On Dynamic R&D Networks

Gian Italo BischiUniversity of Urbinoe-mail: gian.bischi@uniurb.it

Fabio LamantiaUniversity of Calabriae-mail: lamantia@unical.it

Fifth MDEF, Urbino 25-27 September 2008

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: competition and collaboration 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 DistrictsCooperation for sharing of technological knowledge, technological cartelsAccumulated knowledgeR&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

0 1 x

U2

U1

c < a

-c

a+b-c

b

0 1 x

U2

U1

c > a

-c

a+b-cb

Collective efficiency: xU1 + (1-x)U2 = x(ax+bx-c) +(1-x)bx = ax2 + (b-c)xCollective optimum for x = 1

Individual optimal choice different from collective optimual choice

Profit U1 = (a+b)x – c ; Profit U2 = bx

Each player decides by comparing payoff functions

c/a

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

1

( )ii i in

j j

XV C e

X

V = post-innovation profits ei = R&D efforts of firm iXi = effective R&D (including partnerships and spillovers)

jj

i

X

Xprobability to get the patent (technology innovation)

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

•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

•Collusive cartels of firms that maximize joint profits:1

maxi

k

je

j

1

0,1n

i i ij j ijj

X e e

nkeeXk

jjii

1

Cournot Oligopoly games with R&D effortsand spillovers as cost-reducing externalities

D'Aspremont, Jacquemin (1988) "Cooperative and noncooperative R&D duopoly with spillovers”, The American Economic Review, 78, 1133-1137

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, qj )=

),...,(

&),()(max

1

2

nii

iiiiiiq

eeXX

DeffectiveRXwitheXqCqQfi

jij

ii

q

qc

1

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 N1, 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 firmsThe 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 0kjnj-1i.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.

ei = R&D effort of firm ic = marginal cost j[0,1] internal spillovers coefficients (with non-connected firms in sj)-j[0,1] regulate external spillovers

Cournot output, solution ofthe optimization problem

)(max jiq

si

Corresponding max profit of the representative firm in subnetwork sj

Linear cost function of i-th firm belonging to subnetwork sj , with marginal cost

Profit function of the representative firm in subnetwork sj

cost of privateR&D efforts

)()()()()( 2jijiji

ippjiji sesqscqsqbas

)1(

)(

)(Nb

csNca

sq ippji

ji

2

2

)1(

)(

)( iip

pji

ji eNb

csNca

s

j

jjj

s

smmj

s

jjlj

s

ljiji ekneekecsc

in not firmsby effort

in firms nonlinkedby effort in firms linkedby effort

1)(

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

Comparative statics

2

2

1 2

2( (1 ) ( 1) (1 ))( ) 0

1i

i i i i i j je i

n k n nMR s

n n

Hence MR decreases for increasing ki and so marginal profit can become negative and profit decreases as ki exceed a given threshold

2

21 2

2( )

1( ) i i

i

n ke i n n

MR s

If nj=0 and all = 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

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:

2 2

2

1 2

2 ( 1 (1 ) ( 1) (1 ))0

1

i i i i i j in k n n

n n

convex parabola, i

iiiji

nnk

1

1)1()1(min

If i=-j=0 then 1min ik

0min ik min0 1i ik n min 1i ik n

i=0.6 j=0.5 ni=10 nj=10 i=0 j=0.5 ni=10 nj=10 i=0 j= 1 ni=10 nj=10

Cross influence on marginal profits

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) =

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"

hje

tetetej

jjjjj ,...,1 ,))(()()1(

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.

h = 2

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

jijiteCteBANb

tetete jjijj

jjjj

;2,1, ,)()(

)1(

)()()1(

2

2,1 ,))(()()1(

je

tetetej

jjjjj

Three boundary equilibria:

O=(0,0); E1=(-A1/C1,0); E2=(0,-A2/C2)

located along the invariant coordinate axes

An interior equilibrium E3=

2121

1221

2121

2112 ,BBCC

CABA

BBCC

CABA

Effort steady states

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

a=90 b=1 c=6 n1=20 n2=20 k1= k2=121=2=0.3 =9

No spillovers

k1= k2=13

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

e1

e2

E1

E2

E3

E1

E2

E3

e1

e2

Without spillovers, R&D 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 R&D

However network 2 can still make small profits by cutting off R&D expenses

Asymptotic R&D efforts

1

1

e1

e2

1

2

Profits

a=90 b=1 c=6 n1=20 n2=20 k1= k2=121=2=0.3 =9 c.i. (05,.1)

e1

e2

1

2

a=90 b=1 c=6 n1=20 n2=20 k1= k2=131=2=0.3 =9 i.c. e1(0)=0.1, e2(0) = 0.05

Asymptotic R&D efforts

Profits

1

1

Without spillovers, who invests more in R&D in the first period wins the competition

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

e1

e2

E3

E1

E2

Basin of E2 shrinks as 1 increases (here 1=0.2)

a=180 b=1 c=4 n1=20 n2=20 k1= k2=71=2=0.4 =9 No spillovers

E3

E1

E2 e1

e2

•Effect of decreasing k1=3

•Correlated chaotic attractor around the unstable equilibrium E3

•Lakes of B(∞) are nested inside the basin of the chaotic attractor

•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 R&D efforts, a chaotic trajectory in this case

e1E2

E1

E3

Space of effort: Chaos and multistability

Stable equilibrium in a symmetric casea=120 b=1 c=20 n1=20 n2=20 k1= 10 k2=10 1=2=0.5 =45

Again no spillovers

1=0.8 2=0.5 Chaotic attractor with asymmetric speed of adjustment

1=0.8 2=0.5 and k1=8Chaotic attractor and increased complexity of basins of attractors on invariant axis

E3

E1

E2

e1

e2

E2

e1

E3

E1E2e1

e2

e2

e2

E1 e1

e2

e2

E2

Goyal-Moraga (one network and no spillovers) shows that profit is maximized for intermediate levels of connectivity k

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

k1

1

2

Profits

a=140 b=1 c=6 n1=20 n2=20 k2=111=2=0.3 =9No spillovers

e1(0)=0.2, e2(0) = 0.2

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)

0

( ) ( )t

t ki i

k

z t X k

The level of knowledge accumulated up to time t can be modelled as

obsolescence factor which exponentially discounts older info

Xi (t): knowledge gained by firm i at time t, proportional to effective R&D

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.

1

1

0

( ) ( ) ( ) ( 1).t

t ki i i i

k

z t X t X k X t z t

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

Derivation of Cournot equilibrium quantity

Profit function for i-th oligopolist

2( ) ( , , )i i i i i i i iq a bQ q c q e e e

, ( , , ) 0ii i i i i

i

a bQ bq c q e eq

, ( ), 1,...,i i ibq a bQ c q i n

'i ic c

1 1

n n

i ji j

b q bQ na bnQ c

1

1

nj jna c

bQn

11

1 1

nj j ij i j

i i

na c a nc cq a c

b n b n

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:

Space of effort: Possible effect of asymmetric links

e1

e2

E3

E1

E2

a=90 b=1 c=6 n1=20 n2=20 k1= 1 k2=11 1=2=0.3 =9

Again no spillovers

•Inner equilibrium E3 is stable

•Basins of attractor located on invariant axis are in red and green

Specification of aggregate parameters of the map

2

2( )( ( 1) (1 ) )

2 ( 1) (1 ) (1 ) 1 1 ( 1)

2 [ ( 1) (1 ) ] 1 ( 1 ( 1) ) 1

i i i i i j j

i j i i i i j j i j j j j j

i i i i i j j j i i i i i i j j

A a c N n k n

B n N n k n n k n

C N n k n n k n n n N

N=n1+n2