Description
Study on Essays On Issues In New Product Introduction: Product Rollovers, Information Provision, And Return Policies, Product management is an organizational lifecycle function within a company dealing with the planning, forecasting, or marketing of a product or products at all stages of the product lifecycle.
Study on Essays On Issues In New Product Introduction: Product
Rollovers, Information Provision, And Return Policies
Abstract:-
In this dissertation we study several key issues faced by firms while introducing
new products to market. The first essay looks at product rollovers: introduction of
a new product generation while phasing out the old one. We study the strategic
decision of dual vs. single roll jointly with operational decisions of inventory and
pricing during this transitional period. Our results confirm previous findings and
uncover the role and interaction of several parameters that were not examined before.
In the second essay, we investigate the role of information provision and return
policies in the consumer purchasing behavior and on the overall market outcome.
We build a novel model of consumer learning, and we attain significant analytical
findings without making any distributional assumptions. We then fully study the
joint optimization problem analytically under uniform valuations.
In the third essay, we study competition in the framework described in the
second essay and we identify the potential Nash equilibria and associated conditions.
Our findings demonstrate the efect of competition on return policy and information
provision decisions and provide insight on some real-life observations.
ESSAYS ON ISSUES IN NEW PRODUCT INTRODUCTION:
PRODUCT ROLLOVERS, INFORMATION PROVISION, AND
RETURN POLICIES
by
Eylem Koca
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
2011
Advisory Committee:
Professor Gilvan C. Souza, Co-Chair
Professor Yi Xu, Co-Chair
Professor Michael Ball
Professor Peter Cramton
Professor Martin Dresner
c Copyright by
Eylem Koca
2011
Dedication
I dedicate this dissertation to the two women who define me: my guardian an-
gel, my mother, Halime Hatun, who taught me dedication and perseverance among
other things - she's my light, my inspiration; and my dear love, my wife, Glaucia,
without whom this dissertation would neither be possible nor mean anything - she's
my rock, my soulmate.
ii
Acknowledgments
I owe my deepest gratitude to my advisor, Prof. Gilvan C. Souza, for his
invaluable academic and otherwise counsel, and for not giving up on me even in the
hardest times. He has been not only an incredible mentor, but also a true friend.
I would like to give my heartfelt thanks to my dissertation co-chair, Prof.
Yi Xu, who was always there when I needed his guidance and help, and to Prof.
Cheryl Druehl for her support over the years, and for her advice and contributions
especially in the first part of this dissertation.
I would like to express my thanks Prof. Michael Ball, Prof. Peter Cramton,
and Prof. Martin Dresner for agreeing to serve on my dissertation committee and
for taking the time to review the manuscript. My thanks are also due to Prof.
Itir Karaesmen-Aydin and Dr. Barney Corwin, for their continuous support and
guidance.
Lastly, I am greatly indebted to all who helped me through my Ph.D. studies
with their support, advice, and friendship.
iii
Table of Contents
List of Tables vi
List of Figures vii
1 Managing Product Rollovers 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1
1.1.1 Contribution of This Study . . . . . . . . . . . . . . .. . . .
41.2 Related
Literature . . . . . . . . . . . . . . . . . . . . . . . .. . . .
61.3 Model .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 8
1.3.1 Planning Horizon . . . . . . . . . . . . . . . . . . . . .. . . .
81.3.2
Demand Process . . . . . . . . . . . . . . . . . . . . .. . . . 11
1.3.3 Optimization Problem . . . . . . . . . . . . . . . . . .. . . . 17
1.3.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 19
1.4 Comparison of Rollover Strategies: Numerical Analysis . . . .. . . . 21
1.4.1 Parameters Describing the Planning Horizon:t ,o . . .. . . . 22
1.4.2 Parameters Describing the Arrival Process: p, q,u,¸, M
0
. . . 22
1.4.3 Parameters Describing the Reservation Prices: µ, k . .. . . . 24
1.4.4 Auxiliary Parameters and Summary of Runs . . . . . .. . . . 25
1.4.5 Statistical Analysis . . . . . . . . . . . . . . . . . . . .. . . . 28
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 32
2 Return Policies and Seller-Provided Information in Experience Good Markets
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Literature
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Model . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
2.3.1 Consumer Uncertainty . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Market
Demand . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Analysis of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Structural Properties . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Seller's
Optimization Problem . . . . . . . . . . . . . . . . . . 2.4.3 Characterization of
Optimal Information and Refunds . . . . .
2.5 Jointly Optimal Information, Refund and Price Strategy . . . . . . . 2.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
34
38
42
42
45
47
47
51
52
55
59
3 Managing Return Policies and Information Provision under Competition 61
3.1 Introduction . . . . . . . . . . . . . . ......... ........ . 61
3.2 Literature Review . . . . . . . . . . . ......... ........ . 64
3.3 Competition Model . . . . . . . . . . ......... ........ . 65
3.3.1 Market Share Dynamics . . . ......... ........ . 67
3.4 (o, |) Equilibrium . . . . . . . . . . ......... ........ . 69
3.5 Conclusion . . . . . . . . . . . . . . . ......... ........ . 74
iv
A Appendix for Essay 1 75
A.1 Derivation of h(¸) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.3 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A.4 Normalization of| for the Regression . . . . . . . . . . . . . . . . . . 80
B Appendix for Essay 2 82
B.1 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 82
B.2 Structural Properties of the (o, |) Decision Space . . . . . . . . . . . 82
B.2.1 Boundary for Returns: v
u
(o, |) = p
1
| . . . . . . . . . . . . . 82
B.2.2 Boundary for 100% Sales: v
u
(o, |) = 0 . . . . . . . . . . . . . 83
B.2.3 Boundary for No Sales: v
u
(o, |) = 1 . . . . . . . . . . . . . . 83
B.2.4 Redundant Regions . . . . . . . . . . . . . . . . . . . . . . . . 83
B.3 Analysis of theo, | Decision Space . . . . . . . . . . . . . . . . . . . 84
B.3.1 Ex-Ante Efcient Market: v
u
= p
1
. . . . . . . . . . . . . . . . 84
B.3.2 Positive Consumer Surplus: v
u
> p
1
. . . . . . . . . . . . . . . 84
B.3.3 Some Dissatisfied Consumers, No Returns: p
1
> v
u
> p
1
|
B.3.4 Some Dissatisfied Consumers, Some Returns: v
u
< p
1
| .
B.4 Analysis of Candidate Regions and Proof of Proposition 5 . . .
B.4.1 Region (A) . . . . . . . . . . . . . . . . . . . . . . . . . B.4.2 Regions
(B) and (D) . . . . . . . . . . . . . . . . . . . .
B.5 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . .
B.5.1 Region (C) . . . . . . . . . . . . . . . . . . . . . . . . . B.5.2 Region (D)
. . . . . . . . . . . . . . . . . . . . . . . . . B.5.3 Region (E) . . . . . . . . . . . . . .
. . . . . . . . . . . . B.5.4 Deriving the Optimal Strategy . . . . . . . . . . . . .
. .
B.6 On the Value of Optimal Refund Amount . . . . . . . . . . . .
C Appendix for Essay 3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
.
.
.
.
.
.
85
85
88
88
88
89
89
90
91
91
92
94
C.1 Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 94 C.2 Proof of
Proposition 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 95
v
List of Tables
1.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 1.2
Experimental design. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3 Statistics for diference in profits (dual minus single). . . . . . . . . . 27
1.4 Statistics of multiple linear regression: Main efects. . . . . . . . . . . 29
A.1 Statistics of multiple linear regression: Two-way interaction efects. . 81
vi
List of Figures
1.1 Planning horizon and sequence of events. . . . . . . . . . . . . . . . . 10
1.2 Customer arrival intensities for each rollover strategy. . . . . . . . . . 15
1.3 Customer arrival intensities for diferent responsiveness parameters. . 16
1.4 Weibull densities with diferent shape and scale parameters. . . . . . 17
1.5 Arrival rates for diferent difusion parameters. . . . . . . . . . . . . . 23
2.1 Illustration of consumer learning and heterogeneity with information. 44
2.2 Chronology of events in the two-period setting. . . . . . . . . . . . . 45
2.3 Possible cases for v
u
and corresponding market outcomes. . . . . . . . 48
2.4 Seller's (o, |) decision space for V ? U (0, 1) at diferent price points. 50
2.5 Candidate solutions in the seller's (o, |) decision space for V ? U (0, 1). 53
2.6 Critical regions in the (s, c) plane. . . . . . . . . . . . . . . . . . . . . 56
B.1 Candidate regions for optimality for V ? U (0, 1), p
1
= 0.51 and s = 0.4. 87
B.2 Ordering of solutions according to the the regions in Figure 2.6. . . . 93
vii
Chapter 1
Managing Product Rollovers
1.1 Introduction
Firms, particularly in high-tech markets, increasingly see new product introduction
as a tool to gain or maintain market share, to sustain growth, and to create profits.
Accordingly, firms are under constant pressure for faster time-to-market and shorter
life cycles for many products, and face the challenges of managing these. In addi-
tion to traditional new product development issues such as cost, quality, and time-
to-market trade-ofs, more frequent product introductions result in more frequent
product rollovers—the process of phasing out the old generation while introducing
the new to the market. Therefore, successful product introduction requires success-
ful management of product rollovers, which involves several interrelated decisions
including those on timing, pricing, preannouncing, and controlling inventory.
There are two basic product rollover timing strategies available to a firm. In a
dual product rollover (dual roll), the old generation remains in the market for some
time after the launch of the new; in a single product rollover (single roll), the old
generation is discontinued as soon as the new generation arrives (Billington et al.
1998). Both of these strategies have implications on the operational decisions that
a firm must make. In a single roll, sharp price markdowns may be necessary to
clear excess inventory of the old product. Under a dual roll, the old product retards
1
the difusion of the new product into the market; this may be undesirable as new
products typically command higher margins.
There are numerous real-life examples attesting to the interplay and conse-
quences of inventory, pricing, and timing decisions. Intel had scheduled the launch
of its X48 chipset for PC motherboards in January 2008, when the X38 chipset
would be replaced in the high-end market. However, the launch was delayed for
two months due to pressure from the world's largest motherboard manufacturer,
ASUSTEK, on the grounds that it had too much inventory of X38-based parts that
were marginally inferior to X48-based parts. Other manufacturers who had no in-
ventory problems had to wait until March 2008 although they were ready for launch
in January. Another motherboard manufacturer, ASRock, was first to launch its
P43-based motherboards to the mainstream market in June 2008, while all other
manufacturers were struggling with their inventory of older P35-based parts even
with significant price cuts. In November 2007, AMD introduced deep price cuts
for its older Athlon based processors and rushed its long-awaited, quad-core Phe-
nom processors to the market before the holiday season, even though the processors
had a fault which caused unexpected crashes. Further, AMD was unable to meet
the demand at launch and prices remained higher than Intel's competing quad-core
processors that already greatly dominated the market and performed better. In the
end, although the fault was corrected by March 2008, the highly anticipated Phenom
architecture failed to capture the market share expected (various online technology
news sources).
Despite their importance, product rollovers are commonly mismanaged in
2
practice, while understudied in the academic literature. A study of 126 U.S. durable
goods firms reports that 40% of new products failed after launch (Ettlie 1993), one
possible reason being mismanaged rollovers such as the previous examples. Another
study by Greenley and Bayus (1994) indicates that most U.S. and U.K. firms do
not have a formal decision process for product rollovers. Not only is there just a
handful of scholarly papers that discuss product rollover strategies, but there is lit-
tle consensus among them on what rollover strategy to use under what condition.
Saunders and Jobber (1994) identify 11 rollover strategies, which they call "phas-
ing." They survey U.S. and U.K. managers and find that some sort of dual roll was
used in slightly more than half of them. Billington et al. (1998) and Erhun et al.
(2007) present managerial papers that provide understanding and guidelines derived
from intuition and hands-on experience, but no formal treatment of the problem.
While Billington et al. (1998) associate single (dual) roll with low (high) supply and
demand risk, Erhun et al. (2007) state that oftentimes the industry dictates this
decision and that dual roll is an industry standard for high-tech markets even with
low supply and demand risks. The only two papers to our knowledge that attempt
a formal analysis of product rollovers are Levinthal and Purohit (1989) and Lim and
Tang (2006), but neither model incorporates difusion, a key attribute of high-tech
markets. Although they use diferent terminology, Levinthal and Purohit (1989)
consider three alternative strategies: single roll, dual roll, and dual roll with buy-
back of the old generation. They find that single roll is always better than dual roll,
and that single roll is better than dual roll with buy-backs for modest performance
improvements of the new product over the old. Contrast this finding with the rec-
3
ommendation of Billington et al. (1998), who suggest that a large technological gap
between generations (large product risk) favors dual roll. Lim and Tang (2006) find
that dual roll is optimal when marginal costs across generations are similar, using a
linear deterministic demand structure. A few other authors (Carrillo 2005, Li and
Gao 2008, Druehl et al. 2009) simply assume a particular rollover strategy in their
models, regardless of the environment.
1.1.1 Contribution of This Study
We are not aware of an academic study that provides an integrated, formal treatment
of product rollovers that incorporates the dynamics discussed above; we address this
gap using a comprehensive model of product rollovers that includes pricing, inven-
tory, product difusion, and new product preannouncement (before introduction).
More specifically, our key contribution in this essay is to identify the conditions
under which a particular rollover strategy (single vs. dual) is preferred, and which
factors play the most significant role in this strategy decision. We describe our
approach below.
We focus on successive improved generations of a single product by a firm such
as ASUSTEK. The fact that high-tech products are often introduced on a relatively
regular basis supports our model; this notion of (time) pacing of product updates
may also improve a firm's product development capability (Eisenhardt and Brown
1998). For example, the pacemaker company Medtronics has successfully used a
time-pacing strategy (Christensen 1997). We adapt the multi-generation difusion
4
process by Norton and Bass (1987) to model the arrival process of potential cus-
tomers through the life-cycle of a product. Here, however, an arriving customer
buys the product if the price is lower than her reservation price. In addition, the
firm preannounces the new product sometime before its launch and we study dif-
ferent levels of the market's responsiveness to preannouncements to account for the
potential changes in consumer purchasing behavior due to the preannouncement.
The firm first adopts a product rollover strategy, single or dual roll, then decides
on the quantity for the final build of the old product and the price paths for both
products.
We find that the decision between dual and single roll is not trivial and depends
on a number of (exogenous) factors considered in our model. Specifically, dual roll
is preferred to single roll if (i) the time between product introductions is short,
(ii) the preannouncement occurs at the later stages of the life-cycle, (iii) the old
product keeps more of its value at the end-of-life, (iv) the market is less responsive
to preannouncements, (v) the new product is expected to have a slower market
difusion, and/or (vi) performance improvement between the new and old products
is smaller
1
. We also find that the optimal price paths closely follow customer
reservation prices over time.
In the next section, we show how our work relates to and difers from the
existing literature. We then present our model and its analytical solutions in Section
1
Although some of these factors, such as timing of preannouncement, are in reality not exoge-
nous but decided on by the seller, we treat them as exogenous for tractability and to focus on the
two rollover strategies, and we perform a sensitivity analysis to study their impact on profit.
5
1.3, and Section 1.4, a comprehensive numerical analysis of the factors impacting
the optimal rollover strategy. We conclude in Section 1.5.
1.2 Related Literature
We bring together elements from a diverse literature, incorporating rollover strate-
gies, difusion of innovations (Norton and Bass 1987, Bass and Bass 2004), prean-
nouncements (Farrell and Saloner 1986, Manceau et al. 2002, Su and Rao 2008), dy-
namic pricing (Bitran and Mondschein 1997, Smith and Achabal 1998, Elmaghraby
and Keskinocak 2003), and inventory management at the end of life (Cattani and
Souza 2003).
A stream of research has considered the interaction of difusion and new prod-
uct generations. Savin and Terwiesch (2005) model the difusion efects in a duopoly
and find the optimal launch time. Our model difers from theirs in that we study
a multi-generation scenario and the implications of single versus dual roll strate-
gies. Earlier, Wilson and Norton (1989) determined the optimal time to introduce a
product line extension; thus, the rollover strategy is not relevant. They found that
the second product should generally be introduced immediately or not at all, but ig-
nored price and inventory considerations. Mahajan and Muller (1996) extended this
result in a multi-generational scenario where they found that a monopolist should
introduce the next generation either early in the first product's life cycle, or wait
until it has reached maturity (i.e., sales have peaked).
Pricing of a product over its life-cycle has been addressed by a large number of
6
researchers. Several have focused on finding an optimal pricing pattern, assuming
the sales follow the Bass (1969) model (e.g., Robinson and Lakhani 1975, Bass 1980,
Dolan and Jeuland 1981, Kalish 1983, Horsky 1990). However, these studies found
a pricing pattern that follows the sales growth curve, which is not supported by
empirical data (Krishnan et al. 1999). In more recent work, Krishnan et al. (1999)
present a model extending the Generalized Bass Model (Bass et al. 1994), to find an
optimal price path. None of these papers study pricing considering the next product
generation.
The sequence and timing of new product introductions for two or more prod-
ucts with difering quality levels has been considered as a way to alleviate cannibal-
ism (Moorthy and Png 1992, Chen and Yu 2002, Battacharya et al. 2003, Krishnan
and Zhu 2006). Dhebar (1994) examines the pricing and quality level decisions for a
monopolist introducing two generations of products at fixed times. He finds that the
firm may limit the quality (or features) ofered in each generation to minimize con-
sumer regret. In our setting, the efects of cannibalization are modeled, but sequence
is not considered, and higher quality is always valued more by the customers.
A successful rollover requires inventory management for a product (generation
1) at the end of its life. A related stream of literature focuses on determining the
optimal size of a "final buy" (or "final build") for a product nearing the end of its
life when there is uncertain demand (Teunter and Fortuin 1998, Cattani and Souza
2003). Like this stream of research, we also determine the optimal size of the final
build for generation 1, which in our model is being phased out for introduction
of generation 2. Unlike this stream of research, our model considers the demand
7
interactions - cannibalization - between old and new generations.
In summary, although there is a significant body of research analyzing prod-
uct introduction management, modeling life-cycle demand, and considering pricing
implications, no single work demonstrates the role and interaction of these in the
product rollover process. Our contribution to the literature is to investigate all
these aspects of the problem and present an analytical framework for a unified un-
derstanding.
1.3 Model
1.3.1 Planning Horizon
Consider an infinite horizon where, everyt periods, a firm introduces successive new
generations of a certain product, in order to replace the existing old generation. In
such a setting, a transition takes place between two consecutive product generations
everyt periods; our model focuses on one product rollover that is representative of
this repetitive process. The notation used in this essay is explained in Table 1.1.
Let t = 0 be the time when generation 1 is introduced (made available) to the
market; accordingly, generation 2 is introduced at t =t . Throughout the essay, the
following terms are used interchangeably: generation 1 (2), product 1 (2), and old
(new) product. The planning horizon starts at t =ot ,o e (0, 1), which marks the
time when i) generation 2 is preannounced, and ii) the firm produces a final build of
generation 1 and starts concentrating her production capabilities into assuring that
generation 2 is ready to launch at t =t. An immediate extension of separating these
8
Table 1.1: Notation.
Symbol Explanation
i
j
t
ot
T
j
O
¸
h(¸)
ì
ji
(t)
M
i
m
i
F
j
(t)
p
q
|
p
i
(t)
G
it
(-)
Index for product (generation); i = 1, 2
Index for rollover strategy; j = S (single); j = D (dual)
Time between product introductions
Time of final build after launch of a product; 0 T
j
,
0,
if t s ÷(1 ÷o)t
t ÷ (1÷o)t
F
A
(t ÷ e÷|(p+q) e| ÷e| (1.3)
t (1÷o)t
,
if ÷ (1 ÷o)t< t s 0.
1 +
q
e
p
÷|(p+q) e| ÷e
÷
|
Equation (1.1) is similar to N&B except for the multiplier h(¸), such that
h(¸)F
j
(t ÷t ) is the fraction of potential customers of generation 1 who switch
to generation 2 due to its performance improvement. The fraction of potential
customers for a generation at time t, F
j
(t), is higher than or equal to the cor-
responding F (t) in N&B due to the preannouncement efect; p and q are N&B's
coefcients of innovation and imitation, respectively. In N&B, F (t) = 0 ¬ t s 0,
but here, there is adoption of the new generation after preannouncement (but be-
fore introduction time, i.e., for t s 0), which is denoted by F
A
(t). The parameter
| e (0, ·) represents the responsiveness of customers to preannouncements. If
13
| ÷ 0, then customers are not responsive to preannouncements, and the difusion
process approaches N&B starting at t = 0. If, however,| ÷ ·, then customers
are fully responsive to preannouncements, and (1.2) is equivalent to N&B starting
at t = ÷(1 ÷o)t , the announcement of generation 1. Note that the time argu-
ment in Equation (1.3) is negative as F
A
(-) represents the difusion process due to
preannouncement before a generation is introduced; t s 0.
For generation 2, the arrival rate of potential customers is
j
[(M
0
+ m
1
) h(¸)F
j
(t) + m
2
] F
j
(t ÷t ), for t e (t, (1 +o)t ]
ì
2
(t (1.4)
0,
otherwise.
We further assume that market potentials follow a growth pattern according
to the performance improvement: m
i
=¸M
i
÷
1
and M
i
= M
i
÷
1
+ m
i
, where m
i
(M
i
) is the incremental (cumulative) market potential for generation i. Figure 1.2
demonstrates potential customer arrival intensities during the planning horizon for
p + q = 0.3, q/p = 25, M
0
= 100,¸ = 0.5,t = 20,o = 0.5, and| = 6.275. Note
that the arrival rate for generation 1 is independent of the rollover strategy used
for t st , but for t >t ,ì
S
(t) = 0. Because the market is somewhat responsive 1
to preannouncements (|> 0), the arrival rate for generation 2 at the time of its
introduction att is larger than 0 (zero would be the traditional difusion pattern
of N&B). We also haveì
S
(t)>ì
D
(t) because there is some cannibalization of
2 2
generation 2 by generation 1 in a dual roll. In Figure 1.3, the efect of| is illustrated
using p + q = 0.3, q/p = 25, M
0
= 100,¸ = 0.5,t = 20, ando = 0.5. Note that for
| ÷ 0, the market is unresponsive to preannouncements, and thus the difusion of
generation 2 only starts when it is actually introduced at t =t . For| ÷ ·, the
14
market is fully responsive to preannouncements, and difusion starts immediately
after preannouncement atot , as if generation 2 had been introduced at that time.
M2
Dual Roll, gen. 1
Dual Roll, gen. 2
Single Roll, gen. 1
Single Roll, gen. 2
M
1
ot
t
Time (since release of gen. 1)
(1 +o)t
Figure 1.2: Customer arrival intensities for each rollover strategy.
As stated before, the actual sales rate of generation i at time t depends on the
arrival rate of potential customersì
ji
(t), price p
i
(t) of product i, and the distribution
of customer reservation prices. Price will be discussed in the next section. Customers
of product i at time t have reservation prices distributed according to the cumulative
distribution function (cdf) G
it
(-), and probability density function (pdf) g
it
(-). We
assume that G
it
(-) has the shape of a Weibull distribution, as this distribution is
able to capture a variety of consumer behavior and has been used previously in
the literature (Bitran and Mondschein 1997). The Weibull distribution has two
parameters; the mean is mainly determined by the scale parameter|, and the
variance by both| and the shape parameter k. For illustration, Figure 1.4 plots the
15
A
r
r
i
v
a
l
i
n
t
e
n
s
i
t
y
reservation price distribution g
it
(-) for diferent shape and scale parameters. The
firm knows the distributions for both products at any time; this knowledge feature
is common in most marketing and operations models of consumer behavior.
M2
|÷·, gen. 1
|÷·, gen. 2
|= 6.275, gen. 1
|= 6.275, gen. 2
M
1
|÷0, gen. 1
|÷0, gen. 2
ot = 10 t = 20 (1 +o)t = 30
Time (since release of gen. 1)
Figure 1.3: Customer arrival intensities for diferent responsiveness parameters.
We state the assumptions underlying the demand process in this essay as
follows; we comment on these assumptions later in Section 1.5:
(i) There are no explicit competing firms or products or expectation of any.
(ii) Product generations interact only through the arrival process described above.
Once a customer makes a decision to adopt the new generation, her actual
purchase decision is based on the price of the new generation; she does not
re-evaluate her decision (i.e., consider the old generation) if the price of the
new generation is higher than her reservation price.
16
A
r
r
i
v
a
l
i
n
t
e
n
s
i
t
y
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
| = 5, k = 3.6
| = 8, k = 3.6
| = 10, k = 10.8
| = 10, k = 3.6
| = 10, k = 1.2
00
5
10
15
20
25
price
30
35
40
45
50
Figure 1.4: Weibull densities with diferent shape and scale parameters.
(iii) At any time, there are at most two product generations in the market.
(iv) Prices have no in?uence on the customer arrival processes, although they im-
pact actual sales, because an arriving customer only buys if the price is below
her reservation price. Thus, increasing prices decreases sales monotonically.
(v) Customers are neither price strategic nor do they expect a new generation to
be introduced before its announcement.
1.3.3 Optimization Problem
At the start of the planning horizon, the firm decides on the inventory level for prod-
uct 1, denoted by I
0
= I(ot ), and the price paths for both products, p
i
(t), i = 1, 2,
throughout the horizon such that expected discounted profits are maximized. We
17
f
r
e
q
u
e
n
c
y
assume a unit production cost, c
p
, that is constant over time and across genera-
tions. For product 1, there is a constant holding cost of c
h
per unit per time, and
a constant unit salvage value c
s
< c
p
for any remaining inventory at time T
j
. We
assume that the firm uses a continuous review, instantaneous replenishment inven-
tory control policy for product 2; therefore, there are no holding costs and no lost
sales. We find that the inventory control policy for product 2 does not significantly
afect the comparison of rollover strategies, enabling us to make this simplifying
assumption. We have also analyzed periodic review order-up-to policies and found
that the single vs. dual roll comparison was not significantly afected by the num-
ber of inventory reviews. This result is primarily driven by our assumption that
the firm faces no supply constraints for the new generation. Although some firms
face capacity constraints for new products, particularly immediately after introduc-
tion if the product is popular, our model does not capture this efect, and we leave
investigation of supply constraints for future research.
The discount rate iso and a non-stationary Poisson process, with time-depend-
ent arrival intensityì
ji
(t) as its argument, is denoted by N (-). The profit maximiza-
tion problem depends on the rollover strategy and is solved separately for each
strategy. Given the underlying difusion dynamics, the arrival processes for the two
generations are independent from each other; there are no price or inventory inter-
actions between the two arrival processes. Thus, for each strategy, we can partition
the optimization problem into separate problems for each product.
By selecting a rollover strategy j e {S, D}, the firm faces the following
continuous-time stochastic optimization problem for product 1, where G
it
= 1 ÷ G
it
¯
18
denotes the tail distribution:
T
j
e
÷
o(t÷ot
)
p
1
(t)dI(t) + e
÷
o(T
j
÷
ot
)c
s
I(T
j
)
I0,p1(
max
t
)
H
j
1
s.t.
I (T
j
)
=E ÷
0
ot
t
÷c
h
T
j
ot
e
÷
o(t÷ot
)
I(t)dt ÷ c
p
I
0
(1.5)
I(t) = I
0
÷ N
ot
ì
j
1(u)G
1
u
(p
1
)du , t eot, T
j
. ¯
Note that I(t) describes the inventory remaining at time t and we require that
inventory is nonnegative when the product is pulled from the market. The first term
of the expectation is price times sales rate, the second and third terms account for
salvage and holding costs, respectively, and the last is the cost of final build.
The firm's problem for product 2 is:
(1+o)t
max H
j
2 = E
e
÷
o(t÷ot
)
(p
2
(t) ÷ c
p
) N ì
j
2(t)G
2
t
(p
2
) dt ¯
(1.6)
p (t) 2
1.3.4 Solution
t
The optimal price path for product 2 can be determined in a straightforward manner,
as shown in Proposition 1 below.
Proposition 1 The optimal price path for product 2 satisfies:
p
2
(t) ÷ G
2
t((p 2((tt)) = c
p
, ¬t ¯ p ))
g
2
t 2
(1.7)
This price path is unique if and only if
Proof See Appendix A.
(G2
t
)
2
¯
g2t
19
is a decreasing function of p
2
(t), ¬t.
Proposition 1 shows that the price for generation 2 at any point in time depends
simply on the production cost and the consumer's reservation price distribution at
that time. Due to Assumption 2 in the previous section, there is no interaction with
generation 1 customers. Inventory availability also does not afect price due to the
assumptions on reservation prices and inventory replenishment. If we assume that
G
2
t
(-) = G
2
(-) ¬t, then p
2
(t) will be constant.
The optimization problem (1.5) for product 1 is not tractable due to its
stochastic nature and the existence of multiple decision variables. Proposition 2
below shows that the deterministic version of this problem is asymptotically opti-
mal as arrival intensities grow large.
Proposition 2 Solution to the following deterministic optimal control problem is
asymptotically optimal to (1.5) as M
0
grows large.
T
j
max H
j
I0,p1(t) 1
s.t.
I (T
j
)
=
0
ot
e
÷
o(t÷ot
)
p
1
ì
j
1(t)G
1
t
(p
1
) ÷ c
h
I(t) dt + e
÷
o(T
j
÷ot
)
c
s
I(T
j
) ÷ c
p
I
0
¯
dI(t) = ֓
j
(t)G (p ) for t eot, T
j
dt
1
¯
1
t 1
(1.8)
The optimal price path for product 1 from Equation (1.8) satisfies
p
1
(t) ÷ G
1
t((p 1((tt)) = e
o
(t÷ot
)
c
p
+ co
h
÷ co
h
, ¬t ¯ p ))
g
1
t 1
(1.9)
and is unique if and only if
(G1
t
)
2
¯
g 1t
is a decreasing function of p
1
(t), ¬t.
The associated optimal initial inventory is
T
j
I
0
=
ot
ì
j
1(t)G
1
t
(p
1
)dt ¯
20
(1.10)
Proof See Appendix A.
Note that the optimal price path for product 1 closely follows the reservation
price curve, very similarly to the price path for product 2. The only diference is
that the price for product 1 at any time t also accounts for the holding cost of
inventory incurred betweenot and t. If the reservation price curve for product 1 is
decreasing in t, which is a reasonable scenario considering that the product is ending
its life, then the optimal price will also decrease in t accordingly. Because we find
the solution to the asymptotically optimal deterministic problem, the final build
inventory level I
0
will be exactly sufcient to satisfy all demand between [ot, T
j
]
and there will be no leftover inventory.
To find a price path, one needs to solve equations (1.7) and (1.9) for each time
point t. Given the Weibull distributions for G
it
, there are no closed form solutions for
p
i
(t). Numerically, however, this is straightforward: discretize the planning horizon,
and solve (1.7) and (1.9), through any line-search algorithm, for each discrete t. We
study the problem numerically in the next section.
1.4 Comparison of Rollover Strategies: Numerical Analysis
To develop further insight into the choice of rollover strategy, we turn to numerical
analysis and run a full-factorial experimental design with eight model parameters
at three levels each (low, medium and high). This allows us to better understand
under which conditions of parameter values a particular rollover strategy is preferred,
based on maximal profits resulting from the optimization procedure described in
21
Section 1.3. We now describe our experiment.
1.4.1 Parameters Describing the Planning Horizon:t ,o
The planning horizon is [ot, (1 +o)t ]; however, sales horizons for the two products
difer as shown in Figure 1.1. For the old product, length of the sales horizon (during
the planning horizon) for single roll is (1 ÷o)t , while that for dual roll ist . For the
new product, length of the sales horizon does not depend on the rollover strategy
and is always equal toot . Therefore, givent , a small (large)o indicates a long
(short) sales horizon for the old product under single roll. Consequently, we expect
dual roll to result in higher average profits compared to single roll aso increases.
The efect oft , however, is not as straightforward. A longer time horizon means
higher total sales; however, price may decrease more and there may be downward
substitution, negatively afecting the profit rate. In the following numerical studies,
the time unit is months, and we use (10, 20, 30) months fort ; these are typical
times between product introductions in the high tech industry (Druehl et al. 2009).
Given that 0 p|) or return it for refund
(if v < p|). Thus, given|, v
0
= E[max{V, p|}]. Substituting in (2.1), the ex-ante
valuation of a consumer with ex-post valuation v is given by
v
0
0
v
o
= (1 ÷o)E[max{V, p|}] +ov.
v
o
o
Information
v
1
1
0
(2.2)
Figure 2.1: Illustration of consumer learning and heterogeneity with information.
At ano-state, by definition, the consumer purchases the product if v
o
> p.
From (2.2), one can show that this condition is equivalent to v > v
u
, where
v
u
(o, |)
p + (1 ÷o) (p ÷ E[max{V, p|}]) , o
o e (0, 1], | e [0, 1]
(2.3)
is the threshold ex-post valuation for purchase. Note that the definition of v
u
above
presumeso> 0, since we have homogeneity wheno = 0; either all consumers
purchase if v
0
> p, or none of them purchase if v
0
< p.
44
V
a
l
u
a
t
i
o
n
2.3.2 Market Demand
We conceptualize a two-period model to capture the long-run consequences of seller's
decisions. At the beginning of the first period, the seller announces a new product to
be introduced to the market, and sets the first-period price, p
1
and the refund factor,
|. At the same time, the seller also decides ono and accordingly provides tools to
help the consumers make a more informed purchase decision. The seller announces
the second-period price, p
2
, after the first-period arrivals and their purchase decision.
The product is made available for purchase only at the end of each period; there are
thus two purchase points. The chronology of events is summarized in Figure 2.2.
We assume that this product is sufciently distinguished from existing products in
the market to induce an uncertainty in consumer valuations as studied in this essay.
Seller provides information: o Buyers acquire full information.
0 1 2
time
Seller announces product, Product is available for purchase. Second period arrivals.
available at the end of period 1. First period arrivals.
Seller decides on return policy Seller decides on second period price
and first period price.
Figure 2.2: Chronology of events in the two-period setting.
We normalize the initial size of the potential consumer population to 1. These
first-period arrivals are analogous to the "innovators" as described in Bass (1969),
and are uncertain in their valuations as described above. During the second period,
the innovators "spread the word" such that each first-period arrival who is not
dissatisfied creates a second-period consumer base of g > 0; we assume this process
45
to be deterministic for parsimony
3
. Therefore, the more consumers are dissatisfied
in the first period, the less potential buyers in the second period; this dynamic
efectively captures the future consequences of consumer dissatisfaction. We assume
that even with full refunds (| = 1), the consumers who return their items do not
contribute to market growth in the second period; this efectively incorporates the
negative impact of returns to brand and seller image (Lawton 2008). As a result,
in our model, the larger the parameter g, the greater the negative consequences of
causing consumer dissatisfaction and/or returns. Therefore, hereinafter, we aptly
refer to g as "misfit penalty."
We further assume that the second-period arrivals have full information re-
garding their valuations (through owner experiences and reviews as well as seller
provided informational tools); there is no valuation uncertainty in the second pe-
riod. Finally, we assume that any returns occur at the end of the second period and
any returning consumers leave the market.
This setting enables us to study interesting aspects of seller's decisions:
• Through v
u
, a consumer's purchasing decision in the first period is determined
not only by p
1
, but also byo and|.
• Although providing information has a possible cost to the seller, it enables
3
With this definition, we assume that any first-period arrival contributes to the second-period
market even if he leaves without purchasing. The underlying reasoning is that the difusion (of
information) is triggered by being aware of the product, not by purchasing it. While we take it as
given, the value of g is an indicator of the product's market difusion speed which is afected by
market and product characteristics.
46
consumers to make more informed decisions in the first period and therefore
decreases returns.
• Ofering a generous return policy (high|) increases sales in the first period.
However, as we show in the following section, a sufciently high| results
in consumers purchasing and being dissatisfied post-purchase, which in turn
decreases sales in the second period.
In the next section, we show how the interplay of these three decisions determine
the overall outcome both for the seller and the consumers. We analyze the seller's
optimal decision strategy in detail and solve her profit maximization problem under
a more specific setting.
2.4 Analysis of the Model
2.4.1 Structural Properties
We start our analysis by showing some structural properties of the seller's (o, |)
decision space for a given p
1
. First recall that under our setting, the condition for
consumer purchasing in the first period, v
o
> p
1
, is equivalent to v > v
u
; however,
each consumer realizes her own ex-post valuation, v, only after she purchases the
product. If it turns out that v > p
1
, then the consumer is satisfied. If v < p
1
she is dissatisfied; in this case, if her valuation is as low as to be below the refund
amount (v < p
1
|), then she returns the item, otherwise she keeps it although she
is dissatisfied. As a result, there are three possible market outcomes depending on
47
the value of v
u
. Figure 2.3 illustrates these cases, which we explain below:
Case I: v > p
1
Consumers who purchase
in the first period
All buyers are satisfied
0 p
1
| p
1
v
v
1
Case II: p
1
|sv
u
< p
1
Consumers who purchase in the first period
Dissatisfied buyers
0
p
1
| v
who keep
p
1
Satisfied buyers
1
v
Case III: v p
1
Consumers who purchase in the first period
Dissatisfied buyers Satisfied buyers
Dissatisfied buyers Dissatisfied buyers
who return who keep
v
0 v
p
1
| p
1
1
Figure 2.3: Possible cases for v
u
and corresponding market outcomes.
Case I (v
u
> p
1
): If v
u
> p
1
, all consumers who purchase the product have non-
negative surplus (v > p
1
), since they purchase only if v > v
u
. All buyers are
satisfied, and there are no returns.
Case II (p
1
> v
u
> p
1
|): In this case, there are some dissatisfied buyers (v < p
1
),
but all buyers have v > p
1
|: There are some consumers who are dissatisfied
with their purchase but none of them return their item as the refund amount
is not sufciently attractive.
Case III (v
u
< p
1
|): When v
u
< p
1
|, there are some dissatisfied buyers with p
1
>
48
v > p
1
| who keep their items but also some with v < p
1
| who return for a
refund; there are some buyers who are dissatisfied but not all of them return
their items.
The boundary v
u
= p
1
is of particular importance since it marks the condition
for efcient allocation of the product: When v
u
= p
1
, a consumer's purchasing
condition becomes v > p
1
and therefore, 1) all consumers who value the product at
least as much as its price purchase the product, and 2) all consumers who purchase
the product value it at least as much as its price. Note that, when v
u
= p
1
, this
efcient allocation is achieved ex-ante, as opposed to ex-post (which is possible
through a full refund return policy,| = 1). From (2.3), v
u
= p
1
is satisfied foro = 1
(full information) regardless of the value of|. We show that it can also be satisfied
with partial information (0 E[V ], and we state this result in
Proposition 3 below.
Proposition 3 With partial information, i.e. 0 E[V ], there
exists a |
p
e [0, 1] such that ex-ante efcient allocation is achieved, i.e., v
u
(o, |
p
) = p
1
. Furthermore, v
u
(o, |)> p
1
if ||
p
.
Proof See Appendix B.
Proposition 3 is a significant result as it means that, even without providing
full information, the seller can achieve ex-ante efcient allocation of the product
through a partial-refund return policy, i.e. by setting| =|
p
. In other words,
return policies can be used to substitute for full information in order to minimize
consumer dissatisfaction and returns, even when providing information is costless.
49
This means that under costly information provision, the seller has a clear incentive
to design such a return policy in order to minimize consumer dissatisfaction and
returns.
We see from (2.3) that for a given p
1
, any fixed v
u
value results in a relationship
betweeno and|. Therefore, the three cases above translate into three regions on
the (o, |) plane, as shown in Figure 2.4. In essence, v
u
is on the z-axis in Figure 2.4
and each boundary seen on the (o, |) plane represents the curve satisfying the
specified relationship. Although the graph is plotted for uniform valuations and a
particular p
1
value, we show in Appendix B that it is representative of the general
case (in terms of the signs of first and second order derivatives). Note that the cases
p
1
> E[V ] and p
1
< E[V ] result in diferent graphs since in the latter, v
u
> p
1
for
all| e [0, 1] without full information (0 E[V ] p
1
= 0.45< E[V ]
1.0 1.0
A
0.8
0.6
0.4
Y
O
1
p1
Y
O
I
p1
p1
Y
O
II
p
1
B
Y
O
p1B
A
0.8
0.6
0.4
p1
Y
O
II
p1B
Y
O
p1
Y
O
p1B
Y
O
p1B
Y
O
p1B
III
p1
III
0.2
0.0
0.0
Y
O
B
1
B
p
0.2
0.4
B
0.6
Y
O
0.8
0
1.0
0.2
0.0
0.0
A 1
0.2
V
0.4
B
Y
O
0.6
0
0.8
1.0
Figure 2.4: Seller's (o, |) decision space for V ? U (0, 1) at diferent price points.
50
We now turn our attention to the seller's general profit maximization problem.
2.4.2 Seller's Optimization Problem
Facing the market dynamics described in Section 2.3, the seller maximizes two-
period profits by determining the optimal set of decisions, (o
-
, |
-
, p
-
, p
-
). Demand 12
in the first period is (1 ÷ F (v
u
)), since we have unit market size. Demand in the
second period is (g(1 ÷ F (p
2
))(1 ÷ L)), where L max{0, F (p
1
) ÷ F (v
u
)} is the
rate of dissatisfaction in the first-period market. Fraction of returns is given by
M max{0, F (p
1
|) ÷ F (v
u
)} and an amount of p
1
| is refunded for each return.
Production cost per unit is c and each returned product (if any) has a net salvage
value of s. We assume reasonably that s e (0, 1). Note that we allow for s > c,
in which case there is a profitable market for returned items. We write the seller's
optimization problem in general form as follows and characterize the optimal second-
period price in Proposition 4.
max R = (p
1
÷ c) (1 ÷ F (v
u
)) + (÷p
1
| + s) M + g (p
2
÷ c) (1 ÷ F (p
2
)) (1 ÷ L)
o,|,p
s.t.
i
o, | e [0, 1]
(2.4)
Proposition 4 The optimal price in the second period is given by
p
-
= arg max (p ÷ c) (1 ÷ F (p)).
2 p
Proof The solution for p
2
follows since p
2
appears only in the final term in the
objective function and since g (1 ÷ L) > 0 under any circumstances.
51
Proposition 4 shows that the optimization of p
2
is decoupled from the rest
of the decision process. Therefore in the rest of the essay, we continue studying
the joint optimization ofo, | and p
1
. In the next section, we determine strictly
dominated regions and identify conditions for optimality of others.
2.4.3 Characterization of Optimal Information and Refunds
As discussed in Appendix B, boundaries in the (o, |) decision space, which are
critical for consumer dissatisfaction and existence of returns, are functions of p
1
.
Although this complicates the seller's problem of jointly optimizingo, | and p
1
, we
find that a general characterization of the optimalo and| is possible for a given
p
1
. We summarize our findings in Proposition 5. Note that we do not yet make
any assumptions as to how the consumer valuations are distributed. In the rest of
the essay, we use superscripts for association to the indicated region in the (o, |)
decision space, and we use the subscriptu to indicate a threshold value.
Proposition 5 For a given p
1
, the optimal (o, |) corresponds to one of the two
candidate solutions below, depending on the values of p
1
, g, c and s, and on the
distribution F . We depict these solutions in Figure 2.5.
i) Solution (D): (o
-
, |
-
) = (o, |) ,o e [0, 1 ÷
p1
E[max{V,p1|}
]
], p
1
|
=s÷
F (p1|)F
(p1|)
This solution lies on the region where v
u
= 0 and p
1
| s s. All consumers
purchase but the optimal refund amount is less than the salvage value.
ii) Solution (E): (o
-
, |
-
) = {(o, |) , v
u
= p
1
}
52
This solution implies ex-ante efcient allocation, i.e. v
u
= p
1
, which is
satisfied when o = 1 for any |, or if p
1
> E[V], when | =|
p
for any
o> 0.
1.0
E
0.8
A
0.6
0.4
0.2
0.0
Y
O
I
B
1
B
p
D
II
Y
O
p
1
B
Y
O
0
III
s
B
p
1
0.0 0.2 0.4 0.6 0.8 1.0
B
Figure 2.5: Candidate solutions in the seller's (o, |) decision space for V ? U (0, 1).
Specifically, if g > ¯
E
, then Solution (E) is optimal; if g < ¯
E
, then Solution
(D) is optimal, where
g
u
-
(p
1
÷ c) + (s ÷ p
1
|
-
)
F
F(p(1p|) )1
g
u
and |
-
satisfies
¯
E
=
g
u
(p
2
÷ c) (1 ÷ F (p
2
))
, (2.5)
p
1
|
-
= s ÷ F ((p
1
| -)).
F
-
Proof See Appendix B.
p
1
|
Proposition 5 finds that for sufciently small misfit penalty, Solution (D),
which sells to all consumers in the first period, is optimal; equivalently, for suf-
53
ciently large misfit penalty, Solution (E), which suggests no returns through ex-ante
efcient allocation, is optimal. This result has three immediate corollaries. The first
is that there are conditions that makes the ex-ante efcient allocation undesirable
for the seller. Specifically, if the net profit of selling a product to a consumer in the
first period exceeds the expected loss in the second period due to dissatisfying that
consumer, then the seller chooses (o, |) such that every consumer in the first pe-
riod purchases a product, regardless of his valuation. Since the expected loss in the
second period increases with misfit penalty, we conclude that if the misfit penalty
is sufciently small, then it is optimal to sell to all consumers in the first period.
Furthermore, observe that ¯
E
, which is the threshold for absolute dominance of So- g
u
lution (E), increases in s; the larger the salvage value, the narrower the dominance
region of Solution (E).
Second, ofering a refund amount of more than the salvage value of returned
items is not optimal unless it is optimal to provide full information. On the other
hand, in case of full information, the return policy is redundant (since there are no
returns) and the seller is indiferent in choosing a refund amount. Therefore, we say
that it is weakly suboptimal to ofer a refund amount more than the salvage value.
Note that as long as the price is larger than the salvage value, this also means that
it is weakly suboptimal to ofer a full (100%) refund.
Finally, note that in the case of costly information provision, if p
1
> E[V],
providing full information is never optimal since the seller can instead design a
return policy, by setting| =|
p
, to achieve the same efect. If, however, p
1
< E[V],
then the seller would have more incentive to choose Solution (D); that is, Solution
54
(D) would be optimal for a wider range of misfit penalty.
2.5 Jointly Optimal Information, Refund and Price Strategy
In the previous section, we identified the two candidate solutions for optimal (o, |)
without making any distributional assumptions, but under the assumption that p
1
is given. We observe that there is no clear dominance relationship between these
solutions if p
1
is a decision variable as well. In this section, we assume uniform
valuations, F (p) = p, and identify the candidate solutions for jointly optimizing
information, refund and price, and determine the conditions that lead to the opti-
mality of each solution. Uniform valuations is the most common assumption in the
operations management, marketing and economics literatures (Shulman et al. 2009,
Chesnokova 2007, Villas-Boas 2006, Davis et al. 1998, Chu et al. 1998). We employ
this assumption in this essay in order to facilitate closed-form optimal solutions for
better interpretation. This essentially constitutes the optimal (o, |, p
1
, p
2
) strategy
for the seller, contingent on the values of g, c, and s as given in Proposition 6. We
find that there is a boundary, critical for shaping the optimal strategy, in the (s, c)
plane, which we plot in Figure 2.6.
Proposition 6 With uniform valuations, the joint optimal (o, |, p
1
, p
2
) strategy is
such that;
For (s, c) such that c > c
u
(s) - Region (1) in Figure 2.6 - we have g
E,D
s u
g
E,C
s g
D,C
, and
u u
(a) If g > g
E,C
, the optimal solution is Solution (E), u
55
(b) If g < g
E,C
, the optimal solution is Solution (C); u
For (s, c) such that c < c
u
(s) - Region (2) in Figure 2.6 - we have g
D,C
g
E,D
, the optimal solution is Solution (E), u
(b) If g
D,C
< g < g
E,D
, the optimal solution is Solution (D),
u u
(c) If g < g
D,C
, the optimal solution is Solution (C), u
where
c
u
(s) = 2
1 + s
2
÷ 1, 3
g
E,C
= 4(1(1 + s)2÷ 2(1 c+2c) ,
u
+ s
2
)(1 ÷ )
2
g
E,D
= 4 + 12(1++ss)(1÷ 8(12+ c) ,
u
(4
2 2
)
÷ c)
2
g
D,C
= 3(1 4 c)
2
,
1.0
u
÷
0.8
c
0.6
0.4
0.2
0.0
0.0
0.2
1
0.4
s
c
0.6
c
O
s
2
0.8
1.0
Figure 2.6: Critical regions in the (s, c) plane.
and
56
Solution (E): (o
-
, |
-
) = {(o, |) , v
u
= p
-
}, p
-
= p
-
=
1+
c
,
1 1 2 2
Solution (D): (o
-
, |
-
, p
-
, p
-
) = 0,
4+
ss2 ,
4+
8s
2
,
1+
c
, 4
12 2
Solution (C): (o
-
, |
-
, p
-
, p
-
) = 0,
1+
ss2 ,
1+
2s
2
,
1+
c
. 2
12 2
Proof See Appendix B.
We show in Proposition 6 that, for the joint optimization ofo,|, p
1
and p
2
,
there are three solutions among which the seller chooses, depending on the rela-
tionship between market parameters (c, s and g). Solution (E) corresponds to any
(o, |) pair that satisfies v
u
= p
1
. We show in Section 2.4.1 that this implies efcient
allocation of the product at the time of purchase even under valuation uncertainty.
In other words, exactly those consumers who would purchase the product with full
information do purchase the product. We also show that this can be achieved either
by providing full information or, without full information, by setting the refund
amount to a certain level| =|
p
; when providing information is costless, the seller
is indiferent between these two options. However, under costly information provi-
\
sion, Solution (E) reduces too
-
0 and|
-
=|
p
= 2
c
.
1+c
In either case, since the
consumer purchasing behavior is identical to full information case, the seller sets
the classical monopoly prices: p
-
= p
-
=
1+
c
.
1 2 2
Solutions (C) and (D) are both characterized by v
u
= 0 but at diferent prices
and refund factors; Solution (C) suggests higher values in both:|
-
(C)>|
-
(D) and
p
-
(C)> p
-
(D). Therefore, Solution (C) ofers a larger refund amount than Solution
1 1
(D); the former ofers a refund amount equal to the salvage value (p
-
|
-
(C) = s), 1
while the latter ofers half the salvage value (p
-
|
-
(D) =
1
s). In both Solutions (C)
5 7
1 2
and (D), the seller provides no information and ensures that all consumers purchase
in the first period. Solution (C) ofers a larger refund amount than Solution (D) and
attains higher ex-ante consumer valuations in the first period; as a result, the seller
charges more: p
-
(C) = 1
1+s
2
2
> p
-
(D) = 1
4+s
2
.
8
Due to higher price and higher refund
amount, Solution (C) results in both larger dissatisfaction rate (which is equal to
the price), and larger return rate (which is equal to the refund amount).
The optimal strategy states that Solution (E) should be preferred only when
the misfit penalty (g) is sufciently large. In other words, if the misfit penalty
is sufciently small, the seller chooses to maximize her profits at the expense of
consumer satisfaction. We further see that if the cost of production is sufciently
large, Solution (D) is never optimal; higher production costs require higher prices
to compensate for them and Solution (C) is preferred. With low production costs,
there is a range of misfit penalty for which Solution (D) is optimal; for sufciently
low misfit penalty, Solution (C), which is greedier, is optimal.
We attain an intuitive corollary out of Proposition 6, by noting that the opti-
mality threshold for Solution (E) in Region (1) is g
E,C
; in Region (2), the optimality u
threshold for Solution (E) is g
E,D
. Next, we observe that u
cg
E,D
cg
E,C
u
cs
=
4s(1+c)
2
(1+s2)2(1÷c)2
> 0,
and
u
cs
=
16s(5+2c+c
2
)
(4+s2)2(1÷c)2
> 0. Therefore, for a constant c, larger s results in larger
thresholds, which in turn limits the optimality of range for Solution (E). We con-
clude that when salvage value is higher, it is easier for the seller to reject ex-ante
efcient allocation, thus to allow returns.
58
2.6 Conclusion
In this essay, we study the profit maximization problem for a seller who optimizes
information provision, return policy and prices for a new experience good, over a
two-period horizon. With no information, consumers are fully uncertain of their
valuations of the product. However, given more information, they learn more, ap-
proaching to individual valuations; information creates ex-ante heterogeneity among
consumers. On the other hand, being aware of a return option, consumers update
their valuations of the purchase decision.
We make several important contributions with this study. We devise a novel
approach in understanding and modeling the process of a consumer's learning of own
valuation, taking into consideration both partial information and partial-refund re-
turn policies. Our model incorporates two key parameters; 1) market growth rate,
g, which represents the seller's forward channel capability; 2) s, which determines
the value of returned items and therefore points to the seller's reverse channel ca-
pabilities. Building on the dynamics of interaction between information and return
policy in consumer valuations, we treat the seller's optimization problem analyti-
cally, show structural properties of her decision space, and characterize the optimal
solution for the general case. These characterizations lead to three major findings
that are robust to distributional assumptions. First, if the market growth rate—
and hence the future penalty due to consumer dissatisfaction and/or returns—is
sufciently low, then the seller may choose to provide no information to consumers
even if it is costless to do so. We also show that as the salvage value increases, it
59
becomes easier for the seller to withhold information from consumers. This shows
how the seller's reverse channel capabilities interact with her forward channel de-
cisions. Second, we show that even when it is optimal to ensure ex-ante efcient
allocation, Solution (E), it is not necessary to provide full information as this can
be achieved by devising the return policy appropriately (by setting| =|
p
) and
providing only partial information. This is a significant result as it showcases a
situation where the return policy can be used to substitute for informational tools.
Third, we find that ofering a refund amount that is more than the salvage value is
never exercised; the seller can advertise such a refund amount when Solution (E) is
optimal, in which case there are no returns. Lastly, assuming uniform valuations,
we determine the optimal decision strategy for the seller, which dictates the optimal
values for information provision, refund factor and prices given model parameters.
Future work could investigate opportunistic consumer behavior where a con-
sumer "purchases without intention to keep." A seller can be exposed to such
behavior if she ofers lenient returns; tightening return policies for some sellers
is attributed to their losses due to this type of consumer behavior (Davis 2010).
Davis et al. (1995) and Hess et al. (1996) examine consumers who purchase with-
out intention to keep; however, they assume that consumers have no pre-purchase
information, and they do not consider information provision.
60
Chapter 3
Managing Return Policies and Information Provision under
Competition
3.1 Introduction
More than 8% of total retail sales, $185 billion worth of retail merchandise, was
returned to sellers in 2009 in the US, and predictions for the near future indicate
similar outcomes (Davis 2010). While the substantial implications, in terms of direct
and overhead costs, of these returns for the whole supply chain makes the study of
consumer returns valuable, it is further interesting to observe that a significant
amount of these returns have no verifiable defect. For example, they account for
up to 80% of HP printer returns (Ferguson et al. 2006), and 95% of all electronics
purchases (Lawton 2008).
The primary reason for these "false-failure" returns is that the consumers
learn—only after the purchase—that the good is not a perfect fit to their tastes,
preferences, usage norms, established settings, etc. Take for example a consumer
purchasing a new electric razor only to realize that its grip is not as comfortable
as the old one he had, or a curtain set to be brought home only to notice it does
not match the color of furniture at home. When lacking experience with the good
before the purchase, the consumers cannot be certain of the true value of the good
61
for them, and thus, there is a possibility for a misfit. This type of good is commonly
referred to as "experience good" (Nelson 1970). As a result, lack of information—
regarding the value of the good—is the underlying driver of false-failure returns of
experience goods.
Before purchasing an experience good, consumers mainly rely on the seller to
gain access to, and—however limited—experience with the good; this is especially
true in case of a new-to-market good. Consider for example trial versions of com-
mercial software, test-drive events organized by auto manufacturers, fit-rooms that
are a standard in all department stores, electronic stores with items displayed openly
with trained sales personnel present, free samples of cosmetic products made avail-
able through online or physical channels, product samples sent to expert reviewers,
etc. While the level of these eforts by firms vary greatly, their main purpose is to
reduce false-failure returns by providing the consumers information regarding the
true value of the goods. On the other hand, the sellers also ofer customized return
policies that facilitate product returns. For example, Amazon.com has 31 diferent
product-specific return policies with restocking fees of up to 50%, Best Buy has
a customized return policy with restocking fees up to 25%, Nordstrom ofers full
refunds for any return to their stores. In efect, return policies enable consumers
to defer their ownership decisions until after they gain some experience with the
product (for a fee, in the case of partial refunds).
If a seller's objective is to maximize consumer satisfaction, the initial intuition
is that this goal can be achieved either by providing full information to all con-
sumers, or by ofering a full-refund return policy. While providing full information
62
to each and every consumer would cut all false-failure returns, it is also practically
impossible to achieve. Ofering full refunds would enable any misfit alleviated, how-
ever at the cost of the seller—on top of immediate financial costs, negative brand
implications can be substantial; Lawton (2008) reports that 25% of people who
return an item refrain from buying that same brand again, while 14% of such peo-
ple are unlikely to buy from the same seller again. On the other hand, we show
in Essay 2 that a monopoly seller can design a partial-refund return policy to get
rid of false-failure returns, while providing only partial information. In Essay 2,
we also identify conditions where it is in fact optimal for the monopoly seller to
minimize false-failure returns. Then, the question is 'what happens when there is
competition?' Specifically, we pursue the following research questions in this essay:
1) Given competition, is it still possible to design a return policy to efectively
minimize false-failure returns without having to provide full information? If
it is, is such an outcome ever desirable for the sellers?
2) Are there any equilibrium return policy and information provision decisions?
(a) How do they difer from the decisions of a monopoly?
(b) Under what conditions do they exist?
To address these questions, we build on the basic two-period model described
in Essay 2, with the exception of assuming uniform valuations for tractability. In
order to isolate the efect of competition, we conceptualize a perfectly symmetric
duopoly setting, and examine equilibrium information provision and return policy
63
decisions. To our knowledge of the literature, this is the first scholarly work that
analytically studies the efects of competition on joint information provision and
return policy decisions. We identify all potential Nash equilibria and their respective
conditions of existence. Contrasting the results to the monopoly case, we find that,
while competition can induce sellers to withhold information from the consumers
under certain conditions, it forces them to ofer full refunds.
The rest of the essay is organized as follows. Section 3.2 reviews the relevant
literature. In Section 3.3, we describe and analyze the competition model, and we
examine and discuss the equilibrium in Section 3.4. We conclude in Section 3.5.
3.2 Literature Review
Among the very few studies that investigate competition in a similar context as ours,
the paper by Shulman et al. (2011) is the most relevant. In Shulman et al. (2011),
they examine equilibrium prices and return policies in a competitive market where
consumers are not informed of their tastes or valuations. On a single-period horizon,
the sellers ofer horizontally-diferentiated products but provide no information to
consumers, and they extract no value out of returned items. Our competition set-
ting is significantly diferent from theirs in that we look at equilibrium information
provision and return policies incorporating consumer dissatisfaction in the second
period. We show that consumer dissatisfaction and salvage value are critical in
determining the market equilibrium. In direct contrast to our findings, Shulman
et al. (2011) conclude that competition may induce higher restocking fees, whereas
64
we find that sellers typically ofer full refunds in a competitive setting. Their result
can be explained by noting that they do not consider the impact of high restocking
fees on consumer dissatisfaction (given their single period setting), and therefore
the return policy efectively becomes a tool to discourage consumers from returning,
in order to maximize short-term profits. Our findings help explain why full refunds
are observed in competitive retail markets.
Aside from Shulman et al. (2011), Chesnokova (2007) considers a duopoly
where the firms engage in a product reliability and price competition, and returns are
in the form of repairs, not refunds; i.e., the source of returns in her model is product
reliability, and not consumer tastes and preferences as in our model. In the context
of experience goods, Doganoglu (2010), Villas-Boas (2006) and Villas-Boas (2004)
study the price competition of two sellers over an infinite horizon; however, neither
paper considers return policies or pre-purchase information provision. We study a
duopoly case where two identical sellers engage in return policy and information
competition over two periods; this is the first scholarly work to our knowledge to
study the efects of competition on seller decisions on return policies and provision
of information.
3.3 Competition Model
We build the competition setting on the same framework as described in Section 2.3
in Essay 2. Specifically, we conceptualize a consumer's valuation of a product as a
learning process, given information of amounto e [0, 1] by a seller. Further given a
65
return policy with a refund factor of| e (0, 1], such that the refund amount is p|
where p is the purchase price, each consumer has the opportunity to re-consider her
initial purchase decision. As in Essay (2), we assume costless information provision
in our analysis, and we later comment on the impact of costly information provision.
We consider a duopoly case with identical sellers, Y and Z; sellers have identi-
cal unit costs c, net salvage values s, and market growth rates g, and they introduce
new products at the same time. We assume that consumers equally value the prod-
ucts from both sellers; that is, the products are perfect substitutes of each other. In
other words, there is a single, seller-independent distribution F , of consumer valua-
tions V , which we assume to be uniformly distributed: F (p) = p. We further assume
identical period prices, p
1
and p
2
, for both sellers. While we do not assume identi-
cal products, we assume that information provided by a seller on her product does
not contribute to information on the other seller's product; while this assumption
does not hold in general, it is valid for many—if not all—experience goods
4
. These
assumptions help us focus on information and return policy (o and|) competition,
as well as allowing a tractable solution.
We employ the same two-period setting as in the monopoly case, with the
chronology of events shown in Figure 2.2 in Essay 2. As in the monopoly case, we
assume that consumers attain full information on the products in the second period,
regardless of the information provided in the first period. At the start of the first
4
Consider for example two electric razors of diferent brands. Even after having used one of
them, a consumer would have no understanding regarding how well the other product will perform,
how comfortable it will feel in his hand, how comfortable a shave it will provide, etc.
66
period, the sellers simultaneously decide on their respectiveo and|. In the first
period, we assume a unit market size, which is shared between the sellers according
to consumer valuations given the sellers'o and| decisions. Specifically, a consumer
with ex-post valuation v perceives ex-ante valuations of v
o,Y
and v
o,Z
for the sellers'
products, and since prices are equal, chooses the seller with the larger v
o,j
, where
v
o,j
=o
j
v + (1 ÷o
j
)E[max{V, p
1
|
j
}]
=o
j
v + (1 ÷o
j
)1 + (p
1
|
j
) 2
2
for uniform valuations. In the second period, the market size for each seller grows
with rate g in the same manner as in the monopoly case: Consumers who are
dissatisfied or who return their purchases in the first period do not contribute to
market growth. Therefore, similarly, we refer to g as "misfit penalty."
3.3.1 Market Share Dynamics
In preparation for the equilibrium analysis, we here analyze the market share dynam-
ics given the sellers'o and| decisions. Suppose first thato
Y
=o
Z
. Ifo
Y
=o
Z
< 1
and if, without loss of generality,|
Z
>|
Y
, then v
o,Z
> v
o,Y
, ¬v. That is, in the
case of symmetric, partial information, the seller ofering a more lenient return pol-
icy captures the whole market. Ifo
Y
=o
Z
= 1, then v
o,Y
= v
o,Z
, ¬v; if both sellers
provide full information, then the consumers are indiferent between the sellers re-
gardless of the return policies. Suppose without loss of generality that o
Y
v
Y
Z
u u
prefer seller Z to seller Y , while those with v < v
Y
Z
prefer seller Y to seller Z, u
67
where
v
Y
Z
u
(1 ÷o
Y
)(1 + (p
1
|
Y
)
2
) ÷ (1 ÷o
Z
)(1 + (p
1
|
Z
)
2
).
2(o
Z
÷o
Y
)
(3.1)
Therefore, the first-period market share for seller Y is v
Y
Z
; for seller Z, it is (1÷v
Y
Z
).
u u
If|
Y
=|
Z
=|, we have v
Y
Z
= E[max{V, p
1
|}]; that is, in case of identical return u
policies, the threshold valuation is independent of the level of information provided
as long aso
Y
=o
Z
.
Further analysis of the market share dynamics reveals that the seller with a
more lenient return policy can set an appropriate level of information to achieve a
desired market share. Consequently, she can set an appropriate level of information
to achieve 100% market share, that is, drive the other seller out of the market. This
is formalized in Proposition 7 below.
Proposition 7 Suppose, without loss of generality, that o
Y
< 1 and let Z be the
seller ofering a more lenient return policy, i.e., |
Z
>|
Y
. Then, seller Z can
achieve a desired market share,ˆ, by setting o
Z
=o
ˆ
, where
v v
o
Y
+ p
2
(1 ÷o
Y
)2ˆ +|(Zp÷|
Y2
÷ 1 .
o
ˆ
v
1
v
2 1Z 2
|)
In addition, seller Z can drive seller Y out of the market by setting o
Z
e [o,o],
where
o
|
2
÷|
2
and
max 0, o
Y
÷
p
2
(1
1
÷o
Y
)1 ÷Z(p |Y )
2
1Z
,
o
o
Y
+
p
2
(1
1
|
2
÷|
2
.
Proof See Appendix C.
÷
o
Y
)
1
+
Z(p |Y )
2
1Z
68
Regardless of how the market is shared, the dynamics between the consumer
and the seller is identical to the monopoly case: Given that a consumer prefers
seller Z (v > v
Y
Z
), he purchases only if v
o,Z
> p
1
, or equivalently if v > v
u,Z
. u
If p
1
|
Z
s v < p
1
, he is unhappy with the purchase but is not willing to return; if v < p
1
|
Z
, he is unhappy and
would like to return. Therefore, all the findings
regarding the structure of the (o, |) decision space carries on from the monopoly
case. Furthermore, since we assume equal first-period prices, unit costs and net
salvage values, the sellers have identical (o, |) decision spaces.
3.4 (o, |) Equilibrium
In this section, we analyze the sellers' (o, |) decision space in the light of the results
for the monopoly case and the analysis of market shares above, in order to identify
the possible Nash equilibria. In other words, we investigate whether and when there
exists a pair of decisions (o
Y
, |
Y
) and (o
Z
, |
Z
) such that the former is seller Y 's best
response to the latter, which in turn is seller Z's best response to the former. When
a pure-strategy Nash equilibrium does not exist, we identify the mixed-strategy
equilibrium. The intuitive corollary of Proposition 7 suggests the non-existence of
a Nash equilibrium where both sellers set (o
j
< 1, |
j
< 1), since given, without loss
of generality, (o
Y
< 1, |
Y
< 1), seller Z has a potential best response where she sets
a more lenient return policy and an appropriate level of information to capture the
whole market. In fact, we find that capturing the whole market is the best response
to (o
Y
< 1, |
Y
< 1), and we summarize our findings in Proposition 8 below.
69
Proposition 8 In the duopoly where both sellers have identical p
1
, p
2
, c, s and g,
and consumer valuations are uniformly distributed, we identify four thresholds on g
(as functions of other variables) that are critical for the existence and the form of
(o, |) Nash equilibria: ¯
III
, g
IV
, g
V
and g
V
I
. Furthermore, we find that the ordering
g
u
u u u
of these functions is determined by the value of s compared to a threshold function
s
u
. Specifically, the potential (o, |) Nash equilibria and the associated conditions for
their existence are as follows.
For s > s
u
, we have g
V
> ¯
III
, and
u
g
u
(i) If g < ¯
III
, then there is a symmetric pure-strategy Nash equilibrium g
u
where both sellers provide no information but ofer full refund return
policy: (o
j
= 0, |
j
= 1) for both sellers,
(ii) If ¯
III
< g < g
V
, there is no pure-strategy Nash equilibrium. There is a
g
u
u
mixed-strategy Nash equilibrium where both sellers set |
j
= 1 and pick
o e [0, 1] randomly,
(iii) If g > g
V
, there is no pure-strategy Nash equilibrium. There is a mixed- u
strategy Nash equilibrium where both sellers set |
j
= 1 and pick o e
[o, 1] randomly; ˆ
For s < s
u
, we have g
V
< g
V
I
< ¯
III
< g
IV
, and
u u
g
u
u
(i) If g < g
V
, then there is a symmetric pure-strategy Nash equilibrium u
where both sellers provide no information but ofer full refund return
policy: (o
j
= 0, |
j
= 1) for both sellers,
70
(ii) If g
V
< g < g
V
I
, then there is an asymmetric pure-strategy Nash equilib-
u u
rium where one seller provides full information and ofers an arbitrary
return policy, while the other seller provides no information but ofers a
full refund return policy: without loss of generality, (o
Y
= 1, |
Y
e [0, 1])
and (o
Z
= 0, |
Z
= 1),
(iii) If g
V
I
< g < g
IV
, then there is a symmetric pure-strategy Nash equi-
u u
librium where both sellers provide full information and ofer arbitrary
return policies: (o
j
= 1, |
j
e [0, 1]) for both sellers,
(iv) If g > g
IV
, there is no pure-strategy Nash equilibrium. There is a mixed- u
strategy Nash equilibrium where both sellers set |
j
= 1 and pick o e
where
[o, 1] randomly, ˆ
s
u
= 1 ÷ p
1
+ c 2 ÷ p1 1
,
g
u
¯
III
= (p ÷sc÷ c÷ p ),
2
)(1
2
g
IV
= p(p
1
÷ c)(1 ÷ p
1
)),
u
1
(p
2
÷ c)(1 ÷ p
2
g
V
= (1 ÷÷ p
1
(1 ÷ p)(1 ÷)p ),
u
s
p
1
)(p
2
÷ c
1
+c
2
g
V
I
= (2s÷ pc ÷pp
1
(1 c÷ p
1
÷ pc)),
and
u
2÷
1
)(
2
÷ )(1
+
1
(1 ÷ p
1
)
2
(g(1 ÷ p
2
)(p
2
÷ c) + c ÷ s)
o = (c ÷ s)(1 ÷ p )
2
÷ 2p (1 ÷ p )(p ÷ c) + g(1 ÷ p )(p ÷ c)(1 ÷ 2p + 3p
2
). ˆ
1
P
r
o
o
f
S
e
e
A
p
p
endix
C.
1 1 1
7
1
2 2 1 1
We see from Proposition 8 that for sufciently small misfit penalty, g, both
sellers find it the best decision to provide no information to sell to all consumers and
share the market equally, although there are some dissatisfied consumers as well as
some returns. Moreover, we observe that as the salvage value increases, the range of
g, where providing no information is the best decision, grows. Both of these results
are consistent with the monopoly case (Proposition 5).
We observe that for sufciently large misfit penalty (g > ¯
III
for s > s
u
, and g
u
g > g
IV
for s < s
u
), there is no pure-strategy Nash equilibrium since both sellers u
always find a best response where they capture the whole market alone. The mixed-
strategy equilibrium we identify for s > s
u
and ¯
III
< g < g
V
suggests that both
g
u
u
sellers ofer full refunds and picko randomly in [0, 1]. The second mixed-strategy
Nash equilibrium suggests that for sufciently large g, both sellers' best decision
is to ofer full refunds and provide at least partial information; they randomly set
an information level betweeno =o ando = 1. This means that for sufciently ˆ
large g, it is not a best decision to provide little or no information; a result that is
consistent with the monopoly case. Both mixed-strategy equilibria suggest that the
market is not necessarily covered in the first period (i.e., there are some consumers
who leave without purchasing), since full market coverage requireso
j
so(v
u
= 0)
for at least one of the sellers and this is not necessarily the case. Furthermore,
recalling that for a monopoly, the optimal decision given| = 1 and g > ¯
III
is to g
u
provide full information, so that there are no dissatisfied consumers, we note that
both mixed-strategy equilibria imply partial information, resulting in sub-optimal
outcomes where there are some dissatisfied consumers and some returns.
72
For sufciently small net salvage value, s < s
u
, intermediary misfit penalty
values result in two diferent pure-strategy Nash equilibria. In the lower g range
(g
V
< g < g
V
I
), one seller provides full information and ofers an arbitrary return
u u
policy, while the other provides no information but ofers a full refund return policy.
This implies full market coverage in the first period (i.e., all consumers purchase),
and a market allocation such that one seller serves the consumers with higher val-
uations and sees no dissatisfied consumers and no returns, while the other seller
serves the remainders and sees some dissatisfied consumers and some returns. This
is an interesting result given that the sellers are identical. In the higher g range
(g
V
I
< g < g
IV
), both sellers provide full information and ofer arbitrary return
u u
policies. In this case, while consumers are indiferent between the sellers, only the
consumers with ex-post valuations at least as high p
1
purchase in the first period
and there is ex-ante efcient allocation of the goods.
We note that costly information provision may significantly alter the Nash
equilibria, even if the sellers would have the identical cost structure. For example,
it is conceivable that in the mixed-strategy equilibria described above, the range
foro would be capped from above since neither seller would have an incentive to
incur high information provision costs. Furthermore, the pure-strategy equilibria
where one or both sellers provide full information,o = 1, may not exist; in that
case, a mixed-strategy equilibrium where both sellers ofer full refunds,| = 1, with
o randomized over a range may prevail.
Regarding the refund factor,|, recall that in the monopoly solution described
in Proposition 5, full refunds are never exercised; Solution (D) suggests partial
73
refunds, and when Solution (E) is optimal, amount of refund is arbitrary. However,
we see from Proposition 8 that competition changes the picture abruptly. In the
duopoly case, all but one Nash equilibria suggest that at least one seller ofers full
refunds; for s < s
u
and g
V
I
< g < g
IV
, both sellers provide full information and the
u
refund amount becomes arbitrary.
u
Consequently, we conclude, contrasting with the monopoly case, that while
competition results in one or both of the sellers withholding information from con-
sumers in certain cases, it typically forces them to ofer full refunds. That helps
explain why we observe full refunds in practice (e.g., Nordstrom.)
3.5 Conclusion
In this essay, we study competition in the context of information provision and return
policies in experience good markets. In order to isolate the efects of competition
on our results in the monopoly model given in Essay 2, we consider a duopoly
case where two identical sellers engage in information provision and return policy
competition. We identify the possible pure-strategy Nash equilibria, or if none exists,
the mixed-strategy Nash equilibria, and the associated conditions where they take
place. We find, in contrast to the monopoly case, that while competition can cause
the sellers to withhold information under certain conditions, it typically forces them
to ofer full-refund return policies. This finding can shed light on some real-life
phenomena where sellers ofer full refunds and/or they do not put much efort to
provide informational tools to consumers.
74
Appendix A
Appendix for Essay 1
A.1 Derivation of h(¸)
Let R
it
denote the reservation price for product i at time t, a random variable. We
write R
1
t
= u(O)c
1
t
, where u(-) is a (deterministic) linear mapping function andc
1
t is a random variable
with a Weibull distribution (so that R
it
has a Weibull distribu-
tion); similarly R
2
t
= u((1+¸)O)c
2
t
. We define customer utility ç
it
as a log function
of the customer's reservation price ç
it
= ln(R
it
) = ln(u(O)) + ln(c
it
). Becausec
it
has a Weibull distribution, ç
it
has a Gumbel distribution; this is consistent with the
Logit model for choice. As a result, the probability that a customer adopts the new
generation is
h(¸) = e
ln(
u((1+¸
)O))
+ e
ln(
u
(O))
= u(O) ((1u+¸)O) )O) = 1 +¸ .
e
ln(
u((1+¸)O))
A.2 Proof of Proposition 1
u + ((1 +¸ 2+¸ (A.1)
For a non-stationary Poisson process with intensity A(t), E [N (A(t))] = A(t), and
thus (1.6) becomes
(1+o)t
max H
j
2 =
¯
dt
p (t) 2
=
t
(1+o)t e
÷
o(t÷ot
)
(p
2
(t) ÷ c
p
) E N ì
j
2(t)G
2
t
(p
2
)
(A.2)
t
e
÷
o(t÷ot
)
(p
2
(t) ÷ c
p
)ì
j
2(t)G
2
t
(p
2
)dt ¯
75
This is a simple optimal control problem, with the first-order necessary condition
given in (1.7). The solution is similar to that found in Bitran and Mondschein
(1997). For uniqueness of the solution to (1.7), we need
K
2
,t
= p
2
(t) ÷ G
2
t ¯
G
2
t
to be an increasing function of p
2
(t) since
p2(t)÷· lim K
2
,t
= ·. Therefore, we need
0< dK
2
,t
= 1 ÷ dp G
2
t
= 1 ÷ d¯
÷G
2
t ÷ G
2
t
g
2
,t ¯
2
which becomes
dp
2
2
G
2
t
G
2
t 2
÷2G
2
tG
2
t
÷ G
2
tg
2
,t ¯
0>
2
¯
2
d G
2
t
. ¯
G
2
t
2
= dp G2
2 2t
A.3 Proof of Proposition 2
We show through ?uid approximations (Mandelbaum and Pats 1998) that the solu-
tions to the deterministic version of (1.5) is asymptotically optimal as initial max-
imum arrival rate, M
0
, and I
0
grow proportionally large. However, since I
0
is a
decision variable, we first show that it is optimal to select I
0
proportionally large as
M
0
.
Consider a sequence of instances of problem (1.5) indexed by n e Z
+
. Let
M
0
n
denote the initial maximum arrival rate andì
j
1
n
be the resulting arrival rate
intensity function for the n
th
instance. Let
lim M
0
= M
0
. n
n
÷·
n
Thu
s,
we have
n÷·
j
n
1
limìn =ì
j
1.
76
Let I
n
be the decision parameter for the final build and I
n
(t) denote the 0
corresponding inventory trajectory for the n
th
instance, and let all other parameters
be held constant, independent of n. For the n
th
instance, (1.5) becomes
I
n
,p1(t)
T
j
e
÷
o(t÷ot
)
p
1
(t)dI
n
(t)
+
e
÷
o(1÷o)
t
c s
I
n
0
+
T
j
dI
n
(t)
max E ÷ 0
ot
T
j
ot
s.t.
T
j
÷ c
h
ot
e
÷
o(t÷ot
)
I
n
(t)dt ÷ c
p
I
n
0
(A.3)
÷
ot
dI
n
(t) I
n
0
t
I
n
(t) = I
n
÷ N 0
ot
ì
j
1
n
(u)G
1
u
(p
1
)du ¯
T
j
for t eot, T
j
,
where we wrote I(T
j
)
0 as ÷
ot dI(t) I
0
. After dividing the second constraint
by n, taking limits on both sides, and applying Lebesgue's monotone convergence
theorem, we get
n
÷·
n
t
lim 1 I
n
(t) =
n
lim 1 I
n
÷ N
÷·
n
0
ì
j
1(u)G
1
u
(p
1
)du . ¯
ot
Similarly, from the first constraint in (A.3), we have
T
j
n÷· lim ÷
1 n
ot
dI
n
(t)
lim 1 I
n
.
n
÷·
n
0
Therefore, applying the same transformation to the objective function, we can
rewrite (A.3) as
77
lim 1
I
max
t
)
E ÷
T
j
e
÷
o(t÷ot
)
p 1
(t)dI
n
(t)
+
e
÷
o(1÷o)
t
c
s
I
n
0
+
T
j
dI
n
(t)
n
÷·
n
n
,p1( 0
ot
T
j
ot
s.t.
T
j
÷ c
h
ot
e
÷
o(t÷ot
)
I
n
(t)dt ÷ c
p
I
n
0
n÷· lim ÷
1 n
ot
dI
n
(t)
lim 1 I
n
n
÷·
n
0
n
÷·
n lim 1 I
n
(t) =
n
lim 1 I
n
÷ N
÷·
n
0
t
ot
ì
j
1(u)G
1
u
(p
1
)du ¯
for t eot, T
j
.
(A.4)
Suppose (I
-
, p
-
) is an optimal solution to (1.5), with the optimal objective
0 1
function valuet
-
. Then, (I
n
-
, p
-
) is an optimal solution to (A.4) with the objective
1 0 1
function valuet
n
-
, such that I
n
-
andt
n
-
satisfy
n
lim I
n
-
/n = I
-
andt
n
-
=t
-
,
1 0 1
÷· 0
0 1 1
respectively. This follows by observing that (A.4) is equivalent to problem (1.5)
divided by n and taking limits as n ÷ ·. As a result, we have shown that it is optimal to let the
final build, I
0
, grow proportionally large as M
0
in the asymptotic
regime.
Noting that the demand intensity process
t
ot
ì
j
1(u)G
1
u
(p
1
)du ¯
is continuous and uniformly bounded in [ot, T
j
], and we find that in the limit as
n ÷ ·, I
n
(t)/n converges (almost surely and uniformly over a compact set) to I(t),
given by
t
I(t) = I
0
÷
ot
ì
j
1(u)G
1
u
(p
1
)du. ¯
Further details regarding the proof of this convergence result can be found
78
in Mandelbaum and Pats (1998). In this asymptotic regime, the stochastic opti -
mization problem in (1.5) reduces to the optimal control problem in (1.8), where
T
j
I
0
+ ot
dI(t) is replaced with I(T
j
), and the second constraint is substituted into
the first term in the objective function.
The solution to (1.8) can be found as follows. Treating I(t) as the state variable
and p
1
(t) as the control variable, and lettingv ande(t) be the multipliers for the
first and second constraints in (1.8), the Hamiltonian is H = e
÷
o(t÷ot
)
(ì
j
1G
1
t
p
1
÷ ¯
c
h
I)÷eì
j
1G
1
t
, where arguments have been suppressed for simplicity. The optimality ¯
conditions are:
cH = 0?ì
j
e
÷
o(t÷ot
)
÷p G + G +e(t)G = 0,
c p
1
1 1 1t
¯
1
t
1t
(A.5)
cH = ÷ce? c e
÷
o(t÷ot
)
=ce , h
(A.6)
cI ct ct
e(T
j
) =v + e
÷
o(T
j
÷
ot
)c
s
andvI(T
j
) = 0. (A.7)
A first-order condition for I
0
is obtained by considering that the marginal
revenue from the last unit must equal to its marginal cost (including the procurement
cost and cumulative holding costs in time). That is,
T
j
e(T
j
) = c
p
+ c
h
e
÷
o(u÷ot
)
du. (A.8)
ot
Combining (A.7) and (A.8), we get
T
j
v = c
p
+ c
h
ot e
÷
o(u÷ot
)
du ÷ e
÷
o(T
j
÷
ot
)c
s
. (A.9)
However, we must havev> 0, otherwise I
0
÷ · is optimal and the problem in
(1.8) is unbounded. Therefore, from (A.7), I(T
j
) = 0. In other words, the entire
79
initial inventory is depleted during the sales horizon. To find p
1
(t), we proceed as
follows. From (A.5),
e(t) = e
÷
o(t÷ot
)
p
1
÷ G
1
t ¯
G
1
t
On the other hand, (A.7) and (A.8) yield
t
.
(A.10)
e(t) = c
p
+ c
h
e
÷
o(u÷ot
)
du. (A.11)
ot
We combine (A.10) and (A.11) to obtain the necessary condition for the optimal
price pattern for product 1, given in (1.9). The proof of uniqueness follows the same
steps as in the proof for Proposition 1.
Once the optimal price path is determined using (1.9), and given that I(T
j
) =
0, the optimal initial inventory is equal to the total sales through the planning
horizon.
A.4 Normalization of| for the Regression
We normalize the parameter|, for the purposes of running the regression, so that it
takes values between 0 and 1, instead of between 0 and ·. We do this by mapping
| to a new parameteru, according to the normalizing relationship:
| = (1 ÷o)t
1 + W ÷1 · e
÷
1u
,
(A.12)
u u
where W(-) is the Lambert W function. The Lambert W function is the inverse of
f (w) = we
w
and we use the zeroth branch which is single valued and real for the
range ofu considered. It is easily verified that
|
limu = 1, and lim
0
u = 0.
80
÷· |÷
Table A.1: Statistics of multiple linear regression: Two-way interaction efects.
Factor
(intercept)
t
o
log(k)
µ
¸
p+q
log(q/p)
|
t
t : log(k)
t:µ t:¸
t : p+q
t : log(q/p)
t:|
o : log(k)
o: µ o:¸
o : p+q
o : log(q/p)
o:|
log(k) : µ
log(k) :¸
log(k) : p + q
log(k) : log(q/p)
log(k) :|
µ:¸
µ : p+q
µ : log(q/p)
µ:|
¸ : p+q
¸ : log(q/p)
¸:|
p + q : log(q/p)
p+q :|
log(q/p) :|
Adj. R-sq.
t-value
-2.7
-60.9
9.0
-3.8
35.3
-35.7
-43.7
68.3
-94.5
-20.9
-17.7
-1.8
-35.0
-30.4
25.1
-48.1
1.9
6.3
-12.0
-20.0
30.7 -
9.5
10.0
-10.3
-11.3
17.7
-24.4
-4.0
-11.2
15.7
-21.9
-16.3
25.8
-35.2
-6.3
-4.2
-6.0
0.837
--
--- --
- ---
--- --
- ---
--- --
- ---
---
?
--- --
- ---
---
?
--- --
- ---
--- --
- ---
--- --
- ---
--- --
- ---
--- --
- ---
--- --
- ---
--- --
-
Statistical significance codes: '- - -': p ~ 0; '--': 0.001< p < 0.01; '?': 0.05< p < 0.1
81
Appendix B
Appendix for Essay 2
B.1 Proof of Proposition 3
Under partial information, using the definition of v
u
, the condition v
u
= p
1
translates
to p
1
= E[max{V, p
1
|}]. Suppose that p
1
> E[V ]. Then, since p
1
s E[max{V, p
1
}]
and through the intermediate value theorem, there exists|
p
e [0, 1] such that p
1
= E[max{V,
p
1
|
p
}]. Note that if p
1
< E[V ], no such|
p
e [0, 1] exists and there are no first period buyers with
positive surplus foro< 1. Thus, if p
1
> E[V ], setting
| =|
p
is equivalent to providing full information as it completely nullifies the
consequences of valuation uncertainty regardless of the value ofo.
B.2 Structural Properties of the (o, |) Decision Space
B.2.1 Boundary for Returns: v
u
(o, |) = p
1
|
The condition v
u
(o, |) = p
1
| is critical for the existence of returns. Assuming
o> 0, this condition is equivalent to p
1
+
(1
÷o
)
(p
1
÷ E[max{V, p
1
|}]) = p
1
|, which o
reduces to
o =o
r
(|) E[max{V, p
1
|}] ÷ p
1
.
E[max{V, p
1
|}] ÷ p
1
|
Note that, given p
1
, this equation represents a curve in the (o, |) space. Sup-
pose first that p
1
> E[V ]. Then, by Proposition 3,|
p
> 0 exists and therefore
c v
u
co
= ÷o1
2
(p
1
÷ E[max{V, p
1
|}])> 0. Furthermore,o
r
(|)> 0 if and and only if
|>|
p
, since the denominator ino
r
(|) is always positive. Thus, v
u
(o, |)< p
1
|
if and only ifo 0, and
d|
(E[max{V,p1|}]÷p1|)
2
d
2
or
d|2 = p
1
2
(2(1÷F (p1|))(E[max{V,p1|}]÷p1+p1(1÷|)F (p1|))+p1(1÷|)(E[max{V,p1|}]÷p1)F (p1|))
(E[max{V,p1|}]÷p1|)
3
>0
for all|>|
p
. Suppose p
1
< E[V ]. Then, we have
c
cvo
u
> 0,o
r
(|)> 0,
do
r> 0, and d|
82
d
2
or
d|2
> 0 satisfied for all| e [0, 1].
B.2.2 Boundary for 100% Sales: v
u
(o, |) = 0
The condition v
u
(o, |) = 0 reduces to
o =o
0
(|)
1 ÷ E[maxp
1
, p |}]. {V
1
Similar to above, when p
1
> E[V ],|
p
> 0 exists and for|>|
p
,o
0
(|)> 0 is well
defined and we have
c v
u
co
> 0. Finally, the functiono
0
(|) is strictly increasing if F
p
2
F (p1|)
is continuously diferentiable, since
do0
d|
=
1
(E[max{V,p1|}])
2
> 0. If p
1
< E[V ], then we
have
c v
u
co
> 0,o
0
(|)> 0 and
do0
d|
> 0 for all| e [0, 1].
B.2.3 Boundary for No Sales: v
u
(o, |) = 1
Note that v
u
(o, |) = 1 is possible only if p
1
> E[V ], and it reduces to
o =o
1
(|)
p
1
÷ E[max{V, p
1
|}] .
1 ÷ E[max{V, p
1
|}]
Then, it is seen thato
1
(|)> 0 only if 0 s||
p
, we have v
0
= E[max{V, p
1
|}]> p
1
by definition. Therefore, for a
given|>|
p
, the seller is indiferent in choosing ano e [0, o
0
(|)]
83
when information is costless. Consequently, in our analyses in this study, we treat
the conditions v
u
= 0 ando = 0 as equal at a given|>|
p
.
Similarly, the second region is whereo 1; there are
no sales since consumer valuations are in (0, 1). We see that "no sales" is achieved
for anyo e [0, o
1
(|)], which is possible for| p
1
c R
I
(B.2)
We observe that c v
u
= ÷F (v
u
) (p
1
÷ c)< 0; it is optimal to decrease the
purchasing threshold at any price point. We conclude that any (o, |) decision in
84
Region (I) is strictly dominated by Region (E) since v
E
= p
1
< v
I
; the seller never
u u
lets the consumers have positive surplus while there is an option to have efcient
allocation.
B.3.3 Some Dissatisfied Consumers, No Returns: p
1
> v
u
> p
1
|
In the region where p
1
> v
u
> p
1
|, Region (II), there are some dissatisfied buyers
but they are not willing to return their items as the refund amount is not high
enough. We have L = F (p
1
) ÷ F (v
u
), and since there are no returns, M = 0.
Seller's profit maximization problem in this region is
max R
II
= (p
1
÷ c) (1 ÷ F (v
u
)) + (p
2
÷ c) g (1 ÷ F (p
2
)) (1 ÷ F (p
1
) + F (v
u
)) o,|,p i
s.t. p
1
> v
u
> p
1
|
(B.3)
The partial derivative with respect to v
u
is
cR
II
= F (v ) ÷p + c + g (p ÷ c) (1 ÷ F (p ))
c v
u
which is positive for
u 1 2 2
g > g
I
I u
(p
1
÷ c)
(p
2
÷ c) (1 ÷ F (p
2
)),
(B.4)
and negative for g < g
II
. Then, at any given price point (p
1
, p
2
), it is optimal to u
increase v
u
if g > g
II
and it is optimal to decrease v
u
if g < g
II
. Therefore, if g > g
II
,
u u u
Region (II) is dominated by Region (E) since v
E
= p
1
> v
II
; if g < g
II
, Region (II)
u u u
is dominated by the boundary where v
u
= p
1
|. As a result, no internal solution is
optimal in Region (II).
B.3.4 Some Dissatisfied Consumers, Some Returns: v
u
< p
1
|
Region (III) is characterized by v
III
< p
1
|, which means there are some dissatisfied u
consumers (L = F (p
1
) ÷ F (v
u
)) and a portion (M = F (p
1
|) ÷ F (v
u
)) of these
consumers are willing to return their purchases. Thus, the seller's optimization
85
problem in this region is as follows:
max R
III
= (p
1
÷ c) (1 ÷ F (v
u
)) + (÷p
1
| + s) (F (p
1
|) ÷ F (v
u
))
o,|,p
i
+ g (p
2
÷ c) (1 ÷ F (p
2
)) (1 ÷ F (p
1
) + F (v
u
))
(B.5)
s.t. v
u
< p
1
|
In order to solve this problem, we look at the partial derivative of the objective
c R
I
I I
function with respect to v
u
:
which is positive if
c v
u
= F (v
u
) ÷p
1
+c+p
1
| ÷s+g (p
2
÷ c) (1 ÷ F (p
2
))
g > g
I
I I u ((p
1
÷ c) + (s ÷ p
1
|))
(p
2
÷ c) (1 ÷ F (p
2
))
(B.6)
and negative if g < g
III
. Then, since v
II
> v
III
> 0, if g > g
III
, Region (II)
u u u u
dominates Region (III), and if g < g
III
, the boundary where v
u
= 0 dominates u
Region (III). We name the part of this boundary region where|>
s
p1
as Region (B)
and the part where| s
s
p1
as Region (D).
Note that if|>
s
then g
III
< g
II
, and if|<
s
then g
III
> g
II
. Therefore,
p1 u u p1 u u
combining our results so far for Regions (II) and (III), for|<
p1 ,
s
there is a range
g
II
< g < g
III
where Region (E) dominates Region (II) and Region (D) dominates
u u
Region (III) and however it is not obvious which one of the two dominates the other.
In order to identify the threshold g value, we write the profit functions for the two
regions equal, R
E
= R
D
, and solve for g;
g
E
u
c R
E
(p
1
÷ c) + (s ÷ p
1
|)
F
F(pp11|))(
(p
2
÷ c) (1 ÷ F (p
2
))
.
(B.7)
Observing that
cg
>
c R
D
,
cg
we conclude that Region (E) dominates Region
(D) if g > g
E
. Note that for|
p1 , s
w
e
h
a
v
e
g
I
I
I
<
g
I
I
,
a
n
d
u u
86
(i) If g > g
II
, then Region (E) dominates all regions; it is optimal to set u
o = 1.
(ii) If g
III
< g < g
II
, then the boundary region between Regions (II) and
u u
(III) where v
u
= p
1
|, Region (A), dominates all regions.
(iii) If g < g
III
, then Region (B) dominates all regions. u
2. For| s
p1 ,
s
we have g
III
> g
E
> g
II
, and
u u u
(i) If g > g
E
, then Region (E) dominates all regions. u
(ii) If g < g
E
, then Region (D) dominates all regions. u
We see that there are four candidate regions (A, B, D and E) for optimality.
Figure B.1 gives an illustration of these regions for a specific case. Note again that
the findings above are for given|, p
1
, p
2
, c, s, and g values and point to the besto
decision depending on the relationships between these "parameters". For example,
consider a price taking seller for whom the restocking fee, hence|, is also dictated by either the
industry or some trade regulations. Then, the above rules apply directly to find the
optimum amount of information to be provided, assuming it is costless.
1.0
E
0.8
A
0.6
0.4
0.2
0.0
Y
O
I
B
1
B
p
II
Y
O
D
p
1
B
Y
O
0
A
III
B
s
B
p
1
0.0 0.2 0.4 0.6 0.8 1.0
B
Figure B.1: Candidate regions for optimality for V ? U (0, 1), p
1
= 0.51 and s = 0.4.
87
We continue our analysis considering a seller who can set allo,| and p
1
freely,
and use the above rules as a guideline to find the optimum strategy.
B.4 Analysis of Candidate Regions and Proof of Proposition 5
B.4.1 Region (A)
We start with looking at Region (A), which is optimal if g
III
< g < g
II
, which is
u u
in turn possible if|>
p1 .
s
Plugging in the defining constraint, v
u
= p
1
|, to the
objective function and taking partial derivative with respect to|, we get
cR
A
= p F (p |) ÷p + c + g (p ÷ c) (1 ÷ F (p )) ,
c|
1 1 1 2 2
which is negative for g < g
II
. Therefore, it is optimal to decrease| in Region (A), u
where sup(|) =
p1 .
s
As a result, setting| =
s
p1
ando such that v
u
= p
1
| dominates
Region (A). However, from the discussion above, when| =
p1 ,
s
Region (D), where
v
u
= 0, is optimal. This therefore establishes that Region (D) dominates Region
(A).
B.4.2 Regions (B) and (D)
We collectively represent Regions (B) and (D) as v
u
= 0. Thus, the objective
function for these regions is
R
B,D
= (p
1
÷ c) + (÷p
1
| + s) F (p
1
|) + g (p
2
÷ c) (1 ÷ F (p
2
)) (1 ÷ F (p
1
)) .
The first order condition (FOC) for| is
cR
B,D
= p (÷F (p |) + F (p |) (÷p | + s)) = 0,
or equivalently,
c|
1 1 1 1
|
-
= ps ÷ p1 F ((p
1
| -)).
1 1
F
-
p
1
|
(B.8)
Note that since the second term on the right is positive, at optimality, we
have|
-
<
p1 .
s
This means that Region (D) dominates Region (B) since the latter is
defined for|>
p1 .
s
88
Combining our results so far, we see that when g < g
II
, it is never optimal to u
have|>
p1 ;
s
Region (D) dominates both Regions (A) and (B). Therefore, we have
two candidate optimal solutions left: Solution (D), which is defined by v
u
= 0 (or
equivalently,o e [0, o
0
(|)], as shown above) and the FOC given in Equation B.8,
and Solution (E) which is defined by v
u
= p
1
. This constitutes the proof of Propo-
sition 5.
B.5 Proof of Proposition 6
From the analysis of the structural properties of seller's (o, |) decision space, we
know that changing p
1
changes the (o, |) decision space as all the critical boundaries
is a function of p
1
. Therefore, when p
1
is a decision variable as well, the seller has the
tool to change the (o, |) decision space in order to maximize her profits. Considering
this, we observe that the point| =
s
p1
on Region (D) structurally changes the profit
function when solving for the optimal p
1
; when| =
p1 ,
s
refund is equal to salvage
value and each return has zero net after-sales revenue. Therefore, we take this point
explicitly and define Region (C): {o = 0, | =
p 1 }.
s
Profits for Region (C) is given
by R
C
= (p
1
÷ c) + g (p
2
÷ c) (1 ÷ F (p
2
)) (1 ÷ F (p
1
)). Since Region (C) is a single
point on the (o, |) decision space, the seller's profits here is a function of only the
prices.
In the following, we solve for the optimalo,|, p
1
and p
2
for each of the
regions (C), (D), (E), and we identify the optimal strategy for uniformly distributed
consumer valuations; F (p) = p. Using Proposition 4, the optimal second period
price is p
-
= (1 + c)/2 for all regions. 2
B.5.1 Region (C)
In order to solve for the optimal profit function, we take the partial derivative with
c R
C
respect to p
1
,
c p1
= 1 ÷ g (p
2
÷ c) (1 ÷ p
2
), which is positive for g < g
E
. Recall u
that Region (C) is optimal only if g < g
E
; thus, it is optimal to increase p
1
as u
much as possible in the feasible range for Region (C). Constrained by the equality
89
|=
s
p1
for Region (C), the largest value that p
1
can attain is determined by the
smallest value that| can take, which is equal to|
p
. With uniform valuations, we
1+(p1|)
2
have E[max{V, p
1
|}] =
2
, and we find that
\
|
p
= {| , p
1
= E[max{V, p
1
|}]} = 2pp
1
÷ 1 1
from Proposition 3. Therefore, the optimal price should satisfy
\
2p
1
÷ 1 = s, and
it yields p
-
= 1
1+s
2
2
. Then, we find the optimal refund factor as|
-
=
s
p1
=
1+s2 .
2s
As
a result, recalling thato = 0 in Region (C) by definition, Solution (C) is given by
(o
-
, |
-
, p
-
, p
-
)
C
=
2s , 1 + s
2
, 1 + c
12
0, 1 + s
2
2 2
and it yields the net profits of
R
C
-
= 1 g 1 ÷ c 8
2
1 ÷ s
2
+ s2 ÷ c + 1 . 2
2
Note that|
-
=
2s
1+s2
< 1 for all s < 1; it is not optimal to ofer full refunds when
Solution (C) is optimal.
B.5.2 Region (D)
With uniform valuations, the FOC for| yields| =
s
2p1
in Region (D). Plugging this
equality in the objective function, we get
c R
D
c p1
=
c (p1÷c)+
1
(s)
2
+g(p2÷c)(1÷p2)(1÷p1) 4
c p1
=
1 ÷ g (p
2
÷ c) (1 ÷ p
2
). Therefore, profits are increasing in p
1
for g < g
E
, for which u
Region (D) is optimal; i.e., it is optimal to increase p
1
as high as possible in Region
(D). Given the FOC for|, the highest value for p
1
is determined by lowest value of
\
|, which is equal to|
p
=
2
p11÷
1
. Therefore, the optimal price should satisfy p
1
|
p
= p
\
2p
1
÷ 1 =
2
s
, which yields p
-
=
4+
s
2
. The corresponding| is|
-
=
s
=
4
s
.
1 8 2p1 4+s2
Then, these values constitute Solution (D),
2
(o
-
, |
-
, p
-
, p
-
)
D
= 0, 4 4ss
2
, 4 + s , 1 + c ,
12
+ 8 2
which results in the profits of
1
4 ÷ s
2
+ 3s ÷ c + 1 . 2
R
D
-
= 32g 1 ÷ c
2
8 2
Note that|
-
=
(D) is optimal.
4s
4+s2
< 1 for all s < 1; full refunds are not optimal when Solution
90
B.5.3 Region (E)
Given that V ? U (0, 1), the optimal pricing decision in Region (E) is p
-
= p
-
=
(1+c)
In this case, the seller's net profit is equal to
1 2
2
.
R
E
-
= 1(1 ÷ c)
2
(1 + g). 4
As we pointed out above, the seller is indiferent in deciding on an (o, |) point in
Region (E), or more formally, in choosing between (o
-
= 1, |
-
e [0, 1]), and given \
p
1
> E[V ], (o
-
e (0, 1], |
-
=
2p1÷
1
).
p1
B.5.4 Deriving the Optimal Strategy
To summarize our analysis above, there are three solutions that the seller can choose
among, depending on the parameters incorporated in this study:
Solution (C): (o
-
, |
-
, p
-
, p
-
) = 0,
1+
ss2 ,
1+
2s
2
,
1+
c
, with profits R
C
-
. 2
12 2
Solution (D): (o
-
, |
-
, p
-
, p
-
) = 0,
4+
ss2 ,
4+
8s
2
,
1+
c
, with profits R
D
-
. 4
12 2
Solution (E): (o
-
, |
-
) = {(o, |) , v
u
= p
-
}, p
-
= p
-
=
(1+c)
with profits R
E
-
.
1 1 2
2
,
In order to determine the ultimate optimal strategy, we conduct a three-way
comparison of the net profits ofered by these optimal solutions. First, we find the
thresholds on g by conducting three pairwise comparisons between the above net
profits. That is, we determine g
i,j
as a function of c and s by setting R
i
-
= R
j
-
and u
solving for g; g
i,j
u
{g , R
i
-
= R
j
-
}. Through algebraic manipulations, we get
g
D,C
= 3(1 4 c)
2
,
u
÷
g
E,C
= 4(1(1 + )2÷ 2(1 +)
2
c) ),
u
+ s
2
s )(1 ÷ c
2
g
E,D
= 4 + 12(1++ss)(1÷ 8(12+ c) .
u
(4
2 2
)
÷ c)
2
91
Then, we verify that
c(R
D
-
÷ R
C
-
) = 3 (1 ÷ c)
2
s
2
> 0,
c(R
cg
E
-
÷ R
C
-
)
32
c(R
cg
E
-
÷ R
D
-
)
cg
= 1 (1 ÷ c)
2
(1 + s
2
)> 0, 8
1
= 32(1 ÷ c)
2
(4 + s
2
)> 0,
establishing that Solution (i) is preferred to Solution (j) if g > g
i,j
, and Solution (j) u
is preferred to Solution (i) if g < g
i,j
. However, this gives only a partial ordering; u
in order to develop a full ordering, we seek the ordering of these thresholds on g.
Pairwise comparisons of these thresholds show that there is a critical c value that
renders g
D,C
= g
E,C
= g
E,D
: c
u
(s) = {c , g
D,C
= g
E,C
= g
E,D
}. We determine
u u u u u u
and observe that
g
D,C
c
u
(s) = 2
1 + s
2
÷ 1, 3
c
u
= 3(24(12+2s÷)(1 + c))
2
)
2
> 0, 2
g
E,C
u
cc
g
D,C
+s (1 + c
c
u
g
E,D
u
cc
g
E,C
4(4 + s
2
)(1 + c)
= 3(4 + 3s
2
÷ 2(1 + c)
2
)
2
> 0,
c
u
g
E,D
u
cc
= (1 + s
2
s)(4 + 3ss
2
)(1 + c)+ c)
2
)
2
> 0.
2
(4 +
2
÷ 2(1
Therefore, we infer that if c > c
u
(s), then g
E,D
< g
E,C
< g
D,C
, and if c < c
u
(s), then
u u u
g
D,C
< g
E,C
< g
E,D
. Proposition 6 follows by observing the ordering of solutions
u u u
(C), (D) and (E) with respect to g, which is depicted in Figure B.2.
B.6 On the Value of Optimal Refund Amount
It is interesting to investigate if the "refund amount" exceeds the salvage value or
not: does the seller allow returns even when they have negative net revenues? By
definition, Solutions (C) and (D) do not ofer a refund amount more than the salvage value;
they ofer exactly equal to, and exactly half of the salvage value, respectively.
92
Region c>c s
Solution (C) is optimal Solution (E) is optimal
C D E
g
E
,D
C E D
g
E
,C
E C D
g
D
,C
E D C
g
Region c c s
Solution (C) is optimal Solution (D) is optimal Solution (E) is optimal
C D E
g
D
,C
D C E
g
E
,C
D E C
g
E
,D
E D C
g
Figure B.2: Ordering of solutions according to the the regions in Figure 2.6.
In case of Solution (E), if the seller provides full information, there are no
returns and the seller can "advertise" any return policy. On the other hand, if she
\
provides only partial information and sets|
-
=|
p
=
2
p11÷
1
, the refund amount is p
\- \
equal to p
-
|
-
= 2p
1
÷ 1 = c. Therefore, the seller can advertise a refund amount 1
of more than the salvage value if c > s, or c > s
2
. However, since Solution (E) \
does not actually exercise returns, we conclude that a refund amount of more than
the salvage value is never exercised.
93
Appendix C
Appendix for Essay 3
C.1 Proof of Proposition 7
Suppose, without loss of generality, thato
Z
>o
Y
. Then, the condition for seller Z
capturing the whole market is v
Y
Z
= 0. Solving from (3.1) foro
Z
that satisfies this, u
we get
o(v
Y
Z
= 0) =o
Y
+ p
2
(1 ÷o
Y
)1|
Z
(÷|
Y
)
2
.
o
Z
=o
u 1 2
+ p|
2
1Z
Note thato>o
Y
only if|
Z
>|
Y
ando
Y
< 1. Furthermore, we observe that
cv
Y
Z = p
2
(1 ÷o )|
2
÷|
2
> 0
u 1 Y Z Y
co
Z
2(o
Z
÷o
Y
)
2
if|
Z
>|
Y
ando
Y
< 1. Then, anyo
Z
|
Y
,o
Z
e [o
Y
, o]
results in seller Z capturing the whole market.
Suppose now thato
Z
0. In this case,
v
ZY
is defined analogous to v
Y
Z
in (3.1); all consumers with v > v
ZY
prefer seller
u u u
Y to seller Z, while those with v < v
ZY
prefer seller Z to seller Y . Then, seller Z u
captures the whole market if v
ZY
= 1. Solving foro
Z
from (3.1), we find u
o
Y
÷ p
2
(1 ÷o
Y
)1|
Z
(÷|
Y
)
2
.
o
Z
=o(v
ZY
= 1) u
1 2
÷ p|
2
1Z
Observe thato(v
ZY
= 1)|
Y
ando
Y
< 1, and thato(v
ZY
=
u
1)> 0 as long aso
Y
> 0. Moreover, we have
u
cv
ZY
= p
2
(1 ÷o )|
2
÷|
2
> 0
u 1 Y Z Y
co
Z
2(o
Y
÷o
Z
)
2
if|
Z
>|
Y
ando
Y
< 1. Then, anyo
Z
>o(v
ZY
= 1) results in v
ZY
> 1, and
u u
thus seller Z captures the whole market. This means, given thato
Y
e (0, 1) and
|
Z
>|
Y
,o
Z
e [o(v
ZY
= 1), o
Y
] results in seller Z capturing the whole market, or u
94
given thato
Y
= 0 and|
Z
>|
Y
,o
Z
= 0 results in seller Z capturing the whole
market.
Combining the two results above, we conclude that giveno
Y
< 1, seller Z
captures the whole market if she sets|
Z
>|
Y
ando
Z
such thato
Z
e [o,o], where
o
max{0, o(v
ZY
= 1)}. u
Suppose seller Z would like to have only the consumers with valuation greater
than (1 ÷ˆ) prefer seller Z over seller Y ; that is she would like to have v
Y
Z
= 1 ÷ˆ.
v
u
v
From (3.1), we see that this is possible only ifo
Z
>o
Y
and solving foro
Z
, we find
o(v
Y
Z
= 1 ÷ˆ) =o
Y
+ p
2
(1 ÷o
Y
)2ˆ +|(Zp÷|
Y2
÷ 1.
o
Z
=o
ˆ
v
u
v
1
v
2 1Z 2
|)
Then, seller Z can set|
Z
>|
Y
ando
Z
=o
ˆ
>o
Y
to attain v
Y
Z
= 1 ÷ˆ, as long
1÷(p1|Z )
2 v u
v
aso
Y
< 1 and ˆ>v
2
.
Note that v
Y
Z
= 1 ÷ˆ means seller Z has a market vu
share of 1 ÷ v
Y
Z
= ˆ. vu
C.2 Proof of Proposition 8
Before we proceed with the proof, we note that the crucial aspect of the duopoly
case in this essay is that the market is being divided among the sellers, and that there is no
value creation as a result of competition. In other words, there are no win-win scenarios
and the game is rather close to a constant-sum game. In the light of this observation and the
market share dynamics described in Proposition 7, our
first intuition is that setting (o
Z
, |
Z
) to capture the whole market is a potential best
response of seller Z to seller Y 's (o
Y
< 1, |
Y
< 1). Note that once seller Z captures
the whole market, she is efectively a monopoly and the results for the monopoly
case directly apply. Being a monopoly, seller Z clearly prefers to be at the monopoly
optimal solution described in Proposition 5 (recall that in the duopoly case, we as-
sume p
1
and p
2
are given, and therefore Proposition 5 applies). However, we see from
Proposition 7 that conditions for seller Z to become a monopoly is not arbitrary,
and that she is not necessarily able to attain the monopoly optimal solution while becoming
a monopoly. In the proof below, we first identify the cases where seller Z can capture the
whole market at the monopoly optimal solution. Then, we look
95
at the remainder cases step-by-step and investigate whether capturing the whole
market is profitable given that monopoly optimal solution is not attainable. We ul-
timately find that under any condition, seller Z's best response to (o
Y
< 1, |
Y
< 1)
is to appropriately set (o
Z
, |
Z
) to capture the whole market. Since the sellers are identical in terms of
p
1
, p
2
, c, s and g, we conclude that there is no Nash equilibrium where a seller sets (o
j
< 1, |
j
<
1). Given this result, we analyze best responses in
the form of full refund (o
j
< 1, |
j
= 1) and full information (o
Y
= 1, |
Y
e [0, 1]),
and identify the potential Nash equilibria and the associated conditions as given in
Proposition 8.
We start by summarizing Proposition 5 for uniformly distributed valuations:
The optimal (o, |) for a monopolistic seller when p
1
, p
2
, c, s and g are given is that
the seller chooses either Solution (D) if g < ¯
E
, or Solution (E) if g > ¯
E
, where,
g
u
g
u
for uniformly distributed valuations,
Solution(D) : (o
-
, |
-
) =
(o, |) ,o e 0, 1 ÷ 1 +2(p
1
|)
2
, | = 2s
,
Solution(E) : (o
-
, |
-
) = {(o, |) , v
u
= p
1
} , and
p
1
p
1
p
1
÷ c +
s
2
¯
E
= (p ÷ c) (1 ÷ p ). 4p1
g
u
2 2
Suppose first that g > ¯
E
. Then, Solution (E) is optimal for a monopoly g
u
seller and thus, her optimal decision is to seto and| such that v
u
is as close
to p
1
as possible. Therefore in the duopoly case, following Propositions 3 and
7, if g > ¯
E
, then the best response of seller Z to (o
Y
< 1, |
Y
|
p
, however, seller Z cannot capture the whole
market and set v
u,Z
= p
1
at the same time. However, from Proposition 7, ifo
Y
< 1,
she can achieve v
Y
Z
= p
1
by setting|
Z
>|
Y
and u
o
Z
=
o(v
Y
Z u
= p
1
) =o
Y
+
p
2
(1
1
|
2
÷|
2
÷o
Y
)1 + (pZ| )
2
Y÷ 2p .
1Z 1
Note that, since|
Z
>|
Y
>|
p
, we have 1 + (p
1
|
Z
)
2
÷ 2p
1
> 0 and v
u,Z
s p
1
. By
setting v
Y
Z
= p
1
, seller Z ensures that only those consumers with valuation greater u
96
than the price prefer seller Z over seller Y , and since v
u,Z
s p
1
, all such consumers
purchase from seller Z. In other words, seller Z achieves monopoly optimal profits.
We conclude, due to symmetry, that if g > ¯
E
, there is no equilibrium where a seller g
u
sets (o
j
< 1, |
j
< 1), since the other seller can always capture the whole market
profitably.
Suppose that g < ¯
E
. In this case, we know that Solution (D) is optimal for a g
u
monopoly seller, and that she would seto as low as possible so that she can sell to
as many consumers as possible. Thus, if seller Y chooses (o
Y
< 1, |
Y
<
2p1 ),
s
seller
Z can set|
Z
=
s
2p1
ando
Z
= max{0, o(v
ZY
= 1)}, capturing the whole market u
profitably. Therefore due to symmetry, if g < ¯
E
, there is no equilibrium where a g
u
seller sets (o
j
< 1, |
j
<
2p1 ). s
Recall by definition in (2.5) that ¯
E
= g
E
(| =
g
u
u
2p1 )
s
for uniformly distributed
valuations. Then, compare (B.4) and (B.7) in Appendix B to observe that
g
II
= g
E
(| = ps ) = (p ÷pc1)÷ c÷ p )
u u 1 2
(1
2
for uniformly distributed valuations. Given that
c g
E
= u
c|
s ÷ 2p
1
|
(p
2
÷ c) (1 ÷ p
2
)< 0
for all|>
2p1 ,
s
we conclude that g
E
strictly decreases from ¯
E
to g
II
as| goes from
u
g
u
u
s
to
p1 .
s
As a corollary, if g
II
< g < ¯
E
, then there exists a critical return factor,
2p1
u
g
u
s
2p1
g
II
and|>|, it is no longer ˆ
u
optimal to have any dissatisfied buyers for a monopoly. Therefore, while capturing
97
the whole market, seller Z setso
Z
=o(v
Y
Z
= p
1
) and|
Z
>|
Y
, achieving the u
monopoly optimal profits; v
Y
Z
= p
1
means all consumers with v > p
1
prefer seller u
co(v
YZ
=p1)
Z, and since v
u,Z
< p
1
, they all purchase. Note that since
u
c|
> 0, there
exists|
Z
>|
Y
such that 1>o(v
Y
Z
= p
1
)>o
Y
, given thato
Y
< 1 and|
Y
< 1. As u
a result, if g
II
< g < ¯
E
, and given thato
Y
< 1, seller Z has a best response that
u
g
u
enables him to capture the whole market profitably for any|
Y
< 1. We conclude
due to symmetry that if g
II
< g < ¯
E
, there is no equilibrium with (o
j
< 1, |
j
< 1)
for any seller.
u
g
u
So far we showed that if g > g
II
, there is no Nash equilibrium where any seller u
sets (o
j
< 1, |
j
< 1). Then, suppose that g < g
II
. Given this sufciently small u
g, it is optimal for a monopoly seller to sell to all consumers as long as| s
p1 .
s
Therefore in the duopoly case, as response to (o
Y
< 1, |
Y
<
p1 ),
s
seller Z can set
(o
Z
= max{0, o(v
ZY
= 1)}, |
Z
= u
p1 ),
s
and thus capture the whole market selling
to as many consumers as possible. Due to symmetry, we conclude that if g < g
II
, u
neither seller sets (o
j
< 1, |
j
<
p1 )
s
in an equilibrium.
Now recall from (B.6) and the subsequent analysis in Appendix B that if
|>
p1 ,
s
it is optimal for a monopoly seller to sell to all consumers as long as
g
III
> 0 and if g < g
III
. Since g
III
is decreasing in|, if g
III
(| = 1)> 0, then
u u u u
g
III
> 0 for all|< 1; otherwise, since g
III
(| =
p1 )
s
= g
II
> 0, there is a
s
o
Y
and|
Y
=|
Z
= 1, the value ofo
Z
is
irrelevant for both sellers and the value ofo
Y
is irrelevant for seller Z. Therefore, essentially,
seller Z has three potential best responses: 1) (o
Z
p
1
; in the second, seller Y
and seller Z are identical and they equally share the profits; in the third, seller Z
1+p
2
provides full information and we have v
Y
Z
= u
2
1
> p
1
. We write seller Z's net
99
profits under each decision as follows:
R
Z
(o
Z
¯
III
and would like to decrease it if g
u
g < ¯
III
. Suppose g > ¯
III
; then seller Z's optimal decision giveno
Z
R
Z
(o
Z
=o
Y
, |
j
= 1)> R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1) if
g > g
IV
and v
u,Y
>ˆ
u
, otherwise if g > g
IV
and v
u,Y
s
u
and g > g
V
, theno
Z
u o
Y
is the best response
if v
u,Y
>ˆ
u
, ando
Z
>o
Y
is the best response if v
u,Y
s
u
and ¯
III
< g < g
V
; then, due to symmetry, there is no pure-strategy
g
u
u
Nash equilibrium where a seller setso
j
< 1 because providing a marginally less
information than competition is always the best response for both sellers and there
is a continuum of such best responses.
Suppose that s > s
u
and g > g
V
. Then, both sellers' best response is to provide u
marginally less information than the competition until, without loss of generality,
v
u,Z
= ˆ
u,Z
, at which point seller Y 's best response iso
Y
= 1. Seller B's best response v
too
Y
= 1 is either (o
Z
1, |
Z
= 1) with profits R
Z
(o
Z
o
Y
= 1, |
Z
= 1), or
o
Z
= 1 with profits
R
Z
(o
j
= 1) = 1 ((p
1
÷ c)(1 ÷ p
1
) + g(p
2
÷ c)(1 ÷ p
2
)) . 2
We find that R
Z
(o
Z
o
Y
= 1, |
Z
= 1)> R
Z
(o
j
= 1) only if g > g
IV
; however, u
given s > s
u
, we have g
V
> ¯
III
> g
IV
. Therefore, if s > s
u
and g > g
V
, seller Z's
u
g
u
u u
best response too
Y
= 1 is (o
Z
1, |
Z
= 1). Given (o
Z
1, |
Z
= 1), seller Y 's best
response is (o
Y
o
Z
, |
Z
= 1) and the sellers are back in the loop of a continuous
series of best responses where they unilaterally deviate from an equilibrium. As a
result, there is no pure-strategy Nash equilibrium if s > s
u
and g > g
V
. Combined u
with the above result, we conclude that if s > s
u
and g > ¯
III
, there is no pure- g
u
strategy Nash equilibrium in the duopoly.
On the other hand, consider, for s > s
u
and ¯
III
< g < g
V
, the case where
g
u
u
seller Y sets|
Y
= 1 and pickso
Y
e [0, 1] arbitrarily. Without knowing where seller
Y is located in terms ofo, seller Z is forced to randomize his decision as well and her
best response is similarly to set|
Z
= 1 and chooseo
Z
e [0, 1] randomly. Therefore,
given s > s
u
and ¯
III
< g < g
V
, there is a mixed-strategy Nash equilibrium where
g
u
u
both sellers set|
j
= 1, and picko
j
e [0, 1] randomly. Next, consider for s > s
u
and
g > g
V
, the case where seller Y sets|
Y
= 1 and pickso
Y
e [o, 1] randomly, where ˆu
101
oˆ o(v
u
= ˆ
u
) can be found by substituting (C.1) and| = 1 into (2.3). From the v
above analysis, we see that seller Z's best response is to set|
Z
= 1 and randomize
o
Z
e [o, 1]. As a result, given s > s
u
and g > g
V
, there is a mixed-strategy Nash
ˆ
u
equilibrium where both sellers set|
j
= 1 and picko
j
e [o, 1] randomly. ˆ
Consider now s > s
u
and g < ¯
III
. In this case, seller Z's best response to g
u
(o
Y
e (0, 1), |
Y
= 1) giveno
Z
0, |
Z
= 1) =(p
1
÷ c)
2
1
+ (s ÷ p
1
)p
1
+ g(p
2
÷ c)(1 ÷ p
2
)
1 + p
2
÷ p . 1
1
2
We determine that if g < g
V
, then u
R
Z
(o
Z
= 0, o
Y
> 0, |
Z
= 1)> R
Z
(o
Z
=o
Y
, |
j
= 1)> R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1)
and if g > g
V
, then u
R
Z
(o
Z
= 0, o
Y
> 0, |
Z
= 1)< R
Z
(o
Z
=o
Y
, |
j
= 1)< R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1)
for anyo
Y
. Since for s > s
u
we have g
V
> ¯
III
, we conclude that if s > s
u
and
u
g
u
given g < ¯
III
, seller Z's best response to (o
Y
e (0, 1), |
Y
= 1) is (o
Z
= 0, |
Z
= 1). g
u
Consider then (o
Y
= 0, |
Y
= 1); seller Z's best response is either (o
Z
> 0, |
Z
= 1)
with profits equal to R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1), oro
Z
= 0 resulting in
R
Z
(o
j
= 0, |
j
= 1) = 1 ((p
1
÷ c) + (s ÷ p
1
)p
1
+ g(p
2
÷ c)(1 ÷ p
2
)(1 ÷ p
1
)) . 2
We see from above that R
Z
(o
j
= 0, |
j
= 1)> R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1) if s > s
u
and g < ¯
III
; in other words, both sellers' best response to the competition providing g
u
no information and ofering full refund is to provide no information and ofer full
refund. This result leads us to a Nash equilibrium where both sellers provide no
information and ofer a full refund return policy, (o
j
= 0, |
j
= 1), in case of s > s
u
and g < ¯
III
. g
u
Suppose now s < s
u
, in which case we have g
V
< ¯
III
< g
IV
. Therefore, if
u
g
u
u
g < g
V
, then we have g < ¯
III
readily satisfied, and following the analysis above,
u
g
u
1 02
seller Z's best response to (o
Y
e [0, 1), |
Y
= 1) is (o
Z
= 0, |
Z
= 1). As a result,
(o
j
= 0, |
j
= 1) is the only Nash equilibrium if s < s
u
and g < g
V
. u
Given s < s
u
, suppose g > g
V
; then, seller Z's best response to (o
Y
= 0, |
Y
= u
1) is (o
Z
> 0, |
Z
= 1). In other words, both sellers' best response to the competition
setting (o
j
< 1, |
j
= 1) is to provide more information than the competition,
and therefore due to symmetry, there is no equilibrium where both sellers have
(o
j
< 1, |
j
= 1). Consider then (o
Y
= 1, |
Y
e [0, 1]); seller Z's best response is
either (o
Z
= 0, |
Z
= 1) with R
Z
(o
Z
= 0, o
Y
> 0, |
Z
= 1), or (o
Z
= 1, |
Z
e [0, 1])
with R
Z
(o
j
= 1). Comparing the profits, we find that if g < g
V
I
, seller Z's best u
response to (o
Y
= 1, |
Y
e [0, 1]) is (o
Z
= 0, |
Z
= 1), and if g > g
V
I
, then it is u
(o
Z
= 1, |
Z
e [0, 1]), where
g
V
I
u
2s ÷ c ÷ p
1
(1 ÷ p
1
+ c) .
(2 ÷ p
1
)(p
2
÷ c)(1 ÷ p
1
)
We further find that given s < s
u
, we have g
V
< g
V
I
< ¯
III
. Therefore, if s < s
u
and
u u
g
u
g
V
< g < g
V
I
, seller Z's best response to (o
Y
= 1, |
Y
e [0, 1]) is (o
Z
= 0, |
Z
= 1),
u u
to which seller Y 's best response is (o
Y
= 1, |
Y
e [0, 1]). We conclude that, if
s < s
u
and g
V
< g < g
V
I
, there is a Nash equilibrium where one seller provides
u u
full information and ofers an arbitrary return policy, while the other seller provides
zero information but ofers a full refund return policy.
Consider now the case where s < s
u
(for which g
V
I
< ¯
III
< g
IV
) and g
V
I
<
u
g
u
u u
g < g
IV
. We know from above that if g < ¯
III
, then seller Z's best response to
u
g
u
(o
Y
= 1, |
Y
e [0, 1]) giveno
Z
¯
III
, then seller Z's best response giveno
Z
Study on Essays On Issues In New Product Introduction: Product Rollovers, Information Provision, And Return Policies, Product management is an organizational lifecycle function within a company dealing with the planning, forecasting, or marketing of a product or products at all stages of the product lifecycle.
Study on Essays On Issues In New Product Introduction: Product
Rollovers, Information Provision, And Return Policies
Abstract:-
In this dissertation we study several key issues faced by firms while introducing
new products to market. The first essay looks at product rollovers: introduction of
a new product generation while phasing out the old one. We study the strategic
decision of dual vs. single roll jointly with operational decisions of inventory and
pricing during this transitional period. Our results confirm previous findings and
uncover the role and interaction of several parameters that were not examined before.
In the second essay, we investigate the role of information provision and return
policies in the consumer purchasing behavior and on the overall market outcome.
We build a novel model of consumer learning, and we attain significant analytical
findings without making any distributional assumptions. We then fully study the
joint optimization problem analytically under uniform valuations.
In the third essay, we study competition in the framework described in the
second essay and we identify the potential Nash equilibria and associated conditions.
Our findings demonstrate the efect of competition on return policy and information
provision decisions and provide insight on some real-life observations.
ESSAYS ON ISSUES IN NEW PRODUCT INTRODUCTION:
PRODUCT ROLLOVERS, INFORMATION PROVISION, AND
RETURN POLICIES
by
Eylem Koca
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
2011
Advisory Committee:
Professor Gilvan C. Souza, Co-Chair
Professor Yi Xu, Co-Chair
Professor Michael Ball
Professor Peter Cramton
Professor Martin Dresner
c Copyright by
Eylem Koca
2011
Dedication
I dedicate this dissertation to the two women who define me: my guardian an-
gel, my mother, Halime Hatun, who taught me dedication and perseverance among
other things - she's my light, my inspiration; and my dear love, my wife, Glaucia,
without whom this dissertation would neither be possible nor mean anything - she's
my rock, my soulmate.
ii
Acknowledgments
I owe my deepest gratitude to my advisor, Prof. Gilvan C. Souza, for his
invaluable academic and otherwise counsel, and for not giving up on me even in the
hardest times. He has been not only an incredible mentor, but also a true friend.
I would like to give my heartfelt thanks to my dissertation co-chair, Prof.
Yi Xu, who was always there when I needed his guidance and help, and to Prof.
Cheryl Druehl for her support over the years, and for her advice and contributions
especially in the first part of this dissertation.
I would like to express my thanks Prof. Michael Ball, Prof. Peter Cramton,
and Prof. Martin Dresner for agreeing to serve on my dissertation committee and
for taking the time to review the manuscript. My thanks are also due to Prof.
Itir Karaesmen-Aydin and Dr. Barney Corwin, for their continuous support and
guidance.
Lastly, I am greatly indebted to all who helped me through my Ph.D. studies
with their support, advice, and friendship.
iii
Table of Contents
List of Tables vi
List of Figures vii
1 Managing Product Rollovers 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1
1.1.1 Contribution of This Study . . . . . . . . . . . . . . .. . . .
41.2 Related
Literature . . . . . . . . . . . . . . . . . . . . . . . .. . . .
61.3 Model .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 8
1.3.1 Planning Horizon . . . . . . . . . . . . . . . . . . . . .. . . .
81.3.2
Demand Process . . . . . . . . . . . . . . . . . . . . .. . . . 11
1.3.3 Optimization Problem . . . . . . . . . . . . . . . . . .. . . . 17
1.3.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 19
1.4 Comparison of Rollover Strategies: Numerical Analysis . . . .. . . . 21
1.4.1 Parameters Describing the Planning Horizon:t ,o . . .. . . . 22
1.4.2 Parameters Describing the Arrival Process: p, q,u,¸, M
0
. . . 22
1.4.3 Parameters Describing the Reservation Prices: µ, k . .. . . . 24
1.4.4 Auxiliary Parameters and Summary of Runs . . . . . .. . . . 25
1.4.5 Statistical Analysis . . . . . . . . . . . . . . . . . . . .. . . . 28
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 32
2 Return Policies and Seller-Provided Information in Experience Good Markets
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Literature
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Model . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
2.3.1 Consumer Uncertainty . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Market
Demand . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Analysis of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Structural Properties . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Seller's
Optimization Problem . . . . . . . . . . . . . . . . . . 2.4.3 Characterization of
Optimal Information and Refunds . . . . .
2.5 Jointly Optimal Information, Refund and Price Strategy . . . . . . . 2.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
34
38
42
42
45
47
47
51
52
55
59
3 Managing Return Policies and Information Provision under Competition 61
3.1 Introduction . . . . . . . . . . . . . . ......... ........ . 61
3.2 Literature Review . . . . . . . . . . . ......... ........ . 64
3.3 Competition Model . . . . . . . . . . ......... ........ . 65
3.3.1 Market Share Dynamics . . . ......... ........ . 67
3.4 (o, |) Equilibrium . . . . . . . . . . ......... ........ . 69
3.5 Conclusion . . . . . . . . . . . . . . . ......... ........ . 74
iv
A Appendix for Essay 1 75
A.1 Derivation of h(¸) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.3 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A.4 Normalization of| for the Regression . . . . . . . . . . . . . . . . . . 80
B Appendix for Essay 2 82
B.1 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 82
B.2 Structural Properties of the (o, |) Decision Space . . . . . . . . . . . 82
B.2.1 Boundary for Returns: v
u
(o, |) = p
1
| . . . . . . . . . . . . . 82
B.2.2 Boundary for 100% Sales: v
u
(o, |) = 0 . . . . . . . . . . . . . 83
B.2.3 Boundary for No Sales: v
u
(o, |) = 1 . . . . . . . . . . . . . . 83
B.2.4 Redundant Regions . . . . . . . . . . . . . . . . . . . . . . . . 83
B.3 Analysis of theo, | Decision Space . . . . . . . . . . . . . . . . . . . 84
B.3.1 Ex-Ante Efcient Market: v
u
= p
1
. . . . . . . . . . . . . . . . 84
B.3.2 Positive Consumer Surplus: v
u
> p
1
. . . . . . . . . . . . . . . 84
B.3.3 Some Dissatisfied Consumers, No Returns: p
1
> v
u
> p
1
|
B.3.4 Some Dissatisfied Consumers, Some Returns: v
u
< p
1
| .
B.4 Analysis of Candidate Regions and Proof of Proposition 5 . . .
B.4.1 Region (A) . . . . . . . . . . . . . . . . . . . . . . . . . B.4.2 Regions
(B) and (D) . . . . . . . . . . . . . . . . . . . .
B.5 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . .
B.5.1 Region (C) . . . . . . . . . . . . . . . . . . . . . . . . . B.5.2 Region (D)
. . . . . . . . . . . . . . . . . . . . . . . . . B.5.3 Region (E) . . . . . . . . . . . . . .
. . . . . . . . . . . . B.5.4 Deriving the Optimal Strategy . . . . . . . . . . . . .
. .
B.6 On the Value of Optimal Refund Amount . . . . . . . . . . . .
C Appendix for Essay 3
.
.
.
.
.
.
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.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
85
85
88
88
88
89
89
90
91
91
92
94
C.1 Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 94 C.2 Proof of
Proposition 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 95
v
List of Tables
1.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 1.2
Experimental design. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3 Statistics for diference in profits (dual minus single). . . . . . . . . . 27
1.4 Statistics of multiple linear regression: Main efects. . . . . . . . . . . 29
A.1 Statistics of multiple linear regression: Two-way interaction efects. . 81
vi
List of Figures
1.1 Planning horizon and sequence of events. . . . . . . . . . . . . . . . . 10
1.2 Customer arrival intensities for each rollover strategy. . . . . . . . . . 15
1.3 Customer arrival intensities for diferent responsiveness parameters. . 16
1.4 Weibull densities with diferent shape and scale parameters. . . . . . 17
1.5 Arrival rates for diferent difusion parameters. . . . . . . . . . . . . . 23
2.1 Illustration of consumer learning and heterogeneity with information. 44
2.2 Chronology of events in the two-period setting. . . . . . . . . . . . . 45
2.3 Possible cases for v
u
and corresponding market outcomes. . . . . . . . 48
2.4 Seller's (o, |) decision space for V ? U (0, 1) at diferent price points. 50
2.5 Candidate solutions in the seller's (o, |) decision space for V ? U (0, 1). 53
2.6 Critical regions in the (s, c) plane. . . . . . . . . . . . . . . . . . . . . 56
B.1 Candidate regions for optimality for V ? U (0, 1), p
1
= 0.51 and s = 0.4. 87
B.2 Ordering of solutions according to the the regions in Figure 2.6. . . . 93
vii
Chapter 1
Managing Product Rollovers
1.1 Introduction
Firms, particularly in high-tech markets, increasingly see new product introduction
as a tool to gain or maintain market share, to sustain growth, and to create profits.
Accordingly, firms are under constant pressure for faster time-to-market and shorter
life cycles for many products, and face the challenges of managing these. In addi-
tion to traditional new product development issues such as cost, quality, and time-
to-market trade-ofs, more frequent product introductions result in more frequent
product rollovers—the process of phasing out the old generation while introducing
the new to the market. Therefore, successful product introduction requires success-
ful management of product rollovers, which involves several interrelated decisions
including those on timing, pricing, preannouncing, and controlling inventory.
There are two basic product rollover timing strategies available to a firm. In a
dual product rollover (dual roll), the old generation remains in the market for some
time after the launch of the new; in a single product rollover (single roll), the old
generation is discontinued as soon as the new generation arrives (Billington et al.
1998). Both of these strategies have implications on the operational decisions that
a firm must make. In a single roll, sharp price markdowns may be necessary to
clear excess inventory of the old product. Under a dual roll, the old product retards
1
the difusion of the new product into the market; this may be undesirable as new
products typically command higher margins.
There are numerous real-life examples attesting to the interplay and conse-
quences of inventory, pricing, and timing decisions. Intel had scheduled the launch
of its X48 chipset for PC motherboards in January 2008, when the X38 chipset
would be replaced in the high-end market. However, the launch was delayed for
two months due to pressure from the world's largest motherboard manufacturer,
ASUSTEK, on the grounds that it had too much inventory of X38-based parts that
were marginally inferior to X48-based parts. Other manufacturers who had no in-
ventory problems had to wait until March 2008 although they were ready for launch
in January. Another motherboard manufacturer, ASRock, was first to launch its
P43-based motherboards to the mainstream market in June 2008, while all other
manufacturers were struggling with their inventory of older P35-based parts even
with significant price cuts. In November 2007, AMD introduced deep price cuts
for its older Athlon based processors and rushed its long-awaited, quad-core Phe-
nom processors to the market before the holiday season, even though the processors
had a fault which caused unexpected crashes. Further, AMD was unable to meet
the demand at launch and prices remained higher than Intel's competing quad-core
processors that already greatly dominated the market and performed better. In the
end, although the fault was corrected by March 2008, the highly anticipated Phenom
architecture failed to capture the market share expected (various online technology
news sources).
Despite their importance, product rollovers are commonly mismanaged in
2
practice, while understudied in the academic literature. A study of 126 U.S. durable
goods firms reports that 40% of new products failed after launch (Ettlie 1993), one
possible reason being mismanaged rollovers such as the previous examples. Another
study by Greenley and Bayus (1994) indicates that most U.S. and U.K. firms do
not have a formal decision process for product rollovers. Not only is there just a
handful of scholarly papers that discuss product rollover strategies, but there is lit-
tle consensus among them on what rollover strategy to use under what condition.
Saunders and Jobber (1994) identify 11 rollover strategies, which they call "phas-
ing." They survey U.S. and U.K. managers and find that some sort of dual roll was
used in slightly more than half of them. Billington et al. (1998) and Erhun et al.
(2007) present managerial papers that provide understanding and guidelines derived
from intuition and hands-on experience, but no formal treatment of the problem.
While Billington et al. (1998) associate single (dual) roll with low (high) supply and
demand risk, Erhun et al. (2007) state that oftentimes the industry dictates this
decision and that dual roll is an industry standard for high-tech markets even with
low supply and demand risks. The only two papers to our knowledge that attempt
a formal analysis of product rollovers are Levinthal and Purohit (1989) and Lim and
Tang (2006), but neither model incorporates difusion, a key attribute of high-tech
markets. Although they use diferent terminology, Levinthal and Purohit (1989)
consider three alternative strategies: single roll, dual roll, and dual roll with buy-
back of the old generation. They find that single roll is always better than dual roll,
and that single roll is better than dual roll with buy-backs for modest performance
improvements of the new product over the old. Contrast this finding with the rec-
3
ommendation of Billington et al. (1998), who suggest that a large technological gap
between generations (large product risk) favors dual roll. Lim and Tang (2006) find
that dual roll is optimal when marginal costs across generations are similar, using a
linear deterministic demand structure. A few other authors (Carrillo 2005, Li and
Gao 2008, Druehl et al. 2009) simply assume a particular rollover strategy in their
models, regardless of the environment.
1.1.1 Contribution of This Study
We are not aware of an academic study that provides an integrated, formal treatment
of product rollovers that incorporates the dynamics discussed above; we address this
gap using a comprehensive model of product rollovers that includes pricing, inven-
tory, product difusion, and new product preannouncement (before introduction).
More specifically, our key contribution in this essay is to identify the conditions
under which a particular rollover strategy (single vs. dual) is preferred, and which
factors play the most significant role in this strategy decision. We describe our
approach below.
We focus on successive improved generations of a single product by a firm such
as ASUSTEK. The fact that high-tech products are often introduced on a relatively
regular basis supports our model; this notion of (time) pacing of product updates
may also improve a firm's product development capability (Eisenhardt and Brown
1998). For example, the pacemaker company Medtronics has successfully used a
time-pacing strategy (Christensen 1997). We adapt the multi-generation difusion
4
process by Norton and Bass (1987) to model the arrival process of potential cus-
tomers through the life-cycle of a product. Here, however, an arriving customer
buys the product if the price is lower than her reservation price. In addition, the
firm preannounces the new product sometime before its launch and we study dif-
ferent levels of the market's responsiveness to preannouncements to account for the
potential changes in consumer purchasing behavior due to the preannouncement.
The firm first adopts a product rollover strategy, single or dual roll, then decides
on the quantity for the final build of the old product and the price paths for both
products.
We find that the decision between dual and single roll is not trivial and depends
on a number of (exogenous) factors considered in our model. Specifically, dual roll
is preferred to single roll if (i) the time between product introductions is short,
(ii) the preannouncement occurs at the later stages of the life-cycle, (iii) the old
product keeps more of its value at the end-of-life, (iv) the market is less responsive
to preannouncements, (v) the new product is expected to have a slower market
difusion, and/or (vi) performance improvement between the new and old products
is smaller
1
. We also find that the optimal price paths closely follow customer
reservation prices over time.
In the next section, we show how our work relates to and difers from the
existing literature. We then present our model and its analytical solutions in Section
1
Although some of these factors, such as timing of preannouncement, are in reality not exoge-
nous but decided on by the seller, we treat them as exogenous for tractability and to focus on the
two rollover strategies, and we perform a sensitivity analysis to study their impact on profit.
5
1.3, and Section 1.4, a comprehensive numerical analysis of the factors impacting
the optimal rollover strategy. We conclude in Section 1.5.
1.2 Related Literature
We bring together elements from a diverse literature, incorporating rollover strate-
gies, difusion of innovations (Norton and Bass 1987, Bass and Bass 2004), prean-
nouncements (Farrell and Saloner 1986, Manceau et al. 2002, Su and Rao 2008), dy-
namic pricing (Bitran and Mondschein 1997, Smith and Achabal 1998, Elmaghraby
and Keskinocak 2003), and inventory management at the end of life (Cattani and
Souza 2003).
A stream of research has considered the interaction of difusion and new prod-
uct generations. Savin and Terwiesch (2005) model the difusion efects in a duopoly
and find the optimal launch time. Our model difers from theirs in that we study
a multi-generation scenario and the implications of single versus dual roll strate-
gies. Earlier, Wilson and Norton (1989) determined the optimal time to introduce a
product line extension; thus, the rollover strategy is not relevant. They found that
the second product should generally be introduced immediately or not at all, but ig-
nored price and inventory considerations. Mahajan and Muller (1996) extended this
result in a multi-generational scenario where they found that a monopolist should
introduce the next generation either early in the first product's life cycle, or wait
until it has reached maturity (i.e., sales have peaked).
Pricing of a product over its life-cycle has been addressed by a large number of
6
researchers. Several have focused on finding an optimal pricing pattern, assuming
the sales follow the Bass (1969) model (e.g., Robinson and Lakhani 1975, Bass 1980,
Dolan and Jeuland 1981, Kalish 1983, Horsky 1990). However, these studies found
a pricing pattern that follows the sales growth curve, which is not supported by
empirical data (Krishnan et al. 1999). In more recent work, Krishnan et al. (1999)
present a model extending the Generalized Bass Model (Bass et al. 1994), to find an
optimal price path. None of these papers study pricing considering the next product
generation.
The sequence and timing of new product introductions for two or more prod-
ucts with difering quality levels has been considered as a way to alleviate cannibal-
ism (Moorthy and Png 1992, Chen and Yu 2002, Battacharya et al. 2003, Krishnan
and Zhu 2006). Dhebar (1994) examines the pricing and quality level decisions for a
monopolist introducing two generations of products at fixed times. He finds that the
firm may limit the quality (or features) ofered in each generation to minimize con-
sumer regret. In our setting, the efects of cannibalization are modeled, but sequence
is not considered, and higher quality is always valued more by the customers.
A successful rollover requires inventory management for a product (generation
1) at the end of its life. A related stream of literature focuses on determining the
optimal size of a "final buy" (or "final build") for a product nearing the end of its
life when there is uncertain demand (Teunter and Fortuin 1998, Cattani and Souza
2003). Like this stream of research, we also determine the optimal size of the final
build for generation 1, which in our model is being phased out for introduction
of generation 2. Unlike this stream of research, our model considers the demand
7
interactions - cannibalization - between old and new generations.
In summary, although there is a significant body of research analyzing prod-
uct introduction management, modeling life-cycle demand, and considering pricing
implications, no single work demonstrates the role and interaction of these in the
product rollover process. Our contribution to the literature is to investigate all
these aspects of the problem and present an analytical framework for a unified un-
derstanding.
1.3 Model
1.3.1 Planning Horizon
Consider an infinite horizon where, everyt periods, a firm introduces successive new
generations of a certain product, in order to replace the existing old generation. In
such a setting, a transition takes place between two consecutive product generations
everyt periods; our model focuses on one product rollover that is representative of
this repetitive process. The notation used in this essay is explained in Table 1.1.
Let t = 0 be the time when generation 1 is introduced (made available) to the
market; accordingly, generation 2 is introduced at t =t . Throughout the essay, the
following terms are used interchangeably: generation 1 (2), product 1 (2), and old
(new) product. The planning horizon starts at t =ot ,o e (0, 1), which marks the
time when i) generation 2 is preannounced, and ii) the firm produces a final build of
generation 1 and starts concentrating her production capabilities into assuring that
generation 2 is ready to launch at t =t. An immediate extension of separating these
8
Table 1.1: Notation.
Symbol Explanation
i
j
t
ot
T
j
O
¸
h(¸)
ì
ji
(t)
M
i
m
i
F
j
(t)
p
q
|
p
i
(t)
G
it
(-)
Index for product (generation); i = 1, 2
Index for rollover strategy; j = S (single); j = D (dual)
Time between product introductions
Time of final build after launch of a product; 0 T
j
,
0,
if t s ÷(1 ÷o)t
t ÷ (1÷o)t
F
A
(t ÷ e÷|(p+q) e| ÷e| (1.3)
t (1÷o)t
,
if ÷ (1 ÷o)t< t s 0.
1 +
q
e
p
÷|(p+q) e| ÷e
÷
|
Equation (1.1) is similar to N&B except for the multiplier h(¸), such that
h(¸)F
j
(t ÷t ) is the fraction of potential customers of generation 1 who switch
to generation 2 due to its performance improvement. The fraction of potential
customers for a generation at time t, F
j
(t), is higher than or equal to the cor-
responding F (t) in N&B due to the preannouncement efect; p and q are N&B's
coefcients of innovation and imitation, respectively. In N&B, F (t) = 0 ¬ t s 0,
but here, there is adoption of the new generation after preannouncement (but be-
fore introduction time, i.e., for t s 0), which is denoted by F
A
(t). The parameter
| e (0, ·) represents the responsiveness of customers to preannouncements. If
13
| ÷ 0, then customers are not responsive to preannouncements, and the difusion
process approaches N&B starting at t = 0. If, however,| ÷ ·, then customers
are fully responsive to preannouncements, and (1.2) is equivalent to N&B starting
at t = ÷(1 ÷o)t , the announcement of generation 1. Note that the time argu-
ment in Equation (1.3) is negative as F
A
(-) represents the difusion process due to
preannouncement before a generation is introduced; t s 0.
For generation 2, the arrival rate of potential customers is
j
[(M
0
+ m
1
) h(¸)F
j
(t) + m
2
] F
j
(t ÷t ), for t e (t, (1 +o)t ]
ì
2
(t (1.4)
0,
otherwise.
We further assume that market potentials follow a growth pattern according
to the performance improvement: m
i
=¸M
i
÷
1
and M
i
= M
i
÷
1
+ m
i
, where m
i
(M
i
) is the incremental (cumulative) market potential for generation i. Figure 1.2
demonstrates potential customer arrival intensities during the planning horizon for
p + q = 0.3, q/p = 25, M
0
= 100,¸ = 0.5,t = 20,o = 0.5, and| = 6.275. Note
that the arrival rate for generation 1 is independent of the rollover strategy used
for t st , but for t >t ,ì
S
(t) = 0. Because the market is somewhat responsive 1
to preannouncements (|> 0), the arrival rate for generation 2 at the time of its
introduction att is larger than 0 (zero would be the traditional difusion pattern
of N&B). We also haveì
S
(t)>ì
D
(t) because there is some cannibalization of
2 2
generation 2 by generation 1 in a dual roll. In Figure 1.3, the efect of| is illustrated
using p + q = 0.3, q/p = 25, M
0
= 100,¸ = 0.5,t = 20, ando = 0.5. Note that for
| ÷ 0, the market is unresponsive to preannouncements, and thus the difusion of
generation 2 only starts when it is actually introduced at t =t . For| ÷ ·, the
14
market is fully responsive to preannouncements, and difusion starts immediately
after preannouncement atot , as if generation 2 had been introduced at that time.
M2
Dual Roll, gen. 1
Dual Roll, gen. 2
Single Roll, gen. 1
Single Roll, gen. 2
M
1
ot
t
Time (since release of gen. 1)
(1 +o)t
Figure 1.2: Customer arrival intensities for each rollover strategy.
As stated before, the actual sales rate of generation i at time t depends on the
arrival rate of potential customersì
ji
(t), price p
i
(t) of product i, and the distribution
of customer reservation prices. Price will be discussed in the next section. Customers
of product i at time t have reservation prices distributed according to the cumulative
distribution function (cdf) G
it
(-), and probability density function (pdf) g
it
(-). We
assume that G
it
(-) has the shape of a Weibull distribution, as this distribution is
able to capture a variety of consumer behavior and has been used previously in
the literature (Bitran and Mondschein 1997). The Weibull distribution has two
parameters; the mean is mainly determined by the scale parameter|, and the
variance by both| and the shape parameter k. For illustration, Figure 1.4 plots the
15
A
r
r
i
v
a
l
i
n
t
e
n
s
i
t
y
reservation price distribution g
it
(-) for diferent shape and scale parameters. The
firm knows the distributions for both products at any time; this knowledge feature
is common in most marketing and operations models of consumer behavior.
M2
|÷·, gen. 1
|÷·, gen. 2
|= 6.275, gen. 1
|= 6.275, gen. 2
M
1
|÷0, gen. 1
|÷0, gen. 2
ot = 10 t = 20 (1 +o)t = 30
Time (since release of gen. 1)
Figure 1.3: Customer arrival intensities for diferent responsiveness parameters.
We state the assumptions underlying the demand process in this essay as
follows; we comment on these assumptions later in Section 1.5:
(i) There are no explicit competing firms or products or expectation of any.
(ii) Product generations interact only through the arrival process described above.
Once a customer makes a decision to adopt the new generation, her actual
purchase decision is based on the price of the new generation; she does not
re-evaluate her decision (i.e., consider the old generation) if the price of the
new generation is higher than her reservation price.
16
A
r
r
i
v
a
l
i
n
t
e
n
s
i
t
y
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
| = 5, k = 3.6
| = 8, k = 3.6
| = 10, k = 10.8
| = 10, k = 3.6
| = 10, k = 1.2
00
5
10
15
20
25
price
30
35
40
45
50
Figure 1.4: Weibull densities with diferent shape and scale parameters.
(iii) At any time, there are at most two product generations in the market.
(iv) Prices have no in?uence on the customer arrival processes, although they im-
pact actual sales, because an arriving customer only buys if the price is below
her reservation price. Thus, increasing prices decreases sales monotonically.
(v) Customers are neither price strategic nor do they expect a new generation to
be introduced before its announcement.
1.3.3 Optimization Problem
At the start of the planning horizon, the firm decides on the inventory level for prod-
uct 1, denoted by I
0
= I(ot ), and the price paths for both products, p
i
(t), i = 1, 2,
throughout the horizon such that expected discounted profits are maximized. We
17
f
r
e
q
u
e
n
c
y
assume a unit production cost, c
p
, that is constant over time and across genera-
tions. For product 1, there is a constant holding cost of c
h
per unit per time, and
a constant unit salvage value c
s
< c
p
for any remaining inventory at time T
j
. We
assume that the firm uses a continuous review, instantaneous replenishment inven-
tory control policy for product 2; therefore, there are no holding costs and no lost
sales. We find that the inventory control policy for product 2 does not significantly
afect the comparison of rollover strategies, enabling us to make this simplifying
assumption. We have also analyzed periodic review order-up-to policies and found
that the single vs. dual roll comparison was not significantly afected by the num-
ber of inventory reviews. This result is primarily driven by our assumption that
the firm faces no supply constraints for the new generation. Although some firms
face capacity constraints for new products, particularly immediately after introduc-
tion if the product is popular, our model does not capture this efect, and we leave
investigation of supply constraints for future research.
The discount rate iso and a non-stationary Poisson process, with time-depend-
ent arrival intensityì
ji
(t) as its argument, is denoted by N (-). The profit maximiza-
tion problem depends on the rollover strategy and is solved separately for each
strategy. Given the underlying difusion dynamics, the arrival processes for the two
generations are independent from each other; there are no price or inventory inter-
actions between the two arrival processes. Thus, for each strategy, we can partition
the optimization problem into separate problems for each product.
By selecting a rollover strategy j e {S, D}, the firm faces the following
continuous-time stochastic optimization problem for product 1, where G
it
= 1 ÷ G
it
¯
18
denotes the tail distribution:
T
j
e
÷
o(t÷ot
)
p
1
(t)dI(t) + e
÷
o(T
j
÷
ot
)c
s
I(T
j
)
I0,p1(
max
t
)
H
j
1
s.t.
I (T
j
)
=E ÷
0
ot
t
÷c
h
T
j
ot
e
÷
o(t÷ot
)
I(t)dt ÷ c
p
I
0
(1.5)
I(t) = I
0
÷ N
ot
ì
j
1(u)G
1
u
(p
1
)du , t eot, T
j
. ¯
Note that I(t) describes the inventory remaining at time t and we require that
inventory is nonnegative when the product is pulled from the market. The first term
of the expectation is price times sales rate, the second and third terms account for
salvage and holding costs, respectively, and the last is the cost of final build.
The firm's problem for product 2 is:
(1+o)t
max H
j
2 = E
e
÷
o(t÷ot
)
(p
2
(t) ÷ c
p
) N ì
j
2(t)G
2
t
(p
2
) dt ¯
(1.6)
p (t) 2
1.3.4 Solution
t
The optimal price path for product 2 can be determined in a straightforward manner,
as shown in Proposition 1 below.
Proposition 1 The optimal price path for product 2 satisfies:
p
2
(t) ÷ G
2
t((p 2((tt)) = c
p
, ¬t ¯ p ))
g
2
t 2
(1.7)
This price path is unique if and only if
Proof See Appendix A.
(G2
t
)
2
¯
g2t
19
is a decreasing function of p
2
(t), ¬t.
Proposition 1 shows that the price for generation 2 at any point in time depends
simply on the production cost and the consumer's reservation price distribution at
that time. Due to Assumption 2 in the previous section, there is no interaction with
generation 1 customers. Inventory availability also does not afect price due to the
assumptions on reservation prices and inventory replenishment. If we assume that
G
2
t
(-) = G
2
(-) ¬t, then p
2
(t) will be constant.
The optimization problem (1.5) for product 1 is not tractable due to its
stochastic nature and the existence of multiple decision variables. Proposition 2
below shows that the deterministic version of this problem is asymptotically opti-
mal as arrival intensities grow large.
Proposition 2 Solution to the following deterministic optimal control problem is
asymptotically optimal to (1.5) as M
0
grows large.
T
j
max H
j
I0,p1(t) 1
s.t.
I (T
j
)
=
0
ot
e
÷
o(t÷ot
)
p
1
ì
j
1(t)G
1
t
(p
1
) ÷ c
h
I(t) dt + e
÷
o(T
j
÷ot
)
c
s
I(T
j
) ÷ c
p
I
0
¯
dI(t) = ֓
j
(t)G (p ) for t eot, T
j
dt
1
¯
1
t 1
(1.8)
The optimal price path for product 1 from Equation (1.8) satisfies
p
1
(t) ÷ G
1
t((p 1((tt)) = e
o
(t÷ot
)
c
p
+ co
h
÷ co
h
, ¬t ¯ p ))
g
1
t 1
(1.9)
and is unique if and only if
(G1
t
)
2
¯
g 1t
is a decreasing function of p
1
(t), ¬t.
The associated optimal initial inventory is
T
j
I
0
=
ot
ì
j
1(t)G
1
t
(p
1
)dt ¯
20
(1.10)
Proof See Appendix A.
Note that the optimal price path for product 1 closely follows the reservation
price curve, very similarly to the price path for product 2. The only diference is
that the price for product 1 at any time t also accounts for the holding cost of
inventory incurred betweenot and t. If the reservation price curve for product 1 is
decreasing in t, which is a reasonable scenario considering that the product is ending
its life, then the optimal price will also decrease in t accordingly. Because we find
the solution to the asymptotically optimal deterministic problem, the final build
inventory level I
0
will be exactly sufcient to satisfy all demand between [ot, T
j
]
and there will be no leftover inventory.
To find a price path, one needs to solve equations (1.7) and (1.9) for each time
point t. Given the Weibull distributions for G
it
, there are no closed form solutions for
p
i
(t). Numerically, however, this is straightforward: discretize the planning horizon,
and solve (1.7) and (1.9), through any line-search algorithm, for each discrete t. We
study the problem numerically in the next section.
1.4 Comparison of Rollover Strategies: Numerical Analysis
To develop further insight into the choice of rollover strategy, we turn to numerical
analysis and run a full-factorial experimental design with eight model parameters
at three levels each (low, medium and high). This allows us to better understand
under which conditions of parameter values a particular rollover strategy is preferred,
based on maximal profits resulting from the optimization procedure described in
21
Section 1.3. We now describe our experiment.
1.4.1 Parameters Describing the Planning Horizon:t ,o
The planning horizon is [ot, (1 +o)t ]; however, sales horizons for the two products
difer as shown in Figure 1.1. For the old product, length of the sales horizon (during
the planning horizon) for single roll is (1 ÷o)t , while that for dual roll ist . For the
new product, length of the sales horizon does not depend on the rollover strategy
and is always equal toot . Therefore, givent , a small (large)o indicates a long
(short) sales horizon for the old product under single roll. Consequently, we expect
dual roll to result in higher average profits compared to single roll aso increases.
The efect oft , however, is not as straightforward. A longer time horizon means
higher total sales; however, price may decrease more and there may be downward
substitution, negatively afecting the profit rate. In the following numerical studies,
the time unit is months, and we use (10, 20, 30) months fort ; these are typical
times between product introductions in the high tech industry (Druehl et al. 2009).
Given that 0 p|) or return it for refund
(if v < p|). Thus, given|, v
0
= E[max{V, p|}]. Substituting in (2.1), the ex-ante
valuation of a consumer with ex-post valuation v is given by
v
0
0
v
o
= (1 ÷o)E[max{V, p|}] +ov.
v
o
o
Information
v
1
1
0
(2.2)
Figure 2.1: Illustration of consumer learning and heterogeneity with information.
At ano-state, by definition, the consumer purchases the product if v
o
> p.
From (2.2), one can show that this condition is equivalent to v > v
u
, where
v
u
(o, |)
p + (1 ÷o) (p ÷ E[max{V, p|}]) , o
o e (0, 1], | e [0, 1]
(2.3)
is the threshold ex-post valuation for purchase. Note that the definition of v
u
above
presumeso> 0, since we have homogeneity wheno = 0; either all consumers
purchase if v
0
> p, or none of them purchase if v
0
< p.
44
V
a
l
u
a
t
i
o
n
2.3.2 Market Demand
We conceptualize a two-period model to capture the long-run consequences of seller's
decisions. At the beginning of the first period, the seller announces a new product to
be introduced to the market, and sets the first-period price, p
1
and the refund factor,
|. At the same time, the seller also decides ono and accordingly provides tools to
help the consumers make a more informed purchase decision. The seller announces
the second-period price, p
2
, after the first-period arrivals and their purchase decision.
The product is made available for purchase only at the end of each period; there are
thus two purchase points. The chronology of events is summarized in Figure 2.2.
We assume that this product is sufciently distinguished from existing products in
the market to induce an uncertainty in consumer valuations as studied in this essay.
Seller provides information: o Buyers acquire full information.
0 1 2
time
Seller announces product, Product is available for purchase. Second period arrivals.
available at the end of period 1. First period arrivals.
Seller decides on return policy Seller decides on second period price
and first period price.
Figure 2.2: Chronology of events in the two-period setting.
We normalize the initial size of the potential consumer population to 1. These
first-period arrivals are analogous to the "innovators" as described in Bass (1969),
and are uncertain in their valuations as described above. During the second period,
the innovators "spread the word" such that each first-period arrival who is not
dissatisfied creates a second-period consumer base of g > 0; we assume this process
45
to be deterministic for parsimony
3
. Therefore, the more consumers are dissatisfied
in the first period, the less potential buyers in the second period; this dynamic
efectively captures the future consequences of consumer dissatisfaction. We assume
that even with full refunds (| = 1), the consumers who return their items do not
contribute to market growth in the second period; this efectively incorporates the
negative impact of returns to brand and seller image (Lawton 2008). As a result,
in our model, the larger the parameter g, the greater the negative consequences of
causing consumer dissatisfaction and/or returns. Therefore, hereinafter, we aptly
refer to g as "misfit penalty."
We further assume that the second-period arrivals have full information re-
garding their valuations (through owner experiences and reviews as well as seller
provided informational tools); there is no valuation uncertainty in the second pe-
riod. Finally, we assume that any returns occur at the end of the second period and
any returning consumers leave the market.
This setting enables us to study interesting aspects of seller's decisions:
• Through v
u
, a consumer's purchasing decision in the first period is determined
not only by p
1
, but also byo and|.
• Although providing information has a possible cost to the seller, it enables
3
With this definition, we assume that any first-period arrival contributes to the second-period
market even if he leaves without purchasing. The underlying reasoning is that the difusion (of
information) is triggered by being aware of the product, not by purchasing it. While we take it as
given, the value of g is an indicator of the product's market difusion speed which is afected by
market and product characteristics.
46
consumers to make more informed decisions in the first period and therefore
decreases returns.
• Ofering a generous return policy (high|) increases sales in the first period.
However, as we show in the following section, a sufciently high| results
in consumers purchasing and being dissatisfied post-purchase, which in turn
decreases sales in the second period.
In the next section, we show how the interplay of these three decisions determine
the overall outcome both for the seller and the consumers. We analyze the seller's
optimal decision strategy in detail and solve her profit maximization problem under
a more specific setting.
2.4 Analysis of the Model
2.4.1 Structural Properties
We start our analysis by showing some structural properties of the seller's (o, |)
decision space for a given p
1
. First recall that under our setting, the condition for
consumer purchasing in the first period, v
o
> p
1
, is equivalent to v > v
u
; however,
each consumer realizes her own ex-post valuation, v, only after she purchases the
product. If it turns out that v > p
1
, then the consumer is satisfied. If v < p
1
she is dissatisfied; in this case, if her valuation is as low as to be below the refund
amount (v < p
1
|), then she returns the item, otherwise she keeps it although she
is dissatisfied. As a result, there are three possible market outcomes depending on
47
the value of v
u
. Figure 2.3 illustrates these cases, which we explain below:
Case I: v > p
1
Consumers who purchase
in the first period
All buyers are satisfied
0 p
1
| p
1
v
v
1
Case II: p
1
|sv
u
< p
1
Consumers who purchase in the first period
Dissatisfied buyers
0
p
1
| v
who keep
p
1
Satisfied buyers
1
v
Case III: v p
1
Consumers who purchase in the first period
Dissatisfied buyers Satisfied buyers
Dissatisfied buyers Dissatisfied buyers
who return who keep
v
0 v
p
1
| p
1
1
Figure 2.3: Possible cases for v
u
and corresponding market outcomes.
Case I (v
u
> p
1
): If v
u
> p
1
, all consumers who purchase the product have non-
negative surplus (v > p
1
), since they purchase only if v > v
u
. All buyers are
satisfied, and there are no returns.
Case II (p
1
> v
u
> p
1
|): In this case, there are some dissatisfied buyers (v < p
1
),
but all buyers have v > p
1
|: There are some consumers who are dissatisfied
with their purchase but none of them return their item as the refund amount
is not sufciently attractive.
Case III (v
u
< p
1
|): When v
u
< p
1
|, there are some dissatisfied buyers with p
1
>
48
v > p
1
| who keep their items but also some with v < p
1
| who return for a
refund; there are some buyers who are dissatisfied but not all of them return
their items.
The boundary v
u
= p
1
is of particular importance since it marks the condition
for efcient allocation of the product: When v
u
= p
1
, a consumer's purchasing
condition becomes v > p
1
and therefore, 1) all consumers who value the product at
least as much as its price purchase the product, and 2) all consumers who purchase
the product value it at least as much as its price. Note that, when v
u
= p
1
, this
efcient allocation is achieved ex-ante, as opposed to ex-post (which is possible
through a full refund return policy,| = 1). From (2.3), v
u
= p
1
is satisfied foro = 1
(full information) regardless of the value of|. We show that it can also be satisfied
with partial information (0 E[V ], and we state this result in
Proposition 3 below.
Proposition 3 With partial information, i.e. 0 E[V ], there
exists a |
p
e [0, 1] such that ex-ante efcient allocation is achieved, i.e., v
u
(o, |
p
) = p
1
. Furthermore, v
u
(o, |)> p
1
if ||
p
.
Proof See Appendix B.
Proposition 3 is a significant result as it means that, even without providing
full information, the seller can achieve ex-ante efcient allocation of the product
through a partial-refund return policy, i.e. by setting| =|
p
. In other words,
return policies can be used to substitute for full information in order to minimize
consumer dissatisfaction and returns, even when providing information is costless.
49
This means that under costly information provision, the seller has a clear incentive
to design such a return policy in order to minimize consumer dissatisfaction and
returns.
We see from (2.3) that for a given p
1
, any fixed v
u
value results in a relationship
betweeno and|. Therefore, the three cases above translate into three regions on
the (o, |) plane, as shown in Figure 2.4. In essence, v
u
is on the z-axis in Figure 2.4
and each boundary seen on the (o, |) plane represents the curve satisfying the
specified relationship. Although the graph is plotted for uniform valuations and a
particular p
1
value, we show in Appendix B that it is representative of the general
case (in terms of the signs of first and second order derivatives). Note that the cases
p
1
> E[V ] and p
1
< E[V ] result in diferent graphs since in the latter, v
u
> p
1
for
all| e [0, 1] without full information (0 E[V ] p
1
= 0.45< E[V ]
1.0 1.0
A
0.8
0.6
0.4
Y
O
1
p1
Y
O
I
p1
p1
Y
O
II
p
1
B
Y
O
p1B
A
0.8
0.6
0.4
p1
Y
O
II
p1B
Y
O
p1
Y
O
p1B
Y
O
p1B
Y
O
p1B
III
p1
III
0.2
0.0
0.0
Y
O
B
1
B
p
0.2
0.4
B
0.6
Y
O
0.8
0
1.0
0.2
0.0
0.0
A 1
0.2
V
0.4
B
Y
O
0.6
0
0.8
1.0
Figure 2.4: Seller's (o, |) decision space for V ? U (0, 1) at diferent price points.
50
We now turn our attention to the seller's general profit maximization problem.
2.4.2 Seller's Optimization Problem
Facing the market dynamics described in Section 2.3, the seller maximizes two-
period profits by determining the optimal set of decisions, (o
-
, |
-
, p
-
, p
-
). Demand 12
in the first period is (1 ÷ F (v
u
)), since we have unit market size. Demand in the
second period is (g(1 ÷ F (p
2
))(1 ÷ L)), where L max{0, F (p
1
) ÷ F (v
u
)} is the
rate of dissatisfaction in the first-period market. Fraction of returns is given by
M max{0, F (p
1
|) ÷ F (v
u
)} and an amount of p
1
| is refunded for each return.
Production cost per unit is c and each returned product (if any) has a net salvage
value of s. We assume reasonably that s e (0, 1). Note that we allow for s > c,
in which case there is a profitable market for returned items. We write the seller's
optimization problem in general form as follows and characterize the optimal second-
period price in Proposition 4.
max R = (p
1
÷ c) (1 ÷ F (v
u
)) + (÷p
1
| + s) M + g (p
2
÷ c) (1 ÷ F (p
2
)) (1 ÷ L)
o,|,p
s.t.
i
o, | e [0, 1]
(2.4)
Proposition 4 The optimal price in the second period is given by
p
-
= arg max (p ÷ c) (1 ÷ F (p)).
2 p
Proof The solution for p
2
follows since p
2
appears only in the final term in the
objective function and since g (1 ÷ L) > 0 under any circumstances.
51
Proposition 4 shows that the optimization of p
2
is decoupled from the rest
of the decision process. Therefore in the rest of the essay, we continue studying
the joint optimization ofo, | and p
1
. In the next section, we determine strictly
dominated regions and identify conditions for optimality of others.
2.4.3 Characterization of Optimal Information and Refunds
As discussed in Appendix B, boundaries in the (o, |) decision space, which are
critical for consumer dissatisfaction and existence of returns, are functions of p
1
.
Although this complicates the seller's problem of jointly optimizingo, | and p
1
, we
find that a general characterization of the optimalo and| is possible for a given
p
1
. We summarize our findings in Proposition 5. Note that we do not yet make
any assumptions as to how the consumer valuations are distributed. In the rest of
the essay, we use superscripts for association to the indicated region in the (o, |)
decision space, and we use the subscriptu to indicate a threshold value.
Proposition 5 For a given p
1
, the optimal (o, |) corresponds to one of the two
candidate solutions below, depending on the values of p
1
, g, c and s, and on the
distribution F . We depict these solutions in Figure 2.5.
i) Solution (D): (o
-
, |
-
) = (o, |) ,o e [0, 1 ÷
p1
E[max{V,p1|}
]
], p
1
|
=s÷
F (p1|)F
(p1|)
This solution lies on the region where v
u
= 0 and p
1
| s s. All consumers
purchase but the optimal refund amount is less than the salvage value.
ii) Solution (E): (o
-
, |
-
) = {(o, |) , v
u
= p
1
}
52
This solution implies ex-ante efcient allocation, i.e. v
u
= p
1
, which is
satisfied when o = 1 for any |, or if p
1
> E[V], when | =|
p
for any
o> 0.
1.0
E
0.8
A
0.6
0.4
0.2
0.0
Y
O
I
B
1
B
p
D
II
Y
O
p
1
B
Y
O
0
III
s
B
p
1
0.0 0.2 0.4 0.6 0.8 1.0
B
Figure 2.5: Candidate solutions in the seller's (o, |) decision space for V ? U (0, 1).
Specifically, if g > ¯
E
, then Solution (E) is optimal; if g < ¯
E
, then Solution
(D) is optimal, where
g
u
-
(p
1
÷ c) + (s ÷ p
1
|
-
)
F
F(p(1p|) )1
g
u
and |
-
satisfies
¯
E
=
g
u
(p
2
÷ c) (1 ÷ F (p
2
))
, (2.5)
p
1
|
-
= s ÷ F ((p
1
| -)).
F
-
Proof See Appendix B.
p
1
|
Proposition 5 finds that for sufciently small misfit penalty, Solution (D),
which sells to all consumers in the first period, is optimal; equivalently, for suf-
53
ciently large misfit penalty, Solution (E), which suggests no returns through ex-ante
efcient allocation, is optimal. This result has three immediate corollaries. The first
is that there are conditions that makes the ex-ante efcient allocation undesirable
for the seller. Specifically, if the net profit of selling a product to a consumer in the
first period exceeds the expected loss in the second period due to dissatisfying that
consumer, then the seller chooses (o, |) such that every consumer in the first pe-
riod purchases a product, regardless of his valuation. Since the expected loss in the
second period increases with misfit penalty, we conclude that if the misfit penalty
is sufciently small, then it is optimal to sell to all consumers in the first period.
Furthermore, observe that ¯
E
, which is the threshold for absolute dominance of So- g
u
lution (E), increases in s; the larger the salvage value, the narrower the dominance
region of Solution (E).
Second, ofering a refund amount of more than the salvage value of returned
items is not optimal unless it is optimal to provide full information. On the other
hand, in case of full information, the return policy is redundant (since there are no
returns) and the seller is indiferent in choosing a refund amount. Therefore, we say
that it is weakly suboptimal to ofer a refund amount more than the salvage value.
Note that as long as the price is larger than the salvage value, this also means that
it is weakly suboptimal to ofer a full (100%) refund.
Finally, note that in the case of costly information provision, if p
1
> E[V],
providing full information is never optimal since the seller can instead design a
return policy, by setting| =|
p
, to achieve the same efect. If, however, p
1
< E[V],
then the seller would have more incentive to choose Solution (D); that is, Solution
54
(D) would be optimal for a wider range of misfit penalty.
2.5 Jointly Optimal Information, Refund and Price Strategy
In the previous section, we identified the two candidate solutions for optimal (o, |)
without making any distributional assumptions, but under the assumption that p
1
is given. We observe that there is no clear dominance relationship between these
solutions if p
1
is a decision variable as well. In this section, we assume uniform
valuations, F (p) = p, and identify the candidate solutions for jointly optimizing
information, refund and price, and determine the conditions that lead to the opti-
mality of each solution. Uniform valuations is the most common assumption in the
operations management, marketing and economics literatures (Shulman et al. 2009,
Chesnokova 2007, Villas-Boas 2006, Davis et al. 1998, Chu et al. 1998). We employ
this assumption in this essay in order to facilitate closed-form optimal solutions for
better interpretation. This essentially constitutes the optimal (o, |, p
1
, p
2
) strategy
for the seller, contingent on the values of g, c, and s as given in Proposition 6. We
find that there is a boundary, critical for shaping the optimal strategy, in the (s, c)
plane, which we plot in Figure 2.6.
Proposition 6 With uniform valuations, the joint optimal (o, |, p
1
, p
2
) strategy is
such that;
For (s, c) such that c > c
u
(s) - Region (1) in Figure 2.6 - we have g
E,D
s u
g
E,C
s g
D,C
, and
u u
(a) If g > g
E,C
, the optimal solution is Solution (E), u
55
(b) If g < g
E,C
, the optimal solution is Solution (C); u
For (s, c) such that c < c
u
(s) - Region (2) in Figure 2.6 - we have g
D,C
g
E,D
, the optimal solution is Solution (E), u
(b) If g
D,C
< g < g
E,D
, the optimal solution is Solution (D),
u u
(c) If g < g
D,C
, the optimal solution is Solution (C), u
where
c
u
(s) = 2
1 + s
2
÷ 1, 3
g
E,C
= 4(1(1 + s)2÷ 2(1 c+2c) ,
u
+ s
2
)(1 ÷ )
2
g
E,D
= 4 + 12(1++ss)(1÷ 8(12+ c) ,
u
(4
2 2
)
÷ c)
2
g
D,C
= 3(1 4 c)
2
,
1.0
u
÷
0.8
c
0.6
0.4
0.2
0.0
0.0
0.2
1
0.4
s
c
0.6
c
O
s
2
0.8
1.0
Figure 2.6: Critical regions in the (s, c) plane.
and
56
Solution (E): (o
-
, |
-
) = {(o, |) , v
u
= p
-
}, p
-
= p
-
=
1+
c
,
1 1 2 2
Solution (D): (o
-
, |
-
, p
-
, p
-
) = 0,
4+
ss2 ,
4+
8s
2
,
1+
c
, 4
12 2
Solution (C): (o
-
, |
-
, p
-
, p
-
) = 0,
1+
ss2 ,
1+
2s
2
,
1+
c
. 2
12 2
Proof See Appendix B.
We show in Proposition 6 that, for the joint optimization ofo,|, p
1
and p
2
,
there are three solutions among which the seller chooses, depending on the rela-
tionship between market parameters (c, s and g). Solution (E) corresponds to any
(o, |) pair that satisfies v
u
= p
1
. We show in Section 2.4.1 that this implies efcient
allocation of the product at the time of purchase even under valuation uncertainty.
In other words, exactly those consumers who would purchase the product with full
information do purchase the product. We also show that this can be achieved either
by providing full information or, without full information, by setting the refund
amount to a certain level| =|
p
; when providing information is costless, the seller
is indiferent between these two options. However, under costly information provi-
\
sion, Solution (E) reduces too
-
0 and|
-
=|
p
= 2
c
.
1+c
In either case, since the
consumer purchasing behavior is identical to full information case, the seller sets
the classical monopoly prices: p
-
= p
-
=
1+
c
.
1 2 2
Solutions (C) and (D) are both characterized by v
u
= 0 but at diferent prices
and refund factors; Solution (C) suggests higher values in both:|
-
(C)>|
-
(D) and
p
-
(C)> p
-
(D). Therefore, Solution (C) ofers a larger refund amount than Solution
1 1
(D); the former ofers a refund amount equal to the salvage value (p
-
|
-
(C) = s), 1
while the latter ofers half the salvage value (p
-
|
-
(D) =
1
s). In both Solutions (C)
5 7
1 2
and (D), the seller provides no information and ensures that all consumers purchase
in the first period. Solution (C) ofers a larger refund amount than Solution (D) and
attains higher ex-ante consumer valuations in the first period; as a result, the seller
charges more: p
-
(C) = 1
1+s
2
2
> p
-
(D) = 1
4+s
2
.
8
Due to higher price and higher refund
amount, Solution (C) results in both larger dissatisfaction rate (which is equal to
the price), and larger return rate (which is equal to the refund amount).
The optimal strategy states that Solution (E) should be preferred only when
the misfit penalty (g) is sufciently large. In other words, if the misfit penalty
is sufciently small, the seller chooses to maximize her profits at the expense of
consumer satisfaction. We further see that if the cost of production is sufciently
large, Solution (D) is never optimal; higher production costs require higher prices
to compensate for them and Solution (C) is preferred. With low production costs,
there is a range of misfit penalty for which Solution (D) is optimal; for sufciently
low misfit penalty, Solution (C), which is greedier, is optimal.
We attain an intuitive corollary out of Proposition 6, by noting that the opti-
mality threshold for Solution (E) in Region (1) is g
E,C
; in Region (2), the optimality u
threshold for Solution (E) is g
E,D
. Next, we observe that u
cg
E,D
cg
E,C
u
cs
=
4s(1+c)
2
(1+s2)2(1÷c)2
> 0,
and
u
cs
=
16s(5+2c+c
2
)
(4+s2)2(1÷c)2
> 0. Therefore, for a constant c, larger s results in larger
thresholds, which in turn limits the optimality of range for Solution (E). We con-
clude that when salvage value is higher, it is easier for the seller to reject ex-ante
efcient allocation, thus to allow returns.
58
2.6 Conclusion
In this essay, we study the profit maximization problem for a seller who optimizes
information provision, return policy and prices for a new experience good, over a
two-period horizon. With no information, consumers are fully uncertain of their
valuations of the product. However, given more information, they learn more, ap-
proaching to individual valuations; information creates ex-ante heterogeneity among
consumers. On the other hand, being aware of a return option, consumers update
their valuations of the purchase decision.
We make several important contributions with this study. We devise a novel
approach in understanding and modeling the process of a consumer's learning of own
valuation, taking into consideration both partial information and partial-refund re-
turn policies. Our model incorporates two key parameters; 1) market growth rate,
g, which represents the seller's forward channel capability; 2) s, which determines
the value of returned items and therefore points to the seller's reverse channel ca-
pabilities. Building on the dynamics of interaction between information and return
policy in consumer valuations, we treat the seller's optimization problem analyti-
cally, show structural properties of her decision space, and characterize the optimal
solution for the general case. These characterizations lead to three major findings
that are robust to distributional assumptions. First, if the market growth rate—
and hence the future penalty due to consumer dissatisfaction and/or returns—is
sufciently low, then the seller may choose to provide no information to consumers
even if it is costless to do so. We also show that as the salvage value increases, it
59
becomes easier for the seller to withhold information from consumers. This shows
how the seller's reverse channel capabilities interact with her forward channel de-
cisions. Second, we show that even when it is optimal to ensure ex-ante efcient
allocation, Solution (E), it is not necessary to provide full information as this can
be achieved by devising the return policy appropriately (by setting| =|
p
) and
providing only partial information. This is a significant result as it showcases a
situation where the return policy can be used to substitute for informational tools.
Third, we find that ofering a refund amount that is more than the salvage value is
never exercised; the seller can advertise such a refund amount when Solution (E) is
optimal, in which case there are no returns. Lastly, assuming uniform valuations,
we determine the optimal decision strategy for the seller, which dictates the optimal
values for information provision, refund factor and prices given model parameters.
Future work could investigate opportunistic consumer behavior where a con-
sumer "purchases without intention to keep." A seller can be exposed to such
behavior if she ofers lenient returns; tightening return policies for some sellers
is attributed to their losses due to this type of consumer behavior (Davis 2010).
Davis et al. (1995) and Hess et al. (1996) examine consumers who purchase with-
out intention to keep; however, they assume that consumers have no pre-purchase
information, and they do not consider information provision.
60
Chapter 3
Managing Return Policies and Information Provision under
Competition
3.1 Introduction
More than 8% of total retail sales, $185 billion worth of retail merchandise, was
returned to sellers in 2009 in the US, and predictions for the near future indicate
similar outcomes (Davis 2010). While the substantial implications, in terms of direct
and overhead costs, of these returns for the whole supply chain makes the study of
consumer returns valuable, it is further interesting to observe that a significant
amount of these returns have no verifiable defect. For example, they account for
up to 80% of HP printer returns (Ferguson et al. 2006), and 95% of all electronics
purchases (Lawton 2008).
The primary reason for these "false-failure" returns is that the consumers
learn—only after the purchase—that the good is not a perfect fit to their tastes,
preferences, usage norms, established settings, etc. Take for example a consumer
purchasing a new electric razor only to realize that its grip is not as comfortable
as the old one he had, or a curtain set to be brought home only to notice it does
not match the color of furniture at home. When lacking experience with the good
before the purchase, the consumers cannot be certain of the true value of the good
61
for them, and thus, there is a possibility for a misfit. This type of good is commonly
referred to as "experience good" (Nelson 1970). As a result, lack of information—
regarding the value of the good—is the underlying driver of false-failure returns of
experience goods.
Before purchasing an experience good, consumers mainly rely on the seller to
gain access to, and—however limited—experience with the good; this is especially
true in case of a new-to-market good. Consider for example trial versions of com-
mercial software, test-drive events organized by auto manufacturers, fit-rooms that
are a standard in all department stores, electronic stores with items displayed openly
with trained sales personnel present, free samples of cosmetic products made avail-
able through online or physical channels, product samples sent to expert reviewers,
etc. While the level of these eforts by firms vary greatly, their main purpose is to
reduce false-failure returns by providing the consumers information regarding the
true value of the goods. On the other hand, the sellers also ofer customized return
policies that facilitate product returns. For example, Amazon.com has 31 diferent
product-specific return policies with restocking fees of up to 50%, Best Buy has
a customized return policy with restocking fees up to 25%, Nordstrom ofers full
refunds for any return to their stores. In efect, return policies enable consumers
to defer their ownership decisions until after they gain some experience with the
product (for a fee, in the case of partial refunds).
If a seller's objective is to maximize consumer satisfaction, the initial intuition
is that this goal can be achieved either by providing full information to all con-
sumers, or by ofering a full-refund return policy. While providing full information
62
to each and every consumer would cut all false-failure returns, it is also practically
impossible to achieve. Ofering full refunds would enable any misfit alleviated, how-
ever at the cost of the seller—on top of immediate financial costs, negative brand
implications can be substantial; Lawton (2008) reports that 25% of people who
return an item refrain from buying that same brand again, while 14% of such peo-
ple are unlikely to buy from the same seller again. On the other hand, we show
in Essay 2 that a monopoly seller can design a partial-refund return policy to get
rid of false-failure returns, while providing only partial information. In Essay 2,
we also identify conditions where it is in fact optimal for the monopoly seller to
minimize false-failure returns. Then, the question is 'what happens when there is
competition?' Specifically, we pursue the following research questions in this essay:
1) Given competition, is it still possible to design a return policy to efectively
minimize false-failure returns without having to provide full information? If
it is, is such an outcome ever desirable for the sellers?
2) Are there any equilibrium return policy and information provision decisions?
(a) How do they difer from the decisions of a monopoly?
(b) Under what conditions do they exist?
To address these questions, we build on the basic two-period model described
in Essay 2, with the exception of assuming uniform valuations for tractability. In
order to isolate the efect of competition, we conceptualize a perfectly symmetric
duopoly setting, and examine equilibrium information provision and return policy
63
decisions. To our knowledge of the literature, this is the first scholarly work that
analytically studies the efects of competition on joint information provision and
return policy decisions. We identify all potential Nash equilibria and their respective
conditions of existence. Contrasting the results to the monopoly case, we find that,
while competition can induce sellers to withhold information from the consumers
under certain conditions, it forces them to ofer full refunds.
The rest of the essay is organized as follows. Section 3.2 reviews the relevant
literature. In Section 3.3, we describe and analyze the competition model, and we
examine and discuss the equilibrium in Section 3.4. We conclude in Section 3.5.
3.2 Literature Review
Among the very few studies that investigate competition in a similar context as ours,
the paper by Shulman et al. (2011) is the most relevant. In Shulman et al. (2011),
they examine equilibrium prices and return policies in a competitive market where
consumers are not informed of their tastes or valuations. On a single-period horizon,
the sellers ofer horizontally-diferentiated products but provide no information to
consumers, and they extract no value out of returned items. Our competition set-
ting is significantly diferent from theirs in that we look at equilibrium information
provision and return policies incorporating consumer dissatisfaction in the second
period. We show that consumer dissatisfaction and salvage value are critical in
determining the market equilibrium. In direct contrast to our findings, Shulman
et al. (2011) conclude that competition may induce higher restocking fees, whereas
64
we find that sellers typically ofer full refunds in a competitive setting. Their result
can be explained by noting that they do not consider the impact of high restocking
fees on consumer dissatisfaction (given their single period setting), and therefore
the return policy efectively becomes a tool to discourage consumers from returning,
in order to maximize short-term profits. Our findings help explain why full refunds
are observed in competitive retail markets.
Aside from Shulman et al. (2011), Chesnokova (2007) considers a duopoly
where the firms engage in a product reliability and price competition, and returns are
in the form of repairs, not refunds; i.e., the source of returns in her model is product
reliability, and not consumer tastes and preferences as in our model. In the context
of experience goods, Doganoglu (2010), Villas-Boas (2006) and Villas-Boas (2004)
study the price competition of two sellers over an infinite horizon; however, neither
paper considers return policies or pre-purchase information provision. We study a
duopoly case where two identical sellers engage in return policy and information
competition over two periods; this is the first scholarly work to our knowledge to
study the efects of competition on seller decisions on return policies and provision
of information.
3.3 Competition Model
We build the competition setting on the same framework as described in Section 2.3
in Essay 2. Specifically, we conceptualize a consumer's valuation of a product as a
learning process, given information of amounto e [0, 1] by a seller. Further given a
65
return policy with a refund factor of| e (0, 1], such that the refund amount is p|
where p is the purchase price, each consumer has the opportunity to re-consider her
initial purchase decision. As in Essay (2), we assume costless information provision
in our analysis, and we later comment on the impact of costly information provision.
We consider a duopoly case with identical sellers, Y and Z; sellers have identi-
cal unit costs c, net salvage values s, and market growth rates g, and they introduce
new products at the same time. We assume that consumers equally value the prod-
ucts from both sellers; that is, the products are perfect substitutes of each other. In
other words, there is a single, seller-independent distribution F , of consumer valua-
tions V , which we assume to be uniformly distributed: F (p) = p. We further assume
identical period prices, p
1
and p
2
, for both sellers. While we do not assume identi-
cal products, we assume that information provided by a seller on her product does
not contribute to information on the other seller's product; while this assumption
does not hold in general, it is valid for many—if not all—experience goods
4
. These
assumptions help us focus on information and return policy (o and|) competition,
as well as allowing a tractable solution.
We employ the same two-period setting as in the monopoly case, with the
chronology of events shown in Figure 2.2 in Essay 2. As in the monopoly case, we
assume that consumers attain full information on the products in the second period,
regardless of the information provided in the first period. At the start of the first
4
Consider for example two electric razors of diferent brands. Even after having used one of
them, a consumer would have no understanding regarding how well the other product will perform,
how comfortable it will feel in his hand, how comfortable a shave it will provide, etc.
66
period, the sellers simultaneously decide on their respectiveo and|. In the first
period, we assume a unit market size, which is shared between the sellers according
to consumer valuations given the sellers'o and| decisions. Specifically, a consumer
with ex-post valuation v perceives ex-ante valuations of v
o,Y
and v
o,Z
for the sellers'
products, and since prices are equal, chooses the seller with the larger v
o,j
, where
v
o,j
=o
j
v + (1 ÷o
j
)E[max{V, p
1
|
j
}]
=o
j
v + (1 ÷o
j
)1 + (p
1
|
j
) 2
2
for uniform valuations. In the second period, the market size for each seller grows
with rate g in the same manner as in the monopoly case: Consumers who are
dissatisfied or who return their purchases in the first period do not contribute to
market growth. Therefore, similarly, we refer to g as "misfit penalty."
3.3.1 Market Share Dynamics
In preparation for the equilibrium analysis, we here analyze the market share dynam-
ics given the sellers'o and| decisions. Suppose first thato
Y
=o
Z
. Ifo
Y
=o
Z
< 1
and if, without loss of generality,|
Z
>|
Y
, then v
o,Z
> v
o,Y
, ¬v. That is, in the
case of symmetric, partial information, the seller ofering a more lenient return pol-
icy captures the whole market. Ifo
Y
=o
Z
= 1, then v
o,Y
= v
o,Z
, ¬v; if both sellers
provide full information, then the consumers are indiferent between the sellers re-
gardless of the return policies. Suppose without loss of generality that o
Y
v
Y
Z
u u
prefer seller Z to seller Y , while those with v < v
Y
Z
prefer seller Y to seller Z, u
67
where
v
Y
Z
u
(1 ÷o
Y
)(1 + (p
1
|
Y
)
2
) ÷ (1 ÷o
Z
)(1 + (p
1
|
Z
)
2
).
2(o
Z
÷o
Y
)
(3.1)
Therefore, the first-period market share for seller Y is v
Y
Z
; for seller Z, it is (1÷v
Y
Z
).
u u
If|
Y
=|
Z
=|, we have v
Y
Z
= E[max{V, p
1
|}]; that is, in case of identical return u
policies, the threshold valuation is independent of the level of information provided
as long aso
Y
=o
Z
.
Further analysis of the market share dynamics reveals that the seller with a
more lenient return policy can set an appropriate level of information to achieve a
desired market share. Consequently, she can set an appropriate level of information
to achieve 100% market share, that is, drive the other seller out of the market. This
is formalized in Proposition 7 below.
Proposition 7 Suppose, without loss of generality, that o
Y
< 1 and let Z be the
seller ofering a more lenient return policy, i.e., |
Z
>|
Y
. Then, seller Z can
achieve a desired market share,ˆ, by setting o
Z
=o
ˆ
, where
v v
o
Y
+ p
2
(1 ÷o
Y
)2ˆ +|(Zp÷|
Y2
÷ 1 .
o
ˆ
v
1
v
2 1Z 2
|)
In addition, seller Z can drive seller Y out of the market by setting o
Z
e [o,o],
where
o
|
2
÷|
2
and
max 0, o
Y
÷
p
2
(1
1
÷o
Y
)1 ÷Z(p |Y )
2
1Z
,
o
o
Y
+
p
2
(1
1
|
2
÷|
2
.
Proof See Appendix C.
÷
o
Y
)
1
+
Z(p |Y )
2
1Z
68
Regardless of how the market is shared, the dynamics between the consumer
and the seller is identical to the monopoly case: Given that a consumer prefers
seller Z (v > v
Y
Z
), he purchases only if v
o,Z
> p
1
, or equivalently if v > v
u,Z
. u
If p
1
|
Z
s v < p
1
, he is unhappy with the purchase but is not willing to return; if v < p
1
|
Z
, he is unhappy and
would like to return. Therefore, all the findings
regarding the structure of the (o, |) decision space carries on from the monopoly
case. Furthermore, since we assume equal first-period prices, unit costs and net
salvage values, the sellers have identical (o, |) decision spaces.
3.4 (o, |) Equilibrium
In this section, we analyze the sellers' (o, |) decision space in the light of the results
for the monopoly case and the analysis of market shares above, in order to identify
the possible Nash equilibria. In other words, we investigate whether and when there
exists a pair of decisions (o
Y
, |
Y
) and (o
Z
, |
Z
) such that the former is seller Y 's best
response to the latter, which in turn is seller Z's best response to the former. When
a pure-strategy Nash equilibrium does not exist, we identify the mixed-strategy
equilibrium. The intuitive corollary of Proposition 7 suggests the non-existence of
a Nash equilibrium where both sellers set (o
j
< 1, |
j
< 1), since given, without loss
of generality, (o
Y
< 1, |
Y
< 1), seller Z has a potential best response where she sets
a more lenient return policy and an appropriate level of information to capture the
whole market. In fact, we find that capturing the whole market is the best response
to (o
Y
< 1, |
Y
< 1), and we summarize our findings in Proposition 8 below.
69
Proposition 8 In the duopoly where both sellers have identical p
1
, p
2
, c, s and g,
and consumer valuations are uniformly distributed, we identify four thresholds on g
(as functions of other variables) that are critical for the existence and the form of
(o, |) Nash equilibria: ¯
III
, g
IV
, g
V
and g
V
I
. Furthermore, we find that the ordering
g
u
u u u
of these functions is determined by the value of s compared to a threshold function
s
u
. Specifically, the potential (o, |) Nash equilibria and the associated conditions for
their existence are as follows.
For s > s
u
, we have g
V
> ¯
III
, and
u
g
u
(i) If g < ¯
III
, then there is a symmetric pure-strategy Nash equilibrium g
u
where both sellers provide no information but ofer full refund return
policy: (o
j
= 0, |
j
= 1) for both sellers,
(ii) If ¯
III
< g < g
V
, there is no pure-strategy Nash equilibrium. There is a
g
u
u
mixed-strategy Nash equilibrium where both sellers set |
j
= 1 and pick
o e [0, 1] randomly,
(iii) If g > g
V
, there is no pure-strategy Nash equilibrium. There is a mixed- u
strategy Nash equilibrium where both sellers set |
j
= 1 and pick o e
[o, 1] randomly; ˆ
For s < s
u
, we have g
V
< g
V
I
< ¯
III
< g
IV
, and
u u
g
u
u
(i) If g < g
V
, then there is a symmetric pure-strategy Nash equilibrium u
where both sellers provide no information but ofer full refund return
policy: (o
j
= 0, |
j
= 1) for both sellers,
70
(ii) If g
V
< g < g
V
I
, then there is an asymmetric pure-strategy Nash equilib-
u u
rium where one seller provides full information and ofers an arbitrary
return policy, while the other seller provides no information but ofers a
full refund return policy: without loss of generality, (o
Y
= 1, |
Y
e [0, 1])
and (o
Z
= 0, |
Z
= 1),
(iii) If g
V
I
< g < g
IV
, then there is a symmetric pure-strategy Nash equi-
u u
librium where both sellers provide full information and ofer arbitrary
return policies: (o
j
= 1, |
j
e [0, 1]) for both sellers,
(iv) If g > g
IV
, there is no pure-strategy Nash equilibrium. There is a mixed- u
strategy Nash equilibrium where both sellers set |
j
= 1 and pick o e
where
[o, 1] randomly, ˆ
s
u
= 1 ÷ p
1
+ c 2 ÷ p1 1
,
g
u
¯
III
= (p ÷sc÷ c÷ p ),
2
)(1
2
g
IV
= p(p
1
÷ c)(1 ÷ p
1
)),
u
1
(p
2
÷ c)(1 ÷ p
2
g
V
= (1 ÷÷ p
1
(1 ÷ p)(1 ÷)p ),
u
s
p
1
)(p
2
÷ c
1
+c
2
g
V
I
= (2s÷ pc ÷pp
1
(1 c÷ p
1
÷ pc)),
and
u
2÷
1
)(
2
÷ )(1
+
1
(1 ÷ p
1
)
2
(g(1 ÷ p
2
)(p
2
÷ c) + c ÷ s)
o = (c ÷ s)(1 ÷ p )
2
÷ 2p (1 ÷ p )(p ÷ c) + g(1 ÷ p )(p ÷ c)(1 ÷ 2p + 3p
2
). ˆ
1
P
r
o
o
f
S
e
e
A
p
p
endix
C.
1 1 1
7
1
2 2 1 1
We see from Proposition 8 that for sufciently small misfit penalty, g, both
sellers find it the best decision to provide no information to sell to all consumers and
share the market equally, although there are some dissatisfied consumers as well as
some returns. Moreover, we observe that as the salvage value increases, the range of
g, where providing no information is the best decision, grows. Both of these results
are consistent with the monopoly case (Proposition 5).
We observe that for sufciently large misfit penalty (g > ¯
III
for s > s
u
, and g
u
g > g
IV
for s < s
u
), there is no pure-strategy Nash equilibrium since both sellers u
always find a best response where they capture the whole market alone. The mixed-
strategy equilibrium we identify for s > s
u
and ¯
III
< g < g
V
suggests that both
g
u
u
sellers ofer full refunds and picko randomly in [0, 1]. The second mixed-strategy
Nash equilibrium suggests that for sufciently large g, both sellers' best decision
is to ofer full refunds and provide at least partial information; they randomly set
an information level betweeno =o ando = 1. This means that for sufciently ˆ
large g, it is not a best decision to provide little or no information; a result that is
consistent with the monopoly case. Both mixed-strategy equilibria suggest that the
market is not necessarily covered in the first period (i.e., there are some consumers
who leave without purchasing), since full market coverage requireso
j
so(v
u
= 0)
for at least one of the sellers and this is not necessarily the case. Furthermore,
recalling that for a monopoly, the optimal decision given| = 1 and g > ¯
III
is to g
u
provide full information, so that there are no dissatisfied consumers, we note that
both mixed-strategy equilibria imply partial information, resulting in sub-optimal
outcomes where there are some dissatisfied consumers and some returns.
72
For sufciently small net salvage value, s < s
u
, intermediary misfit penalty
values result in two diferent pure-strategy Nash equilibria. In the lower g range
(g
V
< g < g
V
I
), one seller provides full information and ofers an arbitrary return
u u
policy, while the other provides no information but ofers a full refund return policy.
This implies full market coverage in the first period (i.e., all consumers purchase),
and a market allocation such that one seller serves the consumers with higher val-
uations and sees no dissatisfied consumers and no returns, while the other seller
serves the remainders and sees some dissatisfied consumers and some returns. This
is an interesting result given that the sellers are identical. In the higher g range
(g
V
I
< g < g
IV
), both sellers provide full information and ofer arbitrary return
u u
policies. In this case, while consumers are indiferent between the sellers, only the
consumers with ex-post valuations at least as high p
1
purchase in the first period
and there is ex-ante efcient allocation of the goods.
We note that costly information provision may significantly alter the Nash
equilibria, even if the sellers would have the identical cost structure. For example,
it is conceivable that in the mixed-strategy equilibria described above, the range
foro would be capped from above since neither seller would have an incentive to
incur high information provision costs. Furthermore, the pure-strategy equilibria
where one or both sellers provide full information,o = 1, may not exist; in that
case, a mixed-strategy equilibrium where both sellers ofer full refunds,| = 1, with
o randomized over a range may prevail.
Regarding the refund factor,|, recall that in the monopoly solution described
in Proposition 5, full refunds are never exercised; Solution (D) suggests partial
73
refunds, and when Solution (E) is optimal, amount of refund is arbitrary. However,
we see from Proposition 8 that competition changes the picture abruptly. In the
duopoly case, all but one Nash equilibria suggest that at least one seller ofers full
refunds; for s < s
u
and g
V
I
< g < g
IV
, both sellers provide full information and the
u
refund amount becomes arbitrary.
u
Consequently, we conclude, contrasting with the monopoly case, that while
competition results in one or both of the sellers withholding information from con-
sumers in certain cases, it typically forces them to ofer full refunds. That helps
explain why we observe full refunds in practice (e.g., Nordstrom.)
3.5 Conclusion
In this essay, we study competition in the context of information provision and return
policies in experience good markets. In order to isolate the efects of competition
on our results in the monopoly model given in Essay 2, we consider a duopoly
case where two identical sellers engage in information provision and return policy
competition. We identify the possible pure-strategy Nash equilibria, or if none exists,
the mixed-strategy Nash equilibria, and the associated conditions where they take
place. We find, in contrast to the monopoly case, that while competition can cause
the sellers to withhold information under certain conditions, it typically forces them
to ofer full-refund return policies. This finding can shed light on some real-life
phenomena where sellers ofer full refunds and/or they do not put much efort to
provide informational tools to consumers.
74
Appendix A
Appendix for Essay 1
A.1 Derivation of h(¸)
Let R
it
denote the reservation price for product i at time t, a random variable. We
write R
1
t
= u(O)c
1
t
, where u(-) is a (deterministic) linear mapping function andc
1
t is a random variable
with a Weibull distribution (so that R
it
has a Weibull distribu-
tion); similarly R
2
t
= u((1+¸)O)c
2
t
. We define customer utility ç
it
as a log function
of the customer's reservation price ç
it
= ln(R
it
) = ln(u(O)) + ln(c
it
). Becausec
it
has a Weibull distribution, ç
it
has a Gumbel distribution; this is consistent with the
Logit model for choice. As a result, the probability that a customer adopts the new
generation is
h(¸) = e
ln(
u((1+¸
)O))
+ e
ln(
u
(O))
= u(O) ((1u+¸)O) )O) = 1 +¸ .
e
ln(
u((1+¸)O))
A.2 Proof of Proposition 1
u + ((1 +¸ 2+¸ (A.1)
For a non-stationary Poisson process with intensity A(t), E [N (A(t))] = A(t), and
thus (1.6) becomes
(1+o)t
max H
j
2 =
¯
dt
p (t) 2
=
t
(1+o)t e
÷
o(t÷ot
)
(p
2
(t) ÷ c
p
) E N ì
j
2(t)G
2
t
(p
2
)
(A.2)
t
e
÷
o(t÷ot
)
(p
2
(t) ÷ c
p
)ì
j
2(t)G
2
t
(p
2
)dt ¯
75
This is a simple optimal control problem, with the first-order necessary condition
given in (1.7). The solution is similar to that found in Bitran and Mondschein
(1997). For uniqueness of the solution to (1.7), we need
K
2
,t
= p
2
(t) ÷ G
2
t ¯
G
2
t
to be an increasing function of p
2
(t) since
p2(t)÷· lim K
2
,t
= ·. Therefore, we need
0< dK
2
,t
= 1 ÷ dp G
2
t
= 1 ÷ d¯
÷G
2
t ÷ G
2
t
g
2
,t ¯
2
which becomes
dp
2
2
G
2
t
G
2
t 2
÷2G
2
tG
2
t
÷ G
2
tg
2
,t ¯
0>
2
¯
2
d G
2
t
. ¯
G
2
t
2
= dp G2
2 2t
A.3 Proof of Proposition 2
We show through ?uid approximations (Mandelbaum and Pats 1998) that the solu-
tions to the deterministic version of (1.5) is asymptotically optimal as initial max-
imum arrival rate, M
0
, and I
0
grow proportionally large. However, since I
0
is a
decision variable, we first show that it is optimal to select I
0
proportionally large as
M
0
.
Consider a sequence of instances of problem (1.5) indexed by n e Z
+
. Let
M
0
n
denote the initial maximum arrival rate andì
j
1
n
be the resulting arrival rate
intensity function for the n
th
instance. Let
lim M
0
= M
0
. n
n
÷·
n
Thu
s,
we have
n÷·
j
n
1
limìn =ì
j
1.
76
Let I
n
be the decision parameter for the final build and I
n
(t) denote the 0
corresponding inventory trajectory for the n
th
instance, and let all other parameters
be held constant, independent of n. For the n
th
instance, (1.5) becomes
I
n
,p1(t)
T
j
e
÷
o(t÷ot
)
p
1
(t)dI
n
(t)
+
e
÷
o(1÷o)
t
c s
I
n
0
+
T
j
dI
n
(t)
max E ÷ 0
ot
T
j
ot
s.t.
T
j
÷ c
h
ot
e
÷
o(t÷ot
)
I
n
(t)dt ÷ c
p
I
n
0
(A.3)
÷
ot
dI
n
(t) I
n
0
t
I
n
(t) = I
n
÷ N 0
ot
ì
j
1
n
(u)G
1
u
(p
1
)du ¯
T
j
for t eot, T
j
,
where we wrote I(T
j
)
0 as ÷
ot dI(t) I
0
. After dividing the second constraint
by n, taking limits on both sides, and applying Lebesgue's monotone convergence
theorem, we get
n
÷·
n
t
lim 1 I
n
(t) =
n
lim 1 I
n
÷ N
÷·
n
0
ì
j
1(u)G
1
u
(p
1
)du . ¯
ot
Similarly, from the first constraint in (A.3), we have
T
j
n÷· lim ÷
1 n
ot
dI
n
(t)
lim 1 I
n
.
n
÷·
n
0
Therefore, applying the same transformation to the objective function, we can
rewrite (A.3) as
77
lim 1
I
max
t
)
E ÷
T
j
e
÷
o(t÷ot
)
p 1
(t)dI
n
(t)
+
e
÷
o(1÷o)
t
c
s
I
n
0
+
T
j
dI
n
(t)
n
÷·
n
n
,p1( 0
ot
T
j
ot
s.t.
T
j
÷ c
h
ot
e
÷
o(t÷ot
)
I
n
(t)dt ÷ c
p
I
n
0
n÷· lim ÷
1 n
ot
dI
n
(t)
lim 1 I
n
n
÷·
n
0
n
÷·
n lim 1 I
n
(t) =
n
lim 1 I
n
÷ N
÷·
n
0
t
ot
ì
j
1(u)G
1
u
(p
1
)du ¯
for t eot, T
j
.
(A.4)
Suppose (I
-
, p
-
) is an optimal solution to (1.5), with the optimal objective
0 1
function valuet
-
. Then, (I
n
-
, p
-
) is an optimal solution to (A.4) with the objective
1 0 1
function valuet
n
-
, such that I
n
-
andt
n
-
satisfy
n
lim I
n
-
/n = I
-
andt
n
-
=t
-
,
1 0 1
÷· 0
0 1 1
respectively. This follows by observing that (A.4) is equivalent to problem (1.5)
divided by n and taking limits as n ÷ ·. As a result, we have shown that it is optimal to let the
final build, I
0
, grow proportionally large as M
0
in the asymptotic
regime.
Noting that the demand intensity process
t
ot
ì
j
1(u)G
1
u
(p
1
)du ¯
is continuous and uniformly bounded in [ot, T
j
], and we find that in the limit as
n ÷ ·, I
n
(t)/n converges (almost surely and uniformly over a compact set) to I(t),
given by
t
I(t) = I
0
÷
ot
ì
j
1(u)G
1
u
(p
1
)du. ¯
Further details regarding the proof of this convergence result can be found
78
in Mandelbaum and Pats (1998). In this asymptotic regime, the stochastic opti -
mization problem in (1.5) reduces to the optimal control problem in (1.8), where
T
j
I
0
+ ot
dI(t) is replaced with I(T
j
), and the second constraint is substituted into
the first term in the objective function.
The solution to (1.8) can be found as follows. Treating I(t) as the state variable
and p
1
(t) as the control variable, and lettingv ande(t) be the multipliers for the
first and second constraints in (1.8), the Hamiltonian is H = e
÷
o(t÷ot
)
(ì
j
1G
1
t
p
1
÷ ¯
c
h
I)÷eì
j
1G
1
t
, where arguments have been suppressed for simplicity. The optimality ¯
conditions are:
cH = 0?ì
j
e
÷
o(t÷ot
)
÷p G + G +e(t)G = 0,
c p
1
1 1 1t
¯
1
t
1t
(A.5)
cH = ÷ce? c e
÷
o(t÷ot
)
=ce , h
(A.6)
cI ct ct
e(T
j
) =v + e
÷
o(T
j
÷
ot
)c
s
andvI(T
j
) = 0. (A.7)
A first-order condition for I
0
is obtained by considering that the marginal
revenue from the last unit must equal to its marginal cost (including the procurement
cost and cumulative holding costs in time). That is,
T
j
e(T
j
) = c
p
+ c
h
e
÷
o(u÷ot
)
du. (A.8)
ot
Combining (A.7) and (A.8), we get
T
j
v = c
p
+ c
h
ot e
÷
o(u÷ot
)
du ÷ e
÷
o(T
j
÷
ot
)c
s
. (A.9)
However, we must havev> 0, otherwise I
0
÷ · is optimal and the problem in
(1.8) is unbounded. Therefore, from (A.7), I(T
j
) = 0. In other words, the entire
79
initial inventory is depleted during the sales horizon. To find p
1
(t), we proceed as
follows. From (A.5),
e(t) = e
÷
o(t÷ot
)
p
1
÷ G
1
t ¯
G
1
t
On the other hand, (A.7) and (A.8) yield
t
.
(A.10)
e(t) = c
p
+ c
h
e
÷
o(u÷ot
)
du. (A.11)
ot
We combine (A.10) and (A.11) to obtain the necessary condition for the optimal
price pattern for product 1, given in (1.9). The proof of uniqueness follows the same
steps as in the proof for Proposition 1.
Once the optimal price path is determined using (1.9), and given that I(T
j
) =
0, the optimal initial inventory is equal to the total sales through the planning
horizon.
A.4 Normalization of| for the Regression
We normalize the parameter|, for the purposes of running the regression, so that it
takes values between 0 and 1, instead of between 0 and ·. We do this by mapping
| to a new parameteru, according to the normalizing relationship:
| = (1 ÷o)t
1 + W ÷1 · e
÷
1u
,
(A.12)
u u
where W(-) is the Lambert W function. The Lambert W function is the inverse of
f (w) = we
w
and we use the zeroth branch which is single valued and real for the
range ofu considered. It is easily verified that
|
limu = 1, and lim
0
u = 0.
80
÷· |÷
Table A.1: Statistics of multiple linear regression: Two-way interaction efects.
Factor
(intercept)
t
o
log(k)
µ
¸
p+q
log(q/p)
|
t
t : log(k)
t:µ t:¸
t : p+q
t : log(q/p)
t:|
o : log(k)
o: µ o:¸
o : p+q
o : log(q/p)
o:|
log(k) : µ
log(k) :¸
log(k) : p + q
log(k) : log(q/p)
log(k) :|
µ:¸
µ : p+q
µ : log(q/p)
µ:|
¸ : p+q
¸ : log(q/p)
¸:|
p + q : log(q/p)
p+q :|
log(q/p) :|
Adj. R-sq.
t-value
-2.7
-60.9
9.0
-3.8
35.3
-35.7
-43.7
68.3
-94.5
-20.9
-17.7
-1.8
-35.0
-30.4
25.1
-48.1
1.9
6.3
-12.0
-20.0
30.7 -
9.5
10.0
-10.3
-11.3
17.7
-24.4
-4.0
-11.2
15.7
-21.9
-16.3
25.8
-35.2
-6.3
-4.2
-6.0
0.837
--
--- --
- ---
--- --
- ---
--- --
- ---
---
?
--- --
- ---
---
?
--- --
- ---
--- --
- ---
--- --
- ---
--- --
- ---
--- --
- ---
--- --
- ---
--- --
-
Statistical significance codes: '- - -': p ~ 0; '--': 0.001< p < 0.01; '?': 0.05< p < 0.1
81
Appendix B
Appendix for Essay 2
B.1 Proof of Proposition 3
Under partial information, using the definition of v
u
, the condition v
u
= p
1
translates
to p
1
= E[max{V, p
1
|}]. Suppose that p
1
> E[V ]. Then, since p
1
s E[max{V, p
1
}]
and through the intermediate value theorem, there exists|
p
e [0, 1] such that p
1
= E[max{V,
p
1
|
p
}]. Note that if p
1
< E[V ], no such|
p
e [0, 1] exists and there are no first period buyers with
positive surplus foro< 1. Thus, if p
1
> E[V ], setting
| =|
p
is equivalent to providing full information as it completely nullifies the
consequences of valuation uncertainty regardless of the value ofo.
B.2 Structural Properties of the (o, |) Decision Space
B.2.1 Boundary for Returns: v
u
(o, |) = p
1
|
The condition v
u
(o, |) = p
1
| is critical for the existence of returns. Assuming
o> 0, this condition is equivalent to p
1
+
(1
÷o
)
(p
1
÷ E[max{V, p
1
|}]) = p
1
|, which o
reduces to
o =o
r
(|) E[max{V, p
1
|}] ÷ p
1
.
E[max{V, p
1
|}] ÷ p
1
|
Note that, given p
1
, this equation represents a curve in the (o, |) space. Sup-
pose first that p
1
> E[V ]. Then, by Proposition 3,|
p
> 0 exists and therefore
c v
u
co
= ÷o1
2
(p
1
÷ E[max{V, p
1
|}])> 0. Furthermore,o
r
(|)> 0 if and and only if
|>|
p
, since the denominator ino
r
(|) is always positive. Thus, v
u
(o, |)< p
1
|
if and only ifo 0, and
d|
(E[max{V,p1|}]÷p1|)
2
d
2
or
d|2 = p
1
2
(2(1÷F (p1|))(E[max{V,p1|}]÷p1+p1(1÷|)F (p1|))+p1(1÷|)(E[max{V,p1|}]÷p1)F (p1|))
(E[max{V,p1|}]÷p1|)
3
>0
for all|>|
p
. Suppose p
1
< E[V ]. Then, we have
c
cvo
u
> 0,o
r
(|)> 0,
do
r> 0, and d|
82
d
2
or
d|2
> 0 satisfied for all| e [0, 1].
B.2.2 Boundary for 100% Sales: v
u
(o, |) = 0
The condition v
u
(o, |) = 0 reduces to
o =o
0
(|)
1 ÷ E[maxp
1
, p |}]. {V
1
Similar to above, when p
1
> E[V ],|
p
> 0 exists and for|>|
p
,o
0
(|)> 0 is well
defined and we have
c v
u
co
> 0. Finally, the functiono
0
(|) is strictly increasing if F
p
2
F (p1|)
is continuously diferentiable, since
do0
d|
=
1
(E[max{V,p1|}])
2
> 0. If p
1
< E[V ], then we
have
c v
u
co
> 0,o
0
(|)> 0 and
do0
d|
> 0 for all| e [0, 1].
B.2.3 Boundary for No Sales: v
u
(o, |) = 1
Note that v
u
(o, |) = 1 is possible only if p
1
> E[V ], and it reduces to
o =o
1
(|)
p
1
÷ E[max{V, p
1
|}] .
1 ÷ E[max{V, p
1
|}]
Then, it is seen thato
1
(|)> 0 only if 0 s||
p
, we have v
0
= E[max{V, p
1
|}]> p
1
by definition. Therefore, for a
given|>|
p
, the seller is indiferent in choosing ano e [0, o
0
(|)]
83
when information is costless. Consequently, in our analyses in this study, we treat
the conditions v
u
= 0 ando = 0 as equal at a given|>|
p
.
Similarly, the second region is whereo 1; there are
no sales since consumer valuations are in (0, 1). We see that "no sales" is achieved
for anyo e [0, o
1
(|)], which is possible for| p
1
c R
I
(B.2)
We observe that c v
u
= ÷F (v
u
) (p
1
÷ c)< 0; it is optimal to decrease the
purchasing threshold at any price point. We conclude that any (o, |) decision in
84
Region (I) is strictly dominated by Region (E) since v
E
= p
1
< v
I
; the seller never
u u
lets the consumers have positive surplus while there is an option to have efcient
allocation.
B.3.3 Some Dissatisfied Consumers, No Returns: p
1
> v
u
> p
1
|
In the region where p
1
> v
u
> p
1
|, Region (II), there are some dissatisfied buyers
but they are not willing to return their items as the refund amount is not high
enough. We have L = F (p
1
) ÷ F (v
u
), and since there are no returns, M = 0.
Seller's profit maximization problem in this region is
max R
II
= (p
1
÷ c) (1 ÷ F (v
u
)) + (p
2
÷ c) g (1 ÷ F (p
2
)) (1 ÷ F (p
1
) + F (v
u
)) o,|,p i
s.t. p
1
> v
u
> p
1
|
(B.3)
The partial derivative with respect to v
u
is
cR
II
= F (v ) ÷p + c + g (p ÷ c) (1 ÷ F (p ))
c v
u
which is positive for
u 1 2 2
g > g
I
I u
(p
1
÷ c)
(p
2
÷ c) (1 ÷ F (p
2
)),
(B.4)
and negative for g < g
II
. Then, at any given price point (p
1
, p
2
), it is optimal to u
increase v
u
if g > g
II
and it is optimal to decrease v
u
if g < g
II
. Therefore, if g > g
II
,
u u u
Region (II) is dominated by Region (E) since v
E
= p
1
> v
II
; if g < g
II
, Region (II)
u u u
is dominated by the boundary where v
u
= p
1
|. As a result, no internal solution is
optimal in Region (II).
B.3.4 Some Dissatisfied Consumers, Some Returns: v
u
< p
1
|
Region (III) is characterized by v
III
< p
1
|, which means there are some dissatisfied u
consumers (L = F (p
1
) ÷ F (v
u
)) and a portion (M = F (p
1
|) ÷ F (v
u
)) of these
consumers are willing to return their purchases. Thus, the seller's optimization
85
problem in this region is as follows:
max R
III
= (p
1
÷ c) (1 ÷ F (v
u
)) + (÷p
1
| + s) (F (p
1
|) ÷ F (v
u
))
o,|,p
i
+ g (p
2
÷ c) (1 ÷ F (p
2
)) (1 ÷ F (p
1
) + F (v
u
))
(B.5)
s.t. v
u
< p
1
|
In order to solve this problem, we look at the partial derivative of the objective
c R
I
I I
function with respect to v
u
:
which is positive if
c v
u
= F (v
u
) ÷p
1
+c+p
1
| ÷s+g (p
2
÷ c) (1 ÷ F (p
2
))
g > g
I
I I u ((p
1
÷ c) + (s ÷ p
1
|))
(p
2
÷ c) (1 ÷ F (p
2
))
(B.6)
and negative if g < g
III
. Then, since v
II
> v
III
> 0, if g > g
III
, Region (II)
u u u u
dominates Region (III), and if g < g
III
, the boundary where v
u
= 0 dominates u
Region (III). We name the part of this boundary region where|>
s
p1
as Region (B)
and the part where| s
s
p1
as Region (D).
Note that if|>
s
then g
III
< g
II
, and if|<
s
then g
III
> g
II
. Therefore,
p1 u u p1 u u
combining our results so far for Regions (II) and (III), for|<
p1 ,
s
there is a range
g
II
< g < g
III
where Region (E) dominates Region (II) and Region (D) dominates
u u
Region (III) and however it is not obvious which one of the two dominates the other.
In order to identify the threshold g value, we write the profit functions for the two
regions equal, R
E
= R
D
, and solve for g;
g
E
u
c R
E
(p
1
÷ c) + (s ÷ p
1
|)
F
F(pp11|))(
(p
2
÷ c) (1 ÷ F (p
2
))
.
(B.7)
Observing that
cg
>
c R
D
,
cg
we conclude that Region (E) dominates Region
(D) if g > g
E
. Note that for|
p1 , s
w
e
h
a
v
e
g
I
I
I
<
g
I
I
,
a
n
d
u u
86
(i) If g > g
II
, then Region (E) dominates all regions; it is optimal to set u
o = 1.
(ii) If g
III
< g < g
II
, then the boundary region between Regions (II) and
u u
(III) where v
u
= p
1
|, Region (A), dominates all regions.
(iii) If g < g
III
, then Region (B) dominates all regions. u
2. For| s
p1 ,
s
we have g
III
> g
E
> g
II
, and
u u u
(i) If g > g
E
, then Region (E) dominates all regions. u
(ii) If g < g
E
, then Region (D) dominates all regions. u
We see that there are four candidate regions (A, B, D and E) for optimality.
Figure B.1 gives an illustration of these regions for a specific case. Note again that
the findings above are for given|, p
1
, p
2
, c, s, and g values and point to the besto
decision depending on the relationships between these "parameters". For example,
consider a price taking seller for whom the restocking fee, hence|, is also dictated by either the
industry or some trade regulations. Then, the above rules apply directly to find the
optimum amount of information to be provided, assuming it is costless.
1.0
E
0.8
A
0.6
0.4
0.2
0.0
Y
O
I
B
1
B
p
II
Y
O
D
p
1
B
Y
O
0
A
III
B
s
B
p
1
0.0 0.2 0.4 0.6 0.8 1.0
B
Figure B.1: Candidate regions for optimality for V ? U (0, 1), p
1
= 0.51 and s = 0.4.
87
We continue our analysis considering a seller who can set allo,| and p
1
freely,
and use the above rules as a guideline to find the optimum strategy.
B.4 Analysis of Candidate Regions and Proof of Proposition 5
B.4.1 Region (A)
We start with looking at Region (A), which is optimal if g
III
< g < g
II
, which is
u u
in turn possible if|>
p1 .
s
Plugging in the defining constraint, v
u
= p
1
|, to the
objective function and taking partial derivative with respect to|, we get
cR
A
= p F (p |) ÷p + c + g (p ÷ c) (1 ÷ F (p )) ,
c|
1 1 1 2 2
which is negative for g < g
II
. Therefore, it is optimal to decrease| in Region (A), u
where sup(|) =
p1 .
s
As a result, setting| =
s
p1
ando such that v
u
= p
1
| dominates
Region (A). However, from the discussion above, when| =
p1 ,
s
Region (D), where
v
u
= 0, is optimal. This therefore establishes that Region (D) dominates Region
(A).
B.4.2 Regions (B) and (D)
We collectively represent Regions (B) and (D) as v
u
= 0. Thus, the objective
function for these regions is
R
B,D
= (p
1
÷ c) + (÷p
1
| + s) F (p
1
|) + g (p
2
÷ c) (1 ÷ F (p
2
)) (1 ÷ F (p
1
)) .
The first order condition (FOC) for| is
cR
B,D
= p (÷F (p |) + F (p |) (÷p | + s)) = 0,
or equivalently,
c|
1 1 1 1
|
-
= ps ÷ p1 F ((p
1
| -)).
1 1
F
-
p
1
|
(B.8)
Note that since the second term on the right is positive, at optimality, we
have|
-
<
p1 .
s
This means that Region (D) dominates Region (B) since the latter is
defined for|>
p1 .
s
88
Combining our results so far, we see that when g < g
II
, it is never optimal to u
have|>
p1 ;
s
Region (D) dominates both Regions (A) and (B). Therefore, we have
two candidate optimal solutions left: Solution (D), which is defined by v
u
= 0 (or
equivalently,o e [0, o
0
(|)], as shown above) and the FOC given in Equation B.8,
and Solution (E) which is defined by v
u
= p
1
. This constitutes the proof of Propo-
sition 5.
B.5 Proof of Proposition 6
From the analysis of the structural properties of seller's (o, |) decision space, we
know that changing p
1
changes the (o, |) decision space as all the critical boundaries
is a function of p
1
. Therefore, when p
1
is a decision variable as well, the seller has the
tool to change the (o, |) decision space in order to maximize her profits. Considering
this, we observe that the point| =
s
p1
on Region (D) structurally changes the profit
function when solving for the optimal p
1
; when| =
p1 ,
s
refund is equal to salvage
value and each return has zero net after-sales revenue. Therefore, we take this point
explicitly and define Region (C): {o = 0, | =
p 1 }.
s
Profits for Region (C) is given
by R
C
= (p
1
÷ c) + g (p
2
÷ c) (1 ÷ F (p
2
)) (1 ÷ F (p
1
)). Since Region (C) is a single
point on the (o, |) decision space, the seller's profits here is a function of only the
prices.
In the following, we solve for the optimalo,|, p
1
and p
2
for each of the
regions (C), (D), (E), and we identify the optimal strategy for uniformly distributed
consumer valuations; F (p) = p. Using Proposition 4, the optimal second period
price is p
-
= (1 + c)/2 for all regions. 2
B.5.1 Region (C)
In order to solve for the optimal profit function, we take the partial derivative with
c R
C
respect to p
1
,
c p1
= 1 ÷ g (p
2
÷ c) (1 ÷ p
2
), which is positive for g < g
E
. Recall u
that Region (C) is optimal only if g < g
E
; thus, it is optimal to increase p
1
as u
much as possible in the feasible range for Region (C). Constrained by the equality
89
|=
s
p1
for Region (C), the largest value that p
1
can attain is determined by the
smallest value that| can take, which is equal to|
p
. With uniform valuations, we
1+(p1|)
2
have E[max{V, p
1
|}] =
2
, and we find that
\
|
p
= {| , p
1
= E[max{V, p
1
|}]} = 2pp
1
÷ 1 1
from Proposition 3. Therefore, the optimal price should satisfy
\
2p
1
÷ 1 = s, and
it yields p
-
= 1
1+s
2
2
. Then, we find the optimal refund factor as|
-
=
s
p1
=
1+s2 .
2s
As
a result, recalling thato = 0 in Region (C) by definition, Solution (C) is given by
(o
-
, |
-
, p
-
, p
-
)
C
=
2s , 1 + s
2
, 1 + c
12
0, 1 + s
2
2 2
and it yields the net profits of
R
C
-
= 1 g 1 ÷ c 8
2
1 ÷ s
2
+ s2 ÷ c + 1 . 2
2
Note that|
-
=
2s
1+s2
< 1 for all s < 1; it is not optimal to ofer full refunds when
Solution (C) is optimal.
B.5.2 Region (D)
With uniform valuations, the FOC for| yields| =
s
2p1
in Region (D). Plugging this
equality in the objective function, we get
c R
D
c p1
=
c (p1÷c)+
1
(s)
2
+g(p2÷c)(1÷p2)(1÷p1) 4
c p1
=
1 ÷ g (p
2
÷ c) (1 ÷ p
2
). Therefore, profits are increasing in p
1
for g < g
E
, for which u
Region (D) is optimal; i.e., it is optimal to increase p
1
as high as possible in Region
(D). Given the FOC for|, the highest value for p
1
is determined by lowest value of
\
|, which is equal to|
p
=
2
p11÷
1
. Therefore, the optimal price should satisfy p
1
|
p
= p
\
2p
1
÷ 1 =
2
s
, which yields p
-
=
4+
s
2
. The corresponding| is|
-
=
s
=
4
s
.
1 8 2p1 4+s2
Then, these values constitute Solution (D),
2
(o
-
, |
-
, p
-
, p
-
)
D
= 0, 4 4ss
2
, 4 + s , 1 + c ,
12
+ 8 2
which results in the profits of
1
4 ÷ s
2
+ 3s ÷ c + 1 . 2
R
D
-
= 32g 1 ÷ c
2
8 2
Note that|
-
=
(D) is optimal.
4s
4+s2
< 1 for all s < 1; full refunds are not optimal when Solution
90
B.5.3 Region (E)
Given that V ? U (0, 1), the optimal pricing decision in Region (E) is p
-
= p
-
=
(1+c)
In this case, the seller's net profit is equal to
1 2
2
.
R
E
-
= 1(1 ÷ c)
2
(1 + g). 4
As we pointed out above, the seller is indiferent in deciding on an (o, |) point in
Region (E), or more formally, in choosing between (o
-
= 1, |
-
e [0, 1]), and given \
p
1
> E[V ], (o
-
e (0, 1], |
-
=
2p1÷
1
).
p1
B.5.4 Deriving the Optimal Strategy
To summarize our analysis above, there are three solutions that the seller can choose
among, depending on the parameters incorporated in this study:
Solution (C): (o
-
, |
-
, p
-
, p
-
) = 0,
1+
ss2 ,
1+
2s
2
,
1+
c
, with profits R
C
-
. 2
12 2
Solution (D): (o
-
, |
-
, p
-
, p
-
) = 0,
4+
ss2 ,
4+
8s
2
,
1+
c
, with profits R
D
-
. 4
12 2
Solution (E): (o
-
, |
-
) = {(o, |) , v
u
= p
-
}, p
-
= p
-
=
(1+c)
with profits R
E
-
.
1 1 2
2
,
In order to determine the ultimate optimal strategy, we conduct a three-way
comparison of the net profits ofered by these optimal solutions. First, we find the
thresholds on g by conducting three pairwise comparisons between the above net
profits. That is, we determine g
i,j
as a function of c and s by setting R
i
-
= R
j
-
and u
solving for g; g
i,j
u
{g , R
i
-
= R
j
-
}. Through algebraic manipulations, we get
g
D,C
= 3(1 4 c)
2
,
u
÷
g
E,C
= 4(1(1 + )2÷ 2(1 +)
2
c) ),
u
+ s
2
s )(1 ÷ c
2
g
E,D
= 4 + 12(1++ss)(1÷ 8(12+ c) .
u
(4
2 2
)
÷ c)
2
91
Then, we verify that
c(R
D
-
÷ R
C
-
) = 3 (1 ÷ c)
2
s
2
> 0,
c(R
cg
E
-
÷ R
C
-
)
32
c(R
cg
E
-
÷ R
D
-
)
cg
= 1 (1 ÷ c)
2
(1 + s
2
)> 0, 8
1
= 32(1 ÷ c)
2
(4 + s
2
)> 0,
establishing that Solution (i) is preferred to Solution (j) if g > g
i,j
, and Solution (j) u
is preferred to Solution (i) if g < g
i,j
. However, this gives only a partial ordering; u
in order to develop a full ordering, we seek the ordering of these thresholds on g.
Pairwise comparisons of these thresholds show that there is a critical c value that
renders g
D,C
= g
E,C
= g
E,D
: c
u
(s) = {c , g
D,C
= g
E,C
= g
E,D
}. We determine
u u u u u u
and observe that
g
D,C
c
u
(s) = 2
1 + s
2
÷ 1, 3
c
u
= 3(24(12+2s÷)(1 + c))
2
)
2
> 0, 2
g
E,C
u
cc
g
D,C
+s (1 + c
c
u
g
E,D
u
cc
g
E,C
4(4 + s
2
)(1 + c)
= 3(4 + 3s
2
÷ 2(1 + c)
2
)
2
> 0,
c
u
g
E,D
u
cc
= (1 + s
2
s)(4 + 3ss
2
)(1 + c)+ c)
2
)
2
> 0.
2
(4 +
2
÷ 2(1
Therefore, we infer that if c > c
u
(s), then g
E,D
< g
E,C
< g
D,C
, and if c < c
u
(s), then
u u u
g
D,C
< g
E,C
< g
E,D
. Proposition 6 follows by observing the ordering of solutions
u u u
(C), (D) and (E) with respect to g, which is depicted in Figure B.2.
B.6 On the Value of Optimal Refund Amount
It is interesting to investigate if the "refund amount" exceeds the salvage value or
not: does the seller allow returns even when they have negative net revenues? By
definition, Solutions (C) and (D) do not ofer a refund amount more than the salvage value;
they ofer exactly equal to, and exactly half of the salvage value, respectively.
92
Region c>c s
Solution (C) is optimal Solution (E) is optimal
C D E
g
E
,D
C E D
g
E
,C
E C D
g
D
,C
E D C
g
Region c c s
Solution (C) is optimal Solution (D) is optimal Solution (E) is optimal
C D E
g
D
,C
D C E
g
E
,C
D E C
g
E
,D
E D C
g
Figure B.2: Ordering of solutions according to the the regions in Figure 2.6.
In case of Solution (E), if the seller provides full information, there are no
returns and the seller can "advertise" any return policy. On the other hand, if she
\
provides only partial information and sets|
-
=|
p
=
2
p11÷
1
, the refund amount is p
\- \
equal to p
-
|
-
= 2p
1
÷ 1 = c. Therefore, the seller can advertise a refund amount 1
of more than the salvage value if c > s, or c > s
2
. However, since Solution (E) \
does not actually exercise returns, we conclude that a refund amount of more than
the salvage value is never exercised.
93
Appendix C
Appendix for Essay 3
C.1 Proof of Proposition 7
Suppose, without loss of generality, thato
Z
>o
Y
. Then, the condition for seller Z
capturing the whole market is v
Y
Z
= 0. Solving from (3.1) foro
Z
that satisfies this, u
we get
o(v
Y
Z
= 0) =o
Y
+ p
2
(1 ÷o
Y
)1|
Z
(÷|
Y
)
2
.
o
Z
=o
u 1 2
+ p|
2
1Z
Note thato>o
Y
only if|
Z
>|
Y
ando
Y
< 1. Furthermore, we observe that
cv
Y
Z = p
2
(1 ÷o )|
2
÷|
2
> 0
u 1 Y Z Y
co
Z
2(o
Z
÷o
Y
)
2
if|
Z
>|
Y
ando
Y
< 1. Then, anyo
Z
|
Y
,o
Z
e [o
Y
, o]
results in seller Z capturing the whole market.
Suppose now thato
Z
0. In this case,
v
ZY
is defined analogous to v
Y
Z
in (3.1); all consumers with v > v
ZY
prefer seller
u u u
Y to seller Z, while those with v < v
ZY
prefer seller Z to seller Y . Then, seller Z u
captures the whole market if v
ZY
= 1. Solving foro
Z
from (3.1), we find u
o
Y
÷ p
2
(1 ÷o
Y
)1|
Z
(÷|
Y
)
2
.
o
Z
=o(v
ZY
= 1) u
1 2
÷ p|
2
1Z
Observe thato(v
ZY
= 1)|
Y
ando
Y
< 1, and thato(v
ZY
=
u
1)> 0 as long aso
Y
> 0. Moreover, we have
u
cv
ZY
= p
2
(1 ÷o )|
2
÷|
2
> 0
u 1 Y Z Y
co
Z
2(o
Y
÷o
Z
)
2
if|
Z
>|
Y
ando
Y
< 1. Then, anyo
Z
>o(v
ZY
= 1) results in v
ZY
> 1, and
u u
thus seller Z captures the whole market. This means, given thato
Y
e (0, 1) and
|
Z
>|
Y
,o
Z
e [o(v
ZY
= 1), o
Y
] results in seller Z capturing the whole market, or u
94
given thato
Y
= 0 and|
Z
>|
Y
,o
Z
= 0 results in seller Z capturing the whole
market.
Combining the two results above, we conclude that giveno
Y
< 1, seller Z
captures the whole market if she sets|
Z
>|
Y
ando
Z
such thato
Z
e [o,o], where
o
max{0, o(v
ZY
= 1)}. u
Suppose seller Z would like to have only the consumers with valuation greater
than (1 ÷ˆ) prefer seller Z over seller Y ; that is she would like to have v
Y
Z
= 1 ÷ˆ.
v
u
v
From (3.1), we see that this is possible only ifo
Z
>o
Y
and solving foro
Z
, we find
o(v
Y
Z
= 1 ÷ˆ) =o
Y
+ p
2
(1 ÷o
Y
)2ˆ +|(Zp÷|
Y2
÷ 1.
o
Z
=o
ˆ
v
u
v
1
v
2 1Z 2
|)
Then, seller Z can set|
Z
>|
Y
ando
Z
=o
ˆ
>o
Y
to attain v
Y
Z
= 1 ÷ˆ, as long
1÷(p1|Z )
2 v u
v
aso
Y
< 1 and ˆ>v
2
.
Note that v
Y
Z
= 1 ÷ˆ means seller Z has a market vu
share of 1 ÷ v
Y
Z
= ˆ. vu
C.2 Proof of Proposition 8
Before we proceed with the proof, we note that the crucial aspect of the duopoly
case in this essay is that the market is being divided among the sellers, and that there is no
value creation as a result of competition. In other words, there are no win-win scenarios
and the game is rather close to a constant-sum game. In the light of this observation and the
market share dynamics described in Proposition 7, our
first intuition is that setting (o
Z
, |
Z
) to capture the whole market is a potential best
response of seller Z to seller Y 's (o
Y
< 1, |
Y
< 1). Note that once seller Z captures
the whole market, she is efectively a monopoly and the results for the monopoly
case directly apply. Being a monopoly, seller Z clearly prefers to be at the monopoly
optimal solution described in Proposition 5 (recall that in the duopoly case, we as-
sume p
1
and p
2
are given, and therefore Proposition 5 applies). However, we see from
Proposition 7 that conditions for seller Z to become a monopoly is not arbitrary,
and that she is not necessarily able to attain the monopoly optimal solution while becoming
a monopoly. In the proof below, we first identify the cases where seller Z can capture the
whole market at the monopoly optimal solution. Then, we look
95
at the remainder cases step-by-step and investigate whether capturing the whole
market is profitable given that monopoly optimal solution is not attainable. We ul-
timately find that under any condition, seller Z's best response to (o
Y
< 1, |
Y
< 1)
is to appropriately set (o
Z
, |
Z
) to capture the whole market. Since the sellers are identical in terms of
p
1
, p
2
, c, s and g, we conclude that there is no Nash equilibrium where a seller sets (o
j
< 1, |
j
<
1). Given this result, we analyze best responses in
the form of full refund (o
j
< 1, |
j
= 1) and full information (o
Y
= 1, |
Y
e [0, 1]),
and identify the potential Nash equilibria and the associated conditions as given in
Proposition 8.
We start by summarizing Proposition 5 for uniformly distributed valuations:
The optimal (o, |) for a monopolistic seller when p
1
, p
2
, c, s and g are given is that
the seller chooses either Solution (D) if g < ¯
E
, or Solution (E) if g > ¯
E
, where,
g
u
g
u
for uniformly distributed valuations,
Solution(D) : (o
-
, |
-
) =
(o, |) ,o e 0, 1 ÷ 1 +2(p
1
|)
2
, | = 2s
,
Solution(E) : (o
-
, |
-
) = {(o, |) , v
u
= p
1
} , and
p
1
p
1
p
1
÷ c +
s
2
¯
E
= (p ÷ c) (1 ÷ p ). 4p1
g
u
2 2
Suppose first that g > ¯
E
. Then, Solution (E) is optimal for a monopoly g
u
seller and thus, her optimal decision is to seto and| such that v
u
is as close
to p
1
as possible. Therefore in the duopoly case, following Propositions 3 and
7, if g > ¯
E
, then the best response of seller Z to (o
Y
< 1, |
Y
|
p
, however, seller Z cannot capture the whole
market and set v
u,Z
= p
1
at the same time. However, from Proposition 7, ifo
Y
< 1,
she can achieve v
Y
Z
= p
1
by setting|
Z
>|
Y
and u
o
Z
=
o(v
Y
Z u
= p
1
) =o
Y
+
p
2
(1
1
|
2
÷|
2
÷o
Y
)1 + (pZ| )
2
Y÷ 2p .
1Z 1
Note that, since|
Z
>|
Y
>|
p
, we have 1 + (p
1
|
Z
)
2
÷ 2p
1
> 0 and v
u,Z
s p
1
. By
setting v
Y
Z
= p
1
, seller Z ensures that only those consumers with valuation greater u
96
than the price prefer seller Z over seller Y , and since v
u,Z
s p
1
, all such consumers
purchase from seller Z. In other words, seller Z achieves monopoly optimal profits.
We conclude, due to symmetry, that if g > ¯
E
, there is no equilibrium where a seller g
u
sets (o
j
< 1, |
j
< 1), since the other seller can always capture the whole market
profitably.
Suppose that g < ¯
E
. In this case, we know that Solution (D) is optimal for a g
u
monopoly seller, and that she would seto as low as possible so that she can sell to
as many consumers as possible. Thus, if seller Y chooses (o
Y
< 1, |
Y
<
2p1 ),
s
seller
Z can set|
Z
=
s
2p1
ando
Z
= max{0, o(v
ZY
= 1)}, capturing the whole market u
profitably. Therefore due to symmetry, if g < ¯
E
, there is no equilibrium where a g
u
seller sets (o
j
< 1, |
j
<
2p1 ). s
Recall by definition in (2.5) that ¯
E
= g
E
(| =
g
u
u
2p1 )
s
for uniformly distributed
valuations. Then, compare (B.4) and (B.7) in Appendix B to observe that
g
II
= g
E
(| = ps ) = (p ÷pc1)÷ c÷ p )
u u 1 2
(1
2
for uniformly distributed valuations. Given that
c g
E
= u
c|
s ÷ 2p
1
|
(p
2
÷ c) (1 ÷ p
2
)< 0
for all|>
2p1 ,
s
we conclude that g
E
strictly decreases from ¯
E
to g
II
as| goes from
u
g
u
u
s
to
p1 .
s
As a corollary, if g
II
< g < ¯
E
, then there exists a critical return factor,
2p1
u
g
u
s
2p1
g
II
and|>|, it is no longer ˆ
u
optimal to have any dissatisfied buyers for a monopoly. Therefore, while capturing
97
the whole market, seller Z setso
Z
=o(v
Y
Z
= p
1
) and|
Z
>|
Y
, achieving the u
monopoly optimal profits; v
Y
Z
= p
1
means all consumers with v > p
1
prefer seller u
co(v
YZ
=p1)
Z, and since v
u,Z
< p
1
, they all purchase. Note that since
u
c|
> 0, there
exists|
Z
>|
Y
such that 1>o(v
Y
Z
= p
1
)>o
Y
, given thato
Y
< 1 and|
Y
< 1. As u
a result, if g
II
< g < ¯
E
, and given thato
Y
< 1, seller Z has a best response that
u
g
u
enables him to capture the whole market profitably for any|
Y
< 1. We conclude
due to symmetry that if g
II
< g < ¯
E
, there is no equilibrium with (o
j
< 1, |
j
< 1)
for any seller.
u
g
u
So far we showed that if g > g
II
, there is no Nash equilibrium where any seller u
sets (o
j
< 1, |
j
< 1). Then, suppose that g < g
II
. Given this sufciently small u
g, it is optimal for a monopoly seller to sell to all consumers as long as| s
p1 .
s
Therefore in the duopoly case, as response to (o
Y
< 1, |
Y
<
p1 ),
s
seller Z can set
(o
Z
= max{0, o(v
ZY
= 1)}, |
Z
= u
p1 ),
s
and thus capture the whole market selling
to as many consumers as possible. Due to symmetry, we conclude that if g < g
II
, u
neither seller sets (o
j
< 1, |
j
<
p1 )
s
in an equilibrium.
Now recall from (B.6) and the subsequent analysis in Appendix B that if
|>
p1 ,
s
it is optimal for a monopoly seller to sell to all consumers as long as
g
III
> 0 and if g < g
III
. Since g
III
is decreasing in|, if g
III
(| = 1)> 0, then
u u u u
g
III
> 0 for all|< 1; otherwise, since g
III
(| =
p1 )
s
= g
II
> 0, there is a
s
o
Y
and|
Y
=|
Z
= 1, the value ofo
Z
is
irrelevant for both sellers and the value ofo
Y
is irrelevant for seller Z. Therefore, essentially,
seller Z has three potential best responses: 1) (o
Z
p
1
; in the second, seller Y
and seller Z are identical and they equally share the profits; in the third, seller Z
1+p
2
provides full information and we have v
Y
Z
= u
2
1
> p
1
. We write seller Z's net
99
profits under each decision as follows:
R
Z
(o
Z
¯
III
and would like to decrease it if g
u
g < ¯
III
. Suppose g > ¯
III
; then seller Z's optimal decision giveno
Z
R
Z
(o
Z
=o
Y
, |
j
= 1)> R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1) if
g > g
IV
and v
u,Y
>ˆ
u
, otherwise if g > g
IV
and v
u,Y
s
u
and g > g
V
, theno
Z
u o
Y
is the best response
if v
u,Y
>ˆ
u
, ando
Z
>o
Y
is the best response if v
u,Y
s
u
and ¯
III
< g < g
V
; then, due to symmetry, there is no pure-strategy
g
u
u
Nash equilibrium where a seller setso
j
< 1 because providing a marginally less
information than competition is always the best response for both sellers and there
is a continuum of such best responses.
Suppose that s > s
u
and g > g
V
. Then, both sellers' best response is to provide u
marginally less information than the competition until, without loss of generality,
v
u,Z
= ˆ
u,Z
, at which point seller Y 's best response iso
Y
= 1. Seller B's best response v
too
Y
= 1 is either (o
Z
1, |
Z
= 1) with profits R
Z
(o
Z
o
Y
= 1, |
Z
= 1), or
o
Z
= 1 with profits
R
Z
(o
j
= 1) = 1 ((p
1
÷ c)(1 ÷ p
1
) + g(p
2
÷ c)(1 ÷ p
2
)) . 2
We find that R
Z
(o
Z
o
Y
= 1, |
Z
= 1)> R
Z
(o
j
= 1) only if g > g
IV
; however, u
given s > s
u
, we have g
V
> ¯
III
> g
IV
. Therefore, if s > s
u
and g > g
V
, seller Z's
u
g
u
u u
best response too
Y
= 1 is (o
Z
1, |
Z
= 1). Given (o
Z
1, |
Z
= 1), seller Y 's best
response is (o
Y
o
Z
, |
Z
= 1) and the sellers are back in the loop of a continuous
series of best responses where they unilaterally deviate from an equilibrium. As a
result, there is no pure-strategy Nash equilibrium if s > s
u
and g > g
V
. Combined u
with the above result, we conclude that if s > s
u
and g > ¯
III
, there is no pure- g
u
strategy Nash equilibrium in the duopoly.
On the other hand, consider, for s > s
u
and ¯
III
< g < g
V
, the case where
g
u
u
seller Y sets|
Y
= 1 and pickso
Y
e [0, 1] arbitrarily. Without knowing where seller
Y is located in terms ofo, seller Z is forced to randomize his decision as well and her
best response is similarly to set|
Z
= 1 and chooseo
Z
e [0, 1] randomly. Therefore,
given s > s
u
and ¯
III
< g < g
V
, there is a mixed-strategy Nash equilibrium where
g
u
u
both sellers set|
j
= 1, and picko
j
e [0, 1] randomly. Next, consider for s > s
u
and
g > g
V
, the case where seller Y sets|
Y
= 1 and pickso
Y
e [o, 1] randomly, where ˆu
101
oˆ o(v
u
= ˆ
u
) can be found by substituting (C.1) and| = 1 into (2.3). From the v
above analysis, we see that seller Z's best response is to set|
Z
= 1 and randomize
o
Z
e [o, 1]. As a result, given s > s
u
and g > g
V
, there is a mixed-strategy Nash
ˆ
u
equilibrium where both sellers set|
j
= 1 and picko
j
e [o, 1] randomly. ˆ
Consider now s > s
u
and g < ¯
III
. In this case, seller Z's best response to g
u
(o
Y
e (0, 1), |
Y
= 1) giveno
Z
0, |
Z
= 1) =(p
1
÷ c)
2
1
+ (s ÷ p
1
)p
1
+ g(p
2
÷ c)(1 ÷ p
2
)
1 + p
2
÷ p . 1
1
2
We determine that if g < g
V
, then u
R
Z
(o
Z
= 0, o
Y
> 0, |
Z
= 1)> R
Z
(o
Z
=o
Y
, |
j
= 1)> R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1)
and if g > g
V
, then u
R
Z
(o
Z
= 0, o
Y
> 0, |
Z
= 1)< R
Z
(o
Z
=o
Y
, |
j
= 1)< R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1)
for anyo
Y
. Since for s > s
u
we have g
V
> ¯
III
, we conclude that if s > s
u
and
u
g
u
given g < ¯
III
, seller Z's best response to (o
Y
e (0, 1), |
Y
= 1) is (o
Z
= 0, |
Z
= 1). g
u
Consider then (o
Y
= 0, |
Y
= 1); seller Z's best response is either (o
Z
> 0, |
Z
= 1)
with profits equal to R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1), oro
Z
= 0 resulting in
R
Z
(o
j
= 0, |
j
= 1) = 1 ((p
1
÷ c) + (s ÷ p
1
)p
1
+ g(p
2
÷ c)(1 ÷ p
2
)(1 ÷ p
1
)) . 2
We see from above that R
Z
(o
j
= 0, |
j
= 1)> R
Z
(o
Z
= 1, o
Y
< 1, |
Y
= 1) if s > s
u
and g < ¯
III
; in other words, both sellers' best response to the competition providing g
u
no information and ofering full refund is to provide no information and ofer full
refund. This result leads us to a Nash equilibrium where both sellers provide no
information and ofer a full refund return policy, (o
j
= 0, |
j
= 1), in case of s > s
u
and g < ¯
III
. g
u
Suppose now s < s
u
, in which case we have g
V
< ¯
III
< g
IV
. Therefore, if
u
g
u
u
g < g
V
, then we have g < ¯
III
readily satisfied, and following the analysis above,
u
g
u
1 02
seller Z's best response to (o
Y
e [0, 1), |
Y
= 1) is (o
Z
= 0, |
Z
= 1). As a result,
(o
j
= 0, |
j
= 1) is the only Nash equilibrium if s < s
u
and g < g
V
. u
Given s < s
u
, suppose g > g
V
; then, seller Z's best response to (o
Y
= 0, |
Y
= u
1) is (o
Z
> 0, |
Z
= 1). In other words, both sellers' best response to the competition
setting (o
j
< 1, |
j
= 1) is to provide more information than the competition,
and therefore due to symmetry, there is no equilibrium where both sellers have
(o
j
< 1, |
j
= 1). Consider then (o
Y
= 1, |
Y
e [0, 1]); seller Z's best response is
either (o
Z
= 0, |
Z
= 1) with R
Z
(o
Z
= 0, o
Y
> 0, |
Z
= 1), or (o
Z
= 1, |
Z
e [0, 1])
with R
Z
(o
j
= 1). Comparing the profits, we find that if g < g
V
I
, seller Z's best u
response to (o
Y
= 1, |
Y
e [0, 1]) is (o
Z
= 0, |
Z
= 1), and if g > g
V
I
, then it is u
(o
Z
= 1, |
Z
e [0, 1]), where
g
V
I
u
2s ÷ c ÷ p
1
(1 ÷ p
1
+ c) .
(2 ÷ p
1
)(p
2
÷ c)(1 ÷ p
1
)
We further find that given s < s
u
, we have g
V
< g
V
I
< ¯
III
. Therefore, if s < s
u
and
u u
g
u
g
V
< g < g
V
I
, seller Z's best response to (o
Y
= 1, |
Y
e [0, 1]) is (o
Z
= 0, |
Z
= 1),
u u
to which seller Y 's best response is (o
Y
= 1, |
Y
e [0, 1]). We conclude that, if
s < s
u
and g
V
< g < g
V
I
, there is a Nash equilibrium where one seller provides
u u
full information and ofers an arbitrary return policy, while the other seller provides
zero information but ofers a full refund return policy.
Consider now the case where s < s
u
(for which g
V
I
< ¯
III
< g
IV
) and g
V
I
<
u
g
u
u u
g < g
IV
. We know from above that if g < ¯
III
, then seller Z's best response to
u
g
u
(o
Y
= 1, |
Y
e [0, 1]) giveno
Z
¯
III
, then seller Z's best response giveno
Z