ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E20-4

Nash Equilibria in Models of Fiscal Competition
with Unemployment
byToshiki Tamai
Yuya Kikuchi

November 2020

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITYNash Equilibria in Models of Fiscal Competition with Unemployment

Toshiki Tamaia* and Yuya Kikuchiba, bGraduate School of Economics, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan

Abstract
This paper examines two different fiscal competition games under labor market
imperfection. Given that capital moves across regions and affects regional employment,
governments must choose expenditure levels and tax rates on such mobile capital byaccounting for the effects of fiscal variables on both capital and labor. Then, governments
may play these games with either tax rates on mobile capital or with public expenditures.
The presence/absence of absentee ownership of capital and employment externalities areimportant factors for characterizing two distinct Nash equilibria, one that occurs with tax
competition and the other with expenditure competition. In particular, and contrary to theexisting literature, tax rates under tax competition are likely to be more competitive thanunder expenditure competition because of employment externalities. Furthermore, in somecases, governments prefer to choose government expenditure as their strategic variable
rather than tax rates. The presence of employment externalities motivates governments to
use government expenditure as the strategic variable through which it might encourage
strategic effects.
Keywords: Fiscal Competition; Unemployment
JEL Classifications: H21; H71; J64*

Corresponding author. Tel: +81-52-789-2384. E-mail address: tamai@soec.nagoya-u.ac.jp. We thank Mutsumi Matsumoto, Yukihiro
Nishimura, Kimiko Terai, Akihiko Yanase, and participants in the 2020 Annual Meeting of Japanese Institute of Public Finance, the2020 Autumn Meeting of Japanese Economic Association, the 2020 International Conference of International Institute of Public
Finance, and Kansai Public Economics Workshop for their helpful comments. This work was supported by JSPS KAKENHI Grant
Numbers 16K03726, 16KK0077, and 20H01492.

11. Introduction
Fiscal competition has been widely discussed since Zodrow and Mieszkowski (1986) and Wilson(1986) developed theoretical frameworks. 1 One of the important issues in the literature is theexistence of unemployment. Some theoretical studies have attempted to explain the relationshipbetween fisacal competition and unemployment (e.g., Ogawa et al. 2006; Aronsson and Wehke 2008;
Sato 2009; Eichner and Upmann 2012; Exbrayat et al. 2012; Kikuchi and Tamai 2019). Ogawa et al.
(2006) and Kikuchi and Tamai (2019) are based on fixed wages model, whereas Sato (2009)
introduced job search and recruiting friction to fiscal competition model. Aronsson and Wehke (2008),
Eichner and Upmann (2012) and Exbrayat et al. (2012) applied bargaining between unions and firms.
In particular, Ogawa et al. (2006) theoretically examined the issue of fiscal competition andemployment. It is demonstrated in their study that the efficiency of providing public goods financed
by capital taxes depends on the degree of complementarity/substitutability between capital and labor.
They also showed that the mechanism behind the result is based on fiscal and employment externalities.
Most recently, Kikuchi and Tamai (2019) incorporated fiscal transfers into their model and clarified
how externalities affect the motives for choosing tax rates. Their findings imply that the presence of
employment externalities is essential for implementing optimal policies. 2 However, their results
depend on the assumption that governments use only tax rate as policy variable. In practice, tax
competition may no longer exist, and countries are facing intergovernmental/interregional competition
for using other policy variable.3
Wildasin (1988, 1991) developed theoretical framework to answer the question what instruments
governments should implement under the environment of competing for capital with other regions.
Wildasin (1988) considered two possible Nash equilibria, in which governments play games eitherwith tax rates on capital or with public expenditures4. The comparative analysis of the two equilibria
showed that the equilibrium with taxes as the control variable is not identical to the equilibrium withexpenditures. The tax rate under competition in public expenditure is more competitive than that in tax
competition. Wildasin (1991) examined which of the two—tax rates or expenditures—is the best
strategic variable and proved that the tax rate is the better strategic variable.
Subsequent theoretical studies have attempted to clarify the strategic choice on policy instruments(e.g., Hoyt 1993; Bayindir-Upmann 1998; Köethenbüerger 2011a). 5 Hoyt (1993) compared theequilibria of tax competition with the equilibria that of expenditure competition in case of Tibout
model where residents are mobile among jurisdictions in a metropolitan area. He reversed Wildasin(1988)’s conclusion when the government maximize land rent and the demand for housing is elastic.
Bayindir-Upmann (1998) examined two different games of fiscal competition in which local
governments provide public inputs. Contrary to the case of public goods analyzed by Wildasin (1988,
1991), this study shows that competition in productive public expenditure does not necessarily lead tomore competitive tax rates than in tax competition. In relation to the fiscal competition model,
Köethenbüerger (2011a) incorporated fiscal transfers and taxes on labor into the model and examined
the issues related to taxes versus expenditures. His model excludes tax-based mobility and analyzes
only inter-regional dependencies through fiscal transfer. Köethenbüerger (2011a) showed that thepolicy variables chosen by governments vary depending on the structure of fiscal transfers.
1

Recently, Zodrow (2010) and Keen and Konrad (2013) surveyed the theoretical literature on fiscal competition. On the other hand,
Devereux and Loretz (2013) and Revelli (2015) surveyed the recent empirical literature.
2
Some empirical studies have found evidence to support the effect of taxes on employment. Numerous studies have found significant
effects of taxes on employment (e.g., Bettendorf et al. 2009; Feld and Kirchgassner 2002; Feldmann 2011; Felix 2009; Harden andHoyt 2003; Zirgulis and Šarapovas 2017). Although the literature found a negative relationship between taxes and employment,
Feldmann (2011) showed that higher corporate taxes might lower the unemployment rate.
3
Some empirical studies investigated the impact of taxes and public infrastructures on the allocation of private capital. BénassyQuéré et al. (2007) showed that both the corporate taxes and the stock of public capital are significant using a data set of US FDI in
18 EU countries from 1994 to 2003. Hauptmeier et al. (2012) suggested that local governments use both the business taxes and public
inputs to compete for mobile capital using a data set of the 1100 German municipalities in the state of Baden-Wuerttemberg from1998 to 2004.
4
The existence and characteristics of the tax competition equilibrium have been examined by numerous studies (e.g., Laussel and Le
Breton; 1998, Bayindir-Upmann and Ziad, 2005; Petchey and Shapiro, 2009; Taugourdeau and Zaid, 2011; Rota-Graziosi, 2019).
5
Several studies developed comparisons between taxes and transfers and between unit taxes and ad valorem taxes (Hindriks, 1999;
Lockwood, 2004; Akai et al., 2011; Köethenbüerger, 2011b).

2

The purpose of the present study is to examine two different under-employment equilibria. One isthe equilibrium wherein governments compete in terms of tax rates and the other is under competition
for government expenditures. It is shown in this study that there exists a unique Nash equilibrium
regardless of whether the tax rate or expenditure level is the control variable. Comparative analysesof two equilibria demonstrate that the presence/absence of absentee ownership of capital is essential
to characterizing the equilibrium properties and outcomes. If all capital is owned by absentee owners,
the tax rates under expenditure competition are more competitive (i.e., lower) than that under tax
competition. In contrast, the tax rates under expenditure competition are less competitive (i.e., higher)
than under tax competition, depending on the size of fiscal and employment externalities, whenresidents equally share the economy-wide capital stock.
Furthermore, welfare analyses are also conducted in this paper. The analyses show that the welfare
superiority of the possible equilibria depends on the presence/absence of absentee ownership of capitaland the size of the employment externality. With the absentee ownership of capital, government
expenditure is chosen as a strategic variable if the employment externality is sufficiently strong andnegative. However, it will be unlikely that government expenditure is a strategic variable whenresidents equally share the economy-wide capital stock.
Wildasin (1988) compares the equilibrium of tax competition with that of expenditure competitionin an economy without unemployment. We find that with the absentee ownership of capital,
government expenditure is chosen as a strategic variable. Our findings differ from Wildasin (1988),
which finds that tax is a strategic variable. Bayindir-Upmann (1998) investigated two equilibria
similarly to Wildasin (1988) in a situation where the government supplies the public inputs. Unlike
governments supplying public goods, the possibility of negative capital externalities exists when
governments supply public inputs. Thus, similar to Bayindir-Upmann (1998), our results suggest thatpublic expenditure is a desirable strategic variable due to employment externalities.6
The remainder of this paper is organized as follows. Section 2 describes the basic setup of ouranalytical framework. Section 3 characterizes the two different equilibria: the tax competition
equilibrium and the expenditure competition equilibrium. Comparative analysis of two equilibria isalso developed in the section. Finally, Section 4 provides the conclusions of this study.

2. The Model
This section describes our theoretical framework and provides the preliminary results for later analyses.
The basic setup is based on Wildasin (1988) and Ogawa et al. (2006). The economy consists of 𝑁
regions (𝑁 ≥ 2). Each region has identical residents and firms. The population in each region isnormalized to unity and the residents are immobile. There is a single homogenous private good
produced using capital, labor, and land. The production technology is given by a constant-returns-toscale production function 𝑌! = 𝐹(𝐻! , 𝐾! , 𝐿! ), where 𝑌! is the output of the private good, 𝐻! is theland input, 𝐾! is the capital input, and 𝐿! is the labor input employed in region 𝑖 (𝑖 = 1, 2, ⋯ , 𝑁).
It is assumed that each factor reward equals the value of its marginal product. Treating the amount of
0! , 𝐾! , 𝐿! ) ≡ 𝑓(𝐾! , 𝐿! ). We also assume that 𝑓(𝐾! , 𝐿! ) is strictly concave,land input as fixed, 𝑌! = 𝐹(𝐻twice continuously differentiable, and increasing in 𝐾! and 𝐿! .
Capital moves across the region while labor and land are fixed in the locality. Summing up thefactor demand for capital in all regions yields"

0,
3 𝐾! = 𝐾(1)!#$

0 stands for economy-wide capital stock. The resident as worker and the landlord supplies
where 𝐾
one unit of labor and land, respectively. In this paper, some of the labor might be unemployed while
the land is fully employed. Hence, the factor demands for labor and land satisfy 𝐿! ≤ 1 and 𝐻! = 1.
6

Hoyt (1993) showed similar result in some cases using Tiebout model. Köethenbüerger (2011a) showed that the governmentschoose public expenditure under a specific fiscal transfer.

3

As already explained, the rewards for the production factors equal their marginal product. Since
each regional government imposes a per unit capital tax, the cost of using capital is the sum of thecapital price and the unit tax. Taking this into account, the relationship between factor demands andprices are given by𝜌 = 𝑓% (𝐾! , 𝐿! ) − 𝑡! ,(2)𝑤
0! = 𝑓& (𝐾! , 𝐿! ),(3)
where 𝜌 is the common factor price of capital, 𝑡! is the unit tax rate on capital in region 𝑖, 𝑤
0 ! is an
exogenously fixed wage rate in region 𝑖. To ensure the presence of unemployment, we assume that𝑤
0! is higher than the wage rate determined in a competitive labor market.7 Equations (2) and (3)determine the factor demands for capital and labor as functions of their factor prices and capital tax
rate, respectively.
Each regional government provides a public good and finances its supply cost by a per unit capitaltax. The marginal rate of transformation between private and public goods is unity. Local public
expenditure 𝐺! stands for the amount of public goods. Therefore, the budget equation for the regional
government equals
𝑡! 𝐾! = 𝐺! .
(4)
Note that 0 ≤ 𝑡 ≤ 𝑡̅ and 𝑡̅ ≡ 𝑓% must hold from equations (2) and (4).
Since capital moves across regions, the factor price of capital (net return to capital), 𝜌 , isdetermined from equation (1) as a function of the vector of tax rates 𝜏 = (𝑡$ , ⋯ , 𝑡' ). We now verify
that fact. Note that the following equation is derived from equations (2) and (3):𝑑𝐾!
𝑓&& (𝐾! , 𝐿! )
=
< 0,
|𝐽! |
𝑑(𝜌 + 𝑡! )where |𝐽! | ≡ 𝑓%% (𝐾! , 𝐿! )𝑓&& (𝐾! , 𝐿! ) − [𝑓%& (𝐾! , 𝐿! )]( > 0. Using this equation and equation (1), we
obtain8
𝜀! 𝐾!
𝜕𝜌𝜌 + 𝑡!
=−
< 0,
(5)
𝜀𝐾
𝜕𝑡!
∑) ) )
𝜌 + 𝑡)
where𝑑 log 𝐾!
𝜀! ≡< 0.
𝑑 log(𝜌 + 𝑡! )Equations (2) and (5) yield 𝐾! = 𝐾! (𝜌(𝜏) + 𝑡! ) = 𝐾! (𝜏). For a fixed wage level, equation (3)describes the relationship between capital and labor as𝑑𝐿!
𝑓%& (𝐾! , 𝐿! )𝜇! ≡=−⋛ 0 ⇔ 𝑓%& (𝐾! , 𝐿! ) ⋛ 0.
𝑑𝐾!
𝑓&& (𝐾! , 𝐿! )This equation and 𝐾! = 𝐾! (𝜏) provide 𝐿! = 𝐿! P𝐾! (𝜏)Q = 𝐿! (𝜏). Partial differentiation of 𝐾! (𝜏),𝐿! (𝜏), 𝐾) (𝜏), and 𝐿) (𝜏), with respect to 𝑡! leads to𝜀𝐾∑)*! ) )𝜌 + 𝑡)
𝜕𝐾!
𝜀! 𝐾!
=
< 0,
(6)
𝜀𝐾
𝜕𝑡! 𝜌 + 𝑡!
∑) ) )
𝜌 + 𝑡)𝜀) 𝐾)𝜌 + 𝑡)
𝜕𝐿! 𝜀! 𝜇! 𝐾!
=⋚ 0 ⇔ 𝜇! ⋛ 0,𝜀𝐾
𝜕𝑡! 𝜌 + 𝑡!
∑) ) )
𝜌 + 𝑡)∑)*!

7
8

We assume the wage determination by a simple model of efficiency wages (e.g., Yellen 1984).
See the Appendix for the derivation of equation (5).

4

(7)

𝜀! 𝐾! 𝜀) 𝐾)
𝜕𝐾)
𝜌 + 𝑡! 𝜌 + 𝑡)
=−
> 0,
𝜀𝐾
𝜕𝑡!
∑+ + +
𝜌 + 𝑡+(8)

𝜀! 𝜇! 𝐾! 𝜀) 𝐾)
𝜕𝐿)𝜌 + 𝑡! 𝜌 + 𝑡)
(9)=−
⋛ 0 ⇔ 𝜇) ⋛ 0.
𝜀+ 𝐾+
𝜕𝑡!
∑+
𝜌 + 𝑡+
Equations (6) and (8) are essentially the same results derived by Wildasin (1988). Equations (7) and(9) are generalized versions of Ogawa et al. (2006). If the regions are symmetrical in all respects,
equations (6) through (9) then become𝜕𝐾! (1 − 𝑛)𝜀𝐾 𝜕𝐾)𝑛𝜀𝐾 𝜕𝐿! (1 − 𝑛)𝜀𝜇𝐾 𝜕𝐿)
𝑛𝜀𝜇𝐾
=
,
=−
,
=
,
=−
.
𝜕𝑡!
𝑓% (𝐾, 𝐿) 𝜕𝑡!
𝑓% (𝐾, 𝐿) 𝜕𝑡!
𝑓% (𝐾, 𝐿) 𝜕𝑡!
𝑓% (𝐾, 𝐿)
Finally, we consider the ownership of capital and social welfare in each region. Total capital income
0 . Let 𝜃! be the share of the capital stock owned by the residents
in the economy is represented by 𝜌𝐾"
in region 𝑖. Then, we have ∑!#$ 𝜃! ≤ 1. If ∑"!#$ 𝜃! < 1, there are absentee owners of capital. On thecontrary, capital is owned by residents only when ∑"!#$ 𝜃! = 1. With the presence of unemployment,
there are two possible states that might characterize the residents: employed and unemployed. To keep
the model simple, the quasi-linear preference is assumed. Following Ogawa et al. (2006) and Kikuchi
and Tamai (2019), the Benthamite social welfare function is
𝑈! (𝑋! , 𝐺! ) = 𝑋! + 𝑣(𝐺! ),, (𝐺 ),, (𝐺 ), (0)where 𝑣 ! > 0, 𝑣
= ∞, 𝑣 , (∞) = 0, and! < 0, 𝑣
0.
𝑋! = 𝑓(𝐾! , 𝐿! ) − 𝐾! 𝑓% (𝐾! , 𝐿! ) + 𝜌𝜃! 𝐾(10)
The regional government chooses the tax rate or the expenditure level to maximize the social welfare
function, subject to some constraints. The next section provides a detailed analysis.

3. Tax Competition versus Expenditure Competition
This section characterizes two different Nash equilibria for states produced by fiscal competition. One
is a state in which the local government chooses the tax rate to maximize its objective function. The
other is a state in which the regional authority determines the expenditure level in order to maximize
its objective function. In the presence of unemployment, the ownership of capital and the number ofregions significantly influence the equilibrium outcome of each of these two equilibria.

3.1. Tax Competition Equilibrium
First, we define the Nash equilibrium wherein regional governments compete with each other withregard to their tax rates (e.g., Wilson 1986; Zodrow and Mieszkowski 1986). Following Wildasin(1988), the definition is described as follows:
Definition 1. T-equilibrium is a vector 𝜏 ∗ that 𝑡!∗ is the solution to
max 𝑈! (𝑋! , 𝐺! ).!

subject to (4), (10), 𝜌 = 𝜌(𝜏), 𝐾! = 𝐾! (𝜏), 𝐿! = 𝐿! (𝜏), and 𝑡) = 𝑡)∗ (𝑗 ≠ 𝑖).
Solving the optimization problem defined in Definition 1, we have
𝜕𝑈!
𝜕𝐿!
𝜕𝐿!
𝜕𝐾!
𝜕𝜌
𝜕𝐾!
0
=𝑤
0
− 𝐾! 𝑓%& (𝐾! , 𝐿! )− 𝐾! 𝑓%% (𝐾! , 𝐿! )+ 𝜃! 𝐾+ 𝑣′(𝐺! ) _𝐾! + 𝑡!
` = 0.
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
The first and second terms are originated in the presence of unemployment. In particular, the first term

5captures the employment externality shown by Ogawa et al. (2006).9 Here, we introduce a standard
assumption commonly found in the literature on tax competition:
Assumption 1.
1+

𝜕 log 𝐾!
> 0.
𝜕 log 𝑡!

Assumption 1 ensures that the economy is located on the left-hand side of the Laffer curve.
We focus on the symmetrical regions in all respects. Furthermore, two cases of capital ownershipare considered at this point: (a) when absentee owners of capital have full ownership of capital (𝜃! =0) and (b) when residents equally share the economy-wide capital stock (𝜃! = 𝑛).
(a) Absentee owners of capital have full ownership of capital (𝜃! = 0). Then, the first-ordercondition for the regional government’s optimization problem can be rewritten as
𝑤
0 𝜕𝐿!
1−𝑛−
𝐾
! 𝜕𝑡!
𝑣 , (𝐺! ) =,(11)
𝜕 log 𝐾!
1+
𝜕 log 𝑡!
/$where 𝑛 ≡ 𝑁 . Note that 0 < 𝑛 < 1 and 𝑛 → 0 as 𝑁 → ∞.
(b) Residents equally share the economy-wide capital stock (𝜃! = 𝑛). The first-order condition for
the regional government’s optimization problem then becomes
𝑤
0 𝜕𝐿!
1−
𝐾
! 𝜕𝑡!
𝑣 , (𝐺! ) =.
(12)
𝜕 log 𝐾!
1+
𝜕 log 𝑡!
Equations (11) and (12) have similar forms except for the term 𝑛 in the numerator. If the numberof regions is infinitely large (𝑁 → ∞), these two equations can be equalized. Absentee owners ofcapital are crucial to determining the marginal cost of public funds (MCPF).

3.2. Expenditure Competition Equilibrium
Another state of fiscal competition is expenditure competition. From equation (4) and 𝐾! = 𝐾! (𝜏), we
have 𝐺! = 𝐺! (𝜏) = 𝑡! 𝐾! (𝜏). Let 𝑔 = (𝐺$ , ⋯ , 𝐺' ) be the vector of public expenditure. If the systemcomposed of equations (1)-(4) has its inverse matrix, we obtain 𝜏 = 𝜏(𝑔). Then, we can analyze theexpenditure competition equilibrium formally defined as follows:
Definition 2. G-equilibrium is a vector 𝑔⋆ that 𝐺!⋆ is the solution to
max 𝑈! (𝑋! , 𝐺! ).!

subject to (4), (10), 𝜌 = 𝜌P𝜏(𝑔)Q, 𝐾! = 𝐾! P𝜏(𝑔)Q, 𝐿! = 𝐿! P𝜏(𝑔)Q, and 𝐺) = 𝐺)⋆ (𝑗 ≠ 𝑖).
The first-order condition for the regional government’s optimization problem is
𝜕𝑈!
𝜕𝐿!
𝜕𝐾!
𝜕𝐿!
𝜕𝜌 𝜕𝑡
0 ` !
= _𝑤
0
− 𝐾! 𝑓%% (𝐾! , 𝐿! )− 𝐾! 𝑓%& (𝐾! , 𝐿! )+ 𝜃! 𝐾
𝜕𝐺!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡! 𝜕𝐺!
𝜕𝐿!
𝜕𝐾!
𝜕𝐿!
𝜕𝜌 𝜕𝑡
0 d ) + 𝑣′(𝐺! ) = 0+ 3 c𝑤
0− 𝐾! 𝑓%% (𝐾! , 𝐿! )− 𝐾! 𝑓%& (𝐾! , 𝐿! )+ 𝜃! 𝐾𝜕𝑡)𝜕𝑡)𝜕𝑡)𝜕𝑡) 𝜕𝐺!
)*!

The first-bracket term represents the effects of a change in government expenditure in region 𝑖through a change in the capital tax rate in region 𝑖. The second-bracket term captures the strategic

9

In Ogawa et al. (2006), it is referred as an unemployment-exporting externality. A reduction in the capital tax rate in region 𝑖 attracts
capital to the region. If 𝑓!" > 0, the capital inflow to the region decreases employment in the other region while it simultaneously
increases employment in its own region.

6effects of a change in government expenditure in region 𝑖 through changes in the tax rates in otherregions.
Focusing on symmetrical regions, we have10
𝜕𝐾) (𝜏)𝜕𝐾) (𝜏)𝜕𝑡!
1 − 2𝑛= 𝐷/$ c𝐾) (𝜏) + 𝑡)
+f
g 𝑡)
d
𝜕𝐺!
𝜕𝑡)
𝑛
𝜕𝑡!

where

𝜕𝑡)
𝜕𝐾) (𝜏)= −𝐷/$ 𝑡)
,
𝜕𝐺!
𝜕𝑡!

𝜕𝐾) (𝜏)𝜕𝐾) (𝜏)𝜕𝐾) (𝜏) 𝜕𝐾! (𝜏)𝜕𝐾! (𝜏)1 − 2𝑛1−𝑛d c𝐾) (𝜏) + 𝑡)
+f
g 𝑡)
d−f
g 𝑡! 𝑡)
𝜕𝑡!
𝜕𝑡)
𝑛
𝜕𝑡!
𝑛
𝜕𝑡!
𝜕𝑡)
> 0.
(a) Absentee owners of capital have full ownership of capital (𝜃! = 0). The first-order condition canbe reduced to𝜕𝐿! 𝜕𝑡!
𝜕𝐿! 𝜕𝑡)𝑣 , (𝐺! ) = _(1 − 𝑛)𝐾! − 𝑤
0`− 3 c𝑛𝐾! + 𝑤
0
d
.
(13)
𝜕𝑡! 𝜕𝐺!
𝜕𝑡) 𝜕𝐺!
𝐷 ≡ c𝐾! (𝜏) + 𝑡!

)*!

(b) Residents equally share the economy-wide capital stock (𝜃! = 𝑛). The first-order condition thenbecomes𝜕𝐿! 𝜕𝑡!
𝜕𝐿! 𝜕𝑡)𝑣 , (𝐺! ) = _𝐾! − 𝑤
0
`
−𝑤
03
.
(14)
𝜕𝑡! 𝜕𝐺!
𝜕𝑡) 𝜕𝐺!
)*!

In equations (13) and (14), the effects of a change in region i’s public expenditure on tax rates in its
own and other regions are derived (see the Appendix) as𝜕𝑡!
𝑛𝜀𝑡= 𝐷/$ 𝐾 _1 +` > 0,(15)
𝜕𝐺!
𝑓% (𝐾, 𝐿)𝜕𝑡)𝑛𝜀𝑡= 𝐷/$ 𝐾
< 0,
(16)
(𝐾,
𝜕𝐺!
𝑓%
𝐿)
where
𝜀𝑡
𝐷 = _1 +` 𝐾 ( > 0.
𝑓% (𝐾, 𝐿)
Equation (16) captures the strategic effects of a change in government expenditure.

3.3. Comparative Analysis of the Tax and Expenditure Equilibria
We now characterize two different equilibria of fiscal competition through a comparison between the
MCPF (i.e., (11) and (13); (12) and (14)). To conduct the comparative analysis, equations (11) through(14) should be transformed to these comparable forms.
(a) Absentee owners of capital have full ownership of capital (𝜃! = 0). Applying the symmetricalconditions into equations (11) and (13) gives𝜀𝜇_1 − 𝑤` (1 − 𝑛)0𝑓% (𝐾, 𝐿)
T-equilibrium: 𝑣′(𝐺! ) =≡ 𝑇-MCPF,(17)(1 − 𝑛)𝜀𝑡1+
𝑓% (𝐾, 𝐿)𝜀𝜇_1 − 𝑤
0 (𝐾, ` (1 − 𝑛)
𝑓%
𝐿)
(18)
G-equilibrium: 𝑣′(𝐺! ) =≡ 𝐺-MCPF.
𝜀𝑡
1+
𝑓% (𝐾, 𝐿)10

See the Appendix for the derivation of these two equations.

7Under symmetricity in all aspects, 𝜀, 𝜇, 𝑤, 𝐾, 𝐿, and 𝑓% are all positive constants.11 Hence, we candirectly compare equations (17) and (18), evaluated at the given level of the same tax rate:𝑛𝜀𝑡
𝜀𝜇_1 − 𝑤` (1 − 𝑛)0𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
𝑇-MCPF − 𝐺-MCPF =< 0.
(1 − 𝑛)𝜀𝑡𝜀𝑡_1 +` _1 +`
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
This inequality indicates that the marginal cost of supplying a public good in G-equilibrium is larger
than the marginal cost in T-equilibrium. Figure 1 illustrates the relationship between the marginal
benefits of the public good, the marginal cost of public funds at T-equilibrium (T-MCPF), and themarginal cost of public funds at G-equilibrium (G-MCPF). The intersection points A and B,
respectively, correspond to the T-equilibrium and the G-equilibrium. Figure 1 illustrates that the public
good supply (capital tax rate) in G-equilibrium is smaller than the supply (capital tax rate) in Tequilibrium.
Using Nash conjecture, in the state of tax competition equilibrium, the government of region iperceives that a rise in its tax rate increases the supply of public goods in other regions due to theresultant capital flight from its own region to other regions. In contrast, the government presumes thatgovernments in other regions aggressively reduce their tax rates in response to region i’s increase of
its tax rate. Even though the resultant capital flight induces increasing employment or unemployment,
the employment effect has a common impact on the MCPFs; See the numerator of equations (17) and(18). Hence, the key determinant of the relative size of the MCPFs is a strategic effect in thatgovernments in other regions decrease their tax rates in order to maintain a balanced budget.
Formally, we have the following proposition (see the Appendix for the proof of Proposition 1):
Proposition 1. Suppose that absentee owners possess the capital in the economy. There exists a uniquesymmetrical T-equilibrium and G-equilibrium, respectively. Then, the equilibrium tax rate and theexpenditure level satisfy 𝑡 ∗ > 𝑡 ⋆ and 𝐺 ∗ > 𝐺 ⋆ .
Proposition 1 is the extended version of Wildasin (1988), who demonstrated the same result.12 On
the other hand, in our model, the T-equilibrium and the G-equilibrium are not necessary to cause theunder-provision of public goods because of an employment externality. We should investigate whether
the tax rate and the expenditure level are less than optimal. With a lump-sum tax or common tax rates
on capital, the first-order condition for the optimal provision of a public good is 𝑣′(𝐺! ) = 1. Thiscondition yields (𝐺 1 , 𝑡 1 ).13 The optimal conditions and Proposition 1 reveal the following results(see the Appendix for the proof of Proposition 2):
Proposition 2. Suppose that absentee owners possess the capital in the economy. For 𝑡 ∈ [0, 𝑡)̅ , therelationships between the capital tax rate and welfare are(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀(i) 𝑛 +
≤ 0 ⇒ 𝑡 1 ≥ 𝑡 ∗ > 𝑡 ⋆ , 𝑈1 ≥ 𝑈∗ > 𝑈⋆ ,𝑓% (𝐾, 𝐿)
(ii)

0<𝑛+

(iii)

0≤−

(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑛𝜀𝑡<−⇒ 𝑡 ∗ > 𝑡 1 > 𝑡 ⋆ , 𝑈 1 > max(𝑈 ∗ , 𝑈 ⋆ ),𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)(𝑤𝑛𝜀𝑡
0𝜇 + 𝑡)(1 − 𝑛)𝜀<𝑛+⇒ 𝑡 ∗ > 𝑡 ⋆ ≥ 𝑡 1 , 𝑈1 ≥ 𝑈⋆ > 𝑈∗ .
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)( ≡ 𝑘+ holds. Equation (3) leads to 𝑤
In the symmetric economy, 𝐾 = 𝑛𝐾
( = 𝑓" -𝑘+ , 𝐿0 ⇒ 𝐿+ = 𝐿-𝑘+ , 𝑤(0. Capital and labor are bothconstant in equilibrium. Thus, all derivatives of the production function are also constant, as is 𝜇, and𝑑 log 𝐾
𝑓! (𝐾, 𝐿)𝑓"" (𝐾, 𝐿)
𝜀=
=
.
# (𝐾,
𝑑 log(𝜌 + 𝑡) [𝑓!! (𝐾, 𝐿)𝑓"" (𝐾, 𝐿) − 𝑓!"𝐿)]𝐾
12
When 𝜇 = 0, the result reverts to the one derived by Wildasin (1988).
13( , 𝑡 $ = 𝐺 $ /(𝑛𝐾( ).
We have 𝐺 $ = 𝑣 %&' (1). Using equation (4) and 𝐾( = 𝑛𝐾
118

Result (i) of Proposition 2 is parallel to the results obtained by of Wildasin (1988). On the other
hand, results (ii) and (iii) differ from the first result. The interpretation of Proposition 3 is as follows.
From equations (17) and (18), we have:(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑇-MCPF ⋚ 1 ⇔ 𝑛 +⋛ 0,
𝑓% (𝐾, 𝐿)(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑛𝜀𝑡
𝐺-MCPF ⋚ 1 ⇔ 𝑛 ++⋛ 0.
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
The presence of the strategic effect characterizes the difference between T-MCPF and G-MCPF. In
addition to this result, the employment externality influences the MCPFs, depending on the value of𝜇.
If 𝜇 > 0 , there is a positive employment externality. Combining the positive employment
externality with the fiscal externality, the MCPFs tend to be larger than unity. Therefore, public goodsin the T- and G-equilibria will be under-provided. Furthermore, the supply of public goods in the Gequilibrium is smaller than that in the T-equilibrium due to the existence of the strategic effect. When𝜇 < 0, it means that there is a negative employment externality. The relative size of the strategic effect,
the fiscal externality, and the employment externality are all important to maintaining efficiency in theprovision of public goods. If the negative employment externality impacts the economy enough todominate the fiscal externality and the strategic effect, the MCPFs are possibly larger than unity;
resulting in the over-provision of public goods. However, G-MCPF is smaller than T-MCPF due tothe presence of the strategic effect. The public good may be optimally supplied at a certain value of
𝜇. Therefore, the results of Proposition 2 imply that expenditure competition may be preferable to tax
competition.
(b) Residents equally share the economy-wide capital stock ( 𝜃! = 𝑛 ). Using the symmetricalconditions, equations (12) and (14) can be reduced to(1 − 𝑛)𝜀𝜇1−𝑤
0𝑓% (𝐾, 𝐿)
T-equilibrium: 𝑣′(𝐺! ) =,(19)(1 − 𝑛)𝜀𝑡1+
𝑓% (𝐾, 𝐿)(1 − 𝑛)𝜀𝜇𝑛𝜀𝑡+
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)(20)
G-equilibrium: 𝑣′(𝐺! ) =.
𝜀𝑡
1+
𝑓% (𝐾, 𝐿)
The assumption of symmetrical regions leads to a set of constant values for 𝜀, 𝜇, 𝑤, 𝐾, 𝐿, and 𝑓% .
Similar to how we demonstrated Proposition 1, we now have
(1 − 𝑛)𝑛𝜀 ( 𝑡(𝑤
0 𝜇 + 𝑡)[𝑓% (𝐾, 𝐿)](𝑇-MCPF − 𝐺-MCPF = −⋚0⇔𝑤0𝜇 + 𝑡 ⋛ 0.
(1 − 𝑛)𝜀𝑡𝜀𝑡_1 +` _1 +`
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
In contrast to case (a), the strategic effect has no impact on the order of the MCPFs. The strategic
effect appears as the third term of the numerator in equation (20). The strategic effect is offset by theeffect of fiscal externality, which corresponds to the second term of the denominator in equation (19).
Thus, only the sign of (𝑤0𝜇 + 𝑡) is important for determining the inequality between the two MCPFs.
In addition to Assumption 1, we impose the following assumption:
1−𝑤
0Assumption 2.

𝜀̅ ((1 − 𝑛)(𝑤
0𝜇 + 𝑡) f ̅ g
𝑓%𝑣 ,, 𝑘r <.
(𝜀̅𝑡
f1 + ̅ g𝑓%

9This condition displays the infimum of 𝜇 , which represents the employment externality. Under
Assumptions 1 and 2, the analysis developed above leads to the following proposition (see the
Appendix for the proof of Proposition 3):
Proposition 3. Suppose that the ownership of economy-wide capital is equally distributed amongresidents. There exists a unique symmetrical T-equilibrium and a unique symmetrical G-equilibrium,
respectively. These equilibria are characterized as 𝑡 ∗ ≷ 𝑡 ⋆ ⇔ 𝐺 ∗ ≷ 𝐺 ⋆ ⇔ 𝑤
0𝜇 + 𝑡 ≷ 0 for 𝑡 ∈[0, 𝑡̅).
Comparing the results of Propositions 1 and 3, the ownership of capital is crucial to determine theequilibrium tax rate and the supply of the public good. In particular, it affects not only the first-ordercondition of each equilibrium state, but also the strategic effect of the G-equilibrium. When theresidents equally own the capital, the strategic effect is negligible when comparing the tax rates at bothT-equilibrium and G-equilibrium.
According to Proposition 3, a positive employment externality, 𝜇 > 0, leads to 𝑤
0𝜇 + 𝑡 > 0.
Therefore, a positive external effect is generated by fiscal and employment externalities, which areboth positive. A similar result shown by Wildasin (1988) holds in such a case. A negative employment
externality, 𝜇 < 0, yields a non-trivial outcome. Especially, if 𝑤
0𝜇 + 𝑡 < 0, we have the opposite∗⋆
result in comparison to the case where 𝑤
0𝜇 + 𝑡 > 0; 𝑡 < 𝑡 and 𝐺 ∗ < 𝐺 ⋆ . The critical level of the
tax rate, 𝑡 = −𝑤
0𝜇, corresponds to the optimal tax rate derived by Ogawa et al. (2006) when the lumpsum tax is available.14 Whether the equilibrium tax rate is larger or smaller than the critical level isan essential factor in determining whether public goods are over-provided or not. Indeed, we obtainthe following results (see the Appendix for the proof of Proposition 4):
Proposition 4. Suppose that the ownership of economy-wide capital is equally distributed amongresidents. For 𝑡 ∈ [0, 𝑡)̅ , the relationships between the capital tax rate and welfare are(i) 𝑤
0𝜇 + 𝑡 > 0 ⇒ 𝑡 1 > 𝑡 ∗ > 𝑡 ⋆ , 𝑈 1 > 𝑈 ∗ > 𝑈 ⋆ ,(ii) 𝑤
0𝜇 + 𝑡 < 0 ⇒ 𝑡 1 < 𝑡 ∗ < 𝑡 ⋆ , 𝑈 1 > 𝑈 ∗ > 𝑈 ⋆ .
The equilibrium tax rates of case (i) are illustrated in Figure 1 and on the right side of 𝑡 = −𝑤
0𝜇 in
Figure 2. In contrast, the equilibrium tax levels of case (ii) are placed on the left side of 𝑡 = −𝑤
0𝜇 in
Figure 2. Equations (19) and (20), respectively, lead to(𝑤
0 𝜇 + 𝑡)(1 − 𝑛)𝜀𝑓% (𝐾, 𝐿)
𝑇-MCPF − 1 = −⋛0⇔𝑤
0𝜇 + 𝑡 ⋛ 0,
(1 − 𝑛)𝜀𝑡1+
𝑓% (𝐾, 𝐿)(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑓% (𝐾, 𝐿)
𝐺-MCPF − 1 = −⋛0⇔𝑤
0𝜇 + 𝑡 ⋛ 0.
𝜀𝑡
1+
𝑓% (𝐾, 𝐿)
These equations show that the relationship between degree of the employment externality and theequilibrium tax rate is important for maintaining efficiency in providing public goods.
If the employment externality is positive, or not too negative, the MCPFs tend to be larger thanunity. Hence, an under-provision of public goods occurs, and the tax rate in G-equilibrium corresponds
to the lowest one. When the employment externality is negative and has a sufficiently powerful impact,
the MCPFs will be less than unity. This means that an over-provision of public goods occurs. Then,the highest tax rate is that at G-equilibrium. In contrast to case (a), T-equilibrium may be welfaresuperior to G-equilibrium (recall Proposition 2).

14

In our model, we need 𝑡 > 0 because the capital tax is the only financial source for supplying the public good. Hence, 𝜇 < 0 isa required condition. The results shown by Kikuchi and Tamai (2019) imply that the optimal provision of public goods is attainable
regardless of the presence of the intergovernmental transfer if 𝑡 = −𝑤
(𝜇 holds in equilibrium.

104. Conclusion
This paper examined two different fiscal competition games operating under labor market
imperfections. We demonstrated that a unique Nash equilibrium exists for both tax competition and
expenditure competition. Some conditions are required to ensure the existence and uniqueness of the
Nash equilibria, even though they are reasonable, they are not strong. Comparing these two equilibria
revealed that the presence/absence of absentee ownership of capital and employment externalities are
significant factors to be considered in order to characterize the equilibria properties.
Focusing on the most interesting implications, in some cases, this paper showed that tax rates under
expenditure competition are less competitive (i.e., higher) than those under tax competition, which iscontrary to the dominant view in the existing literature. This is only possible when residents equallyshare the capital. If absentee owners own the capital, such a situation does not occur. Furthermore,
welfare analyses demonstrated that governments choose government expenditure as their strategic
variable if the negative employment externality is sufficiently strong and absentee owners possess thecapital. In contrast, the tax rate is likely chosen as the strategic variable if the capital is evenly
distributed among the residents.
Finally, the future direction of this research should be mentioned. In the basic model, a fixed wage
is assumed for analytical simplicity and tractability. This assumption does not vitiate our main findings
under the presence of employment externalities. However, different settings for labor markets mightalter these results. Recent insights about labor economics should be incorporated in order to examine
the robustness of our results. Numerical analyses under specific utility and production functions would
also be helpful to conduct such research and provide a clear relationship between the equilibria anddemonstrate the best response curve. The present study will be the analytical basis for such extensionsof this research.

11

Appendix
A. Derivations of Equations (5) through (9)The total differentiation of equation (2) leads to𝑑(𝜌 + 𝑡! ) = 𝑓%% (𝐾! , 𝐿! )𝑑𝐾! + 𝑓%& (𝐾! , 𝐿! )𝑑𝐿!
|𝐽! |𝑓%% (𝐾! , 𝐿! )𝑓&& (𝐾! , 𝐿! ) − [𝑓%& (𝐾! , 𝐿! )](
=
𝑑𝐾! =
𝑑𝐾 ,
(𝐾
)
(𝐾
𝑓&& ! , 𝐿!
𝑓&& ! , 𝐿! ) !
(where |𝐽! | ≡ 𝑓%% 𝑓&& − 𝑓%&> 0. Hence, we have𝑑𝐾!
𝑓&& (𝐾! , 𝐿! )
=
< 0.
|𝐽! |
𝑑(𝜌 + 𝑡! )The total differentiation of equation (1) yields

(A1)

"

3 𝑑𝐾! = 0.
!#$

From this equation and (A1), we obtain equation (5):

𝑑𝐾!
𝜀! 𝐾!
𝑑𝐾)
𝑑𝐾!
𝜕𝜌
𝜕𝜌
𝜕𝜌
𝑑(𝜌 + 𝑡! )
𝜌 + 𝑡!
f + 1g +
=0⇔
=−
=−
< 0.
3
𝜀) 𝐾)
𝑑𝐾!
𝑑(𝜌 + 𝑡! ) 𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝑑P𝜌 + 𝑡) Q
∑
∑
)*!
!
)𝜌+𝑡
𝑑(𝜌 + 𝑡! ))Using equations (2) and (A1) lead to
|𝐽! |
𝜕𝜌
𝜕𝐾!
+1=
.
𝜕𝑡!
𝑓&& (𝐾! , 𝐿! ) 𝜕𝑡!
Inserting equation (5) into the above equation generates equation (6):
𝑑𝐾)
𝜀𝐾
𝑑𝐾!
∑)*!
∑)*! ) )
𝑑P𝜌
+
𝑡
Q
𝜌
+ 𝑡)
1
𝜕𝐾
𝜕𝐾
𝑑𝐾
𝜀
𝐾
𝑑(𝜌 + 𝑡! )
)
!
!
!
! !
−
+1=
⇔
=
=
< 0.
𝜀) 𝐾)
𝑑𝐾!
𝑑𝐾!
𝑑𝐾)
𝜕𝑡!
𝜕𝑡! 𝑑(𝜌 + 𝑡! )
𝜌 + 𝑡!
∑!
∑)∑)𝑑(𝜌 + 𝑡! )𝑑(𝜌 + 𝑡! )
𝜌 + 𝑡)
𝑑P𝜌 + 𝑡) Q
Equations (3) and (6) yield equation (7):𝜀) 𝐾)∑
𝜕𝐿! 𝑑𝐿! 𝜕𝐾! 𝜀! 𝜇! 𝐾! )*! 𝜌 + 𝑡)==.
𝜀𝐾
𝜕𝑡! 𝑑𝐾! 𝜕𝑡!
𝜌 + 𝑡!
∑) ) )
𝜌 + 𝑡)
Similarly, equations (2), (3), and (A1) make equations (8) and (9), respectively:
𝑑𝐾)
𝑑𝐾!
𝜀! 𝐾! 𝜀) 𝐾)
)
𝑑(𝜌
+
𝑡
𝜕𝐾)
𝑑𝐾)
𝑑P𝜌 + 𝑡) Q
𝜌 + 𝑡! 𝜌 + 𝑡)
𝜕𝜌
!
=
=−
=−
> 0,
𝜀𝐾
𝑑𝐾!
𝜕𝑡! 𝑑P𝜌 + 𝑡) Q 𝜕𝑡!
∑+ + +∑!
𝜌 + 𝑡+𝑑(𝜌 + 𝑡! )𝜀! 𝜇! 𝐾! 𝜀) 𝐾)
𝜕𝐿) 𝑑𝐿) 𝜕𝐾)
𝜌 + 𝑡! 𝜌 + 𝑡)
=
=−
.
𝜀𝐾
𝜕𝑡! 𝑑𝐾) 𝜕𝑡!
∑+ + +
𝜌 + 𝑡+

B. Derivations of equations (11) and (12)The first-order partial derivative of the social welfare function in region 𝑖 with respect to 𝑡! is:
𝜕𝑈!
𝜕𝐿!
𝜕𝐾!
𝜕𝐿!
𝜕𝜌
𝜕𝐾!
0
=𝑤
0
− 𝐾! 𝑓%% (𝐾! , 𝐿! )− 𝐾! 𝑓%& (𝐾! , 𝐿! )+ 𝜃! 𝐾+ 𝑣′(𝐺! ) _𝐾! + 𝑡!
`
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!

12

𝜀) 𝐾)
𝜌 + 𝑡)
𝜕𝜌
𝜕𝐾!
0+ 𝜃! 𝐾+ 𝑣′(𝐺! ) _𝐾! + 𝑡!
`.
𝜀) 𝐾)
𝜕𝑡!
𝜕𝑡!
∑)
𝜌 + 𝑡)
Inserting 𝜕𝑈! /𝜕𝑡! = 0 into above equation and after some calculations, we have𝜀𝐾
𝜀𝐾∑)*! ) )∑)*! ) )0 𝜕𝜌
𝜌 + 𝑡)𝜌+ 𝑡) 𝑤
𝜕𝐿!
0 𝜕𝐿!
𝐾
0 𝜕𝜌
𝐾!
−
𝑤
0
−
𝜃
𝐾
−
−
𝜃
!
!
𝜀𝐾
𝜀𝐾
𝜕𝑡!
𝜕𝑡!
𝐾! 𝜕𝑡!
𝐾! 𝜕𝑡!
∑) ) )∑) ) )(A2)
𝜌 + 𝑡)
𝜌 + 𝑡)
𝑣′(𝐺! ) ==.
𝜕𝐾
𝜕 log 𝐾!
𝐾! + 𝑡! !
1+𝜕𝑡!
𝜕 log 𝑡!
00(a) 𝜃 = 0. Using equation (A2), 𝜃 = 0, 𝐾! = 𝐾/𝑁 = 𝑛𝐾 = 𝐾, 𝑡! = 𝑡) , and 𝜀! = 𝜀) ,
𝑁−1 𝑤
0 𝜕𝐿!
𝑤
0 𝜕𝐿!
−
1−𝑛−
𝑁
𝐾! 𝜕𝑡!
𝐾! 𝜕𝑡!
𝑣′(𝐺! ) ==.
𝜕 log 𝐾!
𝜕 log 𝐾!
1+
1+
𝜕 log 𝑡!
𝜕 log 𝑡!
0/𝑁 = 𝑛𝐾
0 = 𝐾, 𝑡! = 𝑡) , and 𝜀! = 𝜀) ,(b) 𝜃 = 𝑛. Utilizing equation (5), (A2), 𝜃 = 𝑛, 𝐾! = 𝐾
𝑤
0 𝜕𝐿!
𝑤
0 𝜕𝐿
1 − 𝑛 − 𝐾 𝜕𝑡 − (−𝑛)1 − 𝐾 𝜕𝑡 !
!
!
!
!
𝑣′(𝐺! ) ==.
𝜕 log 𝐾!
𝜕 log 𝐾!
1+
1+
𝜕 log 𝑡!
𝜕 log 𝑡!
𝜕𝐿!
=𝑤
0
− 𝐾!
𝜕𝑡!

∑)*!

C. Derivations of equations (13) through (16)The partial derivative of the social welfare function in region i with respect to 𝐺! is
𝜕𝑈!
𝜕𝐿!
𝜕𝐾!
𝜕𝐿!
𝜕𝜌 𝜕𝑡
0 ` !
= _𝑤
0
− 𝐾! 𝑓%% (𝐾! , 𝐿! )− 𝐾! 𝑓%& (𝐾! , 𝐿! )+ 𝜃! 𝐾
𝜕𝐺!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡! 𝜕𝐺!
𝜕𝐿!
𝜕𝐾!
𝜕𝐿!
𝜕𝜌 𝜕𝑡
0 d ) + 𝑣′(𝐺! ).
+ 3 c𝑤
0− 𝐾! 𝑓%% (𝐾! , 𝐿! )− 𝐾! 𝑓%& (𝐾! , 𝐿! )+ 𝜃! 𝐾𝜕𝑡)𝜕𝑡)𝜕𝑡)𝜕𝑡) 𝜕𝐺!
)*!

From this equation, the first-order condition becomes𝑣′(𝐺! )𝜀) 𝐾)𝜀) 𝐾)⎡ ∑)*! 𝜌 + 𝑡⎤⎡
𝜌 + 𝑡)
𝜕𝐿!
𝜕𝜌 𝜕𝑡!
𝜕𝐿!
𝜕𝜌 ⎤ 𝜕𝑡)
)
0
0
⎢
⎥
⎢
= 𝐾!
−𝑤
0
− 𝜃! 𝐾−3 𝑤
0+ 𝐾!
+ 𝜃! 𝐾 ⎥
.
(A3)
𝜀+ 𝐾+
𝜕𝑡!
𝜕𝑡! ⎥ 𝜕𝐺!
𝜕𝑡)𝜕𝑡) ⎥ 𝜕𝐺!
⎢ ∑ 𝜀) 𝐾)
⎢
∑
)*!
+𝜌+𝑡
)𝜌+𝑡
+
⎣
⎦
⎣
⎦
)
0
0
(a) 𝜃 = 0 . Inserting 𝜃 = 0 , 𝐾! = 𝐾/𝑁 = 𝑛𝐾 = 𝐾 , 𝑡! = 𝑡) , and 𝜀! = 𝜀) into (A3), we obtainequation (13).
0/𝑁 = 𝑛𝐾
0 = 𝐾, 𝑡! = 𝑡) , and 𝜀! = 𝜀) ,(b) 𝜃 = 𝑛. Using equation (5), (A3), 𝜃 = 𝑛, 𝐾! = 𝐾𝜕𝐿!
𝜕𝜌 𝜕𝑡!
𝜕𝐿!
𝜕𝜌 𝜕𝑡
0 `
0 d )
𝑣′(𝐺! ) = _(1 − 𝑛)𝐾! − 𝑤
0− 𝜃! 𝐾− 3 c𝑤
0+ 𝑛𝐾! + 𝜃! 𝐾
𝜕𝑡!
𝜕𝑡! 𝜕𝐺!
𝜕𝑡)𝜕𝑡) 𝜕𝐺!
𝜕𝐿! 𝜕𝑡!
𝜕𝐿! 𝜕𝑡)= _𝐾! − 𝑤
0
`
− 3𝑤
0
.
𝜕𝑡! 𝜕𝐺!
𝜕𝑡) 𝜕𝐺!

)*!

)*!

From equation (4), we have𝜕𝑡)𝜕𝐾) (𝜏) 𝜕𝑡++ 𝑡! 3,𝜕𝐺!
𝜕𝑡+ 𝜕𝐺!

(A4)𝜕𝑡)𝜕𝐾) (𝜏) 𝜕𝑡+𝜕𝐾) (𝜏) 𝜕𝑡!
+ 𝑡) 3+ 𝑡).
𝜕𝐺!
𝜕𝑡+ 𝜕𝐺!
𝜕𝑡! 𝜕𝐺!

(A5)

1 = 𝐾) (𝜏)+0 = 𝐾) (𝜏)

+*!

13

Note that the subscripts have been dropped because of symmetry.
Equations (A4) and (A5) lead to𝜕𝐾! (𝜏)𝜕𝐾! (𝜏)𝜕𝑡!
(𝑁 − 1)𝑡!
𝐾! (𝜏) + 𝑡!
𝜕𝑡!
𝜕𝑡)
⎛
⎞ ⎛𝜕𝐺! ⎞1= • ‚.
(A6)⎜
𝜕𝐾) (𝜏)𝜕𝐾) (𝜏)𝜕𝐾) (𝜏)⎟ ⎜ 𝜕𝑡) ⎟
0
(𝜏)
(𝑁
𝑡)
𝐾)
+ 𝑡)
+
− 2)𝑡)𝜕𝑡!
𝜕𝑡)
𝜕𝑡! ⎠ ⎝𝜕𝐺! ⎠⎝In (A6), the determinant of the coefficient matrix is𝜕𝐾) (𝜏)𝜕𝐾) (𝜏)𝜕𝐾) (𝜏) 𝜕𝐾! (𝜏)𝜕𝐾! (𝜏)𝐷 ≡ c𝐾! (𝜏) + 𝑡!
d c𝐾) (𝜏) + 𝑡)
+ (𝑁 − 2)𝑡)d − (𝑁 − 1)𝑡! 𝑡)
.
𝜕𝑡!
𝜕𝑡)
𝜕𝑡!
𝜕𝑡!
𝜕𝑡)
If 𝐷 ≠ 0, by Cramer’s rule, we obtain
𝜕𝐾) (𝜏)𝜕𝐾) (𝜏)𝜕𝑡!
= 𝐷/$ c𝐾) (𝜏) + 𝑡)
+ (𝑁 − 2)𝑡)
d
𝜕𝐺!
𝜕𝑡)
𝜕𝑡!
𝑛𝜀𝑡
= 𝐷/$ 𝐾 _1 +`,𝑓% (𝐾, 𝐿)and𝜕𝑡)𝜕𝐾) (𝜏)𝑛𝜀𝑡= −𝐷/$ 𝑡)= 𝐷/$ 𝐾.
𝜕𝐺!
𝜕𝑡!
𝑓% (𝐾, 𝐿)
D. Proof of Proposition 1
Using (𝜀, 𝜇, 𝑤, 𝐾, 𝐿, 𝑓% ) = (𝜀̅, 𝜇̅ , 𝑤
0, 𝑘r, 𝐿r, 𝑓%̅ ), equation (17) becomes𝜀̅𝜇̅_1 − 𝑤
0 ̅ ` (1 − 𝑛)𝑓%𝑃(𝑡) ≡ 𝑣′P𝑡𝑘rQ −= 0.
(1 − 𝑛)𝜀̅𝑡
1+𝑓%̅
By the assumption 2, the function P is monotonically decreasing in 𝑡 (𝑃, (𝑡) < 0) for−𝑓%̅
𝑡 ∈ „0,….
(1 − 𝑛)𝜀̅Furthermore, we have lim 𝑃(𝑡) = ∞ and.↓3𝜀̅𝜇̅_1 − 𝑤
0 ̅ ` (1 − 𝑛)−𝑓%̅−𝑓%̅ 𝑘r𝑓%𝑃„= −∞.
… = 𝑣′ „… − lim̅(1 − 𝑛)𝜀̅𝑡(1 − 𝑛)𝜀̅(1 − 𝑛)𝜀̅/5".↑1+($/')9:𝑓%̅
∗Therefore, there exists a unique value of 𝑡 satisfying equation (17). Similarly, we have 𝑄, (𝑡) < 0,lim 𝑄(𝑡) = ∞, and
.→<3

where

lim 𝑄(𝑡) = −∞,/5̅
.↑ 9:"𝜀̅𝜇̅_1 − 𝑤
0 ̅ ` (1 − 𝑛)−𝑓%̅𝑓%𝑄(𝑡) ≡ 𝑣′P𝑡𝑘rQ −
=
0
for
𝑡
∈
„0,
….
𝜀̅𝑡
𝜀̅
1+ ̅
𝑓%
Thus, the unique value of 𝑡 ⋆ exists to solve 𝑄(𝑡) = 0.
We now move to show 𝑡 ∗ > 𝑡 ⋆ and 𝐺 ∗ > 𝐺 ⋆ . Taking the difference between equations (17) and(18) yields

14

𝜀𝜇
𝜀𝜇
` (1 − 𝑛) _1 − 𝑤
0 (𝐾, ` (1 − 𝑛)
𝑓% (𝐾, 𝐿)𝑓%
𝐿)
𝑇-MCPF − 𝐺-MCPF =−𝜀𝑡(1 − 𝑛)𝜀𝑡1+
1+
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)𝜀𝑡𝜀𝜇_1 − 𝑤
0 (𝐾, ` (1 − 𝑛)
𝑓% (𝐾, 𝐿)
𝑓%
𝐿)
=
< 0.
(1 − 𝑛)𝜀𝑡𝜀𝑡_1 +` _1 +`
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)_1 − 𝑤
0(A7)

Equations (17) and (18) with (A7) lead to 𝑣 , (𝐺 ∗ ) < 𝑣 , (𝐺 ⋆ ). By the concavity of 𝑣, 𝐺 ∗ > 𝐺 ⋆ holds.
Equation (4) with 𝐺 ∗ > 𝐺 ⋆ yields 𝐺 ∗ = 𝑡 ∗ 𝐾 > 𝑡 ⋆ 𝐾 = 𝐺 ⋆ . Hence, we arrive at 𝑡 ∗ > 𝑡 ⋆ .
E. Proof of Proposition 2
Equations (17) and (18) provide(𝑤
𝜀𝜇
0𝜇 + 𝑡)(1 − 𝑛)𝜀_1 − 𝑤𝑛+0 (𝐾, ` (1 − 𝑛)𝑓%𝐿)𝑓% (𝐾, 𝐿)
𝑇-MCPF − 1 =−1=−,(1 − 𝑛)𝜀𝑡(1 − 𝑛)𝜀𝑡1+
1+
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)(𝑤
𝜀𝜇0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑛𝜀𝑡` (1 − 𝑛)
𝑛+
+
(𝐾,
𝑓%
𝐿)
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
−1=−
.
𝜀𝑡
𝜀𝑡
1+
1+
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)(A8)_1 − 𝑤
0
𝐺-MCPF − 1 =(A9)From equations (A8) and (A9), there are a few possible cases. First, we consider when MCPFs areless than unity. Then, we have(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑛+> 0 ⇒ 𝑇-MCPF < 1,
𝑓% (𝐾, 𝐿)(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑛𝜀𝑡𝑛++≥ 0 ⇒ 𝐺-MCPF ≤ 1.
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
This is true if(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑛𝜀𝑡≥−.
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
Note that the right side of the inequality mentioned above is positive. Under this condition, using
Proposition 1, we obtain 𝑡 ∗ > 𝑡 ⋆ ≥ 𝑡 1 . By the concavity of 𝑣, 𝑈 ∗ < 𝑈 ⋆ ≤ 𝑈 1 holds. These results
show result (iii) in Proposition 2.
Next, we have(𝑤
(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀0𝜇 + 𝑡)(1 − 𝑛)𝜀
𝑛𝜀𝑡
𝑛+
≤0⇒𝑛+
+
< 0.
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
Under the above condition, equations (A8) and (A9) lead to
𝐺-MCPF > 𝑇-MCPF ≥ 1.
Then, we have 𝑡 1 ≥ 𝑡 ∗ > 𝑡 ⋆ and 𝑈 1 ≥ 𝑈 ∗ > 𝑈 ⋆ . Result (i) holds.
The last case is(𝑤(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀0𝜇 + 𝑡)(1 − 𝑛)𝜀𝑛𝜀𝑡𝑛+> 0, 𝑛 ++< 0 ⇒ 𝑇-MCPF < 1 < 𝐺-MCPF.
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
This means that 𝑡 ∗ > 𝑡 1 > 𝑡 ⋆ and 𝑈 1 > max(𝑈 ∗ , 𝑈 ⋆ ). Therefore, result (ii) was proven.
𝑛+15

F. Proof of Proposition 3
In a similar manner to the proof of Proposition 1, 𝑅, (𝑡) < 0, lim𝑅(𝑡) = ∞, andlim 𝑅(𝑡) = −∞,

.↑

.↓3

/5"̅
($/')9:where(1 − 𝑛)𝜀̅𝜇̅−𝑓%̅
𝑓%̅
𝑅(𝑡) ≡ 𝑣′P𝑡𝑘rQ −= 0 for 𝑡 ∈ „0,….
(1 − 𝑛)𝜀̅𝑡(1 − 𝑛)𝜀̅1+𝑓%̅
∗Hence, we obtain the unique value of 𝑡 from the solution of 𝑅(𝑡) = 0. We now introduce thefollowing function 𝑆:(1 − 𝑛)𝜀̅𝜇̅ 𝑛𝜀̅𝑡
1−𝑤
0
+ ̅
𝑓%̅
𝑓%
r
𝑆(𝑡) ≡ 𝑣′P𝑡𝑘Q −
= 0.
𝜀̅𝑡
1+ ̅
𝑓%
We have lim𝑆(𝑡) = ∞,.↓3
(1 − 𝑛)𝜀̅𝜇̅ 𝑛𝜀̅𝑡 𝜀̅𝑛𝜀̅
𝜀̅𝑡_1 + ̅ ` − _1 − 𝑤
0+ ̅ ` ̅
̅
𝑓
𝑓%
𝑓%̅
𝑓% 𝑓%
𝑆 , (𝑡) = 𝑣′′𝑘r − %𝜀̅𝑡 (f1 + ̅ g𝑓%𝜀̅ ((1 − 𝑛)(𝑤
0𝜇 + 𝑡) f ̅ g
𝑓%= 𝑣′′𝑘r −.
𝜀̅𝑡 (f1 + ̅ g𝑓%
Note that 𝐺-MCPF satisfies(1 − 𝑛)𝜀𝜇(𝑤
𝑛𝜀𝑡
0𝜇 + 𝑡)(1 − 𝑛)𝜀1−𝑤+
0
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
𝐺-MCPF − 1 =
−1=−
⋛0
𝜀𝑡
𝜀𝑡
(A10)
1+
1+
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)⇔𝑤
0𝜇 + 𝑡 ⋛ 0.
Three cases should be considered: (i) 𝑤
0𝜇 + 𝑡 > 0. It leads to 𝑆 , (𝑡) < 0 and(1 − 𝑛)𝜀̅𝜇̅ 𝑛𝜀̅𝑡
𝜀̅𝑡𝑤
0𝜇 + 𝑡 > 0 ⇒ 1 − 𝑤
0
+
>
1
+
> 0.
𝑓%̅
𝑓%̅
𝑓%̅
The domain of 𝑆 is given as
−𝑓%̅
0<𝑡<
.
𝜀̅
Then, we obtainlim 𝑆(𝑡) = −∞.
1−𝑤
0

/5̅
.↑ 9:"

These properties of 𝑆(𝑡) show that 𝑆(𝑡) has a unique value of 𝑡 ⋆ such that 𝑆(𝑡 ⋆ ) = 0.
(ii) 𝑤
0𝜇 + 𝑡 = 0. Equation (20) becomes 𝑣 , P𝑡𝑘r Q = 1. By the monotonicity and continuity of 𝑣 , ,the unique solution is derived as𝑣 ,/$ (1)𝑡⋆ =.
𝑘r(iii) 𝑤
0𝜇 + 𝑡 < 0. We have
(1 − 𝑛)𝜀̅𝜇̅ 𝑛𝜀̅𝑡
𝜀̅𝑡𝑤
0𝜇 + 𝑡 < 0 ⇒ 1 − 𝑤
0
+
<
1
+
𝑓%̅
𝑓%̅
𝑓%̅
The domain of 𝑆(𝑡) is16

(1 − 𝑛)𝜀̅𝜇̅−𝑓 ̅
𝑇 = Œ𝑡•𝑡 ∈ f0, 𝑛𝜀̅% _1 − 𝑤
0
`gŽ.
𝑓%̅
Taking this result into account, we obtain lim 𝑆(𝑡) = lim 𝑣′′P𝑡𝑘rQ < 0, where.↑=>? @.↑=>? @

(1 − 𝑛)𝜀̅𝜇̅−𝑓%̅
sup 𝑇 =
c1 − 𝑤0d.
𝑛𝜀̅
𝑓%̅
Therefore, there exists at least one root of 𝑆(𝑡) = 0. To ensure the uniqueness of the root, we needto impose𝜀̅ ((1 − 𝑛)(𝑤
0𝜇 + 𝑡) f ̅ g
𝑓%𝑣 ,, 𝑘r −
< 0.
(
𝜀̅𝑡
f1 + ̅ g𝑓%
If 𝑤
0𝜇̅ + 𝑡 is sufficiently large, a unique value of 𝑡 ⋆ such that 𝑆(𝑡 ⋆ ) = 0.
Equations (19) and (18) yield(1 − 𝑛)𝜀𝜇(1 − 𝑛)𝜀𝜇
𝑛𝜀𝑡
1−𝑤
1−𝑤
+
0
0
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
𝑇-MCPF − 𝐺-MCPF =−𝜀𝑡(1 − 𝑛)𝜀𝑡
1+
1+
(𝐾,
𝑓
𝐿)
(𝐾,
𝑓%
𝐿)
%
(
(1 − 𝑛)𝑛𝜀 𝑡(𝑤
0𝜇 + 𝑡)[𝑓% (𝐾, 𝐿)](=−⋚0⇔𝑤
0𝜇 + 𝑡 ⋛ 0. (A11)(1 − 𝑛)𝜀𝑡𝜀𝑡_1 +` _1 +`
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
G. Proof of Proposition 4
From equation (19), we have
(1 − 𝑛)𝜀𝜇(𝑤
0𝜇 + 𝑡)(1 − 𝑛)𝜀1−𝑤
0
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)
𝑇-MCPF − 1 =
−1=−
⋛0⇔𝑤
0𝜇 + 𝑡 ⋛ 0.
(1 − 𝑛)𝜀𝑡(1 − 𝑛)𝜀𝑡1+
1+
𝑓% (𝐾, 𝐿)
𝑓% (𝐾, 𝐿)(A12)

Equations (A10) and (A12) derive 𝐺-MCPF ⋛ 𝑇-MCPF ⋛ 1 ⇔ 𝑤
0𝜇 + 𝑡 ⋛ 0.
(i) 𝑤
0𝜇 + 𝑡 > 0. Then, 𝐺-MCPF > 𝑇-MCPF > 1 holds. By the concavity of 𝑣, we obtain 𝐺 1 >𝐺 ∗ > 𝐺 ⋆ . Equation (4) and 𝐾! = 𝑘r lead to 𝑡 1 > 𝑡 ∗ > 𝑡 ⋆ . Furthermore, it shows 𝑈 1 > 𝑈 ∗ > 𝑈 ⋆ .
(ii) 𝑤
0𝜇 + 𝑡 < 0. It shows 𝐺-MCPF < 𝑇-MCPF < 1. Then, by the similar way to derive the resultsin the case (i), we obtain 𝐺 1 < 𝐺 ∗ < 𝐺 ⋆ , 𝑡 1 < 𝑡 ∗ < 𝑡 ⋆ , and 𝑈 ⋆ < 𝑈 ∗ < 𝑈 1 .

17

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19

Figures𝑣′(𝐺)MB, MCPF

G-MCPF𝑣′(𝐺 ⋆ )𝑣′(𝐺 ∗ )B

0𝑡⋆

T-MCPF

A𝑡𝑡∗

Figure 1. T-equilibrium and G-equilibrium𝑣′(𝐺)MB, MCPF

𝑣′(𝐺 ∗ )𝑣′(𝐺 ⋆ )0

G-MCPF

T-MCPF

A
B

𝑡∗ 𝑡⋆

−𝑤
0𝜇

𝑡

Figure 2. T-equilibrium and G-equilibrium in the case (b)-(ii)20