Production inefficiency, cross-ownership and regional tax-range coordination

Abstract

Using a simple asymmetric capital-tax competition model where the allocation of mobile capital is distorted in non-cooperative equilibrium, this paper analyzes the welfare impact of regional tax coordination on a range of possible tax rates (a combination of maximum and minimum capital taxes made by a subset of regions). Under the assumption that the ownership of immobile factors (e.g., business land) is diversified across regions, a new possibility of beneficial coordination arises which has not been identified before: tax-range coordination “among capital-exporting regions” or “among capital-importing regions” may improve the welfare of all regions. This is in contrast to the case without cross-ownership where both capital-exporting and capital-importing regions must be involved in tax-range coordination in order to achieve a Pareto improvement.

Appendix

Lemmas 1 and 2. Note from (7) that $k_{i} > E$ if and only if $\tau - t_{i} + \gamma \left( {q_{i} - \chi } \right) > 0$ and that $k_{i} > k_{j}$ if and only if $t_{j} - t_{i} + \gamma \left( {q_{i} - q_{j} } \right) > 0$. Equations (13) and (14) imply that

$$ \tau - t_{i} + \gamma \left( {q_{i} - \chi } \right) = \left( {q_{i} - \chi } \right){\rm Z}, $$

(A1)

$$ t_{j} - t_{i} + \gamma \left( {q_{i} - q_{j} } \right) = \left( {q_{i} - q_{j} } \right){\rm Z}, $$

(A2)

where $Z=\gamma \left( {\frac{N - 1}{N} - \frac{N - 2}{N - 1}\mu } \right) > 0$. (A1) and (A2) prove (15) in Lemma 1. From (A2), the second part of Lemma 2 can be confirmed since $t_{i} - t_{j} = \left( {\gamma - Z} \right)\left( {q_{i} - q_{j} } \right)$ and $\gamma - {\rm Z} = \gamma \left( {\frac{1}{N} + \frac{N - 2}{N - 1}\mu } \right).$

Derivation of (20a, 20b). Differentiating (1) and applying (10) and (11) to the result gives the external impact of tax policy.

$$ \frac{{\partial Y_{i} }}{{\partial t_{j} }} = \frac{{k_{i} - E}}{N} + t_{i} \frac{{\partial k_{i} }}{{\partial t_{j} }} - \gamma \mu q_{i } \frac{{\partial k_{i} }}{{\partial t_{j} }} + \frac{{\gamma \mu q_{j} }}{N - 1}\frac{{\partial k_{j} }}{{\partial t_{j} }} + \gamma \mu \mathop \sum \limits_{h \ne i, j}^{N} \frac{{q_{h} }}{N - 1}\frac{{\partial k_{h} }}{{\partial t_{j} }}. $$

(A3)

In non-cooperative equilibrium, (12) must hold. This equation can be rewritten as

$$ \frac{{k_{i} - E}}{N} = - t_{i} \frac{{\partial k_{i} }}{{\partial t_{i} }} + \gamma \mu q_{i } \frac{{\partial k_{i} }}{{\partial t_{i} }} - \frac{{\gamma \mu q_{j} }}{N - 1}\frac{{\partial k_{j} }}{{\partial t_{i} }} - \gamma \mu \mathop \sum \limits_{h \ne i, j}^{N} \frac{{q_{h} }}{N - 1}\frac{{\partial k_{h} }}{{\partial t_{i} }}. $$

(A4)

Substituting (A4) into (A3) and noting from (10) that $\partial k_{i} /\partial t_{i} = \partial k_{j} /\partial t_{j}$, $\partial k_{i} /\partial t_{j} = \partial k_{j} /\partial t_{i}$ and $\partial k_{h} /\partial t_{j} = \partial k_{h} /\partial t_{i}$, it can be shown that

$$ \frac{{\partial Y_{i} }}{{\partial t_{j} }} = \left[ {t_{i} - \gamma \mu \left( {q_{i } + \frac{{q_{j} }}{N - 1}} \right)} \right]\left( {\frac{{\partial k_{i} }}{{\partial t_{j} }} - \frac{{\partial k_{i} }}{{\partial t_{i} }}} \right). $$

(A5)

Since (10) implies that $\partial k_{i} /\partial t_{j} - \partial k_{i} /\partial t_{i} = 1/\delta$, (A5) is reduced to

$$ \frac{{\partial Y_{i} }}{{\partial t_{j} }} = \frac{1}{\delta }\left[ {t_{i} - \gamma \mu \left( {q_{i } + \frac{{q_{j} }}{N - 1}} \right)} \right]. $$

(A6)

Substituting (13) and (14) into (A6),

$$ \frac{{\partial Y_{i} }}{{\partial t_{j} }} = \frac{{\gamma \left( {q_{i} - \chi } \right)}}{\delta N} + \frac{{\gamma \mu \left[ {2\chi + q_{i} \left( {N - 2} \right)} \right]}}{{\delta \left( {N - 1} \right)}} - \frac{\gamma \mu }{\delta }\left( {q_{i } + \frac{{q_{j} }}{N - 1}} \right), $$

(A7)

or, equivalently,

$$ \frac{{\partial Y_{i} }}{{\partial t_{j} }} = \frac{{\gamma \left( {q_{i} - \chi } \right)}}{\delta N} + \frac{2\gamma \mu }{{\delta \left( {N - 1} \right)}}\left( {\chi - \frac{{q_{i } + q_{j} }}{2}} \right). $$

(A8)

The same procedure yields (20b).

Note that (A7) corresponds to the decomposition of the welfare impact of the present tax-range change that is argued in Sect. 4. The sum of the first and second terms of (A7) is equal to $t_{i} /\delta$. The first term of (A7) and (A8) represents the Peralta–Ypersele effect. The second and third terms of (A7), whose sum is equal to the second term of (A8), constitute the cross-ownership effect: the second term of (A7) captures the impact of cross-ownership on the tax revenue while the third term of (A7) stands for the impact on rent flows between constrained regions.

Derivation of (29a, 29b).

Consider that $\chi > q_{i} > q_{j}$ in which case (28a) shows that $A > 0$, $0 < \varepsilon_{i} < 1$ and $1 < \varepsilon_{j} < 2$. In this case, it can be confirmed from (24a, 24b) that

$$ \frac{N - 1}{2N} < \mu \Rightarrow \frac{{\partial Y_{i} }}{{\partial t_{j} }} > 0, \frac{N - 1}{2N} < \mu < \frac{N - 1}{2N}\varepsilon_{j} \Rightarrow - \frac{{\partial Y_{j} }}{{\partial t_{i} }} > 0, $$

(A9a)

$$ \frac{N - 1}{2N} > \mu \Rightarrow - \frac{{\partial Y_{j} }}{{\partial t_{i} }} > 0 , \frac{N - 1}{2N} > \mu > \frac{N - 1}{2N}\varepsilon_{i} \Rightarrow \frac{{\partial Y_{i} }}{{\partial t_{j} }} > 0, $$

(A9b)

$$ \mu = \frac{N - 1}{2N} \Rightarrow \frac{{\partial Y_{i} }}{{\partial t_{j} }} > 0\quad {\text{and}}\quad - \frac{{\partial Y_{j} }}{{\partial t_{i} }} > 0. $$

(A9c)

What (A9a, A9b) show is that the welfare of regions i and j is improved if

$$ \frac{N - 1}{2N}\varepsilon_{i} < \mu < \frac{N - 1}{2N} or \frac{N - 1}{2N} < \mu < \frac{N - 1}{2N}\varepsilon_{j} . $$

(A10)

These conditions, together with (A9c), prove that $\partial Y_{i} /\partial t_{j} > 0$ and $- \left( {\partial Y_{j} /\partial t_{i} } \right) > 0$ if (29a) holds. Using (28b), the same procedure yields (29b).