When Spillovers Enhance R&D Incentives

Abstract

It is commonly believed that spillover reduces R&D incentives of a firm. This happens because of the appropriability problem. However, some empirical literature shows the possibility of enhanced R&D incentives under spillovers. In the literature this is explained under incomplete information, but we show this theoretically under complete information. We show in particular that in a duopoly there are situations when with no spillovers only one firm invests in R&D, but under spillovers both the firms invest. This occurs when there is complementarity in research and the spillover rate lies in an interval specified by the size of R&D investment.

Appendices

Appendix 1

Given the numerical values of the parameters in Example, further assume that $d = 3$. The we have: $A = 18$, $\varPi \left( { - {\text{D}}} \right) = 9,\;\varPi \left( 0 \right) = 36,\;\varPi \left( {\text{D}} \right) = 81,\;\varPi \left( {2{\text{D}}} \right) = 144,\;\varPi \left( {{\text{D}} + {\text{d}}} \right) = 100$, $\varPi \left( {2{\text{d}} - {\text{D}}} \right) = 25,\;\varPi \left( {2{\text{D}} - {\text{d}}} \right) = 121$

Given above, the payoff matrix when there is no spillover is:

For this payoff matrix, (Y, N) and (N, Y) are two Nash equilibria. The payoff matrix when there is spillover is given below:

Clearly in this game (Y, Y) is the unique Nash equilibrium.

Appendix 2

Let the demand function as faced by firm i be given by $q_{i} = a - p_{i} + \gamma p_{j}$ where $i = 1, 2 (i \ne j)$ and $\gamma \in (0, 1)$. For the marginal costs c₁ and c₂ of firm 1 and firm 2 respectively, the profit function of firm i is $\pi_{i} = (p_{i} - c_{i} )q_{i}$. Then under price competition, the equilibrium prices and quantities are respectively, $p_{i} = \frac{{\left( {2 + \gamma } \right)a + 2c_{i} + \gamma c_{j} }}{{4 - \gamma^{2} }}$ and $q_{i} = \frac{{\left( {2 + \gamma } \right)a - (2 - \gamma^{2} )c_{i} + \gamma c_{j} }}{{4 - \gamma^{2} }}$, therefore, $\pi_{i} = (q_{i} )^{2}$.

Define $K = \left( {2 + \gamma } \right)a - (2 - \gamma - \gamma^{2} )c$ and $\pi \left( x \right) = \left( {\frac{K + x}{{4 - \gamma^{2} }}} \right)^{2}$. Then the payoff matrix under no spillover is given by:

We have $S\left( {WS} \right) = \pi \left( {(2 - \gamma - \gamma^{2} )D} \right) - \pi \left( { - \gamma D} \right) - R$ and $NS\left( {WS} \right) = \pi \left( {(2 - \gamma^{2} )D} \right) - \pi \left( 0 \right) - R$, then $NS\left( {WS} \right) > S(WS)$.

The payoff matrix under spillover is given by:

Here, $S\left( {SS} \right) = \pi \left( {(2 - \gamma - \gamma^{2} )(D + d)} \right) - \pi \left( {\left( {2 - \gamma^{2} } \right)d - \gamma D} \right) - R$, and $NS\left( {SS} \right) = \pi \left( {\left( {2 - \gamma^{2} } \right)D - \gamma d} \right) - \pi \left( 0 \right) - R$. Then, $S\left( {SS} \right) \frac{ < }{ > }NS\left( {SS} \right) \Leftrightarrow d\frac{ < }{ > }\frac{\gamma }{{2 - \gamma^{2} }}D$. Further, at d = 0, we have $NS\left( {SS} \right) > S\left( {SS} \right),$ and at d = D we have $NS\left( {SS} \right) < S\left( {SS} \right)$. Finally, $S\left( {SS} \right)$ is maximized at $\widehat{d} = \frac{{\left[ {\left( {2 - \gamma - \gamma^{2} } \right)^{2} - \gamma \left( {2 - \gamma^{2} } \right)} \right]D - \gamma K}}{{\left( {2 - \gamma^{2} } \right)^{2} - \left( {2 - \gamma - \gamma^{2} } \right)^{2} }}$ iff $\left[ {\left( {2 - \gamma - \gamma^{2} } \right)^{2} - \gamma \left( {2 - \gamma^{2} } \right)} \right]D > \gamma K$, otherwise, $\widehat{d} = 0$.

Following our earlier notation, $E = S\left( {WS} \right) + R$, $B\left( d \right) = S\left( {SS} \right) + R$ and $C\left( d \right) = NS\left( {SS} \right) + R$. Therefore, $B\left( 0 \right) = E$ and $C\left( D \right) < E$.

Finally, if $\left[ {\left( {2 - \gamma - \gamma^{2} } \right)^{2} - \gamma \left( {2 - \gamma^{2} } \right)} \right]D > \gamma K$ holds, then we have a range of R and correspondingly an interval of d for which $(Y, N)$ or $(N, Y)$ is a NE under no spillover but $(Y, Y)$ is the unique NE under spillover. The relevant condition is necessarily satisfied, for example, for $D = 2, K = 4$ and $\gamma = 0.25$.