Diapositiva 1

1
Fifth MDEF, Urbino 25-27 September 2008
On Dynamic RD Networks Gian Italo
Bischi University of Urbino e-mail
gian.bischi_at_uniurb.it Fabio Lamantia University
of Calabria e-mail lamantia_at_unical.it
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• Economic framework competition among firms, role
  of RD
• Firms competing in a market also invest in
  knowledge and new technologies
• RD efforts more effective through collaboration
  information share
• Partnerships, agreements between firms, RD
  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)

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Main research questions

• How to model RD 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?

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• Outline of the talk
• Review of some literature on competition and
  cooperation in RD
• Rent seeking (patent contests) and
  RD networks
• Cornot Oligopoly games with RD
  efforts
• Clusters of firms, industrial Districts
• Cooperation for sharing of technological
  knowledge, technological cartels
• Accumulated knowledge
• RD agreement networks
• A two stage Cournot Oligopoly model with RD,
  spillovers and
• partnership network
• Early results
• Possible extensions of the model (to be done)

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A free-riding dilemma due to spillovers Research
investments or just spillovers?
Population of N firms, each with two strategies
available S1 invest in RD 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
RD) Payoffs are functions U1(x) and U2(x)
defined in 0,1
Profit U1 (ab)x c Profit U2 bx
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Profit U1 (ab)x c Profit U2 bx Each
player decides by comparing payoff functions
c lt a
c gt a
U1
ab-c
U2
b
b
U2
ab-c
U1
x
0
1
c/a
x
0
1
-c
-c
Collective efficiency xU1 (1-x)U2
x(axbx-c) (1-x)bx ax2 (b-c)x Collective
optimum for x 1
Individual optimal choice different from
collective optimual choice
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Some related models in the literature
Rent seeking games (patent contests) with RD
efforts Reinganum, J.F. (1981). "Dynamic Games
of Innovation," Journal of Economic Theory, Vol.
25 Reinganum, J.F. (1982). "A dynamic game for
RD patent protection and competitive behavior,"
Econometrica, Vol. 50
V post-innovation profits ei RD efforts of
firm i Xi effective RD (including
partnerships and spillovers)
probability to get the patent (technology
innovation)
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• Rent seeking games with RD partnership networks
• Peter-J. Jost Product innovation and bilateral
  collaborations. GEABA Discussion paper n. 7/2004
• Effective RD include a network of links due to
  bilateral
• agreements for complete sharing RD results
• 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 RD include both partnership and
  involuntary spillovers
• Collusive cartels of firms that maximize joint
  profits

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Cournot Oligopoly games with RD efforts and
spillovers as cost-reducing externalities D'Aspr
emont, Jacquemin (1988) "Cooperative and
noncooperative RD duopoly with spillovers, The
American Economic Review, 78, 1133-1137 Bisc
hi, Lamantia (2002) Nonlinear duopoly games with
positive cost externalities due to spillover
effects Chaos, Solitons Fractals,
vol.13 f(Q)a?? bQ, Ci(qi, qj )
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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
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Cooperation, deliberate sharing of technological
knowledge, creation of technological
cartels D'Aspremont, Jacquemin (1988)
"Cooperative and noncooperative RD 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 RD 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 RD has
emerged as an important research topic. A clear
understanding of this phenomenon is important for
industrial policies and antitrust
legislation

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Models with RD networks Goyal, S. and Joshi, S
. "Networks of Collaboration in Oligopoly, Games
and Economic Behavior, 2003. Meagher K., Rogers
M., Network density and RD spillovers, Journal
of Economic Behavior Organization, 2004. Goyal
S., Moraga-Gonzales J.L., "RD 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 RD
results. No spillovers are considered. RD
efforts are sunk costs (no knowledge accumulation
is considered). Firms compute the Cournot optimal
quantity and then maximize profits with respect
to RD efforts. The influence of connectivity k
is considered.
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• A two stage Cournot oligopoly model
• A network of N firms divided into subnetworks
  where firms can make bilateral agreements to
  share RD results with some partner firms
• A precompetitive stage where agents commit
  themselves to levels of RD 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 RD, and the
  cost of own RD efforts.
• Each firm can have a cost reduction by means of
• its own RD
• 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.

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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,bgt0
Q ? qi total output in the market. These N
firms are assumed to form a global network
subdivided into h subnetworks, say sj,
j1,...,h, each formed by nj firms Inside each
sj firms can form bilateral agreements for
sharing RD 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.
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Linear cost function of i-th firm belonging to
subnetwork sj , with marginal cost
ei RD effort of firm i c marginal cost
bj?0,1 internal spillovers coefficients
(with non-connected firms in sj) b-j?0,1
regulate external spillovers
cost of private RD efforts
Profit function of the representative firm in
subnetwork sj
Cournot output, solution of the optimization
problem
Corresponding max profit of the representative
firm in subnetwork sj
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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
Hence MR decreases for increasing ki and so
marginal profit can become negative and profit
decreases as ki exceed a given threshold
If nj0 and all bi 0 then
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
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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
As the number of links ki in si increses,
marginal revenue in si declines and this is an
advantage for competitors in network sj
MRei(sj)
convex parabola,
If bib-j0 then
bi0.6 b-j0.5 ni10 nj10
bi0 b-j0.5 ni10 nj10
bi0 b-j 1 ni10 nj10
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The dynamic model of repeated choice of RD
efforts Firms behave myopically, i.e. they
adaptively adjust their RD efforts ej over time
towards the optimal strategy, following the
direction of the local estimate of expected
marginal profits according to "gradient dynamics"

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h 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(t1)-
ej(t)/ ej(t) is assumed to be proportional to
the expected marginal profit.
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 RD and Spillover parameters ?, ?1, ?2,
  ?-1, ?-2

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Effort steady states
Three boundary equilibria O(0,0)
E1(-A1/C1,0) E2(0,-A2/C2) located along the
invariant coordinate axes An interior
equilibrium E3
.
Analytical results on stability are obtainable in
some benchmark cases without spillovers
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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 bi on stability and basins.
• Comparison with the results by Goyal-Montaga, a
  benchmark case obtained for n20 and all ?0
  (influence of k)

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Space of effort Possible effect of symmetric
increment of links
e2
e2
E2
E2
E3
E3
e1
E1
e1
E1
k1 k213 Just one link is added in each
network! Inner equilibrium becomes a saddle whose
stable set (along the diagonal) is the basin
boundary of corner equilibria
a90 b1 c6 n120 n220 k1 k212 a1a20.3
g9 No spillovers
23
e1
Asymptotic RD efforts
a90 b1 c6 n120 n220 k1 k212 a1a20.3
g9 c.i. (05,.1)
Without spillovers, RD investments of networks
converge to a steady state E3 for any i.c. in
B(E3) As ?1 increases, network 1 strongly
increases its efforts whereas network 2
drastically drops its one to zero. Consequently
only network 1 invest in RD However network 2
can still make small profits by cutting off RD
expenses
e2
?1
Profits
p1
p2
?1
24
e1
Asymptotic RD efforts
a90 b1 c6 n120 n220 k1 k213 a1a20.3
g9 i.c. e1(0)0.1, e2(0) 0.05
e2
Without spillovers, who invests more in RD in
the first period wins the competition Bistability
If ?1 exceed a given threshold, network 1 starts
investing in RD and network 2 quits its
effort Discontinuity in efforts and profits
e2
?1
E3
p1
E2
Profits
Basin of E2 shrinks as ?1 increases (here ?10.2)
p2
E1
e1
?1
25
a180 b1 c4 n120 n220 k1 k27 a1a20.4
g9 No spillovers
e2
E1
E1
E3
E3
E2
E2
e1
e1

• Chaotic synchronization E3 is an unstable
  equilibrium and a chaotic attractor exists along
  the diagonal
• Starting from an i.c. outside the diagonal
  competitors will eventually decide the same RD
  efforts, a chaotic trajectory in this case

• Effect of decreasing k13
• Correlated chaotic attractor around the unstable
  equilibrium E3
• Lakes of B(8) are nested inside the basin of the
  chaotic attractor

26
e2
Space of effort Chaos and multistability
e2
a10.8 a20.5 and k18 Chaotic attractor and
increased complexity of basins of attractors on
invariant axis
E2
e2
E3
e2
e1
E1
E3
e2
Stable equilibrium in a symmetric case a120 b1
c20 n120 n220 k1 10 k210 a1a20.5
g45 Again no spillovers
e2
E2
E1
E2
E2
e1
a10.8 a20.5 Chaotic attractor with asymmetric
speed of adjustment
E1
e1
e1
27
p
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 As k1 is below k2
network 1 increases its efforts whereas network 2
decreases its effort to zero
k
p1
a140 b1 c6 n120 n220 k211 a1a20.3
g9 No spillovers e1(0)0.2, e2(0) 0.2
p2
k1
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• 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.
• RD efforts are not sunk costs, as knowledge is
  accumulated over time

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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 firms accumulated RD capital
and consider current investment in RD as a
strategic element. M. L. Petit and B.Tolwinski,
"RD 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
firms potential for innovation depends not omly
on the level of its current investment in RD,
but rather on the accumulated capital invested in
RD 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 RD competition,
Research in Economics, 56 Confessore G., Mancuso
P. (2002). RD spillovers and absorptiove
capacity in a dynamic oligopoly, Operations
Research Proceedings (2003)
30
The level of knowledge accumulated up to time t
can be modelled as
r?0,1 obsolescence factor which exponentially
discounts older info Xi (t) knowledge gained by
firm i at time t, proportional to effective RD
The knowledge capital stock can be obtained
recursively (i.e. inductively) as
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.
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Derivation of Cournot equilibrium
quantity Profit function for i-th oligopolist

where Q is the total industry output
F.O.C.
i.e.
Let the cost function be linear in qi, i.e.
constant marginal cost
. Summing up the n relations
from which
Substituting
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Space of effort Possible effect of asymmetric
links
e2

• Inner equilibrium E3 is stable
• Basins of attractor located on invariant axis are
  in red and green

E2
E3
E1
e1
a90 b1 c6 n120 n220 k1 1 k211 a1a20.3
g9 Again no spillovers
33
Specification of aggregate parameters of the map
Nn1n2