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American Economic Review 2008, 98:4, 1707–1721
http://www.aeaweb.org/articles.php?doi
=
10.1257/aer.98.4.1707
By considering a model with identical rms, Paul Krugman (1980) predicts that a higher elas-
ticity of substitution between goods magnies the impact of trade barriers on trade ows. In this
paper, I introduce rm heterogeneity in a simple model of international trade. When the distribu-
tion of productivity across rms is Pareto, which is close to the observed size distribution of US
rms, the predictions of the Krugman model with representative rms are overturned: the impact
of trade barriers on trade ows is dampened by the elasticity of substitution, and not magnied.
In Krugman (1980), identical countries trade differentiated goods despite the presence of trade
barriers because consumers have a preference for variety. If goods are less substitutable, con-
sumers are willing to buy foreign varieties even at a higher cost, and trade barriers have little
impact on bilateral trade ows. Total exports from country A to country B are given by the fol-
lowing expression:
GDP
A
3 GDP
B
Exports
AB
5 Constant 3 ,
1Trade barriers
AB
2
s
where s is the elasticity of substitution between varieties. A crucial assumption in this model is
that all rms are identical, and that the only form of transportation cost is a variable cost. Under
these assumptions, every rm exports to every country in the world. Trade barriers have a strong
impact on trade ows when the elasticity of substitution between goods is high. Competition is
erce when the elasticity of substitution is high, and any cost disadvantage translates into large
losses of market share.
In this paper, I add rm heterogeneity in productivity, as well as xed costs of exporting.
These simple amendments introduce a new margin of adjustment: the extensive margin. When
transportation costs vary, not only does each exporter change the size of its exports (the intensive
margin), but the set of exporters varies as well (the extensive margin). The main nding of this
paper is that the elasticity of substitution has opposite effects on each margin. A higher elastic-
ity makes the intensive margin more sensitive to changes in trade barriers, whereas it makes the
extensive margin less sensitive. The reason is the following. When trade barriers decrease, new
and less productive rms enter the export market. When the elasticity of substitution is high, a
low productivity is a severe disadvantage. These less productive rms can capture only a small
market share. The impact of those new entrants on aggregate trade is small. On the other hand,
when the elasticity is low, each rm is sheltered from competition. The new entrants capture a
Distorted Gravity: The Intensive and Extensive Margins
of International Trade
By T C*
* Chaney: Department of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637 (e-mail:
tchaney@uchicago.edu). I am grateful to Xavier Gabaix, Marc Melitz, and Daron Acemoglu for their encourage-
ment and advice. I also thank the editor, three anonymous referees, and Richard Rogerson for their comments. For
their suggestions and comments, I also wish to thank Sylvain Chassang, Sam Kortum, David Sraer, and participants
at the Federal Reserve Board of Governors, Boston College, Boston University, Columbia University, University of
Chicago, Chicago GSB, Harvard University, INSEAD, LSE, MIT, the New York Fed, New York University, University
of Pennsylvania, University of California–Berkeley, University of California–San Diego, University of Wisconsin–
Madison, the World Bank, and on the REStud Tour 2005. All remaining errors are mine.
SEPTEMBER 20081708
THE AMERICAN ECONOMIC REVIEW
large market share. The impact of those new entrants on aggregate trade is large. So a higher
elasticity of substitution magnies the sensitivity of the intensive margin to changes in trade bar-
riers, whereas it dampens the sensitivity of the extensive margin.
Which effect dominates? I prove that when the distribution of productivity across rms is
Pareto, which is a good approximation of the observed distribution of US rms,
1
the effect on
the extensive margin dominates. My augmented model predicts that total exports from country
A to country B are given by the following expression:
GDP
A
3 GDP
B
Exports
AB
5 Constant 3 with e91s2 , 0.
1Trade barriers
AB
2
e 1s2
The elasticity of aggregate trade with respect to trade barriers (both variable and xed), e, is
negatively related to the elasticity of substitution, s. Variable trade barriers enter the gravity
equation with an exponent that depends only on the distribution of productivity and not on the
elasticity of substitution, and xed trade barriers with an exponent that is inversely related to the
elasticity of substitution.
The model with heterogeneous rms also predicts that the same trade barriers will have a
larger impact on trade ows than in the model with representative rms. When trade barriers
decrease, each rm exports more. In addition, new rms start exporting. This adjustment on the
extensive margin is quantitatively important. Given the observed distribution of rm size in the
United States, I predict that the elasticity of trade ows with respect to variable trade barriers
such as tariffs is twice as large as it would be in the absence of rm heterogeneity.
The prediction, that the effect of trade barriers on trade ows is magnied by the elasticity of
substitution, is not specic to Krugman’s model of trade. Maurice Obstfeld and Kenneth Rogoff
(2001), for example, explain the six major puzzles in International Macroeconomics by the exis-
tence of trade barriers. The simple model they spell out to illustrate how plausible values for
trade barriers can have a large impact on trade ows relies on the magnication by the elasticity
of substitution. James E. Anderson (1979) presents a theoretical foundation for the gravity equa-
tion based on the Armington assumption of competitive trade in goods differentiated by country
of origin. In both models, a higher elasticity of substitution will magnify the effect of trade bar-
riers on trade ows, even in the absence of increasing returns or monopolistic competition.
The main contribution of this paper is to introduce the extensive margin of trade in a simple and
tractable model with multiple countries and asymmetric trade barriers. The elasticity of aggregate
trade ows with respect to trade barriers is larger than what traditional models would predict. It is
not equal to the elasticity of substitution; it is inversely related to the elasticity of substitution.
In the remainder of this section, I review previous work related to this model, and existing
empirical evidence that supports the predictions of this model.
Marc Melitz (2003) introduces rm heterogeneity in a general equilibrium model of interna-
tional trade. I expand Melitz’s model in the following way. I consider a world with many asym-
metric countries, separated by asymmetric trade barriers. I then study the strategic choice of
rms to export or not, and if they export, which countries to target. I embed my model in a global
equilibrium. Such a model generates predictions for the structure of bilateral trade ows. I can
pin down which rm from which country is able to enter a given market, and how it is affected
by competition from local and other foreign rms, even in the presence of asymmetric bilateral
trade barriers. The presence of xed costs associated with entering foreign markets provides a
simple foundation for the extensive margin of trade.
1
See Erzo G. J. Luttmer (2007) for the most recent evidence, and this introduction for further references.
VOL. 98 NO. 4 1709
CHANEY: DISTORTED GRAVITY
Elhanan Helpman, Melitz, and Yona Rubinstein (forthcoming) develop a similar extension to
the Melitz model with multiple countries. Using bounded support for the productivity shocks,
they can make use of the information contained in the zeros of the trade matrices and improve
on the traditional gravity regressions. They do not, however, generate analytical solutions for the
extensive margin of trade. This gives them more exibility in estimating empirically the prob-
ability that exporters enter a given foreign market. But it prevents them from deriving precise
predictions for the role of variable and xed costs in explaining both the intensive and the exten-
sive margins of international trade.
Kim J. Ruhl (2005) builds a dynamic version of the Melitz model to explain the so-called
elasticity puzzle. He argues that in response to high frequency transitory shocks, most of the
adjustments of exports happen at the intensive margin, whereas in response to permanent shocks
such as trade liberalization, both the intensive and the extensive margins adjust. I abstract from
any dynamic considerations and build a model of the steady-state trade ows between many
countries. Costas Arkolakis (2007) offers an extension to the current model to explain the exis-
tence of small exporters, even in the presence of xed trade barriers. He proposes that rms can
decide what fraction of a market they want to access, where the xed entry cost increases with
the number of consumers reached.
Jonathan Eaton, Sam Kortum, and Francis Kramarz (2007) nd that the current model pro-
vides a good description of rm-level trade using data on French exporters. Among others, the
current model predicts correctly many of the patterns of entry of heterogeneous rms into dif-
ferent markets, and the relationship between the size of a rm on its domestic market, and the
number of foreign markets it enters. The assumption that productivity shocks are Pareto distrib-
uted provides a good t for the rm-level data: it describes precisely the distribution of rm size
within France, as well as which foreign markets a given rm enters.
There is wide empirical evidence that the Pareto distribution is a good approximation of the
upper tail of the distribution of rm sizes. Since exporters are overwhelmingly large rms, and
therefore in the upper tail of the size distribution, this distribution is a good candidate for a
theoretical model of rm selection into export markets. Herbert A. Simon and Charles P. Bonini
(1958) rst noted that the size distribution of rm sizes is well described by a Pareto distribu-
tion. Recent evidence on this empirical regularity for the United States include Robert L. Axtell
(2001) and Luttmer (2007). Xavier Gabaix (2008) provides a survey on the prevalence of “power
law” distributions for rms in the United States and in Europe. Helpman, Melitz, and Stephen
Yeaple (2004) estimate a Pareto distribution for both US and European rms to predict foreign
direct investment in different sectors.
The closest evidence in support of the predictions of the current model are James E. Rauch
(1999), Martin Andersson (2007), Matthieu Crozet and Pamina Koenig (2007), and Koenig
(2005). Rauch nds that trade barriers have a milder impact on trade volumes for goods that
are more homogenous. He denes homogenous goods as goods that are traded on organized
exchanges, or goods that have a reference price. He argues that acquiring information about
differentiated goods is costly, so that effectively differentiated goods face a higher trade barrier.
This reasoning can, however, explain why there should be more trade in homogenous goods, but
not why given trade barriers should have a bigger impact. The current model offers an alternative
explanation for the interaction between product differentiation and trade barriers. I spell out a
clear theoretical channel through which product differentiation affects trade barriers. Andersson
(2007) uses rm-level export data on Swedish rms. He separates out the impact of variable
trade barriers and the impact of xed trade barriers (proxied by measures of “familiarity” of
markets), and separates out their impact on the intensive and extensive margins of trade. First,
he nds that xed costs have a larger impact on the extensive margin than on the intensive
margin of trade. Moreover, the impact of xed trade barriers on the extensive margin is larger
SEPTEMBER 20081710
THE AMERICAN ECONOMIC REVIEW
for differentiated goods than for homogenous goods.
2
Crozet and Koenig (2007) use rm-level
export data on French rms to structurally estimate the current model. The panel dimension
of the data allows them to separate out the distance elasticity of trade costs from the elasticity
of exports with respect to trade barriers. Koenig (2005) uses the same rm-level export data
on French rms. As the current model predicts, she nds that the distance elasticity of indi-
vidual rm exports (the intensive margin) is larger in sectors where goods are more homogenous,
whereas the distance elasticity of the number of rms (the extensive margin) is smaller in sectors
where goods are more homogenous. In addition, she nds that the share of exports explained by
the extensive margin is larger in sectors with more differentiated goods. At the aggregate level,
unlike the predictions of this model, she nds that in sectors with homogenous goods, the dis-
tance elasticity of total exports is mildly larger than in sectors with differentiated goods, but this
difference is not signicant.
Finally, several authors have stressed the quantitative importance of the extensive margin in
explaining aggregate trade ows. David Hummels and Peter J. Klenow (2005) nd that larger
and wealthier countries trade more, and that 60 percent of the difference in aggregate trade ows
comes from differences in the number of goods traded. Along a slightly different line, Kei-Mu Yi
(2003) argues that the increase in trade in intermediate goods, which amounts to trade in more
goods, can help explain the observed increase in international trade in the last decades. All nd
a strong response of the extensive margin to changes in trade barriers or country size that are
consistent with the current model.
The remainder of the paper is organized as follows. Section I introduces a simple model of
international trade with heterogeneous rms and derives partial equilibrium results. In Section II,
I compute the general equilibrium of the world economy. Finally, Section III identies separately
the adjustments of the intensive and the extensive margins of trade, in response to changes in
both variable and xed trade barriers.
I. Setup
In this section, I introduce the basic ingredients of the model. I dene preferences and
technologies, and I characterize the optimal strategies of both rms and consumers in partial
equilibrium.
There are N potentially asymmetric countries that produce goods using only labor. Country n
has a population L
n
. Consumers in each country maximize utility derived from the consumption
of goods from H 1 1 sectors. Sector 0 provides a single homogenous good. The other H sectors
are made of a continuum of differentiated goods. If a consumer consumes q
o
units of good 0, and
q
h
1v 2 units of each variety v of good h, for all varieties in the set V
h
(determined in equilibrium),
she gets a utility U,
(1)
U ; q
o
m
0
q
H
h5 1
q
3
V
h
q
h
1v 2
1s
h
212/s
h
dvr
3s
h
/ 1s
h
2124 m
h
,
where m
0
1
g
H
h5 1
m
h
5 1, and where s
h
. 1 is the elasticity of substitution between two varieties
of good h. I assume that H . 1 so that I can compare sectors characterized by different degrees
of product substitutability.
3
2
The impact of distance is also larger, which suggests that distance proxies for both variable and some xed costs.
3
It is empirically more relevant to compare sectors with different degrees of product substitutability than to com-
pare different economies.
VOL. 98 NO. 4 1711
CHANEY: DISTORTED GRAVITY
Trade Barriers and Technology.—The homogenous good 0 is freely traded and is used as
the numeraire. It is produced under constant returns to scale with one unit of labor in country n
producing w
n
units of good 0. Its price is set equal to 1 so that if country n produces this good,
the wage in country n is w
n
. I shall consider only equilibria where every country produces some
of the numeraire.
4
This assumption greatly simplies the analysis. It allows countries to differ
both in size 1L
n
2 and in productivity 1w
n
2. There are two types of trade barriers, a variable and
a xed cost. The variable cost takes the form of an “iceberg” transportation cost. If one unit of
any differentiated good h is shipped from country i to country j, only a fraction 1/t
h
ij
arrives. The
rest melts on the way. The higher t, the higher the variable trade cost.
5
In addition, if a rm from
country i in sector h exports to country j, it must pay a xed cost f
ij
h
, in units of the numeraire.
All countries have access to the same technology. Due to the presence of xed costs, rms
in the differentiated sectors operate under increasing returns-to-scale technology. Each rm in
sector h draws a random unit labor productivity w. The cost of producing q units of a good and
selling them in country j for a rm with productivity w is
(2) c
ij
h
1q2 5
w
i
t
h
ij
w
q 1 f
ij
h
.
Firms are price setters. Given that demand functions are isoelastic, the optimal price charged in
country j by rm w from country i is a constant mark-up over the unit cost (including transporta-
tion costs): p
ij
h
1w2 5 s
h
/1s
h
2 12 3 w
i
t
h
ij
/w.
As in Helpman, Melitz, and Yeaple (2004), I assume that productivity shocks are drawn from a
Pareto distribution with shape parameter g
h
: productivity is distributed over 31, 1 `2 according to
(3) P 1w
˜
h
, w2 5 G
h
1w2 5 1 2 w
2g
h
,
with g
h
. s
h
2 1. An inverse measure of the heterogeneity in sector h is given by g
h
. Sectors with
a high g are more homogenous, in the sense that more output is concentrated among the smallest
and least productive rms.
6
The assumption that productivity shocks are Pareto distributed is
made rst for analytical tractability, and second because it provides a good approximation of the
distribution of rm sizes in the United States.
7
I assume that the total mass of potential entrants in country n in each differentiated sector is
proportional to w
n
L
n
, so that larger and wealthier countries have more entrants. This assumption
greatly simplies the analysis. It is similar to Eaton and Kortum (2002), where the set of goods
is exogenously given. Since I do not impose free entry, rms generate net prots that have to be
redistributed. I assume that each worker owns w
n
shares of a global fund. The fund collects prof-
its from all rms and redistributes them in units of the numeraire good to its shareholders.
Demand for Differentiated Goods.—The total income spent by workers in country j, Y
j
, is
the sum of their labor income 1w
j
L
j
2 and of the dividends they get from their portfolio 1w
j
L
j
p2,
where p is the dividend per share of the global mutual fund. Given the optimal pricing of rms,
4
As long as the share of the homogenous good, m
0
, is large enough, or trade barriers in the other sectors are large
enough, this condition will hold.
5
t
h
ij
. 1 for any i Z j and t
h
ii
5 1. I also impose a triangular inequality to prevent transportation arbitrages: 5 1i, j, k2,
t
ik
# t
ij
3 t
jk
.
6
ln w has a standard deviation equal to 1/g. The assumption g . s 2 1 ensures that, in equilibrium, the size distri-
bution of rms has a nite mean.
7
See Luttmer (2002) for the most recent evidence.