research is the introduction of a new
methodology to study the relationship dynamics between the customer and the firm,
which helps manufacturers infer the underlying structure of relationship states. The
researcher can dynamically classify customers into the relationship states, and assess
the dynamic effect of interactions between the customer and the firm on the
customer’s relationship state and consequent buying behavior. While the HMM can
also achieve similar results, we show in section 5.3.2 that our model outperforms the
HMM both in terms of its predictive ability as well as the efficiency with which
convergence can be achieved. The number of states of the HMM determines the
number of parameters that need to be estimated, as the researcher has to estimate a set
of parameters for each individual state. On the other hand, the SSMC provides a
major advantage in computational efficiency by allowing the researcher to derive a
continuous probability distribution for the parameters across the states of the
individual customer. Hence the number of parameters to be estimated no longer
increases due to the continuous nature of the state space distribution.
An additional advantage of the SSMC over the HMM is that the researcher no
longer needs to make an assumption about the number of states that the customer
might have. The flexibility of the SSMC lies in the fact that the researcher can predict
88
the most likely relationship state of the customer at a given point in time by simply
studying the probability distribution of the customer at that time across all the
possible states.
The empirical application of the SSMC model is demonstrated in sections 4
and 5. From the results we can see the usefulness of the model in studying dynamic
relationships. The results indicate that CRM interventions do play a role in shifting
the customer from a state of lower loyalty to one of a higher loyalty. Specifically, we
find that CRM interventions that belong to the product mail category and the
relationship mailings category tend to have more of an impact on increasing the
loyalty of the customer than interventions that are more action–oriented, i.e., those
that incentivize the customer into making a purchase immediately through means of a
coupon or sale.
We also find that loyalty plays a positive and significant role in affecting the
customer’s probability of repurchase. The higher the loyalty state of the customer, the
greater is the probability of the customer coming back to the store to make a
purchase. Hence using the SSMC we can test long term impact of the customer-firm
interactions on the relationship between the customer and the firm.
There are two main limitations of this study. Our dataset is comprised of only
those customers who made a purchase in the time period we consider. Customers in
the database who never purchased in the two year span were not included in the
dataset. The exclusion of customers who never make a purchase from the dataset
leads to a selection bias. Such biases can be controlled for by using an indicator
variable that equals 1 if the customer is a purchaser and 0 otherwise. However the
89
lack of data on non-purchasers prevents us from doing so. Hence, due to the
limitation of the dataset, the results derived must be weighted taking the selection bias
into consideration.
The second limitation arises because of the presence of endogeneity in the
model. The endogeneity exists due to the fact that relationship mailings were sent
only to customers who spent >$70 at the store. The problem therefore is that the
marginal distribution of the relationship mailing variable is not independent of the
conditional distribution of the loyalty variable given the relationship mailings. Hence
the relationship mailings variable is endogenous, and this might lead to a bias in the
estimates of the effect of relationship mailings on the loyalty of customer.
The main purpose of this paper is to demonstrate the use of a state space
model to study customer – firm relationship dynamics and illustrate the advantage of
using the state space model over methods prescribed in the previous literature. Hence
our effort was mainly concentrated on using this method to study the customer
relationship characteristics in order to deduce their loyalty states, taking into account
various customer-firm interactions. To keep the model parsimonious we made
simplifying assumptions with respect to the model parameters. A richer dataset,
which includes survey data on variables like customer satisfaction levels, would
provide a better insight into variables that drive loyalty and provide an additional
insight into the effect of these CRM interventions. The use of longitudinal survey data
would be extremely beneficial in determining and shaping a customer’s loyalty state.
Data on channels of purchase would also play an important role in shaping the loyalty
function of the customer.
90
Appendices
Appendix 1: Solving for price and market share in logit demand models.
Solving for price P
i
From (12) in the paper we have,
*
1 (1 )
(1 )
?
?
+ ?
=
?
i i it
i
i it
C S
P
S
. (a)
Simplifying and rewriting (12) we get,
*
1
(1 ) ?
= +
?
i i
i it
P C
S
(b)
Substituting (9) from the paper in (12) we get,
*
1
(1 )
1
?
= +
?
+ +
it
it kt
i i U
i U U
P C
e
e e
, (c)
which can be simplified as
0
*
1
(1 )
? ?
? ?
?
= + +
+
i i i
kt
P
i i U
i i
e
P C
e
. (d)
If we let
kt
U
(1 e ) ? + = , then substituting this in (d) we get,
0
*
1
? ?
? ? ?
?
= + +
i i i
P
i i
i i
e
P C .(e)
Multiplying (e) by
i
? and then subtracting
0
? from both sides, we get
0
*
0 0
1
? ?
? ? ? ?
?
?
? = + + ?
i i i
P
i i i i
e
P C . (f)
Rewriting (f) we have,
0
*
0 0
1
? ?
? ? ? ?
?
?
? + = ? ? +
i i i
P
i i i i
e
P C . (f1)
Taking exponentials on both sides of (f1) and then dividing both sides by ? we have,
* 0
0 0
1
? ?
? ? ? ?
?
? ?
?
? + ? ? +
=
P
i i i
i i i i
e P C
e e
e . (g)
Let
*
0
? ?
?
? +
=
i i
P
e
W , then rewriting (g) we have
0
1 ? ?
?
? ? +
=
i i
C
W
e
We . (h)
It is hence easy to see that (h) is similar to equation (1) in the paper, hence the
solution to (h) is given by
0
1
( )
? ?
?
? ? +
=
i i
C
e
W LambertW . (i.1)
Substituting for W, we have
*
0 0
1
( )
? ? ? ?
? ?
? + ? ? +
=
i i i i
P C
e e
LambertW . (i.2)
Taking the natural logarithms on both sides of (i1) and using the result from equation
(6) we have
0
1
*
0 0
ln( ) 1 ln( ) ( )
? ?
? ? ? ? ? ?
?
? ? +
? + ? = ? ? + ? ?
i i
C
i i i i
e
P C LambertW . (j)
Equation (j) further simplifies to
0
1
*
1 ( )
? ?
?
?
?
? ? +
+ +
=
i i
C
i i
i
i
e
C LambertW
P , (k)
91
which is the closed form solution of P
i
independent of the effect of its own market
share. Substituting for ?, we get the equation (13), which is,
0
0
( 1 )
( )
*
1 ( )
1
? ?
? ?
?
?
? ?
?
+ +
+
=
i i i
k k k
C
i i P
i
i
e
LambertW C
e
P . (k’)
Solving for market share S
it
To simplify notations, we replace 1 ? +
i i
C with ‘?’,
0
? with ‘?’ and
0
1 ? ? ? ? +
i i
C
e with
‘?’. As previously noted, we still maintain the notation(1 ) ? + =
kt
U
e . Thus
substituting this into equation (9) from the paper we have,
S
it
=
*
i i
*
i i
P
P
e
e
?
?
?
? ?
? ?
+
. (m)
Rewriting (m) we have, S
it
=
*
i i
P
e
e e
?
?
?
?
+
. (m1)
Substituting (k) in (m) we have, S
it
=
?
?
?
+e e e
e
LambertW ) (
?
?
?
.
Using the result in equation (1), we can rewrite
as S
it
=
?
?
+
?
?
e
LambertW
e
e
) (
?
?
?
?
. (o)
It is evident that ? = e
-?
e
?
; therefore the (o) becomes,
S
it
=
1
) (
1
1
+
?
?
LambertW
, which can be rewritten as S
it
=
) ( 1
) (
?
?
?
+
?
LambertW
LambertW
. (p)
The equation in (p) is analogous to the result obtained in equation (14). Simply
substituting for the values of ? and ? we get,
0
0
0
0
( 1 )
( )
( 1 )
( )
( )
1
1 ( )
1
? ?
? ?
? ?
? ?
? ?
?
? ?
?
+
=
+
+
i i i
k k k
i i i
k k k
C
P
it C
P
e
LambertW
e
S
e
Lambert W
e
. (p’)
92
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