Discussion Papers In Economics
And BusinessElectoral Commitment in Asymmetric Tax-competition
ModelsYukihiro Nishimura
Kimiko Terai

Discussion Paper 20-21-Rev.
July 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANElectoral Commitment in Asymmetric Tax-competition Models
Yukihiro Nishimura†

Kimiko Terai‡

Osaka University and CESifo

Keio University

March 19, 2021

Abstract
This study examines the political process of tax competition among asymmetric countries,
highlighting the role of the commitment to the electoral promises. The median voters deliberately elect a delegate whose preferences diﬀer from their own (strategic delegation), which
is self-enforcing under symmetric countries. We ﬁrst show that the outcome of strategic delegation is replicated when the candidates do not make binding campaign promises in bothcountries, and the opposite scenario of the binding commitments to the platforms leads to theself-representation by the median voters. We then amplify the model by adding the pre-election
stage where the citizens choose whether the credibility of election promises is critical, throughsubscription numbers of newspapers and social media which determine the cost of betrayal of the
proposed platforms (or the lack of the proposal). We then show that, depending on the type ofasymmetries under consideration, suﬃcient asymmetry or suﬃciently equal income distribution
generates the commitment to the election campaign promises as the equilibrium outcome.
Keywords: Capital-tax competition; Election campaign promises; Asymmetric countries; Voting
JEL classiﬁcation: C72, D72, D78, H23, H87.

1

Introduction

Whether the elected politicians would keep their campaign promises in the oﬃce is a controversial topic. In the globalized economy, political candidates’ (lack of) commitment to tax policies
proposed during the election campaign aﬀects economic activities across countries, which in turnaﬀect citizens’ welfare. Unless the fulﬁllment of the promises is important per se, non-commitment
to the campaign promises is often found useful. Nevertheless, regarding corporate taxes which are
subject to tax competition, elected politicians often fulﬁll their campaign promises, even if theyseek strategic advantages by not doing so. For example, in the U.K., the 2010 Manifesto of the†Corresponding author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama-cho, Toyonaka-shi,
Osaka. Phone & fax: +81 (6)-6850-5230, e-mail: ynishimu@econ.osaka-u.ac.jp‡
Faculty of Economics, Keio University, e-mail: kterai@econ.keio.ac.jp

1

Conservative Party mentioned a reduction in the corporate tax rate. After the 2010 election, theCameron ministry executed the tax cut as speciﬁed in the Manifesto, with the U.K.’s tax rate
eventually falling below the OECD average. In addition, in the 2012 presidential election in the
U.S., the Democrat candidate Barack Obama pledged to reduce the corporate tax rate from 35%
to 28%. This promise was made partly to compete with the Republican candidate Mitt Romney,
who promised a reduction to 25%. After the election, President Obama proposed a reduction in
the corporate tax rate in the Budget Proposal of the United States Government.1
A variety of treatments has been made in the literature of political economics. Downs (1957)(with oﬃce-motivated candidates) and Wittman (1973) (with policy-motivated candidates) suppose that candidates can completely commit to their campaign promises.2 In contrast, in theconventional citizen-candidate models without inter-jurisdictional policy spillovers such as Besley
and Coate (1997) and Osborne and Slivinski (1996), each citizen is allowed to become a candidate
for an election, and the elected candidate who can choose a policy after the election cannot crediblycommit to diﬀerent positions at the time of the election. To have a realistic situation of partiallybinding platforms, Asako (2015) introduced the cost of betrayal for the politician who implementsa policy diﬀerent from his/her platform, which Banks (1990), Grossman and Helpman (2005), andCallander and Wilkie (2007) also employed.
In the context of tax competition, Persson and Tabellini (1992) considered the two-stage gamewith elections and policy-making. In the election stage, each country selects a policymaker throughmajority voting. In the policy-making stage, the elected politicians noncooperatively choose thetax policies of the respective country. Assuming that the voters do not take the foreign policy asgiven in the election stage, they found that the median voter of each country deliberately elects adelegate whose preferences diﬀer from their own (strategic delegation) to aﬀect the decision-makingof another country. Assuming that countries are symmetric, Persson and Tabellini (1992) found thatthe median voters become better oﬀ by this strategic delegation. They concluded that strategicdelegation is self-enforcing (Persson and Tabellini (1992, p. 698)). In the subsequent literatureof the international decision-making, strategic delegation in the domestic politics (representative
democracy)3 is taken for granted (Ihori and Yang (2009), Pal and Sharma (2013), and Ogawa andSusa (2017b)). This paper examines the issue of the commitment to the campaign platforms whencountries are asymmetric.
We ﬁrst show that the outcome of strategic delegation is replicated when the candidates do notmake binding campaign promises in both countries. In contrast, if the candidates make commit1
Needless to say, the structure of corporate income tax is not characterized by the tax rate alone. In both the U.K.
and the U.S.’s case, policies including the enforcement of controlled foreign companies rules and tax enforcement inrelation to multinational corporates were crucial issues. Candidates clearly stated their policy proposals regardingthese issues, which were then executed after the respective elections.
2
Calvert (1985) considered the issue of convergence to the median when the candidates care about the election’s
policy outcomes, and they are also uncertain about voters’ responses.
3
We do not consider the direct democracy in this paper (see Redoano and Scharf (2004) and Ogawa and Susa
(2017a)).

2

ments to their campaign promises, the equilibrium policy-making is equivalent to the case when the
median voter becomes the policymaker in both countries, the case of self-representation following
Segendorﬀ (1998) (Proposition 1.(i) and (ii)). We also show that, depending on the types of asymmetries under consideration, the median voters (and the majority of the citizens of both countries)
can in fact become worse oﬀ as a result of the lack of commitment to election campaign promises(Proposition 3.(ii) and Proposition 5).
Having observed that the lack of commitment is not necessarily desirable for the median voters inasymmetric countries, we consider the following pre-election stage; prior to the election, a majorityof the citizens of each country chooses whether the credibility of election promises is critical (RegimeC) or not critical (Regime N ). Such a choice (inﬂuence) by the public is captured by, for example,
the circulation (subscription) numbers of newspapers and social media that are critical of thepoliticians breaking (or not making) election promises. Accordingly, the candidates at the election
stage make binding campaign promises (commitments) under Regime C, and they do not makecommitments to election promises under Regime N . In hybrid cases when the representative in
one country commits to the campaign promises while the other representative does not commit tothem, the policy outcome is identical to that of Stackelberg tax competition (Proposition 1.(iii)).
We then examine the citizens’ choice at the pre-election stage regarding (the lack of) commitment.
When the countries diﬀer in terms of population size (Bucovetsky (1991), Wilson (1991)) or income
distribution, we show that, regardless of the extent of asymmetry, non-commitment is not only abetter outcome than commitment for the median voters, but also the dominant-strategy outcome(Propositions 2 and 4). However, when the countries are diﬀerent in productivities of capital(Hindriks and Nishimura (2017)) or capital endowments (Hwang and Choe (1995), Peralta andvan Ypersele (2005)), a divergence regarding the preferred tax rate across countries emerges undersuﬃciently large asymmetries or suﬃciently equal income distribution. As a result, contrary to
Persson and Tabellini (1992), commitment to election promises by both countries constitutes theequilibrium outcome (Proposition 3.(ii) and Proposition 5). The determinants of this emergence ofbinding election promises in the equilibrium synonymous to the second-mover incentives in Hindriks
and Nishimura (2017) in the equilibrium of the timing game.
The rest of the paper is organized as follows. Section 2 describes the model and the structureof the game. Section 3 characterizes the equilibrium tax rates in the policy-making stage. Section
4 presents the main results. The proofs of several propositions and lemmas are provided in the
Appendices.

2

The Model

There are two countries, denoted by i = A, B. Let K i and Li (i = A, B) denote the capital andlabor endowment, respectively, of each country. Let k i ≡ K i /Li represent the per capita capitalendowment. Later, we introduce possible asymmetries with respect to population size (LA > LB
and k A = k B ) and the capital–labor ratio (k A > k B and LA = LB ). The former is analyzed by3

Bucovetsky (1991) and Wilson (1991), where country A has a higher population, and the latter isanalyzed by Hwang and Choe (1995) and Peralta and van Ypersele (2005), where country A has ahigher capital endowment.
Capital is perfectly mobile across borders, while labor is perfectly immobile. We refer to country
i’s capital in its per capita form ki ≡ Ki /Li . The production in country i is deﬁned in its per 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 a quadratic production function fi (ki ) = ai ki − ki2 with aA ≥ aB > 0. Thiscase is analyzed by Hindriks and Nishimura (2017).
Each country levies a source tax at rate ti per unit of capital employed. Under capital mobility,
the arbitrage condition involves:
fA0 (kA ) − tA = fB0 (kB ) − tB = r,(1)

where r is the price of capital.
Let si ≡ Li /(LA + LB ) be the share of the population in country i, where we assume sA ≥ sB .
The market-clearing condition is:
sA kA + sB kB = sA k A + sB k B .

(2)

Resident j in country i has θij k i units of capital. θij = 1 means that resident j’s capital holding is
at the average of country i. We denote the position of the median in country i as θim . We assume
that each country has a right-skewed capital endowment distribution with θAm < 1 and θBm < 1.
The government in country i provides an equal lump-sum transfer, γi , to each citizen, which issolely ﬁnanced by taxation on capital: γi = ti ki . A citizen in country i with a capital share of θij
receives (i) labor income fi (ki ) − fi0 (ki )ki , (ii) rent from capital rθij k i , and (iii) γi = ti ki . That is,uij= fi (ki ) − fi0 (ki )ki + rθij k i + ti ki= fi (ki ) + r(θij k 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, a majority of the citizens of each country chooses whether the credibility of election
promises is critical (Regime C) or not critical (Regime N ).
• In Stage 2, a policymaker (delegate) is simultaneously elected under majority rule in bothcountries. Each citizen in country i is allowed to become a candidate for country i’s election.
If Regime C was chosen at the previous stage in country i, then the candidates make binding
commitments to ti . If Regime N was chosen in country i, then the candidates do not makebinding commitments on ti at this stage.
• In Stage 3, if Regime C was chosen before, then country i applies ti announced by the elected
politician in Stage 2. If Regime N was chosen before, then the elected politician of country ichooses ti .
4• In Stage 4, having observed (tA , tB ), private investors in both countries make their investment
decisions, and production takes place. Accordingly, γi = ti ki is determined.
Prior to the election, a majority of the citizens of each country chooses whether the credibility ofelection promises is critical. Such a choice (inﬂuence) by the public is captured by, for example, thecirculation (subscription) numbers of newspapers and social media that are critical of politicians
breaking (or not making) election promises. If the citizens of country i chose to be critical of theelection promises, then in Stage 2, the candidates of country i make binding commitments on ti ;otherwise, the candidates do not commit to ti in Stage 2. Under the international mobility ofcapital, the latter option (non-commitment to ti in the election stage) potentially has a beneﬁt,since it allows to adjust the tax policies and the associated capital inﬂow/outﬂow contingent on
the elected politician of the other country.
The formulation in Asako (2015)4 is helpful. Suppose that the winning candidate in country iannounced a platform tpi in Stage 2. In Stage 3, if the implemented policy ti is diﬀerent from thewinning candidate’s tpi , then the winning candidate incurs the cost of betrayal λi c(ti − tpi ) for λi ≥ 0
and a convex cost function c(·) (c(0) = 0, c(ti − tpi ) > 0, (ti − tpi ) · c0 (ti − tpi ) > 0 and c00 (ti − tpi ) ≥ 0
for all ti 6= tpi ). Accordingly, writing the citizen-candidate’s utility in (3) as uij (ti , t−i ) in terms ofthe tax policies, the winning candidate maximizes uij (ti , t−i ) − λi c(ti − tpi ). Naturally, the citizens
in country i can inﬂuence the parameter λi . To make clear-cut results and to have analytical
simplicity, here we assume that the citizens in each country choose between (a) λi = 0 and (b) avery high λi that induces ti = tpi (λi → ∞).

3

Analysis

The game is solved by backward induction.

3.1

Capital-market Clearing

In Stage 4, the determination of the capital allocation is standard. Let k̂i (tA , tB ) (i = A, B) and∂ k̂ir̂(tA , tB ) be the values of ki (i = A, B) and r that satisfy (1) and (2).5 Then, we have
=
∂ti
s−i
∂ k̂−isi ∂ k̂i∂ r̂
00 ∂ k̂−i< 0 and=−> 0 (i, −i = A, B, −i 6= i). Since
= f−i
< 0,
00
00
s−i fi + si f−i∂tis−i ∂ti∂ti∂ti
the capital importing (exporting) country tends to beneﬁt from a lower (higher) interest rate (seethe second line of (3)). This is called the terms-of-trade eﬀect.

4
5

See also Banks (1990), Grossman and Helpman (2005), and Callander and Wilkie (2007).
A closed-form expression of k̂i (tA , tB ) and other variables are relegated to the Appendix.

5

3.2

Policymaker’s Choice of Tax Rate (Regime N in Country i)

The analysis in Stage 3 depends on the choice of the regimes in each country at Stage 1. If Regime
C was chosen by country i in Stage 1, then the ti announced by the election winner in Stage 2 isapplied. In this subsection, we analyze the cases when Regime N was chosen in country i.
We begin with the case where country −i also chose Regime N in Stage 1. Let us represent
a policymaker’s type in country i by θiP . In Stage 3, given θ−iP , the policymaker with θij = θiPmaximizes his/her utility uiP in (3) by choosing ti , taking account of k̂i (tA , tB ) (i = A, B) andr̂(tA , tB ) determined in the subsequent stage. The ﬁrst-order condition for country i’s policymaker
is:
∂ k̂i
∂ k̂i
∂ r̂
∂ r̂
∂uiP
= (fi0 (ki ) − r)
+ (θiP k i − ki )= ti+(θiP k i − ki )
= 0.
∂ti
∂ti
∂ti
∂ti
∂ti

(4)

For Regime (N, N ), t−i is simultaneously decided with ti . The expression in (4) deﬁnes implicitly the tax reaction function ti ≡ τi (t−i ; θiP ) that depends on the foreign country’s t−i and thepolicymaker’s type θiP . As the intersection of the tax reaction functions, the Stage-2 tax rates(tA (θAP , θBP ), tB (θAP , θBP )) are determined, as functions of (θAP , θBP ).
Next, we examine the case when country i chose Regime N but country −i chose Regime C. In
this case, country i’s choice of ti is represented by (4) but now t−i is given in Stage 2. According to
the commitment structure, the tax choice is made sequentially, equivalent to the Stackelberg tax
choices (see Proposition 1.(iii)).

3.3

The Outcome of the Elections

In Stage 2, country i chooses the tax rate ti under Regime C, and it chooses the policymaker
iP ’s type under Regime N . Here we fully characterize the equilibrium tax rates depending on theregimes. We show our ﬁrst proposition:
Proposition 1 (i) If both countries choose Regime C in Stage 1, the equilibrium tax rates arethose of the Nash equilibrium decided by the median voters of each country.
(ii) If both countries choose Regime N in Stage 1, the median voters strategically elect thepoliticians whose preferences may diﬀer from their own.
(iii) Suppose that country i chooses Regime N in Stage 1, and country −i chooses Regime C.
Then the equilibrium tax rates are equivalent to the Stackelberg tax equilibrium with country −ibeing the Stackelberg leader, where the tax rates are chosen by the median voter of each country.
Proof: (i) Suppose that citizen-candidates in each country make binding commitments on thepreferred tax rate in Stage 2. Then, the election of a policymaker is equivalent to the choice of the
tax rate. Given t−i , which is also credible in Stage 1, the citizen-candidate of type θij announces
ti along the tax-reaction function determined by (4). The utility function (3) is linear in termsof one-dimensional parameter θij . Therefore, as in cases (ii) and (iii) below, the median voter is6pivotal in the voting (see also Lemma 1 below). Therefore, given t−i , the elected representative
announces ti = τi (t−i ; θim ).6 The equilibrium tax rates are characterized by the intersection ofthese tax reaction functions: (tA , tB ) = (tA (θAm , θBm ), tB (θAm , θBm )).
(ii) If election campaign promises by any citizen-candidate are not binding in Stage 1, country
i’s median voter’s problem is given as follows:maxθiPfi (k̂i (ti , t−i )) + r̂(ti , t−i )(θim k i − k̂i (ti , t−i ))s.t. ti = ti (θiP , θ−iP ) and t−i = t−i (θiP , θ−iP ).

(5)

In Stage 2, the voters take the foreign election outcome (θ−iP ) as given in (5), and the elected
policymakers solve (4) to choose ti = ti (θiP , θ−iP ). These policy outcome functions are acknowledged in (5), but the outcomes ti and t−i cannot be given in Stage 2. The ﬁrst-order condition withrespect to θiP derives the following formula (see the Appendix for the derivation):θiP ≶ θim ⇐⇒ θim k i − ki ≶ 0.

(6)

From (2), we conclude that at least one country chooses θiP < θim .
(iii) Suppose that the representative in country −i commits to t−i in Stage 2 and the otherrepresentative (country i) does not commit to ti . As mentioned in Section 3.2, country i’s tax
choice in Stage 3 is determined by (4) (applied to iP = im; again, the tax choice is according to themedian voter’s preference). Then, along ti = τi (t−i ; θim ), country −i can decide on t−i in Stage 2 inorder to maximize u−im : namely, maxt−i f−i (k̂−i (t−i , ti )) + r̂(t−i , ti )(θ−im k −i − k̂−i (t−i , ti )) s.t. ti =τi (t−i ; θim ). These tax choices are equivalent to the Stackelberg tax competition outcome wherecountry −i’s citizen m is the Stackelberg tax leader and country i’s citizen m is the Stackelberg
tax follower. Q.E.D.

3.4

The Structure of the Game and Countries’ Incentives

Table 1 characterizes the outcome of the pre-election stage (Stage 1) with respect to (tA , tB ). Whenboth countries choose commitment C (i.e., to be critical on the election promises), the outcome
is equivalent to the self-representation (following Segendorﬀ (1998)) where θiP = θim for bothcountries. In contrast, when both countries choose non-commitment N , the outcome with strategicdelegation appears where θiP 6= θim at least in one country. Denote ti (θiP , θ−iP ) in Regime (N, N )as tSi . In the hybrid cases where one country chooses C and the other chooses N , the superscript
L and F denote the Stackelberg leader and the follower respectively.7
6

Equivalently, the median voter himself/herself becomes the elected policymaker who announces the preferred tax
rate.
7
The structure of this commitment game is diﬀerent from that of the timing game (e.g., Hindriks and Nishimura(2017) and Ogawa and Susa (2017a)). In particular, diﬀerent from the timing-game literature in the asymmetric tax
competition, we show below that the Stackelberg outcomes never constitute a subgame-perfect equilibrium.

7

If country −i chooses C, then, country i faces a choice between the Nash-tax competition (bychoosing C) and the Stackelberg follower (by choosing N ). Here, the country’s choice depends
on whether there is a second-mover incentive. This issue in the asymmetric tax competition isanalyzed by Hindriks and Nishimura (2017) in the context of aA > aB in the parametric form of
the production function. If country −i chooses N , then country i’s choice depends on whether the
Stackelberg leader (choice of ti ) is more beneﬁcial than the strategic delegation (choice of θiP 6= θim )in Stage 1.

Country A

CommitNo Commit

Country B
Commit(tA (θAm , θBm ), tB (θAm , θBm ))(tFA , tL
B)

No Commit
F(tL
A , tB )SS(tA , tB )Table 1: Outcomes of the Commitment Game
The citizens (public) in Stage 1 inﬂuence the policymakers’ decision-making by the adoption of
the media that are critical of breaking election promises. We assume that the choice of the Regimes
C and N is made by the majority of citizens, which corresponds to the subscription numbersof newspapers and the participants of social media. Then, the following lemma shows that it issuﬃcient to check the preference of the median voter for Stage 1’s choice of regimes:
Lemma 1 For any pair of the outcomes whose tax rates are (t1A , t1B ) and (t2A , t2B ), the preferenceof the median voter is consistent with that of the majority of the same country.
Let us denote the tax rates in each cell of Table 1 as functions of the regime choice of eachFcountry. Namely, t∗i (C, C) ≡ ti (θim , θim ), t∗i (N, N ) ≡ tSi , (t∗A (C, N ), t∗B (C, N )) ≡ (tL
A , tB ), and(t∗A (N, C), t∗B (N, C)) ≡ (tFA , tL
B ). Accordingly, the Stage-4 values are determined by, for example,ki∗ (C, C) ≡ k̂i (t∗i (C, C), t∗−i (C, C)) and r∗ (C, C) ≡ r̂(t∗i (C, C), t∗−i (C, C)). Substituting these values
to (3), uij is denoted as functions of the regime choice of each country.

4Equilibrium Electoral Commitment and Voter Welfare

4.1

Countries with Diﬀerent Population Sizes (sA > sB , aA = aB = a, θAm =θBm = θm , k A = k B = k)

From now on, we explicitly derive the equilibrium outcome and voters’ welfare under the quadratic
production function fi (ki ) = ai ki − ki2 with aA ≥ aB . We ﬁrst consider a framework developedby Bucovetsky (1991) and Wilson (1991) in which the population size diﬀers between countries(sA > sB ).
Proposition 2 Suppose that countries diﬀer in population size.
8(i) The choice of no-commitment is the dominant strategy for both countries.
(ii) Both median voters are made better oﬀ under (N, N ) than under (C, C).
Since (N, C) or (C, N ) never constitutes an equilibrium, here we compare the outcome of nocommitment to commitment. The latter (the tax-competition outcome through self-representation)
is a natural benchmark to recollect the previous results on the asymmetric tax competition. We
have t∗A (C, C) = tA (θAm , θBm ) > t∗B (C, C). When there is no asymmetry in other dimensions(ai = a, k i = k, θim = θm ), the low-population country (country B) obtains higher per capita∗ (C, C) > k ∗ (C, C)). This causes the so-called beneﬁt of smallness in which u∗ (C, C) <capital (kBAAj∗uBj (C, C) for all j such that θAj = θBj , as well as ineﬃciency in capital allocation (the total∑
output i fi (ki )Li is not maximized when fA0 (kA ) > fB0 (kB )). All these properties are shown in
the Appendix.
In contrast, under (N, N ), we show in the Appendix that θAP > θBP for sA > 0.5 > sB . Thatis, under non-commitment to the tax rates, the high-population country sends a policymaker who
is richer than that of the low-population country to counteract the disadvantage generated by thecapital-tax competition. As a result, the diﬀerence and ineﬃciency of the allocation of the capitalis reduced. We show in the Appendix the following:
Lemma 2 Suppose that countries diﬀer in population size. Non-commitment reduces the ineﬃ∗ (C, C))−f 0 (k ∗ (C, C)) > f 0 (k ∗ (N, N ))−f 0 (k ∗ (N, N )) >ciency of the allocation of the capital: fA0 (kA
B B
A A
B B
0.
In this scenario, the low-population country (country B) has higher capital demand elasticity∂ k̂AsB ∂ k̂B∂ k̂B(sensitivity of the tax base to ﬁscal rates): from (1) and (2), −=−<−. This∂tA
sA ∂tB∂tB
eﬀect tends to make country B’s tax rate lower than that of country A. However, the terms-oftrade eﬀect works in the opposite direction because higher tax rates beneﬁt the capital importerand harm the capital exporter through lower interest rates. In conventional models, the terms-oftrade eﬀect is no more important than the tax-base-elasticity eﬀect (Bucovetsky (2009, Lemma 3)).
Here, however, strategic delegation works to increase the capital importer’s tax rate to a greater
extent than that of the capital exporter.
Also, in this subsection, as well as the following subsections, non-commitment reduces the returnof capital (r∗ (N, N ) − r∗ (C, C) < 0: see the Appendix). From (3) and θim < 1, non-commitment
tends to increase the utility of the median voters.
Noncooperative choice of the regime is consistent with the fact that median voters are madebetter oﬀ under non-commitment. Here, there is no beneﬁt of commitment to ti in either country,
regardless of the other country’s choice.

4.2

Countries with Diﬀerent Productivity (aA > aB , sA = sB , θim = θm , k i = k)

In this subsection, we consider the case of aA > aB where country A has higher capital productivity,
keeping symmetry on other dimensions (si = 0.5, θim = θm , k i = k). For convenience, we deﬁne9aA − aB
> 0.
k
In Stage 4, due to the diﬀerence in national productivity (aA > aB ), country A becomes a capital
importer and country B becomes a capital exporter. This eﬀect causes the tax-the-foreigner eﬀect,
which results in tA > tB in the third stage. From (1), we have fA0 (k̂A (tA , tB )) > fB0 (k̂B (tA , tB )),i.e., the allocation of capital is ineﬃcient.
Regarding the second-mover incentives (u∗im (N, C) > u∗im (C, C)), we apply Hindriks and Nishimura(2017) here, in which the likelihood of the second-mover incentives is higher when (a) δ is low and/or(b) θm is low.8 When asymmetry increases, the preferred tax rate in region A (capital importer)and region B (capital exporter) tends to diverge. This divergence of the interest across policymakers makes the sequential move less attractive, and both countries try to take a lead in setting the
tax rates. Thus, simultaneous-move (commitment to ti by both countries) becomes more likely tohappen. Greater capital ownership by the median voter generates the same eﬀect. The similar tendency exists for the other margin (u∗im (N, N ) > u∗im (C, N )): the choice of the strategic-delegation
outcome against the Stackelberg leader is more likely when δ and/or θm are low. As for the tax
choices with non-commitment, a country with lower tax-base elasticity9 is a capital importer, so
the tax gap increases through non-commitment. In the Appendix, we show:
the term δ ≡tA (θAP , θBP ) − tB (θAP , θBP ) =δk− (θAP − θBP )k.
2(7)

Again, we compare the regimes (C, C) and (N, N ) which are the only possible equilibrium regimes.
The ﬁrst term on the right-hand side of (7) captures the tax-the-foreigner eﬀect by the highproductive country, which is present even under commitment (tA > tB when θiP = θm = θ−iP ).
Under non-commitment, the second term strengthens the diﬀerence (θAP < θBP as shown in
the Appendix) so that the tax gap is widened under non-commitment (t∗A (N, N ) − t∗B (N, N ) >t∗A (C, C)−t∗B (C, C) > 0), which reduces capital import from country A to country B and makes the∗ (N, N )) −extent of allocation ineﬃciency worse under non-commitment: from (1), we have fA0 (kA
0
∗
0
∗
0
∗
fB (kB (N, N )) > fA (kA (C, C)) − fB (kB (C, C)) > 0).
Lemma 3 Suppose that countries diﬀer in productivity of the capital. Non-commitment worsens
the ineﬃciency of the allocation of the capital.
Regarding the choice of regimes in Stage 1, we have:
Lemma 4 Suppose that countries diﬀer in productivity of the capital, with aA = aB + δk. Thereexist 0 < δ 1 < δ 2 < δ 3 < δ 4 such that:
8

In Hindriks and Nishimura (2017), θ denotes the share of the domestic ownership of the capital. What mattersis the extent that the policymaker or the decisive voter cares about the value of the capital endowment. The caseof θm → 1 which we analyze below corresponds to that of the full domestic capital ownership addressed in Ogawa(2013).
9
For the case of aA > aB , evaluated at equal tax rates, the capital demand elasticities at tA = tB = t are∂ki tit. Therefore, country A has lower capital demand elasticity than country B.
|εki /ti | = −=∂ti kiai − a−i + 4k

10

(i) For δ ∈ (0, δ 1 ), non-commitment by both countries constitutes the dominance-solvable equilibrium.
(ii) For δ ∈ (δ 1 , δ 2 ), the equilibrium outcomes are (N, N ) and (C, C). Both median voters are
made better oﬀ under (N, N ) than under (C, C).
(iii) For δ > δ 2 , commitment by both countries constitutes the dominance-solvable equilibrium.
(iv) For δ < δ 3 , both median voters are made better oﬀ under (N, N ) than under (C, C). For
δ > δ 4 , both median voters are made better oﬀ under (C, C) than under (N, N ).
δ d (d = 1, 2, 3, 4) are decreasing in θm : with greater θm , commitment by both countries is more
likely to be an equilibrium.
For low values of δ, N is not only a plausible but also a desirable outcome. However, with suﬃciently
high asymmetry or suﬃciently high θm , commitment by both countries becomes not only theequilibrium outcome but also the desirable outcome for the median voters. In some intermediate
range of the degree of asymmetry, there is a discrepancy between the equilibrium outcomes and thewelfare (of the median voters): in case (ii), a welfare inferior outcome of (C, C) appears as anotherequilibrium; moreover, for δ ∈ (δ 2 , δ 3 ) ≈ (9.82(1 − θm ), 9.85(1 − θm )), the equilibrium outcome(C, C) is worse than (N, N ) for median voters: there is a prisoners’-dilemma situation in Stage 1,
since the divergence of the preferred tax rates by median voters makes both countries take a leadthrough commitments on ti .
The following proposition illuminates the determinants of the equilibrium commitment, withthe descriptions of representative limit cases:
Proposition 3 Suppose that countries diﬀer in productivity of the capital, with aA = aB + δk.
(i) Suppose that δ → 0. Then, for all θm < 1, (a) the choice of no-commitment is the dominant
strategy for both countries; and (b) both median voters are made better oﬀ under (N, N ) than under(C, C).
(ii) Suppose that θm → 1. Then, for all δ > 0, (a) the choice of commitment is the dominant
strategy for both countries; and (b) both median voters are made better oﬀ under (C, C) than under(N, N ).
Proposition 3.(i) is consistent with Persson and Tabellini (1992) which dealt with symmetric countries of θm < 1. In contrast, Proposition 3.(ii) refers to the case when the decisive voter cares theaverage of the national capital endowment, or the case where the inequality of income distribution
is not a driving force of determining ti . In short, contrary to Persson and Tabellini (1992), withmore equal distribution of the capital endowment within each country and with suﬃciently highasymmetry across countries, commitment by both countries becomes the equilibrium outcome.

11

4.3

Countries with Diﬀerent Median Voters (θAm > θBm , sA = sB , ai = a, k i = k)

In this subsection, we discuss the case where the median voters’ endowments diﬀer (θBm < θAm <1). The case of diﬀerent θim ’s is due to diﬀerent income distribution,10 which is discussed in thestrategic-delegation literature since Persson and Tabellini (1992).
In this case, the poorer median voter (in country B) aims for higher tax due to stronger
redistributive motive, so that θBP < θAP under the regime (N, N ). Capital ﬂows to the low-tax
country (country A), but the diﬀerence in the redistributive motive dominates the tax-the-foreigner
eﬀect, so non-commitment widens the tax gap (t∗B (N, N ) − t∗A (N, N ) > t∗B (C, C) − t∗A (C, C) > 0)
which leads to the following conclusion:
Lemma 5 Suppose that countries diﬀer in income distribution. Non-commitment worsens the∗ (N, N )) − f 0 (k ∗ (N, N )) > f 0 (k ∗ (C, C)) −ineﬃciency of the allocation of the capital: fB0 (kB
A A
B B
0
∗
fA (kA (C, C)) > 0).
However, we obtain the following result:
Proposition 4 Suppose that countries diﬀer in income distribution. For all 1 > θAm > θBm > 0,(i) The choice of no-commitment is the dominant strategy for both countries.
(ii) Both median voters are made better oﬀ under (N, N ) than under (C, C).
In contrast with Section 4.1 (diﬀerences in population), when the relevant dimension of country
heterogeneity is income distribution, the ineﬃciency of the allocation of capital is worsened by
non-commitment. However, as in Proposition 2, Regime (N, N ) is not only an equilibrium outcome
but also a desirable outcome in terms of the utility of the majority of citizens in both countries.
However, regarding the welfare of the citizens, since non-commitment reduces production in country
B and the equilibrium interest rate, the choice of (N, N ) unambiguously reduces the welfare of richer
citizens in country B.

4.4

Countries with Diﬀerent Capital Endowments (k A > k B , sA = sB , ai =a, θim = θm )Suppose that country A has a higher capital endowment. In this case, country A becomes a capital
exporter and tends to lower the tax rate. However, country A also has a bigger tax base, and thus
can obtain a higher level of tax revenue from the same tax rates. The tax competition leads to
fA0 (kA ) < fB0 (kB ). Parallel to the previous propositions, we obtain the following:
Proposition 5 Suppose that countries diﬀer in capital endowments, with k A = k B (1 + ξ). With
more equal distribution of the capital endowment within each country and with suﬃciently high10

Alternatively, it can result from diﬀerent political participation across citizens in diﬀerent income classes, as in
Benabou (2000).

12

asymmetry across countries, commitment by both countries is more likely to become the equilibriumoutcome.11
As to the tax gap, t∗B (N, N ) − t∗A (N, N ) > t∗B (C, C) − t∗A (C, C) > 0 as in Section 4.2 (see the
Appendix), so that non-commitment leads to more ineﬃcient allocation of capital. With moreequal distribution of the capital and more country asymmetry, the diﬀerence in interests between
the capital importer and the capital exporter becomes more pronounced. These are the drivers tomake commitment by both countries become the equilibrium outcome.

4.5

General Remarks

One of the general features in this section is as follows: regardless of the types of asymmetry
under consideration, asymmetric regimes (namely, (C, N ) and (N, C)), equivalently, the Stackelberg
outcomes, never constitute (pure-strategy) equilibrium. In Sections 4.1 and 4.3, the second-moverincentives (u∗im (N, C) > u∗im (C, C)) and the non-commitment in the other margin (u∗im (N, N ) >u∗im (C, N )) persist, regardless of the degree of asymmetries. In Sections 4.2 and 4.4, the secondmover incentives cease to hold under suﬃcient asymmetry, but the critical degree of asymmetrythat switches the second-mover incentives (δ 1 for country B in Lemma 4)12 is lower than that of theother margin (δ 2 for country B in Lemma 4).13 In short, N is more likely to be the best-response
against N than against C. Starting from (N, N ) being the equilibrium under symmetric countries,
we ﬁnd a suﬃciently high degree of asymmetries (or suﬃciently high θim ) at which both countries
have C to be the best-response to C.
In Sections 4.2 and 4.4, in some intermediate range of the degree of asymmetry, the above
tendency to choose C generates a prisoners’-dilemma situation in which the equilibrium outcome(C, C) is worse than (N, N ) for median voters (δ ∈ (δ 2 , δ 3 ) in Lemma 4).

Appendix
Derivation of (6)
∂uim
∂ k̂i
∂ r̂
∂ r̂
∂uim
∂ k̂i
∂ r̂
= ti
+(θim k i − ki )= (θim −θiP )k i
. Also,= ti+ (θim k i − ki )
.
∂ti
∂ti
∂ti
∂ti
∂t−i
∂t−i
∂t−i
(
)
∂ k̂i
∂ k̂i
∂ r̂
∂ r̂
∂uim
∂ r̂
∂ r̂
Since
=−
and
= − −1, we have
= (θiP k i − ki )− (θim k i − ki )
+1 .
∂t−i
∂ti
∂t−i
∂ti
∂t−i
∂ti
∂ti
∂t−i
∂ti ∂τ−iFromderived below in (A.4), the ﬁrst-order condition of (5) with respect to θiP=∂θiP∂θiP ∂ti
From (4),11
The formal description that supports the statement here is relegated to Lemma 6 in the Appendix. Also, thestatement of Proposition 3 holds here with respect to the asymmetry parameter ξ.
12The same nature is observed for Section 4.4. Here, we write the case of Lemma 4 for a representative illustration.
13
The same features apply to country A, as shown in the Appendix.

13derives:)(∂uim ∂ti∂uim ∂t−i∂τ−i ∂ r̂ ∂ti∂τ−i ∂ti+= (θim − θiP )k i 1 −− (θim k i − ki )= 0. (A.1)∂ti ∂θiP∂t−i ∂θiP∂ti∂ti ∂θiP∂ti ∂θiP()∂τ−i ∂ r̂ ∂ti∂τ−i ∂ti
Below we show that 1 −> 0 and< 0, so that (A.1) derives (6).
∂ti∂ti ∂θiP∂ti ∂θiP∂ r̂
00 si ∂ k̂i
Since= −f−i, (4) is rearranged to∂tis−i ∂ti
(
)
∂uim
∂ k̂i
∂ k̂i
00 si
=
ti − (θiP k i − ki )f−i
≡ βi (ti , t−i )
= 0,
(A.2)
∂ti
s−i ∂ti∂ti
00 si . Note that
where βi ≡ ti − (θiP k i − ki )f−is−i

condition yields14∂ k̂i< 0. A total diﬀerentiation of the ﬁrst-order∂ti∂τi (t−i ; θiP )∂βi /∂t−i=−.
∂t−i∂βi /∂ti
000 = 0, using
When f−i(A.3)

si 00 ∂ k̂i
si 00 ∂ k̂i
∂ k̂i
∂ k̂i
∂τi (t−i ; θiP )s f−i ∂t−i
s−i f−i ∂ti=−, we have= − −i=∈ (0, 1).
∂t−i
∂ti
∂t−i
1 + si f 00 ∂ k̂i
1 + si f 00 ∂ k̂is−i −i ∂tis−i −i ∂ti= 0,
Totally diﬀerentiating the system of βi (ti , t−i ) = 0 and β−i ≡ t−i − (θ−iP k −i − k−i )fi00 ss−ii
we have)()() (00 si dθk i f−i∂βi /∂ti∂βi /∂t−i
dti
iP
s
−i
=
.
∂β−i /∂ti ∂β−i /∂t−i
dt−i
0
We therefore have
) k f 00 si()() k f 00 si (i −i s−ii −i s−i ∂β−i
∂β−i /∂t−i
1∂ti /∂θiP==,∂t−i /∂θiP−∂β−i /∂ti
∂τ−i /∂ti
∆
∆
∂t−i

(A.4)

∂βi ∂β−i∂βi ∂β−i−.
∂ti ∂t−i∂t−i ∂ti
The last equation of (A.4) follows from the formula of the reaction function (A.3)( applied to )∂β−i∂βi ∂β−i∂τi ∂τ−icountry −i. Note that
> 0 from the second-order condition, and ∆ =
1−
>
∂t−i
∂ti ∂t−i∂t−i ∂ti
where ∆ ≡14
In the derivation for (A.1) and (A.4), we do not rely on the quadratic production function. To derive (6), we only
need to assume strategic complementarity in tax rates (∂τ−i /∂ti > 0) which is conventional in the tax-competition
literature. Instead of a mechanical derivation based on the quadratic production function, we adopted a general
illustration here.

14

∂ti
∂τ−i
0. We therefore have
< 0 and 0 << 1, so∂θiP∂ti∂ti ∂τ−i< 0 hold.
∂θiP ∂ti

(

∂τ−i
1−
∂ti

)

∂r̂ ∂ti∂t−i> 0 and=∂ti ∂θiP∂θiP

Proof of Lemma 1
ki and r in (3) are determined by ki = k̂i (tA , tB ) and r = r̂(tA , tB ), so we represent uij = uij (t1A , t1B ).
uij (t1A , t1B ) = uim (t1A , t1B )+r̂(t1A , t1B )·(θij −θim )k i . Therefore, uij (t1A , t1B )−uij (t2A , t2B ) = uim (t1A , t1B )−uim (t2A , t2B )+(r̂(t1A , t1B )− r̂(t2A , t2B ))·(θij −θim )k i . Suppose that uim (t1A , t1B )−uim (t2A , t2B ) > 0. Then,if r̂(t1A , t1B ) > r̂(t2A , t2B ), then uij (t1A , t1B ) > uij (t2A , t2B ) for all θij > θim , and if r̂(t1A , t1B ) < r̂(t2A , t2B ),then uij (t1A , t1B ) > uij (t2A , t2B ) for all θij < θim . If r̂(t1A , t1B ) = r̂(t2A , t2B ), then uij (t1A , t1B ) > uij (t2A , t2B )for all θij . In all cases, the preference of the majority is consistent with that of the median voter.
Q.E.D.

Proof of Proposition 2 and Lemma 2
Begin with the Stage-4 equilibrium allocation of capital. In the quadratic model, solving (1) and(2) gives:s−i((ai − a−i ) − (ti − t−i )) .
k̂i (tA , tB ) = si k i + s−i k −i +
(A.5)
2
Also,
∑ (
)
si (ai − 2k i − ti )r̂(tA , tB ) =
(A.6)
i=A,B

Substituting (kA , kB ) = (k̂A (tA , tB ), k̂B (tA , tB )) into (A.2) (i = A, B) yields (tA , tB ) = (tA (θAP , θBP ),tB (θAP , θBP )).
In the quadratic model, with ki (θiP , θ−iP ) ≡ k̂i (ti (θiP , θ−iP ), t−i (θiP , θ−iP )), (A.1) in Stage 2 at
Regime (N, N ) is equivalent to:()
ki (θiP , θ−iP )s−i
θim − θiP = −
θim −
.
(A.7)
si
ki
Under the suppositions of ai = a−i = a, k i = k −i = k and θim = θ−im = θm , from (A.2) and (A.5),
we deriveti (θiP , θ−iP ) = −(−θ−iP s2−i + θiP s2i + 1)k(−θiP s2−i + θ−iP s2−i + θiP − 1)k, ki (θiP , θ−iP ) =.(A.8)s−i
2si

Solving (A.7) and (A.8) derives the delegate’s type in Regime (N, N ) as
θiP =−(1 − s2i ) + θm (1 + 2s2i )
.
3s2i
15

(A.9)

(2sA − 1)(1 − θm )> 0. In our notation here, ki∗ (N, N ) = ki (θiP , θ−iP )3s2A s2B(where θiP s are those of (A.9)) and ki∗ (C, C) = ki (θm , θm ). From (A.8), we have

We have θAP − θBP =ki∗ (N, N ) =(2si θm + si − θm + 1)k ∗(2si θm − θm + 1)k, ki (C, C) =
,
3si
2si

∗ (C, C) − k ∗ (C, C) > k ∗ (N, N ) − k ∗ (N, N ) > 0. In addition, from (A.6), wefrom which we have kBABA
2 + 2s + 1)(θ − 1)k
(−2s
m
A
A
have r∗ (N, N ) − r∗ (C, C) =< 0.
3sA sB
Regarding the beneﬁt of smallness,
2u∗Aj (C, C) − u∗Bj (C, C) =(1 − θm )2 (1 − 2sA )k< 0 for all θAj = θBj .
4s2A s2Bu∗Aj (N, N ) − u∗Bj (N, N )(1 − θm )2 (1 − 2sA )k< 0 for all θAj = θBj .
3s2A s2B2=∂uim ∂uim ∂τ−i+
= 0 and (4) (equiv∂ti∂t−i ∂tialently, (A.2)) applied to country −i and −iP = −im. We have:
The Stackelberg tax rates are the tax rates that satisfytLi =2k(1 − θm )2k(1 − θm )(1 + s−i ), tF−i =.
si (1 − si )(3 − si )(1 − s−i )(2 + s−i )We proceed to calculate the choice of Stage-1 strategy in Table 1.
2(1 − θm )2 (1 + si )(4 + 3si )k∗∗uim (N, C) − uim (C, C) =>0
4si (1 − si )(2 + si )2(i = A, B).

Moreover,2(1 − θm )2 (2 − si )(6 + 2si − s2i )k∗∗>0
uim (N, N ) − uim (C, N ) =9si (1 − si )(3 − si )(i = A, B).

Therefore, playing N is the dominant strategy for both countries, for all parameter values. Also,2u∗im (N, N ) − u∗im (C, C) =(1 − θm )2 (1 + si )(−4s2i + 8si + 3)k> 0 (i = A, B).
36s2i (1 − si )Thus Proposition 2.(ii) is veriﬁed. Q.E.D.

16

Proof of Lemma 3, Lemma 4 and Proposition 3
Under the suppositions of si = s−i = 0.5, k i = k −i = k and θim = θ−im = θm , from (A.2) and(A.5), we haveti (θiP , θ−iP ) = 2k +)ai − a−i (3θiP + θ−iP )k1(−, ki (θiP , θ−iP ) = k +(ai − a−i ) + 2(θiP − θ−iP )k .
4
2
8
(A.10)

Solving (A.7) and (A.10) derives the delegate’s type in Regime (N, N ) as
θAP = −1 + 2θm −δδ, θBP = −1 + 2θm + .
12
12(A.11)

From (A.6), we have r∗ (N, N ) − r∗ (C, C) = 2k(θm − 1) < 0. In addition, we have θAP < θBP .
(A.10) shows (7) in the text and t∗A (N, N ) − t∗B (N, N ) > t∗A (C, C) − t∗B (C, C) > 0. (1) derives theconclusion of Lemma 3.
The Stackelberg tax rates are, for I(A) ≡ 1 and I(B) ≡ 2:
tL
i =

16k
2
12k
1
(1 − θm ) − (−1)I(i) δk, tF−i =(1 − θm ) + (−1)I(i) δk.
55
55

We proceed to calculate the choice of Stage-1 strategy in Table 1.
(
)
2
88
27k
(δ − 8(1 − θm )) δ +(1 − θm ) ≷ 0u∗Am (N, C) − u∗Am (C, C) = −1600
9⇐⇒ δ ≶ 8(1 − θm ) ≡ δ 0 .

(A.12))
88(δ + 8(1 − θm )) δ −(1 − θm ) ≷ 0
1600
9

27k
u∗Bm (C, N ) − u∗Bm (C, C) = −⇐⇒ δ ≶

2(88(1 − θm ) ≡ δ 1 .
9(A.13))
108(δ − 12(1 − θm )) δ +(1 − θm ) ≷ 0
720
11

11k
u∗Am (N, N ) − u∗Am (C, N ) = −(2⇐⇒ δ ≶ 12(1 − θm ) ≡ δ 5 .

(A.14)

11ku∗Bm (N, N ) − u∗Bm (N, C) = −

(

2

108
(δ + 12(1 − θm )) δ −(1 − θm )720
11⇐⇒ δ ≶

108(1 − θm ) ≡ δ 2 .
11)≷0(A.15)

17

Note that we have δ 0 < δ 1 < δ 2 < δ 5 . Suppose ﬁrst that δ < δ 1 . From (A.13) and (A.15),
non-commitment is the dominant strategy for country B. Then, given country B’s choice of N ,from (A.14), country A chooses N . Therefore, (N, N ) is the dominance-solvable equilibrium. Next,
suppose that δ ∈ (δ 1 , δ 2 ). Then, from (A.12) and (A.13), C is the best-response to C for each other,and from (A.14) and (A.15), N is the best-response to N for each other. Finally, suppose thatδ > δ 2 . From (A.13) and (A.15), commitment is the dominant strategy for country B. Then, given
country A’s choice of C, from (A.14), country A chooses C. Therefore, (C, C) is the dominancesolvable equilibrium. Since δ d (d = 1, 2) are decreasing in θm , with greater θm , (C, C) is more likely
to be an equilibrium.
√u∗im (N, N ) − u∗im (C, C) > 0 for i = A, B
if δ < 24( 15 − 1)(1 − θm )/7 ≡ δ 3 > δ 2 . u∗im (N, N ) −√u∗im (C, C) < 0 for i = A, B if δ > δ 4 ≡ 24( 15 + 1)(1 − θm )/7 ≡ δ 4 > δ 3 . This proves Lemma 4.
For δ < δ 0 , (i) and (iv) of Lemma 4 apply to conclude Proposition 3.(i). In contrast, for θm → 1,then δ d (d = 0, 1, 2, 3, 4, 5) becomes 0, so (iii) and (iv) of Lemma 4 apply to conclude Proposition
3.(ii). Q.E.D.

Proof of Proposition 4 and Lemma 5Under the suppositions of si = s−i = 0.5, ai = a−i = a and k i = k −i = k, from (A.2) and (A.5) we
deriveti (θiP , θ−iP ) = 2k −)(3θiP k + θ−iP k)
1(, ki (θiP , θ−iP ) = k +θiP k − θ−iP k .
2
4(A.16)

Solving (A.7) and (A.16) derives the delegate’s type in Regime (N, N ) as
51
θiP = −1 + θim + θ−im ,3
3(A.17)

from which r∗ (N, N ) − r∗ (C, C) = k(θAm + θBm − 2) < 0. In addition, we have θAP − θBP >θAm − θBm > 0. (A.16) shows t∗B (N, N ) − t∗A (N, N ) > t∗B (C, C) − t∗A (C, C) > 0. (1) derives theconclusion of Lemma 5.
The Stackelberg tax rates are:
tL
i =

4k B
4k B
(4 − 3θim − θ−im ), tF−i =(3 − 2θ−im − θim ).
5
5

We proceed to calculate the choice of Stage-1 strategy in Table 1.
2

u∗im (N, C) − u∗im (C, C) =3(4 − θim − 3θ−im )(44 − 31θim − 13θ−im )k> 0. (i = A, B)
400

Moreover,2u∗im (N, N ) − u∗im (C, N ) =(3 − θim − 2θ−im )(27 − 19θim − 8θ−im )k> 0. (i = A, B)
45
18

Therefore, playing N is the dominant strategy for both countries for all parameter values. Also,for ηi ≡ θim − θ−im , we have:
(
(
√ )
√ )
2
156 12 15
156 12 15
77k
∗
∗
(ηi −
+
(1 − θ−im ))(ηi −−(1 − θ−im )).
uim (N, N ) − uim (C, C) =
144
77
77
77
77
(i = A, B)
Our supposition of θim = θ−im + ηi < 1 is equivalent to ηi < 1 − θ−im . For ηi < 1 − θim , we have:
(
(
√ )
√ )
156 12 15
156 12 15ηi −+(1−θ−im ) < ηi −−(1−θ−im ) ≈ ηi −1.4224(1−θim ) < 0. (i = A, B)
77
77
77
77
Thus Proposition 4.(ii) is veriﬁed. Q.E.D.

Proof of Proposition 5
Under the suppositions of si = s−i = 0.5, ai = a−i = a and θim = θ−im = θm , from (A.2) and (A.5)
we deriveti (θiP , θ−iP ) = k A + k B −)(3θiP k i + θ−iP k −i )11(, ki (θiP , θ−iP ) = (k A + k B ) +θiP k i − θ−iP k −i .
22
4
(A.18)

Solving (A.7) and (A.18) derives the delegate’s type in Regime (N, N ) as
(
)
(
)
1 1
1 1
ξ
− θm , θBP = −1 + 2θm − ξ− θm .
θAP = −1 + 2θm +1+ξ 2 3
2 3

(A.19)

44
ξθm k 2 = (t∗B (C, C) − t∗A (C, C)) > 0. Also, from33(A.6), we have r∗ (N, N ) − r∗ (C, C) = (k A + k B )(θm − 1) < 0.
The Stackelberg tax rates are:From (A.18), we have t∗B (N, N ) − t∗A (N, N ) =

tL
A =

4k B
2k B
(4(1 − θm ) + 2ξ − 3θm ξ), tFB =(6(1 − θm ) + 3ξ − 2θm ξ),
5
5

tL
B =

2k B
4k B
(4(1 − θm ) + 2ξ − θm ξ), tFA =(6(1 − θm ) + 3ξ − 4θm ξ).
5
5

and:

Regarding the choice of regimes in Stage 1, we now prove the following lemma that leads to
Proposition 5:
19

Lemma 6 Suppose that countries diﬀer in capital endowments, with k A = k B (1 + ξ). For 0 <ξ 1 < ξ 2 < ξ 3 < ξ 4 we have:(i) For ξ ∈ (0, ξ 1 ), non-commitment by both countries constitutes the dominance-solvable equilibrium.
(ii) For ξ ∈ (ξ 1 , ξ 2 ), the equilibrium outcomes are (N, N ) and (C, C). Both median voters are
made better oﬀ under (N, N ) than under (C, C).
(iii) For ξ > ξ 2 , commitment by both countries constitutes the dominance-solvable equilibrium.
(iv) For ξ < ξ 3 or suﬃciently low θm , both median voters are made better oﬀ under (N, N ) thanunder (C, C). For ξ > ξ 4 and suﬃciently high θm , both median voters are made better oﬀ under(C, C) than under (N, N ).
With greater θm , commitment by both countries is more likely to be an equilibrium.
Proof of Lemma 6:(k 2 )u∗Am (N, C) − u∗Am (C, C) =2 · 33(((1 − θm ) −

25

31
1
θm −
44
2

1 − θm22
1
1⇐⇒ ξ < 31
1 ≡ ξ or θm < 31 ≡ θ .
θ
−
m
44
2

) )(
)
1
1
ξ
1 − θ m + ξ − θm ξ > 0
2
4(A.20)(() )()(k 2 )2 · 33
311
13
(1 − θm ) −θm −ξ
1 − θ m + ξ − θm ξ > 0
25
4
2
2
44
1 − θm200⇐⇒ ξ < 3(A.21)
1 ≡ ξ or θm < 3 ≡ θ .
θ
−
m
4
2

u∗Bm (C, N ) − u∗Bm (C, C) =

(
(
) )(
)
(k 2 )2 · 9
191
11(1 − θm ) −θm −ξ
1 − θ m + ξ − θm ξ > 0
5
27
2
2
3
1 − θm27
2
2⇐⇒ ξ < 19(A.22)
1 ≡ ξ or θm < 38 ≡ θ .
θ
−
m
27
2

u∗Am (N, N ) − u∗Am (C, N ) =(k 2 )u∗Bm (N, N ) − u∗Bm (N, C) =

2·9

5

(

(
(1 − θm ) −3
1 − θm5
5⇐⇒ ξ < 2
1 ≡ ξ or θm < 4 ≡ θ .
3 θm − 2

2
1
θm −
3
2

) )(
)
1
8
ξ
1 − θ m + ξ − θm ξ > 0
2
27(A.23)We have θ0 < θ1 < θ2 < θ5 .
Suppose ﬁrst that θm > θ2 . We have 0 < ξ 0 < ξ 1 < ξ 2 . Using (A.20)-(A.23), the proof is
analogous to that of Lemma 4 to induce statements (i)-(iii) of Lemma 6 for the cases of (i) ξ < ξ 1 ,(ii) ξ ∈ (ξ 1 , ξ 2 ), and (iii) ξ > ξ 2 respectively. Second, for θm < θ2 , from (A.22) and (A.23), (N, N )20

always constitutes an equilibrium. For θm < θ2 , if θm > θ1 and ξ > ξ 1 (> ξ 0 ), (C, C) constitutes
another equilibrium. Finally, for (a) θm < θ1 or (b) θm ∈ (θ1 , θ2 ) and ξ < ξ 1 , from (A.20), (A.22)
and (A.23), (N, N ) is the dominance-solvable equilibrium.
√
12(13 − 15)(1 − θm )√uim (N, N ) − uim (C, C) > 0 for i = A, B if ξ <≡ ξ 3 > ξ 2 or θm <
77
θ
−
78
+
6
15
m
√
√
78 − 6 15
12(11 − 15)(1 − θm )3
2√≡
≡ θ > θ . uim (N, N ) − uim (C, C) < 0 for i = A, B if ξ >77
53 θm − 66 + 6 15√
66 − 6 15
ξ 4 > ξ 3 and θm >≡ θ4 > θ3 .
53
1 − θmSince ξ d = 2(d = 0, 1, 2, 3, 4, 5, where θd < 1) is decreasing in θm , commitment byθm /θd − 1both countries is more likely to be an equilibrium with greater θm . Q.E.D.

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