incentives to innovate in oligopolies*manc 2131...

INCENTIVES TO INNOVATE IN OLIGOPOLIES*manc_2131 6..28

byPAUL BELLEFLAMME

CORE and Louvain School of Management, Université catholique deLouvain, Belgium

andCECILIA VERGARI†

CORE, Université catholique de Louvain, Belgium, and University ofBologna 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 anoligopoly model with horizontally differentiated products, how much afirm is willing to pay for a process innovation that it would be the onlyone to use. We show that different measures of competition (number offirms, degree of product differentiation, Cournot vs. Bertrand) affectincentives to innovate in non-monotonic, different and potentiallyopposite ways.

1 Introduction

Our objective in this paper is to explore in a systematic way how competitionaffects firms’ incentives to innovate. Thereby, we aim at exploring further therelationship between market structure and innovation incentives, which hasalways been a central, and debated, issue in economics since Schumpeter’sclassic work, Capitalism, Socialism, and Democracy, in 1943.

1.1 Context

Schumpeter’s first conjecture was to stress the necessity of tolerating thecreation of monopolies as a way to encourage the innovation process. Thisargument is nothing but the economic rationale behind the legal protection ofintellectual property and is nowadays widely accepted. Schumpeter’s secondconjecture was that large firms are better equipped to undertake R&D thansmaller ones. The best way to support this conjecture is probably to say thatlarge firms have a larger capacity to undertake R&D, in so far as they candeal more efficiently with the three market failures observed in innovativemarkets, namely externalities, indivisibilities and uncertainty. It is not clear,however, whether large firms, because of their monopoly power, also havelarger 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 commentsand discussion about an earlier draft.

The Manchester School Vol 79 No. 1 6–28 January 2011doi: 10.1111/j.1467-9957.2009.02131.x

© 2010 The AuthorsThe Manchester School © 2010 Blackwell Publishing Ltd and The University of ManchesterPublished 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 theincentives for R&D is Arrow (1962). He compares the profit incentive toinnovate for monopolistic and competitive markets, concluding that perfectcompetition fosters more innovation than monopoly. The intuition behindthis result is that a monopolist has less incentive to innovate because italready makes profit before the innovation, whereas the competitive firm justrecoups 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,1 Arrow’s intuition has made its way through theeconomic press:

It is surely no coincidence that Microsoft’s hidden ability to innovate has becomeapparent 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 amonopoly (in Windows) and a near monopoly (in Office). But in the new marketsof gaming, mobile devices and television set-top boxes, Microsoft has beenunable to exploit its Windows monopoly other than indirectly—it has financedthe company’s expensive forays into pasture new.

1.2 Our Approach

The previous argument seems to suggest that perfect competition alsodominates oligopolistic market structures in terms of innovation incentives.However, this conjecture turns out to be wrong. We argue indeed that anintermediate form of competition may provide a higher incentive to innovatethan 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 bethe only one to use? We also examine under which industry structure thiswillingness 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 thefirm would get without the innovation. That is, we suppose that, if the firmdoes not acquire the innovation, no other firm does. We measure thus thepure ‘profit incentive’ to innovate, i.e. the desire to increase profits inde-pendently of the rival firms. As for the intensity of competition, our modelallows 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 AuthorsThe 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 affectedin qualitatively different ways by an increase in competition. The practicalconsequence is that, depending on the characteristics of the industry of interest,the highest profit incentive can be reached by a competitive firm (Arrow’sclaim), by a monopoly (Schumpeter’s claim) or by an intermediate form ofcompetition. We provide a rule about how to enhance firms’ incentive toinnovate for any industry.

More precisely, our main results can be summarized as follows. First,regarding the effect of product differentiation, the profit incentive isU-shaped whatever the nature of competition, the innovation size and thenumber of firms. Second, the effect of the number of firms depends on theother parameters. Under Cournot competition, the profit incentive eitherdecreases with the number of firms or has a U-inverted shape. Under Ber-trand competition, the profit incentive either decreases or increases with thenumber of firms. In both cases, for the second option to occur, the innovationand the degree of product substitutability must be large enough. Third, inboth natures of competition, there exist ranges of parameters for which theprofit incentive is affected in opposite ways by different measures of compe-tition (it decreases with the number of firms and increases with the degreeof product substitutability). Fourth, Arrow’s result is no longer valid whenproducts are sufficiently differentiated and/or the innovation is not too large;monopoly is then the optimal market structure in terms of profit incentive toinnovate, 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 providingambiguous results. Among others, Blundell et al. (1999) conduct a detailedanalysis of British manufacturing firms and establish a robust and positiveeffect of market share on innovation. In contrast, Aghion et al. (2005) findstrong evidence of non-monotonicity; in particular they find an inverted Ushape for the competition–innovation relationship. Moreover, a recentempirical analysis by Tang (2006), based on Canadian manufacturing firms,shows that different measures of competition affect in different ways theincentives to innovate. In particular, the author argues that ‘both competitionand 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 thatthe relationship between competition and innovation can be positive or nega-tive, ‘depending on specific competition perception and specific innovationactivity’.

The Manchester School8


1.5 Related Literature

Our framework is directly linked to a vast industrial organization literaturewhich, in line with Arrow, assumes that there is only one innovator whichcannot be imitated by competitors.2 Bester and Petrakis (1993) contrast theprofit incentive in Cournot and Bertrand duopolies for different degrees ofhorizontal product differentiation. Bonanno and Haworth (1998) address thesame question as Bester and Petrakis (1993) but under a model of verticalproduct differentiation. Yi (1999) examines the effect of the number of firmson the profit incentive to innovate in Cournot oligopolies with a homoge-neous product. Delbono and Denicolò (1990) compare and contrast staticand dynamic efficiency in oligopolies producing a homogeneous product fordifferent numbers of firms. Like Bester and Petrakis (1993), they use thenature of competition to measure the intensity of competition. They measurethe incentive to innovate by the profit incentive as well as by the ‘competitivethreat’, 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 profitincentive and the loss the firm would face in case any rival firm enjoyed thecost reduction. Hence, the value of the innovation increases because firmswant to pay more in order to protect their position. Boone (2001) also studiesthe competitive threat but in a framework of asymmetric firms where hefollows an axiomatic approach by defining as measure of competition aparameter satisfying particular conditions.

One important conclusion that can be drawn from this literature is thatdifferent dimensions of competition may affect firms’ investment in R&Din non-monotonic and potentially different ways. Our contribution to thisliterature is to provide a unified framework where different sources of com-petition interact and determine firms’ incentives to innovate. We not onlyconfirm the results gathered in the previous literature, but we also extendsome of them and, more importantly, we provide new results about theinteraction between different measures of competition.

Our analysis can be related to two other strands of the literature oninnovation. First, the incentives to innovate can be computed by includingthe revenue the innovator could raise through licensing the innovation. Addi-tional issues arise then about the form the license should take (royalty perunit 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. Ourcreative environment is characterized by what Scotchmer (2004) calls ‘scarce ideas’ todistinguish them from ideas which are common knowledge.



Second, another important question is how much firms are willing toinvest in cost-reducing R&D (and not just how much they are willing topay for an innovation of a given size), especially when all firms have thesimultaneous opportunity to achieve competing innovations. It must thenbe recognized that R&D is like any form of investment in that it precedesthe production stage. As a result, strategic considerations and the extent ofknowledge spillovers play a central role in determining which market struc-ture provides firms with the highest incentives to undertake R&D.3 In thisapproach, as opposed to ours, each firm can obtain some cost reduction byinvesting in R&D. In this symmetric setting, Cournot competition (withstrategic substitutes) always results in higher cost reductions than Bertrandcompetition (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’Aspremontand Jacquemin (1988) presented a seminal analysis, which was thenextended by, for example, Kamien et al. (1992) and generalized by Amiret al. (2003). Finally, we include in this symmetric framework literature arecent paper by Vives (2006). The author analyses a simultaneous-movegame where firms choose an investment–price pair (investment is thus non-strategic). He obtains general and robust results about the relationshipbetween innovation and competition by using general functional forms andconsidering 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 setup the model. In Sections 3 and 4, we examine in turn Cournot and Bertrandcompetition. In Section 5, we define the market structure which maximizesthe incentives to innovate. We conclude in Section 6. The main proofs appearat the end of the paper, while the more technical proofs are relegated to anappendix which is available from the authors upon request.

2 Model

There are n firms (indexed by i = 1, . . . , n) competing in the market. Eachfirm i incurs a constant marginal cost equal to ci and produces a differen-tiated product, qi, sold at price pi. The demand system is obtained fromthe 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 byBrander and Spencer (1983), assuming no R&D spillovers between firms and Cournotcompetition 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.



pi = a - qi - gSj�iqj in the region of quantities where prices are positive. Theparameter g is an inverse measure of the degree of product differentiation:the lower g the more products are differentiated (if g = 1, products areperfect substitutes; if g = 0, products are perfectly differentiated). Thedemand schedule, for g � 1, is then given by qi = a - bpi + dSj�ipj, with

αγ

=+ −( )

an1 1

β γγ γ

= + −( )−( ) + −( )[ ]

1 21 1 1

nn

δ γγ γ

=−( ) + −( )[ ]1 1 1n

In this framework, total quantity Q depends on n as well as on g (this isbecause the representative consumer loves variety). In particular, in thesymmetric case where pi = p "i, we have

Q nn a p

n, γ

γ( ) =

−( )+ −( )1 1

(1)

It follows that n and g influence the individual quantities as well as themarket size.

Under Cournot competition, the equilibrium quantity and profit for firmi are found as

qa n c c

nqi

i jj ii i

C C C=−( ) − + −( )[ ] +

−( ) + −( )[ ]= ( )≠∑2 2 2

2 2 12

γ γ γ

γ γπ (2)

We can also compute the equilibrium price, quantity and profit underBertrand competition:

pn c c

ni

i jj iB =+( ) + − −( )[ ] +

+( ) − −( )[ ]≠∑2 2 2

2 2 1

β δ α β β δ δβ

β δ β δ(3)

q p c p c qi i i i i i iB B B B B= −( ) = −( ) = ( )( )β π β β2 2

1 (4)

Initially, all firms produce at cost ci = c. A new process innovation allowsfirms to reduce the constant marginal cost of production from c to c1 = c - x(with 0 < x < c). We assume that the innovation is non-drastic. That is, thecost reduction does not allow the innovator to behave like a monopolist. Asufficient condition is that the monopoly price corresponding to cl is largerthan the initial cost c, i.e. (a + c - x)/2 > c. Equivalently, assuming without lossof generality that the difference a - c is equal to unity, we have the followingassumption.



Assumption 1: x < a - c = 1.4

Under Cournot competition this definition of non-drastic innovation isequivalent to saying that the innovation does not allow the innovator tothrow the other firms out of the market. As for Bertrand competition this isnot 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 theinnovation is said to be non-drastic, because the innovator cannot price at themonopoly level.

In line with Arrow’s question, we want to find out how much a firm iswilling to pay for acquiring the innovation and being its single user.5 As aconsequence, we concentrate on the profit incentive that can be seen as the‘pure’ incentive to innovate. We denote by pW the profit accruing to theinnovator (the ‘winner’), and by p the pre-innovation profit. For futurereference, we also define pL as the profit accruing to the rivals of the innovator(the ‘losers’). Based on these definitions, we formally define our measure offirms’ 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 incentiveunder 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 degreeof product differentiation (g ) in ways we will now analyze. Finally, note that

PI PI PI PIC C B B1 0 1 02

4, , , ,γ γ( ) = ( ) = ( ) = ( ) = +( )

n nx x

which corresponds to the profit incentive for a monopoly (either becausethere 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 ofcompetition. We consider three measures of the strength of competition: thedegree of product substitutability (g ), the number of firms in the market (n)and the nature of competition (Cournot vs. Bertrand). As for the firstmeasure, we know that, as g decreases, firms’ market power increases becauseproducts become more differentiated. Moreover, given n ex ante symmetricfirms in the market, we can check from expression (1) that total quantity is adecreasing function of g (as products become closer substitutes, the marketsize decreases). Therefore, a change in g affects the degree of competition in

4As shown by Zanchettin (2006) in the duopoly case, this condition is sufficient but not necessaryfor values of g < 1. We also assume that c 3 1, so that Assumption 1 guarantees that theinnovator still has a positive marginal cost.

5We discuss in the conclusion the role of the identity of the innovator (either incumbent oroutside research lab).



two mutually reinforcing ways: as g increases, products become closer sub-stitutes and the market size decreases. In contrast, an increase in the numberof firms affects the degree of competition in two opposite ways: on the onehand, individual firms’ profits decrease but, on the other hand, the marketsize increases.6 However, it can easily be shown that the former effect isstronger than the latter: the pre-innovation, the losers’ and the winner’sprofits are all decreasing in n. We can then take n as another measure of thestrength of competition.7 Finally, as for the nature of competition, we knowfrom Singh and Vives (1984) that going from a Cournot to a Bertrand marketstructure implies an increase in competition.

As we will now explain, the intensity of competition affects the profitincentive to innovate in contrasting ways. The general intuition is that thereis an opposition between a negative competition effect (a more competitivemarket reduces firm i’s profit if either it gets the innovation or it does not) anda positive competitive advantage (the tougher the competition in the market,the larger the innovator’s advantage).

3 Cournot Competition

We study here how the incentives to innovate are affected by the intensity ofcompetition in a Cournot framework. We first investigate the effects of aninfinitesimal cost reduction, which allows us to highlight the various forcesat work. We then consider discrete cost reductions. Finally, we examinewhether the two measures of competition (g and n) affect the incentive toinnovate in a converging or diverging way.

Using expression (2), we find

qn xn

qWC

WC

WC= −( ) + + −( )[ ]

−( ) + −( )[ ]= ( )2 2 2

2 2 12γ γ

γ γπ (5)

qn

qC C C=+ −( )

= ( )12 1

2

γπ (6)

qxn

qLC

LC

LC= −( ) −

−( ) + −( )[ ]= ( )2

2 2 12γ γ

γ γπ (7)

where Assumption 1 is used to guarantee that all equilibrium quantities(especially qL

C ) are always positive. Applying Definition 1, we compute thevalue of the profit incentive as

6One can check from expression (1) that total quantity is an increasing function of n (anadditional firm brings one more variety in the market, which increases total demandbecause consumers like variety).

7In Section 3.2, we discuss how the side-effect of the number of firms on the market size affectsthe sensitivity of the profit incentive to the number of firms.



PIC WC Cn x

n n x

n, γ π π γ γ γ

γ γ( ) = − = + −( )[ ] −( ) + + −( )[ ]{ }

−( ) + −2 2 2 2 2 2

2 22 11 2( )[ ] (8)

In order to analyze how PIC(n, g ) depends on g and n, we can get someintuition by first studying its separate components, πW

C and pC. We summa-rize our results in the following lemma,8 where x̂ n n( ) ≡ +( )1 2 andˆ ,γ n x x x x n x n( ) ≡ + − +( )⎡⎣ ⎤⎦ − −( )[ ]2 1 1 2 .

Lemma 1: (i) The winner’s and pre-innovation profits both decrease with thenumber of firms in the market, (ii) the pre-innovation profit decreases withthe degree of product substitutability, and (iii) the winner’s profit increaseswith the degree of product substitutability if this degree and the size of theinnovation are large enough (i.e. if x x n> ( )ˆ and γ γ> ( )ˆ ,n x ); it decreasesotherwise.

It is worth stressing the third result in Lemma 1. In general, profits aredecreasing in g. Nevertheless, if the innovation is large enough, the innovatormay gain from an increase in g. In particular, πW

C is first decreasing and thenincreasing in g. In fact, as soon as g becomes positive, the firm is no longer amonopolist 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’sprofit decreases. However, as soon as products are sufficiently substitutable,the cost advantage of the innovator becomes more important because theinnovation becomes a sort of substitute for product differentiation and so thewinner’s profit increases. This means that our two measures of competitionhave contrasting effects on the winner’s profit. Note that x̂ n( ) and ˆ ,γ n x( )decrease with n. Therefore, the higher the number of firms in the market, thelarger 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 ofcompetition on the profit incentive.

3.1 Infinitesimal Analysis

In order to single out the forces at work, we first consider an infinitesimalreduction in cost (x → 0). The profit incentive can be seen as the discreteversion of the sensitivity of i’s profits to a reduction in its own cost (i.e.−∂ ∂π i icC evaluated at ci = c). At the Cournot equilibrium, we haveπ γi i i i i j i ip c q a q n q c qC C C C C C= −( ) = − − −( ) −[ ]1 , so that, using the envelopetheorem,

− ∂∂

= + −( ) ∂∂

π γi

ii

j

ii

cq n

q

cq

CC

CC1 (9)

8The proof is relegated to the technical appendix.



In expression (9), the first term measures the direct effect and the second termthe strategic effect of an infinitesimal cost reduction. They are both positive.As ci decreases, two positive forces affect firm i’s profit: first, firm i becomesmore 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 effectswith respect to changes in g and in n. Table 1 decomposes the results.

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

C γ and ∂ ∂q niC are negative: the direct effect decreases as products

become closer substitutes and as the number of firms increases. On the otherhand, 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 threeforces, which are affected in opposite directions by g and n. First, each rivalfirm reduces its quantity in reaction to firm i’s cost reduction ( ∂ ∂ >q cj i

C 0);the larger g and the smaller n, the stronger this individual reaction. Second,the combined effect of these individual reactions increases with productsubstitutability and with the number of rival firms (g (n - 1)). Finally, firm i’sprofits are affected in proportion to its quantity ( qi

C ), which decreases with gand 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 increasesbecause the rival firms’ quantities become more sensitive to a reduction infirm i’s cost. So, as g increases, the direct effect becomes weaker but thestrategic effect becomes stronger. At g = 0, the strategic effect disappears andso the marginal return of a cost reduction decreases with g. At the otherextreme (g = 1), it can be shown that the increase in the strategic effectdominates the decrease in the direct effect and so the marginal return of a costreduction increases with g. There is thus an intermediate value of gfor which the two effects just compensate and where −∂ ∂π i icC reaches aminimum. In other words, the marginal return of a cost reduction is U-shapedwith respect to g. Second, regarding the influence of n, the sensitivity of thestrategic effect remains ambiguous; nevertheless, it can be shown that thetotal effect decreases with n ( −∂ ∂ ∂ <2 0π i ic nC ).

Table 1

Direct effect Strategic effect Total effect

qiC + g (n - 1)

∂∂qc

j

i

C

qiC = − ∂

∂π i

ic

C

Effect of g - + + - ?Effect of n - + - - ?



3.2 Discrete Analysis

We now examine how discrete cost reductions challenge these results. As wehave seen in the infinitesimal analysis, it is impossible to a priori sign thebalance between the different effects at work. Considering discrete costreductions complicates the analysis further as the innovation size affects themagnitude of the various effects. It is thus by direct computations that wehave derived the results stated in the next two propositions; we rely on theinfinitesimal analysis to highlight the intuition behind these results.

Proposition 1: Under Cournot competition: (i) the profit incentive isU-shaped with respect to g, and (ii) the highest profit incentive is reachedunder independent products (g = 0) if the innovation size is below somethreshold (between 2/7 and 2/3), and under homogeneous products (g = 1)otherwise.

Proposition 1 (which is formally stated in the Appendix) establishes thatPIC first decreases with g and then increases. Whether the largest incentive isreached for g = 0 or g = 1 depends on the other parameters (n and x). Weobserve thus that the shape of PIC with respect to g is the same as in theinfinitesimal case. The fact that larger values of g lead to a larger profitincentive to innovate can be seen as a form of substitutability betweenproduct 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 largeenough, 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 theeffect of the number of firms on the profit incentive is also non-monotonic.9

The interesting result is that, when x and g are large enough, the profitincentive to innovate first increases and then decreases with n.10 On the otherhand, from the infinitesimal case, we know that, as x → 0, the profit incentivedecreases with n. At the other extreme, we can show that, when g and x tendto one (i.e. towards homogeneous product and drastic innovation), the profitincentive 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 n for afairly large class of demand functions. He also considers arbitrary innovations under lineardemand 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 therelation between competition and innovation with a UK panel data set.



As already discussed, an increase in the number of firms affects theintensity of competition in two opposite ways: on the one hand, individualfirms’ profits decrease but, on the other hand, the market size increases.Therefore, as a robustness check, in what follows, we try to separate the twoeffects.11 Define

θγ

≡+ −( )

nn1 1

Then we can rewrite the total quantity (1) as Q(n, g ) = q(a - p). We take q asa measure of the size of the market. It decreases with g, and increases with nas long as g < 1 (for g = 1, q = 1 is independent of n). Solving for g in theabove expression and making the proper substitutions, we can express theprofit incentive to innovate as a function of q and n:

PIC = + −( ) + −( ) + − −+( ) − −( )

θ θ θ θ θθ θ θ

22 22

2 4 2 2

2nx n

n n x n n

n n n

The effect of the number of firms on the incentive to innovate can be thusdecomposed as follows:

dPId

PI PI dd

C C C

n n n= ∂

∂+ ∂

∂θθ

The first term is the ‘pure effect’ of the number of firms; it is computed bykeeping q constant, thereby freezing the effect that n has on the size of themarket. The second term is the ‘market-size effect’ of the number of firms: achange in n affects the market size q, which itself affects the incentive toinnovate. 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, onlythe first term remains and we already know that the effect of n can benon-monotonic if the size of the innovation is large enough. This result stillholds for large enough values of g and of x. For instance, it can be shown that,for x = g = 0.95, ∂PIC/∂n > 0 for n 2 10 and ∂PIC/∂n < 0 for n > 10.12 We canthus 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 ofthe market.

11We further check that our non-monotone results on the effect of n on the profit incentive arepreserved under the Shubik and Levitan (1980) linear demand model where the market sizedoes not change with n.

12Moreovcr, looking at the second term, we know that dq/dn > 0, but simulations show that thesign of ∂PIC/∂q can also change with n. For instance, one can check that, for x = g = 1/3,∂PIC/∂q > 0 for n 2 48 and ∂PIC/∂q < 0 for n > 48. In sum, isolating the pure effect of n fromits market-size effect does not shed much light on the non-monotonic relationship betweenthe number of firms and the incentive to innovate.



3.3 Cross-effects

We want now to contrast the ways the two measures of competition affect theprofit incentive. According to Proposition 2, the profit incentive increaseswith the number of firms provided that the innovation size and the degree ofproduct substitutability are large enough. Let us make this statement moreprecise. Considering n as a continuous variable, it is readily shown that

∂∂

( ) > ⇔ > ( ) = − − + +( ) − −( )− + −n

n n xn x n x n

n nx xPIC , ,γ γ γ0

4 4 13 2

2

(10)

where γ n x,( ) > 0 for all admissible n and x, and γ n x,( ) < 1 if and only ifx > (n - 1)/n.

From Proposition 1, we know that the profit incentive is U-shaped withrespect to g. Let �γ n x,( ) denote the value of g for which the profit incentivereaches its minimum (we proved that 0 1< ( ) <�γ n x, for all admissible nand x). We find that γ γn x n x, ,( ) > ( )� for all admissible n and x, which allowsus to state the following proposition.13

Proposition 3: (i) The two measures of competition affect the profit incentivein converging ways either if products are sufficiently differentiated (forγ γ< ( )� n x, , the profit incentive decreases with g and n), or if the innovationis large enough and products are sufficiently substitutes (for x > (n - 1)/n andγ γ> ( )n x, , the profit incentive increases with g and n), and (ii) there alwaysexists a range of parameter values for which the two measures of competitionaffect the profit incentive in diverging ways (for �γ γ γn x n x, min , ,( ) < < ( ){ }1 ,the profit incentive increases with g and decreases with n).

The results of Proposition 3 (which are illustrated in Fig. 1) give us a ruleabout how to enhance the incentives to innovate for any innovation size xand any industry (n, g ). For instance, it tells us for which industries a mergerof two firms (decrease in n) or the introduction of a product innovation(decrease in g ) increases or decreases the incentive to invest in a processinnovation.

Another implication of Proposition 3 is that the same level of profitincentive can be achieved in different industries. In this respect, an instructivethought experiment is to fix the innovation size x and examine which indus-tries give firms the same level of profit incentive as a monopoly (n = 1 and/org = 0). We know from Proposition 1 that, for x < 2/7, the monopoly gives ahigher profit incentive than any other Cournot industry. So, we take forexample x = 1/2 and we compute that the following industries (n, g ) areequivalent 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 themonopoly level, an increase in the number of firms has to be compensated

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



by an increase in product substitutability. In particular, we observe that fivefirms producing a homogeneous product are willing to pay the same amountas a monopoly for a process innovation that decreases the marginal cost ofproduction by x = 1/2.

4 Bertrand Competition

The derivation of the profit incentive is slightly more involved under Bertrandcompetition. Indeed, in contrast with Cournot competition, Assumption 1 isnot sufficient to ensure that the non-innovating firms (the ‘losers’) are allactive on the market after the innovation. In fact, for a loser to set a pricelarger than its marginal cost c (and thus produce a positive quantity), thefirms’ products must be sufficiently differentiated. We need thus to considerthe possibility of corner equilibria in which a number of losers are con-strained to price at marginal cost. The following lemma (which is proved inthe Appendix) characterizes the Bertrand–Nash equilibrium of the post-innovation game for all values of x, n and g. Define

x nnn

B , γ γ γ γγ γ γ

( ) = −( ) − +( )+ −( )

1 2 3 21 2

Lemma 2: The Nash equilibrium of the post-innovation price game is suchthat (i) for x 2 xB(n, g ), all firms price above their marginal cost, and (ii) forx > xB(n, g), the winner is the only firm pricing above its marginal cost.

n1020

1

g

g~

g–

PIC decreases with n and g

PIC increases with n and g

PIC decreases with n andincreases with g

0.8

0.4

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



We have that, for x ∈ (xB, 1), the winner is the only active firm in themarket and behaves like a constrained monopolist (because the innovationis non-drastic). Note that the threshold separating the two cases, xB(n, g ),decreases with n and tends to 2(1 - g )/g for n → • (which is greater than unityfor g < 2/3). Therefore, the corner solution can only be observed for suffi-ciently large values of x and g.

We can now define the profit incentive. First, from expressions (3) and(4), we derive the pre-innovation equilibrium profit:

π γ γγ γ

γγ

B nn

n n,( ) = + −( )

−( ) + −( )[ ]−−( ) +

⎡⎣⎢

⎤⎦⎥

1 21 1 1

13 2

2

(11)

Next, using the proof of Lemma 2, we compute the winner’s profit at theNash equilibrium of the post-innovation game:

π γ βγ γ γ γ γ

γ γWB n

n n n x n

n,( ) =

− +( ) + −( ) +[ ] + −( ) + −( )+ −(

5 5 3 2 2 1 2 2 3

2 3

2 2

)) + −( )⎧⎨⎩

⎫⎬⎭

≤ ( )

( ) = − +⎛⎝⎜

⎞⎠⎟ >

2 2 3

11 1

2

γ γγ

π γγ γ

n

x x n

x x x

for

for

B

WB

,

BB n, γ( )

(12)

Finally, combining (11) and (12), we can express the profit incentive underBertrand competition as

PIPI for

PIB

B WB B

B

B WB

nn n n x x n

n,

, , , ,

,γ

γ π γ π γ γ

γ π( ) =

( ) = ( ) − ( ) ≤ ( )

( ) = γγ π γ γ( ) − ( ) > ( )⎧⎨⎩ B

Bforn x x n, ,(13)

We can now examine how the profit incentive evolves with the twomeasures of competition in a Bertrand framework. As before, in order toassert the intuition behind its sensitivity, we proceed by first examining theeffects of an infinitesimal cost reduction and we then extend the analysis tothe effects of a discrete cost reduction.

4.1 Infinitesimal Analysis

At the Bertrand equilibrium, the following equation holds:

−∂∂

= +−( )

+ −( )−

∂∂

⎛⎝⎜

⎞⎠⎟

+( )

π γγ

i

ii

j

icq

nn

p

cq

BB

direct effect

B

�1

1 2iiB

strategic effect −( )

>� ��

0(14)

As for the Cournot case, the first term of expression (14) measures the directeffect and the second term measures the strategic effect of an infinitesimalcost reduction. The direct effect is positive as firm i becomes 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



complements), which affects negatively firm i’s profit. As above, we areinterested in assessing the sensitivity of these two effects with respect tochanges in g and in n. Table 2 decomposes the results.

In contrast with the Cournot case, we can a priori sign the impact of thenumber of firms: as both the direct effect and the strategic effect decrease withn, the total effect of an infinitesimal cost reduction clearly decreases with thenumber of firms. On the other hand, the effect of a change in g is ambiguous: thedirect effect decreases with g but the strategic effect might increase or decrease.Nevertheless, computations reveal the following: the strategic effect is nil wheng = 0; as soon as g > 0, it first decreases and then increases with productsubstitutability; however, even when this sensitivity is positive, it never com-pensates the negative effect of g on the equilibrium quantity (direct effect). Asa 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 twopropositions (which are both proved in the technical appendix). Startingwith 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 firstdecreases 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 n and x.

We observe thus that the degree of product substitutability affects PIB

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

and PIC in a duopoly model by restricting the range of g to values such thatall firms stay on the market (i.e. x 2 xB(2, g )). They show that there exists acut-off value of g such that PIC > PIB for g below the cut-off and PIB > PIC forg above the cut-off. Using our derivation of PIB 2, γ( ), we can show thatBester and Petrakis’s conclusion extends to larger values of g (i.e. those forwhich 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

qiB − −( )

+ −( )γ

γn

n1

1 2∂∂pc

j

i

B

qiB = − ∂

∂π i

ic

B

Effect of g - - + - ?Effect of n - - - - -



Bertrand competition, the profit incentive reaches its maximum for homoge-neous products independently of the other parameters (n and x). Intuitively,whatever the innovation size and the number of firms, the winner has more towin under homogeneous products because the innovation allows him toremain the only active firm on the market.

On the other hand, the effect of the number of firms on the profit incentiveunder Bertrand competition is monotone: PIB is either decreasing or increasingwith n depending on the innovation size and the degree of product differen-tiation. We show in the technical appendix that there is a unique positivevalue of x, x n, γ( ), such that ∂ ( ) ∂ =PIB n n, γ 0, and that x n x n, ,γ γ( ) < ( )B

for all n and g. Therefore, we have the following proposition.

Proposition 5: Under Bertrand competition, the profit incentive decreaseswith the number of firms for x x n< ( ), γ , and increases otherwise. A neces-sary condition for the latter possibility ( x n, γ( ) < 1) is g > 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 x n, γ( ) as the positive value of x such that ∂PIB(n, g )/∂g = 0, wecan show that x n x n, ,γ γ( ) < ( ), meaning that there are up to three possiblesituations: (i) for x x n< ( ), γ , the profit incentive decreases with both n and g;(ii) for x n x x n, ,γ γ( ) < < ( ), the profit incentive decreases with n but increaseswith g; and (iii) for x x n> ( ), γ , the profit incentive increases with both n andg. As for Cournot competition, it is impossible to have the profit incentiveincrease with n while decreasing with g.

5 Market Structure and Innovation

After studying how different sources of competition affect the profit incentiveto innovate, we can proceed to investigate which market structure maximizesthe profit incentive to innovate.15 Arrow’s (1962) argument, stating thatthe incentive to innovate is larger under perfect competition than undermonopoly, was based on Bertrand competition in a homogeneous market.We check indeed that, with a homogeneous product, the profit incentiveunder Bertrand competition is defined as PIB n,1( ), which increases in nand reaches thus its maximum at PIB ∞( ) =,1 x ; we also compute thatPI PIB B∞( ) = > ( ) = +( ), ,1 1 2 4x x xγ , which establishes Arrow’s result.

Now, Proposition 2 tells us that the previous argument does not holdunder 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’ toinnovate. 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 underdifferent market structures (perfect competition, oligopolies, monopoly). We now wonderunder which conditions this willingness to pay is maximized.



never leads to the highest profit incentive. Also, Proposition 5 shows that assoon as products are differentiated, the profit incentive under Bertrand com-petition might be decreasing in n if the innovation is small enough, meaningthat a monopoly gives higher incentives to innovate.

Combining our previous results, we are in a position to complementArrow’s analysis in a useful way by solving the following exercise: for a givendegree of product differentiation and a given innovation size, what is the com-bination of competition mode and number of firms that yields the highest profitincentive to innovate? As shown in the following proposition, the answer isvery simple to state.

Proposition 6: For sufficiently large values of x and g, Bertrand competitionwith 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 (whichsupposes g > 2/3), the best structure under Bertrand competition (i.e. thenumber of firms maximizing the profit incentive, nB* → ∞ ) yields a higherprofit incentive than the best structure under Cournot competition (i.e. nC* = 1or nC* is (the integer closest to) ˆ ,n x γ( ), depending on the value of x). On theother hand, for x < 2(1 - g )/g, the best market structure is monopoly, whichprovides a profit incentive equal to (x + 2)x/4.

6 Conclusion

In this paper, we have provided a unified framework where different sourcesof competition interact and determine firms’ incentives to invest in a processinnovation. Our results are threefold: first, we confirm the existence of anon-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 Bertrandcompetition with linear demand and horizontally differentiated products byconsidering any number of firms and any degree of product differentiation.Second, we show that different sources of competition can have divergingeffects on the innovation incentives; we characterize the conditions underwhich this occurs, thus providing a rule about how to enhance firms’ incen-tives to innovate. Finally, we provide a rationale for the empirical evidencethat the relationship between competition on the product market and inno-vation is ambiguous and depends on the dimension of competition we aretaking into account. The direct and practical implication of our set-up is thena potential guideline for antitrust authorities.

In the spirit of Arrow (1962), we focus our attention on the profitincentive as a measure of firms’ innovation incentives. This strategy allows usto extend and complement Arrow’s analysis; in contrast, we do not consider



the incentive which comes from the recognition that there exist rival firmscompeting to be the first to innovate, i.e. the competitive threat. The analysisof the competitive threat in such a framework constitutes a first possibleextension of our model. Nevertheless, preliminary results show that the dif-ferent sources of competition considered here affect the competitive threatand the profit incentive in qualitatively similar ways.

A second linked research area we propose to work on deals with theidentity of the innovator. Our profit incentive corresponds to the followingscenario. We can imagine that the innovator is an incumbent firm, whicheither manages to keep its innovation secret, or which is granted a broadpatent of infinite duration and which does not license the innovation to rivalfirms; our question was then to assess how much this incumbent is willing toinvest in R&D. Nevertheless, our approach does not take into account theincentive of the innovator to license his or her cost-reducing innovation northe possibility that the innovator is outside the market.16 In particular,Kamien and Tauman (2002) demonstrate that an outside innovator finds itprofitable to auction more than one license, and that an inside innovator alsohas incentives to license the innovation to its rivals. Their approach is analternative measure of firms’ incentives to innovate, i.e. the innovator’s profitcoming 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 inthe presence of product differentiation. Studying the behavior of the innova-tor in our framework would allow us to have a more complete picture offirms’ 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 n 3 2, there exists�γ C ∈( )0 1, such that ∂PI/∂g < 0 for γ γ> �C , ∂PI/∂g = 0 for γ γ= �C and ∂PI/∂g > 0 forγ γ> �C , and (ii) the highest profit incentive is reached under independent products(g = 0) if x x n n n< ( ) ≡ −( ) +( )C 2 1 3 1 and under homogeneous products (g = 1)if x x n> ( )C ; x nC ( ) increases in n, and is between 2/7 and 2/3.

Proposition 2 is formally stated as follows. Define

ˆ ˆnx

xxγ γ γ γ γ

γ γ γγ γ γ γ( ) = −( ) + −( ) −( )

− −( )( ) = −( ) − − +2 1 3 2 2

22 2 2 42

and(( )

−( )γ γ8 2

(i) Taking n as a continuous variable, the profit incentive reaches a maximum forn n= ( )ˆ γ ,17 and (ii) taking into account that n is an integer, the maximum is reached forn > 1 if and only if x x> ( )ˆ γ , which supposes that g > 0.711.

16The incentive to innovate of an outside inventor who decides to auction a single license wouldbe measured by the competitive threat.

17Computing n̂(g ) at g = 1, we find that the profit incentive decreases with n as long asn > 1/(1 - x); this is equivalent to x < (n - 1)/n, the condition given by Yi (1999).



The proofs of these two propositions are purely technical and can be obtainedfrom the authors upon request.

Proof of Lemma 2

The post-innovation price game is played by one winner (indexed by W ) anda set L of n - 1 losers indexed by i. A typical loser i’s maximization programwrites as

max,

pi j

j L j ii i

ip p p p c p cα β δ δ− + +⎛

⎝⎜⎞⎠⎟

−( ) ≥∈ ≠∑W subject to

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

p p p ci jj L j i

= + + +⎛⎝⎜

⎞⎠⎟∈ ≠

∑12β

α δ δ βW,

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

α δ δ β+ + ≥∈ ≠∑p p cj

j L j iW

,(A1)

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

R p pp p c

ci j j L

j i jj L j iW

W if A is met

oth

, ,( )( ) =+ + +( ) ( )

∈≠ ∈ ≠∑1

21

βα δ δ β

eerwise

⎧⎨⎪

⎩⎪(A2)

As for the winner, the maximization program writes as

maxp

ii L

p p p c x p c xW

W W Wsubject toα β δ− +⎛⎝⎜

⎞⎠⎟ − +( ) ≥ −

∈∑

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

p p p c xi i L ii L

W ( )( ) = + + −( )⎡⎣⎢

⎤⎦⎥∈

∈∑1

2βα δ β (A3)

We show that this value is always larger than c - x. Indeed, the lowest valueof pW, noted pW , is reached when pi = c "i ∈ L. After substitutions, wecompute

p c xx n

nW − −( ) = −( ) + + −( )

+ −>1

21 1 2

1 20

γ γ γγ γ

Let us first characterize the interior Nash equilibrium. Solving the system ofequations given by (A3) and by the top row of (A2) taken n - 1 times, we getequilibrium prices for the winner and the losers respectively given by pW

B and pLB. We

have that



p cn n x

n n

x x

LB − = −( ) − +( ) − + −( )

+ −( ) + −( )≥

⇔ ≤

1 2 3 2 1 22 3 2 2 3

0γ γ γ γ γ γ

γ γ γ γ

BB nnn

, γ γ γ γγ γ γ

( ) = −( ) − +( )+ −( )

1 2 3 21 2

(A4)

Suppose now that condition (A4) is not met. The equilibrium must then be suchthat some losers set pi = c, while the other losers set a price p̂ cL > . Suppose that klosers price above marginal cost, with 0 < k < n - 1. Condition (A1) must be satisfiedfor those k losers (i.e. α δ δ β+ + − −( ) + −( )[ ] ≥p n k c k p cW L1 1 ˆ ) and violated for theother (n - k - 1) ones (i.e. α δ δ β+ + − −( ) +[ ] <p n k c kp cW L2 ˆ ). As p̂ cL > , it is easilyseen that these two inequalities cannot be met simultaneously. We therefore concludethat the only corner equilibrium involves k = n - 1. Now, the winner prices its good atthe maximum level leaving the losers with zero demand at their lowest possible price,i.e. pi = c "i ∈ L. That is, pW

B is such that q n c pLB

W= − − −( )[ ] + =α β δ δ1 0. Aftersubstituting for the values of a, b, d, one gets

p a a c q a p a cWB

WB

WBand= − −( ) = − = −( ) >γ γ 0

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

π γγ γW

B ( ) = − +⎛⎝⎜

⎞⎠⎟1

1 1x

One checks that, when g tends to 1, pWB tends to c (the loser’s marginal cost) and πW

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 offirms nC* and the corresponding profit incentives are

for PI

for PI

C C

C C

x x n x x

x x n n n x

≤ ( ) = ( ) = +( )

> ( ) = ( )

ˆ * ,

ˆ * ˆ ˆ ,

γ γ

γ γ

1 114

2

γγ γγ γ γ

( )( ) =− −( )

⎧

⎨⎪⎪

⎩⎪⎪

,x

x4 2

Under Bertrand competition, we know from Proposition 5 that the profit incentivedecreases with the number of firms for x x n< ( ), γ , and increases otherwise. So, forx x n< ( ), γ , nB* = 1 and PIB(1, g ) = (x + 2)x/4. On the other hand, for x x n> ( ), γ ,nB* = ∞ . The level of the profit incentive depends then on whether x is below or abovexB(n, g ). Computing x n, γ( ) and xB(n, g ) for n → •, we find that they both tend to thesame value, i.e. 2(1 - g )/g. Summarizing, we have

for PI

for PI

B B

B B

x n x x

x n

≤ −( )= ( ) = +( )

> −( )= ∞ ∞(

2 11 1

14

2

2 1

γγ

γ

γγ

γ

* ,

* , )) = − +⎛⎝⎜

⎞⎠⎟

⎧

⎨⎪⎪

⎩⎪⎪

11 1γ γ

x



We compute that x̂ γ γ γ γ γ( ) − −( ) = −( ) >2 1 2 8 02 , which implies that there are threecases to consider:

• for x < 2(1 - g )/g, n nC B* *= = 1 and monopoly should prevail;

• for 2 1−( ) < < ( )γ γ γx x̂ , Bertrand competition should prevail because

PI PIB C∞( ) − ( ) = − −( )⎡⎣⎢

⎤⎦⎥

− >, ,γ γ γγ

γγ

114

2 1 20

3x

x

• for x x> ( )ˆ γ , Bertrand competition should prevail because

PI PIB C∞( ) − ( )( ) = − −( )⎡⎣⎢

⎤⎦⎥

− −− −( )

>, , ,γ γ γ γγ

γ γγ γ γ

n x xxx

2 1 24 2

03

ˆ

References

Aghion, P., Bloom, N., Blundel, R., Griffith, R. and Howitt, P. (2005). ‘Competitionand Innovation: an Inverted-U Relationship’, Quarterly Journal of Economics,Vol. 120, pp. 701–728.

Amir, R., Evstigneev, I. and Wooders, J. (2003). ‘Noncooperative versus CooperativeR&D with Endogenous Spillover Rate’, Games and Economic Behavior, Vol. 42,pp. 183–207.

Arrow, K. (1962). ‘Economic Welfare and the Allocation of Resources for Inven-tions’, in R. Nelson (ed.), The Rate and Direction of Inventive Activity, Princeton,NJ, Princeton University Press.

d’Aspremont, C. and Jacquemin, A. (1988). ‘Cooperative and Noncooperative R&Din Duopoly with Spillovers’, American Economic Review, Vol. 78, pp. 1133–1137.Erratum: American Economic Review, Vol. 80 (1990), pp. 641–642.

Bester, H. and Petrakis, E. (1993). ‘The Incentives for Cost Reduction in a Differen-tiated Industry’, International Journal of Industrial Organization, Vol. 11, pp.519–534.

Blundell, R., Griffith, R. and Van Reenen, J. (1999). ‘Market Share, Market Valueand Innovation in a Panel of British Manufacturing Firms’, Review of EconomicStudies, Vol. 66, pp. 529–554.

Bonanno, G. and Haworth, B. (1998). ‘Intensity of Competition and the Choicebetween Product and Process Innovation’, International Journal of IndustrialOrganization, Vol. 16, pp. 495–510.

Boone, J. (2001). ‘Intensity of Competition and the Incentive to Innovate’, Interna-tional Journal of Industrial Organization, Vol. 19, pp. 705–726.

Brander, J. and Spencer, B. (1983). ‘Strategic Commitment with R&D: the SymmetricCase’, Bell Journal of Economics, Vol. 14, pp. 225–235.

Delbono, F. and Denicolò, V. (1990). ‘R&D Investment in a Symmetric and Homo-geneous Oligopoly’, International Journal of Industrial Organization, Vol. 8, pp.297–313.

Kamien, M. I. (1992). ‘Patent licensing’, in R. J. Aumann and S. Hart (eds), Handbookof Game Theory, Vol. 1, Amsterdam, North-Holland, pp. 332–354.

Kamien, M. I. and Tauman, Y. (2002). ‘Patent Licensing: the Inside Story’, TheManchester School, Vol. 70, pp. 7–15.

Kamien, M. I., Muller, E. and Zang, I. (1992). ‘Research Joint Ventures and R&DCartels’, American Economic Review, Vol. 82, pp. 1293–1306.



Okuno-Fujiwara, M. and Suzumura, K. (1990). ‘Strategic Cost-reduction Investmentand Economic Welfare’, Mimeo. IER, Hitotsuhashi University.

Qiu, L. D. (1997). ‘On the Dynamic Efficiency of Bertrand and Cournot Equilibria’,Journal of Economic Theory, Vol. 75, pp. 213–229.

Schumpeter, J. (1943). Capitalism, Socialism, and Democracy, London, UnwinUniversity Books.

Scotchmer, S. (2004). Innovation and Incentives, Cambridge, MA, MIT Press.Shubik, M. and Levitan, R. (1980). Market Structure and Behavior, Cambridge, MA:

Harvard University Press.Singh, N. and Vives, X. (1984). ‘Price and Quantity Competition in a Differentiated

Duopoly’, RAND Journal of Economics, Vol. 15, pp. 546–554.Spence, M. (1984). ‘Cost Reduction, Competition and Industry Performance’,

Econometrica, Vol. 52, pp. 101–112.Tang, J. (2006). ‘Competition and Innovation Behaviour’, Research Policy, Vol. 35,

pp. 68–82.Vives, X. (2006). ‘Innovation and Competitive Pressure’, CEPR Discussion Paper

4369.Yi, S.-S. (1999). ‘Market Structure and the Incentives to Innovate: the Case of

Cournot Oligopoly’, Economics Letters, Vol. 65, pp. 379–388.Zanchettin, P. (2006). ‘Differentiated Duopoly with Asymmetric Costs’, Journal of

Economics and Management Strategy, Vol. 15, pp. 999–1015.



incentives to innovate in oligopolies*manc 2131...

Documents