Trade and Domestic Production Networks∗
Felix Tintelnota,c , Ayumu Ken Kikkawaa , Magne Mogstada,c , Emmanuel Dhynebab

University of Chicago

National Bank of Belgiumc
NBER
November 26, 2017

Abstract
We use administrative data from Belgium with information on domestic firm-to-firmsales and foreign trade transactions to study how international trade affects firm efficiency and real wages. The data allow us to construct the buyer-supplier network of
the Belgian economy. We document that most firms that do not directly import orexport still have large indirect exposure to foreign trade, and that a firm’s output isaffected by idiosyncratic shocks to its buyers and suppliers. These empirical findings
motivate and guide the development of a model with domestic production networks
and international trade. We obtain new sufficient statistics results for the effects oftrade in a model with fixed network structure, and we develop a tractable model ofendogenous domestic production networks. Comparing our results to those we obtain using existing approaches highlights the importance of data on and modeling ofdomestic production networks in studies of international trade.

∗

We are grateful to Costas Arkolakis, Emmanuel Farhi, Teresa Fort, Matthew Grant, Basile Grassi, Rob
Johnson, Harry Li, Alejandra Lopez Espino, Ferdinando Monte, and Andrei Nagornyi for useful conversations. The views expressed in this paper are those of the authors and do not necessarily reflect the views of
the National Bank of Belgium or any other institution with which one of the authors is affiliated.

1

Introduction

Over the past few decades, the focus of research on international trade has shifted fromcountries and industries towards firms. This shift is in no small part due to the increased
availability of firm-level transaction data on trade. One important insight from this datais that few firms directly import or export goods (Bernard, Jensen, Redding, and Schott,2007). However, the concentration of imports and exports does not necessarily imply thatfew firms benefit from foreign trade. Even if firms themselves do not import or export,
they may still buy from or sell to domestic firms that trade internationally. Capturing thischannel, however, is challenging since domestic firm-to-firm transactions are rarely observed.
In the absence of such data, quantification of the effects of foreign trade on all firms requires
strong assumptions, such as a common intermediate good (Eaton and Kortum, 2002, andBlaum, Lelarge, and Peters, 2016) or the same import shares across importing firms within
broad industries (Caliendo and Parro, 2015, and Costinot and Rodrı́guez-Clare, 2014).
The goal of this paper is to combine data on domestic firm-to-firm sales with information
on foreign trade transactions to study how international trade affects real wages and efficiencyof all firms, including those that do not directly export or import. Our analysis employsa panel dataset with detailed information on Belgian firms for the years 2002-2014. Thisdataset is based on several data sources that we have linked through identifiers. Annual
accounts provide data on input factors and output, custom records and intra-EU declarations
give information on exports and imports, and a value-added tax (VAT) registry provides
information on domestic firm-to-firm transactions. Using this data, we empirically examine
several new dimensions of firms in international trade before developing and estimating amodel of trade and domestic production networks.
In Section 2, we describe the data, construct the domestic production (buyer-supplier)
network of the Belgian economy, and provide two new empirical findings. The first is thatmost firms are exposed to foreign trade through their production network. While only 15%
of firms import directly, 97% of firms obtain foreign inputs either directly or indirectly
through domestic suppliers which use foreign inputs in their production process. Indeed,most firms are heavily dependent on foreign inputs, but only a small number of firms showthat dependence through the direct imports observed in firm-level transaction data on trade.
For example, in a majority of firms, at least 40% of input costs are spent on goods that areimported directly or indirectly.
The second empirical finding is that foreign trade shocks seem to propagate across firms
within production networks. Following Hummels, Jørgensen, Munch, and Xiang (2014), we
measure trade shocks as changes in world export supply and world import demand of countryproduct combinations in which the firm had a previous trade relationship. They argue thatthese shocks are plausibly exogenous and that their impact varies markedly across firms,
2

because the firms — even within the same sector — do not have all inputs in common.
Using our data, we find that positive export shocks to the firm’s buyers and positive import
shocks to the firm’s suppliers both tend to increase the firm’s output, even after controllingfor direct shocks to the firm itself and for shocks to the set of potential buyers and suppliersof the firm.
Taken together, these two empirical findings highlight that information about domestic
firm-to-firm transactions is key to understand the extent to which firms rely on foreign input
and to analyze the propagation of trade shocks. Motivated and guided by this evidence, wedevelop and estimate a model of domestic production networks and international trade. In
our model, firms combine imports, inputs produced by other domestic firms, and labor to
produce differentiated products with a constant elasticity of substitution production function.
Firms are finite and monopolistically competitive.
In Section 3, we assume a fixed network structure (i.e., the buyer-supplier relationships
do not change in response to trade shocks) and quantify how international trade affects
firms’ production costs and consumer prices. The cost reduction for an individual firm due
to international trade depends on two quantities only: the share of input costs that is spenton goods that are imported directly or indirectly and the elasticity of substitution in the
production function. We apply this sufficient statistics formula to our data, and find thatinternational trade is important in reducing the cost of production for plausible values of theelasticity of substitution. For example, with an elasticity of substitution in the production
function of 2, we calculate that shutting down international trade would increase the costof the majority of Belgian firms by at least 70%. To compute the welfare gains from trade,
we combine information on firms’ sales to domestic households with an assumption about
the elasticity of substitution in the utility function. Our baseline results imply that theconsumer price index in Belgium would be 77% higher in the absence of international trade.
While assuming a fixed network structure is convenient to take the model to the data,
it does not allow us to capture how buyer-supplier relationships may change in responseto trade shocks. In Section 4, we therefore develop a model of trade with endogenous
network formation. In particular, we let firms optimally choose their set of suppliers (i.e.
the firm’s sourcing strategy) subject to a buyer-supplier-specific fixed cost for adding a
supplier. Allowing for endogenous network formation is challenging for two reasons. First,
firms face a large discrete choice problem of which suppliers to include in their sourcing
strategy. Second, firms’ sourcing strategies are interdependent, creating a large fixed point
problem: firms take into account the expected sourcing strategies of others in order todetermine their own optimal sourcing strategy, all the while knowing that other firms arethinking in the same way.
Building on Jia (2008) and Antras, Fort, and Tintelnot (2017), we overcome the first3

challenge by using lattice theory to solve firms’ large combinatorial discrete choice problems.
To address the second challenge we consider the formation of an acyclic network, postulating
an ordering of firms and restricting the eligible set of suppliers to firms that appear prior
to the buyer.1 While restrictive, this assumption allows us also to solve a model of firm
trade with endogenous formation of domestic buyer-supplier relationships. We use methodof simulated moments to estimate the model, and then perform counterfactuals to draw
inference about the impact of shutting down international trade with and without endogenous
network formation. Our findings suggest that allowing for endogenous formation of buyersupplier relationships tend to attenuate the effects of banning trade on firms’ cost. Under
endogenous network formation, we find a price index increase around 15% lower than undera fixed network.
Our paper contributes to a growing literature on the economy-wide effects of foreignsourcing.2 Many studies use aggregate data only, relying on the assumption that firms
import intensities are equalized — which is at odds with the data. Using firm-level dataon trade transactions, Blaum et al. (2016) show that accounting for heterogeneity in import
exposure significantly affects the measurement of the gains from international trade. Their
model assumes that firms can import directly and purchase a common intermediate good.
Taking advantage of data on domestic firm-to-firm transactions, we relax the assumptionof a common intermediate good, and derive a parsimonious sufficient statistics formula fora model with a fixed production network. We also go beyond the fixed network structure,solving a model of endogenous network formation with a finite number of firms and fixed
costs for adding suppliers. This contribution builds on the global sourcing model of Antras
et al. (2017). While they distinguish between final good and intermediate good sectors,
we consider a more general input-output structure between firms. In addition, our model
captures not only the firms’ decisions with respect to foreign sourcing but also their choicesof domestic sourcing strategies.3
Our paper also relates to a literature on the formation and consequences of domestic
production networks.4 Bernard, Moxnes, and Saito (2016b) adapt the model of Antras et al.
(2017) to search for domestic suppliers in different locations, where each location has acontinuum of intermediate-good-producing firms. They find significant improvements in firm
1

See Spiegler (2016) for a recent contribution in economics studying belief formation in a directed acyclic
network.
2
See for example Antràs and Helpman (2004), Antràs and Helpman (2008), Rodrı́guez-Clare (2010),
Garetto (2013), Halpern, Koren, and Szeidl (2015), Gopinath and Neiman (2014), Amiti and Konings (2007),
Goldberg, Khandelwal, Pavcnik, and Topalova (2010), and De Loecker, Goldberg, Khandelwal, and Pavcnik(2016).
3
Our work is also related to the analysis in Caliendo and Parro (2015) and Ossa (2015). They find thegains from trade to be larger when taking sectoral input-output linkages into account.
4
A growing body of work studies how firms meet international trading partners. See for example Chaney(2014), Chaney (2016), Morales, Sheu, and Zahler (2015), and Eaton, Kinkins, Tybout, and Xu (2016).

4performance from a reduction in internal search costs in Japan. Furusawa, Inui, Ito, andTang (2017) develop a variant of the global sourcing model of Antras et al. (2017), and use
Japanese buyer-supplier link data to test the model’s predictions. Oberfield (2017) develops
a theory in which the network structure of production forms endogenously among firms thateach purchase a single input. Lim (2015) develops a dynamic model of network formationin which each firm has a continuum of domestic suppliers. With a continuum of suppliers
and buyers, the sales from one firm to another are negligibly small and a link between
two particular firms has no effect on aggregate outcomes. In contrast to these papers,
we develop a model of endogenous network formation with a finite set of suppliers, andincorporate both firm exporting and importing decisions. While our theory assumes simple
solutions for the pricing game between firms, Kikkawa, Magerman, and Dhyne (2017) explore
the segmentation of markets for different buyers, with supplier firms having heterogeneousbargaining power in the supplier-buyer relationships.
Finally, our paper relates to a literature that analyzes how production networks matterfor aggregate effects and transmission of idiosyncratic firm shocks. Gabaix (2011) provides
conditions under which granular shocks can affect aggregate fluctuations. Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2012) study the transmission of shocks along sectoral
input-output networks. Magerman, De Bruyne, Dhyne, and Van Hove (2016) test both channels with the Belgium domestic firm-to-firm data. Barrot and Sauvagnat (2016), Boehm,
Flaaen, and Pandalai-Nayar (2015), and Carvalho, Nirei, Saito, and Tahbaz-Salehi (2016)use natural disasters to study the propagation of shocks in production networks. Carvalho
and Voigtländer (2015) analyze the adoption of inputs by innovators and the evolution of
the domestic production network. Hulten (1978) provides conditions under which the underlying network structure is irrelevant for quantifying the propagation of shocks — up toa first-order approximation — as long as firms’ initial size and the magnitudes of the idiosyncratic shocks are observed.5 In recent work, Baqaee and Farhi (2017) illustrate that thesecond-order effects of shock propagation arising from networks can be large.6 Our paper extends the analysis of shock propagation to foreign trade shocks, while allowing buyer-supplier
relationships to change in response to these shocks.

5

As shown by Kikkawa et al. (2017), to apply the Hulten theorem to a small open economy setting one
would need very strong conditions that are immediately violated in our data: in particular, all firms would
need have a constant ratio of export sales to sales to domestic final consumers.
6
Other recent contributions to determining the effects of networks include Baqaee (2014), Carvalho andGrassi (2017), as well as in the context of financial frictions, Bigio and La’o (2016) and Liu (2016). Atalay,
Hortacsu, Roberts, and Syverson (2011) characterize the buyer-supplier network of the US economy.

5

2

Trade and production networks: Data and evidenceThis section describes the data, documents firms’ direct and indirect exposure to foreign
trade, and shows how the output of a firm is affected by trade shocks to its buyers andsuppliers.

2.1

Data sources and sample selection

Our analysis draws on three administrative data sources from Belgium, accessible only at the
National Bank of Belgium, for the years 2002-2014. These data sources can be linked throughunique identifiers, assigned and recorded by the government for the purpose of collecting
value-added taxes (VAT). Below we briefly describe our data and sample selection, while
additional details are given in Appendix C.
The first data source is the Business-to-Business (B2B) transactions database (see also
Dhyne, Magerman, and Rubinova (2015)). By law, all Belgian firms are required to filethe annual sales to each buyer (provided the annual sales to a given buyer exceeds e250).
Thus, the B2B dataset allows us to measure accurately the identity of the firms’ suppliers
and buyers. The second data source is the annual accounts filed by Belgian firms. These
data contain detailed information from the firms balance sheets on output (such as revenues)
and inputs (such as capital, labor, intermediates) as well as 4-digit (NACE) industry codesand geographical identifiers at the zip code level. In addition, the annual accounts include
information about ownership shares in other enterprises. The third set of data source is theBelgian customs records and the intra-EU trade declarations. These data contain information
about international trade transactions in each year and for every firm. Both imports andexports are disaggregated by product and origin or destination.
One challenge with using the Belgian data is that the information is recorded at the levelof the VAT-identifier. The problem is that a given firm may have several VAT-identifiers(for accounting or tax reasons).7 While organizational choices and transactions across units
within a firm are of interest, our paper is centered on trade between firms. Thus, if a firm
has multiple VAT-identifiers, we aggregate all data up to the firm level using informationfrom the balance sheets about ownership structure. Details of the aggregation are outlined inAppendix C.1. In 2012, for example, the aggregation converts 896,000 unique VAT-identifiers
into 860,000 unique firms. Of these firms, 842,000 had a single VAT-identifier. However, the
18,000 firms with multiple VAT-identifiers are important, accounting for around 60% of thetotal output in the dataset.
7

Existing papers tend to abstract from this issue, analyzing the data at the level of the VAT-identifier.
See e.g. Amiti, Itskhoki, and Konings (2014), Magerman et al. (2016) and Bernard, Blanchard, Van Beveren,
and Vandenbussche (2016a)).

6After constructing a firm-level dataset, we impose a few sample restrictions. We exclude
firms in the government or financial sector. In addition, we restrict the sample to firms
with positive labor costs and employment, tangible assets of more than e100, and positive
total assets in at least one year during our sample period. These criteria are similar tothe ones used by De Loecker, Fuss, and Van Biesebroeck (2014). Applying these criteria
reduces the number of firms significantly. In, 2012, for example, only 139,605 firms satisfy
the above criteria. The large reduction in sample size is mostly driven by the exclusion of
local firms without employees (self-employment) from the sample (687,700 firms in 2012).
Lastly, we drop foreign firms with no local production activity in Belgium from the sample.
These account for a sizable fraction of imports and exports, but have no domestic production
activity in Belgium.
Table 1 illustrates that our selected estimation sample of firms provides relatively goodcoverage of aggregate value added, gross output, exports and imports. However, total sales
in our sample is larger than what are reported in the national statistics. The reason is thatthe output of trade intermediaries in the national statistics is measured by their value added
instead of their total sales. We refer to Appendix C.2 for the same statistics for all Belgian
firms.
Table 1: Coverage of selected sample
GDP
Output(Excl. Gov. & Fin.)
2002 149
411
2007 192
546
2012 212
626
Year

Imports

Exports

210
300
342

229
314
347

Selected sample
Count V.A. Sales Imports
122,460 123
586
179
136,370 157
757
280
139,605 170
829
296

Exports
189
269
295

Notes: All numbers except for Count are denominated in billion Euro in current prices. Belgian GDP and
output are for all sectors excluding public and financial sector. See Appendix C.2 for the same statistics forthe total economy. Data for Belgian GDP, output, imports and exports are from Eurostat.

2.2

Direct and indirect exposure to foreign trade

The Belgian data allow us to construct the buyer-supplier relationships of the Belgian economy, and therefore document firms’ direct and indirect exposure to foreign trade.
We define firm j’s total foreign input share as the sum of firm j’s direct foreign input
share, sF j , and the direct foreign input share of firm j’s suppliers, suppliers’ suppliers, andso forth, each weighted by the firm-pair-specific input shares (sij , ski , ...):

7= sF j +sTF otaljXXsij sF i +i∈ZjD

ski (sF k + · · · ),

(1)

k∈ZiD

|

{z

otal
sT
Fi

}

where ZjD denotes the set of domestic suppliers of firm j, and the denominator of the input
shares is the sum of labor costs, purchases from other firms, and imports. Note that thedefinition of the total foreign input share is recursive: a firm’s total foreign input share issum of its direct foreign input share and the share of its inputs from other firms multiplied bythose firms’ total foreign input shares. While many firm-level datasets contain information
about the direct foreign input share sF j , our data also offer information about firm-pairspecific input shares, sij . As a result, we are able to calculate the total foreign input sharefor every firm. We note that there is one inherent assumption in our definition of the total
foreign input share: When a firm sells its output to multiple firms or final consumers, theforeign input share in the costs of producing these goods is assumed to be the same for all
buyers (i.e., independent of the identity of the buyer). This assumption is consistent withthe model we develop in Sections 3 and 4, where each firm produces a single product.
In Figure 1a, we display a histogram of the total and direct foreign input shares of theBelgian firms. While only 15% of firms import directly, 97% of firms obtain foreign inputs
either directly or indirectly through domestic suppliers which use foreign inputs in theirproduction process. Indeed, most firms are heavily dependent on foreign inputs, but only asmall number of firms show that dependence through the direct foreign input shares observedin firm-level transaction data on trade. In the median firm, for example, the total foreign
input share is 41%. By comparison, the total foreign input shares are 21% and 60% at at
the 20th and 80th percentile. We present direct and total foreign input shares by sector in
Appendix D.1. Even in the service sector, in which firms have a very low share of direct
foreign inputs, the median firm’s total foreign input share is as large as 28%.
Figure 1b performs a similar exercise, but now looks at total export shares and direct
T otalexport shares. We calculate the total export share of firm j, rjF, as the sum of the share ofrevenue of firm j coming from directly export goods, rjF , and the share of revenue comingfrom goods sold to other domestic firms, multiplied by those firms’ total export shares:
T otalrjF= rjF +X

T otalrji riF
,

(2)

i∈Wj

where Wj denotes the set of domestic buyers of firm j, and the denominator of the export
shares is the total revenue of the firm. Direct export is even more concentrated than direct
import, both on the intensive and extensive margin. While only 10% of firms export directly,

8

Figure 1: Histogram of direct and indirect linkages to foreign trade(a) Direct and total foreign input share
100000

Count

4000

3000

2000

1000

0
0

.2

.4
.6
Foreign input share
Direct

.8

1

.8

1

Total

(b) Direct and total export share

129500
61300
9300
6000

Count

5000
4000
3000
2000
1000
0
0

.2

.4
.6
Export share
DirectTotal

P
Notes: Total foreign input share of firm i, sTF otal
is calculated by solving sTF otal
= sF i + j∈Zi sji sTF otal
where
i
i
j
T otalsF i is i’s direct foreign input share, andamong i’s inputs. Total export share firm i, riF
Psji is j’s Tshare
T otalotal
is calculated by solving riF= riF + j∈Wi rij rjF
where riF is i’s share of exports in its revenue, andrij is share of i’s revenue that arises from sales to firm j. The figures are based on the analysis of 139,605
private sector firms in Belgium in 2012. The horizontal lines represent scale breaks on the vertical axis.

9Figure 2: Size premium of direct and indirect linkages to foreign trade-10

Log sales123(a) Foreign inputs

0

.2

.4
.6
Foreign input share
Direct

.8

1

.8

1

Total

0

.5

Log sales
1
1.5

2

2.5

(b) Exports

0

.2

.4

.6
Export share

DirectTotal

Notes: The two figures display the smoothed values with 95% confidence intervals of kernel-weighted local
polynomial regression estimates of the relationship between firms’ sales and their levels of participation inforeign trade. We use the Epanechnikov kernel function with kernel bandwidth of 0.01, pilot bandwidth of0.02, degree of polynomial smooth at 0, and smooth obtained at 50 points. Log sales are demeaned with
4-digit industry fixed effects.

10

82% of firms export either directly or indirectly by selling to domestic buyers which subsequently trade internationally. In terms of trade volume, however, export remains relatively
concentrated even after taking the indirect export into account. The total export share is
only 2% in the median firm, whereas it is 19% at the 80th percentile. In contrast, Figure
1a showed that most firms are heavily dependent on foreign inputs. This difference is partly
driven by the service sector. While many firms in this sector (e.g., restaurants) rely on foreign inputs — often obtained indirectly through domestic suppliers — relatively few export
directly or sell to domestic firms that are exporting directly or indirectly (see Appendix D.2
for direct and total foreign input shares by sector).8
Across a wide range of countries and industries, firms that directly export or importhave been shown to be larger than other firms. A natural question is whether the positive
association between firm size and international trade also carries over to indirect export orimport. Figure 2 investigates this, calculating the average size of firms by direct and total
foreign input shares as well as by direct and total export shares. We demean the log of firmsales using the firm’s four-digit industry average, so that a firm with log sales of zero is thesize of an average firm in its industry. Figure 2 illustrates that average firm size is increasing
in both direct and total foreign input shares. However, firms that import directly tend to bemuch larger than firms that buy foreign inputs through domestic firms. Indeed, firms withless than 60% in total foreign input shares are, on average, of similar size as the average firmin their industry. A similar pattern is evident for size and export. Firms with very high totalexport shares tend to be large. However, over most of the total export share distribution,
there is only a weak relationship between firm size and total export share. Taken together,
the results in Figure 2 suggest that firms do not have to be large to rely heavily on foreign
inputs or to have most of their sales going ultimately to a foreign country.

2.3

Trade shocks and the production network

The analysis in Section 2.2 showed that most firms are exposed to foreign trade through theirproduction network. This finding raises the questions of whether, and to what extent, trade
shocks propagate across firms within production networks. To investigate these questions,
we build on the work by Hummels et al. (2014), who construct measures of trade shocks fromchanges in world export supply and world import demand of country-product combinations
in which the firm had a previous trade relationship. They argue these shocks are plausibly
exogenous, and show that their impact varies markedly across firms, because the firms do
not have all inputs in common. Hummels et al. (2014) use the trade shocks to estimate wage
8

Another possible explanation is that it is difficult to measure all forms of export in the service sector.
For example, when a foreigner eats at a Belgian restaurant, it is technically an export transaction. However,such transactions are not recorded in our data.

11effects of offshoring and exporting in Denmark. We apply the same identification strategy
to the Belgian setting with the goal of examining whether trade shocks to the firm’s actual
buyers or suppliers have stronger impact on its output than trade shock to potential suppliersand customers.
To make the identification strategy precise, consider the following regression model infirst-differences:C
S∆ log Yit =βX∆ log XitC + βM
∆ log MitS
PS
PC
∆ log MitP S∆ log XitP C + βM
+ βX

+ βX ∆ log Xit + βM ∆ log Mit + ϕt + it .

(3)

where Yit denotes the total sales of firm i in year t, and ∆ denotes the change in thevariable from year t − 1 to t. In addition to calender time fixed effects ϕt , we include
three sets of explanatory variables. The first is the measures of import shocks to firm i’s
suppliers ∆ log MitS and export shocks to firm i’s’ buyers ∆ log XitC . Our goal is to consistentlyCS
estimate the coefficients on these variables, βX
and βM. However, there are several threats to
identification. One is that ∆ log MitS and ∆ log XitC are likely to correlate with trade shocks to
i’s potential suppliers and buyers. We therefore include measures of import shocks to firm i’s
potential suppliers, ∆ log MitP S , and export shocks to firm i’s’ potential buyers, ∆ log XitP C .
Another concern is that ∆ log MitS and ∆ log XitC could be correlated with trade shocks that
affect the firm i directly (through it’s direct import demand or supply). To address thisconcern, we control for export shocks, ∆ log Xit , and import shocks, ∆ log Mit , to firm iitself.
To take the regression model to the data, we need to construct the various measures oftrade shocks. To this end, we follow the shift-share approach in Hummels et al. (2014). To
construct an export shock for firm i, ∆ log Xit , we use information about the firm’s productcountry-level exports in year t − 1 (the share variable capturing firm-specific exposure), andthe aggregate shift in world import demand for each country and product:∆ log Xit = log

X k,X
X k,X
ric,t−1 WIDk,c,t − log
ric,t−1 WIDk,c,t−1 .
k,c

|

k,c

{z

}

Xit,t−1|{z

Xit−1,t−1}

k,XThe term ric,t−1
is the share of exports of firm i at year t − 1 that falls on product k soldto country c, and WIDk,c,t is the world import demand (excluding imports from Belgium)of country c for product k.9 We measure the export shock for firms’ buyers in a similar
way. For firm i’s buyers, we construct the weighted average of their export demand shocks,
9

We use NACE 4 digit level to classify products k.

12∆ log XitC , using i’s output share to each buyer in the previous year as the weights:∆ log XitC = log

X

rij,t−1 Xjt,t−1 − logXjrij,t−1 Xjt−1,t−1 .

jFinally, we measure the trade shocks to the firms’ potential buyers, ∆ log XitP C . The potential
buyers of firm i include both the buyers of i’s goods and other firms in the same (4-digit)
sector as the actual buyers. We weight sectors for each firm according to the share of thefirm’s revenue that is sold to firms’ from that sector, riu,t−1 .10 We then construct an export
shock for each sector as a weighted aggregate of export shocks to all firms of that sector.11
We combine these terms to construct an export shock to the potential buyers of firm i:∆ log XitP C = log

X−iriu,t−1 Xut,t−1
− log

X

−i
riu,t−1 Xut−1,t−1.

uu

To construct the import shock variables to the firm itself, its suppliers, and its potential suppliers, we use a similar procedure. These variables use information about changes
in aggregate export supply in foreign countries and the past sourcing of firms from thesecountries. We describe their construction in Appendix E.1.
Table 2 shows the estimation results from the regression model with changes in the firm’s
total sales as the dependent variable (see Table 14 in Appendix E.2 for results with changes indomestic sales and domestic inputs as dependent variables). The results in column 1 suggestthat firms that experience positive trade shocks tend to increase their sales. The estimatesin column 2 suggest that trade shocks to potential buyers and suppliers also affects thefirm’s sales. However, as shown in column 3, shocks to the firms’ actual buyers and suppliers
matter more for the firm’s sales than shocks to the potential buyers and suppliers.12 In terms
P
Salesijt, where Witu denotes
Let sectors, at the NACE 4-digit level, be denoted with u. riu,t = j∈W u TotalSalesititthe set of customers of i producing sector u goods at time t. We fix all weights at the previous year t − 1.
11
For the weights that firm i assigns to each firm within a sector, we use the firms’ sales to domestic
final demand as corresponding weights. We have experimented with different weights and obtained similar
results. These weights vary at the firm i - sector u level, as we exclude firm i’s own exports and imports if
firm i is producing sector u good.
10−iXut,t−1=X
j∈Ut−1 ,j6=i−iXut−1,t−1=

VjHt−1
Xjt,t−1k∈Ut−1 ,k6=i VkHt−1

PX

VjHt−1
Xjt−1,t−1 ,k∈Ut−1 ,k6=i VkHt−1

Pj∈Ut−1 ,j6=iwhere Ut is the set of firms producing sector u good at t, and ViHt is firm i’s sales to domestic final demandat t.
12Note that the actual buyers and suppliers are included in the set of potential buyers and suppliers.
CS
Thus, the coefficients βXand βMshould be interpreted as the additional effect of a trade shocks to the13Table 2: Reduced form results∆ ln Xit∆ ln Mit∆ ln Total Sales
0.106∗∗∗(0.008)0.183∗∗∗(0.010)∆ ln Total Sales
0.103∗∗∗(0.008)0.178∗∗∗(0.010)0.027∗∗∗(0.002)0.040∗∗∗(0.004)

87100

87100

PC
∆ ln Xit∆ ln MitP S
C∆ ln Xit∆ ln MitSN∆ ln Total Sales
0.089∗∗∗(0.009)0.156∗∗∗(0.012)0.025∗∗∗(0.003)0.039∗∗∗(0.005)0.122∗∗∗(0.013)0.041∗∗∗(0.018)87100

Notes: Standard errors are clustered at the firm level. All variables are in terms of yearly log
differences for the period 2002-2012. All specifications include year fixed effects. We truncate
outliers of each variables at the top and bottom 1% level.
* p < 0.10, ** p < 0.05, *** p < 0.01

of magnitudes, the estimates suggest that an 10 percent exogenous increase in the foreign
demand of goods for firm i’s buyers leads to a 1.2 percent increase in the sales of firm i. The
pass through of shocks to firm i’s suppliers is smaller. A 10 percent exogenous increase in
the foreign supply of goods to firm i’s suppliers leads to a .4 percent increase in the sales offirm i.
Taken together, the results in Table 2 suggest that sectoral input-output tables are notsufficient to analyze the propagation of trade shocks. The output of a firm is significantly
affected by idiosyncratic shocks to its buyers and suppliers. This finding is consistent with
Carvalho et al. (2016), who show that the disruption caused by a Japanese earthquake in 2011
propagated through upstream and downstream supply chains. Motivated by this evidence,
we proceed by developing a model of international trade and domestic production networks.

firm’s actual buyers and suppliers as compared to the effect of trade shocks to the firm’s potential buyers
and suppliers14

3

A model of trade with fixed production networks

We now develop a model of trade and domestic production networks, and use it to quantify
how international trade affects firms’ production costs and consumer prices. While thissection assumes a fixed network structure — which is convenient to take the model to thedata — we allow, in Section 4, the buyer-supplier relationships to change in response totrade shocks.

3.1

Model

We describe a small open economy called Belgium. Before describing the model, we briefly
discuss the notation. Since there exist many bilateral directed flows in our model, we willoften have two subscripts. In such cases, the first subscript denotes the origin of the good
and the second subscript denotes the destination of the good.
3.1.1

Preferences and Demand

Each consumer supplies one unit of labor inelastically. Consumers are assumed to haveidentical, homothetic CES preferences over consumption goods:
X

U=(βkH qkH )

σ−1
σ

σ
! σ−1

,

(4)

k∈Ω

where Ω denotes the set of available products in the small open economy, k denotes a product,and H denotes domestic final demand from households. Since all consumers have the same,
homothethic CES preferences for consumption, we can write the aggregate final consumer
demand (in quantities) for product k, given price pkH , as:
p−σ
kH
E,
(5)
P 1−σ
where E denotes the aggregate expenditure in Belgium and P denotes the domestic consumer
price index:
σ−1
qkH = βkH

1
! 1−σ

P =

X

σ−1 1−σ
βjH
pjH

.

(6)

j∈Ω

We assume that final goods are substitutes and therefore σ > 1.
Demand from abroad for product k takes a similar functional form:
σ−1
qkF = βkF

15

p−σ
kF
EF ,
PF1−σ

(7)

where βkF is a product-k-specific foreign demand shifter, pkF is the price of product k abroad,and PF and EF denote the foreign price index and expenditure, respectively .
3.1.2

Production and market structureFirms produce single products. We will use i, j, k to index firms or products. The products
are differentiated across firms. Firms sell the same product to final consumers and to otherfirms as an intermediate input, though not all firms sell to other firms, and not each pairof firms has a buyer-supplier relationship. Note that we allow Belgian firms to sell directly
to foreign consumers, while all foreign products reach Belgian consumers indirectly throughthe importing of inputs by Belgian firms.13
We treat every firm as infinitesimal when selling to final consumers. Hence, when sellingto domestic or foreign final consumers, we assume the market structure is monopolisticcompetition. When selling to other firms, the assumption of infinitesimal size is no longer
reasonable, however, since most firms just have a few selected suppliers. We assume that in
the Nash bargaining between buyer and supplier, the buyer has the full bargaining power.
Given the assumptions on technology described below, this will imply that the supplier sells
at marginal cost to the buyer firm. We note that in this section with exogenous networks,
our main propositions and quantitative results are unchanged if firms charge positive andpossibly heterogeneous mark-ups to customer firms as long as these are fixed. Kikkawa et al.
(2017) analyze a network economy with variable firm-to-firm mark-ups. In our paper, thearguably strong assumption of the bargaining power in firm-to-firm transactions being on
the buyer’s side will be critical for modeling the network formation game in a tractablemanner.14
This section assumes a fixed network structure: we take as given the set of firms, Zj ,from which each firm j is eligible to purchase inputs. For importing firms, Zj contains also
foreign, F , as an eligible supplier. Sometimes we will refer to the set of domestic suppliersof firm j, which we denote by ZjD .
Firms use a CES input bundle of workers and domestic and foreign inputs with elasticity
of substitution ρ > 1 in the production function. We assume that σ > ρ, implying that
consumers are more price-elastic than firms in their purchase of goods. Given the CES
production function, we can write the cost function of firm j as:
13

The assumption that foreign goods reach Belgian consumers only through Belgian firms is reasonablebecause in the data nearly all imports are carried out by firms. We make the assumption that Belgian firms
can reach foreign consumers directly to avoid modeling foreign firms in detail.
14
Also in the industrial organization literature, corner solutions in the bargaining game are sometimes
assumed to obtain tractable solutions for network formation problems. For example, when studying thedeterminants of the hospital networks offered by health plans, Ho (2009) assumes that hospitals make takeit-or-leave-it offers to all health plans in the market.

161/(1−ρ)
cj (Zj ) =1 
ρ−1 1−ρ 
ρ−1 1−ρw`
pkj + α`j
αkj
φj k∈Z
X

.

(8)

j

The first term in the cost function, φj , denotes the exogenous total factor productivity ofP
ρ−1 1−ρρ−1 1−ρw` thepkj + α`jfirm j. Following Antras et al. (2017), we will call Θj (Zj ) = k∈Zj αkj
sourcing capability of firm j, and Zj the sourcing strategy of firm j. The sourcing strategy
may include both domestic and foreign sourcing. The price of labor is denoted by w` . The
share of variable costs by firm j that is spent on intermediate inputs produced by firm k ∈ Zjis:ρ−1 1−ρpkj
αkj
pkj qkjskj ==,cj q jΘj (Zj )(9)where αkj reflects how salient the good produced by firm k is as an input for firm j. Analogously, the share of variable costs by firm j that is spent on labor is:ρ−1 1−ρ
α`j
w`
w` `j
,
s`j =
=
cj q jΘj (Zj )(10)

while the direct foreign input share of firm j (assuming F ∈ Zj ) is:ρ−1 1−ρ

α F j pF j
p F j qF j
sF j ==.
cj q jΘj (Zj )(11)

Before deriving an expression for the total sales of a firm, we discuss the pricing problemof the firm. Due to CES preferences and monopolistic competition, firms charge a constantσmark-up over marginal costs, µ = σ−1, when selling to final consumers at home or abroad.
When selling to other firms, firms engage in Nash bargaining with the full bargaining poweron the side of the buying firm. The buyer will make the supplier just indifferent between
selling to the firm or not, and therefore firms sell at marginal costs to other firms.
In order to sell abroad, firms incur iceberg transport costs, τ . In this section, we takeexport participation, IjF , as given (IjF = 1 for all exporting firms and IjF = 0 otherwise)and endogenize it in Section 4. Firms’ total sales consist of the sum of domestic sales to final
consumers, foreign sales to final consumers, and domestic sales to other firms. Let firm j’s
total sales be denoted by:

17

E
EF
σ−1 1−σ σ−1
σ−1 1−σ σ−1µ φj Θj (Zj )(σ−1)/(ρ−1) τ 1−σ 1−σxj = βjHµ φj Θj (Zj )(σ−1)/(ρ−1) 1−σ + IjF βjF
P }
PF
|
{z
|
{z
}
Domestic sales to final consumers

+

X
|k

ρ−1
I(j ∈ Zk )µ1−ρ φρ−1
αjk
Θ(j)
j

{z

Domestic sales to firms

Exportsxk /µk,Θk (Zk )}(12)

where µk denotes the average mark-up of firm k. Recall that the firm charges a constantmark-up to final consumers and a zero mark-up to other firms. Hence, µk depends on the
distribution of firm k’s sales.
Given that firms make their profits only on sales to final consumers, we can write thevariable profits of firm j given a sourcing strategy, Zj , and export participation, IjF , as1 σ−1 1−σ σ−1Eπjvar (Zj , IjF ) = βjHµ φj Θj (Zj )(σ−1)/(ρ−1) 1−σσP
1 σ−1 1−σ σ−1
EF+ IjF βjF
µ φj Θj (Zj )(σ−1)/(ρ−1) τ 1−σ 1−σ .
σ
PF
3.1.3

(13)

Firms’ dependence on foreign inputs

We now calculate the exposure of firms to foreign inputs, taking into account that the direct
and total foreign input share can be substantially different. Let sF j denote the total foreign
input share of firm j:
sTF otal= sF j +jX

sij sTF otal
i .

(14)

i∈ZjD

The definition of total foreign input share is intuitive in a model with single product firms inwhich each firm uses the same fraction of foreign inputs in the production sold to every buyer.
Proposition 1 shows the link between the total foreign input shares and the cost reduction
from international trade in our model.15 . In the proposition, we also present results basedon two alternative modeling assumptions that researchers often make when they do not haveaccess to domestic firm-to-firm transaction information.
Proposition 1 (Cost increases from banning foreign inputs) Assume ρ > 1.
Given fixed linkages between firms, and leaving domestic nominal wages, w` , unchanged,
the total cost increase from banning foreign inputs is:
15

Note that the assumption made in the proposition that nominal wages are unchanged is not that
restrictive, since nominal wages can be normalized to any value under autarky

18

F · →∞
ĉj |ptotal= 1 − sTF otalj

1/(1−ρ).

(15)

Ignoring linkages and indirect effects (i.e., assuming there is no pass-through of costchanges from domestic suppliers) and leaving domestic nominal wages, w` , unchanged, thedirect cost increase from banning foreign inputs is:
pF · →∞
ĉj |direct= (1 − sF j )1/(1−ρ) .

(16)

Finally, if one assumes an economy with roundabout production in which firms’ outputs
are aggregated to a composite intermediate input according to equation (4) and the composite
intermediate input is the only firm-to-firm input in equation (8), the cost increase fromF · →∞
banning foreign inputs, ĉj |proundabout, is implicitly defined as:
1−ρ
! 1−σ
F · →∞(ĉj |proundabout)1−ρ = s`j + sDj

X

pF · →∞skD (ĉk |roundabout
)1−σ

,

(17)

k

where sDj is the share of firm j’s domestic intermediate good purchases and skD is the shareof firm k in the intermediate good bundle (measured by firm k’s share in total domestic sales).
F · →∞The result on the firm-level cost increase from banning imports, ĉj |ptotal, reflects thata firm’s cost will not only rise according to it’s own direct foreign input share, but also
according to its suppliers’ foreign input shares, suppliers’ suppliers’ foreign input share, andso forth. This firm-level exposure is summarized in the total foreign input share expression.
The elasticity of substitution in the production function, ρ, indicates how easy it is to switch
to alternative inputs, including labor. Given the observed total foreign input share, a lower
value of ρ leads to larger cost increases from banning foreign trade.
F · →∞
In contrast, the expression ĉj |pdirect, yields only the direct effect of on firm-level costfrom banning foreign trade (mirroring the results in earlier work by Arkolakis, Costinot, andRodrı́guez-Clare (2012) and Blaum et al. (2016)). It can be rationalized from our model
under a network in which importers sell all their output to final consumers, and hence thetotal foreign input share of domestic suppliers of other firms in equation (14) is zero. Since
the observed production networks differ from this extreme case, using the formula fromequation (16) leads to cost increases from banning foreign inputs that are too low comparedto the full effect summarized in equation (15).
In the absence of firm-to-firm transaction data, it is possible to approximate the indirect effect by assuming a roundabout production structure (as for example in Blaum et al.
(2016)).16 Under the roundabout production assumption, every firm with the same inter-

16

Blaum et al. (2016) obtain a total cost reduction result from foreign inputs in which each firm buys

19

mediate input share will have the same indirect exposure to foreign goods. However, thisassumption is at odds with our data, and could create bias in the calculation of the firm-level
cost effects from trade.
3.1.4

Aggregation and equilibrium

We now describe the aggregation of our model, discuss how firm profits are redistributed
to consumers and define the equilibrium. In the model with a fixed production network weabstract from fixed costs of linkage formation, and hence πj = πjvar .
We assume that the set of Belgian firms is fixed and that firm profits are distributed toworkers in Belgium. We consider Belgium as a small-open economy and assume that there
are no foreign asset holdings and that trade is balanced. Hence aggregate expenditure in
Belgium is given by
E = w` L +

X

πk .

(18)

k

Balanced trade implies that aggregate exports are equal to aggregate imports:X

σ−1 1−σ σ−1
IjF βjFµ φj Θj (Zj )(σ−1)/(ρ−1) τ 1−σ

j

X 1
EF
s F j xj .
1−σ =
µ
PF
j
j

(19)

Labor market clearing implies that labor income is equal to firms’ labor costs:w` L =X 1jµjs`j xj .

(20)

We next define the equilibrium for the small open economy.
Definition 1 (Equilibrium given a fixed network structure) Given foreign expenditure, EF , foreign price index, PF , and a set of prices by foreign suppliers, {pF j }j , an equilibrium for the model with a fixed network structure and fixed export participation is a wage
level, w` , price index for the consumer, P , and aggregate expenditure, E, such that equations(6), (8), (9), (10), (12), (13), (18), (19), and (20) hold.
We establish uniqueness of the equilibrium in the closed economy in the following lemma.
Lemma 1 (Uniqueness of equilibrium under closed economy) Define a K × K maρ−1trix A where where the (i, j) element is φρ−1αij
if i ∈ Zj and 0 otherwise, and K denotesjthe same CES bundle of intermediate inputs. The production function in their paper is Cobb-Douglas inintermediate inputs and own labor input. Our roundabout cost reduction result in equation (17) is similar
to their result, but derived for a production function which is CES between labor input and intermediates.

20

0
the number of Belgium firms. Assume the matrix I − A is invertible, where I is the identity matrix. Then under a closed economy, for a given nominal wage, there exists a uniqueequilibrium defined in Definition 1.

Lemma (1) is useful because it implies that the counterfactual equilibrium without trade isunique. We next proceed to discuss the change in the aggregate price index arising frombanning international trade.
Proposition 2 (Change in aggregate price index from banning international trade)
Let siH denote firm i’s share in household demand in in the initial equilibrium prior to raising the barriers to trade.
Given a fixed set of firms, network structure and nominal wage, the price index change frombanning international trade can be summarized as follows:1! 1−σ
F · →∞
P̂ |ptotal=X

F · →∞ 1−σsiH (ĉi |ptotal
)

.

(21)

i

If the price of intermediate goods is assumed to be unchanged, the price index change canbe expressed as
1
! 1−σ
X
F · →∞
F · →∞ 1−σP̂ |pdirect.
(22)=
siH (ĉi |pdirect)iFinally, if one assumes roundabout production as defined in proposition 1, then the expression becomes
1
! 1−σ
X
pF · →∞
F · →∞
P̂ |proundabout=.
(23)
siH (ĉi |roundabout)1−σi

The change in the price index from banning international trade is a weighted aggregate ofeach firm’s cost increase with the weight equal to the firm’s share in domestic household
demand, siH , in the initial economy with international trade. Propositions 1 and 2 imply
that the change in the aggregate price index depends on the underlying network structure.
To see this, consider a production network in which all the imports are made by firms that
had no sales to other domestic firms or to the domestic final consumers. In that case, all
the cost increases from banning foreign goods would accrue to firms for which the share indomestic household demand is zero, and therefore the price index effect would be zero as
well. Suppose instead that importers of foreign goods had no sales to other domestic firms,
but sold all their output to domestic final consumers. Then, the price index increase is given
by equations (16) and (22). However, with positive sales to other domestic firms by theimporters, the price index effect is given by equations (15) and (21).
21A corollary of the results from propositions 1 and 2 is that two economies with the sameelasticities of substitutions in production and in the utility functions, and the same levels of
aggregate imports and exports, GDP, and gross production, can have different gains fromtrade. We illustrate this in a simple numerical example in Appendix B.17
The above results illustrates that knowing the underlying micro-structure of the economy
is relevant for the quantitative analysis of the gains from trade. In the following subsection,
we make use of the detailed information about domestic firm-to-firm transactions in our datawhen we calculate the welfare gains from trade for the Belgian economy.

3.2

Empirical results

In this section, we provide a quantitative analysis of how international trade affects firms’
production costs and the consumer price index. As shown above, this analysis requires
information on the observed firm-to-firm transactions, firm-level output, international tradeflows, and labor input, in combination with estimates or assumptions for the elasticity of
substitution in the production function, ρ, and the utility function, σ. Throughout the paper,
the baseline specification assumes σ is equal to 4 and ρ is equal to 2.18 We perform sensitivity
analysis to examine how the results vary with the choice of these parameter values.
To assess the implications of banning foreign inputs, we compute the firm level cost
increases by making use of proposition 1. We use the firm-to-firm network structure as
observed in 2012. Proposition 1 tells us that the total shares of foreign inputs for each firm,sTF otalj , translates to the cost changes that firms face from banning foreign inputs. Figure 3
displays the cost increase of firms from banning international trade (in the red line). This
figure also reports the cost increase under the assumptions of the direct effect (blue line) androundabout economy (green line). In these two cases, it is not necessary to observe domestic
firm-to-firm transactions to calculate the cost increases.
As evident from Figure 3, international trade matters much more for firms’ production
costs if we use our model with fixed domestic production networks than in the direct effectand roundabout economy. For the median firm (in the distribution of cost changes), thecost increase from banning foreign inputs is 70% in our model with domestic production
networks, 41% in the model with the roundabout economy assumption, and zero when
considering only the direct effect. As expected, the cost increases are the lowest under the17This example is related to the discussion in Melitz and Redding (2014) that the gains from trade canbe arbitrarily large in a model with sequential production, as the number of stages of production increases.
However, note that the ratio of gross production to GDP also rises in their example as the number of stages
gets larger. Here we hold the level of gross production and GDP fixed, and illustrate that the gains fromtrade can still differ.
18
Using data for the US, Antras et al. (2017) estimate σ = 3.85 and Oberfield and Raval (2014) estimate
a level of σ between 3 and 5 among various manufacturing industries.

22direct effect assumption. In the roundabout production economy, the non-importing firm’s
cost increase is bounded above by the price increase of the composite intermediate good. By
comparison, when taking the actual production network into consideration, many firms havea cost increase above 65%, while the roundabout model suggest that very few importing
firms have cost increases of that magnitude.

02

Density

4

6

Figure 3: Distributions of ĉ from banning imports

2

4

6

8

Cost change (log scale)
DirectRoundabout

Total

Notes: The parameters used are ρ = 2, σ = 4. See the appendix for derivations and plots for different
parameter values.

In Appendix I, we plot the same distributions for different parameter values of σ and ρ.
Even for alternative parameters, the distribution of cost changes under the actual production
network has a much thicker right tail than the distribution under the roundabout production
assumption. While the distributions become closer to each other if σ = ρ = 2, even in thatcase, the 90th percentile of firm-level cost increases under roundabout is 152%, whereas it is245% when the actual firm-to-firm transactions are used in the calculation (see Table 17 in
the Appendix).19
In Figure 4, we make use of proposition 2 to analyze the implications for the aggregate
consumer price index from banning foreign inputs. This figure reports the changes in aggregate price indices, P̂ , for the three different cases.20 The increase in the price index is muchlarger when one takes into account the domestic production network than when one only19

To read Table 17, recall that in the roundabout economy, σ is the parameter used in the aggregation
F · →∞
to the intermediate input bundle. Note that altering σ does not affect the calculation of ĉj |ptotal, but itpF · →∞
affects the calculation of ĉj |roundabout .
20
Note that siH is not observable in standard datasets. However, firm i’s share of total domestic sales,
siD , is usually observed. As shows in Appendix section A.1, siD and siH are identical in the roundaboutcase. We use siD when calculating the price index changes for the direct effect economy and the roundabout
economy.

23

1

1.2

P hat
1.4

1.6

1.8

Figure 4: Change in aggregate price index, P̂ , from banning imports

DirectRoundabout

Total

Notes: The parameters used are ρ = 2, σ = 4. See the appendix for derivations and plots for different
parameter values.

takes into account firms’ direct exposure to foreign inputs. While the price index increases
by 23% if only the direct effect of banning imports is considered, it rises by as much as 77%
if the full network structure of domestic production is taken into account. Compared tothe model of roundabout production, we also find a larger change in the price index whenusing the information contained in the firm-to-firm transaction data. Under roundabout
production, the domestic price index increases by 67%.
In Appendix I, we report the analogous numbers for P̂ under different parameters of σand ρ. As one would expect, the gains from trade get larger as σ and ρ get lower. We robustly
find that the price index increase from banning international trade is larger in our model thanunder the roundabout economy assumption, as long as σ is sufficiently large relative to ρ. On
the one hand, the assumption in the roundabout economy that each firm obtains the same
exposure to foreign goods through its domestic intermediate input purchases leads to larger
gains from trade in the roundabout economy, as it is harder for consumers to substituteaway from firms that face higher costs. On the other hand, the common assumption inroundabout production economy models (Eaton and Kortum, 2002 and Blaum et al., 2016)is that aggregation of the intermediate input bundle occurs analogous to the aggregation
for final consumption goods (equation (4)), using σ as the elasticity of substitution in that
aggregation. That makes it easier in the intermediate input bundle to substitute from firms
exposed to large imports to non-importing firms, leading to lower cost increases and lower
gains from trade. The second effect dominates when σ is sufficiently large as compared toρ, leading to larger gains from trade when using the actual firm-to-firm production network
data.
Our analysis so far has taken the network structure of firms as fixed. This has allowed
us to derive analytic solutions for the firm-level cost and aggregate price index changes in
the absence of international trade that could then be calculated easily with the firm-to-firm
24transaction and international trade data available to us. In the following section, we analyze
how the network forms endogenously and the implications of endogenous networks for thequantification of the gains from trade.

4

A model of trade with endogenous production networks

This section develops a model of trade with endogenous network formation, allowing buyersupplier relationships to change in response to trade shocks. The model builds on thetheoretical framework presented in Section 3.1. We assume the same preferences, demand
functions, production technology, and market structure as in the model with fixed production
networks. What is new is that firms now optimally choose their set of suppliers (i.e. the firm’s
sourcing strategy) and decide whether to import and export. We first describe the modelwith endogenous network formation and discuss how it can be solved. Then we estimate
the model, and use it to quantify how international trade affects firms’ production costs andconsumer prices with and without endogenous network formation.

4.1

Model

4.1.1

Determination of firm sourcing strategy, import and export participation

We assume that only buyers initiate linkages with other domestic firms. Forming linkages
to suppliers is costly, and firm j incurs a random, firm-pair-specific, fixed cost fkj to add
supplier k. The realization of fixed costs is known to the firm at the time it selects suppliers.
Firms in our model make profits due to positive mark-ups in sales to domestic and foreign
final consumers. Since the buyer is assumed to have all the bargaining power, firms do notmake profits from sales to other firms. Hence, variable profits are proportional to firm-levelsales to final consumers.
Given a sourcing strategy, Zj , and export participation choice, IjF , the profits of firm j arePequal to variable profits minus the fixed costs of domestic and foreign sourcing, k∈Zj fkj w` ,and the fixed costs of exporting, IjF fjF w` :πj (Zj , IjF ) = πjvar (Zj , IjF ) −X

fkj w` − IjF fjF w` .

(24)

k∈Zj

We assume that firm j exogenously meets a set of eligible suppliers, Zj . Firm j then endogenously decides on the set of suppliers and whether or not to export:

25

max πj (Zj , IjF ) s.t. Zj ⊆ Zj , IjF ∈ {0, 1}.

Zj ,IjF(25)

Under the assumption made earlier that final demand is more elastic than the elasticity
of substitution between inputs in the production function, σ > ρ, the sourcing decision iscomplementary across suppliers as well as complementary to the exporting decision. In otherwords, the marginal benefit of adding a supplier is increasing in the set of existing suppliers
and it is higher if the firm exports. We can therefore follow the same approach as Antras
et al. (2017), namely the adaption of the Jia (2008) algorithm, to solve the problem describedin (25), given knowledge about the costs of the eligible suppliers in Zj .
To solve the model with endogenous network formation, we therefore need to determine
the set of eligible suppliers, Zj . When solving the problem for every firm described in(25), a key issue that arises is that each firm needs to guess a set of costs for its eligible
suppliers, where the costs themselves are equilibrium objects and depend on everyone’s
sourcing decision. Not only is it extremely challenging computationally to find a fixed pointin the set of costs for all firms so that these costs are consistent with everyone’s optimal
sourcing decision, but also the uniqueness of such a fixed point is rather unlikely. On the
one hand, if firms guess that suppliers have very high unit costs, this could result in theformation of very few linkages and lead to high unit costs overall. On the other hand, if
firms guess that suppliers have very low unit costs, this could result in the formation of many
linkages and lead to low unit costs overall. To get around these problems, we consider theformation of an acyclic network, postulating an ordering of firms and restricting the eligible
set of suppliers to firms that appear prior to the buyer.
To be concrete, all firms can choose to import foreign inputs and to export their output
abroad. However, the set of eligible domestic suppliers varies across firms. Specifically, weorder firms in a sequence S = {1, 2, 3, ..., N } that restricts the set of eligible suppliers, as
illustrated in Figure 5. Because we assumed buyers has the full bargaining power in any firmto-firm transactions, firms only need to know the choices of the firms prior in the sequence.
Taken together, these assumptions make the network formation tractable, as we describe
below.
Firm 1 is first in the sequence and can only hire labor inputs. To make its decision of
how much labor to hire, firm 1 only needs to know the wage level, w` , and domestic market
Edemand, P 1−σ. Firm 2 is second in the sequence and can hire both labor inputs as well aspurchase the input produced by firm 1. To make these decisions, firm 2 needs to know the
wage and market demand level as well as the cost of its eligible supplier (firm 1). Firm 3 canhire labor and purchase the inputs from firm 1, firm 2, or both. And so on. Given a guessE, one can solve the problems
for equilibrium wages, w` , and domestic market demand, P 1−σof the firms sequentially.
26

Figure 5: Endogenous network formation – eligible connections
Foreign
InputsLabor

Firm 1

Firm 2

Firm 3Firm 4

Other Firms

Domestic
Final Demand

Foreign
Final Demand

Although the above ordering of firms simplifies the problem, we are still limited in thenumber of possible sourcing strategies we can feasibly evaluate. We therefore restrict theset of eligible suppliers for firm j, Zj , to be a random subset from the set of firms prior tofirm j in the sequence. The suppliers for firm j are then optimally chosen as the solution
to the problem in (25). In practice, we choose the cardinality of Zj to be at most 200, so
the firm still chooses among 2200 possible supplier sets. By following Antras et al. (2017)’s
adaption of Jia (2008)’s algorithm, we are then able to tractably solve the discrete choiceproblem.21 The firm’s order in the sequence of supplier choices and its set of eligible suppliers
are becoming attributes of the firm and therefore primitives of the model.
Imposing a tie-breaking rule that in the case of indifference a supplier is included ensuresa unique solution to the problem in (25). As a consequence, the network formation will also
be unique given a set of wages and a guess for the price index. We can then alter wages,
price index, and expenditure to achieve labor market clearing, trade balance, and a fixed
point for the price index and expenditure. Importantly, we are searching here only for a fixed
point in wages and price index (only 2 scalars) as opposed to searching for a large fixed point
vector in every firm’s costs and searching strategies. In other words, the ordering approach
implies that even with a rich micro structure and firm-level heterogeneity, knowing only two
equilibrium variables is sufficient to solve sequentially the firms’ problems. We discuss theaggregation and the equilibrium with endogenous network structure more formally below.
21

We describe the computational algorithm to solve for the firm’s problem in Appendix G. The procedure
is very similar to the one described in Antras et al. (2017). Here, we also develop a greedy algorithm in thecase the the differences in the lower and upper bounds for the optimal solution are too wide to evaluate theprofits of all feasible combinations in between. As in Antras et al. (2017), we find that in about 99% of thecases the lower and upper bounds are perfectly overlapping (see Table 15 in the Appendix).

27

4.1.2Aggregation and equilibrium

The model aggregation and equilibrium are broadly similar to the case with fixed networks.
However, there are a few notable differences. The first is that firms incur fixed costs – paidin units of labor – to add a domestic buyer, import, or export. Therefore, the labor market
clearing condition becomes:w` L =

X µj − 1jµjs`j xj + w`

X


X


j

fkj + IjF fjF  .

(26)

k∈Zj

Additionally, a firm’s profit function now subtracts the incurred fixed costs, and a firm’s
sourcing strategy and export participation are now endogenous choices. However, the tradebalance condition remains unchanged. The following definition formally describes equilibrium with an endogenous network structure.
Definition 2 (Equilibrium with endogenous network structure) Given foreign expenditure, EF , foreign price index, PF , and a set of prices by foreign suppliers, {pF j }j , as well
as set of eligible suppliers, Zj that satisfies acyclicity of the network, an equilibrium for themodel with endogenous network structure and endogenous export participation is a wage level,w` , price index for the consumer, P , and aggregate expenditure, E, as well as a set of sourcing strategies and export participation choices, such that the firm’s optimization problem in(25), and equations (6), (8), (9), (10), (12), (13), (18), (19), and (26) hold.

4.2

Assessing the assumptions about the shape of the network

Given the assumptions we invoked to solve the endogenous network formation, the resulting
network will be acyclic. As shown in Figure 6, in an acyclic firm network, there exists atleast one way to sort firms so that all directed edges face one direction. In contrast, in acyclical network at least one edge will face the opposite direction. This feature of our network
formation mechanism is admittedly restrictive. We now perform two checks to assess how
well the Belgian data can be approximated by an acyclical network.
4.2.1

How cyclic is the production network?

Let ν (i) be an ordering of firms that maps firms {i, j, k, · · · } ∈ Θ into numbers from
{1, · · · N }. To describe how cyclical the Belgian production network is, we want to findthe optimal ν (k) that minimizes the following objective function:min{ν(k)}

X

1 {i ∈ Zj } 1 {ν (i) > ν (j)} ,

i,j

28

Figure 6: Examples of acyclic and cyclic networks

A

A

B

C

D

E

A

B

C

C

Directed acyclic network

Directed cyclic network

B

D

A

E

Directed acyclic network, sorted

C

E

D

B

D

E

Directed cyclic network, sorted

where Zj is the supplier set of firm j. Solving this problem corresponds to minimizing thenumber of directed edges that are facing the direction opposite to that of the sorting order.
In other words, we try to find an ordering that minimizes the number of arrows facing to
the left in the cyclic network in Figure 6.
To solve this problem, which is also known as the feedback arc set problem, we adopt
an algorithm proposed by Eades, Lin, and Smyth (1993). The details of the computationalalgorithm and implementation are presented in Appendix F. The algorithm offers a local
minimum, showing that at most 17% of edges in the whole firm-to-firm network in 2012
violate acyclicity.22 We also search for an ordering that minimizes the value of firm-to-firmsales in violation of acyclicity.23 We find that no more than 22% of firm-to-firm sales are inviolation of acyclicity. We will refer to the former as the unweighted ordering algorithm andthe latter as the weighted ordering algorithm.
A natural question that arises is how different the structure of an economy with anacyclic network is in comparison to the economy observed in the data. One way to make thiscomparison is to calculate input-output tables with and without the firms in buyer-supplier
relationships that violate acyclicity. We find that when calculating input-output tables with72 sectors, the correlation between the input-output table coefficients from the full data andthe data without links in violation of the ordering is 0.92 when using the the unweighted
ordering algorithm output. The correlation is even higher, 0.97, when using the weighted
22

While there is no perfect reference point for this figure, we can compare it to the structure of thedirected social network Twitter. Simpson, Srinivasan, and Thomo (2016) calculate that 23% of edges are inviolation of acyclicity in the Twitter network in the year 2010.
P
23
Specifically, we solve the following problem: min{ν(k)} i,j xij 1 {ν (i) > ν (j)}, where xij is the valueof the sales from firm i to firm j.

29

ordering algorithm output.24
4.2.2

Gains from trade under fixed networks: cyclic versus acyclic production
networks

Another way to assess the assumption of an acyclic network is to examine how the results
based on the exogenous network model change if we exclude transactions that violate acyclicity. It is reassuring to find that estimated effect of international trade on consumer prices
and firms’ cost of production are very similar if we only uses firm-to-firm sales that areconsistent with the acyclic network obtained by the ordering algorithm described in Section
4.2.1. Specifically, we keep the direct import share of each firm the same as in the data, set
all transactions in violation of the ordering to zero, and adjust all other domestic firm-toP
firm input shares such that share of each firm j’s input purchases, i∈Zj sij , is unchanged.25
The results presented in Table 19 in the Appendix show that the gains from trade under an
exogenous network are virtually identical if we only use the subset of transactions for which
the domestic production network is acyclic.

4.3

Estimation of model parameters given endogenous network

When allowing for endogenous network formation, we are not able to analytically solve themodel. Instead, we will structurally estimate the model and provide numerical results forthe counterfactual analysis.
In the estimation of our model, we simulate 100,000 Belgian firms, which is close to the139,605 firms in our sample in 2012. A firm is characterized by a core productivity level,a set of eligible suppliers that satisfies the ordering, a vector of fixed cost draws for all
eligible suppliers, a vector of firm-pair-specific cost shifters, a foreign input cost shifter, fixed
costs of importing and exporting, and a foreign demand shifter. We normalize firms’ laborproductivity shifters, α`j = 1, and firms’ domestic final demand shifters, βjD = 1.
As a first step of the estimation, we recover the productivity distribution of firms (scaledby some general equilibrium objects) from the identity

24

To construct the input-output table coefficients, we aggregate firm-to-firm transactions within the supplying and buying sector. We note that this procedure differs from the national account definition of an
input-output table. First, the rows and columns of our aggregated tables are referring to the main sectors of
the buyers and suppliers, but these firms can also have a significant share of their production in other sectors.
Second, in national account tables, the contribution of the retail and wholesale sectors to the productionof the other goods only refer to the trade margin of retailers and wholesalers. In our data, the retail andwholesale sectors are accounted based on their total sales and total input consumption and not on theirtrade margin.
25
The only exception is when there is no other domestic supplier of a firm, in which case in the data with
only acyclic transactions the domestic firm-to-firm input share is set to zero.

30

1/(σ−1)xiH= φi1/(1−ρ)s`i

P E 1/(σ−1)
.
µw`

(27)
1/(σ−1)

Observing all the terms on the left hand side enables us to estimate the distribution φi P Eµw` .
After visually inspecting the distribution, we assume it is log-normal, and estimate the scaleparameter to be −2.12 and the dispersion parameter to be 1.37.
We next turn to the estimation of the parameters for the distribution of the firm-pairspecific shifter in the production function, αkj , the foreign input cost shifter, αF j , the foreign
demand shifter, βjF , as well as the distribution of fixed cost parameters for domestic firmto-firm purchases, w` fkj , fixed cost of importing, w` fF j , and fixed cost of exporting w` fjF .
We again impose log-normality of the distributions and estimate the scale and dispersion
parameters of those distributions. We assume that αF j , αkj , and βjF are independent draws
from three log-normal distributions which share a common dispersion parameter, Φα,βdisp , and
βF
αdom
αF
have different scale parameters, Φscale , Φscale , Φscale , respectively. Similarly, the fixed costdraws for domestic purchases from other firms, imports, and exports, are drawn indepenfimp
fexp
dom
dently from three log-normal distributions with scale parameters Φfscale, Φscale, Φscale, and afcommon dispersion parameter, Φdisp . The parametric restrictions on the common dispersion
parameters imply we need to estimate only 2 instead of 6 dispersion parameters. Overall,there are 8 parameters to be estimated.
We use method of simulated moments to estimate our parameters. We target three sets ofmoments to match. The first set of moments is helpful in estimating the parameters affecting
domestic-firm-to firm purchases. The draws of the fixed costs govern the extensive margins
of firm-to-firm trade. Thus, we use information about firms’ numbers of suppliers to identify
domΦfscaleand Φfdisp . To do this, we match the model to have the same quartile distribution of
number of suppliers. Following the procedure used by Eaton, Kortum, and Kramarz (2011),
we include in the first vector of moments generated by the model, m̂1 (Φ), the proportionof firms that has a number of suppliers equal to the first, second, third, and fourth quartilein the data. The draws of αkj govern the distribution of both the intensive margin and theextensive margin of firm-to-firm transactions. To identify the parameters Φαscale and Φα,βdisp ,
we target statistics on the labor share of firms. Again, we aim to match the fraction offirms in the data that have labor shares in the first, second, third, and fourth quartile of
the actual labor share in the data. Relatedly, we also aim to match the distribution of
the actual firm-to-firm input shares (conditional on observing trade between firms). Using
the same procedure as above, we include the fraction of firms in the four quartile bins ofthat distribution (using as thresholds the quartiles observed in the data). This generates 12
elements in the vector m̂1 (Φ).
The second set of moments is helpful in estimating the parameters affecting imports and31

exports. We include in m̂2 (Φ) the share of firms that import and export, respectively. We
also include the fraction of firms falling into the bins of the first, second, third, and fourth
quartile of imports in firm-level inputs in the data. Similarly, the fraction of firms that havethe ratio of exports to firm-level domestic sales as in the quartiles in the data. There are
10 elements in the vector m̂2 (Φ). Finally, as a third set of moments we include aggregate
targets such as the ratio of aggregate exports to aggregate final demand, and the weighted
aggregate of firm-level sales to households and foreign input shares, which correspond to thesufficient statistics for the price index increase under fixed networks. There are 3 elementsin the vector m̂3 (Φ).
We describe the difference between the moments in the data and in the simulated modelby ŷ(Φ):

m1 − m̂1 (Φ)
ŷ(Φ) = m − m̂(Φ) =  m2 − m̂2 (Φ)  ,m3 − m̂3 (Φ)
and the following moment condition is assumed to hold at the true parameter value Φ0 :E [ŷ(Φ0 )] = 0.

(28)

The method of simulated moments selects the model parameters that minimize the following objective function:Φ̂ = arg min [ŷ(Φ)]> W [ŷ(Φ)] ,(29)
Φwhere W is a weighting matrix.26
4.3.1

Estimation results

Table 3 shows the values of the estimated parameters.
Table 3: Estimated parameters
Preference and production
F
dom
F
Φ̂α,β
Φ̂αscaleΦ̂αscaleΦ̂βscaledisp-6.69 -2.36 -1.94 2.40

domΦ̂fscale-2.64

Fixed costs
fimp
fexp
Φ̂scaleΦ̂scale8.01
9.47

Φ̂fdisp
8.65

The magnitude of the estimated parameters are difficult to interpret, since their scale isaffected by the choice of normalizations for the foreign market size and the price of theforeign input. We therefore focus on the model fit given these parameter estimates.
26

We weight the moments equally, hence the weighting matrix is the identity matrix.

32

Table 4 shows how well we fit the moments that we target. Note that instead of showing
the moments directly (i.e., the fraction of firms falling into each quartile bin), we show thevalues of the 25th, 50th, and 75th percentiles in both the data and model. The model does apretty good job at fitting the targeted statistics of firm-to-firm transactions. Most statistics
are very close between the model and the data. With respect to fitting firm-level statisticsof trade, the model under-predicts the fraction of firms directly involved into exportingand importing (9.4% and 15.4% in the data and 6.7% and 7.9% in the model, respectively).
However, the model does a decent job of fitting the aggregate importance of trade, illustrated
by a similar ratio of aggregate exports to aggregate sales to domestic final demand in bothmodel and data. The statistics summarizing the gains from trade under fixed networks arealso similar in model and data.
We also examine how well the model fits moments that were not directly targeted in theestimation. Specifically, we have not targeted directly the association of size between buyers
and suppliers that trade with each other. Consistent with the data, the model predicts a weaknegative correlation between the number of suppliers of the buying firms (indegree buyer)
and the number of buyer firms of suppliers (outdegree supplier). Similarly, the correlation
between sales of the buying and selling firm is close to zero both in the data and in themodel.

4.4

Counterfactual with endogenous network formation

Equipped with the parameter estimates of our model, we next turn to the quantitative
analysis of the gains from trade for the Belgian economy under endogenous production
networks. We repeat the same counterfactual experiment as under fixed networks, and ask
how the economic outcomes in the Belgian economy would change if the barriers to tradewere infinite. To shut down trade, we make the costs to import and export prohibitively
large, so that no firm will engage in international trade. We keep the size of the domestic
labor force and all other parameters at their estimated level. We then solve the problem
of a closed economy and normalize the nominal wage to the same level it took in the open
economy. To solve for the counterfactual equilibrium, we find a fixed point in the market
E. After finding the fixed point in market demand, the labor market clears as
demand, P 1−σwell due to Walras’ law.
We are interested in how international trade affects the firms’ production costs, thestructure of the domestic production network, and the aggregate price index. We start by
discussing the change in the overall price index, and then look into the micro-changes thatlead to these price index effects. In our simulated economy, the price index increase by 93
percent with a fixed network structure and 80 percent when we allow for endogenous network
formation. Hence, we find that in our simulated economy with endogenous networks, the33Table 4: Model fit: targeted moments

Number of suppliers 25th percentile
Number of suppliers 50th percentile
Number of suppliers 75th percentile

Data
15
28
49

Model
15
30
53

Share of labor costs 25th percentile
Share of labor costs 50th percentile
Share of labor costs 75th percentile
Firm-to-Firm share 25th percentile
Firm-to-Firm share 50th percentile
Firm-to-Firm share 75th percentile

0.12
0.28
0.50
0.00020.00150.0069

0.21
0.33
0.66
0.00010.00040.0035

Share of firms that export

0.094

0.067

Share of exports in total firm sales 25th percentile
Share of exports in total firm sales 50th percentile
Share of exports in total firm sales 75th percentile

0.008
0.120
0.649

0.001
0.020
0.185

Share of firms that import

0.154

0.079

Share of imports in firm inputs 25th percentile
Share of imports in firm inputs 50th percentile
Share of imports in firm inputs 75th percentile

0.008
0.239
0.653

0.012
0.066
0.317

Ratio of aggregate exports to aggregate sales to domestic final demandP 1−σ  1
T otal 1−ρ 1−σ
F · →∞
P̂ |ptotal
s
=
1
−
s
Fj
i iH
1
P
 1−σ
1−σ
pF · →∞1−ρP̂ |direct =i siH (1 − sF j )

1.02

0.87

1.77

1.93

1.21

1.13

Notes: Percentiles are calculated based on all firms in the sample; Share of labor costs refers to thefraction of labor costs in costs (labor costs + domestic purchases + imports) and the percentiles are calculated based on all firms the sample; Firm-to-Firm share refers to the fraction of costs a firm spends
on one particular supplier and the percentiles are calculated for all firm-to-firm transactions; The percentiles for share of exports in total firm sales are calculated for all firms with positive export sales. The
percentiles for share of imports in firm inputs are calculated for all firms with positive import purchases.

Table 5: Model fit: non-targeted moments

Corr (Indegree Buyer, Outdegree Supplier)Corr (Sales Buyer, Sales Supplier)

Data

Model

-0.05
-0.02

-0.06
0.01

price index rises around 15 percent less than under a fixed network.27
We next look at the distribution of firms’ cost changes. Under fixed networks, we haveseen that every firm’s cost weakly increases when banning international trade. Interestingly,
27

Specifically, 80
93 − 1 = −.14.

34this is no longer the case under endogenous networks. Firms with little direct or indirect engagement in foreign trade actually benefit from banning international trade. This is because
they can expand relatively to the firms’ that are engaged in international trade. Our simulated model suggests that in the absence of international trade, 24 percent of firms actuallywould have weakly lower costs than with international trade.
Compared to the counterfactual analysis with a fixed network structure, most firms havelower cost increases under endogenous network structure. We show the distribution of relative
cost increases under endogenous and exogenous networks in Figure 7a. Note that a small
fraction of firms does have higher cost increases under endogenous networks. This occurs
if a firm not only stops importing and exporting but also drops domestic suppliers from its
sourcing strategy. Figure 7b shows that firms that have relatively large cost increases under
fixed networks tend to have even larger cost increases under endogenous networks.
Figure 7: Cost changes: Endogenous vs. exogenous network(b) Scatter(a) Density
12

1.3

1.2

Relative cost change, endogenous to fixed network

10

8

6

4

2

0
0.3

1.1

1

0.9

0.8

0.7

0.6

0.5

0.4
0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1

Relative cost change, endogenous to fixed network

1.5

2

2.5

3

3.5

4

4.5

Cost change under fixed network

Notes: The left panel shows the density of the ratio of cost changes from banning international trade under
endogenous and fixed networks. For most firms this ratio is smaller than one, implying lower cost increases
under endogenous networks for those firms. The right panel is a scatter plot of the ratio of cost increases
under endogenous and fixed networks (vertical axis) against the cost increases from banning international
trade under fixed networks (horizontal axis). Firms that have high cost increases under fixed networks
sometimes have even larger cost increases under endogenous networks. Observations below the 1st andabove the 99th percentile of cost increases are excluded from the figure.

These heterogeneous cost changes are driven by changes in the domestic linkages between
firms. We find that in the simulated economy, the number of domestic firm-to-firm linkages
increases from 3.70 million to 4.16 million when trade is shut down, an increase of around
12 percent.28 Underlying this net increase, there is churn in firm-to-firm linkages: Around28

In the actual Belgian firm-to-firm data, there are around 6 million domestic firm-to-firm transactions,hence our simulated economy under-predicts the number of linkages between firms.

35

5

2 percent of the domestic linkages that exist under free trade are no longer active in theabsence of trade.

5

Conclusion

In this paper, we used administrative data from Belgium with information on domestic firmto-firm sales and foreign trade transactions to study how international trade affects firm
efficiency and real wages. Our paper offered three sets of results. First, we documented thatmost firms that do not directly import or export still have large indirect exposure to foreign
trade, and that a firm’s output is affected by idiosyncratic shocks to its buyers and suppliers.
These empirical insights motivated the development of a model with domestic production
networks and trade.
Second, we derived new sufficient statistics results for how international trade affects
firms’ production costs. Assuming a fixed network structure, the cost reduction for an individual firm due to international trade depends only on the share of input costs that isspent on goods that are imported directly or indirectly and the elasticity of substitution
in the production function. We applied this sufficient statistics formula to our data, andcompared the results to those we obtain using existing approaches. This comparison highlights the importance of data on and modeling of domestic production networks in studies
of international trade.
Lastly, we developed a novel framework for analyzing the endogenous formation of the
production network. We make the model tractable by focusing on the formation of anacyclic rather than a cyclic production network. While restrictive, this allowed us to solvea model of firm trade with endogenous formation of domestic buyer-supplier relationships.
Reassuringly, we found that the vast majority of buyer-supplier relationships in Belgium canbe described by an acyclic production network. Moreover, both sectoral input-output tables
and the gains from trade under a fixed network structure do not change materially if we
restrict attention to transactions for which the domestic production network is acyclic. Our
approach to endogenous network formation may prove useful in contexts other than trade
where researchers are increasingly interested in the formation and consequences of domestic
production networks.

36

References
Acemoglu, D., V. M. Carvalho, A. Ozdaglar, and A. Tahbaz-Salehi (2012):“The Network Origins of Aggregate Fluctuations,” Econometrica, 80, 1977–2016.
Amiti, M., O. Itskhoki, and J. Konings (2014): “Importers, Exporters, and Exchange
Rate Disconnect,” American Economic Review, 104, 1942–1978.
Amiti, M. and J. Konings (2007): “Trade Liberalization, Intermediate Inputs, and Productivity: Evidence from Indonesia,” American Economic Review, 97, 1611–1638.
Antras, P., T. C. Fort, and F. Tintelnot (2017): “The Margins of Global Sourcing:
Theory and Evidence from U.S. Firms,” American Economic Review, 107, 2514–64.
Antràs, P. and E. Helpman (2004): “Global Sourcing,” Journal of Political Economy,
112, 552–580.
——— (2008): “Contractual Frictions and Global Sourcing,” in The Organization of Firms
in a Global Economy, Cambridge, MA: Harvard University Press.
Arkolakis, C., A. Costinot, and A. Rodrı́guez-Clare (2012): “New Trade Models, Same Old Gains?” American Economic Review, 102, 94–130.
Atalay, E., A. Hortacsu, J. Roberts, and C. Syverson (2011): “Network structureof production,” Proceedings of the National Academy of Sciences, 108, 5199–5202.
Baqaee, D. R. (2014): “Cascading Failures in Production Networks,” mimeo.
Baqaee, D. R. and E. Farhi (2017): “The Macroeconomic Impact of Microeconomic
Shocks: Beyond Hulten’s Theorem,” mimeo.
Barrot, J. N. and J. Sauvagnat (2016): “Input Specificity and the Propagation of
Idiosyncratic Shocks in Production,” Quarterly Journal of Economics, 131, 1543–1592.
Bernard, A. B., E. J. Blanchard, I. Van Beveren, and H. Vandenbussche(2016a): “Carry-Along Trade,” mimeo.
Bernard, A. B., J. B. Jensen, S. J. Redding, and P. K. Schott (2007): “Firms inInternational Trade,” Journal of Economic Perspectives, 21, 105–130.
Bernard, A. B., A. Moxnes, and Y. U. Saito (2016b): “Production Networks, Geography and Firm Performance,” mimeo.
Bigio, S. and J. La’o (2016): “Financial Frictions in Production Networks,” mimeo.
37

Black, P. E. (1999): “Algorithms and Theory of Computation Handbook,” .
——— (2005): “Greedy Algorithm,” Dictionary of Algorithms and Data Structures, 2.
Blaum, J., C. Lelarge, and M. Peters (2016): “The Gains from Input Trade with
Heterogeneous Importers,” mimeo.
Boehm, C., A. Flaaen, and N. Pandalai-Nayar (2015): “Input Linkages and theTransmission of Shocks : Firm-Level Evidence from the 2011 Tohoku Earthquake,” .
Caliendo, L. and F. Parro (2015): “Estimates of the Trade and Welfare Effects of
NAFTA,” Review of Economic Studies, 82, 1–44.
Carvalho, V. M. and B. Grassi (2017): “Large Firm Dynamics and the Business Cycle,”mimeo.
Carvalho, V. M., M. Nirei, Y. Saito, and A. Tahbaz-Salehi (2016): “Supply Chain
Disruptions: Evidence from the Great East Japan Earthquake,” mimeo.
Carvalho, V. M. and N. Voigtländer (2015): “Input Diffusion and the Evolution of
Production Networks,” mimeo.
Chaney, T. (2014): “The Network Structure of International Trade,” American Economic
Review, 104, 3600–3634.
——— (2016): “Networks in International Trade,” The Oxford Handbook of the Economics
of Networks.
Costinot, A. and A. Rodrı́guez-Clare (2014): “Trade Theory with Numbers: Quantifying the Consequences of Globalization,” in Handbook of International Economics, vol. 4,
197–261.
De Loecker, J., C. Fuss, and J. Van Biesebroeck (2014): “International Competition
and Firm Performance : Evidence from Belgium,” NBB Working Paper Series, 269.
De Loecker, J., P. K. Goldberg, A. K. Khandelwal, and N. Pavcnik (2016):“Prices, Markups, and Trade Reform,” Econometrica, 84, 445–510.
Dhyne, E., G. Magerman, and S. Rubinova (2015): “The Belgian Production Network
2002-2012,” National Bank of Belgium Working Paper Series, 288.
Eades, P., X. Lin, and W. F. Smyth (1993): “A Fast and Effective Heuristic for the
Feedback Arc Set Problem,” Information Processing Letters, 47, 319–323.

38

Eaton, J., D. Kinkins, J. Tybout, and D. Xu (2016): “Two-sided Search in International Markets,” mimeo.
Eaton, J. and S. Kortum (2002): “Technology, Geography, and Trade,” Econometrica,
70, 1741–1779.
Eaton, J., S. Kortum, and F. Kramarz (2011): “An Anatomy of International Trade:
Evidence From French Firms,” Econometrica, 79, 1453–1498.
Furusawa, T., T. Inui, K. Ito, and H. Tang (2017): “Global Sourcing and Domestic
Production Networks,” mimeo.
Gabaix, X. (2011): “The Granular Origins of Aggregate Fluctuations,” Econometrica, 79,
733–772.
Garetto, S. (2013): “Input Sourcing and Multinational Production,” American Economic
Journal: Macroeconomics, 5, 118–151.
Goldberg, P. K., A. K. Khandelwal, N. Pavcnik, and P. Topalova (2010): “Imported Intermediate Inputs and Domestic Product Growth: Evidence from India,” The
Quarterly Journal of Economics, 125, 1727–1767.
Gopinath, G. and B. Neiman (2014): “Trade Adjustment and Productivity in Large
Crises,” American Economic Review, 104, 793–831.
Halpern, L., M. Koren, and A. Szeidl (2015): “Imported Inputs and Productivity,”
American Economic Review, 105, 3660–3703.
Ho, K. (2009): “Insurer-Provider Networks in the Medical Care Market,” American Economic Review, 99, 393–430.
Hulten, C. R. (1978): “Growth Accounting with Intermediate Inputs,” The Review of
Economic Studies, 45, 511–518.
Hummels, D., R. Jørgensen, J. Munch, and C. Xiang (2014): “The Wage Effectsof Offshoring : Evidence from Danish Matched Worker-Firm Data,” American Economic
Review, 104, 1597–1629.
Jia, P. (2008): “What Happens When Wal-Mart Comes to Town: An Empirical Analysisof the Discount Retailing Industry,” Econometrica, 76, 1263–1316.
Kikkawa, A. K., G. Magerman, and E. Dhyne (2017): “Imperfect Competition andthe Transmission of Shocks: The Network Matters,” mimeo.
39Lim, K. (2015): “Firm-to-firm Trade in Sticky Production Networks,” mimeo.
Liu, E. (2016): “Industrial Policies and Economic Development,” mimeo.
Magerman, G., K. De Bruyne, E. Dhyne, and Van Hove (2016): “HeterogeneousFirms and the Micro Origins of Aggregate Fluctuations,” National Bank of Belgium Working Paper Series, 312.
Melitz, M. J. and S. J. Redding (2014): “Missing Gains from Trade?” American
Economic Review, 104, 317–21.
Morales, E., G. Sheu, and A. Zahler (2015): “Extended Gravity,” mimeo.
Oberfield, E. (2017): “A Theory of Input-Output Architecture,” mimeo.
Oberfield, E. and D. Raval (2014): “Micro Data and Macro Technology,” mimeo.
Ossa, R. (2015): “Why trade matters after all,” Journal of International Economics, 97,
266–277.
Rodrı́guez-Clare, A. (2010): “Offshoring in a Ricardian World,” American Economic
Journal: Macroeconomics, 2, 227–258.
Simpson, M., V. Srinivasan, and A. Thomo (2016): “Efficient Computation of Feedback Arc Set at Web-Scale,” Proceedings of the VLDB Endowment, 10, 133–144.
Spiegler, R. (2016): “Bayesian Networks and Boundedly Rational Expectations,” Quarterly Journal of Economics, 131, 1243–1290.

40

A

Theoretical Results

A.1

Proof of Proposition 1

Proof.
Total cost increase
We havesTF otal= sF j +jX= sF j +X

sij sTF otal
i

i

"

#

sij sF i +iX

ski (sF k + · · · )

k

and
c1−ρ
=
j

X

ρ−1 ρ−1 1−ρ
ρ−1 ρ−1 1−ρρ−1 1−ραkj
φj ck + α`jφj w` + αFρ−1
pF j .
j φjkConsider the indirect effects (i.e., assuming that suppliers’ cost increases will translate
into price increases for their customers). The unit cost after pF j → ∞ for all j (assumingthat the nominal wage, w` , does not change) is
=
c̃1−ρ
j

X

ρ−1 ρ−1 1−ρ
ρ−1 ρ−1 1−ρ
φj w` ,
αkj
φj c̃k + α`j

k

41

thus
F · →∞ĉ1−ρ|ptotal
=
j

=

c̃1−ρ
j
c1−ρ
j
P ρ−1 ρ−1 1−ρ
ρ−1 ρ−1 1−ρc̃k + α`jφj w`
k αkj φj

= s`j +c1−ρjX

F · →∞
skj ĉ1−ρ|ptotalkk"= s`j +

X#X

skj s`k +sik (s`i +· · · )ik"= 1 − sF j −

X

skj +

X

k

!
1 − sF k −

skj

X

sik

!
+

X

i

k

sik

1 − sF i +i=1−

sF j +!#!

skj sF k +

X

sik −

i

k

=1−

sF j +

X

1 − sF i +

sik

i

"
X

sli

l

"
X

X

X

sli + · · ·l#!

skj sF k +Xsik (sF i + · · · )ik= 1 − sTF otalj .
Therefore, firms’ change in unit costs upon pf → ∞ when considering the full network
effects are as follows:1 1−ρ
F · →∞.
ĉj |ptotal= 1 − sTF otaljDirect cost increase
If only considering the direct effect (i.e., assuming that suppliers’ cost increases will nottranslate into price increases for their customers), with ρ > 1 and pF j → ∞ for all j (i.e.,
autarky), the cost for firm j becomes
c̄1−ρ
=
j

X

ρ−1 ρ−1 1−ρ
ρ−1 ρ−1 1−ρ
φj w` .
αkj
φj ck + α`jkTherefore,
F · →∞ĉ1−ρ|pdirect=jc̄j1−ρ
cj1−ρ

42= 1 − sF j .

!#
+ ···

Re-arranging yields the change in unit cost when considering only the direct effect:

1

F · →∞= (1 − sF j ) 1−ρ .
ĉj |pdirect

Cost increase under roundabout production
In this roundabout production economy, firm j produces its goods with a CES production
technology, using domestic intermediate goods, foreign imports, and labor. The implied unit
cost of firm j becomes
 1
ρ−1 1−ρρ−1 1−ρρ−1 1−ρ 1−ρ
α
P
+
α
p
+
α
w
,
cj = φ−1
j
Dj
D
Fj Fj
`j
`
where PD is a price index of domestic intermediate goods. Associated input shares are
sDj =

1−ρ
φρ−1
αρ−1
j
Dj PD

c1−ρ
j

, sF j =

1−ρ
φρ−1
αρ−1
j
F j pF jc1−ρj, and s`j =ρ−1 1−ρ
φρ−1
α`j
w`
j

c1−ρ
j

.

As in Blaum et al. (2016), we let domestic intermediate goods be produced via roundabout production, with CES substitution parameter σ. The price of an intermediate goodis therefore equal to the CES price index,
1
! 1−σ

PD =

X

σ−1 1−σ
αjD
pjD

,

j

where pjD is the price that firm j charges in the aggregation process. Let pjD = cj . We
ασ−1 p1−σjD, which is the firm j’s contribution to the intermediate
can additionally define sjD = jD
1−σPD
good. We use the firm’s share of domestic sales:

B2B salesj + Sales to HHj.
sjD = Pi (B2B salesi + Sales to HHi )We assume αjD = βjD , so that the two shares sjD and sjH are the same in the model.
Consider a change in cj , upon pF j → ∞ for all j.
 1
ρ−1 1−ρρ−1 1−ρρ−1 1−ρ 1−ρ
cj = φ−1
α
P
+
α
p
+
α
w
j
Dj
D
Fj Fj
`j
`
1

 1−ρ
ρ−1 1−ρρ−1 1−ρP̃D + α`jw`c̃j = φ−1
αDjjX
σ−1 1−σαjD
c̃jP̃D1−σ =j

43Combining these,
1−ρ
! 1−σ

ρ−1
c̃1−ρ
= φρ−1αDjjjXρ−1 1−ρ+ φρ−1α`jw` .
j

σ−1 1−σ
αjD
c̃j

j

Thus,

F · →∞ĉ1−ρ|proundabout
=
j

ρ−1
φρ−1
αDj
j

P

σ−1 1−σj αjD c̃j

ρ−1 1−ρ+ φρ−1
α`j
w`
j

c1−ρ
j
1−ρ
 1−σ

P
= s`j + sDj

1−ρ
 1−σ

σ−1 1−σj αjD c̃j

PD1−ρP

= s`j + sDj

σ−1 1−σ 1−σ pF · →∞ĉj |roundaboutj αjD cj
1−σ
PD

1−ρ
! 1−σ

1−ρ
! 1−σ

= s`j + sDj

X

pF · →∞
sjD ĉ1−σ|roundaboutj.

j
F · →∞The solution to this system of equations ĉ1−ρ|proundabout
is the change in unit costs of eachjfirm, upon autarky.

A.2

Proof of Lemma 1

Using the result that firms sell to other firms are marginal cost and rearranging equation(8), we obtain
X ρ−1 ρ−1 1−ρρ−1 1−ρ
c1−ρ
=φj αkj ck + φρ−1
α`j
w` .
j
j
k

In matrix form, this equation becomes
0 −1
c1−ρ = I − A
φρ−1 ◦ αρ−1
◦ w`1−ρ ,`ρ−1where the (i, j) element of A is φρ−1αij
if i ∈ Zj and 0 otherwise. The assumption thatj0
the matrix I − A is invertible and the fact that under closed economy one can normalize
wage w` , guarantee that there is a unique vector c that solves the equation above. With thecost vector c and constant mark-ups in sales to final consumers, one can compute a uniqueaggregate price index P according to equation (6). Given the cost vector and aggregate
price index, one can then compute a unique aggregate expenditure E from equations (13)44and (18).

A.3

Proof of Proposition 2

Denote post-shock equilibrium variable x with x̃. From equation (6), we have the expression for the price index after the shock,
1
! 1−σ

P̃ =

X

βiσ−1 µ1−σ c̃1−σi.

i

Combining this expression with the pre-shock price index P , we have
P̂ =

P̃
P
P

σ−1 1−σ 1−σ
µ c̃i
i βi
1−σ
P

=
=

X β σ−1 µ1−σ c1−σ
i

i

=

X

i

P 1−σ
1
! 1−σ

1
 1−σ

1
! 1−σ

ĉi1−σsiH ĉi1−σiwhere siH denotes firm i’s share in final consumption. This equation says that the change
in the aggregate price index depends on each firm’s change in cost and its share in final
consumption before the shock.

45

B

Numerical example: gains from trade

Here we demonstrate that economies with identical sets of aggregate exports, aggregate
imports, aggregate gross production and GDP but with different firm-to-firm network structures can potentially generate different gains from trade. We will work with a simple example
comparing two economies, where in both economies aggregate exports and imports are e100,
aggregate gross production is e300, and GDP is e100.
Both economies consist of two firms. Table 6 lays out the details of the two economies. In
economy 1, both firms are identical, and the firm-to-firm network is symmetric in the sensethat firms sell the same amount of goods to each other. In economy 2, firm 1 sells e100 tofirm 2, while firm 2 does not sell any of its goods to firm 1. There is also asymmetry in sales
to household and in exports, where firm 1 sells e10 to households and exports e40, whilefirm 2 sells e90 to households and exports e60. In the two economies, the aggregate valuesof imports, exports, gross production and value added are identical.
Table 6: Two economies
Economy 1
Firm 1 Firm 2
Imports
Exports
Gross production
Firm-to-firm sales
Labor costs / Value added
Sales to households

50
50
150
x12
50
50
50

50
50
150
x21
50
50
50

Economy 2
Firm 1 Firm 2
50
0
150
x12
100
100
50

50
100
150
x21
0
0
50

Now let us compute the direct and total shares of foreign inputs for the two firms in two
economies. Table 7 summarizes the foreign input shares. The direct shares of foreign inputs
for both firms in both economies are 1/3, as they all import e50 while their total inputs
are e150. We can also compute the total shares of foreign inputs by solving the system of
equations, from equation (14). In economy 2, firm 2 has higher exposure to foreign inputs
because not only does the firm import directly, it also relies heavily on goods from firm 1,
which also imports directly.

46

Table 7: Direct and total shares of foreign inputs
Economy 1
Firm 1 Firm 2sF isTF otal
i

1/3
1/2

Economy 2
Firm 1 Firm 2

1/3
1/2

1/3
1/3

1/3
5/9

Finally, let us compute the change in aggregate price index from banning imports. Applying equations (15) and (21), we obtain the changes in price index for economy 1 andeconomy 2:


1
1−
2

 1−σ
1−ρ

1
+
2



P̂1 =

1
2

1
1−
2



1
1−
3

 1−σ
1−ρ

1
+
2



P̂2 =

1
2

5
1−
9

1
! 1−σ
 1−σ
1−ρ

1
! 1−σ
 1−σ
1−ρ

.

If we apply ρ = 2 and σ = 4, then we obtain P̂1 = 2 and P̂2 = 1.73. Even though theaggregate statistics (GDP, gross production, export and import) are identical across the two
economies, the gains from trade in economy 1 turns out to be larger. This is because of theasymmetry in firm-to-firm trade where it creates different sTF otal
for the two firms. While inieconomy 1 the two firms have identical total exposure to foreign inputs, in economy 2, firm
2 has larger exposure. Upon ban of foreign inputs, in economy 2, households will be able
to substitute away toward firm 1’s goods, resulting in a smaller increase in aggregate priceindex.

47

C

Data Appendix

C.1

Grouping VAT-identifiers into firms

As mentioned in the main text, all our datasets are recorded at the VAT-identifier level.
We utilize ownership filings in the Annual Accounts and information from the Balance ofPayments survey in order to aggregate multiple VAT-identifiers into firms. In the ownershipfilings, each enterprise reports a list of all other enterprises of which it has an ownershipshare of at least 10% and the value of the share. In the Balance of Payments survey,
Belgian enterprises with international financial linkages have to report their stock and flows
of financial links. They have to report both the international participation they own and theforeign owners of financial participation in their capital if the participation represents at least
10% of the capital. The survey is designed to cover the population of Belgian enterprises
involved in international financial transactions.
We group all VAT-identifiers into firms if they are linked with more than or equal to 50%
of ownership. In addition, we group all VAT-identifiers into firms if they share the same
foreign parent firm that holds more than or equal to 50% of their shares. We use a “fuzzy
string matching” method to determine whether they share the same foreign parent firm, by
obtaining similarity measures of all possible pairs of foreign firms’ names. Lastly, in order
to correct for misreportings, we also add links to the VAT-identifier pairs if the two werelinked one year before and one year after. We define a firm as the group of VAT-identifiers
that are directly and indirectly linked.
Given these groupings of VAT-identifiers, we then choose the “most representative” VATidentifier for each firm. We use this “head VAT-identifier” as the identifier of the firm.29
Then, in order to make the identifiers consistent over time, we make the following adjustment:
We take firms whose head VAT-identifier was not an identifier of any firm in the previous
year. For such firms, if there exists a VAT-identifier within the firm which was a head
VAT-identifier in the previous year, then we switch the firm identifier to that former head
VAT-identifier.30
Having determined the head VAT-identifier for each firm with multiple VAT-identifiers,
we aggregate all the variables up to the firm level. For variables such as total sales andinputs, we adjust the aggregated variables with the amount of B2B trade that occurred29

The criteria for determining the head VAT-identifier is as follows: (i) If there is only one VAT-identifierin the firm that filed all the full annual accounts, the VAT declarations, and the B2B filings, then thisVAT-identifier is chosen as the head. (ii) If there are no such VAT-identifiers or multiple of them, then wechoose the VAT-identifier that has the largest total assets reported. (iii) If there are no VAT-identifier that
filed the annual accounts, then we choose the VAT-identifier that has the largest amount of total inputs,
which is the sum of labor costs, B2B inputs, and imports.
30
If there are multiple such VAT-identifier, then we choose the “most representative” VAT-identifier, using
the same criteria as above.

48

Table 8: Number of VAT-identifier in firms with multiple VAT-identifiersNum. VAT-identifier

Mean
3

10%
2

25%
2

50%
2

75%
3

90%
4

max
372

within the firm, correcting for double counting. For other non-numeric variables such as
firms’ primary sector, we take the value of its head VAT-identifier.
The number of VAT-identifiers for firms with multiple VAT-identifiers are shown in Table
8.

C.2

Firm selection

Table 9 displays the same numbers for Table 1, with statistics for all Belgian firms added.
Table 9: Coverage of all Belgian firms and selected sample
Year
2002
2007
2012

All Belgian Firms
Count V.A. Sales Imports
714,469 210812
204782,006 274 1080
294
860,373 300 1244
320

Exports
217
282
317

Selected sample
Count V.A. Sales Imports
122,460 123
586
179
136,370 157
757
280
139,605 170
829
296

Exports
189
269
295

Notes: All numbers except for Count are denominated in billion Euro in current prices. Data for Belgian GDP,
output, imports and exports are from Eurostat.

C.3

Reporting thresholds of the international trade dataset

There are different reporting thresholds for the international trade dataset, depending
on if the trade occurred with an extra-EU country or within the EU. The dataset covers
all extra-EU exports and imports by firms with values higher than e1,000 or with weights
bigger than 1,000kg. Nevertheless, we also observe values less than e1,000 as more firms
use electronic reporting procedures. For intra-EU trade prior to 2006, the dataset covers
all exports and imports by firms whose combined imports from intra-EU countries that are
more than e250,000 a year. For intra-EU trade from 2006 onward, the thresholds for exports
and imports changed to e1,000,000 and e400,000, respectively. Import reporting thresholds
became e700,000 per year in 2010. While these reporting threshold for intra-EU trade imply
we miss some trade transaction, they are set to capture at least 93% of aggregate Belgian
trade in the micro-data, hence our data still contains the overwhelming majority of the valueof Belgian trade.

49

C.4

Mapping CN codes into NACE codes

Our international trade dataset records products in Combined Nomenclature (CN) codes,
up to 8 digits. On the other hand, all other datasets that we use record the enterprise’s
primary sector in NACE Rev.2 code. To concord the two classifications, we convert the CN
8 digit codes into NACE Rev.2 codes. As the first 6 digits of CN codes are identical to thecontemporary Harmonized System (HS) codes, we first convert those HS 6-digit codes toClassification of Products by Activity (CPA) codes. We then convert CPA codes to NACE
codes, using the fact that CPA 2008 codes are identical to NACE Rev.2 codes up to 4 digits.
This conversion allows us to convert more than 98% of all international trade recorded inour dataset, in terms of values (in 2012).

50

D

Descriptive statistics

D.1

Direct and Total foreign input shares

In Figures 1a and 8 we present both the direct and total foreign input shares first for theentire sample of private sector firms in Belgium and then differentiated by major sector.
Figure 8: Histogram of direct and total foreign input share by firms’ sector
Agriculture

Manufacturing
15000

3400
90
80
70
60
50
40
30
20
10
0

Count

Count

Count

400
300
200
100
0
0

.2
.4
.6
.8
Foreign input share
Direct

1

0

Total

.2
.4
.6
.8
Foreign input share
Direct

1

Total

0

.2
.4
.6
.8
Foreign input share
Direct1

Total

Other Service
42000
2500

1000

2000

800

Count

Wholesale and Retail
34000
1200
Count

Construction
25000
700
600
500
400
300
200
100
0

1500

600

1000

400
200

500

0

0
0

.2
.4
.6
.8
Foreign input share
Direct

1

Total

0

.2
.4
.6
.8
Foreign input share
Direct1

Total

Notes: The black dot indicates the ending of the bar for the P
total foreign input share. Total foreign input
share of firm i, sTF otal
is calculated by solving sTF otal
= sF i + j∈Zi sji sTF otal
where sF i is i’s direct foreigniijinput share, and sji is j’s share among i’s inputs. The figure is based on the analysis of 139,605 private
sector firms in Belgium in 2012. The horizontal lines represent scale breaks on the vertical axis.

We summarize statistics on the distribution of the the direct and total foreign input shareby firm’s sector in Table 10.

D.2

Direct and Total export share

In Figures 1b and 9 we present both the direct and total export share for the entire
sample of private sector firms in Belgium and then differentiated by major sector.
51

Table 10: Distribution of direct and total foreign input share by firms’ sector

SectorAgriculture
Construction
Manufacturing
Wholesale and Retail
Other Services
Total

Mean
0.03
0.01
0.11
0.10
0.01
0.05

DirectWeighted Mean
0.27
0.11
0.59
0.42
0.18
0.42

Median
0
0
0
0
0
0

Mean
0.49
0.42
0.45
0.52
0.29
0.42

Total
Weighted Mean
0.68
0.48
0.75
0.75
0.41
0.67

Median
0.49
0.42
0.44
0.55
0.28
0.41

Notes: The numbers for the weighted mean are calculated using total input purchases of firms as the weight.

Figure 9: Histogram of direct and total export share by firms’ sector
Agriculture

Manufacturing

Construction
26000
10000
2800
2400
2000
1600
1200
800
400
0

16400

3400
690
150
120
90
60
30
0

1000

Count

Count

Count

5100
750
500
250
0
0

.2

.4
.6
.8
Export share
Direct

1

0

.2

Total

DirectWholesale and Retail
38700
21600
3000
2500
2000
1500
1000
500
0

.4
.6
.8
Export share

1

Total

0

.2

.4
.6
.8
Export share
Direct1

Total

Other Service
42800

Count

Count

22000
2400
1800
1200
600
0
0

.2

.4
.6
.8
Export share
Direct

1

Total

0

.2

.4
.6
.8
Export share
Direct1

Total

Notes: The black dot indicates the ending of thePbar for the total export share. Total export share of firmi, sTiFotal is calculated by solving sTiFotal = s̃iF + j∈Wi s̃ij sTjFotal where s̃iF is i’s share of exports out of its
output, and s̃ij is share of i’s output that went to firm j. Wi is the set of customers of i. The figure isbased on the analysis of 139,605 private sector firms in Belgium in 2012. The horizontal lines represent scale
breaks on the vertical axis.

52

We summarize statistics on the distribution of the the direct and total export share byfirm’s sector in Table 11.
Table 11: Distribution of direct and total export share by firms’ sector

SectorAgriculture
Construction
Manufacturing
Wholesale and Retail
Other Services
Total

Mean
0.04
0.00
0.10
0.04
0.00
0.03

DirectWeighted Mean
0.23
0.08
0.60
0.22
0.14
0.36

Median
0
0
0
0
0
0

Mean
0.31
0.08
0.25
0.10
0.09
0.12

Total
Weighted Mean
0.43
0.19
0.68
0.30
0.26
0.45

Median
0.22
0.02
0.10
0.01
0.01
0.02

Notes: The numbers for the weighted mean are calculated using total sales of firms as the weight.

D.3

Sectoral composition

Table 12 shows the sectoral composition of our selected sample. Values for value added
and output are in billion Euro.
Table 12: Sectoral composition in 2012
SectorAgriculture
Construction
Manufacturing
Wholesale and Retail
Other Services
Other
Total

Count
3,704
26,364
20,38542,99943,495
2,658
139,605

V.A.
1.49
18.3
55.5
31.8
50.3
12.7
170

Output
9.97
46.5
322
245
125
80.5
829

Imports
1.71
5.00
147
85.3
17.6
39.8
296

Exports
2.26
3.65
194
54.5
17.0
24.3
295

D.4

Link survival

E

Transmission of shocks along production chain

E.1

Constructing the exogenous trade shocks

Below, we explain the construction of the variables ∆ log Mit , ∆ log MitS , and ∆ log MitP S .

53

Table 13: 2002 Link Survival
Occurred In...
Count Col % Cum %2002
3,570,077
65.5
65.5
2002 & 2007
912,028
16.7
82.2
2002 & 2012
191,566
3.5
85.7
2002 & 2007 & 2012
778,73414.3
100.0∆ log Mit , an exogenous shock affecting the imports of firm i, is constructed as follows:∆ log Mit = log

X k,M
X k,M
sic,t−1 WESk,c,t − log
sic,t−1 WESk,c,t−1 ,
k,c

k,c

{z

|

}

Mit,t−1{z|

Mit−1,t−1}

where sk,Mic,t−1 is the share of imports of firm i at the initial year t − 1 that falls on productk from country c, and WESk,c,t is the world export supply (excluding sales to Belgium) of
country c for product k.
The import supply shock that i’s suppliers received, ∆ log MitS , is constructed as:∆ log MitS = log

Xski,t−1 Mkt,t−1 − logXski,t−1 Mkt−1,t−1 .

kkFinally, the import supply shock to potential customers of firm i is:∆ log MitP S = log

X−isui,t Mut,t−1
− log

u

X

−i
sui,t−1 Mut−1,t−1.

uWhere the variables on the RHS are constructed analogous to footnotes 10 and 11.31 32
31

The input share from sector u for firm i, siu,t , is defined as the share of inputs of i that came from firms
producing sector u goods:
suit =

Salesjit,
TotalInputsit
u

X

j∈Zit

u
where Zit
denotes the set of suppliers of i producing sector u goods at time t.
32−i
Mut,t−1=X
j∈Ut−1 ,j6=i−i
Mut−1,t−1=

VjHt−1
Mjt,t−1k∈Ut−1 ,k6=i VkHt−1

PX

VjHt−1
Mjt−1,t−1 ,k∈Ut−1 ,k6=i VkHt−1

Pj∈Ut−1 ,j6=iwhere Ut is the set of firms producing sector u good at t, and ViHt is firm i’s sales to domestic final demand

54

E.2

Other reduced form results

Here we report the reduced form results where firms’ changes in domestic sales anddomestic inputs are on the LHS variable. The first column shows the results for firms’changes in totals are on the LHS, and is identical to the third column in Table 2.
Controlling for own shocks and shocks that potential customers and suppliers have received, both positive demand shock on a firm’s actual customers and actual suppliers leadto an increase of the firm’s domestic sales and domestic inputs.
Table 14: Additional reduced form results∆ ln Xit∆ ln MitPC∆ ln Xit∆ ln MitP S
C∆ ln Xit∆ ln MitSN∆ ln Total Sales
0.089∗∗∗(0.009)0.156∗∗∗(0.012)0.025∗∗∗(0.003)0.039∗∗∗(0.005)0.122∗∗∗(0.013)0.041∗∗∗(0.018)87100∆ ln Domestic Sales
0.021(0.013)0.105∗∗∗(0.018)0.037∗∗∗(0.005)0.015∗∗(0.007)0.127∗∗∗(0.022)0.106∗∗∗(0.027)85795∆ ln Domestic Inputs
0.073∗∗∗(0.010)0.093∗∗∗(0.014)0.024∗∗∗(0.004)0.085∗∗∗(0.006)0.086∗∗∗(0.016)0.078∗∗∗(0.022)87363

Notes: Standard errors are clustered at the firm level. All variables are in terms of yearly log
differences from 2002 to 2012. All specifications include year fixed effects. We truncate outliersof each variables at the top and bottom 1% level. Firms’ domestic sales are the sum of theirsales to other domestic firms, and sales to domestic final demand. Firms’ domestic inputs are
the sum of their labor costs and input purchases from other domestic firms.
* p < 0.10, ** p < 0.05, *** p < 0.01

at t.

55

F

Ordering algorithm

In this section we describe the implementation of the ordering algorithm to solve thefeedback arc set problem. We begin by defining some terms and notation.

F.1

Terms and notation• graph / network, G = (V, E) - A collection of a set of edges E and set of vertices V .
Edges describe the relationship between vertices. Two basic classifications of graphs
are based on whether the edges are directed or undirected and whether they are weighted
or unweighted• n = |V |, m = |E|• cycle - A path within a graph where a vertex is reachable from itself• d+ (u) - For a vertex u ∈ V in a directed graph, number of outgoing edges• d− (u) - For a vertex u ∈ V in a directed graph, number of incoming edges• w+ (u) - For a vertex u ∈ V in a directed graph, cumulative sum of weights of outgoingedges• w− (u) - For a vertex u ∈ V in a directed graph, cumulative sum of weights of incoming
edges• sink - A vertex u ∈ V in a directed graph with d+ (u) = 0• source - A vertex u ∈ V in a directed graph with d− (u) = 0• feedback arc set - A set of edges from a directed cyclic graph that when removed makethe graph acyclic• s = slef t sright - Given 2 finite sequences slef t and sright with the indicated notation wesymbolize the concatenation operation. For example, if slef t = (A, B, C) and sright =(X, Y, Z), then s = slef t sright = (A, B, C, X, Y, Z)• bxc is the greatest integer less than or equal to x

56

F.2

Overview

The Belgian B2B data describes a weighted directed graph G = (V, E). Vertices are firms
and edges are sales between firms. The goal of the ordering algorithm is to order firms in away such that a given firm only sells to firms further along in the ordering and only buys
from firms that precede it. The condition desired by this ordering is known in graph theory
as a topological ordering (Black, 1999). A topological ordering exists if and only if a graphis directed and acyclic. The B2B data is cyclic. For the unweighted case our motivation
is to find a feedback arc set of minimal cardinality, that is, what is the minimum numberof transactions that we need to drop (i.e., the “violators”) from our network to satisfy ourordering condition? For the weighted case, we seek to find a feedback arc set such that the
cumulative weight of the violating transactions is minimized. Finding a minimum feedback
arc set is computationally difficult but approximation algorithms exist.

F.3

Unweighted case

The algorithm we use for the paper was first presented by Eades et al. (1993). Thisalgorithm was chosen because it has a linear run time complexity, O(m + n), and becauseof its relative implementation simplicity. The algorithm uses a greedy heuristic throughwhich it builds the proposed ordering s = slef t sright .33 Vertices are initialized into several
buckets: sinks, sources, and δ buckets, where for a vertex u ∈ V , δ(u) = d− (u) − d+ (u).34
At each iteration, the algorithm removes all sinks from the network and prepends them to asequence sright , removes all sources and appends them to a sequence slef t , and then removes
the vertex with the lowest δ score (the most “source”-like vertex) and appends it to slef t .35
Each removal requires updating the buckets to reflect the modified graph. The algorithm
stops when the graph is empty. There will be 2n − 1 buckets, which can be formalized asfollows:36

33

According to Black (2005), a greedy algorithm is “an algorithm that always takes the best immediate, or
local, solution while finding an answer. Greedy algorithms find the overall, or globally, optimal solution forsome optimization problems, but may find less-than-optimal solutions for some instances of other problems.”
34
We have flipped the sign here compared to Eades et al. to be consistent with the diagrams elsewherein our paper.
35
Eades et al. (1993) take the vertex with the maximum δ score.
36
Eades et al. (1993) assume that the graph G is simple (no bidirectional edges), and hence their originalalgorithm only requires 2n − 3 buckets.

57

V−n+1 = Vsources = {u ∈ V | d− (u) = 0; d+ (u) > 0}
Vn−1 = Vsinks = {u ∈ V | d− (u) > 0; d+ (u) = 0}
Vd = {u ∈ V | d = δ(u); d− (u) > 0; d+ (u) > 0}

The bucket V−n+1 contains all the vertices that are only the sources of edges. The bucket
Vn−1 contains all the vertices that are only the sinks of edges (in other words, vertices
that are only receiving edges). Each Vd bucket contains vertices with d net incoming edges(conditional on these vertices having both outgoing and incoming edges).

F.4

Example execution on unweighted network

Consider the following network:

A

B

E

D

C
Let’s trace the execution of the algorithm described by Eades et al.
F.4.1

Initialization

Buckets:A
sources

−3

−2

D

C

B

−1

0

1

Ordering : s = slef t = sright = ()F.4.2

First iteration:

Remove sinks
Updated buckets:

58

E
2

3

sinks

A
sources

−3

−2

−1

C, D

B

0

1

2

3

sinks

2

3

sinks

Updated ordering : slef t = (), sright = (E), s = slef t sright = (E)Remove sources
Updated buckets:

C, D, B
sources

−3

−2

−1

0

1

Updated ordering : slef t = (A), sright = (E), s = slef t sright = (A, E)
Remove vertex with lowest delta score
Updated buckets:BDsources

−3

−2

−1

0

1

2

3

sinks

3

sinks

Updated ordering : slef t = (A, C), sright = (E), s = slef t sright = (A, C, E)
F.4.3

Second iteration

Remove sinks
Updated buckets:

Bsources

−3

−2

−1

0

59

1

2

Updated ordering : slef t = (A, C), sright = (D, E), s = slef t sright = (A, C, D, E)
Remove sources
Updated buckets:

sources

−3

−2

−1

0

1

2

3

sinks

Updated ordering : slef t = (A, C, B), sright = (D, E), s = slef t sright = (A, C, B, D, E)
F.4.4

Final output

Ordering : s = slef t sright = (A, C, B, D, E), Violator edge set: {(D, C)}

A

F.5

C

B

D

E

Weighted case

Simpson et al. (2016) have proposed a modification to adapt the Eades et al. (1993)
algorithm to solve the weighted problem. The required changes are:
1. In the initialization step, all edge weights need to be normalized to be between 0 and
1.
2. δ(u) is redefined as δ(u) = bw− (u) − w+ (u)c.
The key motivation behind these steps is to reformat the network so that the unweighted
version of the algorithm could be used in an identical fashion as before, specifically without
increasing the number of buckets. Without the floor in step 2, for any given network thenumber of buckets could become large.

60

G

Algorithm to solve for the firm’s sourcing strategy
and export participation

A firm is solving the problem described in (25), where profits are defined in equation (24)and variable profits are defined in equation (13). For convenience, we re-state the problemof firm j here:max πj (Zj , IjF ) s.t. Zj ⊆ Zj , IjF ∈ {0, 1},

Zj ,IjF

whereE
1 σ−1 1−σ σ−1πj (Zj , IjF ) = βjHµ φj Θj (Zj )(σ−1)/(ρ−1) 1−σσP
1 σ−1 1−σ σ−1
EF+ IjF βjF µ φj Θj (Zj )(σ−1)/(ρ−1) τ 1−σ 1−σσPFX−
fkj w` − IjF fjF w` .
k∈Zj

=πjvar (Zj , IjF ) −X

fkj w` − IjF fjF w`k∈Zj

In words, the firm is choosing its sourcing strategy, Zj , and export participation, IjF . We
solve the firm’s problem of choosing its sourcing strategy separately for IjF = 0 and IjF = 1.
We then calculate the profits for these two cases and determine the firm is an exporter if
and only if the profits under exporting are higher than under non-exporting.
Below we describe how we solve for the firm’s optimal sourcing strategy for a given export
participation choice.

G.1

Lower and upper bounds for the optimal sourcing strategy

We determine the lower and upper bound for the firm’s sourcing strategy following theprocedure in Jia (2008) and Antras et al. (2017).
G.1.1

Lower bound

We start from a guess of no sourcing from any other domestic supplier and no importing,(0)
Sl . We then check supplier by supplier whether the benefit of adding a supplier (given thecurrent guess of not purchasing from any supplier) is positive. At iteration t, starting from(t)(t)
Sl , we calculate the marginal benefit of adding supplier k ∈/ Sl , k ∈ Zj :

61

(t)

(t)

πjvar (Sl ∪ k, IjF ) − πjvar (Sl , IjF ) − fkj w` .
If the marginal benefit to include supplier k is positive, in the next iteration we include
supplier k in the guess for the sourcing strategy of firm j. Note that given σ > ρ, one isthe least likely to determine the benefit of a supplier is positive when the current guess isno supplier. Hence if it is possible to include a supplier in this iteration, in all the next
iterations the marginal benefit from this supplier will be positive as well.
(t)(t)
Starting from Sl , we consider firm-by-firm if trading with a firm not contained in Sl
brings positive marginal benefit (i.e., the additional variable profits under this sourcing
strategy exceed the additional fixed cost) or not. Then, we add all those firms which bring(t+1)
positive benefit to form Sl
.
(t)
(t+1)
(t)
The process ends when Sl = Slor all eligible suppliers are in Sl already. When
(t)
(t+1)
the process ends (i.e., Sl = Sl), we denote the lower bound of the sourcing strategy for
(t)
(t+1)
∗
firm j as Sl = Sl = Sl.
G.1.2

Upper boundTo determine the upper bound we start from a guess of purchasing from every supplier(0)(incl. foreign), Su . We then check supplier-by-supplier whether the marginal benefit from(t)
dropping the supplier is positive. At iteration t, starting from Su , we calculate the marginal(t)
benefit of dropping supplier k ∈ Su as:

(t)

(t)

πjvar (Sl \ k, IjF ) − πjvar (Sl , IjF ) + fkj w` .
The remainder of the procedure is very similar to the iteration for the lower bound but(t+1)
starting from the opposite direction (i.e., we drop from the next iteration Su
all those
suppliers for which the marginal benefit of dropping is positive). The ending criteria is thesame. We denote the upper bound for the sourcing strategy as Su∗ .

G.2

From lower and upper bounds to optimal sourcing strategy

Once we obtain Su∗ and Sl∗ , we consider 3 alternative cases. Let D = {x ∈ Su∗ | x ∈/ Sl∗ }
denote the set with the elements that are in the upper bound but not in the lower boundfor the sourcing strategy.

62

G.2.1Su∗ = Sl∗If the upper and lower bounds for the sourcing strategy are the same, then we haveobtained the optimal sourcing strategy for the firm (for a given exporting choice).
G.2.2

Su∗ is close to Sl∗

When the cardinality of set D is less than or equal to 15, we consider Su∗ to be close toSl∗ .
In that case we evaluate the profits at all possible combinations of sourcing strategies in
between Su∗ and Sl∗ , including Su∗ and Sl∗ themselves. We choose the combination that yields
the highest total profit as the optimal sourcing strategy for the firm.
G.2.3

Su∗ is far from Sl∗

When the cardinality of set D is larger than 15, then evaluating the profits at all combinations of feasible sourcing strategies in between the two bounds would be too computationallyintensive. For that case, we have developed the following greedy algorithm to determine thefirm’s sourcing strategy:
Starting from Sl∗ , we calculate the marginal benefit from adding separately each supplierin D to the sourcing strategy Sl∗ . Note that by construction the marginal benefit from adding
each of these suppliers individually to Sl∗ is negative (otherwise the algorithm in Section G.1
would have already added those suppliers to the lower bound). We order the suppliers in
D by their marginal benefit of being added to Sl∗ . If the cardinality of D is K, we consider
K − 1 alternative sourcing strategies. We first add the top 2 suppliers in D (those with thehighest marginal benefit of being added evaluated at Sl∗ ) to Sl∗ , then add the top3 suppliers
to Sl∗ , and so forth. Hence, we calculate the profits for K − 1 alternative sourcing strategies.
In addition, we also follow a similar process starting from Su∗ and calculate the marginal
benefit from dropping separately each supplier in D from the sourcing strategy Su∗ . Again,
by construction, the benefit from dropping each of the suppliers individually is negative.
We order the suppliers in D by their marginal benefit of being dropped from Su∗ . We thenconsider K −1 alternative sourcing strategies, in which 2, 3, ..., K suppliers are removed fromSu∗ (following the ranking of their marginal benefit of dropping individually at Su∗ ).
Then, out of these 2K − 2 sourcing strategies we choose the one with the highest total
profit for the firm.
Note that, using the approach in Section G.2.2, the number of sourcing strategies wewould need to calculate profits for would be 2K (growing exponentially in K). The greedy
algorithm developed here, requires evaluations of alternative sourcing strategies that grow63

linearly in K, making it feasible even in the rare case that the difference between Su∗ and Sl∗
is large.
We present statistics on the cardinality of the differences in the bounds in Table 15.

G.3

Statistics on the algorithm
Table 15: Cardinality of differences in the upper and lower bounds

Number of firm draws
× parameter iterations
51,029,600,000

Bounds are
perfectly overlapping
98.95

Percent of cases in which
Differences in bounds Differences in bounds
≤ 15
> 15
0.91
0.14

Notes: During the estimation we have to solve for each firm and parameter guess the firm’s optimal sourcing
strategy and exporting choice. This table presents aggregate statistics on the cardinality of the differences in
the upper and lower bounds for the sourcing strategy summing over the outcomes for each firm, parameter
guess, and exporting choice.

64

H

Algorithm for network formation

Below we describe the algorithm to solve for the network formation in three contexts: In
Section H.1, we describe the algorithm to solve for the network formation and equilibrium
for a given set of parameters. In Section H.2, we describe the algorithm to estimate theparameters of the model. In Section H.3, we describe the algorithm to solve for network
formation and equilibrium in a closed economy.

H.1

Network formation given parameters

Given a set of parameters, size of the labor force, price of foreign goods, and foreign
demand, we follow the steps below to simulate the network formation.
1. Firms with productivities φi are randomly sorted, and indexed with i = 1, 2, 3, · · · .
A firm’s index determines the firm’s set of eligible suppliers, Zi . The set of eligible
suppliers is such that all feasible networks will be acyclic. Firms’ draws of firm-pairspecific fixed cost of sourcing, fixed cost of importing and exporting, export demand,and benefits of importing, and firm-pair-specific cost shifters are also known at thispoint.
2a. All firms make a common guess of the wage level w` .
2b. All firms make a common guess of aggregate demand term: A = EP σ−1 .
3. We assume that firms decide on their sourcing strategies in sequence of i. Firm 1 decides
its sourcing strategy and determines c1 , then firm 2 decides its sourcing strategy anddetermines c2 , and so on. When firms make their sourcing decisions, we assume that
all firms are able to use labor and foreign inputs, but firm i is only able to choose its
suppliers from its eligible supplier set Zi . We determine which suppliers among Zi firmi sources from, using the algorithm described in Section G, and compute ci . After the
final firm i = N decides its sourcing strategy, the whole vector c and the supplier setsof all firms are determined. At this point we have also solved for the firm’s optimalexport participation choice and export sales.
4. Given the network, the guesses for A and w, we are able to compute the equilibrium
variables.
1−σ 1−σσ(a) Sales to domestic final demand of firm i is computed as XiH = σ−1
ci A

1−σ
σ−1 EFσand to foreign final demand is computed as XiF = σ−1
ci1−σ βkF
.
P 1−σ
F

(b) The cost of inputs used for firm i’s sales to domestic final demand is thus CiH =σ−1
XiH and to foreign final demand is CiF = σ−1
XiF .
σ
σ
65

(c) The total input costs of firms, Ci are calculated by solving the system of linear
equations below:
Ci = CiH + CiF +X

sij Cj ,j→ C = (I − S)−1 (CH + CF )where C, CH , and CF are vectors of Ci , CiH , and CiF , respectively, and the i, jelement of matrix S is sij .
(d) The total sales of firm i is then Xi = XiH + XiF + Ci − CiH .
(e) Firm profits and total expenditure on fixed costs.
5. We solve for equilibrium variables of A and w` in the following way: In the outer loop,
we solve for wages such that the labor market clearing condition (26) is solved. In the
inner loop, we iterate over steps 2b-4, such that a fixed point for the market demandlevel, A, is found.

H.2

Parameter estimation and network formation

One possible approach to estimating the parameters of the model is to simulate themodel for each parameter guess according to the algorithm outlined in Section H.1, calculate
the objective function in equation (29), and vary the parameter guesses to maximize theobjective function. However, this requires for each parameter guess finding a fixed point inboth the market demand, A, and a wage level, w` . Below, we describe a more computationallyattractive algorithm to estimate the model.
Throughout the estimation, we set the domestic wage, w` = 1, as well as the domestic
market demand, A = 1. We ensure labor market clearing condition and the fixed point inmarket demand in the following way:
1. Of the 8 parameters to estimate, the mean foreign demand parameter is implicitly
pinned down to take the value that satisfies the trade balance condition.
2. Instead of treating the size of the labor force
as data (note that the units are arbitrary),PAP 1−σ − i πiwe choose its level by setting: L =.
w`
PAP 1−σ −πi iNote that A = w` = 1. Under this level of the size of the labor force, L =,w`
the fixed point for the market demand, A is automatically satisfied. Also, since the tradebalance holds, the domestic labor market clears as well.
fimp
fexp
fdom
f
dom
F
Given a parameter guess for Φ̂αscale, Φ̂αscale, Φ̂α,βdisp , Φ̂scale , Φ̂scale , Φ̂scale , and Φ̂disp , weFvary Φ̂βscaleand go through steps 3 and 4 in Section H.1 to calculate the aggregate trade66

Fbalance. Given the other parameters, the level of Φ̂βscaleis pinned down implicitly such that
aggregate trade balance is equal to zero. Hence, instead of search for a fixed point in both A
and w` , we hold those fixed throughout the estimation and only need to find one fixed pointFin Φ̂βscale, that satisfies trade balance, for each guess of the other seven parameters.

H.3

Network formation in the closed economy

In the closed economy, we can normalize the domestic wage w` = 1. We therefore only
have to follow steps 2b-4 in Section H.1 to pin down the level of domestic market demand,A, in the closed economy.

67

Sensitivity results under exogenous network

Density022

Density

4

4

6

6

Figure 10: Distributions of log ĉ from banning imports, different parameter values

0

I

2
4
Cost change (log scale), rho = 1.5
DirectRoundabout (sigma=2)
Roundabout (sigma=6)

6

8

2
4
Cost change (log scale), rho = 2

Total
Roundabout (sigma=4)

DirectRoundabout (sigma=2)
Roundabout (sigma=6)

6

8

Total
Roundabout (sigma=4)Table 16 shows the median value of cost change ĉi under different parameter values.
Table 16: Median ĉi under different values of σ and ρρ = 1.5
ρ=2

Direct

Total

1
1

2.89
1.70

Roundabout
σ=2 σ=4 σ=6
2.37
1.53
1.33
1.78
1.41
1.28

Table 17: 90th percentile ĉi under different values of σ and ρρ = 1.5
ρ=2

Direct

Total

1.08
1.04

11.93
3.45

68

Roundabout
σ=2 σ=4 σ=6
3.76
1.85
1.50
2.52
1.67
1.42

Table 18: Change in price index P̂ under different values of σ and ρ
ρ σ
1.5 2
1.5 4
1.5 6
2 2
2 4
2 6

P̂ |direct
1.71
1.28
1.17
1.48
1.23
1.15

P̂ |total
3.84
2.25
1.83
2.35
1.77
1.56

P̂ |roundabout
3.80
1.85
1.50
2.56
1.67
1.43

Table 19: Change in price index P̂ under acyclic network
ρ σ
1.5 2
1.5 4
1.5 6
2 2
2 4
2 6

P̂ |total
3.84
2.25
1.83
2.35
1.77
1.56

P̂ |total , acyclic
3.93
2.28
1.84
2.37
1.78
1.57

Notes: The fourth column shows the change in price index from banning imports when taking into accountthe acyclic network structure. We obtain the acyclic network from the algorithm explained in Appendix F, for
the weighted case where we minimize the value of violating transactions.

69