HOW CAN BUYERS ENGAGE SUPPLIERS TO BE MORE SOCIALLY AND
ENVIRONMENTALLY RESPONSIBLE?
by
HOSSEIN RIKHTEHGAR BERENJI
A DISSERTATION
Presented to the Department of Operations and Business Analytics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2019
DISSERTATION APPROVAL PAGE
Student: Hossein Rikhtehgar Berenji
Title: How Can Buyers Engage Suppliers to Be More Socially and Environmentally
Responsible?
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Operations
and Business Analytics by:
Nagesh N. Murthy Co-Chair
Zhibin (Ben) Yang Co-Chair
Pradeep Pendem Core Member
Bruce McGough Institutional Representative
and
Janet Woodruff-Borden Vice Provost and Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2019
ii
©c 2019 Hossein Rikhtehgar Berenji
iii
DISSERTATION ABSTRACT
Hossein Rikhtehgar Berenji
Doctor of Philosophy
Department of Operations and Business Analytics
June 2019
Title: How Can Buyers Engage Suppliers to Be More Socially and Environmentally
Responsible?
A major concern for buyers in a given tier of the supply chain continues to
be the challenge of balancing the economic benefits of outsourcing with the loss in
their ability to control and influence sustainability performance of their suppliers.
The overarching question in this dissertation is how buyers can engage suppliers to
improve social and environmental performance of the supply chain. A combination of
analytical and empirical models are developed and analyzed to offer a buyer guidance
at strategic (i.e., managing trust in buyer-supplier relationships) and tactical (i.e.,
designing suitable contracting mechanisms) levels on how to make suppliers more
socially and environmentally responsible. In the first essay, we consider a buyer
who enjoys the pricing power and also has the ability to commit to contract
terms. We investigate how such a buyer’s commitment to contract terms affects
the sustainability and financial performance of the supply chain. The second essay
focuses on understanding the impact of supplier competition on the buyer’s ability to
influence suppliers’ compliance when suppliers have more parity in contracting power.
Unlike the first essay, wherein the buyer stipulates both price and quantity, this essay
iv
considers situations wherein the supplier offers a wholesale price and the buyer is
limited to only offering the quantity in a wholesale contract. In the third essay, we
propose a framework to investigate the role of specific nature of trust (i.e., calculative
and relational trust) between buyers and suppliers in influencing the impact of their
supplier relationship management strategies on suppliers’ sustainability performance.
This dissertation includes previously unpublished co-authored material.
v
CURRICULUM VITAE
NAME OF AUTHOR: Hossein Rikhtehgar Berenji
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Sharif University of Technology, Tehran, Iran
South Tehran Azad University, Tehran, Iran
DEGREES AWARDED:
Doctor of Philosophy, Operations and Business Analytics, 2019, University of
Oregon
Master of Science, Decision Sciences, 2015, University of Oregon
Master of Business Administration, 2011, Sharif University of Technology
Bachelor of Science,, Industrial Engineering, 2008, South Tehran Azad
University
AREAS OF SPECIAL INTEREST:
Supply Chain Management, Sustainable Operations
GRANTS, AWARDS AND HONORS:
Robin and Roger Best Teaching Award, Lundquist College of Business, 2019
Doctoral Research Grant in Support of Doctoral Dissertation Research,
Lundquist College of Business, 2018
Kimble First Year Teaching Award, University of Oregon, 2017
Robin and Roger Best Teaching Award, Lundquist College of Business, 2017
Robin and Roger Best Research Award, Lundquist College of Business, 2016-
2019
vi
ACKNOWLEDGEMENTS
Undertaking this PhD has been a truly valuable experience for me and it would
not have been possible to do without the support that I received from many people.
I would like to express my deepest appreciation to my committee co-chair and my
advisor Dr. Nagesh Murthy. My success would not have been possible without the
support and nurturing of Nagesh. I truly appreciate him for his continuous support
of my PhD study and related research, for his patience, enthusiasm, motivation, and
immense knowledge. Nagesh helped me to engage with industry and learn more about
real business research questions and to bring those examples to the classroom. I am
so thankful to Nagesh for the countless hours in research meetings, for being always
there to help me, and for his financial support to me to attend many conferences.
On a personal level, Nagesh inspired me by his hardworking and passionate attitude.
I would also like to extend my gratitude to Dr. Zhibin (Ben) Yang, my
committee co-chair. Ben’s excellent skills as a researcher were extremely beneficial
for me and my research in my PhD program. I am thankful to Ben for his advice,
guidance, invaluable comments, careful attention to details, and for asking questions
which incented me to broaden my research from various perspectives.
I would like to thank the rest of my dissertation committee members, Dr. Bruce
McGough and Dr. Pradeep Pendem for their insightful comments.
I gratefully acknowledge the advice and guidance of Dr. Eren Çil, the PhD
program coordinator. I would like to recognize the tremendous support of PhD
program director Dr. Andrew Verner and staffs in the PhD program.
Last but not least, I’m extremely grateful to my family and friends. This journey
would not have been possible without their continued patience, and endless support.
vii
This dissertation is dedicated to my lovely parents, Nahid & Mahdi, for their
endless love, support, and encouragement.
viii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. COMMITTING TO CONTRACT FOR A SUPPLIER’S SOCIAL AND
ENVIRONMENTAL COMPLIANCE . . . . . . . . . . . . . . . . . . 6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
The Benchmark: “No-Commitment Policy” . . . . . . . . . . . . . 21
Partial Commitment Policies . . . . . . . . . . . . . . . . . . . . . 25
Full Commitment Policy . . . . . . . . . . . . . . . . . . . . . . . . 43
Effect of Raising the Standard for the Code of Conduct . . . . . . 49
Summary & Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 54
Bridge to Next Chapter . . . . . . . . . . . . . . . . . . . . . . . . 56
III. IMPACT OF SUPPLIER COMPETITION ON SUPPLIER’S SOCIAL
AND ENVIRONMENTAL COMPLIANCE . . . . . . . . . . . . . . . 57
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Benchmark Model with a Single Supplier When Auditing Precedes
Contracting . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
ix
Chapter Page
Model with Supplier Competition When Auditing Precedes
Contracting . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Effect of Supplier Competition . . . . . . . . . . . . . . . . . . . . 92
Conclusion and Summary . . . . . . . . . . . . . . . . . . . . . . . 94
Bridge to Next Chapter . . . . . . . . . . . . . . . . . . . . . . . . 95
IV. A CONCEPTUAL FRAMEWORK FOR UNDERSTANDING THE
ROLE OF TRUST BETWEEN BUYERS AND SUPPLIERS IN
INFLUENCING SUPPLIERS’ SUSTAINABILITY . . . . . . . . . . . 96
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Conceptual Development . . . . . . . . . . . . . . . . . . . . . . . 104
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
V. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
APPENDICES
A.. TECHNICAL PROOFS - CHAPTER II . . . . . . . . . . . . . . . . . 121
B.. TECHNICAL PROOFS - CHAPTER III . . . . . . . . . . . . . . . . 234
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
x
LIST OF FIGURES
Figure Page
1. Sequence of Actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2. Nash Equilibrium Parameter Space . . . . . . . . . . . . . . . . . . . . . 31
3. Illustration of the Equilibrium Forms in Commitment to Wholesale Price
Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4. From “no-commitment” to a “commitment to only the wholesale price”
when γ > αβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2β−α
5. From “no-commitment” to “commitment to only the wholesale price”
when 0 < γ 6 (4β−α)α
2
− . . . . . . . . . . . . . . . . . . . . . . . . . . . 414β(2β α)
6. From “no-commitment” to a “commitment to only the wholesale price”
when (4β−α)α
2
− < γ <
αβ
− . . . . . . . . . . . . . . . . . . . . . . . . . 424β(2β α) 2β α
7. From “commitment to only the wholesale price” to “full commitment”
2
when (4β−α)α− < γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484β(2β α)
8. Effect of Raising the Standard of Code of Conduct on Supply chain profit 53
9. Comprehensive Research Framework . . . . . . . . . . . . . . . . . . . . . 59
10. Probability Tree of Buyer-Supplier Interaction . . . . . . . . . . . . . . . 64
11. Sequence of Actions in Single Supplier for Contracting Follows Auditing. . 66
12. Presentation of Model Equilibrium of Single Supplier on Plane of Demand
and Market Price. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
13. Sequence of Actions in Competition for Contracting Follows Auditing. . . 74
14. Presentation of Optimal Order Quantities on Plane of Wholesale Prices
When ΦC,NC > ΦNC,C . . . . . . . . . . . . . . . . . . . . . . . . . . 77
15. Presentation of Optimal Order Quantities on Plane of Wholesale Prices
When ΦC,NC 6 ΦNC,C . . . . . . . . . . . . . . . . . . . . . . . . . . 78
16. Presentation of Nash Equilibrium of Wholesale Prices on Plane of θ1 and θ2. 84
xi
Figure Page
17. Presentation of Nash Equilibrium of Compliance Efforts on Plane of
Demand and Market Price. . . . . . . . . . . . . . . . . . . . . . . . . 90
18. Presentation of Equilibrium of Supplier Competition on Plane of Demand
and Market Price. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
19. Conceptual Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
20. Parameter Space for the Buyer’s Optimal Contracting Decision for no-
commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
21. Parameter Space based on the Buyer’s Optimal Quantity Decision for
model of commitment to price . . . . . . . . . . . . . . . . . . . . . . 123
22. Graphical Presentation of Regions of Lemma 3 on plane of (w,D) . . . . 124
23. The Buyer’s Best Response Function in Relation to (D,w) . . . . . . . . . 127
24. The Buyer’s Best Response Function Parameter Space Based on Demand
and Wholesale Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
25. Different Cases for the Supplier’s Best Response Function Based on plane
of (w,D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
26. Possible Forms of the Supplier’s Profit Function in Region A . . . . . . . 147
27. Possible Forms of the Supplier’s Profit Function in Region B-I-1 . . . . . 159
28. Possible Forms of the Supplier’s Profit Function in Region B-I-2 . . . . . 162
29. Possible Forms of the Supplier’s Profit Function in Region B-II . . . . . . 172
30. Possible Forms of the Supplier’s Profit Function in Region C-I . . . . . . 200
31. Possible Forms of the Supplier’s Profit Function in Region C-II . . . . . . 201
32. Illustration of No NE When Intersecting S-BRF 1-2 with B-BRF 2, B-BRF
3, and B-BRF 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
33. Illustration of No NE When Intersecting S-BRF 2-2 with B-BRF 2 . . . . 205
34. Illustration of Different Forms of NE and No NE Cases When Intersecting
S-BRF 2-2 with B-BRF 3 . . . . . . . . . . . . . . . . . . . . . . . . . 207
35. Illustration of NE and No NE Cases When Intersecting B-BRF 4 with
S-BRF 1-4 and S-BRF 1-5 . . . . . . . . . . . . . . . . . . . . . . . . 211
xii
Figure Page
36. Illustration of NE form 1 and Form 2 When Intersecting B-BRF 4 and
S-BRF 2-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
37. Illustration of possible NE cases when intersecting B-BRF 4 with S-BRF
2-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
38. Illustration of possible NE cases when intersecting B-BRF 4 with S-BRF
2-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
39. Division of plane of (D,w) for NE form I and form II . . . . . . . . . . . . 218
40. Illustration of the supplier’s participation constraint for NE region (II) of
Commitment to Wholesale Price . . . . . . . . . . . . . . . . . . . . . 223
41. Decision space of contracting stage for full commitment model . . . . . . 230
42. Graphic Representation of NE (w∗1, w
∗
2) = (Ω2 − p(θ2 − θ1),Ω2) . . . . . . 236
43. Graphic Representation of NE (w∗1, w
∗
2) = (Ω1,Ω1 − p(θ1 − θ2)) . . . . . . 237
44. Presentation of Nash Equilibrium of wholesale Prices on Plane of θ1 and θ2.240
xiii
LIST OF TABLES
Table Page
1. Different Subregions in Region A and Associated Candidate Solutions . . 147
2. Different Subregions in Region B and Associated Candidate Solutions . . 157
3. List of 25 sub-regions of B-I for the range of 0 6 a 6 p − β(p−w) . . . . 184
w Dw(w−c)
4. List of 19 sub-regions of B-II for the range of p − β(p−w)− < a 6 1 . . . . 187w Dw(w c)
5. List of the Supplier’s Best Response Function forms in Region B . . . . . 189
xiv
CHAPTER I
INTRODUCTION
According to United Nations Global Compact, many well-known global
manufacturers and brands are well into sustainability initiatives, and yet face some
significant roadblocks in various tiers of their supply chain. While many brands
outsource their entire production, several major manufacturers outsource a large
fraction of components in their product to lower tiers of the supply chain. Their
ability to influence sustainability in their supply chain is curtailed since they only
have direct control over a fraction of their tiers. Consequently, an overwhelming
majority of sustainability challenges stem from their supplier base, wherein suppliers
can fall seriously behind the curve (UNGC, 2015). Manufacturers are interested in
finding effective ways to push their sustainability efforts upstream using incentives,
business leverage, training programs, and auditing to improve their suppliers’
sustainability performance. A major concern for buyers in a given tier of the supply
chain continues to be the challenge of balancing the economic benefits of outsourcing
with the loss in their ability to control and influence sustainability performance of
their suppliers.
The overarching question in this dissertation is how buyers can engage suppliers
to improve social and environmental performance of the supply chain. A combination
of analytical and empirical models are developed and analyzed to offer a buyer
guidance at strategic (i.e., managing trust in buyer-supplier relationships) and
tactical (i.e., designing suitable contracting mechanisms) levels on how to make
suppliers more socially and environmentally responsible. The dissertation includes
three essays that consider several key questions that together contribute to answering
1
the overarching question of the dissertation. This dissertation has been partially
supported by the Center for Sustainable Business Practices at the Lundquist College
of Business.
The first major set of research questions that are amenable to economic modeling
are addressed using an analytical framework. In the first essay, we investigate
how buyer’s policy of commitment to contract terms affects the sustainability and
financial performance of the supply chain. We consider a framework wherein the
buyer audits the supplier for the supplier’s compliance to a code of conduct. The
degree of the buyer’s commitment to contract terms is defined by his commitment to
price and/or order quantity in a wholesale contract prior to knowing the outcome of
auditing vis-à-vis making it conditional to the outcome of auditing. To the best of our
knowledge, this lever has not been considered hitherto in the sustainability literature.
The specific research questions include 1) How does the buyer’s commitment affect
the supplier’s compliance to the code of conduct and the overall sustainability
performance of the supply chain? 2) How does the buyer’s commitment affect the
financial performance of the buyer and the supplier? 3) What is the interaction
between different ways in which the buyer can progressively commit to price and/or
order quantity? 4) What is the effect of increasing the standard of the code of conduct
on the buyer’s auditing, the supplier’s compliance, and the overall sustainability
performance of the supply chain?
Four multistage game-theoretic models that consider a Nash equilibrium are
developed and analyzed to fully characterize the effect of the buyer’s commitment to
contract terms and the auditing effort on the sustainability and financial performance
of the supply chain. The study provides managers with yet another effective tool
to consider improving compliance in their supplier base. Our results show that
2
commitment provides the buyer with significant opportunity to advance both social
good and profit in a win-win manner. We find that increasing the level of commitment
improves the supplier’s likelihood of compliance to the sustainability standard.
Interestingly, we also find that both contracting opportunity and profit for the buyer
increase monotonically with the degree of commitment. Additionally, committing to
only the price or only the quantity (vis-à-vis no-commitment) has an asymmetric
effect. We show that committing to the price and committing to the quantity are
complementary strategies for the buyer and substitutes for the supplier. This essay
is an unpublished co-authored work with Dr. Nagesh Murthy and Dr. Zhibin (Ben)
Yang.
The second essay focuses on understanding the impact of supplier competition
on the buyer’s ability to influence suppliers’ compliance when suppliers have more
parity in contracting power. Unlike the first essay, wherein the buyer stipulates
both price and quantity, this essay considers situations wherein the supplier offers
a wholesale price and the buyer is limited to only offering the quantity in a
wholesale contract. Depending on whether auditing precedes contracting or follows
contracting, it determines whether suppliers first compete on their compliance efforts
or wholesale prices. We develop a framework to investigate if and how the sequence in
which the supplier competition manifests, influences buyers auditing effort, suppliers
compliance efforts, and financial performance of parties.
We consider two suppliers who are asymmetric in their production and
compliance costs. One supplier has a lower production cost and higher compliance
cost, while the other has a higher production cost and lower compliance cost. We
analyze multi-stage game theoretic models to find the buyer’s optimal auditing effort,
suppliers’ compliance efforts, and profits of parties. Our results indicate that when
3
the auditing precedes the contracting, the buyer does not have to exert significant
auditing effort since suppliers find it beneficial to be compliant with the code of
conduct in the presence of competition. In the process the supplier is also able to
squeeze out the buyer’s profit to zero. When the demand in the market is high,
the supplier with the lower cost of production (but higher compliance cost) wins the
competition. In contrast, when the market price is high while demand is low, the
supplier with the advantage in compliance cost wins the competition. We also show
that the supplier competition leads to increasing the contracting opportunity.
The third essay focuses on a broader research question pertaining to buyer-
supplier relationship management strategies to influence supplier’s sustainability
performance. The primary research question considers how the specific nature of
trust (i.e., calculative vs. relational trust) between the buyer and the supplier
influences the impact of supplier relationship management strategies (transactional
and collaborative) on supplier’s sustainability performance. This research question
is not easily amenable to analytical modeling, and hence addressed by developing
a conceptual and empirical framework. In this essay, we propose a conceptual
framework to lay a foundation for developing a theoretical lens to understand
this phenomena for promoting socially and environmentally responsible behavior
in supply chains. After a careful examination and unification of the literature we
develop a set of hypotheses to enable a formalized study that can address the above
research question.
The remainder of this dissertation is organized as follows. In next chapter
we present our analytical model to examine the role of commitment in enhancing
social and environmental compliance of the supplier. In chapter 3, we investigate
the impact of supplier competition on buyer’s auditing, suppliers’ compliance, and
4
financial performance of parties. In chapter 4, we develop our empirical framework to
examine the role of buyer’s transactional and collaborative approaches on suppliers’
social and environmental performance. Our conceptual framework focuses on the
moderating role of trust in this relationship. Finally, we provide concluding remarks
in chapter 5.
5
CHAPTER II
COMMITTING TO CONTRACT FOR A SUPPLIER’S SOCIAL AND
ENVIRONMENTAL COMPLIANCE
This work was submitted to the journal of Manufacturing & Service Operations
Management with co-authorship of Dr. Nagesh Murthy and Dr. Zhibin (Ben) Yang.
Introduction
Major brands in many industries outsource their production and hence lose
direct control in managing sustainability in their supplier base. It is not surprising
that an overwhelming majority (i.e., as much as 70%) of social and environmental
violations emanate from suppliers (UNGC, 2015). A key finding from an international
survey on different industries in America, Europe and Asia shows that customers
demand more sustainable supply chains (DNVGL, 2014). Citizens at large also
engage in shaping public policy to raise the stringency of national and international
standards for sustainability (Bartley, 2003; Toffel et al., 2015; Thorlakson et al.,
2018). Consumers hold brands and major manufacturers accountable when their
suppliers’ sustainability violations become known in the marketplace. Although
brands may not be legally responsible for their suppliers’ social and environmental
misconduct, they can incur significant costs for their suppliers’ violations (Caro
et al., 2018). The damage to reputation and brand from revelations of social and
environmental misconduct in the supplier base may lead to loss of revenue and market
(Guo et al., 2016; Plambeck and Taylor, 2016).
Given the strategic implications of sustainability in supply chains, the number
of S&P 500 companies publishing sustainability reports has increased from 20% in
6
2011 to 82% in 2016 (Coppola, 2017). Brands and major manufacturers adopt a
code of conduct that sets expectations for their suppliers’ social and environmental
practices. These expectations often increase over time (Nike, 2018; Apple, 2018).
Hence, brands are constantly engaged in enhancing compliance of their suppliers to
an existing code of conduct or to a new, upcoming, and more stringent sustainability
standard. Brands audit their suppliers for compliance to the code of conduct and
proactively or reactively strive to reduce the likelihood of social and environmental
violations in their supplier base (GIIRS, 2014). Yet, the challenge persists as the
compliance in the supply base varies significantly across industries and firms, with
significant consequences for the buyer (Thorlakson et al., 2018).
While auditing is an important tool for buyers to ascertain their suppliers’
compliance with the buyers’ sustainability standard, a key challenge remains: how
to motivate their suppliers to invest, in the first place, in sustainability capabilities
that can enhance the likelihood of their compliance to the requisite (i.e., prevalent
or upcoming) sustainability standard (e.g., the supplier’s social and environmental
code of conduct). Assuring compliance may require an inordinate (i.e., costly) effort
for suppliers. At the same time, catching a non-compliant supplier with certainty
may require an inordinate auditing effort for the buyer. Thus, it becomes important
to design a suitable contracting mechanism that, when used in conjunction with
auditing, can further align the incentives for both parties. Such a mechanism needs
to consider consequences of sustainability violations, while accounting for the costs
associated with varying levels of compliance and costs for varying levels of auditing
effort, including the cost of any corrective action to be undertaken by the supplier
when one fails an audit. One option for a buyer to incentivize the supplier is to
consider committing to contract terms before both parties invest in their respective
7
efforts, and before the auditing outcome (i.e., whether the supplier passed or failed
the audit) is known with regards to the supplier’s compliance status. The buyer’s
commitment to contract terms (i.e., degree of commitment) can manifest in terms
of an a priori commitment to only price, only quantity, or both.
The potential benefit of the buyer’s commitment to similar contract terms to
induce supplier’s investment prior to resolution of a key underlying uncertainty has
been studied under different settings in the operations management literature. Taylor
and Plambeck (2007) study a scenario wherein the supplier has to invest in capacity
for producing a product still under development, hence faces significant demand
uncertainty. Their result indicates that the buyer should commit to both price and
quantity if the production cost is low and either the capacity cost is low or the cost
of capital is high. Otherwise, the buyer should commit to price only. Hu et al.
(2013) study a scenario wherein a buyer is interested in motivating a supplier to
invest for capacity restoration capability when faced with a supply chain disruption.
They find that ex ante (i.e., prior to disruption) commitment to price and quantity
dominates the ex post (i.e., after disruption) commitment, with both parties being
better off. Kim and Netessine (2013) consider a scenario involving development of
an innovative product wherein there exists uncertainty about component production
cost. They examine the effect of commitment to only price, only quantity, and
both on collaborative efforts of parties to reduce the uncertainty about component
production cost and, in turn, lower the expected cost. They find that under all three
types of commitment, neither party exerts collaborative effort.
The potential efficacy of a buyer’s commitment to contract terms has not been
examined in the sustainability literature. Given the promising and yet mixed results
of the benefit of commitment for a buyer, we examine the efficacy of a buyer’s
8
commitment to price and/or quantity when used in conjunction with auditing.
The efficacy for sustainability is measured in terms of the ability to enhance both
the supplier’s likelihood of compliance to the code of conduct prior to auditing
outcome, and overall sustainability compliance in the marketplace (which also
accounts for the corrective action when the supplier fails an audit). We consider the
progression from “no-commitment” to “commitment to only the wholesale price” to
“commitment to both wholesale price and order quantity” to represent the increasing
level of the buyer’s commitment in a wholesale contract. We develop, analyze, and
compare multi-stage game-theoretic models to represent varying levels of a buyer’s
commitment. We solve for the subgame-perfect equilibrium and characterize the
contract terms, the buyer’s auditing effort, and the supplier’s compliance effort at
the equilibrium. Our results show that the commitment provides the buyer with
significant opportunity to advance both social good and profit in a win-win manner.
Interestingly, the transition from “no-commitment” to “commitment to only the
wholesale price” policy vis-à-vis a transition to “commitment to only the quantity”
has an asymmetric effect on sustainability and profit metrics.
We find that increasing the level of commitment improves the supplier’s
likelihood of compliance to the sustainability standard (i.e, the code of conduct). We
show that overall sustainability compliance in the market also improves, in a great
majority of cases, with an increase in the commitment level. Interestingly, we find
that both contracting opportunity and profit for the buyer increase monotonically
with the degree of commitment. Our results indicate that committing to quantity
is financially valuable for the buyer only if the buyer has already committed to the
wholesale price. Thus, committing to the price and committing to the quantity are
complementary strategies for the buyer. In contrast, we find that the supplier’s profit
9
is not changing monotonically by advancing the level of commitment. Our results
show that the buyer’s strategies of commitment to wholesale price and commitment
to quantity have the effect of being substitutes for the supplier. We also find that
when the buyer transitions from “no-commitment” to “commitment to only the
wholesale price” policy, it can result in a win-win-win scenario for both the buyer
and the supplier. In this case not only does the buyer’s profit improve, along with
an improvement in overall sustainability compliance (i.e., win-win for the buyer);
but also, the supplier makes a positive profit (i.e., win for the supplier). Lastly,
we examine the sensitivity of the supplier’s compliance and overall sustainability
compliance to an increase in the standard for the code of conduct. We identify
conditions (i.e., scenarios) in which, notably, the supplier’s compliance and overall
sustainability compliance increase with an increase in the standard for the code of
conduct.
The remainder of chapter is organized as follows. First, we review the relevant
literature. Then, we introduce the model set-up. After that, we analyze the “no-
commitment” policy as a benchmark. Next, we investigate the effect of partial
commitment policies. After that, we analyze the policy of “commitment to both
wholesale price and quantity” and investigate the interplay between the buyer’s
policies that consider increasing level of commitment. Next, we use our analysis in
previous sections and study the effect of raising the standard of the code of conduct
on sustainability and financial metrics. Finally, we provide concluding remarks. The
proofs of all Lemmas and Propositions are provided in Appendix.
10
Literature Review
Our paper is related to research focused on examining the efficacy of sourcing
strategies, supply chain structures, incentives and penalties in contract design,
and auditing schemes as levers for enhancing suppliers’ social and environmental
responsibility.
With regard to the related literature on sourcing strategies and supply chain
structures, Guo et al. (2016) study responsible sourcing in supply chains when faced
with the choice of a responsible supplier who is costly vis-à-vis a less expensive
supplier who may face responsibility violations. Agrawal and Lee (2017) study the
effect of sourcing policies in influencing a supplier’s decision to adopt sustainable
practices. de Zegher et al. (2017a) investigate the interplay between sourcing channel
and contract design to incorporate responsible sourcing strategies while also creating
economic value. Letizia and Hendrikse (2016) study the impact of supply chain
structures using horizontal or vertical alliances among supply chain members to
incentivize suppliers to adopt corporate social responsibility activities. Orsdemir
et al. (2018) investigate vertical integration as a strategy for ensuring corporate
social and environmental responsibility. Huang et al. (2017) study the issue of
how a buyer should manage social responsibility in lower tiers of its supply chain
by examining a scenario wherein a tier 0 buyer either works directly with a tier 2
supplier for enabling its compliance or delegates it to the tier 1 supplier. Zhang et al.
(2017) examine equilibrium sourcing decisions in a three-tier network comprised of
manufacturers, smelters, and mines to understand implications of imposing penalties
on manufacturers for failing to curb usage of conflict minerals.
The next research stream has focused on the effect of incentives and penalties
in contract design for sustainability. Babich and Tang (2012) consider a buyer’s
11
moral hazard problem with hidden action to investigate the effectiveness of using
only deferred payment, only inspection, or both as mechanisms to tackle the
problem of adulteration by its supplier. They find that deferred payment mechanism
is preferable to inspection if the threat of adulteration is low. Chen and Lee
(2016) consider an adverse selection problem with hidden information to study
effectiveness of supplier certification, process audits, and contingent payments as
tools for enhancing a supplier’s compliance to the social and environmental standard.
Their results indicate that while process audit and contingency payment alone can
directly increase the supplier’s compliance, they are not as effective as a supplier
certification instrument for screening suppliers with different levels of ethics. Karaer
et al. (2017) study the effect of offering wholesale price premium and sharing the cost
of a supplier’s sustainability effort in enhancing the supplier’s ability to produce a
more environmentally responsible product. Under a single-supplier setting, they find
that if it is optimal for the buyer to offer a price premium, then the buyer also fully
subsidizes the supplier’s cost for investing in environmental quality. Cho et al. (2018)
investigate the effect of offering price premium, and the effect of whether inspection
information is disclosed, as tools for curbing the use of child labor in supply chains.
They find that the firm’s pricing and inspection strategies behave as substitutes
in curbing child labor. de Zegher et al. (2017b) consider a 3-tier supply chain to
study the efficacy of offering contingent advance payments by palm oil buyers, when
contracting with the mill and farmers, to enhance compliance for sustainable sourcing
commitment.
Another stream of research has focused on examining the efficacy of a variety
of auditing schemes in influencing a supplier’s social or environmental compliance.
Plambeck and Taylor (2016) consider a buyer-supplier setting with an exogenous
12
wholesale contract in order to investigate the degree to which the buyer’s actions
related to increasing auditing effort, to publicizing negative auditing reports, and
to providing a loan to a supplier can backfire, in terms of increased evasive (i.e.,
hiding) action by the supplier. Caro et al. (2018), Fang and Cho (2016), and Chen
et al. (2017) build on the framework in Plambeck and Taylor (2016) to examine the
efficacy of various auditing approaches (e.g., independent, shared, or joint audit) to
influence a supplier’s sustainability effort and ensuing compliance. Caro et al. (2018)
investigate the impact of two non-competing buyers either jointly auditing a common
supplier or sharing results of one’s independent audit with the other buyer. Fang and
Cho (2016) consider blocks of competing buyers sourcing from a common supplier
wherein buyers in each block either jointly audit the supplier or audit the supplier
independently and share results with buyers in the block. However, the contract
with the supplier in this setting is exogenous, and no corrective action is required
by the supplier upon failing an audit. Chen et al. (2017) consider a scenario with
two identical buyers and three suppliers (with an exogenous wholesale price offered
to each supplier). Each buyer sources two components, with one of them being from
the common supplier and the other from a non-common supplier. Motivated by
a budget-constrained setting, they investigate whether buyers should prioritize the
auditing effort based on the degree of supplier’s centrality, i.e., whether to emphasize
the audit for the common supplier vis-à-vis their respective non-common suppliers.
In their setting, the supplier incurs a correction cost upon failing an audit; however
that cost is not linked to the sustainability effort of the supplier.
In contrast to the extant literature, we focus on understanding the efficacy of
commitment to a wholesale contract on enhancing sustainability compliance in the
market. To the best of our knowledge, this lever has not been considered hitherto in
13
the sustainability literature. Thus, with a different focus, we investigate the effect of
the interplay between degree of commitment to an endogenized wholesale contract
and the buyer’s auditing effort on the supplier’s compliance to the code of conduct
and ensuing overall sustainability compliance in the market. We modify the basic
setting in Plambeck and Taylor (2016), wherein the supplier, upon failing the audit, is
expected to take a corrective action that is commensurate with his compliance effort.
Further, we now also solve for the subgame-perfect equilibrium and characterize the
contract terms, the buyers auditing effort, and the supplier’s compliance effort at
the equilibrium. Lastly, we study the effect of raising the standard of the code of
conduct. We present details of the modeling framework in the next section.
Model Setting
We consider a supply chain in which a buyer sources a product from a supplier
and then resells the product to the consumer market. It costs c per unit for the
supplier to manufacture the product. The buyer offers to the supplier a take-it-
or-leave-it wholesale-price contract (w, q), where w is the wholesale price and q is
the order quantity. The market for the product has the demand size of D and the
selling price of p. To focus on the analysis of efficacy of degree of commitment
on sustainability in the supply chain, we simplify p to be exogenous and D to be
deterministic. We assume that c ≤ w ≤ p and 0 ≤ q ≤ D.
The buyer puts in place a supplier code of conduct to guide and audit the
supplier for socially and environmentally responsible business practice. We assume
that the buyer, the supplier, and the market have a common understanding of what
constitutes a violation of this sustainability code of conduct. The market expects
the buyer to source from a compliant supplier. When a supplier is prone to social
14
or environmental misconduct (i.e., violations) in its business practice, it adversely
affects the buyer’s demand in the market.
Without loss of generality, we consider that the supplier is initially non-
compliant with the code of conduct and can make an effort to be compliant. The
supplier’s compliance effort produces a random outcome: the supplier becomes
compliant with the code of conduct with probability e and remains to be non-
compliant with probability 1 − e. Intuitively, a higher compliance effort leads to
a higher probability of being compliant. We follow the literature, such as Plambeck
and Taylor (2016) and Chen et al. (2017), to use the probability e also as a proxy for
the supplier’s compliance effort. The supplier’s compliance effort entails the cost of
Ks(e) for e ∈ [0, 1], which is continuous and strictly increasing in e with Ks(0) = 0
and is differentiable.
The supplier’s compliance status is invisible to the buyer. The buyer has to
audit the supplier to improve its knowledge about the supplier’s compliance status,
but only imperfectly. If the supplier is compliant with the code of conduct, then the
auditing process reliably concludes with a “pass”. With a non-compliant supplier,
the buyer’s auditing process yields a random outcome. With probability a, the
buyer correctly identifies a non-compliant supplier, and the supplier fails the audit.
With probability 1 − a, a non-compliant supplier passes the audit. Intuitively, the
buyer’s probability of correctly identifying a non-compliant supplier increases with
the level of auditing effort, which can be increased, for example, by assembling a more
experienced auditing team. We use the probability of identifying a non-compliant
supplier, a, as a proxy for the buyer’s auditing effort. The buyer’s auditing effort
has the cost of Kb(a), which is continuous and strictly increasing in a ∈ [0, 1] with
Kb(0) = 0 and is differentiable.
15
After the completion of the supplier’s compliance effort and the buyer’s auditing,
the buyer observes “pass” with probability e+(1−e)(1−a) or “fail” with probability
(1 − e)a. We assume that the auditing outcome—“pass” or “fail”—is verifiable by
the buyer and by the supplier, so the buyer can make use of the auditing outcome
in dealing with the supplier. Whenever possible and necessary, the buyer designs
the contract to be contingent on the auditing outcome. We use (wp, qp) and (wf , qf )
to denote the contracts to be used when the auditing outcome is “pass” and “fail”,
respectively. Furthermore, if the buyer’s audit successfully identifies a non-compliant
supplier, the buyer requests the supplier to take corrective action at the supplier’s
own expense. We assume that the buyer’s audit uncovers all violations, and hence, a
corrective action makes a non-compliant supplier perfectly compliant. The supplier’s
cost of correction depends on the level of initial compliance effort and is denoted as
Kc(e), which is continuous and strictly decreasing in e ∈ [0, 1] with Kc(1) = 0 and is
differentiable.
When the buyer fails to identify a non-compliant supplier, we assume that the
market can perfectly identify non-compliance of the supplier in a timely manner,
e.g., via a NGO investigation or via media coverage. The market reacts to non-
compliance with market disruption. Since our focus is on examining the effect of
degree of commitment to contract terms, we make a simplifying assumption that the
buyer loses all demand D with probability 1 due to market disruption.
FIGURE 1. Sequence of Actions.
16
We model a one-period problem, in which the buyer and the supplier contract
for one-time supply and take a sequence of actions. The timing of the actions is
illustrated in Figure 1. First, the supplier exerts the compliance effort e, and the
buyer exerts an auditing effort a. We assume that at the time of auditing the buyer
and supplier cannot observe each other’s effort level. However, after the auditing the
supplier’s compliance effort becomes verifiable to the buyer. Following the literature,
such as Kim and Netessine (2013), we treat the buyer’s and the supplier’s decisions
about their respective levels of effort as a simultaneous-move game and solve for
Nash equilibrium. Next, the supplier takes necessary actions to fulfill the contract
requirement, contingent on the auditing outcome. If the supplier passes the audit,
then the buyer requires the supplier to execute the contract of (wp, qp). If the supplier
fails the audit, then the audit indicates that the supplier is non-compliant. The buyer
invokes the contract of (wf , qf ) and requests the supplier to take corrective action
before executing the contract. The supplier may turn down the buyer’s request for
correction and renege on the contract of (wf , qf ). We assume that the supplier’s
reservation profit of non-participation is zero. Finally, the buyer attempts selling
the product to the market. The outcome of selling depends on the supplier’s true
compliance status. Working with a compliant supplier fully eliminates the risk of
an adverse market response. As such, the buyer sells successfully to the market and
collects the revenue of (p− wi)qi (with qi ≤ D). The subscript i = p if the supplier
passes the audit; and i = f if the supplier fails the audit and is corrected later on
to become compliant. If the supplier remains non-compliant at the time of selling,
market disruption occurs, and the buyer makes zero profit from the market.
As the sequence of actions evolves, the buyer and the supplier incur costs, collect
revenues, and receive information (i.e., auditing outcome and any market disruption).
17
Looking forward at different time epochs, the buyer and the supplier see different
profit functions for the rest of the game. We now elaborate on the buyer’s and the
supplier’s profit-to-go at two time epochs: after the conclusion of auditing and at
the inception of auditing.
After auditing concludes, the buyer observes the outcome to be either “pass”
or “fail”, but not the supplier’s true compliance status. If the supplier passes the
audit, there is a probability that the supplier is, actually, non-compliant and the
buyer experiences market disruption and collects zero revenue. The buyer invokes
the contract of (wp, qp) and anticipates the following expected profit from the rest of
the game:
πp
e
B(wp, qp, e, a) = p q− − p
− wpqp, (2.1)
e+ (1 e)(1 a)
where e− − is the conditional probability that the supplier is compliant ande+(1 e)(1 a)
the market is not disrupted, given that the supplier passes the audit. Note that
the buyer’s expected profit depends on e and a, because they affect the conditional
probability. The supplier collects the following profit from the supply contract:
πpS(wp, qp) = (wp − c)qp. (2.2)
In the case that the supplier fails the audit, the buyer invokes the contract of (wf , qf )
and requests the supplier to take the corrective action to be compliant. Note that
“fail” is an unambiguous indication of the supplier’ non-compliance, so it resolves
the uncertainty in the auditing outcome. The supplier’s corrective action resolves
the uncertainty in the market’s response. The buyer collects the profit of selling to
the market:
πfB(wf , qf ) = (p− wf )qf , (2.3)
18
under the condition that the supplier agrees to take corrective action to be fully
compliant. The supplier’s profit of staying in the contract is its income from the
supply contract less the cost of correction:
πfS(wf , qf , e) = (wf − c)qf −Kc(e). (2.4)
Before auditing begins, the buyer faces the uncertainties of auditing outcome
and the market’s response later on. The buyer’s expected profit is the expectation
of profit to be collected after auditing less the upfront cost of auditing:
Π (w , q , w , q , e, a) = [e+(1−e)(1−a)]πp (w , q , e, a)+[(1−e)a]πfB p p f f B p p B(wf , qf )−Kb(a).
(2.5)
Using the expressions for πpB(wp, qp, e, a) and π
f
B(wf , qf ) in (2.1) and (2.3), one can
transform (2.5) into the more explicit form:
ΠB(wp, qp, wf , qf , e, a) = e(p−wp)qp−[(1−e)(1−a)]wpqp+(1−e)a(p−wf )qf−Kb(a).
(2.6)
This form of ΠB(wp, qp, wf , qf , e, a) helps us understand the buyer’s trade-off in the
face of an uncertain outcome of auditing. The buyer anticipates three possible
situations looking forward. If the supplier is compliant (with probability e), then
the buyer uses the contract of (wp, qp), sells to the market, and collects the profit of
(p − wp)qp. If the supplier is non-compliant but passes the audit (with probability
(1−e)(1−a)), then the buyer pays the amount of wpqp for the supply but experiences
a market disruption. If the supplier is non-compliant and fails the audit (with
probability (1 − e)a), then the buyer uses the contract of (wf , qf ) and collects the
profit of (p− wf )qf .
19
Similarly, the supplier’s expected profit is:
ΠS(wp, qp, wf , qf , e, a) = [e+(1−e)(1−a)]πpS(wp, qp)+[(1−e)a]π
f
S(wf , qf , e)−Ks(e),
(2.7)
which can be written more explicitly as:
[ ]
ΠS(wp, qp, wf , qf , e, a) = [e+(1−e)(1−a)]qp(wp−c)+(1−e)a qf (wf−c)−Kc(e) −Ks(e).
(2.8)
For tractability, we assume that the buyer’s auditing cost, Kb(a) is linear in
a ∈ [0, 1], and that the supplier’s compliance effort cost and correction cost, Ks(e)
and Kc(e) respectively, are linear in e ∈ [0, 1]. Specifically, we assume Kb(a) = γ×a,
Ks(e) = α × e and Kc(e) = β × (1 − e), where γ, α and β are cost coefficients and
positive constants.
The buyer can choose to commit to the wholesale price or to the order quantity
before the inception of auditing, and the buyer has commitment power. In this paper,
the commitment to a contract term means that the buyer decides the contract term
and offers it to the supplier before auditing the supplier. The buyer offers to honor
this a priori contracting decision regardless of the outcome of the audit, so long
as the supplier is willing to take corrective upon failing the audit. There are four
scenarios of contract commitment, depending on whether the buyer commits to the
wholesale price or to the order quantity. In the first scenario, the buyer makes no-
commitment at all. All contracting decisions are made after auditing outcome is
obtained. In the second scenario, the buyer commits to only the order quantity with
q = qp = qf , leaving the wholesale prices wp and wf to be decided after auditing. In
the third scenario, the buyer commits to only the wholesale price with w = wp = wf ,
leaving the order quantities qp and qf to be decided after auditing. Finally, in the
20
fourth scenario, the buyer commits to both the wholesale price and order quantity
with w = wp = wf and q = qp = qf .
The Benchmark: “No-Commitment Policy”
We first analyze the scenario wherein the buyer does not commit to either the
wholesale price or the order quantity. That is, the buyer decides the wholesale price
and the order quantity only after the auditing outcome is revealed, but before the
execution of the contract. We will use this scenario as a benchmark for further
analysis. We analyze this benchmark model backward and solve for the sub-
game perfect Nash equilibrium, starting with the buyer’s contracting decisions after
auditing.
Optimal Contracting Decisions
We first analyze the buyer’s contracting decision, which is made after auditing
is concluded. Given the buyer’s and the supplier’s efforts, respectively a and e,
the buyer decides a menu of two contracts (wp, qp) and (wf , qf ) to be used in the
respective cases where auditing yields the outcome of “pass” and “fail”.
In the case that the supplier passes the audit, the buyer offers the contract of
(wp, qp) to maximize its expected profit:
max πpB(wp, qp, e, a), (2.9)
0≤qp≤D, c≤wp≤p
where πpB(wp, qp, e, a) is defined in (2.1). We present the buyer’s optimal contract in
Lemma 1.
21
Lemma 1. When the buyer makes no contract commitment, and given that the
supplier passes the audit, the buyer’s optimal wholesale price and order quantity
(w∗p, q
∗
p) are:
(1− p
+
e
– If c
)
− ≤ a ≤ 1, or equivalently
1−a
p− ≤ e ≤ 1, then the buyer’s optimal1 e a
c
contract is (w∗p, q
∗
p) = (c,D).
+
≤ (1−
p e)
– If 0 a < c− , or equivalently 0 ≤ e <
1−a
p− , then the buyer does not1 e a
c
contract with the supplier.
At the optimal contract, the supplier’s profit-to-go is πpS(w
∗ ∗
p, qp) = 0.
Proof. All proofs are relegated to the Appendix.
In the case that the supplier fails the audit, the buyer offers the supplier the
contract of (wf , qf ) to maximize its profit of selling to the market, subject to the
supplier taking corrective action:
max πfB(wf , qf )
0≤q ≤D, c≤w ≤p
f f (2.10)
subject to πfS(wf , qf , e) ≥ 0,
where πfB(wf , qf ) and π
f
S(wf , qf , e) are defined in (2.3) and (2.4), respectively. Recall
that, after auditing the supplier’s compliance effort becomes verifiable to the buyer.
Lemma 2 presents the optimal contract for the case that the supplier fails the audit.
Lemma 2. When the buyer makes no contract commitment, and given that the
supplier fails the audit, the buyer’s optimal wholesale price and order quantity (w∗f , q
∗
f )
are: [ ]+
– (If 1− (p−c)D) ≤ e ≤ 1, then the buyer’s optimal contract is (w∗f , q∗f ) =β
c+ β(1−e) , D .
D
22
[ ]+
– If 0 ≤ e < 1− (p−c)D , then the buyer does not contract with the supplier.
β
At the optimal contract, the supplier’s profit-to-go is πf ∗ ∗S(wf , qf , e) = 0.
Lemmas 1 and 2 show that the buyer contracts with the supplier only if the
supplier’s initial compliance effort e or the buyer’s auditing effort a is sufficiently
high. With a high initial compliance effort, the supplier’s ex post correction cost is
low, so the supplier can afford remaining in the contract. Conversely, with a high
upfront auditing effort, there is a low chance that the supplier is non-compliant at
the end of auditing. So, the buyer is confident to contract with the supplier.
Furthermore, Lemmas 1 and 2 show that the supplier collects zero profit after
auditing. This is because the buyer has the contracting power and sets the wholesale
price to drive the supplier’s profit exactly to its reservation profit. When the supplier
passes the audit, the buyer sets the wholesale price to be the production cost. When
the supplier fails the audit, the buyer sets the wholesale price to exactly offset the
production cost and the cost of correction.
Sub-Game Perfect Nash Equilibrium of Auditing and Compliance
Next, we derive the buyer’s decision of auditing effort and the supplier’s decision
of compliance effort, given that the buyer will optimally design the contracts of
(w∗p, q
∗
p) and (w
∗
f , q
∗
f ) after auditing. We solve for Nash equilibrium of a and e. To do
that, we first solve for the buyer’s and the supplier’s best response functions to each
other’s effort level.
From (2.7), the supplier’s total expected profit at the inception of auditing is
Π (w∗, q∗, w∗ ∗S p p f , qf , e, a) = [e+(1−e)(1−a)]π
p
S(w
∗ ∗ f ∗ ∗
p, qp)+[(1−e)a]πS(wf , qf , e)−Ks(e),
(2.11)
23
in which, as Lemmas 1 and 2 show, πpS(w
∗ ∗ f ∗ ∗
p, qp) = πS(wf , qf , e) = 0. Since the
supplier’s profit-to-go after auditing is zero in the optimal contract, his expected
profit is negative for any positive level of compliance effort. Hence, it is optimal
for the supplier to make zero compliance effort upfront, that is, e = 0. As a result,
Π (w∗S p, q
∗ ∗
p, wf , q
∗
f , e, a) ≡ 0 for all a. The buyer extracts all profit of the supply chain
for the entire game.
The buyer chooses the auditing effort a to maximize its expected profit:
max ΠB(w
∗ ∗
p, qp, w
∗ ∗
f , qf , e, a) (2.12)
0≤a≤1
where Π ∗B(wp, q
∗, w∗p f , q
∗
f , e, a) is shown in (2.6). We substitute e = 0, (w
∗
p, q
∗
p), and
(w∗ ∗f , qf ) into (2.12) and solve for the optimal a. Proposition 1 presents the Nash
equilibrium and its outcome.
Proposition 1. Consider that the buyer does not commit to either the wholesale
price or the order quantity. At the equilibrium, the supplier’s compliance effort e∗
and the buyer’s auditing effort a∗, the buyer’s optimal contracts (w∗p, q
∗
p) and (w
∗
f , q
∗
f ),
and the buyer’s and the supplier’s total profit π∗B and π
∗
S are:
– If (p− c)D > β+γ, then we have (e∗, a∗) = (0, 1), (w∗, q∗p p) = (c,D), (w∗, q∗f f ) =
(c+ β , D), π∗ ∗
D B
> 0 and πS = 0.
– If (p− c)D ≤ β + γ, then the buyer does not contract with the supplier.
D(p− c) is the supply chain’s total contribution margin (i.e., excluding cost of
efforts) for producing the product and selling it to the market, and β+γ is the supply
chain’s total cost of efforts (to be eventually borne by the buyer for contracting to
occur) when the supplier does not exert any compliance effort. Proposition 1 shows
24
that the buyer contracts with the supplier if and only if the supply chain’s total
contribution margin is sufficiently high to offset the cost of auditing and correction
when the supplier does not expend any compliance effort upfront.
Furthermore, Proposition 1 shows that, while the supplier has no incentive to
exert any compliance effort (i.e., e∗ = 0), the buyer audits the supplier at the highest
possible level with a∗ = 1. The buyer does so to produce verifiable evidence for
holding the supplier accountable and enforcing the corrective action to avoid market
disruption. Thus, a basic managerial insight for a buyer who does not commit to
contract terms but is able to enforce corrective action when the supplier fails an
audit is that the buyer cannot enhance the upfront compliance of a supplier.
Partial Commitment Policies
In this section we investigate the effect of the buyer’s partial commitment (i.e.,
when the buyer commits to only quantity or only the wholesale price). In this section
first, we briefly present and analyze the policy in which the buyer commits to only
the quantity. We compare this policy to “no-commitment” (benchmark policy) to
characterize the effect of commitment to only the order quantity. We find that
“commitment to only the order quantity” has identical equilibrium outcomes vis-
à-vis the policy of “no-commitment”. Next, we present and analyze the policy in
which the buyer commits to only the wholesale price. Then, we compare this policy
to the “no-commitment” policy to characterize in detail the effect of commitment to
only the wholesale price.
25
Committing to Only Order Quantity
In this section, we analyze the policy in which the buyer commits to only
the order quantity of the contract. Before the buyer and the supplier exert their
respective efforts, the buyer decides the order quantity q and commits to qf = qp = q
to be invoked upon execution of the contract. The buyer decides the wholesale prices
wp and wf after auditing concludes.
The analysis of this policy closely resembles that for the policy without
commitment in §2.4, and the equilibrium outcomes of the two policies are identical.
Hence, we skip all technicalities and refer the reader to Lemmas 1 and 2, and
Proposition 1 for the equilibrium outcomes. Since the buyer still sets the wholesale
price after the conclusion of auditing, similar to that in the policy without
commitment, the buyer extracts the entire profit of the supply chain. So, under
both policies the buyer and the supplier behave identically with the implication
being that “committing to only the order quantity” makes no difference at all. This
offers an added insight for a buyer who is now able to commit to quantity while being
able to enforce corrective action upon the supplier failing an audit. The buyer still
cannot enhance the upfront compliance of a supplier with this commitment.
Committing to Only the Wholesale Price
In this section, we analyze the policy in which the buyer commits to only
the wholesale price of the contract. Before the buyer and the supplier exert
their respective efforts, the buyer decides the wholesale price w and commits to
wf = wp = w to be invoked upon execution of the contract. The buyer decides order
quantities qp and qf after auditing concludes.
26
We analyze the model backward, starting with the buyer’s order quantity
decisions after observing the auditing outcome to be either “pass” or “fail”.
The optimization programs representing the buyer’s decision are identical to
programs (2.9) and (2.10) for the model without contract commitment, except that
wp and wf are dropped out of the buyer’s decision at this time point. We present
the buyer’s optimal qp and qf in Lemma 3.
Lemma 3. Consider the situation wherein the buyer commits to only the wholesale
price.
If the supplier passes the audit, then the buyer’s optimal order quantity qp is such
that
+
(1− p e)
– If w− ≤ a ≤ 1, or equivalently
1−a
p − ≤ e ≤ 1, then the buyer’s optimal1 e a
w
order quantity is q∗p = D; and the supplier’s profit at this stage is (w − c)D.
p +
≤ (1− ew )– If 0 a < − , then the buyer does not order, i.e., q
∗ = 0.
1 e p
If the su[pplier fails t]he audit, then the buyer’s optimal order quantity qf is such that+
– If 1− (w−c)D ≤ e ≤ 1, then the buyer’s optimal order quantity is q∗ = D;
β f
and the sup[plier’s profi]t at this stage is (w − c)D − β(1− e).+
– If 0 ≤ e < 1− (w−c)D , then the buyer does not order, i.e., q∗ = 0.
β f
The insight from Lemma 3 is similar to that from Lemmas 1 and 2 when
there is no-commitment. The buyer orders from the supplier only if the supplier’s
initial compliance effort e or the buyer’s auditing effort a is sufficiently high. A key
difference is that the supplier need not make zero profit, as the buyer may find it
beneficial to set the wholesale price w to be high (in the first stage) and commit to
it.
27
Sub-Game Perfect Nash Equilibrium of Auditing and Compliance
We move one step backward in time to immediately after the buyer’s decision
of w, and solve for the Nash equilibrium of the buyer’s auditing effort a and
the supplier’s compliance effort e. The buyer and the supplier simultaneously
choose a and e to maximize their expected profit: ΠB(wp, q
∗
p, w
∗
f , qf , e, a) and
ΠS(w , q
∗
p p, wf , q
∗
f , e, a). In the buyer’s and the supplier’s profit functions, wp =
wf = w are considered as a given parameter at this stage, and (q
∗, q∗p f ) as optimally
decided according to Lemma 3. There are many combinations of optimal qf and qp
in Lemma 3, depending on the values of a, e, as well as the model parameters of
D, p, w, α, β, and γ. So, to solve for the Nash equilibrium of a and e, there are a
plethora of cases to be considered, each depending on the values of D, p, w, α, β,
and γ. We relegate details of the solution procedure to the Appendix and present
the equilibrium outcome in Lemma 4.
Lemma 4. Consider the scenario that the buyer commits to only the wholesale price
before auditing. At the Nash Equilibrium(NE), the buyer’s auditing effort a∗ and the
supplier’s compliance effort e∗ are:
– When 0 < γ 6 (4β−α)α
2
or p < 8cγβ
2
4β(2β−α) 4β(2β−α)γ−(4β−α)α2
• and under c + (4β−α)α 6 w 6 p − γ then (e∗, a∗) = (1 − α , 1), which is
4βD D 2β
NE form I;
• otherwise, the buyer’s profit is zero and the supplier’s profit is non-positive.
– When γ > (4β−α)α
2
and p > 8cγβ
2
4β(2β−α) 4β(2β−α)γ−(4β−α)α2
• and under c+ (4β−α)α 6 w 6 p− γ and D > 2βγ then (e∗, a∗) = (1− α , 1),
4βD D pα 2β
which is NE form I;
28
• and under ŵ(D, p, c, α, β, γ) 6 w 6 p − γ and D < 2βγ then (e∗, a∗) =
D pα
(1− γ , αDp), which is NE form II;
Dp 2γβ
• otherwise, the buyer’s profit is zero and the supplier’s profit is non-positive.
where ŵ(D, p, c, α, β, γ) =
√
Dγ(4β−α)+2D3p(p+c)− (Dγ(4β−α)+2D3p(p+c))2−16D3p((D3p2C)+(2Dp−γ)βγ)
3 . The4D p
buyer’s profit and the supplier’s profit associated with each forms of NE
are:
– If NE form I holds, which is (e∗, a∗) = (1− α , 1), then
2β
 α
∗ 0 c 6 w < c+ 2Dq =  , q∗ = D,f pD c+ α 6 w 6 p
2D
(1−
α )D(p− w)− γ c 6 w < c+ α
 2β 2DΠB =  , (cor. 1)D(p− w)− γ c+ α 6 w 6 p 2D α α α
and ΠS = 
(1− )D(w − c)− α(1− ) c 6 w < c+
2β 2β 2D
. (cor. 2)
D(w − c)− α(4β−α) c+ α 6 w 6 p
4β 2D
29
– If NE form II holds, which is (e∗, a∗) = (1− γ , αDp), then
Dp 2γβ
 βγ
∗ 0 c 6 w < c+ D2pq =  , q∗ = D,f pD c+ βγ2 6 w 6 pD p
 1
 D(p− w)(2β − α)− γ c 6 w < c+
βγ
2β D2pΠB =
, (cor. 3)
D(p− w)− γ c+ βγ
D2
6 w 6 p
 p 1 D(w − c)(2β − α)− (α−
αγ ) c 6 w < c+ βγ
2β Dp D2pand ΠS =  . (cor. 4)D(w − c)− (α− αγ ) c+ βγ
2Dp D2
6 w 6 p
p
– If NE form III holds, which is (e∗, a∗) = (w , 0), then (q∗, q∗) = (0, 0) and
p f p
(ΠS,Π
w
B) = (−γ , 0).p
– If NE form IV holds, which is (e∗, a∗) = (0, 0), then (q∗, q∗) = (0, 0) and
f p
(ΠS,ΠB) = (0, 0).
Figure 2 shows the graphic representation of Lemma 4. The buyer is able to
justify a full auditing effort in situations when the demand is high while being able
to offer a wholesale price to the supplier that is neither too low nor too high (i.e.,
as represented in region of NE form I in Figure 2-a). In situations when the market
price is low, the buyer exerts full auditing effort regardless of γ (i.e., even when it is
high) when the demand is high, and doing so offers a positive profit by avoiding any
risk of market disruption (i.e., as in region of NE form I in Figure 2-a). When the
auditing cost coefficient is high, the buyer exerts full effort only when both demand
and market price are high (i.e., as in region of NE form I in Figure 2-b). However,
when the demand is moderate, even with a high market price the buyer can no longer
30
FIGURE 2. Nash Equilibrium Parameter Space
where line 1 represents D = (4β−α)α4β(w−c) , line 2 represents D =
2βγ
pα , line 3 represents
2
w = ŵ(D, p, c, α, β, γ) , line 4 represents D = γp−w , and p̄ =
8cγβ
4β(2β−α)γ−(4β−α)α2 .
justify a full auditing effort (i.e., as in region of NE form II in Figure 2-b). In both
regions of NE form I and II, the supplier exerts some level of compliance effort since
he receives a contract with a wholesale price w that is neither too low nor too high.
We next consider the region that is outside of the regions represented by NE
forms I & II. When the wholesale price is high, the buyer has no willingness to exert
any effort, but the supplier exerts some level of compliance effort. The buyer makes
zero profit, and the supplier loses money by exerting compliance effort. When either
demand or wholesale price is low, neither the buyer nor the supplier can justify
their respective efforts. The optimal profit-to-go for both parties is zero. Hence, we
consider the region outside of those NE forms I & II as a no-contracting region in
our further analysis.
31
Optimal Wholesale Price Decision
We move one step backward in time to solve for the buyer’s decision of w. At
this step, before the buyer and the supplier exert their respective efforts, the buyer
decides the wholesale price w and commits to wf = wp = w to be invoked upon
execution of the contract.
Optimization program (2.13) shows that the buyer chooses the wholesale price
w to maximize its profit-to-go, which is presented in function (2.6), subject to
the (ex ante) supplier’s participation in the contracting. The (ex ante) supplier’s
participation constraint is governed by the supplier’s profit-to-go which is presented
in function (2.8). Note that for these functions, we assume that wf = wp = w.
max Π (w , q∗B p p, w , q
∗, e∗f f , a
∗)
c≤w≤p
(2.13)
subject to Π (w , q∗S p p, w
∗ ∗ ∗
f , qf , e , a ) ≥ 0
Proposition 2 presents the buyer’s optimal wholesale price, along with model
equilibrium for committing to only the wholesale price.
Proposition 2. Consider the situation wherein the buyer commits to only the
wholesale price. At the equilibrium, the supplier’s compliance effort e∗ and the buyer’s
auditing effort a∗, the buyer’s optimal contracts (w∗, q∗, q∗f p), and the buyer’s and the
supplier’s respective total profit π∗B and π
∗
S are:
– Form (i): no-contracting;
– Form (ii):(e∗, a∗) = (1 − α , 1), w∗ = c + (4β−α)α , (q∗, q∗) = (D,D),and π∗ >
2β 4βD p f B
0 , π∗S = 0;
32
– Form (iii):(e∗, a∗) = (1− γ ,
αDp), w∗ = ŵ(D, p, c, α, β, γ),
 Dp 2βγ(D, 0), if D < (2β+α)γ
∗ ∗  2pα(qp, qf ) =  , and π∗B > 0 , π∗S > 0.(D,D), otherwise
The above forms are defined by the following conditions:
(4β−α)α2 2– When 0 < γ 6 or p < 8cγβ :
4β(2β−α) 4β(2β−α)γ−(4β−α)α2
- If 0 6 D < (4β−α)α+4βγ− , then form (i) occurs.4β(p c)
- Otherwise, form (ii) occurs.
– When γ > (4β−α)α
2
and p > 8cγβ
2
:
4β(2β−α) 4β(2β−α)γ−(4β−α)α2
- If 0 6 D 6 D̂(p, c, α, β, γ), then form (i) occurs.
- If D > 2γβ , then form (ii) occurs.
pα
- If D̂(p, c, α, β, γ) < D < 2γβ , then form (iii) occurs,
pα
Figure 3 shows the graphic representation of regions of equilibrium forms for
Proposition 2. When the market conditions are not favorable (i.e., either the demand
or the market price is low) we observe a no-contracting region that expands with
increase in auditing cost coefficient (region of equilibrium form i in Figure 3). On the
other hand, when demand and market price are both high, the buyer and the supplier
exert their respective efforts, with the buyer also exerting full auditing effort (region
of equilibrium form ii). Interestingly, when the demand and price are moderate and
2
the auditing cost coefficient is high (i.e., γ > (4β−α)α− ), it leads to a new behavior4β(2β α)
(region of equilibrium form iii) not observed when the auditing cost coefficient is low.
The buyer is now only able to justify a partial auditing effort. In both regions of
equilibrium forms ii and iii, the supplier’s compliance effort is not zero, as the supplier
receives a contract and has incentive to exert some level of compliance effort.
33
FIGURE 3. Illustration of the Equilibrium Forms in Commitment to Wholesale
Price Policy
Where line 1 represents D = (4β−α)α+4βγ 2βγ4β(p−c) , line 2 represents D = pα , line 3 represents
2
D = D̂(p, c, α, β, γ), and p̄ = 8cγβ4β(2β−α)γ−(4β−α)α2 .
The buyer makes a positive profit in all regions wherein contracting exists.
The supplier’s profit is squeezed out to zero in region of equilibrium form ii ; and
surprisingly, the supplier’s profit is positive in region of equilibrium form iii. We
next provide some intuition for it. It should be noted that there are two lower
bounds for wholesale price that are relevant to our game-theoretic analysis to ensure
the supplier’s participation. One lower bound for wholesale price is determined from
the ex ante consideration, to ensure that the supplier is able to justify exerting
compliance effort upfront. The second lower bound for wholesale price is determined
from the ex post consideration, to ensure that that the supplier can justify corrective
action upon failing an audit, even after having exerted compliance effort upfront.
34
When the lower bound for the wholesale price w determined from the ex ante
participation constraint is greater than the lower bound for wholesale price (w)
determined from ex post participation constraint, the buyer can set the wholesale
price in the first step in a manner that squeezes out the supplier’s expected profit
to zero. On the other hand, when the lower bound for the wholesale price for
the supplier’s ex ante participation is less than that for the supplier’s ex post
participation, the buyer is no longer able to squeeze the supplier’s profit to zero
while setting wholesale price in the first step.
To further understand how the above-mentioned lower bounds are affected by
compliance and auditing cost coefficients, we first consider the scenario in which
cost coefficients for compliance α and auditing γ are low (i.e., you are in region
of equilibrium form ii, Figure 3-a). Additionally, α is low enough such that the
equilibrium behavior moves to region of equilibrium form iii in Figure 3-b when the
2
auditing cost coefficient γ is increased to beyond the threshold (i.e., γ > (4β−α)α ),
4β(2β−α)
ceteris paribus.
For α considered in the above scenario, when γ is below the threshold (i.e,
γ 6 (4β−α)α
2
− and wherein you are in region of equilibrium form ii, Figure 3-a),4β(2β α)
the supplier’s compliance effort is high since α is low (see Proposition 2). A high
compliance effort leads to a lower correction cost for ex post consideration. The
supplier’s ex ante compliance cost dominates ex post correction cost. Hence, in region
of equilibrium form ii, the lower bound for the wholesale price for the supplier’s ex
ante participation is greater than that for the supplier’s ex post participation, thus
allowing the buyer to squeeze the supplier’s profit to zero. But when the auditing cost
coefficient is increased to beyond the threshold (i.e., γ > (4β−α)α
2
− ), ceteris paribus,4β(2β α)
the equilibrium behavior is now represented by region of equilibrium form iii in
35
Figure 3-b. The supplier’s optimal compliance effort decreases since γ has increased
(see Proposition 2). As a result, the compliance cost decreases. A lower compliance
effort now leads to a higher correction cost for ex post consideration. The supplier’s
ex ante compliance cost is now dominated by ex post correction cost. Hence in region
of equilibrium form iii the lower bound for the wholesale price for the supplier’s ex
ante participation is lower than that for the supplier’s ex post participation, thus
preventing the buyer from squeezing the supplier’s profit to zero.
Lastly, when the cost coefficient for compliance α is significantly high (and
equilibrium is represented by region of equilibrium form ii in Figure 3), the supplier’s
ex ante compliance cost dominates the supplier’s ex post correction cost despite
reduced level of compliance effort. Hence, the lower bound for the wholesale price
for the supplier’s ex ante participation is greater than that for the supplier’s ex post
participation, thus allowing the buyer to squeeze the supplier’s profit to zero. A
key insight for the buyer from this policy is that, when auditing is not costly, by
committing to only the wholesale price the buyer is always able to justify a full
auditing effort. This ensures full overall sustainability compliance in the market,
resulting from the supplier’s upfront compliance and any subsequent corrective
action. However, when auditing is costly, the buyer can no longer justify a full
auditing effort in all cases; and the buyer loses the ability to ensure full overall
sustainability compliance when the auditing effort is partial.
Effect of Committing to Price Only
In this section we consider the metrics of the supplier’s compliance effort e, the
buyer’s auditing effort a, ensuing overall sustainability compliance in the market Osc,
the buyer’s profit πB, and the supplier’s profit πS in order to compare the benchmark
36
model (i.e., with no-commitment to either wholesale price or order quantity) with
the “commitment to only the wholesale price”. The overall sustainability compliance
in the market is represented by e + (1− e)a, which also accounts for the additional
compliance guaranteed by corrective action when a supplier fails the audit. So, it
measures the likelihood of the buyer’s product not experiencing a market disruption
(i.e., loss of demand) on account of any sustainability violation in its supply base.
The change in each metric is measured based on the value of the corresponding metric
when one transitions from a policy of “no-commitment” to a policy of “commitment
to only the wholesale price”. The change is indicated by using a sign to show whether
it increased, decreased, or remained unchanged.
Based on Proposition 1 and Proposition 2, a comparison of “no-commitment”
policy with “commitment to only the wholesale price” policy for varying levels of
2 2
auditing cost coefficient γ generates three cases (i.e., 0 6 γ 6 (4β−α)α− ,
(4β−α)α <
4β(2β α) 4β(2β−α)
γ < αβ , and γ > αβ− − ). We first present the outcome of comparison under large2β α 2β α
γ (i.e., γ > αβ− ) in Proposition 3, as this case provides a super-set of all scenarios2β α
observed during the transition. We present details for the other cases of γ after
Proposition 3.
Proposition 3. Consider the situation wherein the buyer increases the level of
commitment and transitions from a “no-commitment” policy to a policy with
“commitment to only the wholesale price”. When γ is large (i.e., γ > αβ− ), the2β α
following metrics change as below for all p and D:
– The supplier’s compliance (e) increases monotonically with increase in level of
commitment as stipulated above.
37
– The buyer’s auditing effort (a) increases monotonically with increase in level
of commitment as stipulated above, except when β+γ− < D <
2βγ and p >
p c pα
2cβγ
− − .(2β α)γ αβ
– The overall sustainability compliance Osc increases monotonically with increase
in level of commitment as stipulated above, except when β+γ < D < 2βγ− andp c pα
p > 2cβγ
(2β− .α)γ−αβ
– The buyer’s profit πB and the supplier’s profit πS increase monotonically with
increase in level of commitment as stipulated above.
FIGURE 4. From “no-commitment” to a “commitment to only the wholesale price”
when γ > αβ
2β−α
where line 1 belongs to “no-commitment” model and represents D = β+γp−c . Other lines belong
to “commitment to only the wholesale price”, where line 2 represents D = (4β−α)α+4βγ4β(p−c) , line 3
represents D = 2βγpα , and line 4 represents D = D̂(p, c, α, β, γ).
Figure 4 provides a graphic representation of Proposition 3. We present
the change of metrics in Figure 4 when the auditing cost is high and the buyer
38
advances the level of commitment from “no-commitment” to “commitment to only
the wholesale price”. In region A under both above-mentioned commitment policies,
we do not observe contracting because of low level of demand and market price.
Therefore, all metrics remain unchanged.
In all regions where contracting occurs after committing to price, both the
buyer’s profit and the supplier’s compliance effort increase with the transition to
“commitment to only the wholesale price” policy. This transition also increases the
contracting opportunity. Region B and region E in Figure 4 are regions that now
have contracting. This new contracting opportunity also results in improvement of
overall sustainability compliance in these two regions. The supplier profit is squeezed
to zero in Region B. Interestingly, in region E, we observe a win-win-win situation
in which not only does the buyer’s profit improve, along with an improvement in
overall sustainability compliance; but also, the supplier makes a positive profit. The
intuition for this behavior was discussed in wholesale price setting stage , wherein
the buyer has to set the wholesale price high enough (ex ante) to give the supplier
incentive to participate in corrective action (ex post). In this case, the buyer is thus
unable to squeeze out the supplier’s profit to zero. In region C the buyer exerts
full auditing effort for both policies (with no change). The ensuing perfect auditing
results in maximum overall sustainability compliance for both policies. The supplier’s
compliance increases because committing to price is an incentive for the supplier.
With a commitment to only the wholesale price, the buyer experiences increased
profit by being able to command a lower wholesale price (vis-à-vis no-commitment
policy).
In region D we observe a financially win-win case for the buyer and the supplier,
who see increase in their respective profits. When the buyer commits to only the
39
wholesale price, unlike for the “no-commitment” policy, when in region D, the buyer
is unable to set the contract terms before auditing in a manner that can squeeze out
the supplier’s profit to zero. However, the overall sustainability compliance decreases,
despite an increase in the supplier’s compliance, since the buyer no longer exerts full
auditing effort.
Thus, in all cases where there is contracting, we observe that advancing the
level of commitment policy from “no-commitment” to “commitment to only the
wholesale price” increases the buyer’s profit. Additionally, the supplier’s compliance
effort also increases with the advancement in the level of commitment in all these
cases. However, the supplier’s profit only increases monotonically. In contrast, note
that there is no change in any of the metrics as one transitions from a policy of “no-
commitment” to “commitment to only the quantity” . Interestingly, the transition to
“commitment to only the wholesale price” policy vis-à-vis transition to “commitment
to only the quantity” has an asymmetric effect on sustainability and profit metrics.
Other Cases (i.e., for 0 < γ 6 (4β−α)α
2 2
− and
(4β−α)α < γ < αβ− − ) for Comparing4β(2β α) 4β(2β α) 2β α
Transition from “no-commitment” policy to “Commitment to only the Wholesale
Price” policy
In region A under both above-mentioned commitment policies, we do not observe
contracting because of low level of demand and market price. Therefore, all metrics
remain unchanged.
In all regions where contracting occurs, both the buyer’s profit and the supplier’s
compliance effort increase with the transition to “commitment to only the wholesale
price” policy. This transition also increases the contracting opportunity. Region B
in Figure 5 is the region that now has contracting. This new contracting opportunity
40
FIGURE 5. From “no-commitment” to “commitment to only the wholesale price”
(4β−α)α2when 0 < γ 6
4β(2β−α)
where line 1 belongs to “commitment to only the wholesale price” model and represents
D = (4β−α)α+4βγ4β(p−c) . Line 2 belongs to “no-commitment” and represents D =
β+γ
p−c .
also results in improving overall sustainability compliance in these two regions. The
supplier profit is squeezed to zero in Region B. In region C the buyer exerts full
auditing effort for both policies (hence, no change). The ensuing perfect auditing
results in maximum overall sustainability compliance for both policies. The supplier’s
compliance increases because committing to price is an incentive for the supplier.
With a commitment to only the wholesale price, the buyer experiences increased
profit by being able to command a lower wholesale price (vis-à-vis no-commitment
policy).
2
When auditing cost coefficient is moderate (i.e., (4β−α)α− < γ <
αβ ), we
4β(2β α) 2β−α
observe regions A, B, and C with above-mentioned characteristics in Figure 6.
We also observe a new region which represents the new contracting opportunity.
Interestingly, in this region, region D, we observe a win-win-win situation in which
41
FIGURE 6. From “no-commitment” to a “commitment to only the wholesale price”
when (4β−α)α
2
< γ < αβ
4β(2β−α) 2β−α
where line 1 belongs to “no-commitment” model and represents D = β+γp−c . Other lines belong
to “commitment to only the wholesale price”, where line 2 represents D = (4β−α)α+4βγ4β(p−c) , line 3
represents D = 2βγpα , and line 4 represents D = D̂(p, c, α, β, γ).
not only does the buyer’s profit improve along with an improvement in overall
sustainability compliance but also the supplier makes a positive profit. The intuition
for this behavior was discussed in §2.5, wherein the buyer has to set the wholesale
price high enough (ex ante) to give the supplier incentive to participate in corrective
action (ex post). In this case, the buyer is thus unable to squeeze out the supplier’s
profit to zero.
Thus, in all cases where there is contracting, we observe that advancing the level
of commitment policy from no-commitment to commitment to only the wholesale
price increases the buyer’s profit as well as the supplier’s compliance effort. However
the supplier’s profit only increases monotonically with this transition.
42
Full Commitment Policy
We investigate the effect of the buyer’s full commitment (i.e., the buyer commits
to both wholesale price and quantity) in this section. First , we present and
analyze the “full commitment” policy. Subsequently , we compare this policy to
“commitment to only the wholesale price”, in order to characterize the effect of also
committing to quantity when the buyer has already committed to only the wholesale
price.
The Buyer’s Commitment to Price and Quantity
In this section, we analyze the policy wherein the buyer commits to both the
wholesale price and the order quantity in the contract (i.e., full commitment). The
buyer now decides contract terms (i.e., the wholesale price w and order quantity q)
before the buyer and the supplier exert their respective efforts. The buyer commits
to wf = wp = w and qf = qp = q to be invoked upon execution of the contract.
We analyze the model backward, by first solving for the Nash Equilibrium of the
buyer’s and the supplier’s efforts. Then we analyze the buyer’s problem to choose
the optimal wholesale price and order quantity.
Sub-Game Perfect Nash Equilibrium of Auditing and Compliance
The buyer and the supplier simultaneously choose their respective efforts a
and e to maximize their respective expected profits: ΠB(wp, qp, wf , qf , e, a) and
ΠS(wp, qp, wf , qf , e, a). In the buyer’s and the supplier’s profit functions, wp = wf =
w and qf = qp = q are considered as given at this stage. Details of solution procedures
for finding the best response functions are relegated to the Appendix. We present
the equilibrium outcome in Lemma 5.
43
Lemma 5. Consider the situation wherein the buyer commits to both wholesale price
and order quantity. At the equilibrium, the supplier’s compliance effort e∗ and the
buyer’s auditing effort a∗ are in one of the following cases:
– If 0 6 q < γ , then (e∗, a∗) = (0, 0),
p
– If γ 6 q 6 2γβ , then (e∗, a∗) = (1− γ , αpq ), and
p pα pq 2γβ
– If q > 2γβ , then (e∗, a∗) = (1− α , 1).
pα 2β
Lemma 5 shows that for a low level of order quantity, the supplier has no
willingness to exert any level of compliance effort. The supplier exerts compliance
effort when the order quantity is beyond a threshold (i.e., q > γ ). Rearranging the
p
above cases based on the auditing cost coefficient γ, we can see that the buyer does
not exert any auditing effort when γ is beyond a threshold (i.e., γ > pq). The buyer’s
auditing effort increases with a decrease in γ.
Optimal Contracting Decisions
We move one step backward in time to solve for the buyer’s decision of (w, q).
The optimization program (2.14) shows that the buyer chooses the wholesale price
w and quantity q to maximize its profit-to-go, which is presented in function (2.6),
subject to the supplier’s participation in the contracting. The supplier’s participation
constraint is determined by the supplier’s profit-to-go, which is presented in function
(2.8). Note that for these functions, we assume that wf = wp = w and qf = qp = q.
max ΠB(wp, qp, wf , qf , e
∗, a∗)
06q6D
c6w6p (2.14)
subject to Π (w , q , w , q , e∗ ∗S p p f f , a ) ≥ 0
44
We present the buyer’s optimal decisions, along with model equilibrium, in
Proposition 4.
Proposition 4. Consider the situation wherein the buyer commits to both wholesale
price and quantity. At the equilibrium, the buyer’s optimal wholesale price w∗ and
order quantity q∗, the supplier’s compliance effort e∗ and the buyer’s auditing effort
a∗, and the buyer’s and the supplier’s respective total profits π∗ and π∗B S in the
following regions are:
– Form (i): no-contracting.
– Form (ii): (w∗, q∗) = (c+ (4β−α)α , D), (e∗, a∗) = (1− α , 1), and π∗B > 0, π∗4βD 2β S =
0.
– Form (iii): (w∗, q∗) = (c + α − αγ , D), (e∗, a∗) = (1 − γ , αDp2 ), and π∗D 2pD Dp 2γβ B >
0, π∗S = 0.
Above forms are defined based on following conditions:
2 2
– 0 < γ 6 (4β−α)α or p < 8cγβ
4β(2β−α) 4β(2β−α)γ−(4β− :α)α2
- If 0 6 D < (4β−α)α+4βγ , then form (i) occurs.
4β(p−c)
- Otherwise, form (ii) occurs.
2 2
– When γ > (4β−α)α and p > 8cγβ
4β(2β−α) 4β(2β−α)γ−(4β−α)α2 :
- If 0 6 D 6 D̃(p, c, α, β, γ), then form (i) occurs.
- If D > 2γβ , then form (ii) occurs.
pα
- If D̃(p, c, α, β, γ) < D < 2γβ , then form (iii) occurs,
pα
√
p(α+γ)+ p(2cαγ+p(α2+γ2))
where D̃(p, c, α, γ) =
2p(p− .c)
45
The graphical representation for Proposition 4 (for the full commitment policy)
is identical to Figure 3, which provides a graphical representation for Proposition
3 (commitment to only the wholesale price), with one difference. Line 3 (as seen
in Figure 3) for Proposition 4 would now represent D = D̃(p, c, α, β, γ), instead.
Thus, after accounting for the one aforementioned difference, the conditions for
the corresponding regions for Proposition 4 are identical to that for Proposition
3. The supplier’s compliance effort and the buyer’s auditing effort are identical in
corresponding regions. The rationale for the buyer’s and the supplier’s efforts in
corresponding regions is similar for both policies. Therefore, for the sake of brevity,
we do not discuss the behavior in these regions again.
In all regions wherein contracting exists, the buyer makes a positive profit, and
the supplier’s profit is now squeezed out to zero. The buyer now has to only consider
the ex ante participation constraint for the supplier and is thus able to set the
wholesale price and quantity in a manner that squeezes out the supplier’s profit to
zero, even in region of equilibrium form iii.
Effect of Committing to Quantity in Addition to Price
In this section we consider the metrics that we introduced in §2.5 to compare
the commitment to only the wholesale price policy with the full commitment policy.
Based on Proposition 2 and Proposition 4, a comparison of commitment to only the
wholesale price policy with full commitment policy for varying levels of auditing cost
coefficient γ generates two cases. We present the change of metrics for these two
cases in Proposition 5.
Proposition 5. Consider the situation wherein the buyer increases the level of
commitment and transitions from a “commitment to only the wholesale price” policy
46
to a policy with “full commitment”. The following metrics change as below for all p
and D:
When γ 6 (4β−α)α
2
− ,4β(2β α)
– The supplier’s compliance e, the buyer’s auditing effort a, the overall
sustainability compliance Osc, and the buyer’s and the supplier’s respective
profits πB, πS do not experience any change.
When γ > (4β−α)α
2
,
4β(2β−α)
– The supplier’s compliance e, the buyer’s auditing effort a, the overall
sustainability Osc, and the buyer’s profit πB increase monotonically in level
of commitment, as stipulated above.
– The supplier’s profit πS does not change, except it decreases when
2
D̃(p, c, α, γ) < D < D̂(p, c, α, β, γ) and p > 8cγβ .
4β(2β−α)γ−(4β−α)α2
When the level of commitment is advanced from commitment to only the
wholesale price to full commitment policy, if the auditing cost coefficient γ is low,
the equilibrium outcomes for the two policies are identical. Hence, this transition
does not change any of the metrics considered. In Figure 7, we provide a graphic
representation of Proposition 5 when γ is large. Two regions (i.e., A and B) exhibit
a behavior that is of significant interest to the buyer. In region A, unlike under
commitment to only the wholesale price policy, the buyer is now able to squeeze
the supplier’s profit to zero under full commitment policy. The compliance and
auditing efforts remain unchanged. Interestingly, in this transition the buyer is
able to command a lower wholesale price that leads to higher profit, even without
any change in compliance or auditing efforts. This transition also helps to increase
the contracting opportunity. Region B represents the new contracting opportunity,
47
FIGURE 7. From “commitment to only the wholesale price” to “full commitment”
2
when (4β−α)α− < γ.4β(2β α)
Line 1 and 2 belong to both commitment policies and represent D = (4β−α)α+4βγ4β(p−c) and D =
2γβ
pα
respectively. Line 3 belongs to commitment to only the wholesale price policy and represents
D = D̂(p, c, α, β, γ). Line 4 belongs to full commitment and represents D = D̃(p, c, α, γ).
wherein both parties exert partial effort and the buyer is able to squeeze the supplier’s
profit to zero. The equilibrium outcomes in regions C and D are identical for both
policies. Hence, this transition does not change any of the metrics in these two
regions. An important counter-intuitive managerial insight for the buyer is that
an ability to fully commit to contract terms (vis-à-vis price only) can improve the
contracting opportunity and ensuing profit only when the auditing is costly.
In summary, we find that committing to quantity is valuable for the buyer only
if the buyer has already committed to the wholesale price. Committing to the price
and committing to the quantity are complementary strategies for the buyer. When
the buyer commits to the quantity, the supplier is worse off only if the buyer has
already committed to the wholesale price. Thus, in contrast, the buyer’s strategies of
48
commitment to price and commitment to quantity have the effect of being substitutes
for the supplier. Lastly, the overall sustainability compliance from the buyer’s
commitment to quantity improves only if the buyer has already committed to the
wholesale price.
We present the analysis of the effect of raising the standard of the code of
conduct on sustainability and financial metrics, and ensuing discussion in the next
section.
Effect of Raising the Standard for the Code of Conduct
In addition to increasing suppliers’ compliance effort and overall sustainability
compliance to an existing standard, brands strive to progressively improve
sustainability standards for their suppliers’ processes and facilities. They do so
to meet the increasing expectations for sustainability in the market by increasing
the scope of the code of conduct and/or strengthening the standard for the code
of conduct (Nike, 2018; Apple, 2018). Next, we examine the effect of raising the
standard for the code of conduct on all metrics of interest in this study.
Model Set-up and Notations
We introduce d to represent a quantifiable proxy for the standard of the code
of conduct for the supplier. A higher d implies a higher standard of the code of
conduct. We consider d to be a continuous variable, to model the notion of raising
the standard, and also to modify the definition of cost functions associated with
compliance, auditing, and correction. We modify the supplier’s compliance cost to
Ks(e, d) = α(d)×e, the supplier’s correction cost to Kc(e, d) = β(d)×(1−e), and the
buyer’s auditing cost to Kb(a, d) = γ(d)×a. We allow for more generalized definitions
49
of cost coefficient functions and only assume that α(d), β(d), and γ(d) are increasing
in d. By substituting α(d) for α, β(d) for β, and γ(d) for γ in model sections (i.e.,
§2.4 to §2.6) we are able to obtain the results for this new set-up using the previous
analysis. Next, we develop additional insights regarding when the buyer raises the
standard of the code of conduct (hereafter simply referred to as “standard”).
Results and Discussions
One can see that if the standard increases, the contracting opportunity decreases
for all commitment policies (i.e., region of no-contracting expands). Therefore, we
focus on analyzing how an increase in the standard affects the supplier’s and the
buyer’s effort and the overall sustainability. These metrics remain invariant to an
increase in the standard under “no-commitment” and “commitment to only quantity”
policies. Hence, we focus on the implications for regions of equilibrium forms ii and
iii (i.e., where contracting occurs) under the policies of “commitment to only the
wholesale price” and “full commitment”. In region of equilibrium form ii, we analyze
the implications for compliance effort to a change in standard, since only the supplier
exerts partial effort in this region; and the overall sustainability is maximum, because
of full auditing effort in this region. In region of equilibrium form iii, both the buyer
and the supplier exert partial efforts, with the overall sustainability compliance being
less than the maximum. Hence, for region of equilibrium form iii, we analyze the
effect of raising the standard on the buyer’s auditing, the supplier’s compliance, and
the overall sustainability compliance. We present the results in Proposition 6.
Proposition 6. Consider the situation wherein the buyer commits to “only the
wholesale price” or to “both wholesale price and quantity”. Also, consider associated
50
regions with these commitment policies (referring to regions of equilibrium forms ii
& iii in Propositions 2 and 4).
I) In region of equilibrium form ii, the supplier’s compliance effort is increasing
′ ′
in the standard, if and only if β (d) > α (d) . The buyer’s effort is constant in
β(d) α(d)
the standard.
II) In region of equilibrium form iii, the buyer’s auditing effort is increasing in the
′ ′ ′
standard, if and only if 0 < γ (d) < α (d) − β (d) . Also, the supplier’s compliance
γ(d) α(d) β(d)
effort is always decreasing in the standard.
III) The overall sustainability compliance in region of equilibrium form iii is
′ ′ ′
increasing in the standard, if and only if Dp > 2β(d)γ (d) α (d) β (d)α′(d) β′− (d)
and > .
α(d)( ) α(d) β(d)
α(d) β(d)
In region of equilibrium form ii, the buyer always exerts full auditing effort.
Thus, due to an increase in compliance cost and correction cost coefficients with
an increase in standard, the supplier’s equilibrium compliance effort has to balance
any savings from a reduction in ex ante compliance effort with an increase in ex
post correction cost on account of increased chance of failing the audit. Hence, the
supplier’s compliance effort in equilibrium decreases (increases) with an increase in
standard when the compliance cost is more (less) sensitive to an increase in standard
relative to the correction cost. This offers a valuable insight for the buyer as to when
to expect an increase in the supplier’s compliance effort with an increase in standard,
without any change in auditing effort.
One can see from Propositions 2 and 4 that in region of equilibrium form iii, the
supplier’s effort decreases with an increase in the standard. The buyer can increase
the auditing effort with an increase in standard only in situations wherein the supplier
still finds it beneficial to participate in corrective action, despite an increase in the
51
likelihood of failing the audit. The supplier does so when compliance cost is more
sensitive to an increase in the standard relative to the correction cost. The buyer is
able to do so when increased costs of auditing and the compensation for the supplier’s
correction are more than offset by the savings from market disruption prevented by
the supplier’s corrective action. Hence, the buyer’s auditing effort increases in the
′ ′ ′
standard, if and only if 0 < γ (d) < α (d) − β (d) .
γ(d) α(d) β(d)
Our analysis shows that in region of equilibrium form iii, the overall
sustainability compliance increases with an increase in the standard, if and only
if the market conditions are sufficiently favorable (i.e., the demand or the market
price is high) and the sensitivity of correction cost coefficient to increase in the
standard is less than that for compliance cost coefficient. The supplier finds it more
beneficial to undertake corrective action if compensated for it rather than having to
increase compliance effort. Only when the market is favorable enough is the buyer
able to compensate the supplier for corrective action. The requisite threshold for
favorability of market conditions increases as the relative difference in sensitivity of
the supplier’s cost coefficients decreases. The market thus needs to be even more
favorable when the supplier’s sensitivity for correction cost is not sufficiently lower
than that to compliance cost.
When one is in the region of equilibrium form ii, and sufficiently far from the
boundary (i.e., line D = 2βγ ), the nature of the results are not affected by an
pα
increase in the standard. However, the nature of results changes for points close to
the boundary, as described in Proposition 7 below.
Proposition 7. Consider a situation wherein the buyer commits to only the
wholesale price and you are in the region of equilibrium form ii of Proposition 2.
Select a point very close to and above the line D = 2βγ such that, with an infinitesimal
pα
52
increment in d, the point would now be in the region of equilibrium form iii, if and
′ ′ ′
only if α (d) − β (d) < γ (d) .
α(d) β(d) γ(d)
Proposition 7 states that when the auditing cost is sufficiently sensitive to an
increase in the standard, for the points considered above, the equilibrium behavior
moves from region of equilibrium form ii to region of equilibrium form iii of
Proposition 2 (i.e., for committing to only the wholesale price). This has several
implications. The overall sustainability compliance decreases. The buyer is no longer
able to squeeze the supplier’s profit to zero.
FIGURE 8. Effect of Raising the Standard of Code of Conduct on Supply chain
profit
Considering linear cost function and changing the standard from d = 1 to d = 1.01, with
α = 0.2, β = 0.8, γ = 0.6, c = 0.1, p = 0.5
Additionally, even though both the buyer and the supplier now have positive
profits, our numerical analysis (Figure 8) shows that the buyer is worse off, also,
with a decrease in profit. We know that from Proposition 2, the supplier’s profit
53
in region of equilibrium form ii is zero. As we go to region of equilibrium form
iii, it increases to a positive profit. Therefore, the buyer’s profit in this transition
decreases. In situations wherein correction cost is more sensitive to an increase in
the standard vis-à-vis compliance cost, the buyer is always worse off in the above-
mentioned scenario, regardless of the sensitivity of the auditing cost. However, when
the compliance cost is more sensitive, then the buyer is worse off only when the
sensitivity of the auditing cost is larger than the sensitivity differential between
compliance and correction cost coefficients. This provides a cautionary insight for
the buyer who is considering raising the standard of the code of conduct.
The proofs of all Lemmas and Propositions are provided in Appendix.
Summary & Conclusion
We investigate the efficacy of the buyer’s commitment to contract terms as
yet another tool for the buyer that has not been considered in the literature to
enhance a supplier’s social and environmental compliance. Thus, with a different
focus, we investigate the effect of the interplay between degree of commitment to
an endogenized wholesale contract and the buyer’s auditing effort on the supplier’s
compliance to the code of conduct and ensuing overall sustainability compliance
in the market. We consider a game-theoretic framework to investigate the impact
of policies with varying degrees of commitment on the supplier’s compliance,
sustainability in the marketplace, and the buyer’s and the supplier’s profits. We
solve for the subgame-perfect equilibrium and characterize the contract terms, the
buyer’s auditing effort, and the supplier’s compliance effort at the equilibrium. We
also examine the sensitivity of the supplier’s compliance and overall sustainability
compliance to an increase in the standard for the code of conduct.
54
The interplay between the buyer’s auditing cost and the supplier’s compliance
and correction costs under varying levels of commitment provides significant
managerial insights. Our results show that the commitment provides the buyer with
significant opportunity to advance both social good and profit in a win-win manner.
We find that increasing the level of commitment improves the supplier’s likelihood
of compliance to the sustainability standard. We show that overall sustainability
compliance in the market also improves, in a great majority of cases, with an increase
in the commitment level. Our results indicate that both contracting opportunity and
profit for the buyer increase monotonically with the degree of commitment.
When the buyer commits to only the wholesale price and auditing is not costly,
we find that the buyer is always able to justify a full auditing effort. This ensures full
overall sustainability compliance in the market, resulting from the supplier’s upfront
compliance and any subsequent corrective action. However, when auditing is costly,
the buyer can no longer justify a full auditing effort in all cases and loses the ability to
ensure full overall sustainability compliance. In cases wherein the buyer exerts partial
auditing effort, we find that in spite of experiencing high auditing cost, the buyer
can improve the contracting opportunity and ensuing profit by fully committing to
contract terms. This offers an important counter-intuitive insight for the buyer.
Interestingly, the transition from “no-commitment” to “commitment to only
the wholesale price” policy vis-à-vis a transition to “commitment to only the
quantity” has an asymmetric effect on sustainability and profit metrics. The
overall sustainability compliance from the buyer’s commitment to quantity improves
only if the buyer has already committed to the wholesale price. With regard to
profits, we find that committing to quantity is valuable for the buyer only if the
buyer has already committed to the wholesale price. Committing to the price and
55
committing to the quantity are complementary strategies for the buyer. When the
buyer commits to quantity, the supplier is worse off only if the buyer has already
committed to the wholesale price. Thus, the buyer’s strategies of commitment to
price and commitment to quantity have the effect of being substitutes for the supplier.
Additionally, when the buyer transitions from “no-commitment” to “commitment to
only the wholesale price” policy, we find a win-win-win scenario in which not only
does the buyer’s profit improve along with an improvement in overall sustainability
compliance but also the supplier makes a positive profit. Lastly, we identify
conditions in which, interestingly, the supplier’s compliance and overall sustainability
compliance increase with an increase in the standard for the code of conduct. We also
provide a cautionary insight for the buyer who is considering raising the standard of
the code of conduct. Going beyond our focus in this study, it will be beneficial for
future research to examine the effect of competition on the efficacy of commitment.
Bridge to Next Chapter
In this chapter, we consider a buyer who enjoys the pricing power and also
has the ability to commit to contract terms. We investigate how such a buyer’s
commitment to contract terms affects the sustainability and financial performance
of the supply chain. In the next chapter, we consider a different scenario wherein
the buyer and the supplier(s) have more parity in their contracting power, with the
supplier offering the price. Hence, the buyer’s power is limited. To consider a more
realistic business environment, we also introduce the supplier competition to our
study.
56
CHAPTER III
IMPACT OF SUPPLIER COMPETITION ON SUPPLIER’S SOCIAL AND
ENVIRONMENTAL COMPLIANCE
Introduction
In the previous chapter, we investigated the role of buyer’s commitment to
contract terms in enhancing supplier’s social and environmental compliance. This
represented buyer-supplier dyads wherein the buyer enjoys significant pricing power
in determining the wholesale price and order quantity for the supplier. These same
buyers may not enjoy such pricing power across their other business units. For
example, while major brands in the footwear industry such as Nike and Adidas
enjoy significant pricing power on their captive suppliers in the footwear industry,
these brands share their suppliers for their apparel business with several competitors.
Hence, there is more pricing parity enjoyed by their apparel suppliers. Smaller and
upcoming brands do not even enjoy the pricing power with their major suppliers
since they often account for a small fraction of their supplier’s business. Thus,
it is important to understand how a buyer can influence the supplier to be more
sustainable when a supplier enjoys the pricing power. In this situation, the buyer
can induce competition between suppliers in conjunction with the use of auditing to
influence sustainability of suppliers.
Hence, the second essay focuses on understanding the impact of supplier
competition on the buyer’s ability to influence suppliers’ compliance when suppliers
have more parity in the contracting power. Unlike the first essay, wherein the
buyer stipulates both price and quantity, this essay considers situations wherein the
57
supplier offers a wholesale price and the buyer is limited to only offering the quantity
in a wholesale contract. Depending on whether auditing precedes contracting or
follows contracting, it determines whether suppliers first compete on their compliance
efforts or wholesale prices. We develop a framework to investigate if and how the
sequence in which the supplier competition manifests, influences buyers auditing
effort, suppliers compliance efforts, and financial performance of parties. Based on
the above discussion, we ask following research questions:
1) How does competition between suppliers affect the supplier’s compliance to a code
of conduct and the financial performance of the buyer and suppliers?
2) How do timing of contracting and auditing affect the buyer’s auditing, the
suppliers’ compliance, and the financial performance of the parties?
We introduce a framework to investigate the above research questions. This
framework has two dimensions. In one dimension, we consider the timing of
contracting: Either auditing precedes contracting or auditing follows contracting.
The second dimension presents the structure of supplier base: Single supplier (as
a benchmark) or two competing suppliers. When we consider both dimensions
together, it creates four different scenarios. We show these four scenarios in Figure 9.
The single supplier model is a benchmark model (i.e., we have one buyer who sources
from a supplier) under each timing. We also consider under each timing the scenario
wherein the buyer can source from two competing suppliers. We can compare these
four scenarios in two different ways. First in each timing of contracting, we compare
the benchmark model with the supplier competition model. This comparison shows
the effect of supplier competition on the supplier’s compliance, the buyer’s auditing,
and the financial performance of parties. The second comparison happens within each
structure of supplier base. For instance in a single supplier, we compare model 1 with
58
model 3. It shows the effect of timing of contracting on the supplier’s compliance,
the buyer’s auditing, and the financial performance of parties. By analyzing this
framework, we provide a guideline to a buyer who does not have the leverage to set
the wholesale price, but he can manage the timing of contracting or the structure of
supplier base.
FIGURE 9. Comprehensive Research Framework
In this chapter, we present and analyze models for when auditing precedes
contracting. The analysis for the scenario wherein the contracting precedes auditing
is in progress. We present multi-stage games to analyze models analytically. We
solve for the subgame-perfect equilibrium and characterize the contract terms, the
buyer’s auditing effort, and the supplier’s compliance effort at the equilibrium.
The remainder of this chapter is organized as follows. First, we review the
relevant literature. Then, we introduce the model set-up. After that, we analyze the
single supplier scenario wherein auditing precedes contracting. Next, we investigate
the effect of supplier competition in the above case wherein the auditing precedes
contracting. Then, we compare the above mentioned scenarios to understand the
effect of competition when auditing precedes contracting. Finally, we provide
concluding remarks.
59
Literature Review
Our paper is related to research focused on examining the efficacy of sourcing
strategies, supply chain structures, incentives and penalties in contract design,
auditing schemes, and competition between suppliers as levers for enhancing
suppliers’ social and environmental responsibility. We have already presented the
literature review for the efficacy of sourcing strategies, supply chain structures,
incentives and penalties in contract design, and auditing schemes in the previous
chapter. Here, we review the related literature for competition and point out our
differentiation with it.
One stream of research has focused on investigating the effect of suppliers’
competition in influencing a supplier’s social or environmental compliance. Supplier
competition has been addressed in a variety of papers such as Cachon and Zhang
(2007) and Jin and Ryan (2012). What differentiates our research from these papers
in supply chain management literature is that the wholesale price decision is a
decision for suppliers; and more importantly, the compliance effort of the supplier is
a continuous decision variable, and the consequence of violation is not modeled like
defects in quality literature. In sustainability literature, Karaer et al. (2017) study
the effect of offering a wholesale price premium by the buyer and sharing the cost
of a supplier’s sustainability effort in enhancing the supplier’s ability to produce a
more environmentally responsible product. Under a single supplier setting, they find
that if it is optimal for the buyer to offer a price premium, then the buyer also fully
subsidizes the supplier’s cost for investing in environmental quality. Under supplier
competition, they assume that suppliers are identical except production cost. Also,
they assume that there already exists a wholesale price contract between the buyer
and the supplier, and the buyer is only willing to share costs with the incumbent
60
supplier. However, they do not consider the order quantity allocations and auditing in
their study. Also, the buyer offers the wholesale price in this setting. They find that
as the competition is introduced, cost-sharing is less effective as a lever. Agrawal and
Lee (2017) study the effect of sourcing policies in influencing a supplier’s decision to
adopt sustainable practices. They also investigate the effect of competition between
suppliers who invest in the cost of sustainability. Under this setting, suppliers do
not have the pricing power and the buyer can choose different sourcing policies (i.e.,
sustainable or conventional). They find that the buyer does not find it profitable to
deter a supplier in the presence of supplier competition. However, in our study the
suppliers compete together to quote wholesale prices. Also, we consider the auditing
as a tool that buyer can use to detect any noncompliance.
In contrast to the extant literature, we focus on understanding the efficacy of
timing of contracting, and of competition between suppliers who have the pricing
power, on enhancing sustainability compliance in the market. To the best of our
knowledge, this issue has not been studied in the sustainability literature. Thus,
with a different focus, we investigate the effect of the interplay between different
timing of contracting and competition to an endogenized wholesale contract, and
the buyer’s auditing effort, on the supplier’s compliance to the code of conduct. We
solve for the subgame-perfect equilibrium and characterize the contract terms, the
buyer’s auditing effort, and the supplier’s compliance effort at the equilibrium.
Model Setting
We consider a supply chain in which a buyer sources a product from two suppliers
and then resells the product to the consumer market. It costs ci per unit for the
supplier i to manufacture the product, where i=1,2. Suppliers are not symmetric
61
in production cost, and the second supplier is more costly; i.e., c1 < c2. The buyer
orders quantity qi to supplier i where i = 1, 2. The market for the product has the
demand size of D and the selling price of p. For focusing on the sustainability aspect,
we assume p to be exogenous and D to be deterministic. The buyer and suppliers
coordinate the business by a simple wholesale price contract. This contract has a
wholesale price and an order quantity. We assume the buyer’s order quantity satisfies
0 ≤ qi ≤ D. Two suppliers compete by quoting wholesale prices to the buyer. The
wholesale price that the supplier i offers to the buyer is wi, where i = 1, 2. We
assume that ci ≤ wi ≤ p.
The buyer uses a supplier code of conduct to set the social and environmental
standards for suppliers. The buyer audits suppliers for social and environmental
compliance based on the code of conduct. Consumers, the buyer, and suppliers have
the same understanding of what defines a violation of this social and environmental
code of conduct. The customers expect that the buyer sources from compliant
suppliers. If the buyer sources from a non-compliant supplier, the market will react
negatively to this buyer’s practice.
We assume that suppliers are initially non-compliant and that they can exert
efforts to be compliant with the code of conduct. The outcome of supplier’s
compliance effort is random: The supplier gets to be compliant with the probability
ei and gets to continue to be non-compliant with the probability 1 − ei. We can
explain with this perspective: That a higher compliance effort leads to a higher
probability of being compliant. This is consistent with literature, where probability
ei is used as a proxy for the supplier’s compliance effort (Plambeck and Taylor, 2016).
The supplier incurs the compliance effort cost Ks(ei) as soon as the supplier exerts
62
the compliance effort ei. The cost Ks(ei) for all ei ∈ [0, 1] is continuous and strictly
increasing in ei with Ks(0) = 0 and is differentiable.
The buyer cannot observe the suppliers’ true compliance status. Therefore, the
buyer has to audit suppliers to learn about suppliers’ compliance status, but only
imperfectly. The supplier does pass the audit successfully, if the supplier is compliant
with the code of conduct. If the supplier is not compliant with the code of conduct,
the auditing process will have a random outcome: With probability ai the buyer
catches the non-compliant supplier, and the supplier fails the audit. Also, with
probability 1−ai, the buyer is not able to find any noncompliances, and the supplier
passes the audit. The buyer’s probability of accurately identifying a non-compliant
supplier increases with the level of auditing effort. The buyer’s auditing effort has
the cost of Kb(ai), which is continuous and strictly increasing in ai ∈ [0, 1], where
Kb(0) = 0 and is differentiable.
After suppliers exert their compliance efforts, and the buyer’s auditing is done,
the buyer observes “pass” with probability ei+(1−ei)(1−ai) or “fail” with probability
(1 − ei)ai. We assume that the outcome of compliance effort is not observable by
the supplier, unless the supplier exerts zero or full effort. We come up with two
possible explanations for this. One reason is that the supplier lacks the ability to
audit internally. Perhaps another way to think about it is that a suppliers effort
includes both the compliance effort and any associated internal auditing to reveal
compliance status. However, the suppliers effort still only provides him with a chance
of compliance, and not the true state, as it is never perfect unless one exerts full effort.
We show the probability tree related to buyer-supplier interaction in Figure 10.
Furthermore, if the buyer’s audit successfully catches a non-compliant supplier,
the buyer will not sign a contract with the non-compliant supplier. Instead, the buyer
63
FIGURE 10. Probability Tree of Buyer-Supplier Interaction
invokes a costly fall-back option of cost C, and the buyer collects (p − C)D where
ci < C ≤ p. We assume that C = p, and that, therefore, the buyer’s reservation
profit is zero. If a non-compliant supplier passes the audit, then consumers will
identify non-compliance of the supplier with the probability of 1 in a timely manner
via media or NGO reports. The market disruption is the reaction of consumers to
buyer’s sourcing from a non-complaint supplier. Since our focus is on examining
the effect of competition, we make a simplifying assumption that the buyer loses all
demand D, with probability 1, due to market disruption. If the buyer sources from
two suppliers, and if at least one of them is not complaint, the buyer will lose the
whole market. The non-complaint supplier who gets the order incurs a goodwill cost
GC.
Our model presents a one-period problem, in which the buyer and suppliers
contract for a one-time supply and take a sequence of actions. We divide the sequence
of actions into two stages: auditing stage and contracting stage. Auditing stage has
two phases. In the first phase of the auditing stage, the buyer exerts auditing efforts
a1 and a2 to audit each supplier. In the second phase of the auditing stage, suppliers
choose their compliance efforts e1 and e2 simultaneously. In the contacting stage,
first suppliers compete together to quote wholesale prices to the buyer. In the second
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phase of contracting stage, the buyer allocates order quantities q1 and q2 to each
supplier.
For tractability, we assume that the buyer’s auditing cost, Kb(a) and the
supplier’s compliance effort cost, Ks(e) are linear in a ∈ [0, 1] and in e ∈ [0, 1]
respectively. We assume Kb(a) = γ × ai and Ks(e) = αi × ei, where γ and αi
are cost coefficients and positive constants for i = 1, 2. We assume that it is less
costly to exert compliance effort for the supplier whose production cost is more
expensive, i.e., α1 > α2. Also, we assume that the goodwill cost is pretty high in
comparison with the supplier’s effort cost, when the supplier exerts full compliance
effort, i.e., GC >> αi. In addition the goodwill cost is bigger than maximum
profit channel; and subsequently is bigger than the supplier’s potential profit; i.e.,
GC >> D(p− ci) >> qi(wi − ci).
As we mention in our Introduction section, we define four different scenarios, so
that we can show the effect of competition and timing of contracting on suppliers’
compliance, buyer’s auditing efforts, and financial metrics. In these models, we
investigate the effect of different timing of contracting (i.e., auditing stage before or
after contracting stage), on the buyer’s auditing and profit, and on the suppliers’
compliance effort and profit. We define a benchmark model that has the same
timing of events just with one supplier. We compare the benchmark model with
the competition model which has two suppliers with the same sequence of events. In
the next sections, first we present the benchmark model for the Auditing precedes
Contracting sequence. Next, we present the competition model for this sequence of
action.
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Benchmark Model with a Single Supplier When Auditing Precedes
Contracting
We first analyze the scenario wherein the auditing happens prior to the
contracting stage. That is, the buyer decides the auditing effort, and then the supplier
chooses the compliance effort. Next, in the contracting stage, the supplier sets the
wholesale price, and in the second phase, the buyer sets the order quantity before
the execution of the contract. Figure 13 shows the sequence of events for this model.
FIGURE 11. Sequence of Actions in Single Supplier for Contracting Follows
Auditing.
We use the same parameters that we define in the model setup and omit the
indices of two suppliers, because this model is a single supplier model, i.e., ei = e,
ai = a, wi = w, and ci = c. We use this single supplier scenario as a benchmark
for further analysis. Notice that in this benchmark model that considers a single
supplier, one can still consider either a supplier who has a lower cost of production
and higher compliance cost or vice versa. We solve this benchmark model backward.
We start the analysis with the optimal order quantity decision.
Optimal Order Quantity Decision
We first analyze the buyer’s ordering decision in the contracting stage, which
happens after the supplier’s wholesale price decision. The buyer receives the signal
66
of “pass” or “fail” from the auditing stage. Therefore, the buyer is faced with two
possible situations: when the supplier passes and when the supplier fails the audit.
We define θ = Pr(Complaint|Pass). First, we present the buyer’s profit function
when the supplier passes the audit.
πPB(q) = θ(p− w)q + (1− θ)(−wq) (3.1)
The buyer’s profit function in (3.1) states that if the supplier who passes the audit is
complaint, the buyer makes profit of (p− w)q. If the supplier who passes the audit
is not compliant, then the buyer incurs the cost of sourcing. The buyer chooses the
order quantity q to maximize its profit:
max πPB(q) (3.2)
0≤q≤D
When the supplier fails the audit, the buyer does not order, therefore the buyer
makes zero profit. We assume that the buyer uses a fall-back option to meet the
demand. Lemma 6 presents the optimal order quantity decision of the buyer.
Lemma 6. Consider that the auditing precedes the contracting and the buyer faces
a single supplier. The buyer’s optimal order quantity is:
– When the supplier passes the audit,
∗ if θp ≥ w then the buyer’s optimal order is q∗ = D,
∗ otherwise the buyer’s optimal order is q∗ = 0.
– When the supplier fails the audit, the buyer does not order.
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Proof of Lemma 6. The objective function in the optimization problem (3.2) is linear
in q. We simplify it and it gives us (θp − w)q. When θp − w ≥ 0 then the buyer
orders q∗ = D, otherwise q∗ = 0.
Lemma 6 shows that when the supplier passes the audit and the supplier offers
a low wholesale price (less than the threshold), the buyer has the incentive to order
and sets the order as high as demand. When the buyer observes that the supplier
passes the audit and the supplier offers a high wholesale price, the buyer does not
order. Also, if the supplier fails the audit, the buyer will choose the fall-back option
for sourcing, and the contracting does not happen with the supplier. In the next
section we analyze the supplier’s optimal wholesale price decision.
Optimal Wholesale Price Decision
We go one step back in time and analyze the supplier’s problem for the wholesale
price. The supplier’s profit function depends on whether the supplier passes or fails
the audit. If the supplier passes the audit, the profit function is:
(w − c)q −GC , q > 0
πPS (w) = θ(w − c)q + (1− θ) (3.3)0 , q = 0
In the supplier’s profit function (3.3), if the supplier passes the audit and he is
complaint, then the supplier will collect (w−c)q. If the supplier passes the audit while
he is not complaint, then consumers discover the noncompliance with the probability
one and the supplier incurs a goodwill cost in that case, where he receives a positive
order. In this phase of the contracting stage, the supplier chooses the wholesale price
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w to maximize its profit.
max πPS (w)
c≤w≤p
(3.4)
subject to πPS (w) ≥ 0,
Lemma 7 presents the supplier’s optimal wholesale price under two cases of pass and
fail.
Lemma 7. Consider that the auditing precedes the contracting and the buyer faces
a supplier. The supplier’s optimal wholesale price decision is:
– When the supplier passes the audit, if θ < cD+GC then the supplier does
Dp+GC
not participate and there is no contracting. If θ ≥ cD+GC , then the supplier
Dp+GC
participates and sets the wholesale price w∗ = pθ.
– When the supplier fails the audit, there is no contracting.
Proof of Lemma 7. We simplify the supplier’s profit function when the supplier
passes the audit and write it as (w − c)D − (1 − θ)GC. Based on the supplier’s
participation constraint, if c + (1−θ)GC > θp ⇔ θ < cD+GC , then the supplier does
D Dp+GC
not participate and there is no contracting. If c+ (1−θ)GC ≤ θp⇔ θ ≥ cD+GC , then
D Dp+GC
the supplier participates and sets the wholesale price w∗ = pθ.
Lemma 7 shows that when the probability of being compliant for a supplier
who passes the audit, is low, then the supplier has no willingness to participate in
the contracting. When the probability of being compliant for a supplier who passes
the audit, is high, the supplier participates in the contracting and sets the wholesale
price in a way that squeezes out the buyer’s profit. We can also explain the result
based on demand in the market and goodwill cost. When the demand (goodwill cost)
is high (low) the supplier has the willingness to participate in the contracting and
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quote a wholesale price. When the demand (goodwill cost) is low (high) the supplier
knows that he cannot afford the cost of production and/or the goodwill cost in case
he gets caught by the buyer. In the next section we analyze the supplier’s optimal
compliance effort.
Optimal Compliance Effort Decision
We move one step backward in time and solve for the supplier’s compliance
effort. In this phase of the auditing stage, we find the supplier’s optimal compliance
effort for a given buyer’s auditing effort a. We use our finding in previous sections
to form the supplier’s expected profit in this phase. First, we present the supplier’s
profit function in this phase:
∗ ∗
ΠS(e) = Pr(Pass)π
P
S + Pr(Fail)π
F
S − αe (3.5)
From our analysis in the contracting stage, we know that if the supplier fails the
audit, the supplier will not receive any contract. Therefore, in profit function (3.5),
the second term is zero. The supplier chooses e to maximize its profit. We show this
maximization in optimization program (3.6) :
max ΠS(e)
0≤e≤1
(3.6)
subject to ΠS(e) ≥ 0,
Lemma 8 presents the supplier’s optimal compliance effort in this phase.
Lemma 8. Consider that the auditing precedes the contracting, and the buyer faces
a supplier. The supplier’s optimal compliance effort is:
– If D < α− , then e
∗ = 0.
p c
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– If D ≥ α ∗− , then e = 1.p c
Proof of Lemma 8. We express the supplier’s profit by plugging-in the optimal
solutions from contracting stage. The supplier’s profit is:
 (1−a) cD+GC−αe 0 ≤ e <
pD+GC
 1−a
cD+GC
Π (e) = pD+GCS
(1−a) cD+GC
[e+ (1− e)(1− a)][(pθ − c)D − (1− θ)GC]− αe pD+GC− cD+GC ≤ e ≤ 11 a
pD+GC
(3.7)
(1−a) cD+GC
The supplier’s profit is linear in e, and it is continuous at e = pD+GC− cD+GC . The slope1 a
pD+GC
of the first piece of supplier’s profit is always negative. Therefore, candidates for
optimal solution are e = 0 or e = 1. The value of the function at zero is zero and at
1 is D(p− c)− α. Therefore, if D < α− , then e
∗ = 0, otherwise e∗ = 1.
p c
Lemma 8 shows that when the demand is high, the supplier has the willingness
to exert full compliance effort. When the demand is lower than a threshold, the
supplier is faced with a negative profit, as the cost of compliance is bigger than its
profit. Therefore, the supplier has no willingness to exert the compliance effort. In
the next section, we analyze the buyer’s optimal auditing effort.
Optimal Auditing Effort Decision
We move one step backward in time and solve for the buyer’s optimal auditing
effort a∗. The buyer chooses a to maximize its profit. The outcome of auditing is
one of three different situations. If the supplier is compliant, the supplier passes the
audit. If the supplier is not compliant then either the supplier fails or passes the
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audit. We present the buyer’s profit-to-go in (3.8).
ΠB(a) =Pr(compliant and pass)[(p− w∗)q∗] (3.8)
+Pr(noncompliant and pass)(−w∗q∗)
+Pr(noncompliant and fail)× 0− γa
The buyer’s profit function in (3.8) is linear and decreasing in a, therefore the optimal
decision for the buyer is to exert zero auditing effort. Proposition 8 presents the
buyer’s optimal auditing effort, along with the sub-game prefect equilibrium of the
model.
FIGURE 12. Presentation of Model Equilibrium of Single Supplier on Plane of
Demand and Market Price.
Proposition 8. Consider that the auditing precedes the contracting and the buyer
faces a supplier. The buyer’s optimal auditing effort for the range of all available
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parameters is a∗ = 0. The sub-game prefect equilibrium of model for different range
of parameters is:
– If D < α− , then there is no contracting.p c
– If D ≥ α− , then (e
∗, w∗, q∗) = (1, p,D). Also, the supplier squeezes out the
p c
buyer’s profit to zero.
Proof of Proposition 8. We form the buyer’s profit function by plugging-in the
optimal solutions. The buyer’s profit function for all D < α α
p− is −γa and for D ≥c p−c
is (p−w)D− γa. Both functions are decreasing in a, therefore the optimal solution
is a∗ = 0.
Proposition 8 shows that when the buyer considers a single supplier and the
auditing precedes the contracting, if the demand in the market is high enough, then
the supplier exerts full compliance effort, and there is no need for the buyer to exert
any auditing effort. By considering the benchmark model, if the demand or market
price is low, then the supplier does not participate, and the buyer does not order.
Therefore, we have no contracting. When either the demand or the price is high,
the supplier exerts full effort, and the buyer exerts no auditing effort. The supplier
is able to squeeze out the buyer’s profit to zero. In this benchmark model, when
the buyer works with supplier 2 (i.e., the supplier with a higher production cost)
instead of supplier 1 (i.e., the supplier with a lower production cost), we observe
that the contracting opportunity increases when demand is low and price is high.
We can explain this behavior by looking at the production cost and compliance cost of
supplier 2. Recall that the production cost of supplier 2 is higher, and its compliance
cost is lower vis-à-vis that for supplier 1. Therefore, with the high market price, he
can collect a higher profit margin and exerts full compliance effort. We observe a
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similar behavior in favor of supplier 1, when the price is low and the demand is high.
By switching from supplier 2 to supplier 1, the contracting opportunity increases.
We present the model for supplier competition in the next section. We analyze the
model backwards, similar to our process with the benchmark model.
Model with Supplier Competition When Auditing Precedes Contracting
We analyze the scenario wherein the buyer has two suppliers for sourcing the
product and the auditing stage happens prior to contracting. That is, in the auditing
stage, suppliers decide the compliance efforts after the buyer sets the auditing efforts.
Then, in the contracting stage, suppliers first compete by quoting wholesale prices
to the buyer. Subsequently, the buyer allocates the order quantities between two
suppliers. Figure 13 shows the sequence of events for this model.
FIGURE 13. Sequence of Actions in Competition for Contracting Follows Auditing.
We analyze this model backward and solve for the sub-game perfect Nash
equilibrium, starting with the buyer’s order quantity decisions.
Optimal Order Quantity Decisions
We first analyze the buyer’s ordering decision, which is made after auditing is
concluded. Given the buyer’s and the suppliers’ efforts, respectively (a1, a2) and
(e1, e2), the buyer allocates the order quantity q1 to the supplier 1 and q2 to the
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supplier 2. This decision depends on the outcome of auditing: Suppliers either
“pass” or “fail” the audit. Therefore, we have four different scenarios, based on the
outcome of auditing. First, we consider the case that both suppliers pass the audit.
When both suppliers pass the audit: We show the buyer’s profit function
under this case with πPPB . This profit function depends on whether the supplier who
passes the audit is compliant. Based on this, we have the buyer’s profit-to-go as
following:
πPPB (q1, q2) =ΦC,C [(p− w1)q1 + (p− w2)q2]
(p− w2)q2, q 1
= 0, q2 > 0
+ΦNC,C−w1q1 − w2q2, q1 > 0, q2 > 0
(p− w1)q 1
, q1 > 0, q2 = 0
+ΦC,NC−w1q1 − w2q2, q1 > 0, q2 > 0
+ΦNC,NC(−w1q1 − w2q2) (3.9)
where ΦC,C = Pr(C,C|P, P ), ΦNC,C = Pr(NC,C|P, P ), ΦC,NC = Pr(C,NC|P, P ),
and ΦNC,NC = Pr(NC,NC|P, P ).
Conditional probabilities in the buyer’s profit function (3.9) express the status
of each supplier, given that they both pass the audit. We multiply these probabilities
by the profit that the buyer collects after execution of the contract. For instance,
for the second term in the buyer’s profit function (3.9), the conditional probability
Pr(NC,C|P, P ) represents the case that the first supplier who passes the audit is not
compliant, and that the second supplier who passes the audit is compliant. In this
case, if the buyer sources from the second supplier, the buyer will collect (p−w2)q2.
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Otherwise, the buyer incurs the sourcing cost, because the market reacts to the
non-compliance, and the buyer collects zero revenue.
In the case that both suppliers pass the audit, the buyer allocates order
quantities of (q1, q2) to maximize its expected profit:
max πPP
≤ ≤ ≤ ≤ B
(q1, q2)
0 q1 D, 0 q2 D (3.10)
subject to q1 + q2 6 D,
where πPPB (q1, q2) is defined in (3.9). Lemma 9 summarizes the buyer’s optional order
quantities when both suppliers pass the audit.
Lemma 9. Consider that the auditing precedes the contracting and the buyer faces
two suppliers who both pass the audit. The buyer’s optimal order allocations between
suppliers are:
– If w ∗ ∗2 < p(ΦC,C + ΦNC,C) and w2 6 w1 − p(ΦC,NC − ΦNC,C), then (q1, q2) =
(0, D),
– If w1 < p(Φ
∗ ∗
C,C + ΦC,NC) and w2 > w1 − p(ΦC,NC − ΦNC,C), then (q1, q2) =
(D, 0),
– Otherwise, (q∗, q∗1 2) = (0, 0).
Proof of Lemma 9. Given that the objective function and constraint in the
optimization problem (3.10) are linear in q1 and q2, the optimal solution happens
in the corner solutions. Therefore,
if πPPB (q1, q
PP
2)|(q1,q2)=(D,0) > πB (q1, q2)|(q1,q2)=(0,0) and
πPPB (q1, q2)| PP(q1,q2)=(D,0) > πB (q1, q2)|(q1,q2)=(0,D)
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⇔ w1 < p(ΦC,C + ΦC,NC) and w2 > w1 − p(ΦC,NC − Φ ∗ ∗NC,C) then (q1, q2) = (D, 0).
Similarly, we show that the buyer’s optimal order quantities are (q∗, q∗1 2) = (0, D)
if πPPB (q1, q2)|(q1,q2)=(0,D) is bigger than πPPB (q1, q2)|(q1,q2)=(0,0) and is also bigger than
πPPB (q1, q2)|(q1,q2)=(D,0).
We show the outcome of Lemma 9 in Figures 14 and Figure 15.
FIGURE 14. Presentation of Optimal Order Quantities on Plane of Wholesale Prices
When ΦC,NC > ΦNC,C .
As Figure 14 shows, when the wholesale prices are high enough, the buyer has
no willingness to order at all. When the wholesale price gets lower, then the buyer
orders. Because we are focusing on sustainability, when both wholesale prices are
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equal, the buyer orders from supplier 2, whose cost of being compliant with the code
of conduct is less.
FIGURE 15. Presentation of Optimal Order Quantities on Plane of Wholesale Prices
When ΦC,NC 6 ΦNC,C .
When one of suppliers fails the audit: We show the buyer’s profit function
under this case with πPFB or π
FP
B . This profit function depends on whether the
supplier who passes the audit is compliant. Also, the supplier who fails the audit
will not receive any contract. Based on this, we have the buyer’s profit-to-go for the
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two cases wherein one of the suppliers fails the audit as following:
πPFB = Pr(C,NC|P, F )(p− w1)q1 + Pr(NC,NC|P, F )(−w1q1) (3.11)
πFPB = Pr(C,NC|F, P )(p− w2)q2 + Pr(NC,NC|F, F )(−w2q2) (3.12)
Since the profit functions are symmetric, we explain one of them here in detail. The
first term of buyer’s expected profit function (3.11) shows the profit of the buyer in
case the supplier who passes the audit is compliant. In this case the buyer collects
the profit from the market. The second term of profit function (3.11) shows the profit
that the buyer collects when the supplier who passes the audit is not compliant. In
this case, the buyer faces the market disruption and incurs the cost of sourcing.
As the two events of passing and failing the audit are independent events for
each supplier, we can rewrite the conditional probabilities in the buyer’s expected
profit function (3.11) as following:
Pr(C,NC|P, F ) =Pr(C1|P1)× Pr(NC2|F2)
Pr(Supplier 1 is Compliant & Pass)
= ×
Pr(Pass)
Pr(Supplier 2 is Non-Compliant & Fail)
Pr(Fail)
Pr(Supplier 1 is Compliant & Pass)
= × 1
Pr(Pass)
=Pr(C1|P1)
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Therefore, we can rewrite the buyer’s expected profit in (3.11) and (3.12) as following:
πPFB (q1) = θ1(p− w1)q1 + (1− θ1)(−w1q1) (3.13)
πFPB (q2) = θ2(p− w2)q2 + (1− θ2)(−w2q2) (3.14)
where θ1 = Pr(C1|P1), 1 − θ1 = Pr(NC1|P1), θ2 = Pr(C2|P2), and 1 − θ2 =
Pr(NC2|P2).
In the case that one of the suppliers fails the audit, the buyer allocates the order
quantity of qi to the supplier who passes the audit to maximize its expected profit:
max πPF
≤ ≤ B
(qi) (3.15)
0 qi D
We define a new symbol for the conditional probabilities in πPFB (q1) and π
FP
B (q2).
Lemma 10 presents the buyer’s optimal order quantity decision when one of the
suppliers fails the audit.
Lemma 10. Consider that the auditing precedes the contracting and the buyer faces
two suppliers and one of the suppliers fails the audit. The buyer’s optimal order
allocation decision is:
– When the supplier 2 fails the audit:
∗ if w1 > θ1p then (q∗1, q∗2) = (0, 0).
∗ Otherwise, (q∗1, q∗2) = (D, 0).
– When the supplier 1 fails the audit:
∗ if w ∗ ∗2 > θ2p then (q1, q2) = (0, 0).
∗ Otherwise, (q∗1, q∗2) = (0, D).
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Proof of Lemma 10. We show the proof for one of cases, because they are symmetric.
We simplify the buyer’s objective function πPFB (q1). It gives us (θ1p − w1)q1. This
function is linear in q1, therefore when the coefficient is positive q
∗
1 = D, otherwise
q∗1 = 0. Also, the q
∗
2 = 0 because the second supplier fails the audit.
Lemma 10 shows that when the wholesale price is higher than a threshold, then
the buyer has no willingness to order at all. When the wholesale price is lower than
the threshold, the buyer can make a non-negative profit, and subsequently, the buyer
orders from the supplier who passes the audit.
When both suppliers fail the audit: In this case the buyer will not order at
all from either supplier. Based on our assumption, in this case, the buyer selects his
fall-back option and collects zero profit. In the next section, we present the optimal
wholesale price decisions of suppliers.
Optimal Wholesale Price Decisions
Now, we move one step backward in time and solve for the Nash Equilibrium
of suppliers’ wholesale prices. In fact, both suppliers compete together for quoting
the best wholesale price to the buyer. Each supplier chooses a wholesale price to
maximize its profit for a given competitor’s wholesale price. In this phase of the
contracting stage, we find each supplier’s best response for a given set of buyer’s
auditing efforts (a1, a2), the suppliers’ compliance efforts (e1, e2), and the buyer’s
optimal order quantities (q∗1, q
∗
2). This decision depends on the outcome of auditing:
Suppliers either “pass” or “fail” the audit. Therefore, we have four different scenarios
based on the outcome of auditing. First, we consider the case that both suppliers
pass the audit.
When both suppliers pass the audit: In this scenario, the supplier’s profit
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depends on whether the supplier who passes the audit is compliant. If the supplier is
compliant, then he will collect his profit. If the supplier who passes the audit is not
compliant, then he has to pay the goodwill cost. The supplier incurs the goodwill
cost if and only if the supplier receives a non-negative order from the buyer. Based
on this scenario, each supplier’s expected profit-to-go is:
(w − c )q∗ −GC, q∗i i i i > 0
πPPs = θi(wi − c ∗i i)qi + (1− θi) . (3.16)0, q∗i = 0
Passing and failing the audit for each supplier is an independent event. Therefore,
we can rewrite the conditional probabilities in (3.16): Pr(Ci|P, P ) = Pr(Ci|Pi) = θi
and Pr(NCi|P, P ) = Pr(NCi|Pi) = 1 − θi. Each supplier chooses wi to maximize
its profit. We solve for the best response function for each supplier by running the
following optimization:
max πPPS (wi i)ci≤wi≤p (3.17)
subject to πPPS (wi) > 0,i
We solve for the supplier’s best response while we pay attention to supplier’s
participation constraint in optimization program (3.17). If q∗i = 0, then based on
supplier’s objective function we have no contracting. If q∗i = D, then from supplier’s
participation constraint we can find the minimum acceptable wholesale price that
the supplier can offer. This wholesale price threshold is c + (1−θi)GCi . We defineD
this threshold as Ω = c + (1−θi)DCi i . Therefore, the structure of Nash EquilibriumD
of suppliers’ wholesale price competition has four elements. The NE shows whether
each supplier participates or not, and if the supplier participates, then what wholesale
price is offered by the supplier. We first solve for each supplier’s best response
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function. Then we intersect the best response functions to find NE. Lemma 11
presents Nash Equilibrium of Suppliers’ wholesale prices. Before we present NE,
here we show the relationship between θi, ΦC,C , ΦC,NC , and ΦNC,C :
ΦC,C + ΦC,NC = Pr(C1|P1) = θ1
ΦC,C + ΦNC,C = Pr(C2|P2) = θ2
Lemma 11. Consider that the auditing precedes the contracting and the buyer faces
two suppliers and both of suppliers pass the audit. The suppliers’ optimal decision
whether to participate or not and in case of participation suppliers’ optimal wholesale
prices are:
– If θ < c1D+GC1 and θ2 <
c2D+GC , then both suppliers do not participate and
Dp+GC Dp+GC
there is no contracting.
– If θ < c1D+GC1 and θ
c2D+GC
2 > , then supplier 1 does not participate andDp+GC Dp+GC
Supplier 2 participates and offers w∗2 = pθ2.
– If θ > c1D+GC and θ c2D+GC1 2 < , then supplier 1 participates and offersDp+GC Dp+GC
w∗1 = pθ1 and supplier 2 does not participate.
– If θ > c1D+GC1 and θ2 > θ +
D(c2−c1)
1 , then both suppliers participate andDp+GC DP+GC
optimal wholesale prices are (w∗, w∗1 2) = (Ω1,Ω1 − p(θ1 − θ2)).
– If θ > c2D+GC2 and θ2 < θ +
D(c2−c1)
1 , then both suppliers participate andDp+GC DP+GC
optimal wholesale prices are (w∗1, w
∗
2) = (Ω2 − p(θ2 − θ1),Ω2).
Proof of Lemma 11. We present the proof of this proposition in the Appendix.
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FIGURE 16. Presentation of Nash Equilibrium of Wholesale Prices on Plane of θ1
and θ2.
The Figure 16 is a graphical representation of Lemma 11. Given that both
suppliers pass the audit we do not observe any contracting when the chance of being
compliant with the code of conduct is low for both suppliers. When the chance of
being compliant is low for one of the suppliers but high for the other, then the former
supplier does not participate in the contracting, and the latter offers a wholesale price
in a way so as to squeeze out the buyer’s profit to zero.
Given that both suppliers pass the audit, when the chance of being compliant
with the code of conduct is high for both suppliers, they compete to quote a lower
wholesale price in order to price out the other supplier. We see that when the chance
of being compliant with the code of conduct for supplier 2 is relatively higher than
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for supplier 1, then supplier 2 sets its wholesale price a little lower than the minimum
participation price of supplier 1 to price out the first supplier. On the other hand,
when the chance of being compliant with the code of conduct is higher for supplier 1,
then supplier 1 sets the wholesale price a little lower than the minimum participation
price of supplier 2, to price out supplier 2 and receive the entire order from the buyer.
When one of the suppliers fails the audit: In this scenario, the supplier’s profit
depends on whether the supplier who passes the audit is compliant. If the supplier is
compliant, then he will collect his profit. If the supplier who passes the audit is not
compliant, then he has to pay the goodwill cost. The supplier incurs the goodwill
cost if and only if the supplier receives a non-negative order from the buyer. Also,
the supplier who fails the audit does not receive any contract. We show the profit
function for the case where the supplier 1 passes the audit and the supplier 2 fails the
audit. The other case when supplier 1 fails the audit is similar to this case. Based
on this scenario, the supplier’s expected profit-to-go is:
(w1 − c
∗ ∗
∗  1
)q1 −GC , qPF 1 > 0πs = θ1(w1 − c1)q1 + (1− θ ) . (3.18)1 1 0 , q∗1 = 0
The supplier chooses w1 to maximize its profit-to-go. The optimization program
below shows the supplier’s profit maximization:
max πPFS (w1 1)c1≤w1≤p (3.19)
subject to πPFS (w1 1) > 0,
Lemma 12 presents the optimal wholesale price for the supplier who passes the audit
in this case.
85
Lemma 12. Consider that the auditing precedes the contracting and the buyer faces
two suppliers and one of the suppliers fails the audit. The supplier’s optimal decision
whether to participate or not, and supplier’s optimal wholesale prices in case of
participation are:
– If θ < ciD+GCi , then supplier i does not participate and we have no contracting.Dp+GC
– If θ > ciD+GCi , then supplier i participates and offers the wholesale priceDp+GC
w∗i = pθi.
Proof of Lemma 12. For all θ wii < the supplier does not receive any order andp
we do not have any contracting. When θ wii > we simplify the profit functionp
as (wi − ci)D − (1 − θi)GC. This profit function is increasing in wholesale price.
Also, from constraint we have wi > Ωi. Therefore, whenever Ωi > pθi we have no
contracting. Otherwise the supplier chooses the wholesale price at the maximum
possible level and sets w∗i = pθi.
When one of the suppliers fails, the supplier who passes the audit looks at his
chance of being complaint with the code of conduct. In case, this chance is high,
then the supplier will set the wholesale price in a way to squeeze out the buyer’s
profit. Otherwise, the supplier does not participate, and we have no contracting.
When both suppliers fail the audit: In this scenario, suppliers do not receive
any contracting, and the buyer uses a fall-back option to satisfy the demand and the
buyer makes no profit. In the next section, we will analyze the suppliers’ problem to
find the optimal compliance efforts.
86
Optimal Compliance Efforts Decisions
We move one step backward in time and solve for Nash Equilibrium of suppliers’
compliance efforts . In fact, suppliers move simultaneously to choose their compliance
effort to maximize their profit. In this phase of the auditing stage, we find each
supplier’s best response for a given set of buyer’s efforts (a1, a2). We use our finding
in previous sections to form the supplier’s expected profit in this phase. First, we
present supplier 1’s profit function in this phase:
∗ ∗
ΠS1(e1, e2) =Pr(P
PP PF
1, P2)πS + Pr(P1 1, F2)πS (3.20)1
∗ ∗
+ Pr(F FP1, P2)πS + Pr(F1, F )π
FF
2 S − α1 1 1e1
In profit function (3.20), we have all four combinations of passing and failing of
each supplier. For example, the first term shows the profit of supplier 1 when both
suppliers pass the audit. From our analysis in the contracting stage, we know that
if the supplier fails the audit, the supplier will not receive any contract. Therefore,
in profit function (3.20), the last two terms will be zero. Supplier 1 chooses e1 to
maximize its profit. We show this maximization in optimization program (3.21) :
max ΠS1(e1, e2)
0≤e1≤1 (3.21)
subject to ΠS1(e1, e2) > 0,
Lemma 13 presents the best response function of supplier 1 in this phase.
Lemma 13. Consider that the auditing precedes the contracting and that the buyer
faces two suppliers. Supplier 1’s best response function e∗1 for any given e2 is:
– If 0 6 D 6 α1− , then e
∗
1 = 0 for all e2 in [0, 1].p c1
87
– IF α1 < D < α1− − , then e
∗
1 = 1 for all e2 in [0, ē2], and e
∗
1 = 0 for all ep c c c 2 in1 2 1
(ē2, 1].
– If D > α1− , then e
∗
1 = 1 for all e2 in [0, 1].c2 c1
where ē = (1−a2)C2D+(p−c1)D+(1−a2)GC−α12 .(p−a2c2)D+(1−a2)GC
The supplier does not exert any effort when the cost of compliance is very high.
On the other hand, when the cost of compliance is very low, the supplier exerts full
effort. While the cost is in the mid-range, for all small compliance efforts of the
competitor, the supplier exerts full effort, and for high efforts of the competitor, it
exerts no effort.
The profit function for the second supplier is very similar to the first supplier.
We present the supplier 2’s profit function in (3.22).
∗ ∗
ΠS2(e1, e2) =Pr(P1, P
PP
2)πS + Pr(P1, F2)π
PF
S (3.22)2 2
∗ ∗
+ Pr(F1, P )π
FP
2 S + Pr(F
FF
2 1
, F2)πS − α2e2 2
In profit function (3.22), we have all four combinations of passing and failing of
each supplier. For example, the first term shows the profit of supplier 2 when both
suppliers pass the audit. From our analysis in contracting stage, we know that if the
supplier fails the audit, the supplier will not receive any contract. Therefore, in the
profit function (3.22), the second term and the fourth term will be zero. Supplier
2 chooses e2 to maximize its profit. We show this maximization in optimization
program (3.23) :
max ΠS2(e1, e2)
0≤e2≤1 (3.23)
subject to ΠS2(e1, e2) > 0,
88
To find Nash Equilibrium of suppliers’ compliance efforts, we should find the second
supplier’s best response for any given e1 and then intersect both best response
functions of supplier 1 and supplier 2 to find NE. Notice that supplier 1’s best
response is always happening at e∗1 = 0 or e
∗
1 = 1. Therefore, for finding Nash
Equilibrium, we need to find supplier 2’s best response function at e∗1 = 0 and
e∗1 = 1.
Lemma 14 presents Nash Equilibrium of suppliers’ compliance efforts.
Lemma 14. Consider that the auditing precedes the contracting and the buyer faces
two suppliers. Nash Equilibrium of suppliers’ compliance efforts are:
– If D < α1 and D < α2 , then (e∗, e∗− − 1 2) = (0, 0).p c1 p c2
– If α1 6 D < α2− − or {D >
α1 and D > α2− − }, then (e
∗ ∗
1, e2) = (1, 0).p c1 p c2 c2 c1 p c2
– If α2 α1− 6 D < − , then (e
∗
1, e
∗
2) = (0, 1).p c2 p c1
– If α2 6 D < α1 α1 ∗ ∗ ∗ ∗
p−c2 c2− and D >c1 p− , then (e1, e2) = (1, 0) and (e1, e2) = (0, 1).c1
Figure 17 is a graphic representation of Proposition 14. When the demand or
market price is low, both parties have no incentive to exert any efforts, and we have
no contracting. When the market price increases and the demand is low, supplier 2,
who has a higher cost of production but lower compliance cost, can afford to exert
the compliance cost and exert full effort. While demand increases and it is in the
mid-range (not too high and not too low), both suppliers can participate and exert
efforts. When demand is very high, if the production cost advantage of supplier 1 is
higher than his cost for full compliance, then supplier 1 has the advantage and exerts
full effort, and supplier 2 exerts zero. In the next section, we analyze the buyer’s
problem to find the optimal auditing efforts.
89
FIGURE 17. Presentation of Nash Equilibrium of Compliance Efforts on Plane of
Demand and Market Price.
Optimal Auditing Efforts Decisions
We move one step backward in time and solve for the buyer’s optimal auditing
efforts (a∗1, a
∗
2). The buyer chooses (a1, a2) to maximize its profit. The outcome of
auditing is one of three different situations. If the supplier is compliant, the supplier
will pass the audit. If the supplier is not compliant, then either the supplier fails
or passes the audit. We present the buyer’s profit-to-go in (3.24). As we have two
suppliers, the buyer’s expected profit-to-go function in this phase of the auditing
stage includes nine different terms, as a result of these three-by-three combinations.
Each of these terms shows the expected profit for one of the above combinations.
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ΠB(a1, a2) =Pr(P1C1, P C
∗ ∗ ∗
2 2)[(p− w1)q1 + (p− w2)q∗] 2(p− w∗1)q∗1, q∗1 > 0, q∗2 = 0
+Pr(P1C1, P2NC2)−w∗1q∗1 − w∗q∗, q∗2 2 1 > 0, q∗2 > 0
+Pr(P1C1, F2)(p− w∗ ∗1
)q1
(p− w
∗
2)q
∗
2, q
∗ ∗
 1
= 0, q2 > 0
+Pr(P1NC1, P2C2)−w∗1q∗1 − w∗q∗ ∗ ∗2 2, q1 > 0, q2 > 0
+Pr(P1NC1, P2NC2)(−w∗q∗ − w∗q∗1 1 2 2)
+Pr(P1NC1, F2)(−w∗q∗1 1) + Pr(F1, P2C2)(p− w∗2)q∗2
+Pr(F1, P2NC2)(−w∗ ∗2q2) + Pr(F1, F2)× 0− γ(a1 + a2) (3.24)
Proposition 9 presents the buyer’s optimal auditing efforts, along with sub-game
perfect Nash Equilibrium of the model.
Proposition 9. Consider that the auditing precedes the contracting and the buyer
faces two suppliers. The buyer’s optimal auditing efforts for the range of all available
parameters are (a∗1, a
∗
2) = (0, 0). The sub-game prefect Nash Equilibrium of model
for different range of parameters are:
– If D < α1 α2− and D < − , then (e
∗ ∗
1, e2) = (0, 0) and there is no contracting.p c1 p c2
– If α1 α2 α1 α2− 6 D < − or {D > − and D > − }, then (e
∗, e∗1 2) = (1, 0). Also,p c1 p c2 c2 c1 p c2
(q∗1, w
∗
1) = (D, p) and supplier 2 does not participate. The supplier squeezes out
the buyer’s profit to zero.
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– If α2− 6 D <
α1
− , then (e
∗
1, e
∗
2) = (0, 1). Also, (q
∗ ∗
p c2 p c1 2
, w2) = (D, p) and supplier
1 does not participate. The supplier squeezes out the buyer’s profit to zero.
– If α2− 6 D <
α1 and D > α1− − , then (e
∗
1, e
∗
2) = (1, 0) and (e
∗ ∗
p c2 c2 c1 p c1 1
, e2) = (0, 1).
The optimal wholesale price and order quantity are identical to related cases in
above.
Proposition 9 shows that when the auditing precedes the contracting, supplier
competition puts the pressure on each supplier to exert full effort, whenever the
supplier finds it beneficial to participate. Therefore, the buyer’s auditing does not
need to be as high, because the supplier exerts full effort to be compliant with the
code of conduct. We present the intuition and insights related to comparison of
benchmark model and supplier competition in next section.
Effect of Supplier Competition
In this section we compare the benchmark model with the supplier competition
when the auditing precedes the contracting. Based on the conclusion of the
benchmark model, when demand and price are above a threshold, both suppliers
participate and exert their compliance effort to get the whole order. But, when
demand is high enough, we observe that if the production cost advantage of supplier
1 (i.e., c2− c1) is more than its full effort compliance cost (i.e., α1), then the supplier
1 is the supplier who exerts full compliance effort and participates. When the
demand is in mid-range, both suppliers can participate; and, as our sub-game prefect
equilibrium shows, this case exists for both. Also, when the demand is low and price
is high, supplier 2 has the willingness to exert the effort, because its production cost
is higher and compliance cost lower than the supplier 1. Therefore, its profit margin
increases, and supplier 2 participates in the contracting alone and exerts full effort.
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Our result also shows that the supplier competition leads to creating new
contracting opportunity. When the demand is low and the market price is high,
by switching from supplier 1 to supplier 2 the contracting opportunity increases.
When the demand is high and the market price is low, the buyer can increase the
contracting opportunity by switching supplier 2 with supplier 1. This is driven by
the trade-off between the production cost and the compliance cost of two suppliers.
When the profit of producing the order minus the cost of being fully compliant for
one of suppliers beats the other supplier’s, then buyer’s decision to switch to the
winning supplier will increase the contracting opportunity.
FIGURE 18. Presentation of Equilibrium of Supplier Competition on Plane of
Demand and Market Price.
Our analysis provides managers with another effective decision-making tool
about choice of suppliers to consider to improve compliance in their supplier base
93
structure. We find that when auditing precedes contracting, the buyer’s auditing
effort could be low. The supplier who participates in the contracting has this
willingness to be compliant to the code of conduct; and based on its pricing power,
the supplier squeezes out the buyer’s profit to zero.
Conclusion and Summary
We consider supplier competition in conjunction with auditing as yet another
tool for a buyer to enhance suppliers’ social and environmental compliance, that
has not been considered in the literature. We investigate the effect of supplier
competition particularly in scenarios wherein suppliers determine the wholesale
prices. We consider a game-theoretic framework to investigate the impact of
timing of contracting with two possible supplier base structures (i.e., a single
supplier or two asymmetric competing suppliers) on the suppliers’ compliance, on
the buyer’s auditing effort, and on the buyer’s and the suppliers’ profits. We solve
for the subgame-perfect equilibrium and characterize the contract terms, the buyer’s
auditing effort, and the suppliers’ compliance efforts at the equilibrium.
Our results indicate that when the auditing precedes the contracting, the buyer
does not have to exert significant auditing effort since suppliers find it beneficial
to be compliant with the code of conduct in the presence of competition. In the
process the supplier is also able to squeeze out the buyer’s profit to zero. When the
demand in the market is high, the supplier with the lower cost of production (but
higher compliance cost) wins the competition. In contrast, when the market price is
high while demand is low, the supplier with the advantage in compliance cost wins
the competition. We also show that the supplier competition leads to increasing the
contracting opportunity.
94
Bridge to Next Chapter
In the first essay, we investigated how buyer’s policy of commitment to contract
terms affects the sustainability and financial performance of the supply chain. The
second essay focused on understanding the impact of supplier competition on the
buyer’s ability to influence suppliers’ compliance when suppliers have more parity in
contracting power. In the next chapter, which is the third essay, we focus on a broader
research question pertaining to buyer-supplier relationship management strategies
to influence supplier’s sustainability performance. The primary research question
considers how the specific nature of trust (i.e., calculative vs. relational trust)
between the buyer and the supplier influences the impact of supplier relationship
management strategies (transactional and collaborative) on supplier’s sustainability
performance. This research question is not easily amenable to analytical modeling,
and hence addressed by developing a conceptual and empirical framework.
95
CHAPTER IV
A CONCEPTUAL FRAMEWORK FOR UNDERSTANDING THE ROLE OF
TRUST BETWEEN BUYERS AND SUPPLIERS IN INFLUENCING
SUPPLIERS’ SUSTAINABILITY
Introduction
Social and environmental performance of suppliers of well-known brands has
been in the spotlight over the last decade and has been documented by news media,
agencies, and researchers. For instance, International Labor Organization (ILO)
reports that worldwide, 218 million children are employed by different manufacturing
companies, almost 73 million of whom also work in hazardous conditions (ILO,
2013). Child labor keeps the production costs of local suppliers down, and the benefit
therefrom goes to brands who buy from these suppliers (Locke, 2003). As another
example, a supplier’s apparel factory in Bangladesh collapsed in April, 2013. During
the incident, more than 1000 people were killed. The apparel factory was a supplier
to Walmart and Matalan. Media reported that the building was not safe for workers
(Al-Mahmood et al., 2013). In another incident, a supplier to Toyota in Alabama
forced workers to work overtime at regular pay. The supplier also did not provide
workers with standard workplace safety. As a consequence of this supplier’s behavior,
some employees got injured (Waldman, 2017). In another case, it has been revealed
that BMW and Volkswagen were sourcing from suppliers that employed child labor
(Bengtsen and Kelly, 2016). Such incidents are not limited to only social misconduct
of suppliers. There have also been many incidents of environmental violations. For
instance, it was publicized by NGOs and media that suppliers of Nike and Marks &
96
Spencer released toxic chemicals into the rivers and polluted the air (Hurley et al.,
2017).
Suppliers’ social and environmental misconduct hurt the reputation of, cause
serious problems for, and have threatened the revenue stream of brands. Several
studies show that brands incur huge costs because of such incidents of publicized
misconduct (Guo et al., 2016; Plambeck and Taylor, 2016). Many of these well-
known brands are concerned about suppliers’ social and environmental misconduct
(Lee et al., 2012). Some brands motivate the suppliers to be compliant with social
and environmental standards by incentivizing them, such as Nike’s supplier incentive
program for sustainability (Nike, 2018). Other brands, such as Starbucks, penalize
suppliers upon observing their social and/or environmental misconduct (Lewis et al.,
2012; Porteous and Rammohan, 2013; Porteous et al., 2015). In some cases,
brands choose to employ collaborative approaches to enhance suppliers’ social and
environmental performance. Leading brands try to improve their suppliers’ social
and environmental performance through joint problem-solving and supplier training
(Bai and Sarkis, 2010). IKEA works closely with its suppliers to develop a mutual
understanding of sustainability objectives by working on environmental and social
responsibility goals and codes of conduct (Andersen and Skjoett-Larsen, 2009).
Despite above efforts many buyer firms across range of industries struggle
with achieving social and environmentally responsible behavior in their supply
chains because of their inadequate or ineffective supplier relationship management
practices. Previous research on managing buyer-supplier relationships has examined
how buyers use a specific subset of transactional and collaborative strategies
to influence suppliers’ operational performance (Zajac and Olsen, 1993). Few
studies have extended their focus to also understanding the impact of the above
97
subset of relationship management strategies on suppliers’ sustainability performance
(Klassen and McLaughlin, 1996; Plambeck et al., 2012; Bag, 2018). However, no
previous research has examined the impact of a comprehensive set of transactional
and collaborative relationship management strategies on enhancing suppliers’
sustainability performance.
The notion of trust has been another important construct in understanding
buyer-supplier relationships. Previous research has examined how trust between
buyers and suppliers moderates the impact of transactional and collaborative
approaches on suppliers’ performance, albeit not sustainability performance (Benton
and Maloni, 2005; Ireland and Webb, 2007). While both buyers and suppliers
understand that they are negatively affected when customers penalize a brand for
social and environmental violations in their supply chain, they may yet not be willing
to behave in a responsible manner due to lack of trust and misalignment of incentives
(Gualandris and Kalchschmidt, 2016). Parmigiani et al. (2011) suggest that there
may emerge a growing level of trust between buyer and suppliers, because of pressure
from customers regarding sustainability misconduct. Customer pressure can catalyze
the buyer-supplier relationship towards social and environmental responsibility in the
supply chain. Parties may try to work together by considering to develop trust as
a basis for responding to this pressure. A major question arises as to what extent
is trust important for catalyzing the buyer supplier relationship towards a more
sustainable behavior.
To better understand the effect of buyers’ approaches for enhancing the
suppliers’ social and environmental performance, we propose a conceptual framework
to study the efficacy of a comprehensive set of both transactional and collaborative
supplier relationship management approaches to enhance suppliers’ sustainability
98
performance. Specifically, we propose a framework to investigate the role of
the specific nature of trust (i.e., calculative and relational trust) between buyers
and suppliers in influencing the impact of their supplier relationship management
strategies on suppliers’ sustainability performance.
Some key research questions of interest include:
1) Do supplier relationship management strategies (transactional and collaborative)
influence the supplier’s social and environmental (i.e., sustainability) performance?
2) If so, how do transactional relationship management approaches compare
vis-à-vis collaborative relationship management approaches? Are transactional
(relational) approaches relatively better for enhancing suppliers’ environmental
(social) performance?
3) How does the specific nature of trust between the buyer and the supplier influence
the impact of supplier relationship management strategies on supplier’s sustainability
performance?
In this chapter we propose a conceptual framework to lay a foundation for
developing a theoretical lens to understand this phenomena for promoting social and
environmentally responsible behavior in supply chains. After a careful examination
and unification of the literature we develop a set of hypotheses to enable a formalized
study that can address most of the above research questions. We engage with
managers involved in sustainability roles at firms across multiple industries (e.g.,
footwear and apparel, aerospace, semi-conductors and metals manufacturing) to
conduct a preliminary field-based face validation of the proposed framework. The
actual field-based study that requires extensive data collection with a deeper
engagement with participant firms and industry associations is relegated to post-
99
doctoral work. Requisite scales for measurement of constructs in this proposed
framework will be developed or adapted from the literature.
The remainder of the chapter is organized as follows. In the next section we
review the relevant literature. Then, we introduce our framework and develop the
proposed hypotheses. We conclude with a brief summary.
Literature Review
Our research relates to two streams of literature: (1) supplier relationship
management for the purpose of sustainability; and (2) role of trust in the buyer-
supplier relationship management for enhancing supplier’s sustainability.
In sustainable operations management, there are not many papers which look
at the effectiveness of different supplier relationship management approaches on
social and environmental performance. Scholars point out that there is a need to
understand the value of social and environmental practices that affect the firm’s
value (Servaes and Tamayo, 2013). Bowen et al. (2001) and Gimenez and Tachizawa
(2012) indicate that there is a need to investigate the effect of different supplier
management practices on suppliers’ sustainability performance.
Cousins et al. (2004) focus on activities that a buyer can engage, to improve and
manage the environmental performance of his or her supplier, while considering the
available resources and losses incurred because of incidents of environmental non-
compliance in the market. They suggest that although the supplier monitoring and
offering incentives need more resources, they will help proactive companies who have
taken these actions to gain more competitive advantage, because of improvement
in suppliers’ environmental performance. Zhu et al. (2012) study the effects of
100
collaboration with suppliers and assessment of their performance on improving the
sustainability performance of the firm.
Other researchers look at the green supply chain and green value chains. In these
studies, the focus is on the environmental aspects of the supply chain. For instance,
Bowen et al. (2001) and Corbett and Klassen (2006) characterize different aspects of
practices that can help to improve the supplier’s environmental performance. Most
studies investigate the environmental performance of suppliers. Scholars determine
that there should be additional studies on social aspects of suppliers’ sustainability
performance (Seuring and Müller, 2008; Wu and Pagell, 2011).
Bowen et al. (2001) study the effect of collaborative partnering with suppliers
on supplier’s environmental performance. They find a positive relationship between
them. Locke and Romis (2007) provide a case study on two Mexican factories
and find that having a code of conduct and auditing alone cannot help to improve
some aspects of supplier’s social performance, such as hiring child labor and work
safety. Their study proposes that collaborative strategies, such as training the
supplier or joint investment, can reduce the incidents of misconduct. Locke et al.
(2012) study the efficiency of the supplier program of HP. They find that the
collaboration of the buyer with suppliers to establish a solid national context for
workplace safety is more efficient than regular audit capability building, or supply
chain power. Vachon and Klassen (2006) study the relationship between buyers’
environmental collaboration strategies and logistical and technological changes in a
supply chain. Paulraj (2011) empirically investigates the effect of the buyer firm
participating in environmental collaboration, such as providing the supplier with the
design specification for sustainability as collaboration for cleaner production, on the
supplier’s environmental performance. He finds that this relationship is significantly
101
positive and helps the supplier to improve the sustainability performance. Hollos
et al. (2012) study the effect of collaboration on the green performance of suppliers.
They find that a collaborative approach, such as providing feedback to the supplier
from the buyer, can improve the environmental performance of the supplier, including
demonstrable increases in recycling and in waste reduction.
Some researchers study the effect of incentives and penalties on the supplier’s
social and environmental performance. Klassen and McLaughlin (1996) show that
there exists a positive relationship between environmental management practices,
such as giving awards to suppliers to incentivize them, with suppliers’ environmental
performance. Lewis et al. (2012) investigate the effect of incentives on the
sustainability performance of suppliers by analyzing a game theory model. They
incorporate some realistic features, such as two-part, nonlinear tariff payment
structure, in their analysis. Plambeck et al. (2012) study the behavior and
strategies that leading companies, such as Nike, adopt for improving their suppliers’
environmental performance in China. They find that these brands provide suppliers
with tools and incentives to improve environmental performance. They indicate
that these incentives are more useful than traditional compliance auditing and
supplier’s disclosure. Porteous et al. (2015) study a model that investigates the
relationship between incentive and penalties with suppliers’ social and environmental
performance. They consider several incentive structures, such as increased business,
better terms and conditions, and price premium. For penalties, they consider reduced
business, fines, and termination of contract. They analyze the model based on an
ordinary least squares (OLS) and PROBIT method. They find that the threat
of terminating the contract, with warnings; the incentive of training the supplier;
and public recognition each have a significant positive effect on reducing social and
102
environmental violations in supplier facilities and practices. In a recent work, Bag
(2018) studies supplier management and sustainable innovation in supply networks.
He finds that supplier relationship management positively affects the suppliers’
performance.
Another related stream of research is the importance of trust, within
supplier-buyer relationships, toward improving suppliers’ social and environmental
performance. This segment of the literature is epitomized by couple of studies.
Carter and Jennings (2002) study the effect of buyers’ sustainability practices on
suppliers’ performance. They find that sustainable purchasing leads to buyers’
enhanced trust in suppliers. Sharfman et al. (2009) study the effect of trust upon
buyer-supplier uncertainty and pro-active environmental supplier management, and
on buyers’ cooperative approaches for supply chain environmental management.
Parmigiani et al. (2011) indicate in their study that there may exist a growing level
of trust between firms in the supply chain, due to social and environmental pressure
from customers and buyer firms, in which collaboration and knowledge transfer will
help suppliers meet the buyers’ social and environmental standards. Anisul Huq
et al. (2014) use a multi-case study approach to investigate the adoption and
implementation of socially and environmentally sustainable practices at a supplier
firm in Bangladesh that is a supplier for two brands. They find that moving from
absolute use of power, to collaboration and open dialogues with trust between firms,
can promote the social performance in the supply chain. These studies suggest that
there could be a moderating role for trust between sustainable supplier management
and supplier’s performance. Gualandris and Kalchschmidt (2015) investigate how
the social and environmental performance of the supply chain can be enhanced as
sustainable supply chain management is implemented. They find that trust plays
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a moderating role between sustainability practices and social and environmental
performance. To the best of our knowledge, our unifying conceptual framework lays
the foundation for a study that is the first to examine the efficacy of transactional and
collaborative approaches adopted by buyers in a comprehensive manner to enhance
suppliers’ social and environmental performance. Our conceptual framework also
proposes the need to investigate for the first time with greater specificity the roles
of calculative trust and relational trust between buyer and supplier in influencing
the efficacy of supplier relationship management approaches adopted by the buyer
on supplier’s sustainability performance.
Conceptual Development
In this section we examine the theoretical underpinnings of our proposed
conceptual framework to develop and present the hypotheses. First, we present
the different approaches that a buyer can use to manage its relationship with the
suppliers. Then, we define the social and environmental performance of suppliers.
Next, we use social exchange theory to propose our hypothesis and to show the
relationship between a buyer’s supplier relationship management strategies and
supplier’s social and environmental performance. Finally, we use the literature
and theories to posit our hypothesis about the moderating role of calculative and
relational trust in these relationships. Figure 19 shows our conceptual framework.
Interorganizational Relationships
In recent years, scholars have been paying more attention to interorganizational
relationships (Carey et al., 2011; Cao and Lumineau, 2015). In the supply chain
literature, supplier relationship management is expected to be central to the
104
FIGURE 19. Conceptual Framework
performance of the firms (Amoako-Gyampah et al., 2019). Fynes et al. (2005)
defines supplier relationship management as a purposeful practice that firms engage
in to manage their interactions with suppliers. Supplier relationship management
becomes an important business activity because of market competition, the need
to consider sustainability, and the necessity to reduce the cost in order to be
cost-competitive (Lambert and Schwieterman, 2012). There exist two dominant
approaches for supplier relationship management: (1) transactional (contractual)
approaches; and (2) collaborative approaches (Burt et al., 2003; Spekman and
Carraway, 2006; Whipple et al., 2010).
In the Transactional approach, which is often adversarial, the buyer-supplier
relationship will be categorized based on functions and tasks (Sanders et al., 2007). In
this type of interaction, buyers set the conditions of contracts and are responsible to
check the compliance of suppliers based on those conditions (Spekman and Carraway,
2006). In this relationship, contracts stipulate different terms and conditions based
on the expectations of the buyer (Reyniers and Tapiero, 1995). When buyers
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use transactional approaches, suppliers have willingness to revise their processes
or improve their status, because they are at risk of not winning the contracts
with the buyers (Hahn et al., 1986). Buyers can count on these transactional
strategies, as they engage suppliers, to improve their performance without a high
effort from buyers or much investment (Krause et al., 2000). Coviello et al. (2002)
explain that the transactional approaches would be more impersonal. The choice
of transactional approach depends on opportunistic behavior of the supplier and
the cost of implementation for the buyer (Mellewigt et al., 2007). Buyers can use
different incentives and penalties as transactional approaches in their contracts with
suppliers (Atkinson, 1998).
In the context of sustainability, buyers use different incentives to motivate their
supplier base to be more socially and environmentally responsible. Porteous et al.
(2015) mention price premiums; public recognition of the supplier, e.g., awards;
preferred supplier status; increased business engagements; and better terms and
conditions in the supply contract as different forms of incentives for suppliers. Some
buyers, such as Walmart, Gap, and Puma, offer loans to their suppliers for improving
their social and environmental conditions (Plambeck and Taylor, 2016). Researchers
show that for avoiding incidents of misconduct, buyers can use penalties as another
form of transactional approach (Davidson III and Worrell, 2001). Porteous et al.
(2015) list penalties such as fines, reduction of business, and termination of contract
as different methods of dis-incentivizing suppliers, in order to avoid incidents of social
and environmental misconduct.
The other supplier relationship approach is a collaborative one. Michael Dell,
founder of Dell Computers, said, “Collaboration is the new imperative” (Burt et al.,
2003). In a collaborative relationship, the buyer and the supplier cooperate jointly for
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the long term, and on a continuous basis, to plan and modify their business practices
to improve performance (Coviello et al., 2002; Spekman and Carraway, 2006; Whipple
et al., 2010). Collaborative relationships are longer-term and cooperative. Coviello
et al. (2002) explain that collaborative approaches are more interpersonal and are
plausible to operate in a sustained manner. Collaborative relationships involve both
economic and social elements (Bunduchi, 2008). In a collaborative relationship,
the buyer has the willingness to share technical information, to train the supplier’s
personnel, and to invest in the supplier’s operation if, in return, the supplier improves
its performance and creates benefits for both sides (Zajac and Olsen, 1993). Zhang
and Cao (2018) indicate that firms that engage in collaborative practices are likely
to share resources such as technical expertise and joint training.
Many companies start collaboration with their supplier with the aim to improve
social and environmental sustainability in their supplier’s facility and processes; e.g.
, Nike tries to collaborate with its suppliers to promote workers’ rights and reduce
packaging waste (Gage, 2016). Researchers and industry reports reveal multiple ways
to collaborate with suppliers for promoting environmental and social sustainability
practices. Buyers can offer a variety of supplier training programs (Porteous et al.,
2015; Kainuma and Tawara, 2006). Leading companies try to improve their suppliers’
sustainability status through joint problem-solving (Pimentel Claro et al., 2006;
Canning and Hanmer-Lloyd, 2001; Rickert et al., 2000; Bai and Sarkis, 2010). In
another way, buyers can provide technical know-how to their suppliers for improving
their processes and the sustainability of their facilities (Pedersen and Andersen,
2006; Bai and Sarkis, 2010). In some cases, buyer and supplier jointly invest on
environmentally friendly technology or improvement of social practices (Pedersen
and Andersen, 2006; Simpson et al., 2007; Bai and Sarkis, 2010). Buyers such as Nike
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and IKEA work very closely with their suppliers to develop a mutual understanding
of sustainability objectives, by working on environmental and social responsibility
goals and codes of conduct (Pedersen and Andersen, 2006). In the next section, we
introduce the suppliers’ performance from social and environmental perspectives.
Supplier’s Social and Environmental Performance
Suppliers’ sustainability performance is presented from economic, social, and
environmental perspectives. Scholars define environmental and social performance
in different ways (Maloni and Brown, 2006; Zhu and Sarkis, 2007; Pullman et al.,
2009). Valiente et al. (2012) explain that the supplier’s social and environmental
performance takes into account both the descriptive and the normative dimensions
of corporate responsibility, as well as emphasizing everything that suppliers are
achieving in the domain of environmental and social responsibility policies, practices,
and results. To assess the social and environmental performance of suppliers,
researchers measure the improvement in different metrics. For example, to capture
the environmental performance, Zhu and Sarkis (2007) introduce improvement of
reduction in reduced waste and other pollutants as a way to see environmental
performance. Valiente et al. (2012) assess the occupational safety and health of
workers in the supplier’s firm to capture social performance.
In general, for measuring a supplier’s social and environmental performance,
we can look at the reduction in social and environmental violations. Porteous and
Rammohan (2013) define violations as any kind of noncompliance with or breaches
of national or regional law, industry regulations, and the supplier’s code of conduct.
If the number of these violations is reduced, we can conclude that the supplier’s
compliance advances, and therefore, the supplier’s performance improves.
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To find the above-mentioned violations, most buyers develop a sustainability
code of conduct for their suppliers. The code of conduct sets expectations for
suppliers’ social and environmental practices. Buyers audit their suppliers for
compliance with their code of conduct, utilizing formal, external third party audits
or internal audits, and they strive to reduce the social and environmental violations
in their supplier base (Egels-Zandén, 2007; Awaysheh and Klassen, 2010).
The social dimension of the supplier’s responsibility consists of health, safety,
and working conditions of employees, lawful employment practices, worker benefits
(including fair wages and training), freedom of association, commitment to diversity
and nondiscrimination practices, and improvement in awareness and protection of
the claims and rights of people in the communities served (van der Wiele et al.,
2001; Paulraj, 2011; Porteous et al., 2012; Lee et al., 2012). The environmental
dimension of suppliers’ responsibility consists of responsible use of resources (e.g.,
energy, water, materials, and hazardous substances); emissions reduction; land use;
and waste management of water and materials (Hutchins and Sutherland, 2008;
Porteous et al., 2012; Lee et al., 2012). In the next section, we present our hypothesis
for the relationship between the buyer’s supplier relationship management strategies
and suppliers’ sustainability performance.
Relationship between Buyer’s Supplier Relationship Management Strategies and
Supplier’s Social and Environmental Performance
We rely on social exchange theory as our main theoretical framework to present
the research model and develop our hypotheses. Social exchange theory is a useful
set up to see the effect of interactions on outcomes in a dyadic relationship. Social
exchange theory is composition of principles obtained from different fields, including
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psychology, sociology, and economics (Bandura, 1986; Blau, 2017; Griffith et al., 2006;
Rickert et al., 2000). It introduces a framework in which any interactions between
buyer and supplier can be considered as an exchange of resources (Cropanzano and
Mitchell, 2005). These resources can be economic (e.g., money or goods), but can
also be social in nature, for instance, status or collaboration (Lambe et al., 2001;
Cropanzano and Mitchell, 2005).
An important principle of social exchange theory is that two parties start a
relationship with the expectation that the result of the exchange will be beneficial to
them (Lambe et al., 2001; Blau, 2017). Also, Lambe et al. (2001) point out that the
parties will remain in business together as long as the benefits from the relationship
exceed the alternative relationships.
We use the social exchange theory to propose that both transactional approaches
and collaborative approaches may be effective to improve the supplier’s social and
environmental performance. The supplier’s need to enter or keep the business
relationship is stimulated by motivation to protect his economic outcomes. In the
case of transactional approaches, the supplier knows that, in order to keep the
contract going and to receive some incentives and avoid termination, he needs to
perform well in the areas that the buyer requires. The supplier has the incentive to
fix problems that can improve his performance and that help him to be competitive
enough and not lose the business opportunity (Terpend and Krause, 2015). Taking
advantage of the transactional approaches is an appealing choice for buyers, because
the buyers can use them to improve suppliers’ performance without too much
investment (Krause et al., 2000).
Scholars criticize the use of transactional approaches, because under the pressure
of securing the contract and competition, such approaches encourage suppliers to
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use tricks and short-term fixes. Rather than significant long-term improvements
(Deming, 2018). In our proposed framework we consider buyer’s supplier relationship
strategies in a comprehensive manner and also classify them into transactional and
collaborative strategies. Based on the above discussion, we first focus on buyer’s
transactional relationship management strategies and posit hypotheses stated below:
H1a: Buyer’s use of transactional strategies has a positive relationship with supplier’s
social sustainability performance.
H1b: Buyer’s use of transactional strategies has a positive relationship with supplier’s
environmental sustainability performance.
For buyer’s collaborative approaches, the supplier’s motivation to improve her
performance is not only based on the economic benefits. When the buyer chooses
to use the collaborative approach, a supplier also benefits from the social outcomes
of this relationship (Lambe et al., 2001). These benefits for suppliers include a
desire to identify with the buyer and to obtain knowledge from a buyer (Maloni
and Benton, 2000; Terpend and Krause, 2015). Social exchange theory states that
a supplier who receives economic benefits from the relationship will reciprocate it.
Also, social exchange theory expects that as time goes by and the buyer-supplier
relationship continues, the use of collaborative approaches will improve the efficiency
of the relationship, simultaneously reducing the level of uncertainty (Terpend and
Krause, 2015). Altogether, it initiates a higher level of coordination (Jap, 2001) and
facilitates the information sharing between two parties (Barratt, 2004). As Jap (2001)
shows these activities will benefit both parties and improve performance. Based
on the above discussion we posit hypotheses associated with buyer’s collaborative
strategies:
H2a: Buyer’s use of collaborative strategies has a positive relationship with supplier’s
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social sustainability performance.
H2b: Buyer’s use of collaborative strategies has a positive relationship with supplier’s
environmental sustainability performance.
In the next section, we present our hypothesis related to the moderating role of
trust in the above relationship.
Trust, Calculative Trust, and Relational Trust
Rousseau et al. (1998) defines trust as “a psychological state comprising the
intention to accept vulnerability based upon positive expectations of the intentions
or behavior of another”. Trust is among the most highly-cited dimensions of critical
import in understanding buyer-supplier relationships in the supply chain literature.
In interorganizational relationships, Zaheer et al. (1998a) mention that trust indicates
a partner’s expectation that the other party is reliable, will act as predicted, and will
behave equitably. Other researchers define trust as “the firm’s belief that another
company will perform actions that will result in positive actions for the firm, as
well as not take unexpected actions that would result in negative outcomes for the
firm” (Anderson and Narus, 1990). Trust promotes recognition of stability, improves
mutual coordination, and reduces the performance losses that would occur because
of opportunism (Poppo et al., 2016).
There are different types of trust. Transaction cost economics and game theory
researchers suggest that a system which places incentives with rewards can lead to a
steady and inevitable outcome (Axelrod, 2006; Williamson, 1996). It can be referred
to as a type of trust, which “delimit[s] the elusive notion of trust” (Williamson,
1993). This type of trust is related to the concept of calculative trust, which
means that parties may act in a reliable manner due to commitments that they’ve
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made (Dyer and Chu, 2000; Suh and Kwon, 2006). Calculative trust enlightens
expectations by intentionally assessing progressive conditions: It needs to calculate
the associated value, or costs and benefits, of violations of terms, or of cheating and
cooperation (Bromiley and Harris, 2006; Lewicki et al., 2006; Poppo et al., 2016).
Under calculative trust, firms know possible outcomes of their actions, and if one
company can compete for the transaction with others, then the transaction will be
assigned to the firm which will generate the largest net gain (Williamson, 1993).
Parkhe (1993) points out when calculative trust is high, firms know that failing
to achieve cooperation and performance goals lead to termination of relationship
or penalties. Threats and sanctions maintain collaboration, control exchanges, and
reduce opportunistic behavior (Poppo et al., 2016).
Relational trust is another form of trust which shows a long-established and solid
business relationship (Granovetter, 1985; Ring and Van de Ven, 1994). Repeated
interactions between firms form relational trust (Rousseau et al., 1998). These
interactions let parties form expectations and develop shared values to define the
ways that they can work together better (Bercovitz et al., 2006). Dependability
and reliability in all previous interactions will lead to the generating of positive
expectations about the other party’s intentions (Rousseau et al., 1998). This type
of trust comes into consideration when parties can expect to behave according to
the partner’s priorities and preferences (Lewicki et al., 2006; Saparito et al., 2004).
Successful fulfillment of expectations, and reliable interactions over the course of
time, evoke a high willingness of parties to rely on each other and to expand the
relationship (Rousseau et al., 1998). Lewicki et al. (1996) state that when relational
trust is high, parties have a better understanding of each other, and they can
behave and respond like each other. This mutual understanding improves outcomes,
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decreases risk, increases efficiency, and reduces opportunistic actions (Poppo et al.,
2016). These two types of trust are distinct constructs with different bases (Rousseau
et al., 1998; Poppo et al., 2016). Calculative trust is founded on reasonable evaluation
of well-established rewards and punishments. Additionally, parties regularly make
decisions as to whether to cooperate, based on their costs and benefits calculations
(Saparito et al., 2004). These decisions need to be made accurately and in determined
ways (Poppo et al., 2016). On the other side, relational trust evolves from repeated
interactions in a long relationship. The mutual understanding and shared values
help parties to consider themselves as “us”, and decisions are made based on overall
quality of relationships, rather than single interactions (Rousseau et al., 1998; Uzzi,
2011). Considering calculative trust, wherein interactions will be terminated or
affected by conditions as soon as violations from terms happen, relational trust allows
parties to cooperate and to resolve the unmet expectation via joint efforts (Rousseau
et al., 1998).
Scholars indicate that trust should increase across the supply chain, in order to
boost the social and environmental performance (Parmigiani et al., 2011; Anisul Huq
et al., 2014). Trust in a buyer-supplier relationship, which is considered an extra
safeguard against exploitation, improves the effect of a specific interaction (e.g., a
collaboration) for a buyer (Artz, 1999). Doney and Cannon (1997) study the impact
of supplier trust on a buyer’s current supplier choice and future purchase intentions.
Zaheer et al. (1998b) shows that the level of trust between buyer and supplier is
a powerful influence on supplier performance. Carter and Jennings (2002) show
that buyer’s trust in supplier has a positive impact, both on buyer’s relationship
commitment and on cooperation between parties. Trust provides confidence to
parties that predetermined outcomes will be achieved, and this should lead to greater
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tendency to cooperate (Andaleeb, 1995). Trust plays a moderating role between
practices and performance in the supply chain. Fynes and Voss (2002) show that
trust plays a moderating role between quality practices and performance between
buyer-supplier relationships. Corsten and Felde (2005) show that when trust is
high, the relationship between collaboration strategies and supplier’s performance
will be stronger. Trust provides a safeguard for future interactions between buyer
and supplier, which provides parties with incentives and ways for developing precious
capabilities (Gualandris and Kalchschmidt, 2016). Scholars show that trust can
facilitate communication and information sharing, and that it leads to a higher
performance (Benton and Maloni, 2005; Ireland and Webb, 2007). Therefore, trust
increases the success level of interorganizational activities by bringing motivation
for supplier and buyer to work closely to achieve higher performance capabilities
(Gualandris and Kalchschmidt, 2016). Trust magnifies the ability of buyer and
supplier to combine their resources to achieve their goal (Morgan and Hunt, 1994;
Dyer and Chu, 2003). Thus, when calculative/relational trust is at a high level, we
expect that the buyer’s transactional and collaborative approaches work better, and
that the supplier engages and participates to improve the social and environmental
performance. However, the effect of the buyer’s transactional and collaborative
approaches on the supplier’s environmental and social performance will be of little
value, when the trust is low. We expect this, as the absence of a high level of trust,
and the low level of coordination hinder buyer efforts to completely take advantage
of supplier relationship approaches. Based on the above discussion, we propose the
following hypotheses:
H3a/b: The higher the calculative trust in buyer-supplier relationship, the stronger
the positive relationship between transactional approaches and supplier’s social/
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environmental performance.
H3c/d: The higher the relational trust in buyer-supplier relationship, the stronger
the positive relationship between relational approaches and supplier’s social /
environmental performance.
Control variables
Lastly, we consider several control variables to allow for heterogeneity in our
conceptual framework. These also enable us to have a better understanding of
some boundary conditions for the posited relationships. First, we consider length
of doing business (prior business history), because experience is necessary to support
trust (Lewicki et al., 1996). We consider dependency of buyer on supplier and vice
versa. Dependency may lead to different behavior and affect the trust based on the
literature. We also consider firm size of buyer and supplier; contracting details for
sustainability; and monitoring mechanism based on Lusch and Brown (1996) and
Dahlstrom and Nygaard (1999). Lastly, we consider tools and metrics, standards,
and practices for reporting sustainability adopted by the buyer and supplier firms.
Summary
Suppliers’ social and environmental responsibility has been a major ongoing
concern for leading brands, as awareness about environmental degradation,
publicized suppliers’ social and environmental incidents of misconduct, and climate
change has increased. In addition, activities of NGOs around reporting various social
and environmental issues in developing countries have pushed brands to pay more
attention to their suppliers’ social and environmental performance. For achieving
a higher compliance to social and environmental standards, brands use different
116
strategies. Apte and Sheth (2016) indicate that brands face challenges to improving
social and environmental performance in their supply chain, because they try to do
it solely through monitoring policies and compliance, a strategy that is not helpful
and which fails “time and time again.” Leading companies start employing different
incentive, penalties, and collaboration mechanisms to enhance sustainability of their
suppliers. Thus, there is a strong need to undertake research that can guide firms to
chart their strategies to improve socially and environmentally responsible behavior
in their supply chains.
In this chapter we propose a conceptual framework to lay a foundation for
developing a theoretical lens to understand this phenomena for promoting social and
environmentally responsible behavior in supply chains. After a careful examination
and unification of the literature we develop a set of hypotheses to enable a formalized
study. We propose a conceptual framework to study the efficacy of a comprehensive
set of both transactional and collaborative supplier relationship management
approaches to enhance suppliers’ sustainability performance. Specifically, we propose
a framework to investigate the role of the nature of trust (i.e., calculative and
relational trust) between buyers and suppliers in influencing the impact of their
supplier relationship management strategies on suppliers’ sustainability performance.
117
CHAPTER V
CONCLUSION
Consumers hold brands accountable when their suppliers’ sustainability
violations become known in the marketplace. Brands find it difficult to reduce
the likelihood of social and environmental violations in their supplier base. Also,
an important conundrum for manufacturers today continues to be the challenge of
balancing the economic benefits of outsourcing with the loss in their ability to control
sustainability in their supply chain. This dissertation with three essays investigates
important ways to alleviate this major supply chain sustainability conundrum by
offering insights at both strategic and tactical levels to make suppliers more socially
and environmentally responsible.
In the first essay, we examine the effect of the buyer’s commitment to price
and/or quantity in enhancing the supplier’s compliance to the code of conduct for
sustainability. While the efficacy of commitment to contract terms has been studied
in the operations management literature with mixed results, it has not been hitherto
examined in the literature on sustainability in operations and supply chains. We
analyze and compare multi-stage game-theoretic models to investigate the effect
of varying levels of commitment to contract terms on sustainability and financial
metrics. We find that increasing the level of commitment improves the supplier’s
likelihood of compliance to the sustainability standard. Interestingly, we also find
that both contracting opportunity and profit for the buyer increase monotonically
with the degree of commitment. Additionally, committing to only the price or only
the quantity (vis-à-vis no-commitment) has an asymmetric effect. We show that
committing to the price and committing to the quantity are complementary strategies
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for the buyer and substitutes for the supplier. Our study provides managers with
yet another effective tool to consider to improve compliance in their supplier base.
Our results show that commitment provides the buyer with significant opportunity
to advance both social good and profit in a win-win manner.
The second essay considers a different scenario wherein the supplier competition
is possible. In this chapter, we study the model wherein the buyer and the
supplier(s) have more parity in their contracting power and hence the buyer’s
power is limited. In contrast with the first essay, the supplier offers a wholesale
price and the buyer is limited to only offering the quantity. We consider a game-
theoretic framework to investigate the impact of timing of contracting with two
possible supplier base structures (i.e., single supplier and supplier competition) on
the suppliers’ compliance, on the buyer’s auditing effort, and on the buyer’s and the
suppliers’ profits. We solve for the subgame-perfect equilibrium and characterize the
contract terms, the buyer’s auditing effort, and the suppliers’ compliance efforts at
the equilibrium. We analyze a single supplier model (benchmark) and compare it
with a supplier competition model, wherein the auditing precedes the contracting.
The interplay between the suppliers’ compliance costs and the suppliers’ production
costs under two different supplier base structures provides useful insights. Our results
show that when auditing precedes contracting, the buyer may need to exert low
auditing effort, because the timing of contracting assures that the supplier who can
participate in contracting will exert the effort to be compliant with the code of
conduct. Also, we find that when demand is high enough, if the production cost
advantage of cheaper supplier is greater than its full effort compliance cost, then
this the cheaper supplier is the supplier who exerts the full compliance effort and
participates in contracting.
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In the third essay, we address a broader question that is not easily amenable to
analytical modeling. Therefore, we use an empirical framework. In this chapter, we
propose a conceptual framework to lay a foundation for developing a theoretical lens
to understand this phenomena for promoting social and environmentally responsible
behavior in supply chains. After a careful examination and unification of the
literature we develop a set of hypotheses to enable a formalized study. We propose
a conceptual framework to study the efficacy of a comprehensive set of both
transactional and collaborative supplier relationship management approaches to
enhance suppliers’ sustainability performance. Specifically, we propose a framework
to investigate the role of the specific nature of trust (i.e., calculative and relational
trust) between buyers and suppliers in influencing the impact of their supplier
relationship management strategies on suppliers’ sustainability performance.
120
APPENDIX A
TECHNICAL PROOFS - CHAPTER II
Proof of Lemma 1. Here, we present the proof of optimal contract terms when
the supplier passes the audit in the no-commitment model. We simplify the
objective function of optimization problem (2.9) as follows: ( e− − )pq −we+(1 e)(1 a) p pqp =
( e
e+(1−e)(1− p − wp)qp . The objective function is decreasing in wp. Therefore, whena)
0 ≤ e < 1−ap− , the objective function is maximized by choosing (w
∗ ∗
p, qp) = (c, 0), whicha
c
means no-contracting. When 1−ap− ≤ e ≤ 1, the objective function is maximized bya
c
choosing (w∗p, q
∗
p) = (c,D). We plug the optimal contract terms into supplier’s profit
function and we have πpS(w
∗
p, q
∗
p) = 0.
Proof of Lemma 2. Here, we present the proof of optimal contract terms when
the supplier fails the audit in the no-commitment model. Given the objective function
in optimization problem (2.10) is decreasing in w and increasing in q[, the optim]alf f +
solution happens when the const[raint is bin]ding. Therefore, if 0 6 e < 1− (p−c)D ,β+
then we have no-contracting. If 1− (p−c)D 6 e 6 1, then (q∗, w∗) = (D, c+β(1−e)).
β f f D
We plug the optimal contract terms into the supplier’s profit function and we have
πf (w∗S f , q
∗
f , e) = 0.
Figure 20 summarizes results of Lemma 1 and Lemma 2. It also illustrates
the parameter space for the buyer’s optimal contracting decisions, where ∅ means
no-contracting.
Proof of Proposition 1. We present the proof of NE of efforts in the no-
commitment model in here. Based on Lemma 1, Lemma 2, and the fact that e∗ = 0,
for all D < β− (which we show in Figure 20), the buyer’s profit function in (2.12)p c
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FIGURE 20. Parameter Space for the Buyer’s Optimal Contracting Decision for
no-commitment
is equal to −γa, which is decreasing in a. Therefore, the buyer’s best response is
zero, a∗ = 0. If D > β− , then the buyer’s profit function in optimization programp c
(2.12) is πB(a, e) = aD(p− c)− aβ− aγ. This function is linear in a. The first order
condition of this function with respect to a is ∂πB = D(p−c)−β−γ. It follows from
∂a
linearity of the function that if D(p− c) < β + γ then a∗ = 0, otherwise a∗ = 1.
Proof of Lemma 3. The proof of optimal quantity, when the supplier passes the
audit in the commitment to wholesale price model, is as follows: the buyer’s objective
function for the case when the supplier passes the audit is e− − p qp − wqp =e+(1 e)(1 a)
( e− − p − w)qp, which is linear in qp. Therefore, when
1−a
p − ≤ e ≤ 1, thene+(1 e)(1 a) a
w
q∗p = D. Also when 0 ≤ e < 1−ap − then q
∗
p = 0. The proof of optimal quantity whena
w
the supplier fails the audit is as follows: the buyer’s objective function is (p− w)qf .
The supplier’s participation constraint is (w− c)qf − β(1− e)[> 0. There]fore, based+
on linearity of both objective function and the constraint: if 1− (w−[c)D ≤ e ≤]1,β +
then the buyer’s optimal order quantity is q∗ = D. Also, if 0 ≤ e < 1− (w−c)D ,
f β
then the buyer does not order, i.e., q∗ = 0. we use the result of Lemma 3 to show a
f
parameter space that we use for proof of next Lemmas. We illustrate this parameter
space in Figure 21. We divide our space into 3 regions based on D, p, w, c, and β. In
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each region, we have different patterns of optimal order quantities (q∗ ∗p, qf ). To find
the conditions for each region, we plot e = 1−a and e = 1−D(w−c)p − on a plane of (a, e).a β
w
When a = 0, we find that e = 1−ap − intercept with axis e at
w . Then, we compare
a p
w
the w and 1 − D(w−c) to derive conditions for each regions in Figure 21. When we
p β
are forming the profit functions for the buyer and the supplier in next Lemmas, we
have different regions in which we have multiple piecewise profit functions based on
the different buyer’s ordering pattern in each regions of Figure 21.
FIGURE 21. Parameter Space based on the Buyer’s Optimal Quantity Decision for
model of commitment to price
We present the buyer’s best response function for commitment to wholesale price
in Lemma 15. Next, we present the supplier’s best response function for commitment
to wholesale price in Lemma 16. We solve for the Nash equilibrium using the best
response functions and subsequently present the proof of Lemma 4.
Lemma 15. When the buyer commits to the wholesale price, for any given supplier’s
compliance effort e, the buyer’s best response a∗(e) is:
– Region 1)a∗ = 0 for any e in [0, 1], the buyer’s best response function form 1
(B-BRF 1),
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FIGURE 22. Graphical Presentation of Regions of Lemma 3 on plane of (w,D)
0 0 6 e < e 11 e1 6 e < e2
– Region 2)a∗ =  , the buyer’s best response
any value in [0, 1] e = e2
0 e2 < e 6 1
function form 2 (B-BRF 2),
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
0 0 6 e < e11 e1 6 e < e2any value in [0, 1] e = e2
– Region 3)a∗ = 0 e2 < e 6 e , the buyer’s best response 41 e4 < e < e3
any value in [0, 1] e = e30 e3 < e 6 1
function form 3 (B-BRF 3),0 0 6 e < e 11 e1 6 e < e3
– Region 4)a∗ =  , the buyer’s best responseany value in [0, 1] e = e 30 e3 < e 6 1
function form 4 (B-BRF 4),
0 0 6 e < e41 e 6 e < e
– Region 5)a∗ = 
4 3
, the buyer’s best response
any value in [0, 1] e = e
3
0 e3 < e 6 1
function form 5 (B-BRF 5),
1 0 6 e < e3
– Region 6)a∗ = any value in [0, 1] e = e , the buyer’s best response 30 e3 < e 6 1
function form 6 (B-BRF 6),
125
where e = γ γ1 − , e2 = 1− , e3 = 1−
γ , e4 = 1− D(w−c) . We illustrate regionsD(p w) Dw Dp β
in Figure 23 and region for each best response function is defined based on following
conditions: √
– Region 1: β(p−w) γ− < D < − or {D <
pγ and D < γβ− − and D 6p(w c) p w (p w)w p(w c)
β(p−w)}.
p(w−c) √
– Region 2: pγ γβ√(p− 6 D < √ − .w)w p(w c)
– Region 3: γβ 6 D < γβ−√ and D >
pγ .
p(w c) w(w−c) (p−w)w
– Region 4: D > γβ− and D > Ḋ(p, w, c, β), where Ḋ(p, w, c, β) is thew(w c)
solution of D(p− w)[1− D(w−c) ]− γ = 0.
β √
– Region 5: { γ 6 D < β and β(p−w) 6 D 6 Ḋ(p, w, c, β)} or { γβ
p−w w−c p(w− 6c) p(w−c)
D < β(p−w) pγ
p(w− and D <c) (p− }.w)w
– Region 6: D > β and D > γ− − .w c p w
Also we have the following orders of parameters in each region:
– In Region 2 and Region 3 (B-BRF 2 and B-BRF 3): e1 <
w < e .
p 2
– In Region 4 (B-BRF 4): e1 <
w < e
p 3
.
– In Region 5 (B-BRF 5): w < e4.p
Proof of Lemma 15. For each region, in which we show in Figure 21, we plug the
associated optimal order quantity in optimization max ∗ ∗0≤a≤1 ΠB(wp, qp, wf , qf , e, a)
where wf = wp = w. Then we solve for the buyer’s optimal auditing effort a.
Region A: (0 6 D 6 β(p−w))
p(w−c)
For region A, as we show in Figure 21, we have three subregions for a given e:
126
FIGURE 23. The Buyer’s Best Response Function in Relation to (D,w)
0 6 e 6 w , w < e < 1 − D(w−c) , and 1 − D(w−c) 6 e 6 1. We write the buyer’s
p p β β
objective function for each of sub-cases of e .
def 1−e p
Subregion A-i: 0 6 e 6 w , We define â(e, p, w) = w− . The buyer’s profit in thisp 1 e
region is:
−γa 0 6 a 6 âΠA−iB (a, e) =  .eD(p− w) + (1− e)(1− a)(−wD)− γa â < a 6 1
Whole function is piecewise linear, continuous, and convex in a. The buyer’s profit
is differentiable in each piece, and its FOC with respect to a is decreasing in a until
127
â. Also the FOC of the first piece is smaller than the second piece.

∂ΠA−i −γ 0 6 a 6 âB (a, e) =
∂a Dw(1− e)− γ â < a 6 1
Following the convexity of function ΠA−iB (a, e) in a, by comparing the value of
ΠA−iB (a, e) at zero, Π
A−i
B (a, e)|a=0 = 0 and one, Π
A−i
B (a, e)|a=1 = De(p−w)−γ, we find
γ ⇒ ∗ γ ⇒ ∗ defthat if e > a = 1, and if e < a = 0. We define e = γ
D(p−w) D(p−w) 1 D(p− .w)
The optimal a∗ depends whether e is less or greater than w1 . Therefore, for anyp
given 0 6 e 6 w :
p

w Dw(p− w) 0 0 6 e < e1
If e1 6 , or equivalently γ 6 then a
∗ =
p p  .1 e 6 e 6 w1 p
w Dw(p− w) w
If e1 > , or equivalently γ > then a
∗ = 0 for all 0 6 e 6 .
p p p
Note that a∗(e) can be discontinuous in e (jump up or jump down in e). For example,
the buyer’s best response function is not continuous at e1 in subregion A-i. We can
observe it when we plug e into ΠA−i1 B (a, e1):

ΠA−iB (a, e1) = −aγ 0 6 a 6 â(e1) .(1− a)( pγ− −Dw) â(e1) < a 6 1p w
From above function we can see for any value between (0, 1), value of ΠA−iB (a, e1) is
less than ΠA−i(a, e )| and ΠA−iB 1 a=0 B (a, e1)|a=1.
Subregion A-ii: w < e < 1 − D(w−c) def def, we define e2 = 1 − γ and e4 = 1 − D(w−c) .p β Dw β
128
The buyer’s profit in this region is:
ΠA−iiB (a, e) = eD(p− w) + (1− e)(1− a)(−wD)− γa.
As ΠA−iiB (a, e) is linear in a we compare the values of profit function at zero and one.
ΠA−iiB (a, e)|a=0 = D(ep− w) and Π
A−ii
B (a, e)|a=1 = De(p− w)− γ. If e > 1−
γ ⇒
Dw
ΠA−iiB (a, e)| > Π
A−ii
a=0 B (a, e)| then a∗a=1 = 0, Otherwise a∗ = 1. Note that, a∗(e)
depends on the order of w , e2, and e
w
p 4
. Then for any given 6 e < e
p 4
:
w D2w(w − c) Dw(p− w)
If < e2 < e4 , or equivalently 6 γ 6 then
p β p
1 w < e < ep 2
a∗ = 
any value in [0, 1] e = e .2
0 e2 < e < e4
w Dw(p− w) w
If e2 < < e4 , or equivalently γ > then a
∗ = 0 for all < e < e4.
p p p
w D2w(w − c)
If < e4 < e2 , or equivalently 0 < γ < then
p β
a∗
w
= 1 for all < e < e4.
p
At e in subregion A-ii, πA−ii(a, e )= D(p−w)− pγ2 B 2 . From this function, we can seew
for any value of a ∈ [0, 1], value of ΠA−iiB (a, e2) is constant in a. Therefore, a∗(e2) is
all values of a ∈ [0, 1].
Subregion A-iii: 1 − D(w−c) def6 e 6 1, Note that we define e = 1 − D(w−c)4 . Theβ β
129
buyer’s profit in this region is:
ΠA−iiiB (a, e) = eD(p− w) + (1− e)(1− a)(−wD) + (1− e)aD(p− w)− γa.
The function ΠA−iiiB (a, e) is linear in a. We compare the values of profit function at
zero and one. ΠA−iiiB (a, e)|a=0 = D(ep − w) and Π
A−iii
B (a, e)|a=1 = D(p − w) − γ. If
e > 1− γ then ΠA−iiiB (a, e)|
A−iii
a=0 > ΠB (a, e)|a=1 then a∗ = 0, otherwise a∗ = 1. WeDp
def
define e3 = 1− γ . The optimal a∗ depends on the order of e4 and e3.Dp
D2p(w − c)
If e4 6 e3 , or equivalently γ > then
β

1 e4 6 e < e3
a∗ = any value in [0, 1] e = e . 30 e3 < e 6 1
D2p(w − c)
If e3 < e4 , or equivalently 0 < γ < then a
∗ = 0 for all e4 6 e 6 1.
β
Before we assemble the above conditions to find the buyer’s best response function,
we show the buyer’s best response function is discontinuous at e4 in the boundary of
2 2
subregion A-iii. We plug e into ΠA−iii4 B (a, e ) = [
D p(w−c)
4 −γ]a+D(p−w)− D p(w−c) .β β
Therefore, we see for any value between (0, 1), value of ΠA−iiiB (a, e4) is less than
ΠA−iiiB (a, e
A−iii
4)|a=0 and ΠB (a, e4)|a=1.
We now assemble the results in subregions A-i, A-ii, and A-iii to find the optimal a.
The order of e1, e2, e3, and e4 depends on the value of γ vs. the three thresholds:
D2(w−c)w D2, p(w−c) , and D(p−w)w
2 2
. Note that we always have D (w−c)w < D p(w−c) .
β β p β β
Recall that region A is defined by D 6 β(p−w) . Under this inequality we have
p(w−c)
130
D2(w−c)w 6 D(p−w)w
2
. Then, it remains to compare D p(w−c) vs. D(p−w)w . We find
β p β p
D2p(w−c) 6 D(p−w)w if and only if D 6 wβ(p−w) . Therefore, under D 6 wβ(p−w)
β p p2(w−c) p2 − , we(w c)
2
have D (w−c)w D
2
6 p(w−c) 6 D(p−w)w
2
. Otherwise, D (w−c)w
2
6 D(p−w)w < D p(w−c) . So
β β p β p β
we can assemble the buyer’s best response function in Region A by the value γ vs.
the three thresholds.
A-1: when 0 6 D < wβ(p−w)
p2(w− :c)
2
A-1-a: A-1 and 0 < γ 6 D (w−c)w .
β
w
We have e1 < < e3 and e4 < e2 and e4 < e 3
.
p
0 0 6 e < e 11 e1 6 e < e3
Therefore, a∗ =  .any value in [0, 1] e = e 30 e3 < e 6 1
2 2
A-1-b: A-1 and D (w−c)w < γ 6 D p(w−c) .
β β
w
We have e1 < < e2 < e4 < e3.
p
131

0 0 6 e < e1
1 e1 6 e < e2any value in [0, 1] e = e2
Therefore, a∗ = 0 e < e 6 e . 2 41 e4 < e < e3any value in [0, 1] e = e30 e3 < e 6 1
2
A-1-c: A-1 and D p(w−c) < γ 6 D(p−w)w .
β p
w
We have e1 < < e2 < e4 and e3 < e4.
p
0 0 6 e < e 1
1 e1 6 e < e2Therefore, a∗ =  .any value in [0, 1] e = e 20 e2 < e 6 1
A-1-d: A-1 and D(p−w)w < γ.
p
w
We have e2 < < e1 and e3 < e4. Therefore, a
∗ = 0.
p
A-2: when wβ(p−w) 6 D 6 β(p−w) :
p2(w−c) p(w−c)
132
2
A-2-a: A-2 and 0 < γ 6 D (w−c)w .
β
w
We have e1 < < e3 and e4 < e2 and e4 < e3.
p
0 0 6 e < e
 1
Therefore, a∗ = 
1 e1 6 e < e3
.
any value in [0, 1] e = e 30 e3 < e 6 1
D2A-2-b: A-2 and (w−c)w < γ 6 D(p−w)w .
β p
w
We have e1 < < e2 < e4 < e3.
p


0 0 6 e < e1

1 e1 6 e < e2
any value in [0, 1] e = e2
Therefore, a∗ = 0 e2 < e 6 e .
4
1 e4 < e < e3
any value in [0, 1] e = e3
0 e3 < e 6 1
133
A-2-c: A-2 and D(p−w)w < γ 6 D
2p(w−c) .
p β
w w
We have e2 < < e1 and < e4 < e3.
p p


0 0 6 e < e 41 e4 6 e < e3
Therefore, a∗ = 
.
any value in [0, 1] e = e 30 e3 < e 6 1
2
A-2-d: A-2 and D p(w−c) < γ.
β
w
We have e2 < < e1 and e3 < e4. Therefore, a
∗ = 0.
p
Region B: (β(p−w)− < D <
β )
p(w c) w−c
For region B, as we show in Figure 21, we have three subregions for a given e: 0 6
e 6 1− D(w−c) , 1− D(w−c) < e < w , and w 6 e 6 1. We write the buyer’s objective
β β p p
function for each of sub-cases of e . Note that we previously defined e = 1− D(w−c)4 .β
Subregion B-i: 0 6 e 6 e4, the buyer’s profit is:

ΠB−iB (a, e) = −γa 0 6 a 6 â .eD(p− w) + (1− e)(1− a)(−wD)− γa â < a 6 1
134
Whole function is piecewise linear, continuous, and convex in a. The buyer’s profit
is differentiable in each piece and its FOC with respect to a is decreasing in a until
â and it may be increasing or decreasing in a.

∂ΠB−i(a, e) −γ 0 6 a 6 âB =∂a Dw(1− e)− γ â < a 6 1
We compare the value of function at zero and one. ΠB−iB (a, e)|a=0 = 0 and
ΠB−iB (a, e)|a=1 = De(p − w) − γ. We find that if e 6
γ ∗
− ⇒ a = 1 and ifD(p w)
e > γ ⇒ a∗− = 0. Note that we previously defined e
γ
D(p w) 1
=
D(p− ; then for anyw)
given 0 6 e 6 e4 :
D(w − c)
If 0 < e1 < e4 , or equivalently 0 < γ 6 D(p− w)[1− ] then
β
0 0 6 e < e
a∗
1
=  .1 e1 6 e 6 e4
D(w − c)
If e1 > e4 , or equivalently γ > D(p− w)[1− ] then
β
a∗ = 0 for all 0 6 e 6 e4.
Note that a∗(e) may jump up or jump down in e. For example, the buyer’s best
response function is not continuous at e1 in subregion B-i, simply we can observe it
when we plug e into ΠB−i1 B (a, e1):
−aγ 0 6 a 6 â(e1)ΠB−iB (a, e1) =  .(1− a)( pγ− −Dw) â(ep w 1) < a 6 1
135
From above function we can see for any value between (0, 1), value of ΠB−iB (a, e1)
is less than ΠB−iB (a, e1)|a=0 and Π
B−i
B (a, e1)|a=1. Similarly, we can show that the
buyer’s best response function discontinuity at e4 in subregion B-i with plug e4 into
ΠB−iB (a, e4):

ΠB−iB (a, e4) = −aγ 0 6 a 6 â(e4) .2D(p− w)− D (w−c)(p−aw) − γa â(e4) < a 6 1β
From above function we can see for any value between (0, 1), value of ΠB−iB (a, e4) is
less than ΠB−iB (a, e4)|a=0 and Π
B−i
B (a, e4)|a=1.
Subregion B-ii: e4 < e <
w , the buyer’s profit is:
p
(1− e)aD(p− w)− γa 0 6 a 6 âΠB−iiB (a, e) =  .eD(p− w) + (1− e)(1− a)(−wD) + (1− e)aD(p− w)− γa â < a 6 1
Whole function is piecewise linear and continuous in a. The buyer’s profit is
differentiable in each piece and slope of the second piece is greater than slope of
the first piece .

∂ΠB−ii B (a, e) =
∂a D(1− e)(p− w)− γ 0 6 a 6 âDp(1− e)− γ â < a 6 1
Similar to above regions, we compare the value ΠB−iiB (a, e)|a=0 = 0 and
ΠB−iiB (a, e)|a=1 = D(p − w) − γ. By comparing these two values we find for any
136
given e while e w4 < e < :p
w
If γ 6 D(p− w) then a∗ = 1 for all e4 < e < .
p
w
If γ > D(p− w) then a∗ = 0 for all e4 < e < .
p
Subregion B-iii: w 6 e 6 1, the buyer’s profit is:
p
ΠB−iiiB (a, e) =eD(p− w) + (1− e)(1− a)(−wD) + (1− e)aD(p− w)− γa,
for all 0 6 a 6 1.
∂ΠB−iii(a,e)
This function is linear in a and B = Dp(1−e)−γ. Similar to above subregions
∂a
we compare the value of function at zero and one. ΠB−iiiB (a, e)|a=0 = D(ep−w) and
ΠB−iiiB (a, e)|a=1 = D(p − w) − γ. By comparing these two values we find that if
e 6 1 − γ ⇒ a∗ = 1 and if e > 1 − γ ⇒ a∗ = 0. As we previously defined
Dp Dp
e = 1− γ then for any given w3 6 e 6 1 :Dp p
w
If 6 e3 6 1 , or equivalently 0 < γ 6 D(p− w)
p
1 w 6 e < ep 3
then a∗ = 
any value in [0, 1] e = e .
 3
0 e3 < e 6 1
w w
If e3 < < 1 , or equivalently γ > D(p−w) then a∗ = 0 for all 6 e 6 1.
p p
We assemble above subregions to form the general buyer’s best response function for
Region B. Recall that Region B is defined by β(p−w) < D < β− − . Therefore, the γp(w c) w c
137
thresholds that we find for B-i, B-ii, and B-iii follow this order: D(p−w)[1−D(w−c) ] <
β
D(p − w). We assemble the above cases based on this characteristic and we label
them by B − 1 to B − 3:
B − − − D(w − c)1 : 0 < γ 6 D(p w)[1 ], we have e1 < e4
β
w
and e1 < < e3.p 

0 0 6 e < e
 11 e1 6 e < e3
Therefore, a∗ =  .
any value in [0, 1] e = e3
0 e3 < e 6 1
B − − − D(w − c)2 : D(p w)[1 ] < γ 6 D(p− w), we have e4 < e1
β
w
and e4 < < e3.
p 0 0 6 e < e
 4
1 e 6 e < eTherefore, a∗ = 
4 3
.
any value in [0, 1] e = e 30 e3 < e 6 1
w
B − 3 : γ > D(p− w), we have e4 < e1 and e3 < .
p
Therefore, a∗ = 0 for all 0 6 e 6 1.
Region C: (D > β
w− )c
For region C, as we show in Figure 21 we have two subregions for a given e: 0 6 e 6
w , and w < e 6 1 . We write the buyer’s objective function for each of sub-cases of
p p
138
e .
Subregion C-i: 0 6 e 6 w , the buyer’s profit is:
p
(1− e)aD(p− w)− γa 0 6 a 6 â
ΠC−iB (a, e) =  .eD(p− w) + λ− γa â < a 6 1
where, λ = (1 − e)(1 − a)(−wD) + (1 − e)aD(p − w) .Whole function is piecewise
linear and continuous in a. The buyer’s profit is differentiable in each piece and slope
of the second piece is greater than slope of the first piece.

∂ΠC−i D(1− e)(p− w)− γ 0 6 a 6 âB (a, e) =
∂a Dp(1− e)− γ â < a 6 1
We compare the value of ΠC−iB (a, e) at zero and one. Π
C−i
B (a, e)|a=0 = 0 and
ΠC−iB (a, e)|a=1 = D(p − w) − γ. By comparing these two values we find for any
given e while 0 6 e 6 w :
p
w
If γ 6 D(p− w) then a∗ = 1 for all 0 6 e 6 .
p
If γ > D(p− ww) then a∗ = 0 for all 0 6 e 6 .
p
Subregion C-ii: w < e 6 1, the buyer’s profit is:
p
ΠC−iiB (a, e) = eD(p−w)+(1−e)(1−a)(−wD)+(1−e)aD(p−w)−γa for all 0 6 a 6 1.
∂ΠC−ii(a,e)
This function is linear in a and B = Dp(1 − e) − γ. We compare the value
∂a
of ΠC−iiB (a, e) at zero and one. Π
C−ii
B (a, e)|a=0 = D(ep − w) and Π
C−ii
B (a, e)|a=1 =
139
D(p− w)− γ. By comparing these two values we find that if e 6 1− γ ⇒ a∗ = 1
Dp
and if e > 1 − γ ⇒ a∗ = 0. As we previously set 1 − γ equal to e3, then for anyDp Dp
given w 6 e 6 1 :
p
w
If 6 e3 6 1 , or equivalently 0 < γ 6 D(p− w) then
p
1 w < e < ep 3
a∗ = any value in [0, 1] e = e . 30 e3 < e 6 1
w − wIf e3 < < 1 , or equivalently γ > D(p w) then a∗ = 0 for all 6 e 6 1.
p p
We can assemble above subregions to form the general buyer’s best response function
for Region C. We label these regions by C − 1 and C − 2:
1 0 6 e < e3
C − 1 : if 0 < γ 6 D(p− w) then a∗ = any value in [0, 1] e = e . 30 e3 < e 6 1
C − 2 : if γ > D(p− w) then a∗ = 0 for all e ∈ [0, 1].
We put together final conditions of region A,region B, and region C and we draw
them on the plane of D and w as we show in Figure 23.
One can see that the buyer’s best response structure is the same in the following
regions. Therefore, we combine these regions into regions 1 through 6 using the
following process:
– Region 1: A-1-d ∪ A-2-d ∪ B-3 ∪ C-2
140
FIGURE 24. The Buyer’s Best Response Function Parameter Space Based on
Demand and Wholesale Price
– Region 2: A-1-c
– Region 3: A-1-b ∪ A-2-b
– Region 4: A-1-a ∪ A-2-a ∪ B-1
– Region 5: A-2-c ∪ B-2
– Region 6: C-1
We show these new regions in Figure 23. Hence, we prove the Lemma.
Lemma 16. For any given buyer’s auditing effort a, the supplier’s best response
(S-BRF), e∗(a) is as follows for each of the regions in Figure 25:
141
– In region A-i, e∗(a) is in one of the following forms: S-BRF 1-1 or S-BRF 1-2;
– In region A-ii, e∗(a) has a unique form of S-BRF 2-2. Note that when c >
p(2β−α)2
2 , then region A-ii vanishes;4β
– In region B-i, e∗(a) is in one of the following forms: S-BRF 1-1, S-BRF 1-2,
S-BRF 1-3, S-BRF 1-4, or S-BRF 1-5;
– In region B-ii, e∗(a) is in one of the following forms: S-BRF 2-1, S-BRF 2-3,
S-BRF 2-4, or S-BRF 2-5;
– In region C, e∗(a) is S-BRF 2-1,
where the possible forms of S-BRF e∗(a) are:
142
S-BRF 1-1) e∗ = 0 for all a in [0, 1].
S-BRF 1-2) e∗ = ê for all a in [0, 1].0 0 6 a < aB−I2
S-BRF 1-3) e∗ = ê aB−I 2 6 a < a
B−I .
 10 aB−II1 6 a 6 1
0 0 6 a < a
B−I
 2ê aB−I 6 a < pα
∗  2 2β(p−w)+wαS-BRF 1-4) e = 
.
ē pα
2
− 6 a <
α
 2β(p w)+wα 4β(α−D(w−c)) α20 4β(α−
6 a 6 1
D(w−c))
ê 0 6 a < pα2β(p−w)+wα
S-BRF 1-5) e∗ =  pα B−II
ē .
2β(p− 6 a < aw)+wα 2
ê aB−II2 6 a 6 1
143
ê 0 6 a <
pα
∗  2β(p−w)+wαS-BRF 2-1) e =  .ē pα 6 a 6 1
2β(p−w)+wα

S-BRF 2-2) e∗ =

ê 0 6 a < ȧ
.
ē ȧ 6 a 6 1ê 0 6 a < aB−II1
S-BRF 2-3) e∗ =  .ē aB−II1 6 a 6 1
 pα
ê 0 6 a < 2β(p−w)+wαē pα 6 a < aB−II
∗  2β(p−w)+wα 2S-BRF 2-4) e = 
.
ê aB−II 6 a < aB−II
2 1
ē aB−II1 6 a 6 1
0 0 6 a < aB−I2
S-BRF 2-5) e∗ = 
B−I pα
ê a2 6 a < . 2β(p−w)+wαē pα 6 a 6 1
2β(p−w)+wα
Also, the above-mentioned regions are illustrated in Figure 25 and defined as follows:
2 2
– Region A-i: {c < w 6 p(2β−α)2 and D 6 α(4β−α)− } or {
p(2β−α) < w <
4β 4β(w c) 4β2
p and D 6 β(p−w)}.
p(w−c)
2
– Region A-ii: {c < w 6 p(2β−α) and α(4β−α) < D 6 β(p−w)}.
4β2 4β(w−c) p(w−c)
2
– Region B-i: p(2β−α) β(p−w) α(4β−α)
4β2
< w < p and < D 6 .
p(w−c) 4β(w−c)
144
2 2
– Region B-ii: {c < w 6 p(2β−α)2 and β(p−w) < D < β } or {p(2β−α) < w <4β p(w−c) w−c 4β2
p and α(4β−α) < D < β− − }.4β(w c) w c
– Region C: D > β .
w−c
In addition, ê = 1−ap − , ē = 1−
α ,
w√ a 2aβ
α(4β(p−w)+wα+ w2α2+8(p−w)(wα−2Dp(w−c))β+16(p−w)2β2)
ȧ = ,
8D(p−w)(w−c)β
√
2
aB−I = −(w α−pw(2D(w−c)−α)−β(p−w)
2)− φ
1 2(w2
,
(D(w−c)−α))
√
aB−I = −(w
2α−pw(2D(w−c)−α)−β(p−w)2)+ φ
2 2 ,2(w (D(w−c)−√α))
− −(−[4αβ(p−w)+wα2])+ [(−[4αβ(p−w)+wα2])2−4(4Dβ(p−w)(w−c))(pα2B II )]a1 = ,√2(4Dβ(p−w)(w−c))
B−II −(−[4αβ(p−w)+wα2])− [(−[4αβ(p−w)+wα2])2−4(4Dβ(p−w)(w−c))(pα2)]a2 = , where φ =2(4Dβ(p−w)(w−c))
((w2α−pw(2D(w−c)−α)−β(p−w)2)2−4(w2(D(w−c)−α))(Dp2(w−c)−pwα)).
Also, the supplier’s best response function is continuous at a = pα− and is not2β(p w)+wα
continuous at α
2
− , ȧ, a
B−I , aB−II , and aB−II .
4β(α D(w−c)) 2 2 1
Proof of Lemma 16. Similar to the buyer’s best response function, for each
region of Figure 21, we plug the associated optimal order quantity in optimization
max ∗ ∗0≤e≤1 ΠS(wp, qp, wf , qf , e, a), where wf = wp = w. Then, solve for the supplier’s
optimal compliance effort e. We start with region A, then region B, and finally we
find the e∗ in region C.
Region A: 0 6 D 6 β(p−w)
p(w−c)
For region A of Figure 21, the supplier’s profit ΠAS (a, e) is as following. Note that we
def
define ê = 1−ap − . Note that ê is the inverse function of â in Lemma 3.a
w
−αe 0 6 e < ê
ΠAS (a, e) = 
[e+ (1− e)(1− a)]D(w − c)− αe ê 6 e < e (A.1)4
[e+ (1− e)(1− a)]D(w − c) + Λ− αe e4 6 e 6 1
145
FIGURE 25. Different Cases for the Supplier’s Best Response Function Based on
plane of (w,D)
where Λ = (1 − e)a[D(w − c) − β(1 − e)]. The function ΠAS (a, e) is not continuous
(1− a)wα
at ê, because lim ΠAS = − and ΠAS (a, e)|
(1−a)(wα−Dp(w−c))
e=ê = − − , whilee→ê− p− aw p aw
ΠAS (a, e)|e=ê > lim ΠAS . Also, the supplier’s profit function which is shown in (A.1)
e→ê−
is continuous at e4. Derivative of the supplier’s profit with respect to e is:


∂ΠA(a, e) 
−α 0 6 e < ê
S =
∂e Da(w − c)− α ê 6 e < e (A.2) 42a(1− e)β − α e4 6 e 6 1
146
From (A.2) we find that at e = e4 between the 2nd and 3rd intervals, the derivative
∂ΠA ∂ΠA ∂ΠA
increasing across the boundary, or equivalently, S |e=(e )− < S |e=e4 also S | =∂e 4 ∂e ∂e e=1
−α < 0. Also, the third piece of function (A.1) is concave in e. Therefore, there
are three candidate optimal solutions: e = 0, e = ê, and a local optimal in the third
interval (e = ē = 1− α ).
2aβ
FIGURE 26. Possible Forms of the Supplier’s Profit Function in Region A
Now we can divide the Region A of the supplier’s problem to three sub regions based
∂ΠA ∂ΠA
on order of zero, S | − , and Se=(e ) |e=e4 . We show graphically these three cases in∂e 4 ∂e
Figure 26. Three cases are:
∂ΠA A
I) S | ∂Πe=(e − 6 S | 6 0,∂e 4) ∂e e=e4
∂ΠA A
II) S | ∂ΠS
∂e e=(e
− 6 0 < | , and
4) ∂e e=e4
∂ΠA ∂ΠA
III) 0 < S | Se=(e )− 6 |∂e 4 ∂e e=e4 .
Before we analyze A-I, A-II, and A-III, we present all sub-regions in them and their
associated candidate solutions in Table 1.
Sub-Region Candidate Solutions
A-I 0, and ê
A-II 0, ê, and ē
A-III 0, and ē
TABLE 1. Different Subregions in Region A and Associated Candidate Solutions
147
In our analysis, we assume that at all levels of compliance effort e, sum of the
compliance effort cost and the associated correction cost to subsequently become
compliant (after the fact) is greater than the cost of being fully complaint upfront.
This means αe+ β(1− e) > α, therefore α < β.
Subregion A-I:
∂ΠAS | ∂Π
A
S
e=(e − 6 |
∂e 4
) e=e
∂e 4
6 0 (A.3)
or equivalently, Da(w − c)− α 6 2Da(w − c)− α 6 0.
In this case as we show in Figure 26, ΠAS is decreasing in both intervals of e ∈ (0, ê)
and e ∈ (ê, 1). So, we compare the value of function at 0 and ê. ΠAS |e=0 = 0 and
ΠA| = (1−a)(Dp(w−c)−wα)S e=ê − . From Π
A
S | Ap aw e=ê < πS |e=0 we have:
(1− a)(Dp(w − c)− wα) ⇔ wα< 0 Dp(w − c)− wα < 0⇔ p < .
p− aw D(w − c)
We rewrite the above condition using the definition of e4:
wα ⇔ wα wαp < p < ⇔ p < . (A.4)
D(w − c) β(1− (1− D(w−c))) β(1− e4)
β
The main condition of Region A is D 6 β(p−w)− , we rewrite it this condition as:p(w c)
w w
− 6 p⇔ 6 p. (A.5)1− D(w c) e4
β
Therefore, from (A.4) and (A.5) we conclude that in Region A, the supplier’s profit
at e = 0 is greater the supplier’s profit at e = ê if and only if w 6 p < wα . For
e4 β(1−e4)
148
this condition to be valid we must have w 6 wα
e4 β(1− :e4)
w wα ⇔ αβ6 D 6 . (A.6)
e4 β(1− e4) (w − c)(α + β)
Now we consider the condition in Region A-I, if condition (A.3) holds then a 6
α
− . Note that
α
− can be less or greater than 1 and 0 6 a 6 1. We find the2D(w c) 2D(w c)
condition for order of α vs. 1:
2D(w−c)
α α
If 6 1, then D > . (A.7)
2D(w − c) 2(w − c)
α α
If > 1, then D < . (A.8)
2D(w − c) 2(w − c)
Also, by definition of our parameters, we have this relationship between boundaries
of (A.7) and (A.6): α− <
αβ
− . Therefore, if (A.7) holds, then for all a2(w c) (w c)(α+β)
in [0, α− ] e
∗ = 0. If (A.8) holds, then for all a in [0, 1] e∗ = 0. A-1 and A-2
2D(w c)
summarize conditions of πAS |e=0 > ΠAS |e=ê: 
− w wα A 1 : if 6 p 6 and e4 β(1− e4) 
α 
0 6 D 6 
2(w − c) 
for all 0 6 a 6 1 
∗
− w wαA 2 : if 6 p 6 and  =⇒ e = 0 .
e4 β(1− e4) 
α αβ 
6 D 6 
2(w − c) (w − c)(α + β) 
α 
6 6 

for all 0 a 
2D(w − c)
149
If the condition (A.3) holds and πAS |e=0 6 ΠAS |e=ê then e∗ = ê. Similar to the above
case, we derive the conditions based on our model parameters. A-3 to A-5 present
condition for πAS | Ae=0 6 πS |e=ê: 
− w αβ βA 3 : if 6 p and 6 D 6 
e4 (w − c)(α + β) w − c α 
for all 0 6 a 6 
2D(w − c) 
− wα α αβ

A 4 : if 6 p and 6 D 6 
β(1− e4) 2(w − c) (w − c)(α + β)
α  =⇒ e∗ = ê.
for all 0 6 a 6 
2D(w − c) 
− wα αA 5 : if 6 p and 0 6 D 6
− − β(1 e4) 2(w c) 
for all 0 6 a 6 1 
We can re-write the condition A-3 and label them as:
A-3-i) if w 6 p and αβ 6 D 6 α(4β−α)− for all 0 6 a 6
α .
e4 (w c)(α+β) 4β(w−c) 2D(w−c)
A-3-ii) if w 6 p and α(4β−α)− 6 D 6
β
− for all 0 6 a 6
α .
e4 4β(w c) w c 2D(w−c)
Subregion A-II:
∂ΠAS | ∂Π
A
e=(e )− 6 0 <
S |e=e4 (A.9)∂e 4 ∂e
or equivalently, Da(w − c)− α 6 0 < 2Da(w − c)− α.
∂πA ∂πA
In this case, as S | S
∂e e=e4
> 0 and |
∂e e=1
= −α < 0, thus we can use concavity
characterization of third piece of πAS (a, e) which implies one local maximum(ē =
1 − α ) for function (A.1). In addition we have two more local maxima; we know
2aβ
that as the second piece of function (A.1) is decreasing in e based on condition
(A.9) then ê is another candidate for optimal solution. Likewise as the first piece of
function (A.1) is decreasing in e so e = 0 is the third candidate. Therefore, we can
150
compare the value of the supplier’s profit function for these three candidates to find
the optimal solution under the condition of Region A and condition (A.9).
ΠAS |e=0 =0
ΠA| α(4aβ − α)S e=ē =D(w − c)− 4aβ
A| (1− a)(Dp(w − c)− wα)ΠS e=ê = p− aw
If the condition (A.9) holds and ΠA| A A AS e=0 > ΠS |e=ê and ΠS |e=0 > ΠS |e=ē then e∗ = 0
. Similar to case A-I, we derive the condition based on our model parameters:
− w wα α αβA 10 : if < p < and < D <
e4 β(1− e4) 2(w − c) (w − c)(α + β)
α
for all 6 a 6 1 =⇒ e∗ = 0.
2D(w − c)
151
If the condition (A.9) holds and ΠA| A A AS e=ê > ΠS |e=0 and ΠS |e=ê > ΠS |e=ē then
e∗ = ê . We derive the conditions based on our model parameters as follows:

− w α(4β − α) βA 8 : if < p and < D < 
e4 4β(w − c) w − c α
for all 6 a 6 ȧ 
2D(w − c) 
w 2αβ α(4β − α) 
A− 9 : if < p and < D < 
e4 (w − c)(2β + α) 4β(w − c) 
α α2 
for all 6 a 6 
2D(w − c) 4β(β(e 4 − 1) + α) 
wα α αβ 
A− 11 : if < p and < D < 
β(1− e4) 2(w − c) (w − c)(α + β)
α  =⇒ e∗ = ê,
for all 6 a 6 1 
2D(w − c) 
− w αβ 2αβ

A 12 : if < p and < D < 
e4 (w − c)(α + β) (w − c)(2β + α) 
α 
for all 6 a 6 1 
2D(w − c) 
− w 2αβ α(4β − α)

A 13 : if < p and < D < 
e4 (w − c)(2β + α) 4β(w − c) 
α2 
for all 6 a 6 1 
4β(β(e4 − 1) + α)
√
α(4β(p−w)+wα+ w2α2+8(p−w)(wα−2Dp(w−c))β+16(p−w)2β2)
where ȧ = .
8D(p−w)(w−c)β
If the condition (A.9) holds and ΠA A A A ∗S |e=ē > ΠS |e=0 and ΠS |e=ē > ΠS |e=ê then e = ē .
We derive the conditions based on our model parameters as follows:

− w α β A 6 : if < p and < D < 
e4 (w − c) w − c 
α 
for all ȧ 6 a 6 
D(w − c)
 =⇒ e
∗ = ē.
− w α(4β − α) αA 7 : if < p and < D <
e4 4β(w − c) w − c 
for all ȧ 6 a 6 1 
152
Subregion A-III:
∂ΠA ∂ΠA
0 < S |e=(e )− 6 S |e=e4 (A.10)∂e 4 ∂e
or equivalently, 0 < Da(w − c)− α 6 2Da(w − c)− α.
∂ΠA
In this subregion, second piece of function (A.1) is increasing in e as S |e=(e )− > 0.∂e 4
∂ΠA ∂ΠA
In addition we know that S | > 0 and S | = −α < 0 thus we can
∂e e=e4 ∂e e=1
use concavity characterization of third piece of ΠAS (a, e) which implies one local
maximum(ē = 1 − α ) for function (A.1) for all e ∈ [e4, 1]. Likewise as the first2aβ
piece of function (A.1) is decreasing in e so e = 0 is another candidate for global
maximum. Therefore for finding the global maximum of function (A.1) based on
condition of this subregion, we need to compare the value of function at two points:
ΠAS |e=0 =0,
A| − − α(4aβ − α)ΠS e=ē =D(w c) .4aβ
Holding condition (A.10), we find that D(w − c)− α(4aβ−α) > 0. Therefore,:
4aβ
A− α w14 : for all 6 a 6 1 if < p
D(w − c) e4
α β
and < D < =⇒ e∗ = ē.
w − c w − c
We assemble cases A-1 through A-14 to form general behavior of the supplier’s best
response function with associated conditions in Region A. First, we take the union
of the cases to complete the interval of the a from zero to one. Then, we take the
union of them based on fixed intervals of D and p. We show these steps for each
case as follows.
153
– Combine A-2 with A-10, then combine the outcome with A-1. It gives us:
D < wα− and D <
αβ
− then e
∗ = 0 for all 0 6 a 6 1.
p(w c) (w c)(α+β)
– Combine A-4 with A-11, after that combine the outcome with A-11. It gives
us: if wα− < D <
αβ then e∗ = ê for all 0 6 a 6 1.
p(w c) (w−c)(α+β)
– First, combine A-9 with A-13. Then combine the outcome with A-12. Finally,
combine the result of previous combination with A-3-i. It gives us: if
αβ < D < α(4β−α)− − then e
∗ = ê for all 0 6 a 6 1.
(w c)(α+β) 4β(w c)
– First, combine A-6 with A-14. Combine the outcome with A-7. Then, again
combine the result with α(4β−α)A-8, and after that with A-3-ii. It gives us: if − < 4β(w c)ê 0 6 a < ȧD < β(p−w)− then e∗ =p(w c)  .ē ȧ 6 a 6 1
By looking at the above thresholds, we prove that in region A-i of statement of
Lemma 16, e∗(a) is in one of the following forms: S-BRF 1-1 or S-BRF 1-2; and in
region A-ii, e∗(a) has a unique form of S-BRF 2-2. To provide a general structure for
S-BRF in Region A, we compare the four thresholds wα αβ α(4β−α)
p(w− , − , , andc) (w c)(α+β) 4β(w−c)
β(p−w)
− , and we derive the optimal supplier’s best response function for Region A asp(w c)
following:
w(α + β) β(p− w)
if w < p 6 and 0 6 D 6 =⇒ e∗ = 0 for all a ∈ [0, 1]
β p(w − c)
w(α + β) 4wβ2
if < p <
 β (2β − α)2 and 0 6 D <
wα =⇒ e∗− = 0 for all a ∈ [0, 1] p(w c) and wα 6 D < β(p−w) =⇒ e∗ = ê for all a ∈ [0, 1]
p(w−c) p(w−c)
154

 and 0 6 D <
wα
− =⇒ e
∗ = 0 for all a ∈ [0, 1]
 p(w c)
4wβ2  and
wα 6 D < (4β−α)α =⇒ e∗ = ê for all a ∈ [0, 1]
 p(w−c) 4β(w−c)if 6 p (2β − α)2 

 ê 0 6 a < ȧ and (4β−α)α 6 D 6 β(p−w) =⇒ e∗ =4β(w−c) p(w−c) ē ȧ 6 a 6 1
We prove that in region A-i of statement of Lemma 16, e∗(a) is in one of the following
forms: S-BRF 1-1 or S-BRF 1-2; and in region A-ii, e∗(a) has a unique form of S-BRF
2-2. Before we start analysis of Region B, we present some characteristics of e∗ = ê
and e∗ = ē. The intersection of e∗ = ê and e∗ = ē happens at a = pα :
2β(p−w)+wα
ē = ê⇒ 1− α 1− a pα=
2aβ p
⇒ a = .
− a 2β(p− w) + wα
w
In addition to this, we know that e∗ = ê is decreasing and concave in a and e∗ = ē
is increasing and concave in a for all a ∈ (0, 1):
∂ê −w(p− w) ∂
2ê 2w2(p− w)
= < 0 , = − < 0,
∂a (p− aw)2 ∂a2 (p− aw)3
∂ē α ∂2ē α
= > 0 , = − < 0.
∂a 2a2β ∂a2 a3β
ê 0 6 a < ȧ
In e∗ =  as intersection happen in another point (i.e., a = ȧ)ē ȧ 6 a 6 1
function is not continuous in a. Also we observe a jump in the function at
a = ȧ based on above-mentioned characteristics of ê and ē as ȧ > pα ⇔
√ 2β(p−w)+wα
α(4β(p−w)+wα+ w2α2+8(p−w)(wα−2Dp(w−c))β+16(p−w)2β2) pα
8D(p−w)(w− − > 0 ⇔ 8(p −c)β 2β(p−w)+wα
w)β[wα + 2β(p − w) − 2Dp(w − c)] > 0. In the last inequality if [wα + 2β(p −
w) − 2Dp(w − c)] > 0 then we prove our claim: [wα + 2β(p − w) − 2Dp(w − c)] >
155
6 6 β(p−w)0 D
⇐⇒p(w−c)0 0 < D < 2β(p−w)+wα
2p(w− .c)
Region B: β(p−w) β
p(w− < D <c) w−c
For region B of Figure 21, we define two subregions and the supplier’s profit are
ΠB−IS (a, e) and Π
B−II
S (a, e) in each region respectively. In B-I when 0 6 a 6
p −
w
β(p−w)
− we see three different cases for optimal order quantities including (q
∗, q∗p f ) =Dw(w c)
(0, 0), (q∗p, q
∗
f ) = (0, D), and (q
∗
p, q
∗
f ) = (D,D). In B-II when
p − β(p−w) < a 6 1
w Dw(w−c)
we see three different cases for optimal order quantities including (q∗p, q
∗
f ) = (0, 0),
(q∗ ∗p, qf ) = (D, 0), and (q
∗
p, q
∗
f ) = (D,D).
Sub-region B-I: for all 0 6 a 6 p − β(p−w) B−I− , ΠS (a, e) =w Dw(w c)
−αe 0 6 e 6 e4
a(1− e)[D(w − c)− β(1− e)]− αe e4 < e < ê .[e+ (1− e)(1− a)]D(w − c) + a(1− e)[D(w − c)− β(1− e)]− αe ê 6 e 6 1
(B - I)
Sub-region B-II: for all p − β(p−w)− < a 6 1, Π
B−II
S (a, e) =w Dw(w c)
−αe 0 6 e < ê
[e+ (1− e)(1− a)]D(w − c)− αe ê 6 e < e . 4[e+ (1− e)(1− a)]D(w − c) + a(1− e)[D(w − c)− β(1− e)]− αe e4 6 e 6 1
(B - II)
Before we analyze B-I and B-II, we present all sub-regions in them and their
associated candidate solutions in Table 2.
Subregion B-I: The derivative of the supplier’s profit function with respect to e in
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Region Sub-regions Candidate Solutions
B-I 1 0 and ê
2-i 0 and ê
2-ii 0, ẽ, and ē
B-II 1 0 and ê
2 0, ê, and ē
3 0 and ê
TABLE 2. Different Subregions in Region B and Associated Candidate Solutions
subregion (B - I) is:
−α 0 6 e 6 e4
∂ΠB−

I
S (a, e)

=
∂e a[2β(1− e)−D(w − c)]− α e . (A.11) 4
< e < ê
2a(1− e)β − α ê 6 e 6 1
the supplier’s profit function in region (B - I) is continuous at e = e4, because
lim ΠB−IS (a, e) = Π
B−I
S (a, e)|e=e4 = −e4α. The function (B - I) is not continuouse→e4
at e = ê as lim ΠB−IS (a, e) < Π
B−I
S |e=ê. For showing this we can find the value of
e→ê−
ΠB−I | − lim ΠB−I D(1− a)p(w − c)S e=ê S (a, e) = > 0 where:
e→ê− p− aw
2
ΠB−IS |
aβ(p− w) (1− a)wα
e=ê =D(w − c)− − ,
(p− aw)2 p− aw
B−I (p− aw)[Da(p− w)(w − c)− (1− a)wα]− aβ(p− w)2lim ΠS (a, e) = .
e→ê− (p− aw)2
For each pair of e and a in [0, 1], the third piece of the supplier’s profit function
in (B - I) (defined over interval of ê 6 e 6 1 ) is always greater than second piece
of the supplier’s profit function in (B - I) (defined over interval of e4 < e < ê ) .
Also, ΠB−IS |e=0 = 0 and Π
B−I
S |e=e4 < 0. As we show in (A.11), Π
B−I
S (a, e) for all
e ∈ [0, e4] is linearly decreasing in e. Also second and third pieces of ΠB−IS (a, e) are
157
∂2ΠB−I(a,e)
concave in e because S = −2aβ < 0. From function (A.11) we verify that
∂e2
∂ΠB−IS (a,e)
B−I
| ∂ΠS (a,e) ∂Π
B−I(a,e)
e=1 < 0. We can show that |e=ê− < S |∂e ∂e ∂e e=ê where:
∂πB−IS (a, e) | 2a(p− w)βe=ê− = − α−Da(w − c),
∂e p− aw
∂πB−IS (a, e) | 2a(p− w)βe=ê = − α.
∂e p− aw
def
We define the the local maximum of third piece of function (B - I) as ē = 1 −
α . Then we compare the position of ē with ê to create two sub-regions for B-I as
2aβ
following. Subregions B-I-1 : ē > ê, and in sub-region B-I-2 : ē < ê.
B-I-1) when maximum of the third piece of piecewise function (B - I) happens at
any e ∈ [ê, 1]:
Combining the conditions that define Region B, B-I, and subregion B-I-1 (ē > ê),
then we have:
2β(p− w) + wα β pα p β(p− w)
6 D < and 6 a 6 − .
2p(w − c) w − c 2β(p− w) + wα w Dw(w − c)
(A.12)
We show different possible scenarios of the supplier’s profit function under condition
of B-I-1 (ē > ê) in Figure 27.
Recall that for any pair of e and a in [0, 1], the third piece of the supplier’s profit
function in (B - I) (defined over interval of ê 6 e 6 1 ) is always greater than second
piece of the supplier’s profit function in (B - I) (defined over interval of e4 < e < ê
). Therefore, if ē exists in [ê, 1] , then it is always the local optimal solution for all
e in [e4, 1]. Thus, under the condition of case B-I-1 to find the optimal solution, we
compare ΠB−I | B−I α(4aβ−α)S e=0 = 0 with ΠS |e=ē = D(w − c)− .4aβ
Under condition (A.12), if ΠB−I | < ΠB−I ∗S e=0 S |e=ē then e = ē . The procedure that
158
FIGURE 27. Possible Forms of the Supplier’s Profit Function in Region B-I-1
we follow to derive the e∗ is similar to that for Region A-I. Namely, we first derive
the critical condition for the two candidate solutions e = 0 and e = ē, which is
D(w − c)− α(4aβ−α) > 0 . Next, we identify the condition for this critical condition
4aβ
to hold in region B-I-1, which means comparing the critical condition with definition
of Region B-I-1, 2β(p−w)+wα− 6 D <
β
− and
pα
− 6 a 6
p − β(p−w) .
2p(w c) w c 2β(p w)+wα w Dw(w−c)
Finally, we combine these two conditions to generate various scenarios of optimal
159
best response function. We skip the details and present the best response function.

2αβ α(4β − α) 
(P1) : if 6 D < 
(w − c)(2β + α) 4β(w − c) 
wα(2β − α)
and 6 p < p̂ 
2β[2D(w − c)− α] 

pα α2
for all 6 a 6 
2β(p− w) + wα 4β(β(e 4 − 1) + α)
2αβ α(4β − α) 
(P2) : if 6 D < 
(w − c)(2β + α) 4β(w − c) 
w(2β − α)
and p̂ 6 p <
2βe4 
=⇒ e∗ = ē,
pα p β(p− w)
for all 6 a 6 − 
2β(p− w) + wα w Dw(w − c) 
α(4β − α) β 
(P3) : if 6 D <4β(w − c) w − c 
w(2β − α) 
and w 6 p < 
2βe4
pα p β(p− w) 
for all 6 a 6 − 
2β(p− w) + wα w Dw(w − c)
(A.13)
D(w−c)α2 2
def w[ +4β ]
where p̂ = D(w−c)−α .
4β[β−D(w−c)]
If the condition (A.12) holds and ΠB−IS | > Π
B−I
e=0 S |e=ē then e∗ = 0 . We derive the
conditions based on our model parameters similar to preceding case as follows:
160

α 2αβ 
(P4) : if 6 D < 
2(w − c) (w − c)(2β + α) 
w(2β − α) 
and w 6 p < 
2βe4 
pα p β(p− w) 
for all 6 a 6 − 
2β(p− w) + wα w Dw(w − c) 

2αβ α(4β − α) 
(P5) : if 6 D < 
(w − c)(2β + α) 4β(w − c) 
wα(2β − α) 
and 6 p < p̂ =⇒ e∗ = 0.
2β[2D(w − c)− α]
α2 p − β(p− w)6 6 

for all a
4β(β(e4 − 1) + α) w Dw(w − c)
2αβ α(4β − α) 
(P6) : if 6 D < 
(w − c)(2β + α) 4β(w − c) 
wα(2β − α) 
and w 6 p < 
2β[2D(w − c)− α] 
pα p β(p− w) 
for all 6 a 6 − 
2β(p− w) + wα w Dw(w − c)
(A.14)
B-I-2) when maximum of the third piece of function (B - I) happens at any e ∈
(−∞, ê).
When condition B-I holds, the condition of subregion B-I-2 (ē < ê) is equivalent to:

(1)2β(p−w)+wα 6 D < β and 0 6 a 6 pα2p(w−c) w−c 2β(p−w)+wα
ē < ê⇔  or . (B-I-2)(2)β(p−w) 6 D < 2β(p−w)+wα and 0 6 a 6 p − β(p−w)
p(w−c) 2p(w−c) w Dw(w−c)
def
We define the local maximum of the second piece of function (B - I) as ẽ =
B−I
1 − 1 D(w − c) − α ∂π (a,e). In B-I-2 S |e=ê < 0 and recall that one of the2β 2aβ ∂e
161
∂πB−IS (a,e) ∂π
B−I
| S (a,e)characteristics of function (B - I) is that
∂e e=ê
− < |
∂e e=ê
. Therefore,
∂πB−IS (a,e) |e=ê− < 0 , and ẽ can be either in [e4, ê] or in (−∞, e4). So we divide the∂e
region B-I-2 into two sub-regions based on position of ẽ vs. e4. We illustrate both
sub-regions B-I-2-i and B-I-2-ii in Figure 28. In subregion B-I-2-i, ẽ < e4 and in
sub-region B-I-2-ii, ẽ > e4.
FIGURE 28. Possible Forms of the Supplier’s Profit Function in Region B-I-2
Suppose that the condition B-I-2 holds and in addition the condition of B-I-2-i
(ẽ < e4) holds too, we simplify these conditions and summarize it as follows: The
procedure that we follow to derive the e∗ is similar to that for Region A-I. Namely,
we first derive the critical condition for the two candidate solutions e = 0 and
e = ē, which is D(w − c) − α(4aβ−α) > 0 . Next, we identify the condition for
4aβ
this critical condition to hold in region B-I-1, which means comparing the critical
condition with definition of Region B-I-2, {2β(p−w)+wα 6 D < β and 0 6 a 6
2p(w−c) w−c
pα } or {β(p−w) 6 D < 2β(p−w)+wα− − − and 0 6 a 6
p − β(p−w) }. Finally, we
2β(p w)+wα p(w c) 2p(w c) w Dw(w−c)
combine these two conditions to generate various scenarios of optimal best response
162
function. We skip the details and present the best response function.
− {2β(p w) + wα 2β(p− w) + wα(P1) : 6 D < 
2p(w − c) p(w − c) 
− α pα

and w 6 p < w(2 ) and 0 6 a < } or 
β 2β(p− w) + wα 
{2β(p− w) + wα β

(P2) : 6 D < 
2p(w − c) w − c α pα
and w(2− ) 6 p and 0 6 a < } or 
β 2β(p− w) + wα 
− (B-I-2-i){2β(p w) + wα β

(P3) : 6 D < 
p(w − c) w − c 
α α 
and w 6 p < w(2− ) and 0 6 a < } or 
β D(w − c) 
β(p− w) 2β(p− w) + wα 
(P4) : { 6 D < 
p(w − c) 2p(w − c) 
p β(p− w) 
and w 6 p and 0 6 a 6 − }. 
w Dw(w − c)
In subregion B-I-2-i the supplier’s function (B - I) is decreasing in e for all e ∈ [0, ê).
We show this in Figure 28. We have two candidates for optimal solutions. To find the
2
maximum of function (B - I) we compare ΠB−IS |e=ê = D(w − c)−
(1−a)wα − aβ(p−w)
p−aw (p−aw)2
with ΠB−IS |e=0 = 0.
To compare these two values, we write the supplier’s indifference equation,
ΠB−I | = D(w−c)− (1−a)wα aβ(p−w)
2 B−I 2
S e=ê − − −Π |p aw (p−aw)2 S e=0 = 0, based on a like ηa +νa+ξ =
def def
0 where η = w2(D(w − c)− α) , where ν = w2α− pw(2D(w − c)− α)− β(p− w)2
√
def
and ξ = Dp2(w − c) − pwα. Solutions of this equation are: aB−I = −ν− ∆1 and2η
√
aB−I −ν+ ∆2 = , where ∆ = ν
2 − 4ηξ. Now we look at each component of subregion
2η
B-I-2-i and use the above indifference equation’s characteristic to find the supplier’s
163
best response function in this subregion. For B-I-2-i part (1) 2:
If ∆ < 0⇒ πB−IS |
B−I
e=ê < πS |e=0 =⇒
∗ pαe = 0 for all a ∈ [0, ).
2β(p− w) + wα
(B-I-2-i-P1S1 )
In this case there is no need to check the positivity or negativity of η because ∆ < 0
implies that η will be negative. We prove it by showing ∆ < 0 enforces 2β(p−w)+wα <
2p(w−c)
p2D < β
2+w2(β−α)2−2pwβ(β+α)
− − and η < 0 implies that D <
α
− . If we show that4pwβ(w c) w c
p2β2+w2(β−α)2−2pwβ(β+α)
− <
α then we have the proof:
4pwβ(w−c) w−c
p2β2 + w2(β − α)2 − 2pwβ(β + α) α
<
−4pwβ(w − c) w − c
⇔ p2β2 + w2(β − α)2 − 2pwβ(β + α) > −4pwαβ
⇔ p2β2 + w2(β − α)2 − 2pwβ(β + α) + 4pwαβ > 0
⇔ [βp− w(β − α)]2 > 0.
When ∆ > 0 and we have feasible solutions for the supplier’s indifference equation,
√ √
η > 0
if η > 0 then aB−I B−I1 < a2 . Proof of that is straight forward:
−ν− ∆ < −ν+ ∆ ⇐⇒
2η 2η
√ √
aB−I < aB−I
η < 0
1 2 . Similarly,
−ν− ∆ > −ν+ ∆ ⇐⇒ aB−I1 > aB−I2η 2η 2 .
Consider the case ∆ > 0 and η > 0 then based on condition B-I-2-i part (1) we
can show that pα < aB−I . In this case as η > 0 the value of the supplier’s
2β(p−w)+wα 1
indifference equation will be positive for all a ∈ (−∞, aB−I1 ) which implies πB−IS |e=ê >
2B-I-2-i-P1S1 represents subregion B-I-2-i part (1) sub-case(1)
164
πB−IS |e=0, so
pα
if ∆ > 0 and η > 0⇒ < aB−I1 < aB−I2β(p− w) + wα 2
=⇒ e∗ pα= ê for all a ∈ [0, ). (B-I-2-i-P1S2 )
2β(p− w) + wα
We prove the result B-I-2-S2 by simplifying the condition; ∆ > 0 and η > 0
implies that:
α 2β(p− w) + wα
< D < for all (A.15)
w − c p(w − c)
β α β w(2β − α)
(α > and p < w(2− )) or (α < and p 6 )
2 β 2 2(β − α)
or
2β(p− w) + wα 2β(p− w) + wα
< D < for all (A.16)
2p(w − c) p(w − c)
β α w(2β − α)
α < and p < w(2− ) and p > .
2 β 2(β − α)
Now if we assume that pα > aB−I− , it implies that: D <
α[2β(p−w)−wα] .
2β(p w)+wα 1 4pβ(w−c)
Using condition (A.15), α[2β(p−w)−wα] α− < − . Using condition (A.16),
α[2β(p−w)−wα] <
4pβ(w c) w c 4pβ(w−c)
2β(p−w)+wα pα
2p(w− therefore under all conditions of B-I-2-i-S2 we cannot have >c) 2β(p−w)+wα
aB−I1 .
165
Similarly, while η < 0:
pα
If ∆ > 0 and η < 0 and 0 < aB−I2 < < aB−I2β(p− w) + wα 1
0 0 6 a < a
B−I
∗  2=⇒ e =  .ê aB−I 6 a < pα2 2β(p−w)+wα
(B-I-2-i-P1S3 )
pα
If ∆ > 0 and η < 0 and 0 < aB−I < aB−I2 1 < 2β(p− w) + wα0 0 6 a < aB−I2
=⇒ e∗ = ê aB−I 6 a < aB−I . 2 10 aB−I1 6 a < pα2β(p−w)+wα
(B-I-2-i-P1S4 )
pα
If ∆ > 0 and η < 0 and aB−I2 < 0 < < a
B−I
2β(p− w) + wα 1
=⇒ ∗ pαe = ê for all a ∈ [0, ).
2β(p− w) + wα
(B-I-2-i-P1S5 )
For B-I-2-i-(P2) as 2β(p−w)+wα 6 D < β− − and rest of parameter conditions2p(w c) w c
are identical to (P1) we can derive the optimal supplier’s response like part (1).
Conditions for ∆ and η are similar to (P1) therefore we will have subregions B-I-2-
i-P2S1 to B-I-2-i-P2S5 . For B-I-2-i-(P3), condition implies that ∆ > 0 η > 0. We
166
show this by simplifying ∆ > 0 :
∆ =(p− w)2[p2β2 − 2pwβ(2cD + α + β) + w2((β − α)2 + 4Dpβ)] > 0
⇔ 2pwβ(β + α)− p
2β2 − w2(β − α)2
D > .
4pwβ(w − c)
2 2 2 2
and under model assumptions showing 2pwβ(β+α)−p β −w (β−α) < 2β(p−w)+wα is
4pwβ(w−c) p(w−c)
straightforward where 2β(p−w)+wα− is the lower bound of B-I-2-i part (3). for η > 0p(w c)
we show that:
α
η = Dw2(w − c)− w2α > 0⇔ D > .
(w − c)
In B-I-2-i-(P3), 2β(p−w)+wα− 6 D <
β
− ; based on above inequality and using
α <
p(w c) w c (w−c)
β
− we can conclude that η > 0 under this condition. Similar to previous parts(w c)
of B-I-2-i we can show that with this characteristic aB−I < aB−I1 2 and consequently
0 < α− < a
B−I
1 < a
B−I
2 therefore:D(w c)
2β(p− w) + wα β α
if 6 D < and w 6 p < w(2− )
p(w − c) w − c β
=⇒ e∗ α= ê for all 0 6 a < .
D(w − c)
Last part of B-I-2-i is part (4) which we can divide it to four cases based on the
value of ∆ and η. Similar to proof of part(1), if ∆ < 0 then η < 0 and subsequently
πB−I | < πB−I | then e∗S e=ê S e=0 = 0. We can rewrite this as following:
if ∆ < 0⇒ πB−I | < πB−I | ⇒ ∗ ∈ p − β(p− w)S e=ê S e=0 = e = 0 for all a [0, ].w Dw(w − c)
(B-I-2-i-P4S1 )
167
If we have the case when ∆ > 0 and η > 0 then based on condition B-I-2-i part
(4) we can show that p − β(p−w)− < a
B−I
1 . In this case as η > 0 the value of thew Dw(w c)
supplier’s indifference equation will be positive for all a ∈ (−∞, aB−I1 ) which implies
πB−I | > πB−IS e=ê S |e=0 so:
p β(p− w)
if ∆ > 0 and η > 0⇒ 0 < − < aB−I < aB−I
w Dw(w − c) 1 2
⇒ p β(p− w)= e∗ = ê for all a ∈ [0, − ]. (B-I-2-i-P4S2 )
w Dw(w − c)
Similar to part(1), we have the following sub-cases while η < 0:
p β(p− w)
if ∆ > 0 and η < 0 and 0 < aB−I2 < aB−I1 6 −w Dw(w − c)
0 0 6 a < aB−I2
=⇒ e∗ = ê aB−I 6 a < a
B−I , (B-I-2-i-P4S3 )
 2 10 aB−I1 6 a 6 p − β(p−w)w Dw(w−c)
p β(p− w)
if ∆ > 0 and η < 0 and aB−I2 < 0 < − < aB−Iw Dw(w − c) 1
=⇒ ∗ p β(p− w)e = ê for all a ∈ [0, − ]. (B-I-2-i-P4S4 )
w Dw(w − c)
Next, consider sub-region B-I-2-ii which is when ẽ > e4. Suppose that the condition
B-I-2 holds and in addition the condition of B-I-2-ii (ẽ > e4) holds too. We simplify
168
these conditions similar to Region A and summarize it as follows:
2β(p− w) + wα β α
6 D < and w 6 p < w(2− )
p(w − c) w − c β
α pα
and 6 a < . (B-I-2-ii)
D(w − c) 2β(p− w) + wα
As we illustrate in Figure 28 and based on characteristics of the supplier’s function in
(B - I), we have three candidate solutions. To find the optimal solution, we compare
the value of function (B - I) at e = 0, e = ẽ, and e = ê where:
ΠB−IS |e=0 =0,
2
ΠB−I | − − aβ(p− w) (1− a)wαS e=ê =D(w c) − ,(p− aw)2 p− aw
2
ΠB−I | (Da(w − c) + α)S e=ẽ = − α.4aβ
Under condition B-I-2-ii, we prove below that πB−IS |
B−I B−I
e=ê > πS |e=0 and πS |e=ê >
πB−IS | ∗e=ẽ. Therefore e = ê for all a ∈ [ α
pα
D(w− ,c) 2β(p− ]. Now we show thatw)+wα
πB−IS |
B−I
e=ê > πS |e=0 :
πB−I | > πB−IS e=ê S |e=0
⇔ aβ(p− w)
2 (1− a)wα
D(w − c)− − > 0
(p− aw)2 p− aw
⇔ (1− a)w(p− aw)α + a(p− w)
2β
D > . (A.17)
(w − c)(p− aw)2
169
Based on lower bound of condition B-I-2-ii for D, it suffices to show that
(1−a)w(p−aw)α+a(p−w)2β < 2β(p−w)+wα
(w− :c)(p−aw)2 p(w−c)
(1− a)w(p− aw)α + a(p− w)2β 2β(p− w) + wα
[ √ <(w − c)(p− aw)2 p(w − c) ]
p w(3β − α) + pβ − w2α2 − 2w(p− w)αβ + (p− w)(p+ 7w)β2
⇔ a < [ √ 2w2(2β − α) ]
p w(3β − α) + pβ + w2α2 − 2w(p− w)αβ + (p− w)(p+ 7w)β2
or a > .
2w2(2β − α)
N[ow if we show that the lower bound ]of a in condition B-I-2-ii is less than√
p w(3β−α)+pβ− w2α2−2w(p−w)αβ+(p−w)(p+7w)β2
2 − then the proof is complete.2w (2β α)
[ √ ]
pα p w(3β − α) + pβ − w2α2 − 2w(p− w)αβ + (p− w)(p+ 7w)β2
<
2β(p− w) + wα 2w2(2β − α)
⇔ 0 < 4(p− w)w2(2β − α)β[2p(4β − α)β − w(2β − aα)(α + 4β)]
⇔ 0 < 2p(4β − α)β − w(2β − aα)(α + 4β)
wα2⇔ w − < p (A.18)
2β(4β − α)
Left-hand side of inequality A.18 is less than w, hence the proof is complete.
Similarly, we can prove that πB−I | > πB−IS e=ê S |e=ẽ.
Subregion B-II: when p − β(p−w)− < a 6 1, the supplier’s profit function inw Dw(w c)
region (B - II) is continuous at e = e4, because lim Π
B−II B−II
e→e S
(a, e) = ΠS (a, e4) =
4
D(w − c)[α + β −Da(w − c)] − α. The function (B - II) is not continuous at e = ê
β
as lim ΠB−IIS (a, e) < Π
B−II
S (a, e)|e=ê. For showing this we can find the value of
e→ê−
170
ΠB−IIS (a, e)|e=ê − lim Π
B−II
S (a, e) = Dp(w − c) > 0 where:
e→ê−
B−II (1− a)[Dp(w − c)− wα]ΠS (a, e)|e=ê = ,p− aw
lim ΠB−II
(1− a)wα
S (a, e) =− .
e→ê− p− aw
The derivative of the supplier’s function in with respect to 2 subregion (B - II) is:

−α 0 6 e < ê
∂ΠB−II S (a, e) =
∂e Da(w − c)− α ê 6 e < e . (A.19) 42a(1− e)β − α e4 6 e 6 1
Note that ΠB−IIS |e=0 = 0. As we can observe from (A.19), Π
B−II
S (a, e) for all e ∈
[0, ê) is linearly decreasing in e. Also second piece is linear in e and third pieces of
2 B−II
ΠB−II
∂ Π (a,e)
S (a, e) is concave in e because
S = −2aβ < 0. From function (A.19) we
∂e2
∂ΠB−II(a,e) ∂ΠB−II(a,e) ∂ΠB−II(a,e)
verify that S |e=1 < 0. We can show that S |e=(e )− < S |∂e ∂e 4 ∂e e=e4
where:
∂ΠB−IIS (a, e) |e=(e )− = Da(w − c)− α,
∂e 4
∂ΠB−IIS (a, e) |e=e4 = 2Da(w − c)− α.∂e
Now we can divide the Region B-II of the supplier’s problem to three sub regions
∂ΠB−II(a,e) ∂ΠB−II(a,e)
based on order of zero, S | − , and < S | . We show graphically
∂e e=(e4) ∂e e=e4
these three cases in Figure 29. Three cases are:
∂ΠB−II B−II
Sub-region B-II-1) S
(a,e) |e=(e )− 6
∂ΠS (a,e) |
∂e 4 ∂e e=e4
6 0,
∂ΠB−II B−II
Sub-region B-II-2) S
(a,e) | 6 ∂Π− 0 < S (a,e)
∂e e=(e )
| , and
4 ∂e e=e4
171
∂ΠB−IIS (a,e)
B−II
Sub-region B-II-3) 0 < | − 6 ∂ΠS (a,e) |
∂e e=(e4) ∂e e=e4
.
FIGURE 29. Possible Forms of the Supplier’s Profit Function in Region B-II
∂ΠB−IIS (a,e)
B−II
B-II-1) when | ∂ΠS (a,e)
∂e e=(e )
− 6 |
4 ∂e e=e4
6 0.
∂ΠB−II(a,e)
Under the condition of B-II we simplify the condition of S |
∂e e=(e )
− 6
4
∂ΠB−IIS (a,e) |
∂e e=e4
6 0:
 0 6 D < α− and w 6 p < w and p − β(p−w)− < a 6 12(w c) e4 w Dw(w c) or . α 6 D < β and wβ(2β−α) < p and p − β(p−w) < a < α
2(w−c) w−c 2e4 w Dw(w−c) 2D(w−c)
(B-II-1)
As we illustrate in Figure 29 and as a result of concavity of third piece of function
(B - II) and linearity of other pieces, we have two candidates for optimal solution:
e = 0 and e = ê. So under the condition B-II-1, we compare ΠB−IIS (a, e)|e=0 = 0
with ΠB−II(a, e)| = (1−a)[Dp(w−c)−wα]S e=ê − to find the optimal solution. If the conditionp aw
B-II-1 holds and ΠB−II | 6 ΠB−II | then e∗S e=ê S e=0 = 0 . The procedure that we
follow to derive the e∗ is similar to that for Region A-I. Namely, we first derive
the critical condition for the two candidate solutions e = 0 and e = ê, which is
172
(1−a)[Dp(w−c)−wα]
− > 0 . Next, we identify the condition for this critical condition top aw
hold in region B-I-1, which means comparing the critical condition with definition
of Region B-II-1, {0 6 D < α and w 6 p < w and p − β(p−w)− < a 62(w c) e4 w Dw(w−c)
1} or { α 6 D < β wβ(2β−α) p β(p−w) α
2(w−c) w− and < p and − < a < }.c 2e4 w Dw(w−c) 2D(w−c)
Finally, we combine these two conditions to generate various scenarios of optimal
best response function. We skip the details and present the best response function
as follows.

α w
(P1) : if 0 6 D < and w 6 p < 
2(w − c) e 4 
p β(p− w) 
for all − < a 6 1 
w Dw(w − c) 
α αβ 
(P2) : if 6 D < 
2(w − c) (w − c)(α + β) 
wβ(2β − α) w 
and 6 p < 
2e4 e4 =⇒ e∗ = 0.
p
for all − β(p− w) α < a 6
w Dw(w − c) 2D(w − c)αβ 2αβ
(P3) : if 6 D < 
(w − c)(α + β) (w − c)(2β + α) 
wβ(2β − α) wα 
and 6 p <
2e β(1− e ) 4 4
p 
for all − β(p− w) α < a 6 
w Dw(w − c) 2D(w − c)
173
If the condition B-II-1 holds and ΠB−II B−II ∗S |e=ê > ΠS |e=0 then e = ê . We derive the
conditions based on our model parameters similar to preceding case as follows:

2αβ β
(P4) : if 6 D < 
(w − c)(2β + α) w − c
− wβ(2β α) w 
and 6 p < 
2e4 e4
− 

p β(p w) α 
for all − < a 6 
w Dw(w − c) 2D(w − c)
=⇒ e∗ = ê.
αβ 2αβ 
(P5) : if 6 D < 
(w − c)(α + β) (w − c)(2β + α) 
wα w
and 6 p < 

β(1− e4) e4 
p β(p− w) α 
for all − < a 6 
w Dw(w − c) 2D(w − c)
∂ΠB−II(a,e) ∂ΠB−II(a,e)
B-II-2) when S | − 6 0 < S |
∂e e=(e4) ∂e e=e4
.
Under the condition of B-II we simplify the condition of
174
∂ΠB−II(a,e) ∂ΠB−IIS | 6 S (a,e)e=(e )− 0 < |∂e 4 ∂e e=e4 : 
{ α α wβ(2β − α) w (a) : < D 6 and < p < 
2(w − c) w − c 2e4 e 4 α 
and < a 6 1} 
2D(w − c) 
or 
{ α α wβ(2β − α)

(b) : < D 6 and w < p < 
− 2(w c) w − c 2e 4 
p − β(p− w)

and < a 6 1} 

w Dw(w − c) 
or  (B-II-2 )
α β wβ(2β − α) w 
(c) : { < D 6 and < p < 
w − c w − c 2e4 e4 α α 
and < a 6 } 
2D(w − c) D(w − c) 
or 
{ α β w(β − α) wβ(2β − α)(d) : < D 6 and 6 p <
− − w c w c βe4 2e4 
p β(p− w) α 
and − < a 6 }. 
w Dw(w − c) D(w − c)
We illustrate subregion B-II-2 in Figure B - II. Under condition B-II-2, the supplier’s
function (B - II) is linearly decreasing in e for all e ∈ [0, ê) and it’s not continuous
at e = ê (as we proved it previously). Also function is decreasing linearly in e for
def
all e ∈ [ê, e4) and it is concave with one local maximum (ē = 1− α ). Given these2aβ
characteristics of function under this condition, in each of B-II-2 sub-conditions we
compare the value of function (B - II) at e = 0 , e = ê and e = ē to find the
optimal solution. Thus, we compare ΠB−IIS (a, e)|e=0 = 0, with Π
B−II
S (a, e)|e=ê =
(1−a)[Dp(w−c)−wα] , and with ΠB−II− S (a, e)|e=ē = D(w − c) −
α(4aβ−α) under all sub-
p aw 4aβ
conditions of B-II-2.
175
Given that one of the values is zero, we can write the indifference equation with
two other profit functions as follows: ΠB−IIS (a, e)|
B−II
e=ē − ΠS (a, e)|e=ê = 0, we sort
def
this equation based on a and write it as ρa2 + τa + % = 0 where ρ = 4Dβ(p −
w)(w − def defc) and τ = −[4αβ(p− w) + wα2] and % = pα2. Following the definitions, ρ
which is the coefficient of a2 is positive and τ is negative for all defined parameters.
√ √
Solving this equation for a gives us two solutions aB−II = −τ+ δ and aB−II = −τ− δ1 2ρ 2 2ρ
where δ = τ 2 − 4ρ%. when δ > 0 and we have feasible solutions for the supplier’s
√
indifference equation,then aB−II > aB−II1 2 . Proof of that is straight forward:
−τ− δ <
2ρ
√
−τ+ δ ⇐ρ⇒> 0 aB−II2 < aB−II1 . Now we need to consider all possible cases for the2ρ
supplier’s profit value in each sub-case. Given that we start with B-II-2-part (a), if
πB−IIS (a, e)|e=ē > 0 then we name it condition B-II-2-a-1 and rewrite this condition
as:
(i) : α(4β−α) < D < α and wβ(2β−α) < p < w and α < a < 14β(w−c) w−c 2e4 e4 2D(w−c)
(ii) : 2αβ < D < α(4β−α) and wβ(2β−α) w < p < (w−c)(2β+α) 4β(w−c) 2e4 e4and α α2< a < .
2D(w−c) 4β(β(e4−1)+α)
(B-II-2-a-1 )
If condition B-II-2-a-1-i holds and ΠB−IIS (a, e)|e=ē −Π
B−II
S (a, e)|e=ê > 0 then e∗ = ē,
otherwise e∗ = ê. When B-II-2-a-1-i holds, similar to previous cases we can show
that aB−II2 <
α
− < a
B−II
1 < 1 which implies:2D(w c)
ê
α 6 a 6 aB−II
∗  2D(w−c) 1e =  (B-II-2-a-1-i)ē aB−II1 < a 6 1.
176
Likewise, if condition B-II-2-a-1-ii holds, then
B−II α α
2
a < < < 1 < aB−II2 ,2D(w − c) 4β(β(e4 − 1) + α) 1
∗ α α
2
which implies e = ê for all a in ( , ). (B-II-2-a-1-ii)
2D(w − c) 4β(β(e4 − 1) + α)
If condition B-II-2-a holds and ΠB−IIS (a, e)|e=ē < 0, then we name this condition
B-II-2-a-2 and rewrite it as follows:(i) : α < D < 2αβ and wβ(2β−α) w − − < p < and
α
 2(w c) (w c)(2β+α) 2e4 e4 2D(w−
< a < 1
c)
(ii) : 2αβ < D < α(4β−α) and wβ(2β−α) w α2
(w− − < p < and < a < 1.c)(2β+α) 4β(w c) 2e4 e4 4β(β(e4−1)+α)
(B-II-2-a-2 )
In this case if ΠB−IIS (a, e)|
B−II ∗
e=ē − ΠS (a, e)|e=ê > 0 then e = 0 otherwise we must
find whether ΠB−IIS (a, e)|e=ê is positive or not. If Π
B−II
S (a, e)| ∗e=ê > 0 then e = ê
otherwise e∗ = 0. Solving ΠB−IIS (a, e)|e=ê = 0 for a gives us a = 1, if wα−Dp(w−c) >
0 (which is the coefficient of a in this equation)then for all a 6 1,ΠB−IIS (a, e)|e=ê < 0.
Also, if wα − Dp(w − c) 6 0 then for all a 6 1,ΠB−IIS (a, e)|e=ê > 0 . Considering
condition B-II-2-a-2-i we can show that aB−II < α < 1 < aB−II2 − 1 . Therefore:2D(w c)

αβ 2αβ
(P1) : if < D < 
(w − c)(α + β) (w − c)(2β + α)
wβ(2β − α) wα 
and < p < 
2e4 β(1− e4)  ⇒ ∗ ∈ α e = 0 for all a ( , 1].α αβ 2D(w − c)(P2) : if < D <2(w − c) (w − c)(α + β) 
wβ(2β − α) w 
and < p < 
2e4 e4
177
αβ 2αβ
(P3) : if < D < and
(w − c)(α + β) (w − c)(2β + α)
wα w α
< p < ⇒ e∗ = ê for all a ∈ ( , 1]
β(1− e4) e4 2D(w − c)
2
If condition B-II-2-a-2-ii holds, then aB−II < α < 1 < aB−II2 ,4β(β(e4−1)+α) 1
α2
if wα−Dp(w − c) 6 0 then e∗ = ê for all a in ( , 1].
4β(β(e4 − 1) + α)
(B-II-2-a-2-ii)
Next case is B-II-2-part (b), similar to part(a) if ΠB−IIS (a, e)|e=ē > 0 then we
def
name it condition B-II-2-b-1 and rewrite this condition as following where p̃ =
α2(w−c)
w[ 2
D(w−c)− +4β ]α :
4β(β−D(w−c))
(i) : α(4β−α)− < D < α and w < p < wβ(2β−α) and p − β(p−w) < a 6 14β(w c) w−c 2e4 w Dw(w−c)
(ii) : 2αβ < D < α(4β−α) and p̃ < p <
wβ(2β−α)
 (w−c)(2β+α) 4β(w−c) 2e4and p − β(p−w) α2
w Dw(w− < a < .c) 4β(β(e4−1)+α)
(B-II-2-b-1 )
In B-II-2-b-1 if ΠB−IIS (a, e)| − Π
B−II
e=ē S (a, e)|e=ê > 0 then e∗ = ē otherwise e∗ = ê.
Considering condition B-II-2-a-1-i we prove similar to preceding case that if δ < 0
and as ρ > 0 then based on above mentioned characteristics e∗ = ē, we name this
case B-II-2-b-1-i-P3. On the other hand, if δ > 0 we can have two sub-conditions
178
for this case:
p − β(p− w)(P1) : if δ > 0 and < 1 < aB−II < aB−II
w Dw(w − c) 2 1
=⇒ e∗ p= ē for all a in ( − β(p− w) , 1].
w Dw(w − c)
p β(p− w)
(P2) : if δ > 0 and  − < aB−II < aB−II2 1 < 1w Dw(w − c)ē p − β(p−w) < a 6 aB−IIw Dw(w−c) 2
=⇒ e∗ = ê aB−II < a < aB−II . 2 1ē aB−II1 6 a 6 1
Under the condition of case B-II-2-b-1-ii, δ > 0 and we have:
p β(p− w) α2
(P1) : if 0 < − < < aB−II2 < 1 < aB−IIw Dw(w − c) 4β(β(e4 − 1) + α) 1
2
=⇒ ∗ p β(p− w) αe = ē for all a in ( − , ).
w Dw(w − c) 4β(β(e4 − 1) + α)
p β(p− w) α2
(P2) : if 0 < − < aB−II2 < < 1 < aB−IIw Dw(w − c) 4β(β(e − 1) + α) 1 4ē p − β(p−w) < a 6 aB−II
=⇒ e∗ = 
w Dw(w−c) 2
.
ê aB−II < a < α
2
2 4β(β(e4−1)+α)
179
Under the condition B-II-2-part (b), if ΠB−IIS (a, e)|e=ē < 0 then we name this
condition B-II-2-b-2 and rewrite it as follows :
(i) : 2αβ < D < α(4β−α)− − and w 6 p < p̃ and p − β(p−w) < a 6 1(w c)(2β+α) 4β(w c) w Dw(w−c)

2
(ii) : 2αβ < D < α(4β−α)− − and p̃ < p < wβ(2β−α) and α− 6 a 6 1 . (w c)(2β+α) 4β(w c) 2e4 4β(β(e4 1)+α)(iii) : α < D < 2αβ and w 6 p < wβ(2β−α) and p − β(p−w)− < a 6 12(w c) (w−c)(2β+α) 2e4 w Dw(w−c)
(B-II-2-b-2 )
If B-II-2-b-2 holds, then we need to do the similar analysis that we have done in case
B-II-2-a-2. Under the condition B-II-2-b-2, δ > 0. Now considering condition B-II-
2-b-2-i similar to preceding case, one can show that p− β(p−w) < aB−II < 1 < aB−II
w Dw(w−c) 2 1
and wα−Dp(w− c) 6 0 then e∗ = 0 for all a ∈ ( p − β(p−w)− , 1]. In B-II-2-b-2-ii,w Dw(w c)
B−II α
2
if a < < 1 < aB−II2 then wα−Dp(w − c) 6 04β(β(e4 − 1) + α) 1
α2
which implies e∗ = ê for all a ∈ ( , 1]
4β(β(e4 − 1) + α)
(B-II-2-b-2-ii-P1 )
Also if α
2
< aB−II < 1 < aB−II ,
4β(β(e4−1)+α) 2 1
α2
then wα−Dp(w − c) > 0 which implies e∗ = 0 for all a ∈ ( , 1].
4β(β(e4 − 1) + α)
(B-II-2-b-2-ii-P2 )
Given condition B-II-2-b-2-iii, p − β(p−w) B−II B−II
w Dw(w− < a2 < 1 < a1 then wα−Dp(w−c)
c) > 0 which implies e∗ = 0 for all a ∈ ( p − β(p−w) , 1].
w Dw(w−c)
Next part is B-II-2-(c), under this we can show straightforward that πB−IIS (a, e)|e=ē >
0 and πB−IIS (a, e)|e=ê > 0 and δ > 0, then a
B−II < α < aB−II2 − 1 <
α < 1
2D(w c) D(w−c)
180
which implies:
ê
α < a 6 aB−II
e∗ = 
2D(w−c) 1
.
ē aB−II1 < a 6
α
D(w−c)
Under the condition B-II-2-(d), it is straightforward to show that ΠB−IIS (a, e)|e=ē > 0
and ΠB−IIS (a, e)|e=ê > 0. If δ < 0 as ρ > 0, e∗ = ē for all a ∈ (
p − β(p−w) , α ],
w Dw(w−c) D(w−c)
which we label it as B-II-2-d-P3. In this case if p < w(1−α) we name it B-II-2-d-P3-i,
e4
otherwise B-II-2-d-P3-ii. If δ > 0, we will have two cases:
p − β(p− w) α(P1) : if 0 < < < 1 < aB−II < aB−II
w Dw(w − c) D(w − c) 2 1
⇒ ∗ ∈ p − β(p− w) α= e = ē for all a ( , ).
w Dw(w − c) D(w − c)
p β(p− w) α
(P2) : if 0 < −  < aB−II < aB−II < < 1w Dw(w − c)
2 1 D(w − c)
ē p − β(p−w)− < a 6 aB−IIw Dw(w c) 2
=⇒ e∗ =  B−IIê a2 < a < a
B−II .
 1ē aB−II1 6 a < αD(w−c)
∂ΠB−IIS (a,e) | ∂Π
B−II(a,e)
B-II-3) when 0 < S
∂e e=(e
− < | .
4) ∂e e=e4
∂ΠB−II(a,e)
Under the condition of B-II, we simplify the condition of 0 < S |
∂e e=(e
− <
4)
181
∂ΠB−IIS (a,e) | :
∂e e=e4
 (i) : α 6 D < β and w(β−α) 6 p < w and α− − < a 6 1w c (w c) βe4 e4 D(w−c)
 or (ii) : α 6 D < β and w 6 p 6 w(β−α) and p − β(p−w) < a 6 1.
w−c (w−c) βe4 w Dw(w−c)
(B-II-3)
As we illustrate it in B - II, under condition B-II-3, the supplier’s function (B - II)
is linearly decreasing in e for all e ∈ [0, ê) and it’s not continuous at e = ê (as we
proved it previously). Also function is linearly increasing in e for all e ∈ [ê, e4) and it
def
is concave with one local maximum (ē = 1− α ) for all e ∈ [e4, 1]. Therefore, under2aβ
this condition,in each of B-II-3 sub-conditions, we have two candidate solutions. To
find the optimal solution, we compare the value of function (B - II) at e = 0 and
e = ē. We compare ΠB−IIS (a, e)|e=0 = 0 with Π
B−II(a, e)| = D(w − c) − α(4aβ−α)S e=ē 4aβ
under all sub-regions of B-II-3. Similar to all above sub-regions of (B-II), we can
show that under condition (B-II-3) always ΠB−II(a, e)| < ΠB−IIS e=0 S (a, e)|e=ē which
implies e∗ = ē,. Therefore,
α β w(β − α) w
if 6 D < and 6 p <
w − c (w − c) βe4 e4
α
then e∗ = ē for all a ∈ ( , 1], (B-II-3-i)
D(w − c)
α β w(β − α)
if 6 D < and w 6 p 6
w − c (w − c) βe4
∗ p β(p− w)then e = ē for all a ∈ ( − , 1]. (B-II-3-ii)
w Dw(w − c)
182
Recall that Region B-I is defined as 0 6 a 6 p − β(p−w)− and Region B-II definedw Dw(w c)
as p − β(p−w)− < a 6 1. Now, after we presented the analysis of each case in thew Dw(w c)
above, we want to assemble Region B-I with Region B-II to form the supplier’s best
response function over the range of a ∈[0,1]. To do that, we combine each sub-region
of B-I with each single region of B-II.
Note that in each of the subregion of B-I, the interval of a may be a sub-interval of
0 6 a 6 p − β(p−w)− . In following we list all regions of B-I based on the range of a:w Dw(w c)
– when 0 6 a 6 p − β(p−w)− , we have 4 regions as follows:w Dw(w c)
• B-I-2-i-P4S1 through B-I-2-i-P4S4
– when 0 6 a 6 pα− , we have 11 regions as follows:2β(p w)+wα
• Combination of B-I-2-i-P3 with B-I-2-ii
• B-I-2-i-P1S1 through B-I-2-i-P1S5
• B-I-2-i-P2S1 through B-I-2-i-P2S5
– when pα < a 6 p − β(p−w)− , we have 5 regions as follows:2β(p w)+wα w Dw(w−c)
• Combination of B-I-1-P1 with B-I-1-P5
• B-I-1-P2, B-I-1-P3, B-I-1-P4, and B-I-1-P6
From the above list, we combine regions defined over 0 6 a 6 pα with
2β(p−w)+wα
regions defined over pα < a 6 p − β(p−w)
2β(p−w)+wα w Dw(w− to form the range of a forc)
0 6 a 6 p− β(p−w)− . We combine each region in the above list for 0 6 a 6
pα
w Dw(w c) 2β(p−w)+wα
with each region in the above list for pα− < a 6
p − β(p−w) . There are
2β(p w)+wα w Dw(w−c)
total 55 possible combinations. We find that 34 out of 55 combinations are empty
sets, because they do not have any intersection in term of all other variables but a.
183
Therefore, we have 21 regions that cover the range of 0 6 a 6 p − β(p−w)− afterw Dw(w c)
we consider the combinations. We delegate the details of analysis to Mathematica.
Also, by adding sub-regions B-I-2-i-P4S1 through B-I-2-i-P4S4 from the above to
this number, in total we have 25 regions in B-I for the range of 0 6 a 6 p − β(p−w)
w Dw(w−c)
which we find them based on an exhaustive search over range of 0 6 a 6 p − β(p−w)
w Dw(w−c)
. We provide the list of these 25 regions in Table 3. We label them from B-I-R1
to B-I-R25. Also in the same table, we provide info that shows which regions are
combined together to create each of B-I-R1 to B-I-R25 .
B-I Combination of Sub-Regions
B-I-R1 B-I-2-i-P4S1
B-I-R2 B-I-2-i-P4S2
B-I-R3 B-I-2-i-P4S3
B-I-R4 B-I-2-i-P4S4
B-I-R5 B-I-2-i-P3 and B-I-2-ii, and B-I-1-P3
B-I-R6 B-I-2-i-P1S1, and B-I-1-P4
B-I-R7 B-I-2-i-P1S1, and B-I-1-P6
B-I-R8 B-I-2-i-P1S2, and B-I-1-P3
B-I-R9 B-I-2-i-P1S3 and B-I-1-P1, and B-I-1-P5
B-I-R10 B-I-2-i-P1S3, and B-I-1-P2
B-I-R11 B-I-2-i-P1S3, and B-I-1-P3
B-I-R12 B-I-2-i-P1S4, and B-I-1-P4
B-I-R13 B-I-2-i-P1S4, and B-I-1-P6
B-I-R14 B-I-2-i-P1S5, and B-I-1-P2
B-I-R15 B-I-2-i-P1S5, and B-I-1-P3
B-I-R16 B-I-2-i-P2S1, and B-I-1-P4
B-I-R17 B-I-2-i-P2S1, and B-I-1-P6
B-I-R18 B-I-2-i-P2S2, and B-I-1-P3
B-I-R19 B-I-2-i-P2S3 and B-I-1-P1, and B-I-1-P5
B-I-R20 B-I-2-i-P2S3, and B-I-1-P2
B-I-R21 B-I-2-i-P2S3, and B-I-1-P3
B-I-R22 B-I-2-i-P2S4, and B-I-1-P4
B-I-R23 B-I-2-i-P2S4, and B-I-1-P6
B-I-R24 B-I-2-i-P2S5, and B-I-1-P2
B-I-R25 B-I-2-i-P2S5, and B-I-1-P3
TABLE 3. List of 25 sub-regions of B-I for the range of 0 6 a 6 p − β(p−w)
w Dw(w−c)
184
Next, we list all regions of B-II based on the range of a. Note that in each of the
subregion of B-II, the interval of a may be a sub-interval of p − β(p−w)− < a 6 1.w Dw(w c)
– when p − β(p−w)− < a 6 1, we have 13 regions as follows:w Dw(w c)
• B-II-1
• B-II-3-ii
• B-II-2-b-1-i-P1 through B-II-2-b-1-i-P3
• Combination of B-II-2-b-2-ii-P2 with B-II-2-b-1-ii-P1
• Combination of B-II-2-b-2-ii-P1 with B-II-2-b-1-ii-P2
• B-II-2-b-2-i and B-II-2-b-2-iii
• Combination of B-II-2-d-P1 with B-II-3-i
• Combination of B-II-2-d-P2 with B-II-3-i
• Combination of B-II-2-d-P3-i with B-II-3-i
• Combination of B-II-2-d-P3-ii with B-II-3-i
– when p − β(p−w) < a 6 α , we have 4 regions as follows:
w Dw(w−c) 2D(w−c)
• B-II-1-P2 through B-II-1-P5
– when α− < a 6 1, we have 6 regions as follows:2D(w c)
• B-II-2-a-1-i
• Combination of B-II-2-a-1-ii with B-II-2-a-2-ii
• B-II-2-a-2-i-P1 through B-II-2-a-2-i-P3
• Combination of B-II-3-i-P2 with B-II-2-c
185
From the above list, we combine regions defined over p − β(p−w) < a 6 α
w Dw(w−c) 2D(w−c)
with regions defined over α− < a 6 1 to form the range of a for
p − β(p−w) <
2D(w c) w Dw(w−c)
a 6 1. We pick each region in the above list for p − β(p−w) < a 6 α and
w Dw(w−c) 2D(w−c)
combine it with each region in the above list for α− < a 6 1. There are total 242D(w c)
possible combinations.
We find that 18 out of 24 combinations are empty sets, because they do not have
any intersection in term of all other variables but a. Therefore, after we consider the
combinations, we have 6 regions that cover the range of p − β(p−w)− < a 6 1. Wew Dw(w c)
delegate the details of analysis to Mathematica. Also, by adding 13 sub-regions that
listed in the above in which cover the range of p − β(p−w)
w Dw(w− < a 6 1 without anyc)
combination, in total we have 19 regions in B-II for the range of p − β(p−w)− < a 6 1w Dw(w c)
which we find them based on an exhaustive search over range of p − β(p−w)− < a 6 1.w Dw(w c)
We provide the list of these 19 regions in Table 4.
We label them from B-II-R1 to B-II-R19. Also in the same table, we provide
info that shows which regions are combined together to create each of B-II-R1 to
B-II-R19 . Now, we pick one region from 25 regions in B-I which is defined over
0 6 a 6 p − β(p−w)− and combine it with one region from 19 regions in B-II which isw Dw(w c)
defined over p − β(p−w)− < a 6 1 to create e
∗(a) over the full support of 0 6 a 6 1.
w Dw(w c)
We do the same for all combinations.
There are total 475 possible combinations. After going through each
combination with the aid of Mathematica, we find that: firstly, only 41 combinations
are not empty sets. Also, we find that there exist only 9 distinct forms of the
supplier’s best response function in Region B as we list them in Table 5 by labels of
S-BRF 1-1 to S-BRF 1-5 and S-BRF 2-1, SBRF 2-3 to S-BRF 2-5 and also show the
structure of each of them as follows.
186
B-II Combination of Sub-Regions
B-II-R1 B-II-1
B-II-R2 B-II-3-ii
B-II-R3 B-II-2-b-1-i-P1
B-II-R4 B-II-2-b-1-i-P2
B-II-R5 B-II-2-b-1-i-P3
B-II-R6 B-II-2-b-ii-P2 and B-II-2-b-1-ii-P1
B-II-R7 B-II-2-b-2-ii-P1 and B-II2-b-1-ii-P2
B-II-R8 B-II-2-b-2-i
B-II-R9 B-II-2-b-2-iii
B-II-R10 B-II-2-d-P1 and B-II-3-i
B-II-R11 B-II-2-d-P2 and B-II-3-i
B-II-R12 B-II-3-d-P3-i and B-II-3-i
B-II-R13 B-II-3-d-P3-ii and B-II-3-i
B-II-R14 B-II-1-P2 and B-II-2-a-2-i-P3
B-II-R15 B-II-1-P3 and B-II-2-a-2-i-P2
B-II-R16 B-II-1-P4 and B-II-2-a-1-i
B-II-R17 B-II-1-P4 and B-II-2-a-1-ii and B-II-2-a-2-ii
B-II-R18 B-II-1-P4 and B-II-3-i and B-II-2-c
B-II-R19 B-II-1-P5 and B-II-2-a-2-i-P1
TABLE 4. List of 19 sub-regions of B-II for the range of p − β(p−w)
w Dw(w− < a 6 1c)
187
Specifically, S-BRF 1-1 has 7 regions, S-BRF 1-2 has 2 regions, S-BRF 1-3 has 5
regions, S-BRF 1-4 has 4 regions, S-BRF 1-5 has 2 regions, S-BRF 2-1 has 14 regions,
S-BRF 2-3 has 2 regions, S-BRF 2-4 has 4 regions, and S-BRF 2-5 has 2 regions.
188
Derived from Sub-regions Location
of in Figure
17
S-BRF Form B-I B-II B-i B-ii
B-I-R1 B-II-R1 X
B-I-R1 B-II-R14 X
B-I-R1 B-II-R15 X
S-BRF 1-1 B-I-R6 B-II-R9 X
B-I-R7 B-II-R8 X
B-I-R16 B-II-R9 X
B-I-R17 B-II-R8 X
B-I-R4 B-II-R17 X
S-BRF 1-2 B-I-R4 B-II-R19 X
B-I-R3 B-II-R15 X
B-I-R12 B-II-R9 X
S-BRF 1-3 B-I-R23 B-II-R8 X
B-I-R22 B-II-R9 X
B-I-R13 B-II-R8 X
B-I-R9 B-II-R8 X
B-I-R20 B-II-R6 X
S-BRF 1-4 B-I-R19 B-II-R8 X
B-I-R10 B-II-R6 X
B-I-R14 B-II-R7 X
S-BRF 1-5 B-I-R24 B-II-R7 X
B-I-R5 B-II-R2 X
B-I-R8 B-II-R2 X
B-I-R8 B-II-R10 X
B-I-R8 B-II-R12 X
B-I-R8 B-II-R13 X
B-I-R15 B-II-R3 X
B-I-R15 B-II-R5 X
S-BRF 2-1 B-I-R18 B-II-R2 X
B-I-R18 B-II-R10 X
B-I-R18 B-II-R12 X
B-I-R18 B-II-R13 X
B-I-R25 B-II-R3 X
B-I-R25 B-II-R4 X
B-I-R25 B-II-R5 X
B-I-R2 B-II-R18 X
S-BRF 2-3 B-I-R4 B-II-R16 X
B-I-R8 B-II-R11 X
B-I-R15 B-II-R4 X
S-BRF 2-4 B-I-R18 B-II-R11 X
B-I-R11189 B-II-R3 X
S-BRF 2-5 B-I-R21 B-II-R3 X
TABLE 5. List of the Supplier’s Best Response Function forms in Region B
S-BRF 1-1) e∗ = 0 for all a in [0, 1].
S-BRF 1-2) e∗ = ê for all a in [0, 1].0 0 6 a < aB−I2
S-BRF 1-3) e∗ = ê aB−I 2 6 a < a
B−I .
 10 aB−II1 6 a 6 1
0 0 6 a < a
B−I
 2ê aB−I 6 a < pα
∗  2 2β(p−w)+wαS-BRF 1-4) e = 
.
ē pα
2
− 6 a <
α
 2β(p w)+wα 4β(α−D(w−c)) α20 4β(α−
6 a 6 1
D(w−c))
ê 0 6 a < pα2β(p−w)+wα
S-BRF 1-5) e∗ =  pα B−II
ē .
2β(p− 6 a < aw)+wα 2
ê aB−II2 6 a 6 1
190
ê 0 6 a <
pα
∗  2β(p−w)+wαS-BRF 2-1) e =  .ē pα− 6 a 6 12β(p w)+wα



ê 0 6 a < ȧ
S-BRF 2-2) e∗ = .
ē ȧ 6 a 6 1ê 0 6 a < aB−II1
S-BRF 2-3) e∗ =  .ē aB−II1 6 a 6 1
 pαê 0 6 a < 2β(p−w)+wα pα B−II
∗ 
ē − 6 a < a2β(p w)+wα 2
S-BRF 2-4) e =  .ê a
B−II
2 6 a < a
B−II
 1
ē a
B−II 6 a 6 1

1
0 0 6 a < aB−I2
S-BRF 2-5) e∗ = ê aB−I 6 a < pα . 2 2β(p−w)+wαē pα 6 a 6 1
2β(p−w)+wα
√
2 2
1−a − α α(4β(p−w)+wα+ w α +8(p−w)(wα−2Dp(w−c))β+16(p−w)
2β2)
Also, ê = p − , ē = 1 , ȧ = ,a 2aβ 8D(p−w)(w−c)β
w
√
aB−I = −(w
2α−pw(2D(w−c)−α)−β(p−w)2)− φ
1 ,2(w2(D(w−c)−α))
√
aB−I
2
= −(w α−pw(2D(w−c)−α)−β(p−w)
2)+ φ
2 2 − − ,2(w (D(w c)√α))
−(−[4αβ(p−w)+wα2])+ [(−[4αβ(p−w)+wα2])2−4(4Dβ(p−w)(w−c))(pα2)]
aB−II1 = √2(4Dβ(p−w)(w−
,
c))
B−II −(−[4αβ(p−w)+wα2])− [(−[4αβ(p−w)+wα2])2−4(4Dβ(p−w)(w−c))(pα2)]a2 = − − where φ =2(4Dβ(p w)(w c))
((w2α−pw(2D(w−c)−α)−β(p−w)2)2−4(w2(D(w−c)−α))(Dp2(w−c)−pwα)).
As we show in Table 5, S-BRF 1-1 to S-BRF 1-5 are in Region B-i of Figure 25 in
191
Lemma 16. Also, S-BRF 2-1 to S-BRF 2-5 are in Region B-ii of Figure 25 in Lemma
16. To prove this characteristic, we take each region in Table 5 and if it is in region
B-i, then it satisfies the condition D < α(4β−α)− . Also, if it is in region B-ii, then it4β(w c)
satisfies D > α(4β−α)− . For example, choose S-BRF 1-1 from Region B-i of Figure 254β(w c)
in Lemma 16. Also, from Table 5 we choose one of regions that belongs to S-BRF
1-1. For instance, we pick B-I-R7 and B-II-R8. We follow the following procedure
to show that this region satisfies condition D < α(4β−α) .
4β(w−c)
Region B-I-R7 includes the combination of B-I-2-i-P1S1 and B-I-1-P6. Also, B-II-R8
includes B-II-2-b-2-i.
– In Region B-I-2-i-P1S1 under:
2β(p− w) + wα 2β(p− w) + wα α
6 D < and w 6 p < w(2− ) and
2p(w − c) p(w − c) β
(A.20)
[(w2α− pw(2D(w − c)− α)− β(p− w)2)2 (A.21)
− 4(w2(D(w − c)− α))(Dp2(w − c)− pwα)] < 0,
we have e∗ = 0 for all a in [0, pα ).
2β(p−w)+wα
– In Region B-I-1-P6, under :
2αβ α(4β − α) wα(2β − α)
6 D < and w 6 p < ,
(w − c)(2β + α) 4β(w − c) 2β[2D(w − c)− α]
(A.22)
we have e∗ = 0 for all a in [ pα , p − β(p−w) ].
2β(p−w)+wα w Dw(w−c)
192
– In Region B-II-2-b-2-i, under:
w[ α
2(w−c)
− + 4β22αβ α(4β α) ]D(w−c)−α
< D < and w 6 p < , (A.23)
(w − c)(2β + α) 4β(w − c) 4β(β −D(w − c))
we have e∗ = 0 for all a in ( p − β(p−w) , 1].
w Dw(w−c)
Note that the range of a defined in the above cases are complementary. Next, we
identify the intersection of three regions defined by (A.20), (A.22), and (A.23) to
create e∗(a) for all a in [0, 1]. Their intersection based on D is:
2αβ 2pwβ(β + α)− p2β2 − w2(β − α)2
< D < . (A.24)
(w − c)(2β + α) 4pwβ(w − c)
Finally, we prove that the region defined by (A.24) is sub-region of Region B-i of
2 2 2 2
Figure 25 in Lemma 16 by showing that 2pwβ(β+α)−p β −w (β−α) < α(4β−α) :
4pwβ(w−c) 4β(w−c)
2pwβ(β + α)− p2β2 − w2(β − α)2 α(4β − α)
<
4pwβ(w − c) 4β(w − c)
⇔ 2pwβ(β + α)− p2β2 − w2(β − α)2 < α(4β − α)
⇔ 0 < (p− w)[pβ2 − w(β − α)2]
⇔ (β − α)
2 p
< 1 < .
β2 w
In a similar procedure, we prove that all the supplier’s best response function forms
S-BRF 1-1 to S-BRF 1-5 satisfy the condition D < α(4β−α)− . The proof with more4β(w c)
details is available on software upon request.
To prove one of regions from region B-ii in Figure 25 of Lemma 16, we consider
S-BRF 2-3. Also, from Table 5, as an example we pick B-I-R2 and B-II-R18. The
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next step is to prove that this region satisfies the condition D > α(4β−α)− . The region4β(w c)
B-I-R2 of Table 5 includes B-I-2-i-P4S2. Also, region B-II-R18 includes B-II-1-P4,
B-II-2-c, and B-II-3-i.
– In Region B-I-2-i-P4S2 under
(p− w)2[p2β2 − 2pwβ(2cD + α + β) + w2((β − α)2 + 4Dpβ)] > 0 and
Dw2(w − c)− w2α > 0, (A.25)
we have e∗ = ê for all a ∈ [0, p − β(p−w)− ].w Dw(w c)
– In Region B-II-1-P4 under
2αβ β wβ(2β − α) w
6 D < and 6 p < , (A.26)
(w − c)(2β + α) w − c 2e4 e4
we have e∗ = ê for all a ∈ [ p − β(p−w) α
w Dw(w− , ].c) 2D(w−c)
– In Region B-II-2-c under
α β wβ(2β − α) w
< D 6 and < p < , (A.27)
w − c w − c 2e e
 4 4ê α B−II
∗  2D(w−
< a 6 a
c) 1
we have e =  .ē aB−II α1 < a 6 D(w−c)
– In Region B-II-3-i under
α β w(β − α) w
6 D < and 6 p < , (A.28)
w − c (w − c) βe4 e4
we have e∗ = ē for all a ∈ ( α
D(w− , 1].c)
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We identify the intersection of four regions defined by (A.25), (A.26), (A.27), and
(A.28) to create e∗(a) for all a in [0, 1]. Their intersection is:
{β(p− w) 2β(p− w) + wα wβ< D < and < p} or
p(w − c) 2p(w − c) β − α
{ α 2β(p− w) + wα w(2β − α) wβ< D < and < p < }, or
w − c 2p(w − c) 2(β − α) β − α
{β(2p− w(2β − α)β) 2β(p− w) + wα w(2β − α)β
2
< D < and < p}.
2p(w − c) 2p(w − c) 2(β − α)
We prove that S-BRF 2-3 is always in Region B-ii of Figure 25 in Lemma 16 by
showing that the lower bound of D in above condition is always bigger than α(4β−α) .
4β(w−c)
So if β(p−w) 2β(p−w)+wα wβ
p(w− < D < − and − < p then:c) 2p(w c) β α
α(4β − α) β(p− w)
<
4β(w − c) p(w − c)
⇔ α(4β − α) β(p− w)<
4β p
⇔ − α − wα(1 ) < β(1 )
4β p
⇔ − α wα(1 ) < α < β(1− ). (A.29)
4β p
The inequality (A.29) holds because in this case we have wβ− < p⇔ α < β(1−
w ).
β α p
For the second part of condition which we α < D < 2β(p−w)+wα and w(2β−α)− − − <w c 2p(w c) 2(β α)
p < wβ− , it suffices to show that:β α
α(4β − α) α
<
4β(w − c) w − c
⇔ 4β − α < 1
4β
⇔ − α1 < 1.
4β
195
Also for the last part which we have β(2p−w(2β−α)β) < D < 2β(p−w)+wα
2
− − and
w(2β−α)β <
2p(w c) 2p(w c) 2(β−α)
3
p, it suffices to show that: α(4β−α) < β(2p−w(2β−α)β) ⇔ 2wβ < p.
4β(w−c) 2p(w−c) 2β−α
α(4β − α) β(2p− w(2β − α)β)
<
4β(w − c) 2p(w − c)
⇔ 2wβ
3
< p
2β − α
3 2
If we show that 2wβ < w(2β−α)β− − then the proof is complete:2β α 2(β α)
2wβ3 w(2β − α)β2
<
2β − α 2(β − α)
⇔ 4β(β − α) < (2β − α)2
⇔ 0 < α2.
In a similar procedure, we prove that all the supplier’s best response function forms
S-BRF 2-1 to S-BRF 2-5 satisfy the condition D > α(4β−α) . The proof with more
4β(w−c)
details is available in Mathematica codes in Appendix.
Therefore, we prove that in Region B-i of Lemma 16, possible S-BRF forms are:
S-BRF 1-1, S-BRF 1-2, S-BRF 1-3, S-BRF 1-4, and S-BRF 1-5. Also, we prove that
in Region B-II of Lemma 16, possible S-BRF forms are: S-BRF 2-1, S-BRF 2-3,
S-BRF 2-4, and S-BRF 2-5.
Now, we discuss couple of characteristics of the supplier’s best response functions in
region B which help us in finding NE in the next Lemma. S-BRF is continuous in
a only at point a = pα and it is not continuous in other start and end of
2β(p−w)+wα
domain interval points (e.g. aB−II , aB−II1 2 and etc.) We should know the behavior of
196
ê and ē. Firstly, the intersection of e∗ = ê and e∗ = ē happens at a = pα :
2β(p−w)+wα
α 1− a
ē = ê⇒ 1− = p ⇒
pα
a = .
2aβ − ea 2β(p− w) + wα
w
In addition to this, we know that e∗ = ê is decreasing and concave in a and e∗ = ē
is increasing and concave in a for all a ∈ (0, 1):
∂ê −w(p− w) ∂
2ê 2w2(p− w)
= < 0 , = − < 0,
∂a (p− aw)2 ∂a2 (p− aw)3
∂ē α ∂2ē α
= > 0 , = − < 0.
∂a 2a2β ∂a2 a3β
In S-BRF 2-3 as intersection happens in another point (i.e., a = aB−II1 ) function is
not continuous in a. Also we observe a jump in the function at a = aB−II1 based on
above-mentioned characteristics of ê and ē as pα B−II− < a1 , where a
B−II
√ 2β(p w)+wα 1
=
4αβ(p−w)+wα2+ α2[(4pβ−w(4β−α))2−16Dpβ(w−c)(p−w)]
8Dβ(p−w)(w− :c)
aB−II
pα
1 > 2β(p− w) + wα
√
4αβ(p− w) + wα2 + α2[(4pβ − w(4β − α))2 − 16Dpβ(w − c)(p− w)]
⇔
8Dβ(p− w)(w − c)
− pα > 0
2β(p− w) +√wα
pα(wα2 + α2[(4pβ − w(4β − α))2 − 16Dpβ(w − c)(p− w)])
⇔ > 0
[2β(p− w) + wα]Φ
√
where Φ = (4αβ(p−w)+wα2− α2[(4pβ − w(4β − α))2 − 16Dpβ(w − c)(p− w)]) in
the last inequality, the nomin√ator is positive, and in denominator [2β(p−w)+wα] >
0, if (4αβ(p − w) + wα2 − α2[(4pβ − w(4β − α))2 − 16Dpβ(w − c)(p− w)]) > 0
197
then we prove our claim:
√
(4αβ(p− w) + wα2 − α2[(4pβ − w(4β − α))2 − 16Dpβ(w − c)(p− w)]) > 0
⇔ [4αβ(p− w) + wα2]2 − [α2[(4pβ − w(4β − α))2 − 16Dpβ(w − c)(p− w)]] > 0
⇔ 16Dp(p− w)(w − c)α2β > 0 .
Region C: ( β 6 D)
w−c
Following the structure of optimal order quantities in Figure 21, we form the
supplier’s profit function for Region C :
(1− e)a[D(w − c)− β(1− e)]− αe 0 6 e < êΠCS (a, e) =  .[e+ (1− e)(1− a)]D(w − c) + (1− e)a[D(w − c)− β(1− e)]− αe ê 6 e 6 1
(A.30)
From (A.30), we verify that the supplier’s function in region C is not continuous at
C | − C D(1− a)p(w − c)ê because ΠS (a, e) e=ê lim ΠS (a, e) = > 0 where:
e→ê− p− aw
C (p− aw)[Da(w − c)(p− w)− (1− a)wα]− aβ(p− w)2lim ΠS (a, e) = ,
e→ê−r (p− aw)2
C (1− a)w(p− aw) + a(p− w)2βΠS (a, e)|e=ê =D(w − c)− .(p− aw)2
Also, we can verify that ΠCS (a, e)| Ce=ê > ΠS (a, e)|e=ê− . And in general we prove that
for all e ∈ (0, 1) the second piece of the supplier’s function is greater than the first
198
piece:
− − (1− a)w(p− aw) + a(p− w)
2β
D(w c)
(p− aw)2
− (p− aw)[Da(w − c)(p− w)− (1− a)wα]− aβ(p− w)
2
> 0
(p− aw)2
⇔Dp(1− a)(w − c) > 0.
p− aw
Derivative of the supplier’s profit with respect to e is:

∂ΠC(a, e) a[2β(1− e)−D(w − c)]− α 0 6 e < êS =  . (A.31)∂e 2a(1− e)β − α ê 6 e 6 1
∂ΠC(a,e)
From (A.31), we verify that S |e=1 < 0. In addition we can show that∂e
∂ΠCS (a,e)
C
| ∂ΠS (a,e)e=(ê)− < |e=ê where:∂e ∂e
∂ΠCS (a, e) | 2a(p− w)βê− = −Da(w − c)− α +
∂e p− aw
∂ΠCS (a, e) | 2a(p− w)βê = −α + .
∂e p− aw
Second order derivative of the supplier’s profit function (A.30) with respect to e
is −2aβ for all e ∈ (0, 1) which shows that both pieces of the function (A.30) are
concave in e.
∂ΠC(a,e)
We divide the region C into two sub-regions based on order of S |
∂e e=(ê)
− ,
∂ΠCS (a,e) | , and zero.
∂e e=ê
∂ΠCS (a,e) | ∂Π
C
S (a,e)– Sub-region C-I: e=(ê)− < |e=ê 6 0, and∂e ∂e
∂ΠC(a,e) ∂ΠC(a,e) ∂ΠC(a,e)
– Sub-region C-II:{ S | S S
∂e e=(ê)
− < 0 < |
∂e e=ê
} or {0 6 |
∂e e=(ê)
− <
∂ΠCS (a,e) |
∂e e=ê
}.
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∂ΠCS (a,e) ∂Π
C
| S (a,e)C-I ) when e=(ê)− < |e=ê 6 0.∂e ∂e
If condition of Region C holds, then simplifying the condition of C-I gives us:
pα β
0 6 a 6 and 6 D. (A.32)
2β(p− w) + wα w − c
Under condition (A.32), the second piece of function (A.30) is decreasing concave
∂ΠC(a,e)
in e and first piece is concave in e. Now if S |e=0 < 0, then we name this∂e
new sub-region C-I-i which is defined by: a[2β −D(w − c)] − α < 0. We illustrate
this sub-region in Figure 30. To find optimal solution when condition C-I-i holds,
FIGURE 30. Possible Forms of the Supplier’s Profit Function in Region C-I
we compare ΠCS (a, e)|e=0 with ΠCS (a, e)|e=ê. As we proved similarly in Region B-I,
condition of C-I-i implies that ΠCS (a, e)|e=0 < ΠCS (a, e)|e=ê which means e∗ = ê.
∂ΠC(a,e)
Next sub-region is C-I-ii, which is defined when S | > 0. In this situation
∂e e=0
we have two candidate solution as we illustrate in Figure 30. Thus, to find optimal
solution, we compare ΠCS (a, e)| Ce=ẽ with ΠS (a, e)|e=ê where ẽ is the global maximum
def
of first piece ẽ = 1 − 1 D(w − c) − α . As we proved it in Region B-I, similarly
2β 2aβ
here condition of a[2β −D(w − c)]− α > 0 implies that ΠC(a, e)| < ΠCS e=ẽ S (a, e)|e=ê
200
which means e∗ = ê. Therefore for all sub-region C-I we proved that:
β ⇒ ∗ ∈ pαif 6 D = e = ê for all a [0, ].
w − c 2β(p− w) + wα
∂ΠC{ S (a,e) ∂Π
C(a,e) ∂ΠC(a,e)
C-II ) when | S S
∂e e=(ê)
− < 0 < | } or {0 6 | − <
∂e e=ê ∂e e=(ê)
∂ΠCS (a,e) |
∂e e=ê
}.
If condition of Region C holds, then simplifying the condition of C-II gives us:
pα β
< a 6 1 and 6 D. (A.33)
2β(p− w) + wα w − c
FIGURE 31. Possible Forms of the Supplier’s Profit Function in Region C-II
We show possible cases of this sub-region in Figure 31. When condition C-II holds,
def
the local optimal of second piece (ē = 1 − α ) occurs at e in [ê, 1]. As we proved
2aβ
in characteristics of function (A.30), if the e = ē exist in interval [ê, 1], then it is the
global maximum of function (A.30). Therefore for all sub-regions of C-II, we proved
201
that:
β ⇒ ∗ pαif 6 D = e = ē for all a ∈ ( , 1].
w − c 2β(p− w) + wα
For Region C, combining sub-region C-I and sub-region C-II provides us the final
best response function for this region:

ê 0 6 a 6
pα
∗  2β(p−w)+wαS-BRF 2-1: e =  .ē pα < a 6 1
2β(p−w)+wα
This S-BRF belongs to Region B-ii of Figure 25 in Lemma 16. We prove it by
comparing the lower bound of D in definition of Region C with D = α(4β−α)
4β(w− :c)
α(4β − α) β
< ⇔ (2β − α)2 > 0.
4β(w − c) w − c
Hence, we complete the proof of Lemma 16.
Proof of Lemma 4. We derive the Nash Equilibrium (NE) using the buyer’s best
response function (B-BRF) in Lemma 15 and the supplier’s best response function
(S-BRF) in Lemma 16. Recall that in Lemma 2 we divide the plane of (D,w) into
Regions A, B and C and use them to organize the proof of the B-BFR. B-BFR can
be obtained from Lemma 15 and is illustrated for regions 1 through 6 of the plane
of (D,w) in Figure 22. In Lemma 16, we divide Region A into two subregions A-i
and A-ii, and Region B into B-i and B-ii, in which the S-BRF has a unique form, as
illustrated in Figure 25. We proceed in the order of regions A-i, A-ii, B-i, B-ii, and
C in this proof.
Region A-i: We want to prove that the NE is one of the following three forms:
202
– NE form III: (e∗, a∗) = (w , 0), and
p
– NE form IV: (e∗, a∗) = (0, 0) or no Nash Equilibrium.
Recall from Lemma 16 that region A-i is defined by 0 6 D 6 β(p−w)− and D <
α(4β−α) .
p(w c) 4β(w−c)
In this region we have two possible forms of the supplier’s best response function:
S-BRF 1-1 and S-BRF 1-2, given in Lemma 16. In the same region, there are at most
5 possible forms of the buyer’s best response function, depending on the position of
the intersection of boundary β(p−w)− and boundary
α(4β−α) : B-BRF 1 through B-BRF
p(w c) 4β(w−c)
5, given in Lemma 15. The proof does not depend on how many forms of B-BFR
are selected. We prove with the example of all 5 forms of B-BRF.
First, consider the S-BRF 1-1, which is e∗ = 0 for all a in [0, 1]. The intersection of
S-BRF 1-1 with all forms of B-BRF creates NE (e∗, a∗) = (0, 0), because a∗(0) = 0
in all forms of the B-BRF. (e∗, a∗) = (0, 0) is NE form IV. When (e∗, a∗) = (0, 0),
the buyer’s expected profit ΠB and the supplier’s expected profit ΠS are zero.
Next, consider S-BRF 1-2, which is e∗ = ê(a) for all a in [0, 1], and intersect it with
all 5 forms of B-BRF.
– Intersect S-BRF 1-2 with B-BRF 1 and B-BRF 5, respectively. Recall that
e∗ = ê(a) is a decreasing concave continuous function in a with e∗ = ê(1) = 0
and e∗ = ê(0) = w . In both B-BRF 1 and B-BRF 5, e∗(0) = w and a∗(w ) = 0.
p p p
Therefore, the NE is (e∗, a∗) = (w , 0), which is NE form III.
p
– Intersect S-BRF 1-2 with B-BRF 2. Note that from Lemma 15, e < w1 < e .p 2
There is no intercept between the B-BRF and S-BRF. So, there is no NE. We
illustrate this situation in Figure 32.
– Similar to the above case, there is no NE when we intercept S-BRF 1-2 with the
B-BRF 3 and B-BRF 4, respectively. We illustrate this situation in Figure 32.
203
In the situations without NE, we abuse the notation and treat them as if there exist
the NE of (e∗, a∗) = (0, 0), which is NE form IV, for expositional convenience. Such
treatment does not affect the equilibrium results. No NE means that the buyer and
the supplier take no action, which is (e∗, a∗) = (0, 0), and that there is no continuation
game, that is, (qf , qp) = (0, 0). At the NE of (e
∗, a∗) = (0, 0), as Lemma 2 shows,
the equilibrium of the continuation game is (qf , qp) = (0, 0). So, technically, no NE
is identical to NE of (e∗, a∗) = (0, 0).
FIGURE 32. Illustration of No NE When Intersecting S-BRF 1-2 with B-BRF 2,
B-BRF 3, and B-BRF 4
Region A-ii: We want to prove that the NE is one of the following four forms:
– NE form I: (e∗, a∗) = (1− α , 1),
2β
– NE form II: (e∗, a∗) = (1− γ , αDp),
Dp 2γβ
– NE form III: (e∗, a∗) = (w , 0), and
p
– No Nash Equilibrium (equivalent to NE form IV).
Note that region A-ii is defined by 0 6 D 6 β(p−w)− and D >
α(4β−α)
− . In this regionp(w c) 4β(w c)
there is one possible form of the supplier’s best response function: S-BRF 2-2, given
in Lemma 16. In same region, there are at most 5 possible forms of the buyer’s best
204
response function: B-BRF 1 through B-BRF 5, given in Lemma 15. The proof does
not depend on how many forms of B-BFR are selected. We prove with the example
of all 5 forms of B-BRF. Recall that S-BRF 2-2 is defined as follows: e∗ = ê(a) for
all 0 6 a < ȧ and e∗ = ē(a) for all ȧ 6 a 6 1. S-BRF 2-2 is not continuous at a = ȧ.
For 0 6 a < ȧ, the S-BRF 2-2 is defined as ê(a) which is a decreasing continuous
concave function in a . Also, ê(1) = 0 and ê(0) = w . The second piece of S-BRF
p
2-2, ē(a), is an increasing continuous concave function in a for ȧ 6 a 6 1.
– Intersect S-BRF 2-2 with B-BRF 1. B-BRF 1 is defined as follows: a∗ = 0 for
0 6 e 6 1. e∗(0) = w and a∗(w ) = 0. So, NE is (e∗, a∗) = (w , 0), which is NE
p p p
form III.
FIGURE 33. Illustration of No NE When Intersecting S-BRF 2-2 with B-BRF 2
– Intersect S-BRF 2-2 with B-BRF 2. We illustrate both the S-BFR 2-2 and
B-BRF 2 in Figure 33. The B-BRF 2 is defined as follows: if e1 < e < e2 then
a∗ = 1 , if e = e2 (right edge) a
∗ can take any value in [0, 1], and otherwise
a∗ = 0. Note that from Lemma 15: e w1 < < e2, then there is no intersectionp
between S-BFR 2-2 and B-BRF 2 for all 0 6 a < ȧ. If e2 < ē(ȧ), then there is
205
no intercept between S-BFR 2-2 and B-BRF 2 for ȧ 6 a 6 1. Therefore, there
is no NE. Next we prove ē(ȧ) > e2 for B-BRF 2. Using the expression of ē(ȧ)
and e2 we have:
− α − γ ⇔ Dwα1 > 1 γ > (A.34)
2βȧ Dw 2βȧ
√
Recall that one of the defined conditions of B-BRF 2 is D < γβ− which isp(w c)
D2p(w−c) D2equivalent to γ > . To complete the proof, it suffices that p(w−c) >
β β
Dwα . We plug-in ȧ and we have:
2βȧ
√
α2(w2α2 + 8(p− w)(wα− 2Dp(w − c))β + 16(p− w)2β2)
>
D2(w − c)2(p− w2)β2
−α(pwα + 4(p− w)
2β)
. (A.35)
Dp(p− w)(w − c)β
The right-hand side of inequality (A.35) is negative and the left-hand side is
positive. Hence, we completed the proof.
– Intersect S-BRF 2-2 with B-BRF 3. We illustrate both S-BRF 2-2 and B-BRF
3 in Figure 34. The B-BRF 3 is defined as follows: if e1 < e < e2 or e4 < e < e3
then a∗ = 1; if e = e2 (middle edge) or e = e3 (right edge) then a
∗ can take
any value in [0, 1]; and otherwise a∗ = 0. In this case and from Lemma 15,
e < w1 < e2, thus there is no intersection between S-BRF 2-2 and B-BRF 3p
when 0 6 a < ȧ. Also, in (A.34) we prove that e2 < ē(ȧ). Thereby, there is no
intersection between S-BRF 2-2 and B-BRF 3 for all 0 6 e 6 e2. We illustrate
this in Figure 34. There may exist intersection between S-BRF 2-2 and B-BRF
3 when ȧ 6 a 6 1 and e4 < e 6 1. Note that e∗ = ē(a) is increasing continuous
function in a, ē(a)| αa=1 = 1 − . For an NE to exists, we need to prove that2β
206
FIGURE 34. Illustration of Different Forms of NE and No NE Cases When
Intersecting S-BRF 2-2 with B-BRF 3
ē(a)|a=1 > e4:
ē(a)|a=1 >e4
⇔ − α − D(w − c)1 >1
2β β
⇔ αD > . (A.36)
2(w − c)
In Region A-ii: α(4β−α)− < D. Also,
α < α(4β−α)− − . Therefore,4β(w c) 2(w c) 4β(w c)
inequality (A.36) holds, because: α < α(4β−α)− − < D. To find NE, we2(w c) 4β(w c)
compare the position of ē(ȧ) and ē(a)|a=1 vs. e3. We derive possible NE and
associated conditions in this case as follows and we illustrate them in Figure 34:
• When ē(ȧ) 6 e3 (or equivalently w > ŵ(D, p, c, α, β, γ)):
• under ē(a)|a=1 < e3 (equivalent to D > 2βγ ), it creates NE (e∗, a∗) =pα
(1− α , 1), which is NE form I. We illustrate this in Figure 34-1.
2β
• under ē(a)|a=1 > e3 (equivalent to D 6 2βγ ), it creates NE (e∗, a∗) =pα
(1 − γ , αDp) which is NE form II. We illustrate this in Figure 34-2.
Dp 2γβ
207
Note that ŵ(D, p, c, α, β, γ) =
√
Dγ(4β−α)+2D3p(p+c)− (Dγ(4β−α)+2D3p(p+c))2−16D3p((D3p2C)+(2Dp−γ)βγ)
.
4D3p
• When ē(ȧ) > e3 (or equivalently w < ŵ(D, p, c, α, β, γ)), then there is no
intersection at all between S-BRF 2-2 and B-BRF 3. Therefore, there is
no NE. We illustrate this case in Figure 34-3.
– Intersect S-BRF 2-2 with B-BRF 4. B-BRF 4 is defined as follows: if e1 <
e < e then a∗3 = 1 , if e = e3 (right edge) a
∗ can take any value in [0, 1],
and otherwise a∗ = 0. From Lemma 15, e < w1 < e3. Therefore, we havep
no intersection between S-BRF 2-2 and B-BRF 4 when 0 6 a < ȧ. Similar
to steps that we show in (A.36), one can show that ē(a)|a=1 > e1. Based on
these characteristics of B-BRF 4, To find NE we compare the position of ē(ȧ)
and ē(a)|a=1 vs. e3. Similar to the preceding case, there exist NE form I when
ē(ȧ) 6 e3 and ē(a)|a=1 < e3. Also, there exist NE form II when ē(ȧ) 6 e3 and
ē(a)|a=1 > e3. There is no NE if ē(ȧ) > e3 .
– Intersect S-BRF 2-2 with B-BRF 5. B-BRF 5 is defined as follows: if e4 <
e < e3 then a
∗ = 1 , if e = e3 (right edge) a
∗ can take any value in [0, 1], and
otherwise a∗ = 0. As we proved in (A.36), ē(a)|a=1 > e4, therefore we compare
position of ē(ȧ) and ē(a)|a=1 vs. e3. Similar to both above cases, there exist
NE form I when ē(ȧ) 6 e3 and 1− α < e3. Also, there exist NE form II when2β
ē(ȧ) 6 e and 1 − α3 > e3. There is no NE if ē(ȧ) > e3. This case differs2β
from both above cases in position of w vs. left edge of B-BRF (e4). Based onp
Lemma 15, w < e4, which leads to NE form III (e
∗, a∗) = (w , 0) in this case.
p p
NE form III occurs simultaneously with one of above-mentioned NE form I or
NE form II, no NE (equivalent to NE form IV). When (e∗, a∗) = (w , 0), the
p
208
supplier’s profit ΠS is negative, thus it is Pareto dominated by one of NE I or
NE II, or NE IV.
In Region A (both A-i and A-ii), we find that when D > α(4β−α)− and w >4β(w c)
ŵ(D, p, c, α, β, γ) we have NE forms I and II with the boundary of D = 2βγ which
pα
divides the regions of two forms. Otherwise we have NE forms III and IV.
Region B-i: We want to prove that the NE is one of the following three forms:
– NE form III: (e∗, a∗) = (w , 0), and
p
– NE form IV: (e∗, a∗) = (0, 0) or no Nash Equilibrium.
Note that region B-i is defined by β(p−w) β α(4β−α)
p(w− < D <c) w− and D < . In thisc 4β(w−c)
region there are 5 possible forms of the supplier’s best response function: S-BRF
1-1 through 1-5, given in Lemma 16. In the same region, there are 3 possible forms
of the buyer’s best response function: B-BRF 1, B-BRF 4, and B-BRF 5, given in
Lemma 15. Consider B-BRF 1 and intersect it with all 5 forms of S-BRF:
– Intersect B-BRF 1 with S-BRF 1-1, 1-3, and 1-4, respectively. As e∗(0) = 0
and a∗(0) = 0, NE is (e∗, a∗) = (0, 0), which is NE form IV.
– Intersect B-BRF 1 with S-BRF 1-2 and 1-5, respectively. As e∗(0) = w and
p
e∗(w ) = 0, then (e∗, a∗) = (w , 0), which is NE form III.
p p
Consider B-BRF 4 and intersect it with all 5 forms of S-BRF:
– Intersect B-BRF 4 with S-BRF 1-1. The S-BRF 1-1 is defined as follows:
e∗(a) = 0 for all 0 6 a 6 1. As a∗(0) = 0, then (e∗, a∗) = (0, 0), which is NE
form IV.
209
– Intersect B-BRF 4 with S-BRF 1-2. The S-BRF 1-2 is defined as follows:
e∗(a) = ê for all 0 6 a 6 1. Note that e1 < w < e3 from Lemma 15. There isp
no intercept between the B-BRF 4 and S-BRF 1-2. So, there is no NE.
– Intersect B-BRF 4 with S-BRF 1-3. The S-BRF 1-3 is defined as follows:
e∗(a) = 0 for all 0 6 a 6 aB−I2 ; e
∗(a) = ê for all aB−I2 < a < a
B−I
1 ; and
e∗(a) = 0 for all aB−I1 6 a 6 1. As a
∗(0) = 0 , (e∗, a∗) = (0, 0), which is NE
form IV.
– Intersect B-BRF 4 with S-BRF 1-4. We illustrate both BRFs in Figure 35. The
S-BRF 1-4 is defined by e∗(a) = ê for all aB−I < a < pα ; e∗2 (a) = ē for2β(p−w)+wα
all pα α
2 ∗
2β(p− 6 a 6 , otherwise e (a) = 0. From Lemma 16 and S-w)+wα 4β(α−D(w−c))
2
BRF 1-4 we know, aB−I < pα < α < 1, aB−I < p − β(p−w)2 .2β(p−w)+wα 4β(α−D(w−c)) 2 w Dw(w−c)
Also, similar to (A.34), one can show that ē(a)| 2 < e . In addition,
a= α 3
4β(α−D(w−c))
e∗(0) = 0 and a∗(0) = 0. In conclusion, NE is (e∗, a∗) = (0, 0), which is NE
form IV. We illustrate this case in Figure 35-1.
– Intersect B-BRF 4 with S-BRF 1-5. We illustrate both BRFs in Figure 35.
The S-BRF 1-5 is defined by e∗(a) = ê for all 0 6 a < pα and aB−II 6
2β(p−w)+wα 2
a 6 1; otherwise e∗(a) = ē. In this case, similar to (A.34) we can show
ē(a)|a=aB−II < e3. In addition, from Lemma 15, e <
w
1 < e . Therefore there
2 p
3
is no intersection between B-BRF 4 and S-BRF 1-5. It means there is no NE.
We illustrate this case in Figure 35.
Similar to last 5 cases, intersecting B-BRF 5 with S-BRF 1-1 through S-BRF 1-5
creates NE form III or NE form IV.
Region B-ii: We want to prove that if there exists any NE then the NE should be
one of the following four forms:
210
FIGURE 35. Illustration of NE and No NE Cases When Intersecting B-BRF 4 with
S-BRF 1-4 and S-BRF 1-5
– NE form I: (e∗, a∗) = (1− α , 1),
2β
– NE form II: (e∗, a∗) = (1− γ , αDp),
Dp 2γβ
– NE form III: (e∗, a∗) = (w , 0), and
p
– NE form IV: (e∗, a∗) = (0, 0).
Note that region B-ii is defined by β(p−w) < D < β and D > α(4β−α)− − − . In this regionp(w c) w c 4β(w c)
there are 4 possible forms of the supplier’s best response function: S-BRF 2-1, S-BRF
2-3, S-BRF 2-4, and S-BRF 2-5, given in Lemma 16. In the same region, there are
3 possible forms of the buyer’s best response function: B-BRF 1, B-BRF 4, and B-
BRF 5, given in Lemma 15. Consider B-BRF 1 and intersect it with the S-BRF 2-1,
S-BRF 2-3, and S-BRF 2-4. In these cases e∗(0) = w , then NE is (e∗, a∗) = (w , 0),
p p
which is NE form III. Intersect B-BRF 1 with the S-BRF 2-5. In this case e∗(0) = 0,
the NE is (e∗, a∗) = (0, 0), which is NE form IV. Next, we consider intersecting
B-BRF 4 and B-BRF 5 with 4 forms of S-BRF:
– Intersect B-BRF 4 with S-BRF 2-1. The S-BRF 2-1 is defined as follows: e∗ =
ê(a) for all 0 6 a < pα ; e∗ = ē(a) for all pα
2β(p−w)+wα 2β(p− 6 a 6 1. Note thatw)+wα
211
from Lemma 15, e1 <
w < e3. There is no intersection between B-BRF and S-p
BRF when 0 6 a < pα− . We illustrate this in Figure 36. However, there2β(p w)+wα
exist intersection between B-BRF 4 and S-BRF 2-1 when pα 6 a 6 1.
2β(p−w)+wα
Note that e∗ = ē(a) is increasing continuous function in a, ē(a)| = 1 − αa=1 .2β
Also similar to (A.36), we can show ē(a)|a=1 > e1. Therefore, based on position
of ē(a)|a=1 vs. e3, we derive NE in this case as follows and we illustrate them
in Figure 36:
• under ē(a)|a=1 < e3 (equivalent to D > 2βγ ), it creates NE (e∗, a∗) =pα
(1− α , 1), which is NE form I. We illustrate this in Figure 36-1.
2β
• under ē(a)| 2βγa=1 > e3 (equivalent to D 6 ), it creates NE (e∗, a∗) =pα
(1− γ , αDp), which is NE form II. We illustrate this in Figure 36-2.
Dp 2γβ
FIGURE 36. Illustration of NE form 1 and Form 2 When Intersecting B-BRF 4 and
S-BRF 2-1
– Similar to the above case, intersecting of B-BRF 4 with S-BRF 2-5 creates
NE form I or form II with same conditions. The S-BRF 2-5 is very similar
to S-BRF 2-1, except for all 0 6 a < aB−I , e∗2 = 0. This difference leads to
212
NE (e∗, a∗) = (0, 0), which simultaneously occurs with one of NE form I or NE
form II. However, NE (e∗, a∗) = (0, 0) is Pareto dominated by one of NE form
I or NE form II.
– Intersect BRF-B 5 with S-BRF 2-1 and S-BRF 2-5, respectively. The B-BRF
5 has a jump similar to B-BRF 4 with the same right edge. Therefore, similar
to above cases (B-BRF 4 with S-BRF 2-1 and S-BRF 2-5), it creates NE form
I, II.
– Intersect B-BRF 4 with S-BRF 2-4. The S-BRF 2-4 is defined as follows:
e∗ = ê(a) for all 0 6 a < pα B−II B−II
2β(p− and a 6 a < a ; and otherwisew)+wα 2 1
e∗ = ē(a). In this case from Lemma 15, e1 <
w < e3. Similar to previousp
cases, we can show that ē(a)|a=1 > e1 and ē(a)|a=aB−II < e3. Then, we need to2
compare ē(aB−II1 ) vs. e3:
• When ē(aB−II1 ) 6 e3 (or equivalently w > ŵ(D, p, c, α, β, γ)):
• under ē(a)| < e (equivalent to D > 2βγa=1 3 ), it creates NE (e∗, a∗) =pα
(1− α , 1), which is NE form I. We illustrate this case in Figure 37-1.
2β
• under ē(a)|a=1 > e3 (equivalent to D 6 2βγ ), it creates NE (e∗, a∗) =pα
(1− γ , αDp) which is NE form II. We illustrate this case in Figure 37-
Dp 2γβ
2.
• When ē(aB−II1 ) > e3 (or equivalently w < ŵ(D, p, c, α, β, γ)), then there
is no intersection at all between B-BRF and S-BRF. Therefore, there is
no NE. We illustrate this case in Figure 37-3.
– Similar to above case, intersecting of B-BRF 4 with S-BRF 2-3 creates NE
form I or form II with same conditions.We illustrate NE form I, NE form II,
and no NE for this cases in Figure 38.
213
FIGURE 37. Illustration of possible NE cases when intersecting B-BRF 4 with
S-BRF 2-4
FIGURE 38. Illustration of possible NE cases when intersecting B-BRF 4 with
S-BRF 2-3
– Intersect B-BRF 5 with S-BRF 2-4. In this case from Lemma15, e < w4 <p
e3. We also show in (A.36) ē(a)|a=1 > e4. Similar to (A.36) we can show
ē(a)| B−IIa=aB−II < e3. Then, we need to compare ē(a1 ) vs. e3:2
• When ē(aB−II1 ) 6 e3 (or equivalently w > ŵ(D, p, c, α, β, γ)):
• under ē(a)|a=1 < e3 (equivalent to D > 2βγ ), it creates NE (e∗, a∗) =pα
(1− α , 1), which is NE form I.
2β
• under ē(a)|a=1 > e3 (equivalent to D 6 2βγ ), it creates NE (e∗, a∗) =pα
(1− γ , αDp) which is NE form II.
Dp 2γβ
214
• When ē(aB−II1 ) > e3 (or equivalently w < ŵ(D, p, c, α, β, γ)), then there
is no intersection at all between B-BRF and S-BRF. Therefore, there is
no NE.
– Similar to above case, intersecting of B-BRF 5 with S-BRF 2-3 creates NE
form I or form II with same conditions.
In Region B (both B-i and B-ii), we find that when D > α(4β−α)− and w >4β(w c)
ŵ(D, p, c, α, β, γ) we have NE forms I and II with the boundary of D = 2βγ which
pα
divides the regions of two forms. Otherwise we have NE forms III and IV.
Region C: We want to prove that if there exists any NE then the NE should be one
of the following three forms:
– NE form I: (e∗, a∗) = (1− α , 1),
2β
– NE form II: (e∗, a∗) = (1− γ , αDp), and
Dp 2γβ
– NE form III: (e∗, a∗) = (w , 0).
p
Note that region C is defined by D > β− . In this region there is 1 possible formw c
of the supplier’s best response function: S-BRF 2-1, given in Lemma 16. There
are 2 possible forms of the buyer’s best response function: B-BRF 1 and B-BRF
6, given in Lemma 15. Intersect B-BRF 1 with S-BRF 2-1. As a∗(0) = w , then
p
NE is (e∗, a∗) = (w , 0), which is NE form IV. Next, intersect B-BRF 6 with S-BRF
p
2-1. We know that ē(a)| αa=1 is 1− and S-BRF 2-1 is continuous function in a. If2β
ē(a)|a=1 < e3 (equivalent to D > 2βγ ), then NE is (e∗, a∗) = (1− α , 1), which is NEpα 2β
form I. Otherwise, NE is (e∗, a∗) = (1− γ , αDp), which is NE form II. Now, we find
Dp 2γβ
the buyer’s and the supplier’s profit based on outcome of NE by plugging the value
of NE into the optimal order quantities, the buyer’s profit and the supplier’s profit
215
function in (2.6) and (2.8). The associated optimal order quantities, the buyer’s
profit, and supplier’s profit are:
– If NE form I holds, which is (e∗, a∗) = (1− α , 1), then
2β

0 c 6 w < c+
α
∗  2Dq =  , q∗ = D,f pD c+ α 6 w 6 p
 2D α(1− )D(p− w)− γ c 6 w < c+
α
 2β 2DΠB =  , (cor. 1)D(p− w)− γ c+ α 6 w 6 p 2D(1− α )D(w − c)− α(1− α ) c 6 w < c+
α
 2β 2β 2DΠS =  . (cor. 2)D(w − c)− α(4β−α) c+ α 6 w 6 p
4β 2D
– If NE form II holds, which is (e∗, a∗) = (1− γ , αDp), then
Dp 2γβ
0 c 6 w < c+
βγ
∗  D2pq =  , q∗ = D,f pD c+ βγD2 6 w 6 pp 1 D(p− w)(2β − α)− γ c 6 w < c+
βγ
2β D2pΠB =  , (cor. 3)D(p− w)− γ c+ βγ2 6 w 6 p
 D p

 1 D(w − c)(2β − α)− (α− αγ ) c 6 w < c+ βγ
2β Dp D2pΠS =  . (cor. 4)D(w − c)− (α− αγ ) c+ βγ 6 w 6 p
2Dp D2p
– If NE form III holds, which is (e∗, a∗) = (w , 0), then (q∗, q∗) = (0, 0) and
p f p
(ΠS,ΠB) = (−γ w , 0).p
216
– If NE form IV holds, which is (e∗, a∗) = (0, 0), then (q∗, q∗) = (0, 0) and
f p
(ΠS,ΠB) = (0, 0).
Now we assemble the results of A-i, A-ii, B-i, B-ii, and C to prove Lemma 4. In
Region A-i and B-i (these two regions are defined when D < α(4β−α)− ) the possible4β(w c)
NE are NE form III and Form IV. Also, we prove that the buyer’s profit is always
zero and the supplier’s profit is either zero or negative for both NE form III and
NE form IV. Therefore, we omit these two forms of NE from rest of analysis to find
sub-game prefect equilibrium.
In the rest of regions including Region A-ii, Region B-ii, and Region C (these regions
are defined when D > α(4β−α)− ), when D <
γ
− (where B-BRF is always a
∗ = 0
4β(w c) p w
for all 0 6 e 6 1), NE form III or NE form IV occur, which we exclude them
from our analysis of sub-game prefect equilibrium with the same reason we stated
above. When D > γ− , we have three possible forms of NE including NE formp w
I, NE form II, and NE form IV. As we proved NE form I and NE form II only
occur when w > ŵ(D, p, c, α, β, γ), otherwise NE form IV occurs. We showed when
w > ŵ(D, p, c, α, β, γ), if D > 2βγ then NE form I happens, otherwise NE form II
pα
occurs. NE form I and NE form II are considered to find sub-game prefect equilibrium
in the first stage.
To form the relationship between regions of NE form and NE form II in plane
of (D,w), we find intersections of following lines with each others: D = 2βγ ,
pα
w = ŵ(D, p, c, α, β, γ), D = γ− , and D =
α(4β−α)
− . Intercept of D =
γ and
p w 4β(w c) p−w
w = ŵ(D, p, c, α, β, γ) leads to D̂(p, c, α, β, γ) (minimum point of region of NE
√ √
p(α+2γ)+ p((α−2γ)2+16γ)−8cγ(2β−α)
form II in Figure 2-b) where D̂(p, c, α, β, γ) = √ .
4 p(p−c)
Also, D = (4β−α)α+4βγ is the intercept of D = (4β−α)α γ− − and D = − . Further,4β(p c) 4β(w c) p w
w = ŵ(D, p, c, α, β, γ) and D = (4β−α)α− intersect at D =
2βγ . Note that D = α(4β−α)
4β(w c) pα 4β(w−c)
217
is equivalent to w = c + (4β−α)α . We illustrate two possible positions of above
4βD
FIGURE 39. Division of plane of (D,w) for NE form I and form II
lines in Figure 39. 2βγ
2
> D̂(p, c, α, β, γ) if and only if γ > (4β−α)α and p >
pα 4β(2β−α)
8cγβ2
− − − 2 , therefore it generates the NE parameter space which we shows4β(2β α)γ (4β α)α
in Figure 39-2. In this case, when D > 2βγ then we prove that c + (4β−α)α >
pα 4βD
ŵ(D, p, c, α, β, γ)⇔ D3p(Dpα− 2βγ)[(4β−α)(4Dβ(p− c)− (4β−α)α)− 8γβ2] > 0
2 2
given that γ > (4β−α)α and p > 8cγβ and D > 2γβ , expression in
4β(2β−α) 4β(2β−α)γ−(4β−α)α2 pα
bracket is always positive. As we illustrate in Figure 2-b, whenD > γ , D > α(4β−α)
p−w 4β(w−c)
and w > ŵ(D, p, c, α, β, γ) , if D > 2βγ then NE I occurs, and if D < 2βγ NE II
pα pα
occurs. Also when D > γ− , D >
α(4β−α)
− , and w < ŵ(D, p, c, α, β, γ), then NE IVp w 4β(w c)
occurs. Consequently under the same condition, for all D̂(p, c, α, β, γ) < D < 2γβ ,
pα
we have c+ (4β−α)α > ŵ(D, p, c, α, β, γ).
4βD
2 2
Also 2βγ > D̂(p, c, α, β, γ) if and only if γ 6 (4β−α)α− or p <
8cγβ . In
pα 4β(2β α) 4β(2β−α)γ−(4β−α)α2
this case, NE form II vanishes. We illustrate this case in Figure 39-1. Hence, we
complete the proof.
218
Proof of Proposition 2. Recall that the associated optimal buyer’s profit and the
supplier’s profit for each forms of NE in Lemma 4 are:
– If NE form I holds, which is (e∗, a∗) = (1− α , 1), then
2β
0 c 6 w < c+
α
∗  2Dq =  , q∗ = D,f pD c+ α 6 w 6 p 2D
(1−
α )D(p− w)− γ c 6 w < c+ α
 2β 2DΠB =  , (cor. 1)D(p− w)− γ c+ α 6 w 6 p
 2D α
(1− )D(w − c)− α(1−
α ) c 6 w < c+ α
 2β 2β 2DΠS =  . (cor. 2)D(w − c)− α(4β−α) c+ α 6 w 6 p
4β 2D
– If NE form II holds, which is (e∗, a∗) = (1− γ , αDp), then
Dp 2γβ
0 c 6 w < c+
βγ
D2p
q∗ =  , q∗ = D,f pD c+ βγ2 6 w 6 pD p
 1 βγ D(p− w)(2β − α)− γ c 6 w < c+2β D2pΠB =  , (cor. 3)D(p− w)− γ c+ βγ2 6 w 6 pD p 1 D(w − c)(2β − α)− (α−
αγ ) c 6 w < c+ βγ
2β Dp D2pΠS =  . (cor. 4)D(w − c)− (α− αγ ) c+ βγ
2Dp D2
6 w 6 p
p
– If NE form III holds, which is (e∗, a∗) = (w , 0), then (q∗, q∗) = (0, 0) and
p f p
(ΠS,Π
w
B) = (−γ , 0).p
219
– If NE form IV holds, which is (e∗, a∗) = (0, 0), then (q∗, q∗) = (0, 0) and
f p
(ΠS,ΠB) = (0, 0).
From the above result, the buyer’s profit is zero when NE form III and IV exists.
Also, the supplier’s profit is negative in region III and it’s zero in region IV. Therefore,
we verify that we do not have any contracting for NE regions where (e∗, a∗) = (0, 0)
and (e∗, a∗) = (w , 0). Thus, we exclude them from feasible region of this stage.
p
As we show above in (cor.1) and (cor.3), the buyer’s profit function in NE form I
and NE form II is decreasing in wholesale price. Therefore, the buyer’s would like to
choose the lowest feasible w to maximize his profit. On the other hand, in (cor.2) and
(cor.4) the supplier’s profit function in NE form I and II is increasing in wholesale
price. Thus, if we find the lower bound of w from the supplier’s participation
constraint in optimization problem 2.13 and check its position with our feasible
region then we can find the optimal w. We analyze the optimization problem 2.13
for each condition of Lemma 5 as following.
• When γ > (4β−α)α
2 2
− and p >
8cγβ :
4β(2β α) 4β(2β−α)γ−(4β−α)α2
As we illustrated in Figure 2-b, region associated with NE form I is located between
D = (4β−α)α− (left boundary), D =
γ
− (right boundary), and above the D =
2γβ .
4β(w c) p w pα
The buyer’s profit function for Ne form I is shown in (cor.1). The defined threshold
for piecewise function in (cor.1) is w = c+ α . We rewrite the defined threshold for
2D
piecewise function in (cor.1) as D = α− . We prove that D =
α is less than
2(w c) 2(w−c)
the left boundary ( (4β−α)α− ):4β(w c)
α (4β − α)α ⇔ (4β − α)< 1 < ⇔ α < 2β.
2(w − c) 4β(w − c) 2β
220
Therefore, the first piece of the buyer’s profit function in (cor.1) is out side
of problem’s feasible region and only second piece is in the feasible region for
the maximization program 2.13. The second piece of function (cor.1) which is
(p − w)D − γ, is always positive, because in region NE form I we are above the
curve D = γ− .p w
Now consider the supplier’s profit function in (cor.2). D = α− is the indifferencew c
curve for the first piece of the supplier’s piecewise function in (cor.2). Also, the
defined threshold for the first piece is w < c + α and we rewrite it as D = α− .2D 2(w c)
By showing that α < α , we verify that the first piece of function (cor.2)
2(w−c) w−c
cannot be in the feasible region. The indifference function of the second piece of the
supplier’s function in (cor.2) is D = (4β−α)α− , which is exactly the left boundary of4β(w c)
NE form I in this case. As we proved in the above α < (4β−α)α . Therefore, the
2(w−c) 4β(w−c)
second piece of the supplier’s function is feasible and lies on the left boundary of
region NE form I. So, the supplier’s participation constraint is binding for all region
NE form I. The buyer chooses the lowest w (because the buyer’s profit is decreasing
in w), which in this case is left boundary of region NE form I thus w∗ = c+ (4β−α)α .
4βD
Next, we consider region of NE form II. As we illustrated in Figure 2-a, region
associated with NE form II is located between w = ŵ(D, p, c, α, β, γ) (left boundary),
D = γ− (right boundary), and bellow the D =
2γβ .
p w pα
In region of NE form II, the buyer’s profit function in (cor.3) is a piecewise function
and its defined threshold for pieces is w = c+ βγ2 . We prove that w = c+
βγ
2 is lessD p D p
than the left boundary of region NE form (II) w = ŵ(D, p, c, α, β, γ) which means
first piece of the buyer’s profit function (cor.1) does not cover the region NE form
II and thus only the second piece of function (cor.3) completely covers region of NE
221
form II.
√
pγβ + pγβ[pβ(4β − 2α + γ)− c(2β − α)2]
D̂(p, c, α, β, γ) >
p[2β(p− c) + cα]
⇔ p(α− γ)2[8cαγ + p(γ − 2α)]2 + [p(−2α2 + αγ + γ(8 + γ))− 8cγ]2 > 0
Therefore, we need to use the second piece of the buyer’s profit function for the
optimization program 2.13.
For the supplier’s participation constraint, we consider the supplier’s profit piecewise
function in (cor.4). From the first piece we obtain indifference curve 1 D(w−c)(2β−
2β
α) − (α − αγ ) = 0 and from the second piece we obtain indifference curve D(w −
Dp
c) − (α − αγ ) = 0. Also, the defined threshold of pieces is w = c + βγ . when we
2Dp D2p
solve simultaneously these three equations for D and w, we find that they intersect
4pα2at (w,D) = (c + β , (2β+α)γ2 ). In addition, we prove that indifference curve of(2β+α) γ 2pα
the first piece is greater than indifference curve of the second piece if and only if
w > c+ βγ
D2
:
p
βγ
w > c+ ⇐⇒ 1 D(w − αγ αγc)(2β − α)− (α− ) > D(w − c)− (α− ).
D2p 2β Dp 2Dp
To prove this, we subtract indifference functions and obtain the condition :
1 − − − αγ αγD(w c)(2β α) (α− )− [D(w − c)− (α− )] > 0
2β Dp 2Dp
2
⇔ α[D p(w − c)− βγ] > 0 ⇔ βγD2p(w − c)− γβ > 0 ⇔ w > c+ .
2Dpβ D2p
We illustrate the positions of the pieces of the supplier’s profit function in NE form
II and boundaries of region of NE II in Figure 40. For finding the feasible region
based on the supplier’s constraint, we consider the first piece of function (cor.4) to
222
FIGURE 40. Illustration of the supplier’s participation constraint for NE region (II)
of Commitment to Wholesale Price
be positive 1 D(w− c)(2β−α)− (α− αγ ) > 0 while w < c+ βγ2 which gives us the2β Dp D p
region between green line and red line in Figure 40 for all 0 6 D < (2β+α)γ . Also, the
2pα
second piece of function (cor.4) is positive D(w−c)−α+ αγ > 0, while w > c+ βγ
2Dp D2p
which gives us a region above black line for D > (2β+α)γ and above green line for all
2pα
0 6 D < (2β+α)γ in Figure 40. Therefore, the feasible region derived by the supplier’s
2pα
223
participation constraint for region of NE form II is:
 If 0 6 D <
(2β+α)γ then 1 D(w − c)(2β − α)− (α− αγ ) > 0
 2pα 2β Dp . (A.37)If (2β+α)γ 6 D < 2γβ then D(w − c)− α + αγ > 0
2pα pα 2Dp
Given above-mentioned characteristics of the indifference curves of the supplier’s
profit in (cor.4), we prove that indifference curve of second piece of (cor.4) is always
(less than) bellow of left boundary of region NE form II for all D̂(p, c, α, β, γ) 6
D 6 2γβ . To prove this, we rewrite the indifference curve of second piece in form of
pα
w(D, p, c, α.γ) and then compare it with ŵ(D, p, c, α, β, γ):
α(2Dp− γ)
ŵ(D, p, c, α, β, γ) > c+ ⇔ 8D2(Dpα−2βγ)[2Dp(α−D(p−c))+(Dp−α)γ] > 0,
2D2p
2 2
and based on condition of this case (γ > (4β−α)α and p > 8cγβ− ) the4β(2β α) 4β(2β−α)γ−(4β−α)α2
above inequality always holds. Therefore, we conclude that all region of NE form II is
feasible for our optimization problem, because the supplier’s participation constraint
is not binding. So in region of NE form II, the buyer’s optimal solution happens on
left boundary of the region, which is w∗ = ŵ(D, p, c, α, β, γ).
2
Therefore, we proved that the optimal wholesale price when γ > (4β−α)α and p >
4β(2β−α)
8cγβ2
− − − 2 is as follows:4β(2β α)γ (4β α)α
- If 0 6 D 6 D̂(p, c, α, β, γ), then we have no-contracting.
- If D > 2γβ , then w∗ = c+ (4β−α)α .
pα 4βD
- If D̂(p, c, α, β, γ) < D < 2γβ , then w∗ = ŵ(D, p, c, α, β, γ).
pα
As we proved in the above when D > 2γβ , the supplier’s participation constraint is
pα
binding, thus the supplier’s optimal profit does not make any profit. Also, the buyer’s
224
profit is positive, because D(p−w)−γ > 0. The optimal order quantities associated
with this equilibrium are (q∗p, q
∗
f ) = (D,D). When D̂(p, c, α, β, γ) < D <
2γβ , the
pα
supplier’s participation constraint is not binding, therefore the supplier’s optimal
profit is positive. Also the buyer makes a positive profit in the equilibrium, because
D(p− w)− γ > 0.
• When γ 6 (4β−α)α
2 8cγβ2
− or p < :4β(2β α) 4β(2β−α)γ−(4β−α)α2
As we illustrated in Figure 2-a, region associated with NE form I is located between
D = (4β−α)α− (left boundary) and D =
γ
− (right boundary). These two left and4β(w c) p w
right boundaries intersect in D = (4β−α)α+4βγ− . The buyer’s profit function and the4β(p c)
supplier’s profit function are identical to preceding case. Therefore, we take the
same steps to prove that the supplier’s participation constraint is binding and the
2
optimal wholesale price is w∗ = c + (4β−α)α . Thus, when γ > (4β−α)α and p >
4βD 4β(2β−α)
8cγβ2
4β(2β−α)γ− − the optimal wholesale price is as follows:(4β α)α2
- If 0 6 D < (4β−α)α+4βγ− , then we have no-contracting.4β(p c)
- If (4β−α)α+4βγ− 6 D, then w
∗ = c+ (4β−α)α .
4β(p c) 4βD
In this case, similar to region of NE form I in preceding case, as the supplier’s
participation constraint is binding the supplier’s does not make any profit. Also,
the buyer’s profit is positive, because D(p − w) − γ > 0. The optimal order
quantities associated with this equilibrium are (q∗p, q
∗
f ) = (D,D). We illustrate
optimal wholesale price along with rest of optimal contract terms and efforts in
Figure 3. Hence, we complete the proof.
Proof of Proposition 3. Using Proposition 1 and Proposition 2 and comparing
the change of each mentioned metrics give us the result. D = β+γ− is the onlyp c
threshold (line) for γ in no-commitment. We compare this condition with all curves
225
in Figure 3 to see whether there are any possible intersections. The first curve is
D = (4β−α)α+4βγ− which is shown in Figure 3-a. We have
(4β−α)α+4βγ < β+γ ⇔
4β(p c) 4β(p−c) p−c
(1 − α )α < β, therefore these two curves never intersects. For commitment to
4β
wholesale price, we have two other curves in Figure 3-b. First look at D = 2γβ . We
pα
find conditions for intersection of this curve with the other one as 2γβ < β+γ ⇔ {0 <
pα p−c
γ < αβ− or {γ >
αβ and p < 2cβγ }}, also 2γβ > β+γ ⇔ γ > αβ and p >
2β α 2β−α (2β−α)γ−αβ pα p−c 2β−α
2cβγ
− − , thus if γ <
αβ
− then D =
2γβ does not intersects with D = β+γ ,
(2β α)γ αβ 2β α pα p−c
otherwise intersection would happen at p = 2cβγ− − . This intersection is greater(2β α)γ αβ
2
than p = 8cγβ− − − 2 because γ >
αβ
− , also showing
β+γ
− > D̂(p, c, α, β, γ) is4β(2β α)γ (4β α)α 2β α p c
straightforward and it means that there is no more intersection between curves except
the ones which we proved . Therefore, comparison of no-commitment policy with
commitment to wholesale price policy for varying levels of auditing cost coefficient γ
generates three cases: 0 6 γ 6 (4β−α)α
2
, (4β−α)α
2
− − < γ <
αβ αβ
4β(2β α) 4β(2β α) 2β− , and γ >α 2β− . In theα
next part we show the changes of indexes in transition from “no-commitment” to
“commitment to price”, we start with the case γ > αβ− because it is a super-set of2β α
regions in other cases and we show it in Figure 4. When γ > αβ− , regions A, B and2β α
E are in no-contracting zone in no-commitment case and all index values are zero,
also region A is located in no-contracting zone for commitment to wholesale price.
Therefore, in region A metrics do not change after transition. In region B because of
moving from no-contracting to contracting zone, all indexes increases and the change
in the values of metrics are equal to values of metrics in commitment to wholesale
price model which are positive values, except the supplier’s profit which experiences
no change. The supplier’s profit does not change as the value of it in both models
is zero. In region E because of moving from no-contracting to contracting zone,
all indexes increases and the change in the values of metrics are equal to values of
226
metrics in commitment to wholesale price model which are positive values. In this
region, the supplier’s profit is positive in commitment to wholesale price, therefore by
this transition the supplier’s profit increases. In region C and D, we have contracting
in both policies and we calculate the change. For region C, ∆e = (1− α )− 0 > 0,
2β
∆a = 1 − 1 = 0 and ∆O = [1 − α + (1 − (1 − αsc )) × 1] − [0 + (1 − 0)1] = 02β 2β
and ∆πB = [D(p − c) − (4β−α)α − γ] − [D(p − c) − γ − β] = β − (4β−α)α > 0, also4β 4β
∆πS = 0. For region D, ∆e = (1 − γ ) − 0 = 1 − γ > 0, ∆a = αDp − 1 < 0Dp Dp 2γβ
and ∆Osc = [1 − γ + (1 − (1 − γ ))αDp ] − [0 + (1 − 0)1] = αDp−2βγ < 0 andDp Dp 2γβ 2Dpβ
∆πB = [(p− ŵ(D, p, c, α, β, γ))D − γ]− [D(p− c)− γ − β] > 0. Also πS > 0 in this
region in commitment to wholesale price and πS = 0 at this region in no-commitment.
Therefore, ∆πS > 0.
2
Next case is when 0 6 γ 6 (4β−α)α which we show it in Figure 5. All regions under
4β(2β−α)
this case including regions A, B, and C have identical behavior to the regions A, B,
and C of γ > αβ− case for all metrics. Therefore, we do not discuss them again.2β α
Next case is when (4β−α)α
2
− < γ <
αβ
− which we show it in Figure 6. Regions under4β(2β α) 2β α
this case including regions A, B, and C have identical behavior to the regions A, B,
and C of γ > αβ− for all metrics. Therefore, we do not discuss them again. Region2β α
D in this case has identical behavior to region E of γ > αβ . Therefore, we do not
2β−α
discuss it again.
Proof of Lemma 5. First, note that in this case wp = wf = w and qp = qf = q.
To find NE we first find the buyer’s best response function and the supplier’s best
response function. Then we solve for NE. The buyer’s Best Response Function:
Objective function (2.6) is linear in a and its F.O.C with respect to a is pq(1−e)−γ.
Therefore, for a given the supplier’s compliance effort e, the buyer’s best response is
one of the following cases: if pq < γ then a∗ = 0. If pq > γ then for all 0 6 e 6 1− γ ,
pq
227
a∗ = 1 and for all 1 − γ < e 6 1, a∗ = 0. The supplier’s Best Response: Objective
pq
function (2.8) is concave in e as second order condition of this function with respect
to e is −2aβ. Given F.O.C with respect to e is 2a(1−e)β−α, and ∂πS |e=1 = −α < 0∂e
, and following the concavity characteristic of the function: if 2a(1− e)β − α < 0⇒
a < α ⇒ e∗ = 0 and if 2a(1 − e)β − α > 0 ⇒ a > α ⇒ e∗ = 1 − α . Using
2β 2β 2aβ
above-mentioned the buyer’s and the supplier’s best response functions, if 0 6 q < γ
p
then (a∗, e∗) = (0, 0). If q > γ then we need to compare two values: 1 − α vs.
p 2β
1 − γ . if 1 − α < 1 − γ ⇒ q > 2γβ then (a∗, e∗) = (1, 1 − α ). Otherwise,
pq 2β pq pα 2β
1− α > 1− γ ⇒ γ 6 q 6 2γβ , then (a∗, e∗) = (αpq , 1− γ ) .
2β pq p pα 2γβ pq
Proof of Proposition 4. Plugging each pair of optimal efforts into optimization
problem (2.14), gives us following outcome:
If (a∗, e∗) = (0, 0) then optimization problem would be :
Max − wq
q,w
(A.38)
subject to q(w − c) > 0.
If (a∗, e∗) = (αpq , 1− γ ) then optimization problem would be :
2γβ pq
Max (p− w)q − γ
q,w
(A.39)
αγ
subject to q(w − c)− (α− ) > 0.
2pq
If (a∗, e∗) = (1, 1− α ) then optimization problem would be :
2β
Max (p− w)q − γ
q,w
(A.40)
α(4β − α)
subject to q(w − c)− > 0.
4β
228
For optimization (A.38) as objective function is decreasing in w and q and value
of function is negative, the buyer chooses q = 0 to obtain zero profit. Therefore,
for all 0 6 D 6 γ , we have no-contracting. For simplifying the proof process, we
p
draw the decision space (w, q) in Figure 41. The green curve is objective function
for both optimizations (A.39) and (A.40), the blue curve and orange curve are
respectively the participation constraint in optimizations (A.39) and (A.40), and
the red line is q = 2γβ . From axis q we can find the NE regions across the plane,
pα
where if 0 6 q < γ then (a∗, e∗) = (0, 0) and if γ 6 q 6 2γβ then (a∗, e∗) =
p p pα
(αpq , 1 − γ ) and if q > 2γβ then (a∗, e∗) = (1, 1 − α ), therefore if we were above
2γβ pq pα 2β
q = 2γβ then optimization (A.40) would be in place, if we were in γ 6 q 6 2γβ then
pα p pα
optimization (A.39) would be in place, and finally for all 0 6 q < γ optimization
p
(A.38) would be in place. The intersection of two supplier’s participation in
2
optimizations (A.39) and (A.40) happens at (w, q) = (c + p(4β−α)α , 2γβ2 ) and if8γβ pα
q(w−c)−(α− αγ ) > q(w−c)−α(4β−α) ⇔ q > 2γβ . The buyer wants to make a positive
2pq 4β pα
profit, so he would choose any (w, q) above the green curve (the buyer’s objective
function) and for satisfying the supplier’s participation constraint, if we are above the
red line he needs to choose the combination of contract terms above the orange curve
(the supplier’s participation constraint in (A.40)), and if we are bellow red line he
needs to choose the combination of contract terms above the blue curve (the supplier’s
participation constraint in (A.39)). The green curve (the buyer’s objective function)
intersects with q = 2γβ at (w, q) = (p− pα , 2γβ
2
), whenever p− pα < c+ p(4β−α)α ⇔
pα 2β pα 2β 8γβ2
0 < γ 6 (4β−α)α
2
or {γ > (4β−α)α
2 2
− − and p 6
8cγβ
− − − 2} creates the condition4β(2β α) 4β(2β α) 4β(2β α)γ (4β α)α
of Figure 41-1. In this case, for region above the red line the optimization (A.40) is
in place and feasible region for the problem is region above both green and orange
curves. As objective function in optimization problem (A.40) is decreasing in w
229
FIGURE 41. Decision space of contracting stage for full commitment model
and increasing in q then optimal order quantity is D and optimal wholesale price
is left boundary of feasible region which is orange line (the supplier’s participation
indifference curve for optimization problem A.40), so (w∗, q∗) = (c + (4β−α)α , D), in
4βD
addition as intersection of green and orange curve happens at q = (4β−α)α+4βγ− then4β(p c)
for all 0 6 D < (4β−α)α+4βγ− we have no-contracting as either objective function or4β(p c)
the supplier’s participation constraint would obtain a negative value. In the other
2 2 2
case, if p− pα > c+ p(4β−α)α ⇔ γ > (4β−α)α and p > 8cγβ2 − − − − 2 creates the2β 8γβ 4β(2β α) 4β(2β α)γ (4β α)α
condition of Figure 41-2. In this case, above the red line (D > 2γβ ) we have the same
pα
optimal solutions identical to above case where (w∗, q∗) = (c+ (4β−α)α , D). Bellow the
4βD
red line optimization (A.39) is in place and the supplier’s participation constraint is
blue curve. Similar to above case, objective function at (A.39) is increasing in q and
decreasing in w. the intersection of blue curve (the supplier’s participation constraint
indifference curve) with green curve (the buyer’s objective function indifference
230
√ √
p(2cαγ+p(α2+γ2))−pγ p(α+γ)+ p(2cαγ+p(α2+γ2))
curve) happens at (w, q) = ( , − ) and we√ α 2p(p c)
def p(α+γ)+ p(2cαγ+p(α2+γ2))
define D̃ = − , then for all 0 6 D 6 D̃(p, c, α, β, γ) we have2p(p c)
no-contracting and for all D̃(p, c, α, β, γ) < D < 2γβ then optimal order quantity
pα
is D and optimal wholesale price lies over the supplier’s participation constraint
indifference curve where w∗ = c + α − αγ 2 . Therefore, we prove different formsD 2pD
of equilibrium with their associated conditions and we summarize the equilibrium
forms as follows:
Form (i): no-contracting.
Form (ii): (w∗, q∗) = (c+ (4β−α)α , D), (e∗, a∗) = (1− α , 1), and π∗B > 0, π∗S = 0.4βD 2β
Form (iii): (w∗, q∗) = (c+ α − αγ 2 , D), (e∗, a∗) = (1− γ , αDp), and π∗B > 0, π∗S = 0.D 2pD Dp 2γβ
Above forms are defined based on following conditions:
– 0 < γ 6 (4β−α)α
2 2
− or p <
8cγβ :
4β(2β α) 4β(2β−α)γ−(4β−α)α2
- If 0 6 D < (4β−α)α+4βγ− , then form (i) occurs.4β(p c)
- Otherwise, form (ii) occurs.
2 2
– When γ > (4β−α)α− and p >
8cγβ
4β(2β α) 4β(2β− :α)γ−(4β−α)α2
- If 0 6 D 6 D̃(p, c, α, β, γ), then form (i) occurs.
- If D > 2γβ , then form (ii) occurs.
pα
- If D̃(p, c, α, β, γ) < D < 2γβ , then form (iii) occurs.
pα
Proof of Proposition 5. Using Proposition 2 and Proposition 4 and comparing
the change of indexes give us the results that we show in tables next to Figures
231
of the Proposition 5 . The equilibrium space of both models are very similar and
2
conditions of cases based on γ are identical. We start with when 0 < γ 6 (4β−α)α .
4β(2β−α)
Under this condition, all indexes in both models are the same so transition from
one to another does not lead to any change in indexes. In the other case, when
γ > (4β−α)α
2
− , the supplier’s profit in region A of commitment to wholesale price4β(2β α)
model is positive while transitioning to full commitment model leads to have a
zero profit for the supplier, thus changes in the supplier’s profit is negative and
the supplier’s profit decreases in this transition. Subsequently, the buyer’s profit
increases. The rest of metrics are identical, therefore by this transition they do not
2 2
change. Also, when γ > (4β−α)α− and p >
8cγβ
4β(2β α) 4β(2β− then we prove thatα)γ−(4β−α)α2
D̃(p, c, α, γ) < D̂(p, c, α, β, γ) :
√ √ √
p(α + γ) + p(2cαγ + p(α2 + γ2)) p(α + 2γ) + p((α− 2γ)2 + 16γ)− 8cγ(2β − α)
< √
2p(p− c) √ 4 p(p− c)
⇔ √0 < 4p[2cαγ + p(α2 + γ2)] + (pα− p p((α− 2γ)2 + 16γ)− 8cγ(2β − α))2.
This leads to creation of region B, in which we have no-contracting in commitment
to wholesale price and we have contracting in full commitment. Therefore, in region
B all metrics improve, except the supplier’s profit which still remain zero. In regions
C and D all metrics do not change by transitioning from commitment to wholesale
price to full commitment.
Proof of Proposition 6. We take the derivative of metrics w.r.t. d and evaluate
them:
I) We take the derivative of compliance effort with respect to d and set it greater
′ ′
than zero. ∂e(α(d),β(d)) = −β(d)α (d)+α(d)β (d) > 0⇔ β
′(d) ′> α (d) .
∂d 2β(d)2 β(d) α(d)
II) We take the derivative of auditing effort with respect to d and set it greater
232
[ ]
pD β(d)[γ(d)α′(d)−α(d)γ′∂a(D,p,α(d),β(d),γ(d)) (d)]−α(d)γ(d)β′(d)than zero. = > 0 ⇔
∂d 2β(d)2γ(d)2
β(d)[γ(d)α′(d) − α(d)γ′(d)] − α(d)γ(d)β′(d) > 0 ⇔ α(d) − β(d) > γ(d)
α′(d) β′(d) γ′
> 0. Also,
(d)
∂e(D,p,γ(d)) = −γ
′(d) < 0.
∂d DP
III) We take the derivative of overall sustainability of supply chain with respect to
′ ′ ′
d and set it greater than zero. ∂Ioc(D,p,α(d),β(d),γ(d))) = β(d)α (d)−α(d)β (d) − γ (d) > 0 ⇔
∂d 2β(d)2 pD
Dp > 2β(d)γ
′(d) α′(d) β′(d)
α′(d) β′(d) and > .
α(d)( − ) α(d) β(d)
α(d) β(d)
Proof of Proposition 7. We consider the area above and very close to the line of
D = 2γ(d)β(d) , in Figure 3. If this line moves up a little then those area above the
pα(d)
line will drop bellow it. Therefore, if we take the derivative of the function 2β(d)γ(d)
pα(d)
with respect to d, and [set it greater than zero, then ]it gives us the condition for
2β(d)γ(d)
∂ ′
pα(d) 2 β(d)[γ (d)α(d)−α
′(d)γ(d)]+α(d)γ(d)β′(d) ′ ′
this case: = 2 > 0 ⇔ 0 < α (d) − β (d) <∂d pα(d) α(d) β(d)
γ′(d) α′or (d) β
′
< (d) . We combine these two conditions and rewrite it as follows:
γ(d) α(d) β(d)
α′(d) − β
′(d) < γ
′(d) .
α(d) β(d) γ(d)
233
APPENDIX B
TECHNICAL PROOFS - CHAPTER III
Proof of Lemma 11. As we show in Figure 14 and Figure 15, there exists three
cases for ΦC,NC > ΦNC,C and four cases for ΦC,NC 6 ΦNC,C . We name the former
Case A and the latter Case B. We find the best response function for each supplier
and then intersect those best response functions to find the NE. First, look at the
Case A.
In Case A, we have three sub-cases:
– A-1) pθ2 6 c2 and pθ1 6 c1.
– A-2) pθ2 6 c2 and pθ1 > c1.
– A-3) pθ2 > c2 and pθ1 > c1.
In Case B, we have four sub-cases:
– B-1) pθ2 6 c2 and pθ1 6 c1.
– B-2) pθ2 6 c2 and pθ1 > c1.
– B-3) pθ2 > c2 and pθ1 6 c1.
– B-4) pθ2 > c2 and pθ1 > c1.
Based on the constraint in optimization program 3.17, if Ωi > p then we have no
feasible region. If Ωi 6 p, then we have two cases: 1) for all ci 6 wi 6 Ωi the
supplier i does not participate; 2) for all Ωi 6 wi 6 p the supplier i participates.
When, q∗i = 0, the supplier has no incentive to participate and there is no contracting.
If q∗i = D, then the supplier’s profit function is (wi − ci)D− (1− θi)GC. This profit
234
function is increasing in wi, therefore the supplier chooses the maximum level of wi
which is possible. Now, we look at each sub-cases to find the best response function.
A-1) In this case, (q∗, q∗1 2) = (0, 0), therefore there is no contracting.
A-2) In this case, (q∗1, q
∗
2) = (D, 0) for w1 < pθ1 and (q
∗, q∗1 2) = (0, 0) for w1 > pθ1.
We show this on Figure 14. Therefore, supplier 2 does not participate as he receives
zero order. For supplier 1, we need to check the participation constraint. If Ω1 6 pθ1
then, the supplier 1’s profit will be (wi− ci)D− (1− θi)GC and the supplier chooses
the upper bound of feasible region. Therefore, the optimal wholesale price will be
w∗1 = pθ1. Therefore, if Ω1 6 pθ1, then NE is supplier 1 participates and set wholesale
price w∗1 = pθ1 and supplier 2 does not participate. If Ω1 > pθ1 then (q
∗ ∗
1, q2) = (0, 0)
and we have no contracting.
A-3) Similar to above case, in this case also we need to compare the Ω1 vs. pθ1 and
Ω2 vs. pθ2. There exist 4 combinations of this comparisons as follows:
A-3-i) When Ω1 > pθ1 and Ω2 > pθ2, then none of suppliers participate in the
contracting, because the (q∗1, q
∗
2) = (0, 0).
A-3-ii) When Ω1 > pθ1 and Ω2 6 pθ2, then as supplier 1 does not participate.
Supplier 2’s profit is (w2 − c2)D − (1− θ2)GC, therefore he sets the wholesale price
w∗2 = pθ2.
A-3-iii) When Ω1 6 pθ1 and Ω2 > pθ2, then supplier 2 does not participate as his
participation constraint is not satisfied. Supplier 1 participates and as his profit
function is increasing wholesale price he sets w∗1 = pθ1.
A-3-iv) When Ω1 6 pθ1 and Ω2 6 pθ2, then both suppliers participate as they receive
positive order quantity in some regions. In this situation we need to compare the
intersection of two participation constraint lines vs. w2 = w1− p(θ2− θ1). It creates
two sub-cases:
235
A-3-iv-a) Ω1 6 Ω2−p(θ2−θ1): First we solve for supplier 1’s best response function.
For all w2 ∈ [c2,Ω2), the supplier 2 does not participate, therefore supplier 1 sets
the wholesale price w∗1 = pθ1. For all w2 ∈ [Ω2, pθ2), the supplier 1 sets the price
at w∗1 = Ω2 − p(θ2 − θ1) − . For all w2 ∈ [pθ2, 1], supplier 1 sets the price at
w∗1 = pθ1 − .
Now we solve for supplier 2’s best response function. For all w1 ∈ [c1,Ω1], then
supplier 1 does not participate and supplier 2 sets the wholesale price w∗2 = pθ2.
For all w1 ∈ [Ω1,Ω2 − p(θ2 − θ1) − ], the supplier 2 can set any price which is
w∗2 ∈ [Ω2, p]. For all w1 ∈ [Ω2 − p(θ2 − θ1), pθ1), the supplier 2 sets wholesale price
w∗2 = w1 − p(θ2 − θ1). For all w1 ∈ [pθ1, 1], the supplier 2 sets wholesale price
w∗2 = pθ2 − . When we intersect these two best response functions, we find the NE
as (w∗1, w
∗
2) = (Ω2 − p(θ2 − θ1),Ω2). We show this NE in Figure 42.
FIGURE 42. Graphic Representation of NE (w∗1, w
∗
2) = (Ω2 − p(θ2 − θ1),Ω2) .
A-3-iv-b) Ω1 > Ω2−p(θ2−θ1): First we solve for supplier 1’s best response function.
236
For all w2 ∈ [c2,Ω2], then supplier 2 does not participate and supplier 1 sets the
wholesale price w∗1 = pθ1. For all w2 ∈ [Ω2,Ω1p(θ1 − θ2)], the supplier 1 can set
any price which is w∗1 ∈ [Ω1, p]. For all w2 ∈ [Ω1 − p(θ1 − θ2), pθ2), the supplier 1
sets wholesale price w∗1 = w2 − p(θ2 − θ1). For all w2 ∈ [pθ2, 1], the supplier 2 sets
wholesale price w∗1 = pθ1 − .
Now we solve for supplier 2’s best response function. For all w1 ∈ [c1,Ω1), the supplier
1 does not participate, therefore supplier 2 sets the wholesale price w∗2 = pθ2. For
all w1 ∈ [Ω1, pθ1), the supplier 2 sets the price at w∗2 = Ω2 − p(θ2 − θ1). For all
w1 ∈ [pθ1, 1], supplier 2 sets the price at w∗2 = pθ2 − . When we intersect these two
best response functions, we find the NE as (w∗1, w
∗
2) = (Ω1,Ω1−p(θ1−θ2)). We show
this NE in Figure 43.
FIGURE 43. Graphic Representation of NE (w∗, w∗1 2) = (Ω1,Ω1 − p(θ1 − θ2)) .
B-1) In this case, (q∗, q∗1 2) = (0, 0), therefore there is no contracting.
B-2) In this case, (q∗ ∗ ∗ ∗1, q2) = (D, 0) for w1 < pθ1 and (q1, q2) = (0, 0) for w1 > pθ1.
237
We show this on Figure 15. Therefore, supplier 2 does not participate as he receives
zero order. For supplier 1, we need to check the participation constraint. If Ω1 6 pθ1
then, the supplier 1’s profit will be (wi− ci)D− (1− θi)GC and the supplier chooses
the upper bound of feasible region. Therefore, the optimal wholesale price will be
w∗1 = pθ1. Therefore, if Ω1 6 pθ1, then NE is supplier 1 participates and set wholesale
price w∗1 = pθ1 and supplier 2 does not participate. If Ω1 > pθ1 then (q
∗
1, q
∗
2) = (0, 0)
and we have no contracting.
B-3) In this case, (q∗1, q
∗
2) = (D, 0) for w1 < pθ1 and (q
∗
1, q
∗
2) = (0, 0) for w1 > pθ1.
The optimal solution and NE will be symmetric with respect to previous case for
supplier 2.
B-4) Similar to above case, in this case also we need to compare the Ω1 vs. pθ1 and
Ω2 vs. pθ2. There exist 4 combinations of this comparisons as follows:
B-4-i) When Ω1 > pθ1 and Ω2 > pθ2, then none of suppliers participate in the
contracting, because the (q∗ ∗1, q2) = (0, 0).
B-4-ii) When Ω1 > pθ1 and Ω2 6 pθ2, then as supplier 1 does not participate.
Supplier 2’s profit is (w2 − c2)D − (1− θ2)GC, therefore he sets the wholesale price
w∗2 = pθ2.
B-4-iii) When Ω1 6 pθ1 and Ω2 > pθ2, then supplier 2 does not participate as his
participation constraint is not satisfied. Supplier 1 participates and as his profit
function is increasing wholesale price he sets w∗1 = pθ1.
B-4-iv) When Ω1 6 pθ1 and Ω2 6 pθ2, then both suppliers participate as they receive
positive order quantity in some regions. In this situation we need to compare the
intersection of two participation constraint lines vs. w2 = w1− p(θ2− θ1). It creates
two sub-cases:
B-4-iv-a) Ω1 6 Ω2−p(θ2− θ1): First we solve for supplier 1’s best response function.
238
For all w2 ∈ [c2,Ω2), the supplier 2 does not participate, therefore supplier 1 sets
the wholesale price w∗1 = pθ1. For all w2 ∈ [Ω2, pθ2), the supplier 1 sets the price
at w∗1 = Ω2 − p(θ2 − θ1) − . For all w2 ∈ [pθ2, 1], supplier 1 sets the price at
w∗1 = pθ1 − .
Now we solve for supplier 2’s best response function. For all w1 ∈ [c1,Ω1], then
supplier 1 does not participate and supplier 2 sets the wholesale price w∗2 = pθ2.
For all w1 ∈ [Ω1,Ω2 − p(θ2 − θ1) − ], the supplier 2 can set any price which is
w∗2 ∈ [Ω2, p]. For all w1 ∈ [Ω2 − p(θ2 − θ1), pθ1), the supplier 2 sets wholesale price
w∗2 = w1 − p(θ2 − θ1). For all w1 ∈ [pθ1, 1], the supplier 2 sets wholesale price
w∗2 = pθ2 − . When we intersect these two best response functions, we find the NE
as (w∗1, w
∗
2) = (Ω2 − p(θ2 − θ1),Ω2).
B-4-iv-b) Ω1 > Ω2−p(θ2− θ1): First we solve for supplier 1’s best response function.
For all w2 ∈ [c2,Ω2], then supplier 2 does not participate and supplier 1 sets the
wholesale price w∗1 = pθ1. For all w2 ∈ [Ω2,Ω1p(θ1 − θ2)], the supplier 1 can set
any price which is w∗1 ∈ [Ω1, p]. For all w2 ∈ [Ω1 − p(θ1 − θ2), pθ2), the supplier 1
sets wholesale price w∗1 = w2 − p(θ2 − θ1). For all w2 ∈ [pθ2, 1], the supplier 2 sets
wholesale price w∗1 = pθ1 − .
Now we solve for supplier 2’s best response function. For all w1 ∈ [c1,Ω1), the supplier
1 does not participate, therefore supplier 2 sets the wholesale price w∗2 = pθ2. For
all w1 ∈ [Ω1, pθ1), the supplier 2 sets the price at w∗2 = Ω2 − p(θ2 − θ1). For all
w1 ∈ [pθ1, 1], supplier 2 sets the price at w∗2 = pθ2 − . When we intersect these
two best response functions, we find the NE as (w∗1, w
∗
2) = (Ω1,Ω1 − p(θ1 − θ2)). We
abuse the notation and do not report  in our solution space. We plot the condition
of all above cases with their respective NE on the Plane of θ1 and θ2 in Figure 44.
239
FIGURE 44. Presentation of Nash Equilibrium of wholesale Prices on Plane of θ1
and θ2.
Proof of Lemma 13. By using the parameter space from contracting stage in
Figure 44, we form the supplier’s profit function. Before presenting the function,
we define new symbol. We define θ̃ = c1D+GC , θ̃ = c2D+GC1 2 , ê1 =
(1−a1)θ̂1 , and
pD+GC pD+GC 1−a1θ̂1
ê = (1−a2)θ̂22 1− .a2θ̂2
For all θ2 < θ̂2 ⇔ e2 < ê2 :
−α 1
e1 0 6 e1 < ê1
ΠS(e1, e2) = (e1 + (1− e1)(1− a1))[(pθ1 − c1)D − (1− θ1)GC]− α1e1 ê1 6 e 6 1
(B.1)
240
This function is piecewise linear in e1 and continuous at e1 = ê1. The first piece is
decreasing in e1. Therefore, we have two solution candidates. When 0 < D <
α1
p−c1
then e∗1 = 0 for all e2 ∈ [0, ê2]. Otherwise, e∗1 = 1 for all e2 ∈ [0, ê2].
For all θ2 > θ̂2 ⇔ e2 > ê2 :

−α1e1 0 6 e1 < ê 1
Pr(P1, F2)[(pθ1 − c1)D − (1− θ1)GC]− α1e1 ê1 6 e < ē1ΠS(e1, e2) =  (B.2)Pr(P1, P2)[(w
∗
1 − c1)D − (1− θ1)GC]+Pr(P1, F2)[(pθ1 − c1)D − (1− θ1)GC]− α1e1 ē1 6 e1 6 1
where w∗1 = Ω2 − p(θ2 − θ1), Pr(P1, F2) = [e1 + (1 − e1)(1 − a1)][(1 −
e2)(a2)],Pr(P1, P2) = [e1 + (1− e1)(1− a1)][(1− e2)(1− a2)]. This profit function is
piecewise linear in e1. The function is contentious at e1 = ê1 and e1 = ē1. The slope
of first piece is always negative. The slope of second piece is always less than the
slope of third piece. Therefore, the candidate solutions are e1 = 0 and e1 = 1. By
comparing the value of function at zero and one we have: If 0 < D 6 α1− then e
∗
p c1 1
= 0
for all e α1 α1 ∗ α12 ∈ [ê2, 1]. If − 6 D 6 then e = 0 for all e ∈ [ē , 1]. If < Dp c1 c 2 22−c1 1 c2−c1
then e∗1 = 1 for all e
α1
2 ∈ [ê2, 1]. If − < D <
α1
− then e
∗
1 = 1 for all e2 ∈ [ê2, ē2].p c1 c2 c1
By merging the results of case θ2 < θ̂2 ⇔ e2 < ê2 and θ2 > θ̂2 ⇔ e2 > ê2 :
Supplier 1’s best response function e∗1 for any given e2 is:
– If 0 6 D 6 α1− , then e
∗
1 = 0 for all ep c 2 in [0, 1].1
– IF α1− < D <
α1
− , then e
∗
1 = 1 for all e2 in [0, ē2], and e
∗
1 = 0 for all e2 inp c1 c2 c1
(ē2, 1].
– If D > α1 ∗− , then e1 = 1 for all e2 in [0, 1].c2 c1
241
where ē = (1−a2)C2D+(p−c1)D+(1−a2)GC−α12 .(p−a2c2)D+(1−a2)GC
Proof of Lemma 14. The best response function of supplier 1 is always zero or
one for any given e2. Thus, we can find the Nash equilibrium by finding the best
response function of supplier 2 at e∗1 = 0 and e
∗
1 = 1. First we analyze the case when
e∗1 = 0. In this case, supplier 2’s profit function is :
−α2e2 0 6 e2 < ê2
ΠS2(e1 = 0, e2) = (e2 + (1− e2)(1− a2))[(pθ2 − c2)D − (1− θ2)GC]− α2e2 ê2 6 e 6 1
(B.3)
This function is piecewise linear and continuous at e2. the first piece is decreasing in
e2. Therefore, we can find the optimal solution by comparing the value of function
at zero and one. If D > α2− , then e
∗
2 = 1. Otherwise, e
∗
2 = 0. Next case is whenp c2
e∗1 = 1, in this case the profit function of supplier 2 is always −α2e2. Then, for all
parameters, e∗ ∗2(e1 = 1) = 0. Therefore, the Nash equilibrium of compliance efforts
is:
– If D < α1 and D < α2 , then (e∗− − 1, e
∗
2) = (0, 0).p c1 p c2
– If α1− 6 D <
α2
− or {D >
α1 and D > α2 ∗ ∗
p c1 p c2 c2−c1 p− }, then (e , e ) = (1, 0).c2 1 2
– If α2 6 D < α1 , then (e∗ ∗− − 1, e2) = (0, 1).p c2 p c1
– If α2− 6 D <
α1
− and D >
α1 ∗ ∗
− , then (e1, e2) = (1, 0) and (e
∗, e∗1 2) = (0, 1).p c2 c2 c1 p c1
Proof of Proposition 9. We plug-in the optimal compliance efforts and related
optimal contracting decisions to find the optimal auditing effort. When (e∗, e∗1 2) =
242
(0, 0), there is no contracting. When (e∗1, e
∗
2) = (1, 0) or (e
∗ ∗
1, e2) = (0, 1) then the
buyer’s profit is −γ(a1 + a2), therefore the (a∗1, a∗2) = (0, 0).
243
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