Cross-Border Shopping with Fiscal Externalities

Abstract

In this analysis, we construct a model of asymmetric regions in which the numbers of national borders vary across regions by extending Lucas’s (Reg Sci Urban Econ 34(4):365–385, 2004) one-country model to a two-country model. We consider the following three cases: an integrated world, unitary nations, and decentralization. In the integrated world, a supranational government uniformly implements policy; the outcome in this case is the second-best optimum. In the case of unitary nations, each central government sets a non-coordinated policy. Finally, under decentralization, the central and local governments in both countries set non-coordinated policies. We show that the central governments cannot internalize the fiscal externalities attributed to the existence of a national border in the unitary nations and decentralization cases. Furthermore, in the case of unitary nations, each central government sets a lower tax rate in the region with the national border than in the region without the national border.

Appendices

Appendix

Case of $\tau_{iB} > \tau_{iA}$

If $\tau_{iB} > \tau_{iA}$ and $\tau_{{i^{\prime}B}} > \tau_{iB}$, the central government of country i’s maximization problem is formulated as

$$ \begin{aligned} & \mathop {\hbox{max} }\limits_{{\tau_{iA} ,\tau_{iB} ,g_{iA} ,g_{iB} ,G_{i} }} V\left( {\tau_{iA} } \right) + \int\limits_{{\hat{d}_{iB} }}^{1} {V\left( {\tau_{iB} } \right){\text{d}}d_{iB} } \\ & \qquad + \int\limits_{0}^{{\hat{d}_{iB} }} {V\left( {\tau_{iA} ,d_{iB} } \right){\text{d}}d_{iB} } + \mathop \sum \limits_{j = A,B} b\left( {g_{ij} } \right) + 2B\left( {G_{i} } \right), \\ & s.t.\,g_{iA} + g_{iB} + G_{i} = \tau_{iA} \left( {y_{iA} \left( {\tau_{iA} } \right) + \int\limits_{0}^{{\hat{d}_{iB} }} {y_{iB} \left( {\tau_{iA} , d_{iB} } \right){\text{d}}d_{iB} } } \right) \\ & \qquad + \tau_{iB} \left( {\int\limits_{{\hat{d}_{iB} }}^{1} {y_{iB} \left( {\tau_{iB} } \right){\text{d}}d_{iB} } + \int\limits_{0}^{{\widehat{D}_{{i^{{\prime }} B}} }} {y_{iB} \left( {\tau_{iB} } \right){\text{d}}D_{{i^{{\prime }} B}} } } \right). \\ \end{aligned} $$

(3.52)

From the first-order conditions for this problem and Roy’s identity, we obtain the following conditions:

$$ - \left( {1 + \hat{d}_{iB} } \right)y_{iA} + \lambda_{c} \left\{ {\left( {1 + \hat{d}_{iB} } \right)\left( {y_{iA} + \tau_{iA} \frac{{\partial y_{iA} }}{{\partial \tau_{iA} }}} \right) + \left( {\tau_{iA} y_{iA} - \tau_{iB} y_{iB} } \right)\frac{{\partial \hat{d}_{iB} }}{{\partial \tau_{iA} }}} \right\} = 0, $$

(3.53)

$$ \begin{aligned} & - \left( {1 - \hat{d}_{{iB}} } \right)y_{{iB}} + \lambda _{c} \left\{ {\left( {1 - \hat{d}_{{iB}} } \right)\left( {y_{{iB}} + \tau _{{iB}} \frac{{\partial y_{{iB}} }}{{\partial \tau _{{iB}} }}} \right)} \right. \\ & \quad \left. { + \left( {\tau _{{iA}} y_{{iA}} - \tau _{{iB}} y_{{iB}} } \right)\frac{{\partial \hat{d}_{{iB}} }}{{\partial \tau _{{iB}} }} + \tau _{{iB}} y_{{iB}} \frac{{\partial \widehat{D}_{{i^{\prime } B}} }}{{\partial \tau _{{iB}} }}} \right\} = 0, \\ \end{aligned} $$

(3.54)

$$ b^{{\prime }} \left( {g_{iA} } \right) - \lambda_{c} = 0, $$

(3.55)

$$ b^{{\prime }} \left( {g_{iB} } \right) - \lambda_{c} = 0, $$

(3.56)

$$ 2B^{{\prime }} \left( {G_{i} } \right) - \lambda_{c} = 0, $$

(3.57)

where $\lambda_{c}$ is the Lagrange multiplier that corresponds to the central government’s budget constraint, Eq. 3.52. From Eqs. 3.53–3.57 and the assumption of symmetric countries $\left( {\widehat{D}_{1B} = \widehat{D}_{2B} = 0} \right)$, we obtain the following necessary condition for the local and national public goods¹²:

$$ \begin{aligned} 2B_{i}^{{\prime }} & = b_{iA}^{{\prime }} = b_{iB}^{{\prime }} = \frac{{\left( {1 + \hat{d}_{iB} } \right)y_{iA} }}{{\left( {1 + \hat{d}_{iB} } \right)Y_{iA} + \left( {\tau_{iA} y_{iA} - \tau_{iB} y_{iB} } \right)\frac{{\partial \hat{d}_{iB} }}{{\partial \tau_{iA} }}}} \\ & = \frac{{\left( {1 - \hat{d}_{iB} } \right)y_{iB}}}{{\left( {1 - \hat{d}_{iB} } \right)Y_{iB} + \left( {\tau_{iA} y_{iA} - \tau_{iB} y_{iB} } \right)\frac{{\partial \hat{d}_{iB} }}{{\partial \tau_{iA} }} + \tau_{iB} y_{iB} \frac{{\partial \hat{D}_{{i^{{\prime }} B}} }}{{\partial \tau_{iB} }}}}, \\ \end{aligned} $$

(3.58)

where $Y_{ij} \equiv y_{ij} \left( {\tau_{ij} } \right) + \tau_{ij} \partial y_{ij} \left( {\tau_{ij} } \right)/\partial \tau_{ij} ,\left( {j = A, B} \right).$ These conditions imply that the marginal benefits of public funds must equal the MCPF.

Suppose that the tax rates in region A and B are the same: $\tau_{iA} = \tau_{iB} = \tau_{i}$. In this case, we obtain the following results: $y_{iA} \left( {\tau_{iA} } \right) = y_{iB} \left( {\tau_{iB} } \right) = y\left( {\tau_{i} } \right)$ and $\hat{d}_{iA} = 0$. Substituting these results into Eq. 3.58, the MCPF of local public good A is smaller than that of local public good B. We can see that Eq. 3.58 does not hold if $\tau_{iA} = \tau_{iB} = \tau_{i}$. This result means that $\tau_{iA} \ne \tau_{iB}$. Given the general assumption that the MCPF is increasing in its own tax rate, $\tau_{iA} < \tau_{iB}$ in the optimum. This result is inconsistent with the assumption that $\tau_{iB} > \tau_{iA}$.

Proof of Eqs. 3.50 and 3.51

Substituting Eqs. 3.27 and 3.41 into $\frac{{\partial g_{iB} }}{{\partial t_{iA} }} + \frac{{\partial G_{i} }}{{\partial t_{iA} }} = 0$ and then simplifying the resulting expression, we obtain

$$ \begin{aligned} & \left\{ {\left( {1 - \hat{d}_{{iA}} } \right)\left( {y_{{iA}} + t_{{iA}} \frac{{\partial y_{{iA}} }}{{\partial \tau _{{iA}} }}} \right) - t_{{iA}} y_{{iA}} \frac{{\partial \hat{d}_{{iA}} }}{{\partial \tau _{{iA}} }}} \right\}m_{{iA}} \\ & = \left( {1 - \hat{d}_{{iA}} } \right)T_{{iA}} \frac{{\partial y_{{iA}} }}{{\partial \tau _{{iA}} }} + \left( {\tau _{{iB}} y_{{iB}} - T_{{iA}} y_{{iA}} } \right)\frac{{\partial \hat{d}_{{iA}} }}{{\partial \tau _{{iA}} }}. \\ \end{aligned} $$

(3.59)

Solving Eq. 3.59 with respect to $m_{iA}$, we can obtain Eq. 3.50.

Substituting Eqs. 3.26, 3.28, and 3.42 into $\frac{{\partial g_{iA} }}{{\partial t_{iB} }} + \frac{{\partial G_{i} }}{{\partial t_{iB} }} - \hat{d}_{iA} \frac{{\partial g_{iB} }}{{\partial t_{iB} }} = 0$ and simplifying the resulting expression, we obtain

$$ \begin{aligned}& \left( {1 + \hat{d}_{iA} } \right)\left\{ {\left( {1 + \hat{d}_{iA} + \widehat{D}_{{i^{{\prime }} A}} } \right)\left( {y_{iB} + t_{iB} \frac{{\partial y_{iB} }}{{\partial \tau_{iB} }}} \right) + t_{iB} y_{iB} \left( {\frac{{\partial \hat{d}_{iA} }}{{\partial \tau_{iB} }} + \frac{{\partial \widehat{D}_{{i^{{\prime }} B}} }}{{\partial \tau_{iB} }}} \right)} \right\}m_{iB} \\ & \quad\quad\quad\quad\quad\quad\quad= - \hat{d}_{iA} \left( {1 + \hat{d}_{iA} } \right)\left( {y_{iB} + t_{iB} \frac{\partial y}{{\partial \tau_{iB} }}} \right) + \left( {1 + \hat{d}_{iA} } \right)T_{iB} \frac{{\partial y_{iB} }}{{\partial \tau_{iB} }} \\ & \quad\quad\quad\quad\quad\quad\quad- \hat{d}_{iA} t_{iB} y_{iB} \left( {\frac{{\partial \hat{d}_{iA} }}{{\partial \tau_{iB} }} + \frac{{\partial \widehat{D}_{{i^{{\prime }} B}} }}{{\partial \tau_{iB} }}} \right) + T_{iB} y_{iB} \left( {\frac{{\partial \hat{d}_{iA} }}{{\partial \tau_{iB} }} + \frac{{\partial \widehat{D}_{{i^{{\prime }} B}} }}{{\partial \tau_{iB} }}} \right) - \tau_{iB} y_{iA} \frac{{\partial \hat{d}_{iA} }}{{\partial \tau_{iB} }}. \\ \end{aligned} $$

(3.60)

Because $\widehat{D}_{{i^{{\prime }} B}} = 0$ from the assumption of symmetric countries, we can obtain Eq. 3.51.