Endogenous timing in tax competition withasymmetric governments’ objectives
Shigeo Morita∗
July 15, 2017
Abstract
This study examines a timing game in capital income tax competition between thegovernment as a benevolent maximizer of social welfare and a tax revenue maximizing
Leviathan. When both governments’ objectives are revenue-maximizing, two sequentialmove outcomes are SPNEs. This is true even if there is no absentee capital ownership.
When one government is the Leviathan, outcomes of SPNEs are determined by whether
the benevolent government is a capital importer or a capital exporter. In this model,
the asymmetry in production function is represented by diﬀerent vertical intercepts ofcapital demand. In the case of symmetric objectives, leadership by the large country is
Pareto-dominant equilibrium. However, under the same structure of production function, if the governments’ objectives are asymmetric, the leadership by the small country
is Pareto-dominant.
Keywords: Tax competition; Endogenous timing∗

Faculty of Economics, Fukuoka University, 8-19-1, Nanakuma, Jonan-ku, Fukuoka 814-0180,
Japan. E-mail: shmorita@fukuoka-u.ac.jp1

1

Introduction

The recent studies in the tax competition relax the assumption that the competinggovernments choose their strategic variables simultaneously. In this context, the sourceof heterogeneity between countries or regions is the one of important factors. A typicalexample is the market power in the capital market.Indeed, Kempf and Rota-Graziosi(2010) focus on the gradient of demand for capital, while Hindriks and Nishimura (2015,2017) consider the vertical intercept of the demand for capital. This paper mainly
focuses on the other characteristic, objective functions of the country. Many previous
studies assume that governments are benevolent and maximize social welfare, which isa weighted sum of individuals’ utility functions. In the contrast, Edwards and Keen
(1996) and Wilson (2005) consider a case where the governments maximize their net
tax revenues to increase the government size, in order words, the government acts as a
Leviathan. They may place a low weight on a total surplus, when the governments face
severe revenue shortfall. According OECD (2015), between 2009 and 2014, the ratioof tax revenues to GDP falls in OECD countries, for example, Spain, Norway, Czech
Republic, Luxembourg, and Turkey, while the ratio rises in countries, for example,
Greece, Denmark, Iceland, Estonia, New Zealand. 1 Diﬀerence in objectives of thegovernments may appears among these countries. Theoretically, Pal and Sharma (2013)
construct the model of interregional competition for mobile capital with the stage thatthe governments chooses their objective functions. They show that, in the case of
sequential move competition, the leader chooses to be benevolent, while the follower
chooses to be a Leviathan (See their Proposition 5). These mean that not only themarket power but also the governments’ objectives can be seen as heterogeneity between
countries.
In this paper, we construct the model that not only size of country but also governments’ objectives vary between countries. The goal of government in a country is
to maximize social welfare, while that of the government in another country is tax
revenue. When both governments’ objectives are revenue-maximizing, two sequentialmove outcomes are SPNEs. This is true even if there is no absentee capital ownership.
This is because both governments care about the only their tax revenue, and then their
payoﬀ do not depend on the price of capital. When either government is the Leviathan,outcomes of SPNEs are determined by whether the benevolent government is a capital
importer or a capital exporter. Moreover, we consider a coordination issue. In thismodel, the asymmetry in market power is represented by diﬀerent vertical interceptsof capital demand such as Hindriks and Nishimura (2015, 2017) and Ogawa (2013). In
the case of symmetric objectives, leadership by the large country is Pareto-dominantequilibrium. This is because the reaction functions diﬀer among countries only by thevertical intercepts. But, fixed the structure of production function, if the governments’1

Indeed, cooperate tax revenues have been falling across OECD countries. Average revenues fromcooperate incomes and gains fall from 3.6 % to 2.8 % over from 2007 to 2014.

2

objectives are asymmetric, leadership by the small country is Pareto-dominant. Thisis because the reaction functions diﬀer by not only the vertical intercepts but also gradients. These can be pointed out by interactions between asymmetry of governments’objectives and market power.
The rest of this paper is organized as follows. Section 2 presents a setup of model.
Section 3 analyzes the equilibrium and discusses results. Section 4 consider a coordination issue. Section 5 concludes.

2

Model

There are two countries denoted by i = 1 and i = 2. The production sector in bothcountries utilizes capital and labor, and its technology exhibits constant return to scale.
The production per capita in country i is the following quadratic production function:f i = (Ai − ki )ki where ki denotes the capital per capita in country i and Ai denotes
the parameter for production eﬃciency in country i. It is assumed ∆ ≡ A1 − A2 > 0,
in words, the country 1 is more productive than the country 2. Here, we consider thatthe more (less) productive country corresponds to the larger (smaller) one. The profit
of firm is given by πi = f i (ki ) − rki − Ti ki − ωi where r denotes the market price ofcapital, Ti the unit tax rate on capital, and ωi the wage rate. We define the net tax
revenue T Ri as follows:
T Ri = Ti ki , i = 1, 2

(1)

The total amount of capital in the economy is 2k̄. Following Ogawa (2013), a consumer in country i has an initial endowment of capital, θk̄ where θ ∈ [0, 1] denotes theparameter indicated consumer’s capital ownership. We consider that the capital market
is perfectly competitive. The preference of consumer in the country i is represented byu(ci ) = ci where ci denotes his/her consumption level. The net income of the consumer
consists of labor income, capital income, and the income transfer from the government
in his/her country. The labor and capital income of the consumer in the country i is
IRi = ki2 + rθk̄. The income transfer is financed by the tax revenue T Ri . Thus, thesocial welfare of the country i SWi is given by the sum of incomes of consumer IRi andtax revenue T Ri :SWi = IRi + T Ri i = 1, 2

(2)

The government in each country imposes a unit tax on capital located in its country.
Following Edwards and Keen (1996), the objective function is defined as follows:
Wi = αi SWi + (1 − αi )T Ri i = 1, 2

(3)

where αi ∈ [0, 1] denotes the parameter for motivation of the policymakers. Whenαi = 1, the government maximizes consumer’s utility. However, αi = 0 corresponds to3

a policymaker as revenue-maximizing Leviathans. The market clearing conditions are
2k̄ = k1 + k2 ,and r = fk1 (k1 ) − T1 = fk2 (k2 ) − T2

Under these conditions, the level of capital in each country and the price of capital are
k1 = k̄ +∆ T2 − T1
+
4
4

(4)

k2 = k̄ −∆ T1 − T2
+
4
4

(5)

Ω 4k̄ + T1 + T2
−
2
2

(6)

r=

where Ω ≡ A1 + A2 .
We propose a two stage game with the following timing. In the first stage, thegovernment decides whether to set its tax rate at the first period or at the second
period. Therefore, this is the timing game of Hamilton and Slutsky (1990). If bothgovernments choose their tax rates in the same period, then these governments takedecisions simultaneously. If one government sets its tax rate at the first period and theother at the second period, then tax rates are chosen sequentially. These mean that,in the second stage, the government decides its tax rate simultaneously or sequentially.
To obtain a subgame perfect Nash equilibrium (hereafter, SPNE), we apply the conceptof backward induction to solve the game.

3

Characterizing the tax competition

3.1

Symmetric objectives

As a benchmark case, we consider the case where both governments are revenuemaximizing Leviathans, that is, α1 = α2 = 0. We denote the case in which governments set these tax rates simultaneously by superscript N . On the other hand,when the governments set their tax rates sequentially. The superscript L (F ) meansthat the government is a leader (follower).
First, we solve the case when tax rates are set simultaneously. To insure interior
solutions, we assume
Assumption 1.
k̄ −

∆
>0
12
4

Each government chooses the level of tax rate to maximize equation (1) subject toequation (3) to (5). Solving these yields the following reaction functions:
1
T1 = T2 + 8k̄ + 2∆
2

(7)

1
T2 = T1 + 8k̄ − 2∆2(8)

From equations (7) and (8), the following is obtained:∆ + 12k̄−∆ + 12k̄
, T2N =
3
3
2
(∆
+
12
k̄)
(−∆
+
12k̄)2W1N =, W2N =
36
36
T1N =

Next, we consider the case in which the governments decide their tax rates sequentially.
When the government in the large country acts as a leader, it chooses the tax rate tomaximize (1) taking into account (7). Solving it, we obtain:−∆ + 20k̄
∆ + 12k̄
, T2F =
2
4
2(−∆ + 20k̄)2(∆ + 12k̄), W2F =
W1L =
32
64
T1L =

Similarly, when the small country is a leader, we have:∆ + 20k̄−∆ + 12k̄
, T2L =
4
2
2(−∆ + 12k̄)2(∆ + 20k̄), W2L =
W1F =
64
32
T1F =

It is well known that endogenous timing outcomes depend on crucially the slopeof the reaction functions and the signs of the payoﬀ externalities. From equations (7)and (8), it is easy to show that tax rates are strategic complement (i.e., these reaction
functions functions are upward-sloping). We can also show the existence of positive
externalities:∆∂W1 N N(T1 , T2 ) = k̄ +
> 0,
∂T2
12

∆
∂W2 N N(T1 , T2 ) = k̄ −
>0
∂T1
12

The reason is straightforward. When the government in a country raises its tax rate,Ri
the tax revenue in another country increases, that is, ∂T> 0 for i ̸= j. Therefore,∂Tj
the SPNEs of the timing game are two sequential-move outcomes.
5Kempf and Rota-Graziosi (2010) and Hindriks and Nishimura (2015) show that the
SPNEs of timing game are also two sequential-move outcomes when there is absentee
capital ownership (θ = 0) and all governments are benevolent (α1 = α2 = 1). Under thesetting of their model, the social welfare SWi can be rewritten as SWi = f (ki ) − rki .
When the government in a country raises its tax rate, the price of capital decreases,∂rthat is, ∂T< 0 for i = 1, 2. The tax externality is positive for the governments in bothicountries, and then the equilibria are also two sequential-move outcomes. The pointthat their model and our model have in common is that objective functions exclude theparameter about capital ownership, θ.

3.2

Asymmetric objectives

When we allow the case of asymmetric objectives, that is, α1 ̸= α2 , we have two cases.
First, the government in the large country is the Leviathan, that is, α1 = 0 and α2 = 1.
Second, the opposite case where the government in the small country is the Leviathan,that is, α1 = 1 and α2 = 0.
3.2.1

The case where the large country is the Leviathan

We first solve the case in which tax rates are set simultaneously. The assumption to
insure interior solutions are modified as follows:
Assumption 2.
−∆ + 2k̄(6 + θ) > 0
The large country chooses the level of T1 that maximizes equation (1), but the small
country chooses the level of T2 that maximizes equation (2). Solving these problems,
we obtain the reaction functions in tax rates:
11
T1 = T2 + ∆ + 2k̄
2
2

(9)

1
4(1 − θ)k̄
1
T2 = T1 − ∆ +
3
3
3

(10)

From equations (9) and (10), the following is obtained:
T1N =

W1N =

2{∆ + 2k̄(4 − θ)},5(∆ + 2k̄(4 − θ))2
,
25

W2N =

T2N =

−∆ + 4k̄(3 − 2θ)
53(−∆ + 12k̄)2 + 4k̄θ(12A1 + 13A2 − 144k̄) + 92k̄ 2 θ2100
6

Next, we consider the case in which governments decide their tax rates sequentially,
and the Leviathan decides its tax rate first. It chooses the tax rate that maximizes (1)taking into account equation (10). Solving it yields
T1L =

W1L =

∆ + 2k̄(4 − θ),2(∆ + 2k̄(4 − θ))2
,
24

W2F =

T2F =

−∆ + 2k̄(8 − 5θ)6(−∆ + 16k̄)2 + 4k̄θ(5A1 + 7A2 − 80k̄) + 52k̄ 2 θ248

We turn to the case in which the government in the small country chooses the tax ratethat maximize (2) taking into account equation (9). Solving it yields:
T1F =

W1F =

2{∆ + 2k̄(8 − 3θ)},7(∆ + 2k̄(8 − 3θ))2
,
49

W2L =

T2L =

3{−∆ + 4k̄(3 − 2θ)}7(−∆ + 12k̄)2 + 4k̄θ(4A1 + 3A2 − 48k̄) + 36k̄ 2 θ228

Evaluated at the simultaneous-move outcome, the external eﬀects are∂W1 N N∆ + 2k̄(4 − θ)
(T1 , T2 ) =
>0
∂T2
10

(11)

∂W2 N N−∆ + 4k̄(3 − 2θ)∆(T1 , T2 ) =≷ 0 ⇐⇒≶ 4(3 − 2θ)
∂T1
10
k̄

(12)

Equation (11) means that the large country in the asymmetric objectives receives thesame external eﬀect as in the symmetric objectives. However, the small country suﬀers
from negative external eﬀect, if the small country is a capital exporter (i.e., k2N −θk̄ < 0).
This is because its payoﬀ, SW2 can be rewritten as follows: SW2 = f (k2 ) + r(θk̄ − k2 ).
The first term can be seen as the immobile factor, and the second term is the mobile
factor. If the country is capital exporter, the amount of mobile factor is negative. Thus,
it has the incentive to increase the price of capital. This is because from equation (6),the price of capital is lowered (raised) when the tax rates in a country is raised (lowered).
To characterize the SPNE, we define the first-mover incentive (second-mover incentive): a country has a first-mover incentive (second-mover incentive), if its indirect
utility function in the sequential move game in which it leads (follows) is higher thanin the simultaneous-move game. In Appendix A, we show that
WiL − WiN > 0, i = 1, 2
7

(13)

∆
≶ 4(3 − 2θ)
k̄

(14)

∆
2
≶ (76 − 49θ)
11
k̄

(15)

W1F − W1N ≷ 0 ⇐⇒W2F − W2N ≷ 0 ⇐⇒These are summarized as the following proposition:
Proposition 1. If the government in the large country is the Leviathan, we can derive
three kinds of equilibria:
2
1. When ∆k̄ > 11
(76 − 49θ), a simultaneous-move outcome where both governments
get their tax rate at the first period, is SPNE.
2
2. When 4(3 − 2θ) < ∆k̄ < 11(76 − 49θ), a sequential-move outcome where theLeviathan sets its tax rate at he first period but the benevolent government sets its
tax rate at the second period, is SPNE.

3. When ∆k̄ < 4(3 − 2θ), two sequential-move outcomes are SPNEs.
2
When ∆k̄ > 11
(76 − 49θ), the Leviathan sets a higher tax rate than the benevolent
government in all cases, that is, T1L > T2F , T1F > T2L , and T1N > T2N . This is dueto the fact that the benevolent government is a capital exporter and its social welfare
involves the mobile factor. By setting a lower tax rate, it raises the level of social
welfare. In other words, it seeks to facilitate the ”downward pressure” feature in the
tax competition. However, the Leviathan wants to mitigate the ”downward pressure”
feature for higher tax revenue. When the Leviathan (the benevolent government) isleader, it raises (lowers) its tax rate in comparison to the simultaneous case since it
knows that the benevolent (the Leviathan) also raises (lowers) its tax rate, because
tax rates are strategic complements. These mean that tax rates are ordered as follows:
T1L > T1N > T1F and T2F > T2N > T2L . Therefore, choosing tax rates at the first period
is dominant strategies for governments in both countries. This outcome is related toBárcena-Ruiz (2007), which shows that in a mixed duopoly under the price competition,
the private firm and the public firm set their prices at the first period is the only SPNE.
The pubic firm concerns about the consumer surplus, and then wants to increase themarket competition, while the private firm seeks to reduce the market competition.
These imply that the public firm prefers a lower price than the simultaneous case, but
the private firm prefers a higher price than the simultaneous case. As a result, bothtypes of firms commit to choose their prices in the first period at the SPNE.
Proposition 1 is consistent with results in Hindriks and Nishimura (2017). Their
proposition 1 shows the level of capital ownership switching from the Stackelberg to
the simultaneous-move outcomes, under the symmetry in governments’ objectives (α1 =8α2 = 1) and the asymmetry in the market powers (∆ > 0). Indeed, the left-hand sideof equation (14) and (15) can be rewritten as
W1F − W1N ≷ 0 ⇐⇒ θ ≶3
76 11∆∆− , W2F − W2N ≷ 0 ⇐⇒ θ ≶
−
2 8k̄
49
98k̄

These can be seen as the switching points in which the countries have a second-moverincentives proposed in Hindriks and Nishimura (2017).2
3.2.2

The case where the small country is the Leviathan

We solve the case in which tax rates are set simultaneously. The assumption to insureinterior solutions are modified as follows:
Assumption 3.
−∆ + 2k̄(4 − θ) > 0
The large country chooses the level of T1 that maximizes equation (2), but the small
country chooses the level of T2 that maximizes equation (1). Solving these problems,
we obtain the reaction functions in tax rates:∆ + 4k̄(1 − θ)1
T1 = T2 +
3
3

(16)

1
1
T2 = T1 − ∆ + 2k̄
2
2

(17)

From equations (16) and (17), the following is obtained:
T1N =

W1N =

∆ + 4k̄(3 − 2θ)
,
5

T2N =

−2{∆ + 2k̄(−4 + θ)}
53(−∆ + 12k̄)2 + 4k̄θ(13A1 + 12A2 − 144k̄) + 92k̄ 2 θ2
,
100

W2N =

(∆ − 2k̄(4 − θ))2
25

Next, we consider the case in which governments decide their tax rates sequentially,
and the benevolent government decides its tax rate first. It chooses the tax rate thatmaximizes (2) taking into account equation (17). Solving it yields:
T1L =2

3{∆ + 4k̄(3 − 2θ)}
,
7

T2F =

−2{∆ + 2k̄(−8 + 3θ)}7

By notations used in Hindriks and Nishimura (2017), these correspond to θN and θ2 , respectively.

9

W1L =(∆ + 12k̄)2 + 4k̄θ(3A1 + 4A2 − 48k̄) + 36k̄ 2 θ2
,
28

W2F =

(∆ − 2k̄(8 − 3θ))249

We turn to the case in which the Leviathan decides its tax rate first. The government in
the small country chooses the tax rate that maximizes (1) taking into account equation(16). Solving it, we have
T1F =

W2L =

∆ + 2k̄(8 − 5θ)
,
6

T2L =

−∆ + 2k̄(4 − θ)2(∆ + 16k̄)2 + 4k̄θ(7A1 + 5A2 − 80k̄) + 52k̄ 2 θ2
,
48

W2L =

(∆ − 2k̄(4 − θ))224

Evaluated at the simultaneous-move outcome, the external eﬀects are∂W1 N N∆ + 4k̄(3 − 2θ)∂W2 N N−∆ + 2k̄(4 − θ)
(T1 , T2 ) => 0,(T1 , T2 ) =
>0
∂T2
10
∂T1
10

(18)

When the small country is the Leviathan, it receives a positive spillover. This is thesame as in the case of symmetric objectives. As for the large country, is is the capital
importer (k1N −θk̄ > 0) for any parameter values. Because of the same reason mentioned
above, there is a positive spillover. Therefore, equations (18) for i = 1, 2 are positive.
These are summarized as the following proposition:
Proposition 2. If the government in the small country is the Leviathan, we can derive
two sequential-move outcomes are SPNEs.
Proposition 1 and 2 mean that when either government is the Leviathan, outcomesof SPNEs are related to the mobile factor in the benevolent government’s objectivefunction, that is, the country whose government is benevolent is a capital importer ora capital exporter.

4

A coordination issue

To rank the two SPEs, we apply a criterion: the Pareto-dominance. We introduce
the concept of a first-mover advantage (a second-mover advantage): the government
has a first-mover advantage (a second-mover advantage), if its indirect utility in thesequential-move outcome in which it chooses a tax rate in the first period is higher(lower) than the indirect utility that it chooses a tax rate in the second period. All
derivations of this section are shown in Appendix B.
Under the case of symmetric objectives, we show that
W1L − W1F ≷ 0 ⇐⇒
10

∆
≷ δ1
k̄

(19)

W2L − W2F < 0(20)

From Assumption 1, we discuss the equilibrium selection for 0 < ∆k̄ < 12.
Lemma 1. Consider the case of symmetric objectives.
1. For 0 < ∆k̄ < δ1 , both governments have the second-mover advantage.
2. For δ1 < ∆k̄ < 12, the large country has the first-mover advantage but the small
country has the second-mover advantage.
We turn to the case of asymmetric objectives. In the case in which the government
in the large country is the Leviathan, from equation (12), we discuss the equilibrium
selection for 0 < ∆k̄ < 4(3 − 2θ). We obtain
W1L − W1F < 0
W2L − W2F > 0 ⇐⇒ 0 <∆∆< δ2 or δ3 << 4(3 − 2θ)
k̄
k̄

(21)

(22)

These give
Lemma 2. Consider the case where the government in the large country is a Leviathan.
1. For 0 < ∆k̄ < δ2 or δ3 < ∆k̄ < 4(3 − 2θ), the large country has the second-moveradvantage but the small country has the first-mover advantage.
2. For δ2 < ∆k̄ < δ3 , both governments have the second-mover advantage.
In the case in which the small country is the Leviathan, we obtain
W1L − W1F ≶ 0 ⇐⇒

∆
≶ δ4
k̄

W2L − W2F > 0

(23)

(24)

From Assumption 3, we discuss the equilibrium selection for 0 < ∆k̄ < 2(4 − θ).
Lemma 3. Consider the case where the small country is a Leviathan.
1. For 0 < ∆k̄ < δ4 , the large country has the second-mover advantage but the small
country has the first-mover advantage.
2. For δ4 < ∆k̄ < 2(4 − θ), both governments have the first-mover advantage.
These lemma allows us to find the Pareto-dominant equilibrium.
11

Proposition 3. Suppose that the asymmetry in the market power is represented by
diﬀerent vertical intercepts. We have
1. In the case of symmetric objectives, that is, α1 = α2 = 0, the leadership by thelarge country is Pareto-dominant, if the asymmetry of productivity is suﬃcientlarge.
2. In the case of asymmetric objectives, that is, α1 ̸= α2 , the leadership by the small
country is Pareto-dominant, if the asymmetry of productivity is suﬃcient smallor large.
Under the production function in this model, f (ki ) = (Ai − ki )ki , the country withhigher productivity, that is, the large country has greater market power. Mathematically, it is represented by the vertical intercepts of capital demand. Hindriks andNishimura (2015) show that SPNE where such a country move first is Pareto-dominant
if the asymmetry in market power is suﬃcient. The first argument in Proposition 3 isconsistent with the outcome. In this model, the reaction functions (equations (7) and(8)) have the same gradients but diﬀer the vertical intercepts. Because of the higher
intercept for the large country, the country benefits from higher taxes and thus has afirst-mover advantage. Fixed on the production function, the asymmetry of governments’ objectives translates to the diﬀerence in the gradients (see equations (9), (10),(16), and (17)). In the case in which the large country is the Leviathan, the country
2
1= 21 > ∂T= 31 . The
is more sensitive to the tax rate of the small country, that is, ∂T∂T2∂T1
leadership by the small country is preferable to facilitate a greater positive reaction in
the tax rate of the large country. This leads the small country to move in the first
period. We turn to the case in which the small country is the Leviathan. From theslope of reaction functions, the small country is more sensitive to the tax rate of thelarge country. In contrast, for 0 < ∆k̄ < δ4 , the vertical intercept of the small country’s
reaction function is higher than that of the large country’s reaction function. For therange of parameters, the small country benefits from setting a higher tax rate, and thenthe leadership by the small country is preferable to the leadership by the large country.

5

Conclusion

This paper introduces the diﬀerence of governments’ objectives into the endogenous
timing of tax competition. The government in a country is the revenue-maximizing
Leviathan, but the government in another country is benevolent maximizer of social
welfare. When both governments’ objectives are revenue-maximizing, two sequentialmove outcomes are SPNEs. This is true even if there is no absentee capital ownership.
This is because both governments care about the only their tax revenue, and then their
payoﬀ do not depend on the price of capital. When either government is the Leviathan,outcomes of SPNEs are determined by whether the benevolent government is a capital12

importer or a capital exporter. In this model, the asymmetry in production function isrepresented by diﬀerent vertical intercepts of capital demand. In the case of symmetricobjectives, leadership by the large country is Pareto-dominant equilibrium. However,under the same structure of production function, if the governments’ objectives areasymmetric, the leadership by the small country is Pareto-dominant.

Appendix A
To derive the SPNEs of the timing game, we compare the indirect utility functions of
the governments in both countries. We have
W1L − W1N =(−∆ + 2k̄(4 − θ))2
600

(A.1)

W2L − W2N =(∆ − 4k̄(3 − 2θ)2
175

(A.2)

8
(3∆ + k̄(34 − 11θ))(∆ − 4k̄(3 − 2θ))
1225

(A.3)

1
(∆ + 2k̄(4 − θ))(11∆ − 2k̄(76 − 49θ))
1200

(A.4)

W1F − W1N = −W2F − W2N = −It is easy to show that equations (A.1) and (A.2) are positive. This means that equation(13) is strictly positive. The sign of equation (A.3) depends on the direction of spillover(equation (12)). Then, we get equation (14). According to equation (A.4), under theAssumption 2, the third bracket is negative. Then, we get equation (15).

Appendix B
To determine the first-mover advantage (the second-mover advantage), we compare theindirect utility function of the governments in both countries. When α1 = α2 = 0, we
obtain√1 ∆∆( + 4(1 + 2 2))( − δ1 )
64 k̄
k̄

(B.1)

√
√
∆
1 ∆
( − 4(1 − 2 2))( − 4(1 + 2 2))
64 k̄
k̄

(B.2)

W1L − W2F =W2L − W2F =13

√where δ1 ≡ 4(−1 + 2 2). The sign of equation (B.1) depends on the second bracket.
Then, we get equation (19). Under Assumption 1, the first bracket of equation (B.2)
is positive, but the second bracket is negative. Then, equation (20) is strictly positive.
When α1 = 0 and α2 = 1, we obtain
W1L − W1F = {(∆ + 2k̄(4 − θ)) (∆ + 2k̄(8 − 3θ)) (∆ + 2k̄(4 − θ)) (∆ + 2k̄(8 − 3θ))
√
√
√
√
−
}{
+
(}B.3)
24
49
24
49
W2L − W2F = −√ √k̄ 2 ∆∆( − δ2 )( − δ3 )336 k̄
k̄
√ √

(B.4)

where δ2 ≡ 16−13θ− 3 5192−32θ−57θ and δ3 ≡ 16−13θ+ 3 5192−32θ−57θ . Under Assumption
2, the first bracket of equation (B.3) is negative. Then, equation (21) is strictly positive.
The sign of equation (B.4) depends on both first and second brackets. Then, we getequation (22). When α1 = 1, and α2 = 0, we obtain
2

W1L − W1F =2

k̄ 2 ∆∆( − δ4 )( − δ5 )336 k̄
k̄

(B.5)

√
√
where δ4 ≡ 52 (−16 + 13θ + 2 21(2 − θ)) and δ5 ≡ 52 (−16 + 13θ − 2 21(2 − θ)).
(∆ − 2k̄(8 − 3θ))2 (∆ − 2k̄(4 − θ))2LF+
W2 − W2 = −
49

24

(B.6)

Under Assumption 3, the second bracket is positive. Then, the sign of equation (B.5)
depends on the first bracket. It is obvious that the first term of equation (B.6) is smaller
than the second term. Then, equation (24) is strictly positive.

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