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INCENTIVES TO INNOVATE IN OLIGOPOLIES*manc_2131 6..28

by

PAUL BELLEFLAMME

CORE and Louvain School of Management, Université catholique de

Louvain, Belgium

and

CECILIA VERGARI†

CORE, Université catholique de Louvain, Belgium, and University of

Bologna and University of Roma Tre, Italy

In the spirit of Arrow (The Rate and Direction of Inventive Activity,

Princeton, NJ, Princeton University Press, 1962), we examine, in an

oligopoly model with horizontally differentiated products, how much a

ﬁrm is willing to pay for a process innovation that it would be the only

one to use. We show that different measures of competition (number of

ﬁrms, degree of product differentiation, Cournot vs. Bertrand) affect

incentives to innovate in non-monotonic, different and potentially

opposite ways.

1Introduction

Our objective in this paper is to explore in a systematic way how competition

affects ﬁrms’ incentives to innovate. Thereby, we aim at exploring further the

relationship between market structure and innovation incentives, which has

always been a central, and debated, issue in economics since Schumpeter’s

classic work, Capitalism, Socialism, and Democracy, in 1943.

1.1 Context

Schumpeter’s ﬁrst conjecture was to stress the necessity of tolerating the

creation of monopolies as a way to encourage the innovation process. This

argument is nothing but the economic rationale behind the legal protection of

intellectual property and is nowadays widely accepted. Schumpeter’s second

conjecture was that large ﬁrms are better equipped to undertake R&D than

smaller ones. The best way to support this conjecture is probably to say that

large ﬁrms have a larger capacity to undertake R&D, in so far as they can

deal more efﬁciently with the three market failures observed in innovative

markets, namely externalities, indivisibilities and uncertainty. It is not clear,

however, whether large ﬁrms, because of their monopoly power, also have

larger incentives to undertake R&D.

* Manuscript received 22.1.07; ﬁnal version received 27.11.07.

†We are grateful to Vincenzo Denicolò, Jean Gabsziewicz, Antonio Minniti, Pierre Picard,

Mario Tirelli, Vincent Vannetelbosch and seminar audiences at Kiel for useful comments

and discussion about an earlier draft.

The Manchester School Vol 79 No. 1 6–28 January 2011

doi: 10.1111/j.1467-9957.2009.02131.x

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK, and 350 Main Street, Malden, MA 02148, USA.

6

The pioneering paper studying the effect of market structure on the

incentives for R&D is Arrow (1962). He compares the proﬁt incentive to

innovate for monopolistic and competitive markets, concluding that perfect

competition fosters more innovation than monopoly. The intuition behind

this result is that a monopolist has less incentive to innovate because it

already makes proﬁt before the innovation, whereas the competitive ﬁrm just

recoups its costs. This is the so-called replacement effect: for the monopoly,

the innovation just ‘replaces’ an existing proﬁt by a larger one. As the fol-

lowing quote illustrates,1Arrow’s intuition has made its way through the

economic press:

It is surely no coincidence that Microsoft’s hidden ability to innovate has become

apparent only in a market in which it is the underdog and faces ﬁerce competi-

tion. Microsoft is far less innovative in its core businesses, in which it has a

monopoly (in Windows) and a near monopoly (in Ofﬁce). But in the new markets

of gaming, mobile devices and television set-top boxes, Microsoft has been

unable to exploit its Windows monopoly other than indirectly—it has ﬁnanced

the company’s expensive forays into pasture new.

1.2 Our Approach

The previous argument seems to suggest that perfect competition also

dominates oligopolistic market structures in terms of innovation incentives.

However, this conjecture turns out to be wrong. We argue indeed that an

intermediate form of competition may provide a higher incentive to innovate

than the traditional polar cases (either monopoly or perfect competition).

More generally, we examine how the intensity of competition affects incen-

tives to innovate.

To this end, we consider an oligopoly model with horizontally differ-

entiated products. In this setting, we address the same question as Arrow:

how much is a ﬁrm willing to pay for a process innovation that it would be

the only one to use? We also examine under which industry structure this

willingness to pay reaches a maximum. To measure this willingness to pay,

we compute the difference between the proﬁt the ﬁrm would get by acquir-

ing the innovation (and so reducing its marginal cost) and the proﬁt the

ﬁrm would get without the innovation. That is, we suppose that, if the ﬁrm

does not acquire the innovation, no other ﬁrm does. We measure thus the

pure ‘proﬁt incentive’ to innovate, i.e. the desire to increase proﬁts inde-

pendently of the rival ﬁrms. As for the intensity of competition, our model

allows us to consider different sources: the number of ﬁrms in the market,

the degree of product differentiation and the nature of competition

(Cournot vs. Bertrand).

1Taken from ‘The meaning of Xbox’, The Economist, 24 November 2005.

Incentives to Innovate in Oligopolies 7

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

1.3 Results

The main ﬁnding of this analysis is that different industries are affected

in qualitatively different ways by an increase in competition. The practical

consequence is that, depending on the characteristics of the industry of interest,

the highest proﬁt incentive can be reached by a competitive ﬁrm (Arrow’s

claim), by a monopoly (Schumpeter’s claim) or by an intermediate form of

competition. We provide a rule about how to enhance ﬁrms’ incentive to

innovate for any industry.

More precisely, our main results can be summarized as follows. First,

regarding the effect of product differentiation, the proﬁt incentive is

U-shaped whatever the nature of competition, the innovation size and the

number of ﬁrms. Second, the effect of the number of ﬁrms depends on the

other parameters. Under Cournot competition, the proﬁt incentive either

decreases with the number of ﬁrms or has a U-inverted shape. Under Ber-

trand competition, the proﬁt incentive either decreases or increases with the

number of ﬁrms. In both cases, for the second option to occur, the innovation

and the degree of product substitutability must be large enough. Third, in

both natures of competition, there exist ranges of parameters for which the

proﬁt incentive is affected in opposite ways by different measures of compe-

tition (it decreases with the number of ﬁrms and increases with the degree

of product substitutability). Fourth, Arrow’s result is no longer valid when

products are sufﬁciently differentiated and/or the innovation is not too large;

monopoly is then the optimal market structure in terms of proﬁt incentive to

innovate, which supports Schumpeter’s second conjecture.

1.4 Empirical Evidence

Many empirical studies support our ﬁndings as well as our model. Indeed,

different empirical estimations reach different conclusions about the relation-

ship between product market competition and innovation thus providing

ambiguous results. Among others, Blundell et al. (1999) conduct a detailed

analysis of British manufacturing ﬁrms and establish a robust and positive

effect of market share on innovation. In contrast, Aghion et al. (2005) ﬁnd

strong evidence of non-monotonicity; in particular they ﬁnd an inverted U

shape for the competition–innovation relationship. Moreover, a recent

empirical analysis by Tang (2006), based on Canadian manufacturing ﬁrms,

shows that different measures of competition affect in different ways the

incentives to innovate. In particular, the author argues that ‘both competition

and innovation have many dimensions and that different innovation activi-

ties are associated with different types of competitive pressure’. Accordingly,

he estimates the effect of four types of competition and concludes that

the relationship between competition and innovation can be positive or nega-

tive, ‘depending on speciﬁc competition perception and speciﬁc innovation

activity’.

The Manchester School8

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

1.5 Related Literature

Our framework is directly linked to a vast industrial organization literature

which, in line with Arrow, assumes that there is only one innovator which

cannot be imitated by competitors.2Bester and Petrakis (1993) contrast the

proﬁt incentive in Cournot and Bertrand duopolies for different degrees of

horizontal product differentiation. Bonanno and Haworth (1998) address the

same question as Bester and Petrakis (1993) but under a model of vertical

product differentiation. Yi (1999) examines the effect of the number of ﬁrms

on the proﬁt incentive to innovate in Cournot oligopolies with a homoge-

neous product. Delbono and Denicolò (1990) compare and contrast static

and dynamic efﬁciency in oligopolies producing a homogeneous product for

different numbers of ﬁrms. Like Bester and Petrakis (1993), they use the

nature of competition to measure the intensity of competition. They measure

the incentive to innovate by the proﬁt incentive as well as by the ‘competitive

threat’, i.e. the difference between the proﬁt a ﬁrm obtains with the innova-

tion and the proﬁt it would obtain if a rival ﬁrm discovered the innovation.

Here, the incentive depends on two sources of change of the proﬁt: the proﬁt

incentive and the loss the ﬁrm would face in case any rival ﬁrm enjoyed the

cost reduction. Hence, the value of the innovation increases because ﬁrms

want to pay more in order to protect their position. Boone (2001) also studies

the competitive threat but in a framework of asymmetric ﬁrms where he

follows an axiomatic approach by deﬁning as measure of competition a

parameter satisfying particular conditions.

One important conclusion that can be drawn from this literature is that

different dimensions of competition may affect ﬁrms’ investment in R&D

in non-monotonic and potentially different ways. Our contribution to this

literature is to provide a uniﬁed framework where different sources of com-

petition interact and determine ﬁrms’ incentives to innovate. We not only

conﬁrm the results gathered in the previous literature, but we also extend

some of them and, more importantly, we provide new results about the

interaction between different measures of competition.

Our analysis can be related to two other strands of the literature on

innovation. First, the incentives to innovate can be computed by including

the revenue the innovator could raise through licensing the innovation. Addi-

tional issues arise then about the form the license should take (royalty per

unit of output, ﬁxed fee etc.) and about the number of licenses to be granted.

Moreover, in an oligopoly setting, the identity of the innovator also matters

(does the innovator compete on the product market or not?). Kamien (1992)

surveys this literature and Kamien and Tauman (2002) extend it.

2This literature as well as our paper abstract from the questions related to patent races. Our

creative environment is characterized by what Scotchmer (2004) calls ‘scarce ideas’ to

distinguish them from ideas which are common knowledge.

Incentives to Innovate in Oligopolies 9

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

Second, another important question is how much ﬁrms are willing to

invest in cost-reducing R&D (and not just how much they are willing to

pay for an innovation of a given size), especially when all ﬁrms have the

simultaneous opportunity to achieve competing innovations. It must then

be recognized that R&D is like any form of investment in that it precedes

the production stage. As a result, strategic considerations and the extent of

knowledge spillovers play a central role in determining which market struc-

ture provides ﬁrms with the highest incentives to undertake R&D.3In this

approach, as opposed to ours, each ﬁrm can obtain some cost reduction by

investing in R&D. In this symmetric setting, Cournot competition (with

strategic substitutes) always results in higher cost reductions than Bertrand

competition (with strategic complements), whereas in our framework, Ber-

trand competition leads to larger R&D investments than Cournot compe-

tition when products are close substitutes (as shown by Bester and Petrakis

(1993) for the duopoly case). Regarding R&D cooperation, d’Aspremont

and Jacquemin (1988) presented a seminal analysis, which was then

extended by, for example, Kamien et al. (1992) and generalized by Amir

et al. (2003). Finally, we include in this symmetric framework literature a

recent paper by Vives (2006). The author analyses a simultaneous-move

game where ﬁrms choose an investment–price pair (investment is thus non-

strategic). He obtains general and robust results about the relationship

between innovation and competition by using general functional forms and

considering in turn different measures of competition, which lead to differ-

ent but monotonic relations.

The remainder of the paper is structured as follows. In Section 2, we set

up the model. In Sections 3 and 4, we examine in turn Cournot and Bertrand

competition. In Section 5, we deﬁne the market structure which maximizes

the incentives to innovate. We conclude in Section 6. The main proofs appear

at the end of the paper, while the more technical proofs are relegated to an

appendix which is available from the authors upon request.

2Model

There are nﬁrms (indexed by i=1,...,n) competing in the market. Each

ﬁrm iincurs a constant marginal cost equal to ciand produces a differen-

tiated product, qi, sold at price pi. The demand system is obtained from

the optimization problem of a representative consumer. We assume a qua-

dratic utility function which generates the linear inverse demand schedule

3The idea that R&D generates incentives for ﬁrms to behave strategically was ﬁrst examined by

Brander and Spencer (1983), assuming no R&D spillovers between ﬁrms and Cournot

competition on the product market. Spence (1984), Okuno-Fujiwara and Suzumura (1990)

and Qiu (1997) extended this analysis by considering, respectively, positive spillovers,

Bertrand competition and differentiated products.

The Manchester School10

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

pi=a-qi-gSj⫽iqjin the region of quantities where prices are positive. The

parameter gis an inverse measure of the degree of product differentiation:

the lower gthe more products are differentiated (if g=1, products are

perfect substitutes; if g=0, products are perfectly differentiated). The

demand schedule, for g⫽1, is then given by qi=a-bpi+dSj⫽ipj, with

αγ

=+−

()

a

n11

βγ

γγ

=+−

()

−

()

+−

()

[]

12

11 1

n

n

δγ

γγ

=−

()

+−

()

[]

11 1n

In this framework, total quantity Qdepends on nas well as on g(this is

because the representative consumer loves variety). In particular, in the

symmetric case where pi=p"i, we have

Qn na p

n

,

γγ

()

=−

()

+−

()

11 (1)

It follows that nand ginﬂuence the individual quantities as well as the

market size.

Under Cournot competition, the equilibrium quantity and proﬁt for ﬁrm

iare found as

q

anc c

nq

i

ij

ji

ii

CCC

=−

()

−+ −

()

[]

+

−

()

+−

()

[]

=

()

≠

∑

222

22 1

2

γγ γ

γγ π

(2)

We can also compute the equilibrium price, quantity and proﬁt under

Bertrand competition:

p

nc c

n

i

ij

ji

B=+

()

+−−

()

[]

+

+

()

−−

()

[]

≠

∑

222

22 1

βδαββδ δβ

βδ βδ

(3)

qpc pc q

iiiiii i

BB BB B

=−

()

=−

()

=

()

()

βπβ β

22

1(4)

Initially, all ﬁrms produce at cost ci=c. A new process innovation allows

ﬁrms to reduce the constant marginal cost of production from cto c1=c-x

(with 0 <x<c). We assume that the innovation is non-drastic. That is, the

cost reduction does not allow the innovator to behave like a monopolist. A

sufﬁcient condition is that the monopoly price corresponding to clis larger

than the initial cost c, i.e. (a+c-x)/2 >c. Equivalently, assuming without loss

of generality that the difference a-cis equal to unity, we have the following

assumption.

Incentives to Innovate in Oligopolies 11

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

Assumption 1: x <a-c=1.4

Under Cournot competition this deﬁnition of non-drastic innovation is

equivalent to saying that the innovation does not allow the innovator to

throw the other ﬁrms out of the market. As for Bertrand competition this is

not necessarily the case. As we argue in Section 4, there may be values of x<

1 such that the innovator is the only active ﬁrm in the market, but still the

innovation is said to be non-drastic, because the innovator cannot price at the

monopoly level.

In line with Arrow’s question, we want to ﬁnd out how much a ﬁrm is

willing to pay for acquiring the innovation and being its single user.5As a

consequence, we concentrate on the proﬁt incentive that can be seen as the

‘pure’ incentive to innovate. We denote by pWthe proﬁt accruing to the

innovator (the ‘winner’), and by pthe pre-innovation proﬁt. For future

reference, we also deﬁne pLas the proﬁt accruing to the rivals of the innovator

(the ‘losers’). Based on these deﬁnitions, we formally deﬁne our measure of

ﬁrms’ incentives to innovate as follows.

Deﬁnition 1: The proﬁt incentive is deﬁned as PI =pW-p.

In the next sections, we derive the exact value of the proﬁt incentive

under Cournot and Bertrand competition, respectively, denoted by PIC(n,g)

and PIB(n,g). The proﬁt incentive is clearly increasing in the innovation size,

x. It also depends on the number of ﬁrms in the market (n) and on the degree

of product differentiation (g) in ways we will now analyze. Finally, note that

PI PI PI PI

CC BB

1010

2

4

,,,,

γγ

()

=

()

=

()

=

()

=+

()

nn

xx

which corresponds to the proﬁt incentive for a monopoly (either because

there is a single ﬁrm, n=1, or because products are independent, g=0).

We want to study how the proﬁt incentive changes with the intensity of

competition. We consider three measures of the strength of competition: the

degree of product substitutability (g), the number of ﬁrms in the market (n)

and the nature of competition (Cournot vs. Bertrand). As for the ﬁrst

measure, we know that, as gdecreases, ﬁrms’ market power increases because

products become more differentiated. Moreover, given n ex ante symmetric

ﬁrms in the market, we can check from expression (1) that total quantity is a

decreasing function of g(as products become closer substitutes, the market

size decreases). Therefore, a change in gaffects the degree of competition in

4As shown by Zanchettin (2006) in the duopoly case, this condition is sufﬁcient but not necessary

for values of g<1. We also assume that c31, so that Assumption 1 guarantees that the

innovator still has a positive marginal cost.

5We discuss in the conclusion the role of the identity of the innovator (either incumbent or

outside research lab).

The Manchester School12

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

two mutually reinforcing ways: as gincreases, products become closer sub-

stitutes and the market size decreases. In contrast, an increase in the number

of ﬁrms affects the degree of competition in two opposite ways: on the one

hand, individual ﬁrms’ proﬁts decrease but, on the other hand, the market

size increases.6However, it can easily be shown that the former effect is

stronger than the latter: the pre-innovation, the losers’ and the winner’s

proﬁts are all decreasing in n. We can then take nas another measure of the

strength of competition.7Finally, as for the nature of competition, we know

from Singh and Vives (1984) that going from a Cournot to a Bertrand market

structure implies an increase in competition.

As we will now explain, the intensity of competition affects the proﬁt

incentive to innovate in contrasting ways. The general intuition is that there

is an opposition between a negative competition effect (a more competitive

market reduces ﬁrm i’s proﬁt if either it gets the innovation or it does not) and

a positive competitive advantage (the tougher the competition in the market,

the larger the innovator’s advantage).

3Cournot Competition

We study here how the incentives to innovate are affected by the intensity of

competition in a Cournot framework. We ﬁrst investigate the effects of an

inﬁnitesimal cost reduction, which allows us to highlight the various forces

at work. We then consider discrete cost reductions. Finally, we examine

whether the two measures of competition (gand n) affect the incentive to

innovate in a converging or diverging way.

Using expression (2), we ﬁnd

qnx

nq

W

CW

CW

C

=−

()

++ −

()

[]

−

()

+−

()

[]

=

()

22 2

22 1

2

γγ

γγ π

(5)

qnq

CCC

=+−

()

=

()

1

21

2

γπ

(6)

qx

nq

L

CL

CL

C

=−

()

−

−

()

+−

()

[]

=

()

2

22 1

2

γγ

γγ π

(7)

where Assumption 1 is used to guarantee that all equilibrium quantities

(especially qL

C) are always positive. Applying Deﬁnition 1, we compute the

value of the proﬁt incentive as

6One can check from expression (1) that total quantity is an increasing function of n(an

additional ﬁrm brings one more variety in the market, which increases total demand

because consumers like variety).

7In Section 3.2, we discuss how the side-effect of the number of ﬁrms on the market size affects

the sensitivity of the proﬁt incentive to the number of ﬁrms.

Incentives to Innovate in Oligopolies 13

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

PICW

CC

nx

nnx

n

,

γπ π γγγ

γγ

()

=−= +−

()

[]

−

()

++ −

()

[]

{}

−

()

+−

222222

22

211 2

()

[]

(8)

In order to analyze how PIC(n,g) depends on gand n, we can get some

intuition by ﬁrst studying its separate components,

π

W

Cand pC. We summa-

rize our results in the following lemma,8where ˆ

xn n

()

≡+

()

12

and

ˆ,

γ

nx x xx n xn

()

≡+− +

()

⎡

⎣⎤

⎦−−

()

[]

21 1 2

.

Lemma 1: (i) The winner’s and pre-innovation proﬁts both decrease with the

number of ﬁrms in the market, (ii) the pre-innovation proﬁt decreases with

the degree of product substitutability, and (iii) the winner’s proﬁt increases

with the degree of product substitutability if this degree and the size of the

innovation are large enough (i.e. if xxn>

()

ˆand

γγ

>

()

ˆ,nx); it decreases

otherwise.

It is worth stressing the third result in Lemma 1. In general, proﬁts are

decreasing in g. Nevertheless, if the innovation is large enough, the innovator

may gain from an increase in g. In particular,

π

W

Cis ﬁrst decreasing and then

increasing in g. In fact, as soon as gbecomes positive, the ﬁrm is no longer a

monopolist and, given that the innovation is non-drastic (i.e. the cost reduc-

tion does not allow the innovator to become a monopolist), the winner’s

proﬁt decreases. However, as soon as products are sufﬁciently substitutable,

the cost advantage of the innovator becomes more important because the

innovation becomes a sort of substitute for product differentiation and so the

winner’s proﬁt increases. This means that our two measures of competition

have contrasting effects on the winner’s proﬁt. Note that ˆ

xn

()

and ˆ,

γ

nx

()

decrease with n. Therefore, the higher the number of ﬁrms in the market, the

larger the range of parameters where the winner’s proﬁt is increasing with g.

We now proceed with studying the effects of changes in the toughness of

competition on the proﬁt incentive.

3.1 Inﬁnitesimal Analysis

In order to single out the forces at work, we ﬁrst consider an inﬁnitesimal

reduction in cost (x→0). The proﬁt incentive can be seen as the discrete

version of the sensitivity of i’s proﬁts to a reduction in its own cost (i.e.

−∂ ∂

π

ii

c

Cevaluated at ci=c). At the Cournot equilibrium, we have

πγ

iiii i jii

pcq aq n qcq

CC C C C C

=−

()

=− − −

()

−

[]

1, so that, using the envelope

theorem,

−∂

∂=+ −

()

∂

∂

πγ

i

i

i

j

i

i

cqn

q

cq

C

C

C

C

1(9)

8The proof is relegated to the technical appendix.

The Manchester School14

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

In expression (9), the ﬁrst term measures the direct effect and the second term

the strategic effect of an inﬁnitesimal cost reduction. They are both positive.

As cidecreases, two positive forces affect ﬁrm i’s proﬁt: ﬁrst, ﬁrm ibecomes

more efﬁcient, which allows it to expand its output; second, because quanti-

ties are strategic substitutes, the rival ﬁrms react by reducing their output,

which further beneﬁts ﬁrm i.

We are interested in assessing the sensitivity of these two effects

with respect to changes in gand in n. Table 1 decomposes the results.

Consider ﬁrst the direct effect. From Lemma 1, we know that both

∂∂qi

C

γ

and ∂∂qn

i

Care negative: the direct effect decreases as products

become closer substitutes and as the number of ﬁrms increases. On the other

hand, we cannot a priori sign the sensitivity of the strategic effect. To see this,

it is helpful to note that the strategic effect is itself the combination of three

forces, which are affected in opposite directions by gand n. First, each rival

ﬁrm reduces its quantity in reaction to ﬁrm i’s cost reduction ( ∂∂>qc

ji

C0);

the larger gand the smaller n, the stronger this individual reaction. Second,

the combined effect of these individual reactions increases with product

substitutability and with the number of rival ﬁrms (g(n-1)). Finally, ﬁrm i’s

proﬁts are affected in proportion to its quantity ( qi

C), which decreases with g

and n. As a result, the sensitivity of the total effect is ambiguous.

However, direct computations reveal some clear-cut results. First,

regarding the inﬂuence of g, it appears that the strategic effect is increasing:

it starts from zero when products are independent (g=0) and then increases

because the rival ﬁrms’ quantities become more sensitive to a reduction in

ﬁrm i’s cost. So, as gincreases, the direct effect becomes weaker but the

strategic effect becomes stronger. At g=0, the strategic effect disappears and

so the marginal return of a cost reduction decreases with g. At the other

extreme (g=1), it can be shown that the increase in the strategic effect

dominates the decrease in the direct effect and so the marginal return of a cost

reduction increases with g. There is thus an intermediate value of g

for which the two effects just compensate and where −∂ ∂

π

ii

c

Creaches a

minimum. In other words, the marginal return of a cost reduction is U-shaped

with respect to g.Second, regarding the inﬂuence of n, the sensitivity of the

strategic effect remains ambiguous; nevertheless, it can be shown that the

total effect decreases with n (−∂ ∂ ∂ <

20

π

ii

cn

C).

Table 1

Direct effect Strategic effect Total effect

qi

C+g(n-1) ∂

∂

q

c

j

i

C

qi

C=−∂

∂

π

i

i

c

C

Effect of g-++-?

Effect of n-+--?

Incentives to Innovate in Oligopolies 15

© 2010 The Authors

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3.2 Discrete Analysis

We now examine how discrete cost reductions challenge these results. As we

have seen in the inﬁnitesimal analysis, it is impossible to a priori sign the

balance between the different effects at work. Considering discrete cost

reductions complicates the analysis further as the innovation size affects the

magnitude of the various effects. It is thus by direct computations that we

have derived the results stated in the next two propositions; we rely on the

inﬁnitesimal analysis to highlight the intuition behind these results.

Proposition 1: Under Cournot competition: (i) the proﬁt incentive is

U-shaped with respect to g, and (ii) the highest proﬁt incentive is reached

under independent products (g=0) if the innovation size is below some

threshold (between 2/7 and 2/3), and under homogeneous products (g=1)

otherwise.

Proposition 1 (which is formally stated in the Appendix) establishes that

PICﬁrst decreases with gand then increases. Whether the largest incentive is

reached for g=0org=1 depends on the other parameters (nand x). We

observe thus that the shape of PICwith respect to gis the same as in the

inﬁnitesimal case. The fact that larger values of glead to a larger proﬁt

incentive to innovate can be seen as a form of substitutability between

product innovation (modeled as a reduction of g) and process innovation

(measured by x): the weaker the product innovation, the higher the willing-

ness to pay for a given process innovation.

Proposition 2: (i) The proﬁt incentive is a single-peaked function of n, and (ii)

if the innovation size and the degree of product substitutability are large

enough, then the maximum value of PI is reached for n>1 (competition);

otherwise, the maximum value of PI is reached for n=1 (monopoly).

Proposition 2 (which is formally stated in the Appendix) shows that the

effect of the number of ﬁrms on the proﬁt incentive is also non-monotonic.9

The interesting result is that, when xand gare large enough, the proﬁt

incentive to innovate ﬁrst increases and then decreases with n.10 On the other

hand, from the inﬁnitesimal case, we know that, as x→0, the proﬁt incentive

decreases with n. At the other extreme, we can show that, when gand xtend

to one (i.e. towards homogeneous product and drastic innovation), the proﬁt

incentive is always increasing with the number of ﬁrms.

9This result complements and extends Yi (1999) who focuses on homogeneous product markets.

For an inﬁnitesimal innovation, Yi shows that the proﬁt incentive decreases with nfor a

fairly large class of demand functions. He also considers arbitrary innovations under linear

demand and ﬁnds the same threshold value for the innovation size as we do.

10This result is in line with the empirical evidence found in Aghion et al. (2005) studying the

relation between competition and innovation with a UK panel data set.

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As already discussed, an increase in the number of ﬁrms affects the

intensity of competition in two opposite ways: on the one hand, individual

ﬁrms’ proﬁts decrease but, on the other hand, the market size increases.

Therefore, as a robustness check, in what follows, we try to separate the two

effects.11 Deﬁne

θγ

≡+−

()

n

n11

Then we can rewrite the total quantity (1) as Q(n,g)=q(a-p). We take qas

a measure of the size of the market. It decreases with g, and increases with n

as long as g<1 (for g=1, q=1 is independent of n). Solving for gin the

above expression and making the proper substitutions, we can express the

proﬁt incentive to innovate as a function of qand n:

PIC=+−

()

+−

()

+−−

+

()

−−

()

θθ θθθ

θθθ

2

22

22422

2

nx n nnxnn

nnn

The effect of the number of ﬁrms on the incentive to innovate can be thus

decomposed as follows:

dPI

d

PI PI d

d

CCC

nn n

=∂

∂+∂

∂

θ

θ

The ﬁrst term is the ‘pure effect’ of the number of ﬁrms; it is computed by

keeping qconstant, thereby freezing the effect that nhas on the size of the

market. The second term is the ‘market-size effect’ of the number of ﬁrms: a

change in naffects the market size q, which itself affects the incentive to

innovate. Note that, if ﬁrms produce a homogeneous product (i.e. if g=1),

the second term disappears as the market size is constant. In that case, only

the ﬁrst term remains and we already know that the effect of ncan be

non-monotonic if the size of the innovation is large enough. This result still

holds for large enough values of gand of x. For instance, it can be shown that,

for x=g=0.95,

∂

PIC/

∂

n>0 for n210 and

∂

PIC/

∂

n<0 for n>10.12 We can

thus conclude that a change in the number of ﬁrms may affect in a non-

monotone way the proﬁt incentive independently of its effect on the size of

the market.

11We further check that our non-monotone results on the effect of non the proﬁt incentive are

preserved under the Shubik and Levitan (1980) linear demand model where the market size

does not change with n.

12Moreovcr, looking at the second term, we know that dq/dn>0, but simulations show that the

sign of

∂

PIC/

∂

qcan also change with n. For instance, one can check that, for x=g=1/3,

∂

PIC/

∂

q>0forn248 and

∂

PIC/

∂

q<0forn>48. In sum, isolating the pure effect of nfrom

its market-size effect does not shed much light on the non-monotonic relationship between

the number of ﬁrms and the incentive to innovate.

Incentives to Innovate in Oligopolies 17

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3.3 Cross-effects

We want now to contrast the ways the two measures of competition affect the

proﬁt incentive. According to Proposition 2, the proﬁt incentive increases

with the number of ﬁrms provided that the innovation size and the degree of

product substitutability are large enough. Let us make this statement more

precise. Considering nas a continuous variable, it is readily shown that

∂

∂

()

>⇔>

()

=−−+ +

()

−−

()

−+ −nnnx

nx nx n

nnxx

PIC,,

γγγ

0441

32

2

(10)

where

γ

nx,

()

>0for all admissible nand x, and

γ

nx,

()

<1if and only if

x>(n-1)/n.

From Proposition 1, we know that the proﬁt incentive is U-shaped with

respect to g. Let

γ

nx,

()

denote the value of gfor which the proﬁt incentive

reaches its minimum (we proved that 01<

()

<

γ

nx,for all admissible n

and x). We ﬁnd that

γγ

nx nx,,

()

>

()

for all admissible nand x, which allows

us to state the following proposition.13

Proposition 3: (i) The two measures of competition affect the proﬁt incentive

in converging ways either if products are sufﬁciently differentiated (for

γγ

<

()

nx,, the proﬁt incentive decreases with gand n), or if the innovation

is large enough and products are sufﬁciently substitutes (for x>(n-1)/nand

γγ

>

()

nx,, the proﬁt incentive increases with gand n), and (ii) there always

exists a range of parameter values for which the two measures of competition

affect the proﬁt incentive in diverging ways (for

γγγ

nx nx,min,,

()

<<

()

{}

1,

the proﬁt incentive increases with gand decreases with n).

The results of Proposition 3 (which are illustrated in Fig. 1) give us a rule

about how to enhance the incentives to innovate for any innovation size x

and any industry (n,g). For instance, it tells us for which industries a merger

of two ﬁrms (decrease in n) or the introduction of a product innovation

(decrease in g) increases or decreases the incentive to invest in a process

innovation.

Another implication of Proposition 3 is that the same level of proﬁt

incentive can be achieved in different industries. In this respect, an instructive

thought experiment is to ﬁx the innovation size xand examine which indus-

tries give ﬁrms the same level of proﬁt incentive as a monopoly (n=1 and/or

g=0). We know from Proposition 1 that, for x<2/7, the monopoly gives a

higher proﬁt incentive than any other Cournot industry. So, we take for

example x=1/2 and we compute that the following industries (n,g) are

equivalent to a monopoly in terms of proﬁt incentive: (2, 0.89), (3, 0.94),

(4, 0.98) and (5, 1). We observe that, to maintain the proﬁt incentive at the

monopoly level, an increase in the number of ﬁrms has to be compensated

13The proof of this proposition is relegated to the technical appendix.

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by an increase in product substitutability. In particular, we observe that ﬁve

ﬁrms producing a homogeneous product are willing to pay the same amount

as a monopoly for a process innovation that decreases the marginal cost of

production by x=1/2.

4Bertrand Competition

The derivation of the proﬁt incentive is slightly more involved under Bertrand

competition. Indeed, in contrast with Cournot competition, Assumption 1 is

not sufﬁcient to ensure that the non-innovating ﬁrms (the ‘losers’) are all

active on the market after the innovation. In fact, for a loser to set a price

larger than its marginal cost c(and thus produce a positive quantity), the

ﬁrms’ products must be sufﬁciently differentiated. We need thus to consider

the possibility of corner equilibria in which a number of losers are con-

strained to price at marginal cost. The following lemma (which is proved in

the Appendix) characterizes the Bertrand–Nash equilibrium of the post-

innovation game for all values of x,nand g. Deﬁne

xn n

n

B,

γγγ γ

γγ γ

()

=−

()

−+

()

+−

()

1232

12

Lemma 2: The Nash equilibrium of the post-innovation price game is such

that (i) for x2xB(n,g), all ﬁrms price above their marginal cost, and (ii) for

x>xB(n,g), the winner is the only ﬁrm pricing above its marginal cost.

n

10

2

0

1

g

g

~

g

–

PICdecreases with n and

g

PICincreases with n and

g

PICdecreases with n and

increases with

g

0.8

0.4

Fig. 1 Cross-effects in the Cournot Case (x=9/10)

Incentives to Innovate in Oligopolies 19

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We have that, for x∈(xB, 1), the winner is the only active ﬁrm in the

market and behaves like a constrained monopolist (because the innovation

is non-drastic). Note that the threshold separating the two cases, xB(n,g),

decreases with nand tends to 2(1 -g)/gfor n→•(which is greater than unity

for g<2/3). Therefore, the corner solution can only be observed for sufﬁ-

ciently large values of xand g.

We can now deﬁne the proﬁt incentive. First, from expressions (3) and

(4), we derive the pre-innovation equilibrium proﬁt:

πγ γ

γγ

γ

γ

Bnn

nn

,

()

=+−

()

−

()

+−

()

[]

−

−

()

+

⎡

⎣

⎢⎤

⎦

⎥

12

11 1

1

32

2

(11)

Next, using the proof of Lemma 2, we compute the winner’s proﬁt at the

Nash equilibrium of the post-innovation game:

πγβ γγ γγγ

γγ

W

Bnnn n x n

n

,

()

=−+

()

+−

()

+

[]

+−

()

+−

()

+−

(

55 3 2 2 1 22 3

23

22

))

+−

()

⎧

⎨

⎩

⎫

⎬

⎭

≤

()

()

=−+

⎛

⎝

⎜⎞

⎠

⎟>

22 3

111

2

γγ

γ

πγ γγ

n

xxn

xxx

for

for

B

W

B

,

BB n,

γ

()

(12)

Finally, combining (11) and (12), we can express the proﬁt incentive under

Bertrand competition as

PI PI for

PI

B

BW

BB B

BW

B

nnnn xxn

n

,,,, ,

,

γγπ γπ γ γ

γπ

()

=

()

=

()

−

()

≤

()

()

=

γγπ γ γ

()

−

()

>

()

⎧

⎨

⎩BB

fornxxn,,

(13)

We can now examine how the proﬁt incentive evolves with the two

measures of competition in a Bertrand framework. As before, in order to

assert the intuition behind its sensitivity, we proceed by ﬁrst examining the

effects of an inﬁnitesimal cost reduction and we then extend the analysis to

the effects of a discrete cost reduction.

4.1 Inﬁnitesimal Analysis

At the Bertrand equilibrium, the following equation holds:

−∂

∂=+

−

()

+−

()

−∂

∂

⎛

⎝

⎜⎞

⎠

⎟

+

()

πγ

γ

i

i

i

j

i

cqn

n

p

cq

B

B

direct effect

B

1

12 ii

B

strategic effect −

()

>

0(14)

As for the Cournot case, the ﬁrst term of expression (14) measures the direct

effect and the second term measures the strategic effect of an inﬁnitesimal

cost reduction. The direct effect is positive as ﬁrm ibecomes more efﬁcient.

In contrast, the strategic effect is negative: a reduction of ci, and so a reduc-

tion of pi, leads rival ﬁrms to lower their price pj(prices are strategic

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complements), which affects negatively ﬁrm i’s proﬁt. As above, we are

interested in assessing the sensitivity of these two effects with respect to

changes in gand in n. Table 2 decomposes the results.

In contrast with the Cournot case, we can a priori sign the impact of the

number of ﬁrms: as both the direct effect and the strategic effect decrease with

n,the total effect of an inﬁnitesimal cost reduction clearly decreases with the

number of ﬁrms. On the other hand, the effect of a change in gis ambiguous: the

direct effect decreases with gbut the strategic effect might increase or decrease.

Nevertheless, computations reveal the following: the strategic effect is nil when

g=0; as soon as g>0, it ﬁrst decreases and then increases with product

substitutability; however, even when this sensitivity is positive, it never com-

pensates the negative effect of gon the equilibrium quantity (direct effect). As

a consequence, the inﬁnitesimal proﬁt incentive is always decreasing with g.

4.2 Discrete Analysis

The effects of a discrete cost reduction are described in the following two

propositions (which are both proved in the technical appendix). Starting

with product substitutability, we observe that, in contrast with its inﬁnitesi-

mal counterpart, its effect is non-monotonic.

Proposition 4: Under Bertrand competition, (i) the proﬁt incentive ﬁrst

decreases with g, then reaches a minimum, and ﬁnally increases with g, and (ii)

the highest proﬁt incentive is reached under homogeneous products (g=1)

for all nand x.

We observe thus that the degree of product substitutability affects PIB

and PICin qualitatively similar ways. Bester and Petrakis (1993) compare PIB

and PICin a duopoly model by restricting the range of gto values such that

all ﬁrms stay on the market (i.e. x2xB(2, g)). They show that there exists a

cut-off value of gsuch that PIC>PIBfor gbelow the cut-off and PIB>PICfor

gabove the cut-off. Using our derivation of PIB2,

γ

()

, we can show that

Bester and Petrakis’s conclusion extends to larger values of g(i.e. those for

which an ex post monopoly obtains).14 Moreover, we show that, under

14Simulations indicate that these results still hold for n>2.

Table 2

Direct effect Strategic effect Total effect

qi

B−−

()

+−

()

γ

γ

n

n

1

12

∂

∂

p

c

j

i

B

qi

B=−∂

∂

π

i

i

c

B

Effect of g--+-?

Effect of n-----

Incentives to Innovate in Oligopolies 21

© 2010 The Authors

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Bertrand competition, the proﬁt incentive reaches its maximum for homoge-

neous products independently of the other parameters (nand x). Intuitively,

whatever the innovation size and the number of ﬁrms, the winner has more to

win under homogeneous products because the innovation allows him to

remain the only active ﬁrm on the market.

On the other hand, the effect of the number of ﬁrms on the proﬁt incentive

under Bertrand competition is monotone:PI

Bis either decreasing or increasing

with ndepending on the innovation size and the degree of product differen-

tiation. We show in the technical appendix that there is a unique positive

value of x,xn,

γ

()

, such that ∂

()

∂=PIBnn,

γ

0, and that xn x n,,

γγ

()

<

()

B

for all nand g. Therefore, we have the following proposition.

Proposition 5: Under Bertrand competition, the proﬁt incentive decreases

with the number of ﬁrms for xxn<

()

,

γ

, and increases otherwise. A neces-

sary condition for the latter possibility ( xn,

γ

()

<1)isg>0.65.

As far as the cross-effects of the two measures of competition are con-

cerned, the picture is similar to the one we described for Cournot competi-

tion. Deﬁning xn,

γ

()

as the positive value of xsuch that

∂

PIB(n,g)/

∂

g=0, we

can show that xn xn,,

γγ

()

<

()

, meaning that there are up to three possible

situations: (i) for xxn<

()

,

γ

, the proﬁt incentive decreases with both nand g;

(ii) for xn x xn,,

γγ

()

<<

()

, the proﬁt incentive decreases with nbut increases

with g; and (iii) for xxn>

()

,

γ

, the proﬁt incentive increases with both nand

g. As for Cournot competition, it is impossible to have the proﬁt incentive

increase with nwhile decreasing with g.

5Market Structure and Innovation

After studying how different sources of competition affect the proﬁt incentive

to innovate, we can proceed to investigate which market structure maximizes

the proﬁt incentive to innovate.15 Arrow’s (1962) argument, stating that

the incentive to innovate is larger under perfect competition than under

monopoly, was based on Bertrand competition in a homogeneous market.

We check indeed that, with a homogeneous product, the proﬁt incentive

under Bertrand competition is deﬁned as PIBn,1

()

, which increases in n

and reaches thus its maximum at PIB∞

()

=,1 x; we also compute that

PI PI

BB

∞

()

=>

()

=+

()

,,1124xxx

γ

, which establishes Arrow’s result.

Now, Proposition 2 tells us that the previous argument does not hold

under Cournot competition. Indeed, letting the number of ﬁrms go to inﬁnity

15We are not looking for the welfare-maximizing industry, nor for the ‘welfare incentive’ to

innovate. In contrast, starting from the idea that a process innovation is welfare enhancing

(it implies an increase in total industry proﬁts as well as an increase in consumer surplus),

we have compared how much a ﬁrm is willing to pay for a process innovation under

different market structures (perfect competition, oligopolies, monopoly). We now wonder

under which conditions this willingness to pay is maximized.

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never leads to the highest proﬁt incentive. Also, Proposition 5 shows that as

soon as products are differentiated, the proﬁt incentive under Bertrand com-

petition might be decreasing in nif the innovation is small enough, meaning

that a monopoly gives higher incentives to innovate.

Combining our previous results, we are in a position to complement

Arrow’s analysis in a useful way by solving the following exercise: for a given

degree of product differentiation and a given innovation size, what is the com-

bination of competition mode and number of ﬁrms that yields the highest proﬁt

incentive to innovate? As shown in the following proposition, the answer is

very simple to state.

Proposition 6: For sufﬁciently large values of xand g, Bertrand competition

with an inﬁnite number of ﬁrms yields the highest proﬁt incentive. Otherwise,

(Bertrand or Cournot) monopoly does.

More formally (as proved in the Appendix), for x>2(1 -g)/g(which

supposes g>2/3), the best structure under Bertrand competition (i.e. the

number of ﬁrms maximizing the proﬁt incentive, nB

*→∞) yields a higher

proﬁt incentive than the best structure under Cournot competition (i.e. nC

*=1

or nC

*is (the integer closest to) ˆ,nx

γ

()

, depending on the value of x). On the

other hand, for x<2(1 -g)/g, the best market structure is monopoly, which

provides a proﬁt incentive equal to (x+2)x/4.

6Conclusion

In this paper, we have provided a uniﬁed framework where different sources

of competition interact and determine ﬁrms’ incentives to invest in a process

innovation. Our results are threefold: ﬁrst, we conﬁrm the existence of a

non-monotone and non-unique relationship between the intensity of compe-

tition and the incentives to innovate in a general framework; in particular,

we extend the analysis of the proﬁt incentive to innovate under Bertrand

competition with linear demand and horizontally differentiated products by

considering any number of ﬁrms and any degree of product differentiation.

Second, we show that different sources of competition can have diverging

effects on the innovation incentives; we characterize the conditions under

which this occurs, thus providing a rule about how to enhance ﬁrms’ incen-

tives to innovate. Finally, we provide a rationale for the empirical evidence

that the relationship between competition on the product market and inno-

vation is ambiguous and depends on the dimension of competition we are

taking into account. The direct and practical implication of our set-up is then

a potential guideline for antitrust authorities.

In the spirit of Arrow (1962), we focus our attention on the proﬁt

incentive as a measure of ﬁrms’ innovation incentives. This strategy allows us

to extend and complement Arrow’s analysis; in contrast, we do not consider

Incentives to Innovate in Oligopolies 23

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the incentive which comes from the recognition that there exist rival ﬁrms

competing to be the ﬁrst to innovate, i.e. the competitive threat. The analysis

of the competitive threat in such a framework constitutes a ﬁrst possible

extension of our model. Nevertheless, preliminary results show that the dif-

ferent sources of competition considered here affect the competitive threat

and the proﬁt incentive in qualitatively similar ways.

A second linked research area we propose to work on deals with the

identity of the innovator. Our proﬁt incentive corresponds to the following

scenario. We can imagine that the innovator is an incumbent ﬁrm, which

either manages to keep its innovation secret, or which is granted a broad

patent of inﬁnite duration and which does not license the innovation to rival

ﬁrms; our question was then to assess how much this incumbent is willing to

invest in R&D. Nevertheless, our approach does not take into account the

incentive of the innovator to license his or her cost-reducing innovation nor

the possibility that the innovator is outside the market.16 In particular,

Kamien and Tauman (2002) demonstrate that an outside innovator ﬁnds it

proﬁtable to auction more than one license, and that an inside innovator also

has incentives to license the innovation to its rivals. Their approach is an

alternative measure of ﬁrms’ incentives to innovate, i.e. the innovator’s proﬁt

coming from the mode of licensing and the number of licenses auctioned off.

We leave it to future research to investigate whether their results still hold in

the presence of product differentiation. Studying the behavior of the innova-

tor in our framework would allow us to have a more complete picture of

ﬁrms’ innovation incentives in oligopolistic settings.

Appendix

Propositions 1 and 2

Proposition 1 is stated formally as follows: (i) for all x∈[0, 1] and all n32, there exists

γ

C∈

()

01,such that

∂

PI/

∂

g<0 for

γγ

>C,

∂

PI/

∂

g=0 for

γγ

=Cand

∂

PI/

∂

g>0 for

γγ

>C, and (ii) the highest proﬁt incentive is reached under independent products

(g=0) if xxn n n<

()

≡−

()

+

()

C2131

and under homogeneous products (g=1)

if xxn>

()

C;xn

C

()

increases in n, and is between 2/7 and 2/3.

Proposition 2 is formally stated as follows. Deﬁne

ˆˆ

nx

xx

γγγ γ γ

γγγ γγγ γ

()

=−

()

+−

()

−

()

−−

() ()

=−

()

−− +21 3 22

2

22 2 4

2

and

(( )

−

()

γγ

82

(i) Taking nas a continuous variable, the proﬁt incentive reaches a maximum for

nn=

()

ˆ

γ

,17 and (ii) taking into account that nis an integer, the maximum is reached for

n>1 if and only if xx>

()

ˆ

γ

, which supposes that g>0.711.

16The incentive to innovate of an outside inventor who decides to auction a single license would

be measured by the competitive threat.

17Computing n

ˆ(g)atg=1, we ﬁnd that the proﬁt incentive decreases with nas long as

n>1/(1 -x); this is equivalent to x<(n-1)/n, the condition given by Yi (1999).

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The proofs of these two propositions are purely technical and can be obtained

from the authors upon request.

Proof of Lemma 2

The post-innovation price game is played by one winner (indexed by W) and

a set Lof n-1 losers indexed by i. A typical loser i’s maximization program

writes as

max

,

pij

jLji

ii

i

pp ppc pc

αβ δ δ

−+ +

⎛

⎝

⎜⎞

⎠

⎟−

()

≥

∈≠

∑

Wsubject to

From the ﬁrst-order condition, we ﬁnd that the interior solution to this problem is

given by

pppc

ij

jLji

=++ +

⎛

⎝

⎜⎞

⎠

⎟

∈≠

∑

1

2

βαδ δ β

W

,

This solution holds as long as pi>c, which is equivalent to

αδ δ β

++ ≥

∈≠

∑

ppc

j

jLji

W

,(A1)

We can thus write loser i’s best response function as follows:

Rp p ppc

c

ij

jL

ji j

jLji

W

Wif A is met

oth

,,

()

()

=++ +

()

()

∈

≠∈≠

∑

1

21

βαδ δ β

eerwise

⎧

⎨

⎪

⎩

⎪(A2)

As for the winner, the maximization program writes as

max

pi

iL

pppcx pcx

W

WW W

subject to

αβ δ

−+

⎛

⎝

⎜⎞

⎠

⎟−+

()

≥−

∈

∑

From the ﬁrst-order condition, we ﬁnd that the interior solution to this problem is

given by

pp p cx

iiL i

iL

W

()

()

=++−

()

⎡

⎣

⎢⎤

⎦

⎥

∈∈

∑

1

2

βαδ β

(A3)

We show that this value is always larger than c-x. Indeed, the lowest value

of pW, noted pW, is reached when pi=c"i∈L. After substitutions, we

compute

pcx xn

n

W−−

()

=−

()

++−

()

+− >

1

2

112

12 0

γγγ

γγ

Let us ﬁrst characterize the interior Nash equilibrium. Solving the system of

equations given by (A3) and by the top row of (A2) taken n-1 times, we get

equilibrium prices for the winner and the losers respectively given by pW

Band pL

B.We

have that

Incentives to Innovate in Oligopolies 25

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

pc nnx

nn

xx

L

B−= −

()

−+

()

−+−

()

+−

()

+−

()

≥

⇔≤

12321 2

23223 0

γγ γ γ γ γ

γγ γγ

BB nn

n

,

γγγ γ

γγ γ

()

=−

()

−+

()

+−

()

1232

12 (A4)

Suppose now that condition (A4) is not met. The equilibrium must then be such

that some losers set pi=c, while the other losers set a price ˆ

pc

L>. Suppose that k

losers price above marginal cost, with 0 <k<n-1. Condition (A1) must be satisﬁed

for those klosers (i.e.

αδ δ β

++−−

()

+−

()

[]

≥pnkckpc

WL

11

ˆ) and violated for the

other (n-k-1) ones (i.e.

αδ δ β

++−−

()

+

[]

<pnkckpc

WL

2ˆ). As ˆ

pc

L>, it is easily

seen that these two inequalities cannot be met simultaneously. We therefore conclude

that the only corner equilibrium involves k=n-1. Now, the winner prices its good at

the maximum level leaving the losers with zero demand at their lowest possible price,

i.e. pi=c"i∈L. That is, pW

Bis such that qncp

L

BW

=− − −

()

[]

+=

αβδ δ

10

. After

substituting for the values of a,b,d, one gets

paac qap ac

W

BW

BW

B

and=− −

()

=− = −

()

>

γγ

0

Accordingly, the winner’s proﬁt is computed as (recalling that a-c≡1)

πγ γγ

W

B

()

=−+

⎛

⎝

⎜⎞

⎠

⎟

111

x

One checks that, when gtends to 1, pW

Btends to c(the loser’s marginal cost) and

π

W

B

tends to x. Note also that, in the corner solution, the winner’s proﬁt becomes inde-

pendent of n.

Proof of Proposition 6

Under Cournot competition, we know from Proposition 2 that the optimal number of

ﬁrms nC

*and the corresponding proﬁt incentives are

for PI

for PI

CC

CC

xx n x x

xx n n nx

≤

()

=

()

=+

()

>

()

=

()

ˆ*,

ˆ*ˆˆ

,

γγ

γγ

11

1

42

γγγ γγγ

()()

=−−

()

⎧

⎨

⎪

⎪

⎩

⎪

⎪,x

x42

Under Bertrand competition, we know from Proposition 5 that the proﬁt incentive

decreases with the number of ﬁrms for xxn<

()

,

γ

, and increases otherwise. So, for

xxn<

()

,

γ

,nB

*=1and PIB(1, g)=(x+2)x/4. On the other hand, for xxn>

()

,

γ

,

nB

*=∞. The level of the proﬁt incentive depends then on whether xis below or above

xB(n,g). Computing xn,

γ

()

and xB(n,g) for n→•, we ﬁnd that they both tend to the

same value, i.e. 2(1 -g)/g. Summarizing, we have

for PI

for PI

BB

BB

xn xx

xn

≤−

()

=

()

=+

()

>−

()

=∞ ∞

(

21 11

1

42

21

γ

γγ

γ

γγ

*,

*,

))

=−+

⎛

⎝

⎜⎞

⎠

⎟

⎧

⎨

⎪

⎪

⎩

⎪

⎪111

γγ

x

The Manchester School26

© 2010 The Authors

The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester

We compute that ˆ

x

γγγγγ

()

−−

()

=−

()

>21 2 8 0

2, which implies that there are three

cases to consider:

• for x<2(1 -g)/g,nn

CB

**

==1and monopoly should prevail;

• for 21−

()

<<

()

γγ γ

xx

ˆ, Bertrand competition should prevail because

PI PI

BC

∞

()

−

()

=−

−

()

⎡

⎣

⎢⎤

⎦

⎥−>,,

γγ γ

γ

γ

γ

11

4

21 2 0

3

xx

• for xx>

()

ˆ

γ

, Bertrand competition should prevail because

PI PI

BC

∞

()

−

()()

=− −

()

⎡

⎣

⎢⎤

⎦

⎥−−

−−

()

>,,,

γγγ γ

γ

γγ

γγγ

nx x x

x

21 2

42

0

3

ˆ

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The Manchester School28

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