Regional Science and Urban Economics 36 (2006) 373 – 384
www.elsevier.com/locate/econbase
Validity of the principle of minimum differentiation
under vertical subcontracting
Wen-Jung Liang *, Chao-Cheng Mai
Department of Industrial Economics, Tamkang University, Tamsui, Taipei County 25137, Taiwan
Accepted 7 November 2005
Available online 10 March 2006
Abstract
This paper develops a variant of Hotelling’s [Hotelling, H., 1929. Stability in competition. Economic
Journal 39, 41–57] model involving subcontracting production to explore the validity of the Principle
of Minimum Differentiation. It shows that the competition effect, the subcontracting effect, and the
bargaining power effect jointly determine the equilibrium locations. It also demonstrates that if the ratio
of the transport rates between the subcontracted input and the final product is sufficiently large, the
Principle of Minimum Differentiation arises, but the Principle of Maximum Differentiation occurs if the
condition is reversed. Furthermore, this paper finds that when the consignor takes the whole
subcontracting surplus, the subcontractor will choose vertical foreclosure if this ratio is sufficiently
large.
D 2006 Elsevier B.V. All rights reserved.
JEL classification: R30; L22; D43; C78
Keywords: Vertical subcontracting; Nash bargaining agreement; Vertical foreclosure; Spatial competition; Equilibrium
location
1. Introduction
Hotelling (1929) first proposes that two firms of a homogeneous product agglomerate at the
center of the line market under linear transportation costs, which has been termed the Principle
of Minimum Differentiation. However, D’Aspremont et al. (1979) challenge this principle and
show that the two firms will locate at the endpoints of the line market under quadratic
* Corresponding author. Tel.: +886 2 2625 1863; fax: +886 2 26209731.
E-mail address: [email protected] (W.-J. Liang).
0166-0462/$ - see front matter D 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.regsciurbeco.2005.11.004
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W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
transportation costs.1 This has been termed the Principle of Maximum Differentiation. Over the
past decades, numerous economists have tried to examine the conditions under which the
Principle of Minimum Differentiation is restored. De Palma et al. (1985) and Rhee et al. (1992)
introduce heterogeneity in both consumers and firms; Stahl (1982) considers some harmonious
conjectural variations; Anderson and Neven (1991) assume that firms play Cournot quantity
competition instead of Bertrand price competition in the commodity market; Jehiel (1992) and
Friedman and Thisse (1993) adopt price collusion; Zhang (1995) imposes a price-matching
policy; and Mai and Peng (1999) emphasize the importance of the externality-like benefits
generated from the exchange of information between firms.
It is well recognized that throughout the industrialized world, subcontracting has become an
increasingly popular method for firms to organize their production in order to enhance
competitiveness. The subcontracting of production is a way for firms to seek cheaper suppliers,
thus reducing costs. Indeed, there exist many examples of a firm subcontracting for other firms,
while competing with the same firm in the final goods market. For example, VoIP (Voice over
Internet Protocol) providers compete with the traditional telephone providers in the domestic and
international long distance markets. At the same time, VoIP providers rent the PSTN (Public
Switched Telephone Networks) supplied by the traditional telephone providers to offer the
services of PC-to-Phone and Phone-to-Phone calls;2 another example is webstreaming (i.e., the
delivery of film via the Internet). There are especially two firms in the world that can deliver
webstreaming of the highest quality, Speedera and Akamai. These firms sell webstreaming
services both directly to end consumers, and as an intermediate good to firms, say in Europe,
who offer product lines that may be tailored to specific local demands. Two other examples that
fit the model closely may be worth pointing out: most high-quality watches have a watch
mechanism produced by Swatch, the Swiss watchmaker, and many bakeries sell directly to
consumers while also supplying local supermarkets.3,4
More importantly, there exists a close relationship between subcontracting production and
location agglomeration in reality. Scott (1991) empirically studies the geography of the electronics
assembly subcontract industry in Southern California and finds that electronic assembly
subcontractors are strongly linked in networks of transactional interaction with both suppliers
and customers, and markedly agglomerate with their main markets. This is fully consistent with the
observed location symbiosis between assembly subcontractors and electronics producers. Wang et
al. (2002) use the data of Taiwan’s census in 1991 and show that the level of firms’ subcontracting
activity has a positive relationship with the firms’ regional agglomeration.
1
D’Aspremont et al. (1979) argued that under Bertrand price competition, both firms are forced to choose a zero
equilibrium price when they locate together.
2
VoIP is growing rapidly. According to a study by Info-Tech research entitled bVoIP Is Killing Traditional Telephony:
ReportQ, 23 percent of small- to mid-sized enterprises in the U.S. are already using VoIP technology, and this is expected
to grow to 50 percent by 2008. Another study, entitled bSpecial Report: VoIP and Wireless Plans Could Alter Long
Distance MarketQ, reveals that paid subscribers of VoIP have grown from just over 130,000 in 2003 to over 1 million by
the end of 2004. Readers who are interested in these studies are referred to the following websites: http://
www.tmcnet.com/channels/voip/voip-articles/voip-wireless-plans-alter-long-distance.htm, and http://www.voip-news.
com/art/7x.html.
3
We are grateful to an anonymous referee for providing the above three examples. Still another example quoted from
Kamien et al. (1989, p. 553): McDonnell–Douglas Corporation and Lockheed competed for the contract to produce C-17
Air Force transport planes, and McDonnell–Douglas won. Lockheed then built wing components for the C-17 planes
under subcontract from the McDonnell–Douglas Corporation.
4
One may argue that the final products in the above examples may be differentiated. As the role of product
differentiation in firms’ location decisions has been studied by De Palma et al. (1985) and De Fraja and Norman (1993),
we will not go further here. We will focus on the impact of vertical subcontracting.
W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
375
Unfortunately, to the best of our knowledge, the role of subcontracting production has not
been touched upon theoretically in the location literature. The present paper aims at filling this
gap by developing a variant of Hotelling’s (1929) duopoly model where each firm can alter its
production cost by subcontracting the production of a key intermediate input. We will discuss the
role of subcontracting production in the determination of firms’ optimal location and explore the
validity of the Principle of Minimum Differentiation.
The remainder of the paper is organized as follows. Section 2 develops a basic model where
firms engage in subcontracting production in the intermediate input in which the subcontracting
agreement is signed by firms with a Nash bargaining process. Section 3 explores the sub-game
perfect Nash equilibrium of the firms’ location choices for the cases where either the
subcontractor or the consignor takes the whole subcontracting surplus. In Section 4, the
subcontractor’s decisions of vertical foreclosure and supply are studied. The final section
concludes the paper.
2. The basic model
The basic model is a variant of Hotelling’s (1929) spatial duopoly model. Assume that there is
a linear market, represented by the unit interval [0, 1], on which consumers are uniformly
distributed. Two vertical integrated firms, indexed by 1 and 2, produce a homogenous final
product, Q, using a homogenous intermediate input, q. For simplicity, we assume that the
production of one unit of the final product requires employ one unit of the intermediate input.
Suppose that subcontracted production arises in the intermediate input market. Firm 1 is the
consignor, having a higher production cost of the intermediate input, while firm 2 is the
subcontractor, having a lower production cost of the input. Thus, firm 1 subcontracts out the
whole or part of the production of the intermediate input to reduce costs. The location of firm i is
denoted by x i a [0, 1]. The transportation costs of both the final product and intermediate input
are assumed to take the form of quadratic functions of distance.5 Each consumer buys one unit of
the final product from the firm with the lower full price, i.e., mill price plus transportation cost.
Thus, the full price of the final product for a consumer located at x who buys from firm i is:
p if + t f (x x i )2, where p if denotes the mill price of the final product offered by firm i, and t f is
the transport rate of the final product per unit of distance.
Without loss of generality, we assume x 1 V x 2 throughout this paper. When firms are set up at
x 1 b x 2, the marginal consumer, who is indifferent between purchasing from either firm, is
located at x̂ as given by:
x̂x ¼
p2f p1f
x1 þ x2
þ
:
2tf ðx2 x1 Þ
2
ð1Þ
Using Eq. (1), we can derive aggregate demand for the final product for firms 1 and 2,
respectively, as:
Z x̂x
p2f p1f
x1 þ x2
þ
;
ð2:1Þ
1dx ¼ x̂x ¼
Q1 ¼
2tf ðx2 x1 Þ
2
0
5
For example, D’Aspremont et al. (1979), Tabuchi and Thisse (1995), Mai and Peng (1999), and Zhou and Vertinsky
(2001) employ a quadratic form of the transportation cost function.
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W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
Q2 ¼
Z
1
1dx ¼ 1 x̂x ¼ 1 x̂x
p2f p1f
ðx1 þ x2 Þ
:
2
2tf ðx2 x1 Þ
ð2:2Þ
With respect to the subcontracting decision, we analyze, following Spiegel (1993), ex post
subcontracting in which firms engage in a Bertrand price competition in the final product market
before they decide whether or not to subcontract production of the input.6 As Spiegel (1993) has
argued, this setting is proper and agrees well with reality. In this setting, the game between firms
involves a sub-game perfect equilibrium with three stages of decision. In the first stage, both
firms simultaneously select their locations. In the second stage, the production locations are
known and the firms simultaneously choose their mill prices, p 1f and p 2f, respectively. In the
third stage, following Spiegel (1993), the two firms sign a subcontracting agreement, according
to which firm 2 will produce q s units of subcontracted quantities for firm 1, and in return will
receive a transfer payment. This agreement is assumed to be determined in a bargaining process
that has two properties: first, the subcontracted quantity q s is selected to generate the largest
possible surplus; second, the transfer payment is chosen to divide this surplus between firms 1
and 2 in proportions a and 1 a, respectively, where a a [0, 1]. The parameter a represents firm
1’s bargaining power. The sub-game perfect equilibrium of the model is solved by backward
induction, and we therefore start with the final stage.
In stage 3, given firms’ locations and their mill prices, the two firms sign a subcontracting
agreement aimed at earning the largest possible surplus, which is generated from the joint cost
savings. These savings, denoted S, are the joint cost difference between not having and having
the subcontract.7
n
o
S ¼ c1 c2 ts ðx2 x1 Þ2 qs ;
ð3Þ
where c i (i = 1,2) represents firm i’s marginal cost of producing the intermediate input; t s is the
transport rate of the subcontracted input per unit of distance; q s is the quantity of the intermediate
input subcontracted from firm 1 to firm 2.
Since the subcontracted quantity is determined by the condition of earning the largest surplus,
we differentiate (3) with respect to q s to obtain its optimal quantity as follows:
h
i
BS
¼ c1 c2 þ t s ð x2 x1 Þ 2 :
ð4Þ
Bqs
In order to simplify the analysis, assume that firm 1’s marginal cost, c 1, is always greater than
the sum of firm 2’s marginal cost and the transportation cost of the subcontracted input,
c 2 + t s(x 2 x 1)2. It then follows from (4) that BS/Bq s N 0, so firm 1 will subcontract out the whole
quantity of its demand for the intermediate input, i.e., q s = Q 1.8
We now turn to the second stage to determine the optimal mill prices. Suppose that the
production costs of the final product are the costs of using the intermediate input only. The firms’
6
As Spiegel (1993, p. 571) points out, when there is considerable uncertainty about either the demand for a
downstream product or the marketing cost (which can be learned only when they actually set their downstream
quantities), the firms may wish to postpone their subcontracting decisions ex post. Kamien et al. (1989) also sets up a
game where firms decide downstream price and quantity prior to the subcontracting stage.
7
This is equivalent to maximize the two firms joint profits difference between not having and having the subcontract,
because firms’ locations and mill prices and then the revenues are known in this stage.
8
Since we assume that one unit production of the final product requires one unit of the intermediate input, the
subcontracted quantity is thus equal to the demand for the final product of firm 1, namely Q 1.
W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
377
profit functions can be split into two parts: profits from the final product and those from the
subcontracting deal, and can be expressed, respectively, as:
h
i
ð5:1Þ
p1 ¼ ðp1f c1 ÞQ1 þ a c1 c2 ts ðx2 x1 Þ2 qs ;
h
i
p2 ¼ ðp2f c2 ÞQ2 þ ð1 aÞ c1 c2 ts ðx2 x1 Þ2 qs ;
ð5:2Þ
where p i (i = 1,2) denotes firm i’s total profits.
Substituting the condition q s = Q 1, along with (2.1) and (2.2) into (5.1) and (5.2), respectively,
then differentiating the reduced profit functions with respect to p 1f and p 2f, and finally setting
them both equal to zero yields:9
n
o
p41f ¼ ð1=3Þ tf ðx2 x1 Þðx2 þ x1 þ 2Þ þ ts ð3a 1Þðx2 x1 Þ2 þ 3½ð1 aÞc1 þ ac2 ;
ð6:1Þ
n
o
p42f ¼ ð1=3Þ tf ðx2 x1 Þð4 x2 x1 Þ þ ts ð3a 2Þðx2 x1 Þ2 þ 3½ð1 aÞc1 þ ac2 :
ð6:2Þ
In the first stage, each firm selects an optimal location to maximize its total profits.
Substituting (6.1) and (6.2) into the reduced profit functions, we have:
p14 ¼ ð1=18tf Þðx2 x1 Þftf ð2 þ x2 þ x1 Þ ts ðx2 x1 Þg2 ;
ð7:1Þ
p24 ¼ ð1=18tf Þðx2 x1 Þftf ð4 x2 x1 Þþ ts ðx2 x1 Þg2
n
o
þ ð 1 aÞ c 1 c 2 t s ð x 2 x 1 Þ 2 :
ð7:2Þ
Differentiating (7.1) and (7.2) with respect to each firm’s location, respectively, we obtain:
Bp14 =Bx1 ¼ tf A1 ð2 þ 3x1 x2 Þ þ 3ts A1 ðx2 x1 Þ ¼ 0;
ð8:1Þ
Bp24 =Bx2 ¼ tf A2 ð4 3x2 þ x1 Þ þ 3ts A2 ðx2 x1 Þ 2ð1 aÞts ðx2 x1 Þ ¼ 0;
ð8:2Þ
where
x 2 x 1 z 0;
0 V xi V 1
(i = 1,2);
A 1 = {t f (2 + x 1 + x 2) t s(x 2 x 1)} / (18t f) z 0;10
A 2 = {t f (4 x 1 x 2) + t s(x 2 x 1)} / (18t f) N 0; and 0 V a V 1.
The first term in the right-hand side of (8.1), whose value is non-positive, is named the
competition effect. It indicates that as the two firms become more distant location, their transport
costs at any site become dissimilar and therefore the competition is lessened under Bertrand
price competition. Moreover, the cost deviation will be larger if the transport rate of the final
product is higher. Thus, the higher the transport rate of the final product, the stronger will be the
competition effect. The second term, whose value is non-negative, is called the subcontracting
effect. This effect can reduce the transportation costs of the subcontracted input by moving near
the subcontractor. Moreover, the higher the transport rate of the subcontracted input, the stronger
The second-order conditions are met: B2p 1/Bp 1f2 = B2p 2/Bp 2f2 = 1 / [t f (x 2 x 1)] b 0.
By manipulating, we find that A 1 = p A1* / [6t f (x 2 x 1)] z 0, where p A1* denotes the consignor’s average profit, which is
non-negative.
9
10
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W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
will be the subcontracting effect. Consequently, the consignor’s (i.e., firm 1’s) equilibrium
location thus hinges upon the relative strength of these two effects.
Similarly, the first term in the right-hand side of (8.2) is the competition effect, which also
serves as a centrifugal force to separate the two firms. The second term is the subcontracting
effect. Unlike the counterpart of the consignor, the subcontractor can enjoy a cost advantage by
increasing the consignor’s transportation cost of the subcontracted input if it is located farther
away from the consignor. The third term is an extra effect, denoted the bargaining power effect.
As the subcontractor possesses a stronger bargaining power, it can have a larger fraction of the
subcontracting surplus. Thus, the subcontractor would like to move near the consignor to
increase this surplus via reducing the transportation costs of the subcontracted input. The
bargaining power effect serves as a centripetal force to the subcontractor. Moreover, the stronger
the bargaining power of the subcontractor is, the stronger will be the bargaining power effect.
Consequently, these three effects jointly determine the subcontractor’s equilibrium location.
Assuming that the second-order conditions are satisfied and solving (8.1) and (8.2), we can
derive the firms’ location equilibria as follows:11
x4
1 ¼ 2=½3ðT þ 1Þ þ fð3T þ 1Þ=½3ðT þ 1Þgx4
2;
ð9:1Þ
n
x42 ¼ ð1=16Þ ½38 þ 18T þ 54T ð1 aÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio
½38 þ 18T þ 54T ð1 aÞ2 32½ð63T þ 35Þ 54T ð1 aÞ ;
ð9:2Þ
where T = t s/t f is the ratio of the transport rates between the subcontracted input and the final
product.
In order to highlight the role of firm’s bargaining power over their location choices, in what
follows we shall discuss two polar cases: (i) the subcontractor takes the whole subcontracting
surplus, i.e., a = 0; and (ii) the consignor takes the whole surplus, i.e., a = 1.
3. The firms’ location equilibria
We now proceed to discuss the case in which the consignor has zero bargaining power a = 0.
Substituting a = 0 into (9.2) and then differentiating with respect to T yields:
n
h
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiio
2
Bxs4
b0;
2 =BT ¼ ð9=2Þ 1 ð4T þ 2Þ= 16T þ 16T þ 1
ð10Þ
where the superscript bsQ denotes the variables associated with the case of a = 0.
We see from (10) that the location of the subcontractor falls as the ratio T rises. By
calculating, we find that the subcontractor would locate at the right end of the line segment if the
ratio T is sufficiently small, say lower than 0.07937, but at the point 0.125 if T approaches
infinity. Substituting the solution of the subcontractor’s equilibrium location into (9.1) and then
taking appropriate manipulation, we find that the location of the consignor rises as the ratio T
The second-order conditions require: B2p 1*/Bx 21 = (1 / 9t f)(t f + t s){t f (4 + 3x 1 + x 2) + 3t s(x 2 x 1)} b 0,
2
Bx 2 = (1 / 9t f)(t f + t s){t f(8 3x 2 x 1) + 3t s(x 2 x 1)} 2t s(1 a) b 0.
11
and B2p 2*/
W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
379
increases. The consignor would locate at the left end if T is sufficiently small, say lower than
1.8383, but at the point 0.125 if T approaches infinity. Consequently, the two firms agglomerate
at the point 0.125 if T 6 l, but locate at the opposite endpoints if T is sufficiently small.12
The economic intuition behind the above discussions may be provided by recalling the
analysis in the previous section. We have mentioned that the competition effect, the
subcontracting effect and the bargaining power effect jointly determine subcontractor’s
equilibrium location. The former two effects pull the subcontractor toward the right end of
the line segment, while the third pulls toward the left end. The bargaining power effect is
stronger if the consignor’s bargaining power is weaker. In addition, when the ratio T is larger, the
subcontracting surplus is lower due to a higher transportation cost of the subcontracted input,
such that the subcontractor would like to move leftward to increase the subcontracting surplus.
Consequently, the bargaining power effect reaches its maximum and the subcontractor
agglomerates with the consignor if its bargaining power is (1 a) = 1 (i.e., a = 0) and T 6 l.
On the other hand, both the competition effect and the subcontracting effect determine the
consignor’s equilibrium location. The subcontracting effect of the consignor tends to move its
location toward the subcontractor in order to reduce the transportation cost of the subcontracted
input. This effect attains its maximum and the consignor agglomerate with the subcontractor if
T 6 l. Accordingly, we can establish:
Proposition 1. If the consignor possess zero bargaining power in the subcontracting agreement
and the subcontractor chooses vertical supply, we have:
(i) The larger the ratio of the transport rates between the subcontracted input and the
final product, the less will be the location difference between the two firms.
(ii) If the ratio approaches infinity, the Principle of Minimum Differentiation (i.e.,
x 1s* = x 2s* = 0.125) is valid, and the two firms agglomerate at the point 0.125 instead
of agglomerating at the center of the line market.
(iii) If the ratio is sufficiently small, the Principle of Maximum Differentiation (i.e.,
x 1s* = 0, x 2s* = 1) holds.
Comparing our result with the Principle of Maximum Differentiation as derived by
D’Aspremont et al. (1979), we find that Proposition 1 provides more generalized results by
taking subcontracting production into account. It indicates that the Principle of Minimum
Differentiation may arise when the transportation cost ratio is arbitrarily large, while the
Principle of Maximum Differentiation may occur when the ratio is lower than 0.07937.
Intermediate results may occur when the ratio ranges between 0.07937 and infinity. Moreover,
we find that the Principle of Maximum Differentiation is restored, if the subcontracting
production is absent. This happens because the subcontracting effect and the bargaining power
effect vanish, and only the competition effect, i.e., a centrifugal force, is present.
Next, we move on studying the case in which the consignor takes the whole subcontracting
surplus, i.e., a = 1. Substituting a = 1 into the first-order condition of the subcontractor’s profitmaximization, we obtain:
Bp2c =Bxc2 ¼ A2 tf ð4 3x2 þ x1 Þ þ 3A2 ts ðx2 x1 ÞN0;
ð11Þ
where the superscript bcQ denotes the variables associated with the case of a = 1.
12
s4
This result can also be derived by taking the limit of (9.1) at T 6 l, which yields lim xs4
1 ¼ lim x2 .
T Yl
T Yl
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W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
Since the first-order condition of the subcontractor’ location is definitely positive, the
subcontractor would choose to locate at the right endpoint to maximize its profit. Thus, we have:
xc4
2 ¼ 1:
ð12Þ
Intuitively, (11) shows that the subcontractor’s bargaining power effect vanishes due to the
lack of bargaining power. Since there are only centrifugal forces (i.e., the competition effect and
the subcontracting effect) left to the subcontractor, it will choose to locate as far away from the
consignor as possible to lessen the competition as well as to secure a cost advantage via
increasing its rival’s transportation costs. Therefore, the equilibrium location of the subcontractor
is at the right endpoint as indicated in (12).
Substituting (12) into (9.1) yields:
xc4
1 ¼
3T 1
:
3T þ 3
ð13Þ
It follows from Eq. (13) that the equilibrium location of the consignor is a function of T, i.e.,
the trade-off between a centripetal force caused from the transport rate of the subcontracted input
and a centrifugal force from the transport rate of the final product. To explore the impact of the
ratio T on the equilibrium location of the consignor, we differentiate (13) with respect to T to get:
Bxc4
4
1
¼
N0:
ð14Þ
BT
3ðT þ 1Þ2
It clearly follows from (14) that the larger the ratio, T, the stronger the subcontracting effect
(i.e., the centripetal force) will be. This leads to a result that the consignor will locate closer to
the subcontractor. When the ratio approaches infinity, the consignor will agglomerate with the
subcontractor at the right endpoint of the line market such that the Principle of Minimum
Differentiation is valid. On the other hand, when the ratio is no greater than one third, the
consignor will locate at the other endpoint of the line market, and hence the Principle of
Maximum Differentiation occurs due to a very weak subcontracting effect. Thus, we can
establish the following proposition:
Proposition 2. If the consignor has the whole bargaining power in the subcontracting
agreement and the subcontractor chooses vertical supply, we have:
(i) The larger the ratio of the transport rates between the subcontracted input and the
final product, the less will be the location difference between the two firms.
(ii) If the ratio approaches infinity, the Principle of Minimum Differentiation (i.e.,
x 1c* = x 2c* = 1) is valid. In this case, the two firms will agglomerate at the endpoint of
the line market where the subcontractor locates instead of agglomerating at the
center of the line market.
(iii) If the ratio is no greater than one third, the Principle of Maximum Differentiation
(i.e., x 1c* = 0, x 2c * = 1) holds.
4. Vertical foreclosure vs. vertical supply
In the previous two sections, we have studied firms’ location decisions in the case where the
subcontractor is willing to supply the intermediate input to his rival. Nevertheless, the
W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
381
subcontractor will not provide the input to his rival, when his profits are higher under vertical
foreclosure than under vertical supply. In order to gain new insight into this issue, we intend to
compare subcontractor’s profits under vertical foreclosure with those under vertical supply.
When the subcontractor chooses vertical foreclosure, the consignor is forced to produce his
own intermediate input. Under this circumstance, the subcontracting stage vanishes and the
game in question is reduced to a two-stage game in which the locations are selected in the first
stage and then firms play a Bertrand price competition in the second stage. We can thus express
firms’ profit functions under vertical foreclosure as follows:
p1F ¼ ðp1f c1 ÞQF1 ;
ð15:1Þ
p2F ¼ ðp2f c2 ÞQF2 ;
ð15:2Þ
where the superscript bFQ denotes variables involving vertical foreclosure.
As the analysis for the stage 2 is similar to the one derived in the previous section, we shall
skip this part and concentrate on the stage 1 problem. In stage 1, the first-order conditions for
profit maximization are given, respectively, by:
Bp1F =Bx1 ¼ A1F f ðc1 c2 Þ tf ðx2 x1 Þð2 þ 3x1 x2 Þgb0;
ð16:1Þ
Bp2F =Bx2 ¼ A2F f ðc1 c2 Þ þ tf ðx2 x1 Þð4 3x2 þ x1 Þg ¼ 0;
2
ð16:2Þ
13
where A 1F = [ (c 1 c 2) + t f (x 2 x 1)(2 + x 2 + x 1)] / [18t f (x 2 x 1) ] z 0; x 2 x 1 z 0; 0 V x i V 1;
and A 2F = [ (c 1 c 2) + t f (x 2 x 1)(4 x 2 x 1)] / [18t f (x 2 x 1)2] z 0.14
The first term in the brace of the right-hand side of (16.1) denotes the cost-disadvantage
effect. The consignor tends to stay away from the subcontractor due to its cost disadvantage. The
second term represents the competition effect, which is the same as that discussed in the previous
section. Consequently, there is no centripetal force affecting the consignor’s decision, and it
always locates at the left endpoint of the line market, i.e., x 1F = 0. For the subcontractor, the first
term of (16.2) serves as the cost-advantage effect, which is a centripetal force for the
subcontractor to locate closer to the consignor. The second term is the competition effect, which
is a centrifugal effect. The two effects jointly determine the equilibrium location of the firms.
Assuming that the second-order conditions are satisfied, solving (16.1) and (16.2) yields:
xF1 ¼ 0;
xF2 ¼ 1;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xF2 ¼ ð2=3Þ þ ð4=9Þ ð1=3tf Þðc1 c2 Þ;
xF2 ¼ 0;
ð17:1Þ
if c1 c2 btf ;
if tf Vc1 c2 bð4=3Þtf ;
if c1 c2 Nð4=3Þtf :
ð17:2Þ
We see from (17.1) and (17.2) that when the difference in marginal costs is small, say less
than t f, the two firms locate at the opposite endpoints (i.e., x 1F = 0 and x 2F = 1). In contrast to this,
when the difference in marginal costs is large, say larger than (4/3)t f, the cost-advantage effect
outweighs the competition effect so that the subcontractor locates at the left endpoint. However,
13
By manipulating, we find that A 1F = p A1F / [6t f (x 2 x 1)2] z 0, where p A1F denotes the consignor’s average profit,
which is non-negative.
14
We also find that A 2F = p A2F / [6t f (x 2 x 1)2] z 0, where p A2F denotes the subcontractor’s average profit, which is nonnegative.
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W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
this agglomeration equilibrium is not sustainable because the subcontractor will take
undercutting by charging a mill price slightly lower than c 2 in order to capture the whole
market due to its cost advantage. Therefore, the subcontractor becomes a monopolist and earns a
monopoly profit slightly lower than c 1 c 2. Moreover, the consignor locates at the left endpoint,
while the subcontractor locates at the range [0, 1] when t f V c 1 c 2 V (4/3)t f. Therefore, we find
that the higher the transport rate of the final good is, the stronger will be the competition effect.
Hence, the subcontractor will locate farther away from the consignor. At the extreme case, when
the marginal costs of both firms are identical, this model degenerates into that of D’Aspremont et
al. (1979) and the Principle of Maximum Differentiation occurs. Consequently, we have:
Lemma 1. Under vertical foreclosure, when the difference in marginal costs between the two
firms is large, the subcontractor becomes a monopolist and the Principle of Minimum
Differentiation is not sustainable. However, the Principle of Maximum Differentiation holds
when the difference in marginal costs is small.
We proceed to compare the difference in profits between vertical supply and vertical
foreclosure when the consignor takes the whole subcontracting surplus, i.e., a = 1. The
subcontractor’s profits under vertical foreclosure are calculated as follows:
p2F
¼
¼
¼
ð1=18tf Þ½ðc1 c2 Þ þ 3tf 2 ;
if c1 c2 btf ;
F 2 2
F
ð1=18tf Þ½ðc1 c2 Þ þ 4tf x2 tf x2 ; if tf Vc1 c2 Vð4=3Þtf ;
ðc1 c2 Þ;
if c1 c2 Nð4=3Þtf :
ð18Þ
Substituting (12), (13) and the condition a = 1 into (7.2), we can derive the subcontractor’s
profits under vertical supply as follows:
p2c4 ¼ 200tf =½243ðT þ 1Þ:
ð19Þ
The difference in profits between vertical supply and vertical foreclosure is derivable from
(18) and (19) as follows:
n
o
p2c4 p2F ¼ f200tf =½243ðT þ 1Þg ð1=18tf Þ½ðc1 c2 Þ þ 3tf 2 ; if c1 c2 btf ;
h
F 2 i 2
F
¼ f200tf =½243ðT þ 1Þg ð1=18tf Þ ðc1 c2 Þ þ 4tf x2 tf x2
;
if tf V c1 c2 V ð4=3Þtf ;
¼ f200tf =½243ðT þ 1Þg fðc1 c2 Þg;
if c1 c2 Nð4=3Þtf :
ð20Þ
Note that both c 1 c 2 N 0 and 0 V x 2F V 1 hold. We find from (20) that the difference in profits
is negative and the subcontractor chooses vertical foreclosure if the ratio T is sufficiently large.
As derived in the previous section, the larger the ratio is, the closer the consignor will approach
to the subcontractor and the lower are the profits, which the subcontractor can make under
vertical supply. Thus, when this ratio is sufficiently large, the profits of the subcontractor under
vertical supply are so low that he will choose vertical foreclosure. On the other hand, when this
ratio and the difference in marginal costs are sufficiently small, the difference in profits is
positive and the subcontractor will choose vertical supply. This arises because the cost advantage
that the subcontractor gains is so low that he has less incentive to choose vertical foreclosure.
W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
383
This incentive becomes weaker when both firms separate far enough under vertical supply,
which is the case where this ratio is sufficiently small. Hence, we have:15
Lemma 2. When the consignor takes the whole subcontracting surplus in the subcontracting
agreement, the subcontractor will choose vertical foreclosure if the ratio of the transport rates
between the subcontracted input and the final product is sufficiently large. On the contrary,
when this ratio and the difference in marginal costs are sufficiently small, the subcontractor will
choose vertical supply.
5. Concluding remarks
This paper has developed a variant of Hotelling’s (1929) model involving subcontracted
production to explore the validity of the Principle of Minimum Differentiation. The
subcontracting agreement is signed by firms with a Nash bargaining process in which each
firm captures a fraction of the subcontracting surplus according to its bargaining power. We have
shown that the competition, the subcontracting and the bargaining power effect jointly determine
the equilibrium locations. In particular, we have studied two polar cases in which either the
consignor or the subcontractor takes the whole subcontracting surplus in the subcontracting
agreement, respectively. Several striking results are derived accordingly.
First of all, we have shown that while choosing vertical supply, the Principle of Minimum
Differentiation arises when the ratio of the transport rates between the subcontracted input and the
final product is sufficiently large. By contrast, the Principle of Maximum Differentiation occurs if
this ratio is sufficiently small. It is noteworthy that the agglomeration arises at a point away from
the center of the line market due to asymmetric production costs. This result can be evidenced by
the case where the liquid crystal display (hereafter LCD) manufacturer usually agglomerates with
its rival who is vertically integrated to produce LCD and the glass. Since the glass is so fragile, the
breakage rate is very high during long-distance shipment. This leads to an expensive transport rate
for shipping the glass. Thus, the ratio of the transport rates between the glass and the LCD is so high
that they tend to locate together. Another example is a vertically integrated firm, which owns a
chemical plant and a petroleum refinery, versus a chemical product manufacturer. The intermediate
inputs, such as ethylene, are provided by the refinery directly with pipes because shipping them is
very dangerous and moreover the chemicals may change their molecular structure in transit. Thus,
these chemical plants always locate next to the petroleum refinery. This also reflects the fact that
the transport rates of the intermediate inputs for those chemicals are very high.
Secondly, we have shown that even taking into account the subcontractor’s vertical
foreclosure decision, the Principle of Minimum Differentiation is robust in the case where the
subcontractor takes the whole subcontracting surplus, while it is invalid when the consignor
takes the whole subcontracting surplus.
Lastly, contrasting with the symmetry assumption of the equilibrium locations made in the
early spatial competition literature, for example, Tabuchi and Thisse (1995), and Mai and Peng
15
Following the same procedure, we can show that when the subcontractor takes the whole subcontracting surplus, the
subcontractor will choose vertical supply, and the Principle of Minimum Differentiation is valid if the ratio T and the
difference in marginal costs c 1 c 2 are sufficiently large. This result holds true because the profits of the subcontractor
are slightly lower than c 1 c 2 under vertical foreclosure, which is lower than those profits c 1 c 2 when both firms
agglomerate under vertical supply.
384
W.-J. Liang, C.-C. Mai / Regional Science and Urban Economics 36 (2006) 373–384
(1999), this paper drops this specific assumption and obtains a more generalized solution of the
equilibrium locations, indicating that the equilibrium location may arise at any point within the
line market. This result is reasonable because the production costs of the firms are asymmetric.
Acknowledgements
We are grateful to an anonymous referee and Professor Richard Arnott for very useful
suggestions. The first author would like to thank for the financial support from National Science
Council, Taiwan, and the second author for the National Forum of the Ministry of Education.
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