Profit-splitting rules and the taxation of multinational digital platforms

Abstract

This paper analyzes the strategy of a monopolistic digital platform serving users from two jurisdictions with different corporate tax rates. We consider two profit-splitting rules, Separate Accounting and Formula Apportionment based on the number of users in the two jurisdictions. We show that, even in the absence of transfer pricing, the platform shifts profit from the high-tax to the low-tax jurisdiction exploiting network externalities under Separate Accounting and manipulating the apportionment key under Formula Apportionment. In order to shift profit, the platform distorts prices and quantities. Under Separate Accounting, the direction of the distortions depends on the sign of the externalities. We use a numerical simulation to show that the ranking of fiscal revenues under the two regimes differs in the two jurisdictions: The high-tax jurisdiction prefers Separate Accounting to Formula Apportionment, whereas the low-tax jurisdiction prefers Formula Apportionment to Separate Accounting.

Appendices

Appendix

Proof of Proposition 1

In the absence of externalities, under SA the platform chooses $x_A$ and $x_B$ to maximize the profits separately in the two jurisdictions,

$$\begin{aligned} \frac{\partial V_A}{\partial x_A} = \frac{\partial V_B}{\partial x_B} = 0, \end{aligned}$$

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resulting in the optimal number of users $x_A^*$ and $x_B^*$.

Under FA, the platform chooses $x_A$ and $x_B$ to satisfy the first-order conditions:

$$\begin{aligned} \frac{\partial V}{\partial x_A}\left[ 1 - t_A \frac{x_A}{x_A + x_B} - t_B \frac{x_B}{x_A + x_B}\right]= & {} \frac{V (t_A-t_B) x_B}{(x_A+x_B)^2}, \end{aligned}$$

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$$\begin{aligned} \frac{\partial V}{\partial x_B} \left[ 1 - t_A \frac{x_A}{x_A + x_B} - t_B \frac{x_B}{x_A + x_B}\right]= & {} \frac{V (t_B-t_A) x_A}{(x_A+x_B)^2}. \end{aligned}$$

(15)

If $t_A >t_B$, $\frac{\partial V}{\partial x_A} >0$. As the profit is concave in $x_A$, this implies that $x_A< x_A^*$. Similarly, as $\frac{\partial V}{\partial x_B} <0$, $x_B>x_B^*$. $\square$

Proof of Proposition 2

Let $t'_A > t_A$. Let $z = (x_A,y_A,x_B,y_B)$ denote an optimal choice of the platform when the tax rate in A is $t_A$ and $z'=(x'_A,y'_A,x'_B,y'_B)$ an optimal choice when the rate is $t'_A$. Optimality implies

$$\begin{aligned} (1-t_A) V_A(z) + (1-t_B) V_B(z)\ge & {} (1-t_A) V_A(z') + (1-t_B) V_B(z'), \end{aligned}$$

(16)

$$\begin{aligned} (1-t'_A) V_A(z') + (1-t_B) V_B(z')\ge & {} (1-t'_A) V_A(z) + (1-t_B) V_B (z), \end{aligned}$$

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which can be written as

$$\begin{aligned} (1- t_A) (V_A(z) - V_A(z') )\ge (1-t_B)( V_B(z') -V_B(z)) \ge (1- t'_A) (V_A(z) - V_A(z') ) . \end{aligned}$$

Since $t'_A\ge t_A$, the two extreme inequalities imply $V_A (z) \ge V_A(z')$, which in turn implies $V_B(z') -V_B(z)\ge 0$. When the profit is strictly concave, the optimal choices are unique and the above inequalities are strict.

Consider now the change in the pre-tax profit $\Delta = V_A(z')+ V_B(z') - ( V_A(z)+ V_B(z))$. Writing the post-tax profit $(1-t_A) V_A(z) + (1-t_B) V_B(z)$ as $(1-t_B) V(z) + (t_A-t_B) V_A(z)$, we derive from (16) and (17) again,

$$\begin{aligned} - (t'_A-t_B) (V_A(z) - V_A(z') ) \le (1-t_B) \Delta \le - (t_A-t_B) (V_A(z) - V_A(z') ). \end{aligned}$$

Since $V_A(z) - V_A(z') \ge 0$, $\Delta \le 0$ if $t_A \ge t_B$: Starting with a tax in A at least equal to the level in B, increasing $t_A$ further has a negative effect on the pre-tax profit. The opposite holds, i.e., $\Delta \ge 0$, when $t'_A \le t_B$: In that case, $t_A < t_B$ hence the increase in the tax level diminishes the gap between the tax levels of the countries. $\square$

Proof of Proposition 3

It is convenient to operate a change in variables. Let us define the share of users of type x in A as $\lambda = \frac{x_A}{x_A + x_B}$, the share of users of type y in A as $\mu = \frac{y_A}{y_A+y_B}$, the total number of users of type x as $T= x_A+x_B$ and the total number of users of type y as $Z = y_A + y_B$. We rewrite the profit as

$$\begin{aligned} \Pi ( \lambda , T )= [1 - t_B - ( t_A - t_B ) (\omega \lambda + (1-\omega ) \mu )] V(\lambda T , \mu Z, (1-\lambda ) T, (1-\mu ) Z). \end{aligned}$$

Let $t'_A > t_A$. Let $(\lambda , \mu , T, Z)$ denote an optimal choice of the platform when the tax rate in A is $t_A$ and $(\lambda ' , \mu ', T', Z')$ when the rate is $t'_A$, and $v= V(\lambda T , \mu Z, (1-\lambda ) T, (1-\mu ) Z)$ and $v'= V(\lambda ' T' , \mu ' Z', (1-\lambda ') T', (1-\mu ') Z')$. Optimality implies that

$$\begin{aligned} (1 - t_B - ( t_A - t_B ) (\omega \lambda + (1-\omega ) \mu )) v\ge & {} \\ (1 - t_B - ( t_A - t_B )(\omega \lambda ' + (1-\omega ) \mu ')) v'&\\ (1 - t_B - ( t'_A - t_B )(\omega \lambda ' + (1-\omega ) \mu ') ) v'\ge & {} \\ (1 - t_B - ( t'_A - t_B ) (\omega \lambda + (1-\omega ) \mu )) v,&\end{aligned}$$

which results in:

$$\begin{aligned} \frac{ 1 - t_B - ( t'_A - t_B ) (\omega \lambda ' + (1-\omega ) \mu ') }{ [1 - t_B - ( t'_A - t_B ) (\omega \lambda + (1-\omega ) \mu )] } \ge \frac{ v}{v'} \ge \frac{ 1 - t_B - ( t_A - t_B )(\omega \lambda ' + (1-\omega ) \mu ')}{ 1 - t_B - ( t_A - t_B ) (\omega \lambda + (1-\omega ) \mu ) } \end{aligned}$$

or

$$\begin{aligned}&1+ \frac{ ( t'_A - t_B ) (\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu ')) }{ 1 - t_B - ( t'_A - t_B ) (\omega \lambda + (1-\omega ) \mu ) } \nonumber \\&\quad \ge \frac{ v}{v'} \ge 1+ \frac{ ( t_A - t_B ) (\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu '))}{ 1 - t_B - ( t_A - t_B ) (\omega \lambda + (1-\omega ) \mu ) }. \end{aligned}$$

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Consider the terms on the left and the right of the above equation. Since $t'_A >t_A$, the denominator in the left equation is larger than that on the right. Hence, we must have $( t'_A - t_B )(\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu ')) \ge ( t_A - t_B ) (\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu '))$, which in turn implies $(\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu '))>0$ since $t'_A> t_A \ge t_B$: The share of users in A goes down. This implies $\frac{ V(\lambda T , \mu Z, (1-\lambda ) T, (1-\mu ) Z)}{V(\lambda ' T' , \mu ' Z', (1-\lambda ') T', (1-\mu ') Z') } \ge 1$, so that the pre-tax profit V decreases.

To prove that the tax base in jurisdiction B increases, we need to show (with the change of variables) that $(1- (\omega \lambda + (1-\omega ) \mu )) V(\lambda T , \mu Z, (1-\lambda ) T, (1-\mu ) Z) \le (1- (\omega \lambda ' + (1-\omega ) \mu '))V(\lambda ' T' , \mu ' Z', (1-\lambda ') T', (1-\mu ') Z')$. Using (18), it suffices to show

$$\begin{aligned} \frac{ 1- (\omega \lambda ' + (1-\omega ) \mu ')}{1 - (\omega \lambda + (1-\omega ) \mu )} \ge 1+ \frac{ ( t'_A - t_B ) (\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu ')) }{ 1 - t_B - ( t'_A - t_B ) (\omega \lambda + (1-\omega ) \mu ) }, \end{aligned}$$

which is equivalent to

$$\begin{aligned} \frac{(\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu ')) }{1 -(\omega \lambda + (1-\omega ) \mu )} \ge \frac{ ( t'_A - t_B ) ((\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu ')) }{ 1 - t_B - ( t'_A - t_B )(\omega \lambda + (1-\omega ) \mu ) }, \end{aligned}$$

or, since $(\omega (\lambda -\lambda ') + (1-\omega ) (\mu -\mu ') \ge 0$, to

$$\begin{aligned}{}[1 - t_B - ( t'_A - t_B ) (\omega \lambda + (1-\omega ) \mu )] \ge ( t'_A - t_B ) ( 1 -(\omega \lambda + (1-\omega ) \mu )) \end{aligned}$$

and finally to $1\ge t'_A$, which is of course always satisfied. $\square$

Proof of Proposition 4

As noted in the text, if the profit is concave in $(x_A,x_B)$, the first-order conditions (4) and (5) hold at an interior solution. By the implicit function theorem, $X_A$ and $X_B$ are differentiable in $t_A$ and the derivatives satisfy:

$$\begin{aligned} {\frac{\partial ^2 \Pi }{\partial x_A \partial t_A}} + {\frac{\partial ^2 \Pi }{\partial x_A \partial x_A}} X'_A(t_A) + {\frac{\partial ^2 \Pi }{\partial x_A \partial x_B}} X'_B(t_A) =0, \end{aligned}$$

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$$\begin{aligned} {\frac{\partial ^2 \Pi }{\partial x_B \partial t_A}} + {\frac{\partial ^2 \Pi }{\partial x_A \partial x_B}} X'_A(t_A) + {\frac{\partial ^2 \Pi }{\partial x_B \partial x_B}} X'_B(t_A) =0. \end{aligned}$$

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Solving the system of linear equations, we obtain:

$$\begin{aligned} X'_A(t_A)= & {} \frac{ \frac{\partial ^2 \Pi }{\partial x_A \partial x_B} \frac{\partial ^2 \Pi }{\partial x_B \partial t_A} - \frac{\partial ^2 \Pi }{\partial x_B \partial x_B} \frac{\partial ^2 \Pi }{\partial x_A \partial t_A} }{ \frac{\partial ^2 \Pi }{\partial x_A \partial x_A} \frac{\partial ^2 \Pi }{\partial x_B \partial x_B} - \frac{\partial ^2 \Pi }{\partial x_A \partial x_B} ^2 } \\ X'_B(t_A)= & {} \frac{ \frac{\partial ^2 \Pi }{\partial x_A \partial x_B} \frac{\partial ^2 \Pi }{\partial x_A \partial t_A} - \frac{\partial ^2 \Pi }{\partial x_A \partial x_A} \frac{\partial ^2 \Pi }{\partial x_B \partial t_A} }{ \frac{\partial ^2 \Pi }{\partial x_A \partial x_A} \frac{\partial ^2 \Pi }{\partial x_B \partial x_B} - \frac{\partial ^2 \Pi }{\partial x_A \partial x_B} ^2. } \end{aligned}$$

Factoring out the numerator and denominator by $\frac{\partial ^2 \Pi }{\partial x_A \partial x_A} \frac{\partial ^2 \Pi }{\partial x_B \partial x_B}$, we obtain expressions (8) and (9). Now, under SA,

$$\begin{aligned} \frac{\partial ^2 \Pi }{\partial x_A \partial t_A} = - \frac{\partial V_A}{\partial x_A} \text{ and } \frac{\partial ^2 \Pi }{\partial x_B \partial t_A} = - \frac{\partial V_A}{\partial x_B}. \end{aligned}$$

Using the first-order condition on profit maximization with respect to $x_A$ (4), we have

$$\begin{aligned} - \frac{\partial V_A}{\partial x_A} = \frac{1-t_B}{1-t_A} \frac{\partial V_B}{\partial x_A} = \frac{1-t_B}{1-t_A} x_B \frac{\partial P_B}{\partial x_A}, \end{aligned}$$

which gives the expression for $\delta _A$. Finally,

$$\begin{aligned} - \frac{\partial V_A}{\partial x_B} = - x_B \frac{\partial P_B}{\partial x_A}, \end{aligned}$$

which gives the expression for $\delta _B$. $\square$

Proof of Proposition 5

Consider one of the situations where direct and indirect effects have opposite signs. Suppose that markets are complements and externalities are positive, $\delta _A >0, \delta _B<0$ and $s_A, s_B >0$. All other situations are handled in a similar way. Suppose without loss of generality that the indirect effect dominates the direct effect for $X_A$ so that

$$\begin{aligned} \delta _A + s_A \delta _B < 0, \end{aligned}$$

yielding

$$\begin{aligned} \delta _B < - \frac{\delta _A}{s_A}. \end{aligned}$$

Consider the effect of a change in $t_A$ on $X_B$:

$$\begin{aligned} \delta _B + s_B \delta _A< - \frac{\delta _A}{s_A} + s_B \delta _A = \frac{\delta _A}{s_A} (-1 + s_A s_B) < 0, \end{aligned}$$

where the last inequality is obtained because $\delta _A > 0$ and $s_A s_B < 1$ by strict concavity of the profit. This implies that the direct effect dominates the indirect effect for $X_B$. $\square$

Proof of Proposition 6

The direct effects are still given by the same expression (8) as under SA but computed at an optimal solution under FA. Necessarily, the second derivatives $\frac{\partial ^2 \Pi }{\partial x_A \partial x_A}$ and $\frac{\partial ^2 \Pi }{\partial x_B \partial x_B}$ are negative (even if the profit is not globally concave) so it suffices to compute the sign of $\frac{\partial ^2 \Pi }{\partial x_A \partial t_A}$ and $\frac{\partial ^2 \Pi }{\partial x_B \partial t_A}$.

$$\begin{aligned} \frac{\partial ^2 \Pi }{\partial x_A \partial t_A}= & {} - \frac{\partial V \frac{x_A}{x_A + x_B}}{\partial x_A} \\= & {} - \frac{V x_B}{(x_A + x_B)^2} - \frac{x_A}{x_A + x_B} \frac{\partial V}{\partial x_A}. \end{aligned}$$

By the first-order conditions of profit maximization

$$\begin{aligned} \frac{\partial \Pi }{\partial x_A}= & {} \frac{\partial V}{\partial x_A} \left[ 1 - t_A \frac{x_A}{x_A + x_B} - t_B \frac{x_B}{x_A+x_B}\right] - \frac{V x_B (t_A - t_B)}{(x_A + x_B)^2}, \\= & {} 0 \end{aligned}$$

so that

$$\begin{aligned} \frac{\partial V}{\partial x_A} = \frac{1}{ [1 - t_A \frac{x_A}{x_A + x_B} - t_B \frac{x_B}{x_A+x_B}]} \frac{V x_B (t_A - t_B)}{(x_A + x_B)^2} \ge 0, \end{aligned}$$

showing that $\frac{\partial ^2 \Pi }{\partial x_A \partial t_A} < 0$. Similarly, we compute

$$\begin{aligned} \frac{\partial ^2 \Pi }{\partial x_B \partial t_A}= & {} \frac{V x_A}{(x_A + x_B)^2} - \frac{x_A}{x_A + x_B} \frac{\partial V}{\partial x_B}, \\= & {} \frac{V x_A}{(x_A + x_B)^2} + \frac{1}{ [1 - t_A \frac{x_A}{x_A + x_B} - t_B \frac{x_B}{x_A+x_B}]} \frac{V x_A (t_A - t_B)}{(x_A + x_B)^2} \\> & {} 0, \end{aligned}$$

showing that $\frac{\partial ^2 \Pi }{\partial x_B \partial t_A} > 0.$ $\square$

Online appendix

1.1 Micro-foundation of the prices

In this appendix, we describe a model of consumer behavior which gives rise to the inverse demand functions considered in the text. Every jurisdiction is populated by a continuum of heterogeneous users characterized by their willingness to pay for using the platform. In addition, users experience externalities from the presence of other users on the platform. Formally, letting U and W denote the utilities of users of the two types on the platform,

$$\begin{aligned} U_A= & {}\, \theta _A + u_A(x_A,y_A, x_B,y_B) - p_A, \\ W_A= &\, {} \eta _A + w_A(x_A,y_A, x_B,y_B) - q_A, \\ U_B= &\,{} \theta _B + u_B(x_A,y_A, x_B,y_B) - p_B, \\ W_B= &\, {} \eta _B + w_B(x_A,y_A, x_B,y_B) - q_B, \end{aligned}$$

where $\theta _A$ and $\theta _B$ are distributed according to continuous distributions with full support $F_A$ and $F_B$ on $[{\underline{\theta }}, {\overline{\theta }}]$ and $\eta _A$ and $\eta _B$ are distributed according to continuous distributions with full support $G_A$ and $G_B$ on $[{\underline{\eta }}, {\overline{\eta }}]$.

We now derive the demand associated to the fees $(p_A, q_A, p_B, q_B)$. Let $x_A,y_A, x_B, y_B$ be the common expectation of every user over the number of users in the two jurisdictions. We compute the value of the user of type 1 in jurisdiction A who is indifferent between buying access to the platform or not. This value is given by

$$\begin{aligned} \widehat{\theta _A} = p_A - u_A(x_A,y_A, x_B,y_B), \end{aligned}$$

provided that $p_A - u_A(x_A,y_A,x_B,y_B)$ belongs to the support of $F_A$; otherwise, $\widehat{\theta _A}$ will be equal to one of the extreme values ${\underline{\theta }}$ (if the market is covered) or ${\overline{\theta }}$ (if no user accesses the platform). We can similarly compute the value of all other indifferent users. The two jurisdictions may have different sizes. We normalize the measure of users in jurisdiction B to 1, and let s denote the measure of users in jurisdiction A. Assuming that expectations are rational, the demand thus satisfies

$$\begin{aligned} x_A = s(1-F_A(p_A-u_A(x_A,y_A,x_B,y_B)). \end{aligned}$$

Similarly,

$$\begin{aligned} y_A= & {}\, s(1-G_A(q_A - w_A(x_A,y_A,x_B,y_B)), \\ x_B= &\, {} 1-F_B(p_B-u_B(x_A,y_A,x_B,y_B)), \\ y_B= & \,{} 1-G_B(q_B-w_B(x_A,y_A,x_B,y_B)). \end{aligned}$$

From the computations above, the prices are given by¹⁷

$$\begin{aligned} P_A (x_A,y_A,x_B,y_B)= &\, {} u_A(x_A,y_A,x_B,y_B) + F^{-1}_A (1-\frac{x_A}{s}), \end{aligned}$$

(21)

$$\begin{aligned} Q_A(x_A,y_A,x_B,y_B)= &\, {} w_A(x_A,y_A,x_B,y_B) + G^{-1}_A (1-\frac{y_A}{s}), \end{aligned}$$

(22)

$$\begin{aligned} P_B (x_A,y_A,x_B,y_B)= &\, {} u_B(x_A,y_A,x_B,y_B) + F^{-1}_B (1-x_B), \end{aligned}$$

(23)

$$\begin{aligned} Q_B(x_A,y_A,x_B,y_B)= &\, {} w_B(x_A,y_A,x_B,y_B) + G^{-1}_B (1-y_B). \end{aligned}$$

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The sign of the derivatives $\partial P_A / \partial y_A$, $\partial P_A / \partial x_B$, $\partial P_A / \partial y_B$ depends on the sign of the externalities, $\partial u_A / \partial y_A$, $\partial u_A / \partial x_B$, $\partial u_A/\partial y_B$. For $P_A$ to be decreasing in $x_A$, as required in the paper we need the externalities $\partial u_A/\partial x_A$ to be small relative to the direct effect measured by $\frac{\partial F^{-1}_A (1-\frac{x_A}{s})}{\partial x_A}$. The same computations hold for the inverse demands $Q_A,P_B$ and $Q_B$.

Next, we compute the surplus of users and show that they only depend on the number of users of the same type in the same jurisdiction. Consider the surplus of users of type 1 in jurisdiction A. Taking into account their participation decision, the surplus of users of type 1 in jurisdiction A can be written as

$$\begin{aligned} US_A= & {} \int _{p_A- u(x_A,y_A,x_B,y_B)}^{{\underline{\theta }}} [ \theta _A + u_A(x_A,y_A,x_B,y_B) - p_A ] f_A(\theta _A) \mathrm{{d}} \theta _A, \end{aligned}$$

which, using (21), writes

$$\begin{aligned} US_A= & {} \int _{F^{-1}_A (1-\frac{x_A}{s})}^{{\underline{\theta }}} [ \theta _A - F^{-1}_A (1-\frac{x_A}{s})] f_A(\theta _A) \mathrm{{d}} \theta _A. \end{aligned}$$

The surplus of users of type 1 in jurisdiction A can thus be written as a function of $x_A$. Furthermore, it is easy to check that this function is non-decreasing: Since $F^{-1}_A (1-\frac{x_A}{s})$ is non-increasing in $x_A$ both the domain of integration and the integrand are non-decreasing in $x_A$. Hence, as intuition suggests, an increase in the number of users $x_A$ results in an increase in the user surplus of type 1 users in jurisdiction A. The same reasoning holds for the surplus of all other type of users, showing that measures of number of users and of user surplus are perfectly correlated.

1.2 The linear model

In this appendix, we consider the linear model with two types of users. Let

$$\begin{aligned} P_A(x_A,x_B)= &\, {} 1- {\sigma _A}x_A +\beta x_B, \end{aligned}$$

(25)

$$\begin{aligned} P_B(x_A,x_B)= &\, {} 1-{\sigma _B}x_B +\alpha x_A. \end{aligned}$$

(26)

We first consider the optimal choice of the platform under Separate Accounting. The platform’s profit is given by

$$\begin{aligned} \Pi = (1-t_A)x_A(1-\sigma _A x_A + \beta x_B) + (1-t_B) x_B(1-\sigma _B x_B + \alpha x_A). \end{aligned}$$

The first-order conditions are given by:

$$\begin{aligned} (1-t_A)(1-2 \sigma _A x_A + \beta x_B) + (1-t_B) \alpha x_B= &\, {} 0, \\ (1-t_A) \beta x_A + (1-t_B) (1-2 \sigma _B x_B + \alpha x_A)= & \,{} 0. \end{aligned}$$

Solving the system of linear equations, we obtain the interior solutions:

$$\begin{aligned} X_A= & {} \frac{ 2 \sigma _B (1-t_A)(1-t_B) + (1-t_B) [ \beta (1-t_A) + \alpha (1-t_B)]}{ 4 \sigma _A \sigma _B (1-t_A)(1-t_B) - [\beta (1-t_A) + \alpha (1-t_B)]^2}, \end{aligned}$$

(27)

$$\begin{aligned} X_B= & {} \frac{2 \sigma _A (1-t_A)(1-t_B) + (1-t_A) [ \beta (1-t_A) + \alpha (1-t_B)]}{ 4 \sigma _A \sigma _B (1-t_A)(1-t_B) - [\beta (1-t_A) + \alpha (1-t_B)]^2}. \end{aligned}$$

(28)

The second-order conditions are satisfied if $4 \sigma _A \sigma _B (1-t_A)(1-t_B) > [\beta (1-t_A) + \alpha (1-t_B)]^2$. We need to put additional restrictions (that we do not explicitly spell out) to guarantee that the prices $P_A$ and $P_B$ are positive.

Even in the linear model, the comparative statics effect of changes in $t_A$ on $X_A$ and $X_B$ cannot be signed easily. The decomposition into direct and indirect effects gives

$$\begin{aligned} \delta _A= & {} \frac{1-t_B}{1-t_A} \frac{ \alpha x_B }{2(1-t_A) sigma_A}, \, \delta _B= - \frac{ \beta x_A }{ 2(1-t_B) \sigma _B} \\ s_A= & {} \frac{ (1-t_A) \beta + (1-t_B) \alpha }{2(1-t_A) \sigma _A} , \, \, s_B= \frac{ (1-t_A) \beta + (1-t_B) \alpha }{2(1-t_B) \sigma _B}. \end{aligned}$$

Notice that $s_A$ and $s_B$ are both positive—the linear model captures a market with complements. The direct effects $\delta _A$ and $\delta _B$ depend on the optimal solutions $x_A$ and $x_B$, making a comparison of the magnitude of the direct and indirect effects uneasy. Our next result provides a sufficient condition under which prices $P_A$ and $P_B$ are monotonic in the corporate tax rate $t_A$.

Proposition 7

Suppose that $\sigma _A \sigma _B \ge \max \{ \alpha (\frac{1-t_B}{1-t_A}\alpha +\beta ), \beta (\alpha + \frac{1-t_A}{1-t_B}\beta ) \}$, then at an interior solution, the equilibrium price $P_A$ is decreasing in $t_A$ , while the equilibrium price $P_B$ is increasing in $t_A$.

Proof of Proposition 7

Note that

$$\begin{aligned} P'_A(t_A)= & {}\, - \sigma _A X'_A(t_A) + \beta X'_B(t_A), \\ P'_B(t_A)= &\, {} \alpha X'_A(t_A) - \sigma _B X'_B(t_A). \end{aligned}$$

We use the decomposition of $X'_A(t_A)$ and $X'_B(t_A)$ to write

$$\begin{aligned} P'_A(t_A)= & {} \frac{1}{1-s_A s_B} [ - \sigma _A + \beta s_B] \delta _A + [- \sigma _A s_A + \beta ] \delta _B], \\ P'_B(t_A)= & {} \frac{1}{1-s_A s_B} [ \alpha - \sigma _B s_B ] \delta _A + [\alpha s_A - \sigma _B] \delta _B. \end{aligned}$$

Using the formulas for $\delta _A, \delta _B, \sigma _A$ and $\sigma _B$, the sign of $P'_A(t_A)$ is the same as the sign of

$$\begin{aligned} R'_A= & {} (- 2 \sigma _A \sigma _B (1-t_B) + \beta [(1-t_A) \beta + (1-t_B) \alpha ]) \frac{1-t_B}{1-t_A} \alpha x_B \\&+ ( 2 \beta (1-t_A) \sigma _A - \sigma _A [(1-t_A) \beta + (1-t_B) \alpha ]) (- \beta x_A). \end{aligned}$$

Now, because the price $P_B$ is positive

$$\begin{aligned} \sigma _B \frac{1-t_B}{1-t_A} x_B - \beta x_A > 0. \end{aligned}$$

Hence,

$$\begin{aligned} - \sigma _A \sigma _B (1-t_B) \frac{1-t_B}{1-t_A} \alpha x_B + \sigma _A (1-t_B) \alpha \beta x_A < 0, \end{aligned}$$

so that

$$\begin{aligned} R'_A< & {} (- \sigma _A \sigma _B (1-t_B) + \beta [(1-t_A) \beta + (1-t_B) \alpha ]) \frac{1-t_B}{1-t_A} \alpha x_B \\- & {} \beta \sigma _A (1-t_A) \beta x_A, \\< & {} (- \sigma _A \sigma _B (1-t_B) + \beta [(1-t_A) \beta + (1-t_B) \alpha ]) \frac{1-t_B}{1-t_A} \alpha x_B. \end{aligned}$$

and using the condition $\sigma _A \sigma _B \ge \beta (\alpha + \frac{1-t_A}{1-t_B}\beta )$, $R'_A < 0$.

Similarly, we see that $P'_B(t_A)$ has the same sign as

$$\begin{aligned} R'_B= & {} (2 \alpha (1-t_B) \sigma _B - \sigma _B [(1-t_A) \beta + (1-t_B) \alpha ]) \frac{1-t_B}{1-t_A} \alpha x_B \\&+ (-2 (1-t_A) \sigma _A \sigma _B + \alpha [(1-t_A) \beta + (1-t_B) \alpha ])(- \beta x_A). \end{aligned}$$

Because the price $P_A$ is positive,

$$\begin{aligned} \sigma _A x_A > \alpha \frac{1-t_A}{1-t_B} x_B, \end{aligned}$$

so that

$$\begin{aligned} (1-t_A) \sigma _A \sigma _B \beta x_A > (1-t_A) \sigma _B \beta \alpha \frac{1-t_A}{1-t_B} x_B \end{aligned}$$

and

$$\begin{aligned} R'_B> & {} \alpha (1-t_B) \sigma _B \frac{1-t_B}{1-t_A} \alpha x_B \\&+\, ((1-t_A) \sigma _A \sigma _B - \alpha [[(1-t_A) \beta + (1-t_B) \alpha ]) \beta x_A, \\> &\, {} ((1-t_A) \sigma _A \sigma _B - \alpha [[(1-t_A) \beta + (1-t_B) \alpha ]) \beta x_A. \end{aligned}$$

Using the condition $\sigma _A \sigma _B \ge \alpha (\alpha + \frac{1-t_B}{1-t_A} \beta )$, the result follows. $\square$

Proposition 7 shows that, as in the analysis of Kind et al. (2005, 2008, 2010, 2013), an increase in the corporate tax rate $t_A$ leads to a reduction in the price $P_A$ and an increase in the price $P_B$ in the linear model. Next, we analyze tax competition when platforms are symmetric (Figs. 6, 7, 8, 9, 10, 11).

Proposition 8

Suppose that platforms are symmetric, $\sigma _A = \sigma _B = \sigma$, $\alpha = \beta$. In a model of tax competition, where both countries choose their corporate tax rate to maximize tax revenues, there exists a unique symmetric equilibrium where

$$\begin{aligned} t^* = 1 - \left( \frac{\alpha }{\sigma -\alpha }\right) ^2. \end{aligned}$$

Proof of Proposition 8

Consider the marginal effect of an increase in $t_A$ on $R_A$ at a point where $t_B=t_A =t$. We compute

$$\begin{aligned} \frac{\partial R_A}{\partial t_A} = V_A(t) + (1- 2 \sigma x + \alpha x)X'_A(t_A)+ \alpha x X'_B(t_A). \end{aligned}$$

Now, at $t_A=t_B=t$, $X'_A(t_A) = -X'_B(t_A)$. In addition, $V_A = x(1-\sigma x + \alpha x)$ and $1 - 2 \sigma x + 2 \alpha x = 0$. Hence,

$$\begin{aligned} \frac{\partial R_A}{\partial t_A} = x(1-\sigma x + \alpha x) - 2 \alpha x X'_A(t_A). \end{aligned}$$

Next, using the decomposition formula,

$$\begin{aligned} X'_A(t_A) = \frac{\delta }{1-s} = \frac{\frac{\alpha x}{2(1-t)\sigma }}{1 - \frac{\alpha }{\sigma }} = \frac{\alpha x}{2(\sigma - \alpha )(1-t)}. \end{aligned}$$

Replacing,

$$\begin{aligned} \frac{\partial R_A}{\partial t_A} = x^2 (\sigma - \alpha ) - x^2 \frac{\alpha ^2}{(\sigma -\alpha )(1-t)}. \end{aligned}$$

Hence, at a symmetric equilibrium where $\frac{\partial R_A}{\partial t_A} = 0$,

$$\begin{aligned} t ^*= 1 - ( \frac{\alpha }{\sigma -\alpha })^2. \end{aligned}$$

$\square$

We next consider the optimal choice of the platform under Formula Apportionment. As in the Proof of Proposition 3, let ${\lambda }= \frac{x_A}{x_A + x_B}$ and $T = x_A + x_B$. After this change of variable, the profit of the platform becomes:

$$\begin{aligned} \Pi= & \,{} (1- {\lambda }t_A - (1- {\lambda })t_B) [ {\lambda }T (1- \sigma _A {\lambda }T + \beta (1- {\lambda }T) \\&+ (1-{\lambda }) T (1- \sigma _B (1- {\lambda }) T+ \alpha {\lambda }T)] \\= &\, {} (1- {\lambda }t_A - (1- {\lambda })t_B) T [1 + {\lambda }(1-{\lambda }) (\alpha + \beta ) T - \sigma _A {\lambda }^2 T - \sigma _B (1-{\lambda })^2 T]. \end{aligned}$$

Notice in particular that the optimal choices of the platform under FA only depend on the sum of externalities $\alpha + \beta$. The first-order conditions with respect to ${\lambda }$ and T give rise to a system of quadratic equations:

$$\begin{aligned} 0= & \,{} (t_B - t_A) [1 + {\lambda }(1-{\lambda }) (\alpha + \beta ) T - \sigma _A {\lambda }^2 T - \sigma _B (1-{\lambda })^2 T] \\&+ T[(\alpha + \beta )(1- 2 {\lambda }) - 2 {\lambda }\sigma _A - 2 (1-{\lambda }) \sigma _B] (1- {\lambda }t_A - (1- {\lambda })t_B), \\ 0= &\, {} 1 + 2(\alpha + \beta ) {\lambda }(1-{\lambda }) T - 2 \sigma _A{\lambda }^2 T - 2 \sigma _B (1-{\lambda })^2 T. \end{aligned}$$

Assuming that the parameters are chosen so that the second-order conditions are satisfied, this system of equation gives rise to an interior solution in $({\lambda },T)$ which allows us to compute the optimal number of users $X_A$ and $X_B$. Unfortunately, under Formula Apportionment, even in the linear model, the decomposition of the effect of a change in $t_A$ on the number of users into direct and indirect effects does not give rise to simple formulas that can easily be signed or interpreted.

1.3 Numerical simulations

1.3.1 Asymmetric countries

We suppose that jurisdiction A is three times larger than jurisdiction B, which results in $\sigma _A = \frac{1}{3} -0.1$, whereas $\sigma _B = 0.9$.

Fig. 6

Asymmetric countries: tax base, tax revenues and profits

Fig. 7

Asymmetric countries: users and prices

1.3.2 Asymmetric externalities I: positive from A to B, negative from B to A

We suppose that jurisdictions have the same size, with $\sigma _A = \sigma _B =0.9$ but externalities are asymmetric. There are positive externalities from A to B, reflected in $\alpha = 0.3$, but negative externalities from B to A reflected in $\beta = -0.1$. These numbers capture the situation of a search engine where advertisers are located in the low-tax jurisdiction and users in the high-tax jurisdiction.

Fig. 8

Asymmetric externalities I: tax base, tax revenues and profits

Fig. 9

Asymmetric externalities I: users and prices

1.3.3 Asymmetric externalities II: negative from A to B, positive from B to A

We suppose that jurisdictions have the same size, with $\sigma _A = \sigma _B =0.9$ but externalities are asymmetric. There are negative externalities from A to B, reflected in $\alpha = -0.1$, and positive externalities from B to A reflected in $\beta =0.3$.

Fig. 10

Asymmetric externalities II: tax base, tax revenues and profits

Fig. 11

Asymmetric externalities II: users and prices