Games within borders: are geographically differentiated taxes optimal?

Abstract

The discontinuous tax treatment of sales at borders creates incentives for individuals to cross-border shop. This paper addresses whether it is optimal for a state composed of multiple regions to levy differentiated commodity tax rates across the regions. In a model where states maximize social welfare, a state’s optimal commodity tax system is almost always geographically differentiated. The optimal pattern of geographic differentiation critically depends on fundamental parameters as well as whether the state has a preference for high or low taxes. Under the assumption that utility is linear in consumption and that the elasticity of cross-border shopping is less than unity in absolute value, high-tax states will find it optimal to set a tax rate that is lower in the border region than in the periphery region and low-tax states will find it optimal to set a tax rate that is higher in the border region than in the periphery region. Optimizing high-tax states will set a higher tax rate in the border region if the social welfare measure is sufficiently redistributive. With welfare maximization, it is possible for taxes to be higher in the region near the state border—an outcome that cannot arise when the government cares only about total tax revenue.

Appendix

1.1 7.1 Derivation of MCF terms

The algebraic derivation of Eq. (7) is below. The derivations of all the other marginal cost of fund equations follow a similar process.

Recall that for this case, t _A >t _B and the state is a low-tax state. The government then maximizes $U^{A}(c^{A}, t_{A}c_{A}^{A}+t_{B}c_{B}^{A}+t_{B}c^{B}+t_{B}c_{B}^{N})+U^{B}(c^{B}, t_{A}c_{A}^{A}+t_{B}c_{B}^{A}+t_{B}c^{B}+t_{B}c_{B}^{N})$. Differentiating with respect to t _A and t _B yields

$$t_{A}:\quad U_{C}^{A}\biggl[\frac{\partial c^{A}}{\partial t_{A}} \biggr]+\bigl(U_{G}^{A}+U_{G}^{B}\bigr) \biggl[c_{A}^{A}+t_{A}\frac{\partial c_{A}^{A}}{\partial t_{A}}+t_{B} \frac{\partial c_{B}^{A}}{\partial t_{A}}+t_{B}\frac{\partial c^{B}}{\partial t_{A}}\biggr]=0, $$

Using Eqs. (4) and (5) simplifies:

$$t_{A}:\quad -U_{C}^{A}\biggl[\frac{c_{A}^{A}}{1+t_{A}} \biggr]+\bigl(U_{G}^{A}+U_{G}^{B}\bigr) \biggl[c_{A}^{A}-t_{A}\frac{c_{A}^{A}}{1+t_{A}}-t_{A} \frac{1}{D_{A}''}+t_{B}\frac{1}{D_{A}''}\biggr]=0, $$

Rearranging terms gives

$$t_{A}:\quad U_{G}^{A}+U_{G}^{B}= \frac{U_{C}^{A}\frac{c_{A}^{A}}{1+t_{A}}}{\frac{c_{A}^{A}}{1+t_{A}}-\frac{t_{A}-t_{B}}{D_{A}''}}, $$

The two equations are set equal to each other in order to obtain Eq. (7).

1.2 7.2 Proof of Proposition 1

Proof

Equation (7) or (10) must hold at an optimum. Taking the limit as t _A →t _B implies that the first order conditions are continuous when a state changes from sub-case 1 to sub-case 2. Start from a point where all jurisdictions in State H have equal tax rates (t _A =t _B =t). Because the tax rates are equal, $c_{j}^{i}=0$ for all i≠j. This immediately implies that $c^{A}=c^{B}=c_{A}^{A}=c_{B}^{B}=c$ and that the marginal utilities of consumption are equal. Residents of State N cross-border shop such that $c_{B}^{N}>0$. Using these simplifications, Eq. (7) implies $\mathit{MCF}_{A}(t_{A}=t_{B})-\mathit{MCF}_{B}(t_{A}=t_{B})=\frac{\frac{c}{1+t}}{(\frac{c}{1+t})}-\frac{\frac{c}{1+t}}{(\frac{c}{1+t}-\frac{t}{S_{N}''}+c_{B}^{N})}$. A little algebra will show that this expression is equal to zero if $c_{B}^{N}=\frac{t}{S_{N}''}$, implying equal tax rates. The tax system will be differentiated if MCF _A (t _A =t _B )≠MCF _B (t _A =t _B ). Using (7) and the above substitutions, MCF _A (t _A =t _B )−MCF _B (t _A =t _B )<0 if $c_{B}^{N}<\frac{t}{S_{N}''}$. MCF _A <MCF _B implies the government must lower the tax rate in Region B relative to Region A to equalize the marginal cost of funds. Therefore, $c_{B}^{N}<\frac{t}{S_{N}''}$ guarantees that t _A >t _B is welfare improving. Similarly, it can be shown that MCF _A >MCF _B if $c_{B}^{N}>\frac{t}{S_{N}''}$, which implies t _A <t _B is optimal. □

1.3 7.3 Proof of Corollary 1

Proof

The term of interest in the marginal revenue equation is $-\frac{t_{B}}{S_{N}''}+c_{B}^{N}$. This can be rewritten as $t_{B}\frac{\partial c_{B}^{N}}{\partial t_{B}}\frac{c_{B}^{N}}{c_{B}^{N}}+c_{B}^{N}=(1-\epsilon)c_{B}^{N}$, where ϵ is the elasticity of $c_{B}^{N}$ with respect to t _B and ϵ is positive. □

1.4 7.4 Proof of Proposition 2

Proof

Equation (13) or (16) must holds at an optimum. Taking the limit as t _A →t _B implies that the first order conditions are continuous when a state changes regimes from sub-case 1 to sub-case 2. Start from a point where all jurisdictions in State H have equal tax rates (t _A =t _B =t). Because the tax rates are equal, $c_{j}^{i}=0$ for j≠i when j is a region in State H. Some consumption from Region B is purchased across the state border so that $c_{N}^{B}>0$. This implies that c ^A≤c ^B, $c_{A}^{A}\geq c_{B}^{B}$ and $U_{C}^{A}\geq U_{C}^{B}$. Substituting into Eq. (13), t _A =t _B is optimal if $\mathit{MCF}_{A}(t_{A}=t_{B})-\mathit{MCF}_{B}(t_{A}=t_{B})=U_{C}^{A}-\frac{U_{C}^{B}\frac{c_{B}^{B}}{1+t}}{(\frac{c_{B}^{B}}{1+t}-\frac{t}{S_{B}''})}=0$. The tax system will be differentiated if MCF _A (t _A =t _B )≠MCF _B (t _A =t _B ). MCF _A (t _A =t _B )−MCF _B (t _A =t _B )<0 if $U_{C}^{A}(\frac{c_{B}^{B}}{1+t}-\frac{t}{S_{B}''})<U_{C}^{B}\frac{c_{B}^{B}}{1+t}$ and the government must lower the tax rate in Region B relative to Region A to equalize the marginal cost of funds. Similarly, it can be shown that $U_{C}^{A}(\frac{c_{B}^{B}}{1+t}-\frac{t}{S_{B}''})>U_{C}^{B}\frac{c_{B}^{B}}{1+t}$ implies t _A <t _B . □

1.5 7.5 Derivation of Eq. (19)

To derive this equation, I start with the equality of the marginal cost of funds in a high-tax state, given by Eqs. (13) and (16):

$$\frac{U_{C}^{A}\frac{c_{A}^{A}}{1+t_{A}}}{\frac{c_{A}^{A}}{1+t_{A}}-\frac{t_{A}-t_{B}}{D_{A}''}}=\frac{U_{C}^{B}\frac{c_{B}^{B}}{1+t_{B}}}{\frac{c_{B}^{B}}{1+t_{B}}+\frac{c_{B}^{A}}{1+t_{A}}-\frac{t_{B}-t_{A}}{D_{A}''}-\frac{t_{B}}{S_{B}''}}+\frac{U_{C}^{A}\frac{c_{B}^{A}}{1+t_{A}}}{\frac{c_{B}^{B}}{1+t_{B}}+\frac{c_{B}^{A}}{1+t_{A}}-\frac{t_{B}-t_{A}}{D_{A}''}-\frac{t_{B}}{S_{B}''}}, $$

$$\frac{U_{C}^{A}\frac{c^{A}}{1+t_{A}}}{\frac{c^{A}}{1+t_{A}}+\frac{c_{A}^{B}}{1+t_{A}}-\frac{t_{A}-t_{B}}{D_{B}''}}+\frac{U_{C}^{B}\frac{c_{A}^{B}}{1+t_{B}}}{\frac{c_{B}^{B}}{1+t_{B}}-\frac{t_{B}-t_{A}}{D_{B}''}-\frac{t_{B}}{S_{B}''}}=\frac{U_{C}^{B}\frac{c_{B}^{B}}{1+t_{B}}}{\frac{c_{B}^{B}}{1+t_{B}}-\frac{t_{B}-t_{A}}{D_{B}''}-\frac{t_{B}}{S_{B}''}}. $$

To proceed, I need to compare these expressions when taxes in this state are equal, but still less than taxes in the neighboring state (where $t_{A}\rightarrow t=t_{B}<\bar{t}$). The limit for both equations are the same. Without loss of generality, work with the second equation. There can be no within-state cross-border shopping when t _A =t _B implying that $c_{A}^{B}=0$. Because $c^{B}=c_{A}^{B}+c_{A}^{A}+c_{A}^{N}=c_{A}^{A}+c_{A}^{N}$ and $c^{A}=c_{A}^{B}+c_{A}^{A}=c_{A}^{A}$, it is easy to see that the ability of consumers in B to arbitrage and cross the state border implies that in the limit c ^B≥c ^A and therefore, $U_{C}^{B}\leq U_{C}^{A}$. Substituting into the above equation and letting tax rates converge to t _A =t _B =t, yields

$$\frac{U_{C}^{A}\frac{c^{A}}{1+t}}{\frac{c^{A}}{1+t}-\frac{t-t}{D_{B}''}}\gtreqqless\frac{U_{C}^{B}\frac{c_{B}^{B}}{1+t}}{\frac{c_{B}^{B}}{1+t}-\frac{t-t}{D_{B}''}-\frac{t}{S_{B}''}}, $$

which reduces to

$$\frac{U_{C}^{A}\frac{c^{A}}{1+t}}{\frac{c^{A}}{1+t}}\gtreqqless\frac{U_{C}^{B}\frac{c_{B}^{B}}{1+t}}{\frac{c_{B}^{B}}{1+t}-\frac{t}{S_{B}''}} $$

and simplifying further implies

$$U_{C}^{A}\biggl(\frac{c_{B}^{B}}{1+t}-\frac{t}{S_{B}''} \biggr)\gtreqqless U_{C}^{B}\frac{c_{B}^{B}}{1+t}, $$

which then yields the equation in the text:

$$\frac{(U_{C}^{A}-U_{C}^{B})}{U_{C}^{A}}\frac{c_{B}^{B}}{1+t_{B}}\gtreqqless\frac{t_{B}}{S_{B}''}. $$

1.6 7.6 Proof of Corollary 2

Proof

If marginal utility is constant, then $U_{C}^{A}=U_{C}^{B}$ because the utility functions are identical. To have t _A ≤t _B , then MCF _A (t _A =t _B )≥MCF _B (t _A =t _B ) implies $1\geq\frac{\frac{c_{B}^{B}}{1+t}}{(\frac{c_{B}^{B}}{1+t}-\frac{t}{S_{B}''})}$. But this requires $-\frac{t}{S_{B}''}\geq0$, which is never true. However, MCF _A (t _A =t _B )<MCF _B (t _A =t _B ) is always true because $-\frac{t}{S_{B}''}<0$. This immediately implies t _A >t _B is optimal. □