ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E16-9

Long-Run Growth and Welfare in an
Endogenous Growth Model with Productive
Public Goods and Spillover Effects
byToshiki Tamai
Koji Kitaura
NagoyaUniversity
MakotoHANAZONO

November 2016

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Long-Run Growth and Welfare in an Endogenous Growth Model with
Productive Public Goods and Spillover Effects
Toshiki Tamaia and Koji Kitaurab
a Graduate School of Economics, Nagoya University

Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japanb Faculty of Social Sciences, Hosei University,

4342 Aihara-cho, Machida-shi, Tokyo 194-0298, Japan

Abstract
This paper develops a two-region model of endogenous growth with productive public goods,
perfectly mobile capital, and immobile labor. Productive public goods have cross-border spillover
effects, which mean that the production of one region is affected by the productive public good of
another region. We investigate the interaction between regional fiscal policy, policy tasks of regional
governments, and spillover effects under conditions of capital flight. Our analysis shows that
regional governments, which have the task of maximizing regional welfare, fail to achieve thisbecause of spillover effects. Further, it is shown that fiscal policy, with a common central
government tax rate, also fails to maximize social welfare. Therefore, the Barro tax rule does not
hold.
Keywords: Productive public goods; Economic growth; Welfare
JEL Classifications: O41, H20, E62Corresponding author: tamai@soec.nagoya-u.ac.jp (T. Tamai).

11. Introduction
Empirical literature has investigated the relationship between the fiscal structures of state and local
governments and their economic growth. Miller and Russek (1997) found that state and local taxes
affect economic growth if revenue finances public services, such as transportation. Cohen and Paul
(2004), among others, suggested a positive relationship between regional core infrastructure (e.g.,roads, railways, airports, sewage and water facilities), and output in the US data. More recently, Bom
and Ligthart (2014) found that core infrastructure owned by local governments may be moreproductive than public capital supplied by the federal government, using data from selected OECD
countries. The effects of public investments at the regional level have traditionally been unable toreplicate the effects of public investments at the national level (See, for example, Pereira and Andraz(2013) for an excellent survey). One possible explanation for this paradox is that spillover effects
captured by national level studies are not captured at the regional level (Boarnet, 1998; Mikelbank
and Jackson, 2000). These authors posit that the empirical relevance of spillover effects acrossregions is largely an unresolved issue in theoretical studies. In this paper, we investigate theinteraction between regional fiscal policy, policy tasks of regional governments, and spillover effects,
using a two-region model of endogenous growth with productive public goods.
Existing endogenous growth models with productive public goods has mostly focused on thegrowth and welfare effects of fiscal policies, using only one sector. As shown in Barro (1990),
among others, this increases the marginal product of private capital and leads to economic growth,using a representative agent framework (e.g., Futagami et al., 1993; Turnovsky and Fisher, 1995).
Little theoretical research has attempted to explain the relationship between regional productive
public goods and economic growth. Figuières et al. (2013) is the only theoretical study where
infrastructure effects produced externalities, using a two country model. However, the above authors
did not consider regional growth effects. This study, therefore, analyzes the effect of regional
productive public goods on economic growth and welfare, incorporating tax competition withperfectly mobile capital and immobile labor.
The analysis of tax competition were pioneered by Wilson (1986) and Zodrow and Miezkowski(1986) and has been developed by many other authors. 1 Recently, theoretical studies haveconsidered the relationship between tax competition and economic growth (Lejour and Verbon,
1997; Becker and Rauscher, 2013; Hatfield, 2015), comparing fiscal centralization to
decentralization with imperfect capital mobility. Using perfect mobility, Rauscher (2005) examinedthe effects of a more mobile tax base on economic growth. This issue still seems empiricallycontroversial (e.g., Thornton, 2007).
Reflecting above arguments, our model has two main features. First, and most importantly, we
1

Wildasin and Wilson (2004) and Zodrow (2010) survey recent studies of tax competition.

2

assume that the growth engine, productive public goods, has spillover effects. In this paper,
following Bronzini and Piselli (2009) and Figuières et al. (2013), we assumed that productive public
goods have spillovers. Second, we considered maximizing equilibrium growth rate and regional
welfare by regional governments and compared these results with the benchmark case derived fromsocial welfare maximization or the Barro tax rule. Recently, Chu and Yang (2012) compared theperformance of fiscal decentralization with fiscal centralization in terms of both economic growthand social welfare, using the standard AK model.
The results of this study show that growth-maximizing and regional welfare-maximizing tax rates
set by regional governments are not equal to welfare-maximizing tax rates set by central
governments when there are asymmetric regions in the economy with mobile capital. Although thewelfare-maximizing tax rate is equal to the output elasticity of public input and is the same as thegrowth-maximizing tax rate for symmetric regions with same initial inputs, the welfare-maximizing
tax rate set by central governments differs from the output elasticity of public input for asymmetricregions. Initial capital and labor endowments of each region are different for asymmetric regions.
Capital moves across regions, while labor stays in each region. The government of each region
attracts mobile capital to maximize regional welfare, because capital inflow increases current income
levels. Therefore, the regional government sets the tax rate lower than the efficient level.
The remainder of this paper is organized as follows. In Section 2, a model is presented, anddecentralized equilibrium under fiscal policy by regional governments is characterized in Section 3.
In Section 4, the relationship between decentralized equilibria with different policy tasks is
examined. In particular, we focus on the relationship between regional and social welfare
maximization. Section 5 offers some conclusions.

2. The Model
Consider an economy that consists of two regions denoted by subscript numbers 𝑖 = 1,2. Time iscontinuous and indexed by t. The population of region i is constant over periods and denoted by 𝐿𝑖 .
Each region has a representative immobile resident who lives forever. The capital market is
integrated into the economy, but the labor market is not, and its participants are composed ofworkers living in their own region. We assume that the population of the economy is normalized tounity, i.e., 𝐿 = 𝐿1 + 𝐿2 = 1. Let 𝐾(𝑡) be aggregate private capital: 𝐾(𝑡) = 𝐾1 (𝑡) + 𝐾2 (𝑡).
The lifetime utility function of a representative resident is∞1−𝜃

(𝐶𝑖 (𝑡))− 1 −𝜌𝑡
∫
𝑒 𝑑𝑡,
1−𝜃
0
where 𝐶𝑖 (𝑡) is private consumption, 𝜃 is the inverse of the elasticity of intertemporal substitution,
3

and 𝜌 is the subjective discount rate. The budget constraint of the resident is𝐴̇𝑖 (𝑡) = 𝑟(𝑡)𝐴𝑖 (𝑡) + 𝑤𝑖 (𝑡)𝐿𝑖 − 𝐶𝑖 (𝑡),(1)where 𝐴𝑖 (𝑡) is the asset, 𝑟𝑖 (𝑡) is the equilibrium interest rate, and 𝑤𝑖 (𝑡) is the after-tax wage rateof region i. Solving the optimization problem of the resident, we obtain𝐶𝑖̇ (𝑡) 𝑟(𝑡) − 𝜌=,𝐶𝑖 (𝑡)
𝜃(2)and the transversality condition.
We assume that private goods are produced in each region by perfectly competitive firms using
capital and labor inputs. Regional output is affected by spillover effects of productive public goods;
the production function takes the form of 𝑌𝑖 = 𝐹𝑖 (𝐾𝑖 , ℎ𝑖 (𝐺𝑖 , 𝐺𝑗 )𝐿𝑖 ), where ℎ𝑖 (𝐺𝑖 , 𝐺𝑗 ) captures the
productivity effect of public goods includes spillover effect.
Suppose that regions have same production technology. Specifically, the production function inregion i is assumed to be
𝑌𝑖 (𝑡) = 𝜉[𝐾𝑖 (𝑡)]1−𝛼 [𝐺(𝑡)𝐿𝑖 (𝑡)]𝛼 ,(3)where 𝜉 > 0 is a positive constant, 𝐺(𝑡) represents productive and spillover effects of public
goods, and 0 < 𝛼 < 1.
The term of 𝐺(𝑡) is also specified as
1[(𝐻1(𝑡))−𝜎 + (𝐻2 (𝑡))−𝜎 ]−𝜎𝐺(𝑡) =,
2(4)

where 𝐺𝑖 is the productive public goods provided by region i’s government and 𝜎 ≥ −1. Thus, we
obtain the elasticity of substitution between local government expenditures as 𝜖 = 1/(1 + 𝜎). In
this specification, the productive public goods of one region positively affects not only its own
production but also that of the other region. When 𝜎 = 0, this formulation is consistent withfindings by Bronzini and Piselli (2009), who estimated the long-run relationship between total factor
productivity and public infrastructure between 1980 and 2001 across regions in Italy. They foundthat regional productivity was positively affected by the public infrastructure of neighboring regions.
A similar specification was considered by Figuieres et al. (2013).2
The regional government taxes capital input and uses the tax revenue for productive public goods.
The budget constraint of region i becomes
𝐺𝑖 (𝑡) = 𝜏𝑖 𝑌𝑖 (𝑡),(5)where 𝜏𝑖 denotes the capital tax rate in region i.
The profit maximization conditions for firms are𝑟𝑖 (𝑡) = (1 − 𝜏𝑖 )(1 − 𝛼)𝜉[𝐾𝑖 (𝑡)]−𝛼 [𝐺(𝑡)𝐿𝑖 (𝑡)]𝛼 ,(6)𝑤𝑖 (𝑡) = (1 − 𝜏𝑖 )𝛼𝜉[𝐾𝑖 (𝑡)]1−𝛼 [𝐺(𝑡)𝐿𝑖 (𝑡)]𝛼 /𝐿𝑖 .

(7)

2

Hirshleifer (1983, 1985) assumed the weakest link technology. Although we exclude the case of this extreme case (Leontief case),similar situation are considerable.

4

The factor price equalization holds: 𝑟1 = 𝑟2 because the capital market is integrated. In otherwords, we have
(1 − 𝜏1 )(1 − 𝛼)𝜉[𝐾1 (𝑡)]−𝛼 [𝐺(𝑡)𝐿1 (𝑡)]𝛼 = (1 − 𝜏2 )(1 − 𝛼)𝜉[𝐾2 (𝑡)]−𝛼 [𝐺(𝑡)𝐿2 (𝑡)]𝛼 .

(8)

Let us denote the equilibrium interest rate by 𝑟 . Hence, 𝑟(𝑡) ≡ 𝑟𝑖 (𝑡) holds for 𝑖 = 1,2 inequilibrium.
Let 𝜅𝑖 ≔ 𝐾𝑖 (𝑡)/𝐾(𝑡) and 𝜆𝑖 ≔ 𝐿𝑖 /𝐿. Note that 𝜅1 + 𝜅2 = 1 and 𝜆1 + 𝜆2 = 1 hold. Using
equations (4) and (5), the ratio of public goods to private capital satisfies𝜖𝜖−1 (𝜖−1)

𝜖−1𝑔(𝑡) =[{𝜏1 𝜉𝜅11−𝛼 𝑔(𝑡)𝛼 𝜆1𝛼 } 𝜖 + {𝜏2 𝜉𝜅21−𝛼 𝑔(𝑡)𝛼 𝜆𝛼2 } 𝜖 ],(9)2where 𝑔(𝑡) ≡ 𝐺(𝑡)/𝐾(𝑡). Furthermore, the condition for factor price equalization (7) is rewritten as(1 − 𝜏1 )𝜅1−𝛼 𝜆1𝛼 = (1 − 𝜏2 )𝜅2−𝛼 𝜆𝛼2 .

(10)Equation (10) leads to1

𝜅𝑖 =(1 − 𝜏𝑖 )𝛼 𝜆𝑖
11(1 − 𝜏𝑖 )𝛼 𝜆𝑖 + (1 − 𝜏𝑗 )𝛼 𝜆𝑗

𝜆𝑖

=

1
1 − 𝜏𝑗 𝛼= 𝜅𝑖 (𝜏1 , 𝜏2 ),(11)

𝜆𝑖 + () 𝜆𝑗
1 − 𝜏𝑖where𝜏𝑖 𝜕𝜅𝑖𝜏𝑖=−𝜅,(1
𝜅𝑖 𝜕𝜏𝑖− 𝜏𝑖 )𝛼 𝑗𝜏𝑗 𝜕𝜅𝑖𝜏𝑗=𝜅𝜅𝑖 𝜕𝜏𝑗 (1 − 𝜏𝑗 )𝛼 𝑗

(𝑖, 𝑗 = 1,2, 𝑖 ≠ 𝑗).

For a given 𝜏𝑖 , the ratio of public input to private capital 𝑔 is uniquely determined by equations (9)and (11):𝜖𝜖−1 (𝜖−1)(1−𝛼)𝜖−1{𝜏1 𝜉𝜅11−𝛼 𝜆𝛼1 } 𝜖 + {𝜏2 𝜉𝜅21−𝛼 𝜆𝛼2 } 𝜖𝑔=[]
2= 𝑔(𝜏1 , 𝜏2 ),

(12)

𝜏𝑖 𝜕𝑔
𝜙𝑖
𝛼 − 𝜏𝑖
𝜏𝑖 𝜅𝑖
=
.
[
]+
𝑔 𝜕𝜏𝑖 𝜙𝑖 + 𝜙𝑗 𝛼(1 − 𝛼)(1 − 𝜏𝑖 )(1 − 𝜏𝑖 )𝛼
𝜖−1

𝜖−1

where 𝜙1 ≡ {𝜏1 𝜉𝜅11−𝛼 𝜆1𝛼 } 𝜖 /2 and 𝜙2 ≡ {𝜏2 𝜉𝜅21−𝛼 𝜆𝛼2 } 𝜖 /2.
Equations (6), (8), (11), and (12) yield𝑟 = (1 − 𝜏1 )(1 − 𝛼)𝜉𝜅1−𝛼 𝜆1𝛼 𝑔𝛼 = (1 − 𝜏2 )(1 − 𝛼)𝜉𝜅2−𝛼 𝜆𝛼2 𝑔𝛼
𝜖−1

1−𝛼
𝜖
{𝜏1 (1 − 𝜏1 ) 𝛼 𝜆1}= (1 − 𝛼)𝛼𝜖
𝜖−1 (1−𝛼)(𝜖−1)1−𝛼𝜖+ {𝜏2 (1 − 𝜏2 ) 𝛼 𝜆2 }1

𝜉 1−𝛼 .

2
[

(13)

]

Using equations (3) and (9), the production function in equilibrium is
𝑌𝑖 (𝑡) = 𝜉𝜅𝑖1−𝛼 𝜆𝛼𝑖 𝑔𝛼 𝐾(𝑡).
5(14)The asset market clearing condition is𝐾(𝑡) ≔ 𝐾1 (𝑡) + 𝐾2 (𝑡) = 𝐴1 (𝑡) + 𝐴2 (𝑡).

(15)

By equations (1), (6), (7), (14), and (15), the resource constraint becomes2

2

𝐾̇(𝑡) = 𝑟𝐾(𝑡) + ∑ 𝑤𝑖 (𝑡)𝐿𝑖 − 𝐶(𝑡) = {𝑟 + 𝛼𝜉𝑔 ∑(1 − 𝜏𝑖 )𝜅𝑖1−𝛼 𝜆𝛼𝑖 } 𝐾(𝑡) − 𝐶(𝑡),
𝛼

𝑖=1

𝑖=1

where
𝐶(𝑡) ≔ 𝐶1 (𝑡) + 𝐶2 (𝑡).
Using equations (2), (6), (13), and the definition of 𝐶(𝑡), we obtain the equilibrium growth rate ofprivate consumption as𝐶̇ (𝑡) 𝐶𝑖̇ (𝑡) (1 − 𝜏𝑖 )(1 − 𝛼)𝜉𝜅𝑖−𝛼 𝜆𝛼𝑖 𝑔𝛼 − 𝜌==≡ 𝛾.
𝐶(𝑡) 𝐶𝑖 (𝑡)𝜃(16)Proposition 1. There exists a unique, stationary equilibrium, and the economy is always inequilibrium.

(Proof) See Appendix A.

In stationary equilibrium, all economic variables except for prices and ratios grow at same rate,
giving a balanced growth equilibrium. By Proposition 1, equation (16) stands for the balanced
growth rate. In the balanced growth equilibrium with equation (16), the indirect utility of region 𝑖 isderived as the following (See Appendices A and B):1−𝜃

𝑉𝑖 (0) =1(𝐶𝑖 (0))1− ],[1 − 𝜃 𝜌 + (𝜃 − 1)𝛾 𝜌

(17)

where
1−𝛼
1−𝛼
𝐶𝑖 (0) = [1 + () 𝜇𝑖 ] 𝑟𝐴𝑖 (0) = [𝜂𝑖 + () 𝜅𝑖 ] 𝑟𝐾(0),
𝛼
𝛼
𝜂𝑖 ≡

(18)

𝐴𝑖 (0)𝐾𝑖 (0) 𝜅𝑖, 𝜇𝑖 ≡= .
𝐾(0)𝐴𝑖 (0) 𝜂𝑖

Hereafter, we omit the script t, except for calling readers’ attention to it.
Differentiating equation (17) with respect to 𝜏𝑖 and 𝜏𝑗 , the effects of a change in 𝜏𝑖 and 𝜏𝑗 on
the welfare of region 𝑖 are𝜕𝑉𝑖 (0)1−𝜃 𝜕𝐶𝑖 (0)1−𝜃 𝜕𝛾{𝜌 + (𝜃 − 1)𝛾} + (𝐶𝑖 (0))=[(𝐶𝑖 (0))],
2[𝜌 + (𝜃 − 1)𝛾]
𝜕𝜏𝑖
𝜕𝜏𝑖
𝜕𝜏𝑖

(19a)

𝜕𝑉𝑖 (0)1−𝜃 𝜕𝐶𝑖 (0)1−𝜃 𝜕𝛾{𝜌 + (𝜃 − 1)𝛾} + (𝐶𝑖 (0))=[(𝐶𝑖 (0))].
2[𝜌 + (𝜃 − 1)𝛾]
𝜕𝜏𝑗
𝜕𝜏𝑗
𝜕𝜏𝑗

(19b)

Note that by equation (18), in a general case, we have6𝛼(1 − 𝛼 ) 𝜅𝑖
𝜏𝑖 𝜕𝐶𝑖 (0)𝜂𝑖𝜏𝑖 𝜕𝜂𝑖𝜏𝑖 𝜕𝜅𝑖 𝜏𝑖 𝜕𝑟
=
+
+
,
𝛼
𝛼
𝐶𝑖 (0) 𝜕𝜏𝑖[𝜂𝑖 + (
𝜅𝑖 ] 𝜂𝑖 𝜕𝜏𝑖 [𝜂𝑖 + (𝜅𝑖 ] 𝜅𝑖 𝜕𝜏𝑖 𝑟 𝜕𝜏𝑖
)
)
1−𝛼
1−𝛼

(20a)

𝛼
(1 − 𝛼 ) 𝜅𝑖𝜏𝑗 𝜕𝐶𝑖 (0)𝜏𝑗 𝜕𝜂𝑖𝜏𝑗 𝜕𝜅𝑖 𝜏𝑗 𝜕𝑟
𝜂𝑖
=
+
+
.
𝛼
𝛼
𝐶𝑖 (0) 𝜕𝜏𝑗[𝜂𝑖 + (
𝜅𝑖 ] 𝜂𝑖 𝜕𝜏𝑗 [𝜂𝑖 + (𝜅𝑖 ] 𝜅𝑖 𝜕𝜏𝑗 𝑟 𝜕𝑗
)
)
1−𝛼
1−𝛼

(20b)

3. Local Government Fiscal Policy
In this section, we examine the effects of fiscal policy by local governments. The policy objectivesof local governments are to maximize regional welfare growth. We derive the social
welfare-maximizing tax rates.
The Growth-Maximizing Local Government. We begin our analysis with a case where the local
government has the policy objective of maximizing growth. Then, the first order conditions for
growth-maximizing local governments must satisfy
𝜏𝑖 𝜕𝑟
𝜏𝑖
𝜏𝑖 𝜕𝑔 𝜏𝑖 𝜕𝜅𝑖
𝜙𝑖
𝛼
𝜏𝑖
=−
+𝛼[
−
−
]=0⟺
[
] = 0.
𝑟 𝜕𝜏𝑖
1 − 𝜏𝑖𝑔 𝜕𝜏𝑖 𝜅𝑖 𝜕𝜏𝑖
𝜙𝑖 + 𝜙𝑗 1 − 𝛼 1 − 𝜏𝑖(21)Solving equation (21) with respect to 𝜏𝑖 , we obtain the following proposition:
Proposition 2. Growth-maximizing local governments set the regional income tax rate to𝑔𝑔

𝜏1 = 𝜏2 = 𝛼,(22)Growth maximization is equivalent to maximizing the interest rate. Equation (21) implies that the
local government policy to maximize the interest rate is independent of other local government
policies. Therefore, as in a standard Barro model (1990), the local government must set the tax rate
to satisfy equation (22)3.
A Regional Welfare-Maximizing Local Government. The local government is regionallybenevolent and therefore has the policy objective of maximizing regional welfare. Equations (19a)and (19b) show that the welfare effect of income taxes is composed of two different effects: one is
the growth effect that relates to equation (21), and the other is the effect on initial consumption. By
Proposition 2, an increase in welfare through the growth effect is maximized when equation (22)holds. If
𝜕𝜂𝑖
𝛼
𝜕𝜅𝑖
= −(
,
)
𝜕𝜏𝑖
1 − 𝛼 𝜕𝜏𝑖
3

The relationship between Proposition 2 and the Barro tax rule is discussed in Section 5.

7

(23)

then we have
𝜕𝜂𝑖
𝛼
𝜕𝜅𝑖
𝜏𝑖 𝜕𝐶𝑖 (0) 𝜏𝑖 𝜕𝑟
= −(
⇒
=
.
)
𝜕𝜏𝑖
1 − 𝛼 𝜕𝜏𝑖
𝐶𝑖 (0) 𝜕𝜏𝑖𝑟 𝜕𝜏𝑖With equation (23), the effect on initial consumption is equivalent to the growth effect. Then,regional welfare maximization is equivalent to growth maximization, and equation (21) holds.
Initially, however, regional wealth and total capital stock are fixed. Therefore, equation (21) is
dismissed; ∂𝜂𝑖 /𝜕𝜏𝑖 = 0. Then, by equations (19a)-(22), we obtain𝛼1−𝜃
(1 − 𝛼 )𝜕𝑉𝑖 (0)(𝐶𝑖 (0))
𝜕𝜅𝑖
|
=
< 0.
𝜕𝜏𝑖 𝜏 =𝜏 =𝛼 𝜌 + (𝜃 − 1)𝛾 [𝜂 + ( 𝛼 ) 𝜅 ] 𝜕𝜏𝑖
𝑖
𝑖
𝑗
1−𝛼 𝑖
The inequality mentioned above implies that the regional welfare-maximizing tax rate is less than
the growth-maximizing tax rate, which is equal to the output elasticity of public inputs. A rise in theincome tax rate has a positive effect on output through an increase in public input and negatively
affects welfare through a decrease in disposable income and capital outflow. If the local government
aims to maximize the welfare of its own region, it must take into account not only the positive
welfare effect of economic growth but also the negative welfare effect of capital outflow. Therefore,
the local government maximizes regional welfare by setting the tax rate at a level below thegrowth-maximizing tax rate. This result is summarized as the following proposition:

Proposition 3. The regional government that maximizes regional welfare sets the regional income
tax rate to 𝜏𝑖∗∗ such as𝑔𝑔

max[𝜏1∗∗ , 𝜏2∗∗ ] < 𝜏1 = 𝜏2 = 𝛼.

(24)

4. Central Government Fiscal Policy
In this section, we consider the relationships between the equilibria derived in the previous section
with the central government. We assume that 𝜃 = 1 throughout the remainder of this paper and also
assume that the central government chooses income tax rates for each region to maximize social
welfare by keeping tax rates constant. Define the sum of regional welfare functions as the social
welfare function: 𝑊 = 𝜆1 𝑉1 + 𝜆2 𝑉2 . Then, the partial derivatives of 𝑊, with respect to 𝜏𝑖 , is given
as
𝜕𝑊
𝜕𝑉1
𝜕𝑉2
= 𝜆1
+ 𝜆2
,
𝜕𝜏𝑖
𝜕𝜏𝑖
𝜕𝜏𝑖

(27)

where (19a)-(20b).
The first-order conditions for maximizing social welfare are ∂𝑊/𝜕𝜏𝑖 = 0 (𝑖 = 1,2). Our task
8

focuses on the relationship among tax rates under different government objectives. Thus, we only
have to check the sign of ∂𝑊/𝜕𝜏𝑖 evaluated at the different first-order conditions. At the tax ratethat maximizes regional welfare, we obtain𝛼𝛼(1 − 𝛼 ) 𝜅2 𝜏1 𝜕𝜅2(1 − 𝛼 ) 𝜅1 𝜏1 𝜕𝜅1
𝜕𝑊
𝜆2
=
[
−
] > 0, (28a)|𝜕𝜏1 𝜕𝑉1 =𝜕𝑉2 =0 𝜏1 𝜌 𝜂 + ( 𝛼 ) 𝜅 𝜅2 𝜕𝜏1 𝜂 + ( 𝛼 ) 𝜅 𝜅1 𝜕𝜏1
2
1𝜕𝜏1 𝜕𝜏2
1−𝛼 2
1−𝛼 1
𝛼
𝛼
(1 − 𝛼 ) 𝜅1 𝜏2 𝜕𝜅1(1 − 𝛼 ) 𝜅2 𝜏2 𝜕𝜅2
𝜕𝑊
𝜆1
=
[
−
] > 0. (28b)|𝜕𝜏2 𝜕𝑉1 =𝜕𝑉2 =0 𝜏2 𝜌 𝜂 + ( 𝛼 ) 𝜅 𝜅1 𝜕𝜏2 𝜂 + ( 𝛼 ) 𝜅 𝜅2 𝜕𝜏21
122
𝜕𝜏1 𝜕𝜏21−𝛼1−𝛼Equations (28a) and (28b) show that the regional welfare-maximizing tax rate by a local government
is lower than the social welfare-maximizing tax rate. Each regional government maximizes welfare
in its own region by being concerned about the negative effects of capital outflows. However, capitalflow of one region is capital inflow of another region. Capital inflow brings about financial benefits
and an increase in public expenditures in the destination location. Furthermore, the negative welfare
effect of capital outflows is diminished by simultaneous control of tax rates in two regions.
Therefore, the social welfare maximizing tax rate is set larger than the tax rate of Proposition 3.
On the other hand, we have the following results for the growth-maximizing tax rate:𝜕𝑊
𝜆1 𝜆2
𝜆2 𝜂1 − 𝜆1 𝜂2
=
,
(29a)
|
2
𝛼
𝛼
𝜕𝜏1 𝜏𝑔 =𝜏𝑔=𝛼 (1 − 𝛼) 𝜌 [𝜂 + (
𝜆
]
[𝜂
+
𝜆
]
)
(
)
1
1
2
1−𝛼 1 2
1−𝛼 2
𝜕𝑊
𝜆1 𝜆2
𝜆2 𝜂1 − 𝜆1 𝜂2
=−
.
(29b)
|
(1 − 𝛼)2𝜌 [𝜂 + ( 𝛼 ) 𝜆 ] [𝜂 + ( 𝛼 ) 𝜆 ]𝜕𝜏2 𝜏𝑔 =𝜏𝑔 =𝛼1
12
1−𝛼 1 21−𝛼 2Note that 𝜅𝑖 = 𝜆𝑖 holds when 𝜏1 = 𝜏2 . Equations (29a) shows that the social welfare-maximizing
tax rate is larger (smaller) than the growth-maximizing tax rate if the denominator is positive(negative). Equation (29b) shows the opposite result of (29a). In asymmetric equilibrium, the social
welfare tax rate of Region 1 and Region 2 will differ from each other, because endowment of thelabor force and of capital differ from each other. However, the growth-maximizing tax rate iscommon to the two regions. Therefore, a common tax rate under the heterogeneity of regions
engenders an additional distortion, according to the misdistribution of endowments.
These results provide the following proposition.

Proposition 4. (i) The regional welfare-maximizing tax rate is less than the social
welfare-maximizing tax rate of region i:
𝜏𝑖∗∗ < 𝜏𝑖∗ .

(30a)(ii) The relationship between the social welfare-maximizing tax rate of region i and thegrowth-maximizing tax rate is determined by the relative per-capita wealth of two regions:9𝜏1∗ ⋛ 𝛼, 𝜏2∗ ⋚ 𝛼 ⇔

𝜂1 𝜂2
⋛ .
𝜆1 𝜆2

(30b)

By Proposition 2-4, we summarize the relationship between tax rates under different government
objectives as follows:

Proposition 5. The relationship between the regional welfare-maximizing tax rate, the social
welfare-maximizing tax rate, and the growth-maximizing tax rate are𝜏1∗∗ < 𝛼 < 𝜏1∗ , 𝜏2∗∗ < 𝜏2∗ < 𝛼 ⇔

𝜂1 𝜂2
> ,
𝜆1 𝜆2

(31a)

𝜏1∗∗ < 𝜏1∗ = 𝛼, 𝜏2∗∗ < 𝜏2∗ = 𝛼 ⇔

𝜂1 𝜂2
= ,
𝜆1 𝜆2

(31b)

𝜏1∗∗ < 𝜏1∗ < 𝛼, 𝜏2∗∗ < 𝛼 < 𝜏2∗ ⇔

𝜂1 𝜂2
< .
𝜆1 𝜆2

(31c)

Standard models of endogenous growth with productive government expenditures assume one
sector economy with immobile capital. Since no difference of region and capital flow exists, growth
maximization is equivalent to welfare maximization if the production technology follows a
Cobb-Douglas production function4. However, if there are asymmetric regions in the economy withmobile capital, growth-maximizing and regional welfare-maximizing by regional governments are
not equivalent to welfare-maximizing. If and only if the endowments of capital and labor satisfy(31b), for both region 1 and region 2, will growth maximization be equivalent to social welfare
maximization. Therefore, social welfare maximization is not attainable by means of common income
tax rates.
To characterize this result, we now consider the special case that the central government imposes aflat income tax on a country's residents. The income tax rate is common to all residents in the
country; 𝜏𝑖 = 𝜏 for 𝑖 = 1,2. With 𝜏𝑖 = 𝜏 (𝑖 = 1,2), capital does not move across regions, that is,∂𝜅𝑖 /𝜕𝜏 = 0. Furthermore, we have ∂𝜂𝑖 /𝜕𝜏𝑖 = 0, because the initial wealth share of each region isfixed. Then, the following equation is true for the central government:𝜕𝜂𝑖 𝜕𝜅𝑖𝜏 𝜕𝐶𝑖 (0) 𝜏 𝜕𝑟𝜕𝑉𝑖 (0)
𝜕𝛾
𝜕𝑟
=
=0⇒
=
⇒ sgn
= sgn
= sgn .
𝜕𝜏𝜕𝜏𝐶𝑖 (0) 𝜕𝜏
𝑟 𝜕𝜏
𝜕𝜏
𝜕𝜏
𝜕𝜏
Equation (13), with 𝜏𝑖 = 𝜏 (𝑖 = 1,2), leads to𝜖𝛼𝜖−1 (𝜖−1)(1−𝛼)𝜖−1𝑟 = (1 − 𝜏)(1 − 𝛼) [𝛽(𝜆1) 𝜖 + (1 − 𝛽)(𝜆2 ) 𝜖 ]

4

1

𝛼

𝜉 1−𝛼 𝜏 1−𝛼 .

Misch et al. (2013) investigated the relationship between growth maximization and welfare maximization in the extended Barro(1990) model with the CES production function. They showed that the elasticity of substitution between private capital and public
input is key to the relationship between maximization of growth and welfare. Tamai (2013) derived results similar to Misch et al.
(2013), using a stochastic growth model of Barro (1990) with the CES production technology.

10The partial differentiation of 𝑟 with respect to 𝜏 provides
𝜕𝑟
1
𝛼
=−
+
⋛ 0 ⇔ 𝜏 ⋚ 𝛼.
𝑟𝜕𝜏
1 − 𝜏 (1 − 𝛼)𝜏
We obtain the following tax rule. The welfare-maximizing income tax rate of the central
government is𝜏 ∗ = 𝛼.

(32)This result is the well-known Barro tax rule. With an income tax rate that is common to two regions,a provision of public inputs is independent of the characteristics of the region. Therefore, thewelfare-maximizing income tax rate of the central government is independent of deep parameterssuch as 𝜂𝑖 and 𝜆𝑖 . From (31b) of Proposition 5 and (32), we arrive at the following result:𝜂𝜆Proposition 6. The Barro rule holds if and only if 𝜂1 = 𝜆1 .
2

2Finally, we explain the intuition of Proposition 5 and 6. Proposition 5 implies that residents inregions with large initial per capita assets should be levied a heavy tax on income, while taxes inregions with small initial per capita assets should be low. Focusing on symmetric regions, central
governments have to choose a common tax rate to maximize social welfare, because an economy
with symmetric regions is equivalent to an economy with representative agents; there is no migrationof capital. Maximization of social welfare coincides with maximization of economic growth; thus,
the Barro rule holds.
On the other hand, this is not true for an asymmetric region. Suppose that region 1 has a higher
initial per capita asset than region 2. If income tax rates are the same for these two regions (e.g., theBarro rule is adopted), the rate of return on capital in region 1 is smaller than that of region 2, for agiven, initial per-capita asset before capital moves. Then, capital flight occurs from region 1 toregion 2, bringing about income redistribution from region 1 to region 2. This negatively affects
region 1’s welfare and positively affects region 2’s welfare. Since initial per capita assets in region 1
are larger than in region 2, a positive welfare effect dominates when evaluated by the Barro rule.

5. Conclusion
In this paper, we examined the relationship between regional fiscal policy, policy tasks of regional
government, and spillover effects, using a two-region model of endogenous growth for productive
public goods, mobile capital, and immobile labor. We considered two regional governments’ policy
tasks: 1. that regional governments want to maximize growth, and 2. that regional governments want11

to maximize welfare.
To maximize growth, regional governments set the income tax rate to the output elasticity of
public services. Then, the growth-maximizing tax rate is common for the two regions, and there isno capital flight. However, to maximize welfare, regional governments attract capital into their own
regions, because capital inflow increases current regional income. Then, public services are
insufficiently provided, compared with an optimal level obtained through spillover effects.
In general, a growth-maximizing tax policy does not coincide with a social welfare-maximizing
policy. Welfare maximization by regional governments may not improve the supply of public
services as well as a growth-maximizing policy. However, growth maximization does not provide
the best results for social welfare. Similarly, a social welfare-maximizing policy, by means of acommon tax rate on income, is not suitable for attaining social welfare maximization in an economy
with asymmetric regions. Therefore, the Barro tax rule does not hold in a two-region model withspillover effects. It is necessary initially to introduce asset transfers between two regions to find the
optimal equilibrium. This, however, is a highly controversial issue of redistributive policy.
Finally, we would like to mention the directions of certain extended studies. In this study, we
assume that regional governments have an identical policy task. However, in reality, regional
governments have different policy tasks, leading to a Stackelberg Leader-Follower game.
Incorporating these issues into our basic model will derive additional policy implications. In such acase, our study will provide an analytical basis for these future studies.

12Appendix
A. Proof of Proposition 1 and Derivation of Equation (17)Let 𝑧𝑖 ≡ 𝐶𝑖 /𝐾 (𝑖 = 1,2). Equation (16) and the resource constraint lead to2

𝑧̇𝑖 𝐶𝑖̇ 𝐾̇
= − = 𝛾 − {𝑟 + 𝛼𝜉𝑔𝛼 ∑(1 − 𝜏𝑖 )𝜅𝑖1−𝛼 𝜆𝛼𝑖 } + 𝑧1 + 𝑧2 .
𝑧𝑖 𝐶𝑖 𝐾
𝑖=1

By solving 𝑧̇𝑖 = 0 with respect to 𝑧1 + 𝑧2, we get a unique solution 𝑧1 + 𝑧2 such as 𝑧̇𝑖 = 0:2

𝑧1 + 𝑧2 = {𝑟 + 𝛼𝜉𝑔 ∑(1 − 𝜏𝑖 )𝜅𝑖1−𝛼 𝜆𝛼𝑖 } − 𝛾.
𝛼𝑖=1

The value of 𝑧𝑖 is determined as follows. By (1), (7), (13), (16), and 𝐴̇𝑖 (0) = 𝛾𝐴𝑖 (0), we have
𝐶𝑖 (0) = (𝑟 − 𝛾)𝐴𝑖 (0) + 𝑤𝑖 (0)𝐿𝑖 = (1 − 𝜏𝑖 )𝑌𝑖 (0) + 𝑟[𝐴𝑖 (0) − 𝐾𝑖 (0)]= [(1 − 𝜏𝑖 )𝜇𝑖

𝑌𝑖 (0)+ (1 − 𝜇𝑖 )𝑟] 𝐴𝑖 (0)𝐾𝑖 (0)𝛼𝛼= [1 + () 𝜇𝑖 ] 𝑟𝐴𝑖 (0) = [𝜂𝑖 + () 𝜅 ] 𝑟𝐾(0).
1−𝛼
1−𝛼 𝑖
Therefore, a unique stationary equilibrium exists. The determinant and trace of Jacobian matrix forthe above dynamic system around the stationary equilibrium are tr𝐽 = 𝑧1 + 𝑧2 and det𝐽 = 0. This isbecause the dimension of the dynamic system is substantially one. We now define 𝑧 as 𝑧1 + 𝑧2.
Then, we have2𝑧̇= 𝛾 − {𝑟 + 𝛼𝜉𝑔𝛼 ∑(1 − 𝜏𝑖 )𝜅𝑖1−𝛼 𝜆𝛼𝑖 } + 𝑧.
𝑧
𝑖=1

The characteristic root of this differential equation is positive. Since there is one jumpable variable,
the economy immediately reaches the stationary equilibrium at an initial time and remains there
afterwards.

B. Derivation of Equation (18)Using equation (16), the indirect utility function can be calculated as
∞

1−𝜃

∞
∞
(𝐶𝑖 (𝑡))− 1 −𝜌𝑡1
1−𝜃 −𝜌𝑡
𝑉𝑖 (0) = ∫𝑒 𝑑𝑡 =[∫ (𝐶𝑖 (𝑡))𝑒 𝑑𝑡 − ∫ 𝑒 −𝜌𝑡 𝑑𝑡]
1−𝜃
1−𝜃 0
0
0

=

∞
∞
1
[∫ (𝐶𝑖 (0)𝑒 𝛾𝑡 )1−𝜃 𝑒 −𝜌𝑡 𝑑𝑡 − ∫ 𝑒 −𝜌𝑡 𝑑𝑡]
1−𝜃 0
0

=

∞
∞
1
1−𝜃
[(𝐶𝑖 (0))∫ 𝑒 {(1−𝜃)𝛾−𝜌}𝑡 𝑑𝑡 − ∫ 𝑒 −𝜌𝑡 𝑑𝑡]
1−𝜃
0
0

13

1−𝜃

1
(𝐶𝑖 (0))1=− ].
[1 − 𝜃 𝜌 + (𝜃 − 1)𝛾 𝜌

Acknowledgements
The authors would like to thank Yoshimasa Aoki, Junko Doi, Akira Kamiguchi, Minoru Kitahara,
Kautoshi Miyazawa, Tetsuya Nakajima, Hideki Nakamura, Hikaru Ogawa, Ryosuke Okazawa, Akira
Yakita, Koichi Yotsuya and the participants of PET2015 (Association for Public Economic Theory),
the autumn 2014 meeting of the Japan Association for Applied Economics, the seminar at Osaka
City University, and the Nagoya Macroeconomics Workshop for their helpful comments. Tamai also
thanks the JSPS KAKENHI (Grant Number 24530377) for its financial support.

14

References
Bom, P. R., and Ligthart, J. E., 2014. What have we learned from three decades of research on the
productivity of public capital? J ECON SURV. 28 (5), 889–916.
Barro, R.J., 1990. Government spending in a simple model of endogenous growth, J Polit Econ . 98(5), part 2, S103–S125.
Becker, D., and Rauscher, M., 2013. Fiscal competition and growth when capital is imperfectly
Mobile. Scan J Econ. 115 (1), 211–233.
Boarnet, M. G., 1998. Spillovers and the locational effects of public infrastructure. J. Regional Sci.
38 (3), 381–400.
Bronzini, R., and Piselli, P., 2009. Determinants of long-run regional productivity with geographicalspillovers: the role of R&D, human capital and public infrastructure. REG SCI URBAN ECON.
39 (2), 187–199.
Chu, A. C., and Yang, C. C., 2012. Fiscal centralization versus decentralization: Growth and welfare
effects of spillovers, leviathan taxation, and capital mobility. J Urban Econ. 71 (2), 177–188.
Cohen, J.P. and C.J.M. Paul 2004. Public infrastructure investment, interstate spatial spillovers, andmanufacturing costs, Rev. Econ. Stat. 86 (2), 551–560.
Figuières, C., F. Prieur and M. Tidball 2013. Public infrastructure, non-cooperative investments, andendogenous growth, Can. J. Econ. 46 (2), 587–610.
Futagami, K., Y. Morita and A. Shibata 1993. Dynamic analysis of an endogenous growth modelwith public capital, Scan J Econ. 95 (4), 607–625.
Hatfield, J. W., 2015. Federalism, taxation, and economic growth. J Urban Econ, 87, 114–125.
Hirshleifer, J., 1983. From weakest-link to best-shot: the voluntary provision of public goods, Public
Choice 41 (3), 371–386.
Hirshleifer, J., 1985. From weakest-link to best-shot: correction, Public Choice 46 (2), 221–223.
Lejour, A. M., and Verbon, H. A., 1997. Tax competition and redistribution in a two-country
endogenous-growth model. INT TAX PUBLIC FINANCE. 4 (4), 485–497.
Mikelbank, B. A., and Jackson, R. W., 2000. The role of space in public capital research. Int Reg Sci
Rev. 23 (3), 235–258.
Misch, F., N. Gemmell and R. Kenller 2013. Growth and welfare maximization in models of publicfinance and endogenous growth, JPET. 15 (6), 939–967.
Miller, S. M., and Russek, F. S., 1997. Fiscal structures and economic growth: international evidence.
Econ. Inq. 35 (3), 603.
Pereira, A. and J. Andraz 2013. On the economic effects of public infrastructure investment: asurvey of the international evidence. J. Dev Econ. 38 (4), 1–37.
Rauscher, M., 2005. Economic growth and tax-competing leviathans. INT TAX PUBLIC FINANCE.
12 (4), 457–474.
15

Tamai, T., 2013. The macroeconomic effects of fiscal policy in a stochastically growing economy.
Econ Model. 35, 464–471.
Thornton, J., 2007. Fiscal decentralization and economic growth reconsidered. J Urban Econ. 61 (1),64–70.
Turnovsky, S.J. and H. Fisher 1995. The composition of government expenditure and its
consequences for macroeconomic performance. J Econ Dyn Control. 19 (4), 747–786.
Wilson, J.D., 1986. A theory of inter-regional tax competition. J Urban Econ 19, 296–315.
Wilson, J.D., and Wildasin, D.E., 2004. Capital tax competition: bane or boon. J Public Econ. 88 (6),1065–1091.
Zodrow, G.R., and Mieszkowski, P., 1986. Pigou, property taxation and the underprovision of local
public goods. J Urban Econ. 19, 356–370.
Zodrow, G.R., 2010. Capital mobility and capital tax competition, National Tax Journal 63 (4), Part2, 865–902.

16