Fifth MDEF, Urbino 25-27 September 2008
On Dynamic R&D Networks
Gian Italo Bischi
University of Urbino
e-mail: gian.bischi@uniurb.it
Fabio Lamantia
University of Calabria
e-mail: lamantia@unical.it
Economic framework: competition among firms, role of R&D
 Firms competing in a market also invest in knowledge and new
technologies
R&D efforts more effective through collaboration & information share
Partnerships, agreements between firms, R&D networks
Knowledge spillovers
Trade off between: 8 competition and collaboration
8 knowledge share and protection
Research joint ventures and deliberate sharing of technological knowledge among
firms competing in the same markets have become a fairly widespread form of
industrial cooperation. The economic literature provides strong empirical evidence of
the existence of such arrangements (M.L. Petit,2000)
Main research questions
• How to model R&D choices over time for
firms who share research information but
compete in the marketplace?
• How does competition among different
networks with such a structure look like
and evolve over time?
• What is the effect of knowledge spillovers
on investments decisions?
Outline of the talk
Review of some literature on competition and cooperation in R&D
Rent seeking (patent contests) and R&D networks
Cornot Oligopoly games with R&D efforts
Clusters of firms, industrial Districts
Cooperation for sharing of technological knowledge, technological cartels
Accumulated knowledge
R&D agreement networks
A two stage Cournot Oligopoly model with R&D, spillovers and
partnership network
Early results
Possible extensions of the model (to be done)
A free-riding dilemma due to spillovers
Research investments or just spillovers?
Population of N firms, each with two strategies available:
S1: invest in R&D
S2: just spillovers
Let x = n/N  [0,1] be the fraction of players that choose strategy S1,
(1 x) choose S2
x = 0 : all choose S2 (just spill)
x = 1 : all choose S1 (invest in R&D)
Payoffs are functions U1(x) and U2(x) defined in [0,1]
Profit U1 = (a+b)x – c ;
Profit U2 = bx
Profit U1 = (a+b)x – c ;
Profit U2 = bx
Each player decides by comparing payoff functions
c<a
c>a
U1
a+b-c
b
U2
U2
b
a+b-c
U1
0
-c
c/a
1
x
0
1
x
-c
Collective efficiency: xU1 + (1-x)U2 = x(ax+bx-c) +(1-x)bx = ax2 + (b-c)x
Collective optimum for x = 1
Individual optimal choice different from collective optimual choice
Some related models in the literature
Rent seeking games (patent contests) with R&D efforts
Reinganum, J.F. (1981). "Dynamic Games of Innovation," Journal of
Economic Theory, Vol. 25
Reinganum, J.F. (1982). "A dynamic game for R&D: patent protection
and competitive behavior," Econometrica, Vol. 50
Xi
i  V n
 Ci (ei )
 j 1X j
V = post-innovation profits
ei = R&D efforts of firm i
Xi = effective R&D (including partnerships and spillovers)
Xi
probability to get the patent (technology innovation)
 jXj
Rent seeking games with R&D partnership networks
Peter-J. Jost “Product innovation and bilateral collaborations”. GEABA
Discussion paper n. 7/2004
•Effective R&D include a network of links due to bilateral
agreements for complete sharing R&D results X i  ei 
k
e
j 1
j
kn
•Stability of networks, i.e. the creation/destruction of a new link
increases/decreases profits of partners?
Peter-J. Jost “Joint ventures in patent contests with spillovers and the
role of strategic budgeting”. GEABA Discussion paper n. 7/2006
•Effective R&D include both partnership and involuntary spillovers
n
X i  ei   ij e j ij  0,1
j 1
•Collusive cartels of firms that maximize joint profits:
k
max   j
ei
j 1
Cournot Oligopoly games with R&D efforts
and spillovers as cost-reducing externalities
D'Aspremont, Jacquemin (1988) "Cooperative and noncooperative
R&D duopoly with spillovers”, The American Economic Review, 78,
1133-1137
max f (Q)qi  Ci (qi , X i )  ei2
qi
with X i  effectiveR & D
X i  X i (e1 ,..., en )
Bischi, Lamantia (2002) “Nonlinear duopoly games with positive cost
externalities due to spillover effects” Chaos, Solitons & Fractals, vol.13
f(Q)=a bQ,
ci qi
Ci(qi, qj )=
1   ij q j
Clusters of firms, Industrial Districts
Horaguchi (2008), Economics of Reciprocal Networks: Collaboration in knowledge and
Emergence of Industrial Clusters, Journal Computational Economics, vol. 31
Bischi, Dawid and Kopel (2003), Gaining the Competitive Edge Using Internal and
External Spillovers: A Dynamic Analysis, Journal of Economic Dynamics and Control,
vol. 27.
Bischi, Dawid and Kopel (2003), Spillover Effects and the Evolution of Firm Clusters
Journal of Economic Behavior and Organization, vol. 50.
Location and proximity are important factors in exploiting knowledge spillovers
Audretsch and Feldman (1996), R & D Spillovers and the Geography of Innovation and
Production. American Economic Review vol.86
Head, Ries and Swenson (1995), Agglomeration Benefits and Location Choice:
Evidence from Japanese Manufacturing Investments in the United States. Journal of
International Economics, vol. 38
Cooperation, deliberate sharing of technological knowledge,
creation of technological cartels
D'Aspremont, Jacquemin (1988) "Cooperative and noncooperative R&D
duopoly with spillovers”, The American Economic Review, vo. 78
Baumol, W.J., 1992. Horizontal collusion and innovation. The Economic
Journal 102
Kamien, Mueller and Zang (1992) "Research Joint Ventures and R&D
Cartels." American Economic Review
Petit, M.L., Sanna-Randaccio, F., Tolwinski B. (2000). "Innovation and
Foreign Investment in a Dynamic Oligopoly," International Game
Theory Review, Vol.2
Effects of cooperation in R&D has emerged as an important research topic. A clear
understanding of this phenomenon is important for industrial policies and antitrust
legislation
Models with R&D networks
Goyal, S. and Joshi, S . "Networks of Collaboration in
Oligopoly”, Games and Economic Behavior, 2003.
Meagher K., Rogers M., Network density and R&D spillovers,
Journal of Economic Behavior & Organization, 2004.
Goyal S., Moraga-Gonzales J.L., "R&D Networks", RAND
Journal of Economics, 2001.
A network of N firms, each linked with k firms, 0 k N1, by a bilateral
agreement for a complete share of R&D results.
No spillovers are considered.
R&D efforts are sunk costs (no knowledge accumulation is considered).
Firms compute the Cournot optimal quantity and then maximize profits with
respect to R&D efforts.
The influence of connectivity k is considered.
A two stage Cournot oligopoly model:
A network of N firms divided into subnetworks where firms can make
bilateral agreements to share R&D results with some partner firms
•A “precompetitive stage” where agents commit themselves to levels of
R&D efforts in the direction of increasing profits (following positive
marginal profits by a myopic gradient dynamics)
•A Cournot competitive stage where firms choose the best reply
quantities taking into account the cost-reducing effects of effective R&D,
and the cost of own R&D efforts.
Each firm can have a cost reduction by means of:
- its own R&D
- knowledge by partner firms
- Spillovers (internal and external to the subnetwork)
A natural interpretation of networks may be to consider the subnetworks
as representing different Countries or industrial districts, characterized
by different rules for partnership or different abilities to take advantage
from spillovers.
The static model
A homogenous-product oligopoly with N quantity setting firms
The N firms operate in a market characterized by a linear demand
function
p = a  b Q,
a,b>0
Q =  qi total output in the market.
These N firms are assumed to form a global network subdivided into h
subnetworks, say sj, j=1,...,h, each formed by nj firms
Inside each sj firms can form bilateral agreements for sharing R&D
results.
We assume that each sj is a symmetric network of degree kj, with 0kjnj-1
i.e. every firm in sj has the same number of collaborative ties kj
kj is a parameter that represents the level of collaborative attitude of
subnetwork sj.
Linear cost function of i-th firm belonging to subnetwork sj , with marginal cost
ci ( s j )  c  ei 


  j el n j  1  k j    j

k j el

effort by linked firms in s j
effort by nonlinkedfirms in s j
ei = R&D effort of firm i
c = marginal cost
j[0,1] internal spillovers coefficients (with non-connected firms in sj)
-j[0,1] regulate external spillovers
Profit function of the representative firm in subnetwork sj
e
m
ms j

effort by firms not in s j
cost of private
R&D efforts






 i ( s j )  a  b qi ( s j )   q p   ci ( s j )qi ( s j )  e 2 i ( s j )


p i




Cournot output, solution of
the optimization problem max  i (s j )
qi
qi ( s j ) 
a  Nci ( s j )   c p
p i
b(1  N )
Corresponding max profit of the
representative firm in subnetwork sj
 a  Nci ( s j )   c p 


2
p i
 i (s j )  


e
i

b (1  N )


2

Comparative statics
Let us consider the profit of the representative firm in network si.
If ei increases then the marginal cost is constant (2) and marginal
revenue MR increases, being
MRei ( si ) 
2(ni  ki (1  i )  i (ni  1)  n j (1    j )) 2
1  n1  n2 
2
0
Hence MR decreases for increasing ki and so marginal profit can
become negative and profit decreases as ki exceed a given threshold
If nj=0 and all i = 0 then
MRei (si ) 
2( ni  ki )2
1 n1  n2 
2
the same as in GM
When a firm has more collaborators an increase in its effort not only
lowers its own costs, but also the costs of collaborators, that become
stronger competitors.
The same effect, for similar reasons, is observed as internal or external
spillovers increase
Cross influence on marginal profits
As the representative firm in network si increases ei this has an impact
also on the profit of firms in network sj. The (linear) coefficient of ei in
is:
MRei(sj) =
2ni2 (1  ki (1  i )  i (ni  1)    j (1  ni )) 2
As the number of links ki
 0 in s increses, marginal
2
i
1  n1  n2 
revenue in si declines and
this is an advantage for
  j (ni  1)   i (ni  1)  1
competitors in network sj
min
convex parabola, ki

1  i
If i=-j=0
then kimin  1
kimin  0
i=0.6 j=0.5 ni=10 nj=10
0  kimin  ni  1
i=0 j=0.5 ni=10 nj=10
kimin  ni  1
i=0 j= 1 ni=10 nj=10
The dynamic model of repeated choice of R&D efforts
Firms behave myopically, i.e. they adaptively adjust their R&D efforts ej
over time towards the optimal strategy, following the direction of the
local estimate of expected marginal profits according to "gradient
dynamics"
e j (t  1)  e j (t )   j (e j (t ))
 j
e j
, j  1,..., h
h=2
e j (t  1)  e j (t )   j (e j (t ))
 j
e j
, j  1,2
Two subnetworks s1 and s2 with n1 and n2 firms,
connection degrees k1 and k2 respectively.
We assume linear speeds of adjustment aj(ej) = ajej
i.e. the relative effort change:
[ej(t+1)- ej(t)]/ ej(t)
is assumed to be proportional to the expected marginal profit.
e j (t  1)  e j (t ) 
 j e j (t )
b(1  N )
2
A  B e (t )  C e (t ), i, j  1,2; i  j
j
j i
j
j
Where Aj, Bj and Cj are given functions of the model parameters:
•Oligopoly parameters: n1, n2, a, b (demand); c (marginal cost)
•Network parameters: k1, k2 (subnetwork degrees)
•Cost of R&D and Spillover parameters: , 1, 2, -1, -2
Effort steady states
Three boundary equilibria:
O=(0,0); E1=(-A1/C1,0); E2=(0,-A2/C2)
located along the invariant coordinate axes
 A2 B1  A1C2 A1B2  A2C1 

,
An interior equilibrium E3= 
 C1C2  B1B2 C1C2  B1B2 
Analytical results on stability are obtainable in some benchmark
cases without spillovers
Some results
•Some examples of attracting sets and basins in the space of efforts
•Influence of internal and external spillovers on efforts and profits of
both networks (own network and other network).
•Intra-network and inter-network effects
•Influence of ki and i on stability and basins.
•Comparison with the results by Goyal-Montaga, a benchmark case
obtained for n2=0 and all =0 (influence of k)
Space of effort: Possible effect of symmetric increment of links
e2
e2
E2
E2
E3
E3
E1
e1
k1= k2=13
E1
e1
a=90 b=1 c=6 n1=20 n2=20 k1= k2=12
1=2=0.3 =9
Just one link is added in each network!
No spillovers
Inner equilibrium becomes a saddle whose
stable set (along the diagonal) is the basin
boundary of corner equilibria
e1
Asymptotic R&D efforts
a=90 b=1 c=6 n1=20 n2=20 k1= k2=12
1=2=0.3 =9 c.i. (05,.1)
Without spillovers, R&D
investments of networks
converge to a steady state E3
for any i.c. in B(E3)
e2
1
1
Profits
As 1 increases, network 1
strongly increases its efforts
whereas network 2 drastically
drops its one to zero.
Consequently only network 1
invest in R&D
However network 2 can still
make small profits by cutting
off R&D expenses
2
1
e1
Asymptotic R&D efforts
a=90 b=1 c=6 n1=20 n2=20 k1= k2=13
1=2=0.3 =9 i.c. e1(0)=0.1, e2(0) = 0.05
e2
Without spillovers, who invests
more in R&D in the first period
wins the competition
e2
1 E
2
1
E3
Profits
Bistability
If 1 exceed a given threshold,
network 1 starts investing in R&D
and network 2 quits its effort
Discontinuity in efforts and profits
2
E1
e1
1
Basin of E2 shrinks as 1 increases
(here 1=0.2)
a=180 b=1 c=4 n1=20 n2=20 k1= k2=7
1=2=0.4 =9 No spillovers
e2
E1
E1
E3
E3
E2
e1
E2
•Chaotic synchronization: E3 is an
unstable equilibrium and a chaotic
attractor exists along the diagonal
•Effect of decreasing k1=3
•Starting from an i.c. outside the diagonal
competitors will eventually decide the
same R&D efforts, a chaotic trajectory in
this case
•Lakes of B(∞) are nested inside the
basin of the chaotic attractor
•Correlated chaotic attractor around the
unstable equilibrium E3
e1
e2
Space of effort: Chaos and multistability
e2
1=0.8 2=0.5 and
E2
k1=8
Chaotic attractor and
increased complexity
of basins of
attractors on
invariant axis
e2
E3
e2
E1
e1
E3
Stable equilibrium in a
symmetric case
a=120 b=1 c=20
n1=20 n2=20 k1= 10
k2=10 1=2=0.5
=45
Again no spillovers
e2
e2
E2
E1
e1
E22
1=0.8 2=0.5
Chaotic attractor with
asymmetric speed of
adjustment
E1
e1
e1

Profits
Goyal-Moraga (one network and no spillovers)
shows that profit is maximized for intermediate
levels of connectivity k
The same results is not necessarily
true with multi-network competition
k
1
As k1 is below k2 network 1
increases its efforts whereas
network 2 decreases its effort to
zero
a=140 b=1 c=6 n1=20
n2=20 k2=11
1=2=0.3 =9
No spillovers
2
e1(0)=0.2, e2(0) = 0.2
k1
Possible extensions of the model
Formation of joint ventures (or cartels) where as a result they
maximize the overall profit of the whole subnetwork instead of the
individual profits.
R&D efforts are not sunk costs, as knowledge is accumulated
over time
Accumulated knowledge, Obsolescence of intellectual properties
Spence, M. (1984). "Cost reduction, competition, and industry performance,"
Econometrica, Vol. 52.
Cost-reducing technological innovations is an outcome of the firm’s accumulated R&D capital
and consider current investment in R&D as a strategic element.
M. L. Petit and B.Tolwinski, "R&D cooperation or competition?" European Economic
Review 43 (1999)
Bischi, G.I. and Lamantia, F. (2004) "A Competition Game with Knowledge
Accumulation and Spillovers" International Game Theory Review 6, 323-342
A firm’s potential for innovation depends not omly on the level of its current investment in R&D,
but rather on the accumulated capital invested in R&D over time, a kind of “history
dependence” that requires the use of dynamic models
Absorption capacity
Confessore G., Mancuso P. (2002) "A Dynamic model of R&D competition”,
Research in Economics, 56
Confessore G., Mancuso P. (2002). “R&D spillovers and absorptiove capacity in a
dynamic oligopoly”, Operations Research Proceedings (2003)
The level of knowledge accumulated up to time t can be modelled as
t
zi (t )   t  k X i (k )
k 0
0,1: obsolescence factor which exponentially discounts older info
Xi (t): knowledge gained by firm i at time t, proportional to effective R&D
The knowledge capital stock can be obtained recursively (i.e. inductively) as:
t 1
zi (t )  X i (t )    t 1 k X (k )  X i  t    zi (t  1).
k 0
i.e. the accumulated knowledge at time t is the sum of the effective knowledge Xi(t)
acquired during last round, and a discounted fraction of the knowledge capital stock of
the previous period
Both the cost reduction effect and the capacity to exploit spillovers (i.e. the “absorption
capacity”, see Confessore and Mancuso, 2002) may be assumed to depend on the
accumulated knowledge zi.
Derivation of Cournot equilibrium quantity
 i (qi )   a  bQ  qi  ci (qi , ei , ei )   ei2
Profit function for i-th oligopolist
where Q is the total industry output
F.O.C.
 i
 a  bQ  bqi  ci, (qi , ei , ei )  0
qi
i.e.
bqi  a  bQ  ci, (qi ),
ci'  ci
Let the cost function be linear in qi, i.e. constant marginal cost
.
Summing up the n relations
from which
Substituting:
bQ 
n
n
i 1
j 1
bqi  bQ  na  bnQ  c j
na   nj 1c j
n 1
n

na


1
j 1c j

qi   a 
 ci
b 
n 1
 a  nci   j ic j

b  n  1

i  1,..., n
Space of effort: Possible effect of asymmetric links
e2
•Inner equilibrium E3 is
stable
E2
•Basins of attractor
located on invariant axis
are in red and green
E3
E1
a=90 b=1 c=6 n1=20 n2=20 k1= 1
k2=11 1=2=0.3 =9
Again no spillovers
e1
Specification of aggregate parameters of the map
Ai  2(a  c)( N  (ni  1) i  ki (1  i )  n j   j )
Bi  2n j  N  i (ni  1)  ki (1  i )  n j   j    i (1  n j )  1  k j 1   j     j (n j  1) 

2
Ci  2 [ N  i (ni  1)  ki (1  i )  n j   j ] 1  n j  (1  ki ( i  1)  i  ni i )  ni n j   j    1  N 
N=n1+n2
