DISCUSSION PAPER
SERIES IN
ECONOMICS AND
MANAGEMENT
Market- versus Cost-based Transfer Pricing,
Intra-Company Coordination and Competitive
Behavior on the Intermediate Market
Clemens Löffler & Thomas Pfeiffer
Discussion Paper No. 13-17
GERMAN ECONOMIC ASSOCIATION OF BUSINESS
ADMINISTRATION – GEABA
Market- versus cost-based transfer pricing, intra-company coordination and competitive behavior on the intermediate market
Clemens Loe er and Thomas Pfei er, University of Vienna
Abstract. This paper examines properties and compares the e ectiveness of market- versus costbased transfer pricing. Transfer pricing provides speci c investment incentives, governs competitive
behavior on the intermediate market and guides intra-company trade. In particular, we study
market-based transfer prices that are based either on the rm's price (controllable market-based
transfer pricing) or the competitor's price (uncontrollable market-based transfer pricing), actual
cost-plus and standard cost-based transfer pricing. The optimal controllable market-based and costplus transfer price both entail a markup over (expected) marginal costs and the optimal standard
cost-based transfer price equals expected marginal costs, while the optimal uncontrollable marketbased transfer price entails a discount. Re ning routinely made suggestions in textbooks, we nd
broadly stated that (i) uncontrollable market-based transfer pricing is optimal for intermediate
cost uncertainty when the intensity of competition is not too high, (ii) the controllable marketbased transfer pricing is optimal for high intensity of competition when the cost uncertainty is not
too high, (iii) cost-plus transfer pricing is optimal if cost uncertainty is su ciently high, and (iv)
standard cost-based transfer pricing is optimal if the intensity of competition is not too high and
cost uncertainty is rather low.
1
Introduction
A classical research question refers to the e ectiveness of market- versus cost-based transfer pricing
to coordinate decentralized companies (e.g. Eccles 1983, 1985). Based on neoclassical analysis
(Cook 1955, Hirshleifer 1956), managerial accounting textbooks routinely propose to use marginal
actual costs as the transfer price in the case of a monopolistic intermediate market, and the unadjusted market price in the case of a perfect intermediate market (e.g. Drury 1996, Atkinson et
al. 2001, Horngren et al. 2009). As indicated by several empirical studies, however, rms deviate
from this suggestion using a variety of quite di erent market-based and cost-based transfer pricing
methods that are determined by means of adjustments.1 Since in most cases intermediate markets
are di erentiated, rms can base the market-based transfer price either on the rm's price (controllable market-based transfer price) or on the competitor's price (uncontrollable market-based
transfer price). Besides actual cost-based transfer pricing, standard cost-based transfer pricing is
frequently used in practice.
Up to now a coherent analysis of these frequently discussed market- and cost-based transfer
pricing methods is missing, including uncontrollable and controllable market-based transfer pricing,
actual cost-plus and standard cost-based transfer pricing.2 This paper provides systematic insights
about properties and the e ectiveness of these transfer pricing methods to alleviate intra-company
coordination problems in the presence of an imperfect intermediate market.
To elaborate, we use a simple model of a decentralized
rm with two divisions. Division 1
produces an intermediate good that is sold externally and which is also used by Division 2 to
procure a nal good. While Division 2 acts as a monopolist on the nal market, Division 1 faces
di erentiated Bertrand price competition on the intermediate external market. Division 1 can
reduce its costs by undertaking speci c investments up-front. Lower costs enhance the value of intracompany trade and make competition on the intermediate market more aggressive. Transfer pricing
provides up-front investment incentives, governs competitive behavior on the intermediate market
1
Previous empirical studies have found that a majority of companies apply variants of market- or cost-based
transfer prices. According to Ernst and Young (2011), 39% to 44% of the interviewed rms adopt a market-based
approach and 30% to 36% adopt a cost-based approach. Emmanuel and Mehafdi (1995) provide a rigorous comparison
of various studies.
2
In this vein, for instance, Baldenius (2008, p. 295) states at the end of his survey on internal pricing that it
\would be desirable if future research could develop a comprehensive theory of how alternative pricing mechanisms
perform in settings that account for various external market environments."
1
and guides intra-company trade. Broadly stated, we synthesize previous transfer pricing literature
on incomplete contracting (e.g. Baldenius, Reichelstein and Sahay 1999) with the literature on
competitive behavior on the intermediate market (Arya and Mittendorf 2008).3;4
Market-based transfer prices are frequently calculated by adjusting the external market price
which serves as a reference price (e.g., Atkinson et al. 2001). Under di erentiated Bertrand competition, the rm can either use the competitor's or Division 1's price as a reference price. Under
the uncontrollable market-based transfer price Division 1's pricing decision on the intermediate
market does not in uence its pro t from intra-company trade since the competitor's price cannot be in uenced strategically under simultaneous Bertrand competition. In contrast, under the
controllable market-based transfer price Division 1's pricing decision on the intermediate market
in uences its pro t from intra-company trade which in uences Division 1's behavior on the intermediate market.5 By linking the two markets, controllable market-based transfer pricing induces
a competitive advantage on the intermediate market, but derails trade on the internal market.
Linking the two markets with the controllable market-based transfer price is quite e ective if the
intensity of competition is high, while the uncontrollable market-based transfer price is quite effective if the intensity of competition is low. The controllable market-based transfer price further
has the advantage that Division 1's market-based transfer price re ects more accurately Division
1's actual costs for intra-company trade than the uncontrollable market-based transfer price. Summing up, the uncontrollable market-based transfer price dominates the controllable market-based
transfer price if both the cost uncertainty and the intensity of competition are not too high.
Our analysis provides interesting insights into the determination of additive markups and discounts that are frequently applied in practice to correct for market imperfections (e.g. Drury 1996,
3
Abstracting from an external intermediate market, previous transfer pricing literature on incomplete contracting
has analyzed investment and trade incentives of cost-based and negotiated transfer pricing methods (e.g. Baldenius,
Reichelstein and Sahay 1999, Baldenius 2000, Sahay 2003, Johnson 2006, Pfei er, Schiller and Wagner 2011, and
Baldenius 2008 for an excellent overview). Focusing on strategic delegation, Arya and Mittendorf (2008) have studied
the competitive advantage of controllable market-based versus cost-based transfer pricing thereby abstracting from
intra-company coordination problems arising from speci c investment problems and cost uncertainty (see Arya and
Mittendorf 2010 for an excellent overview on this branch of literature).
4
Consistent with the basic analysis of Arya and Mittendorf (2008), we assume the rm acts as a monopolist on
the nal market. In contrast to previous literature, we do not consider strategic considerations on the nal market
(e.g. Alles and Datar 1998). Arya and Mittendorf (2008, 2010) provide a discussion of key di erences between the
two set-ups and on the strategic delegation literature in general (e.g. Gal-Or 1993, Hughes and Kao 1997) that also
applies to our model.
5
For instance, Feinschreiber (2004) notes that controllable market transfer pricing can induce opportunistic pricing
behavior.
2
Atkinson et al. 2001). Accounting for Division 1's opportunistic behavior, the optimal controllable
market-based transfer price entails a markup over expected marginal costs in order to decrease
Division 1's price on the intermediate market. The markup is decreasing with the intensity of
competition, since the rm has an incentive to soften competition.6 In contrast, the optimal uncontrollable market-based transfer price entails a discount to increase demand on the nal market
and thus investments, since Division 1 has an incentive to underinvest in order to increase the
competitor's price and thus its intra-company pro t. The discount is increasing with the intensity
of competition, since increasing intensity of competition increases Division 1's incentives for such
opportunistic behavior.7
Textbooks frequently suggest using actual marginal cost transfer price for a monopolistic intermediate market, i.e. in the absence of competition. Accounting for the fact that rms frequently
use standard cost-based transfer pricing, we provide rst a performance comparison of these two
cost-based transfer pricing schemes. The cost-plus transfer price determines the transfer price as
actual costs plus a markup, while the standard cost-based transfer price is determined by central
o ce ex-ante. The standard cost-based transfer price separates the pricing decisions of the two markets and provides the same investment decisions as an integrated rm. Under the cost-plus transfer
price, Division 1's pro t from intra-company trade equals the markup times the traded quantity.
Although a positive markup distorts intra-company trade, it is optimal to apply a markup that
provides stronger investment incentives to reduce the costs which in turn increase intra-company
trade. The optimal markup decreases with increasing intensity of competition since the rm has
an incentive to soften competition.8 If costs are deterministic, the cost-plus transfer price induces
less e cient investment and trade incentives so that the standard cost-based transfer price always
dominates. Since in contrast to the standard cost-based transfer price, the cost-plus transfer price
transmits actual cost information to the internal market, cost-plus transfer pricing dominates if
6
Focusing on strategic delegation, Arya and Mittendorf (2008, p. 728) nd that the optimal controllable marketbased transfer price entails a discount over marginal costs in the case of Bertrand competition. Our model comprises
a similar strategic e ect as in Arya and Mittendorf (which decreases the markup), but also entails a countervailing stronger e ect from the intra-company coordination problem (which increases the markup). In the absence of
competition, the optimal controllable market-based transfer price entails a markup. The markup is decreasing with
increasing intensity of competition, since the strategic a ect becomes more important.
7
In the absence of competition, the optimal uncontrollable market-based transfer price equals the expected
marginal costs.
8
Abstracting from an external intermediate market, Sahay (2003) has shown that the optimal cost-plus transfer
price entails a markup over marginal costs to provide cost-reducing investment incentives. Our analysis complements
this nding by showing that the size of the markup should be decreasing with the intensity of price competition.
3
cost uncertainty is su ciently high (and vice versa).
To the best of our knowledge, we are the rst to use an incomplete contracting approach and
systematically provide insights into the structure and the e ciency of market- versus cost-based
transfer pricing combining strategic external competition and intra-company coordination problems.9 Our results re ne suggestions in textbooks to use marginal actual cost-based transfer pricing in the absence of competition on the intermediate market and the unadjusted market price
in the case of a perfect intermediate market. Comparing all these transfer pricing schemes, we
nd broadly stated that (i) the uncontrollable market-based transfer price is optimal for low up
to intermediate intensity of competition and intermediate cost uncertainty, (ii) the controllable
market-based transfer price is optimal for high intensity of competition if cost uncertainty is not
extremely high, (iii) cost-plus transfer pricing is optimal if cost uncertainty is su ciently high, and
(iv) standard cost-based transfer pricing is optimal if the intensity of competition is not too high
and cost uncertainty is rather low. In the absence of competition, the two cost-based transfer prices
weakly dominate the market-based transfer prices. In the absence of cost uncertainty, the standard
cost-based and the controllable market-based transfer prices are the optimal schemes.
Our analysis is related to the following work. In the absence of competition, Baldenius and
Reichelstein (2006) have analyzed the e ectiveness of controllable market-based transfer prices to
coordinate intra-company trade. Abstracting from intra-company coordination problems, Arya and
Mittendorf (2008) have shown that the controllable market-based transfer price always dominates
actual and standard cost-based transfer prices which are identical in the absence of speci c investments and cost uncertainty.10 Our result complements these ndings by showing that standard
cost-based transfer pricing is e ective if cost uncertainty is low and the intensity of competition is
not too high. Actual cost-based transfer pricing is e ective if cost uncertainty is rather high, while
9
To provide a non-trivial performance comparison, all components of our model are necessary, i.e. speci c investments, cost uncertainty and competition on the intermediate market. In fact, we nd (i) that in the absence of
speci c investments standard cost-based transfer pricing is never superior, (ii) that in the absence of cost uncertainty
cost-plus transfer pricing is never superior, and (iii) that in the absence of competition on the intermediate market
the two market-based transfer pricing schemes are never superior.
10
Our analysis also extends previous ndings on the structure of controllable market-based transfer pricing. In fact,
in the absence of competition Baldenius and Reichelstein (2006) have shown that the optimal controllable marketbased transfer pricing entails no markup over expected costs to coordinate optimally the intra-company coordination
problem. In the absence of an intra-company coordination problem Arya and Mittendorf (2008) have shown that
the optimal controllable market transfer pricing entails a discount over (expected) costs to facilitate a strategic
advantage in the case of Bertrand competition. Somewhat surprisingly, our analysis shows that the interrelation of
intra-company coordination and strategic interaction on the intermediate market necessitates a markup.
4
controllable market-based transfer pricing is quite e ective for high intensity of competition and
not too high cost uncertainty. Abstracting from competition on the intermediate market, Pfei er
(2011) has investigated uncontrollable market-based transfer pricing.11 Abstracting from an external intermediate market, previous literature has studied the e ectiveness of various cost-based
transfer pricing methods (Baldenius, Reichelstein and Sahay 1999, Baldenius 2000, Sahay 2003,
Pfei er, Schiller and Wagner 2011). Our analysis complements a key nding of this literature that
standard cost-based transfer pricing dominates cost-plus transfer pricing if cost uncertainty is not
too high.12 In contrast to all these work, in our model the cost-reducing investments generate
strategic e ects on the competitor that di er signi cantly depending on the individual transfer
pricing schemes.13
In an extension section, we study the case of di erentiated Cournot competition on the intermediate market. In contrast to Bertrand competition, the rm facilitates a competitive advantage
by increasing the investments more aggressively. Since the competitor's external price depends on
the quantities on the intermediate market, yet uncontrollable market-based transfer pricing also
links the two markets.14 Nevertheless, our main results concerning the e ectiveness of the individual transfer pricing schemes are similar to those of the Bertrand case. However, in contrast to
Bertrand competition, the controllable market-based transfer price is dominated if the external and
intra-company market have the same size.15
11
Modeling the uncontrollable market-based transfer price as a random variable that is correlated with the rm's
costs, Pfei er (2011) studies uncontrollable market-based transfer pricing using non-linear demand and transfer
pricing functions. While consistent with previous studies our analysis is restricted to simple discounts and linear
demand functions. Our setup explicitly considers competition on the intermediate market and strategic e ects of
investments on the competitor. In the absence of competition, the rm's costs and the uncontrollable market-based
transfer price are uncorrelated in our setting.
12
In particular, our analysis adds the following. (i) In the absence of competition, cost-based transfer pricing is
not dominated by market-based transfer prices. (ii) In the presence of an imperfectly competitive market, actual
cost-based transfer pricing is optimal if cost uncertainty is su ciently high irrespective of the intensity of competition.
(iii) Although standard cost-based transfer prices are the simplest transfer pricing scheme they can be e ective if cost
uncertainty is su ciently low and the intensity of competition is not extremely high.
13
From a broader perspective, our paper relates to the industrial organization literature on R&D competition
which studies oligopolistic market competition when rms can conduct cost-reducing investments before engaging
in competition (e.g. Brander and Spencer 1983, d'Aspremont and Jacquemin 1988, Suzumura 1992). Focusing on
intra-company coordination problems in decentralized companies, we nd that depending on the applied transfer
pricing scheme the strategic bene t of investments can di er signi cantly.
14
In contrast to our previous analysis, the optimal uncontrollable market-based transfer price entails a markup to
facilitate a comparative advantage on the intermediate market. Similarly, the size of the markup for cost-plus transfer
pricing is c.p. increasing with the intensity of competition in order to increase the aggressiveness of competition via
higher investments.
15
Studies on transfer pricing have documented that the magnitude of intra-company trade can vary greatly across
rms (e.g. Emmanuel and Mehafdi 1995, Chapter 3). Our nding predicts that under quantity competition the
controllable market-based transfer price should not be prevalent for ancillary intra-company trade.
5
The reminder of the paper is organized as follows. Section 2 presents the basic set-up and
investigates as a benchmark the case of an integrated rm. Section 3.1 studies the e ectiveness
of market-based transfer pricing. Section 3.2 investigates the e ectiveness of cost-based transfer
pricing. Section 3.3 provides a performance comparison between market- and cost-based transfer
pricing. Section 4 reports brie y ndings for the case of Cournot competition. Section 5 concludes.
2
Model
2.1
Basic set-up
We consider a decentralized rm that consists of two divisions. Division 1 produces a (intermediate)
good that can be sold on an external market or used by Division 2 to produce a nal product for
its external market. Without loss of generality, one unit of the intermediate good is converted into
one unit of the nal product, with the cost of conversion normalized to zero. For simplicity, we
refer to Division 1's external market as intermediate market and to Division 2's external market
as nal market. Division 1 can reduce the production costs by undertaking speci c investments
up-front, when costs are still uncertain.
To elaborate, Division 1 faces di erentiated Bertrand competition on the intermediate market
from a competitor C. To keep the exposition simple, we normalize C's production costs to zero.
From the standard inverse demand function,
pi (qi ; qj )
=
aI
qi
qj
with i; j = 1; C and i 6= j,
(1)
we get the demand system on the intermediate external market (e.g. Singh and Vives 1984)
qi (pi ; pj ) =
aI
1+
pi
1
2
+
pj
1
2
with i; j = 1; C and i 6= j;
(2)
where, p1 and, pC , denote the prices on the intermediate market by Division 1 and C, respectively;
q1 (p1 ; pC ), and, qC (pC ; p1 ), denote the quantities sold by Division 1 and C, respectively; a denotes
the consumers' maximum willingness to pay. The parameter
competition,
2 [0; ] measures the intensity of
< 1. A lower value (higher degree of di erentiation) implies a lower intensity of
competition (and vice versa). If
= 0, Division 1 enjoys monopoly power.
Division 2 enjoys monopoly power in the nal market with the demand function
6
q2 (p2 )
=
aF
p2 ;
(3)
where q2 denotes the quantity sold for a price of p2 and aF denotes the consumers' maximum
willingness to pay. For sake of simplicity, we assume aI = aF = a.
The appendix details the exact conditions that guarantee interior solutions for all equilibria, i.e.
positive quantities and costs and that all second-order conditions are ful lled.
Figure 1 depicts the sequence of events.
- Please insert Figure 1 At date 0, the central o ce speci es the transfer pricing method. We discuss the particular
methods subsequently. At date 1, Division 1 undertakes speci c investments, I 2 [0; I]. The
investments generate non-contractual
xed costs of w(I) = I 2 =2. Given the investment level,
Division 1's cost per unit is given by: C(I) = (c
the productivity of investments I, a > C(I)
c 2 [c; c] and
c
xI), where c denotes basic costs and x denotes
0. The basic costs are uncertain, c =
c
+ ", where
denotes the expected costs that are inferred by an uncorrelated noise term ", i.e.
E["] = 0 and V ar ["] =
2.
At date 2, all parties learn the realization of the costs. At date 3,
Division 1 and the competitor C determine their prices for the intermediate market. Finally, at
date 4, Division 2 makes its pricing decision at the nal market. The pro t of each division is
calculated according to the transfer pricing rule.
Each division is run by a risk-neutral manager who seeks to maximize the pro t of its division.
Given the transfer price t, the pro ts of Division 1 and 2 and the competitor are given by:
1
=
[p1
c(I)] q1 (p1 ; pC ) + [t
2
=
[p2
t] q2 (p2 )
C
=
pC qC (p1 ; pC ):
The rm's corporate pro t is given by:
2.2
Benchmark case: integrated
=
C(I)] q2 (p2 )
w(I)
(4)
1
+
2.
rm
As a convenient benchmark for our subsequent analysis, we
rst abstract from intra-company
coordination problems and investigate the optimal decisions for the integrated rm that maximizes
7
the expected corporate pro t,
=
E [(p1
C(I)) q1 (p1 ; pC ) + (p2
C(I)) q2 (p2 )]
w(I).
(5)
Backward induction provides the subgame perfect equilibrium. At date 4, the
rm determines
the monopoly price yielding the following standard monopoly quantity on the nal market that
depends on the realized costs given the investments I,
a C (I)
:
2
q2 (I) =
(6)
At the intermediate market, the rm and the competitor determine their prices at date 3, leading
to the well-known equilibrium prices in a di erentiated Bertrand duopoly (notice, the reaction
functions are given by: p1 = [a (1
p1 (I)
=
a[2
) + C(I) + pC ]=2 and pC = [a (1
(1 + )] + 2C(I)
2
4
and
pC (I)
a[2
=
) + p1 ]=2):
(1 + )] + C(I)
2
4
(7)
and the corresponding quantities are given by:
q1 (I)
=
a[2
(1 + )]
4
2
2
2
C(I)
2
5
and
qC (I)
=
a[2
(1 + )] + C(I)
:
2
2
4
5
(8)
The rm and the competitor determine their prices simultaneously given their expectations of the
others choice. Since they are not able to strategically in uence each other, the traded quantities
on the intermediate and nal market are independent (see (6) and (8)).
Anticipating the optimal responses in all subsequent stages, central o ce determines the optimal
investment decision that maximizes the expected corporate pro t:
w0 (I )
=
=
@pC
@C(I )
@
(q1 (I ) + q2 (I )) +
@I
@pC @I
"
#
p
(I
)
C(I
)
1
2
xE q1 (I ) + q2 (I )
:
2
2
4
1
E
(9)
The rst term, xE [q1 (I ) + q2 (I )], depicts the rm's investment decision without strategic considerations, taking into account the direct marginal bene t of reducing costs. From the competitor's
perspective, the rm is a Stackelberg leader who can strategically in uence the competitor's price
decision. The second term,
x 2 E [p1 (I )
C(I )] =( 4
8
2
1
2
), re ects the rm's incentive
to reduce the investments strategically in order to soften the price competition with the competitor.
Accordingly, the investments, I; are decreasing in the intensity of competition, , and thus also the
traded quantities, q1 (I ) and q2 (I ). Broadly stated, more intense competition reduces not only
the traded quantity on the intermediate market, but also the one on the nal market.
The rm's maximum expected corporate pro t, E [
( )], can be stated in a convenient simple
form, consisting of a basic value, B ( ), plus a exibly value, F ( )
E[
( )]
1
2
2,
(I )2
2
q1 (I )2 + q2 (I )2
=
E
=:
B ( )+
=:
B ( )+F ( )
2 2
]
[2
[4
2 2
] [1
2
]
+
1
4
(10)
2
2:
The appendix provides the explicit form for B ( ). The basic value, B ( ), re ects the basic
maximum expected corporate pro t that would be attained if the rm ignores the cost information
about the realized state " when determining the prices, p1 , and p2 . The basic value is, ceteris
paribus, increasing with the traded quantities, q1 and q2 , and the productivity of investments, x.
Since trade takes place after costs are known, this generates a exibility value at the intermediate
market of [2
3
2 2 2
] =
[4
2 2
] [1
2
] and a exibility value at the nal market of
2 =4.
Results
In the following, we are interested how frequently discussed market- and cost-based transfer pricing schemes provide investment and pricing incentives on both markets. In contrast to previous
literature (e.g. Baldenius, Reichelstein and Sahay 1999, Baldenius and Reichelstein 2006, Arya and
Mittendorf 2008) we study how external competition a ects intra-company coordination (and vice
versa), i.e. transmitting cost information, providing investment and pricing incentives. We rst
analyze market-based transfer pricing (Section 3.1), then cost-based transfer pricing (Section 3.2).
Finally, we conduct a performance comparison among these methods (Section 3.3).
3.1
Market-based transfer pricing
Under market-based transfer pricing, the transfer price equals a reference price less a discount
that is determined by the central o ce at date 0. Throughout our analysis, we investigate two
9
market-based transfer pricing methods under which either the competitor's or Division 1's market
price serves as reference price. The two methods di er in their implications on the intra-company
coordination problem and on their in uence on the price competition on the intermediate market.
3.1.1
Uncontrollable market-based transfer pricing
Under the uncontrollable market-based transfer price method, the central o ce uses as reference
price the competitor's market price, tu = pC
. Although the uncontrollable market-based transfer
price method is frequently applied in practice and discussed in managerial accounting textbooks
(e.g. Horngren et al. 2009), an investigation of its properties and performance in the presence of
market competition is still missing.
Under simultaneous Bertrand competition the competitor's price cannot be in uenced strategically by Division 1's price behavior. However, Division 1 can in uence with its cost-reducing
investments the competitor's price. As a reaction, the competitor reduces its price which, ceteris
paribus, is detrimental for Division 1's pro t on the external and internal market. This in turn
reduces Division 1's investment incentives. Balancing these forces, central o ce determines the
expected transfer price below the expected marginal costs since a discount increases the demand
on the nal market and thus increases Division 1's investment incentives.
To elaborate, we determine the subgame perfect equilibrium by backward induction. At date 4,
Division 2 determines the monopoly price considering the transfer price, tu , instead of the costs,
C(I), yielding a quantity of:
q2u (pC ; ) =
a tu
2
=
a pC +
:
2
(11)
The quantity equals the one of the benchmark case if the transfer price equals marginal costs,
tu = C(I) for all c.
At date 3, Division 1 determines its price on the intermediate market, considering C's reaction
function, pC (p1 ) = [a (1
) + p1 ]=2 from (7). Division 1's reaction function equals the one of
the benchmark case since Division 2's quantity, q2u , depends only on the competitor's price decision
that cannot be in uenced by Division 1's price decision in the simultaneous Bertrand setting.
Accordingly, the prices and the corresponding quantities are identical to the ones of the benchmark
case, (7) and (8), and do not depend on the discount ,
10
pu1 (I) = p1 (I) ;
puC (I) = pC (I) ;
q1u (I) = q1 (I)
and
u (I) = q (I) :
qC
C
(12)
At date 1, Division 1 determines the speci c investments as follows (abbreviating tu (I u ; ) =
puC (I u )
):
w0 (I u )
@ 1 @puC
@ 1 @pu2 @puC
@C(I u ) u u
(q1 (I ) + q2u (I u ; )) +
+
@I
@pC @I
@p2 @pC @I
#
"
u (I u )
u)
u)
u (I u ; )
p
C(I
a
+
C(I
2t
1
2
:
xE q1u (I u ) + q2u (I u ; )
2
2
2
2 4
4
1
(13)
=
E
=
The rst term, xE [q1u ( ) + q2u ( )], depicts Division 1's investment decision without strategic considerations, accounting for that Division 1 perfectly bene ts from the cost reduction. As Stackelberg
leader, Division 1 can strategically in uence the competitor's price which in turn a ects the price
competition on the intermediate market as well as Division 1's pro t from internal trade. The
second term,
x
2
E [pu1 (I u )
C(I u )] =
2
4
1
2
, represents the strategic e ect of the
investments on the price competition on the intermediate market which is identical to the one of the
benchmark case. The third term,
x
E [a + C(I u )
2tu (I u ; )] =(2 4
2
), re ects Division 1's
incentive to increase its pro t from internal trade. If the transfer price is below the monopoly price
of the nal market, tu (I u ; ) < [a + C (I u )]=2, Division 1 has an incentive to reduce its investments
in order to increase the competitor's price which in turn increases the transfer price. In equilibrium, the transfer price never exceeds the monopoly price of the nal market. The investment level,
I u ( ) ; is increasing in the discount, @I u ( ) =@
discount, @q2u ( ) =@
0, which exceeds Division 1's incentive to increase its pro t from internal
E [a + C(I u )
trade, @(
0, since Division 2's demand is increasing in the
2
2tu (I u ; )] =(2 4
))=@ .
At date 0, considering the decisions of the subsequent stages central o ce determines the optimal
discount,
u
, that maximizes the expected corporate pro t. Using E [tu (I u ;
u
)] = E[puc (I u )]
u
reveals that the expected transfer price comprises a cost minus structure,
E [tu (I u ;
u
)]
=
E [C(I u )
with the expected discount of
11
mu ( )]
(14)
E [mu ( )]
=
x2
(1
2
2
) [4
(2 + )] E[q2u ( )]
x2 [8 + (4
[9 + ([5
(3 + )])])]
(1
2 2
)
)(4
0:
(15)
The discount is, ceteris paribus, increasing in the intensity of competition, , and in the productivity
of the investments, x. The reason is that, by increasing the discount, central o ce increases demand
of Division 2, and in addition increases investment incentives for Division 1. In the absence of
competition,
= 0 (similarly, in the absence of speci c investments, x = 0), the optimal transfer
price equals the expected marginal costs, inducing the same investments and expected quantities
as in the benchmark case.
The maximum expected corporate pro t, E [
a exibility value, F u ( )
E[
u(
)]
u(
)], can be stated as a basic value, B u ( ), plus
2,
2
1
q1u ( )2
=
E
=
Bu( ) +
=:
Bu( ) + F u( )
8 + [4
mu ( )q2u ( ) + q2u ( )2
(9 + [5
2 2
2[4
] [1
I u ( )2
2
(3 + )])]
+
]
4[4
2
2
2
(16)
2 2
]
2:
The intra-company coordination problem yields that the basic value of the uncontrollable marketbased transfer price is always below the one of the benchmark case, B u ( )
exibility value is also below the one of the benchmark situation, F u ( )
B ( ) for all . The
F ( ) for all , since the
uncontrollable market-based transfer price transmits only imperfectly cost information to Division
2. In fact, in the absence of competition the transfer price transmits no cost information to Division
2. The associated exibility value at the nal market is increasing with competition.
The following lemma summarizes key characteristics of the uncontrollable market-based transfer price method. For didactical reasons, we modify slightly our notation emphasizing that the
equilibrium values depend on the level of competition :
Lemma 1 Under the uncontrollable market-based transfer price method,
(i) the expected transfer price is below the expected costs, E[tu ( )]
E[C(I u ( ))] for all :
(ii) The induced investments and the expected quantities on both markets are below the ones of
the benchmark case, I u ( )
I ( ), E[q1u ( )]
12
E [q1 ( )] and E[q2u ( )]
E [q2 ( )] for all .
(iii) The maximum expected corporate pro t is given by: E [
B ( ) and F u ( )
u]
= B u ( )+F u ( )
2
where B u ( )
F ( ) for all :
Our analysis complements Pfei er (2011) who has shown in the absence of competition at the
intermediate market that the optimal uncontrollable market price equals the conditional expected
marginal costs (plus a potential markup that diminishes for linear demand functions). Complementing this nding, our analysis shows that due to the interrelation between competition and
speci c investments the optimal uncontrollable market-based transfer price entails a discount that
is increasing with the intensity of competition.
3.1.2
Controllable market-based transfer pricing
A disadvantage of the uncontrollable market-based transfer price is that it transmits imperfectly
cost information to Division 2 yielding ine cient internal trade. As an alternative, the rm can
use Division 1's market price which might re ect better Division 1's costs. Using the controllable
market-based transfer price, the central o ce ties the transfer price to Division 1's market price,
tc ( ) = p 1 ( )
. By linking the external and internal market, the controllable market-based
transfer price provides for Division 1 a strategic opportunity to soften the competition on the
external intermediate market on the expense of the internal market. The controllable market-based
transfer price provides also the advantage to lower competition on the external market through the
investments and the discount.
To elaborate, at date 4, Division 2 determines the monopoly price given the transfer price, tc ,
yielding the following quantity
q2c (p1 ; ) =
a tc
2
=
a p1 +
:
2
(17)
In contrast to the uncontrollable market-based transfer price, the transfer price is directly in uenceable by Division 1.
At date 3, Division 1 determines its price, considering C's response, pC (p1 ) = [a (1
from (6) and Division 2's response q2 (p1 ; ) from (17),
13
) + p1 ]=2
d 1
dp1
=
=
The rst term, [a (1
@ [p1
C(I)] q1 (p1 ; pC ) @ [tc (p1 )
+
@p1
a (1
) 2p1 + pC
2
1
) 2p1 + pC
C(I)] q2c (p1 ; )
@p1
C(I) a + C(I)
+
2
2
C(I)]= 1
2tc
(18)
:
, represents the standard reaction function of
2tc ] =2, represents the impact of Division 1's
the Bertrand duopoly. The second term, [a + C(I)
pricing decision on internal trade. Increasing the price, p1 , decreases internal trade, q2 , but in turn
increases the transfer price, tc . The second term is positive as long as the transfer price is below the
monopoly price of the nal market, tc < [a + C (I)] =2. In equilibrium the transfer price is always
below the monopoly price of the nal market. Compared to the benchmark case Division 1 has
always an incentive to increase the market price, p1 , even in the absence of competition,
= 0.
The equilibrium prices and corresponding quantities are given by:
pc1 (I; )
=
pcC (I; )
=
[1
2
] [a + 2 (1 + ) (a + )] + 3
3 2
a [1
] [6 + (3
C(I)
2
) ]+
2
3
C(I) + 2
1
2
2
6 2
(19)
q1c (I; )
c
qC
(I; )
=
=
[1
)] [a(3 + )
2 (1 + )]
6 1
[1
] a [(6 + (3
3
2
C(I)
2
) ) + 2 (1 + )] +
6 2
2
3
+
2
3
C(I)
4
:
In contrast to the uncontrollable market-based transfer price, the prices and quantities depend on
the discount, , since the controllable market-based transfer price connects the two markets.
At date 1, anticipating the optimal responses at the previous stages Division 1 determines the
optimal investments as:
w0 (I c )
=
=
E
@C(I c ) c c
(q1 (I ;
@I
"
xE q1c (I c ;
c
c
) + q2c (I c ;
) + q2c (I c ;
c
)
2
c
)) +
3
@ 1
@pC
2
@pcC
@I
pc1 (I c ;
6 2
c
2
C(I c )
)
1
2
#
(20)
:
The rst term, xE [q1c ( ) + q2c ( )], depicts Division 1's investment decision without strategic considerations, accounting for Division 1's direct bene t of reducing the costs. As Stackelberg leader,
14
Division 1 can strategically in uence the competitor's price, which is re ected by the second term,
x
2
3
2
E [pc1 ( )
C( )] =(6 2
2
2
1
). Since increasing investments increase the in-
centives for distorting the subsequent pricing decision, the strategic e ect is less pronounced than
under the controllable market-based transfer price (and the benchmark case).16 In contrast to
the uncontrollable market-based transfer price, Division 1 has no additional incentive to in uence
the competitor's price in order to increase its pro t from internal trade, (@
(@pc2 =@pC )
1 =@p2 )
(@pcC =@I) = 0.
Finally, at date 0, central o ce determines the optimal discount, , that maximizes the expected
c
corporate pro t. Using E [tc ( )] = E[pc1 ( )]
plus a positive markup, mc ( )
E [tc ( )]
=
=
with Dc ( ) =
term, 4 1
2
2
[3
E[C( )] +
, the expected transfer price equals expected costs
0, i.e.
4 1
2
3 2
2
+
2 2 c
x D (
!
)
2
E [q2c ( )]
2E [pc1 ( )
3[2
C( )]
]
2
(21)
E[C( ) + mc ( )]
4
2
+
4
E [q2c ] =(3 2
]=[27[2
2
2 3
]
2
1
3x2 [3
2 2 ]2 6
), re ects the e ect of the discount
(@ =@p2 ) (@p2 =@ ). The second term,
2 2 c
x D (
5
2
+
4
]
0. The rst
on internal trade distortions,
)E [q2c ], re ects the e ect of the discount on the
investments, (@ =@p1 ) (@p1 =@I) (@I=@ ). The third term,
2 2 E [pc1
C(I)] =(3 2
2
), re ects
the strategic e ect of the discount on the competitor, (@ =@pC ) (@pC =@ ). In contrast to the
uncontrollable market-based transfer price, the expected transfer price entails a positive markup
over expected marginal costs. Even in the absence of competition,
= 0, the optimal transfer price
entails an expected markup of 4E [q2c ( )] =6 in order to mitigate Division 1's incentives to distort its
price to gain from internal trade. The markup is increasing in E [q2c ( )], as this increases Division
1's incentives for overpricing.
Finally, the associated maximum expected corporate pro t, E [
16
c(
)], can be stated in terms of
2
The strategic e ect under the controllable market price, x 2 3
E [pc1 (I c ) C(I c )] =(6 2
2
c
2
equals the strategic e ect under the uncontrollable market price, x E [p1 (I c ) C(I c )] =( 4
1
4
c
c
c
2
2
positive term, x E [p1 (I ) C(I )] =(6 4
1
):
15
2
2
2
1
);
); less a
a basic plus a exibility value,
E[
c(
)]
=
E
2
1
q1c ( )2 +
3
=:
Bc( ) +
=:
Bc( ) + F c( )
2 2
2
18[2
2tc ( )
a + C( )
2
3
2
2 2
] [1
2
]
+
3
36[2
q1c ( ) + mc ( )q2c ( ) + q2c ( )2
2 2
2 2
]
!
I c ( )2
2
2
2:
(22)
The controllable market-based transfer price has the advantage that it provides a link to the
competitor's pricing behavior. Division 1 strategically increases its market price, pc1 , above the level
of the benchmark case, pc1 (I) > p1 (I). Although the induced quantity is lower, q1c (I) < q1 (I), the
per unit pro tability of each unit increases, pc1 (I)
the term q1c (I c )[a + C(I c )
2tc (I c ;
c
C(I) > p1 (I)
C(I), which is represented by
)]=2. Since increasing competition alleviates the coordination
problem and, thus, lowers the markup above marginal costs, @mc =@
< 0, this e ect becomes
stronger with increasing intensity of competition. The increased per unit pro tability can lead to
the case that the basic value of the controllable market-based transfer price can exceed the one of
the benchmark case for su ciently intense competition. Technically, B c ( )
and B c ( )
B ( ) for
2[
c
B ( ) for
; ] (notice, whether the cuto value is interior,
c
2 [0;
c
]
, depends
on the parameters). However, since the controllable market-based transfer price transmits only
imperfectly cost information to the nal market, the exibility value of the controllable marketbased transfer price is below the one of the benchmark situation, but exceeds the one of the
uncontrollable market-based transfer price, F u ( )
F c( )
F ( ) for all .
Modifying again our notation, the following Lemma summarizes key characteristics for the controllable market-based transfer price method.
Lemma 2 Under the controllable market-based transfer price method,
(i) the expected transfer price exceeds the expected costs, E[tc ( )] > E[C(I c ( ))] for all :
(ii) The induced investments and expected quantities on both markets are below the ones of the
benchmark case, I c ( )
I ( ), E[q1c ( )]
E [q1 ( )] and E[q2c ( )]
(iii) The maximum expected corporate pro t is given by: E [
F u( )
F c( )
F ( ) for all
and B c ( )
16
B ( ) for
c(
E [q2 ( )] for all .
)] = B c ( ) + F c ( )
2 [0;
c
] and B c ( )
2
where
B ( ) for
2[
c
; ].
Previous work has shown that the optimal controllable market-based transfer price entails no
markup over expected costs in the absence of competition and speci c investments (Baldenius and
Reichelstein 2006), while it entails a discount in the absence of intra-company coordination problems
(Arya and Mittendorf 2008). Somewhat surprising, our analysis shows that the interrelation of
intra-company coordination and strategic interaction on the intermediate market necessitates a
markup that is decreasing with the intensity of competition.
3.1.3
Uncontrollable versus controllable market-based transfer pricing
This section compares the e ciency of the two market-based transfer pricing schemes. As outlined, the uncontrollable market-based transfer price never exceeds the expected marginal costs,
while the controllable market-based transfer price is never below the expected marginal costs. Not
surprisingly, the uncontrollable market-based transfer price induces higher investments and higher
expected traded quantities on both markets compared to the controllable market-based transfer
price. However, the controllable market-based transfer price has the advantage that it is more
e ective in softening competition on the external intermediate market which gets increasingly important with increasing intensity of competition. Hence, the basic value of the uncontrollable
market-based transfer price exceeds the basic value of the controllable market-based transfer price
for low intensity of competition (and vice versa). As outlined, the
exibility value of the con-
trollable market-based transfer price always exceeds the one of the uncontrollable market-based
transfer price. Accordingly, we might suspect that the uncontrollable market-based transfer price
outperforms the controllable market-based transfer price in environments with low cost uncertainty.
The following proposition summarizes our discussion.
Proposition 1 Comparing the two market-based transfer pricing methods, we get:
(i) the expected optimal controllable market-based transfer price is above the expected uncontrollable market-based transfer price, E[tc ( )]
E[tu ( )] for all .
(ii) The induced investments of the uncontrollable market-based transfer price and the associated
expected quantities on both markets exceed the ones of the controllable market-based transfer
price, I u ( ) > I c ( ), E [q1u ( )]
E[q1c ( )] and E [q2u ( )]
17
E[q2c ( )] for all .
(iii) Uncontrollable market-based transfer pricing dominates controllable market-based transfer
pricing for low cost uncertainty (and vice versa),
E[
u(
)]
E[
c(
)] if and only if
b2uc ( ) :=
2
Bu( )
F c( )
Bc( )
:
F u( )
In Section 3.3, we further discuss implications of this result and relations to the literature.
3.2
Cost-based transfer pricing
To provide insights on the e ectiveness of market- versus cost-based transfer pricing, we study in
the following the two most prominent cost-based transfer pricing methods: under cost-plus transfer
pricing the transfer price is based on actual costs plus a markup, while the standard cost-based
transfer price is determined by central o ce ex-ante.
3.2.1
Cost-plus transfer pricing
Under cost-plus transfer pricing, the transfer price is set equal to actual marginal costs plus a
markup, m; that central o ce determines ex-ante, t+ = C(I) + m. Abstracting from competition
on the intermediate market, cost-plus transfer pricing is frequently discussed in several textbooks
(e.g. Solomons 1965, p. 167, Sahay 2003 provides an overview). Cost-plus transfer pricing transmits
actual cost information to the nal market. At rst glance, cost-plus transfer pricing seems to delink
the pricing decisions on the external intermediate and nal market. However, the imposed markup
induces investment incentives and thus links the two markets.
At date 4, Division 2 determines the monopoly price based on the transfer price, t+ , yielding a
monopoly quantity of q2+ (I; m) = [a t+ ]=2 = [a C(I)
m]=2. The quantity equals the one of the
benchmark case if no markup is employed, m = 0. At date 3, Division 1 determines its price. Since
the cost-plus transfer price and the induced quantity cannot be in uenced by the pricing decisions,
the equilibrium prices and induced quantities at date 3 are the same as the ones in the benchmark
+
case, i.e. p+
i (I) = pi (I) and qi (I) = qi (I) for i = 1; C. Summing up, cost-plus transfer pricing
does not connect the pricing decisions on the intermediate and nal market.
At date 1, Division 1's optimal investment strategy is given by:
w0 (I + )
=
xE
"
q1+ (I + )
m
+
2
18
2
#
+
p+
C(I + )
1 (I )
:
2
2
4
1
(23)
C(I)] q2+ = m q2+ , it bene ts only
Since Division 1 receives a pro t from internal trade of [t
indirectly from its investments in that lower costs increase internal trade, @q2+ =@I = x=2. Hence,
the rst term, xE q1+ (I + ) + m=2 , depicts Division 1's investment decision without strategic con+
x 2 E p+
1 (I )
siderations. The strategic e ect,
2
C(I + ) =( 4
1
2
), equals the one of
the benchmark case.
Finally, at date 0, central o ce determines the optimal markup,
2 2
4
m+ = x2
2
1
2 2
4
2
1
2 2
2 2
x2
E q2+ ( ) :
(24)
The markup balances the investment decision, I + , and internal trade decision, q2+ , which both
depend on the markup. In the absence of speci c investments, x = 0, central o ce determines
m+ = 0 and achieves e cient production for both divisions. The markup is decreasing in the
intensity of competition, @m+ =@ < 0, since the central o ces wants to decrease the investments
in order to soften competition on the intermediate market.17
The rm's maximum expected corporate pro t, E [
E[
+(
)]
=
=:
E
1
2
+(
)], is given by:
q1+ ( )2 + m+ ( )q2+ ( ) + q2+ ( )2
B+( ) + F ( )
I + ( )2
2
(25)
2:
While the basic value, B + ( ), is below the one of the benchmark case, B ( ), the exibility value
equals the one of the benchmark situation since cost-plus transfer pricing transmits perfectly cost
information to Division 2.
Lemma 3 Under the cost-plus transfer price method,
(i) the transfer price exceeds the costs, t+ ( ) > C(I c ( )) for all :
(ii) The induced investments and the expected quantities on both markets are below the ones of
the benchmark case, I + ( )
I ( ), E[q1+ ( )]
E [q1 ( )] and E[q2+ ( )]
(iii) The maximum expected corporate pro t is given by: E [
B+( )
+(
E [q2 ( )] for all :
)] = B + ( ) + F ( )
2
where
B ( ) for all :
17
Abstracting from an external intermediate market, a key insight on cost-plus transfer pricing is that the optimal
markup increases with increasing trade at the nal market, q2+ , and the importance of the investment, x, i.e. m+ (0) =
x2 E q2+ (I + ; m+ ) =[2 x2 ] for = 0 (Sahay 2003). Our analysis adds to this key insight by showing that the markup
decreases with the intensity of competition on the intermediate market.
19
3.2.2
Standard cost-based versus cost-plus transfer pricing
A classical research question is whether rms should use actual or standard cost-based transfer
pricing (e.g. Eccles 1983, 1985). While cost-plus transfer pricing incorporates actual cost information, the standard cost-based transfer price, tsc , is determined by central o ce ex-ante. Since the
standard cost-based transfer price does not transmit any cost information to the internal market
and does not link the two markets together, standard cost-based transfer pricing might be seen as
ine ective.
At date 4, Division 2 determines the monopoly price based on the transfer price, tsc , yielding a
deterministic quantity of q2 (tsc ) = [a tsc ] =2. Trade is e cient if by coincidence actual marginal
costs equal the transfer price, tsc = C(I). At date 3, Division 1 has no incentive to strategically
adapt its output decision in order to generate a favorable transfer price. The equilibrium prices
and associated quantities are the same as the one in the benchmark case, i.e. psc
i (I) = pi (I) and
qisc (I) = qi (I) for i = 1; C. Standard cost transfer pricing does not link the pricing decisions of
the two markets.
Division 1's investment strategy at date 1 is given by:
w0 (I sc )
=
"
#
sc )
sc )
psc
(I
C(I
1
:
2
2
4
1
2
xE q1sc (I sc ) + q2sc (tsc )
(26)
The rst term, xE [q1sc (I sc ) + q2sc (tsc )], depicts Division 1's investment decision without strategic
considerations, considering the direct marginal bene t of reducing costs. The strategic e ect on
the competitor,
sc
x 2 E [psc
1 (I )
2
C(I sc )] =( 4
1
2
), equals the one of the benchmark
case. The transfer price has an in uence on the investment level in that it in uences the quantity,
q2sc (tsc ). A higher transfer price, tsc , reduces the quantity, q2sc (tsc ), and thus the investment level.
Finally, at date 0, determining the standard cost-based transfer price equal to expected marginal
costs is optimal, tsc = E[C(I )]. In contrast to cost-plus transfer pricing, the intensity of competition does not impact the structure of the standard costs-based transfer price. However, the expected
costs depend on the induced investment level that is decreasing in the intensity of competition.
20
The rm's maximum expected corporate pro t, E [
E[
sc (
)]
=
E
1
=: B ( ) +
2
)], is given by:
I sc ( )2
2
q1sc ( )2 + q2sc ( )2
(27)
2 2
]
[2
[4
sc (
2 2
] [1
2
2
]
=: B ( ) + F sc ( )
2;
where the basic value equals the one of the benchmark case. Since the standard cost-based transfer
price does not transmit any cost information to Division 2, the exibility value equals the one of
the benchmark case at the intermediate market.
Abstracting from an external intermediate market, previous literature (Pfei er, Schiller and
Wagner 2011) has shown that standard cost-based transfer pricing dominates cost-plus transfer
pricing if cost uncertainty is su ciently low (and vice versa). The following Proposition reveals
that this key insight also carries over to our set-up irrespective of the intensity of competition.18
Proposition 2 Under the standard cost-based transfer price method,
(i) the transfer price equals the expected costs, tsc ( ) = E[C(I ( ))] for all :
(ii) The induced investments and the expected quantities on both markets equal those of the benchmark case, I sc ( ) = I ( ), E[q1sc ( )] = E [q1 ( )] and E[q2sc ( )] = E [q2 ( )] for all :
(iii) The maximum expected corporate pro t is given by: E [
F sc ( )
sc (
)] = B ( ) + F sc ( )
2
where
F ( ) for all :
(iv) Standard cost-based transfer pricing dominates cost-plus transfer pricing for low cost uncertainty (and vice versa),
E[
3.3
sc (
)]
E[
+(
)] if and only if
2
b2+s ( ) =
B ( )
F +( )
B+( )
:
F sc ( )
Performance comparison of market- versus cost-based transfer pricing
This section provides a performance comparison of the market- and cost-based transfer pricing
methods. Our results re ne suggestions in textbooks to use marginal actual cost-based transfer
18
Comparing the two cost-based transfer price methods, we get a similar result as in Proposition 1: (i) the expected
optimal cost-plus transfer price is above the standard cost-based transfer price, E[t+ ( )] tsc ( ) = E[C(I + ( ))] for
all . (ii) The induced investments of the standard cost-based transfer price and the associated expected quantities
on both markets exceed the ones of cost-plus transfer pricing, I sc ( ) > I + ( ), E [q1sc ( )] E[q1+ ( )] and E [q2sc ( )]
E[q2+ ( )] for all .
21
pricing in the absence of competition on the intermediate market and the market price in the case
of a perfect intermediate market (e.g. Drury 1996, Atkinson et al. 2001, Horngren et al. 2009).
Brie y stated, we nd that (i) the uncontrollable market-based transfer price is optimal for low
up to intermediate intensity of competition and intermediate cost uncertainty, (ii) the controllable
market-based transfer price is optimal for high intensity of competition if cost uncertainty is not
extremely high, (iii) cost-plus transfer pricing is optimal if cost uncertainty is su ciently high, and
(iv) standard cost-based transfer pricing is optimal if the intensity of competition is not too high
and cost uncertainty is rather low. In the absence of cost uncertainty,
2
= 0, the standard cost-
based and the controllable market-based transfer price are the optimal schemes. In the absence
of competition,
= 0, the standard cost-based and the uncontrollable market-based transfer price
are identical and the two cost-based transfer prices weakly dominate the two market-based transfer
prices. Before we discuss this issue more in detail, we provide the following formal result, using the
following de nition b2ij ( ) = [B i ( )
B j ( )]=[F j ( )
F i ( )] for i; j = u; c; +; s and i 6= j.
Proposition 3 The other three transfer pricing methods are dominated by:
(i) uncontrollable market-based transfer pricing if min b2uc ( ) ; b2u+ ( )
(ii) controllable market-based transfer pricing if b2u+ ( )
(iii) cost-plus transfer pricing dominates if
2
(iv) standard cost-based transfer pricing if
2
Figure 2 illustrates our nding (Data: a = 1,
2
2
b2us ( ) :
max b2uc ( ) ; b2cs ( ) :
max b2u+ ( ) ; b2c+ ( ) ; b2+s ( ) :
min b2us ( ) ; b2cs ( ) ; b2+s ( ) :
c
= 0:1, x = 0:01,
= 0:99).
- Insert Figure 2 Since the maximum variance, 1:92 10
4,
is below the cuto value, b2c+ ( ), the controllable market-
based transfer price always dominates the cost-plus transfer pricing for high values of . Although
the standard cost-based and the uncontrollable market-based transfer price are identical for
= 0,
their performance di ers signi cantly with increasing intensity of competition.
No competition on the intermediate market,
= 0. Abstracting from an external inter-
mediate market, previous literature has studied the e ectiveness of various cost-based transfer
22
pricing methods (Baldenius, Reichelstein and Sahay 1999, Sahay 2003, Pfei er, Schiller and Wagner 2011). Since for
= 0 standard cost-based and the uncontrollable market-based transfer price
are equivalent, the two cost-based transfer pricing methods dominate (weakly) the market-based
ones. Consistent with previous literature (Pfei er, Schiller and Wagner 2011), cost-plus transfer
pricing dominates standard cost-based transfer pricing for high cost uncertainty (and vice versa).
Previous literature has studied the e ectiveness of controllable market-based transfer prices in the
absence of competition when actual cost-based transfer prices are not available (Baldenius and
Reichelstein 2006).19 If actual cost-based transfer prices are not available, we get immediately from
Proposition 3 that controllable market pricing dominates the other two transfer pricing methods if
2
maxfb2uc (0) ; b2cs (0)g: Otherwise, controllable market-based transfer prices are dominated by
the other transfer pricing methods. Further, as highlighted by Proposition 3, controllable market-
based transfer prices perform pretty well for high intensity of competition if the cost uncertainty
is not too high.
Corollary 1 In the absence of competition,
= 0, cost-plus transfer pricing is optimal if
b2+s (0), while standard cost-based transfer pricing is optimal if
2
2
b2+s (0). In particular, stan-
dard cost-based transfer pricing and uncontrollable market-based transfer pricing are equivalent and
controllable market-based transfer pricing is weakly dominated by the other transfer pricing methods.
No cost uncertainty,
2
= 0. Previous literature has shown that in the absence of an intra-
company coordination problem,
2
= 0 and x = 0, controllable market-based transfer pricing always
dominates cost-based transfer pricing (Arya and Mittendorf 2008). Since for
2
= 0 each transfer
price scheme has a zero exibility value, the expected corporate pro t equals the associated basic
value. As outlined previously, the basic value of standard cost-based transfer pricing exceeds the
one of controllable market-based transfer pricing if the intensity of competition is not too high (and
vice versa), B sc ( ) = B ( )
B c ( ) for
2 [0;
c
] and B sc ( )
B c ( ) for
2[
c
; ]. Standard
cost-based transfer pricing always outperforms cost-plus and uncontrollable market-based transfer
pricing, B sc ( )
B + ( ) and B sc ( )
B u ( ) for all : Summing up, our result complements this
previous nding by showing that in the absence of cost uncertainty cost-plus transfer pricing is
19
Baldenius and Reichelstein (2006) nd that controllable market-based transfer pricing cannot achieve rst-best
for unconstrained capacity, while they show that controllable market-based transfer pricing can induce rst-best for
constrained capacity.
23
indeed always dominated, while standard cost-based transfer pricing is optimal for low intensity of
competition.
Corollary 2 In the absence of cost uncertainty,
optimal for low intensity of competition
c
2
= 0, standard cost-based transfer pricing is
, while controllable market-based transfer pricing
is optimal for high intensity of competition,
c
. In particular, standard cost-based transfer
pricing dominates cost-plus transfer pricing and uncontrollable market-based transfer pricing.
No speci c investments, x = 0. In the absence of speci c investments, cost-plus transfer pricing
allows achieving the same outcome as the benchmark case. Accordingly, cost-plus transfer pricing
dominates standard cost-based and uncontrollable market-based transfer pricing. As before, the
basic value of the controllable market-based transfer price can exceed the one of the benchmark
case for su ciently intense competition. Accordingly, we obtain the following result.
Corollary 3 In the absence of speci c investments, x = 0, cost-plus transfer pricing is optimal if
2
b2c+ ( ), while controllable market-based transfer pricing is optimal for
2
b2c+ ( ). In par-
ticular, cost-plus transfer pricing dominates standard cost-based transfer pricing and uncontrollable
market-based transfer pricing.
Summing up, the corollaries highlight that all three factors, i.e. market competition, cost uncertainty and speci c investments, are necessary to provide a non-trivial performance comparison.
4
Extension: Cournot quantity competition
In this section, we brie y report how our results change when the rm faces di erentiated Cournot
competition on the intermediate market, relaxing aF = aI (see appendix for details). In contrast to Bertrand competition, the rm can improve its competitive position on the intermediate
market by increasing its investments. However, the strategic incentive for overinvestment runs in
the opposite direction to the intra-company coordination problem that induces underinvestment.
Importantly, under Cournot competition, the market prices, p1 and pC , are the residuum of the
quantity decisions, q1 and qC , and not decision variables itself, i.e. pi = aI
qi
qj . Consequently,
the market-based transfer prices become in uenceable by Division 1 irrespective of which marketbased transfer price is used. Despite these di erences, most of our previous key insights carry over
to the Cournot case. For sake of simplicity we use the same notation as in our previous sections.
24
In the case of an integrated rm, the quantity decisions on the two markets are separated. We
obtain the monopoly quantity on the nal market, q2 (I) = [aF
C(I)] =2, and the well-known
equilibrium outputs in a di erentiated Cournot duopoly on the intermediate market, q1 (I) =
[aI [2
]
2
2C(I)] = 4
and qc (I) = [aI [2
2
] + C(I)] = 4
. The rm's investment de-
cision is given by:
2
w0 (I)
=
xE q1 (I) + q2 (I) +
4
2
(28)
q1 (I) :
The term, xE [q1 (I) + q2 (I)], depicts the rm's optimal investments without strategic considerations. The term, x 2 E [q1 (I)] =[4
2
], re ects the strategic e ect to increase the investments.
The rm's maximum expected corporate pro t can be stated as a basic plus a exibility value
E[
4.1
( )]
=
B ( )+
4
2 2
]
[4
+
1
4
2
=
B ( )+F ( )
2:
(29)
Characteristics of the transfer pricing methods
Studying the di erent transfer prices, we show that changing the mode of competition hardly
in uences our insights about cost-based transfer pricing, but alters our insights about market-based
transfer pricing. We thus start with cost-based transfer pricing.
Standard cost-based transfer pricing. As before, standard cost-based transfer pricing separates
the two markets, q2sc (tsc ) = [aF
tsc ]=2, and induces the same quantities on the intermediate market
and the same investments as in the benchmark case, q1sc (I) = q1 (I) and I sc = I . The optimal
standard-cost-based transfer price is set equal to the expected marginal costs, tsc = E[C(I sc )]. The
rm's maximum expected corporate pro t, E [
E[
sc (
)]
=
sc (
)], is given by:
4
B ( )+
4
2
2 2
=
B ( ) + F sc ( )
2:
(30)
As before, the basic value equals the one of the benchmark case.
Cost-plus transfer pricing. Under cost-plus transfer pricing, Division 1 has no incentive to adapt
its output decision on the intermediate market, q1+ (I) = q1 (I) (and q2+ (I; m) = [aF
Division 1's investment decision is given by:
25
C(I) m]=2).
w0 (I + )
xE q1+ (I + ) +
=
m
+
2
4
2
2
q1+ (I + ) :
(31)
As before, Division 1 receives a pro t from internal trade of mq2+ and thus bene ts only indirectly
from its investments in that lower costs increase internal trade, q2+ . The strategic e ect on the
competitor,
2
E[q1+ (I + )]=[4
2
], equals the one of the benchmark case.
Finally, the optimal markup, m, balances investments and trade on the nal market,
2 2
]
[4
m+ = x2
2 2
4
8x2
E q2+ (I + ; m+ ) :
(32)
In contrast to Bertrand competition, the markup is, ceteris paribus, increasing in the intensity of
competition, , since more competition makes higher investments more favorable.
While the basic value, B + ( ), is below the one of the benchmark case, B + ( ) < B ( ), the
exibility value equals the one of the benchmark case. The maximum expected corporate pro t,
E[
+(
)], is given by:
E[
+(
)]
=
B+( ) + F ( )
2:
(33)
Uncontrollable and controllable market-based transfer pricing. In contrast to Bertrand
competition, the market price, pi = aI
qi
qj for i; j = 1; C and i 6= j, is the residuum of the
quantity decisions, q1 and qC . Hence, both market-based transfer prices depend on Division 1's
quantity decision:
@ 1
@q1
@ 1
@q1
=
=
[aI 2q u1
[aI 2q c1
2tu
aF + C(I)
2
u
qC
C(I)]
c
qC
aF + C(I)
C(I)]
2
2tc
=0
(34)
= 0:
In contrast to the Bertrand case, the uncontrollable market-based transfer price also links the two
markets. Under both methods, the traded quantity on the intermediate market is, ceteris paribus,
below the one of the benchmark case.
Similarly, Division 1 determines the investments as
w0 (I l )
=
E
@C(I l ) l
@ 1
q1 ( ) + q2l ( ) +
@I
@pC
@plC
@ 1 @pl2
+
@I
@p2 @pC
@plC
@I
for l = u; c;
(35)
yielding the following investment decision for uncontrollable and controllable market-based transfer
26
pricing, respectively:
2
w0 (I u )
w0 (I c )
= xE q1u ( ) + q2u ( ) +
aF + C ( )
[2 + ] u
q1 ( ) + [2 + ]
8
16
"
3
= xE q1c ( ) + q2c ( ) +
4 3
2
2
q1c ( ) + 3
2
aF + C ( )
8[3
2tu ( )
(36)
#
2tc ( )
:
2
]
Finally, at date 0, central o ce determines the optimal discount for both market-based transfer
prices. In contrast to Bertrand competition, both transfer prices entail a positive markup,
E [tu (I u ;
u
E [C(I u )]
)]
and
E [tc (I c ;
c
E [C(I c )].
)]
(37)
While the controllable market-based transfer price still entails a markup, yet the uncontrollable
market-based transfer price entails a markup and not a discount as in the Bertrand case. The reason
is that although a discount helps mitigating the intra-company coordination problem, Cournot
competition on the intermediate market provides a stronger countervailing e ect. A markup makes
Division 1 more aggressive, strengthening Division 1's position on the intermediate market.
The associated maximum expected corporate pro ts for uncontrollable and controllable marketbased transfer pricing are given by:
E[
E[
u(
c(
)]
)]
=
=
Bu( ) +
Bc( ) +
(2 + ) 16 + 3
[4
2 2
3 6
2 2
32 3
2
(2 + )
2 2
]
2 2
]
9[2
+
64[3
2 2
]
2
+
!
[2 + ]2
256
2
!
2
=: B u ( ) + F u ( )
=: B c ( ) + F c ( )
2
2:
(38)
Since the controllable market-based transfer price responds stronger to changes in Division 1's
quantity decision, the coordination problem and the strategic e ect are more pronounced under
this method. The basic value of the uncontrollable market-based transfer price exceeds the one of
the controllable market-based transfer price if the intensity of competition is not too high (and vice
versa), B u ( )
B c ( ) for
2 [0;
uc ]
and B u ( )
B c ( ) for
2[
is satis ed depends on the parameters). As before, we get: F u ( )
the case of homogeneous Cournot competition,
equivalent, i.e. E [
u (1)]
=E[
uc ;
] (notice, whether
F c( )
uc
F ( ) for all . In
= 1, a single price exists and both methods are
c (1)] :
27
4.2
Performance comparison of market- versus cost-based transfer pricing
As in the Bertrand competition case, broadly stated, cost-plus transfer pricing is optimal in highly
uncertain cost environments, while standard cost-based pricing is optimal if cost uncertainty is
rather low. For intermediate cost uncertainty and low intensity of competition uncontrollable
market-based transfer pricing is optimal, while controllable market-based transfer pricing is optimal for high intensity of competition and not too high cost uncertainty. In the absence of cost
uncertainty,
2
= 0, the standard cost-based and the controllable market-based transfer price are
the optimal transfer pricing schemes. In the absence of competition,
= 0, the uncontrollable
market-based and standard cost-based transfer price are identical and the two cost-based transfer
prices weakly dominate the two market-based transfer prices.
Figure 3 illustrates our nding (Data: aI = 5, aF = 1,
c
= 0:8, x = 0:5 with a maximum variance
of 1:6). In contrast to Figure 2 for the Bertrand competition case, the size of the intermediate
market exceeds the size of the nal market since controllable market-based transfer pricing is never
optimal for aF = aI (a detailed discussion is given below).
{ Figure 3 {
De ning the cut-o values as before, i.e. b2ij ( ) = [B i ( ) B j ( )]=[F j ( ) F i ( )] for i; j = u; c; +; s
and i 6= j, we obtain the following formal result.
Proposition 4 In case of Cournot competition, the other three transfer pricing methods are dominated by:
(i) uncontrollable market-based transfer pricing if min b2uc ( ) ; b2u+ ( )
(ii) controllable market-based transfer pricing if b2u+ ( )
(iii) cost-plus transfer pricing dominates if
2
(iv) standard cost-based transfer pricing if
2
2
2
b2us ( ) :
max b2uc ( ) ; b2cs ( ) :
max b2u+ ( ) ; b2c+ ( ) ; b2+s ( ) :
min b2us ( ) ; b2cs ( ) ; b2+s ( ) :
Size of the two markets and ancillary trade. Studies on transfer pricing have documented that
the magnitude of internal trade can vary extremely across rms (e.g. Emmanuel and Mehafdi 1995,
Chapter 3). Broadly stated, for non-ancillary internal trade, aF >> aI , solving the intra-company
28
coordination problem gets more important than strategic considerations on the intermediate market. In contrast to price competition, even for the case of equal market size, aF = aI , controllable
market-based transfer pricing is never optimal.
Corollary 4 If the market size of the intermediate market exceeds the one of the
aF
nal market,
aI , controllable market-based transfer pricing is never optimal.
Finally, it is instructive to ask how key factors (competition, cost uncertainty, and speci c investments) a ect the optimality of the transfer pricing methods. The following Corollary summarizes
the results that are similar to the Bertrand competition case.
Corollary 5 In the case of Cournot competition, we get:
(i) in the absence of competition,
= 0, cost-plus transfer pricing is optimal if
while standard cost-based transfer pricing is optimal if
2
2
b2+s (0),
b2+s (0). In particular, standard
cost-based transfer pricing and uncontrollable market-based transfer pricing are equivalent and
controllable market transfer pricing is weakly dominated by the other transfer pricing methods.
(ii) in the absence of cost uncertainty,
for low intensity of competition
2
= 0, standard cost-based transfer pricing is optimal
c
, while controllable market-based transfer pricing is
optimal for high intensity of competition,
c
. In particular, cost-plus and uncontrollable
market-based transfer pricing are never optimal.
(iii) in the absence of speci c investments, x = 0, standard cost-based and uncontrollable marketbased transfer pricing are never optimal.
5
Conclusion
While surveys discuss a variety of market- and cost-based transfer price methods applied in practice
(e.g. Emmanuel and Mehafdi 1995, Feinschreiber 2004), properties and e ectiveness of the individual transfer pricing schemes seem to be less well understood. Using an incomplete contracting
framework, we investigate frequently used market- versus cost-based transfer pricing schemes to coordinate intra-company trade when the rm faces competition on the intermediate market. Transfer
29
pricing provides investment incentives up-front, governs competitive behavior on the intermediate
market and guides intra-company trade.
Re ning a frequently made proposal in textbooks to use marginal actual costs as the transfer
price in the absence of competition on the intermediate market, we nd that in this case cost-plus
transfer pricing is optimal if cost uncertainty is su ciently high, while standard cost-based transfer
pricing is optimal if cost uncertainty is su ciently low. The optimal standard cost-based transfer
price is set equal to the expected marginal costs, while in contrast to the textbook proposal the
optimal cost-plus transfer price entails a markup over marginal costs. Our analysis can also be
helpful in extending a frequently made proposal to use the unadjusted market price in the case of
a perfect intermediate market. In fact, we nd for an imperfect intermediate market that using
the competitor's price (uncontrollable market-based transfer price) is optimal for low intensity
of competition and intermediate cost uncertainty. Using the
rm's market price (controllable
market-based transfer price) is optimal for high intensity of competition if cost uncertainty is not
too high. Both market-based transfer price methods necessitate adjustments. While the optimal
controllable market-based transfer price entails a markup over expected marginal costs, the optimal
uncontrollable market-based transfer price entails a discount.
Summing up, our analysis provides new insights into the design of additive discounts and markups
for cost-based and market-based transfer prices and compares the e ectiveness of these commonly
used transfer pricing schemes.
30
Mathematical appendix
A.1
Bertrand competition
Regularity conditions. For simplicity, we restrict our analysis to
C(I)
= 99=100. Positive costs,
0. Since in the benchmark case the investments are the largest which are decreasing in ,
assuring c
xI (0)
0 is su cient to guarantee positive costs, yielding
x2 a:
c
Positive quantity, q1
0. Since the controllable market-based transfer price induces the lowest q1
that is decreasing in , assuring q1c
c
(A1)
c =
0 is su cient to guarantee positive quantities, yielding
a 1679803863894170744002 + 8610685384794589354551x2
10199 8363985317511261998 97757614145369620851x2
329969659359788319479850x4
a
10199 8363985317511261998
To assure c
97757614145369620851x2
(A2)
:
c, x must ful ll
x
x
0:1715:
(A3)
Finally, since the benchmark case induces the highest investments, the associated second order
condition is binding, yielding
xsoc (
x
) =
s
2 2
2 4
32
However, since this condition is ful lled for x
40
2
2
1
+ 13
4
for all :
6
x, we can omit it
Characterization of the benchmark solution. The backward induction process is detailed in
the text. Using the equilibrium values from (6), (7) and (8), yields
E[
( )]
=
E[[p1 ( )
=
E
=
1
C( )] q1 ( ) + [p2 ( )
(I )2
2
(I )2
2
2
E [q1 (I )] + E [q2 (I )]
+
2
2
1
2
C( )] q2 ( )]
(I )2
2
q1 (I )2 + q2 (I )2
31
[2
[4
2 2
]
2 2
] [1
2
]
+
1
4
2;
with
E [q1 (I )]
2 [a
=
E [q2 (I )]
=
x
2
1
]
c 32
2 2
2 4
40
2
1
4
2
+ 13
2a 2
4
6
2
4
6
4a
2
x2
2
32
2 2
2 2
[a
a
2
1
c] 4
2 4
I
2
2
2 2
2 4
[a
=
c]
2
32 40
40 2 + 13 4
2
1
32
+ 13
6
40
2
2
x2
x2
2
4
+ 13
(A4)
x2
6
x2
:
Proof of Lemma 1. We rst characterize the solution for the uncontrollable market-based transfer
price. The backward induction process is detailed in the text. Plugging (11) and (12) into (13),
yields
Iu ( ) = x
2
2
3
2( 1
with k u = 8 + (4
E [q1u ( )]
=
[9 + (5
2a [2
E
[q2u (
2
] [2 + )]2 + x21
)]
=
2( 1
a+
2
8
6
4
2 4
2 c ku
(A5)
x2 k u )
2
+
2
)
5
2
2
4
)
2
2
2
(4
[ + 2])
x2 k u )
4
2
+
]+
4
x2 k u )
a 2
2
(A6)
+
c
2
2
1
2
)
2a [2
2
2( 1
c
2
2
4
2
+ 2a [3 + ] 2
[3 + ]))]). Forward induction yields
] [1
2
2
[1 + ] + 8 1
x2
2
2
4( 1
4
)
x2 k u )
2a [4 + (3
[3 + ])]
(4
4( 1
2
2
4
)
2
[2 + ])
x2 k u )
:
Calculating the optimal discount yields
0 =
d
d
=
=
@ 1 @pu2
@ 2 @puC @I u
+
@p2 @
@pC @I @
E [puC ( )
C( )]
2
+ x2
2( 1
Using the equation above and E [tu ( )] = E[puC ( )
Using (A5), (A6) and (15) yields E [
(i) E[tu ( )]
2
1
u(
)] = E
u
1
(2 + )] E [q2u (I u ;
[4
2
4
] = E [C(I u )
2
q1u ( )2
2
)
2
u
x2 k u )
)]
:
mu ] yields (14) and (15).
mu ( )q2u ( ) + q2u ( )2
(I u ( )2 =2):
E[C(I u ( ))] follows (14) and (15).
(ii) We show I u ( )
I ( ), E[q1u ( )]
E [q1 ( )] and E[q2u ( )]
E [q1 ( )] for all . To induce
I u ( ) = I ( ), (9) and (13), the following equation must be ful lled,
32
#
a + C(I ) 2tu (I ; mu )
:
2
2 4
"
xE [q2 (I )] = xE q2u (I )
Using E [q2 (I )] = (a
E [C(I )]) =2, E [q2u (I )] = (a
E [tu ]) =2, and E [tu (I ; mu )] = E[C(I )]
mu , reduces the above equation to
a
E [C(I )]
2
a
=
yielding as solution, m
b u = (a
E [C(I )])=(4
E[mu ] < m
b u for all , we get: I u ( )
(iii) B u ( )
E [C(I )] 2mu
;
2
2 4
a
2
2
) = E [q2 (I )] =(2 4
I ( ) and thus E[q1u ( )]
Similarly, E [q2u ( )] = E [q2 ( )] requires (a
solution m
e u = E [C(I u )
mu
E [C(I )]
2
E [C(I u )]
). Since
E [q1 ( )] :
mu )=2 = (a
E [C(I )])=2 yielding as
C(I )] : Since E [mu ] < m
e u for all , we get: E[q2u ( )]
B ( ) and F u ( )
2
2
E [q1 ( )].
F ( ) follows as outlined in the text.
Proof of Lemma 2. We rst characterize the solution for the controllable market-based transfer
price. Plugging the equilibrium values of (17) and (19) into (20), yields
Ic ( ) = x
2a [1
2
] [2 + 3] 3
2 3
]
18[2
3
2
2
2
1
2
c
3
2
2 2
2 2 ]2 6
x2 2[3
2
3
2
5
1
4
+
2
4
:
(A7)
Forward induction yields
E [q1c ( )] =
E
[q2c (
)] =
3
36 2
6 2
2
x2 2a
2 2
2 2
36 2
6 2 3 2+
3
36 2
2
1
+
2
2
6
5
2
2
2
4 3 2 2 3
x2
] (a [3 + ] 2 [1 + ]) + c 3
2 2
4
2
2
1
[1
36 2
2
x2 2a
2 2
3
2
1
a(3 + [1
2 2
1
3 2
2
2
2
+
4 3
4 3 2
] )
4 3
3
2
2 2
c
2
2
2 2
2
3
2 2
4
3
2
3
3
3
2
2
x2
+
4
2
x2
2
2
x2
:
Calculating the optimal discount yields
0=
d
d
@ @pcC
@ 1 @pc2 @ 2 @pc1 @ 2 @pc1 @I c
+
+
+
@pC @
@p2 @
@p1 @
@p1 @I @
2
c( )
c )]
c( )
E [q2c ( )]
E
[p
C(I
E
[p
C(I c )] 2 1
1
1
= 2
+
2
2
2
3 2
3 2
4
2
4
c
[3 4 + ]E [q2 ( )]
x2
2 3
2
9[2
] 1
x2 [3 2 2 ]2 6 5 2 + 4
=
33
(A8)
and thus the transfer price (21).
Using (A7), (A8) and (21) yields
E[
c(
)]
=
E[[pc1 ( )
=
E
=:
Bc( ) +
C( )] q1c ( ) + [pc2 ( )
2
1
I c ( )2
2
C( )] q2c ( )]
I c ( )2
2
a + C( ) 2tc ( ) c
q1 ( ) + mc ( )q2c ( ) + q2c ( )2
2
!
2
2
2 2
3 2 2 3
3
2:
2 2
2 +
2 2
18[2
] [1
]
36[2
]
q1c ( )2 +
(i) E[tc ( )] > E[C(I c ( ))] follows directly from (21).
(ii) We show I c ( )
I ( ), E[q1c ( )]
E [q1 ( )] and E[q2c ( )]
E [q2 ( )] for all . To induce
I c = I , using (9) and (20), the following property must be ful lled,
#
"
p
(I
)
C(I
)
1
2
xE q1 (I ) + q2 (I )
2
2
4
1
= xE
"
q1c (I
; mc )
+
q2c (I
2
; mc )
4
#
pc1 (I ; mc ) C(I )
:
2
2
4 2
1
2
Solving the above equation for m
b c and using (A4), (A7), (A8) and (21), reveals
m
bc =
with k c = 384
I c( )
[a
576
2
C(I )] 384
864
4
2
576
4
104
+ 352
6
2
+ 13
4
+ 816
1008
8
2
6
404
+ 652
8
+ 103
4
182
6
10
11
+ 19
a kc
8
. However, since E [mc ] < m
b c for all
;
> 0, we get:
I ( ). From that nding follows directly E [q1c ( )] < E [q1 ( )] and E [q2c ( )] < E [q2 ( )].
(iii) B c ( )
B ( ) and F u ( )
F c( )
F ( ) follows as outlined in the text.
Proof of Proposition 1. (ii) We show I c ( )
I u ( ), E[q1c ( )]
E [q1u ( )] and E[q2c ( )]
E [q2u ( )]
b using (13) and (20), the following equation must be ful lled,
for all . To induce I c = I u = I,
"
#
u (I)
b C(I)
b
b 2tu (I;
b mu )
p
a
+
C(
I)
1
2
u
u
xE q1 Ib + q2 Ib
2
2
2
4
2 4
1
h
i3
2
4
b C(I)
b
pc1 (I)
2
c
c
b
b
4
5:
= xE q1 (I) + q2 (I) +
2
2
6 4
2
Solving the above equation for m
e c , using (A5), (A6), (15), (A7), (A8) and (21), yields
34
m
ec
with g = 4
2
h
=
i
b 768
C(I)
a
2
2
6
) 576
2 2
2
4
548
6
+ 118
8
) 576 1008 2 + 652 4
576 2 + 352 4 104 6 + 13
384
4
+ 1308
2
4
a
2
1584
2
1008
8
2
5
+ 652
182 6 + 19
8
3gmu
;
182 6 + 19 8
4
11
10
8
. Since E [mc ] < m
e c we get: I c ( )
I u ( ). From that
nding follows directly E [q1c ( )] < E [q1u ( )] and E [q2c ( )] < E [q2u ( )].
(i) The relation E[tc ( )]
E[tu ( )] follows from I c ( )
I u ( ), mc ( ) > 0 and mu ( )
0. (iii)
The cut-o value b2uc ( ) follows from (16) and (22).
Proof of Lemma 3. We
rst characterize the solution for the cost-plus transfer price. The
backward induction process is detailed in the text. Since p+
1 (I) = p1 (I), inserting (7) into (23),
yields
I + (m)
2
2
2
4a 2
]2
+ m [1 + ] [2 + ] [2
= x
2
2 1
2 2
4
2 2
4 2
4
c
2
x2
2 2
:
(A9)
Forward induction yields
E q1+ (m)
E q2+ (m)
=
2
4
2
2a 2
2
2 1
=
a
m
2
c
2 2
4
2
x2 2
2
4 1
2 2
2
2 2
4
2
2 2
x2
x2
x2
2 2
+m 2
2 2
4
2
cx
8 2
2 2
4a 2
2
m 2
c
4 2
+
4 2
4 1
2
2 2
8 2
[1 + ] [2 + ]
2 2
(A10)
x2
:
Calculating the optimal markup m, yields (24) as solution of
0=
d
dm
=
@ 1 @p+
@ 2 @I +
2
+
@p2 @m @I @m
=
m
2
2 1
x2 4
2 2
2
4
2 2
2
q1+ ( )2 + m+ ( )q2+ ( ) + q2+ ( )2
2
1
2 2
4 2
x2
E q2+ (I + ; m) :
Using (A9), (A10) and (24) yields
E[
+(
)]
=
E
1
=:
B+( ) +
2 2
]
[2
[4
2 2
] [1
35
2
]
+
1
4
2:
I + ( )2
2
(i) E[t+ ( )] > E[C(I + ( ))] follows directly from (24). (ii) Since m+ =2
q2 ( ), cost-plus transfer
I ( ), implying E[q1+ ( )]
E [q1 ( )] for all . (iii)
pricing yields I + ( )
B+( )
E [q1 ( )] and E[q2+ ( )]
B ( ) follows as outlined in the text.
Proof of Proposition 2. We rst characterize the solution for the standard cost-based transfer
price. The backward induction process is detailed in the text. Since psc
1 (I) = p1 (I), plugging the
equilibrium value of (7) and q2sc (t) = [a tsc ] =2 into (23), yields
I sc (t) = x
[1
2 2
] [1 + ] 4
2 1
[a
2
2
4
q1sc (t) =
4 2
2 2
4
c
2 2
2
(A11)
:
x2
t)=2)
2
2a 2
2 1
2 2
4
Forward induction yields (notice q2sc (t) = (a
2
t] + 4a [2 + ] 2
2
2
+ 2
4
2 2
x2 [a
4 2
t]
2 2
2
c
2
2
:
x2
(A12)
Calculating the optimal transfer price tsc , yields
d
dt
@ 1 @pc2
@p2 @t
=
=
t
E [C (I sc )]
2
= 0 ! tsc = E [C (I sc )] :
(A13)
From tsc = E [C (I sc )] follows I sc ( ) = I ( ), E[q1sc ( )] = E [q1 ( )] and E[q2sc ( )] = E [q1 ( )] for
all . Taking the equilibrium values from (A4) into account, yields
E[
sc (
)] = E
1
2
q1sc (I sc )2 + q2sc (I sc )2
2 2
(I sc )2
[2
]
=: B ( ) +
2
2
2
[4
] [1
2
]
2:
(i) E[tsc ( )] = E[C(I sc ( ))] = E[C(I ( ))] follows I sc ( ) = I ( ). (ii) As shown before. (iii)
F sc ( )
F ( ) follows as outlined in the text. (iv) The cut-o value b2s+ ( ) follows directly from
(25) and (27).
Proof of Proposition 3. Comparing (16), (22), (25) and (27) provides the cut-o values.
Proof of Corollaries 1, 2 and 3. As outlined in the text.
A.2
Cournot competition
Regularity conditions. For didactical reasons, we assume aI =aF > 2. and
C(I)
= 1. Positive costs,
0. Since under the benchmark case the investments are the largest which are decreasing in
36
, assuring c
xI (0)
0 is su cient to guarantee positive costs, yielding
c = x2
c
Positive quantity, q1
(A14)
0. Since the controllable market-based transfer price induces the lowest q1
that is decreasing in , assuring q1c
c
Finally, to assure c
8aI + 9aF
:
18
0 is su cient to guarantee positive quantities, yielding
18x4 aF + aI 36 14x2
72 28x2
c =
9x4
:
(A15)
c, we get for x
x
x = 6
s
a
qI
23aI + 18aF + 9 5a2I + 4a2F
:
4aI aF
Finally, since the benchmark case induces the highest investments, the associated second order
q
2 2
condition is binding, yielding x xsoc ( ) = 2 4
=[ 4 8 2 + 32] for all : However, since
this condition is ful lled for x
x, we can omit it.
Characterization of the benchmark solution. The backward induction process is detailed in
the text. Using the equilibrium values q2 (I) = [aI
C(I)] =2, q1 (I) = [aI [2
]
2C(I)]= 4
2
and (28), yields
E[
( )]
=
E [q1 ( )]
=
with
E [q2 ( )]
I
=
=
(I )2
+
2
E [q1 ( )]2 + E [q2 ( )]2
4
4
x
2
aI [2
2 4
[aF
)] 2
2 2
2 2
2 4
aF 4
4
2
4
32 8 +
2
c ] + 4x (aI [2
32 8 +
2
+ 8aI [2
)]
2 4
2 2
[4
2
32
8
4
1
4
2
c
x2
)]
c
2
+
2 2
]
x2 + 2x2 aF
2 2
2
4
2aF )
x2
32
+
4
8
2
(A16)
+
x2
4
:
Characterization of the standard cost-based transfer price. The backward induction process is detailed in the text. Using E [q1sc ( )] = E [q1 ( )], q2sc ( ) = E [q2 ( )] and I sc = I and (A16)
yields
E[
sc (
)]
=
E
1
2
q1sc ( )2 ] + E[q2sc ( )
37
2
(I sc )2
2
=:
4
B ( )+
4
2 2
2:
Characterization of the cost-plus transfer price. The backward induction process is detailed
in the text. Using q1+ ( ) = q1 ( ) ; a (7) and (23) yields
8aI [2
I + (m) = x
]
16
c
2 2
2( 4
+
2 2
4
:
8x2 )
(A17)
Forward induction yields
E q1+ (m)
=
aI [2
4
q2+ (m)
E
2
4
=
aF
c
2
]
2
c
+ x2
2 2
8x2
8aI [2
+ x2
]
16
c
2 2
4( 4
+
(A18)
2 2
4
:
8x2 )
We obtain the optimal markup, solving
d
dm
=
@ 1 @q2+ @ 2 @I +
+
@q 2 @m @I @m
=
m
2
q2+ (I + ; m+ )
2 2
] E
x2 [4
2
2 2
4
8x2
:
Using (A17), (A18) and (32) yields
E[
+(
)]
= E q1+ ( )2 + m+ ( )q2+ ( ) + q2+ ( )2
I + ( )2
=: B + ( ) +
2
4
[4
2 2
]
+
1
4
2:
Characterization of the uncontrollable market-based transfer price. The backward induction process is detailed in the text. Using q2u (tu ) = [a
I u ( ) = xk u
2aI
with k u = 8 + 3 [2 + ]
E [q1u ( )] =
+
E [q2u ( )] =
+
c [2
aF
2
tu ]=2 and (34) in (36), yields
+ ] 2 + 4aI [1 + ] 2
64 [2 + ] 16 + 3 [2 + ]
8
2
c [2
x2
+ ] + 32 [1 + ]
(A19)
. Forward induction yields
16 (2aI
[aF + 2 ]
c [2 + ])
64 [2 + ] 16 + 3 [2 + ] 2 x2
[2 + ] x2 2 [4 + ] aF aI 4 + 2 + 4 [2 + ]
8 8
64 [2 + ] 16 + 3 [2 + ] 2 x2
aF 16 [2
] (aI
[2 + ])
2
2
2(64
[2 + ] x2 ([16
8
c
[2 + ]
(A20)
x2 )
[2 + ] 16 + 3 [2 + ]
(4
[3 + 5])])aF aI [4 + ] (2
2(64 [2 + ] 16 + 3 [2 + ] 2 x2 )
2
[2 + ] x ( (16
[2 + ] [2 3 ]))
:
2(64 [2 + ] 16 + 3 [2 + ] 2 x2 )
[1
] ))
Calculating the optimal discount , yields
d
d
=
u
@ @qC
@ 1
+
@qC @
@q2
@q2u
@q u @q u
+ 2 C
@
@qC @
+
@ 2 @q1u
+
@q1 @
38
u
@ 2 @q1u
@ 2 @qC
+
@q1 @I
@qC @I
@I u
@
=0
and thus
u
= E [puC ( )
with Du ( ) = 2( + 2)[16
(2 E [ q1u + 2q2u ] x2 Du ( ))E [q2 ( )]
;
4+ 2
C( )]
[3( + 2)
2
x2 [ + 2][3( + 2)
+ 8]]=[64
2
+ 16]]
(A21)
0 for
< 0:975
(and vice versa).
Using the equilibrium values from (A19), (A20) and (A21), yields
E[
u(
)]
aF + C( ) 2tu ( ) u
q1 ( ) + mu ( )q2u ( ) + q2u ( )2
2
!
2
(2 + ) 16 + 3 2 (2 + )
[2 + ]2
2:
+
2 2
256
[4
]
=
E q1u ( )2 +
=:
Bu( ) +
I u ( )2
2
Characterization of the controllable market-based transfer price. The backward induction
process is detailed in the text. Using q2c (tc ) = [a
Ic ( ) = x
3 12
8
2
4
+
tc ]=2 and (34) into (36), yields
aF + 6aI [2
2 2
16 3
2
] 6
3
2 2
3 6
c
2 2
6
4
2
:
x2
(A22)
Forward induction yields
E
[q1c (
)] =
3x2 2 6
2
aF
aI [2
2
] 6
2 2
6 3
E [q2c ( )] =
+
3x2 2aI [2
2
] 6
2
4 6
aF
2
4 3
2
+ 12
2
2
6 3
3 6
2
10 3
aF 2 [2
] (aI
6 3
2 2
2
2 2
3 6
24
6
x2
[2 + ])
2 2
3 6
2
+8 3
2
]
aF
3
c
4
+
3
(2aI [2
x2
c
2
2
x2
:
(A23)
Calculating the optimal discount , yields
d
d
=
c
@ @qC
@ 1
+
@qC @
@q2
@q2c
@q c @q c
+ 2 C
@
@qC @
and thus with Dc ( ) = 3 2 [16
c
= E [pc1 ( )
C( )]
2
]=[16[3
+
@ 2 @q1c
+
@q1 @
2 2
]
3x2 [6
16E [q2c ( )]
2
4[6
]
2
39
c
@ 2 @q1c
@ 2 @qC
+
@q1 @I
@qC @I
2 2
] ]
@I c
@
=0
0
8E [q1c ( )] + x2 Dc ( )E [q2c ( )]
;
2
4[6
]
(A24)
2 )
Using (A22), (A23) and (A24) yields
E[
c(
)]
aF + C( )
2
2 2
3 6
=
E q1c ( )2 +
=:
Bc( ) +
32 3
2 2
2tc ( )
q1c ( ) + mc ( )q2c ( ) + q2c ( )2
!
2 2
9[2
]
2:
+
2 2
64[3
]
I c ( )2
2
Proof of Proposition 4. The reasoning is similar to Propositions 1, 2 and 3.
Proof of Corollary 4. As outlined in the text.
Proof of Corollary 5. The reasoning is similar to Corollaries 1, 2 and 3.
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41
t 0
Central office
specifies transfer
pricing method
t 1
Division 1
undertakes specific
investments
Figure 1. Timeline
t2
t 3
t4
Costs g
get
realized
Division 1 and
competitor
compete in the
intermediate
market
Division 2
determines
price for the
final market
Variance  2
ˆ c2  
0.00015
Cost-plus transfer pricing
0.00010
ˆ u2  
ˆ uc2  
0.00005
Uncontrollable market-based transfer pricing
ˆ
0.2
2
us
 
St d d cost-based
Standard
tb
d ttransfer
f pricing
i i
C
Controllable
e market-b
based tran
nsfer pricin
ng
0.00020
ˆ cs2  
0.4
0.6
0.8

Intensity of Competition 
Figure 2. Performance comparison for the case of Bertrand competition
Variance  2
0.10
Uncontrollable
market-based
transfer pricing
Cost-plus transfer pricing
ˆ u2  
0.08
0.06
̂ 2s  
ˆ us2  
0 04
0.04
ˆ cs2  
Standard cost-based transfer pricing
0.02
0.2
0.4
0.6
0.8
Co
ontrollable market-ba
ased transsfer pricing
g
0 12
0.12
ˆ c2  
1.0
Intensity of Competition 
Figure 3. Performance comparison for the case of Cournot competition

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