American Economic Review 2008, 98:4, 1707–1721
http://www.aeaweb.org/articles.php?doi=10.1257/aer.98.4.1707
Distorted Gravity: The Intensive and Extensive Margins
of International Trade
By Thomas Chaney*
By considering a model with identical firms, Paul Krugman (1980) predicts that a higher elasticity of substitution between goods magnifies the impact of trade barriers on trade flows. In this
paper, I introduce firm heterogeneity in a simple model of international trade. When the distribution of productivity across firms is Pareto, which is close to the observed size distribution of US
firms, the predictions of the Krugman model with representative firms are overturned: the impact
of trade barriers on trade flows is dampened by the elasticity of substitution, and not magnified.
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, consumers are willing to buy foreign varieties even at a higher cost, and trade barriers have little
impact on bilateral trade flows. Total exports from country A to country B are given by the following expression:
GDPA 3 GDPB
ExportsAB 5 Constant 3 s ,
1Trade barriersAB2
where s is the elasticity of substitution between varieties. A crucial assumption in this model is
that all firms are identical, and that the only form of transportation cost is a variable cost. Under
these assumptions, every firm exports to every country in the world. Trade barriers have a strong
impact on trade flows when the elasticity of substitution between goods is high. Competition is
fierce when the elasticity of substitution is high, and any cost disadvantage translates into large
losses of market share.
In this paper, I add firm heterogeneity in productivity, as well as fixed 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 finding of this
paper is that the elasticity of substitution has opposite effects on each margin. A higher elasticity 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 firms enter the export market. When the elasticity of substitution is high, a
low productivity is a severe disadvantage. These less productive firms 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 firm is sheltered from competition. The new entrants capture a
* 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 encouragement 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.
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September 2008
large market share. The impact of those new entrants on aggregate trade is large. So a higher
elasticity of substitution magnifies the sensitivity of the intensive margin to changes in trade barriers, whereas it dampens the sensitivity of the extensive margin.
Which effect dominates? I prove that when the distribution of productivity across firms is
Pareto, which is a good approximation of the observed distribution of US firms, 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:
GDPA 3 GDPB
ExportsAB 5 Constant 3 with e91s2 , 0.
1Trade barriersAB2 e 1s2
The elasticity of aggregate trade with respect to trade barriers (both variable and fixed), 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 fixed trade barriers with an exponent that is inversely related to the
elasticity of substitution.
The model with heterogeneous firms also predicts that the same trade barriers will have a
larger impact on trade flows than in the model with representative firms. When trade barriers
decrease, each firm exports more. In addition, new firms start exporting. This adjustment on the
extensive margin is quantitatively important. Given the observed distribution of firm size in the
United States, I predict that the elasticity of trade flows with respect to variable trade barriers
such as tariffs is twice as large as it would be in the absence of firm heterogeneity.
The prediction, that the effect of trade barriers on trade flows is magnified by the elasticity of
substitution, is not specific to Krugman’s model of trade. Maurice Obstfeld and Kenneth Rogoff
(2001), for example, explain the six major puzzles in International Macroeconomics by the existence of trade barriers. The simple model they spell out to illustrate how plausible values for
trade barriers can have a large impact on trade flows relies on the magnification by the elasticity
of substitution. James E. Anderson (1979) presents a theoretical foundation for the gravity equation 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 barriers on trade flows, 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 flows 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 firm heterogeneity in a general equilibrium model of international trade. I expand Melitz’s model in the following way. I consider a world with many asymmetric countries, separated by asymmetric trade barriers. I then study the strategic choice of
firms 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 flows. I can
pin down which firm from which country is able to enter a given market, and how it is affected
by competition from local and other foreign firms, even in the presence of asymmetric bilateral
trade barriers. The presence of fixed costs associated with entering foreign markets provides a
simple foundation for the extensive margin of trade.
See Erzo G. J. Luttmer (2007) for the most recent evidence, and this introduction for further references.
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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 flexibility in estimating empirically the probability that exporters enter a given foreign market. But it prevents them from deriving precise
predictions for the role of variable and fixed costs in explaining both the intensive and the extensive 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 flows between many
countries. Costas Arkolakis (2007) offers an extension to the current model to explain the existence of small exporters, even in the presence of fixed trade barriers. He proposes that firms can
decide what fraction of a market they want to access, where the fixed entry cost increases with
the number of consumers reached.
Jonathan Eaton, Sam Kortum, and Francis Kramarz (2007) find that the current model provides a good description of firm-level trade using data on French exporters. Among others, the
current model predicts correctly many of the patterns of entry of heterogeneous firms into different markets, and the relationship between the size of a firm on its domestic market, and the
number of foreign markets it enters. The assumption that productivity shocks are Pareto distributed provides a good fit for the firm-level data: it describes precisely the distribution of firm size
within France, as well as which foreign markets a given firm enters.
There is wide empirical evidence that the Pareto distribution is a good approximation of the
upper tail of the distribution of firm sizes. Since exporters are overwhelmingly large firms, and
therefore in the upper tail of the size distribution, this distribution is a good candidate for a
theoretical model of firm selection into export markets. Herbert A. Simon and Charles P. Bonini
(1958) first noted that the size distribution of firm sizes is well described by a Pareto distribution. 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 firms in the United States and in Europe. Helpman, Melitz, and Stephen
Yeaple (2004) estimate a Pareto distribution for both US and European firms 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 finds that trade barriers have a milder impact on trade volumes for goods that
are more homogenous. He defines 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 firm-level export data on Swedish firms. He separates out the impact of variable
trade barriers and the impact of fixed trade barriers (proxied by measures of “familiarity” of
markets), and separates out their impact on the intensive and extensive margins of trade. First,
he finds that fixed costs have a larger impact on the extensive margin than on the intensive
margin of trade. Moreover, the impact of fixed trade barriers on the extensive margin is larger
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for ­differentiated goods than for homogenous goods. Crozet and Koenig (2007) use firm-level
export data on French firms 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 firm-level export data
on French firms. As the current model predicts, she finds that the distance elasticity of individual firm exports (the intensive margin) is larger in sectors where goods are more homogenous,
whereas the distance elasticity of the number of firms (the extensive margin) is smaller in sectors
where goods are more homogenous. In addition, she finds 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 finds that in sectors with homogenous goods, the distance elasticity of total exports is mildly larger than in sectors with differentiated goods, but this
difference is not significant.
Finally, several authors have stressed the quantitative importance of the extensive margin in
explaining aggregate trade flows. David Hummels and Peter J. Klenow (2005) find that larger
and wealthier countries trade more, and that 60 percent of the difference in aggregate trade flows
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 find
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 firms and derives partial equilibrium results. In Sec­tion II,
I compute the general equilibrium of the world economy. Finally, Section III identifies separately
the adjustments of the intensive and the extensive margins of trade, in response to changes in
both variable and fixed trade barriers.
I. Setup
In this section, I introduce the basic ingredients of the model. I define preferences and
technologies, and I characterize the optimal strategies of both firms and consumers in partial
equilibrium.
There are N potentially asymmetric countries that produce goods using only labor. Country n
has a population Ln. 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 qo units of good 0, and
qh 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,
U;
(1) qom0
q q3 qh 1v 2
H
h51
Vh
1sh212/sh
dvr
3sh / 1sh212 4 mh
,
where m0 1 g h51 mh 5 1, and where sh . 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.
H
The impact of distance is also larger, which suggests that distance proxies for both variable and some fixed costs.
It is empirically more relevant to compare sectors with different degrees of product substitutability than to compare different economies.
VOL. 98 NO. 4
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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 wn 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 wn. I shall consider only equilibria where every country produces some
of the numeraire. This assumption greatly simplifies the analysis. It allows countries to differ
both in size 1Ln 2 and in productivity 1wn 2. There are two types of trade barriers, a variable and
a fixed 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 hij arrives. The
rest melts on the way. The higher t, the higher the variable trade cost. In addition, if a firm from
country i in sector h exports to country j, it must pay a fixed cost fijh, in units of the numeraire.
All countries have access to the same technology. Due to the presence of fixed costs, firms
in the differentiated sectors operate under increasing returns-to-scale technology. Each firm 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 firm with productivity w is
(2) cijh
w ithij
1q 2 5
q 1 fijh.
w
Firms are price setters. Given that demand functions are isoelastic, the optimal price charged in
country j by firm w from country i is a constant mark-up over the unit cost (including transportation costs): pijh 1w 2 5 sh / 1sh 2 12 3 wi t hij /w.
As in Helpman, Melitz, and Yeaple (2004), I assume that productivity shocks are drawn from a
Pareto distribution with shape parameter gh : productivity is distributed over 31, 1 `2 according to
(3) P 1w̃h , w 2 5 Gh 1w 2 5 1 2 w2gh,
with gh . sh 2 1. An inverse measure of the heterogeneity in sector h is given by gh. Sectors with
a high g are more homogenous, in the sense that more output is concentrated among the smallest
and least productive firms. The assumption that productivity shocks are Pareto distributed is
made first for analytical tractability, and second because it provides a good approximation of the
distribution of firm sizes in the United States.
I assume that the total mass of potential entrants in country n in each differentiated sector is
proportional to wn Ln , so that larger and wealthier countries have more entrants. This assumption
greatly simplifies 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, firms generate net profits that have to be
redistributed. I assume that each worker owns wn shares of a global fund. The fund collects profits from all firms 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, Yj , is
the sum of their labor income 1wj Lj 2 and of the dividends they get from their portfolio 1wj Lj p2 ,
where p is the dividend per share of the global mutual fund. Given the optimal pricing of firms,
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.
h
t ij . 1 for any i Z j and t hii 5 1. I also impose a triangular inequality to prevent transportation arbitrages: 5 1i, j, k 2 ,
tik # tij 3 tjk.
ln w has a standard deviation equal to 1/g. The assumption g . s 2 1 ensures that, in equilibrium, the size distribution of firms has a finite mean.
See Luttmer (2002) for the most recent evidence.
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and the demand by consumers, exports from country i to country j in sector h, by a firm with a
labor productivity w, are
phij 1 w 2 1 2 sh
(4) x hij 1w 2 5 phij 1w 2 q hij 1w 2 5 mhYj a h b
,
Pj
where P hj is the ideal price index for good h in country j. If only those firms above the productiv–h in country k and sector h export to country j, the ideal price index for good h in
ity threshold w
kj
country j, Pj, and dividends per share, p, are defined as
(5) P hj 5 q a wk Lk 3
N
`
k 51
wkjh
p5
h
1/ 11 2 sh 2
sh w k tkj 1 2 sh
b
dGh 1w 2r
;
sh 2 1 w
h
a a w k L k a 3 h pkl 1 w 2 dGh 1 w 2 b
H
(6) a
N
h51 k, l51
`
wkl
a w nL n
N
,
n51
where phkl 1w 2 5 1 p hkl 1w 2 2 c hkl 1w 2 2 q hkl 1w 2 2 f klh are the net profits that a firm with productivity w
in country k and sector h earns from exporting to country l.
For now, I will consider only sector h. The other sectors are analogous. For notational clarity,
I drop the h subscript and all sectoral variables will refer to sector h when there is no ambiguity.
II. Trade with Heterogeneous Firms
In this section, I compute the global equilibrium of this world economy. To do so, I solve for
the selection of firms into different export markets. I generate predictions for aggregate bilateral
trade flows.
A firm chooses a subset of countries where it sells its output, and sets prices for its good in
each market, taking the strategies of other firms and of consumers as given. Consumers chose
the quantity consumed of each variety available domestically, given prices. All agents move
simultaneously, and an equilibrium is a fixed point to their strategies. Firms decide whether to
enter a given market depending on how much competition they expect to face in that market.
The toughness of competition in turn depends on which firms enter. I derive the solution to this
selection problem, which turns out to be quite tractable.
Productivity Threshold.—Less productive firms are not able to generate enough profits abroad
to cover the fixed cost of entering foreign markets. Exporters are therefore only a subset of
domestic firms. This subset varies with the characteristics of the foreign market. The profits firm
w earns when exporting from i to j are pij 1w 2 5 m/s Yj 3s/ 1s 2 12 1wi tij /w 2 / Pj 4 12s 2 fij. Define
– from p 1w
– 2 5 0 as the productivity of the least productive firm in country i
the threshold w
ij
ij
ij
able to export to country j:
(7) – 5l a
w
ij
1
f ij
Yj
b
1/ 1s212
w itij
Pj
,
– . 1.
with l1 a constant. I assume that trade barriers are always high enough so that 5k, l, w
kl
l1 5 1s/m 2 1/ 1s212 1s/ 1s212 2 .
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Equilibrium Price Indices.—Until now, I have considered aggregate prices as given. They
do adjust, however, depending on country characteristics. Thanks to two simplifying assumptions—wages are exogenously pinned down in the homogenous sector, and the number of potential entrants (not the number of actual entrants) is exogenously given—the set of firms that export
to country j depends only on country j’s characteristics. Plugging the productivity thresholds
from equation (7) into the price index from equation (5), I can solve for the equilibrium price
index:
Pj 5 l2 3 Yj1/g21/ 1s212 3 uj,
(8) where uj2g ; g k 51 1Yk / Y 2 3 1wk tkj 2 2g 3 f kj2 3g/ 1s212214 , Y is world output, and l2 is a constant.
An aggregate index of j’s remoteness from the rest of the world is given by uj.10 It is reminiscent of the “multilateral resistance variable” introduced by Anderson and Eric Van Wincoop
(2003). In addition to their measure, it takes into account the impact of fixed costs and of firm
heterogeneity on aggregate prices.
N
Equilibrium Exports, Thresholds, and Profits.—Exports by an individual firm depend on its
productivity, the trade barriers it must overcome, aggregate demand, the set of competitors it is
facing, and the price they set. Plugging the general equilibrium price index from equation (8) into
the demand function, and into the productivity threshold from equation (7), I can solve simultaneously for firm level exports the productivity thresholds and total world profits. In general equilibrium, exports xij 1w 2 from country i to country j by an individual firm with productivity w, the
– above which firms in i export to j, aggregate output Y , and dividends
productivity threshold w
ij
j
per share p, are given by
Y j 1s212 / g
u j s21
–
(9) x 1w 2 5 • l3 3 a b
3a
b
3 ws21, if w $ w
ij
w itij
ij
Y
0 otherwise,
µ
– 5 l 3 aY b
w
ij
4
Yj
1/g
3a
Yi 5 11 1 l52 3 wi Li,
w itij
uj
b 3 fij1/ 1s212 ,
p 5 l5,
lg2 5 a
g
g 2 1 s 2 1 2 s g/ 1s21221
s
11p
ba b
a
b a
b.
g
m
s21
Y
Note that there is a slight abuse of notations, as dividends per share 1p2 and world output (Y) will be endogenously
determined in equilibrium. However, firms as well as consumers take total world profits and output as a constant. I will
solve for 1p, Y2 in equilibrium.
10
A simple way to interpret this aggregate index is to look at a symmetrical case: when wk tkj 5 wtj and f kj 5 fj for
all k’s, uj 5 fj1/ 1s212 2 1/g 3 wtj. In asymmetric cases, uj is a weighted average of bilateral trade barriers.
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with l3, l4, and l5 as constants.11 Equilibrium variables are functions of fundamentals only: the
size Lj ,12 the productivity wi, the trade barriers fij and tij, and the measure of j ’s remoteness from
the rest of the world, uj.
As expected from this simple monopolistic competition model, exports by individual firms
depend on the transportation cost tij with an elasticity s 2 1. Firm-level trade is very similar to
what a traditional model of trade with representative firms would predict for aggregate bilateral
trade flows. In contrast, because of the selection of firms into the export market, aggregate bilateral trade in the present model will look radically different.
Proposition 1 (Aggregate Trade): Total exports (f.o.b.) X hij in sector h from country i to
country j are given by
(10) Xhij 5 mh 3
Yi 3 Yj
Y
3a
w ithij
u hj
2gh
b
3 1 fijh 2 2 3gh / 1sh212214 .
Exports are a function of country sizes 1Yi and Yj 2 ,13 workers’ productivity 1wi 2 , the bilateral
trade costs, variable 1t hij 2 and fixed 1 fijh 2 , and the measure of j’s remoteness from the rest of the
world 1uhj 2.14
Proof:
See the Appendix.
The gravity structure of trade has been dramatically distorted by the presence of firm
heterogeneity.
First note that the elasticity of exports with respect to variable trade barriers, g, is larger than
in the absence of firm heterogeneity, and larger than the elasticity for each individual firm (both
equal to s 2 1). A reduction in variable costs not only causes an increase in the size of exports
of each exporter, but also allows some new firms to enter. The extensive margin comes on top of
the intensive margin and amplifies the impact of variable costs. This amplification effect is quantitatively important. Anderson and Van Wincoop (2004) argue that if one assumes that trade is
governed by an underlying model of trade with identical firms, trade barriers between the United
States and Canada must be equivalent to a 46 percent tariff in order to explain the observed
bilateral trade flows (for a benchmark case with s 5 8). In the presence of firm ­heterogeneity,
11
l3 5 sl412s, l4 5 3s/m 3 g/ 3g 2 1s 2 12 4 3 1/ 11 1 l 52 4 1/g, and
aa
s h 2 1 mh
b
gh
sh
h51
l5 5
.
H
s h 2 1 mh
12 aa
b
gh
sh
h51
H
I am grateful to Sebastian Krautheim for helpful suggestions on computing aggregate profits.
12
The elasticity of individual firms’ exports with respect to the destination market Yj is less than one. This is due to
the impact of market size on the degree of price competition: as the size of the market grows, more firms enter, which
eat up part of the market shares of existing exporters.
13
GDP is proportional to labor income: Yi 5 11 1 l 52 wi Li.
14
Note that the ratio of i’s market share in k, and j ’s market share in k, depends only on the ratio of i’s trade barriers and j ’s trade barriers. If I define the composite measure of trade barriers kik 5 1wi tik 2 2g 3 f ik2 3g2 1s212 4 / 1s212 , I get:
1Xik / Yi 2 / 1Xjk /Yj 2 5 kik /kjk . Similarly, i ’s market share in k depends only on trade barriers from i relative to trade barriers
from other countries: Xik / Xk 5 3 1Li / L2 kik / gj 1Lj / L2 kjk 4 .
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VOL. 98 NO. 4
1715
and using a heterogeneity parameter estimated from firm-level data 1g/1s 2 12 < 22 ,15 I would
infer from the same trade volume data that trade barriers are equivalent to a 21 percent tariff
11.21 5 !1.462.16 This is far below their 46 percent estimate.
Second, the elasticity of exports with respect to transportation costs depends on the degree of
firm heterogeneity, g. In more homogenous sectors 1g high2 , large productive firms represent a
smaller fraction of firms. The productivity threshold moves in a region where most of the mass
of firms lies. In those sectors, aggregate exports are sensitive to changes in transportation costs
because many firms exit and enter when variable costs fluctuate.
Third and most important, the elasticity of exports with respect to variable costs does not
depend at all on the elasticity of substitution between goods, s,17 and the elasticity of exports
with respect to fixed costs is negatively related to the elasticity s. This prediction is in stark contrast with models with representative firms. In such models, the elasticity of exports with respect
to transportation costs would be equal to s 2 1.
III. Intensive versus Extensive Margins of Trade
In this section, I describe how the elasticity of substitution magnifies the sensitivity of the
intensive margin to trade barriers and dampens the sensitivity of the extensive margin. I prove
that the dampening effect on the extensive margin dominates the magnifying effect on the intensive margin.
Thus far, I have shown that in the presence of firm heterogeneity, the selection of firms into the
export market becomes a key feature of the adjustment of trade flows. The main prediction of the
model is that the extensive margin and the intensive margin are affected in opposite directions by
the elasticity of substitution. If the elasticity of substitution is high, then the impact of trade barriers is strong on the intensive margin and mild on the extensive margin. The reverse holds true
when the elasticity of substitution is low. The dampening effect of the elasticity of substitution on
the extensive margin dominates the magnifying effect on the intensive margin.
Proposition 2 (Intensive and Extensive Margins): The elasticity of substitution 1s2 has no
effect on the elasticity of trade flows with respect to variable trade costs 1z2 , and a negative
effect on the elasticity of trade flows with respect to fixed costs 1j2:
if z ; 2
d ln X ij
d ln tij
and j ; 2
d ln X ij
d ln f ij
, then
'z
'z
5 0 and
, 0.
's
's
15
I measure g/ 1s 2 12 as the regression coefficient of the log of rank (ordering US firms according to their sales in
the United States) on the log of sales, using Compustat data on US listed firms. With an alternative method using the
propensity of French firms to export to multiple markets, Eaton, Kortum, and Kramarz (2007) find a smaller number,
g/ 1s 2 12 < 1.5.
16
To make the two models comparable, despite the presence of fixed entry costs, I assume the entry cost into any
Canadian province is the same for US and Canadian firms.
17
Eaton and Kortum (2002) derive a similar prediction from a different setup. In a Ricardian model of trade, they
find that bilateral trade flows do not depend on the elasticity of substitution between goods, but only on the scaling
parameter of the underlying distribution of productivity shocks. They use Fréchet distributions, which approach Pareto
2u
distributions in their right tails: the distribution for shocks they consider is 1 2 F 1z2 5 1 2 e2Tz
5 Tz2u 1 o 1z2u 2 .
z S 1`
In equilibrium, they predict that the elasticity of trade flows with respect to trade barriers (variable only) is equal to u.
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Proof:
To prove this proposition, I introduce formally the intensive and the extensive margins of
trade. I describe the adjustment of each margin, and the sensitivity of these adjustments to the
elasticity of substitution.
The impact of trade barriers, both variable and fixed, on aggregate trade flows can be decomposed into two different margins. The intensive margin is defined by how much each existing
exporter changes the size of its exports. The extensive margin is defined by how much new
entrants export (in the case of a reduction in trade barriers).
`
Differentiating the expression for aggregate exports, Xij 5 wi Li ewij xij 1w 2 dG 1w 2 , I get the fol18
lowing expressions for each margin:
dXij 5 awi Li 3
`
wij
1 awi Li 3
`
'x ij 1 w 2
'tij
'x ij 1 w 2
–ij 2 G9 1w
–ij 2 3
dG 1w 2 b dtij 2 awi Li x 1w
–ij 2 G9 1w
–ij 2 3
dG 1w 2 b d fij 2 awi Li x 1w
'f ij
8
wij
'wij
'tij
b dtij
'wij
b d fij .
'f ij
8
Intensive margin
Extensive margin
–ij 2 exports more. This is the
Following a reduction of trade barriers, each existing exporter 1w . w
–ij goes down2.
intensive margin. At the same time, higher potential profits attract new entrants 1w
This is the extensive margin.
In elasticity notations, I get the following expression for each margin for changes in the variable cost, tij:19
z;2
d ln X ij
d ln tij
5
1s 2 12
3
Intensive margin
Elasticity
1
1g 2 1s 2 12 2 5 g .
3
Extensive margin
Elasticity
When variable costs move 1s 2 1 increases with s2 , s magnifies the intensive margin, whereas
it dampens the extensive margin 1g 2 1s 2 12 decreases with s2.20 The effect of s on each margin cancels out, so that
'z
5 0.
's
18
I use Leibniz rule to separate the intensive from the extensive margin. I apply Lebesgue’s monotone convergence
theorem to ensure the existence of, and to compute, the intensive margin.
19
See the Appendix for a complete derivation.
20
I have implicitly assumed that changes in both tij and f ij have no significant impact on the general equilibrium.
That is, I have assumed that 0uj /0tij 5 0uj /0 f ij 5 0. This is a fair approximation as long as country i is not too large
compared to the rest of the world 1Yi / Y small). Relaxing this assumption would reinforce my results, but it would make
calculations cumbersome. See the Technical Appendix (available at http://www.aeaweb.org/articles.php?doi=10.1257/
aer.98.4.1707 for a derivation of the general case.
chaney: Distorted Gravity
VOL. 98 NO. 4
1717
In elasticity notation, I get the following expression for each margin for changes in the fixed
costs, fij:
j;2
d ln X ij
d ln f ij
5
0
1
3
Intensive margin
Elasticity
g
21
s21
3
5
g
2 1.
s21
Extensive margin
Elasticity
When only fixed costs move, s has no impact on the intensive margin, whereas it dampens the
impact on the extensive margin 1g / 1s 2 12 2 1 decreases with s2. The impact of s on the elasticity of trade flows with respect to fixed costs is always negative:
'j
, 0.
's
The intuition for these results is the following. When goods are highly differentiated 1s is
low2 , the demand for each individual variety is relatively insensitive to changes in trade costs.
In other words, when s is low, trade barriers have little impact on the intensive margin of trade.
This margin is the only one in the Krugman model of trade with representative firms.
The interaction between the elasticity of substitution and the extensive margin is more complex. When s is low, the market share that each firm is able to capture is relatively insensitive to
differences in productivity. Less productive firms are still able to capture a relatively large market share, despite having to charge a higher price than other firms. As trade barriers decrease,
some firms with a low level of productivity are able to enter. When goods are highly differentiated 1s is low2 , these new entrants are relatively large compared to the firms that are already
exporting. Therefore, the extensive margin is strongly affected by trade barriers when s is low.
The reverse holds when s is high.
In this section, I have explained why the elasticity of substitution has opposite effects on the
intensive and the extensive margins of trade. A higher elasticity of substitution makes the intensive margin more sensitive to changes in trade barriers, whereas it makes the extensive margin
less sensitive. What is the net impact of s on the two margins? I prove in Proposition 2 that with
Pareto distributed productivity shocks, the extensive margin always dominates. Contrary to the
predictions of the Krugman model with representative firms, the elasticity of substitution s
always dampens the impact of trade barriers on trade flows.
IV. Conclusion
Introducing firm heterogeneity leaves many of the predictions of the Krugman (1980) model
of international trade unaffected. Most important, the gravity structure of bilateral trade flows
is preserved. In this paper, I have identified a key difference between the Krugman model with
representative firms and a model with firm heterogeneity. The impact of trade barriers is dampened by the elasticity of substitution, and not magnified by it. I introduce fixed export costs and
adjustments on the extensive margin in a simple model of international trade. A high elasticity
of substitution translates productivity differences into large differences in size. As firm sizes
get more dispersed, fixed costs have a lesser impact on exports: large firms can easily overcome
those fixed costs. Aggregate trade flows are less sensitive to trade barriers when goods are more
substitutable.
1718
September 2008
THE AMERICAN ECONOMIC REVIEW
Appendix
Proposition 1 (Reminded): Total exports (f.o.b.) X hij in sector h from country i to country j
are given by
X hij 5 mh 3
Yi 3 Yj
Y
3a
w ithij
u hj
2gh
b
3 1 f hij 2 2 1gh / 1sh212212 .
Exports are a function of country sizes 1Yi and Yj 2 , workers’ productivity 1wi 2 , the bilateral
trade costs, variable 1t hij 2 and fixed 1 f hij 2 , and the measure of j’s remoteness from the rest of the
world 1uhj 2.
Proof:
Aggregate exports from i to j are defined as the sum of export of each individual firm with
–ijh:
productivity w $ w
X hij 5 wi Li 3 x hij 1w 2 dGh 1w 2.
`
w hij
–ijh 2 and the productivity
From equation (9) we know the size of firm-level exports x hij 1w Z w $ w
h
–
threshold wij . Using the specific assumption about the distribution Gh of productivity shocks
from equation (3), we can rewrite aggregate exports as
X hij
5 wi Li 3
`
w hij
l3h
Yj
3a b
Y
1s 2 12 / g
3a
u hj
w ithij
b
s21
3 wsh21 3
w2gh21
dw,
gh
1/g
wt
–ijh 5 lh4 3 a Y b 3 a i ij b 3 fijh 11/ 1s212 2 ,
with w
Yj
u hj
h
where lh3 and lh4 are some constants.21 Given the assumption of Pareto distributed productivity
and isoelastic preferences, we get a simple solution for the integral. After solving the integral
and rearranging, we get
X hij
5 lh 3
w i L i Yj
Y
3a
w ithij
u hj
2gh
b
3 1 f hij 2 2 1gh / 1sh212212 ,
with lh 5 11 1 l52 3 mh. Noting that Yi 5 11 1 l52 wiLi, we get
X hij
5 mh 3
Yi 3 Yj
Y
3a
w ithij
u hj
2gh
b
3 1 f hij 2 2 1gh / 1sh212212 .
Proposition 2 (Reminded):
(i) The elasticity of the intensive margin of trade with respect to variable trade costs is 1s 2 12.
(ii) The elasticity of the extensive margin of trade with respect to variable costs is 1g 2
1s 2 122.
21 h
l 3 5 s 1lh42 12s, lh4 5 3sh /mh 3 gh / 1gh 2 1sh 2 12 2 3 1/ 11 1 l 52 4 1/gh , and l 5 5 5 3 gHh51 1 1sh 2 12 / gh 2 1mh /sh 2 4 /
312gHh51 1 1sh 2 12 / gh 2 1mh /sh 2 4 6.
chaney: Distorted Gravity
VOL. 98 NO. 4
1719
(iii) The elasticity of the intensive margin of trade with respect to fixed costs is 0.
(iv) The elasticity of the extensive margin of trade with respect to fixed costs is 3g 2
1s 2 124 / 1s 2 12.
Proof:
(i) The definition of the intensive and the extensive margins from Proposition 2 gives us
2
2
d ln X ij
d ln tij
d ln X ij
d ln f ij
5
5
2dX ij/dtij
X ij/tij
2dX ij/df ij
X ij/f ij
tij
awiLi 3
`
'x ij 1 w 2
–ij 2 G9 1w
–ij 2 3
awi Li x 1w
'wij
b
X ij
'tij
8
Intensive margin elasticity
Extensive margin elasticity
f ij
awiLi 3
`
'x ij 1 w 2
dG 1w 2 b 1
tij
X ij
'tij
wij
8
52
dG 1w 2 b 1
f ij
–ij 2 G9 1w
–ij 2 3
awi Li x 1w
'wij
X ij
'f ij
wij
8
b.
X ij
'f ij
8
Intensive margin elasticity
Extensive margin elasticity
52
Using the definition of equilibrium individual exports from equation (9), and assuming that
country i is small enough and/or remote enough, so that 0uj /0tij 5 2g 1Yi / Y2 3 wi2g tij2g21 3
f ij2 3g2 1s212 4/ 1s212 < 0, we get
'x ij 1 w 2
'tij
Integrating over all exporters, we get
5 2 1s 2 12
x ij 1 w 2
.
tij
`
Elasticity of the intensive margin
tij
'x ij 1 w 2
with respect to variable costs 5 2 awiLi 3
dG 1w 2 b
X ij
'tij
wij
wiLi 3 xij 1w 2 dG 1w 2
`
5 1s 2 12
5 1s 2 12
tij
X ij
wij
tij
tij X ij
X ij tij
5 1s 2 12.
(ii) Using the definition of the equilibrium productivity threshold from equation (9), we get
'wij
'tij
5
wij
.
tij
1720
September 2008
THE AMERICAN ECONOMIC REVIEW
Using the definition of firm-level exports xij(w) 5 lijws21 from equation (9), and the definition
of the distribution of productivity shocks G9(w) 5 w2g21 /g from equation (3), we can rewrite
aggregate exports in the following way:
Xij 5 wi Li 3 xij 1w 2 dG 1w 2
`
w hij
5 wi Li 3 lijws21
`
wij
5
5
w 2g21
dw
g
1
–ij1s2122g
w L l s21 w
1 g 2 1 s 2 1 2 2 g i i ij
1
–ij2 G9 1w
–ij2 3 w
–ij .
3 wiLi xij 1w
1
g 2 s 2 12
We therefore get the simple solution for the elasticity:
Elasticity of the extensive margin
tij
'w
–ij2 G9 1w
–ij2 3 ij b
with respect to variable costs 5
awiLi x 1w
X ij
'tij
5
tij w i L i x 1 wij 2 Gr 1 wij 2 3 wij
a
b
tij
X ij
5 1g 2 1s 2 12 2
tij X ij
X ij tij
5 1g 2 1s 2 12 2.
(iii) Using the definition of the equilibrium firm-level exports from equation (9), we get
'x ij 1 w 2
'f ij
5 0.
We directly derive that the elasticity of the intensive margin of trade with respect to fixed costs
is 0.
(iv) Using the definition of the equilibrium productivity threshold from equation (9), we get
'wij
'f ij
5
wij
1
3 ,
s21
f ij
and using the same procedure as in (ii), we prove that the elasticity of the extensive margin with
respect to fixed trade costs is g/ 1s 2 12 2 1.
VOL. 98 NO. 4
chaney: Distorted Gravity
1721
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