Matching, Quality Upgrading, and North-North Trade inIntermediate Goods
Yoichi Sugita
Columbia University
December 12, 2008

Abstract
This paper presents a new mechanism of …rm-level gains from trade in a general equilibrium
model of international team production of …rms heterogeneous in product quality. Trade inintermediate goods between developed countries raises the quality of …nal goods by improving
matching of …rms in production teams. The quality upgrading is decomposed as the short rune¤ect of a reduction in the quality gap among parts and components and the long run e¤ect ofintensi…ed competition among suppliers. With …xed trade costs, the pattern of …rm-level trade
based on this gain is consistent with a variety of stylized facts that have not been presented in
the conventional theories. Firms selectively trade with those with similar characteristics. Bothexporting and importing are concentrated into large and high quality …rms although not alllarge and high quality …rms trade. Trade in intermediate goods improves the quality of even…rms that do not import intermediate goods.
Key Words: matching, heterogeneous …rms, quality, vertical di¤erentiation, trade in intermediate goods, o¤shoring.

E-mail: ys2176@columbia.edu. I am grateful to Donald Davis for his constant encouragement. I also thank Kyle
Bagwell, James Harrigan, Kazuki Konno, Hitoshi Sato, Guru Sethupathy, Kensuke Teshima, Eric Verhoogen, andDavid Weinstein for their helpful comments and suggestions.

11Introduction

One symbolic phenomenon of the globalization is a rise in international team production of …rms
in developed countries. Traditionally, the international division of labor is limited to very simple
forms based on trade in commodities such as wheat for bread. However, the recent development incommunication and transportation technologies allows a more complicated form of division of labor.
Firms in distant countries cooperatively develop a product and trade specially designed parts andcomponents within a team. Boeing 787 dreamliner is a good example. For this new midsize jet,
Boeing arranges a new production team of 43 suppliers mostly selected from developed countries.
The suppliers, which Boeing proudly calls “the world’s most capable top-tier supplier partners”,
produce cutting-edge components newly designed for this airplane.1 Similar team production
is common in other products, especially quality-di¤erentiated products such as automobiles andelectronics.2
One of the most important decisions in the team production is to choose “right”partners froma pool of suppliers at various quality levels. It is well-known that there is a considerable degree ofheterogeneity in …rms’performances within industry. Furthermore, recent empirical studies on theprice data of traded products suggest the observed heterogeneity re‡ects the di¤erence in productquality rather than the di¤erence in productivity.3 Given the prevalence in the heterogeneity of
product quality, the matching of …rms in a production team is an important channel of qualitychange since the combination of the quality of parts and components determines the quality of
12

http://www.boeing.com/commercial/787family/background.html.
Although exports by developing countries are growing, they are likely to contain a considerable amount of indirect

trade between …rms in developed countries. For example, more than 90% of the gross pro…ts of Apple’s iPod, whichis exported from China, are taken by US and Japanese …rms producing cutting-edge components (Linden, Kraemer,
and Dedrick, 2007). At an aggregate level, Koopmans, Wang, and Wei (2008) report foreign value-added containedin China’s exports is 50% on average and even 80% in sophisticated industries such as electronics.
3
Schott (2004) and Hummels and Klenow (2005) are two early studies on the heterogeneity of unit prices oftraded goods within product categories. Hummels and Skiba (2004), Baldwin and Harrigan (2007), Johnson (2008),
and Bernard et al. (2007) observe fob price of traded goods increases in the distance and trade costs, which implies…rms producing higher quality products are more likely to be exported. From plant data, Kugler and Verhoogen(2007) …nd exporting plants tend to have a higher index of output prices.

2

…nished products.
Despite this potential implication for the development in product quality, the matching of
quality-di¤erentiated …rms has been understudied in the literature. Existing models of matchingof …rms focus on a random matching between symmetric …rms rather than heterogeneous …rms
(Casella and Rauch, 2001; Rauch and Casella, 2003; Rauch and Trindade, 2003; Grossman andHelpman, 2005). Existing models of trade by heterogeneous …rms abstract away from matching of…rms. Most studies employ the love of variety as a source of gains from trade, which automatically
implies that all importers trade with all exporters.4 How do …rms choose trading partners? How
does trade change matching of …rms and product quality? The literature lacks a theoreticalframework to analyze these questions.
This paper develops a tractable general equilibrium model of international team production of…rms heterogeneous in product quality. The model combines two well-established models. The …rst
one is a quality-version of Melitz (2003)-type model of heterogeneous …rms extended by Baldwin
and Harrigan (2007) and Kugler and Verhoogen (2008), in which ex ante symmetric …rms become
heterogeneous in their quality as a result of R&D investment. The second one is a competitive
multi-sided matching model of a continuum of …rms by Sattinger (1979), in which …rms compete
for high quality partners. As the simplest model of trade between developed countries, I consider
international matching between two countries that symmetrically di¤er in their technologies.
The model presents a new mechanism of …rm-level gains from trade. Trade in intermediate
goods between developed countries raises the quality of …nal goods by improving matching of…rms in production teams. In the autarky, matching patterns di¤er across countries re‡ecting thedi¤erence in their technologies. In the short run after the opening of trade, the matching patternconverges across countries. This reduces the di¤erence in the quality among parts and components
to improve the quality of …nal goods. In the long run, countries’specialization in low entry cost
sectors increases competition among suppliers to raise the quality of suppliers available for …nal
producers.
Combined with …xed trade costs, the pattern of …rm-level trade based on this new gain is consistent with a variety of stylized facts which are di¢ cult to understand in the conventional models
4

Bernard et al. (2003) use the perfectly competitive model in which …rms do not care about trading partners.

3

based on the love of variety. First of all, …rms selectively trade with those with similar characteristics instead of trading with all …rms. Second, both exporting and importing are concentrated
into a small share of large and high quality …rms within industries.5 While the previous studies
treat heterogeneous exporters and importers in separate frameworks, the current model explainsthem in a single framework. Third, not all large and high quality …rms necessarily trade. While in
the love of variety model, the most productive …rms always choose to trade, in the current model,
some portion of high quality …rms always choose not to trade.6 Finally, the pattern of quality upgrading is consistent with a recent …nding by Amiti and Konings (2007) that a reduction in tari¤son intermediate goods improves the total factor productivity (TFP) of even …rms that do not use
imported intermediate goods. This is puzzling to the conventional model of trade in intermediate
goods such as the love of variety model and the quality-ladder model, in which …rms must importforeign intermediate goods in order to raise the productivity/quality. However, their observation
is totally consistent with the prediction in the current model that trade improves the quality of…nal producers that do not import intermediate goods.
The paper contributes to the literature of so-called heterogeneous …rm trade theories. Many
theories have been developed to analyze exporters heterogeneous in productivity (Bernard, Jensen,
Eaton, and Kortum, 2003; Melitz, 2003), exporters heterogeneous in product quality (Baldwin andHarrigan, 2007; Johnson, 2008; Verhoogen, 2008), and heterogeneous importers (Antràs and Helpman, 2003; Kasahara and Lapham, 2008). However, these studies treat heterogeneous exporters
and importers in separate frameworks. The paper o¤ers the …rst model of trade between …rms
heterogeneous in product quality.
The matching model used in the paper relies on the long history of the matching literaturedeveloped by Gale and Shapley (1962), Becker (1973), and other many studies. Especially, my
5

Bernard, Jensen, Redding, and Schott (2007) survey empirical and theoretical studies on …rm-level trade. See

the papers cited in Bernard et al. (2007) for the concentration of exporting. See Bernard, Jensen, Redding, andSchott (2007), Bernard, Jensen, and Schott (2005), Biscourp and Kramarz (2007), and Kasahara and Lapham (2007)
for the concentrantion of importing.
6
In the Ricardian model by Bernard et al. (2003), it is possible for the most productive …rm to choose not toexport because the high productivity does not assure the comparative advantage. However, their model has noheterogeneous importers.

4model applies Sattinger (1979)’s model of a continuum of agents. My innovation is to let the distribution of …rms at each side of matching endogenously determined, which allows me to analyze thee¤ect of trade liberalization on the distribution of …rms across industries in a general equilibrium
framework.
The paper is closely related with recent studies on international matching of heterogeneous
agents.7 Kremer and Maskin (2006) and Antràs, Garicano, and Rossi-Hansberg (2006) study
North-South matching of heterogeneous workers based on a hierarchical order in the skill intensityof production stages such as managing job and production job. My paper considers matching of…rms between developed countries and does not assume any hierarchical order in the characteristics
of production stage. Furthermore, their models allow workers to move across production stages,
but my model prohibits …rms from moving across production stages. This di¤erence generates
very di¤erent predictions on the distribution of gains from trade across agents. In their models,
the highest skilled managers lose productivity from worse matching after trade, while in my model,
the highest quality …nal producers improve the quality from better matching. Nocke and Yeaple(2008) analyze two-sided matching between a corporate asset and a manager to model international
M&A. While their paper focuses on the existence of two-way international matching under very
weak assumption, the current paper employs a richer structure of …rms’entry and exit to derive
systematic predictions on the pattern of international matching. Furthermore, none of the above
three studies analyzes costs of international matching.
The rest of the paper proceeds as follows. Section 2 sets up the model of a closed economy.
Section 3 analyzes trade between symmetric countries. Section 4 concludes the paper and remarks
on future extensions.
7

A matching model is also becoming a popular tool to study trade between countries with di¤erent distributionsof workers’skill. Grossman and Maggi (2000) model domestic matching between heterogeneous workers. Ohnsornge
and Tre‡er (2007), Costinot (2008), and Costinot and Vogel (2008) study domestic matching between heterogeneous
workers and di¤erent industries.

52

Closed Economy

This section introduces a general equilibrium model of team production in a closed economy
setting. The model introduces Sattinger(1979)-type matching of a continuum of …rms into a Melitz(2003)-type model of heterogeneous with quality di¤erentiation, which is extended by Baldwin andHarrigan (2007), Johnson (2008), and Kugler and Verhoogen (2008). I explain the basic structureof the model and then demonstrate that the distribution of …rms and matching patterns of …rms
re‡ect the technology of the economy.

2.1

Basic Structure

Consider a closed economy endowed with one production factor, labor. Final goods are bothvertically and horizontally di¤erentiated. A representative consumer maximizes a CES utility
function,
U=

Z

1=

q (!) c (!) d!

;

!2

where

is the set of available varieties of …nal goods, ! is a particular variety, c(!) is consumptionof variety !, q(!) is product quality of !, and

2 (0; 1) is a parameter. Let p(!) be a price of !

and I be an aggregate income of this economy. The demand function for ! is derived asc (!) =
where

Iq (!) p(!)
P1

1=(1

;

) > 1 is an elasticity of substitution, and PhR!2p (!)1q(!) d!

i1=(1

)

is

a quality-adjusted price index. The quality parameter q (!) is a demand shifter: a higher q implies
a larger demand at a given price.
There exist three types of …rms: …nal producers in …nal goods sector, suppliers of intermediate
good Z1 (Z1-suppliers) in Z1-sector, and suppliers of intermediate good Z2 (Z2-suppliers) in Z2sector. A …nal producer, a Z1-supplier, and a Z2-supplier form a production team to produce
one variety of …nal good. Intermediate goods are specially designed for a particular variety of…nal good, e.g. car engines and car bodies; therefore, …rms transact them only within a team.
In the following, I use subscripts i and j to denote variables and functions of Z1-suppliers andZ2-suppliers. They always mean that i; j 2 f1; 2g and i 6= j when they are used together.
6Firms are continuum and heterogeneous in production quality. Let x, z1 , and z2 be the quality
parameters of …nal producers, Z1-suppliers, and Z2-suppliers, respectively.8 Quality parameters
can also be interpreted as the quality of components, or product characteristics in a terminology
used in the industrial organization literature. The quality of a …nal good depends on the qualityof team members in the following way,q = xz1 z2 :(1)The “quality production function” in (1) exhibits three properties. First, q is increasing as isnormally expected. Second, q is supermodular. A smooth twice-di¤erentiable function is called(strictly) supermodular if all of its partial cross-derivatives are positive. Suppose a team replaces
one member with the one with higher quality. The supermodularity requires the quality improvement of …nal good caused by this replacement to be increasing in the quality of the other members.
This complementarity in the quality of intermediate goods seems a natural assumption. For instance, the performance of a car engine depends on the quality of the other components such astransmission, body, tires, etc. Finally, q is quasi-concave. Consumers often consider a small gap in
the quality of components is one element of a high quality …nal product. For instance, consumers
might prefer a standard-class car with normal equipment to a luxury-class car with a poor air
conditioner.
The labor market is perfectly competitive and wage is normalized as one. When a teamproduces X unit of a …nal good of quality q, the …nal producer requires LX (q; X) unit of labor,
X unit of intermediate goods Z1, and X unit of intermediate goods Z2. To produce X unit ofintermediate goods Zi designed for …nal goods with quality q, each Zi-supplier requires LZi (q; X)
unit of labor. The labor requirement is symmetric across team members, consists of …xed andvariable components, and increases in q,
Lh (q; X) =qX + ffor h = X; Z1; and Z2:
3(2)

The variable cost increasing in q re‡ects costs of quality control in team production. Since evenone defect component can destroy the whole product, as emphasized by Kremer (1993), productionof high quality …nal goods requires extra costs of quality control for all team members.
8

Readers may call x; z1 ;and z2 “productivity” of quality production.

7

The …rm heterogeneity arises from …rms’entry and exit. There exist in…nitely many potential
entrants of …nal producers and Zi-suppliers. These …rms are ex ante symmetric, but become
heterogeneous as a result of uncertain entry. When …rms enter, each …rm independently draws its
quality parameter from a common Pareto distribution.9 The distribution function is G(s)1 s k

for s 2 [1; 1) where k > 3 is assumed to assure the existence of …nite GDP. Entry requires fXeunit of labor for …nal producers and fZie unit of labor for Zi-suppliers. These entry costs include
not only setup costs but also R&D investment for blueprint of products. Firms are risk neutral sothat they enter until their expected pro…ts are zero.
After knowing quality parameters, …rms form production teams. Matching is frictionless in
two senses: (i) …rms have all information about the other …rms and (ii) they can write complete
contract on the distribution of team’s joint pro…t. The latter assumption abstracts away from a
problem whether a team is formed within or across the boundaries of …rms. For mathematicaltractability, I assume one-to-one matching, i.e. each …rm can join at most one team.10
The model consists of four stages. (i) Pre-entry Stage: a Walrasian auctioneer announces wage
to clear the labor market. (ii) Entry Stage: …rms enter and draw quality parameters by paying…xed entry costs. (iii) Matching Stage: …rms form production teams. (iv) Production Stage: teams
compete in a …nal good market under the monopolistic competition and distribute the joint pro…t.
9

The Pareto distribution is commonly used to characterize empirical distributions of …rm size. See Axtell (2001)

and Helpman, Melitz, and Yeaple (2004), for example. In the current model, the sales of …nal goods follow the
Pareto distributions both in autarky and free trade.
10
The assumption of “one-to-one” is not necessary for the main results, which relies on the assortative matchingof …rms. The important assumption is that each Zi-supplier has a capacity constraint on the number of teams thatthey can join. The existence of the capacity constraint is consistent with a recent study on …rm-level transaction by
Eaton, Eslava, Krizan, Kugler, and Tybout (2008). From the customs transaction data in the US and Colombia,
the authors …nd that most exporters limit their transcations with a small number of buyers. There are also several
theoretical reasons for this capacity constraint: search costs convex in the number of buyers (Eaton et al., 2008),
convex marketing costs (Arkolakis, 2008), and several reasons for vertical foreclosure discussed in the industrialorganization literature. I leave incorporating one of these reasons in the model for future research.

8

2.2

Equilibrium

I derive an equilibrium allocation by backward induction. Although the model has a trivial equilibrium where no …rm enters, I focus on an equilibrium with entry.

Production Stage

Each team sets a …nal good price under the monopolistic competition. Con-sider a team producing a …nal good with quality q. Since team’s marginal cost is q, it follows thatthe optimal price p(q) of …nal goods, the sales r(q) of …nal goods, and team’s joint pro…t(q) allincrease in q,p(q) =
Parameter Aq, r(q) = I (P )1I (optimal output, c =1q, and(q) = Aq

f.

(3)

P)

1

IP

1 , is independent of q. This is because both consumer’s demand andexpresses the market condition exogenous to individual teams. The

marginal costs increase in q and the two e¤ects are balanced under the current speci…cation. Since
the price increases in q, both revenue and pro…t increase in q. From (1) and (3), team’s joint pro…tis increasing and supermodular in the quality of team members,(x; z1 ; z2 ) = Axz1 z2

f.

(4)

Matching Stage Firms choose their partners and decide the distribution of team’s joint pro…t,taking A as given. Two types of functions, pro…t schedules,

X (x) andZi (zi ), and assignment

functions, mZi (x), characterize equilibrium matching. A …nal producer with quality x chooses
Zi-suppliers with quality mZi (x) and receives a residual pro…t

X (x) after paying pro…ts

Zi (zi )for Zi-suppliers. Firms can also choose not to join any team and exit. Following the matching
literature, I focus on stable matching satisfying two conditions: (i) no individual …rm is willing todeviate from the current team; (ii) no trio of a …nal producer, a Z1-supplier, and a Z2-supplier
are willing to deviate from the current teams to form a new team.11 The two conditions require
the following two conditions, respectively: (i) all …rms earn non-negative pro…t,
11

X (x)

0 and

The …rst condition is often called individual rationality and the second condition is pair-wise stability. Roth andSotomayer (1990) is an excellent survey on the literature.

9Zi (zi )0 for all x and zi ; (ii) each …rm in a team is the optimal partner for the other members,
X (x)

=

(x; mZ1 (x) ; mZ2 (x))= max
z1 ;z2Zi (mZi (x))=(x; z1 ; z2 )Z1 (z1 )(x; mZi (x) ; mZj (x))
x0 ; mZi (x) ; zj

= max
0
x ;zj

Z1 (mZ1 (x))

Z2 (mZ2 (x))

Z2 (z2 ) andX (x)Zj (mZj (x))

0
X (x )Zj (zj ).

(5)

(6)

The …rst order conditions for maximization (5) and (6),0X (x) = AmZ1 (x)mZ2 (x) and0Zi (mZi (x)) = AxmZj (x) ;(7)

prove that pro…t schedules increase in quality parameters.
From the supermodularity of joint pro…t in (4), …rms are assortatively matched by quality as
in Becker (1973) and Sattinger (1979). Since a high quality …rm has a higher willingness to pay
for extra quality of partners, a high quality …rm is matched with a high quality …rm.
Lemma 1 mZi (x)mZi (x0 ) if only if xx0 .

Proof. In Appendix.
Production …xed costs allow only teams producing high quality …nal goods to survive. Under
assortative matching, teams producing the lowest quality consist of the lowest quality …rms,(xL ; z1L ; z2L ) = AxL z1L z2L

f = 0;(8)

where xL and ziL are the lowest quality thresholds of …nal producers and Z1-suppliers, respectively,
and satisfy
X (xL ) =Z1 (z1L ) =Z2 (z2L ) = 0:(9)

Firms with lower quality than the lowest quality thresholds choose not to join teams and exit.
Assignment functions must clear the demand and supply for …rms in the matching market. Let
MXe , MZ1e , and MZ2e be the mass of entrants of …nal producers, Z1-suppliers, and Z2-suppliers,
respectively. Under assortative matching, the market clearing condition is written as
MXe [1

G (x)] = MZie [1G (mZi (x))] for all x
10

xL :

(10)

The left hand side is the mass of …nal producers with higher quality than x; the right hand side is
the mass of Zi-suppliers matched with those …nal producers.
Figure 1 describes the market clearing in the matching market. Bars in Figures 1 expresses
the distributions of …nal producers, Z1-suppliers, and Z2-suppliers. The values of the distribution
G(x), G(z1 ), and G(z2 ) are drawn on the horizontal axis. Squares drawn with solid lines express themass of survival …rms, MXe [1G (xL )] and MZie [1G (ziL )], respectively, all of which have thesame area under one-to-one matching. Grey areas are expresses the mass of …rms, MXe [1and MZie [1G (x)]

G (mZi (x))], respectively, all of which must have the same area from (10).

Figure 2 shows that the relative mass of entrants across sectors determines the matchingpattern. As more …rms enter Zi-sector, a …nal producer becomes matched with a Zi-supplierwith better quality. Suppose new Zi-suppliers with mass dMZie enter. A …nal producer withquality x can be matched with better Zi-suppliers only if new entrants have higher quality thanthe current partner. Under the Pareto distribution and other many distributions, the mass of
those high quality Zi-suppliers [1

G (mZi (x))] dMZie falls as mZi (x) rises further; therefore, themarginal improvement is diminishing.12 As drawn in Figure 3, mZi (x) is increasing and concave
in MZie =MXe for given x. Under the Pareto distribution, assignment functions are solved from(10),mZi (x) = x

MZie
MXe

1=k

for all x

xL .

(11)

The pro…t schedules are obtained by integrating the …rst order conditions with initial conditions(9),
X (x) = AZ x

mZ1 (t) mZ2 (t) dt andZi (mZi (x)) = A

xL

for all x

xL and zi

Z xxL

tmZj (t) m0Zi (t) dt;(12)ziL . Pro…ts are increasing in the market size A, the quality parameters of

the partners, and the degree of its advantage in quality over the lowest quality …rm. The cuto¤
condition (9) and the assignment functions (11) further simplify the pro…t schedules as
"
#
"
#
f
x 3
f
zi 3
1 and Zi (zi ) =1 ;X (x) =
3
xL
3
ziL
12

(13)

This concave relationship holds under a wide class of distributions including those exhibiting the non-decreasing

hazard rate g(x)=(1G(x)), which includes uniform, normal, exponential, and other frequently used distributions.

11

for all x

ziL .13 The pro…t schedule is decreasing in the lowest quality thresholdxL and zi

because of two negative e¤ects. When the threshold increases, …rms in that sector become assignedwith lower quality partners from (11), and the market size A shrinks from (8).14

Entry Stage Since …rms are ex ante identical and risk neutral, their expected pro…ts must be
equal to entry costs,
[1
where
and

G (xL )]X and

Zi = [1X = fXe and [1G (ziL )]Zi = fZie ;(14)

Zi are the average pro…ts of …rms in the market,

G (ziL )]

1R1

ziL

X = [1

R1
G (xL )] 1 xL

X (t) g (t) dt

Zi (t) g (t) dt: A straightforward manipulation from (13) proves thatthe average pro…ts turn are constant,
X =

Zi =

f
k

3

:

(15)

This constant average pro…t is consistent with a well-known property of the Melitz-type model(2003) that the average pro…t of …rms becomes constant when …rms’ productivity follows thePareto distribution. In the current model, teams’quality q follows the Pareto distribution.15 To
assure the positive mass of entry, I assume f =(k3)

maxffXe ; fZie g. Then, the lowest quality

thresholds are solved from (14) and (15) asfxL =fXe (k1=k3)

f
and ziL =fZie (k

1=k

3)

.

(16)

The lowest thresholds decrease in entry costs and increases in production …xed costs. The intuition
will be clear below after I solve the mass of consumption varieties, M , and the mass of entrants
in each sector.
Since …rms earn zero expected pro…ts, the aggregate revenue from …nal goods must be equal
to the aggregate labor income, M r = L, where r is the average revenue of survival teams. From
13
14

To derive (13), I use mZi (x) = x (ziL =xL ) derived from (10).
The pro…t schedules in (13) are independent of the quality thresholds in the other sectors because of theconsequence of two opposite e¤ects. When the threshold increases in the other sectors, the …rm becomes assignedwith better partners, but the other rival …rms in the same sector enjoy that improvement, which reduces the market
size A for individual …rms. These two e¤ects are cancelled under the Pareto distribution.
15

The distribution of q is Pr (qs) = 1(qL =s)k=3 , where qL

the market.

12

xL z1L z2L is the lowest quality of …nal goods in

r =

( X+

Z1 +

Z2 + f ) and (15), the mass of consumption varieties is proportional to theratio of labor endowment to production …xed costs,
M=

(k

3)
k

L
f

:

(17)

Under one-to-one matching, the mass of teams is equal to the mass of survival …rms in each sector.
Therefore, the mass of entrants are solved as
MXe =

1

L
M
=
G(xL )fXe k

and MZie =

1

M
L
=
:
G(ziL )
fZie kWhile the relative magnitude of labor endowment to production …xed costs determine the massof varieties consumed, the relative size of entry costs determines the relative size of the mass ofentrants.
The lowest quality thresholds in (16) link the mass of consumption varieties and the mass ofentrants. Lower entry costs attract relatively more entrants, but the total mass of survival …rms(17) is independent of the size of entry costs. Therefore, in the sector with low entry costs, …rms
have to be higher quality to survive than in the other sectors.

Patterns of Parts and Components of Products

Although our main interest is in an open

economy, the closed economy model provides two predictions on observable patterns of the qualityof parts and components. First, the assortative matching implies that more expensive …nal goods
have higher quality in all parts and components. This complementarity of quality of components
is often observed in the literature of hedonic regression as multicollinearity (see e.g. Triplet,2004). Second, a cross-country di¤erence in entry costs creates a horizontal di¤erence in thequality of components. From (11) and (16), the distribution of product characteristics re‡ects the
distribution of entry costs across sectors,mZi (x)
=
x

fXe
fZie

1=a

and

mZ1 (x)=mZ2 (x)

fZ2e
fZ1e

1=a

for all x

xL .

A component produced by a sector with relatively lower entry costs is relatively higher quality
than the other components. National product di¤erentiation is often loosely stated in terms ofhorizontal di¤erences in the quality of components, e.g. “European cars have safety bodies” or“Japanese cars are energy e¢ cient”, even though the most expensive Japanese cars will be safer
13

than the cheapest European cars. In this model, this kind of comparison is valid when …nal goods
with the same price are compared.

3

Open Economy

In this section, I extend the model presented in the last section in a two-country framework.
I analyze the e¤ect of trade intermediate goods, i.e. international matching, on the matchingpatterns and the quality of …nal goods.

3.1

Symmetric Two Countries

As the simplest model of North-North trade in intermediate goods, I consider two symmetriccountries. Two countries, Home and Foreign, are identical except entry costs in the Zi-sectors.
Foreign is a mirror image of Home: Foreign Z1-sector is identical to Home Z2-sector, while Foreign
Z2-sector is identical to Home Z1-sector. Without loss of generality, Home Z1-sector and Foreign
Z2-sector require lower entry costs than the other sectors,
fZ1e = fZ2e < fZ2e = fZ1e ,(18)

where foreign variables and functions are labeled by asterisks. I call Home Z1-sector and Foreign
Z2-sector Low Entry Cost (LEC) sectors and Home Z1-sector and Foreign Z2-sector High Entry
Cost (HEC) sectors.
The horizontal di¤erence in entry costs creates a horizontal di¤erence in the autarky matchingpatterns. Final producers with quality x produce the same quality of …nal goods in both countries,
but in the teams, Zi-suppliers from LEC-sectors are higher quality than Zj-suppliers from HECsectors. Point A and Point A* in Figure 4 represent the quality of Zi-suppliers in a Home teama (x) ; m a (x)), respectively, for given x. The curve(maZ1 (x) ; maZ2 (x)) and in a Foreign team (mZ1
Z2

in Figure 4 is an “iso-q (x) curve,”which depicts a combination of the quality of Zi-suppliers that
produce …nal goods with quality q(x) when they are matched with …nal producers with quality
x. In the following, endogenous variables and functions in the autarky equilibrium are labeled by
“a”.

14

3.2

Trade Costs and Specialization

The opening of trade in intermediate goods allows international matching of …rms between thetwo countries. The mirror-image structure greatly simpli…es the analysis. Wage is equalized acrosscountries and normalized as one. Equilibrium values of functions and variables of Home Zi-sector
are the same as those of Foreign Zj-sector. The other aspects are identical between Home andForeign.
International matching requires fI unit of labor as …xed trade costs, which include transportation costs, communication costs, and costs of adopting foreign standards and regulations, etc. Each…rm in an international team equally shares fI by hiring fI =3 unit of labor. If a …nal producer
forms an international team with two suppliers in the other country, then the team must hire 2fIunit of labor. For simplicity, I assume that trade does not require any variable cost and that …nal
goods are non-tradable.
Three regimes arise depending on the level of fI : autarky, incomplete specialization, andcomplete specialization.

Lemma 2 There exists a threshold level of trade costs f~I : (i) (autarky) if fI = 1, then nointernational matching occurs; (ii) (incomplete specialization) if f~I < fI < 1, then both countries
have positive mass of entrants in all three sectors; (iii) (complete specialization) if fI < f~I , thenboth countries have positive mass of entrants in …nal goods sector and LEC sector, but no entrant
in HEC sector.
Proof. In Appendix.
When fI = 0, countries perfectly specialize in LEC sectors. In the following, I …rst analyze theincomplete specialization equilibrium and then, the complete specialization equilibrium.

3.3

Incomplete Specialization Equilibrium

In this subsection, I examine equilibrium matching patterns of …rms in an incomplete specialization
equilibrium. The trade pattern in the model is not only consistent with stylized facts of …rm-level15

trade, but also provides new predictions. In the next subsection, I will analyze the change in the
product quality.
The incomplete specialization equilibrium is more complicated than the autarky equilibrium
since trade costs prevent some …rms from trading. To distinguish suppliers that can export fromthose that cannot, I introduce the concept of exportability. Home Zi-suppliers with quality zi
are called exportable if

Zi (zi ) + fI=

Zi (zi ), that is, Foreign …nal producers are indi¤erent

between Home Zi-suppliers with quality zi and Foreign Zi-suppliers with the same quality. Home
Zi-suppliers with quality zi are called non-exportable if

Zi (zi ) + fI >Zi (zi ), that is, Foreign…nal producers strictly prefer Foreign Zi-suppliers with zi to Home Zi-suppliers with the same
quality; therefore, non-exportable Zi-supplier cannot join an international team. Notice that while
non-exportable Zi-suppliers never export, exportable Zi-suppliers do not necessarily export. All
Home Zi-suppliers are either exportable or non-exportable because

Zi (zi ) + fI <Zi (zi ) neverholds; otherwise, no …nal producer would choose Foreign Zi-suppliers with zi . Similarly, ForeignZi-suppliers with quality zi are called exportable if
Zi (zi ) + fI >Zi (zi ) + fI =Zi (zi ) and non-exportable if

Zi (zi ).

I prove two lemmas to derive the market clearing conditions in the matching market. I assume
that as in the autarky equilibrium, countries have more entrants in LEC sectors than HEC sectors,
MZ1e = MZ2e > MZ2e = MZ1e ;(19)

and prove this inequality later at the end of this subsection. The …rst lemma shows the quality ofexportable Zi-suppliers are equalized within a team.
Lemma 3 A Z1-supplier is matched with a Z2-supplier with the same quality if either one ofthem is exportable.
Proof. In Appendix.
Lemma 3 follows from the de…nition of exportability and the mirror-image structure of Homeand Foreign. Suppose an exportable Home Z1-supplier with quality z. This …rm may be either
locally matched with Home …rms or internationally matched with Foreign …rms.16 Consider the16

In equilibrium, a …nal producer is never matched with two foreign suppliers because it requires 2fI of trade

costs.

16

international matching …rst. Since the pro…t schedules of Foreign Z2-suppliers is identical to thatof Home Z1-suppliers, the joint pro…t symmetric with respect to the quality parameters predicts
the partner Foreign Z2-supplier also has the same quality z. This logic is also applied for the caseof local matching since …nal producers have the same pro…t schedules across Home and Foreign.
Lemma 3 greatly saves notations. While a complete description of possible matching patterns
requires eight functions, thanks to Lemma 3, only four functions mZi (x) and mZi (x) are enough.
A Home …nal producer with quality x is matched with Zi-suppliers with quality mZi (x) regardlessof the nationality of Zi-suppliers. Similarly, a Foreign …nal producer with quality x is matchedwith Zi-suppliers with quality mZi (x). Notice that the assortative matching continues to hold, i.e.
mZi (x)mZi (x0 ) and mZi (x)mZi (x0 ) if and only if xx0 , since team’s joint pro…t remains

supermodular with …xed trade costs.
The second lemma shows only high quality Zi-suppliers in LEC sectors are exportable.
Lemma 4 There exists a threshold quality zT such that only Home Z1-suppliers and Foreign
Z2-suppliers with higher quality than zT are exportable.
Proof. In Appendix.
The intuition for Lemma 4 is simple. From inequality (19), international matching occurs
only between Home Z1-suppliers and Foreign Z2-suppliers, which are abundant in their countries.
However, exportable suppliers must be high quality since low quality teams cannot a¤ord trade
costs.17
Lemmas 3 and 4 pin down three market clearing conditions for matching. First, since …nal
producers are indi¤erent between domestic Zi-suppliers and foreign Zi-suppliers if they are higher
quality than zT , high quality Home and Foreign …rms are pooled together. The market clearing
condition for high quality …rm becomes(MXe + MXe ) [117

G (x)] = (MZie + MZie ) [1

G (mZi (xT ))] for all x

xT ,

(20)

Lemma 4 crucially depends on the assumption that the distribution of …rm size G is similar across sectors andacross countries. Once the model allows asymmetric distributions of …rms, it will be possible for the highest quality
suppliers to be non-exportable. I do not pursue the case of asymmetric distributions here since the shape of the…rm-size distribution is similar across industries and across countries in data.

17

where xT is de…ned by zT = mZi (xT ). Notice that mZ1 (x) = mZ2 (x) for all xxT from (19).

After the opening of trade, the cross-country di¤erence in matching patterns disappears for highquality teams. For low quality teams that cannot form international teams, the market clearing
condition holds for local …rms,
MXe [G (xT )G (x)] = MZie [G (zT )G (mZi (x))] for all x 2 [xL ; xT ]:(21)

Since the mass of entrants Zi-suppliers di¤ers across countries, international matching mustoccur between high quality …rms. The pattern of international matching is derived from conditions(20), (21), and Lemma 4. Following a tradition in the literature, I focus on an equilibrium thatminimizes the amount of international matching.
Lemma 5 (i) Among …nal producers with given quality xintermediate goods is sX = MXe (MZ1esectors with given quality z

xT , the share of those importing

MZ2e ) = (MZ1e + MZ2e ). (ii)Among Zi-suppliers in LEC

zT , the share of those exporting intermediate goods is sZ = (MZ1e

MZ2e )=2.
Lemma 5 is visually derived from a comparison of Figure 5 with Figure 6. Figure 5 describes
the market clearing conditions (20) and (21). The area of each of six squares with solid lines is
the mass of survival …rms in each sector. Trade costs divide …rms into three groups: high quality…rm in grey area that are matched together and low quality …rms in each stripe area that arematched together. Since the mass of entrants Zi-suppliers di¤ers across countries, international
matching must occur between high quality …rms. Firms that trade are expressed by the shaded
area in Figure 6. Home …nal producers in area A are matched with Foreign Z2-suppliers in areaA’; Foreign …nal producers in area B are matched with Home Z1-suppliers in area B’. Therefore,
only high quality …nal producers can import and only high quality Zi-suppliers in LEC sectors
can export, though not all of them trade.

New Predictions on Firm-level Trade

Quality parameters link several observable character-istics of …rms. First of all, unit price is a proxy for product quality. This is clear for …nal goods
from (3). A unit price of an intermediate good Zi, pZi (zi ), is obtained by dividing the revenue by18

output,pZi (zi )Zi (zi ) + LZi (q (zi ) ; c)c

; where q (zi ) = mZi1 (zi ) mZj mZi1 (zi ) zi .

Since the output c is common for all teams and q is increasing in zi , a unit price is positively
correlated with product quality. From (3), the assortative matching implies that product quality
is positively correlated with revenue, employment, pro…t, and unit prices in each sector.
The model provides new predications on the observable characteristics of trading …rms.
Proposition 1 (1) Firms trade with those with similar characteristics such as revenue, employment, pro…t, and unit prices. (2) When the type of sectors and the nationality of …rms are controlled, the average exporter is larger than the average non-exporter and the average importer islarger than the average non-importer in such common variables as employment, revenue, pro…t,and unit prices. (3) However, …rms that are large in employment, revenue, pro…t, and unit prices
do not necessarily to trade.
Three predictions in Proposition 1 have not been presented in the previous models of heterogeneous …rm trade theories. First, in contrast to standard models of heterogeneous …rms basedon the love of variety such as Melitz (2003) and Kasahara and Lapham (2007), which predicts allexporters trade with all importers, in the current model, high quality exporters selectively trade
only with high quality importers, while low quality exporters selectively trade only with low quality
importers.
Second, Proposition 1 is the …rst demonstration of the concentration of exporting and importing
into large and high quality …rms in a single framework. The concentration of exporting andimporting is one of the most in‡uential empirical …ndings in the last two decades. Although many
papers have been written on this styled fact, none of them studies international trade between
heterogeneous exporters and importers. The in this model naturally explains the similarity ofcharacteristics of exporting …rms and importing …rms.
Finally, the large size and the high quality are necessary conditions for trade, but not su¢ cientconditions. While the standard love of variety models predicts …rms that are larger than a certain
threshold always choose to trade, in data, the correlation between …rm size, or measured produc19

tivity, and trading status is obviously not perfect, (See e.g. Bernard et al. 2003). The current
model predicts the existence of non-trading large …rms without relying on any idiosyncratic shocks.
Before moving to the analysis on the change of the quality in the next subsection, I complete
listing equilibrium conditions for obtaining endogenous variables and prove the inequality (19).
From Lemma 4, fI =Z2 (mZ2 (x))

Z1 (mZ1 (x)) holds if and only if x

xT . Therefore, thethreshold xT is determined byfI=

Z2 (mZ2 (xT ))

= A

Z xT
xL

Z1 (mZ1 (xT ))t mZ1 (t) m0Z2 (t)m0Z1 (t) mZ2 (t) dt;

where pro…t schedules are solved from (9), (12), and (21). Finally, the mass of entrants are solved
from the free entry conditions (14) and the average revenue M r = L. The inequality (19) follows
from the free entry conditions and the inequality of …xed entry costs (18).
Lemma 6 MZ1e = MZ2e > MZ2e = MZ1e .
Proof. In Appendix.

3.4

Quality Upgrading of Final Goods

Trade liberalization a¤ects the quality of …nal goods by changing matching patterns. I analyze
the change of …nal good quality by comparing the autarky equilibrium and the trade equilibrium.
Following Proposition 5, I divide …nal producers by quality at threshold xT . I …rst consider thequality change of high quality …nal producers and, then, that of low quality …nal producers.

Quality Upgrading of High Quality Firms

The opening of trade a¤ects the matching market

in two ways. First, high quality …rms in Home and Foreign are pooled together. Second, …rms
enter and exit under free entry. To separate the former e¤ect from the latter one, I decompose
trade liberalization into short run and long run: in the short run, international matching is allowed,
with the mass of entrants kept at the autarky level; in the long run, the mass of entrants adjusts
to satisfy the free entry conditions.

20

Short Run E¤ect Trade improves the quality of …nal goods in the short run after the
opening of trade. Notice that all propositions and lemmas in the last section holds in the short
run equilibrium in terms of the mass of entrants in the autarky since they are derived from theinequality (19) and do not depend on the particular levels of the mass of entrants. I obtainassignment functions msZi (x) from the market clearing condition (20) by replacing the mass ofentrants with their autarky values,
msZ1 (x) = msZ2 (x) = x

a + Ma
MZ1e
Z1e
a
2MXe

1=k

for x

xT :

(22)

The assignment functions (22) are comparable with those in the autarky equilibrium since they arecommonly expressed in terms of the relative mass of entrants into the Zi-sector and the …nal goods
sector. Figure 7, which replicates Figure 3, draws (11) and (22) for a given x. Since the relative
a + M a ) =2M a , is the average of thosemass of entrants of Z1-supplier and …nal producers, (MZ1eXe
Z1ea =M a and M a =M a , the concave curve implies that ms (x) is higher thanin the autarky, MZ1e
Xe
Z1e
Xe
Zi
a (x). By the quasi-concavity of q, a …nal producer with quality
the average of maZ1 (x) and mZ1xx^ raises the quality of …nal good. This is shown in Figure 8, which draws msZ (x) (Point B)

and maZi (x) and mZia (x) (Point A and Point A*) with iso-q (x) curves.
The source of the short run quality upgrading is the reduction in the di¤erence in the qualityof Zi-suppliers within a team. The competition with foreign …nal producers forces a …nal producer
to be matched with a lower quality Zi-supplier in LEC sector. However, trade also allows it to bematched with a higher quality Zj-supplier in HEC sector than in the autarky. Since consumers
prefer a moderate combination of the quality of components, the latter positive e¤ect compensates
for the former negative e¤ect and improves the overall quality.

Long Run E¤ect

In the long run, the mass of entrants are adjusted to satisfy the free entry

conditions.
Lemma 7 (1)The mass of entrants of …nal producers remains at the autarky level, MXe =a .(2) The mass of entrants of Zi-suppliers in LEC sector rises while that of Zi-suppliers in
MXe

HEC sector falls, i.e.
aaMZ1e > MZ1e
> MZ2e> MZ2e :(23)
21

(3)The relative mass of entrants of Zi-suppliers to …nal producers in the world increases but isbounded by the relative mass in Home autarky.
aa +M aMZie
MZie + MZie
MZie
Zie
>
>
:
a
a
MXe
2MXe
2MXe

(24)

Proof. In Appendix.
After trade, countries increase entrants in LEC sectors, while they reduce entrants in HEC
sectors. This specialization of entry into low entry costs sectors, which may be interpreted as aninternational division of labor in R&D activities between two countries, invites more entrants ofZi-suppliers in the world.
The long run adjustment of …rms’entry and exit further improves the quality of …nal goods.
From (20), the assignment function in the long run is
mZ1 (x) = mZ2 (x) =MZie + MZie
2MXe

1=k

x for x

xT :

(25)

From Figure 7, …nal producers become matched with higher quality Zi-suppliers than in the short
run equilibrium, i.e. mZi (x) > msZi (x). Point C in Figure 8 expresses mZi (x) on a iso-quality
curve q (x) = q l (x), where q l (x) is the quality of …nal good produced by a …nal producer withquality x in the trade equilibrium. From (24) and (25), the upper bound of mZi (x) is maZ1 (x) =a (x). Therefore, Point C is located between Point B and Point D.
mZ2

The source of the long run quality upgrading is competition among Zi-suppliers. Countries’
specialization in LEC sectors increases the mass of entrants of Zi-suppliers in the world. The
intensi…ed competition among Zi-suppliers allows …nal producers to be matched with higher quality
Zi-suppliers.
In sum, trade liberalization improves the quality of high quality …nal goods in two steps. While
trade eliminates the quality gap between Zi-suppliers in the short-run, trade increases the quality
level of Zi-suppliers matched with a given …nal producer in the long-run.
Let

(x)

q l (x) =q a (x) be the degree of quality-change of a …nal producer with x. It is

straight forward to show the quality upgrading is at a constant rate.
Proposition 2(x) = K > 1 for x

xT , where K = (MZ1e + MZ2e )2k = (MZ1e MZ2e )k .

22

Finally, I should remark the quality upgrading does not require importing intermediate goods.
High quality …nal producers equally gain from the opening of trade whether they are matchedwith foreign suppliers or not. I will show this property distinguishes the current model from theconventional models in the later subsection.

Quality Upgrading of Low Quality Firms

Although …nal producers with lower quality thanxT cannot form international teams, it is still possible for them to upgrade the product quality.
From the market clearing condition, the assignment function is
"
"
#
# 1=k
MZje MZie
MZ1e + MZ2e 1=kx kmZi (x) =x 1+ 1
2MXe
xT
2MZie

(26)

for x 2 [xL ; xT ). If x is close to xT , then mZi (x) in (26) is close to the value predicted by (20).
As x becomes smaller, the di¤erence between (26) and (20) becomes wider. Therefore, the degreeof quality-upgrading is increasing in x.
Proposition 3 (i)(xT ) < K and

0 (x) > 0 for x 2 [x ; x ). (ii)
L T

(xaL ) > 1 if MZie =MXe are

a =M a .
su¢ ciently close to MZie
Xe

Proof. In Appendix.
Surprisingly, even …nal producers whose quality is too low to import can upgrade the productquality. The intuition is simple. After the opening of trade, the in‡ow of high quality Z2suppliers from Foreign makes high quality …nal producers to release high quality Z2-suppliers for
low quality …nal producers in Home. However, another opposing e¤ect occurs in the long run. The
specialization into Z1-sectors from Z2-sectors reduces the mass of Home Z2-suppliers. Therefore,
it is possible for all …nal producers with x

xaL to upgrade product quality if the mass of entrants

is close to the autarky level. In general, whether all …nal producers upgrade the product quality
is generally ambiguous.

Testing the Model against the Conventional Models

The model presents the new mecha-

nism of gains from trade in intermediate goods. However, this is not the only model for the positivee¤ect of trade in intermediate goods on the performance of …nal producers. Trade in intermediate
goods improves the productivity/quality in the conventional models such as the love of variety
23

model, e.g. Ethier (1982), and the quality ladder model, e.g. Grossman and Helpman (1991). In
this section, I discuss testable predictions for an empirical test of the current model against theconventional models and introduce an empirical paper supporting the current model.
The prediction on the relationship between the degree of quality-upgrading and the importing
status of …nal producers provides a basis for the test of the current model against the conventionalmodels.
Remark 1 After trade liberalization of intermediate goods, (i) the average degree of qualityupgrading of importing …nal producers is larger than that of non-importing …nal producers. (ii)
The average degree of quality-upgrading of non-importing …nal producers can be positive.
To test the current model against these models, it is su¢ cient to check whether Remark 1 holdin data. In the conventional models, a necessary condition for improving the productivity/quality
is to import intermediate goods. Therefore, the prediction (ii) of Remark 1 should not be observedin these models.
To my knowledge, there is no econometric study that investigates the e¤ect of trade liberalization of intermediate goods on the quality of …nal goods.18 However, a recent study by Amiti
and Konings (2007) on Indonesian plants provides an approximate test. The authors estimate thee¤ect of a reduction in tari¤s on intermediate goods on the total factor productivity (TFP) of
importing …rms and non-importing …rms.
Their …nding supports the current model. Non-importing …rms improve TFPs though importing …rms experience a larger improvement more than non-importing …rms. Their …nding was
puzzling in the conventional love of variety model or the quality ladder model unless there exists some externality between importing …rms and non-importing …rms as the authors suggest.
However, it totally makes sense in the current model.
Although there are two di¤erences between Amiti and Konings (2007) and the ideal test for
Remark 1, it seems reasonable to interpret their exercise as an approximate test for Remark 1.
First, Indonesia is not a developed country which the current model mainly considers. Indonesian18

Verhoogen (2008) investigates the e¤ect of trade liberalization with respect to …nal goods on the quality of …nal

goods in Mexico.

24

manufacturing sector can be regarded as a part of regional production chains among East andSoutheast Asian countries, many of which are developing countries. Therefore, it is likely that tradeliberalization may have increased trade in intermediate goods with those developing countries.
Second, instead of product quality the authors estimate TFP without controlling product quality.
Since it is known that the measured TFP may re‡ect the quality of output as well as true TFP, thechange in measured TFP might re‡ect the change in product quality (Katayama, Lu, and Tybout,
2006; Foster, Haltiwanger, and Syverson, 2008).19

The Lowest Quality Thresholds and the Mass of Consumption Varieties

Finally, I

analyze the e¤ect of trade liberalization on the levels of the lowest quality thresholds and themass of consumption varieties. Consistent with Melitz (2003) and Kasahara and Lapham (2008),
trade liberalization raises the lowest quality thresholds both in exporting sectors and in importing
sectors and reduces the mass of survival teams, i.e. the mass of consumption varieties.
Proposition 4 (i)The lowest quality thresholds of …nal producers and Zi-suppliers in LEC sectors
a ; z ). The lowest quality thresholds of Zi-suppliers in
rise, i.e. xL 2 (xaL ; xT ) and z1L = z2L 2 (z1LT

HEC sectors fall. (ii) The mass of consumption varieties falls.
Proof. In Appendix.
Since high quality …nal producers upgrade the quality of …nal goods at a higher rate than low
quality …nal producers, low quality …nal producers must exit from the market, even though theymight upgrade the product quality. Therefore, the mass of consumption varieties falls. On theother hand, the mass of Zi-suppliers increases in LEC sectors and decreases in HEC sectors from
Lemma 7. Therefore, the lowest quality threshold of Zi-suppliers rises in LEC sectors and falls in
HEC sectors.20
19

It is possible to show Remark 1 can be applied for a coarse measure of team’s productivity, revenue per worker(revenue TFP) RT F P (x)r (q (x)) = [cq (x) + f ]. Alternatively, the quality parameters x, z1 , and z2 can beinterpreted as productivity in such a model where teams produce symmetric goods in a Cobb-Douglass production
technology.
20
From Proposition 3, it is possible that even …nal producers that upgrade the …nal good must exit since the degreeof their quality upgrading is smaller than that of high quality …rms. Therefore, some might think that the quality

25

3.5

Complete Specialization Equilibrium

When fI is su¢ ciently small, countries specialize in the …nal good sector and the LEC sector.
All teams are international teams. Because of the symmetry of Home and Foreign, Home …nal
producers are matched with a half of Home Z1-suppliers and a half of Foreign Z2-suppliers. The
world economy is equivalent with a closed economy with 2L of labor endowment, common entry
costs fZ1e both in Z1-sector and in Z2-sector, and production …xed costs fI + f instead of f . The
assignment functions are
mZ1 (x) = mZ2 (x) = maZ1 (x) =

fXe
fZ1

x for x

xL .

(27)

Point D in Figure 9 expresses mZi (x) with iso-quality curve of q(x). Points A, A*, and C are
those for the Home autarky equilibrium, the Foreign autarky equilibrium, and the incomplete
specialization equilibrium in Figure 8. The quality of …nal goods produced by a …nal producer withgiven quality is higher than in an incomplete specialization equilibrium. Since (27) is independentof fI , trade liberalization does not a¤ect the quality of …nal goods. The mass of entrants of …nal
producers and Zi-suppliers are the same as the autarky, which are also independent of fI . The
lowest quality thresholds in …nal goods sector and in LEC sectors are
xL =

1=k
1=k
f + fIf + fI(k 3), ziL =and M =fXe (k 3)
fZie (k 3)kL
f + fI:

Therefore, trade liberalization increases the mass of consumption varieties, by lowering the lowest
quality thresholds.

4

Concluding Remarks

This paper presents a new mechanism of qualty-upgrading in a tractable general equilibrium
model of matching of …rms heterogeneous in product quality. Trade in intermediate goods between
developed countries raises the quality of …nal goods by improving matching of …rms in a production
upgrading of …rms that cannot import is not important if they cannot survive. However, notice that Proposition 4
heavily depends on the assumption that the total expenditure on …nal goods is constant. If the quality upgrading
expands the total expenditure, e.g. in a multi-industry setting, it would be possible that all …nal producers upgrading
the product quality can survive after trade liberalization.

26

process. The quality upgrading arises both from the short-run e¤ect of convergence in the matchingof …rms and from the long-run competition e¤ect of specialization. The model provides a numberof plausible predictions on …rm-level trade in contrast to the previous model of heterogeneous …rm
models. Firms selectively trade with those with similar characteristics. Trade costs concentrate
both exporting and importing into a small portion of large …rms producing high quality products,
though some portion of large and high quality …rms always choose not to trade. Trade upgrades
the quality of …nal producers that do not use imported intermediate goods, which is supported byan empirical study by Amiti and Konings (2007).
The model presented in the paper is highly simpli…ed. I remark on some extensions. The
current model abstracts away from several frictions in matching, especially search frictions andincomplete contracts. By introducing a dynamic search process into two-sided matching, Shimer
and Smith (2000) con…rm the assortative matching holds on average with some deviations. Since
the main predictions in the current paper are derived from the assortative matching result, an
introduction of search frictions will make the model quantitatively more realistic with maintaining
the qualitative predictions. Second, the hold up problem due to incomplete contracts a¤ects
the organizational form of production teams, FDI or arm’s length. An introduction of contract
costs will allow us to examine the interaction between matching of heterogeneous …rms and …rm’s
boundaries.
An alternative model of vertically di¤erentiated goods might be a market-based model withlinear pricing, in which suppliers announce the price and quality of intermediate goods and wait
for …nal producers to come in an imperfectly competitive market. The market-based model may
be realistic for standardized intermediate goods, e.g. steal. Under the imperfect competition, eventhe best supplier does not usually take all of the market in order to raise a price; therefore, suchmodel will again see a matching problem between suppliers and …nal producers. Whether gains
from international matching found in this paper continue to exit is an interesting question for
future research.
Finally, empirical studies on international matching are necessary. To construct an ideal data
set to directly test assortative matching, one needs to match customs transaction data with data27

on characteristics of exporters and importers in at least two countries. Although international
matching of …rm-level data is currently very di¢ cult, I believe that it will greatly improve ourunderstanding of …rms’trade.

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Arkolakis, Costas. 2008. “Market Penetration Costs and the New Consumers Margin in International Trade.” NBER working paper 14214.
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5

Appendix

Proof for Lemma 1

Without loss of generality, it is su¢ cient to show neither of the following

two cases holds: (i) there exist x and x0 such that x > x0 , mZ1 (x) > mZ1 (x0 ) ; and mZ2 (x0 ) >mZ2 (x); (ii) there exist x and x0 such that x > x0 , mZ1 (x0 ) > mZ1 (x), and mZ2 (x0 ) > mZ2 (x).
(i) Suppose x > x0 and mZ1 (x) > mZ1 (x0 ). By de…nition of mZi (x) andX (x), it follows

that
X (x)

and

X

x0

= AxmZ1 (x) mZ2 (x)f

Z1 (mZ1 (x))

Z2 (mZ2 (x))

AxmZ1 (x) mZ2 x0f

Z1 (mZ1 (x))

Z2

= Ax0 mZ1 x0 mZ2 x0
Ax0 mZ1 x0 mZ2 (x)

f
Z1

32

Z1

mZ1 x0mZ1 x0mZ2 x0Z2;(28)

mZ2 x0Z2 (mZ2 (x)) :(29)

By adding (28) to (29), I obtain
AxmZ1 (x) mZ2 (x) + Ax0 mZ1 x0 mZ2 x0x0 mZ1 x0= A xmZ1 (x)mZ2 (x)The inequality (30) implies mZ2 (x)AxmZ1 (x) mZ2 x0

mZ2 x0Ax0 mZ1 x0 mZ2 (x)

0:

(30)

mZ2 (x0 ).

(ii) Suppose x > x0 and mZ1 (x0 ) > mZ1 (x). By de…nition of mZi (x) andX (x), it follows

that
X (x)

and

X

x0

= AxmZ1 (x) mZ2 (x)f

Z1 (mZ1 (x))

Z2 (mZ2 (x))

AxmZ1 x0 mZ2 x0

f

Z1

mZ1 x0Z2

mZ2 x0= Ax0 mZ1 x0 mZ2 x0

f

Z1

mZ1 x0Z2

mZ2 x0Ax0 mZ1 (x) mZ2 (x)f

Z1 (mZ1 (x))

Z2 (mZ2 (x))

;

(31)

(32)

By adding (31) to (32), I obtain
AxmZ1 (x) mZ2 (x) + Ax0 mZ1 x0 mZ2 x0
= A x

x0

AxmZ1 x0 mZ2 x0

mZ1 x0 mZ2 x0

mZ1 (x) mZ2 (x)The inequality (30) implies mZ2 (x)Ax0 mZ1 x0 mZ2 x0

0

(33)

mZ2 (x0 ). Q.E.D.

(i) Suppose the two countries are in autarky for some …nite fI . Sinceia (z) = z 3 f (z a ) 3
a ) 3 =3 and z a > z a hold from (13) and (16), there must
(z
Z1
1L
1L
1L
1L

Proof for Lemma 2a
Z1 (z)

be some z 0 such thath0Z1 (z ) >0Z1 (z ) + fI .

Then, …nal producers and Z2-suppliers in Foreign

prefer Home Z1-suppliers with quality z 0 to Foreign Z1-suppliers with the same quality. Therefore,
the autarky matching is unstable for any …nite fI .
(ii) Suppose the two countries have positive mass of entrants in …nal good sectors and in LEC
sectors, but no entrant in HEC sectors. I show that when very small mass of Home Z2-suppliers
enter, their expected pro…ts must exceed entry costs when fI is su¢ ciently large. Suppose that themass of the new entrants is so small that it does not change matching pattern. If the new Home
Z2-suppliers draw z2 > z2L (= z1L ), then trade costs allow them to earns
Z1 (z2 ) + fI :33

Z2 (z2 ) =Z2 (z2 )+fI =Notice that the allocation in a complete specialization equilibrium with fI is equivalent withthe one in the free trade equilibrium in which …xed cost f is replaced with f + fI since all teams
must pay fI . From (13) and (16), it follows that
1=k
1=kf + fIf + fI; z1L =; andxL =fXe (k 3)
fZ1e (k 3)(f + fI )Z1 (z) =
3

"

z
z1L

3

#

1 . (34)

Then, I obtain
Z2 (z2 ) =z23 (f + fI )(k 3)=k [(k3Z1 (z2 ) + fI =3) fZ1e ]3=k

+

2fI

f
3

if z2 2 [z1L ; 1). (35)On the other hand, when the new Home Z2-suppliers draw z20 < z1L , they receive approximately
all of the joint pro…t by forming teams with …nal producers with xL
z1L

z1 for some very small

x > 0 andx and Z1-suppliers withz1 > 0 and by paying them only approximately zeropro…t, i.e.
00Z2 z2 ' AxL z1L z2

f =f

z20
z2L

1 :

(36)

I used the cuto¤ condition (8) to obtain (36). From AxL z1L z2L = f + fI , z2L = z1L , and (34), z2L
is obtained asz2L =f
AxL z1L=f(k 1)=k(f + fI )[fZ1e (k3)]1=k:(37)

Since z2L decreases in fI from (37) and Z2 (z2 ) increases in fI for all z2
z2L from (35) andR1(??), [1 G (z2L )] Z2 = z2L Z2 (z) g(z)dz is increasing in fI . Therefore, for su¢ ciently high fI ,the expected pro…t of Z2-suppliers must be strictly positive so that countries must produce bothintermediate goods. Q.E.D.

Proof for Lemma 3

It is su¢ cient to consider two cases. Case (i): an exportable Home Zi-

supplier with quality zi is matched with a Foreign …nal producer with x and a Foreign Zj-supplierwith zj . Suppose zi 6= zj . From the mirror-image structure of Home and Foreign,for all z. Therefore, the Foreign …nal producer earns
X (x)= Axzi zj

Zi (zi )Zj (zj )fIf= Axzi zj

Zj (zi )Zj (zj )

fI

f

Zj

zj0

fI

max
Axzi0 zj0
0 0
zi ;zj

Zj

zi0

34

f:

Zi (z) =Zj (z)

Since the second order condition for maximization requires
Axzi zj

Zj (zi )Zj (zj )fI00
Zj (z) > 0, z

< Ax (z)2

2 Zj (z)= Ax (z)2

Zi (z)f(zi + zj ) =2 satis…esfI

Zj (z)f

fI :The inequality implies that the Home …nal producer with x forms a new team with a Home
Zi-supplier with z and Foreign Zj-supplier with z, which contradicts with stable matching.
Case (ii): an exportable Home Zi-supplier with quality zi is matched with a Home …nal
producer with x and a Home Zj-supplier with zj . Suppose zi 6= zj . From the mirror-image
structure, Foreign Zj-suppliers with quality zi are also exportable. Since

Zi (z) =Zj (z) holds

for all z, the Home …nal producer earns
X (x)= Axzi zj

Zi (zi )Zj (zj )f= Axzi zj

Zi (zi )Zj (zj )f

fI= Axzi zj

Zj (zi )Zj (zj )

f

fI

Zj

zj0

fI

max
Axzi0 zj0
0 0
zi ;zjFrom the second order condition
Axzi zj

Zj (zi )Zj (zj )

zi0

Zj

00
Zj (z) > 0, zfIf:(zi + zj ) =2 satis…es< Ax (z)2

2 Zj (z)= Ax (z)2

Zi (z)ffI

Zj (z)f

fI :The inequality implies that the Home …nal producer with x forms a new team with a Home Zisupplier with z and Foreign Zj-supplier with z, which contradicts with stable matching. Q.E.D.

Proof for Lemma 4

The proof consists of Claims 1 to 3.

Claim 1 Home Z2-suppliers and Foreign Z1-suppliers are non-exportable.
Proof. Suppose Home Zi-suppliers with quality z are exportable. Under the assortative matching,
the market clearing condition for matching between Home Zi-suppliers is expressed
MZ1e

Z 1
z

D
Z1 (t) g(t)dt = MZ2e

Z 1
z

D
Z2 (t) g(t)dt;

35

(38)

whereD
Zi (zi ) is the share of Home Zi-suppliers with quality zi choosing domestic partners.

ByD
Zi (zi ) < 1 holds only when Home Zi-suppliers with quality zi are exportable.

Ade…nition,di¤erentiation of (38) with respect to z leads toD
MZ1e D
Z1 (z) = MZ2e Z2 (z) :(39)

Notice that if Home Zi-suppliers with quality z are exportable,

Zi (z) =Zi (z)

fI , then Home

Zj-suppliers with quality z are non-exportable since under the mirror-image structure, it follows
that

Zj (z) =Zj (z)D
Z1 (z) orD
Z2 (z) can be smaller than unity.

D
Z2 (z) = 1 andD
Z1 (z) = MZ2e =MZ1e < 1 satis…es

fI . Therefore, only one of

From MZ1e > MZ2e , only a combination of

condition (39). Therefore, Home Z2-suppliers are all non-exportable. Q.E.D.
Claim 2 mZ1 (x)Proof. SinceD
Z2 (z) = 1 for all z

Z 1

MZ1e

mZ2 (x) for all x

mZ1 (x)xL .

z2L from Claim 1, the market clearing condition (38) becomesD
Z1 (t) g(t)dt = MZ2e [1G (mZ2 (x))] for all xxL .

A straightforward manipulation yields
MZ1e
MZ2e
Since

Z 1

mZ1 (x)D
Z1 (z)D
Z1 (t)

MZ2e
MZ1e

MZ2e =MZ1e for all x

non-negative for all x

xL . Q.E.D.

Claim 3 (i)Z1 (z) =Z2 (z)

Z1 (z)

g(t)dt = G (mZ1 (x))

G (mZ2 (x)) for all x

xL .

(40)

xL from the proof for Claim 1, the left hand side of (40) is

Z2 (z)Z2 (z) = fI for all z

zT . (ii) 0Z1 (z)

Z1 (z) =Z2 (z) < fI for all z 2 [z1L ; zT ).

Proof. (i) Consider two teams with bundles of quality parameters, (x; z1 ; z2 ) and (x0 ; z10 ; z20 ),respectively. Suppose z2 = z10 (z^). Claim 2 implies that x

x0 and z1z2 = z10

z20 .

Therefore, from the …rst order condition, we obtain
0z ) = Axz1
Z2 (^0z ) = Ax0 z20 .
Z1 (^

(41)

In a complete specialization equilibrium, there exists some exportable Home Z1-supplier withquality z~

zL such that

z)
Z1 (~

z ) = fI .
Z1 (~
36

Suppose there exists z > z~ such that

Z1 (z)

Z1 (z) < fI on the contrary.

Since

Z1 (z) =Z2 (z) holds in equilibrium, the di¤erence in the

pro…t schedules satis…es
Z1 (z)

Z1 (z) = fI +Z z

0
Z2 (u)zT0
Z1 (u)

du:The second term in the right hand side must be non-negative from (41), which it contradicts withZ1 (z)

Z1 (z) < fI .

Z1 (z) = fI

0
Z1 (z )

Therefore, ifholds for all z

0
Z1 (z ) = fI

holds for some z 0 , thenZ1 (z)

z 0 . (ii) Since z2L = z1L < z1L = z2L from MZ2e > MZ1e , thedi¤erence in the pro…t schedules is
Z1 (z)

Z1 (z) =From (41),Z1 (z)

Z1 (z1L ) +

Z z

z1L

0
Z2 (u)0Z1 (u)

du for all z 2 [z1L ; zT ).

(42)

Z1 (z) 2 [0; fI ) for z 2 [z1L ; zT ). Q.E.D.

Finally, I prove LEC sectors have more entrants than HEC sectors. Q.E.D.

Suppose MZ1e = MZ2e < MZ2e = MZ1e , on the contrary. Then, from

Proof for Lemma 6xL ;0 (z) =
Z1

0 (z) for z < z , it requires z
T
1L > z2L , which contradicts with mZ2 (x)Z2mZ1 (x)similar arguments in Claims 1 to 3, it is possible to show mZ2 (x)0 (z) for z
Z2

0 (z)
Z1

zT andmZ1 (x) for x

0 (z) for z < z .
T
Z2

From integration by parts, the free entry condition becomesfZie = [1G (ziL )]

Zi =

Z zT

0
Zi (t) [1ziL

0 (z) =
Z1

Since

fZ2e

0 (z) for z
Z2

fZ1e =

Z zT
z2L

Since

0 (z)
Z1

for x

xL . Q.E.D.

Proof for Lemma 7

G (t)] dt +

Z 1
zT

0
Zi (t) [1zT , the di¤erence in the free entry conditions is

0
Z2 (t) [1

G (t)] dt

Z zT
z1L

(1) From k [10Z1 (t) [1

G (t)] dt > 0.

G (x)] = xg(x) and the integration by parts, the free entry

condition can be rewritten as

1fXe
G (xL )

=
=

1
A
q:
k

G (t)]du:A
G (xL )Z 1

mZ1 (t) mZ1 (t) [1xL

37

G (t)] dtSince A = L= ( M q) from the aggregate zero pro…t, it follows that
MXe =

M
L
=
G (xL )
fXe k

1

a
= MXe
:

(2)(3)The proof for (2) and (3) consists of Claims 5 to 7.
Claim 4 Let

xm0Zi (x) =mZi (x). Then, it follows that

Zi (x)Zi (x) = 1 for x > xT and

Zi (x) =MXe [1 G (x)]:MZie [1 G (mZi (x))]

Proof. Since mZi (x) is linear in x for x > xT ,k

1
mZi (x)

MXe
=
MZie

k

1
x

+

1
zT

From the Implicit Function theorem,
Zi (x) =

MXe
MZie

k

mZi (x)x=k

Zi (x) = 1 for x > xT . From (21), mZi (x) satis…es

MXe
MZie

1
xT

k

for x 2 [xL ; xT ]:Zi (x) is solved as

MXe [1 G (x)]for x 2 [xL ; xT ]:MZie [1 G (mZi (x))](43)

Claim 5 The mass of entrants satis…es
MZ1e fZ1e MZ2e fZ2e+
= 2:
MXe fXeMXe fXe(44)

Proof. From the integration by parts and the …rst order condition, the free entry condition for…nal producers is
fXe =
=

Z 1
x
Z L1
xL

= A

X (t) g(t)dt
0
X (t) [1Z 1G(t)] dt

mZ1 (t) mZ2 (t) [1G(t)] dt:

(45)

xL

From Claim 4, the free entry condition for Zi-suppliers isfZie = A
= A

Z 1
x
Z L1

mZ1 (t) mZ2 (t) Zi (t) [1G(mZi (t))] dt

mZ1 (t) mZ2 (t) [1 G(mZi (t))] dt
Z xT
MXe
+
A
mZ1 (t) mZ2 (t) [1 G(t)] dt:
MZie
xL
xT

38

(46)

Since 2MXe [1G (x)] =P

i=1;2 MZie [1

MZ1e
MZ2e
fZ1e +
fZ2e = 2A
MXe
MXe

Z 1

G (mZi (x))] for x

mZ1 (t) mZ2 (t) [1xL , it follows that

G(t)] dt = 2fXe :xLClaim 6
a
Ma
MZ2e
MZ1e
MZ2e
> Z1e
>
>
:
a
a
MXe
MXe
MXe
MXe
a =M aProof. From fZ1e =fXe = MXe
Z1e and (11), condition (45) is rewritten as
Z 1
fZ1e
fZ1e = fXe=A
mZ1 (t) mZ2 (t) [1 G(maZ1 (t))] dt
fXe
xT
Z xT
a
MXe
+ a A
mZ1 (t) mZ2 (t) [1 G(t)] dt
MZ1exL

Since maZ1 (x) > mZ1 (x) for x(47)xT , the comparison of (46) and (47) proves that MZ1e =MXe >a =M a . From (44), we also obtain M a =M a > M
MZ1eZ2e =MXe .
Xe
Z2e
Xe

Under the constraint of (44), it follows that
MZ1e + MZ2e
2MXe

a + Ma
a
MZ1e
MZ1e
MZ1e
Z2e
if
and
only
if
>
a
a .
2MXe
MXe
MXe

a + M a ) =2M a >Since (MZ1eZ2e
Xe

Proof for Proposition 3"(x) = K 1"

1(i) The degree of quality upgrading ismZi (x)

k

=

MXe
MZie

k

x
xT

for x 2 [xL ; xT ). From (48),1p aa =M a , K > 1 holds. Q.E.D.
MZ1e MZ2e
Xe
#"

x
xT

3

k

#

(MZ1e MZ2e )24MZ1e MZ2e

# 1=k

(48)

(x) is increasing in x. (ii) From (20) and (21), mZi (x) satis…es
1
x

k

+

1
zT

k

MZie MZje
2MZie

if x

a =M a , thenFrom MZ1e > MZ2e , if MZi =MXe are close to MZi
Xe
!k
1 2k
1 2k
1
(MXe )2=
MZ1e MZ2e xaLzT
mZ1 xaL mZ2 xaL

>

a )2
(MXe
a Ma
MZ1e
Z2e

1
xaL

2k

=

1
a
a
z1L z2L

xT .

(MZ1e MZ2e )24MZ1e MZ2ek:a z a > xa maataTherefore, q a (xaL ) = xaL z1L
2L
L Z1 (xL ) mZ2 (xL ) = q (xL ). Q.E.D.

39

!

Let

Proof for Proposition 4

X (xL ) be the average pro…t of …nal producers when the lowest

quality threshold is xL . From the integration by parts and the …rst order condition,
X

=
=

Z 1

x
Z L1

0
X (t)

xL

= A

g (t)
1 G (xL )1 G (t)
1 G (xL )

X (t)

Z 1

mZ1 (t) mZ2 (t)xL

X (xL ) isdt
dt
1 G (t)
1 G (xL )dt:

From the cuto¤ condition (31), it follows that
X

Z 1

Z 1
z1L z2L t2
1 G (t)z1L z2L t2
= A
dt
+
A
m
(t)
m
(t)
Z1
Z2
1 G (xL )(xL )2(xL )2xLxLZ 1
1 G (t)f
z1L z2L t2=+A
mZ1 (t) mZ2 (t)
dt:
2
k 3
1
G (xL )(xL )xL

1 G (t)
1 G (xL )dt

From assignment functions (25) and (26), if xL < xT , then mZ1 (x) mZ2 (x) > z1L z2L x2 = (xL )2
for all x

xL , while if xLxT , then mZ1 (x) mZ2 (x) = z1L z2L x2 = (xL )2 for all xxL .

Therefore, we havea
X (xL ) >

f
k

=

3

fXe
and
G xaL

1

X (xT ) =Therefore, there exists xL 2 (xaL ; xT ) such that

f
k

3

<

fXe
.
G (xT )1

X (xL ) = fXe [1G (xL )] 1 .

a ; z ). The average pro…t of Home Z1-suppliers when
It takes similar steps to prove z1L 2 (z1LT

the lowest quality threshold is z1L ,Z1 (z1L ) ; isZ 1

1 G (t)xL z2L t2
dt
2
1 G (z1L )
z1L (z1L )Z 1
xL z2L t2
1 G (t)+A
mZ11 (t) mZ2 mZ11 (t)dt2
1 G (z1L )(z1L )z1L
Z 1
xL z2L t2f
1 G (t)mZ11 (t) mZ2 mZ11 (t)
=
+A
2
k 3
1
G (z1L )(z1L )z1L

Z1 (z1L ) = A

dt;

where mZ11 ( ) is an inverse function of mZ1 ( ). From assignment functions (25) and (26), it follows
that if z1L < zT , mZ11 (z1 ) mZ2 mZ11 (z1 ) > z1L z2L z12 = (z1L )2 for all z1
mZ11 (z1 ) mZ2 mZ11 (z1 ) = z1L z2L z12 = (z1L )2 for all z1a
Z1 (z1L ) >

f
k

3

=

1

fZ1e
and
a
G z1L

Z1 (zT ) =a ; z ) such that
Therefore, there exists z1L 2 (z1LT40

z1L . Therefore, we havefkz1L ; while if z1L

3

<

fZ1e
.
1 G (zT )Z1 (z1L ) = fZ1e [1G (z1L )] 1 .

zT ,Finally,
Z 1

xL z1L t2
1 G (t)dt
2
1
G (z2L )z2L (z2L )Z 1
xL z1L t2
1 G (t)+A
mZ21 (t) mZ1 mZ21 (t)dt2
1
G (z2L )(z2L )z2L
Z 1
xL z1L t2f
1 G (t)+A
mZ21 (t) mZ1 mZ21 (t)
=
2
k 3
1
G (z2L )(z2L )z2L

Z2 (z2L ) = A

dt:

From assignment functions (25) and (26), it follows that if z1L < zT , mZ21 (z2 ) mZ1 mZ21 (z2 ) <xL z1L z22 = (z2L )2 for all z2
for all z2

zT , mZ21 (z2 ) mZ1 mZ21 (z2 ) = xL z1L z22 = (z2L )2z2L , while if z1Lz2L . Therefore, we haveaZ2 (z2L )

f
k

3

=

1

fZ2e
a
G z2L

1

fZ2e
if z2LG (z2L )az2L.

a must hold.
Therefore, z2L < z2L(ii) The average revenue of teams is r =the mass of international teams. Since

Z1 +

( X+

Z1 +

Z2 + f + fI (MI =M )), where MI

is

Z2 = 2 X holds both in the autarky and in the

trade equilibrium, the mass of consumption varieties is
M=

L
=
r

L
:
(3 X + f + fI (MI =M ))From the proof for (i) of this Lemma,

X is higher than the autarky level.

Q.E.D.

41

Therefore, M < M a .

Figure 1: Matching market equilibrium. Firms with lower quality than the thresholds xL , z1L ,and z2L exit. The mass of the survival …rms MXe [1G (xL )] and MZie [1G (ziL )] , which are

the areas of squares with solid lines must be equalized. The assortative matching implies the massof …nal producers with higher quality than x, which is the area of a grey square at the left, must
be equal to the mass of Zi-suppliers with higher quality than mZi (x), which are the areas of greysquares at the center and the right.

42

43

Figure 2: As the mass of entrants into Zi-sector and …nal goods sector, MZie , increases, a …nal
producer with quality x becomes matched with higher quality of Zi-supplier with mZi (x) though
the marginal improvement is diminishing.

44

zi

mzi(x)MZie
M Xe
Figure 3: The quality of Zi-supplier matched with a …nal producer with quality x, mZi (x), is
increasing and concave in the relative mass of entrants into Zi-sector and …nal goods sector,
MZie =MXe .

45

*

a

m*Z2(x)A*aA

mZ2(x)aq(x)=q(x)am*Z1(x)amZ1(x)*

Figure 4: Autarky matching: The curve depicts a combination of the quality of Zi-suppliers inteams that produces …nal goods of quality q a (x) with a …nal producer with quality x. Point A andA
Point A* expresse the quality of Zi-suppliers in Home autarky teams (mAZ1 (x) ; mZ2 (x)) and inA
Foreign autarky teams (mAZ1 (x) ; mZ2 (x)), respectively. In each team, a Zi-supplier in low entry

cost sector has higher quality than the other Zj-supplier in a symmetric way.

46

Figure 5: Fixed trade costs separate …rms into three groups: high quality …rms in Home andForeign, which are expressed in grey areas, are matched together; low quality …rms in each country,
which are expressed in the same stripe areas, are matched locally.

47

Figure 6: Equilibrium trade patterns. Home …nal producers in area A import from Foreign Z2suppliers in Area A’. Foreign …nal producers in area B import from Home Z1-suppliers in Area
B’.

48

a

mZ1(x)mZ(x)mZs (x)(m Z1a (x)+m*Z1a(x))/2

mz1(x)am*Z1(x)

*a
MZ1e
MXea

a

a
MZ1e+M*Z1ea MZ1e+M*Z1e MZ1e
2MXe
2MXea
MXea

MZ1e
MXe

Figure 7: The quality of Z1-suppliers matched with high quality …nal producers with xxT ishigher than the average of those in two countries in the autarky. In the short run equilibrium,a + M a ) =2M a , is the avthe relative mass of entrants into Z1-sector to …nal producers, (MZ1eZ1e
Xe
a =M a and M a =M a . The quality of Z1-suppliers matchederage of those in the autarky, MZ1e
Xe
Z1e
Xe

with …nal producers with quality x, msZ1 (x), is higher than the average of the two autarky levels, maZ1 (x) and maZ1 (x). In the long run, the relative mass of Z1-suppliers to …nal producer,(MZ1e + MZ1e ) =2MXe , increases so that mZ1 (x) rises further.

49

Figure 8: After the opening of trade, high quality …nal producers with quality xxT upgrade thequality of …nal goods both in the short run, from q a (x) to q s (x), and in the long run, from q s (x)to q l (x). In the short run, …nal producers change Zi-suppliers with maZi (x) to those with msZ (x);in the long run, …nal producers change Zi-suppliers with msZi (x) to those with mZ (x).

50

Figure 9: In a complete specialization equilibrium, a …nal producer with quality x is matched with
Zi-suppliers with quality maZ1 (x) = maZ2 (x) expressed by Point D. The quality of …nal good q (x)is higher than the level in the incomplete specialization equilibrium.

51