Abstract
This paper reconciles two opposite results in the tax competition literature. Kempf and Rota-Graziosi (J Public Econ 94(9–10):768–776, 2010) and Hindriks and Nishimura (J Public Econ 121:66–68, 2015) have shown that the two Stackelberg outcomes prevail as the subgame perfect equilibria when capital is entirely owned by nonresidents. However, Ogawa (Int Tax Public Finance 20(3):474–484, 2013) has shown that the simultaneous-move outcome prevails when capital is entirely owned by residents. We develop a model in which capital ownership can vary freely between these two polar cases. We show that there exists a unique degree of residential capital ownership such that the equilibrium switches from the Stackelberg to the simultaneous-move outcomes. The chance for the simultaneous-move outcome to prevail increases with the extent of production asymmetry between regions. Partial ownership also induces a novel effect of tax leadership that we call the preference reversion effect.
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Notes
In a linear demand for capital, the markup, which measures market power, is independent of the gradient, but is related to the vertical intercept of the demand function. See Hindriks and Nishimura (2015, p.66 and footnote 3).
Kempf and Rota-Graziosi (2015b) allow for a combination of production asymmetry (market power) and ownership asymmetry (financial power). The results they obtain are ambiguous because of the interplay of the two forms of regional asymmetries. We see their analysis as useful extension of our analysis to account for the presence of low-productivity regions with high financial power.
Note that the total supply of capital is \(\bar{K}\) so when \(\theta =1/2\) residents in both regions own together the total stock of capital.
Rewrite the reaction function as \(t_{i}=\tau _{i}(t_{j},\theta )\), treating \(\theta \) as a shifting parameter. Then, one has \({{\partial \tau _{i}}/{\partial \theta }}={F_{j}^{\prime \prime }\bar{K}}/\left[ 1+\left\{ F_{j}^{\prime \prime }-(K_{i}-\theta \bar{K})F_{j}^{\prime \prime \prime }\right\} \frac{\partial K_{i}}{\partial t_{i}}\right] <0\), since the denominator is positive due to the second-order condition. Therefore, \(t^{N}\), the intersection of the reaction functions, is decreasing in \(\theta \).
Suppose otherwise, then \(K_{A}^{N}\le K_{B}^{N}\) at \(\theta =1/2\) so that \(K_{A}^{N}-\bar{K}/2\le 0\le K_{B}^{N}-\bar{K}/2\). Then, (1) implies that \(t_{A}^{N}\le 0\le t_{B}^{N}\). However, \(F_{A}^{\prime }(K_{A}^{N})>F_{B}^{\prime }(K_{A}^{N})\ge F_{B}^{\prime }(K_{B}^{N})\) and \(t_{A}^{N}\le 0\le t_{B}^{N}\) are not compatible with the arbitrage condition. We therefore have \(K_{A}^{N}-\bar{K}/2>0>K_{B}^{N}-\bar{K}/2\) when \(\theta =1/2\).
\(W_{A}^{L}-W_{A}^{N}=b\bar{K}^{2}(\delta +4-8\theta )^{2}/320>0\). \(W_{B}^{L}-W_{B}^{N}=b\bar{K}^{2}(-\delta +4-8\theta )^{2}/320\ge 0\), where the equality holds only at \(\theta =\theta ^{N}\).
Pareto domination in the case of \(\theta =0\) is covered in Hindriks and Nishimura (2015). Indeed, for \(\delta >\sqrt{15}-1\approx 2.872983\), \(\theta ^{1}=0\).
We disregard \(\theta =\theta ^{2}\), which is the borderline case where (Early, Late) and (Early, Early) are the equilibria.
\(\delta =0\) and \(\theta =1/2\) would make \(t_{i}^{N}=t_{i}^{L}=t_{i}^{F}=0\ (i=A,B)\). So here we consider \(\theta <1/2\).
The proof is available upon request.
Hindriks and Nishimura (pp. 67–68) note that the superiority of the leadership by region B comes from the fact that the regional asymmetry translates into difference in the gradients of the reaction functions. However, the argument is valid only when \(\theta \) is small. When \(\theta \) is close to \(\theta ^{N}\), \((t_{A}^{F},t_{B}^{L})\) becomes close enough to \((t_{A}^{N},t_{B}^{N})\), which is Pareto dominated by \((t_{A}^{L},t_{B}^{F})\).
References
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Acknowledgments
The authors would like to thank the Editor-in-Chief (Ronald Davies) and two anonymous referees for their useful comments and suggestions to our paper. Nishimura acknowledges the financial support from the Grants-in-Aid for Scientific Research (C) (the Ministry of Education, Culture, Sports, Science, and Technology, 24530348, 15K03511) and Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation (Japan Society for the Promotion of Science, J2402).
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Appendix
Appendix
1.1 Appendix 1: Proof of Lemma 1
\(W_{A}^{F}-W_{A}^{N}=-3b\bar{K}^{2}(8\theta -4+\delta )(9\delta -88\theta +44)/1600\), with \(9\delta -88\theta +44>0\). So \(W_{A}^{F}\gtrless W_{A}^{N}{\iff } \theta \lessgtr \displaystyle {\frac{1}{2}-\frac{\delta }{8}}= \theta ^{N}.\) Also, \(W_{B}^{F}-W_{B}^{N}=-3b\bar{K}^{2}(-8\theta +4+\delta )(9\delta +88\theta -44)/1600\), with \(-8\theta +4+\delta >0\). Therefore, \(W_{B}^{F}\gtrless W_{B}^{N}{\iff } \theta \lessgtr \displaystyle {\frac{1}{2}-\frac{9\delta }{88}}\equiv \theta ^{2}.\) \(\square \)
1.2 Appendix 2: Pareto dominance
\(W_{A}^{L}-W_{A}^{F}=b\bar{K}^{2}(\delta +(\sqrt{15}+1)(1-2\theta ))(\delta -(\sqrt{15}-1)(1-2\theta ))/50\), so that \(W_{A}^{L}> W_{A}^{F}{\iff } \theta > \max \left\{ 0,\ \displaystyle {\frac{1}{2}-\frac{ \sqrt{15}+1}{28}}\delta \right\} \equiv \theta ^1\). \(W_{B}^{F}-W_{B}^{L}=-b\bar{K}^{2}(\delta +(\sqrt{15}-1)(1-2\theta ))(\delta -(\sqrt{15}+1)(1-2\theta ))/50\), so that \(W_{B}^{F}> W_{B}^{L}{\iff } \theta < \displaystyle {\frac{1}{2}-\frac{ \sqrt{15}-1}{28}}\delta \). \(\displaystyle {\frac{ \sqrt{15}+1}{28}>\frac{ 1}{8}}\) so \(\theta ^1<\theta ^N\), and \(\displaystyle {\frac{ \sqrt{15}-1}{28}<{\frac{1}{8}}} \). Therefore, \(W_{A}^{L}> W_{A}^{F}\) and \(W_{B}^{F}>W_{B}^{L}\) for all \(\theta \in (\theta ^1,\theta ^N]\).
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Hindriks, J., Nishimura, Y. Equilibrium leadership in tax competition models with capital ownership: a rejoinder. Int Tax Public Finance 24, 338–349 (2017). https://doi.org/10.1007/s10797-016-9421-4
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DOI: https://doi.org/10.1007/s10797-016-9421-4