DPRIETI Discussion Paper Series 15-E-023

Comparative Advantage, Monopolistic Competition, andHeterogeneous Firms in a Ricardian Model with a Continuum of Sectors

ARA TomohiroFukushima University

The Research Institute of Economy, Trade and Industryhttp://www.rieti.go.jp/en/

1

RIETI Discussion Paper Series 15-E-023

February 2015

Comparative Advantage, Monopolistic Competition, and Heterogeneous Firms in a Ricardian Model with a Continuum of Sectors*

ARA Tomohiro

Fukushima University

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, Endogenous relative wage

JEL classification: D23, F12, F14, L33, O43

RIETI Discussion Papers Series aims at widely disseminating research results in the form of professional papers, thereby stimulating lively discussion. The views expressed in the papers are solely those of the author(s), and neither represent those of the organization to which the author(s) belong(s) nor the Research Institute of Economy, Trade and Industry.

*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). The author is grateful for helpful comments and suggestions by Jota Ishikawa (Hitotsubashi Univ.) and Discussion Paper seminar participants at RIETI.

1 Introduction

A growing body of empirical evidence using firm-level data has extensively revealed that the extentto which firms participate in exporting varies systematically across countries and sectors. Theseworks have found that, in developed countries, the percentage of firms that export tends to besubstantially higher in sectors where production technology is more complex and customized (suchas chemical products), and this percentage steadily declines as sectors’ production requires simplerand more generic technology (such as textile products). In developing countries, on the other hand,the opposite patterns are typically observed: the ratio of exporting firms to overall firms tends to benotably higher (resp. lower) in simpler (resp. more complex) sectors. At the same time, these studieshave also documented that exporting occurs in every major manufacturing sector of both developedand developing countries: even in strong comparative disadvantage sectors, a small fraction of firmsdo export.1

Why does the fraction of firms that export vary with countries’ comparative advantage? Toaddress this question, combining the recent empirical finding quantified by Levchenko (2007) andNunn (2007) with firm heterogeneity of Melitz (2003), I develop a general-equilibrium Ricardianmodel of North-South trade in which both institutional quality and firm heterogeneity play a keyrole in determining international trade flows. Following Levchenko’s and Nunn’s finding, the modelassumes that each country is different in terms of contracting institutions, and each sector is differentin terms of contract intensity. Furthermore, each firm is different in terms of its productivity a laMelitz. These three-dimensional differences in country, sector, and firm characteristics endogenouslypin down patterns of specialization and trade in equilibrium.

To investigate the role of country and sector characteristics, I build on the concept of “partialcontractibility,” originally developed by Acemoglu, Antras, and Helpman (2007). I consider anenvironment in which North has better institutions for partially ex-ante contractible activities thanSouth, whereas customized sectors make use of relationship-specific investments intensively morethan 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 examinethe interaction between contractual incompleteness and technological complementarity by simplyassuming that production in customized (generic) sectors is more (less) dependent on institutions.Instead, I extend their framework by allowing countries’ institutional quality and sectors’ institutionaldependency to obey the Ricardian law of comparative advantage. This elaboration makes it possibleto capture the aggregate relationship between country and sector characteristics neatly in a way such

1See Bernard, Eaton, Jensen, and Kortum (2003) and Bernard, Jensen, Redding, and Schott (2007) for the UnitedStates, 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 manufacturing sector (an apparelmanufacturing sector), whereby a more customized sector tends to exhibit a higher percentage of exporters across21 sectors. Conversely, Lu (2011, Figure 1) shows that, as of 2005, around 60 percent (less than 20 percent) ofChinese firms export in cloth and fur manufacturing sectors (a chemical manufacturing sector), indicating that thereexists a clear negative relationship between export participation and the capital-labor ratio across 29 sectors. Finally,Bernard et al. (2003, 2007), Tomiura (2007), and Lu (2011) all document that there exist some exporting firms in everymanufacturing sector.

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that North with better institutions produces and exports relatively more in sectors where productionis more institutionally dependent.2

To formalize the higher tendency to export participation in comparative advantage sectors, Iincorporate firm-level differences in productivity. Since exporting requires fixed export costs thatless productive firms cannot cover, only a small fraction of firms are able to export. The variationin this fraction is further reinforced by institution-induced comparative advantage in my setup, be-cause Northern (Southern) firms are relatively better at producing in more (less) contract-dependentsectors, which in turn softens the relative burden of incurring export costs. As a result, comparedto comparative disadvantage sectors of a counterpart country, relatively less productive firms canexport in a country’s comparative advantage sectors. This mechanism explains why the strongereach country’s comparative advantage is, the smaller the productivity cutoff for exporting becomes,thereby leading to a higher exporters’ percentage.

Following the new literature on institutions and trade, I employ contracting institutions – morespecifically contract enforcement – (rather than the classical determinants of international tradesuch as capital or (un)skilled labor) to rationalize the stylized fact of export participation. Thereare at least three reasons for this. First, countries’ abilities to enforce written contracts can havequantitatively larger impacts on comparative advantage than countries’ factor endowments. Forinstance, Nunn (2007) estimates that “contract enforcement explains more of the global patternof trade than countries’ endowments of physical capital and skilled labor combined.” Second, asrigorously 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 independentfactor of the variation in export participation as discussed above. Finally, a Ricardian view of insti-tutional 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 sim-ilar, I show that some phenomena (e.g., home-market effects) are better understood through a lensof Ricardian sources of comparative advantage.

In this paper, I do not attempt to explain why North has better institutions than South, whycustomized sectors depend heavily on institutions more than generic sectors, or why some firmsare more productive than others. Taking these country, sector, and firm characteristics as given,

2There is mounting evidence on the link between countries’ institutions and sectors’ types that affects the patternof trade. For instance, devising a measure of input customization (the share of a sector’s inputs that are not soldin organized exchanges), Nunn (2007) shows that countries with better contract enforcement export relatively morein sectors for which relationship-specific investments are more important. Similarly, Manova (2008) finds evidencethat trade liberalizations induce countries with better financial systems to export relatively more in sectors for whichfinancial requirements are more important. For the other related papers in this literature, see Costinot (2009a).

3As far as log-supermodularity in country and sector characteristics is central, the modeling of technological andinstitutional differences is isomorphic (Costinot, 2009a). This is because a country with better institutions facesless severe underinvestment and has a bigger cost advantage in production, which is a key presumption in Levchenko(2007) and Nunn (2007). Although I interpret institutions as a country’s contractibility to alleviate contractual frictionsbetween firms and suppliers as in Acemoglu et al. (2007), I admit that this channel is at best one of potential sourcesof institution-induced comparative advantage and more nuanced studies along the line of Chor (2010) and Nunn andTrefler (2014) are necessary.

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I instead set out to explore how contracting institutions and heterogeneous firms jointly shape anendogenous pattern of trade. By so doing, the model shows that North with better institutions gainsa comparative advantage in contract-dependent sectors and is a net exporter of customized productsin intra-industry trade. South with worse contracting institutions, on the other hand, is shown to bea net exporter of generic products. Moreover, within sectors in bilateral trade flows, the fraction ofexporters 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 absoluteadvantage in institutional quality in any sector, and there exists no technological difference (in termsof firm productivity distributions) between the two countries.

This paper is closely related to two branches of the recent literature of international trade. Thefirst is an emerging literature on institutions and trade (e.g., Antras, 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 isat the heart of my model as well. In this strand of the papers, however, all firms are generallytreated as identical and therefore every firm is able to export everywhere.4 In the real world, alarge proportion of firms do not export even in strong comparative advantage sectors. The currentpaper 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 bythe existing evidence.

Another branch of the related literature is the so-called heterogeneous-firm model of trade, es-pecially developed by the seminal work of Melitz (2003). While the Melitz model is successful inexplaining the exporters’ behaviors among developed countries (North-North trade), recent empiricalevidence has pointed out that this model is less suitable for the study of bilateral trade flows betweendifferent countries (North-South trade), as exemplified by Lu (2011) who analyzes Chinese firm-levelmanufacturing data. A number of papers – among others, Demidova (2008), Falvey, Greenaway, andYu (2005), Fan, Lai, and Qi (2011), and Okubo (2009) – incorporate the asymmetry of countries inthis setting. My approach differs from these papers, because I focus on the role of wage differentials inNorth-South trade,5 and because most results hold without specifying any parameterization of firmproductivity distributions. More importantly, none of these papers sheds new light on the interplaybetween institutions and comparative advantage. Although I restrict the analysis only to an openeconomy and abstract from welfare implications for expositional simplicity, it is straightforward toextend the current setup to see the impact of trade on inter-/intra-sectoral resource allocations andwelfare gains from trade.

Finally, this paper is also related to the heterogeneous-firm literature on factor-proportions theory,

4Acemoglu et al. (2007) introduce firm heterogeneity in the degree of complementarity among inputs, but all productsare assumed to be freely traded and hence all firms export in their model.

5Wage differentials are one of the most prominent factors that have triggered large trade flows among dissimilarcountries in the past two decades. For instance, noting that in 2006 for the first time the United States did more tradein manufactured goods with developing countries than developed countries, Krugman (2008) asserts that this is largelydue to the wage differentials between the U.S. and developing countries: China’s and Mexico’s wages are respectivelyonly 4 percent and 13 percent of the U.S. level.

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especially to Bernard et al. (2007) as argued above. Using Helpman-Krugman’s (1985) two-factormodel, they provide a rich framework for analyzing distributional consequences from trade, a featuremissing in this Ricardian one-factor model. Their analysis, however, primarily applies to the situationin which two countries are not too different, and numerical simulations are required for outside factor-price-equalization regions. In contrast, it is possible in the current paper to analytically examine tradepatterns 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 inaddressing Krugman’s (1980) home-market effect. I show that, due to selection into domestic andexport markets that varies with comparative advantage, the home-market effect works oppositely forthe extensive/intensive margins of specialization and those of trade between North and South.

2 Setup

Consider a world composing of two large countries, North and South, i ∈ {N, S}. For notationalsimplicity, 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 incomeinto differentiated goods of a continuum of sectors over an interval [0, 1]. The preferences of arepresentative consumer are Cobb-Douglas across sectors and C.E.S. Dixit-Stiglitz within sectors:

U =∫ 1

0λz lnQzdz,

where

Qz =[∫

v∈Vz

qz(v)σ−1

σ dv

] σσ−1

,

is aggregate consumption of varieties in sector z. Vz is the mass of available goods within the sector,which potentially includes both domestic and foreign varieties. Given this aggregate good Qz, itsdual aggregate price is given by

Pz =[∫

v∈Vz

pz(v)1−σdv

] 11−σ

.

λz denotes a constant share of expenditure spent on sector z, which is identical between the twocountries. Letting Rz = PzQz and Y = wL respectively denote aggregate expenditure in sector z

and aggregate labor income in the economy, λz is defined as

λz =PzQz

Y=

Rz

wL,

∫ 1

0λzdz = 1.

Thus, the sum of aggregate sector expenditure equals aggregate labor income (∫ 10 Rzdz = wL).

Letting Xz = λzY denote labor income spent on sector z, the above preferences generate demand

4

functions for each differentiated variety in sector z:

qz(v) = Azpz(v)−σ, Az = XzPσ−1z ,

where Az is the index of aggregate market demand. In the following, I focus on a particular sectorand drop sector subscript z from relevant variables.

Before proceeding further, it is important to note that there is no homogeneous-good sector withnontrade costs, and wage rates w cannot be normalized between North and South.6 This structure ofthe preferences is similar to that of Krugman (1980), and more recently to that of Antras (2005) andOkubo (2009). Note also that while the elasticity of substitution between any two varieties within asector is assumed to be greater than one (σ > 1), the elasticity of substitution between any varietiesacross sectors is unity. The unit elasticity of substitution implies that firm behavior in each sectorcan be analyzed independently.

Production There is a continuum of firms that produce a different variety v in each sector. Laboris the only factor of production to produce a variety and firms face a perfectly elastic supply of laborat each country size L. Since labor is completely mobile across sectors but immobile across countriesas in conventional Ricardian models, a wage rate w is the same across sectors within a country butis different across countries.

Following Krugman (1980) and Melitz (2003), firm technology is summarized in a linear costfunction of output q:

l =

fd + qθ(ϕ,z,µ) = fd + q

ϕµ(z) if domestic production,

fx + τqθ(ϕ,z,µ) = fx + τq

ϕµ(z) if exporting,

where θ(·, ·, ·) is labor productivity, fd is a fixed cost for domestic production, fx is a fixed cost forexporting, and τ(≥ 1) is a iceberg transport cost. These costs are identical across countries andsectors.

A few points are in order for this specification. First, labor productivity θ(·, ·, ·) depends on threefactors: (i) firm-specific ϕ; (ii) sector-specific z; and (iii) country-specific µ. In Melitz (2003), heconsiders symmetric countries, implying that a country-specific factor µ is ignorable. He also focuseson 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, bycontrast, since the two countries are asymmetric and there is a continuum of sectors, the three factorsjointly affect labor productivity. It follows from this cost function that the country-specific factorµ(·) ∈ (0, 1) affects firms’ variable costs only (leaving fixed costs identical) and labor productivity is

6By 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 Ricardianmodels. While introducing a homogeneous good a la Helpman and Krugman (1985) would help to simplify theanalysis, empirical evidence suggests that the bulk of recent trade flows cannot be captured without a terms-of-tradeeffect between developed and developing countries as emphasized in Introduction.

5

0

zi

1

z

1

zS

zN

Figure 1 – Log-supermodularity in contractibility

greater if µ(·) is closer to one. I assume that µ(·) is related to a country’s ability to enforce writtencontracts 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 usedfor 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), andshows that Ricardian sources of comparative advantage hold if 1/a(·, ·) is log-supermodular (i.e.,

∂2

∂z∂µ ln 1a(z,µ) ≥ 0 ⇔ ∂2

∂z∂µ ln a(z, µ) ≤ 0), or equivalently

a(z2, µ1)a(z2, µ2)

≥ a(z1, µ1)a(z1, µ2)

,

for z1 ≥ z2, µ1 ≥ µ2, a(z1, µ2) 6= 0 and a(z2, µ2) 6= 0. My specification is restricted relative toCostinot’s in that θ(ϕ, z, µ) = f(ϕ)/a(z, µ) = ϕµ(z). In addition to applying this reduced form, Ifurther assume that North has partial contractibility strictly superior to South in any sector. Notingthe inverse relationship between µ(·) and a(·, ·), log-supermodularity in terms of µ(·) is given by

1 <µN (z)µS(z)

<µN (z′)µS(z′)

< ∞,

for z > z′, µN > µS , µS(z) 6= 0 and µS(z′) 6= 0. Thus, not only does µ(z) = 1/a(z, µ) satisfylog-supermodularity (or Ricardo’s classic inequality), but North has an absolute advantage in µ(z)in any sector.

Figure 1 depicts µi(z) satisfying the above inequalities with the additional assumptions that

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µN (1) = µS(1) = 1 and µS(0) = 0. As is clear from the figure, log-supermodularity means that thegap between µN and µS is gradually larger as sectors’ production becomes more customized in thisframework. An economic interpretation of this figure is as follows. In a generic sector, the severeholdup problem is less likely irrespective of institutional quality because the production does not relyheavily on relationship-specific investments. As a result, the gap between µN and µS is relativelysmaller and µi is closer to one in a more generic sector. In a customized sector, on the contrary,producers are more likely to suffer from the holdup problem and production efficiency is relativelymore 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 coreof neoclassical trade theory and is also the pivotal element in the empirical evidence reported byLevchenko (2007) and Nunn (2007) – is elegantly captured by log-supermodularity.

Finally, in this decomposition θ(ϕ, z, µ) = ϕµ(z), I refer to a country’s distribution G(ϕ) of firms’productivity draws ϕ as “technologies” as in Melitz (2003), whereas a country’s partial contractibilityµ(z) on firms’ relationship-specific investments as “institutions” as in Levchenko (2007). Further-more, I restrict attention to environments in which all firms have access to the same technologiesacross countries and sectors, i.e., Gi

z(ϕ) = G(ϕ) for i ∈ {N, S} and z ∈ [0, 1].7 Consequently, thereare no technological differences across countries and institutional differences solely give rise to coun-tries’ comparative advantage. In reality, technological and institutional differences coexist and thesetwo differences are not precisely separable. This distinction is not crucial for the analysis below andthe following definition is made for the sake of convenience.

Definition 1 Technologies are a country’s distribution G(ϕ) of firms’ productivity draws ϕ, whichis identical across countries and sectors. Institutions are a country’s partial contractibility µ(z) onfirms’ relationship-specific investments, which varies across countries and sectors.

Each firm chooses its price to maximize profits π = px−wl for domestic production and exporting.Solving profit-maximizing problem yields the following first-order conditions:

p(ϕ) =σ

σ − 1w

ϕµ,

q(ϕ) = A

(σ − 1

σ

ϕµ

w

)σ

,

r(ϕ) = p(ϕ)q(ϕ) = A

(σ − 1

σ

ϕµ

w

)σ−1

,

π(ϕ) =r(ϕ)σ

− wf = B( µ

w

)σ−1ϕσ−1 − wf,

7This means that, if the firm productivity distribution is Pareto, G(ϕ) = 1− (ϕmin/ϕ)k, both the shape and scaleparameters, k and ϕmin, are identical across countries and sectors. While these parameters are more likely to vary withcountry and sector characteristics in evidence (Tybout, 2000), the modeling of sector-variant distributions could comeat the cost of obscuring Ricardian sources of comparative advantage if Gi

z(ϕ) is log-supermodular in sector and firmcharacteristics (Costinot, 2009a).

7

where

B =(σ − 1)σ−1

σσA =

(σ − 1)σ−1

σσXP σ−1,

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 thevariable trade cost (τ > 1), and the supplementary note offers a detailed analysis that incorporatesτ . 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 featuresof the current setup are worth emphasizing. First, wages w cannot be normalized between the twocountries, since they are asymmetric and there is no freely tradable homogeneous-good sector in thismodel. As noted earlier, this assumption is made to examine the role of endogenous factoral termsof trade in the Ricardian model. Second, institutional quality µ enters into these conditions. It isimmediate to see that the pricing rule is higher and the output level is lower in a more customizedsector due to the holdup problem.

Holdup Problem So far, I have not explicitly explored how a country’s institutional quality µ isrelated to the enforcement of contracts between firms and input suppliers, and thus to the holdupproblem 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 aninitial contract ex ante, and they have to renegotiate the contract ex post. In what follows, buildingon seminal work of Antras (2003, 2005) and Antras and Helpman (2004), I show that institutionalquality µ plays a qualitatively similar (but distinct) role with incomplete contracting.

Suppose that, while letting perfect institutions (µ = 1) prevail in any sector of both countries,production of final goods now requires intermediate inputs which each firm cannot manufacture byitself. To produce a variety, every firm asks a domestic input supplier to provide a specialized input.This input is relationship-specific in the sense that it has a higher value within the parties and a thirdparty (such as courts of law) cannot distinguish its true value. Because no enforceable contract willbe signed ex ante in such a circumstance, the firm and its supplier have to bargain over the surplusafter production takes place. Let β and 1− β denote the firm’s and its supplier’s ex-post bargainingpower. 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 break-even 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

p(ϕ) =σ

σ − 1w

ϕ(1− β), q(ϕ) = A

(σ − 1

σ

ϕ(1− β)w

)σ

.

Comparing the two pricing rules reveals µ = 1−β in equilibrium, and the distribution of bargainingpower 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.

8

In such an environment, the input level is ex ante verifiable and the supplier could directly bargainover the profit rather than the revenue. Then, the firm’s profit is π∗F = βπ∗ + T and the supplier’sprofit is π∗S = (1−β)π∗−wl−T , where π∗ = π∗F +π∗S is the joint profit under complete contracting.The supplier would choose its input level to maximize π∗S , so the first-order conditions are

p∗(ϕ) =σ

σ − 1w

ϕ, q∗(ϕ) = A

(σ − 1

σ

ϕ

w

)σ

.

Evidently, the pricing rule is 1/(1−β) times higher under incomplete contracting because the supplierreceives only a fraction of the marginal return to its investment for specialized input, leading theinput level to be (1−β)σ times lower. Given this interpretation, the distribution of ex-post bargainingpower has a direct impact on comparative advantage through ex-ante efficiency of specialized inputproduction, i.e., the holdup problem. In particular, across sectors, a fraction of ex-ante contractibleactivities would be relatively smaller in a more customized sector and the supplier’s holdup problem isrelatively severer in such a sector (Acemoglu et al., 2007). Furthermore, across countries, South wouldhave absolutely worse legal institutions in enforcing contracts and the supplier’s holdup problem inSouth is absolutely severer than North (Antras and Helpman, 2004). This is a theoretical justificationfor log-supermodularity in institutional quality µ, with North having an absolute advantage in it.

The previous literature on organizations and trade typically assumes that the degree of theholdup problem in relationship-specific investments is the same across sectors, but varies acrossorganizational forms (i.e., vertical integration and outsourcing). Instead, I allow this degree to varyacross sectors, while abstracting from the firm boundaries. The model is developed below, keepingin mind the similarity between imperfect institutions and incomplete contracts.

Firm Behavior The current paper analyzes a static version of the Melitz (2003) model. To entera sector in country i ∈ {N,S}, firms bear a fixed cost of entry fe, measured in country i’s laborunits. Upon paying this fixed cost, firms draw their productivity level ϕ from a known distributionGi

z(ϕ) = G(ϕ). After observing this productivity level, each firm decides whether to exit or not. Ifthe firm chooses to produce, it bears additional fixed costs fd for domestic production and fx forexporting, as described before. An entering firm in country i would then immediately exit if πi

d < 0,or would produce and serve its domestic market if πi

d ≥ 0. Moreover, among domestic firms, onlythe most productive firms would earn πi

x ≥ 0 and serve the foreign market in j as exporters underthe assumption fx > fd.

While this firm behavior is similar across countries and sectors, the productivity cutoffs fordomestic production and exporting would vary by reflecting countries’ comparative advantage. Re-garding exporting participation, if πi

x ≥ 0 and πjx ≥ 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 becauseproducts are differentiated and consumers are strictly better off by importing products unavailablein the domestic market. If πi

x ≥ 0 for some firms and πjx < 0 for any firm, on the contrary, one-way

(inter-industry) trade occurs in this sector, whereby exporting from i to j takes place.

9

3 Partial Equilibrium

In this section, I first explore partial equilibrium in which some important variables are exogenouslygiven. The next section embeds this analysis in a general-equilibrium setting.

To see an equilibrium of sector z, consider country j’s market where competition occurs betweendomestic firms in j and exporters from i. From the first-order conditions in the previous section, theprofit functions of these firms are respectively given by8

πjd = Bj

(µj(z)wj

)σ−1

ϕσ−1 − wjfd, πix = Bj

(µi(z)wi

)σ−1

ϕσ−1 − wifx.

Notice that, since these firms compete in j’s market, aggregate demand Bj is common for both profitfunctions. πj

d and πix are measured by different wage rates and contractibility levels, however, because

exporters from i have to use domestic labor and institutions to produce own variety. If there existmultinational enterprises that directly employ local labor and have internal contractibility within thefirm boundaries, this argument is no longer true. See Section 5 for the possibility of foreign affiliateproduction in the current setup.

To compare these two profit functions graphically, they are drawn in (ϕσ−1, π) space with slopeB

( µw

)σ−1 and intercept −wf . Then, πix is steeper (flatter) than πj

d if and only if

µi(z)wi

R µj(z)wj

⇐⇒

µ(z) R ω if i = N,

µ(z) Q ω if i = S,

where µ(z) = µN (z)/µS(z) = a(z, µS)/a(z, µN ) and ω = wN/wS respectively denote the relativecontractibility (or relative labor requirement) and the relative wage in North. Following standardRicardian models, I say that North (South) has an institutional comparative advantage in sectorz if it has a large (small) relative labor productivity in 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 institutionalcomparative advantage is identified as the sectors where the slope of exporters πi

x is steeper thanthat of domestic firms πj

d.

Definition 2 North (South) has an institutional comparative advantage in sector z if µ(z) > ω

(µ(z) < ω), or equivalently if the slope of πNx (πS

x ) is steeper than that of πSd (πN

d ).

Since this definition indicates neither North nor South has a comparative advantage in a sector ofµ(z) = ω where πi

d and πjx are parallel, I first derive the condition under which µ(z) and ω are equal.

From Figure 1, the ratio of contractibility µ(z) = µN (z)/µS(z) has to satisfy µ(z) > 1, µ′(z) <

0, µ′′(z) > 0, limz→1 µ(z) = 1, and limz→0 µ(z) = ∞, where the first condition stems from absoluteadvantage of North and the third stems from log-supermodularity of µi(z). The relative wage in

8Following the literature (e.g. Melitz and Redding, 2014), I assume that all production costs, including the fixedexport cost fx, are measured in terms of a source country labor.

10

0

d

N fw

x

S fw

1

S

x

N

d

S

x

N

d,

N

d

North

S

x

0

d

S fw

x

N fw

1

N

x

S

d

N

x

S

d,

S

d

South

N

x

Figure 2 – Profits from domestic sales and exports

North ω, on the other hand, should be the same for all sectors z ∈ [0, 1] because labor is completelymobile across sectors within a country in the Ricardian model. This suggests that, if ω is greater thanone, these two curves intersect at a unique cutoff z = µ−1(ω) such that: (i) z ∈ [0, z) ⇔ µ(z) > ω;(ii) z = z ⇔ µ(z) = ω; and (iii) z ∈ (z, 1] ⇔ µ(z) < ω. It is then immediate from Definition 2 thatNorth (South) has an institutional comparative advantage in relatively customized (generic) sectorsz ∈ [0, z) (z ∈ (z, 1]).

Next I consider the equilibrium in the cutoff sector z as a benchmark. The left panel of Figure 2depicts the profit functions in Northern market (πN

d , πSx ) under the condition that wSfx > wNfd ⇔

ω < fx

fd. This condition ensures that Southern exporters bear the higher fixed cost (measured by

the local labor wage) than Northern domestic firms. Following the same line of reasoning, the rightpanel depicts the profit functions in Southern market (πS

d , πNx ) under the condition that wNfx >

wSfd ⇔ ω > fdfx

. Figure 2 shows that Northern (Southern) firms with productivity above ϕNx (ϕS

x )export to South (North) and hence two-way trade occurs in the cutoff sector z.

This argument helps understand what happens in sectors other than the cutoff sector z. In sectorswhere North has a comparative advantage (i.e., z ∈ [0, z)), for example, it follows from µ(z) > ω

that the profit functions of Northern firms are steeper relative to those of Southern firms in bothdomestic and export markets of Figure 2. This implies that, as country i’s comparative advantage isstronger, the productivity cutoffs, ϕi

d and ϕix, become relatively smaller than the counterpart cutoffs

in country j and less productive firms are more likely to find it profitable to operate in these sectors.At the same time, the opposite is true for country j: ϕj

d and ϕjx become relatively larger and less

productive firms are more likely to exit these sectors. Moreover, two-way trade would occur in anysector z ∈ [0, 1] as long as the slopes of πN

x and πSx are positive, i.e., B

( µw

)σ−1> 0.

Notice in Figure 2 that, due to wage differentials, domestic firms in j might bear the higher fixedcost than foreign exporters from i in j’s market. Because it is empirically well-known that fx is huge

11

in any manufacturing sector,9 I hereafter assume the following condition for the fixed costs whichwill be shown to be necessarily held in the general-equilibrium setting where ω is endogenous.

Assumption 1 fdfx

< ω < fx

fd.

While the existence of the unique cutoff sector z in the absence of variable trade cost is reminiscentof Dornbusch, Fischer, and Samuelson’s (1977) Ricardian model with a continuum of goods,10 thereexist three noteworthy distinctions between the current paper and theirs. First, since they analyzeperfect competition with homogeneous goods, complete specialization (or inter-industry trade) occursbelow/above the cutoff z; in contrast, this paper studies monopolistic competition with differentiatedgoods 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 intheir neoclassical trade model, but it is endogenously determined in the current framework in whichthe 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 playsa prominent role in shaping countries’ comparative advantage. The result – that a less developedcountry nevertheless exports differentiated goods in customized sectors – seems to be consistent withrecent trade flows (see, e.g., Krugman, 2008).

Proposition 1

(i) If ω > 1, there exists a unique cutoff sector z ∈ (0, 1) such that North (South) has an institu-tional comparative advantage in sectors z ∈ [0, z) (z ∈ (z, 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 interplayamong key variables of the model. To see this, it is useful to go back to Figure 2. The figure depictsϕj

d < ϕix in the cutoff sector z, 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 toincur the higher fixed cost under Assumption 1, and it can be easily formalized by using a partial-equilibrium framework. However, ϕi

d and ϕjd or ϕi

x and ϕjx are not comparable. Obviously, ϕN

d andϕS

d (ϕNx and ϕS

x ) are determined at which πNd = 0 and πS

d = 0 (πNx = 0 and πS

x = 0), but thesevariables depend on the aggregate market demand Bi as well as the wage rate wi, both of whichare exogenous in partial equilibrium. Also, Proposition 1(i) requires that ω should be greater than

9Das, Roberts, and Tybout (2007) econometrically estimate the average costs of foreign market entry among threeColombian manufacturing sectors (leather products, knitted fabrics, and basic chemicals), and find that these fixedcosts are (i) large enough to cause export hysteresis and (ii) remarkably similar across these sectors: the average exportcosts range from $412,000 (in U.S. dollars) for knitted fabrics to $430,000 for leather products.

10The idea of bringing firm heterogeneity into the Dornbusch et al. (1977) model is not entirely new, although themodels’ setups in the existing literature are less general than the current one. For instance, Okubo (2009) uses aspecific distribution of firm productivity levels, whereas Fan et al. (2011) abstract from endogenous wage differentialsbetween countries by introducing the homogeneous-good sector.

12

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 necessaryto endogenize these variables.

4 General Equilibrium

In this section, the partial-equilibrium model is embedded into a general-equilibrium framework toexamine the interaction among the key variables and to see endogenous patterns of specializationand trade.

General-Equilibrium Setup This subsection outlines several equilibrium conditions that playa central role in characterizing the endogenous variables in general equilibrium. In the subsequentsubsections, I solve this general-equilibrium model with some restrictions on the exogenous variables.

First of all, a zero profit condition must hold for the cutoff firms in both domestic and exportmarkets, which is respectively achieved by setting ϕi

d = inf{ϕ : πid(ϕ) > 0} and ϕi

x = inf{ϕ : πix(ϕ) >

0} in the static model. This condition is refereed to as a zero cutoff profit (ZCP) condition:

πid(ϕ

id) = 0 ⇐⇒ Bi

(µi

wi

)σ−1

(ϕid)

σ−1 = wifd, (ZCP id)

πix(ϕi

x) = 0 ⇐⇒ Bj

(µi

wi

)σ−1

(ϕix)σ−1 = wifx, (ZCP i

x)

for i 6= j ∈ {N, S}. In this condition, aggregate market demand of exporters (Bj) should be differentfrom that of domestic firms (Bi) because exporters from country i have to face aggregate marketdemand in country j.

Secondly, a free entry (FE) condition must be satisfied. Since potential entrants are ex anteidentical in the current model, this condition is defined as

∫ ∞

ϕid

πid(ϕ)dG(ϕ) +

∫ ∞

ϕix

πix(ϕ)dG(ϕ) = wife, (FEi)

where the first and second terms in the left-hand side respectively denote the expected operatingprofits from domestic production and exporting earned by potential entrants. The sum of theseexpected profits has to be equal to the fixed entry cost wife.

Finally, a labor market clearing (LMC) condition must be taken into account:

∫ 1

0M i

e

∫ ∞

ϕid

lid(ϕ)dG(ϕ)dz +∫ 1

0M i

e

∫ ∞

ϕix

lix(ϕ)dG(ϕ)dz +∫ 1

0M i

efedz = Li, (LMCi)

where M ie denotes the mass of potential entrants in 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 exportingby potential entrants, whereas the third term is expected labor demands used for investment by

13

potential entrants. Note that lix(ϕ) is summed up over sectors of the economy as a whole becausetwo-way trade can occur in any sector z ∈ [0, 1] as seen in Proposition 1(ii). The sum of theseexpected labor demands has to be equal to the fixed labor supply Li.

Now, it is possible to endogenize the important variables in general equilibrium. Since there arethe eight equations (the ZCP, FE, and LMC conditions that must hold in North and South), theseconditions provide implicit solutions for the following eight unknowns:

ϕNd , ϕS

d , ϕNx , ϕS

x , BN , BS , wN , wS ,

where the LMC condition in South can be omitted by Walras’ law, thereby normalizing wS = 1 as anumeraire. (The mass of potential entrants M i

e can be written as a function of these eight unknownsas will be shown later.)

Relative Equilibrium Conditions This subsection sets forth the characterization of the eightunknowns from the eight equilibrium conditions. It is challenging, however, to solve a full generalequilibrium model with asymmetric countries. In particular, without specifying the functional formof the firm size distribution G(ϕ), explicit solutions of these unknowns cannot be obtained. Inthe following, instead of obtaining the exact values of each of them, the main focus is devoted tocharacterizing 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, supposefirst that ω > 1 in equilibrium. In other words, the LMC conditions are left out from the model asif it were partial-equilibrium. As will be clear, this inequality must be true in general equilibriumbecause 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) inNorth, and thus wages have to be greater in North than South if trade is to occur between the twocountries. 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(ϕd = ϕN

d /ϕSd , ϕx = ϕN

x /ϕSx ) and the relative market demand (B = BN/BS) by focusing on the

ZCP and FE conditions. To do this, using the ZCP conditions, rewrite the FE condition as

fdJ(ϕid) + fxJ(ϕi

x) = fe,

where J(ϕ) =∫∞ϕ [(ϕ/ϕ)σ−1 − 1]dG(ϕ). J(·) is monotonically decreasing with limϕ→0 J(ϕ) = ∞

and limϕ→∞ J(ϕ) = 0. Since the equality of this condition must hold in any sector, changes in z

shift the productivity cutoffs in opposite directions, and thus ϕix/ϕi

d must be strictly increasing ordecreasing in z. Note that these changes in z affect ϕ’s and B’s while they have no impact on w’s aswages are independent of z (as long as labor is completely mobile across sectors), and the equilibriumanalysis here does not rely on the exogenous ω assumption. Moreover, dividing the ZCP condition ofdomestic production by that of exporting for each country, the relative market demand B = BN/BS

14

z xd

, z

)1/(1

B

0

1

1

B

d

x

Figure 3 – Market demand and productivity cutoffs

is given by

B =

(ϕN

x

ϕNd

)σ−1fdfx

if i = N,(

ϕSd

ϕSx

)σ−1fx

fdif i = S.

Using the property of the FE condition derived above, it can be shown that B is strictly increasing inz (see Appendix). The intuition behind this result is explained by recalling that B is proportional tothe relative aggregate price (P = PN/PS). If z is close to one, the institutional differential is almostnegligible whereas there exists the wage differential (i.e., ω > 1). South is thus able to produce goodsrelatively cheaply, thereby leading to PN > PS and B > 1 in the neighborhood of z = 1. If z is closeto zero, on the other hand, the institutional differential is sufficiently large to dominate the wagedifferential (due to log-supermodularity), resulting in PN < PS and B < 1 in the neighborhood ofz = 0. Roughly speaking, this intuition mirrors the idea that country i has a comparative advantagein sectors where the aggregate price P i is relatively lower than P j .11 The first quadrant of Figure 3depicts this relationship in (z,B) space.

Next, dividing the ZCP condition of North by the corresponding condition of South, the relativeproductivity cutoffs ϕd = ϕN

d /ϕSd and ϕx = ϕN

x /ϕSx satisfy the following relative ZCP condition:

ϕd =( ω

B

) 1σ−1 ω

µ, ϕx = (Bω)

1σ−1

ω

µ, (RZCP )

where all variables are represented by the relative terms in North. It is easy to show that ϕd decreaseswith B while ϕx increases with B, and that

ϕx R ϕd ⇐⇒ B R 1.

This equality holds in the cutoff sector z = z because ϕd and ϕx are equal if and only if πid and πj

x

are parallel in that sector (see Figure 2). Thus, the relative market demand and relative productivity

11This statement is not precise because the aggregate price P i includes prices of both domestic and foreign varietiesin the presence of two-way trade. Given log-supermodularity, this would hold in a closed-economy version of the model.

15

N

S

0

0

S

x

N

x

S

d

N

d

zz !!0

s

N

S

0

0

S

x

N

x

S

d

N

d

1 zz

,

Figure 4 – Relationship among productivity cutoffs

cutoffs respectively satisfy B(z) = 1 and ϕd(z) = ϕx(z) = ω1/(σ−1) (using µ(z) = ω) in the cutoffsector. The second quadrant of Figure 3 depicts this relationship in (B, ϕ) space.

Finally, combining the first and second quadrants, Figure 3 highlights the sectoral differenceamong the six endogenous variables (represented in the relative terms) that are derived from theZCP and FE conditions: in any sector z ∈ [0, 1], the relative market demand B is determined in thefirst quadrant, and the relative productivity cutoffs ϕd and ϕx are subsequently determined in thesecond quadrant. It is immediately seen that

0 ≤ z < z ⇐⇒ B < 1 ⇐⇒ ϕd > ω1

σ−1 > ϕx,

z = z ⇐⇒ B = 1 ⇐⇒ ϕd = ω1

σ−1 = ϕx,

z < z ≤ 1 ⇐⇒ B > 1 ⇐⇒ ϕd < ω1

σ−1 < ϕx.

In this relationship, either ϕd or ϕx is necessarily greater than one under the condition that therelative wage ω is greater than one. For instance, ϕd is greater than one in sectors where North hasa comparative advantage z ∈ [0, z); however whether ϕx is greater than one or not is indeterminatein the current setup.

Based on this observation, Figure 4 illustrates the relationship among the productivity cutoffs inthe comparative advantage sectors of North (left panel) and South (right panel). While the figureshows that ϕx = ϕN

x /ϕSx > 1 (left panel) and ϕd = ϕN

d /ϕSd > 1 (right panel), these might not hold

in some sectors. Regardless of whether or not they are greater than one, the gap between ϕix and

ϕid is narrower than the gap between ϕj

x and ϕjd in country i’s comparative advantage sectors. In

addition, this former (latter) gap becomes smaller (bigger) as country i’s comparative advantage isstronger. More formally, it follows from the ZCP conditions that

ϕNx

ϕNd

=(

Bfx

fd

) 1σ−1

,ϕS

x

ϕSd

=(

1B

fx

fd

) 1σ−1

.

Since B is strictly increasing in z, ϕNx /ϕN

d (ϕSx/ϕS

d ) is strictly increasing (decreasing) in z: as countryi’s comparative advantage is stronger,12 the productivity cutoff for exporting ϕi

x is closer to that fordomestic production ϕi

d and a larger portion of firms are able to enter the foreign market. Hence the

12Noting that µ(z) = a(z, µS)/a(z, µN ) is the relative unit labor requirement and is decreasing in z, comparativeadvantage of North (South) is said to be “stronger” if z is closer to zero (one) in this model.

16

model predicts a higher ratio of exporting firms to overall surviving firms in sectors where countriesare relatively more productive.

The above equations also indicate that the productivity cutoff for exporting is bigger than thatof domestic production (ϕi

x > ϕid) in the comparative disadvantage sectors for i ∈ {N,S}.13 In the

comparative advantage sectors, the usual outcome (ϕix > ϕi

d) occurs in both countries if

fd

fx< B <

fx

fd, (1)

whereas the “perverse” outcome (ϕid > ϕi

x) occurs if

B < fdfx

if i = N,

fx

fd< B if i = S.

ϕid > ϕi

x implies that, among surviving firms in i, less productive firms serve only the foreign marketin 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 B) is strongenough relative to the fixed-cost ratio between fd and fx.

Proposition 2

(i) The gap between ϕix and ϕi

d is narrower than ϕjx and ϕj

d in country i’s comparative advantagesectors and this gap is monotonically decreasing in its comparative advantage strength.

(ii) While ϕix is always greater than ϕi

d in country i’s comparative disadvantage sectors, this mightnot hold in extremely strong comparative advantage sectors.

These two findings fit well with recent empirical research. The first finding – among domesticfirms, more firms export in stronger comparative advantage sectors – is consistent with evidence thatwas reviewed in Introduction. The logic of this result comes from the interplay between the Ricardianproductivity difference and relative burden of fixed export costs: log-supermodularity in country andsector characteristics allows relatively less productive firms to incur the fixed export cost relativelymore easily in comparative advantage sectors. Strictly speaking, this observation is not satisfactorysince several important questions cannot be addressed without the mass of varieties produced andexported. For example, how does countries’ comparative advantage affect the extensive and intensivemargins? Does a larger country size lead to a higher export participation ratio in any sector? LaterI investigate what determines the mass of varieties to answer these questions.

13From fx > fd, comparative disadvantage sectors of North, for example, must satisfy

z < z ≤ 1 ⇐⇒ B > 1 =⇒ ϕNx > ϕN

d .

(Note also that Assumption 1 has an influence on the relationship between ϕid and ϕj

x for i 6= j ∈ {N, S}.)

17

The second finding is also in keeping with Lu’s (2011) empirical evidence that manufacturingexporters in China are typically less productive than domestic firms in labor -intensive sectors, butexporters are more productive in capital -intensive sectors. To rationalize this evidence, Lu developsa Heckscher-Ohlin model with heterogeneous firms, emphasizing that allowing factor intensity tovary across sectors is crucial for the Melitz model of North-South trade. Although my theoreticalfocus is apparently different from hers, the central message is surprisingly similar: exporters can beless productive than domestic firms in comparative advantage sectors, whereas exporters are alwaysmore productive in comparative disadvantage sectors. The rationale for this result is as follows. Incomparative disadvantage sectors of country i, by definition, firms in country j are relatively betterat producing than those in i. Thus, exporters from i must be sufficiently productive, not only becausethey have to cover the fixed export cost, but also because they have to compete with more efficientforeign rivals in the export market. In comparative advantage sectors of i, on the other hand, firmsin j are relatively poorer at producing than those in i, which makes relatively less productive firmsin i find it profitable to enter the export market. This is the reason why the partitioning of firmsby export status might not be induced only with the condition fx > fd in comparative advantagesectors. (This holds even with variable trade cost as far as τσ−1fx > fd; see supplementary note.)

If ϕid > ϕi

x were true, all surviving firms in i could export, which is not supported by evidence.14

This result should be interpreted as meaning that aggregate productivity premia of exporters relativeto domestic firms are smaller in stronger comparative advantage sectors. I exclude the possibilityϕi

d > ϕix in the following analysis by assuming that fx is large enough to satisfy (1) in any sector.

Full General Equilibrium The previous subsection provides implicit solutions of firm selection(ϕi

d, ϕix) and aggregate market demand (Bi) for a given wage rate (wi). Now that each sector’s

equilibrium is characterized by these six endogenous variables, this subsection explores full general-equilibrium interactions by explicitly incorporating the LMC conditions in the model. Here I showthat the relative wage in North (ω = wN/wS) is necessarily greater than one, a sufficient conditionof North-South trade required in Proposition 1(i).

Recall from the profit-maximization problem in Section 2 that labor demand of a firm withproductivity ϕ is given by

lid(ϕ) = fd +σ − 1

σ

rid(ϕ)wi

, lix(ϕ) = fx +σ − 1

σ

rix(ϕ)wi

.

Substituting these values into the LMC condition and using the FE condition, the previous LMCcondition is simplified as follows (see Appendix):

∫ 10 Ri

zdz

wi= Li,

where∫ 10 Ri

zdz =∫ 10 P i

zQizdz is aggregate expenditure in country i. Consequently, each country’s

14In Lu’s (2011) dataset, ϕid > ϕi

x occurs because it includes Chinese exporters involved in processing trade. If firmsengage in final-good trade only as in the current model, this possibility would not exist.

18

wage is determined by the equality between aggregate expenditure (∫ 10 Ri

zdz) and aggregate paymentsto labor (wiLi) 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, how-ever, it is easiest to use the balance-of-payments (BOP) condition that is equivalent with the aboveLMC condition in the current model where two-way trade can occur in any sector (see Proposition1(ii)): ∫ 1

0RN

x (z)dz =∫ 1

0RS

x (z)dz, (BOP )

where Rix = M i

e

∫∞ϕi

drix(ϕ)dG(ϕ) is aggregate sector exports. This equivalence stems from that

aggregate expenditure in North is the sum of expenditure spent on domestic products and importsfrom South,

∫ 10 RN

z dz =∫ 10

(RN

d (z) + RSx (z)

)dz, and aggregate income is the sum of domestic and

export revenues, wNLN =∫ 10

(RN

d (z) + RNx (z)

)dz.

Note that due to log-supermodularity, North (South) produces and exports relatively more incustomized (generic) sectors and aggregate exports are strictly increasing in its comparative advan-tage strength. Let z denote a hypothetical cutoff below which North is a net exporter of customizedproducts in two-way trade. Since net aggregate exports in sector z of country i are given by the dif-ference between labor income and expenditure Ri

x(z)−Rjx(z) = wiLi

z−Riz, the above BOP condition

becomes ∫ z

0

(wNLN

z −RNz

)dz =

∫ 1

z

(wSLS

z −RSz

)dz.

This equation simply indicates that North runs a trade surplus in sectors z ∈ [0, z) and a trade deficitin sectors z ∈ (z, 1]. In equilibrium, the relative wage is adjusted so that aggregate trade surplusesare offset by aggregate trade deficits.

It is then straightforward to derive the relative wage ω. Arranging the BOP condition in netexports shown above and using λz = P i

zQiz/Y i defined in Section 2, ω can be explicitly solved as a

function of z:

ω ≡ ξ(z) =

∫ 1z

(LS

z

LS − λz

)dz

L[∫ z

0

(LN

z

LN − λz

)dz

] ,

where L = LN/LS and Liz/Li is a share of labor allocated to sector z. It is easily verified that ξ is an

increasing function of z satisfying limz→0 ξ(z) = 0 and limz→1 ξ(z) = ∞ (see Appendix). Intuitively,these properties follow from the fact that ξ summarizes the LMC condition in each country. If z ishigher 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 z so that Northern labor is moreexpensive, thereby ensuring some Southern labor demands in equilibrium. Figure 5 depicts ξ curvein (z, ω) space.15

15Although ξ curve has some similarity with that in Dornbusch et al. (1977), their neoclassical model allows all

laborers to be allocated to each country’s comparative advantage sectors in trade equilibrium and thusR z

0

LNz

LN dz =R 1

z

LSz

LS dz = 1 in the equation of ξ curve. In contrast, this is not true in the current model due to incomplete specializationthat can occur in any sector.

19

z

0

1

!

!

z 1

Figure 5 – Wage in general equilibrium

The other condition that pins down z = z and ω is the relative partial contactibility µ = µN/µS .Among its properties, µ > 1 (absolute advantage of North) and µ′′ > 0 (log-supermodularity) areof particular importance. Figure 5 depicts µ curve in the same space and the intersection of ξ

and µ curves shows that ω is greater than one. These two curves intersect at (z, ω) because ω =ω(B(z), ϕd(z), ϕx(z)) is expressed in terms of ω in z = z, i.e., B(z) = 1 and ϕd(z) = ϕx(z) = ω1/(σ−1),implying also that ω is endogenous. This ω in turn leads to endogenous solutions of B, ϕd, and ϕx,which completes the characterization of the eight unknowns in equilibrium.

Proposition 3

(i) North (South) is a net exporter of customized (generic) sectors z ∈ [0, z) (z ∈ (z, 1]) in two-waytrade.

(ii) The relative wage in North is greater than one.16

It is worthwhile to stress that the above channel for the relative wage is similar to that developedby Antras (2005), who shows (with representative firms) that, irrespective of the relative country sizeL = LN/LS , better contracting environments in North lead to the higher relative wage in generalequilibrium. As in his model, the equilibrium outcome ω > 1 directly reflects that North has superiorpartial contractibility which helps mitigate the serious holdup problem. This in turn gives rise tobetter production efficiency and overall higher productivity (i.e., wage) in North.

16While one constraint of Assumption 1 (ω > fd/fx) is satisfied as long as fx > fd, the other constraint is written as

Z z

0

λzdz <1− R 1

z

LSz

LS dz + fxfd

LR z

0

LNz

LN dz

1 + fxfd

L.

To see whether ω < fx/fd internally holds, it is enough to check whether the above inequality holds for a sufficiently

large fx/fd that the model goes back to autarkic equilibrium. In this case, it is approximated toR z

0λzdz <

R z

0

LNz

LN dz,which is consistent with the current setup where North has a comparative advantage in sectors z ∈ [0, z).

20

B

z xd

,

B

'B

'

x '

x

d

'd

d

'd

x

'x

'z z

1

0

'

)1/(1

)1/(1'

Figure 6 – Comparative statics with respect to L

Comparative Statics In this subsection, I examine comparative statics with respect to the relativecountry size L in a single unified framework for the analysis of home-market effects on the extensiveand intensive margins that will be addressed in the next subsection.

Figure 6 illustrates a general-equilibrium interaction between good and factor markets by inte-grating the previous analysis. To explore effects of an increase in L, I compare two equilibria in thefigure by adding primes (′) to all variables and functions with thin lines for a new equilibrium. Thefirst and second quadrants come from Figure 3, whereas the fourth quadrant comes from Figure 5.The third quadrant depicts the relationship between ω and ϕ’s, which is derived from the RZCPconditions: ϕd = 1

µB ωσ/(σ−1), ϕx = Bµ ωσ/(σ−1). It is clear that these conditions are increasing in ω,

with ϕd(z) = ϕx(z) = ω1/(σ−1) and z R z ⇔ B R 1 ⇔ ξ R ω ⇔ ϕd R ϕx. The figure shows that:

• In the fourth quadrant, ξ curve shifts to the right (from the LMC/BOP conditions) whilekeeping µ curve unchanged, leading to ω > ω′ and z < z′.

• In the first quadrant, this decreases B for any z ∈ [0, 1] because B(z′) = 1 must be true in anew equilibrium and B is increasing in z (from the ZCP/FE conditions).

• In the second quadrant, ϕd curve shifts inward while ϕx curve shifts upward because B =ωσ/(σ−1)

µ ϕ−1d , B = µ

ωσ/(σ−1) ϕx (from the RZCP condition) and ω is decreasing in L (from thefourth quadrant).

21

• In the third quadrant, both curves shift inward because ϕd = 1µB ωσ/(σ−1), ϕx = B

µ ωσ/(σ−1)

(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 therelative wage ω. Note from Figures 3 and 4 that when ω is sufficiently large, it is more likely thatϕd > 1 and ϕx > 1, and the higher aggregate productivity of domestic firms and exporters in Northis partially reflected by the higher relative wage. Building on this view, the comparative statics heresuggest that an increase in L decreases the Ricardian productivity advantage as well as the firm-levelproductivity advantage in North by lowering ω, although it increases the range of sectors over whichNorth 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 (ϕi

d, ϕix) and aggregate market demand (Bi) through the

relative wage ω in this asymmetric-country model with the Ricardian productivity difference. In asymmetric-country model or a model with a homogeneous-good sector, by contrast, country size hasimpacts 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 charac-terization of the eight equilibrium unknowns. It has not addressed the mass of varieties domesticallyproduced and exported in each sector, which is also endogenously determined in equilibrium. Tocalculate the mass of varieties explicitly, I hereafter assume that firm productivity ϕ follows a Paretodistribution:

G(ϕ) = 1−(

ϕmin

ϕ

)k

, ϕ ≥ ϕmin > 0,

where both the shape and scale parameters, k and ϕmin, are identical across countries and sectors un-der Definition 1. The purpose of this subsection is to investigate the interactions among comparativeadvantage, country size and the extensive/intensive margins.

In what follows, I first derive the mass of potential entrants in a sector of country i, which isdenoted by M i

e. Applying the Pareto distribution to the aggregate price in each country P i, themass of potential entrants is expressed as

M ie =

1σ

(wi

µi

)σ−1 Xi

Bi1

(ϕjd)k−σ+1

− Xj

Bj1

(ϕjx)k−σ+1

∆, 17

where implicit solutions for the eight unknowns are already given in the preceding analysis. Notethat if two-way trade is to occur in any sector as suggested in Proposition 1(ii), M i

e should be strictlypositive for any range of parameters. It directly follows that M i

e > 0 if and only if

Xσ−1

k

(fd

fx

) k−σ+1k

< B < Xσ−1

k

(fx

fd

) k−σ+1k

, (2)

17 ∆ =“

kϕkmin

k−σ+1

”»“1

ϕNd

ϕSd

”k−σ+1

−“

1ϕN

x ϕSx

”k−σ+1–, which is positive under condition (1) and k − σ + 1 > 0.

22

where X = XN/XS = ωL. Although the restriction on B in (2) has different implications fromthat in (1) – i.e. condition (1) by which ϕi

x > ϕid versus condition (2) by which M i

e > 0, the rangeof (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 isextremely strong. To perform a consistent analysis (especially for the LMC/BOP conditions), Ihenceforth assume that the fixed export cost is sufficiently large to satisfy (2) in any sector, but themain analysis would essentially go through even if two types of trade coexist in the model.

I now examine the effects of comparative advantage and country size on the extensive and inten-sive margins. Let aggregate sector exports Ri

x decompose into

Rix = M i

e

∫ ∞

ϕix

rix(ϕ)dG(ϕ)

= [1−G(ϕix)]M i

e ×1

[1−G(ϕix)]

∫ ∞

ϕix

rix(ϕ)dG(ϕ)

= M iex × ri

x,

where M iex is the mass of firms that actually engage in exporting (extensive margin), and ri

x isaverage exports across heterogeneous firms (intensive margin). Similarly, aggregate domestic salesare decomposed into Ri

d = M ied × ri

d. Under the Pareto distribution, these margins are expressed as

M ied =

(ϕmin

ϕid

)k

M ie, M i

ex =(

ϕmin

ϕix

)k

M ie,

rid =

kσ

k − σ + 1wifd, ri

x =kσ

k − σ + 1wifx.

It is obvious that the intensive margins (rid, r

ix) are independent of sector index z and thus comparative

advantage; due to log-supermodularity, however, aggregate domestic sales and exports (Rid, R

ix) are

both increasing in it, suggesting that comparative advantage increases these aggregates solely throughthe extensive margins (M i

ed,Miex). In other words, comparative advantage has two opposing effects

on the intensive margins. First, stronger comparative advantage allows all firms to be relativelybetter at producing and makes each firm’s revenue (ri

d, rix) higher, which increases the intensive

margins. Second, stronger comparative advantage allows less productive firms to enter into domesticand export markets and makes the productivity cutoffs (ϕi

d, ϕix) smaller as seen in Figure 2, which

decreases the intensive margins. 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 therelative terms, rather than the absolute terms. From the equations of the absolute margins, therelative extensive and intensive margins in North are given by

Med =MN

ed

MSed

=Me

(ϕd)k, Mex =

MNex

MSex

=Me

(ϕx)k,

rNd

rSd

=rNx

rSx

= ω > 1,

23

exedMM ,

0

1

z

1 z

edM

exM

XF

XF 1

dz

xz

Figure 7 – Relative extensive margins of specialization and trade

where Me = MNe /MS

e . Hence, regardless of country size, the intensive margins are always higherin North reflecting the absolute advantage assumption. It also follows that the relative extensivemargins of specialization and trade are increasing in L due to the home-market effect (MN

ed/MSed >

LN/LS , MNex/MS

ex > LN/LS) whereas the relative intensive margins are decreasing in it due to thelower relative wage (see Figure 6).18 This effect of country size on the two margins should be oppositein order to restore the trade balance.

To make this point clear, Figure 7 depicts Med and Mex curves in (z, M) space. Using M ie

derived above, these two curves are shown to be downward-sloping with M ′ex(z) < M ′

ed(z) < 0 forz ∈ [0, 1] and their intersection is at

(z, XF−1

ω(F−X)

)where F = (fx/fd)(k−σ+1)/(σ−1). Let zd and zx

respectively denote the cutoffs at which Med(zd) = 1 and Mex(zx) = 1 with zd < zx < z. Thesezd and zx are the cutoffs below which North produces and exports relatively more varieties thanSouth (extensive margin), while z is the cutoff below which North is a net exporter of customizedproducts, i.e., RN

x − RSx = MN

exrNx − MS

exrSx > 0 for z ∈ [0, z). The relationship between the

extensive/intensive margins of trade and aggregate sector exports across sectors are summarized inTable 1. It is important to note that the vertical intersection of the two curves in Figure 7 is smallerthan unity for any L: indeed, if XF−1

ω(F−X) > 1, not only is the intensive margin of trade rix but the

extensive margin of trade M iex is also higher for North in some comparative advantage sectors of

South z ∈ (z, 1], contradicting the previous argument that trade is balanced in the cutoff sector z.Simple inspection also reveals that the vertical intersection between the two curves is increasing inL, and thus this intersection becomes closer to (zx, 1) with the increase in L, whereby the extensivemargin of trade is higher while the intensive margin of trade is lower as claimed above. (This logicalso applies for the extensive/intensive margins of specialization.)

18In contrast to Krugman (1980) who emphasizes the role of the variable trade cost in the home-market effect, thefixed trade cost plays a qualitatively similar role in the current paper. The separate supplementary note develops amore general model in which firms incur both variable and fixed trade costs, and shows that the extensive and intensivemargins are exactly the same as the above even in such a model.

24

Table 1 – Margins of trade and aggregate sector exports

Sector Extensive margin Intensive margin Aggregate sector exports

z ∈ [0, zx) MNx > MS

x rNx > rS

x MNx rN

x > MSx rS

x

z ∈ (zx, z) MNx < MS

x rNx > rS

x MNx rN

x > MSx rS

x

z ∈ (z, 1] MNx < MS

x rNx > rS

x MNx rN

x < MSx rS

x

Regarding the relative intensive margin of specialization and trade in each country,

rNx

rNd

=rSx

rSd

=fx

fd> 1,

suggesting that the intensive margin of exporters is higher than that of nonexporters in any sector ofthe two countries because exporters are on average more productive and earn higher average revenuethan nonexporters. On the other hand, the relative extensive margin of specialization and trade ineach country is given by

M iex

M ied

=(

ϕid

ϕix

)k

=

(1B

fdfx

) kσ−1 if i = N,

(B fd

fx

) kσ−1 if i = S,

which is between zero and unity under condition (1) or (2). Comparative statics on this equationreveal that M i

ex/M ied is increasing in comparative advantage strength (1/B for North and B for South)

and the 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 behindthe last result is as follows. While an increase in L increases the extensive margins in North dueto the home-market effect, it decreases the Ricardian productivity advantage of North through thelower relative wage ω, which decreases the intensive margins there. This decrease in the intensivemargins is bigger for rN

x than rNd (∂rN

x /∂L

∂rNd /∂L

= 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 restorethe trade balance, an increase in the extensive margins is bigger for MN

ex than MNed and thus MN

ex/MNed

is increasing in L. Noting that an increase in L works oppositely for the extensive and intensivemargins in South, this intuition also explains why MS

ex/MSed is decreasing in L.

Proposition 4

(i) The volume of domestic sales and exports increases with comparative advantage strength solelythrough the extensive margins.

25

(ii) The export participation ratio is increasing (decreasing) in the relative country size for North(South).

5 Discussions

In this section, I briefly argue two extensions of the basic model: variable trade costs and multina-tional firms. The separate supplementary note offers a more detailed analysis that incorporates thevariable trade cost.

Transport Costs The model has assumed that only the fixed trade cost fx exists and firms canexport without incurring the variable trade cost τ . If this cost is incorporated into the previoussetting, the profit functions in Northern market are given by

πNd = BN

(µN (z)wN

)σ−1

ϕσ−1 − wNfd, πSx = BN

(µS(z)τwS

)σ−1

ϕσ−1 − wSfx,

and πSx is steeper (flatter) than πN

d if and only if

µS(z)τwS

R µN (z)wN

⇐⇒ ω R τµ(z).

Similarly, from the profit functions in Southern market, πNx is steeper (flatter) than πS

d if and only if

µN (z)τwN

R µS(z)wS

⇐⇒ µ(z) R τω.

In the presence of variable trade cost τ , the cutoff sector z is no longer identical between North andSouth. Instead, there exist two distinct cutoff sectors, namely zN and zS , such that North (South)has an institutional comparative advantage in z ∈ [0, zN ) (z ∈ (zS , 1]), where zN = µ−1(τω) andzS = 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 inz ∈ [zN , zS ] in the current model, i.e., two-way trade can occur in any sector.19 Moreover, the general-equilibrium approach is not qualitatively affected by τ because: (i) changes in z still shift ϕi

d and ϕix in

opposite directions, implying that the relative market demand B = BN/BS increases with z; (ii) therelative productivity cutoffs ϕd = ϕN

d /ϕSd and ϕx = ϕN

x /ϕSx 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’swage 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 suggestthat, even in the presence of variable trade cost τ , there should arise a general-equilibrium interplaybetween good and factor markets that is similar to Figure 6. Therefore, although each absolute termof the eight unknowns is affected by τ , the key characterizations represented in the relative terms

19Just as (2) is required for two-way trade to occur in any sector, so (2’) below is needed in this extension.

26

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 ϕix/ϕi

d in this setting for each countryare respectively given by

ϕNx

ϕNd

= τ

(B

fx

fd

) 1σ−1

,ϕS

x

ϕSd

= τ

(1B

fx

fd

) 1σ−1

,

the usual outcome (ϕix > ϕi

d) occurs in any sector of both countries if

(1τ

) 1σ−1 fd

fx< B < τ

1σ−1

fx

fd. (1’)

On the other hand, two-way trade (M iex > 0) occurs in any sector of both countries if

Xσ−1

k

(1τ

)σ−1 (fd

fx

) k−σ+1k

< B < Xσ−1

k τσ−1

(fx

fd

) k−σ+1k

. (2’)

Evidently, (1’) and (2’) converge to (1) and (2) as τ → 1. While (1) certainly subsumes (2) withoutτ , whether the range of (1’) is greater than the range of (2’) depends on trade cost parameters. (Itis 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 forserving a foreign market. In the real world, firm sales by multinational enterprises have been growingfaster than exporting, and foreign direct investment (FDI) plays a key role in a country where themarket system is less developed.20 As initially raised by Williamson (1985), this issue is of particularimportance when contracts are better enforced within the firm boundaries. In the internationalcontext, 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 internalprincipal-agent relationship. It also implies that the firm’s choice between intra-firm and arm’s-length trade is affected by contractibility. In fact, Bernard, Jensen, Redding, and Schott (2010) findempirical evidence suggesting that the intra-firm fraction of the U.S. imports is significantly higherfor products for which contractibility is more difficult. Because North has a comparative advantagein contract-dependent sectors in the model, it is worth investigating the role played by multinationalfirms in the presence of partial contractibility.

To analyze export versus FDI in the previous environment, suppose that

1 <µN (z)µM (z)

<µN (z′)µM (z′)

< ∞, 1 <µM (z)µS(z)

<µM (z′)µS(z′)

< ∞,

20For simplicity, I have confined attention to the case where imperfect institutions are related to the production sideonly. As surveyed by Dixit (2011), these frictions are also important in other aspects, such as the distribution channelof exported products in foreign countries, and multinationals can alleviate this problem by internalization.

27

for z > z′, µN > µM > µS , µM (z) 6= 0, µM (z′) 6= 0, µS(z) 6= 0 and µS(z′) 6= 0. µM denotes partialcontractibility 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 contractenforcement within the firm boundaries than Southern domestic firms, but their contractibility islower than Northern domestic firms because they have to be at least partially affected by Southerngovernance structure, such as courts of law and social norms (Dixit, 2011): Northern (Southern) firmsreceive negative (positive) feedback from becoming multinationals on contractibility. At the sametime, Northern multinationals are able to exploit wage differentials, while Southern multinationalshave to pay the higher wage. This creates a new tradeoff between wage and contractibility that hasbeen missing in the export-versus-FDI literature. As a result, the profit functions of Northern andSouthern multinationals,

πNm = BS

(µM (z)

wS

)σ−1

ϕσ−1 − wSfm, πSm = BN

(µM (z)wN

)σ−1

ϕσ−1 − wNfm,

are respectively steeper (flatter) than those of Northern and Southern exporters, πNx and πS

x , if andonly if

µM (z)wS

R µN (z)wN

⇐⇒ ω R µ(z),µM (z)wN

R µS(z)wS

⇐⇒ µ(z) R ω,

where µ(z) = µN (z)/µM (z) and µ(z) = µM (z)/µS(z). Thus, FDI undertaken by Northern (South-ern) firms is more likely to emerge in equilibrium if and only if endogenous wage differentials aresufficiently large (small) relative to exogenous contractibility differentials.

Under these circumstances, how does the existence of multinational firms affect the patterns ofspecialization and trade? Is there any systematic relationship between the relative export/FDI flowsand countries’ comparative advantage? If so, what impacts does it have on wage inequality betweenNorth and South? Although these questions have vital implications for consequences of globalization,I leave this extension for my future work.

28

Appendix

A.1 Proof of the Market Demand

I show that the relative market demand B = BN/BS increases with z. Taking the logarithm of theZCP conditions and differentiating them with respect to z gives

BN ′

BN+ (σ − 1)

µN ′

µN+ (σ − 1)

ϕN ′d

ϕNd

= 0, (A.1)

BS′

BS+ (σ − 1)

µS′

µS+ (σ − 1)

ϕS′d

ϕSd

= 0, (A.2)

BS′

BS+ (σ − 1)

µN ′

µN+ (σ − 1)

ϕN ′x

ϕNx

= 0, (A.3)

BN ′

BN+ (σ − 1)

µS′

µS+ (σ − 1)

ϕS′x

ϕSx

= 0. (A.4)

Further, differentiating the FE condition with respect to z and rearranging,

ϕN ′x = −CN ϕN ′

d , (A.5)

ϕS′x = −CSϕS′

d , (A.6)

where Ci = fdJ′(ϕi

d)/fxJ ′(ϕix) > 0. Note that (A.1)–(A.6) are six equations with six unknowns

(ϕN ′d , ϕS′

d , ϕN ′x , ϕS′

x , BN ′, BS′). Substituting (A.5) and (A.6) respectively into (A.3) and (A.4), and

subtracting (A.2) and (A.1) respectively from these yields

CN ϕN ′d

ϕNx

+ϕS′

d

ϕSd

=µ′

µ, − ϕN ′

d

ϕNd

− CSϕS′d

ϕSx

=µ′

µ,

where µ′/µ = µN ′/µN − µS′/µS < 0. These are two equations with two unknowns (ϕN ′

d , ϕS′d ), which

can be solved for

ϕN ′d =

µ′µ

(1

ϕSd

+ CS

ϕSx

)

Ξ, ϕS′

d = −µ′µ

(1

ϕNd

+ CN

ϕNx

)

Ξ,

where

Ξ =1

ϕNd ϕS

d

(ϕN

d ϕSd

ϕNx ϕS

x

CNCS − 1)

.

From the ZCP conditions and Ci defined above, Ξ is positive if and only if

J ′(ϕNd )J ′(ϕS

d )J ′(ϕN

x )J ′(ϕSx )

>

(fx

fd

) 2σσ−1

.

(Assuming a Pareto distribution, for example, the left-hand side is (fx/fd)2(k+1)/(σ−1) and this holdsif k−σ +1 > 0). Under this condition, ϕN ′

d < 0, ϕS′d > 0 and from (A.5) and (A.6) ϕN ′

x > 0, ϕS′x < 0.

From the relative market demand B = BN/BS in the main text, these then imply that B′ > 0. ¤

29

A.2 Proofs of the LMC and BOP Conditions

A.2.1 Proof of the LMC Condition

I show that the LMC condition is simplified as

∫ 10 Ridz

wi= Li.

From the LMC condition, aggregate labor demand in a particular sector of i is given by

M ie

∫ ∞

ϕid

lid(ϕ)dG(ϕ) + M ie

∫ ∞

ϕix

lix(ϕ)dG(ϕ) + M iefe,

where the first two terms are aggregate labor demands for production and the third is aggregatelabor demand for investment by potential entrants in this sector. Using the optimal labor demandof heterogeneous firms, the former aggregate labor demands are

M ie

∫ ∞

ϕid

lid(ϕ)dG(ϕ) + M ie

∫ ∞

ϕix

lix(ϕ)dG(ϕ)

= M ie

[1−G(ϕi

d)]fd +

σ − 1σ

M ie

wi

∫ ∞

ϕid

rid(ϕ)dG(ϕ) + M i

e

[1−G(ϕi

x)]fx +

σ − 1σ

M ie

wi

∫ ∞

ϕix

rix(ϕ)dG(ϕ)

=M i

e

wi

{[1−G(ϕi

d)]wifd +

σ − 1σ

∫ ∞

ϕid

rid(ϕ)dG(ϕ) +

[1−G(ϕi

x)]wifx +

σ − 1σ

∫ ∞

ϕix

rix(ϕ)dG(ϕ)

}

=Ri −Πi

wi,

where Πi denotes aggregate sector profit. On the other hand, the latter aggregate labor demandsare

M iefe =

M ie

wi

{1σ

∫ ∞

ϕid

rid(ϕ)dG(ϕ)− [

1−G(ϕid)

]wifd +

1σ

∫ ∞

ϕix

rix(ϕ)dG(ϕ)− [

1−G(ϕix)

]wifx

}

=Πi

wi,

which is derived from the FE condition:

fe =∫ ∞

ϕid

πid(ϕ)wi

dG(ϕ) +∫ ∞

ϕix

πix(ϕ)wi

dG(ϕ)

=1wi

{∫ ∞

ϕid

rid(ϕ)σ

dG(ϕ)− [1−G(ϕi

d)]wifd +

∫ ∞

ϕix

rix(ϕ)σ

dG(ϕ)− [1−G(ϕi

x)]wifx

}.

Summing up these two kinds of aggregate labor demands gives

M ie

∫ ∞

ϕid

lid(ϕ)dG(ϕ) + M ie

∫ ∞

ϕix

lix(ϕ)dG(ϕ) + M iefe =

M ie

wi

{∫ ∞

ϕid

rid(ϕ)dG(ϕ) +

∫ ∞

ϕix

rix(ϕ)dG(ϕ)

}

=Ri

wi,

30

Finally, integrating the above aggregate sector labor demands over the interval [0,1] completes theproof. ¤

A.2.2 Proof of the BOP Condition

I first show that the BOP condition is written as∫ z

0

(wNLN

z −RNz

)dz =

∫ 1

z

(wSLS

z −RSz

)dz.

From the BOP condition (∫ 10 RN

x (z)dz =∫ 10 RS

x (z)dz) and log-supermodularity, there exists a uniquecutoff z such that

∫ z

0

[MN

e

∫ ∞

ϕNx

rNx (ϕ)dG(ϕ)−MS

e

∫ ∞

ϕSx

rSx (ϕ)dG(ϕ)

︸ ︷︷ ︸RN

x (z)−RSx (z)

]dz

=∫ 1

z

[MN

e

∫ ∞

ϕSx

rSx (ϕ)dG(ϕ)−MS

e

∫ ∞

ϕNx

rNx (ϕ)dG(ϕ)

︸ ︷︷ ︸RS

x (z)−RNx (z)

]dz.

Note that for i 6= j ∈ {N,S}, the terms in the square brackets are

Rix(z)−Rj

x(z) = M ie

Z ∞

ϕix

rix(ϕ)dG(ϕ)−M j

e

Z ∞

ϕjx

rjx(ϕ)dG(ϕ)

= M ie

Z ∞

ϕix

rix(ϕ)dG(ϕ) + M i

e

Z ∞

ϕid

rid(ϕ)dG(ϕ)

| {z }wiLi

z

−„

M ie

Z ∞

ϕid

rid(ϕ)dG(ϕ) + M j

e

Z ∞

ϕjx

rjx(ϕ)dG(ϕ)

| {z }Ri

z

«

= wiLiz −Ri

z,

which is positive (negative) if z ∈ [0, z) and negative (positive) if z ∈ (z, 1] for i = N (i = S). Theproof immediately follows from the above.

Next, I show that the above BOP condition is written as

ω ≡ ξ(z) =

∫ 1z

(LS

z

LS − λz

)dz

L[∫ z

0

(LN

z

LN − λz

)dz

] .

By manipulating the BOP condition,

∫ z

0

(wNLN

z −RNz

)dz =

∫ 1

z

(wSLS

z −RSz

)dz

⇐⇒∫ z

0

(LN

z

LN− RN

z

wNLN

)dz =

∫ 1

z

(wSLS

z

wSLS

wSLS

wNLN− RS

z

wSLS

wSLS

wNLN

)dz

⇐⇒∫ z

0

(LN

z

LN− λz

)dz =

1ωL

∫ 1

z

(LS

z

LS− λz

)dz,

31

where the second equation comes from dividing both sides of the first equation by wNLN , and thethird comes from λz = λN

z = λSz = Ri

z

wiLi . Solving the last equation for ω completes the proof.

Finally, I show that the above BOP condition satisfies

ξ′(z) > 0, limz→0

ξ(z) = 0, limz→1

ξ(z) = ∞.

Let ξ1 =∫ 1z

(LS

z

LS − λz

)dz and ξ2 =

∫ z0

(LN

z

LN − λz

)dz denote the numerator and denominator of ξ(z).

Differentiating these terms with respect to z yields

dξ1

dz=

(−LS

z

LS+ λz

)+

∫ 1

z

∂

∂z

(LS

z

LS− λz

)dz,

dξ2

dz=

(LN

z

LN− λz

)+

∫ z

0

∂

∂z

(LN

z

LN− λz

)dz.

The first term in the right-hand side captures the marginal effect of z on the volume of trade acrosssectors, which is zero because exports from North and South are by definition exactly the same insector z, i.e., LN

z

LN − λz = LSz

LS − λz = 0. The second term captures the marginal effect of z within

sectors, which is∫ 1z

∂∂z

(LS

z

LS − λz

)dz > 0 and

∫ z0

∂∂z

(LN

z

LN − λz

)dz < 0 because log-supermodularity

increases the within-sectoral volume of trade in North (South) as z is close to zero (one). Combinedwith this observation, the desired result follows from noting that both ξ1 and ξ2 are positive for anyz. ¤

A.3 Proofs of the Extensive and Intensive Margins

A.3.1 Derivation of the Extensive Margins

I first show the derivation of M ie. From the C.E.S. preferences, the aggregate price in a sector of

country i is

(P i)1−σ =∫

v∈Vp1−σ(v)dv

= M ie

∫ ∞

ϕid

(σ

σ − 1wi

ϕµi

)1−σ

dG(ϕ) + M je

∫ ∞

ϕjx

(σ

σ − 1wj

ϕµj

)1−σ

dG(ϕ),

Since Bi = (σ−1)σ−1

σσ Xi(P i)σ−1, this aggregate price is also written as (P i)1−σ = (σ−1)σ−1

σσXi

Bi . Com-bining these two relationships gives

M ie

∫ ∞

ϕid

(σ

σ − 1wi

ϕµi

)1−σ

dG(ϕ) + M je

∫ ∞

ϕjx

(σ

σ − 1wj

ϕµj

)1−σ

dG(ϕ) =(σ − 1)σ−1

σσ

Xi

Bi.

32

These two equations, ((PN )1−σ, (PS)1−σ), are the two systems with the two unknowns (MNe ,MS

e ).Rewriting these equations in a matrix form,

(σ

σ−1wN

µN

)1−σ[V (∞)− V (ϕN

d )](

σσ−1

wS

µS

)1−σ[V (∞)− V (ϕS

x )](

σσ−1

wN

µN

)1−σ[V (∞)− V (ϕN

x )](

σσ−1

wS

µS

)1−σ[V (∞)− V (ϕS

d )]

[MN

e

MSe

]=

[(σ−1)σ−1

σσXN

BN

(σ−1)σ−1

σσXS

BS

],

where V (ϕ) =∫ ϕ0 yσ−1dG(y). Applying Cramer’s rule,

M ie =

1σ

(wi

µi

)σ−1 Xi

Bi [V (∞)− V (ϕjd)]− Xj

Bj [V (∞)− V (ϕjx)]

∆,

where ∆ = [V (∞)− V (ϕNd )][V (∞)− V (ϕS

d )]− [V (∞)− V (ϕNx )][V (∞)− V (ϕS

x )] > 0. Note that thisholds for a general distribution function G(·).

Next, I derive condition (2) when G(·) is Pareto. Since V (∞) − V (ϕ) = kϕkmin

k−σ+11

ϕk−σ+1 underPareto, substituting this value into the above M i

e gives

M ie =

1σ

(wi

µi

)σ−1 Xi

Bi1

(ϕjd)k−σ+1

− Xj

Bj1

(ϕjx)k−σ+1

∆,

where ∆ =(

kϕkmin

k−σ+1

)[(1

ϕNd ϕS

d

)k−σ+1−

(1

ϕNx ϕS

x

)k−σ+1]

> 0; and k − σ + 1 > 0 comes from a finite

variance of the truncated Pareto distribution V (ϕ). Then, M ie > 0 if and only if Xi

Bi1

(ϕjd)k−σ+1

>

Xj

Bj1

(ϕjx)k−σ+1

(i 6= j) or

(ϕN

d

ϕNx

)k−σ+1

<B

X<

(ϕS

x

ϕSd

)k−σ+1

⇐⇒(

1B

fd

fx

) k−σ+1σ−1

<B

X<

(1B

fx

fd

) k−σ+1σ−1

.

Arranging this inequality gives condition (2).

Finally, I show that the range of (1) subsumes that of (2). Comparing these two conditions, theabove statement holds if and only if

fd

fx< X = ωL <

fx

fd.

Substituting ωL =

R 1z

„LS

zLS −λz

«dz

R z0

„LN

zLN −λz

«dz

(the BOP condition) into the above inequality yields

1 + fdfx

∫ z0

LNz

LN dz − ∫ 1z

LSz

LS dz

1 + fdfx

≡ Γ1 <

∫ z

0λzdz <

1 + fx

fd

∫ z0

LNz

LN dz − ∫ 1z

LSz

LS dz

1 + fx

fd

≡ Γ2.

Note that Γ1 > 1 − ∫ 1z

LSz

LS dz and Γ2 <(1 + fd

fx

) ∫ z0

LNz

LN dz because the numerator and denominator

33

of ωL are both positive and hence∫ z0

LNz

LN dz +∫ 1z

LSz

LS dz > 1. Therefore, the above inequality satisfies

1−∫ 1

z

LSz

LSdz <

∫ z

0λzdz <

∫ z

0

LNz

LNdz,

which is internally consistent with the current model where North (South) is a net exporter inz ∈ [0, z) (z ∈ (z, 1]). This observation implies that fd

fx< ωL < fx

fdand hence (1) subsumes (2). ¤

A.3.2 Derivation of the Intensive Margins under Pareto

I show that the intensive margins under Pareto are given by

rix =

kσ

k − σ + 1wifx, ri

d =kσ

k − σ + 1wifd.

By definition,

rix =

11−G(ϕi

x)

∫ ∞

ϕix

rix(ϕ)dG(ϕ)

=1

1−G(ϕix)

Bjσ

(µi

wi

)σ−1 [V (∞)− V (ϕi

x)]

(using V (ϕ))

=(

ϕix

ϕmin

)k

Bjσ

(µi

wi

)σ−1kϕk

min

k − σ + 11

(ϕix)k−σ+1

(using Pareto)

=k

k − σ + 1Bjσ

(µi

wi

)σ−1

(ϕix)σ−1

=k

k − σ + 1Bjσ

(µi

wi

)σ−1 1Bj

(µi

wi

)1−σ

wifx (using ZCP ix)

=kσ

k − σ + 1wifx.

By following the similar steps, it is easily confirmed that rid = kσ

k−σ+1wifd. ¤

A.3.3 Derivation of the Extensive Margins under Pareto

I first show that the relative extensive margins are given by

Med =Me

(ϕd)k=

1ω

X −Bk

σ−1 F−1

1−XB− kσ−1 F−1

, Mex =Me

(ϕx)k=

1ω

XB− kσ−1 F − 1

Bk

σ−1 F −X,

where X = XN/XS = ωL and F = (fx/fd)(k−σ+1)/(σ−1) > 1. From M ie under Pareto, Me =

MNe /MS

e is given by

Me =XN

BN1

(ϕNd )k−σ+1 − XS

BS1

(ϕSx )k−σ+1

XS

BS1

(ϕSd )k−σ+1 − XN

BN1

(ϕNx )k−σ+1

.

34

Using the ZCP conditions, this equation can be written as

Me =(

ω

µ

)σ−1

(ϕd)k−σ+1 X −B

kσ−1 F−1

B(1−XB− kσ−1 F−1)

=(

ω

µ

)σ−1

(ϕx)k−σ+1 (XB− kσ−1 F − 1)B

Bk

σ−1 F −X

Dividing the first and second equalities respectively by (ϕd)k and (ϕx)k,

Med =(

ω

µ

1ϕd

)σ−1

︸ ︷︷ ︸Bω

X −Bk

σ−1 F−1

B(1−XB− kσ−1 F−1)

, Mex =(

ω

µ

1ϕx

)σ−1

︸ ︷︷ ︸1

Bω

(XB− kσ−1 F − 1)B

Bk

σ−1 F −X,

where the values in the underbraces come from the RZCP conditions. Arranging these equationsgives the result.

Next, I show that these masses increase more than proportionally to an increase in countrysize, i.e. MN

ed/MSed > LN/LS and MN

ex/MSex > LN/LS . Using Med derived above, it suffices for

MNed/MS

ed > LN/LS to see that the following inequality holds:

Med > L =X

ω⇐⇒ B < X

σ−1k ,

which is true under condition (2). Similarly, for MNex/MS

ex > LN/LS ,

Mex > L ⇐⇒ B <

((X2 − 1) +

√(X2 − 1)2 + (2XF )2

2XF

)σ−1k

≡ Φ.

Since Φ < X(σ−1)/k(fx/fd)(k−σ+1)/k, this inequality is also true under condition (2).

Finally, I show that M ′ex(z) < M ′

ed(z) < 0 for z ∈ [0, 1] and their intersection is at(z, XF−1

ω(F−X)

)

in Figure 7. The first relationship holds from simple inspection of the above expressions of Mex andMed, because only B is a function of z with B′(z) > 0 and k−σ+1 > 0. Regarding the intersection ofthe two curves, it follows from noting that ϕx(z) = ϕd(z) = ω1/(σ−1) and Me(z) = ω

k−σ+1σ−1 XF−1

F−X . ¤

35

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