Discussion Papers In Economics
And Business

The Direction of Strategic Delegation and Voter Welfare in
Asymmetric Tax Competition Models

Yukihiro Nishimura and Kimiko Terai

Discussion Paper 16-27

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Direction of Strategic Delegation and Voter Welfare in
Asymmetric Tax Competition Models

Yukihiro Nishimura and Kimiko Terai

Discussion Paper 16-27

November 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Direction of Strategic Delegation and Voter Welfare in
Asymmetric Tax Competition Models∗
Yukihiro Nishimuraa†and Kimiko Teraib‡aGraduate School of Economics, Osaka University, 1-7 Machikaneyama-cho, Toyonaka-shi, Osaka
560-0043, Japan.
bFaculty of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan

November 7, 2016

Abstract
This paper examined a political process and economic consequences of tax competition
among asymmetric countries. Citizens are endowed with heterogeneous capital incomes. The
median-voters deliberately elect a delegate whose preferences diﬀer from their own, to pursue
advantages in the international tax competition. When the countries have diﬀerent productivity of the capital, the country with a low capital productivity may delegate the tax authority
to a citizen who is richer than the median-voter. As a result, the outcome through strategicdelegation may make the median-voters and the majority of citizens worse-oﬀ than the selfrepresentation outcome, contrary to the previous studies with symmetric countries. Similar
results are obtained when the countries diﬀer in capital endowments. In contrast, when the
countries diﬀer in population size, productive ineﬃciency is reduced, and the median-voters are
better-oﬀ through strategic delegation. We also point out that the role of campaign promises isimportant for the equilibrium tax rates and citizens’ welfare.
Keywords: Capital tax competition; Strategic delegation; Asymmetric countries; Voter welfare
JEL classiﬁcation: C72, D72, D78, H23, H87.
∗
The previous version of this paper (Nishimura and Terai (2013)) was presented at the 2013 Annual Meeting of
the European Public Choice Society, the 2012 Japan Institute of Public Finance, and the workshops in University ofTokyo and Nagoya University. The authors wish to thank Ivo Bischoﬀ, Jin Yeong Kim, Michihiro Kandori, Hikaru
Ogawa and Mutsumi Matsumoto for their useful comments. The usual disclaimer applies. Nishimura acknowledges
the ﬁnancial support from the Grants-in-Aid for Scientiﬁc Research (C) from the Ministry of Education, Culture,
Sports, Science, and Technology (24530348, 15K03511) and Strategic Young Researcher Overseas Visits Program for
Accelerating Brain Circulation from Japan Society for the Promotion of Science (J2402). Terai acknowledges theﬁnancial support from the Grants-in-Aid for Scientiﬁc Research (C) from the Ministry of Education, Culture, Sports,
Science, and Technology (23530382) and the Research Grant from Zengin Foundation for Studies on Economics andFinance.
†
Corresponding author. Tel & fax: +81 (6)-6850-5230. E-mail: ynishimu@econ.osaka-u.ac.jp.
‡
E-mail: kterai@econ.keio.ac.jp.

11Introduction

Globalization in recent years provoked more attention to the nature of international tax competition(see, e.g., Keen and Konrad (2013) for a recent review). In spite of the trend of lowering corporate
income taxes for acquiring mobile capitals, there are several factors to be taken into account. First,
corporate income taxes are widely based on the source principle, because it is convenient for thehost country to keep the primary taxing rights to the foreign investors.1 Second, in the presenceof inequality of capital incomes, conventional political-economy analysis clariﬁes that motive forincome redistribution matters. Using the standard tax-competition framework with intra-national
inequality of capital income, this paper examines international capital allocations, the tax gap andcitizens’ welfare among asymmetric countries.
Countries noncooperatively decide on domestic tax policies which aﬀect international capital
ﬂows. Therefore, in electing the policy-makers, the median-voters have to take economic andpolitical eﬀects into account. We assume that countries are asymmetric, either in terms of capital
productivity, or population size, or capital endowments, so the preferred tax rate diﬀers acrosscountries. Therefore, the ﬁscal competition distorts productive eﬃciency. Domestic politics ischaracterized by the citizen-candidate model where citizens are endowed with heterogeneous capitalincomes. We use the strategic delegation model (e.g., Persson and Tabellini (1992), Segendorﬀ(1998)). The median-voters may deliberately elect a delegate whose preferences diﬀer from his/herown, to aﬀect decision-making of the other country. Alternatively, the median-voters can electthemselves, which is called self-representation, following Segendorﬀ (1998). The comparison of theoutcome under strategic delegation with that of self-representation is our main interest. Analyzing
the interaction between tax competition and the direction of strategic delegation, we show that thepolitical competition through strategic delegation may either aggravate or mitigate the productive
ineﬃciency, depending on the type of asymmetry under consideration.
When the countries have diﬀerent productivity of the capital, the country with a high capital
productivity delegates the tax authority to a citizen who is poorer than the median-voter, aiming
to gain higher tax revenue from the foreigners. However, the other country elects the delegate
who is richer than the median-voter, when the disparity in productivity is suﬃciently large, or
the inequality of capital-income distribution2 is not so high (Proposition 1). Since the direction of
the preferred tax rates diverges with suﬃciently large asymmetry, the outcome obtained throughstrategic delegation may make median-voters (and the majority of citizens) worse-oﬀ than the
self-representation outcome (Proposition 2). These ﬁndings diﬀer from previous results based onsymmetric countries. The reason is because the ineﬃciency of capital allocation due to the tax
competition —which is the focus of the asymmetric tax-competition models— is worsened by thestrategic delegation.
1
A higher foreign ownership share will generally rationalize higher source-based capital income taxes (Huizinga
and Nielsen (1997), Huizinga and Nicodeme (2006)).
2
Here, the income inequality is according to the diﬀerence between the median and the mean capital endowments(the skewness of the distribution), as in Meltzer and Richard (1981).

2

In contrast, when the countries diﬀer in population size (Bucovetsky (1991), Wilson (1991)),
both countries elect the delegate who wishes for higher taxes on capital than under self-representation,and the tax gap is reduced as a result of the strategic delegation. Since productive ineﬃciency is
reduced, the median-voters are unambiguously better-oﬀ through strategic delegation (Propositions
3 and 4). When the countries diﬀer in capital endowments (Hwang and Choe (1995), Peralta andvan Ypersele (2005)), the country with the higher aggregate capital endowment, as the capitalexporting country, may delegate to a richer citizen than the median-voter. However, in this casethere explicitly arises the degree of income inequality above which the direction of strategic delegation is unambiguously left to the median-voter for both countries (Propositions 5 and 6).
In the literature of the international decision-making following Persson and Tabellini (1992),
strategic delegation in the domestic politics (representative democracy) is taken for granted. Instead, we highlight the role of campaign promises, which was not much emphasized in this literature.
If the candidates in each country are committed to the most preferred tax rate at the time of election, the structure of the game is basically one-shot decision-making over the preferred taxes ineach country, and there is no scope for strategic delegation. If, on the other hand, the candidates
do not make binding commitments to the campaign proposal, the game has two stages: voting by
citizens according to the candidates’ type, and ex post decision-making by an elected politician. In
this setting, the scope for strategic delegation arises (Lemma 3). However, in reality, the elected
politicians often fulﬁll their campaign promises, even if they can seek strategic advantages by not
making the promises (see Section 5). Our results suggest that the commitment to the campaign
promises, which brings the self-representation outcome, is sometimes not only plausible but also
desirable among asymmetric countries.
In this paper, unlike Persson and Tabellini (1992) and Ihori and Yang (2009), we consider various
asymmetries among countries to examine international capital allocations, the tax gap and citizens’
welfare.3 On the other hand, the development of asymmetric tax competition models includes
Bucovetsky (1991, 2009), Wilson (1991), Hwang and Choe (1995), Grazzini and van Ypersele (2003),
Peralta and van Ypersele (2005), and Hindriks and Nishimura (2016). The main focus of these
papers is the possibility of tax coordination such as the minimal tax, as well as the welfare eﬀects
of the Stackelberg tax leadership. However, none of these papers examined strategic delegation.4
Nevertheless, our results are in line with previous results. For example, as the citizens’ preference
towards the Stackelberg tax competition resembles citizens’ motive for strategic delegation, our
Proposition 2 has an analogy to Hindriks and Nishimura’s (2016) results. Also, since strategic3

Moreover, the focus of Ihori and Yang (2009) was the optimal number of competing countries consistent withthe optimal provision of public goods. As such, strategic delegation is not indispensable in their model: their results
can be replicated in a model with self-representation. Instead, we explicitly compare strategic delegation and selfrepresentation outcomes. Gottschalk and Peters (2003) used a diﬀerent model that analyzed the case where countries
have diﬀerent skewness of capital incomes. We can readily show the extension of our model to such a case.
4
Exceptions are Ogawa and Susa (2014) and Pal and Sharma (2013). However, we consider a wider variety ofasymmetries than Ogawa and Susa (2014), and we examine the capital allocation, the tax gap and citizens’ welfare.
Pal and Sharma (2013) considered a diﬀerent kind of delegation (namely, the choice of the objective functions), sothat the direction and the nature of the strategic delegation are diﬀerent from ours.

3

delegation is driven by the terms-of-trade eﬀect, it mitigates the eﬀect of the tax-base-elasticity
eﬀect in Bucovetsky (1991, 2009) and Wilson (1991) in the asymmetric population model.
Section 2 describes the model. Section 3 solves the equilibrium under strategic delegation.
Section 4 presents our main results. Section 5 discusses the role of electoral commitment. Section
6 concludes. The proofs of several propositions and lemmas are provided in the Appendix.

2

The Model

There are two countries, denoted by 1 and 2. Let K i and Li (i = 1, 2) denote, respectively, theKibe the per-capita capital endowment.
capital and the labor endowment in each country. Let k i ≡LiLater we introduce possible asymmetries with respect to population size (L1 > L2 and k 1 = k 2 ) or
the capital-labor ratio (k 1 > k 2 and L1 = L2 ). The former is analyzed by Bucovetsky (1991) andWilson (1991) where country 1 has higher population, and the latter is analyzed by Hwang andChoe (1995) and Peralta and van Ypersele (2005) where country 1 has a higher capital endowment.
Capital is perfectly mobile across borders while labor is perfectly immobile. Hereafter we referKi
country i’s capital in its per-capita term ki ≡. The production in country i is deﬁned in its perLi
capita form by the function fi (ki ), with fi (0) = 0 and fi0 (ki ) > 0 > fi00 (ki ). To make our analysis
more explicit, we use later the linear model with the quadratic production function fi (ki ) = ai ki −ki2with a1 ≥ a2 .
Country i sets taxes on capital denoted by ti . Under capital mobility, the arbitrage condition
involves:
f10 (k1 ) − t1 = f20 (k2 ) − t2 = r,(1)
where r is the price of capital.
Let si (i = 1, 2) be the share of the population in country i. Namely, s1 + s2 = 1, where weassumed s1 ≥ s2 . The market clearing condition is:5
s1 k1 + s2 k2 = s1 k 1 + s2 k 2 .
Diﬀerentiating (1) and (2) yields

(2)

sj
∂kj
∂kj
∂ki
si ∂ki∂r=< 0,=−> 0 and= fj00=00
00∂ti
sj fi + si fj∂ti
sj ∂ti∂ti∂ti

si ∂ki< 0 (i, j = 1, 2, j 6= i).
sj ∂ti
Residents within each country have heterogeneity with respect to the share of capital endowments, which is represented by θ, and a resident indexed by θ owns θk i units of per-capita capital. The θ is distributed over the interval (0, θ) with a distribution function Φi (·) such that
−fj00

5

K1 + K2 = K 1 + K 2 ⇐⇒

K1
K2
K1
K2
L1 +
L2 =
L1 +
L2 . Li = si L (i = 1, 2) for the total population L,
L1
L2
L1
L2

which derives (2).

4∫θ0 θdΦi (θ) = 1. We assume that each country has the right-skewed distribution of capital endowments, with the common median θm < 1.

The local government i provides an equal amount of lump-sum transfer γi to each citizen, which
is solely ﬁnanced by taxation on capital. The government’s balanced-budget constraint is given by
γi = ti ki .
A citizen in country i with the capital share θ receives (i) labor income fi (ki ) − fi0 (ki )ki , (ii)rent from capital rθk i and (iii) γi = ti ki . Namely,ui = fi (ki ) − fi0 (ki )ki + rθk i + ti ki= fi (ki ) + r(θk i − ki ),(3)

where the second equality uses the arbitrage condition fi0 (ki ) = r + ti .
Events in the model unfold as follows. In Stage 1, simultaneously in both countries a policymaker (delegate) is elected under majority rule. In Stage 2, each delegate i simultaneously andindependently chooses ti . In Stage 3, having observed (t1 , t2 ), private investors in both countries
make their investment decisions, and productions take place.
∑ Before we proceed, we ﬁrst present an eﬃciency benchmark where the central planner maximizes
i fi (ki )si L subject to (2). The ﬁrst-order condition of the interior optimum with respect to kiyields this ﬁrst-best allocation of capital, denoted by (k1 , k2 ) = (k1∗ , k2∗ ), implicitly deﬁned as follows.
f10 (k1∗ ) = f20 (k2∗ ), s1 k1∗ + s2 k2∗ = s1 k 1 + s2 k 2 .

(4)

The two important departures of our political-economic framework from the ﬁrst-best solution are:(i) the presence of the capital-tax competition in (1); and (ii) the absence of individualized lumpsum transfers in (3). The possible diﬀerence in tax incentives that come from attracting foreign
capital and income redistribution may result in t1 6= t2 where (1) implies f10 (k1 ) 6= f20 (k2 ).

3Equilibrium

3.1

Stage 2: Policy-maker’s Choice of Tax Rate

In this section, we solve our political-economy game backward. Note that the equilibrium conditionsof Stage 3 (the capital-market equilibrium) are characterized by the arbitrage and the market
clearing conditions in (1) and (2).
Let us represent a policy-maker in country i by θi . In Stage 2, given θi and θj , each delegate
selects the tax policy (ti for country i). The policy-maker with θ = θi maximizes his/her utility
ui in (3) by choosing ti , taking account of (1) and (2) in the subsequent stage. The ﬁrst-order5

condition for country i’s policy-maker is:
∂ui
∂ti

∂ki
∂r
+ (θi k i − ki )
∂ti
∂ti
∂ki
∂r
= ti
+(θi k i − ki )
= 0.
∂ti
∂ti
= (fi0 (ki ) − r)

(5)

Given (θi , θj ), this expression deﬁnes implicitly the best response function of country i to the tax ofcountry j: ti = τi (tj ; θi ) (i = 1, 2), which yields the Stage-2 tax rates (t1 , t2 ) = (t1 (θ1 , θ2 ), t2 (θ1 , θ2 )).
The following properties are shown in Appendix 1:
Lemma 1 Suppose that the production function exhibits the form of fi000 (ki ) = 0 (linear model).
∂τi (tj ; θi )∂ti (θi , θj )Then (i)∈ (0, 1). (ii)
< 0: country i’s policy-maker with a higher capital∂tj∂θi∂tj (θi , θj )ownership will choose lower tax rate ti . (iii)< 0: the election of country i’s policy∂θimaker with a higher capital ownership will induce lower tax rate tj .
Part (i) is the conventional strategic complementarity where the reaction function is upward sloping,which is consistent with empirical studies (see, e.g., Brueckner (2003) for an overview). Part (ii) is
intuitively clear. Part (iii) comes from the following formula derived in the Appendix 1:∂tj∂ti ∂τj=.
∂θi∂θi ∂ti

3.2(6)Stage 1: The Median Voter’s Choice of A Policy Maker

The utility function (3) is linear in the type parameter θi which is distributed on the one-dimensional
space. It thus belongs to the class of intermediate preferences, studied by Grandmont (1978) and
Persson and Tabellini (2000). Then we can regard the median-voters as pivotal in selecting thetype of a policy maker since his/her most preferred type is a Condorcet winner. Country i’s
median-voter’s problem is given bymax
θifi (ki ) + r(θm k i − ki )s.t. (1), (2), and ti = ti (θ1 , θ2 ) (i = 1, 2).

(7)

The delegate may be any citizen. A special case is when the decisive voter elects himself/herself,
which is called self-representation, following Segendorﬀ (1998). Let um
i be the utility of country i’s

6median-voter. The ﬁrst-order condition with respect to θi is given by∂um ∂tj
∂um
i ∂ti
+ i
∂ti ∂θi∂tj ∂θi
[
[
]
]
∂tj
∂ki
∂r m
∂ki ∂r m∂ti+ (θ k i − ki )+ ti+
(θ k i − ki )= ti∂ti ∂ti
∂θi
∂tj
∂tj
∂θi
(
)
∂τj ∂r ∂ti∂τj ∂tim− (θm k i − ki )= 0,= (θ − θi )k i 1 −∂ti ∂ti ∂θi∂ti ∂θi(8)

where we made use of (5) and (6).6
The ﬁrst term and the second term in the second line of (8) show the direct eﬀect and thestrategic eﬀect respectively. Under self-representation, the direct eﬀect is proportional to the policymaker’s FOC in (5), which is zero. However, the strategic eﬀect gives scope for(the possibilityof θi 6=
)
∂τ
∂r
∂t
j
i
θm , i.e., strategic delegation. In the context of capital-tax competition, since 1 −>0∂ti ∂ti ∂θi∂τj ∂tiand< 0, we have:∂ti ∂θi
θi ≶ θm ⇐⇒ θm k i − ki ≶ 0.

(9)

To interpret (9), the following decomposition is useful:θm k i − ki = (θm − 1)k i + (k i − ki ).

(10)

The ﬁrst term of the right-hand-side, θm k i − k i , which is negative by θm < 1, represents themedian-voter’s tax incentives (see, e.g., Meltzer and Richard (1981, equation (13))).
The second term of the right-hand-side in (10) is negative for a capital-importing country(k i < ki ) which, other things being equal, beneﬁts from taxing the capital. The capital-exporting
country (k i > ki ) beneﬁts from subsidizing the capital.7 We show in the next section that the direction of capital importing/exporting in the equilibrium is determined by asymmetry of exogenous
parameters.
Therefore, the capital-importer sends a delegate who, as a politician, wishes to tax higher thanunder self-representation. The capital-exporter has an oﬀsetting incentive so that the direction ofstrategic delegation needs to be analyzed.
∂ki∂ki∂r ∂ti
From (5), the ﬁrst term of the second line of (8) becomes (θm − θi )ki. Since=−, the second
∂t
∂θ
∂t
∂ti
i
i
j
»
–
∂r
∂r m
∂tj
∂r
∂r
term of the second line becomes(θi ki − ki ) +(θ k i − ki ). Moreover,+= −1. Using (6) and
∂ti
∂tj
∂θi
∂ti
∂tj
re-arranging yield the third line of (8).
7The capital importer (exporter) beneﬁts from lower (higher) interest rate (see the second term of (3)). Other
things being equal, ∂r/∂ti < 0 derives the preference to the capital taxation. This is called the terms-of-trade eﬀect.
6

7

Solving the system of (8) for i = 1, 2 yields the equilibrium choice of the delegate θi (i = 1, 2)
as a function of θm , denoted by θig . Substituting these to the Stage-2 tax rates and the Stage-3
capital allocations, we obtain ti (θ1g , θ2g ) and ki (t1 (θ1g , θ2g ), t2 (θ1g , θ2g )) ≡ ki (θ1g , θ2g ) (i = 1, 2). From
g g
g g
g g
g g
m
(1) and (3) we deﬁne umi (θ1 , θ2 ) ≡ fi (ki (θ1 , θ2 )) + r(θ1 , θ2 ) · (θ k i − ki (θ1 , θ2 )) as the equilibrium
welfare of country i’s median-voter. The comparison of these values with corresponding values
under self-representation (where θi = θm , i = 1, 2) is our main interest.

3.3

Self-representation vs. Strategic Delegation: A Discussion

g g
We now discuss several reasons why we compare the equilibrium utility (umi (θ1 , θ2 )) with that of
m
m
m
m
m
m
m
m
m
m
self-representation (ui (θ , θ ) ≡ fi (ki (θ , θ )) + r(θ , θ ) · (θ k i − ki (θ , θ ))).
The ﬁrst reason is as follows. Let ui (θ1 , θ2 ; θ) ≡ fi (ki (θ1 , θ2 )) + r(θ1 , θ2 ) · (θk i − ki (θ1 , θ2 )). In
Appendix 2 we show the following:
g g
g g
m m m
Lemma 2 Suppose that r(θm , θm ) > r(θ1g , θ2g ). Then (i) If umi (θ1 , θ2 ) > ui (θ , θ ), then ui (θ1 , θ2 ; θ) >
g g
g g
m
m
m
m
m
m
m
ui (θ , θ ; θ) for all θ ≤ θ . Moreover, (ii) if ui (θ1 , θ2 ) < ui (θ , θ ), then ui (θ1 , θ2 ; θ) <ui (θm , θm ; θ) for all θ ≥ θm . Namely, if country i’s median-voter is made better-oﬀ (worse-oﬀ )under strategic delegation than under self-representation, so is any citizen poorer (richer) than the
median-voter.

The premise of the lemma, r(θm , θm ) > r(θ1g , θ2g ), holds in all the models in Section 4. Therefore, themedian-voter’s welfare is a useful reference for the overall welfare implication of strategic delegtion.
The second reason to analyze the self-representation outcome is to highlight the role of campaign
promises, which was not much emphasized in the literature. If the policy-makers in each country
are committed to the most preferred tax rate, then the structure of the game is basically oneshot decision-making over the preferred taxes in each country. The outcome corresponds to theself-representation (the Nash tax-competition outcome by the median-voters). On the other hand,the current multi-stage model corresponds to the case when the campaign promise by any citizencandidate is not binding. A detailed discussion is given in Section 5 in interpreting our results.

4

Directions of Strategic Delegation and Voter Welfare

4.1

Symmetric Countries (a1 = a2 = a, s1 = s2 = 0.5, k 1 = k 2 = k)

As a benchmark, we start with the case of symmetric countries (a1 = a2 = a, s1 = s2 = 0.5, k 1 =k 2 = k). The equilibrium delegate in an economy is symmetric (θ1g = θ2g ; see, e.g., (13) below) sothat k1 (θ1g , θ2g ) = k2 (θ1g , θ2g ) = k. Substituting ki = k i into (9), the conventional situation of θm < 1(i.e., skewed income distribution) will generate a consequence ofθig < θm (i = 1, 2).

8

From Lemma 1 (ii), (iii), we haveti (θ1g , θ2g ) > ti (θm , θm ), (i = 1, 2).

(11)

g g
We now compare the utilities of median-voters with strategic delegation (umi (θ1 , θ2 )) and under selfm m
m m
m m
m m
representation (umi (θ , θ )). Since t1 (θ , θ ) = t2 (θ , θ ) (see (14) below), we have k1 (θ , θ ) =mm
0
k2 (θ , θ ) = k. From (3), (11) and (1) where r = f (k) − ti , the diﬀerence would eventually be:
g g
g g
m m m
m m
m
um
i (θ1 , θ2 ) − ui (θ , θ ) = −{ti (θ1 , θ2 ) − ti (θ , θ )}(θ k − k) > 0 (i = 1, 2).

(12)That is, in both countries, the median-voters are made better oﬀ by strategic delegation. Also, asshown in Lemma 2 (note that r(θm , θm ) − r(θ1g , θ2g ) > 0 here), the citizens who have lower income
than the median-voter are better oﬀ under strategic delegation.
As discussion in Persson and Tabellini (1992) and Ihori and Yang (2009),8 the political eﬀectm(θ < 1) and the strategic delegation (θig < θm ) mitigate the conventional tax-competition eﬀect
towards the lower tax rates. This result is driven purely by the income-redistribution motive.
The departure from the symmetric countries illuminates rich implications for international capital
allocations, the tax gap and citizens’ welfare.

4.2

Countries with Asymmetric Productivities (a1 > a2 , s1 = s2 = 0.5, k 1 = k 2 =k)In this subsection we consider the case of a1 > a2 where country 1 has higher capital productivities.
It turns out that country 1’s median-voter takes advantage of the national rent of higher productivity
and aggravates so-called tax-the-foreigner eﬀect.
In order to see the diﬀerence in the tax incentives and its eﬀect on strategic delegation, we
overview the structure of the equilibrium backwards. In Stage 3, the diﬀerence in national productivities (a1 > a2 ) has direct eﬀects in which country 1 tends to have higher capital employment.
Given k 1 = k 2 and s1 = s2 , country 1 becomes the capital importer and country 2 becomes thecapital exporter. These eﬀects in turn cause the tax-the-foreigner eﬀect demonstrated in the second
term of (10), which causes t1 > t2 in the second stage.
a1 − a2
It is convenient to deﬁne the term δ ≡> 0. In Appendix 3, we show that the equilibriumkdelegation is given by9
θ1g = −1 + 2θm −δδ< θm , θ2g = −1 + 2θm + .
12
12

8

(13)

Note that, unlike Ihori and Yang (2009, Proposition 1), we do not have their issue of possible undersupply of
the public-good since ti ki is a lump-sum transfer which only causes redistribution. Also, the structure of our model
is richer than Persson and Tabellini (1992) so that we will examine various dimensions of the political tensions andeconomic eﬀects in the following subsections.
9
Formally, (13) is written as, for example, when δ = 0, θig = max{−1 + 2θm , 0}. To highlight the main featuresof the model, we suppress a full analysis taking account of corner solutions, assuming that the parameters are in a
relevant range to satisfy the interior solution.

9

In addition to the eﬀect mentioned in Section 4.1 (θig = −1 + 2θm < θm when δ = 0), a country
with high (low) productivity tends to send a low (high) capital owner. We therefore have θ1g < θmfor country 1. As to the relationship between θ2g and θm , there are oﬀsetting eﬀects between
intranational political competition (that directs for higher taxes) and the incentive as a capital
exporter (that directs for lower taxes).
Proposition 1 Suppose that a1 > a2 , s1 = s2 = 0.5, k 1 = k 2 = k.
(i) θ1g < θm for all parameter values, and θ2g ≶ θm ⇐⇒ δ ≶ 12(1 − θm ) ≡ δ 0 .10(ii) k1 (θ1g , θ2g ) < k1 (θ1m , θ2m ) < k1∗ , k2 (θ1g , θ2g ) > k2 (θ1m , θ2m ) > k2∗ , and k1 (θ1g , θ2g ) > k2 (θ1g , θ2g ).
The reason for part (ii) is as follows. In Appendix 3 we show:
t1 (θ1 , θ2 ) − t2 (θ1 , θ2 ) =δk− (θ1 − θ2 )k.
2(14)

The ﬁrst term of the right-hand side captures the diﬀerence in country’s tax incentives due to the
tax-the-foreigner eﬀect, which is present even under self-representation (θi = θm ). We therefore
have t1 (θm , θm ) > t2 (θm , θm ) (the more productive country levies the higher tax rates). Under
strategic delegation, the second term strengthens the diﬀerence (θ1g < θ2g from (13)), so that the
tax gap is widened under strategic delegation (t1 (θ1g , θ2g ) − t2 (θ1g , θ2g ) > t1 (θm , θm ) − t2 (θm , θm )),which induces the capital ﬂow from country 1 to country 2. This contrasts with the conclusion ofSection 4.1 where the tax-competition eﬀect is mitigated through the strategic delegation. There,
the increase of the median-voter’s welfare is solely by income redistribution, whereas here theallocation of capital matters. The allocation of ki is ineﬃcient under tax competition, and theineﬃciency (divergence from the ﬁrst-best allocation) is worsened by strategic delegation.
Proposition 2 Suppose that a1 > a2 , s1 = s2 = 0.5, k 1 = k 2 = k. There exist δ 1 > δ 0 ≡δ2
δ1
12(1 − θm ) and δ 2 ∈ (0, δ 0 ), 0 ≈ 0.82085 and 0 ≈ 1.3923 such that:
δ
δ
g g
m m m
m g g
m m m
(i) For δ ∈ (0, δ 2 ), um
1 (θ1 , θ2 ) > u1 (θ , θ ) and u2 (θ1 , θ2 ) > u2 (θ , θ ): both median-voters
are made better-oﬀ under strategic delegation than under self-representation.
g g
m m m
m g g
m m m
(ii) For δ ∈ (δ 2 , δ 1 ), um
1 (θ1 , θ2 ) > u1 (θ , θ ) and u2 (θ1 , θ2 ) < u2 (θ , θ ): the median-voterof country 1 (high-productivity country) is made better-oﬀ under strategic delegation than underself-representation, whereas the opposite happens for country 2’s median-voter.
g g
m m m
m g g
m m m
(iii) For δ > δ 1 , um
1 (θ1 , θ2 ) < u1 (θ , θ ) and u2 (θ1 , θ2 ) < u2 (θ , θ ): both median-voters
are worse-oﬀ under strategic delegation than under self-representation.
When the diﬀerence in productivities is suﬃciently low, the eﬀect in (12) (i.e., greater redistributive taxes through strategic delegation) is a dominant force for median-voters’ utilities understrategic delegation. However, for suﬃciently high productivity diﬀerences, the beneﬁt of strategicdelegation diminishes for both countries, as the divergence of the preferred tax rates represented by
10

Proposition 1 is originally shown in Nishimura and Terai (2013) in an augumented model with a public-input.

10(13) and (14) increases. The low-productivity country has lower threshold value towards preferring
self-representation. We can also interpret the proposition with keeping δ and varying 1 − θm . As
δ 2 and δ 1 are decreasing in θm , lower θm (greater skewness of the income distribution) would makecase (i) of Proposition 2 more likely to happen.11
The above observations are similar to a recent contribution by Hindriks and Nishimura (2016)
who analyzed the beneﬁt of tax leadership (Stackelberg tax competition). They considered aframework where the citizens have uniform capital ownership12 (corresponding to our θm ) andshow that: (i) there exists a threshold level of θm below which both countries prefer the Stackelberg
outcomes to the conventional Nash tax-competition outcome, and vice versa for suﬃciently highθm ; and (ii) the threshold levels are decreasing with respect to production asymmetry (δ). In
essence, the citizens’ preference towards the Stackelberg leadership resembles that of the strategicdelegation, in the sense that the median-voter here takes account of the direct eﬀect (the ﬁrst termof (8)) and the indirect eﬀect (the second term of (8)). On the other hand, self-representation does
not have the indirect eﬀect. See Section 5 for further discussions.

4.3

Countries with Diﬀerent Populations (s1 > s2 , a1 = a2 = a, k 1 = k 2 = k)

We now consider a framework by Bucovetsky (1991) and Wilson (1991) in which the population
is diﬀerent across countries (s1 > s2 ). Following the conventional framework, we can show thatt1 (θm , θm ) > t2 (θm , θm ) under self-representation. When there is no asymmetry in other dimensions(a1 = a2 = a, k 1 = k 2 = k) the low-population country (country 2) obtains higher per-capita capital(k2 (θm , θm ) > k1 (θm , θm )). This causes so-called beneﬁt of smallness in which u1 (θm , θm ; θ) <u2 (θm , θm ; θ) for all θ, as well as ineﬃciency of the capital allocation (k2 (θm , θm ) > k2∗ = k1∗ >k1 (θm , θm )). In the following, we investigate the implications of strategic delegation in this model.
In this case, we obtain the following (θ1g , θ2g ):θig =−(1 − s2i ) + θim (1 + 2s2i )
.
3s2i

(15)

One can verify θ1g > θ2g for s1 > 0.5 > s2 . Namely, the high-population country sends a policy-makerthat is more right, in order to counteract the disadvantage from the capital-tax competition.
Proposition 3 Suppose that s1 > s2 , a1 = a2 = a, k 1 = k 2 = k.
(i) θ1g > θ2g and θ1g < θm for all parameter values.
(ii) k1 (θ1m , θ2m ) < k1 (θ1g , θ2g ) < k1∗ = k, k2 (θ1m , θ2m ) > k2 (θ1g , θ2g ) > k2∗ = k.
As to the direction of strategic delegation, θ2g < θm for country 2, as a capital-importer. The capitalexporter has oﬀsetting eﬀects between redistribution and capital-exporting as shown in (10), but
From (A.9) in Appendix 3, δ ≤ 4 is necessary to obtain an interior solution. Therefore, when θm is suﬃciently
low, the cases of θ2g > θm in Proposition 1 (ii) and δ > δ i (i = 1, 2) in Proposition 2 (ii) and (iii) may not appear.
12
In their model, a part of capitals is owned by the residents of the third country.
11

11

in the linear model we obtain θ1g < θm unambiguously. As to the allocation of capital, unlike
Proposition 1 (ii), the tax gap is reduced under strategic delegation13 (0 < t1 (θ1g , θ2g ) − t2 (θ1g , θ2g ) <t1 (θm , θm ) − t2 (θm , θm )), which mitigates the capital ﬂight to the low-population country andeﬃciency loss under self-representation.
When countries diﬀer in population size, the low-population country (country 2) has higher∂k1=
capital demand elasticity (sensitivity of the tax base to ﬁscal rates): from (1) and (2), −∂t1
s2 ∂k2∂k2−< −. This eﬀect tends to make country 2’s tax rate lower than that of country 1.
s1 ∂t2∂t2
However, the terms-of-trade eﬀect works in the opposite direction, where higher tax rates beneﬁt thecapital importer and harm the capital exporter through the lower interest rate. In the conventionalmodels (Bucovetsky (1991, 2009), Wilson (1991)), the terms-of-trade eﬀect is not more important
than the tax-base-elasticity eﬀect. Here, on the other hand, the strategic delegation works toincrease the capital importer’s tax rate to the greater extent than the capital exporter. Thisobservation is in contrast with the case of Section 4.2 where a1 > a2 , in which a country with lower
tax-base elasticity14 is a capital importer, so the tax gap increases by strategic delegation.
Proposition 4 Suppose that s1 > s2 , a1 = a2 = a, k 1 = k 2 = k.
g g
m m m
m g g
m m m
(i) For all s1 ∈ (0.5, 1), um
1 (θ1 , θ2 ) > u1 (θ , θ ) and u2 (θ1 , θ2 ) > u2 (θ , θ ): both medianvoters are made better-oﬀ under strategic delegation than under self-representation.
(ii) u1 (θ1g , θ2g ; θ) < u2 (θ1g , θ2g ; θ) for all θ: the beneﬁt of smallness is preserved under strategicdelegation.
Under population asymmetry, contrary to Proposition 2, the beneﬁt of strategic delegation formedian-voters is preserved for all parameter values. The reason is that the allocation of capital
becomes closer to the ﬁrst-best, compared with the self-representation level. The extent of the
population who beneﬁts from strategic delegation is in fact large in this case. From the derivationof Appendix 6, one can show that, for the citizens who hold average level of capital (θi k i = k i ), (i)u2 (θ1g , θ2g ; 1) > u2 (θm , θm ; 1) for all s1 ∈ (0.5, 1); and (ii) u1 (θ1g , θ2g ; 1) > u1 (θm , θm ; 1) for suﬃciently
high s1 . With the same logic as above, in respective cases, ui (θ1g , θ2g ; θ) > ui (θm , θm ; θ) holds for allθ ≤ 1. These features are surprising, given that, for other dimentions of asymmetry in Sections 4.2
and 4.4, ui (θ1g , θ2g ; 1) < ui (θm , θm ; 1) always holds at least for the capital-exporting country.15
13

From

(A.10)

inAppendix

5

we

have

t1 (θ1g , θ2g ) − t2 (θ1g , θ2g )=2 k(1 − θm )(2s1 − 1)
3
s1 s2

=

2
(t1 (θm , θm ) − t2 (θm , θm )) > 0.
3 14
For the case of a1 > a2 , evaluated at equal tax rates, the capital demand elasticities at t1 = t2 = t are∂ki tit. Therefore, country 1 has lower capital demand elasticity than country 2.
|εki /ti | = −=∂ti kiai − aj + 4k2
(7δ 2 + 48δ(1 − θm ))k15
In Section 4.2, u2 (θ1g , θ2g ; 1) − u2 (θm , θm ; 1) = −< 0 for all δ > 0, and Proposition 2 (ii)576
and Lemma 2 suggest that u1 (θ1g , θ2g ; 1) < u1 (θm , θm ; 1) for suﬃciently high δ. Similarly in the model of Section 4.4
below, for the capital exporter (country 1), u1 (θ1g , θ2g ; 1) − u1 (θm , θm ; 1) < 0 whenever k1 > k2 .

12

4.4

Countries with Asymmetric Capital Endowments (a1 = a2 = a, s1 = s2 =0.5, k 1 > k 2 )Suppose now that country 1 has a higher capital endowment. In this case, country 1 becomes
capital-exporter who tends to lower the tax rate. However, country 1 also has a bigger tax base,which can receive higher tax revenue from the same tax rates.
Let ξ > 0 be the value such that k 1 = k 2 (1 + ξ). The equilibrium delegation is given by
(
)
(
)
1 1 m
ξ
1 1 m
g
g
m
m
θ1 = −1 + 2θ +− θ , θ2 = −1 + 2θ − ξ− θ< θm .
(16)
1+ξ 2 3
2 3
Parallel to the previous propositions, we obtain the following:
Proposition 5 Suppose that a1 = a2 = a, s1 = s2 = 0.5, k 1 > k 2(.
(i) θ2g < θm and θ2g < θ1g for all parameter values. When θm ∈)3
, 1 , then θ1g ≶ θm ⇐⇒ ξ ≶4

36(1 − θm )≡ ξ 0 (θm ). When θm ≤ , then θ1g < θm .
4θm − 3 g g
4(ii) k1 (θ1 , θ2 ) > k1 (θ1m , θ2m ) > k1∗ , k2 (θ1g , θ2g ) < k2 (θ1m , θ2m ) < k2∗ = k1∗ .

Proposition 6 Suppose that a1 = a2 = a, s1 = s2 = 0.5, k 1 > k 2 . There exists θ1 ≈ 0.71120 andθ2 ≈ 0.80683 such that:
g g
m m
1 m
(i) For θm ∈ (θ1 , 1), thete exists ξ 1 (θm ) such that um
θ2 ) ≷) um
1 (θ1 , (1 (θ , θ ) ⇐⇒ ξ ≶ ξ (θ ).
1
m
m
1
m
m
1
1
m
0
m
ξ (θ () is decreasingin θ , with ξ (θ ) > 0 for all θ ∈ θ , 1 and ξ (θ ) < ξ (θ ) for all
)
3
g g
m m m
, 1 . For θm ≤ θ1 , um
θm ∈1 (θ1 , θ2 ) > u1 (θ , θ ) for all ξ > 0.
4
g g
m m m
2 m
(ii) For θm ∈ (θ2 , 1), thete exists ξ 2 (θm ) such that um
2 (θ1 , θ2 ) ≷ u2 (θ , θ ) ⇐⇒ ξ ≶ ξ (θ ).
g g
2
m
m
2
m
0
m
m
2
m
2
m
ξ (θ ) is decreasing in θ with ξ (θ ) > ξ (θ ) for all θ ∈ (θ , 1). For θ ≤ θ , u2 (θ1 , θ2 ) >m mum
2 (θ , θ ) for all ξ > 0.
Proposition 5 (i) shares the basic features with Proposition 1, in that only the capital-exporter may
delegate to a richer citizen than a median-voter. However, in this case there explicitly arises a valueof θm below which the direction of strategic delegation is unambiguously left to the median-voter
for both countries. As to the tax gap, t2 (θ1g , θ2g ) − t1 (θ1g , θ2g ) > t2 (θm , θm ) − t1 (θm , θm ) > 0,16 whichbrings Proposition 5 (ii). Also in Proposition 6, there explicitly arises a value of θm below which
the strategic delegation is unambiguously beneﬁcial to the median-voters.

5

The Role of Campaign Promises

In this section we discuss the role of campaign promises in interpreting our results.
16

From (A.12) in Appendix 7 we have t2 (θ1g , θ2g ) − t1 (θ1g , θ2g ) =

13

4 m
4
ξθ k2 = (t2 (θm , θm ) − t1 (θm , θm )) > 0.
3
3

Suppose that the policy-makers in each country are committed to the most preferred tax rate inStage 1. Then the election of a policy maker is equivalent to the choice of his/her most preferred tax
rate. Given tj (j 6= i) which is also credible at Stage 1, the citizen-candidate announces the most
preferred tax rate along the tax reaction function. As in conventional models, the median-voter ispivotal in the voting. The equilibrium tax choices are given by the Nash equilibrium of (5) with
θi = θm = θj , which is the self-representation outcome.
If, on the other hand, the campaign promise by any citizen-candidate is not binding, country
i’s median-voter’s problem is given by (7). The reason is as follows. In Stage 1, the voters takethe foreign election outcome (θj , j 6= i) as given in (7). In Stage 2, the elected policy-makers
solve (5) and choose ti = ti (θ1 , θ2 ) (i = 1, 2). These policy outcome functions are acknowledged as
constraints in (7), but the outcomes ti and tj are not given in Stage 1. Although the median-voter
is pivotal in selecting θi , he/she may not choose θm (and, accordingly, he/she may not choose(t1 , t2 ) = (t1 (θm , θm ), t2 (θm , θm ))), in the equilibrium. We therefore conclude the following:
Lemma 3 (a) If citizen-candidates are committed to their proposal of tax rates at Stage 1, theequilibrium tax rates are (t1 (θm , θm ), t2 (θm , θm )), i.e, those of self-representation. Accordingly, them mequilibrium utility of the median-voters in country i corresponds to umi (θ , θ ). (b) If citizencandidates do not commit to their proposal of tax rates at Stage 1, the equilibrium tax rates are(t1 (θ1g , θ2g ), t2 (θ1g , θ2g )), i.e, those of strategic delegation. Accordingly, the equilibrium utility of theg gmedian-voters in country i corresponds to umi (θ1 , θ2 ).
An interesting hybrid case is when the representative in one country commits to ti in Stage 1 andthe other representative does not commit to tj . The equilibrium outcome of this case is identicalto that of the Stackelberg tax competition.17
Persson and Tabellini (1992) examined the case of the symmetric countries. Assuming that thevoters do not take the foreign policy as given in Stage 1, they found that the median-voters become
better-oﬀ by strategic delegation (our (12)). They concluded that strategic delegation is selfenforcing (Persson and Tabellini (1992, p. 698)). In the subsequent literature of the international
decision-making, strategic delegation in the domestic politics (representative democracy) is taken for
granted.18 Lemma 3 implies that there is lack of commitment in Stage 1. However, our Propositions
17

Suppose that the representative in one country (say, country 1) commits to t1 in Stage 1 and the other representative (country 2) does not commit to t2 . Since country 2 cannot inﬂuence the choice of t1 through strategic delegation,
country 2’s choice is determined by (5) with θi = θm . Then country 1 can decide on t1 along τ2 (t1 ; θm ). The choice
∂um
∂um
1
1 ∂τ2
of t1 is characterized by+= 0 (which is, through (6), equivalent to (8) with θ2 = θm ). Therefore, the∂t1∂t2 ∂t1
equilibrium is identical to the Stackelberg tax competition outcome where country 1 is the Stackelberg-tax leader
and country 2 is the Stackelberg-tax follower.
18
On the other hand, the self-representation outcome is typically perceived as the result of direct democracy. Ogawa
and Susa (2014), for example, considered nations’ choice between the direct democracy (self-representation) and therepresentative democracy which carries strategic delegation. They show that shifting from direct democracy to representative democracy is the dominant strategy of the median-voters in each country, irrespective of the productivity
diﬀerences. Thus their focus is put on the shift of the political regime, not on the credibility of electoral promises.

14

2 and 6 show that, when asymmetry (the productivity gap or the diﬀerences in capital endowments)
is suﬃciently large, contrary to the suggestion by Persson and Tabellini (1992), the median-voter
and the majority of the citizens of both countries can be worse-oﬀ by the lack of commitment toa campaign promise on the policies. Even though each country aims to gain through a choice ofθig 6= θm in (8), the median-voter of one (or both) country can eventually become worse-oﬀ thanui (θm , θm ). Thus, strategic delegation can constitute a Prisoners’ Dilemma. As in Hindriks andNishimura (2016) which we quoted after Proposition 2, further tax changes along country j’s tax
reaction function (the indirect eﬀect in (8)) may cause negative externalities to country j undersuﬃcient asymmetry, when the direction of preferred tax rates diverges.19
In reality, the elected politicians often fulﬁll their campaign promises, even if they can seek
strategic advantages by not making the promises.20 Our results justify such real-world cases. In
some cases, the commitment to the campaign promise is not only plausible but also desirable.

6

Conclusion

This paper examined a political process and economic consequences of tax competition among
asymmetric countries. Various types of asymmetries across countries are examined. When the
countries have diﬀerent productivity of the capital, strategic delegation increases the divergenceof tax rates across countries. As a result, strategic delegation may make the majority of citizens
worse-oﬀ than the self-representation outcome. These ﬁndings diﬀer from the previous results basedon symmetric countries. Notably, strategic complementarity in the tax-reaction functions is not
suﬃcient to make the median-voters better-oﬀ than the self-representation outcome. The patterns
are diﬀerent in a model with diﬀerent population. A high-population country elects a representative more right than the low-population country. This eﬀect not only reduces the capital ﬂight tothe low-population country, but also increases the median-voter’s utility. Since it also reduces theineﬃciency of capital allocation associated with the tax competition, strategic delegation is beneﬁcial to the low-population country, too. When the countries diﬀer in capital endowments, there
19

In the tax-leadership game (see footnote 17), as a result of the tax leader’s choice, the tax follower may become
worse-oﬀ than the outcome of the Nash-tax competition. Similarly, in the strategic delegation game, each country
tries to take advantage of the indirect eﬀect, which may not be beneﬁcial to the other country.
20
For example, in the U.K., 2010 Manifesto of the Conservative Party mentioned the reduction in the corporate
tax rate. After the 2010 election, the Cameron ministry executed the tax cut as speciﬁed in the Manifesto, with the
U.K.’s tax rate over the OECD average eventually falling below it. Also, in the 2012 presidential election in the U.S.,
the Democrat candidate Obama pledged to reduce the corporate tax rate from 35% to 28%. This promise is partly tocompete with the Conservative candidate Romney who promised the reduction to 25%. After the election, President
Obama proposed a reduction in the corporate tax rate as he swore, in the Budget Proposal of the United States
Government. Needless to say, the structure of the corporate income tax is not characterized by the tax rate alone.
In the U.K.’s case, policies including the enforcement of Controlled Foreign Companies (CFC) rules are crucial toproceed the territorial system. In the U.S., on the other hand, compatibility between the worldwide system and tax
enforcement towards multinational corporates is a crucial issue. The candidates have clearly stated policy proposals
for these issues, which are executed after the respective elections.

15

explicitly arises the degree of income inequality (skewness) above which the strategic delegation isunambiguously beneﬁcial to the median-voters.
We also highlighted the role of campaign promises. In the global economy, one country’s choiceof a tax rate generates ﬁscal externalities. Therefore, political candidates’ commitment to the tax
rate during the election is crucial for the determinants of the equilibrium outcome and citizens’
welfare. The electoral competition model traditionally assumed that electoral promises are binding,following Downs (1957), but Alesina (1988) examined the case where policy-motivated politicians,
once they are elected, would not care their initial promise. These polar cases correspond to selfrepresentation and strategic delegation of the current study, respectively. We showed in Section
5 that a hybrid case in a multinational setting yields a Stackelberg tax-competition outcome.
Given public concerns about politicians’ honesty as well as the criticism by the media, campaign
promises are not wholly irrelevant in reality. Whereas recent studies on the electoral competition
model endogenized the commitment decision incorporating the cost of betrayal,21 extension of thesestudies to the multinational setting would be an intriguing subject for further studies.

Appendix
Appendix 1: Proof of Lemma 1
Since

∂r
si ∂ki= −fj00, (5) is rearranged to∂ti
sj ∂ti()∂ui
si ∂ki∂ki=ti − (θi k i − ki )fj00≡ βi (ti , tj )= 0,∂ti
sj ∂ti∂ti

where βi ≡ ti − (θi k i − ki )fj00 ssji .
∂ki< 0. Total diﬀerenciation of the ﬁrst-order condition yields
Note that
∂ti
[
]
si
00 ∂ki − (θ k − k )f 000 ∂kj
fi ii j ∂tjj ∂tjsj∂τi (tj ; θi )∂βi /∂tj
[
]
=−
= −
∂tj
∂βi /∂ti
000 ∂kji
1 + ssji fj00 ∂k
−
(θ
k
−
k
)f
i i
i j ∂ti
∂ti
[
]
si
si
00
000 ∂kisj fj + sj (θi k i − ki )fj
∂ti
[
]
=
,
i
1 + ssji fj00 + ssji (θi k i − ki )fj000 ∂k∂ti
wehere we used21sj
sj ∂kj∂kj∂kisi ∂ki
∂ki
∂ki
=
=
< 0,
=−
and
=−
.
00
00
∂ti
sj fi + si fj
si ∂tj∂ti
sj ∂ti∂tj∂tiSee, e.g., Austen-Smith and Banks (1989), Terai (2009), and Asako (2015).

16

(A.1)

(A.2)

∂τi (tj ; θi )∂ki
< 0, when fj000 = 0 (linear model), we have 0 << 1. That is, tax rates∂ti∂tj
are strategic complements in both countries with the slopes less than unity. This proves part (i) of
Lemma 1.
s
Totally diﬀerentiating the system of βi (ti , tj ) = 0 and βj ≡ tj − (θj k j − kj )fi00 sji = 0, we have
)
) (
)(
(
k i fj00 ssji dθidti∂βi /∂ti ∂βi /∂tj
=
.
s
dtj
∂βj /∂ti ∂βj /∂tj
k j fi00 sji dθj
Since

We therefore have(
) k f 00 si () k f 00 si()
i j sji j sj ∂βj∂ti /∂θi∂βj /∂tj1==,∂tj /∂θi−∂βj /∂ti∂τj /∂ti
∆
∆ ∂tj

(A.3)

∂βi ∂βj∂βi ∂βj−.
∂ti ∂tj∂tj ∂ti
The last equation of (A.3) follows from the formula of the reaction function (A.2)( applied to)∂βj∂βi ∂βj∂τi ∂τjcountry j. Note that
> 0 from the second-order condition, and also, ∆ =1−>∂tj∂ti ∂tj∂tj ∂ti∂tj∂ti∂ti ∂τi
0. We therefore have
< 0 (part (ii)) and=< 0 (part (iii) and (6)). Q.E.D.
∂θi∂θi∂θi ∂tj
where ∆ ≡Appendix 2: Proof of Lemma 2
g g
m
m m
m g g
ui (θ1 , θ2 ; θ) = umi (θ1 , θ2 )+r(θ1 , θ2 )·(θ −θ )k i . Therefore ui (θ1 , θ2 ; θ)−ui (θ , θ ; θ) = ui (θ1 , θ2 )−
g g
g g
m m
m
m m
m m
m
um
i (θ , θ ) + (r(θ1 , θ2 ) − r(θ , θ )) · (θ − θ )k i . Note that (r(θ1 , θ2 ) − r(θ , θ )) · (θ − θ ) > 0
g g
m
m
m
m
m
for all θ < θ , and (r(θ1 , θ2 ) − r(θ , θ )) · (θ − θ ) < 0 for all θ > θ . Then the result of Lemma
2 (i) (ii) follows. Q.E.D.

Appendix 3: Proof of Proposition 1
Begin with the Stage-3 equilibrium allocation of capital. In the linear model, solving (1) and (2)gives:sj(A.4)
ki (t1 , t2 ) = si k i + sj k j + ((ai − aj ) − (ti − tj )) .
2
Another benchmark, the ﬁrst-best solution in (4) is:sjki∗ = si k i + sj k j + (ai − aj ) .
2

(A.5)

In Stage 2, from (A.1) we have:ti = fj00

si· (θi k i − ki (t1 , t2 )), j 6= i.
sj
17

(A.6)

Under the suppositions of si = sj = 1/2 and k i = k j = k, from (A.4) and (A.6), we haveti (θ1 , θ2 ) = 2k +ai − aj(3θi + θj )k−,4
2

from which we derive (14). Also,ki (θ1 , θ2 ) ≡ ki (t1 (θ1 , θ2 ), t2 (θ1 , θ2 )) = k +)1((ai − aj ) + 2(θi − θj )k .
8

(A.7)

In the linear model, (8) is equivalent to:θ m − θi = −

sj
si

(
θm −

ki (θ1 , θ2 )
ki

)
.

(A.8)

Solving (A.7) and (A.8) derives (13) of the text, from which we derive Proposition 1 (i). In turn,from (A.5) and (A.7) we haveki (θ1g , θ2g ) = k −(−1)i δk(−1)i δk, ki (θm , θm ) = k −,128ki∗ = k −(−1)i δk(i = 1, 2). (A.9)
4

These equations lead to Proposition 1 (ii). Also, r(θ1g , θ2g ) − r(θm , θm ) = 2k(θm − 1) < 0, so theconclusion of Lemma 2 holds. Q.E.D.

Appendix 4: Proof of Proposition 2
Under the suppositions of si = sj = 1/2 and k i = k j = k, the diﬀerence of the median-voter’sutilities under strategic delegation and under self-representation is
(
)(
)
√
√
2
24( 15 − 1)
24( 15 + 1)
7k
m
m
m g g
m m m
δ+
(1 − θ )δ−(1 − θ ) ≷ 0
u1 (θ1 , θ2 ) − u1 (θ , θ ) = −
576
7
7
√
24( 15 + 1)⇐⇒ δ ≶(1 − θm ) ≡ δ 1
7
7k
g g
m m m
um
2 (θ1 , θ2 ) − u2 (θ , θ ) = −

2

(

)(
)
√
√
24( 15 + 1)
24( 15 − 1)mm
δ+
(1 − θ )δ−(1 − θ ) ≷ 0
7
7

576
√
24( 15 − 1)⇐⇒ δ ≶(1 − θm ) ≡ δ 2 .
7√√
δ22( 15 − 1)
δ1
2( 15 + 1)
Since 0 =(≈ 0.82085) < 0 =(≈ 1.3923), we obtain Proposition 2. Q.E.D.
δ
7
δ
7
18

Appendix 5: Proof of Proposition 3
Under the suppositions of ai = aj = a and k i = k j = k, from (A.4) and (A.6), we deriveti (θ1 , θ2 ) = −(−θi s2j + θj s2j + θi − 1)k(−θj s2j + θi s2i + 1)k, ki (θ1 , θ2 ) =
,
sj
2si

Solving (A.8) and (A.10) derives (15) of the text. θig −θm = −and θ1g − θ2g =(A.10)(1 − si )(1 + si )(1 − θm )< 0 (i = 1, 2)
3s2i(2s1 − 1)(1 − θm )> 0, so we obtain Proposition 3 (i). From (A.10) we have
3s21 s22ki (θ1g , θ2g ) =(2si θm − θm + 1)k(2si θm + si − θm + 1)k, ki (θm , θm ) =
.
3si
2si

(A.11)

From (A.5) we have k1∗ = k2∗ = k. Note thatki (θ1g , θ2g ) − ki (θm , θm ) =(1 − θm )(2si − 1)k(1 − θm )(2si − 1)k, ki∗ − ki (θ1g , θ2g ) =(i = 1, 2).
6si
3si

Since 2s1 − 1 > 0 and 2s2 − 1 < 0, we obtain Proposition 3. Also, r(θ1g , θ2g ) − r(θm , θm ) =(−2s21 + 2s1 + 1)(θm − 1)k< 0, so the conclusion of Lemma 2 holds. Q.E.D.
3s1 s2

Appendix 6: Proof of Proposition 4
Under the suppositions of ai = aj = a and k i = k j = k, the assertion in Proposition 4 (i) follows
from22
2

g g
m m m
um
i (θ1 , θ2 ) − ui (θ , θ ) =(1 − θm )2 (1 + si )(−4s2i + 8si + 3)k> 0 (i = 1, 2).
36s2i sj

Moreover,2

u1 (θ1g , θ2g ; θ) − u2 (θ1g , θ2g ; θ)=

(1 − θm )2 (1 − 2s1 )k< 0 for all θ.
3s21 s22

Thus Proposition 4 (ii) is veriﬁed. Q.E.D.
2(1 − θm )2 (2si − 1)(10s2i − 5si − 3)k(i = 1, 2).
36s2i sj
g
g
g
g
m
m
m
m
u2 (θ1 , θ2 ; 1) − u2 (θ , θ ; 1) > 0 for all s1 ∈ (0.5, 1), and u1 (θ1 , θ2 ; 1) − u1 (θ , θ ; 1) > 0 for suﬃciently high s1 .
22

For the citizens with θi = 1, ui (θ1g , θ2g ; 1) − ui (θm , θm ; 1) =19

Appendix 7: Proof of Proposition 5
Under the suppositions of ai = aj = a and si = sj = 1/2, from (A.4) and (A.6), we derive)(3θi k i + θj k j )1
1
1(, ki (θ1 , θ2 ) = k 1 + k 2 +θi k i − θj k j .
(A.12)
22
2
4
ξ(4θm − 3) − 6(1 − θm )Solving (A.8) and (A.12) derives (16) of the text. θ2g < θm and θ1g − θm =,6(1 + ξ)so we obtain Proposition 5 (i). From (A.12) we haveti (θ1 , θ2 ) = k 1 + k 2 −

1
1
1
1
1
1
ki (θ1g , θ2g ) = k 1 + k 2 − (−1)i ξθm k 2 , ki (θm , θm ) = k 1 + k 2 − (−1)i ξθm k 2 (i = 1, 2).(A.13)
2
2
3
2
2
4
1
1
From (A.5) we have k1∗ = k2∗ = k 1 + k 2 . These equations lead to Proposition 5 (ii). Also,22r(θ1g , θ2g ) − r(θm , θm ) = (k 1 + k 2 )(θm − 1) < 0, so the conclusion of Lemma 2 holds. Q.E.D.

Appendix 8: Proof of Proposition 6
Under the suppositions of ai = aj = a and si = sj = 1/2, the diﬀerence of the median-voter’s√
78 − 6 15mutilities under strategic delegation and under self-representation is, for θ >≡ θ1 ≈
77
0.71120,
)
(
)
√ )
√
(k 2 )2 ((
g g
m m m
m
m
um
(θ
,
θ
)
−
u
(θ
,
θ
)
=
−
15
−
77
θ
ξ
+
12
13
+
15
(1
−
θ
)
×
78
+
6
1
1
1 2
((144 · 77
)
(
)
)
√
√
77 θm − 78 + 6 15 ξ − 12 13 − 15 (1 − θm ) ≷ 0√
12(13 − 15)(1 − θm )√⇐⇒ ξ ≶≡ ξ 1 (θm ).
77 θm − 78 + 6 15
)
(
3
1
m
m
1
1
m
0
m
m
,1 .
ξ (θ ) > 0 for all θ ∈ (θ , 1), and ξ (θ ) < ξ (θ ) for all θ ∈4√66 − 6 15
For θm >≡ θ2 ≈ 0.80683,
53
)
(
)
√
√ )
(k 2 )2 ((
g g
m m m
um
66 + 6 15 − 53 θm ξ + 12 11 + 15 (1 − θm ) ×2 (θ1 , θ2 ) − u2 (θ , θ ) = −((144 · 53
(
)
√ )
√ )
53 θm − 66 + 6 15 ξ − 12 11 − 15 (1 − θm ) ≷ 0√
12(11 − 15)(1 − θm )√≡ ξ 2 (θm ).
⇐⇒ ξ ≶
53 θm − 66 + 6 15
g g
m m m
ξ 2 (θm ) > ξ 0 (θm ) for all θm ∈ (θ2 , 1). For θm ≤ θi , umi (θ1 , θ2 ) − ui (θ , θ ) > 0 for all ξ > 0 (i =1, 2). We therefore obtain Proposition 6. Q.E.D.

20

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