Economics Performance Under Endogenous Knowledge Spillovers

Abstract

The aim of this paper is to examine the impact of growing R&D cooperation networks on R&D knowledge spillovers and equilibrium outcomes. We find that as the cooperation in the network grows, the graph theoretic distances between competitors decrease. This allows the R&D spillovers between non-cooperators to increase. Moreover, we find that increasing the cooperation in the network improves the payoffs of firms. This indicates that developing the cooperation network is individually desirable to remain active in the industry.

Appendix

1.1 Appendix 1: Proofs of Propositions

Proof of Proposition 1

Since spillover $\upbeta \in [0, 1)$ is applied between non-linked firms i and k, then the length of the geodesic between the two firms is $d_{ik} \ge 2$. To continue applying the spillover after decreasing the distance, we assume $d_{ik} > 2$. Thus, the function ${\mathbf {B}} \ = \ e^{d_{ik} \ln \upbeta }$ increases as the distance $d_{ik}$ decreases. $\square$

Proof of Proposition 2

Assume firms i and k are connected by a path such that $d_{ik} > 2$. Let firm $\ell$ be linked to k and let $G'$ be a network generated by linking firms i and $\ell$ (see Fig. 11). Then, the distance between firms i and k in $G'$ reduces to $d_{ik} = 2$ which implies $\upbeta ^{d_{ik} (G')} > \upbeta ^{d_{ik} (G)}$. $\square$

Fig. 11

An illustrative example for proof of Proposition 2

Proof of Proposition 3

Assume firms i and k are connected by a path such that $d_{ik} > 2$. Let firms i and $\ell$ be connected in the cooperation network G by a path such that $d_{i \ell }>2$. Let $G'$ be a network generated by linking firms i and $\ell$ (see Fig. 12). Then, we have two cases:

case 1: if $d_{ik} (G') = d_{ik} (G) \Rightarrow \upbeta ^{d_{ik} (G')} = \upbeta ^{d_{ik} (G)}$,

case 2: if $d_{ik} (G') < d_{ik} (G) \Rightarrow \upbeta ^{d_{ik} (G')} > \upbeta ^{d_{ik} (G)}$.

This implies $\upbeta ^{d_{ik} (G')} \ge \upbeta ^{d_{ik} (G)}$. $\square$

Fig. 12

An illustrative example for proof of Proposition 3

1.2 Appendix 2: Numerical Simulation of the Original Model

In the following, we repeat Example 2 under the model of Jackson (2008).

Example 5

Assume six firms sell homogeneous goods and compete by their quantities and assume the cooperation of firms in R&D forms network $G_4$ given in Fig. 5. Let network $G_5$ generated from $G_4$ by adding a link between firms 2 and 4, and between firms 3 and 6. Figure 13 shows the equilibrium outcomes in the two networks $G_4$ and $G_5$ under the model of Jackson (2008).

Fig. 13

The R&D investment and the profit of firms in the networks ${G}_{4}$ and ${G}_{5}$ given in Fig. 5. The parameters used to plot the figure are $a=12, \ \overline{c}=10$ and $\gamma =2$