Multi-Product Firms and Trade Liberalization

Andrew B. Bernardy
Tuck School of Business at Dartmouth & NBER
Stephen J. Reddingz
London School of Economics, Yale & CEPR
Peter K. Schottx
Yale School of Management & NBER
First Version: December 2006
This Version: January 2009
Abstract
This paper develops a general equilibrium model of international trade that features selection
across …rms, products and countries. Firms’export decisions depend on a combination of …rm“productivity” and …rm-product-country “consumer tastes”, both of which are stochastic andunknown prior to the payment of a sunk cost of entry. Higher-productivity …rms export a wider
range of products to a larger set of countries than lower-productivity …rms. Trade liberalizationinduces endogenous reallocations of resources that foster productivity growth both within andacross …rms. Empirically, we …nd key implications of the model to be consistent with U.S. tradedata.
Keywords: heterogeneous …rms, endogenous product scope, love of variety, core competency
JEL classi…cation: F12, F13, L11

Bernard and Schott (SES-0241474) and Schott (SES-0550190) thank the National Science Foundation,
and Redding thanks Princeton University and the Centre for Economic Performance (CEP), for researchsupport. We thank Jim Davis from Census for timely disclosure, and Evan Gill, Justin Pierce and Yanhui
Wu for excellent research assistance. We thank Costas Arkolakis, Jonathan Eaton, Gordon Hanson, Marc
Melitz, Guy Michaels, Peter Neary, Henry Overman and Esteban Rossi-Hansberg for insightful comments.
The empirical research in this paper was conducted at the Boston, New York and Washington U.S.
Census Regional Data Centers. Any opinions, …ndings, and conclusions or recommendations expressedin this material are those of the authors and do not necessarily re‡ect the views of the National Science
Foundation or the U.S. Census Bureau. Results have been screened to insure that no con…dential data
are revealed.
y
100 Tuck Hall, Hanover, NH 03755, USA, tel: (603) 646-0302, fax: (603) 646-0995, email: andrew.b.bernard@dartmouth.eduzHoughton Street, London. WC2A 2AE UK. tel: (44 20) 7955-7483, fax: (44 20) 7831-1840, email:s.j.redding@lse.ac.uk
x
135 Prospect Street, New Haven, CT 06520, USA, tel: (203) 436-4260, fax: (203) 432-6974, email:peter.schott@yale.edu

Multi-Product Firms and Trade Liberalization

1.

2

Introduction

While multiple-product, multiple-destination …rms dominate international trade, comparatively little research examines their production and export decisions or how thesedecisions are in‡uenced by globalization. We develop a tractable, general equilibrium
model of endogenous product and destination-country selection that o¤ers a natural andintuitive explanation for key features of U.S. trade data. Our model and empirical analysis demonstrate the importance of exporters’product and country selection for aggregate
trade ‡ows as well as for …rm- and industry-level productivity growth in response to tradeliberalization.
We emphasize a distinction between three “extensive”margins of trade –the number ofexporting …rms, the number of exported products and the number of export destinations –and an “intensive”margin of average exports per …rm-product-destination. Our empiricalevidence reveals that these extensive margins account for much of the observed variationin export shipments across countries and …rms, and that they also behave quite di¤erently
from the intensive margin. We show that the well-known negative e¤ect of distance onbilateral trade ‡ows, for example, is due primarily to extensive margins: while the numberof exporting …rms and the number of exported products decline signi…cantly with distance,
the relationship between distance and the intensive margin is, if anything, positive. In
our model, variation in extensive margins is driven by selection, with the most able …rms
exporting more products to more destinations. As trade costs fall, the weakest …rms exit
and, within surviving …rms, the least-pro…table products are dropped. These reactions
induce revenue-based productivity growth both across and within …rms, with the latterproviding an additional source of welfare gains from trade. The model thus provides anintuitive micro-foundation for the much-discussed idea that international trade spurs …rms
to rationalize production, but one that does not imply money being left on the table prior
to liberalization.
Our theoretical setup is straightforward. Following the closed-economy model ofBernard, Redding and Schott (2009), we allow …rms to produce multiple products ofvarying pro…tability. We assume that …rm pro…tability in a particular product increases
with two stochastic and independent draws. The …rst, for …rm “productivity”, governs
the amount of labor that must be used to produce a unit of output. The second, for…rm-product-country “consumer tastes”, regulates demand for a …rm’s product in a particular destination country. We assume both draws are revealed to …rms after incurring
a sunk cost of entry. If …rms decide to enter after having observed these draws, they face…xed and variable costs for each good they choose to supply to a market as well as a …xed
cost of serving each market that is independent of the number of goods supplied. On
the demand side, we assume consumers possess Dixit-Stiglitz (1977) constant elasticity of
substitution preferences.
The model yields several distinctive predictions that …nd support in U.S. trade data.

Multi-Product Firms and Trade Liberalization

3

The …rst is that trade liberalization induces …rms to shift resources towards their relatively
high-pro…t (i.e., “core-competency”) products.1 As in the Melitz (2003) model, our model
has falling trade costs heightening competition and causing the lowest-productivity …rms
to exit. In our model, a similar reallocation also occurs within …rms, as greater competition induces all surviving …rms to drop their least-pro…table, domestically sold products.
Evidence in favor of this reallocation is found in several existing studies of …rm diversi…cation during trade liberalization. Baldwin and Gu (2009), for example, …nd a sharp
decline in product diversi…cation among Canadian …rms following the tari¤ reductions
mandated by the Canada-U.S. Free Trade Agreement (CUSFTA). Here, we provide evidence of a similar retrenchment among U.S. manufacturing …rms in response to reductions
in Canadian tari¤s on U.S. imports.
A second set of predictions relates to the response of the extensive and intensive margins of trade to changes in variable trade costs. In our model, increases in the variable
trade costs of serving a particular destination country decrease the country-level extensive margins of exports, i.e., the number of exporting …rms and the number of exported
products. The e¤ect of rising trade costs on the intensive margin of average exports per…rm-product-country, in contrast, is ambiguous. On the one hand, higher variable trade
costs decrease a given …rm’s exports of a given product to a given destination. On theother hand, higher variable trade costs induce …rms to export fewer products and serve
fewer export destinations with lower values of consumer tastes, raising average exports
per …rm-product-destination. We investigate the empirical relevance of these predictions
cross-sectionally, examining how U.S. exports vary across trading partners using partners’ distance from the United States as a proxy for variable trade costs. In line withthe model’s predictions, we …nd that an increase in distance has a negative and statistically signi…cant association with the extensive margins and a positive but statisticallyinsigni…cant e¤ect on the intensive margin.
A third set of predictions relates …rm productivity to …rms’ extensive margins, i.e.,
the number of products …rms export and the number of countries to which they export.
In the model, more productive …rms have more active extensive margins because they areable to earn higher revenue and therefore cover the …xed costs of exporting at lower levelsof consumer demand. Here, too, there is ambiguity with respect to the intensive margin:
while higher productivity raises a …rm’s exports of a given product to a given destination,it also extends their reach into additional products and destinations with lower valuesof consumer tastes and therefore lowers exports. We evaluate this implication of themodel using cross-sectional data on U.S. …rm exports. We …nd a positive and statisticallysigni…cant relationship between …rms’ estimated productivity and both the number of
products they export and the number of export destinations they serve. In contrast, we
1

The concept of core competency appears in both the theory of the …rm and the business strategy
literature (see for example Porter 1998 and Sutton 2005). As formalized here, core competency refers to…rms’most pro…table activity.

Multi-Product Firms and Trade Liberalization4…nd that average exports per product and destination are generally ‡at with respect to…rm productivity.
Our …nal set of empirical results demonstrate the model’s usefulness for understanding
how the extensive margins of trade contribute to the extreme inequality of U.S. exports
across exporting …rms and destinations. We compare the actual distributions of exports
across …rms and countries to hypothetical distributions that successively eliminate thein‡uence of extensive margins. In both cases, we …nd substantial changes in the patternof exports, with exports considerably less concentrated across …rms and countries in the
hypothetical distributions than in the actual distributions.
Our analysis contributes to three literatures. First, we extend the in‡uential heterogeneous…rm model of Melitz (2003) to incorporate multiple-product, multiple-destination …rms
as well as heterogeneity within …rms. Our e¤ort is related to other generalizations of theMelitz framework, including Eaton, Kortum and Kramarz (2008), who investigate the
distribution of export shipments across countries, Chaney (2008), who analyzes the relationship between trade costs, the elasticity of substitution and trade ‡ows, and Bernard,
Redding and Schott (2007) and Melitz and Ottaviano (2008a), who study the behavior ofheterogeneous …rms in settings incorporating endowment-driven comparative advantage
and market power, respectively.
Second, our paper is related to the closed-economy industrial organization literature on
multiple-product …rms. Early research in this literature, reviewed by Bailey and Friedlaender (1982), emphasizes supply-side economies of scope. Subsequent analysis, by Brander
and Eaton (1984), Shaked and Sutton (1990), Eaton and Schmidt (1994) and Johnson
and Myatt (2003), introduces demand-side forces favoring production of multiple goods
as well as strategic interaction between …rms. More recently, Klette and Kortum (2004)
and Lentz and Mortenson (2005) have developed innovation-based models of multipleproduct …rms, while Bernard, Redding and Schott (2009) develop a model emphasizing…rm heterogeneity and selection.
Finally, our paper relates to more recent open-economy models of multiple-product…rms that have appeared in the international trade literature. One strand of this research,e.g., Eckel and Neary (2006), focuses on strategic interaction between a …xed number ofsymmetric multiple-product …rms. Another, which includes Agur (2006) and Feenstra andMa (2008), examines the relationship between trade liberalization and the proliferationof product variety with multiple-product …rms. Several other contributions vary in termsof how …rms’ extensive margins are determined. Nocke and Yeaple (2006) consider amodel in which …rms with higher organizational capability produce a larger number of
products but where the devotion of this capability to larger numbers of goods raises the(common) cost of producing every product. Melitz and Ottaviano (2008b) entertain asetting where …rms choose endogenous ranges of varieties within products subject to aladder of progressively higher costs as they move away from their ideal variety. Arkolakis

Multi-Product Firms and Trade Liberalization

5and Muendler (2008) model an environment in which …rms supply endogenous rangesof varieties subject to market entry costs that are convex in the number of products
exported to a destination. Here, we seek a tractable general equilibrium framework formodeling the distribution of exports across …rms, products and destinations that does
not impose speci…c functional forms on probability distributions and which incorporates
demand heterogeneity across countries within …rms and products.
The remainder of the paper is structured as follows. Section 2 provides motivation
for key assumptions of the model. Section 3 develops the model and solves for generalequilibrium. Section 4 examines the comparative statics of trade liberalization and theproperties of the open-economy equilibrium. Section 5 provides evidence of the model’sconsistency with key features of U.S. trade data. Section 6 concludes. Appendices A
through C contain data and theoretical appendices.
2.

Motivation

Key attributes of the data that we seek to incorporate in our model are variation inexports across …rms, variation in exports across products within …rms, and variation inexports across countries within …rm-products. In our model we interpret this variation asarising from an interaction of productivity that varies across …rms and consumer tastes
that vary across …rm-product-country combinations. In this section we document sourcesof variation in U.S. exports across …rms, products and countries and also show that U.S.
…rms’export markets do not appear to follow a strict “hierarchy”. The data used here
are described in greater detail in Section 5 and Appendix A.
We evaluate the importance of …rm, product and destination-country attributes in
U.S. exports by regressing the logarithm of …rm-product-country exports on three sets of…xed e¤ects and comparing their explanatory power. Table 1 reports results using datafor 2002, but we note that we obtain similar results across the years spanned by our data.
Our …rst step is to use country-product …xed e¤ects to sweep out heterogeneity acrosscountries and products that is common to all …rms, such as comparative advantage, trade
costs or markets’and products’overall size. In our regressions, products correspond toten-digit Harmonized System categories. As noted in the …rst row of the table, countryproduct …xed e¤ects explain 26 percent of the variation in log exports.
The second row of the table investigates the importance of …rm attributes via theaddition of …rm …xed e¤ects. Addition of these …xed e¤ects increases the R2 of the modelto 43 percent, indicating that …rm-level factors that are common across products andcountries explain roughly 23 percent (17/74) of the remaining variation in log exports.
This relative importance of …rm factors motivates our use of a …rm “productivity”draw.
The …nal regression, reported in the third row of Table 1, uses …rm-product rather than…rm …xed e¤ects. This regression allows for product variation within …rms that is constantacross countries. The R2 of 89 percent highlights the importance of product selection

Multi-Product Firms and Trade Liberalization

6

Table 1: Common and Idiosyncratic Components of Firm-Product Shipments
Cumulative RSquared
Fixed Effects
Country-Product
0.26Country-Product + Firm
0.43
Country-Product + Firm-Product0.892Notes: Table reports R s of a sequence of OLS
regressions of the logarithm of U.S. firm-productcountry exports in 2002 on the fixed effects noted ineach row.

within …rms in explaining U.S. export ‡ows. At the same time, it reveals that roughly
15 percent (11/74) of the variation in U.S. exports that remains after including countryproduct …xed e¤ects is due to idiosyncratic heterogeneity across countries within …rmproducts.2 This …nding motivates our use of a stochastic …rm-product-country “taste”
draw.3Further evidence of idiosyncratic heterogeneity across countries within …rm-products,
which we refer to as “…rm-product-country heterogeneity”, comes from considering hierarchies of export markets. If …rm-product-country heterogeneity were unimportant,…rms’products would exhibit a strict hierarchy across destinations, i.e., the markets inwhich their less pro…table products appear would be subsets of the markets in which theirmost pro…table product appears, and rankings of …rms’products by size would be identical across all of the markets in which they jointly appear.4 We examine each of these
implications in turn.
First, for each multiple-product …rm in the U.S. export data in 2002, we compare
the markets to which it exports its “largest” (i.e., in the model, its most pro…table)
product to the markets to which it exports all of its other (“smaller”) products. Absent…rm-product-country heterogeneity, the destinations to which …rms export their smaller
products should be subsets of the destinations to which they export their largest product.
Empirically, we de…ne a …rm’s largest product as the one it sends to the largest numberof export destinations, with ties going to the product with the greatest export value. Let
nkf be the number of export destinations for smaller product k in …rm f and jkf be thenumber of destinations smaller product k has in common with the …rm’s largest product.
Then N estedkf = jkf =nkf represents the share of “smaller”product k’s markets that are2

As might be expected, the explanatory power of …rm and …rm-product …xed e¤ects declines whensingle-export-product …rms are excluded. Restricting the sample to …rms that export ten or more products, for example, yields cumulative R2 s of 0.28, 0.37 and 0.79, respectively.
3
In the model developed below, …rm-product-country exports are log linear in the …rm “productivity” draw and the …rm-product-country “taste” draw, which is consistent with the log linear empirical
speci…cation employed here.
4
In the model developed below, a strict hierarchy of markets is observed in the absence of …rm-productcountry heterogeneity in “taste.”Multi-Product Firms and Trade Liberalization

7

common to the …rm’s largest product. If …rms’products exhibit a strict hierarchy, one
would expect N estedkf = 1. Instead, we …nd a mean value of 0.67 across the 151,204…rm-product observations in our sample, which bootstrap standard errors reveal to besigni…cantly di¤erent from unity.5
Examination of bilateral correlations of country rankings across “smaller” products
within …rms conveys a similar message. Preparing the data in the same manner justdescribed, we compute the rank correlations of all …rms’smaller products with their largest
product across all of the destinations to which both products are jointly exported. Absent…rm-product-country heterogeneity, all bilateral correlations would be unity. Instead, we…nd an average correlation of 0.17 across all …rm-product combinations, with 38 percent of
the bilateral correlations being both positive and statistically signi…cant and an additional
24 percent being positive but not signi…cant. This …nding of imperfectly correlated within…rm product rankings across destination markets is consistent Munch and Nguyen’s (2008)
analogous demonstration of imperfectly correlated …rm rankings across export markets in
Danish export data.
3.

The Model

We consider a world consisting of many countries and many products. Firms decide whether to produce, what products to make, and where to export these products.
Products are imperfect substitutes in demand, and within each product …rms supply horizontally di¤erentiated varieties. The model can therefore be interpreted as capturing an
economy consisting of many products or as capturing an industry (e.g. consumer electronics) with many products (e.g. DVD-players, Televisions and MP3-players) of which …rms
supply distinct varieties (e.g. Sony, JVC and Panasonic).6 For simplicity, we develop
the model for the case in which products are symmetric and the world consists of n + 1
symmetric countries.7
3.1. Preferences and Endowments
The world consists of a home country and a continuum of n foreign countries, each ofwhich is endowed with Ln units of labor that are supplied inelastically with zero disutility.
The representative consumer in each country derives utility from the consumption ofa continuum of products that we normalize to the interval [0; 1]. There is a constant5

Averaging N estedkf …rst across products within …rms and then across …rms yields a value of 0.59,
which bootstrapping also reveals is signi…cantly di¤erent from unity. Results are similar if restricted toa sample of …rms exporting ten or more products.
6
We develop a multi-industry version of the model in a web-based technical appendix.
7
While the model does not in general have a closed form solution with asymmetric countries andproducts, it is possible to characterize the equilibrium numerically if speci…c functional forms for …rm
productivity and consumer tastes are assumed. With Pareto distributions for …rm productivity andconsumer tastes, a number of analytical results are possible for asymmetric countries.

Multi-Product Firms and Trade Liberalization

8

elasticity of substitution across products so that the utility function for the representative
consumer in country j takes the standard Dixit-Stiglitz (1977) form:
Uj =

Z 1

1

Cjk dk

;

0<

(1)

< 1;

0

where k indexes products. Within each product, a continuum of …rms produce horizontallydi¤erentiated varieties of the product. Hence Cjk is a consumption index, which also takes
the constant elasticity of substitution form, and depends on the varieties consumed fromeach country in the world:
"Z
#1
Z
n+1

Cjk =

;

[ ijk (!) cijk (!)] d!di

!2

0

0<

< 1;

(2)

ijk

where i and j index countries, ! indexes varieties of product k supplied from country i tocountry j and ijk denotes the endogenous set of these varieties.8 The demand parameter0 captures the strength of the representative consumer’s tastes for …rm varietyijk (!)! and re‡ects demand heterogeneity, as discussed further below. While not central to
our results, we make the natural assumption that the elasticity of substitution acrossvarieties within products is greater than the elasticity of substitution across products:
1
1>> 1. Similarly, we assume for simplicity that the elasticity of1
11
substitution across varieties within products,, is the same for all products. The
1corresponding price index dual to (2) is:
Pjk =

"Z

0

n+1 Z

!2

ijk

pijk (!)
ijk (!)

1

d!di

#11

:

(3)

As products are symmetric, we suppress from here forward the implicit dependence on
products k. Furthermore, countries are also symmetric, and the only di¤erence between
the domestic market and each export market is that a common value of trade costs has
to be incurred for each export market. Therefore, instead of indexing variables in termsof country of production, i, and market of consumption, j, we distinguish between thedomestic market, d, and each export market, x, except where otherwise indicated.
3.2. Production Technology
The speci…cation of entry and production follows Melitz (2003). However, we augmentthat model to allow …rms to manufacture multiple products and to allow for demandheterogeneity across products and countries. There is a competitive fringe of potential
8

Our model focuses on …rms that supply multiple products for …nal consumption. While vertical
integration provides another reason why …rms can produce multiple products (intermediate and …nal),
many …rms supply multiple products for …nal consumption.

Multi-Product Firms and Trade Liberalization

9…rms who are identical prior to entry. In order to enter, …rms must incur a sunk entry costof fei > 0 units of labor in country i. Incurring the sunk entry cost creates a …rm brand anda blueprint for one horizontally di¤erentiated variety of each product that can be produced
using this brand.9 Firms are de…ned by their production technology (productivity ') andproduct attributes that in‡uence demand (consumer tastes ). Both productivity andconsumer tastes are uncertain prior to entry. Thus, only once the sunk entry cost has
been incurred does a …rm observe its productivity, ', and consumer tastes for each of its
products k in each of the countries j, jk .10
Firm productivity, ' 2 [0; 1), is common across all of a …rm’s products and is drawn
from a continuous distribution g (') with cumulative distribution function G ('). In contrast, consumer tastes for a …rm’s varieties, jk 2 [0; 1), vary across both products k
and countries j, and are drawn separately for each product and country from a continuous distribution z ( ) with cumulative distribution function Z ( ). This conceptualization
captures the idea that some …rm characteristics (productivity) enhance pro…tability for
the …rm as a whole, while other …rm characteristics (product attributes) a¤ect pro…tability
unevenly across both products and countries. Our formulation allows for demand heterogeneity across countries for a given product, as well as demand heterogeneity acrossproducts for a given country.11
To make use of law of large numbers results, we make the simplifying assumption thatthe productivity and consumer tastes distributions are independent across …rms. For thesame reason, we also assume that the productivity and consumer tastes distributions are
independent of one another and that the consumer tastes distributions are independent
across products and countries. While these simplifying assumptions provide a tractable
way to introduce heterogeneity across products and countries within …rms, it is straightforward to extend the analysis to introduce dependence in consumer tastes. For example,
the model can be augmented with a common component of consumer tastes, which varies
across products within …rms but is the same across countries. This common component ofconsumer tastes plays a similar role in the model to …rm productivity, which is the sameacross both countries and products within …rms.12 Similarly, the model can be augmented
9

Our framework can be also extended to allow for endogenous measures of varieties within eachproduct, introducing the potential for strategic interaction since …rms are no longer of measure zero
relative to each product.
10
As the focus of our analysis is the cross-section distribution of exports across …rms, products andcountries, we abstract from stochastic variation over time in …rm productivity and consumer tastes, andhence from steady-state adding and dropping of products and destinations. However, the model can beextended to capture these dynamics by incorporating features from the closed-economy model of Bernard,
Redding and Schott (2008), as shown in the web-based technical appendix.
11
The three features that are important for our analysis are (a) variation in pro…tability across products
within …rms, (b) variation in pro…tability across countries within products within …rms, and (c) variationin average pro…tability across …rms. While our formulation captures these features in an intuitive andtractable way, one could also generate idiosyncratic variation in …rm-product-country pro…tability frominteractions between …rm, product and country characteristics.
12
See the web-based technical appendix for further discussion of this extension. A special case of ourMulti-Product Firms and Trade Liberalization

10

with common components of consumer tastes for groups of related products within …rms.
More generally, explicit correlation in consumer tastes draws can be allowed, although at
the cost of reduced analytical tractability.
While these extensions complicate the analysis, they do not alter the model’s central
mechanism of selection within …rms, which is driven by heterogeneity across products
within …rms. Even with our simplifying assumptions, a …rm’s pro…tability is correlated
across products and countries, because higher productivity raises a …rm’s pro…tability
across all products and countries. These correlations are, however, imperfect because of
the stochastic variation in consumer tastes.
Once the sunk cost has been incurred, and productivity and consumer tastes are observed, …rms decide whether to enter and what products to make. Labor is the sole factor
of production.13 We assume that …rms face …xed costs of supplying each market, which
are Fx > 0 for each foreign market and Fd > 0 for the domestic market. These marketspeci…c …xed costs capture among other things the costs of building distribution networks.
In addition, we assume that …rms face …xed costs of supplying each product to a market,
which are fx > 0 for each foreign market and fd > 0 for the domestic market. These
product and market-speci…c …xed costs capture the costs of market research, advertisingand conforming to foreign regulatory standards for each product. As more products aresupplied to a market, total …xed costs rise, but average …xed costs fall, since the …xed
cost of serving each market is spread over a larger number of products.
In addition to these …xed costs, there is also a constant marginal cost of production
for each product that depends on …rm productivity, such that qjk ('; jk ) =' units of labor
are required to produce qjk ('; jk ) units of output of product k for market j. Finally, weallow for variable costs of trade, such as transportation costs, which take the standard“iceberg” form, whereby a fraction > 1 of a variety must be shipped in order for one
unit to arrive in a foreign country.14 We note that to simplify the characterization of a…rm’s decision to a supply a market, we have set …xed production costs to zero and haveassumed instead a …xed cost of supplying each market.15 For simplicity, we have also
assumed that the …xed costs of serving each market are incurred in terms of labor in themodel is where there is a single …rm-product draw that is the same for all countries, which corresponds
to perfect correlation of consumer tastes across countries.
13
The multiple-industry version of the model considered in the web-based technical appendix also introduces multiple factors of production, which gives rise to Heckscher-Ohlin based comparative advantage,as in the single-product heterogeneous-…rm model of Bernard, Redding and Schott (2007).
14
For evidence on the magnitude of overall trade costs, see Anderson and van Wincoop (2004) andHummels (2001). For evidence on the …xed costs of exporting, see Roberts and Tybout (1997), Bernard
and Jensen (2004), Eaton, Kortum and Kramarz (2008).
15
In our setting with demand heterogeneity across countries, a …rm can choose to export a productbut not supply it domestically, as found empirically by Iacovone and Javorcik (2008). Setting …xed
production costs equal to zero simpli…es the …rm’s problem, because each market can be consideredseparately. While introducing …xed production costs is straightforward, as shown in Section B3. of theappendix, it complicates the analysis by introducing interdependence across markets.

Multi-Product Firms and Trade Liberalization

11

country of production, although it is straightforward to instead consider the case where
they are incurred in the market supplied.
3.3. Firm-Product Pro…tability
Demand for a product variety depends on the own-variety price, the price index forthe product and the price indices for all other products. If a …rm is active in a productmarket, it manufactures one of a continuum of varieties and so is unable to in‡uence theprice index for the product. At the same time, the price of …rm’s variety in one productmarket only in‡uences the demand for its varieties in other product markets through theprice indices. Therefore, the …rm’s inability to in‡uence the price indices implies thatits pro…t maximization problem reduces to choosing the price of each product variety
separately to maximize the pro…ts derived from that product variety.16 This optimization
problem yields the standard result that the equilibrium price of a product variety is aconstant mark-up over marginal cost:
1w;(4)
px ('; x ) = pd ('; d ) ='where equilibrium prices in the export market are a constant multiple of those in thedomestic market due to the variable costs of trade; d varies across products with idiosyncratic realizations of consumer tastes; for the same reason x varies across both products
and export markets. We choose the wage in one country as the numeraire, which together
with country symmetry implies w = 1 for all countries.
Substituting for the pricing rule, equilibrium revenue in each export market and thedomestic market are respectively:
1rx (';x) =1x

rd (';d) ;rd (';d) = E (P ' d)

1

;

(5)

d

where E denotes aggregate expenditure on a product in each market. The equilibrium
pro…ts from a product in each export market and the domestic market are therefore:
x (';

x) =

rx (';

x)

fx ;

d (';

d) =

rd (';

d)

fd :

(6)

From equations (5) and (6), …rm productivity enters the equilibrium revenue andpro…t functions in exactly the same way as consumer tastes, because prices are a constantmark-up over marginal costs and demand exhibits a constant elasticity of substitution.17
16

The structure of our model eliminates strategic interaction within or between …rms. This choice ofmodel structure enables us to focus purely on the implications of introducing multi-product …rms into amodel of industry equilibrium with ongoing entry, exit and …rm heterogeneity.
17
Therefore di¤erences in consumer tastes have the same e¤ect on equilibrium …rm revenue as di¤erences in productivity. As a result, has an equivalent interpretation as a component of …rm productivity
that is speci…c to products and countries. While consumer tastes are perhaps a more natural source of heterogeneity across countries, the determination of general equilibrium under this alternative interpretation
remains unchanged.

Multi-Product Firms and Trade Liberalization

12

Additionally, equation (5) implies that the relative revenue from two varieties of the same
product within a given market depends solely on relative productivity and consumer
tastes:00

rij ('00 ;) = ('00 ='0 )1( 00 = 0 )1

rij ('0 ; 0 ) :(7)

Similarly, as countries are symmetric, equation (5) implies that the relative revenue derived from two varieties of the same product with the same values of productivity andconsumer tastes in the export and domestic markets depends solely on variable trade
costs: rx ('; 0 ) =rd ('; 0 ) = 1 .
A …rm with a given productivity ' and consumer taste draw decides whether or not
to supply a product to a market based on a comparison of revenue and …xed costs forthe product. For each …rm productivity ', there is a zero-pro…t cuto¤ for consumer
tastes for the domestic market, d ('), such that a …rm supplies the product domestically
if it draws a value of d equal to or greater than d ('). This value of d (') is de…ned by
the following zero-pro…t condition:rd (';

d (')) =

fd :

(8)

Using this product zero-pro…t cuto¤ condition for each …rm productivity (8) together withrelative variety revenues in (7), d (') can be expressed relative to its value for the lowest
productivity supplier, d ('d ):d (') ='d'd ('d ) ;(9)

where 'd is the lowest productivity at which a …rm supplies the domestic market. As a…rm’s own productivity increases, its zero-pro…t cuto¤ for consumer tastes falls because
higher productivity ensures that su¢ cient revenue to cover product …xed costs is generated
at a lower value of consumer tastes. In contrast, an increase in the lowest productivity at
which a …rm supplies the domestic market, 'd , or an increase in the zero-pro…t consumer
tastes cuto¤ for the lowest productivity supplier, d ('d ), raises a …rm’s own zero-pro…t
consumer tastes cuto¤. The reason is that an increase in either 'd or d ('d ) enhances
the attractiveness of rival …rms’products, which intensi…es product market competition,and hence increases the value for consumer tastes at which su¢ cient revenue is generated
to cover product …xed costs.
Following a similar line of reasoning, there is an exporting cuto¤ for consumer
tastes x (') for each …rm productivity ', such that the …rm will export the product if
it draws a value of x equal to or greater than x ('). This value of x (') is de…ned byan analogous zero-pro…t condition:
rx (';

x (')) =

fx :

(10)

Multi-Product Firms and Trade Liberalization

13

Using this product exporting condition for each …rm productivity together with relative
variety revenues in (7), the product exporting cuto¤ for each …rm productivity, x ('), canbe expressed relative to the product exporting cuto¤ for the lowest productivity exporter,x ('x ):
x (') ='x'

x ('x ) ;(11)

where 'x is the lowest productivity at which a …rm exports. Following a parallel logic
as above, a …rm’s product exporting cuto¤ is increasing in its own productivity but isdecreasing in the lowest productivity at which a …rm exports, 'x , and is decreasing in the
product exporting cuto¤ of the lowest productivity exporter, x ('x ).
As consumer tastes are independently and identically distributed across the unit continuum of products, the law of large numbers implies that the fraction of products supplied
to the domestic market by a …rm with a given productivity ' equals the probability ofdrawing a consumer taste above d ('), that is [1 Z ( d ('))]. Similarly, the fraction of
products exported by a …rm with a given productivity ' equals the probability of drawing
a consumer taste above x ('), that is [1 Z ( x ('))]. Finally, as consumer tastes arealso independently and identically distributed across the continuum of countries, the lawof large numbers implies that the fraction of foreign countries to which a given productis exported equals [1 Z ( x ('))].
The relative values of the exporting and zero-pro…t cuto¤s for consumer tastes depend
on …xed and variable trade costs. From relative revenue in the domestic and export
markets (5), the zero-pro…t cuto¤ condition (8) and the exporting cuto¤ condition (10),1(fx =fd ) > 1.
x (') > d (') for values of …xed and variable trade costs for which
For parameter values satisfying this inequality, there is probabilistic selection into export
markets within …rms: a …rm is more likely to supply a product domestically than toexport the product. Nevertheless heterogeneity in demand across countries implies thateven for parameter values satisfying this inequality, a su¢ ciently high consumer taste
draw in an export market, combined with a su¢ ciently low consumer taste draw in thedomestic market, can induce a …rm to export the product but not supply the product tothe domestic market.18
3.4. Firm Pro…tability
Having examined equilibrium revenue and pro…ts from each product, we now turn tothe …rm’s equilibrium revenue and pro…ts across the continuum of products as a whole.
As consumer tastes are independently distributed across the unit continuum of symmetricproducts, the law of large numbers implies that a …rm’s expected revenue across the unit
18

For empirical evidence of …rms exporting products not supplied to the domestic market, see Iacovone
and Javorcik (2008).

Multi-Product Firms and Trade Liberalization

14

continuum of products equals its expected revenue for each product. Expected revenue
for each product is a function of …rm productivity ' and equals the probability of drawing a consumer taste above the cuto¤ times expected revenue conditional on supplying
the product. Therefore total …rm revenue across the unit continuum of products in thedomestic and export markets is:
Z 1
rj ('; j ) z ( j ) d j ;j 2 fd; xg :rj (') =j (')

Similarly, total pro…ts in the domestic and export markets equal expected pro…ts fromeach product minus the relevant …xed costs:
Z 1
rj ('; j )fj z ( j ) d j Fj
j 2 fd; xg :(12)j (') =
j (')

Consider the relationship between expected pro…ts from each product and …rm productivity in (12). Equilibrium revenue from each product within the domestic market, rd ('; d ),is increasing in …rm productivity and consumer tastes. Hence the lower a …rm’s productivity, ', the higher its zero-pro…t consumer tastes cuto¤, d ('), and the lower its probabilityof drawing a consumer tastes high enough for a product to be pro…table. Therefore …rms
with lower productivities have lower expected pro…ts from individual products and supply
a smaller fraction of products to the domestic market, [1 Z ( d ('))]. For su¢ ciently low…rm productivity, the excess of domestic market revenue over product …xed costs in thesmall range of pro…table products falls short of the …xed cost of supplying the domestic
market, Fd . The requirement that total pro…ts in the domestic market are greater thanor equal to zero therefore de…nes a zero-pro…t cuto¤ for …rm productivity, 'd , such that
only …rms that draw a productivity equal to or greater than 'd enter:(13)('d ) = 0:

Combining this …rm zero-pro…t condition with the product zero-pro…t condition (8), we
can determine the zero-pro…t cuto¤ for consumer tastes for the lowest productivity …rm,d ('d ). Using these two zero-pro…t conditions together with total …rm pro…ts (12) andrelative variety revenues (7), we obtain:
"
#
Z
1

d

('d )

1

d

d ('d )

1 fd z ( d ) d d = Fd ;(14)which determines a unique equilibrium value of d ('d ) as a function of the …xed cost of
supplying the domestic market, Fd , and other parameters.
A directly analogous line of reasoning applies for each export market. For su¢ ciently
low …rm productivity, the excess of export market revenue over product …xed exporting
costs in the small range of pro…table products falls short of the …xed costs of serving a

Multi-Product Firms and Trade Liberalization15

foreign destination, Fx . The requirement that total …rm pro…ts from an export market are
greater than or equal to zero therefore de…nes an exporting cuto¤ for …rm productivity,'x , such that only …rms that draw a productivity equal to or greater than 'x export toeach foreign destination:(15)

x ('x ) = 0:

Combining this …rm exporting condition with the product exporting condition (10), we candetermine the exporting cuto¤ for consumer tastes for the lowest productivity exporter,x ('x ). Using these two zero-pro…t conditions together with total …rm pro…ts from theexport market (12) and relative variety revenues (7), we obtain:#"Z1

x ('x )

1x

x ('x )

(16)

1 fx z ( x ) d x = Fx ;which determines a unique equilibrium value of x ('x ) as a function of the …xed cost ofserving an export market, Fx , and other parameters.
The value of the exporting productivity cuto¤, 'x , relative to the zero-pro…t productivity cuto¤, 'd , depends on …xed and variable trade costs. From relative revenue in thedomestic and export markets (5), relative revenue within the same market for di¤erent
values of productivity and consumer tastes (7), the product zero-pro…t cuto¤ condition(8) and the product exporting cuto¤ condition (10), the exporting and zero-pro…t productivity cuto¤s are related as follows:'x = 'd ;

fx
fd

1
1

d ('d )x ('x );

(17)

where d ('d ) and x ('x ) were determined as a function of the …xed cost of serving thedomestic market, Fd , and the …xed cost of serving an export market, Fx , above.
For values of …xed and variable trade costs in (17) for which > 1, there is selection
into export markets across …rms, where only the most productive …rms export and lowerproductivity …rms serve only the domestic market.19 Therefore, although idiosyncraticdi¤erences in product demand across countries can induce a …rm to export a productbut not supply it to the domestic market, the law of large numbers implies that theseidiosyncratic di¤erences in demand average out across products. As a result, for theparameter values satisfying the inequality in (17), no …rm ever serves an export market
without also serving the domestic market.
As for simplicity we have developed the model for the case of symmetric countries,
the exporting productivity cuto¤ is the same for each export market. Therefore each…rm either serves only the domestic market or both the domestic market and all export
19

A large empirical literature …nds evidence of selection into export markets across …rms, e.g., Bernard
and Jensen (1995, 1999) and Roberts and Tybout (1997).

Multi-Product Firms and Trade Liberalization

16

markets. Clearly, introducing country asymmetries would result in di¤erent exporting
productivity cuto¤s for each export market, so that more productive …rms would export
to more destinations. Nonetheless, even with symmetric countries, the number of export
markets to which a particular product is exported varies with …rm productivity. Moreproductive …rms have lower consumer-taste cuto¤s and therefore export a given productto more countries.
3.5. Free Entry
Having examined a …rm’s ex post decision whether to supply markets, we now consider
the …rm’s ex ante decision whether to incur the sunk cost of entry. Firms from thecompetitive fringe decide whether or not to enter based on a comparison of the expectedvalue of entry and the sunk entry cost. We assume that there is a constant exogenous
probability of …rm death, , due to force majeure events beyond the …rm’s control, which
generates ongoing …rm entry and exit. Therefore the expected value of entry, Vi , is theex ante probability of successful entry multiplied by average …rm pro…ts conditional on
entry, i , and discounted by the probability of …rm death. Free entry requires:V =[1

G ('d )]= fe ;(18)

where the ex ante probability of successful entry is [1 G ('d )]. Expected pro…ts conditional on entry, i , equal expected pro…ts in the domestic market plus the probabilityof exporting times the number of foreign markets times expected pro…ts in each export
market:
Z 1
Z 1
g (')
1 G ('x )
g (')
=
d' +
n
d':
d (')
x (')
1 G ('d )1 G ('d ) 'x
1 G ('x )'dUsing total …rm pro…ts in the domestic and export markets (12), product pro…ts in thedomestic and export markets (6) and the relationship between relative variety revenues(7), the free entry condition can be written solely in terms of the zero-pro…t and exportingcuto¤s for productivity and consumer tastes:##
Z "Z 1 "
1
1 1
d
V =
1 fd z ( d ) d d Fd g (') d'
(19)
'd
d (')
d (')
#
#
Z "Z 1 "1
n 1
x+
1 fx z ( x ) d x Fx g (') d' = fe ;
'x
x (')
x (')
where the …rst term captures expected pro…tability in the domestic market and the second
term captures expected pro…tability in export markets.

Multi-Product Firms and Trade Liberalization

17

3.6. Goods and Labor Markets
The steady-state equilibrium is characterized by a constant mass of …rms entering eachperiod, Me , and a constant mass of …rms producing, M . The mass of …rms supplying agiven product to the domestic market, Md , is a constant fraction of the mass of …rms, M ,which is determined by the probability of supplying a product to the domestic market,[1 Z ( d ('))]. Similarly, the steady-state mass of …rms supplying a given product toeach export market, Mx , is a constant fraction of the mass of …rms, M , which is determinedby the probability of exporting a product, [1 Z ( x ('))]. The steady-state equilibriumalso features stationary distributions of …rm productivity and consumer tastes in thedomestic and export markets, which are determined by the zero-pro…t and exportingcuto¤s, 'd , 'x , d (') and x (').
We begin by characterizing the mass of …rms and the fraction of these …rms that supply
a given product to the domestic market and each export market. For the mass of …rms
to be constant in steady-state, the mass of successful entrants that draw a productivitysu¢ ciently high to produce must equal the mass of existing …rms that die, which yields
the following steady-state stability condition:[1

G ('d )] Me = M:

(20)

This steady-state stability condition can be combined with the free entry condition (18) toshow that total payments to labor used in entry equal aggregate pro…ts: M = Me fe = Le .
Total payments to labor used in production, on the other hand, equal aggregate revenue
minus aggregate pro…ts, Lp = R M . Combining these two expressions, it follows that
total payments to labor equal aggregate revenue: R = L. Thus the labor market clears:
Lp + Le = L;

(21)

Having solved for aggregate revenue, the mass of …rms can be determined as follows:
R
M= ;
(22)
r
where average revenue can be expressed solely in terms of the zero-pro…t and exportingcuto¤s for productivity and consumer tastes, as shown in equation (36) in the appendix.
Of the mass of …rms with productivity ', the fraction [1 Z ( d ('))] supply a given
product to the domestic market, while the fraction [1 Z ( x ('))] supply the given product to each export market. Hence the mass of …rms supplying a product to the domestic
market is:
"Z
#
1
g (')
d' M;
(23)
Md =
[1 Z ( d ('))]
1 G ('d )'dwhile the mass of …rms supplying a product to each export market is:
Z 1
g (')
Mx =
[1 Z ( x ('))]
d' M;
1 G ('x )'x(24)Multi-Product Firms and Trade Liberalization

18

where heterogeneity in consumer tastes across countries implies that a …rm can export agiven product to one foreign country but not to another.
Using the equilibrium pricing rule (4) and country symmetry, the aggregate price index
for each product can be written as a function of the mass of …rms supplying each productto the domestic and export markets, as well as the prices of varieties with a weighted
average of …rm productivity and consumer tastes in the domestic and export markets:
#11
"
1
1
1
+ nMx
;
(25)
P = Md'~d'~xwhere the weighted average in the domestic market, '~ d , is pinned down by the zero-pro…tcuto¤s of productivity and consumer tastes, 'd and d ('), as speci…ed in Section B1. ofthe appendix. Similarly, the weighted average in the export market, '~ x , depends on theexporting cuto¤s of productivity and consumer tastes, 'x and x ('), as also speci…ed in
Section B1. of the appendix.
3.7. General Equilibrium
The open-economy general equilibrium is referenced by a vector of six variables: thezero-pro…t productivity cuto¤, 'd , the exporting productivity cuto¤, 'x , the zero-pro…t
consumer tastes cuto¤ of the least productive …rm, d ('d ), the exporting consumer tastes
cuto¤ of the least productive exporter, x ('x ), the price index for each product, P , andaggregate revenue, R. All other endogenous variables can be written as functions of thesesix elements of the equilibrium vector.
Proposition 1 There exists a unique open-economy equilibrium referenced by the sextuple{'d , 'x , d ('d ), x ('x ), P , R}.
Proof. See Appendix.
In Proposition 1, we characterize general equilibrium for arbitrary continuous distributions of …rm productivity and consumer tastes, g (') and z ( ). In the web-based technicalappendix, we report solutions for the special case of Pareto distributions of …rm productivity and consumer tastes, which we use below in examining the model’s comparative
statics.
4.

Properties of the Open-Economy Equilibrium

One of the central features of our model is heterogeneity and self-selection within …rms.
This heterogeneity exists in a …rm’s exports across products within a given country andacross countries for a given product, and in both cases re‡ects idiosyncratic variation indemand for a …rm’s varieties. As a result of this heterogeneity, trade liberalization results

Multi-Product Firms and Trade Liberalization

19

in reallocations of resources within …rms that in‡uence standard revenue-based measuresof …rm productivity.
Proposition 2 (Core Competencies) The opening of the closed economy to trade results in within-…rm resource reallocation: all surviving …rms drop products with low
values of consumer tastes from the domestic market, which raises their revenue-based productivity. In addition, high-productivity surviving …rms begin to export, and therefore add
products with high values of consumer tastes in the export market, which further raises
their revenue-based productivity.
Proof. See Appendix.
The key to understanding this result is recognizing that opening a closed economy
to trade has uneven e¤ects across …rms depending on whether or not they begin to export and uneven e¤ects across products within …rms depending on whether or not theseproducts begin to be exported. Since there is a positive ex ante probability of drawing
a productivity high enough to export, the opening of trade increases the expected valueof entry. As a result, there is increased entry, which enhances product-market competition in the domestic market and raises the zero-pro…t productivity cuto¤ 'd below which…rms exit. This rise in 'd elevates the average productivity of the varieties supplied bycompeting …rms, strengthens product-market competition, and so induces surviving …rms
to drop lower-consumer-tastes products.
This concentration on core competencies that follows the opening of trade leads toa change in the composition of …rm output that increases revenue-based …rm productivity. Marginally viable low-consumer-tastes products are dropped; output of all surviving
products for the domestic market contracts; and entry into exporting generates new output for export markets of higher-consumer-tastes products. Each of these responses shifts
the composition of …rm output towards higher-consumer-tastes products that have higher
revenue per unit of labor input. Therefore, even though …rm productivity and consumer
tastes are themselves parameters that are …xed by assumption, revenue-based …rm productivity rises as a result of the within-…rm reallocation of resources across products.20
This within-…rm reallocation following the opening of trade contrasts with the standard model of heterogeneous …rms and trade, in which …rms are implicitly restricted to
production of just a single product. Our results also contrast with the predictions ofan alternative framework in which products are randomly allocated to …rms. In such a20

We follow Foster, Haltiwanger and Syverson (2008) in di¤erentiating between revenue- and quantitybased measures of productivity. Since both …rm varieties and products are horizontally di¤erentiatedfrom one another, the revenue-based measures of productivity typically used in empirical work are thetheoretically appropriate measures for examining the impact of trade liberalization on …rm productivity.
As discussed in footnote 17, consumer tastes and …rm productivity have the same e¤ects on equilibrium
revenue, which implies that they have the same e¤ects on revenue-based measures of productivity, as
shown in Section B2 of the Appendix.

Multi-Product Firms and Trade Liberalization

20

model, only those …rms who happen to have products with a value of consumer tastes
above the old and below the new zero-pro…t cuto¤ would drop those products after the
opening of trade. In contrast, in our model, each …rm chooses optimally the range of
products to supply, and therefore all …rms focus on their core competencies in higher
consumer tastes products after the opening of trade. This optimal choice of the range of
products to supply has systematic implications for the extensive and intensive margins oftrade, which we examine further below.
While we have developed the core competencies prediction of the model in the context
of opening the closed economy to international trade, related but more nuanced predictions
hold for reductions in variable trade costs in the open-economy equilibrium of the model.21
Proposition 3 Reductions in variable trade costs also result in within-…rm reallocationthat leads surviving multiple-product …rms to focus on their core competencies:(a) surviving …rms that continue to supply only the domestic market drop products withlow values of consumer tastes, which raises their revenue-based productivity,(b) surviving …rms that enter the export market drop products with low values of consumer
tastes from the domestic market and add products with higher values of consumer tastes
in the export market, which raises their revenue-based productivity,(c) surviving exporters drop products with low values of consumer tastes from the domestic market and add products in the export market. As these products added in the export
market have higher values of consumer tastes than those dropped from the domestic market, but have lower values of consumer tastes than those previously exported, the e¤ect on
their revenue-based productivity is ambiguous.
Proof. See Appendix.
The dropping of products from the domestic market following reductions in variable
trade costs is a source of revenue-based productivity growth for all …rms. However, …rms
that enter the export market experience an additional source of revenue-based productivity growth. New exporters not only drop lower-consumer tastes products from thedomestic market, but also expand output of higher-consumer tastes products for the export market, which further shifts the composition of …rm output towards products withhigher revenue per unit of labor input. This prediction is consistent with a number of
empirical studies that …nd signi…cant increases in revenue-based productivity in the yearof transition to exporting (see, for example, Bernard and Jensen 1999).
For continuing exporters, the implications of reductions in variable trade costs for
productivity are more subtle. Continuing exporters experience revenue-based productivity growth from the expansion of output at previously exported products, which shifts
21

While we concentrate on reductions in variable trade costs, reductions in the …xed costs of exporting
products have similar e¤ects, except where otherwise indicated, as long as there remains selection into
export markets: x (') > d (') and 'x > 'd .

Multi-Product Firms and Trade Liberalization

21

the composition of output towards higher-consumer tastes products. But continuing exporters also begin to export products that have lower values of consumer tastes than thosepreviously exported, which reduces average revenue per unit of labor input in export markets. Combining these two e¤ects, it is ambiguous whether reductions in variable trade
costs lead to higher or lower revenue-based productivity growth at continuing exporters
than at non-exporting …rms. This is consistent with empirical …ndings of no signi…cantdi¤erences in revenue-based productivity growth between continuing exporters and nonexporters (see for example Bernard and Jensen 1999).
Whereas the previous two propositions have concentrated on the model’s core competencies predictions and their impact on …rm revenue-based productivity, we now examine
more closely the model’s predictions for the relationship between the extensive and intensive margins and variable trade costs.
Proposition 4 A reduction in variable trade costs:(a) increases aggregate exports through the share of products exported to a given country
by incumbent exporters (within-…rm product extensive margin),(b) increases aggregate exports through the share of countries to which a given productis exported by incumbent exporters (within-…rm country extensive margin),(c) increases aggregate exports through the share of …rms that export (across-…rm extensive margin),(d) has an ambiguous e¤ect on aggregate exports through average exports per …rmproduct-country (intensive margin).
Proof. See Appendix.
The intuition for these results is as follows. A reduction in variable trade costs reduces
the price of products in export markets, which with elastic demand increases revenue andvariable pro…ts in export markets. As a result, some lower-consumer-tastes products thatwere previously only supplied to the domestic market can now be pro…tably exported.
Therefore reductions in variable trade costs raise aggregate exports through the share of
products exported to a given country and the share of countries to which a given product isexported (the within-…rm extensive margins of products and destinations). Additionally,a reduction in variable trade costs induces some lower productivity …rms that previously
only served the domestic market to enter export markets. Hence aggregate exports also
increase because of a rise in the share of …rms that export (the across-…rm extensive
margin).
In contrast, the reduction in variable trade costs has an ambiguous e¤ect on averageexports per …rm-product-country (the intensive margin). This result arises from twocounteracting forces. On the one hand, the reduction in variable trade costs reduces
the price of products that are already exported in each foreign country, which increases
average exports per …rm-product-country. On the other hand, the reduction in variable

Multi-Product Firms and Trade Liberalization

22

trade costs causes products with lower values of consumer tastes to be exported. As
these lower-consumer-taste products are exported in smaller amounts, this reduces averageexports per …rm-product-country through a change in export composition. The net e¤ectof these two forces is ambiguous and depends on the functional form of the distribution
for consumer tastes. For the special case of a Pareto distribution of consumer tastes, thetwo forces exactly o¤set one another, so that average exports per …rm-product-country
are independent of variable trade costs, as shown in the web-based technical appendix.22
Having characterized the relationship between the margins of trade and variable trade
costs, we now turn to examine their relationship with …rm productivity.
Proposition 5 Higher …rm productivity ':(a) increases total …rm exports through the share of products exported by the …rm to agiven country (within-…rm product extensive margin),(b) increases total …rm exports through the share of countries to which a given productis exported by the …rm (within-…rm country extensive margin),(c) has an ambiguous e¤ect on total …rm exports through the intensive margin of averageexports per …rm-product-country.
Proof. See Appendix.
The intuition for these comparative statics follows a similar logic as for the reduction invariable trade costs. More-productive …rms charge lower prices for their products, whichwith elastic demand increases revenue and variable pro…ts for a given value of consumer
tastes. As a result, more productive …rms …nd it pro…table to export a product at a lower
value of consumer tastes, and hence export more products to each country (the within…rm product extensive margin) and export each product to more countries (the within-…rm
country extensive margin). Both extensive margin responses induce endogenous variation
across …rms in the share of exports in …rm output.
In contrast, higher …rm productivity has an ambiguous e¤ect on average exports per…rm-product-country (the intensive margin). On the one hand, higher …rm productivity
increases exports of a given product with a given value of consumer tastes for a given
destination, which increases average exports per …rm-product-country. On the other
hand, higher …rm productivity causes products with lower values of consumer tastes to beexported. As these lower-consumer-taste products are exported in smaller amounts, thisreduces average exports per …rm-product-country through a change in export composition.
22

While a reduction in the …xed costs of exporting increases aggregate exports through the extensive
margins, it has a di¤erent e¤ect on the intensive margin from a reduction in variable trade costs. As the…xed costs of exporting fall, a lower value of consumer tastes is required to generate su¢ cient revenue
to pro…tably export, which unambiguously reduces average exports per …rm-product-country through achange in export composition, as shown in the web-based technical appendix for a Pareto distribution of
consumer tastes.

Multi-Product Firms and Trade Liberalization

23

Again the net e¤ect of these two counteracting forces depends on the functional form of
the distribution for consumer tastes. For the special case of a Pareto distribution of
consumer tastes, the two forces exactly o¤set one another, so that average exports per…rm-product-country are independent of …rm productivity, as shown in the web-based
technical appendix.23
As the model’s ambiguous predictions for the intensive margin arise from changes inexport composition, they have an immediate corollary. The model implies that exports ofa given product with a given value of consumer tastes to a given destination (which holds
export composition constant) increase more rapidly with …rm productivity than averageexports per product to the same destination. We examine this further implication of themodel in our empirical analysis below.
5.

Empirical Evidence

In this section, we compare key predictions of the model with U.S. trade data. We also
provide a detailed examination of the distribution of export shipments across products
and destinations within …rms and present evidence on the quantitative importance ofthe extensive margins of trade for the distribution of U.S. exports across …rms and trade
partners. We begin with a brief description of our data, deferring a more detailed summary
to Appendix A below.
5.1. Data Description
Our analysis makes use of two datasets: the U.S. Linked/Longitudinal Firm Trade
Transaction Database (LFTTD) and the U.S. Census of Manufactures (CMF). The LFTTD
captures all U.S. international trade transactions from 1992 through 2004 and is derived
from customs forms. For each export transaction, we observe the identity of the …rm responsible for the export, the export product’s ten-digit Harmonized System (HS) productclassi…cation, the value shipped, the date of the shipment and the destination country.
The quinquennial CMF collects information on manufacturing establishments’inputs
and output in each census year. We use this information along with industry price de‡ators
provided by Bartelsman et al. (2000), to compute manufacturing establishments’revenuebased labor productivity and total factor productivity (TFP).24 The CMF also tracks
23

The lack of correlation between the intensive margin and …rm productivity requires both a Pareto
distribution of consumer tastes and a product …xed cost of exporting, fx , that is independent of consumer
tastes, x . Even with a Pareto distribution of consumer tastes, the correlation between the intensive
margin and …rm productivity is negative if fx is increasing in x , and is positive if fx is decreasing in x ,as shown in the web-based technical appendix.
24
We measure …rm TFP as the shipment-weighted average TFP of its plants. Plant TFP in a given
census year is measured relative to other plants in its main industry in percentage terms using the multifactor superlative index number of Caves et al. (1982). This index accounts for plants’ use of capital,
production workers, non-production workers and materials. Plant shipments, capital and materials are

Multi-Product Firms and Trade Liberalization24

the set of …ve-digit Standard Industrial Classi…cation (SIC) categories manufacturing
establishments produce over time. We use this information to examine changes in …rm
scope between census years. We link manufacturing establishments in the CMF to …rms
in the LFTTD using the bridge developed by Bernard, Jensen and Schott (2009).
We interpret the approximately 8,000 ten-digit HS and 1,500 …ve-digit SIC codesused to classify exports and production as discrete partitions of the model’s continuumof products, which become coarser as one increases the level of aggregation. With thisinterpretation, the model provides a natural explanation for the coexistence of singleand multiple-product …rms. We think of …rms producing a single product as those whose
range of products falls within a single classi…cation code. Multiple-product …rms, on theother hand, are those whose product range is wide enough to span several classi…cation
codes.
5.2. Core Competencies
One of our central theoretical results in Propositions 2 and 3 is that trade liberalization induces …rms to focus on their core competencies, which raises revenue-based …rm
productivity. As a large empirical literature has provided evidence of …rm revenue-based
productivity growth following trade liberalization, we concentrate on the model’s morenovel prediction that trade liberalization leads to a focusing on core competencies throughreallocations of resources within …rms.25
In the model, trade liberalization induces …rms to drop products with low values ofconsumer tastes from the domestic market, but induces them to add products with highvalues of consumer tastes in the export market. In a setting without …rm-product demandheterogeneity across countries, the range of products exported is a subset of the rangeof goods produced and sold at home for parameter values for which there is selection
into export markets. Therefore, without …rm-product-country demand heterogeneity, themodel yields the unambiguous prediction that trade liberalization reduces the range of
products produced by a …rm.
In contrast, in our setting with demand heterogeneity across countries, trade liberalization can induce a …rm to add new products in export markets that it does not supply
to the domestic market. These are products with high values of consumer tastes in export
markets but low values of consumer tastes in the domestic market, and their addition can
increase the range of products produced by a …rm. Therefore, with demand heterogeneity,
trade liberalization has an ambiguous e¤ect on the number of products produced, dependde‡ated according to the four-digit SIC de‡ator of its major industry using de‡ators provided by Bartelsman et al. (2000). Wages are de‡ated by the U.S. consumer price index available at www.bls.gov. A
plant’s main industry is the four-digit SIC in which it has the largest value of shipments.
25
For evidence on …rm productivity growth following trade liberalization, see among others Tybout andWestbrook (1995), Bernard, Jensen and Schott (2006), Pavcnik (2002), Tre‡er (2004), Fernandes (2008),
and Park et al. (2008).

Multi-Product Firms and Trade Liberalization25

ing on whether the deletion of products from the domestic market exceeds the addition of
new products in export markets that are not supplied domestically. Thus, evidence thatthe number of products produced falls following trade liberalization is consistent with thefocusing on core competencies emphasized by the model, even though with demand heterogeneity focusing on core competencies does not necessarily imply a fall in the numberof products produced.
To examine the impact of trade liberalization on the number of products …rms produce, we use the Canada-U.S. Free Trade Agreement (CUSFTA) as a natural experiment.
CUSFTA, signed in 1988, came into e¤ect on January 1, 1989 and involved substantial
tari¤ reductions for a number of goods (see, for example, Tre‡er 2004). In contrast to
many trade liberalizations in developing countries, CUSFTA involved a clearly de…nedchange in trade policy that did not come as part of larger package of reforms. Additionally, the mechanism behind the focusing on core competencies in our model is an
expansion of export opportunities, and CUSFTA was a reciprocal liberalization that enhanced export opportunities for …rms in both Canada and the United States. Consistent
with our model, Baldwin and Gu (2009) …nd a sharp decline in product diversi…cation
among Canadian …rms following implementation of CUSFTA. Here, we provide evidenceof a similar response among U.S. …rms.
Our empirical analysis combines data from the CMF on the number of …ve-digit SIC
goods …rms produced in the years 1987 and 1992, before and after the introduction of
CUSFTA, with data on reductions in Canadian tari¤s on U.S. manufacturing imports
measured at the four-digit SIC level over this period.26 We measure a …rm’s exposure
to CUSFTA as the domestic-shipment weighted average tari¤ reduction in the four-digit
SIC industries in which it was active in 1987,
P 87
vf i ( Tari¤ i )i
P 87;Tari¤ f =vf iiwhere f and i index …rms and four-digit SIC industries, respectively; vf87i represents …rm
domestic shipments in industry i in 1987; and Tari¤ i is the percentage point change
in the Canadian tari¤ rate on U.S. manufacturing imports in industry i between 1989
and 1992.27 Across industries, the mean tari¤ reduction is 4.0 percentage points with astandard deviation of 2.9 percentage points. Across surviving …rms in our data, the meanand standard deviation of Tari¤ f is 3.1 and 2.4 percentage points, respectively. Note26

Five-digit SIC products are de…ned consistently across years as in Bernard, Redding and Schott(2008). As the LFTTD data on exports are not available until 1992, our analysis of the impact of CUSFTA
is restricted to the CMF, which records the number of …ve-digit SIC products U.S. manufacturing …rms
produce in 1987 and 1992.
27
Industry-level Canadian tari¤ data are from Tre‡er (2004) and are available from 1989 to 1992. We
note that we obtain –as expected for a largely reciprocal liberalization –similar results when using U.S.
tari¤ changes on Canadian four-digit SIC imports over the same period.

Multi-Product Firms and Trade Liberalization

26

that in the regressions below we use
Tari¤ f as a covariate so that increases in thisvariable represent greater tari¤ reductions, i.e., greater increases in export opportunities.
Our empirical strategy involves a “di¤erences-in-di¤erences”speci…cation, in which wecompare the number of products produced by …rms before and after CUSFTA (…rst difference) in industries experiencing above versus below-median Canadian tari¤ reductions(second di¤erence). This “di¤erences-in-di¤erences” speci…cation can be represented as
the following regression:
Products f t =(P ostt

Exposuref ) +f + dt + uf t

(26)

where f , again, denotes …rms, Products f t is the number of …ve-digit SIC products produced by a …rm in 1987 and 1992; P ostt is a dummy variable which equals zero in 1987
prior to CUSFTA and one in 1992 afterwards; Exposuref is a dummy variable whichequals one if a …rm experienced above-median Canadian tari¤ reductions between 1989
and 1992 and zero otherwise; f are …rm …xed e¤ects that control for unobserved heterogeneity in the number of products across …rms; dt are time dummies that control forcommon macro shocks; and uf is a stochastic error.28
As we have two cross-sections of data in 1987 and 1992, the …xed e¤ects speci…cationin (26) has an equivalent representation in …rst di¤erences. Taking …rst di¤erences in(26), the left-hand side variable becomes the change in the number of products between
the two years, the right-hand side variable becomes the Exposuref dummy variable forwhether a …rm experienced above or below-median Canadian tari¤ reductions, the …rm…xed e¤ects f di¤erence out, and the time dummies dt become a constant. We cluster
the standard errors in this …rst di¤erences speci…cation by …rms’ main four-digit SIC
industry to account for the fact that our …rm-speci…c measure of exposure to CUSFTA isconstructed using four-digit SIC data on Canadian tari¤ reductions.
Results are reported in the …rst row of Table 2. In column (1), we …nd that …rms
experiencing above-median Canadian tari¤ reductions reduce the number of products
they produce relative to …rms experiencing below-median Canadian tari¤ reductions. In
columns (2) and (3), we show that this result is robust to including additional controls
for major …rms’four-digit industry and log 1987 employment as a measure of initial …rmsize.
In the second row, we replace the number of products on the left-hand side of (26)with an alternative measure of …rm diversi…cation used by Baldwin and Gu (2009). ThisP
“entropy” measure is de…ned as i sf kt ln(sf kt ), where sf kt represents the share of …rm
shipments accounted for by …ve-digit SIC product k. It captures the extent to which a…rm’s output is skewed towards its largest rather than its smallest products. Estimating28

We …nd similar results using alternative cuto¤s, e.g., comparing …rms experiencing Canadian tari¤
reductions above the 75th percentile to those experiencing Canadian tari¤ reductions below the 25th
percentile. Results are also robust to including the number of products …rms produce in 1987, and tousing the log di¤erence in the number of products produced rather than levels.

Multi-Product Firms and Trade Liberalization

27

Table 2: U.S. Manufacturing Firm Diversi…cation During the Canada-U.S. Free Trade
Agreement
[1]

[2]

[3]

Change in Products-0.0590.015-0.6240.101-0.572
0.096

Change in Entropy

0.011
0.003

0.156
0.026

0.153
0.026

Firm Observations

66,472

66,472

66,472

No

Yes

Yes

Major Industry Dummy Variables

Log 1987 Employment
No
No
Yes
Notes: Table reports mean difference in noted variable between surviving firms
experiencing above- and below-median changes in Canadian export opportunities
from 1987 and 1992. Each cell reports the mean difference and associatedstandard error from a seperate OLS regression. Change in products refers tochange in number of five-digit SIC categories produced in the United States.
Change in entropy is defined in the text. Change in export opportunities refers to the
output-weighted average change in Canadian tariffs across the four-digit SIC
industries produced by the firm. Standard errors are clustered according to firms'
main four-digit SIC industry. Additional covariates are included as noted.

the regression speci…cation again in …rst di¤erences, column (1) shows that …rms experiencing above-median Canadian tari¤ reductions exhibit a rise in entropy, i.e., an increased
concentration of production in their largest products, relative to …rms experiencing belowmedian Canadian tari¤ reductions. Columns (2) and (3) show that this …nding is robust
to controlling for major four-digit industry and initial …rm size. Overall, both sets of
empirical results provide support for the idea that trade liberalization induces …rms tofocus on their core competencies.29
5.3. Margins of Trade and Trade Costs
Proposition 4 relates the extensive and intensive margins of exports to variable trade
costs. Here, we use a “gravity equation” speci…cation to examine empirically how themargins of trade vary across countries using their distance from the United States as aproxy for variable trade costs.30
We begin by decomposing aggregate U.S. exports to each destination country c (xc )into the number of …rms exporting to the destination (fc ), the number of ten-digit HS
products exported to the destination (pc ), the fraction of …rm-product combinations withpositive exports which we refer to as the “density”of trade (dc ), and the average value of29

Additional empirical support comes from Iacovone and Javorcik (2008), who …nd a decline in thenumber of goods produced and a rise in the number of goods exported at Mexican …rms following
the North American Free Trade Agreement (NAFTA) that superseded CUSFTA in 1994. For further
supportive evidence using import penetration data, see Bowen and Wiersema (2005) and Liu (2006).
30
Distance is a widely-used proxy for trade costs in the large gravity equation literature, as reviewed
by Disdier and Head (2008). For direct evidence on the relationship between trade costs and distance,
see, for example, Hummels (2001) and Limao and Venables (2001).

Multi-Product Firms and Trade Liberalization

28

exports per …rm-product-country conditional on positive exports (xc ),(27)

xc = fc pc dc xc ;where dc = oc =(fc pc ), xc = xc =oc , and oc is the number of …rm-product observations withpositive exports for country c:31 In this speci…cation, the density extensive margin captures
the extent to which …rm-product cells for a given export destination are “…lled-in”. It
ranges from minf1=fc ; 1=pc g to unity.
We regress total exports to trade partners, as well as each component of total exports,on the great-circle distance of trade partners from the United States. To control for the
potential in‡uence of market size, we also include trade partners’GDP,ln Zc =

0+

1 ln distance c +2 ln GDPc + "c ;(28)

where Zc 2 fxc ; fc ; pc ; dc ; xc g.32 For brevity, we report regression results using 2002 data,
but note that results for other years are similar.
The …rst column of Table 3 echoes the well-known result that destination-country
exports decline with distance and increase with market size. Results in the next fourcolumns decompose these overall e¤ects into the contributions of the extensive and intensive margins. Consistent with the model, the number of exporting …rms and the numberof exported products decline with distance. In contrast, density rises with distance. This
result is also intuitive: as the number of exporting …rms and the number of exported products both increase, the number of possible …rm-product observations expands multiplicatively. If each …rm is active in a limited subset of products, the number of …rm-productobservations with positive trade expands less than proportionately. As a result, density
is negatively correlated with the …rm and product extensive margins, and hence withaggregate U.S. exports. By summing the coe¢ cients for density and number of products,
we …nd that the number of products with positive exports per …rm, oc =fc , declines with
distance. Finally, in contrast to the strong negative e¤ect of distance on the extensive
margins, we …nd that distance has a positive but statistically insigni…cant e¤ect on theintensive margin of average exports per …rm and product to a country. Re-estimating thesame speci…cation for other years, we …nd that the coe¢ cient on distance for the intensive
margin is positive but statistically insigni…cant throughout. This ambiguous pattern ofresults is consistent with the ambiguous intensive-margin predictions of the theory.
31

Including a term for the density of trade augments the gravity speci…cation considered in Bernard,
Jensen, Redding, and Schott (2007) and isolates the intensive margin.
32
Distance data are from CEPII (see Mayer and Zignago 2006). GDP data are from the World Bank’s
World Development Indicators database. In contrast to many gravity equation speci…cations, speci…cation (28) includes importer but not exporter GDP, because with data on only U.S. exports, exporter
characteristics are controlled for in the regression constant.

Multi-Product Firms and Trade Liberalization

29

Table 3: Gravity and the Margins of U.S. Exportsln(Valuec)ln(Firmsc)ln(Productsc)

ln(Densityc)ln(Intensivec)ln(Distancec)

-1.37
0.17

-1.17
0.15

-1.10
0.15

0.84
0.13

0.05
0.10

ln(GDPc)

1.01
0.04

0.71
0.03

0.55
0.03

-0.48
0.03

0.23
0.02

Constant
Observations

7.82

0.52

3.48

-2.20

6.03

1.83

1.59

1.55

1.37

1.07

175

175

175

175

175

2

R
0.82
0.76
0.68
0.66
0.37
Notes: Table reports results of country-level OLS regressions of U.S. exports or theircomponents on trading-partners' GDP and great-circle distance (in kilometers) from theUnited States. Standard errors are noted below each coeficient. Data are for 2002.

5.4. Margins of Trade and Firm Productivity
Our …nal set of theoretical results, from Proposition 5, link …rms’extensive margins
to their productivity. In this section we examine the extent to which these margins
increase with estimates of …rm productivity and with other variables that, in the model,
are monotonically related to …rm productivity.
The …rst …ve columns of Table 4 summarize the results of regressing the log number often-digit HS products exported by …rms on a series of covariates using OLS. The second…ve columns report analogous regressions using the log number of destinations served by…rms as the dependent variable. As only the CMF contains the information on inputs
needed to estimate …rms’total factor productivity (TFP) and labor productivity (domestic
shipments per worker), used in the …rst two columns of each panel, for consistency werestrict our analysis throughout the table to the subset of exporting …rms from the LFTTD
that also appear in the CMF. Data are for 1997, the latest year for which merged data
is available.33 All regressions include dummies for …rms’ main four-digit SIC industry
to focus on variation across …rms within the same major industry, and we also cluster
standard errors on this dimension of the data. We note that measuring TFP for multipleproduct …rms is problematic unless data on inputs, outputs and prices is available at the…rm-product level (see also De Loecker 2007). Since this requirement is not met by ourdata, we also consider total …rm exports and other variables that in the model are related
monotonically to …rm productivity.
The …rst two columns of each panel examine the relationship between …rms’extensive
margins and their revenue-based TFP and log labor productivity. Consistent with the33

We note that we …nd results very similar to those reported in the …nal three columns of each panel
when we re-estimate the same speci…cation on the larger set of …rms that appear in the LFTTD, both in1997 and across years.

Multi-Product Firms and Trade Liberalization

30

model, we …nd positive and statistically signi…cant correlations in all four regressions.
Table 4: Correlation of U.S. Firms’Extensive and Intensive Margins
ln(Productsf)
TFPf

[1]
0.071
0.022

ln(Outputf/Workerf)

[2]

[3]

ln(Countriesf)
[4]

[6]
0.076
0.022

0.474
0.019

ln(Exportsf)thln(5 -Largest Productf)

[9]

[10]

0.329
0.006
0.345
0.005

0.405
0.004
0.436
0.096

[8]

0.347
0.006
0.345
0.003

1.894
0.006

[7]

0.426
0.020
0.384
0.004

ln(Largest Productf)

Constant[5]-3.022 -2.300
0.053 0.061

0.405
0.004

1.292
0.006

-0.797 -3.141 -2.714 -1.733
0.101 0.072 0.078 0.051

Observations
27,987 27,987 27,987 27,987 16,215 27,987 27,987 27,987 27,987 16,215
2
R
0.13
0.18
0.69
0.56
0.50
0.21
0.24
0.60
0.55
0.53
Notes: Table reports results of firm-level OLS regressions of the log number of ten-digit HS products exportedby the firm, or log the number of destination countries served by the firm, on noted covariates. All regressions
include dummies for firms' main four-digit SIC industry, and we standard errors are clustered on thisdimension of the data. Results in columns 5 and 10 are restricted to firms exporting at least five products. Data
are for 1997.

The last three columns of each panel of Table 4 consider the relationship between …rms’
product and country extensive margins and less-direct manifestations of …rm productivity.
In the model, variation across …rms in the size of their largest product to a particular
destination is due solely to variation in …rm-level productivity. With a continuum of
symmetric products and independent and identical consumer taste distributions, each…rm draws the same distribution of consumer tastes across products within a destination.
Thus, while the identities of …rms’ largest (or second-largest, etc.) products will varyacross …rms, their size, conditional on …rm-level productivity, will not. As a result, moreproductive …rms have greater exports of their largest product than less productive …rms
supplying the same destination. Additionally, with a continuum of symmetric countries in
the model, more productive …rms also have greater total exports of their largest productand greater total exports across all products and destinations.34
The last three columns of both panels of Table 4 show strong and statistically significant positive correlations between …rms’extensive margins and their total exports, theexports of their largest product across destinations and the exports of their …fth-largest
34

In practice, products and countries can be asymmetric, which implies that a …rm could have hightotal exports of its largest product and high total exports across all products either because it has ahigh productivity or because it happens to draw high consumer tastes for large countries and products.
Nonetheless, even with asymmetric products and countries, exports of a given product to a given country
are monotonically increasing in …rm productivity. Therefore, even in the presence of asymmetries, exports
of a …rm’s largest product and total …rm exports are positively correlated with …rm productivity.

Multi-Product Firms and Trade Liberalization

31

product across destinations. Results are similar using other-ranked products (e.g., thesecond-largest). We note that these relationships need not be positive. Diseconomies ofscope, for example, could provide more productive …rms with greater incentives to specialize in their most pro…table products and destinations, potentially inducing negative
relationships in Table 4.
We also investigate a related prediction of the model, which has exports of a given…rm-product – e.g., the …rm’s largest product – increasing more rapidly with productivity than average exports per product. This gap arises from the di¤erences in export
composition between less- and more-productive …rms highlighted in Proposition 5. As
productivity rises, exports of a given product increase but the …rm extends itself into
additional products with lower values of consumer taste.
To examine this prediction, we again use cross-sectional data on …rms exporting different numbers of products. The left panel of Figure 1 displays the mean size of …rms’
largest export product versus the mean size of their average export product across …rms
exporting up to ten products.35 The right panel reports the same statistics but for amuch narrower sample restricted to …rms’exports of Machinery and Electrical products(HS 84-85) to Canada. Vertical axes in both panels use logarithmic scales, and both…gures are truncated at ten products to conform with Census disclosure requirements.
Data are for 2002, but we note that we …nd similar patterns in other years as well as forother combinations of countries and products. Consistent with the compositional e¤ects
emphasized in the model, both panels show that largest-product exports increase sharply
with the number of products exported, while average exports per product increase moremodestly.36
5.5. Within-Firm Heterogeneity
A key feature of our model is heterogeneity in export shipments across products anddestinations within …rms. As examined above, this variation has aggregate implications forboth revenue-based productivity and the relationship between the extensive and intensive
margins of trade. In this section, we present additional evidence on within-…rm productand destination heterogeneity and the extent to which it conforms to the predictions of
a Pareto distribution. For brevity we report results for …rms exporting ten products orexporting to ten destinations, but note that …ndings are similar for …rms producing up
to nine products and serving up to nine destinations, where we again restrict the analysis
to ten or less products and destinations to conform with Census disclosure requirements.
35

Firms exporting up to ten products account for roughly 85 percent of U.S. exporting …rms and 10
percent of U.S. export value, respectively.
36
As discussed in footnote 23, Pareto-distributed consumer tastes and product …xed costs of exportingthat are independent of consumer tastes together imply that average exports per product are uncorrelatedwith the number of products exported. This implication is at odds with the moderate but statisticallysigni…cant rise in average exports per product displayed in Figure 1.

Multi-Product Firms and Trade Liberalization

All Products

32

HS 84-85 Products to Canada
1,000

10,000

Value ($)

Value ($)

1,000
100

100

10

10
1

2

3

4

5

6

7

8

9

Products Exported by Firm
Largest ProductAverage Product

10

1

2

3

4

5

6

7

8

9

10

Products Exported by Firm
Largest ProductAverage ProductNotes: Figures display the mean value of firms' largest and average export product across all firms exporting the noted number of products.
The left panel displays trends for all exports. The right panel is restricted to exports of Machinery and Electrical products (HS 84-85) to
Canada. Data are for 2002. Y-axes are logged.

Figure 1: Exports of Firms’Largest and Average Products by Number of Products Firms
Export, 2002.

As before, data are for 2002, but results are comparable across years.
The …rst two columns of Table 5 report the average share of each product and each
destination in total …rm exports for …rms exporting ten ten-digit HS products and serving
ten destinations, respectively. As apparent from the …rst two columns, the distributionof exports across products and destinations is highly skewed, with the largest of a …rm’s
products or destinations accounting for roughly one half of its exports. A similar skewness
in the distribution of exports across products is observed even within destinations, as
shown in the third column of the table, which restricts analysis to …rms’ exports to
Canada. Comparable skewness is also apparent within …rms’ Machinery and Electrical
product exports to Canada, in the fourth column.
In the model, exports are log linear in consumer tastes and only vary across a …rm’s
products within destinations as a result of heterogeneity in consumer tastes. Under theassumption of a Pareto distribution of consumer tastes, the model implies a linear relationship between the log rank of products and their log share of …rm exports within
destinations.37 To provide evidence on this relationship, we estimate an OLS regressionof the log rank of products exported to Canada on their log share of …rm exports to
Canada using the data reported in the third column of Table 5. The …tted and actual37

If the distribution of …rm exports across products within destinations is Pareto with minimum value kaand shape parameter a, we have Prob(x > x0 ) = (k=x0 ) , where x denotes exports. Taking logarithms inthis expression and rearranging terms yields the following relationships: log (Rankp ) = A a P
log (xp ) =B a log (Sharep ), where Rankp is the rank of xp , Sharep = xp =X, and A, B and X = p xp areconstants.

Multi-Product Firms and Trade Liberalization

33

Table 5: Distribution of Firm Exports Across Products and Destinations, 2002

% of Firm Exports

HS 84-85
Products
Products
ExportExportExported to Exported to
Rank Products
Destinations
CanadaCanada
1
49.0
47.7
47.4
47.9
2
18.6
19.5
19.4
19.3
3
10.5
11.1
11.1
11.0
4
6.7
7.1
7.0
7.0
5
4.6
4.9
4.8
4.7
6
3.4
3.4
3.4
3.3
7
2.5
2.4
2.5
2.4
8
1.9
1.7
1.9
1.8
9
1.5
1.2
1.5
1.4
10
1.1
0.9
1.1
1.1
Notes: First two columns display mean percent of firm exports
represented by product or country with noted rank (from high tolow) across firms exporting ten ten-digit HS products or exportingto ten destinations, respectively, in 2002. Third and fourth columns
restricted to firms exporting ten products to Canada, and ten
Machinery and Eletrical products (HS 84-85) to Canada,
respectively. Sample sizes across the four columns are 2791,
1641, 983 and 322 firms, respectively.

values for log rank and log share are displayed in Figure 2. As indicated in the …gure,actual values lie above the regression line in the middle of the distribution and below the
regression line in the tails, implying thinner tails than the Pareto distribution. Including
a quadratic term in the log share of …rm exports in the regression, we …nd that the null
hypothesis of linearity implied by a Pareto distribution is rejected at conventional levelsof statistical signi…cance.
We note that we …nd a similar pattern of results for …rms exporting between four andnine products to Canada, and for …rms’exports to other destinations. We also …nd similar
rejections of a Pareto distribution for the distribution of exports across products for all
destinations (…rst column of Table 5), the distribution of exports across destinations for
all products (second column of Table 5), and the distribution of exports across products
within HS codes 84-85 for Canada (fourth column of Table 5).38
Taken together, these results suggest that the rejection of a Pareto distribution is afeature of the distribution of …rm sales across products within destinations and is not
driven by aggregation across destinations or products. Furthermore, the distribution ofexports within …rms exhibits the same features as the distribution of sales across …rms in
the …rm-size literature, which also …nds evidence of thinner tails than a Pareto distribution(see, for example, Rossi-Hansberg and Wright 2007).
38

These results are consistent with those in Arkolakis and Muendler (2008), who …nd departures froma Pareto distribution in the upper and lower tails of the distribution of exports within destinations in
Brazilian data.

Multi-Product Firms and Trade Liberalization

34

Export-Product Rank vs Size

1

2

Product Rank
4

6

10

Ten-Product Exporters, 2002

FittedActual5
10Product Size (%)

20

40

60

Notes: The solid line plots average within-firm product rank (low to high) against within-firm
product size across the 1641 firms exporting ten products to Canada. The fitted line is theestimated relationship from an ordinary-least-squares regression of log product rank on log
product size.

Figure 2: Within-Firm Export Product Size versus Product Rank for Firms Exporting
Ten Products to Canada in 2002

5.6. Quantitative Importance of the Extensive Margins
One of the more striking features of micro data on international trade is the overwhelming importance of multiple-product, multiple-destination …rms. U.S. …rms exporting …veor more ten-digit Harmonized System (HS) products, for example, account for 98 percent of total U.S. exports, while …rms exporting to …ve or more destinations represent 93
percent of exports. Moreover, these activities are highly correlated: 92 percent of U.S.
exports are due to …rms that export …ve or more products while also exporting to …ve or
more destinations.39
In our framework, extensive margins magnify inequality in the …rm-size and countrysize export distributions compared with a setting in which these margins do not exist.
Relative to less-productive …rms, more-productive …rms export more products to a given
destination, export a given product to more destinations, and export more value per given
product-country and level of consumer taste. Analogously, countries with lower variable
trade costs receive more products from a given …rm, receive a given product from more…rms, and receive more value per given …rm-product and level of consumer taste.
39

These …gures are reported by Bernard, Jensen and Schott (2009). Bernard, Redding and Schott (2009)
and Goldberg, Khandelwal, Pavcnik and Topalova (2008) show that multiple-product …rms account forcorrespondingly large shares of production.

Multi-Product Firms and Trade Liberalization

35

Here, we quantify the importance of …rms’product and country selection for the …rmsize distribution by constructing counterfactual distributions that sequentially eliminate
their extensive margins. These counterfactuals are based on a decomposition of total …rm
exports (xf ) into the number of countries to which a …rm exports (cf ), the number of
products which the …rm exports (pf ), the density of …rm exports (df = of =(cf pf )) andaverage exports per country-product conditional on positive exports (xf ):xf = cf pf df xf(29)

where xf = xf =of and of is the number of country-product observations with positive
exports for …rm f . In our …rst counterfactual (C1), we exclude the country extensive
margin by assuming that all …rms export to just a single country. That is, we use equation(29) to recalculate total …rm exports for all …rms by using cf = 1 in place of its actual
value. In our second counterfactual (C2), we further assume that …rms export only asingle product, setting both pf = 1 and df = 1.40
Actual and counterfactual …rm export distributions across …rm-export-size deciles arereported in Figure 3. As indicated in the …gure, excluding extensive margins results inconsiderably less concentrated distributions of exports across …rms. Comparing the actual, C1 and C2 distributions, we …nd that the top 10 percent of …rms account for 96
percent of actual exports but only 84 percent of hypothetical exports when the country
margin is eliminated and only 76 percent of exports when both extensive margins are
eliminated. To explain the extreme concentration of actual exports, standard models ofheterogeneous …rms and trade require either a highly-skewed distribution of productivity
and/or a high elasticity of substitution. Figure 3 suggests that endogenous product anddestination-market selection can rationalize this concentration with less skewed productivity distributions or lower elasticities of substitution. As a result of the positive correlation
between …rms’ productivity and their extensive margins, a given skewness of the productivity distribution and a given elasticity of substitution results in more inequality ofexports across …rms than in conventional settings.
To quantify the impact of the extensive margins of trade on the distribution of exports
across U.S. export destinations, Figure 4 plots U.S. export destinations’total export value
against the intensive-margin component of that value, i.e., xc versus xc from the decomposition of country exports in (27). While by construction actual exports are greater thanor equal to the contribution of the intensive margin, the ratio of actual exports to theintensive margin increases substantially with the value of actual exports. As the axes ofthe …gure are in logarithms, this …nding implies that the percentage contribution of theextensive margin to actual exports increases with the value of actual exports. An immediate corollary of this …nding is that the pattern of actual U.S. exports across countries is40

As predicted by our model, the number of countries to which a …rm exports is highly skewed: roughly
two-thirds of …rms export to a single country. We also …nd skewness in the number of products …rms
export: roughly two-…fths of …rms export a single product to a single country.

Multi-Product Firms and Trade Liberalization

36

Firm-Size Distributions by Decile, 2002
100
90
80
70
60
50
40
30
20
10
0
10

20

30

40

50

60

70

Actual

C1

C2

80

90

100

Notes: Figure plots actual and two counter-factual distributions of
U.S. exports across firm-size deciles, from low to high. The first
counter-factual distribution (C1) assumes firms serve only a single
export destination. The second counter-factual distribution (C2)further assumes that firms export only a single product. Firms are reranked according to size for each distribution. Data are for 2002.

Figure 3: Actual versus Counterfactual Firm-Size Distributions, 2002

substantially di¤erent from what it would be with only intensive-margin variation. The
greater contribution of the extensive margin at higher values of actual exports magni…es
inequality in the distribution of U.S. exports across countries.
The data displayed in Figure 4 exhibit a starkly di¤erent pattern from the predictionsof representative-…rm new trade theory models such as Krugman (1980), which predictthat all of the response to di¤erences in trade costs across countries occurs on the intensive margin. Thus, Figure 4 reinforces the message that the types of extensive-margin
adjustments emphasized by our model are quantitatively important in the aggregate aswell as across …rms.
6.

Conclusion

Firms exporting multiple products to multiple destinations dominate production andtrade but have received comparatively little theoretical attention. This paper develops a
tractable general equilibrium model of multiple-product, multiple-destination …rms that is
consistent with key features of the distribution of exports across …rms, products and countries. In contrast to existing approaches, our model incorporates idiosyncratic variation

Multi-Product Firms and Trade Liberalization

37

Total Exports versus Intensive Margin, by Country.1

Total Exports
1

10

100

Billions of USD

.001

.01
.1
Intensive Margin

1

10

100

Notes: Figure displays U.S. export destinations' total export value against their intensivemargin of average exports per firm-product observation. Sample restricted to countries
receiving exports from at least 100 U.S. firms. Data are for 2002.

Figure 4: The Contribution of the Intensive Margin to U.S. Exports, by Destination
Country

in exports within …rms across both products and destinations through heterogeneity inconsumer tastes. Incorporating such heterogeneity within …rms not only brings the model
closer to the data, but also has important aggregate implications for the determinants oftrade and the impact of trade liberalization on …rm and industry productivity.
We show that U.S. trade data provide strong empirical support for key predictions of
the model. Across countries, we …nd that the negative e¤ect of distance on bilateral tradeis accounted for entirely by the extensive margins, with distance having, if anything, apositive association with the intensive margin. Across …rms, we …nd that more productive…rms export more products to more destinations, that trade liberalization is accompanied
by …rms focusing on their core competencies, and that the export value of …rms’largest
products increase more rapidly with the number of products exported than average exports
per product, consistent with …rm productivity in‡uencing export composition.
More broadly, our analysis suggests that the reallocation of resources in response toevents such as trade liberalization may be even more important than hitherto thought
insofar as it occurs within as well as across …rms. Relatedly, our …ndings point to therole of the within-…rm organization of economic activity across products and countries inin‡uencing …rm, industry and aggregate outcomes.

Multi-Product Firms and Trade Liberalization

38

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Multi-Product Firms and Trade LiberalizationA

42

Data Appendix

Data on …rm’s domestic output, domestic factor use and the number of …ve-digit
SIC goods produced domestically are from the U.S. Census of Manufactures (CMF).
Data on export value, ten-digit HS export products and export destinations is from theLinked/Longitudinal Firm Trade Transaction Database (LFTTD).
A1. Census of Manufactures (CMF) Data
Manufacturing Censuses are conducted every …ve years and we make use of data fromthe 1987, 1992 and 1997 censuses. The sampling unit for each Census is a manufacturing“establishment”, or plant, and the sampling frame in each Census year includes information on the mix of products produced by the plant. Very small manufacturing plants(referred to as Administrative Records) are excluded from the analysis unless otherwise
noted because data on their mix of products are unavailable. Because product-mix decisions are made at the level of the …rm, we aggregate the data to that level for ouranalysis.41
Our de…nition of “product” is based upon 1987 Standard Industrial Classi…cation(SIC) categories, which segment manufacturing output generally according to its end use.
We refer to …ve-digit SIC categories as products or goods.42 In our sample, aggregate
manufacturing contains 1848 products. For each …rm in each Census year, we record theset of products in which the …rm produces. We also observe …rms’ total and productlevel output. There are an average of 141,561 surviving …rms in each Census year forwhich such extensive-margin adjustments can be observed. For more detail, see Bernard,
Redding and Schott (2009).
A2. Linked/Longitudinal Firm Trade Transaction Database (LFTTD)
This dataset has two components. The …rst, foreign trade data assembled by the
U.S. Census Bureau and the U.S. Customs Bureau, captures all U.S. international tradetransactions between 1992 and 2004 inclusive. For each ‡ow of goods across a U.S. border,
this dataset records the product classi…cation and the value. Products in the LFTTD are
tracked according to ten-digit Harmonized System (HS) categories, which break exported
goods into 8572 products. The second component of the LFTTD is the Longitudinal
41

Firm identi…ers are derived from …rms’ legal identities, and …rms can consist of one or many establishments. Census uses an annual Company Organization Survey both to determine how new …rms
are organized and to keep track of changes in incumbent …rms’ownership structure over time, e.g., thebuying and selling of plants, the creation of new plants or the closing of existing plants.
42
SIC categories undergo minor revisions in each census year but experienced a major revision in 1987.
Census uses an internally generated concordance to map product codes collected in censuses after 1972
to the 1987 revision. To be conservative, we drop the roughly 1 percent of …ve-digit codes (representing
roughly 5 percent of total value) that do not appear in the 1987 to 1997 censuses.

Multi-Product Firms and Trade Liberalization

43

Business Database (LBD) of the U.S. Census Bureau, which records annual employment
and survival information for most U.S. establishments.43 Employment information foreach establishment is collected in March of every year and we aggregate the establishment
data up to the level of the …rm. Matching the annual information in the LBD to thetransaction-level trade data yields the LFTTD. We note that our ability to match tradetransactions to …rms is imperfect: across 1992 to 2004, we match transactions representing
76 and 82 percent of export and import value, respectively. For further details about theconstruction of the dataset, see Bernard, Jensen and Schott (2009).
A3. Other Data Sources
Industry-level Canadian tari¤ data are from Tre‡er (2004). To match Canadian SIC
manufacturing industries to U.S. SIC manufacturing industries we use a concordancedeveloped by Statistics Canada. Using this concordance, we can observe tari¤ changes
for 40 percent (174) of U.S. manufacturing industries representing 50 percent of total
manufacturing shipments in 1987. For industries where tari¤ information is missing, weassign the average of the two-digit SIC industry in which it belongs.
B

Theory Appendix

B1. Weighted Average Productivity
Weighted-average productivity in the domestic market is:

'
~d

"

1

1
G ('d )

Z 1

' ~ d (')

1g (') d'

'd

# 11

(30)

;

where ~ d (') denotes weighted-average consumer tastes in the domestic market for a …rm
with productivity ':~ d (') ="1

1
Z ( d ('))

Z 1

1
d

z ( d) d d

d (')

# 11

:

Similarly, weighted-average productivity in the export market is:
'
~x
43

1

1
G ('x )

Z 1

' ~ x (')

1

1

1g (') d'

;

(31)

'x

This dataset excludes the U.S. Postal Service and …rms in agriculture, forestry and …shing, railroads,
education, public administration and several smaller sectors. See Jarmin and Miranda (2002) for an
extensive discussion of the LBD and its construction.

Multi-Product Firms and Trade Liberalization

44

where ~ x (') denotes weighted-average consumer tastes in the export market for a …rm
with productivity ':~ x (') =1

1
Z ( x ('))

Z 1

1

1
x

1

z ( x) d x

:

x (')

B2. Revenue-Based Productivity
Following standard revenue-based measures of productivity (see for example Kletteand Griliches 1996 and De Loecker 2006), we de‡ate a …rm’s revenue from an individual
product by the aggregate price index for that product. The …rm’s real revenue from a
product in market j is therefore:
rj (';Pjj)=

pj (';j ) qj (';

j)

Pj

:

Using the inverse CES demand curve to substitute for pj (';rj (';
Pj

j)

1

=

j

qj (';

1

j)

Ej
Pj

j ), we obtain:
1:

Using the production technology and taking logarithms, we obtain:
log rj (';j)

log Pj =1log (' j ) +1

log lvj (';

j) +

1

log

Ej
Pj

;

(32)

where lvj ('; j ) denotes variable labor input and Ej =Pj captures aggregate product market conditions for market j.
From the “revenue production function” (32), the revenue-based measure of a …rm’s
productivity in a product for market j is ' j , which captures both true productivity,', and demand, j . Given data on revenue and inputs by …rm, product and market,
and given appropriate instruments for variable labor input, lvj ('; j ), and controls for
aggregate product market conditions, Ej =Pj , a …rm’s revenue-based productivity in a
product and market can be estimated from (32).
Following standard empirical methods for productivity aggregation, we de…ne productivity for the …rm as a whole as the revenue-share weighted average of productivity foreach product and market:
"Z
#
Z 1
1
'
r~d ('; d ) d d d + n
r~x ('; x ) x d x ;
(33)
= '

Z 1d

(')

x (')

r~ ('; ) dd (')

where r~j ('; j ) = rj ('; j ) z ( j ) =r (') for j 2 fd; xg andr~ ('; ) = [rd ('; ) + nrx ('; )] z ( ) =r (').

Multi-Product Firms and Trade Liberalization

45

While the exposition in this section of the appendix concentrates on measures ofrevenue-based productivity derived from production function estimation, similar results
hold for other revenue-based measures such as labor productivity. Although our assumption of constant elasticity of substitution preferences and monopolistic competition implies
that prices are inversely related to the …rm productivity draw ', revenue-based labor productivity is monotonically increasing in ' because of …xed production costs. As ' rises,
variable labor input and output rise, but prices fall, leaving revenue per variable input
unchanged. Nonetheless, as ' rises and variable labor input and revenue increase, the…xed labor input is spread over more units of revenue, which raises revenue-based laborproductivity.
B3. Fixed Production Costs
To simplify the characterization of a …rm’s decision to supply a market, we set …xed
production costs to zero in the main text and instead assume a …xed cost of supplying
each market. In this section of the Appendix, we show that the introduction of …xed
production costs is straightforward, but merely complicates the analysis without adding
additional insight, because it introduces interdependence across markets.
In a setting with demand heterogeneity across countries and …xed production costs,
the decision to whether supply a product to an individual market depends on whether that
product is produced, which depends on pro…tability across all markets. In the presenceof a product …xed production cost of # > 0, a …rm’s production and market supply
decisions can be characterized as follows. First, denote the (endogenous) set of products
produced by a …rm by (!) [0; 1], where we index …rms by ! because …rms with same
productivity ' can produce di¤erent sets of products depending on their idiosyncratic
draws for consumer tastes. Product k is supplied to market j by …rm ! with productivity' and consumer tastes jk if:
rjk (';

jk ; !)

fj ;

and

k2

(!) .

(34)

Second, denote the (endogenous) set of markets to which product k can be pro…tably supplied by …rm ! by k (!) [0; n + 1]. Product k is produced by …rm ! with productivity' and realizations of consumer tastes jk that vary across markets j if:Z
rjk ('; jk ; !)fj dj #:
(35)
j2

k (!)

Together these two equations characterize the …rm’s market supply and production decisions. Intuitively, when there are …xed production costs, a …rm will only supply a productto a market if it both draws a consumer tastes above the product cuto¤ (34) for thatmarket and also can generate su¢ cient variable pro…ts across all markets to cover …xed
production costs (35). The analysis remains exactly as in the main text except that inMulti-Product Firms and Trade Liberalization

46

addition to the product cuto¤ condition being satis…ed ((8) for the domestic market and(10) for each export market), we also require the pro…table production condition (35) to
be satis…ed.
B4. Proof of Proposition 1
Proof. As the model has a recursive structure, the determination of general equilibrium
is straightforward. We begin by determining {'d , 'x , d ('d ), x ('x )}. The zero-pro…tcuto¤ condition for the least productive …rm (14) determines d ('d ) independently ofthe other equations of the model. Similarly, the exporting cuto¤ condition for the least
productive exporter (16) determines x ('x ) independently of the other equations of themodel. Having characterized { d ('d ), x ('x )}, 'd can be determined from the free entry
condition independently of the other equations of the model. Substituting for d ('),
x ('), and 'x using (9), (11) and (17), the free entry condition (19) can be written
solely in terms of 'd and the already determined d ('d ). As ' ! 0, V ! 1, while
as ' ! 1, V ! 0. Moreover, V is monotonically decreasing in 'd . Therefore the freeentry condition alone determines a unique equilibrium value of 'd . Having characterized{ d ('d ), x ('x ), 'd }, 'x follows immediately from (17). Given { d ('d ), x ('x ), 'd ,'x }, we can determine d (') for all ' 'd and x (') for all ' 'x from (9) and (11).
We next determine the price index and aggregate revenue for each product {P , R}.
Aggregate revenue for the economy as a whole can be determined from the steady-statestability and free entry conditions, which as discussed in the main text imply R = L.
The price index for each product, P , depends on weighted average productivity and themass of …rms in the domestic and export markets (equation (25)). Having characterized{'d , 'x , d ('d ), x ('x )}, and hence the functions d (') and x ('), weighted average
productivity in the domestic and export market follow from (30) and (31). To determine
the mass of …rms supplying each product to the domestic and export markets, note that
average …rm revenue can be written as a function of {'d , 'x , d ('d ), x ('x )} alone using(7):#
Z 1 "Z 1
1
g (')
d
r =
fd z ( d ) d d
d'
(36)
(')
1
G
('
)
'd
(')
d
d
d
#
Z 1 "Z 1
1
1 G ('x )
g (')
x
+
d'
fx z ( x ) d x
1 G ('d )
(')
1
G
('
)
'x
(')
x
x
x
The mass of …rms follows immediately from aggregate revenue and average revenue: M =R=r. Combining the mass of …rms with {'d , 'x , d ('d ), x ('x )} and the functions
d (') and x (') determines the mass of …rms supplying a given product to the domestic
market, Md , in (23) and the export market, Mx , in (24). This completes the determinationof the product price index. Revenue for each product follows immediately from the CESMulti-Product Firms and Trade Liberalization47

revenue function, the product price index and aggregate revenue. This completes thecharacterization of the equilibrium sextuple {'d , 'x , d ('d ), x ('x ), P , R}.
B5. Proof of Proposition 2
Proof. The free entry condition in the open economy (19) takes the same form as in theclosed economy except that the expression for the expected value of entry, V , includes anadditional positive term (in the closed economy 'x ! 1 and x (') ! 1). From theproof of Proposition 1, V can be written solely in terms of 'd and d ('d ), where from
(14) d ('d ) is unchanged by the opening of trade. As V is monotonically decreasing in'd , it follows that 'd must be higher in the open economy than in the closed economy inorder to equate V with the unchanged sunk entry cost.
As d (') = ('d =') d ('d ) is monotonically increasing in 'd , and d ('d ) is unchanged by
the opening of trade, the opening of trade raises d (') for each value of ', and induces…rms to drop low-consumer-tastes products from the domestic market. In addition, somesurviving …rms begin to export for …nite values of 'x and x (') = ('x =') x ('x ) in the
open economy. As there is selection into export markets, these new entrants into exportingare high-productivity …rms, 'x > 'd , and the products that they add in export markets
have high consumer tastes, x (') > d (').
The e¤ect of the opening of trade on …rm revenue-based productivity (33) depends onthe change in the …rm revenue share of products with di¤erent values of consumer tastes.
Under autarky, the …rm revenue share of products with consumer tastes 2 [ dA (') ; 1)is:(' ) 1 z ( )
;
1
('
)
z
(
)
d
A
(')

r~A ('; ) = R 1

2 [ dA (') ; 1);

(37)

d

where r~ ('; ) r ('; ) z ( ) =r (') is de…ned as above; the superscript A denotes autarky;
and the superscript T will be used to denote the open economy below. To characterize the…rm revenue share of products in the open economy, consider non-exporters and exporters
in turn.
(a) For non-exporters, products with consumer tastes 2 [ dA (') ; dT (')) are dropped
from the domestic market and hence experience a decline in their share of …rm revenue.
In contrast, products with consumer tastes 2 [ dT (') ; 1) experience a rise in theirshare of …rm revenue. Therefore the distribution r~ ('; ) in the open economy …rst-orderstochastically dominates that in the closed economy, and revenue-based productivity (33)
for non-exporters rises.
(b) For exporters, products with consumer tastes
2 [ dA (') ; dT (')) are dropped
from the domestic market and hence experience a decline in their share of …rm revenue.
Products with consumer tastes 2 [ dT (') ; xT (')) experience an ambiguous change inMulti-Product Firms and Trade Liberalization48

their share of …rm revenue:(' ) 1 z ( )r~ ('; ) = R 1(' ) 1 z ( ) dAd (')
Z dT (')(' ) 1 z ( ) d
1
T

A
d (')

2 [ dT (') ;

;
1

n

1

Z 1

(' )

1

T
x ('));

z( )d :Tx (')

As the sign of 1 is in general ambiguous, r~T ('; ) ? r~A ('; ) for 2 [ dT (') ; xT (')).
Finally, products with consumers tastes 2 [ xT (') ; 1) experience a rise in their shareof …rm revenue:r~T ('; ) =2[1 + n 1 ] (' ) 1 z ( );2 [ xT (') ; 1);
R1
1
1[1 + n] A (') (' )
z( )d
2
d
2
3
T
R
(')
[1 + n 1 ] dA (') (' ) 1 z ( ) d4
5 > 0:R T (') d+ xT (') n 1 (' ) 1 z ( ) ddwhere we have re-written the denominator of r~T ('; ) in a di¤erent form. As 2 > 0,r~T ('; ) > r~A ('; ) for 2 [ xT (') ; 1).
Irrespective of whether the revenue share of products with intermediate consumer tastes
2 [ dA (') ; dT (')) rises or falls, the di¤erence between the open economy value ofr~ ('; ) and the closed economy value goes from being negative at low values of2ATT[ d (') ; d (')) to being positive at high values of 2 [ x (') ; 1). This is a su¢ cientcondition for the distribution r~ ('; ) in the open economy to …rst-order stochasticallydominate that in the closed economy. Therefore revenue-based …rm productivity (33) for
exporters rises.
B6. Proof of Proposition 3
Proof. We …rst characterize d'd =d . From the free entry condition (19), de…ne=
V fe . By the implicit function theorem, d'd =d = (d =d ) = (d =d'd ). Substituting
for 'x , d (') and x (') in (19) using (9), (14), (11), (16), and (17), we obtain dV =d < 0and dV =d'd < 0. Therefore, we have established that d'd =d < 0.
We next characterize d'x =d . Di¤erentiating with respect to in equation (17), we
obtain:
d'x
d 'x

=1+

d'd
d 'd

;

(38)

It follows that to establish d'x =d > 0, it su¢ ces to show that (d'd =d ) = ( ='d ) > 1. To
do so, we again use the implicit function theorem to evaluate d'd =d = (d =d ) = (d =d'd ).
Additionally, equations (9), (14), (11), (16), and (17) imply the following: d x (') =d =x (') = , d x (') =d'd = x (') =' , d'x =d = 'x = and d'x =d'd = 'x ='d . Combining

Multi-Product Firms and Trade Liberalization

49

these results with d'd =d = (d =d ) = (d =d'd ), we obtain (d'd =d ) = ( ='d ) > 1.
Therefore we have established that d'x =d > 0.
Since d (') = ('d =') d ('d ), where d ('d ) is invariant to , and since d'd =d < 0, we
have established that d d (') =d < 0. Additionally, since x (') = ('x =') x ('x ), wherex ('x ) is invariant to , and since d'x =d > 0, we have established that d x (') =d > 0.
The remainder of the proof follows a similar structure as for Proposition 2 above. Therefore, to conserve space, we sketch the remainder of the proof here and present the complete
derivation in the web-based technical appendix.
(a) As d d (') =d < 0, reductions in variable trade costs induce domestic …rms to drop
low consumer tastes products from the domestic market, which raises their revenue-based
productivity (33).
(b) As d d (') =d < 0, reductions in variable trade costs also induce new exporters todrop low-consumer-tastes products from the domestic market, which raises their revenuebased productivity (33). In addition, new exporters add high consumer tastes productsin the export market (since x (') > d (') ), which further raises their revenue-based
productivity.
(c) As d d (') =d < 0, reductions in variable trade costs also induce continuing exporters to drop low consumer tastes products from the domestic market, which raises
their revenue-based productivity (33). In addition, as d x (') =d > 0, reductions in variable trade costs induce continuing exporters to add additional products in export markets.
As these additional products have low values of consumer tastes relative to the products
exported prior to the reduction in variable trade costs, adding these additional products
can reduce the revenue-based productivity of continuing exporters. The overall change inthe revenue-based productivity of continuing exporters is therefore ambiguous.
B7. Proof of Proposition 4
Proof. (a) and (b): The share of products exported to a given country by existing
exporters with productivity ' 'x is [1 Z ( x ('))]. Similarly, the share of countries towhich a given product is exported by existing exporters of that product is [1 Z ( x ('))].
Additionally, from (11) and (16), we have x (') = ('x =') x ('x ) where x ('x ) is invariant to . Therefore, to establish (a) and (b), it su¢ ces to show d'x =d > 0, which
has already been established in the proof of Proposition 3. Hence reductions in variable
trade costs reduce 'x , reduce x ('), and raise [1 Z ( x ('))].
(c) and (d): The share of …rms that export equals[1 G ('x )] = [1 G ('d )]. Since
the proof of Proposition 3 established d'd =d < 0 and d'x =d > 0, it follows thatd =d < 0. Hence reductions in variable trade costs increase the share of …rms thatexport, .

Multi-Product Firms and Trade Liberalization

50(e) Average exports per product-country for an existing exporter are:
Z 1
1
1
x
fx z ( x ) d x :rx (') =1 Z ( x (')) x (')
x (')

(39)

Since x (') = ('x =') x ('x ), where x ('x ) is invariant to , and since d'x =d > 0, we
have established that d x (') =d > 0. Now note that from (39):32drx (') 6dZ ( x (')) =d x (')=6rx (')
4d
1 Z ( x ('))
|
{z
}

(

1)
rx (')x (')

fx z ( x (')) 7
7 d x (') ;1 Z ( x (')) 5 | d{z }>0>0which is in general ambiguous in sign depending on the value of x (') and the functional
form of the cumulative distribution function for consumer tastes.
B8. Proof of Proposition 5
Proof. (a) and (b): The share of products exported to a given country by existing
exporters with productivity ' 'x is [1 Z ( x ('))]. Similarly, the share of countries towhich a given product is exported by existing exporters is [1 Z ( x ('))]. As from (11)x (') = ('x =') x ('x ) is monotonically decreasing in ', and as Z ( ) is a continuous
cumulative distribution function that is increasing in , it follows that [1 Z ( x ('))] is
increasing in ', which establishes parts (a) and (b) of the proposition.
(c) Average exports per product-country for an existing exporter are (39). Therefore:23drx (') 6dZ ( x (')) =d x (')=6rx (')
4d'
1 Z ( x ('))
|
{z
}
>0

(

1)
rx (')x (')

fx z ( x (')) 7
7 d x (')
1 Z ( x (')) 5 d'| {z }<0which is in general ambiguous in sign depending on the value of x (') and the functional
form of the cumulative distribution function for consumer tastes.