Comparative Advantage, Monopolistic Competition,
and Heterogeneous Firms in a Ricardian Model with a
Continuum of Sectors∗
Tomohiro Ara†
Fukushima University
July 2015
Abstract
Why does the fraction of firms that export vary with countries’ comparative advantage? To
address this question, I develop a general-equilibrium Ricardian model of North-South trade in
which both institutional quality and firm heterogeneity play a key role in determining international trade flows. Because of contractual frictions that vary across countries and sectors, North
with better institutions produces and exports relatively more in sectors where production is more
institutionally dependent. In addition, institution-induced comparative advantage makes it relatively easier for Northern heterogeneous firms to incur export costs in more contract-dependent
sectors, thereby leading to a higher exporters’ percentage.
Keywords: Comparative advantage, firm heterogeneity, log-supermodularity, relative wage
JEL Classification Numbers: D23, F12, F14, L33, O43
∗
This study is conducted as a part of the Project “Trade and Industrial Policies in a Complex World Economy”
undertaken at Research Institute of Economy, Trade and Industry (RIETI). I am deeply grateful to Arghya Ghosh
and Hodaka Morita for invaluable guidance and encouragement, and to Richard Baldwin, Andrew Bernard, Arpita
Chatterjee, Qingyuan Du, Taiji Furusawa, Jota Ishikawa, Keith Maskus, Phillip McCalman, Devashish Mitra, Raymond
Riezman, Nicolas Schmitt, and seminar and conference participants at various institutions for helpful comments and
suggestions. I have also benefited from financial support from UNSW, RIETI and JSPS under grant #25780156. All
errors are my own.
†
Faculty of Economics and Business Administration, Fukushima University, Fukushima 960-1296, Japan. Email
address: [email protected].
1
Introduction
A growing body of empirical evidence using firm-level data has extensively revealed that the extent
to which firms participate in exporting varies systematically across countries and sectors. These
works have found that, in developed countries, the percentage of firms that export tends to be
substantially higher in sectors where production technology is more complex and customized (such
as chemical products), and this percentage steadily declines as sectors’ production requires simpler
and more generic technology (such as apparel products). In developing countries, on the other hand,
the opposite patterns are typically observed: the ratio of exporting firms to overall firms tends to be
notably higher (resp. lower) in simpler (resp. more complex) sectors. At the same time, these studies
have also documented that exporting occurs in every major manufacturing sector of both developed
and developing countries: even in strong comparative disadvantage sectors, a small fraction of firms
do export.1
Why does the fraction of firms that export vary with countries’ comparative advantage? To
address this question, combining the recent empirical finding quantified by Levchenko (2007) and
Nunn (2007) with firm heterogeneity of Melitz (2003), I develop a general-equilibrium Ricardian
model of North-South trade in which both institutional quality and firm heterogeneity play a key
role in determining international trade flows. Following Levchenko’s and Nunn’s finding, the model
assumes that each country is different in terms of contracting institutions, and each sector is different
in terms of contract intensity. Moreover, each firm is different in terms of its productivity à la Melitz.
These three-dimensional differences in countries’, sectors’ and firms’ characteristics endogenously pin
down patterns of specialization and trade in equilibrium. Although the current paper focuses on
institutional factors as a source of countries’ comparative advantage, they can be interpreted more
broadly. All what matters for the main results is that labor productivity stemming from countries’
and sectors’ characteristics satisfies log-supermodularity.
To investigate the role of countries’ and sectors’ characteristics, I build on the concept of “partial
contractibility,” originally developed by Acemoglu, Antràs, and Helpman (2007). I consider an
environment in which North has better institutions for partially ex-ante contractible activities than
South, whereas customized sectors make use of relationship-specific investments intensively more
than generic sectors. Because of these contractual frictions that vary across countries and sectors,
aggregate output differences emerge. In contrast to Acemoglu et al., I do not explicitly examine
the interaction between contractual incompleteness and technological complementarity by simply
assuming that production in customized (generic) sectors is more (less) dependent on institutions.
1
See Bernard, Jensen, Redding, and Schott (2007) for the United States, Tomiura (2007) for Japan, and Lu (2011)
for China, respectively. For example, Bernard et al. (2007, Table 2) report that, as of 2002, 36 percent (8 percent) of
U.S. firms export in a chemical sector (an apparel sector), whereby a more customized sector tends to exhibit a higher
percentage of exporters across 21 sectors. Conversely, Lu (2011, Figure 1) shows that, as of 2005, around 60 percent
(less than 20 percent) of Chinese firms export in cloth and fur sectors (a chemical sector), indicating that there exists a
clear negative relationship between export participation and the capital-labor ratio across 29 sectors. While the current
paper tries to rationalize this relationship from log-spermodularity in countries’ and sectors’ characteristics, theoretical
results should be interpreted with caution as the sectors in question admittedly differ along several other dimensions,
and the differences in export participation observed in these data cannot be purely attributed to this property.
1
Instead, I extend their framework by allowing countries’ institutional quality and sectors’ institutional
dependency to obey the Ricardian law of comparative advantage. This elaboration makes it possible
to capture the aggregate relationship between country and sector characteristics neatly in a way such
that North with better institutions produces and exports relatively more in sectors where production
is more institutionally dependent.2
To formalize the higher tendency to export participation in comparative advantage sectors, I
incorporate firm-level differences in productivity. Since exporting requires fixed export costs that
less productive firms cannot cover, only a small fraction of firms are able to export. The variation
in this fraction is further reinforced by institution-induced comparative advantage in my setup, because Northern (Southern) firms are relatively better at producing in more (less) contract-dependent
sectors, which in turn softens the relative burden of incurring export costs. As a result, compared
to comparative disadvantage sectors of a counterpart country, relatively less productive firms can
export in a country’s comparative advantage sectors. This mechanism explains why the stronger
each country’s comparative advantage is, the smaller the productivity cutoff for exporting becomes,
thereby leading to a higher exporters’ percentage.
Modeling the share of firms that export helps to understand whether differences in aggregate
output due to comparative advantage are achieved through differences in the extensive and intensive
margins. Since the share of exporters is defined as the ratio of the mass of exporters to the mass of
domestic firms, calculating this share requires both aggregate domestic sales and aggregate exports
to be decomposed into the mass of firms (extensive margin) and average sales per firm (intensive
margin). While it is not surprising that total domestic sales and total exports increase with countries’
comparative advantage strength, this increase comes mainly from the extensive margins (net change
in the intensive margins is generally ambiguous and is specially independent under the case of a Pareto
distribution). Further, the increase in these two extensive margins is not the same magnitude; indeed,
the fact that the share of exporters is increasing in the degree of comparative advantage suggests
that the extensive margin of exports increases with comparative advantage more significantly than
the extensive margin of domestic production. Clearly, these insights cannot be obtained when simply
analyzing how aggregate exports vary with comparative advantage.
Following the new literature on institutions and trade, I employ contracting institutions – more
specifically contract enforcement – (rather than the classical determinants of international trade
such as capital or (un)skilled labor) to rationalize the stylized fact of export participation. There
are at least three reasons for this. First, countries’ abilities to enforce written contracts can have
quantitatively larger impacts on comparative advantage than countries’ factor endowments. For
instance, Nunn (2007) estimates that “contract enforcement explains more of the global pattern
of trade than countries’ endowments of physical capital and skilled labor combined.” Second, as
2
There is mounting evidence on the link between countries’ institutions and sectors’ types that affects the pattern
of trade. For instance, devising a measure of input customization (the share of a sector’s inputs that are not sold
in organized exchanges), Nunn (2007) shows that countries with better contract enforcement export relatively more
in sectors for which relationship-specific investments are more important. Similarly, Manova (2008) finds evidence
that trade liberalizations induce countries with better financial systems to export relatively more in sectors for which
financial requirements are more important.
2
rigorously demonstrated by Costinot (2009a), this empirical evidence is appropriately captured by
(Ricardian) institutional differences with log-supermodularity in country and sector characteristics.3
In this specification, the characteristics of firms in terms of their productivity are an independent
factor of the variation in export participation as discussed above. Finally, a Ricardian view of institutional differences can give a complementary explanation for Bernard, Redding, and Schott’s (2007)
factor-endowment-driven comparative advantage theory. Although the main result is strikingly similar, I show that some phenomena (e.g., home-market effects) are better understood through a lens
of Ricardian sources of comparative advantage.
In this paper, I do not attempt to explain why North has better institutions than South, why
customized sectors depend heavily on institutions more than generic sectors, or why some firms
are more productive than others. Taking these country, sector, and firm characteristics as given,
I instead set out to explore how contracting institutions and heterogeneous firms jointly shape an
endogenous pattern of trade. By so doing, the model shows that North with better institutions gains
a comparative advantage in contract-dependent sectors and is a net exporter of customized products
in intra-industry trade. South with worse contracting institutions, on the other hand, is shown to be
a net exporter of generic products. Moreover, within sectors in bilateral trade flows, the fraction of
exporters is monotonically increasing in countries’ comparative advantage strength. These results,
both of which are consistent with firm-level empirical research, hold even if North has an absolute
advantage in institutional quality in any sector, and there exists no technological difference (in terms
of firm productivity distributions) between the two countries.
This paper is closely related to two branches of the recent literature of international trade. The
first is an emerging literature on institutions and trade (e.g., Antràs, 2005; Acemoglu et al., 2007;
Costinot, 2009b). These papers show that, even in the absence of inherent technological differences,
cross-country institutional differences can endogenously generate comparative advantage, which is
at the heart of my model as well. In this strand of the papers, however, all firms are generally
treated as identical and therefore every firm is able to export everywhere.4 In the real world, a
large proportion of firms do not export even in strong comparative advantage sectors. The current
paper demonstrates that not only is comparative advantage endogenously induced by institutions,
but the fraction of exporters is higher in stronger comparative advantage sectors, as suggested by
the existing evidence.
Another branch of the related literature is the so-called heterogeneous-firm model of trade, especially developed by the seminal work of Melitz (2003). While the Melitz model is successful in
explaining the exporters’ behaviors among developed countries (North-North trade), recent empirical evidence has pointed out that this model is less suitable for the study of bilateral trade flows
3
As far as the pattern of specialization and trade is central, the modeling of technological and institutional differences
is isomorphic (Costinot, 2009a). While I interpret “institutions” as a country’s ability to alleviate contractual frictions
between firms and suppliers as in Acemoglu et al. (2007), the results do not change at all even if they are interpreted as
a country’s ability to use a given “technology” as in usual Ricardian models. All what matters is that labor productivity
stemming from countries’ and sectors’ characteristics satisfies log-supermodularity.
4
Acemoglu et al. (2007) introduce firm heterogeneity in the degree of complementarity among inputs, but all products
are assumed to be freely traded and hence all firms export in their model.
3
between different countries (North-South trade), as exemplified by Lu (2011) who analyzes Chinese
firm-level manufacturing data. A number of papers – among others, Demidova (2008), Fan, Lai, and
Qi (2011), and Okubo (2009) – incorporate the asymmetry of countries in this setting. My approach
differs from these papers, because I focus on the role of wage differentials in North-South trade,5 and
because most results hold without specifying any parameterization of firm productivity distributions.
More importantly, none of these papers sheds new light on the interplay between institutions and
comparative advantage. Although I restrict the analysis only to an open economy and abstract from
welfare implications for expositional simplicity, it is straightforward to extend the current setup to
see the impact of trade on inter-/intra-sectoral resource allocations and welfare gains from trade.
Finally, this paper is also related to the heterogeneous-firm literature on factor-proportions theory,
especially to Bernard et al. (2007) as argued above. Using Helpman-Krugman’s (1985) two-factor
model, they provide a rich framework for analyzing distributional consequences from trade, a feature
missing in this Ricardian one-factor model. Their analysis, however, primarily applies to the situation
in which two countries are not too different, and numerical simulations are required for outside factorprice-equalization regions. In contrast, it is possible in the current paper to analytically examine trade
patterns between any two countries of arbitrary country size (with endogenous wage differentials)
by sacrificing distributional issues via trade liberalization. A further distinction of this paper is in
addressing Krugman’s (1980) home-market effect. I show that, due to selection into domestic and
export markets that varies with comparative advantage, the home-market effect works oppositely
for the extensive/intensive margins of domestic production and those of exports between North and
South, leading the share of exporters to change oppositely to the relative country size.
2
Setup
Consider a world composing of two large countries, North and South, i ∈ {N, S}. For notational
simplicity, country superscript i is dropped unless needed in this section.
Demand Each country is populated by a mass L of identical consumers who devote their income
into differentiated goods of a continuum of sectors over an interval [0, 1]. The preferences of a
representative consumer are Cobb-Douglas across sectors and C.E.S. Dixit-Stiglitz within sectors:
Z
U=
1
λ(z) ln Q(z)dz,
0
where
"Z
Q(z) =
#
q(z, v)
σ−1
σ
dv
σ
σ−1
,
v∈V (z)
5
Wage differentials are one of the most prominent factors that have triggered large trade flows among dissimilar
countries in the past two decades. For instance, noting that in 2006 for the first time the United States did more trade
in manufactured goods with developing countries than developed countries, Krugman (2008) asserts that this is largely
due to the wage differentials between the U.S. and developing countries: China’s and Mexico’s wages are respectively
only 4 percent and 13 percent of the U.S. level.
4
is aggregate consumption of varieties in sector z. V (z) is the mass of available goods within the
sector, which potentially includes both domestic and foreign varieties. Given this aggregate good
Q(z), its dual aggregate price is given by
"Z
#
1
1−σ
p(z, v)1−σ dv
P (z) =
.
v∈V (z)
λ(z) denotes a constant share of expenditure spent on sector z, which is identical between the two
countries. Letting R(z) = P (z)Q(z) and Y = wL respectively denote aggregate expenditure in sector
z and aggregate labor income in the economy, λ(z) is defined as
P (z)Q(z)
R(z)
=
,
Y
wL
λ(z) =
which must satisfy
Z
1
λ(z)dz = 1.
0
R1
Therefore, the sum of aggregate sector expenditure equals aggregate labor income ( 0 R(z)dz = wL).
Letting X(z) = λ(z)Y denote labor income spent on sector z, the above preferences generate demand
functions for differentiated variety v in sector z:
q(z, v) = A(z)p(z, v)−σ ,
where A(z) = X(z)P (z)σ−1 is the index of aggregate market demand. In the following, I focus on a
particular variety in sector z and drop variety subscript v from relevant variables.
Before proceeding further, it is important to note that there is no homogeneous-good sector with
nontrade costs, and wage rates w cannot be normalized between North and South.6 This structure of
the preferences is similar to that of Krugman (1980), and more recently to that of Antràs (2005) and
Okubo (2009). Note also that while the elasticity of substitution between any two varieties within a
sector is assumed to be greater than one (σ > 1), the elasticity of substitution between any varieties
across sectors is unity. The unit elasticity of substitution implies that firm behavior in each sector
can be analyzed independently.
Production There is a continuum of firms that produce a different variety in each sector. Labor is
the only factor of production to produce a variety and firms face a perfectly elastic supply of labor
at each country size L. Since labor is completely mobile across sectors but immobile across countries
as in conventional Ricardian models, a wage rate w is the same across sectors within a country but
is different across countries.
6
By excluding a homogeneous-good sector, it is possible to explicitly investigate the role of the relative wage or
“factoral terms of trade” (Matsuyama, 2008) in comparative advantage, which is an orthodox practice in Ricardian
models. While introducing a homogeneous good à la Helpman and Krugman (1985) would help to simplify the
analysis, empirical evidence suggests that the bulk of recent trade flows cannot be captured without a terms-of-trade
effect between developed and developing countries as emphasized in Introduction.
5
Following Krugman (1980) and Melitz (2003), firm technology is summarized in a linear cost
function of output q:

q
q
f +
d
θ(ϕ,z,µ) = fd + ϕµ(z)
l=
f + τ q = f + τ q
x
x
θ(ϕ,z,µ)
ϕµ(z)
if domestic production,
if exporting,
where θ(·, ·, ·) is labor productivity, fd is a fixed cost for domestic production, fx is a fixed cost for
exporting, and τ (≥ 1) is a iceberg transport cost (note in particular that the fixed costs, fd and fx ,
are being incurred in units of labor). These costs are identical across countries and sectors.
A few points are in order for this specification. First, labor productivity θ(·, ·, ·) depends on three
factors: (i) firm-specific ϕ; (ii) sector-specific z; and (iii) country-specific µ. In Melitz (2003), he
considers symmetric countries, implying that a country-specific factor µ is ignorable. He also focuses
on one sector within each country, leading a sector-specific factor z to be absent from his analysis.
Therefore, only a firm-specific factor ϕ is important in the Melitz model. In the current model, by
contrast, since the two countries are asymmetric and there is a continuum of sectors, the three factors
jointly affect labor productivity. It follows from this cost function that the country-specific factor
µ(·) ∈ (0, 1) affects firms’ variable costs only (leaving fixed costs identical) and labor productivity is
greater if µ(·) is closer to one. I assume that µ(·) is related to a country’s ability to enforce written
contracts between firms and suppliers (as will be shown in the next subsection) and is referred to as
“partial contractibility” in this paper.
Second, I adopt a reduced form of labor productivity: θ(ϕ, z, µ) = ϕµ(z). While this form is used
for simplicity, one can justify this simplification from Costinot’s (2009a) log-supermodular argument.
He defines Ricardian technological differences as labor productivity that satisfies
θ(ϕ, z, µ) =
f (ϕ)
,
a(z, µ)
where a(·, ·)(> 1) is the unit labor requirement (defined as the inverse of labor productivity), and
shows that Ricardian sources of comparative advantage hold if 1/a(·, ·) is log-supermodular (i.e.,
∂2
∂z∂µ
1
ln a(z,µ)
≥0⇔
∂2
∂z∂µ
ln a(z, µ) ≤ 0), or equivalently
a(z 2 , µ1 )
a(z 1 , µ1 )
≥
,
a(z 2 , µ2 )
a(z 1 , µ2 )
for z 1 ≥ z 2 , µ1 ≥ µ2 , a(z 1 , µ2 ) 6= 0 and a(z 2 , µ2 ) 6= 0. My specification is restricted relative to
Costinot’s in that θ(ϕ, z, µ) = f (ϕ)/a(z, µ) = ϕµ(z).7 In addition to applying this reduced form, I
further assume that North has partial contractibility strictly superior to South in any sector. Noting
the inverse relationship between µ(·) and a(·, ·), log-supermodularity in terms of µ(·) is given by
1<
7
µN (z)
µN (z 0 )
<
< ∞,
µS (z)
µS (z 0 )
Note that µ is used in ϕµ(z) as a function of z, and countries’ characteristic µ is henceforth dropped from arguments.
6
Pi
1
PN
PS
z
1
0
Figure 1 – Log-supermodularity in contractibility
for z > z 0 , µN > µS , µS (z) 6= 0 and µS (z 0 ) 6= 0.8 Thus, not only does µ(z) = 1/a(z, µ) satisfy
log-supermodularity (or Ricardo’s classic inequality), but North has an absolute advantage in µ(z)
in any sector.
Figure 1 depicts µi satisfying the above inequalities with the additional assumptions that µN (1) =
µS (1) = 1 and µS (0) = 0. As the figure indicates, log-supermodularity means that the gap between
µN and µS is gradually larger as z is closer to zero. In the following, z is interpreted as the degree
of customization in sector z: the smaller is z, the greater is customization of production in sector z.
An economic interpretation of this figure is as follows. In a more generic sector (i.e., a sector with
greater z), the severe holdup problem is less likely irrespective of institutional quality because the
production does not rely heavily on relationship-specific investments. As a result, the gap between
µN and µS is relatively smaller and µi is closer to one in a more generic sector. In a more customized
sector (i.e., a sector with smaller z), producers are more likely to suffer from the holdup problem and
production efficiency is relatively more sensitive to institutional quality. Although µi is significantly
less than one for both countries, superior contractibility gives North a relatively bigger cost advantage
in a more customized sector. As formally established by Costinot (2009a), this “relatively more”
property – which lies at the core of neoclassical trade theory and is also the pivotal element in
the empirical evidence reported by Levchenko (2007) and Nunn (2007) – is elegantly captured by
log-supermodularity.
Finally, in this productivity decomposition θ(ϕ, z, µ) = ϕµ(z), I refer to a country’s distribution
of firms’ productivity draws as “technologies” as in Melitz (2003), whereas a country’s contractibility
on firms’ relationship-specific investments as “institutions” as in Levchenko (2007). Further I restrict
8
It is known that if labor productivity is log-supermodular, aggregate output is log-supermodular (Costinot, 2009a);
and in the current setup with the C.E.S. preferences, aggregate revenue is also log-supermodular. Thus, for z > z 0 ,
QN (z 0 )
QN (z)
<
,
QS (z)
QS (z 0 )
RN (z)
RN (z 0 )
<
.
RS (z)
RS (z 0 )
This property plays a key role in examining the aggregate relationship between countries’ and sectors’ characteristics
in the general equilibrium analysis.
7
attention to environments in which all firms have access to the same technologies across countries
and sectors, i.e., Gi (ϕ, z) = G(ϕ) for i ∈ {N, S} and z ∈ [0, 1], where G(·) denotes a cumulative
distribution from which firms draw their productivity.9 Thus, there are no technological differences
across countries and institutional differences solely give rise to countries’ comparative advantage. In
reality, technological and institutional differences coexist and these two differences are not precisely
separable. To facilitate the analysis below, the following definition is made for the sake of convenience.
Definition 1 Technologies are a country’s distribution G(ϕ) of firms’ productivity draws ϕ, which
is identical across countries and sectors. Institutions are a country’s partial contractibility µ(z) on
firms’ relationship-specific investments, which varies across countries and sectors.
Each firm chooses its price to maximize profits π = pq−wl for domestic production and exporting.
Solving profit-maximizing problem yields the following first-order conditions:
σ
w
,
σ − 1 ϕµ(z)
µ
¶
σ − 1 ϕµ(z) σ
q(ϕ, z) = A(z)
,
σ
w
p(ϕ, z) =
¶
µ(z) σ−1 σ−1
ϕ
,
r(ϕ, z) = p(ϕ, z)q(ϕ, z) = σB(z)
w
µ
¶
r(ϕ, z)
µ(z) σ−1 σ−1
π(ϕ, z) =
− wf = B(z)
ϕ
− wf,
σ
w
where
B(z) =
µ
(σ − 1)σ−1
(σ − 1)σ−1
A(z)
=
X(z)P (z)σ−1 ,
σσ
σσ
is aggregate market demand. For analytical simplicity, I assume that the variable trade cost is zero
(τ = 1) and thus p = pd = px . Section 5 shows that the main result qualitatively holds even with the
variable trade cost (τ > 1), and the supplementary note offers a detailed analysis that incorporates
τ . It is also assumed that the fixed trade cost is higher than the fixed production cost (fx > fd ) and,
under this assumption, only a subset of firms are able to export even in the absence of τ .
While the above first-order conditions are similar to those in the existing literature, two features
of the current setup are worth emphasizing. First, wages w cannot be normalized between the two
countries, since they are asymmetric and there is no freely tradable homogeneous-good sector in this
model. As noted earlier, this assumption is made to examine the role of endogenous factoral terms
of trade in the Ricardian model. Second, institutional quality µ enters into these conditions. It is
immediate to see that the pricing rule is higher and the output level is lower in a more customized
sector due to the holdup problem.
This means that, if firm productivity is drawn from a Pareto distribution, G(ϕ) = 1 − (ϕmin /ϕ)k , both shape and
scale parameters, k and ϕmin , are identical across countries and sectors. While both parameters are likely to vary with
countries’ and sectors’ characteristics in evidence (Tybout, 2000), the modeling of sector-variant distributions could
come at the cost of obscuring Ricardian sources of comparative advantage if these distributions are log-supermodular
in sectors’ and firms’ characteristics (Costinot, 2009a).
9
8
Holdup Problem So far, I have not explicitly explored how a country’s institutional quality µ is
related to the enforcement of contracts between firms and input suppliers, and thus to the holdup
problem in relationship-specific investments. As originally proposed by Grossman and Hart (1986),
the holdup problem occurs because the parties cannot specify every unforeseeable contingency into an
initial contract ex ante, and they have to renegotiate the contract ex post. In what follows, building
on seminal work of Antràs (2003, 2005) and Antràs and Helpman (2004), I show that institutional
quality µ plays a qualitatively similar (but distinct) role with incomplete contracting.
Suppose that, while perfect institutions (µ = 1) prevail in any sector of both countries, production
of final goods now requires intermediate inputs which firms cannot manufacture by themselves. To
produce a variety, each firm has to ask a domestic input supplier to provide a specialized input. This
input is relationship-specific in that it has a higher value within the parties and a third party cannot
distinguish its true value. Since no enforceable contract will be signed ex ante in such a circumstance,
the firm and its supplier have to bargain over the surplus after production takes place. Let β and
1 − β denote the firm’s and its supplier’s ex-post bargaining power, which vary across countries and
sectors. Then, the firm’s profit is πF = βpq + T and the supplier’s profit is πS = (1 − β)pq − wl − T ,
where T is a transfer from the supplier to the firm. This transfer works to make the supplier breakeven and the firm’s ex-post profit is π = πF + πS = pq − wl in a subgame-perfect Nash equilibrium.
The supplier chooses its input level to maximize πS , so the first-order conditions are
w
σ
,
p(ϕ, z) =
σ − 1 ϕ(1 − β(z))
µ
q(ϕ, z) = A(z)
σ − 1 ϕ(1 − β(z))
σ
w
¶σ
.
Comparing the two pricing rules reveals µ = 1 − β in equilibrium, and the distribution of bargaining
power between the firm and its supplier is directly related to institutional comparative advantage.
To see this in more detail, imagine what happens if the agents were able to sign complete contracts.
In such an environment, the input level is ex ante verifiable and the supplier could directly bargain
over the profit rather than the revenue. Then, the firm’s profit is πF∗ = βπ ∗ + T and the supplier’s
profit is πS∗ = (1 − β)π ∗ − T , where π ∗ = πF∗ + πS∗ is the joint profit under complete contracting. The
supplier would choose its input level to maximize πS∗ , so the first-order conditions are
σ w
,
p (ϕ) =
σ−1ϕ
∗
µ
∗
q (ϕ) = A
σ−1ϕ
σ w
¶σ
.
Evidently, the pricing rule is 1/(1−β) times higher under incomplete contracting because the supplier
receives only a fraction of the marginal return to its investment for specialized input, leading the
input level to be (1−β)σ times lower. Given this interpretation, the distribution of ex-post bargaining
power has a direct impact on comparative advantage through ex-ante efficiency of specialized input
production, i.e., the holdup problem. In particular, across sectors, a fraction of ex-ante contractible
activities would be relatively smaller in a more customized sector and the supplier’s holdup problem is
relatively severer in such a sector (Acemoglu et al., 2007). Furthermore, across countries, South would
have absolutely worse legal institutions in enforcing contracts and the supplier’s holdup problem in
South is absolutely severer than North (Antràs and Helpman, 2004). This is a theoretical justification
9
for log-supermodularity in institutional quality µ, with North having an absolute advantage in it.10
Although I formalize µ from incomplete-contract perspectives, there are alternative mechanisms
that may account for it. One of such candidates is financial constraints that exporters have to face
when covering the fixed costs. As demonstrated by Nunn (2007), sectors that are “contract-intensive”
tend to be sectors that produce more complex and technologically advanced products that are more
likely to require larger relationship-specific investments, relative to “non-contract-intensive” sectors
that produce simpler and technologically less advanced products. Now, to the extent that countries
differ in the development of their financial markets, cross-country differences in financial development
can lead to differences in comparative advantage (Manova, 2013). Moreover, constraints in financial
markets can also reduce the share of exporters who must pay some entry costs in order to access
foreign markets, with potential larger impacts in comparative disadvantage sectors (Chaney, 2013).
In this context, µ may as well alternatively capture the degree of financial development, and a more
contract-intensive sector (i.e., lower z sector) requires relatively better developed financial markets
(i.e., higher µ country) which in turn confers North a comparative advantage in these sectors.
As is clear from this alternative explanation, the mechanisms at work are not necessarily restricted
to partial contractibility. While the current paper focuses mainly on institutional factors as a source
of countries’ comparative advantage, they can be interpreted more broadly. All what matters for the
main results is that labor productivity stemming from countries’ and sectors’ characteristics satisfies
log-supermodularity.
Firm Behavior The current paper analyzes a static version of the Melitz (2003) model. To enter
a sector in country i ∈ {N, S}, firms bear a fixed cost of entry fe , measured in country i’s labor
units. Upon paying this fixed cost, firms draw their productivity level ϕ from a known distribution
Gi (ϕ, z) = G(ϕ). After observing this productivity level, each firm decides whether to exit or not.
If the firm chooses to produce, it bears additional fixed costs fd for domestic production and fx for
exporting, as described before. An entering firm in country i would then immediately exit if πdi < 0,
or would produce and serve its domestic market if πdi ≥ 0. Moreover, among domestic firms, only
the most productive firms would earn πxi ≥ 0 and serve the foreign market in j as exporters under
the assumption fx > fd .
While this firm behavior is similar across countries and sectors, the productivity cutoffs for
domestic production and exporting would vary by reflecting countries’ comparative advantage. Regarding exporting participation, if πxi ≥ 0 and πxj ≥ 0 for some firms in i 6= j ∈ {N, S}, well-known
two-way (intra-industry) trade occurs in this sector: trade occurs even in the same sector because
products are differentiated and consumers are strictly better off by importing products unavailable
in the domestic market. If πxi ≥ 0 for some firms and πxj < 0 for any firm, on the contrary, one-way
(inter-industry) trade occurs in this sector, whereby exporting from i to j takes place.
10
I am aware that the concept of partial contractibility, µ, here is not exactly the same as that developed by Acemoglu
et al. (2007): they use µ as the fraction of tasks that are contractible for input provision, whereas I use it as the fraction
of revenue that firms share at a bargaining stage. Most important for the pattern of trade, however, is not this modeling
difference but the fact that µ varies with countries’ and sectors’ characteristics, which also holds in their work.
10
3
Partial Equilibrium
In this section, I first explore partial equilibrium in which some important variables are exogenously
given. The next section embeds this analysis in a general-equilibrium setting.
To see an equilibrium of sector z, consider country i’s market where domestic firms in i and
foreign exporters from j engage in monopolistic competition. From the first-order conditions in the
previous section, the profit functions of these firms are respectively given by11
µ
πdi (ϕ, z)
i
= B (z)
µi (z)
wi
¶σ−1
µ
σ−1
ϕ
i
− w fd ,
πxj (ϕ, z)
i
= B (z)
µj (z)
wj
¶σ−1
ϕσ−1 − wj fx .
Notice that, since these firms compete in i’s market, aggregate demand B i is common for both profit
functions. πdi and πxj are measured by different wage rates and contractibility levels, however, because
exporters from j have to use domestic labor and institutions to produce own variety. If there exist
multinational enterprises that directly employ local labor and have internal contractibility within the
firm boundaries, this argument is no longer true. See Section 5 for the possibility of foreign affiliate
production in the current setup.
To compare these two profit functions graphically, they are drawn in (ϕσ−1 , π) space with slope
¡ µ ¢σ−1
B w
and intercept −wf . Then, πdi is steeper than πxj in i’s market (or πxi is steeper than πdj in
j’s market) if and only if
µj (z)
µi (z)
R
wi
wj

µ(z) R ω
⇐⇒
µ(z) Q ω
if i = N,
if i = S,
where µ(z) = µN (z)/µS (z) = a(z, µS )/a(z, µN ) and ω = wN /wS respectively denote the relative
contractibility (or the relative labor requirement) and the relative wage in North. Following standard
Ricardian models, I say that North (South) has an institutional comparative advantage in sector z if
it has a large (small) relative labor productivity in terms of partial contractibility µ(z), and/or a small
(large) relative wage ω, i.e., µ(z) > ω (µ(z) < ω). Under this definition, country i’s institutional
comparative advantage is identified as sectors where π i is steeper than π j in each country’s market.
Definition 2 North (South) has an institutional comparative advantage in sector z if µ(z) > ω
(µ(z) < ω), or equivalently if π N (π S ) is steeper than π S (π N ) in each country’s market.
Since this definition indicates neither North nor South has a comparative advantage in a sector
where µ(z) = ω, I first derive the condition under which this equality holds. From Figure 1, the ratio
of contractibility µ(z) = µN (z)/µS (z) has to satisfy µ(z) > 1, µ0 (z) < 0, µ00 (z) > 0, limz→1 µ(z) =
1 and limz→0 µ(z) = ∞, where the first condition stems from absolute advantage of North and the
third condition stems from log-supermodularity of µi (z). The relative wage in North ω, on the other
11
Following the literature (e.g. Melitz and Redding, 2014), I assume that all production costs, including the fixed
export cost fx , are measured in terms of a source country labor.
11
S dS , S xN
S dN , S xS
S dS
S dN
S xS
0
M N V 1
d
M S V 1
x
S xN
M V 1
0
M M S V 1
x
S V 1
d
M V 1
wS f d
wN fd
wS f x
wN f x
North
South
Figure 2 – Profits from domestic sales and exports
hand, should be the same for all sectors z ∈ [0, 1] because labor is completely mobile across sectors
within a country in the Ricardian model. This suggests that, if ω is greater than one, these two curves
intersect at a unique cutoff z̄ = µ−1 (ω) such that: (i) z ∈ [0, z̄) ⇔ µ(z) > ω; (ii) z = z̄ ⇔ µ(z) = ω;
and (iii) z ∈ (z̄, 1] ⇔ µ(z) < ω. It is then immediate from Definition 2 that North (South) has an
institutional comparative advantage in relatively customized (generic) sectors z ∈ [0, z̄) (z ∈ (z̄, 1]).
Next I consider the equilibrium in the cutoff sector z̄ as a benchmark. Figure 2 illustrates the
profit functions of πdi and πxj in this sector for North (left panel) and South (right panel). As is
well-known, this figure depicts productivity sorting: firms with productivity ϕ > ϕ̄id produce in i,
whereas firms with productivity ϕ > ϕ̄ix export to j. Since these two profit functions must be parallel
in this sector, if some domestic firms produce in i (i.e., ϕ̄id is finite), some foreign firms export from j
(i.e., ϕ̄jx is also finite). Thus, two-way trade occurs in sector z̄: trade occurs even in the same sector,
because products are differentiated and consumers are strictly better off by importing products that
are not available in the domestic country.
This argument helps understand what happens in sectors other than the cutoff sector z̄. In sectors
z ∈ [0, z̄) where North has a comparative advantage, for example, πdN and πxN are respectively steeper
than πxS and πdS in both domestic and export markets. This however does not always imply that the
N
productivity cutoffs, ϕ̄N
d and ϕ̄x , become smaller than the counterpart cutoffs in the cutoff sector
z̄ and less productive firms are more likely to find it profitable to operate in these sectors. This is
because the slope of the profit functions depends not only on µ/w but also on aggregate demand B,
which is so far exogenously given. I will show in the next section that North’s comparative advantage
allows the profit functions to change as depicted by the dotted lines in Figure 2. As a consequence,
N
while ϕ̄N
x becomes smaller and less productive firms can enter the export market, ϕ̄d becomes larger
and only more productive firms can survive in the domestic market. Similarly, two-way trade would
12
occur in any sector z ∈ [0, 1] as long as the slope of πxN and πxS is positive, although this might not
always true once B is taken into account.
Note in Figure 2 that the intercept of πxi lies below the intercept of πdj for both countries, i.e.,
wS fx > wN fd ⇔ ω <
fx
fd
in North and wN fx > wS fd ⇔ ω >
fd
fx
in South, ensuring that foreign
exporters from i must bear the higher fixed cost (measured by the source country labor wage) than
domestic firms in j. Because it is empirically well-known that fx is huge in any manufacturing sector,
I hereafter assume that these two inequalities hold in any sector z ∈ [0, 1], which will be shown to
be necessarily satisfied in the general-equilibrium setting where ω is endogenous.
Assumption 1
fd
fx
<ω<
fx
fd .
While the existence of the unique cutoff sector z̄ in the absence of variable trade cost is reminiscent
of Dornbusch, Fischer, and Samuelson’s (1977) Ricardian model with a continuum of goods,12 there
exist three noteworthy distinctions between the current paper and theirs. First, since they analyze
perfect competition with homogeneous goods, complete specialization (or inter-industry trade) occurs
below/above the cutoff z̄; in contrast, this paper studies monopolistic competition with differentiated
goods and incomplete specialization (or intra-industry trade) can occur in all manufacturing sectors.
Secondly, the mass and size of domestic firms and exporters are both indeterminate and irrelevant in
their neoclassical trade model, but it is endogenously determined in the current framework in which
the mass of varieties exported is only a subset of the mass of varieties produced in the home market.
Finally, this paper’s focus is on North-South trade where the difference in economic development plays
a prominent role in shaping countries’ comparative advantage. The result – that a less developed
country nevertheless exports differentiated goods in customized sectors – seems to be consistent with
recent trade flows (see, e.g., Krugman, 2008).
Proposition 1
(i ) If ω > 1, there exists a unique cutoff sector z̄ ∈ (0, 1) such that North (South) has an institutional comparative advantage in sectors z ∈ [0, z̄) (z ∈ (z̄, 1]).
(ii ) Two-way trade can occur in any sector z ∈ [0, 1].
It is important to emphasize that this partial-equilibrium analysis cannot clarify the interplay
among key variables of the model. To see this, it is useful to go back to Figure 2. The figure depicts
ϕ̄jd < ϕ̄ix in the cutoff sector z̄, indicating that foreign exporters from i are more productive than
domestic firms in j. This outcome is not wholly surprising because foreign exporters are assumed to
incur the higher fixed cost under Assumption 1, and it can be easily formalized by using a partialequilibrium framework. However, ϕ̄id and ϕ̄jd or ϕ̄ix and ϕ̄jx are not comparable. Obviously, ϕ̄N
d and
12
The idea of bringing firm heterogeneity into the Dornbusch et al. (1977) model is not entirely new, although the
models’ setups in the existing literature are less general than the current one. For instance, Okubo (2009) uses a
specific distribution of firm productivity levels, whereas Fan et al. (2011) abstract from endogenous wage differentials
between countries by introducing the homogeneous-good sector.
13
S
N
S
N
S
ϕ̄Sd (ϕ̄N
x and ϕ̄x ) are determined at which πd = 0 and πd = 0 (πx = 0 and πx = 0), but these
variables depend on the aggregate market demand B i as well as the wage rate wi , both of which
are exogenous in partial equilibrium. Also, Proposition 1(i) requires that ω should be greater than
one if North-South trade is to occur, but it is not clear whether this holds or not. In this sense,
the above partial-equilibrium setting is restricted, and a general-equilibrium approach is necessary
to endogenize these variables.
4
General Equilibrium
In this section, the partial-equilibrium analysis is embedded into a general-equilibrium framework
to examine the interaction among the key variables and to see endogenous patterns of specialization
and trade.
General-Equilibrium Setup This subsection first outlines several equilibrium conditions that play
a central role in characterizing the endogenous variables in general equilibrium. In the subsequent
subsections, I solve this general-equilibrium model with some restrictions on the exogenous variables.
Firstly, a zero profit condition must hold for all sectors z ∈ [0, 1] of domestic and export markets.
The productivity cutoff that satisfies this condition is identified by ϕ̄id (z) = inf{ϕ : πdi (ϕ, z) > 0}
and ϕ̄ix (z) = inf{ϕ : πxi (ϕ, z) > 0} for domestic firms and exporters respectively. This condition is
refereed to as a zero cutoff profit (ZCP) condition:
µ
πdi (ϕ̄id , z)
i
= 0 ⇐⇒ B (z)
µ
πxi (ϕ̄ix , z)
µi (z)
wi
j
= 0 ⇐⇒ B (z)
µi (z)
wi
¶σ−1
(ϕ̄id (z))σ−1 = wi fd ,
(ZCPdi )
(ϕ̄ix (z))σ−1 = wi fx ,
(ZCPxi )
¶σ−1
for i 6= j ∈ {N, S}. In this condition, aggregate market demand of exporters (B j ) should be different
from that of domestic firms (B i ) because exporters from country i have to face aggregate market
demand in country j.
Secondly, a free entry (FE) condition must be satisfied for all sectors. Since potential entrants
are ex ante identical in the current model, this condition is defined as
Z
∞
ϕ̄id (z)
Z
πdi (ϕ, z)dG(ϕ)
∞
+
ϕ̄ix (z)
πxi (ϕ, z)dG(ϕ) = wi fe ,
(F E i )
where the first and second terms in the left-hand side respectively denote the expected operating
profits from domestic sales and exports earned by potential entrants in sector z. The sum of these
expected profits has to be equal to the fixed entry cost wi fe .
Finally, a labor market clearing (LMC) condition must be taken into account:
Z
0
1
Z
Mei (z)
∞
ϕ̄id (z)
Z
ldi (ϕ, z)dG(ϕ)dz +
0
1
Z
Mei (z)
∞
ϕ̄ix (z)
14
Z
lxi (ϕ, z)dG(ϕ)dz +
0
1
Mei (z)fe dz = Li , (LM C i )
where Mei denotes the mass of potential entrants, which varies across sectors in country i. In this
equation, the first and second terms in the left-hand side are the sum of expected labor demands
used for domestic production and exporting by potential entrants, whereas the third term is expected
labor demands used for investment by potential entrants. Note that lxi is summed up over sectors
of the economy as a whole because two-way trade can occur in any sector z ∈ [0, 1] as seen in
Proposition 1(ii). The sum of these expected labor demands that aggregate the use of labor across
all sectors in i has to be equal to the fixed labor supply Li .
Now, it is possible to endogenize the important variables in general equilibrium. Since there are
the eight equations (the ZCP, FE, and LMC conditions that must hold in North and South), these
conditions provide implicit solutions for the following eight unknowns:
S
N
S
N
S
N
S 13
ϕ̄N
d (z), ϕ̄d (z), ϕ̄x (z), ϕ̄x (z), B (z), B (z), w , w ,
where the LMC condition in South can be omitted by Walras’ law, thereby normalizing wS = 1 as a
numéraire. (The mass of potential entrants Mei can be written as a function of these eight unknowns
as will be shown later.)
Relative Equilibrium Conditions This subsection sets forth the characterization of the eight
unknowns from the eight equilibrium conditions. It is challenging, however, to solve a full general
equilibrium model with asymmetric countries. In particular, without specifying the functional form
of the firm size distribution G(ϕ), explicit solutions of these unknowns cannot be obtained. In
the following, instead of obtaining the exact values of each of them, the main focus is devoted to
characterizing the relative terms of these unknowns.
Recall from Proposition 1(i) that North-South trade occurs only if the relative wage in North ω
is greater than one. Although this relative wage is endogenously determined in the model, suppose
first that ω > 1 in equilibrium. In other words, the LMC conditions are left out from the model as
if it were partial-equilibrium. As will be clear, this inequality must be true in general equilibrium
because I assume that North has an absolute advantage in partial contractibility µ in any sector.
This means that the marginal product of labor is higher (on average across heterogeneous firms) in
North, and thus wages have to be greater in North than South if trade is to occur between the two
countries. This intuition will be confirmed later by integrating the LMC conditions.
Under the circumstance, I first examine the sectoral difference in the relative productivity cutoffs
N
S
S
N
S
(ϕd = ϕN
d /ϕd , ϕx = ϕx /ϕx ) and the relative market demand (B = B /B ) by focusing on the
ZCP and FE conditions (recall these two conditions must hold for all sectors). To do this, using the
ZCP conditions, rewrite the FE condition as
fd J(ϕ̄id (z)) + fx J(ϕ̄ix (z)) = fe ,
13
As is evident from the dependence on z, the first six endogenous variables are allowed to vary across sectors in this
model; in contrast, wages are independent of z due to perfect intersectoral mobility of labor. Related to this, it would
be more accurate to say that the ZCP and FE conditions are the six equations that hold for each sector, and the LMC
conditions are the additional two equations that aggregate the use of labor in all sectors for each country.
15
B
Md
Mx
B
1
M d ,M x
Z 1/(V 1)
0
z
1
z
Figure 3 – Market demand and productivity cutoffs
where J(ϕ̄) =
R∞
ϕ̄
[(ϕ/ϕ̄)σ−1 − 1]dG(ϕ). J(·) is monotonically decreasing with limϕ̄→0 J(ϕ̄) = ∞
and limϕ̄→∞ J(ϕ̄) = 0.14 Since the equality of this condition must hold in any sector, changes in
z shift the productivity cutoffs in opposite directions, and thus ϕ̄ix /ϕ̄id must be strictly increasing
or decreasing in z. Note that these changes in z affect ϕ̄’s and B’s while they have no impact on
w’s as wages are independent of z (so long as labor is completely mobile across sectors), and the
equilibrium analysis here does not rely on the exogenous ω assumption. Moreover, dividing the ZCP
condition of domestic production by that of exporting for each country, the relative market demand
B = B N /B S is given by
³
´σ−1
N

fd
 ϕ̄xN (z)
fx
ϕ̄d (z)
B(z) = ³ S ´σ−1

fx
 ϕ̄dS (z)
fd
ϕ̄x (z)
if i = N,
if i = S.
Using the property of the FE condition derived above, it can be shown that B is strictly increasing in
z (see Appendix). The intuition behind this result is explained by recalling that B is proportional to
the relative aggregate price (P = P N /P S ). If z is close to one, the institutional differential is almost
negligible whereas there exists the wage differential (i.e., ω > 1). South is thus able to produce goods
relatively cheaply, thereby leading to P N > P S and B > 1 in the neighborhood of z = 1. If z is close
to zero, on the other hand, the institutional differential is sufficiently large to dominate the wage
differential (due to log-supermodularity), resulting in P N < P S and B < 1 in the neighborhood of
z = 0. Roughly speaking, this intuition mirrors the idea that country i has a comparative advantage
in sectors where the aggregate price P i is relatively lower than P j .15 The first quadrant of Figure 3
depicts this relationship in (z, B) space.
14
I have so far assumed that the fixed cost parameters – fd , fx and fe – are common across all sectors z ∈ [0, 1].
As indicated earlier, it would be more realistic to allow the fixed costs to be systematically correlated by the degree
of customization z: “contract-intensive” sectors tend to require large relationship-specific investments in evidence.
In Appendix, I show that the following result continues to hold if these fixed costs increase proportinately with the
customization measure, i.e. f (az) = af (z) for any positive number of a.
15
This statement is not precise because the aggregate price P i includes prices of both domestic and foreign varieties
in the presence of two-way trade. Given log-supermodularity, this would hold in a closed-economy version of the model.
16
N
d
N
x
N
d
N
x
N
N
0
0
s
,
S
0
S
S
x
S
d
0
S
x
S
d
z
0! z! z
z 1
Figure 4 – Relationship among productivity cutoffs
Next, dividing the ZCP condition of North by the corresponding condition of South, the relative
S
N
S
productivity cutoffs ϕ̄d = ϕ̄N
d /ϕ̄d and ϕ̄x = ϕ̄x /ϕ̄x satisfy the following relative ZCP condition:
µ
ϕ̄d (z) =
ω
B(z)
¶
1
σ−1
ω
,
µ(z)
1
ϕ̄x (z) = (B(z)ω) σ−1
ω
,
µ(z)
(RZCP )
where all variables are represented by the relative terms in North. It is easy to show that ϕ̄d decreases
with B while ϕ̄x increases with B, and that
ϕ̄x (z) R ϕ̄d (z) ⇐⇒ B(z) R 1.
Further, simple geometry in Figure 2 suggests that ϕ̄d and ϕ̄x are equal if and only if πdi and πxj are
parallel in the cutoff sector z̄. Thus, the relative market demand and relative productivity cutoffs
respectively satisfy B(z̄) = 1 and ϕ̄d (z̄) = ϕ̄x (z̄) = ω 1/(σ−1) (using µ(z̄) = ω) in that sector. The
second quadrant of Figure 3 depicts this relationship in (B, ϕ̄) space.
Finally, combining the first and second quadrants, Figure 3 highlights the sectoral difference
among the six endogenous variables (represented in the relative terms) that are derived from the
ZCP and FE conditions: in any sector z ∈ [0, 1], the relative market demand B is determined in the
first quadrant, and the relative productivity cutoffs ϕ̄d and ϕ̄x are subsequently determined in the
second quadrant. It is immediately seen that
1
0 ≤ z < z̄ ⇐⇒ B(z) < 1 ⇐⇒ ϕ̄d (z) > ω σ−1 > ϕ̄x (z),
1
z = z̄ ⇐⇒ B(z) = 1 ⇐⇒ ϕ̄d (z) = ω σ−1 = ϕ̄x (z),
1
z̄ < z ≤ 1 ⇐⇒ B(z) > 1 ⇐⇒ ϕ̄d (z) < ω σ−1 < ϕ̄x (z).
In this relationship, either ϕ̄d or ϕ̄x is necessarily greater than one under the condition that the
relative wage ω is greater than one. For instance, ϕ̄d is greater than one in sectors where North has
a comparative advantage z ∈ [0, z̄); however whether ϕ̄x is greater than one or not is indeterminate
in the current setup.
Based on this observation, Figure 4 illustrates the relationship among the productivity cutoffs in
the comparative advantage sectors of North (left panel) and South (right panel). While the figure
S
N
S
shows that ϕ̄x = ϕ̄N
x /ϕ̄x > 1 (left panel) and ϕ̄d = ϕ̄d /ϕ̄d > 1 (right panel), these might not hold
17
in some sectors. Regardless of whether or not they are greater than one, the gap between ϕ̄ix and
ϕ̄id is narrower than the gap between ϕ̄jx and ϕ̄jd in country i’s comparative advantage sectors. In
addition, this former (latter) gap becomes smaller (bigger) as country i’s comparative advantage is
stronger. These can be seen more formally in terms of the productivity cutoff ratio:
ϕ̄N
x (z)
=
ϕ̄N
d (z)
µ
¶ 1
fx σ−1
B(z)
,
fd
ϕ̄Sx (z)
=
ϕ̄Sd (z)
µ
1 fx
B(z) fd
¶
1
σ−1
.
Comparing these two ratios immediately reveals that ϕ̄ix /ϕ̄id is smaller than ϕ̄jx /ϕ̄jd in country i’s
N
S
S
comparative advantage sectors. Also since B is strictly increasing in z, ϕ̄N
x /ϕ̄d (ϕ̄x /ϕ̄d ) is strictly
increasing (decreasing) in z, which is achieved through the following channel (see Appendix):
0
0
0
0
N
S
S
ϕ̄N
d (z) < 0, ϕ̄x (z) > 0, ϕ̄d (z) > 0, ϕ̄x (z) < 0.
This is what I have stressed by the dotted lines in Figure 2: as country i’s comparative advantage
0
0
S
is stronger,16 not only are domestic firms more productive (ϕ̄N
d (z) < 0, ϕ̄d (z) > 0), but also less
0
0
S
productive firms are more likely to export (ϕ̄N
x (z) > 0, ϕ̄d (z) < 0). In Figure 2, this is equivalent
with that, according to comparative advantage strength, the slope of the profit function is steeper
for πx but flatter for πd . Hence comparative advantage works oppositely for the productivity cutoffs
of domestic production and exporting: ϕ̄id increases and ϕ̄ix decreases with the degree of comparative
advantage. It is through this channel that the model predicts a higher ratio of exporting firms to
overall surviving firms in sectors where countries are relatively more productive.
The above equations also indicate that the productivity cutoff for exporting is necessarily bigger
than that of domestic production (ϕ̄ix > ϕ̄id ) in the comparative disadvantage sectors for i ∈ {N, S}.17
In the comparative advantage sectors, the usual outcome (ϕ̄ix > ϕ̄id ) occurs in both countries if
fx
fd
< B(z) < ,
fx
fd
(1)
whereas the “perverse” outcome (ϕ̄id > ϕ̄ix ) occurs if

B(z) <
B(z) >
fd
fx
fx
fd
if i = N,
if i = S.
ϕ̄id > ϕ̄ix implies that, among surviving firms in i, less productive firms serve only the foreign market
in j, while more productive firms serve both the foreign market in j and the home market in i. Clearly,
this occurs in sectors where countries’ comparative advantage (measured by aggregate demand B)
is strong enough relative to the fixed-cost ratio between fd and fx .
16
Noting that µ(z) = a(z, µS )/a(z, µN ) is the relative unit labor requirement and is decreasing in z, comparative
advantage of North (South) is said to be “stronger” if z is closer to zero (one) in this model.
17
From fx > fd , comparative disadvantage sectors of North, for example, must satisfy
N
z̄ < z ≤ 1 ⇐⇒ B(z) > 1 =⇒ ϕ̄N
x (z) > ϕ̄d (z).
18
Proposition 2 Taking the relative wage ω as given, the following holds:
(i ) The cutoff ratio ϕ̄ix /ϕ̄id is smaller than ϕ̄jx /ϕ̄jd in country i’s comparative advantage sectors and
this ratio is monotonically decreasing in its comparative advantage strength.
(ii ) While ϕ̄ix is always greater than ϕ̄id in country i’s comparative disadvantage sectors, this might
not hold in extremely strong comparative advantage sectors.
These two findings fit well with recent empirical research. The first finding – among domestic
firms, more firms export in stronger comparative advantage sectors – is consistent with evidence that
was reviewed in Introduction. The logic of this result comes from the interplay between the Ricardian
productivity difference and relative burden of fixed export costs: log-supermodularity in country and
sector characteristics allows relatively less productive firms to incur the fixed export cost relatively
more easily in comparative advantage sectors. Strictly speaking, this observation is not satisfactory
since several important questions cannot be addressed without the mass of varieties produced and
exported. For example, how does countries’ comparative advantage affect the extensive and intensive
margins? Does a larger country size lead to a higher export participation ratio in any sector? Later
I investigate what determines the mass of varieties to answer these questions.
The second finding is also in keeping with Lu’s (2011) empirical evidence that manufacturing
exporters in China are typically less productive than domestic firms in labor -intensive sectors, but
exporters are more productive in capital -intensive sectors. To rationalize this evidence, Lu develops
a Heckscher-Ohlin model with heterogeneous firms, emphasizing that allowing factor intensity to
vary across sectors is crucial for the Melitz model of North-South trade. Although my theoretical
focus is apparently different from hers, the central message is surprisingly similar: exporters can be
less productive than domestic firms in comparative advantage sectors, whereas exporters are always
more productive in comparative disadvantage sectors. The rationale for this result is as follows. In
comparative disadvantage sectors of country i, by definition, firms in country j are relatively better
at producing than those in i. Thus, exporters from i must be sufficiently productive, not only because
they have to cover the fixed export cost, but also because they have to compete with more efficient
foreign rivals in the export market. In comparative advantage sectors of i, on the other hand, firms
in j are relatively poorer at producing than those in i, which makes relatively less productive firms
in i find it profitable to enter the export market. This is the reason why the partitioning of firms
by export status might not be induced only with the condition fx > fd in comparative advantage
sectors. (This holds even with variable trade cost as far as τ σ−1 fx > fd ; see supplementary note.)
If ϕ̄id > ϕ̄ix were true, all surviving firms in i could export, which is not supported by evidence.18
This result should be interpreted as meaning that aggregate productivity premia of exporters relative
to domestic firms are smaller in stronger comparative advantage sectors. I exclude the possibility
ϕ̄id > ϕ̄ix in the following analysis by assuming that fx is large enough to satisfy (1) in any sector.
18
In Lu’s (2011) dataset, ϕ̄id > ϕ̄ix occurs because it includes Chinese exporters involved in processing trade. If firms
engage in final-good trade only as in the current model, this possibility would not exist.
19
Full General Equilibrium The previous subsection provides implicit solutions of firm selection
(ϕ̄id , ϕ̄ix ) and aggregate market demand (B i ) for a given wage rate (wi ). Now that each sector’s
equilibrium is characterized by these six endogenous variables, this subsection explores full generalequilibrium interactions by explicitly incorporating the LMC conditions in the model. Here I show
that the relative wage in North (ω = wN /wS ) is necessarily greater than one, a sufficient condition
of North-South trade required in Proposition 1(i).
Recall from the profit-maximization problem in Section 2 that the amount of labor required to
produce q is
ldi (ϕ, z) = fd +
σ − 1 rdi (ϕ, z)
,
σ
wi
lxi (ϕ, z) = fx +
σ − 1 rxi (ϕ, z)
.
σ
wi
Substituting these values into the LMC condition and using the FE condition, the previous LMC
condition is simplified as follows (see Appendix):
R1
0
R1
R1
P i (z)Qi (z)dz is aggregate expenditure (or revenue). Thus country i’s wage wi
R1
is determined by the equality between aggregate expenditure ( 0 Ri (z)dz) and aggregate payments
where
0
Ri (z)dz =
Ri (z)dz
= Li ,
wi
0
to labor (wi Li ) as in usual general-equilibrium trade models.
By Walras’ law, I can focus on the LMC condition in North. To derive the relative wage ω, it
is easiest to use the balance-of-payments (BOP) condition, which is equivalent with the above LMC
condition in the current model where two-way trade can occur in any sector (see Proposition 1(ii)):
Z
0
where Rxi (z) = Mei (z)
R∞
1
Z
RxN (z)dz
i
ϕ̄id (z) rx (ϕ, z)dG(ϕ)
1
=
0
RxS (z)dz,
(BOP )
is aggregate sector exports. This equivalence stems from
the fact that aggregate expenditure in North is the sum of expenditure spent on domestic products
¢
R1
R1¡
and imports from South, 0 RN (z)dz = 0 RdN (z) + RxS (z) dz, and aggregate income is the sum of
¢
R1¡
domestic and export revenues, wN LN = 0 RdN (z) + RxN (z) dz.
Due to log-supermodularity, North (South) to produces and exports relatively more in customized
(generic) sectors and aggregate exports are strictly increasing in its comparative advantage strength
N
x (z)
(R
<
RS (z)
x
N (z 0 )
Rx
S (z 0 )
Rx
for z > z 0 ). Let z̃ denote a hypothetical cutoff below which North is a net exporter of
customized products in two-way trade. Since net aggregate sector exports are given by the difference
between aggregate sector labor income and expenditure, the above BOP condition becomes
Z
z̃
¡ N N
¢
w L (z) − RN (z) dz =
0
Z
1¡
¢
wS LS (z) − RS (z) dz,
z̃
where Li (z) is aggregate sector labor supply. This equation simply indicates that North runs a trade
surplus in sectors z ∈ [0, z̃) and a trade deficit in sectors z ∈ (z̃, 1]. In equilibrium, the relative wage
is adjusted so that aggregate trade surpluses are offset by aggregate trade deficits.
20
!
!
1
0
z
1
z
Figure 5 – Wage in general equilibrium
It is then straightforward to derive the relative wage ω. Arranging the BOP condition above and
using λ(z) = P i (z)Qi (z)/Y i defined in Section 2, ω can be explicitly solved as a function of z̃:
R 1 ³ LS (z)
´
−
λ(z)
dz µ 1 ¶
z̃
³
´
ω ≡ ξ(z̃) = R
,
z̃ LN (z)
− λ(z) dz L
0
LN
LS
where L = LN /LS is relative country size. In the current setting with free entry, aggregate sector
N
(z)
labor supply is equal to aggregate sector revenue and it is also log-supermodular ( LLS (z)
<
z >
z 0 );
LN (z 0 )
LS (z 0 )
for
thus free trade allows laborers to be allocated relatively more in sectors where countries’
comparative advantage is relatively stronger. Using this property, it is verified that ξ is an increasing
function of z̃ and under mild conditions limz̃→0 ξ(z̃) = 0 and limz̃→1 ξ(z̃) = ∞. Intuitively, this result
follows from noticing that ξ summarizes the LMC condition in each country. If z̃ is higher for a given
ω, there are more labor demands in North and Southern production is less likely. For North-South
trade to occur, therefore, ξ must be increasing in z̃ so that Northern labor is more expensive, thereby
ensuring some Southern labor demands in equilibrium. Figure 5 depicts ξ curve in (z, ω) space.19
The other condition that pins down z̄ = z̃ and ω is the relative partial contactibility µ = µN /µS .
Among its properties, µ > 1 (absolute advantage of North) and µ00 > 0 (log-supermodularity) are of
particular importance. Figure 5 depicts µ curve in the same space, showing that the intersection of
the two curves determines ω, which is necessarily greater than one. In addition, this ω also satisfies
Assumption 1 (see Appendix). These two curves intersect at (z̄, ω) because ω ≡ ξ is equal to µ if and
only if z = z̄, i.e., ω = µ(z̄). This endogenous solution of ω in turn leads to endogenous solutions of
B, ϕ̄d and ϕ̄x , which completes the characterization of the eight unknowns in equilibrium.
19
Although ξ curve has some similarity with that in Dornbusch et al. (1977), their neoclassical model allows all
R z̃ N
laborers to be allocated to each country’s comparative advantage sectors in trade equilibrium and thus 0 LLN(z) dz =
R 1 LS (z)
dz = 1 in the equation of ξ curve. In contrast, this is not true in the current model due to incomplete
z̃
LS
specialization that can occur in any sector.
21
Proposition 3
(i ) North (South) is a net exporter of customized (generic) sectors z ∈ [0, z̄) (z ∈ (z̄, 1]) in two-way
trade.
(ii ) The relative wage in North is greater than one.
It is worthwhile to stress that the above channel for the relative wage is similar to that developed
by Antràs (2005), who shows (with representative firms) that, irrespective of the relative country size
L = LN /LS , better contracting environments in North lead to the higher relative wage in general
equilibrium. As in his model, the equilibrium outcome ω > 1 directly reflects that North has superior
partial contractibility which helps mitigate the serious holdup problem. This in turn gives rise to
better production efficiency and overall higher productivity (i.e., wage) in North.
Note also that, in the current model with identical tastes across countries and no distortions, the
relative wage is a measure of welfare of the representative worker in North relative to that in South.
Thus the fact that ω > 1 implies that welfare per worker is higher in North.
Comparative Statics In this subsection, I examine comparative statics with respect to the relative
country size L in a single unified framework for the analysis of home-market effects on the extensive
and intensive margins that will be addressed in the next subsection.
Figure 6 illustrates a general-equilibrium interaction between goods and labor markets by integrating the previous analysis. To explore effects of an increase in L, I compare two equilibria in the
figure by adding primes (0 ) to all variables and functions with thin lines for a new equilibrium. The
first and second quadrants come from Figure 3, whereas the fourth quadrant comes from Figure 5.
The third quadrant depicts the relationship between ω and ϕ̄’s, which is derived from the RZCP
1
σ/(σ−1) , ϕ̄ = B ω σ/(σ−1) . It is clear
x
µB ω
µ
1/(σ−1)
(z̄) = ω
and z R z̄ ⇔ B R 1 ⇔ ξ R
conditions: ϕ̄d =
that these conditions are increasing in ω,
with ϕ̄d (z̄) = ϕ̄x
ω ⇔ ϕ̄d R ϕ̄x . The figure shows that:
• In the fourth quadrant, ξ curve shifts to the right (from the LMC/BOP conditions) while
keeping µ curve unchanged, leading to ω > ω 0 and z̄ < z̄ 0 .
• In the first quadrant, this decreases B for any z ∈ [0, 1] because B(z̄ 0 ) = 1 must be true in a
new equilibrium and B is increasing in z (from the ZCP/FE conditions).
• In the second quadrant, ϕ̄d curve shifts inward while ϕ̄x curve shifts upward because B =
ω σ/(σ−1) −1
ϕ̄d ,
µ
B =
µ
ϕ̄
ω σ/(σ−1) x
(from the RZCP condition) and ω is decreasing in L (from the
fourth quadrant).
• In the third quadrant, both curves shift inward because ϕ̄d =
1
σ/(σ−1) ,
µB ω
ϕ̄x =
B σ/(σ−1)
µω
(from the RZCP condition) and B is decreasing in L (from the first quadrant).
As is clear, the endogenous variables respond to changes in the exogenous variable L through the
relative wage ω. Note from Figures 3 and 4 that when ω is sufficiently large, it is more likely that
22
B
Mx
Mx '
Md
Md '
B
B'
1
Z 1/(V 1)
M d ,M x
Z '1/(V 1)
z
z'
z
0
P
Z'
Md '
Z
Md
Mx '
['
Mx
[
Z
Figure 6 – Comparative statics with respect to L
ϕ̄d > 1 and ϕ̄x > 1, and the higher aggregate productivity of domestic firms and exporters in North
is partially reflected by the higher relative wage. Building on this view, the comparative statics here
suggest that an increase in L decreases the Ricardian productivity advantage as well as the firm-level
productivity advantage in North by lowering ω, although it increases the range of sectors over which
North has a comparative advantage and is a net exporter of customized products.
One of key insights arising from the comparative statics is that, even with the C.E.S. preferences,
country size does affect firm selection (ϕ̄id , ϕ̄ix ) through the relative wage ω in this asymmetric-country
model with the Ricardian productivity difference µ. In a symmetric-country model or a model with
a homogeneous-good sector, by contrast, country size has impacts only on the mass of potential
entrants without affecting the firm-level variables.
Margins of Specialization and Trade Up until now the analysis has mainly dealt with the
characterization of the eight equilibrium unknowns. It has not addressed the mass of domestic firms
and exporters, which is also endogenously determined in equilibrium. To calculate the mass of firms
explicitly, I hereafter assume that firm productivity ϕ is drawn from a Pareto distribution:
µ
G(ϕ) = 1 −
ϕmin
ϕ
¶k
,
23
ϕ ≥ ϕmin > 0,
where both shape and scale parameters, k and ϕmin , are identical across countries and sectors under
Definition 1. The purpose of this subsection is to investigate the impact of comparative advantage
and country size on the extensive and intensive margins.
In what follows, I first derive the mass of potential entrants. Applying the Pareto distribution
to the aggregate sector price in each country P i , this mass is expressed as
µ
1
Mei (z) =
σ
wi
µi (z)
¶σ−1
X i (z)
1
B i (z) (ϕ̄j (z))k−σ+1
d
−
X j (z)
1
B j (z) (ϕ̄jx (z))k−σ+1
∆(z)
, 20
where implicit solutions for the eight unknowns are already given in the preceding analysis. Note
that if two-way trade is to occur in any sector as suggested in Proposition 1(ii), Mei should be strictly
positive in any sector of country i. It directly follows that Mei > 0 for z ∈ [0, 1] if and only if
µ
X(z)
σ−1
k
fd
fx
¶ k−σ+1
k
µ
< B(z) < X(z)
σ−1
k
fx
fd
¶ k−σ+1
k
,
(2)
where X = X N /X S = ωL. Although the restriction on B in (2) has different implications from that
in (1) – i.e. (1) by which ϕ̄ix > ϕ̄id versus (2) by which Mei > 0, the range of (1) is certainly greater
than the range of (2) in the current setup (see Appendix). Accordingly, only with (1) does one-way
trade take place in sectors where countries’ comparative advantage is extremely strong. To perform
a consistent analysis (especially for the LMC/BOP conditions), I henceforth assume that the fixed
export cost is sufficiently large to satisfy (2) in any sector, but the main analysis would essentially
go through even if two types of trade coexist in the model.
I first examine the effect of comparative advantage on the extensive and intensive margins. Let
aggregate sector exports Rxi decompose into
Z
Rxi (z)
=
Mei (z)
= [1 −
∞
ϕ̄ix (z)
rxi (ϕ, z)dG(ϕ)
G(ϕ̄ix (z))]Mei (z)
1
×
[1 − G(ϕ̄ix (z))]
Z
∞
ϕ̄ix (z)
rxi (ϕ, z)dG(ϕ)
i
= Mex
(z) × r̄xi (z),
i is the mass of firms that engage in exporting (extensive margin), and r̄ i is average exports
where Mex
x
across heterogeneous firms (intensive margin). Similarly, aggregate domestic sales are decomposed
i × r̄ i . Under the Pareto distribution, these margins are expressed as
into Rdi = Med
d
µ
i
Mex
(z)
=
µ
i
Med
(z)
20
“
∆=
kϕk
min
k−σ+1
” »“
1
ϕ̄N
ϕ̄S
d
d
”k−σ+1
=
“
−
ϕmin
ϕ̄ix (z)
ϕmin
ϕ̄id (z)
1
S
ϕ̄N
x ϕ̄x
¶k
¶k
Mei (z),
r̄xi =
kσ
wi fx ,
k−σ+1
Mei (z),
r̄di =
kσ
wi fd .
k−σ+1
”k−σ+1 –
, which is positive under (1) and k − σ + 1 > 0.
24
M ed , M ex
M ex
M ed
1
XF 1
Z F X 0
zd z x
z
1
z
Figure 7 – Relative extensive margins of specialization and trade
It is obvious that the intensive margins (r̄di , r̄xi ) are independent of sector index z and thus comparative
advantage; due to log-supermodularity, however, aggregate domestic sales and exports (Rdi , Rxi )
are both increasing in it, suggesting that comparative advantage increases these aggregates solely
i , M i ). As a result, the extensive margins are log-supermodular
through the extensive margins (Med
ex
M N (z)
( MdS (z) <
d
MdN (z 0 )
MdS (z 0 )
and
MxN (z)
MxS (z)
<
MxN (z 0 )
MxS (z 0 )
for z > z 0 ).
Intuitively, comparative advantage has two opposing effects on the intensive margins that can be
seen in Figure 2. First, comparative advantage affects the productivity cutoffs: in case of exporters,
stronger comparative advantage in i allows ϕ̄ix to be smaller, and since less productive firms enter
the export market, this decreases the average export revenue r̄xi . Second, comparative advantage
also affects the slope of the profit functions: in case of exporters, stronger comparative advantage
i
µ
in i allows B j ( w
i ) to be steeper, and since all surviving firms earn the higher profit from the export
market, this increases r̄xi . Following the same line of reasoning, these two opposing effects similarly
i
µ
arise for domestic firms, with only differences that ϕ̄id becomes greater and B i ( w
i ) becomes flatter
according to i’s comparative advantage. Under the Pareto distribution, these two effects are exactly
offset, leaving the intensive margins independent of comparative advantage strength.
As in the previous subsections, the current paper’s primary interest lies in the analysis of the
relative terms, rather than the absolute terms. From the equations of the absolute margins, the
relative extensive and intensive margins in North are given by
Med (z) =
N (z)
Med
Me (z)
=
,
S
(ϕ̄d (z))k
Med (z)
Mex (z) =
N (z)
Mex
Me (z)
=
,
S
Mex (z)
(ϕ̄x (z))k
r̄dN
r̄xN
=
= ω > 1,
r̄xS
r̄dS
where Me = MeN /MeS is relative mass of entrants. Thus, regardless of country size, the two intensive
margins are always higher in North which reflects the absolute advantage assumption. Regarding the
two extensive margins, log-supermodularity allows both Med and Mex to be strictly decreasing in z.
0 (z) < M 0 (z) < 0
Figure 7 depicts Med and Mex curves in (z, M ) space. These two curves satisfy Mex
ed
25
Table 1 – Margins of trade and aggregate sector exports
Sector
Extensive margin
Intensive margin
Aggregate sector exports
z ∈ [0, z̄x )
z ∈ (z̄x , z̄)
z ∈ (z̄, 1]
N > MS
Mex
ex
N
S
Mex < Mex
N < MS
Mex
ex
r̄xN > r̄xS
r̄xN > r̄xS
r̄xN > r̄xS
N r̄ N > M S r̄ S
Mex
ex x
x
N
S r̄ S
Mex r̄xN > Mex
x
N r̄ N < M S r̄ S
Mex
x
ex x
³
for z ∈ [0, 1] and their intersect is at
XF −1
z̄, ω(F
−X)
´
where F = (fx /fd )(k−σ+1)/(σ−1) . Note that as
countries’ comparative advantage is stronger, Mex becomes relatively greater than Med . This finding
suggests that the degree of comparative advantage has different impacts of the two extensive margins:
the mass of exporting firms increases relatively more significantly than the mass of domestic firms in
comparative advantage sectors. In other words, the share of firms that export is log-supermodeular
M N (z)/M N (z)
( MxS (z)/MdS (z) <
x
d
MxN (z 0 )/MdN (z 0 )
MxS (z 0 )/MdS (z 0 )
for z > z 0 ).
Figure 7 is also useful for understanding whether differences in aggregate output due to comparative advantage are achieved through differences in the extensive and intensive margins. Let z̄d and
z̄x respectively denote the cutoffs at which Med (z̄d ) = 1 and Mex (z̄x ) = 1 with z̄d < z̄x < z̄. These
z̄d and z̄x are the cutoffs below which North produces and exports relatively more varieties than
South (extensive margin), while z̄ is the cutoff below which North is a net exporter of customized
N r̄ N − M S r̄ S > 0 for z ∈ [0, z̄). Since the intensive margin is always
products, i.e., RxN − RxS = Mex
x
ex x
higher in North, the relationship between the extensive/intensive margins of trade and aggregate
sector exports across sectors are summarized in Table 1. It is important to note that the vertical
intersection of the two curves in Figure 7 is smaller than unity for any relative country size L: indeed,
if
XF −1
ω(F −X)
i
> 1, not only is the intensive margin of trade r̄xi but the extensive margin of trade Mex
is also higher for North in some comparative advantage sectors of South z ∈ (z̄, 1], contradicting the
previous argument that trade is balanced in the cutoff sector z̄. Thus, North does not always have
the larger mass of exporters in its comparative advantage sectors to compensate the higher relative
wage. Although the above result holds under the special case of a Pareto distribution, the extensive
margin is generally more important than the intensive margin in capturing aggregate trade flows
because comparative advantage has the two opposing effects on the intensive margin as mentioned
above.
This analysis has direct implications for the effect of country size L on the extensive and intensive
margins. From the expression of the relative margins, it follows that the relative intensive margins
increase with L due to the lower relative wage (see Figure 6). For the relative extensive margins,
in contrast, the mass of entrants first increases with L due to the home-market effect (MeN /MeS >
N /M S > LN /LS
LN /LS ),21 and then the mass of domestic firms and exporters increases with it (Med
ed
N /M S > LN /LS ). This effect of country size on the two margins should be opposite because
and Med
ed
21
In contrast to Krugman (1980) who emphasizes the role of the variable trade cost in the home-market effect, the
fixed trade cost plays a qualitatively similar role in the current paper. The separate supplementary note develops a
more general model in which firms incur both variable and fixed trade costs, and shows that the extensive and intensive
margins are qualitatively similar to the above even in such a model.
26
N /M S must increase with it; otherwise trade would not balanced
r̄xN /r̄xS = ω decreases with L, Mex
ex
at the cutoff sector z̄.
I conclude this subsection by noting the relative extensive/intensive margin of specialization and
trade in each country. Regarding the relative intensive margin, I have
r̄xN
r̄xS
fx
> 1,
=
=
N
S
fd
r̄d
r̄d
suggesting that the intensive margin of exporters is higher than that of nonexporters in any sector
of both countries because exporters are on average more productive and earn higher average revenue
than nonexporters (note that this ratio is independent of both comparative advantage strength z or
relative country size L). On the other hand, the relative extensive margin of specialization and trade
in each country is given by
µ
¶k
ϕ̄id (z)
ϕ̄ix (z)
³
´ k

 1 fd σ−1
B(z) fx
= ³
´ k

 B(z) fd σ−1
fx
i (z)
Mex
=
i (z)
Med
if i = N,
if i = S,
which is between zero and unity under (1) or (2). Comparative statics on this equation reveal that
i /M i is increasing in comparative advantage strength (1/B for North and B for South) and the
Mex
ed
fixed-cost ratio (fd /fx ), and is decreasing in the degree of firms’ productivity dispersion (k/(σ − 1)).
Further, this ratio is increasing (decreasing) in the relative country size L for North (South) because
B decreases with L (recall the comparative statics in Figure 6). The intuition behind the last result is
as follows. While an increase in L increases the extensive margins in North due to the home-market
effect, it decreases the Ricardian productivity advantage of North through the lower relative wage
ω, which decreases the intensive margins there. This decrease in the intensive margins is bigger for
N
r̄xN than r̄dN ( ∂∂ r̄r̄xN /∂L
=
/∂L
d
fx
fd
> 1): a decrease of the Ricardian productivity advantage is more serious
for exporters because they have to incur the higher fixed cost. To restore the trade balance, an
N than M N and thus M N /M N is increasing in L.
increase in the extensive margins is bigger for Mex
ex
ed
ed
Noting that an increase in L works oppositely for the extensive and intensive margins in South, this
S /M S is decreasing in L.
intuition also explains why Mex
ed
Proposition 4 Under the Pareto distribution of firm productivity, the following holds:
(i ) The volume of domestic sales and exports increases with comparative advantage strength solely
through the extensive margins.
(ii ) The export participation ratio is increasing (decreasing) in the relative country size for North
(South).
27
5
Discussions
In this section, I briefly argue two extensions of the basic model: variable trade costs and multinational firms. The separate supplementary note offers a more detailed analysis that incorporates the
variable trade cost.
Transport Costs The model has assumed that only the fixed trade cost fx exists and firms can
export without incurring the variable trade cost τ . If this cost is incorporated in the previous setting,
the profit functions in country i’s market are given by
µ
πdi (ϕ, z)
i
= B (z)
µi (z)
wi
¶σ−1
µ
σ−1
ϕ
i
− w fd ,
πxj (ϕ, z)
i
= B (z)
µj (z)
τ wj
¶σ−1
ϕσ−1 − wj fx ,
and πdi is steeper than πxj if and only if
µi (z)
µj (z)
R
wi
τ wj
⇐⇒ µ(z) R
ω
.
τ
Similarly, from the profit functions in j’s market, πxi is steeper (flatter) than πdj if and only if
µi (z)
µj (z)
R
τ wi
wj
⇐⇒ µ(z) R τ ω.
In the presence of variable trade cost τ , the cutoff sector z̄ is no longer identical between North and
South. Instead, there exist two distinct cutoff sectors, namely z̄ N and z̄ S , such that North (South)
has an institutional comparative advantage in z ∈ [0, z̄ N ) (z ∈ (z̄ S , 1]), where z̄ N = µ−1 (τ ω) and
z̄ S = µ−1 (ω/τ ) (see Definition 2). While this result bears a resemblance to that in Dornbusch et al.
(1977), Figure 2 shows that the variable trade cost τ does not allow nontraded sectors to exist in
z ∈ [z̄ N , z̄ S ] in the current model, i.e., two-way trade can occur in any sector.22 Moreover, the generalequilibrium approach is not qualitatively affected by τ because: (i) changes in z still shift ϕ̄id and ϕ̄ix in
opposite directions, implying that the relative market demand B = B N /B S increases with z; (ii) the
S
N
S
relative productivity cutoffs ϕ̄d = ϕ̄N
d /ϕ̄d and ϕ̄x = ϕ̄x /ϕ̄x are exactly the same as those examined in
Section 4; and (iii) the variable trade cost does not affect a mechanism through which each country’s
wage is determined by the equality between aggregate expenditure and aggregate payments to labor,
giving rise to ω > 1 under the absolute advantage assumption. These observations jointly suggest
that, even in the presence of variable trade cost τ , there should arise a general-equilibrium interplay
between goods and labor markets that is similar to Figure 6. Therefore, although each absolute term
of the eight unknowns is affected by τ , the key characterizations represented in the relative terms
generally continue to hold.
One of interesting implications of this extension is that the introduction of τ alters the relationship
between (1) and (2). Since the relative productivity cutoffs ϕ̄ix /ϕ̄id in this setting for each country
22
Just as (2) is required for two-way trade to occur in any sector, so (2’) below is needed in this extension.
28
are respectively given by
ϕ̄N
x (z)
=τ
ϕ̄N
d (z)
µ
¶ 1
fx σ−1
B(z)
,
fd
ϕ̄Sx (z)
=τ
ϕ̄Sd (z)
µ
1 fx
B(z) fd
¶
1
σ−1
,
the usual outcome (ϕ̄ix > ϕ̄id ) occurs in any sector of both countries if
µ ¶ 1
1 fx
1 σ−1 fd
< B(z) < τ σ−1 .
τ
fx
fd
(1’)
i > 0) occurs in any sector of both countries if
On the other hand, two-way trade (Mex
X(z)
σ−1
k
µ ¶σ−1 µ ¶ k−σ+1
µ ¶ k−σ+1
k
k
σ−1
1
fx
fd
σ−1
< B(z) < X(z) k τ
.
τ
fx
fd
(2’)
Evidently, (1’) and (2’) converge to (1) and (2) as τ → 1. While (1) certainly subsumes (2) without
τ , whether the range of (1’) is greater than the range of (2’) depends on trade cost parameters. (It
is not possible to directly compare (1) and (1’) or (2) and (2’) because B is a function of τ .)
Multinational Enterprises The main analysis has assumed that exporting is only one option for
serving a foreign market. In the real world, firm sales by multinational enterprises have been growing
faster than exporting, and foreign direct investment (FDI) plays a key role in a country where the
market system is less developed.23 As initially raised by Williamson (1985), this issue is of particular
importance when contracts are better enforced within the firm boundaries. In the international
context, this means that Northern firms are able to respond to poor contract enforcement by FDI
(while employing local workers in South), thereby replacing weak external governance with an internal
principal-agent relationship. It also implies that the firm’s choice between intra-firm and arm’slength trade is affected by contractibility. In fact, Bernard, Jensen, Redding, and Schott (2010) find
empirical evidence suggesting that the intra-firm fraction of the U.S. imports is significantly higher
for products for which contractibility is more difficult. Because North has a comparative advantage
in contract-dependent sectors in the model, it is worth investigating the role played by multinational
firms in the presence of partial contractibility.
To analyze export versus FDI in the previous environment, suppose that
1<
µN (z 0 )
µN (z)
<
< ∞,
µM (z)
µM (z 0 )
1<
µM (z)
µM (z 0 )
<
< ∞,
µS (z)
µS (z 0 )
for z > z 0 , µN > µM > µS , µM (z) 6= 0, µM (z 0 ) 6= 0, µS (z) 6= 0 and µS (z 0 ) 6= 0. µM denotes partial
contractibility within the boundaries of multinational firms, lying between µN and µS for z ∈ [0, 1]
in Figure 1. These inequalities mean that multinational enterprises have strictly better contract
23
For simplicity, I have confined attention to the case where imperfect institutions are related to the production side
only. As surveyed by Dixit (2011), these frictions are also important in other aspects, such as the distribution channel
of exported products in foreign countries, and multinationals can alleviate this problem by internalization.
29
enforcement within the firm boundaries than Southern domestic firms, but their contractibility is
lower than Northern domestic firms because they have to be at least partially affected by Southern
governance structure, such as courts of law and social norms (Dixit, 2011): Northern (Southern) firms
receive negative (positive) feedback from becoming multinationals on contractibility. At the same
time, Northern multinationals are able to exploit wage differentials, while Southern multinationals
have to pay the higher wage. This creates a new tradeoff between wage and contractibility that has
been missing in the export-versus-FDI literature. As a result, the profit functions of Northern and
Southern multinationals,
µ
N
πm
(ϕ, z)
S
= B (z)
µM (z)
wS
¶σ−1
µ
ϕ
σ−1
S
− w fm ,
S
πm
(ϕ, z)
N
= B (z)
µM (z)
wN
¶σ−1
ϕσ−1 − wN fm ,
are respectively steeper than those of Northern and Southern exporters, πxN and πxS , if and only if
µM (z)
µN (z)
R
⇐⇒ ω R µ̃(z),
wS
wN
µM (z)
µS (z)
R
⇐⇒ µ̂(z) R ω,
wN
wS
where µ̃ = µN /µM and µ̂ = µM /µS . Thus, FDI undertaken by Northern (Southern) firms is more
likely to emerge in equilibrium if and only if endogenous wage differentials are sufficiently large
(small) relative to exogenous contractibility differentials.
Under these circumstances, how does the existence of multinational firms affect the patterns of
specialization and trade? Is there any systematic relationship between the relative export/FDI flows
and countries’ comparative advantage? If so, what impacts does it have on wage inequality between
North and South? Although these questions have vital implications for consequences of globalization,
I leave this extension for my future work.
6
Conclusions
This paper develops a general-equilibrium Ricardian model of North-South trade in order to address
the empirical fact that the share of firms that export increases with countries’ comparative advantage.
Since the share of firms that export is defined as the ratio of the mass of exporters to the mass of
domestic firms, this share cannot be obtained without decomposing both aggregate domestic sales and
aggregate exports into the extensive and intensive margins. Separating these two margins explicitly,
I find that an increase in these two aggregates due to comparative advantage can be largely explained
by the extensive margins, because the intensive margins receive two opposing effects that make net
change generally ambiguous. Regarding the extensive margin of exporting, for example, comparative
advantage increases the intensive margin by allowing all operating firms to be relatively better at
exporting, but it simultaneously decreases this margin by allowing less productive firms to enter the
export market. I also find that while the extensive margins of aggregate domestic sales and aggregate
exports increase with the degree of comparative advantage, the extensive margin of exports increases
with comparative advantage more significantly than the extensive margin of domestic production.
30
As repeatedly emphasized throughout the paper, institutional factors are one of potential sources
of countries’ comparative advantage. Indeed they can be interpreted more broadly and there would
be alternative mechanisms at work, such as the degree of financial development. All what matters for
the results is that labor productivity stemming from countries’ and sectors’ characteristics satisfies
log-supermodularity (or Ricardo’s classic inequality). Applying this concept to a general-equilibrium
Ricardian model featured with monopolistic competition and firm heterogeneity, I show that if labor
productivity is log-supermodular, not only is aggregate output but also other endogenous aggregates
– including aggregate labor supply and the share of firms that export – are also log-supermodular.
In this way, free trade allows resources to be allocated relatively more in the sectors where countries’
comparative advantage is relatively stronger, thereby leading to a higher exporters’ percentage in
these sectors.
That log-supermodularity is only important is not to imply, however, that the distinction between
technological and institutional sources of comparative advantage is not crucial for international trade.
In particular, as argued by Levchenko (2007), when institutional sources of comparative advantage
are formalized in the incomplete-contract framework, factor rewards may diverge as a result of trade
opening, which in turn may affect the gain from trade. Thus, if intersectoral wage inequality exists,
modeling institutional sources of comparative advantage in the standard Ricardian setting might
ignore key insights arising from the incomplete-contract literature. Although the trade literature has
extensively assumed wage equality in the presence of contracting imperfections (e.g., Antràs, 2005),
it would be interesting to know more about the extent to which this channel could affect the labor
allocation and the share of firms that export in the current Ricardian model.
31
Appendix
A.1
Proof of the Market Demand
I show that the relative market demand B = B N /B S increases with z. Taking the logarithm of the
ZCP conditions and differentiating them with respect to z gives
0
0
0
ϕ̄N
B N (z)
µN (z)
d (z)
+
(σ
−
1)
+
(σ
−
1)
= 0,
B N (z)
µN (z)
ϕ̄N
d (z)
0
(A.1)
0
0
ϕ̄Sd (z)
B S (z)
µS (z)
+
(σ
−
1)
+
(σ
−
1)
= 0,
B S (z)
µS (z)
ϕ̄Sd (z)
0
0
(A.2)
0
B S (z)
µN (z)
ϕ̄N
x (z)
+
(σ
−
1)
+
(σ
−
1)
= 0,
S
N
B (z)
µ (z)
ϕ̄N
x (z)
0
0
(A.3)
0
B N (z)
µS (z)
ϕ̄Sx (z)
+
(σ
−
1)
+
(σ
−
1)
= 0.
B N (z)
µS (z)
ϕ̄Sx (z)
(A.4)
Further, differentiating the FE condition with respect to z and rearranging,
0
0
N N
ϕ̄N
x (z) = −C ϕ̄d (z),
S0
(A.5)
S0
ϕ̄x (z) = −C S ϕ̄d (z),
(A.6)
where C i = fd J 0 (ϕ̄id )/fx J 0 (ϕ̄ix ) > 0. Note that (A.1) – (A.6) are six equations with six unknowns
0
0
0
0
0
0
S
N
S
N
S
(ϕ̄N
d , ϕ̄d , ϕ̄x , ϕ̄x , B , B ). Substituting (A.5) and (A.6) respectively into (A.3) and (A.4), and
subtracting (A.2) and (A.1) respectively from these yields
0
0
ϕ̄Sd (z)
C N ϕ̄N
µ0 (z)
d (z)
=
,
+
ϕ̄N
µ(z)
ϕ̄Sd (z)
x (z)
0
0
ϕ̄N (z) C S ϕ̄Sd (z)
µ0 (z)
− dN
−
,
=
ϕ̄Sx (z)
µ(z)
ϕ̄d (z)
0
0
S
where µ0 /µ < 0. These are two equations with two unknowns (ϕ̄N
d , ϕ̄d ), which can be solved for
0
ϕ̄N
d (z) =
where
µ0 (z)
µ(z)
³
1
ϕ̄S
d (z)
+
CS
ϕ̄S
x (z)
´
Ξ
1
Ξ= N
ϕ̄d (z)ϕ̄Sd (z)
µ
,
0
ϕ̄Sd (z) = −
µ0 (z)
µ(z)
³
1
ϕ̄N
d (z)
+
Ξ
CN
ϕ̄N
x (z)
´
,
¶
S
ϕ̄N
d (z)ϕ̄d (z) N S
C C −1 .
S
ϕ̄N
x (z)ϕ̄x (z)
From the ZCP conditions and C i defined above, Ξ is positive if and only if
0 S
J 0 (ϕ̄N
d (z))J (ϕ̄d (z))
>
0 S
J 0 (ϕ̄N
x (z))J (ϕ̄x (z))
µ
fx
fd
¶
2σ
σ−1
.
(Assuming a Pareto distribution, for example, the left-hand side is (fx /fd )2(k+1)/(σ−1) and this holds
0
0
0
0
S
N
S
if k − σ + 1 > 0). Under this condition, ϕ̄N
d < 0, ϕ̄d > 0 and from (A.5) and (A.6) ϕ̄x > 0, ϕ̄x < 0.
From the relative market demand B = B N /B S in the main text, these then imply that B 0 > 0.
32
¤
A.2
Proofs of the LMC and BOP Assumptions
A.2.1 Proof of the LMC Assumption
I show that the LMC condition is simplified as
R1
0
Ri (z)dz
= Li .
wi
From the LMC condition, aggregate sector labor supply in country i is given by
Z
i
L (z) =
Mei
Z
∞
ϕ̄id (z)
ldi (ϕ, z)dG(ϕ)
+
Mei (z)
∞
ϕ̄ix (z)
lxi (ϕ, z)dG(ϕ) + Mei (z)fe ,
where the first two terms are aggregate sector labor supply for production and the third is aggregate
sector labor supply for investment by potential entrants. Substituting the amount of labor ldi and lxi
(and dropping argument z for notational simplicity), aggregate sector labor supply for production is
Z
Mei
∞
ϕ̄id
Z
ldi (ϕ)dG(ϕ) + Mei
∞
ϕ̄ix
lxi (ϕ)dG(ϕ)
Z
Z
£
¤
£
¤
σ − 1 Mei ∞ i
σ − 1 Mei ∞ i
i
i
= Mei 1 − G(ϕ̄id ) fd +
r
(ϕ)dG(ϕ)
+
M
1
−
G(
ϕ̄
)
f
+
r (ϕ)dG(ϕ)
x
e
x
σ wi ϕ̄id d
σ wi ϕ̄ix x
(
)
Z
Z
¤ i
£
¤ i
Mei £
σ−1 ∞ i
σ−1 ∞ i
i
i
= i
1 − G(ϕ̄d ) w fd +
rd (ϕ)dG(ϕ) + 1 − G(ϕ̄x ) w fx +
rx (ϕ)dG(ϕ)
w
σ
σ
ϕ̄id
ϕ̄ix
=
Ri − Πi
,
wi
where Πi denotes aggregate sector profit. Aggregate sector labor supply for investment is
Mei fe
Mi
= ie
w
=
(
1
σ
Z
∞
ϕ̄id
rdi (ϕ)dG(ϕ)
£
¤
1
− 1 − G(ϕ̄id ) wi fd +
σ
Z
∞
ϕ̄ix
rxi (ϕ)dG(ϕ)
£
¤
− 1 − G(ϕ̄ix ) wi fx
)
Πi
,
wi
which is derived from the FE condition:
Z ∞ i
Z ∞ i
πd (ϕ)
πx (ϕ)
fe =
dG(ϕ) +
dG(ϕ)
i
w
wi
ϕ̄id
ϕ̄ix
(Z
)
Z ∞ i
∞ i
£
¤
£
¤
rd (ϕ)
1
r
(ϕ)
x
= i
dG(ϕ) − 1 − G(ϕ̄id ) wi fd +
dG(ϕ) − 1 − G(ϕ̄ix ) wi fx .
w
σ
σ
ϕ̄id
ϕ̄ix
Summing up these, it follows that aggregate sector labor supply is equal to aggregate sector revenue:
M i (z)
Li (z) = e i
w
=
(Z
∞
ϕ̄id (z)
Z
rdi (ϕ, z)dG(ϕ) +
Rdi (z) + Rxi (z)
.
wi
33
∞
ϕ̄ix (z)
)
rxi (ϕ, z)dG(ϕ)
Finally, integrating the above aggregate sector labor supply over the interval [0,1], aggregate labor
R1
supply Li = 0 Li (z)dz is given by
R1
R1
Rdi (z)dz + 0 Rxi (z)dz
L =
wi
R1 i
R1 j
0 Rd (z)dz + 0 Rx (z)dz
=
wi
R1 i
R (z)dz
= 0
.
wi
i
0
(using the BOP condition)
This completes the proof.
¤
A.2.2 Proof of the BOP Condition
I first show that the BOP condition is written as
Z
z̃
Z
¡ N N
¢
w L (z) − RN (z) dz =
0
1¡
¢
wS LS (z) − RS (z) dz.
z̃
Rewrite the BOP condition as
Z
0
z̃
¡ N
¢
Rx (z) − RxS (z) dz =
Z
1¡
¢
RxS (z) − RxN (z) dz.
z̃
Note that aggregate sector income in i is the sum of domestic and export revenues, wi Li = Rdi + Rxi ,
and aggregate sector expenditure is the sum of expenditure spent on domestic products and foreign
imports, Ri = Rdi + Rxj . The result directly follows from subtracting Ri from wi Li .
Next, I show that the above BOP condition is written as
R 1 ³ LS (z)
´
−
λ(z)
dz µ 1 ¶
z̃
³
´
ω ≡ ξ(z̃) = R
,
z̃ LN (z)
L
−
λ(z)
dz
N
0
L
LS
By manipulating the BOP condition derived above,
Z
Z 1
¡ N N
¢
¡ S S
¢
N
w L (z) − R (z) dz =
w L (z) − RS (z) dz
0
z̃
¶
¶
Z z̃ µ N
Z 1µ S
RS (z) wS LS
L (z) RN (z)
L (z) wS LS
⇐⇒
− N N dz =
− S S N N dz
LN
w L
LS wN LN
w L w L
0
z̃
¶
¶
Z 1µ S
Z z̃ µ N
1
L (z)
L (z)
− λ(z) dz =
− λ(z) dz,
⇐⇒
N
L
ωL z̃
LS
0
z̃
where the second equation comes from dividing both sides of the first equation by wN LN , and the
third comes from λ(z) = λN (z) = λS (z) =
P i (z)Qi (z)
Yi
the result.
34
=
Ri (z)
.
w i Li
Solving the last equation for ω yields
Finally, I show that ξ is increasing
in z̃, ´satisfying limz̃→0 ³ξ(z̃) = 0 and ´limz̃→1 ξ(z̃) = ∞ under
R z̃ ³ LN (z)
R1 S
N
mild conditions. Let ξ = 0
− λ(z) dz and ξ S = z̃ LL(z)
− λ(z) dz respectively denote
S
LN
the denominator and numerator of ξ. Substituting aggregate sector labor supply into these, ξ i can
be expressed in terms of expenditure only:
¢
R z̃ ¡ N
S
0 Rx (z) − Rx (z) dz
ξ (z̃) =
,
R1
N (z)dz
R
0
¢
R1¡ S
N
R
x (z) − Rx (z) dz
S
z̃
.
ξ (z̃) =
R1
S (z)dz
R
0
N
Note that the numerator of ξ i represents aggregate net export in country i. Then suppose for
example what happens if z̃ increases and North’s comparative advantage sectors expand. To restore
¢
R z̃ ¡
the trade balance, aggregate net exports in North 0 RxN (z) − RxS (z) dz must decrease and those
¢
R1¡
0
0
in South z̃ RxS (z) − RxN (z) dz must increase, suggesting that ξ N (z̃) < 0 and ξ S (z̃) > 0. Since
ξ=
ξS 1
,
ξN L
this proves that ξ 0 (z̃) > 0. To show that limz̃→0 ξ(z̃) = 0 and limz̃→1 ξ(z̃) = ∞, I assume
0
0
0
0
that limz̃→0 ξ N (z̃) = ∞, limz̃→0 ξ S (z̃) = 0, limz̃→1 ξ N (z̃) = 0 and limz̃→1 ξ S (z̃) = ∞. The desired
result is immediately obtained under these mild conditions.
¤
A.2.3 Proof of Assumption 1
I show that the relative wage ω necessarily satisfies Assumption 1. It is immediately seen that one
constraint of this assumption (ω > fd /fx ) is satisfied as long as
f > fd because
ω is greater than one.
„ x
«
R1
For the other constraint (ω < fx /fd ), substituting ω =
LS (z)
−λ(z)
LS
z̄
R z̄ “ LN (z)
LN
0
evaluated at z̃ = z̄) into the latter constraint yields
µZ
¶µ
z̄
λ(z)dz
0
dz
”
−λ(z) dz
¡1¢
L
(the BOP condition
¶
Z z̄ N
Z 1 S
fx
fx
L (z)
L (z)
1+ L − L
dz < 1 −
dz.
N
fd
fd
L
LS
0
z̄
(A.7)
As derived above, aggregate sector labor supply is equal to aggregate sector revenue, and from the fact
R z̄
R z̄
that North (South) is a net exporter in sectors z ∈ [0, z̄) (z ∈ (z̄, 1]) (i.e., 0 RxS (z)dz < 0 RxN (z)dz
R1
R1
and z̄ RxN (z)dz < z̄ RxS (z)dz), Li (z) must satisfy
Z
1−
z̄
1
LS (z)
dz <
LS
Z
Z
z̄
λ(z)dz <
0
0
z̄
LN (z)
dz.
LN
(A.8)
Note that (A.7) is internally consistent with (A.8) in that, in the range of (A.8), (A.7) satisfies
µZ
¶µ
z̄
λ(z)dz
0
¶
Z z̄ N
Z z̄
fx
fx
L (z)
1+ L − L
dz <
λ(z)dz,
fd
fd
LN
0
0
which also satisfies (A.8). Thus, if each country is a net exporter above/below the cutoff sector z̄,
the relative wage ω in equilibrium is smaller than fx /fd . This completes the proof.
35
¤
A.3
Proofs of the Extensive and Intensive Margins
A.3.1 Proof of the Extensive Margins
I first show the derivation of Mei . From the main text, the aggregate sector price in country i is
Z
i
(P (z))
1−σ
=
Mei (z)
µ
∞
ϕ̄id (z)
σ
wi
σ − 1 ϕµi (z)
(σ−1)σ−1 i
X (P i )σ−1 ,
σσ
Since B i =
¶1−σ
Z
dG(ϕ) +
Mej (z)
µ
∞
ϕ̄jx (z)
σ
wj
σ − 1 ϕµj (z)
¶1−σ
dG(ϕ).
(σ−1)σ−1 X i
.
σσ
Bi
this aggregate sector price is also written as (P i )1−σ =
Combining these two relationships gives
Z
Mei (z)
µ
∞
ϕ̄id (z)
wi
σ
σ − 1 ϕµi (z)
¶1−σ
Z
dG(ϕ)+Mej (z)
µ
∞
ϕ̄jx (z)
σ
wj
σ − 1 ϕµj (z)
¶1−σ
dG(ϕ) =
(σ − 1)σ−1 X i (z)
.
σσ
B i (z)
These two equations, ((P N )1−σ , (P S )1−σ ), are the two systems with the two unknowns (MeN , MeS ).
Rewriting these equations in a matrix form,
³
"
#  (σ−1)σ−1 X N (z) 
[V (∞) − V (ϕ̄Sx (z))] MeN (z)
σσ
B N (z) 
³

´1−σ
³
´1−σ
=  (σ−1)
,
σ−1 X S (z)
N
S
S
σ
w
σ
w
N (z))]
S (z))]
Me (z)
σ
S (z)
[V
(∞)
−
V
(
ϕ̄
[V
(∞)
−
V
(
ϕ̄
σ
N
S
B
x
d
σ−1 µ (z)
σ−1 µ (z)
σ
wN
σ−1 µN (z)
´1−σ
where V (ϕ) =
³
[V (∞) − V (ϕ̄N
d (z))]
Rϕ
0
Mei (z)
σ
wS
σ−1 µS (z)
´1−σ
y σ−1 dG(y). Applying Cramer’s rule,
1
=
σ
µ
wi
µi (z)
¶σ−1
X i (z)
[V
B i (z)
(∞) − V (ϕ̄jd (z))] −
X j (z)
[V
B j (z)
(∞) − V (ϕ̄jx (z))]
∆(z)
,
S
N
S
where ∆ = [V (∞) − V (ϕ̄N
d )][V (∞) − V (ϕ̄d )] − [V (∞) − V (ϕ̄x )][V (∞) − V (ϕ̄x )] > 0. Note that this
holds for a general distribution function G(·).
Next, I derive (2) when G(·) is a Pareto distribution. Since V (∞) − V (ϕ) =
the Pareto distribution, substituting this value into the above
1
Mei (z) =
σ
³
where ∆ =
kϕkmin
k−σ+1
´ ·³
1
S
ϕ̄N
d ϕ̄d
µ
wi
µi (z)
´k−σ+1
¶σ−1
−
1
S
ϕ̄N
x ϕ̄x
´k−σ+1 ¸
X j (z)
1
B j (z) (ϕ̄jx )k−σ+1
,
> 0; and k − σ + 1 > 0 comes from a finite
variance of the truncated Pareto distribution V (ϕ). Then, Mei > 0 if and only if
Xj
1
B j (ϕ̄jx )k−σ+1
µ
ϕ̄N
d (z)
ϕ̄N
x (z)
under
gives
∆(z)
³
−
X i (z)
1
B i (z) (ϕ̄j (z))k−σ+1
d
Mei
kϕkmin
1
k−σ+1 ϕk−σ+1
Xi
1
B i (ϕ̄j )k−σ+1
d
>
(i 6= j) or
¶k−σ+1
B(z)
<
<
X(z)
µ
ϕ̄Sx (z)
ϕ̄Sd (z)
¶k−σ+1
µ
⇐⇒
Arranging this inequality gives (2).
36
1 fd
B(z) fx
¶ k−σ+1
σ−1
B(z)
<
<
X(z)
µ
1 fx
B(z) fd
¶ k−σ+1
σ−1
.
Finally, I show that the range of (1) subsumes that of (2). Comparing these two conditions, the
above statement holds if and only if
fd
fx
< X(z) = ωL(z) < .
fx
fd
«
dz
z̄
”
R z̄ “ LN (z)
N −λ(z) dz
0
R1
Substituting ωL =
„
LS (z)
−λ(z)
LS
(the BOP condition at z̃ = z̄) into the above inequality yields
L
1+
fd
fx
R z̄
0
R 1 LS (z)
LN (z)
dz
−
N
z̄ LS dz
L
1 + ffxd
Z
≡ Γ1 <
z̄
λ(z)dz <
1+
0
fx
fd
R z̄
0
R 1 LS (z)
LN (z)
dz
−
N
z̄ LS dz
L
1 + ffxd
≡ Γ2 .
³
´R N
z̄
and Γ2 < 1 + ffxd 0 LLN(z) dz because the numerator and denominator
R z̄ N
R1 S
of ωL are both positive and hence 0 LLN(z) dz + z̄ LL(z)
S dz > 1. Thus, the above inequality satisfies
Note that Γ1 > 1−
R1
LS (z)
z̄ LS dz
(A.8), which shows that North (South) is a net exporter in z ∈ [0, z̄) (z ∈ (z̄, 1]). This observation
implies that
fd
fx
< ωL <
fx
fd
and hence (1) subsumes (2).
¤
A.3.2 Proof of the Intensive Margins under Pareto
I show that the intensive margins under Pareto are given by
kσ
wi fx ,
k−σ+1
r̄xi =
r̄di =
kσ
wi fd .
k−σ+1
By definition,
r̄xi =
1
1 − G(ϕ̄ix (z))
Z
∞
ϕ̄ix (z)
rxi (ϕ, z)dG(ϕ)
µ i ¶σ−1
£
¤
µ (z)
1
j
B (z)σ
=
V (∞) − V (ϕ̄ix (z))
i
i
1 − G(ϕ̄x (z))
w
µ i
¶k
µ i ¶σ−1
kϕkmin
ϕ̄x (z)
µ (z)
1
=
B j (z)σ
ϕmin
wi
k − σ + 1 (ϕ̄ix (z))k−σ+1
µ i ¶σ−1
µ (z)
k
j
B (z)σ
=
(ϕ̄ix (z))σ−1
k−σ+1
wi
µ i ¶σ−1
µ i ¶1−σ
k
µ (z)
1
µ (z)
j
B (z)σ
=
wi fx
i
j
k−σ+1
w
B (z)
wi
kσ
=
wi fx .
k−σ+1
By following the similar steps, it is easily confirmed that r̄di =
37
kσ
i
k−σ+1 w fd .
(using V (ϕ))
(using Pareto)
(using ZCPxi )
¤
A.3.3 Proof of the Relative Extensive Margins under Pareto
I first show that the relative extensive margins are given by
k
k
Me (z)
1 X(z)B(z)− σ−1 F − 1
Mex (z) =
=
,
k
(ϕ̄x (z))k
ω B(z) σ−1
F − X(z)
Me (z)
1 X(z) − B(z) σ−1 F −1
Med (z) =
=
,
k
(ϕ̄d (z))k
ω 1 − XB(z)− σ−1
F −1
where X = ωL and F = (fx /fd )(k−σ+1)/(σ−1) > 1. From Mei under Pareto, Me = MeN /MeS is
µ
Me (z) =
ω
µ(z)
X N (z)
1
k−σ+1
B N (z) (ϕ̄S
d (z))
−
X S (z)
1
k−σ+1
B S (z) (ϕ̄S
x (z))
X S (z)
1
k−σ+1
B S (z) (ϕ̄N
d (z))
−
X N (z)
1
k−σ+1
B N (z) (ϕ̄N
x (z))
¶σ−1
.
Using the ZCP conditions, this equation can be written as
µ
Me (z) =
µ
=
ω
µ(z)
ω
µ(z)
¶σ−1
k
k−σ+1
(ϕ̄d (z))
¶σ−1
X(z) − B(z) σ−1 F −1
k
B(z)(1 − XB(z)− σ−1 F −1 )
k
k−σ+1
(ϕ̄x (z))
(X(z)B(z)− σ−1 F − 1)B(z)
k
B(z) σ−1 F − X(z)
Dividing the first and second equalities respectively by (ϕ̄d )k and (ϕ̄x )k ,
µ
1
ω
Med (z) =
µ(z) ϕ̄d (z)
|
{z
µ
B(z)
ω
ω
1
Mex (z) =
µ(z) ϕ̄x (z)
|
{z
¶σ−1
k
X(z) − B(z) σ−1 F −1
k
−
−1
} B(z)(1 − X(z)B(z) σ−1 F )
¶σ−1
,
k
(X(z)B(z)− σ−1 F − 1)B(z)
k
B(z) σ−1 F − X(z)
}
,
1
B(z)ω
where the values in the underbraces come from the RZCP conditions. Arranging these equations
gives the result. Note that since r̄dN /r̄dS = r̄xN /r̄xS = 1/ω, aggregate sector domestic sales and exports
in the relative terms, RdN /RdS and RxN /RxS , are respectively proportional to their relative extensive
N /M S and M N /M S :
margins, Med
ex
ex
ed
Rd (z) =
RdN (z)
= ωMed (z),
RdS (z)
Rx (z) =
RxN (z)
= ωMex (z).
RxS (z)
Thus, Med and Mex are log-supermodular if and only if Rd and Rx are log-supermodular.
³
´
0 (z) < M 0 (z) < 0 for z ∈ [0, 1] and their intersection is at z̄, XF −1
Next, I show that Mex
ed
ω(F −X) in
Figure 7. The first relationship holds from simple inspection of the above expressions of Mex and
Med , because only B is a function of z with B 0 (z) > 0 and k −σ +1 > 0. Regarding the intersection of
the two curves, it follows from noting that ϕ̄x (z̄) = ϕ̄d (z̄) = ω 1/(σ−1) and Me (z̄) = ω
38
k−σ+1
σ−1
XF −1
F −X .
¤
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40
Supplementary Note (Not for Publication)
This Note offers a detailed analysis that incorporates the variable trade cost τ in the basic model.
The purpose in this Note is to show that all the results in the main text qualitatively continue to
hold even in this setup.
With the variable trade cost τ , the eight equilibrium conditions (the ZCP, FE and BOP conditions
for country i ∈ {N, S}) are given by
µ
i
(ZCPdi )
(ϕ̄ix (z))σ−1 = wi fx ,
(ZCPxi )
fd J(ϕ̄id (z)) + fx J(ϕ̄ix (z)) = fe ,
Z 1
Ri (z)dz = wi Li ,
(F E i )
µ
B (z)τ
¶σ−1
(ϕ̄id (z))σ−1 = wi fd ,
B (z)
j
µi (z)
wi
1−σ
µi (z)
wi
¶σ−1
(BOP i )
0
where J(ϕ̄) =
R∞
ϕ̄
[(ϕ/ϕ̄)σ−1 − 1]dG(ϕ). Note that only the ZCP condition of exporting is directly
affected by the variable trade cost τ . In particular, τ does not enter the FE condition even with the
variable trade cost (see Melitz and Redding (2014) for the similar formulation). It is also immediate
to see that each country’s wage wi is determined by the equality between aggregate expenditure and
aggregate payments to labor, and that the LMC condition is equivalent with the BOP condition for
the determination of the relative wage ω even with the variable trade cost τ .
The characterization of the eight unknowns can be solved from these eight equilibrium conditions
by following the same steps as in the main text. I first analyze the sectoral differences in the relative
S
N
S
N
S
productivity cutoffs (ϕ̄d = ϕ̄N
d /ϕ̄d , ϕ̄x = ϕ̄x /ϕ̄x ) and the relative market demand (B = B /B ),
focusing on the ZCP and FE conditions (and leaving the BOP/LMC conditions out from the model).
After characterizing each sector’s equilibrium variables, I embed the sectoral equilibrium into the full
general-equilibrium framework by integrating the BOP/LMC conditions to show that the relative
wage (ω = wN /wS ) is greater than one.
As noted above, the variable trade cost τ does not enter the FE condition. Since the equality of
this condition must hold in any sector, changes in z shift ϕ̄id and ϕ̄ix in opposite directions (as long
as these cutoffs shift), and ϕ̄ix /ϕ̄id must be increasing or decreasing in z. Moreover, dividing the ZCP
condition of domestic production by that of exporting, the relative market demand B = B N /B S is

³ N ´σ−1

fd
τ 1−σ ϕ̄xN (z)
fx
ϕ̄d (z)
B(z) =
³ S ´σ−1
ϕ̄d (z)

fx
1
 1−σ
S
τ
ϕ̄x (z)
fd
if i = N,
if i = S.
Using the property of the FE condition derived above, it can be shown that B is increasing in z (its
proof is similar to that in the main text). The first quadrant of Figure S.1 depicts this relationship
in (z, B) space.
S-1
B
Md
Mx
B
1
M d ,M x
§Z ·
Z 1 /(V 1) ¨¨ ¸¸
©P¹
0
zN
1
zS
z
Figure S.1 – Market demand and productivity cutoffs
Next, dividing the ZCP condition of North by the corresponding condition of South, the relative
S
N
S
productivity cutoffs ϕ̄d = ϕ̄N
d /ϕ̄d and ϕ̄x = ϕ̄x /ϕ̄x satisfy exactly the same relative ZCP condition
as before:
µ
ϕ̄d (z) =
ω
B(z)
¶
1
σ−1
ω
,
µ(z)
1
ϕ̄x (z) = (B(z)ω) σ−1
ω
,
µ(z)
(RZCP )
and hence
ϕ̄x (z) R ϕ̄d (z) ⇐⇒ B(z) R 1.
The second quadrant of Figure S.1 depicts this relationship in (B, ϕ̄) space. Note that the horizontal
intersection between the ϕ̄d and ϕ̄x curves is not ω 1/(σ−1) in this case. To see this, recall that ϕ̄d and
ϕ̄x are equal if and only if the profit functions πdi and πxj are parallel. As discussed in section 5, when
firms have to incur the variable trade cost, πxN is parallel with πdS if and only if ω = µ(z)/τ , whereas
πxS is parallel with πdN if and only if ω = µ(z)τ . Thus, there exist the two cutoff sectors, namely
z̄ N = µ−1 (τ ω) and z̄ S = µ−1 (ω/τ ) with z̄ N < z̄ S (see Figure S.2). Obviously, ϕ̄d (z̄ i ) = ϕ̄x (z̄ i ) in
both cutoff sectors of country i ∈ {N, S}, and the vertical intersection between the ϕ̄d and ϕ̄x curves
is still B(z̄ i ) = 1 (using the RCZP condition). Since ω and µ are not equal in the cutoff sectors, the
horizontal intersection is ω 1/(σ−1) (ω/µ). Given that B increases with z, this also implies that B is
flat between z̄ N and z̄ S (i.e., B is weakly increasing in z) in the presence of variable trade cost, as is
drawn in the first quadrant of Figure S.1.
Finally, combining the first and second quadrants, Figure S.1 highlights the sectoral difference
among the six endogenous variables (represented in the relative terms) as follows:
1
0 ≤ z < z̄ N
⇐⇒ B(z) < 1 ⇐⇒ ϕ̄d (z) > ω σ−1
z̄ N ≤ z ≤ z̄ S
⇐⇒ B(z) = 1 ⇐⇒ ϕ̄d (z) = ω σ−1
1
1
z̄ S < z ≤ 1 ⇐⇒ B(z) > 1 ⇐⇒ ϕ̄d (z) < ω σ−1
S-2
µ
¶
ω
> ϕ̄x (z),
µ(z)
µ
¶
ω
= ϕ̄x (z),
µ(z)
µ
¶
ω
< ϕ̄x (z).
µ(z)
Z
PW
P
W
Z
1
0
z
zS
zN
1
Figure S.2 – Two cutoff sectors
In this relationship, ω 1/(σ−1) (ω/µ) – the intersection between the ϕ̄d and ϕ̄x curves – is greater than
one in z ∈ [z̄ N , z̄ S ] as the following holds from ω = µ(z̄ N )/τ = µ(z̄ S )τ for any µ(z) > 1 and τ > 1:
1
¶
ω
[µ(z̄ N )/τ ] σ−1
=
ϕ̄d (z̄ ) = ϕ̄x (z̄ ) = ω
µ(z̄ N )
τ
µ
¶
1
1
ω
= ϕ̄d (z̄ S ) = ϕ̄x (z̄ S ) = ω σ−1
= [µ(z̄ S )τ ] σ−1 τ > 1.
S
µ(z̄ )
N
N
1
σ−1
µ
Thus, there arises a relationship among the productivity cutoff similar to Figure 4 even with variable
trade cost. Moreover, dividing the ZCP conditions gives
ϕ̄N
x (z)
=τ
N
ϕ̄d (z)
¶ 1
µ
fx σ−1
B(z)
,
fd
ϕ̄Sx (z)
=τ
ϕ̄Sd (z)
µ
1 fx
B(z) fd
¶
1
σ−1
.
N
S
S
Since B is weakly increasing in z, ϕ̄N
x /ϕ̄d (ϕ̄x /ϕ̄d ) is weakly increasing (decreasing) in z. This
proves Proposition 2(i) in the case with variable trade cost.
The above equations also indicate that the usual outcome (ϕ̄ix > ϕ̄id ) occurs in the comparative
disadvantage sectors of both countries, and in the sectors where neither country has a comparative
advantage that arise in the presence of variable trade cost, z ∈ [z̄ N , z̄ S ]. In contrast, the “perverse”
outcome (ϕ̄id > ϕ̄ix ) can occur in the comparative advantage sectors if

B(z) < ¡ 1 ¢σ−1
τ
B(z) > τ σ−1 fx
fd
fx
fd
if i = N,
if i = S.
Although this outcome is less likely in the presence of variable trade cost, it is still possible to occur
in sectors where countries’ comparative advantage is strong enough relative to the variable and fixed
trade costs. (Note the partitioning of firms by export status requires τ σ−1 fx /fd > 1 in the model.)
This proves Proposition 2(ii) in the case with variable trade cost.
S-3
Given implicit solutions of firm selection (ϕ̄id , ϕ̄ix ) and aggregate market demand (B i ), I turn to
integrating the LMC/BOP condition. As noted earlier, the BOP condition does not directly affected
by the variable trade cost and thus the property of ω = ξ(z̃) does not change, i.e., it is an increasing
function of z̃ and under the mild conditions limz̃→0 ξ(z̃) = 0 and limz̃→1 ξ(z̃) = ∞. Together with
µ, it is possible to solve for the unique relative wage ω, showing that ω > 1 in equilibrium. Further,
the comparative static result continues to hold in this setup. In particular, an increase in relative
country size L = LN /LS still decreases ω and country size does affect the firm-level variables even
if the C.E.S. preferences are imposed. This in turn implies that B is weakly decreasing in L because
ω = µ(z̄ N )/τ = µ(z̄ S )τ and a decrease in ω necessarily leads to an increase in z̄ i (recall that µ is a
decreasing function of z).
Having shown that the equilibrium characterizations of the eight endogenous variables are similar
to those in the main text, I next derive the extensive/intensive margins of specialization and trade
to check the robustness of Proposition 4. With the variable trade cost and Pareto distribution, the
mass of potential entrants in country i is expressed as
1
Mei (z) =
σ
˜ =
where ∆
³
kϕkmin
k−σ+1
µ
wi
µi (z)
¶σ−1
´ ·³
in the current model,
´k−σ+1
1
S
ϕ̄N
d ϕ̄d
i
Me should be
−
X i (z)
1
B i (z) (ϕ̄j (z))k−σ+1
d
τ 2(1−σ)
³
1
S
ϕ̄N
x ϕ̄x
j
(z)
− τ 1−σ X
B j (z)
1
(ϕ̄jx (z))k−σ+1
˜
∆(z)
´k−σ+1 ¸
,
. For non-traded sectors not to exist
strictly positive in any sector of country i. It directly follows
that Mei > 0 for z ∈ [0, 1] if and only if condition (2’) holds as shown in section 5.
I can now examine the effects of comparative advantage and country size on the extensive and
intensive margins. Under the Pareto distribution with the variable trade cost, it is possible to show
that both margins are expressed in exactly the same form as the main text:
µ
i
Mex
(z) =
µ
i
Med
(z)
=
ϕmin
ϕ̄ix (z)
ϕmin
ϕ̄id (z)
¶k
¶k
Mei (z),
r̄xi =
kσ
wi fx ,
k−σ+1
Mei (z),
r̄di =
kσ
wi fd .
k−σ+1
i , M i ) are indirectly affected by variable trade cost τ through
Although the extensive margins (Med
ex
the productivity cutoffs and the mass of entrants, the intensive margins (r̄di , r̄xi ) are independent of
τ (for a given ω). This influence of the variable trade cost on these two margins under the Pareto
distribution is analogous to that in the previous literature (see, e.g., Melitz and Redding (2014)).
While the existing models do not explicitly deal with the role of comparative advantage, the current
paper shows that the intensive margins are independent of both comparative advantage strength and
variable trade cost. Since aggregate domestic sales and aggregate exports increase with comparative
advantage strength under log-supermodularity, this finding suggests that Proposition 4(i) still holds
even with τ : the volume of domestic sales and exports increases with comparative advantage strength
solely through the extensive margins. The above equations also indicate that the relative extensive
S-4
N /M S , M N /M S ) and the relative intensive margins (r̄ N /r̄ S , r̄ N /r̄ S ) are exactly the
margins (Med
ex
ex
x
x
ed
d
d
same as the main text. Thus, the two margins respond oppositely to country size L even with τ : the
relative intensive margins are decreasing in L whereas the relative extensive margins of specialization
and trade are increasing in it in order to restore the trade balance.
Regarding the relative intensive margin of specialization and trade in each country (r̄xN /r̄dN , r̄xS /r̄dS ),
it is again exactly the same as the main text. On the other hand, the relative extensive margin of
specialization and trade in each country is given by
µ
¶k
ϕ̄id (z)
ϕ̄ix (z)
 ³
´ k

1 fd σ−1
 1k
B(z) fx
τ
=
³
´ k

 1 B(z) fd σ−1
fx
τk
i (z)
Mex
=
i (z)
Med
if i = N,
if i = S.
While the export participation ratio is depressed relative to the main text, this ratio still increases
(decreases) with the relative country size L for North (South) because B weakly decreases with L.
(Recall the comparative statics examined above.) Therefore, Proposition 4(ii) qualitatively continues
to hold even with the variable trade cost.
S-5