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This paper provides a survey on studies that analyze the macroeconomic effects of intellectual property rights (IPR). The first part of this paper introduces different patent policy instruments and reviews their effects on R&D and economic growth. This part also discusses the distortionary effects and distributional consequences of IPR protection as well as empirical evidence on the effects of patent rights. Then, the second part considers the international aspects of IPR protection. In summary, this paper draws the following conclusions from the literature. Firstly, different patent policy instruments have different effects on R&D and growth. Secondly, there is empirical evidence supporting a positive relationship between IPR protection and innovation, but the evidence is stronger for developed countries than for developing countries. Thirdly, the optimal level of IPR protection should tradeoff the social benefits of enhanced innovation against the social costs of multiple distortions and income inequality. Finally, in an open economy, achieving the globally optimal level of protection requires an international coordination (rather than the harmonization) of IPR protection.
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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-gSjiqjin 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 g1, is then given by qi=a-bpi+dSjipj, 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 (x0). 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
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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 x0, 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.
The Manchester School16
© 2010 The Authors
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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
© 2010 The Authors
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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.
The Manchester School18
© 2010 The Authors
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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
© 2010 The Authors
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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
The Manchester School20
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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
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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.
The Manchester School22
© 2010 The Authors
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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
© 2010 The Authors
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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).
The Manchester School24
© 2010 The Authors
The Manchester School © 2010 Blackwell Publishing Ltd and The University of Manchester
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"iL. 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"iL. 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-c1)
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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