The Impact of Cooperative R&D and Advertising on Innovation and Welfare

Abstract

This paper studies the impact of cooperative R&D and advertising on innovation and welfare in a duopolistic industry. The model incorporates two symmetric firms producing differentiated products. Firms invest in R&D and advertising in the presence of R&D spillovers and advertising spillovers. Advertising spillovers may be positive or negative. Four cooperative structures are studied: no cooperation, R&D cooperation, advertising cooperation, R&D and advertising cooperation. R&D spillovers and advertising spillovers always increase innovation and welfare if products are highly differentiated and/or spillovers are sufficiently high. The ranking of cooperation settings in terms of R&D, profits and welfare depends on product differentiation, R&D spillovers and advertising externalities. Firms always prefer cooperation on both dimensions, which is socially beneficial only when advertising and R&D spillovers are sufficiently high.

Appendix

A.1 Proof of Proposition 1

$$ \begin{aligned}\pi_{i} &= \frac{{\left[ {\left( {2 - b} \right)\left( {k - \alpha } \right) - b\theta a_{i} - b\beta x_{i} + 2\left( {a_{i} + x_{i} } \right) + a_{j} \left( {2\theta - b} \right) + x_{j} \left( {2\beta - b} \right)} \right]^{2} }}{{\left( {4 - b^{2} } \right)^{2} }}\\&\quad - \frac{1}{2}\lambda a_{i}^{2} - \frac{1}{2}\gamma x_{i}^{2} \quad i,j = 1, 2\quad i \ne j.\end{aligned} $$

(A.1)

Taking the derivative of Eq. (A.1) with respect to own R&D and advertising and the competitor’s to evaluate the strategic interaction shows that:

• (1) $\frac{{\partial^{2} \pi_{i} }}{{\partial x_{i} \partial x_{j} }} > 0$ if $\left( {2\beta - b} \right) > 0$; otherwise, it is negative.

• (2) $\frac{{\partial^{2} \pi_{i} }}{{\partial x_{i} \partial a_{j} }} > 0$ if $\left( {2\theta - b} \right) > 0$; otherwise, it is negative.

• (3) $\frac{{\partial^{2} \pi_{i} }}{{\partial a_{i} \partial x_{j} }} > 0$ if $\left( {2\beta - b} \right) > 0$; otherwise, it is negative.

• (4) $\frac{{\partial^{2} \pi_{i} }}{{\partial a_{i} \partial a_{j} }} > 0$ if $\left( {2\theta - b} \right) > 0$; otherwise, it is negative.

A.2 Proof of Proposition 2

$$ \frac{{\partial X^{NN} }}{\partial \beta } = \frac{{4\lambda \left( {k - \alpha } \right)\left( {\left( {2\left( {2 - b\beta } \right)^{2} - 2b\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + b\gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right)} \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.2)

$$ \frac{{ \partial a^{NN} }}{\partial \beta } = \frac{{8\gamma \left( {k - \alpha } \right)\left( {2 - b - 2b\beta } \right)\left( {2 - b\theta } \right)}}{{\left( {\gamma \left( {2 - b} \right)\left( {2 + b} \right)^{2} - 2\left( {1 + \beta } \right)\left( {2 - b\beta } \right) - 2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right)} \right)^{2} }} $$

(A.3)

$$ \frac{{\partial \pi^{NN} }}{\partial \beta } = \frac{\begin{aligned} & (4\gamma \lambda^{2} \left( {k - \alpha } \right)^{2} \left[ {2\gamma \left( {2 - b\theta } \right)\left( {4 + b\left( {b\theta \left( {1 + \beta } \right) - \beta \left( {4 + b} \right)} \right)} \right)} \right. \\ & \quad \left. { + 2\lambda \left( {2 - b\beta } \right)^{3} + \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} \left( {4 + b\left( {2 - b + \beta \left( {4 - b} \right)} \right)} \right)} \right] \\ \end{aligned} }{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right)^{3} }} \qquad $$

(A.4)

$$ \frac{{ \partial Y^{NN} }}{\partial \beta } = \frac{{4\gamma \left( {4 - b^{2} } \right)\left( {k - \alpha } \right)\left( {2 - b - 2b\beta } \right)}}{{\left( {2\left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \left( {2 - b} \right)\left( {2 + b} \right)^{2} + 2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right)} \right)^{2} }} $$

(A.5)

$$ \frac{{\partial W^{NN} }}{\partial \beta } = \frac{\begin{aligned} & 4\gamma \lambda^{2} \left( {k - \alpha } \right)^{2} \left[ {4\gamma \left( {2 - b\theta } \right)\left( {4 + b\left( {b\theta \left( {1 + \beta } \right) - \beta \left( {4 + b} \right)} \right)} \right) + 4\lambda \left( {2 - b\beta } \right)^{3} } \right. \\ & \quad \left. { - \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} \left( {12 - b\left( {4 + b - b^{2} + 2\beta \left( {6 - b^{2} } \right)} \right)} \right)} \right] \\ \end{aligned} }{{2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} )^{3} }} $$

(A.6)

$$ p^{NN} = \frac{{2\alpha \gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2k\left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \left( {4 - b^{2} } \right)\left( {k + \alpha + b\alpha } \right)} \right)}}{{2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} }} $$

(A.7)

$$ \frac{{\partial X^{RN} }}{\partial \beta } = \frac{{4\lambda \left( {2 - b} \right)\left( {k - \alpha } \right)\left[ {\lambda \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} + \gamma \left( {2 + b} \right)^{2} } \right) - 2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right)} \right]}}{{(2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.8)

$$ \frac{{\partial a^{RN} }}{\partial \beta } = \frac{{16\gamma \left( {2 - b} \right)\left( {k - \alpha } \right)\left( {1 + \beta } \right)\left( {2 - b\theta } \right)}}{{\left( {\left( {2 - b} \right)\left[ {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right] - 2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right)} \right)^{2} }} $$

(A.9)

$$ \frac{{\partial \pi^{RN} }}{\partial \beta } = \frac{\begin{aligned} & 4\gamma \lambda^{2} \left( {2 - b} \right)\left( {k - \alpha } \right)^{2} \left( {1 + \beta } \right) \\ & \left[ {2\gamma \left( {2 + b} \right)\left( {1 - \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2 - b} \right)^{2} \left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)} \right] \\ \end{aligned} }{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2 - b} \right)\left[ {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right]} \right)^{3} }} $$

(A.10)

$$ \frac{{\partial Y^{RN} }}{\partial \beta } = \frac{{8\gamma \left( {2 - b} \right)^{2} \left( {2 + b} \right)\left( {k - \alpha } \right)\left( {1 + \beta } \right)}}{{\left( {\left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right) + 2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right)} \right)^{2} }} $$

(A.11)

$$ \frac{{\partial W^{RN} }}{\partial \beta } = \frac{\begin{aligned} & 8\gamma \lambda^{2} \left( {2 - b} \right)\left( {k - \alpha } \right)^{2} \left( {1 + \beta } \right) \\ & \left[ {2\lambda \left( {2 - b} \right)^{2} \left( {1 + \beta } \right)^{2} + \gamma \left( {2 + b} \right)\left( {2\left( {1 - \theta } \right)\left( {2 - b\theta } \right) - \lambda \left( {4 - b^{2} } \right)^{2} } \right)} \right] \\ \end{aligned} }{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)} \right)^{3} }} $$

(A.12)

$$ p^{RN} = \frac{{\left( {k - \alpha } \right)^{2} \gamma \lambda \left( {\lambda \left( {2 - b} \right)^{2} \left( {\gamma \left( {2 + b} \right)^{2} - 2\left( {1 + \beta } \right)^{2} } \right) - 2\gamma \left( {2 - b\theta } \right)^{2} } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.13)

$$ \frac{{\partial X^{NA} }}{\partial \beta } = \frac{{4\lambda \left( {k - \alpha } \right)\left( {2b\gamma \left( {2 - b} \right)\left( {1 + \theta } \right)^{2} + \lambda \left[ {2\left( {2 - b\beta } \right)^{2} - b\gamma \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right]} \right)}}{{\left( {2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left[ {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} } \right]} \right)^{2} }} $$

(A.14)

$$ \frac{{\partial a^{NA} }}{\partial \beta } = \frac{{8\gamma \left( {2 - b} \right)\left( {k - \alpha } \right)\left( {2 - b - 2b\beta } \right)\left( {1 + \theta } \right)}}{{\left( {2\left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.15)

$$ \frac{{\partial \pi^{NA} }}{\partial \beta } = \frac{{4\gamma \lambda^{2} \left( {k - \alpha } \right)^{2} \left( {2\lambda \left( {2 - b\beta } \right)^{3} + \gamma \left( {2 - b} \right)\left( {4 - b\left( {2 - b + \beta \left( {4 - b} \right)} \right)} \right)\left( {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} } \right)} \right)}}{{\left( {2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} } \right)} \right)^{3} }} $$

(A.16)

$$ \frac{{\partial Y^{NA} }}{\partial \beta } = \frac{{4\gamma \left( {4 - b^{2} } \right)\left( {k - \alpha } \right)\left( {2 - b - 2b\beta } \right)}}{{\left( {2\left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} + \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.17)

$$ \frac{{\partial W^{NA} }}{\partial \beta } = \frac{\begin{aligned} & 4\gamma \lambda^{2} \left( {k - \alpha } \right)^{2} \left( {4\lambda \left( {2 + b\beta } \right)^{3} + \gamma \left( {2 - b} \right)} \right. \\ & \left. {\left[ {4\left( {4 - b\left( {2 - b - \beta \left( {4 - b} \right)} \right)} \right)\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} \left( {12 - b\left( {4 + b - b^{2} + 2\beta \left( {6 - b^{2} } \right)} \right)} \right)} \right]} \right) \\ \end{aligned} }{{\left( {2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} } \right)} \right)^{3} }} $$

(A.18)

$$ p^{NA} = \frac{{\left( {2\alpha \gamma \left( {2 - b} \right)\left( {1 + \theta } \right)^{2} - \left( {2k\left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \left( {4 - b^{2} } \right)\left( {k + \alpha + b\alpha } \right)\gamma } \right)\lambda } \right)}}{{\left( {2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} } \right)} \right)}} $$

(A.19)

$$ \frac{{ \partial X^{RA} }}{\partial \beta } = \frac{{4\lambda \left( {k - \alpha } \right)\left( {2\lambda \left( {1 + \beta } \right)^{2} - 2\gamma \left( {1 + \theta } \right)^{2} + \gamma \lambda \left( {2 + b} \right)^{2} } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\lambda \left( {1 + \beta } \right)^{2} - \gamma \lambda \left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.20)

$$ \frac{{\partial a^{RA} }}{\partial \beta } = \frac{{16\gamma \left( {k - \alpha } \right)\left( {1 + \beta } \right)\left( {1 + \theta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\left( {1 + \beta } \right)^{2} - \left( {2 + b} \right)^{2} \gamma } \right)^{2} }} $$

(A.21)

$$ \frac{{\partial \pi^{RA} }}{\partial \beta } = \frac{{4\gamma \left( {k - \alpha } \right)^{2} \left( {1 + \beta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.22)

$$ \frac{{\partial Y^{RA} }}{\partial \beta } = \frac{{8\gamma \left( {k - \alpha } \right)\left( {2 + b} \right)\left( {1 + \beta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\left( {1 + \beta } \right)^{2} - \left( {2 + b} \right)^{2} \gamma } \right)^{2} }} $$

(A.23)

$$ \frac{{\partial W^{RA} }}{\partial \beta } = \frac{{8\gamma \lambda^{2} \left( {k - \alpha } \right)^{2} \left( {1 + \beta } \right)\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\lambda \left( {1 + \beta } \right)^{2} - \gamma \lambda \left( {2 + b} \right)^{3} } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\lambda \left( {1 + \beta } \right)^{2} - \gamma \lambda \left( {2 + b} \right)^{2} } \right)^{3} }} $$

(A.24)

$$ p^{RA} = \frac{{2\alpha \gamma \left( {1 + \theta } \right)^{2} + 2k\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)\left( {k + \alpha + b\alpha } \right)}}{{2\gamma \left( {1 + \theta } \right)^{2} + 2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} }}. $$

(A.25)

Note that to simplify the analysis, $\lambda$ is fixed at $\lambda = 1$ by assumption. Output is positive if $k > \alpha$. Considering $k = \alpha \;{\text{and}}\; \lambda = 1$ and evaluating $p$ at different parameter values indicates the validity of $\gamma$ and shows that it meets the required thresholds where required.

Evaluating the above equations for any value of parameters shows that R&D spillovers have a positive impact on innovation, output, profit, and welfare when firms cooperate in R&D. Otherwise, the impact is positive if R&D spillovers are low enough.

A.3 Proof of Proposition 3

$$ \frac{{\partial X^{NN} }}{\partial \theta } = \frac{{8\gamma \left( {k - \alpha } \right)\left( {2 - b\beta } \right)\left( {2 - b - 2b\theta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \left( {2 - b} \right)\left( {2 + b} \right)^{2} \gamma } \right)^{2} }} $$

(A.26)

$$ \frac{{\partial a^{NN} }}{\partial \theta } = \frac{{4\gamma \left( {k - \alpha } \right)\left( {2\gamma \left( {2 - b\theta } \right)^{2} + 2\lambda b\left( {1 + \beta } \right)\left( {2 - b\beta } \right) - b\gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.27)

$$ \frac{{\partial \pi^{NN} }}{\partial \theta } = \frac{\begin{aligned} & 4\gamma^{2} \lambda \left( {k - \alpha } \right)^{2} \left[ {2\gamma \left( {2 - b\theta } \right)^{3} - \lambda \gamma \left( {2 - b} \right)\left( {2 + b} \right)^{2} \left( {4 - b\left( {2 - b + \theta \left( {4 - b} \right)} \right)} \right)} \right. \\ & \left. { + 2\lambda \left( {2 - b\beta } \right)\left( {4 - 4b\theta + b^{2} \left( {\beta - \theta \left( {1 - \beta } \right)} \right)} \right)} \right] \\ \end{aligned} }{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right)^{3} }} $$

(A.28)

$$ \frac{{\partial Y^{NN} }}{\partial \theta } = \frac{{4\gamma^{2} \left( {4 - b^{2} } \right)\left( {k - \alpha } \right)\left( {2 - b - 2b\theta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.29)

$$ \frac{{\partial W^{NN} }}{\partial \theta } = \frac{\begin{aligned} & 4\gamma^{2} \lambda \left( {k - \alpha } \right)^{2} \left[ {4\gamma \left( {2 - b\theta } \right)^{3} - \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} \left( {12 - b\left( {4 + b - b^{2} - 2\theta \left( {6 - b^{2} } \right)} \right)} \right)} \right. \\ & \quad \left. { + 4\lambda \left( {2 - b\beta } \right)\left( {4 - 4b\theta + b^{2} \left( {\beta - \theta \left( {1 - \beta } \right)} \right)} \right)} \right] \\ \end{aligned} }{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + 2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) - \gamma \lambda \left( {2 - b} \right)\left( {2 + b} \right)^{2} } \right)^{3} }} $$

(A.30)

$$ \frac{{\partial X^{RN} }}{\partial \theta } = \frac{{8\gamma \left( {2 - b} \right)\left( {k - \alpha } \right)\left( {1 + \beta } \right)\left( {2 - b - 2b\theta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} + \gamma \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.31)

$$ \frac{{\partial a^{RN} }}{\partial \theta } = \frac{{4\gamma \left( {k - \alpha } \right)\left( {2\gamma \left( {2 - b\theta } \right)^{2} + \lambda b\left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)} \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.32)

$$ \frac{{\partial \pi^{RN} }}{\partial \theta } = \frac{{4\gamma^{2} \lambda \left( {k - \alpha } \right)^{2} \left[ {2\gamma \left( {2 - b\theta } \right)^{3} + \lambda \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)\left( {4 - b\left( {2 - b - \theta \left( {4 - b} \right)} \right)} \right)} \right]}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)} \right)^{3} }} $$

(A.33)

$$ \frac{{\partial Y^{RN} }}{\partial \theta } = \frac{{4\gamma^{2} \left( {4 - b^{2} } \right)\left( {k - \alpha } \right)\left( {2 - b - 2b\theta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.34)

$$ \frac{{\partial W^{RN} }}{\partial \theta } = \frac{\begin{aligned} & 4\gamma^{2} \lambda \left( {k - \alpha } \right)^{2} \left[ {4\gamma \left( {2 - b\theta } \right)^{3} + 4\lambda \left( {2 - b} \right)\left( {1 + \beta } \right)^{2} \left( {4 - b\left( {2 - b + \theta \left( {4 - b} \right)} \right)} \right)} \right. \\ & \quad \left. { - \lambda \gamma \left( {2 - b} \right)\left( {2 + b} \right)^{2} \left( {12 - b\left( {4 + b - b^{2} - 2\theta \left( {6 - b^{2} } \right)} \right)} \right)} \right] \\ \end{aligned} }{{\left( {2\gamma \left( {1 + \theta } \right)\left( {2 - b\theta } \right) + \lambda \left( {2 - b} \right)\left( {2\left( {1 + \beta } \right)^{2} - \left( {2 + b} \right)^{2} \gamma } \right)} \right)^{3} }} $$

(A.35)

$$ \frac{{\partial X^{NA} }}{\partial \theta } = \frac{{16\gamma \left( {2 - b} \right)\left( {k - \alpha } \right)\left( {2 - b\beta } \right)\left( {1 + \theta } \right)}}{{\left( {2\left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.36)

$$ \frac{{\partial a^{NA} }}{\partial \theta } = \frac{{4\gamma \left( {2 - b} \right)\left( {k - \alpha } \right)\left( {\gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} + \lambda \left( {2 + b} \right)^{2} } \right) - 2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right)} \right)}}{{\left( {2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.37)

$$ \frac{{\partial \pi^{NA} }}{\partial \theta } = \frac{{4\gamma^{2} \lambda \left( {2 - b} \right)\left( {k - \alpha } \right)^{2} [\left( {1 + \theta } \right)\left( {2\lambda \left( {2 + b} \right)\left( {1 - \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)^{2} \left( {\lambda \left( {2 + b} \right)^{2} - 2\left( {1 + \theta } \right)^{2} } \right)} \right]}}{{\left( {2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} } \right)} \right)^{3} }} $$

(A.38)

$$ \frac{{\partial Y^{NA} }}{\partial \theta } = \frac{{8\gamma^{2} \left( {k - \alpha } \right)\left( {1 + \theta } \right)\left( {2 - b} \right)^{2} \left( {2 + b} \right)}}{{\left( {2\left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \left( {2 + b} \right)^{2} } \right)} \right)^{2} }} $$

(A.39)

$$ \frac{{\partial W^{NA} }}{\partial \theta } = \frac{{8\gamma^{2} \lambda \left( {2 - b} \right)\left( {k - \alpha } \right)^{2} \left( {1 + \theta } \right)\left[ {2\lambda \left( {2 + b} \right)\left( {1 - \beta } \right)\left( {2 - b\beta } \right) - \gamma \left( {2 - b} \right)^{2} \left( {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{3} } \right)} \right]}}{{\left( {2\lambda \left( {1 + \beta } \right)\left( {2 - b\beta } \right) + \gamma \left( {2 - b} \right)\left( {2\left( {1 + \theta } \right)^{2} - \lambda \left( {2 + b} \right)^{2} } \right)} \right)^{3} }} $$

(A.40)

$$ \frac{{\partial X^{RA} }}{\partial \theta } = \frac{{16\left( {k - \alpha } \right)\left( {1 + \beta } \right)\gamma \left( {1 + \theta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.41)

$$ \frac{{\partial a^{RA} }}{\partial \theta } = \frac{{4\gamma \left( {k - \alpha } \right)\left( {2\gamma \left( {1 + \theta } \right)^{2} - 2\left( {1 + \beta } \right)^{2} \lambda + \left( {2 + b} \right)^{2} \gamma \lambda } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\left( {1 + \beta } \right)^{2} \lambda - \left( {2 + b} \right)^{2} \gamma \lambda } \right)^{2} }} $$

(A.42)

$$ \frac{{\partial \pi^{RA} }}{\partial \theta } = \frac{{4\gamma^{2} \left( {k - \alpha } \right)^{2} \left( {1 + \theta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.43)

$$ \frac{{\partial Y^{NA} }}{\partial \theta } = \frac{{8\gamma^{2} \left( {2 + b} \right)\left( {k - \alpha } \right)\left( {1 + \theta } \right)}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\left( {1 + \beta } \right)^{2} - \gamma \left( {2 + b} \right)^{2} } \right)^{2} }} $$

(A.44)

$$ \frac{{\partial W^{RA} }}{\partial \theta } = \frac{{8\gamma^{2} \lambda \left( {k - \alpha } \right)^{2} \left( {1 + \theta } \right)\left[ {2\gamma \left( {1 + \theta } \right)^{2} + 2\lambda \left( {1 + \beta } \right)^{2} - \gamma \lambda \left( {2 + b} \right)^{3} } \right]}}{{\left( {2\gamma \left( {1 + \theta } \right)^{2} + 2\lambda \left( {1 + \beta } \right)^{2} - \gamma \lambda \left( {2 + b} \right)^{2} } \right)^{3} }}. $$

(A.45)

Evaluating the above equations for any value of parameters shows that advertising spillovers have a positive impact on innovation, output, profit, and welfare when firms cooperate in advertising. Otherwise, the impact is positive if advertising spillovers are low enough.

A.4 Proof of Proposition 4

Section 3 provides the formula for $x_{i}$, $\pi_{i}$, and $W$ as a function of other parameters. Taking the deduction of each pair of equations shows that for any parameter set that yields a valid price value ($p > 0$) the following holds.

• (1) If $b = \beta = 0$ and $\theta < 0$, then $X_{NN} = X_{RN} > X_{NA} = X_{RA}$, $\pi_{NN} = \pi_{RN} > \pi_{NA} = \pi_{RA}$, and $W_{NN} = W_{RN} > W_{NA} = W_{RA}$

• (2) If $b = 0$, $0 < \beta < 0.1$, and $\theta < 0,$ then $X_{RN} > X_{NN} > X_{RA} > X_{NA}$ and $W_{RN} > W_{NN} > W_{RA} > W_{NA}$.

• (3) If $b = 0$ and $\beta > 0.1,$ and $\theta < 0,$ then $X_{RN} > X_{RA} > X_{NN} > X_{NA}$ and $W_{RN} > W_{RA} > W_{NN} > W_{NA}$.

• (4) $X_{NA} > X_{RA}$ if $\theta < - b/2$ and $\beta < b/2$

• (5) $X_{RN} > X_{NA}$ if $\theta < - b/2$ and $\beta < b/2$

• (6) $X_{RA} > X_{RN}$ if $\theta > b/2$

• (7) $X_{NA} > X_{NN}$ if $\theta > b/2$

• (8) $X_{RN} > X_{NN}$ if $\beta > b/2$

• (9) $X_{RN} > X_{NA}$ if $\theta < \beta$

• (10) $X_{NA} > X_{RA}$ if $\theta < b/2$ and $\beta < b/2$

• (11) $X_{NN} > X_{RA}$ if $\theta < b/2$ and $\beta < b/2$

• (12) $\pi_{RA} > \pi_{NA} > \pi_{RN} > \pi_{NN}$ if $\beta \ne 0$.