Firm Locations, Consumption Pollution and TaxI

Haitao Cheng

Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan

Abstract

In this paper, I extend the footloose capital model to investigate how trade liberalization

and consumption taxes affect the firm locations and GHG emissions from consumption. As

trade costs decline, global GHG emissions at first decrease and then rise. An increase in

the consumption tax in one country reduces its emissions and raises the other country’s

emissions; as a consequence, global emissions always decrease. When trade costs are high,

identical taxes are socially desirable. When trade costs are low, full agglomeration in the

larger country maximizes the global welfare. In the extension part, I show that a unilaterally

prohibitive tariff on dirty goods never reduces the tariff country’s GHG emissions and may

increase global emissions.

Keywords: Trade liberalization; Consumption pollution; Consumption tax; Footloose

capital model

JEL classification: F12; F18; Q56; Q58.

1. Introduction

Consumption is an important source of GHG emissions. According to the US Environ-

mental Protection Agency, direct GHG emissions from residents and businesses (excluding

agricultural and industrial activities) accounted for approximately 11.6 percent of total US

II would like to thank Professors Jota Ishikawa, Yoichi Sugita and Tomiura Eiichi for helpful sugges-tions. Thanks also go to seminar participants at Hitotsubashi University. Financial support from HasegawaInternational Scholarship Foundation is gratefully acknowledged. All remaining errors are my own. E-mailaddresses: ed185013@g.hit-u.ac.jp.

Preprint submitted to JSIE September 21, 2019

GHG emissions in 2017. However, the previous research strongly focuses on the emissions

from the production side, leaving the consumption side exploited so slightly. In this paper,

I intend to fill up the gap to investigate the relationship between trade and GHG emissions

from consumption.1 Specifically, I will show how trade liberalization and environmental

regulations affect the firm locations and GHG emissions and how the social planner designs

its optimal taxes in the presence of the agglomeration effect.

Channels through which trade liberalization affects GHG emissions are different with

perspective to the production and consumption sides. Production emissions occur where

the dirty goods are produced, while consumption emissions appear where they are con-

sumed. Location matters a lot when probing the impacts of the two sides. Suppose firms

fully agglomerate in one country, production emissions are all from that country. However,

consumption emissions also occur in other countries as long as the residents consume dirty

goods. In the former case, the countries without production emissions do not want to im-

pose environmental policy. In the latter case, all the countries should take responsibilities

because they are all polluters. Several environmental instruments are prevailing in the pre-

vious papers such as emission taxes, emission quotas, and emission standards. However,

they are not feasible enough to study consumption emissions since the governments may

feel challenging to measure them. Instead, I pick up ad valorem consumption tax on con-

sumers as the instrument. It is easy to implement and tractable to study how environmental

regulation affects emissions.2

In the main part of this paper, I decompose the total effect of trade liberalization and

consumption tax on GHG emissions into the firm-relocation effect and demand effect. The

firm-relocation effect describes the changes in GHG emissions due to the relocation of firms.

It differs from the composition effect in the previous research in the sense that the composi-

tion effect measures the reallocation of production factors across relatively clean and dirty

1To concentrate on the consumption side, Markusen (2017) shows an alternative way to go by introduc-ing non-homothetic preferences and involving no pollution-from-consumption-versus-production issue in hismodel.

2An example of consumption tax functioned as environmental regulation is the tax on the consumptionof fossil fuel. Governments like Norway and the UK often impose fuel tax as an ecotax to control CO2

emissions from vehicles.

2

goods. The net firm-relocation effect is always neutral in the model. The demand effect

captures the changes in GHG emissions due to the changes in the consumption of dirty

goods. It plays an essential rule in the following analyses. Given consumption tax, as trade

costs decline, the demand effect tends to decrease global emissions at first and then increase

them. Given trade costs, as consumption tax increases in either country, global emissions

always decline since the demand effect is always negative. In the welfare analysis, I focus on

the global optimum instead of national optimum to see what countries should do coopera-

tively but not what countries will do individually. The analysis shows how the supranational

regimes (e.g., EU) should design environmental tax to control GHG emissions.3 The finding

is striking: when trade costs are high, identical taxes are socially desirable although the

countries own different market sizes; when trade costs are low, the social planner has an

incentive to induce all firms to locate in the larger country.

The literature on trade and pollution traces back to the 1970s (e.g., Markusen, 1975;

Pethig, 1976) and develops prosperously around the establishment of WTO (e.g., Grossman

and Krueger, 1991; Markusen et al., 1993; Copeland and Taylor, 1994; Markusen et al.,

1995; Copeland and Taylor, 1995b; Ulph, 1996). Since then, both theoretical and empirical

research keeps growing fast. This paper follows the strand of previous literature on trade

and pollution in new economic geography.4 The traditional method employed in the issue is

Heckscher-Ohlin model where the emission of pollution is treated as an input of production,

and it aims to emphasize the role of comparative advantage in the traditional trade theory.

This paper uses the footloose capital model to concentrate on the preference towards varieties

of goods, monopolistic market, location patterns of firms and environmental issues, following

Zeng and Zhao (2009), Ishikawa and Okubo (2011), Ishikawa and Okubo (2016) and Ishikawa

3In EU’s case, the environmental tax rate can be viewed as the cap of GHG emissions in the emissionstrading system.

4More generally, there exist many studies investigating the relationship between trade, pollution andenvironmental policy when the firm or plant locations are endogenous (e.g., Markusen et al., 1993; Mottaand Thisse, 1994; Markusen et al., 1995; Rauscher, 1995; Hoel, 1997; Dijkstra et al., 2011; Dong et al.,2012). However, they only have a few firms in their models and concentrate on endogenous market structuredepending on the fixed costs of export and FDI. This paper is feasible to study a large number of firmsin the economy and focuses on firm relocation due to different capital rents across countries. For a surveyabout FDI and environmental regulations, please refer to Erdogan (2014).

3

and Okubo (2017). The basic setup in this paper is very close to Ishikawa and Okubo (2011).5

However, they only analyze the unilaterally environmental product standards and do not

look further into the welfare analysis. Instead, I investigate bilateral consumption taxes

in both countries and derive the social optimum, which is the main contribution to the

previous literature. The examination of how consumption taxes affect the firm behaviors

and GHG emissions is quite similar to the analysis of full border tax adjustment in Ishikawa

and Okubo (2017) in the sense that environmental regulation appears in the consumption

country. However, their focus is on the comparison of unilateral emission quotas and emission

taxes in the presence of GHG emission from production. In the extension part, I also probe

a unilaterally prohibitive tariff on the imports of dirty goods and find that such a policy is

never good to the local environment.

Although few, there still exist some papers about trade and consumption pollution. Be-

sides Ishikawa and Okubo (2011), Krutilla (1991) derives a series of second-best consumption

taxes with the existence of the production and consumption pollution respectively and con-

cludes that environmental production and consumption taxes affect the world price and

trade balance in the opposite way. Hu and McKitrick (2016) compare the production and

consumption pollution with a model similar to Antweiler et al. (2001) and find that trade

liberalization affects the two kinds of pollution differently through the trade-induced com-

position effects. Ishikawa and Okubo (2010) examine the effects of environmental and trade

policies on the consumption pollution in an international duopoly model. They also compare

the efficiency of different policies and demonstrate that tariff could mitigate consumption

pollution more effectively than emission taxes. Copeland and Taylor (1995a) employ a tra-

ditional Heckscher-Ohlin model and verify that dirty industry migration hypothesis is still

valid in the presence of local consumption pollution. Compared to their findings, I demon-

strate that whether firms migrate to the countries with laxer environmental regulations also

depends on the market sizes.

5The remained three papers are all about production pollution. Note that Zeng and Zhao (2009) inves-tigate the effect of production pollution on agricultural productivity, while Ishikawa and Okubo (2016) andIshikawa and Okubo (2017) investigate the effect of production pollution on residents.

4

The paper is organized as follows. Section 2 introduces the basic setups of the model.

Section 3 shows two benchmark cases of autarky and trade in goods only. Section 4 derives

the equilibrium in trade with footloose capital and investigates how trade liberalization

and consumption taxes affect the firm locations and GHG emissions. Section 5 studies the

social optimum and discusses the consumption tax competition. Section 6 considers the

unilaterally prohibitive tariffs on dirty goods as an extension. Section 7 concludes.

2. Basic Model

The basic model extends the footloose capital model in Martin and Rogers (1995) by

involving the GHG emissions from consumption and consumption taxes. Consider a world

with two countries called Home and Foreign and two factors named capital and labor.

Individuals in each country consume two kinds of goods: dirty manufacturing good (M)

with different varieties and clean agricultural good (A). Each firm produces a variety of

good M with one unit of capital as the fixed cost. Besides, one unit of labor is required to

produce one unit of good M . Dirty goods are traded with symmetric trade costs τ in the form

of “iceberg” such that τ units of dirty goods are traded for one unit is consumed eventually.

Good A is the numeraire. It is homogeneous and freely tradeable. The production of

good A is subject to constant returns to scale, and each unit of it is produced by one

unit of labor. The wage rate is normalized to 1. Denote labor and capital stock in Home

as L and K, those in Foreign as L∗ and K∗. The two production factors are distributed

proportionately across countries, and capital is distributed uniformly across residents within

each country. Denote s as the share of world’s labor and capital belonging to Home, i.e.,

L/(L + L∗) = K/(K + K∗) = s. Without loss of generality, the total levels of capital

and labor are both normalized to unity and assume that Home owns a larger market size

(s > 0.5). Labor is immobile across countries but can freely move across sectors and firms.

Capital is mobile across countries when there is no restriction on it. Capital rents are

payed to the local owners. GHG emissions are generated during the consumption of dirty

goods with one unit of the dirty goods emitting one unit of GHG emissions. Governments

impose ad valorem consumption taxes on consumers to control GHG emissions under the

5

requirements of environmental agreements such as the Paris Agreement.

The utility of a representative individual in Home is described as a quasi-linear function:

U = µ lnM + A−D(EW ) (1)

where

M ≡

(n∑i=1

xHHi1− 1

σ +n∗∑j=1

xFHj1− 1

σ

) 1

1− 1σ

(2)

is the CES composite consumption of dirty manufacturing varieties and A is the consump-

tion of clean good. D(·) is the damage function of GHG emissions. EW = E+E∗ represents

the global level of GHG emissions with E and E∗ referred to the local levels in each country.

n is the number (or the ratio) of differentiated varieties in Home, n∗ is that in Foreign. xHHi

denotes the quantity of Home consumption of variety i produced in Home, xFHj the coun-

terpart produced in Foreign. Note that the first capital letter in the superscript represents

the location of production and the second represents the location of consumption. σ > 1

is the constant elasticity of substitution between two varieties as usually seen in the CES

function. µ is the intensity of preference towards good M. The larger the value of µ, the

more the individual spends on dirty goods.

The budget constraint of the representative individual in Home is

n∑i=1

pi(1 + t)xHHi +n∗∑j=1

p∗jτ(1 + t)xFHj + A =I

L. (3)

I = L + rK + T denotes the total income where r is Home capital rents and T =

L(t∑n

i=1 pixHHi + τt

∑n∗

j=1 p∗jx

FHj

)is the total tax revenue with t representing Home ad

valorem consumption tax. The term xFHj but not xHFi enters into the total tax revenue

because the consumption tax is imposed on consumers but not producers. pi and p∗j are

the pre-trade prices of dirty goods in each country, τp∗j is the import price of Foreign dirty

goods in Home adjusted by trade costs. Similarly, the utility function and corresponding

variables (with asterisk) for Foreign can be derived. For simplicity, I assume µ and σ are

the same across countries. The aggregate price index facing the individual in Home is

6

P =

(n∑i=1

[pi(1 + t)]1−σ +n∗∑j=1

[τp∗j(1 + t)]1−σ

) 11−σ

. (4)

Different from tax on producers, a country’s consumption tax cannot affect the aggregate

price in the other country directly. However, the indirect channel still exists through the

effect of consumption tax on firm locations.

The markup pricing rule of Dixit-Stiglitz monopolistic competition model in Dixit and

Stiglitz (1977) still holds here, the identical pre-tax prices across varieties are derived as

p = pi = p∗j =1

1− 1σ

=σ

σ − 1.

3. Two Benchmark Cases

Globalization is characterized not only by the decrease in trade costs but also by the

mobility of capital. With footloose capital, capital owners can build plants in any nation to

purchase benefits. The consumption structure of residents changes due to firm relocation,

which, in turn, changes the location of GHG emissions. I pick up three special cases to

investigate how trade and trade liberalization affects GHG emissions. This section deals

with two cases: autarky and trade in goods without capital mobility. The next section

examines the case with both trade in goods and footloose capital.

3.1. Equilibrium in Autarky

In autarky, the number of firms equals the level of capital in each country implying that

nA = s and n∗A = 1 − s. The consumption of each variety is xHHA = µ(σ−1)sσ(1+t)

and xFFA =

µ(σ−1)(1−s)σ(1+t∗) . GHG emissions are EA = snxH = µs(σ−1)

σ(1+t), E∗A = (1− s)n∗xFF = µ(1−s)(σ−1)

σ(1+t∗)and

EWA = EA +E∗A = µ(σ−1)

σ

(s

1+t+ 1−s

1+t∗

). In this case, consumption tax affects local and global

GHG emissions through the changes in demand for domestic dirty goods. Suppose the two

countries impose consumption taxes at the same level, then GHG emissions from Home are

larger than those from Foreign due to the larger market size in Home.

7

3.2. Trade Equilibrium without Capital Mobility

With the development of transport tools, trade in goods is feasible across long distance;

but the mobility of capital is still difficult because of high fixed costs, e.g., building plants

and adopting to the local legislation. In this case, the number of firms still equal the level

of domestic capital, while the consumption involves the imported goods. The consumption

of each variety becomes

xHH =µ(σ − 1)

σ(1 + t)

τσ

sτσ + (1− s)τ; xFH =

µ(σ − 1)

σ(1 + t)

1

sτσ + (1− s)τ;

xHF =µ(σ − 1)

σ(1 + t∗)

1

sτ + (1− s)τσ; xFF =

µ(σ − 1)

σ(1 + t∗)

τσ

sτ + (1− s)τσ.

The levels of GHG emissions are

E = s[sxHH + (1− s)xFH ] =µs(σ − 1)

σ(1 + t)

sτσ + (1− s)sτσ + (1− s)τ

;

E∗

= (1− s)[sxHF + (1− s)xFF ] =µ(1− s)(σ − 1)

σ(1 + t∗)

s+ (1− s)τσ

sτ + (1− s)τσ;

EW

=µ(σ − 1)

σ

{s

1 + t

sτσ + (1− s)sτσ + (1− s)τ

+1− s1 + t∗

s+ (1− s)τσ

sτ + (1− s)τσ

}.

Trade in goods enlarges the number of varieties consumed by the residents while shrinking

the consumption of each variety, leading to lower local and global GHG emissions. Allowing

trade in goods is beneficial to the environment. As trade costs decline, the consumption of

each domestic variety decreases, and that of each imported variety increases. Both Home

and Foreign GHG emissions decline at first and then rise.

4. Trade Equilibrium with footloose Capital

Further development of information and communication techniques enhances the mo-

bility of capital, firms nowadays are able to relocate to other countries with higher capital

rents. In this section, I study how trade liberalization affects pollution in the presence of

footloose capital. The consumption of each Home- and Foreign-produced variety in Home

8

and Foreign is derived as

xHH =µ(σ − 1)

σ

τσ

(1 + t)(nτσ + n∗τ); xFH =

µ(σ − 1)

σ

1

(1 + t)(nτσ + n∗τ);

xHF =µ(σ − 1)

σ

1

(1 + t∗)(nτ + n∗τσ); xFF =

µ(σ − 1)

σ

τσ

(1 + t∗)(nτ + n∗τσ).

Given trade costs and consumption tax, as more firms locate in Home, competition becomes

harsher in the local market, Home aggregate price goes down, which leads to more total

Home demand for dirty goods. However, Home consumption of each Home and Foreign

variety decreases. Note that the share of expenditure on domestic dirty goods for each

individual in Home is

Share ≡ npxHH

PM=

n

n+ (1− n)τ 1−σ

which is increasing in the mass of firms in Home. The elasticity of the share with respect to

the number of firms in Home is

dShare/Share

dn/n=

1

1 + n(τσ−1)< 1.

The share of expenditure on domestic dirty goods increases less proportionately than the

increase in the mass of firms in Home. Therefore, the consumption of each domestic variety

decreases. Analogously, the share of expenditure on Foreign dirty goods and the correspond-

ing elasticity are derived as

Share∗ ≡ n∗τp∗xFH

PM=

(1− n)τ 1−σ

n+ (1− n)τ 1−σ;

dShare∗/Share∗

dn/n= − n

1− n1

τ 1−σ + n(1− τ 1−σ)< −1.

The share of expenditure on Foreign dirty goods decreases more proportionately than the

increase in the number of firms in Home. Hence, Home demand for each imported variety

declines. In the following, I will examine how location patterns of firms affect the consump-

tion and the resulting GHG emissions when they are endogenously determined by trade

9

costs and consumption tax.

Based on the market clearing condition, the total demand for each variety in Home and

Foreign equals the total production of each firm. So the total production of each variety is

Y = sxHH + (1− s)τxHF

=µ(σ − 1)

σ

[sτσ

(1 + t)(nτσ + n∗τ)+

(1− s)τ(1 + t∗)(nτ + n∗τσ)

];

(5)

Y ∗ = sτxFH + (1− s)xFF

=µ(σ − 1)

σ

[sτ

(1 + t)(nτσ + n∗τ)+

(1− s)τσ

(1 + t∗)(nτ + n∗τσ)

].

(6)

Since the capital is footloose across countries, no arbitrage exists. Therefore, the capital

rents should be the same (r = r∗), which, in turn, equalizes the total production of each

variety (Y = Y ∗).6 Along with the condition where n + n∗ = K + K∗ = 1, the number of

firms (or varieties) in Home and Foreign are derived as

n =1

τσ−1 − 1

s1−sτ

σ−1 − 1+t1+t∗

s1−s + 1+t

1+t∗

; n∗ =1

τσ−1 − 1

1+t1+t∗

τσ−1 − s1−s

s1−s + 1+t

1+t∗

. (7)

Suppose the two countries impose the same consumption taxes, then the equations above

collapse into

n = s+2s− 1

τσ−1 − 1; n∗ = 1− s− 2s− 1

τσ−1 − 1

displaying the home market effect in Helpman and Krugman (1985): firms disproportionately

locate in the country with larger market size and further trade liberalization enhances the

tendency until full agglomeration occurs in the larger market. If s ≥ 11+τ1−σ

, firms fully

agglomerate in Home; if 12< s < 1

1+τ1−σ, firms disperse in the two countries.

Describe the relationship between the number of firms in Home and relative consumption

taxes in Fig. 1. Intuitively, firms all locate in the low-tax country if the other country’s tax

is too high. When the tax difference is not so large among the two countries, an increase in

6Remind that r and r∗ are capital rents in Home and Foreign respectively and they are supposed to besufficiently large to make sure that the consumption of the numeraire to be positive at the equilibrium.

10

one country’s tax induces firms to run away.

Fig. 1. Number of Home Firms and Relative Consumption Taxes.

For 1+t1+t∗≤ s

1−sτ1−σ, all firms agglomerate in Home, Foreign specializes in the production

of the clean good and imports the dirty goods from Home. In this case, n = 1, n∗ = 0;

xHH = µ(σ−1)σ

11+t

, xHF = µ(σ−1)σ

1τ(1+t∗)

, xFH = xFF = 0; E = µ(σ−1)σ

s1+t

, E∗ = µ(σ−1)σ

1−sτ(1+t∗)

and EW = µ(σ−1)σ

[s

1+t+ 1−s

τ(1+t∗)

].

For 1+t1+t∗≥ s

1−sτσ−1, all firms agglomerate in Foreign, Home specializes in the production

of the clean good and imports the dirty goods from Foreign. In this case, n = 0, n∗ = 1;

xFF = µ(σ−1)σ

11+t∗

, xFH = µ(σ−1)σ

1τ(1+t)

, xHH = xHF = 0; E∗ = µ(σ−1)σ

1−s1+t∗

, E = µ(σ−1)σ

sτ(1+t)

and EW = µ(σ−1)σ

[1−s1+t∗

+ sτ(1+t)

].

Agglomeration tends to occur in the country whose market size is relatively larger and

consumption tax is relatively lower. As trade costs decline, the threshold for full agglom-

eration in Home becomes higher and that for full agglomeration in Foreign becomes lower.

Trade liberalization makes full agglomeration easier to occur. If there are no trade costs,

the downward-sloping curve in Fig. 1 disappears, firm locations are determined only by the

market sizes and consumption taxes.

For s1−sτ

1−σ < 1+t1+t∗

< s1−sτ

σ−1, firms disperse across the two countries. Before investi-

gating the effects of trade liberalization and consumption taxes on GHG emissions, I first

11

examine how firms respond to such changes —– the firm-relocation effect.

Since∂n

∂t= − 1

(1 + t∗)(τσ−1 − 1)

s1−s(τ

σ−1 + 1)(s

1−s + 1+t1+t∗

)2 < 0, (8)

∂n∗

∂t= −∂n

∂t> 0, (9)

the model indicates that given the consumption tax in Foreign, firms relocate from Home

to Foreign as the consumption tax in Home increases. However, whether firms relocate to

Foreign when Home consumption tax is higher under trade liberalization depends on the

strengths between market sizes and consumption taxes:

∂n

∂τ=

(σ − 1)τσ−2

(τσ−1 − 1)2(

s1−s + 1+t

1+t∗

) ( 1 + t

1 + t∗− s

1− s

). (10)

Suppose Home imposes a relatively higher consumption tax. Then the equation is likely to

be negative when Home also owns a relatively larger market size. As long as t > t∗ and

s1−s >

1+t1+t∗

hold simultaneously, firms relocate to the country with a higher consumption

tax as trade costs decline because home market effect is dominant.7 On the other hand, if

s1−s = 1+t

1+t∗stands, firms equally disperse across countries and trade liberalization has no

impact on the location of firms because r = r∗ = 1σ−1Y is independent of τ in this case.

The specific levels of each type of consumption are

xHH =µ(σ − 1)

σ

τσ

τσ + τ

(1

1 + t+

1− ss

1

1 + t∗

), (11)

xFH =µ(σ − 1)

σ

1

τσ + τ

(1

1 + t+

1− ss

1

1 + t∗

), (12)

xHF =µ(σ − 1)

σ

1

τσ + τ

(1

1 + t∗+

s

1− s1

1 + t

), (13)

xFF =µ(σ − 1)

σ

τσ

τσ + τ

(1

1 + t∗+

s

1− s1

1 + t

). (14)

7See Forslid et al. (2017) for more details about the relationship between asymmetric emission taxes (onproduction) and home market effect.

12

The consumption of domestic dirty goods is always larger than that of imported ones due

to trade costs. Besides, an increase in the consumption tax of either country always reduces

the consumption of dirty goods because the after-tax price indices P and P ∗ rise. Trade

liberalization raises the consumption of each imported variety and decreases that of each

domestic variety since the prices of imported varieties become relatively cheaper as trade

costs decline.

Local GHG emissions are the sum of emissions from domestic and imported consumption

shown as follows

E = nsxHH + n∗sxFH =µ(σ − 1)

σ(τ 2σ−1 − τ)

[(τ 2σ−1 − 1)

s

1 + t− (τσ − τσ−1) 1− s

1 + t∗

]; (15)

E∗ = n(1− s)xHF + n∗(1− s)xFF

=µ(σ − 1)

σ(τ 2σ−1 − τ)

[(τ 2σ−1 − 1)

1− s1 + t∗

− (τσ − τσ−1) s

1 + t

];

(16)

global GHG emissions are the sum of emissions from the two countries

EW = E + E∗ =µ(σ − 1)

σ

τσ + 1

τσ + τ

(s

1 + t+

1− s1 + t∗

). (17)

When firms disperse across countries under trade in dirty goods, allowing footloose cap-

ital raises the mass of firms and decreases the demand for each variety of dirty goods in

Home, and vice versa in Foreign. The total effect of allowing footloose capital on global

GHG emissions is ambiguous due to the existence of consumption taxes in each country.

However, when the countries impose the same consumption taxes, allowing footloose capital

necessarily reduces global GHG emissions.8

In the following, I show how trade liberalization affects global GHG emissions. Equations

(15), (16) and (17) tell that GHG emissions depend strongly on trade costs, and the model

provides a special phenomenon where trade liberalization itself may increase global GHG

emissions.9

8See Appendix for proof.9Ishikawa and Okubo (2017) find a neutral effect of trade on global production pollution in the absence

13

Proposition 1. Trade is always good for the global environment. The effect of trade

liberalization on global GHG emissions is not monotonic. As trade costs decline, global

GHG emissions decrease at first and then rise.

Proof: See Appendix.

Whether capital is immobile or footloose, trade always reduces GHG emissions due to

the preference towards varieties, no income effect, and the existence of trade costs. Trade

enlarges the choices of dirty goods, and individuals consume more varieties with fixed total

expenditure. The total amount of their consumption decreases because of trade costs. Thus,

global GHG emissions decrease.

The impact of trade liberalization on GHG emissions can be decomposed into the firm-

relocation effect and demand effect.10

∂EW

∂τ=∂E

∂τ+∂E∗

∂τ

=∂n

∂τsxHH +

∂n∗

∂τsxFH +

∂n

∂τ(1− s)xHF +

∂n∗

∂τ(1− s)xFF︸ ︷︷ ︸

net firm−relocation effect=0

+

∂sxHH

∂τn+

∂(1− s)xFF

∂τn∗︸ ︷︷ ︸

demand effect of domestic dirty goods

+∂sxFH

∂τn∗ +

∂(1− s)xHF

∂τn︸ ︷︷ ︸

demand effect of imported dirty goods

=µ(σ − 1)

σ

(s

1 + t+

1− s1 + t∗

)(σ − 1)τσ − στσ−1 − 1

(τσ + τ)2.

(18)

The relocation effect measures the impact of firm relocation, and the demand effect cap-

of any environmental intervene. This is because the total demand for each variety in the original footloosecapital model is irrelevant to trade costs (τ). For more details, refer to the equations (7c), (7d) and (8) inMartin and Rogers (1995). However, in the presence of environmental policies, Ishikawa and Okubo (2017)also provide a non-monotonic effect of trade liberalization on global GHG emissions.

10The global GHG emissions can be written in a minimalist way as EW = NX where N is the mass

of firms and X is the level of consumption of each variety. Taking logs and differentiating yields dEW

EW =dNN + dX

X . The changes in global GHG emissions are decomposed into the changes in firm locations (thefirm-relocation effect) and the changes in consumption levels (the demand effect). To examine the effect

of trade liberalization, the decomposition changes into ∂EW

∂τ = ∂N∂τ X + ∂X

∂τ N . Analogously, in the case of

consumption tax below, the decomposition becomes ∂EW

∂t = ∂N∂t X+ ∂X

∂t N . (More accurately, ∂EW

∂τ and ∂EW

∂t

should be written as ∂EW

∂τ |dt=dt∗=0 and ∂EW

∂t |dτ=0.)

14

tures the impact of changes in consumption. The firm-relocation effect in each country can

be positive, negative or neutral depending on the relationship between consumption taxes

and market sizes. However, the net firm-relocation effect of trade liberalization is always

neutral because the consumption scales of dirty goods are the same in the two countries.

A change in the GHG emissions in one country caused by the firm relocation is accurately

offset by the change of emissions in the other country. Thus, how trade liberalization affects

global GHG emissions is determined by the demand effect of dirty goods. As trade costs de-

cline, the consumption of each imported variety increases and that of each domestic variety

decreases. At the beginning of trade liberalization, trade costs are still high. The increase of

GHG emissions from more consumption of imported goods is dominated by the decrease of

them from less consumption of domestic goods; the total emissions tend to decrease. How-

ever, as trade liberalization proceeds, GHG emissions from imported goods increase more

than the reduction of GHG emissions from domestic goods; the total emissions begin to

increase. Specifically, the level of GHG emissions in free trade is equal to that in autarky

because there are no trade costs in both cases.

I now turn to examine how consumption tax affects GHG emissions. In this part, con-

sumption tax is still exogenous. The endogenous case follows in section 5.

Proposition 2. An increase in the consumption tax in one country reduces the emissions

in that country and raises the emissions in the other country. Global GHG emissions always

decrease.

Note that∂E

∂t=∂n

∂tsxHH +

∂n∗

∂tsxFH︸ ︷︷ ︸

−

+∂sxHH

∂tn+

∂sxFH

∂tn∗︸ ︷︷ ︸

−

< 0; (19)

∂E∗

∂t=∂n

∂t(1− s)xHF +

∂n∗

∂t(1− s)xFF︸ ︷︷ ︸

+

+∂(1− s)xHF

∂tn+

∂(1− s)xFF

∂tn∗︸ ︷︷ ︸

−

> 0. (20)

The impact of consumption tax on GHG emissions can also be decomposed into the firm-

relocation effect and demand effect. The demand effect is always negative since an increase

in consumption tax decreases the demand for both the domestic and imported dirty goods.

15

The firm-relocation effect is negative in one country and positive in the other country due to

the relatively larger scale of consumption of domestic goods compared to imported goods.

The net firm-relocation effect of consumption tax is also neutral because the consumption

scales of dirty goods are the same in the two countries. Thus, the total effect of consumption

tax on global GHG emissions is always negative because of the decreasing demand for dirty

goods. Besides, the total effect of consumption tax on GHG emissions is negative for Home

and positive for Foreign, which verifies the existence of “carbon leakage” under consumption

pollution. The model provides a case where unilateral consumption tax always reduces the

global GHG emissions, which is different from the previous literature showing that unilateral

environmental regulation may increase GHG emissions through carbon leakage.11

5. Endogenous Environmental Tax

5.1. Social Optimum

This section studies how the social planner maximizes social welfare by deciding on the

consumption taxes in each country. Since the firm-relocation effect on global GHG emissions

is neutral in the model, the social planner imposes consumption taxes to control the demand

for dirty goods. For simplicity, I assume a linear damage function of emissions and consider

a case where the firms disperse across countries when there is no consumption tax, i.e.,

D(EW ) = EE and τ 1−σ < s1−s < τσ−1.12 Define the social welfare as the sum of individual

welfare around the world:

WG = sW + (1− s)W ∗. (21)

Solving the social optimum gives

t = t∗ =σ − 1

σ

τσ + 1

τσ + τ− 1

σ. (22)

11See Ishikawa and Okubo (2017).12The environmental damage function is traditionally assumed to be an increasing and convex function of

pollution levels so that the marginal willingness to pay for pollution reduction is increasing and the interiorsolution of environmental tax exists. However, welfare analysis with firm relocation becomes complex due tothe existence of discontinuities. Therefore, a linear environmental damage function is adopted for simplicityand tractability. See, for example, Markusen et al. (1993) and Markusen et al. (1995).

16

If σ is sufficiently small, the social planner imposes no consumption taxes in the countries

(t = t∗ = 0). If σ is sufficiently large, consumption taxes are positive and identical.13 For

simplicity, I only consider the case of sufficiently large σ. Then, the global welfare and global

emissions are derived as

W = µ lnµ+µ

σ − 1ln(1 + τ 1−σ) + 1− µ ln

[1 +

τσ + 1

τσ + τ

]+

µs

σ − 1ln s

+µ(1− s)σ − 1

ln(1− s)− µ, (23)

EW =µ(τσ + 1)

2τσ + τ + 1. (24)

As long as firms disperse, identical consumption taxes are socially optimal, and firms do

not change their locations. However, as shown in the traditional NEG literature without

disutility from pollution, social welfare becomes higher as more firms locate in the larger

country. Thus, the social planner in this paper may also have an incentive to induce full

agglomeration in Home by manipulating the consumption taxes.14 If the social planner does

so, the consumption taxes in each country are

tH =s(σ − 1)

σ(τ 1−σ + τ−σ + 2)− 1; t∗H =

(1− s)(σ − 1)

σ(2τσ−1 + 1 + τ−1)− 1; (25)

the social welfare is

WH =µ lnµ− µ(1− s) ln τ + 1− µ ln

[1 +

τσ + 1

τσ + τ

]− µs ln

[s(1 + τ 1−σ)

]− µ(1− s) ln

[(1− s)(1 + τσ−1)

]− µ;

(26)

and the global emissions are

EWH =

µ(τσ + 1)

2τσ + τ + 1. (27)

13See Appendix for details of the derivation of global optimum and a more general case where the damagefunction is convex.

14The social planner never induces full agglomeration in the smaller country. See details in Appendix.

17

The comparison between the social welfare levels in the two cases is shown by

W −WH =µσ

σ − 1ln ∆ (28)

where

∆ =(1 + τ 1−σ)

1σ s

sσ (1− s) 1−s

σ

τ(1−s)(1−σ)

σ ss(1−σ)σ (1 + τ 1−σ)

s(1−σ)σ (1− s)

(1−s)(1−σ)σ (1 + τσ−1)

(1−s)(1−σ)σ

= ss(1− s)(1−s)(1 + τ 1−σ)s(σ−1)+1

σ (τ−1 + τσ−2)(1−s)(σ−1)

σ .

(29)

In the first line of the equation above, the numerator measures the total utility from dirty

goods in W and the denominator measures that in WH . Total incomes are not revealed in

∆ because they equal 1 + µ(τσ+1)2τσ+τ+1

in both cases. If ∆ ≥ 1, then W ≥ WH ; if ∆ < 1, then

W < WH .

Proposition 3. For ∆ ≥ 1, identical consumption taxes are socially optimal regardless of

the market sizes. For ∆ < 1, full agglomeration in the larger country is socially desirable.

Since global GHG emissions are the same in the two cases and the firm-relocation effect

is neutral, the demand effects are also the same. Whether the social planner has an incentive

to induce full agglomeration (or to manipulate the location of firms) depends only on the

utility of consumption given the level of dirty goods. In other words, the question is how the

social planner maximizes the residents’ utilities by distributing the firms (and thus the dirty

goods) in the presence of trade costs and different sizes of markets. Note that trade costs

and the mass of firms affect the utility through the changes in the aggregate price index. If

∆ ≥ 1, then W ≥ WH holds, which is likely to occur when the trade costs are high. The

social planner has no incentive to induce full agglomeration because the total utilities are

too low in the no agglomeration country due to the high price of the imported dirty goods

there. If ∆ < 1, then W < WH holds, which is likely to occur when trade costs are low and

the market sizes are very close to each other. The social planner manipulates consumption

taxes to raise full agglomeration in the larger country because the increase of utilities in the

18

agglomeration country is higher than the decrease in the other country.

5.2. Discussion: Consumption Tax Competition

Due to the messy forms of the national welfare functions in new economic geography

models, environmental tax competition is not often seen in the previous literature.15 This

section discusses how to derive the Nash Equilibrium in a two-stage game when the differ-

ence of market sizes is modest, i.e., τ 1−σ < s1−s < τσ−1. At the first stage, the two countries

simultaneously decide on the levels of consumption taxes. At the second stage, firms deter-

mine their location and production patterns as a response to the environmental regulation.

In the main part of this paper, I have already investigated the firm behaviors under con-

sumption taxes. Here, I concentrate on the first stage of the game. In the traditional tax

competition without pollution, nations usually “race to the bottom” or “race to the top”

depending on whether there exists agglomeration effect. In this paper, new consequences

may occur in the presence of GHG emissions. On the one hand, agglomeration means more

consumption; on the other hand, more consumption results in more emissions. Governments

may “race to the bottom” if the utility from consumption is determinant, and “race to the

top” if the damage from emissions is sufficiently huge.

15An exception is Forslid et al. (2017). However, they concentrate on production pollution. Environmentaltax competition is different under different kinds of pollution. For instance, when a country becomes aperiphery, its domestic environmental regulation does not affect the production pollution any more sincethere is no firm and production there. However, it still affects the consumption pollution as long as theresidents buy the import goods.

19

Fig. 2. Firm Location Patterns and Consumption Taxes.

Fig. 2 describes the relationship between firm locations and consumption taxes based

on Fig. 1. The coordinate system consists of three areas; in area (i), full agglomeration

occurs in Home, in area (iii), full agglomeration occurs in Foreign, and in area (ii), firms

disperse across countries. Denote the welfare function in each area as W(i), W(ii) and W(iii)

respectively. Given 1 + t∗, Home chooses among areas (iii) and (ii) if 1 ≤ 1 + t∗ < 1−ssτ and

chooses among the three areas if 1 + t∗ ≥ 1−ssτ . Analogously, I can derive the best responses

of Foreign given Home consumption taxes. Fig. 3 provides a simple example where σ = 2.

The red lines are the best responses of Home and the blue ones of Foreign. Countries race

to the bottom, and t = t∗ = 0 is the only Nash equilibrium.16

16See Appendix for the derivation of Nash equilibrium.

20

Fig. 3. Nash Equilibrium when σ = 2.

6. Extensions

6.1. Unilaterally Prohibitive Tariff on Dirty Goods

In this section, I investigate how firm locations and GHG emissions change if a country

imposes a prohibitive tariff unilaterally. Even as a member of WTO, a country may prohibit

the import of dirty goods from the other countries according to GATT Article XX, the

general exceptions clause, e.g., “... necessary to protect human, animal or plant life or

health.”17 However, I show that such a policy is never beneficial to the tariff country

concerning its GHG emissions due to trade costs. In the following, assume that Home

prohibits the import of dirty goods from Foreign unilaterally and capital is still footloose

across countries. The variables, in this case, are denoted with tildes and 0 < s < 1.18

First, examine the effect of this prohibitive tariff on the location patterns of firms. For

s1−s ≥ τσ−1 − 1, all the firms agglomerate in Home. This case occurs only when Home

17See Fischer and Fox (2012) for details about the compatibility of different kinds of environmental policieswith laws and rules in WTO.

18Since environmental regulation in this section is unilateral, no restriction is imposed on s in case thatmarket size matters.

21

market is sufficiently large and trade costs are sufficiently low so that Foreign firms move

to Home to serve the bigger domestic market and export the dirty goods back to Foreign.

In this case, n = 1, n∗ = 0; xHH = µ(σ−1)σ

, xHF = µ(σ−1)σ

1τ, xFH = xFF = 0; E = µs(σ−1)

σ,

E∗ = µ(1−s)(σ−1)στ

and EW = µ(σ−1)σ

(s+ 1−sτ

).

For s1−s < τσ−1 − 1, derive the mass of firms in each country and compare them to the

case without policy intervention.19

n =sτσ

τσ − τ, n∗ =

(1− s)τσ − ττσ − τ

; (30)

n− n =1− s

τσ−1 − 1> 0, n∗ − n∗ = − 1− s

τσ−1 − 1< 0. (31)

After Home prohibits the import of dirty goods from Foreign, some of the Foreign firms

relocate to Home to serve the market there. The unilaterally prohibitive tariff on the import

from Foreign makes the full agglomeration easier to occur in Home. The intuition behind

this phenomenon is that Foreign firms lose the Home market totally as long as they stay in

Foreign, so the prohibitive tariff leads the Foreign firms to be thirstier to relocate to Home.

The consumption levels of each variety in each country are derived as follows:

xHH =µ(σ − 1)

sσ

τσ − ττσ

; xHF =µ(σ − 1)

(1− s)σ1

τσ; xFF =

µ(σ − 1)

(1− s)σ. (32)

Note that xFH = 0 and xFF are independent of trade costs since the import of dirty goods

from Foreign to Home is prohibited. Besides, comparing the consumption levels to those

in the case with no environmental regulation gives xHH − xHH < 0, xHF − xHF > 0,

xFF−xFF > 0. The unilaterally prohibitive tariff on the imports shrinks Home consumption

of each domestic variety and enlarges Foreign demand for each domestic and imported

variety. Intuitively, the prohibitive tariff induces Foreign firms to relocate to Home. The

19We can simply get the results in the case without policy intervention by setting t = t∗ = 0 in the mainpart of this paper.

22

mass of Home firms increases, and the mass of Foreign firms decreases. Note that

xHH =µ(σ − 1)

σ

1

n; xHF =

µ(σ − 1)

σ

1

nτ + n∗τσ; xFF =

µ(σ − 1)

σ

τσ

nτ + n∗τσ. (33)

As Foreign firms relocate to Home, more varieties are available in Home, the consumption

of each variety decreases. In Foreign, domestic firms become fewer, and they are able to

provide more for the residents due to less competition; on the other hand, residents consume

more imported dirty goods from Home to compensate for fewer domestic varieties.

Derive the local GHG emissions in each country and sum them up to get the global level:

E =µs(σ − 1)

σ; E∗ =

µ(σ − 1)

σ

(1− τσ − 1

τσ − τs

); EW =

µ(σ − 1)

σ

(1− τ − 1

τσ − τs

). (34)

Since the individuals in Home only consume the domestic dirty goods, Home GHG emissions

only depend on its market size and are irrelevant to that in Foreign. Compared to those in

full agglomeration, global and Foreign emissions decrease and Home emissions remain the

same.

Comparing the emissions to those without environmental regulation, we have

Proposition 4. The prohibitive tariff on the imports of dirty goods never reduces the local

GHG emissions of the tariff country and always reduces those in the other country. The

effect on global GHG emissions is ambiguous. However, as long as s1−s <

12(τσ−1 − 1) and

τσ−1 > 2, global emissions always increase due to the prohibitive tariff.

Due to market sizes and trade costs, firm location patterns may change after the imposi-

tion of tariffs. In the following, consider the case where firms disperse in the two cases and

trade costs are sufficiently high, i.e., τ 1−σ − 1 < s1−s < τσ−1 − 1 and τσ−1 > 2.20

The differences of emissions between the cases with and without prohibitive tariff are

20See Appendix for more details and welfare analysis.

23

shown as

E − E =µ(σ − 1)

σ

1

τ 2σ−1 − τ[τσ(1− s)− τs]︸ ︷︷ ︸

incremental emissions from domestic goods

+µ(σ − 1)

σ

1

τ 2σ−1 − τ[s− τσ−1(1− s)

]︸ ︷︷ ︸

reduced emissions from imported goods

=µ(σ − 1)

σ

τ − 1

τ 2σ−1 − τ[τσ−1 − (1 + τσ−1)s

]> 0

(35)

since s1−s < τσ−1 − 1;

E∗ − E∗ =µ(σ − 1)

σ

1

τ 2σ−1 − τ︸ ︷︷ ︸incremental emissions from imported goods

+µ(σ − 1)

σ

1

1− τ 2σ−2︸ ︷︷ ︸reduced emissions from domestic goods

= −µ(σ − 1)

σ

τ − 1

τ 2σ−1 − τ< 0;

(36)

EW − EW =µ(σ − 1)

σ

τ − 1

τσ − τ

(τσ − ττσ + τ

− s). (37)

For Home, the incremental emissions from the domestic dirty goods are larger than the

reduced emissions from the imported goods. Therefore, Home emissions rise. For Foreign,

the outcomes are on the reverse. How global GHG emissions change in response to the

unilaterally prohibitive tariff is ambiguous and strongly dependent of the market sizes and

trade costs. If the market size is sufficiently small or trade costs are sufficiently high, then the

prohibitive tariff is likely to increase the global emissions because the incremental emissions

from domestic goods in the tariff country are dominant and become higher as the market

size becomes smaller and the trade costs become higher.

7. Conclusion

In this paper, I mainly studied the relationship between trade liberalization, GHG emis-

sions from consumption and consumption taxes in the footloose capital model. Trade is

good to the global environment. However, trade liberalization tends to decrease the GHG

emissions at first and increase them later. An increase in the consumption tax forces firms to

move to the country with laxer regulation. How trade liberalization affects the firm location

24

patterns is ambiguous and depends on the strengths between consumption taxes and market

sizes. If market size is large enough, firms may relocate to the country with more stringent

consumption taxes. In the social optimum, the social planner imposes identical consumption

taxes if trade costs are high and tends to induce full agglomeration in the larger country if

trade costs are low. By studying the unilaterally prohibitive tariff on the imports of dirty

goods, I showed that such a policy targeting to control the local GHG emissions does not

work and it might be harmful to the global abatement as well. Although I investigated both

environmental and trade policies, I did not examine the relationship between them. The

questions of which policy is more efficient and how to design an efficient mechanism with

the combination of the two policies are left for future research.21

As an essential source of consumption pollution, gasoline is always in the center of this

issue. Gasoline is crucial not only because the consumption of it is massive but also be-

cause the changes in its price have been demonstrated to be an important channel of carbon

leakage (e.g., Kiyono and Ishikawa, 2013). Further analysis should be done to study how

policy-makers design their policies to control the GHG emissions with the existence of car-

bon leakage stemming from the price changes of natural resources. Another issue is the

endogenous choice of firms between abatement investment and offshoring. If environmental

regulation is not stringent compared to abatement costs, firms may prefer to pay for their

emissions. However, if environmental regulation is strict but not so strict as the costs of off-

shoring, firms may abate their emissions by investing in expensive equipment and machines.

If environmental regulation is sufficiently stringent, firms may choose to do offshoring instead

of abatement investment. Such endogenous choices are important especially when firms are

heterogeneous concerning their productivity, mobility and emission intensities.

21See Chen and Woodland (2013) for a review about environmental and trade policies under climatechange.

25

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27

Appendix

A1. Show that allowing for footloose capital reduces global GHG emissions when

consumption taxes are the same across countries.

This finding directly comes from the fact that the difference between GHG emissions in

the two cases is

EW − EW=µ(σ − 1)

σ

(2s− 1)(τ − 1)τσ

τσ + τ

(1− s1 + t∗

1

sτ + (1− s)τσ− s

1 + t

1

sτσ + (1− s)τ

)= −µ(σ − 1)

σ(1 + t)

(2s− 1)2(τ − 1)τσ+1

[sτσ + (1− s)τ ][sτ + (1− s)τσ](τσ + τ)

< 0

where the second line is derived based on the assumption that the two countries impose the

same consumption taxes.

A2. Proof of Proposition 1.

The difference of GHG emissions between autarky and trade without capital mobility is

EW − EW

A = −µ(σ − 1)

σ

(s

1 + t

(1− s)(τ − 1)

sτσ + (1− s)τ+

1− s1 + t∗

s(τ − 1)

sτ + (1− s)τσ

)< 0.

It implies that trade without capital mobility is good for the global environment. Further-

more, the difference in the presence of footloose capital is

EW − EWA = −µ(σ − 1)

σ

τ − 1

τσ + τ

(s

1 + t+

1− s1 + t∗

)< 0

showing that trade with footloose capital is good for the global environment. Therefore,

openness to trade is good to the global environment.

Differentiating global GHG emissions in trade with footloose capital with respect to τ

gives∂EW

∂τ=µ(σ − 1)

σ

(s

1 + t+

1− s1 + t∗

)(σ − 1)τσ − στσ−1 − 1

(τσ + τ)2.

28

Note that ∂(σ−1)τσ−στσ−1−1∂τ

= σ(σ−1)τσ−2(τ −1) > 0 always holds, so (σ−1)τσ−στσ−1−1

is an increasing function of τ for τ > 1. When τ = 1, the polynomial is -2; when τ = σ+1σ−1 ,

the polynomial is equal to τσ−1 − 1 > 0. Thus, ∂EW

∂τ= 0 exists for 1 < τ < σ+1

σ−1 . Denote

the certain point as τ0, then EW decreases at 1 < τ < τ0 and increases at τ > τ0. In other

words, as trade costs decline from infinity to unity, global GHG emissions decrease at first

and then increase.

A3. Derivation of Social Optimum

Define the social welfare as the sum of individual welfare around the world:

WG = sW + (1− s)W ∗

where

W = µ lnM + I/s− µ− EW

= µ lnµ− µ lnP + 1 + r + T/s− µ− EW

= µ lnµ−{µ ln(

σ

σ − 1)− µ

σ − 1ln(1 + τ 1−σ) + µ ln(1 + t) +

µ

σ − 1ln

(1 +

1− ss

1 + t

1 + t∗

)}+ 1

+µ

σ

(s

1 + t+

1− s1 + t∗

)+ µ

t

1 + t− µ− µ(σ − 1)

σ

τσ + 1

τσ + τ

(s

1 + t+

1− s1 + t∗

).

and

W ∗ = µ lnµ− µ ln(σ

σ − 1) +

µ

σ − 1ln(1 + τ 1−σ)− µ ln(1 + t∗)− µ

σ − 1ln

(1 +

s

1− s1 + t∗

1 + t

)+ 1

+µ

σ

(1− s1 + t∗

+s

1 + t

)− µ 1

1 + t∗− µ(σ − 1)

σ

τσ + 1

τσ + τ

(s

1 + t+

1− s1 + t∗

).

Taking the derivatives of the social welfare with respect to t and t∗ respectively gives

∂WG

∂t= µζ

(−1 + φζ − 1

σ − 1

ζ∗

ζ + ζ∗+

1

σ − 1

1− ss

ζ

ζ + ζ∗

),

29

∂WG

∂t∗= µζ∗

(−1 + φ∗ζ − 1

σ − 1

ζ

ζ + ζ∗+

1

σ − 1

s

1− sζ∗

ζ + ζ∗

),

where

ζ =s

1 + t, ζ∗ =

1− s1 + t∗

,

φ =1

s

σ − 1

σ+

1

s

σ − 1

σ

τσ + 1

τσ + τ,

φ∗ =1

1− sσ − 1

σ+

1

1− sσ − 1

σ

τσ + 1

τσ + τ.

Note that φ∗ = s1−sφ.

Solving the social optimum gives

ζ =1

φ, ζ∗ =

1

φ∗

indicating identical consumption taxes as below

t = t∗ =σ − 1

σ

τσ + 1

τσ + τ− 1

σ.

In the following, I show that the conclusion of identical consumption taxes across coun-

tries still holds in the case where the damage function is increasing and convex (D′(EW ) > 0

and D′′(EW ) > 0). In this case, taking the derivatives of social welfare concerning t and t∗

separately gives

∂WG

∂t= µζ

(−1 + φζ − 1

σ − 1

ζ∗

ζ + ζ∗+

1

σ − 1

1− ss

ζ

ζ + ζ∗

),

∂WG

∂t∗= µζ∗

(−1 + φ

∗ζ − 1

σ − 1

ζ

ζ + ζ∗+

1

σ − 1

s

1− sζ∗

ζ + ζ∗

),

where

ζ =s

1 + t, ζ∗ =

1− s1 + t∗

,

φ =1

s

σ − 1

σ+

1

s

σ − 1

σ

τσ + 1

τσ + τD′,

30

φ∗

=1

1− sσ − 1

σ+

1

1− sσ − 1

σ

τσ + 1

τσ + τD′.

Note that φ∗

= s1−sφ. Let ∂WG

∂t= 0 and ∂WG

∂t∗= 0, manipulate the equations to get

(ζ

s− ζ∗

1− s

)(1

σ − 1+ φs(ζ + ζ∗)

)= 0

implying the identical consumption tax

t = t∗.

Bringing the equation back to ∂WG

∂t= 0 shows

1 + t︸︷︷︸LHS

=σ − 1

σ

σ − 1

σ

τσ + 1

τσ + τD′︸ ︷︷ ︸

RHS

.

According to the conditions that D′′ > 0 and ∂EW

∂t< 0, ∂D′

∂t= ∂D′

∂EW∂EW

∂t= D′′ ∂E

W

∂t< 0

holds. Thus, as long as RHS|t=0 > 1, there exists t = t∗ 6= 0 satisfying the global optimum

as shown in Fig. A.1.

Fig. A.1. Existence of Environmental Tax at SO when RHS|t=0 > 1.

31

A4. Social optimum: full agglomeration in Home

If the social planner induces full agglomeration in Home, the social welfare becomes

WH = µ lnµ− µ ln(σ

σ − 1)− µ(1− s) ln τ + 1− µs ln(1 + t)− µ(1− s) ln(1 + t∗)

− µ(σ − 1)

σ

(s

1 + t+

1− s1 + t∗

)− µ(σ − 1)

σ

[s

1 + t+

1− sτ(1 + t∗)

].

Solving the first order conditions with respect to t and t∗ gives

t =σ − 2

σ; t∗ =

σ − 1− τστ

.

Note that1 + t

1 + t∗=

2τ

1 + τ> 1 >

s

1− sτ (1−σ).

The optimal taxes are not available since the full agglomeration in Home never happens if

the consumption taxes are imposed. WH is an increasing function with respect to 1+t1+t∗

for

1+t1+t∗≤ s

1−sτ(1−σ), so the social planner acts at the threshold value where 1+t

1+t∗= s

1−sτ(1−σ).22

Then the consumption taxes are derived as

1 + tH =s(σ − 1)

σ(τ 1−σ + τ−σ + 2);

1 + t∗H =(1− s)(σ − 1)

σ(2τσ−1 + 1 + τ−1);

the global welfare is

WH =µ lnµ− µ ln(σ

σ − 1)− µ(1− s) ln τ + 1− µ ln

{σ − 1

σ

[1 +

τσ + 1

τσ + τ

]}− µs ln

[s(1 + τ 1−σ)

]− µ(1− s) ln

[(1− s)(1 + τσ−1)

]− µ;

and global GHG emissions are

EWH =

µ(τσ + 1)

2τσ + τ + 1.

22See Appendix A5.

32

The comparison between the global welfare levels in the two cases is

W −WH =µσ

σ − 1ln ∆

where

∆ = ss(1− s)(1−s)(1 + τ 1−σ)s(σ−1)+1

σ (1 + τσ−1)(1−s)(σ−1)

σ τ(1−σ)(1−s)

σ .

If ∆ ≥ 1, then W ≥ WH ; if ∆ < 1, then W < WH . In Fig. A.2, I do the simulations for

σ = 6 and find that the former case tends to hold when τ is large and the latter case tends

to hold when τ is small and s is very close to 0.5.

Fig. A.2. ∆ max when σ = 6, s ∈ (0.5, 0.99) and τ ∈ (1, 5).

A5. Show that 1+t1+t∗

= s1−sτ

(1−σ) is satisfied when WH is maximized at full

agglomeration in Home.

Assume that α = 1+t1+t∗

. Note that α ≤ s1−sτ

(1−σ). The social welfare at full agglomeration

in Home becomes

WH = 2µ lnµ− 2µ ln(σ

σ − 1)− µ ln τ + 2− µs ln(1 + t)− µ(1− s) ln(1 + t∗)

− µ(σ − 1)

σ

(s

1 + t+

1− s1 + t∗

)− µ(σ − 1)

σ

[s

1 + t+

1− sτ(1 + t∗)

].

33

Deriving the first order condition of the global welfare with respect to 1 + t∗ gives

∂WH

∂1 + t∗=

µ

(1 + t∗)2

{−(1 + t∗) +

σ − 1

σ

[2s

α+ (1− s)(1 + τ−1)

]}= 0;

thus, 1 + t∗ = σ−1σ

[2sα

+ (1− s)(1 + τ−1)]. Taking it back to the welfare function implies

WH = 2µ lnµ− 2µ ln(σ

σ − 1)− µ ln τ + 2− µs lnα− µ ln(1 + t∗)

− µ(σ − 1)

σ

(2s

α+ (1− s)(1 + τ−1)

)1

1 + t∗.

Differentiating the equation with respect to α gives

∂WH

∂α=µs

α

{2σ

(σ − 1)[2s+ (1− s)(1 + τ−1)α]− 1

};

Since the polynominal in the parenthesis is decreasing in α and

∂WH

∂α|α= sτ1−σ

1−s=µs

α

{2σ

(σ − 1)s(2 + τ 1−σ + τ−σ)− 1

}>µs

α

{2 + τ 1−σ + τ−σ + στ−σ(τ − 1)

(σ − 1)s(2 + τ 1−σ + τ−σ)(1 + τ 1−σ)

}> 0

due to s1−s < τσ−1. Thus, ∂WH

∂α> 0 always holds for α ≤ s

1−sτ(1−σ). So 1+t

1+t∗= s

1−sτ(1−σ) is

satisfied when WH is maximized at full agglomeration in Home.

A6. Social optimum: full agglomeration in Foreign

If the social planner induces full agglomeration in Foreign, the social welfare becomes

WF =µ lnµ− µ ln(σ

σ − 1)− µs ln τ + 1− µ ln

{σ − 1

σ

[1 +

τσ + 1

τσ + τ

]}− µs ln

[s(1 + τσ−1)

]− µ(1− s) ln

[(1− s)(1 + τ 1−σ)

]− µ.

34

Note that WH −WF = µ(2s− 1) ln(

τ+τσ

1+τ1−σ

)> 0 always holds for s > 0.5, the social planner

never has an incentive to raise full agglomeration in the smaller country.

A7. Environmental Tax Competition with σ = 2

I first investigate how Home levies its tax given the consumption tax in Foreign. The

welfare functions in the three areas are shown respectively as

W(i) = µ lnµ− µ ln 2 + 1− µ ln(1 + t) +µ

2

(s

1 + t+

1− s1 + t∗

)− µ

1 + t

− µ

2

[s

1 + t+

1− sτ(1 + t∗)

];

W(ii) = µ lnµ− µ ln 2 + µ ln(1 + τ−1) + 1− µ ln(1 + t)− µ ln

(1 +

1− ss

1 + t

1 + t∗

)+µ

2

(s

1 + t+

1− s1 + t∗

)− µ

1 + t− µ

2

τ 2 + 1

τ 2 + τ

(s

1 + t+

1− s1 + t∗

);

W(iii) = µ lnµ− µ ln 2− µ ln τ + 1− µ ln(1 + t) +µ

2

(s

1 + t+

1− s1 + t∗

)− µ

1 + t− µ

2

[s

τ(1 + t)+

1− s1 + t∗

].

Differentiating the welfare functions with respect to 1 + t separately gives

∂W(i)

∂(1 + t)=

µ

(1 + t)2[−(1 + t) + 1] ;

∂W(ii)

∂(1 + t)=

µ

(1 + t)2

[−(1 + t)− (1− s)(1 + t)2

s(1 + t∗) + (1− s)(1 + t)− s

2+ 1 +

1

2

τ 2 + 1

τ 2 + τs

];

∂W(iii)

∂(1 + t)=

µ

(1 + t)2

[−(1 + t)− s

2+ 1 +

s

2τ

].

For 1 ≤ 1 + t∗ < 1−ssτ , Home chooses among areas (iii) and (ii). Note that

∂W(iii)

∂(1+t)< 0 in

35

area (iii) and∂W(ii)

∂(1+t)< 0 in area (ii), so Home best responses to Foreign choices are

1 + t =

s

1−sτ(1 + t∗) if maxW(iii) > maxW(ii)

1 if maxW(iii) ≤ maxW(ii)

,

where maxW(iii) = W(iii)|1+t= s1−s τ

σ−1(1+t∗), maxW(ii) = W(ii)|1+t=1.

The difference between the optimal welfare in the two cases is

∆ max = maxW(ii) −maxW(iii)

=µs

2− µ− µ

2

τ 2 + 1

τ 2 + τs+ µ ln(

s

1− sτ)− µ ln(1 +

1− ss

1

1 + t∗) + µ ln(1 + t∗)

− µ(1− s)1 + t∗

{−1

sτ−1 +

1

2

τ 2 − ττ 2 + τ

}+ µ ln(1 + τ−1) + µ ln τ,

In order to examine how the difference changes as consumption tax in Foreign increases,

I differentiate the difference with respect to 1 + t∗ and get

∂∆ max

∂(1 + t∗)=

1− ss(1 + t∗)2 + (1− s)(1 + t∗)

+1

1 + t∗+

1− s(1 + t∗)2

{− 1

sτ+

1

2

τ − 1

τ + 1

}>

1− ss

1

(1 + t∗)2 + (1 + t∗)+

1

1 + t∗+

1− s(1 + t∗)2

{− 1

sτ+

1

2

τ − 1

τ + 1

}=

1

(1 + t∗)2

{1− ss

1 + t∗

2 + t∗+ 1 + t∗ + (1− s)

(− 1

sτ+

1

2

τ − 1

τ + 1

)}

>1

(1 + t∗)2

1− ss

1

2+ 1− 1− s

sτ︸ ︷︷ ︸(+)

+1− s

2

τ − 1

τ + 1

> 0.

The second positive symbol stands because the terms in brace at the third line are increasing

in (1 + t∗). ∆ max is positive from the simulations in Fig. A.3 where µ is normalized to

unity and t∗ is set to be 0.23

23The specific value of µ does not affect the symbol of ∆ max.

36

Fig. A.3. ∆ max when σ = 2, s ∈ (0.5, 0.99) and τ ∈ (1, 200).

For 1 + t∗ ≥ 1−ssτ , Home chooses among the three areas. According to the facts that (1)

∂W(i)

∂(1+t)< 0, (2) W(i) > W(ii) holds for s

1−sτ−1(1 + t∗) < 1 + t < s

1−sτ(1 + t∗), (3) W(i) > W(iii)

holds for 1 + t > s1−sτ(1 + t∗), the best response of Home is to choose 1 + t = 1.

Analogously, I derive the best responses of Foreign given Home consumption taxes. De-

scribe the best responses of each country in Fig. 3 in the main part where the red lines

represent those of Home and the blue ones Foreign. There is only one Nash Equilibrium

at which both countries impose no consumption taxes and firms diverse across countries.

Based on the main part of the paper, t = t∗ = 0 holds at global optimum. Thus, when σ = 2

(or σ is sufficiently small), imposing no environmental regulation on GHG emissions from

consumption is both nationally and globally optimal.

37

A8. Unilaterally Prohibitive Tariff

A8.1. Conclude how the unilaterally prohibitive tariff affects global GHG emissions in the

following tables.

(i) For τ 1−σ < 12(τσ−1 − 1) (or τσ−1 > 2):

s1−s (0, τ1−σ) (τ1−σ, 12(τσ−1 − 1)) (12(τσ−1 − 1), τσ−1 − 1) (τσ−1 − 1, τσ−1) (τσ−1,+∞)

No PolicyFA in

ForeignDispersion across countries

FA in

Home

Prohibitive

TariffDispersion across countries FA in Home

Comparison

of Emissions

E > E

E∗ < E∗

EW > EW

E > E

E∗ < E∗

EW > EW

E > E

E∗ < E∗

EW < EW

E > E

E∗ < E∗

EW < EW

E = E

E∗ = E∗

EW = EW

Table A.1. Comparisons of Emissions when τ 1−σ < 12(τσ−1 − 1).

(ii) For 12(τσ−1 − 1) < τ 1−σ < τσ−1 − 1 (or 1+

√5

2< τσ−1 < 2):

s1−s (0, 12(τσ−1 − 1)) (12(τσ−1 − 1), τ1−σ) (τ1−σ, τσ−1 − 1) (τσ−1 − 1, τσ−1) (τσ−1,+∞)

No Policy FA in Foreign Dispersion across countriesFA in

Home

Prohibitive

TariffDispersion across countries FA in Home

Comparison

of Emissions

E > E

E∗ < E∗

EW < EW

E > E

E∗ < E∗

EW < EW

E > E

E∗ < E∗

EW < EW

E = E

E∗ = E∗

EW = EW

Table A.2. Comparisons of Emissions when 12(τσ−1 − 1) < τ 1−σ < τσ−1 − 1.

38

(iii) For τσ−1 − 1 < τ 1−σ < τσ−1 (or 1 < τσ−1 < 1+√5

2):

s1−s (0, 12(τσ−1 − 1)) (12(τσ−1 − 1), τσ−1 − 1) (τσ−1 − 1, τ1−σ) (τ1−σ, τσ−1) (τσ−1,+∞)

No Policy FA in Foreign DispersionFA in

Home

Prohibitive

TariffDispersion across countries FA in Home

Comparison

of Emissions

E > E

E∗ < E∗

EW < EW

E > E

E∗ < E∗

EW < EW

E > E

E∗ < E∗

EW < EW

E = E

E∗ = E∗

EW = EW

Table A.3. Comparisons of Emissions when τσ−1 − 1 < τ 1−σ < τσ−1.

A8.2. Welfare analysis when τ 1−σ − 1 < s1−s < τσ−1 − 1 and τσ−1 > 2.

In this condition, firms always disperse. Home welfare when there is no environmental

regulation is derived as

WN = µ lnµ−µ ln(σ

σ − 1)+

µ

σ − 1ln(1+τ 1−σ)+

µ

σ − 1ln s+1− µ(σ − 1)

σ

(1 +

τσ + 1

τσ + τ

).

Home welfare with the prohibitive tariff is derived as

WPT = µ lnµ−µ ln(σ

σ − 1)+

µ

σ − 1ln

τσ−1

τσ−1 − 1+

µ

σ − 1ln s+1−µ(σ − 1)

σ

(2− τ − 1

τσ − τs

).

Thus,

WPT −WN = − µ

σ − 1ln[1− τ 2(1−σ)

]+µ(σ − 1)

σ

(s− τσ−1 − 1

τσ−1 + 1

).

The first term measures the change in utility of dirty goods consumption which is always

positive. The second term measures the change in GHG emissions. When the market size

is sufficiently large (s > τσ−1−1τσ−1+1

), GHG emissions decrease due to the prohibitive tariff as

explained in the main part; therefore, Home welfare improves.

39