Doctoral Dissertation

Study on the Optimal Matching Grant Rates in the
Models of Tax Competition among Jurisdictional
Governments

March, 2021

Graduate School of Social Sciences,
Hiroshima University
Tong Yang

Doctoral Dissertation

Study on the Optimal Matching Grant Rates in the
Models of Tax Competition among Jurisdictional
Governments

Supervisor

Toshikazu Ohsawa, Professor
Yasunori Ouchida, Professor
Takashi Senda, Professor

March, 2021

Graduate School of Social Sciences,
Hiroshima University
Department of Economics
Tong Yang

Acknowledgements

First of all, I would like to thank my advisor, Professor Toshikazu Ohsawa, not only
for his helpful comments on my research but also for his patient instruction throughout
my one year research and five years graduate student life.
Second, I would like to express my deep gratitude to Hikaru Ogawa of The Universityof Tokyo for the valuable comments and suggestions.
Finally, I am also thankful to Professor Hiroshi Futamura, Professor Takashi Senda
and Professor Yasunori Ouchida for their stimulating discussion and constructive
suggestions to improve the quality of this dissertation.

Table of Contents
Chapter 1 Introduction ......................................................................................................... 1
1.1. Fiscal Externalities and Benefit Spillovers .............................................................. 1
1.2. The Matching Grant Program ................................................................................... 2
1.3. Agency Problems and Capital Tax Competition....................................................... 3
1.4. Transaction Costs and the heterogeneity of jurisdictions ....................................... 4
1.5. Overview of This Dissertation ................................................................................... 5
Chapter 2 Productive Effects of Public Spending, Spillover and Optimal Matching
Grant Rate ............................................................................................................................. 7
2.1. Introduction ................................................................................................................ 7
2.2. The Model ................................................................................................................. 10
2.3. Results of Comparative Statics ............................................................................... 14
2.4. Perfectly Mobile Private Capital ............................................................................. 17
2.5. Conclusions ............................................................................................................... 23
Chapter 3 Effect of Agency Costs on the Optimal Matching Grant Rate in a Model of Tax
Competition with Benefit Spillovers ................................................................................. 24
3.1. Introduction .............................................................................................................. 24
3.2. The Model ................................................................................................................. 26
3.3. The Introduction of Tax Competition...................................................................... 29
3.4. Conclusions ............................................................................................................... 34
Chapter 4 A Corrective Device for Large Heterogeneous Jurisdictions in a Two-Period
Economy with Spillover Effects ......................................................................................... 36
4.1. Introduction .............................................................................................................. 36
4.2. The Model ................................................................................................................. 39
4.3. The Stackelberg Equilibria...................................................................................... 41
4.3.1. The Second Period ............................................................................................. 41
4.3.2. The First Period ................................................................................................. 44
4.4. Discussion ................................................................................................................. 48
4.5. Conclusions ............................................................................................................... 51
Chapter 5 Conclusions ........................................................................................................ 53
Appendix 2.1 ........................................................................................................................ 55
Appendix 2.2 ........................................................................................................................ 57
TABLE 2.1 ........................................................................................................................... 58
TABLE 2.2 ........................................................................................................................... 59
TABLE 2.3 ........................................................................................................................... 60
TABLE 2.4 ........................................................................................................................... 61
TABLE 2.5 ........................................................................................................................... 62
Mathematical Note: Derivation of (2-22) .............................................................................. 63

TABLE 3.1. The Relationship between

andTABLE 3.2. Some Simulated Values for,............................................................. 64

and

͛ ͛

........................................ 65

References ........................................................................................................................... 66

Chapter 1 Introduction

1.1. Fiscal Externalities and Benefit Spillovers
This dissertation addresses the relation between the optimal matching grant rates
and different kinds of externalities in the models of tax competition among
jurisdictional governments.
The problem of the under-provision of local public goods in a tax competition model,
in which private capital is mobile among countries, is examined by the literature on
public finance. The under-provision of local public goods arises from different kinds ofexternalities, such as benefit spillovers and fiscal externalities.
Wilson (1986), Zodrow and Mieszkowski (1986), Bjorvatn and Schjelderup (2002)
examined the under-provision of public goods in a tax competition model, in which
private capital is mobile among countries. The public inputs and tax rate are bothinefficiently low resulting from the fiscal externalities among jurisdictions. In thesestudies, in addition to the benefit spillovers of public goods, as a result of tax
competition (race to the bottom), the under-provision of public goods also resulted fromfiscal externalities.
In the literature, four types of externalities have been assumed: external effect ofconsumption, pecuniary externalities, tax externalities, and external effect of public
inputs. The external effect of consumption is that local public goods that one
jurisdiction provides can raise other jurisdictional inhabitants’ utility. Pecuniary
externalities occur when actions by jurisdictional governments affect the profit, or
pecuniary gains, of a third party but not their ability to produce. Tax externalities
mean that if a jurisdictional government finances its local public goods by taxing
movable capital among jurisdictions, the inhabitants in other jurisdictions are also
affected by the tax rate in that jurisdiction. The external effect of public inputs is that
local public inputs that one jurisdiction provides can attract the movable factors of
production in other jurisdictions. However, the external effect of production, which canbe called the fifth type of externality, has not yet been considered. Therefore, in
chapter 2 of this dissertation, the productive externalities (the productive spillover
effects) that directly raise other jurisdictions’ productivities are also considered, whichhave not been analyzed in the literature. For example, if roads, which are goods withthe characteristics of both public goods and public inputs, are built in one jurisdiction,
they can also work for productivity, procuring raw materials, and logistics in otherjurisdictions.
1

In the traditional small-jurisdiction tax competition model, the after-tax return tocapital is a parameter for each jurisdiction. However, there are large-jurisdiction
models in the literature, including Hauer and Wooton (1999), Kanbur and Keen (1993)
and many others. For example, Burbidge and Myers (1994) find that the pecuniary
externality and the fiscal externality, which work in opposite directions, can becancelled out if the capital importer subsidises capital, while the capital exporter taxes
capital. Conversely, DePater and Myers (1994) confirm that the capital importer taxes
capital while the capital exporter provides a subsidy on capital if a lump-sum tax isavailable for the jurisdictions. In addition, DePater and Myers (1994) demonstrate thatthe pecuniary externality among large heterogeneous jurisdictions derived from achange in the capital price, which is affected by distortionary capital taxes, should be
moderately internalised by a corrective device. In chapter 4 of this dissertation, we
follow DePater and Myers (1994) because the strategy of manipulating the terms oftrade is incentive compatible for the jurisdictions.

1.2. The Matching Grant Program
It has been considered that the voluntary provision of public goods and the provisionof local public goods with spillovers are insufficient even when using the ‘Lindhal
mechanism’ because of the existence of free-riders and, therefore, that a matching
grant from a central government to persons or to local governments for (local) public
goods is required to solve the problem.
Both direct and indirect fiscal externalities can be corrected by matching revenue,
matching expenditure or equalisation grants devised by the state and federal
governments. Matching grants are a very particular policy device and this is relevantin the context of the vast literature. The seminal article by Boadway et al. (1989)
shows the relationship between the efficient provision of public goods and an optimal
matching grant rate. Roberts (1992) uses the same model and analyses issues
including the efficiency of subsidies. Akai and Ihori (2002) replace individuals withlocal governments and examine the welfare effects of the central government’s
subsidies for local public goods in a Nash equilibrium model with two types of public
goods, local and central. Using a model without tax competition and a model with tax
competition separately, Ogawa (2006) analyzed how the optimal matching grant ratedepends on the degree of benefit spillovers of public goods, the number of regions, andthe private capital demand elasticity with respect to the tax rate.
In the discussion of the spillover effect of local public goods among jurisdictions, theprevailing view is that spillover will aggravate the under-provision of such goods (seee.g. Boadway and Pestieau and Wildasin 1989). However, an alternative opinion on2this problem is that the spillover effect of local public goods may alleviate theunder-provision of such goods in some situations. Ogawa (2006) argues that thematching grant rate may decrease with spillover effect if the elasticity of capital withrespect to the capital tax rate is significant in a tax competition model. Kawachi andOgawa (2006) find that local governments are more inclined to provide local public
goods efficiently given a sufficient spillover effect in a repeated-game model with large
homogeneous jurisdictions. Additionally, the role of matching grants as a commitment
device has been considered in recent research (see Akai and Sato 2019). Most of the key
assumptions of this dissertation correspond with the conventional wisdom presented in
the studies above.

1.3. Agency Problems and Capital Tax Competition
Agency problems arise in any environment involving principal-agent relationships.
Following the theory of agency, if the principal hopes to ensure that the agent will
make decisions that are optimal for the principal rather than the agent themselves,the differing objectives of the pair make agency costs inevitable. This issue not only
applies to firm ownership structure, but also to the political agency process. Numerous
recent studies have analysed agency problems. Jensen and Meckling (1976) explainthat agency costs arise in any environment without cooperation between agent andprincipal. Besley and Case (1995) demonstrate that the incumbent will improve his
welfare by investing less effort, contradicting the objectives of voters. The marginal
disutility of effort incurring agency costs can be resolved by the threats to thegovernment’s re-election (see, e.g., Seabright 1996). Dahlby (1996) has analysed agency
problems for the case of fiscal externalities. Both direct and indirect fiscal externalities
can be corrected by matching revenue, matching expenditure or equalisation grants
devised by the state and federal governments. Within the vertical fiscal externalities
framework, Dahlby and Wilson (2003) show that the jurisdictional government will
underprovide or overprovide local public goods depending on the assumptions made.
Furthermore, when one considers a model of fiscal federalism featuring both verticaland horizontal fiscal externalities, as Brülhart and Jametti (2019) have observed, tax
competition can represent second-best efficient Leviathan taming by constraining thescope for public-sector revenue maximisation.
The literature (see, e.g., Belleflamme and Hindriks 2005; Besley and Case 1995)
emphasises agency problems and confirms that agency costs can raise the marginal
cost of local public goods provided in jurisdictions. Consequently, jurisdictions will
experience an undersupply of local public goods if the marginal benefit of local publicfunds is unaffected by a given issue. However, few studies deal with agency problems
3

in situations involving horizontal tax externalities. One example of such a study is
Nishigaki and Kato (2016), who show that yardstick competition in the small
jurisdictions model generates additional costs of financing public goods and increases
the seriousness of the under-provision of public goods caused by tax competition. In
contrast to their approach, this dissertation finds that tax competition mitigates theunder-provision of local public goods if agency costs are large enough. Accordingly, theoptimal matching grant rate devised by the federal government should decrease with
the intensiveness of tax competition, considering the large agency costs.
In this dissertation, we introduce tax competition and benefit spillovers into themodel to generalise the effect of horizontal fiscal externalities and benefit spillovers
leading to under-provision of public goods on the agency cost problem.

1.4. Transaction Costs and the heterogeneity ofjurisdictions
The costs of moving faced by private capital, which are also referred to astransaction costs (see, for example, Lee 1997), should not be ignored in a tax
competition model. When the private capital investor has decided to locate in one
jurisdiction and invest in some projects, these projects will usually last for a long
period of time. Once the private capital is invested, it is usually quite difficult toabandon the projects and leave the jurisdiction because of the large moving costs. Even
if the private capital can move freely among the jurisdictions in the initial stage,
imperfect mobility is inevitable in the later stages. Therefore, we must consider bothtransaction costs and inter-temporal effects in a tax competition model. There areseveral relevant studies that consider such issues. For example, Lee (1997) considers
the imperfect mobility of private capital arising from transaction costs in a two-period
tax competition model. He shows that a jurisdictional government will over-provide
local public goods in the second period because of transaction costs and that thejurisdictions may choose a lower capital tax rate than that chosen in a one-period tax
competition model to increase capital stock in the first period. Furthermore, byintroducing a head tax into the model, Ogawa (2000) confirms that the jurisdictional
government may subsidise private capital in the first period to increase capital stock inthe second period when a lump-sum tax is available to a hyperopic jurisdictional
government. This result is compatible with that of the repeated game explained by
Coates (1993). There are also some two-period-model constructions that are relevant toour study (for example, King, McAfee and Welling 1993). However, most of the relevant
literature analysing the transaction costs and dynamic effects does not clarify the4

important roles played by the spillover effects of public goods and the heterogeneity ofjurisdictions in a repeated-game model. Hence, the focus of this dissertation is to
examine these roles.
Furthermore, Ogawa (2007) confirms that, in a tax competition model with large
heterogeneous jurisdictions, the jurisdiction with less efficient production technology islikely to increase its capital tax rate to drive out private capital and obtain substantial
spill-in effects from the other jurisdiction with more efficient production technology.
This means that a distortional capital tax may lead to a more efficient level of local
public goods funding. Thus, by introducing spillover effects into our analysis, we verify
that the jurisdiction with the less efficient production technology may choose to tax
private capital in the first period, assuming that a lump-sum tax is available to it, andreceive substantial spillover benefits from the other jurisdiction with more efficient
production technology in the second period when the jurisdiction is hyperopic andbenevolent, which is quite different from Ogawa (2000). In other words, these
constructions are put together to model an interesting phenomenon and not simply toarrive at predetermined results.

1.5. Overview of This Dissertation
In chapter 2, we analyze the optimal matching grant rate for local public spendingthat has characteristics of both public goods and public inputs. Its spillover effects toother regions are assumed to be regarding consumption and/or production. Contrary tothis approach, traditional analyses of matching grants for public spending have only
focused on public goods and consumptive spillover. By considering these factors,we
obtain some more generalized conclusions and intriguing results; for example, even if
private capital is completely immobile, the productive effect of public expenditure
lowers the optimal matching grant rate when the production spillover is zero, or
smaller than that of consumption, and vice versa.
In chapter 3, we examine the effect of horizontal fiscal externalities on the optimal
matching grant rate in a model where agency costs are inevitable. Agency problems
arise in any environment involving a principal-agent relationship. Because thischapter takes agency costs into account, the main results should differ from thestandard conclusions of the tax competition literature. This chapter finds that thedegree of agency costs and benefit spillovers determine the relationship between tax
competition and the optimal matching grant rate. If agency costs are relatively small,and benefit spillover is zero, the optimal matching grant rate should increase with thefactors of production demand elasticities with respect to the factor tax rate and viceversa. Tax competition thus may ease the inefficiency arising from agency costs only if
5the disutility of effort is so large that the benefits from tax competition exceed the costs
when benefit spillover is zero.
In chapter 4, we introduce the spillover effect of public goods and the heterogeneityof jurisdictions to the capital tax competition literature using a two-period economy. A
clear result is that the revision of a corrective device used by the central government in
the first period to ensure an optimal level of a local public good is provided by a
hyperopic jurisdictional government, significantly depends on the relative size of theincome and spill-in effects in the second period. The relative size of the two effects,
which work in opposite directions, is determined by the tastes and endowments of thejurisdictions, the form of their production functions and the degree of spillovers, among
other factors.

6

Chapter 2 Productive Effects of Public
Spending, Spillover and Optimal Matching
Grant Rate1

2.1. Introduction
In this chapter, the optimal matching grant rate from the central government for
local public spending is analyzed with the characteristics of both public goods andpublic inputs funded by jurisdictional governments.
A matching grant from central government to persons is a strong instrument tosolve the problem of an insufficiently voluntary provision of public goods (see e.g.,
Boadway et al. 1989). However, that study does not assume tax competition formovable private capital; that is, the reason for the under-provision of public goods isonly free-riders arising from the benefit spillovers of public goods2.
Wilson (1986), Zodrow and Mieszkowski (1986), Bjorvatn and Schjelderup (2002)
examined the under-provision of public goods in a tax competition model, in which
private capital is mobile among countries. In these studies, in addition to the benefit
spillovers of public goods, as a result of tax competition (race to the bottom), theunder-provision of public goods also resulted from fiscal externalities.
Zodrow and Mieszkowski (1986) presented a tax competition model in which public
expenditure enters a production function as a public input and increases the marginal
productivity of capital. The public inputs and tax rate are both inefficiently low
1This chapter is based on Ohsawa and Yang (2020).

2

Other studies that have analyzed the matching grant of the central government to the voluntary provision of

public goods funded by persons or jurisdictional governments in a model without fiscal competition are
Feldstein(1980), Warr(1982), Drissen(1987), Feldstein(1987), Glazer and Konrad(1996), Andreoni andBergstrom(1996), Kirchsteinger and Puppe(1997), Roberts (1992), and Akai and Ihori (2002). For example,
Feldstein (1980), Drissen (1987), and Feldstein (1987) examined whether tax subsidies are more efficient thandirect expenditure on a voluntary activity favored by public policy. In addition, Warr (1982), Glazer and Konrad(1996), Andreoni and Bergstrom (1996), and Kirchsteinger and Puppe (1997) analyzed the effect of increasing amatching grant rate for an act of charity or the voluntary provision of public goods. Roberts (1992) showed theboundary of efficient subsidies. Akai and Ihori (2002) replaced individuals with local governments and examined
the welfare effects of central government subsidies for local public goods in a Nash equilibrium model with two
types of public goods: local and central.

7

resulting from the fiscal externalities among jurisdictions. However, Noiset (1995)
indicated that the assumption made by Zodrow and Mieszkowski (1986) is meaningless,and the public inputs and tax may be either too high or too low. Matsumoto (1998)
showed the relationship between the type of production function and theunder-provision of public inputs in a tax competition model3. Keen and Marchand(1997) sorted local public expenditure into local public goods and local public inputs,
and stated that local public inputs are over-provided relative to local public goods inlocal public expenditure.
Zodrow and Mieszkowski (1986) and Keen and Marchand (1997) analyzed public
goods and public inputs separately as different public expenditure. However, as Keenand Marchand (1997) stated, it may be quite difficult to definitely differentiate local
public goods and local public inputs from local public expenditure 4 , for example,
education, harbors, roads, and clean air. Specifically, roads are necessary forfacilitating production (local public inputs), in addition to the enjoyment of leisure forresidents in the jurisdiction and other jurisdictions (local public goods)5.
Using a model without tax competition and a model with tax competition separately,
Ogawa (2006) analyzed how the optimal matching grant rate depends on the degree of
benefit spillovers of public goods, the number of regions, and the private capitaldemand elasticity with respect to the tax rate. However, local public inputs were notconsidered in the two models.
Therefore, in this chapter, we assume that jurisdictional governments provide local
public expenditure with the characteristics of both public goods and public inputs; thatis, local public goods for which the provision level is a variable in both the residents’utility function and the regional production function, and the spillover effects of which
can influence the utility of inhabitants in other regions. First, we analyze the case inwhich private capital cannot move among jurisdictions. Then, we analyze the case in
3

Matsumoto (1998) showed that public inputs are over-provided if they are factor-augmenting public inputs, andunder-provided or over-provided if they are firm-augmenting public inputs, but public inputs are under-provided if
the number of firms is endogenous, even if they are firm-augmenting public inputs. Furthermore, in a tax
competition model, Dhilion, Wooders, and Zissimos (2007) and Bénassy-Quéré, Gobalraja, and Trannoy (2007),and Hindriks and Myles( (2013) chapter 20 ) analyzed the condition in which public inputs are under-provided byjurisdictional governments.
4

Keen and Marchand (1997) stated: “In practice, of course, many items of public expenditure have bothconsumption and production effects, as indeed is clear from the examples just given.” (p 34 note 3)
5

Education can be seen as raising productivity, and inhabitants’ utility directly and subsequently improves byreceiving it. Its spillover effects of consumption and production can also be expected in other regions. Concerning
environmental policies, if one region’s air becomes cleaner, the air in other regions also becomes cleaner, to somedegree, and inhabitants’ utility and productivity improve because they become healthier.

8which private capital can move freely among jurisdictions. By assuming a good withthe characteristics of both public goods and public inputs, we obtain new conclusions
that cannot be obtained from the above-mentioned studies in which local public goods
and local public inputs were considered separately. For example, even if private capitalis assumed to be immovable, the effect of raising productivity as a public input may
alleviate the under-provision of public goods because of the external effect ofconsumption.
Moreover, in this chapter, the productive externalities (the productive spillover
effects) that directly raise other jurisdictions’ productivities are also considered, whichhave not been analyzed in the literature. For example, if roads, which are goods withthe characteristics of both public goods and public inputs, are built in one jurisdiction,
they can also work for productivity, procuring raw materials, and logistics in otherjurisdictions. Of course, this effect may be zero in many cases6.
In the literature, three types of externalities have been assumed: external effect ofconsumption, tax externality, and external effect of public inputs. The external effect ofconsumption is that local public goods that one jurisdiction provides can raise otherjurisdictional inhabitants’ utility. Tax externality means that if a jurisdictional
government finances its local public goods by taxing movable capital among
jurisdictions, the inhabitants in other jurisdictions are also affected by the tax rate inthat jurisdiction. The external effect of public inputs is that local public inputs that one
jurisdiction provides can attract the movable factors of production in otherjurisdictions. However, the external effect of production, which can be called the fourth
type of externality, has not yet been considered.
If we separately assume that the characteristics of public goods or public inputs andbenefit spillovers are zero, this model can be regarded as that in Zodrow andMieszkowski (1986), which introduced a matching grant from central government tojurisdictional governments for local public goods or local public inputs. For example, if
jurisdictional governments provide local public expenditure without the characteristics
of public goods (for example, both the effect that directly raises the jurisdictional
inhabitants’ utility and the external effect of consumption are zero) and the external
effect of production, this model can be thought of as that in Zodrow and Mieszkowski(1986), which introduced a matching grant from central government to jurisdictional
governments for local public inputs. Conversely, if jurisdictional governments provide
local public expenditure without the characteristics of public inputs (i.e., both theeffect that directly raises jurisdictional production and the external effect of production
6

Even if we consider that the productive spillovers are irrelevant or we estimate them as zero, many new results

can still be obtained in the chapter.

9

are zero) and the external effect of consumption, this model can be thought of as that in
Zodrow and Mieszkowski (1986), which introduced a matching grant from central
government to jurisdictional governments for local public goods.
The remainder of this chapter is organized as follows: In Section 2.2, we present themodel in the absence of factor mobility, and derive the optimal matching grant rate for
local public expenditure with the characteristics of both public goods and public inputs.
In Section 2.3, we show the generalized and intriguing results, and discuss theintuition behind them. In Section 2.4, we introduce perfectly mobile private capital
among regions, and achieve more generalized results. Finally, in section 2.5, we
provide conclusions.

2.2. The ModelIn our model, there are

identical jurisdictions, and in each jurisdiction( = 1,2, … , ), there is a single immobile resident, with preferences defined by a strictly
quasi-concave utility function u( ,numeraire good and) , whereis the level of a local public good that the resident consumes.

The local public good consumption in jurisdiction=+∑is the consumption of a private
is defined as

,

(2-1)

where

is the provision of the local public good by a jurisdictional government , and(0 ≦≦ 1 ) is a parameter that indicates the degree of benefit spillover fromconsumption.
We also assume thatis a public good and/or a public input (public capital), which

may have the external effects of production, for example, environmental policies, or
policies for the maintenance and improvement of items such as public security,
education, and roads7.
Therefore, we can see thathas the effect of directly increasing the utility of its

jurisdiction’s resident, and/or has the effect of increasing its jurisdiction’s productivity.
Additionally,

may have the effect of increasing the utilities of other jurisdictions’

residents and/or have the effect of increasing other jurisdictions’ productivity.
We assume that the aggregate production function in jurisdictionis, whereis the level of the aggregate amount of local public input (private capital is assumed
to be fixed in this and the next section, and therefore suppressed in these sections). We

7

For the reason why we use the assumption about the good that local governments provide, see section 2.1 in thischapter.

10also assume that 0 ≦

(

) and

(

) ≦ 0 are satisfied8.

is defined as
=
where

,

+ ∑
(0 ≦

(2-1 )≦ 1) is a parameter for indicating the degree of benefit spillover fromproduction. Jurisdictional governments impose lump-sum taxes on the resident as thesource of revenue for local public good provision (in this and the next section, we
assume that private capital is fixed so that we can see that tax is levied on private
capital, as in Section 2.4, in which private capital is mobile), and central government
imposes lump-sum taxes on the resident as the source of revenue for the matching
grant. The resident’s budget constraint is given by
=
where

−

(2-2)

− ℎ,

is the lump-sum tax imposed by jurisdiction

and ℎ is the tax imposed bycentral government, which is assumed to be identical for all jurisdictions.9 The budget
constraint of jurisdictional government
denotes89can be given by.

Boadway et al. (1989) and Lee (1995) showed that even if a uniform subsidy rate is assumed for all identicalindividuals, it may be possible for lump-sum taxes to differ.
However, in reality, at least in developed countries, it must be impossible for a government to impose different
lump-sum taxes on identical individuals in a district that it governs because such a tax policy may conflict withthe tax principle of “horizontal equity;” therefore, its bill will fail to pass. Regarding “horizontal equity,” forexample, see Rosen and Gayer (2010).
It is also impossible for a government to impose different lump-sum taxes, even on heterogeneous individuals ina district that it governs because, from the tax principle of equity, there is no reason to impose such discriminatory
taxes.
In Japan, for example, there are and will be some types of lump-sum taxes. First, at present, all local
governments impose lump-sum taxes on all their taxpayers (prefectures: 1,000 yen, and cities: 3,000 yen per head,in principle). Second, 35 prefectures and Yokohama city impose forest environmental taxes on all their taxpayers(300–1,200 yen per head), which have been uniform lump-sum taxes since 2017. For example, it is 500 yen perhead in Hiroshima prefecture and 1,200 yen in Miyagi prefecture, for all taxpayers. In Kanagawa prefecture only,
it includes a part that depends on taxpayers’ income. Finally, central government will impose the forest
environmental tax (1,000 yen per head) from 2024. Local and central governments can change the tax rate, butcannot impose a discriminatory tax rate. The revenue of national and local forest environmental taxes must be
used for environmental policies. Therefore, these taxes and policies can be viewed as similar to the lump-sum
taxes and local public goods that we analyze in this chapter.

11

+

=

,

(2-3)

is the matching grant that jurisdictional government

where

government. Hence, matching grant

receives from central

can be given by

,

=
where

(2-4)

is the rate of the uniform matching grant10. Fiscal revenue serves to financethe provision of local public goods with the characteristics of local public input. ℎ in(2-2) should satisfy the central government’s budget constraint, which is given by
∑

.

= ℎ=∑

(2-5)

This model has two stages:
In stage1, central government chooses the national tax ℎ and the matching grantrate.

In stage 2, the jurisdictional government
public goods, taking ℎ andchooses the local tax

and local

as given.

Jurisdictional governments may not be necessarily efficient because they only careabout their inhabitants’ welfare, and are not interested in the externalities of local
public goods, if

and ℎ are not optimal.

Central government cares about all individuals in the country; however, it canneither choose private goods

nordirectly, but it wants to determine

and ℎ,

which leads to an efficient provision of local public goods by jurisdictional governments
in non-cooperative equilibrium, according to the Pareto-optimal condition derived
below.
Jurisdictional government

wishes to maximize the utility of its resident subject to(2-1), (2-1c), (2-2), (2-3), (2-4), taking the tax rates

and the provision of local public

goods with the characteristics of local public capital of other jurisdictions as given.
Then, the maximization problem is defined as

,

10

=

( , ),

Boadway et al. (1989) and Ogawa (2006) assumed such local and central lump-sum taxes and the matching

grant rate similar to, ℎ and

.

12

s.t.

∑

,

=

−

− ℎ,

=

+ ∑

,

=

+

+

,

=
.

=

We use the substitution method and differentiatewith respect to, and thefirst-order condition can be written as=− (1 −+(2-6)) = 0.

The Pareto-optimal condition, however, is derived as,( = 1,2, … , )∑s.t. ∑

Let

+∑

=∑

(

).

denotes the Lagrange multiplier of the constraint above. Then, the Lagrangefunction is given by( , )=∑

Differentiating+∑−∑( , ) with respect to

and

=

+

∑

=

+

= 0,

+

+

∑

1−

− ∑

which can be rewritten as
13

(

)㸬

, we obtain() = 0,

∑
(

)㸫 ∑

(

)

=

(2-7)

.

We have considered that all jurisdictions are identical. Therefore, (2-7) at thesymmetric equilibrium can be rewritten as

(
(

[

)
)]

(

,

=

)

(2-8)

where the jurisdiction-specific subscripts

andare omitted.

A comparison of (2-6) and (2-8) shows that the optimal matching grant rate, whichcentral government should choose, is given by

=

where

(

(

)
(

)

+

(

)(

)

(

(

)

(2-9)

),

) is the marginal productivity of the local public good with thecharacteristics of local public capital. If (2-9) holds, then the budget constraint ofcentral government ∑government∑,is satisfied, and the goal of central= ℎ=∑( , ) is reached.

=∑

From (2-9), it should be noted that if
can be rewritten as

(

=

)
(

) = 0, the optimal matching grant rate(

. In this case, m is the optimal matching grant rate)with respect to the normal local public good Ogawa (2006) indicated. Furthermore,when= 1, m is the same as the optimal matching grant rate Boadway et al. (1989)

derived.
(2-9) can be rewritten as

=

(

)
(

)

[1 −

(

)] +

(

)
(

)

(

(2-9’)).

2.3. Results of Comparative Statics
The following results can be obtained in consideration of the preceding analysis.
First, differentiating (2-9), we obtain the relationships between
14

and

,

and

:

=

=

=

(

){

(

)[
(

[

(

)]}

,

)]

(2-10)

)
) (
,
(
)

(

(

[
[

(2-11)

(

)]
(

)]

)

.

(2-12)

㸼0 can be determined easily. From (2-8), if the rate of marginal substitution
between G and x is positive, then 1 −and()[1 + ( − 1)] > 0. Therefore,

>0

> 0.

Ogawa (2006) showed that

> 0, and> 0 in the case where

(

)=

= 0.

Therefore, we obtain the following results based on the analysis above.
Proposition 2.1 Even if a good is not only a public good but also a public input, its
optimal rate of the matching grant increases with the external effect of consumption
and production, and with the number of jurisdictions.
Second, (2-9) shows that if the local public good has the characteristics of local public
input, then compared with the case without those characteristics,> , lower when

< , and the same whenis higher when

= .

The results suggest the following. We assumed that the local public good has thecharacteristics of local public input, so it has a positive effect on regional productivity.
Each jurisdictional government has been aware of this point and will spontaneously
increase the provision of the local public good. For this reason, the matching grant thatjurisdictional governments receive from central government should be reduced, even if
weconsider

(

)
(

)

1㸫 (

the

entiresocialwelfare.

Furthermore,

we

can

see

from

) in (2-9’) that the effect of lowering the optimal rate of thematching grant increases with.

However, as the characteristics of local public capital also generate the external
15

effect of production, we can see from

(

)
(

)

) in (2-9c) that m increases with(.

Hence, we obtain the following result.
Proposition 2.2

Compared with the case without the characteristics of local public

input, the optimal matching grant rate with respect to the local public good with thecharacteristics is higher when the external effect of production is larger than that ofconsumption, lower when that effect is smaller than that of consumption, and the same
when that effect is the same as that of consumption.
(See the column “immobile private capital” in Table 2.1.)

Third, when

= 0, we can obtain m =

(

)
(

)

) from (2-9c). The increase1㸫 (of regional productivity reduces the distortion of resource allocation caused by theeffect of. Consequently, the optimal matching grant rate with respect to the local

public good with the characteristics of local public input is lower than that rate
regarding the normal local public good without the characteristics11. Furthermore, thehigher the marginal productivity of this type of public good(), the lower the rate.
Proposition 2.3

When there is no external effect of production ( = 0), the optimal

matching grant rate with respect to the local public good with the characteristics ofpublic input is lower than that rate without the characteristics. Additionally, thehigher the marginal productivity of such a public good, the lower the optimal rate withrespect to the local public good with those characteristics.
If the good that is both a public good and public input raises productivity in theregion ( () > 0), the optimal matching grant rate m should be higher than, lower

than, or equal to the rate with respect to the public good without those characteristics
when

> ,

<

, or

㸻 , respectively. This is because the way in which thecharacteristics of public input that raise productivity affect m is different throughthan through .
The effect of raising productivity (productive effect) through. Therefore, whether the effect raises or lowers m depends

effect throughlowerson whether

is larger thaneffect through
11

If

(

) = 0,

onraises m, but theor not, or we may say the direction of the productiveis the opposite of that throughis the same as Ogawa (2006) obtains.

16.

)㸻0),

raises m because the public goodraises utilities in other regions directly through. However, if the productive effect isIf the productivity effect is zero ( () > 0), then the region recognizes that the public good raises the

positive ( (productivity in the region, and voluntarily provides more public good. Therefore,
central government does not need to correct the insufficient provision of public goods
via a matching grant to some degree.
However, the effect througheffect is zero,is different from that through . If the productive

does not affect welfare in other regions, and it is meaningless. Thus,

the productive effect through raises, provided the productive effect is not zero.

This relationship has not been recognized before now. For example, Zodrow andMieszkowski (1986) used the Nash equilibrium model to show that a tax on movable
capital results in the insufficient provision of a public good and public input. However,they analyzed them separately, that is, not simultaneously. Moreover, they neither
assumednor. Keen and Marchand (1997) analyzed them simultaneously.

However, they considered them as separate goods, not as a good that is both a public
good and public input.

2.4. Perfectly Mobile Private CapitalThus far, we have demonstrated the public input in the aggregate production
function, but have not explicitly shown private capital. Additionally, private capital
was immobile among regions, and therefore, was suppressed. However, we can see
models of tax competition among the jurisdictions in many studies, such as Zodrow andMieszkowski (1986) and Ogawa (2006 sec.3), in which private capital is perfectly
mobile among regions, and jurisdictional governments can raise revenue only with adistortional capital tax in their own regions. Therefore, in the present chapter, we now
rebuild the model of tax competition in which private capital is also put into production
explicitly and is perfectly mobile among regions, and jurisdictional governments canraise revenue, not with a lump-sum tax but rather a distortional capital tax12.
As before, we assume that there arejurisdiction

identical jurisdictions, and in each

( = 1,2, … , ), there is a single immobile resident and the resident

provides labor,= 1. The production functions in all jurisdictions are identical, andlabor is not shown explicitly in those functions; that is, the production function inregionis simply given by
=

12

,

,

(2-13)

As we noted, the lump-sum tax used before can be seen as a tax on immobile private capital.

17

is the private capital in region . There is perfect private capital mobility. In

whereequilibrium, therefore, the after-tax return to capital is equalized across jurisdictions:k ,for all

−

=

,

−

( = 1,2, … , ), where

(

=
,

≡∂

tax rate per unit of capital in jurisdiction

,
and

≠

)

/∂

(2-14)> 0,< 0,

is theis the after-tax return to private

capital in the country.
The total supply of private capital in the country is fixed atsuch that.

=∑(2-15)The budget constraint of the resident requires
=

,

−

,

+

− ℎ,

(2-16)

is the initial endowment of private capital in jurisdiction

whereand ℎ is the

lump-sum tax that central government imposed. Substituting (2-14) into (2-16), (2-16)
can be rewritten as
=

,

−

+

−

(2-16’)− ℎ.

The jurisdictional government budget constraint is given by+=.

(2-17)As before, jurisdictional government

receives a matching grant from central

government to provide the local public good with the characteristics of local public
input. Hence, the following condition holds:=.

(2-18)The lump-sum tax imposed by central government ℎ will be chosen to satisfy that
central government’s following budget constraint:
∑

= ℎ=∑

.

Jurisdictional government(2-19)
aims to maximize the following utility function of its
18

:

resident by choosing its tax rate

㸩 ∑

=

㸫

㸪

,

+ (

㸫

)㸫ℎ .

(2-20)

The first-order condition yields

(

)

where

(

,

(

)

,

㸧[

(

)]

≡∂

,

=

13,

/ ∂

(2-21)and E is the private capital demand elasticity

with respect to the tax rate in jurisdiction , that is,∀ (see Appendix≡

2.1 for derivations)14.
Because in this model, the good that jurisdictional governments provide is same as
Zodrow and Mieszkowski's (1986) type of public input, if=== 0, the capitaltax not only drives out private capital but also attracts it through the characteristics of
input (tax revenue is used to produce the public good with the characteristics, whichraises the marginal productivity in the jurisdiction and attracts private capital)15.
However, as

rises, the latter effect is weakened because the marginal productivity ofother jurisdictions rises. Ultimately, if= 1, the effect vanishes. Therefore,

positive or negative. Moreover, we assume that

< 0, and that if

can be= 1,

is anegative fixed value, whether the public good has the characteristics of public input or
not16.

13

Bjorvatn and Schjelderup (2002) shows showed that there is no incentive for tax competition, if= 1, without

the characteristics of input or the productive spillover. In this case, the tax competition distortion is eliminated, if=
14= 1㸪or if

Note that= 1 and= 0㸬

is not an absolute value but an ordinary value because

can be positive or negative, whilecommonly, elasticity is expressed by an absolute value.
15

Zodrow and Mieszkowski (1986) excluded the possibility of the over-provision of public input, that is, thepossibility thatwill raise, as a result of their “stable”conditions. However, Noiset (1995) indicated that theassumption made by them does not have the meaning, and the tax and the public input may be either too high or
too low. Matsumoto (1998) showed the relationship between the type of production function and theunder-provision. See footnote 2.
16

We may easily understand this if

is defined as≡ 㸩(1 − )19

concretely, whereIf private capital is fixed, we have

= 0 in (2-21) and it is the same as (2-6).

(2-21) can be compared with the Pareto-optimal condition (2-8), and the optimal rateof the matching grant is given by17㸻If

㸫1 㸫 (1 − ) +( 㸫 )( − ) .

㸩

(2-22)= 0, (2-22) is the same as the form that Ogawa (2006) derived, where private

capital is perfectly mobile among regions. Additionally, if= 0, (2-22) is the same

form as (2-9). Note that from (2-8), if the rate of marginal substitution between
andis positive, then 1 −()[1 + ( − 1)] > 0.

Now, we can derive the following proposition from (2-22) directly.
Proposition 2.4 Whether(the optimal matching grant rate with respect to the

local public good with the characteristics of the local public input) is higher than(the rate in the case without the characteristics) or not is shown in Table 2.1.
is lower thanFor example,
㸼 , but

is higher than

, when 㸼, when= 1 andand㸼< , or when

, unless<and= 1 (because when

= 1, the tax effect of attracting private capital vanishes).
Proposition 2.5 Even if private capital is perfectly mobile among regions, when㸼(1 − )/ , we still can derive Proposition 2.3. However, we cannot do so if≦ (1 − )/ .
(See the rows "0 ≤<and(< 0) ≡
䠙= 㸯", "0 ≤(> 0) ≡<

< 1" , and " 0 =∀ . Therefore, ε is the private capital demand elasticity.

with respect to the tax rate, when public inputs are fixed, andis the private capital demand elasticity withrespect to the tax rate through the characteristics of public input, when
17

When

=

= 0,

in Table 2.1.)== 㸫 . Therefore, if< 0, then=0 (> 0. The condition=).

< 0 is the same as the “stablecondition” in Zodrow and Mieszkowski (1986). They explain this as follows: “We also assume the model is stable inthe sense that each jurisdiction perceives that raising taxes will drive out capital … otherwise, taxes would always
be raised”(Zodrow and Mieszkowski (1986) p. 363). Noiset (1995) indicated that, under that assumption, there isan under-provision of public input. In such a case, central government will lead each jurisdiction to increase its
public input, which means

> 0.

20

with

Moreover, using (2-22), we obtain the relationships of
we assume that if we change

and, then([From (2-8), we have1 −> 0 is

>

㸫

㸫

and

(2-23)[1 + ( − 1)] > 0. Therefore, a sufficient condition for㸯(㸩 +( − ))Therefore, if> 0 and≧ , or㸼− (1 − )㸫We also have the relationship between㸩 () ∙[ ([(> 0 is<0= 1)19.
andFurthermore, we can derive the relationship between

where 1 −:). This condition is the same as that in Ogawa (2006), if(for example, if 㸻0 or=, where> 0, 18. Additionally, the necessary and sufficient condition for(i.e., − <

=

, and

[1 + ( − 1)]}.

{1 −

)]

,

will not change with them, that is,

is constant. We have the following relationship between

=

,

)
)]

> 0 because

as follows:.

(2-24)

and≧ , thenandas follows:> 0 because

]

,

< 0.

(2-25)< 1 to satisfy the positive denominator on theleft-hand side of (2-8). Hence, the numerator on the right-hand side of (2-25) is positive,when>or when= 1, unlessthe sufficient condition for== 0. For 1 > ∀ ≧ 0,

≦ −1. Therefore,

㸼0 is 㸼 − 120. We can derive the following proposition

from the above.

18

In Ogawa (2006), E is necessarily negative, whereas in this model, it may be positive or negative because

jurisdictional governments provide the good, which may have the characteristics of the public input.
19

Strictly speaking, Ogawa(2006) argues that if − >20This result is the same as that in Ogawa (2006), if, then m is not a monotonous function of< 0 (for example, if

21= 0 or= 1).

.

Proposition 2.6 If private capital is perfectly mobile among regions, the optimal
matching grant rateshould increase with the external effect of consumption

long as the private capital demand elasticity with respect to the tax rateor larger than (1 − )/ . However, whenand> (1 − )/Additionally,
=

=

when

should decrease with> 1/( − 1), or whenhas no effect on= 0,

is positive,. Ifincreases with the external effect of production≧ , thenshould increase with

= 0. When< (1 − )/ ,

, as

.

㸼 − 1, unless.

The proposition above is shown in Tables 2.2, 2.3, and 2.4.
Next, we derive the relationship between
toand. Differentiatingwith respect, we obtain

(

=

where

)

(
(

)
)

= 0 only if(2-26)≤ 0,= 1; that is, the benefit spillovers on the external effect of=

production and consumption are perfect and equivalent (see Appendix 2.2 forderivations). This case indicates the following result: as the degree of spillovers
increases, the jurisdictional government takes more of the external effect into account(see Ogawa [2006]). For== 1, there are perfect spillovers and there is no tax

competition at all (see Bjorvatn and Schjelderup [2002]). Hence,

has no effect on

One may think that it is not appropriate to differentiatewith respect tobecauseis assumed to be an endogenous variable. However, even if we use theassumption≡ 㸩 (1 − ) , as in note 16, and differentiate m with respect

to , , or (1 − ) , the results can be obtained easily and are the same as≤ 0,.

≤ 0, and

(

)

≤ 0.

Proposition 2.7 The optimal matching grant rateshould decrease as the private

capital demand elasticity with respect to the tax rate
has no effect on; that is,(or ,, (1 − ) ). However,

only if the benefit spillovers on the external effect of production andconsumption are perfect and equivalent; that is,
The above proposition is shown in Table 2.5.

22

=

= 1.

2.5. Conclusions
In this chapter, we analyzed the relationship between the optimal matching grantrate, the local public expenditure with the characteristics of public goods and public

inputs, and the degree of spillovers:and. In the literature, thus far, public goods

or public inputs have only been analyzed separately. Thus, by considering thecharacteristics,

and,we obtained some intriguing results. For example, whether

the effect of public inputs on the optimal matching rate for the expenditure is positiveor negative depends on whether

is larger than: if

⋚ , then⋚, whereis the optimal matching grant rate with respect to the public goods with thecharacteristics of public inputs andis the rate with respect to the public good

without the characteristics. We also obtained some more generalized conclusions21.
These are shown in Propositions 2.1–2.7 and Tables 2.1–2.5. Even if the production
spillover is estimated to be zero, most of the results remain valid.
Finally, we should more specifically present the implication of policies from theabove results. As the characteristics of their inputs and their productive spillover effecthave not been considered until now, if we consider those, the matching grant rate forpublic goods, such as environmental policies, should be lower, and the rate for public
goods, such as education or canals, should be higher.

21

For example, if the public goods lack the characteristics of public inputs and the external effect of production,

the results are the same as those in Ogawa (2006), irrespective of whether private capital is perfectly mobile
among regions.

23

Chapter 3 Effect of Agency Costs on the
Optimal Matching Grant Rate in a Model of
Tax Competition with Benefit Spillovers22

3.1. Introduction
Agency problems arise in any environment involving principal-agent relationships.
Following the theory of agency, if the principal hopes to ensure that the agent will
make decisions that are optimal for the principal rather than the agent themselves,the differing objectives of the pair make agency costs inevitable. This issue not only
applies to firm ownership structure, but also to the political agency process. Numerous
recent studies have analysed agency problems. Jensen and Meckling (1976) explainthat agency costs arise in any environment without cooperation between agent andprincipal. Besley and Case (1995) demonstrate that the incumbent will improve his
welfare by investing less effort, contradicting the objectives of voters. The marginal
disutility of effort incurring agency costs can be resolved by the threats to thegovernment’s re-election (see, e.g., Seabright 1996). Belleflamme and Hindriks (2005)
observe that yardstick competition between jurisdictions exerts both discipline andsorting effects within the political agency framework. Voters can mitigate agency costs
below the Leviathan level by adopting a successful voting strategy (as noted by Wrede
2001).
Dahlby (1996) has analysed agency problems for the case of fiscal externalities. Bothdirect and indirect fiscal externalities can be corrected by matching revenue, matchingexpenditure or equalisation grants devised by the state and federal governments.
Within the vertical fiscal externalities framework, Dahlby and Wilson (2003) show thatthe jurisdictional government will underprovide or overprovide local public goods
depending on the assumptions made. Furthermore, when one considers a model offiscal federalism featuring both vertical and horizontal fiscal externalities, as Brülhart
and Jametti (2019) have observed, tax competition can represent second-best efficientLeviathan taming by constraining the scope for public-sector revenue maximisation.
Following Zodrow and Mieszkowski (1986), local public goods provided by local
governments are underprovided through interregional property tax competition.
Therefore, if lump-sum tax cannot be operated by the local government, the22This chapter is based on Yang (2020).

24

under-provision of local public goods financed by the distortionary tax is inevitable in astandard model of tax competition. To solve this issue, it is often argued that anintergovernmental matching grant devised by the federal government can be
introduced to ease the exorbitant marginal cost of local public funds deriving from thedownward pressure on tax rates, assuming that the federal government is benevolent
and omniscient (see, e.g., Dahlby 1996).
In the discussion of the spillover effect of local public goods among jurisdictions, theprevailing view is that spillover will aggravate the under-provision of such goods (seee.g. Boadway and Pestieau and Wildasin 1989). However, an alternative opinion on
this problem is that the spillover effect of local public goods may alleviate theunder-provision of such goods in some situations. Ogawa (2006) argues that thematching grant rate may decrease with spillover effect if the elasticity of capital withrespect to the capital tax rate is significant in a tax competition model. Kawachi andOgawa (2006) find that local governments are more inclined to provide local public
goods efficiently given a sufficient spillover effect in a repeated-game model with large
homogeneous jurisdictions.
The literature (see, e.g., Belleflamme and Hindriks 2005; Besley and Case 1995)
emphasises agency problems and confirms that agency costs can raise the marginal
cost of local public goods provided in jurisdictions. Consequently, jurisdictions will
experience an undersupply of local public goods if the marginal benefit of local publicfunds is unaffected by a given issue. However, few studies deal with agency problems
in situations involving horizontal tax externalities. One example of such a study is
Nishigaki and Kato (2016), who show that yardstick competition in the small
jurisdictions model generates additional costs of financing public goods and increases
the seriousness of the under-provision of public goods caused by tax competition. In
contrast to their approach, this chapter finds that tax competition mitigates theunder-provision of local public goods if agency costs are large enough. Accordingly, theoptimal matching grant rate devised by the federal government should decrease with
the intensiveness of tax competition, considering the large agency costs.
In this chapter, we introduce tax competition and benefit spillovers into the model togeneralise the effect of horizontal fiscal externalities and benefit spillovers leading tounder-provision of public goods on the agency cost problem.
This chapter is organised as follows. The inefficiency of the agency cost problem in aclosed economy with consumption spillover effect among jurisdictions is set out insection 3.2. We introduce tax competition into the model in section 3.3 to generalise theeffect of horizontal fiscal externalities leading to under-provision of public goods on theagency cost problem. Finally, Section 3.4 draws conclusions.

25

3.2. The ModelIn this section, we start with the case where private capital is not mobile. There are
identical local jurisdictions, where each jurisdiction( = 1,2, … , ) has a single

immobile resident, and preferences are defined by a strictly quasi-concave utility), wherefunction u( ,is the consumption of a private numeraire good andthe local public good. The local public good consumption in jurisdiction
=
where
and

+

is defined by

,

∑

is

(3-1)

is the provision of the local public good by the jurisdictional government ,≦1㸧indicates the degree of benefit spillover. We assume that the㸦0≦jurisdictional governments are partly self-interested. Therefore, the welfare function ofjurisdictional governmentis written by=

are the agency costs, with[ ( )] > 0,

[ ( )] < 0,( , ), where[ ( )] +[ ( )][ ( )] < 0. Additionally,( ) denotes the variation in effort, which reflects differences in incumbent types,( ) > 0,with( ) > 0,( ) > 023 . Although the jurisdictional governments

desire more excess rents, they must consider the utility of residents in theirjurisdictions to ensure their own re-election and access to rents. However, the central
government is assumed to be benevolent. For simplicity, we assume that the aggregate
output is exogenous in each jurisdiction, and can be expressed by. The jurisdictional

governments impose lump-sum taxes on the resident to fund local public good
provision, and the central government imposes lump-sum taxes on the resident to fundthe matching grant. The budget constraint of the resident is given by
=
where

−

(3-2)

− ℎ,

is the lump-sum tax imposed by jurisdiction

and ℎ is the tax imposed bycentral government, which is assumed to be identical for all jurisdictions. The budget
constraint of the government of jurisdiction
+
where

=

can be given by

,

(3-3)

is the matching grant the government of jurisdiction

central government. Hence, the matching grant=
23,

Note that agency costs decrease inreceives from thecan be given by
(3-4)

as

< 0.

26

is the rate of the uniform matching grant. We assume a uniform matchingwheregrant rate because the jurisdictions are homogeneous at the symmetric equilibrium24.
Fiscal revenue finances the provision of local public goods. In (3-2) ℎ should satisfy
the central government’s budget constraint, which is given by
.

= ℎ=∑

∑

(3-5)

This model involves two stages:
In stage 1, the central government chooses national tax rate ℎ and matching
grant rate.

In stage 2, the jurisdictional government
public good, taking ℎ andchooses local tax rateand local

as given.

The central government cares about all individuals in the country, and wants to set
and ℎ so as to realise efficient provision of local public goods by jurisdictional
governments in a non-cooperative equilibrium, according to the Pareto-optimal
condition derived below.
The jurisdictional government

wishes to maximise the objective function

subject to (3-1), (3-2), (3-3), (3-4), taking the tax rates
public goods of other jurisdictions

and the provision of local

as given. The maximisation problem is thendefined as,=[ ( )] +

s.t.

=

−

+

=

(3-6)

( , )

−ℎ

=
=

+

∑

.

We use the substitution method and differentiatewith respect to, and thefirst-order condition can be written as25=+

− (1 −

)

(3-7)

= 0.

24

See Boadway et al (1989) and Ogawa(2006).

25

The second-order condition is assumed to be satisfied according to the objective function.

27

We have considered all jurisdictions to be identical. That is,
and

=

,

=

. Therefore, (3-7) at the symmetric equilibrium can be rewritten as+− (1 −

)

=

(3-8)

= 0,

where the jurisdiction-specific subscripts

andare omitted.

The Pareto-optimal condition, however, is derived by( = 1,2, … , ),∑,s.t. ∑

Let

+∑

=∑

,

denotes the Lagrange multiplier of the constraint above. Then, the Lagrangefunction is given by( , )=∑+

Differentiating

=

+

∑

=

+

= 0,

∑

+∑

−∑

( , ) with respect to

+

(

and

).

gives us

= 0,

which can be rewritten as

+

∑

=

.

(3-9)

28

We have considered all jurisdictions to be identical. Therefore, (3-9) at the symmetricequilibrium can be rewritten as[1 + ( − 1)]=,(3-10)

where the jurisdiction-specific subscripts

andare omitted.

Comparison of (3-8) and (3-10) shows the optimal matching grant rate which thecentral government should choose is given by26

m=

(

)
(

)

−

Recalling that

(3-11).

< 0,> 0, we know from (3-11) that the optimal matching grant

rate increases with agency costs (⁄< 0). Besides benefit spillovers, note thatagency costs may also aggravate the under-provision of local public goods in theabsence of tax competition among local jurisdictions27. This finding corresponds withconclusions from the literature (e.g., see Belleflamme and Hindriks 2005; Besley andCase 1995). Notably, this result is an extension of Boadway et al. (1989) and Ogawa(2006) with reference to a particular case.

3.3. The Introduction of Tax Competition
Thus far, we have shown the inefficiency of the agency cost problem in a closed
economy with consumption spillover effect among jurisdictions. However, in practice,private capital is movable among jurisdictions and jurisdictional governments canraise revenue only or partly via distortional capital taxes within their own
jurisdictions 28 . Therefore, in the present study, we now rebuild the model of tax
competition in which private capital is put into production and is perfectly mobile

26

Equation (3-11) can be easily derived by substituting (3-10) into (3-8).

27

One might argue that an incentive contract using yardstick evaluation may realise the efficient provision of

local public goods in full information equilibrium. Furthermore, Nishigaki, Nishimoto and Yasugi et al. (2016) also
provide a preliminary empirical test that supports these results. However, these cases are explicitly neglected in
the analysis.
28

See Zodrow and Mieszkowski (1986), Ogawa (2006), Kikuchi and Tamai (2019).

29

among jurisdictions, and jurisdictional governments raise revenue not with alump-sum tax but rather a distortional capital tax29.
As before, we assumeidentical jurisdictions, where each jurisdiction

1,2, … , ) has a single immobile resident providing the labour,( == 1. The production

functions in all jurisdictions are identical, and the labor is not shown explicitly in thosefunctions. That is, the production function in jurisdiction

is simply given by,=(3-12)

is the private capital in jurisdiction . Private capital has perfect mobility.

where

Therefore, in equilibrium the after-tax return to capital is equalised acrossjurisdictions−
for all

=

−

(

=

( = 1,2, … , ), wherek≡∂

per unit of capital in jurisdiction

and

≠
/∂

)
> 0,

(3-13)< 0,is the tax rateis the after tax return to private capital in

the country.
The total supply of private capital in the country is fixed atsuch that.

=∑(3-14)The budget constraint of the resident is as follows
=
where

−

− ℎ,

+

(3-15)

is the initial endowment of private capital in jurisdiction

and ℎ is the

lump-sum tax imposed by the central government. Substituting (3-13) into (3-15),(3-15) can be rewritten as
=

−

+

−

(3-15’)− ℎ.

The jurisdictional government budget constraint is given by
+

29

=

.

(3-16)

This is similar to Ogawa (2006). However, agency costs have not been considered in a model of tax competition

with benefit spillovers.

30

As before, the jurisdictional government

receives a matching grant from thecentral government to provide the local public good. Hence, the following condition
holds
.

=

(3-17)The lump-sum tax imposed by the central government ℎ will be set to satisfy thebudget constraint of the central government, as follows30
∑

= ℎ=∑

(3-18).

This model involves two stages:
In stage 1, the central government chooses the national tax ℎ and the matching
grant rate.

In stage 2, the jurisdictional government
public good, taking ℎ and

The jurisdictional government

chooses the local tax

and the local

as given.

wishes to maximise the objective function

subject to (3-13), (3-14), (3-15), (3-16), and (3-17), taking the tax rates
provision of local public goods of other jurisdictions

and theas given. Then, themaximisation problem is defined as,=[ ( )] +( , ),

s.t.

k

−

k

=

=
−

+
=
=∑

=

−
k +

(3-19)
= ,

(

≠

)

− ℎ,

,

,
.

The first-order condition gives31

30

We assume that the central government’s budget constraint must satisfy the hard budget constraint. Thus, thematching grants from central government are only funded by the lump-sum tax.
31

The second-order condition is assumed to be satisfied according to the objective function.

31

=

+

+

+

∑

+

−

+(

−

+

= 0.

)
(3-20)We have considered all jurisdictions to be identical. Therefore, (3-20) at thesymmetric equilibrium can be rewritten as[1 − (1 − )](1 − ) +where− (1 −)= 0,(3-21)

is the private capital demand elasticity with respect to the tax rate injurisdiction , that is,∀.

≡−

As for comparison of (3-8) and (3-10), comparison of (3-21) and (3-10) shows the
optimal matching grant rate that the central government should choose is given by

m=

(

)

(

(

)
)

(

−

)

.

(3-22)

We assume that we are on the left side of the Laffer curve, that is, 1 − > 0.
Recalling once again that

> 0, we know from (3-22) that the optimal< 0,

matching grant rate increases with agency costs even given horizontal fiscal
externalities among jurisdictions (< 0). Notably, this result is an extension⁄of Ogawa (2006) with reference to a particular case.
Next, we derive the relationship between

and . Differentiatingwith respect

to , we obtain

=

(

)

+

.

(3-23)To study the effects of tax competition and benefit spillovers on the optimal matching
grant rate, we present the following basic rationale: Generally, the larger the factors of
production demand elasticities with respect to the factor tax rates are, the more
intense tax competition becomes. The marginal cost of public funds is widely
considered to be larger for local jurisdictions facing a situation of tax competition andbenefit spillovers. Therefore, jurisdictional governments are inclined to provide fewer
local public goods under such a situation. Consequently, agency costs will decrease32

because[ ( )] < 0 and( ) > 0 . However, the welfare of residents will bedecreased owing to less provision of local public goods. This means that the utilitarian
form of welfare considering agency costs in each jurisdiction significantly depends on
the two effects working in the opposite direction32. For example, if agency costs andbenefit spillovers are small33, the decrease of agency costs is smaller than the decreaseof the welfare of residents. The net effect is that the utilitarian form of welfare
considering agency costs in each jurisdiction is decreased. Restated, the inefficiency ofeach jurisdiction is increased. For that reason, the matching grant rate from thecentral government should be increased to eliminate the inefficiency resulting from tax
competition and benefit spillovers. Conversely, if agency costs are large enough, thedecrease of those agency costs exceeds the decrease in the welfare of residents. The net
effect is an increase in the utilitarian form of welfare considering agency costs in eachjurisdiction. The inefficiency of each jurisdiction thus is mitigated. Accordingly, thematching grant rate from the central government should be decreased.
Assuming agency costs are inevitable34, we obtain the following proposition:
Proposition 3.1: We assume that 1 − > 0. The following results can be obtained.
(1) If agency costs are relatively small, that is, 1 +> 0, and= 0, the optimal

matching grant rate should increase with the private capital demand elasticities withrespect to the capital tax.
(2) If agency costs are relatively small, that is, 1 +> 0, and= 1, the optimal

matching grant rate should decrease with the private capital demand elasticities withrespect to the capital tax.
(3) If agency costs are relatively small, that is, 1 +> 0, and 0 << 1, therelationship between the optimal matching grant rate and the private capital demand
elasticities with respect to the capital tax is ambiguous.
(4) If= −1 and= 0, no relation exists between the optimal matching grant

rate and the private capital demand elasticities with respect to the capital tax.
32

Under information asymmetry, economic competition among jurisdictional governments also has both positive

and negative effects on the underinvestment in jurisdictional infrastructure within a complex decentralized
system (see Gorbachuk, Dunaievskyi and Suleimanov 2019).
33

Bjorvatn and Schjelderup (2002) shows that there is no incentive for tax competition, if= 1. In this case, thetax competition distortion is eliminated㸬
34

Where no agency costs exist in the jurisdictions, the results will be in accordance with the existing literature.

33

(5) If= −1 and> 0, the optimal matching grant rate should decrease withthe private capital demand elasticities with respect to the capital tax. This is especiallyso if= −1 and

= 1,

= −1.

(6) If agency costs are sufficiently large, that is, 1 +< 0, the optimal matching

grant rate should decrease with the private capital demand elasticities with respect tothe capital tax, independent of benefit spillovers.
The above proposition is shown in TABLE 3.1.
Note that horizontal fiscal externalities originating from tax competition and benefit
spillovers result in under-provision of local public goods (inefficiency). However, theseexternalities also can ease the under-provision of local public goods resulting fromagency costs (inefficiency correction). These two effects simultaneously work in
opposite directions. If agency costs are small and benefit spillover is zero, the former
effect exceeds the latter one, meaning horizontal fiscal externalities aggravate theunder-provision of local public goods. Conversely, when agency costs are small andbenefit spillovers are perfect, or when agency costs are large enough, the latter effectexceeds the former one, which means horizontal fiscal externalities may ease theunder-provision of local public goods. In particular, when agency costs are small andbenefit spillovers are imperfect, the magnitude of the two effects will be ambiguous.
Additionally, when= −1 and benefit spillover is zero, the two effects cancel eachother out, which means horizontal fiscal externalities cannot affect the provision of
local public goods.

3.4. Conclusions
This chapter has focused on the effect of horizontal fiscal externalities on the optimal
matching grant rate in a model where agency costs are inevitable. When benefit
spillover is zero, the relationship between the optimal matching grant rate and private
capital demand elasticities with respect to capital tax depends on agency costs. Thismeans that the inefficiency arising from agency costs may be eased by tax competition
only if the disutility of effort is so large that the benefits resulting from tax competition
exceed its costs when benefit spillover is zero. However, if benefit spillovers occur
among jurisdictions, the results will be ambiguous.
Note that the analysis abstracts from dynamic aspects. The ratchet effect is34

frequently introduced in a two-period setting where timing and commitment of policies(including capital taxes imposed by sub-central governments and matching grants
from central government) matter and should also be addressed in future research.
Evidently agency costs (the disutility of effort) in providing local public goods should
be set to equal the marginal increase in the probability of re-election multiplied by thevalue of being re-elected35. By ignoring the problems associated with re-election, we
obtain some succinct results in our paper. Therefore, the robustness of the results
should be further tested by introducing incumbent politicians into the model. In
particular, we choose a simple form of social welfare function to formulate themaximisation problem. Therefore, employing a more general form of objective function
and considering re-election may provide intriguing insights, an avenue that is left to
future research. In addition, our model also needs the verification of empirical analysis
in the future research.
Finally, we should more concretely present the policy implications of the above
results. Agency costs may aggravate the under-provision of local public goods when notax competition exists among local jurisdictions. Therefore, the matching grant ratefrom central government to local governments with relatively large agency costs
should be relatively high. Although the matching grant rate increases with agency
costs even given horizontal fiscal externalities and benefit spillovers among
jurisdictions, the horizontal fiscal externalities among jurisdictions decrease theseriousness of the under-provision of public goods caused by agency costs if agency
costs in the political agency process are sufficiently large. Accordingly, the matching
grant rate from central government to local governments with relatively large agency
costs, ceteris paribus, should be appropriately reduced.

35

See Seabright (1996).

35

Chapter 4 A Corrective Device for Large
Heterogeneous Jurisdictions in a
Two-Period Economy with Spillover
Effects36

4.1. Introduction
This chapter reconsiders the provision of a local public good by a jurisdictional
government in a two-period economy with spillover effects when the jurisdictional
government is assumed to be hyperopic or farsighted. The corrective device used by thecentral government to ensure the optimal level of the local public good is provided by
the jurisdictional government should be adjusted accordingly.
The literature analysing capital tax competition is relevant to this study (see, forexample, Zodrow and Mieszkowski 1986; DePater and Myers 1994). The basic idea of
Zodrow and Mieszkowski (1986) is that perfect mobility of private capital among small
homogeneous jurisdictions results in under-provision of a local public good, which isfinanced by a distortionary property tax because a lump-sum tax is unavailable.
However, DePater and Myers (1994) demonstrate that the pecuniary externality
among large heterogeneous jurisdictions derived from a change in the capital price,which is affected by distortionary capital taxes, should be moderately internalised by acorrective device. In the traditional small-jurisdiction tax competition model, theafter-tax return to capital is a parameter for each jurisdiction. In this chapter, there
are only two jurisdictions, so the after-tax return to capital is endogenous, which,unsurprisingly, leads to tax exporting. There are large-jurisdiction models in theliterature, including Hauer and Wooton (1999), Kanbur and Keen (1993) and manyothers. This chapter is based on a large-jurisdiction model that is similar to those in
the literature above.
The costs of moving faced by private capital, which are also referred to astransaction costs (see, for example, Lee 1997), should not be ignored in a tax
competition model. When the private capital investor has decided to locate in one
jurisdiction and invest in some projects, these projects will usually last for a long
period of time. Once the private capital is invested, it is usually quite difficult to36

This chapter is based on Yang (2021).

36

abandon the projects and leave the jurisdiction because of the large moving costs. Even
if the private capital can move freely among the jurisdictions in the initial stage,
imperfect mobility is inevitable in the later stages. Therefore, we must consider bothtransaction costs and inter-temporal effects in a tax competition model. There areseveral relevant studies that consider such issues. For example, Lee (1997) considers
the imperfect mobility of private capital arising from transaction costs in a two-period
tax competition model. He shows that a jurisdictional government will over-provide
local public goods in the second period because of transaction costs and that thejurisdictions may choose a lower capital tax rate than that chosen in a one-period tax
competition model to increase capital stock in the first period. Furthermore, byintroducing a head tax into the model, Ogawa (2000) confirms that the jurisdictional
government may subsidise private capital in the first period to increase capital stock inthe second period when a lump-sum tax is available to a hyperopic jurisdictional
government. This result is compatible with that of the repeated game explained by
Coates (1993). There are also some two-period-model constructions that are relevant toour study (for example, King, McAfee and Welling 1993). However, most of the relevant
literature analysing the transaction costs and dynamic effects does not clarify theimportant roles played by the spillover effects of public goods and the heterogeneity ofjurisdictions in a repeated-game model. Hence, the focus of this study is to examine
these roles.
The problem of capital tax competition may be solved by making a transfer from one
jurisdiction to another jurisdiction when a lump-sum tax is not available in a capitaltax competition model with imperfect population mobility among large heterogeneous
jurisdictions (see Burbidge and Myers 1994). Burbidge and Myers (1994) find that thepecuniary externality and the fiscal externality, which work in opposite directions, canbe cancelled out if the capital importer subsidises capital, while the capital exporter
taxes capital. Conversely, DePater and Myers (1994) confirm that the capital importer
taxes capital while the capital exporter provides a subsidy on capital if a lump-sum tax
is available for the jurisdictions. In this chapter, we follow DePater and Myers (1994)
because the strategy of manipulating the terms of trade is incentive compatible for thejurisdictions.
In the discussion on the spillover effects of local public goods among different
jurisdictions, the prevailing view is that such spillover effects will aggravate theunder-provision of local public goods (see, for example, Boadway, Pestieau andWildasin 1989). However, another quite different view is that the spillover effects oflocal public goods may alleviate the under-provision of local public goods in somesituations. In a repeated-game model with large homogeneous jurisdictions, Kawachi
and Ogawa (2006) find that the jurisdictional governments are more inclined to37

provide an efficient level of local public goods when the degree of the spillover effects issufficient. Furthermore, Ogawa (2007) confirms that, in a tax competition model withlarge heterogeneous jurisdictions, the jurisdiction with less efficient production
technology is likely to increase its capital tax rate to drive out private capital andobtain substantial spill-in effects from the other jurisdiction with more efficient
production technology. This means that a distortional capital tax may lead to a more
efficient level of local public goods funding. This finding is a key motivation andimplication for the chapter.
This chapter is closely related to the literature on fiscal federalism. It has been
considered that the voluntary provision of public goods and the provision of local public
goods with spillovers are insufficient even when using the ‘Lindhal mechanism’
because of the existence of free-riders and, therefore, that a matching grant from acentral government to persons or to local governments for (local) public goods isrequired to solve the problem. Matching grants are a very particular policy device andthis is relevant in the context of the vast literature. The seminal article by Boadway etal. (1989) shows the relationship between the efficient provision of public goods and an
optimal matching grant rate. Roberts (1992) uses the same model and analyses issues
including the efficiency of subsidies. Akai and Ihori (2002) replace individuals withlocal governments and examine the welfare effects of the central government’s
subsidies for local public goods in a Nash equilibrium model with two types of public
goods, local and central. Furthermore, Ogawa (2006) argues that the matching grant
rate may decrease with spillover effects if the elasticity of capital with respect to thecapital tax rate is significant in a tax competition model. Additionally, the role ofmatching grants as a commitment device has been considered in recent research (seeAkai and Sato 2019). Most of the key assumptions of this chapter correspond with theconventional wisdom presented in the studies above. Finally, we note that matching
grants are especially empirically relevant for China and Japan.
By introducing spillover effects into our analysis, we verify that the jurisdiction withthe less efficient production technology may choose to tax private capital in the first
period, assuming that a lump-sum tax is available to it, and receive substantial
spillover benefits from the other jurisdiction with more efficient production technology
in the second period when the jurisdiction is hyperopic and benevolent, which is quite
different from Ogawa (2000). In other words, these constructions are put together tomodel an interesting phenomenon and not simply to arrive at predetermined results.
The remainder of the chapter is organised as follows. The basic model is set out insection 4.2, in which we introduce the spillover effects of public goods and theheterogeneity of jurisdictions into a two-period economy. In section 4.3, we show the
Stackelberg equilibria by employing backward induction to obtain the optimal
38

corrective device to be employed by the central government in the two periods. In
section 4.4, we discuss our findings based on the derived optimal corrective device.
Section 4.5 draws conclusions.

4.2. The Model
The model that we use is similar to that used in Ogawa (2000). There are twoheterogeneous jurisdictions in a two-period game37 and, in each jurisdiction( =1,2)38, the immobile resident is normalised to unity, with preferences defined by astrictly quasi-concave utility function39
private numeraire good in period
public good in period
=
where

+

,

, where( = 1,2) and. The local public goodis the consumption of ais the consumption of a local

is defined by:

,

㸦4-1㸧

is the provision of the local public good by jurisdictional government

(0 ≦

and

≦ 1 ) is a parameter indicating the degree of spillover benefits fromjurisdiction to jurisdiction .
We assume that the well-behaved aggregate production function in jurisdiction, and that

(

)

and

(

)

can be rewritten as

is

and

,

is the private capital employed by jurisdiction in period .
respectively, where
The production function can be assumed to take the quadratic form, for
k =
− 0.5
, which is also used by Wildasin (1991) and Ogawa
example,(2007) in their numerical analyses, because the marginal productivity of private
capital can take a linear and concise form, that is,= −. The production
and . The private
technology in the jurisdiction depends on the parameters
capital is perfectly mobile in the first period and perfectly immobile in the second
period. We assume that the private capital is myopic, following Ogawa (2000). The
reason behind this is that the jurisdictional governments cannot commit tosecond-period taxes given the immobility of private capital in the second period. Even
if the private tax rate was 100% in the second period, the private capital could not
move to another jurisdiction. Thus, the private capital providers consider the tax rate
only in the first period when making the location decision because they do not believein the jurisdictional government’s commitment to the tax rate in the second period. In
other words, the capital owners do not take into account second-period taxation in
their location decision.
37

The model can be written in a simpler way using only one period (see, for example, Ogawa 2007). However, as

this chapter focuses on how the degree of governmental hyperopia and asymmetry in capital ownership affect the
optimal redistribution mechanism, the dynamic effects must be considered.
38

For simplicity, we assume that there are only two jurisdictions in the model. It can be confirmed that most of

the results in this paper will not change qualitatively even if there are more than two jurisdictions.
39

The properties of the utility functions are similar to those in the extant literature (for example, Ogawa 2000).

39

The total supply of private capital in the country is fixed at=+such that:

.

(4-2)

In equilibrium, therefore, the after-tax return to capital in the first period isequalised across jurisdictions as follows:
−
where

=

−

=

,

(

≠

)

is the tax rate per unit of capital employed by jurisdiction

(4-3㸧
and

is theafter-tax return to private capital in the country in the first period. Based on theestablished conventions, for example, see Bucovetsky (1991) and Ogawa (2007), we
obtain the effect of changes in the first-period tax rate on the after-tax return toprivate capital and the location of private capital by taking total derivatives of (4-2)
and (4-3), as follows:

=

(4-4)

<0

=−

>0

(4-5)

=−

<0

(4-6)

The budget constraint of the resident in the first period requires that:
=

−

−ℎ ,

+

(4-7)

where
is the initial endowment of private capital in jurisdiction with=and ℎ is the uniform lump-sum tax that the central government has imposed.
Following Ogawa (2000), we postulate thatis a fraction of the capital stock ownedby the resident in jurisdiction
and that it does not change with time, where+= 1. Local and central governments can change the tax rate, but cannot impose
a discriminatory tax rate. Moreover, it is required that the revenue from the national
and local forest environmental taxes must be used for environmental policies.
Therefore, these taxes and policies can be viewed as similar to the lump-sum taxes andthe local public goods that we analyse in this article.
Substituting (4-3) into (4-7), (4-7) can be rewritten as:
=

−

+

−

(4-7’)−ℎ .

During the second period, the after-tax return to capital may differ between the two40

jurisdictions because of the immobility of private capital. Therefore, the budget
constraint of the resident in the second period requires that:
=

−

+

−

+

−

−ℎ .

(4-8)

The jurisdictional government budget constraint is given by:
=

+

.

(4-9)

The central government establishes a corrective device to encourage thejurisdictional government

to provide the local public good. Hence, the following

condition holds:
=
where

,

(4-10)

is the matching grant received by the jurisdictional government

central government in period

andthe jurisdictional government

from the central government in period

from theis the rate of the matching grant received by.

The lump-sum tax (subsidy) imposed (offered) by the central government ℎ will bechosen to satisfy the following budget constraint of that central government:+=ℎ +ℎ =+.

(4-11)

Modelling intergovernmental transfer/taxes in such a way is well-established in theliterature (see, for example, Ogawa 2006 and Boadway et al. 1989).

4.3. The Stackelberg Equilibria
We assume that the central government and the jurisdictional governments play a
Stackelberg game with centralised leadership and that there is a unique Stackelberg
equilibrium in each period. As this two-period game is a subgame perfect equilibrium,
we employ backward induction to solve the problem for each jurisdictional
government.

4.3.1. The Second Period
In the second period, there are two stages:
In stage 1, the central government chooses the national lump-sum tax (subsidy)
41

ℎ and the matching grant rate (the Pigovian tax rate)

as a Stackelberg

leader.
In stage 2, the jurisdictional government
local public goodchooses the capital tax rate

as a Stackelberg follower, taking ℎ and, and theas given.

In the second period, the jurisdictional government

maximises the utility of theand. Although the jurisdictional

residents by choosing, givenandgovernments cannot commit to second-period taxes, some facts (for example, the laws
and policies in the jurisdictions) stop the capital-importing country from taxing away
all capital and redistributing it to its citizens. Therefore, the capital owners would
foresee this in their location decision. Following Ogawa (2000), the optimisation
problem for jurisdictional government
,

,

,

s.t.

=

−

=

can be written as:

+

−

+

−ℎ ,

,

+

=

+

=

.

,

The first-order condition for jurisdictional government

=

−

+

−

is given by:(4-12)= 0,

where the jurisdictional government takesasin the first period. It can bederived that the second-order condition is satisfied under some realistic functional
assumptions and the properties of the equilibria are fully determined (see Ogawa
2007). Rearranging (4-12), we have:

=

.

1−

(4-13)The optimal corrective device that the central government should choose is given by:=1−(4-14).

The Pareto-optimal condition is derived by:

42

,

,

s.t.

Let

+

+

,

,

+

+

=

.

+

denotes the Lagrange multiplier of the constraint above. Then, the Lagrangefunction is given by:

,

=

+

Differentiating

+

−

−

㸬

with respect to,and, yields:

+

,

=

+

=

+

= 0,

=

+

+

+

+

= 0,

+

−

+

−

= 0,

which can be rewritten as:+=(4-15).

A comparison of (4-13) and (4-15) shows that the optimal matching grant rate (thePigovian tax rate) that the central government should choose is given by:

=1−

(1 −

(4-16)).

This finding corresponds with the conclusions from the existing literature (forexample, see Bjorvatn and Schjelderup 2002). Notably, this result is an extension of
Ogawa (2000) with reference to a particular case.
43

The inefficiency here arises first from the under-provision of local public goods
resulting from the spillover effects. This is determined by the degree of spillovers. The
larger are the degree of spillovers, the larger is the positive externality. In addition,inefficiency arises from the over-provision of local public goods resulting from tax
exporting40. This effect is determined by the proportion of the capital stock owned by
the jurisdiction’s residents. The larger this proportion is, the larger is the negative
fiscal externality ignored by the jurisdictional government. If the former positive
externality is larger than the latter negative fiscal externality, the net effect is that
local public goods are under-provided by the jurisdictional government in the second
period, which means that the optimal corrective device provided by the central
government is a matching grant. Conversely, if the former positive externality issmaller than the latter negative fiscal externality, the net effect is that local public
goods are over-provided by the jurisdictional government in the second period, whichmeans that the optimal corrective device that the central government should provide isa Pigovian tax. This finding may be summarised in the following proposition.

Proposition 4.1: If the spillover effect is larger than the tax-exporting effect in thesecond period, the central government should choose the matching grant as acorrective device. In this sense, the local public good is under-provided. On the contrary,
if the spillover effect is smaller than the tax-exporting effect in the second period, thecentral government should choose the Pigovian tax as a corrective device. In this case,the local public good is over-provided.
This proposition mainly restates what the prior literature has found in similar
contexts (see, for example, Bjorvatn and Schjelderup 2002).

4.3.2. The First Period
In the first period, there are two stages:
In stage 1, the central government chooses the national lump-sum tax (subsidy) ℎand the matching grant rate (the Pigovian tax rate)

as a Stackelberg

leader.
In stage 2, the jurisdictional government
local public goodas a Stackelberg follower, taking ℎ , ℎ ,given.

40

chooses the capital tax rate

See Noiset (2003).

44

and theandasThe capital tax rate in the second period depends on the amount of private capital
located in jurisdictionin the second period. Owing to the immobility of private

capital in the second period, the amount of private capital located in jurisdiction
the second period is equal to the amount in the first period, that is,same time, the amount of private capital located in jurisdiction=in. At thein the first perioddepends on the capital tax rate, which is chosen by the jurisdictional government in
the first period. Therefore, following Ogawa (2000), we assume thatis expressed as a function of=, where. This means that how the jurisdictional government

chooses the optimal capital tax rate in the second period is significantly determinedby the capital tax rate that it chose in the first period. Note that this does not meanthatis predetermined. The jurisdictional government chooses

to maximise thediscounted sum of the utilities in the two periods, given the variables for jurisdictional
government . If the jurisdictional government is hyperopic, the maximisation problem
for jurisdictional government

in the first period can be written as:

,

=

,

+

s.t.

=

−

+

=

−

+

=

+

,

=

+

,

=

+

,

=

+

,

=

,

=

,

=

,

,
−ℎ ,
−

+

−ℎ ,

−

,

= ( ),
by assuming that the discount factor for the jurisdictional government is≥ 0. To

derive the first-order condition, we use the substitution method and differentiatewith respect to. Substituting (4-1), (4-3), (4-7), (4-8), (4-9) and (4-10) into theobjective function, we obtain:

=

+
+

+

+
−

− 1−

45

−

−
−

.

+

+

+

(4-17)

Substituting (4-13) into (4-17), (4-17) can be rewritten as:

=

+

+

+

1
1−

+

−

−

−

1−

−

1
1−

.

+

+

+

−

(4-18)

Rearranging (4-18) with cancellation, we have:

=

+

+

+

+

−

−

Using (4-2) and the assumption that

−
.

+

=

(4-19), the first-order condition can be written

as:
.
=

+
+

−

+
−

−
−

−
(4-20)= 0.

It can be derived that the second-order condition is satisfied under some realistic
functional assumptions and the properties of the equilibria are fully determined (seeOgawa 2007).
The optimal corrective device that the central government should choose is given by:=1−.

The Pareto-optimal condition is derived by:46(4-21)

,

,

s.t.

+

,

+

,

+

+

+

+

=

+

,

+

+

+

=

+

,

,

,

where we assume that the discount factor for the central government isand≥ 0. Letdenote the Lagrange multipliers of the constraints above, respectively. Then,the Lagrange function is given by:

,

,

,

=

+

+

+

+

Differentiating

=

+

=

+

+

,

,

+

+

=

+

= 0,

+

+

=φ

+

+

+

−

−

+

and

gives us:

.

+

−

−

,

with respect to

,

,

,

,

= 0,
+

+

+

= 0,

=

=

+

+

= 0,

−
+

−
−

= 0,
−

= 0,

which can be rewritten as:+=(4-22-1)

,
47

+

=

(4-22-2).

A comparison of (4-20), (4-22-1) and (4-22-2) shows that the optimal matching grant
rate (the Pigovian tax rate) that the central government should choose is given by:=1−.
(4-23)


Notably, this result is an extension of Ogawa (2000) with reference to a particularcase. It may appear that it is a kind of extensive form game in which the central
government does not have to commit in the second period to the matching grant that it
chose in the first period. However, the matching grant can be used as a commitment
device in some situations (see, for example, Akai and Sato 2019). Therefore, to simplify
the analysis in the model, we assume that the central government can commit in thesecond period to the matching grant that it chose in the first period.

4.4. Discussion
To sign, we differentiate the optimal corrective device with, yielding:=−.
(4-24)

We assume that we are on the left-hand side of a Laffer curve,< 0, the sign of+> 0. As

depends only on the bracketed term in the numerator. On the

one hand, the hyperopic jurisdictional government has an incentive to decrease the tax
rate in the first period to attract the private capital because the jurisdictional
government considers the income in the second period (the income effect). The firstterm in the bracketed term of the numerator is positive. However, if the spillover effectin the second period is taken into account by the hyperopic jurisdictional government,
the jurisdictional government has an incentive to increase the tax rate in the first
period to drive out the private capital and obtain the spillover benefits from the other48

jurisdiction (the spill-in effect). The second term in the bracketed term of thenumerator is negative. The relationship between the corrective device in the first
period and the degree of hyperopia of the jurisdictional government significantly
depends on the relative size of the two effects that are working in the opposite directionin the second period, as stated succinctly in the following proposition.
Proposition 4.2: When the income effect is larger than the spill-in effect in the second
period, the optimal matching grant rate (the Pigovian tax rate) in the first period fromthe central government to a more hyperopic jurisdictional government should be
increased (decreased). Conversely, when the spill-in effect is larger than the income
effect in the second period, the optimal matching grant rate (the Pigovian tax rate) in
the first period from the central government to a more hyperopic jurisdictional
government should be decreased (increased).
Notice that the external validity of this proposition depends on a political strategy ofthe politicians. The benefits that the politicians can obtain in one jurisdiction (there-election rent) equals the marginal increase in the probability of re-election
multiplied by the value of being re-elected. Of course, these factors are seen as theexogenous variables in this model. If the politicians would like to stand for election for
the next term, the conclusion would be valid and could also be a benchmark for someextensions in the future. However, if the politicians would like to stand down, theywould be myopic and their discount factor might be zero in the first period. The result
would collapse into the finding in Ogawa (2007).
To see the properties of the capital allocation among the two jurisdictions in such anequilibrium, differentiation of the optimal corrective device with respect to−shows that:

=
1+

−
1+

1

> 0.

1

(4-25)

This equation corresponds with that of Bucovetsky (1991) and Lee (1997). It is
obvious that the equilibrium is a symmetric equilibrium when−

= 0, whichmeans the capital does not move at all. The jurisdiction is a capital exporter if−> 0, and it is a capital importer if
49−< 0. As> 0, we have thefollowing relationships for jurisdiction :
if
if
if
where

∗

−
−
−

> 0 then= 0 then< 0 then

>
=
<

∗

,

∗

,

∗

,

is the optimal matching grant rate from the central government in asymmetric equilibrium. This conclusion is a generalisation of that derived by Ogawa(2000) in a strategic tax competition model. We obtain the third result as follows.
Proposition 4.3: In the first period, the optimal matching grant rate (the Pigovian tax
rate) from the central government to a capital-exporting jurisdictional government islarger (smaller) than that in the symmetric equilibrium. However, the optimal
matching grant rate (the Pigovian tax rate) from the central government to acapital-importing jurisdictional government is smaller (larger) than that in thesymmetric equilibrium.
The intuition behind this result is interpreted as follows. The capital exporter
desires a high after-tax return to private capital to increase the capital income arisingfrom exporting capital. Thus, in the first period, the capital exporter would choose alower tax rate and a lower level of local public goods than in the symmetric equilibrium
to manipulate the terms of trade41. For that reason, in the first period, the optimal
matching grant rate (the Pigovian tax rate) from the central government to acapital-exporting jurisdictional government is larger (smaller) than that in thesymmetric equilibrium. Conversely, the capital importer desires a low after-tax returnto private capital to reduce the capital costs arising from importing capital. Thus, inthe first period, it would choose a higher tax rate and a higher level of local public
goods than in the symmetric equilibrium to manipulate the terms of trade. Accordingly,
in the first period, the optimal matching grant rate (the Pigovian tax rate) from thecentral government to a capital-importing jurisdictional government is smaller (larger)
than that in the symmetric equilibrium.
Now, we state the boundaries of the research and applications of the model. In somesuburban areas, for example, less populated areas surrounding a metropolitan area
but of lower socioeconomic status, beneficial spillovers of local public goods from theurban core are necessary and essential for the suburban residents. If the politicians inthese kinds of jurisdictions place a significant weight on the distant future, theunder-provision of local public goods might be eased to some extent. Accordingly, the41

See Ogawa (2007) and Burbidge and Myers (1994).

50

central government should decrease the current period’s optimal matching grant rate
to some extent. However, in some urban areas, for example, a densely populated urban
core in a metropolitan area with high socioeconomic status, benefit spillovers of local
public goods from the surrounding territories are unnecessary and negligible for theseurban residents. If the politicians in these kinds of jurisdictions place a significant
weight on the distant future, the under-provision of local public goods might beaggravated to some extent. Accordingly, the central government should increase thecurrent period’s optimal matching grant rate to some extent.
This result is quite different from Ogawa (2000). One jurisdictional government
might tax private capital in the first period to receive more benefit spillovers fromother jurisdictions in the second period even if a lump-sum tax is available for the
benevolent and hyperopic jurisdictional government. Of course, the robustness andexternal validity of this research requires further analysis and the incorporation ofother key assumptions such as, for example, political re-election motivations.

4.5. Conclusions
This chapter has focused on the effect that the degree of hyperopia of jurisdictional
government has on the optimal corrective device in a two-period model in which
spillover effects are considered. We have obtained the following results.
(1) If the spillover effect is larger than the tax-exporting effect in the second period,
the central government should choose a matching grant as a corrective device.
Conversely, if the spillover effect is smaller than the tax-exporting effect in the second
period, the central government should choose a Pigovian tax as a corrective device.
(2) When the income effect is larger than the spill-in effect in the second period, forexample, if the production technology in the jurisdiction is significantly higher than inother jurisdictions and the spillover benefits received by the jurisdiction are not verylarge, the optimal matching grant rate (the Pigovian tax rate) in the first period, which
is set by the central government and directed to the more hyperopic jurisdictional
government, should be increased (decreased). Conversely, when the spill-in effect islarger than the income effect in the second period, for example, the production
technology in the jurisdiction is significantly lower than in other jurisdictions and thespillover benefits received by the jurisdiction are relatively large, the corresponding
optimal matching grant rate (the Pigovian tax rate) in the first period should bedecreased (increased).
(3) In the first period, the optimal matching grant rate (the Pigovian tax rate) fromthe central government to a capital-exporting jurisdictional government is larger(smaller) than that in the symmetric equilibrium. However, the optimal matching51

grant rate (the Pigovian tax rate) set by the central government in relation to acapital-importing jurisdictional government is smaller (larger) than that in thesymmetric equilibrium.
For simplicity, it has been assumed that capital is perfectly immobile in the second
period. If we introduce transactions costs into the model in the second period, ourresults may be adjusted quantitatively. However, our findings about the provision of
local public goods will not be changed even if capital is imperfectly mobile in the second
period.
There is no inter-temporal redistribution via public debt or public investment as
would usually be relevant when determining public finances across time (for example,
Barro 1979; Jensen and Toma 1991). In a dynamic model, the timing of andcommitment to policies (for example, see Wildasin 2003) matter. However, forsimplicity’s sake, these issues are not addressed in the present research. In addition,
alternative ways of redistributing resources between jurisdictions (for example,
sharing the tax revenue, as in Hindriks et al. 2008) may lead to more efficient
outcomes. This topic is left for future research.
It is worth noting that the degree of hyperopia of jurisdictional government isdetermined by the probability of re-election and the rent from being re-elected for thepoliticians. If the politicians in the jurisdiction can obtain high rent from being
re-elected or if the probability that the politicians will be re-elected in the next term isvery high, the jurisdictional government may be possess greater foresight in this term,and vice versa. Although these factors are seen as the exogenous variables in this study,
the finding could provide a benchmark for some extensions in the future. Therefore, in
future work, it would be interesting to take into account these issues concerning theincumbents and the anti-incumbency factors.

52

Chapter 5 Conclusions
This dissertation analyzed the relationship between the optimal matching grant
rates and different kinds of externalities, such as benefit spillovers and fiscal
externalities, in the models of tax competition among jurisdictional governments.
This dissertation first surveyed the existing literature on some different kinds ofexternalities in the standard models of capital tax competition. The problem of theunder-provision of local public goods in a tax competition model, in which private
capital is mobile among countries, is examined by the literature on public finance. It isconsidered that the matching grant program from a central government to thejurisdictional governments is a strong instrument to solve the problem of an
insufficiently provision of local public goods.
Second, this dissertation analyzed the optimal matching grant rate for local public
spending that has characteristics of both public goods and public inputs. Its spillover
effects to other regions are assumed to be regarding consumption and/or production.
Contrary to this approach, traditional analyses of matching grants for public spending
have only focused on public goods and consumptive spillover. By considering thesefactors,we obtain some more generalized conclusions and intriguing results; forexample, even if private capital is completely immobile, the productive effect of public
expenditure lowers the optimal matching grant rate when the production spillover iszero, or smaller than that of consumption, and vice versa.
Third, this dissertation examined the effect of horizontal fiscal externalities on the
optimal matching grant rate in a model where agency costs are inevitable. Agency
problems arise in any environment involving a principal-agent relationship. Because
this chapter takes agency costs into account, the main results should differ from thestandard conclusions of the tax competition literature. This chapter finds that thedegree of agency costs and benefit spillovers determine the relationship between tax
competition and the optimal matching grant rate. If agency costs are relatively small,and benefit spillover is zero, the optimal matching grant rate should increase with thefactors of production demand elasticities with respect to the factor tax rate and viceversa. Tax competition thus may ease the inefficiency arising from agency costs only if
the disutility of effort is so large that the benefits from tax competition exceed the costs
when benefit spillover is zero.
Finally, this dissertation introduced the spillover effect of public goods and theheterogeneity of jurisdictions to the capital tax competition literature using a
two-period economy. A clear result is that the revision of a corrective device used by the53

central government in the first period to ensure an optimal level of a local public goodis provided by a hyperopic jurisdictional government, significantly depends on therelative size of the income and spill-in effects in the second period. The relative size ofthe two effects, which work in opposite directions, is determined by the tastes andendowments of the jurisdictions, the form of their production functions and the degreeof spillovers, among other factors.
Note that the robustness of the above-mentioned results should be further tested byintroducing incumbent politicians into the model. In particular, we choose a simple
form of social welfare function to formulate the maximisation problem. Therefore,
employing a more general form of objective function and considering re-election may
provide intriguing insights, an avenue that is left to future research. In addition, ourmodel also needs the verification of empirical analysis in the future research.

54

Appendix 2.1
Differentiating (2-20), we obtain

㸩

=
+

㸩 ∑

㸫 −

,

䠇

−

䠉

䠇

(

,

)

= 0.
(A2.1-1)

From (2-14),,

㸫 − = 0, and because of the assumption that all jurisdictions

are identical, that is,䠉䠙0, we therefore obtain

㸩 ∑

䠙

㸩

−

,

(A2.1- 2).

From (2-1’), (2-17), and (2-18) we obtain

=

䠙

+ ∑

.

+ ∑

(A2.1-3)

Differentiating (A2.1-3) we obtain

䠙

㸩

∑

+

(A2.1-4).

Substituting (A2.1-4) into (A2.1-2), we obtain

㸩 ∑

+

䠙

−

,

+

+

∑

.
(A2.1- 5)

Differentiating (2-15), we obtain

∑

䠙㸫

.

(A2.1-6)55

Using (A2.1-6) and㸩
䠙−= , we obtain(1 − ),

Multiplying (A2.1- 7) by+(1 − ), we obtain (2-21).

56.

(A2.1- 7)

Appendix 2.2
It is clearly shown from (2-26) that if<0 is derived. Furthermore, if=< ,= 0 is derived, and if= 1,< 0 is also derived. We now consider< 1,> . Substituting

the case where

=

with

(

)

into (2-26) using

<

(

,

)

which we derived because of the positive denominator on the left-hand side of (2-8), we
obtain(where, if

(

)

) (

(

=

as

Regarding

<

)
)

(

<

)

(

)

) (

(
(

)

)

=

)

(A2.2-1)≦ 0,, the numerator in (A2.2-1) is larger than it is when

> . Therefore, we argue that
≦ 0,(

≦ 0, and()< 0 if> .

≦ 0, the logic is the same as that for

57

=

≦ 0.

TABLE 2.1
immobile

mobile private capital

private capital
=0

>

=

<

>

=

<

<

<

<

<

1=

>

>

1>

>

>

1>

=

=

1=

=

<
=

= ( − 1)/

0≤

<

=1

<

<

=

0≤

<

<1

<

<

<

0<γ=β <1

=

<

<

<

<

<

<

0=

=

=

=0

>

is the optimal matching grant rate with respect to the public good with thecharacteristics of the public input andis the rate with respect to the public good

without the characteristics.
is the private capital demand elasticity with respect to the tax rate in thejurisdiction.
Note that we assumed above that if= 1,

is a negative fixed value, whether the

public goods have the characteristics of the public input or not.

58

TABLE 2.2>
the private capital
is not mobile among

1−

<

1−

=

1−

> 0 ( = 0)

regions
the private capitalis mobile perfectly

>0

<0

59

=0

TABLE 2.3
  =0the private capital
is not mobile among

and

>
≧

>0

regions
the private capital>0

is mobile perfectly

60TABLE 2.4
besides==0>
besides

the private capital
is not mobile among

>0

=
>0

regions

=

1
−1

=0

=0
=0

( = 0)

the private capital>0

is mobile perfectly
among regions

61=0TABLE 2.5
besides

=

=

=1

=1

the private capital isnotmobileamong

regions
the private capital is< 0,mobile perfectly(1 − )

< 0,

< 0,

62

≦0

=
=

(1 − )==0Mathematical Note: Derivation of (2-22)
From (2-8), (2-21), and

(

(
(

)–

)
)]

)=

(

, we obtain

,

)
(

,

(M2-1)

where the jurisdiction-specific subscript

is omitted. From (M2-1) we can reach the[solution for

=(

[

)]

.

63

TABLE 3.1. The Relationship between

=0
0<

<1
=1

< −1

= −1

<0

=0

<0

<0

<0

= −1

0>

> −1

>0
=

and

=0

=1

0

>0

<0

=0

For instance, some simulated values for the parameters above can be considered as
TABLE 3.2 below.

64

TABLE 3.2. Some Simulated Values for=0= 21= 0.05= 21=

1(0.067)
15

,

and

= −2

= −1

= −0.4

=0

= −1

=0

= 0.6

=1

= −1.525= −0.525= 0.075= 0.475= −1.6= −0.6=0= 0.4= −1.875= −0.875= −0.275= 0.125

= −2

= −1

= −0.4=0= 21= 0.25
= 21
=1
= 21

65

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