Public Investment Competition, Spillover Effects and Regional Integration
Center for Science, Technology and Innovation Policy Studies, Kyushu University Tadahisa Ohno

1. Introduction
Recently, there has been a trend to discussion of decentralization and the common problem of autonomy intaxation and public investment. Accordingly, it is important to discuss whether decentralized
decision-making with regard to a tax and public investment is socially efficient. Recently, regional activation
has become an extremely important problem in all regions of Japan. It is important to consider local public
investment as a local public policy to enhance regional activation. Herein, we especially discuss decentralized
decision making with regard to local public investment because we specifically examine public finance inJapan. When we examine competition in public investment among local governments, the spillover effect for
the local public good must also be considered because not only residents in one region but also residents inother regions might enjoy benefits from one region’s local public good or service.
As described herein, we analyze whether decentralized decision making for public investment is sociallyefficient when each local government chooses the level of public investment in its region under a situation inwhich capital is perfectly mobile among regions and a local public good has a spillover effect. Furthermore,
municipal mergers have been enhanced in Japan. Therefore, we also analyze regional integration.
Theoretical studies of public investment competition include those of Keen and Marchand (1997), Hindriks
et al. (2008), and Dembour and Wauthy (2009).
Keen and Marchand (1997) analyze regional competition with regard to capital tax and public investment.
They assumed simultaneous tax and investment choices. They show undertaxation and overinvestment inequilibrium.
Hindriks et al. (2008) consider a model of federation with two heterogeneous regions that seek to attract
capital by competing in capital income taxes and public investment that enhance capital productivity. Theirfeature is that regions make public investments before tax decisions so that public investments have astrategic effect on tax choices and so that regions can attract capital by investing more or taxing less. Theyfind that underinvestment and undertaxation exist in equilibrium.
Dembour and Wauthy (2009) produce analyses showing that local governments choose infrastructurelevels in a first stage and compete in taxes in a second stage, which is similar to the result reported byHindriks et al. (2008). However Dembour and Wauthy (2009) examine the properties of Subgame PerfectNash equilibria in this stage game depending on the extent to which the benefits of infrastructure spill overfrom one region to the other. They show that the presence of inter-regional spillovers allows jurisdictions tocontrol the intensity of tax competition and therefore affects the optimal levels of infrastructure selected at
equilibrium.
These theoretical studies of public investment competition specifically examine the timing of choice forpublic investment and capital tax rate and conclude various results. However, these studies do not consider
spillover effects for local public goods in their model framework. Actually, it is important to consider spillover
effects for local public goods. For example, spillover effects for local public goods occur in mobile factors, suchas public traffic.
1

This paper is based on the framework of Hindriks et al. (2008). In Hindriks et al. (2008), each local
government chooses the capital tax rate and public investment under a situation in which the spillover
effects with the provision of local public good do not exist. In contrast, in this paper, each local government
chooses the level of public investment. The central government controls the capital tax rate under a situationin which the local public good has spillover effects.
Results show that public investment in equilibrium is more socially desirable if either the spillover effect islarge, or the capital tax rate is high. Furthermore, regional integration can achieve a socially optimal level ofinvestment for any degree of economies of scale.

2. ModelConsider a country comprising two regions i=1,2. In each region, the local government chooses a level of
public investment Ii that enhances domestic capital productivity. The regions’ choices, denoted as I =(I1, I2 ), determine the allocation of capital x1 and x2 across regions, the precise mechanism of which will be
described below. The production in each region is given by the function Fi (xi , Ii ), which is increasing, twice
continuously differentiable, and concave in the level of capital xi for i = 1,2. Naturally, the private capitaland public investment are complements, so that the cross derivative of ∂2 Fi⁄∂xi ∂Ii is positive. The cost of
public investment is given as the convex function ci (Ii ), which is assumed to be quadratic for analyticaltractability: ci (Ii ) = Ii2⁄2. In each region, the local government levies on the mobile tax base (capital) andsupplies a local public good that has spillover effects. Therefore, the budget constraint of the local
government is txi = gi . Here, t is a capital tax that is equal among regions. gi is the local public good inregion i.
Under perfect mobility, the allocation of capital across regions must equate its net return in two regions. We
then obtain the following equality.
∂F1 (x1 ,I1 )∂x1=∂F2 (x2 ,I2 )
∂x2

(1)

.

We assume that the regions correctly anticipate how their public investment decision will affect the capital
allocation. By normalizing the total stock of capital to 1, the arbitrage condition (1) determines the amount of
the capital in each region, x1 = x1 (I)and x2 = x2(I). Each region maximizes welfare function Wi , the sum of the return to the immobile factor,
and the benefit from consumption of local public good, net of the investment costs.
Wi = Fi (xi , Ii ) −∂Fi (xi ,Ii )∂xi

I2

xi + gi + λgj − 2i ,(2)In that equation, i ≠ j and λ is the spillover effect (0 ≤ λ ≤ 1). We assume no domestic ownership ofcapital. Regions tax capital because it is simple to extract rents from the capital owners.
Next, we specify the production functions. The production functions are given asx2

Fi (xi , Ii ) = (γ + Ii )xi − δ 2i ,(3)where parameter δ ≥ 1 is the rate of decline of the marginal product of capital with the amount of capital
invested in the region. Consequently, the regional production functions exhibit decreasing returns to capitaland constant returns to investment. The welfare in the region i simplifies to the following.
2

δx2

I2

Wi = 2i + gi + λgj − 2i .

(4)

We first provide a benchmark by deriving the efficient outcome that maximizes the sum of the two regional
welfare levels.
Lemma 1The socially optimal level of capital is x1o = x2o =12

and socially optimal level of investment is I1o = I2o = t −

δ

γ+2.

3. Competing in Public Investment
In this section, we analyze the local government decision of the level of investment. The timeline is thefollowing. In the first stage, each local government chooses its investment independently. In the next stage,
the firm in each region chooses its demand for capital. We solve this game backwards.
First we analyze the equilibrium of capital market. Eq. (1) and the constraint of amount of capital yields
the following levels of capital in respective regions.
xir =Ii −Ij +δ(5)2δ

Given the public investment of another region, each local government i anticipates the allocation of capitaland independently chooses its investment Ii to maximize Wi .
We can derive the first-order condition as presented below.
δxir
Here,

dxri
dIi

=

1
2δ

and

dxrj
dIi

=−

1
2δ

dxri
dIi

+t

dxri
dIi

= Ii − λt

dxrj
dIi

(6)

. The left-hand side of eq. (6) is the sum of the marginal benefits ofconsumption of private good and public good from increase of capital by the investment in region i.
Consequently, the left-hand side of eq. (6) is the marginal benefit from the investment in region i. The firstterm of the right-hand side of eq. (6) is the marginal increase of the cost from investment in region i. The
second term of the right-hand side of eq. (6) is the marginal decrease of the benefit from the spillover effectfrom the decrease of another region’s capital by investment in region i. Therefore, the right-hand side of eq.
(6) is the marginal cost from the investment in region i. Consequently, eq. (6) is the condition under which
the marginal benefit from the investment equals the marginal cost from the investment in region i. The local
government in region i determines its investment to meet eq. (6) given the investment in the other region.
The investment in region i which meets eq. (6) in each region is the following.
Iir =Ij −δ−2t+2λt
1−4δHere, the investment in region i denotes Iir. Equation (7) denotes the best reaction function of the local
government in region i on the investment, which is decided by the local government in the other region.
From eq. (7), we can derive the equilibrium level of investment as presented below.
3(7)

I1∗ = I2∗ =4δ2 −2δ+8tδ(1−λ)−4t(1−λ)(8)−8δ+16δ2

4. Efficiency of Public Investment
In this section, we analyze the efficiency of public investment from comparison equilibrium of public
investment with the socially optimal level of public investment. We subtract the socially optimal level ofinvestment from the equilibrium level of public investment, which yields the following.
I∗ − Io =Here, we assume that γ >4δ3 −3δ+1
8δ2 −4δ

4δ2 +8tδ−δ+2tλ(1−2δ)−2t−8tδ2 −4γδ+8γδ2 −4δ3(9)−4δ+8δ2. From this assumption of γ, when the spillover effect does not exist(λ = 0), the sign of eq. (9) is positive (I ∗ − I o > 0). Therefore, we obtain the following result.
Proposition 1
Let γ >

4δ3 −3δ+1
8δ2 −4δ

. Then when the spillover effect does not exist, a unique symmetric equilibrium involving

overinvestment prevails in each region.
Next we analyze the effect by which spillover affects the equilibrium. We differentiate λ from eq. (9).
Therefore, we can derive the following equation.
d(I∗ −I )dλ=2t(1−2δ)−4δ+8δ2(10)

0

From eq. (10), we can obtain the following result.
Proposition 2
Public investment in equilibrium is more socially desirable if the spillover effect is large.
The interpretation of proposition 2 is the following. When public investment in region i increases, thecapital in the other region j decreases. Consequently, the public good provision in the other region j decreases
because of the decrease of the capital in its region.
According to these circumstances, the benefit of spillover from the other region j to the region i will decrease.
This situation is the marginal cost from the public investment in region i. Here, if the spillover effect is large,
then the local government in region i will overestimate the marginal cost from the public investment.
Consequently, the greater the spillover effect, the more socially desirable is public investment in equilibrium.
Next we analyze the effect by which the increase of the capital tax rate affects the equilibrium. We
differentiate between t and eq. (9). Therefore, we can derive the following equation.
d(I∗ −I )dt=8δ(1−δ)+2λ(1−2δ)−2−4δ+8δ2

From eq. (11), we can obtain the following result.

4

0

(11)

Proposition 3
Public investment in equilibrium is more socially desirable if the capital tax rate is high.
The interpretation of proposition 3 is the following. When the public investment in region i increase, thecapital in its region will increase. Consequently, the public good provision in the region i increases because of
the increase of the tax revenue from increase of capital. This situation illustrates the marginal benefit frompublic investment in region i.
However, when the public investment in region i increases, the capital in the other region j decreases.
Consequently, the public good provision in the other region j decreases because of the decrease of tax revenue
from the decrease of capital in its region.
According to this situation, the benefit of spillover from the other region j to region i will decrease. Thissituation is the marginal cost from the public investment in region i.
The marginal benefit of public investment and the marginal cost of public investment will increase if thecapital tax rate is high. Here, because the increase of marginal cost of public investment is greater than theincrease of marginal benefit of public investment, the higher the capital tax rate, the greater the degree towhich public investment in equilibrium is socially desirable.

5. Regional Integration
This section presents analysis of regional integration. Next, we consider the situation in which each local
government mutually integrates as one government. Here, if each local government integrates, theneconomies of scale might occur. However, integrated government might not be able to realize the residents’actual preference of local public good in each region. Therefore, we assume the following welfare for region i.
Wi = Fi (xi , Ii ) −∂Fi (xi ,Ii )∂xi

I2

xi + α(gi + λgj ) − β 2i

(12)Here, α denotes the degree of comprehension of the residents’ actual preference of local public good in eachregion (0scale(0αβ1). β signifies the degree of infrastructural cost reduction effect from economies of1). Integrated government decides public investment in each region to maximize the sum of

the revised two regional welfare eq. (12).
We can derive the level of investment as presented below.
δ

I1I = I2I = t − γ + 2(13)Therefore, we obtained the following result.
Proposition 4
Regional integration can achieve a socially optimal level of investment for any degree of economies of scaleand comprehension of the residents’ actual preference of the local public good in each region.

Appendix
Proof of Lemma 1: To determine a Pareto optimal allocation, we consider the following problem.

5δx12 δx22
I12 I22++ (1 + λ)g1 + (1 + λ)g2 − −xi ,Ii22
2 2
xs.t. x1 + x2 = 1
tx1 = g1 , tx2 = g2
t−

∂F1
∂F2
= 0, t −
=0
∂x1
∂x 2

The Lagrangian function is the following.
=δx12 δx22
I12 I22++ (1 + λ)tx1 + (1 + λ)tx2 − − + (x1 + x2 − 1)222 2
+ 1 (t − γ − I1 + δx1 ) +Here, the variables , 1 and

2 (t − γ − I2 + δx 2 )2 are Lagrangian variables. Consequently, the first-order condition is thefollowing.
∂= δxi + (1 + λ)t + +∂xi∂= −Ii −
∂Iiiδ = 0i =0∂= x1 + x 2 − 1 = 0∂∂= t − γ − Ii + δxi = 0,
∂ i(i = 1,2)

The socially optimal allocations which meet the first-order condition above are the following. The socially
1

δ

2

2

optimal level of capital is x1o = x2o = . The socially optimal level of investment is I1o = I2o = t − γ + .

Main References[1] Dembour, C. and X. Wauthy (2009), “Investment in public infrastructure with spillovers and tax
competition between contiguous regions,” Regional Science and Urban Economics, 39, pp.679-687.
[2] Hindriks, J., S. Peralta and S. Weber (2008), “Competing in taxes and investment under fiscal
equalization,” Journal of Public Economics, 92, pp.2392-2402.
[3] Keen, M. and M. Marchand (1997), “Fiscal competition and the pattern of public spending,” Journal of

Public Economics, 66, pp.33-53.

6