The Role of Transfer Pricing Schemes in
Coordinated Supply Chains
Kashi R. Balachandran
Stern School of Business
New York University
212-998-0029
kbalacha@stern.nyu.edu
Shu-Hsing Li
College of Management
National Taiwan University, Taiwan
886-2-2363-0231 ext. 2997
shli@mba.ntu.edu.tw
Taychang Wang
College of Management
National Taiwan University, Taiwan
886-2-2363-0231 ext. 2960
tcwang@ccms.ntu.edu.tw
Hsiao-Wen Wang
College of Management
National Changhua University of Education, Taiwan
886-4-723-2105 ext. 7511
hwwang@cc.ncue.edu.tw
February 2006
The Role of Transfer Pricing Schemes in Coordinated Supply Chains
Abstract
The objective of the paper is to study how transfer pricing schemes interact with
subcontractors’ opportunistic behaviors to affect supply chain coordination. We model
the supply chain incorporating asymmetric information among all the parties,
contractor’s innovation activities, subcontractors’ misappropriation, and transfer
pricing schemes. We examine the impact of various transfer pricing schemes on
supply chain efficiency. Specifically, we conduct a performance comparison between
the variable-cost transfer pricing scheme and the full-cost transfer pricing scheme. We
find that the subcontractor’s choice of a transfer pricing scheme affects the
contractor’s sourcing decisions and the supply chain performance, and the
variable-cost transfer pricing scheme performs better in achieving supply chain
coordination.
Keywords: Transfer pricing scheme, Coordinated supply chains, Nash bargaining
solution, Misappropriation.
1
1. Introduction
Recent research focus on inter-firm trades has introduced new challenges and
opportunities for accounting researches (see Baiman and Rajan, 2002a; Dekker, 2003).
Issues in supply chain management have attracted considerable interests in accounting
field1. These studies are devoted to scenarios where the authors exploit accounting
information and examine their impact on supply chain performance. However, they do
not analyze how the interaction between parties’ proprietary information and
accounting systems affects supply chain performance.2 Specifically, these papers do
not explore the role a choice of a transfer pricing scheme can play in inter-firm
relationships, examining the distinguishing benefits of various transfer pricing
schemes to supply chain coordination. In addition, extant supply chain literature has
emphasized the importance of information sharing in coordinating supply chains (e.g.,
Cachon and Fisher, 2000; Lee et al., 2000; Chopra and Meindl, 2006). However, Li
(2002) suggests that the well-known biggest obstacle to information sharing within
supply chains is a lack of trust between parties. Moreover, supply chain practitioners
indicate that accounting systems, such as inventory management systems and transfer
pricing schemes, have significant effects on supply chain performance.3 Somewhat
1
2
3
Specific issues addressed by accounting literature are as follows: 1. outsourcing and make/buy
decisions (e.g., Anderson et al., 2000), 2. inter-organizational cost management (e.g., Cooper and
Slagmulder, 2004), 3. strategic alliances and networks (i.e., Baiman et al., 2001), 4. value chain
analysis (i.e., Baiman et al., 2000, Baiman and Rajan 2002b) and quality issues (i.e., Balachandran
and Radhakrishnan 2006).
Except for Kulp (2002). Kulp’s study focuses on the properties of information that the retailer shares,
the manufacturer’s use of this information, and the resulting inventory management contract
(traditional inventory system vs. Vendor Managed Inventory system) and how these elements interact
to affect supply chain performance. Compared to our work, however, Kulp ignores the incentive
effects of accounting systems on parties’ up-front decisions.
Tata Consultancy Services (TCS) suggests that in the whole gamut of supply chain management,
companies act as a value hub integrating some key perceptions. For example, one aspect in the
perceptions is about relationship, partnership and alliances. TCS indicates that the related issues
include inter-company transfer pricing and strategic alliances. On the other hand, Vidal and
Goetschalckx (2001) indicate that most researchers on global logistics have taken transfer pricing as
a typical accounting problem rather than an important decision opportunity that significantly affects
the management of a global supply chain. However, this is not the case in real global logistics system
since management can decide the transfer price with some degree of flexibility within given limits.
Several researches have addressed the transfer pricing problem as an integral component of the
2
surprisingly, little attention has been paid to analyzing how the interaction between
subcontractors’ opportunism and transfer pricing schemes affects the efficiency of
supply chains.
In current supply chains practice, the prevailing organizational structure in
industry is based on decentralized decision making (see Sahin and Robinson, 2002).
Clearly, there is a need to build in performance measurement mechanisms to facilitate
efficient supply chain coordination. Transfer pricing scheme is an instrument to
coordinate the actions of divisional managers and to evaluate their performance in a
decentralized firm.
We model the subcontractor as a decentralized firm and study
the role of transfer pricing schemes within this firm on coordinated supply chains.
In another line of research, increasing attention has been paid to obtaining a
better transfer pricing scheme to facilitate internal trades and align the interests of
subunits with those of headquarters. Baldenius, Reichelstein and Sahay (1999)
compare the effectiveness of standard-cost and negotiated transfer pricing schemes in
firms where divisional managers possess symmetric information. They show that the
negotiated transfer pricing often performs better than the standard-cost transfer
pricing scheme. Lambert (2001) suggests that future work should consider a transfer
pricing model that divides production costs into a fixed and a variable cost to study
more meaningful issues.4 We develop a model for the supply chain and abstract from
managerial compensation issues, by focusing on analyzing the commonly used
cost-based schemes in practice and compare the variable-cost and the full-cost
transfer pricing schemes.5
4
5
The coordinating activities include the headquarters (HQ)
optimization of a supply chain; see, for example, Canel and Khumawala (1997).
Lambert (2001) indicates that many of the more recent studies in transfer pricing have moved away
from deriving the optimal transfer pricing mechanism. Instead, these researches have concentrated on
comparing alternative transfer pricing schemes.
Some surveys (see Horngren et al., 2006, p. 767; Kaplan and Atkinson, 1998, p. 458) indicate that for
domestic transfer pricing, managers in all countries are inclined to adopt cost-based transfer pricing
schemes. The surveys also show that the most popular method of determining transfer price in
3
of the decentralized subcontractor firm stipulating the two divisional managers to
make relationship-specific investments and efforts in anticipation of the contractor’s
R&D activity.
The HQ coordinates the activities of his divisions by adopting a
transfer pricing scheme.
The major questions are: given the divergent incentives of
all the parties in the supply chain, what role does transfer pricing scheme play in
inter-firms’ relationships and in supply chain performance?
Which transfer pricing
scheme performs better in achieving supply chain coordination?
The objective of the paper is to model the supply chain and analyze the above
questions. Specifically, the supply chain is modeled with asymmetric information,
incorporating the contractor’s R&D innovation, the subcontractor’s misappropriation
possibility and accounting choices with respect to the choice of a transfer pricing
scheme. We, specifically, examine the following. First, in the absence of incentive
problems (e.g., the subcontractor would not choose to misappropriate), whether the
contractor strictly prefers to establish the coordinated supply chain rather than end the
sourcing relationship to increase his surplus. Second, considering the divergent
incentives, we identify the determinants of the contractor’s innovation disclosure
strategy and examine how the contractor’s relationships decisions are affected. Third,
we examine whether the individual party’s investments and efforts decisions are
sub-optimal. We further explore the impact of the choice of transfer pricing schemes
on the up-front decisions. Lastly, we conduct a performance comparison between the
variable-cost transfer pricing scheme and the full-cost transfer pricing scheme.
The results are as follows. First, the first-best solution in the absence of incentive
problems shows that the contracting parties can benefit from organizing the
coordinated supply chain. Second, the contractor’s relationship choices and each
practice is a full-cost pricing scheme. According to a global transfer pricing survey by Ernst & Young
(2003) the cost-plus method is the most common method for pricing intra-company services in all
countries.
4
party’s investments decisions are distorted in the presence of incentive problems.
Third, with all the divergent incentives present, we find information sharing
distortions, inefficient trades, and holdup problems in the supply chain. Our results
are consistent with the transaction cost economic theory in that contractor firms will
take the magnitude of transaction costs into account in deciding on outsourcing the
“new” product. Finally, we find that the variable-cost transfer pricing scheme
performs better than the full-cost transfer pricing scheme for transfers between
divisions in the decentralized subcontractor firm.
The paper contributes several results. For the supply chain studies, in addition to
the subcontractor’s misappropriation possibility, we show that the subcontractor’s
accounting choices affect the contractor’s willingness to share information on his new
innovation. More precisely, we provide new results about the effect of accounting
choices on the strategic behaviors of parties in the supply chain. Specifically, we find
the choice of a transfer pricing scheme for internal transfers in the subcontractor firm
has differential impact on supply chain collaboration. For the transfer pricing
literature, we extend its impact on inter-firm collaborations. We show that the choice
of a transfer pricing scheme affects not only the division’s decisions within a firm but
also the strategies of other parties in the supply chain. In addition, we find that the
variable-cost transfer pricing scheme dominates the full-cost transfer pricing scheme.
The neo-classical literature on transfer pricing suggests that trade distortion can be
avoided if firms adopt a variable-cost transfer pricing scheme. However, we find that
trade distortion still exists even if the subcontractor adopts the variable-cost transfer
pricing scheme. Overall, our analysis highlights that supply chains need to consider
the incentive implications of accounting choices within a subcontractor firm.
The remainder of the paper is organized as follows. In section 2, we describe and
formulate the analytical model. In section 3, we characterize the bargaining game. In
5
section 4, we use a numerical example to compare the variable-cost transfer pricing
scheme and the full-cost transfer pricing scheme. We conclude in section 5.
2. The model
2.1 General description of problems
We study a one-period supply chain coordination problem that consists of a
subcontractor, a contractor and a consumer (see figure 1). We assume that the
subcontractor is a fully decentralized firm consisting of a headquarters (HQ) and two
divisions.6 The supplying division (D1) produces and transfers goods demanded by
the buying division (D2).
D2 further assembles the transferred-in intermediate
goods into finished goods and delivers them to the contractor.7 We assume that both
the divisional managers are risk-neutral and effort-averse. A transfer pricing scheme is
set by the HQ to price the intra-firm transfer between D1 and D2.
Subcontractor
HQ
D1
D2
Contractor
Consumer
TP schemes
Figure 1. The supply chain framework
We assume that the contractor must first purchase products from the
subcontractor before selling in the consumer market.8 We assume that the
subcontractor and D2 have necessary incentives to fulfill the demand of the contractor.
6
7
8
Throughout our analysis, the term “subcontractor” represents the decentralized firm as a whole.
We assume that one unit of intermediate good can only be processed into one unit of finished
product.
The subcontractor has comparative advantages in producing products. Obviously, such assumption
demonstrates and explains contractors’ motivations of outsourcing, i.e., reducing costs (see
Narasimhan and Jayaram, 1998; Logan, 2000). Also, we assume that the contractor has comparative
advantages in selling the finished goods.
6
D1 will choose the quantities of intermediate goods to manufacture and transfer out to
maximize her objective. Both D1 and D2 do not have a market to sell this line of
goods outside. (This may not be the only product for the divisions and hence it is
important to set prices for internal transfers.) Consequently, D1 will have no
incentives to produce in excess of the number demanded by D2.
At the start of the process, the contractor invests in R&D activities to bring out a
“new” product. In anticipation of participating in the new product, D1 will make a
process-related investment and choose an effort level to exert in order to enhance the
quality of the new process; D2 will choose an effort to enhance the assembly process.
Assume that HQ cannot directly observe the divisions’ actions and investments. Since
neither the process investment nor the efforts are publicly observable9 plus each
division only focuses on maximizing its surplus, divisional managers may be driven
to adopt dysfunctional behaviors. Specifically, D1’s optimal choice may diverge from
that of HQ, and consequently its investments and efforts may result in
underproduction. The quantity the subcontractor can transfer to the contractor is
constrained by the number that D1 decides to transfer to D2. This may influence the
effort level choice of D2 on assembly maintenance.
The contractor’s R&D
investment incentives may be reduced due to the shirking of the subcontractor
divisions.
The maximum benefits accrue to the supply chain from the full exchange of
proprietary information on the innovation by the contractor and the efficient transfer
of (intermediate) product between the subcontractor divisions. The supply chain may
experience information distortions, inefficient trades, and holdup problems if each
party in the supply chains only optimizes their individual objective. In our scenario,
9
As a result, HQ cannot sign complete contingent contracts with the divisions. Also, an upfront
contract across the divisions is not viable.
7
the contractor acquires innovation via up-front R&D investment. This innovation
information needs to be shared with the subcontractor in order to process the new
product through a coordinated supply chain. However, the subcontractor may decide
to use this information opportunistically and misappropriate for his own benefit10. In
order to simplify our analysis and to focus on our main issues of the paper, we assume
the subcontractor’s misappropriation is the result of a coordinated decision of the
parties HQ, D1 and D2 in the decentralized firm. The benefits from such opportunistic
behaviors cannot be contracted on and hence this game is incomplete contracting.11
In addition to suffering from the potential misappropriation, the contractor also
faces “architectural” risk due to the subcontractor’s accounting choices (specifically,
choice of transfer pricing scheme) prior to outsourcing. These risks together may
influence the contractor’s sourcing decisions and innovation disclosure strategies.
The misappropriation risk alone may prompt the contractor to sacrifice the efficiency
benefits of design and production of the innovated product and end the supply chain
relationship.
The time line of the model is as follows (see figure 2). At date 0, HQ selects
either a variable-cost or a full-cost transfer pricing scheme to guide internal trades.
(We do not consider any optimal transfer price schemes but rather study the impact of
the choice of a scheme on the supply chain coordination.) At date 1, the divisions D1
and D2 decide individually on their private levels of investments and efforts. The
contractor invests in R&D activities. The state variables are realized at date 2. The
contractor privately observes whether an innovation occurs or not. In addition, the
10
11
As the contractor voluntarily discloses his innovation information to the subcontractor, and the latter
fully fulfills the production need of the former, it is reasonable to expect that the total surplus shared
by the supply chain will increase.
The contractor can deter the subcontractor from misappropriating by seeking legal protection for his
invention. In reality, however, the procedure of patent protection or lawsuit is long, expensive and
often not viable. That is, the property rights over patents are difficult to identify and defend. As a
result, if the subcontractor can misappropriate even parts of the contractor’s innovation, it will be
consistent with our model.
8
contractor will rationally take into account not only the value of the innovation but
also the risk of potential misappropriation to decide on whether to disclose the
innovation and adopt the coordinated supply chain. HQ, D1, and D2 observe the
realized production costs. At date 3, HQ decides whether to accept the contractor’s
offer and whether to misappropriate if the contractor reveals the innovation
information. At this stage, the contractor does not know the subcontractor’s decision
on misappropriation. Both the contractor and HQ bargain over sharing of the total
surplus and sign a contract. At date 4, D1 decides on the quantity of the intermediate
goods to transfer out to D2. The internal trade is finished, and D1 receives her transfer
price. At date 5, D2 assembles the intermediate goods and delivers the product to the
contractor. The inter-firm transaction is completed and HQ obtains the surplus from
collaboration. D2 receives the residual surplus from HQ.
0
HQ selects either variable-cost or full-cost transfer pricing scheme.
1
The contracting parties decide on their investments and efforts.
2
~ ~
The state variables ~,  1 ,  2


realize.
The contractor decides on disclosure of his innovation.
HQ, D1, and D2 observe the realized costs.
3
HQ decides on misappropriation
Both the contractor and HQ bargain over the total surplus and sign a contract.
4
D1 decides on the quantity of the intermediate goods to transfer out to D2.
D1 receives her transfer price.
5
D2 assembles the goods and delivers them to the contractor.
D2 receives the residual surplus from HQ.
Figure 2. The time line of the model
2.2 The model formulation
9
2.2.1 Relationship-specific investment and production description
At date 1, the contractor strategically chooses an R&D investment, r to develop
an innovation. At date 2, the innovation v ( v  0 ),12 is generated according to the
known relationship, v~  V ( r, ~ ) , where ~ , a random variable is distributed according
to a continuous distribution function F ( ) with a density function of f ( ) over the
support (0,  ] . We assume V ( r , ~ ) is differentiable and increasing in r for v  0 .
Let


0
f ( )d denote the probability of an innovation occurring.13 The contractor
privately observes the innovation occurrence together with its underlying value
including the market response.
At date 1, D1 strategically makes a process-related investment  and exerts a
process quality enhancing effort X , at a personal cost, w( X ) , on the potential “new”
product. In essence, such investment and effort give D1 the following benefits. First,
she only has to spend a fixed setup cost, C ( ) , to begin producing the intermediate
goods when the contractor adopts the coordinated supply chain. We assume that
C ()  0 , C ()  0 for all  .14 Second, the effort of quality enhancing activity,
X reduces the variable production cost of the intermediate good, C1 ( q, ) , incurred by
D1, produce q units of the intermediate good of the “new” product with value v .
The variable cost C1 ( q, ) is given as follows:
~
C1 ( q, )  (k  ~)q ,
(1)
where k and ~ denote nonrandom and random components respectively of the
12
13
The intrinsic value of innovation is normalized here at unity. Therefore, v represents not only the
physical innovation but also the value of the innovation.
We employ such probability measure to capture a fact that in most R&D activities, there is always a
significant possibility that nothing will be manifested. The probability of no innovation is

14
1   0 f ( )d  1  F ( ) .
The intuition behind the assumption is that the specific investment decreases C () , but cost savings
decline with increasing  .
10
variable production cost. Specifically, ~ is the cost of production heterogeneity
~
generated by ~  Y ( X ,1 ) .15
~
We assume that Y ( X ,1 ) is differentiable and
~
~
decreasing in X for all 1 . Assume further that  1 has a cumulative distribution
function G  1  with density function of g ( 1 ) and support [ 1 , 1 ] . Let E~   e
be common knowledge among HQ, D1, and D2. Division D2 chooses a level
of assembly maintenance effort Z , at a personal cost w(Z ) , at date 1.
The effort helps to reduce the variable assembly costs of the intermediate good
borne by D2, C2 ( q, ) .16 The relation between Z and C2 ( q, ) is:
~
~
C2 ( q, )   q ,
(2)
~
~
~
where  is the cost of assembly heterogeneity generated by   K ( Z , 2 ) .
~
~
Assume that K ( Z , 2 ) is differentiable and decreasing in Z for all  2 . We further
~
assume that  2 has a cumulative distribution function H  2  with density function
~
of h  2  and support [ 2 , 2 ] . We also assume that  > 0 even if Z   .
To summarize, the total production cost of q units of the “new” product
incurred by the subcontractor as a whole, TC s ( q, ) is:
~
~
TC s ( q, )  C ()  C1 ( q, )  C2 ( q, )
.
~
~
 C ( )  ( k   ) q   q
(3)
2.2.2 The description of transfer pricing schemes


15
~
~
~
As a result, C1 ( q, k , X , ~, 1 )  k  Y ( X , 1 ) q  ( k  ~) q . The stochastic component, ~
16
accounts for the costs of all the uncontrollable events that can affect production yields and cycle
time.
After D1 has transferred out the intermediate goods to D2, D2 needs to entail a variable cost to
assemble the intermediate goods and transform those goods into the “new” products.
11
HQ has access to all actual costs of the “new” product via accounting reports, but
remains uninformed of the divisions’ investments and efforts, and the realizations of
~
~
1 and  2 . In effect, all realized production costs are common knowledge among
HQ, D1, and D2 at date 2. HQ chooses either the variable-cost or the full-cost based
transfer pricing scheme to guide intra-firm transfers.17 We assume the following
forms for the transfer price schemes.
The variable-cost transfer-pricing scheme is specified as:
Tˆvc (qˆ )  (1   )( k   )qˆ  12  (  e)qˆ 2 ,
(4)
where Tˆvc ( qˆ ) represents transfer revenue that D1 can receive from transferring
q̂ units of the intermediate goods to D2 in the variable-cost transfer pricing scheme;
  0 denotes the markup ratio of the variable-cost scheme that is determined by HQ,
( k   ) denotes D1’s realized variable production cost per unit,
1
2
 (  e)qˆ 2
denotes a penalty (compensation) for D1 when the variable production cost per unit
deviates from the expected variable cost per unit,   0 is the parameter of
punishment (reimbursement).18
The full-cost transfer pricing scheme is specified as:
Tˆ fc ( qˆ )  (1   )[C ( )  ( k   ) qˆ ]  12  (  e) qˆ 2 ,
(5)
where Tˆ fc ( qˆ ) is the transfer revenue under the full-cost transfer pricing
17
18
Unlike a mechanism design approach adopted by several authors in the transfer pricing literature
(e.g., Ronen and Balachandran, 1988; Balachandran and Ronen, 1989; Amershi and Cheng, 1990),
and instead of taking a laissez-faire way that the central office is assumed to only specify generic
rule about the rights and obligations of divisions (e.g., Baldenius et al., 1999; Baldendius, 2000), we
assume that HQ will design transfer pricing schemes with a harmonized perspective. That is, in our
scenario, HQ sets the transfer prices and allows the divisions to trade according to the transfer
prices.
Such a transfer pricing function can partly capture the spirit of controllability as described in
management accounting textbooks.
12
scheme quantity q̂ is transferred, C ( ) is D1’s realized fixed setup cost, and
  0 denotes the markup ratio of the full-cost scheme that is determined by HQ.19
We introduce a penalty or reward component, similar to the well known New
Soviet incentive scheme, in the transfer pricing schemes (4) and (5) to align the
incentives of D1 so that he or she can make production decision and exercise efforts
desired by D2 and the contractor. If the penalty or reward component is not added, D1
will has no incentive to control the costs and will like to produce as many as possible
regardless of costs. This clearly is not the best interest of the contractor. Given the
variable-cost transfer-pricing scheme, at optimum 12  (  e) q, in fact, shall be set equal
to bq in (6) of the following section.
Then, we have  
2b
for each realized ε.
(  e)
2.2.3 The description of consumer market
The inverse demand function of the “new” product, p (q ) , is assumed to be:
p( q)  v  bq ,
(6)
where q denotes the quantity of the “new” product demanded by the consumer,
v is the quality or value of the innovated “new” product,20 and b is a positive
constant. The revenue p is realized by the contractor if he finds an innovation v
(with probability F ( ) ) and truthfully releases the information, and the supply chain
is implemented.
2.2.4 The description of the subcontractor’s misappropriation risk
We assume that the subcontractor HQ and the divisions act together in their
decision to misappropriate. Once the decision to misappropriate is made, the
subcontractor makes production decisions together. We assume that the subcontractor
19
20
HQ doesn’t set any standard for C () . That is, C () will be entirely passed on to D2 under the
full-cost scheme.
The innovation can be transformed into a higher quality product via subcontractor’s production
process. As a result, the terms “innovation” and “quality” are interchangeable in our scenario.
13
can use the misappropriated information to produce and then sell units of quality v
directly to the market at lower prices and obtain a contribution profit that is 
( 0    1 ) multiplied by the total contribution margin to the whole chain when the
contractor adopts the coordinated supply chain.21
3. The equilibrium analysis
In this section, we solve the game for the Nash bargaining equilibria by the
method of backward induction. The contractor and the subcontractor will share the
total surplus following the Nash bargaining solution (Nash, 1950).
In addition, we
use three indexes: efficient trades, efficient investments, and the extent of information
sharing to measure the efficacy of the supply chain coordination.
3.1 Equilibrium without incentive problems: the first-best scenario
Upon developing an innovation, the contractor has two options. He can fully
disclose the innovation and organize the coordinated supply chain; or, he can withhold
the information and end the relationship. The contractor’s status-quo surplus
subsequent to the innovation disclosure (i.e., his no-agreement utility) is assumed to
be zero. We construct a baseline, first best, case where there exist no incentives
problems to which we will compare our asymmetric information case.
3.1.1 First-best transferred quantity
In the absence of any incentive problem (e.g., the subcontractor would not
choose to misappropriate, both the contractor and D1 would make adequate
investments, and D1 and D2 would choose the first best effort levels), the contractor
would fully disclose his proprietary information on innovation and adopt the
21
The subcontractor’s contribution margin of the misappropriated “new” product per unit is
[(v  bq)  (k     )] . We use this for ease of analysis. (See Baiman and Rajan, 2002a).
Alternatively, we can use  times the market price of the “new” product,  (v  bq) , but we
believe the qualitative nature of our results will still hold. We assume that   1 because the
subcontractor is not in the business of selling the products to the consumer.
14
coordinated supply chain. The quantities to be exchanged are given by maximizing
the profit for the entire chain:
Max (v  bq)q  (k     )q  C() ,
(7)
q
where ( v  bq ) q is the revenue to the supply chain and (k     ) q  C () is
the supply chain’s total production costs of manufacturing and selling q units of the
“new” product. Notation v here represents the contractor has made the first-best
level of R&D investment r FB ; namely, v  V ( r FB ,  ) . Similarly,  , C ( ) , and 
denote the divisions have made the first-best levels of the investments and efforts.
FB
qmax

v  k     22
.
2b
(8)
The total surplus profit in the first-best scenario is  * 
(v  k     ) 2
 C ( ) .
4b
The Nash bargaining solution implies that the contracting parties divide the
surplus equally. In the absence of any incentive problems, the contractor’s optimal
surplus when adopting the coordinated supply chain is  c* =
(v  k     ) 2 1
 2 C ( ) .
8b
Clearly, in the absence of incentive problems, the transfer pricing schemes play
no role in affecting D1’s choices of quantity. All parties can attain their respective
FB
maximum profits at units of qmax
. We term this the efficient trade. Yet, a transfer
price needs to be established for all transfers since divisions may engage in multiple
products and are evaluated based on their divisional profits. The HQ will choose
transfer prices so that the divisions are indifferent to the specific choice. That is the
mark up ratios for the two schemes are chosen so that the resultant divisional profits
for transferring the stipulated first best quantity are the same. The relationship
between the markup ratios for the two policies under consideration is given in lemma
22
Without loss of generality, we assume qmax  0 . That is, we assume v  k      0 .
FB
15
1.
Lemma 1. HQ will choose the markup ratios of the cost-plus transfer pricing schemes
according to the rule of  
FB
 (k  e)qmax
 C ( )
FB
C ()  (k  e)qmax
.
Proof: Follows readily by equating the expected profits under the two schemes and
noting in (4) that the expected penalty is zero.
Observation:23
1.  is strictly decreasing with C (  ) for each  (i.e.,

 0 ). That is, the
C ()
smaller C (  ) , the larger  .
2.

0.
C ()
3.    for all C ( ) .
For example, HQ can choose full cost as the transfer price scheme (i.e.,   0 ).
Equivalently, use the variable cost scheme with

C ( )
FB
(k  e)qmax
The markup is the ratio of fixed setup cost per unit to the variable cost per unit.
3.2 Equilibrium with asymmetric information: in the presence of
incentive problems
In the presence of incentive problems, the contractor should outsource
SB
qmax

v  k  
units of the “new” product based on the whole chain’s
2b
maximization program. The total surplus of the entire chain in the second-best
scenario is 
**
(v  k     ) 2

 C ( ) . Notation v here represents that the
4b
contractor has made the second-best level of R&D investment r SB ; namely,
23
In what follows, the arguments of   , C ()  and   , C ( )  are omitted for the purpose of easier
exposition.
16
v  V ( r SB ,  ) . Similarly,  , C ( ) , and  represent that the divisions have
undertaken the second-best levels of the investments and efforts (i.e.,   Y ( X SB , 1 ) ).
However, the contractor decision to outsource needs to be analyzed. (The variables
here refer to second best scenario and it is reflected in their arguments. For
simplicity we have not appended sb to each of the variables.)
3.2.1 Second-best transferred quantity
In the presence of incentive issues, D1’s decision on the quantity to be
transferred in alternative transfer price schemes will depend on her maximization
programs. Specifically, D1’s maximization program under the variable-cost transfer
pricing scheme is:
Max  1vc ( qˆ )
qˆ
 (1   SB )( k   vc ) qˆ  12  ( vc  e) qˆ 2  ( k   vc ) qˆ  C (  SB
)
vc
  SB ( k   vc ) qˆ   ( vc  e) qˆ  C (  vc )
1
2
2
SB
,24
(9)
SB
s.t. qˆ  qmax
where (1  SB )( k   vc )qˆ  12  ( vc  e)qˆ 2 denotes the transfer prices that D1 can
receive by transferring q̂ units of the intermediate goods to D2, (k   vc )qˆ  C ( SB
)
vc
is D1’s total costs of producing q̂ units of the intermediate goods, and  SB denotes
the markup ratio of the variable-cost scheme that is determined by HQ. The constraint
in equation (9) implies that D1 will have no incentives to overproduce.
By the
first-order condition approach, D1’s optimal choice of transferred quantity under the
variable-cost transfer pricing scheme in the presence of incentive problems (i.e., the
second-best transferred quantity), qˆ1*vc , is:
24
SB
SB
 vc  Y ( X vc , 1 ) , where X vc represents D1’s second-best level of enhancing process quality
effort in the variable-cost transfer pricing scheme.
17
qˆ1*vc 
 SB ( k   vc )
if  vc  e .
 ( vc  e)
(10)
SB
Otherwise, qˆ1*vc  qmax
if  vc  e . D1’s maximum profit under the variable-cost
transfer-pricing scheme is as follows:

 1vc
 SB2 (k   vc ) 2

 C (  SB
) if  vc  e .25
vc
2 ( vc  e)
(11)
Under the full-cost transfer-pricing scheme, D1’s maximization program is:
Max  1 fc ( qˆ )
qˆ
 (1   SB )[C (  SBfc )  ( k   fc ) qˆ ]  12  ( fc  e) qˆ 2  ( k   fc ) qˆ  C (  SBfc )
  SB [C (  fc )  ( k   fc ) qˆ ]  12  ( fc  e) qˆ
SB
2
.26
(12)
SB
s.t. qˆ  qmax
and the optimum qˆ1* fc is given by:
qˆ1* fc 
 SB (k   fc )
if  fc  e .
 ( fc  e)
(13)
SB
Otherwise, qˆ1*fc  qmax
.
D1’s corresponding maximized profit is:

 1 fc
 SB2 (k   fc ) 2

  SB C (  SBfc ) if  fc  e .
2 ( fc  e)
(14)
Note particularly, HQ sets transfer prices based on which D1 and D2 trade.
Since realized variable costs may differ from the most efficient amounts, setting the
transfer prices based solely on the actual costs will distort incentives for divisions to
make up-front investments and efforts. Therefore, HQ will punish D1 when the
realized variable costs are higher than the expected. If lower, HQ will reward D1.
25
26

Without loss of generality, we assume that  1vc  0 .
 fc  Y ( X fc , 1 ) ,where X fc represents D1’s second-best level of enhancing process quality effort in
SB
SB
the full-cost transfer pricing scheme.  S B denotes the markup ratio of the full-cost scheme that is
determined by HQ in the second-best scenario.
18
Proposition 1. D1, acting according to self-interests, will choose underproduction
relative to the first-best benchmark and the quantity desired by the contractor
regardless of the transfer pricing schemes adopted.
Proof: See Appendix.
Proposition 1 suggests that the contractor and the supply chain experience
inefficient trades when HQ adopts decentralization and the incentive issues are
recognized. We find that with a decentralized subcontractor firm, divergences of
preferences between the divisions will cause trade distortions not only in intra-firm
relationships but also in inter-firm relationships. In addition, our results can partly
capture the implications of bullwhip effect for supply chain management if one can
view the parties as independently operated firms.27 Below we confine our attention in
the case of  i  e i  vc, fc, and proceed to analyze how D1’s cost behaviors,
affected by her up-front decisions, influence supply chain performance.
The underproduction discussed above reduces the total profit in the supply chain.
It will be interesting to assess the loss in profit. The loss is given by:
 i   **  i
SB
 b( qmax
 qˆ1*i ) 2
i  vc, fc
,
(15)
where i  vc, fc represents the transfer pricing scheme, i represents the
whole supply chain’s profit after taking the impact of inefficient trade into account,
SB
and (qmax
 qˆ1*vc ) is the extent of trade distortion under the corresponding scheme.
The loss in (15) for the variable costing scheme can be shown as follows:
SB
If D1 transfers the quantity desired by the contractor (i.e., qmax
) the whole chain’s
maximizum profit is:
27
The bullwhip effect is essentially the phenomenon of demand variability amplification along a
supply chain, from the retailers, distributors, manufacturers, and the suppliers, and so on (see Lee et
al., 2000).
19
 **  (v  bqmSBa x)qmSBa x (k     )qmSBa x C() .
Plugging D1’s second-best quantities (i.e., qˆ1*vc ) into the whole chain’s maximization
program yields: vc  (v  bqˆ1*vc )qˆ1*vc  (k     )qˆ1*vc  C() . Subtracting vc from
SB
 ** gives  vc  b( qmax
 qˆ1*vc ) 2 . The proof for the fixed cost scheme is similar.
The magnitude of the loss depends on the extent of trade distortion (i.e.,
SB
(qmax
 qˆ1*vc ) ) and the amplitude of demand (i.e., b ). Clearly, the transfer pricing
schemes influence the contracting parties’ behaviors of investing, and in turn, affect
the supply chain coordination. In addition, the Nash bargaining solution implies that
the contracting parties will have to bear the impact of the inefficient trade together
and split the surplus equally.
3.2.2 Characterization of the contractor’s innovation sharing strategies
Assume a situation where the contractor truthfully discloses the innovation in
anticipation of establishing a collaborated supply chain with the subcontractor. In this
situation, the contractor’s status-quo surplus subsequent to innovation disclosure is
zero. However, the subcontractor, based on self-interests, may decide to unilaterally
misappropriate the revealed innovation and supply the consumer market. The related
maximization program of the subcontractor with misappropriation is as follows:
Max [( v  bq)q  (k     )q]  C() ,
q
(16)
where  ( v  bq  k     ) q is the total contribution margin that the
decentralized firm can obtain from misappropriation. Optimal q , and the
corresponding maximum profit ˆ are:
q* 
28
v  k     28
,
2b
Note that q 
*
v  k  
2b
(17)
SB
is equal to q max .
20
ˆ 
 (v  k     ) 2
4b
 C ( ) .29
(18)
This implies that the subcontractor’s status-quo surplus is ˆ .30
The Nash bargaining solution implies that for i  vc, fc, the transfer pricing
scheme adopted by HQ, the contractor’s optimal surplus from disclosing information
and entering into the coordinated supply chain, ci , is his status-quo utility plus
one-half of the additional utility created by reaching an agreement and forming the
coordinated supply chain. That is,


ci    12 (i  ˆ   )
2
  (v  k     ) 2

1  (v  k     )
0 
 C (  )   i  
 C (  ) 
2
4b
4b

 ,31

2
(1   )( v  k     )

 12  i
i  vc, fc
8b

where  is the contractor’s status-quo surplus.
(19)
The subcontractor’s optimal surplus from joining the coordinated supply chain
under i  vc, fc, transfer pricing scheme when the possibility of misappropriation
exists, si , is generated in the same manner as the contractor’s. That is,

si  ˆ  12 (i  ˆ   )
(1   )( v  k     ) 2

 C    12  i
8b
.32
i  vc, fc
(20)
After paying D1’s transfer prices, HQ allocates the residual surplus, obtained
29
30
31
32
We assume ˆ  0 so that the misappropriation is a viable option for the subcontractor.
Note particularly, the subcontractor’s status-quo surplus subsequent to the contractor’s innovation
disclosure is her unilaterally misappropriated surplus. Therefore, the threat point of the
subcontractor cannot include producing for the contractor.
Without loss of generality, we assume that ci  0 .
We assume si  0 v  0 and i  vc, fc to ensure that the subcontractor is always willing to
join the coordinated supply chain arrangement.
21
from the contractor, to D2. That is, D2’s optimal surplus is,
2i 
(1   )( v  k     ) 2

 C    12  i   1i i  vc, fc.33
8b
(21)
We compare the contractor’s surplus with and without disclosing the innovation
information and obtain the following results.
Proposition 2. Consider all the incentive issues present (that is, the second best
solution). The optimal disclosure strategy for the contractor is to share his proprietary
information with the subcontractor when the innovations v fall between two
thresholds, vi  v  vi  , where
vi  k     


2baSB (k   i )
1 1  ,
 ( i  e)
vi   k     

(22)

2baS B (k   i )
1 1  .
 ( i  e)
(23)
i  vc , a   when the variable cost transfer scheme is adopted and i  fc , a  
when the fixed cost scheme is adopted.34
Proof: See Appendix.
Proposition 2 demonstrates that the contractor’s innovation sharing strategy
depends on the innovation falling in three distinct regions. Specifically, when
innovation v falls in vi  v  vi for i vc, fc, the contractor will disclose the
innovation and strategically outsource the “new” product to the subcontractor. That is,
the contractor rationally shares his proprietary information with the subcontractor and
correctly anticipates that the rational subcontractor will not misappropriate but rather
willingly join the coordinated supply chain to create higher surplus. Otherwise, the
contractor rationally withholds his innovation and ends the relationship with the
33
34
We assume 2 i  0 v  0 and i  vc, fc to ensure that D2 is always willing to join the
coordinated supply chain arrangement.
Subscript i =vc for variable cost scheme and i=fc for fixed cost scheme. a   for variable cost
scheme and a   for fixed cost scheme in all the discussions to follow.
22
subcontractor. In essence, the solutions presented in Proposition 2 are inefficient
because some transactions that should be implemented as the coordinated supply
chain (i.e., when v  vi ) are inefficiently foregone. Clearly, part of the innovation
value is lost and is not shared by the whole chain.
4. Comparison between the two transfer pricing schemes: A numerical example
We have shown that in addition to the misappropriation, the subcontractor’s
transfer pricing also affect the supply chain coordination. We now compare the
performances under the variable-cost transfer pricing scheme and the full-cost transfer
pricing scheme and discuss the implications for supply chain management. To
~
facilitate the comparison, we assume explicit functional forms for V ( r , ~ ) , Y ( X ,1 ) ,
~
K ( Z , 2 ) and C ( ) . We further assume probability distributions for the state
~
~
variables ~ ,  1 and  2 .
Let V ()  r   , Y ()  1  X , K ()   2  Z , and
C ()     . Also let  ~ U [  r,   r ] , 1 ~ U [ 1  X , 1  X ] , and
 2 ~ U [ 2  Z , 2  Z ] to denote that  ,  1 and  2 are uniformly distributed over
the corresponding intervals. The expected value of ~ (the random component in D1’s
variable production costs) is normalized as zero (e.g., E (~ )  0 ). Also, we let k  0
~
so that C1 ()  ~q .
The analysis will start with comparing the contracting parties’ levels of
investments and efforts between the schemes.
Proposition 3. Consider all the incentive issues among the contracting parties. All
the four incentives given below are lower for the fixed-cost transfer pricing scheme as
compared to the variable cost scheme.
23
(1) The contractor’s incentive for R&D investment.
(2) D1’s incentive for the process-related investment.
(3) D1’s effort incentive on enhancing the process quality.
(4) D2’s effort incentive on assembly maintenance.
Proof: See Appendix.
Proposition 3 illustrates (given the specific example) that in addition to the
subcontractor’s misappropriation, the accounting choices of the decentralized firm
also affect the parties’ investment and effort decisions. In general, we find that the
variable-cost transfer pricing scheme performs better in directing the parties’ up-front
decisions. Below, we give a brief discussion of the results.
D1’s transfer prices under the variable-cost transfer pricing scheme (i.e.,
 (k   vc )qˆ  12  ( vc  e)qˆ 2 ) are made of a compensated component (  (k   vc )qˆ )
SB
SB
and a penalty component ( 12  ( vc  e)qˆ 2 ). D1 by choosing a process quality effort
level influences  vc , and hence balances the two components. In the full-cost transfer
pricing scheme, the positive term in the transfer price has the component C ( ) that
can be entirely passed on to D2. This provides D1 incentives to exert less effort on the
process quality in the full-cost transfer pricing scheme.
D1 has incentives to invest more on the new process in the variable-cost transfer
pricing scheme as it can reduce C ( ) . In the fixed cost scheme this is passed on to D2.
Proposition 4. Considering all the incentive issues among the contracting parties,
the quantity of the intermediate goods that D1 decides to transfer to D2 in the
variable-cost transfer pricing scheme is higher than those in the full-cost transfer
pricing scheme. That is, qˆ1*fc  qˆ1*vc .
Proof: See Appendix.
24
Corollary 1: For the whole supply chain the trade distortion (stated in (15) in the
variable-cost transfer pricing scheme is smaller than that in the full-cost transfer
SB
SB
pricing scheme. That is,  vc  b( qmax
 qˆ1*vc ) 2   fc  b( qmax
 qˆ1*fc ) 2
Proof: Follows from Proposition 4 and hence omitted.
Proposition 4 and Corollary 1 show that the supply chain experiences inefficient
trades regardless of the transfer pricing scheme adopted. In particular, the extent of
trade distortion is more in the full-cost transfer-pricing scheme.
Proposition 5. The contractor’s innovation disclosure region is larger in the
variable-cost transfer-pricing scheme.
Proof: See Appendix.
Corollary 2. The contractor’s innovation disclosure region depends on the magnitude
of markup. Specifically, the larger is the markup ratio chosen by HQ, the larger the
contractor’s disclosure region.
Proof: See Appendix.
Corollary 3. The contractor’s innovation disclosure region depends on the possibility
of misappropriation.
Proof: See Appendix.
Proposition 5 suggests that the contractor’s innovation disclosure strategy is
affected by the subcontractor’s accounting choices, and in turn, influences the supply
chain coordination. Corollary 2 and 3 suggest that the contractor’s willingness to
share innovation is affected by the magnitude of the markup and the possibility of
misappropriation. From the whole supply chain perspective, the variable-cost transfer
pricing scheme performs better than the full-cost transfer pricing scheme in the
presence of incentives problems. In essence, the supply chain in the variable-cost
scheme will have higher level of investments, higher level of efforts, higher extent of
25
information sharing, lower extent of trade distortion, and larger surplus than under the
full-cost transfer pricing scheme.
To illustrate Proposition 5 and Corollary 2, assume that the relationship between
 and  follows Lemma 1, and let b  0.5 ,   0.4 ,   5 , and   [0,1] . For this
example, Figure 3 shows that the disclosure set of the variable-cost transfer pricing
scheme, S vc , is higher than that of the full-cost transfer pricing scheme, S fc . Notice
Innovation Disclosure S
that the disclosure willingness is strictly increasing with the markup ratio.
Svc
0.6
0.4
Sfc
0.2
Markup Ratio
0.2
0.4
0.6
0.8
1
 ( )
Figure 3. The contractor’s optimal disclosure strategy under various markup
ratios
We also provide an example to illustrate Proposition 5 and Corollary 3. We
assume that the relationship between  and  follows Lemma 1, b  0.5 ,   0.6 ,
and   5 . For this example, Figure 4 shows the same result like figure 3. In addition,
the disclosure region is strictly decreasing with the possibility of misappropriation.
26
Innovation Disclosure S
20
15
Svc
10
5
Sfc
0.2
0.4
0.6
0.8
1
Misappropriation Possibility 
Figure 4. The contractor’s optimal disclosure strategy under various possibility
of misappropriation
In a seminal analysis of the transfer-pricing problem, Hirshleifer (1956)
identifies conditions under which pricing intermediate good at its actual marginal cost
maximizes firms’ total profit. The neo-classical literature on transfer pricing hence
suggests that trade distortions can be avoided if firms adopt a variable-cost transfer
pricing scheme. However, marginal-cost pricing is rare in practice. That is, most firms
set their transfer price above marginal cost. To reconcile the discrepancy between
theory and practice, accounting theorists suggest that the marginal-cost scheme will
be optimal only in firms where all decision makers have symmetric information and
no incentive problems exist (see Vaysman, 1996). However, we find that trade
distortions cannot be avoided even if the subcontractor adopts the variable-cost
transfer pricing scheme in which transfer price is set above the marginal cost of D1.
We also find that the variable-cost transfer pricing scheme performs better even if
incentive problems exist among the contracting parties.
On the other hand, several researchers, such as Lambert (2001) and Bockem and
Schiller (2004), suggest that setting transfer price at marginal cost is suboptimal in
27
situations where parties can make relationship-specific investments that lower the
marginal cost of production. Our findings are consistent with their arguments.
5. Conclusion
One main and interesting finding in this paper is that in addition to the possibility
of misappropriation, the subcontractor’s accounting system will influence the
contractor’s innovation disclosure strategies and, in turn, affect the efficiency of the
supply chain. Further, we show information distortions, inefficient trades and holdup
problems in the supply chain when incentives problems exist among the contracting
parties. Another interesting finding, we show using a specific example, is that the
variable-cost transfer pricing scheme performs better than the fixed cost scheme in
achieving supply chain coordination.
Appendix
Proof of Proposition 1:
SB
The quantities qmax that the contractor wants to outsource in the asymmetric information scenario
is smaller than (or equal to) those in the first-best scenario qmax . That is, qmax  qmax according to
FB
the literature. However, we need to prove
FB
SB
SB
qmax
 qˆ1*i .
SB
 qˆ1*vc
(1) To prove q max
For a meaningful supply chain coordination, total contribution margin after
considering D1 and D2’s maximum variable cost under the variable-cost transfer
pricing scheme should be larger than zero for all the realized ε. For this to be true,
the following shall hold:

(v   max  bq)q  (1   )( k   max )q  12  ( max  e)q 2  0 ,
(A1)
v   max  bq  (1   )( k   max )  12  ( max  e) q .
(A2)
Where  max is D1’s maximum cost of production heterogeneity, and  max denotes
D2’s maximum variable cost of assembly.
SB
SB
To show qmax

 qˆ1*vc ( qmax
 ( k   vc )
v  k  
> qˆ1*vc  SB
), equivalently prove
 ( vc  e)
2b
28
equation (A3):
1

 ( k   vc )
v  k  
 1  1 S B
,
bq
2  ( vc  e ) q
(A3)
 bq  v  k      12  ( vc  e) q   SB ( k   vc )

.
1
bq
2  ( vc  e ) q
(A4)
The inequality for numerator holds due to equation (A2). In addition, since
2b
SB
in section 2.2,2., we prove qmax

 qˆ1*vc .
(  e)
SB
(2) To prove qmax
 qˆ1*fc
Similarly, total contribution margin under the full-cost transfer pricing scheme
should be larger than zero. That is,
(v   max  bq)q  (1   )[C ()  (k   max )q]  12  ( max  e)q 2  0 ,
 v   max  bq 
(A5)
(1  )C ()
 (1  )( k   max )  12  ( max  e)q .
q
(A6)
SB
To show qmax
 qˆ1*fc , equivalently prove equation (A7):
1

 ( k   vc )
v  k  
 1  1 S B
,
bq
2  ( vc  e ) q
(A7)
 bq  v  k      12  ( vc  e) q   SB ( k   vc )

.
1
bq
2  ( vc  e ) q
The inequality for numerator holds due to (A7). Given  
(A8)
2b
, we prove
(  e)
SB
qmax
 qˆ1*fc .
Q.E.D.
Proof of Proposition 2:
When the variable cost transfer scheme is adopted by the HQ, the contractor’s
innovation disclosure region in the presence of incentive problems is determined by
the following rule:
(A9)
cvc  0 ,
where cvc represents the contractor’s optimal surplus when disclosing the
innovation and agreeing to organize the coordinated supply chain. Plugging the
corresponding surplus into equation (A9) gives:
v  k     SB ( k   vc ) 2

]
2b
 ( vc  e)
0,
8b
(1   )( v  k     )2  4b2 [
29
(A10)
Solving with respect to v , we obtain the following results:
vvc  v  vvc  ,
(A11)
where,
vvc  k     


2bSB ( k   vc )
1 1  ,
 ( vc  e)

(A12)

2bS B ( k   vc )
1 1  .
 ( vc  e)
The proof when the fixed cost scheme is adopted is similar.
vvc   k     
(A13)
Q.E.D.
Proof of Proposition 3:
We first consider the contractor’s incentive of up-front R&D investment r at

date 1. The contractor’s expected ex ante profit  ci , is as follows:
 2 1
vi (1 ,  2 )
 2 1
vi (1 ,  2 )

 ci  
 

where
vi (1 ,  2 )
vi (1 ,  2 )
ci f ( ) g (1 )h( 2 )dd1d 2  r ,
(A14)
f ( )d represents the probability that the contractor will
SB
employ the coordinated supply chain and outsource qmax
units of the “new” product.
The contractor’s optimal investment level in the presence of incentive problems, ri S B ,
is given by:


  2  1 vi ()  (1   )( v  k     ) 2 f ( ) g (1 ) h( 2 )

dd1d 2
 2  1 vi ()

SB
ri


1 

  1 . (A15)
 2 v  k     a(k   ) 2

8b 

4
b
(

)
f
(

)
g
(

)
h
(

)


1
2

 2  1 vi () 
2b
 (  e)
 dd d 

1
2
  2  1 vi ()

ri SB
Given the contractor’s disclosure strategies, D1’s expected surplus with

process-related investment,  , and enhancing process quality effort, X ,  1i , is:
 2 1
vi ()
 2 1
vi ()

 1i  
 

1i f ( ) g (1 )h( 2 )dd1d 2    w( X ) .
(A16)
This implies a unique optimal choice of investment,  Si B , and an optimal level of
effort X iS B :
2
1
  
2
1
vi  ( )
vi ( )
  C () f ( ) g (1 )h( 2 ) 
dd1d 2  1 ,
 Si B
 a 2 (k   ) 2


f ( ) g (1 )h( 2 ) 

 2  1 vi ( )  2 (  e)
dd d  w( X ) .
1
2
 2  1 vi ()
X iSB
30
(A17)
(A18)
Given the contractor’s disclosure strategies, D2’s expected surplus with assembly

maintenance effort, Z ,  2 i , is:
 2 1
vi ()
 2 1
vi ()

 2i  
 
2i f ( ) g (1 )h( 2 )dd1d 2  w(Z ) .
(A19)
The necessary first-order condition for an optimum Z vcSB is:


  2  1 vi ()  (1   )( v  k     ) 2 f ( ) g (1 ) h( 2 )

dd1d 2
 2  1 vi ()

SB
Z i


1 

  w( Z ) . (A20)
 2 v  k     a(k   ) 2

8b 
  4b (

) f ( ) g (1 ) h ( 2 ) 


 2  1 v ( ) 
2b
 (  e)
 dd d 
    i
1
2
  2  1 vi ()

Z iSB
We use V ()  r   , Y ()  1  X , K ()   2  Z ,  ~ U [  r,   r ] ,
1 ~ U [ 1  X , 1  X ] , and  2 ~ U [ 2  Z , 2  Z ] to complete the proof.
(1) Comparison between rvcSB and rfcS B :
we rewrite equation (A15) as follows:


   Z   X vi ()  (1   )( r    1  X   2  Z ) 2 f ( ) g (1 ) h ( 2 )

d
  Z   X vi ()

r

1 


 2 r   1  X  2  Z a 2
   1 , (A21)
8b 
  4b (
 ) f ( ) g (1 ) h ( 2 )  
2b
 Z
 X
vi ()


 d 

   Z   X vi ()

r
2
1
2
1
2
2
1
1
where vi ()  (1  X )  ( 2  Z )  a
1
2b

(1  1   )  r
, vi  ()  (1  X )  ( 2  Z )  a
2b

(1  1   )  r
1
1
, f ( 2 ) 
and d  dd1d 2 .

( 1   1 )
( 2   2 )
Solving equation (A21) with respect to r yields:


1
 2b


ri SB  A

 ( 1   1   2   2 ) ,
 2a 1   ( 1   )( 2   ) 2


1
2


f ( ) 
,
, f (1 ) 
where A 
1
1
1
 ( 1   1 ) ( 2   2 )
(A22)
, and a  SB , i  vc for variable cost scheme
and a   SB , i  fc for fixed cost scheme.
Comparing the results for the two schemes, we get rvcSB  rfcSB by S B   S B .
(2) Comparison between SBvc and  SfcB :
31
We rewrite equation (A17) as:


1
1
1

 (    )

 ( 1   1 ) ( 2   2 ) 
 2  Z  1  X vi () 
dd1d 2  1 .
 2  Z  1  X vi ()

(A23)
Solving equation (A23) with respect to  yields:
 4ba 1   ( 1   1 )( 2   2 ) 
 SBi  A
,



(A24)
where a  SB , i  vc for variable cost scheme and a   SB , i  fc for fixed
cost scheme.
Comparing the results for the two schemes we obtain  SBvc   SBfc by S B   S B ,
0  SB  1 and 0   SB  1 .
(3) Comparison between X vcSB and X SfcB :
We rewrite equation (A18) as follows:
 a 2 (1  X ) 1

1
1



2
 ( 1   1 ) ( 2   2 ) 
 2  Z  1  X vi () 
dd1d 2  1 .
 2  Z  1  X vi ()
X
Solving equation (A25) with respect to X yields:
3
 2ba 1   ( 1   1 )( 2   2 ) 
X iS B  A
,
 2


(A25)
(A26)
Comparing the expressions for the two schemes, we obtain X vcSB  X SB
by
fc
  .
SB
SB
(4) Comparison between Z vcSB and Z SfcB :
We rewrite equation (A20) as follows:


   Z   X vi ()  (1   )( r    1  X   2  Z ) 2 f ( ) g (1 ) h ( 2 )

d
  Z   X vi ()

Z

1 

 2 r   1  X  2  Z a 2
  1.
8b 
  4b (
 ) f ( ) g (1 ) h ( 2 )  
2b
 Z
 X
vi ()


 d 
 

  Z   X  vi ( )
Z


Solving equation (A27) with respect to Z yields:



 

1 2 1    4ba

SB


Z i  A
A
(



)



12
12   .


  1 2 1  
 





where 1  ( 1   1 ) ,  2  ( 2   2 ) and 12  ( 1   1   2   2 ) .
2
1
2
1
2
2
(A27)
1
1
Comparing the results for the two schemes, we obtain Z vcSB  Z SB
by S B   S B .
fc
32
(A28)
Q.E.D.
Proof of Proposition 4:
With respect to qˆ1*vc :
Plugging equation (A26) into equation (10), we get:
qˆ1*vc 

.

(A29)

.

(A30)
SB
Similarly,
qˆ1* fc 
SB
It follows that qˆ1*vc  qˆ1* fc by S B   S B .
Q.E.D.
Proof of Proposition 5:
The disclosure region of the variable-cost transfer pricing scheme, S vc , is measured
by the distance between the two thresholds. That is, Svc  vvc  vvc . That is,
S vc 
4 SBb 1  

.
(A31)
Similarly, the disclosure region of the full-cost transfer pricing scheme, S fc , is:
S fc 
4 SBb 1  

.
(A32)
It follows that Svc  S fc .
Q.E.D.
Proof of Corollary 2:
Differentiating S vc 
4 SBb 1  

with respect to  S B gives:
S vc 4b 1  

 0.
 SB

Similarly,
(A33)
S fc 4b 1  

 0.
 SB

Q.E.D.
Proof of Corollary 3:
Differentiating S vc 
4 SBb 1  

with respect to  gives:
33
Svc
4bSB 1  
2bSB


 0.

 1  
 2
Similarly,
(A34)
S fc
4b SB 1  
2b SB


 0.

2

1



Q.E.D.
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