Tohoku University Policy Design Lab Discussion Paper

TUPD-2022-001

Two-sided Heterogeneity:
New Implications for Input Trade

Tomohiro Ara
Faculty of Economics and Business Administration, Fukushima University

Policy Design Lab, Tohoku University

February 2022

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Two-sided Heterogeneity:
New Implications for Input Trade∗
Tomohiro Ara†
Fukushima University
February 17, 2022

Abstract
How diﬀerent are the impacts of trade cost on trade flows between intermediate goods and final goods?
How large are the welfare gains from trade for intermediate goods relative to final goods? To address thesequestions, we develop a heterogeneous firm model in which selection into exporting and importing play a
key role in industry productivity of vertically-related sectors. We show that the impact of trade cost on
trade flows is is greater for intermediate goods than for final goods, due to an extra adjustment through theextensive margin. We also find that the impact of trade cost on welfare is greater for intermediate goods
than for final goods if and only if the domestic input share is smaller than the domestic output share.
Keywords: Two-sided heterogeneity, input trade, selection
JEL Classification Numbers: F12, F14∗ This study was conducted as a part of the Project “Analyses of Oﬀshoring” undertaken at the Research Institute of Economy,
Trade and Industry (RIETI). A part of this paper is extended and is published as Ara and Zhang (2020) but the work presented
here is a much generalized development and has not been published elsewhere. The author would like to thank Joaquin Blaum forvaluable comments and discussions which improve the quality of the paper significantly. Financial support from the Japan Societyof the Promotion of Science under grant numbers 19K01599, 20H01492 and 20H01498 is gratefully acknowledged. This paper waspreviously circulated under the title “Complementarity between Firm Exporting and Firm Importing on Industry Productivity andWelfare.”† Faculty of Economics and Business Administration, Fukushima University, Fukushima 960-1296, Japan. Email address:
tomohiro.ara@gmail.com

1

Introduction

How diﬀerent are the impacts of trade cost on trade flows between intermediate goods and final goods? Howlarge are the welfare gains from trade for intermediate goods relative to final goods? Though the recent trade
literature has devoted enormous eﬀort to developing new trade models that match with empirical evidence, few
theoretical work has explored a distinctive feature of intermediate good trade that is absent in final good trade
when the two types of goods are costly traded subject to selection. The present paper tries to fill this important
gap in the literature by deriving a gravity equation of intermediate goods and by relating the trade elasticity
obtained from that gravity equation with the welfare gains from trade.
There is mounting evidence suggesting that intermediate good trade has been growing faster and its sharein world trade is larger than final good trade, due to “outsourcing” or “oﬀshoring” that fragments production
processes across the globe (Hummels et al., 2001; Hanson et al., 2005; Johnson and Noguera, 2012). Some recent
empirical work has revealed that when firms import intermediate goods from foreign countries, firm importing
displays a number of the same performance diﬀerences as firm exporting. A series of work by Bernard et al.
(2007, 2012, 2018a) unveil empirical regularity on firm importing that, just as in exporters, importers are larger
and more productive than non-importers within the same industries, and only a small fraction of firms import.
Another line of empirical work has also documented that the impact of trade liberalization is quite diﬀerent
between intermediate goods and final goods. For example, Amiti and Konings (2007) find that input tariﬀ
reductions increase industry productivity twice greater than output tariﬀ reductions in Indonesia; Topalova andKhandelwal (2011) similarly find that firms’ gains from input tariﬀ reductions can be ten times greater thanthose from output tariﬀ reductions in India. Further, input tariﬀ reductions give rise to diﬀerent productivity
gains from output tariﬀ reductions by expanding technological possibilities of firms, as shown by Goldberg etal. (2010).
This paper develops a heterogeneous firm model in which selection into exporting and importing play a keyrole in industry productivity. In our model, an industry is composed of two production sectors (i.e., upstream
and downstream sectors) in which the former sector produces and exports intermediate goods by using labor,
and the latter sector imports intermediate goods to produce and export final goods by using intermediates andlabor. While intermediate good suppliers in the upstream sector are modeled in a similar way to Melitz (2003),
one of main departures from Melitz (2003) is that final good firms in the downstream sector incur additional
fixed cost when using intermediate goods imported from the foreign market. As a result, selection occurs not
only in the upstream sector but also in the downstream sector, which allows us to capture the empirical patternthat only more productive firms access imported intermediate goods. This framework also allows us to show
diﬀerent impacts of tariﬀ reductions on resource reallocations in vertically-related sectors, which could help us
explain why input tariﬀ reductions can increase industry productivity more than output tariﬀ reductions. For
the sake of parsimony, we focus on two symmetric countries; nonetheless, our model identifies a new channel
through which selection in the downstream and upstream sectors jointly aﬀects the equilibrium outcomes in aconsistent manner with empirical evidence.
Our first contribution is in showing that the impact of trade cost on trade flows is greater for intermediate
goods than for final goods. The result is most clearly seen in a special case of a Pareto distribution with freeentry in the vertically-related sectors. Let Tj denote the value of trade between country 1 and country 2 where
the subscript j is either final goods (j = X) or intermediate goods (j = M ). Then Tj is given by
Tj = ψj ×

1

Y1a × Y2b,ετj jwhere ψj is a constant term, Y1 , Y2 are each country’s GDP and τj is trade barriers between the two countries.
As with a usual gravity equation, the value of trade is positively aﬀected by sizes of exporting and importing
countries but is negatively aﬀected by trade barriers for both types of goods. In that sense, the gravity equation
applies to intermediate goods let alone final goods. We find, however, that the elasticity of the value of tradewith respect to trade barriers is endogenously greater for intermediate goods than for final goods, i.e., εM > εX .
The diﬀerence in the trade elasticity arises through which trade barriers have a diﬀerent impact on selection
into exporting and importing in the vertically-related sectors.
In the case of intermediate good trade, reductions in trade cost allow not only intermediate good suppliers
to export intermediate goods more easily, but also final good firms to import these goods that are used fortheir production more easily. As new and less productive suppliers (firms) start exporting (importing) in the
upstream (downstream) sector, the eﬀect on trade flows is magnified in the current setting relative to a single
production sector setting. In the case of final good trade, in contrast, such reductions induce only new final goodfirms to start exporting since intermediate good suppliers do not import final goods for their production. Thissuggests that there exists an extra adjustment in the set of importers (extensive margin) in intermediate goodtrade that is absent in final good trade, which elevates the trade elasticity of intermediate goods relative to that
final goods.1 The finding could help us better understand why intermediate good trade has been growing fasterthan final good trade in globalization (Hummels et al., 2001; Hanson et al., 2005; Johnson and Noguera, 2012).
To demonstrate our contribution most sharply, we focus on bilateral trade flows between symmetric countries;
however, our result would hold for those flows between asymmetric countries by introducing an outside goodthat equalize wages.
Our second contribution is in showing that the impact of trade cost on welfare is greater for intermediate
goods than for final goods if and only if the domestic input share is smaller than the domestic output share.
In our vertical production model where both final goods and intermediate goods are costly traded subject toselection, trade liberalization can magnify the standard selection eﬀect relative to a single production model:
resource reallocations occur in the two production sectors, forcing the least productive firms (suppliers) to exit
the downstream (upstream) sector. As in Melitz (2003), the welfare gains from trade liberalization are directly
related to selection that occurs in the vertically-related sectors. However, trade liberalization in final goods andthat in intermediate goods trigger such two-sided reallocations on a diﬀerent scale, which makes it ambiguous
to compare the absolute magnitude of welfare changes. Given complex interactions between the two production
sectors, it is generally diﬃcult to figure out economic factors that generate diﬀerent welfare changes associatedwith trade liberalization in final goods and intermediate goods separately. We find however that welfare changes
are again clearly seen in a special case of a Pareto distribution with free entry in the two production sectors.
Let λj denote the domestic share where j is either final goods (j = X) or intermediate goods (j = M ). Then
welfare changes associated with trade cost τj are− 1

b εj ,c=λWjwhere a “hat” is proportional changes in a variable. Thus, welfare changes associated with trade liberalizationin each type of goods can be captured by only the the domestic share and the trade elasticity estimated fromthe gravity equation above. In other words, the welfare formula by Arkolakis et al. (2012) applies even in the
1 In the model with roundabout production, it is known that the presence of importing leads to an expansion of both exporters
and importers (Gopinath and Neiman, 2014; Blaum, 2019). While this channel works in our model with vertical production in thesense that reductions in output trade cost indirectly induce entry of final good exporters and raise the extensive margin elasticity(relative to that in a single production sector model), the direct eﬀect is strong enough that εM > εX .

2presence of two-sided heterogeneity of trade.2 Though the result may not be very surprising by itself, note that
not only is the trade elasticity εj but also the domestic share λj is far away from equality between final goods andintermediate goods: the share of intermediate goods is greater than that of final goods in the real world. Thismeans that the welfare evaluation holding the two domestic shares equal might lead to wrong understanding of
globalization where fragmentation of production processes plays a prominent role in improving welfare in eachcountry. The finding that the welfare gains from trade are greater for intermediate goods under rapidly rising
input trade is also consistent with empirical evidence that input tariﬀ reductions increase industry productivity
more than output tariﬀ reductions (Amiti and Konings, 2007; Topalova and Khandelwal, 2011), because suchproductivity improvement is typically associated with the higher welfare gains from input trade liberalization
than from output trade liberalization.
This paper contributes to the literature that examines the impact of imported input on industry productivity,
trade flows and welfare gains in a setup with input-output linkages. Antràs et al. (2017) develop a multi-country
sourcing model in which more productive final good firms import intermediate goods from a larger number ofcountries. Under the condition that final goods are non-tradable, they find, like ours, that the aggregate tradeelasticity with respect to variable trade cost tend to be higher than the firm-level trade elasticity in a gravity
equation of intermediate good trade. However, they employ the Eaton-Kortum (2002) framework for sourcing
intermediate goods, which implies that the upstream sector is characterized by perfect competition and selection
into the export market is not operative for intermediate good suppliers. Further, they do not address diﬀerencesin the welfare gains from trade between intermediate goods and final goods, which is one of the main focusesin our paper.
Bernard et al. (2018b) develop a model of two-sided heterogeneity in terms of productivity in intermediate
good suppliers and final good firms. Under the condition that final goods are non-tradable, they find the negative
degree assortivity among final good firms and intermediate good suppliers, i.e., more productive intermediate
good suppliers tend to match with less productive final good firms. While intermediate good suppliers self-selectinto exporting by incurring fixed trade cost, selection of final good firms is made by matching without paying
fixed trade cost. In practice, however, such fixed cost is empirically relevant for explaining why a small fraction offirms import (Kasahara and Lapham 2013; Halpern et al. 2015). Abstracting from matching between final goodfirms and intermediate good suppliers, we show that trade liberalization gives rise to two-sided reallocations as
a result of self-selection into exporting and importing, which turns out to be crucial to derive the diﬀerence inthe gravity structure of intermediate good trade and final good trade.
Our result of the welfare gains from intermediate good trade is related to that in Melitz and Redding (2014).
They show that when non-traded final goods are produced from traded intermediate goods, the welfare gains
from trade liberalization are magnified by raising domestic productivity. They consider, however, only perfectlycompetitive markets in every production sector in which all the firm-level variables do not play a key role inresource reallocations by trade liberalization. This paper, while allowing final goods to be produced from tradedintermediate goods, focuses on two-sided heterogeneity and identifies a potential channel through which tradeliberalization in intermediate goods can have a diﬀerent impact from that in final goods due to selections in thetwo production sectors.
2 There are two classes of models with input-output linkages of production. First is the model with “roundabout” production inwhich output is sold to consumers as final goods and also to firms as intermediate goods. This modeling approach is often employedin the recent literature, including Arkolakis et al. (2012, section IV). Second is the model with “vertical” production in which final(intermediate) goods are produced in the downstream (upstream) sector. This modeling approach used in this paper is found in
the IO literature and is applied to the trade literature; see for example Ishikawa and Spencer (1999), Ghosh and Morita (2007), andAra and Ghosh (2016). The diﬀerence in the modeling approach has a non-trivial consequence for trade flows and welfare gains
through selection in the vertically-related sectors.

3

Table 1 – Gravity in China’s imports
Overall

FinalIntermediate

Total

Extensive

Intensive

Total

Extensive

Intensive

Total

Extensive

Intensive

Distance−0.758(0.015)−0.549(0.007)−0.208(0.010)−0.689(0.023)−0.508(0.010)−0.181(0.016)−0.797(0.020)−0.574(0.009)−0.223(0.013)Tariﬀ−0.148(0.018)−0.085(0.007)−0.064(0.014)−0.106(0.022)−0.072(0.008)−0.034(0.016)−0.184(0.029)−0.100(0.011)−0.084(0.023)

No. of obs
Adj. R2

576, 509
0.403

576, 509
0.498

576, 509
0.389

220, 693
0.443

220, 693
0.478

220, 693
0.446

354, 976
0.372

354, 976
0.510

354, 976
0.343

Source: Ara and Zhang (2020, Tables 3 and 4)
Notes: Standard errors clustered at product-level are in brackets. Product and year fixed eﬀects are included. All results are
statistically significant at the 1% level.

The current paper is most closely related to Ara and Zhang (2020). In that paper, we extend our setup to a
multiple-industry, asymmetric-country framework in order to empirically investigate its theoretical prediction.
While the two papers have a similar flavor, the scope of the papers is diﬀerent. Ara and Zhang (2020) focus oncountry asymmetry to allow for potential diﬀerences in trade cost across countries, but complexity of the modelleads them to study a special case where either type of goods is only tradable. In contrast, we develop a moregeneral model in which both final goods and intermediate goods are costly traded and address the role playedby the interaction between these two types of goods in the welfare gains from trade. Nevertheless, one of themain findings is the same: the trade elasticity is greater for intermediate goods than for final goods. Below weprovide some evidence supporting this theoretical prediction.
To assess empirically this pattern, we first divide China’s imports in terms of U.S. dollar into final goods
and intermediate goods by applying the UN Broad Economic Categories classification to the China Customs
database (at the 6-digit HS product level) in 2000-2007. For each product imported from each trading partner ineach year, the value of total imports of two types of goods is further decomposed into the number of importing
firms with positive trade flows (extensive margin) and the average import value conditional on positive tradeflows (intensive margin). We then estimate a gravity equation of our model derived under the condition that
only either type of goods are tradable in order to see how the trade elasticity with respect to variable trade cost
(distance and tariﬀ) is diﬀerent across the two types of goods. We also estimate the gravity of overall goods
without distinguishing among final goods and intermediate goods.
Table 1 reports the estimates for the trade elasticities that are decomposed into the extensive and intensive
margins for each type of goods where all variables are measured in logs. In the estimates of overall goods,
the negative relationship between variable trade cost and the total import value is largely accounted for by theextensive margin, which accords well with the previous findings (e.g., Bernard et al., 2007). However, comparing
the estimates between the two types of goods, the coeﬃcient of variable trade cost on the total import value isgreater for intermediate goods than for final goods, while retaining the major role of the extensive margin in
the gravity for both types of goods. To check whether there is a statistically significant diﬀerence in the tradeelasticity between the two types of goods, we regress the gravity equation with an input dummy, and find thatthe coeﬃcient diﬀerence is significant only for the extensive margin (see Table 5 in Ara and Zhang (2020)).
These pieces of evidence highlight a new prediction of our model: the trade elasticity is significantly greater forintermediate goods than for final goods due mainly to the extensive margin.

4

2

Model

Consider a model in which two symmetric countries costly trade both input and output. There is one industry
composed of upstream and downstream sectors in which suppliers produce diﬀerentiated input and firms produce
diﬀerentiated output (using input) in monopolistically competitive markets. Labor is only a factor of productionand each country is endowed with L units of labor which is chosen as a numeraire of the model.

2.1

Consumers

Consumers’ preference is represented by a CES utility function with elasticity σ > 1:
(∫

σ
) σ−1

σ−1

U=

qi σ di,iwhere qi is output quantity produced by firm i. Utility maximization yields consumers’ output demand:σ−1
qi = p−σR,
i P

where pi is an output price, R is consumers’ aggregate output expenditure and
(∫
p1−σ
di
i

P =

1
) 1−σ

i

is the price index associated with the output bundle. Defining an aggregate output Q ≡ U , we have P Q = R.

2.2

Firms

Firms’ technology is linear combination of firms’ productivity φi and input bundle xi :qi = φi xi .
The productivity level φi is randomly drawn from a fixed distribution G(φi ) with unbounded upper support,
while the input bundle xi is produced by a CES production function with elasticity σ:
(∫
xi =
v

σ−1
σ

xDiv dv + 1M i

∫
v

σ−1
σ

xM iv dv

σ
) σ−1

,

where xDiv and xM iv are domestic and imported input quantity used by firm i and provided by supplier v,3
and 1M i is an indicator function which takes the value of one if firm i uses imported input and zero otherwise.
To facilitate the analysis below, we follow Bernard et al. (2018b) in assuming that only input is used for output
production and the elasticity of substitution between input in firms’ technology is identical with the elasticity of
substitution between output in consumers’ preference, but these simplifications would not aﬀect the qualitative
results of the paper. Cost minimization yields firm i’s input demand for variety v:
σ−1
xDiv = p−σ
ei ,
Dv ci
σ−1
xM iv = p−σei ,M v ci
3 The subscripts i and v are attached to relevant variables to firms and suppliers respectively.

5where pDv and pM v are domestic and imported input prices set by supplier v (common to all firms i) and
(∫

1
) 1−σ
∫
1−σ
p1−σ
dv
+
1
p
dv
,
Mi
Dv
Mv
v
v
∫
∫
ei = eDiv dv + 1M i eM iv dv,

ci =

v

v

where eDiv = pDv xDiv and eM iv = pM v xM iv are domestic and imported input expenditure incurred by firm iand provided by supplier v. A few points are in order for this specification. First, substituting xDiv and xM iv
into the CES production function and rearranging, firm i’s total input expenditure ei is expressed as
ei =

ci
qi .
φi

(1)

Thus, firm i’s input expenditure increases with its output quantity qi but decreases with its productivity level
φi . Second, from the input pricing rules set by suppliers and selection into exporting among suppliers, it follows
that the price index associated with the input bundle (referred to as firm i’s unit cost hereafter) is expressed as1−σc1−σ= c1−σi
D (1 + 1M i τM ∆)where c1−σ
=
D

∫
v

(2)

p1−σ
Dv dv, τM is variable trade cost of input and ∆ is the market share of exporting suppliers

defined later. To understand this, suppose that τM is suﬃciently high that no supplier profitably exports. Then
∆ = 0 and the unit cost is the same across all firms. Evidence suggests however that firms using both domestic
and imported input have a cost advantage over firms using only domestic input (Halpern et al., 2015). Further,even if τM is not prohibitively high, there is selection into exporting in the upstream sector and not all suppliers
export. Then ∆ < 1 and the unit cost of importing firms is lower than that of non-importing firms. Intuitively,
firms using both domestic and imported input can exploit a “love-of-variety” eﬀect for output production andraise their production eﬃciency.
Given the sourcing strategy and associated unit cost, firm i’s domestic profit is given by
(
πDi =

pDi −

ci
φi

)

σ−1
p−σ
R − fDi ,Di P

where fDi is firm i’s fixed production cost that satisfies fDi = fD + 1M i fDM . If firm i sources input only fromthe domestic market, it incurs fixed cost of domestic sourcing: fDi = fD . In contrast, if firm i also sources
input from abroad, it incurs additional fixed cost of foreign sourcing: fDi = fD + fDM . This assumption follows
from the firm importing literature in which importing firms incur higher fixed sourcing cost than non-importing
firms to serve the domestic market (e.g., Antràs and Helpman, 2004). Firm i chooses its domestic output pricepDi to maximize domestic profit. Observing that firm i takes the term P σ−1 R as given, profit maximization
yields the following pricing rule for domestic output:
pDi =

σ ci
.
σ − 1 φiThis in turn gives us the following expression of domestic revenue:
rDi = pDi qDi = σBc1−σ
φσ−1
i
i

6

$&!

$%!

!%"#$

!&"#$

!!"#$

"#$
!%(

"#$
!&(

!!"#$

−#%
−#&

−#% − #%(−#& − #&((a) Domestic profit(b) Export profit

Figure 1 – Domestic and export profit in downstream sector

where
B=

(σ − 1)σ−1 σ−1PR
σσis the index of output market demand. Domestic profit is
πDi =

rDi
− fDi = Bc1−σ
φσ−1− fDi .
ii
σComparing the unit cost ci and fixed cost fDi , firm i chooses its domestic sourcing strategy so that
{

}

σ−1
1−σ
σ−1
πDi = max 0, Bc1−σ− fD , B(1 + τM∆)c1−σ− fD − fDM .
D φi
D φi

Figure 1(a) draws domestic profit. In (φσ−1, πDi ) space, the slope of πDi is Bc1−σ
for non-importing firmsiD
1−σ
while Bc1−σ
D (1 + τM ∆) for importing firms, reflecting that variable profit is greater for importing firms due tolower unit cost. However, the intercept of πDi is −fD for non-importing firms while −fD − fDM for importing
firms, reflecting that fixed cost is greater for importing firms due to additional sourcing cost. From this tradeoﬀ,
there are productivity cutoﬀs at which domestic profit of importing firms exceed that of non-importing firms,
σ−1
σ−1
namely φσ−1
DM > φD . This ensures that a fraction of firms above φDM use both domestic and imported input

while others between φσ−1and φσ−1D
DM use only domestic input to serve the domestic market.
Among operating firms in the domestic market, more eﬃcient firms above some cutoﬀ can enter the export
market. Additional profit from exporting is given by()τ X ciσ−1πXi = pXi −p−σ
R − fXi ,
Xi P
φi

7

where fXi is firm i’s fixed trade cost that satisfies fXi = fX + 1M i fXM . The fixed cost structure is similar tothat in the domestic market in that importing firms incur higher fixed sourcing cost than non-importing firms
to serve the export market. On top of fixed trade cost, exporting firms also incur variable trade cost of output
τX and hence unit cost is higher for exporting firms than for domestic firms.4 Profit maximization yields thefollowing pricing rule for exported output:pXi =σ τX ci.
σ − 1 φiThis in turn gives us the following expression of export revenue:
rXi = pXi qXi = σB(τX ci )1−σ φσ−1.
iExport profit isrXi− fXi = B(τX ci )1−σ φσ−1
− fXi .
iσπXi =Comparing the unit cost τX ci and fixed cost fXi , firm i chooses its export sourcing strategy so that
{

}
1−σ

πXi = max 0, B(τX cD )φσ−1
− fX ,i1−σ
B(1 + τM∆)(τX cD )1−σ φσ−1
− fX − fXMi.

Figure 1(b) draws export profit. Comparison to Figure 1(a) reveals a similar pattern between the domestic
and export markets: there are productivity cutoﬀs at which export profit of importing firms exceed that of nonσ−1
σ−1
importing firms, namely φσ−1
XM > φX . This ensures that a fraction of exporting firms above φXM use both

domestic and imported input, which emerges only if they incur higher fixed cost to serve the export market.
To characterize the equilibrium of the downstream sector, we need to identify the productivity cutoﬀs. From
Figure 1 and the domestic and export profit above, these cutoﬀs are given by
σ−1
Bc1−σ
= fD ,D φD

B(τX cD )1−σ φσ−1
= fX ,X(3)

B(τM cD )1−σ ∆φσ−1
DM = fDM ,B(τX τM cD )1−σ ∆φσ−1
XM = fXM .

From the productivity cutoﬀ φc for c ∈ {D, X, M, XM }, we have the following selection patterns that arise in
the downstream sector. First, it follows from (3) that
(

φX
φD

)σ−1
=

σ−1
τX
fX
,
fD

(

φXM
φDM

)σ−1
=

σ−1
τX
fXM
.
fDM

(4)

Assuming variable trade cost τX and fixed trade cost fX , fXM are large so that φX > φD and φXM > φDM ,(4) means firm selection into exporting, which holds not only for firms using domestic input but also for firms
using imported input; thus not all importing firms export. Moreover,
(

φDM
φD

)σ−1
=

(

σ−1
fDM
1 τM
,
∆
fD

φXM
φX

)σ−1
=

σ−1
fXM
1 τM
.
∆
fX

4 The subscripts X and M are attached to relevant variables to output trade and input trade respectively.

8(5)Assuming variable trade cost τM and fixed trade cost fDM , fXM are large so that φDM > φD and φXM > φX ,(5) means firm selection into importing, which holds not only for the domestic market in Figure 1(a) but also
for the export market in Figure 1(b); thus not all exporting firms import. These selection patterns accord wellwith empirical evidence that firm selection is ubiquitous for both exporting and importing (Bernard et al., 2007,2012, 2018a). As a result, the productivity cutoﬀs in the downstream sector satisfy
φD < min{φDM , φX } < φXM .
The ranking of the productivity cutoﬀs means that, among operating firms, those with the lowest productivity
between φD and min{φDM , φX } use only domestic input and sell their output in only the domestic market,
whereas those with the highest productivity above φXM use both domestic and imported input and sell their
output in both the domestic and export markets.
In addition to the zero profit cutoﬀ condition, we impose a free entry condition. Upon incurring fixed entry
cost fE , a mass of entrants ME observe their productivity level φi drawn from a fixed distribution G(φi ). Thenfirm i decides whether to enter the downstream sector by choosing markets from which to source input as wellas to which to provide output, or to exit without producing. Obviously the former outcome occurs wheneverφi is greater than the domestic productivity cutoﬀ φD in (3). Hence the free entry condition is defined as∫ ∞πi dG(φi ) = fE ,φD

where the left-hand side is expected profit among operating firms which is equivalent to M1E∫iπi di. Using the

productivity cutoﬀs in (3), this condition can be expressed asfD J(φD ) + fX J(φX ) + fDM J(φDM ) + fXM J(φXM ) = fE ,where J(φc ) =∫ ∞ [( φi )σ−1
φc

φc

(6)

]
− 1 dG(φi ) is a strictly decreasing function of φc . It is worth emphasizing thatthe zero profit cutoﬀ condition (3) and the free entry condition (6) cannot characterize the downstream sector.
As in (3), the import productivity cutoﬀs φDM and φXM are aﬀected by the market share of input exporters ∆that is endogenously determined in the upstream sector. This implies that any trade shocks that induce changesin ∆ in the upstream sector has an impact on firm selection in the downstream sector through the availabilityof input used to output production by firms.
We conclude this section by deriving the domestic output share as well as the domestic input share. First,
from the fact that the aggregate revenue of firms equals the aggregate expenditure of consumers, the domestic
output share (from a viewpoint of consumers) is given by∫rDi di1
λX = ∫i
=
,
1−σ
r
di
1 + τX ΛX
i i
where
ΛX =
∫∞

(7)

1−σ
V (φX ) + τM
∆V (φXM )1−σV (φD ) + τM
∆V (φDM )φσ−1dG(φi ) is a strictly decreasing function of φc . It can be shown that the numerator andi∫∫denominator of ΛX are proportional to i rXi di and i rDi di respectively. Following Melitz and Redding (2015),
and V (φc ) =φcΛX is referred to as the market share of exporting firms, though we recognize that there are lots of definitions

9

of the market share in the literature. From this, if variable trade cost of input τM is suﬃciently high so that nosupplier profitably exports (∆ = 0), the domestic output share (7) collapses to that in the plain Melitz model.5
Second, from the fact that aggregate revenue of suppliers equals aggregate expenditure of firms, the domestic
input share (from a viewpoint of firms) is given by∫eDi di1=λM = ∫i,1−σe di
1 + τM ∆ΛM
i i
where
ΛM =

(8)

1−σ
V (φDM ) + τX
V (φXM ).
1−σ
V (φD ) + τX V (φX )∫∫It can be shown that the numerator and denominator of ΛM are proportional to i eM i di and i eDi di respectively,and ΛM is referred to as the market share of importing firms. Not surprisingly, if variable trade cost of inputτM is suﬃciently high so that no supplier profitably exports (∆ = 0) and no firm profitably imports (ΛM = 0),
the domestic input share (8) collapses to unity. To make the analysis more interesting, we restrict the range ofτM under which ∆ < 1 and ΛM < 1.

2.3

Suppliers

Suppliers’ technology is represented by a linear cost function of labor that involves fixed cost and marginal costwhere the latter cost is inversely related to productivity:
lvp = k +

xv 6
.
ϕv

All suppliers incur the same fixed cost k but it varies with the markets to which suppliers provide their input.
The productivity level ϕv is randomly drawn from a fixed distribution G(ϕv ) with unbounded upper support,∫
while xv = i xiv di is total input quantity provided by supplier v where xiv is firm i’s input demand for variety v
as in the last section. These features of input production implies that the upstream sector is also characterized
by a monopolistically competitive market and self-selection in exporting among suppliers as in the downstream
sector.
Given the production technology and firms’ input demand, supplier v’s domestic profit is given by
(
πDv =

pDv −

1
ϕv

)

p−σ
Dv

∫
cσ−1
ei di − kD ,iiwhere kD is suppliers’ fixed production cost which is common for all operating suppliers in the domestic market.
Supplier v chooses its domestic price pDv to maximize domestic profit. Observing that supplier v takes the term
∫ σ−1
ei di as given, profit maximization yields the pricing rule for domestic input:
c
i i
pDv =

σ 1
σ − 1 ϕv

and the following expression of domestic revenue:
rDv = pDv xDv = σAϕσ−1v
5 See, for example, eq (18) in Melitz and Redding (2015) who study the special case without importing.
6 The superscript p is attached to stress production worker. Labor is also used for entry which is denoted by le as shown later.
v

10where(σ − 1)σ−1 σ−1
A=
cD
σσ

∫
eDi dii

is the index of input market demand. Domestic profit is
πDv =

rDv
− kD = Aϕσ−1− kD .
v
σ

Supplier v chooses its domestic production strategy so that{}πDv = max 0, Aϕvσ−1 − kD .
Like Figure 1(a), domestic profit can be drawn in (ϕσ−1, πDv ) space where the slope of πDv is A and the interceptv
is −kD . In contrast to firms who decide not only to serve the domestic market but also to import input fromabroad in their domestic strategy, suppliers decide only to serve the domestic market in their domestic strategy
as they do not import firms’ output to produce their input in the presence of vertical linkages between upstream
and downstream sectors. As a result, there is a unique productivity cutoﬀ at which domestic profit of operating
σ−1
suppliers is zero, namely ϕσ−1
provide domestic input to
D . This ensures that a fraction of suppliers above ϕD

firms while others below ϕσ−1
immediately exit.
D
Among operating suppliers in the domestic market, more eﬃcient suppliers above some cutoﬀ can enter theexport market. Additional profit from exporting is given by(πM v =

pM v −

τM
ϕv

)

p−σ
Mv

∫
ciσ−1 ei di − kM ,iwhere kM is suppliers’ fixed export cost which is common for all exporting suppliers. On top of fixed trade cost,
exporting suppliers also incur variable trade cost of input τM .7 Clearly trade cost structure is similar to firms
in the downstream sector. Profit maximization yields the pricing rule for exported input:
pM v =σ τM.
σ − 1 ϕv

Note that the input prices satisfy pM v = τM pDv just like the output prices satisfy pXi = τX pDi . As seen in (2),this gives rise to firms’ unit cost in such a way that a markup on diﬀerentiated input (set by suppliers) is fully
passed through to the output price bundled from diﬀerentiated input (set by firms), i.e., double marginalization
takes place between firms and suppliers. Combining the input pricing rule with the imported input demand,
we get the following expression of export revenue:
1−σrM v = pM v xM v = σAτM
ΛM ϕσ−1.
v

It is important to see that the market share of importing firms ΛM enters the revenue expression of exporting
suppliers. This is because, so long as τM is not prohibitively high, trade cost limits the set of importing firms towhich exporting suppliers sell their input. As a result of this additional channel operating through selection in
importing among firms, variable trade cost of input τM aﬀects supplier’s export revenue rM v not only through1−σsupplies’ shipment (i.e., through τM) directly, but also through firms’ market share who profitably make use

7 Recall that the subscript M is attached to relevant variables to input trade in this paper.

11

of imported input (i.e., through ΛM ) indirectly. Export profit isπM v =rM v
1−σ− kM = AτM
ΛM ϕσ−1− kM .
v
σ

Supplier v chooses its export strategy so that
{

}

1−σ
πM v = max 0, AτM
ΛM ϕσ−1− kM .
v

Like Figure 1(b), export profit can be drawn in (ϕσ−1, πDv ) space and there is a unique productivity cutoﬀs,v
namely ϕσ−1
M , above which suppliers profitably export input to the foreign market.
To characterize the equilibrium of the upstream sector, we identify the productivity cutoﬀs of suppliers.
From the domestic and export profit above, these cutoﬀs are given by
Aϕσ−1= kD ,D1−σ
AτM
ΛM ϕσ−1= kM .
M(9)

From the productivity cutoﬀ ϕc for c ∈ {D, M }, we have the following selection pattern in the upstream sector:
(

ϕM
ϕD

)σ−1
=

σ−1
kM
1 τM
.
ΛM
kD

(10)

Assuming variable trade cost τM and fixed trade cost kM are large (while keeping ΛM < 1) so that ϕM > ϕD ,(10) means supplier selection into exporting. Note the similarity between (5) and (10) in the selection patterns.
Whereas (5) imposes firm selection into importing in the downstream sector, (10) imposes supplier selection
into exporting in the upstream sector.
In addition to the zero profit cutoﬀ condition, we impose a free entry condition. Upon incurring fixed entry
cost kE , a mass of entrants NE observe their productivity level ϕv drawn from a fixed distribution G(ϕv ). Then
supplier v decides whether to enter the upstream sector by choosing markets to which to provide input or toexit without producing, and the former outcome occurs whenever ϕv is greater than the domestic productivity
cutoﬀ ϕD in (9). Hence the free entry condition is defined as∫ ∞πv dG(ϕv ) = kE ,ϕDwhere the left-hand side is equivalent to N1E∫v

πv dv. Using the productivity cutoﬀs in (9), we getkD J(ϕD ) + kM J(ϕM ) = kE ,(11)where the functional form of J(ϕc ) is the same as that of J(φc ). Note that (9) and (11) cannot characterize theupstream sector. The export productivity cutoﬀ ϕM is aﬀected by the market share of output importers ΛMwhich means that the range of input exporters depends on the range of output importers. As a result of this,the impact of trade cannot be examined without taking into account the interaction between the upstream anddownstream sectors.
We conclude this section by deriving the domestic input share. Since suppliers’ input revenue equals firms’
input expenditure, the domestic input share (8) is alternatively defined as the ratio of the aggregate revenue

12

of domestic input to the aggregate revenue of total input earned by suppliers. Using the domestic and export
revenue above, the domestic input share (8) is given by∫
rDv dv1
λM = ∫v
=
,
1−σ
r
dv
1
+
τ
v
v
M ∆ΛM
where

V (ϕM ).
V (ϕD )∆=

We can show that the numerator and denominator of ∆ are proportional to∫r dv andv Mv∫

r dv respectively,v Dv

and ∆ is referred to as the market share of exporting suppliers. Notice that ∆ < 1 so long as supplier selection
into exporting (10) holds. The expression of unit cost (2) follows from this market share ∆ and the pricing rules
set by suppliers pDv , pM v .

2.4

EconomyTo close the model, we impose the labor market clearing condition. Noting that labor is used in both downstream
and upstream sectors, the condition is expressed as∫∫li di +

lv dv = L,ivwhere li = lie +lip and lv = lve +lvp denote labor used for entry and production by firm i and supplier v respectively.
Substituting labor used by firms and suppliers, the labor market clearing condition is simply expressed as
R = L.
Moreover, aggregate labor used in each production sector is given by∫li di =
i

L
,
σ

(

∫
lv dv =
v

σ−1
σ

)
L,

and hence the labor allocation between the two production sectors is exogenously fixed. Using the labor market
clearing condition as well as the zero profit cutoﬀ and free entry conditions, the mass of entrants ME , NE canbe written as a function of labor endowment L and the productivity cutoﬀs φc , ϕc .
Welfare per worker (equivalent to real wage) is expressed as(2

W = σ 1−σσ−1σ

) 2σ−1σ−1

1σ−1
L σ−1 (fD kD )− σ−1 φD ϕD λM
.
2

1

(12)

As is standard in the literature, welfare rises with country size L and falls with fixed production cost fD , kD .
More important however are the suﬃcient statistics for welfare in the presence of vertical linkages: when both
input and output are costly traded subject to selection, welfare is endogenously determined not only by thedomestic productivity cutoﬀ of firms φD in the downstream sector but also by the domestic productivity cutoﬀ
of suppliers ϕD and the domestic input share λM in the upstream sector.
This completes the characterization of the model. The next section solves for the equilibrium to address theimpact of trade on resource reallocations, trade flows and welfare gains.

13

3

Equilibrium

Since there are the eight equilibrium conditions in the model ((3), (6), (9), (11)), these eight conditions jointly
provide implicit solutions for the following eight unknowns:
φD , φX , φDM , φXM , B, ϕD , ϕM , A,
where the labor market clearing condition is omitted by choosing labor as a numeraire of the model. Once these
unknowns are determined, the other endogenous variables can be written as a function of them.

3.1

Resource Reallocations

We start with examining the impact of trade liberalization on the productivity cutoﬀs. Recall that the market
share of exporting suppliers ∆ and that of importing firms ΛM enter the zero profit cutoﬀ conditions in (3) and(9) respectively. As a result, changes in the equilibrium variables by trade liberalization depend critically on how
these market shares are aﬀected by such liberalization. For example, totally diﬀerentiating ∆ = V (ϕM )/V (ϕD ),changes in the market share of exporting suppliers in the upstream sector are given byd ln ∆ = −θM d ln ϕM + θD d ln ϕD ,where θc ≡ −d ln V (ϕc )/d ln ϕc can be thought of as the extensive margin elasticity in the upstream sector.8 By
definition, the extensive margin elasticity θc is a function of the productivity cutoﬀ ϕc , and changes in ∆ comenot only from changes in ϕc directly but also from changes in θc indirectly. To make the following analysis assimple as possible, we hereafter restrict our attention to a subset of the general productivity functions where theextensive margin elasticity satisfies θc = θ for any c. This means that the extensive margin elasticity is the sameacross all productivity cutoﬀs taking a constant value regardless of suppliers’ global status. In a similar vein, letϑc ≡ −d ln V (φc )/d ln φc denote the extensive margin elasticity in the downstream sector and we assume thatϑc = θ for any c. Admittedly, this is a restrictive assumption but it holds under one of the most commonly-used
distributions: Pareto. Specifically, if φi and ϕv are distributed Pareto with a common shape parameter γ, theextensive margin elasticities in the two production sectors are given byθc = ϑc = γ − (σ − 1) ≡ θ.

(13)

Although recent empirical work reports that the extensive margin elasticities are less likely to be constant (e.g.,
Bas et al., 2017), the Pareto distribution is nonetheless a good approximation of observed micro-level data andwe follow the standard practice in the literature.
With this restriction, we first consider the impact of input trade liberalization on the productivity cutoﬀs.
Diﬀerentiating and solving the equilibrium conditions simultaneously yields the following expressions of changes
in the domestic productivity cutoﬀs φD , ϕD (see Appendix A.1 for proof):()(σ − 1)(1 − λM )d ln φD = −d ln τM ,σ−1−θ
()(σ − 1)(1 − λM )d ln ϕD = −d ln τM .
σ−1−θ
8 See Arkolakis et al. (2012, p.110) for the single production sector model.

14

(14)

(14) shows that reductions in τM increase φD in the downstream sector as well as ϕD in the upstream sector,
forcing the least productive firms and suppliers to exit the respective production sector if and only if
σ − 1 > θ.

(15)

In the inequality, σ −1 is the (common) intensive margin elasticities under the CES utility/production functions,whereas θ is the (common) extensive margin elasticities in the two production sectors under the distributional
assumption. Thus, (15) requires that the extensive margin elasticities are not too large relative to the intensive
margin elasticities. We impose the inequality in the analysis below, since (14) implies that less productive firms
and suppliers otherwise enter the respective production sector by input trade liberalization, which would be lesslikely in reality.
It is possible to see the shift in the other productivity cutoﬀs. In the upstream sector, from the free entry
condition (11), it follows that the export productivity cutoﬀ ϕM shifts in the opposite directions to ϕD , andhence input trade liberalization generates the Melitz-type resource reallocations:d ln ϕDd ln ϕM<0<.
d ln τM
d ln τM
In the downstream sector, from selection into exporting (4), the export productivity cutoﬀ φX (φXM ) change
proportionately to φD (φDM ). Moreover, from the free entry condition (6), the import productivity cutoﬀ φDM(φXM ) shifts in the opposite direction to φD (φX ). Together with (14), we haved ln φDd ln φXd ln φDMd ln φXM=<0<=.
d ln τM
d ln τM
d ln τM
d ln τM
This means that resource reallocations arise even within exporting firms: more productive firms sourcing input
from multiple markets expand by input trade liberalization, whereas less productive firms sourcing input fromonly a single market shrink by such liberalization. As a result, firms that simultaneously export and import are
more likely to benefit from input trade liberalization, magnifying the eﬀect of initial productivity diﬀerences
and leading to sales concentration toward these most globalized firms (Bernard et al., 2018a).

Lemma 1: If both input and output are costly traded subject to selection under (15),(i ) Input trade liberalization gives rise to resource reallocations not only in the upstream sector but also in thedownstream sector.
(ii ) Input trade liberalization simultaneously induces such reallocations even within exporting firms: more (less)productive exporting firms sourcing input from multiple markets (a single market) expand (shrink).

Let us next investigate the impact of output trade liberalization. As shown in Appendix A.2, reductions inτX have the following impact on the domestic productivity cutoﬀs φD , ϕD :()θ(µD − µM )(1 − λM )d ln φD = − 1 − λX +d ln τX ,σ−1−θ
()(σ − 1)(µD − µM )(1 − λM )d ln ϕD = −d ln τX ,
σ−1−θ
15

(16)

where µD (µM ) is the domestic output share of firms using only domestic input (both domestic and imported
input).9 (16) shows that reductions in τX increase ϕD in the upstream sector as well as φD in the downstream
sector, forcing the least productive suppliers and firms to exit in each production sector if and only if (15) andµD − µM > 0 where the latter condition is equivalent to
V (φXM )V (φX )>.
V (φDM )V (φD )

(17)

Noticing that V (φc ) is proportional to the aggregate output produced by firms above the productivity cutoﬀ φc ,V (φXM )/V (φDM ) is the output share of exporting firms conditional on also importing, whereas V (φX )/V (φD )is the output share of exporting firms conditional on producing. The inequality would be likely to hold in thecircumstance where exporting and importing exhibit some complementarity: firms engaging in both exportingand importing increment their output relatively more than firms engaging in only one of these global activities.
Based on the recent empirical finding that intense importers tend to be also intense exporters (Blaum, 2019),
we will assume that not only is (15) but also (17) is satisfied in the following analysis.
We can see the shift in the other productivity cutoﬀs. On the one hand, reductions in τX increase ϕD but
decrease ϕM , giving rise to resource reallocations from less productive suppliers to more productive suppliers in
the upstream sector, just as in reductions in τM . On the other hand, from selection into importing (5) and thefree entry condition (6), reductions in τX increase φD and φDM but decrease φX and φDM in such a way thatd ln φDd ln φDMd ln φXd ln φXM=<0<=.
d ln τXd ln τXd ln τXd ln τX
This means that, in contrast to input trade liberalization, resource reallocations arises within importing firms:
more productive firms providing output to multiple markets expand by output trade liberalization, while lessproductive firms providing output in only a single market shrink by such liberalization. Despite this diﬀerence,
however, firms that simultaneously export and import are more likely to benefit from output trade liberalization(Bernard et al., 2018a). While the finding looks similar to the existing result in the literature exploring firms’
export and import decisions, it operates through diﬀerent channels. For example, Bernard et al. (2018a) focus on
strategic market power across a small number of global firms where input is produced under perfect competition.
In contrast, we focus on endogenous selection across measure-zero producers in the vertically-related sectors
where input is produced under imperfect competition.

Lemma 2: If both output and input are costly traded subject to selection under (15) and (17),(i ) Output trade liberalization gives rise to resource reallocations not only in the downstream sector but also
in the upstream sector.
(ii ) Output trade liberalization simultaneously induces such reallocations even within importing firms: more(less) productive importing firms providing output to multiple markets (a single market) expand (shrink).

The diﬀerence in the impacts of trade shows that input trade liberalization may require less strict conditions
than output trade liberalization to trigger the resource reallocations in the vertically-related sectors: input trade
9µ

D =

1
V (φ

)

1−σ
X
1+τX
V (φ )
Dand µM =

1
V (φ

)

1−σ
XM
1+τX
V (φ
)

. Note that these always satisfy 1 > µD − µM .

DM

16liberalization requires only (15) while output trade liberalization requires both (15) and (17). More importantly,
the diﬀerence in (14) and (16) illustrates potential channels through which input trade liberalization has moresignificant impacts on the trade-induced reallocations than output trade liberalization found by empirical work
(e.g., Amiti and Konings, 2007; Topalova and Khandelwal, 2011). In fact, the comparison of the impact on thedomestic productivity cutoﬀ in the upstream sector reveals thatd ln ϕDd ln ϕD>,d ln τM
d ln τX
and input trade liberalization always gives rise to greater resource reallocations among suppliers. As for changes
in the domestic productivity cutoﬀ in the downstream sector,
d ln φDd ln φD>d ln τM
d ln τX⇐⇒(σ − 1 − θ)(λX − λM ) + θ(1 − µD + µM )(1 − λM ) > 0.

Hence, the suﬃcient condition for this inequality is λX ≥ λM , i.e., the domestic output share is greater than orequal to the domestic input share. This would be satisfied in current globalization where the input trade shareis larger than the output trade share in the world trade volumes.

3.2

Trade Flows

Having shown the impact of trade on resource reallocations in the vertically-related sectors, let us turn to theimpact on trade flows. We will continue to examine the impact of input trade liberalization and output tradeliberalization separately, but note that trade liberalization in either type of goods aﬀects trade flows of bothtypes of goods. For example, input trade liberalization aﬀects not only input trade flows directly but also output
trade flows indirectly. Thus we first analyze the direct eﬀect of trade liberalization on each type of goods, andthen analyze the indirect eﬀect of such liberalization.
Consider first the impact of input trade liberalization. To see the sensitivity of input trade flows to changes
in variable trade cost of input, we derive the (full) trade elasticity with respect to variable trade cost below.
Following Melitz and Redding (2015) and using the domestic input share (8), this trade elasticity is given by(εM = −
=

d ln

1−λM
λM

)

d ln τM((σ − 1)| {z }+

Intensive margin elasticity)
d ln ∆−
d ln τM{z}|
Exporter extensive margin elasticity
in upstream sector()
d ln ΛM−
d ln τM
{z
}
|

+

.

Importer extensive margin elasticity
in downstream sector

The fact that the extensive margin elasticity stems from the upstream and downstream sectors indicates that
reductions in variable trade cost of input allow not only suppliers to export input more easily, but also firms
to import input that are used for their production more easily. In other words, the impact of trade on input
trade flows can be magnified in the presence of vertical linkages, due to additional entry that takes place in therespective production sector. Moreover, with a constant extensive margin elasticity θc = ϑc = θ, the exporterand importer extensive margin elasticities are the same for each other taking a constant value at−d ln ∆
d ln ΛM
θ(σ − 1)=−=.
d ln τM
d ln τM
σ−1−θ17Summing up the three terms, the input trade elasticity is expressed asεM =(σ − 1)(σ − 1 + θ)
.
σ−1−θ

(18)

As for output trade, using the domestic output share (7), the trade elasticity with respect to variable trade
cost of output is given by(
εX = −
=

d ln

)

1−λX
λX

d ln τX((σ − 1)| {z }−+|

Intensive margin elasticity)
d ln ΛXd ln τX{z}.

Exporter extensive margin elasticity
in downstream sector

Since suppliers do not import output produced by firms in input production, output trade liberalization does not
induce suppliers to enter and there is no importer extensive margin elasticity in the upstream sector. However,this does not mean that output trade liberalization has no impact on suppliers in the upstream sector at all. An
expansion of exporting firms (by output trade liberalization) triggers an expansion of importing firms, which inturn leads to an expansion of exporting suppliers through changes in the market demand A, B. In other words,
the output trade elasticity can be also magnified relative to that in the single production sector setting due tothe joint interaction between the vertically-related sectors. Although the result seems natural, (16) shows that
output trade liberalization leads to resource reallocations in the two production sectors if and only if (17) holds,
which is related to the extensive margin. In fact, the exporter extensive margin elasticity is[()]
d ln ΛX
σ−1+θ
−
=θ 1+
(ηD − ηX )(µD − µM ) ,d ln τX
σ−1−θwhere ηD (ηX ) is the domestic (foreign) input share of firms using only domestic input, satisfying ηD − ηX > 0
under (17).10 Hence the output trade elasticity is
[
(
)
]
σ−1+θ
εX = σ − 1 + θ 1 +(ηD − ηX )(µD − µM ) .
σ−1−θ

(19)

As shown by Chaney (2008), the output trade elasticity can be decomposed into the intensive margin elasticity
σ − 1 and the extensive margin elasticity θ when firm heterogeneity is present. While the decomposition applies,
the extensive margin elasticity can be larger in the multi-production sector model than in the single production
sector model. Obviously, so long as (17) holds, the exporter extensive margin elasticity has an additional term
which captures the feedback eﬀect from an expansion of the upstream sector production to an expansion of thedownstream sector production triggered by output trade liberalization. In this way, the impact of output tradeliberalization on output trade flows can be magnified with vertical production, even though such liberalization
does not directly induce entry of suppliers in the upstream sector.
Which trade elasticity is greater when both input and output are costly traded subject to selection? Simple
comparison of the trade elasticities in (18) and (19) immediately reveals that
εM > εX .
10 η

D =

1
V (φ

1−σ
DM
1+τM
∆ V (φD))and ηX =

1
V (φ

1−σ
XM
1+τM
∆ V (φX)). Note that these always satisfy 1 > ηD − ηX .

18

Thus, we can say that the trade elasticity is always greater for input than for output. This finding accords withthe widely-known empirical fact that input trade has been growing faster than output trade in the real world(e.g., Hummels et al., 2001; Hanson et al., 2005; Johnson and Noguera, 2012).
The finding suggests that the gravity structure is drastically diﬀerent between input trade and output trade.
Suppose that φi and ϕv are distributed Pareto with a scale parameter φmin = ϕmin = 1 and a shape parameter∫∫γ. Using (13), output trade flows RX = i rXi di and input trade flows EM = v rM v dv can be decomposed into
)σ−1+θ
(1 σ(σ − 1 + θ)
1
RX =
fX ×
ME ,
η
θ
φX
|X
{z
}
|
{z
}
Average sales per firm

and
EM =

Mass of firms

σ(σ − 1 + θ)
kM ×
θ {z
|
}
Average sales per supplier

(
|

1
ϕM

)σ−1+θ{z

NE .
}Mass of suppliers

Moreover, it can be shown that the mass of entrants in each production sector is proportional to country size
L but independent of the productivity cutoﬀs φc , ϕc under the distribution:
ME =

σ−1
L
,
σ(σ − 1 + θ) fE

NE =(σ − 1)2L.
σ 2 (σ − 1 + θ) kE

Substituting φX from (3) and ϕM from (9) as well as ME , NE derived above, we can express these trade flows
as a gravity equation form:
RX =

σ−1+θ
ψX
− θ
LB σ−1 (τX cD )−(σ−1+θ) fX σ−1 ,
ηX
σ−1+θ
σ−1

EM = ψM ΛM

LA

σ−1+θ
σ−1

(20)

θ
−(σ−1+θ) − σ−1τMkM ,where ψX and ψM are some constant term.11 As in a usual gravity equation, trade flows in either type of goods
are a function of exporting country size L, importing country demand B, A, and bilateral trade barriers, bothvariable τX , τM and fixed fX , kM . The functional form is very similar to the gravity equation in Chaney (2008)
in terms of the elasticity of trade flows with respect to trade barriers; however, output trade flows include theforeign input share of firms using only domestic input ηX while input trade flows include the market share of
output importers ΛM , which work to elevate the elasticity of trade flows relative to that in the single production
sector model. For example, applying the Pareto distribution to the market share of output exporters, we get
[
σ−1+θ
σ−1

ΛM

−(σ−1+θ)

= τM

(

µD
µM

) σ−1
(
θ

fDM
fD

)−1 (

kM
kD

θ
θ ] σ−1−θ
)− σ−1.

which is of course negatively aﬀected by both variable and fixed trade cost. Thus variable trade cost τM decrease
input trade flows not only through supplies’ shipment with elasticity σ −1+θ(= γ) directly as in Chaney (2008),
but also through firms’ market share who profitably use imported input with elasticity θ(σ−1+θ)
σ−1−θ indirectly bychanges in ΛM . The same claim applies to output trade flows in the sense that they are indirectly aﬀected bychanges in ηX .
2

(σ−1)
σ−1
11 ψ
and ψM = σθk
X = θf
E

E.

19

To obtain the trade elasticity from the gravity equation (20), it follows from the domestic shares λX , λM
that the trade elasticities can be alternatively written as follows:12
εX = −
Applying Pareto to RD =d ln(RX /RD ),d ln τX∫r di, ED =i Di

εM = −d ln(EM /ED ).
d ln τM∫

r dv and using ΛM , we can show that the closed-form solutions ofv Dv

εX and εM are the same expressions in (19) and (18) respectively. Defining the trade elasticities with respect tofixed trade cost fX , kM similarly, these elasticities are also greater for input trade than for output trade. Hencethe model predicts that, when estimating the elasticity of the value of trade with respect to trade barriers fromthe gravity equation, the trade elasticity is endogenously greater for input trade than for output trade.
The result on trade flows is our first main proposition of the paper.

Proposition 1: If trade liberalization induces resource reallocations in the vertically-related sectors, the impactof trade barriers on trade flows is greater for input trade than for output trade.

It is worth stressing that our result does not come from a CES production function where output is produced
by a variety of input. As is evident from the above decompositions, the result comes from the diﬀerence in theimpact of trade liberalization on the extensive margin: reductions in variable trade cost induce entry of new
firms and suppliers into the respective production sector in a diﬀerent way. From this reason, it is possible toempirically test our theoretical prediction taking account of the diﬀerent impact on the extensive margin. In a
companion paper (Ara and Zhang, 2020), we study this channel by estimating the gravity equation derived in a
multiple-industry, asymmetric-country setting under the Pareto distribution. As overviewed in the Introduction,
we find empirical support for the theoretical prediction in China’s imports.
We conclude this section by briefly mentioning the eﬀect of input (output) trade barriers on output (input)
trade flows. The trade elasticities capturing this indirect eﬀect are defined as(ε̃X = −

d ln

1−λX
λX

d ln τM(),ε̃M = −

d ln

1−λM
λM

d ln τX).

It is clear that both ε̃X and ε̃M have no intensive margin elasticity, implying that trade cost have the impact
on each trade flows only through changes in the market share of firms and suppliers, i.e., the extensive margin
elasticity. Using the impact of trade liberalization on the productivity cutoﬀs in Lemmas 1 and 2, we get
(

)
σ−1+θ
ε̃X = (σ − 1)(ηD − ηX ),
σ−1−θ
(
)
σ−1+θ
(µD − µM ).
ε̃M = (σ − 1)
σ−1−θ

(21)

Thus, so long as there is some complementarity between exporting and importing under (17), both ε̃X and ε̃Mhave the positive value. As a result, input trade liberalization not only increases input trade flows directly, butalso increases output trade flows indirectly, and vice versa.
12 This is similar to that in Arkolakis et al. (2012) who consider the “partial” trade elasticity focussing only on the direct eﬀect ofvariable trade cost, while we consider the “full” trade elasticity taking into account all eﬀects of that cost. Though these elasticities
are generally diﬀerent, they are the same when the extensive margin elasticity is constant. See Melitz and Redding (2015).

20

3.3

Welfare Gains

We finally examine welfare implications of trade liberalization in the presence of the vertically-related sectors.
Totally diﬀerentiating (12), we know that changes in welfare per worker W can be captured by changes in thedomestic productivity cutoﬀs φD , ϕD as well as the domestic input share λM in this model:(
d ln W = d ln φD + d ln ϕD +

1
σ−1

)
d ln λM .

(22)

Consider first the impact of input trade liberalization on welfare. On the one hand, the impact of variable
trade cost of input on the domestic productivity cutoﬀs are given by (14). Then (22) shows that reductions inthis trade cost improve welfare by raising the domestic productivity cutoﬀs in the two production sectors. The
welfare changes come from the Melitz-type resource reallocations among firms and suppliers seen in Lemma 1:
input trade liberalization magnifies the standard selection eﬀect by inducing the least productive firms as wellas the least productive suppliers to exit the respective production sector so long as condition (15) is satisfied.
This two-sided selection magnifies the welfare gains from trade relative to the single production sector setting.
On the other hand, totally diﬀerentiating the domestic input share in (8) and noting the definition of εM in thelast section, the impact of variable trade cost of input on the domestic input share is given byd ln λM = (1 − λM )εM d ln τM .
Then (22) shows that reductions in this trade cost deteriorate welfare by reducing the domestic input share.
The reason for the welfare changes is explained as follows. Using φD in (3) and the output market demand B,
welfare is defined as an inverse of the output price index:
1
=
P

(

σ−1
σ

)(

L
σfD

1
) σ−1

φD
.
cD

Thus, for given φD , welfare is negatively aﬀected by the unit cost of firms using only domestic input cD because
the higher is this unit cost, the less eﬃcient are these firms and the higher is the output price index. Note that
input trade liberalization endogenously aﬀects the unit cost by changing the range of input available to firms.
Using ϕD in (9) and the input market demand A, the unit cost is expressed as
1
=
cD

(

σ−1
σ

σ
) σ−1
(

L
σkD

1
) σ−1

1
σ−1
ϕD λ M.

Thus, for given ϕD , the unit cost is negatively aﬀected by the domestic input share λM because the higher is thisshare, the more available is input for firms using only domestic input. This means that when there is selection
into importing among firms, input trade liberalization can have a negative impact on welfare by increasing theunit cost of firms who cannot access imported input. However, this negative eﬀect is always dominated by the
positive eﬀect from selection. Substituting εM in (18) and summing up the three terms in (22), the changes in
welfare with respect to input trade liberalization are simply given byd ln W = −(1 − λM ) ln τM ,(23)

which indicates that input trade liberalization is always welfare-enhancing. More important is that the elasticity
of welfare with respect to input trade barriers is equivalent to the foreign input share 1 − λM .
21

Next we examine the impact of output trade liberalization on welfare. The impact of variable trade cost of
output on the domestic productivity cutoﬀs are given by (16). Then, from Lemma 2, (22) shows that reductionsin this trade cost have a similar impact as above in the sense that such reductions improve welfare by raising
the domestic productivity cutoﬀs in the two production sectors, so long as conditions (15) and (17) are satisfied.
On the other hand, the impact on the domestic input share operates through which output trade liberalization
indirectly increases input trade flows, due to the complementarity between exporting and importing under (17).
Observing that the corresponding trade elasticity is ε̃M in (21), the impact of variable trade cost of output on
the domestic input share is given byd ln λM = (1 − λM )ε̃M d ln τX .
Then (22) shows that reductions in this trade cost deteriorate welfare by reducing the domestic input share,
which is explained by noting the impact of τX on cD . While each of the welfare changes is qualitatively similar
between input trade and output trade, the magnitude of these changes is diﬀerent between them because thedomestic productivity cutoﬀs φD , ϕD rise on a diﬀerent scale (see (14) and (16)), which in turn gives a diﬀerent
impact on the domestic input share λM . Substituting ε̃M in (21) and summing up the three terms in (22), thechanges in welfare with respect to output trade liberalization are simply given byd ln W = −(1 − λX ) ln τX .

(24)

It is important to note again that the elasticity of welfare with respect to output trade barriers is equivalent to
the foreign output share 1 − λX .
We are ready to compare the changes in welfare by input trade liberalization and output trade liberalization.
It follows immediately from (23) and (24) thatd ln Wd ln W>d ln τM
d ln τX⇐⇒

λX > λM

Thus, the welfare gains from input trade liberalization are greater than those from output trade liberalization if
and only if the domestic output share is greater than the domestic input share. Recall that this is the suﬃcientcondition under which input trade liberalization induces greater changes in the domestic productivity cutoﬀ in
the downstream sector than output trade liberalization. In this sense, this welfare result is consistent with theresource-reallocation result in Lemmas 1 and 2.
We conclude this section by relating our welfare result to the Arkolakis et al. (2012) welfare formula. Using
the relationship between the changes in the domestic input share and the changes in variable trade cost, we can
express the changes in welfare by input trade liberalization (23) as
d ln W = −d ln λM.
εM

Thus, the welfare changes by input trade liberalization can be captured by only their two suﬃcient statistics:
the domestic input share λM and the input trade elasticity εM , even in the presence of the vertically-related
sectors. Similarly, totally diﬀerentiating the domestic output share in (7) and noting the definition of εX , theimpact of variable trade cost of output on the domestic output share is given byd ln λX = (1 − λX )εX d ln τX .
22

Using this relationship, the changes in welfare by output trade liberalization (24) are expressed as
d ln W = −d ln λX.
εX

Defining proportional changes of a variable by a “hat” (e.g., x
b = dx/x), the changes in welfare associated withtrade liberalization in each type of goods are given by
−

1

− 1

b εM = λb εX .
c=λ
W
M
X

(25)

Therefore, the welfare formula by Arkolakis et al. (2012) applies here: conditional on the two suﬃcient statistics,
the welfare gains from trade are the same between input trade and output trade. More importantly, our welfare
comparison displays the main emphasis of Arkolakis et al. (2012) in a very clear manner. Holding the domestic
share equal between input and output (λX = λM ), Lemmas 1 and 2 show that input trade liberalization always
has a greater eﬀect on the two-sided resource reallocations than output trade liberalization. Given this diﬀerencein the impact of trade, it is natural to imagine that the welfare gains from trade are greater for input trade thanfor output trade. The welfare expression (25) indicates that this is not the case. As stressed in the last section,
the fact that the trade elasticities are diﬀerent between the two types of trade reflects that there exists an extra
adjustment in the extensive margin for input trade that is absent for output trade. However, conditional on
the two suﬃcient statistics, this extra margin only aﬀects the composition of the welfare gains from trade, nottheir total size.
Though useful, the equivalence in the welfare changes by trade liberalization in input and output holds under
the condition that the domestic share is equal between input and output. There is however mounting evidence
suggesting that input trade has been growing faster and its share in world trade is larger than output trade(Hummels et al., 2001; Hanson et al., 2005; Johnson and Noguera, 2012). This piece of evidence implies that thewelfare evaluation holding the domestic share in input and output equal might lead to wrong understanding of
globalization where fragmentation of production processes plays a prominent role in improving welfare in eachcountry. The finding that the welfare changes are greater for input trade than output trade under rapidly rising
input trade is also consistent with empirical evidence that input tariﬀ reductions increase industry productivity
more than output tariﬀ reductions (Amiti and Konings, 2007; Topalova and Khandelwal, 2011), because suchproductivity improvement is typically associated with the higher welfare gains from input trade liberalization
than from output trade liberalization.
Besides this caveat, (25) depends on a constant extensive margin elasticity which makes the mass of entrants
in each production sector invariant to any trade shocks. If the extensive margin elasticity is variable, however,trade liberalization aﬀects welfare from changes in the mass of entrants, yielding an additional welfare channel.
Given recent empirical work reporting a non-constant extensive margin elasticity (Bas et al., 2017), we should
be careful about interpreting our welfare result.13
The result on welfare gains is our second main proposition of the paper.
Proposition 2: If trade liberalization induces resource reallocations in the vertically-related sectors, the impactof trade barriers on welfare gains is greater for input trade than for output trade if and only if the domestic
output share is greater than the domestic input share.

13 See Head et al. (2014) and Melitz and Redding (2015) for welfare implications with a variable extensive margin elasticity.

23

4Conclusion

This paper has presented a heterogeneous firm model in which selection into exporting and importing play a
key role in industry productivity of vertically-related sectors. We show that reductions in trade cost increase
trade flows for intermediate goods more than for final goods, due to an extra adjustment through the extensive
margin. We also find that reductions in trade cost induce greater welfare changes for intermediate goods thanfor final goods if and only if the domestic output share is greater than the domestic input share. These findings
could help us to obtain better understanding about rapidly rising growth of intermediate good trade and large
productivity gains associated with trade liberalization in intermediate goods reported by empirical work. One
of broader policy implications from this paper is that the diﬀerence in the trade elasticities between final goods
and intermediate goods is crucial for understanding the mechanism that generates the diﬀerence in the welfare
gains from trade.
To highlight an extra adjustment through the extensive margin in intermediate good trade, we have resorted
to a two-symmetric-country setting. While our model can be extended to a many-symmetric-country setting,it is challenging to develop a many-asymmetric-country setting where reductions in trade cost potentially havean asymmetric impact on trading partners, but such an extension could provide an important channel throughwhich to magnify a positive correlation between productivity levels of firms and the number of source countries.
It is also interesting to explore the impact of country asymmetry on the trade pattern between final goods andintermediate goods, and general-equilibrium consequences of such specialization patterns for the welfare gains.
Does a larger country host disproportionally more final good firms and become a net exporter of final goods invertical linkages? Does unilateral trade liberalization in final goods induce agglomeration of final good firms ina liberalizing country and give rise to the greater welfare gains there relative to unilateral trade liberalization
in intermediate goods? We leave these questions to feature work.

24

A

ProofsA.1

Proof of Lemma 1

To show the derivation of (14), we first derive changes by variable trade cost of input in the downstream sector.
Taking the log and diﬀerentiating the zero profit cutoﬀ conditions (4) and (5) with respect to τM ,d ln φX − d ln φD = 0,d ln φXM − d ln φDM = 0,1
d ln ∆ + d ln τM ,σ−1
1d ln φXM − d ln φX = −d ln ∆ + d ln τM .
σ−1
d ln φDM − d ln φD = −(A.1)

Using the definition of ∆ and θc = θ,d ln ∆ = −θ(d ln ϕM − d ln ϕD ).
Moreover, diﬀerentiating the free entry condition (6) with respect to τM ,∑fc J ′ (φc )φc d ln φc = 0.

cSolving this for d ln φDM and d ln φXM by using d ln φX = d ln φD and d ln φXM = d ln φDM from (A.1),
d ln φDM = −αd ln φD ,
where
α≡

(A.2)

fD J ′ (φD )φD + fX J ′ (φX )φX.
fDM J ′ (φDM )φDM + fXM J ′ (φXM )φXM

Just like (3) and (6) cannot characterize the levels in the downstream sector, (A.1) and (A.2) cannot characterize
the changes in the downstream sector through the changes in the market share d ln ∆.
Next we calculate changes by variable trade cost of input in the upstream sector. Taking the log anddiﬀerentiating the zero profit cutoﬀ condition (10) with respect to τM ,d ln ϕM − d ln ϕD = −1d ln ΛM + d ln τM .
σ−1

(A.3)

Using the definition of ΛM and ϑc = θ as well as d ln φX = d ln φD and d ln φXM = d ln φDM from (A.1),
d ln ΛM = −θ(d ln φDM − d ln φD ).
Moreover, diﬀerentiating the free entry condition (11) with respect to τM ,∑kc J ′ (ϕc )ϕc d ln ϕc = 0.

cSolving this for d ln ϕM , we haved ln ϕM = −βd ln ϕD ,

25

(A.4)

whereβ≡kD J ′ (ϕD )ϕD.
kM J ′ (ϕM )ϕM

Note the similarity between (A.1) and (A.3) as well as (A.2) and (A.4).
Finally, we solve for the changes in the economy by variable trade cost of input by taking account of thejoint interaction between the two production sectors. Substituting (A.2) and (A.4) into the third equation of(A.1), and substituting (A.2) and (A.4) into (A.3) respectively,θ(β + 1)d ln ϕD + d ln τM ,σ−1
θ−(β + 1)d ln ϕD = −(α + 1)d ln φD + (σ − 1) + d ln τM .
σ−1−(α + 1)d ln φD = −Solving these for d ln φD and d ln ϕD yields()
σ−1
d ln τM ,(σ − 1 − θ)(α + 1))(
σ−1
d ln τM .
d ln ϕD = −(σ − 1 − θ)(β + 1)

d ln φD = −1
1
It remains to show α+1= β+1= 1 − λM . Diﬀerentiating J(φc ) =′(J (φc ) = −From V (φc ) =

∫∞
φc

σ−1
φc

∫ ∞ [( φi )σ−1
φc

φc

]
− 1 dG(φi ) with respect to φc ,)[J(φc ) + 1 − G(φc )].

V (φc ) and hence
φσ−1dG(φi ), it follows that J(φc ) + 1 − G(φc ) = φ1−σciJ ′ (φc ) = −(σ − 1)φ−σc V (φc ),(A.5)

which also holds for suppliers’ productivity cutoﬀ ϕc . Substituting (A.5) into the definition of α and β above,and subsequently using the zero profit cutoﬀ conditions (3) and (9),
α=β=

1
.
1−σ
τM
∆ΛM

(A.6)

The result follows immediately from using (A.6) in the definition of λM .

A.2

Proof of Lemma 2

To show the derivation of (16), we closely follow the steps in Appendix A.1. Taking the log and diﬀerentiating
the zero profit cutoﬀ condition (4) and (5) with respect to τX ,d ln φX − d ln φD = d ln τX ,d ln φXM − d ln φDM = d ln τX ,1
d ln φDM − d ln φD = −d ln ∆,σ−1
1d ln ∆,
d ln φXM − d ln φX = −
σ−1

26

(A.7)

where d ln ∆ is the same expression in Appendix A1. Moreover, diﬀerentiating the free entry condition (6) withrespect to τX also yields the same expression in Appendix A.1. Solving this for d ln φDM and d ln φXM bynoting d ln φX = d ln φD + d ln τX and d ln φXM = d ln φDM + d ln τX in (A.7),
d ln φDM = −αd ln φD − γd ln τX ,
where
γ≡

(A.8)

fX J ′ (φX )φX + fXM J ′ (φXM )φXM.
fDM J ′ (φDM )φDM + fXM J ′ (φXM )φXM

As for the equilibrium in changes in the upstream sector, taking the log and diﬀerentiating the zero profit
cutoﬀ condition (10) with respect to τX ,d ln ϕM − d ln ϕD = −1d ln ΛM ,
σ−1

(A.9)

where
d ln ΛM = −θ(d ln φDM − d ln φD ) − (σ − 1 + θ)(µD − µM )d ln τX .
Moreover, diﬀerentiating the free entry condition (11) with respect to τX yields
d ln ϕM = −βd ln ϕD .

(A.10)

Finally, substituting (A.8) and (A.10) into the third equation of (A.7), and substituting (A.8) and (A.10)
into (A.9) respectively,θ(β + 1)d ln ϕD + γd ln τX ,σ−1θθγ − (σ − 1 + θ)(µD − µM )−(β + 1)d ln ϕD = −(α + 1)d ln φD −d ln τX .
σ−1σ−1−(α + 1)d ln φD = −Solving for d ln φD and d ln ϕD yields()
γ(σ − 1 − θ) + θ(µD − µM )d ln φD = −d ln τX ,(σ − 1 − θ)(α + 1))(
(σ − 1)(µD − µM )d ln ϕD = −d ln τX .
(σ − 1 − θ)(β + 1)
Moreover, using (A.5) into the definition of γ above,
γ = (1 − λX )(α + 1).
The result follows immediately from using this and (A.6) in the definition of λM .

27

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29

S

Supplementary Note (Not for Publication)

This Note contains detailed derivations for the expressions omitted in the main text due to the space constraint.
The same section names are attached to the following sections where the detailed derivations are required.

S.1

Firms

To derive firm i’s input expenditure in (1), substituting xDiv and xM iv into the CES production function,
ei
.
ci

xi =

Rewriting this as ei = ci xi and using firm i’s technology qi = φi xi yields (1).
∫
To derive firm i’s unit cost in (2), note that c1−σ
= v p1−σDDv dv can be expressed as
c1−σ
= NE
D

∫ ∞
ϕD

p1−σ
Dv dG(ϕv ).

Using the input pricing rule pDv and the definition of V (ϕc ), we get
(
c1−σ
= NE
D
Similarly, c1−σ
=
M

∫
v

σ
σ−1

)1−σ
V (ϕD ).

p1−σ
M v dv can be expressed as
(
c1−σ
= NE
M

στM
σ−1

)1−σ
V (ϕM ).

Substituting these into c1−σ
= c1−σ+ 1M i c1−σ
and using ∆ = V (ϕM )/V (ϕD ) yields (2).
i
D
M
The free entry condition can be expressed as∫ ∞∫ ∞πDi dG(φi ) +φD

πXi dG(φi ) = fE .
φX

From domestic profit πDi , the expected profit from the domestic market is given by
∫ ∞

∫ φDM
πDi dG(φi ) =

φD

φD

(

)
σ−1
Bc1−σ
− fD dG(φi ) +
D φi

∫ ∞
φDM

(

)
1−σ
σ−1
B(1 + τM∆)c1−σ− fD − fDM dG(φi ).
D φiRearranging and substituting φD and φDM in (3) into the above equality,∫ ∞ [(∫ ∞πDi dG(φi ) = fD
φD

φD

φi
φD

)σ−1

]
− 1 dG(φi ) + fDM∫ ∞ [(
φDM

φi
φDM

)σ−1

]
− 1 dG(φi ).

Similarly, the expected profit from the export market is given by∫ ∞ [(∫ ∞πXi dG(φi ) = fX
φX

φX

φi
φX

)σ−1

]
− 1 dG(φi ) + fXM∫ ∞ [(φXM

Using the definition of J(φc ) yields (6).

S-1

φi
φXM

)σ−1

]
− 1 dG(φi ).

∫To derive the domestic output share in (7), note that R =∫R=r di isi i∫
rDi di +rXi di.

iiThis aggregate revenue can be expressed as∫ ∞R = ME∫ ∞rDi dG(φi ) + ME

φD

rXi dG(φi ).
φX

From domestic revenue rDi , the aggregate revenue from the domestic market RD =∫ φDM
RD = ME
φD

(

)
σ−1
dG(φi ) + ME
σBc1−σ
D φi

∫ ∞

(

φDM

∫

r di is given by
i Di

)
1−σ
σ−1
dG(φi ).
σB(1 + τM∆)c1−σD φiUsing the definition of V (φc ) and rearranging, this revenue is rewritten as1−σ
RD = ME σBc1−σ
D (V (φD ) + τM ∆V (φDM )).

Similarly, the aggregate revenue from the export market RX =∫r di can be expressed as
i Xi

1−σ
RX = ME σB(τX cD )1−σ (V (φX ) + τM
∆V (φXM )).

Substituting these in λX = RD /R yields (7).
To derive the domestic input share in (8), note that E =
∫
E=
∫
v

eDiv dv and eM i =

∫
v

(S.2)

e di isi i∫
eDi di +iwhere eDi =

∫

(S.1)

eM i di,ieM iv dv. This aggregate expenditure can be expressed as∫ ∞E = ME∫ ∞eDi dG(φi ) + ME

φDeM i dG(φi ).
φDM

σ−1
σ−1
Note that eDiv = p1−σ
ei and eM iv = p1−σ
ei where ei is written from (1) as
Dv ci
M v ciei =ci(qDi + 1Xi τX qXi ),φiwhere 1Xi is an indicator function which takes the value of one if firm i exports their output and zero otherwise.
Substituting consumers’ output demand qDi , qXi into ei and subsequently using this in eDiv , eM iv ,σ−11−σeDiv = (σ − 1)Bp1−σ(1 + 1Xi τX
),
Dv φi
σ−1
1−σ
eM iv = (σ − 1)Bp1−σ(1 + 1Xi τX).
M v φi

Recalling that

∫
v

1−σ
p1−σ
and
Dv dv = cD

∫
v

1−σ
p1−σ
M v dv = cM , aggregation of eDiv , eM iv yileds

σ−11−σ
eDi = (σ − 1)Bc1−σ(1 + 1Xi τX
),
D φi
σ−1
1−σ
eM i = (σ − 1)Bc1−σ(1 + 1Xi τX
).
M φi

S-2

(S.3)

Then, the aggregate expenditure of domestic input ED =∫e di is expressed as
i Di

1−σ
ED = ME (σ − 1)Bc1−σ
D (V (φD ) + τX V (φX )).

Similarly, the aggregate expenditure of imported input EM =

(S.4)

∫

e di is expressed as
i Mi

1−σ
EM = ME (σ − 1)Bc1−σ
M (V (φDM ) + τX V (φXM )).

(S.5)

Substituting these and (2) into λM = ED /E yields (8).

S.2

Suppliers

Using (S.3), (S.4) and (S.5), supplier v’s revenue rDv =∫e di and rM v =
i Div

∫

e
di is given byi M iv

σ−1
rDv = p1−σ
Dv cD ED ,σ−1
rM v = p1−σ
M v cM EM .

Moreover, it follows from (S.4) and (S.5) that
(
EM =

cM
cD

)1−σ
ΛM ED ,and hence
σ−1
rM v = p1−σ
M v ΛM cD ED .

Using the input pricing rule pDv , pM v and the definition of A yields the expression of rDv , rM v . The zero profit
cutoﬀ condition (9) follows immediately from that expression.
The free entry condition can be expressed as∫ ∞∫ ∞πDv dG(ϕv ) +ϕDπM v dG(ϕv ) = kE ,ϕMwhere the first (second) term is expected profit from the domestic (export) market. From domestic profit πDv ,the expected profit from the domestic market is given by∫ ∞∫ ∞πDv dG(ϕv ) =

ϕD

(

)
Aϕσ−1
− kD dG(ϕv ).
v

ϕD

Substituting ϕD in (9) into the above equality,∫ ∞ [(∫ ∞πDv dG(ϕv ) = kD
ϕD

ϕD

ϕv
ϕD

)σ−1

]
− 1 dG(ϕv ).

Similarly, the expected profit from the export market is given by∫ ∞ [(∫ ∞πM v dG(ϕv ) = kM
ϕM

ϕM

Using the definition of J(ϕc ) yields (11).
S-3

ϕv
ϕM

)σ−1

]
− 1 dG(ϕv ).

To derive the domestic input share in (8), note that E =∫∫ ∞E = NE∫r dv + v rM v dv can be expressed as
v Dv
∫ ∞

rDv dG(ϕv ) + NE
ϕD

rM v dG(ϕv ),ϕM

where the first (second) term is aggregate revenue from the domestic (export) market. From domestic revenue∫
rDv , the aggregate revenue from the domestic market ED = v rDv dv is given by∫ ∞ED = NE()σAϕvσ−1 dG(ϕv ).

ϕDUsing the definition of V (ϕc ), this revenue is rewritten as
ED = NE σAV (ϕD ).
Similarly, the aggregate revenue from the export market EM =

(S.6)

∫

r dv can be expressed as
v Mv

1−σ
EM = NE σAτMΛM V (ϕM ).

(S.7)

Substituting these in λM = ED /E yields (8).

S.3

EconomyTo define the labor market clearing condition of the economy, let us first consider the aggregate amount of labor∫used for entry and production in the downstream sector. Denoting this by Li = i lie di + lip di,∫ ∞Li = ME fE + ME

∫ ∞fDi dG(φi ) + ME

φD

fXi dG(φi ).
φX

Every entrant incurs fixed entry cost fE , and firm i above φD (φX ) also incurs fixed production (export) costfDi (fXi ) to serve the domestic (export) market. Note that production worker is used only for fixed cost in thedownstream sector since firms purchase input from the market and hence labor is not used for variable cost (i.e.,
for transforming input into output). From domestic profit πDi , export profit πXi and the free entry condition(6) in the downstream sector, Li is expressed as
Li = R − E.

(S.8)

We also consider the aggregate amount of labor in the upstream sector. Denoting this by Lv =∫ ∞Lv = NE kE + NE
ϕD

p
lDv
G(ϕv ) + NE

∫ ∞
ϕM

∫

le dv + lvp dv,
v v

p
lM
v dG(ϕv ),ppwhere lDv
= kD + xDv /ϕv and lMv = kM + τM xM v /ϕv . Every entrant incurs fixed entry cost kE , and supplier v
pp
above ϕD (ϕM ) also employ labor used for production lDv(lMv ) to serve the domestic (export) market. Thus,

production worker is used for both variable cost and fixed cost in the upstream sector. From domestic profit
πDv , export profit πM v and the free entry condition (11) in the upstream sector, Lv is expressed as
Lv = E.
S-4(S.9)

Substituting (S.8) and (S.9) into Li + Lv = L, we get the standard labor market condition R = L. Moreover,noting R = RD + RX in (S.1) and (S.2) as well as E = ED + EM in (S.4) and (S.5),
(

σ−1
σ

E=

)
R.

(S.10)

Substituting this into (S.8) and (S.9) gives us aggregate labor allocated to each production sector.
To get the mass of entrants in each production sector, we first consider the mass of entrants in the downstream
sector. Substituting φc in (3) into (S.1) and (S.2), R = RD + RX can be expressed as
R = ME σ∑

fc φ1−σV (φc ).
ccMoreover, using R = L and rewriting this gives us the mass of entrants in the downstream sector:
ME =

σ

L
.
1−σ V (φ )
f
(φ
c)
c
c c

∑

(S.11)

Similarly, substituting ϕc in (9) into (S.6) and (S.7), E = ED + EM can be expressed as
E = NE∑kc ϕ1−σV (ϕc ).
ccMoreover, using (S.10) and rewriting this gives us the mass of entrants in the upstream sector:
(
NE =

σ−1
σ

)
σ

L
.
1−σ V (ϕ )cc kc (ϕc )∑(S.12)

Regarding welfare per worker defined as W ≡ U/L, it follows from Q ≡ U and P Q = R = L that
1
Q
= .
L
P

W =

Noting that w = 1, welfare per worker is equivalent to the real wage. To get the output price index, using φDin (3) and the definition of the output market demand B,
1
=
P

(

σ−1
σ

)(

L
σfD

1
) σ−1

φD
,
cD

which depends not only on firms’ domestic productivity cutoﬀ φD but also on firms’ unit cost cD , both are an
endogenous variable of the model. Using ϕD in (9), (S.10) and the definition of the input market demand A,
1
=
cD

(

σ−1
σ

σ
(
) σ−1

L
σkD

1
) σ−1

1
σ−1
ϕD λ M.

Combining these two expressions,
(
W =

σ−1
σ

(
) 2σ−1
σ−1

L
σfD

1
(
) σ−1

Rewriting this gives us the welfare expression in (12).

S-5

L
σkD

1
) σ−1

1
σ−1
φD ϕD λM.

S.4

Trade Flows

Under the Pareto distribution, J(φc ) =∫ ∞ [( φi )σ−1
φc]∫∞
− 1 dG(φi ) and V (φc ) = φc φiσ−1 dG(φi ) are simple

φcpower functions of the productivity cutoﬀ φc :
σ−1
1
,
σ−1+θ
θ (φc )
σ−1+θ 1V (φc ) =.
θ(φc )θ
J(φc ) =(S.13)

To compute the mass of entrants ME , NE under the Pareto distribution, using (S.13) in the mass of entrants
in the downstream sector (S.11),
ME =σ(σ−1+θ)θ[∑L

c fc (φc )−(σ−1+θ)].

Further, applying the Pareto distribution to the free entry condition (6),
[
]
σ−1 ∑
−(σ−1+θ)fc (φc )= fE .
θc
Combining the two expressions, we have that the mass of entrants is proportional to labor endowment L:
L
σ−1.
σ(σ − 1 + θ) fE

ME =(S.14)

Similarly, using (S.13) in the mass of entrants in the upstream sector (S.12) and applying the Pareto distribution
to the free entry condition (11), we also haveNE =L(σ − 1)2.
σ 2 (σ − 1 + θ) kE

To derive the gravity equation (20), let output trade flows RX =per firm and the mass of firms:
RX =
=

1
1 − G(φX )∫ ∞(S.15)∫

r di decompose into the average sales
i Xi

rXi dG(φX ) × [1 − G(φX )]MEφX

1 σ(σ − 1 + θ)
fX ×
ηX
θ

(

1
φX

)σ−1+θ
ME ,where the second equality follows from using (S.2), (S.13) and the definition of ηX . This decomposition in turncan be rearranged as)(
1 σ(σ − 1 + θ)L
σ−1
fX × (φX )−(σ−1+θ)ηX
θσ(σ − 1 + θ) fE)− σ−1+θ
(
)
(
σ−1
σ−1
fX
1
LfX
=
1−σ
ηX
θfE
B(τX cD )
σ−1+θ
ψX
− θ
=
LB σ−1 (τX cD )−(σ−1+θ) fX σ−1 .
ηX

RX =

S-6

(using (S.14))(using (3))

Similarly, input trade flows are)(σ(σ − 1 + θ)(σ − 1)2L
σ−1+θ
k M × ϕMθσ 2 (σ − 1 + θ) kE()()− σ−1+θ
σ−1(σ − 1)2
kM
=
LkM
1−σ
σθkE
AτM ΛM

EM =

σ−1+θ

(using (S.15))(using (9))θ−(σ−1+θ) − σ−1
kM .

σ−1+θ

= ψM ΛMσ−1 LA σ−1 τMThis gives us the gravity equation expressions in (20).

To get the output trade elasticity from the gravity equation, applying (S.13) to RD in (S.1) and subsequently
dividing RX derived above by RD ,RX−(σ−1+θ) ηD
= τX
RD
ηX

(

fX
fD

θ
)− σ−1.

Taking the log and diﬀerentiating RX /RD with respect to τX ,(εX = σ − 1 + θ −d ln(ηD /ηX )d ln τX).

Moreover, from the definition of ηD , ηX , it follows that(−d ln(ηD /ηX )d ln τX

)

(
=θ

σ−1+θ
σ−1−θ

)
(ηD − ηX )(µD − µM ).

This gives us the same expression of the output trade elasticity in (19).
To get the input trade elasticity from the gravity equation, applying (S.13) to ED in (S.6) and subsequently
dividing EM derived above by ED ,σ−1+θEM−(σ−1+θ)
= τM
ΛMσ−1
ED

(

kM
kD

θ
)− σ−1.

)Moreover, noting that ΛM = µµDMVV(φ(φDMand solving for ΛM and ∆,
D)

[
σ−1+θ
σ−1

ΛM

(

= τM

µD
µM

[

(

−(σ−1+θ)−(σ−1+θ) µDσ−1+θ

∆ σ−1 = τM

µM

(
) σ−1
θ

fDM
fD

fDM
fD

θ
θ ] σ−1−θ
)− σ−1kM,kD
θ) ] σ−1−θ

)−1 (θ
)− σ−1(

kM
kD(S.16)−1.

Taking the log and diﬀerentiating EM /ED with respect to τM ,(εM = σ − 1 + θ +σ − 1 + θ d ln ΛM
σ − 1 d ln τM).

While ΛM in (S.16) depends not only on τM directly but also on µD /µM indirectly, it follows from the definition
D /µM )of µD , µM that d ln(µ= 0 and only the direct changes of τM remains. This gives us the same expression
d ln τMof the input trade elasticity in (18).
S-7

To show the trade elasticities capturing the indirect eﬀect, note thatd ln(RX /RD )d ln ΛX=−,d ln τM
d ln τMd ln(EM /ED )d ln ∆
d ln ΛMε̃M = −=−−,
d ln τXd ln τXd ln τXε̃X = −D V (φX )which shows that there is only the extensive margin elasticity. Applying (S.13) to ΛX = ηηX
V (φD ) ,

ηD
ΛX =
ηX

θ
( σ−1 )− σ−1
τX fX
,
fD

while the closed-form solutions of ∆ and ΛM are given in (S.16). It can be easily shown that()
d ln ΛX(σ − 1)(σ − 1 + θ)=(ηD − ηX ),d ln τMσ−1−θ()
d ln ∆
θ(σ − 1)−=(µD − µM ),d ln τXσ−1−θ
()
d ln ΛM(σ − 1)2−=(µD − µM ),d ln τX
σ−1−θ−This gives us the expressions of the trade elasticities in (21).

S-8