Competition, R&D, and the Cost of Innovation∗

Philippe Askenazy† Christophe Cahn‡ Delphine Irac‡

June 2007version 2.0

Abstract

This paper proposes a model of distance to technological frontier that encompasses themagnitude of the impact of competition on R&D according to the cost or size of the inno-vation. The effect of competition on R&D is inverted U-shaped. But, the shape is flatterand thus competition policy is less relevant for innovation, when innovations are relativelycostly. Intuitively, if innovations are costly for a firm, competitive shocks have to be largeto change its innovation decisions. Using a unique panel dataset from the Bank of France,we test this model. Empirical findings are consistent with the theoretical predictions.

Keywords: Competition, R&D, innovation.

JEL Classification: O40.

∗We thank Philippe Aghion, Gilbert Cette, and participants at workshop at GREQAM. We are also gratefulto Nicolas Berman and Laurent Eymard for valuable research assistance. We retain sole responsability for anyremaining errors. The views expressed herein are those of the authors and do not necessarily reflect those of theaffiliated institutions.

†Paris School of Economics, IZA, and Banque de France. askenazy@pse.ens.fr.‡Banque de France, DGEI-DAMEP-SEPREV. christophe.cahn@banque-france.fr,

delphine.irac@banque-france.fr.

1

1 Introduction

The debate on the impact of market structure, and thus competition policy on innovation is still

vigorous. The classic opposition is between Schumpeter (1942), Arrow (1962), Dasgupta and

Stiglitz (1980), etc. Schumpeter (1942) argues that a firm is incited to innovate if it enjoys a

monopoly position in order to avoid the entry of potential rivals; a monopoly position also ensures

a long-term view and can push to some risky investments in R&D. Some empirical evidences

seem to support the Schumpeterian view. For example, recently, Blundell et al. (1999) find a

positive relationship between individual ex ante market share and innovation.

However, following Arrow (1962), under perfect ex-post appropriation, the profit margins

could be larger in an ex-ante competitive industry than under monopoly. Aghion et al. (2002,

2005) tend to reconcile the two approaches in a model that captures both mechanisms. This

model exhibits an inverted U-shape relationship between innovation and competition. In this

model, competition could increase innovation profit margin for firms closed to the technological

frontier (escape competition) but high competition could also reduce incentive to innovate for

laggards (disincentive effect).

The goal of our paper is to extend this combined approach by studying the magnitude (and

not the sign) of the impact of competition on innovation behaviour. We propose a model that

takes into account the firm size and/or the size of the innovation in the sector. Intuitively, if

innovations are large and costly in the sector of the firm, competitive shocks have to be large to

change its innovation choices. Therefore, the inverted U-shape is flatter and thus competition

policy is less relevant for innovation, when innovations are costly (or firm size is small). Using

a unique panel dataset from the Bank of France, we test this model. Empirical findings are

consistent with the theoretical predictions. When the sectoral size of innovation relative to the

size of the firm is large, changes in the competitive position of the firm does not seem robustly

associated with changes in R&D intensity. But when it is low, individual firm R&D effort is

strongly related to its competitive situation.

These results are also related to the literature on innovation decision that stresses the role of

firm size (e.g. Cohen and Klepper, 1996). It could be easier to finance innovative investment in

large firms which are well-known and enjoy better relations with external investors or banking

lenders. In addition, because of the importance of sunk costs linked to innovation investments,

large firms are more incited to engage in innovative activities. Empirical evidences seem to

support such views. For example, Crepon et al. (1998) find a positive significant effect of firm

2

size on the likelihood to undertake R&D.

The paper is organized as follows. Section 2 is devoted to detail the model and the main

empirical result. Econometric strategies and data are presented in section 3, that also provides

the key empirical findings. Some perspectives are given in a last section.

2 Theoretical framework

2.1 Basic elements of the model

We present in this section a simple model that can encompass the roles of the cost of the

innovations in the industry of the firm and the market or firm sizes for the impact of competition

on R&D. This model is based on the standard Aghion et al. (2005)’s framework in which we

introduce additional parameters.

Households and final goods

We assume that a unit mass of homogenous households supplying labor inelastically seeks to

maximize discounted sum of logarithmic instantaneous utility flows with a constant rate r. The

argument of these utility functions is the consumption good y which is produced according to

the following production function ln(yt) =∫ 10 Φj ln(xjt)dj with Φj > 0 ∀j and

∫ 10 Φjdj = 1,

where Φj represents the weight in the utility function of intermediate output xj .1 Intermediate

goods are yield by duopolists according to the relation xj = xAj + xBj . The assumptions upon

which the model is built allow us to choose the numeraire for the prices of intermediate goods

in each sector by normalizing the households’ current expenditure in good j proportionally to

its weight in the utility function, such as pjxj = 2Φj .2

Intermediate production

Intermediate firms produce goods from labor with constant returns to scale taking the wage

rate as given, which leads to independent of quantities produced unit cost of production for each

duopolist. Contrary to Aghion et al. (2005), we can not normalize labor to one because Φj’s

are heterogenous. We assume that one unit of labour employed by each intermediate duopolist

generates outflow equal to:

Γi = γki , i = {A,B}, (1)

where ki is the technology level of duopoly firm i in a certain sector, and γ > 1 is the size of the

1Note that Aghion et al. (2005) take a particular form for which Φj = 1 ∀j.2In the remaining part of the paper, sectorial subscript j is omitted as long as it does not create confusion.

3

leading-edge innovation. The total output of a duopolist is

LiΓi = Liγki , i = {A,B}, (2)

where Li is the amount of labor devoted to production by the firm. By the same way, we define

πm and π−m to be the equilibrium profit flow by employee of a firm m steps ahead, respectively

behind, its rival. Hence, the economy is composed of two types of sector: either leveled (neck-

and-neck) type, where there is no technology gap, or unleveled where a leader and a follower

coexist.

R&D

We assume a R&D cost function ψ(n) = βn2/2, where β is a increasing function of the size of

innovation γ with β(γ = 1) = 0. ψ(n) defines the total cost that a leading firm have to spend to

gain one technological step according to a Poisson process of parameter n. This cost can also be

spent by a following firm to move a step forward with hazard rate n+h, where h represents the

opportunity gain to copy the leader, even if no R&D efforts are made. From now, n0 denotes

the R&D efforts of each firm in a neck-and-neck sector and nm (resp. n−m) those of a leader

(resp. follower) firm in a unleveled industry.3

Product market competition

To complete the model, we describe the profit flow for each type of intermediate firms. In each

unleveled sector, all profits are kept by the leader firm, say A for instance, so that it receives the

difference between its revenue pAxA and the total cost of production cAxA, where pA and cA are

respectively the price and production cost of one unit produced by the leader firm. The market

structure a la Bertrand inside the sector implies that the leader’s price equals production cost

of the follower. Based on the no profit condition for the follower, we have pA = pB = p and

consequently, xA = xB = x/2. Hence, leader’s revenues are given by pAxA = Φ. As a result,

profits of the leader firm are given by πm = πmΦ with πm = 1−γ−m.4 On the contrary, follower

firm makes no profit so that π−m = 0. As regards neck-and-neck industries, the profit flows

depend on which extent the duopolists collude, according to the assumption of a competition a

la Bertrand. As a result, the profit of a leveled firm is comprised between zero and the half of

what a monopolist could earn, which leads to π0 = π0Φ = επ1Φ, with 0 6 ε 6 1/2 and ν = 1− ε3It is worthwhile to bear in mind that in the the catch-up process prevents the leader firm to innovate, so that

its R&D efforts in the case where the maximum sustainable gap is m = 1, n1, are zero.4Based on our normalization, we have cAxA = cAΦ/pA = cAΦ/cB . Since one unit of labour can produce γki

unit of goods for i ∈ {A, B}, firms need xi/γki unit of labour to produce xi so that the unit cost of productionis ci = ω/γki , where ω is the wage rate assumed to be the same among the firms. Hence, we have cAxA =ΦγkB−kA = Φγ−m.

4

is a global measure of product market competition.

Furthermore, it is worth noting that the size of duopolists of sector j is directly related

—more precisely proportionnal— to the magnitude of its nominal demand. Indeed, according

to the model and assuming a constant wage rate ω, the size of the follower firm and firms in

leveled sector is given by:5

LB = Φ/ω, (3)

whereas the leader firm’s size can by written as

LA = γ−mΦ/ω. (4)

2.2 Equilibrium research efforts

Bellman equations

This subsection determines the equilibrium conditions in the model. Let V−m, Vm, and V0 denote

respectively the steady state value of being currently a follower, a leader, and a neck-and-neck

firm. We have standard Bellman asset equations:

rVm = πmΦ + nm(Vm+1 − Vm) + (nm−1 + h)(Vm−1 − Vm) − βn2

m

2, (5)

rV−m = π−mΦ + nm(V−m−1 − V−m) + (n−m + h)(V−m+1 − V−m) − βn2−m

2, (6)

rV0 = π0Φ + n0(V1 − V0) + n0(V−1 − V0) − βn2

0

2, (7)

where ex post n0 = n0 represents the R&D intensity by the other duopolist in a leveled sector,

which is identical in the Nash equilibrium. The first order conditions give:

Vm+1 − Vm = βnm, (8)

V−m+1 − V−m = βn−m, (9)

V1 − V0 = βn0. (10)

The one-step case

As Aghion et al. (2005), we have to restrict m = 1 in order to be able to obtain a closed-

form solution for the model. This means that the maximum sustainable gap is one: if a leader

innovates, then the follower can imitate the leader’s past technology.

5The no-profit condition for the firm B implies ωLB = pBxB = px/2 = Φ. As xA = xB, we have LA =LBΓB/ΓA = LBγ−m.

5

In the case where m = 1, we can rewrite the previous systems as follows:

rV1 = π1Φ + (n−1 + h)(V0 − V1), (11)

rV−1 = π−1Φ + (n−1 + h)(V0 − V−1) − βn2−1

2, (12)

rV0 = π0Φ + n0(V1 − V0) + n0(V−1 − V0) − βn2

0

2, (13)

Then, the first order conditions give:

V0 − V−1 = βn−1, (14)

V1 − V0 = βn0. (15)

These conditions combined with the Bellman equations imply the following reduced form R&D

equations :

βn2

0

2+ β(r + h)n0 − (π1 − π0)Φ = 0 (16)

βn2−1

2+ β(r + h+ n0)n−1 − π0Φ − β

n20

2= 0 (17)

Neck-and-Neck sector

The solution of equation (16) gives the equilibrium research intensity for the leveled firm:

n0 = −(r + h) +

√

(r + h)2 +2νπ1Φ

β(18)

Taking into account the extend of innovation, we have the following proposition:6

Proposition 1 Assuming a convex shape in the cost of innovation for β, the slope of the re-

lationship between innovation and competition in the neck-and-neck sectors is a non negative,

decreasing function of the size of innovation γ.

Proof. See Appendix A

The size of the demand for intermediate good could improve the relationship established in

Proposition 1, as described in the following corollary:

Corollary 1 For a given size of innovation, the lower the size of the demand Φ, the lower the

effect of competition on research activities in the leveled sector.

6This property includes the Aghion et al. (2005)’ result that the R&D efforts in neck-and-neck sectors, n0, isan increasing function in the competition index.

6

Proof. See Appendix A

The intuition behind proposition 1 and corollary 1 relies on the fact that for industries

where costly innovations take place, gains in terms of research efforts are mitigated by the

limited impact of competition on firms’ decision to innovate. On the contrary, R&D activities

are stimulated in sectors related to cheap innovations in order to escape from competition with

less difficulties. Furthermore, these effects are more pronounced in large firms related to sectors

for which the demand is large.

Unleveled sector

As for the neck-and-neck sectors, we find in a similar way the equilibrium research intensity for

the laggard firm from equation (17):

n−1 = −(r + h+ n0) +

√

(r + h)2 + n20 +

2π1Φ

β(19)

Once again, taking into account the cost of innovation, we have the following proposition:7

Proposition 2 The slope of the relationship between innovation and competition in the un-

leveled sectors is a non positive, increasing function of the size of innovation γ.

Proof. See Appendix A

This proposition comes together with the following corollary:

Corollary 2 For an either small or large innovation, the lower the extend of the demand Φ,

the lower the effect of competition on research activities in the unleveled sector.

Proof. Demonstration follows proof of corollary 1.

Aggregate innovation

We now derive the aggregate flow of innovations I from µ1 and µ0 which represents the steady-

state probability of being an unleveled and a neck-and-neck industry respectively, with µ1 +

µ0 = 1. The steady-state probability that a sector moves from an unleveled to leveled state

is µ1(n−1 + h). The reverse move appears with a steady-state probability of 2µ0n0. In the

steady-state we have µ1(n−1 +h) = 2µ0n0. Hence, the aggregate flow of innovations is given by:

I =

∫

{Φj}1

0

4n0(n−1 + h)

2n0 + n−1 + h. (20)

7This property includes the Aghion et al. (2005)’ result that the R&D efforts in unleveled sectors, n−1, is adecreasing function in the competition index.

7

Implicitly I is a function of ν and γ. For a given firm, the expectation of its flow of innovation

is proportional to I(ν, γ). This last equation (20) leads to the inverted-U relationship between

competition and innovation as stated by Aghion et al. (2005). Nevertheless, based on the two

previous propositions, the shape of this relationship is dependant of the overall cost of innovations

that occur in the economy. Hence, the following theorem holds:

Theorem 1 The more the cost of global innovation in a given economy, the flatter the inverted-

U relationship between competition and research activities in this economy.

Proof. The theorem is directly established according to both propositions 1 and 2.

Figure 1 depicts the shape of the relationship when the size of innovation varies, for Φ = 1,

r = 0.05, and h = 2.8 The black dots represent the maxima for each inverted-U curves.9

Theoretical counterpart of lerner index at the firm level

Based on this model, the theoretical counterpart of our empirical measure of competition, the

Lerner index, is given by

λ0 = (1 − ν)(1 − γ−1) (21)

λ−1 = 0 (22)

λ1 = 1 − γ−1. (23)

where λ0, λ−1, and λ1 are related to leveled, follower, and leader firm respectively. Hence, the

Lerner index decreases with competitive pressure for neck-and-neck firms, whereas the Lerner

index depends positively on the size of innovation. For a given firm, the expected Lerner index

is then given by

E(λ) = µ0λ0 +µ1

2λ−1 +

µ1

2λ1

= (1 − µ1)(1 − ν)(1 − γ−1) +µ1

2(1 − γ−1) (24)

The marginal effect of a change in the lerner index on innovation is decreasing with γ.

3 Empirical illustration

The goal of this section is to illustrate the theorem 1 on French firms panel data.

8Setting Φ to unity implies a uniform distribution among sectors. Each curve correponds to a particulareconomy for which γ is constant among sectors.

9One can derive the analytical expression for the maximum as it leads to solve a 3rd-order polynomial equation.As these computations are cumbersome and could alter the clarity of our main purpose, we prefer to shownumerical simulation.

8

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

4.9

5

5.1

5.2

5.3

5.4

5.5

5.6

1.04

1.045

1.05

Φ = 1, r = 0.05, h = 2

ν

Agg

rega

te fl

ow o

f Inn

ovat

ions

(I)

Figure 1: The inverted-U relationship with a varying size of innovation (bullets indicate themaximum)

9

3.1 Data

We use 2 main different datasets: first data from the observatory of firms at the Banque de

France; second, the French R&D survey from the French ministry of research.

a.) The two main variables of interest in the regressions, namely the R&D effort and the

competition index, come from the Fiben and Centrale des Bilans databases (Banque de France

Balance Sheet dataset). They are collected on a voluntary basis. Clerks in the different local

establishments of the Bank of France contact firm to complete a survey. The Fiben database is

based on firms tax forms and includes all businesses with more than 500 employees and a fraction

of smaller firms. Its number of employees coverage rate is 57% for industry but smaller for service

sectors. The Bank of France uses these data (plus information from banks including payment

incidents) for computing the firm score, which is used by commercial bank for evaluating the

financial risk for each firm (see Bardos, 1998). Consequently, firms that are interested to access

to the credit market are incited to provide these data. The advantage of this base is to include

firm that have episodic R&D activities or novel firms. We focus on firms that have conducted

observable R&D activities at least one year since their creation. Our sample includes about

16,000 firms from 1990 to 2004. But it is not balanced. The sectoral distribution of observations

is given in appendix.

These data allow us to build a lerner index for each firm. We only observe sectoral price

provided by the INSEE, but we have detailed information on costs. The lerner index is supposed

to measure the market power of the firm by the difference between price and marginal costs

(which equals the negative inverse of demand elasticity). Since neither price nor marginal costs

are available at the firm level, we compute the index using operating profits net of depreciation

and provisions minus the financial cost of capital (cost of capital*capital stock) over sales (in line

with Aghion et al., 2005). The Fiben database contains very detailed balance sheet information

that enables to compute these Lerner indicators10.

We use immobilized R&D (line KC of French tax forms) divided by the gross value added

as a proxy of the R&D effort of firms.

b) In addition, we exploit the R&D survey from the French Ministry of research. It includes

information about total R&D expenditure and the number of patents for around 3000 firms. The

question about the number of patents exists since 1999 only. The survey targets firms that are

10Lerner=(VA-depreciation-cost of capital.capital stock-provision)/salesUsing the standard mnemonics of French tax forms: Lerner=[VA-(AQ+AS+AU+AW+AY- AQ-1-AS-1-AU-1-AW-1-AY-1)-0.085.capital-(DR-DR-1)]/FL

10

likely to do research and development. One flaw of this survey is that it covers well known firms

that do research on a continuous basis pretty well whereas its quality is much smaller regarding

firms that do research on an occasional basis only. We assume that this survey provides relevant

information about the average size of a patent for a given sector (3 first digits of the NAF 700).

We compute the sectoral cost of innovation as the average of the ratio of total R&D expenditures

divided by the number of patents, over 1999-2002 (see below for details).

3.2 Econometric strategy

In order to take the model to the data, we first have to estimate the flow of innovations. Only

the amount of R&D expenditure (current prices) is directly available in our databases. To

circumvent this difficulty, we do the following approximation. From the R&D survey from the

French Ministry of Research we derive an estimate for the innovation cost:

γs =R&D exp enditure in the sec tor s

number of patents in the sec tor s

We have estimates of the innovation size for 200 sectors (average over 1999-2002), which appears

as way too much to summarize in the appendix. Innovation sizes are consistent with some

priors and anecdotal evidence. Among the sectors in which γ is inferior to 2.000 euros per

patent, we find: edition, leather, wood. Sectors with high γ (superior to 10.000 euros) are

for instance: transportation, pharmaceutical goods, software etc. It is then straightforward to

derive a potential flow of innovation, nj, for each firm j by the ratio:

nj =R&Dj expenditures

γs

with R&D expenditure being observed at the firm level from Fiben.

Our sample includes innovative firms that have implemented R&D at least once in their

life time. These firms are identified by the fact that they exhibit positive R&D assets in their

balance sheet.

To approximate the inverted-U shape, we use a quadratic form. Precisely we estimating the

model :

I = nj = λ2j (−α1 + α2.ln(innovation size)) + λj(β1 − β2.ln(innovation size)) + cj + years + ε,

where, λj is the firm lerner index, years are dummies for year in order to correct for the overall

business cycle, and is firm fixed effect that encompasses fixed firm characteritics (sector, average

employment or value-added etc.). Two measures of innovation size are alternatively considered.

11

First the logarithm of the absolute value of innovation cost, namely γ at the sector level. Second

the innovation cost relatively to the size of the firm (measured by its value added.)11

All the α’s and β’s are expected to be found positive according to the theoretical predictions.

Via the introduction of the coefficients α1 and β1, innovation and lerner entertains an inverted

U relationship. If α2 and β2 are positive and significant then this inverted U relationship gets

flatter and flatter when the absolute or relative size of innovation increases. We need to have

two caveats in mind when working with this specification. Quadratic function implies that the

inverted U relationship is symmetrical around its maximum, which is not what the model gives.

We will see that this hypothesis turns out innocuous.

3.3 Results

Restricting the sample to firms with R&D (Table 1) gives results that stick to the model. Leaving

aside the interaction terms (with innovation size), a quadratic function seems a good descrip-

tion of how innovation and competition are related to each other: α1 and β1 are positive and

significant. Especially in sectors where innovations can be implemented with small magnitude

in an incremental way, the interaction terms can be assumed as negligible and the inverted U

relationship fully applies. However, since α2 and β2 are found positive and significant too, this

result does not apply for bigger innovation sizes.

From the regression coefficients in column (1) Table 1, the U shape curve converges to a curve

that is not statiscally significant from a simple horizontal line for values of γ/VA (in logarithm)

above 4.3, which corresponds to firms for which the cost of a patent is above 7.6% of their value

added. Above this value, both the coefficients of the linear and of the quadratic terms are almost

statistically zero. Interestingly, 85% of the firms are in this situation, the others having relative

innovation size that is sufficiently low for the inverted U relationship not to be degenerated to

a line. The same findings apply when looking at the absolute size of innovation (column (2)

Table 1). It is straightforward to check that the curve is flat, for γ above 7.1, namely sectors

for which the cost of a patent is above 1200 euros (85% of the sectors are in this case, (see the

distribution of γ in appendix)).

For the few sectors with small relative cost of innovations (i.e.γ/VA < 4.3), the maximum

of the curve is obtained for a lerner equals to 0.20 for the median firm (with γ/VA = 3.7).

Interestingly, 75% of the firms in this subsample have lerner that is below this optimum value

(see appendix). To put it another way, if we restrict the analysis to the few sectors for which an

11alternatively we take the number of employees as a measure of size. Results are not qualitativel altered.

12

Table 1: The cost of innovation and the magnitude of the U-inverted shape - French firms,1989-2004

Dependant variable: potential flow of innovation, n

(1) (2)

lerner2t−1 -0.0020∗∗∗ -0.00040∗∗

(-4.02) (-2.51)

lerner2t−1.ln(γ) 0.0003∗∗∗ —(4.02) —

lerner2t−1.ln(γ/va) — 0.0001∗∗∗

— (2.83)lernert−1 0.0021∗∗∗ 0.0008∗∗∗

(4.79) (3.70)lernert−1.ln(γ) -0.0003∗∗∗ —

(-5.47) —lernert−1.ln(γ/va) — -0.0002∗∗∗

— (-5.84)Years yes yesFirm fixed effects yes yes

Number of obs. 100089 100043Number of firms 15592 15586R2 0.73 0.73

Source: Bank of France (Fiben). ∗∗ significant at 5%, ∗∗∗ significant at 1%.

13

inverted U shape exists for France, these sectors appear mainly in a neck-to-neck situation for

which competition has a positive impact on innovation.

4 Perspectives

Both theoretical predictions and empirical illustration support the mechanism that competition

less impacts firm decision when the cost of the innovation in its sector is high on absolute or

relatively to its value-added.

These results may lead to significant political prescriptions. The inverted U-shape already

suggests modulating competitive policies according to the state of the industry (Aghion et al. ,

2005). A second dimension of differentiation should be the nature of innovations in the industry.

If they are costly, policy changes have to be massive for expecting an impact; at the limit, in

such sectors, the shape is so flat that competitive policy is not a relevant tool to boost research

effort of firms.

Because this conclusion is strong, deeper research could be worthwhile. On the theoretical

side, the size of innovations or the size of firms are partly endogenous to the competitive envi-

ronment. To endogenize them may alter our arguments. An interesting avenue would also be

to explore the effects of credit constraints that particularly concern small innovative firms. The

roch data of the Banque de France on these constraints could help to directly test the associated

predictions.

In addition, recent papers have highlighted the joint effects of product market regulation and

labor market regulation on economic performances (see OECD, 2006, for a review); therefore

the predictions of the model have also to be tested for countries with less or more restrictive

labor market regulations than France.

A Appendix: Proofs

A.1 Proof of proposition 1

Let’s note σ0 and σ−1 the first derivative of the R&D intensity with respect to product market

competition respectively in the leveled and unleveled sectors. As regards neck-and-neck sectors,

we have:

σ0 =∂n0

∂ν=

χ

2√

(r + h)2 + νχ(25)

14

with χ ≡ 2π1Φ/β. The first derivative of σ0 with respect to γ gives:

∂σ0

∂γ=∂χ

∂γ

2(r + h)2 + νχ

4((r + h)2 + νχ)3/2(26)

which is of the sign of ∂χ∂γ .

Since we have:

∂χ

∂γ=

∂

∂γ

(

2(1 − 1γ )L

β(γ)

)

=2L

β2(γ)

(

1

γ2β(γ) − (1 − 1

γ)β′(γ)

)

(27)

hence sign(∂χ∂γ ) = sign( 1

γ2β(γ) − (1 − 1γ )β′(γ)). Introducing Φ defined as:

Φ(γ) =1

γ2β(γ) − (1 − 1

γ)β′(γ)

we have Φ′(γ) = −2β(γ)γ3 −β′′(γ)(1− 1

γ ), which is non positive for each γ if β is convex. Since we

have Φ(1) = 0, Φ, and consequently ∂χ∂γ is negative for each γ > 1. As a result, σ0 is a decreasing

function of the size of innovation. This establishes Proposition 1 related to neck-and-neck sectors.

A.2 Proof of corollary 1

Since we have ∂σ0

∂χ > 0 according to the notation from the previous section, and as χ is increasing

with respect to L, the slope σ0 is an increasing function of the firm’s size. This establishes

Corollary 1.

A.3 Proof of proposition 2

We first prove the following cases: assuming that the R&D cost function is convex and such

as β(γ) =1o(1 − 1/γ), for large or costly innovations (γ � 1) as well as for small or cheap

innovations (γ ∼ 1), the slope of the relationship between innovation and competition in the

unleveled sectors is a non positive, increasing function of the size of innovation γ.

For the size of innovation being in the neighborhood of 1, which means that innovations are

incremental, and under the assumption that β(γ) =1o(1 − 1/γ), we have:

χ =2(1 − 1

γ )L

β(γ)� 1.

So we find an equivalent of n0 for large χ’s:

n0 ∼χ�1

(νχ)1

2 ,

15

and so the research intensity in leveled sectors:

n−1 ∼χ�1

(√

1 + ν −√ν)χ

1

2 ,

As a consequence, we have the slope of the relationship between competition and R&D intensity

for small χ’s:

σ−1 =∂n−1

∂ν≈

χ�1

√ν −

√1 + ν

2√

1 + ν√νχ

1

2

which is a non positive, increasing function of γ.

In a similar way, for costly innovations, i.e. large γ, we have:

χ =2(1 − 1

γ )L

β(γ)� 1.

Hence, the first order approximation of n0 for χ sufficiently small gives:

n0 =νχ

2(r + h)+ o(χ).

It follows that the linear approximation of n−1 is:

n−1 =(1 − ν)χ

2(r + h)+ o(χ).

As a consequence, we have the slope of the relationship between competition and R&D intensity

for small χ’s:

σ−1 =∂n−1

∂ν≈

χ∼0− χ

2(r + h)

which is a non positive, increasing function of γ.

The system of equations giving n0 and n−1 is quasi-homogenous in r + h. It can be thus

rewritten as a non-parametric system as following.

Let m0 = n0/(r + h), m−1 = n−1/(r + h) and X = π1L/[β(r + h)]2. The system becomes

m0 = −1 +√

1 + νX, (28)

m−1 = −1 −m0 +√

1 +m20X. (29)

Proving the proposition (i.e. showing that ∂2n−1

∂γ∂ν > 0 or ∂2n−1

∂L∂ν < 0) is then equivalent to

show that ∂2n−1

∂X∂ν < 0, with ν ∈]0; 1[ and X ∈]0;+∞[.

It is simple to show that it is true for large or small value of X. But formal calculus does not

allow proving this property for all X. Therefore, we use a numerical representation (see figure 2)

of ∂2n−1

∂X∂ν on the field (ν,X) ∈]0; 1[×]0;+∞[. This completes the proof of Proposition 2.

A.4 Description of the sample

16

00.2

0.40.6

0.81 0

20

40

60

80

100

−0.5

−0.4

−0.3

−0.2

−0.1

0

X

ν

∂2 n−

1/∂ X

∂ ν

Figure 2: 0,1[

17

Table 2: Distribution of observations by main sectors (NAF 16) in %

Agriculture, hunting and forestry 1.1Fishing 5.4Mining and quarrying 8.2Manufacturing 1.7Electricity, gas and water supply 17.0Construction 20.7Wholesale and retail trade; repair of motor vehicles, motor-cycles and personal and household goods

0.5

Hotels and restaurants 4.7Financial intermediation 21.3Real estate, renting and business activities 3.0Education 0.7Health and social work 14.0Activities of households 1.9

Table 3: Descriptive statistics for the main variables

Mean St. dev. d1 d9 Num. of Obs.

Lerner 0.32 0.17 0.12 0.54 100089ln(γ/va) -1.69 1.57 -3.69 0.24 100043Flow of innovation, n 1.62E-4 1.99E-3 0 1.79E-4 100089

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