ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E04−4

Tax Competition, Spillovers, and Subsidies
byHikaru Ogawa

September 2004

ECONOMIC RESEARCH CENTERSCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Tax Competition, Spillovers, and Subsidies

Hikaru Ogawa∗

Abstract
In the literature of local public finance, one of the well-known
property of optimal matching grant program is that the matching
grant rate should increase as the degree of benefit spillovers of public
goods increases. This paper presents the re-examination of properties of optimal matching grant program, using the model of Bjorvatn
and Schjelderup (2002). The result formally captures a quite counterintuitive property of matching grant that optimal matching grant rate
might decrease with the degree of spillover externality.

[keywords] tax competition, spillover externality, matching grant.
[JEL classification code] H4, H7, R5* Associate Professor, School of Economics, Nagoya University,
email: ogawa@soec.nagoya-u.ac.jp1

1

Introduction

Traditional analyses had suggested that a benefit spillover of local public
goods would cause local governments to provide suboptimal level of public goods. Local provision of public goods generating spillover externality
should be enhanced by a Pigovian subsidies made by the central government[see Williams (1966), Pauly (1970), and Oates (1972) among others].
The important implication of optimal subsidy policy derived in the literature is that as the degree of spillovers increases, the optimal matching
grant rate increases. In the context of regional governments within a country,
benefit spillover of public goods is an inevitable phenomenon since the jurisdictional boundaries do not coincide with the range that the benefit spreads
to. In the international context, looking at the development of global environmental issue as a simple example, we find that a correlation among
the national governments’ activities is fairly strengthened compared withthe past in the world economy. Moreover, political eﬀorts to create broad
economic unions is another recognition of the importance of international
spillovers. The integration such as European Union must have promoted
the diﬀusion of benefit originated from each country’s public service.
Facing these situations, we find from conventional argument that we needan extensive subsidy program that would be implemented by supra-regional
government. Many studies, including King (1984), Lee (1995), Lockwood(1999), Figuieres and Hindriks (2002), Akai and Ihori (2002) among others
follow the conventional argument and use the matching grant to stimulate
regional public goods provision.
The purpose of this paper is to provide the reexamination of the properties of optimal matching grant program. While most of the traditionalmodels are highly simplified to obtain clear-cut results, this paper extends
their studies to incorporate some realistic factors into the model, focusing2

on the production aspect. More specifically, this paper incorporates the
production sector, capital mobility, and distorting taxation explicitly into
the model. This generalization leads our analytical framework to the tax
competition model with spillover externality developed by Wildasin (1991)
and Bjorvatn and Schjelderup (2002). Specifically, this paper is closely related to Bjorvatn and Schjelderup’s paper which derives a result that there
arises no fiscal externality problem when the benefits of local public goods
perfectly spillout to the other regions. Using the model of Bjorvatn andSchjelderup (2002), this paper formally determines the optimal matchingrate to reexamine the property of optimal matching grant policy.
By this extension of the model, somewhat counter-intuitive result is obtained for the eﬀect of changes in the degree of benefit spillovers of public
goods on the optimal matching grant policy. This paper shows that theconventional argument regarding the eﬀect of increase in the degree of benefit spillover on the matching grant rate might not hold. In this paper, byintroducing the factor mobility into the model, there arises another sourceof ineﬃciency, called fiscal externality. The existence of fiscal externality
acts on the direction that regional governments underprovide public goods.
Then, we argue that the increase in spillover actually reduces the distortion due to the fiscal externality, so that optimal matching grant rate might
decrease.
In the next section, we redevelop a basic spillover model in the absence ofa production sector. In section 3, the production sector and factor mobility
are introduced into the basic model to show that there are reasons to consider
that an optimal matching grant rate should be reduced as the degree of
benefit spillovers of public goods increases.

32

Basic Model

We start from the basic model used in Oates (1972), Lee (1995), and Boadway et al. (1989). There are n identical regions, and in each region i, there isa single immobile resident, with strictly quasi-concave preferences u(xi , Gi )defined over consumption of a private numeraire good x and a public good
G. The assumption of identical regions allows us to isolate the eﬃciency
problem of public good provision from the equity issues. The public good
consumption in i is defined byGi ≡ gi + β

X

gj ,

(1)

j6=i

where gi is the provision of public good by regional government i, and β ∈(0.1] is the degree of benefit spillover.
We initially assume that each region is endowed with an exogenous income yi , and the governments impose lump-sum taxes on the resident. The
resident’s budget constraint will be given by

xi = yi − τi − zi ,(2)

where τi and zi are the taxes imposed by the regional and central governments, respectively. The regional government i raises revenue τi and receives
a grant from the central government,si = mi gi ,(3)

where mi is the rate of the matching grant. Fiscal revenue serves to financethe provision of local public goods. The private goods can be used as an
input to produce local public goods, and units can be chosen so that the
public good provision in region i can be measured in terms of units of private4goods. Hence, the budget constraint of the regional government can be givenbyτi + si = gi .

(4)

zi in (2) will be chosen so as to satisfy the budget constraint of the central
government,Pi mi gi =Pi zi .

The regional government i wishes to maximize the utility of its resident
subject to (1)-(4) taking the tax rates of other regions as given. Then, themaximization problem will be defined as




X τj
τi

+β
max ui = u yi − τi − zi ,τi
1 − mi
1
−
m
j
j6=i

The first-order condition gives us

uig = uix (1 − mi ).

(5)

This condition can be compared with the Pareto optimal condition,
uig + βX

ujg = uix ,

(6)

j6=i

which is derived by the maximization ofsource constraint,Pi xi +

Pi gi =

Pi yi .

Pii u subject to the national re-

We now assume that the ratesof the matching grant are the same for all regions1 . Then, since we haveconsidered the case of identical regions, (5) and (6) can be, respectively,
rewritten as
ug
=1−m
ux
1

(7)

As shown in Lee (1995), there exists at least one policy combination of (m, zi ) thatis Pareto improving. We can easily show that there is a large number of eﬃcient resource
allocations if we allow the grant rate vary across the regions. See Boadway et al. (1989).

5

and
ug
1
=
,
ux
1 + β(n − 1)(8)

where the region specific subscript i is omitted. Comparison of (7) and (8)shows that an optimal rate of the matching grant is simply given bym=β(n − 1),
1 + β(n − 1)(9)which reproduces the following well known result.

Proposition 1. ∂m/∂β > 0 and ∂m/∂n > 0. That is, the optimal
matching grant rate increases with the degree of spillovers and thenumber of regions.

3

Tax Competition and Spillovers

So far, we have assumed that the regional income is exogenously given andgovernments are allowed to use a lump-sum tax. In this section, we now
introduce the production sector and factor (capital) mobility explicitly andassume regional government can raise revenue only with the distortional
capital tax. This causes so-called tax competition among the regions. The
basic tax competition model we use to derive the optimal matching grantpolicy here is the same form as that used in Wildasin (1988,1991), Hoyt
(1991), Bjorvatn and Schjelderup (2002)2 .
2In the capital tax competition literature, an optimal subsidy program to correct anineﬃcient resource allocation has been studied previously by Wildasin (1989) and DeParterand Myers (1994). The framework used in this paper is similar in some respects to thosestudies, but their interests are not on the eﬀects of changes in β and n on the optimal
subsidy rate.

6

Production of private goods in region i is conducted by a large numberof identical firms and requires using capital and labor. The aggregate production function in region i is f (ki ) (fk (ki ) > 0, fkk (ki ) < 0), where ki is theamount of capital located in region i and labor is omitted. The economy has
a fixed stock of capital, k̄, which is perfectly mobile among regions. In theequilibrium, therefore, the net (after tax) return to capital will be equalized
across the regions,fk (ki ) − ti = ρ,

∀i

(10)

where ti is the capital tax rate and ρ is the net return to capital. The total
supply of capital is fixed at k̄ such that

k̄ =

n
X

ki .

(11)

i=1

The resident in region i receives labor income, f (ki ) − fk (ki )ki , and therent from capital, ρk̄i , where k̄i is the initial endowment of capital in region i.
Hence, the budget constraint of the resident requires xi = f (ki ) − fk (ki )ki +ρk̄i − zi , which can be rewritten as
xi = f (ki ) − ti ki + ρ(k̄i − ki ) − zi ,(12)

by using (10).
Now, the local public good in region i, gi , is financed by capital taxation,
ti ki , and the subsidy from the central government,si = mi gi .

(13)

Therefore, the regional government budget constraint is given byt i k i + s i = gi
7(14)

Substituting (12), (13) and (14) into the resident’s utility function, themaximization problem of the regional government i is now defined as

X tj kjti ki
,max ui = u f (ki ) − ti ki + ρ(k̄i − ki ) − zi ,
+β
ti
1−m
1−m
j6=i

where we have already assumed that the matching grant rates are the samefor all regions. Assuming that regional governments act under the Nashassumption, the tax rate will be implicitly solved by∂ρ(1 − m)[ki − ∂t(k̄i − ki )]
uig
³
´ i P
=
i
∂k
ux
ki 1 + ∂ki ti + β
tj j∂ti ki

j6=i

(15)

∂ti

Since we have assumed that regions are identical, it makes sense to focus
on a symmetric equilibrium where ti = t and ki = k̄i = k. Furthermore,using (10) and (11), we have the following comparative static results in thesymmetric equilibrium.

∂ki
∂ti
∂kj
∂ti
∂ρ
∂ti

n−1
<0
nfkk
1
= −
>0
nfkk
1
= − <0
n
=

(16)
(17)
(18)

Using (16)-(18), (15) will be rewritten as
1−m
ug
=
,
ux
1 − ²(1 − β)(19)where the subscript ‘i’ is omitted again and ² is the capital demand elasticity
with respect to the tax rate evaluated at the symmetric equilibrium, ² ≡−(∂ki /∂ti )(ti /ki ) ∀i. In the tax competition literature, where there exists
no benefit spillovers, ² < 1 is often assumed. However, in this paper, ² is8

allowed to take a value greater than 1, since the assumption we need in thispaper is ²(1 − β) < 1.
The optimal provision of public good can be expressed by the same
condition as in (8), which can be found by maximizing the sum of utilities
with respect to gi . From (8) and (19), we find that an equilibrium will be
Pareto optimal whenm=β(n − 1) + ²(1 − β).
1 + β(n − 1)

(20)

For ² = 0, (20) reduces to (9). From (20), we obtain the following result.

Proposition 2. If the elasticity of capital ² is strictly higher than (n−1)/n,then the matching grant rate is not a monotonous increasing functionof the degree of benefit spillovers.

Proof. As an extreme case, compare the matching grant rates in the casesof β = 0 and β = 1. m(β = 1) < m(β = 0) if3

²>

n−1
.
n

(21)

The intuition of Proposition 2 is as follows. In the economy describedabove, there are two sources of ineﬃciencies; (i) benefit spillovers of public
good and (ii) fiscal externality. As often explained in the literature, the causeof fiscal externality lies in that the regional government ignores the external
eﬀects of its tax change on the other regions; Although a tax increase inregion i aﬀects the other regions’ fiscal budget asP

j6=i tj (∂kj /∂ti ), regional

government i does not take it into account [Wildasin (1989)]. However, since3

Assuming ∂²/∂β = 0, (21) can be obtained by taking derivatives of m with respect toβ since ∂m/∂β = [n − 1 − ²n + (1 − β)[1 + β(n − 1)](∂²/∂β)]/[1 + β(n − 1)]2 .

9there exist benefit spillovers of public goods in the model described above,
the regional government takes a part of the external eﬀects into account,
and as the degree of spillovers increases the regional government becomesto consider the external eﬀect more. For instance, as shown in the recent
article of Bjorvatn and Schjelderup (2002), if β = 1 regional government
perfectly accounts for the fiscally external eﬀect. Hence, β plays a role inmeasuring how much fiscal externality the regional governments account for.
Though an increase in β increases the spillover externality, it reduces the
distortion due to the fiscal externality, so that the optimal matching grant
rate might decrease.
The combination of (n, ²) where (21) holds is shown by the shaded areain Figure 1. For instance, when n = 2, the rate of the matching grant should
be decreased as the degree of spillovers increases if ² > 0.5. Parry (2003,
p.46-47) states that ‘based on the empirical evidence, a range of about 0.1
to 0.8 seems the most plausible”, by referring Bartik (1991) and Brueckner(2001).
For further illustration, consider the familiar Cobb-Douglas production
function: yi = kia (0 < a < 1). Now, we define the capital demand elasticity,ηi ≡ −[∂ki /∂(ρ + ti )][(ρ + ti )/ki ] > 0, which is given by
1
.
1−a

(22)

n−1
ti
n−1
t
,
=
n (ρ + ti )(1 − a)n a(1 − a)k a−1

(23)

ηi =

In the symmetric equilibrium, using (10) and (22), we have4²i =which implies that (21) holds if

4

See Wildasin (1988, 1989).

t> 1.
a(1 − a)ka−1

10

(24)

To provide some insight as to when the matching grant rate should be decreased as the degree of benefit spillover increases, rewrite (24) as

θ > a(1 − a),(25)

where θ ≡ tk/k a denotes the share of local public spending in the regional product. Consider the Cobb-Douglas type of utility function, ui =Gbi (0 < b < 1). In the symmetric equilibrium, we have b = θ, whichx1−bireveals that (25) is likely to hold to the extent that the resident puts highweight on the public goods. This example is worth noting because it showsthat Proposition 2 might hold irrespective of the number of regions.
If we assume the Cobb-Douglas type of functional forms, we have anotherresult.

Proposition 3. As the number of regions increases, the optimal matching
grant rate increases.

Proof. Taking derivatives of m with respect to n in (20), we have∂²β[1 − ²(1 − β)] + (1 − β)(1 + β(n − 1)) ∂n
∂m
.
=
∂n
(1 + β(n − 1))2b
As we assume the Cobb-Douglas functions, we have ² = n−1
n a(1−a) ,implying ∂²/∂n > 0. Hence, ∂m/∂n > 0 since 1 > ²(1 − β).

Proposition 3 is a simple extension of the result of Hoyt (1991) if we
consider a reduction in n. As shown in Hoyt (1991), the decrease in thenumber of regions reduces the distortion due to the fiscal externality. In
addition, in this paper, the reduction in n reduces the spillover externality.
11

Two factors act on the direction that reduces optimal grant rate.

4

Conclusion

The conventional wisdom is that the optimal size of grant program increases
as the degree of benefit spillovers of public goods increases. However, in thispaper, we identify another role of spillover that works in the opposite direction. In this paper, we argue that the increase in spillover actually reduces
the distortion due to the fiscal externality, so that an optimal matching grant
rate might decrease with the increase in the degree of benefit spillovers.
The benefit spillover of public goods is an inevitable phenomenon bothin the contexts of regional governments within a country and of national
governments within the world economy. Our finding is significant in thepolicy aspect since it tells that we do not have to have an extensive subsidy program, implemented by supra-regional government, when we face theincrease in the benefit spillover of public policies.
Finally, it should be noted that some of the assumptions can be relaxedwithout changing the main result (Proposition 2) of this paper. Specifically,
the symmetry assumption allows us to derive clear cut result but some relaxation of this assumption would only weaken the result in a quantitative
sense; Allowing asymmetric regions only changes the square of the shaded
area in Figure 1.

Acknowledgement
This research was carried out while the author was on leave at the Universityof Kentucky. The author is very grateful to Martin School of Public Policy
and Administration for its support. Earlier version of this paper has been
12presented at the University of Kentucky, Institute of Statistical Research,
University of Tokyo, and the 43rd Annual meeting of Western Regional Science Association. The author would like to thank David Wildasin, William
Hoyt, Marco Castaneda, James Marton, Pierre Picard, Nobuhiro Okuno,
Dan Sasaki, Toshihiro Matsumura, Toshihiro Ihori, Masayoshi Hayashi, andJohn Carruthers for helpful comments. The author thanks the Shikishima
foundation, Japan Society for the Promotion of Science, and Japan Center
for Economic Research for financial support.

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14

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15

²
1
1−β

1
² = n−1
n

0.5

n
0

2

Figure 1. The area where m(β = 1) < m(β = 0) holds.
Note: 0 < β < 1, n ≥ 2 and ² < 1/(1 − β).

16