Does Endogenous Timing Matter in Implementing
Partial Tax Harmonization?
Jun-ichi Itaya and Chikara Yamaguchi
Does Endogenous Timing Matter in Implementing
Partial Tax Harmonization? ∗
Jun-ichi Itaya†and Chikara Yamaguchi‡
April 6, 2015
Abstract
The endogenous timing of moves is analyzed in a repeated game setting of capital tax
competition, where a subgroup of countries implementing partial tax harmonization and
outside countries choose whether to set capital taxes sequentially or simultaneously. It is
shown that the simultaneous-move outcome prevails in every stage game of the repeated
tax-competition game as its subgame perfect equilibrium if a tax-union consists of similar countries, whereas either a simultaneous-move or the (Stackelberg) sequential-move
outcome can arise in every stage game when a tax-union consists of dissimilar countries.
This is due to the fact that asymmetry among countries in terms of productivities makes
the tax-union members of tax union have opposite incentives towards the terms of trade
in order to manipulate the price of capital. Although Ogawa (2013) points out that
there exists only a simulataneous-move Nash equilibrium, our tax coordination model
implies that the participation constraint for such dissimilar countries does not allow for
a simultanoues-move outcome. and so the only seuential-move exists.
JEL classification: H30; H87
Keywords: Tax competition; Tax harmonization, Endogenous timing; Repeated game;
∗
The first and second authors gratefully acknowledge the financial support provided by the Grant-in-Aid
for Scientific Research from the Japan Society for the Promotion of Science (#21530290) and (#24530379),
respectively.
†
Corresponding author. Graduate School of Economics and Business Administration, Hokkaido University,
Sapporo 060-0809, Japan. Tel: +81-11-706-2858; Fax: +81-11-706-4947; E-mail: [email protected]
‡
Faculty of Economic Sciences, Hiroshima Shudo University, 1-1-1 Ozukahigashi, Asaminami-ku, Hiroshima
731-3195, Japan. Tel: +81-82-830-1238; Fax: +81-82-830-1313; E-mail: [email protected]
1
Introduction
This paper examines under what conditions partial tax coordination (i.e., tax coordination
implemented by a subset of the existing countries) is sustained in a repeated tax-competition
model in the presence of endogenous timing in the orders of plays. The coordination of
tax policies among sovereign jurisdictions has often been considered as a remedy against
inefficiently low taxes on mobile tax bases or production inefficiency induced by asymmetric tax
rates in the literature on tax competition. Although tax coordination among all the countries
in the economy is desirable, generally, it is difficult to achieve full tax coordination among
all existing countries because some countries may prefer a lower tax status for commercial
reasons (i.e., the so-called tax haven) and because the differences in social, cultural, and
historical factors or economic fundamentals such as endowments and technologies may prevent
the countries from accepting a common tax rate. Therefore, partial tax harmonization, rather
than global or full tax harmonization, is politically more acceptable, and thus one could be
compelled to resort to partial tax coordination.
Various papers have studied partial tax harmonization from different perspectives (see,
e.g., Keen and Konrad, 2013 for a useful survey). In particular, there are substantial literatures to examine how the formation of a partial tax union which harmonizes capital taxation
affects the welfare of countries inside and outside a tax union. Konrad and Schjelderup (1999)
demonstrate that in the standard static tax competition framework with symmetric countries,
based on the assumption of strategic complementarity between the tax rates of a tax union
and outside countries, partial harmonization can improve not only the welfare of the union
but also that of the outside countries. This line of research agenda would provide one possible answer to the question as to whether or not the countries such as the ECA (Enhance
Cooperation Agreement) in the EU or the EU in the world economy have a motivation to
form a coalition group of countries which implements tax harmonization. Bucovetsky (2009)
considers an economy consisting of heterogenous population-sized jurisdictions and presents a
sufficient condition under which the grand coalition Pareto-dominates all other coalition structure. Vrijburg (2009) sets up a three-country model with heterogenous population size, and
1
shows that partial harmonization unambiguously increases welfare for outside countries, but
the welfare of inside countries might either increase or decrease, although their coordinated
taxes unambiguously increase. Furthermore, Vrijburg and de Mooij (2009) compare between
the case of fully non-cooperative Nash competition and the case where a subset of countries
that implement partial tax harmonization behaves as a Stackelberg leader in the same threecountry model, and shows that when the tax rates of the larger outside country and the tax
union consisting of two smaller countries are strategic substitutes, partial harmonization increases the welfare of the outside country and decreases the welfare of the tax union under
Nash competition, and vice versa under Stackelberg competition. The analysis of Vrijburg
and de Mooij (2009) is not only quite interesting from a theoretical point of view, but also the
first attempt to introduce the Stackelberg competition in the literature on tax coordination
among a subset of countries, since most studies have exclusively employed a simultaneous tax
competition between a tax union and the rest of individual countries. Their framework has
been quite frequently used in the field of international policy coordination such as international environmental agreements (e.g., Kyoto protocol). Indeed, the literature on self-forcing
IEAs examines both of the cases where all countries (signatories and non-signatories) make
their decisions simultaneously (Carraro and Siniscalco 1993), and where the countries that
have ratified the IEAs (signatories) act as a leader, whose decisions proceed the decision of
the countries that remain outside the IEA (Barrett 1994, Rubio and Ulph 2006, Diamantoudi
and Sartzetakis 2006). These two approaches not only have become something of a workhouse
tool to study IEAs, but also offer significantly important policy implications for implementing
a stable IEA. Given this background, it is quite natural and especially relevant to investigate
a tax competition model in which a tax union behaves as a Stackelberg leader or follower.
Nevertheless, there is a usual criticism of Stackelberg equilibrium in that the order of moves
of the players is exogenously given ex-ate. The current paper is to incorporate an endogenous
determination of the timing of moves between a tax union and outside countries in the setting
of repeated tax-competition games.
Our research is closely related to the recent contribution of Kempf and Rota-Graziosi
(2010) which questions the assumption that the governments compete for mobile capital in
2
a simultaneous-move Nash equilibrium in which competing governments simultaneously and
independently select their tax rates, and shows that Stackelberg equilibria are subgame perfect
equilibria (hereafter, SPNEs) in a two-stage timing game involving a pre-play stage in which
the governments commit themselves to move early or late before they choose their capital tax
rates, while simultaneous-move Nash equilibria are not commitment robust (i.e., not SPNEs).
More recently, Ogawa (2013) demonstrates that if capital is owned by the residents of the
countries, the simultaneous-move Nash equilibrium emerges as a SPNE of the above two-stage
game. Although those authors did not consider the interaction between tax coalition groups
and outside countries, the extension we consider could be not only justified by the abovementioned literature on IEAs, but also motivated by the fact that the timing of decision
making is an essential strategic variable pertaining to competing governments that affects the
consequences of fiscal competition in tax competition games involving a coalition group of
countries which harmonizes their fiscal policies.
The purpose of the present paper is to investigate the tax competition between a tax union
(such as the ECA in the EU or the EU in the world economy) which implements tax harmonization and the rest of individual countries. The noteworthy feature of the present analysis
is to employ a heterogenous three-country version of the repeated tax-competition model developed by Itaya et al. (2014). The repeated tax-competition model would be a more relevant
and satisfactory workhouse to explain voluntary and self-enforcing cooperation [see, e.g., Cardarelli, at al. (2002), Catenaro and Vidal (2006), and Itaya et al. (2008, 2014) in more details].
In particular, it should be stressed the fact that national tax authorities always have a strong
incentive to unilaterally deviate from even a Pareto-improving coordinated tax rate in the hope
of reaping gains such as more tax revenues or higher welfare levels; hence, the countries generally end up in failure in implementing such implicit collusion or an explicit agreement for tax
coordination without supernational agency that could enforce it. The maintenance of collusive
agreements, explicitly or implicitly, needs repeated interaction that allows union-members to
punish a deviator from an implicit agreement or a tax union in the future. Another notable
feature of our model is to allow for the pre-play (or communication) stage of the subsequent
repeated tax-competition game in order to endogenously determine the order of moves of a
3
tax union and outside countries. More precisely, the timing of moves in every stage game of
the repeated tax-competition game will be endogenously determined at its outset.
This paper demonstrates the following. First, the capital exporting (importing) tax union
is willing to raise (lower) the price of capital to increase (decrease) the income (payment) from
capital trade. Their opposed incentives to manipulate the price of capital in its favor result
in the simultaneous-move equilibrium of every stage game of the repeated tax-competition
game. In other words, the simultaneous-move outcome prevails in every stage game of the
repeated tax-competition game as its subgame perfect equilibrium if a tax-union consists of
similar countries. Second, Stackelberg or the sequential-move outcome can arise in every stage
game when a tax-union consists of dissimilar countries. Third, when the medium-productivity
country, which is an outside country, leads while the tax union consisting of high- and lowproductivity countries follows, partial tax coordination is most likely to prevail as the SPNE
of the subsequent repeated tax-competition game compared to any other combination of the
countries comprised of a tax union.
The remainder of the paper is organized as follows. Section 2 presents an analytical model
and characterizes its one-shot Nash solution. Section 3 investigates the equilibrium outcomes
when a tax union implementing partial tax harmonization and an outside country move simultaneously, when a tax union behaves as the Stackelberg leader, or when it behaves as the
Stackelberg follower. In section 4, we investigate the likelihood of sustaining tax harmonization among a subset of countries in a repeated tax-competition game. Section 5 concludes the
paper with a discussion on extending our model.
2
The model
There are three countries which are indexed by i = H, M, L. We consider an infinite horizon
model with repeated interactions between a tax union comprised of any two countries and an
outside country (i.e., a remaining country). To make the problem tractable, we assume that
the environment is stationary, so that our dynamic game (or supergame) is reduced to an
infinitely repetition of the one-shot (stage) game. In other words, this stationary assumption
4
eliminates the saving made by households and investment decisions made by firms.1 In each
country there are a continuum of homogenous households which are normalized to be 1; they
are immobile across the countries, while their capital endowments, denoted by k, are identical,
fixed through time and perfectly mobile across the countries. These factors are required in
the production of a single private good. For analytical simplicity, we employ the quadratic
production function: fi (ki ) ≡ (Ai − ki )ki , where Ai represents the level of productivity and
ki is the per capita amount of capital used in country i.2 Without any loss of generality, we
further assume that country H, M and L, respectively, use the different technologies whose
productivities are ordered such as AH ≥ AM ≥ AL , and that Ai > 2ki , i = H, M, L, in order
to ensure a positive but diminishing marginal productivity of capital. Since the government of
country i levies a unit tax on capital at rate τ i , the after-tax profit of firms located in country
i is given by πi = fi (ki ) − (r + τ i )ki in each period, where r is the net return on capital. The
firm’s first-order conditions for profit maximization in country i are
r = fi0 (ki ) − τ i ,
i = H, M, L.
(1)
Since the entire amount of capital in the economy is fixed at 3k through time, the market
for capital is perfectly competitive and hence capital flows across the countries until the net
returns on capital, fi0 (ki ) − τ i , are equalized. The market clearing conditions in each period
are characterized by
3k = kH (r + τ H ) + kM (r + τ M ) + kL (r + τ L ),
1
(2)
If the investment and saving decisions made by economic agents are explicitly introduced into the model,
capital accumulation takes place over time so that the stationary assumption is no longer valid. To eliminate
the saving decision made by households, we may implicitly assume that they live in only two periods. When
young, the household receives wage income. Nevertheless, since it is assumed that the household consumes
only when old, she transfers her whole income from the first to the second period of her life by investing in
either the home or the foreign country. We further assume that firms do not incur any adjustment costs when
undertaking investment activity, their intertemporal profit maximization problems simplify to an infinitely
repetition of a static profit maximization one.
2
This specification has been commonly adopted by Bucovetsky (1991, 2009), Grazzini and van Ypersele
(2003), Kempf and Rota-Graziosi (2010), Ogawa (2013) and Wildasin (1991).
5
where the capital demand function ki (r + τ i ) = (1/2)(Ai − r − τ i ) is obtained by solving (1)
for ki . Further, solving (2) for r, together with ki (r + τ i ) derived above for i = H, M, L, yields
the equilibrium net return r∗ and the equilibrium amount of capital employed by country i,
ki∗ :
r∗ = A − 2k − τ ,
¢
¡
A
−
A
− (τ − τ i )
i
,
ki∗ = k −
2
(3)
(4)
P
P
where A ≡ ( Ai ) /3 and τ ≡ ( τ i ) /3 are the average productivity level and the average
tax rate over all existing countries, respectively. By differentiating (3) and (4), we have
∂kj∗
∂r∗
∂ki∗
1
1
=
= − < 0 and
= > 0, i 6= j.
∂τ i
∂τ i
3
∂τ i
6
(5)
Each household inelastically supplies one unit of labor to the domestic firms and invests its
own fixed capital holdings in the home and foreign countries. Since there are no opportunities
of savings, they spend their entire income on the consumption of the numéraire good ci .
Accordingly, the budget constraint of the household resided in country i is expressed by
ci = wi + r∗ k. To get closed-form expressions for the tax reaction functions of the respective
countries, we need to assume that the utility function of country i’s residents is linear in ci
¡
¢
and gi , i.e., ui (ci , gi ) ≡ ci + gi = fi (ki∗ ) + r∗ k − ki∗ , where gi can be viewed as either a
publicly provided good or lump-sum transfer from the production sector to the households,
and the government’s budget constraint of country i is given by gi = τ i ki∗ .3
The government chooses an optimal tax rate so as to maximize the discounted present
value of total utilities of country i’s residents over an infinite horizon, given the expectation
regarding the behavior of the other governments. Under the stationarity assumption, the
government ends up simply playing an infinitely repetition of the following one-shot stage
game. In the fully noncooperative Nash equilibrium of each stage game, therefore, taking (3),
(4), and the tax rates chosen by the other countries as given, the government of country i
3
The assumption of a linear utility function has been commonly used in the literature on tax competition.
See Bucovetsky (1991, 2009) and Peralta and van Ypersele (2005).
6
independently and individually chooses τ i to maximize the one-period’s utility of country i’s
residents. The first-order condition of country i is as follows:4
∂ui
∂k ∗
∂r∗
= [fi0 (ki∗ ) − r∗ ] i + (k − ki∗ )
= 0.
∂τ i
∂τ i
∂τ i
(6)
By substituting (5) into (6) and rearranging, we obtain the following best-response function
of country i:
τi =
τ j + τ h + (2Ai − Aj − Ah )
,
8
(7)
which reveals that the tax rates of inside and outside countries are strategic complement
(i.e., ∂τ i /∂τ j > 0 for i 6= j). By solving (7) for the tax rates set by all countries, and then
substituting the resulting expressions into (3) and (4), we obtain the one-shot Nash equilibrium
N
N
tax rate τ N
i , the price of capital r , and the amount of capital demand ki , respectively:
=
τN
i
(Ai − Aj ) + (Ai − Ah )
2Ai − Aj − Ah
=
,
9
9
rN = A − 2k,
kiN = k +
(8)
(9)
2Ai − Aj − Ah
= k + τN
i .
9
(10)
It follows from (8) and (10) that a country with the most productive technology (i.e., country
H) imports capital with taxation τ N
H > 0 (since (AH − AM ) + (AH − AL ) > 0), while a country
with the least productive technology (i.e., country L) exports capital with subsidies τ N
L < 0
(since (AL − AH ) + (AL − AM ) < 0). This is the terms of trade effect; i.e., capital importers
(exporters) are willing to levy positive (negative) tax rates on capital in order to lower (raise)
the capital remuneration r∗ in (3). As a result, every country engages in manipulating the
price of capital in its favor thus leading to the zero average tax rate in the Nash equilibrium,
i.e., τ N = 0 (which is confirmed by summing (8) over all i). By making use of (8), (9), and
(10), we show that country i (i = H, M, L) can achieve the following one-period’s utility level
4
In what follows, we restrict the equilibrium to be an interior solution.
7
in every stage game:
uN
i = (Ai − k)k +
3
2 (2Ai − Aj − Ah )2
.
81
(11)
Partial coordination in a stage game
Let G ( {H, M, L} represent a subset of the existing countries; we consider three possible
subsets forming a tax union which implements partial tax harmonization which lasts over an
infinite horizon period, such as G ∈ {{H, M}, {M, L}, {H, L}}.
Since the repeated game we consider can be viewed simply as an infinitely repetition of
an identical stage game, in order to endogenize the order of moves chosen by competing governments we simply add to the repeated tax-competition game an initial stage (which we
may call a “pre-play stage game” or a “timing game”) in which every government simultaneously decides whether to move first or later in every stage game of the subsequent repeated
game.5 More precisely, following Kempf and Rota-Graziosi (2010), the tax union G and the
outside country h both simultaneously determine their timings for the choice of tax rates at
the pre-play stage; hence the strategy set for each country is given by {lead, follow}. If their
announced timings coincide with each other; i.e., either the strategy profile (lead, lead), or
(follow, follow) is announced, a simultaneous-move game between the tax union G and the
outside country h will be played in every stage game, while a sequential-move stage game will
be played if their announced timings are different; i.e., either the strategy profile (lead, follow)
or (follow, lead) is announced and played in the subsequent stage games. The structure of the
extended repeated tax-competition game is summarized as follows:
• Pre-play stage: A tax union and an outside country simultaneously announce at the
pre-play stage whether to set tax rates earlier or later in every stage game of the subsequent repeated tax-competition game.
• Every stage of the repeated game: A tax union and an outside country repeatedly
set taxes in every stage game of the repeated game according to the moves predetermined
5
This stage game corresponds to the subgroup Nash equilibrium analyzed by Konrad and Schjelderup
(1999).
8
at the pre-play stage.
Their announced timings have to be credibly committed. To this end, first, a Nash equilibrium in timing choices is selected at the pre-play stage game. Second, in every stage game
of the subsequent repeated tax-competition game, a tax union and an outside country find it
to be in their best interest to select their strategies according to their announced moves or
knowing when the other chooses its tax rate, which should be comprised of a subgame perfect
Nash equilibrium (SPNE) of the repeated tax-competition game.
3.1
Simultaneous-move stage game
Let us begin with the case where each stage game is a simultaneous-move one between a partial
tax union G consisting of countries i and j.6 Those member countries have agreed to jointly
choose their tax rates in order to maximize the sum of their one-period’s utilities represented
¢
¡
by W (G) ≡ ui + uj = fi (ki∗ ) + fj (kj∗ ) + r∗ kh∗ − k , while the outside country h independently
chooses its tax rate in order to maximize uh period by period.7 The first-order conditions of
/ G are given by
the tax union G with respect to τ i and τ j for i, j ∈ G; h ∈
∗
¢ ∂r∗
∂kj∗
∂W (G)
∂k∗ ¡
0
∗ ∂ki
0 ∗
= fi (ki )
+ fj (kj )
+ r∗ h + kh∗ − k
= 0,
∂τ i
∂τ i
∂τ i
∂τ i
∂τ i
and ∂W (G) /∂τ j = 0, respectively. By substituting (3), (4), and (5) into the above conditions,
the best-response functions of the tax union members are derived as follows:
τi =
2τ j + 2τ h + (Ai + Aj − 2Ah )
, i, j ∈ G; h ∈
/ G,
7
(12)
while τ j is obtained by switching i and j in (12). Those symmetry forms lead to τ i = τ j ≡ τ C ;
namely, the tax union G optimally chooses a common coordinated tax rate. The reason is that
the equalization of the tax rates internalizes the externalities arising from the tax asymmetries
among union-member countries, thus leading to achieving production efficiency within the tax
6
Konrad and Schjelderup (1999) call this equilibrium as a subgroup Nash equilibrium.
It is assumed throughout the paper that country j is a member of the tax union, but country h is not
whenever country i is a member of that tax union.
7
9
union. As a result, the best-response function of the tax union can be expressed by
τC =
2τ h + (Ai + Aj − 2Ah )
.
5
(13)
On the other hand, the outside country h individually and independently chooses its bestreplied tax rate according to (7) with i being exchanged for h and setting τ i = τ j ≡ τ C . By
solving these best-response functions, we obtain the harmonized tax rate for the tax union,
S
τ C (GS ), and τ C
h (G ) for the outside country (the superscript S of G stands for a case where
the tax union G takes a simultaneous-move), the equilibrium net return, rC (GS ) (using (3)),
the amount of capital demanded for each tax-union member, kiC (GS ) for ∀i ∈ G, and khC (GS )
for the outside country h (using (4)), respectively:
Ai + Aj − 2Ah
Ai + Aj − 2Ah
S
, τC
,
h (G ) = −
6
12
Ai + Aj + 2Ah
rC (GS ) =
− 2k,
4
7Ai − 5Aj − 2Ah
Ai + Aj − 2Ah
kiC (GS ) = k +
, khC (GS ) = k −
,
24
12
τ C (GS ) =
(14)
(15)
(16)
It follows from (14) and (16) that the tax union G imports (exports) capital with taxes
(subsidies); i.e., τ C (GS ) > 0 (< 0), while the outside country exports (imports) with subsidies
S
C
S
C
S
(taxes) τ C
h (G ) < 0 (> 0). More importantly, the signs of τ (G ) and τ h (G ) in (14) depend
on the difference in productivity between the members of the tax union and the outside
country; i.e., on the sign of (Ai − Ah ) + (Aj − Ah ). On the other hand, if there are no
productivity asymmetries; i.e., Ai = Aj = Ah , all countries have no incentive to trade capital
S
and thus τ C (GS ) = τ C
h (G ) = 0 obtains (see (14)); consequently, the overall production
efficiency in the economy is always attained without tax harmonization. However, when their
production technologies are heterogeneous (i.e., Ai 6= Aj 6= Ah ), the tax union and the outside
country usually have opposite incentives to manipulate the price of capital in its favor, thus
inducing them to set the tax rates of opposite signs (recall (14)). It is also important to note
that there is another conflict between the tax-union members since the formation of a tax
union leads to force all member countries to accept a common coordinated tax rate, although
10
the most favorable tax rates for the respective members are usually different from this common
tax rate due to the asymmetry of productivity; for example, when countries H and L form a
tax union, capital-importing country H prefers a positive tax rate, whereas capital-exporting
country L prefers a negative tax rate. Then if that tax union were to set a common positive
tax rate (i.e., if (AH − AM ) + (AS − AM ) > 0), capital-exporting country L is forced to make
transfers to country H, and thus incurs losses.
The utility levels of any member i ∈ G and country h are respectively given by
(7Ai − 5Aj − 2Ah ) (11Ai − Aj − 10Ah )
,
576
(Ai + Aj − 2Ah )2
S
.
(G
)
=
(A
−
k)k
+
uC
h
h
72
S
uC
i (G ) = (Ai − k)k +
(17)
(18)
By making use of (11) and (17), the participation constraint for country i (i.e., ∀i ∈ G) is
given by:8
S
N
uC
i (G ) − ui =
3.2
243(Ai − Aj )(Ai − Ah ) − (2Ai − Aj − Ah )(31Ai − 83Aj + 52Ah )
≥ 0. (19)
5184
Tax union acts as a leader
In this subsection, we consider a sequential-move stage game in which a tax union G chooses
its coordinated tax rate before the outside country h has chosen its optimal tax rate. After
observing the tax rate set by the tax union, country h chooses its tax rate in accordance with
(7). In this case, the governments of countries i and j belonging to the tax union solve the
sequence of the following one-period’s maximization problem period by period:
¢
¡
max W (G) = fi (ki∗ ) + fj (kj∗ ) + r∗ kh∗ − k , s.t. (7) for h.
τ i ,τ j
8
S
N
2
In contrast, for the outside country h, we obtain that uC
h (G ) − uh = −7(Ai + Aj − 2Ah ) /648 ≤ 0. This
implies that the welfare of country h unambiguously falls as a result of the formation of tax union, unless
Ai + Aj = 2Ah .
11
The first-order conditions with respect to τ i and τ j are given by
¶
¶
µ ∗
µ ∗
∂kj
∂kj∗ ∂τ h
∂W (G)
∂ki
∂ki∗ ∂τ h
0
∗
0 ∗
+ fj (kj )
= fi (ki )
+
+
∂τ i
∂τ i ∂τ h ∂τ i
∂τ i ∂τ h ∂τ i
µ ∗
¶
¶
µ
¡ ∗
¢ ∂r∗ ∂r∗ ∂τ h
∂kh ∂kh∗ ∂τ h
∗
+r
+ kh − k
= 0,
+
+
∂τ i
∂τ h ∂τ i
∂τ i ∂τ h ∂τ i
and, similarly, ∂W (G) /∂τ j = 0 for i, j ∈ G; h ∈
/ G. By making use of (3), (4), (5), and (7)
for h and rearranging, we obtain the coordinated tax rate, τ i = τ j = τ C (GL ), and the tax
L
rate set by the outside country h, τ C
h (G ), and substituting them into (3) and (4) gives the
equilibrium price of capital, rC (GL ), and the capital demands kiC (GL ) for ∀i ∈ G, and khC (GL )
for the outside country:
3 (Ai + Aj − 2Ah ) C L
Ai + Aj − 2Ah
, τ h (G ) = −
,
14
14
3Ai + 3Aj + 8Ah
rC (GL ) =
− 2k,
14
4Ai − 3Aj − Ah C L
Ai + Aj − 2Ah
kiC (GL ) = k +
, kh (G ) = k −
,
14
14
τ C (GL ) =
(20)
(21)
(22)
where the superscript L of G stands for a sequential-move stage game in which the tax union
G acts as a leader. The one-period’s utility levels of the tax-union members i (i.e., ∀i ∈ G) and
the outside country h, when the tax union G moves first and the outside country h follows,
are given by:
(Ai − Ah ) (4Ai − 3Aj − Ah )
,
28
(Ai + Aj − 2Ah )2
L
.
(G
)
=
(A
−
k)k
+
uC
h
h
98
L
uC
i (G ) = (Ai − k)k +
(23)
(24)
Using (11) and (23), the participation constraints for ∀i ∈ G in this game are given by
L
N
uC
i (G ) − ui =
162(Ai − Aj )(Ai − Ah ) − (2Ai − Aj − Ah )(31Ai − 56Aj + 25Ah )
≥ 0. (25)
2268
12
3.3
Tax union acts as a follower
Finally, consider a sequential-move stage game, where the outside country h leads and a
tax union G follows (which we may call GF ). In this case, the tax union G first sets its
tax rate according to (13), while the leader (i.e., the outside country h) solves the following
maximization problem in every stage game:
¡
¢
maxuh (GF ) = fh (kh∗ ) + r∗ k − kh∗ , s.t. (13),
τh
The first-order condition is
∂uh
= [fh0 (kh∗ ) − r∗ ]
∂τ h
Ã
X ∂k ∗ ∂τ l
∂kh∗
h
+
∂τ h l=i,j∈G ∂τ l ∂τ h
!
¡
¢
+ k − kh∗
Ã
X ∂r∗ ∂τ l
∂r∗
+
∂τ h l=i,j∈G ∂τ l ∂τ h
!
= 0.
Combining the above condition with (3), (4), (5), and (13) yields the coordinated tax rate,
F
C
F
τ C (GF ), the tax rate set by h, τ C
h (G ), the equilibrium capital remuneration, r (G ), and
/ G:
the capital demands, kiC (GF ) for ∀i ∈ G, and khC (GF ) for h ∈
Ai + Aj − 2Ah C F
3 (Ai + Aj − 2Ah )
, τ h (G ) = −
,
8
16
5Ai + 5Aj + 6Ah
rC (GF ) =
− 2k,
16
9Ai − 7Aj − 2Ah C F
Ai + Aj − 2Ah
kiC (GF ) = k +
, kh (G ) = k −
.
32
16
τ C (GF ) =
(26)
(27)
(28)
The corresponding one-period’s utility levels for the tax union members (∀i ∈ G) and the
outside country h when h moves first and the tax union G follows in every stage game, are as
follows:
(9Ai − 7Aj − 2Ah ) (13Ai − 3Aj − 10Ah )
,
1024
(Ai + Aj − 2Ah )2
F
.
(G
)
=
(A
−
k)k
+
uC
h
h
64
F
uC
i (G ) = (Ai − k)k +
13
(29)
(30)
In this game, the participation constraints for ∀i ∈ G when the tax union follows is expressed
by:
F
N
=
uC
i (G ) − ui
1
[81(7Ai + 6Aj − Ah )(11Ai − 2Aj − 9Ah )
82944
−(2Ai − Aj − Ah )(2476Ai − 1319Aj − 1157Ah )] ≥ 0.
3.4
(31)
Comparison of tax rates
Straightforward comparison of the tax rates obtained so far (i.e., (8), (14), (20) and (26))
yields the following lemma:
Lemma 1 The ranking of the tax rates set by the tax union {H, L} and at the Nash equilibrium
is given by
< 0 < τ C (GF ) < τ C (GS ) < τ C (GL ) < τ N
τN
L
H , for εH > εL ,
(32)
< τ C (GL ) < 0 < τ C (GS ) < τ C (GF ) < τ N
τN
L
H , for εH < εL .
The ranking of the tax rates set by the tax union {H, M} and at the Nash equilibrium is given
by
N
C
F
C
S
C
L
0 < τN
M < τ H < τ (G ) < τ (G ) < τ (G ), for εH < εL .
(33)
The ranking of the tax rates set by the tax union {M, L} and at the Nash equilibrium is given
by
N
τ C (GL ) < τ C (GS ) < τ C (GF ) < τ N
L < τ M < 0, for εH > εL .
(34)
Note that as shown in the following subsection, the participation constraints for the
respective tax unions depend on the relative productivity ratio, denoted by εH /εL , where
AH − AM ≡ εH and AM − AL ≡ εL , as shown later. As pointed out by Kempf and RotaGraziosi (2010), under the strategic complementary property of taxes and the positive externalities created by asymmetric tax rates the Stackelberg leader and follower are able to set
their favorable tax rates compared to those at the Nash equilibrium. The Stackelberg leader in
general tends to exploit the gain associated with the "first-mover advantage". More precisely,
14
if εH > εL (εH < εL ), the tax union {H, L} which acts as the Stackelberg leader becomes a
capital importer (exporter), and thus prefers a higher positive (lower negative) harmonized
tax. As indicated by (32), therefore, the tax union can set the second-highest (second-lowest)
tax rate. Similarly, since the tax union {H, M} is a capital importer, it prefers the highest tax
rate so that the Stackelberg leader (=the tax union) is willing to set the highest (see (33)),
while since the tax union {M, L} is a capital exporter, it prefers a lower (negative) tax rate;
consequently, the tax union ends up setting the lowest tax rate (see (34)).
3.5
Timing selection
The payoff matrix depicted in Table 1 represents the payoffs to each player for each combination
of strategies that are chosen at each stage game:
outsider h \ union member i
Lead
Follow
Lead
Follow
S
C
S
uC
h (G ), ui (G )
F
C
F
uC
h (G ), ui (G )
L
C
L
C
S
C
S
uC
h (G ), ui (G ) uh (G ), ui (G )
Table 1. A payoff matrix of each stage game.
To start with, we compare between the utility levels of outside country h in the stage games
with different timing of moves described so far, which yields the following result:
Lemma 2 An outside country is always willing to move first.
Proof. Comparing between (18), (24) and (30) yields:
13 (Ai + Aj − 2Ah )2
≥ 0,
3528
(Ai + Aj − 2Ah )2
C
F
C
S
≥ 0.
uh (G ) − uh (G ) =
576
S
C
L
uC
h (G ) − uh (G ) =
The outside country h has a dominant strategy unless Ai +Aj = 2Ah ; it always has an incentive
to be a leader regardless of the timing chosen by the tax union.
15
Notice that in the degenerate case Ai + Aj = 2Ah , the countries are indifferent to the
different timings of moves.9
From (17), (23), and (29), furthermore, we obtain the following results for all tax unionmember countries i ∈ G:
(Ai + Aj − 2Ah )(37Ai − 35Aj − 2Ah )
R 0,
4032
(Ai + Aj − 2Ah )(179Ai − 109Aj − 70Ah )
F
C
S
R 0.
uC
i (G ) − ui (G ) = −
9216
L
C
S
uC
i (G ) − ui (G ) =
(35)
(36)
Although the signs of (35) and (36) appear to be ambiguous, their signs would be determined
depending on the productivity parameters of all three countries. Before doing this, since the
outside country h always chooses to be a leader (recall Lemma 1), the game GL is never played;
hence, we ignore (35). A further inspection of (36) yields the following lemmas:
Lemma 3 The tax union {H, M} is possible in the simultaneous-move game when εH /εL ≤
√
(72 3 − 106)/83 ' 0.225, while the tax union {M, L} is possible in the simultaneous-move
√
game when εH /εL ≥ (53 + 36 3)/26 ' 4.437 > 1.557.
Proof. Denote AH −AM ≡ εH > 0 and AM −AL ≡ εL > 0 (so that AH −AL = εH +εL > 0).
Using these notations, we can rewrite (36) for the respective members of the tax union {H, M}
as:
(εH + 2εL )(179εH + 70εL )
> 0,
9216
εH
(εH + 2εL )(−109εH + 70εL )
70
S
C
F
R 0, iff
' 0.642. (37)
Q
uC
M (G ) − uM (G ) =
9216
εL
109
S
C
F
uC
H (G ) − uH (G ) =
To make it possible to form the tax union {H, M} in the simultaneous-move game GS , the
S
participation constraints (19) for the union-members {H, M} must be satisfied; i.e., uC
H (G ) ≥
9
However, we do not further investigate the degenerate case where Ai + Aj = 2Ah (i.e., Ai = Aj = Ah )
because in this case the tax rates set by all countries end up being equal to zero in equilibrium so that there
is no need for tax harmonization.
16
C
S
N
uN
H and uM (G ) ≥ uM . Since the second inequality can be rewritten as
S
N
uC
M (G ) − uM =
52ε2L − 212εH εL − 83ε2H
≥ 0,
5184
(38)
√
it is easy to see that (38) holds only when εH /εL ≤ (72 3 − 106)/83 ' 0.225. Since this
upper bound is smaller than the upper bound given by (37), i.e., εH /εL ≤ 70/109 ' 0.642,
S
C
F
the inequality uC
H (G ) > uH (G ) is satisfied. In addition, the first participation constraint for
S
N
country H, uC
H (G ) ≥ uH , is always satisfied, because
S
N
uC
H (G ) − uH =
181ε2H + 316εH εL + 52ε2L
> 0.
5184
Put together, the tax union {H, M} in the simultaneous-move game GS is possible as long as
√
εH /εL ≤ (72 3 − 106)/83 ' 0.225.
√
On the other hand, when εH /εL > (72 3 − 106)/83 ' 0.225, (38) does not hold due to
the above result. Moreover, the participation constraint (31) does not hold also, because
F
N
uC
M (G ) − uM =
−347ε2L − 2060εL εH − 42εH
< 0.
82944
As a result, country M never wants to participate in a tax union regardless of whether the tax
√
union acts as a leader or a follower when εH /εL > (72 3 − 106)/83 ' 0.225. Hence, the tax
union is not possible for this range of εH /εL .
Next, consider the tax union {M, L}. We can show that
εH
(2εH + εL )(109εL − 70εH )
109
R 0, iff
' 1.557,
R
9216
εL
70
(2εH + εL )(179εL + 70εH )
S
C
F
uC
> 0.
L (G ) − uL (G ) =
9216
S
C
F
uC
M (G ) − uM (G ) = −
S
C
F
When εH /εL > 109/70 ' 1.557, it holds that uC
M (G ) > uM (G ), and thus both member
countries are willing to choose simultaneous-moves. Moreover, the participation constraints
17
(19) for the union members M and L also hold, because
S
uC
M (G )
−
uN
M
S
N
uC
L (G ) − uL
√
εH
52ε2H − 212εH εL − 83ε2L
53 + 36 3
≥ 0, iff
' 4.437,
=
≥
5184
εL
26
52ε2H + 316εH εL + 181ε2L
> 0.
=
5184
(39)
Put together, the tax union {M, L} which opts for acting as a leader is possible, when εH /εL ≥
√
(53 + 36 3)/26 ' 4.437(> 1.557).
√
When εH /εL < (53 + 36 3)/26 ' 4.437, however, it is also confirmed that
F
N
uC
M (G ) − uM =
−428ε2H − 2060εL εH − 347ε2L
< 0,
82944
thus implying that country M does not want to participate in a tax union regardless of
whether it acts as a leader or follower. Consequently, the tax union {M, L} is not possible
√
when εH /εL < (53 + 36 3)/26 ' 4.437.
Lemma 4 The tax union G = {H, L} which chooses simultaneous moves is possible when
√
√
(72 3 − 23)/181 ≤ εH /εL ≤ (23 + 72 3)/831, while tax union G = {H, L} which chooses
√
√
sequential moves is possible if (288 11 − 683)/1285 ≤ εH /εL ≤ (683 + 288 11)/347.
Proof. For the tax union {H, L}, the expressions in (36) can be rewritten as follows:
(εH − εL )(179εH + 109εL )
,
9216
(εH − εL )(109εH + 179εL )
S
C
F
.
uC
L (G ) − uL (G ) = −
9216
S
C
F
uC
H (G ) − uH (G ) =
(40)
(41)
Since the above expressions display the opposite signs depending on the difference εH − εL ,
the most preferable timings of the respective members are always opposed except for the case
of εH = εL .
First, suppose that the tax union {H, L} acts a leader, i.e., a simultaneous-move game
arises. Then we have to examine whether the participation constraints (19) and (31) for the
respective members of this tax union are satisfied or not. The participation constraint for
18
country H is given by
S
uC
H (G )
−
uN
H
√
εH
181ε2H + 46εH εL − 83ε2L
72 3 − 23
≥ 0, iff
' 0.562,
=
≥
5184
εL
181
while that for country L is given by
S
uC
L (G )
−
uN
L
√
εH
−83ε2H + 46εH εL + 181ε2L
23 + 72 3
≥ 0, iff
' 1.78.
=
≤
5184
εL
83
√
it is immediate that both participation constraints are simultaneously satisfied, when (72 3 −
√
23)/181 ≤ εH /εL ≤ (23 + 72 3)/83 (i.e., approximately, 0.562 ≤ εH /εL ≤ 1.78). Hence, the
tax union {H, L} in the sequential game is possible for this overlapping range of εH /εL .
Consider the tax union {H, L} which acts as a follower. In this case the participation
constraints for countries L and H are given by
F
N
uC
H (G ) − uH
F
N
uC
L (G ) − uL
√
εH
1285ε2H + 1366εH εL − 347ε2L
288 11 − 683
≥ 0, iff
' 0.212,
=
≥
82944
εL
1285
√
−83ε2H + 46εH εL + 181ε2L
683 + 288 11
εH
=
≤
≥ 0, iff
' 4.721.
5184
εL
347
A close inspection of these two conditions reveal that there exists a common overlapping range
√
in which both participation constraints are simultaneously satisfied as long as (288 11 −
√
683)/1285 ≤ εH /εL ≤ (683 + 288 11)/347 (i.e., approximately, 0.212 ≤ εH /εL ≤ 4.721).
Hence, the tax union {H, L} in the sequential-move game is possible for this range of εH /εL .
To sum up, we have the following important result:
Proposition 1 The tax unions {H, M} and {M, L} take a simultaneous-move, while the tax
union {H, L} takes either a simultaneous-move or sequential-move.
Three remarks are in order. First, when the tax union {H, L} acts as a follower, it is easier
for the participation constraints for the respective union members to be satisfied compared
to those when it acts as a leader (see Lemma 4). In other words, it is less likely for the
participation constraints to be satisfied under the simultaneous-move game, the sequential19
move equilibrium prevails in a wider range of the relative productivity ratio εH /εL compared
to the range in which the simultaneous-move equilibrium emerges. The intuitive reason for
this outcome is as follows. In the simultaneous-move game the positive harmonized tax rate
is higher than that in the sequential-move game, thus leading to a lower net return. The
resulting lower net return raises the utility of capital-importing country H, while it reduces
that of capital-exporting country L. Secondly, it follows from Lemma 2 that the capitalimporting and capital-exporting tax unions {H, M} and {M, L} both prefer simultaneous
moves to sequential moves. In other words, the simultaneous-move equilibrium prevails as
a SPNE of our repeated tax-competition game played by either the tax union {H, M} and
the outside country L or the tax union {M, L} and the outside country H. This result is
consistent with Ogawa (2013) which uses a two-stage game of tax competition. Capitalexporting tax union {M, L} (capital-importing tax union {H, M}) is willing to raise (lower)
the price of capital to augment the income (diminish the payment) from capital trade, while
the outside country wishes to manipulate it in the other direction. These opposed incentives
of the tax union and the outside country toward the price of capital give rise to the only
simultaneous-move equilibrium. Ogawa (2013) finds that when there are no absentee capital
owner, asymmetric countries have opposite incentives to manipulate the price of capital (the
terms of trade effect) in the canonical one-shot tax competition model, so that the only
simultaneous-move equilibrium of a two-stage tax competition game emerges.
In contrast, our result shows that there is a possibility of the sequential-move equilibrium
in a stage game with the tax union {H, L} even if the tax union and outside country have
opposite incentives. In our model, since the tax union {H, L} can become a capital exporter
or importer depending on the degrees of productivity asymmetries measured by εH and εL ,
the most preferable timings of the respective member countries do not coincide each other.
S
C
F
C
S
C
F
As a result, if εH > εL , it follows from (41) that uC
H (G ) > uH (G ), but uL (G ) < uL (G ).
However, it may be possible for country L to join a capital-importing tax union which acts as
a follower, as long as the participation constraints for both union members hold.
20
4
Sustainability
In this section, we investigate under which timing of plays it is more likely to sustain the tax
union {H, L} in the repeated tax-competition game (we relegate the other types of tax union
such as {M, L} and {H, M} to the Appendix). Assume that in every period, the tax union
{H, L} jointly sets a common capital tax rate on condition that the other member follows
it in the previous period. If at least one country deviates from it, then their cooperation
collapses, thus triggering the punishment phase that results in the Nash equilibrium, which
persists forever. Provided that all countries posses a common discount factor δ ∈ [0, 1), the
following conditions must be satisfied so as to sustain cooperation:
δ
1 C m
m
ui (G ) ≥ uD
uN , i = H, L, m = F, S.
i (G ) +
1−δ
1−δ i
(42)
The LHS of (42) is the discounted total utility for a resident in the union-member country
i = H, L, when the tax harmonization supported by the tax union is sustained over an infinite
period, while the RHS of (42) represents the sum of the utility resulting from the deviation in
m
the current period uD
i (G ) and the discounted value of total utility resulting from the Nash
phase in all following periods.
4.1
Sustainability under simultaneous-move
C
S
C
S
In a simultaneous-move stage game, by setting τ C
h = τ M (G ) and τ j = τ (G ) (see (14)), we
S
can obtain the best-deviation tax rate for country i, τ D
i (G ) (using (7)), the resulting capital
remuneration, riD (GS ) (using (3)), the amount of capital demanded, kiD (GS ) (using (4)), and
S
the corresponding one-period’s utility level, uD
i (G ) for i ∈ {H, L}, respectively, as follows:
25Ai − 11Aj − 14AM
,
96
7Ai + 11Aj + 14AM
riD (GS ) =
− 2k,
32
25Ai − 11Aj − 14AM
kiD (GS ) = k +
,
96
(25Ai − 11Aj − 14AM )2
S
.
(G
)
=
(A
−
k)k
+
uD
i
i
4608
S
τD
i (G ) =
21
(43)
Figure 1: Minimum discount factors for G = {H, L} in a simultaneous-move game.
Substituting (11), (17), and (43) into the equality of (42) yields the minimum discount factors
of country i ∈ {H, L}, above which they find it to be in their interest to cooperate:
δ i (GS ) ≡
S
C
S
uD
81 (Ai − 3Aj + 2AM )2
i (G ) − ui (G )
=
,
N
S
(11Ai − Aj − 10AM ) (139Ai − 65Aj − 74AM )
uD
i (G ) − ui
(44)
It is straightforward to show that δ H (GS ) ≷ δ L (GS ) if and only if εH ≶ εL , while δ H (GS ) =
δ L (GS ) = 9/17 when εH = εL . The tax harmonization implemented by the tax union {H, L}
is sustainable as a SPNE of the repeated tax-competition game only if the actual (common)
discount factor of both countries δ is larger than the threshold discount factor defined by
δ ∗ (GS ) ≡ max{δH (GS ), δL (GS )}. It is easy to confirm that δ ∗ (GS ) < 1 holds only when
√
√
(72 3 − 23)/181 < εH /εL < (72 3 + 23)/83 (i.e., approximately, 0.562 < εH /εL < 1.78).
Furthermore, differentiating (44) with respect to εH /εL yields
dδ H (GS )
dδ L (GS )
< 0 and
> 0,
d (εH /εL )
d (εH /εL )
which implies that the locus of the minimum discount factor of country H (L) is sloped
downward (upward). Taken together, we can draw Fig. 1 in which δ H (GS ) < δ L (GS ) holds
for εH /εL > 1, while δ H (GS ) > δ L (GS ) holds for εH /εL < 1. An intuitive explanation is as
follows. The net capital exporting position of the tax union is an importer with a positive
coordinated tax when εH > εL , while that of the tax union is an exporter with a negative
coordinated tax when εH < εL (recalling τ C (GS ) = (εH − εL ) /6)). As a result, the large is
22
the ratio εH /εL when εH > εL , the more productive is the production function of country H
and the more amount of capital -importing country H is to demand (i.e., import), thus leading
to a larger coordinated (positive) tax rate. The increased τ C (GS ) lowers the net return thus
S
C
reducing uC
H (G ). To make clear how a higher τ affects the sustainability of cooperation, we
rewrite the sustainability condition (42) as follows:
¢
δ ¡ C S
S
C
S
ui (G ) − uN
≥ uD
i
i (G ) − ui (G ), i = H, L,
1−δ
which says that the tax union is sustainable as long as the discounted future losses inflicted
by the punishment (i.e., the opportunity cost from deviation) appearing on the left-hand side
should be greater than the immediate gain from deviation appearing on the right-hand side.
C
C
increases the opportunity cost from deviation (i.e.,
Since uC
H increases with τ , a higher τ
N
D
C
C
uC
H − uH ). Although the immediate gain (i.e., uH − uH ) also increases with τ , it can be
verified from (44) that the minimum discount factor of country L falls so that its incentive
to cooperate will be enhanced by increasing the ratio εH /εL (i.e., countries H and L become
more heterogenous), and vice versa for country L. When εH < εL , on the other hand, the
entire results are reversed.
4.2
Sustainability under sequential-move
In a sequential-move stage game, country i = H, L can deviate from their cooperation by
F
setting its best-deviation tax rate τ D
i (G ) given the tax rates set by the other union-member
C
F
j and the outside country h which behaves as a Stackelberg leader. Setting τ C
h = τ M (G ) and
F
τ j = τ C (GF ) in (26) yields the following best-deviation tax rate, τ D
i (G ), the equilibrium net
return, riD (GF ), the amount of capital demanded, kiD (GF ), and the corresponding one-period’s
23
Figure 2: Minimum discount factors for G = {H, L} in a sequential-move game.
F
utility level, uD
i (G ) for i ∈ {H, L}, respectively:
31Ai − 17Aj − 14AM
,
128
35Ai + 51Aj + 42AM
riD (GF ) =
− 2k,
128
31Ai − 17Aj − 14AM
kiD (GF ) = k +
,
128
(31Ai − 17Aj − 14AM )2
F
uD
(G
)
=
(A
−
k)k
+
.
i
i
8192
F
τD
i (G ) =
(45)
Substituting (11), (29), and (45) into the equality of (42) yields the minimum discount factors
of country i ∈ {H, L} for the repeated tax-competition game with sequential-moves as follows:
δ i (GF ) ≡
F
C
F
uD
81 (5Ai − 11Aj + 6AM )2
i (G ) − ui (G )
=
.
N
F
(23Ai − 25Aj + 2AM ) (535Ai − 281Aj − 254AM )
uD
i (G ) − ui
(46)
It is straightforward to show that δ H (GF ) ≷ δL (GF ) if and only if εH ≶ εL , while δ H (GF ) =
δ L (GF ) = 9/17 when εH = εL . If the actual discount factor δ is greater than the threshold
discount factor defined by δ ∗ (GF ) ≡ max{δ H (GF ), δ L (GF )}. It turns out that δ ∗ (GF ) < 1
√
√
holds only when (288 11 − 683)/1285 < εH /εL < (288 11 + 683)/347 (which precisely coincides with the range in which the participation constraints of countries L and H are satisfied);
consequently, their cooperation is sustainable as a SPNE of the repeated tax-competition
game. Differentiation of (46) with respect to εH /εL confirms that dδ H (GF )/d(εH /εL ) < 0 and
dδ L (GF )/d(εH /εL ) > 0. Put together, we can draw Fig. 2 which shows that δ H (GF ) < δ L (GF )
holds for εH /εL < 1, and thus country L has a stronger incentive to deviate than country H
24
Figure 3: Minimum discount factors of G = {H, L} for their partial tax harmonization.
does, while δ H (GF ) > δ L (GF ) holds for εH /εL > 1, and thus country H has a stronger incentive to deviate than country L does for the reason stated before. Fig. 2 further shows
that the larger is the difference between εH and εL (i.e., countries H and L become more
heterogenous), the harder it is to sustain their cooperation.
4.3
Result
Comparing (44) with (46) reveals that δ∗ (GF ) < δ∗ (GS ) for either range of εH /εL except for
the case of δ ∗i (GF ) = δ ∗i (GS ) = 9/17 at εH /εL = 1. This is illustrated in Fig. 3. An inspection
of Fig. 3 immediately reveals the following results for the behavior of the tax union {H, L}:
Proposition 2 In the repeated tax-competition game for the tax union {H, L},
(i) there exists a SPNE in which the coordinated tax rate is sustained if countries H and L
are sufficiently patient;
(ii) as asymmetry between εH and εL becomes smaller (i.e., εH /εL → 1), it is easier to
sustain the partial tax harmonization for both simultaneous and sequential-moves, whereas as
asymmetry between εH and εL becomes larger, the opposite result holds; and
(iii)when the tax union acts as a follower, it can sustain tax harmonization in a wider range
25
of the asymmetry between εH and εL compared to the tax union which plays the simultaneousmove game except for the case of εH = εL .
To understand the economic logic behind Proposition 2 (see also Figs. 1 and 3), we suppose
that the ratio εH /εL is increased (maintaining εH > εL ). In this range, the tax union {H, L}
imports capital from country M regardless of the timing of moves. As seen from (14) and
(26); i.e., τ C (GS ) = (εH − εL )/6 > 0 and τ C (GF ) = (εH − εL ) /8 > 0, the tax union will
choose a positive harmonized tax rate in order to reduce the capital payments borne by the
capital-importing tax union. However, country L is frustrated by the positive harmonized
tax rate, because low-productivity country L which is a capital-exporting country is forced to
make income transfers to country H as a result of a positive tax rate (recall that a capitalexporting country prefers a negative tax rate). As the ratio εH /εL becomes larger, the union
wants to import more capital from the outside country M, which induces the union to set a
higher harmonized tax rate. As a consequence, country L will be more frustrated, thereby
discouraging the incentive of country L to cooperate. In other words, the more heterogenous
are the productivities of countries, the more fragile is the sustainability of the tax union.
Conversely, the more homogenous (i.e., the ratio εH /εL approaches 1) are the productivities of countries, the more robust is the sustainability of the tax union, while capital-importing
(high-productivity) country H will be frustrated by the negative tax rate, because of higher
interest payments. Accordingly, country H’s incentive to deviate is enhanced as the ratio
εH /εL becomes smaller, leading to a more negative harmonized tax rate. It is, therefore,
concluded that the likelihood of sustainability for their tax harmonization relies crucially on
the degree of asymmetry, as in Itaya et al. (2008).
Why does the tax union which acts as a follower can sustain tax harmonization in a
wider range of εH /εL compared to the tax union which plays in the simultaneous-move
game? The reason is as follows. When εH > εL , straightforward comparison of (15) with (27)
reveals that rC (GF ) > rC (GS ); hence, the capital-importing tax union as a whole prefers a
lower capital price rC (GS ) and thus the simultaneous-move outcome. Nevertheless, the union
member country L (a capital-exporter), which is a key player in the sense of determining
the size of the critical discount factor, prefers moving late, not only because its capital
26
Figure 4: Minimum discount factors of the respective coutries under simultaneous-move games.
reward from capital trade in the sequential-move game is larger than in the simultaneous£
¤
£
¤
move game; i.e., rC (GF ) k − kLC (GF ) > rC (GS ) k − kLC (GS ) ,10 but also because the above
positive effect on capital remuneration (i.e., the terms of trade effect) associated with moving-
late overweighs its decreased domestic production (recall that kLC (GF ) < kLC (GS ) caused by
rC (GF ) > rC (GS ) leads to f (kLC (GF )) < f (kLC (GS ))). The resulting net impact unambiguously
deters the incentive of country L to deviate from tax harmonization when the tax union acts
as a follower. This is manifest in the observation of δ L (GF ) < δ L (GS ) for εH /εL > 1 in
Fig. 3. In contrast, country H prefers a simultaneous-move, because the increased domestic
production overweighs higher capital payments.11 This results is manifest in the observation
of δH (GF ) > δ H (GS ) in Fig. 3. Hence, as illustrated in Fig. 3, the sustainability of the
capital-importing tax union when εH > εL is dictated by the incentive of the capital exporter
L. Similarly, due to the terms of trade effect, capital-importing country H is more beneficial
when the tax union is a capital-importing one than when it is a capital-exporting one because
of the higher price of capital which prevails in the simultaneous-move game. This is shown as
δ H (GF ) < δ H (GS ) for εH < εL in Fig. 3.
10
C
C
This is confirmed from that rC (GF ) − rC (GS ) = (εH − εL )/16£ and kL
(GS )¤ − kL
(GF ) =
εL )/96.
£ (εH −
¤
11
C
C
C
C
This is shown by that kH
(GS ) > kH
(GF ) and rC (GF ) k − kH
(GF ) − rC (GS ) k − kH
(GN ) =
£
¤
(εH − εL ) 23εH + 25εL + 16(AM − 2k) /1536 > 0.
27
As seen from Fig. 4, which illustrates the minimum discount factors of the member countries consisting of all possible partial tax unions, it is found that there is no range of εH /εL
that is consistent with any partial tax harmonization (see also the Appendix). Although comparing Fig. 3 with Fig. 4 reveals that there always exists a certain range for εH /εL in which
no partial tax coordination is formed if the tax union {H, L} were not to allow for choosing
late. In other words, if the timing of moves is exogenously fixed, this may mislead to conclude
not only that no tax coordination is impossible for some range of the degree of asymmetry,
but also that the presence of asymmetry, together with the exogenously determined timing of
moves, may prevent from the formation of tax coordination. However, if the timing of moves
is endogenously determined, it is always possible to form some partial tax union depending on
values of εH /εL .
The sustainability of partial tax harmonization crucially depends on the productivity asymmetry between countries H and M relative to that between countries L and M, i.e., the ratio
(εH /εL ) (= (εH /εM )/ (εL /εM )). More precisely, the similar the ratio between these two asymmetries (i.e., εH /εL → 1), the more likely is the partial tax harmonization excluding country
M to be sustained. In contrast, if the median country M were to join the tax union, it always
has a stronger incentive to deviate compared to the other member country such as the high or
lower-productivity country because country M is always forced to make income transfers to
its partner as a result of the common harmonized tax rate, while when country M is not in
the union, the closer its productivity to the average productivity level, the less the amount of
trading and, consequently, the less the amount of income transfers between the union-member
countries H and L.
Most notably, the sustainable range for εH /εL can be enlarged when the tax union behaves
as the Stackelberg follower than it does as the Stackelberg leader. This is because the Stackelberg follower cannot exploit “the first-mover advantage”, since it (=the tax union) could
not choose the most favorable tax rate (recall Lemma 1). Although this choice places the tax
union that follows at a disadvantage relative to the tax union which leads, it would improve
the position of an unfavorable partner, which is a potential deviator from tax harmonization
within the tax union, in a way that reduces income transfers from a favorable country to an
28
unfavorable one. This reduction mitigates the losses borne by the unfavorable partner in the
tax union, thus deterring the incentive of deviation.
5
Concluding remarks
This paper reveals that the sustainability of partial tax coordination in a repeated taxcompetition model is affected by the endogenous timing of moves. It shows that allowing
the pre-play stage of the repeated tax-competition game yields a possibility of the sequentialmove equilibrium even if a stage game of our repeated game has the same structure as that
of the standard static tax competition model such as Ogawa (2003). When the preferences of
tax union members toward the price of capital are completely opposite, it is impossible that
their most preferable timings cannot coincide each other. We show that when the mediumproductive country leads and the tax union consisting of high- and low-productive countries
follows, partial tax coordination is more likely to prevail as the SPNE of the repeated taxcompetition game. Ogawa (2013), on the other hand, stresses the importance of ownership
of capital, while Eichner (2014) introduces heterogenous preferences for public goods in a tax
competition model. The results of our paper indicate not only that some of their results do
not survive over to our model implementing partial tax harmonization in a subset of countries,
but also that we would provide another important channel to essentially affect the timing of
moves chosen by the tax union implementing partial tax harmonization. The key difference
is that in our setting we have to impose the participation constraints for potential member
countries to guarantee the formation of a tax union implementing tax harmonization.
These results have quite important policy implications. The first implication is that the
insights obtained from the two-stage timing game of tax competition such as Kempf and
Rota-Graziosi (2010) and Ogawa (2013) may not survive to the repeated tax-competition
model between a tax-coalition group and outside countries. In particular, unlike the results of
those authors, our result indicates that a tax union tends to resist against becoming a first
mover in tax competition games, which appears to be counter-intuitive. As the most likely
scenario, the ECAs are expected to move the first on the ground that one would expect the
29
ECAs to take a leadership for establishing full tax harmonization among EU’s member countries. Unfortunately, our paper indicates that this optimistic expectation is never voluntarily
materialized; accordingly, other motivations, such as political motivations, are called upon for
the ECA to play a role of Stackelberg leadership in establishing comprehensive reforms for
the cooperate tax systems of EU Member States, such as a so-called Common Consolidated
Corporate Tax Base for European multinational enterprises.
The results obtained in this paper critically rely on the restrictive structure of the present
model; e.g., a linear utility function and a quadratic production function in a three-country
setting. To ascertain the robustness of our results, we have to conduct the same analysis
under more general functions and/or more than three countries. Due to its complexity, we
need to resort to a numerical analysis. In particular, it is quite interesting to check the
robustness of our results in a model featuring an arbitrary number of heterogenous countries.
It also deserves further study to investigate whether our conclusion is still robust or not in a
heterogenous model with respect to population size, initial capital endowments or valuation
of local pubic goods.
Appendix
The minimum discount factors of country i, j ∈ G = {H, M} or {M, L} are as follows:
δ i (GS ) =
S
C
S
uD
81 (Ai − 3Aj + 2Ah )2
i (G ) − ui (G )
.
=
N
S
(11Ai − Aj − 10Ah ) (139Ai − 65Aj − 74Ah )
uD
i (G ) − ui
Using the notations AH − AM = εH and AM − AL = εL as before and differentiating δ i (GS )
with respect to the ratio εH /εL yields:
∂δ H (HM S )
εH
εH
∂δ M (HM S )
46
R 0 if
R 0 if
R 2, and
Q ,
∂ε
εL
∂ε
εL
19
S
S
∂δ M (ML )
εH
∂δ L (ML )
εH
19
1
R 0 if
R 0 if
Q , and
R .
∂ε
εL
46
∂ε
εL
2
√
By making use of the results that δ H (HM S ) < δ M (HM S ) ≤ 1 for εH /εL ∈ (0, 2(36 3 −
53)/83] ' 0.225, δ H (HM S ) = δ M (HM S ) = 81/185 at εH /εL = 0, 1 ≥ δ M (LM S ) > δ L (MLS )
30
√
for εH /εL ∈ [(36 3+53)/26, ∞) ' 4.437, and δ M (MLS ) = δ L (MLS ) = 81/185 as εH /εL → 0,
we obtain Fig. 4.
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