ASYMMETRIC SPILLOVERS AND INVESTMENTS IN RESEARCH AND DEVELOPMENT OF LEADERS AND FOLLOWERS Leader and follower R&D investments by Jan Vandekerckhove* and Raymond De Bondt** Forthcoming in Economics of Innovation and New Technology March 1st, 2007 *
**
Catholic University of Leuven
Catholic University of Leuven
Department of Managerial Economics,
Department of Managerial Economics,
Innovation and Strategy
Innovation and Strategy
Naamsestraat 69
Naamsestraat 69
B – 3000 LEUVEN
B – 3000 LEUVEN
Belgium
Belgium
Phone: +32 16 32 69 04
Phone: +32 16 32 69 02
Fax: +32 16 32 67 32
Fax: +32 16 32 67 32
[email protected]
[email protected]
Jan Vandekerckhove
Raymond De Bondt
Responsible for correspondence
The authors gratefully acknowledge the Editor and three anonymous referees for their useful comments on the paper, but remain responsible for all remaining shortcomings. .
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ABSTRACT The focus of this paper is on research and development with sequential moves. Leading firms move before followers in investment and in output choices in a four stage game setting. There may be spillovers between leaders and between followers and also between these two groups of players. The impact of symmetric spillovers is similar but not identical to tendencies in two stage models with simultaneous R&D. A wide set of circumstances is identified where followers invest more than leaders. Situations are detailed where consumer surplus and static welfare are best served by cooperation of followers rather than cooperation of leaders.
Keywords: Cost-Reducing R&D, Sequential Game, Cooperation, Asymmetric Spillovers
J.E.L. Classification: D72, D43, L13
2
1. INTRODUCTION Firms tend to be frequently involved in strategic investments in their attempt to achieve or maintain sustainable competitive advantages. Strategic investments may take many forms, such as expenditures to increase business and technological knowledge accumulation, advertising or service outlays to develop or maintain goodwill in the market, and investments directed at modifying product characteristics, production processes or features of the internal organization and/or the external institutional environment.
A number of the main characteristics of these investments are fairly well understood and are helpful in inspiring competitive analysis. Strategic investments, for example, tend to change the parameters of the market rivalry outcomes, they may hurt or benefit rivals and firms may have an incentive to temper or exaggerate efforts for strategic reasons. Some investments may involve special features. Asymmetric information between parties involved, knowledge spillovers and cooperation between some or all of the players involved, for example, will influence innovative strategic efforts in research and development.
While broader interpretations of the analysis are possible, the focus hereafter is on strategic innovative activities. Investment means research and development outlays that produce new knowledge which in turn allows to reduce a constant unit variable cost of production. This somewhat narrow approach has hopefully the advantage of clarifying the intuition of reported tendencies. At the same time, it allows to relate the findings to earlier ones reported in the already abundant literature following the seminal papers of d’Aspremont and Jacquemin (1988) and Kamien et al. (1992) (see e.g. De Bondt 1997; Sena 2004 and Motta 2004).
The main idea motivating the analysis here is that innovative firms tend to be heterogeneous and thus may work with different business models and strategies (Röller and Sinclair-Desgagné 1996). It is well known that some players may attempt to be technological leaders to exploit so called first-mover advantages. Others may use a second mover approach and rely on there ability to quickly adopt what other firms demonstrate as valuable (Barney 2002; Schnaars 1994). In the computer and pharmaceutical industry, for example, some firms play a ‘me too’ or a duplication strategy while others actively search for new blockbusters and drastic innovations. Imitators often can duplicate first movers’ patent based
3
advantages at a lower cost. One well known study reports that 60% of all patents are imitated within 4 years of being granted, without legally violating patent rights of first movers (Mansfield et al. 1981).
The simplest way to capture these and related heterogeneities is to look at an industry with leaders and followers, with strategic R&D investments taking place in an ongoing dynamic process. In some stages of this process some players will be leading and others will observe the moves of the leaders and will follow with their own efforts. Leaders may or may not have a sufficiently strong or long patent protection and in some industries reverse engineering learning or duplication may be difficult and in others it may be easy. After some time followers may sometimes learn something from leaders. In many cases relevant new knowledge can also spill over between leaders and between followers again through reverse engineering or through contacts with common suppliers or with supporting outside laboratories. Changing jobs, professional exchanges at conferences and visits, and other features likewise may also generate transfer of relevant knowledge. It is probably typical that the heterogeneity between firms also means differences in spillovers. Spillovers of knowledge from leaders to followers need not be equal to the spillovers from followers to leaders. Leaders may differ in size, technology and product portfolio from followers, so that spillovers between leaders need not be the same as spillovers between second movers. The differences in leading versus following strategies thus need to be looked at in environments that allow asymmetric spillovers.
Some aspects of leader follower behavior and asymmetric spillovers have already been looked at in earlier contributions. It is known, for example, that such role playing does affect the incentives of the players in innovation races (Reinganum 1985; Doraszelski 2003) and in output games (Daughety 1990; Kamien and Zang 1990). Etro (2004) shows, for example, that a leader will invest more than the followers when there is free entry of followers.
Most of the theoretical insights in strategic R&D investments are however strongly biased towards settings with firms choosing their R&D simultaneously. A limited set of more recent work has begun to focus on the implications of heterogeneity. De Bondt and Henriques (1995), Amir and Wooders (1999) and Amir et al. (2000) look at asymmetric spillovers and role playing in R&D investments in a duopoly with simultaneous output decisions. Halmenschlager (2004) extends this setting to one leader and two
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followers, but only the latter invest in R&D while Atallah (2005) looks in detail at asymmetric spillovers but in a duopoly with simultaneous moves. Goel (1990) looks at only one Stackelberg leader investing and followers benefiting, and also Crampes and Langinier (2003) show in some specific settings the importance of extending this asymmetry to leader follower situations.
To further the understanding of leader follower settings in innovative industries it thus appears worthwhile to investigate how leadership and asymmetric spillovers change the predictions of simultaneous move models with symmetric or asymmetric spillovers. The group of followers and leaders may have more than one member. Therefore, this paper looks at an oligopoly with n firms with k leaders and n-k followers, investing in R&D, with possible asymmetric spillovers and subsequent Stackelberg output competition. Spillovers between leaders may differ from spillovers between followers and from spillovers between leaders and followers. If knowledge also spills over from followers to leaders, an industry with symmetric spillovers can be looked at. This will be done only to compare tendencies with existing symmetric settings. The setting has an early entrance character in which both leaders and followers invest before the output commitments1.
R&D investments in a symmetric two stage oligopoly with simultaneous choices have some clear characteristics. Spillovers discourage efforts and stimulate cooperative R&D choices. Cooperation stimulates efforts provided that the symmetric spillover exceeds a critical value. The impact of R&D cooperation on firm profits, total output and static welfare is driven by the same critical value. In this paper, it is questioned whether these tendencies are also present in a four stage setting with sequential moves. First, the relation between spillovers and investments of leaders and followers is looked at. To continue, investments of leaders and followers are compared. Finally, the impact of cooperation on R&D investments, profits and welfare is analyzed.
The main results of the paper are the following. With regard to the impact of spillovers on R&D investments, tendencies can differ from the two stage models. More specifically, the investments of cooperating leaders or cooperating followers can be decreasing in the symmetric spillover, which is
1
A “late entrance” setting with leaders choosing investment and output before the followers choose their investments and outputs has also been investigated, with similar results, but is not reported here.
5
caused by, respectively, the presence of the spillover from leaders to followers and the spillover from the followers to the leaders. Secondly, the model predicts that, if the spillover from the leaders to the followers is sufficiently high, the followers spend more resources on cost-reducing R&D than the leaders. Thirdly, the comparison of competitive and cooperative R&D yields similar results as in the two stage models. Again, critical spillovers drive the tendencies concerning R&D investments, profits and static welfare. However, these critical spillover values are not the same as in the two stage models.
The theoretical oligopoly setting will be detailed in the next section. In the subsequent sections results, are provided on:
-
the influence of spillovers on R&D efforts (section 3);
-
the comparison of leader and follower efforts (section 4);
-
the impact of cooperation of leaders or followers2 on R&D efforts (section 5);
-
the impact of cooperation of leaders or followers on profits (section 6) and
-
the impact of cooperation of leaders or followers on static welfare (section 7).
Finally, a conclusion and some policy implications are provided (section 8).
2. THE MODEL The model is an extension to sequential moves of earlier settings with simultaneous moves (see d’Aspremont and Jacquemin 1988). The focus is on an oligopoly market consisting of n firms competing non-cooperatively on the output market with homogenous products. Of these n firms, k firms behave as leaders while the remaining n-k firms are followers. It is assumed that leaders move before the followers in the investment stage and that they also move before the followers in the output stage. The inverse demand function is:
p = a−
k i =1
qiL −
n j = k +1
q Fj ,
(1)
2
Cooperation of both leaders and followers has also been investigated but will not be reported here. Results on this setting can be found in Vandekerckhove and De Bondt (2007).
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with i = 1,...,k and j = k+1,...,n. The choke price is a,
k i =1
qiL and
n j = k +1
q Fj denote the total output of
respectively the leaders (L) and the followers (F). Leaders and followers can commit to strategic investments, xiL and x Fj , in an attempt to maintain or improve the own competitive position. These commitments can take many forms, focused for example on branding, quality improvements or cost reductions. Throughout the paper (and in line with previous literature), they will be equated with innovative investments, say in R&D, that reduce unit costs. However, results could as well be applied to demand enhancing efforts that increase the parameter a.
A four stage early entrance setting is considered. The sequence of moves runs as follows. In the first stage, all k leaders decide on their strategic investment levels, knowing that each of the n-k followers will observe their efforts. In the second stage, followers decide on their commitments to innovative efforts. In the third stage, each leader commits to an output level, after observing the investments of the followers, and anticipating the subsequent output choices of the followers. In the final stage, followers decide on their output observing the results of the previous stages. The game is solved through backward induction.
Spillovers may occur in the investment stages one and two, between respectively the leaders and the followers. There may also be spillovers between the leaders in stage one and between the followers in stage two. This means that four groups of spillovers are looked at:
LL;
-
leader-specific spillover
-
follower-specific spillover
-
spillover from leaders to followers
LF;
-
spillover from followers to leaders
FL.
FF;
Insert Figure I
All spillovers are given and symmetric in each category. Figure 1 explains the notation. The asymmetries are thus limited to possible differences between the four mentioned groups. The last
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group, spillovers from followers to leaders, is only used here to explain the consequences of overall symmetry (
LL=
LF=
FF=
FL=
). So, in the symmetric spillover case, it is assumed that there is also
knowledge spilling over from followers to leaders in order to relate results with previous findings of the two-stage models. With asymmetric spillovers, it is assumed that there is no spillover from the followers to the leaders. This assumption is not only simplifying the analysis but is also in line with Amir and Wooders (1999) and Amir et al. (2000).
In modern market economies spillovers will depend on the type of strategic investments, as well as on the industry, the cultural, economic and legal environment. Some polar cases that will receive attention are blue print diffusion (
LF=1)
and idea diffusion (
LF=0).
The cases of blue print diffusion include
cases without effective patent protection and with knowledge developed by leaders that is a pure public good. With idea diffusion, the leaders are able to completely appropriate their knowledge because of, for example, patent protection, the absence of reverse engineering, different suppliers of inputs, little job rotation between firms, no developed professional contacts, etc. In this case followers only learn that it is possible to improve on production technologies and observe investment levels of leaders. Their R&D moves have therefore to follow those of the leaders. Many real world spillovers may lie between these extremes that nevertheless are useful as benchmarks. The terminology serves as a metaphor and is borrowed from technology diffusion studies, where the wheel serves as an example of blue print diffusion, since it presumably was easy to copy just by looking at it. Other technologies such as the production of porcelain were reinvented around 1700 in Europe, given the Chinese examples from the 7th century that reached the West in the 14th century. The German follower chemists only knew for certain that it could be done and thus faced idea diffusion (Diamond 1997).
All firms enter stage 1 or 2 with an ex ante unit cost equal to c. The leaders enter stages three with an ex post unit cost ciL . The followers enter stage four with an ex post unit cost c Fj . Both ex post values are the difference between the ex ante unit costs and the amount of effective knowledge that the player has accumulated in the previous stages. The effective value is the sum of the own efforts and the imported knowledge from other firms that results from the spillovers. The ex post unit costs of leaders and followers are therefore given by the following equations (2) and (3):
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ciL = c − xiL + β LL
k l =1 l ≠i
c Fj = c − x Fj + β FF
xiL + β FL
n f = k +1 f≠j
n f = k +1
x Ff + β LF
x Fj
k l =1
xlL
i = 1,...,k
(2)
j = k+1,...,n
(3)
with c the ex ante unit costs of the leaders and the followers. On the basis of these ex post unit costs, a Stackelberg equilibrium is obtained in stages 3 and 4. The equilibrium output levels depend, given (2) and (3), on the investments to be made in the earlier stages. These investments have for all firms a cost g(x) with diminishing returns:
g ( x) =
τ 2
× x 2 with a given parameter and > 0.
(4)
There is no discounting. Leaders and followers choose their efforts looking at profits:
(
)
( )
i = 1,...,k
(5)
(
)
( )
j = k+1,...,n
(6)
π iL = p − ciL × qiL − g xiL
π Fj = p − c Fj × q Fj − g x Fj
With qiL and q Fj the Stackelberg equilibrium values of the last stages. The followers choose their investments in stage 2 and the leaders choose in stage 1, knowing the effect on the rest of the game.
The output choices of the leaders in stage three and of followers in stage four are always simultaneous Cournot-Nash strategies within each group and stage. Leaders may choose investments independently from other leaders, and likewise for the followers. This game is labelled somewhat loosely the competition game (indicated by COMP.LF) as both leaders and followers compete in R&D. Note that the competition refers to simultaneous competitive R&D choices within the group of leaders and followers in respectively the first and second stage of the game. The followers however observe the efforts of the leaders. When there are several leaders and several followers, cooperation among the players can be introduced. Here only cooperation, which is defined as the coordination of R&D
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strategies without increasing knowledge transfers3, in the investment stages is analyzed. Mathematically, this is captured by the maximization of joint profits. Cooperation among leaders but not among followers is looked at, while also cooperation among followers but not among leaders is analyzed. One may think of situations where leaders and followers are operating in different geographic regions or business cultures. They may also face different discounting of the future so that cooperation is sustainable for one group but not for the other (Kesteloot and Veugelers 1995). The case where leaders cooperate in stage one, while followers compete among each other in stage two, is labelled COOP.L. The game where leaders compete among each other in stage one and followers cooperate in stage two is indicated by COOP.F. Table 1 summarizes the investment behaviour of the three possibilities4.
Table 1. Three possibilities of investment behaviour of leaders and followers in respectively stage one and two. In all games, a, Stackelberg equilibrium in output is anticipated in the stages three and four. Choice of R&D investment
By leaders vis à vis other leaders
By followers vis à vis other followers
COMP.LF
Nash
Nash
COOP.L
Cooperation
Nash
COOP.F
Nash
Cooperation
The solutions for the optimal investment levels and consequent output and profit levels of leaders and followers are rather complex. The analysis and results were therefore obtained by executing numerous numerical simulations5. When possible, analytical solutions are presented6.
3
The only exception to this assumption is introduced in section 7.2. Cooperation of leaders and of followers and between leaders and followers is not looked at, see also footnote 2. 5 The simulations have been conducted with the following parameter values: n=10, a=1000, c=100 and =400. However, results are robust for other parameter values. 4
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3. THE IMPACT OF SPILLOVERS ON R&D EFFORTS Competitive simultaneous choices of R&D tend to be discouraged by larger symmetric spillovers, while cooperative choices tend to be stimulated by symmetric transfers (see e.g. De Bondt 1997). In the leader follower model these tendencies appear as well, although some interesting variations apply. First symmetric and then asymmetric spillovers can be looked at.
3.1. Symmetric spillovers The case of symmetric spillovers with
LF= FL= LL= FF=
is somewhat artificial in the present setting
with two groups of players. The underlying heterogeneity of leaders and followers will typically result in some asymmetries in the possibilities for transfer of knowledge within or between the groups. Simulations indicate that an increase in such a symmetric spillover
tends to discourage R&D of all
players in each of the three games. Cooperative efforts no longer can be stimulated by larger spillovers as in the simultaneous two stage models. The only exception to these idiosyncratic results concerns industries with a large number of leaders (k>n/2). In such settings, an increase in the spillover
again
will stimulate cooperative efforts of the leading firms. Of course this exception may not be that relevant as the case of numerous leaders is less frequent than the situation with a few leaders and many followers.
Allowing for some asymmetries in spillovers produces more convincing tendencies, summarized in Proposition 1. Proposition 1. Suppose leaders learn nothing from followers ( symmetric -
LF =
LL= FF
FL=0)
and let the other spillovers be
= °. An increase in this symmetric spillover ° will:
discourage R&D by competing leaders (in COMP.LF and COOP.F) and competing followers (in COMP.LF and COOP.L);
-
stimulate R&D by a large number of cooperating leaders (k>n/2 in COOP.L);
-
discourage R&D by a small number of cooperating leaders (k n/2 in COOP.L);
-
stimulate R&D by cooperating followers (in COOP.F).
6
With cooperating leaders (game COOP.L), their investments decrease with the outgoing spillover LF. For high values of LF and low values of LL, the solutions can yield negative investments of the leaders, which are impossible. It is then assumed that the leaders do no invest at all (xL=0).
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These tendencies are almost the same as the ones that appear in symmetric oligopolies with simultaneous choices of R&D preceding Cournot-Nash rivalry. Indeed, the investments of cooperating followers are now increasing in the spillover. In other words, the negative relation between the overall symmetric spillover
and investments of cooperating followers is due to the outgoing spillover from
the followers to the leaders
FL,
which is assumed to be equal to 0 in Proposition 1. The only difference
with the two stage models is the third claim. The spillover ° may discourage efforts of a small number of cooperating leaders that anticipate imitation by competing followers. The reason is that the symmetric change embodies three effects: -
an increase in leakage from the leaders to the followers (
LF)
that discourages leaders’ R&D
efforts, -
an increase in the leader-specific spillover (
LL)
-
an increase in the follower-specific spillover (
that stimulates cooperating leaders’ R&D efforts, FF)
that can both discourage or stimulate leader’s
R&D efforts.
For a small number of leaders the first effect dominates. Proposition 2 confirms that the leakage from leaders to followers is, just like the spillover from followers to leaders, also a driving force in the reversal of typical tendencies. Proposition 2. Suppose leaders learn nothing from followers ( from leaders (
LF=0).
Assume
LL= FF=
FL=0)
and followers learn nothing
°°. An increase in this symmetric group specific spillover
°° will: -
discourage R&D of competing leaders (in COMP.LF and COOP.F) and competing followers (in COMP.LF and COOP.L);
-
stimulate R&D by cooperating leaders (in COOP.L);
-
stimulate R&D by cooperating followers (in COOP.F).
3.2. Asymmetric spillovers If all spillovers are different, it is possible to investigate the effect of changing one variable at the time. The tendencies are given in Proposition 3. Proposition 3. Suppose leaders learn nothing from followers ( specific spillover
LL
FL=0).
An increase in the leader-
among leaders will:
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-
discourage R&D of competing leaders (in COMP.LF and COOP.F);
-
stimulate R&D by cooperating leaders (in COOP.L);
-
have ambiguous effects on R&D of followers (in all games).
Similarly, an increase in the follower-specific spillover
FF
among followers will:
-
discourage R&D of competing followers (in all games);
-
stimulate R&D by cooperating followers (in COOP.F);
-
have ambiguous effects on R&D of leaders (in all games).
An increase in the one-way outgoing spillover
LF
from leaders to followers tends to:
-
discourage R&D of leaders (in all games);
-
have ambiguous effects on followers (in all games).
These results are in line with the tendencies of symmetric spillovers of Proposition 2. An increase in one-way spillovers from leaders to followers decreases the appropriability of the advantages of the innovative efforts and lowers the levels of R&D of the leaders. Cooperation among leaders or among followers does not change this tendency. This generalizes the finding that one-way spillovers from an innovator to an imitator tend to lower the effort of the former (as found by Amir and Wooders 1999). In a simple setting with one leader and follower, the follower is stimulated by such an increase in received knowledge, but the reverse tendency appears also possible7.
4. IMPACT OF LEADING OR FOLLOWING ON R&D INVESTMENTS It is well known that incumbent established firms often fail to remain technological leaders. A number of factors can contribute to the tendency for innovative performance to slow down, related to for example the fear of cannibalization, the sunk nature of existing technology and the inappropriate internal organisation of the evaluation and the operation of new innovative ventures. Economists have detected in racing and other models the disincentives caused by high current profits and have called this the replacement effect. In many markets with vertical differentiation, buyers will tend to focus on the best quality and will leave even slightly lower quality versions. Newcomers may not be inhibited by a desire to protect current success, and have strong incentives to engage in larger efforts to introduce 7
In game COOP.L, it is possible that followers’ investments decrease in LF. More specifically, this tends to happen when the leader-specific spillover is low. Note that there is only a decrease in followers’ investments when LF is high.
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superior imitations that hurt leaders’ profits. Empirical tendencies in some very large samples have detected that the lower efforts of the incumbent and the larger efforts of the challengers seem to prevail on average (Czarnitzki and Kraft 2004). In some markets, all of this may result in so called technological leapfrogging, as has been detailed for video game console industry (Schilling 2003). Followers that surpass pioneers may also be involved in more intensive innovative efforts for a variety of reasons, including second mover advantages (Schnaars 1994). But perhaps the rivalry that is modelled here also provides incentives that support these tendencies.
The purpose of this section is to analyse the role of spillovers with regard to the comparison of R&D investments of leaders and followers in all three games of the early entrance setting. The findings here, both with symmetric and asymmetric spillovers, are again extracted from numerical simulations.
4.1. Symmetric Spillovers With symmetric spillovers, comparing leaders’ and followers’ efforts yields Proposition 4. Proposition 4. With symmetric spillovers, competing leaders always invest more than followers (COMP.LF and COOP.F). However, a limited number of competing followers may invest more than cooperating leaders (COOP.L) provided that the symmetric spillover is low and more than half of the industry firms take the role of leader (k>n/2).
Tendencies described in Proposition 4 can be observed, in a more general interpretation of strategic investments, in industries in which competing leading firms are using different strategies and business models compared to smaller fringe firms. In the beer sector, for example, large multinational players focus on global advertising and intensive branding, while smaller ones rely on local specialized beer with little or no advertising efforts. Spillovers of goodwill and specific knowledge from one player to the other tend to be rather low and thus a scenario with a (low) symmetric spillover scenario applies. However, when a lot of leaders were then to cooperate, followers would realize higher investments. A similar argument could be made for high tech industries with symmetric high information flow between all players.
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4.2. Asymmetric Spillovers Sharper differences are detected with asymmetric spillovers as Proposition 5 details. However, these tendencies are somewhat complicated. The comparison between leader and follower efforts is mainly driven by the magnitude of the spillover from the leaders to the followers, Proposition 5. For each game
in the early entrance setting with
number of firms and the number of leaders and given values for
LF. FL=0, LL
there exists, given the
and
which is called the equalizer spillover, for which the following applies, with
FF,
a critical
e LF [
],
indicating the game
being played (COMP.LF, COOP.L, COOP.F): -
if
e LF< LF [
] then xL[ ]>xF[ ];
-
if
e LF= LF [
] then xL[ ]=xF[ ];
-
if
e LF> LF [
] then xL[ ]1,
when
e LF [
] are given. Note that
the investments of the leaders are always higher than the investments of the followers.
Analogously, when
e LF LFe). n=10 n=10 FF FF k=2
LL
0
1
0
e LF >1
e LF >1
1
e LF =0,764
e LF =0,856
k=5
LL
0
1
0
e LF >1
e LF >1
1
e LF =0,058
e LF =0,178
Table 2b. Numerical examples for the equalizer spillover LFe in the game with cooperating leaders and competing followers (COOP.L). Leaders invest more (less) than followers if and only if LF< LFe ( LF> LFe). n=10 n=10 FF FF k=2
LL
0
k=5
1
0
e LF =0,382
e LF =0,428
1
e LF =0,944
e LF =0,988
0
LL
1
0
e LF =0,007
e LF =0,033
1
e LF =0,966
e LF =0,988
Table 2c. Numerical examples for the equalizer spillover LFe in the game with competing leaders and cooperating followers (COOP. F). Leaders invest more (less) than followers if and only if LF< LFe ( LF> LFe). n=10 n=10 FF FF k=2
LL
0
1
0
e LF >1
e LF FFC ( FF< FFC). n=10 n=10 LL LL k=2
LF
0 0
C FF
1
C FF
k=5
1
=0,05
C FF
=0,05
C FF
=0,05
LF
=0,05
0 0
C FF
1
C FF
=0,0313
C FF
=0,0313
=0,0313
C FF
=0,0313
As can be derived from table 3a, the critical leader-specific spillover number of leaders k when LF=1
(
LL
C
/ k0 for
LF=0)
1
LL
C
is (slightly) increasing in the
and decreasing in the number of leaders when
Combining this with numerical tendencies with
FL=0
allows stating
Proposition 8. Proposition 8. Suppose leaders learn nothing from followers (
FL=0).
Let
LL
C
be the critical
spillover that is needed for leaders to invest more when they cooperate, see Proposition 7. -
With idea diffusion (
LF =
0), the critical spillover
the number of leaders increases (
9
LL
C
LL
C
tends to be small and to increase as
/ k > 0).
C
stands for the critical (C) leader-specific spillover, LL, for which investments of leaders are the same with competition and cooperation. FFC stands for the critical (C) follower-specific spillover, FF, for which investments of followers (F) are the same with competition and cooperation. LL
18
-
With blue print diffusion (
LF=1),
the critical spillover
decrease as the number of leaders increases (
LL
LL
C
tends to be large and to
C
/ k < 0).
From Proposition 8 and table 3a, it is clear that R&D cooperation among leaders is more likely to yield higher R&D investments compared to competition when the spillover from the leaders to the followers is small, or, more generally patent protection is good.
As can be seen in Table 3b, the values of the critical follower-specific spillover
C FF
are very low.
Consequently, cooperation among followers is almost always yielding higher R&D investments than with R&D competition. Only when there is no knowledge sharing at all among the followers (
FF=0),
R&D competition results in higher investments than with R&D cooperation among followers. Moreover, differentiation reveals that sign
C FF /
k = sign [2k–(1+n)].
Once again, externalities are the rationale for these results. For small (large) values of (
LL> LL
C
LL,
i.e.
LL< LL
C
), investments of one leader create a negative (positive) externality on the other leaders, by
which overinvestment (underinvestment) occurs. With cooperation, the internalization of the externalities leads to Proposition 8. An analogous reasoning applies to the followers.
6. IMPACT OF COOPERATION ON PROFITABILITY Cooperation of leaders in COOP.L or of followers in COOP.F not surprisingly results in an increase of profits of the cooperating players, compared to their profits in the game COMP.LF in which they compete. With spillovers equal to the critical spillovers that drive the comparison of cooperative and competitive efforts, the profits remain unchanged. A similar result is obtained in symmetric simultaneous models where profits are unaffected by cooperation if and only if the symmetric spillover equals ½ (assuming homogenous products). The reason is that for this spillover level, the R&D efforts are unchanged by a cooperative strategy. The same reason and tendency are also present in the sequential setting considered in this paper, both with symmetric or asymmetric spillovers.
With symmetric spillovers and
=
LC
the leaders’ profits do not change with R&D cooperation
(COOP.L) compared to R&D competition (COMP.LF). Analogously, for =
FC
, the followers’ profits
19
do not change with R&D cooperation (COOP.F) compared to R&D competition (COOP.F). With asymmetric spillovers, an analogous tendency applies with the critical values being respectively and
C FF ,
LL
C
as defined in Proposition 7.
7. IMPLICATIONS FOR STATIC WELFARE Finally, the impact of cooperation in R&D of leaders or followers on static welfare is analyzed. Static welfare (SW) is defined as the sum of consumer surplus (CS)10 and producer surplus (PS)11. Both symmetric and asymmetric spillovers are looked at.
Static welfare is, just like in the two stage models, following the tendencies of consumer surplus. Cooperation among the leaders (COOP.L) results in higher static welfare than with competing leaders (COMP.LF) if and only if the symmetric spillover LC
value
(
LL
C
(asymmetric spillover
LL)
is larger than a critical
). Analogously, cooperation among the followers (COOP.F) results in higher static
welfare than with competing followers (COMP.LF) if and only if the symmetric spillover (asymmetric spillover
FF)
is larger than a critical value
FC
(
C FF ).
In order to see which game yields the highest CS, PS and SW, the analysis is limited to some polar cases as the comparison of CS, PS and SW across games is very sensitive to the parameters of the model.
7.1. Symmetric spillovers In this section, it is analyzed which game yields the highest CS, PS and SW when spillovers are symmetric. This analysis is done for two scenarios, namely no information sharing ( =0) and full information sharing ( =1), see Table 4.
k
10 11
CS = PS =
i =1
qiL +
n j :k +1
q Fj
2
2 k i =1
π iL +
n j :k +1
π Fj
20
Table 4. Summary of games yielding the highest CS, PS and SW with symmetric spillovers. =0 =1
All k Small k Intermediate or High k
Highest CS COMP.LF COOP.F COOP.L
Highest PS COOP.L COOP.F COOP.L
Highest SW COMP.LF COOP.F COOP.L
As can be seen in Table 4, when there is no information sharing, CS is the highest when both leaders and followers cooperate. With a maximum spillover, a distinction needs to be made between a small number of leaders and an intermediate or high number or leaders. With a small number of leaders, the game with cooperating followers yields both the highest CS and PS. By contrast, when there are a lot of leaders, CS and PS are highest in game COOP.L12
Tendencies of static welfare can be summarized in Proposition 9. Proposition 9 When there is no information sharing among firms ( =0), static welfare is the highest when both leaders and followers compete in R&D. With full information sharing ( =1), the game with cooperating followers (COOP.F) yields the highest SW when the number of leaders is small and the game with cooperating leaders (COOP.L) results in the highest SW when the number of leaders is intermediate or high.
However, it is again not appropriate to give too much weight to the benchmark case of symmetric spillovers. Still it is interesting to observe the possibility that the leader may be better of letting imitators cooperate. The question then is if similar tendencies apply with asymmetries. The next result summarizes some findings.
12
Note that the notions of ‘small’ and ‘intermediate or high’ are not exactly the same for the analysis of CS and PS. For example, when n=10 and k=2 with =1, CS is the highest in game COOP.F, but with k 3, game COOP.L leads to the highest CS. However, when n=10 and k 3 with =1, PS is the highest with in game COOP.F, but with k 4, game COOP.L leads to the highest PS.
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7.2. Asymmetric spillovers With asymmetric spillovers, it is assumed that there are no within group spillovers with competition among members and maximum spillovers in case of cooperation among member firms. In table 5, it is summarized which game results in the highest CS, PS and SW.
Table 5. Summary of games yielding the highest CS, PS and SW with asymmetric spillovers Small k Intermediate k with Intermediate k with High k
LF=0
LF=1
Highest CS COOP.F COOP.L COOP.F COOP.L
Highest PS
Highest SW COOP.F COOP.L COOP.F COOP.L
COOP.L
On the one hand, the game with the cooperating followers yields the highest consumer surplus when there is either a small number of leaders or when there is an intermediate number of leaders combined with blue print diffusion from leaders to followers (
LF=1).
On the other hand, the game with
cooperating leaders results in the highest CS when there are a lot of leaders or there is blue print diffusion (
LF=0)
from an intermediate number of cooperating leaders to the followers. Producer
surplus is, however, always the highest when leaders cooperate.
Tendencies of static welfare are formulated in Proposition 10. Proposition 10 Suppose leaders learn nothing from followers ( COMP.LF;
LL=1
and
FF=0
in COOP.L; and
FL=0)
LL=0
and
and suppose that FF=1
LL=0
and
FF=0
in
in COOP.F. In this case, the game
that is resulting in the highest static welfare depends on the number of leaders and the level of the spillover from the leaders to the followers. o
With a low number of leaders, the game with cooperating followers is yielding the highest static welfare.
o
With an average number of leaders and idea diffusion (
LF=0)
from leaders to followers,
the game with cooperating leaders yields the highest static welfare. o
With an average number of leaders and idea diffusion (
LF=1)
from leaders to followers,
the game with cooperating followers yields the highest static welfare.
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o
With a high number of leaders, the game with cooperating leaders yields the highest static welfare.
Static welfare is characterized by the same tendencies of consumer surplus. In other words, the CS tendencies always dominate the tendencies of PS, which is in line with results of two stage models. It is important to notice that in some cases cooperation by the followers yields the highest welfare.
8. CONCLUSION AND SOME POLICY IMPLICATIONS This paper analyzes the strategic investments of leaders and followers in a four stage setting with sequential investment decisions followed by sequential output decisions, given symmetric or asymmetric spillovers
Firstly, it has been shown that the impact of symmetric spillovers on R&D investments of leaders and followers is not that straightforward as in the two stage models with simultaneous moves. More specifically, the symmetric spillover is usually having a negative impact on cooperative investments of leaders and followers, which is due to, respectively, the spillover from the leaders to the followers and the spillover from the followers to the leaders. With asymmetric spillovers, tendencies are, however, the same as in the two stage models and help us to understand tendencies with symmetric spillovers.
Secondly, the investments of leaders and followers have been compared for both symmetric and asymmetric spillovers. It has been described that with symmetric spillovers, the leaders always invest more than the followers, except for some limited (and irrelevant) cases. With asymmetric spillovers, the spillover from leaders to followers plays a crucial role in the comparison of leaders’ and followers’ investments. After all, if this spillover is sufficiently high, the followers tend to realize higher R&D investments than the followers.
Thirdly, the economic performance of R&D cooperation has been compared with R&D competition. This analysis yields similar findings as in the two stage models with simultaneous moves as critical spillovers drive this comparison. With symmetric spillovers, investments of leaders with R&D cooperation are higher than with R&D competition if the spillover is larger than (n-k+1)/(n-k+2).
23
Similarly, followers realize higher investments with R&D cooperation than with R&D competition if the spillover is larger than ½. When analyzing the case with asymmetric spillovers, critical leaderspecific (follower-specific) spillovers determine whether R&D cooperation enhances R&D investments compared with R&D cooperation.
This paper contributes to a better understanding of strategic investment incentives and the resulting implications for profitability and welfare when decisions on R&D and output are made sequentially. Specifically, the paper is able to analyze when followers tend to invest more than leaders. Moreover, the paper also covers cooperation among first and second movers. Consequently, this study could help to develop policy guidelines dealing with R&D cooperation.
A detailed comparison of static welfare, however, in the different games is highly dependent on the parameters of the model. Therefore, it is better to look at those scenarios in which the static welfare is the highest. On this basis, some basic insights can be obtained by which some policy guidelines could be formulated. When only a few firms dominate, static welfare is the highest when followers cooperate. But when an industry is dominated by a high number of firms, the leaders should cooperate in order to have the highest static welfare. With an intermediate number of leading firms, the knowledge flow from leaders to followers determine when static welfare is the highest. More specifically, with blue print diffusion, static welfare is the highest with followers cooperating while with idea diffusion, it would be recommended to allow for cooperation among leaders.
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LIST OF FIGURES Figure I: Early Entrance Setting
25
FIGURES Figure I: Early Entrance Setting The early entrance setting for k = 2 and n = 5. Extension to the general case of 1 < n is obvious. The spillover from followers to leaders in the benchmark case of symmetric spillovers (
LL=
FL
LF=
k
is set equal to zero, except FF=
FL=
).
26
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