Intellectual property and taxation of digital platforms

Abstract

I study the impact of ad-valorem and unit taxes on the intellectual property policies of two-sided digital platforms. First, I address the monopoly case, in which I show that the effects of taxes depend on which side they are levied on. If developers are taxed, I find that ad-valorem taxes reduce the platform openness and the exclusivity period granted to developers to exploit their innovations. The opposite is true when taxes are levied on users. On the other hand, the effect of unit taxes is ambiguous in general. Then, I extend the model to address the duopoly case, and I find that competition may increase welfare, but it is not guaranteed. The effects of taxes on welfare are similar in both regimes. In general, they are ambiguous, but I characterize those cases which are welfare-enhancing unambiguously. I conclude highlighting the potential impact of the Digital Service Tax (DST) proposed by the European Commission on platform openness and digital innovation in Europe.

Appendices

Appendices

A: Parker and Van Alstyne model with horizontally differentiated developers

In the duopoly setting, developers are horizontally differentiated. To prove that this change does not qualitatively change the conclusions, let’s extend Parker and Van Alstyne’s model to accommodate such an assumption. Eq. 3.1 becomes,

$$\begin{aligned} \max _{\sigma , \delta } \varPi = V(1-\sigma )+\frac{1}{2}M^{d} p \end{aligned}$$

where $M^{d}= \frac{p^{e}k(\sigma V)^{\alpha }+\delta p^{e}M^{\alpha }_{e}k^{\alpha +1}(\sigma V)^{\alpha ^2}}{2 \lambda }$, the monopoly demand that the platform faces on the developer’s side. Likewise the original model, platforms profits are well behaved, and there exists a tuple $\left\langle \sigma ^*,\delta ^* \right\rangle$ that maximizes $\varPi _{j}$. Therefore, the optimal lengths of the exclusivity period and the degree of openness are respectively,

$$\begin{aligned} \delta ^{*}= & {} {\left\{ \begin{array}{ll} \frac{1}{2}\left( 1- \frac{\pi _{1} }{\pi _{2}} \right) &{} \quad {\text{if}}\; \pi _{2}> \pi _{1} \\ 0 &{} \quad {\text{otherwise}} \end{array}\right. } \\ \sigma ^{*}= & {} {\left\{ \begin{array}{ll} \frac{ \alpha v(1-\delta )(\pi _{1}+\delta \alpha \pi _{2})}{4V \lambda } &{} \quad {\text{if}}\; \delta ^* > 0 \\ \frac{(v p^{e} \alpha k/ 4 \lambda )^{1/(1- \alpha )}}{V} &{} \quad {\text{if}}\; \delta ^* = 0. \end{array}\right. } \end{aligned}$$

Note that the optimal $\delta$ does not change, but the optimal $\sigma$ is expressed differently. The transportation cost ($\lambda$) appears in the denominator, reducing the optimal $\sigma$. On the other hand, in the numerator, $v(1-\delta )$ is increasing the optimal $\sigma$. This value was included in $\pi$ in the original model because it considers that the platform and developers can perfectly predict the price. In this extended version, I distinguish between the expected price that developers take into account to form their profit expectations ($p^e$) and the price ($v(1-\delta )$) that platforms know that developers will set under their intellectual property policies. Lastly, by the implicit function theorem, it is possible to prove that $\frac{\partial \sigma }{\partial \delta }<0$ like in Parker and Van Alstyne’s model,

$$\begin{aligned} \frac{\partial \sigma }{\partial \delta }=\frac{v \alpha [\pi _{1}-\pi _{2} \alpha (1-2\delta )]}{-\sigma 4 \lambda V+v(1-\delta )\alpha ^{2}[\pi _{1}+\alpha ^{3}\delta \pi _{2}]}<0 \end{aligned}$$

With respect to taxes, the optimal $\delta$ and $\sigma$ become,

$$\begin{aligned} \delta ^{*}_{vat}= & {} {\left\{ \begin{array}{ll} \frac{1}{2}\left( 1- \frac{\pi _{1} }{\pi _{2}} \right) &{} \quad {\text{if}}\; \pi _{2}> \pi _{1} \\ 0 &{} \quad {\text{otherwise}} \end{array}\right. } \\ \delta ^{*}_{sp}= & {} {\left\{ \begin{array}{ll} \frac{1}{2}\left( 1- \frac{\pi _{1} }{\pi _{2}} - \frac{\tau ^{sp} }{v}\right) &{} \quad {\text{if}}\; \pi _{2}> \pi _{1} \\ 0 &{} \quad {\text{otherwise}} \end{array}\right. } \\ \sigma ^{*}_{vat}= & {} {\left\{ \begin{array}{ll} \frac{\alpha v (1- \delta ) (\pi _{1}+\delta \alpha \pi _{2})}{V4 \lambda (1+\tau ^{vat})} &{} \quad {\text{if}}\; \delta ^*> 0 \\ \frac{(v \alpha k p^{e}/ 4 \lambda (1+\tau ^{vat}))^{1/(1- \alpha )}}{V} &{} \quad {\text{if}}\; \delta ^* = 0 \end{array}\right. } \\ \sigma ^{*}_{sp}= & {} {\left\{ \begin{array}{ll} \frac{\alpha (v(1- \delta ) - \tau ^{sp}) (\pi _{1}+\delta \alpha \pi _{2})}{V4 \lambda } &{} \quad {\text{if}}\; \delta ^* > 0 \\ \frac{(\alpha (v-\tau ^{sp}) k p^{e}/ 4 \lambda )^{1/(1- \alpha )}}{V} &{} \quad {\text{if}}\; \delta ^* = 0 \end{array}\right. } \end{aligned}$$

In this general model, the conclusions remain the same as in Sect. 3.1.

B: Optimal solutions for $\delta$ and $\sigma$: Proof Proposition 2

Taking Eq. 4.3, the first-order conditions on platform profits with respect to $\delta _j$ are

$$\begin{aligned} \frac{\partial \varPi _j}{\partial \delta _{j}}= \frac{[1 - \delta _j]v \pi ^{d}_{2,j}}{8 \lambda } - v \left( 1/4+ \frac{ \pi ^{d}_{1,j} +\delta _j \pi ^{d}_{2,j} -\pi ^{d}_{1,-j} -\delta _{-j}\pi ^{d}_{2,-j}}{8 \lambda }\right) =0 \end{aligned}$$

(B.1)

If I rearrange terms in Eq. B.1, I arrive at the first case of Eq. 4.5. Note that the second-order conditions are

$$\begin{aligned} \frac{\partial ^2 \varPi _j}{\partial \delta _j^2}=\frac{-k^{1+\alpha }vm^{\alpha }_{e,j}p^{e}_{j}(\sigma _{j} V_{j})^{\alpha ^{2}}}{4 \lambda }<0 \end{aligned}$$

which implies that $\varPi _j$ is concave in $\delta _j$, and by solving the first-order conditions, I obtain the maximum. Also, note that this interior solution only exists if $2\pi ^{d}_{j,2} + \pi ^{d}_{-j,2} > \varDelta _{1}+6 \lambda$. This expression can be easily derived from Eq. 4.5. Otherwise, $\delta _j=0$, the corner solution. Taking Eq. 4.3, the first-order conditions on platform profits with respect to $\sigma _j$ are

$$\begin{aligned} \begin{aligned} \frac{\partial \varPi _j}{\partial \sigma _j}&= -V_{j}n_{1,j} +V_{j}(1-\sigma _j)\frac{\partial n_{1,j}}{\partial \sigma _j}\\&\quad + \frac{(1-\delta )v\left( \alpha \sigma _{j}^{\alpha - 1} V_{j}^{\alpha } p^{e}_j k + p^{e}_j m^{\alpha }_{e,j} \alpha ^{2}\delta _j k^{1+\alpha } \sigma _{j}^{\alpha ^{2}-1}V_{j}^{\alpha ^{2}} \right) }{8 \lambda } =0 \end{aligned} \end{aligned}$$

(B.2)

Again, if I rearrange terms in Eq. B.2, I arrive at the first case of Eq. 4.6. In this case, the second-order conditions are

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 \varPi _j}{\partial \sigma _j^2}=&-2V_j \frac{\partial n_{1,j}}{\partial \sigma _j}+V_j(1-\sigma _j)\frac{\partial ^2 n_{1,j}}{\partial \sigma _j^2}\\&\frac{(1-\delta _j)v \left( (\alpha -1)\alpha k p^{e}_{j} \sigma _j^{\alpha -2} V_j^{\alpha }+ \alpha ^{2} (\alpha ^{2} -1) \delta _j k^{1+\alpha }m^{\alpha }_{e,j} p^{e}_{j} \sigma _j^{\alpha ^{2}-2}V_j^{\alpha ^{2}} \right) }{8 \lambda } <0 \end{aligned} \end{aligned}$$

(B.3)

it implies that $\varPi _j$ is concave in $\sigma _j$, and by solving the first-order conditions, I obtain the maximum. Also, note that this interior solution only exists if $\delta _j>0$. Otherwise, I only have to substitute $\delta _j=0$ in Eq. B.2, and $\sigma$ would be equal to the second case of Eq. 4.6, the corner solution.

Eq. 4.6 shows that $\sigma$ depends on $\pi _{i,j}$, $i=1,2$ that also depends on $\sigma$. This situation may raise concerns about the uniqueness of the solution, but the proof is similar to the one in Parker and Van Alstyne (2018). Nonetheless, it is a little bit more tedious. For simplicity’s sake, I omit it here, but it is available upon request. Lastly, Eq. 4.5 depends on $\sigma$ that depends on $\delta$, by the implicit function theorem, $\frac{\partial \sigma }{\partial \delta }<0$. Note that $\sigma ^*$ is similar in both the duopoly and the monopoly frameworks.

C: Optimal solutions for $\delta$ and $\sigma$: Proof Proposition 3

Taking Eq. 4.8, the first-order conditions on platform profits with respect to $\delta _j$ are the same as before. Therefore, the previous proof applies here. On the other hand, the first-order conditions on platform profits with respect to $\sigma _j$ are

$$\begin{aligned} \begin{aligned} \frac{\partial \varPi _j}{\partial \sigma _j}=&-V_{j}n_{1,j} +V_{j}(1-\sigma _j)\frac{\partial n_{1,j}}{\partial \sigma _j}\\&+ \frac{(1-\delta )v\left( \alpha \sigma _{j}^{\alpha - 1} V_{j}^{\alpha } p^{e}_j k + p^{e}_j m^{\alpha }_{e,j} \alpha ^{2}\delta _j k^{1+\alpha } \sigma _{j}^{\alpha ^{2}-1}V_{j}^{\alpha ^{2}}\right) }{8 \lambda (1+\tau ^{vat})} =0 \end{aligned} \end{aligned}$$

(C.1)

If I rearrange terms, I arrive at the first expression of Eq. 4.11. In this case, the second-order conditions also verify that the interior equilibrium is the maximum. Nonetheless, it also exists a corner solution that is reached when $\delta _j=0$, as in the previous case. To prove whether or not $\sigma _j$ is decreasing with respect to $\tau ^{vat}$, I proceed in two parts. First, there is a direct and negative effect of $\tau ^{vat}$ on $\sigma _j$, but a change in $\sigma _j$ triggers a change in $\delta _j$ that influence $\sigma _j$ again and so on. By the implicit function theorem, $\frac{\partial \sigma }{\partial \delta }<0$, and by differentiating Eq. 4.5, $\frac{\partial \delta }{\partial \sigma }<0$. Therefore, the impact of ad-valorem taxes on $\sigma _j$ is negative when $\delta ^*>0$. If $\delta ^*=0$, there an unambiguously negative effect of $\tau ^{vat}$ on $\sigma _j$.

Lastly, if I take Eq. 4.9, the first-order conditions on platform profits with respect to $\delta _j$ are

$$\begin{aligned} \frac{\partial \varPi _j}{\partial \delta _j}= \frac{([1 - \delta _j]v-\tau ^{sp}) \pi ^{d}_{2,j} }{8 \lambda } - v \left( 1/4+ \frac{ \pi ^{d}_{1,j} +\delta _j \pi ^{d}_{2,j} -\pi ^{d}_{1,-j} -\delta _{-j}\pi ^{d}_{2,-j}}{8 \lambda }\right) =0 \end{aligned}$$

(C.2)

In this case, the second-order conditions also verify that the interior equilibrium is a maximum. Nonetheless, it also exists a corner solution that is reached when $(v-\tau ^{sp})2\pi ^{d}_{j,2} + \pi ^{d}_{-j,2} > \varDelta _{1}+6 \lambda v$. Lastly, if I take Eq. 4.9, the first-order conditions on platform profits with respect to $\sigma _j$ are

$$\begin{aligned} \begin{aligned} \frac{\partial \varPi _j}{\partial \sigma _j}=&-V_{j}n_{1,j} +V_{j}(1-\sigma _j)\frac{\partial n_{1,j}}{\partial \sigma _j}\\&\quad + \frac{ ((1-\delta )v-\tau ^{sp})\left( \alpha \sigma _{j}^{\alpha - 1} V_{j}^{\alpha } p^{e}_j k + p^{e}_j m^{\alpha }_{e,j} \alpha ^{2}\delta _j k^{1+\alpha } \sigma _{j}^{\alpha ^{2}-1}V_{j}^{\alpha ^{2}}\right) }{8 \lambda } =0 \end{aligned} \end{aligned}$$

(C.3)

If I rearrange terms, I arrive at the first expression of Eq. 4.14. The second-order conditions also verify that the interior equilibrium is a maximum, and it also exists the corner solution that is reached when $\delta _j=0$, like in the previous cases. To prove whether or not $\sigma _j$ is decreasing with respect to $\tau ^{sp}$, the procedure is the same as before. The effect is ambiguous when $\delta ^*>0$ and negative if $\delta ^*=0$.