# Games within borders: are geographically differentiated taxes optimal?

## Authors

- First Online:

DOI: 10.1007/s10797-012-9235-y

- Cite this article as:
- Agrawal, D.R. Int Tax Public Finance (2012) 19: 574. doi:10.1007/s10797-012-9235-y

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## 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.

### Keywords

Commodity taxationCross-border shoppingTax competitionPreferential tax rates### JEL Classification

H21H25H73H77R12## 1 Introduction

“Jeux Sans Frontières”—the title of a classic commodity tax competition paper (Kanbur and Keen 1993)—translates from French to “Games Without Borders.” What happens if countries play games *within* borders by setting geographically differentiated tax rates?

Geographic borders between different states and countries create a discontinuous tax policy. Forty-five American states impose a sales tax ranging between 2.9 % and 7.25 %. Among European Union member states, the Value Added Tax (VAT) ranges from 15 % to 27 %. Given these disparities in tax rates, consumers have large incentives to engage in cross-border shopping. As cross-border shopping is inefficient, it is important to know if a uniform tax policy is socially optimal compared to a policy that differentiates tax rates based on geographic location.

The Mexican VAT features a differentiated tax rate depending on proximity to the United States border. The standard tax rate in Mexico is 16 %, but goods purchased within twenty kilometers of the lower tax United States border are assessed a rate of 11 %. Davis (2011) finds that there is also a modest, but statistically significant distortion from Mexicans living in the high-tax zone who shop or locate in the preferred tax zone. If the size of the distortion induced within Mexico is small relative to the decrease in the size of the distortion at the international border, then such a policy may be optimal for Mexico. In addition, the Netherlands differentiated gasoline taxes near the German border in the 1990s and Italy differentiated gas taxes near the Slovenian border in the early 2000s (Kessing 2008). Duty-free shopping at airports is the most visible representation of reduced tax border-zones—where the border-zone is sufficiently small and restricted to cross-border shoppers.

In the tax competition literature, differences in tax rates arise because of differences in size or public good preferences. However, in all these models, the state competes over only *one* sales tax rate. In this paper, acknowledging that tax competition will imply differentials in tax rates at borders, I consider what a state’s optimal policy is when the state can select *two* sales tax rates within its borders—a rate near the state border and a rate away from the border.^{1} The evidence suggests that in addition to country size and population, the spatial location of regions within a state is an essential factor to consider when setting tax rates.

In the context of a Haufler (1996) model that combines elements from both Mintz and Tulkens (1986) and Kanbur and Keen (1993), I demonstrate that from a welfare maximizing state planner’s perspective, the optimal tax system in the presence of borders is almost always a geographically differentiated tax. The model that I will outline in this paper is a partial equilibrium model where one state selects an optimal tax policy taking as given an exogenous tax rate in the neighboring state. If regions within a state are homogeneous, then tax differentiation within a state will most likely arise when there are differences between states. Thus, solving for a Nash equilibrium when states each choose two tax rates would not be interesting if states were identical and is uninformatively complicated when states are different. This simplification has the advantage of allowing for welfare maximization.

Geographic differentiation of tax rates is not surprising, but the specific pattern of geographic differentiation is novel. A shocking result is that a higher tax rate in the border region is sometimes optimal—a result not attainable with Leviathan governments. For a low-tax state, if the elasticity of cross-border shopping is sufficiently small, the social planner will want to raise taxes in the region closest to the border in an effort to *export* the tax burden to foreign consumers. In high-tax states, higher taxes in the border region follows from a desire to *equalize* consumption across the two regions that outweighs the desire to *expand* the tax base. Inequality in consumption is a concern for the social planner in a high-tax state because residents near the border have the opportunity to increase consumption through cross-border shopping—but it would play no role if the government cared solely about revenue.

Policymakers often discuss equality, expanding the tax base, and exporting the tax burden to “foreigners” in the context of the tax system. As is noted by Slemrod (2007), tax evasion is not uniform across the entire population and this can create serious inequities as the people who evade taxes keep more income. Applied to commodity taxes, cross-border shopping is primarily done by residents near borders and results in higher consumption for those who evade. Further, examples of the desire to export taxes and to expand the tax base are also common. For example, municipalities on the low-tax side of a border set higher local option sales tax rates than municipalities on the high-tax sides of borders. In doing so, municipalities on the low-tax side of the border can export some of the sales tax burden to cross-border shoppers. Additionally, high hotel taxes in cities that tourists frequent are another mechanism to export the tax burden. This exporting effect contrasts with the desire of jurisdictions to expand the tax base. In 2009, the Arkansas state legislature passed a law that requires the cigarette tax in border towns be equal to the cigarette tax in the neighboring town outside of the state. The law only applies if the resulting rate is less than the statutory Arkansas state rate and if the border town in Arkansas and the neighboring state have a population of more than 5000 people. The law also applies to retailers less than 300 feet from the border.^{2} The legislation implemented in Arkansas is a clear example of a state trying to reduce cross-border shopping in order to expand the tax base.

## 2 Tax competition and cross-border shopping

The literature on tax competition has developed with an emphasis on trying to explain asymmetries in tax rates among competing jurisdictions. Haufler (1996) implies that a country may have higher tax rates than its neighbor because of a stronger preference for public good provision. This is in contrast to models such as Kanbur and Keen (1993), Nielsen (2001) and Trandel (1994), which focus on country size or population as an explanation for variation in equilibrium tax rates. These models only allow for one tax rate within a state.

Several studies have focused precisely on the role of distance to a competing jurisdiction as a key variable of interest. Lovenheim (2008), Merriman (2010), and Harding et al. (forthcoming) show that the elasticity of demand, tax evasion, and tax incidence are a function of distance from the border. These papers clearly indicate that geographic space shapes the responsiveness of individuals. The results, together with the theoretical results that tax differentials will exist at state borders, provide a powerful argument for geographic differentiation of tax rates.

Although discriminatory taxation has been studied in the context of multiple tax bases (Janeba and Peters 1999; Keen 2001), relatively few papers have studied the geographic differentiation of taxes. Two known exceptions are Kessing (2008) and Nielsen (2010), which study geographically differentiated taxes in the context of the Nielsen (2001) model. Kessing (2008) and Nielsen (2010) differ from this paper by considering the problem in the context of revenue maximizing governments and only for high-tax states.

This paper builds upon the model in Haufler (1996) because differences in public goods provision is a natural starting point for explaining how differences in tax rates arise across borders and, more importantly, because a model of welfare maximizing governments (without a separable utility function) will allow for a rich treatment of the problem.

## 3 Model

### 3.1 Setup of the model

*k*=

*H*,

*N*. The home state (H) has two identical regions indexed

*i*=

*A*,

*B*denoting the away region (A) and border region (B).

^{3}The neighboring state (N) is identical in terms of the number of regions, but will never set differentiated tax rates. The geographic setting is depicted in Fig. 1.

### 3.2 Consumers and firms

A representative consumer lives at the center of each region. Each consumer is endowed with *M* dollars of income. The consumer has the choice to purchase a quantity of the consumption good, *c*, in her home region or in a neighboring region or state. Let *superscripts* on *c* index where the person lives and *subscripts* index the region that the good is purchased in. Thus, \(c_{j}^{i}\) denotes the quantity of the consumption good that the resident of *i* purchases in region *j*. Let *c*^{i} denote the total consumption of a resident of region *i*. Then, \(c^{i}={\displaystyle {\textstyle \sum}_{j}}c_{j}^{i}\) for all *i* and \(c^{i}=c_{i}^{i}\) if \(c_{j}^{i}=0\) for all *j*.

Firms providing the private good are assumed to be located exogenously at any point where consumers shop. Firms are perfectly competitive and set price equal to the marginal cost of production. Increases in demand in a particular region resulting from cross-border shopping will not alter the pre-tax price. The pre-tax price is normalized to one.

Transportation costs constrain consumers from purchasing goods in another jurisdiction. If the individual decides to purchase in her own region, she simply goes to the store at the point of the line corresponding to where she lives and no transportation costs are incurred. If shopping in a neighboring region, the individual faces a transportation cost function denoted \(D_{i}(c_{j}^{i})\) for *j* not equal to *i*.^{4} I assume that the marginal cost of purchasing the first unit abroad is zero for residents of each region. This will guarantee that if taxes are not equal, the individual in the high-tax region will immediately begin to purchase some of the goods in the low-tax region. In addition, \(D_{i}'(c_{j}^{i})>0\) and \(D_{i}''(c_{j}^{i})>0\) for \(c_{j}^{i}>0\). Additionally, *D*_{i}(0)=0, \(D_{i}'(0)=0\) and \(D_{i}''(0)>0\). These assumptions on the transportation cost function guarantee that as the tax differential between the regions increase, the individual will purchase more of his consumption from the low-tax region. For notational convenience, let *S*_{i} denote the transportation cost function for crossing a state border. I assume *S*_{i}=*D*_{i}. Individuals are willing to travel, at most, one jurisdiction over to make cross-border purchases.

A convex transportation cost function can be justified by a composite consumption good that represents consumption goods that are heterogeneous in terms of the ease of their transportability. More importantly, a convex transportation cost makes the representative agent problem behave like a Hotelling-style model where consumers are along a continuum. Convex transportation cost functions implicitly underlie models of cross-border shopping where agents live along a continuum and are heterogeneous in their distance to the border. The convex transportation cost function is a modeling technique for mimicking a Kanbur and Keen (1993) model, while also allowing for welfare maximizing governments with a representative agent. Thus, the representative agent will proxy for heterogeneous consumers located at different distances from the border.

Consumers have preferences over consumption and a publicly provided good. The functional form of the utility function is identical across all individuals. Preferences are given by the utility function *U*^{i}(*c*^{i},*G*), which is strictly quasi-concave. Individuals can only choose how much of the consumption good to purchase at home and abroad.

### 3.3 The state planner’s problem

The home state sets taxes, *t*_{i}, on goods purchased in region *i* to fund a public good. Taxes are levied according to the origin principle, which implies that if an individual crosses a border, he will pay the tax to the jurisdiction of purchase.^{5} The paper assumes that states provide a state-level public good, meaning that all the revenue raised within the state is aggregated to provide a public good that is uniform across all regions.^{6} The assumption of a state public good merits some discussion. The model considers whether a state should implement discriminatory taxation in geographic space. The model does not consider what are the optimal geographically differentiated tax rates when towns are allowed to set tax rates and keep revenue earned in the town. As discussed, examples of border-zones include the Mexican Value Added Tax, the Netherlands’ and Italy’s historical gas taxes, duty-free shops, and the Arkansas cigarette tax. In these examples, taxes are differentiated within the state, but the provision of public goods is not or was not related to the revenue intake from each region within the state. Furthermore, the models of Kessing (2008) and Nielsen (2010) make the same assumption. For these reasons, I assume a state provided public good common to both regions.

Letting *R*_{i} denote the total revenue raised in jurisdiction *i* and letting 0≤*ρ*≤1 denote the rate of transformation between revenues and expenditures, \(G={\textstyle {\displaystyle {\textstyle \sum}_{i=A,B}}}\rho R_{i}(t_{i})\). I assume that the production technology is unity, so that *ρ*=1. The home state and the neighboring state differ in their preferences for public goods.

Because the utility function is concave, the social welfare function captures equity concerns. The social welfare function must be continuous and strictly quasi-concave in the strategies. The first order conditions of the government’s constrained maximization problem must be continuous when a region switches from being a high-tax jurisdiction to a low-tax jurisdiction. This will be the case if the transportation cost function, described above, is identical across all regions. I assume weak conditions hold such that the first order conditions are concave in the strategies. Continuous and concave first order conditions and quasi-concavity of the objective function will guarantee an interior solution.

*t*

_{i}≥

*t*

_{j}): and for the low-tax region (

*t*

_{i}<

*t*

_{j}):

## 4 Results

### 4.1 Solving the model

I derive the optimal tax rates assuming that State N sets a fixed and uniform tax rate of \(\bar{t}\) and that this state does not respond competitively to geographic differentiation in the home state. Therefore, the equations that follow characterize the home *state’s optimal response* to a fixed tax rate rather than a *Nash equilibrium*. Tax competition, especially with geographically differentiated rates on both sides of the border would add additional complexities and cases. Introducing tax competition would require additional simplifications—such as revenue maximization—that will eliminate the interaction of the effects presented.^{7}

In all of the cases, State H selects *t*_{A} and *t*_{B} taking as given and fixed \(\bar{t}\) by maximizing Eq. (1) subject to the constraints below.

### Case Low—The case of a state with low public good preferences:

\(t_{B}<\bar{t}\)

In this scenario, the neighboring state is a high-tax state and the home state has a preference for lower levels of public goods. The problem facing the low-tax state’s social planner is that she wishes to determine the optimal combination of regional tax rates given that the neighbor has selected a high fixed rate of \(\bar{t}>t_{B}\).

*Sub-Case 1—Preferential tax rates at the border:**t*_{A}≥*t*_{B}

*Sub-Case 2—Higher taxes at the border:**t*_{B}≥*t*_{A}

### Case High—The case of a state with high public good preferences:

\(t_{B}>\bar{t}\)

In this scenario, the neighboring state is a low-tax state, while the home state has a stronger preference for public goods. The problem facing a high-tax state’s social planner is that she wishes to determine the optimal combination of regional tax rates given that the neighbor has selected a low fixed rate of \(\bar{t}<t_{B}\).

*Sub-Case 1—Preferential tax rates at the border:**t*_{A}≥*t*_{B}

*Sub-Case 2—Higher taxes at the border:**t*_{B}≥*t*_{A}

^{8}

### 4.2 The marginal cost of funds in a federation

Equations (7), (10), (13), and (16) equate the marginal cost of funds (or *MCF*) across jurisdictions within a state such that *MCF*_{A}=*MCF*_{B}.^{9} As in Dahlby and Wilson (1994), in a federation, the marginal cost of funds must be equal across all jurisdictions at an optimum.^{10}

The *MCF* can be decomposed into two parts, which I call the “within cost of funds” (*WCF*) and the “private consumption externality” (*PCE*). The *WCF* would be the *MCF* without any externalities and it specifically measures the direct cost in jurisdiction *i* of changing tax rate *t*_{i}. The *WCF* is present on both sides of the equalities in Eqs. (7), (10), (13), and (16). On one side of each equality, a second term is also present. This term is the *PCE* and is only present in the marginal cost of funds for the low-tax region of each sub-case.^{11} This term captures how changes in the low-tax region’s tax rate distort the consumption decisions of individuals in the high-tax region. As the tax differential between the two regions becomes larger, the high-tax region will have a larger fraction of its goods purchased in the neighboring jurisdiction—resulting in relative distortions to the consumption profile. The social planner accounts for this effect on consumption when choosing the tax rate in the low-tax region. The choice of the high-tax region’s rate never affects consumption of the low-tax region because the residents of the low-tax region always purchase commodities at home.

Marginal revenue is a key component of the *MCF*. Suppose *i* denotes the relatively high-tax jurisdiction and *j* denotes the relatively low-tax jurisdiction. Then, the government can possibly expand the base by attracting additional cross-border shoppers with a lower rate. The terms *MR*_{i} and *MR*_{j} in the sub-cases above contain a \(-\frac{t_{i}}{D_{i}''}\) and \(-\frac{t_{j}}{D_{i}''}\), respectively. I will refer to this as the “tax base effect” or *TBE*.^{12} This is the change in revenue in a jurisdiction resulting from changes in the amount of cross-border shopping due to a tax rate change. Increases in the tax rate decrease the amount of consumption purchased within the region via changes in cross border shopping. Alternatively, the government can increase the tax rate, thereby increasing the revenue raised from those shoppers who continue to shop within the jurisdiction. In the low-tax regions for the sub-cases above, *MR*_{j} contains a \(c_{j}^{i}\) term. I will call this the “tax exporting effect” or *TEE*.^{13} As more residents of *i* cross the border to shop in *j*, the larger are the incentives of *j* to raise its tax rate to extract additional revenue from non-residents. This effect moves in the opposite direction of the “tax base effect.” The “tax exporting effect” is only present in the relatively low-tax jurisdictions.

### 4.3 Characterizing a pattern to the optimal tax rates

At an optimum, it must be the case that the marginal cost of funds is equal in all jurisdictions within a federation. Therefore, the optimal tax rate will be geographically differentiated if *MCF*_{A}≠*MCF*_{B} when the tax rates are equal in Region A and B. The exact conditions for geographic differentiation are specified below.

### Proposition 1

*For a low*-*tax state with open borders*, *the optimal tax system depends on the relative size of the tax base effect and tax exporting effect*. *If*, *at the state border*, *the tax base effect is larger in absolute value than the tax exporting effect*, *then**t*_{A}>*t*_{B}*is the optimal response to a neighboring state’s high tax rate*. *If*, *at the state border*, *the tax exporting effect is larger in absolute value than the tax base effect*, *then**t*_{A}<*t*_{B}*is optimal*. *Only if the tax base effect exactly offsets the tax exporting effect will uniform taxation be optimal*.

### Proof

See Appendix. □

Intuitively, because the regions are located in the relatively low-tax state, the residents do not have any opportunity to shop in the other state. Starting from a position where tax rates are equal across regions, the consumption profiles are equal in both regions and there is no within-state cross-border shopping. Differences in the marginal cost of funds will be determined solely by the relative magnitudes from any efficiency losses to the public good, which are driven entirely by the differences in marginal revenue across regions. Differences in marginal revenue result from the tax base and tax exporting effect in the border region.

### Corollary 1

*The tax exporting effect will exactly offset the tax rate effect if the elasticity of cross*-*state shopping with respect to the border region’s tax rate is unit elastic*. *If**ϵ*>1, *the tax base effect will dominate the tax exporting effect and then**t*_{A}>*t*_{B}*is optimal*. *The tax exporting effect will dominate if**ϵ*<1 *and then**t*_{A}<*t*_{B}*is optimal*.

### Proof

See Appendix. □

If cross-state shopping is very price responsive, then small adjustments down in the border region’s tax rate will result in larger quantities of cross-border shopping and additional revenue gains from the neighboring state. On the other hand, if cross-state shopping is not very responsive with respect to the neighboring jurisdiction’s tax rate, then the government can increase revenue by raising the tax rate and exporting the tax to foreign residents.

Therefore, on the low-tax side, the optimal tax is geographically differentiated if cross-border shopping of the other state’s resident’s is not unit elastic. In the following section I will subsequently return to discuss what the magnitude of this elasticity is likely to be.

Now, I consider a state with a high preference for public goods. On the high-tax side, cross-border shopping allows border-zone residents to obtain additional consumption. This implies that the proof defining the conditions for geographic differentiation must account for the benefit some consumers receive from cross-border shopping.

### Proposition 2

*For a high*-*tax state with open borders*, *the optimal tax system depends on the relative size of the tax base effect and differences in the marginal utility of consumption across the two regions*. *If the tax base effect is sufficiently large in absolute value and differences in the marginal utility of consumption are sufficiently small* (\(U_{C}^{A}\thickapprox U_{C}^{B}\)), *then**t*_{A}>*t*_{B}*is the optimal response to a neighboring state’s low tax rate*. *If the tax base effect is sufficiently small in absolute value and differences in the marginal utility of consumption are sufficiently large* (\(U_{C}^{A}\ggg U_{C}^{B}\)), *then**t*_{A}<*t*_{B}*is optimal*. *If the tax base effect exactly offsets the differences in the marginal utility of consumption*, *then uniform taxation is optimal*.

### Proof

See Appendix. □

*t*

_{B}=

*t*

_{A}, the direction of the inequality:

^{14}

Lowering taxes in the border region will reduce revenue leakage. This will be optimal if the right side of (19) is larger than the left side. If the tax base effect is large, it is more likely to be optimal to lower taxes in the border region because individuals are very responsive. The *TBE* will be larger if the elasticity of cross-state shopping is highly responsive to the border-zone’s own tax rate. Inequalities in the marginal utilities of consumption—the left side of (19)—may be small depending on how concave the utility function is, as well as how much of the gains to consumption are offset by the transportation cost. If the tax base effect is significantly responsive and the consumption profiles in the two regions are similar or if the utility function is not very concave so that even large differences in consumption do not concern the social planner, then a preferential tax rate near the border will be optimal.

Contrarily, if inequality is large and the *TBE* is sufficiently small, then *t*_{A}<*t*_{B}. If the resident of Region B has much more total consumption than the resident of Region A or if the utility function is very concave so that even small differences in consumption concern the social planner, it will be possible for tax rates to be higher in the border region. This will be especially true if the tax base effect is sufficiently small in absolute value. The planner will, therefore, account for the both the elasticity of cross-state shopping with respect to the border-zone’s tax rate and the elasticity of total consumption in the border-zone with respect to its own tax rate.

### Corollary 2

*If the utility function is linear in consumption*, *i*.*e*. *has a marginal utility that is constant*, *then**t*_{A}≤*t*_{B}*is never optimal*. *Preferential tax rates in the border region will always be optimal*.

### Proof

See Appendix. □

If individuals have a utility function that is linear in consumption (for example utility that is quasi-linear with respect to the consumption good), then the social planner does not care about how consumption is allocated across all individuals and the problem is similar to revenue maximizing governments. Because utility is no longer concave with respect to consumption, equity concerns vanish from the planner’s problem. With no equity concerns, raising the tax rate near the border is never optimal because it will only increase the revenue leakage across state lines, while also inducing a distortion within the state.

It is reasonable that utility is linear in consumption if cross-border shopping is a matter of a few easily carried items purchased in bulk. Differences in the marginal utilities across these goods seem unlikely to be large, unlikely to have significant complementarity or substitutability with public goods, and unlikely to be a major policy concern.

## 5 What determines the outcome?

It is clear that a strong taste for public goods will lead to Case High and that a weak taste for public goods will lead to Case Low. But what about the sub-cases?

### 5.1 A state with low public good preferences

If the home state is acting in autarky and it is restricted to choosing a single tax rate to maximize domestic welfare, the optimal commodity tax is simply a head tax. At an optimum, the home state will choose the tax rate \(\tilde{t}\) such that \(2\frac{U_{G}}{U_{C}}=1\) (the Samuelson rule holds), which corresponds to private consumption of \(\tilde{c}\). Recalling that the state has two agents, in autarky, social welfare is \(\tilde{W}=2U(\tilde{c},2\tilde{t}\tilde{c})\).

*ϵ*, a lower tax could raise more or less revenue from foreigners. The magnitude of

*ϵ*is determined endogenously. Rearranging Eq. (20) yields

Because the public good is over-provided, the right side of Eq. (21) is unambiguously positive. Therefore, the first order conditions imply that *ϵ* must be less than 1 for an optimizing low-tax state.^{15} This need not be the case for a high-tax state.

The implication of Eq. (21) is that the solution to a low-tax state’s optimal tax rate given an exogenous neighboring tax rate must be a tax rate at which *ϵ*<1. I must determine, however, if *ϵ* can be less than one. Totally differentiating Eq. (18) yields \(\frac{\partial\epsilon}{\partial\tilde{t}}>0\) if \(S_{N}'''\geq0\). Noting that as \(\tilde{t}\rightarrow0\), *ϵ*→0, it must be that there exists a critical tax rate, \(\hat{t}\), such that *ϵ*<1 for all *t* in the interval \([0,\hat{t})\) and *ϵ*>1 for all *t* in \((\hat{t},\bar{t})\) where \(\hat{t}\) may be greater than \(\bar{t}\). If *ϵ*=1, then \(\hat{t}=c_{B}^{N}S_{N}''\). Thus, if \(c_{B}^{N}\) or \(S_{N}''\) are sufficiently large and \(S_{N}'''\geq0\), then *ϵ* will be less than 1.

Proposition 1 is proven by comparing the marginal cost of public funds when the tax rates are equal in both regions. Without loss of generality, the proof can compare the *MCF* across regions with the equality of any tax rate—including \(\tilde{t}\). All that matters is that the preference for public goods in State H is low relative to the neighboring state and thus that \(\tilde{t}<\bar{t}\). Therefore, under the assumption that \(S_{N}'''\geq0\) and \(\hat{t}\) is sufficiently large, the optimal tax must be (and will be) a tax rate where *ϵ* is less than 1, which implies *t*_{A}<*t*_{B}.

A higher tax rate at the border can only arise in the low-tax state if the elasticity of *cross-border shopping* with respect to the *border-zone’s* tax rate is less than unity in absolute value. Whether the assumptions above guaranteeing that *ϵ*<1 hold is an empirical question. The literature has focused on estimating the elasticity of *total* consumption with respect to the *own*-jurisdiction tax rate. Estimates of this elasticity vary from −5.9 (Walsh and Jones 1988) to −1.38 (Tosun and Skidmore 2007). Fox (1986) shows that the elasticity in response to a one percent increase in the tax rate varies by location, suggesting the size and density of the border region may influence the relevant elasticity for a social planner. The literature has not focused on estimating the elasticity of cross-border shopping with respect to the neighboring jurisdiction’s tax rate. However, the heterogeneity of responses across jurisdictions in Fox (1986) suggests that the size of *ϵ* will depend critically on the characteristics of the neighboring jurisdiction’s border-region.

To summarize, if a state has a preference for low public good provision, the optimal response of the state depends on the relative size of the tax base and tax exporting effects at the border. Starting from \(t_{A}=t_{B}=\tilde{t}\), considering the following policy change: the low-tax state raises the tax rate at the border and lowers it away from the border in a manner that keeps tax revenues from domestic consumers constant. This policy will raise welfare if and only if the tax change will raise tax collections from foreigners. Starting from *t*_{A}=*t*_{B}, the policy change will have no first order effect on domestic revenue because *c*^{A}=*c*^{B} initially. If *ϵ*<1, the policy will raise tax collections from foreigners. If *ϵ*>1, tax revenue from foreigners will fall. Thus, if *ϵ*<1 when *t*_{A}=*t*_{B}, the government can “export” a large portion of its revenue raising capabilities to non-residents by increasing the tax rate at the expense of losing relatively little cross-border shopping. When \(t_{A}=t_{B}=\tilde{t}<\bar{t}\), for a low-tax state to be maximizing welfare, I have demonstrated that *ϵ* must be less than 1 under relatively mild assumptions. This will not be the case for the high-tax state.

### 5.2 A state with high public good preferences

Up to now, I have assumed that the social welfare function, *W*=∑_{i=A,B}*U*^{i}(*c*^{i},*G*), gives equal weight to consumers in both regions. Proposition 2 implies that the concavity of the social welfare function will help to determine whether taxes are higher or lower in the border-zone. This proposition suggests that the weight that the social planner will give to each agent is also important. If one consumer obtains excessively large amounts of consumption, but the social planner gives this region additional weight, the outcome will be very different than if the planner gives this region zero weight. One reason the planner may want to give extra weight to one region is if populations of in the two regions are different—but that all people in a region are identical and equidistant from the region border.

Consider the monotonic concave transformations of the utility functions, *Ω*_{i}[*U*^{i}(*c*^{i},*G*)], where \(\varOmega_{i}'(\cdot)>0\) and \(\varOmega_{i}''(\cdot)\leq0\) for all *i*. The social welfare function now becomes \(\hat{W}=\sum_{i=A,B}\varOmega_{i}[U^{i}(c^{i},G)]\). If *Ω*_{A}=*Ω*_{B}, maximizing \(\hat{W}\) is equivalent to maximizing Eq. (1). If *Ω*_{A}≠*Ω*_{B}, then the magnitude of \(\frac{\varOmega_{A}'}{\varOmega_{B}'}\) is essential to determining the pattern of geographic differentiation. If \(\frac{\varOmega_{A}'}{\varOmega_{B}'}<1\), the planner is giving additional weight to the resident of Region B. It is as if Region B has a larger population.

As \(\frac{\varOmega_{A}'}{\varOmega_{B}'}\) becomes smaller and less than 1, the *MCF*_{A}(*t*_{A}=*t*_{B}) grows smaller relative to *MCF*_{B}(*t*_{A}=*t*_{B}). This suggests that even if the consumption differences between the residents of Region A and Region B are very large, if \(\frac{\varOmega_{A}'}{\varOmega_{B}'}\) is sufficiently small to overcome these inequality differences, the social planner will always want to lower taxes in the border region. As \(\frac{\varOmega_{A}'}{\varOmega_{B}'}\) becomes larger and greater than one, *MCF*_{A}(*t*_{A}=*t*_{B}) becomes larger relative to *MCF*_{B}(*t*_{A}=*t*_{B}). As a result, the social planner will more likely want higher taxes in the border region even if consumption differences are small between the two regions. Therefore, the monotonic concave transformation of the utility function suggests that in addition to the sizes of the tax base effect and the inequality in consumption across the two regions, the social planner’s weight to each region is also important. The transformation is useful to help clarify when the unintuitive case that regional tax rates should rise near a low-tax border is possible. Higher taxes in the border-zone become more likely when the border region is sparsely populated relative to the away region.^{16} With this transformation, such a pattern of geographic differentiation may arise even if the marginal utility of consumption is equal across agents.

To summarize, if a state has a preference for high taxes, it can never capture cross-border shoppers from the neighboring low-tax state. Starting from \(t_{A}=t_{B}<\bar{t}\), considering the following policy change: the high-tax state lowers the tax rate at the border and raises it away from the border in a manner that keeps tax revenues raised from domestic consumers constant. By reducing the tax gap with the neighboring state, there is a reduction in cross-state shopping. Starting at *t*_{A}=*t*_{B}, the reduction in cross-state shopping is large relative to the added cross-border shopping within the state. It must be that the policy change will increase total private consumption (*c*^{A}+*c*^{B}) in the state. As a result of the increase in consumption, if the marginal utility of private consumption were the same in both regions and each region received equal weight in the welfare function, the policy change must increase overall welfare. However, at *t*_{A}=*t*_{B}, residents of the border region have *c*^{A}>*c*^{B}. Thus, when the social welfare function is concave, the marginal utility of private consumption is higher for residents of the region away from the border. If the welfare measure is sufficiently “redistributive”, then the distributional effect will create incentives for the planner to reduce *t*_{A} relative to *t*_{B}. Which of these two offsetting effects dominates also depends on how much weight the social planner gives to each region—and therefore, the relative populations.

## 6 Extensions and discussion

### 6.1 Extensions

Many states utilize multiple tax instruments such as income taxes and sales taxes. This raises the question as to whether geographically differentiated sales taxes are optimal if the state has two tax instruments. Intuitively, a state with a high preference for public goods could equalize its sales tax rate to the rate of the neighboring state and then assess a higher income tax. In the absence of migration and labor supply distortions, such a solution would eliminate any possibility of cross-border shopping while obtaining the desired level of public services. However, uniform sales tax rates across the states are rare. One reason for this is that sales tax revenues and income tax revenues are imperfect substitutes as states often seek to rely on multiple taxes to different degrees. Taking as given that these differences in the sales tax rate will exist and assuming that the state cannot geographically differentiate its income tax rate, then the above results will remain applicable.

Second, I assume that producers fully pass forward the tax to the consumer. Firms may adjust their prices depending on how far they are located from the border. Harding et al. (forthcoming) find that the incidence of taxation varies depending on a firm’s distance to the nearest low-tax border. As such, firms near low-tax borders pass through less of the tax to consumers. So long as the individual firms do not change prices in a manner that completely reduces the after-tax price at borders, then geographic differentiation of tax rates will still be optimal. The degree of geographic differentiation will be different than if the incidence of the tax is entirely on consumers because the firms are making some of the price adjustments that the social planner would do through geographic differentiation.

Finally, geographic differentiation of tax rates presents issues relating to horizontal equity. Although consumers in both regions have equal incomes, the tax burden that they face will vary depending on their residence. Such a violation of horizontal equity is a concern, however, it would also be present if the tax system were uniform within a state. Under uniform taxation, residents with equal incomes have heterogeneous opportunities for cross-border shopping in the neighboring state. Thus, the uniform tax system will be horizontally unequal on the basis of some residents cross-border shopping while other residents will be unable to cross-border shop because of how far they reside from the border.

### 6.2 Comparison to the existing literature

Although the above results are derived under welfare maximizing governments, a large portion of the tax competition literature has relied on revenue maximizing governments. In the context of this model, revenue maximizing governments are characterized by the marginal utility of consumption being zero. Therefore, a model with governments as revenue maximizers is nested within the model presented above. As noted above, revenue maximizing governments is the subject of two working papers: Kessing (2008) and Nielsen (2010). Revenue maximization allows for the characterization of an equilibrium—when it does exist.

In Kessing (2008), the equilibrium policy is to raise tax rates in the region away from the border if the size of the border-region is sufficiently large. Consider the case where the home country is a high-tax country and where the border-zone is sufficiently large. In this case, the neighboring country will set the same tax rate as in the non-preferential case. The home country will set the same tax rate in the border-region as it would select in the non-preferential case, but will add an additional mark-up to the tax rate in the region away from the border. In Nielsen (2010), no Nash equilibrium will exist in pure strategies when the government can select both the tax rates and the size of the border-region. The endogenous width of the border-zone in Nielsen (2010) provides an interesting contrast. Selecting the size of the border-region will make it so that the away region is composed of some “swing-shoppers.” These shoppers are willing to cross the border-zone and purchase the good in the foreign country. The neighboring country can lower tax rates to obtain a discrete jump in revenues. However, in a Stackelberg equilibrium with the neighboring country as a leader, an equilibrium will exist with preferential tax rates at the border. These results suggest that depending on the nature of the strategic interaction and whether the size of the border-region is exogenous, tax competition may or may not be intensified.

Unlike Kessing (2008) and Nielsen (2010), I assume that tax competition will not occur in response to border-zones. The results of this paper differ from the revenue maximizing models discussed above in that tax rates may be higher in the border-region—suggesting that differentiated tax rates need not result in a race to the bottom. In addition, I do not restrict differentiated tax rates to only high-tax states. As I suggest in this paper, preferential or discriminatory rates may be desirable for low-tax jurisdictions as well. Determining if tax competition, in a model of welfare maximizing governments, induces a race to the bottom will inevitably depend on the structure of the game and the size of the border-region. Whether such tax competition occurs in response to border-zones remains an open question.

## 7 Conclusion

The model presented here suggests that when the tax system is characterized by a line resulting from geographic borders, uniform within-state taxation is not an optimal policy under most conditions. The border-line encourages cross-border shopping. This distortion is indicative of deadweight loss, but benefits the low-tax state. When states differ in preferences for public goods, tax differentials will naturally arise at borders.

The model above presents an administratively feasible tax system, where discrete changes in the tax system based on location will be welfare enhancing.^{17} Different rates within a state will induce additional discontinuities in the tax system, but the state planner can utilize these small discontinuities to increase revenue or smooth consumption. Shopping in a different region within the state incurs inefficiencies in transportation, but it has the possibility of obtaining additional revenues from residents outside of the state. The result is a first order gain with a second order loss.

The results presented in this paper suggest that as states seek ways to increase revenue and as the European Union continues its process of integration, additional study of geographically differentiated taxes should be an important part of future research. In the presence of tax differentials, a state can improve the social welfare of residents through geographic differentiation of the tax rate if neighboring states do not respond strategically. The principle of geographic differentiation within a state is likely to apply to other types of non-tax policy where similar distortions result from policy differentials at the border.

The word “optimal” in this paper will always refer to what is “optimal from the prospective of a state planner” rather than what is “globally optimal” unless specifically noted otherwise.

The law providing for this provision is Act 180 of the Regular Session of the 87th General Assembly.

The regions could be two equally sized jurisdictions. The word “region” need not imply a non-governing entity. As an alternative, the two regions can be viewed as two towns—with one town being the preferred tax region—as in the case of Arkansas.

The transportation cost is independent of distance. Each consumer lives an identical amount of distance from the nearest region or state border.

The sales tax in the United States is levied de facto according to the origin principle. The use tax is notoriously under-enforced. Because the use tax is often evaded, taxes are implicitly paid based on the location of purchase rather than the destination of the sale.

When the public good is provided at the state level, this means that the size of the tax base in one region is independent of the level of public good provision that the region receives.

An alternative way to proceed would be to maintain welfare maximization, start from a symmetric equilibrium where all tax rates are identical and then shock the public good preferences in one state. With only one tax rate, these results are extremely complex.

Note I need to add one more derivative to Eq. (4) to solve this problem. This derivative can be obtained by totally differentiating Region B’s individual budget constraint.

A large literature on the marginal cost of funds has emerged, including Slemrod and Yitzhaki (1996) and Slemrod and Yitzhaki (2001).

The intuition for this can be seen in an example. If the *MCF* is 1.5 in Region A but is 1.1 in Region B, then raising an additional dollar of revenue in Region B is less costly than raising an additional dollar in Region A. Raising additional revenue from Region B is welfare enhancing for the state even though it is not a Pareto improvement for Region B.

The equations are arranged so that it is always the second additive term on its side of the equality.

Both sides of the equation in the text are positive because \(U_{C}^{A}\geq U_{C}^{B}\). The right side illustrates the revenue change through the tax base effect. The left side illustrates the social planner’s concern for equality. The derivation of this equation is in the Appendix.

In this model, the only decision is over where to shop. Thus, commodity taxation in a closed economy will amount to a head tax. In an open economy with a high-tax neighbor, commodity taxation imposes no other distortions. Of course, this would not necessarily hold if other distortions existed in the economy, such as an endogenous labor-leisure choice.

When interpreting this result, it is important to remember that all agents in each region are identical and located the same distance from borders. The interpretation of the results would change if the population of the region were distributed heterogeneously with respect to distance to borders.

It is worth recognizing that there are significant administrative difficulties of implementing border-zones under a VAT. For example, it raises the question of how inter-region purchases of intermediate goods are treated. Although similar issues arise with sales and excise taxes, such concerns are not as significant.

## Acknowledgements

I am especially grateful to my dissertation committee chair Joel Slemrod along with the members of my committee—David Albouy, Robert Franzese and James Hines. I also wish to thank Paul Courant, Lucas Davis, Charles de Bartolomé, Dhammika Dharmapala, Reid Dorsey-Palmateer, Marcel Gérard (discussant), Michael Gideon, Makoto Hasegawa, Ravi Kanbur, Sebastian Kessing, Michael Lovenheim (discussant), Byron Lutz (discussant), Søren Bo Nielsen, Ben Niu, Stephen Salant, Nathan Seegert, Jeff Smith, Caroline Weber, and David Wildasin for helpful suggestions and discussions. Suggestions from conference participants at the 2011 International Institute of Public Finance Annual Congress, the 2011 Association of Public Economic Theory Conference and the Michigan Tax Invitational contributed to the paper. Two anonymous referees and the editor, Ruud de Mooij, improved the paper. Any remaining errors are my own.