HIAS-E-111

Border Carbon Adjustments with
Endogenous Assembly Locations
Haitao Cheng (a)(a) Hitotsubashi Institute for Advanced Study, Hitotsubashi University

This version: August, 2021

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Border Carbon Adjustments with Endogenous Assembly Locations?
Haitao Cheng
Hitotsubashi Institute for Advanced Study, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan

This version: August 25, 2021

Abstract
In this study, we develop a two-country model to examine whether border carbon adjustments(BCAs) are more effective than emission tax alone in preventing carbon leakage and decreasing
global emissions with endogenous assembly locations. Specifically, we explore three policy regimes:i) emission taxes alone (no BCAs), ii) emission taxes and carbon-content tariffs (partial BCAs),
and iii) emission taxes, carbon-content tariffs, and tax rebates on exports (full BCAs). We findthat the effectiveness of BCAs depends on whether BCAs induce assembly relocation. If assembly
relocation does not occur, BCAs prevent carbon leakage and decrease global emissions. However,
if BCAs induce assembly relocation, carbon leakage may occur with partial BCAs, and global
emissions may be higher with full BCAs.
Keywords: Abatement Investments; Border Carbon Adjustments; Carbon Leakage; Endogenous
Assembly Locations; International Oligopoly
JEL classification: F18; H23; Q54.

?
I wish to thank two anonymous referees, an anonymous associate editor, and the editor, Professor Angus Chu,
for their insightful and helpful suggestions. I am deeply indebted to Professor Jota Ishikawa for his invaluable advice
and constant encouragement. I also thank Professors Yoichi Sugita, Kensuke Teshima, and Eiichi Tomiura and theparticipants at the seminars at Hitotsubashi University for their helpful comments. This study is based on Chapter
5 of my doctoral thesis submitted to Hitotsubashi University. All remaining errors are my own. E-mail address:
haitaoecon@gmail.com

1. Introduction
A country’s ambition to deal with climate change may be thwarted by the existence of carbon
leakage. That is, stricter environmental regulations in a country decrease its greenhouse gas (GHG)
emissions yet increase other countries’ emissions (e.g., Aichele and Felbermayr, 2015). To cope withcarbon leakage, policymakers are inclined toward border carbon adjustments (BCAs) in addition
to domestic emission taxes. For instance, the American Clean Energy and Security Act of 2009
(also known as the Waxman-Markey Bill) proposed a cap-and-trade system requiring importers topurchase emission permits as domestic producers do. On July 14, 2021, the European Commission
adopted a proposal for the carbon border adjustment mechanism, which would put a carbon priceon the imports of some targeted products. Policymakers believe that emission tax, together with
BCAs, can internalize the environmental costs of production across countries and, therefore, canbe more effective in mitigating carbon leakage and controlling global emissions than emission tax
alone. Many studies have demonstrated the effectiveness of BCAs, including Veenendaal et al.
(2008), Elliott et al. (2010), Böhringer et al. (2012), and Fischer and Fox (2012). However, no
study has investigated the limitations of BCAs with vertical linkages.
Vertically related markets deserve more attention when examining the effectiveness of BCAs.
First, intermediate goods account for roughly 60 percent of international trade (Johnson, 2014).
Second, the production of intermediate goods can be dirtier than the production of final goods, suchas in the production of tires and bodyshells versus the assembly of automobiles. Third, a country
may find it easier to regulate direct emissions rather than indirect emissions because of the highadministration costs of data collection for emissions in each stage of production, especially whensome parts are completed in foreign countries (e.g., Lockwood and Whalley, 2010; Kortum andWeisbach, 2017). Based on the latter two points, producers may escape environmental regulations
through exports, FDI, or assembly relocation under BCAs.1 For instance, a final goods producer
can assemble the dirty inputs in a country with more lax environmental regulations and export theclean final goods to a country implementing BCAs, which may result in carbon leakage and moreglobal emissions. We examine this possibility in the present study.
We build a two-country model with several intermediate goods producers and a final goods
1

The concern about environmental tax evasions under BCAs is also mentioned in Monjon and Quirion (2010),
McAusland and Najjar (2015) and Cosbey et al. (2019). For instance, McAusland and Najjar (2015) find thatthe pipeline decision of Canada’s oil sand producers would be affected by the destination-based carbon tax. The
producers export polluting oil and crude oil to unregulated overseas markets, while the stakeholders would import conventionally extracted relatively clean oil. In addition, regarding the SB 775 proposal for the California
cap-and-trade program that seeks full BCAs, Meredith Fowlie argues that “... this asymmetric treatment of whatstays home and what gets sent outside of California creates an incentive to re-allocate more emission-intensive production to the export market to avoid the carbon price.” See https://energyathaas.wordpress.com/2017/05/22/
californias-carbon-border-wall/2

producer. The final goods producer is a monopolist in the final goods market that chooses its
assembly location endogenously, which is the key point of our study. The production of intermediate
goods emits GHG and is subject to oligopolistic competition because some polluting industries,
such as chemicals and cement, have a strong feature of oligopoly.2 Intermediate goods producers
have identical abatement technologies, and their production becomes clean after this abatement.
This assumption, together with the existence of the trade costs of final goods, is meaningful indetermining the timing of assembly relocation.
Countries differ in their awareness of environmental protection. Country 1, a developed country,
imposes emission tax on the domestic production of dirty goods. However, country 2, a developing
country, does not implement environmental regulations. To avoid carbon leakage and decrease
global emissions, country 1 is inclined to implement BCAs, similar to what the EU intends to do.
We specifically examine three environmental policy regimes to examine the effectiveness of BCAs:i) emission taxes alone (Regime α), ii) emission taxes and carbon-content tariffs on the imports ofdirty inputs (Regime β), and iii) emission taxes, carbon-content tariffs, and rebates on the exportsof dirty inputs (Regime γ).
With this model, we find that whether BCAs are effective in dealing with carbon leakage andglobal emissions depends on whether BCAs induce assembly relocation. If trade costs are high and(or) abatement costs are low, the assembly is located in the taxing country under BCAs. Carbon
leakage is eliminated and global emissions decrease because BCAs reduce the imports of dirty
inputs. In this case, BCAs are effective. However, if trade costs are low and (or) abatement costs
are high, then the final goods producer relocates its assembly to the non-taxing country, which canlead to carbon leakage in Regime β and more global emissions in Regime γ. In this case, BCAs
induce assembly relocation, which in turn makes them invalid. According to Article XX of GATT,
the general exceptions clause, countries may impose BCAs if they are “necessary to protect human,
animal or plant life or health.” Our findings indicate that Regime γ may be incompatible withthese related rules.
We mainly follow two strands of the literature in our study. The first strand examines theeffectiveness of BCAs in preventing carbon leakage and controlling emissions (Yomogida and Tarui,
2013; Jakob et al., 2013; Eichner and Pethig, 2015; Ishikawa and Okubo, 2017; Cheng and Ishikawa,2021). Yomogida and Tarui (2013) show that carbon leakage can occur under BCAs throughthe trade of dirty goods when the emissions per unit of dirty goods in the taxing country are
sufficiently higher than those in the non-taxing country. Jakob et al. (2013) adopt a two-sector,
two-country model to compare the effectiveness between consumption-based and production-based
2
International oligopoly model has been extensively employed to investigate environmental issues in international
trade. See Higashida and Jinji (2006), Riveiro (2008), Ishikawa and Okubo (2010), and Celik and Orbay (2011).

3

carbon pricing systems in mitigating carbon leakage. They show that carbon leakage can still
occur through terms-of-trade effects and may even be larger under consumption-based carbon
pricing (i.e., carbon pricing with full BCAs). Eichner and Pethig (2015) develop a two-period
two-country general equilibrium model with fossil fuel as the production input to examine how
carbon tax with full BCAs (also called a consumption-based carbon tax) affects each country’s
emissions in each period through changes in fossil fuel prices. They find that such tax in the first
period may increase both countries’ emissions as long as the income effect is sufficiently strong. In
contrast to these studies, we focus on how firm relocation leads to carbon leakage under BCAs.
Ishikawa and Okubo (2017) and Cheng and Ishikawa (2021) extend the footloose capital modeland the strategic trade policy model to examine the same channel as ours. Ishikawa and Okubo(2017) show that BCAs do not affect firm locations while decreasing each firm’s production in the
non-taxing country; therefore, no carbon leakage exists under BCAs. Cheng and Ishikawa (2021)
also find that firm relocation can lead to carbon leakage under BCAs; however, they do not consider
vertical linkages. In contrast, we demonstrate that carbon leakage may still exist under BCAs due
to assembly relocation.
The second strand concerns environmental issues in the presence of vertical linkages. Examples
include Hamilton and Requate (2004), Greaker (2006), Bushnell and Mansur (2011), and Wan
et al. (2018). Similar to our study, Hamilton and Requate (2004) and Wan et al. (2018) introduce
an upstream industry comprising polluting producers, while Greaker (2006) considers an upstream
market for environmental innovation. However, the three papers do not consider endogenous
assembly locations and BCAs, and their interests are not in policy effectiveness in dealing withcarbon leakage. Bushnell and Mansur (2011) discuss a BCA-equivalent policy in the upstream
market, as we do, and show that such a policy can prevent carbon leakage. Their findings are basedon the assumption that the downstream firm locations are fixed. By relaxing this assumption, we
can investigate how the final goods producer evades BCAs through assembly relocation, which canlead to carbon leakage and more global emissions.
The remainder of this paper is organized as follows. Section 2 describes the basic componentsof the proposed model. Section 3 examines and compares the three policy regimes. Section 4
discusses the welfare implications and assumptions of the model. Finally, Section 5 concludes thepaper.
2. Setup
The basic setup of our study is shown in Figure 1. A final goods producer, which is a monopolist,
produces homogeneous final goods by assembling homogeneous inputs from countries 1 and 2. The
4

Figure 1: Basic setup of the model

assembly location is endogenous. The numbers of intermediate goods producers in countries 1 and
2 are n1 and n2 , respectively, with n1 + n2 = N . Intermediate goods are freely traded acrosscountries; however, each unit of the final goods incurs τ units of trade costs.3 For simplicity, we
assume that only country 1 consumes final goods and that the inverse demand function is linear.
P = b − X/2,

(1)

where b is a parameter measuring the market size in country 1 and is assumed to be sufficiently
large; thus, trade is always positive after relocation throughout our analysis.4
The production of final goods is clean; however, one unit of intermediate goods emits one unitof GHG during production.5 In order to control GHG emissions, country 1 imposes environmental
regulations. We specifically explore three environmental policy regimes: Regime α, which includes
solely an emission tax on the domestic production of dirty goods; Regime β, which pairs a carboncontent tariff on imports of dirty inputs with the emission tax; and Regime γ, which includes anemission tax, a carbon-content tariff, and a tax rebate on the exports of its dirty goods. The ratesof the three instruments are assumed to be the same and are denoted by t.
3
Our main findings do not change much even if each unit of the intermediate goods also incurs τ units of trade
costs when transported across countries. We discuss the changes in footnotes 9 and 10.
4The denominator “2” in the demand function is imposed for simplicity and has no impact on our results.
5EU’s carbon border adjustment mechanism covers the power sector and energy-intensive industrial sectors in thefirst phase, such as aluminum, cement, iron, steel, and electricity, which have strong features of intermediate goods.
See https://ec.europa.eu/taxation_customs/green-taxation-0/carbon-border-adjustment-mechanism_en5For simplicity, the production costs of intermediate goods are simplified to zero. Intermediate
goods producers are identical and subject to Cournot competition. They can employ abatement forclean production, which generates no emissions.6 The unit abatement cost is denoted by c. Afterabatement, the producer’s marginal cost becomes c. If the producers are regulated by emission tax
higher than the abatement costs, then they will choose to employ abatement instead of paying thetax.
One unit of final goods requires only one unit of intermediate goods for assembly. Therefore,z1G=z + τ
2

if A = 1,(2)

if A = 2

where A denotes the assembly location, G is the unit cost of final goods, z1 and z2 are the prices
of intermediate goods given the assembly in countries 1 and 2, respectively.
The model has three stages of decision-making. In the first stage, considering country 1’s
environmental policies as given, the final goods producer decides on its assembly location. The
location decision on assembly is put in the first stage because it is a relatively long-term behavior
compared to the decisions on production. In the second stage, N intermediate goods producers
commit to the quantities of intermediate goods supplied to the final goods producer. In the third
stage, taking the price of the intermediate goods as a given, the final goods producer decides on
its monopolist price and the corresponding quantity of final goods.
Before moving on, we discuss the game-theoretic structure regarding the final goods producer.
It is a price taker in the input market but a price maker in the final goods market. The assumptionof this formulation can be justified if we treat the inputs as commonly used materials across a large
number of industries such as steel so that any monopsony power of the final goods producer in a
specific industry vanishes.7
As usual, we solve the game backward. In the last stage, the profit of the final goods producer
isπF = (P − G)X.

(3)

With the inverse demand function in equation (1), the price of the monopolist is derived as P ∗ =6

The assumption of no emissions after abatement can be understood from two perspectives. First, some emissions
remain after the abatement, but carbon emissions are captured, utilized, and stored (CCUS) with an investment
cost of c. Second, we consider an example of a switch from coal to renewable electricity. We thank the associate
editor for the helpful suggestions. In addition, our main results do not change, even if we assume lower yet positive
emissions per unit of input production after abatement, because assembly relocation can still occur before abatementin Regimes β and γ, leading to carbon leakage and more global emissions, respectively.
7
We would like to thank the associate editor for this very helpful suggestion. For more discussions about thisassumption, refer to Ishikawa and Spencer (1999) and Ino and Matsumura (2021).

6

(b + G)/2.8 Therefore, the equilibrium quantity of final goods is X ∗ = 2(b − P ∗ ) = b − G. Taking
the price and quantity into the equation (3), we obtain the monopolist’s profit as
πF = (b − G)2 /2.

(4)∗ (A = 1 or 2) can be
In the second stage, the input price at the Cournot-Nash equilibrium zA

obtained from the market-clearing condition, where the demand by the final goods producer isequal to the total supply of the inputs produced by the N producers. The details are discussed in
the following sections.
In the first stage, the final goods producer chooses its assembly location by comparing its profits
with the assembly in countries 1 and 2. This comparison can be transformed into an equivalent
comparison of the unit costs of the final goods.
1 if z ∗ ≤ z ∗ + τ
12A=.
2 if z ∗ > z ∗ + τ
1
2

(5)

Thus, the assembly is located in a country with a lower unit cost. If the final goods producer decides
to locate its assembly in country 1, it saves trade costs. However, the final goods producer may
be incentivized to locate its assembly in country 2 to evade environmental regulations, especially
under BCAs.
3. Three Policy Regimes
3.1. Regime α: Emission Tax Alone
This section introduces the benchmark case in which country 1 imposes only emission tax on thedomestic production of intermediate goods. We first analyze the behaviors of intermediate goods
producers given the assembly in countries 1 and 2, respectively, and then show how the final goods
producer chooses its assembly location endogenously.
Assuming that the assembly is located in country 1, the inverse demand for intermediate goods
by the final goods producer can be derived from the equilibrium quantity of final goods in the laststage:
X ∗ = b − G =⇒ z1α = G = b − X ∗ = b − n1Xxα11i +i=1

8

The asterisks denote the variables at the equilibria throughout the paper.

7

n2
X
j=1


xα21j  .

(6)

xα11i and xα21j are the production of intermediate goods producers i and j in countries 1 and 2,
respectively, with the first number in the subscripts referring to the location of the intermediate
Pn1 αgoods producers, and the second number is the location of the assembly.9 X ∗ =i=1 x11i +Pn2 α
j=1 x21j stems from the assumption that one unit of final goods requires one unit of intermediate
goods for assembly.
For each intermediate goods producer in countries 1 and 2, the profit isααπ11i= [z1α − min(t, c)]xα11i ; π21j
= z1α xα21j .

(7)

Producers in country 1 either choose to employ abatement or pay the emission tax directly, depending on the relative values of tax rates and abatement costs. Producers in country 2 face noenvironmental regulation; therefore, they neither pay the emission tax nor employ abatement.
The market outcome is obtained by a Nash equilibrium of the game in which the N intermediate
goods producers compete in quantity, given the inverse demand function in equation (6). Solving
the profit maximization problems, we obtain the equilibrium quantities and price of intermediate
goods as follows:
xα∗
11 =

b − (1 + n2 ) min(t, c)b + n1 min(t, c)
α∗
; xα∗
.
21 = z1 =
1+N
1+N

(8)

Emission tax always increases the equilibrium price when it is lower than the abatement costs
but does not affect the price when it becomes higher than the abatement cost because that iswhen producers in country 1 employ abatement. Correspondingly, when the emission tax is lower
than the abatement costs, as t increases, producers in country 1 produce less because of higher
production costs. Producers in country 2 then provide more dirty inputs because they are notregulated by the emission tax, which can make them more competitive.
Assuming that the assembly is located in country 2, the inverse demand for intermediate goods
is derived as:



n1
n2
X
X
z2α = b − τ − 
xα12i +
xα22j  .
i=1

(9)

j=1

xα12i and xα22j are the production of intermediate goods producers i and j in countries 1 and 2,
respectively.

9
Since the intermediate goods producers are identical, we suppress the letters i and j in the subscripts when
there is no ambiguity.

8

The profits of producers areααπ12i= [z2α − min(t, c)]xα12i ; π22j
= z2α xα22j .

(10)

Producers in country 1 are still regulated by the emission tax.
Analogously, we can derive the equilibrium quantities and price of intermediate goods as follows
xα∗
12 =

b − τ + n1 min(t, c)b − τ − (1 + n2 ) min(t, c)
α∗
; xα∗
.
22 = z2 =
1+N
1+N

(11)

As shown in equation (5), the final goods producer locates its assembly in the country with alower unit cost of intermediate goods. Note that
α∗
α∗
Gα∗
2 = z2 + τ > z1

(12)always holds. Therefore, we have the following lemma.
Lemma 1: The assembly is always located in country 1 under Regime α.
The rationale behind this finding is as follows: By locating its assembly in country 1, the final
goods producer can save trade costs. However, even if the assembly is located in country 2, theintermediate goods producers in country 1 still have to pay the emission tax.
We derive each country’s and global emissions based on the equilibrium quantities of intermediate goods:
n1 [b − (1 + n2 )t] n2 (b + n1 t) N b − n1 t

,,
if t ≤ c

1+N
1+N 
1+N
α∗
(E1α∗ , E2α∗ , EW)= .
n2 (b + n1 c) n2 (b + n1 c)
,
if t > c
 0,
1+N
1+N

(13)

The dashed lines in Figures 2 and 3 describe the relationship between the emission tax and GHG
emissions.10 When the emission tax is not high, an increase in the tax rate decreases country
1’s emissions while increasing country 2’s emissions, which verifies the existence of carbon leakage.
When the emission tax becomes sufficiently high, it has no impact on emissions because production
in country 1 becomes clean.

10
Figures 2, 3 and 4 are drawn with the variable values a = 3, N = 1, n1 = 0.3, n2 = 0.7, c = 0.5. Trade costs
are manipulated to derive different cases: τ = 0.25 in Figures 2 and 4, and τ = 0.4 in Figure 3.

9

3.2. Regime β: Emission Tax + Carbon Tariff
In Regime β, aside from emission tax on domestic production, country 1 imposes a carbon-content
tariff on imports of intermediate goods from country 2 so that all dirty goods consumed in country
1 are regulated at the same level. The final goods producer may have an incentive to escape thecarbon-content tariff by locating its assembly in country 2, which would not be possible under
Regime α.
Assuming that the final goods producer places its assembly in country 1, the intermediate
goods producers in both countries face the same environmental regulations. The inverse demandfor intermediate goods is:
z1β = b − n1Xxβ11i +

i=1

n2
X


xβ21j  .

(14)

j=1

The profit of an intermediate goods producer in countries 1 and 2 then becomes
ββ
π11i= [z1β − min(t, c)]xβ11i ; π21j
= [z1β − min(t, c)]xβ21j .

(15)

The intermediate goods producers in country 2 also choose to either employ abatement or pay thetariff directly in Regime β.
The equilibrium quantities and price of intermediate goods are given by:β∗xβ∗
11 = x21 =b − min(t, c)b + N min(t, c); z1β∗ =
.
1+N
1+N

(16)

Assuming that the final goods producer locates its assembly in country 2, the analysis becomesexactly the same as that in Regime α, because only the intermediate goods producers in country
1 are regulated by the emission tax.
When deciding on the location of assembly, the final goods producer faces a trade-off between
saving trade costs and avoiding the carbon-content tariff. It saves trade costs but has to pay the
carbon-content tariff on the import of inputs by locating the assembly in country 1. However, it
has to pay the trade costs instead of the carbon-content tariff if the assembly is located in country2.
If trade costs are high and/or abatement costs are low, that is, τ ≥ n2 c/N , then z2β + τ ≥ z1β
holds, and the assembly is located in country 1 because saving the high trade costs is dominant.

10Each country’s and global emissions at equilibrium are derived as follows:β∗
E1β∗ , E2β∗ , EWn (b − t) n2 (b − t) N (b − t)
 1
if t ≤ c
,
,
1+N
1+N
1+N
=
.

(0, 0, 0)
if t > c(17)

Carbon leakage is prevented because the dirty inputs from country 2 face a carbon-content tariff.
As a result, global emissions are lower than those in Regime α.
If trade costs are low and/or abatement costs are high, that is, τ < n2 c/N , then z2β + τ < z1βholds for t > tβ1 ≡ N τ /n2 . The final goods producer relocates its assembly to country 2 at t = tβ1
because avoiding the carbon-content tariff becomes dominant as t increases.
Lemma 2. If τ < n2 c/N , the assembly is relocated from country 1 to country 2 as t increases inRegime β.
In this case, each country’s and global emissions at equilibrium are given by:
n1 (b − t) n2 (b − t) N (b − t),,
if t ≤ tβ1



1+N
1+N
1+N
 n [b − τ − (1 + n )t] n (b − τ + n t) N (b − τ ) − n t 
 

1
2
2
1
1
β∗ 0
β∗ 0
β∗ 0
E1 , E2 , EW =,,if tβ1 < t ≤ c .

1
+
N
1
+
N
1
+
N





n2 (b − τ + n1 c) n2 (b − τ + n1 c)
,
if t > c
 0,
1+N
1+N
(18)
As shown in Figure 2, carbon leakage occurs for tβ1 < t ≤ c because the intermediate goods
producers in country 2 no longer pay the carbon tariff, which makes them more competitive.
Assembly relocation invalidates Regime β’s role in preventing carbon leakage. However, global
emissions are still lower than those in Regime α because the existence of trade costs decreases thedemand for final goods and, thus, the quantity of dirty inputs used for their production.11
We conclude our findings in the following proposition:12
Proposition 1. If τ ≥ n2 c/N , Regime β is more effective than Regime α in preventing carbon
leakage; however, if τ < n2 c/N , carbon leakage occurs because of assembly relocation induced byRegime β. Global emissions are always lower in Regime β than in Regime α.
11
If we allow trade costs of intermediate goods, global emissions in Regime β can be higher than that in Regime
α for t > c if trade costs are low and/or abatement costs are high.
12
Please see Appendix A for proofs of the propositions.

11Figure 2: Comparison of the emissions under Regime β (solid lines) and Regime α (dashed lines) for τ ≤ n2 c/N .

3.3. Regime γ: Emission Tax + Carbon Tariff + Tax Rebate
Compared with Regime β, country 1 then imposes a tax rebate on its exports of intermediate
goods to country 2 to eliminate their cost disadvantage in the foreign market. Given the assemblyin country 1, the analysis becomes exactly the same as that in Regime β. When the assembly is
in country 2, the inverse demand for intermediate goods becomes:z2γ = b − τ − n1Xxγ12i +

i=1

n2
X


xγ22j  .

(19)

j=1

The profits of intermediate goods producers in countries 1 and 2 areγγ
π12i= z2γ xγ12i ; π22j
= z2γ xγ22j .

(20)

The equilibrium quantities and price of intermediate goods are obtained as
γ∗
γ∗
xγ∗
12 = x22 = z2 =

12

b−τ
.
1+N

(21)

The unit cost of the final goods given A = 2 isz2γ∗ + τ =b + Nτ
.
1+N(22)

When deciding on the assembly location, the final goods producer faces a trade-off between
trade costs and higher input costs due to environmental regulations. If τ ≥ c, then the final goods
producer always locates its assembly in country 1. This is more likely to occur when abatement
costs are low and/or trade costs are high. Our rationale is that the final goods producer saves
trade costs greatly when they are high and benefits from lower production costs of inputs after
employing abatement; producers are willing to employ abatement when abatement costs are low.
In this case, each country’s and global emissions at equilibrium are
n1 (b − t) n2 (b − t) N (b − t),,
if t ≤ c
γ∗ 
1+N
1+N
1+N
.
E1γ∗ , E2γ∗ , EW=(0, 0, 0)
if t > c(23)

Carbon leakage is prevented and global emissions are lower in Regime γ than in Regime α.
If τ < c, then the assembly is relocated to country 2 before abatement. The threshold tax rateistγ2 = τ.

(24)

This is more likely to occur when trade costs are low and/or abatement costs are high. In Regime
γ, the final goods producer has a stronger incentive to locate the assembly in country 2 than
Regime β, that is, tγ2 < tβ1 , because the intermediate goods exported from country 1 are exempt
from environmental regulation due to a rebate.
Lemma 3. If τ < c, the final goods producer has a stronger incentive to relocate the assembly incountry 2 in Regime γ than in Regime β.
In this case, each country’s and global emissions at the equilibrium are


n
(b
−
t)
n
(b
−
t)
N
(b
−
t)

1
2
,
,
if t ≤ tγ2


 
1
+
N
1
+
N
1
+
N
γ∗ 0
γ∗ 0
γ∗ 0

E1 , E2 , EW = .
n1 (b − τ ) n2 (b − τ ) N (b − τ ),,
if t > tγ2

1+N
1+N
1+N

(25)

The solid lines in Figure 3 depict the effect of Regime γ on GHG emissions when trade costs arelower than abatement costs. As the assembly is relocated to country 2, each producer’s production13

in countries 1 and 2 does not change. Therefore, emissions do not change at the threshold tax
rate, implying no carbon leakage.13 In addition, we find that global emissions can be higher in
Regime γ than in Regime α. If the emission tax rate is higher than the abatement costs, Regime
γ relieves the production of intermediate goods in country 1 from environmental regulation by
inducing assembly to be located in country 2. Intermediate goods producers in country 1 thus
become more competitive and produce more without abatement. Therefore, Regime γ increases
country 1’s emissions while decreasing country 2’s emissions. Consequently, global emissions may
be higher if the former effect dominates. This case occurs when b > (N τ + n1 n2 c)/n1 .
Next, we compare Regime γ with Regime β. If τ > c, the assembly is always located in country
1 in both regimes. The tax rebate on exports of dirty goods to country 2 plays no role. The
emissions are the same in the two regimes for a given t. If n2 c/N < τ ≤ c, the assembly is always
located in country 1 in Regime β, but is relocated to country 2 at t = tγ2 in Regime γ. Eachcountry’s emissions are the same in the two regimes for t ≤ tγ2 ; however, they are higher in Regime
γ for t > tγ2 because of assembly relocation. If τ < n2 c/N , the assembly is relocated from country
1 to country 2 at t = tβ1 in Regime β and at t = tγ2 in Regime γ. As shown in Figure 4, country 1’s
emissions are the same in the two regimes for t ≤ τ and are higher in Regime γ for t > τ because
the tax rebate increases country 1’s exports of intermediate goods with assembly in country 2.
In contrast, country 2’s emissions are the same in the two regimes for t ≤ τ , higher in Regime γ
for τ < t ≤ tβ1 , and lower in Regime γ for t > tβ1 . Consequently, global emissions are the samein the two regimes for t ≤ τ ; however, they are always higher in Regime γ for t > τ because of
the existence of a tax rebate. Assembly relocation invalidates Regime γ’s role in decreasing global
emissions.
We conclude our findings in the following proposition:
Proposition 2. If τ ≥ c, Regime γ is more effective than Regime α in dealing with carbon leakage
and global emissions. If τ < c, global emissions can be higher in Regime γ than in Regimes α andβ.

13This finding is based on the assumption that intermediate goods can be freely transported across countries.
Assuming that there are trade costs for intermediate goods, carbon leakage can also occur in Regime γ.

14

Figure 3: Comparison of the emissions under Regime γ (solid lines) and Regime α (dashed lines) for τ ≤ c.

Figure 4: Comparison of the emissions under Regime γ (solid lines) and Regime β (dashed lines) for τ ≤ n2 c/N .

15

4. Discussions
4.1. Welfare Analysis
In this part, we briefly discuss how emission tax with and without BCAs affects each componentof the countries’ welfare.14 We assume that country 1’s welfare consists of consumer surplus, theprofits of the n1 intermediate goods producers and the final goods producer, tax revenue, andenvironmental damage from global warming. Country 2’s welfare consists of only the profits of then2 intermediate goods producers and environmental damage from global warming. The analysis
below implies that which policy regime is preferable strongly depends on how country 1 evaluates
its environmental damage from global warming.
In Regime α, before the abatement, an increase in the emission tax harms consumers, the n1
intermediate goods producers, and the final goods producer in country 1 while benefiting the n2
intermediate goods producers in country 2, and the global environment. The effect on tax revenue
in country 1 is ambiguous. Country 2’s welfare then increases. If the positive effect from fewer
emissions dominates the negative effects, country 1’s and global welfare improves as well. At the
point of abatement, the consumer surplus and profits of producers do not change. Tax revenue
decreases to zero and global emissions decrease. The abatement improves country 2’s welfare but
necessarily worsens country 1’s welfare if the benefits from fewer global emissions are smaller thanthe loss of tax revenue. Country 1 has an incentive to impose an emission tax lower than theabatement cost (t < c) if the environmental damage from global warming is low in country 1.
In Regime β, an increase in tax harms consumers, the n1 intermediate goods producers, andthe final goods producer before abatement regardless of whether assembly relocation occurs. In
contrast, a tax increase hurts the n2 intermediate goods producers in country 2 when the assemblyis located in country 1 but benefits them before their abatement when it is located in country 2.
The effect on tax revenue is ambiguous. As long as the assembly is located in country 1, the shift
from Regime α to Regime β for a given tax rate is harmful to the consumers in country 1 andthe n2 intermediate goods producers in country 2 but is beneficial to the n1 intermediate goods
producers and the tax revenue (before abatement) in country 1. As global emissions decrease, thetwo countries may benefit from the regime shift. However, if the assembly is in country 2 under
Regime β, the shift can be harmful to the n1 intermediate goods producers and the tax revenue; forexample, tβ1 < t ≤ c in Figure 2. As a result, the shift from Regime α to Regime β can worsen all
welfare components except environmental quality. Country 1 has an incentive to impose Regime
β when the improvement in environmental quality dominates all losses, although the regime shift
would cause firm relocation and carbon leakage.
14

Please see Appendix B for mathematical calculations on the welfare comparisons.

16

In Regime γ, an increase in the tax hurts the N intermediate goods producers and the final
goods producer but improves the environment before assembly relocation or abatement. However,it does not affect them after the relocation and abatement. The effect on tax revenue is ambiguous.
Analogously, we can investigate how the shift from Regime α to Regime γ affects welfare components. If the assembly is in country 1 under Regime γ, the effects are qualitatively the same as theshift from Regime α to Regime β. However, if the assembly is in country 2 under Regime γ, thenthe shift benefits the n1 intermediate goods producers in country 1 but harms the n2 intermediate
goods producers in country 2; how the shift affects country 1’s tax revenue, final goods producer’s
profits, and global emissions is ambiguous. If global emissions are higher in Regime γ than in
Regime α and country 1 has a strong awareness of environmental protection, then country 1 has
no incentive to impose Regime γ, although doing so benefits its polluting producers.
Regimes β and γ are the same when the assembly is located in country 1. If the assembly is
in country 1 under Regime β but in country 2 under Regime γ (e.g., tγ2 < t ≤ tβ1 in Figure 4),
the shift from Regime β to Regime γ benefits consumers and all producers, but harms country 1’s
tax revenue and the environment. If the assembly is in country 2 under both regimes, then theregime shift harms the n2 intermediate goods producers. The effects on the other components are
qualitatively the same as in the former case, except that country 1’s tax revenue is not affected fort > c because they are zero under both regimes. Whether country 1 has an incentive to shift fromRegime β to Regime γ depends on how much it suffers from more global emissions under Regimeγ.
4.2. Existence of Country 2’s Market
Although the assumption that only country 1 consumes the final goods is useful for us to focus on
the trade-off between trade costs and abatement investment, as shown in equation (5), one might
wonder how our results would change if country 2 also consumes the final goods. The essence of
our findings still holds. That is, the effectiveness of BCAs still depends on the assembly location,
as BCAs are more effective if the assembly is always located in country 1 and may not be effective
if it is located in country 2. Note that whether country 2 consumes the final goods affects thetiming of the assembly relocation. If the market size in country 2 is sufficiently small, then theanalysis is very similar to that of the main part. If it is sufficiently large, the assembly is likely toalways be located in country 2 to save trade costs in all three policy regimes. In the latter case,
Regime β is the same as Regime α; however, global emissions are higher in Regime γ than in theother two regimes.

174.3. Endogenous Locations of Input Producers
In the main part of this paper, the locations of the intermediate goods producers are fixed, with n1
producers in country 1 and n2 producers in country 2. In this section, we discuss their relocation.
That is, the number of firms in each country is determined endogenously. For simplicity, we
assume that there is no relocation cost. In Regime α, all intermediate goods producers are locatedin country 2 to avoid emission tax. The assembly is always located in country 1 to save trade
costs. Carbon leakage occurs because of the relocation of n1 producers. In Regime β, as longas the assembly is located in country 1, all intermediate goods producers are regulated by theenvironmental tax wherever they are. This happens when abatement costs are low and (or) trade
costs are high. The n1 producers have no incentive to relocate to country 2, thus eliminating carbon
leakage. However, when trade costs are sufficiently low, the n1 intermediate goods producers andassembly are relocated to country 2 at a specific tax rate to evade BCAs. In this case, carbon
leakage occurs and becomes even more serious than in the case of the fixed locations of input
producers. The analysis of Regime γ is similar to that of the main part. The intermediate goods
producers have no incentive to change their locations because they cannot avoid the environmental
tax if the assembly is in country 1, and are not regulated at all if the assembly is located in country
2. However, global emissions are lower in Regime γ than in Regime α because of the trade costsof the final goods.
4.4. Bertrand Competition in the Input Market
In this section, we discuss how the results would change if we consider Bertrand competition insteadof Cournot competition in the input market. In contrast to Cournot competition, the final goods
producer procures intermediate goods from the country at a lower cost. In Regime α, the final
goods producer always locates its assembly in country 1 and imports all the inputs from country
2 because the inputs in country 1 are regulated by the emission tax. In Regime β, if the assemblyis located in country 1, the intermediate goods producers are identical because they have the same
production and environmental costs. However, as the tax rate increases in country 1, the final
goods producer has an incentive to relocate its assembly to country 2 before the intermediate
goods producers invest in abatement, which is the same as in the case of Cournot competition.
When the assembly is in country 2, the final goods producer purchases all inputs from the n2
producers in country 2 because n1 producers in country 1 have to pay emission tax and therefore
have higher input costs. Assembly relocation thus causes carbon leakage, which becomes evenworse than in the case of Cournot competition. In Regime γ, the N intermediate goods producers
face the same environmental regulation, regardless of the assembly location. Assembly relocation
does not lead to carbon leakage, as explained in the case of Cournot competition. Compared with18

Regime α, country 1’s emissions are higher, and country 2’s and global emissions are lower in
Regime γ.
4.5. Regulations on Final Goods and Carbon Footprint Tax
We made two critical assumptions regarding the final goods in the main part. First, the productionof final goods is clean and, thus, unregulated by environmental taxes. In the following, we showhow our results change if we loosen this assumption. Suppose the final goods emit GHG during
their production and are taxed in the three regimes. In Regime α, the final goods producer
may be incentivized to locate its assembly in country 2 to avoid emission tax on the final goods.
Thus, Lemma 1 may not hold. However, in Regimes β and γ, the final goods producer’s decision
on assembly relocation is not incentivized by the tax avoidance of final goods. In other words,
assembly relocation frees input production in country 2 from environmental regulation in Regime
β and input production in both countries in Regime γ. Therefore, carbon leakage and more global
emissions can still occur, even if the final goods are dirty and regulated.
Second, we assumed that environmental policies are imposed only on direct emissions, which incentivizes the final goods producer to relocate assembly to country 2. If indirect emissions were also
regulated, the incentive would be eliminated, and the assembly would always be located in country
1. Therefore, a carbon footprint tax is more effective in preventing carbon leakage and decreasing
global emissions in our model.15 However, as shown in Appendix C, if we modify our model byassuming a market of final goods in country 2, a carbon footprint tax can backfire.16 Under acarbon footprint tax, the production of inputs for the final goods consumed in country 1 is taxed,
while that for the final goods consumed in country 2 is exempt from environmental regulations.
Compared with production under emission tax alone, country 2’s input production decreases, butcountry 1’s input production increases. It is likely that the latter strength dominates the former,
increasing global emissions. Contrastingly, we find that a carbon footprint tax eliminates carbon
leakage even if it causes assembly relocation in the presence of country 2’s market.
5. Concluding Remarks
We developed a simple model to examine and compare how BCAs affect production and GHG
emissions in the presence of intermediate goods and endogenous assembly locations. We found

15

Different from our result, Jakob et al. (2013) and Eichner and Pethig (2015) find that carbon leakage can still
occur under a carbon footprint tax as shown in Section 1. Please refer to McAusland and Najjar (2015) for morediscussions on carbon footprint taxes.
16
Please see Appendix C for a formal analysis about the comparison of an emission tax alone and a carbon
footprint tax in the presence of country 2’s market of final goods.

19that BCAs avoid carbon leakage and decrease global emissions by reducing the imports of dirty
inputs when the assembly is located in a taxing country. However, BCAs may induce assembly
relocation, which in turn invalidates BCAs. That is, carbon leakage can still occur because of
assembly relocation in a regime with emission tax and a carbon-content tariff. In addition, theintroduction of a tax rebate can lead to more global emissions than other regimes after assembly
relocation. Our findings imply that BCAs may not be more effective in dealing with carbon leakage
and controlling global emissions than emission tax alone when the assembly location is endogenous.
However, as long as the assembly is located in the taxing country, BCAs become more effective.
To ensure this, the taxing country can subsidize abatement activities and encourage technology
transfer across countries so that production costs decline even without assembly relocation.
We conclude by discussing an important issue for future research. Although we introduce
vertically related markets in our analysis, our model is still too naive to describe today’s trade andenvironmental problems. With the development of information and communication technologies,
production has entered into the era of global value chains (GVCs), where the final goods producers
segment their production into parts and source them worldwide. Consequently, emissions are
generated along GVCs in different countries. In particular, inputs are transported across countries
multiple times before they are assembled into final goods, during which emissions are also generated
from shipping. It would be worthwhile to investigate how BCAs affect the patterns of GVCs andthus the emissions from both production and transport.17

17

See Cheng et al. (2021) for an example of environmental taxes in GVCs but without BCAs.

20

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22

Appendix
A: Proofs of Propositions 1 and 2
Proof of Proposition 1:
In Regime α, carbon leakage occurs for t ≤ c. This can be verified from equation (13):
α∗
∂EW
∂E1α∗
n1 (1 + n2 )
∂E2α∗
n1 n2
n1
=−
< 0;
=
> 0;
=−
< 0.
∂t
1+N
∂t
1+N
∂t
1+N

An increase in the tax rate decreases country 1’s emissions but increases country 2’s emissions.
In Regime β, if τ ≥ n2 c/N , the assembly is always located in country 1. From equation (17),for t ≤ c, we have
β∗

∂EW
∂E1β∗
n1
∂E2β∗
n2
N
=−
< 0;
=−
< 0;
=−
< 0.
∂t
1+N
∂t
1+N
∂t
1+N
An increase in the tax rate decreases country 1’s, country 2’s and global emissions. Carbon leakage
does not occur in Regime β. Therefore, we conclude that if τ ≥ n2 c/N , Regime β is more effective
than Regime α in preventing carbon leakage. In this case, global emissions are lower in Regime βthan in Regime α becausen t 2 >0
if t ≤ cβ∗α∗.
EW− EW = 1n +(bN+ n c)1

 2
> 0 if t > c
1+N
In Regime β, if τ < n2 c/N , the final goods producer relocates its assembly to country 2 at
t = N τ /n2 ≡ tβ1 . Country 1’s emissions decrease but country 2’s emissions increase at t = tβ1 ,which can be verified from equation (18):
∆E1β∗

0

=−
β

t=t1

0
n1 (τ + n2 tβ1 )n2 [(1 + n1 )tβ1 − τ ]< 0; ∆E2β∗== n1 τ > 0.
1+N
1+N
t=tβ
1

In addition, an increase in t decreases country 1’s emissions but increases country 2’s emissions fortβ1 < t ≤ c because we have the following relationships from equation (18):
0

β∗ 0

0

∂EW
∂E1β∗
n1 (1 + n2 )
∂E2β∗
n1 n2
n1
=−
< 0;
=
> 0;
=−
< 0.
∂t
1+N
∂t
1+N
∂t
1+N
To conclude, if τ < n2 c/N , then carbon leakage occurs because of assembly relocation induced by23

Regime β. In this case, global emissions are lower in Regime β than in Regime α because
 n t
2

>0



1
+
N

Nτ
β∗ 0
α∗
EW
− EW
=
>0
1 + N n2 τ > 0
1+N

if t ≤ tβ1
if tβ1 < t ≤ c .
if t > cHence, we conclude that global emissions are always lower in Regime β than in Regime α.
Proof of Proposition 2:
In Regime γ, if τ ≥ c, the assembly is always located in country 1. From equation (23), fort ≤ c, we have
γ∗

∂EW
∂E1γ∗
n1
∂E2γ∗
n2
N
=−
< 0;
=−
< 0;
=−
< 0.
∂t
1+N
∂t
1+N
∂t
1+N
An increase in the tax rate decreases country 1’s, country 2’s and global emissions. Carbon leakage
does not occur in Regime γ. In this case, global emissions are lower in Regime γ than in Regime
α becausen t
 2 >0
if t ≤ cγ∗α∗
EW− EW = 1n +(bN+ n c)
.
1

 2
> 0 if t > c
1+N
Therefore, we conclude that if τ ≥ c, Regime γ is more effective than Regime α in dealing withcarbon leakage and global emissions.
In Regime γ, if τ < c, the final goods producer relocates its assembly to country 2 at t = τ .
For t > c, the difference of global emissions in Regime γ and Regime α is
0

γ∗
α∗
EW
− EW
=

n1 b − N τ − n1 n2 c.
1+N

Global emissions in Regime γ can be higher than those in Regime α if b > (N τ + n1 n2 c)/n1 .
For t > c, the difference of global emissions in Regime γ and Regime β is
0

γ∗
β∗
EW
− EW
=

n1 (b − τ − n2 c)
> 0.
1+N

Global emissions in Regime γ are higher than those in Regime β.
Based on the results above, we conclude that if τ < c, global emissions can be higher in Regime
γ than in Regimes α and β.
24

B: Welfare Analysis
In this part, we briefly discuss how emission tax with and without BCAs affects each componentof the countries’ welfare. We assume that country 1’s welfare consists of consumer surplus (CS),the profits of the n1 intermediate goods producers (n1 π1 ) and the final goods producer (πF ), tax
revenue (T R), and environmental damage from global warming (D1 (EW )). Country 2’s welfare
consists of only the profits of the n2 intermediate goods producers (n1 π1 ) and environmental
damage from global warming (D2 (EW )). D1 and D2 are increasing in EW . The analysis below
implies that which policy regime is preferable strongly depends on how country 1 evaluates its
environmental damage from global warming.
In Regime α, we derive each component of the countries’ welfare:
1
1
1
α∗ 2
CS = X α∗ 2 = (n1 xα∗
11 + n2 x21 ) =
4
4
4
α



N b − n1 min(t, c)
1+N

2
;


b − (1 + n2 ) min(t, c) 2
;
1+N


1 N b − n1 min(t, c) 212α;πF = (b − G) =
2
2
1+N

2
n1 π1α = n1 (xα∗11 ) = n1 n1 t[b − (1 + n2 )t] ,αα∗1+NT R = tE1 =0,2
n2 π2α = n2 (xα∗21 ) = n2

For t ≤ c,

if t ≤ c;
if t > cb + n1 min(t, c)
1+N

2
.

∂π α
∂CS α
∂n1 π1α∂n2 π2α< 0;< 0; F < 0;
> 0.
∂t
∂t
∂t
∂t

An increase in the emission tax harms consumers, the n1 intermediate goods producers, and thefinal goods producer in country 1 while benefiting the n2 intermediate goods producers in country 2,
and the global environment. Country 2’s welfare increases. ∂T Rα /∂t is positive for t ≤ b/2(1 + n2 )but negative for t > b/2(1 + n2 ). The effect of a tax increase on tax revenue in country 1 isambiguous. However, if the positive effect from fewer emissions dominates the negative effects,
country 1’s and global welfare improves as well. At the point of abatement, the consumer surplus
and profits of producers do not change. Tax revenue decreases to zero and global emissions decrease.
The abatement improves country 2’s welfare but necessarily worsens country 1’s welfare if the25benefits from fewer global emissions are smaller than the loss of tax revenue. Country 1 has anincentive to impose an emission tax lower than the abatement cost (t < c) if the environmental
damage from global warming is low in country 1.
In Regime β, if τ ≥ n2 c/N , the assembly is located in country 1, the welfare components areshown as follows:
2 1
1
1
2
β∗
CS = X β∗ =n1 xβ∗
+
n
x
=
2
11
21
4
4
4
β

n1 π1β = n1



xβ∗
11

2


= n1

1
1
πFβ = (b − G)2 =
2

2





T Rβ = t E1β∗ + E2β∗n2 π2β = n2



xβ∗
21

2





N b − N min(t, c)
1+N

b − min(t, c)
1+N

2
;

2
;

N b − N min(t, c)
1+N

2
;



 N t(b − t) , if t ≤ c
1+N
=
;

0,
if t > c= n2b − min(t, c)
1+N

2
.

Note that CS β < CS α , n1 π1β > n1 π1α , πFβ < πFα , T Rβ > T Rα for t ≤ c, n2 π1β < n1 π2α . The shift
from Regime α to Regime β for a given tax rate is harmful to the consumers in country 1 andthe n2 intermediate goods producers in country 2, but is beneficial to the n1 intermediate goods
producers and the tax revenue (before abatement) in country 1. As global emissions decrease, thetwo countries may benefit from the regime shift.
If τ < n2 c/N , the final goods producer relocates its assembly to country 2 at t = tβ1 . The
welfare components are 
1 Nb − Nt 2




4
1+N



 
0
1
N
(b − τ ) − n1 t 2
β
CS =

4
1+N


2


1
N
(b
−
τ
)
−
n
c
1



4
1+N

26

if t ≤ tβ1
if tβ1 < t ≤ c ;if t > c

 


b−t 2


n1



 1 + N
b − τ − (1 + n2 )t 2β0
n1 π1 = n1

1+N






b
−
τ
− (1 + n2 )c 2


n1
1+N
 


1 Nb − Nt 2




2
1+N



 
1 N (b − τ ) − n1 t 2
β0
πF =
2
1+N






1
N
(b
− τ ) − n1 c 2



2
1+N

if t ≤ tβ1
if tβ1 < t ≤ c ;if t > c

if t ≤ tβ1
if tβ1 < t ≤ c ;if t > cN t(b − t)




 1+N
β0
T R = n1 t[b − τ − (1 + n2 )t]

1+N



0
 


b−t 2


n2


1+N



 
0
b − τ + n1 t 2βn2 π 2 = n2
 1 + N 2b − τ + n1 c


n2
1+N

if t ≤ tβ1
if tβ1 < t ≤ c ;if t > c
if t ≤ tβ1
if tβ1 < t ≤ c .
if t > c

For t ≤ tβ1 , the assembly is in country 1 in both regimes. The shift from Regime α to Regime
β for a given tax rate is harmful to the consumers in country 1 and the n2 intermediate goods
producers in country 2, but is beneficial to the n1 intermediate goods producers and the tax revenue
in country 1.
For t > tβ1 , the assembly is in country 1 in Regime α but in Regime β in Regime β. The shift is0

harmful to the n1 intermediate goods producers because n1 π1β < n1 π1α . The shift is also harmful to0

the tax revenue in country 1 before abatement because T Rβ < T Rα for tβ1 < t ≤ c. As a result, theshift from Regime α to Regime β can worsen all welfare components except environmental quality.
Country 1 has an incentive to impose Regime β when the improvement in environmental quality
dominates all losses, although the regime shift would cause firm relocation and carbon leakage.
In Regime β, an increase in tax harms consumers, the n1 intermediate goods producers, and thefinal goods producer before abatement regardless of whether assembly relocation occurs, i.e., for000t ≤ c, ∂CS β /∂t < 0, ∂CS β /∂t < 0, ∂n1 π1β /∂t < 0, ∂n1 π1β /∂t < 0, ∂πFβ /∂t < 0, ∂πFβ /∂t < 0. In
27contrast, a tax increase hurts the n2 intermediate goods producers in country 2 when the assemblyis located in country 1 but benefits them before their abatement when it is located in country 2,00i.e., ∂n2 π2β /∂t < 0 for t ≤ c, ∂n2 π2β /∂t < 0 for t ≤ tβ1 , ∂n2 π2β /∂t > 0 for tβ1 < t ≤ c. The effect on
0

tax revenue is ambiguous because the signs of ∂T Rβ /∂t and ∂T Rβ /∂t depend on the parametervalues and the tax rates.
In Regime γ, if τ ≥ c, the assembly is located in country 1, the welfare components are thesame as those in Regime β:
1
1
1
γ∗ 2
n1 xγ∗=CS = X γ∗ 2 =11 + n2 x21444γ

n1 π1γ = n1

2
xγ∗
= n1
11

1
1
πFγ = (b − G)2 =
2

2







N b − N min(t, c)
1+N

b − min(t, c)
1+N

2
;

2
;

N b − N min(t, c)
1+N

2
;



 N t(b − t) , if t ≤ c
γ∗ 
γ∗
γ
1+N
T R = t E1 + E2 =;0,
if t > cn2 π2γ = n2 xγ∗
21

2


= n2

b − min(t, c)
1+N

2
.

If τ < c, the final goods producer relocates its assembly to country 2 at t = tγ2 . The welfare
components are 
1 Nb − Nt 2

4 1+N 
CS γ 0 =
1 N (b − τ ) 2


4
1+N
 

b−t 2


n1
0
 1 + N 2
n1 π1γ =

b−τ

n1
1+N
 

1 Nb − Nt 2

2 1+N 
0
πFγ =

1
N (b − τ ) 2


2
1+N
28

if t ≤ tγ2;
if t > tγ2

if t ≤ tγ2;
if t > tγ2

if t ≤ tγ2;
if t > tγ2 N t(b − t)0
1+N
T Rγ =

0
 

b−t 2


n2
0
 1 + N 2n2 π2γ =

b−τ

n2
1+N

if t ≤ tγ2;

if t > tγ2
if t ≤ tγ2.
if t > tγ2

In Regime γ, an increase in the tax hurts the N intermediate goods producers and the final
goods producer but improves the environment before assembly relocation or abatement, i.e., for00t ≤ c, ∂n1 π1γ /∂t < 0, ∂n2 π2γ /∂t < 0, ∂πFγ /∂t < 0; for t ≤ tγ2 , ∂n1 π1γ /∂t < 0, ∂n2 π2γ /∂t < 0,0∂πFγ /∂t < 0. However, it does not affect them after the relocation and abatement. The effect on
tax revenue is ambiguous because the signs of ∂T Rγ /∂t and ∂T Rγ /∂t depend on the parametervalues and tax rates. In the following, we can investigate how the shift from Regime α to Regime
γ affects welfare components. If the assembly is in country 1 under Regime γ, the effects are
qualitatively the same as the shift from Regime α to Regime β. However, if the assembly is incountry 2 under Regime γ, then the shift benefits the n1 intermediate goods producers in country00

1 but harms the n2 intermediate goods producers in country 2, i.e., n1 π1γ > n1 π1α , n2 π2γ > n2 π2α .
How the shift affects country 1’s tax revenue, final goods producer’s profits, and global emissions
is ambiguous. If global emissions are higher in Regime γ than in Regime α and country 1 has astrong awareness of environmental protection, then country 1 has no incentive to impose Regimeγ, although doing so benefits its polluting producers.
Regimes β and γ are the same when the assembly is located in country 1. If the assembly is
in country 1 under Regime β but in country 2 under Regime γ (e.g., tγ2 < t ≤ tβ1 in Figure 4),
the shift from Regime β to Regime γ benefits consumers and all producers, but harms country000

1’s tax revenue and the environment, i.e., for tγ2 < t ≤ tβ1 , we have CS γ 0 > CS β , πFγ > πFβ ,
0

0

0

0

0

n1 π1γ > n1 π1β , n2 π2γ > n2 π2β , T Rγ 0 < T Rβ . If the assembly is in country 2 under both regimes,
00then the regime shift harms the n2 intermediate goods producers (n2 π2γ < n2 π2β ). The effects on
the other components are qualitatively the same as in the former case, except that country 1’s tax
revenue is not affected for t > c because they are zero under both regimes, i.e., for t > tβ1 , we have
0

0

0

0

CS γ 0 > CS β , n1 π1γ > n1 π1β , T Rγ 0 = T Rβ = 0.

29

C: Comparison of Emission Tax and Carbon-footprint Tax
In this section, we examine and compare two policy regimes of emission tax alone (Regime ζ) andcarbon footprint tax (Regime ψ) in the presence of country 2’s market. The game of decisionmaking is the same as in the main part. For simplicity, we assume away the abatement choices of
the intermediate goods producers.18 The inverse demand functions in the two countries are
P1 = b1 − X1 /2; P2 = b2 − X2 /2.

Regime ζ: emission tax alone
In Regime ζ, country 1 imposes only an emission tax on the domestic production of intermediate
goods. We first analyze the behaviors of intermediate and final goods producers in the second andthird stages given the assembly in countries 1 and 2, respectively, and then show how the final
goods producer chooses its assembly location endogenously.
Assuming that the assembly is located in country 1, in the third stage, the final goods producer’s
profit isζ= (P1ζ − Gζ1 )X1ζ + (P2ζ − Gζ1 − τ )X2ζ .
πF,1

Gζ1 is the unit cost of final goods consumed in country 1. The unit cost of final goods consumed incountry 2 is therefore Gζ1 + τ .
Solving the profit maximization problem of the final goods producer, we can derive the equilibrium quantity of final goods consumed in country 1 and country 2:
X1ζ∗ = b1 − Gζ1 ; X2ζ∗ = b2 − Gζ1 − τ.
In the second stage, the inverse demand for intermediate goods by the final goods producer canbe derived from the the equilibrium quantity of final goods in the last stage:

z1ζ =



1
b1 + b2 − τ − 
2

n1
X
i=1

xζ11i +n2Xxζ21j  .

j=1

b1 and b2 are assumed to be sufficiently large so that the consumption of final goods is positive ineach market, i.e., min(b1 , b2 − τ ) > z1ζ .

18
The main finding in this section would not change even if we consider abatement. That is, global emissions canbe higher under carbon footprint tax than under emission tax alone.

30

For each intermediate goods producer in countries 1 and 2, the profit isζζ
π11i= (z1ζ − t)xζ11i ; π21j
= z1ζ xζ21j .

The market outcome is obtained by a Nash equilibrium of the game in which the N intermediate goods producers compete in quantity given the inverse demand function. Solving the profit
maximization problems, we obtain the equilibrium quantities and price of intermediate goods as
xζ∗
11 =

b1 + b2 − τ + 2n1 t
b1 + b2 − τ + 2n1 t
b1 + b2 − τ − 2(1 + n2 )t; xζ∗; z1ζ∗ =
.
21 =
1+N
1+N
2(1 + N )Assuming that the assembly is located in country 2, in the third stage, the final goods producer’s
profit isζπF,2= (P̂1ζ − Gζ2 − τ )X̂1ζ + (P̂2ζ − Gζ2 )X̂2ζ .

Gζ2 is the unit cost of final goods consumed in country 2. The unit cost of final goods consumed incountry 1 is therefore Gζ2 + τ .
Solving the profit maximization problem of the final goods producer, we can derive the equilibrium quantity of final goods consumed in country 1 and country 2:
X̂1ζ∗ = b1 − Gζ2 − τ ; X̂2ζ∗ = b2 − Gζ2 .
In the second stage, the inverse demand for intermediate goods by the final goods producer canbe derived from the the equilibrium quantity of final goods in the last stage:

z2ζ =



1
b1 + b2 − τ − 
2

n1
X
i=1

xζ12i +n2Xxζ22j  .

j=1

b1 and b2 are assumed to be sufficiently large so that the consumption of final goods is positive ineach market, i.e., min(b1 − τ, b2 ) > z2ζ .
For each intermediate goods producer in countries 1 and 2, the profit is


ζ
ζ
π12i
= z2ζ − t xζ12i ; π21j
= z2ζ xζ22j .
Solving the profit maximization problems, we obtain the equilibrium quantities and price ofintermediate goods as
xζ∗
12 =

b1 + b2 − τ + 2n1 t
b1 + b2 − τ + 2n1 t
b1 + b2 − τ − 2(1 + n2 )t; xζ∗; z2ζ∗ =
.
22 =
1+N
1+N
2(1 + N )31

The equilibrium quantities and price of intermediate goods are the same as those when the assemblyis in country 1.
In the first stage, the final goods producer decides on its assembly location by comparing the
profits with the assembly in country 1 and country 2.
If the assembly is in country 1, its profit is
ζ∗
πF,1
=

2
2 1 
1
b1 − z1ζ∗ +b2 − z1ζ∗ − τ .
2
2

If the assembly is in country 2, its profit is
ζ∗
πF,2
=

2
2 1 
1
b1 − z2ζ∗ − τ +b2 − z2ζ∗ .
2
2

The profit difference is
ζ∗
ζ∗
πF,1
− πF,2= (b1 − b2 )τ.

Therefore, the final goods producer will locate its assembly in the country with a larger market
size to save on trade costs of final goods.
Regime ψ: carbon footprint tax
In Regime ψ, country 1 imposes an emission tax on the dirty inputs used for the final goods
consumed in country 1. In our setting, if the assembly is located in country 1, the intermediate
goods producers in country 1 need to pay emission tax for their production and the intermediate
goods producers in country 2 must pay a carbon-content tariff for their exports to country 1. The
final goods producer sells the final goods in both countries. For the final goods consumed in country
1, there are no emission taxes on them because the assembly does not generate emissions. For the
final goods consumed in country 2, country 1 refunds the emission taxes paid for the emissions
embedded in them. If the assembly is located in country 2, the intermediate goods producers incountry 1 face a tax refund when they export the inputs to country 2. The intermediate goods
producers in country 2 do not need to pay the emission tax. The final goods producer must pay a
carbon-content tariff for the emissions embedded in exports to country 1. Under a carbon footprinttax, all the dirty inputs used for the final goods consumed in country 1 are regulated by a carbon
tax t, but the dirty inputs used for the final goods consumed in country 2 are exempt. In thefollowing, we first analyze the behaviors of intermediate and final goods producers in the second
and third stages given the assembly in countries 1 and 2, respectively, and then show how the final
goods producer chooses its assembly location endogenously.
Assuming that the assembly is located in country 1, in the third stage, the final goods producer’s
32

profit is
ψ
ψ
ψ
ψ
ψ
πF,1
= (P1ψ − Gψ1 )X1 + (P2 − G1 − τ + t)X2 .

Gψ
1 is the unit cost of final goods consumed in country 1. The unit cost of final goods consumed
in country 2 is therefore Gψ
1 + τ − t.
Solving the profit maximization problem of the final goods producer, we can derive the equilibrium quantity of final goods consumed in country 1 and country 2:ψ∗ψX1ψ∗ = b1 − Gψ
1 ; X2 = b2 − G1 − τ + t.

In the second stage, the inverse demand for intermediate goods by the final goods producer canbe derived from the the equilibrium quantity of final goods in the last stage:



n1
n2
X
X
1
 .
z1ψ = b1 + b2 − τ + t − 
xψ
xψ
11i +
21j
2
i=1

j=1

For each intermediate goods producer in countries 1 and 2, the profit is




ψ
ψ
ψ
−
t
xψ
;
π
=
z
π11i
= z1ψ − t xψ
1
21j .
11i
21j
Solving the profit maximization problems, we obtain the equilibrium quantities and price ofintermediate goods asψ∗xψ∗
11 = x21 =b1 + b2 − τ − t
b1 + b2 − τ + (1 + 2N )t; z1ψ∗ =.
1+N
2(1 + N )Assuming that the assembly is located in country 2, in the third stage, the final goods producer’s
profit is
ψ
ψ
ψ
ψ
ψ
πF,2
= (P̂1ψ − Gψ
2 − τ − t)X̂1 + (P̂2 − G2 )X̂2 .

Gψ
2 is the unit cost of final goods consumed in country 2. The unit cost of final goods consumed
in country 1 is therefore Gψ
2 + τ + t.
Solving the profit maximization problem of the final goods producer, we can derive the equilibrium quantity of final goods consumed in country 1 and country 2:ψ∗ψX̂1ψ∗ = b1 − Gψ
2 − τ − t; X̂2 = b2 − G2 .

In the second stage, the inverse demand for intermediate goods by the final goods producer can

33be derived from the the equilibrium quantity of final goods in the last stage:



n1
n2
X
X
1
 .
z2ψ = b1 + b2 − τ − t − 
xψ
xψ
12i +
22j
2
i=1

j=1

b1 and b2 are assumed to be sufficiently large so that the consumption of final goods is positive ineach market, i.e., min(b1 − τ − t, b2 ) > z2ψ .
For each intermediate goods producer in countries 1 and 2, the profit is
ψ
ψ
ψ ψ
π12i
= z2ψ xψ12i ; π21j = z2 x22j .

Solving the profit maximization problems, we obtain the equilibrium quantities and price ofintermediate goods asψ∗xψ∗
12 = x22 =b1 + b2 − τ − t
b1 + b2 − τ − t; z2ψ∗ =.
1+N
2(1 + N )The equilibrium quantities of intermediate goods are the same as those when the assembly is incountry 1.
In the first stage, the final goods producer decides on its assembly location by comparing the
profits with the assembly in country 1 and country 2.
If the assembly is in country 1, its profit is
ψ∗
=
πF,1

2 1 
21
b1 − z1ψ∗ +b2 − z1ψ∗ − τ + t .
2
2

If the assembly is in country 2, its profit is
ψ∗
πF,2
=

2 1 
21
b1 − z2ψ∗ − τ − t +b2 − z2ψ∗ .
2
2

The profit difference is
ψ∗
ψ∗
πF,1
− πF,2= (b1 − t − b2 )τ.

The carbon footprint tax plays a role in shrinking the market in country 1 because only the
inputs used for the production of final goods consumed in country 1 are regulated by the carbon
tax. If b1 is sufficiently large so that b1 − t − b2 > 0 always holds, then the assembly is locatedin country 1. If b2 is sufficiently large so that b1 − t − b2 < 0 always holds, then the assemblyis located in country 2. Assembly relocation does not occur in the above two cases. However, if
b1 > b2 but the difference is small, then the assembly is located in country 1 for t ≤ b1 − b2 but in34

country 2 for t > b1 − b2 . Although assembly relocation occurs, there is no carbon leakage in thiscase because each intermediate goods producer’s production does not change.
Comparing the global emissions in the two policy regimes, we have
ψ∗
ζ∗
EW
− EW
=

(2n1 − N )t.
1+N

If n1 > N/2, global emissions are higher under carbon footprint tax than under emission tax alone.
The intuition is as follows. Under carbon footprint tax, the production of inputs for the final goods
consumed in country 1 is taxed, while that for the final goods consumed in country 2 is exempt fromenvironmental regulations. Compared with production under emission tax alone, country 2’s inputζ∗ψ∗ζ∗
production decreases (n2 xψ∗21 < n2 x21 ), but country 1’s input production increases (n1 x11 > n1 x11 ).

If n1 > N/2, the latter strength dominates the former, increasing global emissions.
To conclude, in the presence of country 2’s market, carbon footprint tax eliminates carbon
leakage even if it causes assembly relocation; however, global emissions can be higher under carbon
footprint tax than under emission tax alone.

35