Content uploaded by Fouad El Ouardighi

Author content

All content in this area was uploaded by Fouad El Ouardighi on Aug 11, 2015

Content may be subject to copyright.

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

Your article is protected by copyright and all

rights are held exclusively by Springer Science

+Business Media New York. This e-offprint is

for personal use only and shall not be self-

archived in electronic repositories. If you wish

to self-archive your article, please use the

accepted manuscript version for posting on

your own website. You may further deposit

the accepted manuscript version in any

repository, provided it is only made publicly

available 12 months after official publication

or later and provided acknowledgement is

given to the original source of publication

and a link is inserted to the published article

on Springer's website. The link must be

accompanied by the following text: "The final

publication is available at link.springer.com”.

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

Author's personal copy

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 [2–4]. 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 [12–16]. A supportive empirical evaluation of this theory can be found in [17].

Author's personal copy

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. [19–21]. 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 [27–29]. 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. [22–26].

5Open-loop strategies provide a useful benchmark for assessing the strategic effects related to Markovian

strategies [30].

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 629

•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., [31–35].

7A related stream of literature, not discussed here, investigates the impact of R&D competition on market

structure; e.g., [36–40].

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.

Author's personal copy

630 J Optim Theory Appl (2014) 161:626–647

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.

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 631

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 t≥0, 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 ≤ε≤1 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].

Author's personal copy

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):= αi−βipi(t) +γipj(t ) ≥0,

where αi>0, βi>0 and 0 ≤γi<β

iare parameters, i, j =1,2,i = j. We assume

symmetric parameters for both ﬁrms, that is, αi≡α, βi≡β,γi≡γ,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 )) =ci−Zi(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,ui≥0Πi=T

0pi(t) −c+Zi(t)Si(t) −ui(t )2/2dt +miSi(T ), (3)

where mi>0 denotes the marginal salvage value of ﬁrm i’s 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].

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 633

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=(pi−c+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 i’s 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=0⇒pi=(2β+γ )(α +βc) −β[(2β+γε)X

i+(2βε +γ)X

j]

4β2−γ2,(5)

Hi

ui=0⇒ui=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 i’s 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)[α−(β −γ)c]+(2β2−γ2−βγε)Xi+(2β2ε−γ2ε−βγ)Xj}

4β2−γ2(7)

in an OLNE, and

˙

λi=−Hi

Xi−Hi

pjpjXi=−Si+βγ(2βε +γ)π

i

4β2−γ2(8)

in a CLNE, where

πi≡pi−c+Zi

=(2β+γ)[α−(β −γ)c]+(2β2−γ2−βγ ε)Xi+(2β2ε−γ2ε−βγ)Xj

4β2−γ2,(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.

Author's personal copy

634 J Optim Theory Appl (2014) 161:626–647

The transversality conditions are

λi(T ) =mβ (2β2−βγ ε −γ2)

4β2−γ2,(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 ˙

λi≤0in(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

2β2−γ2,1]. 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(2β2−γ2−βγ ε)

4β2−γ2Si≤0 in a CLNE,

where 0 <2(2β2−γ2−βγ ε)

4β2−γ2<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 0≤γ<βand 0≤ε≤1, ﬁrm i’s R&D investment and

R&D stock are,respectively,

uol

i|γ<β,ε(t) =1

cos(ΦT ) β[α−(β −γ)c]sin[Φ(T −t)]

√β(1+ε)(β −γ)(2β−γ) +Λcos(Φt),(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(Γ T ) 2β(2β2−γ2−βγε)[α−(β −γ)c]sin[Γ(T −t)]

(1+ε)(β −γ)(2β+γ)(2β−γ)

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)(β−γ)

2β−γ,Ψ:= α−(β−γ)c

(1+ε)(β−γ),Λ:= mβ (2β2−βγ ε−γ2)

4β2−γ2,

Γ:= 2β(1+ε)(β−γ )(2β2−γ2−βγ ε)

(2β+γ )(2β−γ)

2,and ol and cl denote “open-loop” and “closed-

loop,” respectively,i=1,2.

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 635

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

⎤

⎥

⎥

⎦=⎡

⎢

⎢

⎣

00−f−g

00−g−f

10 0 0

01 0 0

⎤

⎥

⎥

⎦⎡

⎢

⎢

⎣

uol

1

uol

2

Xol

1

Xol

2

⎤

⎥

⎥

⎦−⎡

⎢

⎢

⎣

e

e

0

0

⎤

⎥

⎥

⎦,(15)

where e:= β[α−(β−γ)c]

2β−γ,f:= β(2β2−γ2−βγε)

4β2−γ2, and g:= β(2β2ε−γ2ε−βγ)

4β2−γ2. A general

solution is

⎡

⎢

⎢

⎣

uol

1

uol

2

Xol

1

Xol

2

⎤

⎥

⎥

⎦=⎡

⎢

⎢

⎢

⎣

C1C2−C3−C4

C1C2C3C4

C2

√f+g−C1

√f+g−C4

√f−g

C3

√f−g

C2

√f+g−C1

√f+g

C4

√f−g−C3

√f−g

⎤

⎥

⎥

⎥

⎦⎡

⎢

⎢

⎣

sin(t√f+g)

cos(t√f+g)

sin(t√f−g)

cos(t√f−g)

⎤

⎥

⎥

⎦

−⎡

⎢

⎢

⎣

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

⎤

⎥

⎥

⎦=⎡

⎢

⎢

⎣

00−2f2/β −2fg/β

00−2fg/β −2f2/β

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

−D1−D2D3D4

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(t√2(f −g)/β)

cos(t√2(f −g)/β)

sin(t√2(f +g)/β)

cos(t√2(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).

Author's personal copy

636 J Optim Theory Appl (2014) 161:626–647

By assumption, both Ψand {cos[Φ(T−t)]

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 0≤γ<βand 0≤ε≤1, ﬁrm i’s pricing policy,i=1,2,

is given by

pol

i|γ<β,ε(t) =1

(2β−γ)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

(2β−γ)α+βc−(1+ε)Ψcos[Γ(T −t)]

cos(Γ T ) −1

+Λsin(Γ t)

Γcos(Γ 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 i’s 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 ≤t≤Tand x<π/2T, and hence Fis well-deﬁned, and all of its

trigonometric components are non-negative.

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 637

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 0≤γ<βand 0≤ε≤1, we have

Φ≥Γ. (27)

Proof By the deﬁnitions of Φand Γ(which are positive), we obtain

Φ

Γ2

=(2β+γ)(2β−γ)

2(2β2−γ2−βγε) =4β2−γ2

4β2−2γ2−2βγε ≥4β2−γ2

4β2−2γ2≥1.(28)

Author's personal copy

638 J Optim Theory Appl (2014) 161:626–647

Given α,βand ε, we now assume that the time horizon Tis set such that T<

π/2Φ≤π/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 i’s 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+√2−1−1/√2)β such that for any γ≤γ

,it holds that

Zol

i|γ<β,ε=0(t) ≤Zol

i|γ<β,ε=1(t), i =1,2,(29)

for 0≤t≤Twith T<π/2√β.Conversely,if γ>γ

,(19)still holds if T>2mβ/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=(√3−1)β such 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 0≤t≤Twith T<π/2√β.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) =α−(β −γ)c

β−γf1t,β(β −γ)

2β−γ−1

+m√β(2β2−γ2)

(2β+γ)

√(β −γ)(2β−γ)f2t,β(β −γ)

2β−γ,(31)

Zol

i|γ<β,ε=1(t) =α−(β −γ)c

β−γf1t,2β(β −γ)

2β−γ−1

+m√2β(β −γ)

√2β−γf2t,2β(β −γ)

2β−γ.(32)

If 0 ≤γ≤(7/2+√2−1−1/√2)β , then m√β(2β2−γ2)

(2β+γ)

√(β−γ )(2β−γ) ≤m√2β(β−γ)

√2β−γ.

In this case, Zol

i|γ<β,ε=0(t) ≤Zol

i|γ<β,ε=1(t). Conversely, if γ>(

7/2+√2−

1−1/√2)β, we get: limγ→β−Zol

i|γ<β,ε=0(t) =α(T −t/2)t +mβ t / 3 and

limγ→β−Zol

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 t≤2(T −mβ/3α).ForT≥2mβ /3α, we get 2(T −mβ /3α) > T , that is,

t≤2(T −mβ/3α) for every t≤T. 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+√2−1−1/√2)β and t≤2(T −

mβ/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

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 639

Zcl

i|γ<β,ε=0(t) =α−(β −γ)c

(β −γ) f12β(β −γ)(2β2−γ2)

(2β+γ)(2β−γ)

2−1

+mβ(2β2−γ2)

2(β −γ)(2β+γ)f22β(β −γ)(2β2−γ2)

(2β+γ)(2β−γ)

2,(33)

Zcl

i|γ<β,ε=1(t) =α−(β −γ)c

(β −γ) f12√β(β −γ)

(2β−γ) −1

+mβf22√β(β −γ)

(2β−γ) .(34)

If 0 ≤γ≤(√3−1)β, then 2β(β−γ)(2β2−γ2)

(2β+γ )(2β−γ)

2≤2√β(β−γ)

(2β−γ) as well as

mβ(2β2−γ2)

2(β−γ )(2β+γ) ≤m√β, and therefore Zcl

i|γ<β,ε=0(t) ≤Zcl

i|γ<β,ε=1(t). Other-

wise, 2β(β−γ)(2β2−γ2)

(2β+γ )(2β−γ)

2≥2√β(β−γ)

(2β−γ) and mβ(2β2−γ2)

2(β−γ )(2β+γ) ≥m√βhold. Both

2β(β−γ)(2β2−γ2)

(2β+γ )(2β−γ)

2and 2√β(β−γ)

(2β−γ) are decreasing in γ, and they coincide at γ=

(√3−1)β. Thus, their maximal value is obtained when γ=0 is substituted in

2√β(β−γ)

(2β−γ) , that is, √β.IfT< π

2√β, then √β< π

2T, and thus 2β(β−γ )(2β2−γ2)

(2β+γ )(2β−γ)

2and

2√β(β−γ)

(2β−γ) are necessarily lower than π/2Tfor every 0 ≤γ<β.

Note that T<π/2√βis 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., γ≤γ

≈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≤˜γ≈0.73β).

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

Author's personal copy

640 J Optim Theory Appl (2014) 161:626–647

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 γ≤γ

or both α>

4m3

√β/3πand T>2mβ /3α,it holds that

pol

i|γ<β,ε=0(t) ≥pol

i|γ<β,ε=1(t), i =1,2,(35)

for 0≤t≤Twith T<π/2√β.

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 0≤t≤Twith T<π/2√β.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 0≤t≤T.For every 0≤γ<βand 0≤ε≤1, 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 ≤t≤T,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 ≤ε≤1), 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

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 641

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 0≤t≤T.For every 0≤γ<βand 0≤ε≤1, then

pol

i(t) ≤pcl

i(t), i =1,2.(39)

Proof By formulas (19) and (20), for every 0 ≤t≤T,wehave

pol

i(t) =1

2β−γα+βc−(1+ε)F(Φ,t ),i=1,2,(40)

pcl

i(t) =1

2β−γα+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 i’s 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

2mβ/3α<T <π/2√β.

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

Author's personal copy

642 J Optim Theory Appl (2014) 161:626–647

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

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 643

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:

Author's personal copy

644 J Optim Theory Appl (2014) 161:626–647

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

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 645

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.

References

1. Spence, M.: Cost reduction, competition and industry performance. Econometrica 52, 101–121

(1984)

2. D’Aspremont, C., Jacquemin, A.: Cooperative and noncooperative R&D in duopoly with spillovers.

Am. Econ. Rev. 78, 1133–1137 (1988)

3. D’Aspremont, C., Jacquemin, A.: Cooperative and noncooperative R&D in duopoly with spillovers:

Erratum. Am. Econ. Rev. 80, 641–642 (1990)

4. Kamien, M.I., Muller, E., Zang, I.: Research joint ventures and R&D cartels. Am. Econ. Rev. 82,

1293–1306 (1992)

5. Kogut, B.: The stability of joint-ventures: reciprocity and competitive rivalry. J. Ind. Econ. 38, 183–

198 (1989)

6. Das, T.K., Teng, B.S.: Instabilities of strategic alliances: an internal tensions perspective. Organ. Sci.

11, 77–101 (2000)

7. Deeds, D.L., Hill, C.W.L.: An examination of opportunistic action within research alliances: evidence

from the biotechnology industry. J. Bus. Venturing 14, 141–163 (1998)

8. Oxley, J.E., Sampson, R.C.: The scope and governance of international R&D alliances. Strateg.

Manag. J. 25, 723–749 (2004)

9. Amaldoss, W., Staelin, R.: Cross-function and same-function alliances: how does alliance structure

affect the behavior of partnering ﬁrms? Manag. Sci. 56, 302–317 (2010)

10. Barney, J., Hansen, M.: Trustworthiness as a source of competitive advantage. Strateg. Manag. J. 15,

175–190 (1994)

Author's personal copy

646 J Optim Theory Appl (2014) 161:626–647

11. Utterback, J.M., Abernathy, W.J.: A dynamic model of process and product innovation. Omega 3,

639–656 (1975)

12. Hayes, R.H., Wheelwright, S.C.: Link manufacturing process and product life cycles. Harv. Bus. Rev.

57, 133–140 (1979)

13. Hayes, R.H., Wheelwright, S.C.: The dynamics of process-product life cycles. Harv. Bus. Rev. 57,

127–136 (1979)

14. Ettlie, J.E.: Product-process development integration in manufacturing. Manag. Sci. 41, 1224–1237

(1995)

15. Pisano, G.: The Development Factory: Unlocking the Potential of Process Innovation. Harvard Busi-

ness School Press, Boston (1997)

16. Damanpour, F., Gopoalakrishnan, S.: Organizational adaptation and innovation: the dynamics of

adopting innovation types. In: Brockhoff, K., Chakrabarti, A., Hauschild, J. (eds.) The Dynamics

of Innovation, pp. 57–80. Springer, Berlin (1999)

17. Butler, J.E.: Theories of technological innovation as useful tools for corporate strategy. Strateg.

Manag. J. 9, 15–29 (1988)

18. Martin, S.: R&D joint ventures and tacit product market collusion. Eur. J. Polit. Econ. 11, 733–741

(1995)

19. Reinganum, J.F.: Dynamic games of innovation. J. Econ. Theory 25, 21–41 (1981)

20. Doraszelski, U.: An R&D race with knowledge accumulation. Rand J. Econ. 34, 20–42 (2003)

21. Kim, B., El Ouardighi, F.: Supplier–manufacturer collaboration on new product development. In:

Jørgensen, S., Vincent, T., Quincampoix, M. (eds.) Advances in Dynamic Games and Applications to

Ecology and Economics, pp. 527–545. Birkhauser, Boston (2007)

22. Erickson, G.M.: R&D spending and product pricing. Optim. Control Appl. Methods 11, 269–276

(1990)

23. Bayus, B.L.: Optimal dynamic policies for product and process innovation. J. Oper. Manag. 12, 173–

185 (1995)

24. Saha, S.: Consumer preferences and product and process R&D. Rand J. Econ. 38, 250–268 (2007)

25. Lambertini, L., Mantovani, A.: Process and product innovation by a multiproduct monopolist: a dy-

namic approach. Int. J. Ind. Organ. 27, 508–518 (2009)

26. Chenavaz, R.: Dynamic pricing, product and process innovation. Eur. J. Oper. Res. 222, 553–557

(2012)

27. Ba¸sar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM, Philadelphia (1998)

28. Dockner, E.J., Jørgensen, S., Van Long, N., Sorger, G.: Differential Games in Economics and Man-

agement Science. Cambridge University Press, Cambridge (2000)

29. Long, N.V.: A survey of dynamic games. In: Economics. World Scientiﬁc, Singapore (2010)

30. Fudenberg, D., Levine, D.K.: Open-loop and closed-loop equilibria of dynamic games with many

players. J. Econ. Theory 44, 1–18 (1988)

31. Kamien, M.I., Schwartz, N.L.: Timing of innovations under rivalry. Econometrica 40, 43–60 (1972)

32. Reinganum, J.F.: On the diffusion of new technology: a game theoretic approach. Rev. Econ. Stud.

48, 395–405 (1981)

33. Fudenberg, D., Tirole, J.: Preemption and rent equalization in the adoption of new technology. Rev.

Econ. Stud. 52, 383–401 (1985)

34. Riordan, M.H.: Regulation and preemptive technology adoption. Rand J. Econ. 23, 334–349 (1992)

35. Hoppe, H.C., Lehmann-Grube, U.: Innovation timing games: a general framework with applications.

J. Econ. Theory 121, 30–50 (2005)

36. Loury, G.C.: Market structure and innovation. Q. J. Econ. 93, 395–410 (1979)

37. Flaherty, M.T.: Industry structure and cost-reducing investment. Econometrica 48, 1187–1209 (1980)

38. Lee, T., Wilde, L.L.: Market structure and innovation: a reformulation. Q. J. Econ. 94, 429–436 (1980)

39. Lach, S., Rob, R.: R&D, investment, and industry dynamics. J. Econ. Manag. Strategy 5, 217–249

(1996)

40. Laincz, C.A.: R&D subsidies in a model of growth with dynamic market structure. J. Evol. Econ. 19,

643–673 (2009)

41. Amir, R., Evstigneev, I., Wooders, J.: Noncooperative versus cooperative R&D with endogenous

spillover rates. Games Econ. Behav. 42, 183–207 (2003)

42. Kamien, M.I., Zang, I.: Meet me halfway: research joint ventures and absorptive capacity. Int. J. Ind.

Organ. 18, 995–1012 (2000)

43. Cohen, W.M., Levinthal, D.A.: Innovation and learning: the two faces of R&D. Econ. J. 99, 569–596

(1989)

Author's personal copy

J Optim Theory Appl (2014) 161:626–647 647

44. Grünfeld, L.A.: Meet me halfway but don’t rush: absorptive capacity and strategic R&D investment

revisited. Int. J. Ind. Organ. 21, 1091–1109 (2003)

45. Leahy, D., Neary, J.P.: Absorptive capacity, R&D spillovers, and public policy. Int. J. Ind. Organ. 25,

1089–1108 (2007)

46. Milliou, C.: Endogenous protection of R&D investments. Can. J. Econ. 42, 184–205 (2009)

47. Hammerschmidt, A.: No pain, no gain: an R&D model with endogenous absorptive capacity. J. Inst.

Theor. Econ. 165, 418–437 (2009)

48. Dockner, E.J., Feichtinger, G., Mehlmann, A.: Dynamic R&D competition with memory. J. Evol.

Econ. 3, 145–152 (1993)

49. Cellini, R., Lambertini, L.: Dynamic R&D with spillovers: competition vs. cooperation. J. Econ. Dyn.

Control 33, 568–582 (2009)

50. Amir, R., Wooders, J.: One-way spillovers, endogenous innovator/imitator roles, and research joint

ventures. Games Econ. Behav. 31, 1–25 (2000)

51. Lambertini, L., Lotti, F., Santarelli, E.: Infra-industry spillovers, and R&D cooperation: theory and

evidence. Econ. Innov. New Technol. 13, 311–328 (2004)

52. Griliches, Z.: Patent statistics as economic indicators: a survey. J. Econ. Lit. 28, 1661–1707 (1990)

53. Griliches, Z.: The search for R&D spillovers. Scand. J. Econ. 94, 29–47 (1992)

54. Engwerda, J.: LQ Dynamic Optimization and Differential Games. Wiley, Chichester (2005)

55. Singh, N., Vives, X.: Price and quantity competition in a differentiated duopoly. Rand J. Econ. 15,

546–554 (1984)

56. Dockner, E.J., Feichtinger, G., Jørgensen, S.: Tractable classes of nonzero-sum open-loop Nash dif-

ferential games: theory and examples. J. Optim. Theory Appl. 45, 179–197 (1985)

57. Léonard, D.: The signs of the co-state variables and sufﬁciency conditions in a class of optimal control

problems. Econ. Lett. 8, 321–325 (1981)

Author's personal copy