HIAS-E-62

A Matching Theory of Global Supply Chains
Yoichi Sugita
Graduate School of Economics, Hitotsubashi University, Japan

December 2017

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A Matching Theory of Global Supply Chains
Yoichi Sugita ⇤Hitotsubashi University

December 2017

Abstract: This paper develops a simple general equilibrium model of global supply chains (GSCs)
that jointly addresses three key decisions of firms forming GSCs, namely selection (whether to form
a GSC), location (where to find GSC partners), and matching (with which firms to form a GSC). The
model develops a Becker type assortative matching model of final producers and suppliers both ofwhich are heterogeneous in capability (productivity/quality) of their tasks, and integrates it with aMelitz type model of selection and a Ricardian comparative advantage model of location. The model
presents a new mechanism of gains from trade associated with firm heterogeneity. Namely, tradeliberalization causes rematching of firms toward positive assortative matching at the world level as arecent empirical study on exporter-importer matching data observes.
JEL classification: F12.
Keywords: Global supply chains, firm heterogeneity, two-sided heterogeneity, matching, trade inintermediate goods, quality differentiation.
Acknowledgments: The old version was circulated under the title “Matching, Quality, and Comparative Advantage: A Unified Theory of Heterogeneous Firm Trade.” I thank Don Davis for his
constant encouragement. I also thank JaeBin Ahn, Kyle Bagwell, Ron Findlay, Takakazu Honryo,
Jota Ishikawa, Amit Khandelwal, Alex McQuoid, Hitoshi Sato, Guru Sethupathy, Hitoshi Shigeoka,
Kensuke Teshima, Catherine Thomas, Eric Verhoogen, Jon Vogel, David Weinstein and seminar participants at Boston College, Bank of Japan, Columbia University, the EEA meeting in Oslo, Erasmus
University, Hitotsubashi University, Hong Kong University of Science and Technology, the Midwest
International Trade Meeting in Madison, Stockholm School of Economics, and Waseda University.
Financial support from the Wallander Foundation is gratefully acknowledged.

⇤

Graduate School of Economics, Hitotsubashi University, Japan. (E-mail: yoichi.sugita@r.hit-u.ac.jp.

1

Introduction

International trade has been traditionally modeled as cross-border transaction between consumers
and firms representing industries, but the last decade has witnessed rises in two strands of literature challenging this traditional model. The first one is the “global supply chains” (GSCs) literature(e.g., Jones and Kierzkowski, 1990; Yi, 2003; Grossman and Rossi-Hansberg, 2007).1 A production process in modern manufacturing is rarely finished within one country and often takes a formof team or “chain” of firms across different countries specializing in different tasks. The majority
of international trade are nowadays firm-to-firm transactions of intermediate goods rather than firmto-consumer transactions of final goods.2 The second “heterogeneous firm trade” (HFT) literatureemphasizes that a representative firm model may not fully capture the impact of trade liberalization on industries. A robust finding that only firms that have high productivity/quality engage inexporting outputs and/or importing inputs within industries has developed new theories that tradeliberalization improves industrial performance by reallocating resources toward more capable firms
within industries (e.g. Melitz, 2003; Bernard, Jensen, Eaton, and Kortum, 2003).3 These advancessuggest that to better understand modern international trade and its consequences, we need a modelon how heterogeneous firms form GSCs.
This paper develops an integrated model of GSCs with firm heterogeneity. The model jointly
addresses three key decisions of firms forming GSCs, namely “selection” (whether to form a GSC),“location” (where to find GSC partners), and “matching” (with which firms to form a GSC) in a simple general equilibrium framework. Among these three decisions, the GSC and HTF literatures haveaddressed selection and location. A major innovation of this paper is to model trade in intermediate
goods in GSCs as two-sided matching of heterogeneous importers (final producers) and heterogeneous exporters (suppliers) and to integrate it with standard models of selection and location. The
model demonstrates a new mechanism of gains from trade associated with firm heterogeneity andreplicates a variety of facts on exporters and importers.
The empirical trade literature has recently started studying matching of exporters and importers
using transaction level data. Though the literature is still in an early stage, several studies alreadyfound evidence that firms do not act in anonymous markets and that matching matters for firms.
First, firms trade with few partners in contrast to the standard love of variety model where every
exporter trades with every importer. In a narrowly defined product (e.g. HS 6 digit product) in one
1

Global supply chains are called by many other names, e.g. fragmentation of production process, offshoring, outsourcing, and vertical integration.
2
For instance, Bems, Johnson and Yi (2010) estimate that trade in intermediate goods account for 64% of the world
trade.
3
See, for example, Bernard and Jensen (1995, 1999) and Pavcnik (2002) for such findings that motivated the theories.

1

country, a typical exporter trades with only few importers, typically median one (Blum, Claro andHorstmann, 2010, 2012; Eaton, Eslava, Jinkins, Krizan, and Tybout, 2012; Carballo, Ottaviano, andVolpe Martincus, 2013; Sugita, Teshima, and Seira, 2015). Furthermore, a companion paper, Sugita
et al. (2015), found that even firms trading one product with multiple partners in one country conductmost of trade (more than 80%) with the single main partners in Mexican textile/apparel exports to
the US. Since firms trade with few firms, choosing right partners should be crucial for both exporters
and importers. Second, matching seems systematically related with firm’s characteristics instead ofpurely random matching. Sugita et al. (2015) investigated how matching of Mexican exporters and
US importers changed when the US removed quota on textile/apparel products in 2005 and had ahuge entry of Chinese exporters. They found US importers switched their Mexican partners to thosemaking greater pre-shock exports whereas Mexican exporters switched their US partners to thosemaking fewer pre-shock imports.
James Rauch (1996) has pioneered modeling firm-to-firm trade as two-sided matching and developed a series of models with his coauthors. Following Rauch, I consider matching of exporters andimporters as an assignment model where firms can trade with a given number of partners for someexogenous reasons (e.g. transaction costs). To extend Rauch’s models where firms are symmetric,
the model incorporates firm heterogeneity in capability, which can be interpreted as productivity or
quality, and the complementarity of firm capability within matches. That is, the model is the Becker(1973) marriage market model where final producers and suppliers match positive assortatively bycapability.
I integrate the Beckerian matching model with standard models of selection and location: aMelitz (2003) type model of firm selection and a Ricardian comparative advantage model of GSC
location. Each firm specializes in one task in a supply chain and becomes heterogeneous in capability by drawing random capability parameters at their entry as in Melitz (2003). After seeing theircapabilities, final producers and suppliers match in one-to-one under perfect information. Thanks tocomplementarity within matches, a stable matching becomes positive assortative matching (PAM) by
capability. Highly capable final producers match with highly capable suppliers while low capability
final producers match with low capability suppliers.
I first consider international matching without any trade costs in a two country setting. Countries
differ in entry costs into intermediate good sectors, which gives a rise to the Ricardian comparativeadvantage for international matching. Each country has relatively more high capability suppliers inthose sectors with relatively low entry costs (“comparative advantage” sectors). Final producers arewilling to match suppliers in foreign comparative advantage sectors (i.e. own country’s comparative
disadvantage sectors). The opening of international matching causes rematching of firms toward
2

PAM at the world level. High capable firms match with high capable firms in foreign countries.
This rematching toward PAM leads to an efficient use of technology exhibiting complementarity andraises the world welfare. When production technology of a supply chain exhibits quasi-concavity
in the capability of firms in each stage, this rematching improves technology for production of eachvariety of final goods, leading to higher quality or cheaper prices. Since trade volume is increasing infirm capability, these predicted rematching is consistent with the finding on Mexico-US rematchingby Sugita et al. (2005).
The free trade model incorporates matching and location of GSCs, but not selection. To incorporate selection, I introduce fixed costs of international matching, which allow only high quality firms
to trade. Some of high quality firms in the comparative advantage sectors form GSCs with foreign
firms, while other firms form local supply chains (LSCs) with domestic firms. PAM naturally explains why exporters and importers are on average larger and more capable than nontrading firms.
It is because firms trade with those with similar capability ranks. With fixed trade costs, the opening of international matching improves production technology of highly capable final producers, butworsen that of low capable final producers. Low capable final producers reduce product quality or increase prices. Some of the least capable final producers exit and the total number of varieties of final
goods decrease. Despite of these negative effects on the consumer welfare, the model predicts thatthe opening of trade increases the aggregate welfare by improving matching among high capability
firms.

Related Literature
The current paper is related to three strands of literature.
GSC Location

First of all, the current model is related to the GSC models that emphasize compar-

ative advantage as an important determinant of location of GSCs (e.g. Dixit and Grossman, 1982;
Sanyal and Jones, 1982; Sanyal, 1983; Feenstra and Hanson, 1996; Yi, 2003; Grossman and RossiHansberg, 2008; Costinot, Vogel, and Wang, 2013; Baldwin and Venables, 2013). These models do
not analyze heterogeneous firms, but have richer structures such as multiple stages, multiple countries, and multiple factors than the current model. Therefore, the current model should be regarded
as complement to these models rather than substitute.
Firm Selection

Second, the current model is related to the HFT models that has mainly focused on

selection of firms in exporting or importing. The literature has started from models of selection ofexporters on productivity (Melitz, 2003; Bernard et al., 2003). Verhoogen (2008), Baldwin and Har3

rigan (2011) and Johnson (2012) developed models with selection of exporters on quality. Bernard,
Redding and Schott (2007) and Okubo (2010) developed multi-industry versions of the Melitz models featuring a Heckscher-Ohlin type and a Ricardo type of comparative advantage, respectively. To
address facts on selection of importers, Halpern, Koren, and Szeidl (2009) and Kasahara and Lapham(2013) introduced productivity heterogeneity of final producers in the love of variety model of trade
in intermediates.
Though these previous studies on firm selection only considered the heterogeneity of either exporters or importers, a recent study by Bernard, Moxnes, and Ulltveit-Moe (2013) developed models
of selections of both heterogeneous exporters and importers based on productivity.4 Their modeland the current model consider different types of trade in intermediate goods: in Bernard et al.
(2013), exporters produce horizontally differentiated goods, the reason for forming GSCs is the loveof varieties of intermediate goods, while in the current model, exporters produce vertically (quality)
differentiated goods and the reason for forming GSCs is comparative advantage.5
Matching and Trade

James Rauch (1996) has proposed modeling firm-to-firm trade as two-sided

matching and developed a series of models with his coauthors (Casella and Rauch, 2001; Rauch andCasella, 2003; Rauch and Trindade, 2003). Since these pioneering works were developed before
the HFT literature has emerged, the early models analyze matching of symmetric and horizontallydifferentiated exporters and importers. These studies introduce random matching to express uncertainties in international trade and investigate how institutional and technological devices overcome
uncertainties.6
The previous studies in the trade literature applied the assortative matching model mainly formatching of heterogeneous workers within/across countries.7 Grossman and Maggi (2000), Ohnsorge and Trefler (2007), Costinot (2009), Costinot and Vogel (2010) and Grossman, Helpman, andKircher (2103) considered local matching of workers and industries to generalize the comparative advantage theory into the case that worker’s skills are of continuous type. Kremer and Maskin (2006)
and Antras, Garicano, and Rossi-Hansberg (2006) developed international matching of Southern production workers and Northern managers by their skills. Costinot, Vogel, and Wang (2013) analyzed
international matching of workers to form global supply chains in a multi-country model. There4

As a related study, Carballo et al. (2013) developed a model of trade between symmetrically heterogeneous consumers
and heterogeneous exporters where consumers have different tastes as in ideal variety models.
5
To see the difference between horizontal and vertical differentiations, suppose that goods were sold under an identicalprice and with no trade costs. Then, in Bernard et al. (2013), all importers would be willing to buy from all exporters,
while in the current model, all importers would be willing to buy only from the highest quality exporters.
6
Casella and Rauch (2001) and Rauch and Casella (2003) considered ethnic group ties, Rauch and Trindade (2003)
information technology, and Grossman and Helpman (2005) internalization through vertical integration.
7
An important exception is Nocke and Yeaple (2008) who analyze international M&As as matching of corporate assets
and managers.

4are two important differences between these worker-matching models and the current firm-matchingmodel. First, in the former models, workers are heterogeneous in skills but all firms (or production
teams of workers) use identical technology, while in the current model, workers are homogeneous but
firms that are heterogeneous in technology (i.e. task quality). Second, in the former models, workers
are heterogeneous in skills ex ante before they choose tasks, while in the current model, firms arehomogeneous in capability ex ante and becomes heterogeneous after they enter.8
The rest of the paper is organized as follows. Section 2 sets up a model in a closed economy.
Section 4 introduces free trade in intermediate goods in a two country setting. Section 4 introduces
trade cost. Section 5 offers some concluding comments. Appendix presents calculations and proofs
in more detail.

2

Closed Economy

2.1

Basic Setting

Consider an economy that produces differentiated final goods from the only production factor, labor.
The labor endowment is given by L and the competitive wage is normalized to one.
Production of a final good requires three tasks, 1, 2, and X. Task i = 1, 2 produces different
intermediate goods Zi by using labor. Task X produces a final good by combining intermediate
goods Z1 , Z2 and labor. Each firm specializes in one task. I call three types of firms final producers,
Z1 suppliers, and Z2 suppliers, respectively, according to their specializations.
The model has three stages. In stage 1, firms become ex post heterogeneous in capability ina Melitz (2003) style. Capability represents productivity and/or quality, depending on parametersspecified below. There exist infinitely many potential entrants that are ex ante symmetric. Let x,
z1 , and z2 be the capability of final producers, Z1 producers, and Z2 producers, respectively. Each
entrant decides a task to specialize in, pays fixed entry costs, and independently draws capability
from an identical Pareto distribution. The density function is given by g(s) ⌘ k/sk+1 and the(1/s)k for s 2 [1, 1), where k > 3 is assumed

cumulative distribution function is by G(s) ⌘ 1

to ensure a finite GDP. While the probability of capability draws is symmetric across sectors, entry
costs are asymmetric: entry requires fXe units of labor for final producers and fie units of labor for
Zi producers (i = 1, 2). I assume f1e < fXe < f2e for expositional purposes, though all the main
results depend only on the difference of f1e and f2e . Firms are risk neutral so they enter until their
8

The two classes of models differ in the main predictions, e.g. on who receive a better match from the opening ofinternational matching. For instance, in Antras et al. (2006), the productivity and output of the highest skill managers
decreases in North, who could be interpreted as final producers importing intermediates, while in the current model, theproduct quality and output of the highest quality final producers always increase.

5

expected profits become zero.
In stage 2, after knowing their own capability, a final producer, a Z1 supplier, and a Z2 supplierform a team that produces one variety of final goods. For simplicity, matching is one-to-one(-to-one),so each firm joins only one team. Firms choose their partners and decide how to divide team’s total
profits in stage 3. Team capability ✓ depends on member’s capability. Thus, matching endogenously
determines the distribution of team capability ✓ across teams.
In stage 3, teams compete in a monopolistically competitive market as firms in Melitz (2003) do.
The representative consumer maximizes the following CES utility function:
U=Z✓ (!)↵ q (!)⇢ d!

1/⇢

subject to!2⌦

Zp(!)q (!) d! = I!2⌦

where ⌦ is the set of available varieties of final goods, q(!) is the consumer’s consumption of variety!, ✓(!) is the capability of the team producing !, and I is the aggregate income. Consumer’s demand
for a variety with price p and capability ✓ is derived as:q(p, ✓) =
where

I✓↵ p
P1

⌘ 1/ (1is the price index.

(1),⇢) > 1 is the elasticity of substitution and P ⌘⇥R!2⌦ p(!)

1✓ (!)↵ d!

⇤1/(1

)

Team capability depends on the capability of team members as follows:(2)✓ = xz1 z2 .

Team capability (2) is increasing, supermodular, and quasi-concave.9 The supermodularity capturescomplementarity among the quality of parts and components, following Kremer (1993) and Kugler
and Verhoogen (2008). For instance, if a car producer upgrades the quality of engine, the supermodularity implies that the marginal quality improvement of the car is positively related to the quality ofother components such as transmission, body, tires, etc. The quasi-concavity means that consumers
prefer final goods with moderate combinations of parts quality. For instance, consumers may prefera standard-class car with normal equipment to a luxury-class car with a poor air conditioner.10
Production technology is of Cobb-Douglass type. When a team produces q units of final goods,
Zi supplier in the team produces qi units of intermediate good Zi using li units of labor. Then,
combining these Z1 and Z2 with lx units of labor, the final producer assembles q units of final goods.
9

An increasing twice-differentiable function with positive cross-derivatives is called (strictly) supermodular.
An alternative interpretation of the quasi-concavity is that a moderate combination of the quality of intermediate goods
allows final producers to develop high quality final goods.
106The production functions of final producers and Zi suppliers are given as:
q=x

1/3 1/3

fx )1/3 q1 q2(lx

and qi = zi 3 (lifi ) ,The labor requirement for a team with capability ✓ producing q units of final goods becomes:(3)l(✓, q) = ✓ q + f,
where f ⌘ fx + f1 + f2 .

Team capability ✓ may represent productivity and/or quality, depending on parameters ↵ and .

When ↵ = 0 and< 0, ✓ may be called productivity. As firms in the Melitz model, all teams facesymmetric demand functions, while a team with high ✓ has lower marginal costs. When ↵ > 0 and> 0, ✓ may be called quality. As firms in Baldwin and Harrigan (2011) and Johnson (2012), teams
with high ✓ face a large demand at a given price but incur high marginal costs.

2.2

Autarky Equilibrium

This section solves for an autarky equilibrium by backward induction. Appendix presents calculations in more detail. I use subscripts i and j to denote variables and functions of Z1 suppliers and Z2
suppliers throughout the paper. They always mean that i, j 2 {1, 2} and that i 6= j when i and j areused together.

Stage 3: Production Stage

Since team’s marginal costs is ✓ , team’s optimal price is p(✓) =✓ /⇢. Hence, team’s revenue R(✓), total costs C(✓), and joint profits ⇧ (✓) areR(✓) = A✓ , C(✓) = (where A ⌘ I (⇢P )1. Parameter

1) A✓ + f, and ⇧ (✓) = A✓
⌘ ↵

(

(4)

1) summarizes how capability affects teamprofits. Since comparative statics on parameters ↵, , andnormalizef,is not the main interest of the paper, I= 1 by choosing the unit of ✓. This normalization greatly simplifies calculation. Then,team profits become:⇧(xz1 z2 ) = Axz1 z2

(5)

f.

Stage 2: Matching Stage Firms choose their partners and decide the distribution of team profits, taking A as given. Two types of functions, profit schedules, ⇡X (x) and ⇡i (zi ), and matching

7

functions, mi (x), characterize equilibrium matching. A final producer with capability x matcheswith Zi suppliers with capability mi (x) and receives the residual profit ⇡X (x) after paying profits⇡i (mi (x)) for the partners. Following the matching literature, I focus on stable matching that satisfies two conditions: (i) (individual rationality) no firm is willing to deviate from the current team toexit; (ii) (pair-wise stability) no trio of a final producer, a Z1 supplier, and a Z2 supplier is willingto deviate from their teams to form a new team.11 These two conditions are mathematically statedas follows: (i’) all firms earn non-negative profits, ⇡X (x)0 and ⇡i (zi )0 for all x and zi ; (ii’)

each firm is the optimal partner for the other team members:12⇡X (x) = ⇧ (xm1 (x) m2 (x))= max ⇧ (xz1 z2 )z1 ,z2⇡1 (z1 )⇡i (mi (x)) = ⇧ (xmi (x) mj (x))= max
⇧ x0 mi (x) zj
0x ,zj⇡1 (m1 (x))⇡2 (m2 (x))⇡2 (z2 ) and(6)⇡X (x)⇡j (mj (x))⇡X (x0 )⇡j (zj ).

(7)

It is known that stable matching is positive assortative matching (PAM) by capability, m0i (x) > 0,(e.g., Becker, 1973; Sattinger, 1979).13 Since team profits are supermodular, a high capability firm
has a greater willingness to pay for the extra capability of its partners. Therefore, high capability
firms outbid others for high capability firms, while low capability firms match with remaining low
capability firms.
The first order conditions for the maximization in (6) and (7) are:⇡i0 (mi (x)) = Axmj (x) > 0 and ⇡x0 (x) = Am1 (x)m2 (x) > 0.

(8)

Conditions (8) imply that profit schedules ⇡X (x) and ⇡i (zi ) are increasing functions of own capabilities.
Because of fixed costs, there exists a cutoff of team capability ✓L such that only teams withcapability ✓✓L produce on the market. PAM implies ✓L = xL z1L z2L where xL and ziL arecapability cutoffs for final producers and Zi suppliers, respectively, such that only final producers
with capability xxL and Zi suppliers with capability zi

11ziL participate in the matching market.

Roth and Sotomayor (1990) is an excellent textbook on the matching literature.
Although this definition of stability is commonly used in the literature, it should be noted that it implicitly assumes
firms can write complete contracts on the distribution of team profits. Therefore, the model is silent on issues related withincomplete contracts, which include the choice of boundaries of firms.
13
See e.g., Legros and Newman (2007) for the formal proof.
128

These capability cutoffs satisfy:A✓L = AxL z1L z2L = f and ⇡X (xL ) = ⇡1 (z1L ) = ⇡2 (z2L ) = 0.

(9)Equilibrium matching functions satisfy matching market clearing conditions:
MXe [1

G (x)] = Mie [1G (mi (x))] for all x

(10)

xL ,

where MXe , M1e , and M2e are the mass of entrants of final producers, Z1 suppliers, and Z2 suppliers,
respectively. The left-hand side of (10) is the mass of final producers whose capability is higher thanx, while the right-hand side is the mass of Zi suppliers whose capability is higher than mi (x).
Because these two sets of firms match with each other under one-to-one PAM, the equality in (10)must hold for all xxL .

Equilibrium matching patterns reflect the relative mass of entrants across sectors. Figure 2.2
describes this relation. The width of each rectangle is equal to the mass of entrants in each sector
under the assumption M1e > M2e , which will be proved later from f1e < f2e , while the vertical axis
expresses the value of G(s). The three gray areas express the mass of final producers with higher
capability than x (the left area), that of Z1 suppliers with higher capability than m1 (x) (the centerarea), and that of Z2 suppliers with higher capability than m2 (x) (the right area), respectively. The
matching market clearing condition (10) implies that all of the three areas must have the same size.
The figure shows that a final producer matches with a higher capability Zi supplier from the sector
with more entrants (i.e. Z1 ) and a lower capability one from the sector with fewer entrants (i.e. Z2 ).

G(s)
1
G(x)

MXe

=

M1e

=

M2e G(s)1=G(m1(x))G(m2(x))G(z1L )G(xL )

Exit
Exit
Exit

0

G(z2L )
0

Figure 1: The autarky matching pattern reflects the relative size of entrants across sectors. A firmfrom a sector with more entrants has high capability in a team.

9

Matching functions can be obtained from (10) in simple forms as:mi (x) = x

✓

Mie
MXe

◆1/k

=x

✓

ziL
xL

◆

for all x

(11)

xL .

As drawn in Figure 2.2, mi (x) is increasing and concave in M1e /MXe when x is fixed. As therelative mass of entrants in the Z1 sector to the final sector increases (M1e /MXe "), a final producer

with given capability x becomes able to match with a Z1 supplier with higher capability, but final
producer’s gain in matching is diminishing because high capability firms are scarce. This concave
relationship in Figure 5 depends on the shape of G, but is not specific to the Pareto distribution. It
holds under a wide class of distributions including uniform, normal, exponential, and other frequently
used distributions with the non-decreasing hazard rate g(x)/(1G(x)).

m1(x)1/k( MM ) x
1e

Xe

M1e
MXe

Figure 2: The capability mi (x) of a Zi supplier matching with a final producer with capability x isincreasing and concave in the mass of entrants in the Z1 sector relative to the final sector, Mie /MXe .
Profits of individual firms can be obtained by integrating the first order conditions (8) with initialconditions (9):⇡X (x) = A

Z x

m1 (t)m2 (t)dt and ⇡i (mi (x)) = A

xL

Z x

tmj (t)dt.

(12)

xL

Firm’s profits are increasing in the team capability and the firm’s capability advantage over the leastcapable firms and increasing in the market condition A.14 From (11) and (12), profit schedules in(12) are simplified as:f⇡X (x) =
3

"✓

x
xL

◆3

1

#

f
and ⇡i (zi ) =
3

"✓

zi
ziL

◆3

#

1 .

(13)

14
The stability condition alone determines the distribution of profits within teams. This is a virtue of this class ofmatching models with continuums of agents (Sattinger, 1979). We do not need to specify “bargaining power parameters”
on how to split the matching surplus within matches.

10

From (11), the profit schedules in (13) also implies ⇡X (x) = ⇡i (mi (x)), that is, each member
receives exactly one third of the team profits, reflecting symmetry in production and capability draws.
Entry Stage

Firms’ entry conditions determine the capability cutoffs and the mass of entrants.

Because risk-neutral firms enter until their expected profits become zero, free entry conditions are
given by[1

G (xL )] ⇡¯X = fXe and [1(14)

G (ziL )] ⇡¯i = fie ,R1
where ⇡
¯X and ⇡¯i are the average profits of firms in the market, ⇡¯X = [1 G (xL )] 1 xL ⇡X (t) g (t) dt
R1
and ⇡¯i = [1 G (ziL )] 1 ziL ⇡i (t) g (t) dt. A manipulation from (13) [see Appendix] shows thatthe average profits are constant as follows15⇡¯X = ⇡
¯i =

f
k

3

(15)

.

Because firms earn zero expected profits, the aggregate revenue of teams must be equal to the aggregate income, M r̄ = L, where r̄ is the average revenue of surviving teams and M is the mass ofsurviving teams, i.e. the mass of varieties. From r̄ =(¯⇡X + ⇡¯1 + ⇡¯2 + f ) and (15), the mass of

consumption varieties is proportional to the ratio of labor endowment to production fixed costs as in
the standard Dixit-Stiglitz model:
M=

(k

3)
k

I assume f /(k

✓ ◆
L
.
f
3)

(16)
max{fXe , fie } to ensure firms’ entry. Then, the capability cutoffs are

obtained from (14) and (15) as follows:


f
xL =
fXe (k

1/k

3)



f
and ziL =fie (k

1/k

3)

.

(17)

Since the mass of teams is equal to the mass of surviving firms in each sector, the mass of entrants
are obtained as:
MXe =

1

M
L
=
and Mie =G(xL )
fXe k
1

M
L
=
.
G(ziL )
fie k

(18)

The relative size of labor endowment to entry costs determines the mass of entrants in each sector.
From f1e < fXe < f2e , the mass of entrants follows M1e > MXe > M2e .
15

The constant average profit also holds in the Melitz-type (2003) model with the Pareto distribution.

11

The capability cutoffs in (17) link the mass of consumption varieties and the mass of entrants.
Lower entry costs attract more entrants, but the total mass of surviving firms (16) is independentof the size of entry costs. Therefore, the capability cutoffs are negatively related to entry costs andpositively to production fixed costs.

2.3

Welfare Theorem

This section presents welfare properties of an autarky equilibrium. Suppose the social planner maximizes the welfare W ⌘ U ⇢ by choosing output q(✓) of production teams, matching, capability

cutoffs (xL , z1L , z2L ), and mass of entrants (MXe , M1e , M2e ). Matching may be either deterministic or stochastic and determines the distribution function H(✓) of team capability ✓ and the capability
cutoff ✓L . I call this problem the long run planner’s problem and also consider the short run planner’s
problem where the mass of entrants are fixed. A long run decentralized equilibrium is the one considered in the previous section. I consider a short run decentralized equilibrium where the mass ofentrants are fixed and the aggregate profits are lump sum transferred to the representative consumer.
In this setting, I establish the following welfare theorem.
Proposition 1. A decentralized autarky equilibrium achieves the solution to the social planner’s
problem both in the short run and in the long run..
The proof is given in Appendix. Proposition 1 is related to two known welfare theorems. The
first theorem is a classic result of Koopmans and Beckmann (1957) and Shapley and Shubik (1972)
that a stable matching in a frictionless market maximizes the total payoffs of agents when the distributions of agents are fixed. Gretsky, Ostroy and Zame (1992) prove the theorem for the case ofcontinuums of agents. The second theorem is recently proved by Dhingra and Morrow (2014) thatthe monopolistic competition equilibrium maximizes the welfare in the Melitz model with a CES
utility and a continuum of firms in autarky.
I present a sketch of the proof here. Following Dhingra and Morrow (2014), for given matching,capability cutoffs (xL , z1L , z2L ) and mass of entrants (MXe , M1e , M2e ), I first obtain optimal production q(✓), which corresponds to production in a decentralized equilibrium because markups are
identical across varieties. Then, substituting optimal q(✓) to the welfare, I obtain:


W = M

Z 1
✓L

1

✓dH(✓)

2

4L

X

MXe fXeMie fie

MXe [1i=1,2

Notice that the welfare (19) is increasing in the aggregate team capability M

3

G(xL )]f 5
R1

1

.

(19)

✓L ✓dH(✓) when cut-offs and mass of entrants are given. From the Koopmans-Beckmann-Shapley-Shubik theorem, a12stable matching, that is PAM, maximizes the aggregate profits and the aggregate team capability
R1
M ✓L ✓dH(✓). Once H(✓) is optimally chosen, it is possible to show cutoffs and mass of entrants
are chosen optimally. This is because of similar logic behind the Dhingra-Morrow theorem: a con-tinuum of firms and monopolistic competition eliminates the rent-steeling motive of entries to makeentries at the socially optimal level.

3Free Trade

This section introduces trade in intermediate goods, i.e. international matching of firms, in a twocountry setting. To focus on new aspects of the model, I assume final goods are nontradable. Section
3 considers free trade where international matching requires no costs. Section 4 introduces costs ofinternational matching.

3.1

Comparative Advantage

I introduce another country, Foreign, as a mirror image of Home on entry costs for Zi sectors:⇤⇤
f1e = f2e
< f2e = f1e.

(20)

Foreign variables and functions are labeled by “⇤”. Home and Foreign are identical in other aspects.
Introducing differences in entry costs is a simple way to formulate technological differences acrosscountries, which gives a rise to Ricardian comparative advantage.16 I refer to Home Z1 sector andForeign Z2 sector as the CA (comparative advantage) sectors and Home Z2 sector and Foreign Z1
sector as the CD (comparative disadvantage) sectors. This mirror-image structure greatly simplifies
the analysis: equilibrium functions and variables in the Home Zi sector are the same as those in
the Foreign Zj sector and other aspects are identical between Home and Foreign. Because the twocountries have the same wage, I continue to normalize it to one.
It is informative to compare Home and Foreign in autarky. First, from (11) and (18), autarky
matching differs between the two countries:
ma1 (x) = m⇤a
2 (x) =

✓

fXe
f1e

◆1/k

x>

✓

fXe
f2e

◆1/k

x = ma2 (x) = m⇤a
1 (x) for all xxaL .

Variables and functions of an autarky equilibrium are labeled by superscript “a”. Figure 3 draws
autarky matching of Home and Foreign with an iso-✓ curve for a final producer with capability x.
16
An alternative approach is to assume symmetric entry costs and asymmetric Pareto distributions Gt (s) = 1
that are allowed to have different means across sector t 2 {1, 2, X}.

13(vt /s)kThis iso-✓ curve describes a combination of capability of Z1 and Z2 suppliers that a final producer
with capability x needs to match in order to achieve team capability ✓. Points A and A⇤ represent
capability of Z1 and Z2 suppliers in Home and Foreign that match with final producers with capability
x in autarky, respectively. In autarky, the CA sectors have a relatively larger number of entrants thanthe CD sectors compared to the foreign country. Therefore, Home final producers match with relative
more capable Z1 suppliers and less capable Z2 suppliers compared to Foreign final producers withthe same capability.

m*2a(x)

A*

A

a

m2(x)

a
X

m*1a(x)am1(x)Figure 3: Capability of Z1 and Z2 suppliers matching with final producers with capability x in Home
autarky (Point A) and Foreign autarky (Point A⇤)
When member capability within teams is interpreted as quality of components, or more generally
product characteristics, Figure 3 predicts a cross-country difference in product characteristics. For
example, consider car production requiring two input suppliers, designers (Z1 ) and parts makers(Z2 ), in Italy (Home) and Japan (Foreign). Suppose Italy has lower entry costs for designers andJapan has for parts makers. When cars sold in similar prices (i.e. similar ✓) are compared, Italiancars have better design (Paint A) while Japanese cars have more durable parts (Point A⇤ ).
Second, the profits of Zi suppliers also differ between the two countries. From (13), Zi suppliersin the CD sectors receive higher profits than foreign Zi suppliers with the same capability:⇡2a (z)⇡2⇤a (z) = ⇡1⇤a (z)f⇡1a (z) =

3

"✓

z
a
z2L

◆3

✓

z
a
z1L

◆3 #

> 0.

(21)

The difference in autarky profits (21) implies that the autarky matching is unstable under free trade.
Final producers in each country are willing to match with foreign Zi suppliers from the foreign CA
sectors. This international matching motivated by autarky profits differences is reminiscent of theories of comparative advantage where autarky prices differences leads to international trade. Notice14

that the profit difference (21) is increasing in capability x. Therefore, highly capable firms are more
willing to participate in international trade than low capability firms.
The opening of trade causes two adjustments: (1) re-matching of existing firms and (2) new
entry and exit of firms. To understand the role of each adjustment, section 3.2 analyzes a short-run
equilibrium in which re-matching occurs only among firms that enter in an autarky equilibrium. For
simplicity, I assume firms that entered but did not form any match in an autarky equilibrium canparticipate the matching market at no cost. Section 3.3 analyzes a long run equilibrium in whichfirms adjust entry and exit to satisfy the free entry conditions.

3.2

Short-run Equilibrium

After the opening of trade, matching remains unstable as long as the profit schedules of Zi suppliers
differ across countries. Therefore, in a new equilibrium, the profit schedules of Zi suppliers mustsatisfy:⇡1 (z)⇡1⇤ (z) = ⇡2 (z)⇡2⇤ (z) = 0.

(22)

Under free trade, the matching market is globally integrated so that matching functions are equalized
across countries. New matching functions satisfies the global matching market clearing condition:
a
⇤a
(MXe
+ MXe
) [1

a
⇤a
G (x)] = (Mie
+ Mie
) [1

G (mi (x))] for all x

(23)

xL .

Notice that the mass of entrants are fixed at autarky levels in a short run equilibrium. Since the CA
sectors have more entrants than the CD sectors, condition (23) implies that some suppliers in the CA
sectors must match with foreign final producers. Let sX (x) be the share of importers among Home
final producers with capability x and si (zi ) be the share of exporters among Home Zi suppliers withcapability zi . From (18) and (23), they are obtained as:
a
⇤a
⇤
M1e
M1e
f1e
f1e
=
a
⇤a
⇤ for xM1e + M1e
f1e + f1e
M a M ⇤af⇤
f1e
s1 (z1 ) = 1e a 1e = 1e ⇤for z12M1e
2f1e

sX (x) =xL ,z1L , and s2 (z2 ) = 0 for z2⇤
The share of trading firms increases in the extent of comparative advantage, f1e

describes the distributions of firms engaging in exporting and importing.

15

z2L .

(24)

f1e . Figure 4

G(s)

a

To
Foreign

M 1ae

M Xe

a

M2 e

From
Foreign

G(s)
1

1

E*

E

I
a

I
E
E*

: Home Importers: Home Exporters: Foreign Exporters

s1 M

s1 M

sXMXea

0

a
1e

a
1e

{

G(xL )

{

{

a )
G(z1L

Comparative
Disadvantage(CD sector)

Comparative
Advantage(CA sector)

G(z1L )aG(z2L )
0

Figure 4: The distribution of exporters and importers in Home in a short run free trade equilibrium
From condition (23), Home matching functions are obtained as follows:
m1 (x) = m2 (x) = x✓

a + M ⇤aM1e
1e
a + M ⇤a
MXe
Xe

◆1/k

=x

✓

ziL
xL

◆

for x

xT .

(25)

Matching functions (25) are similar to autarky matching functions (11). Both functions are increasing
and concave in the relative mass of entrants in the Zi sector to the final sector. Figure 5 draws thisrelationship for m1 (x). As Figure 5 shows, the capability of partner Zi suppliers mi (x) under trade
is higher than the average of the two partners in autarky, ma1 (x) and ma2 (x) (= m⇤a
1 (x)).
The concave curve in Figure 5 represents a source of product-level gains from international
matching. Rather than having many entrants in one sector and few in other sectors, having a moderatecombination of entrants allow final producers to match with higher capability suppliers on average.
Thus, since team capability ✓ is quasi-concave, final producers increase team capability. Figure 6
draws iso-✓X (x) curves for a Home final producer with x. Point A expresses the capability of autarky partners, mai (x), and Point S expresses the capability of new partners in a short run tradeequilibrium, mi (x). From Figure 5, the capabilities of new partners are higher than the averagecapability of its autarky partners. Therefore, final producers increases team capability.
The new capability cutoffs are obtained from the labor market clearing condition:a
L = MXe
fXe +

X

i=1,2

a
a
Mie
fie + MXe

Z 1

C(✓X (t))g(t)dt.

(26)

xL

where ✓X (x) ⌘ xm1 (x)m2 (x) and C(✓X (x)) are team capability and team labor costs, respectively,16

z1
M1e 1/k
x
MXe

( )

a

m1(x)
m1(x)=m2(x)aam1(x)+ m2(x)
2
a

m2(x)
a
= m*1(x)

*a
1e
*a
Xe

M
M
Foreign
Autarky

a
1e
a
Xe

*a
1e
*a
Xe

M +M
M +M
Short-run
Free Trade

a
1e
a
Xe

M
M
Home
Autarky

M1e
MXe

Figure 5: Under trade a final producer with x matches Z1 and Z2 suppliers that have higher capability
than the average of the two autarky partners.
for a final producer with capability x. The first two terms in the right hand side of (26) represent laborfor entry, while the last term represents labor for production. Since C(✓X (x)) = (1) A✓X (x)+ffrom (4), a final producer with capability x increases its labor inputs from autarky if and only ifA✓X (x) increases from autarky. Since A✓X (x) = ✓X (x)/✓X (xL ) = (x/xL )3 both in autarky andin a short run free trade equilibrium, if xL = xaL , then A✓X (x) does not change from autarky. Thismeans xL = xaL since if xL = xaL , then all teams use the same amount of labor and the labor market
clearing condition (6) holds. Therefore, the cutoff for final producers does not change. On the other
hand, the capability cutoffs change for Zi suppliers. They fall in the CA sectors and rise in the CD
a >zasectors: z1L
1L = z2L > z2L .

The change in the cutoffs leads to the change in profits. From (15) and (25), profit functions
continue to be (13). From the changes in the capability cutoffs, final producers receive the sameprofits as in autarky, Zi suppliers in the CA sectors increase profits, and Zi suppliers in the CD
sectors decrease profits.
Welfare Theorem

The welfare theorem continues to hold after the opening of trade. Consider thesocial planner who maximizes an increasing symmetric concave function of U and U ⇤ . Then, theshort run free trade equilibrium maximizes the world welfare. Since an autarky equilibrium allocation
is feasible under free trade, a corollary is that the opening of free trade increases the world welfare
in the short run. Since U = U ⇤ always holds from symmetry, this corollary also implies the opening

17

L
S

m2(x)

C
(Long runfree trade)

X

A

a

m2(x)
a

X
a
X

(Short run
free trade)(Autarky)am1(x) m1(x)
m2(x)
a
a
m2(x)+ m1(x)2Figure 6: Final producers improve team capability under trade.
of free trade increases the welfare of each country in the short run.
Proposition 2. (1) The short run free trade equilibrium maximizes the world welfare. (2) The openingof free trade increases the welfare of each country in the short run.
The proof is in Appendix. The key intuition is again that the welfare function (1) is increasingin the aggregate team capability, which is maximized under stable matching. After the openingof trade, autarky matching becomes unstable since it is not PAM at the world level. International
trade systematically changes matching so that PAM holds at the world level, raising the aggregate
capability and the welfare.
Evidence for Welfare Improving Rematching

The key welfare improving mechanism is that

trade liberalization causes rematching toward PAM at the world level. Define that a firm upgrades
partner if the firm switches its partner to the one with higher capability, and that a firm downgrades
partner if the firm switches its partner to the one with lower capability. Then, the rematching toward PAM is summarized as follows. First, final producers have partner upgrading in the CD sectors
and partner downgrading in the CA sectors. Second, Zi suppliers in the CA sectors have partner
upgrading. Third, Zi suppliers in the CD sectors have partner downgrading.
The Model’s prediction on rematching in trade liberalization is consistent with a recent empiricalfinding from exporter-importer matching data. Sugita, Teshima, and Seira (2015) investigate customs
transaction records of Mexican textile/apparel exports to the US where they can observe identities
of Mexican exporters and US importers and transaction volume for each hs 6 digit product in each18

year from 2004 to 2009. They find two pieces of evidence for PAM of exporters and importers bycapability.
First, they found in each hs 6 digit product and year, matching of Mexican exporters and US
importers is approximately one-to-one. Even though there are firms trading with multiple partners,
these firms concentrate their product-level trade with the single main partners. They calculate tradevolume of “main-to-main” transactions where the exporter and the importer are both the main partnerof each other for the product and find that the share of “main-to-main” trade in the aggregate textile/apparel trade volume is more than 80%. This finding justifies the matching approach to modelinginternational trade.
Second, Sugita et al. (205) find that Mexican textile/apparel exports to the US offers a unique natural experiment for testing the key prediction of PAM, systematic rematching in trade liberalization.
Before 2005, the US imposed import quota on some textile/apparel products under the Mutil-fibrearrangement (MFA). Since Mexican exports to the US enjoy quota-free access through the North
American Free Trade Agreement (NAFTA), the MFA quota protected Mexican exports to the US as
if they were domestic transactions. In 2005, the US removed the MFA quota, which results in a hugeincrease in Chinese exports and a corresponding drop in Mexican exports in the US market. They findthe following systematic partner changes occur more often in products for which Chinese exports to
the US were subject to binding quotas. These partner changes are that US importers switched theirmain Mexican partners to those making greater pre-shock exports; that Mexican exporters switched
their main US partners to those making fewer pre-shock imports.
The second findings of Sugita et al. (2015) are consistent with the current model by interpretingtrade between US firms and Mexican firms within NAFTA as if they were domestic matching. Sincefirm’s trade volume is increasing in its own capability in the model, their findings can be interpreted
as partner upgrading of US final producers and partner downgrading of Mexican suppliers in the CD
sectors where China has comparative advantages.

3.3Long-run Equilibrium

Free trade equalizes profits of Zi suppliers between the CA and CD sectors, but entry costs differ
between them. Therefore, in a long run free trade equilibrium, no firm enters in the CD sector.
Countries completely specialized in the CA sectors.
A long run free equilibrium is obtained as follows. First, from complete specialization, matchingmarket clearing conditions become⇤(MXe + MXe) [1G(x)] = M1e [1⇤G(m1 (x))] = M2e[1

19

G(m2 (x))]

and matching functions become
m1 (x) = m2 (x) =

✓

M1e
⇤
MXe + MXe

◆1/k

x=

✓

z1L
xL

◆

x for all x

(27)

xL .

Since the From (12) and (27), profits functions continue to be (13). Furthermore, it is possible toshow that expected profits also continue to be (15). Thus, the mass of varieties and the capability
cutoffs continue to be (16) and (17), respectively. From (27), 2M = 2MXe [1⇤ [1
G(z1L )] = M2e

G(xL )] = M1e [1⇤ )] hold. Then, the mass of firms are obtained as MaG(z2L
Xe = MXe , MZ1e =⇤ , and M
2MZ1e
Z2e = 0.
a + M a implies that the relative mass of entrants in the Z sectors to the
Notice that MZ1e > MZ1i
Z2
a /M a for i = 1, 2. Point L in Figure 6 expresses matching
final sector in the world increases to MZ1e
Xe

in a long run free trade equilibrium, showing each final producer further improves team capability
from a short run equilibrium.

4

Costly Trade

The heterogeneous firm trade literature has established two facts: (1) only large and highly capablefirms engage in exporting and importing; (2) trade liberalization leads the least capable firms to exit.
The model in section 3 fails to address these two facts, predicting no correlation of capability andparticipation in trade, and no exit of final producers after trade liberalization. This section shows thata simple extension of introducing fixed trade costs allows the model to predict these two facts.
International teams requires fixed trade costs: fM units of local labor for an importing final
producer and fE units of local labor for an exporting Zi supplier. For simplicity, I assume traderequires no variable trade cost. Let fT ⌘ fM + fE be total fixed trade costs for an international team.

Then, team profits are expressed as:⇧ (x, z1 , z2 )fT II = Axz1 z2

8< 1 for an international teamfT II , where II =.
: 0 for a domestic teamIn the following, I consider a long run equilibrium where fT is sufficiently high that the CD sector
has positive entry.
Trade Pattern With fixed trade costs, only firms with high capability engage in international
matching. Notice that the autarky profit difference in (21) increases in capability z. There exist
thresholds for capability xT and zT such that only final producers with x

20xT and Zi suppliers

in the CA sectors with z

zT are willing to form international teams. In equilibrium, the profit

difference for highly capable Zi suppliers between Home and Foreign must be equal to fixed trade
costs.
Lemma 1. There exists an export capability cutoff zT > ziL for i = 1, 2 such that profit schedules
for Zi suppliers satisfy the following no arbitrage condition:⇡1⇤ (z)⇡1 (z) = ⇡2 (z)Proof. In Appendix.

⇡2⇤ (z)8< = fT
: <f

if

z

zT

(28)

otherwise.

T

Since the no arbitrage condition (28) holds only for highly capable firms, the matching market isglobally integrated for high capability firms, while it is segmented across countries for low capability
firms.
Lemma 2. There exist an import cutoff for capability xT > xL such that Home matching functions
mi (x) satisfy (i) zT = mi (xT ), (ii) global matching market clearing conditions:⇤(MXe + MXe) [1⇤G (x)] = (Mie + Mie
) [1

(29)

G (mi (x))]

for all x 2 [xT , 1) and (iii) local matching market clearing conditions:
MXe [1

G (x)] = M1e [1G(m1 (x))]

MT = M2e [1G(m2 (x))] + MT

(30)

for all x 2 [xL , xT ), where MT is the mass of Home Z1 suppliers (Foreign Z2 suppliers) engaging
in international matching. Foreign matching functions are obtained from m⇤i (x) = mj (x).
Proof. In Appendix.
⇤ and M = M ⇤ under the mirror image assumption, the global matching
Because MXe = MXeie
jemarket clearing condition (29) implies that final producers with xmatch with exactly a half of Zi suppliers with z

xT in each country as a wholezT in the world. Therefore, Home and Foreign

have identical matching functions, mi (x) = m⇤i (x) for x

xT . From these properties, I obtain thenext lemma on the mass of firms that engage in international matching.
Lemma 3. The share of importers among Home final producers and the share of exporters among

21

Home Zi suppliers satisfysX (x) =8⇤< M1e M1e⇤ ⌘ sX:

M1e +M1e

0

for x

xT

otherwise; s1 (z1 ) =8⇤< M1e M1e
⌘ s1
2M1e

:

0

for z1zTotherwise(31)

and s2 (z2 ) = 0 for all z2 . The corresponding shares of Foreign firms are given by s⇤X (x) = sX (x),s⇤1 (z) = s2 (z) , and s⇤2 (z) = s1 (z). The mass of exporters and the mass of importers are equal to
MT = sX MXe [1G(xT )] = s1 M1e [1G(zT )].

Proof. In Appendix.
Figure 7 expresses the distribution of capability of Home exporters and importers. Area E expresses Home exporters in the Z1 sector that match with Foreign final producers; Area E ⇤ expresses
Foreign exporters in the Z2 sector that match with Home importers in the final sector expressed in
Area I. The size of each of Areas E, E ⇤ , and I is equal to MT .

G(s)

To
Foreign

M 1e

M Xe

From
Foreign

M2 e

G(s)
1

1

s1 M1e

sXMXe

{

{

{

G(xT )E*

E

I

s1 M1eG(zT )G(z1L )G(xL )0I
E
E*: Home Importers: Home Exporters: Foreign Exporters

Comparative
Advantage(CA sector)

Comparative
Disadvantage(CD sector)

G(z2L )
0

Figure 7: International matching occurs among high capability firms
As Figure 7 shows, only highly capable firms engage in exporting and importing. Since highcapable firms have larger revenue and employment than low capable firms, the model predicts a wellestablished fact on firms engaging in international trade. That is, exporters and importers are on
average more capable and larger than non-trading firms.
Proposition 3. In each sector, (i) exporters have on average higher capability and larger revenue
than non-exporters; (ii) importers have on average higher capability and larger revenue than nonimporters.
22

After liberalization, Zi suppliers in the CA sectors gain in matching and Zi suppliers in in the CD
sectors lose. Therefore, in the long run, entry increases in the CD sectors relative to the CA sectors.
Thanks to this specialization into sectors with lower entry costs, the total mass of entrants of Zi
suppliers to final producers in the world increase.
Lemma 4. In the long run after the opening of trade from autarky: (i) the mass of entrants in thea ; (ii) the mass of entrants in the Z sectors increases in the
final sector does not change MXe = MXeia > Ma > M .
CA sectors, but decreases in the CD sectors, M1e > M1e2e2eProof. In Appendix
The opening of trade with fixed trade costs improves team capability of highly capable final
producers, but at the same time, it decreases team capability of the lowest capable final producers.
From (29), matching functions for high capability final producers take a form similar to (25) as
follows:m1 (x) = m2 (x) =✓⇤Mie + Mie⇤MXe + MXe

◆1/kx for x

xT .

(32)

Lemma 4 implies that the mass of entrants in the Zi sectors relative to the final sector increases
M +M ⇤

M a +M a⇤

ie
ie
ie
ie
from autarky, MXe+M ⇤ > M a +M a⇤ . Therefore, a comparison of (25) and (32) shows that a final
Xe

Xe

Xe

producer with higher capability than xT matches suppliers with higher capability than in a short run
free trade equilibrium. Figure 6 describes new matching as Point LC and shows team capability
increases from autarky.
The specialization toward the CA sectors, however, hurts final producers with low capability.
Because the CA sector already has more entrants than the CD sector, further entry into the CA sector
widens the imbalance of entrants in the two intermediate good sectors. The next lemma shows that
final producers with the lowest capability reduce their team capability below the autarky level since
they are forced to match with suppliers having more extreme combinations of capability than inautarky.
Lemma 5. In a long run costly trade equilibrium, there exists a threshold x0 > xL : (i) final producers
a (x), if x > x0 andwith capability x increase team capability compared to autarky, ✓X (x) > ✓Xa (x), if x < x0 ; (ii) the capability of Z suppliers in a team converges
decrease it, ✓X (x) < ✓Xicompared to autarky, m1 (x) < ma1 (x) and m2 (x) > ma2 (x), if x > x0 , while they diverge,
m1 (x) > ma1 (x) and m2 (x) < ma2 (x), if x < x0 .
Proof. In Appendix.
23

The free entry conditions (14) determine the capability cutoffs in a long run equilibrium. The
opening of trade intermediate goods changes expected profits across sectors for given domestic cutoffs. Expected profits increase more strongly in the CA sectors than in the final sectors and morestrongly in the final sectors than in the CD sectors. To satisfy to satisfy the free entry conditions, thedomestic cutoffs increase more strongly in the CA sector than in final sector and more strongly in the
final sector than in the CD sector. 17
Lemma 6. In the long run after the opening of trade in intermediate goods from autarky, the capability cutoffs increase more strongly in the CA sector than in final sector and more strongly in the
final sector than in the CD sector:
z1L
xL
z2L
> a >1> a .
a
z1L
xL
z2L
Proof. In Appendix.
Aggregate Welfare After the opening of trade, the mass of entrants in the final producers do notchange but the cutoff for final producers rise. Thus, the mass of varieties of final goods decrease andsome final goods become sold with lower quality or higher prices, both of which tend to decrease
the consumer welfare. What is the total impact of trade on the aggregate welfare? Appendix showsthat the welfare theorem continues to hold under costly trade. Therefore, the opening of costly trade
increases the welfare of each country.
Proposition 4. (1) The costly trade equilibrium maximizes the world welfare. (2) The opening ofcostly trade increases the welfare of each country.

5

Conclusion

This paper has analyzed location, selection and matching of global supply chains in a tractable general equilibrium model. The model demonstrates a new gain from trade associated with firm heterogeneity. Trade improves the world welfare by changing matching of firms in the world positive
assortative by capability. A companion empirical paper, Sugita, Teshima and Seira (2015), identifies
this welfare-improving rematching in an actual trade liberalization episode.
The model employs several simplifying assumptions for tractable analysis. Relaxing some ofthem will make it possible to study further implications of firm-to-firm matching in international
17
The larger increase in the domestic capability cutoff in the CA sector compared to the CD sector is similar to Bernard,
Redding and Schott (2007), in which the domestic productivity cutoff rises more in the CA sector than in the CD sector.

24

trade. The first one is to allow countries asymmetric so that wages differ across countries. Whenfactor prices are different across countries, firm’s capability depends not only on its technology suchas product quality and productivity, but also on factor prices. At the same time, factor prices will
depend on matching patterns, that is, with whom local firms form global supply chains. The second
possible extension is to allow randomness in meeting. Since Shimer and Smith (2000), the Becker
type assortative matching model is extended to include randomness in meeting (see Smith, 2011,
for survey). These studies find positive assortative matching does not exactly hold, but does holdon average with some deviations. This will be a more realistic picture of actual matching in data,
though I guess many of the current results shown in a model without randomness will continue tohold in a model with randomness. The third is to formulate a micro-foundation of ✓. An example is
to introduce supplier’s learning technology from buyers. This extension allows us to analyze how theextent of learning by exporting depends on the characteristics of importers and exporters. Finally, thecurrent model is silent about boundaries of firms. For instance, the current model implicitly assumes
complete contracts in the matching stage. To incorporate the theory of multinational firms by Antras(2003) based on contract incompleteness would be an interesting task for future research.

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30

Appendix (Not for Publication)
A1. Solving for an Autarky Equilibrium
Production Stage
Utility Maximization
The maximization problem for the representative consumer is:

q(·)

Z

1/⇢

✓(!)↵ q(!)⇢ d!
!2⌦
Z
subject top (!) q (!) d! = I,

max U =

(A.1)

!2⌦

where q (!) is quantity demanded for variety ! of final good, p (!) is the price of !, ✓ (!) is theteam capability of ! and I is the aggregate income. Setting up a Lagrangian
L=

Z

1/⇢
↵

⇢

✓(!) q(!) d!

+

!2⌦



Z

I

p (!) q (!) d! ,!2⌦

I obtain the first order conditions as:@L= ✓ (!)↵ q(!)⇢ 1 U (1 ⇢)/⇢@q(!)

p (!) = 0 for all ! 2 ⌦.

(A.2)

For any two varieties ! and ! 0 , we have✓

✓ (! 0 )
✓ (!)

◆↵ ✓

q(! 0 )
q (!)

◆⇢ 1

p(! 0 )
p(!)
✓
◆ 1 ✓
◆ ↵
q(! 0 )p(! 0 ) ⇢ 1 ✓ (! 0 ) 1 ⇢
=
q (!)
p(!)
✓ (! 0 )✓◆✓◆p(! 0 )q(! 0 )p(! 0 ) 1
✓ (! 0 ) ↵=p(!)q (!)
p(!)
✓ (!)
✓
◆1 ✓
◆
0
p(! )
✓ (! 0 ) ↵00p(! )q(! ) =p(!)q (!) .
p(!)
✓ (!)
=

Integrating both sides with respect to ! 0 2 ⌦, we obtain
Z

Z
q(!)
p(! )q(! )d! =✓(! 0 )↵ p(! 0 )1
p (!) ✓ (!)↵ !0 2⌦! 0 2⌦q(!)
I=
P1
p (!) ✓ (!)↵
0

0

0

q(!) =

I✓ (!)↵ p (!)
P1 ,

,

1

d! 0

where P ⌘⇥R!2⌦ p(!)✓ (!)↵ d!

1

I✓a p
P1

q(p, ✓) =

⇤1/(1

)

is the price index. Therefore,(A.3)

Profit Maximization
Facing the demand function (A.3), a team with capability ✓ sets the price to maximize the teamprofits:max(ppf subject to (A.3).

c✓ )q(p, ✓)The first order condition is(p(p@q(p, ✓)+ q(p, ✓) = 0@pq(p, ✓)c✓ )+ q(p, ✓) = 0
p

c✓ )

(p

c✓ ) + p = 0✓
p=A team chooses the optimal price p(✓) = c✓⇢ .
⇣ ⌘ 1
Let A ⌘ I ⇢P
and ⌘ ↵
(
c

1

◆

c✓ =

1). Team’s output q (✓), revenue R(✓), costs C(✓),

and profits ⇧(✓) become:
1

⇣⇢⌘

✓(↵

)

✓(↵

) +

C(✓) = c✓ q (✓) + f
✓
◆ 1
I ⇢P
=
✓(↵
⇢
c

) +

q (✓) = IPcR(✓) = p(✓)q (✓)
✓
◆
⇢P
=I
c

;

1

= A✓ ;=(⇧ (✓) = R(✓)
I normalize+f1) A✓ + f ;C(✓) = A✓

c✓
⇢

f.

= 1.

2

Matching StageIn stable matching, matching functions, mi (x) , and profit functions, ⇡X (x) and ⇡i (zi ), satisfy two
conditions: (1) (individual rationality) all firms receive non-negative profits; (2) (pair-wise stability)each firm is the optimal partner for the other team members. The (pair-wise) stability condition
implies that a final producer maximizes its residual profit:⇡X (x) = max ⇧ (xz1 z2 )z1 ,z2⇡1 (z1 )(A.4)⇡2 (z2 ).

The first order condition for (A.4) becomes:⇡i0 (mi (x)) = Axmj (x) > 0.

(A.5)

The stability condition also implies that Zi suppliers maximize residual profits:⇡i (mi (x)) = max⇧(x0 mi (x)zj )0x ,zj⇡X (x0 )⇡j (zj ).

The first order condition becomes:
0⇡X(x) = Am1 (x)m2 (x) > 0.

(A.6)

As will be shown in section A2 below, stable matching is positive assortative, m0i (x) > 0. There0 (x) > 0 and ⇡ 0 (z ) > 0. The
fore, from (A.5) and (A.6), profit functions are increasing: ⇡Xi iindividual rationality condition implies that capability cutoffs exists, xL , z1L , and z2L , such thatziL = mi (xL ), ✓L = xL z1L z2L and ⇡X (xL ) = ⇡i (ziL ) = 0.

(A.7)

Integrating the first order condition (A.6) with an initial condition (A.7), I obtain final producer’s
profits as:
Z x

0⇡X (x) = ⇡X (xL ) +
⇡X
(t)dt
xL
Z x
0
=
⇡X
(t)dt
xL
Z x
=A
m1 (x)m2 (x) dt.

(A.8)

xL

Integrating the first order condition (A.5) with an initial condition (A.7), I obtain Zi supplier’s profits

3

as:
Z x
⇡i (mi (x)) = ⇡i (mi (xL )) +⇡i0 (mi (t))m0i (t)dtxL
Z x=⇡i0 (mi (t))m0i (t)dt
xL
Z x
=A
xmj (t)m0i (t)dt
x
Z Lx
=A
m1 (t)m2 (t) ⌘i (t)dt,

(A.9)

xL

where ⌘i (x) ⌘ xm0i (x)/mi (x).

Stable matching functions satisfy the matching market clearing conditions:
MXe [1

G (x)] = Mie [1G (mi (x))] for all xxL .

(A.10)The left hand side in (A.10) is the mass of final producers with higher capability than x, while theright hand side in (A.10) is the mass of Zi suppliers with higher capability than mi (x) . PAM requires
equation (A.10) to hold for all final producers with higher capability than the domestic cutoff.
From the Pareto distribution, the matching market clearing condition is written as: for all x
MXe [1

G (x)] = Mie [1xL ,G (mi (x))]

MXe
Mie
=
k
x
mi (x)k✓◆Mie 1/k, mi (x) =
x.
MXe
,

(A.11)

Substituting xL into (A.11), I obtainziL = mi (xL ) =

✓

Mie
MXe

◆1/k

mi (x)
x
=
.
ziL
xL

xL and(A.12)

From (A.12), I obtain✓X (x)
xm1 (x)m2 (x)==✓L
xL zIL z2L

✓

x
xL

◆3

(A.13).

The elasticity of matching function is ⌘i (x) = 1. Then, (A.8) and (A.9) imply ⇡X (x) = ⇡i (mi (x)) =4

⇧(✓X (x))/3, which is further simplified from (13) as:
1⇡X (x) = ⇡i (mi (x)) = ⇧(✓X (x))
3
1
= [A✓X (x) f ]3
f ✓X (x)
=
1
3
✓L
"✓ ◆
#
f
x 3
1
=
3
xL

Entry Stage
Since firms are risk neutral, the expect profit from entry is equal to entry costs in equilibrium:[1

G (xL )] ⇡¯X = fXe and [1R1
G (xL )] 1 xL ⇡X (t) dG (t) and ⇡¯i = [1where ⇡
¯X = [1(A.14)G (ziL )] ⇡¯i = fie ,average profits of active firms.

R1
G (ziL )] 1 ziL ⇡i (t) dG (t) are the(x)
From integration by parts and mziiL= xxL , the average profits is simplified as:

Z 1
1
⇡
¯X =
⇡X (t) g(t)dt
1 G(xL ) xL
Z 1
h
1
1
0
=
⇡X
(t) [1 G(t)] dt⇡X (t) [1
1 G(xL ) xL
1 G(xL )Z 1
1=⇡ 0 (t) [1 G(t)] dt
1 G(xL ) xL X
Z 1
1
=
A
m1 (t)m2 (t) [1 G(t)] dt
1 G(xL )xLZ 1ftm1 (t)m2 (t) 1 G(t)=dt
1 G(xL ) xL xL z1L z2Lt
Z 1 ✓ ◆3 ft
1 G(t)=dt
1 G(xL ) xL xLt
Z 1= xkL 3 f
t k+2 dt
xL
!
3 k
x
L
= xkL 3 f
k 3
=

f
k

3

.

5

G(t)]|1xLi

Since✓g(mi (x))
1 G(ziL )

◆

◆
k
kziL
m0i (x)mi (x)k+1✓ ◆✓◆k ✓ 0◆kziL
mi (x)x=x
mi (x)mi (x)✓ ◆⇣ ⌘kxL k=from (A.12) and ⌘i (x) = xm0i (x)/mi (x) = 1
x
x
kxkL
= k+1
x
g(x)
=
1 G(xL )m0i (x) =✓and ⇡X (x) = ⇡i (mi (x)), the average profit of Zi suppliers is written as:
Z 1
1⇡i (zi ) g(zi )dzi
1 G(ziL ) zL
Z 1
1=⇡i (mi (x)) g(mi (x))m0i (x)dx (from zi = mi (x))
1 G(ziL ) xL✓◆
Z 1g(mi (x))m0i (x)=⇡i (mi (x))
dx.
1 G(ziL )
xL
✓
◆
Z 1
g(x)
=
⇡X (x)dx
1 G(xL )
xL

⇡
¯i =

=⇡
¯X
=

f
k

3

Free entry also implies M r̄ = L, where M is the mass of active teams and r̄ is the average teamrevenue. Since the average revenue is:
r̄ =
=

1
1

=
=

Z 1
1
r(✓X (x))g(x)dx
G(xL ) xL
Z 1
1[⇧ (✓X (x)) + f ] g(x)dx
G(xL ) xL(¯⇡X + ⇡¯1 + ⇡¯2 + f )

k

kf
,
3

(A.15)The mass of varieties is
M=

(k

3) L
.
k f

6

The cutoff capabilities are obtained from the free entry as
xL =
ziL =

✓
✓

1
1

◆

1/k
⇡
¯X 1/kf==fXefXe (k 3)◆1/k ✓ ◆1/k 
1
⇡
¯i
f
=
=
.1/k
G(ziL )fiefie (k 3)
1
G(xL )

From M = MXe [1
MXe =
Mie =

1
1

◆1/k

✓

G(xL )] = Mie [1G(ziL )], the mass of entrants are obtained as:

M
L
=
and
G(xL )fXe (k 3)
M
L=.
G(ziL )
fie (k 3)

From these, matching functions become:mi (x) =

✓

Mie
MXe

◆1/k

x=

✓

fXe
fie

◆1/k

(A.16)x.

A2. Welfare Theorem
I prove the welfare theorem for a costly trade equilibrium and consider the cases of autarky andfree trade as its special cases when fT = 1 and fT = 0. Suppose the social planner maximizes asymmetric increasing and concave function of U and U ⇤ . Because Home and Foreign are symmetric,

the optimal solution always requires U = U ⇤ . Therefore, I consider a problem maximizing U bychoosing allocation in the Home market.
In the long run problem, the planner maximizes U chooses the following variables: (V1) Production of teams with capability ✓, q(✓); (V2) the distribution function H(✓) of ✓ that is generated byfeasible matching; (V3) Mass of international teams MT , capability cut-offs, xL , z1L and z2L , andMass of entrants MXe , M1e , and M2e . In the short run problem, the planner chooses (V3’) Mass ofinternational teams MT and capability cut-offs, xL , z1L and z2L for given MXe , M1e , and M2e instead of (V3). Notice that because of the mirror image structure, finding these variables for Home issufficient. Following Dhingra and Morrow (2014), I solve the problem in step by step. Step 1 finds an
optimal (V1) for given (V2)-(V3) and substitute it to the objective function. Step 2 finds an optimal(V2) maximizing the new objective function for given (V3) and substitute it to the objective function.
Step 3 find an optimal (V3). Then, I compare the solution to a decentralized market equilibrium.
In Step1, the planner faces the resource constraint.
L = Le + M

Z 1h
✓L

where Le ⌘ MXe fXe +iq(✓)✓ + f dH(✓) + MT fT .

P

i=1,2 Mie fie is labor for entry costs, the second term is labor for produc-tion, and the third term is fixed trade costs. The team capability cutoff ✓L and the mass of Home7

international teams MT are chosen in Step 2. The mass of varieties M = MXe [1G(xL )] is chosenin Step 3. Entry costs Le is chosen in Step 4. Therefore, the social planner solves⇢

max W ⌘ U =M
subject to M

Z 1
x
Z L1
✓L

✓↵ q(✓)⇢ dH(x)q(✓)✓ dH(✓)  L

Le

Mf(A.17)

MT f T .

The first order condition is⇢✓↵ q(✓)⇢ 1
where✓ = 0 for all ✓✓L ,> 0 is the Lagrange multiplier. From the first order condition, I obtainq(✓) =

⇣⇢⌘

✓(↵

)

(A.18).

Substituting this into the resource constraint (A.17) and
⇣⇢⌘

M

Z 1

✓(↵

) +

✓L

dH(✓) = LLe⇣⇢⌘

L e M f MT f T
R1.
M ✓L ✓dH(✓)

Substituting (A.18) into W and using
W =

⇣⇢⌘

1

M

Z 1

=

L

= (↵

Mf

= (↵

) +

) +

= 1 lead to

MT f T= 1, I obtain✓dH(x)✓L! 1
Z 1
L e M f MT f T
R1=
M✓dH(x).
M ✓L ✓dH(✓)
✓L
✓ Z 1
◆1
1
= M
✓dH(x)(L Le M f MT fT ).
L

✓L

In Step 2, when M = MXe [1 G(xL )], MT , and mass of firms are given, the planner chooses
R1H(✓) and ✓L to maximize M ✓L ✓dH(x). Since Koopmans and Beckmann (1957) and Shapley
and Shubik (1972), a stable matching, that is, PAM is known to maximize the aggregate payoffs
R1
M ✓L ✓dH(x) M f MT fT . Gretsky, Ostroy and Zame (1992) prove this result when agents arecontinuums as in the current model. Since M and MT are given in this step, PAM actually maximizes
R1
the aggregate team capability M ✓L ✓dH(x). Therefore, the social planner chooses PAM.

When the mass of international teams is given at MT , PAM is expressed as matching functions

mi (x) satisfying market clearing conditions (A.36) and (A.37) in the costly trade equilibrium. I write

8these conditions again:
2MXe [1⇤G(x)] = (Mie + Mie) [1

MXe [1

G(x)] = M1e [1G(mi (x))] for x

G(m1 (x))]

xT

MT = M2e [1G(m2 (x))] + MT for x 2 [xL , xT ],(A.19)

where xT is given by MT = sX MXe [1G(xT )].

Since the aggregate team capability becomes M
objective function for Step 3 becomes
W =

✓

MXe

Z 1

✓X (x) g(x)dx

xLLong run problem

◆1

(L

Le

R1

R1
✓L ✓dH(x) = MXe xL ✓X (x) g(x)dx, theMX [1G(xL )]f

MT f T )1.

I first consider Step 3 for the long run problem. Define vi ⌘ Mie /MXe andvT ⌘ MT /MXe . Then, conditions (A.19) are expressed as✓

v1 + v2
2

1

G(x) =

1

G(x) = v1 [1◆[1G(mi (x))] for x

G(m1 (x))]

vT = v2 [1xT
G(m2 (x))] + vT for x 2 [xL , xT ].

These conditions determine matching functions as m1 (x, v1 , v2 , vT ) and m2 (x, v1 , v2 , vT ), which isdifferentiable at x 2 (xL , xT ) [ (xT , 1).

For x > xT , total differentiation leads to the partial derivatives of mi
@mi
2g(x)
=
>0
@x
(v1 + v2 ) g(mi (x))@mi@mi@mi [1[1 G(mi (x))]
=
=
=
@v1
@v2
(v1 + v2 ) g(mi (x))
@x
@mi
= 0.
@vT

G(mi (x))]
>0
2g(x)

For x < xT , total differentiation leads to the partial derivatives of mi
@mi
g(x)
=
>0
@x
vi g(mi (x))@mi
1 G(mi (x))@mi [1 G(mi (x))]
=
=
.
@vi
vi g(mi (x))
@x
g(x)
@m1
1
1 @m1
=
=
@vT
v1 g(m1 (x))g(x) @x
@m2
1
1 @m2
=
=
.
@vT
v2 g(m2 (x))g(x) @x
Define9⇥ (xL , v1 , v2 , vT ) ⌘Z 1✓ (x, m1 (x, v1 , v2 , vT ), m2 (x, v1 , v2 , vT )) g(x)dx.

xL

Then, the partial derivatives of ⇥ are obtained as follows:@⇥= ✓ˆ (xL , v1 , v2 , vT ) g(xL ) = ✓L g(xL )
@xL
Z 1 ˆ
@⇥
@ ✓ (t, v1 , v2 , vT )=g(t)dt
@v1
@v1
xL
◆
Z xT
Z ✓
@✓ @m11 1 @✓@✓ @m1=[1 G(mi (t))] dt ++[1
2 xT @z1 @z2@xxL @z1 @x
Z 1@✓ @m1@✓@✓=[1 G(mi (t))] dt since=for x xT
@z1
@z2
xL @z1 @x
Z 1@⇥@✓ @m2=[1 G(m2 (t))] dt@v2
xL @z2 @x
Z xT 
@⇥@✓ @m2@✓ @m2=dt.
@vT@z2 @x@z2 @xxL

G(m1 (t))] dt(A.20)

Using vi , vT and ⇥, the objective function becomesln W =

1+[ln MXe + ln ⇥ (xL , v1 , v2 , vT )]
2
0
X
1 4
ln L MXe @fXe +vi fie + [113G(xL )]f + vT fT A5 .

i=1,2

I first find an optimal MXe maximizing this function. The first order condition is
✓

1

◆

L

P
fXe + i=1,2 vi fie + [1 G(xL )]f + vT fT
1
h
i+
= 0.
P
MXe
MXe fXe + i=1,2 vi fie + [1 G(xL )]f + vT fT

The mass of entrants of final producers is obtained as
MXe =

h

fXe +

Since
L

2

MXe 4fXe +P

L
i=1,2 vi fie + [1X

G(xL )]f + vT fT

3

G(xL )] f 5 = L

vi fie + vT fT + [1

i=1,2

10

(A.21)

i.

L

=

✓

1

◆

L,

the objective function becomesln W = ln ⇥ (xL , v1 , v2 , vT )2

ln 4fXe +X

vi fie + [1

i=1,2

3

G(xL )]f + vT fT 5 + constant.
(A.22)

Finally, I find xT , xL , v1 , and v2 that maximize (A.22). The first order conditions are@ ln W
1 @⇥g(xL )f
P
=
+
=0
@xL
⇥ @xL fXe + i=1,2 vi fie + [1 G(xL )]f + vT fT@ ln W
1 @⇥
f1e
P
=
=0
@v1
⇥ @v1 fXe + i=1,2 vi fie + [1 G(xL )]f + vT fT@ ln W
1 @⇥
f2e
P
=
=0
@v2
⇥ @v2 fXe + i=1,2 vi fie + [1 G(xL )]f + vT fT@ ln W
1 @⇥
fT
P
=
= 0.
@vT
⇥ @vT
fXe + i=1,2 vi fie + [1 G(xL )]f + vT fTUsing (A.20) and (A.21), these conditions become@ ln W✓L g(xL )=+ g(xL )f@xL⇥
Z 1@ ln W1@✓ @m1=[1 G(mi (x))] dx f1e@v1⇥ xL @z1 @xZ@ ln W
1 1 @✓ @m2=[1 G(m2 (x))] dx f2e@v2⇥ xL @z2 @xZ@ ln W
1 xT @✓ @m2@✓ @m2=dx fT@vT⇥ xL @z2 @x@z2 @x
These conditions are further simplified a

Z 1
L@✓ @m1[1
MXe ⇥ xL @z1 @x
Z 1
L@✓ @m2[1
MXe ⇥ xL @z2 @x
Z xT 
L@✓ @m2
MXe ⇥ xL @z2 @x

L✓L = fMXe ⇥G(mi (x))] dx = f1e
G(m2 (x))] dx = f2e@✓ @m2dx = fT .
@z2 @x

11

MXe
=0
L
MXe
=0
L
MXe
=0
L
MXe
= 0.
L

Decentralized equilibrium

Finally, I compare these first order conditions with conditions charac-terizing a decentralized equilibrium. First, the price index becomes
P =

Z

p(!)1

✓ (!)↵ d!

1/(1

)

!2⌦



Z 1
1/(1
1
=
MXe
✓X (x) g(x)dx⇢xL
1= [MXe ⇥]1/(1 ) .
⇢)The market condition A is written as:

A=
=

I

(⇢P )

1

L
.
MXe ⇥

(A.23)

From integration by parts, the average profits of Zi suppliers can be written as
Z 1
fie =
⇡i (z) g(z)dz
ziL
Z 1
=
⇡i (mi (t)) g(mi (t))m0i (t) dt ( substitution of z = mi (t))
xZ L1=⇡i0 (mi (t)) m0i (t) [1 G(mi (t))] dt (integration by parts)x
Z L1@✓ @mi=A[1 G(mi (t))] dt
xL @z1 @x
The no arbitrage condition (28) can be written as
fT = ⇡2 (zT ) ⇡1 (zT )
Z xT
Z xT
0
0
=
⇡2 (m2 (t))m2 (t)dt⇡10 (m1 (t))m01 (t)dt
xLxL◆
Z xT ✓@✓ @m2@✓ @m2=A
dt.
@z2 @x@z2 @xxL
Using (A.23), the cutoff condition, the free entry conditions, and the no arbitrage conditions become

Z 1
L@✓ @m1[1
MXe ⇥ xL @z1 @x
Z 1
L@✓ @m2[1
MXe ⇥ xL @z2 @x
Z xT 
L@✓ @m2
MXe ⇥ xL @z2 @x

L✓L = fMXe ⇥G(mi (x))] dx = f1e
G(m2 (x))] dx = f2e@✓ @m2dx = fT .
@z2 @x12

These are the first conditions for the social planner’s problem. Therefore, a decentralized long run
costly trade equilibrium maximizes the world welfare. In autarky, this means that a decentralized
equilibrium maximizes the welfare in each country.
Short run problem

Finally, I consider Step 3 for the short run problem. When the mass of entrants

are fixed, the planner’s problem is to choose xL and vT maximizing
1

W = (MXe ⇥(xL , v1 , v2 , vT )) (LLe

MXe (vT fT + [1G(xL )] f ))1.

This is equivalent to maximize
ln W = ln ⇥(xL , v1 , v2 , vT ) + (1) ln[LLe

MXe (vT fT + [1G(xL )] f )]The first order condition is@ ln W✓L g(xL )MXe g(xL )f
=
+(
1)
=0
@xL
⇥
L Le MXe (vT fT + [1 G(xL )] f )Z xT 
@ ln W1@✓ @m2@✓ @m2
MXe fT
=
dx (
1)
@vT
⇥ xL @z2 @x@z2 @xL Le MXe (vT fT + [1G(xL )] f )(A.24)The resource constraint implies
LLe

MXe vT fT = MXe

Z 1

C(✓X (x))g(x)dx
Z 11) AMXe✓X (x)g(x)dx + MXe [1xL=(

G(xL )] fxL=(1) AMXe ⇥ + MXe [1G(xL )] f.

Therefore, it holds thatLLe

MXe [vT fT + [1G(xL )] f ] = (1) AMXe ⇥.

Substituting the first order conditions (A.24) leads toA

Z xT xL

A✓L = f@✓ @m2@z2 @x@✓ @m2dx = fT .
@z2 @x

These conditions are the cutoff condition and the no arbitrage condition in a short run costly tradeequilibrium. Therefore, the solution to the social planner’s problem and a decentralized costly tradeequilibrium achieve the same allocation.

13=0

A3. Proofs for Lemmas and Propositions
Preparation for the proofs for Lemmas 1, 2, and 3
In the following, I assume M1e > M2e and prove it in Lemma 4. I first introduce the concept ofexportable Zi suppliers.
Definition 1. Home Zi suppliers with capability zi are called “exportable” if ⇡i (zi ) + fT = ⇡i⇤ (zi )and “non-exportable” if ⇡i (zi ) + fT > ⇡i⇤ (zi ). Similarly, Foreign Zi suppliers with capability zi arecalled exportable if ⇡i⇤ (zi ) + fT = ⇡i (zi ) and non-exportable if ⇡i⇤ (zi ) + fT > ⇡i (zi ).
The exportability is a necessary (but not sufficient) condition for Zi suppliers to export. When
Home Zi suppliers with capability zi are exportable, Foreign final producers are indifferent between
Home Zi suppliers with capability zi and Foreign Zi suppliers with the same capability. When HomeZi suppliers with capability zi are non-exportable, Foreign final producers strictly prefer Foreign Zi
suppliers with capability zi to Home Zi suppliers with the same capability. Therefore, non-exportable
Zi suppliers never export.
Proofs for Lemmas1, 2, and 3 use the following Claims 1, 2, 3, 4, and 5.
Claim 1. All Zi suppliers are classified as either exportable or non-exportable.
Proof. Suppose ⇡i (zi ) + fT < ⇡i⇤ (zi ) holds. Then, no final producer would choose Foreign Zi suppliers with capability zi . Therefore, ⇡i (zi ) + fT < ⇡i⇤ (zi ) never holds and all Home final producers
are classified as either exportable or non-exportable. From the symmetry of the two countries, allForeign final producers are also classified as either exportable or non-exportable.
Claim 2. A Z1 supplier matches with a Z2 supplier with the same capability if either one of them isexportable.
Proof. From the symmetry of Home and Foreign, it is sufficient to consider the following two cases:(i) an exportable Home Zi supplier matches with a Foreign Zj supplier; (ii) an exportable Home Zi
supplier matches with a Home Zj supplier.
Case (i): Suppose an exportable Home Zi supplier with capability zi matches with a Foreign Zj
supplier with capability zj 6= zi and a final producer with capability x. From the stability condition(6), the profit of the final producer satisfies the following inequality:⇤⇡X (x) = ⇡X(x) = ⇧ (xzi zj ) ⇡i (zi ) ⇡j⇤ (zj ) fTmax
⇧ xzi0 zj0
0 0
zi ,zj⇡i zi0⇡j⇤ zj0

fT .

From ⇡j⇤ (z) = ⇡i (z), the above inequality holds if and only if :⇧ (xzi zj )⇡i (zi )⇡i (zj )max⇧ xzi0 zj0
0 0
zi ,zj14⇡i zi0⇡i zj0 .

(A.25)

Because the second order condition for maximization in the right hand side of (A.25) requires⇡i00 (z) > 0, i.e., ⇡i (z) is a convex function. Let z̃ ⌘ (zi + zj ) /2. The convexity of ⇡i and thequasi-concavity of ⇧ (xzi zj ) imply:⇧ (xzi zj )⇡i (zi )⇡i (zj ) < ⇧ (xz̃ z̃)2⇡i (z̃) .

This inequality contradicts with (A.25). Therefore, only zi = zj satisfies (A.25).
Case (ii): an exportable Home Zi supplier with zi matches with a Home Zj supplier with zj .
Suppose zi 6= zj and they match with a final producer with x. If the final producer is from Home,
the profit satisfies:⇡X (x) = ⇧ (xzi zj )⇡i (zi )⇡j (zj )max⇧ xzi0 zj0
0 0
zi ,zj⇡i zi0⇡j zj0 .

If the final producer is from Foreign, the profit satisfies:⇤⇡X(x) = ⇧ (xzi zj )⇡i (zi )⇡j (zj )2fTmax⇧ xzi0 zj0
0 0
zi ,zj⇡i zi0⇡j zj02fT .

From the mirror-image symmetry, Foreign Zj supplier with zi is also exportable, i.e. ⇡j⇤ (zi ) +fT = ⇡j (zi ). Because ⇡j⇤ (zi ) = ⇡i (zi ), this implies ⇡j (zi ) = ⇡i (zi )fT . Then, the last two

inequalities hold if and only if (A.25) holds. From the argument in case (i), zi = zj must hold tosatisfy (A.25).
Claim 3. All suppliers in the CD sectors are non-exportable.
Proof. Suppose Home Z2 suppliers with capability z are exportable. From Claim 2, Home Z2 suppliers with capability z match with Home Z1 suppliers with the same capability. The matching market
clearing condition between Home Z1 suppliers and Home Z2 suppliers is expressed as:
M1e

Z 1
z

sD
1 (t) g(t)dt = M2e

Z 1
z

sD
2 (t) g(t)dt,(A.26)

where sDi (zi ) is the share of Home Zi suppliers matching with Home Zj suppliers among Home Zi
suppliers with capability zi . From the definition of exportability, sDi (zi ) < 1 holds only when HomeZi suppliers with capability zi are exportable.
D
I first show that for given z, only one of either sD
1 (z) or s2 (z) can be smaller than unity. Supposethat Home Zi suppliers with capability z are exportable, i.e. ⇡i⇤ (z) = ⇡i (z) + fT . The mirror-image
structure implies that ⇡j (z) = ⇡j⇤ (z) + fT . This means that Home Zj suppliers with capability z
are non-exportable because ⇡j (z) + fT = ⇡j⇤ (z) + 2fT > ⇡j⇤ (z) . Therefore, if sD
i (z) < 1, thensDj (z) = 1.

15

A differentiation of (A.26) with respect to z leads to:DM1e sD
1 (z) = M2e s2 (z) .

(A.27)

D
From M1e > M2e , only a combination of sD
2 (z) = 1 and s1 (z) = M2e /M1e < 1 satisfies condition(A.27). Therefore, all Home Z2 suppliers are non-exportable. From the mirror-image structure, allForeign Z1 suppliers are also non-exportable.
Claim 4. An inequality m1 (x)m2 (x) holds for all x

xL . A strict inequality m1 (x) > m2 (x)holds if there exist a positive mass of non-exportable Z1 suppliers with higher capability than m1 (x).
Proof. Because sD
2 (z) = 1 for all z

z2L from Claim 3, the market clearing condition (A.26)

becomes:M1e

Z 1

m1 (x)sD
1 (t) g(t)dt = M2e [1G (m2 (x))] for all x

Dividing both sides by M2e and adding G(m1 (x))
M1e
M2e

Z 1

m1 (x)

Because sD
1 (z)

✓

sD
1 (t)

M2e
M2e

◆

(A.28)

1 to both sides, I obtain:
G (m2 (x)) for all x

g(t)dt = G (m1 (x))

M2e /M1e for all x

xL .

xL .

(A.29)

xL from (A.27) with sD
2 (z) = 1 from Claim 3, the left-handside of (A.29) is non-negative for all xxL and strictly positive if there exist a positive mass of

non-exportable Z1 suppliers with higher capability than m1 (x).
Claim 5. An inequality ⇡20 (z)⇡10 (z) hold for all z. The inequality becomes strict if there exist a

positive mass of non-exportable Z1 suppliers with higher capability than z.
Proof. (Case 1) Suppose z

max{z1L , z2L }. Consider two teams with bundles of capability pa-

rameters (x, z1 , z2 ) and (x0 , z10 , z20 ), respectively. Suppose z2 = z10 (⌘ ẑ). Claim 4 implies thatz1z2 = z10

z20 . PAM implies x

x0 and ✓X (x)✓X (x0 ). Therefore, from the first orderconditions (8), we obtain:⇡20 (ẑ = m2 (x)) = Axm1 (x) =A✓X (x)ẑ

⇡10 ẑ = m1 x0= Ax0 m2 (x0 ) =A✓X (x0 ). (A.30)ẑSuppose there exist a positive mass of non-exportable Z1 suppliers with higher capability than z.
From Claim 4, x > x0 and ✓X (x) > ✓X (x). Therefore, inequality (A.30) becomes strict.
(Case 2) Suppose z1L > z

z2L . Then, ⇡1 (z) = ⇡10 (z) = 0 and ⇡20 (z)order condition (8).
Now I am ready to prove Lemmas 1, 2, and 3.

16

0 hold from the firstProof for Lemma 1
Proof. There must exist a positive mass of exportable Home Z1 suppliers in a trade equilibrium.
Consider exportable Home Z1 suppliers with capability z̃ > zL in a trade equilibrium. Suppose there
exists z > z̃ such that ⇡1⇤ (z)⇡1 (z) < fT on the contrary to the Lemma. Because ⇡1⇤ (z) = ⇡2 (z)for all z and ⇡1⇤ (z̃)⇡1⇤ (z)⇡1 (z̃) = fT , the difference in the profit schedules satisfies:
Z z
⇥ ⇤0
⇤
⇡1 (z) = ⇡1⇤ (z̃) ⇡1 (z̃) +⇡1 (u) ⇡10 (u) du
z̃
Z z
⇥ 0
⇤
= fT +⇡2 (u) ⇡10 (u) du.
z̃

The second term in the right-hand side must be non-negative from (A.30), which contradicts ⇡1⇤ (z)⇡1 (z) < fT . Therefore, if ⇡1⇤ (z 0 )⇡1 (z 0 ) = fT holds for some z 0 , then ⇡1⇤ (z)⇡1 (z)

fT holds

for all z z 0 . From Claim 1, this means that ⇡1⇤ (z) ⇡1 (z) = fT for all z z 0 .
⇤ < z
⇤
⇤
Notice that z2L = z1L
1L = z2L from M1e > M2e . From ⇡1 (z) = ⇡2 (z) for all z, thedifference in the profits of Z1 suppliers between Home and Foreign is:⇡1⇤ (z)⇡1 (z) = ⇡1⇤ (z2L )From Clam 5, ⇡1⇤0 (z)⇡10 (z)⇡1 (z2L ) +

Z z

z2L

⇥ ⇤0
⇡1 (u)

0 for all z and ⇡1⇤ (z2L )threshold zT such that ⇡1⇤ (z) ⇡1 (z) = ⇡2 (z)⇡2 (z) ⇡2⇤ (z) < fT for all z < zT .

⇤
⇡10 (u) du.
⇡1 (z2L ) = 0. Therefore, there exists a⇡2⇤ (z) = fT for all z

zT and ⇡1⇤ (z)⇡1 (z) =Proof for Lemma 2
Proof. From Claim 3 and Lemma 1, Home Z1 suppliers with z1

zT are exportable, but other

Home Z1 suppliers and all Home Z2 suppliers are non-exportable. From PAM and Claim 2, there
exists xT such that mi (xT ) = zT and m1 (x) = m2 (x) for x

xT .

The market clearing condition for matching between Home final producers and Z1 suppliers is:MXe [1

G (x)] = M1e [1G(m1 (x))]

M1e

Z 1

s1 (t) g(t)dt for all xxL ,(A.31)

m1 (x)

where s1 (z) is the share of exporters among Home Z1 suppliers with capability z. The second termin A.31 is the mass of Home Z1 suppliers matching with Foreign final producers. The same condition
for Foreign final producers is
⇤
MXe
[1

⇤
G (x)] = M1e[1

G (m⇤1 (x))] + M1e

Z 1m⇤1 (x)s1 (t) g(t)dt for all x

xL . (A.32)

The first term in the right hand side is the mass of Foreign Z1 suppliers and the second term isthe mass of Home Z1 suppliers. Since m1 (x) = m⇤1 (x) (= m2 (x)) for all x

17xT from themirror-image structure and Claim 2, adding (A.31) and (A.32) up obtains the global matching market
clearing conditions (29). From the mirror-image structure, a similar condition holds for Z2 producers.
Let MT be the mass of Home Z1 suppliers matching with Foreign final suppliers, which is given
R1
by MT = M1e m1 (xT ) s1 (t)g(t)dt. Substituting MT in (A.31) and (A.32) and using the mirror
image symmetry obtain local matching market clearing conditions (30) .
Proof for Lemma 3
Proof. Let sX (x) be the share of importers among Home final producers with capability x and si (z)be the share of exporters among Home Zi suppliers with capability with z. The corresponding sharesfor Foreign s⇤X (x) and s⇤i (z) are similarly defined.
Since only final producers with x

xT and Zi suppliers in the comparative sectors with z

zT

are exportable, sX (x) = s⇤X (x) = 0 for all x  xT , s1 (z) = s⇤2 (z) = 0 for all z  zT , ands2 (z) = s⇤1 (z) = 0 for all z.
A differentiation of (29) with respect to x > xT j leads to⇤
2MXe g(x) = (M1e + M1e) g(m1 (x))m01 (x) for x > xT

(A.33)

and a differentiations of (A.31) with respect to x > xT leads to
MXe g(x) = M1e (1s1 (m1 (x))) g(m1 (x))m01 (x) for x > xT .

A comparison of (A.33) and (A.34) proves s1 (m1 (x)) = (M1ewhich means s1 (z) = (M1e⇤ ) / (2M ) for all z
M1e1e(A.34)⇤ ) / (2M ) for all x > x ,
M1e
1e
T

zT .

The market clearing condition for matching between Foreign final producers and Home Z1 suppliers is
⇤
MXe

Z 1
x

s⇤X (t) g(t)dt = M1e

Z 1m⇤1 (x)s1 (t) g(t)dt for x

xT⇤ , m (x) = m⇤ (x) and s (x) = s⇤ (x) for all x
From MXe = MXe
1X
1X

xT , the above inequality

becomes:
MXe

Z 1
x

sX (t) g(t)dt = M1e

Z 1s1 (t) g(t)dt for x

xT

m1 (x)

A differentiation of both sides with respect to x leads to
MXe sX (x) g(x) = M1e s1 (m1 (x)) g(m1 (x))m01 (x) for x > xT .

18

(A.35)

A comparison of (A.34) and (A.35) leads to⇤s1 (m1 (x))
M1e M1e=⇤ for all x
1 s1 (m1 (x))
M1e + M1esX (x) =xT .

Preparation for proofs for Lemmas 4 and 5 I first prove the following Claims 6 to 8.
Claim 6. For any x > 1 and any n < k,

1

1
G (x)

Z 1 ✓ ◆ntkg(t)dt =.
xknx

Proof. It holds that

1

1
G (x)

Z 1 ✓ ◆nZ 1
t
k n
g(t)dt = kx
t (k n) 1 dtxxx!
x (k n)
k n
= kx
k n
=

k
k

n

.

Claim 7. Let ✓X (x) ⌘ xm1 (x)m2 (x) and ✓i (zi ) ⌘ zi mi 1 (zi )mj mi 1 (zi ) be the team capability
for final producers with capability x and Zi suppliers with capability zi , respectively.

✓X (x)
>
✓L

✓

x
xL

◆3

✓1 (z1 )
,
>
✓L

✓

z1
z1L

Proof. For final producers with capability x◆3✓2 (z2 )
, and
<
✓L

✓

z2
z2L

◆3

.

xT , from Lemma 3, matching market clearing condi-

tions (29) can rewritten as
M1e [1

G (m1 (x))] = MXe [1G (x)] + sX MXe [1G(x)]M2e [1

G (m2 (x))] = MXe [1G (x)]

G(x)].

sX MXe [1(A.36)The second term of the right hand side in the first equation represents the mass of Foreign final
producers matching with Home Z1 suppliers with higher capability than m1 (x). The second term in
the right hand side in the second equation represents the mass of Home final producers matching with
Foreign Z2 suppliers with higher capability than m2 (x). For final producers with capability x  xT ,19

from MT = sX MXe [1G(xT )], matching market clearing conditions (30) can rewritten as

M1e [1

G (m1 (x))] = MXe [1G (x)] + sX MXe [1G(xT )]

M2e [1

G (m2 (x))] = MXe [1G (x)]G(xT )].

sX MXe [1(A.37)

Define⇢1 G(xT )(x, xT ) ⌘ max 1,
1 G(x)( ✓ ◆ )
x k
= max 1,.
xT

Then, it is possible to summarize (A.36) and (A.37) as
M1e [1

G (m1 (x))] = MXe [1G (x)] [1 + sX (x, xT )]M2e [1

G (m2 (x))] = MXe [1G (x)] [1 + sX (x, xT )] .

(A.38)

From these, matching functions are obtained as
m1 (x) =✓M1e
MXe (1 + sX (x, xT ))◆1/k

x and m2 (x) =✓M2eMXe (1 sX (x, xT ))

◆1/k

x,

and a capability function,
0✓X (x) = @

2
MXe

⇣

M1e M2e1s2X(x, xT )

2

11/k

⌘A

x3 .

(A.39)

Then, it holds that
✓X (x)
=
✓L

✓

x
xL

◆3 "

1 (sX (xL , xT ))2
1 (sX (x, xT ))2

#1/k

>

✓

x
xL

◆3

.

The market clearing conditions (A.38) can also be written as
Mie [1G(zi )] (1Mie [1G(zi )] (1where si ⌘ (Miemi1si (zi , zT )) = MXe [1⇥2si (zi , zT )) = Mje 1

G(mi 1 (zi ))]G mj mi 1 (zi )⇤,Mje ) /2Mie . From these, I obtain matching functions as




1/k
MXe
1
(zi ) =
zi and mj mi (zi ) =Mie (1 si (zi , zT ))Mie (1

20

1/k
Mje
zi .
2si (zi , zT ))and a capability function as✓i (zi ) =



2 (1
Mie

1/k
MXe Mjezi3 .
si (zi , zT )) (1 2si (zi , zT ))Then, it holds that
✓i (zi )(1 si (ziL , zT )) (1=✓L(1 si (zi , zT )) (12si (ziL , zT )) 1/k
2si (zi , zT ))

✓

zi
ziL

◆3

.

From s1 > 0 and s2 < 0, I obtain that ✓1 (z1 )/✓L > (z1 /z1L )3 for z1 > z1L and ✓2 (z2 )/✓L <(z2 /z2L )3 for z2 > z2L .
a ,⇡Claim 8. In the long run trade equilibrium, it holds that ⇡¯X > ⇡¯X¯1 > ⇡¯1a and ⇡¯2 < ⇡¯2a .

Proof. From (14) and [1G(x)] = xg(x)/k,

Z 1
1⇡ 0 (t)[1 G(t)]dt
1 G(xL ) xL X
✓
◆
Z 1
1
g(t)
0
=
⇡ (t)tdt
1 G(xL ) xL X
k

Z 1
1
1
=
Atm1 (t)m2 (t)g(t)dt .
k 1 G(xL ) xL

Z 1
f
1
✓X (t)=
g(t)dt .
k 1 G(xL ) xL ✓L⇡¯X =(A.40)

From Claim 6, the average profits of final producers in autarky become
f
a
⇡
¯X
=

k

=

f
k

"
"


1
1

#
a (t)
✓X
g(t)dt✓La
xa
L#
Z 1 ✓ ◆3
1
tg(t)dtG(xaL ) xaL xL
1
G(xaL )

Z 1

f
k
k k 3
"
#
Z 1 ✓ ◆3f
1
t=g(t)dtk 1 G(xL ) xL xL=a .
From Claim 7, (A.40) and (A.41), ⇡¯X > ⇡¯X

The first order condition (8) is written as⇡i0 (zi ) = A✓i (zi )f ✓i (zi )=.
zizi ✓ L21(A.41)

From this, the average profit for Zi suppliers becomes
Z 1
1⇡ 0 (t)[1 G(t)]dt
1 G(ziL ) ziL i
✓
◆
Z 1
1
f ✓i (t) tg(t)=dt
1 G(ziL ) ziL t ✓L
k
✓ ◆
◆
Z 1✓
f
1
✓i (t)=
g(t)dt.
k 1 G(ziL ) ziL
✓L

⇡
¯i =

(A.42)

From Claim 6, the average profit of Zi suppliers in autarky is✓ ◆Z 1✓ a ◆f1✓i (t)
g(t)dtaa
k 1 G(ziL ) ziL
✓La
✓ ◆
◆
Z 1✓
f
1
t 3
=
g(t)dta )aa
k 1 G(ziL
ziL
ziL
✓ ◆✓
◆
f
3
=
k
k 3
✓ ◆
◆
Z 1✓
f
1
t 3
=
g(t)d
k 1 G(ziL ) ziL ziL

⇡
¯ia =

(A.43)

From Claim 7, (A.42), and (A.43), it holds that ⇡¯1 > ⇡¯1a and ⇡¯2 < ⇡¯2a .
Claim 9. The free entry conditions are written as
fXe =

Z 1
xL

0
⇡X
(t)[1

G(t)]dt and fie =

Z 1
xL

0
⇡X
(t)⌘i (t)[1(A.44)

G(t)]dt.

Proof. From integration by parts, the free entry condition for final producers becomes:
fXe =
=
=

Z 1
x
Z L1
x
Z L1
xL

⇡X (t) g(t)dt
0⇡X(t) [1

G(t)] dt + lim ⇡X (x) [10⇡X(t) [1

G(t)] dt.

x!1

Since from f.o.c. (8)⇡i0 (mi (x))m0 (x) = Ami (x)mj (x)0= ⇡X(x)⌘i (x),✓xm0i (x)mi (x)

22

◆

G(x)]

⇡X (xL ) [1

G(x)L ]

the free entry condition for Zi suppliers becomes:
Z 1
fie =
⇡i (z) g(z)dz
ziL
Z 1
=
⇡i (mi (t)) g(mi (t))m0i (t) dt ( substitution of z = mi (t))
xZ L1=⇡i0 (mi (t)) m0i (t) [1 G(mi (t))] dt (integration by parts)
xL
Z 1
0
=
⇡X
(t) ⌘i (t) [1 G(mi (t))] dt.
xL

Proof for Lemma 4(i) From (A.44) and [1G(x)] = xg(x)/k, the free entry condition for final producers becomes:

Z 1
1
⇡
¯X =
⇡ 0 (t) [1 G(t)]dt
1 G (xL ) xL X
Z 1 0t⇡X (t)1=g(t)dt.
1 G (xL ) xL
k
✓ ◆
Z 1
1
1
=
A✓X (t)g(t)dt
k 1 G (xL ) xL
✓ ◆
Z 1
1
1
=
r (✓X (t)) g(t)dtk
1 G (xL ) xL
✓ ◆
1
=
r̄
k
✓ ◆
1
L
=
.
k
M
From (18), the mass of entrants becomes:
MXe =(ii)(iii)

1

M
⇡
¯X
L
=
M=
G(xL )
fXe
fXe k

a
= MXe
.

I first prove (ii) and (iii) using Claims 10 to 12.

Claim 10. Let ⌘i (x) ⌘ xm0i (x) /mi (x). (i) In autarky, ⌘i (x) = 1 for all x⌘i (x) = 1 if x > xT and ⌘i (x) =xL . (ii) Under trade,MXe [1 G (x)]if x 2 [xL , xT ).
Mie [1 G (mi (x))]

Proof. (i) Because autarky matching functions (11) are linear in x, ⌘i (x) = 1 for all x

xL . (ii)

From (32), matching functions under trade are also linear and ⌘i (x) = 1 for x > xT . From (30),mi (x) satisfies:✓1
mi (x)

◆k

MXe
=
Mie

✓ ◆k ✓ ◆k
1
1
+
x
zT

MXe
Mie
23

✓

1
xT

◆k

for x 2 [xL , xT ].

From the implicit function theorem, the derivatives of both sides by x isk✓1
mi (x)◆k+1

m0i (x) =

MXe
k
Mie

✓

mi (x)x✓ ◆k+1
1
for x 2 [xL , xT ].
x

From this,m0 (x)xMXe⌘i (x) = i=mi (x)Mie◆k=

MXe [1 G (x)]for x 2 [xL , xT ).
Mie [1 G (mi (x))](A.45)

Claim 11. The mass of entrants satisfies:(A.46)

M1e f1e + M2e f2e = 2MXe fXe .

Proof. From Claim 9 and Claim 10, the free entry condition for Zi suppliers (A.44) becomes
Z 1
0
fie =
⇡X
(t) ⌘i (t) [1 G(mi (t))] dtx
Z L1Z
MXe xT 00=⇡X (t) [1 G(mi (t))] dt +⇡ (t) [1 G(t)] dt.
(A.47)
Mie xL X
xT
P
Because 2MXe [1 G (x)] = i=1,2 Mie [1 G (mi (x))] for all x xT , it follows that:
Z 10
M1e f1e + M2e f2e = 2MXe⇡X(t) [1 G(t)] dt (from (A.47))
xL= 2MXe fXe (from (A.44)).

Claim 12.
M1e
Ma
Ma
M2e
M1e + M2e
Ma + Ma
> a1e > a2e >and> 1e a 2e .
MXe
MXe
MXe
MXe
2MXe
2MXe
Proof. Notice thataMXea from (18)
M1e
Z
1
Ma
0
= Xe
⇡X
(t)[1 G(t)]dt from Claim 9
a
M1e
xL
✓ a ◆
Z 1
a Z xT
MXe
MXe
0
0
=
⇡X (t)[1 G(t)] dt +⇡X(t) [1 G(t)] dt
a
a
M1e
M1e
xT
xL
Z 1
Z xT
Ma
0
0
=
⇡X
(t) [1 G(ma1 (t))] dt + Xe⇡X(t) [1 G(t)] dt from (10).
a
M
xT
1e xL

f1e = fXe

Because ma1 (x) > m1 (x) for x(A.48)

xT , a comparison of (A.47) and (A.48) proves that M1e /MXe >a /M a . From M
a
a
a
a
a
M1e
Xe = MXe , M1e > M1e . Since M1e f1e + M2e f2e = MXe fXe also holds fromXe

24

(18), it holds
M2e f2e = 2MXe fXea
MXe f1e > 2MXea
M1e f1e = M2e
f2e .

a.
Thus, M2e > M2e

Preparation for Proofs for Lemma 5 and Lemma 6
Claim 13. In a long run equilibrium, ⇡¯1 > ⇡¯X > ⇡¯2 .
Proof. From (29), (A.47) and Claim 10, the free entry condition for Zi suppliers becomes✓◆ Z xTMXe
0
G (mi (t))] dt +⇡X(t) [1 G (t)] dt
M
ie
xT
xL
✓
◆Z 1
✓
◆ Z xT
2MXe
MXe
0
0
=
⇡X (t) [1 G (t)] dt +⇡X(t) [1 G (t)] dt from (29)M1e + M2e
M
ie
xT
xL
✓
◆Z 1
✓
◆Z 1
MXe
2MXe
MXe
0
0
=
⇡X (t) [1 G (t)] dt +⇡X(t) [1 G (t)] dt
MieM1e + M2e
Mie
xL
xT
✓
◆ Z 1✓◆Z 1
Mie MjeMXe00=⇡X (t) [1 G (t)] dt +⇡X(t) [1 G (t)] dt
MieM1e + M2e
xL
xT

fie =

Z 1

0
⇡X
(t) [1

Using MT = sX MXe [1M2e )/(M1e + M2e ), and M = MXe [1

G (xT )], sX = (M1e

G(xL )],

the above equation becomes
Z 1
✓
◆
✓
◆
Z
M
1 G (t)sX (1 G(xT )) 1 0
1 G (t)
0
⇡X
(t)
dt +
⇡X (t)dt
M1e xL
1 G(xL )1 G(xL )1 G(xT )
xT

M
MT
=
⇡
¯X +
⇡
¯T
(A.49)
M1e
M

f1e =

and
Z 1
✓
◆
M
1 G (t)
0
f2e =
⇡ (t)
dt
M2e xL X
1 G(xL )

M
MT
=
⇡
¯X
⇡
¯T ,
M2e
M
where ⇡
¯T ⌘ [1R1 0
G(xT )] 1 xT ⇡X(t)[1sX (1 G(xT ))1 G(xL )

Z 1
xT

0
⇡X
(t)

✓

1 G (t)
1 G(xT )◆dt(A.50)

G(t)]dt. From the free entry conditions, equation (A.49)

25

is written asMT
M1e f1e
⇡
¯T =
M
✓M
◆
fXe
MT
M1e f1e
fXe
+
⇡
¯T =
1 G(xL )M
MXe fXe 1 G(xL )✓◆MT
M1e f1e
⇡
¯T =
1 ⇡
¯X
M
MXe fXe✓◆✓◆MT
M1e f1e M2e f2e
MT
⇡
¯T =
⇡
¯X from (A.46)
M
2MT fXe
MXe
MT
MT
⇡
¯T =
⇡
¯X from (A.56)
M
MXe [1 G(x0 )]
MT
MT
⇡
¯T =
⇡
¯X ,
M
M0
⇡
¯X +

0

where M ⌘ MXe [1(A.51)

G(x0 )] .

From (A.49), (A.51), and M1e [1G(z1L )] = M + MT , the free entry condition for Z1 suppliers

becomes:f1e
G(z1L )

M
MT
=
⇡
¯X +
⇡
¯T
M1e [1 G(z1L )]
M
✓
◆✓ 0
◆
M
M + MT=⇡¯X .
M + MT
M0
✓
◆✓ 0
◆
M
M + MT=⇡¯X .
0M
M + MT⇡¯1 =1From (A.50), (A.51), and M2e [1G(z2L )] = M

MT , the free entry condition for Z2 suppliers

becomesf2e
G(z2L )

M
MT
=
⇡
¯X +
⇡
¯T
M2e [1 G(z2L )]
M
✓
◆✓ 0
◆
M
M
MT
=
⇡
¯X .
M0
M MT

⇡
¯2 =

1

Since
M 0 + MTM + MT

M0
M

MT
2MT (M M 0 )
=
> 0,
MT
(M + MT ) (M MT )⇡¯1 > ⇡¯X > ⇡¯2 holds.
Proof for Lemma 526

Proof. Defining x01 , x02 , z10 , and z20 such that z10 = m1 (x01 ) = ma1 (x01 ) and z20 = m2 (x02 ) = ma2 (x02 ),I will show x01 = x02 ⌘ x0 and x0 > xL . Notice that from (32), x01 and x02 are smaller than xT . Fromdefinition, these variables satisfy:
⇥
MXe 1
⇥
a
MXe
1
⇥
MXe 1
⇥
a
MXe
1

⇥
= M1e 1
⇤
⇥
a
G x01
= M1e
1
⇤
⇥
G x02
= M2e 1
⇤
⇥
a
G x02
= M2e
1
G x01

⇤

G z10

⇤

⇤
0

G z1

⇤
0

G z2
G z20

⇤

MT ,
,

(A.53)+ MT ,(A.54).

(A.55)

From straightforward manipulations, we obtain:
✓
◆
⇥
⇤ M1e
⇥
⇤
a
0
MT = M1e 1 G z1
MXe 1 G x01 from (A.52)
aM✓ 1e ◆
⇥
⇤
⇥
⇤
M1e
a
= MXe
1 G x01
MXe 1 G x01 from (A.53)
a
M
 1e
⇥
⇤ M1e MXea= MXe
1 G x01
a
a
M1e
MXe
and

MT

⇥

= MXe 1⇥
= MXe 1
⇥
a
= MXe
1

◆
M2e
G
1 G
from (A.54)
Ma✓ 2e ◆
⇤
⇥
⇤ M2e
a
G x02
MXe
1 G x02
from (A.55)aM2e⇤ MXe M2e
0
G x2
a
a .
MXe
M2e
x02

⇤

From (18) and (A.46), we have:✓◆ ✓M1e MXeMXeaaa
M1e MXe
MXe

a
M2e

M2e
a
M2e

⇥

◆

z20

⇤

✓


a
a
MXe M1e MXe
M2e MXe
=
+
2
a
a
a
MXe
MXe M1e
MXe M2e
MXe M1e f1eM2e f2e=+2
a
MXe MXe fXe MXe fXe= 0.

Therefore, x01 = x02 ⌘ x0 .

a (x0 ) = ✓ (x0 ) = x0 z 0 z 0 . From autarky matching functions,
By definition of x0 , ✓X
X
1 2

a
✓X
(x) = x3✓

M1e M2e
2
MXe

◆1/k

a
= ✓X
(x0 )

(A.52)⇣ x ⌘3x0

In a costly trade equilibrium, it holds from (A.39):

27.

✓X (x) =✓M1e M2e
2
MXe

◆1/k

x

3



1/k
1
1 (sX (x, xT ))2#1/k(sX (x0 , xT ))2(sX (x, xT ))2"⇣ x ⌘3 1= ✓X (x0 ) 0x
1⇣ x ⌘3a= ✓X(x0 ) 0
(x) ,
x
a
= ✓X
(x) (x) ,whereh(x) ⌘ 1(sX (x0 , xT ))2

i1/k h/ 1(sX (x, xT ))2

x < xT , 0 (x) > 0 for x < xT . From (x0 ) = 1,i1/k

. Since (x, xT ) = (x/xT )k for(x) > 1 for x > x0 and (x) < 1 for x < x0 .

a (x) for x > x0 and ✓ (x) < ✓ a (x) for x < x0 .
Therefore,✓X (x) > ✓X
X
X

Finally, I prove x0 > xL . From (A.52) and (A.53),
⇥

MXe 1

1

⇤

✓

a
MXe
a
M1e

◆

⇤
G x0
MT
✓ a ◆
MXe
0
MT = [1 G(x )]M1e MXe
a
M1e

✓ a ◆
MXe
1
1
=
MXe
M1e
a
G (x0 )
MTM1e

1 M1e f1e
=
MXe
MT
fXe

1 2M1e f1e 2MXe fXe
=
MT
2fXe
M1e f1e M2e f2e=.
2MT fXe

G x

0

= M1e

⇥

1

(A.56)

Let the mass of surviving Home Zi suppliers be Mi . Since Mi = Mie [1[1

G (ziL )] and fie =G (ziL )] ⇡¯i , the numerator of (A.56) becomesM1e f1eM2e f2e = M1✓

1

= M1 ⇡¯1

f1e
G (z1L )

◆

M2

✓

1

f2e
G (z2L )◆(A.57)

M2 ⇡¯2 .

Let the mass of surviving Home final producers be M . From (A.46), it holds that2M ⇡¯X = 2MXe [1G(xL )] ⇡¯X= 2MXe fXe= M1e f1e + M2e f2e(A.58)= M1 ⇡¯ 1 + M2 ⇡¯228

Since the mass of surviving Zi suppliers satisfy M1 = M + MT and M2 = M

MT , the share ofinternational teams becomes
MT
M1 M2
=
.
M
M1 + M2(A.59)

From (A.56) and the free entry conditions, I obtain

1

1
G (x0 )

1

1
M1e f1e M2e f2e⇡¯X=
G(xL )
2MT fXefXeM⇡¯X M1e f1e M2e f2e MT=
MT fXe
2M ⇡
¯X
M

M⇡
¯ X M1 ⇡¯ 1 M2 ⇡¯ 2 M1 M2=from (A.56), (A.58), and (A.59)
MT fXe M1 ⇡¯ 1 + M2 ⇡¯ 2 M1 + M2
M⇡
¯X
2M1 M2 (¯
⇡1 ⇡
¯2 )
=
> 0.
(A.60)
MT fXe (M1 ⇡¯ 1 + M2 ⇡¯2 ) (M1 + M2 )From Claim 13, ⇡¯1 > ⇡¯2 , and the inequality (A.60), I obtain x0 > xL .
Proof for Lemma 6
Proof. The free entry conditions imply
1
⇡
¯X
=
a
⇡
¯X
1

G (xaL )=G (xL )

✓

xL
xaL

◆k

and

1
⇡
¯i
=
a
⇡
¯i
1

a )
G (ziL=G (ziL )

✓

ziL
a
ziL

◆k

.

a =⇡
From Claim 13 and ⇡¯1a = ⇡¯X
¯2a , I obtain

⇡
¯1
⇡
¯X
⇡
¯2
z1L
xL
z2L
> a > a ) a > a > a .
a
⇡
¯1
⇡
¯X
⇡
¯2
z1L
xL
z2L
a and ⇡
Since ⇡¯X > ⇡¯X¯2 < ⇡¯2a from Claim 8, the free entry conditions imply xxLa > 1 > zz2L
a .
L

29

2L