Discussion Papers In Economics
And Business
Tariﬀ policies, variable markups, and within-sector
misallocation∗

Atsushi Tadokoro

Discussion Paper 20-16

December 2020

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANTariﬀ policies, variable markups, and within-sector
misallocation∗
Atsushi Tadokoro †December 2020

Abstract
This study examines the inﬂuence of import tariﬀ policies on welfare in a twocountry model with heterogeneous ﬁrms and variable markups. In this model, themarket outcome under free trade provides too small (large) output level for more (less)productive ﬁrms and too many varieties due to markup pricing, which can be partiallycompensated by import tariﬀs. If countries cooperatively adopt a symmetric importtariﬀ, the eﬃcient tariﬀ that maximizes the total welfare level of the two countries ispositive when the introduction of the small symmetric import tariﬀ suﬃciently improves
this within-sector misallocation.

Keywords: Variable markups, Misallocation, Firm heterogeneity
JEL Classiﬁcation: F12, F13, L11∗

I thank Koichi Futagami, Kazuhiro Yamamoto, Hayato Kato, and Koji Shintaku for helpful comments
and suggestions. All remaining errors are my own.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka, 560-0043,
JAPAN; e-mail address: sifurywhizagbal@gmail.com

1

Introduction

In trade negotiations between WTO members, as the number of members increases, the coordination of interests between member countries tends to become more complicated, making
it diﬃcult to respond quickly to new challenges and rule-making. In such cases, bilateralor regional trade agreements still play an important role today, as they allow for relatively
easy coordination of interests. When a country enters into a bilateral trade agreement, howshould the agreement be design: should it be concluded between similar countries in terms ofmarket size and technology, or should it be free trade agreement? Although much attention
has been devoted to the study of international trade models that consider heterogeneousﬁrms since the seminal work of Melitz (2003), little is known about the eﬀects of bilateral
trade policy in such models while these studies focus on the eﬀect of unilateral trade policy.
The impact of bilateral trade agreement is deeply related to the model structure thatdeal with it. In particular, the canonical models of monopolistic competition with constantelasticity of substitution (CES) preferences, which generates constant markups, opposite
externalities related to the ineﬃciency of the product variety oﬀset each other. That is,
the market equilibrium under free trade is eﬃcient (when there is no other sector than themonopolistically competitive one). Therefore, there is no room for welfare improving policy
intervention. As Dhingra and Morrow (2019) show, the market outcome is ﬁrst best under
CES preferences and demand-side elasticity determines how resources are misallocated.
In this study, I analyze the eﬀects of uncooperative and cooperative trade policies on
welfare with a focus on import tariﬀ policies in a monopolistically competitive model withheterogeneous ﬁrms á la Melitz and Ottaviano (2008), who incorporate endogenous markups
by introducing the linear demand system into the Melitz model. Incorporation of such a
linear demand system makes the markups of ﬁrms endogenous. The presence of endogenous
markups aﬀects the eﬃciency of resource allocation. Based on the closed economy setting of
the Melitz-Ottaviano model, Nocco et al. (2014) show that, in addition to the ineﬃciency ofthe product variety that arise under constant markups, endogenous markups create withinsector misallocation and market outcome becomes excessively ineﬃcient.1
Then, the following question arises: how does bilateral import tariﬀ policy aﬀect thiswithin-sector misallocation and welfare in the model of monopolistic competition with ﬁrm
1

Using the multi-country setting of Melitz and Ottaviano (2008), Nocco et al. (2019) also show that free
trade allocation of resources fails to be eﬃcient due to this within-sector misallocation.

1

heterogeneity and variable markups? The present study shows that the market outcome
under free trade creates this within-sector misallocation even in the model without ﬁrm
entry, and that it can be partially improved by bilateral import tariﬀ policy.
I modify the Melitz-Ottaviano model by incorporating ad valorem import tariﬀs without considering free entry in the diﬀerentiated goods sector.2 In this model, there are twocountries with two sectors: the ﬁrms in a sector produce diﬀerentiated varieties under monopolistic competition while ﬁrms in the other sector produce homogeneous good under perfectcompetition. In this setting, an increase in a country’s import tariﬀ aﬀects its welfare throughchanges in cross-sector and within-sector allocations. With respect to the impact on crosssector allocation, an increase in a country’s import tariﬀ increases income by increasing its
tariﬀ revenue and domestic proﬁts and raising the average price of diﬀerentiated goods soldin that country, which shifts the production of goods sold in the country from the diﬀerentiated good sector to the homogeneous good sector. Moreover, it also aﬀects within-sector
allocation in the diﬀerentiated good sector through three channels: product variety, productselection, and product mix. As for the ﬁrst channel, an increase in the import tariﬀ decreases
the number of varieties: it intensiﬁes export competition among exporters in the country’s
trading partner and thereby forces some of exporters to stop exporting. Although it also
increases the number of domestic ﬁrms in the country, the former eﬀect (which decreases
exporters) dominates the latter eﬀect (which increases domestic ﬁrms). As for the second
channel, by an increase in the import tariﬀ, product selection becomes tougher for exporters
in the country’s trading partner and more relaxed for domestic ﬁrms in the country. This isbecause ﬁrms that stop (start) producing for the country by this tariﬀ increase are the least
productive exporters (domestic ﬁrms). The impact of an increase in the ad valorem importtariﬀ on the last channel, product mix, can be divided into two eﬀects. Firstly, an increasein the import tariﬀ directly decreases the output level of exporters in the country’s trading
partner. This decrease is greater for less productive exporters, which in turn causes some of
the least productive exporters to stop exporting. Secondly, this decrease in the number ofexporters increases the output level of all surviving ﬁrms uniformly. As a result, an increasein the import tariﬀ increases (decreases) the output level of more (less) productive exporters
in the country’s trading partner and increases the output level of domestic ﬁrms in the country. Thus, the welfare eﬀect of a country’s import tariﬀ is determined by a combination of2

In the next section, I show that the number of ﬁrms changes with trade policies even without free entry.

2these eﬀects.
This study characterizes the uncooperative and cooperative import tariﬀ policies, resulting in the following main ﬁndings. First, the Nash tariﬀs are positive and, when countries
are symmetric, lower than the eﬃcient import tariﬀ that the countries adopt uniformly tomaximize the total welfare level of the countries.3 An increase in a country’s import tariﬀ
increases its welfare mainly by increasing its income (and thereby the output level of homogeneous good sold in the country) when the import tariﬀ is suﬃciently small. By contrast,
an increase in the country’s import tariﬀ harms its trading partner by decreasing its income
owing to a decrease in export proﬁts. As a result, the Nash tariﬀs are positive due to theeﬀect of increasing income in both countries, and higher than the eﬃcient tariﬀ because, in a
global welfare perspective, this eﬀect is partially oﬀset by the eﬀect of the decreasing income
in each country’s trading partner.
Second, if countries cooperatively adopt a symmetric import tariﬀ, the eﬃcient tariﬀ thatmaximizes the total welfare level of the two countries is positive under the following situation:
the symmetric import tariﬀ is introduced when global free trade prevails and its introduction
suﬃciently improves within-sector misallocation. Otherwise, global free trade is desirable. In
this model, the free trade allocation of resources is ineﬃcient in both cross-sector and withinsector allocation due to markup pricing. Compared to the ﬁrst-best allocation set by a social
planner to maximize the total welfare level of the countries, in the market allocation under free
trade, too small total output level is produced in the diﬀerentiated good sector (cross-sector
misallocation); too many varieties are sold (ineﬃcient product variety); too low-productivity
ﬁrms, both domestic ﬁrms and exporters, remain in the market (ineﬃcient product selection);and too small (large) output level is produced by more (less) productive ﬁrms, both domesticﬁrms and exporters (ineﬃcient product mix), in both countries. Within-sector misallocationconsists of ineﬃcient product variety, product selection, and product mix. Thus, underfree trade, markup pricing creates not only cross-sector misallocation due to undersupply of
the total output level in the diﬀerentiated good sector, but also within-sector misallocation:
more productive ﬁrms do not pass on their entire cost advantage to consumers by raising theirmarkups and they end up selling less than ﬁrst-best output level, which leaves room for lessproductive ﬁrms to end up being oversupplied and for the least productive ﬁrms to survive
3

In this model, the optimal import tariﬀ, which is set by a country to maximize its welfare with the importtariﬀ set by the country’s trading partner as given, is consistent with the Nash tariﬀ. This is because theoptimal import tariﬀ is determined independently of the import tariﬀ set by the country’s trading partner.

3

ineﬃciently.4 The introduction of the symmetric import tariﬀ shifts the production fromthe diﬀerentiated good sector to the homogeneous good sector and thereby distorts crosssector allocation.5 Meanwhile, the introduction of the symmetric import tariﬀ increases thetotal welfare level through partially improving within-sector misallocation.6 As a result, theeﬃcient tariﬀ that maximizes the total welfare level of the two countries is positive if andonly if the latter eﬀect dominated the former. This result diﬀers from that of Melitz andOttaviano (2008), who treat trade liberalization as a decrease in transportation costs andshow that bilateral trade liberalization leads to welfare gain for the two countries in both thelong run and short run. Unlike transportation costs, changes in import tariﬀs have tariﬀrevenue impacts. The diﬀerence between transportation costs and import tariﬀs in thesemodels results in a diﬀerence in the eﬀects of trade policies on welfare.
Third, I analyze under what circumstances both countries can mutually beneﬁt from theintroduction of a symmetric import tariﬀ from the initial situation of global free trade whenthe eﬃcient tariﬀ is positive. I ﬁnd that the introduction of the symmetric import tariﬀ
improves welfare of both countries not only when countries are close to symmetric, but also
when the degree of asymmetry across countries is large: when one country has a relatively
larger population size and number of high-productivity ﬁrms than the other country.7 In
these cases, the introduction of the symmetric import tariﬀ has little impact on the welfare
level of such countries through changes in cross-sector allocation. Therefore, it improves thewelfare level of the countries through improving within-sector misallocation when the eﬃcient
tariﬀ is positive. This result indicates that it is crucial for countries that participate in a
trade agreement to decide their import tariﬀs based on their relative size.

4

Unlike this case, in the canonical models of monopolistic competition with CES preferences which includes
the outside good sector, the number of varieties are ineﬃciently small because the market outcome underfree trade provides ineﬃciently small total output level in the monopolistically competitive sector due tomarkup pricing but eﬃcient output level of each ﬁrm in the sector due to their constant markups (Melitz
and Redding, 2014 and 2015).
5
It also distorts within-sector allocation through product selection and product mix of domestic ﬁrms by
allowing the least productive domestic ﬁrms to produce and less productive domestic ﬁrms to increase theoutput level.
6
It decreases the number of varieties in both countries by shutting out the least productive exporters,
increases (decreases) the output level of more (less) productive exporters, and increases the output level ofmore productive domestic ﬁrms.
7
If countries are symmetric and the eﬃcient tariﬀ is positive, then the welfare of both countries is maximized at the eﬃcient import tariﬀ.

4

1.1

Related literature

Some studies incorporate tariﬀ policies into the Melitz-Ottaviano model. Bagwell and Lee(2020) incorporate import and export tariﬀs into the Melitz-Ottaviano model and study the
impact of trade policy in a symmetric two-country economy. They show that starting at
global free trade, the impact of introducing the total tariﬀ (the sum of tariﬀs imposed whenexporting from one country to the other) and its symmetric increase on joint welfare depends
on a simple relationship among parameters. The present study obtains complementary results
to this ﬁnding, even in a model allowing asymmetric countries and without free entry, andshows analytically the condition under which the introduction of a symmetric import tariﬀ
increases not only joint welfare, but also welfare in both countries. While Bagwell andLee (2020) also show that, under some assumptions, symmetric Nash tariﬀ is higher thanthe eﬃcient tariﬀ when the introduction of the symmetric tariﬀ increases joint welfare, thepresent study can provide this result without requiring the assumptions they impose.8
Nocco et al. (2019) consider the eﬃciency properties of the market outcome in a multicountry setting of Melitz and Ottaviano (2008) and characterize the policy tools that national
policy makers can use cooperatively to make the market achieve the eﬃcient outcome. Underan unconstrained choice of tools, which include domestic and trade policies, and countryspeciﬁc and ﬁrm-speciﬁc production subsidies/taxes, Nocco et al. (2019) show that themarket can achieve the ﬁrst-best outcome. When ﬁrm-speciﬁc production subsidies/taxes
are unavailable, they consider a second-best scenario in which a per-unit production subsidy
is oﬀered to all ﬁrms and ﬁnanced by a lump-sum tax on consumers. Relative to this work,
the present study diﬀers from theirs in policy instruments and model structure. In thepresent study, governments cooperatively choose the eﬃcient ad valorem import tariﬀ, whichis uniform across countries and ﬁrms, without using domestic policy instruments. With
respect the model structure, the model in this study, which does not consider free entry, hassimilar characteristics to the model in Nocco et al. (2019) in terms of how the free market
outcome departs from the ﬁrst-best outcome. In the present study, the market level of thenumber of varieties sold in each country is above that in the ﬁrst-best outcome. This result
8

To obtain this result, Bagwell and Lee (2020) assume that the symmetric Nash and eﬃcient tariﬀs areinterior solutions and that the joint-welfare function is quasi-concave in the symmetric tariﬀ. By contrast,
the present study can show that Nash tariﬀs are interior solutions; the eﬃcient tariﬀ is an interior solution
when the introduction of the symmetric import tariﬀ increases joint welfare; and joint welfare function isquasi-concave in the symmetric import tariﬀ when the introduction of the symmetric import tariﬀ increases
joint welfare and the countries are symmetric.

5

implies that, even without free entry, the eﬀect that Nocco et al. (2019) discuss as the entry
externality created by endogenous markups is inherent in this model, where, by keeping
price above marginal cost more than higher marginal cost rivals, lower marginal cost ﬁrms
leave ineﬃciently larger room for a fringe of the highest marginal cost ﬁrms to produce.9
Moreover, depending on whether per-unit subsidies or ad valorem taxes are used, the impactof the policy instrument on within-sector allocation varies: in the Melitz-Ottaviano model,
while per-unit subsidies increase the output level of all ﬁrms uniformly, ad valorem taxes
increase (decrease) the output level of more (less) productive ﬁrms.
Demidova (2017) removes the outside good from the Melitz-Ottaviano model and characterizes optimal unilateral import tariﬀs for small and large countries. She shows thatthe optimal tariﬀs are positive for both small and large countries. Compared to Demidova(2017), the present study analyzes the eﬀect of tariﬀ policies on cross-sector allocation as
well as within-sector allocation and characterizes the eﬃcient tariﬀ which maximizes thetotal welfare level of two countries.
The present study is also related to the following studies. Demidova and Rodrı́guez-Clare(2009) and Haaland and Venables (2016) consider a small-country version of the Melitz modeland characterize a unilateral trade policy that achieves ﬁrst-best allocation. Felbermayr et al.
(2013) extend Demidova and Rodriguez-Clare (2009) to the case of two large countries andcharacterize the optimal tariﬀ. Campolmi et al. (2020) characterize the Nash equilibrium asconsisting of ﬁrst-best level labor subsidies that achieve production eﬃciency, and ineﬃcientimport subsidies and export taxes aimed at improving the domestic terms of trade whenboth domestic and trade policies are available. Costinot et al. (2020) consider the case ofavailable domestic and trade policy instruments and characterize optimal unilateral tariﬀs
both when tariﬀs are ﬁrm-speciﬁc and when they are uniform in a canonical model of intraindustry trade with monopolistic competition and ﬁrm-level heterogeneity. Bagwell andLee (2018) incorporate a homogeneous good sector in the Melitz model in a symmetric twocountry economy and show that, starting at global free trade, the introduction of a symmetric
import tariﬀ lowers joint welfare. These studies build on the model with CES preference,
which generates constant markups. By contrast, the present study analyzes how tariﬀ policies
aﬀect the within-sector misallocation created by endogenous markups and shows the case in
9

In the model in Nocco et al. (2019), whether the number of varieties sold in each country is above or
below that in the ﬁrst-best outcome depends on relationship between this externality and other entry-related
externalities.

6which the introduction of a symmetric import tariﬀ improves welfare in both countries.

1.2

Organization of the article

The rest of the paper is organized as follows. Section 2 describes our model. Section 3 showsthat the Nash tariﬀs are positive, and conducts comparative statics. Section 4 characterizes
the eﬃcient tariﬀ that maximizes the total welfare level of the two countries. Section 5 showsthat both countries can simultaneously gain by imposing a symmetric import tariﬀ compared
to global free trade, even when the degree of asymmetry across countries is large. Section 6
concludes.

2

Model

In this study, it is assumed that there are two countries labeled H (home) and F (foreign),
two sectors, and one production factor, which is labor in this model. Labor Li (i = H, F ) isinelastically supplied by households in each country and is immobile between countries.

2.1

Households

The preferences of households are deﬁned over a continuum of diﬀerentiated varieties ofgoods and a homogeneous good. The diﬀerentiated goods are indexed by ω ∈ Ωi and thehomogeneous good is chosen as the numeraire. All households in country i share the same
preference and each household maximizes the following utility function:Zc
Ui = q0,i+αγ
qic (ω)dω −

2

Ωi

Z

η
qic (ω)2 dω −

2

Ωi

2

Z
qic (ω)dω

,

(1)

Ωi

subject to the budget constraint
Z
c
q0,i
+

pi (ω)qic (ω)dω = Ii ,

(2)

Ωi
c
where Ωi is the set of all available diﬀerentiated goods varieties in country i; q0,iand qic (ω)

are the individual consumption of the numeraire good and each variety ω in country i,
respectively; pi (ω) is the price of variety ω in country i; and Ii is the income of households in7

country i. Income consists of wage, proﬁts, and the lump-sum transfer from a government.
I assume that the households are stockholders of domestic ﬁrms. The parameters α, η, andγ are positive constants. A lower γ indicates that the diﬀerentiated varieties become closer
substitutes and in the limit case of γ = 0, households care only about the total amountof diﬀerentiated goods they consume. η represents the degree of non-separability. When ηequals zero, the utility function becomes separable across the diﬀerentiated varieties.
c
I assume that the households have positive demand for the numeraire good (q0,i> 0).10

Using the ﬁrst-order conditions for utility maximization, the inverse demand for each variety
ω is given bypi (ω) = α − γqic (ω) − ηQi , ∀ω ∈ Ω∗i ,where Ω∗i ⊂ Ωi represents the subset of varieties in which qic (ω) > 0, and Qi ≡

(3)
R
Ω∗i

qic (ω)dω isthe aggregate consumption of all diﬀerentiated goods. By integrating both sides of (3) overΩ∗i , I obtainQi =Ni(α − p̄i ),γ + ηNi(4)

R
where Ni is the number of consumed (domestic and imported) varieties, and p̄i = (1/Ni ) Ω∗ pi (ω)dωiis the average price of consumed varieties in country i. Using (3) and (4), I obtain the following market demand for variety ω in country i, qi (ω):
qi (ω) = Li qic (ω) =Li max(p− pi (ω)),
γ i

(5)

where
pmax
≡
i

γα + ηNi p̄i
γ + ηNi(6)

represents the threshold price in country i at which demand for a variety is driven to zero.
Note that (3) implies pimax ≤ α.

2.2

Firms

Perfect competition prevails in the homogeneous good market. One unit of production ofhomogeneous good requires one unit of labor input. The homogeneous goods are freely traded
between the countries. Thus, the wage becomes one in both countries.
10c
In Appendix 7.5, I show the suﬃcient condition for q0,i
> 0 in equilibrium.

8In the diﬀerentiated goods sector, there is a continuum of Ki potential ﬁrms in country i,where Ki is assumed to be constant. They produce the diﬀerentiated goods under monopolistic competition. Each ﬁrm requires c units of labor to produce one unit of the diﬀerentiatedgood. I assume that the unit labor requirement c follows Pareto distribution:c ∼ Gi (c) =

c
cM
i

θ
, c ∈ [0, cMi ], θ ≥ 1,

where Gi (c) is cost distribution in country i, θ is an index of the dispersion of the cost, andcM
i is the upper bound of the cost in country i. In addition, when ﬁrms in country i export
their goods to country j, they face an iceberg trade cost τij and an ad valorem import tariﬀ
tij , where τii = tii = 1, τij ≥ 1, and tij ≥ 1 for i, j ∈ {H, F } and i ̸= j.
Potential ﬁrms in country i determine whether they produce or shut down for domestic
and foreign markets after governments in the countries set their tariﬀs. The ﬁrms produce
for the country in which they can earn positive operating proﬁts, and otherwise shut down.
Then, the proﬁt maximization problem for ﬁrms in country i with cost c that sells their goods
to consumers in country j is given by
max(

pij
− τij c)qij ,tijs.t. qij =Lj max(p− pij ),
γ jwhere pij and qij are the tariﬀ inclusive price and the quantity sold in country j, respectively.
Let pij (c) and qij (c) denote the proﬁt-maximizing price and quantity set by country i’s ﬁrms
with cost c to sell their goods to country j. The proﬁt-maximizing price and quantity areτij tij pmaxjpij (c) =(+ c),
2 τij tij
Lj τij tij pmaxjqij (c) =
(
− c).
2γ
τij tij
Next, I deﬁne the cost cutoﬀs. Let cij be the upper bound of the cost for ﬁrms in country
i that sell in the market of country j. Firms with cost cij are indiﬀerent to producing formarket j.
cii = sup{c : πii (c) > 0} = pmax
,
i

pmax
j
.
cij = sup{c : πij (c) > 0} =
τij tij

9(7)

As described in Melitz and Ottaviano (2008), the cost cutoﬀ represents the toughness ofcompetition in a market. In Appendix 7.2, I show that cii > cij in equilibrium, which meansthat there are no ﬁrms that export but do not produce domestically. I call cii the domestic
cost cutoﬀ and cij for i ̸= j the export cost cutoﬀ in country i, and assume that cM
isi
suﬃciently high to be above cii .11
Using these cost cutoﬀs, I obtain the price, quantity, revenue, proﬁt, and markup of ﬁrms
in country i that sell their goods in country j:pij (c) =qij (c) =rij (c) =πij (c) =µij (c) =
τij tij
1(cij + c) = (cjj + τij tij c) ,22Lj τij tij
Lj(cij − c) =(cjj − τij tij c) ,2γ
2γ
Lj τij2 tij 2(cij − c2 ),4γ
Lj τij2 tij(cij − c)2 ,4γ
tij cijpij (c)= ( + 1).
τij c
2 c

(8)
(9)

Lower-cost ﬁrms set lower prices but also higher markups. This generates the within-sector
misallocation distortion, because more productive ﬁrms do not pass on their entire costadvantage to households by raising their markups and end up selling too little, while lessproductive ﬁrms end up being oversupplied.

2.3

Government

The government in country i imposes the tariﬀ tji on exporters in country j and transfers
the tariﬀ revenue to households. A ﬁrm in country j produces for export if it can earn nonnegative proﬁts from sales in the foreign country. Among Kj Gj (cjj ) ﬁrms in country j, ﬁrms
with cost c (≤ cji ) export, so that the number of exporters is described by Kj Gj (cji ). The
budget constraint of the government is given by
Ti = Kj Gj (cji )(tji − 1)r̄ji
Li kj τji −θ=(tji − 1)tji −(θ+1) cii θ+2 ,2γ(θ + 2)
11

In Appendix 7.1, I show the suﬃcient condition for cii < cM
i in equilibrium

10

(10)

where r̄ji = Gj (c1 ji )

R cji
0

τ

rji (c)dGj (c) = GjL(ciji ) ji

−θ t −(θ+1) c θ+2jiiiθ2γ(θ+2)cM
jis the average revenue of ﬁrms

θin country j from sales in country i and kj ≡ Kj /cM
j . Let kj be the productivity index ofcountry j, which measures the number of productive ﬁrms in country j.12

2.4

Equilibrium

The number of sellers in country i, Ni , is composed of domestic producers and exporters incountry j, that is,
Ni = Ki Gi (cii ) + Kj Gj (cji )= ki cii θ + kj cji θ= (ki + kj (τji tji )−θ )cii θ .

(11). Using this relationship,
From (7), the domestic cost cutoﬀ cii , is expressed as cii = pmaxi
I rewrite the threshold price condition (6) as follows:1(γα + ηNi p̄i )ηNi + γ2(θ + 1) α − cii
=
,
A
cii

cii =
⇔ Ni

(12)

where A ≡ η/γ andRc
Rc
Ki 0 ii pii (c)dGi (c) + Kj 0 ji pji (c)dGj (c)2θ + 1
p̄i =
=
cii .
Ni
2(θ + 1)

(13)

Thus (11) and (12) determine the domestic cost cutoﬀ cii and Ni . From (11), and (12), cii isdetermined by the following equation:2(θ + 1) α − ciiAcii θ+1⇔ A(ki + kj (τji tji )−θ )cii θ+1 + 2(θ + 1)cii = 2(θ + 1)α.
ki + kj (τji tji )−θ =(14)

For instance, the number of ﬁrms below cost c′ (which satisﬁes c′ < cji and c′ < cij ) in both countriesθθis Ki Gi (c′ ) = ki c′ and Kj Gj (c′ ) = kj c′ . Then, the number of ﬁrms above cost c′ is ki (cθii − c′θ ) andθ′θ
kj (cjj − c ). As explained in Lemma 4, an increase in ki or kj decreases domestic cutoﬀs, cii and cjj , so thatit does not necessarily increase the number of ﬁrms above cost c′ while increasing the number of ﬁrms below
cost c′ .
1211Note that the import tariﬀ set by country j’s government does not aﬀect the domestic cost
cutoﬀ in country i: dcii /dtij = 0.
Totally diﬀerentiating (14), I show the following:dcii
θAkj τji −θ tji −(θ+1) cii θ+2
=
> 0.
dtji
2(θ + 1) (θ + 1)α − θcii

(15)

Since the domestic cost cutoﬀ is an increasing function of tji , the range of cii is expressed ascFii T ≤ cii < cAUii (< α),(16)where cFii T ≡ limtji →1 cii and cAUii ≡ limtji →∞ cii .
Next, I consider the impacts of an import tariﬀ on the number of varieties and the export
cost cutoﬀ. From (12) and (15), I obtain2(θ + 1)α dcii
dNi
=−
< 0.
dtji
Acii 2 dtji

(17)

Noting that cji = cii /(τji tji ) by (7), I can show that the export cost cutoﬀ in country j, cji ,is decreasing with tji :dcji
cji θAki (τji tji )θ cji θ + A(ki (τji tji )θ + kj )cji θ + 2(θ + 1)
< 0.
=−
dtji
tji
(θ + 1)A(ki (τji tji )θ + kj )cji θ + 2(θ + 1)
Thus, from this equation and (15), I obtain the following lemma:
Lemma 1. (Product selection) For countries i and j with i, j ∈ {H, F } and i ̸= j, an increasein country i’s import tariﬀ increases the domestic cost cutoﬀ in country i and decreases theexport cost cutoﬀ in country j.
As for the number of varieties, I obtain the following lemma:
Lemma 2. (Product variety) For countries i and j with i, j ∈ {H, F } and i ̸= j, an increasein country i’s import tariﬀ results in an increase in the number of domestic ﬁrms in country
i and a decrease in the number of exporters in country j, which leads to a decrease in thenumber of varieties sold in country i.
An increase in country i’s import tariﬀ intensiﬁes the export competition in country j and12relaxes the domestic competition in country i, which makes the least exporters stop exporting
and the least domestic ﬁrms start producing domestically (product selection eﬀect). Since theformer eﬀect which decreases exporters dominates the latter eﬀect which increases domestic
ﬁrms, an increase in the import tariﬀ decreases the number of varieties sold in country i(product variety eﬀect).
In addition to product selection eﬀect and product variety eﬀect, an increase in country
i’s import tariﬀ also aﬀects prices and output levels set by exporters in country j. This
impact diﬀers depending on their productivity. Using (8) and (9), the eﬀects of country i’simport tariﬀ on the price and output level set by country j’s exporters with cost c are givenbydpji (c)
1
=
dtji
2



dcii
+ τji c
dtji


> 0,

dqji (c)
Li
=
dtji
2γ






>0

(0 ≤ c <1 dcii)τji dtjidcii− τji c.(18)
1 dcii

dtji

<0 (
< c ≤ cji )τji dtji

The impact of an increase in the ad valorem import tariﬀ on price and output level can bedivided into two eﬀects. The second terms in (18) represent the direct eﬀect of an increase in
the import tariﬀ. The impact of this eﬀect depends on their productivity: more productive(lower c) exporters have a smaller increase in price and a smaller reduction in their exports.
The ﬁrst terms in (18) represent the eﬀect of an increase in the domestic cutoﬀ due toa decrease in the number of exporters, which causes all surviving ﬁrms to increase theirprice and output level uniformly.13 As a result, an increase in the import tariﬀ causes moreproductive exporters to increase their output level with smaller increase in their price, which
decreases the output level of less productive exporters (see Appendix 7.3). Since these moreproductive exporters set lower price than less productive exporters (see (8)), an increase in
the import tariﬀ increases the output level of low-priced goods. The impact of an increasein the import tariﬀ on product mix can be summarized as the following lemma:
Lemma 3. (Product mix) For countries i and j with i, j ∈ {H, F } and i ̸= j, an increasein country i’s import tariﬀ increases (decreases) the output level of more (less) productive
exporters in country j and increases the output level of domestic ﬁrms in country i.
13

The impact of an increase in country i’s import tariﬀ on the price and output level set by domestic ﬁrms
dcii
in country i is determined only by this eﬀect and do not depend on their productivity: dpdtiiji(c) = 12 dt>0
ji
i dciiand dqdtiiji(c) = L
2γ dtji > 0.

13

Thus, an increase in the import tariﬀ aﬀects within-sector misallocation through threechannels as explained in Lemmas 1–3, and these channel can be expressed by the domestic
cost cutoﬀs: the domestic cost cutoﬀs determine the export cost cutoﬀs, the number ofvarieties, the output level, and thereby the welfare level. The domestic cost cutoﬀs vary withchanges in the characteristics of goods, technology, and transportation costs. At the end of
this section, I summarize the eﬀects of these parameters on the domestic cost cutoﬀ in thefollowing lemma.
Lemma 4. The domestic cost cutoﬀ declines as varieties are closer substitutes (lower γ), thedegree of non-separability is higher (higher η), the number of high-productivity ﬁrms in bothcountries is larger (higher ki and kj ), and the transportation cost from country j to country
i is lower (lower τji ).
Proof. Totally diﬀerentiating (14), I obtain(ki + kj (τji tji )−θ )cii θ+2
dcii
= −
< 0,
dA
2(θ + 1) ((θ + 1)α − θcii )dcii
Acii θ+2
= −
< 0,
dki
2(θ + 1) ((θ + 1)α − θcii )A(τji tji )−θ cii θ+2
dcii
= −
< 0,
dkj
2(θ + 1) ((θ + 1)α − θcii )θAkj τji −(θ+1) tji −θ cii θ+2
dcii
=
> 0,
dτji
2(θ + 1) ((θ + 1)α − θcii )where A = η/γ.

3Uncooperative tariﬀ policy

In this section, I examine the Nash tariﬀs and characteristics of them.

3.1

Nash tariﬀ

Substituting (3) into (2) yields
Z

Z
c
q0,i
= Ii − αΩ∗iqic (ω)dω + γ

!2

Z

Ω∗i

qic (ω)2 dω + η

14

Ω∗i

qic (ω)dω.

(19)

Then, plugging (18) into (1), I obtain the following welfare measure:
Ui = Ii + CSi + ACi ,where CSi ≡ γ2RΩ∗i

qic (ω)2 dω and ACi ≡ η2

RΩ∗i

qic (ω)dω2(20)= η2 Qi 2 . CSi represents the sumof consumer surplus for each variety, because γ2 qic (ω)2 corresponds to the triangular region
under the demand curve for variety ω where qic (ω) = (pmax− pi (ω))/γ by (5). ACi representsi
consumer surplus for individual aggregate consumption of all diﬀerentiated goods, Qi .14 CSi
and ACi are given by

CSi

" Z 
#
2
2
Z cji cii
cii − cτji tji (cji − c)γKidGi (c) + Kj
dGj (c)
=
2
2γ
2γ
0
0
(α − cii )cii,
2η(θ + 2)
η 2 (α − cii )2
=
Q =
,
2 i
2η
=

ACi

(21)
(22)

where
Z cii
Q i = Ki0cii − cdGi (c) + Kj
2γ

Z cji
0

τji tji (cji − c)α − ciidGj (c) =
.
2γ
η

(23)

Income consists of wage, proﬁts from domestic and export sales, and the transfer. Income
in country i is given by1
Ti[Ki Gi (cii )π̄ii + Ki Gi (cij )π̄ij ] +
Li
Li


ki
Lj −θ −(θ+1) θ+2
θ+2
cjj
= 1+
cii
+ τij tij2γ(θ + 1)(θ + 2)Li
kj τji −θ+(tji − 1)tji −(θ+1) cii θ+22γ(θ + 2)

Ii = 1 +−(θ+1)

ki + (θ + 1)kj τji−θ (tji − 1)tji −(θ+1) θ+2 Lj ki τij−θ tijcii+
cjj θ+2= 1+
2γ(θ + 1)(θ + 2)
Li 2γ(θ + 1)(θ + 2)
where π̄ij = Gi (c1 ij )

R cij
0

L

τ

−θ t −(θ+1) c θ+2ijjjθ2γ(θ+1)(θ+2)cMiπij (c)dGi (c) = Gi (cjij ) ij

14

(24)

is the average proﬁt of country

When γ approaches 0 in the utility function (1), households care only about the aggregate consumptionof diﬀerentiated goods, Qi .

15

i’s ﬁrms from sales in country j.
The welfare eﬀect of its own tariﬀ change can be decomposed into three terms:
dIi
dCSi dACi
dUi
=
+
+
.
dtji
dtji
dtji
dtji

(25)

As for the ﬁrst term in (25), I obtain following lemma.
Lemma 5. The relationship between income and an import tariﬀ in country i follows ahump-shaped curve.
Proof. See Appendix 7.4.
Changes in country i’s import tariﬀ aﬀect its income through changes in tariﬀ revenue
and proﬁts from domestic sales. An increase in country i’s import tariﬀ increases the proﬁts
from domestic sales in country i. The relationship of tariﬀ revenue in country i and its
import tariﬀ follows a hump-shaped curve due to a decrease in the number of exporters incountry j and an increase in the average tariﬀ revenue. Therefore, an increase in country i’simport tariﬀ increases its income when that import tariﬀ is suﬃciently small. This change inincome reﬂect the consumption level of homogeneous good (see Appendix 7.5). An increasein country i’s import tariﬀ increases the consumption level of homogeneous good in thatcountry when tji is suﬃciently small.
Next, I state the eﬀects of import tariﬀs on the second term and the third term in (25)in the following lemma.
Lemma 6. An increase in the import tariﬀ in country i lowers ACi , and raises (lowers) CSi
when α ≥ (<)2cii .
Proof. Diﬀerentiating (21) and (22) with respect to tji , I obtaindACi
α − cii dcii
= −= −2Bi tji (θ + 2)(α − cii ) < 0dtjiη dtjiα − 2cii dcii
dCSi=
= Bi tji (α − 2cii )dtji
2η(θ + 2) dtji
θk τ

−θ

t

(26)
(27)

−(θ+2) c θ+2

j ji
ji
ii
where Bi ≡ 4γ(θ+1)(θ+2)> 0.
(θ+1)α−θcii

An increase in country i’s import tariﬀ increases the domestic cost cutoﬀ, which increases
the average price and decreases the aggregate output level in country i given by (13) and16

(23), respectively. Thus, the consumer surplus for individual aggregate consumption, ACi ,decreases. However, an increase in the country’s import tariﬀ does not necessarily decrease
the sum of consumer surplus for each variety, CSi . Since the combination of diﬀerentiatedgoods consumed by households is determined by product selection, product variety, andproduct mix, the impact of an increase in country i’s import tariﬀ on CSi is determined bychanges in this within-sector allocation as in Lemmas 1–3. The net eﬀect of the import tariﬀ
on CSi is determined by these relationship and is positive when the domestic cost cutoﬀ is
suﬃciently small: when α − 2cii > 0 by (27).
Next, I show that the Nash tariﬀs are positive. From (26), (27), and (42) in Appendix
7.4, I obtain

dIi
dACi
dCSi
+
+
dtji
dtji
dtji



θ+1
= Bi
− tji ψi (cii ) + tji (α − 2cii ) ,
θ

dUi
=
dtji



(28)

where ψi (cii ) ≡ θ(θ+2)
Aki cii θ+1 + 2θcii + 2α > 0. The ﬁrst term in brackets in (28) represents
θ+1
the eﬀects of changes in the import tariﬀ on welfare by changes in cross-sector allocation:
increasing the import tariﬀ increases income and the average price, which leads to an increasein the consumption level of homogeneous good and a decrease in the aggregate consumption
level of diﬀerentiated goods.15 It can be easily shown that this eﬀect raises welfare fortji < (θ + 1)/θ. The second term in brackets in (28) represents the eﬀects of changes in
the import tariﬀ on welfare by changes in within-sector allocation. This eﬀect raises welfare
when α − 2cii > 0. The welfare eﬀect of the import tariﬀ is determined by the sum of these
two terms in (28). As a result, I obtain the following proposition.
Proposition 1. In the two-country economy, the Nash tariﬀs, (tnF H , tnHF ), are positive.16
They are given by
#

"
1
α
−
2c
ii
tnji = 1 +
1 + θ(θ+2)θ+1θ
Aki cii+ 2(θ + 1)cii + α

i, j ∈ (H, F ), i ̸= j,

(29)

θ+1

15

From (41) and (52) in Appendix, I conﬁrm that the impacts of the import tariﬀ on income and theconsumption level of homogeneous good are positive for tji ∈ [1, θ+1θ ].
16
In this model, since the import tariﬀ in country i that maximizes its welfare with the import tariﬀ set
by country j as given is determined independently of the import tariﬀ set by country j (see Appendix 7.6),
this optimal import tariﬀ in country i is consistent with the Nash tariﬀ.

17

where cii is endogenously determined and 1 < tnji < 2(1 + 1/θ). The solution of (29) uniquely
exists if θ ≤ θ̄ holds.
Proof. See Appendix 7.6.
Intuitively, the import tariﬀ policy aﬀects welfare through changes in cross-sector andwithin-sector allocation. The ﬁrst component in (29) shows that the welfare eﬀects of theimport tariﬀ through changes in cross-sector allocation are neutralized when tji = 1 + 1/θ(> 1). This is derived from the ﬁrst term in brackets in (28). Then, the second component
shows that the welfare eﬀect of the import tariﬀ through changes in within-sector allocationis positive (negative) when the second component is greater (less) than 1. As a result, even
if the second component is less than 1, the ﬁrst component dominates the second component
and the Nash tariﬀ is positive.
Although the Nash tariﬀ is always positive, an increase in a country’s import tariﬀ decreases income in its trading partner owing to a decrease in its export proﬁts:
Lemma 7. In the two-country economy, an increase in a country’s import tariﬀ decreases
income and does not aﬀect the consumer surplus for individual aggregate consumption andthe sum of consumer surplus for each variety in the country’s trading partner.
Proof. As for income, see Appendix 7.4. As for ACi and CSi , diﬀerentiating (21) and (22)with respect to tij , I obtain
dACi
dCSi
=
=0
dtij
dtij
θk τ

−θ

t

(30)

−(θ+2) c θ+2

ji
ii
j ji
where Bi ≡ 4γ(θ+1)(θ+2)> 0.
(θ+1)α−θcii

From this lemma, I obtain the following proposition.
Proposition 2. An increase in country i’s import tariﬀ generates welfare loss in country j:
dUj
dIj
Li 1
=
= − Bi ψi (cii ) < 0.
dtji
dtji
Lj θ

(31)

Thus, an increase in country i’s import tariﬀ decreases income in country j, which decreases the consumption level of homogeneous good in country j (see Appendix 7.5), thereby
decreasing the welfare level of country j.
18

3.2

Eﬀects of a decrease in transportation costs on Nash tariﬀs

Next I examine how the Nash tariﬀs given by (29) is aﬀected by changes in transportation
costs: changes with τji and τij . In the rest of this section, I assume θ ≤ θ̄. Then, I establish
the following lemma.
Lemma 8. In the two-country economy, a decrease in transportation cost from country j tocountry i increases tnji , but does not change by a decrease in transportation cost from country
i to country j.
Proof. See Appendix 7.6.
Intuitively, a decrease in τji softens export competition in country j. Then, more productive exporters in country j increase their markups and decrease their output levels tomaximize their export proﬁts. Therefore, there is room for improving the welfare level ofcountry i by increasing its import tariﬀ, because it increases income (due to an increase in tariﬀ revenue) and the output level of relatively low-priced goods produced by more productive
ﬁrms.

4

Eﬃcient trade policy

In this section, I consider the case in which both countries adopt a symmetric import tariﬀ,
tji = tij ≡ tw , and examine how the total welfare level of two countries, W ≡ LH UH + LF UF ,is aﬀected by changes in the symmetric import tariﬀ. The transportation costs are assumed
to be zero, that is, τji = τij = 1 to focus on the eﬀects of the symmetric import tariﬀ.
To characterize the eﬀect of the symmetric import tariﬀ, I ﬁrst compare the market
allocation under free trade and the ﬁrst best allocation set by a social planner to maximize
the total welfare level of the countries. Then, I obtain the following lemma:
Lemma 9. Compared to the ﬁrst-best allocation, in the market allocation under free trade,(i) the total output level of diﬀerentiated goods sold in each country is below the ﬁrst-best; (ii)the domestic and export cost cutoﬀs in each country are above the ﬁrst-best; (iii) the numberof varieties sold in each country is above the ﬁrst-best; and (iv) the output level of more (less)productive ﬁrms, both domestic ﬁrms and exporters, sold in each country is below (above) theﬁrst-best.
19Proof. See Appendix 7.7.
Markup pricing causes the total output level in the diﬀerentiated good sector to be undersupplied, and thereby creates cross-sector misallocation. Moreover, it also creates withinsector misallocation. More productive ﬁrms do not pass on their entire cost advantage toconsumers by raising their markups and they end up being undersupplied. This gives room
for less productive ﬁrms to end up being oversupplied and for the least productive ﬁrms
to survive ineﬃciently. Thus, with respect to within-sector allocation, the market outcome
under free trade provides ineﬃciently high cost cutoﬀs (ineﬃcient product selection), ineﬃciently large number of varieties (ineﬃcient product variety), and ineﬃciently small (large)
output level of more (less) productive ﬁrms (ineﬃcient product mix).
Next, the impact of an increase in a symmetric import tariﬀ on these allocations are
summarized as follows.
Lemma 10. An increase in a symmetric import tariﬀ (i) decreases the total output level ofdiﬀerentiated goods sold in each country; (ii) increases (decreases) the domestic (export) cost
cutoﬀ in each country; (iii) decreases the number of varieties sold in each country; and (iv)increases (decreases) the output level of more (less) productive exporters and increases theoutput level of domestic ﬁrms sold in each country.
Proof. See Appendix 7.8.
Therefore, from Lemmas 9 and 10, if countries start at global free trade and introduce asymmetric import tariﬀ, its introduction distorts cross-sector allocation by shifting the production from the diﬀerentiated good sector to the homogeneous good sector. It also distorts
within-sector allocation through product selection and product mix of domestic ﬁrms by allowing the least productive domestic ﬁrms to produce (product selection) and less productive
domestic ﬁrms to increase the output level (product mix). Meanwhile, the introduction of
the symmetric import tariﬀ improves within-sector misallocation: it decreases the numberof varieties sold in each country (product variety) by shutting out the least productive exporters (product selection), increases the output level of more productive domestic ﬁrms andexporters, and decreases the output level of less productive exporters (product mix). Thus,
the introduction of the symmetric import tariﬀ distorts cross-sector allocation and improves
within-sector allocation through product variety eﬀect. The impacts of its introduction on

20

within-sector allocation through product selection and product mix eﬀects have both improving and distorting eﬀects. As a result, the net eﬀect of the symmetric import tariﬀ on the
total welfare level is determined by the sum of these eﬀects.
As shown in Appendix 7.9, diﬀerentiating W with respect to tw yields
dW
dtw


X
dIi
dACi
dCSi
=
+
+
Li
Li
dtw
dtw
dtw
i∈{H,F }i∈{H,F }X
X= (1 − tw )Li Bi ψi (cii ) + tw
Li Bi (α − 2cii )Xi∈{H,F }−(θ+2)=i∈{H,F }θtw(1 − tw )4γ(θ + 1)(θ + 2)

X

Li bi ψi (cii ) + twi∈{H,F }XLi bi (α − 2cii ) ,i∈{H,F }(32)
k cθ+2j ii> 0. The ﬁrst term in brackets in (32) represents the total welfare eﬀect
where bi ≡ (θ+1)α−θcii

of an increase in the symmetric import tariﬀ by changes in cross-sector allocation in bothcountries. The second term in brackets in (32) represents the eﬀect from changes in withinsector allocation in both countries. In contrast to (28), the ﬁrst term in (32) is negative fortw > 1 and is zero with global free trade, tw = 1. Then, if countries start at global free trade,the second term in (32) determines whether the introduction of a small symmetric importtariﬀ is desirable. Substituting tw = 1 into (32), I obtainXdWθ
=Li bi (α − 2cFii T )dtw tw =14γ(θ + 1)(θ + 2)i∈{H,F }=θ(LH bH + LF bF )(α − 2cFw T ).
4γ(θ + 1)(θ + 2)

(33)

T
Note that from (14), cFHH
= cFF TF ≡ cFw T , because τF H = τHF = 1. The introduction ofa symmetric import tariﬀ increases the total welfare level when it improves within-sector
misallocation: when α − 2cFw T > 0. As a result, I obtain the following proposition:
Proposition 3. In the two-country economy, consider the case in which countries adopt asymmetric import tariﬀ. The eﬃcient tariﬀ tew that maximizes the total welfare level of two
countries is positive if and only if α − 2cFw T > 0. Otherwise, a global free trade policy is21

desirable:

tew =



 1+ ∑ 1∑

i∈{H,F } [Li bi (α−2cii )]θ(θ+2)Lb
Aki cii θ+1 +2(θ+1)cii +αiii∈{H,F }
θ+1

[

(

)]


α − 2cFw T > 0 ,α − 2cFw T ≤ 0(34)

where cii is endogenously determined and 1 ≤ tew < 2.
Proof. See Appendix 7.9.
Global free trade is ineﬃcient when cFw T is low: when α − cFw T > 0.17 In other words,
from Lemma 4, when there are several productive ﬁrms in both countries (large kH and kF )or varieties are close substitutes (large A). The intuition behind this result is as follows.
An increase in the symmetric import tariﬀ increases (decreases) the output level of more(less) productive exporters in both countries, which improves the within-sector misallocation
through improving product mix. Then, the greater the number of more productive ﬁrms
in both countries, the greater the number of exporters (varieties) that increase the output
level by an increase in that tariﬀ. These more productive exporters set lower price thanless productive exporters, so that an increase in that tariﬀ has a greater eﬀect on increasing
the total welfare level by improving this misallocation. Similarly, as varieties become closer
substitutes, an increase in the symmetric import tariﬀ works on increasing the total welfare
level by improving this misallocation, because households come to prefer consuming more
low-priced goods by increasing that tariﬀ than consuming more varieties by decreasing thattariﬀ.
To better understand the characteristics of the eﬃcient tariﬀ, consider the case in whichMcountries are symmetric (KH = KF , cM
H = cF , LH = LF , and thus, cHH = cF F ). Then, theeﬃcient tariﬀ (34) can be written as

tew =



 1 + θ(θ+2)

 1

θ+1

α−2ciiAki cii θ+1 +2(θ+1)cii +α17
α − 2cFw T > 0 .
FTα − 2cw ≤ 0(35)This inequality is quite similar to the condition for too much entry at the market equilibrium in Bagwelland Lee (2020), where they show that this distortion can be corrected by the introduction of a symmetrictariﬀ (see Propositions 4 and 8 in their paper). The model in the present study, which does not consider freeTentry, has similar characteristics to the model in Bagwell and Lee (2020) in that the inequality, α − cF
w > 0,
holds when there are too many potential ﬁrms in both countries, and that the introduction of a symmetric
import tariﬀ increases the total welfare level when this inequality holds.

22

The eﬃcient tariﬀ given by (35) has the same form as the second component of the Nash tariﬀ
given by (29).18 In contrast to the Nash tariﬀ, the eﬃcient tariﬀ is positive only when the
introduction of an import tariﬀ partially improves within-sector misallocation: α − 2cFw T > 0.
Comparing the Nash tariﬀs and the eﬃcient tariﬀ, I obtain the following proposition:
Proposition 4. Assume θ ≤ θ̄. If countries are symmetric, the Nash tariﬀs are higher thanthe eﬃcient tariﬀ.
Proof. See Appendix 7.9.
An increase in a country’s import tariﬀ increases its welfare mainly by increasing its
income (and thereby the output level of homogeneous good sold in the country) when theimport tariﬀ is suﬃciently small. By contrast, an increase in the country’s import tariﬀ
harms its trading partner by decreasing its income (see Proposition 2). As a result, the Nashtariﬀs are positive due to the eﬀect of increasing income in the countries, and higher thanthe eﬃcient tariﬀ because, in a global welfare perspective, this eﬀect is partially oﬀset by theeﬀect of the decreasing income in each country’s trading partner. Unlike the Nash tariﬀs, theeﬃcient tariﬀ is positive if and only if the introduction of a symmetric import tariﬀ improves
within-sector misallocation.

5

Mutual gains by symmetric tariﬀ policy

In the previous section, I show that the eﬃcient tariﬀ is positive if and only if the introductionof a symmetric import tariﬀ partially improves the within-sector misallocation: if and only
if α − 2cFw T > 0. In an asymmetric-country setting, however, there is a case in which theintroduction of the symmetric import tariﬀ decreases the welfare of a county while increasing
the welfare of the other country even when the eﬃcient tariﬀ is positive.
Thus, in this section, I analyze what kind of asymmetric countries can mutually gain by
the introduction of the symmetric import tariﬀ from the initial situation of global free trade
when the eﬃcient tariﬀ is positive. L ≡ Li /Lj and k ≡ ki /kj denote the relative population
size and the relative size of the productivity index for country i, respectively. In this section,
I assume α − 2cFw T > 0 to focus on the case in which the eﬃcient tariﬀ is positive. Since from18

Since the cost cutoﬀs are determined endogenously, the level of the eﬃcient tariﬀ (35) is diﬀerent fromthat of the second component in (29).

23

(14) the level of cFw T depends on the aggregate size of productivity index, ki + kj , I assume
that ki + kj is constant and takes a value that ensures α − 2cFw T > 0. Then, changes in therelative size k do not aﬀect the aggregate size ki + kj and thereby cFw T : dcFw T /dk = 0.
The welfare eﬀect of the introduction of a symmetric import tariﬀ in country i is given
bydLi Ui>0
dtw tw =1
2 β1 k + β2
⇔ L>k ≡ L(k),θ + 2 β3 k + α(36)where β1 ≡ α + θcFw T , β2 ≡ (θ + 1)((θ + 1)α − θcFw T ), and β3 ≡ (2θ + 1)α − 2θcFw T are positive.
Since L(k) is an increasing function with respect to k, it can be shown as depicted in Figure
1 (see Appendix 7.10).
From Figure 1, country i can gain by introducing a small symmetric import tariﬀ when
the pair of (k, L) is above L(k). In other words, the introduction of a symmetric importtariﬀ improves the welfare of a country that has a larger population or a smaller productivity
index. The intuition behind this ﬁnding is as follows. While the introduction of a symmetric
import tariﬀ increases income due to an increase in domestic proﬁts and tariﬀ revenue, it also
decreases income due to a decrease in export proﬁts in both countries. If a country has a larger
population, ﬁrms in that country earn less proﬁts from export sales than from domestic sales
because the market size in its trading partner is small. Thus, the introduction of a symmetric
import tariﬀ improves welfare of the country with larger population by an increase in income
through an increase in domestic proﬁts. If a country has a smaller productivity index, there
are fewer exporters in that country than exporters in its trading partner. In such a country,
the share of tariﬀ revenue in income is greater than that of export proﬁts, so that the
introduction of a symmetric import tariﬀ improves the welfare of the country with a smaller
productivity index by an increase in income through an increase in tariﬀ revenue.
Next, I derive the welfare eﬀect of introducing a symmetric import tariﬀ in country j. Ina similar way to (36), I obtaindLj Uj>0
dtw tw =1
θ + 2 αk + β3
k ≡ L(k).
⇔ L<
2 β2 k + β1
24

(37)

Since L(k) is an increasing function with respect to k and L(k) < L(k) for k > 0 holds (seeAppendix 7.10), L(k) can be shown as in Figure 1. Country j can gain by introducing asmall symmetric import tariﬀ when the pair of (k, L) is below L(k).
As a result, I obtain the following proposition.
Proposition 5. Assume α −2cFw T > 0. In a two-country economy, if countries start at global
free trade, the introduction of the symmetric import tariﬀ improves welfare in both countries
if and only if L(k) < L < L(k).
Proof. See appendix 7.10.
The introduction of a symmetric tariﬀ improves welfare in both countries even when
the degree of asymmetry across countries is large: one country has a larger populationand productivity index than the other. The intuition behind this result is as follows. As
mentioned above, imposing a symmetric tariﬀ increases income in a country with a larger
population owing to an increase in domestic proﬁts. By contrast, it decreases income in acountry with a larger productivity index owing to a decrease in export proﬁts. Then, if acountry has a larger population and productivity index, these opposite eﬀects on income areoﬀset.19 As a result, the introduction of a symmetric import tariﬀ increases the welfare ofthat country by improving the within-sector misallocation (because α − 2cFw T > 0 holds).
Similarly, it increases the welfare level of its trading partner by improving the within-sector
misallocation. This is because following opposite eﬀects are oﬀset: the eﬀect of imposing thesymmetric tariﬀ that decreases income in a country with a smaller population owing to adecrease in export proﬁts and the eﬀect that increases income in a country with a smaller
productivity index owing to an increase in tariﬀ revenue.

6

Conclusion

By incorporating ad valorem import tariﬀs and not considering free entry, I modify Melitz andOttaviano (2008), who develop a trade model with heterogeneous ﬁrms and variable markups.
In this model, the variable markups create within-sector misallocation: more productive ﬁrms
do not pass on their entire cost advantage to consumers by raising their markups and they
19

Speciﬁcally, changes in cross-sector allocation by the introduction of a symmetric import tariﬀ have little
impact on the welfare of that country compared to changes in within-sector allocation by the introduction ofthat tariﬀ.

25

end up selling below the eﬃcient output level, which causes less productive ﬁrms to end up
being oversupplied and the least productive ﬁrms to survive ineﬃciently. Thus, the market
level of the number of varieties is above that in the ﬁrst-best outcome. This fact implies that,even without free entry, the eﬀect that Nocco et al. (2019) discuss as the entry externality
created by endogenous markups is inherent in this model.
I examine the inﬂuence of uncooperative and cooperative import tariﬀ policies on welfare
in a two-country model. The conclusions of this study are summarized as follows. First,
the Nash tariﬀs are positive and, when countries are symmetric, lower than the eﬃcientimport tariﬀ that the two countries adopt uniformly to maximize the total welfare level ofthe countries. Second, the eﬃcient tariﬀ is positive when, starting at global free trade, theintroduction of the small symmetric import tariﬀ suﬃciently improves within-sector misallocation. This result is diﬀerent from that of Melitz and Ottaviano (2008), who consider that
trade liberalization decreases transportation costs and show that bilateral trade liberalization
leads to welfare gain for both countries. The result in the present study is also diﬀerent fromthat of Bagwell and Lee (2018), who adopt a CES preference setting that generates constantmarkups and show that the introduction of a small symmetric import tariﬀ lowers the total
welfare level of the two countries. Third, starting at global free trade, the introduction ofa small symmetric import tariﬀ improves the welfare level of both countries if and only if
the countries are close to symmetric or one country has a larger population and number ofhigh-productivity ﬁrms than the other. This result indicates that it is crucial for countries
that enter a trade agreement to determine their import tariﬀs based on their relative size.
Finally, the present study focuses on the role of import tariﬀ policies and therefore,
does not use domestic policies and other trade policies as tools for bilateral trade policy. Aninteresting question is, by combining import tariﬀs with other policy instruments, whether themarket can achieve the ﬁrst best outcome without ﬁrm-speciﬁc production subsidies/taxes.
I leave this question for consideration in future research.

26

7

Appendix

7.1

Suﬃcient condition for cii < cM
i

AU
M
AU
From (15) and (16), cii < cM
i holds if cii < ci holds. By using (14), cii is determined by

the following equation:
Aki cii θ+1 + 2(θ + 1)cii = 2(θ + 1)α.

(38)

Since the left-hand side (LHS) of (38) is an increasing function of cii , the suﬃcient condition
for cii < cM
i is
θ+1

Aki cM+ 2(θ + 1)cM
i > 2(θ + 1)α

 i
AKi
+ 1 cM
⇔i > α,2(θ + 1)θ(see Figure 2).
where ki ≡ Ki /cM
i7.2

Proof of cii > cij

cii is determined by (14). Similarly, noting that the relationship cjj = tij τij cij , cij is determined byA(kj + ki (tij τij )−θ )cjj θ+1 − 2(θ + 1)cjj = 2(θ + 1)α⇔ A ki + kj (tij τij )θ tij τij cij θ+1 + 2(θ + 1)tij τij cij = 2(θ + 1)α.
Comparing coeﬃcients of the LHS of (14) and (39),A ki + kj (tji τji )−θ < A ki + kj (tij τij )θ tij τij
2(θ + 1) < 2(θ + 1)tij τij .
Thus cii > cij (see Figure 3).

27

(39)

7.3

Product mix eﬀect

From (9), the output level of country j’s exporters with cost c isqji (c) =Li τji tji(cji − c).
2γ

(40)

The relationship between productivity and the output level of exporters can be depicted as
the dotted line in Figure 4, where the intercepts along the vertical and horizontal axes areLi τji tji
Li
cji =
cii ,
2γ
2γ

cji

respectively. Since from Lemma 1 an increase in tji increases cii and decreases cji , it shifts(40) from the dotted line to the solid line in Figure 4, which implies that an increase incountry i’s import tariﬀ causes more (less) productive exporters in country j to increase(decrease) their output level.

7.4

Proof of Lemmas 5 and 7

Income is given by (24):−(θ+1)

ki + (θ + 1)kj τji−θ (tji − 1)tji −(θ+1) θ+2 Lj ki τij−θ tijcii+
cjj θ+2 .
Ii = 1 +2γ(θ + 1)(θ + 2)
Li 2γ(θ + 1)(θ + 2)28

Diﬀerentiating both sides of this equation with respect to tji , I obtain
dIi
dtji

=

=

=

=

=


−(θ+2) θ+2
tjicii(θ + 1)kj τji−θ (−θtji + θ + 1)
2γ(θ + 1)(θ + 2)−θ
θ+2 −1 dcii−(θ+1)+ ki + (θ + 1)kj τji (tji − 1)tji(θ + 2)tji cii
dtji



−θ
−(θ+2) θ+2
θkj τjitjicii
θ+1
2(θ + 1)− tji ((θ + 1)α − θcii )4γ(θ + 1)(θ + 2) (θ + 1)α − θcii
θ


−θ
θ+1
−(θ+1) θ + 2+ ki + (θ + 1)kj τji (tji − 1)tji
Atji cii
θ+1



θ+1
Bi 2(θ + 1)
− tji ((θ + 1)α − θcii )
θ

θ+2
θ+1
−θ −θ θ+1+
tji Aki cii + (θ + 2)(tji − 1)Akj τji tji cii
(41)
θ+1



θ+1
Bi 2(θ + 1)
− tji ((θ + 1)α − θcii )
θ


θ+2
θ+1
θ+1
tji Aki cii + (θ + 2)(tji − 1) 2(θ + 1)(α − cii ) − Aki cii
+
θ+1


θ+1
θ+1
Bi (θ + 2)Aki cii
α − tji zi (cii ) ,(42)+ 2(θ + 1)cii + 2
θ
θk τ

−θ

t

−(θ+2) c θ+2

ii
j ji
ji
> 0 and zi (cii ) ≡ θ(θ+2)
where Bi ≡ 4γ(θ+1)(θ+2)
Aki cii θ+1 + 4(θ + 1)cii − 2(θ + 1)α.
(θ+1)α−θcii
θ+1

θ+1
I use the relationship Akj τji−θ t−θ= 2(θ + 1)(α − cii ) − Aki cθ+1from (14) at the fourth
ji ciiii

line. The characteristics of zi (cii ) is as follows: dciidzi (cii )= θ(θ + 2)Aki cθii + 4(θ + 1)
> 0 for tji > 0
dtji
dtji
lim zi (cii ) = zi (0) = −2(θ + 1)α < 0

tji →0AU
lim zi (cii ) = 2θ(θ + 2)(α − cAUii ) + 4(θ + 1)cii − 2(θ + 1)α


AU
> 0.
= 2 θ2 (α − cAUii ) + (θ − 1)α + 2cii

(43)
(44)

tji →∞In deriving (45), I use the relationship Aki cAU
ii

θ+1

(45)

= 2(θ + 1)(α − cAUii ) from (38). Next, I

deﬁne tzi which satisﬁes zi (cii )|tji =tz = 0. From (43) – (45) it is straightforward to show thati0 < tzi < ∞ and the sign of (42) is positive if tji ≤ tzi :dIi> 0 if tji ≤ tzi .
dtji
29

(46)

If tji > tzi , the sign of (42) is described asα(θ + 2)Aki cii θ+1 + 2(θ + 1)cii + 2 θ+1
dIi
θ⋛ 0 ⇔ tji ⋚dtji
zi (cii )
= 1+

2
(θ+2)
Aki cii θ+1 − 2(θ + 1)cii + 2 (θ+1)αθ+1θzi (cii )(θ+2)= 1 + θ+1

(47)

Aki cii θ+1 + 2(θ + 1)( θ+1
α − cii )θ(> 1),zi (cii )where the characteristics of the right hand side (RHS) of (47) are
RHS47 > 1,

lim RHS47 = ∞ for tji > tzi .

tji →tzi(48)

In addition, diﬀerentiating the RHS of (47) with respect to tji , I obtaindRHS47
dRHS47 dcii=
dtji
dcii dtji


θ+2
dcii
(θ + 1)2θ= −α<02(θ + 2)Aki cii ((θ + 1)α − θcii ) + 42zi (cii )
θ
dtji
(49)
for tji > tzi . From (46)–(49), the relationship between income and an import tariﬀ in country
i follows a hump-shaped curve (see Figures 5 and 6).
Next, I show the eﬀects of an increase in the import tariﬀ set by country j’s government
on income in country i. Diﬀerentiating (24) with respect to tij , I obtain
dIi
dtij

=
=
=
=


−θ −(θ+2) θ+2 
cjj
Lj ki τij tij−1 dcjj−(θ + 1) + (θ + 2)tij cjj
Li 2γ(θ + 1)(θ + 2)dtij−θ −(θ+2) θ+2
cjj
12(θ + 1)
θ+2
Lj θki τij tij−θ −θ θ+1−((θ + 1)α − θcjj ) +Aki τij tij cjj
Li 4γ(θ + 1)(θ + 2) (θ + 1)α − θcjj
θ
θ+1



Lj
2(θ + 1)
θ+2
θ+1
Bj −
((θ + 1)α − θcjj ) +2(θ + 1)(α − cjj ) − Akj cjj
Li
θ
θ+1


Lj 1 θ(θ + 2)
Lj 1
θ+1
− Bj
Akj cjj + 2θcjj + 2α = − Bj ψj (cjj ) < 0,(50)
Li θ θ + 1
Li θ
θk τ

−θ

t

−(θ+2) c

θ+2

i ij
ij
jj
where Bj ≡ 4γ(θ+1)(θ+2)
Akj cjj θ+1 + 2θcjj + 2α > 0.
> 0 and ψj (cjj ) ≡ θ(θ+2)(θ+1)α−θcjj
θ+1

30

7.5

Homogeneous good

Here, I show the eﬀects of an import tariﬀ on the consumption of a homogeneous good inccountry i and the suﬃcient condition for q0,i
> 0. Plugging (8), (9), and (24) into (2), I

obtainZcq0,i= Ii −Ω∗ipi (ω)qic (ω)dω= Ii −

ki + kj (τji tji )−θ θ+2
cii
2γ(θ + 2)= 1−

θki + (θ + 1)kj τji −θ tji −(θ+1) θ+2 Lj ki τij −θ tijcii +cjj θ+2 ,2γ(θ + 1)(θ + 2)
Li 2γ(θ + 1)(θ + 2)−(θ+1)

where

R
Ω∗i

c
dq0,i
dtji

pi (ω)qic (ω)dω =

=

=

=

=

=

(51)

ki +kj (τji tji )−θ θ+2
cii . Diﬀerentiating (51) with respect to tji , I obtain2γ(θ+2)
1−(θ+2) θ+2−
cii−(θ + 1)2 kj τji−θ tji
2γ(θ + 1)(θ + 2)
θ+1 dcii−θ−(θ+1)+ θki + (θ + 1)kj τji tji(θ + 2)ciidtji
−(θ+2) θ+2
kj τji−θ tji
cii−−2(θ + 1)2 ((θ + 1)α − θcii )4γ(θ + 1)(θ + 2) ((θ + 1)α − θcii )θ2 (θ + 2)
θ+1−θ −θ θ+1+
tji Aki cii + θ(θ + 2)Akj τji tji cii
θ+1
−(θ+2) θ+2
cii
kj τji−θ tji−−2(θ + 1)2 ((θ + 1)α − θcii )4γ(θ + 1)(θ + 2) ((θ + 1)α − θcii )
θ2 (θ + 2)
θ+1
θ+1
+
tji Aki cii + θ(θ + 2) 2(θ + 1)(α − cii ) − Aki cii
θ+1
−(θ+2) θ+2
kj τji−θ tji
cii
2(θ + 1) (α + θcii ) + θ(θ + 2)Aki cθ+1ii
4γ(θ + 1)(θ + 2) ((θ + 1)α − θcii )θ2 (θ + 2)−
tji Aki cθ+1
ii
θ+1


−(θ+2) 2θ+3 θ2 Aki kj τji −θ tji
cii
2(θ + 1)2 α + θcii θ + 1+− tji . (52)4γ(θ + 1)2 ((θ + 1)α − θcii )θ2 (θ + 2)Aki cθ+1
θ
ii

31

The sign of (52) is determined bycdq0,i2(θ + 1)2 α + θcii θ + 1⋛ 0 ⇔ tji ⋚ 2+(> 1),dtjiθ (θ + 2)Aki cθ+1
θ
ii

(53)

where the RHS of (53) is a decreasing function of tji :dRHS532(θ + 1)2 (θ + 1)α + θ2 cii dcii
=− 2
< 0.
dtji
θ (θ + 2)Aki
dt
cθ+2
ji
ii

(54)

From (53) and (54), the relationship between the consumption level of homogeneous goods
and the import tariﬀ in country i follows a hump-shaped curve.
Meanwhile, diﬀerentiating (51) with respect to tij , I obtain


c
dq0,i
Lj
θ+2
2
dIi
θ+1
= − Bj< 0.
Akj cjj + 2cjj + α =
dtij
Li
θ+1
θ
dtij

(55)

c
c
Next, I show the suﬃcient condition for q0,i
> 0. Since the relationship between q0,iandc
is a decreasing function ofcountry i’s import tariﬀ tji follows a hump-shaped curve and q0,ic
> 0 holds if
country j’s import tariﬀ tij , from (51) q0,i

θki + (θ + 1)kj τji −θ F T θ+2
θki
θ+2
cii
cAU
> 0 and 1 −>0ii
2γ(θ + 1)(θ + 2)2γ(θ + 1)(θ + 2)
1
! θ+2
 1

2γ(θ + 1)(θ + 2) θ+2
2γ(θ
+
1)(θ
+
2)
AU
FT
FT
≡ cii
and cii <≡ cAU⇔ cii <iiθki
θki + (θ + 1)kj τji−θ
1−c> 0 ishold. Therefore, by using (14), the suﬃcient condition for q0,iA ki + kj τji −θ F T θ+1
cii+ cFii T > α and2(θ + 1)

32

θ+1
Aki
cAU
+ cAU
ii
ii > α.
2(θ + 1)

(56)

7.6

Proof of Proposition 1 and Lemma 8

7.6.1

Proof of Proposition 1

Putting (26), (27), and (42) into (25), I obtain
dUi
dtji

dIi
dCSi dACi
+
+
dtji
dtji
dtji

θ+1
= Bi (θ + 2)Aki cii θ+1 + 2(θ + 1)cii + 2αθθ(θ + 2)
θ+1
−tji
Aki cii + 2(θ + 1)cii + α .
θ+1
=

(57)

The sign of (57) is determined byα(θ + 2)Aki cii θ+1 + 2(θ + 1)cii + 2 θ+1dUiθ⋛ 0 ⇔ tji ⋚θ(θ+2)dtji
Aki cθ+1+ 2(θ + 1)cii + α
ii
θ+1

= 1+

θ(θ+2)
Aki cii θ+1 + (θ + 2)α1
θ+1≡ Φi (cii )(> 1).
θ θ(θ+2) Aki cii θ+1 + 2(θ + 1)cii + α

(58)

θ+1

Since from (15) tji and cii have one-to-one correspondence, tji can be expressed as a functionof cii . Rearranging (14), I obtain
1/θ
1
Akj cii θ+1≡ tji (cii ).
tji =τji 2(θ + 1)(α − cii ) − Aki cii θ+1(59)

Substituting (59) into the LHS of (58), both sides of (58) are expressed as functions of cii :dUi⋛ 0 ⇔ tji (cii ) ⋚ Φi (cii ).
dtji

(60)

From (15) and (16), the characteristics of tji (cii ) are expressed as follows.
dtji (cii )
> 0,
dcii

tji (cFii T ) = 1,

33

lim tji (cii ) = ∞cii →cAU
ii(61)

Next, I establish the shape of Φi (cii ). Diﬀerentiating Φi (cii ) with respect to cii , I obtaindΦi (cii )=dciiθ(θ+2)
θ+1(θ + 2)ϕi (cii )Aki cii

θ+1 + 2(θ + 1)c

2 ,

(62)

ii + αwhere ϕi (cii ) ≡ (2θcii − (θ + 1)α) Aki cii θ − 2 θ+1
α. Then, the sign of dΦi (cii )/dcii can beθwritten asdΦi (cii )⋛ 0 ⇔ ϕi (cii ) ⋛ 0.
dcii

(63)

α
α
It is straightforward to show that ϕi (cii ) < 0 if cii < θ+1. If cii ≥ θ+1, the characteristics
θ 2
θ 2of ϕi (cii ) are as follows:

ϕi


θ+1α
θ+1
= −2
α < 0,θ 2
θ
dϕi (cii )= 2θAki cii θ + (2θcii − (θ + 1)α)θAki cii θ−1 > 0
dcii

(64)
(65)

α
for cii ≥ θ+1. I deﬁne c˜ii which satisﬁes ϕi (c˜ii ) = 0. Then, from (63) – (65), I obtain
θ 2





 ≤ 0 (cii ≤ c˜ii )dΦi (cii )  ≤ 0 (cii ≤ c˜ii )⇔ϕi (cii )dcii 
 > 0 (cii > c˜ii ) > 0 (cii > c˜ii )(66)

Taking the limits of Φi (cii ) as cii → 0 and cii → ∞ gives
lim Φi (cii ) = 2(

cii →0θ+1),θ

lim Φi (cii ) = 1 +cii →∞

1θ

(67)

respectively. From (66), (67), and Φi (cii ) > 1, I obtain1 < Φi (cii ) < 2(

θ+1
) for cii ∈ (0, ∞).
θ

(68)

Therefore, from (61) and (68), Φi (cii ) and tji (cii ) intersect in cii ∈ [cFii T , cAUii ) (see Figure7). Since cii is an increasing function of tji , there is a welfare-maximizing import tariﬀ thatsatisﬁes (58) with equality.
34

Next, I show the suﬃcient condition for the uniqueness of the solution of (29). From(61) and (66), the solution of (29) uniquely exists if Φi (cii ) is a decreasing function of cii forcii ∈ [cFii T , cAU
ii ):
cAU
ii ≤ c˜ii .

(69)

From (66), (69) holds if and only if
ϕi (cAUii ) ≤ 0


θ+1
AU θ
2θcAU
−2
α≤0
ii − (θ + 1)α Aki ciiθi2(θ + 1) h 2 AU 2
2
AU
2
2θ
c
−
(3θ
+
θ
−
1)αc
+
θ(θ
+
1)α
≥0
⇔
ii
ii
θcAU
ii

⇔

2

⇔ 2θ2 cAU
− (3θ2 + θ − 1)αcii AU + θ(θ + 1)α2 ≥ 0ii
2(3θ2 + θ − 1)2 − 8θ3 (θ + 1) 2
(3θ2 + θ − 1)α2AU⇔ 2θ cii −−α ≥ 0.
4θ2
8θ2
θ

(70)

α−cAU

In deriving (70), I use the relationship Aki cAU= 2(θ + 1) cAUii from (38). Thus, (70) holdsii
ii

for all cAU
ii if(3θ2 + θ − 1)2 − 8θ3 (θ + 1) ≤ 0
p√√1+ 5+4 2⇔ θ ≤ 2+
≡ θ̄,
2

(71)

where θ̄ ≈ 3.55. Therefore, the suﬃcient condition for the uniqueness of the solution of (29)is (71).

7.6.2

Proof of Lemma 8

I assume θ ≤ θ̄ to ensure the uniqueness of the solution of (29). As shown above, tji isdetermined at the intersection of tji (cii ) and Φi (cii ) in Figure 7. It is straightforward toshow that a decrease in τji aﬀects only tji (cii ) by shifting it up, as shown in Figure 8, which
immediately proves that the Nash tariﬀ tnji increases and the domestic cost cutoﬀ cii decreases
as τji falls. However, a decrease in τij does not aﬀect tji (cii ) and Φi (cii ) so that the Nashtariﬀ tnji does not change.
35

7.7

Proof of Lemma 97.7.1First-best outcome

To evaluate the eﬃciency of the market outcome under free trade, I consider the problem
faced by a benevolent social planner who maximizes the total welfare of the countries, taking
as given the endowment of labor (Li ) and potential ﬁrms (Ki ), the production functions of
the two goods, and the distribution of c (c ∼ Gi (c)). Then, the planner chooses the quantity
cof the homogeneous good (q0,i = Li q0,i
) and the quantity of each diﬀerentiated good producedin country i and sold in country j (qij (c) = Lj qijc ).
Accordingly, given (1), the planner’s problem is given bymax

c ,q c ,q (c),q (c),q (c),q (c)
q0,i
ij
jj
ji
0,j ii

W = Li Ui + Lj Uj ,(72)

subject to the resource constraint:
Z cM
i
q0,i + Ki

Z cMicqii (c)dGi (c) + Ki

0
Z cM
j

(73)

τji cqji (c)dGj (c) = Lj

(74)

Z cM
j
cqjj (c)dGj (c) + Kj

q0,j + Kj

τij cqij (c)dGi (c) = Li ,
0

0

0

36

for i ̸= j, where!
Z cM
j q (c)qii (c)ji
Li Ui = q0,i + αLi Ki
dGi + Kj
dGj
L
Li
i
0
0
!

2

2
Z cM
Z cM
i
j
γ
qii (c)qji (c)− Li Ki
dGi + Kj
dGj
2
Li
Li
0
0
!2
Z cM
Z cM
i
j q (c)qii (c)ηji
dGi + Kj
dGj
− L i Ki
2
Li
Li
0
0
!
Z cM
Z cM
i
j
= q0,i + α Ki
qii (c)dGi + Kj
qji (c)dGj
Z cM
i

0

−

γ
2Li

η
−
2Li

Z cM
i
Ki

0

qii (c)2 dGi + Kj

Z cM
j

0

0

Z cM
i

Z cM
j

Ki

qii (c)dGi + Kj
0!
qji (c)2 dGj!2qji (c)dGj

.

(75)

0

Substituting (73) and (74) into (72), this problem can be rewritten asmax
qii (c),qij (c),qjj (c),qji (c)W = Li Ui + Lj Uj ,

(76)

where
Li Ui = Li −

Z cM
i
Ki

cqii (c)dGi + Ki
0

η
−
2Li

!

Z cM
j
qii (c)dGi + Kj

0

γ
−
2Li

τij cqij (c)dGi
0

Z cM
i
+ α KiZ cM
iKi

qji (c)dGj
0qii (c)2 dGi + Kj

Z cM
j

0

0

Z cM
i

Z cM
j
0

0

37

!
qji (c)2 dGj!2qji (c)dGj

qii (c)dGi + Kj

Ki

!

Z cM
i

.

(77)

The ﬁrst order conditions with respect to qii (c) and qji giveLiη(α − Qi − c)γLiLi= (csii − c),γqiis (c) =Liη(α − Qi − τji c)γLiLi τji s(cji − c),
=
γ

s
qji
(c) =

(78)
(79)

(80)
(81)

respectively, where ‘s’ labels ﬁrst best optimum variables andηcsii ≡ α − Qsi ,
L
 i

η s
cs
1
s
α − Qi = ii ,
cji ≡
τji
Li
τji
!
Z cM
Z cM
j
i
s
s
qji (c) dGj .
qii (c) dGi + Kj
Qsi = Ki

(82)
(83)
(84)

0

0

csii , csji and Qsi represent domestic cost cutoﬀ in country i, the export cost cutoﬀ in countryj, and the total output level of diﬀerentiated varieties sold in country i, respectively.
Integrating (78) and (80) gives
Z csii
Ki

Li
qiis (c)dGi =γ0Z csji
Kj




η s
θ
sθ
s θ+1
ki c
(α − Qi )ki cii −,Liθ + 1 ii



η s
θ
s θ+1
s θ
kj c
(α − Qi )kj cji − τji,γLiθ + 1 jiLisqji(c)dGj =

0

(85)

(86)

θ

respectively, where ki = Ki /cM
i . Then, using (85), (86), and (83) to (84), I obtain
Qsi =

θ s
Li (ki + kj τji−θ )csii θ (α − θ+1cii )γ + η(ki + kj τji−θ )csii θ

38

(87)

Putting this into (82), the domestic cost cutoﬀ is determined by the following equation:
A(ki + kj τji−θ )csii θ+1 = (θ + 1)(α − csii ),(88)

where A = η/γ. Then, using this equation, (87) can be written as
Li
Qsi =
η

θ s
A(ki + kj τji−θ )csii θ+1 (α − θ+1cii )csii + A(ki + kj τji−θ )ciis θ+1θ
Li (θ + 1)(α − csii )(α − θ+1 csii )ηcsii + (θ + 1)(α − csii )θLi (θ + 1)(α − csii )(α − θ+1 csii )=θ sη(θ + 1)(α − θ+1
cii )

=

=

Li (α − csii )
.
η

(89)

The number of varieties in country i, Ni , is composed of domestic producers and exporters
in country j, that is
Nis = Ki Gi (csii ) + Kj Gj (csji )= (ki + kj τji−θ )csii θ .

(90)s(c) by (88), (83), (89),Thus, I obtain the ﬁrst best level of csii , csji , Qsi , Nis , qiis (c), and qji(90), (79), and (81), respectively.

7.7.2

Market outcome

Substituting tji = tij = 1 into (14), the domestic cost cutoﬀ at the market equilibrium isdetermined by
θ+1
A(ki + kj τji−θ )cm= 2(θ + 1)(α − cm
ii
ii ),

(91)

where ‘m’ labels the equilibrium values under free trade. Then, the export cost cutoﬀ incountry j is given by
cm
ji =

39

cm
ii
.
τji

(92)

From (23), (11), and (9), I obtain
Li (α − cm
ii )
,
η

(93)

θ
Nim = (ki + kj τji −θ )cm
ii ,

(94)

Qm
i =

Li m
(c − c),2γ ii
Li τji m
m
(c) =
qji
(c − c).
2γ jiqiim (c) =

7.7.3

(95)
(96)

Market failure

I examine how the free market outcome departs from the ﬁrst-best outcome. Comparing thedomestic cost cutoﬀs given by (88) and (91), it is straightforward to show that (see Figure
9)csii < cm
ii .

(97)

Then, from (83), (89), (90), (92), (93), and (94), the following relationships hold:csii
cm< ii = cm
ji ,
τji
τji
Li (α − cm
Li (α − csii )
ii )
s
Qi =
>
= Qm
i ,
η
η
csji =

θm
Nis = (ki + kj τji−θ )csii θ < (ki + kj τji −θ )cm
ii = Ni .

(98)
(99)
(100)

From (79) and (95), the gap between qiis (c) and qiim (c) evaluates toqiis (c) − qiim (c) =Li s(2c − cmii − c).
2γ ii

(101)

s
m
To conﬁrm that the sign of 2csii − cm
ii is positive, replace cii with 2cii on the LHS of (91)

to obtain
A(ki + kj τji−θ )(2csii )θ+1 = 2θ+1 (θ + 1)(α − csii ) ≡ LHS91 (2csii ),swhere I use (88). Similarly, replacing cm
ii with 2cii on the RHS of (91) gives2(θ + 1)(α − 2csii ) = 2(θ + 1)(α − csii ) − 2(θ + 1)csii ≡ RHS91 (2csii ).
40

It is straightforward to show that
LHS91 (2csii ) > RHS91 (2csii ),which implies 2csii − cm
ii > 0 (see Figure 10).
Therefore, the sign of (101) can be expressed as follows: ≥ 0 (c ∈ [0, 2cs − cm ])
ii
ii
s
m
qii (c) − qii (c).
 < 0 (c ∈ (2cs − cm , cm ])
ii

ii

(102)

ii

s
m
From (81) and (96), the gap between qji(c) and qji(c) is given by

Li τji s(2cji − cmji − c)2γ
Li s=(2c − cmii − τji c).
2γ ii

m
s
(c) − qji
(c) =
qji

In the same way as (102), I obtain ≥ 0 (τji c ∈ [0, 2cs − cm ])
ii
ii
s
m
.
qji (c) − qji (c) < 0 (τ c ∈ (2cs − cm , cm ])
ji

ii

ii

(103)

ii

From (97)–(100), (102), and (103), I obtain Lemma 9.

7.8

Proof of Lemma 10

Consider the case of tji = tij = tw . Since from (14) tij dose not aﬀect cii , totally diﬀerentiating(14) gives
dciiθAkj τji −θ tw −(θ+1) cii θ+2
dcii
=
=
> 0.
dtw
dtji
2(θ + 1) (θ + 1)α − θcii

(104)

Then, in the same way as Lemma 1, I obtain
dcji
dcji
=
< 0.
dtw
dtji

41

(105)

From (12), (23) and (104), it is straightforward to show that
dNi
< 0,
dtw

dQi
< 0.
dtw

(106)

From (9), diﬀerentiating qii (c) and qji (c) with respect to tw gives
dqii (c)Li dcii=> 0,dtw
2γ dtwdqji (c)
Li
=
dtw
2γ






>0

(0 ≤ c <(107)
1 dcii)τji dtwdcii− τji c,1 dcii

dtw

<0 (
< c ≤ cji )τji dtw(108)

respectively (see Appendix 7.3).
From (104)–(108), I obtain Lemma 10.

7.9

Proof of Propositions 3 and 4

7.9.1

Proof of Proposition 3

Diﬀerentiating W = LH UH + LF UF with respect to tw , I obtain


X
X
dIi
dACi
dCSi
dW
=
Li
+
.
+
Li
dtw
dtw
dtw
dtw
i∈{H,F }(109)i∈{H,F }Since CSi and ACi depend only on tji from Lemmas 6 and 7, I obtain
dCSi= Bi tw (α − 2cii )
dtw


dACi
= Bi −2tw (θ + 2)(α − cii )
dtw

42

(110)
(111)

From (24), Ii is divided into terms that depend on tji and tij , respectively. Then, from (42)and (50), the derivative of Ii with respect to tw is
dIi
dtw


θ+1
= Bi (θ + 2)Aki cii θ+1 + 2(θ + 1)cii + 2αθθ(θ + 2)
θ+1
−tw
Aki cii+ 4(θ + 1)cii − 2(θ + 1)αθ+1
Lj 1− Bj ψj (cjj ),
Li θ

(112)

Akj cjj θ+1 + 2θcjj + 2α > 0. Using (111) and (112), I obtain
where ψj (cjj ) = θ(θ+2)
θ+1

Li

dACi
dIi
+
dtw
dtw




θ+1
= Li Bi (θ + 2)Aki cii θ+1 + 2(θ + 1)cii + 2αθθ(θ + 2)
θ+1
Aki cii+ 4(θ + 1)cii − 2(θ + 1)α
−tw
θ+1
1
−Lj Bj ψj (cjj ) − Li Bi (2tw (θ + 2)(α − cii ))

θ
1
θ+1
− tw ψi (cii ) − Lj Bj ψj (cjj )= Li Biθθ(113)

Thus, the welfare eﬀects of tw through changes in cross-sector allocation in both countries
are given by
X
i∈{H,F }


Li

dACi
dIi
+
dtw
dtw




1
θ+1
− tw ψH (cHH ) − LF BF ψF (cF F )= LH BH
θ
θ


θ+1
1
+LF BF− tw ψF (cF F ) − LH BH ψH (cHH )θθ= LH BH (1 − tw )ψH (cHH ) + LF BF (1 − tw )ψF (cF F )XLi Bi ψi (cii )= (1 − tw )(114)
i∈{H,F }43

Substituting (110) and (113) into (109), I obtain
dW
dtw


X
dIi
dACi
dCSi
=
+
+
Li
Li
dtw
dtw
dtw
i∈{H,F }i∈{H,F }X
X= (1 − tw )Li Bi ψi (cii ) + tw
Li Bi (α − 2cii )Xi∈{H,F }=i∈{H,F }−(θ+2)
θtw

4γ(θ + 1)(θ + 2)X(1 − tw )X

Li bi ψi (cii ) + twi∈{H,F }
Li bi (α − 2cii ) ,i∈{H,F }(115)
k cθ+2

j ii
where bi ≡ (θ+1)α−θc> 0. Substituting tw = 1 into (115), I obtain
ii

θ
dW
=
dtw tw =1

P

FTi∈{H,F } Li bi (α − 2cii )4γ(θ + 1)(θ + 2)

.

(116)

T
Note that from (14) cFHH
= cFF TF ≡ cFw T because τji = τij = 1. Then I can rewrite (116) as

follows:dW
θ(bH LH + bF LF )(α − 2cFw T ).
=dtw tw =14γ(θ + 1)(θ + 2)

(117)

If α − 2cFw T ≤ 0, it is straightforward to show that α − 2cHH ≤ 0 and α − 2cF F ≤ 0
for tw ∈ [1, ∞) because the domestic cutoﬀs are increasing functions of tw . Therefore, if
α − 2cFw T ≤ 0, then dW/dtw ≤ 0 for tw ∈ [1, ∞):dW< 0 ifdtw

α − 2cFw T ≤ 0.

(118)

If α − 2cFw T > 0, (117) is positive. In addition, rearranging (115), I obtain−(θ+2)Xθtw
θ(θ + 2)dW
θ+1=
Aki cii + 2(θ − tw (θ + 1))cii + (2 − tw )α ,Li bi (1 − tw )dtw
4γ(θ + 1)(θ + 2)θ+1i∈{H,F }(119)
where dW /dtw < 0 for tw ≥ 2.

44

Therefore, if α − 2cFw T > 0, there is an eﬃcient tariﬀ, tew that maximizes the total welfare
level of the two countries. From (119), the eﬃcient tariﬀ satisﬁes
θ+1e θ(θ + 2)ee
Li bi (1 − tw )Aki cii + 2(θ − tw (θ + 1))cii + (2 − tw )α = 0θ+1i∈{H,F }hiPθ(θ+2)
θ+1
L
b
Ak
c
+
2θc
+
2α
i i
i ii
ii
i∈{H,F }θ+1hi⇔ tew = Pθ(θ+2)
θ+1
Aki cii+ 2(θ + 1)cii + αi∈{H,F } Li bi
θ+1P[Li bi (α − 2cii )]hi .
i∈{H,F }⇔ tew = 1 + P(120)
θ(θ+2)
θ+1 + 2(θ + 1)c + α
L
b
Ak
c
i i
i ii
ii
i∈{H,F }
θ+1

dW
=0 ⇔
dtw

7.10

X

Proof of Proposition 5

From (110)–(112), country i’s welfare eﬀect of a symmetric import tariﬀ can be written as
dLi Ui
dtw




dIi
dCSi dACi
= Li
+
+
dtw
dtw
dtw


−(θ+2)
θtw
θ+1=
Li bi (θ + 2)Aki cθ+1α+ 2(θ + 1)cii + 2
ii
4γ(θ + 1)(θ + 2)θθ(θ + 2)1
θ+1
−tw
Aki cii + 2(θ + 1)cii + α− Lj bj ψj (cjj ) ,
θ+1
θ

45

(121)

k cθ+2

j ii
where bi ≡ (θ+1)α−θc> 0 and ψj (cjj ) ≡ θ(θ+2)
Akj cjj θ+1 +2θcjj +2α > 0. Substituting tw = 1
θ+1
ii

into (121), I obtain

θ+2
θ+1
θ
cFw T
AcFw T(Li − Lj )(θ + 2)ki kj
4γ(θ + 1)(θ + 2) (θ + 1)α − θcFw T
θ+1


θ+2
2
+Li kjα − Lj ki 2cFw T + α
θ
θ

θ+2
θ
Li
Lj ki cFw T
kj=
2( − 1)(θ + 2)
(α − cFw T )F
T
4γ(θ + 1)(θ + 2) (θ + 1)α − θcw
Ljki + kj
2
Li kj θ + 2
FT
α − 2cw − α+
Lj ki θθ
θ+2
θLj ki cFw T
α − cFw T
=
2(L
−
1)(θ
+
2)
4γ(θ + 1)(θ + 2) (θ + 1)α − θcFw T
k+1

Lθ+2
2
+
α − 2cFw T − α
k θ
θ

θ+2
1
θ
Lj ki cFw T=
2(L − 1)θ(θ + 2)(α − cFw T )k4γ(θ + 1)(θ + 2) (θ + 1)α − θcFw T θk(k + 1)

FT
(122)
+L(θ + 2)α(k + 1) − 2θcw k(k + 1) − 2k(k + 1)α ,dLi Ui=dtw tw =1

F T θ+1w
where L ≡ Li /Lj and k ≡ ki /kj . I use the relationship Acθ+1

FT

w )from (14) at the= 2(α−cki +kjsecond line. The sign of (122) is positive if the following inequality holds:
dLi Ui>0
dtw tw =1⇔ 2(L − 1)θ(θ + 2)(α − cFw T )k + L(θ + 2)α(k + 1) − 2θcFw T k(k + 1) − 2k(k + 1)α > 0⇔ (θ + 2) (2θ + 1)α − 2θcFw T k + α L > 2(α + θcFw T )k 2 + 2(θ + 1)((θ + 1)α − θcFw T )k⇔ L>

(α + θcFw T )k + (θ + 1)((θ + 1)α − θcFw T ) 2k
2 β1 k + β2=
k ≡ L(k),((2θ + 1)α − 2θcFw T ) k + α
θ+2θ + 2 β3 k + α(123)

where β1 ≡ α + θcFw T , β2 ≡ (θ + 1)((θ + 1)α − θcFw T ), and β3 ≡ (2θ + 1)α − 2θcFw T are positive.
In the same way, the introduction of a small symmetric import tariﬀ increases the welfare46

of country j if the following inequality holds:
dLj Uj>0
dtw tw =1
θ + 2 αk + β3
⇔ L<k ≡ L(k).
2 β2 k + β1

(124)

I assume that ki + kj is constant and takes a value that ensures α − 2cFw T > 0. Then,changes in k do not aﬀect ki + kj and thereby cFw T : dcFw T /dk = 0. Diﬀerentiating L(k) andL(k) with respect to k, I obtaindL(k)2 β1 β3 k 2 + 2αβ1 k + αβ2
=
>0
dk
θ+2
(β3 k + α)2

(125)

dL(k)
θ + 2 αβ2 k 2 + 2αβ1 k + β1 β3
> 0.
=
dk
2
(β2 k + β1 )2(126)

The diﬀerence between L(k) and L(k) isθ (2θ2 + 5θ + 4)α − 2θ(θ + 1)cFw T (α − 2cFw T )L(k) − L(k) =(k + 1)2 k > 0.
2(θ + 2)(β2 k + β1 )(β3 k + α)(127)Thus, L(k) > L(k) for k > 0. From (123)–(127), I obtaindLi Ui> 0 anddtw tw =1

dLj Uj> 0 ⇔ L(k) < L < L(k).
dtw tw =1(128)

Finally, substituting k = 1 into L(k) and L(k), I obtain2 β1 + β2
θ(α − 2cFw T )=1−<1
θ + 2 β3 + α(θ + 2) ((θ + 1)α − θcFw T )θ + 2 α + β3
θ(α − 2cFw T )L(1) ==1+ 2> 1.
2 β2 + β1(θ + 2θ + 2)α − θ2 cFw T

L(1) =

Using (123)–(130), I depict Figure 1.

47

(129)
(130)

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49

𝐴𝑘𝑖 𝑐𝑖𝑖𝜃+1 + 2 𝜃 + 1 𝑐𝑖𝑖

𝐿ⅆ𝐿𝑖 𝑈𝑖ቤ>0
𝑑𝑡𝑤 𝑡 =1
𝑤

ⅆ𝐿𝑗 𝑈𝑗ቤ<0
𝑑𝑡𝑤 𝑡 =1

𝐿ത (𝑘)𝑤

2 𝜃+1 𝛼ⅆ𝐿𝑖 𝑈𝑖ቤ>0
𝑑𝑡𝑤 𝑡 =1
𝑤

ⅆ𝐿𝑗 𝑈𝑗ቤ>0
𝑑𝑡𝑤 𝑡 =1𝑤

1

𝐿(𝑘)ⅆ𝐿𝑖 𝑈𝑖ቤ<0
𝑑𝑡𝑤 𝑡 =1
𝑤
ⅆ𝐿𝑗 𝑈𝑗
ቤ
>0
𝑑𝑡
𝑤

𝑡𝑤 =1

𝑘

1

0

0

Figure 1: Welfare eﬀect of the introduction ofa small symmetric import tariﬀ

𝑐𝑖𝑖𝐴𝑈 𝑐𝑖𝑀

𝑐𝑖𝑖

Figure 2: Suﬃcient condition for cii < cMi𝑞𝑗𝑖 (𝑐)
𝐿𝐻𝑆39

𝐿𝐻𝑆14

2 𝜃+1 𝛼

0

𝐿𝑖
𝑐
2𝛾 𝑖𝑖

𝑐𝑖𝑗

𝑐𝑖𝑖

0

𝑐𝑖𝑖 , 𝑐𝑖𝑗

Figure 3: Proof of cij < cii𝑐𝑗𝑖𝑐Figure 4: Eﬀects of an increase in tji on qji (c)50𝐿𝐻𝑆47

𝐿𝐻𝑆47

𝑅𝐻𝑆47

𝑅𝐻𝑆47

1

1

0

𝑡𝑧𝑖

1

𝜕𝐼𝑖
>0
𝜕𝑡𝑗𝑖

𝑡𝑗𝑖

0

Figure 5: Proof of Lemma 3 when tzi ≤ 1

2

𝜃+1
𝜃

Φ𝑖 (𝑐𝑖𝑖 )

𝑡𝑧𝑖 1

𝑡𝑗𝑖
𝜕𝐼𝑖
>0
𝜕𝑡𝑗𝑖

𝜕𝐼𝑖
<0
𝜕𝑡𝑗𝑖

𝜕𝐼𝑖
<0
𝜕𝑡𝑗𝑖

Figure 6: Proof of Lemma 3 when tzi > 1

𝑡𝑗𝑖 (𝑐𝑖𝑖 )
2

𝜃+1
𝜃

Φ𝑖 (𝑐𝑖𝑖 )𝑡𝑗𝑖 (𝑐𝑖𝑖 )

𝑡𝑗𝑖𝑛
𝑡𝑗𝑖𝑛

1

1

0

𝑐𝑖𝑖𝑛

𝑐𝑖𝑖𝐹𝑇
𝜕𝑈𝑖
>0
𝜕𝑡𝑗𝑖

𝑐𝑖𝑖𝐴𝑈

𝑐𝑖𝑖

𝜕𝑈𝑖
<0
𝜕𝑡𝑗𝑖

0

Figure 7: Proof of Proposition 1

𝑐𝑖𝑖𝐹𝑇

𝑐𝑖𝑖𝑛

𝑐𝑖𝑖𝐴𝑈

𝑐𝑖𝑖

Figure 8: Proof of Lemma 8

51

𝐿𝐻𝑆88 (= 𝐿𝐻𝑆91 )𝐿𝐻𝑆91

2 𝜃+1 𝛼2 𝜃+1 𝛼

𝑅𝐻𝑆91

𝑅𝐻𝑆91

𝜃+1 𝛼

0

𝑅𝐻𝑆88

𝑐𝑖𝑖𝑠 𝑐𝑖𝑖𝑚

𝛼

0

𝑐𝑖𝑖

Figure 9: Proof of Lemma 9

𝑐𝑖𝑖𝑚

2𝑐𝑖𝑖𝑠

𝑐𝑖𝑖

Figure 10: Proof of Lemma 952