Jurisdictional Tax Competition and the Division of Nonrenewable Resource Rents

Abstract

This paper presents a model of nonrenewable resource extraction across multiple jurisdictions which engage in strategic tax competition. The model incorporates rents due to both resource scarcity and capital scarcity as well as intra-region Ricardian rents. Regions set taxes on nonrenewable resource production strategically to balance tax revenues and local benefits from investment conditional on other regions’ tax rates. A representative extraction firm then allocates production capital across regions and time to maximize the present value of profits. Generally, we find that the division of resource rent between firms and regional governments ultimately depends on the relative scarcity of natural and production capital, relative costs across space, and the value regional governments place on economic activity. This theoretical result provides policymakers with information on the determinants of optimal tax rates and motivates future empirical research on the factors influencing the division of resource rent in practice.

Appendix

Derivation of 3-region model with 2 competitive regions.

$$\begin{aligned} H= & {} \left( 1-\gamma _{1} \right) pq_{1}-\frac{A_{1}q_{1}^{2}}{2x_{1}}+\left( 1-\gamma _{2} \right) pq_{2}-\frac{A_{2}q_{2}^{2}}{2x_{2}}+\left( 1-\gamma _{3} \right) p\left( \bar{q}-q_{1}-q_{2} \right) \\&-\frac{A_{3}\left( \bar{q}-q_{1}-q_{2} \right) ^{2}}{2x_{3}}+\lambda _{1}\left( -q_{1} \right) +\lambda _{2}\left( -q_{2} \right) +\lambda _{3}\left( -\left( \bar{q}-q_{1}-q_{2} \right) \right) \\ \frac{dH}{dq_{1}}= & {} \left( 1-\gamma _{1} \right) p-\frac{A_{1}q_{1}}{x_{1}}-\left( 1-\gamma _{3} \right) p+\frac{A_{3}\left( \bar{q}-q_{1}-q_{2} \right) }{x_{3}}-\lambda _{1}+\lambda _{3}=0\\ \frac{dH}{dq_{2}}= & {} \left( 1-\gamma _{2} \right) p-\frac{A_{2}q_{2}}{x_{1}}-\left( 1-\gamma _{3} \right) p+\frac{A_{3}\left( \bar{q}-q_{1}-q_{2} \right) }{x_{3}}-\lambda _{2}+\lambda _{3}=0\\ \dot{\lambda }_{1}= & {} r\lambda _{1}-\left( \frac{A_{1}q_{1}^{2}}{2x_{1}^{2}}+ \right) \\ \dot{\lambda }_{2}= & {} r\lambda _{2}-\left( \frac{A_{2}q_{2}^{2}}{2x_{2}^{2}} \right) \\ \dot{\lambda }_{3}= & {} r\lambda _{3}-\left( \frac{A_{3}\left( \bar{q}-q_{1}-q_{2} \right) ^{2}}{2x_{3}^{2}} \right) \end{aligned}$$

Obtain system of ODEs in state and co-state:

Step 1 Use $q_{2}{} { foctosolvefor}\hat{q}_{2}\left( q_{1} \right)$

$$\begin{aligned} \hat{q}_{2}=\frac{\left( p\left( \gamma _{3}-\gamma _{2} \right) +\frac{A_{3}\bar{q}}{x_{3}}-\frac{A_{3}q_{1}}{x_{3}}-\lambda _{2}+\lambda _{3} \right) }{\frac{A_{3}}{x_{3}}+\frac{A_{2}}{x_{2}}} \end{aligned}$$

Step 2 Plug $\hat{q}_{2}{} { into}q_{1}{} { focandsolvefor}q_{1}^{*}{}$.

$$\begin{aligned} q_{1}^{*}=\, \frac{\left( \frac{A_{3}\bar{q}}{x_{3}}-\frac{A_{3}}{x_{3}}\left( \frac{\left( p\left( \gamma _{3}-\gamma _{2} \right) +\frac{A_{3}\bar{q}}{x_{3}}-\lambda _{2}+\lambda _{3} \right) }{\frac{A_{3}}{x_{3}}+\frac{A_{2}}{x_{2}}} \right) -\lambda _{1}+\lambda _{3}+p\left( \gamma _{3}-\gamma _{1}\right) \right) }{\frac{A_{1}}{x_{1}}+\frac{A_{3}}{x_{3}}-\frac{\frac{A_{3}^{2}}{x_{3}^{2}}}{\frac{A_{3}}{x_{3}}+\frac{A_{2}}{x_{2}}}} \end{aligned}$$

Step 3 Plug $q_{1}^{*}{} { into}\hat{q}_{2}{} { toget}q_{2}^{*}$:

$$\begin{aligned} q_{2}^{*}=\, =\frac{\left( p\left( \gamma _{3}-\gamma _{2} \right) +\frac{A_{3}\bar{q}}{x_{3}}-\frac{A_{3}q_{1}^{*}}{x_{3}}-\lambda _{2}+\lambda _{3} \right) }{\frac{A_{3}}{x_{3}}+\frac{A_{2}}{x_{2}}} \end{aligned}$$

System of ODEs with 3 regions:

$$\begin{aligned} \dot{x}_{1}= & {} -q_{1}^{*}\\ \dot{x}_{2}= & {} -q_{2}^{*}\\ \dot{x}_{3}= & {} -\left( \bar{q}-q_{1}^{*}-q_{2}^{*} \right) \\ \dot{\bar{q}}= & {} \alpha \left( \left( 1-\gamma _{1} \right) p-\frac{A_{1}}{x_{1}}q_{1}^{*}-\lambda _{1} \right) \\ \dot{\lambda }_{1}= & {} r\lambda _{1}-\left( \frac{A_{1}q_{1}^{2}}{2x_{1}^{2}}\right) \\ \dot{\lambda }_{2}= & {} r\lambda _{2}-\left( \frac{A_{2}q_{2}^{2}}{2x_{2}^{2}}\right) \\ \dot{\lambda }_{3}= & {} r\lambda _{3}-\left( \frac{A_{3}\left( \bar{q}-q_{1}-q_{2} \right) ^{2}}{2x_{3}^{2}} \right) \\&\lambda _{T1},\, \lambda _{T2},\, \lambda _{T3},\\&x_{01},\, x_{02},\, x_{03},\, \bar{q}_{0}\, \, \, { given} \end{aligned}$$