Enforcement Cooperation and Tax
Competition Among Asymmetric Countries∗
Jean HINDRIKS1 and Yukihiro NISHIMURA2

July 26, 2017
Abstract
Using a proﬁt shifting model with multinational enterprises that
operate in two countries, large and small, we analyze the determinantsof voluntary cooperation on enforcement eﬀort and information sharing arrangements. The beneﬁt from tougher enforcement by one country is the mitigation of the tax competition, so that enforcement choice
has a public-good nature. In a framework that countries decide tax
rates noncooperatively, we compare the equilibria of the noncooperative and cooperative choice of enforcement. With suﬃcient disparity
in the country sizes, the low-tax country is not willing to participate
enforcement coordination. We then consider two extensions. The ﬁrst
extension is complementarity (imperfect substitutability) of countries’
enforcement eﬀorts, which comes from administrative or legal natureof the collective decision-making. We show that cooperation is more
viable with greater enforcement complementarity. In the second extension, we consider the Stackelberg tax competition where the large
country leads. Compared with the simultaneous tax choice, both the
tax leader and the tax follower will exert more enforcement eﬀorts inthe sequential tax choice. As a result, the enforcement cooperation
will be more viable under the Stackelberg tax competition.
JEL Classiﬁcation: C72, F23, F68, H25, H87.
Keywords: Proﬁt shifting, Tax competition, Information sharing,Tax leadership∗

Yukihiro Nishimura acknowledges the ﬁnancial support from the Grants-in-Aid for
Scientiﬁc Research (C) from the Ministry of Education, Culture, Sports, Science, andTechnology (24530348, 15K03511) and Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation from the Japan Society for the Promotion of
Science (J2402).
1
CORE, Université catholique de Louvain. jean.hindriks@uclouvain.be2
Graduate School of Economics, Osaka University. ynishimu@econ.osaka-u.ac.jp1

1

Introduction

Globalization has prompted the emergence of multinational enterprises (MNEs)
with divisions in diﬀerent countries. Under the widely used source-based
taxation (also known as territorial system),1 these MNEs found that they
can reduce their overall tax liabilities by shifting proﬁt and income between
branches, which caused a well-known problem of Base Erosion and Proﬁt
Shifting (BEPS, hereafter referred to as “proﬁt shifting”). The presenceof proﬁt shifting gets convincing empirical support.2 To address this issue,OECD’s (2015) report proposed global action plans, which contains a seriesof speciﬁc enforcement eﬀorts to be taken by the governments.3 However,a real challenge to international coordination is the absence of enforceable
global institution, and, in addition, the diﬀerence across countries regarding
preferred tax rates on the corporate incomes. Indeed, in reality, the enforcement of legislations diﬀer signiﬁcantly across countries: some legislations only
loosely acknowledge the “arm’s length principle”, whereas others ask ﬁrms tosubmit detailed transfer pricing reports for strict tax compliance purposes.
In this paper, we analyze the determinants of voluntary cooperation on
enforcement eﬀort and information sharing arrangements. To address thepresent issues in a tax competition framework, we develop a simple twocountry model with diﬀerent market sizes, which follows Kanbur and Keen
(1993), Hindriks et al. (2014) and Keen and Konrad (2013). MNEs shift
1

Most OECD countries adopted territorial systems, since it is relatively easy for taxpayers to avoid resident-based taxes on capital income. Bacchetta and Espinosa (1995) showedthat countries do not choose the residence-based principle when information transmission
can be strategic. Proﬁt shifting is also observed under the tax system of the UnitedStates; see, for example, Swenson (2001), Clausing (2003), Klassen and Laplante (2012),
and Grubert and Altshuler (2013).
2
Swenson (2001) and Clausing (2003) found that taxation has a signiﬁcant eﬀect on
intra-ﬁrm prices of the MNEs in the United States. Mintz and Smart (2003) found that
Canadian ﬁrms which operate in multiple jurisdictions have a high elasticity of taxable
income with respect to corporate tax rates. Bartelsmann and Beetsma (2003) used thedata from 22 OECD countries and found that more than 65 percent of the additional
revenue resulting from a unilateral tax increase is lost because of proﬁt shifting. Huizinga
and Laeven (2008) examined European MNEs and found that proﬁt shifting leads to a
substantial redistribution of corporate tax revenues. IMF (2011) used Bankscope dataon banks to ﬁnd that 1 percent-point higher tax rate reduces reported ﬁscal proﬁts by
between 6 and 8.5 percent. In the ﬁnancial sector, Næss-Schmidt et al. (2012) found thata one percent increase in the tax rate leads to a 1.9 percent decrease in an MNE’s aﬃliates.
See also Devereux and Griﬃth (2003) and Auerbach et al. (2010).
3
These proposals include countering harmful tax practices (Action 5), mandatory disclosure of suspected aggressive tax planning (Action 12), transfer pricing documentationand country-by-country reporting (Action 13), and eﬀectiveness of dispute resolution (Action 14).

2

proﬁts from the high to the low tax country’s division, subject to a concealment cost. Countries choose the level of enforcement eﬀorts, such as
more eﬃcient information sharing, strict monitoring and inspection, and theeﬀort for negotiation and agreement with the other country’s tax authority. The enforcement eﬀorts incur administration costs to each country. But
its beneﬁt, the reduction of aggressive tax planning that mitigates the tax
competition, is diﬀused to the other country. Therefore, enforcement choicebecomes a game of voluntary contribution to a public good. We compare theequilibria of the noncooperative and cooperative choice of enforcement. In
the latter scenario, countries choose the enforcement levels so as to maximize
their joint welfare, but countries still decide tax rates noncooperatively.4 In
the absence of lump-sum transfers to share the gain of cooperation, with
suﬃcient disparity in the country sizes, the low-tax country is not willing toparticipate enforcement coordination (Proposition 1). It is of central concern
in the application of coordination proposals to OECD, G20 and the European
Union (EU), since the adoption of such proposals often requires unanimity.
We will then identify two diﬀerent drivers of cooperation. The ﬁrst driveris complementarity in the enforcement choices. Enforcement eﬀorts increase
the cost of proﬁt shifting. However, in reality, dispersed (unilateral) enforcement eﬀorts by the involved countries are not eﬀective. For instance, thelack of exchange of tax-relevant information by host country makes the taxable income unclear to the home country. Indeed, for recommendations of
the OECD BEPS Project, including mandatory disclosure of cross-border tax
planning (Action 12) and advance pricing arrangement (Action 14), countries
are free to choose whether to introduce them or not, based on each country’s domestic tax regime.5 Also, during the mutual agreement procedure ontransfer prices and taxable incomes of the MNEs, the capital-importing country may well be inclined to favor the MNE’s ﬁnancial and legal expert’s view
towards lower transfer prices and higher taxable incomes to its own. Given
that double taxation is not allowed by the tax treaty, the capital-importing
country can thus exercise a veto power towards tighter enforcement.6 To formalize the notion of imperfect substitutability (complementarity), we apply
Hirshleifer’s (1983) social composition function. The case of veto power corresponds to the weakest-link formula, where the capital-exporting country has
to compromise to the minimum of the enforcement standard. If in contrast,
4

Host country retains the primary taxing rights over source-based incomes. Accordingly, preferred tax rates diﬀer across countries.
5
Such weak forms of recommendation are called “Best Practices”. Action 14 (eﬀectiveness of dispute resolution) is amended with eleven Best Practices for implementation.
6
This scenario in a collective decision-making was mentioned by Hirshleifer (1983, p.
373).

3

relevant information including the transfer pricing documentation from eachcountry (Article 13 of OECD (2015)) can be added up towards better tax
enforcement, then the associated eﬀorts have more substitutable (additive)
nature. By parametrically treating the degree of complementarity, we showthat cooperation is more viable with greater enforcement complementarity(Proposition 2).
The second driver of cooperation is tax leadership. The analysis of endogenous leadership has attracted research interest since Kempf and RotaGraziosi (2010).7 Following Hindriks and Nishimura (2015), we consider thecase in which the large country leads.8 Compared with simultaneous tax
choice, sequential tax choice makes both the tax leader and the tax follower
exert more enforcement eﬀort. Moreover, the Stackelberg tax competition
reduces the dispersion of the enforcement eﬀorts, provided that asymmetryis suﬃciently large (Lemma 5). As a result, the enforcement cooperation will
be more viable than under the Nash tax competition (Proposition 3).
The main interest of the previous studies was to understand how transfer
prices are aﬀected by international tax diﬀerences and tax systems (for example, Swenson (2001), Kind et al. (2005), Devereux et al. (2008), Nielsen
et al. (2008), Huizinga and Laeven (2008), Amerighi and Peralta (2010),
and Klassen and Laplante (2012)). Some evidence also showed that transfer
pricing regulations signiﬁcantly mitigate proﬁt shifting (for example, Bartelsmann and Beetsma (2003) and Lohse and Riedel (2013)). However, thesestudies did not examine the choice (determinants) of enforcement eﬀorts bycountries with diﬀerent economic structure.9 In a recent paper, Hindriks etal. (2014) proposed a solution based on a system of voluntary tax sharing
agreement among asymmetric countries. In this paper, we endogenize the
MNE’s concealment cost and examine policy issues with respect to enforcement complementarity and tax leadership. Cremer and Gahvari (2000) andStöwhase and Traxler (2005) examined the implications of tax evasion for7

Altshuler and Goodspeed (2015) demonstrated that European countries set their corporate tax rates following the United States 1986 Tax Reform Act. Also, Redoano (2007)
and Chatelais and Peyrat (2008) provided evidence of sequential tax decisions among European countries. Stöwhase (2013) adopted Stackelberg setting, simply to avoid the difﬁculties resulting from possible discontinuities in the payoﬀ functions with simultaneous
tax setting. As such, in his framework, the Nash and the Stackelberg tax competitions are
not comparable. Also, unlike ours, a public-good nature of the tax enforcement is absent.
8
In our model, the large leadership equilibrium Pareto dominates the small leadershipequilibrium under suﬃcient asymmetry, as in Hindriks and Nishimura (2015).
9
Exceptions are Bacchetta and Espinosa (1995), Peralta et al. (2006), and Bucovetsky
and Hauﬂer (2008). However, our formal model diﬀers from these papers. More importantly, none of these papers discussed the determinants of the emergence of voluntarycooperation on enforcement eﬀort.

4

tax competition. However, unlike ours, the tax evasion and the tax enforcement (auditing) in their papers do not have horizontal interactions.10 In
contrast, in the BEPS problem, the shifted incomes from one branch (netof the concealment cost) become taxable incomes of other countries. As
such, one country’s enforcement eﬀorts directly aﬀect the tax base of othercountries. Using spatial econometric approach, Durán-Cabré et al. (2015)
conﬁrmed positive horizontal interactions between regional administrationsin audit policies among Spanish regional governments.
The rest of the paper is organized as follows. Section 2 describes themodel and the timing of the game. Section 3 derives the equilibrium under noncooperative and cooperative enforcement. Section 4 examines two
extensions, enforcement complementarity and tax leadership. Section 5 concludes. The proofs of several propositions and lemmas are provided in the
Appendices.

2

Framework

2.1

The Model

The model used follows Hindriks et al. (2014) and Keen and Konrad (2013).
There are two countries denoted by 1 and 2. Each country is characterized by
the linear (inverse) demands for a homogeneous good pi (qi ) = αi − βqi (i =1, 2). Two multinational enterprises (MNEs), a and b, have branches in eachcountry and compete à la Cournot in each domestic market. For each ﬁrm,
production incurs the country-speciﬁc unit cost ci ≥ 0 (i = 1, 2). We assume
α1 − c1 ≡ γ1 ≥ α2 − c2 ≡ γ2 > 0.
Namely, Country 1 is the large country, either because (i) αi that represents
population size or income per capita is higher, or because (ii) the supply costci is lower.
Associated with production decisions (qia , qib ), in country i = 1, 2, ﬁrm
k = a, b generates πik = {pi (qia + qib ) − ci }qik in country i. Then it may, atsome cost, shift proﬁts between branches so as to minimize their total tax
liability. Namely, it decides how much proﬁt to report, π̃ik in country i, where
total reported proﬁt must equal total realized proﬁt (π̃1k + π̃2k = π1k + π2k ).
Given country i’s source-based tax rate ti on( the reported
proﬁt, the ﬁrm k’s)
proﬁt becomes (1 − t1 )π̃1k + (1 − t2 )π̃2k − C πik , π̃ik . We follow Hauﬂer and
10

In Cremer and Gahvari (2000) and Stöwhase and Traxler (2005), a ﬁrm decides on
concealment depending solely on the tax rates of the home country, and the convex auditing
cost does not depend on the intensity of auditing of the other countries.

5Schjelderup’s (2000) transfer-pricing modeland() introduce a convex and nonﬁscally deductible concealment cost C πik , π̃ik (widely used by, for example,
Swenson (2001), Kind et al. (2005), Peralta et al. (2006), Devereux et al.
(2008), Nielsen et al. (2008), and Keen and Konrad (2013)):11)2)((
C πik , π̃ik = 2δ(e) πik − π̃ik , i = 1, 2 and k = a, b.

(1)

Several explanations are in order. First, δ(e) is a scaling factor of resource
costs associated with proﬁt shifting or transfer pricing. It reﬂects the costof hiring accounting experts, or the expected market sanction when caught
cheating on tax liabilities. As a standard assumption in the literature, theconcealment cost is an increasing and convex function of the amount of proﬁt
shifting, |πik − π̃ik |, independent of the direction of shifting (outward andinward shiftings are cost-equivalent).
Second, δ(e) = δ(ei , ej ) depends on the governments’ enforcement eﬀorts
ei , ej , such as more eﬃcient information sharing, tougher monitoring, and theeﬀort for negotiation and agreement with the other country’s tax authority.
δ(e) is an increasing function of ei and ej , so that stricter enforcement implies
higher δ(e). It can either be additively separable or there is interdependence
between ei and ej . In the former case, the cost (1) can be interpreted as
the tax compliance costs separately required by the two governments (forexample, documentation requirements).12
Tax revenue in country i is()
Ri = ti π̃ia + π̃ib
We assume that governments seek to maximize their ﬁscal revenue net ofthe enforcement cost (the tax administration costs). Adding the consumer
surplus in the governmental objective function will not aﬀect the analysis
because the consumer surplus is independent of the tax choices. This feature
is synonymous to widely used model by Kanbur and Keen (1993). We assume
11

See also Huizinga and Laeven (2008) and Amerighi and Peralta (2010) for a slightly
diﬀerent speciﬁcation. We do not expect our results to be inﬂuenced by this alternative
speciﬁcation. In a related model of international debt shifting by MNEs, Huizinga etal. (2008) considered the model of allocating tax allowances of debt ﬁnance, whereby
allocation of the debt aﬀects the expected bankruptcy costs which are not tax-deductible.
As in Lemma 1 below, the ﬁrm utilizes international tax rate diﬀerences on the debt
allocation. The concealment cost may reﬂect penalties to be paid to the government.
However, as a standard assumption in this literature, we consider the scenario that theconcealment is a pure waste of resources.
12
Klassen and Laplante (2012) showed that proﬁt shifting in a given country depends
not only on the enforcement of the regulations in the home country but also on theimplementation of the regulations in the host country.

6that ti ≤ 1, for i = 1, 2. Assuming quadratic cost of enforcement (c(ei ) =e2
η 2i ), welfare in country i is
Wi = Ri − η( e )2
i

2

,

(2)

where η > 0 is a parameter for the enforcement cost. To ensure interior
solution (ti ≤ 1 (i = 1, 2)) in the equilibrium, we assume throughout therest of the paper that η ≥ 3.

2.2

Proﬁt Shifting by Firms

The sequence of events is as follows. First, both countries set their enforcement eﬀorts and tax rates. Subsequently, given enforcement and tax choices,multinational enterprises compete à la Cournot in each local market andchoose a level of production in each country and the proﬁt to be shifted.
The issue of tax and enforcement timing will be described and discussed
in the next subsection. In this subsection, we ﬁrst analyze ﬁrms’ decisions ineach country, given the tax t = (t1 , t2 ) and enforcement e = (e1 , e2 ) choices
made earlier. Firm k (k = a, b) chooses the quantities to produce in eachmarket, (q1k , q2k ) and the proﬁt to report, (π̃1k , π̃2k ), so as to maximize theafter-tax proﬁt net of the shifting cost
(1 − t1 )π̃1k + (1 − t2 )π̃2k − 2δ(e)(π1k − π̃1k )2 ,√√subject to π̃1k + π̃2k = π1k + π2k . Let γ1 = 23 β(1 + ²), γ2 = 23 β(1 − ²) with
² ∈ [0, 1), and let π̃i = π̃ia + π̃ib be the total reported proﬁt in country i.
Hindriks et al. (2014) showed the following:13
γ12
1+²
1−²
≡
, π2k =
(k = a, b), π̃1 = π̃1a + π̃1b =9β
4
4
1 + ² t1 − t2
1 − ² t1 − t2−
, π̃2 = π̃2a + π̃2b =
+
.
2
2δ(e)
2
2δ(e)

Lemma 1 π1k =Proﬁts are independent of taxes, and the total proﬁt is normalized to 1. Note
that, as in standard models, the amount proﬁt shifting πi − π̃i is proportional
to the tax diﬀerence ti −tj and inversely proportional to the enforcement level
δ(e). Also, the tax base elasticity (sensitivity of the tax base to ﬁscal rates) ishigher in the small country,14 which is consistent with standard approaches.
The ﬁrms’ production decisions are qia = qib = γi /(3β), pi − ci = γi /3, so thatkBased
( k on)this,kthe ﬁrst-order condition for π̃i (k = a, b, i = 1, 2)kbecomes −ti + tj − ∂C πi , π̃i /∂ π̃i = 0. They derive Lemma 1.
∂ π̃1 t1
14=
Evaluated at equal tax rates, the tax base elasticities at t1 = t2 = t are −∂t1 π̃113

πia = πib = γi2 /(9β).

7

2.3

Tax and Enforcement Timing

From Lemma 1, we have π̃i = π̃i (t, e) and Ri (t, e) = ti π̃i (t, e). The net tax
revenue (2), taking account of the ﬁrms’ behavior, is:()( e )2( e )2
1 + ²i ti − tjii
Wi = ti π̃i (ti , tj , e) − η
−η
= ti
−
,
2
2
2δ(e)
2
where ²1 = ² = −²2 . Before we analyze the noncooperative choices of taxes
and enforcement eﬀorts, in this subsection, we discuss the issue of tax andenforcement timing.
We ﬁrst describe the case where the tax ti and the enforcement ei arechosen simultaneously and noncooperatively. In the Appendix we show thefollowing:
Lemma 2 Suppose that ti and ei (i = 1, 2) are chosen simultaneously andnoncooperatively. Then, when δ(e) = 0.5e1 + 0.5e2 , there is no equilibrium.15
Proof: See the Appendix.
An intuition of Lemma 2 is as follows. If ti and ei are chosen simultaneously, from the point that satisﬁes the local ﬁrst-order conditions, there
always exist deviations to increase the tax revenue whereby managing the tax
leak due to the proﬁt shifting (mathematically, the Hessian matrix associatedwith the ﬁst-order conditions does not satisfy negative deﬁniteness).
For the rest of this paper, we adopt the following sequentiality of enforcement and tax decisions: (i) First, both countries set their enforcement eﬀorts.
(ii) Second, both countries choose their tax rates. The reasons we adopt thissetup are as follows. First, the level of enforcement eﬀorts is determined by
speciﬁc rules and laws of monitoring, inspection and information sharing,which are less reversible in nature than the tax rates which can be changed
more easily, so they are assumed to be chosen before taxes (see Bacchetta
and Espinosa (1995), Peralta et al. (2006) and Keen and Konrad (2013)).16
17
Second, by treating tax-enforcement decisions as a commitment device,t∂ π̃2 t2
t
and −
=
.
δ(e)(1 + ²)
∂t2 π̃2δ(e)(1 − ²)15
The enforcement function in Lemma 2 is used in our benchmark analysis.
16
See also Bucovetsky and Hauﬂer (2008) where governments decide the degree of tax
preferences in favor of mobile capital before they set taxes on capital. The choice of suchvariables determines long-run reputation, based on which the governments decide the tax
rates.
17
When the sequence of the decisions is reversed so that taxes are chosen ﬁrst andthe enforcement is chosen later, then: (i) the subgame-perfect equilibrium exists only8

this structure allows us to examine a practical issue of the possibility of enforcement cooperation under international tax competition. As to the tax
timing, we ﬁrst assume the conventional framework where countries choose
tax rates simultaneously. As an extension, we examine sequential tax choicesà la Hindriks and Nishimura (2015) in Section 4.2.

3

Tax and Enforcement Choices

3.1

Tax choices

The modiﬁed sequence of events is as follows. First, both countries set theirenforcement eﬀorts. Second, they compete in taxes. Third, MNEs choose alevel of production and the proﬁt to be shifted. The model is solved by backward induction. Hence, given the ﬁrms’ decisions characterized in Lemma 1,
in this section, we derive the governments’ choices of taxes and enforcement
eﬀorts. As to the enforcement eﬀorts, we compare the equilibria of the noncooperative and cooperative choice of enforcement to see whether the latter
scenario is preferred unanimously or not in Section 3.3.
In the second stage, given the enforcement e = (ei , ej ), each country
noncooperatively chooses its own tax rate ti (i = 1, 2) to maximize the tax
revenue:18)(
1 + ²i ti − tj
Ri (ti , tj , e) = ti π̃i (ti , tj , e) = ti
−
,
2
2δ(e)
where ²1 = ² = −²2 . The ﬁrst-order conditions are:∂Wi
1 + ²i ti − tj
−1
=
−
+ ti
= 0.
∂ti
2
2δ(e)
2δ(e)

(3)

The second-order conditions are satisﬁed. They yield the following equilibrium taxes:
(
(
)
)
3+²
3−²
N
N
t1 (e) = δ(e)and t2 (e) = δ(e).
33As discussed in Hindriks and Nishimura (2015, p.66), the large country has
greater market power.19 In turn, the large country taxes higher with greaterif ² ≥ 0.2, (ii) only country 1 exerts enforcement eﬀorts, and (iii) both the equilibrium
taxes and enforcement eﬀorts are lower than those obtained in Sections 3.1 and 3.2. The
derivation is available upon request to the authors.
18
As such, ei in (2) is treated as a sunk cost when countries choose taxes.
19This is an analogy to Amir and Stepanova (2006) in which the ﬁrms’ cost asymmetry
implies diﬀerent markups.

9δ(e)²> 0. From Lemma 1, both ﬁrms shift
3
1+²
proﬁts from the large to the small (low-tax) country: we have π̃1N =
−
2
²
3+²
3−²
N
=
and π̃2N =. The tax revenues RiN (e) = tNi (e)π̃i (i = 1, 2)
3
6
6
are:
(
)2
(
)2
δ(e) 3 − ²
δ(e) 3 + ²
NN≥ R2 (e) =for all ² ≥ 0.
(4)
R1 (e) =23
2
3
N
asymmetry: tN
1 (e) − t2 (e) = 2

Note that, given e, total revenue is increasing in the extent of country asymmetry.

3.2

Noncooperative enforcement choices

In the ﬁrst stage, taking account of the behavior in the subsequent stages, thegovernment in each country chooses enforcement eﬀorts, either noncooperatively or cooperatively. In this subsection, we ﬁrst examine noncooperative(decentralized) enforcement choices.
As a benchmark, we consider the case of perfect substitutability between
enforcement choices. Total enforcement level is given byδ(e) = 0.5e1 + 0.5e2 .

(5)Enforcement eﬀorts increase the cost of proﬁt shifting which in turn inﬂuences the choice of taxes. So country i, given ej , maximizes WiN (ei , ej ) =e2
RiN (ei , ej ) − η 2i where RiN (e)’s are given in (4). The ﬁrst-order conditions
N
20
yield the equilibrium enforcement eN = (eN
1 , e2 ) given by:
(
)2
(
)2
1 3+²
1 3−²
9 + ²2 N
N
N
Lemma 3 e1 =, e2 =, δ(eN ) ≡ δ N =
.δ
η
6
η
6
36η
is increasing in ²: higher asymmetry induces higher enforcement in equilibrium.
20

See also (8) below. A general treatment is as follows:(titi − tj2(δ(e))2)∂δ(ei , ej ) ∂RiN (ti , tj , e) ∂tNj+− ηei = 0.
∂ei
∂tj
∂ei

The ﬁrst two terms capture the marginal beneﬁt of the enforcement. When taxes andenforcement eﬀorts are decided simultaneously, as shown in (11) in the Appendix, the lowtax country never has an incentive for enforcement. In contrast, the second term appears
under our sequential structure, which gives an incentive to a more strict enforcement.

10

Lemma 3 illustrates the synergy between market power and enforcement.
From (4), the marginal beneﬁt of enforcement is proportional to tax revenue.
With higher asymmetry, country 1 exhibits higher equilibrium taxes. Inturn, the country is playing for high stakes in the enforcement game, whichNN
causes higher equilibrium level of enforcement: eN
1 > e2 , and e1 increases
N
quadratically in ². This increase of eN
so that is δ N is
1 in ² is dominant in δ
also increasing in ². This eﬀect is similar to that of the total revenue noted
above. We will come back to this point in Section 4.1 under enforcement
complementarity.
Plugging the equilibrium enforcement into the welfare function, we have
2
(eN
i )
N
N N
N
Wi (eN
,
e
)
=
R
(e
,
e
)
−
η
(i = 1, 2).
1
2
i
1
2
2

3.3

Beneﬁt of enforcement cooperation

We now examine cooperative enforcement choices in the ﬁrst stage, and see
whether the cooperative framework is adopted unanimously or not. Here,both countries choose their enforcement levels so as to maximize their joint
welfare. This reﬂects an agreement for the level of information exchange
in the tax treaty. However, reﬂecting a real-world situation that most tax
treaties do not have cooperative or harmonized tax rates, we assume thatcountries still decide taxes (ti ) noncooperatively. Therefore, countries chooseNe = (e1 , e2 ) anticipating the noncooperative tax game (tN
1 (e), t2 (e)) and tax
revenues (R1N (e), R2N (e)) in (4). That is,
)
)
(
∑(
e2j
(3 + ²)2 (3 − ²)2e2e2iN= max δ(ei , ej )·+−η i −η .
max
Ri (ei , ej ) − η
ei ,ejei ,ej
2
18
18
2
2
i
(6)
N∗
N∗ N∗
The ﬁrst-order conditions derive cooperative eﬀort levels e = (e1 , e2 )given by:
9 + ²2
∗
N∗
eN
=
e
=
1
2
18η
Enforcement eﬃciency requires both countries to exert the same enforcement
eﬀort, due to the quadratic (convex) cost function. Also, cooperative enforcement is increasing in the market asymmetry. Compared with the noncooperative solution, enforcement cooperation doubles the total level of enforcement.
This is because the positive ﬁscal externality of enforcement ∂RjN /∂ei > 0
for i 6= j is now internalized. Enforcement cooperation induces the welfare
∗ 2
(eN
i )
N N∗ N∗
∗ N∗
)
−
η
,
e
(e
)
=
R
(i = 1, 2). As to the beneﬁt of,elevels Wi (eN
2
1
i
2
1
2
enforcement cooperation, the large country unambiguously gains by cooperaNN∗ N∗
tion (W1 (eN
1 , e2 ) > W1 (e1 , e2 ) for all ² > 0), due to higher enforcement for11internalization of the enforcement externality. However, for country 2 there∗Nare oﬀsetting eﬀects. As eN
2 increases in ² but e2 decreases in ² (Lemma 3),
the cost of coordinated enforcement increases in ². As a result, even though
the tax revenue increases (R2N (eN ∗ ) − R2N (eN ) > 0), country 2 may prefer thenoncooperative regime. The following proposition tells us that this is true
under suﬃcient asymmetry.
∗ N∗
NN
Proposition 1 For the large country, W1 (eN
1 , e2 ) > W1 (e1 , e2 ) for allN∗ N∗
parameter values. For the small country, on the other hand, W2 (e1 , e2 ) >NN
W2 (eN≈0.2592817. Therefore the small country prefers
1 , e2 ) iﬀ ² < ²noncooperative enforcement when ² > ²N .

Proof: See the Appendix.
In the following section, we consider two diﬀerent forces that can either
promote or discourage enforcement eﬀorts and cooperation. The ﬁrst is related to the form of the enforcement technology (enforcement complementarity). The second is related to the form of tax competition (tax leadership).

4

Drivers of cooperation

4.1

Enforcement complementarity

Enforcement eﬀorts increase the cost of proﬁt shifting. However, in reality, dispersed (unilateral) enforcement eﬀorts among the involved countries
are not eﬀective. For instance, the lack of tax-relevant information by hostcountry makes the taxable income unclear to the home country. To formalize
the notion of imperfect substitutability of enforcement eﬀorts, we modify theenforcement function of (5) in the benchmark case. As an extension, we takethe following CES formula:()− 1δ(e1 , e2 ) = 0.5 e1 −ρ + 0.5 e2 −ρ ρ , ρ ≥ −1.

(7)

The representative cases are: (i) ρ = −1 (perfect substitutes in (5)); (ii)0.5
ρ → 0 (Cobb-Douglas case where δ(e1 , e2 ) = e0.5
1 e2 ); and (iii) ρ → ∞ (theweakest-link case where δ(e1 , e2 ) = min[e1 , e2 ]). If for example, the low-tax
country can exercise the veto power during the mutual agreement procedureon the transfer price and taxable incomes of the MNEs, then the composition
function becomes closer to the weakest-link formula.
The equilibrium strategy of the second stage (tax choice) is in fact thesame as Section 3.1. That is, given δ(e), we obtain Ri = RiN (e) (i = 1, 2)

12

in (4) as the equilibrium tax revenue. Recall that R1N (e) ≥ R2N (e) with thestrict inequality when ² > 0.
We then move on to examine the noncooperative choice of ei and ej inthe ﬁrst stage, when δ(e1 , e2 ) takes the form of (7). Country i, given ej ,e2maximizes WiN (ei , ej ) = RiN (ei , ej ) − η 2i :∂δ(e) 1∂ei 2(

3 + ²i3)2− ηei = 0 ⇒) 1+ρ
1 −ρ−1 (−ρ −ρ
ei
0.5e−ρ
+
0.5e
1
2
η

(

3 + ²i6)2= ei , (8)with ²1 = ² = −²2 .
(8) constitutes the enforcement reaction function of country i. For ρ =()2
1 3 + ²i
−1, (8) is= ei . Namely, the reaction functions are orthogonalη6
at ei = eN
i in Lemma 3 as a dominant strategy. However, for ρ > −1,
the enforcement reaction functions are upward-sloping, i.e., there is strategic
complementarity in enforcement eﬀorts. For the Cobb-Douglas case (ρ = 0),()2
1 0.5 3 + ²i
for example, (8) is ej= ei1.5 . Figure 1 illustrates the cases ofη6(ρ = −0.2, ² = 0.25) and (ρ = 5, ² = 0.25).
Figure 1 around here.
From (8), we obtain the following:
) −ρ
−2ρ
−2ρ
2
1 (
Lemma 40.5 (² + 3) 2+ρ + 0.5 (3 − ²) 2+ρ
(² + 3) 2+ρ ,
36η
(
) 1+ρ
−2ρ
−2ρ
2
1
−ρ
2+ρ + 0.5 (3 − ²) 2+ρ
0.5
(²
+
3)
(3 − ²) 2+ρ , and δ N (ρ) =
eN
(ρ)
=
2
36η
(
) 2+ρ
−2ρ
−2ρ
1
−ρ
2+ρ
2+ρ
0.5 (² + 3)
+ 0.5 (3 − ²)
.
36η
N
N
N
N
eN
1 (−1) > e1 (ρ) > e2 (ρ) > e2 (−1) for all ρ > −1. δ (ρ) is decreasing2in ρ for all ρ ≥ −1. δ N (ρ) is increasing in ² when ρ < − , and δ N (ρ) is3
2
decreasing in ² when ρ > − .
3
1+ρ

eN
1 (ρ) =

Proof: See the Appendix.
N
In Figure 1, the dotted curve WP illustrates the trajectory of (eN
1 (ρ), e2 (ρ))
Non (e1 , e2 )-space (with ² = 0.25). It starts with P=(eN
1 (−1), e2 (−1)). Givene2 < e1 , greater complementarity (larger ρ) reduces the marginal beneﬁt ofcountry 1 (LHS of (8)) and increases that of country 2. As a result, eN
1 (ρ)

13

N
N
decreases in ρ, whereas eN
2 (ρ) increases in ρ. As ρ increases, (e1 (ρ), e2 (ρ))
N
N
converges to W=(e1 (∞), e2 (∞)), where:eNi (∞) =

(3 − ²)2 (3 + ²)236η
²2 + 9Figure 2 around here.
N
Figure 2 illustrates eN
1 (ρ) and e2 (ρ) for diﬀerent ²’s. Given ρ, whenasymmetry increases, there are two eﬀects for the determination of eN
1 (ρ).
First, asymmetry per se tends to increase e1 . Second, on the other hand,the decrease of eN
2 (ρ) tends to reduce e1 , due to strategic complementarity. From these eﬀects, eN
1 (ρ) at ² = 0.25 (the dashed curve located above)Nsingle-crosses e1 (ρ) at ² = 0.1 (the solid curve located above) from above:
there is a critical value of ρ above (below) which higher asymmetry induces
lower (higher) enforcement by country 1 in equilibrium. In this way, there
is interaction between asymmetry and complementarity. As to the enforcement level, higher asymmetry induces higher equilibrium enforcement as inLemma 3 if and only if the technology is suﬃciently substitutable, whereas
higher ² induces lower δ N (ρ) when ρ is suﬃciently high. (
)
∑
e2i
N
Next, cooperative eﬀort levels are given by maxei ,ej i Ri (ei , ej ) − η 2 ,which is the same as (6) of Section 3.3. With the enforcement)function (7),
(
2
1+ρ
(
)
9
+
²
−ρ −ρ
the ﬁrst-order conditions are now 0.5e−ρ
= e2+ρ
(i =1 + 0.5e2i18η∗1, 2). Invariant with respect to imperfect substitutability, we obtain eN
=
1
2
9
+
²
∗
eN
as in the basic model. With greater complementarity, nonco=
2
18η
operative solution becomes less attractive, as δ N (ρ) decreases in ρ. Recall
that, when ρ = −1, the large country unambiguously gains by cooperation∗ N∗
NN(W1 (eN
1 , e2 ) > W1 (e1 , e2 ) for all ² > 0) but it is not the case for the
N
N
N
∗ N∗
small country: W2 (eN
1 , e2 ) > W2 (e1 , e2 ) if and only if ² < ² ≈0.2592817(Proposition 1). In comparison to the basic case, when ρ > −1, we show thefollowing:

Proposition 2 Suppose that the enforcement function is represented by (7)NNN∗∗with ρ > −1. For the large country, W1 (eN
1 (ρ), e2 (ρ)) > W1 (e1 (ρ), e2 (ρ))
N∗∗
for all parameter values. For the small country, on the other hand, W2 (eN
1 (ρ), e2 (ρ)) >NN
N
W2 (eN1 (ρ), e2 (ρ)) iﬀ ² < ² (ρ). ² (ρ) is an increasing function of ρ, withNN
² (−1) = ² ≈ 0.2592817 in Proposition 1 and ²N (∞) ≈ 0.4094092130 < 1.
14

Proof: See the Appendix.
Figure 3 around here.
For given level of complementarity (ρ), Figure 2 above tells us two eﬀects
Nof the increase in asymmetry (²). First, the gap of eN
1 (ρ) and e2 (ρ) tends
to increase. Second, there is an ambiguity of the increase of country 1’s enforcement eﬀort (as well, the free-riding beneﬁt of country 2) with respect
to an increase in asymmetry. Proposition 2 tells us that the ﬁrst eﬀect is
crucial for the beneﬁt of cooperation by country 2, in that country 2 prefers
the noncooperative regime under suﬃcient asymmetry: as discussed abovein Proposition 1, a driving force of country 2’s deviation is the conﬂict of interest with respect to the desired level of the enforcement. However, cooperation is more viable with greater enforcement complementarity: for example,²N (−0.2) ≈ 0.3114695738, ²N (0) ≈ 0.319521, and ²N (∞) ≈ 0.4094092130.
NThis is because the diﬀerence in eN
1 (ρ) and e2 (ρ) decreases as ρ increases.
On (ρ, ²)-space, Figure 3 divides the area where cooperation is viable (below/right of the curve ²N (ρ)). Note, however, that even with perfect comNN
plementarity (ρ = ∞ at which eN
1 (ρ) = e2 (ρ)), we have ² (∞) < 1: with
suﬃcient asymmetry, country 2 prefers the noncooperative regime with lower
enforcement eﬀorts.

4.2

Tax leadership

In reality, even if the countries do not reach in cooperation in tax decisions,the countries can enhance welfare within a noncooperative framework byadopting tax leadership (namely, Stackelberg tax competition). Following
Hindriks and Nishimura (2015), we consider the case in which (large) country
1 leads. At the end of this subsection, we brieﬂy discuss the case of the small
country’s leadership.
Keeping our sequential structure of enforcement and tax choices, we solve
the game by backward induction. Given the enforcement choices e = (ei , ej ),the (small) country 2, as the tax follower, chooses t2 , given t1 . Country 1, as
the Stackelberg leader, maximizes R1 (t1 , t2 , e) with respect to t1 , along thecountry 2’s tax reaction function t2 = tˆ2 (t1 ; e). This gives the equilibriumtax rates denoted by (tS1 (e), tS2 (e)) and the tax revenues RiS (e) = tSi (e)π̃iS (i =1, 2) as follows (see the proof of Lamma 5 for derivation):
(
(
)
)
3+²
δ(e)
5−²
δ(e)
S
S
t1 (e) = δ(e)(3+²)2 , R2S (e) =(5−²)2 .
, t2 (e) = δ(e), R1S (e) =
2
4
16
32
(9)
15

Several explanations are in order. First, with the given level of e, tSi (e) >ˆtNi (e) (i = 1, 2), due to the strategic complementarity in tax choices (∂ t2 (t1 ; e)/∂t1 >S
SN
N0). Second, the tax diﬀerential is t1 (e) − t2 (e) > t1 (e) − t2 (e). And third,
we have:
R1S (e) ≷ R2S (e) ⇐⇒ ² ≷ ²̂ ≈ 0.31370850.
Namely, when ² is small enough, the beneﬁt of the second-mover advantage
in the tax competition dominates the initial exogenous asymmetry, so thatthe tax leader collects less tax revenue than the tax follower (R1S (e) < R2S (e)),
and vice versa when ² is large enough. As discussed later, this feature aﬀects
the incentive for tax enforcement.
We then move backward to compute noncooperative enforcement choices,
followed by Stackerlberg taxation, eS = (eS1 , eS2 ). As to the enforcement
function, in this subsection we go back to the benchmark formulation ofδ(e1 , e2 ) = (0.5 e1 + 0.5 e2 ) (equation (5)). Indeed, the cost of proﬁt shifting,measured by δ(e), aﬀects equilibrium tax levels: higher δ(e) lowers tax competition and increases equilibrium taxes. The question is how the enforcement
eﬀorts will be changed under tax leadership.
(3 + ²)2(5 − ²)2
S
S S
S
N
> eN
,
e
=
> eN
1
2
2 , δ(e1 , e2 ) ≡ δ > δ .
32η
64η
Namely, under the Stackelberg tax competition, both the leader and the tax
follower exert more eﬀort than under the Nash tax game.
NN N(ii) RiS (eS ) > RiN (eN ) (i = 1, 2) and tS1 (eS ) − tS2 (eS ) > tN
1 (e ) − t2 (e ).
N(iii) |eS1 − eS2 | ≶ eN⇐⇒ ² ≷ ²̃ ≈ 0.16094072 < ²̂. Namely, the1 − e2
Stackelberg tax competition reduces the dispersion of the enforcement eﬀorts,
provided that asymmetry is suﬃciently large.

Lemma 5 (i) eS1 =Proof: See the Appendix.
Both countries get higher revenue in the Stackelberg subgame given thesame eﬀort as in the Nash subgame (RiS (e) > RiN (e) (i = 1, 2) from (4) and(9)). As a result, the marginal beneﬁt of eﬀort is higher in the Stackelberg
situation (eSi > eNi (i = 1, 2) in part (i)). Endogenous enforcement further
increases the relative beneﬁt of the Stackelberg outcome (RiS (eS ) > RiS (eN ) >RiN (eN ) (i = 1, 2)). Also, from δ S > δ N in part (i) and (9), the tax gap is alsoN
N N
increased: tS1 (eS ) − tS2 (eS ) > tS1 (eN ) − tS2 (eN ) > tN
1 (e ) − t2 (e ). Similar toa “minimum tax” over the Nash (simultaneous) tax competition in Kanbur
and Keen (1993), the beneﬁt by the large country comes from the higher
tax revenue along the tax reaction function, and at the same time, the small
country beneﬁts from the higher tax diﬀerence, due to ∂ tˆ2 (t1 ; e)/∂t1 < 1,16that allows more proﬁt shifting in our context.21 As emphasized in Kanbur
and Keen (1993), this type of coordination strategy is diﬀerent from tax
harmonization that eliminates the beneﬁt of the proﬁt shifting to the small
country.
The reason for part (iii) is as follows. In contrast with the Nash tax
Ncompetition where eN
1 > e2 for all parameter values, the large country may
exert lower enforcement eﬀort if the asymmetry is small enough: from part(i) we have eS1 ≷ eS2 ⇐⇒ ² ≷ ²̂. This means that the dispersion of
enforcement eﬀorts is lower under Stackelberg competition than under Nash
tax competition if ² is close to ²̂, and in fact, the range of asymmetries whereN|eS1 − eS2 | < eN
1 − e2 holds is quite large.
We next examine the cooperative solution and compare it with the noncooperative enforcement decisions, where the tax choices take place under
the Stackelberg competition. As in Section 3.3, both countries choose theirenforcement levels e = (e1 , e2 ) so as to maximize their joint welfare but
countries compete in taxes subsequently. Therefore, given e,( countries choose)∑e2(tS1 (e), tS2 (e)). The choice of ei and ej is given by maxei ,ej i RiS (ei , ej ) − η 2i .
In the Appendix we show the following:
Proposition 3 Under the Stackelberg tax competition, the enforcement coS∗
operation is more viable than under the Nash tax competition: Wi (eS∗
1 , e2 ) >Wi (eS1 , eS2 ) (i = 1, 2) for ² < ²S ≈ 0.6542911 with ²S > ²N .
Proof: See the Appendix.
The tax leadership (by the large country) facilitates the participation of
the small country in the enforcement cooperation. There are two reasons.
First, RiS (eS ) > RiN (eN ) (i = 1, 2) as in Hindriks and Nishimura (2015),
so that the beneﬁt of internalization of enforcement externality is larger.
Second, the enforcement gap is reduced as we showed in Lemma 5. (iii). As
in our previous propositions, the second eﬀect reduces the conﬂict of interest
with respect to the desired level of the enforcement.
Lastly, we discuss the issue of the leadership.22 Let RiM (e) (i = 1, 2)
be the country i’s revenue under country 2’s leadership with given level of21

The diﬀerence between the minimum tax and the large country’s leadership is thatthe former entails a move along the large country’s reaction function, whereas we consider
here the move along the small country’s reaction function.
22
A background literature is as follows. Following Kanbur and Keen’s (1993) crossborder shopping model, Wang (1999) assumed that the large country behaves as a Stackelberg leader, and showed that both countries become better oﬀ by the tax leadership. In
our model where asymmetry is deﬁned by the market size, when δ is ﬁxed, the equilibriumtax timing à la Hamilton and Slutsky (1990) is the large country’s leadership. See Hindriks
and Nishimura (2015, p.68).

17

e. Then we have R1M (e) > R1S (e) > R1N (e) and R2S (e) > R2M (e) > R2N (e).
Namely, country 2’s leadership also increases countries’ enforcement incentives compared with the Nash tax competition. However, regarding equiM
M
N
N
librium enforcement eﬀort eMi (i = 1, 2), we have e1 − e2 > e1 − e2 forall ² (enforcement gap is increased by the leadership). As a result, for the
∗ M∗
M∗ M∗
M
M
cooperative eﬀort level (eM
1 , e2 ), W2 (e1 , e2 ) < W2 (e1 , e2 ) if and only if
M² > ² ≈ 0.01692095340. That is, the threshold value of viable cooperation iseven smaller than ²N . Moreover, we can show that the large leadership equilibrium Pareto dominates the small leadership equilibrium under suﬃcient
asymmetry, as in Hindriks and Nishimura (2015).23 According to Proposition 3, the former equilibrium leads to higher tax revenues to both countries
as well as more viable enforcement coordination than under the Nash tax
competition.

5

Conclusion

Using a proﬁt shifting model with multinational enterprises that operate in
two countries with diﬀerent market sizes, we analyze the determinants of voluntary cooperation on enforcement eﬀort and information sharing arrangements. When the enforcement eﬀorts exhibit complementarity, we showedthat cooperation is more viable with greater enforcement complementarity(less additive/substitutable nature). The exact speciﬁcation comes from administrative or legal nature of the collective decision-making. However, we
can infer in which situations the enforcement coordination (by unanimity)
is more viable; namely, in the situation where the capital-importing country
can exercise a veto power towards tighter enforcement, and where bilateral(less dispersed) eﬀorts by host and home countries are required for eﬀective
information exchange.
In the Stackelberg tax competition where the large country leads, theequilibrium taxes become higher, and the proﬁt shifting increases. As a result, both the tax leader and the tax follower will exert more enforcement
eﬀorts than under the simultaneous tax choice, and the enforcement cooperation will be more viable under the Stackelberg tax competition. Interestingly,
this type of coordination strategy is diﬀerent from tax harmonization thateliminates the beneﬁt of the proﬁt shifting to the small country.
The action programs of the OECD BEPS Project have various degree ofcommitment, from the strong commitment to consistent implementation of
the program across countries to the weak form where countries are free to
23

M
S S
M M
W2 (eM
1 , e2 ) < W2 (e1 , e2 ) for all ², and when ² > 0.6618315754, W1 (e1 , e2 ) <S S
W1 (e1 , e2 ).

18

assess and pick. Looking forward, this paper has two contributions. First,
developing a simple model that captures the central features of proﬁt shifting, we produced sharp insights into the costs and beneﬁts of enforcement
cooperation. Second, given the rapid development of the empirical literature,
the analysis here guides us what empirical quantities to look for, towards apractical prediction of the viability of the enforcement cooperation.

Appendix
Proof of Lemma 2: When ti and ei are chosen simultaneously and independently, the ﬁrst-order conditions of the welfare maximization with respect
to ti and ei are:∂Wi
1 + ²i ti − tj
−1
=
−
+ ti
= 0,
∂ti
2
2δ(e)
2δ(e)
)
(
∂δ(ei , ej )ti − tj∂Wi= ti− ηei ≤ 0.
∂ei
2(δ(e))2
∂ei

(10)
(11)

(10) derives the preferred tax level by country i, where t1 > t2 for ² ∈ (0, 1)
and t1 = t2 for ² = 0. Taking these into account and given ∂δ(ei , ej )/∂ei > 0,(11) implies that country 2 chooses e2 = 0. When δ(e) = 0.5e1 + 0.5e2 , from(10) and (11), the set of values that satisfy the ﬁrst-order necessary)( condition
²
(3
+
²)
3+²
n
n
n
n
is uniquely given by e1 =, e2 = 0 and t1 = 0.5e1
and
18η
3
)
(
3−²
tn2 = 0.5en1. However, evaluated at (en2 , tn2 ), the Hessian matrix of
3
country
1 associatedwith (10) and (11) does not satisfy negative deﬁniteness:¯ 2¯¯ ∂ W1 ∂ 2 W1 ¯¯ (∂t1 )2 ∂t1 ∂e1 ¯ −2²2 (3 − ²)2 (3 + ²)2 + (36ηe1 )3² (3 − ²) a¯¯. For ea1 =(e1 <4¯ ∂ 2 W1 ∂ 2 W1 ¯ =223328η e136η
¯
2 ¯∂e1 ∂t1(∂e1 )²(5² + 9)(3 − ²)
, we have W1 (ea1 , ta1 , en2 , tn2 ) − W1 (en1 , tn1 , en2 , tn2 ) =432η
1² (1 + ²)2 (1 − ²)
, which is positive for ² > 0. For ² = 0, we set eb1 =
192η
36η
1
n n n nbn
b b n nb
, and we have W1 (e1 , t1 , e2 , t2 ) − W1 (e1 , t1 , e2 , t2 ) =(e1 > e1 ) and t1 =144η7
> 0. Therefore, the equilibrium does not exist. Q.E.D.
5184η
(
)(
)2
9 + ²23−²
N N
N N∗
Proof of Proposition 1: R2 (e ) − R2 (e ) =is18η
6
en1 ) and ta1 =

19

)2 (
)
3+²
9 + ²2 − 2²
is in624η
81 + ²4 − 36²3 + 54²2 − 324²
creasing in ². From these, W2N (eN ∗ )−W2N (eN ) =,2 · 362 ηwhich is positive (negative) when ² < ²N (² > ²N ). The analogous calcula81 + ²4 + 36²3 + 54²2 + 324²
tion shows W1N (eN ∗ ) − W1N (eN ) => 0 for all ².
2 · 362 η
Q.E.D.
∗ 2
2
(eN
(eN
2 )
2 )
decreasing in ², whereas η
−η
=
2
2

(

Proof of Lemma 4: Solving the system of (8) for i = 1, 2, we obtainNNNN
the expression of (eN
1 (ρ), e2 (ρ)) and δ (ρ). Clearly e1 (ρ) > e2 (ρ). Let
1+ρ
(
)
−2ρ
−2ρ
−ρ
f (x, y) ≡ 0.5 x 2+ρ + 0.5 y 2+ρ
. Since f is increasing in both x and y,
one has f (3 + ², 3 + ²) > f (3 + ², 3 − ²) and f (3 + ², 3 − ²) > f (3 − ², 3 − ²),NNN
which is equivalent to eN
1 (−1) > e1 (ρ) and e2 (ρ) > e2 (−1) respectively.
NDiﬀerentiating δ (ρ) with respect to ρ, we obtain
) 2+ρ
−1 2
−2ρ
−2ρ
∂δ N (ρ)
1 (
−ρ
2+ρ
2+ρ
+ 0.5 (3 − ²)=
0.5 (3 + ²)
×
∂ρ
36η
ρ2
) (
)
{(
−2ρ
−2ρ
−2ρ
−2ρ
0.5 (3 + ²) 2+ρ + 0.5 (3 − ²) 2+ρ ln 0.5 (3 + ²) 2+ρ + 0.5 (3 − ²) 2+ρ
}
−2ρ
−2ρ
−2ρ
−2ρ
−0.5 (3 + ²) 2+ρ ln (3 + ²) 2+ρ − 0.5 (3 − ²) 2+ρ ln (3 − ²) 2+ρ < 0,from Jensen’s inequality, since the function g(x) = x ln x is convex withrespect to x.
Diﬀerentiating δ N (ρ) with respect to ², we obtain) −ρ −1 ()−3 ρ−2−2ρ
−2ρ
−3 ρ−2∂δ N (ρ)
1 (
2+ρ
2+ρ
2+ρ
2+ρ
(3 + ²)=0.5 (² + 3)
+ 0.5 (3 − ²)
− (3 − ²)
,
∂²
36η
2+ρ

2
2
2
which is positive if ρ < − , and negative if ρ > − . For ρ = − ,
3
3
3
( )
1
2
=
for all ² ≥ 0. Q.E.D.
δN −
3
4η
(
)(
)
9 + ²2
9 + ²2 − 12²N∗ N∗
Proof of Proposition 2: W2 (e1 , e2 ) =
as
18η
36
(
)2
3−²
N
N
in the benchmark case, whereas W2 (e1 (ρ), e2 (ρ)) =
×
6
)
(
−2ρ
−2ρ
2+ρ + 2 (3 + ²) 2+ρ
2+2ρ
(
)
(3
−
²)
−2ρ
−2ρ
−ρ
0.5 (² + 3) 2+ρ + 0.5 (3 − ²) 2+ρ
. From these,72η
we numerically show that W2N (eN ∗ ) − W2N (eN (ρ)) is positive (negative) when20

² < ²N (ρ) (² > ²N (ρ)). ²N (ρ) is shown in Figure 3. The analogous calculationshows W1N (eN ∗ ) − W1N (eN (ρ)) > 0 for all ². Q.E.D.
Proof of Lemma 5: From (3), the(tax reaction
function of country 2 is
)
1−²
t1
∂ tˆ2
arg maxt2 R2 (t2 , t1 , e) ≡ tˆ2 (t1 ; e) = δ(e)+ . We have
∈ (0, 1)2
2
∂t1
as in conventional models of tax competition. The ﬁrst-order condition of∂R1 ∂R1 ∂ tˆ2 1 + ² t1 − tˆ2 (t1 )
1 − (1/2)+=
country 1 is given by
−
− t1
= 0,
∂t1
∂t2 ∂t1
2
2δ(e)
2δ(e)
which yields t1 = tS1 (e) and tS2 (e) = tˆ2 (tS1 (e); e) in (9).24 tS1 (e) − tS2 (e) =1 + 3²
1 + ² 1 + 3²
3+²
5−²
δ(e)
. From Lemma 1, π̃1S =−=< π̃2S =
.
4
2
2·4
8
8
We then have RiS (e) = tSi (e)π̃iS (i = 1, 2) as in (9).
From (9) and (5) in the text, the maximization of WiS (ei , ej ) = RiS (ei , ej )−(3 + ²)2(5 − ²)2e2
η 2i with respect to ei yields eS1 =and eS2 =. eS1 − eN
1 =
32η
64η
1
1
(3 + ²)2 > 0, eS2 − eN
(3 + ²)(27 − 7²) > 0. Therefore, δ S > δ N ,
2 =
288η
576η
so part (i) of the lemma holds. The proof of part (ii) of the lemma was
7 − ²2 − 22²<
shown in the text. For ² < ²̂ ≈ 0.31370850, 0 < eS2 − eS1 =
64η
²
N
eN
if and only if ² > ²̃ ≈ 0.16094072. For all ² ≥ ²̂, 0 ≤ eS1 − eS2 =1 − e2 =3η
−7 + ²2 + 22²N< eN
1 − e2 . Therefore, part (iii) of the lemma holds. Q.E.D.
64η
Proof of Proposition 3: The welfare levels under the noncooperative
enforcement choice (followed by Stackelberg tax competition) are:(W1 (eS1 , eS2 ) =

3+²
4

)2 (

17 + ²2 − 2²
64η

)

(
, W2 (eS1 , eS2 ) =

5−²
8

)2 (

61 + 5²2 + 14²
128η

The joint welfare maximizing enforcement iseS∗
1 =3²2 + 2² + 43
= eS∗
2
64η

S S∗ S∗S∗
The cooperative welfare levels, Wi (eS∗
1 , e2 ) = Ri (e1 , e2 ) − η(
24Note that the revenue function is concave in the tax rate, ∂−1/(2δ(e)) < 0.

21

2
(eS∗
i )
(i =
2

)
∂R1
∂R1 ∂ tˆ2+/∂t1 =∂t1∂t2 ∂t1),

1, 2), are:)()
29 + 5²2 + 46²
3²2 + 2² + 43
,
64η
128
( 2
)( 2
)
3² + 2² + 43
² − 42² + 57
S∗ S∗
W2 (e1 , e2 ) =
.
64η
128
(

S∗
W1 (eS∗
1 , e2 ) =23 + 7²4 + 116²3 + 282²2 + 1364²>
2 · 642 η
463 − ²4 − 44²3 + 42²2 − 716²
0 for all ². For the small country, W2N (eS∗ )−W2N (eS ) =,642 ηwhich is positive when ² < ²S ≈ 0.6542911, and negative when ² > ²S . This
critical value of asymmetry is greater than ²N ≈ 0.2592817 in the Nash game.
Q.E.D.
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25

Figure 1: Enforcement reaction functions for ρ = −0.2 (solid curves) andNρ = 5 (dashed curves). The curve WP is the trajectory of (eN
1 (ρ), e2 (ρ)).

Figure 2: Noncooperative enforcement level for ² = 0.1 and ² = 0.25

Figure 3: Critical value of asymmetry ²N (ρ)