Upstream Outsourcing, Economies of Scale, and
Incentives for Downstream R&D
Yutian Chen∗
Debapriya Sen†
May 6, 2010
Abstract
We investigate the impact of outsourcing upstream intermediate goods on the incentives for downstream cost-reducing R&D investment, in the context that upstream
production exhibits economies of scale and R&D investment entails spillover. Through
its impact on downstream competition, outsourcing imposes opposing distortions on
downstream R&D efforts. On the one side, it partly internalizes R&D externality
through efficiency improvement, which enhances R&D investment. On the other side,
it mitigates downstream competition, which can reduce R&D investment. We find
that, when firms compete in R&D, outsourcing enhances R&D investment only when
spillover is relatively large; whereas when firms cooperate in R&D by maximizing joint
profit, outsourcing always enhances R&D investment. Moreover, unless spillover is
small in R&D cooperation, upstream outsourcing combined with downstream R&D
cooperation leads to the largest R&D investment, the lowest price for final product,
and the highest profit for each firm.
Keywords: Outsourcing; Economies of Scale; R&D Spillover
JEL Classification: D41, L13, L14
∗
Department of Economics, California State University, Long Beach, CA 90840, USA. Email:
[email protected]
†
Department of Economics, Ryerson University, Toronto, Ontario M5B 2K3, Canada.
[email protected]
1
Email:
1
Introduction
Nowadays, outsourcing has attracted a huge volume of attention from public media as well as
research community. However, few research has been devoted to the impact of outsourcing on
outsourcing firms’ incentive for undertaking innovative activities, notwithstanding the public
attention on this issue.1 In this work, we investigate how the outsourcing of upstream goods
will affect outsourcing firms’ cost-reducing R&D expenditure for downstream production, in
a context where upstream production exhibits economies of scale.
Cost-saving is recognized as the first and foremost reason for outsourcing, which in many
cases is rooted in scale economies in upstream production/services.2 For example, economies
of scale play a large role in the business process outsourcing (BPO) arena, including shared
services centers and call centers. IT outsouricng by and large is because “a technologically
strong vendor specialized in IT offers attractive advantages including economies of scale”
(Rost (2006), p. 19). Many business outsource their payroll functions to specialized companies like Paychex Inc. because “a company that specializes in payroll processing can
probably do payroll processing at a much lower cost than a company that makes ball bearings” (Hira and Hira (2005), p. 72). More examples include the outsourcing of clinic trail
by pharmaceutical firms; and the outsourcing of enterprise resource planning.
Intuitively, economies of scale in upstream production can be best exploited when downstream producers congregate their demand of the intermediate good on a common upstream
provider. However, for downstream firms to benefit from economies of scale, the price they
pay for the intermediate good needs to drop when upstream production entails a lower average cost. In this work, we model the upstream market as constituting a monopolist, where
1
For example, there is concern that outsourcing may hamper innovation. Pisano and Shin (2009) argue
that, as a result of off-loading non-core activities to specialists aborad, “the U.S. has lost or is in the process
of losing the knowledge, skilled people, and supplier infrastructure needed to manufacture many of the
cutting-edge products it invented”, which could impede R&D efforts for core activities. Hira and Hira (2005,
page 117) argue that “While there is a consensus that the most recent growth in wages and the economy was
driven by technology—by technological innovation—the current exporting of jobs could well kill the seeds of
future innovation.”
2
“The importance of economies of scale is the most talked about aspect of outsourcing”(Outsourcing
Journal, May 2001. Source: http://www.outsourcing-journal.com/may2001-insights.html.) According to
Burkholder (2005), a top reason for firms to outsource is to take advantage of “an outside provider’s lower
cost structure, which may be the result of a greater economies of scale” (p. 52).
2
new entrants face zero entry cost.3 Such a perfectly contestable market forces the monopolist provider to charge its average cost for downstream firms and earns a zero profit. As
an implication, the larger the total quantity demanded by downstream firms, the lower the
upstream price set by the monopolist provider.
We first analyze an outsourcing game, where the upstream monopolist and n ≥ 2 downstream producers have the same technology in producing the intermediate good, for which
economies of scale exists. Each downstream firm chooses either to outsource the intermediate
good to the monopolist, or produce it in-house. Not surprisingly, we find that in equilibrium,
all downstream firms outsource so that they can better exploit scale economies. Outsourcing
leads to two consequences in downstream competition. First, each downstream firm enjoys
an efficiency gain from outsourcing, as the upstream price is lower than the average cost of
in-house production. Second, and more implicitly, each downstream firm also competes less
aggressively in the final-good market, because with outsourcing it faces a constant marginal
cost instead of a decreasing one as with in-house production.
After that, we incorporate downstream cost-reducing R&D decisions into the model in
order to investigate the impact of upstream outsourcing on downstream R&D investment.
In line with D’Aspremont and Jacquemin (1988) and Kamien et al. (1992), we assume that
each downstream firm chooses its R&D level before engaging in quantity competition for
the final good. Moreover, R&D entails spillover. The problem of our central interests is to
compare the equilibrium R&D investment in the case when all downstream firms produce
in-house the intermediate good, to the case when all of them outsource to the upstream
monopolist as predicted by the outsourcing game. We also consider two scenarios for R&D
environment: whether firms compete with each other in R&D, or they cooperate in R&D
by maximizing joint profit. The size of spillover is no smaller in R&D cooperation than in
R&D competition, since there can be voluntary knowledge transmission among innovating
firms when they cooperate in R&D.
We have three major findings. First, when firms compete in R&D, outsourcing enhances
downstream R&D investment only when spillover is large. The reason takes root in the effect
of outsourcing on downstream competition. On the one side, when spillover generates the
negative externality to outsourcing firms, efficiency gain of outsourcing partly internalizes
3
This assumption guarantees that upstream price decreases in downstream demand, without incurring
extra complexity which does not add insights to our model. The same result will arise if instead, there are
two identical upstream firms who produce homogeneous good and compete in prices.
3
such externality and increases individual incentive to undertake R&D. The reason is, a
decrease in competitors’ costs also help to increase the demand of upstream good hence
lowers the upstream price. On the other side, the mitigation of downstream competition
leads to mixed effects on R&D spending. While an individual firm tends to increases its R&D
investment when it faces softer competitors, it also tends to decrease its R&D investment as
itself is now a softer competitor. The aggregate effect of outsourcing on R&D expenditure
thus depends on the size of spillover. Only when spillover is large enough, the benefit from
efficiency gain and from facing softer competitors can dominate the disadvantage from itself
being a softer competitor, making R&D investment larger under outsourcing for each firm.
Second, when firms cooperate in R&D, outsourcing always enhances R&D investment for
each downstream firm. In R&D cooperation, each firm cares not only its individual profit but
also profits of its competitors. Outsourcing enhances R&D investment because it improves
total profit of downstream firms through two channels: the easy-to-see one is the efficiency
gain associated with outsourcing, when each firm faces a lower cost for the upstream good.
The second one is the mitigation of downstream competition by outsourcing. In oligopoly
competition, a more aggressive action taken by one firm improves its profit at the expense of
its competitors. The increase in its profit is less than the decrease in profit of all other firms,
leading to a lowered industry profit. Thus when outsourcing makes each firm compete less
aggressively, industry profit is larger. Therefore, when downstream firm cooperate in R&D,
they have larger incentive to invest in R&D under outsourcing.
Third, a comparison of all cases we consider shows that, upstream outsourcing combined
with downstream R&D cooperation yields each downstream firm the largest profit. Moreover,
unless spillover is small under R&D cooperation, such a strategy also produces the largest
R&D investment and the lowest price for the final good.
Our work is closed related to the literature on firms’ incentives for undertaking R&D
activities. D’Aspremont and Jacquemin (1988) and Kamien et al. (1992) focus on the effect
of R&D cooperation on R&D investments. They show that in the presence of R&D spillover,
individual incentive for R&D investment is less than the social optimal level, and R&D cooperation in certain circumstances can partly solve this problem. Barnerjee and Lin (2003) also
consider the downstream R&D incentive when downstream firms purchase from a upstream
monopolist for the intermediate good. However, in their model the monopolist increases its
price as a response to a larger downstream demand. Moreover, there is no spillover in R&D.
They find that, the supplier’s opportunistic behavior in pricing may hamper downstream
4
innovation. More related literature includes Milliou (2004), which investigates the impact of
R&D information flow between integrated and disintegrated firm on the incentives for R&D.
Theoretical literature on strategic outsourcing has been confined to its impacts on competition (see, e.g., Cachon and Harker (2002), Shy and Stenbacka (2003), Arya et al. (2008),
Chen et al. (2009)). Among them, Cachon and Harker also reveals the competition-softening
effect of outsourcing when production exhibits economies of scale. They show that, even
when upstream product is specified so that outsourcing can not generate efficiency gain, still
downstream firms have incentive to outsource in order to benefit from a mitigated downstream competition. Chen and Sen (2010) consider the role of scale economies in driving
outsourcing in a scenario when vertically integrated downstream firm also has incentive to
supply its rivals. They find that, even when upstream market constitutes a unique pure
provider, competition from the vertically integrated firm can again force the upstream price
to be at a low level, at which both the integrated firm as well as the its disintegrated rivals
outsource to the upstream pure provider.
The paper is organized as follows. Section 2 solves an outsourcing game when upstream
production is characterized by economies of scale, in order to gain insights on equilibrium
sourcing strategies of downstream firms. Section 3 incorporates downstream R&D decisions
into the outsourcing game, and compare equilibrium R&D investment of the four models
under consideration. Major findings are developed. Section 4 then concludes.
2
Model Setup
We posit an industry consisting of n ≥ 2 identical firms producing a homogeneous final good
P
α. The quantity firm i produces is qi , with Q ≡ i qi the total quantity produced for α. The
inverse demand function for good α is P = max{a − Q, 0} with a > 0. Let N = {1, ..., n}
the set of firms in the market α.
An intermediate good η is required in order to produce good α. For i ∈ N, firm i can
convert one unit of η into one unit of α at a constant average cost, initially given by m > 0.
Each firm i ∈ N can produce η by itself. There also exists a monopolist firm U which
produces only η. The upstream market is perfectly contestable, characterized by free entry
and exit. Thus firm U is vulnerable to “hit-and-run” strategy of potential entrants for any
subnormal profit it earns. As a result, in equilibrium firm U gets zero profit from providing
downstream firms with good η. We assume that economies of scale prevail in the production
5
of good η and technology is the same for each producer. For j ∈ {N, U }, firm j’s production
cost of η is given by

 bq − cq 2 for q ≤ b/2c
j
j
j
C(qj ) =
 b2 /4c
for qj > b/2c
Here b > 0, c > 0. For 0 < qj ≤ b/2c, firm j’s average production cost of η is AC(qj ) =
C(qj )/qj = b − cqj and its marginal cost is M C(qj ) = b − 2cqj , both are linearly decreasing
in qj . Moreover, M C(qj ) < AC(qj ). We make the following assumption:
b/2c > a > b + m > 0.
(1)
The first inequality guarantees that in equilibrium, production of good η must entail
positive marginal cost. Note that (1) implies
c < b/2a < 1/2.
It guarantees that the production cost for good η is not “too cancave” so that the existence
and uniqueness of Cournot equilibrium is guaranteed.
For i ∈ N, each of firm i has two alternative sources for η: either outsources η exclusively
to firm U or produces η exclusively in-house.4 Before we investigate the impact of outsourcing
on incentives for downstream firms to invest in R&D, we shall first analyze an outsourcing
game which does not incorporate the R&D spending by each downstream firm. Solving the
outsourcing game serves mainly two goals: first, we show that in the unique SPNE, every
downstream firm outsources η to firm U at a price equivalent to U ’s average production
cost. It confirms that the problem of our major interests, namely, the impact of outsourcing
on downstream firms’ R&D investments, is relevant. Second, the mechanism for upstream
outsourcing to affect downstream R&D efforts is imbedded in the impact of outsourcing
on downstream competition. Analyzing the outsourcing game offers foundations for us to
understand findings from comparisons of R&D efforts in the presence and in the absence of
upstream outsourcing.
2.1
An Outsourcing Game without Downstream R&D Investments
In this part, we look for equilibrium sourcing mode for good η of each downstream firm by
solving an outsourcing game. Denote the sourcing mode of firm i ∈ N by δi , with δi = 1
4
For simplicity, we assume the exclusivity in firm i, i ∈ N’s sourcing mode. In fact, exclusivity will
endogenously arise due to firm i’s incentive to pursue economies of scale.
6
if firm i outsources η to U and δi = 0 if i produces η in-house. According to each firm’s
sourcing mode, the set of N = {1, ..., n} firms is partitioned into two subsets: the set K of k
firms who outsource η to U and its complement N/K of n − k firms who produce η in-house.
Consider a three-stage game, denoted as game G:
Stage one. Firm U announces its price w of good η.
Stage two. For i ∈ N, firm i simultaneously chooses sourcing mode δi of good η.
Stage three. For i ∈ N, firm i simultaneously chooses quantity qi for good α.
Solution concept to Game G is Subgame Perfect Nash Equilibrium in pure strategy
(SPNE). Since the upstream market is perfectly contestable, in equilibrium the price of U
must be its average production cost whenever it supplies positive quantity, so that profit of
U from supplying downstream firms is zero.5 For i ∈ N, firm i’s profit at the terminal node
of game G is
πi (w, (δ1 , q1 ), ..., (δn , qn )) =

 P (Q)qi − wqi
if δi = 1
 P (Q)q − C(q ) if δ = 0
i
i
i
At given w and k, there exists a unique Cournot equilibrium in stage three. If all
downstream firms are producing positive quantities in the market α, for i ∈ N, firm i’s
Cournot quantity is

(a − m)(1 − 2c) + (n − k)b − (n − k + 1 − 2c)w


if δi = 1
 q O (w, k) =
n − 2ck + 1 − 2c
qi (w, δi , k) =

a − m − b(k + 1) + wk

 q I (w, k) =
if δi = 0
n − 2ck + 1 − 2c
Since w < b must hold for any downstream firm to outsource to U , it is guaranteed that
q O (w, k) > 0 in equilibrium. Moreover, q O (w, k) is strictly decreasing in w. On the other
side, q I (w, k) > 0 requires w > b −
a−m−b
.
k
πi (w, δi , k) =
Firm i’s Cournot profit is

 [q O (w, k)]2
if δi = 1
 [q I (w, k)]2 (1 − c) if δ = 0
i
Suppose k > 1 in stage two, then total quantity outsourced to U is kq O (w, k). There exists
a unique w which solves w = AC(kq O (w, k)) = b − ckq O (w, k), given by
w(k) =
5
1
{b(n + 1 − 2c) − ck[a(1 − 2c) + b(n − k + 2)]},
H
A formal proof is given in the proof to Theorem 1 below. This finding is consistent to the necessary
conditions for a “sustainable equilibrium” to a perfectly contestable monopolist. See Proposition 2B6 in
page 28, Baumol et al. (1982).
7
where H ≡ n + 1 − 2c + (2c2 − cn − 3c)k + ck 2 . Consider two extreme cases:
¥ All downstream firms produce good η in-house, i.e., k = 0. The upstream price w
becomes irrelevant. For i ∈ N, firm i’s Cournot quantity and profit are
q I ≡ q I (w, 0) =
¥
a−b−m
,
n + 1 − 2c
π I ≡ π I (w, 0) = [q I ]2 (1 − c).
All downstream firms outsource, i.e., k = n. At given w, for i ∈ N, firm i’s Cournot
quantity and profit are
q O (w, n) =
a−w−m
,
n+1
π O (w, n) = [q O (w, n)]2 .
There exists a unique w which equals to the average cost of U when n downstream firms
outsource to U . I.e., w = AC(nq O (w, n)) = b − cnq O (w, n) is solved as
w ≡ w(n) =
b(n + 1) − nc(a − m)
> 0.
n + 1 − nc
It is verifiable that w is the lowest price of η for U to get a non-negative profit through
producing the demand of downstream firms.6 At w = w, the Cournot quantity and profit of
each firm i ∈ N are
q O ≡ q O (w, n) =
a−b−m
,
n + 1 − nc
π O ≡ π O (w, n) = [q O (w)]2 .
The following theorem summarizes our major finding to game G. Since entry pressure forces
firm U to charge the lowest price at which it is willing to produce the quantity demanded by
6
To see this, first, suppose all downstream firms are active in the market α. In any equilibrium where
k ≥ 1 firms outsource, the lowest price of U for a non-negative profit is w = w(k). At w(k), Cournot quantity
of firm i, i ∈ N is

1

 q O (w(k), k) = (1 − 2c)(a − b) if
H
qi (w(k), δi , k) =

 q I (w(k), k) = 1 (1 − ck)(a − b) if
H
For q I (w(k), k) > 0, it must be 1 − ck > 0. Then
dw(k)
dk
=
−1
H c(1
δi = 1
δi = 0
− 2c)(a − b)(n + 1 − 2c − ck 2 ) < 0. The
minimum value of w(k) is given by w(n) = w. Second, suppose only the set of K outsourcing firms produce
positive quantity while firms in N/K produce zero quantity. Cournot quantity of firm i is

a−w

 q̃ O (w, k) =
if δi = 1
k+1
q̃i (w, δi , k) =

 q̃ I (w, k) = 0
if δi = 0
The lowest value of w for a non-negative profit for U is solved from AC(k q̃ O (w, k)) = w as w̃(k) =
b(k+1)−ack
k+1−kc ,
which is decreasing in k. The minimum of w̃(k) is w̃(n) = w. Thus w is the lowest price of U at which it
gets non-negative profit by producing the demand of downstream firms.
8
downstream firms, outsourcing to this common provider U allows downstream firms to best
exploit economies of scale. The existence of an SPNE where all downstream firms outsource
to U is not surprising. Moreover, it is the unique SPNE to game G.
Theorem 1 There exists a unique SPNE to game G, where U sets w = w for good η, then
each of firm i, i ∈ N outsources good η to U . In the market α, qi = q O , πi = π O .
Proof: See the Appendix.
Whenever there exists an upstream market of η, economies of scale induce all downstream
firms to outsource to a common upstream provider. Instead, if the upstream market does
not exist, then downstream firms have to rely on in-house production for η. From now on,
unless particularly specified, the term “outsourcing” or “in-house production” refers to the
sourcing mode for all downstream firms N as either they all outsource η to U , or they all
produce η in-house.
To understand the impact of outsourcing on downstream competition, we compare the
average cost and quantity of each downstream firm in the presence and in the absence of
upstream outsourcing. With in-house production, each firm faces a constant average cost of
η given by AC(q I ). It is verifiable that AC(q I ) > w. Outsourcing has an explicit effect: it
improves production efficiency for downstream firms.
Proposition 1 AC(q I ) > w. As a result, q O ≥ q I where equality holds only at n = 2; and
πO > πI .
In addition, there is an implicit effect of outsourcing on downstream competition. Outsourcing converts a decreasing marginal cost into a constant one for each downstream firm,
leading each to compete less aggressively in the market α. It holds that
q O (w, n)|w=AC(qI ) < q I , π O (w, n)|w=AC(qI ) > π I .
That is, if each downstream firm is equally efficient under outsourcing and in-house production, it produces more under in-house production. The reason is, firms produce according
to marginal cost instead of average cost. Under outsourcing, firm i, i ∈ N faces a constant marginal cost of η given by the upstream price. Instead, under in-house production,
economies of scale imply a decreasing marginal cost of η, which gives rise to an incentive
for firm i to further expand its quantity in the market α. Outsourcing softens downstream
9
competition by diminishing such an incentive of quantity expansion, which again contributes
to the larger profit each downstream firm acquires under outsourcing.
We then move on to consider models which incorporate the R&D expenditure xi by each
firm i, i ∈ N, where xi works to reduce firm i’s marginal cost of α initially given by m > 0.
3
Models with Downstream R&D Investments
To investigate the impact of outsourcing on downstream R&D, we impose an R&D stage
at the beginning of game G, where each of firm i ∈ N simultaneously chooses its R&D
expenditure xi ≥ 0 for production technology of α. These decisions determine each firm’s
unit cost of α. The magnitude of unit cost reduction realized by firm i ∈ N in good α
is f (Xi ), where f is an R&D production function, and Xi is firm i ∈ N’s effective R&D
investment, which is determined by the combined individual R&D expenditure of all firms
N and a spillover parameter. We consider two scenarios in the R&D stage: either firms N
compete by investing x1 , ..., xn to maximize individual profit, or they collaborate in R&D
decision to maximize joint profit. In the former case, the spillover is due to involuntary
knowledge leakage measured by β ∈ [0, 1]. Firm i’s effective R&D investment is
Xi = xi + β
X
xj
i ∈ N.
(2)
j6=i
Instead, if they cooperate in R&D stage, the spillover can include voluntary knowledge
exchange among downstream firms and is no less than β, characterized by βc ∈ [β, 1]. Thus
each firm i ∈ N’s effective R&D investment in R&D cooperation is
Xi = xi + βc
X
xj
i ∈ N.
(3)
j6=i
Unit cost of firm i ∈ N in converting good η into good α is m − f (Xi ). We make the
following assumptions.
Assumption 1: f (X) is twice differentiable, f (0) = 0, f (X) ≤ m, f 0 (X) > 0, f 00 (X) < 0 for
all X ≥ 0.
Assumption 2: f (X) satisfies (i) limX→∞ f (X) =
(a−b−m)(1−2c)
n−1
and (ii) f 0 (0) >
(n+1−2c)2
.
2(a−b−m)(1−c)
Assumption 2 is needed to guarantee that all participating firms find it optimal to be
active at the game’s production stage (condition (i)) and to invest in R&D in the R&D
10
investment stage (condition (ii)). Note that the concavity of f in Assumption 1 and condition
(i) in Assumption 2 imply
lim f 0 (X) = 0.
X→∞
This property guarantees existence of equilibria in which R&D investment decisions are
bounded from above. For the uniqueness of symmetric equilibria, we need assumption on the
unimodality properties of the profit functions. Consider the case of a monopoly. Its quantity
is a function of its R&D investment X, defined as q M ≡
a−b−m+f (X)
.
2(1−c)
The corresponding
monopoly profit net of R&D investment is π M − X, given by
[a − b − m + f (X)]2
− X.
4(1 − c)
Assumption 3: π M −X is a strictly concave function for X ≥ 0, or equivalently, its derivative,
given by the function [a − b − m + f (X)]f 0 (X), decreases in X.
Given any symmetric cost structure achieved by firms N in the R&D stage, game G
ensues. To investigate the impact of upstream outsourcing on downstream R&D, we compare
the scenario when downstream firms outsource to firm U to the scenario when they produce
η in-house, given that they may compete or cooperate in the R&D stage. In particular, we
consider four models as described below.
Case NI: R&D competition and in-house production. Firms N compete in the R&D stage.
After that, each produces good η in-house, then compete in the market α.
Case NO: R&D competition and outsourcing. Firms N compete in the R&D stage. After
that, each outsources good η to U at price w, then compete in the market α.
Case CI: R&D cooperation and in-house production. Firms N cooperate in the R&D
stage by maximizing joint profit. After that, each produces good η in-house, then compete
in the market α.
Case CO: R&D cooperation and outsourcing. Firms N cooperate in the R&D stage by
maximizing joint profit. After that, each outsources good η to U at price w, then compete
in the market α.
Given the assumed sourcing mode for η of firms N and the R&D environment as either
they cooperate or compete in R&D, each model is reduced to be including two stages: the
R&D stage, where each i ∈ N simultaneously chooses R&D investment xi ; followed by the
production stage, where each i ∈ N simultaneously chooses quantity qi for α. For each
11
model, we solve for the symmetric equilibrium R&D expenditure. We do backward induction
starting from the production stage for equilibrium quantity, then moving back to the R&D
stage for equilibrium R&D expenditure.
We shall first analyze Case NI and Case NO to gain insights on the effect of upstream
outsourcing on downstream R&D spending when firms compete with each other in R&D
decisions. After that, we analyze Case CI and Case CO to see the impact of outsourcing
when firms collaborate in R&D. When all four models are solved, we compare among them
the level of R&D investment, the market price of α, and the profit of each downstream firms.
3.1
Downstream R&D Competition
In this subsection, we solve for the equilibrium R&D expenditure under in-house production
and outsourcing of good η given that downstream firms compete in the R&D stage. A
comparison between these two cases is given by the end of this subsection.
3.1.1
Case NI: R&D Competition and In-house Production
We starts from the case when downstream firms produce η in-house. Let q = {q1 , ..., qn }. In
the production stage, for i ∈ N, firm i’s profit is given by
πi (x1 , ..., xn ; q) = P (Q)qi − C(qi ) − [m − f (Xi )]qi − xi .
By maximizing πi , the Cournot quantity for each of firm i ∈ N is
P
a − b − m (n + 1 − 2c)f (Xi ) − i f (Xi )
qi (x1 , ..., xn ) =
+
,
n + 1 − 2c
(n + 1 − 2c)(1 − 2c)
(4)
and firm i ∈ N’s Cournot profit is
πi (x1 , ..., xn ) = [qi (x1 , ..., xn )]2 (1 − c) − xi .
(5)
In the R&D stage, each of firm i ∈ N simultaneously solves
max πi (x1 , ..., xn ).
xi
The first order necessary condition is
∂πi
2qi (1 − c)
=
{(n − 2c)f 0 (Xi ) −βΣj6=i f 0 (Xj )} − 1 = 0, i, j ∈ N.
{z
}|
{z
}
∂xi
(n + 1 − 2c)(1 − 2c) |
direct effect
12
externality
(6)
The first term in braces stands for the positive direct effect of firm i’s R&D investment on
its profit through reducing its own cost of good α. The second term in braces stands for
the negative externality firm i’s R&D investment has on its profit through reducing its rival
firms’ cost of good α. Under Assumption 3, this negative externality acts to reduce R&D
investment. We consider symmetric solutions only, that is, xi = xN I , Xi = X N I for all i ∈ N:
∂πi
n − 2c − β(n − 1)
}f 0 (X) − 1 = 0, i, j ∈ N.
= 2qi (1 − c){
∂xi
(n + 1 − 2c)(1 − 2c)
Inserting (4) and reorganizing the equation gives
[a − b − m + f (X N I )]f 0 (X N I ) =
(n + 1 − 2c)2 (1 − 2c)
.
2(1 − c)[n − 2c − (n − 1)β]
(7)
At the symmetric SPNE, quantity and profit of each firm are
qiN I =
We have
a − b − m + f (X N I )
, πiN I = [qiN I ]2 (1 − c) − xN I ,
n + 1 − 2c
i∈N
(8)
∂πiN I
[2(1 − c)β − 1]f 0 (X N I )
= 2(1 − c)
, j ∈ N, j 6= i
∂xj
(n + 1 − 2c)(1 − 2c)
1
. In Kamien et al.
2(1−c)
1
β is 21 . Notice that 2(1−c)
which is positive if and only if β >
(1992) where production cost of
α is linear, the threshold value of
> 12 . The presence of economies
of scale makes it harder for one firm to benefit from other firms’ R&D expenditure. When
upstream production exhibits economies of scale, each firm competes more aggressively in
the downstream market. If firm j invests in R&D, the benefit conferred by economies of
scale on firm i, i 6= j by making firm i more aggressive upon its cost reduction, is not big
enough to cover its loss when it faces a more efficient and aggressive competitor firm j. As
a result, it becomes harder for firm i to get a higher profit due to firm j’s R&D activity.
3.1.2
R&D Competition and Outsourcing (Case NO)
We now consider the case when firms N outsource to firm U for good η, given that they
compete in R&D stage. In the upstream market, the equilibrium price w of good η is
determined by firm U ’s average production cost. We first solve for the Cournot quantities
in the production stage at a given w.
In the production stage, for i ∈ N, firm i’s profit is
πi (x1 , ..., xn ; w; q) = P (Q)qi − wqi − [m − f (Xi )]qi − xi .
13
For i ∈ N, i’s Cournot equilibrium quantity is
a − w − m + nf (Xi ) −
qi (x1 , ..., xn ; w) =
n+1
P
j6=i
f (Xj )
.
The equilibrium price w is solved from w = AC(Q(x1 , ..., xn ; w)) = b − c
P
i qi (x1 , ..., xn ; w)
as
P
b(n + 1) − nc(a − m) − c i f (Xi )
w=
, i ∈ N.
n + 1 − nc
which is decreasing in Xi . By inserting w into qi (x1 , ..., xn ; w), each of firm i ∈ N produces
P
(a − b − m) − (1 − c) i f (Xi )
qi (x1 , ..., xn ; w) =
+ f (Xi ),
(9)
n + 1 − nc
and its profit is
πi (x1 , ..., xn ; w) = [qi (x1 , ..., xn )]2 − xi .
(10)
In the R&D stage, each of firm i ∈ N simultaneously solves
max πi (x1 , ..., xn ; w).
xi
The first order necessary condition is
∂πi
∂qi (w) ∂qi (w) ∂w
= 2qi [
+
]−1
∂xi
∂xi
∂w ∂xi
P
P
β j6=i f 0 (Xj ) cf 0 (Xi ) + cβ j6=i f 0 (Xj )
nf 0 (Xi )
−
+
} − 1.
= 2qi {
n
+
1
n
+
1
(n
+
1)(n
+
1
−
nc)
| {z } |
{z
} |
{z
}
direct effect
externality
(11)
upstream efficiency gain
Three terms are in braces. The first term stands for the direct effect of firm i’s R&D
investment on its profit through reducing its average cost of good α, which is positive. The
second term is the negative externality of firm i’s R&D investment on its profit through
reducing competitors’ average cost of good α. The last term is positive. It stands for the
upstream efficiency gain when firm i’s R&D investment decreases w, the upstream price of
good η. There are two channels for such efficiency gain: one is due to the increase in firm
i’s demand of η when it is more efficient in producing α, reflected by cf 0 (Xi ); the other is
due to the increase in all other firms’ demand of η when i’s R&D spillover reduces their cost
P
of α, reflected by cβ j6=i f 0 (Xj ). Thus under outsourcing, the externality of firm i’s R&D
can be partly internalized through its upstream efficiency gain.
Consider symmetric solutions in R&D investment, i.e. Xi = X N O for all i ∈ N. (11)
gives
2(a − b − m + f (X N O )) 0 N O
f (X )[n(1 − c) + c − β(n − 1)(1 − c)] = 1,
(n + 1 − nc)2
14
rewritten as
[a − b − m + f (X N O )]f 0 (X N O ) =
(n + 1 − nc)2
.
2[n(1 − c) + c − β(n − 1)(1 − c)]
(12)
At the symmetric SPNE, quantity and profit of each firm are
qiN O =
We have
a − b − m + f (X N O )
O
, πiN O = [qiN O ]2 − xN
i ,
n + 1 − nc
i ∈ N.
(13)
[(2 − c)β − (1 − c)]f 0 (X N O )
∂πiN O
=2
, j ∈ N, j 6= i
∂xj
n + 1 − nc
which is positive if and only if β >
1−c
.
2−c
Note that
1−c
2−c
< 12 . Although the existence of
economies of scale makes it harder for one firm to benefit from other firms’ R&D investment,
outsourcing over offsets this effect. First, outsourcing converts a decreasing marginal cost
into a constant one for downstream firms, thus eliminates the extra incentive for them to
compete harshly. Second, outsourcing also converts the R&D efforts of any downstream firm
into a lower upstream price, which benefits all of the downstream firms. Therefore, under
outsourcing, it becomes easier for one firm to benefit from other firms’ R&D expenditure.
3.1.3
Comparing Case NI and Case NO
Comparing the equilibrium R&D level in Case NI and Case NO shows that, outsourcing
enhances R&D investment when R&D spillover is relatively severe.
1
Lemma 1 X N O T X N I ⇔ β T β̄, with β̄ ∈ [ 2(1−c)
, 1) defined by
β̄ ≡1 −
(1 − 2c)[3n2 (1 − c) + nc(nc − 2) + 4c − 3]
.
(1 − c)(n − 1)(n2 c − 8nc + 6n + 8c2 − 12c + 6)
Proof: Since rhs((8)) T rhs((13)) ⇔ β T β̄, by Assumption 3, the result follows.
Note that if c = 0, we have X N O = X N I . Under our assumption that firm U is equally
efficient as downstream firms in producing good η, outsourcing has no impact on downstream
R&D investment when there is no scale economies. Moreover,
1
, limn→∞
2(1−c)
dβ̄
dn
> 0 for n > 2, β̄(n = 2) =
β̄ = 1. The more competitive the downstream market, the harder it is for
outsourcing to improve downstream R&D investment.
The intuition of Lemma 1 can be illustrated through the comparison of (6) and (11). First,
(1−c)(n−2c)
(n+1−2c)(1−2c)
>
n
,
n+1
implying a smaller positive direct effect of R&D under outsourcing.
Outsourcing converts a decreasing marginal cost of firm i into a constant one, hence makes
15
firm i less aggressive in the market α. Because firm i benefits less upon the same level
of cost reduction, it has less incentive to invest in R&D. We call this as the incentivehampering effect of outsourcing. Second,
β(n−1)
n+1
<
β(n−1)(1−c)
,
(n+1−2c)(1−2c)
implying a smaller negative
externality of R&D under outsourcing. The reason is, outsourcing also changes the cost
structure of firm i’s competitors, making them compete less aggressively in the market
α. Upon the same cost reduction for competitors due to spillover, firm i is disadvantaged
less. Therefore, it has a stronger incentive to invest in R&D. We call this the spillovermitigating effect of outsourcing. Finally, the third term in (11) is absent in (6), implying
that the upstream efficiency gain is an extra benefit from i’s R&D expenditure conferred by
outsourcing. Channelled through outsourcing, firm i’s R&D investment reduces upstream
price of good η, leaving firm i a stronger incentive to invest in R&D. We call this the
efficiency-improving effect of outsourcing. Note that the first effect is independent of β,
whereas the second and third effects are strengthened by a larger β. Lemma 1 then shows
that, for spillover big enough, the aggregate of the second and third effects dominates the
first effect, leading to an enhanced R&D investment.
3.2
Downstream R&D Cooperation
We consider Case CI and Case CO where firms N cooperate in the R&D stage. R&D
cooperation differs from R&D competition in two aspects. First, each firm i ∈ N maximizes
joint profits when choosing xi in the R&D stage; second, spillover is βc ∈ [β, 1]. If βc = 1,
firms fully share their technique knowledge; if βc = β, there is no voluntary knowledge share
so that spillover is kept the same as in R&D competition.
3.2.1
R&D Cooperation and In-house Production (Case CI)
We first consider the case when firms N produce η in-house. In the production stage, equilibrium quantity and profit are given by (4) and (5) as in Case NI. In the R&D stage, firm
i, i ∈ N chooses xi to maximize
T (x1 , ..., xn ) ≡
X
πj (x1 , ..., xn ).
j∈N
The first order necessary condition is
∂T
∂πi X ∂πj
=
+
= 0,
∂xi
∂xi
∂xi
j6=i
16
where
X
∂πj
dqj
2(1 − c)qj
= 2(1 − c)qj
=
f 0 (Xv )}.
{(n − 2c)βc f 0 (Xj ) −f 0 (Xi ) − βc
{z
}
∂xi
dxi
(1 − 2c)(n + 1 − 2c) |
v6=i,j
externality
|
{z
}
competitive effect
The first part in the braces is positive, standing for the beneficial externality of firm i’s R&D
investment on firm j’s profit due to spillover. The second part is negative, standing for the
competitive effect of firm i’s R&D investment on firm j when all firms other than firm j are
more efficient in good α. Assuming symmetry, the first order necessary condition gives
2qi (1 − c){
1 + (n − 1)βc 0 CI
}f (X ) − 1 = 0.
n + 1 − 2c
(14)
By inserting qi (x, ..., x), it is rewritten as
[a − b − m + f (X CI )]f 0 (X CI ) =
(n + 1 − 2c)2
.
2(1 − c)[1 + βc (n − 1)]
(15)
At the symmetric equilibrium, quantity and profit of each firm are
qiCI =
Moreover,
a − b − m + f (X CI )
, πiCI = [qiCI ]2 (1 − c) − xCI
i ,
n + 1 − 2c
X ∂πjCI
j6=i
∂xi
=
i ∈ N.
(16)
2(1 − c)(n − 1)qj
{2(1 − c)βc − 1}f 0 (X CI ).
(1 − 2c)(n + 1 − 2c)
Firm j’s profit increases in i’s R&D expenditure if and only if βc >
1
.
2(1−c)
When spillover
is sufficiently large, the beneficial externality dominates the competitive effect, so that firm
i’s R&D expenditure benefits firm j.
3.2.2
R&D Cooperation and Outsourcing (Case CO)
We then consider the case when downstream firms cooperate in R&D stage and outsource
η to firm U . In the production stage, equilibrium quantity and profit are given by (9) and
(10) as in Case NO. In the R&D stage, each of i ∈ N chooses xi to maximize
T (x1 , ..., xn ; w) ≡
X
πj (x1 , ..., xn ; w) =
j∈N
X
X
[qj (x1 , ..., xn ; w)]2 −
xj .
j∈N
The first order necessary condition of joint profit maximization is
∂πi X ∂πj
∂T
=
+
= 0,
∂xi
∂xi
∂xi
j6=i
17
j∈N
where
dqj (w)
∂πj
∂qj (w) ∂qj (w) dw
= 2qj
= 2qj {
+
}
∂xi
dxi
∂xi
∂w dxi
P
P
nβc f 0 (Xj ) f 0 (Xi ) + βc v6=i,j f 0 (Xv ) cf 0 (Xi ) + cβc j6=i f 0 (Xj )
= 2qj {
−
+
}.
+ 1 }|
n{z+ 1
+ 1 − nc)
| n {z
} | (n + 1)(n{z
}
externality
competitive effect
upstream efficiency gain
In braces, the first term is the positive externality of firm i’s R&D investment on firm j due
to spillover. The second term is the negative competitive effect of firm i’s R&D investment
on firm j through improving j’s competitors’ efficiency in good α. The last term stands for
j’s upstream efficiency gain when i’s R&D investment reduces upstream price of good η.
Same as in Case CO, there are two channels for such efficiency gain: one is through firm i’s
enhanced demand of η when its own R&D reduces its cost; the other is through all other
firms’ enhanced demand of η when i’s R&D reduces their costs.
Assuming symmetry in R&D investment, the first order necessary condition of joint profit
maximization is
2qi {
1
(n − 1)βc
n[c + c(n − 1)βc ]
+
+
}f 0 (X) − 1 = 0.
n+1
n+1
(n + 1)(n + 1 − nc)
(17)
Inserting qi (x, ..., x) and reorganizing the equation gives
[a − b − m + f (X CO )]f 0 (X CO ) =
(n + 1 − nc)2
.
2[1 + βc (n − 1)]
(18)
The equilibrium quantity and profit of each firm are
qiCO =
Moreover,
a − b − m + f (X CO )
, πiCO = [qiCO ]2 − xCO
i ,
n + 1 − nc
i ∈ N.
(19)
∂πjCO
[(2 − c)βc − (1 − c)] 0 CO
= 2(n − 1)qj
f (X ),
∂xi
n + 1 − nc
which is positive if and only if βc >
1−c
.
2−c
For spillover large enough, externality and upstream
efficiency gain dominate the competitive effect, leading firm j a larger profit when firm i
invests in R&D.
3.2.3
Comparing Case CI and Case CO
Lemma 2 compares equilibrium R&D expenditure in Case CI and Case CO. It shows that,
outsourcing always enhances downstream R&D investment when firms cooperate in R&D.
18
Lemma 2 X CO > X CI .
Proof: It follows (15), (18) and Assumption 3.
In R&D cooperation, each firm takes into account the externality its R&D confers on all
of its competitors when deciding its R&D investment. As a result, the second term in braces
of both (14) and (17) is positive. Notice that both the first and second terms are larger in (17)
since
1
n+1
>
1−c
.
n+1−2c
The reason is, the competition softening effect of outsourcing benefits
the downstream industry, hence enhances R&D spending when there is R&D cooperation.
Recall that in Cournot competition, it is the incentive of each firm to expand quantity at
the expense of other firms which leads to a low equilibrium profit of each firm. Therefore,
when outsourcing softens downstream competition, total profit in the market of α is larger.
As a result, each firm has a stronger incentive to invest in R&D when it cares not only own
profits but also rivals’ profits. Moreover, notice that the upstream efficiency gain of R&D
investment, given by the third term in (17), is absent in (14). The total effect is, outsourcing
always enhances R&D investment when firms cooperate in the R&D stage.
3.3
Comparison of Four Cases
The following lemma shows that, with or without outsourcing, downstream R&D cooperation
enhances incentives for R&D investment if spillover under R&D cooperation is not too small.
Lemma 3
(I) X CI T X N I ⇔ βc T
1−β
;
1−2c
(II) X CO T X N O ⇔ βc T (1 − c)(1 − β).
Proof: (I) follows (7) and (15); (II) follows (12) and (18).
If βc = β, i.e., there is only involuntary knowledge leakage, R&D cooperation enhances
incentives for R&D investment if β >
1
2(1−c)
under in-house production, and β >
outsourcing. These are also the conditions for
∂πj
∂xi
1−c
2−c
under
> 0, when R&D expenditure benefits rival
firms. Note that if c = 0, both X CO > X N O and X CI > X N I hold if and only if β > 12 ,
the same result as in D’Aspremont and Jacquemin (1988) and Kamien et al. (1992). The
more general result in Lemma 3 sheds new light on how R&D cooperation enhances R&D
expenditure. It shows that, the larger the value of β, the larger the range of βc where R&D
cooperation has a positively effect on R&D spending. Intuitively, a larger β in R&D competition makes each firm to be more inclined to free ride on others’ R&D investment, which
19
hampers incentives to do R&D. Instead, a larger βc in R&D cooperation saves more of the
duplication in R&D efforts, which enhances incentives to do R&D. Therefore, a larger value
of β supports a larger range of βc at which R&D cooperation enhances R&D investment.7
Define
Z(β) ≡
(1 − c)[n − 2c − β(n − 1)](n + 1 − nc)2
1
−
.
2
(n − 1)(1 − 2c)(n + 1 − 2c)
n−1
Remark: Z(β) strictly decreases in β, with Z(β) <
1−β
,
1−2c
Z(β) T (1 − β)(1 − c) ⇔ β S β̄.
1
1−c
, 2(1−c)
).8
Moreover, Z(β̂) = β̂, with β̂ ∈ ( 2−c
Lemma 4 X CO T X N I ⇔ βc T Z(β).
From a comparison on the equilibrium effective R&D investment of all four models, our
major finding is given in the following proposition. (A more comprehensive result is given
by Proposition 2A in the Appendix.)
Proposition 2
I. If βc > Z(β), then X CO > X ξ , ξ = N O, CI, N I.
II. If βc < Z(β), then X N I > X ξ , ξ = N O, CI, CO.
Proof: I. By Lemma 2 and Lemma 4, X CO > X CI , X CO > X N I . By Lemma 3 and the fact
(a) both Z(β) and (1 − β)(1 − c) strictly decrease in β; (b) Z(β) > (1 − β)(1 − c) for β < β̄
(see the remark) and (c) β̄ > β̂, we conclude that X CO > X N O .
II. By Lemma 2 and Lemma 4, X N I > X CO > X CI . By Lemma 3 and the fact Z(β̄) < β̄,
it holds that X N I > X N O .
In Figure 1, the shadowed area gives the range of parameters where equilibrium R&D
expenditure in Case CO exceeds what it is in all other cases.
7
Interestingly, notice that in Lemma 3, (1 − c)(1 − β) is decreasing in c whereas
1−β
1−2c
is increasing in
c. The size of economies of scale imposes opposite distortions on the range of βc for R&D cooperation to
improve R&D investment with and without upstream outsourcing. With outsourcing, a larger size of scale
economies works for a larger upstream efficiency gain to the whole industry. As such efficiency gain is taken
into account when firms cooperate in R&D, there is a larger range of βc where R&D cooperation improves
R&D investment. Instead, with in-house production, a larger size of scale economies works only for a harsher
downstream competition. Because
∂πj
∂xi
> 0 only when βc >
1
2(1−c)
with the right-hand-side increasing in c,
larger economies of scale make it harder for one firm’s R&D spending to benefit other firms. As a result, the
range of βc , where &D cooperation improves R&D investment, will shrink when scale economies are more
significant.
8
β̂ ≡
(1+n−4c−8c3 −3nc+4n2 c+4nc3 −n2 +10c2 +3n3 c+6n2 c3 −2n2 c4 −8n2 c2 −3n3 c2 +n3 c3 −n3 )
.
(n−1)(5n2 c−3n2 c2 −2n2 +n2 c3 −4n−10nc2 +12nc−2−12c2 +7c+8c3 )
20
Figure 1: Z(β)
Proposition 3 A sufficient condition for pCO < pξ , ξ = N O, CI, N I is βc > Z(β).
Proof: By n ≥ 2 and (8), (13), (16), (19), it is clear that when X CO > X ξ , it holds that
pCO < pξ , ξ = N O, CI, N I. The proposition then follows Proposition 2.
Proposition 4 π CO ≥ π ξ , ξ = N O, CI, N I; π CI ≥ π N I .
Proof: Case CO vs. Case NO. In case CO it is feasible, although not necessarily optimal,
that each of i ∈ N sets xi =
XNO
,
1+(n−1)βc
such that xi + (n − 1)βc xi = X N O . Thus
a − b − m + f (X N O ) 2
XNO
) −
n + 1 − nc
1 + (n − 1)βc
NO
a − b − m + f (X ) 2
XNO
≥(
) −
n + 1 − nc
1 + (n − 1)β
π CO ≥ (
= πN O
21
Case CO vs. Case CI. In case CO it is feasible, although not necessarily optimal, that
each of i ∈ N sets xi =
X CI
,
1+(n−1)βc
such that xi + (n − 1)βc xi = X CI . Thus
a − b − m + f (X CI ) 2
X CI
) −
n + 1 − nc
1 + (n − 1)βc
CI
a − b − m + f (X ) 2
X CI
) −
≥(
n + 1 − 2c
1 + (n − 1)βc
CI
a − b − m + f (X ) 2
X CI
>(
) (1 − c) −
n + 1 − 2c
1 + (n − 1)βc
π CO ≥ (
= π CI
Case CO vs. Case NI. In case CO it is feasible, although not necessarily optimal, that
each of i ∈ N sets xi =
XNI
,
1+(n−1)βc
such that xi + (n − 1)βc xi = X N I . Thus
a − b − m + f (X N I ) 2
XNI
) −
n + 1 − nc
1 + (n − 1)βc
NI
a − b − m + f (X ) 2
XNO
≥(
) −
n + 1 − 2c
1 + (n − 1)β
NI
a − b − m + f (X ) 2
XNO
>(
) (1 − c) −
n + 1 − 2c
1 + (n − 1)β
π CO ≥ (
= πN O
Case CI vs. Case NI. In case CI it is feasible, although not necessarily optimal, that each
of i ∈ N sets xi =
XNI
,
1+(n−1)βc
such that xi + (n − 1)βc xi = X N I . Thus
a − b − m + f (X N I ) 2
XNI
) (1 − c) − n
n + 1 − 2c
1 + (n − 1)βc
NI
XNI
a − b − m + f (X ) 2
≥ n[(
) (1 − c) −
]
n + 1 − 2c
1 + (n − 1)β
nπ CI ≥ n(
= nπ N I
Theorem 2 Among all the cases considered, upstream outsourcing combined with downstream R&D cooperation (Case CO) leads to the highest equilibrium profit. Moreover, unless
technique spillover under R&D cooperation is small (i.e. βc < Z(β)), Case CO dominates
other cases in that it also yields the largest R&D investment and the lowest price for the
final product.
22
4
Conclusion
In this work, we investigate the impact of outsourcing upstream good on the incentives
for outsourcing firms to invest in downstream cost-reducing R&D, in a circumstance where
outsourcing is driven by economies of scale in upstream production. A critical property we
find from analyzing a pure outsourcing game is that, the upstream price is decreasing in the
quantity demanded from downstream firms.
Literature reveals that when R&D entails spillover, R&D investment is downward distorted from social optimal level because of the appropriability problem in R&D investment.
Here we show that, outsourcing of the upstream intermediate good partly solves this problem. When one firm invests in downstream R&D, the spillover helps its competitors to
also reduce downstream production cost. As a result, the total downstream production is
boosted, which in turn enhances the demand of the upstream good and reduces its price.
Through such an upstream efficiency gain, the R&D externality to the innovating firm is
partly internalized, meaning that each firm can possess a larger incentive to invest in R&D.
At the same time, the effect of outsourcing in mitigating downstream competition leads
to rather mixed effect on incentives for innovation. On the one side, when all other firms
compete less aggressively in the final good market, each individual firm benefits more from
a reduced cost. Therefore, incentives to invest in R&D become larger. On the other side,
when each individual firm competes less aggressively in the final good market, each benefits
less from a reduced cost and will have a smaller incentive to invest in R&D.
Which effect can dominate then depends on the size of spillover, and on whether firms
cooperate or compete in R&D. If firms compete in R&D decisions, the spillover needs to be
large enough for outsourcing to improve downstream R&D investment. However, if firms
cooperate in R&D decisions by maximizing joint profit, then outsourcing always improves
downstream R&D investment regardless of the size of spillover. A comparison of all cases
under consideration then reveals that, R&D cooperation for downstream production combined with upstream outsourcing (Case CO) yields the largest profit for each downstream
firm. Moreover, as long as the spillover under R&D cooperation is not too small, Case CO
generates the largest R&D investment as well as the lowest price for the final product.
23
Appendix
Proof of Theorem 1.
First, w.l.o.g., given that w = w for η and all firms i, i ∈ N, i 6= 1 outsource to U , we show
that in stage two firm 1 will not deviate from outsourcing to producing in-house. When all
a−b−m 2
) > 0. Suppose firm 1 deviates to in-house
n firms outsource, each gets profit π O = ( n+1−nc
production in stage two. Then in stage three each of firm i ∈ N, i 6= 1 produces qiO (w, n−1) =
(a−m−b)(n+1−nc−2c)
(a−m−b)(n+1−n2 c)
whereas firm 1 produces q1I (w, n − 1) = (n+1−nc)(n+1−2nc)
.
(n+1−nc)(n+1−2nc)
. If c ≥ n+1
, q1I (w, n − 1) = 0. Firm 1
condition for q1I (w, n − 1) > 0 is c < n+1
n2
n2
and is worse off. If c <
n+1
,
n2
Note that the
gets zero profit
it is verifiable that q O (w, n)−q1I (w, n−1) ≥ 0, with equality holds
only at n = 2. Firm 1’s profit under deviation is π1I (w, n − 1) = [q1I (w, n − 1)]2 (1 − c) < π O .
It is again worse off. Therefore, firm 1 will not deviate to in-house production.
Second, given that w = w, we show that it must be k = n in SPNE. Suppose k ≤ n − 1 in
equilibrium. If quantity produced by each of j ∈ N/K is zero, then firm j ∈ N/K will deviate
from in-house production to outsourcing because by doing so, firm j produces qjO (w, k + 1) =
(n+1)(a−b−m)
(n+1−cn)(k+2)
> 0 and gets profit πjO (w, k+1) = [qiO (w, k+1)]2 > 0. If all n firms are producing
positive quantities, then each of j ∈ N/K produces qjI (w, k) =
[n+1−nc−ckn](a−b−m)
(n+1−nc)(n−2ck+1−2c)
and gets
profit πjI (w, k) = [qjI (w, k)]2 (1 − c). Notice that qjI (w, k) > 0 requires nck < n + 1 − nc.
Let firm j ∈ N/K deviates from in-house production to outsourcing. It can be verified that
c(n+1)(n−2)(n−2c−2ck)(a−b−m)
(n+1−nc)(n−2ck+1−2c)(n+1−4c−2ck)
1) > πjI (w, k), firm j will again
qjO (w, k + 1) − qjI (w, k) =
≥ 0, where equality holds only at
n = 2. Since πjO (w, k +
deviate to outsourcing. We prove
that any k ≤ n − 1 can not be in equilibrium.
Third, we show that U will not deviate to either w < w or w > w. If w < w, all
downstream firms will continue to order from U . Because AC(nq O (w, n)) is strictly increasing
in w and
dAC(nq O (w,n))
dw
=
nc
<
n+1
O
1, AC(nq O (w, n)) intersects the 45◦ line from above. Since
AC(nq O (w, n)) = w, AC(nq (w, n)) > w for w < w. Firm U gets negative profit and will
not deviate. Instead, if w > w, then there is a positive-profit entry plan for a potential
entrant to the market η: by undercutting w, it can get the demand of all downstream firms
and a positive profit, leaving U zero profit. Thus U has no incentive to deviate either. This
finishes the proof that the strategy stated in the theorem constitutes an SPNE.
Last, we show that the SPNE is unique. It is clear that w > w can not be in SPNE since
it will be undercut by a potential entrant. It then remains to show that there does not exist
a price w0 of η lower than w, such that it is no smaller than the average cost of producing
24
the output demanded at that price. As long as it is true, there is no space for a potential
entrant to undercut w for a non-negative profit. Based on the proof in the second step, if
w0 < w, all n downstream firms will outsource to the entrant. It is sufficient to show that
there does not exist w0 such that w0 < w and w0 ≥ AC(nq O (w0 , n)). It immediately follows
the fact that AC(nq O (w, n)) > w for w < w.
Proposition 2A
I. If β > β̂, then X CO > X δ , δ = N O, CI, N I. Moreover, the following hold.
(a) X N O > X N I if and only if β > β̄.
1
,
2(1−c)
1−β
.
1−2c
(b) If β >
βc >
1
then X CI > X N I . If β ∈ (β̂, 2(1−c)
), then X CI > X N I if and only if
II. If β < β̂, then X CO > X δ , δ = N O, CI, N I if and only if βc > Z(β); X N I > X δ ,
δ = N O, CI, CO if and only if βc < Z(β). Moreover, the following hold.
(a) X N I > X N O .
(b) X CI > X N I if and only if βc >
1−β
.
1−2c
, β̂), then X CO > X N O . If β <
(c) If β ∈ ( 1−c
2−c
1−c
,
2−c
then X CO > X N O if and only if
βc > (1 − c)(1 − β).
Proof: First, we compare X CO to equilibrium effective R&D in all the other cases. By (18)
and (15), it holds that X CO > X CI . Comparing (18) and (12) shows that X CO > X N O if and
only if βc > (1−c)(1−β). For β >
1−c
,
2−c
it holds that (1−c)(1−β) < β, so that this condition is
automatically satisfied. Thus this condition is relevant only for β <
1−c
.
2−c
Note that
1−c
2−c
< β̂.
Finally, comparing (18) and (7) shows that X CO > X N O if and only if βc > Z(β). Since
Z(β) > β if and only if β < β̂, it follows that for β > β̂, βc > Z(β) is automatically
satisfied, which guarantees that X CO > X N I ; yet for β < β̂, there exists range of β such
that βc < Z(β) and X CO < X N I . Furthermore, it is true that Z(β) > (1 − c)(1 − β) for
β < β̄. Therefore, for β > β̂, or β < β̂ and βc > Z(β), we have X CO > X δ , δ = N O, CI, N I.
Second, we compare X CI to X N I . By (15) and (7), it holds that X CI > X N I if and only
if βc > (1 − β)/(1 − 2c). Since (1 − β)/(1 − 2c) < β if and only if β <
is relevant only when β <
1
.
2(1−c)
Note that β̂ <
25
1
2(1−c)
< β̄.
1
,
2(1−c)
this condition
Third, we compare X N O and X N I . By (12) and (7), we have that X N O > X N I if and
only if β > β̄. Note that β̄ > β̂. Then for β < β̂, it is always true that X N I > X N O . In
fact, when β < β̂ and βc < Z(β), it holds that X N I > X N O , X N I > X CO > X CI .
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