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Abstract

This paper investigates the research and development accumulation and pricing strategies of two firms competing for consumer demand in a dynamic framework. A firm’s research and development is production-cost-reducing and can benefit from part of the competitor’s research and development stock without payment. We consider decisions in a game characterized by Nash equilibrium. In this dynamic game, a player’s action depends on whether the competitor’s current research and development stock are observable. If the competitor’s current research and development stock are not observable or observable only after a certain time lag, a player’s action can be solely based on the information on the current period t (open-loop strategy). In the converse case, it can also include the information on the competitor’s reaction to a change in the current value of the state vector (closed-loop strategy), which allows for strategic interaction to take place throughout the game. Given the cumulative nature of research and development activities, a primary goal of this paper is to determine whether, regardless of the observability of the competitor’s current research and development stock, free research and development spillovers generate a lower level of scientific knowledge than research and development appropriability. A second objective of the paper is to determine how the observability of the rival’s current research and development stock affects a firm’s research and development and pricing decisions and payoffs under imperfect research and development appropriability.
1 23
Journal of Optimization Theory and
Applications
ISSN 0022-3239
Volume 161
Number 2
J Optim Theory Appl (2014) 161:626-647
DOI 10.1007/s10957-013-0433-2
Research and Development with
Stock-Dependent Spillovers and Price
Competition in a Duopoly
Fouad El Ouardighi, Matan
Shnaiderman & Federico Pasin
1 23
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J Optim Theory Appl (2014) 161:626–647
DOI 10.1007/s10957-013-0433-2
Research and Development with Stock-Dependent
Spillovers and Price Competition in a Duopoly
Fouad El Ouardighi ·Matan Shnaiderman ·
Federico Pasin
Received: 17 April 2013 / Accepted: 11 September 2013 / Published online: 25 September 2013
© Springer Science+Business Media New York 2013
Abstract This paper investigates the research and development accumulation and
pricing strategies of two firms competing for consumer demand in a dynamic frame-
work. A firm’s research and development is production-cost-reducing and can benefit
from part of the competitor’s research and development stock without payment. We
consider decisions in a game characterized by Nash equilibrium. In this dynamic
game, a player’s action depends on whether the competitor’s current research and de-
velopment stock are observable. If the competitor’s current research and development
stock are not observable or observable only after a certain time lag, a player’s action
can be solely based on the information on the current period t(open-loop strategy).
In the converse case, it can also include the information on the competitor’s reaction
to a change in the current value of the state vector (closed-loop strategy), which al-
lows for strategic interaction to take place throughout the game. Given the cumulative
nature of research and development activities, a primary goal of this paper is to de-
termine whether, regardless of the observability of the competitor’s current research
and development stock, free research and development spillovers generate a lower
level of scientific knowledge than research and development appropriability. A sec-
ond objective of the paper is to determine how the observability of the rival’s current
research and development stock affects a firm’s research and development and pricing
decisions and payoffs under imperfect research and development appropriability.
Communicated by Gustav Feichtinger.
F. El Ouardighi (B)
ESSEC Business School, Cergy Pontoise, France
e-mail: elouardighi@essec.fr
M. Shnaiderman
Bar-Ilan University, Ramat-Gan, Israel
F. Pa sin
HEC Montréal, Montréal, Canada
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J Optim Theory Appl (2014) 161:626–647 627
Keywords Research and development competition ·Research and development
spillovers ·Price competition ·Duopoly market
1 Introduction
It is well known that both cost-reducing research and development (R&D) invest-
ments and consumer surplus decline with free spillovers among competing firms1[1].
Moreover, R&D cooperation prior to competition in the product market has been de-
scribed as an efficient means of internalizing such spillovers [24]. Although this
option can theoretically improve R&D and social welfare levels, in practice it may
prove neither desirable nor feasible.
First, numerous joint ventures, including those involving R&D cooperation, show
high rates of early termination (30–70 %) [5,6]. One explanation for this trend is op-
portunistic behaviors [7]. As suggested in [8], opportunistic actions that undermine
the position of a direct competitor provide greater benefits than opportunistic actions
at the expense of a non-competing firm. This contention is supported in [9], which
shows that an increase in inter-alliance competition reduces investments in the focal
alliance but increases investments in competition outside the scope of the alliance.
This increased competition implies that the risk of opportunism in alliances between
competitors may substantially increase the monitoring costs of direct competitor al-
liances. These costs will limit the potential of the alliance to achieve a competitive
advantage [10]. Further, governance structures such as equity joint ventures provide
insufficient protection to induce extensive knowledge sharing among alliance partic-
ipants when the partner firms are direct competitors in the product market. Hence,
partners have the choice of either abandoning the potential gains from R&D cooper-
ation altogether or limiting “the scope of alliance activities to those that can be suc-
cessfully completed with limited (and carefully regulated) knowledge sharing” [8].
As a result, R&D cooperation may not be desirable.
Nonetheless, cost-reducing R&D and pricing decisions can hardly be adopted se-
quentially. As suggested by the well-known product-process life cycle theory of Ut-
terback and Abernathy [11], the rate of process innovation induced by cost-reducing
R&D activities depends on the present stage of a product’s life cycle.2That is, after
the emergence of a dominant product design, which corresponds to the segmental
stage of the industry, specialized production equipment is introduced and the rate
of innovation related to the production process is important. In this stage, product
innovations requiring radical changes to the production process are voided, and pro-
duction cost minimization becomes an important goal. Concomitantly, firms seek to
reduce their production costs through R&D accumulation while engaging in price
competition on the product market. Unless tacit collusion on the product market is
1R&D spillovers can arise through channels such as R&D personnel movement, formal and informal net-
works and meetings, publications related to research output, patent applications and reverse engineering.
2See also [1216]. A supportive empirical evaluation of this theory can be found in [17].
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628 J Optim Theory Appl (2014) 161:626–647
assumed,3this simultaneity of decisions makes it difficult for competitors to initiate
R&D cooperation at this stage. As a result, R&D cooperation may not be feasible.
Since firms ultimately face a scenario of competitive R&D along with product
market competition, this paper will not compare the sequential R&D outcomes re-
lated to the competitive and cooperative scenarios, as is usually done in the literature.
Instead, it assumes that R&D and pricing decisions are made simultaneously in a
fully competitive setting. As a key component of the problem, R&D is interpreted
as a stock variable, e.g. [1921]. Due to the cumulative nature of R&D activities,
we consider a dynamic analysis of simultaneous, competitive R&D and pricing de-
cisions to determine whether R&D accumulation over time generates less scientific
knowledge under free R&D spillovers than under R&D appropriability, as the lit-
erature based on static or two-stage game models argues. To this end, firms’ R&D
and pricing strategies under perfect R&D spillovers and full R&D appropriability are
determined and compared in the setup of a dynamic, non-cooperative duopoly game.4
In dynamic games, players can choose from a variety of strategies [2729]. A strat-
egy is a decision rule that a player selects before the beginning of the game. The strat-
egy specifies what action the player should take at each time t, based on the current
information available to the player. A player’s action can be solely based on infor-
mation available on the present time t(open-loop strategy), or it can also include
information regarding the opponent’s competitive reaction to a change in the current
value of the state vector, if it is available (closed-loop strategy).5
In the context of R&D competition, closed-loop strategies rest on the observability
of a competitor’s R&D stock. If a firm cannot (or can do so only after a certain
time lag) observe its rival’s R&D stock, then it cannot condition its actions on the
state vector, and an open-loop strategy might be more appropriate than a closed-loop
strategy. A second objective of this paper is to determine whether the observability of
the rival’s current R&D stock affects a firm’s R&D and pricing decisions under free
spillovers in a duopoly. To this end, we evaluate and compare open- and closed-loop
R&D and pricing strategies under free spillovers among duopolist firms.
We consider two firms that produce and sell competing products on the demand
market. Bertrand demand functions are assumed. Firms can invest in R&D to lower
their marginal production costs and therefore potentially increase their market de-
mand via price reductions. Given costless mutual spillovers, each firm simultaneously
selects its R&D investment and consumer price over a finite time horizon. The dy-
namical system states are represented by the firms’ R&D stocks. We investigate the
following issues:
How are R&D accumulation, pricing strategies and payoffs affected by the magni-
tude of R&D spillovers over time for the open-loop and closed-loop strategies?
3As noted in [18], “common assets create common interests, and common interests make it more likely
that firms will noncooperatively refrain from rivalling behaviour”.
4Prior studies simultaneously combining R&D with pricing in dynamical setting have been limited to a
monopoly context, e.g. [2226].
5Open-loop strategies provide a useful benchmark for assessing the strategic effects related to Markovian
strategies [30].
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How are R&D accumulation, pricing strategies and payoffs affected by the observ-
ability of the rival’s current R&D stock, i.e., by the choice of decision rules?
The paper proceeds as follows. Section 2presents the literature to which this pa-
per contributes, and Sect. 3develops the differential game model. In the two sub-
sequent sections, we determine the decision rules that reflect different forms of the
Nash equilibrium, open-loop Nash equilibrium (OLNE) and closed-loop Nash equi-
librium (CLNE), and compare the results. Section 6provides numerical illustrations.
Section 7offers concluding remarks.
2 Literature Review
Our research concerns two categories of models: non-tournament games without
R&D accumulation6and tournament games with R&D accumulation.
In the first category, firms are not engaged in a race but can all simultaneously
succeed at “producing innovations” in a context in which the results of the inno-
vation process exhibit imperfect appropriability.7Spence [1] was among the first
to formally account for the issues of knowledge spillovers and R&D subsidies in
a one-stage game. Each firm’s unit production cost is assumed to decline with the
firm’s own R&D stock. It is shown that R&D incentives decrease with reduced ap-
propriability, and cooperative R&D may be suitable to raise welfare. D’Aspremont
and Jacquemin [2,3] made the first attempt to analyze spillovers with respect to re-
search joint ventures (RJVs) in a two-stage game model, wherein nearly symmetric
firms first choose R&D either cooperatively or non-cooperatively, costlessly acquire
spillovers from rivals, and engage in quantity competition in the product market. For
large spillovers, the cooperative R&D level has been shown to be higher than the non-
cooperative one. Kamien et al. [4] propose different R&D organizational models in-
volving free spillovers in an oligopolistic market, that is, R&D expenditure carteliza-
tion and/or full information sharing. Cartelized RJVs have been demonstrated to be
the most desirable organizational model because they provide both the highest R&D
level and social welfare (see also [41]). In [42], a three-stage game is formulated
where the duopolist firms choose the extent of outgoing spillovers generated by their
R&D activities before simultaneously deciding on R&D and allowing for involun-
tary spillovers. Each firm must acquire absorptive capacity8to benefit from its rival’s
R&D. It is shown that when firms cooperate in R&D, they choose identical R&D
approaches and fully share their knowledge, whereas they choose firm-specific R&D
approaches and keep at least part of their knowledge private if R&D budgets are set
6In contrast, in tournament games without R&D accumulation, the timing of new technology adoption
plays a central role. In the relevant literature, technological competition may take the form of a pre-emption
game (i.e., first-mover advantage) due to rivalry in the product market, or a waiting game (i.e., late-mover
advantage) resulting from technological uncertainty and informational spillovers; e.g., [3135].
7A related stream of literature, not discussed here, investigates the impact of R&D competition on market
structure; e.g., [3640].
8Cohen and Levinthal [43] introduced the concept of absorptive capacity in R&D, which is defined as the
ratio of “usable” to “actual” rival R&D and depends on a firm’s own level of investment in R&D.
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non-cooperatively. Grünfeld [44] shows that the absorptive capacity of a company’s
own R&D does not drive up the incentive to invest in R&D when the market size
is small or the absorptive capacity effect is weak. Leahy and Neary [45] assert that
costly absorption enhances the effectiveness of a firm’s own R&D and lowers the ef-
fective spillover coefficient, such that R&D cooperation becomes less attractive even
under full information sharing. Milliou [46] finds that imperfect R&D appropriabil-
ity results in increased R&D investments. Hammerschmidt [47] demonstrates that
greater R&D spillovers lead firms to invest more in R&D to strengthen their absorp-
tive capacity.
Regarding the category of tournament games that allow for R&D accumulation,
firms are engaged in a race in which the winner receives a greater payoff than the
second adopter. Reinganum [19] formulated a differential game model in which two
firms are involved in a technology adoption race, where the probability of winning
the race by any given date is a function of the R&D stock accumulated by that date.
The game, which allows for R&D spillovers, has a special, exponential structure that
implies that the open-loop and feedback Nash equilibria are equivalent. At date 0, the
firms must simultaneously pre-commit to an adoption date, and the exponential dif-
ferential game admits a unique, symmetric, subgame-perfect, open-loop equilibrium,
such that each firm can choose its equilibrium level of research effort independent of
its accumulated R&D.9It is shown that under full R&D appropriability, innovation
by competing rivals occurs, on average, sooner than innovation by cooperative firms
(referred to as the optimization of the joint objective criterion). When participating
in full information sharing, cooperating firms are the first to innovate.10 An exten-
sion of this model by Doraszelski [20] assumes that the R&D stocks that firms have
acquired as a result of their past R&D efforts are relevant to firms’ current R&D ef-
forts and thus to the outcome of the race. Unlike the model in [19], Doraszelski [20]
omits the existence of spillovers but allows for history-dependence, in that each firm
chooses its equilibrium R&D effort based on its accumulated knowledge. In addi-
tion, the R&D stocks are assumed to have a positive depreciation rate over an infinite
time horizon. Steady-state feedback Nash equilibrium strategies are computed using
approximation methods. Firms are shown to have an incentive to reduce R&D expen-
ditures as their R&D stock increases. Further, a firm will respond aggressively to an
increase in its rival’s R&D stock if it has a sufficiently large R&D stock, and weakly
otherwise. In other words, “the response in the firm’s investment in R&D to a change
in its own knowledge stock swamps the response to a change in its rival’s knowledge
stock” [20].
The present paper extends the literature by interpreting R&D competitive deci-
sions in a duopoly as a non-tournament game with R&D accumulation. A similar
9Dockner et al. [48] show that closed-loop strategies can be implemented as a subgame-perfect equilibrium
in Reinganum’s [19] model.
10Fudenberg and Tirole [33] note that an open-loop solution reflects infinitely long information lags, and a
first-mover advantage is not supported by subgame-perfect strategies if firms are unable to pre-commit to
future actions. Assuming negligible information lags (closed-loop solution), the authors obtain two equi-
libria: a maturation equilibrium, in which a later innovation yields a higher payoff, and a pre-emption
equilibrium, in which the two firms invest on two different dates but their rents are equalized. There-
fore, when firms cannot pre-commit themselves to adopt technology on specific dates, timing competition
reduces the initial delay in new technology adoption.
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interpretation is found in [49], where a firm’s R&D investments seek to accumulate
cost reductions. Cellini and Lambertini’s [49] paper differs from ours in that it pro-
ceeds as in non-tournament games without R&D accumulation, whereas we analyze
alternative decision rules (OLNE and CLNE) that apply only when firms behave fully
competitively. As in tournament games with R&D accumulation [19,20], we use a
differential game instead of a multi-stage one. However, an important difference of
our paper is that we focus on how firms’ R&D accumulation strategies can affect
price competition on the product market. Our formulation specifically differs from
that of Doraszelski [20] because we consider the existence of free spillovers between
the firms. The formulation also differs from those of Reinganum [19] and Cellini and
Lambertini [49] in that spillovers apply to the rival’s R&D stock rather than to effort.
3 Model
We consider a differential game in which two identical firms sell differentiated rather
than homogeneous products and compete on price for demand. Each firm has the
ability to reduce its unit production cost by increasing its stock of R&D. Due to free
spillovers, each firm can benefit from its competitor’s R&D stock. At each period,
each firm controls two decision variables: its sales’ price and its investment in R&D
accumulation.
Time tis continuous such that t0, and the time horizon Tisassumedtobefixed
and finite. Firm i’s R&D stock is denoted by Xi(t),i=1,2, and evolves over time
according to the transition equation
˙
Xi(t) =ui(t), Xi(0)=0,(1)
where ui(t) 0, i=1,2, is firm i’s effort rate of R&D accumulation. Due to the
finite planning horizon, the depreciation of the R&D stock need not be considered.
At each point in time, the scientific knowledge of each firm is given by Zi(t), that
is,
Zi(t) :=Xi(t ) +εXj(t), (2)
where 0 ε1 is a parameter for the costless spillover effect, i, j =1,2,i =j, such
that ε=0 reflects the case of full R&D appropriability and ε=1 the case of perfect
R&D spillovers. The case in which 0 <ε<1 reflects the assumption that the effect
of a firm’s own R&D outweighs the benefits accruing freely from the rival firm when
both firms spend identical amounts on R&D. As in most theoretical models of R&D
and spillovers, we assume that the duopolist firms generate and receive symmetric
spillovers.11
As suggested above, spillovers apply here to R&D stock rather than to R&D effort.
This assumption relies on the simple analogy that patent protection generally applies
to R&D outcomes rather than instantaneous efforts (e.g., [46]). Since R&D spillovers
inversely reflect the level of patent protection [52,53], it therefore seems more plau-
sible to assume that spillovers are more likely to arise from R&D outcomes than
11Notable exceptions to symmetric spillovers are those presented in [50,51].
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632 J Optim Theory Appl (2014) 161:626–647
are instantaneous efforts. In this regard, R&D spillovers are stock-dependent rather
than effort-dependent [19,49]. This assumption gives our model a linear-quadratic
property, which generally allows for differences between OLNE and CLNE [54].
Unlike non-tournament games without R&D accumulation, we assume that firms’
consumer demand is determined by (Bertrand) price rather than (Cournot) quantity
competition, such that it decreases in firm i’s price and increases in the rival’s price.
Consumer demand Si(t) is specified as a linear function, that is,
Sipi(t ), pj(t):= αiβipi(t) +γipj(t ) 0,
where αi>0, βi>0 and 0 γi
iare parameters, i, j =1,2,i = j. We assume
symmetric parameters for both firms, that is, αiα, βiβ,γiγ,i=1,2. The
parameter αdenotes firm i’s potential market, which we assume to be large enough
to make the decision to invest in R&D sufficiently attractive. The parameter βdenotes
firm i’s demand’s sensitivity to its own price, and the parameter γinversely reflects
the degree of product substitutability between firm iand its rival.12
Next, we define a payoff functional for each firm. Given the finite planning hori-
zon, the discounting of future profits is omitted. Each firm’s objective includes its
profits over the planning period and a salvage value of terminal sales.
The scientific knowledge is assumed to linearly reduce the unit production cost
Ci(Zi(t)), that is, Ci(Zi(t )) =ciZi(t ) 0. This formulation is consistent with
the formulations of D’Aspremont and Jacquemin [2] and Kamien et al. [4]. For sim-
plicity, we assume ci=c>0, i=1,2, where cis sufficiently large both to pre-
vent negative marginal production cost and to act as an incentive to invest in R&D.
However, we suppose that the cost of effort in R&D accumulation is an increasing
quadratic function: aiui(t)2/2, ai>0, i=1,2. Note that the results would remain
qualitatively identical if a general function were adopted for firm i’s cost of effort in
R&D accumulation. For simplicity, we assume that ai=1, i=1,2, without loss of
generality.
Thus, the objective function of firm iis as follows:
Max
pi,ui0Πi=T
0pi(t) c+Zi(t)Si(t) ui(t )2/2dt +miSi(T ), (3)
where mi>0 denotes the marginal salvage value of firm is terminal sales. Invoking
the symmetry property, we assume that mi=m>0, i=1,2.
4 Non-cooperative Equilibrium Strategies
This section will first derive OLNE and CLNE strategies in terms of R&D accumula-
tion and pricing policies. In an OLNE, the firms are assumed to know their own R&D
stock but not a rival’s R&D stock. Accordingly, they use a minimum amount of infor-
mation because the control variables depend on the current period exclusively. The
OLNE is weakly time-consistent, which allows for the continuation of equilibrium
actions and ensures the credibility of the players’ commitments at any time along the
12Note that the demand function above can be easily rewritten in a Cournot setting [55].
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equilibrium path. In a CLNE, because each firm knows its own R&D stock as well
as the rival’s, its decisions depend on the history of the game as summarized by the
initial and current values of the state variables, which allows for strategic interaction
throughout the game.
Firm i’s Hamiltonian is
Hi=(pic+Xi+εXj)(α βpi+γpj)u2
i
2+λiui,(4)
i, j =1,2,i = j, where λi(t ) is firm i’s costate variable associated with the firm’s
R&D stock, i=1,2. Note that the redundant costate variable is omitted in firm i’s
Hamiltonian (see [56]); that is, only the costate variable related to the state variable
that is effectively controlled by firm iis included.13 Since firm is R&D stock has a
positive influence on its objective function, its costate variable λiis expected to be
positive [57].
Now consider the necessary conditions for the maximization of the Hamiltonian
Hi
pi=0pi=(2β+γ )(α +βc) β[(2β+γε)X
i+(2βε +γ)X
j]
4β2γ2,(5)
Hi
ui=0ui=λi,(6)
i, j =1,2, i= j. It is readily shown that the Hamiltonian is strictly concave in
(pi,u
i), which guarantees a unique maximum of Hi,i=1,2. Equations (5) and (6)
demonstrate the following: a firm’s price is negatively affected by the firm’s own
stock and its rival’s stock of R&D. Equation (5) shows that, all things being equal, a
marginal increase in the spillover effect results in a decrease in firm is sales price.
A non-negative value of firm i’s effort rate in R&D accumulation requires a non-
negative value of the costate variable λi.
The costate equations are as follows (e.g., [26]):
˙
λi=−Hi
Xi
=−β{(2β+γ)[αγ)c]+(2β2γ2βγε)Xi+(2β2εγ2εβγ)Xj}
4β2γ2(7)
in an OLNE, and
˙
λi=−Hi
XiHi
pjpjXi=−Si+βγ(2βε +γ)π
i
4β2γ2(8)
in a CLNE, where
πipic+Zi
=(2β+γ)[αγ)c]+(2β2γ2βγ ε)Xi+(2β2εγ2εβγ)Xj
4β2γ2,(9)
and Siis given on the right-hand side of (7), i, j =1,2, i=j.
13Intuitively, one would expect the costate variable associated with the rival’s stock of R&D to be zero,
and hence a change in such a stock would have no impact on firm i’s optimal profit.
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The transversality conditions are
λi(T ) =mβ (2β2βγ ε γ2)
4β2γ2,(10)
which are non-negative, i, j =1,2, i= j. Note that the transversality conditions
in (10) reflect the terminal value imputed by each firm to its own R&D stock because
each firm’s sales price assigns an instantaneous value to its own R&D stock in (5),
which impacts terminal sales.
In the case of an OLNE, assuming demand functions are non-negative, it should
hold that ˙
λi0in(7), and λishould thus be non-increasing over the entire planning
horizon because λi>0 for all t∈[0,T], which implies that the controls ui(t ) in (6)
are strictly positive and non-increasing for all t∈[0,T]. This result implies that the
state variables are also strictly positive and non-decreasing for all t>0.
In the case of a CLNE, firm i’s R&D stock Xihas a positive effect on its own unit
profit margin πiin (9). However, firm j’s R&D stock Xjhas a non-negative influence
on firm i’s unit profit margin only if the spillover effect from the former to the latter
is sufficiently large, that is, if ε∈[ βγ
2β2γ2,1]. If the products are not substitutes, then
for any ε∈[0,1],firmj’s R&D stock Xjalways has a positive influence on firm
i’s unit profit margin. It can be shown that ˙
λi=−2(2β2γ2βγ ε)
4β2γ2Si0 in a CLNE,
where 0 <2(2β2γ2βγ ε)
4β2γ2<1. Since λi(T ) are equivalent in OLNE and CLNE, λi(t)
is lower in a CLNE than in an OLNE, t<T,i=1,2. In either case, the controls
ui(t) in (6) are non-negative and non-increasing for all t∈[0,T], which implies that
the state variables are also non-negative and non-decreasing t>0.
Proposition 4.1 For every 0γ<βand 0ε1, firm i’s R&D investment and
R&D stock are,respectively,
uol
i|γ<β,ε(t) =1
cos(ΦT ) β[αγ)c]sin[Φ(T t)]
β(1+ε)(β γ)(2βγ) +Λcost),(11)
Xol
i|γ<β,ε(t) =Ψcos[Φ(T t)]
cos(ΦT ) 1+Λsin(Φt )
Φcos(ΦT ) (12)
in an OLNE,and
ucl
i|γ<β,ε(t) =1
cos T ) 2β(2β2γ2βγε)[αγ)c]sin[Γ(T t)]
(1+ε)(β γ)(2β+γ)(2βγ)
2
+Λcos(Γ t),(13)
Xcl
i|γ<β,ε(t) =Ψcos[Γ(T t)]
cos(Γ T ) 1+Λsin(Γ t)
Γcos(Γ T ) (14)
in a CLNE,where Φ:= β(1+ε)(βγ)
2βγ,Ψ:= αγ)c
(1+ε)(βγ),Λ:= mβ (2β2βγ εγ2)
4β2γ2,
Γ:= 2β(1+ε)(βγ )(2β2γ2βγ ε)
(2β+γ )(2βγ)
2,and ol and cl denote “open-loop” and “closed-
loop,” respectively,i=1,2.
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Proof To solve firm i’s R&D effort rate and stock in an OLNE, we form the associ-
ated differential system in the control-state space. Differentiating (6) with respect to
time and substituting for ˙
λifrom (7), i=1,2, and simplifying, we use (1) to obtain
the following system of 4 linear differential equations:
˙uol
1
˙uol
2
˙
Xol
1
˙
Xol
2
=
00fg
00gf
10 0 0
01 0 0
uol
1
uol
2
Xol
1
Xol
2
e
e
0
0
,(15)
where e:= β[αγ)c]
2βγ,f:= β(2β2γ2βγε)
4β2γ2, and g:= β(2β2εγ2εβγ)
4β2γ2. A general
solution is
uol
1
uol
2
Xol
1
Xol
2
=
C1C2C3C4
C1C2C3C4
C2
f+gC1
f+gC4
fg
C3
fg
C2
f+gC1
f+g
C4
fgC3
fg
sin(tf+g)
cos(tf+g)
sin(tfg)
cos(tfg)
0
0
e
f+g
e
f+g
,(16)
where C1,...,C
4are constants of integration. Using the boundary conditions
uol
i(T ) =Λand Xi(0)=0, i=1,2, the resolution of the system gives (11) and
(12). To solve firm i’s R&D effort rate and stock in a CLNE, we differentiate (6) with
respect to time and substitute for ˙
λifrom (8), i=1,2, and simplify, and recall (1), to
get
˙ucl
1
˙ucl
2
˙
Xcl
1
˙
Xcl
2
=
002f22fg/β
002fg/β 2f2
10 0 0
01 0 0
ucl
1
ucl
2
Xcl
1
Xcl
2
2ef /β
2ef /β
0
0
.(17)
A general solution of the system is given by
ucl
1
ucl
2
Xcl
1
Xcl
2
=
D1D2D3D4
D1D2D3D4
D2
2(f g)/β D1
2(f g)/β
D4
2(f +g)/β D3
2(f +g)/β
D2
2(f g)/β
D1
2(f g)/β
D4
2(f +g)/β D3
2(f +g)/β
×
sin(t2(f g)/β)
cos(t2(f g)/β)
sin(t2(f +g)/β)
cos(t2(f +g)/β)
0
0
e
f+g
e
f+g
,(18)
where D1,...,D
4are constants of integration. Using the boundary conditions
ucl
i(T ) =Λand Xi(0)=0, i=1,2, the resolution of the system gives (13)
and (14).
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By assumption, both Ψand {cos[Φ(Tt)]
cos(ΦT ) 1}are non-negative on the right-hand
side of (12). Therefore, a sufficient condition for a feasible value of Xol
i|γ<β,ε(t)
in (12) is that the planning horizon should not be too large, that is, T<π/2Φ. Simi-
larly, a sufficient condition for a feasible value of Xcl
i|γ<β,ε(t) in (14)isT<π/2Γ.
Corollary 4.1 For every 0γ<βand 0ε1, firm i’s pricing policy,i=1,2,
is given by
pol
i|γ<β,ε(t) =1
(2βγ)α+βc(1+ε)Ψcos[Φ(T t)]
cos(ΦT ) 1
+Λsin(Φt )
Φcos(ΦT )  (19)
in an OLNE,and
pcl
i|γ<β,ε(t) =1
(2βγ)α+βc(1+ε)Ψcos[Γ(T t)]
cos(Γ T ) 1
+Λsin(Γ t)
Γcos T )  (20)
inaCLNE.
Proof Plugging the RHS of (12) and (14) respectively for both players in (5) and
simplifying gives (19) and (20).
Note that if cis very close to α/(β γ) in (19) and (20), then Ψ→ 0. Since
neither Λnor Φdepends on α, then for high values of α,pol
i(t) is positive for every
t>0. That is, a feasible solution for the sales price is guaranteed whenever the firms’
potential market and initial marginal production cost are both sufficiently large.
5 Comparative Analysis
In this section, we prove the following:
(i) When the products are not close substitutes, firm i’s scientific knowledge is
greater under perfect spillovers than under full appropriability in both an OLNE
and in a CLNE. Correspondingly, firm is pricing policy is lower under perfect
spillovers than under full appropriability in both an OLNE and in a CLNE.
(ii) Firm i’s scientific knowledge in an OLNE is higher than in a CLNE. Moreover,
firm i’s pricing policy in an OLNE is lower than in a CLNE.
Consider the following function:
F(x,t):=Ψcos[x(T t)]
cos(xT ) 1+Λsin(xt )
xcos(xT ) .(21)
We assume that 0 tTand x<π/2T, and hence Fis well-defined, and all of its
trigonometric components are non-negative.
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Lemma 5.1 Fis non-negative for every value (x, t ).
Proof Since x(T t) xT < π/2, then cos[x(T t)]≥cos(xT ), and (21) is non-
negative.
Lemma 5.2 Fis increasing in xfor every fixed value of t.
Proof Differentiate Fin x, then
∂F(x,t)
∂x =Ψ
∂xcos x(T t)
cosxT +Λ
∂xsin xt
xcosxT .(22)
The first derivative in (22)is
∂x
cos[x(T t)]
cos(xT )
=(T t)sin[x(T t)]cos(xT ) +Tsin(x T ) cos[x(T t)]
cos2(xT ) .(23)
The denominator is positive, and the numerator satisfies
(T t)sinx(T t)cos(xT ) +Tsin(x T ) cosx(T t)
(T t)sin(xT ) cosx(T t)sinx(T t)cos(x T )
=(T t)sin(xT ) 0.(24)
The second derivative in (22)is
∂xsin(xt)
xcos(xT ) =xt cos(xt ) cos(xT ) sin(xt)[cos(xT ) xT sin(xT )]
x2cos2(xT ) .(25)
The denominator is positive, and the numerator satisfies
xt cos(xt ) cos(xT ) sin(xt )cos(x T ) xT sin(xT )
=xt cos(xt ) cos(xT ) sin(xt ) cos(xT ) +xT sin(x t ) sin(xT )
xtcos(x t) cos(xT ) +sin(x t) sin(xT )sin(x t) cos(xT )
=xt cosx(T t)sin(x t) cos(xT )
>xt
cosx(T t)cos(xT )xtcos(xT ) cos(xT )=0.(26)
Since Ψ, Λ > 0, then from (24) and (26), the derivative (22) is positive.
Lemma 5.3 For every 0γ<βand 0ε1, we have
ΦΓ. (27)
Proof By the definitions of Φand Γ(which are positive), we obtain
Φ
Γ2
=(2β+γ)(2βγ)
2(2β2γ2βγε) =4β2γ2
4β22γ22βγε 4β2γ2
4β22γ21.(28)
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Given α,βand ε, we now assume that the time horizon Tis set such that T<
π/2Φπ/2Γ, and hence, the results of Lemmas 5.1 and 5.2 are valid for x=Φ
and x=Γin F.
The two following results show that firm is scientific knowledge and pricing pol-
icy are, respectively, greater and lower under perfect spillovers than under full appro-
priability in both an OLNE and a CLNE.
Theorem 5.1 Consider the OLNE case.For any given α,β,εand c,there exists a
threshold γ
=(7/2+211/2such that for any γγ
,it holds that
Zol
i|γ<β,ε=0(t) Zol
i|γ<β,ε=1(t), i =1,2,(29)
for 0tTwith T<π/2β.Conversely,if γ>γ
,(19)still holds if T>2/3α,
which is only possible if α>4m3
β/3π.
Consider the CLNE case.For any given α,β,εand c,there exists a threshold
˜γ=(31such that for any γ≤˜γ,it holds that
Zcl
i|γ<β,ε=0(t) Zcl
i|γ<β,ε=1(t), i =1,2,(30)
for 0tTwith T<π/2β.Otherwise,the converse inequality holds.
Proof We first consider the OLNE case. As shown in Lemma 5.2, the functions
f1(x) =cos[x(T t)]
cos(xT ) and f2(x ) =sin(xt)
xcos(xT ) are increasing in xand are greater than
1at[0,π/2T).Wehave
Zol
i|γ<β,ε=0(t) =αγ)c
βγf1t,β(β γ)
2βγ1
+mβ(2β2γ2)
(2β+γ)
γ)(2βγ)f2t,β(β γ)
2βγ,(31)
Zol
i|γ<β,ε=1(t) =αγ)c
βγf1t,2β(β γ)
2βγ1
+m2β(β γ)
2βγf2t,2β(β γ)
2βγ.(32)
If 0 γ(7/2+211/2 , then mβ(2β2γ2)
(2β+γ)
γ )(2βγ) m2β(βγ)
2βγ.
In this case, Zol
i|γ<β,ε=0(t) Zol
i|γ<β,ε=1(t). Conversely, if γ>(
7/2+2
11/2, we get: limγβZol
i|γ<β,ε=0(t) =α(T t/2)t +mβ t / 3 and
limγβZol
i|γ<β,ε=1(t) =2α(T t/2)t. Thus, for γ=β,Zol
i|ε=0(t) Zol
i|ε=1(t)
iff t2(T /3α).ForT2 /3α, we get 2(T /3α) > T , that is,
t2(T /3α) for every tT. The difference Zol
i|γ<β,ε=1(t) Zol
i|γ<β,ε=0(t)
is decreasing in γfor every tif γ>(
7/2+211/2and t2(T
/3α). Thus, given γ<β, for every twe have Zol
i|γ<β,ε=1(t) Zol
i|γ<β,ε=0(t)
Zol
i|γ=β,ε=1(t) Zol
i|γ=β,ε=0(t) 0. We now turn to the CLNE case. Also, we have
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Zcl
i|γ<β,ε=0(t) =αγ)c
γ) f12β(β γ)(2β2γ2)
(2β+γ)(2βγ)
21
+mβ(2β2γ2)
2γ)(2β+γ)f22β(β γ)(2β2γ2)
(2β+γ)(2βγ)
2,(33)
Zcl
i|γ<β,ε=1(t) =αγ)c
γ) f12β(β γ)
(2βγ) 1
+mβf22β(β γ)
(2βγ) .(34)
If 0 γ(31, then 2β(βγ)(2β2γ2)
(2β+γ )(2βγ)
22β(βγ)
(2βγ) as well as
mβ(2β2γ2)
2γ )(2β+γ) mβ, and therefore Zcl
i|γ<β,ε=0(t) Zcl
i|γ<β,ε=1(t). Other-
wise, 2β(βγ)(2β2γ2)
(2β+γ )(2βγ)
22β(βγ)
(2βγ) and mβ(2β2γ2)
2γ )(2β+γ) mβhold. Both
2β(βγ)(2β2γ2)
(2β+γ )(2βγ)
2and 2β(βγ)
(2βγ) are decreasing in γ, and they coincide at γ=
(31. Thus, their maximal value is obtained when γ=0 is substituted in
2β(βγ)
(2βγ) , that is, β.IfT< π
2β, then β< π
2T, and thus 2β(βγ )(2β2γ2)
(2β+γ )(2βγ)
2and
2β(βγ)
(2βγ) are necessarily lower than π/2Tfor every 0 γ<β.
Note that T<π/2βis a feasibility condition as defined above in an OLNE
and a CLNE, where βis the maximum value of Φand Γ, respectively, which is
obtained for γ=0 and ε=1. A necessary and sufficient condition for full R&D
appropriability to generate lower scientific knowledge than perfect R&D spillovers
in an OLNE requires imperfect product substitutability (i.e., γγ
0.51β). Con-
versely, a sufficient condition for perfect R&D spillovers to generate greater scientific
knowledge than full R&D appropriability is that the time horizon is not too short.
This condition is possible whenever the potential market is sufficiently large, that
is, if α>4m3
β/3π. However, a necessary and sufficient condition for full R&D
appropriability to generate less scientific knowledge than perfect R&D spillovers
in a CLNE is that the products should not be excessively close substitutes (i.e.,
γ≤˜γ0.73β).
In an OLNE, a firm’s incentive to invest in R&D depends (positively) on the firm’s
additional profit resulting from a marginal decrease in its production costs due to an
increase in its R&D stock. This additional profit is equivalent to the firm’s consumer
demand. For imperfect product substitutes, a firm’s demand due to free-riding on
its rival’s R&D stock under perfect spillovers is greater than that due to the firm’s
own R&D stock under full appropriability. As a result, the greater its rival’s R&D
stock, the greater the firm’s incentive to invest in R&D. This pattern of strategic
interactions is akin to the action–reaction concept, which is well supported in the
available empirical literature on R&D competition (for a review, see [20]). A similar
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pattern can be observed for close product substitutes due to R&D accumulation over
a sufficiently large time horizon.
In a CLNE, a firm’s additional profit resulting from a marginal decrease in its own
production costs due to an increase in the firm’s own R&D stock is partially offset by
the reduction in consumer demand that results from a price reduction induced by the
free-riding behavior of the rival firm. For imperfect product substitutes, the overall
additional profit is still positive, and hence the firm’s incentive to invest in R&D
is greater under perfect spillovers than under full appropriability. For close product
substitutes, however, perfect spillovers are more detrimental to the firm’s consumer
demand than full appropriability, and hence the greater the spillover effects, the lower
a firm’s incentive to invest in R&D. The dynamic effects related to the OLNE case are
therefore absent, and both firms set their respective R&D efforts close to the levels
that would prevail in a static equilibrium.
Corollary 5.1 Consider the OLNE case.Whenever either γγ
or both α>
4m3
β/3πand T>2 /3α,it holds that
pol
i|γ<β,ε=0(t) pol
i|γ<β,ε=1(t), i =1,2,(35)
for 0tTwith T<π/2β.
Consider the CLNE case.For any γ≤˜γ,it holds that
pcl
i|γ<β,ε=0(t) pcl
i|γ<β,ε=1(t), i =1,2,(36)
for 0tTwith T<π/2β.Otherwise,the converse inequality holds.
Proof Inequalities (35) and (36) follow from the proof of Theorem 5.1.
Therefore, if products are not close (not too close) substitutes, firm i’s pricing
policy is lower under perfect spillovers than under full appropriability in an OLNE
(in a CLNE).
We now demonstrate that firm i’s scientific knowledge and pricing policy in an
OLNE are, respectively, higher and lower than in a CLNE.
Theorem 5.2 Let 0tT.For every 0γ<βand 0ε1, then
Zol
i(t) Zcl
i(t), i =1,2.(37)
Proof Using Lemmas 5.1,5.2 and (27), we get
F(Φ,t)F(Γ,t)0,(38)
for every t.From(12) and (14), for every 0 tT,wehaveXol
i|γ<β,ε(t) =F(Φ,t)
and Xcl
i(t) =F(Γ,t),i=1,2, respectively. Thus, using (2), we obtain (37)for0
γ<β.
Whatever the R&D spillovers (0 ε1), if the products are not substitutes, the
OLNE and CLNE do not differ in terms of R&D accumulation policies because there
is no competitive interaction between the firms. As a result, the firms should obtain
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equivalent payoffs in OLNE and CLNE. Conversely, if the products are imperfect
substitutes (i.e., 0 <γ <β), the duopolist firms develop greater scientific knowledge
with OLNE rather than CLNE strategies. In a CLNE, each firm updates its investment
decision over time by accounting for the fact that a marginal increase in its own R&D
stock reduces the competitor’s sales price, which in turn is detrimental to the former’s
demand, which can be observed by comparing firm i’s costate equations in (7)for
OLNE and (8) for CLNE. As a result, the firms have a greater incentive to free-ride
when they play CLNE strategies than when they play OLNE strategies. A similar
result is reported in [28] in the context of a public good capital accumulation model.
Corollary 5.2 Let 0tT.For every 0γ<βand 0ε1, then
pol
i(t) pcl
i(t), i =1,2.(39)
Proof By formulas (19) and (20), for every 0 tT,wehave
pol
i(t) =1
2βγα+βc(1+ε)F(Φ,t ),i=1,2,(40)
pcl
i(t) =1
2βγα+βc(1+ε)F(Γ, t),i=1,2,(41)
respectively. The coefficients of Fin the formulas of prices are negative, and there-
fore by (38), inequality (39) is satisfied for 0 γ<β.
Firm i’s sales price is lower in an OLNE than in a CLNE whenever the products
are imperfect substitutes. That is, a greater investment in R&D results in a lower sales
price, which is offset by increased demand. In the OLNE case, firm is sales’ price
converges to a level below that in the CLNE case. Therefore, the competitors gen-
erate more sales if they pre-commit to their respective plans rather than make plans
contingent on the value of the state variables. An OLNE can then be considered as a
commitment by the competitors to maintaining relatively high R&D efforts and low
price levels. In this regard, the adoption of an OLNE is clearly more pro-competitive
than that of a CLNE. Only when the products under consideration are not substitutes,
regardless of the level of spillovers among the competitors do the OLNE and CLNE
lead to identical pricing policies.
6 Numerical Illustrations
To compare the firms’ terminal scientific knowledge levels and payoffs under perfect
R&D spillovers and full R&D appropriability, we select the following parameters
(Table 1). The game duration is set to a sufficiently large extent, 20 periods, such that
2/3α<T <π/2β.
Figure 1presents the difference between scientific knowledge under perfect R&D
spillovers and full R&D appropriability over time in an OLNE (a) and in a CLNE (b).
In an OLNE, perfect spillovers always generate greater scientific knowledge than full
appropriability over time, notably for low product substitutability. In contrast, perfect
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Tab l e 1 Baseline parameters Parameter αβγ εc m
Value 5 000 0.5 [0, 0.475] [0, 1] 5 000 50
Fig. 1 Differences in scientific knowledge between perfect spillovers and full appropriability over time in
OLNE and CLNE
Fig. 2 Differences in payoffs between perfect spillovers and full appropriability over time in OLNE and
CLNE
spillovers only yield greater scientific knowledge than full appropriability in a CLNE
for limited product substitutability.
Figure 2presents the differences in the cumulative payoffs under perfect R&D
spillovers and full R&D appropriability over time in an OLNE (Fig. 2.a) and in a
CLNE (Fig. 2.b). In an OLNE, perfect spillovers result in greater payoffs than in
the case of full appropriability after a short initial time interval, notably for lower
levels of product substitutability. In contrast, perfect spillovers give rise to greater
payoffs than in the case of full appropriability only in a CLNE for limited product
substitutability.
In Fig. 3.a, the terminal levels of scientific knowledge in an OLNE and a CLNE
are equivalent if the products are not substitutes, regardless of the level of spillovers
among the competitors. If the products are imperfect substitutes, the two decision
rules lead to quite different levels of terminal scientific knowledge. That is, the greater
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Fig. 3 Differences in terminal scientific knowledge and cumulative payoffs between OLNE and CLNE
the substitutability between the competing products is, the greater the difference be-
tween OLNE and CLNE terminal scientific knowledge becomes. The difference is
substantially amplified by the magnitude of spillovers; that is, the larger the spillover
effect, the lower the terminal scientific knowledge in a CLNE compared with that in
an OLNE. For close substitute products with perfect spillovers, the CLNE terminal
scientific knowledge is less than 10 % of the OLNE terminal scientific knowledge.
In Fig. 3.b, the percentage of variation between OLNE payoffs and CLNE payoffs
is highest for imperfect substitute products, notably for perfect R&D spillovers. In
this case, the cumulative profits in an OLNE can reach a level 14 % above those in a
CLNE. As expected, if the products are not substitutes, the payoffs in an OLNE and
a CLNE are equal.
We further illustrate our results by focusing on the case in which the difference
between OLNE and CLNE payoffs is maximal, with γ=0.325, and comparing the
OLNE (CLNE) solutions with full R&D appropriability and perfect R&D spillovers,
respectively denoted OL;1 and OL;0 (CL;1 and CL;0). Figure 4.a shows that firm i’s
R&D stock increases concavely over time in both OLNE and CLNE. Firms accumu-
late a slightly greater R&D stock in an OLNE under perfect spillovers than under
full appropriability. Conversely, firms develop a much larger R&D stock in a CLNE
under full appropriability than under perfect spillovers.
Overall, firm i’s production cost and price are both significantly (slightly) lower in
an OLNE (CLNE) under perfect R&D spillovers than under full R&D appropriability.
In the case of imperfectly substitutable products, R&D spillovers are more profitable
to customers than full R&D appropriability in both an OLNE and a CLNE. However,
an OLNE is more socially efficient than a CLNE because it is more profitable than
a CLNE for both customers and competing firms, most notably under perfect R&D
spillovers.
7 Conclusions
A primary goal of the present study is to investigate the simultaneous pricing and
R&D decisions of competing firms in a duopoly market. Due to the cumulative na-
ture of R&D activity, we adopt a dynamic framework to examine how competitors’
decisions change over time. The suggested model contains four key assumptions:
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Fig. 4 Firm i’s R&D stock, production cost and sales price with OLNE and CLNE over time
spillovers are free and stock-dependent, firms are identical and R&D efforts do not
interact with pricing. We compare two decision rules based on the Nash equilibrium
concept (OLNE and CLNE) with respect to their ability to generate greater payoffs.
The results obtained in this study indicate that the dynamic nature of the R&D
accumulation process has been underestimated in the literature on R&D competi-
tion. For imperfectly substitutable products, free R&D spillovers generate greater
scientific knowledge than full R&D appropriability in both an OLNE and a CLNE.
Consumer surplus and payoffs are also greater. These results clearly contradict the
intuition of simpler static or two-stage models, according to which R&D incentives
decrease with lower appropriability, which is always invalid under imperfect product
substitutability. However, our results moderate those of [46] because we show that
greater spillovers actually preclude higher R&D investments for close product sub-
stitutes in an OLNE under sufficiently short time horizons, and the same holds in a
CLNE.
Our results also show that the decision rule adopted by firms is of primary impor-
tance in ensuring that the results of [49], according to which dynamic effects may re-
sult in lower production costs in the competitive setting under perfect spillovers than
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under full appropriability, hold. Under imperfect product substitutability, an OLNE
strategy for R&D effort and pricing is more suitable than a CLNE strategy to raise
scientific knowledge, firms’ payoffs and consumer surplus. That is, if the competitors
pre-commit to their respective plans rather than make plans contingent on the value of
the rival’s R&D stock, their R&D stock should be greater than if they decide to do the
opposite. The lower R&D stocks and higher prices, notably under perfect spillovers,
mean that the CLNE strategies are more myopic than OLNE strategies.
By making its current R&D stock observable by its rival firm over the entire game,
a firm faces reduced incentives to accumulate R&D, which weakens its ability to
compete on price on the product market. Hence, it is not in the interest of a firm to
adopt a CLNE if the other firm plays an OLNE, as it is a trivially dominated op-
tion. If both firms adopt CLNE strategies, the price competition is softened, which
is neither in their joint interest nor in the interest of the consumers. Due to the exis-
tence of spillovers in R&D stocks, the mutual observability of current R&D stocks
therefore involves a loss in payoffs that makes Markovian CLNE strategies irrele-
vant. A more efficient strategy for a firm to develop a price competitive advantage
is thus to maintain the secrecy of its own R&D stock level over time. Therefore, an
OLNE represents a beneficial option for both firms and consumers. To a certain ex-
tent, commitment strategies in R&D competition can be considered as substitutes for
cooperation whenever the latter cannot be envisioned.
As an important topic for future research, our non-tournament game with R&D
accumulation can be extended to account for uncertain R&D outcomes, and therefore
for uncertain benefit from R&D spillovers.
Acknowledgements The authors are grateful to Gary Erickson and Steffen Jørgensen for constructive
suggestions on an early draft. The paper was written while the first author was visiting the Department of
Logistics and Operations Management at HEC-Montreal in Canada.
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... Expression (15) verifies that each firm's non-cooperative investments for abatement technology innovation depends on the degree of the spillover β, market competition intensity λ and emission tax τ . Note that, the condition in equation (15) is obtained by rearranging Π i (k n , k n ) ≥ 0. Furthermore, if the condition in (15) is not satisfied, then the symmetric solution k n is not unique. ...
... Let Π n denotes the firm i ′ s relative profit function under an optimal noncooperative investments for abatement technology innovation. Substituting both expressions (15) and (16) into (8), yields ...
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