Post on 09-May-2020
DPRIETI Discussion Paper Series 20-E-035
Transfer Pricing Regulation and Tax Competition
CHOI, Jay PilMichigan State University
FURUSAWA, TaijiUniversity of Tokyo
ISHIKAWA, JotaRIETI
The Research Institute of Economy, Trade and Industryhttps://www.rieti.go.jp/en/
RIETI Discussion Paper Series 20-E-035
April 2020
Transfer Pricing Regulation and Tax Competition*
Jay Pil Choi
Michigan State University
Taiji Furusawa
University of Tokyo
Jota Ishikawa
Hitotsubashi University & RIETI
Abstract
The paper analyzes multinational enterprises' incentives to manipulate internal transfer prices to take
advantage of tax differences across countries, and implications of transfer-pricing regulations as a
countermeasure against such profit shifting. We find that tax-motivated foreign direct investment
(FDI) may entail inefficient internal production but may benefit consumers. Thus, encouraging
transfer-pricing behavior to some extent can enhance social welfare. Furthermore, we consider tax
competition between two countries in order to explore the interplay with transfer-pricing regulations.
We show that the FDI source country will be willing to set a higher tax rate and tolerate some profit
shifting to a tax haven country if the regulation is tight enough. We also indicate a novel mechanism
through which it is the larger country that undertakes tax-motivated FDI, the pattern we often observe
in reality.
Keywords: Multinational enterprise; Corporate tax; Transfer pricing; Foreign direct investment; Arm's
length principle; Tax competition
JEL classification: F12, F23, H21, H26, L12, L51
The RIETI Discussion Paper Series aims at widely disseminating research results in the form of professional
papers, with the goal of stimulating lively discussion. The views expressed in the papers are solely those of
the author(s), and neither represent those of the organization(s) to which the author(s) belong(s) nor the
Research Institute of Economy, Trade and Industry.
* This paper is an extensively revised version of a part of Choi et al. (2018) and is presented as a part of the Project
“Analyses of Offshoring” undertaken at the Research Institute of Economy, Trade and Industry (RIETI). The authors
wish to thank Arnaud Costinot (coeditor) and two anonymous referees for helpful comments and suggestions. We
also thank Hayato Kato, Yuka Ohno, Pascalis Raimondos, Martin Richardson, Jay Wilson and participants in various
conferences and seminars for their valuable discussions and comments. This research was initiated during Choi's
visit to Hitotsubashi Institute for Advanced Study whose hospitality is greatly appreciated. Choi acknowledges
financial support from the Ministry of Education of the Republic of Korea and the National Research Foundation of
Korea: Grant Number NRF-2016S1A5A2A01022389. Furusawa and Ishikawa acknowledge financial support from
the Japan Society of the Promotion of Science through the Grant-in-Aid for Scientific Research (A): Grand Numbers
19H00594 and 17H00986.
1 Introduction
Corporate tax rates substantially differ across countries, and some countries adopt prefer-
ential tax measures.1 Since multinational enterprises (MNEs) actively engage in intrafirm
transactions across borders,2 they have an incentive to manipulate internal transfer prices
to save tax payments, an activity often called “transfer pricing”. MNEs tend to shift
their profits from high-tax countries to low-tax jurisdictions (see Hines and Rice, 1994;
Huizinga and Laeven, 2008; Bauer and Langenmayr, 2013; Davies et al., 2018). For in-
stance, inspections by the Vietnamese tax authorities have found that “the most common
trick played by FDI enterprises to evade taxes was hiking up prices of input materials
and lowering export prices to make losses or reduce profits in books”.3 In addition, Egger
et al. (2010) find that the average subsidiary of an MNE pays about 32% less tax than
similar local firms in high-tax countries. According to Goldman Sachs, the tax saving by
U.S. MNEs amounts to $2 trillion, equivalent to four years’ worth of U.S. corporate tax
revenues (Nikkei, August 31, 2016).
If governments do not regulate transfer pricing, MNEs may shift a large proportion
of their profits away from their countries to low-tax or no-tax jurisdictions, narrowing
their tax bases significantly. Governments thus impose transfer-pricing rules to control
transfer-price manipulation. In particular, the Organisation for Economic Co-operation
and Development (OECD) proposed that internal transfer prices follow the so-called arm’s
length principle (ALP) in its guidelines for transfer pricing published in 1995 and revised
in 2010.4
The basic approach of the ALP is that the headquarters and affiliates of an MNE
should be treated as “operating as separate entities rather than as inseparable parts of
a single unified business” and the controlled internal transfer price should mimic the
market price that would be obtained in comparable uncontrolled transactions at arm’s
length. This kind of comparative analysis is at the heart of the application of the ALP.
Currently, the ALP is the international transfer-pricing principle to which OECD member
1Preferential tax measures are observed in many countries. For example, Vietnam grants special orpreferential tax rates to attract foreign direct investment (FDI). Thailand provides tax incentives forFDI. The Taiwanese corporate tax rate is lower by 7 percentage points for foreign firms with some fixedplaces of business or some business agents in Taiwan.
2Bernard et al. (2010) report that over 46% of U.S. imports comprised intrafirm transactions in 2000.3http://vietnamlawmagazine.vn/transfer-pricing-unbridled-at-fdi-enterprises-4608.html4The OECD also initiated a project in 2012 to create an action plan to address base erosion and profit
shifting.
1
countries have agreed.
Against this background, the purpose of this study is twofold. The first purpose is
to closely examine MNEs’ incentive to manipulate transfer prices to avoid tax payments,
and governments’ regulatory responses to such practices. We specifically investigate the
transfer pricing resulting from tax-motivated FDI and the implication of the ALP, when
imperfect competition prevails in the final-good market. To this end, we consider a
stylized set-up of two countries with different corporate tax rates, where a monopolist
that produces and sells its final products in the high-tax country can set up its subsidiary,
which produces intermediate goods, in the low-tax country to engage in tax-saving transfer
pricing. We show that this monopolistic MNE shifts all its profits to the low-tax country
by choosing the standard monopoly price as the transfer price, replicating the monopoly
outcome in the market.
To counter such extreme transfer-pricing activities, the MNE’s home country may
tighten transfer-pricing regulation by lowering the transfer-price cap. In this study, we
consider transfer-price caps to be a regulatory measure.5 Tax authorities audit tax-
avoidance behaviors by comparing the prices used in intrafirm transactions with those of
similarly uncontrolled transactions between independent parties (i.e., arm’s length prices);
this method of the ALP is called the comparable uncontrolled price (CUP) method. In
practice, however, it is often difficult to find a comparable transaction of similar products
between independent enterprises. In such cases, other ALP methods such as the cost
plus (CP) method, are applied.6 The price cap which we consider in this study can be
thought of as a CP method, as regulatory authorities often regard the cost plus a reason-
able mark-up as a transfer price that meets the ALP.7 The CP method is also a natural
ALP in the case of monopoly, because in many monopolistic environments it is hard to
find other transactions of comparable inputs.
We show that transfer pricing entails lowering what the MNE perceives as its marginal
cost of production. The “perceived marginal cost” declines as a result of transfer pricing,
because the marginal tax saving that arises from an additional shipment of intermediate
goods serves as the marginal benefit of production for the MNE. Not surprisingly, there-
5This assumption has often been made to investigate transfer-pricing regulations. See Raimondos-Moler and Scharf (2002), Peralta et al. (2006), Becker and Fuest (2012), and Matsui (2012), amongothers.
6Other suggested methods include the resale price method, transactional net margin method, andtransactional profit split method. See OECD (2010) for more details.
7Other ALP methods (except for the CUP method) indirectly regulate the ceiling of the transfer price.
2
fore, FDI occurs even if it is less efficient to produce the intermediate goods internally
at its foreign subsidiary, because it can be used as a vehicle to lessen its tax burden
with an inflated internal price. Interestingly, however, profit shifting with the regulation
may also benefit consumers, because transfer pricing lowers the monopolist’s perceived
marginal cost, thereby leading to more production which alleviates allocative inefficiency
due to market power. Thus, the MNE’s home country may want to encourage its firm’s
tax-saving, transfer-pricing activities.
The second purpose of the paper is to identify the economic environment in which a
country is indeed willing to select a higher tax rate than another country to encourage
its monopolistic firm to engage in FDI and transfer pricing, hoping that the resulting
alleviation of allocative inefficiency enhances its social welfare. To this end, we first
investigate a sequential-move, tax-competition game (i.e., a Stackelberg tax-competition
game) played by two welfare-maximizing governments within our baseline model with
transfer-pricing regulation.8 We show that the nature of tax competition can depend
on the tightness of transfer-pricing regulation. In particular, the MNE’s home country
is willing to set a higher tax rate and tolerate some profit shifting to an endogenously-
determined, tax-haven country, if the regulation is sufficiently tight. However, if regulation
is too lax, tax competition leads to a “race to the bottom” that eliminates any incentives
for tax-motivated FDI. Interestingly, this implies that a tax haven country does not always
prefer lax transfer-pricing regulation. Thus, the incentives of the host and FDI source
countries can be aligned to set up global regulatory standards for transfer pricing.
Finally, we extend the sequential-move game to a simultaneous-move, tax-competition
model with multiple industries, played by two countries with different population sizes.9
This set-up allows us to endogenously determine the FDI source country that selects
a higher tax rate and investigate the characteristics of such countries. We show that
a pure-strategy, subgame-perfect equilibrium exists if countries are sufficiently different
in size. The large country sources FDI, choosing a higher tax rate than the smaller
country. It is willing to create an environment in which its firm engages in FDI to save
tax payments because, for the large country, the benefit of the resulting increase in the
8For example, Baldwin and Krugman (2004) and Stowhase (2013) also analyze tax competition in asequential-move game.
9We treat transfer-pricing regulation as given in our analysis, because the ALP is agreed as the transfer-pricing regulation among OECD member countries. By contrast, Mansori and Weichenrieder (2001) andRaimondos-Moler and Scharfy (2002) focus on the strategic use of transfer-pricing rules rather than taxeswhen analyzing transfer pricing in a single industry. In their analyses, governments non-cooperativelychoose their optimal transfer-pricing rule, taking the tax rates as given.
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consumer surplus is large relative to the tax revenue that it would have earned if it hosted
FDI. The smaller country, on the contrary, undercuts the large country’s tax rate, because
the tax revenue collected from the subsidiary of the firm operating in the large market
is large. This novel result is consistent with the fact that most low-tax countries or tax
havens are small countries.10 The result also gives some justification to use a model
with the Stackelberg tax-setting nature, because it is reasonable to think that in reality
small tax-haven countries set their tax rates after observing large countries’ tax rates.
The welfare impacts of transfer pricing, which are readily obtained in the sequential-move
game, can also extend to the setting of the simultaneous-move game.
Our study is thus at the intersection of international trade and public economics.
Horst (1971) initiated the theory of MNEs in the presence of different tariff and tax rates
across countries and explored the profit-maximizing strategy of a monopolistic firm selling
to two national markets (i.e., how much it should produce in each country and what the
optimal transfer price for goods exported from the parent to the subsidiary would be).
Horst (1971) and subsequent studies (e.g., Itagaki, 1979) show that an MNE’s optimum
price would be either the highest or the lowest possible allowed by the limits of government
rules and regulations, depending on tax and tariff schedules among countries. Samuelson
(1982) is the first study to point out that for an MNE subject to the ALP, the arm’s
length reference price itself can be partially determined by the firm’s activities.
Lee (1998) analyzes the effects of profit taxes on firm behaviors and shows that a
monopolist engaged in tax avoidance may produce either more or less than it would in
the absence of a tax.11 The focus of his analysis is on deriving the conditions under which
profit taxes are non-neutral in terms of the monopolist’s output decision. It is shown
that the effects of tax avoidance on the consumer surplus depend crucially on the shape
of the audit probability and penalty rate. Our model identifies a different channel (i.e.,
transfer pricing to shift profits across jurisdictions with different tax rates) through which
the consumer surplus is affected when the monopolist aims to minimize the overall tax
burden.
In our model, the welfare of the high-tax country can improve because the consumers
there benefit from regulated transfer-pricing activities. In particular, welfare can improve
even if intermediate-good production is less efficient with FDI than without FDI. Matsui
(2012) is closely related to our study in that he considers internal transfer prices in MNEs
10These countries include Ireland, Luxembourg, Singapore, Switzerland and the Netherlands.11We thank an anonymous referee for directing our attention to this study.
4
as the channel for tax avoidance and shows that an MNE’s transfer pricing can benefit
consumers through an increase in its output. In a Dixit–Stiglitz model of monopolistic
competition, he shows that tax authorities face a trade-off between consumer welfare and
tax revenue.12 However, he deals with neither tax-motivated FDI nor tax competition.
We also show that the relationship between the transfer-price cap and consumer price is
not monotonic.
Tax competition between countries of different size has been explored extensively
(Bucovetsky, 1991; Wilson, 1991; Haufler and Wooton, 1999; Stowhase, 2005, 2010). The
literature also shows that the smaller country is likely to set a lower tax rate in various
settings. Stowhase (2005, 2010) is related to our analysis in the sense that profit shifting
is explicitly taken into account in a tax-setting game between two tax-revenue-maximizing
governments. In our analysis, however, the governments maximize social welfare, caring
about not only tax revenue but also profits and consumer welfare. Moreover, Stowhase
(2005, 2010) does not deal with transfer-pricing regulation. We identify an aforementioned
novel logic of an equilibrium outcome that the larger country is willing to set a higher
corporate tax rate than the smaller country in the presence of transfer-pricing regulation.
Although tax competition has been examined in the presence of profit shifting (Stowhase,
2005, 2010; Bucovetsky and Haufler, 2008; Slemrod and Wilson, 2009; Johannesen, 2010;
Marceau et al., 2010), only a few theoretical studies take transfer-pricing regulation into
consideration. Elitzur and Mintz (1996) study tax competition between the MNE’s home
country and its host country, both of which try to maximize individual tax revenues. The
MNE operates under rigid transfer-pricing regulation along the lines of the CP method.
In their model, however, there is no consideration of whether the firm sets up a subsidiary
in the host country. Hence, they are unable to discuss the possible inefficiency caused
by tax-motivated FDI to a high-cost country. Moreover, in their model, transfer pricing
is used not only for profit shifting but also for an incentive scheme to improve the sub-
sidiary’s efficiency. Peralta et al. (2006) identify an interesting role of transfer-pricing
regulation in the context of bidding for a firm: loosening the control of profit shifting
can mitigate tax competition so that winning becomes more beneficial in the presence of
indigenous firms that also pay corporate taxes.13 In some cases, therefore, the winning
country does not strictly enforce the regulation even though it entails tax revenue leak-
12Matsui (2012) focuses on the long-run equilibrium in which the MNE’s post-tax profits are zero.13Becker and Fuest (2012) introduce a model that does not require the presence of indigenous firms
and obtain qualitatively the same result as Peralta et al. (2006).
5
age. Hosting a firm (or the headquarters of a firm in our context) is different from getting
hold of its profits. Because of its reduced-form structure of the model, however, they
are unable to examine the effect of transfer pricing on the firm’s output and its welfare
consequences through the effect on the consumer surplus. In our study, this channel is
crucial to identifying the importance of market size in the endogenous determination of
the FDI source country and a tax haven that hosts FDI.
The rest of the paper is organized as follows. Section 2 introduces the basic set-up
of a monopoly model with transfer pricing. In the model in which the monopolist has
already set up a subsidiary in the foreign country, we examine this monopolistic MNE’s
transfer-pricing behavior when its home tax rate is exogenously given at a higher rate
than the foreign rate. In particular, we show that the MNE increases its output from
the monopolistic production level when its home country imposes a binding cap on the
transfer price. In Section 3, we model a sequential-move, tax-competition game, played by
two countries, still assuming that only one firm, in the first-mover country, sources inputs
from the foreign country. We show that the home country selects a higher tax rate than
the foreign country if and only if transfer-pricing regulation is sufficiently tight. Section 4
extends the tax-competition game to a simultaneous-move game with a monopoly in each
country operating in each of the two industries. Moreover, we allow the two countries
to differ in size. Then, we show that there is a pure-strategy equilibrium in which the
larger country sets a higher tax rate than the smaller one to become the source country
of tax-motivated FDI. Section 5 concludes.
2 Monopoly Model with FDI: Transfer Pricing and Regulations
2.1 Baseline Model
There are two countries, Country 1 and Country 2, with possibly different corporate tax
rates, t1 and t2. In Country 1, there is a monopolistic final-good producer, which we often
call the MNE. The MNE’s headquarters that produces the final good is immobile and tied
to Country 1, while its intermediate input can be procured either from the open market
at a price of ω or from its subsidiary in Country 2.14 We assume that one unit of the input
is converted into one unit of the final good without incurring any additional costs and
that input production is subject to constant-returns-to-scale technology. We also assume
14Alternatively, we can assume that ω is the headquarters’ constant marginal production cost of theintermediate input.
6
away the costs associated with FDI.15 The subsidiary’s unit cost of input production (i.e.,
the marginal production cost) is given by c. The final good is consumed only in Country
1.
The MNE can choose an internal transfer price of γ, when its foreign subsidiary sup-
plies its input to the headquarters. Without any tax rate differential between the two
countries, the MNE’s optimal internal transaction price for the input is simply its marginal
cost, c, to eliminate the double marginalization problem. However, as the two countries
have different tax rates, the MNE can engage in transfer pricing such that it chooses γ to
optimally allocate its profits between its headquarters and subsidiary and thus minimize
its tax burden.
In this section, we assume that the two countries’ corporate tax rates are exogenously
given such as t1 > t2, meaning that the MNE in Country 1 has an incentive to manipulate
its internal transfer price to avoid tax payments. We also assume throughout this section
that the MNE has already set up its subsidiary in Country 2 to procure intermediate
goods, thereby focusing the analysis on the MNE’s transfer-pricing strategy and its home
country’s regulatory response.
2.2 The Benchmark Case: No Regulation
As a benchmark case, we first consider the choice of transfer price by the MNE without
any transfer-pricing regulation. In this case, the MNE selects its transfer price, γ, and
output, q, to maximize the following global profits after tax:
Π = (1− t1) [P (q)− γ]q︸ ︷︷ ︸Downstream Profits
+ (1− t2) (γ − c)q︸ ︷︷ ︸Upstream Profits
, (1)
where P (q) is the downward-sloping inverse demand function faced by the monopolist
(with the corresponding demand function q = D(p)). Decision making is assumed to be
centralized. That is, with the firm’s objective function above, the headquarters producing
the final good makes an output decision that maximizes the overall profit of the firm, not
only the profit of the downstream division.
We can rewrite the objective function of the monopolist, expressed by (1), as
Π = (1− t1)[P (q)− ξ]q, (2)
15If a fixed cost of setting up the subsidiary exists, FDI becomes less likely. However, the essence ofour results would not change.
7
where
ξ ≡ (1− t2)c− (t1 − t2)γ
1− t1. (3)
That is, the MNE facing different tax rates across countries behaves as if its marginal pro-
duction cost were ξ, which can be considered as its perceived marginal cost of production.
Since the MNE’s profit decreases as ξ increases, the monopolist’s optimal choice of γ is
to minimize ξ regardless of its output level. Note that ξ is decreasing in its transfer price
γ because it can be used as a vehicle to shift profit from the high-tax country (Country
1) to the low-tax country (Country 2).
As pointed out by Horst (1971), it immediately follows that the optimal choice is to
set γ as high as possible with the constraint that the downstream profit is non-negative.
This would imply that all profits from a high-tax country are shifted toward to a low-
tax country in the absence of any regulation. This simple model illustrates the need for
regulations to counter such MNE’s profit-shifting motives to reduce the tax burden.
To show this rigorously, we first derive q∗(γ), which we define as the MNE’s optimal
output when its transfer price is set at γ. Let qm(ξ) denote the optimal monopoly quantity
that maximizes the monopolist’s profits when its perceived marginal cost is ξ. Since
qm(ξ(γ)) is increasing in γ and hence P (q∗(γ)) is decreasing in γ, there is a unique γ = γ0
such that P (q∗(γ0)) = γ0. In other words, if γ < γ0, the optimal output by the MNE
is given by q∗(γ) = qm(ξ(γ)) with the downstream division in the high-tax Country 1
making a positive profit. However, if γ > γ0, the non-negative profit constraint for the
headquarters is binding because P (q(γ)) < γ. Therefore, the MNE’s optimal output q∗(γ)
cannot be qm(ξ(γ)). The maximum output that can be used to transfer profit is capped
by P−1(γ). That is, if γ > γ0, we have q∗(γ) = P−1(γ) = D(γ), which is decreasing in γ.
We can conclude that
q∗(γ) =
{qm(ξ(γ)) if γ < γ0
P−1(γ) = D(γ) if γ ≥ γ0.
The following lemma helps us determine the MNE’s optimal choice of γ.
Lemma 1. Without any transfer-pricing regulation, the MNE’s optimal choice of γ can-
not be less than γ0. This implies that all the downstream profits are transferred to the
subsidiary located in the country with a lower tax rate, that is, P (q∗(γ)) = γ.
Proof. Suppose, on the contrary, that γ < γ0. Then, the optimal output by the MNE is
given by q∗(γ) = qm(ξ(γ)) With an infinitesimal increase in γ, the new transfer price is
8
still less than γ0 while reducing ξ. As a result, the MNE’s profit increases, resulting in a
contradiction.
Given Lemma 1, we can rewrite the optimal choice of γ by the MNE as
maxγ
(1− t2)(γ − c)D(γ). (4)
This is mathematically equivalent to the standard monopoly pricing problem where γ can
be considered as the monopolist’s price. This implies that the MNE chooses a monopoly
price equal to P (qm(c)). Proposition 1 summarizes our discussion in the case of no reg-
ulation and calls for regulations to counter the MNE’s profit-shifting motives to reduce
the tax burden.
Proposition 1. (Full Profit Shifting with the Monopoly Outcome under No Regulation)
In the absence of any transfer-pricing regulation, the MNE shifts all its profits to the
country with the lower tax rate by choosing the transfer price at the standard monopoly
price, replicating the monopoly outcome in the market.
2.3 Transfer-pricing Regulation and the MNE’s Decisions
In reality, regulations would prevent the choice from being a corner solution and limit the
MNE’s profit-shifting motives. The most-widely adopted and agreed-upon standard prac-
tice is the ALP, which requires intrafirm transfer prices to mimic the market prices that
would be obtained in comparable uncontrolled transactions at arm’s length, as discussed
in the Introduction. Although this principle is conceptually sound and straightforward,
its implementation as a regulatory policy may be difficult and subject to different inter-
pretations. For instance, in the monopoly context, such a comparative analysis is unlikely
to be feasible, simply because there is no comparable transactions as it is the only firm
that produces the final good; no other firms acquire similar inputs. Even if similar inputs
are transacted in the market by other firms for different purposes, a firm may argue that
the available inputs are not suitable to meet its specifications, which explains why they
are engaged in its production in the first place. In other words, what constitutes a similar
input may not be clear-cut and could be subject to disputes unless comparable inputs are
identical.
In our analysis, we simply model Country 1’s transfer-pricing regulation as a price cap
for the transfer price to limit the extent to which the MNE can shift its profits to the
9
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qm(c) =1 � c
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P (qm(c))<latexit sha1_base64="Q5pfn6PClAnStJTq1O80qPcW5+s=">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</latexit><latexit sha1_base64="Q5pfn6PClAnStJTq1O80qPcW5+s=">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</latexit><latexit sha1_base64="Q5pfn6PClAnStJTq1O80qPcW5+s=">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</latexit><latexit sha1_base64="Q5pfn6PClAnStJTq1O80qPcW5+s=">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</latexit>
�̄<latexit sha1_base64="myiteofBN2Ixrg5ybcZRjhlzkeg=">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</latexit><latexit sha1_base64="myiteofBN2Ixrg5ybcZRjhlzkeg=">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</latexit><latexit sha1_base64="myiteofBN2Ixrg5ybcZRjhlzkeg=">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</latexit><latexit sha1_base64="myiteofBN2Ixrg5ybcZRjhlzkeg=">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</latexit>
q = P�1(�̄)<latexit sha1_base64="nA5Oi59UXZV1p0CqtxXmTHWk1Yg=">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</latexit><latexit sha1_base64="nA5Oi59UXZV1p0CqtxXmTHWk1Yg=">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</latexit><latexit sha1_base64="nA5Oi59UXZV1p0CqtxXmTHWk1Yg=">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</latexit><latexit sha1_base64="nA5Oi59UXZV1p0CqtxXmTHWk1Yg=">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</latexit>
q = qm(⇠(�̄))<latexit sha1_base64="jG3sAYOYYLWRVytn21rHqrgkl+c=">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</latexit><latexit sha1_base64="jG3sAYOYYLWRVytn21rHqrgkl+c=">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</latexit><latexit sha1_base64="jG3sAYOYYLWRVytn21rHqrgkl+c=">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</latexit><latexit sha1_base64="jG3sAYOYYLWRVytn21rHqrgkl+c=">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</latexit>
P�1(c) =1 � c
b<latexit sha1_base64="jjiXgRXONq8D1N5SQRw7F0zQyyA=">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</latexit><latexit sha1_base64="jjiXgRXONq8D1N5SQRw7F0zQyyA=">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</latexit><latexit sha1_base64="jjiXgRXONq8D1N5SQRw7F0zQyyA=">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</latexit><latexit sha1_base64="jjiXgRXONq8D1N5SQRw7F0zQyyA=">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</latexit>
Figure 1: MNE’s optimal output level
low-tax country. Country 1 caps the transfer price at γ̄. Under this regulation, the MNE
can only select γ that is less than or equal to γ̄.
We derive the MNE’s optimal choice of γ under the restriction and its optimal output
level as a function of γ̄. As shown in the previous subsection, the MNE raises γ as much as
it can, all the way up to the monopoly price P (qm(c)) if possible. If γ < γ0, for example,
the MNE can raise its profit by increasing γ, thereby decreasing the perceived marginal
cost ξ. If γ ≥ γ0, it can also raise its profit by increasing γ because γ here serves as
the price of the products and increasing γ is equivalent to bringing the price closer to
the monopoly price P (qm(c)), as shown in (4). Therefore, the MNE chooses the largest
possible γ under the price cap of γ̄, as long as this regulation is binding, that is, as long
as γ̄ ≤ P (qm(c)). If γ̄ > P (qm(c)), the MNE simply chooses γ at the monopoly price and
produces qm(c) units of the good in such a case. Figure 1 illustrates the MNE’s optimal
output level as a function of the price cap γ̄ (in the case of linear demand, as given by
(5)-(7) below). An increase in γ̄ increases the optimal output if c ≤ γ̄ ≤ γ0 but decreases
it if γ0 < γ̄ ≤ P (qm(c)).
Lemma 2. With the given transfer-price cap γ̄, the MNE chooses γ = γ̄ if γ̄ ≤ P (qm(c)),
and γ = P (qm(c)) if γ̄ > P (qm(c)). The optimal output level is given by
q∗(γ̄) =
qm(ξ(γ̄)) if c ≤ γ̄ ≤ γ0,
P−1(γ̄) if γ0 < γ̄ ≤ P (qm(c)),
qm(c) if γ̄ > P (qm(c)).
10
As an illustration and for later purposes, we explicitly derive some of the above key
functions and variables in the case of linear demand, such that P (q) = 1− bq:
qm(ξ(γ̄)) =1− c
2b+
(t1 − t2)(γ̄ − c)2b(1− t1)
, (5)
P−1(γ̄) =1− γ̄b
, (6)
γ0 =(1− t1) + (1− t2)c
(1− t1) + (1− t2). (7)
2.4 Optimal Taxation and Transfer-pricing Regulation
Figure 1 shows the possibility that Country 1, the home country of the MNE, can mitigate
the monopoly distortion by relaxing transfer-pricing regulation and hence inducing the
MNE to produce more. Here, we investigate whether Country 1 can indeed enhance
its social welfare by optimally adjusting the transfer-price cap. In addition, we derive
its optimal policy when both the transfer price cap and the corporate tax rate can be
freely chosen, for the given tax rate in Country 2 (i.e., the country that hosts the MNE’s
subsidiary).
Country 1’s social welfare is defined as the sum of the consumer surplus (CS), producer
surplus (PS), and tax revenue (TR). It follows from
CS =
∫ q∗(γ̄)
0
P (q)dq − P (q∗(γ̄))q∗(γ̄),
PS = Π1 = (1− t1)[P (q∗(γ̄))− ξ(γ̄)]q∗(γ̄),
TR = t1[P (q∗(γ̄))− γ̄]q∗(γ̄)
that Country 1’s social welfare can be written as
W1 =
∫ q∗(γ̄)
0
[P (q)− ξs(γ̄)]dq, (8)
where the social marginal cost ξs is defined and can be written with the use of (3) as
ξs(γ̄) = (1− t1)ξ(γ̄) + t1γ̄
= (1− t2)c+ t2γ̄. (9)
Obviously, W1 increases with the MNE’s output level, q∗(γ̄), and decreases with the social
11
marginal cost, ξs(γ̄).
Lemma 3. The optimal transfer price cap for Country 1 is greater than γ0. It equals γ0
if t2 is small, while it is less than γ0 if t2 is sufficiently large to be close to t1.
Proof. As Figure 1 shows, q∗(γ̄) increases by lowering γ̄ when γ̄ > γ0. Because a decrease
in γ̄ also decreases ξs(γ̄) as (9) shows, W1 increases by lowering γ̄ in this range of γ̄. Thus,
the optimal γ̄ is greater than γ0.
In the range that c ≤ γ̄ ≤ γ0, a reduction in γ̄ involves a trade-off between the
beneficial effect of lowering ξs(γ̄) and adverse effect of decreasing q∗(γ̄). If t2 is small,
the latter adverse effect outweighs the former effect. Indeed, if t2 = 0 at the extreme, (9)
shows that ξs(γ̄) = c, and hence decreasing γ̄ from γ0 only reduces q∗(γ̄) leaving ξs(γ̄)
as it is. Hence, it is optimal for Country 1 to set γ̄ = γ0. However, if t2 is not so small,
the beneficial effect of lowering ξs(γ̄) outweighs the adverse effect at γ̄ = γ0, making it
optimal for Country 1 to set γ̄ below γ0. At the extreme where t2 is so large to be equal
to t1, we know from (3) that ξ(γ̄) = c and thus q∗(γ̄) = qm(c). Hence, the optimal γ̄ is as
small as c to minimize ξs(γ̄).
What if Country 1 can adjust t1, for the given t2, as well as choose an arbitrary
γ̄? Redefining ξ as a function of t1 as well as γ̄ (i.e., ξ(γ̄, t1)), we obtain from (3) that
∂ξ/∂t1 = −(γ̄ − c)(1 − t2)/(1 − t1)2 < 0. Since qm is decreasing in ξ, this means that
qm increases as t1 rises, except at γ̄ = c. This relationship is easily verified from (5) in
the linear demand case, as illustrated by the counterclockwise rotation of the qm line in
Figure 2. As this figure shows, an increase in t1 coupled with a decrease in γ̄ (from γ0 to
γ0′) enables Country 1 to increase q∗(γ̄) and decrease ξs(γ̄) at the same time.
Proposition 2. The MNE’s home country can enhance its social welfare by increasing its
corporate tax rate and tightening transfer-pricing regulation simultaneously. The monopoly
distortion is completely eliminated, meaning social welfare is maximized at the limit where
the tax rate approaches 1, whereas the transfer-price cap approaches the marginal cost of
the intermediate good.
Proof. As t1 increases, so does qm(ξ(γ̄, t1)) for any γ̄. It also follows from γ0 = P (qm(γ0, t1)),
the definition of γ0, that γ0 decreases accordingly, as Figure 2 illustrates. Thus, Country 1
can induce the MNE to produce more and lower the social marginal cost, as seen in (9), by
raising t1 and simultaneously lowering γ̄ to keep γ̄ = γ0. That is, Country 1 can increase
its social welfare by raising t1 and lowering γ̄, while keeping P−1(γ̄) = qm(ξ(γ̄, t1)). As
12
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qm(c) =1 � c
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P (qm(c))<latexit sha1_base64="Q5pfn6PClAnStJTq1O80qPcW5+s=">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</latexit><latexit sha1_base64="Q5pfn6PClAnStJTq1O80qPcW5+s=">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</latexit><latexit sha1_base64="Q5pfn6PClAnStJTq1O80qPcW5+s=">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</latexit><latexit sha1_base64="Q5pfn6PClAnStJTq1O80qPcW5+s=">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</latexit>
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q = P�1(�̄)<latexit sha1_base64="nA5Oi59UXZV1p0CqtxXmTHWk1Yg=">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</latexit><latexit sha1_base64="nA5Oi59UXZV1p0CqtxXmTHWk1Yg=">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</latexit><latexit sha1_base64="nA5Oi59UXZV1p0CqtxXmTHWk1Yg=">AAACeXichVHLLgRBFD3T3uM1HguJDSZkSExuIyESicTGcjwGiWFS3cro6JfunkmY+AE/YGGDRASfYeMHLHyCWJJYsHCnpxNBcCtVderUPbdOVWmuafgB0UNMqamtq29obIo3t7S2tSc6Old8p+jpMqs7puOtacKXpmHLbGAEplxzPSkszZSr2u5cZX+1JD3fcOzlYN+VG5Yo2Ma2oYuAqXyie28ms1keVQ9TOU14uYKwLDGcTyQpTWH0/QRqBJKIIuMkLpHDFhzoKMKChI2AsQkBn9s6VBBc5jZQZs5jZIT7EoeIs7bIWZIzBLO7PBZ4tR6xNq8rNf1QrfMpJnePlX0YpHu6ome6oxt6pPdfa5XDGhUv+zxrVa108+1HPUuv/6osngPsfKr+9BxgG1OhV4O9uyFTuYVe1ZcOjp+XphcHy0N0Tk/s/4we6JZvYJde9IsFuXiCOH+A+v25f4KVsbRKaXVhIjmbjr6iEb0YQIrfexKzmEcGWT73AKe4wnXsTelXUspINVWJRZoufAll/AN3C5FO</latexit><latexit sha1_base64="nA5Oi59UXZV1p0CqtxXmTHWk1Yg=">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</latexit>
q = qm(⇠(�̄))<latexit sha1_base64="jG3sAYOYYLWRVytn21rHqrgkl+c=">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</latexit><latexit sha1_base64="jG3sAYOYYLWRVytn21rHqrgkl+c=">AAACe3ichVHLSiNBFD3pUUczPqKCCG5kghJFwu1hxEEYENy49DF5gNFQ3VYyhf2yuxPGycwPzA+4cKUgIvoXbvwBF36CzDKCGwVvOg2iot6iqk6duufWqSrDs1QQEl0ltA8dnV0fu3uSn3r7+gdSg0P5wK35psyZruX6RUME0lKOzIUqtGTR86WwDUsWjO3F1n6hLv1Auc6PcNeTG7aoOqqiTBEyVU6N7Hzf2bQzpV8qUzKEX6oK2xZTU+VUmrIUxfhLoMcgjTiW3dQxStiCCxM12JBwEDK2IBBwW4cOgsfcBhrM+YxUtC/xF0nW1jhLcoZgdpvHKq/WY9bhdatmEKlNPsXi7rNyHBN0SSfUpAs6pWu6e7VWI6rR8rLLs9HWSq888G907fZdlc1ziJ+Pqjc9h6jgW+RVsXcvYlq3MNv6+u+95tr86kRjkg7pP/s/oCs65xs49RvzaEWu7iPJH6A/f+6XIP8lq1NWX/maXsjGX9GNMXxGht97DgtYwjJyfO4fHOIUZ4l7La1NazPtVC0Ra4bxJLTZB2Djkig=</latexit><latexit sha1_base64="jG3sAYOYYLWRVytn21rHqrgkl+c=">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</latexit><latexit sha1_base64="jG3sAYOYYLWRVytn21rHqrgkl+c=">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</latexit>
P�1(c) =1 � c
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Figure 2: The effect of an increase in the source country’s tax rate
shown in (3), at the limit where t1 approaches 1, γ̄ approaches c, meaning that ξ remains
finite to satisfy the above equality. Noting that ξs(γ̄) also approaches c, we conclude that
Country 1 can maximize its social welfare by raising t1 and lowering γ̄, thereby attaining
a market outcome equivalent to that under perfect competition at the limit where t1 = 1
and γ̄ = c.
In the current scenario in which the MNE’s home country has two policy tools, namely
the transfer-price cap and corporate tax rate, it can manage to increase the MNE’s output
and minimize tax leakage at the same time. Even though the country lowers γ̄, which
dampens the MNE’s incentive to produce more, it can increase t1 sufficiently large to
more than offset the incentive; hence the MNE is induced to increase its output because
of the increased incentive to save tax payments. This benefits the country because it can
increase the MNE’s output while reducing tax leakage. Indeed, this proposition indicates
that the country can lower the price of the good all the way to the MNE’s actual marginal
cost of production by tailoring its tax rate and price-cap.
3 Tax Competition between the MNE’s Home Country and the
Host Country
We have thus far analyzed how the MNE reacts to transfer-pricing regulation and corpo-
rate taxation in its home country, and discussed the optimal policy for the home county.
13
Proposition 2 reveals that the home country can completely eliminate the monopoly dis-
tortion through extreme transfer-pricing regulation and taxation. In reality, however, the
transfer price is regulated by the ALP, which suggests that it should be set at the level
that provides reasonable markups to the marginal production cost of intermediate goods.
On the contrary, sovereign countries have more freedom to set their individual corporate
tax rates. In this section, therefore, we fix γ̄, and discuss how the tax rates are determined
between the MNE’s home government and host government.
We consider a sequential-move game in which Country 1 (the MNE’s home country)
first sets its tax rate, and then Country 2 tries to undercut Country 1’s tax rate to
attract FDI, given the transfer-pricing regulation represented by the price cap of γ̄. We
show that the nature of tax competition can depend on the tightness of transfer-pricing
regulation. In particular, if γ̄ is not too large, Country 1 selects a relatively high tax rate
to allow Country 2 to undercut its tax rate in a unique subgame perfect equilibrium. The
monopolist in Country 1 sets up a subsidiary in Country 2 and engages in transfer pricing
to save tax payments. Country 1 is willing to let its own firm do so, because it leads to
an expansion of production and helps increase its social welfare. If regulation is too lax,
(i.e., γ̄ is sufficiently large), however, tax competition leads to the “race to the bottom”
and eliminates any incentive for tax-motivated FDI.
The next section considers a simultaneous-move game between the two countries,
where the direction of FDI is endogenously determined. Although the model setting in
the next section may be considered as more general than the one here, it entails multiple
equilibria in some cases and no pure-strategy equilibrium in other cases. Thus, it is
meaningful to consider a sequential-move game that pins down a unique equilibrium, to
gain insights into the tax competition when a monopolistic firm may engage in transfer
pricing. Moreover, the timing assumption seems rather realistic, especially when Country
2 merely acts as a tax haven.
Here, we relax the assumption that the MNE has already set up its subsidiary in
Country 2. That is, we now explicitly incorporate the firm’s choice of FDI for intermediate-
goods procurement, while maintaining the assumption that the firm’s headquarters and
its production facility for the final goods is located in Country 1.
Social welfare for either country depends on whether the firm engages in FDI. In the
case where the firm engages in FDI, Country 1’s social welfare can be written as
W1 =
∫ q∗(t1,t2)
0
[P (q)− ξs(t2)]dq, (10)
14
where ξs(t2) = c+ t2(γ̄− c), and the MNE’s output is now written as a function of t1 and
t2. When the firm does not engage in FDI, then it loses the opportunity of tax avoidance
while its marginal cost is ω. Country 1’s social welfare in this case is given by
W1 =
∫ qm(ω)
0
[P (q)− ω]dq =3(1− ω)2
8.
Country 2’s social welfare, by contrast, only consists of the tax revenue from the
MNE’s subsidiary, and can be written as
W2 = t2(γ̄ − c)q∗(t1, t2), (11)
if the firm engages in FDI, whereas it equals 0 otherwise.
With sequential moves, Country 1 maximizes W1 subject to
t2 ∈ BR2(t1) ≡ arg maxt2
t2(γ̄ − c)q∗(t1, t2),
taking into account that Country 2 will undercut its tax rate to obtain tax revenue. We
solve this problem by backward induction as usual.
For the sake of concreteness, we henceforth assume that the demand for the final good
is linear such that P (q) = 1− bq.
3.1 Country 2’s Best Response
Let us first consider the best response for Country 2, whose social welfare is given by
(11). Now, we see from (5) that qm decreases with t2. Therefore, if t2 is smaller than
the threshold given by (7), the firm’s output is constant at P−1(γ̄). Hence, Country 2 is
better off by increasing t2 up to the threshold. If t2 is sufficiently large that qm(ξ(t1, t2)) ≤P−1(γ̄), we have q∗(t1, t2) = qm(ξ(t1, t2)). Therefore, we can use (5) to write (11) as
W2 = t2(γ̄ − c)[
1− c2b
+(t1 − t2)(γ̄ − c)
2b(1− t1)
].
It is readily verified that the t2 that maximizes W2 in this case is given by
t2 =1− c− (1− γ̄)t1
2(γ̄ − c) . (12)
15
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t̄<latexit sha1_base64="AOro0pgCD0dNvkaM7Wc1ymZA6/E=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8Lbsi6DHgxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN//GSbIHTSxoKKq66e6KMyksBsG3t7a+sbm1Xdop7+7tHxxWjo6bVueG8QbTUpt2TC2XQvEGCpS8nRlO01jyVjy6nfmtJ26s0OoBxxmPUjpQIhGMopOa3Zgagr1KNfCDOcgqCQtShQL1XuWr29csT7lCJqm1nTDIMJpQg4JJPi13c8szykZ0wDuOKppyG03m107JuVP6JNHGlUIyV39PTGhq7TiNXWdKcWiXvZn4n9fJMbmJJkJlOXLFFouSXBLUZPY66QvDGcqxI5QZ4W4lbEgNZegCKrsQwuWXV0nz0g8DP7y/qtb8Io4SnMIZXEAI11CDO6hDAxg8wjO8wpunvRfv3ftYtK55xcwJ/IH3+QMoqI7H</latexit><latexit sha1_base64="AOro0pgCD0dNvkaM7Wc1ymZA6/E=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8Lbsi6DHgxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN//GSbIHTSxoKKq66e6KMyksBsG3t7a+sbm1Xdop7+7tHxxWjo6bVueG8QbTUpt2TC2XQvEGCpS8nRlO01jyVjy6nfmtJ26s0OoBxxmPUjpQIhGMopOa3Zgagr1KNfCDOcgqCQtShQL1XuWr29csT7lCJqm1nTDIMJpQg4JJPi13c8szykZ0wDuOKppyG03m107JuVP6JNHGlUIyV39PTGhq7TiNXWdKcWiXvZn4n9fJMbmJJkJlOXLFFouSXBLUZPY66QvDGcqxI5QZ4W4lbEgNZegCKrsQwuWXV0nz0g8DP7y/qtb8Io4SnMIZXEAI11CDO6hDAxg8wjO8wpunvRfv3ftYtK55xcwJ/IH3+QMoqI7H</latexit><latexit sha1_base64="AOro0pgCD0dNvkaM7Wc1ymZA6/E=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8Lbsi6DHgxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN//GSbIHTSxoKKq66e6KMyksBsG3t7a+sbm1Xdop7+7tHxxWjo6bVueG8QbTUpt2TC2XQvEGCpS8nRlO01jyVjy6nfmtJ26s0OoBxxmPUjpQIhGMopOa3Zgagr1KNfCDOcgqCQtShQL1XuWr29csT7lCJqm1nTDIMJpQg4JJPi13c8szykZ0wDuOKppyG03m107JuVP6JNHGlUIyV39PTGhq7TiNXWdKcWiXvZn4n9fJMbmJJkJlOXLFFouSXBLUZPY66QvDGcqxI5QZ4W4lbEgNZegCKrsQwuWXV0nz0g8DP7y/qtb8Io4SnMIZXEAI11CDO6hDAxg8wjO8wpunvRfv3ftYtK55xcwJ/IH3+QMoqI7H</latexit><latexit sha1_base64="AOro0pgCD0dNvkaM7Wc1ymZA6/E=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8Lbsi6DHgxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN//GSbIHTSxoKKq66e6KMyksBsG3t7a+sbm1Xdop7+7tHxxWjo6bVueG8QbTUpt2TC2XQvEGCpS8nRlO01jyVjy6nfmtJ26s0OoBxxmPUjpQIhGMopOa3Zgagr1KNfCDOcgqCQtShQL1XuWr29csT7lCJqm1nTDIMJpQg4JJPi13c8szykZ0wDuOKppyG03m107JuVP6JNHGlUIyV39PTGhq7TiNXWdKcWiXvZn4n9fJMbmJJkJlOXLFFouSXBLUZPY66QvDGcqxI5QZ4W4lbEgNZegCKrsQwuWXV0nz0g8DP7y/qtb8Io4SnMIZXEAI11CDO6hDAxg8wjO8wpunvRfv3ftYtK55xcwJ/IH3+QMoqI7H</latexit>
Figure 3: Country 2’s reaction curve
This t2 is a valid best response only if the resulting γ0 is greater than or equal to γ̄. To
see the condition for this requirement, we substitute (12) into (7) and find that γ0 ≥ γ̄ if
and only if
t1 < t̄ ≡ 3(1− γ̄)− (γ̄ − c)3(1− γ̄)
.
If this condition is violated, t2 is chosen to satisfy γ0 = γ̄, i.e.,
t2 = 1− 1− γ̄γ̄ − c (1− t1).
Another condition that is required for the t2 in (12) to be the best response is t2 ≤ t1,
which reduces to
t1 ≥ t ≡ 1− c2(γ̄ − c) + (1− γ̄)
.
Consequently, Country 2’s best response function is given by
B2(t1) =
t1 if t1 ≤ t,
1−c−(1−γ̄)t12(γ̄−c) if t < t1 ≤ t̄,
1− 1−γ̄γ̄−c (1− t1) if t̄ < t1 < 1,
(13)
where it is readily verified that t < t̄. Figure 3 depicts Country 2’s best response function.
The intuition for this result is as follows. When t1 is low (more precisely, when t1 ≤ t),
it is optimal for Country 2 to just undercut Country 1’s tax rate, as reducing the tax rate
16
further and inducing more output does not raise tax revenue. Thus, in this range, Country
2’s tax rate is a strategic complement to Country 1’s tax rate. However, if t1 becomes
moderately large (i.e., t < t1 ≤ t̄), then it can be optimal for Country 2 to lower the tax
rate to induce more output and hence tax revenue in response to an increase in Country
1’s tax rate. In this case, Country 2’s tax rate is a strategic substitute to Country 1’s
tax rate. Finally, if the tax rate in Country 1 becomes sufficiently large (i.e., t̄ < t1 < 1),
the incentive to shift profit by raising production becomes too large, while the zero profit
condition for the MNE’s headquarters is binding. As the output is capped and cannot
increase any more, Country 2 follows suit as Country 1 increases its tax rate. Once again,
Country 2’s tax rate becomes a strategic complement to Country 1’s tax rate.
3.2 Country 1’s Optimal Tax Rate
Now, we analyze Country 1’s optimal tax rate when it considers the undercutting threat
by Country 2. To avoid unnecessary complication in the exposition, we assume henceforth
(1 − c)2/2b > 3(1 − ω)2/8b, which means that the highest social welfare Country 1 can
achieve through regulation and tax policies in the presence of FDI, which is the same as
welfare under perfect competition as Proposition 2 shows, is greater than its social welfare
without FDI. If this condition is violated, Country 1 will never set a higher tax rate than
Country 2. In the subgame perfect equilibrium, both countries thus set their tax rates
to 0, whereas the monopolist engages in FDI if and only if c < ω. Here, we make the
following assumption
(1− c)2 >3
4(1− ω)2, (14)
which is equivalent to (1− c)2/2 > 3(1−ω)2/8 to focus on the interesting and meaningful
case.
As Country 2 will always choose a point on its reaction curve, Country 1’s optimal
choice can be considered as the choice of a point on Country 2’s reaction curve, as shown
in Figure 3. Then, we have the following proposition.
Proposition 3. The subgame perfect equilibrium of tax competition can be summarized
as follows. There is a critical level of γ̄, denoted by γ̄∗(< (1 + c)/2), such that (i) the
equilibrium tax profile is (t1, t2) = (t̄, 2/3) if γ̄ < γ̄∗ and (ii) (t1, t2) = (0, 0) otherwise.
Proof. We consider three segments on Country 2’s reaction curve: the segments between
A = (0, 0) and B = (t, t), B and C = (t̄, 2/3), and C and D = (1, 1) (Figure 3). First,
17
we argue that Country 1 prefers point A to any other points in the segment AB because
along this segment, ξ = [(1− t2)c− (t1− t2)γ]/(1− t1) = c. Thus, the monopolist’s output
is fixed at q∗(t1, t2) = qm(c) = (1+ c)/2. By contrast, ξs = c+ t2(γ̄− c) is increasing as we
move from A to B as t2 increases with t1. Thus, point A is preferred to any other points in
the segment AB. Moreover, point C is preferred to any other points in the segment BC.
Along the BC segment, output is increasing from qm(c) to P−1(γ̄). In addition, Country
2 responds by decreasing its tax rate as t1 increases. Thus, on both accounts, Country 1
is better off, and hence it prefers point C to any other points in the segment BC. Finally,
in the segment CD, output is once again fixed at the level of P−1(γ̄). Thus, as in the
segment AB, ξs = c+ t2(γ̄ − c) increases as we move from C to D as t2 increases with t1.
This leaves us two candidate equilibrium points, A and C. When point A is chosen,
Country 1’s welfare is given by
W1(A) = W1(0, 0) = max
{3(1− ω)2
8b,3(1− c)2
8b
}.
In this case, the monopolist’s FDI decision is always efficient from the viewpoint of global
production efficiency. When point C is chosen by the monopolist, it can be easily verified
that
W1(C) = W1(t̄, 2/3) =(3− 2c− γ̄)(1− γ̄)
6b. (15)
Observe that W1(C) is decreasing in γ̄, while W1(C)|γ̄=c = (1 − c)2/2b > W1(A) under
the assumption expressed by (14), and W1(C)|γ̄= 1+c2
= 5(1− c)2/24b < W1(A). Therefore,
there is a unique γ̄∗ such that W1(C)|γ̄=γ̄∗ = W1(A). This implies that in the subgame
perfect equilibrium, (i) (t1, t2) = (t̄, 2/3) if γ̄ < γ̄∗ and (ii) (t1, t2) = (0, 0) otherwise.
Proposition 3 reveals that Country 2 will undercut Country 1’s tax rate so that the firm
in Country 1 establishes a foreign subsidiary to engage in transfer pricing, if and only if
transfer-pricing regulation is tight. Otherwise, the two countries race to the bottom in tax
competition, and the firm undertakes FDI if and only if c < ω (i.e., no tax-motivated FDI
occurs). It is rather counter-intuitive that loosening regulation deters transfer pricing. If
the price cap is high, however, so is the resulting price for the final good. Then, Country
1 would obtain an insufficient benefit from allowing its firm to engage in transfer pricing.
Consequently, it selects zero tax rate to prevent Country 2 from undercutting the tax
rate; then the firm has no incentive to engage in FDI to avoid tax.
18
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no FDI<latexit sha1_base64="gtVRJ4BzJiCD2YA4w2hDNhq8oOU=">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</latexit><latexit sha1_base64="gtVRJ4BzJiCD2YA4w2hDNhq8oOU=">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</latexit><latexit sha1_base64="gtVRJ4BzJiCD2YA4w2hDNhq8oOU=">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</latexit><latexit sha1_base64="gtVRJ4BzJiCD2YA4w2hDNhq8oOU=">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</latexit>
W1 #, W2 "W1 + W2 #
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W1 #, W2 "W1 + W2 "
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W1 ", W2 "W1 + W2 "
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Figure 4: Welfare impacts
3.3 Welfare Impacts
Next, we assess the welfare impacts of opening the possibility of the firm’s FDI. We
examine the impact on world welfare, W1 + W2, as well as on each country. It follows
from (10) and (11) as well as t2(γ̄ − c)− ξs = −c that world welfare can be written as
W1 +W2 =
∫ q∗(t1,t2)
0
[P (q)− c]dq, (16)
when the firm engages in FDI. As the expression in (16) indicates, there is a trade-
off between enhancing allocative efficiency, namely, the efficiency attained by increasing
quantity from the monopolistic production level, and sacrificing production efficiency,
namely, the cost-minimizing production location. As we have seen, Country 1 can induce
the firm to produce more, only by encouraging it to engage in FDI in Country 2, which
may have a higher cost of production. It is readily shown, on the contrary, that W2 = 0
and hence world welfare equals W1 = 3(1− ω)2/8, if the firm does not undertake FDI.
Let us first consider the case in which γ̄ < γ̄∗. In this case, the firm engages in FDI
if and only if ω > ξ, as illustrated in Figure 4. Since Country 2 enjoys a positive welfare
level if and only if the firm sets up its subsidiary there, Country 2’s welfare improves if
and only if ω > ξ. As Figure 4 illustrates, Country 1’s welfare increases if and only if
ω > ω∗1, and world welfare enhances if and only if ω > ω∗.
Country 1’s welfare increases, if and only if W1(C) > 3(1 − ω)2/8, which is readily
shown to be equivalent to ω > ω∗1, where
ω∗1 ≡ 1− 2
3
√(1− γ̄)(3− γ̄ − 2c).
The threshold ω∗1 lies between ξ and c. We know from the definition of γ̄∗ that if ω = c,
then W1(C) > 3(1 − ω)2/8. That is, under tight transfer-pricing regulation, Country 1
prefers an equilibrium with FDI because of the increase in allocative efficiency, and this
is particularly so in the absence of production inefficiency when ω ≥ c. As ω falls and
19
hence c− ω rises, the gain in allocative efficiency decreases, while the loss in production
efficiency increases. The open-market marginal cost ω reaches the threshold before it
reaches ξ, where the gain in allocative efficiency disappears.
Similarly, world welfare increases, if and only if W1(C) +W2(C) > 3(1−ω)2/8, where
W1(C) +W2(C) is calculated from (16) with C = (t̄, 2/3). Deriving the threshold
ω∗ ≡ 1− 2√3
√(1− γ̄)(1 + γ̄ − 2c)
from
W1(C) +W2(C) ≡ (1− γ̄)(1 + γ̄ − 2c)
2b=
3(1− ω∗)2
8b,
it is readily shown that ω∗ < ω∗1, and W1(C)+W2(C) > 3(1−ω)2/8 if and only if ω > ω∗.
Now, both Country 1 and Country 2 enjoy gains from allocative efficiency. Consequently,
the threshold for world welfare is less than that for Country 1’s welfare.
Under tight transfer-pricing regulation, both countries benefit from the firm’s FDI and
transfer pricing, even though it involves a loss of production efficiency. However, these
gains diminish as foreign production costs c rise above home production costs ω.
In the case where γ̄ > γ̄∗, (t1, t2) = (0, 0) in the equilibrium, and the firm engages in
FDI, if and only if c < ω. Production efficiency is guaranteed with FDI. The countries,
however, do not enjoy gains from allocative efficiency with FDI because the firm sets γ = c
and the standard monopoly equilibrium is realized.
The above analysis is summarized in the following proposition.
Proposition 4. (i) Suppose γ̄ < γ̄∗. Then, there are three welfare level thresholds:
ξ < ω∗ < ω∗1. FDI arises and Country 2’s welfare improves if and only if ω > ξ, Country
1’s welfare increases if and only if ω > ω∗1, and the world welfare increases if and only
if ω > ω∗. (ii) Suppose γ̄ > γ̄∗. Then, FDI arises if and only if ω > c. FDI improves
Country 1’s welfare but does not affect Country 2’s welfare.
Before closing this section, we should mention an interesting, counter-intuitive effect
of transfer-pricing regulation on Country 2. Since Country 2, as a tax haven, can generate
tax revenue from the MNE only when γ̄ < γ̄∗, it does not always prefer laxer transfer-
pricing regulation, once we take into consideration that Country 1’s willingness to allow
transfer pricing depends on the tightness of such regulation. Thus, there is some room
for harmonizing and setting up global regulatory standards in transfer pricing. As Figure
4 indicates, tighter transfer-pricing regulation benefits both countries as long as the loss
20
in production efficiency is small.
4 Endogenous Selection of the FDI Source Country
We have thus far assumed that the MNE’s headquarters is located in Country 1 and
thus Country 1 always sources FDI. What characteristics does the FDI source country
have? Does it always have an incentive to select a higher tax rate to become an FDI
source country? To answer these questions, we consider here a simultaneous-move, tax-
competition game, played by two countries, either of which can be an FDI source country.
Each country has its own industry: Industry 1 in Country 1 and Industry 2 in Country
2.16 In Country 1, a monopolist, Firm 1, sells its products only in Country 1. Similarly, in
Country 2, Firm 2 in Industry 2 produces goods and sells them only in Country 2. Each
firm, however, can set up its subsidiary in the other country to produce intermediate
goods, which are shipped to its headquarters for final-goods production. Demand for
Firm 1’s final goods is characterized by the inverse demand function P1(q1) = 1− q1/L1,
whereas that for Firm 2’s goods is given by P2(q2) = 1−q2/L2, where L1 and L2 represent
the two countries’ populations, respectively.17
We suppose L1 ≥ L2, and show that otherwise symmetric countries have different tax
strategies in the equilibrium when L1 is sufficiently larger than L2: the larger country,
Country 1, will select a tax rate higher than Country 2’s, to deliberately allow its own
firm to engage in transfer pricing to save tax payments. Thus, this section provides a
theoretical foundation for the commonly observed phenomenon that tax-haven countries
tend to be small and that firms in large countries set up subsidiaries there to engage in
transfer pricing.
Let us first derive Country 1’s best response function. If Country 1 chooses a higher
tax rate than Country 2, it becomes an FDI source country that maximizes its social
welfare, given by (10), for the given t2. We find immediately that t1 can affect W1 only
through q∗(t1, t2). As shown in Figure 1, if t1 is sufficiently large that γ0 ≤ γ̄, Firm 1
selects q∗ = P−1(γ̄), which is independent of t1. On the contrary, if t1 is so small that
γ0 > γ̄, Firm 1 chooses qm(ξ(γ̄)), which is smaller than P−1(γ̄) and hence less favorable
for Country 1. Thus, Country 1 chooses any t1 that yields γ0 ≤ γ̄. It follows from (7)
16We thank Arnaud Costinot (the co-editor of this journal) for suggesting this extended model.17The specified linear demand functions can be derived if we assume that consumers’ valuations are
uniformly distributed on [0,1] in each country.
21
that the threshold t1 that yields γ0 = γ̄ should satisfy
1− t1 =γ̄ − c1− γ̄ (1− t2), (17)
so that Country 1’s best response function as an FDI source country is a correspondence
given by
Bs1(t2) =
{t1
∣∣∣∣1− γ̄ − c1− γ̄ (1− t2) ≤ t1 < 1
}. (18)
Any effective price cap must satisfy γ̄ < (1 + c)/2 = P (qm(c)), which directly means that
(γ̄ − c)/(1− γ̄) < 1. Thus, we have Bs1(t2) > t2 for any t2 < 1.
If Country 1 chooses a lower tax rate than Country 2, on the contrary, it becomes an
FDI host country and its best response is given by
Bh1 (t2) =
t2 if t2 ≤ t,
γ̄−c+(1−γ̄)(1−t2)2(γ̄−c) if t < t2 ≤ t̄,
1− 1−γ̄γ̄−c (1− t2) if t̄ < t2 < 1,
(19)
as it was derived as a best response function for Country 2, shown in (13), in the last
section.
The following lemma shows the two countries’ individual best response functions.
Lemma 4. For the given L1/L2, there is a threshold t̃2 such that Country 1’s best response
function is
B1(t2) =
{Bs
1(t2) if t2 < t̃2,
Bh1 (t2) otherwise,
(20)
and t̃2 is increasing in L1/L2. Country 2’s best response function is symmetric to Country
1’s. The only substantial difference is that the threshold t̃1 decreases with L1/L2.
Proof. See the Appendix.
Figure 5 illustrates Country 1’s reaction curve, which is characterized by (20).
Given the above lemma, we are ready to derive the Nash equilibrium of the simultaneous-
move, tax-competition game played by the two countries. We consider two cases. In the
first case, γ̄ is sufficiently small that t̃1 = t̃2 ≥ 2/3 holds when L1 = L2. In the second
case, γ̄ is large, meaning that t̃1 = t̃2 < 2/3 when L1 = L2. The threshold of 2/3 here
is derived from Bhi (t̄) = 2/3 for i = 1, 2. As a preliminary investigation, consider the
22
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Bs1
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Figure 5: Country 1’s reaction curve
knife-edge case where t̃1 = t̃2 = 2/3 holds when L1 = L2. The two countries are perfectly
symmetric when L1 = L2. If t2 = t̃2 = 2/3, one of Country 1’s best responses to t2 = 2/3
is to select t1 = t̄ to become the FDI source country, as Figure 6 shows, while Country
2’s best response to t1 = t̄ is to select t2 = 2/3. Similarly, if t1 = t̃1 = 2/3, one of
Country 2’s best response is to select t2 = t̄, while Country 1’s best response to t2 = t̄
is to select t1 = 2/3. Consequently, as Figure 6 illustrates, there are two Nash equilibria
(t1, t2) = (t̄, 2/3), (2/3, t̄).
When t̃1 = t̃2 ≥ 2/3 holds with L1 = L2, there is a continuum of equilibria; however,
when t̃1 = t̃2 = 2/3, as shown above, Country 1 becomes the source country and the host
in each of the cases. As illustrated in Figure 7, the Nash equilibrium set is then given by{(t1, t2)|1− t2 =
1− γ̄γ̄ − c (1− t1), t1 ≥ t̃1, t2 ≤ t̃2
}∪{
(t1, t2)|1− t1 =1− γ̄γ̄ − c (1− t2), t2 ≥ t̃2, t1 ≤ t̃1
}. (21)
In the first equilibrium set, Country 1 plays the role of the source country, whereas
Country 2 plays the role of the host. The roles are switched in the second equilibrium set.
In this case, each country is willing to be the source because γ̄ is so small that the tax
leakage to the host country is small. Each country also has an incentive to be the host
country because the source country’s tax rate is so high that it can maintain a relatively
high tax rate while undercutting the other country’s.
23
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Figure 6: Nash equilibrium when L1 = L2 and t̃1 = t̃2 = 23
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Figure 7: Nash equilibrium when L1 = L2 and t̃1 = t̃2 >23
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Figure 8: Nash equilibrium when L1 > L2
As L1/L2 increases, however, the equilibrium set under which Country 1 is the host
shrinks and eventually disappears. As Figure 8 depicts, t̃1 decreases and t̃2 increases as
L1/L2 rises. Consequently, the first equilibrium set in (21) expands, whereas the second
set shrinks. The second equilibrium set eventually disappears when t̃1 becomes smaller
than 2/3. In this asymmetric case, the equilibrium set is{(t1, t2)|1− t2 =
1− γ̄γ̄ − c (1− t1), t1 ≥ t̄1, t2 ≤ t̃2
}. (22)
In the second case where t̃1 = t̃2 < 2/3 holds with L1 = L2, there is no pure-strategy
Nash equilibrium as long as the two countries are completely symmetric. As L1/L2 rises,
however, the Nash equilibrium set, as described by (22) and graphically represented in
Figure 8, will appear.
We summarize the above results as a proposition.
Proposition 5. If Country 1 is sufficiently large relative to Country 2, there is a Nash
equilibrium set of the tax-competition game, in which Country 1 sets a higher tax rate than
Country 2 so that the firm in Country 1 sets up a subsidiary in Country 2 and engages
in transfer pricing. If the two countries are of a similar size and regulation is so tight
that the transfer-price cap is low, then in addition to this Nash equilibrium set, there is a
Nash equilibrium set in which Country 2, the smaller country, selects a higher tax rate to
become the FDI source country.
25
Since the transfer pricing caused by tax-motivated FDI shifts profits from the FDI
source country to the FDI host country, tax revenue decreases in the former country
but increases in the latter country. This tax-revenue increase becomes larger as the FDI
source country becomes larger. Furthermore, transfer pricing with regulation increases
output in the FDI source country, implying that the consumers in this country gain. This
gain also becomes larger as the country becomes larger. Thus, the larger country has an
incentive to be the FDI source country, whereas the smaller country has an incentive to
be the FDI host country. Hence, as was discussed in the sequential-move game, it may
not be optimal for the smaller country (i.e., the FDI host country) to “just” undercut the
tax rate of the large country (i.e., the FDI source country), because the tax revenue may
not be maximized for the smaller country.
5 Concluding Remarks
We have analyzed MNE’s incentives to manipulate an internal transfer price to take ad-
vantage of tax differences across countries and discussed implications of transfer-pricing
regulation as a countermeasure against such profit shifting. We found that tax-motivated
FDI may entail inefficient internal production but could benefit consumers. Thus, en-
couraging transfer-pricing behavior to some extent can enhance social welfare.
We have also considered tax competition between (exogenously determined) source
and host countries to explore the interplay between tax competition and transfer-pricing
regulation. In tax competition, each government non-cooperatively sets the tax rate to
maximize its social welfare. We showed that the nature of tax competition can depend on
the tightness of transfer-pricing regulation. In particular, the source country is willing to
set a higher tax rate and tolerate profit shifting to a tax-haven country under sufficiently
tight regulation. However, if regulation is too lax, tax competition leads to a “race to
the bottom” and eliminates any incentives for tax-motivated FDI. This finding implies
that a tax-haven country does not always prefer lax transfer-pricing regulation. Thus, the
incentives of the host and FDI source country can be aligned to set up global regulatory
standards for transfer pricing.
Finally, we have extended our tax competition model to endogenously determine the
identity of the source country in a set-up with multiple industries. This extended set-up
allows us to rationalize our basic model by deriving an equilibrium outcome that the
larger country is willing to set a higher corporate tax rate than the smaller country in
26
the presence of transfer-pricing regulation. We often observe in reality that firms in large
countries establish subsidiaries in small tax-haven countries and engage in transfer pricing.
The result that the large country sources FDI also provides some justification for using
a model with the Stackelberg tax-setting nature because it is reasonable to think that in
reality small tax-haven countries set their tax rates after observing large countries’ tax
rates. The welfare impacts of transfer pricing obtained in the sequential-move game can
also be extended to the setting of the simultaneous-move game.
Our study has mainly focused on a simple the monopoly setting. With oligopolistic
market competition, additional issues could arise, however. For instance, with oligopolistic
competition in the final-good market of the FDI source country, the internal transfer price
has additional strategic effects that further strengthen the incentive to inflate the transfer
price at the expense of rivals’ profits. Tax-motivated FDI by the MNE has spillover effects
that reduce tax revenue from other final-good producers as well as the MNE. Moreover,
with the presence of competitors that use similar inputs, the CUP method may be adopted
as an application of the ALP. In such a case, we can also uncover a novel mechanism for
input foreclosure when the input market is also imperfectly competitive. The MNE may
have an incentive for input foreclosure even if it is a more efficient input producer. The
new mechanism stems from the dependence of the transfer price on the market price
of a “comparable” input, which is endogenously determined. Some of these issues are
analyzed by Choi et al. (2018). In addition, with oligopolistic competition, each firm’s
FDI decision may depend on other firms’ FDI decisions. These issues represent potential
areas for future research.
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29
Appendix: Proof of Lemma 4
To show that Country 1’s best response is given by (20), we derive Country 1’s social
welfare when it sources and hosts FDI, respectively.
When County 1 sources FDI, it chooses t1 so that the price equals γ̄ regardless of t2
as we have seen. Country 1’s social welfare in this case consists only of the total surplus
in Industry 1. Thus, it follows from q∗ = L1(1− γ̄) and ξs = c+ (γ̄ − c)t2 that its social
welfare as a source country is given by
W s1 =
L1
2(1− γ̄)2 + (γ̄ − ξs)L1(1− γ̄)
=L1
2(1− γ̄)2 + L1(γ̄ − c)(1− γ̄)(1− t2),
for any t2. It is readily verified that W s1 is decreasing in γ̄, meaning that W s
1 takes the
largest value of L1(1− c)2/2 when γ̄ = c.
Country 1 compares this welfare W s1 with that when it selects a lower tax rate than
Country 2 and hence hosts a subsidiary of Firm 2. Social welfare as an FDI host country
consists of the total surplus in Industry 1 and that in Industry 2, and it thus takes a
different form depending on the level of t2 as (19) suggests.
When t2 ≤ t, the best response of Country 1 is to slightly undercut t2, or t1 is set equal
to t2 while Country 1 hosts FDI. In this case, Firm 2’s perceived marginal cost becomes
ξ2 = c and hence q∗2 = L2(1− c)/2, meaning that we have
W h1 =
3L1(1− ω)2
8+ t2(γ̄ − c)L2(1− c)
2.
Consequently, we have W s1 ≥ W h
1 if and only if
L1
L2
[(1− γ̄)2
2+ (γ̄ − c)(1− γ̄)(1− t2)− 3(1− ω)2
8
]≥ t2(γ̄ − c)(1− c)
2. (23)
It follows from the assumption expressed by (14) that as γ̄ decreases to c, the left-hand
side of this inequality approaches to a positive number, while the right-hand side converges
to 0. Thus, this inequality is satisfied for any t2 if γ̄ is sufficiently small. If γ̄ is not that
small, there exists a t̃2 such that W s1 ≥ W h
1 if and only if t2 ≤ t̃2. It is easy to see that
the threshold t̃2 increases with L1/L2.
We obtain a similar result when t < t2 < t̄. In this case, we have t1 = [1 − c + (1 −
30
γ̄)(1− t2)]/[2(γ̄ − c)] as indicated in (19). Then, we have
ξ2 =(1− t1)c− (t2 − t1)γ̄
1− t2= γ̄ − γ̄ − c− (1− γ̄)(1− t2)
2(1− t2),
and hence
q∗2 =L2(1− ξ2)
2=L2[γ̄ − c+ (1− γ̄)(1− t2)]
4(1− t2).
Country 1 obtains a tax revenue of
t1(γ̄ − c)q∗2 =L2[γ̄ − c+ (1− γ̄)(1− t2)]2
8(1− t2),
meaning that social welfare as an FDI host country is given by
W h1 =
3L1(1− ω)2
8+L2[γ̄ − c+ (1− γ̄)(1− t2)]2
8(1− t2).
Thus, we have W s1 ≥ W h
1 if and only if
L1
L2
[(1− γ̄)2
2+ (γ̄ − c)(1− γ̄)(1− t2)− 3(1− ω)2
8
]≥ L2[γ̄ − c+ (1− γ̄)(1− t2)]2
8(1− t2).
(24)
As γ̄ falls to c, the right-hand side of this inequality converges to L2(1 − c)2(1 − t2)/8.
However, we haveL2(1− c)2(1− t2)
8<
L2(1− c)2(γ̄ − c)8[2(γ̄ − c) + (1− γ̄)]
under t2 > t, while the right-hand side of this inequality converges to 0 as γ̄ falls to c.
This implies that W s1 > W h
1 for any t2 if γ̄ is small enough. In addition, the right-hand
side of (24) is increasing in t2 if and only if 1− t2 < (γ̄ − c)/(1− γ̄), which is true when
t2 > t since 1− t2 < (γ̄− c)/[2(γ̄− c) + (1− γ̄)] < (γ̄− c)/(1− γ̄). This implies that when
the threshold t̃2 falls in this range, t̃2 rises with L1/L2.
We also obtain a similar result in the final case where t2 ≥ t̄. In this case, we have
t1 = [γ̄ − c− (1− γ̄)(1− t2)]/(γ̄ − c) and q∗2 = L2(1− γ̄). Thus, we have
W h1 =
3L1(1− ω)2
8+ t1(γ̄ − c)L2(1− γ̄)
=3L1(1− ω)2
8+ L2(1− γ̄)[γ̄ − c− (1− γ̄)(1− t2)].
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We have W s1 ≥ W h
1 if and only if
L1
L2
[(1− γ̄)2
2+ (γ̄ − c)(1− γ̄)(1− t2)− 3(1− ω)2
8
]≥ (1− γ̄)[γ̄ − c− (1− γ̄)(1− t2)].
It is readily verified that this inequality is satisfied if γ̄ is sufficiently small, and that if t̃2
falls in this region it increases with L1/L2.
Before turning to Country 2’s best response, we show the relationship between γ̄ and
t̃2 more rigorously. We restrict γ̄ to the range (c, (1 + c)/2), that is, the transfer price
lies between the marginal cost and monopoly price. As we have shown, if γ̄ is sufficiently
close to c, we have W s1 > W h
1 under the assumption (14), meaning that t̃2 = 1. As γ̄
increases, W s1 decreases while W h
1 increases. At a certain level of γ̄, these two payoffs
become equal, and t̃2 starts to decrease as γ̄ further increases. Tentatively assuming that
t̃2 becomes smaller than t when γ̄ reaches its upper bound, the equation W s1 = W h
1 , which
is characterized by (23) with equality when t2 = t̃2 and γ̄ = (1 + c)/2, can be written as
L1
L2
[3(1− c)2
8− 3(1− ω)2
8− t̃2(1− c)2
4
]=t̃2(1− c)2
8.
If c < ω, the t̃2 that satisfies this equation is positive and thus the threshold t̃2 is positive
even at the limit. This is because FDI itself is beneficial for the source country, because
of the gains in production efficiency. We may rather focus on the more interesting case
where c ≥ ω, while satisfying the assumption (14). In that case, t̃2 reaches 0 before γ̄
reaches its upper bound: the assumption that t̃2 < t is thus satisfied in this case.
Country 2’s best response is similarly derived. Recall that Bs1 and Bh
1 do not de-
pend on L1/L2, as (18) and (19) show, and that L1/L2 only affects Country 1’s best
response function B1 through t̃2, as (20) shows. Country 2’s best response function is
therefore symmetric to Country 1’s. The only substantial difference is that the threshold
t̃1 decreases with L1/L2, whereas t̃2 increases.
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