ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E19-3

Unemployment, Tax Competition, and Tax
Transfer PolicybyToshiki Tamai
Gareth MylesMay 2019

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Unemployment, Tax Competition, and Tax Transfer Policy
Toshiki Tamaiy and Gareth Mylesz
May 31, 2019

Abstract
The reduction of capital tax rates witnessed over the past wo decades has been motivated by thewish to boost investment and reduce unemployment. Previous models of tax competition have
explored the consequences for capital allocation in great detail, but have mostly been silent on theemployment e¤ects due to the assumption of a perfect labor market. To address the impact of tax
competition on employment and public good provision this paper reconsiders the analysis in thepresence of a labor market imperfection that generates unemployment. We incorporate a wage
rigidity and intergovernmental transfers …nanced by the labor income tax into the standard tax
competition model to explore the outcome of federal-state policy interaction. In this setting there
is an employment externality of taxation in addition to the standard …scal externality. The key
factors in determining the e¢ ciency are the substitutability/complementary between capital andlabor in production (which determines the magnitude of the employment externality) and the costof the e¢ cient level of public good supply relative to the ability of the labor income to generate
revenue. If there is complementarity the equilibrium is Pareto e¢ cient when the labor income tax
can …nance the public good. When the labor income tax is not su¢ cient for …nancing, there canonly be e¢ ciency if the aggregate externality is negative - but this outcome is only one of multiple
equilibria. We also that federal government leadership generally improves (and does not worsen)social welfare in comparison to the equilibrium policies under simultaneous policy choice.
Keywords: H21; H71; J64
JEL classi…cation: Tax competition; Unemployment; Transfer policy

This work was supported by JSPS KAKENHI Grant Number 16K03726 and 16KK0077.
y 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.
z Address: School of Economics, The University of Adelaide, Adelaide, South Australia 5005, Australia. Tel: +61-88313-4768. E-mail: gareth.myles@adelaide.edu.au.

1

1

Introduction

International and interregional tax competition have pressured regional and central governments into
decreasing taxes over the past two decades. In particular, the statutory corporate income tax rate
has decreased signi…cantly in many countries. According to OECD (2019), the average combined(central and sub-central government) statutory tax rate for all covered jurisdictions was 21.4% in2018, compared to 28.6% in 2000. The motivation for the tax reforms in OECD countries was thepromotion of investment and employment via reductions in the corporate income tax rate, and many of
the countries that lowered the tax rate did so via a series of cuts over several years (OECD 2017). The
reforms are supported by empirical evidence which shows that the corporate income tax rate a¤ects
the employment rate (e.g. Feld and Kirchgassner 2002; Feldmann 2011).
The conventional theory of tax competition assumes perfect capital and labor markets, and predicts
a race to the bottom. When capital is fully mobile among regions, an increase in the capital tax rateby the government of one region causes capital ‡ight from that region to other regions. This expands
the tax bases of other governments and constitutes the …scal externality that drives tax competition.
As a consequence of the …scal externality, all governments have an incentive to levy an ine¢ ciently low
capital tax rate which results in underprovision of public goods (Wilson 1986; Zodrow and Mieszkowski1986).1 Hence, a policy such as a Pigouvian tax/subsidy at the federal level is necessary to correct for
the …scal externality (Wildasin 1989; DePeter and Myers 1994).
The standard description of the tax competition process does not address the employment consequences. This is because previous models make the assumption of a perfect labor market which
precludes them from exploring whether the reforms can have an employment-boosting e¤ect. To investigate this issue requires the incorporation of a labor market imperfection into the analysis in order tocapture the stated motivation behind observed policy choices. An imperfect labor market introduces
a new dimension into the analysis because the allocation of mobile capital determined by tax rates
in‡uences employment levels. The direction in which employment changes is determined by the complementarity/substitutability between capital and labor: when they are substitutes (complements) aloss of mobile capital raises (reduces) employment. Hence, if there is an imperfect labor market, a state
government can a¤ect the employment level through choice of the capital tax rate. If one state loses
mobile capital then the other gains, so an imperfect labor market creates an employment externality ofcapital taxation (Ogawa et al. 2006a).2 This employment externality is additional to the standard …scal externality. Unlike the …scal externality which is always positive (a tax increase boosts the revenueof other jurisdictions due to capital ‡ight), the employment externality can be positive or negative.
It is negative if capital is a substitute for labor so more capital reduces employment and positive if
capital and labor are complements. This second externality is an additional source of ine¢ ciency intax setting.
Two recent studies have analyzed the e¢ ciency of public good/public input provision in the presenceof labor market imperfections.3 Gillet and Pauser (2018) show that public inputs …nanced entirely by
lump-sum tax are ine¢ ciently overprovided because of the employment and …scal externalities. They
also demonstrate that an additional tax (subsidy) on capital will improve e¢ ciency through its strategicin‡uence on the interest rate. Kikuchi and Tamai (2019) investigate the e¤ects of an equalizing transfer
on the e¢ ciency of public good supply. They focus on some representative equalization transfers (e.g.,
tax base equalization, revenue equalization) without treating the federal government as an explicit
decision maker. In contrast, this paper explores the interaction between federal and state policies,
with the federal government as a strategic actor. Within this framework we clarify the e¤ectiveness ofintergovernmental transfers at resolving the ine¢ ciency resulting from tax competition. Furthermore,
we consider the nature of interaction between federal and state governments (e.g. Boadway and Keen
1 Numerous empirical studies support that the governments compete each other in the tax rates (For instance, Devereux et al.2008).
2 They refer it as the unemployment-exporting externality (Ogawa et al. 2006a, b;Sato 2009). Following Gillet andPauser (2018), we call it employment externality.
3 Some studies also examine the relation between unemployment and tax competition (e.g., Aronsson and Wehke 2008;
Eichner and Upmann 2012; Exbrayat et al. 2012).

2

1996)4 since the timing of policy decisions a¤ects the equilibrium outcome. Hayashi and Boadway(2001) provide empirical support for the federal government acting as a Stackelberg leader, and federal
government leadership is commonly assumed in the existing literature (Keen and Konrad 2013).5
Therefore, it is insightful to contrast the outcome with simultaneous policy choice to the outcome
with centralized leadership when considering the e¤ects of interregional transfers with a labor market
imperfection.
We assume that there are two state governments and a federal government in the economy. The
state governments posses the right to levy a tax on capital and have the responsibility for providing apublic good. The federal government levies a tax on the labor incomes of all residents in the country
and distributes the tax revenue between the two states as an intergovernmental transfer. Capital isperfectly mobile between states, while labor is immobile. A labor market imperfection is introducedby assuming that the wage rate is …xed above the competitive level which creates underemployment inboth states. The labor income tax has no direct distortionary e¤ect since we assume labor is suppliedinelastically. The mobility of capital implies that any policy instrument that a¤ects the return tocapital will have a distortionary e¤ect due to capital relocation between states. The production sideof our model is similar to the setting developed by Köthenbüerger (2008), except for the assumptions
about labor supply and the …xed wage: Köthenbüerger assumes elastic supply of divisible labor whereas
we assume inelastic supply. However,in our model capital taxes a¤ect the amount of employment asthey do in Köthenbüerger (2008) because a change in the tax rate in‡uences the factor demand forlabor via a change in the factor price.
The main results of our analysis are as follows: First, we derive the equilibrium policies undersimultaneous policy choice and characterize how the tax rates and public goods supply are determinedby the …scal and employment externalities. If the employment externality is positive, the equilibriumtransfer to the states (and the labor income tax) is positive and enhances social welfare. It is noteworthy
that the equilibrium policy is Pareto e¢ cient when the total cost of supplying the public good canbe covered by the labor income tax alone. However, a non-negative transfer policy does not improvesocial welfare in all cases. This is because a positive transfer gives state governments an incentive todecrease their capital tax rates. Hence, it worsens the equilibrium ine¢ ciency when there is a negative
employment externality since this decreases the capital tax rates chosen by the state governments.
Second, we analyze the equilibrium policies under federal government leadership. In this case, thefederal government chooses the intergovernmental transfers, as well as the tax rate on labor income,taking the state governments’responses into account. Surprisingly, when the employment externality
is positive, the outcomes are equivalent to those obtained with simultaneous policy choice. In contrast,
the e¢ ciency of public good supply is improved for some combinations of parameter values if theemployment externality is negative. Federal government leadership enables a negative transfer from thefederal government to the state governments. Therefore, the federal government sets an appropriately
negative transfer and succeeds in ensuring the optimal level of public goods by the state governments.
The remainder of this paper is organized as follows. The basic model is developed in Section
3. Section 4 analyzes the equilibrium outcomes when federal and state governments simultaneously
choose their policies. The economic performance under federal government leadership is investigatedin Section 5. Finally, Section 6 provides some conclusions.
4 Boadway and Keen (1996) and Boadway et al. (1998) analyze the federal-state policy interaction at the early stage.

These studies focus on an overlapping tax base and shed light on the presence of vertical externality through it. However,this paper do not treat the issue of overlapping tax base.
5 Altshuler and Goodspeed (2015) …nd that smaller countries follow the beggest country when they choose the tax
policy. It seems to be true on the relation between federal and state governments. Numerous studies investigate thedecentralized leadership and the e¢ ciency (e.g., Köethenbüerger 2004, 2008; Breuille et al. 2010; Caputo and Silva
2014, 2015, 2017). It is natural to analyze the impact of the decentralized leadership on the e¢ ciency because a positivetransfer has the incentive e¤ect to reduce the state governments’tax rates. We can rely on their contributions.

3

2

The basic model

Consider an economy which is composed of two states. Each state has a representative …rm and acontinuum of size one of residents. The production technology in state i; i = 1; 2, is described byyi = F (ki ; li ; zi ) where F is a constant-returns-to-scale function, yi is the output of single good, ki isthe capital input, li is the labor input, and zi is the land input. Assuming that land size is …xed atunity, the production function can be rewritten as
yi = f (ki ; li );(1)where f (ki ; li ) F (ki ; li ; 1). It is assumed that the function f is strictly concave and at least twice
continuously di¤erentiable. Pro…t maximization yields@f (ki ; li )= r + ti ;(2)@ki
@f (ki ; li )
(3)
fli
= w;
@li
where r is the interest rate, ti is the source-based capital tax rate, and w is the …xed wage rate.
Furthermore, the return to land, i ; is given byfki

i = f (ki ; li )(r + ti )ki

wli :

(4)

Following Ogawa et al. (2006a), we assume that the …xed wage rate is higher than the wage rate
determined in a competitive labor market.
Each resident in state i has a capital endowment of i k units, a labor endowment of one unit, anda land endowment of one unit. Note that k is the entire capital endowment of the economy and iis the share for the residents in state i. The residents supply these endowments inelastically. The
utility function is Ui = xi + v(gi ) where xi and gi denotes private good and public good consumption
respectively. We specify v(gi ) = log gi ; where > 0. Since w exceeds the competitive level, there isunemployment, so the residents can be divided into two groups: the employed and the unemployed.
An employed resident must pay a labor income tax at rate and so has the budget constraint xi =r i k + i + (1) w. In contrast, an unemployed resident does not pay the labor income tax so theirbudget constraint is xi = r i k + i . Consequently, the utilities of the employed (e) and the unemployed(u) are respectively
UieUiu= r ik += r ik +i + (1) w + v(gi );i + v(gi ):

(5)
(6)

To ensure that the residents of both states want to work, the following condition is imposed:
Uie

Uiu = (1)w

0; i = 1; 2:(7)

The capital market clearing condition is2X

ki = k:(8)

i=1

On the other hand, there is unemployment in the labor market in each state, so li < 1; i = 1; 2.
Equations (2)–(4) and (8) lead to an equilibrium described by r = r(t), ki = ki (t), and li = li (t)where t = (t1 ; t2 ). The responses of r, ki , and li to a change in ti are given by (see Appendix A forderivation):
@ki
@ti

=

@li
@ti

=

@r
@ti

=

fll1 fll2
i j
flk
fll

flli

< 0;

@ki
=
@tj

fll1 fll2i
R 0 , flk
R 0;
j< 0; (i 6= j)

> 0;

@li
=
@tj

(9)
i j
flk
fll

i
Q 0 , flk
R0

(10)
(11)

4

wherei
fkk
i@ 2 f (ki ; li )< 0; flli ;
@ki2
i
fkk
flli

i
flk

2

@ 2 f (ki ; li )i
< 0; fkl
@li2
fll2

> 0;

1
1 + fll@ 2 f (ki ; li ) i; flk@ki @li@ 2 f (ki ; li );@li @ki

2 < 0:

Using equations (9)–(11), we obtain
@ i
=
@ti

1+

@r
@ti

ki ;

@ i
=
@tj

@r
ki :
@tj

(12)

@li
is the same as that of the cross derivative fkl .6 As reported in OECD
From (10) the sign of @ti(2017), both the public and politicians seem to believe that reductions in the corporate tax will
reduce unemployment, which implies an increase in capital tax rate must have a negative impaction employment. This belief is only valid in the model if flk< 0. However, empirical evidence on
the relation of employment to the corporate tax rate is controversial: Feld and Kirchgassner (2002),
Bettendorf et al. (2009), Zirgulis and Šarapovas (2017) found that a rise in the corporate tax rate
signi…cantly increased unemployment levels while Feldmann (2011) showed that higher corporate taxes
may lower the unemployment rate. Hence, we need to consider both possibilities in the analysis below.
Integrating the utility function over the continuum of residents, the welfare function for region i is

Vi = r i k +

i + (1) wli + v(gi ):(13)

Equation (13) is the objective function for state government i when choosing its tax policy. Each state
government taxes the return to the capital, receives a tax transfer from the federal government andprovides a public good for its residents. Consequently, the budget constraint for state government i is
gi = ti ki + si ;(14)

where si denotes the tax transfer from the federal government. The federal government income tax is
imposed on all residents in the country so the budget constraint for the federal government is
w

2
X

li =

2
X

si

(15)

i=1

i=1

The Pareto e¢ cient allocations for the economy are used as a comparison point in the analysis
that follows. To obtain the characterization consider the maximization of state 1’s welfare subject toequations (1)–(4), (14) and (15) for given a given level of state 2’s welfare. Solving the problem yields(see Appendix B)
v 0 (gi ) = 1 , g = ;(16)where g stands for the e¢ cient level of public good supply. The optimal tax rates at this equilibrium
are
(t ;

)

=

(t ;

)

=

0;

for

wl
wl
k

;1

wl;
for

> wl:

(17)
(18)

Since the labor income tax has no distortionary e¤ect, it can be treated as a lump-sum tax.7 Whenwl the public good can be …nanced entirely through the income tax, so the distortionary capital
tax is not used. In contrast, when > wl the labor income tax is set at its maximum and the capitaltax …nances the remainder of the public good.
6 One example form of the production function that satis…es f

kl < 0 is y = sin (kl), where k; l 2 [1; 1:5]. Anotherexample is y = [1 exp ( k l 1)].
7 If the labor supply generates a (…xed unit of) disutility , the upper bound of labor income tax will be lowered.
Hence, the planner needs to impose the capital tax even if < wl.

5

3

Simultaneous policy choice

The analysis in this section focuses on the equilibrium outcome with simultaneous policy choice.
The state and federal governments simultaneously choose their policy instruments to maximize theirobjective functions, subject to constraints, under the assumption of Nash behavior. The game has
the following two stages. At the …rst stage, each state government chooses their policy instruments
to maximize their objective function subject to their budget equation taking the …rm’s and residents’behavior into account. At the second stage, …rms choose their factor demands to maximize pro…t given
the policy instruments ft1 ; t2 ; s1 ; s2 ; g.
The behavior of the …rms is characterized by the factor demand functions (9)–(11). Taking thesedemands into account, state government i faces the maximization problem:
max Vi = r i k +

i + (1

fti 0g) wli + v(gi );(19)

subject to (4), (14), r = r(t), ki = ki (t), and li = li (t) for given tj , si ; sj , (j 6= i). Exploiting thesymmetry of the two states, the partial derivative of Vi with respect to i is
@Vi
=
@ti

)w

ki + (1@li@ki:+ v 0 (gi ) ki + ti
@ti
@ti

(20)

From (20), the …rst-order condition for an interior solution of (19) isv 0 (gi ) =1 + (1
)
1+ i

i

;

(21)

where
ti @kiQ 0 , ti R 0;
ki @tiw @li
iR 0 , flkR 0:
ki @tii

i

The terms i and i represent the …scal externality and employment externality of the capital tax
respectively. The …scal externality is always positive when the tax rate is positive (Wildasin 1989).
The employment externality can be positive or negative (Ogawa et al. 2006a) depending on the crossderivative fkl of the production function.8 In contrast, the …rst-order condition for a corner solution(ti = 0) can be obtained from (20) as
v 0 (gi )1 + (1

)

i:

(22)

0

We impose the following assumption to ensure v (gi ) > 0 at the optimum irrespective of whether it isinterior or at a corner.
Assumption 1.

i >

1 and

i >

1.

Equations (21) and (22) determine the best response functions for the state governments. Again
appealing to the symmetry of the states, the equilibrium properties of the reaction function are summarized as follows.
Lemma 1. Suppose that the regions are symmetric. With t
for each state government is given by

0, the equilibrium capital tax function[1 + (1) ] wl;+ [1 + (1) ]kt = max 0;
where
t

1 @k
> 0:
k @t

=

8 Sato (2009) also shows the unemployment-exporting externality (positive employment externality) using job-searchmodel.

6

Proof. For the interior solution, equations (14), (15), and (21) yieldtk + wl=

1 + (1
1

)

)t=

t

[1 + (1) ] wl:+ [1 + (1) ]kNote that k and l are independent of both t and in the symmetric equilibrium. The capitaltax rate derived above can be negative; imposing the lower bound of 0 for the tax rate gives thetax function in Lemma 1.
The terms
and that appear in the tax function of Lemma 1 are associated with the …scal
externality and employment externality, respectively. An increase in the labor income tax has twodi¤erent impacts on the resident’s utility level: it has a negative impact via a decrease in the levelof disposable labor income and a positive impact via the increase in the transfer from the federal
government for public good provision. The balance of these two impacts determines the capital tax
rates chosen by the state governments. In the absence of a labor income tax ( = 0) and, consequently,
no transfers from the federal government (si = 0), the capital tax rate istO =+ (1 + ) k> 0:

An increase in decreases the capital tax rate because a large positive value of indicates a signi…cantnegative e¤ect of a rise in the capital tax rate on employment. In contrast, with the maximum labor
income tax ( = 1), the terms involving are eliminated from the tax function. A labor income taxof this magnitude eliminates the e¤ect of the employment externality.
As shown in Ogawa et al. (2006a), the equilibrium supply of public good is a¤ected by the presenceof the employment externality. When there are no transfers from the federal government to the states,
the following result applies:
Lemma 2. Without transfers ( = 0 =) si = 0; i = 1; 2), the supply of public good in each state isequal to
k
gO =
Rg ,
+ k Q 0:+ (1 + ) k= 0 lead to g O = tk. Hence, we obtain

Proof. Equations (14) and (15) with s =gO

g = tk

=

(

+ k)
R0,
+ (1 + ) k+ k Q 0:

The term+ k captures the aggregate external e¤ect derived from the …scal and employment
externalities. The public good is overprovided (underprovided) if the aggregate externality is negative(positive). In the standard setting without the employment externality, the positive …scal externality
always results in underprovision. The employment externality can reverse this conclusion if it issu¢ ciently negative. The consequence of Lemma 2 is that there will be ine¢ ciency unless the …scaland employment externalities neutralize each other. When there is ine¢ ciency, there is a potential
role for transfers from the federal government to provide a resolution and achieve e¢ ciency.
The federal government chooses the transfers and the labor income tax to maximize a utilitarian
welfare function de…ned on state welfare levels subject to equations (4), (7), (14), (15), r = r(t),ki = ki (t), and li = li (t) for given ti . Formally, the optimization of the federal government is
#
"
2
2
2
X
X
X
li
si
+ si i + (1
)w ;
(23)
max L =
Vi + w
fs1 ;s2 ; g

i=1

i=1

i=1

where is the Lagrange multiplier for the constraint (15), i is the Kuhn-Tucker multiplier for nonnegativity of transfers, and is the Kuhn-Tucker multiplier for the constraint (7). The …rst-order7

conditions are: v 0 (gi ) +
"

si

:

:

i

(

1)

(1

i
2
X

li

i=1

= 0; si#0; iw = 0;

(25)

0;

1:

(26)

) w = 0;0; si i = 0;(24)Equations (14), (15), (21), and (24)–(26) constitute the description of the Nash equilibrium forsimultaneous choice. We use the notation g N for the supply of public goods, tN for the capital tax
rate, and N for the labor income tax at the Nash equilibrium. The best response functions for thestate governments were stated in Lemma 1. The best response functions for the federal government
are subject to the boundary conditions requiring non-negative tax rates and a tax rate on labor incomethat does not exceed 1. The presence of the boundary conditions create kinks in the graphs of thebest response functions. The full set of possible cases is illustrated in Figure 1. The nature of theequilibrium is dependent on the value of relative to wl; so we treat the cases < wl and > wl
separately.
First, the equilibrium with < wl is characterized in the following result.
Proposition 1. When < wl;(i) If fkl 0 the unique Nash equilibrium given bytN ;N; g N = 0;

wl

;g

:

(ii) If fkl < 0 the unique Nash equilibrium given by
(a)

tN ;

N

; gN

=

(b)

tN ;

N

; gN

=

(

+ k
wl)
;
;g
wl + kwl + k+ (1 + ) k; 0; g O

for j kj <for j kj >;:Proof. See Appendix C.1.
If fkl0, the aggregate e¤ect of the …scal and employment externalities is positive. Hence,the state governments have an individual incentive to reduce their tax rates to attract capital. In
response, the federal government increases the transfer to the state governments via an increase in
the labor income tax. The transfer from the federal government to the state governments provides anadditional incentive to decrease the capital tax rate because the transfer covers the decrease in thecapital tax revenue. Therefore, the equilibrium policy is a zero capital tax with a positive transfer(Figure 1a).
When fkl < 0, there are two possible cases: j kj <(the aggregate external e¤ect is positive) andj kj >(the aggregate external e¤ect is negative). For j kj <the same mechanism as describedabove is at work. However, the unemployment-importing externality (i.e., a negative employment
externality) partially o¤sets the …scal externality and the incentive to reduce the capital tax rates isdiminished. Therefore, both the capital tax rate and the transfer are positive in equilibrium (Figure1b). If j kj >each state government wishes to increase its capital tax rate above the level when
j kj <. In response, the federal government reduces the labor income tax and the transfer tostate governments. The equilibrium capital tax rates are positive without a transfer (Figure 1c). In
particular, the state governments overprovide public goods under the equilibrium policies. The transferfrom federal to state governments fails to decrease the capital tax rate which implies that a transferfrom state governments to federal government would be e¢ cient, if it were permitted.
Second, we characterize the equilibrium for > wl:

8

t

t

Federal government
Overprovision

Overprovision

Federal government

Nash equilibrium

Underprovision

Nash equilibrium
Underprovision
State governmentState government
0

0

a.

> 0;

> j kj ;

< wl

b.

< 0;

> j kj ;

< wl

t

t

Federal government

Overprovision

Nash equilibrium

Overprovision

State government
Nash equilibrium
Underprovision

Underprovision
State government

Federal government
0

0

c.

< 0;

< j kj ;

< wl

d.

> 0;

> j kj ;

> wl

t

t

Nash equilibrium(stable)Overprovision

Overprovision
Nash equilibrium(unstable)

Federal government

Federal governmentState governmentState government
Underprovision

Underprovision

Nash equilibrium(stable)Nash equilibrium
0

0

e.

< 0;

> j kj ;> wl

f.
Figure 1.

9< 0;< j kj ;> wl

Proposition 2. When > wl,(i) If fkl 0 the unique Nash equilibrium is given bytN ;(ii) If fkl < 0(a) When j kj <

tN ;

N

( wl + k)
wl
; 1;
+k
+k

; gN =:, the unique Nash equilibrium is given bytN ;(b) When j kj >

N

N

wl
( wl + k)
; 1;
+k
+k

; gN =:, there exist three Nash equilibria given by:wl( wl + k)
; 1;
+k
+k

; gN =

;

(

wl)
+ k
;
;g
wl + kwl + k;+ (1 + ) k; 0; g O:

Proof. See Appendix C.2.
The mechanisms through which tax policies operate are similar to those of Proposition 1, andonly the presence of the labor income tax wedge creates a di¤erence to the analysis with < wl. If
j kj <(which includes the case fkl 0), the upper bound for the labor income tax rate constrains
the federal government’s choice of tax policy. The equilibrium policy requires both a positive tax
and a transfer (Figure 1d and 1e). The case of j kj >is signi…cantly di¤erent with three Nashequilibria. For a low (high) tax rate on labor income, the state governments choose high (low) tax
rates on capital, and then the federal government responds with a lower (higher) tax rate on labor
income. The process converges either to one of the two equilibria at the boundary for the income tax,
or to an interior equilibrium. The interesting observation is that the e¢ cient provision of public goodis attained at the interior equilibrium (Figure 1f). The condition in case (ii) of Proposition 2 can be
rewritten as9
j kj R

,

t @l
l @t
t @k
k @t

=

Elasticity of employment rate to capital tax rate
R:
Elasticity of capital to capital tax rate
wl

(27)

Since the proposition applies when < wl, the multiple equilibria arise when the cross derivative of
the production function is negative and the elasticity of employment is large relative to the elasticity
of capital.
Focusing on the capital tax rate and the level of public good supply, the results discussed above
for < wl are summarized in the next proposition.
Proposition 3. When < wl,(i) If fkl 0, then:tN = t < tO and g O < g N = g .
(ii) If fkl < 0, then:(a) j kj <implies t < tN < tO and g O < g N = g .
(b) j kj >implies t < tN = tO and g < g N = g O .
Proof. See Appendix D.1.
The corresponding results for> wl are as follows.

9 This relation is derived fromj kj R

,

w @l
k
k @t

R

1 @k
@l
,w
R
k @t
@t

10

1 @k
t @l
, wl
R
k @t
l @t

t @k
:
k @t

(a)

< wl
gN
tN
N

(b)

> wl
gN
tN
N

Table 1: Equilibrium tax policy under simultenous policy choice
fkl 0
fkl < 0; j kj <fkl < 0; j kj >gO < gN = g
gN = gO > gNNOt =t =0
0<t <ttN = tO > 0
N
N
N
=
<1
0<
<
=0<
fkl 0
fkl < 0; j kj <fkl < 0; j kj >gO < gN < g
g N R g O (g N < g ) g N < g < g O g N = g < g O
t N R tO
tN < t O
tN < tO
tN < tO
NN==1=
0< N <

gN = gO > gt N = tO > 0
N=0<

Proposition 4. When > wl;(i) If fkl 0, then:t < tN < tO or t < tO < tN and g O < g N < g .
(ii) If fkl < 0, then:(a) j kj <implies t < tN < tO and g O < g N < g or g N g O < g .
(b) j kj >implies t < tN tO and g N g < g O or g < g N = g O .
Proof. See Appendix D.2.
Equation (17), Lemma 1, Lemma 2, and Proposition 1 enable us to compare the equilibrium tax
policies. Table 1 (a) shows the relationship between the state government’s policy without transfers andthe Nash equilibrium in the case where < wl. For fkl 0, the outcome is based on the mechanism
explained in connection with Proposition 1. If there is an unemployment-importing externality (fkl <0), Kikuchi and Tamai (2019) show that tax base equalization with a negative coe¢ cient improves
the e¢ ciency of public good provision. When the governments simultaneously choose the tax rates,
the federal government has to set a non-negative transfer because there is a possibility of the state
governments choosing non-positive capital tax rates. Thus, a transfer from the state governments tothe federal government is impossible when there is simultaneous policy choice.
Table 1 (b) displays the outcomes for > wl derived from Lemma 1, Lemma 2, and Proposition 2.
When fkl 0, a transfer from the federal government to the state governments improves the e¢ ciencyof public good provision even though it is not su¢ cient to ensure the e¢ cient level of public good
supply. Then, the capital tax rate with the transfer can be either higher or lower than the tax rate
without the transfer. In particular, the capital tax rate with the transfer is smaller than the tax rate
without transfer when takes a su¢ ciently large positive value.10
Depending on the overall external e¤ect, two cases are possible when fkl < 0. If the …scal externality
dominates the unemployment importing externality (j kj <), the supply of public good can beeither improved or worsened when compared to the case where = 0; but the public good is always
below the e¢ cient level. In this case, the capital tax rate with transfers is lower than the tax rate
without transfers.

4

Federal government leadershipThis section investigates the equilibrium outcome with sequential policy choice. We consider federal
government (centralized) leadership, which implies pre-commitment to policy by the federal government. The game structure under federal government leadership can be described as follows. At the
1 0 Contrasting the income tax rate in Proposition 2 and
N

R

O

,

O = 0, we an see that

+k
k

Q

11

wl
wl

:

…rst stage, the federal government takes the best responses of the state governments and the …rms into
account, and chooses transfers and an income tax, fs1 ; s2 ; g ; to maximize social welfare subject tothe federal budget constraint. At the second stage, each state government chooses its capital tax rate
to maximize (13) subject to (14) and the reaction of …rms. Finally, at the third stage, …rms in eachstate, i, determine input use, fki ; li g ; to maximize their pro…ts for given fti ; si ; g (i = 1; 2).
Backward induction is applied to solve the game. The reactions of …rms are represented by equations(11)–(12). The optimization problem for each state government yields equation (21). Equations (14)and (21) provide the optimal response of ti to a change in si and . The reaction function of state i’s
government has the following properties (See Appendix E):
@ti
@si
@ti
@si

=
=

(1 + i ) ki v 00 (gi )
@ti
@ti
w @li
< 0;
= 0;
=
R 0 , fkl R 0 for ti > 0;Di
@sj@
Di @ti@ti@ti== 0 for ti = 0;
@sj
@

where

(28)
(29)

@ 2 Vi
< 0:
@t2i

Di

An increase in the transfer to state i has a positive e¤ect on state i’s government revenue but no impact
on state j’s government revenue. The increase in revenue enables the state governments to decrease
their capital tax rates. The income tax rate a¤ects the welfare gain/cost of the capital tax throughjob creation/loss. Hence, the e¤ect of a rise in the income tax rate on the capital tax rate depends
upon the e¤ect of the capital tax rate on employment.
Taking the relationships r = r(t), ki = ki (t), li = li (t), i = 1; 2; and equation (28) or (29) into
account, the federal government solves
"
#
2
2
2
X
X
X
max L =
Vi + w
li
si
+ (1
)w :
(30)
fs1 ;s2 ; g

i=1

i=1

i=1

Using the symmetry of regions, the …rst-order conditions are composed of equations (26), and
si

:

:

2
X
@Vj @tj@tj @sij=1

2X@Vi @ti
i=1

@ti @

+ [(1

+

2
X

)

[(1

@ti
0
+ v 0 (gi )i v (gi )] ki
@si

i

)

i

i=1

@ti
+(
i v (gi )] ki
@
0

= 0;

1) w

2
X

(31)

li

w = 0:(32)

i=1

Solving equations (14), (15), (21), (26), (31), and (32) yields the equilibrium under federal government
leadership.
Let tF be the capital tax rate, g F be the public good supply, and F be the labor income tax underfederal government leadership. The nature of the equilibrium for < wl is summarized as follows.
Proposition 5. When < wl,(i) If fkl 0, the equilibrium policies are
tF ;F; g F = 0;

wl

;g

:

(ii) If fkl < 0 the equilibrium policies aretF ;F; gF =

(

wl)
+ k
;
;g
wl + kwl + kProof. See Appendix F.1.
12:

Case (i) in Proposition 5 is identical to that in Proposition 1. Case (ii) shows that federal government leadership achieves an e¢ cient provision of public good even though the tax rates di¤er from theoptimal tax rates. In each case, the federal government decides the labor income tax rate to maximizesocial welfare by considering the reactions of the state governments. Since the federal government isaware of the substitutability between the capital tax and the transfer in the state governments’response
functions, the e¢ cient level of public good supply is appropriately chosen. Based on this anticipation,
the federal government should pre-commit to set F to be the optimal outcome for fkl > 0.
On the other hand, the federal government does not choose the e¢ cient level of labor income tax
rate for fkl < 0. Since the state governments prefer to choose higher capital tax rates for fkl < 0,
the federal government has to set the labor income tax rate higher than the e¢ cient level to keep ane¢ cient supply of public good. In the same way as Proposition 1, we obtain11
j kj < wl ,

t @l
l @t
t @k
k @t

Elasticity of employment rate to capital tax rate
< 1:
Elasticity of capital to capital tax rate

=

(33)

For the case of > wl, the equilibrium under federal government leadership is described in thefollowing proposition.
Proposition 6. When > wl,(i) If fkl 0, then the equilibrium policies aretF ;F( wl + k)
wl
; 1;
+k
+k

; gF =:(ii)If fkl < 0,(a) For j kj < wl, the equilibrium policies aretF ;F( wl + k)
wl
; 1;
+k
+k

; gF =:(b) Forj kj > wl, the equilibrium policies aretF ;F; gF =

(

+ k
wl)
;
;g
wl + kwl + k.

Proof. See Appendix F.2.
When the overall external e¤ect is positive ((i) and (a) of (ii)), the federal government pre-commits
to choose F = 1 because there is no alternative to improve welfare. After that, the state governments
prefer to choose the tax rate lower than the level that enables the e¢ cient provision of public good.
This is because the state governments have an incentive to reduce their tax rates to attract capital.
However, the equilibrium policies are di¤erent when the overall external e¤ect is negative. If > wl andwl < j kj <((b) of (ii)), the pre-commitment works well. As stated above, the state governments
are motivated to increase the tax rates. Hence, federal government accomplishes the e¢ cient level of
public good supply by pre-committing to choose appropriate (negative) tax rate on labor income.
From the analysis, we establish the following propositions.
Proposition 7. When < wl,(i) If fkl 0, the equilibrium policies satisfy tF = t and g O < g F = g .
(ii) If fkl < 0. then(a) When j kj << wl, the equilibrium policies satisfy t < tF < tO and g O < g F = g .
(a) When< j kj < wl, the equilibrium policies satisfy t < tO < tF and g F = g < g O .
1 1 The condition is derived fromj kj < wl ,

w @l
k
k @t

<

wl

1 @k
(*
k @t

13

< 0) , w

@l
<
@t

wl

1 @k
t @l
,
<
k @t
l @t

t @k
k @t

(a)

< wl
gF
tF
F

(b)

> wl
gF
tF
F

Table 2: Equilibrium tax policy underfederal government leadershipfkl 0
fkl < 0; j kj << wl
fkl < 0;
< j kj < wl
gO < gF = g
gF = g < gO
tF = t = 0
0 < tF < tO
0 < tO < tF
F
F
F
=
<1
0<
<
<0<
fkl 0
fkl < 0; j kj < wl <fkl < 0; wl < j kj <fkl < 0; wl << j kjgO < gF < g
gO < gF = g
gF = g < gO
tF R t O
0 < tF < tO
0 < tO < tF
0 < tF < tO
F
F
=
=1
<0<
0< F <

Proof. See Appendix G.1.
Proposition 8. When > wl,(i) fkl 0 the equilibrium policies satisfy tF Q tO and g O < g F < g .
(ii) fkl < 0.
(a) If j kj << wl, the equilibrium policies satisfy t < tF < tO and g O < g F < g .
(b) If wl < j kj <, the equilibrium policies satisfy t < tO < tF and g O < g F = g .
(c) If wl << j kj, the equilibrium policies satisfy t < tF = tN < tO and g F = g N =O
g <g .
Proof. See Appendix G.2.
Using lemmas 1 and 2, plus propositions 5 and 6, we can provide a summary of the equilibrium
policies with federal government leadership. For< wl the results are displayed in Table 2 (a)and for> wl in Table 2 (b). A comparison between the results in tables 1 and 2 reveals someinteresting policy implications. The analysis is also helpful for understanding the e¤ectiveness offederal government leadership.
Consider …rst the case of< wl. We discuss the policies for fkl < 0 in detail to focus on thee¤ectiveness of federal government leadership.12 The federal government knows that there is substitutability between the capital tax and the transfer in the state governments’reactions, and that thelabor income tax has no distortionary e¤ect. Hence, the e¢ cient provision of public good is attainablewith federal government leadership because a negative tax rate on labor income is allowed in thisregime. If the federal government pre-commits to an appropriate value of F < 0, the state governments have to choose capital tax rates tF > 0 that result in the e¢ cient provision of the public good
because they are willing to increase the tax rates. Therefore, federal government leadership enables thee¢ cient supply of the public good regardless of the unemployment exporting/importing externality.
Now consider > wl. If fkl 0; or fkl < 0 and j kj << wl, the equilibrium policies underfederal government leadership coincide with the Nash equilibrium policies. The federal government
has no choice (it has to set = 1). Therefore, the interpretation of the results is identical to those of
Proposition 4. The interesting points of Proposition 8 are that the e¢ cient supply of the public goodis attainable even if j kj <(it cannot be under simultaneous policy choice), and that three Nashequilibrium policies are reduced to one equilibrium policy if j kj >.
As explained above, the federal government can choose a negative tax rate on labor income undercentralized leadership. This is the key to deriving the e¢ cient supply of the public good for wl <j kj <. With simultaneous policy choice, the (repeated) anticipation of the governments split
into two opposite directions at the middle equilibrium as the boundary for j kj >: two extreme
equilibrium policies are possibly chosen. However, the federal government leadership excludes suchextreme choices. Only the policy to attain the e¢ cient supply of the public good is a unique choicefor the federal government.
1 2 See the interpretation of Proposition 1 and 5 for the case where fkl= =wl.

14

0. Federal government has no alternative to5Conclusion

This paper has investigated the interaction of tax policies, intergovernmental transfers, and strategic
timing in the presence of a labor market imperfection. The equilibrium policies are determined by thecost of the public good relative to the maximum revenue attainable from the labor income tax, and thebalance of the …scal externality and employment externality of the capital tax. The …scal externality
is always positive, but the sign of the employment externality depends on the substitutability orcomplementarity between capital and labor in the production function.
First, consider the case where the cost of supplying the public good can be covered by using only
the labor income tax. The equilibrium tax policies under simultaneous choice are Pareto e¢ cient if
capital and labor are complementary. However, when capital and labor are substitutes, the equilibrium
policy may not be the e¢ cient policy even if the labor income tax …nances the cost of the public good.
Second, assume the labor income tax alone cannot …nance the cost of providing the public good.
Regardless of the substitutability/complementarity between capital and labor, the equilibrium taxes
will not be at the e¢ cient levels. However, there remains a possibility that the public good is e¢ ciently provided when the aggregate externality (the sum of the …scal and employment externalities)
is negative. On the other hand, this outcome is one of multiple equilibria and may not be realized - inwhich case one of the ine¢ cient equilibria will be established.
The timing of policy choices by the governments is an important determinant of the e¢ ciencyof public good provision and the comparison of timing structures developed in the paper has somesigni…cant policy implications: First, the equilibrium policies under simultaneous policy choice are
identical to those under federal government leadership if capital and labor are complements: the federal
government has no need to take a leadership role to improve social welfare. Second, federal government
leadership improves the e¢ ciency of providing public goods if capital and labor are substitutes. In
this case, the federal government must act as a leader to achieve an e¢ cient outcome.
Finally, we should note some possible future extensions of this research. First, it is insightful
to incorporate asymmetry of regions into our basic model. Even though symmetric regions lead toanalytically clear implications, the di¤erences in wage levels, the capital and the labor endowments
are important determinants of intergovernmental transfer. Second, public inputs should be considered
because employment and pecuniary externalities are associated with public input provision. Then, thestate governments tend to overprovide public input exceed its optimal level.13

1 3 See Gillet and Pauser (2018) for the detail. Ogawa et al (2006b) examined the small open economy (i.e., there is no
pecuniary externality).

15

Appendix
A. Derivation of equations (9)–(11)Total di¤erentiation of (2), (3), and (8) gives
0 1
1
fkk
0 fkl
0
2
2
B 0 f
0
f
kk
kl
B 1
1
B f
0
fll
0
B lk
2
@ 0
flk
0 fll2
1
1
0
0

1
1 0
10
dt1
dk1
1
C
C B
B
1 C
C B dk2 C B dt2 CB dl1 C = B 0 C :
0 C
C
C B
CB
0 A @ dl2 A @ 0 A0dr0The determinant of the coe¢ cient matrix is
1
fkk
0
1
flk
0
1

0
2
fkk
0
2
flk
1

1
fkl
0
fll1
0
0

0
2
fkl
0
fll2
0

1
1
0
0
0

h
1 1
fll
= fll2 fkk

1
flk

2

i

h
2 2
fll
+ fll1 fkk

2
flk

2

i

< 0:

Applying Cramer’s rule, we obtain equations (9)–(11).

B. Derivation of equation (16)The condition for Pareto e¢ ciency can be obtain by solving
max L = r 1 k +s;t;+

1 + (1r 2k +) wl1 + v(t1 k1 + s1 )2 + (1) wl2 + v(t2 k2 + s2 ) +

"

w

2
X
i=1

li

2
X

#

si :

i=1

The …rst-order conditions are
@L
dl1
@V1
@V2
dl2
=
+
+ w
+
=0
@ti
@ti
@ti
dti
dti
@L
= 0 ) v 0 (g1 ) = ; v 0 (g2 ) = ;@si@L= 0 ) l1 + l2 = (l1 + l2 ) ;
@

(34)
(35)
(36)

Equation (36) yields=

l1 + l 2:
l 1 + l2(37)

Furthermore, equation (34) can be reduced to
dr
dt1
dr
dt2

1k

1k

k1 +
k1 +

2k

2k

k2

k1 + (1

) (1

k2

k2

) (1

(1

dl1
+
dt1
dl2
)w
+
dt2
)w

k1 + (t1k2 + (t2

dk1
dt1
dk2
t1 )
dt2
t2 )

Due to symmetry, we have t1 = t2 . Then, equations (38) and (39) lead to = 1. Inserting
equation (37), we obtain = 1. These results and equation (35) indicate v 0 (gi ) = 1.

16=0;(38)=

0:(39)= 1 into

C. Proof of Propositions 1 and 2
C.1. Proof of Proposition 1
Interior solutions for federal government’s optimization problem (23): s > 0 and 0 < < 1 ( =0). Equations (24)–(26) yield = 1 and v 0 (g) = 1. If t > 0, equation (21) and v 0 (g) = 1 lead to
(1

)

= ,t=

(1

@l
@t

)w

@k
=
@t

(1

)

:

=

(40)

(i) fkl 0. Let be fkl > 0. Then, t < 0 is derived from equation (40). This contradicts t > 0. Let
be t = 0. Equations (14), (15), t = 0, and v 0 (g) = 1 yield
= wl:

(41)

=

(42)

g =
From equation (41), we obtain
N

wl

:

When < wl, equation (22) does not contradict 0 < N < 1 and v 0 (g) = 1. Thus, there is a Nashequilibrium with tN = 0, equation (42), and g . We now assume fkl = 0. Equation (40) leads to t = 0.
If < wl, equations (41) and (42) hold. These equations derive the same result shown above.
(ii) fkl < 0. From equation (40), t > 0 is derived. Equations (14), (15), (16), and (40) yield
N

=

+ k
:
wl + k(43)

wl):
wl + k

(44)

From equations (40) and (43), we obtain
tN =

(

When
< wl, equation (44) shows that wl > j kj is required to ensure tN > 0. Since
< wl,
wl >
> j kj holds. Equation (43) yields 0 < N < 1 if
> j kj. There exists a Nashequilibrium which satis…es equations (41), (43), and (44).
Corner solutions for federal government’s optimization problem (23): s = wl and = 1 ( = 0,
P2
> 0). Equation (25) becomes (1) i=1 li = > 0; > 1 holds. By equation (24), v 0 (g) = > 1.
For t > 0, equations (24) and (21) providev 0 (g) =Equations (15), (45), symmetry of regions,
tN

1
1+

=

> 1:

(45)

= 1, and v 0 (g) = =g give
wl
;
+k
+k
:
wl + k

=
=

(46)
(47)

Note that these equations hold for fkl 2 R. When < wl, tN < 0 and > 1 are derived from equations(46) and (47). These results contradict tN > 0 and equation (45). There is no Nash equilibrium with
= 1 if < wl.
Corner solutions for federal government’s optimization problem (23): s = 0 and = 0 ( > 0,
= 0). v 0 (g) = 1< 1 is derived from from equations (24) and (25).
(i) fkl > 0. For fkl > 0, equation (21) indicates v 0 (g) > 1 for t > 0. These results contradict eachother. For t = 0, g = 0 holds because of = s = 0. This is also inconsistent with v 0 (g) = 1< 1.
17

For fkl = 0, equation (21) is v 0 (g) > 1 for t > 0. When t = 0, we obtain v 0 (0) = 1. These results
contradict v 0 (g) < 1.
(ii) fkl < 0. For t > 0, equation (21) isv 0 (g) =1 + (1
1+

)

1+
1 t

=

R1,j jQt :When > 0 or < 0 and j j < t , the equation mentioned above contradicts v 0 (g) > 1. If
t < j j, 0 < v 0 (g) < 1 holds. Equations (14), (15), and = 0 providetN =Equations (14), (15), (48), Lemma 2 and+ (1 + ) k< 0 and= tO > 0:(48)< j kj :(49)= 0 giveg N = g O > g forC.2. Proof of Proposition 2

If > wl, equations (46) and (47) lead to tN > 0 and > 1, which are consistent with tN > 0 andequation (45). Equations (14), (15), (46), and N = 1 provide
g N = tN k + wl =( wl + k)
:
+k

(50)

Therefore, the triplet of equations (46), (50), and N = 1 is a Nash equilibrium for fkl 2 R.
In equation (44), wl < j kj is necessary to be tN > 0 for > wl. Note that> wl holds
if > wl. Equations (43) and (44) lead to tN > 0 and 0 < N < 1 if j kj >. Note that (41)holds. Therefore, a Nash equilibrium with tN > 0, 0 < N < 1, and g exists if < 0 and j kj >.
Furthermore, equations (48), (49), and N = 0 hold if < 0 and j kj >. These equations also givea Nash equilibrium. Therefore, three Nash equilibria exist when j kj >.

D. Proof of Propositions 3 and 4
D.1. Proof of Proposition 3
Case (i). From lemma 1, Lemma 2, and Proposition 1, we obtain tN = 0 < tO , N => 0, andg = > gO .
Case (ii). j kj <: Lemma 2 and Proposition 1 lead to N > 0 and g => g O . SinceNN
t >0=t ,<holds. Using Lemma 1 and Proposition 1, we have
tN

tO

=
=
=

(

[ + (1 + ) k] (wl)
wl)
( wl + k)=
wl + k+ (1 + ) k( wl + k) [ + (1 + ) k]
( + k)[ + (1 + ) k] wl
wl( wl + k) [ + (1 + ) k]
( + k) [(1 + ) wl]< 0 , tN < tO :( wl + k) [ + (1 + ) k]

Furthermore, equation (17) and Proposition 1 yieldNj kj >: Since we have

=
N

+ k
wl + k

wl

=

(

wl) k
<0,
( wl + k) wl= 0, tN = tO > 0, and g N = g O .

18

N

<

:

D.2. Proof of Proposition 4
Case (i). From Lemma 1 and Proposition 2, we obtain
tN

tO

wl
+k
wl) k
(
( + k) [==[ + (1 + ) k] (=+ (1 + ) k( + k) [( + k) wl
R 0:+ (1 + ) k]wl) ( + k)
+ (1 + ) k]=k(

[ + (1 + ) k] wl+ k) [ + (1 + ) k]
(51)

Equation (51) shows that tN can be either larger or smaller than tO . Lemma 2 and Proposition 2
provide g N < g and
gN

gO

=
=

k
[ + (1 + ) k] ( wl + k)
( wl + k)( + k) k=+k+ (1 + ) k( + k) [ + (1 + ) k]
[ + (1 + ) k] wl + k 2
f( + k) wl + ( wl + k) kg=:(52)( + k) [ + (1 + ) k]
( + k) [ + (1 + ) k]

Equation (52) is positive for > 0. Therefore, we have g N > g O .
Case (ii). j kj <: Equations (51) and (52) are derived from Lemma 1, Lemma 2, and Proposition 2. Equation (51) gives tN < tO for < 0. From equation (52), the magnitude relation between
g N and g O is ambiguous. j kj >: There are three equilibria. First, we consider the equilibrium
where N == 1. From equation (51), tN < tO holds. Equation (52) shows g N < g O . Next, we
examine the equilibrium with N 2 (0; 1). N < 1 =holds. Lemma 1 and Proposition 2 lead to
tN

tO

=
=

(

wl)
wl + k( + k) [(( wl + k) [[ + (1 + ) k] (wl)
( wl + k)=+ (1 + ) k( wl + k) [ + (1 + ) k]wl)wl]
< 0 , tN < tO :+ (1 + ) k]

From Lemma 2 and Proposition 2, we have g N = g < g O . The last case is the equilibrium withN
= 0. Then, the result is same as the one in Proposition 3.

E. Derivation of reaction functions
Equation (21) yields
D1 dt1=D2 dt2

=

@l1
d ;
@t1
@l2
(1 + 2 ) k2 v 00 (g2 )ds2 + w
d :@t2(1 + 1 ) k1 v 00 (g1 )ds1 + w

These equations provide equation (28).

F. Proof of Propositions 5 and 6
F.1. Proof of Proposition 5Interior solution for state government’s optimization problem (19): Using Lemma 1, we canverify the capital tax rate as[1 + (1) ] wl:t=+ [1 + (1) ]kFrom Assumption 1, t > 0 ,we obtain[1 + (1(1) l) ] wl > 0. Combined equation (32) with equation (31),

2

w

@t
= [v 0 (g)
@s
19

]

@t
:
@

(53)

Interior solution for federal government’s optimization problem (30):< 1 ( = 0). For theinterior solution (t > 0) of (19), equations (21), (31), and (32) with = 0 yield simultaneous (nonlinear) equations system composed of equation (21),[(1
2X) + t] k[(1@t+ 1 + (1
@s

)

(1

@t
+ (1
@

t) (

1) w

) + t] k

i=1

t)
2
X

l

=

0;

(54)

=

0:

(55)

i=1

One of the solution for the system is (1) + t = 0.
(i) fkl 0. First, we consider the case where fkl > 0. Equation (53) provides v 0 (g) R , Q 1.
From t > 0, < 1, and > 0, (1) + t > 0 holds. From equation (21), we obtain v 0 (g) > 1 > .
This result cannot be an equilibrium because they create a contradiction: equations (28), (54), and(55) produce
0

k @t
@t
+1+
@s
wl @
(1 + i ) k 2 v 00 (gi )k 1 @li+1+> 0:
Di
l Di @ti= k=

We now consider the case where fkl = 0. Equation (31) and (32) leads to v 0 (g) =equation (31), fkl = 0, and = 1, we havev 0 (g) == 1. Using

1
:
@t
k @s

1

When fkl = 0, t = 0 is the interior solution of (19). From the above equation and equation (21) witht = 0, we have v 0 (g) = 1. Equations (14), (15), (16), tF = 0, and v 0 (g) = 1 yield
F

=

wl

:

(56)

From equation (56), we obtain F < 1 for < wl. This result shows (i) in Proposition 5 for fkl = 0.
(ii) fkl < 0. From equation (53), we have v 0 (g) R , R 1: Note that equation (40) holds.
Hence, we have
tF =

(

wl)
;
wl + kF=+ k:wl + k(57)

Equation (57) and j kj << wl lead to tF > 0 and F 2 (0; 1). If< j kj < wl, equation (57)F
F
yields t > 0 and< 0. For< wl < j kj, tF < 0 is obtained from equation (57). This makes acontradiction.
Corner solution for federal government’s optimization problem (30): = 1 ( > 0). From Lemma
1, equations (31), and (32), we have v 0 (g) > 1, and
wl
.
+k

tF =
For

(58)

< wl, equation (58) is negative. Therefore, this contradicts the assumption of t > 0.
)
Corner solution for state government’s optimization problem (19): v 0 (g) 1 + (1holds and the optimization problem (30) can be reduced to (23) with = 0. Since t = 0,
0 isnecessary to be g 0.
Interior solution for federal government’s optimization problem (30): < 1 ( = 0). From equations (24)–(26), we obtain = 1 and v 0 (g) = 1. Equation (56) and tF = 0 are derived from equations(14) and (15). (i) fkl > 0. Then, we have v 0 (g)1
1 + (1) . This inequality is consistent20

with v 0 (g) = 1. Therefore, the result (i) in Proposition 1 for fkl > 0 is alive if < wl. (ii) fkl < 0.
v 0 (g) 1 + (1
) < 1 holds. v 0 (g) = 1 contradicts this inequality.
Corner solution for federal government’s optimization problem (30):= 1 ( > 0). For t = 0,v 0 (g)1 from (22). On the other hand, equations (31) and (32) lead to v 0 (g) > 1. This makes acontradiction.
F.2. Proof of Proposition 6
Interior solution for state government’s optimization problem (19). Interior solution forfederal government’s optimization problem (30): < 1 ( = 0).
(i) fkl
0. For fkl > 0, the result is same as the one shown in the proof of Proposition 5. If
fkl = 0, equation (56) leads to > 1. This makes a contradiction.
(ii) fkl < 0. Equation (57) holds. If> j kj > wl, we have tF > 0 and F < 0. If
F> wl > j kj, t < 0 from equation (57). This result contradicts the assumption of t > 0. For
j kj >> wl, equation (57) provides tF > 0 and F 2 (0; 1).
Corner solution for federal government’s optimization problem (30): = 1 ( > 0). From Lemma
1, equations (31), and (32), we have v 0 (g) > 1, equation (58),
@t
@s
@t
0
v (g)k@v 0 (g)k= v 0 (g)

;

=

(

1) wl

=
@t

1
1+

,

w
:
2

These equations yieldv 0 (g) =(i) fkl

1

k @s

=

1

@t
k @s
:
1+

0. For fkl > 0, equation (21) provides
tF =wl + k+kwl> 0 and g F =+k< g for> wl:Then, the values of and are determined by equations (31) and (32). Furthermore, this result holds
if fkl = 0. (ii) fkl < 0. All results of (i) holds.
Corner solution for state government’s optimization problem (19). As shown in Proof of
Proposition 5, there is no equilibrium outcome when > wl.

G. Proof of Propositions 7 and 8
G.1. Proof of Proposition 7
Case (i). The outcome is same as that of Proposition 3. Case (ii). From Lemma 1 and Proposition 5,
we have(wl)( + k) [(1 + ) wl]
tF tO ==:
(59)
wl + k+ (1 + ) k( wl + k) [ + (1 + ) k]
When j kj << wl, equation (59) is negative: 0 < tF < tO . If
< j kj < wl, equation (59) is
positive 0 < tO < tF . Furthermore, Lemma 2 and Proposition 5 lead tog F = g R g O , j kj R

21

:

(60)

G.2. Proof of Proposition 8
Case (i). The results of Proposition 4 hold. Case (ii). Lemma 1 and Proposition 6 provide equation(59) for wl < j kj and
tF

tO

=
=

wl
+k
(
wl) k
( + k) [[ + (1 + ) k] (=+ (1 + ) k( + k) [( + k) wl
for wl > j kj :+ (1 + ) k]wl) ( + k)
+ (1 + ) k]=k(

[ + (1 + ) k] wl+ k) [ + (1 + ) k]
(61)

For wl < j kj <, equation (59) is positive: 0 < tO < tF . For wl << j kj, equation (59) isFO
negative: 0 < t < t . If j kj < wl <, equation (61) yields 0 < tF < tO . From Lemma 2 andProposition 6, we obtain equation (59) for wl < j kj and
n
o
wl
[
+
(1
+
)
k]
+
k
k
k
k
( wl + k)
gF gO ==+k+ (1 + ) k( + k) [ + (1 + ) k]
k
wl
wl
=
( + k)+ 1+
k for wl > j kj : (62)( + k) [ + (1 + ) k]kk
If > 0, equation (62) is positive: g F > g O . If
gF > gO .

22< 0 andwl > j kj, equation (62) is positive:References[1] Akai, N. and M. Sato (2008), Too big or too small? A synthetic view of the commitment problem
of interregional transfers, Journal of Urban Economics 64, 551–559.
[2] Altshuler R. and T.J. Goodspeed (2015), Follow the Leader? Evidence on European and US tax
competition, Public Finance Review, 43 (4), 485–504.
[3] Aronsson, T. and S. Wehke (2008), Public goods, unemployment and policy coordination, Regional
Science and Urban Economics 38, 285–298.
[4] Baskaran, T. and M. Lopes da Fonseca (2013), The economics and empirics of tax competition: asurvey, Discussion Papers, Center for European Governance and Economic Development Research,
No. 163.
[5] Bettendorf, L., A. Horst and R. De Mooij (2009), Corporate tax policy and unemployment inEurope: an applied general equilibrium analysis, World Economy 32, 1319–1347.
[6] Boadway, R. (2006), Intergovernmental redistribute transfers: e¢ ciency and equity, In: Ahmad,
E. and G. Brosio (eds.) Handbook of Fiscal Federalism, Edward Elgar, Northampton.
[7] Boadway, R. and M.J. Keen (1996), E¢ ciency and the optimal direction of federal-state transfers,
International Tax and Public Finance 3, 137–155.
[8] Boadway, R., M. Marchand, and M. Vigneault (1998), The consequences of overlapping tax bases
for redistribution and public spending in a federation, Journal of Public Economics 68, 453–478.
[9] Boadway, R. and A. Shah (eds.) (2007), Intergovernmental Fiscal Transfers: Principles and Practices, The World Bank, Washington DC.
[10] Breuillé, M.-L., T. Madiès, and E. Taugourdeau (2010), Gross versus net equalization scheme ina federation with decentralized leadership, Journal of Urban Economics 68, 205–214.
[11] Bucovetsky S. and M. Smart. (2006), The e¢ ciency consequences of local revenue equalization:
tax competition and tax distortions, Journal of Public Economic Theory 8, 119-144.
[12] Caputo, E. and D. Silva (2014), Selective decentralized leadership, Journal of Urban Economics
83, 1–5.
[13] Caputo, E. and D. Silva (2015), E¢ cient earmarking under decentralized …scal commitments,
International Tax and Public Finance 22, 683–701.
[14] Caputo, E. and D. Silva (2017), Tax competition and federal equalization schemes with decentralized leadership, International Tax and Public Finance 24, 164–178.
[15] Chirinko, R.S. and D.J. Wilson (2008) State investment tax incentives: a zero-sum game? Journalof Public Economics 92, 2362–2384.
[16] Dahlby, B. (1996), Fiscal externalities and the design of intergovernmental grants, International
Tax and Public Finance 3, 397–412.
[17] Dahlby, B. and N.A. Warren. (2003), Fiscal incentive e¤ects of the Australian equalization system,
Economic Record 79, 434–445.
[18] DePeter, J. and G.M. Myers. (1994), Strategic capital tax competition: a pecuniary externality
and a corrective device, Journal of Urban Economics 36, 66–78.
[19] Devereux, M.P., B. Lockwood, and M. Redoano (2008), Do countries compete over corporate tax
rates? Journal of Public Economics 92, 1210–1235.
23

[20] Egger, P., M. Köethenbüerger, and M. Smart (2010), Do …scal transfers alleviate business tax
competition? Evidence from Germany, Journal of Public Economics 94, 235–246.
[21] Eichner, T. and T. Upmann (2012), Labor markets and capital tax competition, International
Tax and Public Finance 19, 203–215.
[22] Exbrayat, N., C. Gaign, and S. Riou (2012), The e¤ects of labour unions on international capitaltax competition, Canadian Journal of Economics 45, 1480–1503.
[23] Feld, L. and G. Kirchgassner (2002), The impact of corporate and personal income taxes on thelocation of …rms and on employment: some panel evidence for the Swiss cantons, Journal of Public
Economics 87, 129–155.
[24] Feldmann, H. (2011), The unemployment puzzle of corporate taxation, Public Finance Review
39, 743–769.
[25] Felix, R.A. (2009), Do state corporate income taxes reduce wages? Economic Review, Federal
Reserve Bank of Kansas City 94 (2), 77–102.
[26] Figuieres, C., J. Hindriks, and G.D. Myles (2004), Revenue sharing versus expenditure sharing ina federal system, International Tax and Public Finance 11, 155–174.
[27] Gillet, H. and J. Pauser (2018), Public input provision in asymmetric regions with labor market
imperfections, German Economic Review 19, 466–492.
[28] Harden, J.W. and W.H. Hoyt (2003), Do states choose their mix of taxes to minimize employment
losses? National Tax Journal 56, 7–26.
[29] Hayashi, M. and R. Boadway (2001), An empirical analysis of intergovernmental tax interaction:
The case of business income taxes in Canada, Canadian Journal of Economics 34, 481–503.
[30] Keen, M. and K.A. Konrad (2013), The theory of international tax competition and coordination:
In Handbook of Public Economics 5, Auerbach, A.J., Chetty, R., Feldstein, M. and E. Saez (ed.)[31] Kikuchi, Y. and T. Tamai (2019), Tax competition, unemployment, and intergovernmental transfers, forthcoming in International Tax and Public Finance.
[32] Köethenbüerger, M. (2002), Tax competition and …scal equalization, International Tax and Public
Finance 9, 391–408.
[33] Köethenbüerger, M. (2004), Tax competition in a …scal union with decentralized leadership, Journal of Urban Economics 55, 498–513.
[34] Köthenbüerger, M. (2008), Federal tax-transfer policy and intergovernmental pre-commitment,
Regional Science and Urban Economics 38, 16–31.
[35] OECD (2017), Tax Policy Reforms 2017: OECD and Selected Partner Economies, OECD Publishing, Paris.
[36] Ogawa, H., Y. Sato, and T. Tamai (2006a), A note on unemployment and capital tax competition,
Journal of Urban Economics 60, 350–356.
[37] Ogawa, H., Y. Sato, and T. Tamai (2006b), Tax competition and public input provision withimperfect labor markets, DEE Discussion Paper No. 06-2, Nagoya University.
[38] Sato, Y. (2009), Capital tax competition and search unemployment, Papers in Regional Science88, 749–764.

24

[39] Wildasin, D.E. (1989), Interjurisdictional capital mobility: …scal externality and a corrective
subsidy, Journal of Urban Economics 25, 193–212.
[40] Wildasin D.E. and J.D. Wilson (2004), Capital tax competition: Bane or boon? Journal of Public
Economics 88, 1065–1091.
[41] Wilson J.D. (1986), A theory of inter-regional tax competition, Journal of Urban Economics 19,
296–315.
[42] Zodrow G.R. (2010), Capital mobility and tax competition, National Tax Journal 63, 865–901.
¼[43] Zirgulis, A. and T. Sarapovas(2017), Impact of corporate taxation on unemployment, Journal ofBusiness Economics and Management 18, 412–426.
[44] Zodrow G.R. and P. Mieszkowski (1986), Pigou, Tiebout, property taxation, and the underprovision of local public goods, Journal of Urban Economics 19, 356–370.

25