OSIPP Discussion Paper : DP-2008-E-011

When Market Competition Benefits Firms
September 18, 2008
Junichiro Ishida
Osaka School of International Public Policy (OSIPP), Osaka University

Toshihiro Matsumura
Institute of Social Science, University of Tokyo

Noriaki Matsushima
Graduate School of Business Administration, Kobe University【キーワード】competition, oligopoly, R&D, heterogeneity, entry【 要 約 】 A conventional wisdom in economics posits that more intense market
competition, measured in almost any way, reduces firm profit. In this paper, we
challenge this conventional wisdom in a simple Cournot model with strategic R&D
investments wherein an efficient firm (dominant firm) competes against less efficientfirms (fringe firms). We find that an increase in the number of fringe firms canstimulate R&D by the dominant firm, while it always reduces R&D by each of thefringe firms. More importantly, this force can be strong enough to compensate for theloss that arises from more intense market competition: the dominant firm's profit may
indeed increase with the number of fringe firms, quite contrary to the conventional
wisdom. An implication of this result is far-reaching, as it gives dominant firms to
help, rather than harm, fringe competitors. We relate this implication to a practiceknow as open knowledge disclosure, especially Ford's strategy of disclosing its
know-how publicly and extensively at the beginning of the 20th century.

Correspondence to: Junichiro Ishida, OSIPP, Osaka University, 1-31 Machikaneyama, Toyonaka,
Osaka 560-0043, Japan. Phone: +81-6-6850-5634. E-mail: jishida@osipp.osaka-u.ac.jp

When Market Competition Benefits Firms∗
Junichiro Ishida†
Osaka School of International Public Policy, Osaka University
Toshihiro Matsumura‡Institute of Social Science, University of Tokyo
Noriaki Matsushima§
Graduate School of Business Administration, Kobe University
September 18, 2008

Abstract
A conventional wisdom in economics posits that more intense market competition,measured in almost any way, reduces firm profit. In this paper, we challenge this conventional wisdom in a simple Cournot model with strategic R&D investments wherein
an efficient firm (dominant firm) competes against less efficient firms (fringe firms). We
find that an increase in the number of fringe firms can stimulate R&D by the dominantfirm, while it always reduces R&D by each of the fringe firms. More importantly, thisforce can be strong enough to compensate for the loss that arises from more intense
market competition: the dominant firm’s profit may indeed increase with the number offringe firms, quite contrary to the conventional wisdom. An implication of this result is
far-reaching, as it gives dominant firms to help, rather than harm, fringe competitors.
We relate this implication to a practice know as open knowledge disclosure, especiallyFord’s strategy of disclosing its know-how publicly and extensively at the beginning of
the 20th century.

JEL classification: L13, O32, M21
Key words: competition, oligopoly, R&D, heterogeneity, entry∗
We would like to thank Hiroaki Ino, Hiroshi Kitamura, Shinjiro Miyazawa, Keizo Mizuno, and seminar participants at Kwansei Gakuin University for their helpful comments. Of course, any errors are theresponsibility of the authors.
†
Corresponding author: Junichiro Ishida, Osaka School of International Public Policy, Osaka University, 1-31 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. Phone: +81-6-6850-5634. E-mail:jishida@osipp.osaka-u.ac.jp‡
Toshihiro Matsumura, Institute of Social Science, University of Tokyo, 7-3-1, Hongo, Bunkyo, Tokyo,
113-0033, Japan. Phone: +81-3-5841-4932. E-mail: matsumur@iss.u-tokyo.ac.jp§
Noriaki Matsushima, Graduate School of Business Administration, Kobe University, 2-1 Rokkodai, Nada,
Kobe, Hyogo, 657-8501, Japan. Phone: +81-78-803-6981. E-mail: nmatsush@kobe-u.ac.jp

1

Introduction

A conventional wisdom in economics posits that more intense market competition, measured
in almost any way, reduces firm profit. Consequently, it should be in the best interest of
profit-maximizing firms to reduce the degree of market competition, especially if it can bedone through some legal means. There are indeed several means to achieve this end. One
way is to gain some market power through product differentiation. Another, though it issometimes subject to some legal restrictions, is to reduce the number of competitors in agiven market through means such as collusion, entry deterrence, predation and horizontal
merger. In any event, firms are supposed to earn higher profits if they can place themselvesin less competitive environments.
In this paper, we challenge this conventional wisdom, in particular asking the following
question: do firms always dislike intense market competition? At a glance, this negative relationship between market competition and firm profit in fact seems to stand fairly robustly instandard oligopoly models. As a typical example, consider an n-firm Cournot model. In thissetup, one can easily show that each individual firm’s profit declines as n increases, thereby
lending strong support to the conventional wisdom that firms dislike intense competition.
This conclusion also seems to be robust when the baseline model is augmented with costreducing R&D investments. The intuition behind this is simple and goes as follows. Since
an increase in R&D investments lowers the marginal cost at the expense of a rise in the fixed
cost, the investing firm can naturally gain more when it produces more. Since an increase inn generally lowers each firm’s output, that makes it harder for each firm to exploit the scaleof economies. As it stands, therefore, an increase in n unambiguously lowers the investment
level and, as a natural consequence of this, the equilibrium profit.
Despite this seemingly convincing intuition, however, we claim in this paper that thisis not a result that generally holds true; we rather argue that this is a mere artifact of thespecification that the firms are symmetric with respect to the initial efficiency (which may
be defined broadly).1 In this paper, we provide a simple model which defies the conven1

In this paper, the initial productivity is defined in terms of the ex ante marginal cost (before any

investment takes place), but a similar conclusion is expected to arise when it is defined in a variety of other1tional wisdom that firms always dislike intense market competition in an otherwise standard
Cournot environment with strategic R&D investments. Our model does not rely on strategic
market preemption or collusion; we instead take a more direct route, focusing on the (somewhat neglected) nature of cost-reducing R&D investments. The basic setup is deceptively
simple and standard except for one twist: the firms are asymmetric with respective to theinitial productive efficiency.2 More precisely, we consider a market consisting of one dominant firm and many fringe firms that are equally less efficient. Within this setup, we explore
how the intensity of market competition, measured by an increase in the number of (fringe)
firms, affects each firm’s incentive for R&D investments as well as its resulting equilibrium
profit.
This simple and seemingly standard setting yields several counterintuitive results, twoof which are particularly illuminating and worth emphasizing here. First, we show that anincrease in the number of firms increases the dominant firm’s incentive for R&D investments
for a wide range of parameter values. Second, when this effect is strong enough, an increasein the number of firms also increases the dominant firm’s equilibrium profit, quite contrary
to the conventional wisdom. These results are in stark contrast to the standard setup where
an increase in the number of firms typically reduces the amount of R&D investments, notto mention the equilibrium profits. Our model suggests that more intense competition, inthe sense that there are more competitors in the market, is not always a bad news for firms
with advanced technologies. This fact leads to implications that are rather far-reaching,because it gives those dominant firms a reason to help, rather than harm, fringe competitors
in the market. We later relate our results to a practice known as open knowledge disclosure,especially focusing on a compelling case of Ford Motor Company back at the turn of the20th century, when we discuss this implication of the model.
ways. We make some comments on this in the concluding section.
2

The standard model of strategic R&D is formulated by Brander and Spencer (1983), and the litera-ture dealing with strategic R&D competition is now fairly abundant. See Spence (1984), d’Aspremont andJacquemin (1988), Suzumura (1992), Lahiri and Ono (1999), and Kitahara and Matsumura (2006). The lasttwo papers also focus on initial cost difference among firms but dos not investigate the relationship between
market competition and R&D.

2

The key to our argument is the ex ante productivity differential among competing firms
and its impact on the strategic incentive for R&D investments. The investment to reduce its
own marginal cost works as a commitment to expand its production, which crowds out theother firms’ output by some margin. We show that this strategic gain is actually increasingin the number of competitors, as a unit decrease in the marginal cost can affect more firms
when more of them are around. When this gain is sufficiently large, it can actually induce
some firms to invest more and consequently makes them more efficient. As we will see, this
could indeed happen for the dominant firm, but never for the fringe firms, implying that
more intense market competition tends to widen the dispersion in the ex post marginal costs.
As the market becomes more competitive, each fringe firm becomes less motivated and lessefficient, consequently leaving more rents to be exploited by the dominant firm. This gives
competitive edges to the dominant firm and, when this effect is strong enough to compensate
for the loss that arises from more intense market competition, so does its profit.
Several recent studies have raised instances where an increase in the number of firms may
actually increase firm profit. Coughlan and Soberman (2005), Chen and Riordan (2007),and Ishibashi and Matsushima (Forthcoming) belong to this strand, but the underlying
mechanism of our model differs substantially from that in theirs.3 In those previous studies,market entry works as a commitment device to soften market competition, so that themarket actually becomes less competitive as firms enter into it. In contrast, the dominantfirm in our model benefits more directly from intense market competition. This difference issummarized most succinctly by the following (possibly empirically testable) feature of ourmodel: the equilibrium market price is decreasing in the number of firms in our model while
it is increasing in theirs. To the best of our knowledge, this paper is the first to suggest
a channel though which more intense market competition per se benefits individual firms.4
Although we do not know a priori which scenario is more plausible, as it certainly depends
3

There is also a line of works which argue that having weak competitors might help because it keeps

stronger competitors out of the market. See, for instance, Ashiya (2000) on this point.
4

In a bilateral oligopoly model, Naylor (2002) shows that an increase in the number of downstream firms

could increase aggregate industry profits because more intense downstream competition shifts bargaining
power in favor of the downstream firms.

3

on the minute details of the underlying structure, this fact may be used to discriminate
between our mechanism and other existing ones when examining more specific cases.
The paper proceeds as follows. The next section outlines the basic environment and illustrates fundamental forces behind our setup. Building on this intuition, section 3 provides
a more detailed analysis of the model and derives main results. Section 4 discusses implications of the model, especially relating it to a practice known as open knowledge disclosure
and what we call the Ford story. Finally, section 5 offers some concluding remarks.

2

Cournot with strategic R&D investments revisited

2.1

Setup

In order to make our point as emphatically as possible, we stick with a simple and standardized version of Cournot competition, augmented with cost-reducing R&D investments,
as much as possible. Consider an industry consisting of n firms, each denoted by i ∈{1, 2, ..., n} ≡ N . The model has two stages: each firm first chooses the investment level,
which subsequently determines its marginal cost, and then engages in Cournot competition,
given the realized marginal costs.
More precisely, in the first stage, each firm determines how much to invest in cost-reducing
R&D activities. Let xi denote the investment level chosen by firm i. A unit increase in theinvestment decreases the firm’s marginal cost by the same margin. The total production cost
incurred by the firm is thus given by (zi − xi )qi , where qi denotes the output level chosen byfirm i. In this specification, zi signifies the ex ante marginal cost (before the investment),
while ci ≡ zi −xi the ex post marginal cost (after the investment). The cost of the investment
is denoted by I(xi ) and assumed to satisfy the usual properties: I ′ > 0, I ′′ > 0, I ′ (0) = 0
and limx→∞ I ′ (x) = ∞.
In the second stage, upon observing xi for all i ∈ N , the firms engage in standard Cournot(quantity) competition. The inverse demand function is specified as
p = 1 − Q,
where Q =Pi∈N qi(1)

is the total output. Each firm simultaneously chooses qi so as to4

maximize its own profit.
As can be seen, the basic setup is a text-book Cournot model with strategic R&D investments, only with one exception: we allow zi to differ across firms in a particular way. Morespecifically, we consider a situation where there is one dominant firm and n − 1 fringe firms
by assuming that0 < z1 < zf < 1, where zf ≡ z2 = . . . = zn .
For expositional clarity, in what follows, we refer to firm 1 as dominant while all of theothers as fringe (each of which is often denoted by subscript f in the subsequent analysis).
Evidently, the key factor here is that the firms are ex ante asymmetric, not that there is one
firm that stands out.5

2.2

Equilibrium

In this subsection, we go through the optimization problems faced by each firm. Since themodel is standard enough, we only briefly sketch the outline of the analysis.
In the second stage, given the realized marginal costs for all i ∈ N , each firm chooses its
output qi . Each firm’s problem in this stage can be written as!
nXmax πi ≡ 1 −qi − ci qi .
qi

i=1

Assuming that the interior solutions exist, the equilibrium output is obtained asP
1 + j6=i (zj − xj ) − n(zi − xi )
qi =
.
n+1

(2)

It follows from this that the optimal output is a function of the investment levels, and wewrite qiE (x1 , x2 , . . . , xn ), where superscript E henceforth denotes the equilibrium value of
any respective endogenous variable. This immediately leads us to obtain the equilibrium(gross) profit of each firm:
πiE (x1 , x2 , . . . , xn ) =

1−

n
X
i=1

qiE − ci

!

qiE =

(1 +

P

2
j6=1 (zj − xj ) − n(zi − xi )),(n + 1)2(3)

which is also a function of the investment levels.
5

Our main results hold as long as the firms are asymmetric with respect to the ex ante marginal cost,

although this specification illustrates our point in perhaps the most striking way.

5

In the first stage, each firm simultaneously decides how much to invest in reducing themarginal cost. The first-period problem faced by each firm is defined as choosing xi tomaximize the net profit Πi , taking the other firms’ choices as given:maxxi

Πi (x1 , x2 , . . . , xn ) ≡ πiE (x1 , x2 , . . . , xn ) − I(xi ).

For expositional purposes, we dissect each firm’s gains into two segments and term them(somewhat loosely) as strategic and non-strategic. The investment to reduce the marginal
cost works as a strategic commitment to expand its production, which crowds out the otherfirms’ output by some margin. This in turn raises the equilibrium price and consequently
benefits the investing firm even with its output fixed. We refer to this gain as the strategic
gain of the R&D investment. In contrast, we refer to any other benefits that accrue from the
R&D investment broadly as the non-strategic gain, mostly for expositional purposes. Using
these terms, the first-order condition can be decomposed asnqiE + (pE − ci − qiE )|{znon-strategic gain

∂qiE X ∂qjE E−
q = I ′ (xi ).
∂xi
∂xi i
j6=i
}|
{z
}

(4)

strategic gain

For the moment, we suppose that I ′′ (xEi ) is large enough to satisfy the second-order condition.

2.3

Conventional wisdom?

It is pervasively believed that, in standard oligopoly models, an increase in the number offirms strictly decreases firm profit. It is also believed that this conclusion is robust to theaddition of cost-reducing R&D investments because an increase in the number of firms lowers
each firm’s output, which only reduces the incentive to invest in reducing the marginal cost.
Despite this seemingly robust intuition, however, we claim that this is not a result thatgenerally holds true. Here, we illustrate how this slight alteration to the basic structure of
the model potentially changes the market outcomes.
Our focus is on the marginal gain of the R&D investment, which is captured by theleft-hand side of (4), and especially how it responds to a change in the number of firms n.
Although the marginal gain is believed to be strictly decreasing in n, this does not necessarily
hold when the firms are inherently asymmetric. We show that a situation may arise where6

an increase in n actually raises the marginal value of the R&D investment for the dominantfirm and hence its investment level. More importantly, when this investment-stimulating
effect is strong enough, the dominant firm can benefit from an increase in n, i.e., intensecompetition is not always a bad news for firms with advanced technologies.
We now examine more closely whether and when the marginal gain increases with n. To
see this, we fix ci (i = 1, 2, . . . , n) (so that they are not necessarily the equilibrium values)
and see how an exogenous increase in n affects the marginal gain. Since the fringe firms are
all symmetric, the marginal gain for the dominant firm can be written as∂qfE E∂π1E∂q E= q1E + (pE − c1 − q1E ) 1 − (n − 1)
q ,
∂x1
∂x1
∂x1 1

(5)

where subscript f represents each fringe firm. Since ∂qiE /∂xi = n/(n + 1) and ∂qfE /∂xi =−1/(n + 1) from (2), the marginal gain is further reduced toq E + n(pE − c1 ) (n − 1)q1E
∂π1E
= 1
+
.
∂x1
n {z
+1
|
} | n{z+ 1 }non-strategic gain

(6)

strategic gain

A unit decrease in the marginal cost lowers the fringe firms’ total output by (n − 1)/(n + 1)
and hence raises the dominant firm’s profit by q1E (n − 1)/(n + 1). The strategic gain is
increasing in n since a unit decrease can affect more firms when n is large, which proves tobe critical in giving rise to our main results.
Substituting pE = 1 − q1E − (n − 1)qfE and qfE = (1 + c1 − 2cf )/(n + 1) into (6) yields
En(1 − (n − 1)qf − c1 )2n(1 − cf + n(cf − c1 ))
∂π1E
=
=
,
∂x1
n+1
(n + 1)2(7)

from which we obtain−2(n − 1)(1 − cf ) + 4n(cf − c1 )∂ 2 π1E=.
∂x1 ∂n(n + 1)3(8)

This is positive if and only if
cf − c1 >(n − 1)(1 − cf )
.
2n

(9)

This condition illuminates when the presence of fringe firms stimulates the dominant firm’s
investment. To see this, suppose that the firms are ex post symmetric, i.e., c1 = cf . (9) then7

becomes c1 > 1, which evidently never holds (as it is the necessary and sufficient condition
for the dominant firm to produce strictly positive output). In this symmetric case, therefore,
we end up with the result that we are all accustomed to: any increase in n always reduces themarginal gain and hence the investment level. Examining (9) more closely reveals, however,that this conclusion (that the marginal gain is decreasing in n) is not something that always
holds true. As can easily be seen, (9) is more likely to hold for any given n when thedispersion in the ex post marginal costs, i.e., cf − c1 , is sufficiently large.
The workings of the model perhaps become more transparent when we do the same
exercise for the fringe firms. The marginal gain for each fringe firm can be written as
!
E
E
E
∂q
∂q
∂πfE
∂q
f
f
1
= qfE + (pE − cf − qfE )−+ (n − 2)
(10)
qfE ,
∂xf
∂xf
∂xf
∂xf
where, with a slight abuse of notation, ∂πfE /∂xf denotes the marginal gain of the investment
for each fringe firm. This is further simplified to
∂πfE
∂xf

=

qfE + n(pE − cf )(n − 1)qfE

n {z
+1

n{z+ 1 }|non-strategic gain

+
}|

.

(11)

strategic gain

Since each fringe firm is smaller than the dominant firm, i.e., q1E > qfE , the strategic gain is
in general substantially weakened. Going through the same steps as above, we obtain
∂πfE
∂xf

=

n(1 − q1E − (n − 2)qfE − cf )n+1=

2n(1 + c1 − 2cf ),(n + 1)2(12)

which is decreasing in n regardless of c1 and cf , as long as 1 + c1 − 2cf > 0 (which is thenecessary and sufficient condition for each fringe firm to produce strictly positive output).
The conventional wisdom thus generally holds true for the fringe firms whose investment
always decreases with n.
In sum, we can make the following two observations concerning the impact of the numberof competitors on the incentive to invest:
Observation 1. The dominant firm’s investment may increase with n.
Observation 2. Each fringe firm’s investment always decreases with n.
An increase in n always reduces each fringe firm’s investment while it may increase thedominant firm’s. When this happens, intense market competition widens the dispersion in8

the ex post marginal costs. This is further reinforced by the fact that the investments are
strategic substitutes: the dominant firm responds to this by increasing the investment evenfurther to take advantage of the situation. When this effect works strongly enough, thedominant firm’s profit may also increase with n. In the next section, we will explore this
aspect of strategic investments in more depth, in an attempt to throw a new light at therelationship between competition and market outcomes.

3Competition and market outcomes in asymmetric oligopoly

3.1

Competition stimulates investments by the rich

The driving force of our model is the endogenous nature of the marginal costs. We thus
start with examining how an increase in n affects each firm’s incentive to invest in reducing
its marginal cost. Throughout this section, we work with a more tightly specified version of
the setup described above: in particular, we assume that I(xi ) = γxi 2 . This specification
allows us to parameterize the importance of the R&D investment where a small γ meansthat endogenous cost reduction is an important part of the production process.
With this cost function, we can obtain a closed-form solution as the first-order condition(4) for the investment level is now modified as
=
xE
1
xE
=
f

n((n + 1)γ − n + (n2 − 1)γzf − n((n + 1)γ − 1)z1 ),((n + 1)γ − n)((n + 1)2 γ − n)n((n + 1)γ − n + (n + 1)γz1 − (2(n + 1)γ − n)zf ).
((n + 1)γ − n)((n + 1)2 γ − n)

(13)
(14)

Notice that the second-order conditions are satisfied if and only ifn2− γ < 0.
(n + 1)2(15)

We assume that γ ≥ 1 so that the second order condition is satisfied for any given n.
We also need to check whether the interior solutions indeed exist, which we have thus
far taken for granted. To check this, we only need to look at the fringe firms since q1E > 0EE
Eand xE1 > 0 if qf > 0 and xf > 0. For the fringe firms, qf > 0 for any given n if and only ifz1 >((2γ − 1)n + 2γ)zf − ((γ − 1)n + γ)
≡ z1.
γ(n + 1)9(16)

Notice that a firm would not invest in cost reduction if it is to produce no output; therefore,
this is also the necessary and sufficient condition for xk > 0. If (16) fails to hold, xE
f =ΠE
f = 0, i.e., fringe firms virtually cease to exist. In this case, the dominant firm’s behavior
naturally becomes independent of n where
1 − zf − 2(1 − 2zf )γ1 − 2zf + z1 , if
< z1 ≤ z 1 ,
2γ
xE
=
1

 1 − z1 ,otherwise,
4γ − 1
1 − zf − 2(1 − 2zf )γ< z1 ≤ z 1 ,(1 − zf )2 − (1 − 2zf + z1 )2 γ, if
2γ
ΠE
=
1
γ(1 − z1 )2,otherwise,4γ − 1(17)(18)

Since this case is apparently uninteresting, in the subsequent analysis, we restrict our attention to the case where z1 > z 1 so that fringe firms have some role to play.
We have seen that an increase in n may induce the dominant firm to invest more when
the dispersion in the ex post marginal costs are large. As can easily imagined, we can make asimilar statement in terms of the dispersion in the exogenously given ex ante marginal costs.
Proposition 1 For any n > 1 and γ ≥ 1, (i) there exists some nonempty interval Z x ≡x
E(z 1 , z1x ) such that xE
1 is increasing in n if and only if z1 ∈ Z ; (ii) xf is decreasing in n.

Proof See Appendix.

The proposition makes two claims. The first claim is the more illuminating part, whichsays that the dominant firm’s R&D investment can be increasing in n, quite contrary to theconventional wisdom. The key factor turns out to be the relative location of zf and z1 , i.e.,the dispersion in the ex ante marginal costs between the dominant and the fringe firms. The
dominant firm is induced to make more investment as n increases when the fringe firms are
ex ante sufficiently (but not too) inefficient in a relative sense. Moreover, it says that Z x is
generally nonempty, meaning that we can always find some z1 such that ∂xE
1 /∂n > 0. These
results are in stark contrast to the standard setup where an increase in n typically reduces
the investment level because it implies smaller rents for innovating firms.

10These results imply that more intense competition tends to make the dominant firm evenmore dominant. To see this, note that the fringe firms always invest less as n increases, as in
the standard setup (the third claim). More intense competition thus always discourages thefringe firms but that is not necessarily the case for the dominant firm. As the market become
more competitive, therefore, the dispersion in the ex post marginal costs may become evenlarger. As we will see next, because of this effect, the dominant firm may actually benefit
from an increase in n.
More intense competition is more likely to induce the dominant firm to invest more when
the upperbound z1x is larger. Not surprisingly, this is the case when γ is relatively small,i.e., when R&D investments are sufficiently important in the production process. The next
proposition is a formal representation of this fact.
Proposition 2 For any n > 1 and γ ≥ 1, ∂z1x /∂γ < 0.
Proof Differentiating z1x with respect to γ, we have−(1 − zf )(n − 1)(n + 1)2 ((γ − 1)n + γ)((n + 1)2 (3n − 2)γ − n(n2 + 3n − 2))∂z1x=. (19)∂γn(2n − (n + 2)(n + 1)2 γ + 2(n + 1)3 γ 2 )2Note that (n + 1)2 (3n − 2)γ − n(n2 + 3n − 2) > 0 for any γ > 1 because, when γ = 1,
(n + 1)2 (3n − 2)γ − n(n2 + 3n − 2) = 2n3 + n2 + n − 2 > 0.
This means that the numerator of ∂z1x /∂γ is always negative.

(20)
Q.E.D.

The range for which the dominant firm’s investment increases with n can be best seengraphically. Figure 1 illustrates the range for two different values of γ. These examples
indicate that the dominant firm’s investment increases with n for a wide range of parametervalues.
[Figure 1 here]

3.2

Competition makes the rich get richer

When z1 ∈ Z x , more intense competition makes the dominant firm even more dominant in
the sense that the dispersion in the ex post marginal costs becomes wider. When this effect is11

strong enough to compensate the loss from more intense market competition, the dominantfirm’s profit may actually increase as more fringe firms enter the market. This can indeed
happen, as the next proposition indicates.
Proposition 3 For any n > 1 and γ ≥ 1, (i) there exists some nonempty interval Z P ≡PE(z 1 , z1P ) such that ΠE
1 is increasing in n if and only if z1 ∈ Z ; (ii) Πf is decreasing in n.

Proof See Appendix.

Proposition 3 is the main result of the paper, which basically runs parallel to Proposition
1, making two claims of similar nature. In particular, it again shows that Z P is nonempty
for any given n > 1 and γ ≥ 1, and hence we can always find z1 such that ∂ΠE
1 /∂n > 0.
As the market becomes more competitive, the dominant firm invests more (if z1 ∈ Z x ) andthe fringe firms invest less. As a consequence, the dispersion in the marginal costs gets evenlarger at the ex post stage when the firms engage in Cournot competition. The dominantfirm can benefit from this, even though ex post market competition becomes more severe asn increases.
Competition induces the dominant firm to invest more when γ is relatively small, andthis same logic apparently carries over to the dominant firm’s profit. The next proposition
is a sequel to Proposition 2, showing that the dispersion in the ex ante marginal costs needs
to be small in environments where R&D investments are sufficiently important.
Proposition 4 For any n > 1 and γ ≥ 1, ∂z1P /∂γ < 0.
Proof See Appendix.

Figure 2 illustrates the range of z1 for which the dominant firm’s profit is increasing in
n, again for two different values of γ. Naturally, the range for which the dominant firm’s
profit increases is narrower than that for which its investment increases. In fact, as can be
expected, the dominant firm’s profit increases with n only if its investment increases with n,i.e., the former is a necessary condition of the latter.
12

Proposition 5 For any γ ≥ 1 and n > 1, Z P ⊂ Z x .
Proof With some algebra we obtainz1x − z1P=

z1P − z 1 =(1 − zf )(n + 1)((n + 1)γ − n)2 ((n + 1)2 γ − n)2 ((n + 1)2 γ − (n2 + 1))> 0,n[2n − (n + 2)(n + 1)2 γ + 2(n + 1)3 γ 2 ]H
(1 − zf )n(n − 1)((n + 1)γ − n)((n + 1)2 γ − n)> 0,γ(n + 1)H
where H ≡ [n2 (1 − n) + n3 (n + 1)2 γ − (n + 1)3 (2n2 + 1)γ 2 + (n + 1)5 γ 3 ],for any γ ≥ 1 and n > 1, which proves the proposition (and also the nonemptiness of Z x
and Z P .

Q.E.D.

[Figure 2 here]

3.3

What happens at the aggregate level?

Several recent studies have raised instances where incumbents benefit from a new market
entry. In those previous cases, however, the equilibrium price actually rises with a new
entrant coming into the market: market competition becomes less severe despite the factthat there are now more firms in the market. What is common among those previous studies
is therefore that they find a channel through which a market entry somehow makes the market
less competitive. In Ishibashi and Matsushima (Forthcoming), for instance, a new entry into
the low-end market works as a commitment device for the incumbents not to supply to thatmarket (hence not to lower the price to accommodate low-end consumers). This is actually
profit-enhancing for the incumbents as they can focus on the high-end market, allowing themto charge a higher price to high-end consumers.
In contrast, our model works in a totally different way. The difference is succinctly summarized by a distinguishing, and empirically testable, feature of our model: the equilibrium
price is indeed decreasing in n so that the market becomes truly more competitive as nincreases. In our model, there is nothing unusual about the impact that an increase in n has
on the equilibrium market price. To see this, the equilibrium price is computed aspE =(n + 1)(1 + (n − 1)zf + z1 )γ − n,(n + 1)2 γ − n
13

(21)

which straightforwardly leads to the next result.
Proposition 6 For any n > 1 and γ ≥ 1, pE is decreasing in n.
Proof Differentiating pE with respect to n, we haveγ[(n + 1)2 (1 − 2zf + z1 )γ − (1 − zf )n2 + (zf − z1 )]
∂pE
=−
.
∂n
((n + 1)2 γ − n)2(22)

For z1 ≥ z 1 , the partial derivative is maximized at z1 = z 1 . Substituting z1 = z 1 into thepartial derivative, we have
1 − zf∂pE< 0.
=−∂n z1 =z(n + 1)((n + 1)2 γ − n)

(23)

1

Therefore, pE is decreasing in n.

Q.E.D.

To the best of our knowledge, our model is the first to show that an incumbent canbenefit from a market entry in an environment where the equilibrium price is decreasing in
n. Of course, our intention is not to exclude those existing views because there can be manychannels through which an incumbent benefits from a market entry. Which view is moreplausible depends certainly on the underlying structure and is purely an empirical matter
left for future research. We argue, however, that this fact provides a nice discriminating
factor when one examines what forces are more likely to be at work in each specific case.
The equilibrium price is decreasing in n necessarily means that the total quantity supplied
is increasing. This naturally raises a question about the composition of the total output (orthe market share). Let QE−1 denote the total quantity supplied by the fringe firms, which is
given byQE
−1 =γ(n2 − 1)[(1 − 2zf + z1 )(n + 1)γ − (1 − zf )n].
((n + 1)2 γ − n)((n + 1)γ − n)(24)

Surprisingly, the total output produces by the fringe firms may decrease with n.
Proposition 7 (i) For any n > 1 and γ ≥ 1, (i) there exists some nonempty interval
QPQx
Z Q ≡ (z 1 , zlQ ) such that QE−1 is decreasing in n if z1 ∈ Z ; (ii) Z ⊂ Z ⊂ Z .

14

Proof See Appendix.

It directly follows from Proposition 6 that the total quantity supplied is increasing inEn, as in standard Cournot models. This implies that if QE−1 decreases with n, then q1

must increase – yet another unusual property. This also implies that z1 ∈ Z Q is a sufficientcondition for the dominant firm’s market share to increase with n.
Finally, we briefly discuss the impact of market competition on the industry-wide rateof innovation. The question we ask here is whether market competition spurs or inhibits
innovations in our setup. To this end, suppose that we measure the industry-wide rate ofinnovation by the (effective) total investment M :M ≡ γx21 + (n − 1)γx2f .

(25)

Figure 3 provides some numerical examples of the relationship between M and n. As thefigure indicates, almost anything goes in our setup, and no clear prediction can hence bemade from this. First, if the dispersion in the ex ante marginal costs is small and z1 > zlx ,the dominant firm’s investment decreases with n, and the total investment is more likely todecrease (see two examples on the left-hand side of Figure 3). On the contrary, when thedispersion is large, the dominant firm invests more as n increases, even to the extent thatit more than offset a decrease in the total investment of the fringe firms. As a consequence,
the total investment is more likely to be increasing in n (see two examples on the right-handside). If the dispersion is in some intermediate range, the total investment is non-monotonewith respect to n (see examples at the middle). The last case where the dispersion falls intosome intermediate range is perhaps most interesting, as recent evidence seems to suggestthat the relationship between market competition and innovation is inverted U-shape, e.g.,
Aghion and Griffith (2005).
[Figure 3 here]
The literature examining the relationship between market competition and innovation isvery large and diverse, as it is certainly an old issue which have attracted attention of many15

economists (at the very least from the days of Schumpeter), and many models have been
proposed to account for observed patterns of the relationship between market competition
and innovation, e.g., Aghion et al. (2005). We thus do not intend to make too much out of
this, because we only analyze a very specific industry structure. If there is anything we caninsist on this, though, at least this much is certain: the distribution of (initial) productivity
matters, even in a simple Cournot framework like ours. This is an insight which, in our view,has not received enough attention, and it is of some interest to approach this issue from thisperspective, both theoretically and empirically.

4

Open knowledge disclosure: the Ford story

In the previous sections, we have analyzed a simple Cournot model with strategic R&D investments when the firms are inherently asymmetric. The most important message here isthat firms with more advanced technologies may benefit from having more (fringe) competitors. We argue that an implication that this leads to is rather far-reaching, because thosedominant firms actually have an incentive to help, rather than harm, minor competitors sothat they can remain just sufficiently competitive to stay in the market. In this section, we
focus on this implication of the model, relating it to a practice known as open knowledge
disclosure and especially the case of Ford back at the turn of the 20th century.
To discuss this issue, let us start with the following question: why are some firms moreproductive than others? There is one straightforward answer to this: if there is any sure way
to gain market power, it is to make new innovations over the existing ways of production
and commercialization. In fact, innovations, in a broad sense, are the typical and sometimes
the only source of competitive edges that a firm can gain over its existing or potential
competitors. For instance, a firm may drive competitors out of the market if it can attain alevel of efficiency that no one can catch up to. New ideas are also indispensable to invent aunique, differentiated, product that cannot easily be imitated. The problem, as it has beenclearly recognized, is that although innovative ideas are surely hard to come by, they arevery easy to copy once they are created: in fact, most ideas can be copied at almost no
cost. It is hence critical for innovating firms to establish and protect their innovative ideas,
16

provided that they can benefit from being in less competitive environments.
In reality, though, firms do not seem overly concerned about keeping “secrets” to themselves – at least not always. For instance, it is often suggested that a large fraction ofpatentable innovations are not patented (Mansfield, 1986).6 Firms are not simply recklesswith their secrets; they often go beyond just that as they intentionally and freely disclose
what appears to be critical knowledge, even to their direct competitors. Informal know-how
trading between competitors is very active, often though informal networks of process engineers (von Hippel, 1988). Employees frequently give technical information to colleagues inother firms, including direct competitors (Schrader, 1991). All of these seem to suggest thatfirms do not protect their innovations as carefully as the theory predicts.
Of course, some fraction of know-how trading occurs in a closed setting – the practiceoften referred to as closed knowledge disclosure.7 In this case, some reciprocal agreements,
either explicitly or implicitly, can in principle be made between the giver and the recipients,
barring many difficulties associated with trading ideas.8 In many instances, though, firms
not just freely and but also publicly give away critical knowledge through open channels suchas publications in scientific journals, presentation in conferences or more informal communication – a practice often referred to as open knowledge disclosure.9 This practice is highly
puzzling because, when a firm makes its critical knowledge publicly accessible, it inevitably
loses some, if not all, control over the diffusion of the know-how. Once the know-how becomes publicly available, it is virtually impossible for the disclosing firm to take it back or
to make any profit from it. No future exchange of favors can be expected either because it is6

A typical view on this is that patents are not always the best way to protect their ideas. This might bedue to the limited power of Intellectual Property (IP) rights. Anton and Yao (2004) focus on this aspect.
7

There is a scope for know-how trading especially if the industry profit increases with the number offirms. In this case, the concerned parties could reach an agreement if they can somehow manage to find away to appropriately divide the surplus (perhaps through some explicit contracts).
8

Trading an idea is difficult especially when its quality is uncertain: a recipient must see the idea toevaluate the quality but, once the idea is observed, there is no reason to buy it. Partial disclosure may be
optimal in a situation like this. See Anton and Yao (2004) for this.
9

A seminal work on open knowledge disclosure is Allen (1983) who argues that many new production

technologies have been developed by a process called “collective invention.” See Penin (2007) for further
evidence on open knowledge disclosure.

17

nearly impossible to identify the beneficiaries of the know-how when it is disclosed via open
channels.
One story stands out in this respect. At the beginning of the 20th century, Ford was
the dominant automobile producer which had attained an unprecedented level of efficiency
with radically new technologies. The pace of growth was astonishing, especially considering
the fact that the entire industry had also been growing at a rapid pace. Ford’s share of all
automobile production grew from 9.4% in 1908 to 48% in 1914.10 In it main filed of competition, i.e., the cheap car filed, Ford’s share was 96%, practically making it a monopolist. The
figures suggest Ford’s incredible presence in the industry, given that the industry was not
as concentrated as it is today.11 There were certainly many factors that had contributed toFord’s success, but the main source of its competitive edges was undoubtedly its production
efficiency made possible by several innovations and inventions such as the moving assemblyline system and Henry Ford’s scientific management (or so-called “Fordism”): in 1914, Ford,
only with 13,000 employees, manufactured 260,720 while all of the other companies, with
66,350 employees combined, manufactured mere 286,770. There was virtually no competition; Ford was simply too good. Surprisingly, though, Ford had no intention of “hiding its
secrets.” Nevins (1954, p.508) notes:
Engineers came from all over America and Europe to study this achievementin efficiently standardized and specialized production. Nothing was concealed.
Indeed, Henry Ford and his associates this year cooperated with the editors ofEngineering in laying before the world, in the technically detailed and richly
illustrated pages of Arnold and Faurote’s Ford Methods and the Ford Shops, ...
It is hard to believe that Ford expected something in return from those fringe competitors
which seemed to have nothing worthwhile to offer at the time. Why did Ford give away its
critical knowledge so generously? Our model provides a partial answer to this and, more
broadly, a sensible reason for open knowledge disclosure: there is a channel through which10

All figures are taken from Nevins (1954).

11

At the time, the number of manufacturers exceeded well above one hundred.

18

the innovating firm can benefit from disseminating its innovative ideas publicly. The Ford
story fits our description particularly well for two reasons: (i) endogenous cost reduction –a driving force of our model – is evidently an important factor in the automobile industry;(ii) Ford possessed apparently superior technologies and cost advantages. In a situation like
this, market competition tends to make a dominant firm even more dominant, and hence
eliminating fringe competitors was not necessarily in its best interest. When there are more
fringe competitors, competition among them becomes severer and that discourages them.
Fringe competitors make less investment and, consequently, become less efficient, which
actually works for the dominant firm. According to this logic, Ford had every reason to help,
rather than harm, fringe competitors so that they can remain (just sufficiently) competitivein the market.

5

Conclusion

In this paper, we revisit a fundamental question of market competition: do firms always
dislike intense market competition? We identify a situation under which the conventional
wisdom that intense market competition decreases firm profit fails to ascertain itself. In amarket consisting of inherently asymmetric firms, firms with advanced technologies can infact benefit from having more (fringe) competitors, indicating that the conventional wisdom
may not be as robust as generally believed. Our model also implies that in some cases, there
is a reason for dominant firms to help, rather than harm, fringe competitors, just for theirown sake (even without any spillover or network effects). This implication of the model
provides a plausible explanation for open knowledge disclosure, especially Ford’s strategy atthe beginning of the 20th century.
While our analysis takes the number of (fringe) firms as exogenous, we can easily extend
our model to incorporate free entry by fringe firms. Suppose that firms enter the market attheir own discretion by incurring some fixed entry cost. In a setup like this, an equilibrium
number of firms is determined by the entry cost, and a question in the spirit of the present
paper is whether a decrease in the entry cost, which typically intensifies market competition,
ever raises the dominant firm’s profit. In the context of strategic entry deterrence, Etro (2004,
19

2006) show that in free entry markets, a leaders engaging in strategic commitment prior to
the entry of followers make aggressive investments to deter their entry. Note, however, thatin his setting, the leader’s profit is never decreasing in the entry cost of followers: in thissense the models of entry deterrence do not yield our implication on market competition andfirm profit.12 In contrast, in our setup, one can show that a decrease in the entry cost canindeed raise the dominant firm’s profit, even when firms first enter and then simultaneously
determine the investment levels.
It is our view that the present analysis provides only a first step to better understandthe nature of market competition in asymmetric oligopoly, and hence that there are several
avenues to extend the current analysis. First, our results should hold even the firms are ex
ante asymmetric in more broad senses. More specifically, although the initial efficiency isdefined only in terms of the ex ante marginal cost of production, a similar conclusion holds
when it is defined, for instance, in terms of the efficiency of cost reduction, (measured by γ
in the model). It is hence an important task to see how far we can push the logic present in
our model.
Second, we only examine a particular industry structure – one dominant firm and n − 1equally inefficient fringe firms – to make our points in a relatively clear manner, but themodel’s implications are certainly not restricted to this structure. Our main contention
is rather that the distribution of initial productivity matters, for the incentive for R&D
investments and the resulting equilibrium profits. We believe that this is an important insightespecially when we examine the relationship between market competition and innovation,and it is of some interest to pursue this aspect, both theoretically and empirically, in future.
12

For the discussion of R&D competition in Stackelberg settings, see also Ino and Kawamori (Forthcoming).

20

Appendix: proofs
Proof of Proposition 1 (i) From (13) we haveγ[n2 (n2 + 1) − 3n2 (n + 1)2 γ + (n + 1)3 (3n − 1)γ 2 ]zf
∂xE
1
=
,
∂n
((n + 1)γ − n)2 ((n + 1)2 γ − n)2γ[(n2 − 1)((n + 1)γ − n)2 + (2n2 − n(n + 1)2 (n + 2)γ + 2n(n + 1)3 γ 2 )z1 ]−, (26)((n + 1)γ − n)2 ((n + 1)2 γ − n)2which is positive if and only ifz1 <n2 (1 − n2 ) + 2(n − 1)n(n + 1)2 γ − (n − 1)(n + 1)3 γ 2n(2n − (n + 2)(n + 1)2 γ + 2(n + 1)3 γ 2 )[n2 (1 + n2 ) − 3n2 (n + 1)2 γ + (3n − 1)(n + 1)3 γ 2 ]zf+≡ z1x ; (27)n(2n − (n + 2)(n + 1)2 γ + 2(n + 1)3 γ 2 )For the nonemptiness, see the Proof of Proposition 5.
(ii) From (14) we have
∂xE
f
∂n

= −γ



2[(n − 1)(n + 1)3 γ 2 − n2 (n + 1)2 γ + n2 ]z1
2((n + 1)γ − n)2 ((n + 1)2 γ − n)2n[n(n2 − 2) − (n − 2)(n + 1)2 γ]+2((n + 1)γ − n)2 ((n + 1)2 γ − n)2[2(n − 1)(n + 1)3 γ 2 − n(n + 1)2 (3n − 2)γ + n4 ]zf−, (28)
2((n + 1)γ − n)2 ((n + 1)2 γ − n)2which is maximized at z1 = z 1 (because ∂xE
f /∂n is strictly decreasing in z1 ). Evaluating thepartial derivative at this value, we have
∂xE
f
∂n

=−
z1 =z 1n(1 − zf )< 0.
2(n + 1)((n + 1)γ − n)((n + 1)2 γ − n)(29)

Therefore, under the maintained assumptions, xE
f is strictly decreasing in n as long as z1 > z 1
and xEf > 0.

Q.E.D.

Proof of Proposition 3 (i) Given the equilibrium investment levels, we can obtain the
net profits as follows:γ((n + 1)2 γ − n2 )((n + 1)γ − n + (n2 − 1)γzf − n((n + 1)γ − 1)z1 )2, (30)((n + 1)γ − n)2 ((n + 1)2 γ − n)2γ((n + 1)2 γ − n2 )((n + 1)γ − n + (n + 1)γz1 − (2(n + 1)γ − n)zf )2ΠE=. (31)
f((n + 1)γ − n)2 ((n + 1)2 γ − n)2
ΠE
1 =

21

From (30), we have∂ΠE1= 2γ 2 ((n + 1)γ − n + (n2 − 1)γzf − n((n + 1)γ − 1)z1 )×∂n 
n2 (n3 + 1) − n(n + 1)2 (3n2 − n + 2)γ + (n + 1)3 (3n2 + n + 1)γ 2 − (n + 1)5 γ 3((n + 1)γ − n)3 ((n + 1)2 γ − n)3[n2 (1 − n) + n3 (n + 1)2 γ − (n + 1)3 (2n2 + 1)γ 2 + (n + 1)5 γ 3 ]z1−((n + 1)γ − n)3 ((n + 1)2 γ − n)33
2
2[n (1 + n ) − n(n + 1) (4n2 − n + 2)γ + (n + 1)3 (5n2 + n + 2)γ 2 − 2(n + 1)5 γ 3 ]zf,−((n + 1)γ − n)3 ((n + 1)2 γ − n)3(32)
which is positive if and only if
z1 < −(n + 1)((n + 1)γ − n)2 ((n + 1)2 γ − (n2 − n + 1))[n2 (1 − n) + n3 (n + 1)2 γ − (n + 1)3 (2n2 + 1)γ 2 + (n + 1)5 γ 3 ]+[−n3 (n2 + 1) + n(n + 1)2 (4n2 − n + 2)γ]zf[n2 (1 − n) + n3 (n + 1)2 γ − (n + 1)3 (2n2 + 1)γ 2 + (n + 1)5 γ 3 ][−(n + 1)3 (5n2 + n + 2)γ 2 + 2(n + 1)5 γ 3 ]zf
+ 2
≡ z1P
[n (1 − n) + n3 (n + 1)2 γ − (n + 1)3 (2n2 + 1)γ 2 + (n + 1)5 γ 3 ](33)For the nonemptiness, see the Proof of Proposition5.
(ii) From (31) we have
∂ΠE
f
∂n

= −2γ 2 (γ(n + 1)z1 + ((γ − 1)n + γ) − ((2γ − 1)n + 2γ)zf )K(n, γ),

(34)

where
[(n + 1)5 γ 3 − (n + 1)3 (2n2 + n + 2)γ 2 + n(n + 1)2 (n2 + 2)γ − n3 ]z1((n + 1)γ − n)3 ((n + 1)2 γ − n)353
32[2(n + 1) γ − (n + 1) (5n + 2n + 3)γ 2 + n(n + 1)2 (4n2 − n + 4)γ − n2 (n3 + n + 1)]zf−((n + 1)γ − n)3 ((n + 1)2 γ − n)3(n + 1)((n + 1)γ − n)2 ((n + 1)2 γ − (n2 − n + 1))+. (35)((n + 1)γ − n)3 ((n + 1)2 γ − n)3
K(n, γ) =For any z1 ≥ z 1 , (γ(n + 1)z1 + ((γ − 1)n + γ) − ((2γ − 1)n + 2γ)zf ) is nonnegative. We
can thus prove the proposition if K(n, γ) > 0. Note that since K(n, γ) is increasing in z1 , itsuffices to show this at z1 = z 1 :(1 − zf )((γ − 1)n2 + 2γn + γ)
> 0.
(n + 1)γ((n + 1)γ − n)2 ((n + 1)2 γ − n)222This shows that for any n, ΠE
f is decreasing in n.

Q.E.D.

Proof of Proposition 4 Differentiating z1P with respect to γ, we have
(1 − zf )n(n + 1)2 ((γ − 1)n + γ)
∂z1P
= J(n, γ) 2,
∂γ[n (1 − n) + n3 (n + 1)2 γ − (n + 1)3 (2n2 + 1)γ 2 + (n + 1)5 γ 3 ]2

(36)

where
J(n, γ) ≡ n(n5 − 3n3 + 5n2 − 3n + 2) − (n + 1)2 (3n4 − 3n3 + 2n2 + n + 2)γ+ (n + 1)4 (3n2 − 3n + 4)γ 2 − (n + 1)6 γ 3 .

(37)Note that ∂z1P /∂γ < 0 if and only if J(n, γ) < 0. To show this, we first obtain∂J(n, γ)∂γ

= −(n + 1)2 (3n4 − 3n3 + 2n2 + n + 2)+2(n + 1)4 (3n2 − 3n + 4)γ − 3(n + 1)6 γ 2 ,∂ 2 J(n, γ)∂γ 2∂ 3 J(n, γ)∂γ 3= 2(n + 1)4 (3n2 − 3n + 4) − 6(n + 1)6 γ,= −6(n + 1)6 < 0.

Substituting γ = 1 into J(n, γ), ∂J(n, γ)/∂γ, and ∂ 2 J(n, γ)/∂γ 2 yield
J(n, 1) = −(7n4 + 7n3 + 9n2 − 4n − 1) < 0,∂J(n, γ)= −3(n + 1)2 (n3 + 6n2 + n − 1) < 0,∂γγ=1
∂ 2 J(n, γ)= −2(n + 1)4 (9n − 1) < 0,∂γ 2γ=1which assures that J(n, γ) is always negative.

Q.E.D.

Proof of Proposition 7 (i) From (24), we have∂QE((n + 1)γ − n)(2(n + 1)2 γ − (n2 + 1))
−1
=γ×
∂n
((n + 1)γ − n)2 ((n + 1)2 γ − n)22
2[n (n + 1) − n(n + 1)(5n2 + n + 4)γ + (n + 1)2 (7n2 + 6n + 3)γ 2 − 4(n + 1)4 γ 3 ]zf+((n + 1)γ − n)2 ((n + 1)2 γ − n)2
(n + 1)γ[n(n2 − n + 2) − 2(n + 1)(n2 + n + 1)γ + 2(n + 1)3 γ 2 ]z1+. (38)((n + 1)γ − n)2 ((n + 1)2 γ − n)223This is positive if and only if
z1 <((n + 1)γ − n)2 (2(n + 1)2 γ − (n2 + 1))(n + 1)γ[n(n2 − n + 2) − 2(n + 1)(n2 + n + 1)γ + 2(n + 1)3 γ 2 ][−n2 (n2 + 1) + n(n + 1)(5n2 + n + 4)γ]zf+(n + 1)γ[n(n2 − n + 2) − 2(n + 1)(n2 + n + 1)γ + 2(n + 1)3 γ 2 ][−(n + 1)2 (7n2 + 6n + 3)γ 2 + 4(n + 1)4 γ 3 ]zf≡ z1Q . (39)+(n + 1)γ[n(n2 − n + 2) − 2(n + 1)(n2 + n + 1)γ + 2(n + 1)3 γ 2 ]For the nonemptiness, see the next part (ii) of this proof.
(ii) With some algebra we obtainz1x − z1Q =z1Q − z1P=2(1 − zf )((n + 1)γ − n)2 ((n + 1)2 γ − n)2 ((n + 1)2 γ − (n2 + 1))> 0,
n(n + 1)γ[2n − (n + 2)(n + 1)2 γ + 2(n + 1)3 γ 2 ]H ′(1 − zf )(n − 1)((n + 1)γ − n)2 ((n + 1)2 γ − n)2 ((n + 1)2 γ − (n2 + 1))> 0,
γ(n + 1)HH ′where H = [n2 (1 − n) + n3 (n + 1)2 γ − (n + 1)3 (2n2 + 1)γ 2 + (n + 1)5 γ 3 ],H ′ ≡ [n(n2 − n + 2) − 2(n + 1)(n2 + n + 1)γ + 2(n + 1)3 γ 2 ],which proves the proposition.

Q.E.D.

24

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von Hippel, Eric. 1988. The Source of Innovation. New York: Oxford University Press.

26

0.45

zl
0.35

0.4

0.25

0.35

0.15

zl

0.3
0.2
0.1
5

10

15

20

25

30

n

0.05

0.25

5

(γ = 3/2)

10

15

20

(γ = 10)Figure 1: The range of z1 (∂x1 /∂n > 0, zf = 1/2).

27

25

30

n

zl

zl

0.42

0.27
5

10

15

20

25

0.38

30

n

0.26
0.25
0.24

0.36

0.23

0.34

0.22

0.32

5

(γ = 6/5, zf = 1/2)

10

15

20

25

(γ = 9/5, zf = 1/2)

Figure 2: The range of z1 (∂Π1 /∂n > 0).

28

30

n

M
0.00555
0.00550.005450.00540.00535
0.00530.00525

M
0.00350.00325
0.003
0.00275
0.00250.00225
5

10

15

20

25

30

M
0.00760.00740.00725n5(z1 = 2/5, γ = 10)

10

15

20

25

30

n

15

20

25

30

n

0.0066(z1 = 3/10, γ = 10)M

100.0068(z1 = 1/4, γ = 10)M
0.0306M

0.0304

0.035

0.026
0.024
0.0302

0.034

0.022
5

10

15

20

25

30

5

n

10

15

20

25

30

n
0.033

0.02980.018
0.0296(z1 = 0.45, γ = 3/2)

5(z1 = 0.428, γ = 3/2)

10

15

25

(z1 = 0.42, γ = 3/2)

Figure 3: The relation between M and n (zf = 1/2).

29

20

30

n