ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E21-2

Tax Competition, Economic Growth,and Social Welfare
byToshiki Tamai

February 2021

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Tax Competition, Economic Growth, and Social Welfare
Toshiki Tamai
February 10, 2021

Abstract
Tax competition is one of the central issues of the interregional and public …nance analyses. We
address this issue by focusing on the dynamic aspects of tax competition and developing an endogenous growth model. This model incorporates knowledge spillover, perfectly mobile capital,and local public goods …nanced by capital tax. We show that equilibrium tax rate negatively associates with the degree of tax competition. Hence, escalating tax competition stimulates capital
accumulation in a dynamic model and enhances economic growth by lowering tax rates. The tax
rates and economic growth may have a hump-shaped relationship through a change in the numberof regions if the congestion e¤ect on common knowledge access is su¢ ciently large. Regardinge¢ ciency in a decentralized economy, …erce tax competition worsens social welfare even if it raises
equilibrium growth rate. Thus, undersupply of public goods due to intense competition must be
addressed, in which tax coordination is desirable. However, mild tax competition is superior totax coordination. Therefore, resolving insu¢ cient investment due to knowledge spillover should
be prioritized.
Keywords: Tax competition; Economic growth; Social welfare; Knowledge spillover
JEL classi…cation: H23; H71; H72; O41

Address: Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8601, Japan.
Tel: +81-52-789-4933. E-mail: tamai@soec.nagoya-u.ac.jp. I am grateful to Akira Kamiguchi and Hikaru Ogawa for
their comments. This work was supported by JSPS KAKENHI Grant Numbers 16K03726, 18H00865, 20H01492, theNitto Foundation, and the Zengin Foundation for Studies on Economics and Finance.

11

Introduction

Tax competition has been long examined in the literature as one of the core issues of international andregional economics and local public …nance. The consequences of tax competition are controversial andhave attracted public attention. In the outstanding study, Oates (1972) argues that tax competition
is characterized by race to the bottom, which leads to ine¢ ciently low supply of public goods. Most
literature has asserted that …scal externalities through a mobile tax base causes under-provision of
public goods (e.g., Wilson 1986; Zodrow and Mieszkowski 1986), whereas several studies have revealed
that ine¢ ciency due to other distortions may be removed.1
Recent studies have focused on the dynamic aspects of tax competition and its impact on economic growth.2 Optimal taxation theory holds that capital taxation impedes capital accumulationand therefore economic growth (e.g., Chamley 1986; Judd 1985). Numerous empirical studies foundevidence supporting such view.3 Hat…eld (2015) theoretically shows that capital tax cuts by interregional competition enhances economic growth by raising the rate of return on capital. Köthenbüerger
and Lockwood (2010) demonstrate that tax competition may have positive growth e¤ects through“extreme” tax competition in a deterministic growth model without stochastic shocks. Furthermore,they show that stochastic shocks ease tax competition and raise tax rates above a centralized level.
The relationship between tax competition and economic growth has an alternative question. Enhancing economic growth through tax cuts may increase social welfare through permanent increases
in future incomes. However, tax decrease reallocates consumption resources at the time when the incident occurs. A decrease in current consumption negatively impacts social welfare. Hence, the growthe¤ects of tax competition should be examined from the viewpoint of not only “economic growth” butalso “optimal growth” to improve social welfare. The goal of this study is to investigate whether tax
competition promotes social welfare, in addition to examining its growth e¤ects.
We address the issue by developing a multi-region endogenous growth model with regional knowledge spillover and local public goods …nanced by capital tax. Knowledge spillover is based on MarshallArrow-Romer (MAR) externalities and engine for sustainable growth (Arrow 1962; Romer 1986).4
Positive externalities in the model prevent all or nothing of capital movement and “extreme” taxcompetition suggested by Köthenbüerger and Lockwood (2010). Therefore, our framework is parallel
to standard models of static tax competition, such as Zodrow and Mieszkowski (1986), except for
endogenous determination of capital stock and economic growth through capital accumulation.
Miyazawa et al. (2019) recently present a two-region endogenous growth model similar to ourframework to examine the …scal sustainability under tax competition. Our theoretical approach isconsistent with their contribution in the sense that we treat spillover e¤ects as economic growth driver.
However, our study has di¤erent aims. We demonstrate that equilibrium tax rate negatively depends
on degree of tax competition; therefore tax competition enhances economic growth. Furthermore,
given that our model assumes multiple regions, we address the relationship between tax competition,
economic growth, and the number of regions, which Hoyt (1991) examined using a static model.
We begin by reexamining the relationship between tax competition and economic growth. Then,
we focus on dynamic equilibrium, namely, tax competition equilibrium, in our analyses to illustratethat governments compete with each other in tax rates such that they behave along with the Nash
1 See Zodrow (2010) and Baskaran and Lopes da Fonseca (2013) for a survey on recent tax competition literature.
2 Lejour and Verbon (1997) pioneered the studies on tax competition and economic growth. They develop a twocountry endogenous growth model with imperfect capital mobility and two classes of workers and capitalists. Following
Lejour and Verbon (1997), Rauscher (2005) presents an endogenous growth model with productive public input andimperfect capital mobility and where the government acts as Leviathan.
3 Kneller et al. (1999) and Lee and Gordon (2005) …nd a negative relationship between corporate tax rate andeconomic growth using multi-country panel data. Djancov et al. (2010) verify that corporate tax rates negatively impact
aggregate investment, FDI, entrepreneurial activity, and economic growth. Ferede and Dahlby (2012) show that a 1
percentage point cut in corporate tax rate corresponds to a 0.1-0.2 percentage point increase in annual growth rate using
the panel data of Canadian provinces.
4 Bronzini and Piselli (2009) estimate the long-term relationship between total factor productivity, R&D, humancapital and public infrastructure using the panel data of Italian regions. Their empirical evidence shows that R&D andneighboring regions’public infrastructure positively a¤ect regional productivity.

2

conjectures. Equilibrium growth rate is negatively associated with equilibrium capital tax rate. Furthermore, a degree of …scal externalities, that is, tax competition intensity, decreases the equilibrium
growth rate. These results verify a positive relationship between tax competition and economic growth.
The number of regions a¤ects equilibrium outcome through scale and congestion e¤ects on knowledge spillover, which is discussed in Miyazawa et al. (2019). Our quantitative analysis shows that
decreased number of regions alleviates tax competition, except for certain cases, such as strong scalee¤ect (weak congestion e¤ect) and high elasticity of capital-labor substitution. We also numericallyfound that economic growth and the number of regions may have a non-monotonic (especially humpshaped) relationship, even though tax rate decreases with the number of regions. In certain cases,
increasing competitors reduces not only the tax rate but also economy-wide productivity through congestion e¤ect. Our results imply that tax competition and economic growth have a non-monotonic
relationship based on the number of regions.
Chu and Yang (2012) examine the relative merits of decentralized and centralized …scal systems on
economic growth and social welfare in an endogenous growth model with imperfect capital mobility.
They show that certain tax competition is desirable because it has an optimal degree. Our result also
shows that tax competition is desirable in particular cases. This result is due to the combination of…scal externality and knowledge spillover, though Chu and Yang (2012)’s result is attributed to capital
mobility and government behavior.
We consider the welfare impact of tax competition by comparing tax competition equilibriumwith another equilibrium concept namely, tax coordination equilibrium.5 Tax coordination perfectly
removes the e¤ects of …scal externalities. Consumption resource is e¢ ciently allocated between private
and public goods. However, tax coordination equilibrium has lower growth rate than tax competition
equilibrium. Thus, the welfare order of the two equilibria changes depending on the degree of …scal
externalities and knowledge spillover. Hence, the welfare in the tax coordination equilibrium is above(below) that under “extreme” (mild) tax competition.
Tax competition and tax coordination equilibria yield welfare inferior to centralized equilibrium.
Tax competition eases ine¢ ciency caused by knowledge spillover, whereas competition arises from …scal
externality. Tax coordination removes …scal externalities, though adversely impacts the solution ofine¢ ciency caused by knowledge spillover. Therefore, centralization is the best way to raise growth andwelfare if the central government has policy instruments such as investment subsidy and interregionaltransfers. This result on the relationship between economic growth and decentralization is contrary to
Hat…eld (2015).
The remainder of this paper is organized as follows. Section 2 describes our theoretical framework
and provides the preliminary consideration for e¤ects of changes in capital tax rates. Section 3 presents
two concepts of equilibrium, namely, tax competition equilibrium and tax coordination equilibria, andcharacterizes these equilibria. Section 4 conducts a comparative analysis of the two equilibria andexamines their optimality. Section 5 examines the relationship between equilibrium outcomes and thenumber of regions. Finally, Section 6 concludes this paper by showing future research directions.

2

The model

Consider a dynamic competitive economy consists of n regions with continua of identical residents and…rms. The residents’population and mass of …rms in each region are respectively normalized to unity.
The time in the model is discrete and indexed by t. Time-dependent economic variables are notated,such as x(t), and we describe them as x. The notation x0 is used for the variable in the next time
period t + 1, that is, x(t + 1) x0 at period t.6 Final good production has two inputs, namely, capitaland labor, of which only capital moves across regions in an economy.
5 The present study characterizes the two types of equilibria of tax competition and coordination to address theissue. The task is motivated by Batina (2009), Wildasin (2003), and Tamai (2008), who study such issues using dynamic
models with a neoclassical production function. This paper complements the previous studies by incorporating knowledge
spillover to ensure sustainable growth and investigating the relationship between tax competition and economic growth.
6 At the period t + 1, x0 is x(t + 2).

3

2.1

Basic setup

Firms produce homogenous goods through capital and labor inputs, where MAR type spillover exists.
Each …rm controls only its capital and labor inputs for pro…t maximization. Such …rm-speci…c behaviors, particularly capital use, positively a¤ect a …rm’s productivity including that of other …rms at theeconomy-wide level. Thus, the production function in the region i is formulated as
Yi = F (Ki ; Li );where is the labor productivity. Following Romer (1986), knowledge spillover a¤ects the labor productivity. Furthermore, accessing knowledge may entail congestion costs. Thus, the labor productivity
function is formulated as7
K= v;nwhere v is the degree of congestion and
n
X
Ki K:
i=1

We assume that the production function F is continuously twice di¤erentiable, strictly concave withrespect to each input, and constant-returns-to-scale. By the assumption of the linearly homogenous
production function, we obtainyi = f (xi ) ;
where
yi

Yi
; xi
Li

ki

Ki
; f (xi )Li; ki

F (xi ; 1) :Let r and wi be the economy-wide interest rate and the wage rate of the region i, respectively.
Physical capital used in region i is taxed by the regional government at the rate of i . Each region’s
representative …rm subject to production technology maximizes its pro…t, taking r; wi , k, and as
given. In a competitive economy, pro…t maximization of the representative …rm in region i leads torwi= f 0 (xi )=[f (xi )i;xi f 0 (xi )] :

(1a)
(1b)

The regional government imposes capital tax to …nance government expenditures for public good
provision. Hence, the government’s budget equation isi ki =

i i k = gi ;(2)

where gi denotes public goods supplied to region i’s residents and
ki
:
k

i

Furthermore, we assume that the government is democratically elected by the residents as the tax ratedecision-maker. Thus, the tax rate will be determined as the level that is preferred by median residents.
The government’s decisions follow a representative resident because all residents are identical.
The preference of the representative resident is speci…ed as
1Xt[log ci + log gi ] ;

(3)

t=0

7 Lejour and Verbon (1997) assume that productivity depends on domestic capital spillover. Irman and Wigger (2006)

and Miyazawa et al. (2019) also adopt the assumption that labor productivity depends on the economy-wide capitalstock, without scale e¤ects and incorporating scale and congestion e¤ects, respectively, in a two-country endogenous
growth model. We will discuss this point in Section 5.

4

where is the discount factor ( = 1= (1 + ), > 0, > 0), ci is the private good consumption, andis the taste for public good consumption ( > 0). The resident earns labor and capital income fromsupplying his/her labor and capital he/she holds. Total income is allocated between consumption andinvestment. Then, the resident’s budget equation is
a0i = (1 + r) ai + wi

ci :

(4)

Capital and labor market clearing conditions are
n
X

Ki =

i=1

n
X

ai Li and Li = 1:i=1Capital market clearing condition holds that aggregate capital demand is equal to aggregate capital
supply (i.e., aggregate asset holding). In labor markets, regional labor demand equals regional labor
supply.

2.2Preliminary results

We now characterize the equilibrium system for given tax policies
xi and i is given by i = zxi (z n1 v ). Equation (1a) yields
i =

i (r +

i) ;

where
0
i (r +i) = zx0i (r +i . Note that the relationship between

i) =

(5)
z
< 0:
f 00 (xi )The ratio of regional capital input to average capital stock decreases with the rental cost of private
capital.
By the de…nition of i , we obtainnX(6)i = n:
i=1

The system composed of Equations (1a)–(6) is given asrwigi= r ( i;
= wi ( i ;= gi ( i ;

i) ;

(7a)
(7b)
(7c)

i; ) ;
i ; k) :Total di¤erentiation of Equations (7a)–(7c) yield (see Appendix A for the details)
@r
@ i
@wi
@ i
@gi
@ i

=
=
=

0
(r + i )Pi 0< 0;
i (r + i )@r
1+i k < 0;
@ i

1

i
i

(8a)
(8b)

i k;

(8c)

where
i @ i

i
i

@ i
@ i

i @ i

= z

> 0;

@xi
z
= 00
@ i
f (xi )
5

(9a)
1+

@r
@ i

< 0:

(9b)

In Equations (7a)–(7c),i denotes a vector of taxes, except for the tax rate in region i. Theseequations are similar to those derived by Wildasin (1989). Regional capital in our model is measured
relative to its average level.
Base on Zodrow and Mieszkowski (1986), we impose the following assumption:
Assumption 1. 0 < ii < 1.
The economy is situated on the left side of the La¤er curve. Under Assumption 1, we have
@gi
= 1
@ i

i
i

i k > 0:To clarify the relationship between tax competition, economic growth, and welfare, we initiallyconsider an economy without scale e¤ect.
Assumption 2. v = 1:
Under Assumption 2, scale and congestion e¤ects are o¤set each other. Thus, we have z = 1,
xi = i , and = k. Scale and congestion e¤ects quantitatively impact economic variables, though they
do not qualitatively a¤ect the relationship between tax competition, economic growth, and welfare.
Analyzing the relationship between equilibrium outcomes and the number of regions will be insightful.
Hoyt (1991) demonstrates that tax rate increases as the number of regions decreases using a static
model of tax competition. From the dynamic analysis perspective, the impact of a change in thenumber of regions on economic growth is important for the examination of the relationship between
tax competition and economic growth. We will set aside Assumption 2 in Section 5 and examine thee¤ects of the number of regions.

3

Equilibrium analysis

Following the conventional theory of tax competition, we focus on two comparable equilibrium concepts.
The …rst concept holds that each government chooses the tax rate in their region taking the tax rates inother regions, whereas the second one asserts that all governments change their tax rates simultaneously
at the same margin from the initially identical level. The analysis developed in this section characterizes
growth and welfare e¤ects of tax competition.

3.1

Tax competition equilibrium

The government’s objective function coincides with the present value of the residents’ utility, (3).
Each government needs the information about other regions’capital tax rates to determine its region’s
tax rate. We focus on governments that choose their tax rates through the Nash behavior as anatural extension of static tax competition models. The Nash equilibrium is de…ned as tax competition
equilibrium as follows:
De…nition 1. Tax competition equilibrium is a set of sequences f i (t);Pi (t); k(t)g satisfying Equations
1(1a)–(6) and the clearing condition for factor markets and maximizing t=0 t [log ci + log gi ] subject
to Equation (4) taking
i as given.
We can derive the tax competition equilibrium by solving the optimization problem,

V (ai ; k) = max0 flog ((1 + r ( i ;i;i )) ai + wi ( i ;i ; k)

a0i ) + log gi ( i ;i ; k) +V (a0i ; k 0 )g :i

(P1)
The …rst-order conditions are@V (ai ; k)@r ( i ; i )@wi ( i ;
=
ai +
@ i
@ i
@ i
@V (ai ; k)1
@V (a0i ; k 0 )
=
+
= 0:
@a0i
ci
@a0i
6

i ; k)

1@gi ( i ; i ; k)
+
= 0;
ci
gi
@ i

(10a)
(10b)

Equation (10a) shows that the marginal cost of a rise in the tax rate must equal its marginal bene…t.
It yields the equilibrium condition for supplying public good. Equation (10b) shows that the marginal
cost of an increase in investment must equal the marginal increase of the discounted value of utility.
Furthermore, the transversality condition is required to ensure that the utility function is bounded.
By the optimality condition, we have
@V (ai ; k)
@ai

1 @ci
@V (a0i ; k 0 ) @a0i+
ci @ai@a0i@ai@a0i 1@V (a0i ; k 0 ) @a0i
1 + r ( i;
1 + r ( i; i)+=0@ai ci
@ai
@ai
ci

=
=

(11)
i)

:

Thus, Equations (10a)–(11) yield
ci
gi

=

c0i
ci

=

@r( i ;@ ii)ai + @wi ( @i ; i i ;k)@gi ( i ;
@ i
0
i;

1+r

0

i ;k)

;

(12a)

:

i

(12b)

Equation (12a) corresponds to the marginal cost of public funds (MCPF) in the tax competition
equilibrium. It consists of distortionary e¤ects through factor price and government revenue changes.
Equation (12b) denotes the consumption growth rate, referred as the Eular equation. From the symmetricity in all aspects, we have i = 1, ai = ki = k,@r ( i ;
@ i

i)

=

1 @w ( i ; i ; k)
;
=
n
@ i

n

1
n

k < 0; and@gi ( i ; i ; k)
= (1
@ i

) k:

Equations (12a), and (12b) establish the following results (see Appendix B for the proof of Lemma
1):
Lemma 1. A unique tax competition equilibrium exists, which satis…es
c
g

=
=
=

1
1
[(1

;

) (1 + A) + (1
1 + (1) (1(1 + A) ;where

f 0 (1)2 (0; 1) ; A
f (1)) A] (1));

f (1) > 0:

Note that is the equilibrium capital share (the output elasticity of capital) and A is the totalfactor productivity in the equilibrium. Let be 1
f 00 (1) > 0. Then, is the equilibrium marginale¤ect of a rise in the capital cost on from the de…nitions. The level of or measures the degree of…scal externality. By the de…nition of and with the symmetricity of regions, we have
=

n

1
n

:

(13)

Equation (13) illustrates the one-to-one correspondence between and and between and . In the
tax competition equilibrium, the equilibrium tax rate, , in Lemma 1 and Equation (13) yield
( ; )

(1

)

2

+ [1 + (1

7

) +

]

= 0;

Figure 1. Tax competition equilibrium
where
[(1

) (1 + A) + (1

) A]

n

1
n

> 0:

Figure 1 illustrates the graph of ( ; ). A unique level of exists, which is consistent with Assumption 1. A rise in moves the graph of ( ; ) downward for < 1. Indeed, the total di¤erentiationof ( ; ) and implicit function theorem lead to@@=

1 + (1

(1
) (1

) [(1
) (1 + A) + (1) A]2 ) + [(1) (1 + A) + (1

) A]

> 0:

(14)

Therefore,
is positively associated with . If is large, the equilibrium marginal e¤ect of a rise in
the capital cost on is also large. It brings about intense response to a change in the capital tax rate.
Hence, the elasticity of with respect to increases with an increase in .
In Lemma 1, the equilibrium condition of public good provision shows that the MCPF increases
with an increase in . Fiscal externality relatively reduces public good supply to private goods. Thee¤ects of …scal externality are commonly shown in static models (e.g., Zodrow and Mieszkowski 1986).
As shown in the dynamic model of tax competition with imperfect capital mobility, a degree of capital
mobility signi…cantly determines equilibrium tax rate on capital (e.g., Lejour and Verbon 1997; Tamai2008). With perfect capital mobility, equilibrium tax rate depends on the elasticity of substitution.
Thus, we have
@
@

=

@
@

=

[(1

) (1 + A) + (1[1 + (1

@
@

@
@

) (1

< 0:

) A]
2

) ]

< 0;

(15a)
(15b)

Equation (15a) implies that …scal externality generates tax competition. The mechanism behind
this result is similar to that clari…ed by static models (e.g., Zodrow and Mieszkowski 1986; Wildasin1989). Equilibrium tax and economic growth rate illustrate the dynamic aspect of the result. Basedon Lemma 1 and Equation (15a), tax competition reduces equilibrium capital tax rate and raises
equilibrium growth rate. Hat…eld (2015) and Miyazawa et al. (2019) also show a positive relationship8between tax competition and economic growth. Their result is similar to that of our model because
their models assume a linear production function with respect to capital.8 In addition, we provide
new insights into the relationship between tax competition and economic growth. The equilibriumtax rate due to tax competition depends on the intensity of …scal externality. Then, capital tax rateis endogenously determined and negatively associated with the degree of tax competition. Therefore,
intense tax competition reduces the equilibrium tax rate on capital and enhances economic growth.
These results are summarized as the following proposition:
Proposition 1. Equilibrium growth rate in tax competition equilibrium increases with …scal externality
intensity.
Speci…ed production functions are useful for interpreting the impacts offollowing CES production function:and . We consider the1f (x) =

x

+ (1

)

A;

(16)

where denotes the substitution parameter (1). Note that x = under Assumption 2 andelasticity of substitution between capital and labor is given by 1= (1 + ). Under Equation (16) with
v = 1 and symmetricity of regions, we obtainf (1) = A; f 0 (1) = A; and=1(1 + ) (1) A:

High (low) elasticity of capital and labor substitution leads to high (low) elasticity of with respect
to because is negatively associated with . The analysis mentioned above shows that the intensityof …scal externality is linearly correlated with elasticity of substitution. For higher elasticity of capitaland labor substitution, a rise in the capital tax rate reduces capital demand. Hence, the impact on
the tax base depends on the elasticity of substitution. Higher substitution elasticity strengthens the…scal externality and lowers the equilibrium tax rate. The following result is derived from Proposition
1 with equation (16):Corollary 1. Equilibrium growth rate is positively associated with the elasticity of capital and labor
substitution if the production function is Equation (16).
The elasticity of substitution between capital and labor is essential for capital and labor share
distribution. When the elasticity of substitution is larger than unity, a decrease in the tax rate on
capital will reduce labor share. Therefore, Corollary 1 and the fundamental result on factor sharesand capital–labor substitution imply that capital tax competition declines labor share but stimulates
economic growth. The recent …ndings suggest that technological progress, which is related to thesubstitutability between capital and labor, a¤ects capital and labor share distribution (e.g., OECD2012). Corollary 1 implies that tax competition in recent decades may be one of the reasons for labor
share decline.
The following result is helpful to conduct a welfare analysis (see Appendix C for the proof of Lemma
2):
Lemma 2. The indirect utility function of (P1) is given by
V (k; k)=[log (1 + A) + log ] (1 + )
2+

log ((1(1)) (1 + A) + (1
1) A) + log + (1 + ) log k:

8 Hat…eld (2015) assumes a linear production function presented by Barro (1990). Regional tax policies in the modelcause all or none of capital ‡ow because there is no interregional spillover e¤ect. In contrast, Miyazawa et al. (2019)
assume that the production function takes the form of the Cobb-Douglas function with spillover e¤ects and without
productive government expenditure. We use the generalized production function used in Miyazawa et al. (2019),
including the CES production function.

9

Lemmas 1 and 2 yield the indirect utility function in the tax competition equilibrium for a given k.
Welfare levels vary depending on the equilibrium tax rate. Di¤erentiating V with respect to provides
dV
1
=
d
1

(1

(1 + )) (1 + A)

1) (1 + A(1) + (1

)A

R0,

Q b;

(17)

where b 2 (0; 1 + A). The second-order derivative of V with respect to becomes"#
2
1(1 + )
(1
)
d2 V
=
+
2 +
2 < 0:
2d 2
1(1) (1 + A)[(1) (1 + A) + (1) A]
These derivatives show that the welfare curve exhibits an inverted-U shape.
Evaluating Equation (17) at the equilibrium capital tax rate characterizes the relationship between
tax competition and welfare. Given that
! 0 as ! 1, we havesgn
Furthermore,
sgn

dV
d

> 0 when=! +1:= 0. Thus, we obtain

=
=

=

dV
d

(1

(1 + )(1) (1 + A)
1) (1 + A) + (1

(1

)

)A

(1

1+
) (1 + A)< 0:

The equilibrium tax rate monotonically decreases with an increase in . Therefore, a unique level ofexists, where the welfare level in tax competition equilibrium equals that when = b. Let be b a
critical value. Figure 2 illustrates the economic situation. Then, we have the following result:Lemma 3. Welfare level is increasing in
only if < b.

if and only if> b, whereas it is decreasing in

if and

Welfare level is nonlinearly associated with , which is the measure of …scal externality. In the
tax competition equilibrium, capital taxation has a distortionary e¤ect on public good supply. Then,
capital tax rate is ine¢ ciently set to a low level. The low tax rate stimulates capital accumulation
through enhancing private capital investment. Thus, the low tax rate positively impacts welfare given
the positive growth e¤ect. Smaller (larger) strengthens (weakens) the negative welfare e¤ect. The
positive (negative) welfare e¤ect dominates over the negative (positive) welfare e¤ect.

3.2

Tax coordination equilibrium

Another equilibrium concept is required for comparison to tax competition equilibrium. Suppose thatall governments simultaneously change the tax rate by the same margins initially at the same tax
rate. Then, no capital movements occur. Formally, we consider the equilibrium based on the following
de…nition:
De…nition 2. Tax coordination equilibrium is a set of sequences f i (t);Pi (t); k(t)g satisfying Equations
1(1a)–(6) and the clearing condition for factor markets and maximizing t=0 t [log ci + log gi ] subject
to Equation (4) and i = i = .
The value function corresponding to the tax coordination equilibrium is
V (ai ; k) = max0 flog (r ( ) ai + wi (k)
i ;ai

10

a0i ) + log ( k) + V (a0i ; k 0 )g :(P2)

Figure 2. Indirect utility, tax rate, and tax elasticity of capital ratio
The …rst-order conditions are@V (ai ; k)@@V (ai ; k)
@a0i

=
=

@r ai
+ = 0;@ ci@V (a0i ; k 0 )
1
+
= 0:
ci
@a0i

(18a)
(18b)

A comparison between the …rst-order conditions demonstrates that a marginal increase in the capitaltax rate has di¤erent e¤ects on the value function. In the tax coordination equilibrium, changes in
the capital tax rates do not result in capital ‡ow and income loss through capital out‡ow. The costof capital tax …nancing is only the tax burden. Hence, the cost of capital tax in the tax coordination
equilibrium is lower than that in the tax competition equilibrium. On the other hand, Equations (10b)and (18b) show that marginal increase in the asset holding in the next period has the same impact on
the value function if it has an identical functional form.
The derivative of the value function with respect to ai and Equations (18a) and (18b) lead to
1 @ci
1 @
@V (a0i ; k 0 ) @a0i
1+r
@V (ai ; k)
=
+
+
=
:
@ai
ci @ai
@ai
@a0i
@ai
ci

(19)

Equation (19) has an identical structure to that of equation (11). However, the di¤erence between
Equations (10a) and (18a) provides tax rate di¤erential. This condition a¤ects the level of marginale¤ect of ai on the value function even if the functional forms are identical.
11Equations (18a), (18b), (19) and (P2) yield the following lemma (see Appendix D for the proof of
Lemma 4):
Lemma 4. There exists a unique tax coordination equilibrium, satisfying
c?
g?

=

?

=

?

=

1;
[(1

) (1 + A) + (1
1 + (1)?
(1 + A
) :

) A]

;

Tax coordination equilibrium coincides with tax competition equilibrium if and only if = 0. Furthermore, the form of the value function in the tax coordination equilibrium is identical to that in the tax
competition equilibrium.
Fiscal externality does not exist because of no tax rate di¤erential (i.e., i = i = ). The choiceof does not a¤ect the allocation cost between private and public good, and MCPF is equal to unity.
Therefore, private and public goods are e¢ ciently allocated for a given resource. Tax coordination
equilibrium is one of the special cases in tax competition equilibria. When = 0, tax competition does
not exist. Hence, this situation equals the tax coordination equilibrium. In the other words, as ! 0,
tax competition equilibrium converges to tax coordination equilibrium.

4

Tax competition vs tax coordination

This section characterizes the properties of and di¤erences between two equilibria. First, we consider
their MCPF and equilibrium tax rates. At the equilibrium capital tax rate in the tax coordination
equilibrium, the marginal welfare e¤ect of a rise in the tax rate becomes
sgn

dV
d

=
= ?

[(1
) A + ?](1
) (1 + A

?)

< 0:

(20)

Equation (20) and Lemmas 2–4 show that ?
holds because of the concavity and continuity of theindirect utility function. For a given k, higher tax rate leads to lower MCPF. The following proposition
is derived from Lemmas 1–5:
Proposition 2. MCPF in the tax competition equilibrium is larger than that in the tax coordination
equilibrium, that is
c?
c
1= ?
:
g
g
Then, the capital tax rate in the tax competition equilibrium is smaller than that in the tax coordination
equilibrium, ?
.
Without tax coordination, a rise in a region’s capital tax rate generates capital out‡ow that decreases the tax base in one region and increases that of other regions. Hence, this …scal externalitye¤ect increases MCPF. Similar to the static models of tax competition, tax coordination improves the
allocation e¢ ciency of private and public goods for a given k. However, this e¤ect does not simply
imply that tax coordination improves welfare because of intertemporal decision-making of consumption
and investment with positive externality of knowledge spillover e¤ect.
Based on Lemma 2, larger intensity of tax competition leads to lower tax rate on capital. Equilibrium growth rate monotonically decreases with the tax rate because the net return on capital ismonotonically decreases with the tax rate. Lemmas 1 and 5 hold that tax competition equilibrium
converges to tax coordination equilibrium with less intensity of tax competition. Therefore, Lemmas
1, 2, and 5 provide the following proposition:
12

Proposition 3. The balanced growth rate in the tax competition equilibrium is higher than that in the?
tax coordination equilibrium, that is:
Capital taxation has a distortionary e¤ect on private investment because the public good in thismodel is unproductive. It negatively works on capital accumulation. Hence, the equilibrium growthrate decreases with an increase in the capital tax rate. Proposition 2 illustrates that the capital tax
rate in the tax coordination equilibrium dominates over that in the tax competition equilibrium. Thesee¤ects show that the equilibrium growth rate in the tax competition equilibrium is larger than that in
the tax coordination equilibrium. Proposition 2 holds that tax coordination improves the allocatione¢ ciency between private and public goods, which positively impacts welfare. In contrast, Proposition
3 shows that tax coordination reduces equilibrium growth rate because it increases equilibrium capitaltax rate. A decrease in the equilibrium growth rate negatively a¤ects welfare level, and tax coordination
does not always improve welfare.
Lemmas 1–5 lead to the following result:
Proposition 4. Let be e as the value of , satisfying V = V ? . Then, there exists a unique value of e
and V > V ? holds for 0 < < e.

Proposition 4 enables us to consider the e¢ ciency of decentralized economies. As mentioned above,a decentralized economy have two sources of the ine¢ ciency, namely, …scal externality and knowledge
spillover e¤ects. An e¤ective interregional cooperation of tax policies is required to remove …scal
externality e¤ect. However, such cooperative tax policies raise the capital tax rate and reduce theequilibrium growth rate. When knowledge spillover exists, the growth rate in the decentralized economy
is lower than that without a positive externality. Hence, the cooperative tax policies do not always
improve economic e¢ ciency.
To analyze the …rst best outcome, the social planner’s problem should be de…ned.
De…nition 3.
Symmetric centralized equilibrium is a set of sequences f (t); k(t)g satisfying
P1
maximizing t=0 t [log c + log g] subject to Equation (4).

= 1 and

The value function of the optimization problem mentioned above is
W (k) = max flog ((1 + A) kk;kk 0 ) + log ( k) + W (k 0 )g :(P3)

Solving the optimization problem (P3) leads to the following results (see Appendix E for the proof of
Lemma 6):
Lemma 5. In centralized equilibrium, optimal condition for public good provision, capital tax rate,and economic growth rate are given by
co
go

=

o

=

o

=

1;
(1 + A) (11+(1 + A) :

)

o

o

;

Furthermore, the value function isWo =log ((1 + A) (1)1) + log+

1+
1

log k:The social planner maximizes social welfare without any distortionary e¤ects, though the tax
instrument takes the form of capital tax. Socially optimal MCPF is equal to unity because one unit ofhomogenous goods can be transformed into one unit of private and public goods. Lemmas 4 and 5 showthat MCPF in the tax coordination equilibrium is equal to that in the socially optimal equilibrium.
13

However, this condition does not imply that the tax rate in the tax coordination equilibrium is thesocially optimal rate. Indeed, we haveo?

(1 + ) (1) A + (1 + A) (1(1 + ) [1 + (1
) ]

=

)

< 0:

The tax di¤erence between the tax competition equilibrium and socially optimal equilibrium canbe positive or negative: o R
derived from Lemmas 1 and 5. The magnitude of relationship varies
depending on production and utility function parameters. In particular, the elasticity of capital ratiowith respect to the tax rate is important to determine the di¤erence. When ! 1,
becomes zero.
Then, o >holds. In contrast, o <when ! 0. Hence, there exists , which satis…es o = .
However, public good supply is not socially optimal, though the tax rate is identical to the optimallevel.
In the socially optimal equilibrium, distortion does not exist and the rate of return on capital ishigher than those with distortionary taxes and positive externality. Therefore, the growth rate in
the socially optimal equilibrium is higher than those in the other two equilibria. Thus, we obtain?
<< o from Lemmas 1, 4, and 5. Welfare di¤erence depends on tax and growth rates. The
welfare in the socially optimal equilibrium is naturally the highest, and a high growth rate is essential
to raising the welfare level.
As a result of the analysis mentioned above, we establish the following proposition on the di¤erence
between three equilibria regarding tax rates, growth rates, and welfare levels (see Appendix F for the
proof of Proposition 4):
Proposition 5. Suppose that the regions are symmetric in all aspects. Then, (i) o << o < ? for > . (ii) ? << o holds. (iii) V ? < V < W o for< while?
o
V < V < W for > e. The welfare di¤ erence between W o and V is decreasing in .

< ? for< e while

Proposition 5 holds that tax competition and tax coordination equilibria are inferior to centralized
equilibrium in the welfare level. The result implies that tax competition is better (worse) than tax
coordination as the second best policy if competition is mild (intense). Furthermore, Proposition 5
indicates the importance of the spillover e¤ect to welfare. Parameter (1) stands for the degree of
spillover e¤ect. When = 1, the spillover e¤ect is vanished. Then, the negative welfare e¤ect of tax
competition is minimized. In contrast, smaller strengthens negative welfare e¤ect.

5

Number of regions and equilibrium outcomesThis section analyzes the relationship between equilibrium outcomes and number of regions. We
focus on tax competition equilibrium without Assumption 2 and several endogenous variables, suchas the elasticity of with respect to tax rate, equilibrium tax rate, and economic growth rate. In thesymmetric tax competition equilibrium ( = 1), we have x = z 1 = nv 1 . Thus, the equilibrium valueof satis…es( ; n)(1) 2 + [1 + (1
) + ]
= 0;
where

= (n),[(1sk
B) (1 + sk B) + (1sk ) B]

n

1
nv

> 0;

f 0 (z)= sk (n) ;zf (z)
zf (z) = B (n) :If v = 1, then we obtain sk = and B = A. Hence, = and =and B are the capital share and output-capital ratio, respectively.
14

hold for v = 1. Note that sk

Total di¤erentiation ofyieldsd=dn
1 + (112 ) (1

d
:
dn

) +

(21)

The e¤ect of a change in n on depends on d =dn, which is composed of the e¤ects on (c + g)=k and. The sign of d =dn is undetermined without any assumptions. Depending on and v, d =dn could
be positive or negative under Equation (16). Based on the conventional view of tax competition, is
expected to positively associate with n. If v = 1, then we have
dB
d
dsk
=
=
= 0:
dn
dn
dn

(22)

Hence, Equation (21) becomes9(12 ) (1d=dn[1 + (1

)
) +

] (n

1) n

> 0 for v = 1:

Equation (13) and equilibrium growth rate are rewritten as
=
=

n

1
nv
(1 + sk B

;

(23)
) :

Using the above equations, we obtain
n d
dn
d
dn

=
=

nd
n
nd
+v
;
dn n 1
dn
dsk
d
dB
B
+ sk
:
dn
dn
dn

(24)
(25)

The e¤ect of a change in n on
depends on that on . Hence, the shape of production function
and the degree of congestion are the key determinants of the sign of Equation (24). The e¤ect of achange in n on
coincides with that on post-tax interest rate. The number of regions a¤ects economic
growth rate through changes in the capital share, output-capital ratio, and tax rate. These e¤ects ofa change in n are ambiguous. Thus, we rely on a numerical analysis to clarify the signs of Equations(21), (24), and (25). However, we can obtain analytical results in certain case. When v = 1, Equations(21)–(25) lead to
d
dn
d
dn

=
=

1
nd
+
(n 1) n
dn
d
> 0:
dn

n

< 0;

The analyses developed above provide the following results:
Proposition 6. Suppose that Assumption 2 holds. Regardless of the elasticity of capital and labor
substitution, an increase in the number of regions raises the elasticity of capital ratio with respect tocapital tax rate, reduces the equilibrium capital tax rate, and increases the equilibrium growth rate intax competition equilibrium.
As Proposition 6 is our analytical benchmark, we now examine Equations (21), (24), and (25)through numerical computations. Suppose that Equation (16) is the production function. We set( ; ; A) = (0:3; 0:01; 2). Then, approximates 0:99. We consider three cases in the CES production
function ( = 0:5; 0; 1) and two cases in the degree of congestion (v = 0:8; 1:2).
15

(a) v = 0:5(b) v = 2
Figure 3. Number of regions and economic variables when=

0:5

In case 1 ( = 0:5), the elasticity of capital-labor substitution equals 2. Three panels (a) in
Figure 3 illustrate the e¤ects of a change in n on , , and if scale e¤ect dominates the congestione¤ect. For v = 0:8, sk , and decrease with n, whereas B and r increase with n through the scalee¤ect. Given that the number of regions positively impacts , an increase in n raises (see Equation(21)). From equation (23), the equilibrium tax rate follows a motion opposite to for a given n. An
increase in n a¤ects through a change in . and its coe¢ cient in Equation (23) oppositely respondto an increase in n. The e¤ects on , , and its coe¢ cient in Equation (23) generate a non-monotonic
relationship between and n. Regarding -n relationship, bottom panel (a) of Figure 3 shows that theequilibrium growth rate monotonically increases with the number of regions because the scale e¤ect
dominates the congestion and tax-distortion e¤ects.
Three panels (b) in Figure 3 show the impacts of a change in n on , , and when the congestione¤ect dominates the scale e¤ect. monotonically increases with n, whereas monotonically decreases
with n. If v = 1:2, all of the e¤ects on sk , r, and are opposite to those of v = 0:8. They generatea non-monotonic e¤ect of n on . monotonically decreases with n. Contrary to panel (a), panel (b)shows that there exists a hump-shaped relationship between equilibrium growth rate and number ofregions. For a small n, an increase in n has a strong marginal impact on economic growth through adecrease in the tax rate. However, the congestion e¤ect overweighs the positive growth e¤ect of thedecreased tax rate for a large n.
In case 2 ( = 0), the elasticity of capital-labor substitution is equal to unity. Hence, the production
function takes the form of the Cobb-Douglas function. The curves in Figure 4 are similar to thosein case 1. The intuitions are identical to those of case 1, except for upper panel of (b). However, its
quantitative e¤ects di¤er from those in case 1. For v = 0:8, is smaller than that in case 1, whereas
and are larger than those in case 1. For v = 1:2, and are larger than those in case 1, though is
9 Note that the numerator and denominator are positive based on Assumption 1.

16

(a) v = 0:8(b) v = 1:2
Figure 4. Number of regions and economic variables when=0

smaller than that in case 1. These di¤erences are due to the presence of the e¤ects on capital share.
In case 3 ( = 1), the elasticity of capital-labor substitution is 0:5. For v = 0:8, the e¤ect of achange in n on sk is opposite to that of case 1, though the impacts on B, , and r have the samesigns of those in case 1. If n is small, then an increase in n positively impacts through an increasein capital income. On the other hand, the positive e¤ect on capital income is overweighed by othernegative e¤ects on , such as a decrease in . Hence, a non-monotonic relationship between and n
exists. Economic growth rate monotonically increases with n because the qualitative impact of nonmonotonicity is small, though r and oppositely respond to a change in n. For v = 1:2, and curves
become similar to those of case 1. However, the relationship between and n di¤ers from that of case
1 because the e¤ect of n on sk is opposite to that of case 1.
The results of the three cases are summarized below:
Remark 1. (a) When the scale e¤ ect dominates the congestion e¤ ect, the equilibrium tax rate decreases
with the number of regions for a small n though it increases with n for a large n. (b) When thecongestion e¤ ect dominates the scale e¤ ect, the equilibrium tax rate monotonically decreases with thenumber of regions.
Remark 2. (a) When the scale e¤ ect dominates the congestion e¤ ect, economic growth rate monotonically increases with the number of regions. (b) When the congestion e¤ ect dominates the scale e¤ ect,
there exists a hump-shaped relationship between economic growth rate and number of regions.
Hoyt (1991) theoretically showed that a decrease in the number of regions increases the tax rate andpublic service level using a static model of tax competition. Several empirical studies also found thatthe number of regions negatively impacts local tax rates (e.g., Breuillé and Zanaj 2013). Regarding -nrelationship, Panels (b) derive the same result as Hoyt (1991). Panels (a) di¤er from the pioneering

17(a) v = 0:8(b) v = 1:2
Figure 5. The number of regions and economic variables when=1

study. However, for a small n, panels (a) also yield a negative relationship between tax rate and numberof regions. Hence, these results are consistent with the empirical evidence under certain conditions.
Our main …nding from the numerical simulation shows that economic growth and number of regions
have a non-monotonic relationship regardless of elasticity of substitution. Previous studies demonstratethat an increase in n decreases capital tax rate (e.g., Hoyt 1991), and that a decreased tax rate enhances
economic growth as growth rate is negatively associated with the tax rate (e.g., Miyazawa et al. 2019).
This view holds if v = 1 (Proposition 6). However, if v 6= 1, the number of regions is not monotonicallyassociated with the degree of tax competition and equilibrium tax rate. In certain cases, scale andcongestion e¤ects complicate -n relationship. Even if tax rate monotonically decreases with numberof regions, the congestion e¤ect overweighs the positive growth e¤ect of decreased tax rate for a large
n. Then, increasing competitors cannot positively impact economic growth.

6Conclusion

This paper examined the relationship between tax competition, economic growth, and social welfare in
an endogenous growth model with local public goods and freely mobile capital. Following Arrow (1962)
and Romer (1986), knowledge spillover engines for sustainable growth. Jurisdictional governments tax
capital that moves among regions for local public good supply. MAR and …scal externalities occur
because of knowledge spillover and mobile tax base. These two externalities have di¤erent e¤ects on
economic growth and social welfare. Knowledge spillover causes less capital investment and negatively
impacts economic growth, whereas …scal externality brings about capital tax rate competition ina decentralized economy and positively a¤ects economic growth by lowering the tax rates. Fiscal
externality ine¢ ciently lowers capital tax rate, resulting in the underprovision of local public goods.
Ine¢ cient resource allocation worsens social welfare even if higher growth rate enhances welfare. Hence,18

the welfare e¤ect of tax competition could be positive or negative.
Tax competition intensity is positively associated with the elasticity of capital-labor substitution.
Strong …scal externality with high elasticity of substitution intensi…es capital tax rate competition.
On the one hand, intense tax competition increases investment by reducing the tax rate and boosts
economic growth. On the other hand, intense tax competition brings about extremely low supply of
local public goods. The negative welfare e¤ect of underprovision of public goods dominates the positive
welfare e¤ect of economic growth. Intense tax competition is undesirable; thus tax coordination
improves social welfare. In contrast, mild tax competition enhances economic growth and increasessocial welfare because the positive welfare e¤ect of tax competition through accelerated economic
growth is larger than its negative welfare e¤ect through undersupply of public goods. These results
imply that tax competition is preferable depending on the degree of …scal externality (i.e., the elasticity
of substitution). However, tax competition and coordination are the second best policies because tax
competition and coordination equilibria are inferior to social optimum in the welfare level.
The number of regions a¤ects equilibrium outcomes through scale and congestion e¤ects on knowledge spillover as discussed in Miyazawa et al. (2019). Our quantitative analysis shows that a decreased
number of regions alleviate tax competition, except for certain cases, such as strong scale e¤ect (equivalently, weak congestion e¤ect) and high elasticity of capital-labor substitution. We also numericallyfound that economic growth and the number of regions may have a non-monotonic (especially humpshaped) relationship though tax rates decrease with the number of regions. In certain cases, increasing
competitors reduces not only tax rate but also economy-wide productivity of knowledge spillover
through the congestion e¤ect. Our results imply that tax competition and economic growth have anon-monotonic relationship in the aspect of the number of regions.
Finally, we provide future direction of this research. Our study reveals that tax coordination isdesirable under intense tax competition, suggesting that centralization improves social welfare in sucha case. We assumed only an identical tax rate common among all regions in the tax coordination that
corresponds to centralization. However, as shown in the previous studies, the central government plays
an active role in improving economic ine¢ ciency through instruments such as interregional transfer
and subsidy. Incorporating the role and policy instruments of the central government can explain
issues of centralization and decentralization in local public …nance. Our main results are based on theanalyses of symmetric equilibrium, though it shed light on the dynamic aspects of tax competition from
economic growth and social welfare perspectives. We need to model asymmetric regions to illustrate
the strategic relationship among jurisdictional governments. This study provides an analytical basis
for these extensions.

19Appendix
A. Derivations of Equations (8a)–(9b)Total di¤erentiation of Equations (5) and (6) provide
0

=

n
X

x0i (r +

i)

i=1

,

@r
=
@ i

@ (r + i )@r X 0=x (r +@ i@ i i j0j ) + xi (r +

i)

1+

@r
@ i

0

x0 (r + i )Pn i 0=i=1 xi (r + i )(r + i )2 ( 1; 0):0i=1 i (r + i )Pn i

Partial di¤erentiation of Equation (5) with respect to
@ i
@xi
=z
= x0i (r +
@ i
@ i

i)

(A1)

and Equation (A1) yield
1+

@r
@ i

z < 0:(A2)

Using Equations (1b), (2), and (A2), we obtain
@wi
@ i
@gi
@ i

@xi
=
@ i

1+

@r
@ i

@ i
k= 1
@ i

i
i

i k:

=

xi f 00 (xi )

=

i+

i

i k < 0;

(A3)
(A4)

B. Proof of Lemma 1
Applying Equations (1a) and (A1)–(A4) with symmetricity of regions to Equations (12a) and (12b)
leads to
c
g
c0
c

c
1
=
;
k
1

=
=

(1 + A

(A5)

) :

(A6)

Summing up both sides of Equation (4) and inserting Equations (1a) and (1b) into it, we have
k0
=1+A
k

c
:
k

(A7)

Equations (A6) and (A7) provide
c
k

=
,

1+A

(1 + A0)= (1) (1 + A) + (1

)A

(A8)

0

c
k
= :
c
k

Using Equations (A5) and (A8), we obtain=[(1) (1 + A) + (1
1 + (1) (1Equation (A9) becomes the quadratic equation
the quadratic polynomial is[1 + (1

) +

2

]

4 (1

)

) A] (1
)

)

:

(A9)

( ; ) = 0 using Equation (13). The discriminant of= 1 + 2 (1
20

) +

+ [(1

)

2

] > 0:

( ; ) = 0 have two real positive roots. Furthermore, the calculation shows(1; ) =

(1

) + 1 + (1) += 1 > 0:

Therefore, only one root of ( ; ) = 0 (smaller one) is unique solution to ( ; ) = 0, which isconsistent with 0 < < 1 (Assumption 1). The unique value of determines the unique value offrom Equation (13). These results prove Lemma 1.

C. Proof of Lemma 3
Suppose that the value function takes the form ofV (ai ; k) =0+

1 log k +2 log (ai +

k) ;

(A10)

where i and are undetermined coe¢ cients (i = 0; 1; 2). Then, the partial derivative of Equation(A10) and the symmetricity of regions provide@V (ai ; k)@V (ai ; k)
2
2
= 1+
)
:
= 1+
@k
k
ai + k
@k
k
(1
+
)k
ai =k

(A11)

The following equation is derived from (P1):@V (ai ; k)
1 @ci
1 @gi
@V (a0i ; k 0 ) @k 0
=
+
+
:
@k
ci @k
gi @k
@k 0
@k

(A12)

Using the symmetricity of regions and Equation (A10), Equation (10b) becomes@V (a0i ; k 0 )
=
@a0i

1
=
ci

2

a0i + k 0

:

(A13)

Di¤erentiating Equation (A13) leads to
1 @ci
=
ci @k

@k 0
a0i +

k 0 @k)k @ci=
ci @k ai =k
1+

:

(A14)

From Equations (A11)–(A14), we obtain
1+

2

=

1+

+ +

1+

1+

2

1+

:

A comparison between the coe¢ cients of this equation yields
1 =

;

1

2 =

1
1

:

Evaluating Equation (A13) at ai = k, we have
@V (a0i ; k 0 )
1
2
=
= :
0
@a0i
(1
+
)
k
c
ai =k

(A15)

Inserting Equation (A8) into Equation (A15) leads to
2k(1 + ) k 0

=
)

1
(1

) (1 + A
2

1+=) + (1)A
1+ A) (1 + A) + (1

(1
21

)A

:

Therefore, we have
1+

=

(1

) (1 + A) + (1(1) (1 + A))A:

Equations (P1) and (A10) give0+( 1+

2 ) log k +2 log (1 +log ((1 + A) kk k 0 ) + (log + log k)+ [ 0 + ( 1 + 2 ) log k 0 + 2 log (1 + )]= log ((1) (1 + A) + (1) A) + [1 + + ( 1 + 2 ) ] log k+ log + f 0 + ( 1 + 2 ) [log (1 + A) + log ] + 2 log (1 + )g :)=Comparing the coe¢ cients, we obtain( 1+2 ) log k = [1 +

+( 1+

2)

] log k , 1 + = (1)( 1 +

2) ;

and
(1

)

Hence,

0=log ((1) (1 + A) + (1) A) + log+ ( 1 + 2 ) [log (1 + A) + log ] (1) 2 log (1 + )[log (1 + A) + log ] (1 + )=log ++ log (1) + log (1 + A
1):0 is determined as

0 =

1
1

log +

[log (1 + A) + log ] (1 + )+ log (11

) + log (1 + A) :Inserting all coe¢ cients into Equation (A10), we arrive at
V

=
=

0+

1
1

1 log k +2 [log (1 +) + log k][log (1 + A) + log ] (1 + )log +1+ log ((1) (1 + A) + (1) A) + (1 + ) log k :D. Proof of Lemma 4
Using the symmetricity of regions and Equation (2), Equation (18a) becomes
c
c
=1, =
g
k

:

(A16)

From Equations (1a), (18b), and (19), Equation (A6) holds in the tax coordination equilibrium. The
resource constraint is identical to Equation (A7). Using Equations (A6), (A7), and (A16), we have
(1 + A
Solving this equation with respect to
?

=

)

=1+A

1+

:

leads to[(1

) (1 + A) + (1
1 + (1)) A]:

Equations (A5) and (A16) are coincided each other if and only if = 0. Furthermore, the functional
form of the value function is common between these two equilibria because the tax rate is constantover time.

22

E. Proof of Lemma 5
The …rst-order conditions are@W (k)@@W (k)@k 0

k(1 + A) k
1(1 + A) k

=
=

k

k0

k

k0

+

= 0;

(A17)

+

@W (k 0 )
= 0:
@k 0

(A18)

Using Equation (2) and the resource constraint, Equation (A17) becomes
c
=
g

c
= 1:
k

(A19)

We have@W (k)
@k

=
=

1 @c
1 @g
@W (k 0 ) @k 0
+
+
c @k
g @k
@k 0 @k
1
@g
@k 0
@g
@k 0
1+A
1+A
+
+
+
=
:
c
@k
@k
@k
@k
c

(A20)

Using Equations (A18), (A20), and the resource constraint, we obtainc0= (1 + A) :
c

(A21)

The resource constraint becomes
k0
=1+A
k

c
=1+A
k

1+

(A22)

Equations (A21) and (A22) lead too=(1 + A) (1
1+

)

()

c0
k0
= :
c
k

We consider that the value function takes the form of
W (k) =
Equations (P3), (A22), and (A23) with
0 +(1

)

1 log k = log ((1 + A) (1o0+

1 log k:

(A23)

yield

)

)+(1 + ) log k+ log + f 0 +1 [log (1 + A) + log

]g :
(A24)

A comparison between the coe¢ cients of Equation (A24) provides
=

1

=

0

1+
;
1
log ((1 + A) (1))) + log + (1+[log (1 + A) + log ]1

1:Hence, we arrive at
Wo==log ((1 + A) (1)o) + log1(1 + ) [log (1 + A) + log (1+(1 + ) [log (1 + A) + log ]
(1

2

)

o

+

)] + log
1
:

23

1+
1

log k
(1 + ) log (1 + ) + (1 + ) log k

F. Proof of Proposition 5
The partial derivatives of the value functions yieldd (W o
d

V )

=

(1

[(1
) (1

) (1 + A) (1 + A

24) + (1 + ) (1) [(1) (1 + A) A] A) + (1) A]< 0:

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