Commodity Taxes Under Fiscal Competition

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Commodity Taxes Under Fiscal Competition:

Stackelberg Equilibrium and Optimality

By YOU-QIANG WANG*

A major issue in research on tax competition ishow competition between neighboring jurisdic-tions influences commodity taxes. The pioneeringstudy of Jack Mintz and Henry Tulkens (1986),extended by Alain de Crombrugghe and Tulkens(1990), shows that strategic behavior, as well asdifferences in tastes and endowments, can leadneighboring regions to impose differing taxes.Since Mintz and Tulkens use a very general

model, however, their analysis does not describeany particular pattern that such commodity taxdifferentials can be expected to take. In contrast,focusing on the relative population densities oftwo adjoining regions, Ravi Kanbur and MichaelKeen (1993) use a model of spatial competition toshow that unequal population densities produce aparticular pattern of equilibrium taxes: a sparselypopulated nation has an incentive, ceteris paribus,to impose commodity taxes that are lower thanthose imposed by an adjoining densely populated

nation. By incorporating both the possibility thatfirms possess market power and the possibilitythat governments maximize resident welfarerather than tax revenue, Gregory A. Trandel(1994) extends the Kanbur and Keen analysis andshows that the basic result in the Kanbur and Keenpaper is quite robust.

As in much previous work1 (see Mintz andTulkens, 1986; George R. Zodrow and PeterMieszkowski, 1986; David E. Wildasin, 1988;John D. Wilson, 1991; Ben Lockwood, 1993)on tax competition, Kanbur and Keen (1993)examine a noncooperative (Nash) equilibrium

in a tax-setting model. They demonstrate that theNash equilibrium tax rates are unambiguously toolow and provide a complete characterization ofmutually advantageous coordinated domestic taxreforms. Their analysis depends crucially on thedifferences in population size. However, they donot consider the impact of such differences on thesequence of decisions in a tax-setting model. It isnatural and conceivable that, in a real-world situ-

ation of tax-setting, the large region moves firstand the small region moves second. Thus, forexample, we might consider a metropolitan areawith a central city surrounded by suburbs, pro-vided that local governments have some freedomto set taxes. Because of the presence of cross-border shopping and the disparity of domesticmarkets, such sequential (Stackelberg) tax-settingschemes turn out to be incentive compatible,which will be specified below.

The present paper investigates the Stackel-

berg equilibrium in the Kanbur and Keen (1993)model and compares the Stackelberg equilib-rium with the Nash equilibrium. In particular,the paper shows that the Stackelberg equilib-rium strictly dominates the Nash equilibrium, inthe sense that each of the two neighboring re-gions under consideration could collect highertax revenues if both chose the correspondingStackelberg tax rates. It also demonstrates thatthe Stackelberg tax rates of both regions arehigher than their corresponding Nash tax rates,and the extent to which the small region under-cuts the large region is larger in the Stackelbergequilibrium than in the Nash equilibrium.

Harmonization and minimum tax rules areimportant policy concerns in the EuropeanCommunity, which now has minimum indirecttax rates and continues to be interested in har-monization.2 Using the Nash equilibrium as abenchmark, Kanbur and Keen (1993) show thatthe large region benefits from the strategy of

* Department of Economics, Chinese University ofHong Kong, Shatin, NT, Hong Kong. I thank RobertSchwab for his enthusiastic encouragement and helpful ad-vice. I am greatly indebted to two anonymous referees forproviding valuable advice and suggesting interesting exten-sions. Any errors are, of course, mine.

1 Roger H. Gordon (1992) and C. Sausman (1994) con-sider capital income tax competition in the Stackelberg case.However, their models are structurally rather different fromthe Kanbur and Keen (1993) model. I am grateful to areferee for pointing this out.

2 I thank a referee for raising the issues that motivate thediscussion of Section III presented in the text.

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harmonization if and only if the harmonized taxrate is high enough and that imposing a min-imum tax is strictly Pareto improving. Here Iexamine the two types of reform while choosingthe Stackelberg equilibrium as a benchmark. In

contrast with their results, I find that in theStackelberg case, both types of reform certainlyharm the small region and benefit the largeregion. More strikingly, the imposition of aminimum tax rate in the Stackelberg case willactually lead the large region to cut its tax rate.

This paper closely follows the Kanbur andKeen model and proceeds as follows. Section Ipresents both the basic Kanbur and Keen modeland the Stackelberg equilibrium of the model.Section II discusses the comparison between the

Stackelberg equilibrium and the Nash equilib-rium. Section III examines two strategies of taxcoordination: harmonization and the impositionof a minimum tax rate. Finally, Section IVoffers concluding comments.

I. The Model and the Stackelberg Equilibrium

The formal structure of the Kanbur and Keenmodel may be summarized as follows.

Consider two adjoining regions, home andforeign, which lie on the interval [1, 1] with

a border between them at the origin. Withineach region, the population is homogeneous anddistributed uniformly, but the two populationsmay differ in size: there are h individuals in thehome region and H in the foreign region. Theratio h/H is defined as the relative size ofthe home region and h is called small if andonly if 1. Throughout the paper, lowercaseletters refer to the home region and uppercaseletters refer to the foreign region; the superscripts refers to the Stackelberg case and the super-

script n refers to the Nash case.There is a single taxed good in the two regions.

The producer price of the commodity is bothconstant and the same in both regions. Stores arelocated everywhere within the two regions. Eachstore is assumed to charge its customers the pro-ducer price plus the tax of the jurisdiction in whichit is located. Each consumer buys one unit of thecommodity if its cost to her is less than or equal toher reservation price; otherwise, she buys none. Aconsumer can purchase the commodity in eitherregion at her will.

Consider the decision problem of a consumer

in the home region. She pays the producer priceplus the home excise tax t if she buys in thehome region. However, if she purchases in theforeign region, the total cost is the sum of theproducer price, the foreign excise tax T, and the

transportation cost s, where 0 is the costof per-unit distance from the border and s is thedistance at which she is located from the border.So a consumer in the home region will buy inthe foreign region if and only if the followingtwo conditions are satisfied:

(1) t T / s

and

(2)v

T

s

0

where v is the consumers reservation price netof the producer price.

The objective of each government is taken tobe the maximization of its tax revenue;3 allsubsequent references to optimality and incen-tive compatibility are to be interpreted in thatsense. Consider the case in which the reserva-tion prices of the consumers are infinite, so thatreservation prices do not constrain governmentsin their tax-setting and condition (2) vanishes.

Then, given (1), the revenue of the home regionis:

(3) r t, T th 1 t T / t Tth tH T t / t T.

Given T, the home region maximizes r(t, T) bychoosing t. Let t(T) be the resulting home re-gions best-response correspondence. If thehome region is the smaller of the two ( 1),then4

3 See Kanbur and Keen (1993) for the justifications ofthis assumption.

4 The reaction functions (4) and (5) are identical with (5)and (6) in Kanbur and Keen (1993) respectively. The deri-vation of t(T) in (4) can be outlined as follows. Given T,differentiating the two expressions in (3) with respect to tand subsequently solving the two resulting first-order con-ditions yield that t ( T)/2 ifT and t ( T)/2 ifT . For any T in the overlapping interval [,], t(T) is thus derived by comparing the two revenuesobtained at t ( T)/2 and t ( T)/ 2 respec-tively. It turns out that t(T) shifts from ( T)/2 to (T)/2 at T .

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(4) t T T /2 T T /2 T

while if it is the larger ( 1),

(5) t T T /2 T T T T /2 T .

For the purpose of our analysis, I assume 1 throughout the rest of the paper; that is, thehome region is small and the foreign region islarge. Here I examine the outcome when theborder is open and tax competition is charac-terized in the framework of the Stackelberggame. More precisely, I consider the follow-

ing scenario: the large region (leader) choosesits own tax rate T to maximize its tax revenuewhile bearing in mind that the small region(follower) will observe T before choosing itsown tax rate t to maximize its tax revenue.5 Inaddition, the tax revenue functions of bothregions are common knowledge. The follow-ing proposition presents the Stackelberg equi-librium of this game.

PROPOSITION 1: Assuming the reservationprices of all consumers in the two regions areinfinite, there exists a unique Stackelberg equi-librium. The equilibrium taxes are

(6) ts 1

2

3

4

Ts 1

2.

PROOF:6

By analogy with (3), the foreign regionsrevenue is found to be

(7) R T, t TH 1

T

t / T

tTH Th t T / T t.

Then, given the home regions reaction functiont(T) described in (4), the foreign region maxi-mizes R(T, t(T)) by choosing T. To solve thismaximization problem, we need to consider twocases.

Case 1: T .

By (4), t(T) ( T)/ 2 and t(T) T. So,in this case, the maximization problem reducesto

MaxT

TH TH T / 2 .

Let R0 R( , ( )/2). Then one

can easily verify that R0 MaxT TH

(TH( T)/(2))} H 1 (1 )/2).

Case 2: T .

Similarly, by (4), t(T) ( T)/2 andt(T) T. Hence, in this case, the maximizationproblem reduces to

MaxT

TH TH T / 2 .

Solving the above problem yields the optimal

T

s

(1 (/ 2)), and hence the correspond-ing optimal ts ((1/2) (3/4)). Obvi-ously, these solutions are feasible and consistentwith respect to the constraints. Furthermore,notice that R(Ts, ts) R0.

7 This completes theproof of the proposition.

5 I am grateful to a referee for his/her suggestion to studythe Stackelberg case in which the small region behaves asthe leader of tax-setting. Interestingly, the central results ofthis paper continue to hold in this case; that is, both regionslevy higher taxes and collect higher tax revenues in theStackelberg equilibrium than in the Nash equilibrium. Fur-thermore, in this case, both regions set the same tax rate if 12; the large region undercuts the small region if 12. Compared with the tax revenues in (11) presented be-low, the large regions tax revenue is higher for every (0, 1], while there exists a (12, 1) such that the smallregions tax revenue is higher for every (0, ), equal at and lower for every (, 1].

6 I thank a referee for pointing out a simplification in theproof.

7 Note that R(Ts, ts) R0 H(2 ( 2)2/4

( 2) )/ 2. Then R(Ts, ts) R0 follows from (2

( 2)2/4)2 (( 2)

)2 (164 ( 2)2(32)2)/16 0.

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II. The Stackelberg Equilibrium

vs. the Nash Equilibrium

In this section, I explore the properties of theStackelberg equilibrium characterized in (6)

and compare them with the Nash equilibrium.For the convenience of exposition, I restate (8)and (10) of Kanbur and Keen (1993) as thefollowing proposition.

PROPOSITION 2: Assuming the reservationprices of all consumers in the two regions areinfinite, there exists a unique Nash equilibrium.The equilibrium taxes are

(8) tn

1

3

2

3Tn

2

3

3.

The equilibrium tax revenues are

(9) rn H1

3

2

3

2

Rn H 23

3

2

.

Consider first the comparison between theStackelberg equilibrium with the Nash equilib-rium. From (6) and (8),

(10) Ts Tn 2 ts tn

31

2

Ts ts 12 4 13 3 Tn tn 0

and hence we state the following proposition.

PROPOSITION 3: Both regions levy highertaxes in the Stackelberg equilibrium than in theNash equilibrium, i.e., Ts Tn and ts tn.Furthermore, in the Stackelberg equilibrium,the small region undercuts the large region

more severely than in the Nash equilibrium, i.e.Ts ts Tn tn.

Figure 1 presents the Stackelberg and Nashequilibrium taxes and the large regions tax

revenues in the two equilibria. The thin andthick lines represent the best-response corre-spondences of the large and small regions re-spectively. The two lines intersect at the point(Tn, tn), giving rise to the Nash equilibrium.The two (convex) curves are iso-revenue curvesof the large region: the lower one passesthrough the point (Tn, tn) and hence corre-sponds to the Nash equilibrium tax revenue Rn;being tangent to the best-response correspon-dence line of the small region at the point (Ts,

t

s

), the upper one corresponds to the Stackel-berg equilibrium tax revenue Rs.8 Note that (Ts,ts) is the Stackelberg equilibrium.

The result of Proposition 3 is clearly illus-trated in Figure 1. The intuition is straightfor-ward. For a given market, raising the tax ratewill increase tax revenue. With an open borderbetween the two regions, however, some homeconsumers could be lost to the foreign market asa consequence of the higher tax rate at home.Hence a higher tax rate will have two opposingeffects on revenue. In the Nash problem, when

choosing its tax rate, each region ignores thebeneficial effect that raising its tax rate wouldhave on the revenues of the other region bypushing cross-border shopping in its direction.Thus the Nash tax rates are unambiguously toolow. However, in the Stackelberg problem, thesmall region will raise its own tax rate if itobserves a high tax rate set by the large region;the large region knows this and consequentlyraises its own tax rate with less concern aboutthe impact on cross-border shopping. As a re-

sult, the Stackelberg tax rates are higher thanthe Nash tax rates.9 Moreover, the Stackelberg

8 Note that the iso-revenue curve corresponding to Rn

has a slope of zero at the point (Tn, tn). Keeping this inmind, one can easily see why the iso-revenue curve corre-sponding to Rs must be higher. I am thankful to a referee forpointing this out.

9 One could also see this via the standard theory ofoptimal taxation. Since the small region will increase its taxrate in response to an increase in the large regions tax rate,the large regions tax base elasticity is perceived to besmaller in the Stackelberg equilibrium. I am grateful to areferee for pointing this out.



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tax-setting scheme allows the small region tofurther undercut the large region in maximizingits own tax revenue.

Substituting (6) into (3) and (7) respectively,shows that the Stackelberg equilibrium tax rev-enues are

(11) rs H1

2

3

4

2

Rs

H

2 1

22

.

Now, comparing (9) with (11) yields the fol-lowing proposition.10

PROPOSITION 4: Both regions collect highertax revenues in the Stackelberg equilibriumthan in the Nash equilibrium. The percentage oftax revenue increase in the small region is atleast 56.25 percent, while it is exactly 12.5

percent in the large region.

The large region plays a leadership role and setsa high tax rate. This creates an opportunity for thesmall region to raise its own tax rate. Because ofthe presence of cross-border shopping, each re-gion benefits from the high tax rate set by the

other region. Thus, in the end, both regions gainhigher tax revenues as a result of higher tax ratesset by both regions (see Figure 1 for the taxrevenue increase in the large region).11 The impli-cation of Proposition 4 is that the Stackelberg

tax-setting schemes are incentive compatible andthe Stackelberg equilibrium strictly dominates theNash equilibrium. Therefore, in the presence ofcross-border shopping, both regions would preferto play the Stackelberg game.

III. Tax Coordination in the Stackelberg Game

In this section, using the Stackelberg equilib-rium tax revenues (11) as benchmarks, I analyzethe impact of tax harmonization and a minimum

tax constraint on the tax revenues of the tworegions. In doing so, I maintain the assumptionthat the reservation prices of all consumers inthe two regions are infinite.

Consider the case of tax harmonization to anybetween the Stackelberg equilibrium tax rates;that is, the two regions set a common tax rate where ts Ts. Then the large region andsmall region collect tax revenues R H andr h respectively. At Ts, r r(Ts,Ts) r(ts, Ts) rs since ts is the unique bestresponse of the small region to Ts.12 On the

other hand, one can readily verify that Rs R

at ts. Since R and r are strictly increasingin , we thus have the following.

PROPOSITION 5: Harmonization to any be-tween the Stackelberg equilibrium tax rates iscertain to harm the small region and benefit thelarge region.

Tax harmonization eliminates the possibilityof the small region collecting any revenue from

those residents of the large region who cross-shop in the Stackelberg equilibrium. Eventhough the small region could gain revenuefrom its own residents due to harmonization, itwould always end up with a negative net gain.On the contrary, since the lower of the Stack-elberg taxes is relatively high, harmonization

10 Direct algebraic manipulation shows that (Rs Rn)/Rn 0.125 and (rs rn)/rn (( 2 )(10 17 )) /(16(1 2)2). Notice that (rs rn)/rn is a decreasingfunction ofon the interval (0, 1], and is equal to 0.5625 at 1.

11 In Figure 1, one can also add the iso-revenue curves ofthe small region at the Nash and Stackelberg equilibria tosee the tax revenue increase in the small region.

12 The argument here is the same as in Kanbur and Keen(1993).

FIGURE 1. THE STACKELBERG AND NASH EQUILIBRIA



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between the Stackelberg taxes would certainlyleave the large region better off.13

Now suppose that some lower bound isimposed on the tax rate that a region maychoose, where ts Ts. Under this con-

straint, the small regions best-response corre-spondence t(T) changes in the following way(as depicted in Figure 2). Here we need toconsider two cases: and .14 If , then

(12)

t T T /2 T T

T T 2

T /2 T 2

while if ,

(13)

t T T 2

T /2 T 2

where T (2 2 (1 )( ))/.15

Recall the unconstrained best-response corre-spondence of the small region in Figure1. Clearly, the impact of the minimum tax con-straint on the best-response correspondence ofthe small region is shown in Figure 2. The jumppoint moves to T if [see Figure2(a)], while no jump point exists if [seeFigure 2(b)]. In both cases, the left part of thelower segment rotates clockwise to yield a por-tion horizontal at .

What about the properties of a Stackelbergequilibrium if one exists? In the case of ,if the relative size of the small region is largeor the imposed minimum tax rate is relativelyhigh, then the large region might be better off

by undercutting the small region. On the otherhand, if the large region undercuts the smallregion, then its tax revenue rises with its owntax rate and hence it will always choose thehighest undercutting tax rate T, which would

lead the small region to eventually undercut thelarge region by setting t(T) [see Figure2(a)]. Therefore, if there does exist a Stackel-berg equilibrium in the case of , then itmust be at a point where the small region un-dercuts the large region.16 Matters are muchsimpler for the case of . The minimum taxrate now is high enough so that the large regionwill maximize its tax revenue by setting theminimum tax rate, which will in turn force thesmall region to choose the minimum tax rate as

well [see Figure 2(b)]. Formally, the followingresult can be established.

PROPOSITION 6: Let be a minimum taxrate such that ts Ts, where ts and Ts areas in (6).

(i) If , then there can be no more than oneStackelberg equilibrium. Taxes at such anequilibrium are:

(14) tm

Tm

2.

(ii) If , then there exists a unique Stack-elberg equilibrium. The equilibrium taxesare

(15) tm

Tm .

(iii) In the Stackelberg equilibrium under the

minimum tax constraint, the small regionis certainly worse off, while the large re-gion is definitely better off than in theunconstrained Stackelberg equilibrium.

(See the Appendix for the proof.)Figure 2 exhibits the (possible) Stackelberg

equilibria and the gain in the large regions tax

13 Kanbur and Keen (1993) show that the large countrybenefits from harmonization to if and only if T, whereT is characterized in their Proposition 10. Noting that ts T, one can see that this is indeed consistent with theirconclusion.

14 Note that, in the analysis of Kanbur and Keen (1993),the case of is the only one that needs to be consid-ered. For this case, the reaction function (12) given below isthe same as in Kanbur and Keen.

15 The derivation of T and the proof of T ( )/ 2 are available from the author upon request. Note that ts implies 2 ( )/2 and hence T 2 . Also, refer to footnote 4 to see (12) and (13).

16 Thus, in the case where the large region is better off byundercutting the small region, a Stackelberg equilibriumfails to exist.



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revenue.17 The minimum tax constraint limits the

scope for the small region to undercut the largeregion. In contrast with raising its tax rate in theNash case, the large region now takes full advan-tage of its leadership role in setting taxes andoptimally cuts its tax rate to gain tax revenue. Thetax cut in the large region adversely affects the taxrevenue of the small region. In the end, since therevenue loss due to diminished cross-shoppingoutweighs the gain on sales to its own residents,the small region unambiguously loses from theminimum tax strategy.

IV. Conclusion

This paper has focused on the comparisonbetween the Stackelberg and the Nash equilibriain the Kanbur and Keen (1993) model. Themain result of the paper is that, given two neigh-boring jurisdictions with different populationsizes, both jurisdictions collect less tax revenuesif they adopt the Nash tax-setting schemesrather than the Stackelberg ones where the rel-atively densely populated jurisdiction behaves

as the leader. Thus, in the presence of cross-border shopping, the Stackelberg tax rates in-crease both regions tax revenues. Based on theStackelberg tax rates, however, both tax harmo-nization and the imposition of a minimum taxrate will harm the sparsely populated region andbenefit the densely populated region. Althoughthese results are derived in a simple and limitedmodel, they do shed some light on the optimal

policy design of tax competition and tax coor-

dination among neighboring jurisdictions.

APPENDIX:18 PROOF OF PROPOSITION 6

Consider first the case of . By (7) and(12), the large regions tax revenue R(T, t(T))is an increasing function ofT on the interval [,T). Thus, in Stackelberg equilibrium, Tm and tm by the minimum tax constraint.Then it follows that the Nash and Stackelbergequilibria coincide in this case. Hence, tm andTm must be as in (14) for the case of [seeProposition 11 of Kanbur and Keen (1993) andfootnote 14].

At tm and Tm ( )/2, the taxrevenues of the small region and large region arer(tm, Tm) H((2 1) )/(2) and R(Tm,tm) H( )2/(4) respectively. As functionsof, r(tm, Tm) 2(2 1)2/4 and R(Tm, tm) 9H(1 /2)2/16 on the interval (ts, Ts). Then,rs r(tm, Tm) follows from rs 2(2 1)2/4H(2 4 2)/16 0, and (11) implies thatR(Tm, tm) Rs.

For the case of , (13) implies that aStackelberg equilibrium must be at a point where T 2 . Furthermore, by (7) and (13),R(, ) H Max

T2 TH TH(

T)/}. Thus there exists a unique Stackelbergequilibrium, and taxes at the equilibrium are char-acterized in (15). Finally, by Proposition 5, wehave the conclusion on the tax revenue changes inboth regions.

17 The tax revenue loss of the small region can be shownin Figure 2 as well. Refer to footnote 11.

18 I thank a referee for providing insightful suggestionsfor improving the Appendix.

(a) (b)

FIGURE 2. A MINIMUM TAX



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REFERENCES

de Crombrugghe, Alain and Tulkens, Henry. OnPareto Improving Commodity Tax Changesunder Fiscal Competition. Journal of Public

Economics, April 1990, 41(3), pp. 33550.Gordon, Roger H. Can Capital Income Taxes

Survive in Open Economies? Journal of Fi-nance, July 1992, 47(3), pp. 115980.

Kanbur, Ravi and Keen, Michael. Jeux SansFrontieres: Tax Competition and Tax Coor-dination When Countries Differ in Size.American Economic Review, September1993, 83(4), pp. 87792.

Lockwood, Ben. Commodity Tax Competitionunder Destination and Origin Principles.

Journal of Public Economics, September1993, 52(2), pp. 14162.Mintz, Jack and Tulkens, Henry. Commodity

Tax Competition Between Member States ofa Federation: Equilibrium and Efficiency.

Journal of Public Economics, March 1986,29(2), pp. 13372.

Sausman, C. Tax Competition and DynamicGames. M.A. dissertation, University of Es-sex, U.K., 1994.

Trandel, Gregory A. Interstate Commodity TaxDifferentials and the Distribution of Resi-dents. Journal of Public Economics, March1994, 53(3), pp. 43557.

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Wilson, John D. Tax Competition with Interre-gional Differences in Factor Endowments.Regional Science and Urban Economics, No-vember 1991, 21(3), pp. 42351.

Zodrow, George R. and Mieszkowski, Peter. Pi-gou, Tiebout, Property Taxation, and the Un-derprovision of Local Public Goods.Journal of Urban Economics, May 1986,19(3), pp. 35670.


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