On the incentive compatibility of universal adoption of destination-based cash flow taxation

Abstract

We analyze the incentives for an individual country to deviate from destination-based cash flow taxation (DBCFT) in a two-country model in which both countries have adopted DBCFT. A change in a country’s corporate tax rate, degree of taxation of capital income, and/or level of border adjustment generates welfare effects through fiscal effects, a price level effect, and relative price effects. We establish that at least one country will have an incentive to deviate from universal DBCFT by reducing the deduction for capital investments, even with asymmetric countries. For a deviation involving a reduction in border adjustments, we show that both countries will have an incentive to deviate in the symmetric case. Universal DBCFT will not be incentive compatible in a one-shot tax setting game, so commitment mechanisms will be required to sustain it as an equilibrium.

Appendix

Proof of Lemma 1:

To establish the results in Lemma 1, we calculate comparative statics using the necessary conditions in (11)–(14). We totally differentiate these conditions with respect to $\lambda$ and $\lambda ^*$ and evaluate at $\delta =\delta ^*=0$, $\lambda =1-t$, $\lambda ^*=1-t^*$, taking into account the impact of changes in outputs on consumer prices. Recalling that in a market equilibrium with universal DBCFT that $f_1(k,m)=f_1^*(K-k,M-m))$ and $f_2(k,m)=f_2^*(K-k,M-m),$ we have

$$\begin{aligned}&\pi _{kk}\mathrm{d}K+\pi _{km}\mathrm{d}m+\pi _{kK}\mathrm{d}K+\pi _{kx_1}\mathrm{d}x_1 =p_2 \mathrm{d}\lambda -\mathrm{d}\lambda ^*, \end{aligned}$$

(30)

$$\begin{aligned}&\pi _{km}\mathrm{d}K+\pi _{mm}\mathrm{d}m+\pi _{mK}\mathrm{d}K+\pi _{mx_1}\mathrm{d}x_1=0, \end{aligned}$$

(31)

$$\begin{aligned}&\pi _{kK}\mathrm{d}K+\pi _{mK}\mathrm{d}m+\pi _{KK}\mathrm{d}K+\pi _{Kx_1}\mathrm{d}x_1 \nonumber \\&\qquad +(1-t^*)h^{*\prime } (x_1^*)f_1^*(K-k,M-m)\frac{\mathrm{d}p_1^*}{\mathrm{d}x_1^*}(f_1^*(K-k,M-m)\mathrm{d}K-\mathrm{d}x_1)= \mathrm{d}\lambda ^*, \nonumber \\ \end{aligned}$$

(32)

and

$$\begin{aligned}&\pi _{kx_1}\mathrm{d}K+\pi _{mx_1}\mathrm{d}m+\pi _{Kx_1}\mathrm{d}K+\pi _{x_1x_1}\mathrm{d}x_1+ (1-t)h^\prime (x_1)\frac{d p_1}{d x_1}\mathrm{d}x_1 \nonumber \\&\quad \quad - (1-t^*)h^{*\prime }(x_1^*)\frac{\mathrm{d}p_1^*}{\mathrm{d}x_1^*}(f_1(K-k,M-m)\mathrm{d}K-\mathrm{d}x_1) = 0 \end{aligned}$$

(33)

as $\mathrm{d}x_1^*=f_1^*(K-k,M-m)\mathrm{d}K-\mathrm{d}x_1$, $\mathrm{d}p_1/\mathrm{d}x_1=p_2u^{\prime \prime }(c_1)h^\prime (x_1)$, and $\mathrm{d}p_1^*/\mathrm{d}x_1^*=u^{*\prime \prime }(c_1^*)h^{*\prime }(x_1^*)$. The subscripts on $\pi$ in (30)–(33) refer to derivatives of the multinational’s profit function holding prices constant.

Direct calculation yields, at a universal DBCFT market equilibrium, that $\pi _{kk}= (1-t^*)p_1^*h^{*\prime }(x_1^*)(f_{11}(k,m)+f_{11}^*(K-k,M-m))<0$, $\pi _{km}=(1-t^*)p_1^*h^{*\prime }(x_1^*)(f_{12}(k,m)+f_{12}^*(K-k,M-m))>0$, $\pi _{kK}=-(1-t^*)p_1^*h^{*\prime }(x_1^*)f_{11}^*(K-k,M-m)>0$, $\pi _{kx_1}=\pi _{mx_1}=0$, $\pi _{mm}=(1-t^*)p_1^*h^{*\prime }(x_1^*)(f_{22}(k,m)+f_{22}^*(K-k,M-m))<0$, $\pi _{mK}=-(1-t^*)p_1^*h^{*\prime }(x_1^*)f_{12}^*(K-k,M-m)<0$, $\pi _{KK}=(1-t^*)p_1^*[h^{*\prime \prime }(x_1^*)f_1^{*2} (K-k,M-m)+h^{*\prime }(x_1^*)f_{11}^*(K-k,M-m)]<0$, $\pi _{Kx_1}=-(1-t^*)p_1^*h^{*\prime \prime }(x_1^*)f_1^*(K-k,M-m)>0$, and $\pi _{x_1 x_1}=(1-t)p_1 h^{\prime \prime }(x_1)+(1-t^*)p_1^*h^{*\prime \prime }(x_1^*)<0$. It is straightforward but tedious to show that the leading principal minors of the Hessian matrix associated with the multinational firm’s optimization problem at fixed prices alternate in sign, with those of odd order negative under Assumption 1. Therefore, the Hessian is negative definite and the optimization problem of a representative firm is strictly concave in $\{k, m, K, x_1\}$.

The effect of changes in K and $x_1$ on prices is captured by the terms

$$\begin{aligned} a_1 & = (1-t^*)u^{*\prime \prime }(h^*(x_1^*))h^{*\prime }(x_1^*)^2 f_1^*(K-k,M-m)^2<0,\\ a_2 & = -(1-t^*) u^{*\prime \prime }(h^*(x_1^*)) h^{*\prime }(x_1^*)^2 f_1^*(K-k,M-m)>0, \text {and} \\ a_3 & = (1-t^*)u^{\prime \prime }(h(x_1))h^\prime (x_1)^2+(1-t^*)u^{*\prime \prime }(h^*(x_1^*))h^{*\prime }(x_1^*)^2<0. \end{aligned}$$

We then have

$$\begin{aligned} \left( \begin{array}{cccc} \pi _{kk} & \pi _{km} & \pi _{kK} & 0\\ \pi _{km} & \pi _{mm} & \pi _{mK} & 0\\ \pi _{kK} & \pi _{mK} & \pi _{KK}+a_1 & \pi _{Kx_1}+a_2 \\ 0 & 0 & \pi _{Kx_1}+a_2 & \pi _{x_1 x_1}+a_3 \end{array} \right) \left( \begin{array}{c}\mathrm{d}K\\ \mathrm{d}m\\ \mathrm{d}K\\ \mathrm{d}x_1 \end{array} \right) = \left( \begin{array}{c} p_2 \mathrm{d}\lambda -\mathrm{d}\lambda ^*\\ 0\\ \mathrm{d}\lambda ^*\\ 0 \end{array} \right) . \end{aligned}$$

(34)

We can also establish using the properties of the Hessian of the firm’s optimization problem and the fact that $a_1 a_3 -a_2^2>0$ that the 4x4 matrix in (34), which we denote by $\nabla ^2 \Pi$, is negative definite. We thus have $|\nabla ^2 \Pi |>0$.

To solve efficiently for $\mathrm{d}K/d \lambda$ and $\mathrm{d}x_1/d \lambda$, we define

$$\begin{aligned} |AK\lambda |=\left| \begin{array}{cccc} \pi _{kk} & \pi _{km} & p_2 & 0 \\ \pi _{km} & \pi _{mm} & 0 & 0\\ \pi _{kK} & \pi _{mK} & 0 & \pi _{Kx_1}+a_2 \\ 0 & 0 & 0 & \pi _{x_1 x_1}+a_3\end{array} \right| =p_2(\pi _{x_1 x_1}+a_3)((1-t^*) p_1^*h^{*\prime })^2 \Delta <0 \end{aligned}$$

(35)

and

$$\begin{aligned} |Ax\lambda |=\left| \begin{array}{cccc} \pi _{kk} & \pi _{km} & \pi _{kK} &p_2\\ \pi _{km} & \pi _{mm} & \pi _{mK} & 0\\ \pi _{kK} & \pi _{mK} & \pi _{KK}+a_1 & 0 \\ 0 & 0 & \pi _{Kx_1}+a_2 & 0\end{array} \right| =-p_2(\pi _{Kx_1}+a_2)((1-t^*) p_1^*h^{*\prime })^2 \Delta <0. \end{aligned}$$

(36)

where $\pi _{mk}\pi _{Km}-\pi _{Kk}\pi _{mm}=((1-t^*) p_1^*h^{*\prime })^2\Delta$ from the definition in Assumption 1. This establishes $\mathrm{d}K/d \lambda = |AK\lambda |/|\nabla ^2 \Pi |<0$ and $\mathrm{d}x_1/d \lambda =|Ax\lambda |/|\nabla ^2 \Pi |<0$. Using the fact that $\frac{\mathrm{d}x_1^*}{\mathrm{d}\lambda }=f_1^*(K-k,M-m) \frac{\mathrm{d}K}{\mathrm{d}\lambda }-\frac{\mathrm{d}x_1}{\mathrm{d}\lambda }$ in the neighborhood of a universal DBCFT market equilibrium, $\mathrm{d}x_1^*/d \lambda$ can be shown to be negative. One can also evaluate the good 1 price changes as

$$\begin{aligned} \frac{\mathrm{d}p_1}{\mathrm{d}x_1} \frac{\mathrm{d}x_1}{\mathrm{d}\lambda }=\frac{p_2^2 u^{\prime \prime } h^\prime ((1-t^*) p_1^*h^{*\prime } )^3 \Delta (h^{*\prime \prime } u^{*\prime }+u^{*\prime \prime } h^{*\prime })}{|\nabla ^2 \Pi |}>0 \end{aligned}$$

(37)

and

$$\begin{aligned} \frac{\mathrm{d}p_1^*}{d x_1^*} \frac{\mathrm{d}x_1^*}{\mathrm{d}\lambda }=\frac{u^{*\prime \prime } ((1-t^*)p_1^*)^3 (h^{*\prime })^4 \Delta (h^{*\prime \prime } u^{*\prime }+u^{*\prime \prime } h^{*\prime })}{|\nabla ^2 \Pi |}>0. \end{aligned}$$

(38)

Analogously,

$$\begin{aligned} |AK\lambda ^*|=\left| \begin{array}{cccc} \pi _{kk} & \pi _{km} & -1 & 0 \\ \pi _{km} & \pi _{mm} & 0 & 0\\ \pi _{kK} & \pi _{mK} & 1 & \pi _{Kx_1}+a_2 \\ 0 & 0 & 0 & \pi _{x_1 x_1}+a_3\end{array} \right| =(\pi _{x_1 x_1}+a_3)((1-t^*) p_1^*h^{*\prime })^2 \Delta ^*<0 \end{aligned}$$

(39)

and

$$\begin{aligned} |Ax\lambda ^*|=\left| \begin{array}{cccc} \pi _{kk} & \pi _{km} & \pi _{kK} &-1\\ \pi _{km} & \pi _{mm} & \pi _{mK} & 0\\ \pi _{kK} & \pi _{mK} & \pi _{KK}+a_1 & 1 \\ 0 & 0 & \pi _{Kx_1}+a_2 & 0\end{array} \right| =-(\pi _{Kx_1}+a_2)((1-t^*) p_1^*h^{*\prime })^2 \Delta ^*<0 \end{aligned}$$

(40)

where we use the fact that $\pi _{kk} \pi _{mm}-\pi _{km}^2=(1-t^*) p_1^*(h^{*\prime })^2(\Delta + \Delta ^*)$. Thus, $\mathrm{d}K/d \lambda ^*= |AK\lambda ^*|/|\nabla ^2 \Pi |<0$ and $\mathrm{d}x_1/d \lambda ^*=|Ax\lambda ^*|/|\nabla ^2 \Pi |=\frac{\Delta ^*}{p_2 \Delta } \frac{\mathrm{d}x_1}{\mathrm{d}\lambda }<0$. Similarly, we have $\frac{\mathrm{d}x_1^*}{\mathrm{d}\lambda ^*}=f_1^*\frac{\mathrm{d}K}{\mathrm{d}\lambda ^*}-\frac{\mathrm{d}x_1^*}{\mathrm{d}\lambda ^*}=\frac{\Delta ^*}{p_2 \Delta } \frac{\mathrm{d}x_1^*}{\mathrm{d}\lambda }<0$. $\square$

Combining these results, we have $\frac{\mathrm{d}p_1^*}{\mathrm{d}\lambda ^*}=\frac{\Delta ^*}{p_2 \Delta } \frac{\mathrm{d}p_1^*}{\mathrm{d}\lambda }$ and $\frac{\mathrm{d}p_1}{\mathrm{d}\lambda ^*}=\frac{\Delta ^*}{p_2 \Delta } \frac{\mathrm{d}p_1}{\mathrm{d}\lambda }$.

Proof of Lemma 2:

Part (a) follows from direct differentiation of (7). To prove part (b), we need to calculate the effect of a change in the border adjustments $\{\delta ,\delta ^*\}$ on the equilibrium values of $\{k,m,K,x_1 \}$. Totally differentiating (11)–(14) and evaluating the expressions at universal DBCFT yields

$$\begin{aligned} \left( \begin{array}{cccc} \pi _{kk} & \pi _{km} & \pi _{kK} & 0\\ \pi _{km} & \pi _{mm} & \pi _{mK} & 0\\ \pi _{kK} & \pi _{mK} & \pi _{KK}+a_1 & \pi _{Kx_1}+a_2 \\ 0 & 0 & \pi _{Kx_1}+a_2 & \pi _{x_1 x_1}+a_3 \end{array} \right) \left( \begin{array}{c}\mathrm{d}K\\ \mathrm{d}m\\ \mathrm{d}K\\ \mathrm{d}x_1 \end{array} \right) = \left( \begin{array}{c} t^*(t \mathrm{d}\delta -t^*\mathrm{d}\delta ^*)\\ qf_2(t \mathrm{d}\delta -t^*\mathrm{d}\delta ^*)\\ 0\\ -t^*q (t \mathrm{d}\delta -t^*\mathrm{d}\delta ^*) \end{array} \right) , \end{aligned}$$

(41)

where the left matrix on the left-hand side is the negative definite matrix $\nabla ^2 \Pi$ from Lemma 1. Notice that each term in the column vector on the right-hand side of (41) is multiplied by $t\mathrm{d} \delta - t^*\mathrm{d} \delta ^*$. Thus, for each $y \in \{k,m,K,x_1 \}$, it must be that $\frac{\mathrm{d}y}{t \mathrm{d}\delta }=\frac{\mathrm{d}y}{-t^*\mathrm{d} \delta ^*}$. To establish (c), we use the fact that $\frac{\mathrm{d}c_1}{\mathrm{d}x_1}=h^\prime (x_1) \mathrm{d}x_1$ and (b) to obtain $sign \frac{\mathrm{d}c_1}{\mathrm{d} \delta }=-sign \frac{\mathrm{d}c_1}{\mathrm{d} \delta ^*}$. The result for $c_1^*$ follows similarly from $\frac{\mathrm{d}x_1^*}{\mathrm{d}\delta }=-\frac{\mathrm{d}x_1}{\mathrm{d}\delta }$ from (1) at an initial equilibrium with DBCFT. $\square$

Proof of Lemma 3:

This lemma requires that we solve (41) explicitly for $\mathrm{d}K/\mathrm{d}\delta$ and $\mathrm{d}x_1/\mathrm{d}\delta$. To do so define

$$\begin{aligned} AK\delta =\left( \begin{array}{cccc} \pi _{kk} & \pi _{km} & tt^*& 0 \\ \pi _{km} & \pi _{mm} & tqf_2 & 0\\ \pi _{kK} & \pi _{mK} & 0 & \pi _{Kx_1}+a_2 \\ 0 & 0 & -tt^*q & \pi _{x_1 x_1}+a_3\end{array} \right) \end{aligned}$$

(42)

and

$$\begin{aligned} Ax\delta =\left( \begin{array}{cccc} \pi _{kk} & \pi _{km} & \pi _{kK} & tt^*\\ \pi _{km} & \pi _{mm} & \pi _{mK} & tqf_2\\ \pi _{kK} & \pi _{mK} & \pi _{KK}+a_1 & 0 \\ 0 & 0 & \pi _{Kx_1}+a_2 & -tt^*q\end{array} \right) . \end{aligned}$$

(43)

Then, $\mathrm{d}K/\mathrm{d} \delta = |AK\delta |/|\nabla ^2 \Pi |$ and $\mathrm{d}x_1/\mathrm{d} \delta =|Ax\delta |/|\nabla ^2 \Pi |$.

In a symmetric economy, $|AK\delta |=t^2 q(\pi _{Kx_1}+a_2)|\nabla _2^2 \pi |+t^2 (\pi _{x_1 x_1}+a_3) (\pi _{km}\pi _{mK}-\pi _{kK}\pi _{mm}),$ where $\nabla _k^2 \pi$ is the $k^{th}$ order principal minor of $\nabla ^2 \pi$ and $\pi _{kk}\pi _{mK}-\pi _{kK}\pi _{km}=0.$ Further simplification yields

$$\begin{aligned} |AK\delta |=2t^2(1-t)^2 p_1^2 (h^\prime )^2 \nabla ^2 f[2q(\pi _{K x_1}+a_2)+\pi _{x_1 x_1}+a_3] =0, \end{aligned}$$

(44)

so, at $\delta =\delta ^*=0$ and $t=t^*$, $\mathrm{d}K/\mathrm{d} \delta =0$ and $\mathrm{d}x_1^*/\mathrm{d} \delta =-\mathrm{d}x_1/\mathrm{d} \delta$ . It is also the case in a symmetric economy that

$$\begin{aligned} |Ax\delta | & = -t^2(\pi _{Kx_1}+a_2)(\pi _{km}\pi _{mK}-\pi _{mm}\pi _{kK})-t^2 q|\nabla _3^2\Pi | \nonumber \\ & = 2t^2(1-t)^3p_1^2(h^\prime )^2 \nabla ^2 f f_1 [u^\prime h^{\prime \prime }-u^{\prime \prime }(h^\prime )^2]-t^2q|\nabla _3^2 \pi | \end{aligned}$$

(45)

so

$$\begin{aligned} |Ax\delta |=-2t^2(1-t)^3p_1^2(h^\prime )^2\nabla ^2 f \left( p_1 f_1 h^{\prime \prime } +(h^\prime )^2 f_1 u^{\prime \prime }+q p_1 h^\prime f_{11} \right) >0. \end{aligned}$$

(46)

$\square$

Notation List: (*) denotes Foreign values
Consumer variables/functions
$L (L^*)$	Labor endowment
$c_1 (c_1^*), c_2 (c_2^*)$	Good 1 and good 2 consumption
$U (U^*)$	Consumer utility function/Country welfare
$u (u^*)$	Good 1 subutility function
$\nu (\nu ^*)$	Public good subutility function
$\beta (\beta ^*)$	Multinational ownership share
Multinational variables/functions
M	Managerial skill endowment
$k (k^*)$	Affiliate capital employment
$m (m^*)$	Affiliate managerial skill usage
K	Aggregate capital
$x_1 (x_1^*)$	Intermediate good production
$f (f^*)$	Intermediate good production function
$h (h^*)$	Good 1 production function
e	Net Home exports of the intermediate good
q	Intermediate good transfer price
$\pi$	Global after-tax multinational profit
$\pi ^h$	Home affiliate profit
Prices
$p_1 (p_1^*)$	Good 1 price
$p_2 (p_2^*)$	Good 2 and capital price
$p_{2x} (p_{2x}^*)$	Capital export price
$w (w^*)$	Wage rate
Country variables
$g (g^*)$	Public good expenditure
$t (t^*)$	Corporate tax rate
$\lambda (\lambda ^*)$	After-tax cost of a dollar of capital expenditure
$\delta (\delta ^*)$	Border adjustment
$T_1 (T_1^*)$	Affiliate corporate taxes
$T_2 (T_2^*)$	Capital trade corporate taxes
$z_2$	Home net capital exports