DISCUSSION PAPERS IN ECONOMICS
Working Paper No. 99-17
R&D Joint Ventures:
The Case of Asymmetric Firms
Antje Baerenss
Department of Economics, University of Colorado at Boulder
Boulder, Colorado
October 1999
Center for Economic Analysis
Department of Economics
University of Colorado at Boulder
Boulder, Colorado 80309
© 1999 Antje Baerenss
R&D Joint Ventures: The Case of Asymmetric Firms
Antje Baerenss
Department of Economics
University of Colorado
October, 1999
Abstract
In this paper, a simple model of R&D cooperation is developed to investigate the private
versus the social incentives for R&D cooperation in the case of asymmetric firms. The
Cournot model with R&D and stochastic innovation before production yields asymmetric
equilibria in the production and research stage. The main findings confirm the result of
previous studies that R&D cooperation increases welfare for most parameter values. However,
welfare may be reduced if cooperation partners are very dissimilar. In the case of strong initial
asymmetries the joint venture has an anti-competitive effect, as it tends to preserve the nearmonopoly position of the low cost firm. In addition, this study shows that the welfare results
and the willingness to cooperate do not only depend on the spillover parameter but also on the
degree of the cost reduction and the initial cost difference between the firms. The results
indicate that the firms have different motives for cooperation. In contrast to the symmetric
case, the desire of the two firms to cooperate does not coincide for a wide range of parameters.
2
I. Introduction
Antitrust regulation, beginning with the Sherman Antitrust Act of 1890, declares illegal every
contract or conspiracy in restraint of trade (Sherman Act, §1) and every attempt to monopolize
any part of the trade or commerce (Sherman Act, §2). In the 1970s and early 1980s, concerns
arose that the antitrust horizontal restraints doctrine unduly discourages desirable collaborative
R&D activity1. In response to these concerns, Congress in 1984 enacted the National
Cooperative Research Act, which was extended to the National Cooperative Research and
Production Act in 1993 to include joint production ventures. The act protects R&D joint
ventures (RJVs) and certain qualifying joint production ventures from the strict application of
antitrust laws. RJVs are judged under the rule of reason, “taking into account all relevant facts
affecting competition, including, but not limited to, effects on properly defined, relevant
research, development, product, process, and service markets.”2
The current treatment of RJVs under antitrust regulation is partly based on research that
shows that R&D joint ventures, in contrast to general collusive behavior, can increase welfare.
Some of that research, such as Ordover and Willig (1985), Grossman and Shapiro (1996), and
Jorde and Teece (1989, 1990), approaches the problem from the legal perspective. A relatively
large body of industrial organization literature relies on economic models to analyze the effects
of cooperative research. The existing models capture the basic characteristics of R&D that
make it different from other firm activities. Knowledge as the outcome of R&D can easily be
transferred to other users. Even with patent protection there is a certain amount of spillover to
rival firms. The innovating firm faces a free-rider problem that tends to reduce R&D effort.
The increase in welfare resulting from RJVs is mainly driven by the internalization of these
spillover effects.
D’Aspremont and Jacquemin (1988) find that in a linear duopoly model with
deterministic R&D process cooperation increases welfare for large spillover parameters only.
Benefits from information sharing such as eliminating duplicate effort are not considered in
their model. Suzumura (1992) generalizes d’Aspremont and Jacquemin’s results for a wide
1
Gellhorn, Kovacic (1994)
2
Areeda, Kaplow (1997)
3
class of oligopoly models. He finds that with large spillovers, not only the non-cooperative
equilibrium R&D level but also the cooperative one is socially insufficient at the margin. In
the absence of spillovers, the cooperative equilibrium R&D level remains socially too small at
the margin, but the non-cooperative outcome is socially excessive. Kamien, Muller, and Zang
(1992) allow for R&D coordination, information sharing, or both. They show that RJVs
increase welfare also for low spillover parameters if the firms share the R&D output in addition
to coordinating their expenditures. These models were designed to study the welfare effects of
RJVs without addressing the firms’ willingness to cooperate. Choi (1993) studies the private
versus the social incentives to form RJVs and introduces stochastic R&D outcomes. Choi
argues that the social benefits from RJVs always exceed the private benefits. In the literature,
there are numerous related models that vary in complexity. However, most of them share one
common feature: firms are identical.
This paper extends previous work in two respects. First, the model allows firms to be
heterogeneous in the initial stage of the game. Second, instead of allowing firms to choose the
optimal size of the innovation, this paper looks at the effect of a given cost reduction on the
incentives to cooperate.
Heterogeneity of firms is captured by ex ante differences in marginal costs. Out of a
variety of aspects in which real-life firms may differ, marginal cost differences were chosen to
keep the model tractable. Adding this feature into a model of R&D cooperation changes the
duopoly game in important respects. Initial asymmetry does not only result in asymmetric
equilibria. It also leads to different motives for R&D spending and introduces new incentives
for and against R&D cooperation. For example, the high cost firm has a smaller market share
at the onset of the game. It is eager to catch up with the low cost firm but it is not obvious
which strategy the firm would choose. It could spend more on R&D trying to outperform the
low cost firm. Alternatively, it could spend less trying to free-ride on the innovation of its rival
or it could try to combine forces with the leader in a cooperative agreement.
The model also takes into account that firms might be faced with the decision on how
much to spend in order to reach a predetermined lower cost level. In other words, a given
innovation can vary in its importance, i.e. the cost reduction could be large or small, and the
firms can decide how intensively to pursue this innovation. Keeping in mind that there is an
industry leader and a follower, reaching the same technological level implies different “prices”
4
from innovation for the two firms. Firm asymmetry and varying innovation size are realistic
characteristics of the innovation process. Incorporating them into the model widens the
spectrum of equilibrium outcomes and therefore makes the results more applicable to antitrust
and technology policy. Furthermore, the confirmation of existing results for the symmetric
case with a different model adds to the robustness of these results.
In this paper, the simple Cournot model with R&D and stochastic innovation before
production yields asymmetric equilibria in the production and research stage, some of which
include corner solutions. The main findings, obtained by simulation, confirm the welfare
results from the literature for the symmetric case. In the asymmetric case, however, several
aspects of the R&D process interact to generate new results on welfare and firm behavior. The
two main forces driving the model are the spillover effect and the sharing of R&D results in the
RJV.
If firms compete in R&D, the spillover effect tends to reduce R&D effort. Cooperation
in a RJV allows to internalize the positive externality. If the internalization effect dominates,
firms will increase their R&D effort with cooperation. This effect reaches its maximum when
the spillover parameter is equal to one. It does not exist in the case of no spillovers.
The second important feature of the model, the sharing of information that was gained
in the R&D process, only takes effect if firms cooperate in R&D. Setting the spillover
parameter equal to one within the joint venture has two effects. The reduction of duplicate
effort and the resulting cost savings put a downward pressure on R&D expenditure.
Alternatively, for any given R&D expenditure the probability of attaining the lower cost
increases. The cost reduction by itself affects firm profits positively. On the other hand,
information sharing eliminates the asymmetry between the firms if one of them succeeds. This
increases competition between the firms in the production stage. Therefore, the sharing of
information has a cost reducing effect but at the same time it can reduce industry profit by
eliminating the asymmetry. Without this negative effect on profits, firms could always find a
cooperative agreement that weakly increases industry profit. At worst, they could duplicate the
non-cooperative outcome with cooperation. When the spillover parameter in the industry is
close to one, the information sharing effects disappears and the results are only driven by the
internalization of the spillovers.
5
The model shows that the initial cost difference between the firms, the size of the
spillover parameter, and the degree of the cost reduction together determine how information
sharing and internalization of spillovers affect welfare and firm profits. Depending on the
dominance of the effects, parameter regions can be identified in which both firms prefer to
compete in R&D, both want to cooperate, or only one firm wants to cooperate. In contrast to
the symmetric case, the desire of the firms to cooperate does not coincide for a wide range of
parameters. The welfare analysis shows that there are cases in which cooperation decreases
welfare. If the initial asymmetry is large, the high cost firm overinvests in R&D. In this case,
a RJV has an anti-competitive effect by inducing the high cost firm to reduce its R&D effort.
The goal of the joint venture is to preserve the existing asymmetry rather than to facilitate
innovation. In all other cases, R&D cooperation increases welfare. Comparing the willingness
to cooperate and the effect of cooperation on welfare allows drawing policy implications. If
side-payments or other forms of compensation are not feasible, the firms will often fail to agree
on a RJV even if the effect on industry profit and welfare would be positive.
The next section introduces the general setup of the model. Sections 3 and 4 describe
the non-cooperative and the cooperative game, respectively. Welfare comparisons are made in
section 5 and section 6 concludes.
II. The Model
This study builds on the existing literature on research joint ventures. Many aspects of
the model are standard formulations, which helps to make the results comparable. Two firms
play a two-stage Cournot-Nash game. In stage I, firms invest in R&D to reduce production
cost in stage II. In stage II, firms compete in quantities non-cooperatively. Both firms have
constant marginal cost (ci), which differ in the beginning of the game. Firm 1 is the low cost
firm such that initially c1 < c2.
I follow d’Aspremont and Jacquemin (1988), Kamien et al. (1992,1993), and others in
using linear demand. For simplicity, the inverse demand is represented by P=1-Q where
Q=q1+q2 denotes total industry output. Many authors have chosen a deterministic R&D
process, which greatly simplifies the analysis. However, stochastic R&D outcomes are an
essential feature of research and development activity. Similar to Choi (1993), I adopt a
6
framework in which firms can invest in R&D to reduce marginal cost from high cost (ciH) to
low cost (ciL). Firms either succeed or fail to innovate. The probability of success for firm i is
given by xi with 0  x i  1 . This probability is the choice variable of the firm in stage I and it
could also be interpreted as the chosen research intensity. The cost of obtaining probability xi
is C(x i )  k
x i2
. This R&D cost function is convex and represents decreasing returns to scale
2
for R&D expenditures. The parameter k is a positive constant that has a scaling effect. In the
current model, k does not vary over firms, which implies equal R&D capabilities of firms. The
initial cost difference does not need to originate from differing R&D capability. It could be the
case that the firms’ environment differs, e.g. if they are located in different countries. The cost
asymmetry could also stem from firm specific dissimilarities such as patents, firm history, and
management. It is straightforward to extend the model to differing R&D capabilities.
However, differing R&D capabilities might force the results in a way that the low cost firm
with higher R&D capability is simply more competitive and drives the opponent out of the
market.
As in most models on RJVs, there exists a spillover effect that captures the
uncompensated flow of knowledge from one firm to the other. In particular, if only one firm
succeeds, the other will benefit from the innovator’s cost reduction and reduce its cost by
 c where c is the cost reduction and 0    1 is the spillover parameter.
In addition to the initial cost difference, the nature of the cost reduction has to be
specified. Two scenarios come to mind. It could be the case that the firms attempt to improve
their existing technology, which implies a cost reduction by a certain amount or a percentage
of the initial cost. If both firms succeed, the low cost firm would still be the industry leader.
Alternatively, the firms could discover an entirely new technology, implying that both firms
can reach the same low cost and the asymmetry can be eliminated. I adopt the latter approach
and assume that c2H>c1H and c2L=c1L=cL. Given this specification, the spillover c is
asymmetric. If firm 1 fails and 2 succeeds, c1 is the amount by which firm 1 will reduce its
cost. In this case, c 1 will equal c1H-cL. If firm 1 is the innovator, c 2 is firm 2’s cost
reduction of c2H-cL. As c1H<c2H, c 1< c 2 which implies that firm 2 benefits more at any
given spillover parameter  that is greater than zero.
7
III. The non-cooperative equilibrium
Let me first consider the case in which the firms compete in both the production and the
research stage. To find the Cournot-Nash equilibrium of the two-stage game, I solve the game
backwards. In stage II, firms know the outcome of the R&D process. There are 4 possible
outcomes:
i
SS
denotes the profit of firm i if both firms succeed,
i
SF
if firm i succeeds, the other fails,
i
FS
if firm i fails, the other succeeds, and
i
FF
if both fail.
Given the linear demand function, profit for firm i in the output game is:
 1  c j  2c i
 i  (P  ci )q i which becomes  i  
3

2

 given the optimal output for firm i.

The profits for firm 1 in each possible outcome of the R&D process are:

SS
1
 1  c 2 L  2c1L 
1  cL 

 

3


 3 

FF
1
 1  c 2 H  2c1H 


3


2

 1  (c 2 H  c 2 )  2c1L 


3


2
SF
1

 1  c 2 L  2(c1H  c1 ) 


3


2
FS
1
2
2
Similar equations hold for firm 2.
To ensure an interior product market equilibrium in the beginning of the game, the cost
of the high cost firm c2H has to be less than
1  c1H
. This condition allows for corner solutions
2
in production after one firm innovates. All simulation results are for a low cost cL equal to
zero. Allowing for a higher cL narrows the solution space because cL has to be lower than c1H.
This choice of the low cost does not change the results qualitatively. Diagram 1 illustrates how
the numerical results were obtained.
8
Taking the output competition in stage II as given, each firm chooses xi in stage I to
maximize expected profit. Firm i’s maximization problem is:
x i2
E
SS
SF
FS
FF
max

(
x
,
x
,

)

x
x


x
(
1

x
)


(
1

x
)
x


(
1

x
)(
1

x
)


k
(1)
i
i
j
i j
i
i
j
i
i
j
i
i
j
i
2
xi
The first order condition is:
(2)
 iE
FF
SF
FS
SF
FF
 x j ( SS
i   i   i   i )   i   i  k x i 0
x i
FF
SF
FS
SF
FF
(3) [ x j ( SS
i   i   i   i )   i   i  k x i]x i  0
The equilibrium probability xi* is given by
(4) x *i 
B j A i  Bi k
i 1,2 i  j
k 2  Ai A j
where
for
B j A i  Bi k
 0 and xi*=0 otherwise.
k 2  Ai A j
FF
SF
FS
A i   SS
i  i  i  i
FF
SF
FS
A j   SS
j  j  j  j
FF
B i   SF
i  i
FF
B j   SF
j  j
The simulation shows that for the high cost firm x2 is always greater than zero and the firm’s
reaction function is downward sloping. For the low cost firm, the slope of the reaction
function and the intercept change sign depending on the initial cost difference. For   0 , low
initial cost of firm 1 and relatively high cost of firm 2, firm 1’s optimal probability is a corner
solution with x*1 = 0. The solution moves closer to the corner the higher the cost ratio c2H /c1H.
The three possible scenarios are illustrated in diagrams 2 through 4. All equilibria are unique
and stable. The stability criterion used is
 2  iE
x i x j
1 i, j 1,2 i  j . This criterion was used by Henriques (1990) in a two-stage R&D
 2  iE
(x i ) 2
game.
Comparing x1* and x2*, i.e. comparing B2A1+B1k and B1A2+B2k yields the following
condition:
(5) x1* < x2* iff
B1  SS  k   1FS

B 2  SS  k   FS
2
9
It is easy to show that the right-hand side of the equation is greater than 1 for 0    1 and
equal to 1 for   1 . Whether B1 is less than B2 depends on the spillover parameter and on the
cost distribution. The simulation shows that condition (5) is satisfied for a wide range of
parameters.
Result 1:
The low cost firm’s probability of success is less than the high cost firm’s probability for most
spillover parameters and cost combinations. X1 exceeds x2 by a small amount only if both
firms have high initial cost (c1H>0.75) and the spillover parameter lies between 0.3 and 0.7.
(See Figure 1)
It was noted earlier that the spillover effect benefits the high cost firm more than its
rival because the small firm can absorb knowledge from the industry leader. One might expect
that with R&D competition the high cost firm therefore reduces its R&D effort to free-ride on
the low cost firm. On the other hand, the expected gain from innovation is larger for the high
cost firm. It has a low profit in the beginning of the game but it can potentially catch up with
or overtake the leader. Result 1 indicates that for the high cost firm the incentive to win the
higher “price” tends to be dominant.
In a 1997 discussion paper, Roeller et al. develop a deterministic model with
heterogeneous firms that does not allow for spillovers. They find that the low cost firm spends
more on R&D. My results show that the Roeller et al. finding is specific to their model.
IV. R&D cooperation
This section describes the two-stage game with a RJV instead of R&D competition. In
Stage I firms can now cooperate in R&D, taking the product market competition as given. I
define a RJV as a contractual agreement between the two firms that specifies each firm’s
contribution and the distribution of the research results. In this model, I follow Choi (1993)
and Kamien et al. (1992, 1993) in fixing the spillover parameter at unity if firms cooperate.
This means that members of the RJV share R&D results completely. The assumption on the
10
SF
FS
cost reduction c1L=c2L=cL plus   1 also implies that  SS
i   i   i for both firms. The
RJV members now choose xi and xj to maximize joint profit. The RJV’s maximization
problem is:
(6) max 
x1 , x 2
E
RJV

 ( x 1  x 2  x 1 x 2 ) 2  (1  x 1 )(1  x 2 ) 
SS
FF
1

FF
2

 x 12 x 22 
 k  
2 
 2
The first order conditions for this maximization problem are:
(7)
 ERJV
 2 SS  2 SS x 2  (1  x 2 ) 1FF   FF
2  0
x 1
(8)
 ERJV
 2 SS  2 SS x 1  (1  x 1 ) 1FF   FF
2  0
x 2
Solving for x1 and x2 gives a symmetric solution with:
(9) x 1  x 2  x 
2 SS   1FF   FF
2
k  2 SS   1FF   FF
2
x0
Comparing the cooperative and the non-cooperative equilibrium probabilities several
observations can be made. For the benchmark case in which firms are identical, the numerical
calculations show that for   0 the non-cooperative probability exceeds the cooperative one
for each firm. In that case, there are no externalities to internalize. The cost effect of reducing
duplicate effort dominates. For   1 , the opposite is true. This confirms previous results that
firms underinvest if they compete in R&D and the spillover is high. Holding the spillover
parameter constant, the range in which the firms decrease their cooperative R&D effort
compared to the non-cooperative outcome increases with the size of the innovation. This
indicates that the firms compete more intensely in R&D if the innovation is drastic. In that
case, sharing of the R&D results under cooperation eliminates the threat of being driven out of
the market and the firms can cut back on their spending. Nevertheless, as will be seen in the
next section, the reduction in R&D spending does not outweigh the positive effects of
information sharing in welfare terms.
Looking at the asymmetric case, the calculations show that the high cost firm tends to
reduce its R&D effort with cooperation. For   0 , the cooperative x2 is lower than x2 under
11
competition for all cost combinations. With increasing spillover effect, the cooperative x2 is
higher if the firms are relatively similar. If the high cost firm has a very high cost relative to
the leader, its competitive R&D effort is always higher than under cooperation. The low cost
firm tends to increase its R&D effort with cooperation. The higher the spillover parameter, the
larger the cost range (moving from minor to major innovation) for which this is true.
Using the equilibrium success probabilities from the cooperative and the noncooperative model, expected profits for each scenario can be computed and compared.
Result 2:
Expected industry profit is always larger under cooperation for high spillover parameters
(   0.6 ). For lower spillover parameters this is only true for “non-drastic” innovation, i.e.
when the potential cost reduction is not too large.
The increase of industry profit with cooperation for high spillover parameters is driven
by the internalization of the spillovers. Without a RJV, the free-rider problem has a strong
decreasing effect on R&D effort. When firms cooperate, R&D effort is chosen to maximize
joint profit, thus allowing for a larger expected cost reduction. For low spillover parameters,
the externality is less problematic. The information sharing within the RJV becomes more
important. With R&D competition and large cost reductions it is possible for one firm to drive
its rival out of the market. The firms give up the chance of becoming a monopoly if they
decide to cooperate. Therefore, the firms are only willing to form a RJV if the cost reductions
are relatively low. For large cost reductions, the negative profit effect of sharing R&D
outcomes dominates.
When looking at industry profit, willingness to cooperate can be determined by the
change in industry profit. If industry profit under cooperation exceeds that under R&D
competition, there is a gain to be split between the firms. However, in order to reach a
cooperative agreement that both firms can accept, both firms need to gain by choosing the
cooperative research intensity or they have to be allowed to bargain. The next result indicates
that bargaining and side-payments are important in the case of firm asymmetries.
12
Result 3:
The gains from cooperation are distributed unevenly (See Figure 2). The cases in which firm 1
and firm 2 gain from cooperation often do not coincide.
The regions in Figure 2 are obtained by computing and comparing individual firm profit under
R&D competition and under cooperation. In the cooperative case, both firms have the same
research probability from equation (9) and they also pay for this probability themselves. If
under this condition cooperation yields a higher profit for the firm, it is said to be willing to
cooperate. Side-payments from one firm to the other are not considered in this case. If firms
were allowed to set up any agreement on a RJV that they like, they would always cooperate if
industry profit increases with a RJV. The firms could bargain in the first stage to split the
expected gain from cooperation. This is indicated in Figure 2 as “agreement area”.3 The
importance of the parameter regions in Figure 2 lies in the implications for the legal treatment
of RJVs.
Obviously, in the symmetric case firms either both want to form a RJV or not. The
model reproduces previous results which state that firms are more likely to cooperate if the
spillover parameter is high. By forming a RJV, firms can internalize the R&D spillover. In
addition, the simulation shows that the willingness to cooperate also depends on the size of the
innovation. For low spillover parameters, the firms are only willing to cooperate if the
innovation is small. To make cooperation possible for larger potential cost reductions, the
spillover parameter has to increase.
Moving away from the symmetric case, the low cost firm becomes less willing to
cooperate. For a relatively wide range of parameters, only firm 2 wants to cooperate. When
the firms are very different, only firm 1 prefers cooperation. The regions in which only one
firm wants to form a RJV indicate that the motives for cooperation can be different for the two
3
A natural extension at this point would be to find the solution, e.g. the Nash bargaining solution, to the firms’
bargaining problem. Furthermore, it could be investigated how firms might develop alternative agreements under
the constraint that side-payments are illegal ex ante. Then firms could choose cooperative probabilities that
maximize industry profit under the condition that both firms have to be at least weakly better off. As a result,
asymmetric R&D efforts might increase to area in which both firms are willing to cooperate even if side-payments
are not permitted. This approach, however, exceeds the scope of this paper.
13
firms. Both firms can internalize the spillover effect with cooperation. The high cost firm can
reduce its R&D effort relative to the potential cost reduction. The most interesting case arises
when firm 1’s cost is very low. Under competition, it would choose not to spend money on
R&D. Its expected profit as well as industry profit would increase if the firms would cooperate
and firm 1 would engage in R&D. The effect of this cooperation would be to “pull down” firm
2’s investment and to decrease the overall success probability. Without side-payments, firm 2
would not cooperate but firm 1 would be willing to pay firm 2 to enter into an agreement.
Under these circumstances, a RJV has an anti-competitive effect. As will be shown in the next
section, the welfare effect of such an agreement would be negative.
Analyzing the private incentives for cooperation if firms differ in marginal costs
unfolds a larger spectrum of firm behavior. As result 3 suggests, side-payments or other forms
of compensation are likely to play an important role in R&D cooperation. If legal restrictions
prevent side-payments, the firms may fail to reach a cooperative agreement even if industry
profit would increase.
V. Welfare comparison
Before comparing the non-cooperative and the cooperative game in terms of social
welfare, I will present the social optimum as a benchmark. Expected social welfare (WE) is
equal to expected consumer surplus (C) plus expected industry profit   . In particular,
(10) W E  x 1 x 2 CSS   SS  x 1 (1 x 2 )CSF   SF  (1 x 1 )x 2 C FS   FS 
 x i2 x 2j 
 (1 x 1 )(1 x 2 )C    k  
2 
 2
FF
FF
where CSS and  SS are consumer surplus and industry profit if both firms succeed, C SF
and  SF are consumer surplus and industry profit if firm 1 succeeds and firm 2 fails, and so on.
Several authors used the first-best welfare criterion as a comparison to the firms’ game.
Suzumura (1992) questions the relevance of the first-best welfare function as a yardstick to
evaluate the performance of RJVs. He argues that the enforcement of the first-best
arrangement is hardly feasible for any government and that gains from cooperative R&D
14
should be evaluated within the alternative feasible arrangements. Consequently, Suzumura
suggests the use of the second-best welfare function as a benchmark. In that case, the social
planner takes the output as given by the firms and maximizes over the probabilities x1 and x2
only. In this paper, both the first-best and the second-best solutions were considered.
The first best solution requires that the firms sell their product at marginal cost,
implying that only one firm remains in the market if their costs differ. This solution assumes
that the social planner can shut down the high cost firm if both firms fail to innovate and have
the low cost firm produce at a price equal to marginal cost. The solution for the success
probabilities is symmetric and equal to:
(1  c1H ) 2  (1  c L ) 2
2
(11) x 1  x 2  x SP1 
2
(1  c1H )  (1  c L ) 2
k
2
Solving for the second best outcome yields:
 C FS   FS  C SF   SF 
C FF   FF  C FS   FS  k

A SP  k


SP 2
(12) x 1 
A SP  k
 C FS   FS  C SF   SF 
C FF   FF  C SF   SF  k

A SP  k


SP 2
(13) x 2 
A SP  k
where: A SP  CSS   SS  C FF   FF  CSF   SF  C FS   FS
The social planner’s second-best solution is only symmetric if the firms have the same cost.
Comparing the non-cooperative and the cooperative probabilities to the first-best and second
best solutions can be summarized as follows:
Result 4:
In the non-cooperative equilibrium, the high cost firm overinvests in R&D if the industry
leader’s cost is very low. The result holds for any spillover parameter and in comparison to
both the first and the second-best solution. The low cost firm always underinvests compared to
both the first and the second-best solution.
15
The cooperative equilibrium probability falls short of the first-best solution except for
c1H=0. In that case, the social planner would not invest in R&D but a RJV would invest a
small amount. Both solutions are independent of the spillover parameter. For the second-best
probabilities, the same result holds except for low spillovers and very low cost for firm 1. In
this region the cooperative solution exceeds the social planners x1.
In contrast to d’Aspremont and Jacquemin’s model, the comparison of R&D effort does
not directly translate into welfare results. The R&D intensity change interacts with the
information sharing effect of RJVs to affect the welfare ranking of the regimes. In addition, in
this model of asymmetric R&D effort the allocation of research activity matters. For a given
total success probability (the probability that at least one firm finds the new technology) R&D
expenditure is minimized if both firms spend the same amount. Due to asymmetry, this never
holds for the competitive outcome or the second-best welfare outcome.
Result 5:
Comparing the cooperative and the non-cooperative R&D scenario, the simulation shows that
welfare is higher under cooperation for all spillover parameters and all cost combinations as
long as c2H /c1H.< 5. Figure 3 shows that the area in which firm asymmetry leads to a welfare
reduction under cooperation (c2H /c1H.> 5) is small relative to the size of the parameter space.
As mentioned above, only when the low cost firm’s cost is very low, cooperation
decreases welfare. With the specific setup of the modelFor low values of the spillover
parameter, there exists a relatively large cost range in which cooperation would increase
welfare but firms are unwilling to cooperate. This tends to be the case for “drastic
innovations” when the potential cost reduction is large.
Naturally, welfare under a
cooperative agreement is always lower than under the first-best outcome. However, for low to
medium spillover values, the cooperative outcome results in a higher welfare level than the
social planner’s second-best. This result might seem surprising because the firms maximize
industry profit without taking consumers into account. The welfare increase is driven by the
information sharing effect and the R&D cost savings. In the second-best setting in this model,
the social planner can set x but not  . With cooperation, the firms share their R&D results,
16
which increases the probability that both firms will produce with the low cost which in turn
increases expected consumer surplus. The overall success probability tends to be lower with
cooperation. However, the information sharing combined with a lower total R&D expenditure
increases welfare above the competitive level and the social planner’s second best. For high
spillover parameters, the information sharing effect disappears and the social planner’s second
best exceeds the cooperative welfare level.
VI. Conclusions
In this paper, a simple model of R&D cooperation was developed to investigate the private
incentives for R&D cooperation. It was also examined whether the welfare results of previous
studies hold in the case of asymmetric firms. It was confirmed that in most cases R&D
cooperation increases welfare. The only exceptions are RJVs in which the industry leader with
a very low cost pays the small firm to cooperate. Allowing for initial cost differences showed
that the firms often do not have the same incentives to cooperate. In order to reach a profit and
welfare increasing agreement in these cases, one firm has to be able to compensate the other
firm for the decrease in its expected profit. If this is not feasible, many welfare improving joint
ventures might never form.
The present model is very simple and policy implication should therefore be drawn with
caution. Keeping the assumptions of the model in mind, in particular the complete information
sharing in RJVs, three suggestions could be made. (i) Competing firms should be allowed to
form R&D joint ventures unless the innovation is small and the leader has already low costs. If
the cooperating firms are very dissimilar, the effect on research activity should be monitored.
(ii) In all other cases, side-payments should be allowed to facilitate cooperation if one firm is
not willing to cooperate. (iii) In cases where the spillover parameter is low, the innovation is
drastic, and the firms are unwilling to cooperate, information sharing should be encouraged.
Subsidizing the small firm’s R&D does not seem an appropriate measure to increase welfare.
Instead, cautiously placed financial incentives for cooperation might be welfare improving.
17
Diagram 1
The Simulation
In order to be able to compare the equilibrium success probabilities, profits, and welfare
results in the cooperative and the non-cooperative game, values were computed for the relevant
cost range in 0.05 size intervals. This was done for spillover parameters between zero and one
in 0.1 size intervals. The range of the calculations was determined by two conditions being
satisfied. The costs have to lie between zero and one and both firms need to be able to produce
in the beginning of the game, i.e. the costs have to be similar enough that if both firms fail to
innovate the outcome will be a duopoly.
0  c1H  c 2H  1
and
q FF
2  0  c1H  2c 2 H  1
c2H
c1H
0 0.05 0.1 0.15
…
0.5
…
0.8 0.85 0.9 0.95 1
Firm 1
0
Monopoly
0.05
0.1
0.15
Symmetric Case:
Firms are identical
along the diagonal.
.
0.5
.
0.85
0.9
0.95
Firm 2
Monopoly
1
To maximize the solution space, the low cost cL is equal to zero for both firms. The diagonal in
the diagrams represents the symmetric case. Moving along the diagonal from the upper left to
the lower right corner illustrates an increase in the size of the innovation. Above the diagonal,
firm 1 is the low cost firm, below the diagonal firm 2 has the lower cost. The case c2H < c1H
was included in the diagrams to give a better visual impression of the results.
18
Reaction Functions
Diagram 2
x1=R(x2)
x2
x2=R(x1)
0
x1
Strategic Substitutes
Diagram 3
C2H/C1H ratio is increasing
x1=R(x2)
x2
x2=R(x1)
0
x1
High Cost Firm: Substitute
Low Cost Firm: Complement
x2
x1=R(x2)
Diagram 4
x2=R(x1)
0
x1
Corner Solution
19
Probabilities under R&D Competition
Beta = 0
Probabilities X1 (Low Cost Firm)
Figure 1
Probabilities X2 (High Cost Firm)
0.25
0.25
0.2
0.2
0.15
Beta = 0
0.15
X1
X2
0.1
1
0.1
0.9
0.75
0.8
0.6
0.05
0.45
Beta = 1
0.25
0.25
0.225
0.2
0.2
0.175
0.175
0.15
0.15
X1 0.125
X2 0.125
0.1
0.4
0.8
0.9
0.6
0.4
0.7
1
c1 H
0.4
0.025
0
0
0.9
0.6
c2 H
0.75
0.3
0
0.45
0
0.2
1
0.8
0.9
0.6
0.7
0.5
0.3
0.4
0.6
0.05
c1 H
0.2
c2 H
1
0.8
0.15
0
Beta = 1
0.075
0.6
0.05
0.025
0
0.1
1
0.8
0.075
0.1
0.5
Probabilities X2 (High Cost Firm)
0.225
0.2
0.2
0.1
c2 H
Probabilities X1 (Low Cost Firm)
0
0.15
1
0.8
0
0
0
0.9
0.6
0.7
0.4
0.5
0.2
0.3
0.1
0
0.2
c2 H
c1 H
0.3
0.3
0
0.6
0.05
c1 H
0.4
Difference in Probabilities
Difference in Probabilities X2 - X1
Difference in Probabilities X2 - X1 Beta = 0.5
Beta = 0
0.14
0.12
0.12
0.1
X1 > X2
0.1
0.08
0.08
X2 - X1
X2 - X1 0.06
1
0.04
0.06
1
0.04
0.8
0.8
0.02
-0.02
0
1
0.9
0.7
0.5
0.8
c1 H
0.6
0.3
0.1
0.2
0.4
0
0.4
1
0.9
0.7
0.8
0.6
0.5
0.3
c1 H
0.4
0.1
0.2
0.2
0
0
0.6
0
0.2
0.4
c2 H
0
0.6
0.02
20
c2 H
Competition versus Cooperation
Figure 2
Figure 3
Difference in Expected Profits for the Firms
Difference in Industry Profits and Welfare
Low Cost Firm
High Cost Firm
Cooperate
Cooperate
1
With Cooperation:
Industry Profit Welfare
Compete
Cooperate
2
Increases
Decreases
A
Cooperate
Compete
3
Decreases
Increases
B
Compete
Compete
4
Increases
Increases
C
Industry profit increases with cooperation
(Agreement Area)
Industry profit increases with cooperation
(Agreement Area)
C2H
C1H
C2H
0
.05
.1
.15
.2
.25
.3
.35
.4
.45
.5
0
0
2
1
1
1
3
3
3
3
3
3
.05
.55
.6
.65
.7
.75
.8
.85
.9
.95
1
Beta = 0
C1H
0
.05
.1
.15
.2
.25
.3
.35
.4
.45
.5
0
0
C
C
C
C
A
A
A
A
A
A
.65
2
1
1
1
1
1
1
1
3
3
3
.05
C
C
C
C
C
C
C
C
C
A
A
.1
1
1
1
1
1
1
1
1
1
3
3
3
.1
C
C
C
C
C
C
C
C
C
C
C
A
1
1
1
1
1
1
1
1
1
1
1
3
.15
C
C
C
C
C
C
C
C
C
C
C
C
.2
1
1
1
1
1
1
1
1
1
1
1
1
3
.2
C
C
C
C
C
C
C
C
C
C
C
C
C
.25
3
1
1
1
1
1
1
1
1
1
1
1
1
.25
A
C
C
C
C
C
C
C
C
C
C
C
C
.3
3
1
1
1
1
1
1
1
1
1
2
1
1
3
.3
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.35
3
1
1
1
1
1
1
1
1
2
2
1
3
3
.35
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.7
.75
.8
.85
.9
.95
.4
3
3
1
1
1
1
1
1
1
2
2
4
3
3
3
.4
A
C
C
C
C
C
C
C
C
C
B
B
B
C
C
.45
3
3
3
1
1
1
1
2
2
4
4
4
3
3
3
.45
A
A
C
C
C
C
C
C
C
B
B
B
B
B
B
.5
3
3
3
1
1
1
2
2
2
4
4
4
4
3
3
3
.5
A
A
C
C
C
C
C
C
B
B
B
B
B
B
B
B
3
3
1
1
1
1
4
4
4
4
4
4
4
3
A
C
C
C
C
C
B
B
B
B
B
B
B
B
3
1
1
3
3
3
4
4
4
4
4
4
4
.6
C
C
C
C
B
B
B
B
B
B
B
B
B
3
3
3
3
3
4
4
4
4
4
4
.65
C
C
C
B
B
B
B
B
B
B
B
3
3
3
4
4
4
4
4
4
4
C
B
B
B
B
B
B
B
B
B
3
3
4
4
4
4
4
4
B
B
B
B
B
B
B
B
4
4
4
4
4
4
4
.8
B
B
B
B
B
B
B
4
4
4
4
4
.85
B
B
B
B
B
4
4
4
4
.9
B
B
B
B
4
4
.95
B
B
.6
.65
.7
.75
.8
.85
.9
.55
.95
.7
.75
1
4
1
C2H
.05
.1
.15
.2
.25
.3
.35
.4
.45
.5
0
0
2
1
1
1
3
3
3
3
3
3
.05
2
1
2
2
2
1
1
1
3
3
3
B
.55
.6
.65
.7
.75
.8
.85
.9
.95
1
Beta = 0.5
C1H
0
.05
.1
.15
.2
.25
.3
.35
.4
.45
.5
0
0
C
C
C
C
A
A
A
A
A
A
.05
C
C
C
C
C
C
C
C
A
A
A
.55
.6
.65
.7
.75
.8
.85
.9
.95
.1
1
2
1
2
2
2
2
1
1
1
3
3
.1
C
C
C
C
C
C
C
C
C
C
C
A
.15
1
2
2
1
1
2
2
2
1
1
1
1
.15
C
C
C
C
C
C
C
C
C
C
C
C
.2
1
2
2
1
1
1
1
2
2
1
1
1
1
.2
C
C
C
C
C
C
C
C
C
C
C
C
C
.25
3
1
2
2
1
1
1
1
2
2
1
1
1
.25
A
C
C
C
C
C
C
C
C
C
C
C
C
.3
3
1
2
2
1
1
1
1
1
1
2
1
1
1
.3
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.35
3
1
1
2
2
1
1
1
1
1
1
2
2
1
.35
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.4
3
3
1
1
2
2
1
1
1
1
1
1
2
2
2
.4
A
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.45
3
3
1
1
1
2
1
1
1
1
1
1
1
2
2
.45
A
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.5
3
3
3
1
1
1
2
1
1
1
1
1
1
1
2
2
.5
A
A
C
C
C
C
C
C
C
C
C
C
C
C
C
C
3
1
1
1
1
2
1
1
1
1
1
1
1
2
.55
A
C
C
C
C
C
C
C
C
C
C
C
C
C
1
1
1
2
2
1
1
1
1
1
1
2
2
.6
C
C
C
C
C
C
C
C
C
C
C
C
C
1
1
2
2
1
1
1
1
1
1
2
.65
C
C
C
C
C
C
C
C
C
C
C
2
2
2
1
1
1
1
1
2
2
.7
C
C
C
C
C
C
C
C
C
C
2
2
2
1
1
1
2
2
.75
C
C
C
C
C
C
C
C
2
2
2
2
1
2
2
C
C
C
C
C
B
B
2
2
2
4
4
C
C
B
B
B
2
4
4
4
.9
B
B
B
B
4
4
.95
B
B
.6
.65
.7
.75
.8
.85
.9
.95
.8
.85
1
4
1
C2H
.05
.1
.15
.2
.25
.3
.35
.4
.45
.5
0
2
1
1
1
3
3
3
3
3
3
.05
2
1
2
2
2
2
2
1
3
3
3
B
.55
.6
.65
.7
.75
.8
.85
.9
.95
1
Beta = 1
C1H
0
.05
.1
.15
.2
.25
.3
.35
.4
.45
.5
0
0
C
C
C
C
A
A
A
A
A
A
.05
C
C
C
C
C
C
C
C
A
A
A
.55
.6
.1
1
2
1
2
2
2
2
2
2
1
1
3
.1
C
C
C
C
C
C
C
C
C
C
A
A
.15
1
2
2
1
2
2
2
2
2
2
2
1
.15
C
C
C
C
C
C
C
C
C
C
C
C
.2
1
2
2
2
1
2
2
2
2
2
2
2
2
.2
C
C
C
C
C
C
C
C
C
C
C
C
C
.25
3
2
2
2
2
1
2
2
2
2
2
2
2
.25
A
C
C
C
C
C
C
C
C
C
C
C
C
.65
.7
.75
.8
.85
.9
.95
.3
3
2
2
2
2
2
1
2
2
2
2
2
2
2
.3
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.35
3
1
2
2
2
2
2
1
1
2
2
2
2
2
.35
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.4
3
3
2
2
2
2
2
1
1
1
2
2
2
2
2
.4
A
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.45
3
3
1
2
2
2
2
2
1
1
1
2
2
2
2
.45
A
A
C
C
C
C
C
C
C
C
C
C
C
C
C
.5
3
3
1
2
2
2
2
2
2
1
1
1
2
2
2
2
.5
A
A
A
C
C
C
C
C
C
C
C
C
C
C
C
C
3
1
2
2
2
2
2
2
1
1
1
1
2
2
.55
A
C
C
C
C
C
C
C
C
C
C
C
C
C
2
2
2
2
2
2
2
1
1
1
1
2
2
.6
C
C
C
C
C
C
C
C
C
C
C
C
C
2
2
2
2
2
1
1
1
1
1
1
.65
C
C
C
C
C
C
C
C
C
C
C
2
2
2
2
1
1
1
1
1
1
.7
C
C
C
C
C
C
C
C
C
C
2
2
2
1
1
1
1
1
.75
C
C
C
C
C
C
C
C
2
1
1
1
1
1
1
.8
C
C
C
C
C
C
C
1
1
1
1
1
.85
C
C
C
C
C
1
1
1
1
C
C
C
C
1
1
C
C
.6
.65
.7
.75
.8
.85
.9
.95
1
1
C2H
0
0
.55
1
C2H
0
.55
C1H
.6
.15
.55
C1H
.55
.9
.95
1
1
1
C
21
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