MANAGEMENT SCIENCE
Vol. 51, No. 1, January 2005, pp. 133–150
issn 0025-1909 eissn 1526-5501 05 5101 0133
inf
orms
®
doi 10.1287/mnsc.1040.0212
© 2005 INFORMS
Sell the Plant? The Impact of Contract
Manufacturing on Innovation, Capacity,
and Profitability
Erica L. Plambeck
Terry A. Taylor
I
n the electronics industry and others, original equipment manufacturers (OEMs) are selling their production
facilities to contract manufacturers (CMs). The CMs achieve high capacity utilization through pooling (sup-
plying many different OEMs). Meanwhile, the OEMs focus on innovation: research and development, product
design, and marketing. We examine how this change in industry structure affects investment in innovation and
capacity, and thus profitability. In particular, innovation is noncontractible, so OEMs will invest less in inno-
vation than is ideal for the industry as a whole. Hence, although contract manufacturing improves capacity
utilization, it may reduce the profitability of the industry as a whole by weakening the incentives for innovation.
Contract manufacturing is not the only means to achieve capacity pooling. Alternatively, the OEMs can pool
capacity with one another through supply contracts or a joint venture. This may result in underinvestment or
overinvestment in innovation and capacity, but always increases profitability. We find that the sale of production
facilities to a CM improves profitability for the industry as a whole if and only if OEMs are subsequently in a
strong bargaining position vis-à-vis the CM. If the OEMs are indeed very strong, the gain from pooling capacity
via contract manufacturing is maximized in industries with moderate cost of capacity.
Key words: biform games; contract manufacturing; capacity pooling
History: Accepted by Fangruo Chen and Stefanos A. Zenios, special issue editors; received September 30, 2001.
This paper was with the authors 9 months for 2 revisions.
1. Introduction
The use of contract manufacturing is important and
growing in a range of industries, including electron-
ics, pharmaceuticals, automotive, and food and bev-
erage production (Tully 1994). Increasingly, firms that
traditionally manufactured their own products are
outsourcing production and focusing instead on prod-
uct design, development, and marketing. In the elec-
tronics industry, contract manufacturing grew at a
compounded average growth rate of roughly 25%
from 1989 to 1998. Contract manufacturing’s share
of total electronics production grew from 9% in 1994
to 17% in 1998 (Francois 1999). Original equipment
manufacturers (OEMs) outsourced $75 billion to con-
tract manufacturers (CMs) in 2000, representing 10%
of total electronics production. The bulk of electron-
ics outsourcing is in computers (personal comput-
ers, workstations, peripherals) and communications
(network and telecommunication equipment). Despite
pessimism about the near term, the long-term growth
rate for electronics contract manufacturing is pro-
jected to be 25% per year (Boase 2001). In pharmaceu-
ticals, outsourcing accounted for 20% of production in
1988. In 1998, the figure was 50%–60%, and it is pro-
jected to grow to 60%–70% by 2005 (van Arnum 2000).
Historically, firms have done both innovation
(research and development, product design, and mar-
keting) and production in-house. Under such vertical
integration, because each firm fills its demand from
its own production capacity, inefficiency in the use of
capacity can result. For example, the pharmaceutical
industry is characterized by long development cycles
(roughly 12 years) and intense time-to-market pres-
sure. Consequently, a company that wants to man-
ufacture its own product must make a large capital
investment in a plant before the drug has completed
regulatory trials. If the drug fails, the plant may have
little value (Tully 1994). The prospect of inefficient
capacity use has implications for how firms invest in
innovation and production capacity.
In industries where production asset specificity is
low, contract manufacturing offers the potential for
improved capacity utilization because CMs can pool
demand from a diverse set of OEMs. A common
belief is that contract manufacturing increases the
rate of innovation by reducing the cost of production
133
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
134 Management Science 51(1), pp. 133–150, © 2005 INFORMS
capacity and allowing OEMs to focus their financial
and managerial resources on product development
and marketing. Surprisingly, we find that outsourcing
may reduce innovation. We characterize the sensitivity
of capacity and innovation to industry structure and
the bargaining confidence (i.e., the anticipated bar-
gaining strength) of OEMs.
The first step in our analysis is to compare the tra-
ditional firm model with a pooled model in which
firms share their production capacity and act to maxi-
mize total system profit. The effect of such ideal pool-
ing on innovation depends on how innovation affects
demand: If innovation increases the potential market
size, then pooling increases innovation; if innovation
increases the probability that a product is successful,
then pooling can either increase or reduce innovation.
Second, we show how innovation and capacity invest-
ments deviate from their ideal levels when pooling is
achieved by outsourcing to a CM. The basic problem
is that OEMs will invest in innovation only what they
expect to recoup by negotiating a favorable supply
contract with the CM.
We evaluate the effect of outsourcing production to
a CM on the profitability of the firms. Morgan Stanley
reports that in the electronics industry, the set of OEM
assets that are candidates for divestiture is substantial,
with potential for more than $10 billion per year in
sales (Fleck and Craig 2001). Managers of such assets
must decide whether to sell the plant and outsource
production or keep manufacturing in-house. While
outsourcing manufacturing can increase profit by
improving capacity utilization, it may instead reduce
profit by weakening the incentives for innovation.
For an OEM, an alternative to outsourcing produc-
tion to a CM is to retain production and pool capac-
ity with other OEMs through supply contracts or a
joint venture. Although it has received less attention
in the business press, outsourcing among OEMs is
widespread. In electronics, the OEM outsourcing mar-
ket, at $115 billion in 2000, is over 50% larger than the
CM outsourcing market (Boase 2001). For example,
in addition to manufacturing its own computer prod-
ucts, Taiwan’s Acer manufactures for Compaq and
IBM. Pharmaceutical companies use excess capacity
to manufacture for competitors (Tully 1994). AMD
and Fujitsu have been highly successful in their joint
venture to produce flash memory chips (Mahon and
Decker 2001). We prove that OEM outsourcing can
result in excessive innovation relative to the ideal
pooled case. If the bargaining power of an OEM vis-à-
vis the CM is large, then total system profit is greater
under CM outsourcing than OEM outsourcing. How-
ever, if OEMs anticipate being in a weak bargaining
position vis-à-vis the CM, the OEMs would do bet-
ter to outsource among themselves than to sell their
production facilities to a CM.
Literature Survey
While OEMs may contract for capacity, they are un-
likely to make these contracts contingent on their early
investments in innovation or on the resulting market
conditions. In the economics literature on incomplete
contracts, Klein et al. (1978) and Williamson (1979)
were the first to argue that if a buyer and a sup-
plier cannot write complete, contingent contracts and
if the values of their assets depend on collaboration,
then they will make inefficient investments. Gross-
man and Hart (1986) investigate how changes in asset
ownership affect the incentives for investment. In this
spirit, our paper shows how industry structure—who
owns the production facilities—influences the incen-
tives for investment in innovation and capacity. Hart
and Moore (1990) further illustrate how ownership
influences incentives for employees as well as owner-
managers. An implication is that consolidating our
OEMs and CM into a single firm would not necessar-
ily achieve the optimal levels of innovation and capac-
ity because of difficulties in providing incentives to
employees within a firm.
The economics literature on incomplete contracts
is based on cooperative game theory. However, the
use of cooperative game theory to analyze problems
in operations and supply chain management is very
new. Several papers assume that capacity/inventory
is noncontractible and examine how firms invest in
capacity and subsequently bargain cooperatively over
its use. In contrast to the literature (all the papers ref-
erenced below assume that demand is exogenous), we
assume that demand is endogenous, and that demand-
stimulating innovation is noncontractible. Anupindi
et al. (2001) and Granot and Sosic (2003) consider
a network of retailers with independent stochas-
tic demands: Each chooses his inventory level; then
demand is realized; and the retailers bargain coop-
eratively over the transshipment of excess inventory
to meet excess demand. Anupindi et al. (2001) pro-
pose an allocation mechanism (a rule for shipping
inventory and sharing the gains from trade) that is in
the core and creates incentives for retailers to choose
system-optimal inventory levels in the initial stage.
Granot and Sosic (2003) point out that this allocation
mechanism will fail to achieve the first best if retail-
ers can hold back some of their residual inventory.
They propose two alternative allocation mechanisms
that induce the retailers to share their residual inven-
tory efficiently, but may not be in the core. In general,
there are no core allocation rules that induce efficient
sharing of residual inventories. Van Mieghem (1999)
considers a setting in which one OEM can purchase
capacity from a second OEM after demand occurs.
To induce first-best capacity investments, the OEMs
must write contracts with payments contingent on the
state of demand. In Chod and Rudi (2003), two OEMs
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
Management Science 51(1), pp. 133–150, © 2005 INFORMS 135
invest in capacity, update their demand forecasts, and
then trade capacity. With demand uncertainty formu-
lated as a multiplicative shock to the price, the poten-
tial for trade increases the initial capacity investment.
A simplifying assumption in our analysis is that
the OEMs do not compete for customers. Other
researchers (Parlar 1988, Lippman and McCardle
1997, Mahajan and van Ryzin 2001) consider the
inventory-stocking decisions made by competitive
retailers. A common insight is that this competition
causes the retailers to carry excess stock. Anupindi
and Bassok (1999) and Müller et al. (2002) consider
the case where retailers instead cooperate in ordering
and pooling inventories. Lee and Whang (2002) and
Rudi et al. (2001) consider settings in which retailers
trade stock after experiencing demand.
Another key assumption in our analysis, and in the
aforementioned OM and economics papers, is that
all parties have common information. Spulber (1993)
considers a monopolist manufacturer that must allo-
cate its scarce capacity across a set of buyers, but
does not know their individual or aggregate demand
precisely. He derives optimal pricing schemes for
the manufacturer, and proves that information asym-
metry results in underinvestment in capacity by
the manufacturer and inefficient allocation among
the buyers. Cachon and Lariviere (1999) also con-
sider capacity choice and allocation under informa-
tion asymmetry, assuming a constant wholesale price.
Tunca and Mendelson (2001) model multiple suppli-
ers and OEMs that contract for supply and then, after
obtaining private demand information, trade in an
exchange.
Recent economics and industrial organization
papers on outsourcing are surveyed by Sturgeon
(2002) and Mookherjee (2003).
This paper is organized as follows. Section 2 evalu-
ates the impact of pooling capacity. Section 3 explores
the setting where OEMs outsource production to a
CM. Section 4 explores the issues in the previous two
sections using a distinct innovation-demand model.
Section 5 provides concluding remarks.
2. The Impact of Ideal Pooling on
Investment and Profit
This section compares the traditional model in which
each firm builds its own capacity to meet its own
demand with a pooled model in which firms share
their production capacity and chose their investments
to maximize joint profits. This ideal pooled scenario
provides a benchmark for the system with contract
manufacturing considered in §3.
Consider an OEM that is developing a new prod-
uct. Assume that the price per unit when q units are
sold is M − q With probability e, the product is suc-
cessful and M = H; otherwise, M = L where L ≤ H.
The OEM must invest in production capacity c at
a cost of k>0 per unit before the demand for the
new product is realized. After the capacity cost is
incurred, the marginal cost of production is, for sim-
plicity, assumed to be negligible. The expected opera-
tional profit for the OEM is
0
= max
c≥0
eH − cc + 1 − e max
q∈0c
L − qq− kc
(1)
Padmanabhan and Png (1997) employ a similar
demand model, where e H and L are exogenous. We
extend this model to consider the role of investment
in innovation to stimulate demand.
Early investments by the OEM in innovation (mar-
ket research and product development) may influence
demand through either the potential market size H
or the success probability e For example, in the phar-
maceutical industry, basic research and clinical trials
aimed at expanding a drug’s therapeutic range (i.e.,
the range of medical conditions for which the drug is
proven to be efficacious) increase the potential market
size. Drug development investments aimed at increas-
ing the probability that a drug passes clinical trials
increase the success probability. For any specific drug
candidate, innovation investments may be directed
primarily at one of these two objectives.
We consider the case where innovation influences
the potential market size first and examine the case
where innovation influences the success probability
in §4. Assume that the total cost of innovation fH
is an increasing function of the market size H, twice
differentiable and convex, and satisfies fH→ as
H →
H<. The OEM chooses a market size that
maximizes his total expected profit, which is given by
V
0
= max
H∈L
H
0
H − f H (2)
Consider two identical OEMs that pool their pro-
duction capacity. The maximum expected profit that
they can achieve jointly is
V
P
= max
H
1
H
2
∈L
H
P
H
1
H
2
− fH
1
− fH
2
(3)
where
P
H
1
H
2
= max
c≥0
R
P
c H
1
H
2
− 2kc (4)
R
P
c H
1
H
2
= e
2
max
q
1
q
2
≥0
q
1
+q
2
≤2c
H
1
− q
1
q
1
+ H
2
− q
2
q
2
+ e1 − e
i=12
max
q
H
q
L
≥0
q
H
+q
L
≤2c
H
i
− q
H
q
H
+ L − q
L
q
L
+ 1 − e
2
2 max
q∈0c
L − qq
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
136 Management Science 51(1), pp. 133–150, © 2005 INFORMS
Here, c denotes the production capacity per OEM,
and H
i
is the potential market size for OEM i, for i =
1 2. Henceforth, we assume the OEMs should exert
some effort to innovate, i.e., that the optimal solution
to (3) satisfies H
P
i
>L; i = 1 2
Subsequent sections evaluate the effect of pooling
on the optimal levels of capacity, market size, suc-
cess probability, and expected profit. Throughout, the
superscripts P and 0 indicate pooling or the absence
of it. For example, the optimal capacity investment
per OEM in the system with pooling is denoted c
P
,
whereas optimal capacity for the isolated OEM is c
0
.
The profit function
0
H is convex, so problem (2)
may fail to have a unique optimal solution. Similarly,
problem (3) may fail to have a unique solution. Fur-
thermore, an optimal solution to problem (3) is not
necessarily symmetric because
2
/H
1
H
2
P
H
1
H
2
takes both positive and negative values. However,
imposing a restriction on the innovation cost func-
tion ensures uniqueness and symmetry. In particular,
assume for the remainder of this paper that
d
2
fH
dH
2
>
e
2
(5)
for H ∈ L
H In many contexts, investments in inno-
vation (research and development, product design,
marketing) yield diminishing returns in terms of their
impact on the market size (equivalently, the marginal
cost of innovation is increasing). Condition (5) simply
requires that the marginal cost of effort be increasing
sufficiently rapidly. While this need not hold in every
context, focusing on the case of unique, symmet-
ric solutions greatly facilitates analytical comparisons.
The following proposition compares the solutions to
problems (2) and (3). Define
H = L + 2k/e
2
Proposition 1. The optimization problem (2) has a
unique solution H
0
. The optimization problem (3) has a
unique solution, and it is symmetric:
H
P
1
= H
P
2
= H
P
Furthermore,
H
0
≤ H
P
where the inequality is strict if and only if
H
0
<
H (6)
Thus, innovation is larger when capacity is pooled.
Two different insights underlie the proof of this result.
First, for any fixed level of capacity, an increase
in market size (either H or L)ismore beneficial
when capacity is pooled. Therefore, pooling drives
the OEMs to invest more in innovation. Second, we
have made a structural assumption that innovation
increases not only the expected market size, but also
its variability. Variability is highly detrimental to the
isolated OEM. However, in the pooled scenario, the
OEMs are better equipped to handle variability, so
they exert more innovation effort. If L = 0 then the
decision of whether to set H>0 can be interpreted as
a market entry decision. One scenario that may occur
is H
P
>H
0
= 0 That is, by improving the efficiency
with which capacity is used, pooling stimulates entry
into a market that would otherwise be unserved.
Although pooling increases innovation, pooling
may either increase or decrease the level of capac-
ity. The proof of this result relies on the observation
that when OEMs are faced with the same poten-
tial market sizes, capacity in the pooled case takes
on more moderate values. To see this, consider the
case where H (and consequently the optimal capac-
ity) is very large. Then, the incremental value of addi-
tional capacity is larger for the isolated OEM because
the probability that capacity will be used is greater
(e>e
2
). In contrast, if H (and consequently the opti-
mal capacity) is small, then the incremental value
of additional capacity is greater in the pooled case
because the probability that capacity will be used is
greater (e
2
+ 2e1 − e>e). The precise result is that
if H ≤
H then c
P
H H ≥ c
0
H
if H>
H then c
P
H H < c
0
H
where
H = maxL + k/e3k/2e This result, in con-
junction with Proposition 1, yields:
Proposition 2.
If H
0
< H then c
0
<c
P
if H
0
>
H then c
0
>c
P
An implication of Proposition 2 is that if innova-
tion and capacity are costly (cheap), then pooling
increases (reduces) the optimal capacity. For exam-
ple, innovation and capacity may be relatively costly
for pharmaceutical companies, especially firms devel-
oping highly novel drugs (e.g., biopharmaceuticals).
Propositions 1 and 2 suggest that in this context, com-
panies that develop and produce their own drugs
underinvest (relative to the pooled ideal) in both inno-
vation and capacity. In recent years the pharmaceuti-
cal industry has been marked by considerable merger
activity. Mergers of drug companies may not achieve
the ideal pooled outcome due to incentive problems
within the combined firm. However, to the extent
that such problems are overcome and that mergers do
result in the ideal pooled outcome, our results sug-
gest that mergers may lead drug companies to make
larger investments in innovation and capacity.
Finally, we are concerned with the effect of pooling
on expected profit for the system as a whole. Define
the gain from pooling as
=
P
H
P
H
P
− 2fH
P
− 2
0
H
0
− fH
0
(7)
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
Management Science 51(1), pp. 133–150, © 2005 INFORMS 137
Clearly, the opportunity for pooling cannot reduce
expected profit for the integrated system, so must
be positive. However, the magnitude of is very sen-
sitive to the cost of capacity k and the success proba-
bility e. One might expect to be strictly increasing in
k because pooling allows the OEMs to utilize capacity
more efficiently. However, by repeated application of
the envelope theorem, we have
k
= 2c
0
H
0
− c
P
H
P
H
P
Combining this with Proposition 2 yields
k
> 0ifH
0
>
H
< 0ifH
0
< H
Surprisingly, the gain from pooling is maximized at
an intermediate value of the cost of capacity k.Ifk is
small, then H
0
is large and capacity is smaller in the
pooled case; hence, the gain from pooling is increas-
ing in the capacity cost. Conversely, if k is large, then
H
0
is small, and the gain from pooling is decreasing
in the capacity cost. By a similar analysis, we find that
e
> 0ife ≤
1
2
and H
0
< H
< 0ife ≥
1
2
and H
0
>
H
The gain from pooling is maximized at an interme-
diate value of the success probability e, which corre-
sponds to high variability in the market sizes.
3. Impact of Contract Manufacturing
on Investment and Profit
To address the question of who should own the plant
(i.e., the capability to produce), we first, in §3.1, com-
pare the traditional model of §2 with a system in
which an independent CM exclusively possesses the
capability to produce. As before, both OEMs make
early investments in innovation. Then, they negoti-
ate supply contracts with the CM. The CM invests
in capacity at a cost of k per unit. Finally, demand
is realized, and the CM allocates capacity between
the OEMs. In making their investment decisions, all
parties seek to maximize their own expected prof-
its. This model illustrates the incentive problems that
arise due to an organizational split between innova-
tion and production. These problems arise because
innovation is not contractible: Each OEM invests in
innovation only insofar as he expects to recoup this
investment later by negotiating a favorable supply
contract. Hence, joint profit is lower than in the
ideal pooled model of §2. We characterize conditions
under which the system with contract manufacturing
achieves greater innovation and profit than the tradi-
tional model.
Second, in §3.2, we explore how the OEMs can
retain plant ownership, yet pool capacity through
supply contracts. We find that this yields greater
profit than the traditional model without pooling.
However, total profit may be either lower or higher
than in the system with contract manufacturing. Pre-
sumably, if the introduction of a CM will increase
the total system profit, OEMs should sell their pro-
duction facilities to a CM at a price that reflects
the future operational profits of the CM. However,
we will not model the sale of the production facili-
ties explicitly. Instead, we will focus on the individ-
ual product-based innovation and quantity decisions,
which occur relatively frequently and set the stage for
the asset divestiture decision.
To model the negotiations over supply contracts
and capacity allocation, we first introduce some con-
cepts from cooperative game theory. A cooperative
game consists of a finite set of N players, and a charac-
teristic value function v 2
N
→ R which specifies the
maximum value that can be created by each subset
of players. An allocation for the cooperative game is
a vector x ∈ R
N
specifying how this value is divided
among the N players. An allocation satisfies the added
value principle if each player realizes a fraction of the
value he individually adds to the group
x
i
≤ vN − vN\i for each player i ∈ N
and if the allocation is efficient
i∈N
x
i
= vN
Osborne and Rubinstein (1994) provide an excel-
lent introduction to cooperative game theory. Stuart
(2001) proposes the biform game, in which each player
i has a set of possible strategies A
i
and associated
cost function f
i
A
i
→ R . The profile of strategies cho-
sen by the players a ∈
×
i∈N
A
i
determine the charac-
teristic value function v
a
2
N
→ R for a cooperative
game. As in a strategic form noncooperative game,
the players simultaneously choose strategies. How-
ever, in addition to the immediate cost associated with
his own strategy, each player takes into account the
cooperative game that results from the strategy pro-
file. From the added value principle and conditional
on the strategy profile a, player i expects a net pay-
off of
V
i
a = #
i
v
a
N − v
a
N \i − f
i
a
i
where #
i
∈ 0 1 is his bargaining confidence index.
That is, he believes that he will appropriate the frac-
tion #
i
of his added value in the ensuing cooperative
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
138 Management Science 51(1), pp. 133–150, © 2005 INFORMS
game.
1
In the simple case with N = 2, the Nash bar-
gaining solution implies that # = 1/2: The two parties
will split the gain from cooperation (Nash 1953). The
noncooperative solution concept of Nash equilibrium
extends naturally to the biform game: Each player
chooses a strategy that maximizes his own net payoff
V
i
, given the strategic choices of the other players.
3.1. Pooling Capacity with a Contract
Manufacturer
The two OEMs play the following biform game. First,
they strategically choose market sizes H
N
i
∈ L
H and
incur the associated costs fH
N
i
. All parties observe
the demand distribution and cost of production, and
then a cooperative game ensues as the OEMs and CM
negotiate over supply contracts. OEM i has an added
value of
P
H
i
H
j
−
0
H
j
where
P
H
i
H
j
is the
expected operational profit for the integrated system
with pooling and
0
H
j
is the expected operational
profit that the CM and OEM j can achieve alone; i = j.
Each OEM has a bargaining confidence of # ∈ 0 1.
Therefore, the OEMs’ strategic market sizes H
N
1
H
N
2
constitute a Nash equilibrium if
H
N
1
= argmax
H∈L
H
#
P
HH
N
2
−
0
H
N
2
−f H (8)
H
N
2
= argmax
H∈L
H
#
P
H
N
1
H−
0
H
N
1
−f H (9)
We have assumed that the three parties bargain over
transfer payments, but ultimately adopt supply con-
tracts that will induce the CM to make the capacity
investment that is optimal for the pooled system and
allocate that capacity optimally between the OEMs
after demand is realized. We now justify this assump-
tion by considering tradable capacity options
2
as a
supply contract.
After observing the demand distributions, the CM
sells c
P
H
1
H
2
tradable options to each OEM with
an exercise price of zero. Each OEM is guaranteed
c
P
H
1
H
2
units of production capacity and the legal
right to sell this production capacity to the other OEM
after demand is realized. The investment of the CM
in production capacity, c, must be at least 2c
P
H
1
H
2
because he faces severe penalties for failure to fulfill
the capacity options. After the CM chooses his capac-
ity investment and demand is realized, the capacity
is traded in a cooperative game. The outcome is a
1
The set of all players’ confidence indices #
i
i=1 N
may not be con-
sistent with an efficient allocation. The confidence indices represent
the players’ beliefs, not the actual outcome of the cooperative game.
2
The world’s largest semiconductor contract manufacturer, Taiwan
Semiconductor Manufacturing Corporation, sells tradable capacity
options to OEMs. Each OEM is guaranteed the legal right to sell
his reserved production capacity to another OEM after demand is
realized (Economist 1996).
capacity allocation that maximizes total revenue. The
CM will participate in this trading only if he builds
excess, speculative capacity; i.e., c>2c
P
H
1
H
2
.In
the capacity allocation, the CM expects to obtain #
m
of his value-added. In the event that both OEMs have
high demand, this is
max
q
1
q
2
≥0
q
1
+q
2
≤c
H
1
− q
1
q
1
+ H
2
− q
2
q
2
− max
q
1
q
2
≥0
q
1
+q
2
≤2c
P
H
1
H
2
H
1
− q
1
q
1
+ H
2
− q
2
q
2
In the event that OEM i has high demand and OEM
j has low demand, the CM’s value-added is
max
q
H
q
L
≥0
q
H
+q
L
≤c
H
i
− q
H
q
H
+ L − q
L
q
L
− max
q
H
q
L
≥0
q
H
+q
L
≤2c
P
H
1
H
2
H
i
− q
H
q
H
+ L − q
L
q
L
In the event that both OEMs have low demand, the
CM’s value-added is
2 max
q∈0c/2
L − qq−
i=1 2
max
q∈0c
P
H
1
H
2
L − qq
Therefore, the CM chooses the production capacity
that maximizes his expected profit by solving the fol-
lowing problem:
max
c∈2c
P
H
1
H
2
#
m
R
P
c/2H
1
H
2
− kc
Because R
P
·H
1
H
2
is concave and #
m
≤ 1 his opti-
mal production capacity is 2c
P
H
1
H
2
. That is, the
CM will not build excess, speculative capacity. Thus,
tradable options induce optimal capacity investment
and allocation.
We proceed to investigate how the Nash equilib-
rium levels of innovation (8)–(9) differ from the ideal
derived in §2. Consider the optimization problem
max
H
1
H
2
∈L
H
#
P
H
1
H
2
− fH
1
− fH
2
(10)
From Proposition 1 we know that it has an optimal
solution, which shall be denoted H
N
1
# H
N
2
#. This
is a Nash equilibrium because it satisfies conditions
(8)–(9). Furthermore, when the OEMs have complete
bargaining confidence (i.e., each anticipates capturing
his entire value-added),
H
N
1
1 = H
N
2
1 = H
P
This follows from the observation that when # = 1,
problems (10) and (3) are identical. On the contrary,
if the OEMs have zero bargaining confidence, then
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
Management Science 51(1), pp. 133–150, © 2005 INFORMS 139
the OEMs have no incentive to invest in innovation
because f is strictly increasing,
H
N
1
0 = H
N
2
0 = L
Although in principle the Nash equilibrium H
N
1
#
H
N
2
# need not be either unique or symmetric,
Proposition 3 establishes that this Nash equilibrium
is unique and is symmetric (the proof relies on condi-
tion (5)).
Our main result is that market sizes increase with
the bargaining confidence of the OEMs. This is highly
intuitive: As the bargaining confidence of an OEM
increases, he has a stronger incentive to invest in inno-
vation.
Proposition 3. H
N
1
# H
N
2
# is the unique Nash
equilibrium, and it is symmetric:
H
N
1
# = H
N
2
# = H
N
#
Furthermore, there exists ˆ# ∈ 0 1 such that
H
N
# = L for # ∈ 0 ˆ# (11)
H
N
# is strictly increasing in #
for # ∈ ˆ# 1 with H
N
1 = H
P
(12)
By comparing Propositions 1 and 3, we conclude
that if the OEMs’ bargaining confidence is sufficiently
high, the level of innovation effort (and resulting mar-
ket size) will increase due to outsourcing. On the con-
trary, if the OEMs have little bargaining confidence,
then each OEM—anticipating that much of the value
created by his investment in innovation will be expro-
priated by the CM in the later contracting stage—will
invest less in innovation.
Let c
N
# = c
P
H
N
1
# H
N
2
# denote the optimal
capacity in the Nash equilibrium. Because the opti-
mal pooled capacity is monotone in the market size,
(11)–(12) imply that
c
N
# = L − k
+
/2 for # ∈ 0 ˆ#
c
N
# is strictly increasing in #
for # ∈ ˆ# 1 with c
N
1 = c
P
Intuitively, as the OEMs become more confident in
their bargaining position and the expected market
size grows, production capacity also increases. Recall
that if H
0
is sufficiently large, then c
P
<c
0
, and there-
fore c
N
#<c
0
(i.e., outsourcing reduces the level of
capacity investment). This tends to occur when the
cost of capacity is low or the success probability is
high. Otherwise, if c
P
>c
0
and the OEMs’ bargain-
ing confidence is high, then outsourcing increases the
level of capacity investment.
Outsourcing manufacturing can increase profit by
improving the efficiency with which capacity is used.
However, as demonstrated above, outsourcing can
weaken the incentives for innovation, eroding system
profit. On balance, which factor dominates depends
on how much of his added value the OEM expects
to capture in the contracting stage. Let # denote
the total system gain in expected profit due to
outsourcing:
# =
P
H
N
1
# H
N
2
# − fH
N
1
#
− fH
N
2
# − 2
0
H
0
− fH
0
(13)
It is straightforward to show that 1>0 and
0 ≤ 0 Further, # is increasing in #. Hence, if the
OEMs’ bargaining confidence is sufficiently high, then
sufficient incentives for innovation exist that the gain
from outsourcing is positive. If the OEMs’ bargaining
confidence is low, then outsourcing reduces system
profit. Hence, OEMs that anticipate being in a weak
bargaining position vis-à-vis a CM should retain their
production facilities rather than outsource.
The gain due to outsourcing is also sensitive to
the cost of capacity. The cost of capacity varies by
industry. For example, the cost of capacity for soft-
ware media replication is small relative to the cost
of innovation (software development) and the antic-
ipated sales revenue. This is not the case for semi-
conductor manufacturing, where fabrication capacity
is very expensive. Because the cost of capacity varies
by industry, the relative attractiveness of outsourc-
ing varies by industry. Consider capacity costs suf-
ficiently small that contract manufacturing is viable
(i.e., it results in nonzero profit). For small, fixed #,
the gain from contract manufacturing # is increas-
ing in k. (When # is small, contract manufacturing
reduces the total system capacity, c
N
# < c
0
, and thus
reduces the impact on profit of an increase in the unit
cost of capacity.) However, if # is sufficiently large, the
gain from contract manufacturing # is maximized
at an intermediate value of k as is the gain from ideal
pooling. In conclusion, outsourcing is most attractive
in industries where the capacity cost is moderate and
OEMs are in a strong bargaining position vis-à-vis
the CM.
We have demonstrated how contract manufactur-
ing may fail to achieve ideal pooling. Alternatively,
the OEMs may retain ownership of their production
facilities and pool capacity among themselves, elimi-
nating the CM. The next subsection investigates how
this shifts the incentives for innovation and capacity
investment.
3.2. Pooling Capacity Among Original
Equipment Manufacturers
The two OEMs first strategically choose market sizes
H
n
i
∈ L
H and incur the associated costs fH
n
i
.
As before, a cooperative game ensues as the OEMs
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
140 Management Science 51(1), pp. 133–150, © 2005 INFORMS
bargain over the level of capacity investment. We
assume that they achieve an optimal capacity invest-
ment and allocation of that capacity between the
OEMs after demand is realized, either through supply
contracts or a joint venture.
3
The gain from cooper-
ation
P
H
1
H
2
−
0
H
1
+
0
H
2
will be evenly
divided, according to the Nash bargaining solution.
Therefore, the OEMs’ strategic market sizes H
n
1
H
n
2
constitute a Nash equilibrium if
H
n
1
= arg max
H∈L
H
0
H +
1
2
P
H H
n
2
−
0
H +
0
H
n
2
− fH
(14)
H
n
2
= arg max
H∈L
H
0
H +
1
2
P
H
n
1
H
−
0
H
n
1
+
0
H − fH
(15)
We proceed to investigate how the Nash equilib-
rium level of innovation differs from the ideal derived
in §2 and the level of innovation with contract man-
ufacturing. Consider the optimization problem
max
H
1
H
2
∈L
H
1
2
P
H
1
H
2
+
0
H
1
+
0
H
2
− fH
1
− fH
2
and let H
n
1
H
n
2
denote the optimal solution. This
is a Nash equilibrium because it satisfies (14)–(15).
Proposition 4 establishes that this is the unique Nash
equilibrium and it is symmetric (the proof relies on
condition (5)). Our main result is that when the OEMs
outsource among themselves, the resulting market
size is greater than would be adopted by an isolated
OEM, but lower than in the ideal pooling scenario.
That is, the OEMs underinvest in innovation.
Proposition 4. H
n
1
H
n
2
is the unique Nash equilib-
rium, and it is symmetric:
H
n
1
= H
n
2
= H
n
Furthermore,
H
0
≤ H
n
≤ H
P
(16)
where the second inequality is strict if and only if (6) holds.
The insight behind the proof of Proposition 4 is that
by strengthening his alternative option, each OEM
can improve his bargaining outcome. Therefore, in
deciding how much to invest in innovation, each
3
AMD and Fujitsu have a joint venture to produce flash memory
chips. The firms cooperatively choose the capacity level of the joint
venture’s production facilities. Each firm has the nominal right to
purchase 50% of the output at cost, and is penalized for taking less
than 40%. However, as the demand for flash memory fluctuates,
the two parties frequently renegotiate the contract, which specifies
“cost” and “penalty” to ensure an efficient allocation of capacity
among themselves (Doran 2001).
OEM places weight on the expected operational profit
0
in the scenario without pooling (where the optimal
level of innovation is lower than in the pooling sce-
nario), although pooling is inevitable. This qualitative
insight remains true even if the OEMs fail to antic-
ipate the Nash bargaining outcome; insofar as the
OEM anticipates that his profit allocation will increase
with his “go-it-alone” alternative
0
, he will underin-
vest in innovation.
Proposition 4 implies that OEMs that own their pro-
duction facilities increase their profit (relative to the
traditional model) by outsourcing among themselves.
To see this, observe that
1
2
P
H
n
H
n
− fH
n
≥
1
2
P
H
0
H
0
− fH
0
≥
0
H
0
− fH
0
where the first inequality follows by Proposition 4
and the observation that 1/2
P
H H − fH is
concave in H (which is established in the proof of
Proposition 1). By comparing Propositions 3 and 4, we
conclude that if the OEMs anticipate being in a weak
bargaining position vis-à-vis the CM, then outsourc-
ing among OEMs results in greater innovation and
expected OEM profit than outsourcing to a CM. How-
ever, when the OEMs’ bargaining confidence is high
(# is close to unity), contract manufacturing results in
greater system profit.
4. The Success Probability
Innovation Model
Section 2 points out that in some industry contexts
innovation investments may be directed primarily at
increasing the probability that a product is success-
ful rather than increasing the size of the potential
market. This section explores the setting where the
OEM’s early investment in innovation influences the
product’s success probability e, rather than its mar-
ket size H. We demonstrate that the effect of either
pooling or outsourcing on innovation depends impor-
tantly on how innovation affects demand. When inno-
vation influences the market size, pooling increases
innovation, and outsourcing leads to underinvest-
ment in innovation. In contrast, when innovation
influences the success probability, pooling may reduce
innovation, and outsourcing (among OEMs) can
result in overinvestment in innovation.
Nonetheless, the primary qualitative results regard-
ing the effect of outsourcing on system and OEM
profit are similar under both innovation models:
OEMs that own their production facilities increase
their profits by outsourcing among themselves. Out-
sourcing to a CM increases system profit (relative to
either setting in which the OEMs own their produc-
tion facilities) if and only if the OEMs’ bargaining con-
fidence is high. If the OEMs’ bargaining confidence is
low, then outsourcing to a CM reduces OEM profit.
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
Management Science 51(1), pp. 133–150, © 2005 INFORMS 141
4.1. The Impact of Ideal Pooling on Investment
and Profit
This subsection compares the traditional model—in
which each firm builds its own capacity to meet
its own demand—with a pooled model in which
firms share their production capacity and choose
their investments to maximize joint profits. The for-
mulation is analogous to the case where innovation
influences the market size. For any given success
probability, the OEM’s expected operational profit
0
is given by (1). Assume that the total cost of inno-
vation ge is twice differentiable, strictly increasing,
convex, and satisfies ge → as e → 1. The OEM
chooses a success probability e that maximizes his
total expected profit, which is given by
V
0
= max
e∈0 1
0
e − ge (17)
Consider two identical OEMs that pool their pro-
duction capacity. The maximum expected profit that
they can achieve jointly is
V
P
= max
e
1
e
2
∈0 1
P
e
1
e
2
− ge
1
− ge
2
(18)
where
P
e
1
e
2
= max
c≥0
R
P
c e
1
e
2
− 2kc
R
P
c e
1
e
2
= e
1
e
2
2H − cc + e
1
+ e
2
− 2e
1
e
2
· max
q
H
q
L
≥0
q
H
+q
L
≤2c
H − q
H
q
H
+ L − q
L
q
L
+ 1 − e
1
1 − e
2
2 max
q∈0c
L − qq
Section 2 demonstrates that when innovation influ-
ences the size of the market, pooling increases innova-
tion. One might conjecture that a similar result holds
when innovation influences the success probability.
The main result of this subsection is that this conjec-
ture need not hold: When innovation influences the
success probability, the optimal innovation level in the
pooled system may be larger or smaller than the inno-
vation level without pooling.
In general, problem (17) may fail to have a unique
solution. Similarly, problem (18) may fail to have a
unique solution, and further, a solution need not be
symmetric. The following proposition gives sufficient
conditions for uniqueness and symmetry. Thereafter,
we simply assume that problem (17) has a unique
solution e
0
and that problem (18) has a unique and
symmetric solution e
P
.
Proposition 5. The optimization problem (17) has a
solution e
0
. A sufficient condition for e
0
to be the unique
solution is that
d
2
ge
de
2
> max
H − L
2
2
H − L
3
2k
for e ∈ 0 1
(19)
The optimization problem (18) has a solution e
P
1
e
P
2
. Sup-
pose that
d
2
ge
de
2
>
H − L
2
4
for e ∈ 0 1 (20)
Then, the solution is symmetric, and furthermore, if
d
2
ge
de
2
>
1
2
2
P
e e
e
2
1
for e ∈ 0 1 (21)
then the solution is also unique:
e
P
1
= e
P
2
= e
P
The functional form of
2
/e
2
1
P
e e is given in
the proof of Proposition 5 in the appendix.
We now proceed to characterize the impact of
pooling on innovation (Proposition 6) and capacity
(Proposition 7). The following lemma evaluating the
effect of pooling on capacity for a fixed success proba-
bility is needed to prove Propositions 6 and 7. Define
e
P
= 1 −
H − k
+
/H − L
e
0
= k − L
+
/H − L ∧ 1
e =
2k + H − 3L
+
/2H − 2L ∧ 1ifH<3L
3k/2H∧ 1ifH ≥ 3L
¯
e =
k/H − L ∧ 1ifH<3L
3k/2H∧ 1ifH ≥ 3L
Lemma 1. For e ∈ 0 1 if e ≤
e
P
then c
P
e e =
c
0
e = 0;ife
P
<e≤ e
0
then c
P
ee>c
0
e = 0;if
e
0
<e<e then c
P
ee>c
0
e > 0 if e ≤ e ≤
¯
e then
c
P
e e = c
0
e > 0 if e>
¯
e, then 0 <c
P
e e < c
0
e
Proposition 6 characterizes the impact of pooling
on the optimal success probability. The proposition
demonstrates that when innovation influences the
success probability, pooling may either increase or
decrease innovation. If innovation is costly so that e
0
is small, pooling increases innovation. If innovation
is cheap so that e
0
is close to unity, pooling decreases
innovation. Define
ekHL
= inf
e ∈
e
0
1
d
0
e
de
≥
P
e
1
e
2
e
1
e
1
=e
2
=e
Proposition 6. If e
0
>ekHLthen e
P
<e
0
if 0 <
e
0
<ekHL then e
P
>e
0
where 0 <ekHL<1
Because market size is stochastically increasing in
innovation, the OEM’s expected operational profit is
increasing in e whether capacity is pooled or not.
However, variability in the market size is more detri-
mental to an isolated OEM, and this variability is a
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
142 Management Science 51(1), pp. 133–150, © 2005 INFORMS
strictly concave function of e maximized at e = 1/2
Hence, the isolated OEM tends to set his innovative
effort at high or low rather than intermediate lev-
els, whereas pooling motivates the OEM to adopt an
intermediate level of innovation. This explains why
the level of innovation may be greater or smaller
under pooling. Propositions 1 and 6 highlight that
the effect of pooling on innovation depends on how
innovation affects the demand distribution. Although
Propositions 1 and 6 report distinct results regarding
the effect of pooling on innovation, both are consistent
with the observation that the pooled OEM is more
tolerant of market-size variability.
The proof of Proposition 6 is based on showing
that ekH L is the unique crossing point of d
0
/de
and
P
/e
1
, at which both functions are positive.
The stated inequalities then follow from the first-order
conditions for problems (17) and (18). In an extensive
computational study, we observed that ek H L is
universally increasing in the cost of capacity k Typ-
ically, when the cost of capacity k is low, pooling
reduces the optimal success probability (i.e., e
P
<e
0
,
and for large k, pooling increases the optimal success
probability. Figure 1 illustrates this phenomenon.
The next proposition characterizes the impact of
pooling on the capacity level.
Proposition 7.
If e
0
< minek H L e then c
0
<c
P
if e
0
> maxek HL
¯
e then c
0
>c
P
The result is analogous to Proposition 2: Under both
innovation models, if innovation and capacity are
costly so that the level of innovation for an isolated
OEM is small, then the optimal capacity investment
is larger under pooling. On the contrary, if innova-
tion and capacity are cheap, then pooling reduces
the optimal capacity investment. Pooling may simul-
taneously lead to reduced innovation and increased
capacity. For example, if H = 3L= 1k= 12, and
ge = 055e
2
+ 1
e>04
e − 04/101 − e then e
0
=
Figure 1 Optimal Success Probability with Pooling e
P
and Without
Pooling e
0
as a Function of Capacity Cost k, for the System
with H = 15L= 10 and ge =−7 log1 − e
Capacity Cost k
Success Probability
e
0
e
P
0.6
0.4
0.2
0
5.0 5.5 6.0 6.5 7.0 7.5
0.
055c
0
= 045e
P
= 054, and c
P
= 046 Here, the
reduced innovation of the pooled OEM corresponds
to more variability in the market size, and the pooled
OEM increases his capacity to compensate.
Finally, we turn to the impact of pooling on
expected profit. We define the gain from pooling as
in (7), but substitute the decision variable e for H and
the cost function g for f . As previously observed,
is increasing in the cost of capacity when k is small,
but decreasing when k is large:
k
> 0ife
0
> maxek HL
¯
e
< 0ife
0
< minek H L e
The result follows from Proposition 7 and the
repeated application of the envelope theorem to
/k
One might expect the gain from pooling to increase
with the market size H, as this introduces variability.
Surprisingly, when the market size H is large, the gain
from pooling actually decreases with H. The precise
result, obtained by similar analysis, is that
H
> 0ife
0
< minek H L e
< 0ife
0
> maxek HL
¯
e and
H>L+ 2k/e
P
2
For large H, we have e
0
> maxek H L
¯
e, which
guarantees that an isolated OEM invests more in
capacity (c
0
>c
P
) and more in innovation (e
0
>e
P
). In
this state of affairs, increasing the market size H adds
more expected revenue for the isolated OEM than in
the pooling scenario. We therefore conclude that the
gain from pooling is maximized at some intermediate
value of H.
4.2. Impact of Contract Manufacturing on
Investment and Profit
Having considered the impact of pooling when the
OEMs make decisions to maximize the profit of the
total system, we now consider the impact of pool-
ing when independent OEMs seek to maximize their
own profit. The traditional model in which each OEM
builds its own capacity to meet its own demand
serves as the base case for comparison. Section 4.2.1
considers the case where OEMs outsource produc-
tion to an independent CM and characterizes when
outsourcing increases system profit. An alternative
means of achieving pooling is for the OEMs to retain
their production facilities and outsource among them-
selves through supply contracts or a joint venture; this
is explored in §4.2.2.
4.2.1. Pooling Capacity with a Contract Manu-
facturer. Consider the setting in which the OEMs
outsource production to a CM. The formulation is
analogous to the case where innovation influences the
market size. First, the OEMs strategically choose suc-
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
Management Science 51(1), pp. 133–150, © 2005 INFORMS 143
cess probabilities e
N
i
and incur the associated costs
ge
N
i
. Then, a cooperative game ensues as the OEMs
and CM negotiate over supply contracts. As in §3.1,
the CM and OEMs negotiate supply contracts that
will induce the CM to make the capacity investment
that is optimal for the pooled system and allocate that
capacity optimally between the OEMs after demand is
realized. In making his innovation investment, OEM i
anticipates that in the subsequent contract negotiation
stage he will capture # of his value-added,
P
e
i
e
j
−
0
e
j
i = j
Hence, the OEMs’ strategic success probabilities
e
N
1
e
N
2
constitute a Nash equilibrium if
e
N
1
= arg max
e∈0 1
#
P
e e
N
2
−
0
e
N
2
− ge (22)
e
N
2
= arg max
e∈0 1
#
P
e
N
1
e−
0
e
N
1
− ge (23)
Consider the problem
max
e
1
e
2
∈0 1
#
P
e
1
e
2
− ge
1
− ge
2
From Proposition 5 we know that it has an opti-
mal solution, which shall be denoted e
N
1
# e
N
2
#.
This is a Nash equilibrium in success probabilities
because it satisfies (22)–(23). Results for the extreme
cases where either the OEMs have complete or zero
bargaining confidence are similar to the case where
innovation influences market size. When the OEMs
have complete bargaining confidence, they exert the
level of effort that is optimal for the pooled system:
e
N
1
1 = e
N
2
1 = e
P
When the OEMs have zero bar-
gaining confidence, they have no incentive to invest
in innovation, and consequently e
N
1
0 = e
N
2
0 = 0
In general, the Nash equilibrium e
N
1
# e
N
2
# is
not symmetric and is not the unique Nash equilib-
rium. This makes it difficult to perform compara-
tive statics and analyze how outsourcing affects the
levels of innovation effort and capacity investment.
Techniques for comparative statics in systems with
multiple Nash equilibria typically require monotonic-
ity in the best-response function (cf. Lippman et
al. 1987). Unfortunately, because
2
/e
1
e
2
P
e
1
e
2
takes both positive and negative values, the optimal
success probability for one OEM is not monotone as a
function of the success probability of the other OEM.
Fortunately, if the function g is sufficiently convex, the
Nash equilibrium e
N
1
# e
N
2
# is unique and sym-
metric. Better yet, for a range of parameter values, the
equilibrium is in dominant strategies: One OEM need
not anticipate the success probability of the other.
Hence, for a wide range of parameter values, we have
comparative statics.
Proposition 8. Suppose that
d
2
ge
de
2
>
#H − L
2
4
for e ∈ 0 1 (24)
Then, every Nash equilibrium is symmetric. If
d
2
ge
de
2
>#
2
P
e e
e
2
1
for e ∈ 0 1 (25)
then there exists a unique Nash equilibrium: e
N
1
# =
e
N
2
# = e
N
#, and for some ˜# ∈ 0 1,
e
N
# is strictly increasing for # ∈ ˜# 1
e
N
# = 0 for all # ∈ 0 ˜# (26)
In particular, if
k ∈ H − L 3L − H/2 (27)
then 24 implies that e
N
# is a unique equilibrium in
dominant strategies and satisfies 26.
The functional form of
2
/e
2
1
P
e e is given in
the proof of Proposition 5 in the appendix. Proposi-
tion 8 shows that, just as in the case where innova-
tion influences the market size, innovation is increas-
ing in the OEMs’ bargaining confidence. The opti-
mal capacity in the Nash equilibrium is c
N
# =
c
P
e
N
1
# e
N
2
# Consistent with the previous model,
because the optimal pooled capacity is monotone in
the success probability, the level of capacity invest-
ment also increases with #.
Combining these results with Proposition 6, we
conclude that if e
0
> maxek H L
¯
e then innova-
tion and capacity are always lower with outsourcing
than for an isolated OEM; i.e.,
e
N
# < e
0
and c
N
# < c
0
for all # ∈ 0 1
This tends to occur when either innovation or capac-
ity is cheap. On the contrary, if innovation and capac-
ity are expensive, then e
0
< minek H Le. In this
case, there exist thresholds 0 <#
< ¯#<1 such that
e
N
# < e
0
and c
N
# < c
0
if #<#
e
N
# > e
0
and c
N
# > c
0
if #> ¯#
i.e., just as when innovation influences market size,
outsourcing leads to higher innovation and capacity
investment if and only if the OEMs’ bargaining con-
fidence is high.
Finally, we characterize the impact of outsourcing
manufacturing on system profit. Define # as in
(13), with the obvious substitution of decision vari-
ables e for H. # represents the change in expected
system profit that results when OEMs outsource pro-
duction to a CM rather than build capacity to meet
their own demand. As in the previous section, #
is increasing in # and becomes positive if and only
if the bargaining confidence of the OEM is suffi-
ciently large. Thus, regardless of how innovation
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
144 Management Science 51(1), pp. 133–150, © 2005 INFORMS
Figure 2 Equilibrium Success Probability and Production Capacity
OEM Bargaining Confidence
Success Probability e
N
(α)
k =7.5k =5
0.6
0.6 0.8
0.4
0.4
0.2
0.2
0
1.0
OEM Bargaining Confidence
Capacity c
N
(α)
k =7.5
k =5
0.6 0.80.40.20 1.0
2
3
4
0.
α
α
affects the demand distribution, if the OEMs’ bargain-
ing confidence is sufficiently small, then outsourc-
ing will reduce expected profit for the system as a
whole; weakened incentives for innovation outweigh
the benefits of efficient capacity utilization. The results
in §3.1 regarding the impact of k on # when inno-
vation influences the market size also apply when
innovation influences the success probability.
These insights are best illustrated and extended
with a numerical example. As in Figure 1, we con-
sider the system with H = 15, L = 10, and ge =
−7 log1 − e. Figure 2 shows how the capacity cost
and the OEMs’ bargaining confidence affect the equi-
librium success probability and subsequent capac-
ity investment. The parameter values satisfy (27), so
we have a unique equilibrium success probability in
dominant strategies. Figure 3 shows the increase in
expected profit due to contract manufacturing, as a
proportion of the expected profit without pooling:
#/2
0
e
0
− 2ge
0
. The case with high produc-
tion cost is motivated by observations from the phar-
maceutical and semiconductor industries, whereas the
case with low capacity cost is typical of the soft-
ware industry, where outsourcing of media produc-
tion is common. When the cost of capacity is large, the
potential gain from outsourcing is great, but OEMs
will not invest in innovation unless they expect suc-
cess in bargaining over supply contracts (#>075). In
contrast, when the cost of capacity is small, the OEMs
invest in innovation over a wider range (#>045).
Figure 3 Increase in Expected Profit Due to Contract Manufacturing
OEM Bargaining Confidence
Increase in Expected
Profit Due to Contract
Manufacturing
k =7.5
k =5
0.4
0.4 0.8 0.8 1.0
0.2
0.2
0.0
–0.2
0.0
α
However, pooling adds little value; indeed, unless
#>068 outsourcing reduces expected profit for the
system as a whole. The examples demonstrate that,
consistent with the theoretical results, the success
probability, capacity, and gain from outsourcing are
all increasing in #.
Instead of outsourcing production to a CM, the
OEMs may alternatively retain ownership of their
production facilities and pool capacity among them-
selves. The next subsection investigates how this
shifts the incentives for innovation and capacity
investment.
4.2.2. Pooling Capacity Among Original Equip-
ment Manufacturers. Consider the setting in which
the OEMs outsource among themselves. The formu-
lation is analogous to the case where innovation
influences the market size in §3.2. First, the OEMs
strategically choose success probabilities e
n
i
and incur
the associated costs ge
n
i
. Then, a cooperative game
ensues as the OEMs bargain over the level of capacity
investment. As in §3.2, the firms achieve the capacity
investment that is optimal for the pooled system and
allocate the capacity optimally. In making his inno-
vation decision, OEM i anticipates capturing half of
the subsequent gain from cooperation, so his expected
revenue is
0
e
i
+
1
2
P
e
i
e
j
−
0
e
i
+
0
e
j
i = j
Thus, the OEMs’ strategic success probabilities e
n
1
e
n
2
constitute a Nash equilibrium if
e
n
1
= arg max
e∈0 1
0
e +
1
2
P
e e
n
2
−
0
e +
0
e
n
2
− ge
e
n
2
= arg max
e∈0 1
0
e +
1
2
P
e
n
1
e
−
0
e
n
1
+
0
e − ge
This subsection demonstrates that the effect of OEM
outsourcing on innovation depends importantly on
how innovation affects the demand distribution.
Section 3.2 demonstrates that when innovation influ-
ences the market size, OEMs that outsource among
themselves underinvest in innovation. The main
result of this subsection is that when innovation influ-
ences the success probability, the OEMs may overin-
vest in innovation.
Consider the problem
max
e
1
e
2
∈01
1
2
P
e
1
e
2
+
0
e
1
+
0
e
2
− ge
1
− ge
2
and let e
n
1
e
n
2
denote its optimal solution. This is
a Nash equilibrium, but it is not necessarily unique
or symmetric. Our main result is that when the two
OEMs do make similar investments in innovation, the
resulting success probabilities may be larger than in
the ideal pooled case.
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
Management Science 51(1), pp. 133–150, © 2005 INFORMS 145
Proposition 9. Suppose that
d
2
ge
de
2
>
H − L
8k
2
k + 2maxkH − L for e ∈ 0 1
(28)
Then, every Nash equilibrium is symmetric, and further-
more, if
d
2
ge
de
2
>
1
2
2
P
e e
e
2
1
+
d
2
0
e
de
2
for e ∈ 0 1
(29)
then there exists a unique Nash equilibrium
e
n
1
= e
n
2
= e
n
and it satisfies
e
P
<e
n
<e
0
if e
0
>ekHL
e
0
<e
n
<e
P
if 0 <e
0
<ekHL
(30)
(The functional form of
2
/e
2
1
P
e e and an
upper bound on d
2
/de
2
0
e are given in the proof
of Proposition 5 in the appendix.) As in the setting
where innovation increases the market size, the Nash
equilibrium innovation level falls between the opti-
mal innovation level for the ideal pooled system and
the optimal innovation level for the isolated OEM.
However, contrast with the setting in which innova-
tion increases the market size emerges when capac-
ity and innovation are cheap. Then, e
P
<e
n
<e
0
.
The OEMs overinvest in innovation (and hence capac-
ity). Therefore, introduction of a CM would reduce
innovation, regardless of the OEMs’ bargaining con-
fidence. Alternatively, when capacity and innovation
are expensive so that e
0
<e
n
<e
P
, introduction of a
CM will increase innovation if and only if the OEMs
anticipate being in a strong bargaining position.
Although the two innovation models yield distinct
results for the impact of outsourcing on innovation,
both models yield similar results for the effect of
outsourcing on system and OEM profit. Section 3.2
establishes that when innovation influences the mar-
ket size, OEMs that own their production facilities
increase their profit (relative to the traditional model)
by outsourcing among themselves. Further, outsourc-
ing to a CM increases system profit (relative to either
setting in which the OEMs own their production facil-
ities) if and only if the OEM’s bargaining confidence
is high. Both results hold for the success probability
model as well, provided that the relevant restrictions
on the innovation cost function hold; the results fol-
low from Propositions 8 and 9.
5. Discussion
Should an OEM sell the plant? Our analysis indicates
that OEMs might do better to trade capacity among
themselves rather than to outsource to a CM, particu-
larly if the OEMs anticipate being in a weak bargain-
ing position vis-à-vis the CM. Plant ownership gives
the OEM a strategic alternative to supply contracts,
and thus improves his bargaining position. Unfor-
tunately, this leads the OEM to make investments
in innovation that improve his strategic alternative
(going it alone) at the expense of overall system effi-
ciency. In particular, the OEM will overinvest to ensure
that a new product will be successful when capacity
and innovation are cheap, and will underinvest when
such investments are costly. With contract manufac-
turing, the OEM will always underinvest in innova-
tion, spending only what he expects to recoup by
negotiating a favorable supply contract with the CM.
From a system perspective, contract manufacturing
performs well if and only if OEMs anticipate being in
a strong bargaining position vis-à-vis the CM.
We have analyzed a stylized model with two sym-
metric OEMs and common information to obtain a
tractable expression for the Nash equilibrium in inno-
vation. However, our main insight, that with contract
manufacturing an OEM will underinvest in innova-
tion, is very robust. This insight relies only on the
assumed sequence of events: innovation, contracting,
then capacity investment. The OEM cannot generate
revenue without the CM, and in negotiating a sup-
ply contract his innovation is a sunk cost. Therefore,
the OEM cannot capture the full increase in expected
contribution that flows from investment in innova-
tion, and will spend less on innovation than would
be optimal for the system as a whole. This is a classic
hold-up problem.
Our assumption that OEMs make investments
in innovation before they contract for capacity is
motivated by examples from the electronics and
automotive industries. However, an OEM with low
bargaining confidence or a low value-added should
consider purchasing capacity options before invest-
ing in innovation. A follow-on paper (Plambeck and
Taylor 2004) describes how small biotechnology firms
contract for manufacturing capacity three years in
advance of production, before investing in clinical tri-
als. Contracting for capacity in advance strengthens
the incentive for innovation. However, advance con-
tracts are renegotiated with high probability, which
can weaken the incentive for investment. Contracting
in advance and subsequent renegotiation impose high
transaction costs, and may therefore be undesirable.
As an alternative to a formal contract, an OEM
and CM may adopt an informal agreement, sus-
tained by the potential for repeat business. There-
fore, our single-period model may underestimate
the efficiency of contract manufacturing. In practice,
successful CMs seek collaborative, long-term partner-
ships with their OEM customers. Solectron Vice Presi-
dent Eddie Maxie reports that Solectron scrupulously
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
146 Management Science 51(1), pp. 133–150, © 2005 INFORMS
avoids “price gouging” of loyal OEM customers when
total industry demand outstrips supply Maxie (2000).
If our single-period game is extended to a repeated
game, the OEMs and CM can adopt a subgame perfect
Nash equilibrium that gives strict Pareto improvement
over their single-period strategies. Each OEM invests
more in innovation, and the CM, on observing aus-
picious demand conditions, offers a favorable supply
contract (cf. Taylor and Plambeck 2003).
We have assumed that the OEMs operate in sepa-
rate markets. This is appropriate, for example, when
a pharmaceutical CM contracts with drug compa-
nies that serve separate markets, or an electronics
CM produces subassemblies for PCs and network-
ing equipment. However, CMs also supply competing
OEMs. For example, Flextronics produces cellphones
for Ericsson and Motorola (Thurm 2001). Competing
OEMs may even pool capacity among themselves; the
sales forces of AMD and Fujitsu compete vigorously
in Europe to sell the flash memory produced in their
joint venture. Modeling competition between OEMs
that outsource to a CM or pool their own capac-
ity may be a promising direction for future research.
Because the benefit of pooling is closely related to
the degree of demand correlation, explicitly modeling
demand correlation may shed additional light on the
impact of industry structures that involve pooling.
Another natural extension of our model would be
to incorporate learning. To the extent that product
research and development benefit from concurrent
design for manufacturing, OEMs may prefer to keep
manufacturing (and hence manufacturing expertise)
in-house. On the contrary, insofar as the production
cost per unit decreases with cumulative output, con-
tract manufacturing is advantageous.
Although we do not model dynamics of the evolu-
tion of contract manufacturing over time, our results
suggest that contract manufacturing might become a
victim of its own success. Specifically, as OEMs sell
off productive assets and transfer production to CMs,
CMs will grow in size relative to their OEM cus-
tomers. Daiwa Research Institute reports that in elec-
tronics, because of the emergence of a small number
of large CMs, bargaining power has shifted to these
CMs (Sarmah 2000). Further, as OEMs lose produc-
tion expertise and capabilities, control may further
shift to CMs. Our results suggest that an erosion in
OEM power (bargaining confidence) will lead to less
innovation, lower total industry profit, and thus, per-
haps, a return to vertical integration. This hypoth-
esis is consistent with the historical pattern in the
computer, automotive, and bicycle industries, where
industry structure has cycled from vertical integration
to outsourcing and back again (Fine 1998).
A technical appendix (Plambeck and Taylor 2003) to
this paper is available at http://mansci.pubs.informs.
org/ecompanion.html.
Acknowledgments
The authors are grateful to Enis Kayis for identifying the
equilibrium in dominant strategies in Proposition 8, and to
their referees for a host of helpful suggestions.
Appendix
Throughout, we use the notation
F
ˆ
x
ˆ
y
x
to denote
F xy
x
x=
ˆ
xy=
ˆ
y
Proof of Proposition 1. We begin by establishing exis-
tence of optimal solutions to (2). Because the function f is
continuous and fH→ as H →
H, the objective func-
tion in (2) is continuous and bounded above by
H
2
/4, and
achieves its maximum in the region L
H. Note that
d
2
0
H
dH
2
=
0 if H ∈ LL + k − L
+
/e
e
2
/2 if H ∈ L + k − L
+
/e L + k/e
e/2 if H ∈ L + k/e
H
(A.1)
so (5) implies that (2) has a unique optimal solution
because the objective is strictly concave. Existence of an
optimal solution to (3), H
P
1
H
P
2
, follows by an analogous
argument.
Similarly, the objective function in (3) is continuous and
bounded above by
H
2
/2, and achieves its maximum in
L
H× L
H. To prove symmetry, let us assume without
loss of generality that H
P
1
≥ H
P
2
. With a little effort one can
show that
P
H
P
1
H
P
2
H
1
−
P
H
P
1
H
P
2
H
2
≤
e
2
H
P
1
− H
P
2
(A.2)
(see Plambeck and Taylor 2003 for the details). We have
assumed that H
P
i
>L i= 1 2, so from the first-order con-
ditions for (3),
df H
P
1
dH
−
df H
P
2
dH
=
P
H
P
1
H
P
2
H
1
−
P
H
P
1
H
P
2
H
2
≤
e
2
H
P
1
− H
P
2
If H
P
1
>H
P
2
, this contradicts (5) so we must have H
P
1
= H
P
2
.
To establish uniqueness, we observe that
d
2
P
H H
dH
2
=
2 − e
2
e
2
1 + 21 − ee
if c
P
H H ∈
0 min
L
2
H − L
4
2 − e
2
e
4 − 3e
if c
P
H H ∈
L
2
H − L
4
e1 + e
2
if c
P
H H ∈
H − L
4
L
2
e4 − 3 − ee
22 − e
if c
P
H H ∈
max
L
2
H − L
4
H + L
4
e if c
P
H H ∈
H + L
4
H − k
2
≤ e
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
Management Science 51(1), pp. 133–150, © 2005 INFORMS 147
so (5) implies concavity of the objective function
P
H H − 2fH and, hence, uniqueness of the optimal
solution to (3).
To complete the proof of Proposition 1, we need to show
that H
P
>H
0
when H
0
<
H and that H
P
= H
0
when H
0
≥
H.
We have assumed that H
P
>L, so the desired result holds
trivially when H
0
= L. Suppose that H
0
>L. We observe that
H
0
infH
0
H > 0H ≥ L = L + k − L
+
/e
and because f
> 0, H
0
> H
0
. Therefore, it will be sufficient
to show that
P
H H
H
1
>
d
0
H
dH
if H ∈
H
0
H
=
d
0
H
dH
if H ∈
H
(A.3)
For the case H ≤ 3L, we have
P
H H
H
1
−
d
0
H
dH
=
e − e
2
3e − 3e
2
H + 1 − 5e + 3e
2
L + 2e − 1k
2 + 4e − 4e
2
if c
P
H ∈
0
H − L
4
e − e
2
H − L
4
if c
P
H ∈
H − L
4
L
2
1 − e2k − e
2
H − L
8 − 4e
if c
P
H ∈
L
2
H + L
4
0ifc
P
H ∈
H + L
4
H − k
2
where c
P
H is the optimal capacity for the pooled system,
the maximizer in (4). From the first-order optimality con-
ditions, c
P
H ≥ H + L/4 if and only if H ≥
H. Thus, it
remains to show that /H
1
P
HH>d/dH
0
H if
H<
H. This is immediate if c
P
H ∈ H − L/4H + L/4.
If e ≤ 1/2ork>L, then H>
H
0
implies that
3e − 3e
2
H + 1 − 5e + 3e
2
L + 2e − 1k > 0 (A.4)
If c
P
H<H− L/4, then
c
P
H =
2e − e
2
H − L + L − k
2 + 4e − 4e
2
If e>1/2, then c
P
H<H− L/4 implies that H<2k +
2e − 3L/2e − 1, which in turn implies k>L. Conse-
quently, H>
H
0
implies (A.4). Therefore, (A.3) is satisfied
for H ≤ 3L. The proof for H>3L is similar (see Plambeck
and Taylor 2003).
Proof of Proposition 3. First, we will assume (5) and
prove that all Nash equilibria are symmetric (by contra-
diction). Suppose that HG is a Nash equilibrium with
H>G. Then,
#
P
H G
H
1
−
df H
dH
= 0 (A.5)
#
P
H G
H
2
−
df G
dH
= 0ifG>L
≤ 0ifG = L
(A.6)
From (A.2),
#
P
H G
H
1
−
P
H G
H
2
≤
#e
2
H − G (A.7)
Combining (A.5)–(A.7) we find that
df H
dH
−
df G
dH
≤
#e
2
H − G
which contradicts (5).
To prove uniqueness, we observe that a symmetric Nash
equilibrium H
N
H
N
must satisfy
#
P
H
N
H
N
H
1
−
df H
N
dH
= 0ifH
N
>L
≤ 0ifH
N
= L
(A.8)
and that if (5) is satisfied, (A.8) has a unique solution.
Finally, the result (11)–(12) is obtained by applying the
implicit function theorem to (A.8).
Proof of Proposition 4. First, we will prove (by contra-
diction) that (5) implies that all Nash equilibria are symmet-
ric. Suppose that H
1
H
2
is a Nash equilibrium in market
sizes with H
1
>H
2
. Then, the following first-order condi-
tions must be satisfied:
1
2
P
H
1
H
2
H
1
+
d
0
H
1
dH
−
df H
1
dH
= 0 (A.9)
1
2
P
H
1
H
2
H
2
+
d
0
H
2
dH
−
df H
2
dH
= 0ifH
2
> 0
≤ 0ifH
2
= 0
(A.10)
From (A.1),
d
0
H
1
dH
−
d
0
H
2
dH
≤
e
2
H
1
− H
2
(A.11)
Combining (A.9)–(A.11) with (A.2), we find that
df H
1
dH
−
df H
2
dH
≤
e
2
H
1
− H
2
which contradicts (5) if H
1
>H
2
.
To establish uniqueness, we observe that any symmetric
Nash equilibrium must satisfy the first-order condition
1
2
P
H
n
H
n
H
1
+
d
0
H
n
dH
−
df H
n
dH
= 0ifH
n
>L
≤ 0ifH
n
= L
(A.12)
and condition (5) guarantees that for all H ≥ L,
1
2
2
P
H H
H
2
1
+
d
2
0
H
dH
2
−
d
2
fH
dH
2
< 0
so (A.12) has a unique solution. Finally, (16) follows imme-
diately from (A.3) which was shown in the proof of Propo-
sition 1.
Proof of Proposition 5. By arguments analogous to the
proof of Proposition 1, problems (17) and (18) have optimal
solutions. We also observe that
d
2
0
e
de
2
≤ max
H − L
2
2
H − L
3
2k
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
148 Management Science 51(1), pp. 133–150, © 2005 INFORMS
and therefore (19) implies that the objective in (17) is strictly
concave, so (17) has a unique optimal solution.
Next, we will prove that any optimal solution e
P
1
e
P
2
to
(18) must be symmetric. Without loss of generality, suppose
that e
P
1
≥ e
P
2
. From the first-order conditions,
dge
P
1
de
−
dge
P
2
de
=
P
e
P
1
e
P
2
e
1
−
P
e
P
1
e
P
2
e
2
= 2
max
q
H
+q
L
≤2c
P
H −q
H
q
H
+L−q
L
q
L
−H −c
P
c
P
− max
q∈0c
P
L−qq
e
P
1
−e
P
2
≤
1
4
H −L
2
e
P
1
−e
P
2
(A.13)
(See Plambeck and Taylor 2003 for additional details as to
why the last inequality holds.) Condition (20) implies that
e
P
1
= e
P
2
.
Finally, (21) implies that the objective function
P
e e −
2ge is a strictly concave function of e, and problem (18)
has a unique optimal solution. Using the envelope theorem,
it is straightforward to obtain that
2
/e
2
1
P
e e has the
following form:
2
P
ee
e
2
1
=
1−e+e
2
H +k−2ek−2−3e+e
2
L
2
1+2e−2e
2
3
if c
P
ee ∈
0min
L
2
H −L
4
e
2
H +2k−3ek
2
4e−3e
2
3
if c
P
ee ∈
L
2
H −L
4
H −L
2
4
if c
P
ee ∈
H −L
4
L
2
2k−2ek+e
2
H −L
2
4e
3
2−e
3
if c
P
ee ∈
max
L
2
H −L
4
H +L
4
k
2
e
4
if c
P
ee ∈
H +L
4
H −k
2
2
/e
2
1
P
e e is written as a function of e instead of
c
P
e e in Plambeck and Taylor (2003).
Proof of Lemma 1. See Plambeck and Taylor (2003).
Proof of Proposition 6. It is helpful to define R
0
c e =
eH −cc+1−e max
q∈0c
L−qq. We will begin by show-
ing that ek HL ∈ 0 1. By the envelope theorem,
P
1 1
e
1
=
e
1
R
P
c
P
1 1 1 1
<
d
de
R
0
c
0
1 1 =
d
0
1
de
and therefore, ek H L < 1. If L<k, then ek H L ≥
e
0
> 0 On the contrary, if L ≥ k, then
P
0 0
e
1
=
e
1
R
P
c
P
0 0 0 0>
d
0
0
de
which implies that ekH L must be strictly positive.
To complete the proof, it is sufficient to establish the sin-
gle crossing property
P
e e
e
1
>
d
0
e
de
if e < ek H L
<
d
0
e
de
if e > ek H L
(A.14)
We will need the following results. If e ≥
2k/H − L, then
d
0
e
de
−
P
e e
e
1
=
2 − ek
2
4e
3
> 0
From the envelope theorem and the definition of ekH L,
R
P
c
P
e e ek H L ek H L
e
1
=
R
0
c
0
e ek H L
e
(A.15)
We also observe that for any c,
R
P
cee
e
1
−
R
0
c e
e
> 0ife<
1
2
≤ 0ife ≥
1
2
(A.16)
2
R
0
c e
ec
> 0 (A.17)
It is easy to verify
e ≤
2k/H − L ∧ 1 and that for all
e ∈ 1/2 ∨
e
0
e,
2
R
0
c
0
ee
ec
>
2
R
P
c
P
eee e
e
1
c
> 0
dc
0
e
de
>
dc
P
e e
de
> 0
Suppose ek H L > 1/2. Then, (A.15)–(A.17) imply
c
0
ek H L < c
P
ek H L ek H L. An immediate
corollary of Lemma 1 is that if c
0
e<c
P
e e, then
e<
e; hence, ek H L < e. The functions d/de
0
e and
/e
1
P
e e are differentiable with respect to e almost
everywhere, and for e ∈ ek H L
e,
d
de
d
0
e
de
−
P
e e
e
1
=
2
R
0
c
0
ee
e
2
−
2
R
P
c
P
eee e
e
2
1
+
2
R
0
c
0
ee
ec
·
dc
0
e
de
−
2
R
P
c
P
eee e
e
1
c
·
dc
P
e e
de
> 0
It is easy to verify
d
0
e
de
−
P
e e
e
1
> 0 for e ∈
e
2k
H − L
∧ 1
Thus, we have shown (A.14) if ek H L > 1/2. A related
argument establishes (A.14) if ekHL ≤ 1/2 (see Plam-
beck and Taylor 2003).
Proof of Proposition 7. If e
0
< minek H Le, then
c
0
e
0
≤ c
P
e
0
e
0
<c
P
e
P
e
P
, where the first inequality
Plambeck and Taylor: Impact of Contract Manufacturing on Innovation, Capacity, Profitability
Management Science 51(1), pp. 133–150, © 2005 INFORMS 149
holds because e
0
< e (by Lemma 1) and the second inequal-
ity holds because c
P
· · is increasing and e
0
< ekHL
implies e
0
<e
P
(by Proposition 6). If e
0
> maxek H L
¯
e,
then c
0
e
0
>c
P
e
0
e
0
>c
P
e
P
e
P
, where the first inequal-
ity holds because e
0
>
¯
e and the second inequality holds
because c
P
· · is increasing and e
0
> ekHL implies
e
0
>e
P
.
Proof of Proposition 8. From (A.4),
#
P
x y
e
1
−
P
x y
e
2
≤
#
4
H − L
2
x − y (A.18)
As in the proof of Proposition 3, symmetry follows from
the bounds (24) and (A.18). Uniqueness follows from (25)
by an argument analogous to that in the proof of Propo-
sition 3. Define Fea= #
P
e a − ge. Given that e
N
#
is the unique Nash equilibrium, application of the implicit
function theorem to the OEM’s first-order condition.
F ea
e
e=a=e
N
#
= 0
establishes (26).
It remains to show that for the parametric region (27),
e
N
# is an equilibrium in dominant strategies and (24)
ensures that it is the unique equilibrium. (27) implies
H<3L.Fore ∈ e
e, with
e
=
H + 2k − 3L
H − L
− a and
e =
2k
H − L
− a
the optimal capacity is
c
P
e a =
H − Le + a + 2L − k
4
(A.19)
Condition (27) implies that e
≤ 0 and e ≥ 1. Hence, c
P
e a
is given by (A.19) for every e ∈ 0 1. Restatement of (22)
yields e
N
1
# = arg max
e∈0 1
Fea
a=e
N
2
#
. With a little effort
one can show that the first-order condition is
F ea
e
= #
H − L
2
4
e+
H − L3L + H − 4k
8
−
dge
de
= 0
(A.20)
Further,
2
Fea
e
2
=
#H − L
2
4
−
d
2
ge
de
2
< 0 (A.21)
where the inequality follows from (24). (A.21) ensures that
the OEM’s optimal success probability is the unique solu-
tion to (A.20). Because (A.20) is independent of a, the
OEM’s optimal success probability e does not depend on
the success probability of the other OEM, a. We conclude
that (A.20) characterizes a unique equilibrium in dominant
strategies.
Proof of Proposition 9. The results on symmetry and
uniqueness are obtained by arguments analogous to the
proof of Proposition 4 (see Plambeck and Taylor 2003 for a
detailed proof), and (30) is a straightforward extension of
the proof of Proposition 6.
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