L. Lambertini

R. Cellini

English

Research Papers in Economics

R&D Incentives and Market
Structure: A Dynamic Analysis1
Roberto Cellini§ - Luca Lambertini#
§ Dipartimento di Economia e Metodi Quantitativi
Universit` a di Catania
Corso Italia 55, 95129 Catania, Italy
phone 39-095375344, fax 39-095-370574
cellini@unict.it
# Dipartimento di Scienze Economiche
Universit` ad iB o l o g n a
Strada Maggiore 45, 40125 Bologna, Italy
phone 39-051-2092600, fax 39-051-2092664
lamberti@spbo.unibo.it
February 9, 2004
1We thank Jeroen Hinloopen and George Leitmann for useful comments and
discussion. The usual disclaimer applies.Abstract
We investigate dynamic R&D for proc e s si n n o v a t i o ni na no l i g o p o l yw h e r e
ﬁrms invest in cost-reducing activities. We focus on the relationship between
R&D intensity and market structure, proving that the industry R&D invest-
ment monotonically increases in the number of ﬁrms. This Arrowian result
contradicts the established wisdom acquired from static games on the same
topic.
J.E.L. Classiﬁcation: C73, D43, D92, L13, O31
Keywords: diﬀerential games, process innovation, R&D, spillovers1I n t r o d u c t i o n
We propose a dynamic analysis of the relationship between market power
and R&D eﬀorts, in order to reassess a well-known issue in the theory of in-
dustrial organization, that can be traced back to the debate between Schum-
peter (1942) and Arrow (1962). The so-called Schumpeterian hypothesis
maintains that there exists an inverse relationship between the intensity of
competition and the pace of technical progress. That is, according to Schum-
peter, monopoly is the market structure that should ensure the fastest and
largest technical progress. This relies upon the idea that monopoly ensures
the highest proﬁt level and therefore the larger internal sources for funding
R&D activities. Exactly the opposite view is expressed by Arrow, since he
focuses upon the replacement eﬀect, according to which a monopolist should
be induced to rest on his laurels, while a ﬁrm operating in a competitive
environment should strive for new technologies or new products, in order to
throw her rivals out of business.1
In order to assess this issue, we take a diﬀerential game perspective,
proposing a dynamic version of a model ﬁrst introduced in a static framework
by d’Aspremont and Jacquemin (1988). We consider an oligopoly where
n ﬁrms sell a homogeneous product and compete in quantities. Moreover,
they also invest at each point in time in R&D for process innovation, i.e.,
reducing the marginal cost of production of the ﬁnal good. The R&D activity
is characterized by positive externalities, entailing that each ﬁrm receives a
positive spillover from the investments carried out by all other ﬁrms in the
industry.
Our model has the desirable property of being state-redundant or per-
fect, so that the open-loop solution is a Markovian equilibrium. We proceed
1For an exhaustive overview of the related literature, see Reinganum (1989) and Martin
(2001).
1in two steps. First, we characterize the individually optimal path of R&D
investment for a given level of marginal production cost. Second, we obtain
the steady state levels of investment and marginal cost. With respect to both
the optimal path and the steady-state level of R&D investment, the following
conclusions hold. The individual eﬀo r ti sa l w a y sd e c r e a s i n gi nt h en u m b e r
of ﬁrms while the opposite holds for the aggregate R&D investments. This
result has an Arrowian ﬂavour, since as the degree of competition becomes
tougher, the aggregate investment becomes larger. This is in sharp contrast
with the conclusions drawn from the static version of the same model (Hin-
loopen, 2000) where a non-monotone relationship exists between aggregate
R&D investment and market structure. Under this perspective, our model
highlights the value added of a properly dynamic analysis over the static
approach based upon a multistage game.
The remainder of the paper is structured as follows. Section 2 illustrates
the basic setup. The solution of the open-loop game is investigated in section
3, while the industry R&D performance is assessed in section 4. Section 5
contains concluding remarks.
2 The setup
We consider an oligopoly with n ﬁrms selling a homogeneous goods over
continuous time, t ∈ [0,∞). In every instant, the market demand function
writes as follows:
p(t)=A − qi(t) − Q−i(t). (1)
where Q−i(t) is the output supplied by all ﬁrms other than i.E a c hﬁrm sup-
plies the market through a technology characterized by a constant marginal
cost, ci. Accordingly, her instantaneous cost function for the production of
the ﬁnal good is Ci (ci,q i,t)=ci(t)qi(t). The marginal cost borne by ﬁrm i
2evolves over time according to the following kinematic equation:
dci(t)
dt
≡
·
ci = ci (t)[−ki(t) − βK−i(t)+δ] , (2)
where ki(t)i st h eR & De ﬀort exerted by ﬁrm i at time t, while K−i(t)i st h e
aggregate R&D eﬀort of all other ﬁrms and parameter β ∈ [0,1] measures the
positive technological spillover that ﬁrm i receives from the R&D activity of
the rivals. Parameter δ ∈ [0,1] is a constant depreciation rate measuring the
instantaneous decrease in productive eﬃciency due to the ageing of technol-
ogy. Equation (2) is indeed a dynamic version of the linear R&D technology
employed by d’Aspremont and Jacquemin (1988).
The instantaneous cost of running R&D activity is:
Γ(ki,t)=b[ki(t)]
2 , (3)
where b is a positive parameter. Throughout the game, ﬁrms discount future
proﬁts a the common and constant discount rate ρ>0.
Firms adopt a strictly noncooperative behaviour in choosing both the out-
put levels and the R&D eﬀorts, each ﬁrm operating her own R&D division.2
The objective of ﬁrm i consists in maximizing discounted proﬁts:
Πi =
Z ∞
0
©
[A − qi(t) − Q−i(t) − ci(t)]qi(t) − b[ki(t)]
2ª
e
−ρtdt (4)
subject to the set of dynamic constraints (2). The corresponding Hamiltonian
function is:
Hi(q,k,c,t)=e
−ρt{[A − qi(t) − Q−i(t) − ci(t)]qi(t) − b[ki(t)]
2 +( 5 )
−λii(t)ci(t)[ki(t)+βK−i(t) − δ]+
−
X
j6=i
λij(t)cj(t)
"
kj(t)+β
Ã
ki(t)+
X
l6=i,j
kl(t)
!
− δ
#)
2For a discussion of R&D cooperation in the same model, see Cellini and Lambertini
(2003).
3where λij(t)=µij(t)eρt is the co-state variable (evaluated at time t) associ-
ated with the state variable cj(t), and q,k,c are the vectors of control and
state variables.
3T h e o p e n - l o o p s o l u t i o n
Here we characterize the Nash equilibrium under the open-loop information
structure. As a ﬁrst step, we prove the following result:
Lemma 1 The open-loop Nash equilibrium of the game is subgame (or Markov)
perfect.
Proof. We are going to show that the present setup is a perfect game in
the sense of Leitmann and Schmitendorf (1978) and Feichtinger (1983). In
summary, a diﬀerential game is perfect whenever the closed-loop equilibrium
collapses into the open-loop one, the latter being thus strongly time consis-
tent, i.e., subgame perfect.3 Consider the closed-loop information structure.
The relevant ﬁrst order conditions (FOCs) are:
∂Hi (.,t)
∂qi(t)
= A − 2qi(t) − Q−i(t) − ci(t)=0; ( 6 )
∂Hi (.,t)
∂ki(t)
= −2bki(t) − λii(t)ci(t) − β
X
j6=i
λij(t)cj(t)=0. (7)
As a ﬁrst step, observe that (6) only contains ﬁrm i’s state variable, so that in
choosing the optimal output at any time during the game ﬁrm i may disregard
the current eﬃciency of the rival. That is, there is no feedback eﬀect in the
3The label ‘perfect game’ is due to Fershtman (1987), where one can ﬁnd a general tech-
nique to identify any such games. Another class of games where open-loop equilibria are
subgame perfect is investigated by Reinganum (1982). For further details, see Mehlmann
(1988, ch. 4) and Dockner et al. (2000, ch. 7).
4output choice. Conversely, at ﬁrst sight there seem to be a feedback between
the R&D decisions, as (7) indeed contains all state variables, at least for any
positive spillover eﬀect.4 The core of the proof consists in showing that no
feedback eﬀect are actually present, even for positive spillover levels.
Taking the above considerations into account, the adjoint or co-state
equations are:
−
∂Hi (.,t)
∂ci(t)
−
X
j6=i
∂Hi (.,t)
∂kj(t)
·
∂k∗
j (.,t)
∂ci(t)
=
∂λii(t)
∂t
− ρλii(t) ⇔ (8)
∂λii(t)
∂t
= qi (t)+λii(t)[ki(t)+βK−i(t)+ρ − δ]+ (9)
−
β
2b
X
j6=i
λji(t)
"
βλii(t)ci(t)+λij(t)cj(t)+β
X
l6=i,j
λil(t)cl(t)
#
−
∂Hi (.,t)
∂cj(t)
−
∂Hi (.,t)
∂ki(t)
·
∂k∗
i (.,t)
∂cj(t)
−
X
l6=i,j
∂Hi (.,t)
∂kl(t)
·
∂k∗
l (.,t)
∂cj(t)
=
∂λij(t)
∂t
−ρλij(t) ⇔
(10)
∂λij(t)
∂t
= λij(t)
"
kj(t)+βki(t)+β
X
l6=i,j
kl (t)+ρ − δ+( 1 1 )
−
β
2b
Ã
2bki (t)+λii(t)ci(t)+β
X
j6=i
λij(t)cj(t)
!#
+
−
β
2b
X
l6=i,j
λlj (t)
"
βλii (t)ci (t)+λil (t)cl (t)+β
X
j6=i,l
λij(t)cj(t)
#
where each term
∂Hi (.,t)
∂kj(t)
·
∂k∗
j (.,t)
∂ci(t)
(12)
captures the feedback eﬀect from j to i, and partial derivatives ∂k∗
j (.,t)/∂ci(t)
are calculated using the optimal values of investments as from FOC (7):
k
∗
j (.,t)=−
λjj(t)cj(t)+βλji(t)ci(t)
2b
. (13)
4Intuitively, if β =0 , then the investment plans are completely independent and there-
fore it is apparent that no feedback eﬀect operates.
5These conditions must be evaluated along with the initial conditions
{ci(0)} = {c0,i} and the transversality conditions
lim
t→∞ λij(t) · cj(t)=0,i ,j=1 ,2. (14)
Note that, on the basis of ex ante symmetry across ﬁrms, λlj(t)=λij(t)
for all l. Accordingly, from (11), we have ∂λij(t)/∂t =0i nλij(t)=0 . Then,
using this piece of information, we may rewrite the expression for the optimal
investment of ﬁrm i as follows:
k
∗
i (.,t)=−
λii(t)ci(t)
2b
, (15)
which entails that ∂k∗
i (.,t)/∂cj(t)=0f o ra l lj 6= i, i.e., feedback (cross-
)eﬀects are nil along the equilibrium path. Accordingly, the open-loop equi-
librium is a degenerate closed-loop one, and it is strongly time consistent, or
equivalently, subgame perfect. It is also worth observing that this procedure
shows that FOCs are indeed unaﬀected by initial conditions as well. The
property whereby the FOCs on controls are independent of states and initial
conditions after replacing the optimal values of the co-state variables is known
as state-redundancy, and the game itself as state-redundant or perfect.
On the basis of Lemma 1, we can proceed with the characterization of
the open-loop solution. The FOCs on controls as well as the transversality
conditions are the same as above, while the co-state equations simplify as
follows:
−
∂Hi (.,t)
∂ci(t)
=
∂λii(t)
∂t
− ρλii(t) ⇔ (16)
∂λii(t)
∂t
= qi (t)+λii(t)[ki(t)+βK−i(t)+ρ − δ]
−
∂Hi (.,t)
∂cj(t)
=
∂λij(t)
∂t
− ρλij(t) ⇔ (17)
∂λij(t)
∂t
= λij(t)[kj(t)+βK−j(t)+ρ − δ]
6From FOCs (6-7) we have, respectively:
q
∗
i(t)=
A − Q−i(t) − ci(t)
2
, (18)
ki (t)=−
λii(t)ci(t)
2b
, (19)
since λij(t)=0f o ra l lj 6= i, at any t ∈ [0,∞). While (18) has the usual
appearance of a standard Cournot best reply function, the optimal R&D
eﬀort in (19) depends upon i’s co-state variable. Such expression can be
diﬀerentiated w.r.t. time to get the dynamic equation of ki(t):
dki(t)
dt
≡
·
ki = −
1
2b
·
ci(t)
·
λii(t)+λii (t)
·
ci(t)
¸
(20)
with
·
λii(t) obtaining from (9). Then, (20) can be further simpliﬁed by using
λii (t)=−
2bki (t)
ci (t)
(21)
which obtains from (7). This yields:
·
ki = −
1
2b
[ci (t)qi (t) − 2bki (t)] . (22)
The next step consists in imposing the symmetry conditions cj(t)=ci(t),
kj(t)=ki(t)a n dqj(t)=qi(t) for all j, and solve the system of the best reply
functions (18), yielding the Cournot-Nash output level of each ﬁrm:
q
N(t)=
A − c(t)
n +1
(23)
which can be plugged into (22). Accordingly, we may simplify the dynamics
of the R&D eﬀort of any single ﬁrm as follows:
·
k = −
1
2b
·
c(t)[A − c(t)]
n +1
− 2bρk (t)
¸
. (24)
Imposing the stationarity condition
·
k =0 , we obtain:
k
N(t)=
c(t)[A − c(t)]
2bρ(n +1 )
≥ 0f o ra l lc(t) ∈ [0,A], (25)
7where the superscript N stands for Nash equilibrium.
The steady state level of marginal cost c(t) can be found by solving:
·
c = −c(t)
£
k
N(t)(1+β (n − 1)) − δc(t)
¤
=0 ( 2 6 )
which yields c =0a n d
c =
A(1 + β (n − 1)) ±
p
(1 + β (n − 1))[A2 (1 + β (n − 1)) − 8bδρ(n +1 ) ]
2(1+β (n − 1))
(27)
All solutions in (27) are real if and only if δρ ≤ A2 (1 + β (n − 1))/[8b(n +1 ) ].
If so, they also satisfy the requirement c ∈ [0,A]. By checking the stability
conditions, we may prove the following:
Proposition 2 Provided that δρ ≤ A2 (1 + β (n − 1))/[8b(n +1 ) ], the steady
state point
c
ss =
A(1 + β (n − 1)) −
p
(1 + β (n − 1))[A2 (1 + β (n − 1)) − 8bδρ(n +1 ) ]
2(1+β (n − 1))
k
ss =
δ
1+β (n − 1)
i st h eu n i q u es a d d l ep o i n te q u i l i b r i u m .
Proof. Omitted for brevity.
4C o m p a r a t i v e s t a t i c s
Now we focus on the interplay between market structure (as measured by
the number of ﬁrms) and the industry incentive to invest in process R&D.
To this aim, we examine eﬀect of a change in n on individual and aggregate
R&D eﬀorts, both along the equilibrium path (expression (25)) and in steady
state.
8This discussion revisits the debate between Schumpeter (1942) and Arrow
(1962). Their respective views can be summarized as follows. According to
the Schumpeterian hypothesis, R&D investments and technical progress are
positively related to the ﬂow of proﬁts and therefore we should expect to
observe higher R&D eﬀorts and a faster innovation process under monopoly
than any other market form. Conversely, Arrow claims that the incentive
to generate technical progress is negatively aﬀected by market power, being
then maximized under perfect competition. The Arrowian position relies
upon the idea that innovation is more attractive for a competitive ﬁrm than
f o ram o n o p o l i s tw h o ,b yd e ﬁnition, can not improve his market power.
In order to assess this issue in the present model, we proceed as follows.
The aggregate R&D investments along the equilibrium path and in steady
state are, respectively:
K
N (t)=
c(t)[A − c(t)]n
2bρ(n +1 )
; K
ss =
δn
1+β (n − 1)
(28)
It is immediate to verify that:
∂KN (t)
∂n
=
2bρc(t)[A − c(t)]
4[bρ(n +1 ) ]
2 > 0( 2 9 )
∂Kss
∂n
=
δ (1 − β)
[1 + β (n − 1)]
2 ≥ 0
which entails that the behaviour of the industry is clearly Arrowian. If in-
stead we examine the individual investment, we obtain ∂kN (t)/∂n,∂kss/∂n <
0 everywhere. This entails that any increase in the number of ﬁrms brings
about a decrease in individual R&D eﬀort. This is caused by two facts: on
the one hand, tougher market competition reduces proﬁts and therefore that
funds available to any given ﬁrm for conducting R&D activity; on the other, a
larger population of ﬁrms means a larger amount of positive externality that
any ﬁrm receives from the rivals. On the aggregate, a scale eﬀect prevails,
9so that the overall expenditure of the industry is monotonically increasing in
n.5
Hinloopen (2000) has solved the oligopoly equilibrium with n ﬁrms in
the static case, ﬁnding that both aggregate and individual R&D eﬀorts are
non-monotone (ﬁrst increasing and then decreasing) w.r.t. n. Under this
respect, the static approach proves to fall short of appropriately accounting
for the inherently dynamic nature of research and development which is not
captured by multistage game modelling.
5 Concluding remarks
We have analyzed dynamic R&D investments for cost-reducing innovation in
a Cournot oligopoly in order to evaluate the inﬂuence of market structure on
R&D incentives.
The setup employed in the present paper is a dynamic version of the
static game examined in d’Aspremont and Jacquemin (1988). Two features
are worth stressing. First, the game is state-redundant, so that the open-loop
solution is Markovian. Second, an Arrowian conclusion is established, as the
aggregate R&D eﬀort is everywhere increasing in the number of ﬁrms. The
drastic diﬀerence between our results and the ambiguous conclusions drawn
from the static model relies upon smoothing the investment eﬀorts over a
long time horizon, a perspective which is ruled out by deﬁnition in a static
setting.
5For a similar result concerning the incentives towards R&D for product innovation,
see Cellini and Lambertini (2002, 2004).
10References
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R&D Incentives and Market Structure: A Dynamic Analysis

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