Discussion Papers In Economics
And Business

Endogenous Choice on Tax Instruments
in a Tax Competition Model: Unit Tax
versus Ad Valorem Tax
Nobuo Akai

Hikaru Ogawa

Yoshitomo Ogawa

Discussion Paper 10-01

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEndogenous Choice on Tax Instruments
in a Tax Competition Model: Unit Tax
versus Ad Valorem Tax
Nobuo Akai

Hikaru Ogawa

Yoshitomo Ogawa

Discussion Paper 10-01
January 2010

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEndogenous Choice on Tax Instruments
in a Tax Competition Model: Unit Tax
versus Ad Valorem Tax
Nobuo Akai
Osaka School of International Public Policy, Osaka University,
1-31 Machikaneyama, Toyonaka, Osaka 560-0043, Japanemail:akai@osipp.osaka-u.ac.jp,

Hikaru Ogawa
School of Economics, Nagoya University,
Furocho Chikusaku, Nagoya 464-8601, Japan
Tel.:+81-52-789-2379, Fax:+81-52-789-4924, email:ogawa@soec.nagoya-u.ac.jp

Yoshitomo Ogawa
Faculty of Economics, Kinki University,
3-4-1 Kowakae, Higashiosaka, Osaka 577-8502, Japan,
email:ogawa@eco.kindai.ac.jp,

Abstract
This paper analyzes an endogenous choice problem with regard to tax
instruments in a capital tax competition model. Considering a symmetricand two-region model of tax competition, where each region is allowed tochoose either unit or ad valorem tax, we show that selecting unit tax as
a policy instrument is the dominant strategy of governments. An interpretation of this result is clearly explained by the properties of the best
response curves.

JEL Classification: H20, H21, H77
Keywords: Tax competition, Unit tax, Ad valorem tax

1

1

Introduction

As surveyed by Wilson (1999), there are many literatures on tax competition.
However, attention about whether it matters if the taxes are unit or ad valorem
is not so popular as the topic. In almost literatures in the context of the ZodrowMieszkowski-Wilson (ZMW) model, the tax is levied per unit of capital (i.e., it
is a unit tax), whereas in reality, taxes are on capital income and are therefore
always ad valorem (i.e., a proportion of income from the capital).
The pioneering work on this problem is Lockwood (2004), which shows that
the Nash equilibrium in unit taxes is generally diﬀerent from the Nash equilibrium in ad valorem taxes.
"If countries are symmetric, and both private and public goods are normal,then (i) the symmetric Nash equilibrium in taxes exists and is unique in eachcase; and (ii) equilibrium taxes and public good provision are always lower whencountries compete with ad valorem taxes." (Lockwood, 2004)
Given this interesting result, one simple straightforward extension of thismodel is to introduce a stage for the selection of the tax system, a unit tax or an
ad valorem tax, before they compete with each other. Even if the equilibriumwith a unit tax is superior, this system may not be selected in the dynamic twostage strategic game. Therefore, it is valuable to explore this strategic behavior of
the government in a dynamic game with the strategic selection of tax instruments.
This paper analyzes an endogenous choice problem of tax instruments, unit
or ad valorem, in a capital tax competition model. Considering a symmetricand two-region model where each region maximizing its tax revenue is allowed
to choose either a unit tax or an ad valorem tax,1 we show that both regions
choose unit taxes in a Nash equilibrium.
In order to interpret this result, we can point out the following two mechanisms. First, the reason for the diﬀerence in Nash equilibrium outcomes withunit and ad valorem taxation. Second, the reason for the adoption of unit tax
in the dynamic game with strategic selection of tax instruments.
First, as clearly explained in Lockwood (2004), the diﬀerence in Nash equilibrium outcomes with unit and ad valorem taxation depends on the elasticity of
tax increase on capital. The elasticity in the case where ad valorem taxation isadopted in another region is greater in absolute value than that in the case withunit tax.
The simple intuition for the greater elasticity when competing with the region
under ad valorem taxation comes from how the eﬀective tax rate in anotherregion is changed with an inflow of capital. When an ad valorem tax is adopted
1

Our model is the same as the Zodrow-Mieszkowski-Wilson (ZMW) model, except for thenumber of regions and the revenue-maximizing governments.

2

in another region, any outflow of capital by the tax increase lowers the eﬀective
unit tax in another region, given that the ad valorem tax in region 2 is fixed,
because this eﬀective rate is the fixed ad valorem tax multiplied by the marginal
product of capital that falls with an inflow of capital. The lower eﬀective tax
rate in another region induces more inflow of capital into another region, whichmeans that the capital elasticity by the tax increase becomes greater.
Second, we consider why the unit tax is selected in the dynamic game withthe strategic selection of tax instruments. As long as the eﬀective tax rate is thesame, the own reaction curve remained unchanged, given the tax instrument inanother region. This means that the reaction curve becomes the same in bothcases with unit and ad valorem taxation, given the tax instrument in anotherregion. The crucial fact is that the selection of the own tax instruments changes
the reaction of another region through the change of the tax elasticity on capitalin another region. The change of the reaction curve aﬀects the equilibrium.
When the own tax instrument is unit, the eﬀective tax rate in another region
becomes higher, which is better for the own region because its capital becomes
relatively large through the lower tax elasticity in another region. As a result,both governments adopt unit tax and obtain a larger revenue as compared withad valorem taxation.
This paper is organized as follows. The model is explained in the next section,and the main proposition is derived in Section 3. Section 4 concludes the paper.

2

The model

We consider a symmetric and two-region (regions 1 and 2) model.2 Region i is
populated by Li identical residents, and has Ki endowments of capital. Labor isimmobile, whereas capital is mobile, involving no transportation costs between
the two regions. Firms in each region are perfectly competitive and produce
an identical product. The profit of each firm is zero in equilibrium owing tofree entry. The output of region i is given by F i (Li , Ki ), which has technology
involving constant returns to scale. The symmetric assumption implies that
L1 = L2 and K1 = K2 . We set Li = 1 without loss of generality and conduct theanalysis by using the production function per capita, f i (ki ), where ki ≡ Ki /Li .
Governments of both regions impose taxes on the return on capital, and canmake a choice between a unit tax and an ad valorem tax. Their aims are to
maximize tax revenue, and they behave in a Nash manner.3 The game played2

Our model is the same as the Zodrow-Mieszkowski-Wilson (ZMW) model except for thenumber of regions and the revenue-maximizing governments.
3
This paper deals with a government that aims to maximize tax revenue, and hence does
not describe the behaviors of consumers and the uses of tax revenue. This assumption could bejustified, as Kanbur and Keen (1993), Janeba (2000), Kothenburger (2005), Keen and Ligthart3

between the two governments is constructed by two-stage decision making.4 In
the first stage, they choose the tax instruments; they choose either unit or advalorem taxation to maximize the tax revenue. Observing the tax instrument,
the regions choose the rate of tax in the second stage.
Let us denote the unit tax of region i by Ti and the ad valorem tax by ti .
Noting that the two regions face the same post-tax rate of return on capital (r)as a result of the perfect mobility of capital, the market equilibrium conditionsof capital are described by½

r = fki − Ti ,r = fki (1 − ti ),unit tax,
ad valorem tax,

k = k1 + k2 ,

(1)
(2)

where fki (≡ ∂f i /∂ki ) is the marginal product of capital and k is the total amountof capital in the economy.
The tax revenue (gi ) of region i is½

3

gi = Ti ki ,gi = ti fki ki ,unit tax,
ad valorem tax.

(3)

The competition in tax instruments and tax
levelsThis section analyzes a tax competition when there is a choice of tax instruments.
To avoid analytical complexity and derive solutions explicitly, we adopt the following quadratic function for capital, f i (ki ) = [ai − (bi ki /2)] ki, where ai > 0 andbi > 0. This function is often used in the literature on this field, for example,
Wildasin (1991), Brueckner (2004), and Itaya, Okamura and Yamaguchi (2008).
The marginal product of capital is fki = ai − bi ki . From the assumption of thesymmetric regions, a1 = a2 (≡ a) and b1 = b2 (≡ b).
We consider a two-stage game of tax competition. In stage 1 of the game,the governments of the two regions noncooperatively and simultaneously choose(2006), and among others, if we assume a Leviathan-type of government. Alternatively, thegovernments’ objective of revenue maximization can be justified when the tax-competing governments face severe revenue shortfalls, such that tax revenue is suﬃciently more important
than private good consumption. This setting, admittedly, is made only for technical reasonsand makes it possible for us to analytically solve the equilibrium when two governments impose
a diﬀerent type of tax on capital.
4
Wildasin (1991) is the pioneering research that has applied this type of two-stage game toa tax competition analysis.

4the tax instruments: either unit or ad valorem taxation. In the second stage,observing the tax instruments, they determine the levels of tax rates. Taking
tax instruments and tax levels of the two regions, firms maximize their profits.
We derive a tax instrument in the subgame perfect Nash equilibrium by using a
backwards induction.

3.1

Tax level (stage 2)

This section examines the tax rates determined by the two regions in stage 2,
given the tax instruments of the regions: (i) both regions choose unit tax; (ii)both regions choose ad valorem tax; and (iii) one region chooses unit tax, while
the other chooses ad valorem tax. We show the equilibrium tax rates in eachcase by the following:
Lemma 1 The equilibrium tax rates in Stage 2 are as follows. (i) When bothregions choose unit tax,
Ti = bk,i = 1, 2,(ii) when both regions choose ad valorem tax,
ti =

bk
,
a

i = 1, 2,(iii) when region i chooses unit tax, while region j chooses ad valorem tax,bk H
H+ , and tj =,2
42(a − bk)√
where H ≡ 6a − bk − 25b2 k2 + 36a2 − 36abk.
Ti =i, j = 1, 2 and i 6= j,

Proof. See Appendix A.
To make our solution meaningful (for capital demands and the post-tax rateof return on capital to be positive in all cases), we make the following assumption.
Assumption 2a > 3bk.
This guarantees that tax rates and tax revenues are positive in all cases (seeAppendix A).
The conversion of unit tax into the eﬀective ad valorem rate enables thecomparison of the tax rates between diﬀerent tax instruments; the best responseof unit tax is converted into ad valorem tax by using ti = Ti /fki . Let us definetmh
as the eﬀective ad valorem tax rate of region i in the Nash equilibrium wheniregion i chooses tax instrument m for m = U, A and the other region chooses tax
5instrument h for h = U, A, where U indicates the choice of unit tax and A, thatof the ad valorem tax. The eﬀective ad valorem tax rates in the Nash equilibrium
are given by
2bk
tUi U =,
i = 1, 2,(4)
2a − bkbktAA= ,i = 1, 2.
(5)ia(2bk + H) [4(a − bk) − H],i = 1, 2,(6)
tUi A =
4a [4(a − bk) − H] − 2(H + 2bk)(a − bk)
tAU
=
i

H
,
2(a − bk)i = 1, 2,(7)Note that (5) is given in Lemma 1-(ii) and that (7) is given in Lemma 1-(iii).
and tUi A . From these expressions, we
See Appendix B for the derivation of tUUiprovide the following proposition.
Proposition 1 The equilibrium taxes evaluated in terms of the eﬀective ad valorem rates in Stage 2 satisfy tAA< tUi A < tAU< tUi U .
ii
Proof. See Appendix C.
We characterize the equilibrium tax rate in Proposition 1 by using the governments’ reaction curves evaluated in terms of the eﬀective ad valorem tax rate.
Such reaction curves are obtained by using ti = Ti /fki . (i) When both regions
choose unit tax, the best response evaluated in terms of the eﬀective ad valorem
tax rate of each region is given byti =2tj (a − bk) + 2bk,
4a − bk − tj (2a − bk)i, j = 1, 2 and i 6= j,(8)(ii) when both regions choose ad valorem tax, it is given byti =(2 − tj )[bk + (a − bk)tj ],4a − bk − (3a − bk)tj

i, j = 1, 2 and i 6= j,(9)(iii) when region i chooses unit tax, while region j chooses ad valorem tax, thebest responses of regions i and j are, respectively
ti =
tj =

(2 − tj )[bk + (a − bk)tj ], and4a − bk − (3a − bk)tj

2(a − bk)ti + 2bk,
4a − bk − (2a − bk)ti
6i, j = 1, 2 and i 6= j.

(10)See Appendix D for the derivation of these equations. In Figure 1, the ad valorem
tax rate of government i, ti ∈ (0, 1), is taken on the axis. Government i’s reaction
function when both governments employ the ad valorem tax system is plottedby the curve RiAA , and the stable equilibrium is given by point A. The reaction
function when both governments employ the unit tax system can by plotted inthis figure since we can replace the unit tax by the eﬀective ad valorem tax. The
curve RiU U represents government i’s reaction function when both governments
employ the unit tax system. Point B shows the (converted) ad valorem tax
rates in the equilibrium when both governments employ unit tax. Furthermore,government i’s reaction function when it employs the ad valorem tax system
and government j employs the unit tax system is plotted by the curve RiAU .
Symmetrically, the curve RiU A represents government i’s reaction curve when it
employs unit tax but government j employs ad valorem tax. The points C and
D are the equilibria obtained in the tax competition in which one government
employs unit tax, while the other government employs ad valorem tax.
Two features of reaction curves should be mentioned in Figure 1. First,
government i’s reaction curve is unchanged when it changes its tax instrumentfrom unit to ad valorem tax, and vice versa. Second, government i’s reaction
curves when government j employs ad valorem tax is located further upwards
from its reaction curve when government j employs unit tax. Here, we give a
proof of the identity of RiAA and RiU A . When government i chooses the unit
tax and government j chooses the ad valorem tax, from (1), we have fki − Ti =(1 − tj )fkj , which yields,
1
dki
= i
j ,
dTi
fkk + (1 − tj )fkk

(11)i
≡ ∂fki /∂ki . When both governments choose ad valorem tax, from (1),where fkk
we have (1 − ti )fki = (1 − tj )fkj , which yields

fki
dki
=
j .
i
dti
(1 − ti )fkk
+ (1 − tj )fkk

(12)

When government i employs ad valorem tax, from gi = ti fki ki , the first-ordercondition for revenue maximization is expressed asdkidki+ ti fki
.
dti
dti

(13)

Substituting (12) for dki /dti in (13) yields£ ij ¤ki + ti fki = 0.
fkk + (1 − tj )fkk

(14)i
0 = fki ki + ti ki fkk

7

When government i employs the unit tax, from gi = Ti ki the first-ordercondition for revenue maximization is expressed as
0 = ki + Ti

dki
.
dTi

Substituting (11) for dki /dTi in (15) yields¤£ ijki + Ti = 0.
fkk + (1 − tj )fkk

(15)

(16)

The eﬀective unit tax rate can be described by the ad valorem tax rate as
Ti = ti fki .
From (16) and (17) we obtain£ ij ¤fkk + (1 − tj )fkk
ki + ti fki = 0,

(17)

(18)

Eq. (18) is identical to (14). Therefore, we find that RiAA is identical to RiU A .
Furthermore, since this argument can be applied for any tax instrument government j employs, we prove that RiUU is identical to RiAU .
We now turn to the consideration of the geometric positioning of the reaction
curves. The intuition for the positioning of the reaction curve follows from theconsideration of the welfare cost of increasing the tax rate in the jurisdiction.
Taking government 1, we first consider the positioning of R1U U and R1UA . Let
us assume that both governments employ unit tax. Government 1 expects someoutflow of capital to occur if it increases its unit tax, while the other keeps its
unit tax rate fixed. This capital outflow is recognized as the cost of raising unit
tax. If government 2 employs ad valorem tax, government 1 expects an evenlarger outflow as it raises its unit tax rate and thereby the cost of raising theunit tax. This is because the other jurisdiction, 2, lowers the eﬀective unit tax
rate in response to the tax increase in region 1, and this magnifies the flow ofcapital from region 1 to 2. As a description of this point, remember that theeﬀective unit tax in region 2 is equivalent to the ad valorem tax rate multipliedby the marginal product of capital. Although the unit tax rate of government 2
is given when government 1 increases its unit tax rate, the capital inflow in region
2, accompanied by the tax increase in region 1, reduces the marginal productivityof capital; hence, the eﬀective unit tax rate will be lowered. The decrease in theeﬀective unit tax rate in region 2 leads to further capital outflow from region 1,
so that the tax base is more elastic for government 1 when the other employs
the ad valorem tax system. Therefore, the curve R1U U lies on the upper side of
the curve R1U A . The same explanation will be applicable to the positioning of
R2U U and R2U A . These properties of the reaction curves give rise to the patternof equilibrium outcome in Figure 1.
8

We now return to Proposition 1, in which the inequalities tUA
< tUU
and
i
i
AA
AU
ti < ti show that the tax rates of region i when the partner region (regionj) uses ad valorem tax are lower than those when region j chooses unit tax
regardless of whether the tax instrument of region i is unit or ad valorem. The
inequalities tAA< tUA
and tAU< tUU
show that the tax rates of region i when itiiiichooses ad valorem tax are lower than those when it chooses unit tax regardlessof whether the tax instrument of region j is unit or ad valorem. These findings
can be interpreted by considering the case of i = 1 and j = 2. When region 2changes tax instruments from unit to ad valorem, the reaction curve of region 1
shifts downward and that of region 2 does not, so that the equilibrium tax rateof region 1 decreases. When region 1 changes tax instruments from unit to ad
valorem, the reaction curve of region 2 shifts upward and that of region 1 does
not, so that the equilibrium tax rate of region 2 decreases.

3.2

Tax instrument (stage 1)

This section solves the equilibrium tax instruments in Stage 1. We use gimh todenote the tax revenue of region i when region i chooses tax instrument m for
m = U, A and the other region chooses tax instrument h for h = U, A, where U
indicates the choice of unit tax and A, that of the ad valorem tax.
Lemma 2 (i) When both regions choose unit tax, the tax revenues of both region
arebk2
giU U =,
i = 1, 2,(19)2(ii) when both regions choose ad valorem tax, these are
giAA =(2a − bk)bk 2,
4ai = 1, 2,(20)(iii) when region i chooses unit tax, while region j chooses ad valorem tax, thetax revenues of regions i and j are, respectively
giU A =
gjAU =(H + 2bk)2 (a − bk), and8b[4(a − bk) − H]

H[(a − bk)(6bk − H) − 2bkH][2(4a − 3bk) − H],
8b [4(a − bk) − H]2(21)i, j = 1, 2 and i 6= j.

Proof. See Appendix E.
Table 1 shows the payoﬀ matrix in the first stage, in which gimh is given by(19)-(21). Lockwood (2004) simply compares the symmetric equilibrium between
9

giUU and giAA and shows giU U > giAA . This result is easily confirmed in Table 1
since giU U − giAA = b2 k3 /4a > 0. The characteristic feature is that in the casewhere the two regions choose diﬀerent tax instruments, the tax revenue is notequal between the two regions (g1AU 6= g2U A and g1U A 6= g2AU ).
We provide the following proposition.
Proposition 2 Choosing unit tax is a dominant strategy; two regions choose
unit tax as a policy instrument in a Nash equilibrium.
Proof. See Appendix F.
This result shows that unit tax is selected when the tax instrument can beselected before the governments compete with regard to the tax rate. Given the
result from Lockwood (2004) that the equilibrium with unit tax is superior, this
proposition suggests that this desirable equilibrium with unit tax is derived evenif we allow for endogenous choice on tax instruments.
Region 1 tries to make region 2 set a higher tax rate in order to attract
capital to region 1. To make region 2 set a high tax rate, it is an eﬀective
strategy for region 1 to employ unit tax. This is because while the employmentof ad valorem tax produces a side eﬀect that gives region 2 an incentive to lower
the tax rate, the employment of unit tax does not have such an unfavorable
eﬀect. To demonstrate this, we consider the tax setting of region 2. Whenregion 1 employs unit tax in the first stage, given region 1’s unit tax rate in thesecond stage, region 2 sets its tax rate to equate the marginal benefits of tax
increase (an increase in tax revenue) with marginal cost (a decrease in regional
output accompanied by tax outflow). When region 1 employs ad valorem tax,
region 2 sets its tax rate to equate the marginal benefit and marginal cost. In
this case, however, the marginal cost region 2 faces is greater than that it faces
when region 1 employs unit tax. Since the eﬀective (unit) tax rate of region 1 isgiven by T1 = t1 fk1 (k1 ), the tax increase in region 2 increases k1 and decreases
fk1 , which lowers the eﬀective tax rate of region 1 even if t1 is given. This willcause further capital outflow from region 2, resulting in region 2 choosing a lower
tax rate. Owing to the presence of the unfavorable side eﬀect in the case of advalorem tax, region 1 commits itself to employing unit tax.

4

Conclusion

Extending the result obtained from Lockwood (2004), we have shown in thispaper that a superior equilibrium can be achieved with the selection of the unit
tax, even in the case where the tax instrument can be selected before tax competition. In this paper, we focus on which tax instruments, unit or ad valorem,10

are selected in the capital tax competition model. This analysis can be applied
to various models with tax instruments, for example, tax exporting competition
or tariﬀ competition.
Our model, which is in the line of a basic one in the literature on tax competition, generates a tax-cutting war as each region avoids the outflow of capitalto the partner’s country. As explained below Proposition 2, the outflow eﬀectof capital owing to the imposition of tax can be repressed when the partner’s
region employs unit tax as compared to ad valorem tax. Therefore, the use ofunit tax leads to a higher eﬀective tax rate than ad valorem tax, and hence, unit
taxes are beneficial to both countries. However, in a optimal tariﬀ frameworkwith two symmetric countries, Lockwood and Wong (2000) showed that ad valorem tariﬀs lead to less tariﬀ-induced distortion in Nash equilibrium, and bothcountries choose the ad valorem tariﬀs. The choice of tax instruments, unit orad valorem forms, depends on the structure of a model.
Finally, it should also be noted that the results are derived within the context
of a model that is very general in some respects, but they obviously depend onother less general assumptions. For example, the model qualifies the governments
as Leviathans, but one could easily imagine that the governments are benevolent
and compete for mobile capital so as to maximize the utility of residents, or more
specifically, they could be moderate Leviathans so that they take both resident’s
welfare and budget size into account. Furthermore, the production functionis specified by the quadratic form. On the one hand, these assumptions clear
the way for analytical examination, on the other they leave room for additional
generalization for future research.

Appendix A: the proof of Lemma 1.
(i) Unit vs. Unit
Noting the symmetry (a1 = a2 (≡ a) and b1 = b2 (≡ b)), from a quadratic-type
production technology and (1), we obtainr = a − bki − Ti ,i = 1, 2.

(A-1)

The revenue maximization problem for government i is given by
max gi = Ti ki ,Ti

i = 1, 2.

The first-order condition gives∂gi∂ki= ki + Ti
= 0,
∂Ti
∂Ti
11

i = 1, 2.

(A-2)

From (2) and (A-1), we obtainki =bk + Tj − Ti
,
2b

(A-3)

i = 1, 2.

which yields
∂ki
1
=− ,
∂Ti
2b

(A-4)

i = 1, 2.

Using (A-2)-(A-4), we obtain the best response of region i:Ti =bk + Tj,
2i, j = 1, 2 and i 6= j,(A-5)

which yields the tax rate in the Nash equilibrium:(A-6)

T1 = T2 = bk.

From the symmetry, k1 = k2 = k/2 in the equilibrium. From this, (A-1), and(A-6), the post-tax rate of return on capital in the equilibrium is given byr=

2a − 3bk> 0,
2

where the inequality follows from the Assumption.

(ii) Ad valorem vs. ad valorem
In this case, it follows from (1) thatr = (a − bki )(1 − ti ),(A-7)

i = 1, 2.

The revenue maximization problem for government i is given by
max gi = ti (a − bki )ki ,ti

i = 1, 2.

The first-order condition gives∂gi∂ki= ki (a − bki ) + ti (a − 2bki )
= 0,
∂ti
∂ti

i = 1, 2.

(A-8)

Using (2) and (A-7), we obtaina − bki
∂ki
,
=−
∂ti
b(2 − ti − tj )i, j = 1, 2 and i 6= j.

12

(A-9)

From (2), (A-8), and (A-9), we obtain the best response of region i:ti =(2 − tj ) [bk + (a − bk)tj ],4a − bk − (3a − bk)tj

i, j = 1, 2 and i 6= j.

(A-10)

which yields the tax rate in the Nash equilibrium:t 1 = t2 =bk.
a

(A-11)

Note that ti < 1 from the Assumption.
From the symmetry, k1 = k2 = k/2 in the equilibrium. From this, (1), and(A-11), the post-tax rate of return on capital in the equilibrium is given by(2a − bk)(a − bk)> 0,2awhere the inequality follows from the Assumption.
r=(iii) Unit vs. ad valorem
Without loss of generality, we consider the case where region 1 chooses unit tax
and region 2 chooses ad valorem tax. In this case,r = a − bk1 − T1 = (a − bk2 ) (1 − t2 ).

(A-12)The revenue maximization problem in each region is given by
max g1 = T1 k1 ,T1

max g2 = t2 (a − bk2 )k2 .
t2

The first-order conditions are∂k1∂g1= k1 + T1
= 0,
∂T1
∂T1
∂k2
∂g2
= k2 (a − bk2 ) + t2 (a − 2bk2 )
= 0.
∂t2
∂t2
From (2) and (A-12), we havek1 =(A-13)(A-14)

at2 + (1 − t2 )bk − T1,b (2 − t2 )(A-15)−at2 + T1 + bk.
b(2 − t2 )(A-16)k2 =13

These show that
∂k1
∂k2
1
a − bk2
, and
.
=−
=−
∂T1
b(2 − t2 )∂t2b(2 − t2 )(A-17)

From these, (A-13), (A-14), and (A-17), we obtain the best responses:
T1 =at2 + (1 − t2 )bk,
2

2 (T1 + bk).
(4a − T1 − bk)
By using these, the tax rates in the Nash equilibrium are given by
t2 =

T1 =

bk H
+ ,
2
4

(A-18)(A-19)(A-20)

H.
(A-21)2(a − bk)
To confirm that T1 > 0 and t2 > 0, we prove that H > 0. By noting that
6a − bk > 0 and 25b2 k2 + 36a2 − 36abk > 0 from the Assumption, H > 0 if andonly if (6a − bk)2 − (25b2 k 2 + 36a2 − 36abk) > 0. It follows that¡¢
(6a − bk)2 − 25b2 k2 + 36a2 − 36abk = 24bk(a − bk),t2 =which shows that H > 0 from the Assumption. Thus, (A-20) and (A-21) showthat T1 > 0 and t2 > 0.
Next, we confirm that k1 > 0, k2 > 0 and r > 0. From (A-12), (A-15),(A-16), (A-18), and (A-19), the capital demand in each region and the post-tax
rate of return on capital in the equilibrium are given by(H + 2bk)(a − bk),
2b [4(a − bk) − H]

(A-22)(a − bk)(6bk − H) − 2bkH,
2b [4(a − bk) − H]

(A-23)

k1 =
k2 =
r=

[2(4a − 3bk) − H] [2(a − bk) − H].
4 [4(a − bk) − H]

(A-24)Let us prove that 2(a − bk) − H > 0 under the Assumption. It follows that√2(a − bk) − H = −(4a + bk) + 25b2 k2 + 36a2 − 36abk.
Note that 2(a − bk) − H > 0 if and only if
25b2 k2 + 36a2 − 36abk − (4a + bk)2 > 0.
14

We obtain
25b2 k2 + 36a2 − 36abk − (4a + bk)2 = (10a − 7bk)(2a − 3bk) + 3b2 k2 > 0,
where the inequality immediately follows from the Assumption. Thus,2(a − bk) − H > 0.

(A-25)

This and the Assumption imply that2(4a − 3bk) − H > 0, and 4(a − bk) − H > 0.

(A-26)

From the Assumption, H > 0, (A-22) and (A-24), to (A-26), we obtain k1 > 0
and r > 0.
Next, we define(a − bk)(6bk − H) − 2bkH = M1 − M2 ,p
M1 ≡ (a + bk) 25b2 k2 + 36a2 − 36abk),
M2 ≡ 5b2 k2 + 6a2 − abk.

As M1 > 0 and M2 > 0, (a − bk)(6bk − H) − 2bkH > 0 if M12 − M22 . It follows
that
M12 − M22 = 8abk (3a − bk) (2a − 3bk) + 15a2 (bk)2 > 0,
where the inequality follows from the Assumption. Thus, (a − bk)(6bk − H) −2bkH > 0. From this and (A-23), k2 > 0.

Appendix B: the derivations of (4) and (6)From fki = a − bki and ti = Ti /fki , we have
ti =

Ti
.
a − bki(B-1)

We obtain (4) from (A-6), (B-1) and ki = k/2, and (6) from (A-20), (A-22), and(B-1).

Appendix C: the proof of Proposition 1
To prove Proposition 1, we show that (i) tAA< tUi A , (ii) tUA< tAU
and (iii)
i
i
i
AU
UU
ti < ti .
15

(i) the proof of tAA< tUi Ai
From (5) and (6),
tUA
− tAA
=
i
i

bk
(2bk + H) [4(a − bk) − H]− .
4a [4(a − bk) − H] − 2(H + 2bk)(a − bk)aRearranging this yields¡¢a (O1 − O2 ) tUA− tAA= (O3 − O4 ) ,
i
i

where

(C-1)

√
O1 ≡ (3a − bk) 25b2 k2 + 36a2 − 36abk > 0,O2 ≡ abk + 10a2 − b2 k2 > 0,
¡ 2¢√
O3 ≡ 4a − abk + b2 k2 25b2 k 2 + 36a2 − 36abk > 0,
O4 ≡ 24a3 − 18a2 bk + b3 k3 + 13ab2 k2 > 0,

where the ineqaulities follows from the Assumption.
It follows that£¤
O12 − O22 = 8b2 k 2 (a − bk) 8a(a − bk)2 + 3b2 k2 (a − bk) + 2a2 bk > 0,£¤
O32 − O42 = 8b2 k 2 (a − bk) 8a(a − bk)2 + 2a2 bk + 3b3 k3 (a − bk) > 0,

where the inequalities follow from the Assumption. These show that O1 −O2 > 0
and O3 − O4 > 0. From this and (C-1), we have 0 < tUi A − tAAi .

(ii) the proof of tUi A < tAUi
From (6) and (7),
tAU
− tUA
=
i
i
This yields

where(2bk + H) [4(a − bk) − H]H−.
2(a − bk) 4a [4(a − bk) − H] − 2(H + 2bk)(a − bk)¡¢− tUi A(a − bk) (O1 − O2 ) tAUi= N1 − N2 ,
2

(C-2)

¡
¢√
N1 ≡ 5a2 − abk + b2 k 2 25b2 k2 + 36a2 − 36abk > 0,
N2 ≡ 30a3 + 15ab2 k2 − 21a2 bk + b3 k3 > 0,

where the inequalities follow from the Assumption.
It follows that

N12 − N22 = 4b2 k2 (a − bk)2 (5a − 6bk)(2a − bk) > 0,in which the inequality follows from the Assumption. This shows that N1 > N2 .
Since O1 − O2 > 0 and N1 − N2 > 0, (C-2) shows that tAU> tUi A .
i
16

(iii) the proof of tAU< tUi Ui
From (4) and (7),
tUU
− tAU
=
i
i
This yields

2bk
H
−
.
2a − bk 2(a − bk)¢¡= Q1 − Q2 ,2(a − bk) (2a − bk) tUi U − tAU
i

where

√
Q1 ≡ (2a − bk) 25b2 k 2 + 36a2 − 36abk,
Q2 ≡ −12abk + 5b2 k2 + 12a2 .

From the Assumption (a − bk) (2a − bk) > 0, Q1 > 0 and Q2 > 0.
It follows that
Q21 − Q22 = 16ab2 k2 (a − bk) > 0,in which the inequality follows from the Assumption. Therefore, tUi U > tAUi .

Appendix D: the derivations of (8), (9), and (10)From (2), (A-1), and Ti = (a − bki )ti (= fki ti ), we have(a − bki ) (1 − ti ) = [a − b (k − ki ) (1 − tj )] ,i, j = 1, 2 and i 6= j.

From this, we obtainki =ati − atj − bk + bktj,
−2b + bti + btj

i, j = 1, 2 and i 6= j.

Substituting this for ki in Ti = (a − bki )ti yields
Ti =(2a − bk)(tj − 1)ti,
ti + tj − 2

i, j = 1, 2 and i 6= j.

From this and (A-5), we obtain (8).
Expression (9) is given by (A-9).
From (2), (A-12), and T1 = (a − bk1 )t1 , we haveki =ati − atj − bk + bktj,
−2b + bti + btj

i, j = 1, 2 and i 6= j.

(D-1)

From (A-13), (A-17), and (D-1), we obtain the first equation in (10). From
(A-14), (A-17), and (D-1), we obtain the second equation in (10).

17Appendix E: the proof of Lemma 2
Expression (19) is derived from (3), (A-6) and ki = k/2, and (20) from (3), (A11), and ki = k/2. In the case of (iii), without loss of generality, we assume that
region 1 chooses unit tax and region 2 chooses ad valorem tax. We obtain thefirst equation in (21) from (3), (A-20), and (A-22) and the second equation from(3), (A-21), and (A-23).

Appendix F: the proof of Proposition 2
To prove Proposition 2, we show that giUU > giAU and giUA > giAA .

(i) The proof of giU U > giAU
From Lemma 2, we obtainbk 2 H [(a − bk) (6bk − H) − 2bkH] [2 (4a − 3bk) − H]
. (F-1)−2
8b [4 (a − bk) − H]2√
Let us define s ≡ bk/a and h ≡ H/a(= 6 − s − 25s2 − 36s + 36). Note thats < 1 from the Assumption. Using this definition, (F-1) can be rewritten as¢¡= giU U − giAU = 4s2 [4(1 − s) − h]2 − h[(1 − s)(6s − h) − 2sh][2(4 − 3s) − h]
= V1 − V2 ,giUU − giAU =where= ≡ 8b [4 (a − bk) − H]2 /a4 > 0√
V1 ≡ (6 + 5s + 2s2 + 2s3 ) 36 − 36s + 25s2 > 0,V2 ≡ (36 + 12s + 5s2 + 14s3 + 8s4 ) > 0,in which V1 > 0 follows from s < 1. It is easily shown from 1 − s > 0 that
V12 − V22 = 12s3 (3s + 4)(1 − s)2 (2 − s)2 > 0.
This shows that V1 − V2 > 0 and, hence, giU U − giAU > 0.

(ii) The proof of giU A > giAA
From Lemma 2, we obtain(H + 2bk)2 (a − bk) (2a − bk)bk2
UA
AA
−
.
gi − gi =8b[4 (a − bk) − H]
18

4a

(F-2)

Using the definition of s ≡ bk/a and h ≡ H/a, (F-2) can be rewritten as¡¢< giUA − giAA = (h + 2s)2 (1 − s) − 2 [4(1 − s) − h] (2 − s)s2
= (36 − 48s + 29s2 − 9s3 − 3s4 )√−(6 − 5s + s2 − s3 ) 36 − 36s + 25s2 ,where < ≡ 8b[4 (a − bk) − H]/a3 > 0, where the inequality follows from (A-25)
and a − bk > 0.
Let us define the following:
W1 ≡ (36 − 48s + 29s2 − 9s3 − 3s4 )
µ
¶2
24
36
2
+
> 0,
= 3s (4 + s)(1 − s) + 17 s −17
17√
W2 ≡ (6 − 5s + s2 − s3 ) 36 − 36s + 25s2£¤√
= 1 + (1 − s)(5 + s2 ) 36 − 36s + 25s2 > 0,

where the inequalities follow from 1 > s. It follows from 1 − s > 0 that
W12 − W12 = 4s3 (1 − s)[(1 − s)(11 − 4s) + 1](2 − s)2 > 0.
This shows that W1 − W1 > 0 and, hence, giU U − giAU > 0.

References
Brueckner J. K. (2004), Fiscal decentralization with distortionary taxation:
Tiebout vs. tax competition, International Tax and Public Finance 11,133-153.
Cardarelli, R., Taugourdeau, E., and Vidal, J. (2002), A repeated interactions
model of tax competition, Journal of Public Economic Theory 4, 19-38.
Itaya, J., Okamura, M., and Yamaguchi, K. (2008), Are regional asymmetries
detrimental to tax coordination in a repeated game setting?, Journal of
Public Economics 92, 2403-2411.
Janeba, E. (2000), Tax competition when governments lack commitment: Excess capacity as a countervailing threat, American Economic Review 90,
1508-1519.
Kanbur, R. and Keen, M. (1993), Jeux sans Frontieres: Tax competition andtax coordination when countries diﬀer in size, American Economic Review
83, 877-892.
19

Kawachi, K. and Ogawa, H. (2006), Further analysis on public good provision
in a repeated game setting, FinanzArchiv 62, 339-352.
Keen, M. and Ligthart, J. (2006), Incentives and Information exchange in international taxation, International Tax and Public Finance 13, 163-180.
Kothenburger, M. (2005), Leviathans, Federal transfers, and the cartelization
hypothesis, Public Choice 122, 449-465.
Lockwood, B. (2004), Competition in unit vs. ad valorem taxes, International
Tax and Public Finance 11, 763-772.
Lockwood, B and Wong, K. (2000), Specific and ad valorem tariﬀs are notequivalent in trade wars, Journal of International Economics 52, 183-195.
Wildasin, D. E. (1991), Some rudimentary ’duopolity’ theory, Regional Scienceand Urban Economics 21, 393-421.
Wilson J. D. (1999), Theories of tax competition, National Tax Journal 52,
269-304.

20

Unit tax

Ad valorem tax

g1U U , g2U U

g1U A , g2AU

Ad valorem tax g1AU , g2U A

g1AA , g2AA

Unit tax

Table 1.

21

R 1 U U (=R 1 A U )t1

R 1 A A (=R 1 U A )R 2 U U (=R 2 A U )R 2 A A (=R 2 U A )

1

B
C
D

bk
4 a  2 bk

A

1

bk
4 a  2b k

Figure 1. Reaction Curves22t2