Competition, Darwinian Selection and Growth - CiteSeerX

Competition, Darwinian Selection and Growth∗ Vincenzo Denicolò Department of Economics, University of Bologna Piercarlo Zanchettin Department of Economics, University of Leicester May 26, 2006

Abstract We study the effect of product market competition on the incentive to innovate and the economy’s rate of growth in an endogenous growth model. We extend earlier endogenous growth models by accounting for the possibility that in each period many asymmetric firms (i.e., an endogenously determined number of successive innovators) are simultaneously active. We identify the competition effect, the selection effect, and the front loading of profits associated with a change in competitive pressure. When the intensity of competititon increases, the competition effect reduces the incentive to innovate, but both the front loading of profits and the selection effect raise the incentive to innovate. We demonstrate circumstances in which the selection effect dominates the competition effect. In these circumstances, the front loading of profits and the fact that the selection effect dominates the competition effect make the equilibrium rate of growth increase with the intensity of competition.

∗

We thank, without implicating, Giuseppe Bertola, Guido Cozzi, Steve Martin, Peter Neary and Marco Pagnozzi for useful comments. We are also grateful to seminar audiences at the EARIE conference in Madrid, University College Dublin, Rome, European University Institute, Milan, Salerno, Paris I, Leicester and the RES conference in Nottingham.

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1

Introduction

To reconcile the Schumpeterian view that the search for temporary monopoly rents is the primary engine of growth and empirical evidence that competition is good for growth,1 a recent literature has modeled step-by-step innovations, non-tournament technical progress, or nonprofit-maximizing innovative firms (see Aghion and Griffith (2005) for an excellent survey). This paper shows that a positive relationship between competition and growth can also emerge in the standard leapfrogging model with profit-maximizing firms and tournament technical progress popularized by Aghion and Howitt (1992), Segerstrom, Anant and Dinoupolos (1990) and Grossman and Helpmann (1991). We depart from the standard model only to allow for several asymmetric firms (i.e., an endogenously determined number of successive innovators) to be simultaneously active in each industry. The presence of at least two active firms is, arguably, a minimal ingredient needed to analyze the effects of varying the intensity of product market competition,2 and naturally emerges as an equilibrium property when innovations are non-drastic and firms engage in Cournot competition. In early quality ladder models, in contrast, only the technological leader is active in the product market at each point in time because innovations are drastic (Aghion and Howitt (1992)),3 or else firms compete a là Bertrand (Grossman and Helpman (1991); Segerstrom, Anant and Dinoupolos (1990)). To analyze the effect of product market competition on innovation and growth, we contrast Bertrand with Cournot competition. The analysis of a quality ladder 1

Nickell (1996), Blundell, Griffith and van Reenen (1995) and Aghion et al. (2005) find that competiton tends to be positively associated with R&D investment, or that that the relationship between the two variables is inverted U-shaped. Aghion and Griffith (2005) discuss the empirical literature more extensively. 2 Early work on competition and growth has often measured the intensity of competition by the elasticity of demand, which determines the size of the innovator’s mark-up. With several active firms, we can disentangle the effects of a change in the degree of competition from those associated with changes in structural (taste and/or technology) parameters that ultimately determine the elasticity of demand. 3 An innovation is drastic if the innovator is unconstrained by outside competition.

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endogenous growth model with Cournot competition between asymmetric firms is a contribution that may be interesting in its own right, since there is ample empirical evidence of micro-level heterogeneity in active firms’ efficiency.4 We also use a conjectural variations reduced-form model that encompasses the Bertrand and Cournot equilibria as special cases and yields a continuous index of the intensity of competition. Our model possesses a steady state in which m + 1 firms are simultaneously active, i.e., the latest innovator and m past innovators. An innovator that does not conduct any further research will be displaced gradually, remaining active, and reaping positive profits, for m + 1 periods (a period is the random time interval between two innovations). As new innovations arrive, the old innovator’s market share shrinks but he will not exit the market until after m + 1 successive innovations have occurred. The number of active firms and their respective market shares are endogenously determined as a function of the intensity of competition, the elasticity of demand, and the size of innovations. Within this framework, more intense competition has a Darwinian selection effect in addition to the standard allocative efficiency effect. That is, with tougher competition less efficient firms have lower market shares and are driven out of the market more quickly. Whereas the allocative efficiency effect of more intense competition is bad for growth — creating the well known Schumpeterian trade-off between static and dynamic efficiency — the Darwinian selection effect unambiguously enhances the economy’s growth rate. More precisely, our model encompasses three effects of more intense competition on the value of innovations and hence on the incentive to innovate — a competition 4

Dinopoulos and Segerstrom (1999) develop a Schumpeterian quality ladder model with Cournot competition, but they assume that patents expire when further innovation occurs. That is, innovation k falls into the public domain as soon as innovation k +1 arrives. The equilibrium price with Cournot competition is then the same as with Bertrand competition, even though the leader does not produce all of the output. As with Bertrand competition, each innovator reaps positive profit for one period only.

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effect, a selection effect, and a front-loading effect.5 To understand these effects, note first of all that when innovators are displaced gradually, the value of an innovation is a weighted average of all active firms’ profits, where the weights reflect the expected length of time periods, discounting, and growth.6 Then, the competition effect is the change in industry profits that would obtain if all active firms shared the same technology. This is bad for growth: if firms were symmetric, more intense competition would lead to lower industry profits and hence a lower incentive to innovate. However, when innovative technological knowledge is proprietary, firms have access to different technologies. With asymmetric firms, a rise in the intensity of competition raises the market shares of low-cost firms and lowers those of high-cost firms. This selection effect improves the productive efficiency of the industry. For example, under Bertrand competition all of the output is produced by the most efficient firm — the latest innovator; whereas under Cournot competition high-cost firms produce a positive share of total output. For any given price, greater productive efficiency translates into higher industry profits. Thus, when the selection effect outweighs the competition effect, competition is good for industry profits, and hence for growth. Finally, more intense competition makes profits accrue sooner to the innovator. For example, with Cournot competition each innovator collects its rents over various time periods, whereas with Bertrand competition an innovator obtains all of his rents in the first period after the innovation is achieved. In a stationary environment with no discounting, such a front loading of profits would have no effect on the incentive to innovate. In a growth model, however, delayed profits increase over time periods as the economy grows, but firms discount future rents. The transversality condition implies that discounting prevails over growth, and so the front loading of profits raises the incentive to innovate. 5

We borrow this terminology from Aghion and Schankerman (2004) — who however focus on the profits of the most efficient firms only — and Segal and Whinston (2005). 6 When incumbents invest in R&D (Section 6) a similar conclusion holds, but the weights are different and there is also a positive weight on monopoly profits, that of course are independent of the intensity of competition.

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The competition effect, the selection effect, and the front-loading effect together determine whether more intense competition increases or decreases the equilibrium rate of R&D in our simple model (and also operate in more highly structured models). In particular, we identify two circumstances in which the selection effect outweighs the competition effect: large innovations and tough competition. First, when innovations are almost drastic, the equilibrium price is close to the monopoly price irrespective of the mode of competition. In this case, the competition effect is second order, but the selection effect is first order. Thus, with large innovations industry profits are monotonically increasing in the intensity of competition. Second, when the price is close to the Bertrand equilibrium price, it turns out that the selection effect can be remarkably large. For example, in a conjectural variations equilibrium, starting from the Bertrand equilibrium price a unit increase in price raises the industry average cost by as much as one. Therefore, when competition is strong a fall in the intensity of competition can lower industry profits. Although these qualitative results are local, numerical calculations show that the selection effect can prevail over the competition effect in a sizeable region of parameter values. In this region, the front loading of profits and the fact that the selection effect dominates the competition effect make the equilibrium rate of growth increase with the intensity of competition.

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Literature review

The debate on the effect of competition on the incentive to innovate goes back to Schumpeter (1943) and Arrow (1962). Schumpeter (1943) claims that there exists a positive correlation between innovation and market power. He argues that for a variety of reasons a monopoly may likely develop and employ a more advanced technology than a competitive industry. This claim has been countered by Arrow (1962), who argues that the incentive to innovate is higher in competitive industries, because a monopolist’s post-innovation profits replace his pre-innovation profits, whereas this replacement effect vanishes under competition. Moving to the case of oligopoly, a 5

more recent industrial organization literature has shown that the effect of product market competition on the value of an innovation depends on the nature of technical progress (tournament or non-tournament) and on who conducts the research (incumbents or outside firms): see Boone (2000) for an excellent survey. This literature has focused on the case of a single innovation, and has identified a market share effect of competition that favours the innovator, i.e. more intense competition reallocates output from less efficient to more efficient firms. This market share effect may or may not outweigh the negative effect of more intense competition on the equilibrium price. Aghion and Schankerman (2004) call these effects the selection effect and the competition effect, respectively. In a model with low- and high-cost firms, they identify circumstances where the low-cost firms’ profits increase with the intensity of competition. This literature focuses on a single innovation framework and therefore equates the incentive to innovate to the (increase in the) profits of the technological leader. We differ from this literature in that we model an infinite sequence of innovations where innovators are displaced gradually. As a consequence, in our framework the incentive to innovate is a weighted average of active firms’ profits rather than the leader’s profits, and the positive effect of more intense competition on the leader’s market share does not translate directly into higher incentives to innovate, but operates indirectly via the effect of more intense competition on the productive efficiency of the industry and the front loading of profits. In this respect, our analysis is close to Segal and Whinston (2005), who study a partial equilibrium model of successive innovations in which each innovator stays active for two periods. However, their model differs from ours in other respects; for example, it posits an inelastic demand, a fixed timing of innovations, and an exogenously given number of firms (m = 1). Moreover, they do not compare Bertrand and Cournot competition, but focus on various practices that may or may not be anti-competitive. In the endogenous growth literature, Aghion et al. (2001) and Aghion et al. 6

(2005) develop models of step-by-step innovations with two active firms, and show that more competition (as measured by an increase in the degree of product substitutability or the extent to which firms can collude in neck-and-neck industries)7 may be beneficial to growth. Whether it is more appropriate to model innovations as proceeding step-by-step or to allow for leapfrogging perhaps depends on how innovations are protected. If innovators rely on patent protection, it seems reasonable to assume that laggards can leapfrog the leader because all innovative knowledge must in principle be disclosed in the patent specification. If instead innovators rely on secrecy, then laggards must duplicate the leader’s innovative knowledge before moving forward on the quality ladder and we have step-by-step innovations. A major difference between step-by-step and leapfrogging models is that in stepby-step innovations models the incentive to innovate is given by the difference between the innovator’s prospective profits and his current profits. A rise in product market competition lowers both, but can raise the difference. In particular, in step-by-step innovations models more intense competition strengthens firms’ incentive to innovate when they are neck-and-neck (the escape competition effect); and the fraction of industries in which firms are neck-and-neck decreases with the intensity of competition (the composition effect). The escape competition effect and the composition effect combine to generate a inverted U-shaped relationship between product market competition and innovation. Another strand of the literature introduces agency issues into the picture (Aghion, Dewatripont and Rey (1999)). In these models, non-profit maximizing managers delay the adoption of new technologies until profits fall below a threshold level. The effect of tougher competition is to reduce profits thereby speeding up the adoption process. In the non-tournament models of van de Klundert and Smulders (1997) and 7

In a simplified version of the model, Aghion, Harris and Vickers (1997) parametrize the intensity of competition also as a switch from Cournot to Bertrand competition. See also Encaoua and Ulph (2004).

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Peretto (1999), tougher competition reduces the equilibrium number of varieties and increases the size of active firms, which raises their incentive to innovate. In a related contribution, d’Aspremont, Dos Santos Ferreira and Gerard-Varet (2002) consider the case in which in each period there are several firms which have successfully innovated, and others that have access only to the prior art (which is in the public domain). They compare the Cournot and the Bertrand equilibrium, and also analyze an intermediate case in which all successful innovators co-operatively engage in limit pricing. They show that growth is fastest in this intermediate case, and conclude that the relationship between competition and growth is inverted-U shaped. The main difference between these endogenous growth literature and our paper is that we do not make any special assumption: we use the standard leapfrogging model with profit-maximizing firms and tournament technical progress. The novelty of our analysis lies in that we allow for several firms to be simultaneously active — which requires that innovations are non-drastic and competition is weaker than Bertrand competition. The rest of the paper is organized as follows. In Section 3, we develop a textbook quality ladder model of endogenous growth. In Section 4 we cover new ground by allowing for several firms to be simultaneously active in the same industry. We solve the model under Cournot competition, determining the value of an innovation when innovation is sequential and innovators are displaced by the occurrence of subsequent innovations gradually. We show that the incentive to innovate depends both on industry profits and the distribution of profits across active firms. We also derive the model’s solution using a conjectural variations equilibrium. Section 5 studies how the intensity of product market competition impacts on the incentive to innovate. Section 6 develops the analysis of the case in which incumbents innovate repeatedly, and Section 7 offers some concluding remarks.

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3

Model assumptions

In this section we present a textbook quality ladder model of endogenous growth. In particular, we use a one-sector version of the model of Barro and Sala-i-Martin (2003, ch. 7), but our results are more general and can be reproduced in many other models, with or without scale effects. Preferences and technology The economy is populated by identical individuals whose mass is normalized to 1. Each individual has linear intertemporal preferences: u(c) =




∞

c(t)e−rt dt

0

so that the rate of time preference r coincides with the equilibrium rate of interest. Each individual inelastically offers one unit of labor. The final good y is produced in a perfectly competitive market using labor (which is in fixed supply) and an intermediate good the quality of which increases over time because of technical progress. We normalize at 1 the quality of the intermediate good at time 0, and we denote by q > 1 the size of each innovation. We assume that innovations are non-drastic,8 which requires that q ≤

1 α.

Time is continuous

but can be divided into periods, where a period is the random time interval between two innovations, as in Aghion and Howitt (1992). In period k, where k − 1 is the number of past innovations, the final good can be produced according to the following constant-returns-to-scale production function: kα , 0 < α < 1, yk = X

(1)

where labor input is set equal to one, (1 − α) is the share of labor’s income, and k = k−1 q k−1−i xi,k is the quality-adjusted index of a composite good which comX i=0 bines all past generations of intermediate goods. Here xi,k denotes the input of the 8

If innovations were drastic, the model’s equilibrium would be independent of the intensity of competition in the product market.

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k intermediate good of vintage k − 1 − i in period k. It is convenient to rewrite X k = q k−1 Xk , where Xk = k−1 q−i xi,k measures the input of the composite as X i=0

intermediate good in efficiency units relative to the last vintage (i.e., vintage k − 1).

From the production function (1) one obtains the demand for the intermediate good (measured in efficiency units) 1

1 − 1−α

Xk (pk ) = α 1−α pk

α

q 1−α (k−1)

where pk is its price. The demand function (2) has a constant elasticity

(2) 1 1−α :

the

greater α, the more elastic is demand. The final good may be consumed, used to produce intermediate goods, or used in research. Independently of its quality, the intermediate good is produced using the final good with a constant marginal rate of transformation that is normalized to 1. Technical progress Innovative activity happens at a rate determined by R&D efforts. In period k there is a patent race for innovation k. Patent races come in a sequence in the sense that only after one innovation is achieved can the race for the next one begin. The size of innovations is exogenous, q, but the timing of innovations is a probabilistic function of the amount invested in R&D by research firms. Initially we assume that incumbents do no research; research is conducted by outsiders, and so in each period the current leader is systematically replaced.9 In period k, each firm ℓ participating in the race decides its R&D effort nℓk (in terms of the final good) to obtain the kth innovation. The R&D effort determines the expected time of successful completion of the R&D project according to a Poisson discovery process with a hazard rate equal to λk nℓk , with λk > 0. The projects of different firms are independent, so that the aggregate instantaneous probability of  success is simply the sum of the individual probabilities. Let nk = ℓ nℓk denote 9

In Section 6 below we extend the analysis to the case in which incumbents invest in R&D so that technological leadership can persist over time periods.

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aggregate R&D investment in period k. Then, the innovation occurs according to a Poisson process with a hazard rate zk = λk nk .10 There is perfect, infinitely-lived patent protection, so that nobody can imitate the innovation without infringing the patent. Because innovative technological knowledge is proprietary, in period k only the (k − i)th innovator, who holds a patent on the (k − i)th innovation, can practice it.11 Under the assumption that all innovations are obtained by outsiders, nobody holds multiple patents. Steady state In a steady-growth equilibrium the price of the (latest vintage of the) intermediate good, in terms of the consumption good, will be constant. This implies that Xk will α

grow at rate q 1−α . It then follows immediately from (1) that yk will also grow at α

α

rate q 1−α . This is the growth factor between periods, and we denote it by g ≡ q 1−α . In a steady state, output, consumption, the input of intermediate goods, profits, and R&D investment will all grow at rate g between periods. In order to guarantee the existence of a steady state with positive growth we assume that λk = λg −k .12 Because in a steady state nk grows at rate g across periods, under this assumption the hazard rate zk = λk nk can be constant across periods. Finally, note that the following transversality condition must hold: r > z(g − 1). Our results immediately extend to the case where zk = λk nβk , with 0 < β ≤ 1 . The case β < 1 may reflect the presence of external diseconomies in research. 11 We follow the vast majority of endogenous growth models in ruling out patent licensing. The standard justification for this assumption is that licensing agreements between successive innovators would have anti-competitive effects and thus would be prohibited by antitrust authorities. In our model, however, patent licensing agreements could be arranged so as to improve productive efficiency with no anticompetitive effects, and so ruling them out is not as innocent an assumption as in the early models. If patent licensing was ubiquitous, all of the output would be produced with the most efficient technology and so the selection effect would vanish. As a matter of fact, however, innovative technological knowledge may be difficult to codify and transmit to others and a variety of transaction costs impede licensing agreements. These transaction costs may likely result in an equilibrium outcome in which some active firms do not use the latest generation technology. To the extent that the product market equilibrium exhibits some productive inefficiency, our qualitative results may continue to hold even if we allow for some licensing agreements. 12 This assumption is standard in quality ladders models: compare Barro and Sala-i-Martin (2003). 10

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If this condition is violated, the utility u is unbounded.

4

Equilibrium growth

In this section we derive the model solution. We cover new ground by accounting for the possibility that in each period many firms are simultaneously active. Recall that only the kth innovator, who holds a patent on his vintage of the good, can produce the good of vintage k. Independently of its quality, the intermediate good is produced using the final good on a one-to-one basis. However, in period k it takes qi−1 units of the intermediate good of vintage k − i to make one unit of the intermediate good k − 1 in efficiency units. Innovator k − i’s unit cost of producing the intermediate good, measured in efficiency units, is therefore q i−1 . Thus we can proceed as if the intermediate good was homogeneous but firms had different production costs, i.e. 1 for the latest innovator, q for the penultimate innovator, q 2 for the third latest innovator and so on. Let m denote the number of past innovators that are active in addition to the technological leader (with Bertrand competition m = 0). Each innovator will then remain active, and reap positive profits, for m + 1 periods: in the first period after his innovation is achieved, he is the technological leader, in the second period he is the second most efficient firm, in the third period he is the third most efficient amongst active firms, and so on. The number of active firms is endogenously determined as a function of the intensity of competition, the elasticity of demand, and the size of innovations. Equilibrium in the product market To begin with, consider the standard case of Bertrand competition. Because innovations are non-drastic and in a leapfrogging equilibrium the next best quality is available to a firm other than the current leader, with Bertrand competition the leader is constrained by outside competition. The outcome is a limit-pricing equilib-

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rium where the leader prices at pB = q (the unit cost of the leader’s most efficient competitor) and drives all its competitors out of the market. At this limit-pricing equilibrium, there is no productive inefficiency. Let πi,k be the flow of profit earned by innovator k − 1 − i in period k. Thus, π0,k is the technological leader’s profit; π1,k is the profit of the second most efficient firm (i.e., innovator k − 2); and so on. With Bertrand competition, 1

1

− 1−α 1−α k−1 πB α g 0,k = (q − 1) q

(3)

and πB s,k = 0 for s ≥ 1. Next, suppose that firms compete à la Cournot. Given the demand function (2), the Cournot equilibrium price is constant across periods and can be easily calculated as C

pC =

1 + q + q2 + ... + q m mC + α

(4)

where mC is the largest integer such that pC ≥ q m+1 .13 It can be checked that pC > q: with Cournot competition, the equilibrium price is greater than under Bertrand competition.14 Aggregate output Xk is then given by Xk (pC ) and individual outputs xi,k are: xi,k

  Xk (pC ) pC − q i = . (1 − α) pC

(It is understood that xi+1,k = 0 when pC ≤ q i ). 13

mk +1 To show that mC is constant across periods, note that inequality pC rewrites as k ≥ q C

C 1 + q + q 2 + ... + q mk ≥ q mk +1 . C mk + α

Because all the parameters in this inequality are constant, mC k must be constant across periods. 14 Indeed, we have C

pC

=

1 + q + q 2 + ... + q m mC + α

C

> > =

αq + q + q 2 + ... + q m because innovations are non-drastic mC + α C αq + m q because q > 1 mC + α B q(= p ).

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(5)

Inspection of (5) confirms that low-cost firms hold larger market shares than high-cost firms. More precisely, in the Cournot equilibrium, the ratio of any two active firms’ market shares equals the ratio of their respective price-cost margins. This relationship trivially holds also in the Bertrand equilibrium. However, in the Cournot equilibrium different vintages of the intermediate good will be simultaneously produced, even if older vintages are less productive: the product market equilibrium exhibits productive inefficiency. To facilitate the comparison between the Bertrand and the Cournot equilibrium, and to obtain a continuous index of the intensity of competition, we now present a more general solution that encompasses these equilibria as special cases. Intermediate cases between Bertrand and Cournot emerge in various industrial organization models. For concreteness, we use a conjectural variations equilibrium under the assumption that the conjectural variations parameter is the same for all firms. Such conjectural variations equilibria can be seen as reduced forms of more highly structured models with flexible and endogenously determined capacity constraints or partial collusion.15

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Let ρ denote the common conjectural variations parameter. Standard calculations show that the equilibrium price becomes17 p(ρ) =

1 + q + q 2 + ... + q m(ρ) [m(ρ) + 1] − (1 − α)(1 − ρ)

(6)

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While most of the industrial organization literature has focused on the the symmetric case, Dockner (1992) shows that a conjectural variations equilibrium can be seen as the long-run (steady state) equilibrium of a dynamic game where asymmetric firms face adjustment costs when scaling up or down output. More recently, D’Aspremont, Dos Santos Ferreira and Gerard Varet (2003) rationalize asymmetric conjectural variations equilibria using the more general notion of oligopolistic equilibria. They also show that there is a one-to-one correspondence between conjectural variations equilibria and supply function equilibria. 16 When firms are symmetric (i.e, neck-and-neck), Aghion et al. (2005) also use a reduced form model of partial collusion that can be interpreted as a conjectural variations equilibrium. When firms are asymmetric, however, they assume that there is Bertrand competition. 17 In a conjectural variations equilibrium, the first-order conditions for profit maximization become (1 − ρi )p′ (X)xi + p = q i

i = 0, ..., m

where ρi is the conjectural variations parameter of firm i. Provided that ρi is the same for all firms, equation (6) follows.

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for m(ρ) ≥ 1, where m(ρ) is the largest integer such that p(ρ) ≥ q m+1 . The equilibrium price decreases with ρ:18 the Cournot solution is obtained for ρ = 0 and the Bertrand solution for ρ =

1−αq 19 (1−α)q .

Hence, the parameter ρ can be taken as a

measure of the intensity of competition. Equilibrium output becomes Xk [p(ρ)], and individual outputs are:   p(ρ) − q i Xk [p(ρ)] xi,k (ρ) = . (1 − α) (1 − ρ) p(ρ)

(7)

In any conjectural variations equilibrium, the ratio of any two active firms’ market shares equals the ratio of their respective price-cost margins. All that matters for our main results, however, is that more efficient firms hold larger market shares, and a rise in competition reallocates output from less efficient to more efficient firms (see Appendix 2). These properties would hold under most specification of product market competition. Proposition 1. The number of active firms decreases with the intensity of competition, the step size of innovations, and the elasticity of demand. Proposition 1 highlights the Darwinian selection effect of more intense competition. When competitive pressure is more intense, past innovators are driven out of the market more quickly and the number of active firms decreases. This Darwinian selection effect has been emphasized in the industrial organization literature since Demsetz (1972), but is relatively neglected in the growth literature. For any given level of the intensity of competition, Proposition 1 tells us that an increase in q raises the degree of cost asymmetry between successive innovators and leads to a swifter replacement of past innovators. A similar effect is associated with an increase in the elasticity of demand, that lowers the price-cost margin and leaves 18

Because m(ρ) jumps down at certain critical points as ρ increases, the equilibrium price is only piecewise differentiable. Consequently, caution should be used in differentiating those variables with respect to ρ. Note, however, that individual outputs, total output and the equilibrium price are continuous in ρ at these critical points. Therefore, to show that p(ρ) is a decreasing function, it suffices to check that the derivative is negative whenever the function is differentiable. 19 In the symmetric case q = 1, the Bertrand solution is obtained when ρ = 1.

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α

1 drastic innovations (m = 0)

qα = 1 m=1 m>3

m=2 m=3

q=1

q

Figure 1: The equilibrium number of active firms as a function of the elasticity of demand α and the size of innovations q with Cournot competition. less room for inefficient firms to operate. To get a feeling for the size of these effects, Figure 1 illustrates how the number of active firms varies with the elasticity of demand and the step size of innovations under Cournot competition. Stokey (1995) notes that if innovations occur every few years, a reasonable range for q is 1.02 to 1.04; if innovations occur only a couple of times per century, then a reasonable range for q is 1.25 to 1.50. Barro and Sala-iMartin (2003) note that reasonable values for α, the share of capital out of national income, range from 0.30 if capital is interpreted as physical capital to 0.70 if capital includes human capital. The grey area in Figure 1 corresponds to the “reasonable” range q ∈ [1.02, 1.50] and α ∈ [0.30, 0.70]. Over this range, with Cournot competition the displacement of incumbents is typically gradual: an innovator may not be driven out of the market until after three or four new innovations have arrived.

Incentives to innovate We next focus on the research sector. The expected discounted profit of an

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outsider firm that invests nℓk units of the final good in period k to obtain innovation k, as of the start of the race and given that aggregate investment in R&D is nk , is λk nℓk E(Vk ) − nℓk r + nk λk where E(Vk ) is the expected value of the kth innovation, and will be determined presently. Because there is free entry by outsiders, in equilibrium the following zeroprofit condition must hold: λk E(Vk ) = 1

(8)

This zero-profit condition determines aggregate R&D effort nk (with constant returns to scale in research, individual R&D efforts are indeterminate). To determine the expected value of innovation k, E(Vk ), one must take into account that innovators are displaced gradually. This means that the kth innovator’s rents will not be terminated by the occurrence of the (k + 1)th innovation. Thus, E(Vk ) is determined by the following asset condition:   rE(Vk ) = π 0,k+1 − zk+1 E(Vk ) − E(Vk1 )

(9)

This equation says that securities issued by the leader pay the flow profit π0,k+1 in   period k +1, less the expected capital loss zk+1 E(Vk ) − E(Vk1 ) that will be incurred

when the next innovation is achieved. Such a capital loss is the difference between the value of being leader and that of being the second most efficient firm in the market,

i.e. E(Vk ) − E(Vk1 ), where E(Vkh ) is the value of innovation k after h periods, i.e. in period k + h, and E(Vk ) ≡ E(Vk0 ). The value of being the second most efficient firm in the market, E(Vk1 ), is in turn determined by the asset condition   rE(Vk1 ) = π1,k+2 − zk+2 E(Vk1 ) − E(Vk2 ) , and so on. Eventually, after m+1 innovations, the kth innovator will exit the market, so that E(Vkm+1 ) = 0. Consequently, we have rE(Vkm ) = πm,k+m+1 − zk+m+1 E(Vkm ). 17

These m + 1 equations can be solved recursively yielding π0,k+1 π1,k+2 zk+1 + + ... r + zk+1 (r + zk+1 ) (r + zk+2 ) m

zk+i πm,k+m+1 + (r + zk+i ) (r + zk+m+1 ) i=1 

m i  πi,k+i+1 zk+h = (r + zk+i+1 ) (r + zk+h )

E(Vk ) =

i=0

(10)

h=1

When m = 0, this expression reduces to the standard formula E(Vk ) =

π 0,k+1 (r+zk+1 ) ,

that

says that the value of the kth innovation is obtained by discounting the innovator’s flow profits, with an interest rate augmented by the factor zk+1 that captures the expected duration of the innovator’s leadership. In general, equation (10) says that the value of the kth innovation is the expected present value of all future profits that the innovator will get in the m + 1 periods for which he is active in the product market. In each period, the discount factor is augmented to keep into account the probability that the current flow of profits is terminated by the occurrence of the next i zk+h innovation. Moreover, future profits are weighted by the factors (r+zk+h ) , which h=1

can be interpreted as the “discounting-adjusted probabilities” that future innovations

are achieved: with a Poisson discovery process, each future innovation eventually occurs with probability one, but since there is discounting, a delayed success counts less than instant success. More intuition on the determinants of the incentive to innovate can be gained by focusing on the case of a stationary environment in which zk and π i,k are constant across periods. In the limiting case in which z tends to zero, the value of the innovation will then depend only on the technological leader’s profit, π0 . In the polar case in which r tends to 0, the value of the innovation would depend only on the sum m  total of firms’ profits, Π = πi . In general, both industry profits and the profit i=0

distribution across firms matter. Equilibrium

With free entry by outsiders, in equilibrium the zero-profit condition λk E(Vk ) = 1 18

must hold. In a steady state, z is constant and profits grow at rate g between periods: πi,k = πi g k , where πi ≡ πi,0 . Equation (10) then reduces to: E(Vk ) =

m  z i g k+i πi i+1 i=0 (r + z)

(11)

Equilibrium in the research industry is then determined by inserting (11) into the zero-profit condition (8) obtaining H(z) =

1 λ

(12)

where H(z) ≡

m  i=0

z i gi πi . (r + z)i+1

Equation (12) determines the equilibrium hazard rate, z ∗ . This in turns fully determines the economy’s rate of growth. To see this, note that the growth factor between periods, g, is constant. This means that the equilibrium rate of growth is entirely determined by the expected length of each period, which in turn depends on the speed of technical progress: with an exponential distribution of the timing of success, the expected waiting time for each innovation is 1z .20 Consequently, an increase in z is associated with faster growth. Lemma 1. Assume that

π0 r

>

1 λ.

Then a unique, strictly positive steady state

equilibrium hazard rate z ∗ exists. Condition

π0 r

>

1 λ

ensures that research is sufficiently profitable that some research

is conducted at equilibrium. With Bertrand competition, equation (12) can be explicitly solved to get 1

1

z ∗ = λα 1−α (q − 1) q − 1−α − r It is immediate to check that the steady-state level of research is a decreasing function of the rate of time preference r and an increasing function of both the productivity of R&D effort λ and the step size between innovation q. Tedious algebra confirms 20

The economy’s expected rate of growth is z log g, i.e. approximately z(g − 1).

19

that these properties continue to hold for any level of the intensity of competition between Betrand and Cournot. Our next task is to analyze the impact of a switch from Cournot to Bertrand competition, or, more generally, a rise in competitive pressure, on the economy’s rate of growth.

5

Competition and growth

In this section we analyze the effect of an increase in the intensity of competition on the model’s equilibrium. We focus on how competition affects the incentive to innovate and hence the rate of growth. In Section 4 we have shown that the incentive to innovate, and hence the rate of growth, depends both on industry profits and their distribution across firms. Accordingly, we analyze the effect of more intense product market competition on industry profits and the distribution of profits across firms. We decompose the effect of product market competition on industry profits into a competition effect and a selection effect. We demonstrate circumstances in which the selection effect dominates the competition effect and thus more intense competition leads to higher industry profits. Finally, we identify the front loading of profits associated with a change in competitive pressure. Although in our general equilibrium, endogenous growth model we have assumed an isoelastic demand and a constant size of innovations in order to ensure that the model has a steady state, the effects we identify in this Section are more general. In a partial equilibrium framework, Appendix 2 extends our results by assuming generic demand functions and distributions of unit costs across firms and by weakening the conjectural variations assumption. Competition and industry profits To proceed, note that there is a one-to-one relationship between ρ and p, and thus one can take p as an inverse measure of the intensity of competition. To this end, it

20

is convenient to write individual outputs and profits as a function of the equilibrium price p. Using (7), individual outputs can be rewritten as xi,k (p) =

Xk (p) (p − qi ) (m + 1)p − [1 + q + q2 + ... + q m ]

(13)

where it is understood that m depends on p via ρ, and individual profits become πi,k (p) =

Xk (p) (p − q i )2 (m + 1)p − [1 + q + q 2 + ... + qm ]

Industry profits are Π =

s 

πi , or Π = [p(X) − c¯]X, where c¯ =

i=0

(14)

m

xi i i=0 X q

is the

industry average cost, i.e. the weighted average of firms’ marginal costs. The change in industry profits associated with a change in the intensity of competition is dΠ dX = X + (p − c¯) dp dp

  competition effect

−

d¯ c X dp

 

(15)

selection effect

The competition effect is the change in industry profits that would obtain if the number of active firms and their respective markets shares stayed constant. If Π(X) is concave and the price is lower than the monopoly price — two properties that always hold in our model — the competition effect is negative. However, the number of active firms and their market shares change with the equilibrium price and thus c¯ changes with the intensity of competition. The associated change in industry profits is the selection effect. The selection effect is positive: an increase in the intensity of competition, that is to say, improves the productive efficiency of the industry.21 The intuitive reason is that a rise in competitive pressure raises the market shares of low-cost firms and lowers the market shares of high-cost firms. This reduces the total cost at which any given industry output is produced. For example, with Bertrand competition all of the output is produced by the low-cost firm, whereas under Cournot competition high-cost firms have positive market shares. Thus, a switch from Cournot to Bertrand competition improves the productive efficiency of the industry. Can the selection effect dominate the competition effect, so that more intense competition raises industry profits? We identify two cases in which the answer is in 21

See Appendix 2 for a formal proof of this claim.

21

the affirmative: large innovations and tough competition.22 When innovations are large, the price is close to the monopoly price irrespective of the intensity of competition. As a consequence, the competition effect is second order and is dominated by the selection effect. Proposition 2. If innovations are sufficiently large (i.e., if q is close to

1 α)

and there are at least two active firms, industry profits increase with the intensity of competition. An immediate corollary of Proposition 2 is that when q is close to

1 α,

industry

profits are greater with Bertrand competition than with Cournot competition. The intuition is that at q =

1 α

both Bertrand and Cournot competition yield the monopoly

solution. Starting from q = α1 , consider now the effect of decreasing q. With Bertrand competition, the presence of firm 1 now constrains the low-cost firm (i.e., firm 0) that must price at p = q, but when q is close to the monopoly price the effect of competition on the low-cost firm’s profit is second order — the profit function is flat at pM . With Cournot competition a fall in q reduces the equilibrium price less than under Bertrand competition, but now it also raises the high-cost firm’s market share. Since q > 1, with Cournot competition the negative effect on industry profits of a fall in q is first order, whence the result follows. Consider next a fall in the intensity of competition starting at Bertrand competition — the equilibrium which most quality ladder models of endogenous growth with non-drastic innovations focus on. Proposition 3. Starting at the Bertrand equilibrium, a small decrease in the intensity of competition lowers industry profits. In fact, starting at the Bertrand equilibrium, a small decrease in the intensity of competition effectively leaves the industry average price-cost margin (p − c¯) un22

Our results are reminiscent of Aghion and Schankerman’s (2004), who however look for conditions under which a rise in competition raises the profits of the low-cost firms. Such conditions are necessary but not sufficient for a rise in competition to raise industry profits.

22

industry profits

ΠB

ΠC

q=1

q=2

q

Figure 2: Industry profits under Bertrand and Cournot competition as a function of the size of innovations for α = 12 . changed. When p = q, a unit increase in price raises the market share of firm 1 (the high-cost firm) by

1 q−1 ,

and this causes a unit increase in c¯. Because the industry

average price-cost margin (p − c¯) does not increase, the increase in price must necessarily lower industry profits. This extreme case illustrates the fact the the selection effect can be quite large indeed. Figures 2 and 3 illustrate Propositions 2 and 3, respectively. With α = 12 , Figure 2 plots industry profits under Bertrand and Cournot competition as q ranges from 1 to 2 (the monopoly price). Industry profits are greater with Bertrand competition for q > 1.35. Figure 3 displays the regions in which industry profits increase or decrease with the intensity of competition as q ranges from 1 to 2 and p ranges from pB = q to pC =

2(1+q) 5 .

Although our qualitative results are local, Figures 2 and 3 show

that the selection effect can prevail over the competition effect in a sizeable region of parameter values. Competition and the distribution of profits In Section 4 we have shown that the incentive to innovate depends on both the

23

p dΠ 0 dp

p=1 q=1

2

q

Figure 3: The effect of tougher competition on industry profits as a function of the size of innovation and the intensity of competition. sum total of active firms’ profits and the distribution of profits across active firms. Accordingly, we now focus on how a rise in the intensity of competition affects the profit distribution, for any given level of industry profits. Let the profit distribution be the mC +1-dimensional23 vector Π = (π0 , π 1 , ..., πmC ). Because the effective unit cost increase with firms’ rank order, we have π0 ≥ π1 ≥ ... ≥ πmC , with strict inequalities whenever profits are strictly positive. The Lorenz m C πi curve of the profit distribution is LΠ ( mCh+1 ) = Π , where Π is industry i=mC −h+1

profits. It shows the proportion of industry profits earned by any given percentage of the population of firms, starting from the least profitable firm.

Our next result is that the profit distribution becomes more unequal according to the Lorenz dominance criterion as competition becomes more intense. That is to say, the Lorenz curve of the profit distribution shifts down as the intensity of competition increases. Lorenz dominance implies that profit inequality would increase as the intensity of competition increases according to a wide set of inequality measures. 23

Since m drecreases with the intensity of competition, there are at most mC + 1 active firms.

24

Lemma 2. If there are at least two active firms, an increase in the intensity of competition makes the profit distribution more unequal according to the Lorenz dominance criterion. Lemma 2 follows from the simple fact that low-cost firms gain, and high-cost firms lose in relative terms when the market becomes more competitive. The reason is twofold: first, the market shares of low-cost firms increase with the intensity of competition; second, when the equilibrium price falls, the percentage decrease in the price-cost margin is larger for high-cost firms. In our dynamic model of successive innovations, over time periods each innovator reaps total industry profits, irrespective of the intensity of competition. However, the Lorenz dominance result means that as the intensity of competition increases the innovator will get a larger proportion of his prospective rents in the first i periods for which he is active, for all i = 1, ..., mC (over mC + 1 periods at most, he always get 100 per cent of industry profits). This is the front loading of profits associated with more intense competition. We now show that the front loading of profits tends to increase the incentive to innovate, for any given level of industry profits. Proposition 4. If industry profits are weakly increasing in the intensity of competition, an increase in the intensity of competition raises the equilibrium rate of growth. Competition and growth Propositions 2 and 4 lead to the following corollary. Corollary 1. If innovations are sufficiently large (i.e., if q is close to

1 α ),

then

the rate of growth is monotonically increasing in the intensity of competition. In particular, the rate of growth under Bertrand competition is higher than the rate of growth under Cournot competition if the size of the innovations is large. Clearly, as q falls firms become more symmetric and eventually ΠB (q) < ΠC (q). By Proposition 4, this means that the rate of growth can (but need not) be greater with 25

α

1

Π B < ΠC

ΠB > ΠC αq = 1

1

q

Figure 4: Industry profits under Bertrand and Cournot competition as a function of the elasticity of demand α and the size of innovations q Cournot competition if the size of innovations is sufficiently small. Numerical calculations show that the interval in which aggregate profits are greater under Bertrand competition, and thus more competition is surely associated with faster growth, can be quite large. Figure 4 illustrates. The shaded area in Figure 4 again corresponds to the “reasonable” ranges q ∈ [1.02, 1.50] and α ∈ [0.30, 0.70]. Over this range, more intense competition may well be associated with faster growth. Similarly, Propositions 3 and 4 lead to the following corollary. Corollary 2. If the intensity of competition is sufficiently high (i.e., p is close to the Bertrand equilibrium price q), then the rate of growth increases with the intensity of competition. Competition and welfare Although a detailed welfare analysis is outside the scope of this paper, a few remarks are in order. An increase in the intensity of competition has two effects on social welfare, a static effect and a dynamic effect. The static effect is unambiguously positive. Indeed, for any given state of the technology, the price is lower and output 26

is greater with tougher competition. Further, if competition is Bertrand, only the most efficient firm is active and so only the highest quality good is produced in equilibrium, ensuring that productive efficiency is achieved. The dynamic effect, that operates via the incentive to innovate and the rate of growth, is more complex. In general, competition can be growth-enhancing or growth-reducing. In addition, the equilibrium rate of growth can exceed the socially optimal rate, which means that faster growth is not necessarily socially beneficial. Therefore, the overall welfare effect of tougher competition is generally ambiguous.

6

Persistent leadership

So far we have assumed that incumbents do no research and are systematically replaced by outsiders. If the size of innovations is not too small, Arrow’s replacement effect implies that in a simultaneous moves R&D game between incumbents and outsiders, incumbents have a lower incentive to innovate than outsiders. With constant returns to scale in research, the hypothesis that incumbents do no research then follows from our primitive assumptions. However, there is ample evidence that incumbents account for much of the research done and the resulting pattern of persistent leadership is well documented in many industries. To allow for repeated innovations by the incumbents, we now depart from the simultaneous moves framework by assuming that the current leader has a move advantage in the patent race starting after the latest innovation, as in Gilbert and Newbery (1982). As it turns out, this leads to a pre-emption equilibrium in which all the research is done by the leader, but the amount of research is still determined by the outsiders’ zero-profit condition (see Denicolò (2001) and Etro (2004)). To simplify matters, we focus on the case in which each innovation is non-drastic but a two-step lead gives the leader monopoly pricing power, that is αq 2 ≥ 1 > αq.

27

(16)

Under assumption (16), if the current leader obtains the next innovation, he can charge the monopoly price pM and earn monopoly profits independently of the industry’s past history. If instead the innovation is obtained by an outsider, he will have to compete with the previous leader. To proceed, we first show that on the equilibrium path, the leader will conduct all of the research so as to pre-empt outsiders.24 Proposition 5. When the latest innovator has a move advantage in the next patent race, he will conduct all of the research and the amount of research is sufficiently large as to pre-empt outsiders. The intuition is as in Gilbert and Newbery (1982). The outsiders’ zero-profit condition λk E(VkO ) = 1,

(17)

where E(VkO ) is the expected value of the kth innovation to an outsider, fully determines the aggregate investment in R&D; this means that the outsiders’ aggregate best response curve has slope -1. In a Stackelberg equilibrium, the leader anticipates that its current leadership is terminated by the occurrence of the next innovation with a constant instantaneous probability anyway, and this eliminates Arrow’s replacement effect. But a successful leader would have at least a two-step quality lead and so can extract more profits from the next innovation than a successful outsider, who would face a competitor (i.e., the current leader itself) that is more efficient than the one faced by a successful leader. This leads to a pre-emption equilibrium. Having shown that all the research is conducted by the leader, and the amount of research is determined by the outsiders’ zero profit condition (17), it remains to determine the value of an innovation to a outsider. Denote the expected value of the L kth innovation to the current leader by E(VkL ). Given that nO k = 0 and nk = nk , 24

Arrow’s replacement effect implies that the second most efficient active firm will conduct no research in any case, because the value of the next innovation to it would be the same as to any outsider, less its current (positive) profit.

28

O L where nL k+1 is the R&D investment of the leader, and nk+1 = nk+1 −nk+1 is outsiders’

aggregate R&D investment, E(VkL ) is implicitly determined by the following arbitrage condition:25   L L rE(VkL ) = πM k+1 + λk+1 nk+1 E(Vk+1 ) − E(Vk ) − nk+1 . This condition differs from (11) because a successful leader now reaps transitory   L L monopoly profits π M k+1 , and gets a capital gain E(Vk+1 ) − E(Vk ) if he wins the   next race (he would suffer a capital loss E(VkL ) − E(Vk1 ) if the race was won by an outsider, which however never happens on the equilibrium path). In contrast, the expected value of the kth innovation to an outsider, E(VkO ), is given by   L rE(VkO ) = π0,k+1 + λk+1 nk+1 E(Vk+1 ) − E(VkO ) − nk+1 O Here π0,k+1 replaces πM k+1 , and capital gains or losses are calculated using E(Vk )

rather than E(VkL ) as a reference point. To understand this formula, recall that an outsider anticipates the ensuing preemptive equilibrium and therefore recognizes that if he wins the current race, he will conduct all the research in the subsequent races. L ), Note also that the value of innovation k + 1 to the winner of the kth race, E(Vk+1

is the same for the leader and for the outsiders in race kth, i.e. is independent of the current cost advantage. This is true because a firm leading by three or more steps will get the same profit as a firm leading by only two steps. Because in a steady growth equilibrium

L ) E(Vk+1 E(VkL )

=

O ) E(Vk+1 E(VkO )

= g, from the above

equations we get: E(VkO )

πM − π0,k+1 π0,k+1 = + gz k+1 r+z (r + z)2

For the sake of comparison, recollect that in the corresponding leapfrogging equilibrium the value of an innovation to an outsider was E(VkO ) =

π0,k+1 π1,k+1 + gz r+z (r + z)2

25

See the proof of Proposition 5 in Appendix A for a more general statement of the leader’s and the outsiders’ value functions, that holds also out of the equilibrium growth path.

29

The first term is the same; the second term is greater with persistent leadership than in the leapfrogging model because πM −π0 > π1 . (The difference πM −π0 −π1 reflects the extra-value of being leader when persistent leadership rather than leapfrogging is anticipated.) The incentive to innovate is now a weighted average of the profit of the most-efficient firm and monopoly profit. The latter is independent of the mode of competition, but the former strictly increases with the intensity of competition whenever industry profits (weakly) increase (Proposition 4); therefore, Corollaries 1 and 2 continue to hold. Because our main result holds for sufficiently large innovations, it seems natural to focus on the case in which inequality (16) holds. This assumption can however be relaxed using the method exposed in Denicolò (2001) and Etro (2004). When an s-step lead is required to allow the leader to engage into monopoly pricing, the value of an innovation to an outsider becomes a weighted average of monopoly profits and the profits of a leader leading by h steps, with h = 1, ..., s − 1, with declining weights. All of our results would therefore continue to hold.

7

Concluding remarks

In this paper, we have re-considered the relationship between competition and growth in a standard neo-Schumpeterian model with improvements in the quality of products. We have shown that competition is good for growth either if the size of (non-drastic) innovations is large, or competition is strong. This follows from two effects — the front loading of profits and the selection effect — that arise when innovators are only gradually displaced by the occurrence of the next innovation so that two or more asymmetric firms are simultaneously active in the same industry. Although we have demonstrated these effects using a simple model in which the good is effectively homogeneous and there are no fixed costs, the front loading of profits and the selection effect operate under more general conditions. Using the Singh and Vives (1984) model of duopoly with differentiated products, Zanchettin 30

(2006) shows that if firms are sufficiently asymmetric, the more efficient firm’s profits and industry profits increase with the intensity of competition, as measured by the degree of substitutability of the products. Although they focus on the profits of the most efficient firms, even industry profits can raise with the intensity of competition (as captured by the transportation cost parameter) in the Aghion and Schankerman (2004) model. The presence of fixed costs generally speeds up the process whereby innovators are gradually displaced by the occurrence of subsequent innovations, but to the extent that several asymmetric firms are simultaneously active our effects continue to operate even with positive fixed costs. Our results underscore that more intense competition has a Darwinian selection effect in addition to the standard allocative efficiency effect. Both effects improve static efficiency. However, whereas the allocative efficiency effect is bad for growth, creating the well known Schumpeterian trade-off between static and dynamic efficiency, the Darwinian selection effect of more intense competition also enhances the economy’s growth rate.

31

References [1] Aghion, P., R.W. Blundell, N. Bloom and R. Griffith, 2005, Competition and innovation: An inverted-U relationship, Quarterly Journal of Economics CXX, 701-728. [2] Aghion, P., M. Dewatripont and P. Rey, 1999, Competition, financial discipline, and growth, Review of Economic Studies 66, 825-852. [3] Aghion P. and R. Griffith, 2005, Competition and growth. Reconciling theory and evidence, Cambridge, MA, MIT Press. [4] Aghion, P., C. Harris and J. Vickers, 1997, Competition and growth with stepby-step innovation: An example, European Economic Review 41, 771-782. [5] Aghion, P., C. Harris, P. Howitt and J. Vickers, 2001, Competition and growth with step-by-step innovation, Review of Economic Studies 68, 467-492. [6] Aghion, P. and P. Howitt, 1992, A model of growth through creative destruction, Econometrica 60, 323-351. [7] Aghion, P. and M. Schankerman, 2004, On the welfare effects and political economy of competition-enhancing policies, Economic Journal 114, 800-824. [8] Arrow, K., 1962, Economic welfare and the allocation of resources for invention, in R. Nelson, ed., The rate and direction of innovative activity, Princeton, Princeton University Press. [9] Barro, R. J. and X. Sala-i-Martin, 2003, Economic growth, Cambridge, MA, MIT Press. [10] R. Blundell, Griffith R. and van Reenen J., 1995, Dynamic count data models of technological innovations, Economic Journal 105,

32

[11] J. Boone, 2000, Competitive Pressure: The Effects on Investments in Product and Process Innovation, RAND Journal of Economics 31, 549-569. [12] d’Aspremont, C., Dos Santos Ferreira, R. and L. A. Gerard Varet, 2002, Strategic R&D investment, competition toughness and growth, mimeo, CORE. [13] d’Aspremont, C., Dos Santos Ferreira, R. and L. A. Gerard Varet, 2003, Competition for market share or for market size: oligopolistic equilibria with varying competitive toughness, Document de Travail 2003/10, CORE. [14] Demsetz H., 1972, Industry structure, market rivalry, and public policy, Journal of Law and Economics 16, 1-10. [15] Denicolò V., 2001, Growth with non-drastic innovations and the persistence of leadership, European Economic Review 45, 1399-1413. [16] E. Dinopoulos and P.S. Segerstrom, 1999, A Schumpeterian model of protection and relative wages, American Economic Review 89, 450-472. [17] Dockner E.J., 1992, A dynamic theory of conjectural variations, Journal of Industrial Economics 40, 377-395. [18] Encaoua, D. and D. Ulph, 2004, Catching up or leapfrogging? The effects of competition on innovation and growth, mimeo, University of Paris I. [19] Etro F., 2004, Innovation by leaders, Economic Journal 114, 281-303. [20] Gilbert, R. and Newbery, D., 1982, Preemptive patenting and the persistence of monopoly, American Economic Review 72, 514—526. [21] Grossman, G.M., and E.Helpman, 1991, Innovation and Growth in the Global Economy, Cambridge (Mass.), MIT Press. [22] S. Nickell, 1996, Competition and corporate performance, Journal of Political Economy 104, 724-46. 33

[23] Peretto P., 1999, Cost reduction, entry, and the interdependence of market structure and economic growth, Journal of Monetary Economics 43, 173-195. [24] Schumpeter, J., 1943, Capitalism, socialism and democracy, London, Allen and Unwin. [25] Y. Segal and M. Whinston, 2005, Antitrust in innovative industries, Stanford Law School WP #312. [26] Segerstrom, P., Anant, C. and Dinopoulos, E., 1990, A Schumpeterian model of the product cycle, American Economic Review 80, 1077—1091. [27] Singh, N., and Vives, X., 1984, Price and quantity competition in a differentiated duopoly, RAND Journal of Economics 15, 546-554. [28] van de Klundert, T. and S. Smulders, 1997, Growth, competition and welfare, Scandinavian Journal of Economics 99, 99-118. [29] Zanchettin, P., 2006, Differentiated duopoly with asymmetric costs, Journal of Economics and Management Strategy, forthcoming.

34

Appendix A This Appendix contains all proofs that are omitted in the text. Proof of Proposition 1. Define Ω(m) ≡

1+q+q 2 +...+qm (m+1)−(1−α)(1−ρ)

− q m+1 and m ˜ as the

(necessarily unique) solution to Ω(m) = 0. Then, the equilibrium number of firms is the maximum integer not larger than m ˜ and the comparative statics properties of m(ρ) mirror those of m. ˜ Next, note that qm ∂Ω |m=m {q − log q + q log q [(m + 1) − (1 − α)(1 − ρ)]} ˜= − ∂m (m + 1) − (1 − α)(1 − ρ) and is always negative for q > 1. It follows that for any other arbitrary parameter a that influences m, ˜ the sign of

∂m ˜ ∂a

equals the sign of

∂Ω ∂a .

Using this fact, it is

immediate to verify that m, ˜ and hence m, decreases with α and ρ. As for q, note that ∂Ω 1 + 2q + 3q 2 + ... + mq m−1 = − (m + 1)q m < 0 ∂q (m + 1) − (1 − α)(1 − ρ whence the result follows.
Proof of Lemma 1. Define H(z) ≡

m

z i gi πi i=0 (r+z)i+1 .

First of all, we show that

H(z) is monotonically decreasing in z. Differentiating H(z) we get: m m−1  (i + 1)z i g i (πi − gπi+1 ) (m + 1)z m g m πm d  z i gi π i H (z) = = − − dz (r + z)m+2 (r + z)i+1 (r + z)i+2 i=0 i=0 ′

A sufficient condition for H ′ (z) to be negative is that πi ≥ gπi+1 , or (p − qi )2 ≥ g(p − q i+1 )2 A sufficient condition for this inequality to hold is (p − q i ) ≥ g(p − q i+1 ) or p(g − 1) ≤ q i (qg − 1) 35

This inequality is satisfied for all i = 0, ..., m − 1 if it holds for i = 0, that is if p(g − 1) ≤ (qg − 1) which reduces to 1

p≤

q 1−α − 1 α

q 1−α − 1

Clearly, it suffices to show that this inequality holds true when p equals the monopoly price

1 α,

which always exceeds the Cournot equilibrium price pC . Therefore, we must

prove that 1

1 q 1−α − 1 ≤ α α q 1−α − 1 that reduces to  1  α q 1−α − 1 − α q 1−α − 1 ≤ 0 At q = 1, the weak inequality is satisfied as an equality. To conclude the proof, it suffices to show that the derivative with respect to q of the left hand side of the above inequality is negative. Differentiating we get  1   1  α α d  1−α α q − 1 − α q 1−α − 1 = − q 1−α − q 1−α ≤ 0. dq (1 − α) q

This completes the proof that H ′ (z) < 0, which implies that the equilibrium, if it exists, is unique. To show existence, note that H(0) =

1 λ

π0 r

>

1 α

is the monopoly price that

and limz→∞ H(z) = 0.

Because H(z) is continuous, an equilibrium exists.
Proof of Proposition 2. Note first of all that would be charged by an unconstrained leader, and so the equilibrium price. If q is sufficiently close to

1 α,

1 α

places an upper bound on

q 2 exceeds

1 α

and so at most two

firms are active at equilibrium. Thus, suppose there are exactly two active firms, i.e. the latest innovator and the penultimate innovator. This implies that Next, note that when q is close to

1 α,

p must also be close to

dΠ 1 dX d¯ c = X + [ − 1] − X dp α dp dp

  =0

36

1 α.

1 α

> p > q.

It follows that

where X + [ α1 − 1] dX dp = 0 because p is close to the monopoly price c¯ =

m  xi i=0

X

1 α.

Since

qi

with two active firms we have c¯ =

(p − 1) + q(p − q) 2p − (1 + q)

whence d¯ c (q − 1)2 >0 = dp [2p − (1 + q)]2 It follows that dΠ (q − 1)2 =− X < 0. dp [2p − (1 + q)]2 Thus, industry profits monotonically decrease with price; in other words, industry profits monotonically increase with the intensity of competition.
Proof of Proposition 3. When the equilibrium price is close to the Bertrand equilibrium price q, only two firms will be active, i.e. the latest innovator and the penultimate innovator. Using (9), industry profits become π0,k + π1,k =

(p − 1)2 + (p − q)2 Xk . 2p − (1 + q)

Differentiating with respect to p and evaluating the derivative at p = pB we get: 2−α 1 dπk (p) |p=pB = −α 1−α (1 − α)−1 (q − 1)q − 1−α g k−1 < 0. dp

This means that a raise in the equilibrium price (i.e., a fall in the intensity of competition) reduces industry profits at p = pB .
Proof of Lemma 2. Let p and p′ denote two price levels, with p < p′ . The move from p′ to p corresponds to an increase in the intensity of competition. We must show h h that LΠ(p) ( s+1 ) ≤ LΠ(p′ ) ( s+1 ) for all h, with at least one strict inequality. Note that

πi+1 = πi



p − q i+1 p − qi

2 37

i = 0, 1, ..., m − 1

whenever firms i and i + 1 are active at equilibrium. Differentiating we get d dp



πi+1 πi



=2

(p − q i+1 ) i+1 (q − qi ) > 0 (p − q i )3

whence it immediately follows that πi+1 (p) πi+1 (p′ ) < for all i = 0, 1, ..., m − 1 > i. πi (p) πi (p′ )

(A1)

Inequalities (A1) imply π0 (p) π0 (p′ ) > Π(p) Π(p′ ) provided that there are at least two active firms at p′ , whence it follows that πs (p) + πs−1 (p) + ... + π1 (p) πs (p′ ) + πs−1 (p′ ) + ... + π1 (p′ ) < Π(p) Π(p′ ) s s ) < LΠ(p′ ) ( s+1 ). i.e., LΠ(p) ( s+1 1 1 Clearly, (A3) also implies LΠ(p) ( s+1 ) ≤ LΠ(p′ ) ( s+1 ). Now suppose to the contrary i i that there exists i such that LΠ(p) ( s+1 ) > LΠ(p′ ) ( s+1 ). Let h be the minimum value i i of i for which inequality LΠ(p) ( s+1 ) > LΠ(p′ ) ( s+1 ) holds, so that

πs (p) + πs−1 (p) + ... + π s−h (p) π s (p′ ) + πs−1 (p′ ) + ... + πs−h (p′ ) > . Π(p) Π(p′ ) and πs (p) + πs−1 (p) + ... + πs−h−1 (p) π s (p′ ) + πs−1 (p′ ) + ... + πs−h−1 (p′ ) ≤ . Π(p) Π(p′ ) These inequalities imply πs−h−1 (p) πs−h−1 (p′ ) < Π(p) Π(p′ )

(A2)

and that there exists at least one j > s − h − 1 such that πj (p) πj (p′ ) > Π(p) Π(p′ ) Combining (A2) and (A3) we get πs−h−1 (p) πs−h−1 (p′ ) < πj (p) πj (p′ ) but this violates (A1). This contradiction establishes the result.
38

(A3)

Proof of Proposition 4. The incentive to innovate is a weighted average of m z i gi all active firms’ profits, H = i=0 ω i π i , where the weights ω i = (r+z)i+1 reflect

the expected length of time periods, discounting, and growth. The transversality condition r > z(g − 1) implies that discounting prevails over growth, i.e. r+z ωi = >1 ω i+1 zg

and so the weights are decreasing in i : ω 0 ≥ ω 1 ≥ ... ≥ ω m . The Lorenz dominance result (Lemma 2) shows that a rise in competitive pressure shifts profits from the least profitable firms to the most profitable ones. With declining weights, such a front  loading of profits implies that the incentive to innovate H = m i=0 ωi π i increases with the intensity of competition, provided that industry profits do not fall. Thus we have ∂H ∂p

< 0, whence the result follows.
Proof of Proposition 5. Denote the expected value of the kth innovation to

the current leader by E(VkL ). This is implicitly determined by the following arbitrage condition:     L O L 1 L L L rE(VkL ) = πM k+1 −λk+1 nk+1 E(Vk ) − E(Vk ) +λk+1 nk+1 E(Vk+1 ) − E(Vk ) −nk+1 . O L where nL k+1 is the R&D investment of the leader, and nk+1 = nk+1 − nk+1 is out-

siders’ aggregate R&D investment. This condition differs from (1) because a successful leader now reaps transitory monopoly profits π M k+1 , and gets a capital gain     L ) − E(V L ) if he wins the next race while suffering a capital loss E(V L ) − E(V 1 ) E(Vk+1 k k k if the race is won by an outsider. The expected value of the kth innovation to an

outsider (or to the second most efficient active firm, whose incentive to innovate is however diminished by its current profits), E(VkO ), is given by     L O 1 L L O rE(VkO ) = π0,k+1 −λk+1 nO k+1 E(Vk ) − E(Vk ) +λk+1 nk+1 E(Vk+1 ) − E(Vk ) −nk+1 O Here π0,k+1 replaces πM k+1 , and capital gains or losses are calculated using E(Vk )

rather than E(VkL ) as a reference point. Note that the value of innovation k + 1 to 39

L ), is the same for the leader and for the outsiders the winner of the kth race, E(Vk+1

in race kth, i.e. is independent of the current cost advantage; this follows because a firm leading by three or more steps will get the same profit as a firm leading by only two steps. Solving for E(VkL ) and E(VkO ) and subtracting one obtains

E(VkL ) − E(VkO ) =

πM k+1 − π 0,k+1 r + zk+1

The outsiders’ zero profit condition is λk E(VkO ) = 1

(A4)

and completely determines aggregate R&D effort nk+1 ; this means that the outsiders’ aggregate best-response curve has slope -1. In a Stackelberg equilibrium, the leader perceives that nk+1 is implicitly given by (A4), and therefore the leader’s value function reduces to     L L 1 L L 1 rE(VkL ) = πM k+1 −λk+1 nk+1 E(Vk ) − E(Vk ) +λk+1 nk+1 E(Vk+1 ) − E(Vk )) −nk+1 . If the leader does all the research, E(Vk1 ) =

π 1,k+2 r+zk+2 ;

moreover,

1 λk+1

O ) and = E(Vk+1

so the last two terms in the above expression reduce to M   L 1 O L π k+2 − π 0,k+2 − π 1,k+2 >0 λk+1 nL k+1 E(Vk+1 ) − E(Vk ) − E(Vk+1 ) = λk+1 nk+1 r + zk+2

This means that E(VkL ) is strictly increasing in nL k+1 : therefore the leader will, indeed, do all the research.


40

Appendix B In this Appendix we generalize our analysis of the competition effect, the selection effect and the front loading of profits. We assume generic demand functions and distributions of marginal costs among firms and we develop the analysis in a partial equilibrium framework. Consider an industry comprising s + 1 asymmetric firms, indexed by i = 0, 1, ..., s, producing a homogeneous product. Let firms’ marginal cost be constant at ci per unit, and label firms so that c0 < c1 < ... < cs . Thus, firm 0 is the technological leader (e.g. the latest innovator), firm 1 is the leader’s most efficient competitor (e.g. the penultimate innovator) and so on. The number of firms that are active in equilibrium, m + 1, is determined endogenously; firm m is the least efficient amongst active firms. Let demand be given by X(p), where p is price, X is output, and X(.) is a strictly decreasing and twice differentiable function on [0, p¯] and is zero on [¯ p, ∞). It follows that inverse demand, p(X), is decreasing and twice differentiable on [0, X(0)]. For simplicity, we assume that the following regularity condition holds: 2p′ (X) + p′′ (X)X < 0 on [0, X(0)]. This assumption of decreasing marginal revenue simplifies the exposition but is not needed for most of our results. The individual firm’s profit function is πi = [p(X) − ci ]xi , where xi is the individual firm’s output. To keep the analysis interesting, assume that c1 is lower than the monopoly price associated with c0 , pM (c0 ) = arg max(p − c0 )X(p). If this assumption failed, firm 0 could engage in monopoly pricing without fear of being displaced by its competitors. Initially we parametrize the degree of competition by a switch from Cournot to Bertrand competition. With Bertrand competition, the outcome is a limit-pricing equilibrium in which price equals the marginal cost of the second most efficient firm and all of the output is produced by the low-cost firm: pB = c1 , mB = 0, and

41

B 26 xB 0 = X = X(c1 ). In a Cournot equilibrium, the first-order conditions are C p′ (X C )xC i + p = ci

where X C =

m C

i=0

i = 0, ..., mC

(B1)

C C xC i and m is the greatest integer such that p ≥ cm holds. As is

well known, with Cournot competition both the equilibrium price and the number of active firms are higher than under Bertrand competition: pC > pB and mC ≥ 1. Therefore, a switch from Cournot to Bertrand competition is associated with an increase in the intensity of competition. To facilitate the comparison between the Bertrand and the Cournot equilibrium we now present a general solution that encompasses these equilibria as special cases, and also allows us to obtain a continuous index of the intensity of competition. We use a reduced-form specification in which the intensity of competition is simply identified with the (inverse of the) equilibrium price. Thus, let p range from the Cournot equilibrium price pC to the Bertrand equilibrium price pB = c1 as product market competition increases. Intermediate cases between Bertrand and Cournot emerge, indeed, in various strands of the industrial organization literature, like models with flexible and endogenously determined capacity constraints, or in partially collusive equilibria. The number of active firms, m(p) is determined as a function of p as the largest integer such that p > cm . Industry output is X(p). To pin down the industry equilibrium, we assume that relative market shares depend on price and firms’ unit costs: xi+1 = Si (p, c0 , ..., cs ). xi

i = 0, ..., m − 1.

(B2)

s+1 For analytical convenience, we assume that each function Si : [ci+1 , pC ] × R+ → R+

is continuous and that it is piecewise differentiable in p when p > ci+1 (it is understood that xi+1 = 0 when p ≤ ci+1 ). The active firms’ equilibrium outputs and profits are 26

With constant marginal costs, the assumption 2p′ (X) + p′′ (X)X < 0 ensures that the second order conditions are satisfied and the Cournot equilibrium is unique.

42

m(p)

then uniquely determined by the adding up condition



xi = X(p).27 To reproduce

i=0

the Cournot equilibrium for p = pC , we assume that Si (pC , c0 , ..., cs ) =

pC −ci+1 ; pC −ci

the

Bertrand equilibrium is always reproduced at p = pB = c1 provided that xi = 0 when p ≤ ci . Without making any more specific assumption on the nature of product market competition, we simply assume that more efficient firms hold larger market shares, and that a rise in competition reallocates output from less efficient to more efficient firms. This means that Si < 1 and

∂Si ∂p

> 0 for all i = 0, ..., m − 1.

Some of our later results require a more specific condition, namely: CONDITION

I.

Si (p, c0 , ..., cs ) ≥

p − ci+1 p − ci

i = 0, ..., m − 1.

When Condition I holds as an equality, the ratio of any two active firms’ market shares equals the ratio of their respective price-cost margins. This corresponds to the conjectural variations equilibrium analyzed in the paper. Consider now an increase in the intensity of competition, i.e. a fall in the equilibrium price. If the number of active firms and their respective markets shares stayed constant, the fall in the equilibrium price would unambiguously reduce industry profs  πi . This is the competition effect. The reason why the competition effect its Π = i=0  xi is negative is that industry profits are Π = [p(X) − c¯]X, where c¯ = m i=0 X ci is the industry average cost, i.e. weighted average of firms’ marginal costs. With constant

market shares, c¯ is constant; since Π(X) is concave, if the price is lower than the monopoly price any fall in price must then reduce industry profits. However, the number of active firms and their market shares change with the equilibrium price. As a consequence, c¯ changes with the intensity of competition, and the associated change in industry costs and profits is the selection effect. 27

To show existence and uniqueness of the solution, note that for any given p, m(p) is uniquely determined. Equations (B2) provide m(p) independent conditions. Together with the adding up condition, they comprise a system of m(p) + 1 linearly independent equations in the m(p) + 1 unknowns x0 , ..., xm , the solution of which exists and is unique.

43

Formally, the change in industry profits associated with a change in the intensity of competition is28 dΠ dX = X + (p − c¯) dp dp

  competition effect

d¯ c − X dp

 

(B3)

selection effect

We now show that an increase in the intensity of competition improves the productive efficiency of the industry; a fall in the equilibrium price, that is to say, lowers the industry average cost. Result 1. The selection effect is positive. Proof. From (B2) we get xi+1 xi = Si X X whence it follows that d  xi  xi ∂Si d  xi+1  = Si + . dp X dp X X ∂p m d  xi  i Since ∂S i=0 dp X = 0, from the above formula it is clear that if there ∂p > 0 and   d xi are at least two active firms, there exists k < m such that dp X < 0 for i ≤ k and   d xi dp X > 0 for i > k. In words, an increase in price reallocates output from more efficient to less efficient firms. The selection effect is m  d  xi  d¯ c X = ci X dp dp X i=0

=

m  i=0

d  xi  X>0 dp X   d xi dp X = 0, and the inequality follows be-

(ci − ck )

 where the equality follows because m i=0   d xi cause each term (ci − ck ) dp X is positive, except the kth term which is zero.


Next we show that Propositions 2 and 3 in the text hold for any demand function

and any distribution of costs across firms. 28

Because m(p) jumps up at certain critical points c1 , c2 , ..., cm as p increases, variables like outputs, profits etc. are piecewise differentiable. Consequently, caution should be used in differentiating those variables with respect to p. Any such derivative calculated at ci , where ci is the critical value of p at which m jumps up from i − 1 to i, must be interpreted as the right derivative under the assumption xi (ci ) = 0.

44

Result 2. When the marginal cost of the second most efficient firm c1 is close to the monopoly price pM (c0 ), industry profits increase with the intensity of competition. Proof. Note first of all that when the marginal cost of the second most efficient firm c1 is close to the monopoly price pM (c0 ), only two firms will be active at equilibrium. As c1 approaches pM (c0 ), by continuity of S0 it follows that x1 must tend to zero (the reason is that p cannot exceed pC which in turn cannot exceed pM (c0 ), and so it must converge to c1 ). As a consequence, c¯ approaches c0 as c1 approaches pM (c0 ). Equation (B3) therefore reduces to dΠ dp

dX d¯ c = X[pM (c0 )] + [pM (c0 ) − c0 ] − X[pM (c0 )] dp dp

  =0

d¯ c = −X[pM (c0 )] 0 dp dp (p − c ˆ ) i=0

(B5)

where σ2c is the variance of active firms’ marginal costs. To prove the first equality in (B5), i.e. m

 d¯ c dxi X= (ci − c¯) dp dp i=0

45

note that m

d  xi ci dp X i=0 m m dxi dX i=0 ci X dp − i=0 ci xi dp

d¯ c = dp =

X2 m xi − dX i=0 ci X dp

m

dxi i=0 ci dp

=

m

X  − m i=0

dxi i=0 ci dp

=

dxi ¯ dp c

X

whence the result follows immediately. Next, note that m m  dxi  dxi dX = − c¯ (ci − c¯) ci dp dp dp i=0

i=0

From (B4) we get dxi xi dX ci − cˆ 1 ≥ + X(p) 2 dp X dp (p − cˆ) [m(p) + 1] Substituting into the above expression we get m  dX dxi − c¯ ci dp dp i=0

m 

m

dX xi dX  ci − cˆ 1 ≥ + − c¯ ci X(p)ci 2 X dp dp (p − cˆ) [m(p) + 1] i=0

i=0  =¯ c

=

m  X(p) ci (ci − cˆ) [m(p) + 1] (p − cˆ)2 i=0

because the first and third term on the right hand side cancel out. But

m

c) i=0 ci (ci −ˆ [m(p)+1]

is the variance of active firms’ marginal costs, σ2c , and so the inequality in (B5) is obtained. Having established that (B5) holds true, from (B3) and (B5) we get dΠ dX σ 2c ≤ X + (p − c¯) −X dp dp (p − cˆ)2 Now suppose that p is slightly increased starting at p = pB = c1 . Clearly, only two firms, 0 and 1, will then be active. Consequently, σ 2c =

(c1 −ˆ c)2 +(c0 −ˆ c)2 2

(p − cˆ)2 , whence we get dΠ dX |p=pB ≤ (p − c¯) < 0.
dp dp 46

= (c1 − cˆ)2 =

Next, we now focus on how a rise in the intensity of competition affects the profit distribution, for any given level of industry profits. Result 4. If there are at least two active firms, an increase in the intensity of competition makes the profit distribution more unequal according to the Lorenz dominance criterion. Proof. Let p and p′ denote two price levels, with p < p′ . The move from p′ to p corresponds to an increase in the intensity of competition. We must show that h h LΠ(p) ( s+1 ) ≤ LΠ(p′ ) ( s+1 ) for all h, with at least one strict inequality. Note that

πi+1 p − ci+1 = Si πi p − ci

i = 0, 1, ..., m − 1

whenever firms i and i + 1 are active at equilibrium. Differentiating we get d dp



πi+1 πi



= Si

(ci+1 − ci ) p − ci+1 ∂Si + >0 (p − ci )2 p − ci ∂p

whence it immediately follows that πi+1 (p) πi+1 (p′ ) < for all i = 0, 1, ..., m − 1 > i. πi (p) πi (p′ ) The proof then proceeds as in the proof of Lemma 2 in Appendix A.


47