Indirect taxes in a cross-border shopping model: a monopolistic competition approach

Abstract

In a model of cross-border shopping with monopolistic competition, we examine the relative merits of an ad valorem (ADV) tax and a unit (specific) tax. Our focus is on the effects of the opening of borders that enables consumption outside the country of the residence. The result shows that the increased cross-border shopping may strengthen or weaken the superiority of either of the two tax methods, depending on the relative weight that the government places on tax revenue and welfare. Specifically, while cross-border shopping always increases the welfare of all countries in ADV tax competition, welfare increases only when the governments place a large weight on tax revenue in unit tax competition. Cross-border shopping lowers welfare when governments with high weight on welfare compete in unit tax.

Appendix

1.1 Proof of Lemma 1

From (28) and (29), $R^{AC}-R^{UC}=0$ and $U^{AC}-U^{UC}=0$ when $\theta =1/\sigma$. In addition, $R^{AC}-R^{UC}$ is an increasing function of $\theta$ and $U^{AC}-U^{UC}$ is a decreasing function of $\theta$. Therefore, $R^{AC}>(<)R^{UC}$ and $U^{AC}<(>)R^{UC}$ for $\theta >(<)1/\sigma$. Furthermore,

$$\begin{aligned} \frac{d({W^{+}}^{AC}-{W^{+}}^{UC})}{d \theta }&= \frac{d(R^{AC}-R^{UC})}{d \theta } + \frac{d(U^{AC}-U^{UC})}{d \theta } \\&= \frac{\mu }{\sigma -1} -\frac{\mu }{(1-\theta )(\sigma -1)} \le 0, \end{aligned}$$

suggesting that ${W^{+}}^{AC}-{W^{+}}^{UC}$ is a decreasing function of $\theta$. Notice that $W^{AC}-W^{UC}=0$ at $\theta =1/\sigma$. Then we have ${W^{+}}^{AC}<(>){W^{+}}^{UC}$ for $\theta >(<) 1/\sigma$.

1.2 Proof of Lemma 3

If $\theta = \hat{\theta }^{O}$, $t^{UO} = r^{AO} = 0$. Therefore, $p^{AO}=p^{UO}$, $R^{AO}=R^{UO}$, and $U^{AO} = U^{UO}$. Thus, ${W^{+}}^{AO} = {W^{+}}^{UO}$ if $\theta = \hat{\theta }^{O}$. From (39) and (40), $p^{AO}-p^{UO}$ and $R^{AO}-R^{UO}$ are the increasing function of $\theta$. Thus, $p^{AO} >(<)p^{UO}$ and $R^{AO}>(<)R^{UO}$ for $\theta >(<)\hat{\theta }^{O}$. Since

$$\begin{aligned}&\frac{d(U^{AO} - U^{UO})}{d \theta }\\&\quad = -\frac{\sigma \mu }{\sigma -1}\cdot \frac{(1-\theta ) [ 2\tau (\sigma -1)-\mu ]^2}{ \{ (\sigma -1)\mu + (1-\theta )[2 \tau (\sigma -1)-\mu ] \} \{ (\sigma -1)^2 \mu + (1-\theta )\sigma [2 \tau (\sigma -1)-\mu ] \}} \le 0, \end{aligned}$$

$U^{AO} - U^{UO}$ is a decreasing function of $\theta$. Thus, $U^{AO}<(>)U^{UO}$ for $\theta >(<)\hat{\theta }^{O}$. Furthermore,

$$\begin{aligned}&\frac{d({W^{+}}^{AO}-{W^{+}}^{UO})}{d \theta } = \frac{d(R^{AO}-R^{UO})}{d \theta } + \frac{d(U^{AO}-U^{UO})}{d \theta } \\&\quad = \frac{2 \tau (\sigma -1)-\mu }{(\sigma -1) \{ 2[\tau (\sigma -1)-\mu ] +\sigma \mu \} \{ 2[\tau (\sigma -1)-\mu ] +(\sigma -1)\mu \}}\mu \\&\qquad -\frac{\sigma \mu }{\sigma -1}\cdot \frac{(1-\theta ) [ 2\tau (\sigma -1)-\mu ]^2}{ \{ (\sigma -1)\mu + (1-\theta )[2 \tau (\sigma -1)-\mu ] \} \{ (\sigma -1)^2 \mu + (1-\theta )\sigma [2 \tau (\sigma -1)-\mu ] \}}, \end{aligned}$$

which is positive at $\theta =1$ and negative at $\theta =0$ and $\theta =\hat{\theta }^{O}$. Furthermore, the second-order derivative shows

$$\begin{aligned}&\frac{d^2({W^{+}}^{AO}-{W^{+}}^{UO})}{d \theta ^2}\\&\quad =\frac{\sigma \mu }{\sigma -1}\cdot \frac{[2\tau (\sigma -1)-\mu ]^2 \{ (\sigma -1)^3 \mu ^2 -(1-\theta )^2 \sigma [2\tau (\sigma -1)-\mu ]^2 \} }{ \{ (\sigma -1)\mu + (1-\theta )[2 \tau (\sigma -1)-\mu ] \}^2 \{ (\sigma -1)^2 \mu + (1-\theta )\sigma [2 \tau (\sigma -1)-\mu ] \}^2}, \end{aligned}$$

which implies that ${W^{+}}^{AC}-{W^{+}}^{UC}$ has at most one extremum for $0 \le \theta \le 1$. Therefore, ${W^{+}}^{AC}-{W^{+}}^{UC}$ is a decreasing function of $\theta$ for $0 \le \theta < \hat{\theta }^{O}$. Hence, ${W^{+}}^{AO}>{W^{+}}^{UO}$ for $\theta <\hat{\theta }^{O}$.

1.3 Proof of Lemma 4

We have

$$\begin{aligned} t^{UC}-t^{UO}&=\frac{(\sigma -1)[\theta (\sigma -1)^2 + (1-\theta )]}{(1-\theta ) \sigma \{ (\sigma -1)^2 \mu + (1-\theta )\sigma [2\tau (\sigma -1)-\mu \} ]} \mu c > 0, \\ r^{AC}-r^{AO}&=\frac{\theta \sigma \mu }{[2 \tau (\sigma -1) -\mu ]+(\sigma -1)\mu } \ge 0, \end{aligned}$$

which derives Lemma 4.

1.4 Proof of Proposition 2

$$\begin{aligned}&(R^{AC} - R^{AO})-(R^{UC} - R^{UO} ) \nonumber \\&\quad = \frac{\{ 2(2\sigma -1)[(\sigma -1)\tau -\mu ] +\sigma (\sigma -1)\mu \} -(1-\theta )\sigma \{ 4[(\sigma -1)\tau -\mu ] +\sigma \mu \} }{\sigma \{ 2[(\sigma -1) -\mu ]+(\sigma -1)\mu \} \{ 2[(\sigma -1) -\mu ]+\sigma \mu \} }\mu ^2.\nonumber \\ \end{aligned}$$

(42)

Thus, $(R^{AC} - R^{AO})-( R^{UC} - R^{UO} )$ increases with an increase in $\theta$. Furthermore, $(R^{AC} - R^{AO})-( R^{UC} - R^{UO} )=R^{UO}-R^{AO}>0$ if $\theta =1/\sigma$ and $(R^{AC} - R^{AO})-( R^{UC} - R^{UO} )<0$ if $\theta =0$. Therefore, Proposition 2 is proved.

1.5 Proof of Proposition 4

$$\begin{aligned} (U^{AO} - U^{AC}) -( U^{UO} - U^{UC}) = -\mu \ln \alpha g(\theta ) h(\theta ) \end{aligned}$$

(43)

where

$$\begin{aligned} \alpha&= \frac{[2(\sigma -1)\tau -\mu + (\sigma -1)\mu ]^{\sigma /(\sigma -1)}}{2[(\sigma -1)\tau -\mu ] + (\sigma -1)\mu }, \\ g(\theta )&= \frac{(1-\theta )\sigma [2(\sigma -1)\tau -\mu ]+\mu (\sigma -1)^2}{(1-\theta )\sigma [2(\sigma -1)\tau -\mu ]+\mu \sigma (\sigma -1)}, \\ h(\theta )&= \left[ 2(\sigma -1)\tau -\mu + \frac{\sigma -1}{1-\theta }\mu \right] ^{-1/(\sigma -1)}. \end{aligned}$$

As $g(\theta )$ and $h(\theta )$ are the decreasing function of $\theta$, the RHS of (43), that is, $-\mu \ln \alpha g(\theta ) h(\theta )$, is an increasing function of $\theta$. Moreover, we have $-\mu \ln \alpha g(1/\sigma ) h(1/\sigma ) = U^{AO}-U^{UO}>0$ and

$$\begin{aligned} -\mu \ln \alpha g(0) h(0)= - \mu \ln \frac{2\sigma (\sigma -1)\tau +(\sigma ^2 -3\sigma + 1)\mu }{2\sigma (\sigma -1)\tau +(\sigma ^2 -3\sigma )\mu } < 0. \end{aligned}$$

Therefore, $(U^{AO} - U^{AC}) -( U^{UO} - U^{UC}) > 0$ if $0\le \theta <\tilde{\theta }$ and $(U^{AO} - U^{AC}) -( U^{UO} - U^{UC}) < 0$ if $\tilde{\theta } < \theta \le 1$. Accordingly, Proposition 4 is proved.

1.6 Proof of Proposition 5

We have

$$\begin{aligned} \left. \frac{d W^+_j}{d t_j}\right| _{l_j=1}&=\left. \frac{d R_j}{d t_j}\right| _{l_j=1} + \left. \frac{d U_j}{d t_j}\right| _{l_j=1} \\&=-\frac{\mu }{(c+t_j)^2}\left( \frac{c}{\sigma } + t_j \right) , \\ \left. \frac{d W^+_j}{d r_j}\right| _{l_j=1}&=\left. \frac{d R_j}{d r_j}\right| _{l_j=1} + \left. \frac{d U_j}{d r_j}\right| _{l_j=1} \\&=-\frac{\mu }{1-r_j}\left( \frac{1}{\sigma -1} + r_j \right) . \end{aligned}$$

These show that domestic welfare is maximized at $t^{UC}|_{\theta =0}$ and $r^{UC}|_{\theta =0}$. Furthermore, they show that domestic welfare decreases as the tax rate moves away from $t^{UC}|_{\theta =0}$ and $r^{UC}|_{\theta =0}$.

Domestic welfare is maximized at $\theta =0$ in the equilibrium of the closed-border economy and in the equilibrium of the open-border economy with ADV tax competition. In the equilibrium of the open-border economy with unit tax competition, domestic welfare is maximized at $\theta =\mu /[2\tau \sigma (\sigma -1)-\sigma \mu ]$ because of $t^{UC}=t^{UO}$. These derive Proposition 5.

1.7 Proof of Proposition 6

$$\begin{aligned}&\lim _{\theta \rightarrow 1} [({W^+}^{AO}-{W^+}^{AC})-({W^+}^{UO}-{W^+}^{UC})]\nonumber \\&\quad = \lim _{\theta \rightarrow 1}[(U^{AO} - U^{AC}) -( U^{UO} - U^{UC})] \nonumber \\&\qquad -\lim _{\theta \rightarrow 1}[(R^{AC} - R^{AO})-( R^{UC} - R^{UO} )] \end{aligned}$$

(44)

Substituting (42) and (43) into (44), we obtain $\lim _{\theta \rightarrow 1} [({W^+}^{AO}-{W^+}^{AC})-({W^+}^{UO}-{W^+}^{UC})] = \infty$, that is ${W^+}^{AO}-{W^+}^{AC}>{W^+}^{UO}-{W^+}^{UC}$ if $\theta$ is sufficiently close to 1. Further, it is obvious, by Proposition 5, that if $\theta$ is sufficiently close to 0, $W^{AO}-W^{AC}\ge 0>W^{UO}-U^{UC}$ and $|W^{AO}-W^{AC}|<|W^{UO}-U^{UC}|$.