INCENTIVES TO INNOVATE IN OLIGOPOLIES*manc_

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1 The Manchester School Vol 79 No January 2011 doi: /j x INCENTIVES TO INNOVATE IN OLIGOPOLIES*manc_ by PAUL ELLEFLAMME CORE and Louvain School of Management, Université catholique de Louvain, elgium and CECILIA VERGARI CORE, Université catholique de Louvain, elgium, and University of ologna 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. ertrand) affect incentives to innovate in non-monotonic, different and potentially opposite ways. 1 Introduction 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 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 ; final version received 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. Published by lackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK, and 350 Main Street, Malden, MA 02148, USA. 6

2 Incentives to Innovate in Oligopolies 7 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 following quote illustrates, 1 Arrow 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 competition. Microsoft is far less innovative in its core businesses, in which it has a monopoly (in Windows) and a near monopoly (in Office). ut 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 incentives to innovate. To this end, we consider an oligopoly model with horizontally differentiated 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 acquiring 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 independently 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. ertrand). 1 Taken from The meaning of Xbox, The Economist, 24 November 2005.

3 8 The Manchester School 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 ertrand 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 competition (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 relationship between product market competition and innovation thus providing ambiguous results. Among others, lundell et al. (1999) conduct a detailed analysis of ritish 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 activities 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 negative, depending on specific competition perception and specific innovation activity.

4 Incentives to Innovate in Oligopolies 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. 2 ester and Petrakis (1993) contrast the profit incentive in Cournot and ertrand duopolies for different degrees of horizontal product differentiation. onanno and Haworth (1998) address the same question as ester 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 homogeneous product. Delbono and Denicolò (1990) compare and contrast static and dynamic efficiency in oligopolies producing a homogeneous product for different numbers of firms. Like ester 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 innovation 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. oone (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 competition 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. Additional 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. 2 This 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.

5 10 The Manchester School 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 structure provides firms with the highest incentives to undertake R&D. 3 In 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 ertrand competition (with strategic complements), whereas in our framework, ertrand competition leads to larger R&D investments than Cournot competition when products are close substitutes (as shown by ester 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 nonstrategic). 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 different 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 ertrand 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. 2 Model There are n firms (indexed by i = 1,...,n) competing in the market. Each firm i incurs a constant marginal cost equal to c i and produces a differentiated product, q i, sold at price p i. The demand system is obtained from the optimization problem of a representative consumer. We assume a quadratic utility function which generates the linear inverse demand schedule 3 The idea that R&D generates incentives for firms to behave strategically was first examined by rander 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, ertrand competition and differentiated products.

6 Incentives to Innovate in Oligopolies 11 p i = a - q i - gs j iq j in the region of quantities where prices are positive. The parameter g is an inverse measure of the degree of product differentiation: the lower g the more products are differentiated (if g = 1, products are perfect substitutes; if g = 0, products are perfectly differentiated). The demand schedule, for g 1, is then given by q i = a - bp i + ds j ip j, with a α = 1 + γ ( n 1 ) 1+ γ ( n 2) β = ( 1 γ) [ 1+ γ ( n 1) ] γ δ = ( 1 γ) [ 1+ γ ( n 1) ] In this framework, total quantity Q depends on n as well as on g (this is because the representative consumer loves variety). In particular, in the symmetric case where p i = p "i, we have na ( p) Q( n, γ )= (1) 1+ γ ( n 1) It follows that n and g influence the individual quantities as well as the market size. Under Cournot competition, the equilibrium quantity and profit for firm i are found as ( 2 γ) a [ 2+ γ ( n 2) ] ci + γ c j i j C C C 2 qi = π i = ( qi ) (2) ( 2 γ) [ 2+ γ ( n 1) ] We can also compute the equilibrium price, quantity and profit under ertrand competition: p i = ( 2β+ δ) α + β[ 2β δ( n 2) ] ci + δβ c j i ( 2β+ δ) [ 2β δ( n 1) ] ( ) = ( ) = ( )( ) 2 2 i i i i i i i q = β p c π β p c 1 β q (4) Initially, all firms produce at cost c i = c. A new process innovation allows firms to reduce the constant marginal cost of production from c to c 1 = 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 c l is larger than the initial cost c, i.e. (a + c - x)/2 > c. Equivalently, assuming without loss of generality that the difference a - c is equal to unity, we have the following assumption. j (3)

7 12 The Manchester School 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 ertrand 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. 5 As a consequence, we concentrate on the profit incentive that can be seen as the pure incentive to innovate. We denote by p W the profit accruing to the innovator (the winner ), and by p the pre-innovation profit. For future reference, we also define p L as the profit accruing to the rivals of the innovator (the losers ). ased on these definitions, we formally define our measure of firms incentives to innovate as follows. Definition 1: The profit incentive is defined as PI = p W -p. In the next sections, we derive the exact value of the profit incentive under Cournot and ertrand competition, respectively, denoted by PI C(n, g) and PI (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 xx ( + 2) PIC( 1, γ)= PIC( n, 0)= PI( 1, γ)= PI( n, 0)= 4 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. ertrand). As for the first measure, we know that, as g decreases, 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 g affects the degree of competition in 4 As shown by Zanchettin (2006) in the duopoly case, this condition is sufficient but not necessary for values of g < 1. We also assume that c 3 1, so that Assumption 1 guarantees that the innovator still has a positive marginal cost. 5 We discuss in the conclusion the role of the identity of the innovator (either incumbent or outside research lab).

8 Incentives to Innovate in Oligopolies 13 two mutually reinforcing ways: as g increases, products become closer substitutes 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. 6 However, 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 n as another measure of the strength of competition. 7 Finally, as for the nature of competition, we know from Singh and Vives (1984) that going from a Cournot to a ertrand 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). 3 Cournot 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 (g and n) affect the incentive to innovate in a converging or diverging way. Using expression (2), we find q q ( 2 γ)+ [ 2+ γ ( n 2) ] x = ( 2 γ) [ 2+ γ ( n 1) ] W C W C W C 2 π = ( q ) (5) 1 2 = π = ( q ) (6) 2 + γ ( n 1 ) C C C ( 2 γ) γx ql C = L C ql C 2 π = ( ) (7) ( 2 γ) [ 2+ γ ( n 1) ] where Assumption 1 is used to guarantee that all equilibrium quantities (especially q LC ) are always positive. Applying Definition 1, we compute the value of the profit incentive as 6 One 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). 7 In 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.

9 14 The Manchester School [ 2+ γ ( n 2) ]{ 2( 2 γ)+ [ 2+ γ ( n 2) ] x} (, γ)= π π = x ( 2 γ) 2 [ 2+ γ ( n 1) ] 2 (8) C C PI C n W In order to analyze how PI C(n, g) depends on g and n, we can get some intuition by first studying its separate components, π WC and p C. We summarize our results in the following lemma, 8 where ˆx ( n) 1 ( n+ 2) and ˆ γ ( nx, ) 2 x+ 1 xx ( + n) [ 1 xn ( 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 x> xˆ ( n) and γ > ˆ γ ( 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 WC is first decreasing and then increasing in g. In fact, as soon as g becomes positive, the firm is no longer a monopolist and, given that the innovation is non-drastic (i.e. the cost reduction 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 ˆx ( n) and ˆ γ ( nx, ) decrease with n. Therefore, the higher the number of firms in the market, the larger the range of parameters where the winner s profit is increasing with g. We now proceed with studying the effects of changes in the toughness of competition on the profit incentive. 3.1 Infinitesimal Analysis In order to single out the forces at work, we first consider an infinitesimal reduction in cost (x 0). The profit incentive can be seen as the discrete version of the sensitivity of i s profits to a reduction in its own cost (i.e. C π i c i evaluated at c i = c). At the Cournot equilibrium, we have C C C C C C πi = ( pi ci) qi = [ a qi γ ( n 1 ) qj ci] qi, so that, using the envelope theorem, πi c C i = + ( ) C C q j qi γ n i c q C 1 (9) 8 The proof is relegated to the technical appendix. i

10 Incentives to Innovate in Oligopolies 15 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 c i decreases, two positive forces affect firm i s profit: first, firm i becomes more efficient, which allows it to expand its output; second, because quantities 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 g and in n. Table 1 decomposes the results. Consider first the direct effect. From Lemma 1, we know that both C q i γ and qi C n are 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 g and n. First, each rival C firm reduces its quantity in reaction to firm i s cost reduction ( qj ci > 0); the larger g and 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 ( q C i ), 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 g increases, 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 C for which the two effects just compensate and where π i c i reaches 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 2 C total effect decreases with n ( c n< 0). π i i Table 1 Direct effect Strategic effect Total effect C q i + g(n - 1) C q j ci C q i = Effect of g ? Effect of n ? C π i c i

11 16 The Manchester School 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 PI C first decreases with g and then increases. Whether the largest incentive is reached for g = 0org = 1 depends on the other parameters (n and x). We observe thus that the shape of PI C with respect to g is the same as in the infinitesimal case. The fact that larger values of g lead 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 willingness 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 x and g are large enough, the profit incentive to innovate first increases and then decreases with n. 10 On the other hand, from the infinitesimal case, we know that, as x 0, the profit incentive decreases with n. At the other extreme, we can show that, when g and x tend to one (i.e. towards homogeneous product and drastic innovation), the profit incentive is always increasing with the number of firms. 9 This 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 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. 10 This 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.

12 Incentives to Innovate in Oligopolies 17 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 n θ 1 + γ ( n 1 ) Then we can rewrite the total quantity (1) as Q(n, g) = q(a - p). We take q as 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 g in the above expression and making the proper substitutions, we can express the profit incentive to innovate as a function of q and n: 2 n( θ + n 2) x+ 4θn 2n 2θ PI C = θ nx( θ + n 2) 2 2 ( θ + n) ( 2θn n θ) The effect of the number of firms on the incentive to innovate can be thus decomposed as follows: C C C dpi PI PI d = + θ dn n θ dn The first term is the pure effect of the number of firms; it is computed by keeping q constant, thereby freezing the effect that n has on the size of the market. The second term is the market-size effect of the number of firms: a change in n affects 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 n can be non-monotonic if the size of the innovation is large enough. This result still holds for large enough values of g and of x. For instance, it can be shown that, for x = g = 0.95, PI C / n > 0 for n 2 10 and PI C / n < 0 for n > We can thus conclude that a change in the number of firms may affect in a nonmonotone way the profit incentive independently of its effect on the size of the market. 11 We further check that our non-monotone results on the effect of n on the profit incentive are preserved under the Shubik and Levitan (1980) linear demand model where the market size does not change with n. 12 Moreovcr, looking at the second term, we know that dq/dn > 0, but simulations show that the sign of PI C / q can also change with n. For instance, one can check that, for x =g= 1/3, PI C / q > 0forn 2 48 and PI C / q < 0forn > 48. In sum, isolating the pure effect of n from its market-size effect does not shed much light on the non-monotonic relationship between the number of firms and the incentive to innovate.

13 18 The Manchester School 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 n as a continuous variable, it is readily shown that ( )> > ( )= + ( + ) 2 n x 4 n x 4 n n x ( n 1 PI ) C, γ 0 γ γ, (10) n n 3+ nx 2x where γ ( nx, )> 0 for all admissible n and x, and γ ( nx, )< 1 if 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 g for which the profit incentive reaches its minimum (we proved that 0< γ ( nx, )< 1 for all admissible n and x). We find that γ ( nx, )> γ ( nx, ) for all admissible n and 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 γ < γ ( nx, ), the profit incentive decreases with g and n), or if the innovation is large enough and products are sufficiently substitutes (for x > (n - 1)/n and γ > γ ( nx, ), the profit incentive increases with g and 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 γ ( nx, )< γ < min { γ ( nx, ), 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 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 x and examine which industries 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 13 The proof of this proposition is relegated to the technical appendix.

14 Incentives to Innovate in Oligopolies 19 g 1 PI C increases with n and g g 0.8 PI C decreases with n and increases with g 0.4 PI C decreases with n and g g~ 0 2 Fig. 1 Cross-effects in the Cournot Case (x = 9/10) 10 n 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. 4 ertrand Competition The derivation of the profit incentive is slightly more involved under ertrand 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 constrained to price at marginal cost. The following lemma (which is proved in the Appendix) characterizes the ertrand Nash equilibrium of the postinnovation game for all values of x, n and g. Define x ( 1 γ) ( 2γn 3γ + 2) ( n, γ )= γ ( 1+ γn 2γ) Lemma 2: The Nash equilibrium of the post-innovation price game is such that (i) for x 2 x (n, g ), all firms price above their marginal cost, and (ii) for x > x (n, g), the winner is the only firm pricing above its marginal cost.

15 20 The Manchester School We have that, for x (x, 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, x (n, g), decreases with n and tends to 2(1 - g)/g for n (which is greater than unity for g < 2/3). Therefore, the corner solution can only be observed for sufficiently 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: 1+ γ ( n 2) γ π ( 1 n, γ)= (11) ( 1 γ) [ 1+ γ ( n 1) ] γ ( n 3)+ 2 Next, using the proof of Lemma 2, we compute the winner s profit at the Nash equilibrium of the post-innovation game: γ γ γ γ γ πw [ ( 5 5n+ n ) + 3( n 2) + 2] x+ ( 1 )( 2+ 2 n 3 ) ( n, γ)= β ( 2+ γn 3γ) ( 2+ 2γn 3γ) for x x ( n, γ ) (12) π ( γ)= x 1 W for x > x ( n, γ ) γ γ Finally, combining (11) and (12), we can express the profit incentive under ertrand competition as PI ( n, γ)= πw ( n, γ) π ( n, γ) for x x ( n, γ) PI ( n, γ )= (13) PI ( n, γ)= πw ( γ) π ( n, γ) for x > x ( n, γ) We can now examine how the profit incentive evolves with the two measures of competition in a ertrand 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 ertrand equilibrium, the following equation holds: πi c i ( ) = + γ n p 1 j qi + ( ) qi > 0 1 γ n 2 c direct effect ( + ) i strategic effect ( ) 2 (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 i becomes more efficient. In contrast, the strategic effect is negative: a reduction of c i, and so a reduction of p i, leads rival firms to lower their price p j (prices are strategic

16 Incentives to Innovate in Oligopolies 21 Table 2 Direct effect Strategic effect Total effect q i ( ) γ n 1 pj 1+ γ ( n 2) ci q i = Effect of g ? Effect of n π i c i 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 g and 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 g is ambiguous: the direct effect decreases with g but 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 compensates the negative effect of g on 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 infinitesimal counterpart, its effect is non-monotonic. Proposition 4: Under ertrand 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 n and x. We observe thus that the degree of product substitutability affects PI and PI C in qualitatively similar ways. ester and Petrakis (1993) compare PI and PI C in a duopoly model by restricting the range of g to values such that all firms stay on the market (i.e. x 2 x (2, g)). They show that there exists a cut-off value of g such that PI C > PI for g below the cut-off and PI > PI C for g above the cut-off. Using our derivation of PI ( 2, γ ), we can show that ester 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 14 Simulations indicate that these results still hold for n > 2.

17 22 The Manchester School ertrand competition, the profit incentive reaches its maximum for homogeneous products independently of the other parameters (n and 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 ertrand competition is monotone:pi is either decreasing or increasing with n depending on the innovation size and the degree of product differentiation. We show in the technical appendix that there is a unique positive value of x, x( n, γ ), such that PI ( n, γ ) n= 0, and that x( n, γ)< x ( n, γ) for all n and g. Therefore, we have the following proposition. Proposition 5: Under ertrand competition, the profit incentive decreases with the number of firms for x< x( n, γ ), and increases otherwise. A necessary condition for the latter possibility ( x( n, γ )< 1) isg > As far as the cross-effects of the two measures of competition are concerned, the picture is similar to the one we described for Cournot competition. Defining xn, ( γ ) as the positive value of x such that PI (n, g)/ g = 0, we can show that xn (, γ)< xn (, γ), meaning that there are up to three possible situations: (i) for x< x( n, γ ), the profit incentive decreases with both n and g; (ii) for xn (, γ)< x< xn (, γ), the profit incentive decreases with n but increases with g; and (iii) for x> x( n, γ ), the profit incentive increases with both n and g. As for Cournot competition, it is impossible to have the profit incentive increase with n while decreasing with g. 5 Market 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 ertrand competition in a homogeneous market. We check indeed that, with a homogeneous product, the profit incentive under ertrand competition is defined as PI( n,1), which increases in n and reaches thus its maximum at PI(,1)= x; we also compute that PI(, 1)= x> PI( 1, γ )=( x+ 2) x 4, 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 15 We 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.

18 Incentives to Innovate in Oligopolies 23 never leads to the highest profit incentive. Also, Proposition 5 shows that as soon as products are differentiated, the profit incentive under ertrand competition might be decreasing in n if 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 combination 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 x and g, ertrand competition with an infinite number of firms yields the highest profit incentive. Otherwise, (ertrand 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 ertrand competition (i.e. the number of firms maximizing the profit incentive, n * ) yields a higher profit incentive than the best structure under Cournot competition (i.e. n C * = 1 or n C * is (the integer closest to) ˆ nxγ (, ), 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. 6 Conclusion 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 competition and the incentives to innovate in a general framework; in particular, we extend the analysis of the profit incentive to innovate under ertrand 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 incentives to innovate. Finally, we provide a rationale for the empirical evidence that the relationship between competition on the product market and innovation 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

19 24 The Manchester School 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 different 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 innovator 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 n 3 2, there exists γ C ( 01, ) such that PI/ g < 0 for γ > C γ, PI/ g = 0 for γ = C γ and PI/ g > 0 for γ > C γ, and (ii) the highest profit incentive is reached under independent products (g = 0) if x< xc ( n) 2( n 1) ( 3n+ 1) and under homogeneous products (g = 1) if x> xc ( n) ; xc( n) increases in n, and is between 2/7 and 2/3. Proposition 2 is formally stated as follows. Define 2 2γ ( 1 γ) x + ( 3γ 2) ( 2 γ) 22 ( γ) ( γ 2γ + 4) nˆ( γ )= and xˆ( γ )= 2 γ ( 2 γ xγ) γ ( 8 γ ) (i) Taking n as a continuous variable, the profit incentive reaches a maximum for n= nˆ ( γ ), 17 and (ii) taking into account that n is an integer, the maximum is reached for n > 1 if and only if x> xˆ ( γ ), which supposes that g > The incentive to innovate of an outside inventor who decides to auction a single license would be measured by the competitive threat. 17 Computing nˆ(g) at g = 1, we find that the profit incentive decreases with n as long as n > 1/(1 - x); this is equivalent to x < (n - 1)/n, the condition given by Yi (1999).

20 Incentives to Innovate in Oligopolies 25 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 L of n - 1 losers indexed by i. A typical loser i s maximization program writes as max α βpi + δp + δ pj pi c pi c pi W ( ) subject to j L, j i From the first-order condition, we find that the interior solution to this problem is given by 1 pi = + p + pj + c 2β α δ W δ β j L, j i This solution holds as long as p i > c, which is equivalent to α + δp + δ p βc W j L, j i j (A1) We can thus write loser i s best response function as follows: ( ) ( ) 1 j i + pw + pj + c if A is met j L, j i Ri( pw, ( pj) j L) 1 = 2β α δ δ β c otherwise As for the winner, the maximization program writes as max α βpw + δ pi pw c x subject to pw c x pw ( + ) i L (A2) From the first-order condition, we find that the interior solution to this problem is given by 1 pw (( pi ) i L)= + pi + ( c x) 2β α δ β i L (A3) We show that this value is always larger than c - x. Indeed, the lowest value of p W, noted p W, is reached when p i = c "i L. After substitutions, we compute 1 ( 1 γ)+ x( 1+ γn 2γ) pw ( c x)= > γn 2γ 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 p W and p L.We have that