Appendix A
1. Transport costs are infinite (\(\phi = 0\))
Scenario I
When the transport costs are infinite (\(\phi = 0\)), it is found that symmetric configuration \(h_{1} = h_{2} = h_{3} = 1/3\) will yield \(P_{1}^{\mu } = P_{2}^{\mu }\) from (23) and \(w_{1} = w_{2}\) form (45). This ensures this configuration is an equilibrium from (52) with the same tax rate for each region, \(t_{1} = t_{2} = t_{3} = t\). For the stability condition, \({\text{d}}\rho_{21} /{\text{d}}h_{2}\) < 0 is needed when it is evaluated at the symmetric equilibrium point. It is derived that
$$\frac{{\partial \rho_{21} }}{{\partial h_{2} }} = - \frac{{9\left[ {\left( {1 + 2r} \right)\left( {\sigma - \mu - 1} \right) - \delta \mu \left( {1 - r} \right)} \right]}}{{2\left( {2r + 1} \right)\left( {\sigma - 1} \right)}}.$$
(52)
The stability is then obtained as \(\left( {1 + 2r} \right)\left( {\sigma - \mu - 1} \right) - \delta \mu \left( {1 - r} \right) > 0\). This condition will be satisfied as \(\sigma\) is large and \(\mu\) is small.
Scenario II
When the transport costs are infinite (\(\phi = 0)\), suppose that symmetric configuration \(h_{1} = h_{2} = h_{3} = \frac{1}{3}\) is an equilibrium. The wage rate will be the same among regions: \(w_{1} = w_{2} = w_{3}\). Then the real income will also be the same among regions due to the price index in each region will be the same. In addition, the public good provision will be the same among regions: \(g_{1} = g_{2} = g_{3}\) when the tax rates are the same among regions. It then yields that \(G_{2} > G_{1} = G_{3}\) as long as \(r > 0\) from (33) to (37). This implies that \(V_{2} > V_{1} = V_{3}\). We thus conclude the symmetric configuration \(h_{1} = h_{2} = h_{3} = \frac{1}{3}\) is not an equilibrium.
2. Transport costs are zero (\(\phi = 1)\)
Scenario I
When the transport costs are zero (\(\phi = 1)\), the agglomeration is ensured when \(h_{2} = 1\), \(h_{1} = h_{3} = 0\), with the same tax rate and \(\rho_{21}\) > 1 for the agglomeration point. It is derived that
$$\rho_{21} = \left[ {\frac{\sigma + 2r\sigma + 2\mu (1 - r)}{\sigma + 2r\sigma - \mu (1 - r)}} \right]^{\delta } > 1.$$
(53)
Scenario II
When the transport costs are zero (\(\phi = 1)\), the agglomeration at median location is ensured when \(h_{2} = 1\), \(h_{1} = h_{3} = 0\), and \(\rho_{21}\) > 1 for the agglomeration point with the same tax rate. It is derived that
$$\rho_{21} = \left[ {\frac{\sigma + 2r\sigma + 2\mu (1 - r)}{{\sigma + \left( {r + r^{2} } \right)\sigma + \mu r\left( {1 - r} \right) - \mu (1 - r)}}} \right]^{\delta } > 1.$$
(54)
The agglomeration at non-median location is ensured when \(h_{1} = 1\), \(h_{2} = h_{3} = 0\), and \(\rho_{12}\) > 1 for the agglomeration point with the same tax rate. It is derived that
$$\rho_{12} = \left[ {\frac{{\sigma + r\left( {1 + r} \right)\sigma + 2\mu - \mu r(1 + r)}}{\sigma + 2r\sigma + \mu r - \mu }} \right]^{\delta } .$$
(55)
The stability condition is obtained as \(\left( {3 - 2r - r^{2} } \right)\mu - r\left( {1 - r} \right)\sigma > 0\). It is satisfied when \(\mu\) is large and \(\sigma\) is small enough. This condition is very unlikely to be satisfied.
Appendix B
Scenario I
The indirect utility of each region is as follows:
$$V_{1} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left( {1 - t_{1} } \right)\left( {t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } ,$$
(56)
$$V_{2} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left( {1 - t_{2} } \right)\left( {rt_{1} Y_{1} P_{1}^{ - \mu } + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } ,$$
(57)
$$V_{3} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left( {1 - t_{3} } \right)\left( {rt_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } .$$
(58)
To solve the maximum of indirect function of each region is to differentiate \(V_{1}\), \(V_{2}\), and \(V_{3}\) with respect to \(t_{1}\), \(t_{2}\), and \(t_{3}\), respectively.
$$\begin{aligned} \frac{{\partial V_{1} }}{{\partial t_{1} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left( {t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {t_{1} Y_{1} P_{1}^{ - \mu } } \right.} \right. \\ & \quad \left. {\left. { + rt_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right) + \delta \left( {1 - t_{1} } \right)Y_{1} P_{1}^{ - \mu } } \right], \\ \end{aligned}$$
(59)
$$\begin{aligned} \frac{{\partial V_{2} }}{{\partial t_{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left( {rt_{1} Y_{1} P_{1}^{ - \mu } + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {rt_{1} Y_{1} P_{1}^{ - \mu } } \right.} \right. \\ & \quad \left. {\left. { + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right) + \delta \left( {1 - t_{2} } \right)Y_{2} P_{2}^{ - \mu } } \right], \\ \end{aligned}$$
(60)
$$\begin{aligned}
\frac{{\partial V_{3} }}{{\partial t_{3} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left( {rt_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ - \left( {rt_{1} Y_{1} P_{1}^{ - \mu } } \right. \right. \\ & \quad \left. \left. + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } \right)+ \delta \left( {1 - t_{3} } \right)Y_{3} P_{3}^{ - \mu } \right].
\end{aligned}$$
(61)
Letting (59)–(61) be zero yields the first-order condition for this problem:
$$\left( {1 + \delta } \right)t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } = \delta Y_{1} P_{1}^{ - \mu } ,$$
(62)
$$rt_{1} Y_{1} P_{1}^{ - \mu } + \left( {1 + \delta } \right)t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } = \delta Y_{2} P_{2}^{ - \mu } ,$$
(63)
$$rt_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + \left( {1 + \delta } \right)t_{3} Y_{3} P_{3}^{ - \mu } = \delta Y_{3} P_{3}^{ - \mu } .$$
(64)
Solving (62)–(64) yields the Nash tax rate for each region in (48). Note that the second-order condition is satisfied as follows:
$$\begin{aligned} \frac{{\partial^{2} V_{1} }}{{\partial t_{1}^{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left\{ { - \left( {1 - \delta } \right)Y_{1} P_{1}^{ - \mu } \left( {t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 2} } \right. \\ & \quad \cdot \left[ { - \left( {1 + \delta } \right)t_{1} Y_{1} P_{1}^{ - \mu } - rt_{2} Y_{2} P_{2}^{ - \mu } - rt_{3} Y_{3} P_{3}^{ - \mu } + \delta Y_{1} P_{1}^{ - \mu } } \right] + \left( {t_{1} Y_{1} P_{1}^{ - \mu } } \right. \\ & \quad \left. {\left. { + rt_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {1 + \delta } \right)Y_{1} P_{1}^{ - \mu } } \right]} \right\} < 0, \\ \end{aligned}$$
(65)
after substituting (62) into it.
$$\begin{aligned} \frac{{\partial^{2} V_{2} }}{{\partial t_{2}^{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left\{ { - \left( {1 - \delta } \right)Y_{2} P_{2}^{ - \mu } \left( {rt_{1} Y_{1} P_{1}^{ - \mu } + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 2} } \right. \\ & \quad \cdot \left[ { - rt_{1} Y_{1} P_{1}^{ - \mu } - \left( {1 + \delta } \right)t_{2} Y_{2} P_{2}^{ - \mu } - rt_{3} Y_{3} P_{3}^{ - \mu } + \delta Y_{2} P_{2}^{ - \mu } } \right] + \left( {rt_{1} Y_{1} P_{1}^{ - \mu } } \right. \\ & \quad \left. {\left. { + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {1 + \delta } \right)Y_{2} P_{2}^{ - \mu } } \right]} \right\} < 0, \\ \end{aligned}$$
(66)
after substituting (63) into it.
$$\begin{aligned} \frac{{\partial^{2} V_{3} }}{{\partial t_{3}^{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left\{ { - \left( {1 - \delta } \right)Y_{3} P_{3}^{ - \mu } \left( {rt_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 2} } \right. \\ & \quad \cdot \left[ { - rt_{1} Y_{1} P_{1}^{ - \mu } - rt_{2} Y_{2} P_{2}^{ - \mu } - \left( {1 + \delta } \right)t_{3} Y_{3} P_{3}^{ - \mu } + \delta Y_{3} P_{3}^{ - \mu } } \right] + \left( {rt_{1} Y_{1} P_{1}^{ - \mu } } \right. \\ & \quad \left. {\left. { + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {1 + \delta } \right)Y_{3} P_{3}^{ - \mu } } \right]} \right\} < 0, \\ \end{aligned}$$
(67)
after substituting (64) into it.
Scenario II
The indirect utility of each region is as follows:
$$V_{1} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left( {1 - t_{1} } \right)\left( {t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + r^{2} t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } ,$$
(68)
$$V_{2} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left( {1 - t_{2} } \right)\left( {rt_{1} Y_{1} P_{1}^{ - \mu } + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } ,$$
(69)
$$V_{3} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left( {1 - t_{3} } \right)\left( {r^{2} t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } .$$
(70)
To solve the maximum of indirect function of each region is to differentiate \(V_{1}\), \(V_{2}\), and \(V_{3}\) with respect to \(t_{1}\), \(t_{2}\), and \(t_{3}\), respectively.
$$\begin{aligned} \frac{{\partial V_{1} }}{{\partial t_{1} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left( {t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + r^{2} t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {t_{1} Y_{1} P_{1}^{ - \mu } } \right.} \right. \\ & \quad \left. { + rt_{2} Y_{2} P_{2}^{ - \mu } + r^{2} t_{3} Y_{3} P_{3}^{ - \mu } } \right) + \left. {\delta \left( {1 - t_{1} } \right)Y_{1} P_{1}^{ - \mu } } \right], \\ \end{aligned}$$
(71)
$$\begin{aligned} \frac{{\partial V_{2} }}{{\partial t_{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left( {rt_{1} Y_{1} P_{1}^{ - \mu } + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {rt_{1} Y_{1} P_{1}^{ - \mu } } \right.} \right. \\ & \quad \left. {\left. { + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right) + \delta \left( {1 - t_{2} } \right)Y_{2} P_{2}^{ - \mu } } \right], \hfill \\ \end{aligned}$$
(72)
$$\begin{aligned} \frac{{\partial V_{3} }}{{\partial t_{3} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left( {r^{2} t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {r^{2} t_{1} Y_{1} P_{1}^{ - \mu } } \right.} \right. \\ & \quad \left. { + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } + \delta \left( {1 - t_{3} } \right)Y_{3} P_{3}^{ - \mu } } \right]. \hfill \\ \end{aligned}$$
(73)
Letting (71)–(73) be zero yields the first-order condition for this problem:
$$\left( {1 + \delta } \right)t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + r^{2} t_{3} Y_{3} P_{3}^{ - \mu } = \delta Y_{1} P_{1}^{ - \mu } ,$$
(74)
$$rt_{1} Y_{1} P_{1}^{ - \mu } + \left( {1 + \delta } \right)t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } = \delta Y_{2} P_{2}^{ - \mu } ,$$
(75)
$$r^{2} t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + \left( {1 + \delta } \right)t_{3} Y_{3} P_{3}^{ - \mu } = \delta Y_{3} P_{3}^{ - \mu } .$$
(76)
Solving (74)–(76) yields the Nash tax rate for each region in (49)–(51). Note that the second-order condition is satisfied as follows:
$$\begin{aligned} \frac{{\partial^{2} V_{1} }}{{\partial t_{1}^{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left\{ { - \left( {1 - \delta } \right)Y_{1} P_{1}^{ - \mu } \left( {t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + r^{2} t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 2} } \right. \\ & \quad \cdot \left[ { - \left( {1 + \delta } \right)t_{1} Y_{1} P_{1}^{ - \mu } - rt_{2} Y_{2} P_{2}^{ - \mu } - r^{2} t_{3} Y_{3} P_{3}^{ - \mu } + \delta Y_{1} P_{1}^{ - \mu } } \right] + \left( {t_{1} Y_{1} P_{1}^{ - \mu } } \right. \\ & \quad \left. {\left. { + rt_{2} Y_{2} P_{2}^{ - \mu } + r^{2} t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {1 + \delta } \right)Y_{1} P_{1}^{ - \mu } } \right]} \right\} < 0, \hfill \\ \end{aligned}$$
(77)
after substituting (74) into it
$$\begin{aligned} \frac{{\partial^{2} V_{2} }}{{\partial t_{2}^{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left\{ { - \left( {1 - \delta } \right)Y_{2} P_{2}^{ - \mu } \left( {rt_{1} Y_{1} P_{1}^{ - \mu } + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 2} } \right. \\ & \quad \cdot \left[ { - rt_{1} Y_{1} P_{1}^{ - \mu } - \left( {1 + \delta } \right)t_{2} Y_{2} P_{2}^{ - \mu } - rt_{3} Y_{3} P_{3}^{ - \mu } + \delta Y_{2} P_{2}^{ - \mu } } \right] + \left( {rt_{1} Y_{1} P_{1}^{ - \mu } } \right. \\ & \quad \left. {\left. { + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {1 + \delta } \right)Y_{2} P_{2}^{ - \mu } } \right]} \right\} < 0, \hfill \\ \end{aligned}$$
(78)
after substituting (75) into it.
$$\begin{aligned} \frac{{\partial^{2} V_{3} }}{{\partial t_{3}^{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left\{ { - \left( {1 - \delta } \right)Y_{3} P_{3}^{ - \mu } \left( {r^{2} t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 2} } \right. \\ & \quad \cdot \left[ { - r^{2} t_{1} Y_{1} P_{1}^{ - \mu } - rt_{2} Y_{2} P_{2}^{ - \mu } - \left( {1 + \delta } \right)t_{3} Y_{3} P_{3}^{ - \mu } + \delta Y_{3} P_{3}^{ - \mu } } \right] + \left( {r^{2} t_{1} Y_{1} P_{1}^{ - \mu } } \right. \\ & \quad \left. {\left. { + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta - 1} \left[ { - \left( {1 + \delta } \right)Y_{3} P_{3}^{ - \mu } } \right]} \right\} < 0. \hfill \\ \end{aligned}$$
(79)
after substituting (76) into it.
Appendix C
Scenario I
The indirect utility of each region is as follows:
$$V_{1} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left( {1 - t_{1} } \right)\left( {t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } ,$$
(80)
$$V_{2} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left( {1 - t_{1} } \right)\left( {rt_{1} Y_{1} P_{1}^{ - \mu } + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } ,$$
(81)
$$V_{3} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left( {1 - t_{1} } \right)\left( {rt_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } .$$
(82)
To simply the calculation for analytical solution, \(\delta = 1\) is use for the following analysis. To maximize the sum of the indirect utility of each region, \(V = V_{1} + V_{2} + V_{3}\), is to differentiate V with respect to \(t_{1}\), \(t_{2}\), and \(t_{3}\)
$$\begin{aligned} \frac{\partial V}{{\partial t_{1} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left( { - 2t_{1} Y_{1} P_{1}^{ - \mu } - 2rt_{2} Y_{2} P_{2}^{ - \mu } - 2rt_{3} Y_{3} P_{3}^{ - \mu } } \right. \\ & \quad \left. { + Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } \right), \\ \end{aligned}$$
(83)
$$\begin{aligned} \frac{\partial V}{{\partial t_{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left( { - 2rt_{1} Y_{1} P_{1}^{ - \mu } - 2t_{2} Y_{2} P_{2}^{ - \mu } - 2rt_{3} Y_{3} P_{3}^{ - \mu } } \right. \\ & \quad \left. { + rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } \right), \\ \end{aligned}$$
(84)
$$\begin{aligned} \frac{\partial V}{{\partial t_{3} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left( { - 2rt_{1} Y_{1} P_{1}^{ - \mu } - 2rt_{2} Y_{2} P_{2}^{ - \mu } - 2t_{3} Y_{3} P_{3}^{ - \mu } } \right. \\ & \quad \left. { + rY_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } } \right). \\ \end{aligned}$$
(85)
Letting (83)–(85) be zero yields the first-order condition for this problem:
$$2t_{1} Y_{1} P_{1}^{ - \mu } + 2rt_{2} Y_{2} P_{2}^{ - \mu } + 2rt_{3} Y_{3} P_{3}^{ - \mu } = Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } ,$$
(86)
$$2rt_{1} Y_{1} P_{1}^{ - \mu } + 2t_{2} Y_{2} P_{2}^{ - \mu } + 2rt_{3} Y_{3} P_{3}^{ - \mu } = rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } ,$$
(87)
$$2rt_{1} Y_{1} P_{1}^{ - \mu } + 2rt_{2} Y_{2} P_{2}^{ - \mu } + 2t_{3} Y_{3} P_{3}^{ - \mu } = rY_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } .$$
(88)
Solving (86)–(88) yields the optimal tax rate:
$$t_{i}^{*} = \frac{{\left| {D_{i} } \right|}}{\left| D \right|},\quad i = 1,2,3,$$
(89)
where
$$\left| D \right| = 8Y_{1} P_{1}^{ - \mu } Y_{2} P_{2}^{ - \mu } Y_{3} P_{3}^{ - \mu } \left| {\begin{array}{*{20}c} 1 & r & r \\ r & 1 & r \\ r & r & 1 \\ \end{array} } \right|,$$
(90)
$$\left| {D_{1} } \right| = \left| {\begin{array}{*{20}c} {Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } & {2rY_{2} P_{2}^{ - \mu } } & {2rY_{3} P_{3}^{ - \mu } } \\ {rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } & {2Y_{2} P_{2}^{ - \mu } } & {2rY_{3} P_{3}^{ - \mu } } \\ {rY_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } } & {2rY_{2} P_{2}^{ - \mu } } & {2Y_{3} P_{3}^{ - \mu } } \\ \end{array} } \right|,$$
(91)
$$\left| {D_{2} } \right| = \left| {\begin{array}{*{20}c} {2Y_{1} P_{1}^{ - \mu } } & {Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } & {2rY_{3} P_{3}^{ - \mu } } \\ {2rY_{1} P_{1}^{ - \mu } } & {rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } & {2rY_{3} P_{3}^{ - \mu } } \\ {2rY_{1} P_{1}^{ - \mu } } & {rY_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } } & {2Y_{3} P_{3}^{ - \mu } } \\ \end{array} } \right|,$$
(92)
$$\left| {D_{3} } \right| = \left| {\begin{array}{*{20}c} {2Y_{1} P_{1}^{ - \mu } } & {2rY_{2} P_{2}^{ - \mu } } & {Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } \\ {2rY_{1} P_{1}^{ - \mu } } & {2Y_{2} P_{2}^{ - \mu } } & {rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } \\ {2rY_{1} P_{1}^{ - \mu } } & {2rY_{2} P_{2}^{ - \mu } } & {rY_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } } \\ \end{array} } \right|.$$
(93)
Calculating for (89) yields the optimal tax rate of each region being 0.5. The second-order condition is satisfied for the maximum of V as follows:
$$\frac{{\partial^{2} V}}{{\partial t_{1}^{2} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1}^{2} P_{1}^{ - 2\mu } ,$$
(94)
$$\frac{{\partial^{2} V}}{{\partial t_{2}^{2} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2}^{2} P_{2}^{ - 2\mu } ,$$
(95)
$$\frac{{\partial^{2} V}}{{\partial t_{3}^{2} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3}^{2} P_{3}^{ - 2\mu } ,$$
(96)
$$\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{2} }} = \frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{1} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } r Y_{1} P_{1}^{ - \mu } Y_{2} P_{2}^{ - \mu } ,$$
(97)
$$\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{3} }} = \frac{{\partial^{2} V}}{{\partial t_{3} \partial t_{1} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } r Y_{1} P_{1}^{ - \mu } Y_{3} P_{3}^{ - \mu } ,$$
(98)
$$\frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{3} }} = \frac{{\partial^{2} V}}{{\partial t_{3} \partial t_{2} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } r Y_{2} P_{2}^{ - \mu } Y_{3} P_{3}^{ - \mu } .$$
(99)
$$\left| B \right| = \left| {\begin{array}{*{20}l} {\frac{{\partial^{2} V}}{{\partial t_{1}^{2} }}} \hfill & {\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{2} }}} \hfill & {\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{3} }}} \hfill \\ {\frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{1} }}} \hfill & {\frac{{\partial^{2} V}}{{\partial t_{2}^{2} }}} \hfill & {\frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{3} }}} \hfill \\ {\frac{{\partial^{2} V}}{{\partial t_{3} \partial t_{1} }}} \hfill & {\frac{{\partial^{2} V}}{{\partial t_{3} \partial t_{2} }}} \hfill & {\frac{{\partial^{2} V}}{{\partial t_{3}^{2} }}} \hfill \\ \end{array} } \right|,$$
(100)
$$\left| {B_{1} } \right| = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1}^{2} P_{1}^{ - 2\mu } < 0,$$
(101)
$$\left| {B_{2} } \right| = \left| {\begin{array}{*{20}l} {\frac{{\partial^{2} V}}{{\partial t_{1}^{2} }}} \hfill & {\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{2} }}} \hfill \\ {\frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{1} }}} \hfill & { - \frac{{\partial^{2} V}}{{\partial t_{2}^{2} }}} \hfill \\ \end{array} } \right| = 4\mu^{2\mu } \left( {1 - \mu } \right)^{{2\left( {1 - \mu } \right)}} Y_{1}^{2} P_{1}^{ - 2\mu } Y_{2}^{2} P_{2}^{ - 2\mu } \left( {1 - r^{2} } \right) > 0,$$
(102)
$$\left| {B_{3} } \right| = \left| B \right| = 8\mu^{3\mu } \left( {1 - \mu } \right)^{{3\left( {1 - \mu } \right)}} Y_{1}^{2} P_{1}^{ - 2\mu } Y_{2}^{2} P_{2}^{ - 2\mu } Y_{3}^{2} P_{3}^{ - 2\mu } \left( { - 1 + 3r^{2} - 2r^{3} } \right) < 0.$$
(103)
Scenario II
The indirect utility of each region is as follows:
$$V_{1} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left( {1 - t_{1} } \right)\left( {t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + r^{2} t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } ,$$
(104)
$$V_{2} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left( {1 - t_{1} } \right)\left( {rt_{1} Y_{1} P_{1}^{ - \mu } + t_{2} Y_{2} P_{2}^{ - \mu } + rt_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } ,$$
(105)
$$V_{3} = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left( {1 - t_{1} } \right)\left( {r^{2} t_{1} Y_{1} P_{1}^{ - \mu } + rt_{2} Y_{2} P_{2}^{ - \mu } + t_{3} Y_{3} P_{3}^{ - \mu } } \right)^{\delta } .$$
(106)
To simply the calculation for analytical solution, \(\delta = 1\) is use for the following analysis. To maximize the sum of the indirect utility of each region, \(V = V_{1} + V_{2} + V_{3}\), is to differentiate V with respect to \(t_{1}\), \(t_{2}\), and \(t_{3}\)
$$\begin{aligned} \frac{\partial V}{{\partial t_{1} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1} P_{1}^{ - \mu } \left( { - 2t_{1} Y_{1} P_{1}^{ - \mu } - 2rt_{2} Y_{2} P_{2}^{ - \mu } - 2r^{2} t_{3} Y_{3} P_{3}^{ - \mu } } \right. \\ & \quad \left. { + Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + r^{2} Y_{3} P_{3}^{ - \mu } } \right), \\ \end{aligned}$$
(107)
$$\begin{aligned} \frac{\partial V}{{\partial t_{2} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2} P_{2}^{ - \mu } \left( { - 2rt_{1} Y_{1} P_{1}^{ - \mu } - 2t_{2} Y_{2} P_{2}^{ - \mu } - 2rt_{3} Y_{3} P_{3}^{ - \mu } } \right. \\ & \quad \left. { + rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } \right), \hfill \\ \end{aligned}$$
(108)
$$\begin{aligned} \frac{\partial V}{{\partial t_{3} }} & = \mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3} P_{3}^{ - \mu } \left( { - 2r^{2} t_{1} Y_{1} P_{1}^{ - \mu } - 2rt_{2} Y_{2} P_{2}^{ - \mu } - 2t_{3} Y_{3} P_{3}^{ - \mu } } \right. \\ & \quad \left. { + r^{2} Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } } \right). \hfill \\ \end{aligned}$$
(109)
Letting (107)–(109) be zero yields the first-order condition for this problem:
$$2t_{1} Y_{1} P_{1}^{ - \mu } + 2rt_{2} Y_{2} P_{2}^{ - \mu } + 2r^{2} t_{3} Y_{3} P_{3}^{ - \mu } = Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + r^{2} Y_{3} P_{3}^{ - \mu } ,$$
(110)
$$2rt_{1} Y_{1} P_{1}^{ - \mu } + 2t_{2} Y_{2} P_{2}^{ - \mu } + 2rt_{3} Y_{3} P_{3}^{ - \mu } = rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } ,$$
(111)
$$2r^{2} t_{1} Y_{1} P_{1}^{ - \mu } + 2rt_{2} Y_{2} P_{2}^{ - \mu } + 2t_{3} Y_{3} P_{3}^{ - \mu } = r^{2} Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } .$$
(112)
Solving (110)–(112) yields the optimal tax rate:
$$t_{i}^{*} = \frac{{\left| {E_{i} } \right|}}{\left| E \right|},\quad i = 1,2,3.$$
(113)
where
$$\left| E \right| = 8Y_{1} P_{1}^{ - \mu } Y_{2} P_{2}^{ - \mu } Y_{3} P_{3}^{ - \mu } \left| {\begin{array}{*{20}c} 1 & r & {r^{2} } \\ r & 1 & r \\ {r^{2} } & r & 1 \\ \end{array} } \right|,$$
(114)
$$\left| {E_{1} } \right| = \left| {\begin{array}{*{20}c} {Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + r^{2} Y_{3} P_{3}^{ - \mu } } & {2rY_{2} P_{2}^{ - \mu } } & {2r^{2} Y_{3} P_{3}^{ - \mu } } \\ {rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } & {2Y_{2} P_{2}^{ - \mu } } & {2rY_{3} P_{3}^{ - \mu } } \\ {r^{2} Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } } & {2rY_{2} P_{2}^{ - \mu } } & {2Y_{3} P_{3}^{ - \mu } } \\ \end{array} } \right|,$$
(115)
$$\left| {E_{2} } \right| = \left| {\begin{array}{*{20}c} {2Y_{1} P_{1}^{ - \mu } } & {Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + r^{2} Y_{3} P_{3}^{ - \mu } } & {2r^{2} Y_{3} P_{3}^{ - \mu } } \\ {2rY_{1} P_{1}^{ - \mu } } & {rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } & {2rY_{3} P_{3}^{ - \mu } } \\ {2r^{2} Y_{1} P_{1}^{ - \mu } } & {r^{2} Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } } & {2Y_{3} P_{3}^{ - \mu } } \\ \end{array} } \right|,$$
(116)
$$\left| {E_{3} } \right| = \left| {\begin{array}{*{20}c} {2Y_{1} P_{1}^{ - \mu } } & {2rY_{2} P_{2}^{ - \mu } } & {Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + r^{2} Y_{3} P_{3}^{ - \mu } } \\ {2rY_{1} P_{1}^{ - \mu } } & {2Y_{2} P_{2}^{ - \mu } } & {rY_{1} P_{1}^{ - \mu } + Y_{2} P_{2}^{ - \mu } + rY_{3} P_{3}^{ - \mu } } \\ {2r^{2} Y_{1} P_{1}^{ - \mu } } & {2rY_{2} P_{2}^{ - \mu } } & {r^{2} Y_{1} P_{1}^{ - \mu } + rY_{2} P_{2}^{ - \mu } + Y_{3} P_{3}^{ - \mu } } \\ \end{array} } \right|.$$
(117)
Calculating for (113) yields the optimal tax rate of each region being 0.5. The second-order condition is satisfied for the maximum of V as follows:
$$\frac{{\partial^{2} V}}{{\partial t_{1}^{2} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1}^{2} P_{1}^{ - 2\mu } ,$$
(118)
$$\frac{{\partial^{2} V}}{{\partial t_{2}^{2} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{2}^{2} P_{2}^{ - 2\mu } ,$$
(119)
$$\frac{{\partial^{2} V}}{{\partial t_{3}^{2} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{3}^{2} P_{3}^{ - 2\mu } ,$$
(120)
$$\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{2} }} = \frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{1} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } r Y_{1} P_{1}^{ - \mu } Y_{2} P_{2}^{ - \mu } ,$$
(121)
$$\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{3} }} = \frac{{\partial^{2} V}}{{\partial t_{3} \partial t_{1} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } r^{2} Y_{1} P_{1}^{ - \mu } Y_{3} P_{3}^{ - \mu } ,$$
(122)
$$\frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{3} }} = \frac{{\partial^{2} V}}{{\partial t_{3} \partial t_{2} }} = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } r Y_{2} P_{2}^{ - \mu } Y_{3} P_{3}^{ - \mu } .$$
(123)
$$\left| F \right| = \left| {\begin{array}{*{20}c} {\frac{{\partial^{2} V}}{{\partial t_{1}^{2} }}} & {\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{2} }}} & {\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{3} }}} \\ {\frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{1} }}} & {\frac{{\partial^{2} V}}{{\partial t_{2}^{2} }}} & {\frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{3} }}} \\ {\frac{{\partial^{2} V}}{{\partial t_{3} \partial t_{1} }}} & {\frac{{\partial^{2} V}}{{\partial t_{3} \partial t_{2} }}} & {\frac{{\partial^{2} V}}{{\partial t_{3}^{2} }}} \\ \end{array} } \right|,$$
(124)
$$\left| {F_{1} } \right| = - 2\mu^{\mu } \left( {1 - \mu } \right)^{1 - \mu } Y_{1}^{2} P_{1}^{ - 2\mu } < 0,$$
(125)
$$\left| {F_{2} } \right| = \left| {\begin{array}{*{20}c} {\frac{{\partial^{2} V}}{{\partial t_{1}^{2} }}} & {\frac{{\partial^{2} V}}{{\partial t_{1} \partial t_{2} }}} \\ {\frac{{\partial^{2} V}}{{\partial t_{2} \partial t_{1} }}} & { - \frac{{\partial^{2} V}}{{\partial t_{2}^{2} }}} \\ \end{array} } \right| = 4\mu^{2\mu } \left( {1 - \mu } \right)^{{2\left( {1 - \mu } \right)}} Y_{1}^{2} P_{1}^{ - 2\mu } Y_{2}^{2} P_{2}^{ - 2\mu } \left( {1 - r^{2} } \right) > 0,$$
(126)
$$\left| {F_{3} } \right| = \left| F \right| = 8\mu^{3\mu } \left( {1 - \mu } \right)^{{3\left( {1 - \mu } \right)}} Y_{1}^{2} P_{1}^{ - 2\mu } Y_{2}^{2} P_{2}^{ - 2\mu } Y_{3}^{2} P_{3}^{ - 2\mu } \left( { - 1 + 2r^{2} - r^{4} } \right) < 0.$$
(127)
