Technology, Competition, and Multi-Product Firms∗
Akira Sasahara†
Hitotsubashi University
Very preliminary and incomplete
December 1, 2011

Abstract
This paper examines the impacts of trade on ﬁrms’ behavior and welfare by introducing multi-product ﬁrms
into Krugman’s (1979) homogeneous ﬁrm model. In a world consisting of two otherwise identical countries
diﬀering only in technology level of their ﬁrms, the impact of bilateral and unilateral trade liberalizationis considered. It is shown that, in a high-tech country, bilateral trade liberalization induces ﬁrms to dropless eﬃcient products from the domestic market and start exporting most eﬃcient products. On theother hand, in a low-tech country, it is suggested that ﬁrms’ within-ﬁrm-adjustment of product mix is lessﬂexible. Moreover, it is shown that unilateral trade liberalization hurts the liberalizing country but beneﬁts
its trading partner. However, it is suggested that the result is not valid once heterogeneity across ﬁrms is
introduced.

Key Words; Multi-product ﬁrms; Technology; Trade liberalization; New economic geography
JEL codes: F12, F13, F15, L11∗ The author is deeply grateful to Prof. Jota Ishikawa for continuous guidance and encouragement. Thanks are also given
to seminar participants at Hitotsubashi University for helpful comments and suggestions. Financial support from the Ministryof Education, Culture, Sports, Science and Technology of Japan under the Global Center of Excellence Project is gratefully
acknowledged. Of course, all remaining errors are the author’s responsibility.
† Corresponding Address: Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, Japan 186-8601.
e-mail: akirasasahara@gmail.com

1

Introduction

A considerable development in international theory during last twenty years has added microeconomic foundation on the theory of international trade. Krugman (1979, 1980, 1991)
illustrates trade between nations that share same factor endowment and technology by incorporating increasing returns to scale. Melitz’s (2003) seminal work make it possible to explain
how aggregate trade ﬂow constitutes of exports by individual ﬁrms. Besides, a recent growing
body of literature in trade theory focuses on how aggregate export of each ﬁrm constitutes ofexports of individual products within ﬁrms.1 They incorporate multi-product ﬁrms and examine how tougher competition due to trade aﬀects ﬁrms’ optimal product-scope, the amount of
production and exports of individual products within ﬁrms. However, due to the complexityof the model on multi-product ﬁrms, most of the papers fail to capture how a bilateral or unilateral reduction of trade costs between asymmetric countries aﬀect individual ﬁrms’ behavior
and welfare.
This paper succeeds in introducing multi-product ﬁrms into a model of new economic geography and examines how progressive trade liberalization aﬀect ﬁrms’ product mix, location
decision, price index, and welfare. Most of literature on multi-product ﬁrms introduces twodimensional ﬁrm heterogeneity into their model. For example, Bernard et al. (2010) embed
heterogeneous ﬁrm’ ability and product attributes, and numerous models incorporate diﬀerent
marginal costs across products within ﬁrms and ﬁrm heterogeneity (e.g., Nocke and Yeaple,
2008; Arkolakis and Muendler, 2009; Mayer et al., 2010). However, they fail to capture themechanism that the interaction of asymmetric countries induced by falling trade costs aﬀects
the behavior of multi-product ﬁrms. On the other hand, a simple homogeneous ﬁrm model
developed in this paper makes it possible to analyze the impacts of the asymmetric trade liberalization between asymmetric countries.
Related LiteratureThis paper contributes to three strands of literature. First, it relates to existing theoreticalresearch on multi-product ﬁrms in international trade literature. Bernard et al. (2010), Eckel
and Neary (2010) show that trade liberalization induces ﬁrms to drop less productive products
and makes them ‘leaner and meager.’ These work emphasis new gains from trade that is
attained by reallocating resources from less productive products to productive products withinﬁrms. Regarding the within-ﬁrm product selection, Arkolakis and Muendler (2009) and Mayer
et al. (2008) examines the impact of competition in the export markets on the product mix andshow that only the most productive products are exported to competitive markets. Chatterjee
et al. (2011) examines the eﬀect of exchange rate shocks on the markups of products producedby multi-product ﬁrms. And Feenstra and Ma (2010), and Agur (2010) theoretically suggests
variety proliferation due to trade by introducing multi-product ﬁrms even though the result iscannot obtained in a single-product ﬁrm models.2
1 Eckel and Neary (2010) call the within-ﬁrm adjustments in the range of goods produced by multi-product ﬁrms as ”intra-ﬁrm

extensive margin.”
2 Baldwin and Forslid (2010) and Arkolakis et al. (2009) show that the mass of available variety for a consumer decreases due
to trade when ﬁxed export cost is greater than the ﬁxed cost for domestic operation.

1

Second, this work contributes to a literature on new economic geography. Ottaviano (2011)
argues that future research on new economic geography should look more deeply into ﬁner
micro-heterogeneity across people and ﬁrms. Heterogeneous ﬁrms are introduced in new economic geography models by Baldwin and Okubo (2006, 2009), Okubo et al. (2008), and Okubo(2009). Even though this paper introduces homogeneous ﬁrms, they produce a continuum ofvertically diﬀerentiated products. Therefore, in the sense that each ﬁrm produces heterogeneous products, this paper oﬀers an alternative way to add heterogeneity across products to a
new economic geography model.
Finally, this paper relates to a literature on the role of technology in international trade.
The traditional Ricardian model explained how a technology diﬀerence causes trade between
nations. Lately, Eaton and Kortum (2002) embed technology diﬀerence across countries andinvestigate the impact of trade on welfare and labor reallocation. Furusawa (2011) identiﬁes
a ﬂying geese type technology development using a North-South trade model. The model
developed in this paper makes it possible to introduce technology diﬀerence across countries,and examine how ﬁrms in high-tech and low-tech countries respond to progressive trade liberalization.
Model description and results
The model embodies a representative consumer that has a CES preference and two sectors:
the monopolistically competitive diﬀerentiated goods sector and the perfectly competitive homogeneous goods sector. Contrary to the traditional models such as Krugman (1980), eachﬁrm produces a mass of vertically diﬀerentiated products. The mass of products produced byeach ﬁrm is aﬀected by a degree of competition that is represented in the price index. The
tougher competition, a decrease in price index, reduces the mass of products produced by eachﬁrm.
In a world consisting two symmetric countries, it is shown that a bilateral reduction of trade
costs decreases the CES price index in both countries since it bring a tougher competition. The
ﬁerce competition induces ﬁrms in each country continuously drop less eﬃcient products fromthe domestic market and start exporting most eﬃcient products to other country. Moreover,welfare that is deﬁned by the inverse of CES price index increases due to the bilateral reductionof trade costs.
On the other hand, in a world consisting two countries diﬀering only the technology levelof their ﬁrms, it is shown that a bilateral reduction of trade costs continuously decreases
the CES price index in the high-tech country, which induces ﬁrms in the high-tech country
to continuously drop less eﬃcient products from the domestic market and continuously add
most eﬃcient products as export products. However, interestingly, the behavior of low-tech
country’s variables is non-monotonic if the technology gap is suﬃciently large. In the low-tech
country, the CES price index (welfare) ﬁrstly decreases (increases) because of a reduction of
trade cost, but after further liberalization, it increases (decreases). This non-monotonicityof the price index bring a non-monotonic behavior of ﬁrms in the low-tech country: ﬁrstly,
they drop less eﬃcient domestic products and start exporting most eﬃcient products, but
2

after further liberalization, they start adding and dropping previously dropped domestic andpreviously added exporting products, respectively. These results can explain existing empiricalevidences that ﬁrms in U.S. and Japan (that are high-tech countries) signiﬁcantly reduces thenumber of domestic products after the tariﬀ reduction (see Bernard et al., 2010; Kawakami
and Miyagawa, 2010), and ﬁrms in India (that would be a low-tech country) do not reduce thenumber of domestic products after the reduction of tariﬀ at statistically signiﬁcant level (seeGoldberg et al., 2010).3
In addition, surprisingly, it is shown that mass of operating ﬁrms in each country does notdepend on trade cost. Therefore, in equilibrium, the mass of ﬁrms doesn’t change due to tradeliberalization. All the eﬀects that aﬀect the mass of ﬁrms are completely absorbed by thewithin-ﬁrm-adjustment of product mix.
The tractableness of the model makes it possible to examine the impact of a unilateral reduction of trade cost on ﬁrms’ product mix and welfare. One might think that a unilateral
reduction of trade costs by a country bring a tougher competition to the country that unilaterally reduce its trade cost. However, it is not the case in this general equilibrium model.
Suppose that country 1 reduces its trade cost but country 2 does not. The unilateral reductionof trade cost by country 1 make country 2 better export base, which oﬀer an incentive ﬁrms toagglomerate country 2. As a result of tougher competition induced by agglomeration, country
2 gains from trade. Firms in country 2 drop their domestic products and increase the mass ofexported products. Firms in country 1 add domestic products and reduce mass of exported
products. The unilateral reduction of trade cost reduces welfare of the liberalizing country but
its trading partner experiences welfare gain. This result is similar to previous work such as
Melitz and Ottaviano (2008).
These eﬀects of a unilateral reduction of trade cost are crucially depends on the settingthat ﬁrms can instantly enter to, and exit from the market. A simple way to introduce thefriction that makes ﬁrms to be slow to move across countries is making ﬁrms heterogeneous inproductivity. In heterogeneous ﬁrm model, ﬁrms do not have an incentive move across countries
because they earn strictly positive proﬁts contrary to the homogeneous ﬁrm model that eachﬁrm earn zero proﬁt. This simple extension of Krugman-type multi-product ﬁrm model into
Melitz-type multi-product ﬁrm model gives diﬀerent results. In this heterogeneous ﬁrm model,
it is shown that unilateral trade liberalization beneﬁts both countries, and the increase in
welfare in the liberalizing country is greater than that in its trading partner. Therefore,
this paper may contributes to a literature in the line of Chaney (2008) that contrasts thediﬀerence between Krugman-model and Melitz-model with regard to the impact of elasticity
of substitution on the trade ﬂow.
The remainder of the paper is organized as follows. Next section describes the basic structureof the model. Section 3 derives equilibrium solutions of the closed-economy model. Section
4 extends the model into open-economy model and the comparative statics are exercised.
3 Though Iacovone and Javorcik (2010) shows that ﬁrms in Mexico, possibly a low-tech country, signiﬁcantly reduces their
number of domestic products after trade liberalization, it is important that this model points out the possibility of productproliferation in a low-tech country if the technology gap between nations is suﬃciently large.

3

Section 5 introduces heterogeneous ﬁrms and contrasts the results with homogeneous ﬁrm
model. Section 6 concludes.

2

The Model

Consider an economy that produces homogeneous goods and diﬀerentiated goods with only
labor. The market of the homogeneous goods is perfectly competitive and that of diﬀerentiatedgoods is monopolistically competitive. One unit of a homogeneous good is produced by one
unit of labor, which pin down the wage rate. The wage rate is chosen as numéraire: w = 1.
The model does not need homogeneous goods sector in a closed and an open-economy consistsof symmetric countries. However, the model of asymmetric countries discussed later needs
homogeneous goods sector in order to satisfy the balance of trade condition by assuming thathomogeneous goods are freely traded.
Preference
The representative consumer has a two tier utility function:,
U = uµ x1−µ0where x0 is the consumption of homogeneous goods and
µZ

u=

x(ω)

σ−1
σ

¶ σ

σ−1

dω

,

σ>1

ω∈Ω

is the utility from the consumption of diﬀerentiated products where x(ω) is the consumption ofa variety ω of a diﬀerentiated good; Ω is the mass of available varieties; and σ is the elasticity
of substitution. The utility maximization problem subject to budget constraint gives us thedemand for a variety ω as:p(ω)−σ µLx(ω) =,
P 1−σwhere P is the CES price index dual to the utility function deﬁned as follows:
µZ

¶ 1
1−σ

P =

p(ω)

dω

1−σ

.

ω∈Ω

Production
Each ﬁrm produces a mass of products, h. The marginal cost of the production of a productω is: 1/λ(ω) where λ(ω) is the eﬃciency of the product ω. Therefore, the proﬁt maximizing
ﬁrms set their price of product λ(ω) as:p(ω) =1,ρλ(ω)

where ρ relates to elasticity of substitution σ as following: σ = 1/(1 − ρ). The larger λ(ω)
enables ﬁrms to produce the product with lower cost, which enables them to set lower price.
4

As products with the same eﬃciency behave symmetrically, we index products from now onby λ alone. The eﬃciency of a product, λ is distributed within ﬁrms according to Pareto
distribution:µ ¶θ
TG(λ) = 1 −,λwhere T and θ are the technology parameters that are to be asymmetric across countries in
the later section. T captures the minimum eﬃciency of products within ﬁrms. Therefore, if aﬁrm have higher T , the ﬁrm has superior technology.4 The introduction of the vertical productdiﬀerentiation within ﬁrms is motivated by existing empirical studies. For example, Iacovone
and Javorcik (2010) ﬁnd an intense product churning within ﬁrms due to trade, suggesting theexistence of within-ﬁrm product heterogeneity.5 For simplicity, it is assumed that all ﬁrms ina country share same technology.
Given the marginal cost and the optimal price, the revenue of producing product ω is writtenas:µL
r(λ) =.
(ρλP )1−σ
Now the proﬁt function of each ﬁrm can be written as:
π=

Z ∞"
λ∗

#

µL
− f dG(λ) − F,σ(ρλP )1−σ

where f is ﬁxed cost for the production of each product and F is the ﬁxed cost for operation.
Each ﬁrm produces a product if its proﬁtability is greater than the ﬁxed cost, f . The proﬁt
is increasing in the price index P , which corresponds to an inverse measure of the degree ofcompetition in a market. Product-cutoﬀ of the product λ is given by setting the inside of thebracket zero:Ã! 1
1 µL 1−σ
∗
λ =
.
(1)
ρP σf
The product-cutoﬀs are decreasing in P , which means that if the market is competitive (Pis lower), each ﬁrms produces fewer products. This mechanism is crucial in deriving main
results in this paper. Once λ∗ is determined, the mass of products produced by each ﬁrm isdetermined by:h ≡ 1 − G(λ∗ ).
The production technology is very similar to that of Bernard et al. (2010). However, theyinterpret λ as the preference of consumers which is exposed to idiosyncratic shock from theviewpoint of ﬁrms. Contrary to their model, λ is interpreted as product eﬃciency which isperfectly known by ﬁrms. Therefore, proﬁt maximizing ﬁrms choose to produce products withhigher λ that give them non-negative proﬁt. In the sense, this model embed ‘core eﬃciency’that implies the existence of product ladders: ﬁrms are more eﬃcient in the production of
products near to the core eﬃciency and less eﬃcient in the production of products far from
4 Arkolakis (2010) introduces technology diﬀerence across countries using the exactly same way.

Eaton and Kortum (2002)
embed technology of a country by a similar way. The latter assumes that each country’s eﬃciency is distributed following Fréchet
distribution function.
5 See also, Baldwin and Gu (2009) and Bernard et al. (2010) for similar empirical results.

5their core eﬃciency as in Mayer et al. (2009), Arkolakis and Muendler (2009), and Eckel and
Neary (2010).6

3

Closed economy

This section characterize the equilibrium of a closed-economy model. The model has three
unknown variables, the mass of ﬁrms, M , the CES price index, P , and the product-cutoﬀ ofproducts, λ∗ . Therefore, we need three equilibrium conditions. Equation (1) is the one of threeequilibrium conditions. The second condition is free entry condition that set the proﬁt of eachﬁrm to zero. Applying Pareto distribution to the above proﬁt function and letting it zero, freeentry condition is written as:π=

T θ γLλ∗−θ+σ−1 − f T θ λ∗−θ − F = 0,σ(ρP )1−σ(2)θ. The last equilibrium condition is labor market clearing condition that
where γ ≡ θ−σ+1
equates labor demand with labor supply L:








 σ − 1 T θ γL ∗−θ+σ−1
θ ∗−θ

M
λ
+
fT λ
+ |{z}F 
 = µL,1−σ| {z } σ (ρP )
ﬁxed cost|{z}ﬁxed costs of products(3)

labor demand for production

where inside of the bracket represents labor demand from each ﬁrms; the ﬁrst term in thebracket is the labor demand for production,7 the second term is the labor demand for the ﬁxed
cost for products, and last term is the ﬁxed operation cost. The labor demand from each ﬁrms
multiplied by the mass of operating ﬁrms must equal to total labor supply, L.
There are three unknowns, M , P , and λ∗ and three equilibrium conditions. Therefore, themodel is solvable. An unique set of equilibrium values is summarized as follows:
M =µL

F σγ ,
γ−1

1
P =
ρT

Ã

(4)
µL
σf

! 1 Ã
1−σ

"

λ

∗

#1

F 1
f γ−1

θ
f
= T
(γ − 1) .
F

!1
θ

,

(5)
(6)

The mass of operating ﬁrms, M does not depend on the technology parameter T . The intuition
behind the result is following. The higher T increase the proﬁt of each ﬁrms, which decreases
the available labor force and decreases the mass of ﬁrms. On the other hand, the higher T
6 Mayer et al. (2010) pin down the optimal range of production by assuming each ﬁrm faces additional production cost for a newvariety. In the paper by Arkolakis and Muendler (2009), incremental local entry cost and costs from declining eﬃciency determines
the optimal product scope. Eckel and Neary (2010) assume that marginal cost varies across products and the combination of the
product ladder and the cannibalization eﬀect determines the optimal range of product scope.
7 Let the revenue and the cost for production of each ﬁrm as r and l, respectively. From the basic property of monopolisticcompetition models, it follows that:rr−l = .
σ
Therefore, the labor demand for production can be written as l = σ−1
r because wage rate is one.
σ

6

decrease the proﬁt of each ﬁrms because the higher T make the market more competitive (seethe equilibrium price index). This increases the available resource and increases mass of ﬁrms.
These oppositely working forces are in balance in this model.
Technology parameter T is appeared in P and λ∗ . Because the higher T make each ﬁrms
more eﬃcient in production, it make the market more competitive. Therefore, the equilibrium
price index has negative correlation with T . This eﬀect of T on P is also appeared in theequilibrium product-cutoﬀ, λ∗ . The competitiveness of the market because of the higher T ,make the product-cutoﬀ higher, then reducing the mass of products produced by each ﬁrms.
The results establish following proposition on the relationship between market sides, Li and(Mi , Pi , λ∗ii and λ∗ij ).
Proposition 1: The larger country has the larger mass of ﬁrms and the lower price index,
but the market size does not aﬀect mass of products produced by each ﬁrm.
Proof. It immediately follows, from (4), (5) and (6) that ∂M/∂L > 0, ∂P/∂L < 0, and∂λ∗ /∂L = 0, respectively. ¤In the next section, the model is extended to the two country-open-economy model. The
equilibrium values obtained here are compared with those in the open-economy model.

4Open economy

Consider the world consisting two countries indexed by i = 1, 2 and populated by Li identicalhouseholds, each of which has a unit of labor supplied inelastically and immobile across countries. Suppose that the homogeneous goods are freely traded, which equates wage rates in two
countries. Firms in each country are allowed to export their products to the other country.
Firms from i have to pay additional iceberg trade cost τij to sell one unit of a product inmarket j. The iceberg trade costs are τij ≥ 1 for i 6= j and assume that τii = 1 for i = 1, 2.
Similarly, as the ﬁxed cost for exporting, each products require ﬁrms in i to pay the ﬁxed cost,fij ≥ fii to sell the products to market j.
Firms in each country has asymmetric technology represented by exogenous distributionfunction of product eﬃciency, Gi (λ) = 1 − (Ti /λ)θ . Here we assume that the countries haveasymmetric technology parameters, Ti for i = 1, 2.
In the open-economy, the price of a product λ supplied by a ﬁrm in country i to country i,
pii and the price of a product λ supplied by a ﬁrm in country i to country j is written as:
pii =

1
,
ρλ

pij =

τij
,
ρλ

i = 1, 2

respectively. The proﬁt function of each ﬁrm in the open economy is given as:
πi =

2 Z ∞
X
∗
k=1 λik"#τik1−σ µLk− fik dGi (λ) − F,σ(ρλPk )1−σ7i = 1, 2where
1
=
ρPi

λ∗ii

Ã

µLi
σfii

! 1

1−σ

,

i = 1, 2(7)is the product-cutoﬀ for the domestic market below which products would make negative
proﬁts if they are produced, and hence they are not produced.
τij
=
ρPj

λ∗ij

Ã

µLj
σfij

! 1

1−σ

,

i = 1, 2(8)is the product-cutoﬀ for the domestic market below which products would make negative proﬁts
if they are exported, and hence they are not exported. Therefore, for suﬃciently high values ofﬁxed and variable trade costs, the model features selection into export markets. Only the mosteﬃcient products are exported, while intermediate eﬃciency products are supplied to only thedomestic market, and least eﬃcient products are not produced. However, if the foreign market
is suﬃciently large and less competitive, it might be the case that some products are exported
by not supplied to the domestic market.8 These equations consists a subset of equilibrium
conditions.
The remaining equilibrium conditions are, as in closed economy, free entry conditions andlabor market clearing conditions. The former conditions are written as:
πi =

" 1−σ
2
X
τik Tiθ γµLk

σ(ρPk )1−σk=1#

λ∗−θ+σ−1− fik Tiθ λ∗−θ− F = 0.
ik
ik

i = 1, 2(9)Lastly, the labor market clearing requires:
Mi

( 2 "
1−σ θ
X σ − 1 τik
Ti γµLk(ρPk )1−σ

σ

k=1

#

λ∗−θ+σ−1+ fik Tiθ λ∗−θ
ik
ik

)

+F

= µLi .

i = 1, 2(10)

To summarize, there are eight unknown equilibrium variables: Mi , Pi , λ∗ii , λ∗ij for i = 1, 2.
And we have eight equilibrium conditions: four equations that determine product-cutoﬀ, (7)and (8), two free entry conditions, (9), and two labor market conditions, (10). Therefore, as
in closed-economy model, the model is solvable. However, in order to simplify the solutions it= ff22.
is assumed that ff11
12
21
An unique set of equilibrium values are summarized as follows:
Mi =

Pi

µL

F σγ ,
γ−1

i = 1, 2

Ã

! 1 "

1
=
ρ
"

λ∗ii

=

µLi
σfii

1−σ

1
1
F
−θ −θ 2
fii 1 − τij τji Φ γ − 1F11−θ −θ 2
fii 1 − τij τji Φ γ − 1Ãλ∗ij = τij

fii
fij

! 1 "
1−σ

(11)

Ã

τij−θ Φ
1
−
Tiθ
Tjθ

1
F
1
−θ −θ 2
fjj 1 − τji τij Φ γ − 1Ãτij−θ Φ
1
−
Tiθ
Tjθ

!# θ1

,

i = 1, 2(12)!#− θ1.
Ãi = 1, 2τji−θ Φ
1
−
Tjθ
Tiθ

(13)

!#− θ1.

i = 1, 2(14)

8 This is consistent with empirical evidence conﬁrmed by Eckel et al. (2009). They show that multi-product ﬁrms sell fewer
products in their export than their home markets, though they earn higher proﬁts abroad when the foreign market is larger.

8³´ θ−σ+1

where Φ ≡ ffijii 1−σ ≤ 1 if fij ≥ fii . Using λ∗ii and λ∗ij , the mass of products supplied by eachcountry i’s ﬁrm to market i is expressed as:
F
1
1
hii =
−θ −θ 2
fii 1 − τij τji Φ γ − 1Ã!

τij−θ Φ
1
−
,
Tiθ
Tjθ

i = 1, 2

and the mass of products supplied by each country i’s ﬁrm to market j is expressed as:Ã

hij = τij−θ

fii
fij

!− θ

1−σ

F
1
1
−θ −θ 2
fjj 1 − τji τij Φ γ − 1Ã!

τji−θ Φ
1
−
.
Tjθ
Tiθ

i = 1, 2

In order to make the mass of products non-negative, hii ≥ 0 and hij ≥ 0, we need the following
assumption:
Assumption 1
τji ≥

Tj 1
Φθ .
Ti

i = 1, 2

In order to understand the meaning of this assumption, as later in this paper, suppose thatcountry 1 has a superior technology than country 2: T1 > T2 . To assure non-negative mass of
1
products exported by country 2 to country 1, h21 , the model needs the assumption: τ12 ≥ TT21 Φ θ .
However, if the technology gap between countries is suﬃciently large and τ12 is low, it might be1
the case that τ12 < TT21 Φ θ . In this case h21 is negative. The reason is following. Because country
1 has a very superior technology than country 2, ﬁrms in country 1 are more proﬁtable thanthose in country 2. Moreover, the small τ12 make the country 1 a better export base, whichmake the country 1 more attractive than country 2 from the viewpoint of ﬁrms. These two oﬀer
a strong incentive to ﬁrms to agglomerate country 1. This competitive environment sharply
decreases P1 , which sharply decreases the exports by country 2 into country 1, h21 . Therefore,
at an extreme case, h21 can be negative. In order to exclude such a strange situation, we need
1
Assumption 1. The meaning of the assumption: τ21 > TT12 Φ θ can be explained by same logic.9

4.1

Bilateral reduction of the trade cost

Because our model allows asymmetry between countries, the impact of an asymmetric reduction of trade costs can be examined. However, before illustrating the consequences of asymmetric liberalization, we ﬁrst quickly describe the case of symmetric liberalization among symmetric countries. Therefore, it is assumed that τ12 = τ21 = τ , L1 = L2 = L, and T1 = T2 = T .
The comparative statics with regard to a reduction of τ give following proposition:
Proposition 2: In a world consisting two identical countries, bilateral trade liberalization in
the form of reduction of τ :9 In order to assure non-negative mass of products exported by country 1 to country 2, h , we need the assumption: τ12
21 ≥T2 /T1 Φ1/θ . This condition always hold if the technology gap between countries is suﬃciently large such that τ21 ≥ 1 > T2 /T1 Φ1/θ .
In this case, the mass of products exported by country 1 to country 2, h12 cannot be negative for all τ ≥ 1. Therefore, the restrictionon τ21 is non-binding. The inferior technology of country 2 cannot oﬀer an incentive to ﬁrms to agglomerate in country 2, then,P2 is relatively higher. This less competitive environment in country 2 assures positive h12 .

9(i) monotonically decreases price indices, Pi and Pj , which improves welfare in both countries;(ii) monotonically reduces (does not aﬀect) mass of available variety for a consumer if fij > fii(fij = fii );(iii) induces ﬁrms to consecutively drop less eﬃcient products from domestic market;(iv) induces ﬁrms to start exporting most eﬃcient products.
Proof. See the Appendix. ¤Result (i) is driven by a similar mechanism with Melitz (2003). An increase in the domestic
product-cutoﬀ and the import of foreign inexpensive products lower the price index, which
improves welfare in both countries. Result (ii) is very similar to the result obtained by Arkolakis
et al. (2009) and Baldwin and Forslid (2010). In their single-product Melitz-type models, it
is shown that trade liberalization decreases the mass of available variety if the ﬁxed cost forexporting is greater than the ﬁxed cost of domestic operation.10
Results (iii), (iv) are supported by several existing empirical studies of variety diversiﬁcation
during trade liberalization. For example, Baldwin and Gu (2009) and Barnard et al. (2010)
ﬁnd sharp decline in the number of products among Canadian and U.S. ﬁrms, respectively.
Next, the eﬀects of bilateral liberalization in the asymmetric countries are examined. The
asymmetric market size, L1 6= L2 , does not make a new result, therefore, it is assumed that
L1 = L2 = L. However, we assume that the two countries have asymmetric technology
parameters. More speciﬁcally, let’s assume that country 1 has a superior technology thancountry 2: T1 > T2 . In this world, a bilateral reduction of trade costs gives us following
results:
Proposition 3: If ﬁrms in country 1 has a superior technology than those in country 2, i.e.,
T1 > T2 , and the technology gap is suﬃciently large, bilateral trade liberalization in the formof reduction of τ12 = τ21 = τ :(i) monotonically decreases P1 , which monotonically improves welfare of country 1;(ii) ﬁrstly decreases P2 then increases it, which ﬁrstly improves then reduces welfare of country2;(iii) induces ﬁrms in country 1 consecutively drop less eﬃcient products from domestic market;(iv) induce ﬁrms in country 1 consecutively add most eﬃcient products as exporting products;(v) induces ﬁrms in country 2 ﬁrstly drop domestic less eﬃcient products, but after further
deeper liberalization, start producing previously dropped products in the domestic market;(vi) induces ﬁrms in country 2 ﬁrstly add most eﬃcient products as exporting products, but
after further liberalization, drop previously added products from the export market.
Proof. See the Appendix. ¤These results are essentially driven by free entry conditions of two countries. Because ﬁrms
in country 1 have a superior technology, their proﬁtability is higher than ﬁrms in country 2.
10 Baldwin and Foslid (2010) call this eﬀect as “anti-variety eﬀect” of trade liberalization. However, it is shown that the positiveeﬀect owing the productivity improvement of the industry dominates the negative “anti-variety” eﬀect.

10

Moreover, bilateral trade liberalization makes ﬁrms in country 1 more proﬁtable than ﬁrms
in country 2 (even though 2’s ﬁrms are also beneﬁted by trade liberalization). Therefore, inorder to restore equilibrium, that is, in order to equalize ﬁrms’ expected proﬁts in country 1
and 2, P1 must sharply decrease after the reduction of τ . The sharp reduction of P1 also has
negative eﬀect on the proﬁtability of ﬁrms in country 2. Therefore, again, in order to equalize
expected proﬁts of ﬁrms in country 1 and 2, P2 must slightly increase. This eﬀect is strong
when τ is low because the eﬀect of τ on ﬁrms’ proﬁt is magniﬁed as τ decreases.11
The non-monotonic behavior of ﬁrms in country 2 is explained by same logic. In the country
2’s market, when τ is suﬃciently low, because P2 is increased by the reduction of τ , now that
2’s ﬁrms can produces larger mass of products due to the relaxed competition. Therefore, they
can add previously dropped domestic products. On the other hand, the progressive decreasein τ continuously has induced 2’s ﬁrms to export, however, due to the sharp decrease in P1 ,ﬁrms in country 2 begin to stop exporting previously exported products.

Figure 1: Mass of products in country 1

Figure 2: Mass of products in country 2

11 Recall that θ > 4. This means that the value of τ −θ is not linear with respect to τ .

11

4.2

Unilateral reduction of the trade cost

We now describe the eﬀects of a unilateral reduction of trade cost by country 1 (a decrease inτ21 , holding τ12 constant). Since asymmetric market sizes and technologies do not make theresults insightful, we assume L1 = L2 = L and T1 = T2 = T . A unilateral reduction of τ21
gives following results:
Proposition 4: At any values of T1 , T2 , L1 and L2 , the progressive unilateral trade liberalization by country 1 in the form of reduction of τ21 :(i) monotonically increases P1 , which monotonically reduces welfare of country 1;(ii) monotonically decreases P2 , which monotonically raises welfare of country 2;(iii) induces ﬁrms in country 1 consecutively add products for the domestic market;(iv) induce ﬁrms in country 1 consecutively drop products from the export market;(v) induces ﬁrms in country 2 consecutively drop products from the domestic market;(vi) induces ﬁrms in country 2 consecutively add products for the export market.
Proof. See the Appendix. ¤Again, the essence of the results is in the free entry conditions. A reduction of τ21 makes
country 2 better export base, which increases the expected proﬁt from being country 2. Therefore, the reduction of τ21 oﬀers an incentive ﬁrms to agglomerate in country 2. Even thoughmass of ﬁrms in country 2 does not change in equilibrium, this threat in country 2 decreases
P2 . The decrease P2 make the market in country 2 more competitive, inducing 2’s ﬁrms todrop domestic products and 1’s ﬁrms to quite exporting previously exported products. This
competitive environment inhibits the formation of agglomeration that should occur in a new
economic geography model with single-product ﬁrms.
On the other hand, a decrease in τ21 increases P1 in order to restore equilibrium: country
1 must be more proﬁtable by relaxing competition (with an increase in P1 ). Because ﬁrms
in country 1 have an incentive to move to country 2, this raises P1 , which increases mass of
products produced by ﬁrms in country 1. Now since ﬁrms in country 1 can produce larger massof products, it prevents 1’s ﬁrms from leaving country 1. Therefore, in this multi-product ﬁrm
model, the de-industrialization that occurs in single-product ﬁrm model does not happens. Allof the eﬀects on mass of ﬁrms are absorbed by within-ﬁrm adjustment.
In addition, it is clear that the liberalizing country experiences a welfare loss while its trading
partner experiences a welfare gain.12 These results are driven by the setting that ﬁrms canfreely move across countries. The lowering trade costs in a country make the other country
better export base, which generates an incentive to agglomerate in the country. Even though
they do not move in the equilibrium, the possibility of moving across countries aﬀects priceindices, and then aﬀecting welfare.
Even though the direct eﬀect of a decrease in τ21 lowers P1 , the indirect eﬀect through within12 The similar results are derived in Melitz and Ottaviano (2008) with a setting of linear demand and Demidova (2008) with asetting of CES preference. The mechanism of the welfare loss of liberalizing country in their model is essentially same with thismodel.

12

ﬁrm-adjustment increases P1 . The source of the within-ﬁrm-adjustment is the footlooseness ofﬁrms. Therefore, if the model introduces a friction that sticks ﬁrms in the locating countries,
the eﬀect of unilateral trade cost could be changed: the direct eﬀect would dominate theindirect eﬀect. With this motivation, I introduce heterogeneity in productivity across ﬁrms
because it is a way of introducing the friction that weakens the footlooseness of ﬁrms.

5

Heterogeneous Firms

This section shows that, once we introduce heterogeneity across ﬁrms into the model, unilateral
reduction of trade cost beneﬁts a liberalizing country as in Demidova and Rodriguez-Clare(2011) and also beneﬁts its trading partner.
Suppose that, by paying ﬁxed entry cost, fe , ex ante identical ﬁrms draw their productivity,ϕ, from the Pareto distribution function:
H(ϕ) = 1 − ϕ−k ,where k is the shape parameter that is assumed to be greater than θ to make the ﬁrms’ proﬁt
ﬁnite. Assume that ϕ is the inverse of ﬁrm’s marginal cost that aﬀects all of the products
within a ﬁrm in a same way. Therefore, ﬁrm’s pricing rules are:
1
τij
pii =
, pij =,
i = 1, 2
ρϕλρϕλ
which implies that the eﬃcient ﬁrm can set the lower price. Now the proﬁt of a ﬁrm whose
productivity is ϕ is written as:
πi (ϕ) =

"

2 Z ∞
X
∗
k=1 λik (ϕ)= (γ − 1)2X#

τik1−σ µLk− fik dGi (λ) − Fσ(ρϕλPk )1−σ
Ã

fik

k=1

where
λ∗ii (ϕ)

1
=
ρϕPi

Ã

Ti
∗
λik (ϕ)

L
σfii

!θ

− F,

i = 1, 2

! 1

1−σ

,

i = 1, 2(15)

is the product-cutoﬀ in domestic market below which products would make negative proﬁts ifthey are produced, and hence they are not produced. Similarly,
τij
λ∗ij (ϕ) =
ρϕPj

Ã

L
σfij

! 1

1−σ

,

i = 1, 2(16)

is the product-cutoﬀ for foreign markets below which products would make negative proﬁts ifthey are exported hence they are not exported. These imply that the more productive ﬁrms
can produce and export larger mass of products.
It is clear that all ﬁrms participate in exporting because any ﬁrms have at least some very
eﬃcient products that can make the positive proﬁt even in the foreign market. 13
13 Contrary to the model developed here, in the model by Bernard et al. (2010), a subset of ﬁrms participate in exporting as in

Melitz (2003) because each ﬁrm have to pay ﬁxed exporting cost, Fx . The introduction of such cost is straightforward. However,in order to compare the heterogeneous model with the homogeneous ﬁrm model developed in previous sections, we assume thatthe ﬁxed exporting cost is only appeared in fij and we do not introduce Fx .

13

Firms’ proﬁt function is re-written as:Ãϕ
πi (ϕ) = F  ∗
ϕi



!θ

− 1 ,

i = 1, 2

where
ϕ∗θ
i =

F
γ−1

·

P2
k=1

Ti ρPk
τik

³

σfik
Lk

´ 1 ¸θ

,i = 1, 2

1−σis the cutoﬀ productivity above which ﬁrms can make non-negative proﬁts. Now we can obtain
expected proﬁt of ﬁrms as:
π̄i =

Ã !

θ
ϕ
F
− 1

Z ∞
ϕ∗i

= F

dH(ϕ)
1 − H(ϕ∗i )

ϕ∗i

θ
.
k−θ

i = 1, 2

It is clear that ﬁrms’ expected of proﬁt is same in country 1 and country 2 even though they
are asymmetric. As pointed out in Helpman et al. (2003) and Baldwin and Forslid (2010),
free entry of ﬁrms ensure that the expected proﬁt is same in both markets.
Now I have done all preparations to derive equilibrium. The heterogeneous multi-productﬁrm model has 10 unknowns: Pi , Mi , ϕ∗i , λ∗ii , and λ∗ij for i = 1, 2. There are 10 equilibrium
conditions: four product-cutoﬀ conditions ((15) and (16)), two free entry conditions, two labormarket conditions, plus two CES price index equations. The free entry conditions are writtenas:[1 − H(ϕ∗i )]π̄i /δ = fe ,i = 1, 2(17)

where LHS is expected proﬁt from entry and RHS is the cost of entry. Labor market clearing
requires:
( 2 
X

Mi
k=1

|

Ã

(σ − 1)τik1−σ Tiθ γµLk
Ti
+
f
ik
∗
∗
σ(ρϕ̃Pk )1−σ [λik (ϕ̃)]θ−σ+1
λik (ϕ̃)
{z

!θ 
+
}

F
|{z}

Fθ
k−θ
| {z }+

ﬁxed cost)= µLi,(18)

ﬁxed entry fee

labor demand for production plus ﬁxed costs of products

for i = 1, 2.14 LHS is labor demand and RHS is labor supply. ϕ̃ is the average productivity ofRdH(ϕ)
heterogeneous ﬁrms written as: ϕ̃ = ϕ∞∗ ϕ 1−H(ϕ∗ ) . Lastly, the CES price indices are:
Pi1−σ

=

2
X
k=1

Mi

Ã

Z ∞Z ∞ϕ∗k

λ∗ki (ϕ)

τki
ρϕλ

!1−σ

dGk (λ)dH(ϕ),
1 − H(ϕ∗i )i = 1, 2(19)

Therefore, the model is solvable. However, it is too complicated to obtain the analytical result.
Then, a sketch of a way of solving the model is described and a numerical example is shown.
14 The labor demand for ﬁxed entry fee, L , is written as: L = f M where M is mass of entrants. Using the free entry
e
e
e e
e
condition, it can be written that:Le = Me fe =θ
δfe M
= M π̄ = M F.
1 − H(ϕ∗ )k−θ14The calculation process would be following:
Step 1: Using (17), derive equilibrium cutoﬀ productivities, ϕ∗i for i = 1, 2.
Step 2: Using equilibrium cutoﬀ productivities, (15), (16), (18) and (19), equilibrium mass ofﬁrms, Mi for i = 1, 2, are obtained.
Step 3: Using equilibrium cutoﬀ productivities, equilibrium mass of ﬁrms, (15), (16) and (19),obtain equilibrium price indices, Pi for i = 1, 2.
Step 4: Using equilibrium price indices, (15) and (16), obtain product-cutoﬀs, λ∗ii , and λ∗ij fori = 1, 2.
Following these steps, equilibrium values are characterized. But only equilibrium cutoﬀ productivities are analytically solvable. Regarding other endogenous variables, the simultaneous
equations that give equilibrium values are presented in the following.
Step 1: Equilibrium cutoﬀ productivities are given as:
Ã

ϕ∗i =

F θ
δfe k − θ

!1

k

,

i = 1, 2which implies that the two countries have same cutoﬀ productivity, then the subscript i is
dropped from the cutoﬀ productivity: ϕ∗1 = ϕ∗2 = ϕ∗ . Equilibrium price indices are determined
as solutions of following simultaneous equations:15
Step 2: Equilibrium mass of ﬁrms in each country is obtained by solving following simultaneous equations:"Mi

Tiθ [(σ − 1)γ + 1]
σγϕ∗k

Ã

τij−θ Lj Φ
Li+
Tiθ Mi + τji1−σ Tjθ ΦMj Tjθ Mj + τij1−σ Tiθ ΦMi

!

#

Fθ
= µL,(20)+F +k−θ

for i = 1, 2. It is diﬃcult to obtain analytical solutions of Mi for i = 1, 2. Therefore, anumerical example is shown below.
Step 3: Once equilibrium mass of ﬁrms, Mi for i = 1, 2 are determined, equilibrium priceindices are obtained as functions of Mi for i = 1, 2:k−θ
1Pi = ∗θγ(ρϕ ) k

Ã

Li
σfii

1/θ

!− σ−θ−11

1−σTiθ Mi + τji1−σ Tjθ ΦMj

. i = 1, 2

(21)

Step 4: Once equilibrium price indices, Pi for i = 1, 2 are determined, equilibrium productcutoﬀs are determined by (15) and (16).
Because our primary interest is the eﬀects of unilateral trade liberalization, the eﬀect of unilateral reduction of trade costs by country 1 (a decrease in τ21 , holding τ12 ) is examined. Using
15 These equations are obtained by inserting (15) and (16) into (19).

15(21), it is numerically shown that, when the technology levels are similar between countries,
unilateral trade liberalization by country 1 increases M1 and M2 as opposed to standard Melitz(2003)-type models.16 In addition, the increase in M1 is greater than the increase in M2 .
Figure 3 shows the result of the numerical exercise.17 The dashed lines are the equations(21) when τ12 = τ21 = 5 and the solid lines are those when τ12 = 5 and τ21 = 2. It is shownthat a decrease in τ21 increases M1 and M2 and the increase in M1 is greater than the increasein M2 .
The intuition behind the result is following. A decrease in τ21 make the market in country 1
more competitive, which reduces the sales of each products and the mass of products producedby each ﬁrm in country 1. Therefore, labor required for production decreases. This decreasein labor demand allows additional entry. Then, a decrease in τ21 increases mass of ﬁrms incountry 1.
On the other hand, in country 2, a decrease in τ21 reduces sales in country 1 and mass ofexported products because the decreases in τ21 makes country 1 more competitive. This eﬀect
reduces labor demand as in ﬁrms in country 1. However, a decrease in τ21 gives ﬁrms in country
2 export opportunity. That is, as a result of a decrease in τ21 , more products are exported tocountry 1, which increases labor demand from each ﬁrm in country. This eﬀect reduces massof ﬁrms in country 2. Therefore, the increase in M2 is smaller than the increase in M1 .18
One might think that decreases in revenue of each ﬁrm decreases proﬁt of each ﬁrm, which
Fθ. However, a
is inconsistent with the result that expected proﬁt of ﬁrms is constant: π = k−θdecrease in revenue does not mean decrease in expected proﬁt. Because ﬁrms drop products,
now ﬁrms pay fewer ﬁxed costs for production, which has positive impacts on ﬁrms’ proﬁt. The
positive impact and the negative impact from tougher competition is in balance. Therefore, adecrease in revenue is not against the fact that ﬁrms’ expected proﬁt is constant.
Now the eﬀect of a decrease in τ21 on equilibrium price indices can be identiﬁed. From
∂P1
∂P1
∂P2
equation (21), it follows that ∂M< 1, ∂M< 1, and ∂τ> 1. In addition, it is suggested by
1
2
21
∂M1
∂M2
the numerical exercise that ∂τ21 < 1 and ∂τ21 < 1. Therefore, a decrease in τ21 reduces P1 . Thisimplies that liberalizing country beneﬁts from the unilateral trade liberalization. Moreover,since a decrease in τ21 increases M1 and M2 , the unilateral reduction of trade costs by country
1 improves welfare in country 2.
Result 1: In the heterogeneous multi-product ﬁrm model, if the technology level is similar
between countries, unilateral trade liberalization by country 1 in the form of reduction of τ21 :(i) monotonically decreases P1 and P2 , which monotonically raises welfare of country 1 and 2;(ii) country 1 beneﬁts more than country 2;
16 In standard Melitz (2003)-type models, a decrease in trade costs induces ﬁrms to begin exporting, which requires additional
resources for paying higher ﬁxed cost for exporting. However, in this model, such eﬀect is excluded because all ﬁrms are exportingsome most eﬃcient products even if trade costs are very high.
17 It is assumed that L = L = 400, F = 1, µ = 0.5, [(σ − 1)γ + 1]/σγϕ∗k = 0.5, T = T = 1, k = 5, and θ = 4.
12
12
18 If the liberalizing country has a inferior technology than its trading partner and the technology gap is suﬃciently larger,
it could be the case that unilateral trade liberalization increases mass of ﬁrms in the country that is not liberalizing. This isbecause the increase in labor demand due to additional export opportunity dominates the decrease in labor demand due tointensiﬁed competition in the liberalizing country since country 1 is less competitive due to their inferior technology. Conversely,
the liberalizing country has a superior technology than country 1, it could be the case that mass of ﬁrms in the country that is
not liberalizing sharply increases.

16

Figure 3: The eﬀect of unilateral reduction of trade costs(iii) induces ﬁrms in country 1 and 2 consecutively drop products for the domestic market;(iv) induce ﬁrms in country 1 and 2 consecutively drop products from the export market.
From this numerical exercise, it is suggested that the introduction of heterogeneity acrossﬁrms alters the results obtained in homogeneous ﬁrm model developed in previous section.
That is, it is suggested that unilateral reduction of trade costs beneﬁts liberalizing country. In
addition, not only the liberalizing country but also its trading partner could beneﬁts from a
unilateral reduction of trade costs.
One might be interested in the impacts of bilateral trade liberalization among asymmetriccountries. However, its impact on welfare is ambiguous. For the discussion on the eﬀects ofbilateral trade liberalization, see the Appendix.

6

Conclusion

This paper has examined the impact of trade on the market and ﬁrms’ behavior by introducing
multi-product ﬁrms into a model of new economic geography. The simple model developed
here enables us to investigate how the interaction of asymmetric countries induced by tradeliberalization inﬂuences the ﬁrms’ behavior, price index and welfare.
It is shown that symmetric bilateral trade liberalization aﬀects diﬀerently ﬁrms in high-techand low-tech countries. In a high-tech country, it is suggested that ﬁrms could signiﬁcantly
reduces their domestic products and start exporting the most eﬃcient products. On the other
hand, in the low-tech country, it is suggested that ﬁrms do not reduce domestic products somuch. In addition, in a setting of new economic geography with homogeneous ﬁrms, ﬁrms
in a low-tech country take a non-monotonic behavior: they ﬁrst drop and add their domestic

17and exporting products, respectively, but after further liberalization, they add and drop previously dropped domestic and added exporing products, respectively. These diﬀerence in ﬁrms’
behavior in high-tech and low-tech countries can explain diﬀerent existing empirical results indeveloped and developing countries.
Moreover, regarding the eﬀects of unilateral trade liberalization, results diﬀer in Krugmantype homogeneous multi-product ﬁrm model and Melitz-type heterogeneous ﬁrm model. Countrary to the homogeneous ﬁrm model that unilateral reduction of trade costs hurts the liberalizing country, in the heterogeneous ﬁrm model, it is shown that not only the liberalizing
country but also its trading partner gains from trade and the gains in the liberalizing country
is greater.
This framework develops a new and very tractable way of describing how diﬀerence in technology and trade costs across countries aﬀects ﬁrms’ behavior and welfare diﬀerently. I hopethat this provides a useful foundation for future empirical investigations on ﬁrms’ behavior
and its international comparison.

A
A.1

Appendix
Proof of Proposition 2

Since two countries share same technology parameter, let T1 = T2 = T .
Proof of (i) and (iii)
The partial diﬀerentiation of hii with respect to τ12 = τ21 = τ is:∂hii
τ −θ−1 θΦ F= θ
∂τT (γ − 1) fiiÃ

1 − τ −θ Φ
1 − τ −2θ Φ2

!2

> 0.

for all τ ≥ 1

Therefore, the mass of products supplied by ﬁrms in country i to county i monotonicallydecreases as tau decreases. This proves part (iii) of Proposition 2. Then part (i) follows
because hii has a perfect negative relationship with λ∗ii that has a perfect negative relationshipwith Pi .
Proof of (iv)
The partial diﬀerentiation of hij with respect to τ12 = τ21 = τ is:
Ã

fii
∂hij
=−
∂τ
fij

!− θ

1−σ

τ −θ−1 θ F
T θ (γ − 1) fiiÃ

1 − τ −θ Φ
1 − τ −2θ Φ2

!2

< 0.

for all τ ≥ 1

Therefore, the mass of products exported by ﬁrms in country i to country j increases as τ
decreases.
Proof of (ii)In the equilibrium, the mass of operating ﬁrms in each country does not change due to a18

reduction of τ . Moreover, all ﬁrms are engaging in exporting in an open-economy. Therefore,
the change in the mass of variety for a consumer is fully captured by the change in the massof products supplierd by domestic ﬁrms and foreign ﬁrms:Ã

hii + hji = 1 + τ −θ

fji
fii

!− θ 
1−σ1F 1 − τ −θ Φ.
T θ (γ − 1) fii 1 − τ −2θ Φ2

The partial diﬀerentiation of hii + hji with respect to τ12 = τ21 = τ is:


Ã

∂
fji
(hii + hji ) = Φ −∂τfii!− θ 
1−σ

τ −θ−1 θ F
T θ (γ − 1) fii

Therefore, hii + hji decreases as τ decreases if Φ >
Φ=

³

´ θ
fji − 1−σ.
fii

Recall that Φ ≡

³

fij
fii

´ θ−σ+11−σ³Ã

1 − τ −θ Φ
1 − τ −2θ Φ2

!2

.

´ θ
fji − 1−σ; and τ does not aﬀect hii + hji if
fii∂. Hence it follows that ∂τ(hii + hji ) > 0 if ffjiii > 1;∂
and ∂τ
(hii + hji ) = 0 if ffjiii = 1.
Now all statements in Proposition 2 are proved. ¤A.2

Proof of Proposition 3

Proof of (i), (ii), (iii) and (v)Suppose that, as in the main text, country 1 has superior technology than country 2: T1 > T2 . Ishow that the mass of domestic products produced by each ﬁrm in country 1, h11 , monotonicallyincreases and that in country 2, h22 , has non-monotonic behavior under a moderate parameter
restriction. By showing this, the results, (i), (ii), (iii) and (v) in Proposition 3 follows because
hii and Pi have a perfect positive correlation, and hii and λ∗ii have a perfect negative correlation
for i = 1, 2.
The partial diﬀerentiation of hii with respect to τij = τji = τ is:
Ã

!

∂hii
1 F
θΦτ −θ−12τ −θ Φ 1 + τ −2θ Φ2
=
−
+
.
∂τ
γ − 1 fii (1 − τ −2θ Φ2 )2Tiθ
Tjθ
Therefore, the sign of ∂hii /∂τ is determined as:Tθ∂hiiτ θ Φ−1 + τ −θ Φ
R 0 ⇔ jθ Q
.
∂τ
Ti
2
Therefore, in country i, the non-monotonic behavior of ﬁrms arises when:Tjθτ θ Φ−1 + τ −θ Φ
<
,
Tiθ
2
Tjθ
τ θ Φ−1 + τ −θ Φ
>
.
Tiθ
2
Recall that Φ ≡

³

fij
fii

´ θ−σ+1
1−σfor high value of τ
for low value of τ

. Therefore, Φ−1 ≥ 1 ≥ Φ when fij ≥ fii . Then τ Φ
θ

θ −1
−θ
increasing in τ and fij . As τ → 1 and fij → fii , τ Φ 2+τ Φ → 1.

19

−1 +τ −θ Φ

2isNow suppose that country 1 has superior technology than country 2. The results, (i), (ii),(iii) and (v) arise if the following condition is satisﬁed:τ θ Φ−1 + τ −θ ΦTθTθ> 1θ > 2θ , for high value of τ
2
T2
T1
θ
θ −1
−θ
T1
τ Φ +τ ΦT2θ
>>. for low value of τ
T2θ
2
T1θ

(22)
(23)

This condition implies that non-monotonic behavior is more likely to occur when technology
gap is larger and the diﬀerence between fij and fii is smaller. On the contrary, the nonmonotonicity in country 2 would not occur when fij is extremely high or the technology gap is
suﬃciently small. However, P1 and h11 always monotonically decrease as τ decreases becauseTθτ θ Φ−1 +τ −θ Φ
≥ 1 > T2θ for all τ ≥ 1.
2
1

Proof of (iv) and (vi)
I show that the mass of products exported by country 1, h12 , monotonically increases as τ
decreases: ∂h∂τ12 < 0 for all τ ≤ 1, and the mass of exported products by country 2, h21 , has
the non-monotonic behavior: ∂h∂τ21 < 0 for high τ and ∂h∂τ21 > 0 for low τ .
The partial diﬀerentiation of hij with respect to τ = τij = τji is:∂hij
1 Fθτ −θ−1=∂τγ − 1 fii (1 − τ −2θ Φ2 )2

Ã

fii
fij

!− θ Ã
1−σ!

2τ −θ Φ 1 + τ −2θ Φ2
−
.
Tjθ
Ti−θ

Therefore, the sign of ∂hij /∂τ is determined as:Tθ∂hijτ θ Φ−1 + τ −θ Φ
R 0 ⇔ jθ R
.
∂τ
Ti
2
It follows that country 1 monotonically increases the mass of exporting products because
θ −1
−θ
T2θ
< 1 ≤ τ Φ 2+τ Φ for all τ ≥ 1. By contrary, ﬁrms in country 2 ﬁrst increase their mass ofT1θ
exporting products, h21 , but reduce it after further reduction of τ if technology gap between
countries is large enough and fij is low enough so that:τ θ Φ−1 + τ −θ ΦTθ> 1θ , for high value of τ
2
T2
θ
θ −1
T1
τ Φ + τ −θ Φ
. for low value of τ>
T2θ2
Therefore, the results (iv) and (vi) hold if the results (i), (ii), (iii), and (v) hold.
Now I can conclude this proof by maintaining that all results in Proposition 3 occur whentechnology gap between the countries are large enough and fij is not so high so as to satisfy(15)-(16). ¤A.3

Proof of Proposition 4

Proof of (i) and (iii)
The partial diﬀerentiation of h11 with respect to τ21 is:−θ −θ−1 2
τ21 Φ
1 F θτ12∂h11=−−θ −θ 2 2
∂τ21
γ − 1 f11 (1 − τ12 τ21 Φ )

20

Ã

!

−θ
1
Φ
τ12
.
−
θ
θ
T1
T2

11
Under Assumption 1, it always follows that ∂h< 0. Therefore, a reduction of τ21 always∂τ21
increases the mass of products supplied by ﬁrms in country 1 to country 1. Now part (iii) of
Proposition 4 has been proved. This also proves part (i) because h11 has a perfect negative
relationship with λ∗11 and λ∗11 has a pefect negative relationship with P1 .

Proof of (ii) and (iv)
The partial diﬀerentiation of h22 with respect to τ21 is:
−θ
1 F
τ21
θΦ
∂h22
=−
−θ −θ 2 2
∂τ21
γ − 1 f22 (1 − τ12 τ21 Φ )

Ã

!

−θ
τ12
Φ
1
−
.
θ
θ
T1
T2

22
Under Assumption 1, it always follows that ∂h> 0. Therefore, a reduction of τ21 always∂τ21
decreases the mass of products supplierd by ﬁrms in country 2 to country 2. Now part (iv) of
Proposition 4 has been proved. This also proves part (ii) because h22 has a perfect negative
relationship with λ∗22 that has a pefect negative relationship with P2 .

Proof of (v)The partial diﬀerentiation of h12 with respect to τ21 is:Ã∂h12−θ f22
= τ12
∂τ21
f12

!− θ

1−σ

−θ−1
1 F
θΦτ21
−θ −θ 2 2γ − 1 f22 (1 − τ21τ12 Φ )

Ã

!

−θ
1
τ12
−
Φ .
T1θ
T2θ

12
Under Assumption 1, it always follows that ∂h> 0. Therefore, a reduction of τ21 always∂τ21
decreases the mass of products exported by ﬁrms in country 2 to country 1.

Proof of (vi)The partial diﬀerentiation of h21 with respect to τ21 is:
∂h21
1
=−
∂τ21
γ−1

Ã

f11
f21

!− θ

1−σ

Ã

−θ−1
−θ 2
F
τ21
θ
τ21
Φ
1
+
−θ −θ 2−θ −θ 2
f11 1 − τ12 τ21 Φ
1 − τ12 τ21 Φ

!Ã

!

−θ
1
τ12
−
Φ .
T1θ T2−θ12Under Assumption 1, it always follows that ∂h> 0. Therefore, a reduction of τ21 always∂τ21
decreases the mass of products exported by ﬁrms in country 2 to country 1.
Now all of the statements in Proposition 4 are proved.¤

A.4

Bilateral trade liberalization among asymmetric countries inheterogeneous ﬁrm model

Suppose that country 1 has a superior technology than country 2. It can be numerically shownthat bilateral reduction of trade cost (a decrease in τ21 = τ12 = τ ) increases M2 more than M1
as shown in Figure 4. Figure 4 represents equations (20) by assuming L1 = L2 = 400, F = 1,
µ = 0.5, (σγ + 1)/σγϕ∗k = 0.5, T1 = 3.5, T2 = 1. The dashed lines are equations (20) in caseof τ21 = τ12 = 5 and the solid lines are equations (20) in case of τ21 = τ12 = 2. The intuition
is following:21

Why there are more ﬁrms in country 2 than country 1?
Because ﬁrms in country 2 cannot make larger proﬁts which reduces labor demand in country
2, then allowing more entry in country 2.
Why bilateral reduction of trade costs increases mass of ﬁrms?
A decreases τ intensify the competition in both markets, which reduces sales and then reduces
labor demand, which increases M1 and M2 .
Why bilateral reduction of trade costs increases M2 more than M1 ?
Additional export opportunity by decreases in τ increases sales of ﬁrms in country 1 morethan those in country 2 because country 1 has a superior technology. On the other hand, ﬁrms
in country 2 demands fewer labor because they cannot make larger proﬁt due to a inferior
technology. Then, more ﬁrms can enter country 2 than country 1 as τ decreases.

Figure 4: The eﬀect of biilateral reduction of trade costs among asymmetric countries

P1 and P2 decreases due to bilateral trade liberalization, which improves welfare in bothcountry 1 and 2. However, it is not clear whether country 1 beneﬁts more or country 2 beneﬁts
more. Since M2 increases more than M1 , one might presume that bilateral trade liberalization
beneﬁts country 2 more than country 1. But it might not be the case. It is suggested fromequation (21) that the eﬀect of a increase in M2 on P2 is weak because T2 is small and theeﬀect of a increase in M1 on P1 is strong because T1 is large. Intuitively, even if mass of ﬁrms
in country 2 increased, its eﬀect on P2 is weak because they produces expensive goods due totheir inferior technology. Conversely, even if the increase in M1 is smaller than the increase M2 ,its impacts on P1 is strong because they produces low-priced goods. Therefore, the diﬀerence
between the decrease in P1 and P2 is unclear. Then, the diﬀerence between the increase in P1
22and P2 is also unclear.

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