ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E05-5

Role of Spillover in the RepeatedInteraction Model of Tax CompetitionbyHikaru Ogawa
Keisuke Kawachi

May 2005

ECONOMIC RESEARCH CENTERSCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Role of Spillover in the RepeatedInteraction Model of Tax Competition
Hikaru Ogawa and Keisuke Kawachi
School of Economics, Nagoya University, Japan

Abstract
This paper generalizes the model of Cardarelli et al. (Journal of
Public Economic Theory, 2002, 4(1), 19-38) by adding the benefit
spillover of local public goods. Traditional public finance literaturesuggests that a benefit spillover is ‘harmful’ since it causes the choicesof local governments to be ineﬃcient from the viewpoint of society as
a whole. This paper, however, shows that the spillovers can act as abeneficial factor in achieving an eﬃcient outcome.

Keywords: repeated tax competition, spillover externality

Correspondence to:
Hikaru Ogawa
School of Economics, Nagoya University,
Furocho Chikusa-ku, Nagoya 464-8601,
Japan
Tel.: +81-52-789-2379; Fax: +81-52-789-4924;
e-mail: ogawa@soec.nagoya-u.ac.jp1

1

Introduction

The literature on tax competition analyzes the ineﬃcient local tax settingin the presence of source-based taxes on mobile capital. Most of the earlystudies have examined tax competition in a single-period framework and,have shown that local governments choose ineﬃciently low levels of public
goods1 .
More recent studies, notably Coates (1993) and Cardarelli, Taugourdeau
and Vidal (2002; CTV hereafter), have departed from the traditional singleperiod framework of tax competition analysis. Coates presents a repeatedgame model of tax competition with perfectly mobile capital to generate aresult that local governments choose negative tax rates in the equilibrium.
CTV examine the conditions under which policy coordination can result
from the repeated interactions among governments. They show that aneﬃcient outcome would prevail when regional asymmetries on the income
and preferences are weak.
The aim of this paper is similar to CTV’s, and we follow it in agreeingthat local governments are in the game of (infinitely) repeated tax competition. The features that diﬀerentiate our model from CTV’s analysis is that
we add the spatial externality (often called spillover) of local public goods.
In the context of regional governments within a country, benefit spillover ofpublic goods is an inevitable phenomenon since jurisdictional boundaries do
not coincide with the limits to which the benefit extends. In the international context, taking the development of global environmental issues as asimple example, we find that a correlation among national governments’ activities is robust, compared with the past in the world economy. Moreover,political eﬀort to create broad economic unions is another recognition of theimportance of international spillovers. Integration such as that of European
Union must have promoted the diﬀusion of benefits originating from each
1

There is a large body of research on single-period tax competition. A partial list
includes Zodrow and Mieszkowski (1986), Wilson (1986), and Wildasin (1989). See Wilson(1999), Zodrow (2003) and Wildasin and Wilson (2004) for a general review of the tax
competition model.

2

country’s public service.
One clear result in the local public finance literature is that when there
is a benefit spillover (spatial externality) of local public goods, local governments underestimate the true social benefit and demand too little of the
public goods. In this sense, the spatial externality of local public goods isthe reason why local governments’ choices about taxes tend to be socially
ineﬃcient. The classical economic solution to the spillover problem is toforce local governments to consider the true social benefits. A simple way to
do this is to provide a Pigovian subsidy made by a higher-level government,or to make the government’s jurisdiction broad enough to include consumers
who enjoy benefits [see, for example, Oates (1972), Boadway and Wildasin(1984), and Fischer (1996)].
This paper addresses the integration of repeated tax competition analysis with spillover externality to reach a somewhat surprising result: The
existence of spillover externality works to make local governments choose
eﬃcient tax rates within the context of infinitely repeated tax competition. Given the presence of spillover externality with capital mobility, local
governments are concerned with the spill-in eﬀects of public goods from surrounding regions. This entails reducing the local government incentive todeviate from an eﬃcient outcome.
This paper is organized as follows. Section 2 introduces the basic modelof repeated tax competition developed by CTV. We incorporate a benefit
spillover of local public goods into the model. In section 3, we turn to theanalysis of one-shot game. In section 4, we derive our main result, i.e., as thedegree of benefit spillover increases, local governments are likely to choose
an eﬃcient tax rate. Section 5 oﬀers concluding remarks.

2

Model

This section develops a simple framework for two regions. Since the analysis
is based on a model of repeated tax competition, as formalized by CTV, the3description of the model will be brief. A nation consists of two regions withan infinite time period. Regions and time periods are denoted by subscripts
i = 1, 2 and t = 0, 1, 2, ..., respectively.

2.1

Consumer

In each region, a two period-lived immobile consumer is born every periodt. The utility of a consumer born at period t in region i is defined overconsumption of a private numeraire good c and a public good consumption
G at time t + 1, and is simply given by

ui,t = ci,t+1 + γi Gi,t+1 ,(1)where γi > 1 and Gi,t+1 ≡ gi,t+1 + βji gj,t+1 . gi,t+1 is the provision of public
good by local government i, and βji ∈ (0.1) is the degree of benefit spilloverfrom region j to i. When βji → 0, there exists no spillover externality, andthat our model reduces to that of CTV. If βji is positive, local public goods
provided by region j yield benefits not only to the local residents of j but
also those of region i as well. βji → 1 corresponds to complete or perfectspillovers. By allowing for interregional spillovers, our model generalizes
that of CTV2 .
The consumer is endowed with wi units of the good when young, invests
it either at home or abroad, and consumes the after-tax gross return frominvestment when he/she is old. The consumer transfers all his/her endowments to the second period of life since we assume the consumer derives
utility only when he/she is old3 .
When the consumer born at period t is young, he/she chooses the allocation of an initial endowment between home, si,t , and, investment abroad,
σi,t :

2

For further analysis of spillover in the tax competition model, see Wildasin (1991),
Bjorvatn and Schjelderup (2002), and Ogawa (2005).
3
See also Keen and Kotsogiannis (2003) for this type of consumer behavior.

4wi,t = si,t + σi,t .

(2)

Following Persson and Tabellini (1992), Bacchetta and Espinosa (1993),
and CTV, we assume there is a net cost from investing abroad which isrepresented by a strictly convex function, η(σi,t ) = 0.5(σi,t )2 /µ, where µ > 0
is a measure of capital mobility.
The budget constraint for the consumer when he/she is old requires:

ci,t+1 = (1 − τi,t+1 )si,t + (1 − τj,t+1 )σi,t − η(σi,t ),(3)where τi,t+1 ∈ [0, 1] is the tax rate levied by local government i in periodt + 1. As assumed in CTV, we also assume here that the rate of interest is
constant and set equal to zero4 .
The consumer maximizes (1) with respect to si,t and σi,t , subject to (2)and (3), which yields

σi,t = max{0, µ(τi,t+1 − τj,t+1 )}.

2.2

(4)

Governments

Local government provides local public goods that yield benefits to its own
residents and generate positive spillovers across the region. Private goods
can be used as an input to produce local public goods, and units can bechosen so that the public goods provision in region i at each period t canbe measured in terms of units of private goods. The budget constraint on
local government requires that the cost of providing local public goods must
be equal to the sum of the revenue from capital tax. Thus, the budget
constraint is given by

gi,t+1 = (si,t + σj,t )τi,t+1 .
4

Assuming positive-exogenous interest rates would not modify our results.

5(5)

By assuming that local governments precommit their tax policies at periodt + 1 to the policies announced at period t, we avoid the time consistency
issues.
Now we consider the eﬃcient behavior of local government. Local governments control capital tax rate. CTV assumes each local government maximizes the discounted utility of the residents in the model without spillover
externality:

Vi =

∞
X

δit (ci,t+1 + γi gi,t+1 ).

(6)

t=0

Then they derive the autarkic social optimum tax rate as τi,t+1 = 1 ∀i. We
can easily show that the introduction of spillover externality strengthens
their result, and that the optimal tax rates still equal 15 .

3

One-shot game

A one-shot game is formalized as follows. Local governments at period t are
assumed to be myopic in the sense that they only consider the welfare of
consumers in their own region born at period t, ui,t = ci,t+1 + γi (gi,t+1 +βji,t+1 gj,t+1 ). Then the maximization problem for local government i in the
one-shot game is formulated as

max ui,t = ci,t+1 + γi (gi,t+1 + βji,t+1 gj,t+1 )
s.t.

(2), (3), (4), (5).

After some straightforward manipulations, the reaction function is givenby5

To clarify the distinct result, in this paper, we have used a linear utility function. The readers may feel that the optimal tax rate and the degree of spillover are
independent of each other. In our model, if we allow non-linear preferences and assume CES type of utility function, we can easily derive the optimal tax rate that is
aﬀected by the degree of spillover. Assuming that the social welfare function is given by
1
P P∞ t £ αα¤ α, where γi > 1, α < 1, i 6= j, we have theit=0 δi ci,t+1 + γi (gi,t+1 + βji,t+1 gj,t+1 )1αthe optimal tax rate as τ = [1 + γ α−1 (1 + β) α−1 ]−1 < 1. CTV provides a numerical
exercise that shows CES utility function would be chosen without aﬀecting the nature of
the main results.

6

( (γ −1)w +γ µτ (1+β )i

τi,t+1 =

i

i

j

ji

2µγi
(γi −1)wi +µτj [γi (1+βji )−1]µ(2γi −1)if

τi,t+1 < τj,t+1

if

τi,t+1 > τj,t+1 .

(7)

The following two assumptions are made in the following analysis:

wj (γj − 1)
wi (γi − 1)≤
γi (1 − βji )γj (1 − βij )γi µ(1 − βji )γj µ(1 − βij )A2. wi <, wj <.
γi − 1γj − 1A1.

A1 is made no loss of any generality, and A2 ensures an interior solution.
Under A1 and A2, we find the following Nash equilibrium tax rates:

τiN

=

τjN

=

(2γj − 1)(γi − 1)wi + γi (γj − 1)(1 + βji )wj,
γi µ{(γj − 1)(1 − βji ) + γj [2 − βij (1 + βji )]}(γj − 1)[2γi wj + (γi − 1)wi ] + (γi − 1)γj wi βij≥ τiN .
γi µ{(γj − 1)(1 − βji ) + γj [2 − βij (1 + βji )]}

(8)
(9)

Since it is our intention to analyze the eﬀects of benefit spillovers on theequilibrium, we simply assume that regions are symmetric, except for the
degree of spillovers (γi = γj = γ, wi = wj = w, βji 6= βij ). This implies that
A1 and A2 reduce to

A10 . βji ≤ βij
γµ(1 − βji )γµ(1 − βij )A20 . w <
,w <
.
γ−1
γ−1
In this case, the equilibrium tax rates are given by

τiN(γ − 1)(3γ − 1 + γβji )w
γµ{(γ − 1)(1 − βji ) + γ[2 − βij (1 + βji )]}(γ − 1)(3γ − 1 + γβij )w≤ τjN =,
γµ{(γ − 1)(1 − βji ) + γ[2 − βij (1 + βji )]}=so that capital is driven out from region j to i if βji < βij :

7

(10)

σi = 0,
σj =

(11)

(γ − 1)(βij − βji )w.
(γ − 1)(1 − βji ) + γ[2 − βij (1 + βji )](12)This result shows that region j, which receives a relatively high level ofbenefit spillover from region i has an incentive to choose a relatively hightax rate, τi < τj , when βji < βij . The intuition underlying this result isunderstood when we consider the behavior of local government j, which
considers that if it drives out capital by choosing a high tax rate, capital
escaping from region j contributes to an increase in the level of local public
goods in region i, so that region j receives an abundance of beneficial spill-in
eﬀects. This implies that the marginal cost of increasing taxes in region j isless than that in region i.
Next, the welfare comparison gives us the following result.

Proposition 1. Assume symmetric regions, except for the degree of
spillovers (γi = γj = γ, wi = wj = w, βji < βij ). The residents of region j
can be shown to be better oﬀ than those of region i, ui < uj , in the one-shot
equilibrium.

proof. The equilibrium values allow us to calculateNuN
j − ui12= (τjN − τiN )(γ − 1)w + µ(τjN − τiN ) + γ(βij τiN − βji τjN )w
2+γµ(τjN − τiN )(βij τiN + βji τjN ) − γµ(τjN − τiN )(τjN + τiN )=(γ − 1)(βij − βji )w2 J,
2µH 2

where H ≡ (γ −1)(1−βji )+γ[2−βij (1+βji )] > 0 and J ≡ 4(3−βji −βji βij −βij )γ 2 − (4 − 5βji + βij )γ + (βij − βji ). Since γ > 1 and 0 < βji , βij < 1, itNis shown that J > 0, so that uN
j > ui as βij > βji .

84Repeated Tax Competition

In this section, we discuss the traditional infinitely repeated game setting.
We now consider the possibility of local governments choosing eﬃcient tax
rates, τi = 1. Assume now that regions are perfectly symmetric (γi = γj =γ, wi = wj = w, βji = βij = β), so that A1’ and A2’ reduce to

A3.

w<

γµ(1 − β).
γ−1

An eﬃcient outcome is supported through the use of a trigger strategy represented by

τi,t+1 = 1

if

= τiN

τj,t = 1otherwise.

In the trigger strategy of an infinitely repeated game, each local government chooses a tax rate equal to one in the current period if another local
government choose an eﬃcient tax rate in the previous period. If any local
government defected in the previous period, then other governments would
revert to the single-shot Nash equilibrium forever.
We follow the usual assumption in the infinitely repeated game literaturethat the game continues for infinite periods, t = 0, 1, 2, ..., ∞, and thateach local government discounts its future residents’ welfare by the discountfactor 0 < δi ≤ 1. In period t, each government i decides to choose either
cooperation or defection. As usual, δ̂ exists as the critical value of δ for local
governments so that those local governments choose eﬃcient tax rates; forall δ ≥ δ̂, local governments choose the eﬃcient tax rate, while for δ < δ̂, aneﬃcient outcome can not be supported.
Since computing the full solution is laborious, we relegate the details tothe appendix and summarize the main result as follows.

9Proposition 2. Under A3, δ̂ is a monotonous decreasing function of β.
That is, as the degree of spillover increases local governments are likely tochoose eﬃcient tax rates.

The intuition behind this result is explained as follows. Region i has
an incentive to not provide cooperation since it can promote capital inflow
by reducing its tax rate. Region i recognizes that when it reduces its tax
rate, capital flows in, so that reducing tax rates raises revenue for the public
good provision in region i. However, it also accounts for the fact that it
can benefit less from the public goods provision in region j when region isiphons oﬀ region j’s capital. For region i, the gain from the defection fromcooperation is small when βji is large. Since the single-period incentive todefect declines as the degree of benefit spillover increases, the critical valueof the discount parameter, δˆi , diminishes as βji increases.

5

Concluding Remarks

The conventional wisdom is that the presence of spillover externality is‘harmful’ in the sense that it induces local governments to choose inefficient levels of capital tax. However, we have identified another role ofspillover that works in the opposite direction. In this paper, we argue thatthe increase in spillover actually reduces local government’s incentives to
not participate in the cooperative outcome, so that local governments are
more likely to choose eﬃcient tax rates when the degree of benefit spillovers
increases.
The benefit spillover of public goods is an inevitable phenomenon in
the context both of regional governments within a country and of national
governments within the world economy. Our finding is significant from a
policy aspect since it tells us that we need not have an extensive subsidy
program implemented by a supra-regional government when we encounter
an increase in the benefit spillover of public policies.
Finally, it should be noted that some of our assumptions could be re10

laxed without changing the main result of this paper. Specifically, although
our assumption of a linear utility function allows us to derive clear-cut results, some relaxation of this assumption would only change the results in aquantitative sense.

Appendix
The critical value of δ would be obtained by

δ̂ =

C
uD
i − ui,NuD
i − ui

where

D
D
D
D
D
uD
i (τi , 1) = (1 − τ )w + γ[(τ + β)w + µ(1 − τ )(τ − β)],
N
N
N
N
uN
i (τi , τj ) = w[1 − τ + γτ (1 + β)],uCi (1, 1) = γw(1 + β),(γ − 1)w + γµ(1 + β)
τiD =
,
2γµ
(γ − 1)w
τN =.
γµ(1 − β)Solving δ̂ explicitly, we haveδ̂ =(1 − β)[γµ(1 − β) − w(γ − 1)].
(1 − β)[γµ(1 − β) − w(γ − 1)] + 4w(γ − 1 + γβ)Under A3, a straightforward numerical check shows that δ̂ is a monotonous
decreasing function of the degree of benefit spillovers, β;

4w{[γµ(1 − β) − w(γ − 1)](2γ − 1) + γµ(1 − β)(γ − 1 + γβ)}
∂ δ̂
< 0.
=−
∂β
{(1 − β)[γµ(1 − β) − w(γ − 1)] + 4w(γ − 1 + γβ)}2
Furthermore, we prove here that δ̂ is in [0,1]. Denote the resident’s utility
level in region i when it deviates (D) and cooperate (C) as viD and viC ,respectively
11D
viD = uDi (τi , 1) +

viC

=

1
uC .
1−δ i

δ
uN ,
1−δ i

Now we define ψ(δ) ≡ (1 − δ)(viC − viD ). Then if we have δ̂ ∈ [0, 1] whichyields ψ(δ) > 0 ∀δ ∈ (δ̂, 1), regions keep cooperating. We can easily provethat ∃δ̂ ∈ [0, 1], since the following relationship holds:Nψ 0 (δ) = uDi − uii
1 − τiN h=
2w(γ − 1)(1 + β) + 4wγβ + γµ(1 − β)2 (τiN + 1) > 0,4
[γµ(1 − β) − (γ − 1)w]2Dψ(0) = uC
−
u
=
−
< 0,
i
i
4γµ
N
N
ψ(1) = uC
i − ui = w(1 − τi ) [(γ − 1) + γβ] > 0.

Acknowledgement
Earlier version of this paper has been presented at Kobe University of Commerce, the 2004 Spring Meeting of Japanese Economic Association, PET 04
Peking, and Kyushu University. The authors would like to thank Tomoya
Ida, Nobuo Akai, Minoru Kunizaki, Motohiro Sato, and Moriki Hosoe for
helpful comments.

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13

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14