Discussion Paper No. 983

STRATEGIC PERILSOF OUTSOURCING:
SOURCING STRATEGYAND PRODUCT POSITIONING

Noriaki Matsushima
Cong Pan

October 2016

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Strategic Perils of Outsourcing: Sourcing Strategy andProduct Positioning∗
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Cong Pan‡
Graduate School of Economics, Osaka University.
October 25, 2016

Abstract
Although outsourcing input production has long been considered as an important
approach to help downstream manufacturers enhance structural eﬃciency, we provide atheoretical explanation for why outsourcing may negatively aﬀect downstream ﬁrms’ profitability. We consider a duopoly model wherein downstream manufacturers endogenously
determine their input sourcing and product positioning strategies. We show that when
inputs from outside suppliers are not perfectly compatible with downstream manufacturers’ requests, outsourcing causes downstream manufacturers to pursue aggressive productpositioning behavior, leading to the prisoner’s dilemma—even though both downstream
manufacturers could be better oﬀ producing inputs in-house, they may still choose outsourcing.

JEL classifications code: C72, L13, M31.
Keywords: Outsourcing; Product diﬀerentiation; Suppliers; Hotelling model.
∗

We gratefully acknowledge ﬁnancial support from JSPS KAKENHI Grant Numbers JP15H03349 and
JP15H05728, Grant-in-Aid for JSPS Fellows Grant Number JP15J05223, and the program of the Joint Usage/Research Center for Behavioral Economics at ISER, Osaka University. The usual disclaimer applies.
†
Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki,
Osaka 567-0047, Japan. Phone: (81)-6-6879-8571. Fax: (81)-6-6879-8583. E-mail: nmatsush@iser.osakau.ac.jp‡
Corresponding author: Cong Pan, Graduate School of Economics, Osaka University, Machikaneyama
1-7, Toyonaka, Osaka 560-0043, Japan. Phone: (81)-6-6850-6111. Fax: (81)-6-6850-5205. E-mail:pge042pc@student.econ.osaka-u.ac.jp.

11Introduction

Outsourcing inputs has long been an important part of manufacturing. In general, it brings
downstream manufacturers a trade-oﬀ between avoiding ﬁxed costs of holding their own input
facilities and incurring wholesale prices oﬀered by outside suppliers (e.g., Shy and Stenbacka,
2003; Xiao et al., 2014; Liu and Tyagi, 2016). That is, outsourcing has a trade-oﬀ between
eliminating ﬁxed cost and adding marginal costs. In addition, compared with insourcing, outsourcing helps downstream manufacturers save resources for producing inputs, and thus, helps
them concentrate better on quality control, although this happens at the cost of procuring
inputs (Xiao et al., 2014).1 That is, outsourcing also has a trade-oﬀ between a higher quality
and a poorer procuring condition.
By taking into account the above trade-oﬀs, we also consider the interdependence between a downstream manufacturer’s product positioning and sourcing strategies (Novak andEppinger, 2001; Ulrich and Ellison, 2005). Product positioning, which is related to product diﬀerentiation, always involves speciﬁc knowledge and/or knowhow. It is possible that
outsourcing puts a downstream manufacturer with high product distinctiveness at a disadvantage because of the growing diﬃculty in ﬁnding a suitable outside input supplier. To
investigate this matter, we employ a simple Hotelling model that enables us to discuss the relation between downstream manufacturers’ product positioning and sourcing strategies.2 We
consider a duopoly case wherein downstream manufacturers endogenously determine sourcing
strategies by anticipating the following decisions on product positioning represented by theirlocations along a linear city. That is, they determine their sourcing strategies, anticipating
1

For example, Techtronic Industries (TTI), a leading investment holding company whose products include

well-known brands like Milwaukee, AEG, Ryobi, Homelite, Hoover US, Dirt Devil, and Vax, outsources its
production to external suppliers, and focuses on product design and other speciﬁcations in-house (Zhu et al.,2007). For additional information on theoretical demonstrations, see Grossman and Helpman (2002).
2

As in related theoretical papers, ﬁrms’ product positioning strategies are captured by locations along the

Hotelling line (Tyagi, 2000, 2001; Guo, 2006; Kim and Serfes, 2006; Meza and Tombak, 2009; Sajeesh andRaju, 2010; Liu and Tyagi, 2011; Sajeesh, 2016). Also see papers in the marketing context (e.g., Hauser, 1988;
Moorthy, 1988; Desai, 2001; Kuksov, 2004; Xiang and Sarvary, 2007; Zhang, 2011).

2

the subsequent location choices in the Hotelling line. Two upstream ﬁrms, whose locations
are exogenously given at each edge on the linear city, engage in price competition for thedownstream business. Each upstream ﬁrm incurs a per unit transportation cost to supply its
input to a downstream manufacturer. This transportation cost reﬂects an adjustment cost to
meet the request made by a downstream manufacturer. When a downstream manufacturer
chooses outsourcing, it procures inputs from the upstream ﬁrm that oﬀers the lowest input
price for it.3
As noted, outsourcing inputs enables a downstream manufacturer to focus on quality control, leading to more premium ﬁnal products than those under insourcing.4 In the absenceof outsourcing, because downstream manufacturers do not procure inputs from the outside,
their product characteristics do not aﬀect the cost structures. Thus, manufacturers makeproducts that are suﬃciently diﬀerentiated to alleviate the downstream competition. Whenoutsourcing is employed, on the other hand, the product positioning of an outsourcing downstream manufacturer aﬀects the wholesale price charged by the trading upstream ﬁrm. In
reality, the wholesale price is determined by the per unit cost of the upstream ﬁrm, which
does not supply to this outsourcing downstream manufacturer as in the standard Bertrand
competition. Compared with the location under insourcing, the downstream manufacturer
tends to locate geographically closer to the less compatible upstream ﬁrm under outsourcing
in order to create a better outside option for itself, and it procures inputs at a lower wholesale price. Thus, the downstream price competition becomes more intense. When only one
downstream manufacturer chooses outsourcing, it can enlarge demand by oﬀering premium
3

In the theoretical context of vertical integration and separation, Matsushima (2004, 2009) considers similar

settings in the sense that downstream ﬁrms determine their locations while upstream ﬁrms engage in pricecompetition in the Hotelling line. However, Matsushima (2004, 2009) restricts downstream ﬁrms’ locations to
the line segment, while the present study considers the location problem along the whole line. Besides, Matsushima (2004, 2009) assesses whether a downstream ﬁrm’s choice on vertical structure (integration/separation)is based on the joint proﬁts of the downstream and its pairing upstream ﬁrms, while in our case, a downstreamﬁrm decides whether to outsource by evaluating its own proﬁt change only.
4

As an extension of the main model, we also consider another possible beneﬁt of outsourcing—eliminating/reducing ﬁxed costs necessary for input production.

3

ﬁnal products and locating at a good position and thus obtains better proﬁts than the otherﬁrm choosing insourcing. However, when both downstream manufacturers outsource their
input production, the relative advantage in their ﬁnal products’ quality no longer exists, andonly the losses from intensiﬁed price competition remain. Therefore, our analysis generates a
counterintuitive result: outsourcing may bring no beneﬁts and only losses. More speciﬁcally,
although both downstream manufacturers are better oﬀ by insourcing input production, they
may still choose outsourcing.
Our results have important managerial implications in ﬁrms’ sourcing strategies. Until
now, large numbers of real-world cases have described outsourcing to be a beneﬁcial management strategy from two points of view: converting ﬁxed cost to variable cost and chasing
high-quality input products.5 However, researchers have paid scant attention to a quite general situation in vertical relations, namely, one where the intermediate products supplied by
an upstream ﬁrm are not perfectly compatible with the downstream ﬁrms’ requests. In someindustries, in order to supply diﬀerent downstream manufacturers, an input supplier always
has to develop a basic product model, which is then customized for a particular manufacturer.6
Our study shows the dark side of outsourcing, namely that, in the presence of such imperfectcompatibility, downstream manufacturers sometimes coordinate themselves to produce inputs
independently regardless of the supposed beneﬁts of outsourcing.
Downstream manufacturers’ sourcing strategies have received considerable attention fromthe theoretical aspect. To capture players’ strategic incentives, some studies have endogenizedproducers’ sourcing patterns and have obtained results similar to ours in the sense that three
types of sourcing outcomes can exist in equilibrium—none, either, or both producers outsource
5

Liu and Tyagi (2016) list many real-world cases wherein outsourcing helps manufacturers save ﬁxed costs.

Zhu et al. (2007) summarize examples about how outsourcing boosts quality-improvement eﬀorts in supply
chains.
6

An example of this type can easily be found in the aircraft and automobile industries (Ahmadjian and

Lincoln, 2001; Bonaccorsi and Giuri, 2001). Needless to say, economists have also theoretically focused on theimperfect compatibility of inputs in vertically related industries (e.g., Eaton and Schmitt, 1994; Norman andThisse, 1999; Matsushima, 2004, 2009).

4their input production (e.g., Shy and Stenbacka, 2003; Gilbert et al., 2006; Xiao et al., 2014;
Liu and Tyagi, 2016). In these studies, however, as long as the output producer chooses
outsourcing in equilibrium, it always beneﬁts from either a better procurement condition ormore premium outputs. Our ﬁnding contrasts with such results by showing that even if both
output producers’ outsourcing may exist in equilibrium, both of them lose beneﬁts.
Liu and Tyagi’s (2011) research is closely related to our study as their research motivation
also stems from how production positioning aﬀects producers’ sourcing strategies. However,
Liu and Tyagi (2011) model each upstream ﬁrm as an exclusive supplier, so that the inputs
are perfectly compatible for each downstream manufacturer. The downstream manufacturer
who chooses outsourcing tends to further diﬀerentiate its products from its rival’s so as toalleviate double marginalization. The resultant downstream price competition is so relaxedthat in equilibrium, both downstream manufacturers always outsource their input production.
Our study considers the imperfect compatibility of inputs so that outsourcing results in lessdiﬀerentiation in ﬁnal products and more intense downstream price competition. Thus, owing
to the diﬀerence in the sourcing environment, our study shows contrasting results and adds
several new insights to the existing literature.7
The remaining paper is organized as follows. Section 2 explains the model. Section 3 shows
the main result. Section 4 brieﬂy discusses the extension of the model. Section 5 concludes
the paper.

2

The Model

There are two downstream ﬁrms, D1 and D2 , competing in a linear city wherein consumers are
uniformly distributed along the interval [−1/2, 1/2] and with a mass of 1. Each downstream
ﬁrm, Di , is located at li ∈ (−∞, ∞) (i = 1, 2). A consumer living at y ∈ [−1/2, 1/2] incurs
a transportation cost of t(li − y)2 when purchasing a product from Di . The consumers have
7

Although Marjit and Mukherjee (2008) show a similar result, namely that outsourcing creates the prisoner’s

dilemma, their deﬁnition of “outsourcing” diﬀers from ours. In that study, outsourcing is deﬁned as aninvestment for marginal cost reduction and thus involves no structural change in the supply chain.

5unit demands; that is, each consumer purchases either one or zero units of the product. Eachconsumer derives a surplus from consumption (gross of price and transportation costs) equal
to v + ei when purchasing a product from Di (i = 1, 2), where v is the beneﬁt from the basic
characteristics of each product, and ei is the product quality of Di . Let v be large enough suchthat every consumer gains positive utility from purchasing. Two upstream ﬁrms, UA and UB ,supply inputs to the downstream market with no production costs. For simplicity, we assume
that UA (resp. UB ) is exogenously ﬁxed at point −1/2 (resp. 1/2). When UA (resp. UB )supplies one unit of input to Di , UA (resp. UB ) incurs a transportation cost, τ (−1/2 − li )2(resp. τ (1/2 − li )2 ).8
We assume that each downstream ﬁrm can determine whether (i) it procures inputs fromeither upstream ﬁrm, Uj (j = A, B), at an input price, wij , and produces ﬁnal products withhigh quality (ei = αt(> 0) under outsourcing), or (ii) it produces inputs by itself at no cost,and produces ﬁnal products with low quality (ei = 0 under insourcing). The assumption captures a scenario in which vertical decentralization beneﬁts the decentralized downstream ﬁrm,
because it can concentrate on its retail activities and improve its product quality (Grossman
and Helpman, 2002).9
Following Liu and Tyagi (2011), we set the time line of the game as follows. In the ﬁrst
stage, the downstream ﬁrms simultaneously decide their sourcing patterns—either an outsourcing one or an insourcing one. If both downstream ﬁrms choose insourcing, the upstreamﬁrms are driven out of the market. In the second stage, the downstream ﬁrms simultaneously
decide their locations. In the third stage, if at least one downstream ﬁrm chooses outsourcing,each upstream ﬁrm, Uj , simultaneously oﬀers its input prices, wij ∈ [0, ∞), to the outsourcing
downstream ﬁrm(s), Di . The upstream ﬁrms engage in price competition for the downstream
business. If both downstream ﬁrms choose insourcing, the game skips to the last stage. In the8

Matsushima (2004, 2009) does not assume the quality advantage of a downstream ﬁrm even under out-

sourcing (a vertically separated downstream ﬁrm in his models).
9

In the context of outsourcing, Grossman and Helpman (2002) explain why decentralized downstream ﬁrms

have some advantages with regard to eﬃciency and/or quality.

6

last stage, the downstream ﬁrms set their prices pi ∈ [0, ∞) simultaneously, and the demandis created.

3

Results

Hereafter, we use the superscripts “ii,” “oo,” and “oi” to denote the three types of subgames respectively: “Insourcing-Insourcing,” “Outsourcing-Outsourcing,” and “OutsourcingInsourcing.”
Before deriving the equilibrium outcome in each subgame, we ﬁrst derive the marginal
consumer who is indiﬀerent between purchasing from D1 and D2 :10x(p1 , p2 , l1 , l2 , e1 , e2 ) =l 1 + l 2 p 2 − p 1 + e1 − e2+.
22t(l2 − l1 )(1)Thus, the demand faced by ﬁrm 1, Q1 , and that faced by ﬁrm 2, Q2 , are given by
Q1 (p1 , p2 , l1 , l2 , e1 , e2 ) = min{max(x(p1 , p2 , l1 , l2 , e1 , e2 ) + 1/2, 0), 1},Q2 (p1 , p2 , l1 , l2 , e1 , e2 ) = 1 − Q1 (p1 , p2 , l1 , l2 , e1 , e2 ).

(2)

We set the assumption seen below and deﬁne the following symbol to simplify the expositions of the outcomes in lemmas and propositions:
Assumption 1 τ is not large enough. Concretely,√µ ≡ τ /t < 3(4 2 − 5)/2 ≃ 0.985.
This inequality in Assumption 1 is needed for the existence of equilibrium in the subsequentdiscussion. Besides, we also need Assumption 2 to guarantee that each type of sourcing
outcome exists in equilibrium:
Assumption 2 The quality enhancement from outsourcing is not too large. Concretely,α < 0.129.
If α is too large, at least one of the downstream ﬁrms will choose outsourcing, and thus the“ii” type sourcing outcome fails to exist.
10Equation (1) is derived using equation v + e1 − t(l1 − x)2 − p1 = v + e2 − t(l2 − x)2 − p2 .

7

3.1

The Outsourcing-Insourcing Case

Without losing generality, we consider the case that D1 chooses outsourcing, while D2 chooses
insourcing.
In the last stage, the downstream ﬁrm i decides its price by solving the following optimization problem:
max π1oi (p1 , p2 , l1 , l2 , w1 ) ≡ (p1 − w1 )Q1 (p1 , p2 , l1 , l2 , αt, 0),(3)max π2oi (p1 , p2 , l1 , l2 ) ≡ p2 Q2 (p1 , p2 , l1 , l2 , αt, 0).

(4)

p1

p2

The equilibrium prices are as follows:poi
1 (l1 , l2 , w1 ) =poi
2 (l1 , l2 , w1 ) =2w1 + e + (l2 − l1 )(3 + l1 + l2 )t,3
w1 − e + (l2 − l1 )(3 − l1 − l2 )t
.
3

(5)
(6)

In the third stage, the upstream ﬁrms engage in price competition for only D1 . The
conditions in Assumption 1 and Assumption 2 guarantee that the location of D1 is always
on the left-hand side of the center (l1 < 0). Thus, U1 wins the Bertrand competition with awholesale pricew1oi = τ (1/2 − l1 )2 ,(7)

which is the transportation cost of U2 .
Substituting equations (5), (6), and (7) into equations (3) and (4), we obtain the maximization problems of the downstream ﬁrms in the second stage:[αt − τ (1/2 − l1 )2 + t(l2 − l1 )(3 + l1 + l2 )]2,
l1
18(l2 − l1 )t[τ (1/2 − l1 )2 − αt + t(l2 − l1 )(3 − l1 − l2 )]2max π2oi (l1 , l2 ) =.
l2
18(l2 − l1 )tmax π1oi (l1 , l2 ) =Solving the ﬁrst-order conditions, we derive the following lemma:

8

(8)
(9)

Lemma 1 When one downstream ﬁrm chooses outsourcing and the other one chooses insourcing, the following location pattern is an equilibrium outcome:−3 − 4µ + 3µ2 + H oi −3 + 5µ + H, l2 =,
6µ(1 + µ)
6µ√where H ≡ 9 − (3 − 12α)µ + (4 + 12α)µ2 .
l1oi =At the equilibrium outcome, each ﬁrm’s proﬁt is
π1oi =
π2oi =[−9 + (21 + 12α)µ + (14 + 12α)µ2 + (3 + 7µ)H]2 t,
243µ2 (1 + µ)(6 + 2µ + H)
[9 + (33 − 12α)µ + (4 − 12α)µ2 − (3 − 2µ)H]2 t.
243µ2 (1 + µ)(6 + 2µ + H)

(10)
(11)

The proﬁt when α = 0 and µ = 0 is π1oi |α=0,µ=0 = π2oi |α=0,µ=0 = 3/4.
A simple comparison between π1oi and π1oi leads to the following proposition:
Proposition 1 For any µ and α, π1oi > π2oi . That is, the proﬁt of the outsourcing downstreamﬁrm is relatively higher than that of the insourcing downstream ﬁrm.

Proof:The diﬀerence between π1oi and π2oi is given as()
24µ(1 + µ)α − 2(9 + 6µ − 5µ2 ) + (6 + 5µ)H toioi
π1 − π2 =27µ(1 + µ)()√
2−2(9 + 6µ − 5µ ) + (6 + 5µ) 9 − 3µ + 4µ2 t>> 0.
27µ(1 + µ)

The ﬁrst inequality is derived by setting α = 0 because ∂(π1oi − π2oi )/∂α > 0. The second()2√inequality is derived by the following comparison: (6 + 5µ) 9 − 3µ + 4µ2 − (2(9 + 6µ −5µ2 ))2 = 405(1 + µ)µ2 > 0.

Q.E.D.

The outsourcing ﬁrm has an incentive to move toward the center to reduce its input pricew1oi in (7). Anticipating the incentive of the outsourcing ﬁrm, the insourcing ﬁrm locates far
away from the center. In fact, |l2oi | > |l1oi | (see Lemma 1). Note that in standard Hotellingmodels, it is quite well-known that a downstream ﬁrm’s quality advantage is equivalent with
its cost advantage. Here, even though the quality adjusted marginal cost of the outsourcing
9ﬁrm, w1oi − αt, is higher than that of the insourcing ﬁrm, zero, the location advantage ofthe former ﬁrm dominates its cost disadvantage.11 As a result, the outsourcing ﬁrm earns a
relatively higher proﬁt than the insourcing ﬁrm. Notice that l2oi > 1/2, implying that D1 ’s
outsourcing forces D2 to locate out of the line segment. This fact follows from our assumptionthat downstream ﬁrms can locate along the whole line, not necessarily within the line segment.
As in Matsushima (2004, 2009), if downstream ﬁrms’ locations are restricted within the line
segment, D1 ’s outsourcing would then cause D2 ’s location at the end point, 1/2. If this is thecase, Proposition 1 holds only when µ < α. Because both downstream ﬁrms locate at the end
points, the outsourcing ﬁrm does not have location advantage over the insourcing one. Hence,
the outsourcing ﬁrm achieves a higher proﬁt only when its quality advantage is suﬃciently
high. In our case, Proposition 1 stems from a diﬀerent mechanism because it is independentof µ and α.
One might consider that outsourcing is better than insourcing when product positioning
matters because of the relative proﬁtability of outsourcing in Proposition 1. However, as
discussed in the following subsections, outsourcing can appear as a result of the prisoner’s
dilemma.

3.2

The Outsourcing-Outsourcing CaseIn the last stage, the downstream ﬁrm, i, decides its price by solving the following optimization
problem:
max πioo (p1 , p2 , l1 , l2 , w1 , w2 ) ≡ (pi − wi )Qi (p1 , p2 , l1 , l2 , αt, αt).
pi

(12)

The wholesale price, wi , is determined through price competition between the upstream ﬁrms,
wi = min{wiA , wiB }. The equilibrium prices are as follows:poo
1 (l1 , l2 , w1 , w2 ) =poo
2 (l1 , l2 , w1 , w2 ) =11(l2 − l1 )(3 + l1 + l2 )t + 2w1 + w2,3
(l2 − l1 )(3 − l1 − l2 )t + w1 + 2w2.
3

Because Proposition 1 holds for any positive α, w1oi − αt is larger than zero when α is quite small.

10

(13)
(14)

Substituting pi (l1 , l2 , w1 , w2 ) in (13) and (14) into Equation (12), we obtain
π1oo (l1 , l2 , w1 , w2 ) =π2oo (l1 , l2 , w1 , w2 ) =[(l2 − l1 )(3 + l1 + l2 )t − w1 + w2 ]2,18(l2 − l1 )t[(l2 − l1 )(3 − l1 − l2 )t + w1 − w2 ]2.
18(l2 − l1 )t

(15)
(16)

In the third stage, each upstream ﬁrm sets its price at the rival’s transportation cost if its
own cost is lower than its rival’s.12 The wholesale prices set by Uj are as follows:
1
1
1
1
oo
oo
w1j
= max{τ (− − l1 )2 , τ ( − l1 )2 }, w2j= max{τ (− − l2 )2 , τ ( − l2 )2 }.
2
2
2
2
Following the results of Matsushima (2004), we only consider a subgame in which l1 ≤ 0 andl2 ≥ 0. In the subgame, UA and UB supply to D1 and D2 respectively, and set their wholesale
prices at
1
1
oo
oo
w1A
= τ ( − l1 )2 , w2B= τ (− − l2 )2 .
2
2

(17)

Using Equations (15), (16), and (17), we obtain the maximization problems of the downstream ﬁrms in the second stage:[(l2 − l1 )(3 + l1 + l2 )t + (1 − l1 + l2 )(l1 + l2 )τ ]2,
l1
18(l2 − l1 )t[(l2 − l1 )(3 − l1 − l2 )t − (1 − l1 + l2 )(l1 + l2 )τ ]2.
max π2oo (l1 , l2 ) =l2
18(l2 − l1 )tmax π1oo (l1 , l2 ) =

(18)
(19)

The ﬁrst-order conditions lead to the following lemma:
Lemma 2 When both downstream ﬁrms choose outsourcing, the following location pattern is
an equilibrium outcome:l1oo = −3 − 2µ3 − 2µ, loo =.
4(1 + µ) 24(1 + µ)At the equilibrium outcome, each ﬁrm’s proﬁt is
πioo =

3 − 2µt.
4(1 + µ)Each ﬁrm’s proﬁt when µ = 0 is πioo |µ=0 = 3t/4. πioo is strictly decreasing in µ.
12

As noted by Matsushima (2004), it is also possible that when τ is large enough, an upstream ﬁrm optimallysets its wholesale price below its rival’s transportation cost. However, this situation does not appear within
the parameter ranges wherein our main propositions hold.

11

As discussed in the previous subsection, each outsourcing ﬁrm has an incentive to moveoo in (17)). The force is stronger when the
toward the center to reduce its input price (see wij

per unit transportation cost of each upstream ﬁrm, µ(= τ /t), is larger. In fact, lioo decreases
with µ, leading to tougher downstream competition. As a result, πioo monotonically decreases
with µ. Although outsourcing works as a credible driver to locate closer to the center, it canaccelerate downstream competition.

3.3

The Insourcing-Insourcing CaseIn the last stage, the downstream ﬁrm, i, decides its price by solving the following optimization
problem:
max πiii (p1 , p2 , l1 , l2 ) ≡ pi Qi (p1 , p2 , l1 , l2 ).
pi

(20)

The equilibrium prices are as follows:pii
1 (l1 , l2 ) =(l2 − l1 )(3 − l1 − l2 )t(l2 − l1 )(3 + l1 + l2 )t ii, p2 (l1 , l2 ) =
3
3

(21)

ii
Substituting pii
1 (l1 , l2 ) and p2 (l1 , l2 ) in (21) into Equation (20), we obtain the maximization

problems of the downstream ﬁrms in the second stage:(l2 − l1 )(3 + l1 + l2 )2 t,
l1
18
(l2 − l1 )(3 − l1 − l2 )2 t
max π2ii (l1 , l2 ) ≡.
l2
18max π1ii (l1 , l2 ) ≡

(22)
(23)

The ﬁrst-order conditions lead to the following lemma:
Lemma 3 When both downstream ﬁrms choose insourcing, the following location pattern is
an equilibrium outcome:3
3
l1ii = − , l2ii = .
4
4
At the equilibrium outcome, each ﬁrm’s proﬁt is
πiii =

12

3t.
4

This result is equivalent with that of Tabuchi and Thisse (1995).
From Lemmas 2 and 3, we have the following result:
Proposition 2 For any positive µ, πiii > πioo . That is, the proﬁts when both downstream
ﬁrms’ choose insourcing are strictly larger than when they select outsourcing.
Interestingly, although outsourcing provides relative advantages (including location andquality) to a downstream ﬁrm employing it, these advantages are completely cancelled out if
both downstream ﬁrms employ outsourcing. As a result, outsourcing can merely give rise to
the prisoner’s dilemma.

3.4

Sourcing Patterns in Equilibrium

In the ﬁrst stage, each downstream ﬁrm simultaneously decides the sourcing pattern. Although we need to derive the threshold values of µ in which πiii = π1oi and π2oi = πioo , wecannot explicitly derive these values of µ owing to the mathematical complexity. We therefore check these values numerically. Let µii (α) and µoo (α) be the threshold values of µ suchthat πiii = π1oi and π2oi = πioo respectively. Under the notation, we have the following relations:• πiii < π1oi if and only if µ < µii (α). That is, given that one of the two downstream ﬁrms
insources, the other prefers outsourcing if and only if µ < µii (α).
• π2oi < πioo if and only if µ < µoo (α). That is, given that one of the two downstream ﬁrms
outsources, the other prefers outsourcing if and only if µ < µii (α).
To secure the possibility that πiii ≥ π1oi , we impose the upper value of α at 0.129 as inAssumption 2. We set α at 0.01 intervals and check the threshold values of µ at each α. The
result is summarized in Table 1. The relation between the proﬁts and µ(= τ /t) is summarized
in Figure 1.
[Table 1 and Figure 1 about here]13

We ﬁnd that µii (α) > µoo (α) for any α.13 From the argument, we obtain the following
proposition.
Proposition 3 Under the parameter restrictions imposed above, the equilibrium sourcing pattern is summarized as follows:
1. Both downstream ﬁrms choose outsourcing if and only if µ ≤ µoo (α).
2. One downstream ﬁrm chooses outsourcing while the other chooses insourcing if and only
if µoo (α) ≤ µ ≤ µii (α).
3. Both downstream ﬁrms choose insourcing when µii (α) ≤ µ.
From Proposition 3, when the input procurement condition is loose (τ is small enough),both downstream manufacturers choosing outsourcing appears as an equilibrium outcome.
This ﬁnding is also supported by some empirical studies demonstrating that geographicalagglomeration reduces costs of input delivery and thus makes downstream manufacturers more
willing to outsource their input production (Holl, 2008; Belso-Martinez, 2010; Figueiredo etal., 2010).
From Propositions 2 and 3, we have the following corollary:
Corollary 1 When µ < µoo (α), although both downstream ﬁrms would be better oﬀ insourcing, both still choose outsourcing.

4

Extension

We have considered the situation in which outsourcing beneﬁts a downstream ﬁrm througha quality advantage, ei = αt. We now consider another possible beneﬁt of outsourcing—eliminating/reducing ﬁxed costs, F , necessary for input production. We can directly apply
the result in the previous section to the new situation by substituting ei = 0 and introducing
ﬁxed costs F for insourcing. We show the result of outsourcing decisions in graphical terms in
13

This relation holds true even if we set α at 0.001 intervals and check the threshold values of µ at each α.

14

Figure 2. The equilibrium property in the new situation is similar to that in the main modelin the sense that the higher the value of µ, the stronger the tendency to insource.
[Figure 2 about here]
As in the previous section, we set F at 0.01 intervals and check the threshold values of µ
at each F . The result is summarized in Table 2. The relation between the proﬁts and µ is
summarized in Figure 3.
[Table 2 and Figure 3 about here]
We ﬁnd that µii (F ) > µoo (F ) for any F , and there always exists the parameter range
wherein prisoner’s dilemma occurs.14 From the argument, we obtain the following proposition.
Proposition 4 Under the parameter restrictions imposed above, the equilibrium sourcing pattern is summarized as follows:
1. Both downstream ﬁrms choose outsourcing if and only if µ ≤ µoo (F ).
2. One downstream ﬁrm chooses outsourcing while the other chooses insourcing if and only
if µoo (F ) ≤ µ ≤ µii (F ).
3. Both downstream ﬁrms choose insourcing when µii (F ) ≤ µ.

5Concluding Remarks

In this study, we demonstrate the strategic perils of outsourcing. We consider a linear citywherein two upstream suppliers’ locations are exogenously given while two downstream manufacturers endogenously determine their respective location and sourcing pattern—outsourcing
or insourcing. We show that under outsourcing, although each downstream manufacturer will
choose the supplier whose inputs are more compatible, it tends to locate closer to the othersupplier for better outside options, and thus obtains input products at a lower wholesale
14This relation holds true even if we set F at 0.001 intervals and check the threshold values of µ at each F .

15price. This feature may create a prisoner’s dilemma: both downstream manufacturers choose
outsourcing and thus locate too close to each other, leading to overly intense downstream
competition.
For tractability, we assume that the ﬁnal products’ quality level under outsourcing isgiven exogenously. Even though endogenizing the quality level may make our model morecomprehensive, we ﬁnd the process to be too complicated technically. Because our results arerobust as long as the ﬁnal products’ quality takes positive values, we believe endogenizing thequality level would not lead to further contributions.

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Table 1: Threshold values of µµ\αµii (α)µoo (α)µ\αµii (α)µoo (α)

0.01
0.017
0.016
0.07
0.154
0.122

0.02
0.034
0.033
0.08
0.188
0.141

0.03
0.054
0.050
0.09
0.228
0.161

19

0.04
0.075
0.085
0.10
0.277
0.180

0.05
0.098
0.085
0.11
0.341
0.201

0.06
0.125
0.104
0.12
0.437
0.221

Table 2: Threshold values of µµ\Fµii (F )µoo (F )µ\Fµii (F )µoo (F )0.01
0.02480.02460.07
0.22180.20200.02
0.05130.05040.08
0.26750.23760.03
0.07990.07760.09
0.31960.275420

0.04
0.11070.10620.10
0.38060.31550.05
0.14420.13640.11
0.45450.35800.06
0.18100.16830.12
0.55000.4033

(a) The equilibrium proﬁts(b) The downstream ﬁrm’s equilibrium locations

Figure 1: The equilibrium outcomes (α = 1/12)

21

Figure 2: Equilibrium outcomes and equilibrium proﬁts(µ = 0.6 at the right-hand side ﬁgure)

22

πi

πi
πiii

πiii
π1oi

F

π1oi
π2oi

π2oi
πioo

πioo
Prisoner’s
dilemma

µ

µ
0

0

oo

oi
(F = 1/16)(F = 0)Figure 3: Equilibrium proﬁts (F = 0 and F = 1/16)

23

ii