Diversity and O¤shoring�

Hiroshi Gotoy, Yan Maz, and Nobuyuki Takeuchix

September 6, 2019

Abstract

How does the distribution of human capital a¤ect o¤shoring? To study this question,we extend Grossman and Maggi (2000) to allow o¤shoring, which means that workersin di¤erent countries can collaborate in teams. We �rst examine how changes in theaverage skill level and skill diversity a¤ect matching patterns and wage schedules underautarky. Next, we investigate the e¤ects of o¤shoring and compare the equilibria undero¤shoring with its corresponding equilibria under free trade. We demonstrate that if twocountries di¤er in skill diversity but share the same average skill level, the equilibrium undero¤shoring is the same as the equilibrium under free trade. The wages of workers with thesame skill level are equalized under free trade, and thus, there is no incentive for o¤shoringto occur. However, if two countries only di¤er in the average skill, the equilibrium undero¤shoring is di¤erent from the equilibrium under free trade. The wages of workers with thesame skill level under free trade are not equalized across countries, and thus o¤shoring canoccur. Furthermore, we demonstrate that o¤shoring and free trade have di¤erent e¤ectson the welfare of the workers with the highest and lowest skill levels.

1 Introduction

In recent years, production processes have increasingly involved multiple countries, with eachcountry specializing in certain tasks.1 This phenomenon of o¤shoring has attracted consid-erable attention from policy makers and economists. Our goal in this paper is to investigatehow the distribution of human capital a¤ects o¤shoring and income distribution. To do so, weintroduce o¤shoring into the seminal study by Grossman and Maggi (2000), which highlights

�Yan Ma is grateful to �nancial support from JSPS KAKENHI Grant (No. 16H03617). We are indebtedto co-editor Araud Costinot for his insightful suggestions. Valuable comments from Eric Bond, Pao-li Chang,Fumio Dei, Taiji Furusawa, Gene Grossman, Jota Ishikawa, Konstantin Kucheryavyy, Yi Lu, Joel Rodrigue,Koji Shintaku, Vladimir Tyazhelnikov, and Yoichi Sugita are greatly appreciated.

yFaculty of Commerce and Economics, Chiba University of Commerce, Japan; E-mail: hgoto@cuc.ac.jp;zGraduate School of Business Administration, Kobe University, Japan; E-mail: mayan003@kobe-u.ac.jp;

phone: (+81) 078-803-6980.xFaculty of Economics, University of Marketing and Distribution Sciences; E-mail: Nobuyuki_Takeuchi @

red.umds.ac.jp1This phenomenon is referred to as fragmentation in Jones and Kierzkowski (1990), trade in tasks in Gross-

man and Rossi-Hansberg (2008), and vertical specialization in Hummels, Ishii, and Yi (2001).

1

how skill diversity a¤ects the pattern of trade. This question is important because countriesdi¤er in the distribution of skills, as found by Bombardini, Gallipoli, and Pupato (2012).2

To investigate how the distribution of skills a¤ects the possibility of o¤shoring, we �rstanalyze how changes in skill diversity and the average skill level a¤ect the relative price,matching patterns and wage schedules under autarky. Following Grossman and Maggi (2000),density function is symmetric and there are two sectors: The technology of one sector exhibitssupermodularity, i.e, a technology that requires complementarity between tasks; the technologyof the other sector exhibits submodularity, i.e, tasks are substitutable. We demonstrate thatan increase in skill diversity increases the relative price of the supermodular good becausethe productivity of the submodular sector increases due to the increase in skill diversity, andthus, the relative supply of supermodular good decline. Moreover, an increase in skill diversityincreases the wages of high-skilled workers while decreases wages of low-skilled workers dueto two e¤ects of an increase in skill diversity on wages. One is matching e¤ects, which meansworkers change their team partners and sectors to work; and the other is the price e¤ect, whichmeans in the numerare sector, wages of workers with extreme skills go in opposite directions. Inaddition, we demonstrate that an increase in the average skill level decreases the relative priceof supermodular sector and increases the low-skilled workers because the low-skilled workersbecome relatively scarce.

As in Antras, Garricano and Rossi-Hansberg (2006), o¤shoring occurs when workers indi¤erent countries collaborate in teams. If two countries have the same average skill level butdi¤erent skill diversity, as in Grossman and Maggi (2000), o¤shoring will not occur becausewages are equalized under free trade without o¤shoring. In other words, the free-trade equi-librium is the same as the equilibrium under o¤shoring. However, if two countries have thesame skill diversity but di¤erent average skill levels, the wages of workers with the same skilllevel are not equalized and o¤shoring can occur. Even in the case that two countries havethe same coe¢ cient of variation, wage are not equalized in the equilibrium, which features notrade in �nal goods. However, o¤shoring can occur.

We also separately examine the e¤ects of trade and o¤shoring on the income distribution.We demonstrate that free trade without o¤shoring and o¤shoring have di¤erent e¤ects onworkers�welfare in the submodular sector, in which workers with the highest and lowest skillscollaborate. Trade a¤ects the welfare of workers in the same direction: All gain from trade,or all lose from trade. In the free-trade equilibrium, the country with the higher average skillexports the good produced in the supermodular sector, because the country with a higheraverage skill level has a higher relative supply of supermodular good. However, o¤shoringa¤ects the welfare of workers with the highest skills and that of workers with the lowest skillsin opposite directions. Intuitively, workers with the highest skills in the country with thehigher average skill level have the opportunity to form teams with better partners in theforeign country, and thus, they gain from o¤shoring. It follows that workers who are replaced

2See Figure 1 in Bombardini, Gallipoli, and Pupato (2012). They use scores on the International Adult Lit-eracy Survey (IALS), an internationally comparable measure of work-related skills, to document the di¤erencesin the mean and standard deviation of skills among 19 countries during the period 1994�1998. Their resultssupport that skill dispersion a¤ects the pattern of trade, as predicted by Grosman and Maggi (2000).

2

by foreign workers with lower skills lose. These results are consistent with the �ndings of arecent study by Hummels, Jorgensen, Munch, and Xiang (2014), who estimate how o¤shoringand exporting a¤ect wages by skill type using matched worker-�rm data from Denmark.3

O¤shoring a¤ects workers in the foreign country in opposite directions, as workers with thelowest skills �nd better partners, and workers with the highest skills lose their best partners.

Our paper is related to two strands of the trade literature. One is the literature on o¤-shoring, which is large and diverse.4 Several papers, mostly notably Feenstra and Hanson(1996), Yi (2003), and Grossman and Rossi-Hansberg (2008), have investigated the e¤ects ofo¤shoring on trade volumes, trade patterns and the income distribution. Costinot, Vogel, andWang (2013) highlight how global or local technological changes a¤ect countries participatingin the same global supply chain. Antras and Chor (2013) emphasize the optimal allocation ofownership rights along the global value chain. Baldwin and Venables (2013) reveal the impli-cations of production processes for o¤shoring. Our paper is closely related to, among others,Antras, Garricano and Rossi-Hansberg (2006) and Costinot and Vogel (2010). Antras, Garri-cano and Rossi-Hansberg (2006) study the impact of the formation of cross-country teams withone manager and several workers on the organization of production and wages. They assume auniform distribution of skills in both the North and the South, and two countries always sharethe same coe¢ cient of variation in their paper. In contrast, our paper investigates how theaverage skill level and skill diversity a¤ect o¤shoring with a symmetric distribution functionand we allow the di¤erence in the coe¢ cient of variation across countries. Costinot and Vogel(2010) develop a one-good model with heterogenous workers who di¤er in skill levels. Theproduction of the �nal good requires many tasks di¤ering in skill intensity. Worker�s produc-tivity is assumed to be log supermodular in task intensity and skill, and thus their matchingis positive assortative matching between workers and tasks. The key feature of their modelis that they assume away spillovers between workers and factor prices are always equalizedacross countries. Our paper considers two types of production technologies, which lead topositive assortative matching in the supermodular sector and negative assortative matchingin the submodular sector. In addition, our matching features matching workers with workersto sectors.

Moreover, our paper is related to a growing literature that uses matching and assign-ment models in an international context, for example, Grossman and Maggi (2000), Nockeand Yeaple (2008), Costinot (2009), and Ngienthi, Ma, and Dei (2011). Our article is closelyconnected with, among others, studies on talent (human capital) and trade pioneered by Gross-man and Maggi (2000), who analyze how the skill distribution a¤ects countries�comparativeadvantages and trade patterns. Research in this area includes Grossman (2004), Bougheasand Riezman (2007), Ohnsorge and Tre�er (2007), Bombardini, Giovanni, and Germán (2012,2014), and Chang and Huang (2014). None of these articles, however, consider o¤shoring.Ngienthi, Ma, and Dei (2011) focus on o¤shoring among countries at di¤erent stages of devel-

3Denmark has a relatively higher average skill level than most of the countries in Figure 1 of Bombardini,Gallipoli, and Pupato (2012). Thus, our results are consistent with the following �ndings in Hummels et al.(2014, p. 1). That is, within job spells, (1) o¤shoring tends to increase the high-skilled wage and decrease thelow-skilled wage; (2) exporting tends to increase the wages of all skill types.

4See Antras and Rossi-Hansberg (2009) for a review.

3

opment. In contrast, our paper studies how the distribution of human capital a¤ects o¤shoring.Our paper is organized as follows. Section 2 describes the basic set-up and the equilibrium

under autarky. In section 3, we investigate how changes in skill diversity and the average skilllevel a¤ect the relative price of supermodular good, matching patterns and wage schedulesunder autarky. We examine the e¤ects of o¤shoring and compare the equilibria under o¤shoringwith its corresponding equilibria under free trade in section 4. In section 5, we considerhow o¤shoring and free trade a¤ects the income distribution. Section 6 provides concludingremarks.

2 The Closed Economy

In this section, we introduce the setup and some results in Grossman and Maggi (2000) (here-after, GM (2000)). There is a continuum of workers who di¤er in their skill levels, t. Theskill distribution is represented by a cumulative distribution function �(t), t 2 [tmin; tmax],where tmin and tmax denote the minimum and maximum skill levels. Let L and �t represent themeasure of labor force and the average skill level, respectively.

There are two sectors: One is sector C, where C represents �complementarity�, for example,the car industry. The other is sector S, where S represents �substitutability�, for example,the software industry. Production in each sector requires a team of two workers. Output bya pair of workers in sector i (i = C;S) is F i(t1; t2), where tj (j = 1; 2) represents the skilllevel of the worker performing task j. F i is assumed to be monotonically increasing in tj ,symmetric, and to exhibit constant returns to skills. In sector C, a pair of workers performcomplementary tasks, and the production function exhibits supermodularity such that thecross derivative FC12 > 0; i.e, the marginal product of an individual�s skill is greater the moreable his co-worker is. In sector S, the workers exert e¤ort on substitutable tasks, and theproduction function exhibits submodularity such that FS12 < 0; i.e., the marginal product ofskills is higher the less capable his partner is. Given constant returns to skills, the productionfunctions in sectors C and S imply decreasing returns to �individual�skill (FCjj < 0) in sectorC, but increasing returns to individual skill (FSjj > 0) in sector S. All markets are perfectlycompetitive. Since the two sectors di¤er in the nature of the production technology, theindustry output of sector C and that of sector S are maximized through self-matching andcross-matching of workers�skill levels, respectively.

Lemma 1 (GM (2000), Lemmas 1 and 3) For any given output YS of good S, the output YCof good C is maximized by 1) allocating all workers with skill levels t � t̂ and all workers withskill levels m(t̂) � t to sector S, where m(t) is de�ned implicitly by �[m(t)] = 1� �(t) and t̂solves YS = L

R t̂tmin

FS (t;m(t)) d�(t); (2) allocating the remaining workers with t 2 [t̂; m(t̂)]to sector C such that t1 = t2 in all teams, i.e., YC =

�CL2

Rm(t̂)t̂

td�(t), where �C = FC(1; 1).5

5We borrow the approach of summarizing Lemmas 1 and 3 in GM (2000) from Chang and Huang (2014).

4

Figure 1: Matching Rules

Following GM (2000), we assume that the density function � � d�=dt is symmetric aboutits mean, �t, which implies that m(t) = 2�t� t. Figure 1 shows the matching rules described byLemma 1: workers with skill t 2 (t̂; m(t̂)) will match workers with the same skill levels intoC sector, while workers with skill t 2 [tmin; t̂] will match workers with skill t 2 [m(tmin);m(t̂)]into S sector.6 Thus, matching in sector C presents the positive assortative matching whilematching in sector S presents the negative assortative matching.

In a competitive equilibrium, all �rms maximize their pro�ts and all markets clear. Asstated in Lemma 1, workers with skills between t̂ and m(t̂) are allocated to sector C and otherworkers are cross-matched into S sector. Thus, the outputs of goods in two sectors as follows:

YC =L

2�C�t

Z 2�t�t̂

t̂�(t)dt; (1)

YS = L

Z t̂

tmin

FS (t; 2�t� t)�(t)dt: (2)

Following GM (2000), we assume that the preference is homothetic. Thus, the ratio of quan-tities consumed of good S (XS) and good C (XC) depends only on the relative price of goodC as in equation (3)

XSXC

= f(p); (3)

where p = pC=pS denote the relative price of good C. Let MRT � � dYS=dt̂

dYC=dt̂denote the

marginal rate of transformation (MRT) of the production possibility frontier (PPF), and we

6Note that m(tmin) = tmax:

5

have

MRT =FS�t̂; 2�t� t̂

��C�t

: (4)

Clearly, MRT depends on the average skill level �t and the least skilled level in sector C, i.e., t̂.In the competitive equilibrium, the relative price p is equal to MRT. Thus, the relative pricep and t̂ are determined as follows:

p =FS�t̂; 2�t� t̂

��C�t

; (5)

YSYC

= f(p): (6)

Since each member of a matched pair contributes equally to output in sector C, the mem-bers of a team are paid the same. Since there are constant returns to scale in both sectors andwages paid to a team are equal to the revenue it earn, we have

w(t) =p�Ct

2, for t 2 [t̂; m(t̂)]. (7)

Firms in sector S also earn zero pro�ts in the competitive equilibrium. Thus, we have

w(t) + w[(m(t)] = FS [t;m(t)], for t 2 [tmin; t̂]. (8)

Pro�t maximization implies that FS1 [s;m(s)] = w0(s) for all s � t̂.7 Moreover, equation (7)yields the wage for the marginal worker in sector S with skill t̂. For workers with less skillthan t̂, we have

w(t) =p�C t̂

2�Z t̂

tFS1 (�; 2�t� �) d� , for t 2 [tmin; t̂): (9)

The wages for workers with skills at the upper end of the distribution can be obtained asfollows by combining (8) and (9):

w(t) = FS (t; 2�t� t)� p�C t̂2

+

Z t̂

2�t�tFS1 (�; 2�t� �) d� , for t 2 (2�t� t̂; tmax]: (10)

Thus, a competitive equilibrium is a set of equations of (1), (2), (5), (6), (7), (9) and (10). Weprove in Appendix A that a unique equilibrium exists.

3 Comparative Statics in the Closed Economy

In this section, we investigate how exogenous changes in skill diversity and the average skilllevel a¤ect the relative price of good C, the matching rules that are captured by t̂ and m(t̂),and the wage schedule. Such changes in the distribution of skill may capture the e¤ects ofeducational reforms.

7See details in GM (2000, page 1266).

6

Figure 2: Skill Diversity and the Relative Price

3.1 A Change in Skill Diversity

We begin with examining how an increase in skill diversity with the average skill level �xeda¤ects the relative price of good C, p, and the matching rules, t̂ and m(t̂). Our de�nitionof skill diversity follows GM (2000). That is, the distribution of skill � is more diverse than�� if �(t) > ��(t) for t < t

0and �(t) < ��(t) for t > t

0, for some t

0> tmin. This de�nition

of diversity captures the idea that there are relatively more workers with extreme skill levels(either high or low) under � than under ��.

An increase in skill diversity relatively increases the supply of workers with extreme skills,whom are assigned to S sector. Thus, the relative supply of good S increases with t̂ �xed,which leads to a rise in the relative price of good C, as shown in Figure 1.8 This in turnenlarges sector C and shrinks sector S. As a result, t̂ decreases while m(t̂) increases due tothe symmetry of density function. We prove in Appendix B that the relative price of goodC in the equilibrium increases. It follows that the equilibrium relative supply of good S, YSYC ,

increase, which implies that the direct e¤ect of in increase in skill diversity on YSYCdominates

the indirect e¤ect brought about by a fall in t̂. We can see from Figure 2 that an increase inskill diversity leads to an enlargement of sector C as t̂ decreases and m(t̂) increases. Hence,we have Lemma 2.

Lemma 2 An increase in skill diversity leads to an increase in the relative price of the goodproduced in supermodular sector, p. In addition, t̂ decreases while m(t̂) increases.

Next, we turn to the e¤ects of an increase in skill diversity on the wage schedule as shownin Figure 3. For workers in sector C before and after skill diversity increases, their nominal

8PPF with a higher skill diversity can be obtained as in GM (2000).

7

Figure 3: Skill Diversity and Wage Schedules

wages increase in proportion to the increase in p due to (7). For workers in sector S, thereare two e¤ects of an increase in skill diversity on their wages: one is matching e¤ect andthe other is the price e¤ect. Matching e¤ect means that the increase in p due to an increasein skill diversity leads to a fall in t̂ and an increase in m(t̂), which in turn leads to workerschanging their team partners and sectors, for example, workers with t 2 [t̂; t̂�] and thosewith t 2 [m(t̂);m�(t̂�)]. The price e¤ect means that wages of low-skilled workers and wagesof high-skilled workers go in opposite directions, which comes from no change in the price ofthe numeraire good. Since the technology of submodularity implies increasing returns to skill,wages of high-skilled workers increase at the cost of low-skilled workers (see Appendix C forproof).

3.2 A Change in the Average Skill Level

We investigate how an increase in the average skill level with skill diversity constant a¤ects therelative price, p, and matching patterns re�ected by t̂ and m(t̂). We assume that the averageskill level �t increases from �t�, i.e., �t = ��t�, where � > 1. Thus, an increase in � re�ects a risein the average skill level. Keeping skill diversity constant means that the new density function�(t) and the original density function ��(t) satis�es �(t) = ��[t� (� � 1)�t�] for t 2 [tmin; tmax]and I = tmax � tmin = t�max � t�min.

An increase in the average skill level with skill diversity kept constant raises the outputsof both goods, and thus the PPF is on the outside of the original one. From (1), we know ifall workers are inputted into sector C, the output will increase � times than before. However,if all workers are assigned to sector S, the output will increase less than � times than before.Note that if the skill diversity increases � times, the output caused by assigning all workers toS sector will increase � times. As shown by Figure 4, an increase in the average skill leads to afall in the relative price of C, p. Intuitively, an increase in the average skill level relatively raises

8

Figure 4: Average Skill Level and the Relative Price

the productivity of sector C, and thus the relative price of good C declines. It follows thatsector C will shrink and sector S will expand. Therefore, there are two e¤ects of an increasein the average skill level on both t̂ and m(t̂). One e¤ect is the direct e¤ect of an increase inthe average skill level, which increases both t̂ and m(t̂). The other e¤ect is the indirect e¤ectthrough the fall in the relative price p, which increases t̂ but decreases m(t̂). We demonstratein Appendix D that in the equilibrium t̂ increases and under an uniform distribution functionof skills, m(t̂) increases in the equilibrium, which implies that the direct e¤ect dominates theindirect e¤ect. 9

Lemma 3 An increase in the average skill level leads to a decrease in the relative price of thegood that is produced in the supermodular sector, p. Moreover, t̂ increases due to an increasein the average skill level. Under an uniform distribution function, an increase in the averageskill level leads to an increase in m(t̂).

The e¤ects of an increase in the average skill level on wages of workers are shown in Figure5 . Intuitively, an increase in the average skill provides more relatively skilled workers and lessunskilled workers. As a result, the wages of workers with higher skill levels go down while thewages of workers with lower skill levels go up.

4 The World Economy

In the remainder of this paper we consider a world economy comprising two countries, Home(H) and Foreign (F ). Workers are internationally immobile and it is assumed that two coun-

9With a general density function, it is di¢ cult to pin down the relative change of t̂ to �t. However, thesimulations show that m(t̂) increase in the equilibrium.

9

Figure 5: The Average Skill Level and Wage Schedules

tries share the same preference. In each country we assume that production is as described insection 2. Let �i(t) and Li, i = H;F , represent the density function of skills and the laborsize in country i, respectively. We �rstly examine the e¤ects of o¤shoring between two coun-tries that either di¤er in skill diversity or di¤er in the average skill level. Following Antras,Garricano and Rossi-Hansberg (2006), o¤shoring means that workers in di¤erent countries cancollaborate in teams. For simplicity, we assume that there are no o¤shoring costs. In eithercase, we focus on the equilibrium with the least amount of o¤shoring. Next, we compareeach equilibrium under o¤shoring with its corresponding equilibrium under free trade withouto¤shoring. Hereafter, free trade means free trade in �nal goods and no o¤shoring.

4.1 Skill Diversity and the World Economy

We begin with exploring the e¤ects of o¤shoring between two countries that di¤er in skilldiversity. We next turn to investigating the e¤ects of free trade between these two countriesand compare the equilibrium under o¤shoring with the equilibrium under free trade.

4.1.1 Skill Diversity and O¤shoring

We assume that two countries share the same average skill level but Home has a more diversedistribution of skills than Foreign. Since workers can collaborate in international teams, weneed to consider the global distribution of skills. Let �W (t) represent the density function ofskills under o¤shoring, which is given by

�W (t) =LH

LH + LF�H(t) +

LFLH + LF

�F (t); (11)

where t 2 [tWmin; tWmax], with tWmin = min[tHmin; tFmin] and tWmax = max[tHmax; tFmax]. Since the averageskill level in Home, �tH , is equal to that in Foreign, �tF , and the density function of skills in each

10

country is symmetric about its mean level of skill, �W (t) is symmetric about the average skilllevel worldwide, �tW � (�tH +�tF )=2 = �tH = �tF . It follows that Lemma 1 holds under o¤shoringwith t̂W and mW (t̂W ) representing the marginal skill levels in sector C.10 Thus, we have

Y WC =�C2(LH + LF )�t

W

Z 2�tW�t̂W

t̂W�W (t)dt; (12)

Y WS = (LH + LF )

Z t̂W

tWmin

FS�t; 2�tW � t

��W (t)dt; (13)

where Y WC and Y WS are denoted by the outputs of good C and good S in the world undero¤shoring, respectively.

The relative price of good C under o¤shoring, pW , and t̂W are determined by the followingtwo equations:

pW =FS�t̂W ; 2�tW � t̂W

��C�tW

; (14)

Y WSY WC

= f(pW ): (15)

The �rst equation represents that the marginal rate of transformation is equal to the priceunder o¤shoring and the second equation re�ects that the relative supply is equal to the relativedemand in the world.

Wages of workers in sector C is

w(t) =pW�Ct

2, for t 2 [t̂W ;mW (t̂W )]. (16)

For workers with less skill than t̂W , we have

w(t) =pW�C t̂

W

2�Z t̂W

tFS1��; 2�tW � �

�d� , for t 2 [tWmin; t̂W ): (17)

The wages for workers with skills at the upper end of the distribution are obtained as follows:

w(t) = FS�t; 2�tW � t

�� p

W�C t̂W

2+

Z t̂W

2�tW�tFS1��; 2�tW � �

�d� , for t 2 (2�tW � t̂W ; tWmax]: (18)

Thus, a competitive equilibrium under o¤shoring is a set of equations of (12), (13), (14),(15), (16), (17) and (18). As proved in Appendix A, there exists an unique equilibrium undero¤shoring as the world economy under o¤shoring can be considered as a closed economy witha larger labor size.

10The density function �W (t) may not be continuous at tHmin and tFmax. However, the cumulative distribution

function is everywhere continuous, and thus, Lemma 1 holds.

11

Let pH and pF represent the autarky relative prices of good C in Home and Foreign,respectively. If the distribution of skill in Home is more diverse than that in Foreign, it isclear from (11) that Home has a more diverse distribution of skills relative to the world andthe world has a more diverse distribution of skills relative to Foreign. Following Lemma 2, wehave

Lemma 4 Suppose that the distribution of skill in Home is more diverse than that in Foreign,pW satis�es that pH > pW > pF . In addition, we have t̂H < t̂W < t̂F and mF (t̂F ) <mW (t̂W ) < mH(t̂H).

Since o¤shoring decreases the relative price of good C in Home, which leads to t̂W > t̂H

and mW (t̂W ) < mH(t̂H), then workers move out of sector C in Home under o¤shoring. InForeign, since the relative price of good C increases, workers exit sector S.

4.1.2 Skill Diversity and Trade

We consider a situation that workers cannot collaborate in international teams and goods aretraded freely between two countries as described above. In the free-trade equilibrium, therelative prices of good C in both countries are equalized. Using (5), we obtain

FS�t̂H ; 2�t

H � t̂H�

�C�tH=FS�t̂F ; 2�t

F � t̂F�

�C�tF,

where t̂H and t̂F represent marginal skill levels of C sector under free trade in Home and inForeign, respectively. Since �tH is equal to �tF and FS(�) is homogeneous of degree one, weobtain that t̂H is equal to t̂F . In other words, marginal skill levels of C sector are equalizedacross countries under free trade. It follows that the equations of (12), (13), (14), (15), (16),(17) and (18) also hold in the free-trade equilibrium with t̂W = t̂H = t̂F . Thus we have

Proposition 1 If Home and Foreign share the same average skill level but di¤er in skilldiversity, the equilibrium under o¤shoring is the same as the free-trade equilibrium.

Since t̂H is equal to t̂F , the wages of workers with the same skill level are equalized acrosscountries. In addition, the wage schedule under o¤shoring is same as that under free trade.11

If two countries have the same labor size, this free-trade equilibrium is same as the one in GM(2000). It follows that the e¤ects of o¤shoring on welfare is the same as the e¤ects of freetrade on welfare in GM (2000).

4.2 Average Skill Level and the World Economy

First, we investigate the e¤ects of o¤shoring between two countries that di¤er in the averageskill level. Next, we examine how free trade a¤ects these two countries and compare theo¤shoring equilibrium with the free-trade equilibrium.11GM (2000) investigated how free trade a¤ect the wage schedule in each country.

12

4.2.1 Average Skill Level and O¤shoring

We assume that two countries share the same skill diversity but di¤er in the average skill level,i.e., Home has a higher average skill level than Foreign. For simplicity, we assume that bothcountries have the same labor size, i.e., LH = LF .12 When two countries di¤er in the averageskill level, the density function of skills under o¤shoring �W (t) is given by

�W (t) =

8>>>>>><>>>>>>:

�F (t)

2t 2�tFmin; t

Hmin

��H(t)

2+�F (t)

2t 2�tHmin; t

Fmax

��H(t)

2t 2�tFmax; t

Hmax

�:

Clearly, �W (t) is symmetric about the mean level of skills worldwide, �tW � (�tH + �tF )=2. Itfollows that Lemma 1 holds under o¤shoring with t̂W and mW (t̂W ) being the marginal skilllevels in sector C and the equations of (12), (13), (14), (15), (16), (17) and (18) also hold inthis case.

We compare the relative price of good C under o¤shoring with the relative price underautarky in each country. O¤shoring has two e¤ects on the relative price: one is the meane¤ect because the density function under o¤shoring has a lower average skill level than Home�swhile a higher average skill level than Foreign�s; the other is the diversity e¤ect because thedensity function under o¤shoring is more diverse than both Home�s and Foreign�s. Accordingto Lemma 2 and Lemma 3, both the mean e¤ect and the diversity e¤ect lead to a higher relativeprice under o¤shoring pW than Home�s autarky relative price pH , which follows that t̂H > t̂W .However, the comparison between mW (t̂W ) and mH(t̂H) is ambiguous. The diversity e¤ectleads tomW (t̂W ) > mH(t̂H) from Lemma 2, while the mean e¤ect leads tomW (t̂W ) < mH(t̂H)under an uniform distribution function of skills from Lemma 3.

The comparison of the relative price under o¤shoring with Foreign�s autarky relative priceis ambiguous under general density functions. The mean e¤ect leads to a lower relative priceunder o¤shoring than Foreign�s autarky relative price while the diversity e¤ect leads to a higherrelative price under o¤shoring than Foreign�s autarky relative price. It follows that both therelationship between t̂W and t̂F , and between m(t̂W ) and m(t̂F ) are ambiguous under generaldensity functions. Under an uniform distribution of skills in each country, both the mean e¤ectand the diversity e¤ect lead to m(t̂F ) < m(t̂W ). With additional conditions that t̂W 2 [tHmin,tFmax], which implies that the gap between �tH and �tF is small enough, and the consumptionshare of good C is not too high, we prove in Appendix E that pF > pW and t̂W > t̂F , whichimplies skill diversity e¤ect dominates the mean e¤ect.

Lemma 5 Suppose that Home has a higher average skill level than Foreign. We have pW > pH

and t̂H > t̂W . With an uniform distribution function of skills in each country, we havem(t̂F ) < m(t̂W ), pF > pW and t̂W > t̂F if the gap between �tH and �tF is small enough and theconsumption share of good C is not too high.

12 If LH 6= LF , �W (t) will not be symmetric about �tW .

13

Figure 6: The Average Skill Level and Relative Price Comparison

If the gap between �tH and �tF is big enough or the consumption share of good C is highenough, we can obtain with simulations under a beta distribution function that pW > pF > pH

as shown in Figure 6 and Figure 7, respectively. When the gap between �tH and �tF is largeebough or the consumption share of good C is high enough, the production of good S isconducted by workers with extreme skills. Thus, the skill diversity e¤ect dominates the meane¤ect to raise the relative price under o¤shoring even higher than the autarky relative price ofForeign.

Next, we turn to the e¤ects of o¤shoring on nominal wages. Clearly, the wages of workerswith the same skill level are equalized, as shown in Figure 8. In the country with the higher(lower) average skill level, o¤shoring increases (decreases) wages of workers with high skillswhile lowers (increases) wages of workers with low skills.

4.2.2 Average Skill Level and Free Trade

We investigate the pattern of trade between Home and Foreign if two countries di¤er in theaverage skill level. We assume that the average skill level in Home �tH is �(> 1) times that inForeign �tF , i.e., �tH = ��tF . Thus, the pattern of trade between two countries can be determinedby examining the e¤ects of an increase in � on the relative supply of good S, YS=YC , holdingMRTH equal to MRTF . Substituting �tH = ��tF into (4) to obtain

MRTH =FS�t̂H ; 2��tF � t̂H

��C��tF

.

Since FS�t̂H ; 2��tF � t̂H

�is homogeneous of degree one, holding MRTH equal to MRTF

implies that t̂H = �t̂F , where t̂F satis�es MRTF =FS�t̂F ;2�t

F�t̂F�

�C�tF. In addition, we obtain 13

13See Appendix D for the calculation.

14

Figure 7: Consumption Share of Good C and Relative Price Comparison

Figure 8: The Average Skill Level and Wage Schedules under O¤shoring

15

Figure 9: The Average Skill Levels and Wages Under Free Trade

d(YS=YC)

d�jMRT=const< 0: (19)

Hence, Home has a comparative advantage in producing good C. Since GM (2000) demon-strates that if two countries have a common average skill level, the country with a higher skilldiversity has a comparative advantage in submodular good. we have the following proposition:

Proposition 2 A higher average skill level and a higher skill diversity a¤ect comparativeadvantage in opposite directions.

An increase in the average skill level has two e¤ects on the output in sector C: one isthat the range of skill levels assigned to sector C expands from t 2 [t̂F ; 2�t

F � t̂F ] to t 2[�t̂F ; 2��t

F � �t̂F ], which increases the output of good C, and the other is that workers insector C are more productive due to an increase in their skills, which also increases the outputof good C. However, an increase in the average skill level has two con�icting e¤ects on theoutput in sector S: one is that sector S shrinks, which leads to a decline in the output ofS, and the other is that workers in sector S become more productive due to an increase intheir skills, which increases the sector�s output. As a result, an increase in the average skilllevel leads to a decline in YS=YC with MRT held constant. It follows that Home, which hasa higher average skill level, has a comparative advantage in good C.

We next examine wage schedules of two countries under free trade. Since t̂H is equal to �t̂F

in the free-trade equilibrium, nominal wages are not equalized between countries as illustratedin Figure 8 (see proof in Appendix F). Thus, we have the following Proposition.

Proposition 3 If two countries di¤er in skill diversity, the wages of workers with the samelevel of skills are equalized across countries under free trade. If two countries only di¤er in theaverage skill level, the wages of workers with the same level of skills are not equalized underfree trade.

16

Interestingly, in the case in which there is no trade in the free-trade equilibrium, the wagesof workers with the same level of skills are not equalized if two countries have di¤erent averageskills.

Corollary 1 Suppose that �H(�t) = �F (t), � 6= 0, for all t 2 [tFmin; tFmax], then the free-tradeequilibrium has no trade in �nal goods. However, the wages of workers with same level of skillsare not equalized in the free-trade equilibrium.

Whether wages are equalized depends solely on the average skill levels under free trade. Iftwo countries have the same average skill levels, all workers have paired with their best partnersunder free trade, and thus, there is no incentive for o¤shoring. However, if two countries havedi¤erent average skill levels, there exist incentives for workers in the submodular sector to pairwith Foreign workers under free trade. Hence, o¤shoring can occur between two such countriesin the submodular sector.

5 The Income Distribution

We use the indirect utility function, V (pC ; pS ;W (t)), to measure welfare. Since the preferenceis homothetic, we have V (pC ; pS ;W (t)) = v(pC ; pS)W (t), where W (t) is the nominal wage ofworkers with skill level t. Since V (pC ; pS ;W (t)) is homogeneous of degree 0 with respect topC ; pS ;W (t), we obtain

V (pC ; pS ;W (t)) = V (p; 1; w(t)) = �(p)w(t);

where �(p) � v(p; 1), p � pC=pS , and w(t) �W (t)=pS .In this section, we will compare welfare under o¤shoring with that under free trade. First,

we investigate the e¤ects of o¤shoring on welfare. Next, we examine the e¤ects of free tradeon welfare.

5.1 The Income Distribution under O¤shoring

The e¤ects of o¤shoring on the income distribution are shown in Figure 9.14 The e¤ectsof o¤shoring on welfare are similar to those on the nominal wages. Interestingly, o¤shoringa¤ects the workers in the submodular sector in opposite directions: workers with the highestskill levels bene�t from o¤shoring in the country with a higher average skill level, while workerswith the lowest skills lose. We have opposite results in the country with a lower average skilllevel.14We use a beta distribution function with both shape parameters equal to 1.6. Moreover, L = L� = 1,

tmin = 7, tmax = 17, t�min = 5, and t�max = 15. Thus, we have �t = 12, �t

� = 10 and standard deviation � = 2:44 inboth countries. In addition, we use the CES production function F (t1; t2) = (t�1 + t

�2)

1� with � = 4 in sector S

and � = 0:5 in sector C. We use a Cobb-Douglas utility function with the consumption share of good C � = 0:6to plot the wage schedule and explore the e¤ects of o¤shoring on the income distribution for � 2 [0:005; 0:965].We obtain similar results by using di¤erent parameters of the beta distribution function, which includes auniform distribution function.

17

Figure 10: O¤shoring and Welfare

Figure 11: The Income Distribution under Free Trade

5.2 The Income Distribution under Free Trade

We turn to the income distribution under free trade. Since home country has a higher averageskill level and thus export good C, trade bene�ts the middle-skilled workers (t 2 [t̂; m(t̂)])who work in the exportable sector both before and after opening to trade because their realwages increase in terms of either good. We also know that workers who work in the importablesector before and after opening to trade are worse o¤. The lowest skilled workers lose becausetheir real wage decreases in terms of either good. The highest skilled workers also lose becausethe increases in their nominal wages are less than the increase in the relative price. Whetherthe remaining workers who switch from the importable to the exportable sector are better orworse o¤ depends on the consumption shares. Therefore, we can see that o¤shoring and freetrade have di¤erent e¤ects on the welfare of workers in the submodular sector.

18

Proposition 4 O¤shoring and free trade have di¤erent e¤ects on the welfare of workers inthe submodular sector. Trade a¤ects the welfare of workers in the same direction: All gainfrom trade, or all lose from trade. However, o¤shoring a¤ects the welfare of workers with thehighest skills and workers with the lowest skills in opposite directions.

O¤shoring o¤ers an incentive for workers in the submodular sector to form internationalteams, which leads to labor reallocation. Workers with the highest skills in the country withthe higher average skill have the opportunity to form teams with better partners in the foreigncountry. Therefore, those workers gain from o¤shoring, and workers with lower skills who arereplaced by foreign workers lose. O¤shoring a¤ects workers in the foreign country in oppositedirections, as workers with the lowest skills �nd better partners and workers with the highestskills lose their best partners.

6 Concluding Remarks

In this paper, we investigate how skill diversity and the average skill level a¤ect the possibilityof o¤shoring. We demonstrate that if two countries have the same skill diversity but di¤erentaverage skills, the wages of workers with the same level of skills are not equalized, and thus,o¤shoring is possible. When two countries have the same average skill level but di¤erent skilldispersions, as in Grossman and Maggi (2000), the wages of workers with the same level of skillsare equalized under free trade, and thus, o¤shoring is impossible. Moreover, we demonstratethat free trade and o¤shoring have di¤erent e¤ects on the wages of workers with highest andlowest skill levels. Speci�cally, the results in the country with a higher skill level are consistentwith the �ndings in Hummels et al. (2014).

We show that o¤shoring leads to an increase in wage inequality in the country with a higheraverage skill. This result is consistent with the fact that developed countries have experienceda rise in wage inequality since the 1980s (see Feenstra and Hanson (1996, 1997, 1999)).

Our paper considers two important factors related to workers�skills, the average skill leveland the skill dispersion, and is useful to interpret several observations on o¤shoring in the realworld. However, we abstract from o¤shoring costs and imperfect observation of workers�skillsin the current setup. These issues are left for future work.

Appendix A: The Existence of Unique EquilibriumWe prove that there exists an unique equilibrium under autarky. We de�ne D(t̂) � XS

XC.

Using (3), (4), and (5), we obtain

D(t̂) � XSXC

= f

FS�t̂; 2�t� t̂

��C�t

!; t̂ 2 [tmin; �t]:

Di¤erentiating the above equation to obtain

D0(t̂) � f 0 FS�t̂; 2�t� t̂

��C�t

!FS1�t̂; 2�t� t̂

�� FS2

�t̂; 2�t� t̂

��C�t

:

19

Since f 0(�) > 0, the sign of D0(t̂) depends on the sign of FS1�t̂; 2�t� t̂

�� FS2

�t̂; 2�t� t̂

�, which

is proved to be negative below.Without loss of generality, we prove FS1 (t1; t2)� FS2 (t1; t2) � 0, if t1 � t2. Since the tasks

are symmetric, we have FS(t1; t2) = FS(t2; t1) and FS2 (t1; t2) = FS1 (t2; t1). It follows that

FS1 (t1; t2)� FS2 (t1; t2) = FS1 (t1; t2)� FS1 (t2; t1) = FS1�t1t2; 1

�� FS1

�t2t1; 1

�:

The last equality is obtained because FS(t1; t2) is homogeneous of degree one and thusFS1 (t1; t2) is homogeneous of degree zero. Since F

S(t1; t2) exhibits submodularity, we haveFS11 � 0, which gives that FS1 (t1=t2; 1) � FS1 (t2=t1; 1) � 0, if t1 � t2. It follows thatFS1�t̂; 2�t� t̂

�� FS2

�t̂; 2�t� t̂

�< 0 if t̂ 2 [tmin; �t], and thus D(t̂) is decreasing in t̂, which

leads to D(tmin) � D(�t) > 0.Next, we de�ne �(t̂) � YS

YC. Using (1) and (2), we obtain

�(t̂) � YSYC

=2

�C�t

R t̂tmin

FS (s; 2�t� s)�(s)dsR 2�t�t̂t̂ �(s)ds

; t̂ 2 [tmin; �t]:

Di¤erentiating the above equation yields

�0(t̂) =2�(t̂)Z 2�t�t̂

t̂�(s)ds

�p+

YSYC

�> 0:

Therefore, �(t̂) is increasing in t̂. In addition, we have �(tmin) = 0 and limt!�t �(t) = +1.Since D(t̂) is decreasing in t̂ and �(t̂) is increasing in t̂, D(tmin) > �(tmin), and D(�t) <

limt!�t �(t) = +1, there exists a unique t̂ satisfying D(t̂) = �(t̂). Since FS1 (t; 2�t� t) �FS2 (t; 2�t� t) < 0 and p =

FS(t̂;2�t�t̂)�C�t

, there exists a unique autarky equilibrium (t̂; p).

Appendix B: Proof for Lemma 2 (Skill Diversity)

We �rst prove that an increase in skill diversity leads to a decrease in t̂ while an increasein m(t̂). Let t < �t� be a point in the support of ��and let t00 < t0. Following GM (2000), wede�ne a single, symmetric mean-preserving spread (SSMPS) of �� as an arbitrary decreasein density d�� on [t0; t0 + dt] and on [m(t0);m(t0 + dt)], accompanied by an equal increase indensity on [t00; t00 + dt] and [m(t00);m(t00 � dt)]. As stated in GM (2000), if �(t) and ��(t) aresymmetric, �t = �t�, and � is more diverse than ��, � can be generated from �� by a sequenceof SSMPS�s. Similar to proof for proposition 4 in GM (2000), we can demonstrate that therelative supply of good S, YSYC , increases in skill diversity. Thus, together with (1) and (2), therelative supply of good S is determined as follows:

YSYC

= �(t̂; �D); (B1)

20

where �D represents skill diversity. Clearly, we have @�@�D

> 0 and @�@t̂> 0.

From (4), (5) and (6), we obtain

YSYC

= f(MRT (t̂)): (B2)

Combining (B1) and (B2) yields

f(MRT (t̂)) = �(t̂; �D): (B3)

Di¤erentiating (B3), we obtain

dt̂

d�D=

@�@�D

f 0(p)@MRT@t̂

� @�@t̂

:

Since f 0(p) > 0, @�@�D

> 0, @�@t̂> 0, and @MRT

@t̂(=

FS1 (t̂;2�t�t̂)�FS2 (t̂;2�t�t̂)�C�t

) < 0 due to FS1 (t1; t2)�FS2 (t1; t2) � 0, if t1 � t2, we obtain dt̂

d�D< 0. Because m(t̂) = 2�t � t̂, we have dm(t̂)

d�D> 0.

Clearly, an increase in skill diversity leads to an increase in the relative price p due to p =MRT and @MRT

@t̂< 0.

Appendix C: Proof for the E¤ects of Skill Diversity on Wage Schedules

We �rstly focus on the e¤ects of an increase in skill diversity on wages of workers whowork in sector C and sector S before and after this change in skill diversity, respectively. Anincrease in the relative price p due to an increase in skill diversity raises wages of workers whowork in sector C before and after this change in skill diversity, i.e., t 2 [t̂�;m�(t̂�)], as shownin equation (7). For workers with skills at the lower end of distribution, i.e., t 2 [tmin; t̂], thee¤ects of an increase in skill diversity on their wages are obtained as follows from (9):

@w(t)

@�D=

��C2

�@p

@t̂t̂+ p

�� FS1

�t̂; 2�t� t̂

�� dt̂

d�D:

Substituting @p@t̂=

FS1 (t̂;2�t�t̂)�FS2 (t̂;2�t�t̂)�C�t

and (5) into the above equation to obtain

@w(t)

@�D=

"�C2

FS1�t̂; 2�t� t̂

�� FS2

�t̂; 2�t� t̂

��C�t

t̂+FS�t̂; 2�t� t̂

��C�t

!� FS1

�t̂; 2�t� t̂

�# dt̂

d�D

=

"FS1�t̂; 2�t� t̂

�(t̂� 2�t)� FS2

�t̂; 2�t� t̂

�t̂+ FS

�t̂; 2�t� t̂

�2�t

#dt̂

d�D

=�t� t̂�t

�FS2�t̂; 2�t� t̂

�� FS1

�t̂; 2�t� t̂

�� dt̂

d�D;

21

where the third equality is obtained because FS is homogeneous of degree one. Since FS2�t̂; 2�t� t̂

��

FS1�t̂; 2�t� t̂

�> 0 and dt̂

d�D< 0, we have @w(t)

@�D< 0. It follows that the wages for workers with

skills at the upper end of the distribution increase as in (10).Next, we turn to the wages of workers who change the sector. Since t̂ decreases from t̂�

due to an increase in skill diversity, workers with skill levels t 2 [t̂; t̂�] shift from sector S tosector C. The di¤erence between their wages before and after sector adjustment is

wD(t) =�C2pt� �C

2p�t̂� +

Z t̂�

tFS1 (�; 2�t� �) d� , (C1)

where p� represents the relative price before the change in skill diversity. Clearly, we havewD(t̂

�) > 0 and wD(t̂) < 0, because workers with skill t̂� and t̂ work in the supermodular andsubmodualr sector, respectively, before and after the change in skill diversity. Di¤erentiating(C1) and using (5) to obtain

w0D(t) =FS�t̂; 2�t� t̂

�2�t

� FS1 (t; 2�t� t) .

If t = t̂, we have w0D(t̂) =(2�t�t̂)2�t [FS2

�t̂; 2�t� t̂

�� FS1

�t̂; 2�t� t̂

�] > 0. If t = t̂�, we have

w0D(t̂�) =

FS�t̂; 2�t� t̂

�� 2�tFS1

�t̂�; 2�t� t̂�

�2�t

�FS�t̂�; 2�t� t̂�

�� 2�tFS1

�t̂�; 2�t� t̂�

�2�t

=(2�t� t̂�)2�t

[FS2�t̂�; 2�t� t̂�

�� FS1

�t̂�; 2�t� t̂�

�] > 0;

where the second inequality is obtained from FS�t̂; 2�t� t̂

�> FS

�t̂�; 2�t� t̂�

�due to FS1 (t; 2�t� t)�

FS2 (t; 2�t� t) < 0. In addition, we have

w00D(t) = F

S12 (t; 2�t� t)� FS11 (t; 2�t� t) < 0:

Therefore, w0D(t) > 0, for t 2 [t̂; t̂�]. Since wD(t̂�) > 0 and wD(t̂) < 0, there exists a unique ~t,at which wD(~t) = 0.

Workers with skill levels t 2 [2�t � t̂�; 2�t � t̂] also shift from sector S to sector C. Thedi¤erence between their wages before and after sector adjustment is

wD(t) =�C2pt� FS (t; 2�t� t) + �C

2p�t̂� �

Z t̂�

2�t�tFS1 (�; 2�t� �) d�

=tFS

�t̂; 2�t� t̂

�� 2�tFS (t; 2�t� t)2�t

+t̂�FS

�t̂�; 2�t� t̂�

�2�t

�Z t̂�

2�t�tFS1 (�; 2�t� �) d� (C2)

22

When t = 2�t� t̂, we have

wD(2�t� t̂) =t̂�FS

�t̂�; 2�t� t̂�

�� t̂FS

�t̂; 2�t� t̂

�2�t

�Z t̂�

t̂FS1 (�; 2�t� �) d�

=

Z t̂�

t̂[d(�FS (�; 2�t� �)

2�t� FS1 (�; 2�t� �)]d�

=

Z t̂�

t̂[FS (�; 2�t� �) + �(FS1 (�; 2�t� �)� FS2 (�; 2�t� �))� 2�tFS1 (�; 2�t� �)

2�t]d�

=

Z t̂�

t̂

(2�t� 2�)[FS2 (�; 2�t� �)� FS1 (�; 2�t� �)]2�t

d� > 0;

where the third equality is obtained by using that FS is homogeneous of degree one. Di¤er-entiating (C2) yields

w0D(t) =FS�t̂; 2�t� t̂

�� 2�t[FS1 (t; 2�t� t)� FS2 (t; 2�t� t)]� 2�tFS1 (2�t� t; t)

2�t

=t̂FS1

�t̂; 2�t� t̂

�� 2�tFS1 (t; 2�t� t) + (2�t� t̂)FS2

�t̂; 2�t� t̂

�2�t

:

It follows that w00D(t) = F

S12(t; 2�t� t)� FS11(t; 2�t� t) < 0. When t = 2�t� t̂�, we have

w0D(2�t� t̂�) =t̂FS1

�t̂; 2�t� t̂

�� 2�tFS2

�t̂�; 2�t� t̂�

�+ (2�t� t̂)FS2

�t̂; 2�t� t̂

�2�t

=t̂[FS1

�t̂; 2�t� t̂

�� FS2

�t̂; 2�t� t̂

�] + 2�t[FS2

�t̂; 2�t� t̂

�� FS2

�t̂�; 2�t� t̂�

�]

2�t< 0;

where FS2�t̂; 2�t� t̂

��FS2

�t̂�; 2�t� t̂�

�< 0 is due to FS21 (t; 2�t� t)�FS22 (t; 2�t� t) < 0. Therefore,

we have w0D(t) < 0 for t 2 [2�t � t̂�; 2�t � t̂]. Since wD(2�t � t̂) > 0, we have wD(t) > 0 fort 2 [2�t� t̂�; 2�t� t̂].

Appendix D: Proof for Lemma 3 (Average Skill)

Since the average skill level �t increases from �t�, i.e., �t = ��t�, equations of (1), (2), and (5)are rewritten as follows:

YC =L

2�C��t

�Z 2��t��t̂

t̂�(t)dt; (D1)

YS = L

Z t̂

t�min+(��1)�t�FS (t; 2��t� � t)�(t)dt; (D2)

p =FS�t̂; 2��t� � t̂

��C��t�

; (D3)

23

where �(t) = ��[t� (� � 1)�t�], tmin = t�min + (� � 1)�t�, and tmax = t�max + (� � 1)�t�.From (D1), (D2), and (6), we have

YSYC(t̂; �) = f(p): (D4)

From (D3), we obtaint̂ = t̂(p; �): (D5)

Di¤erentiating (D3), we have

@t̂

@p(p; �) =

�C��t�

FS1�t̂; 2��t� � t̂

�� FS2

�t̂; ��t� � t̂

� < 0;@t̂

@�(p; �) =

t̂

�> 0:

due to FS1�t̂; 2��t� � t̂

�� FS2

�t̂; 2��t� � t̂

�< 0.

Substituting (D5) into (D4) to obtain

YSYC(t̂(p; �); �) = f(p): (D6)

Di¤erentiating (D6) to obtain

dp

d�= �

@YS=YC

@t̂

@t̂

@�+@YS=YC@�

@YS=YC

@t̂

@t̂

@p� f 0(p)

: (D7)

From (D1) and (D2), we obtain

@(YS=YC)

@t̂=

2�(t̂)Z 2�t�t̂

t̂�(t)dt

�p+

YSYC

�> 0

@(YS=YC)

@�=

2

�C�t

Z 2�t�t̂

t̂�(t)dt

�(�t�Z t̂

tmin

�2FS2 �� FS�0

�dt� �t�FS(tmin; 2�t� tmin)�(tmin)

��C�t�

2

YSYC

"2�t�(t̂) +

Z 2�t�t̂

t̂�(t)dt� �t�

Z 2�t�t̂

t̂�0(t)dt

#)

=2�t�

�C�t

Z 2�t�t̂

t̂�(t)dt

�(Z t̂

tmin

�FS1 + F

S2

��(t)dt� FS(t̂; 2�t� t̂)�(t̂)

��C2

YSYC

"2�t�(t̂) +

Z 2�t�t̂

t̂�(t)dt

#):

24

where note that �(t) = �(2�t � t) and �0(t) = (��)0(t � (� � 1)�t�). Substituting the aboveequation into (D7), the numerator of (D7) is obtained as follows:

@YS=YC

@t̂

@t̂

@�+@YS=YC@�

= � 2

�

Z 2�t�t̂

t̂�(t)dt

�((�t� t̂)�(t̂)

�p+

YSYC

�� 1

�C

Z t̂

tmin

(FS1 + FS2 )�(t)dt+

1

2

YSYC

Z 2�t�t̂

t̂�(t)dt

)

= � 2

�

Z 2�t�t̂

t̂�(t)dt

�((�t� t̂)�(t̂)

�p+

YSYC

�� 1

�C

Z t̂

tmin

(FS1 + FS2 )�(t)dt+

1

�c�t

Z t̂

tmin

FS�(t)dt

)

= � 2

�

Z 2�t�t̂

t̂�(t)dt

�((�t� t̂)�(t̂)

�p+

YSYC

�+

1

�C�t

Z t̂

tmin

(�t� t)(FS2 � FS1 )�(t)dt)< 0:

Since @(YS=YC)=@t̂ > 0, f 0 > 0, and @t̂=@p < 0, the denominator of (D7) is negative. Sincethe numerator of (D7) is negative, we obtain dp=d� < 0.

From (D5), we have

dt̂ =@t̂

@pdp+

@t̂

@�d�:

Since @t̂=@p < 0, dp=d� < 0, and @t̂=@� > 0, an increase in � leads to an increase in t̂.Di¤erentiating m(t̂) = 2��t� � t̂, we obtin

dm(t̂)

d�=d(2��t� � t̂)

d�= 2�t� � dt̂

d�=2�t� t̂�

� @t̂

@p

dp

d�:

Because we have the following relationship:

dp

d�= �

@YS=YC

@t̂

@t̂

@�+@YS=YC@�

@YS=YC

@t̂

@t̂

@p� f 0(p)

> �

@YS=YC

@t̂

@t̂

@�+@YS=YC@�

@YS=YC

@t̂

@t̂

@p

;

it is satis�ed that

dm(t̂)

d�=2�t� t̂�

� @t̂

@p

dp

d�

>2�t� t̂�

+@t̂

@�+@YS=YC@�

�@YS=YC@t̂

=1

@YS=YC

@t̂

�2�t

�

@YS=YC

@t̂+@YS=YC@�

�;

25

where

2�t

�

@YS=YC

@t̂+@YS=YC@�

=2

�

Z 2�t�t̂

t̂�(t)dt

"�t�(t̂)

�p+

YSYC

�� 1

�C�t

Z t̂

tmin

(�t� t)(FS2 � FS1 )�(t)dt#:

If � follows an uniform distribution, we have �(t) = 1I for t 2 [tmin; tmax] and I =

1tmax�tmin . It

follows that

�t�(t̂)

�p+

YSYC

�� 1

�C�t

Z t̂

tmin

(�t� t)(FS2 � FS1 )�(t)dt

=1

I

"�tp+

1

�C(�t� t̂)

Z t̂

tmin

FSdt� 1

�C�t

Z t̂

tmin

(�t� t)(FS2 � FS1 )dt#

=1

I

"�tp+

1

�C�t

Z t̂

tmin

�t̂

�t� t̂FS + �t(FS1 + F

S2 )

�dt

#> 0:

Appendix E: Proof for pH < pW < pF and t̂F < t̂W < t̂H

In order to compare the relative price under o¤shoring with each country�s autarky relativeprice, the equations that determine the relative price of good C and marginal skill level in sectorC either under autarky or under o¤shoring are rewritten as follows

Y TSY TC

= K(t̂T ; ) � 2

�C

Z t̂T

tFmin

FS�t; 2�tT � t

��T (t)dt

�tTZ 2�tT�t̂T

t̂T�T (t)dt

;

pT =FS�t̂T ; 2�tT � t̂T

��C�tT

=) t̂T = t̂T (pT ; );

Y TSY TC

= f(pT ); f 0 > 0; pT � pTCpTS;

where �tT � (1� )�tF + �tH , 0 � � 1, and

�T (t) �

8><>:(1� )�F (t) t 2 [tFmin; tHmin)(1� )�F (t) + �H(t) t 2 [tHmin; tFmax) �H(t) t 2 [tFmax; tHmax]

:

26

When = 0, we obtain the autarky equilibrium of Foreign and when = 1, we obtain theautarky equilibrium of Home. The equilibrium under o¤shoring is captured by = 1=2.

Combining (E1), (E2), and (E3), we have

K�t̂T�pT ;

�; �= f(pT ) =) @K

@t̂T

�@t̂T

@pTdpT +

@t̂T

@ d

�+@K@ d = f 0dpT :

It follows that

dpT

d = �

@K@t̂T

@t̂T

@ +@K@

@K@t̂T

@t̂T

@pT� f 0

;

dt̂T

d =@t̂T

@pTdpT

d +@t̂T

@ :

From (E1) and (E2), we have

@t̂T

@pT=

�C�tT

FS1�t̂T ; 2�tT � t̂T

�� FS2

�t̂T ; 2�tT � t̂T

� < 0;@t̂T

@ =t̂T��tH � �tF

��tT

> 0;

@K@t̂T

=2�T (t̂T )Z 2�tT�t̂T

t̂T�T (t)dt

�pT +

�T (2�tT � t̂T ) + �T (t̂T )2�T (t̂T )

Y TSY TC

�> 0:

Thus, if@K@t̂T

@t̂T

@ +@K@

< 0; (E4)

we obtain dpt=d < 0 and dt̂=d > 0.We prove (E4) in the following. Suppose t̂T 2 [tHmin; tFmax] and 2�tT � t̂T 2 [tHmin; tFmax].

K(t̂T ; ) is rewritten as

K(t̂T ; ) = 2

�C

(1� )Z t̂T

tFmin

FS�t; 2�tT � t

��F (t)dt+

Z t̂T

tHmin

FS�t; 2�tT � t

��H(t)dt

�tTZ 2�tT�t̂T

t̂T�T (t)dt

:

Di¤eretiating K with respect to yields@K@

=1Z 2�tT�t̂T

t̂T�T (t)dt

�A;

27

where

A = 2

�C�tT

"Z t̂T

tHmin

FS�H(t)dt�Z t̂T

tFmin

FS�F (t)dt+ 2(�tH � �tF )Z t̂T

tFmin

FS2 �T (t)dt

#

� YTS

Y TC

"�tH � �tF�tT

Z 2�tT�t̂T

t̂T�T (t)dt+

Z 2�tT�t̂T

t̂T�H(t)dt�

Z 2�tT�t̂T

t̂T�F (t)dt+ 2(�tH � �tF )�T

�2�tT � t̂T

�#

=2

�C�tT

"Z t̂T

tHmin

FS�H(t)dt�Z t̂T

tHmin

FS�F (t)dt�Z tHmin

tFmin

FS�F (t)dt

#

+2

�C�tT

�tH � �tF�tT

"Z tHmin

tFmin

t(FS2 � FS1 )�T (t)dt+Z t̂T

tHmin

t(FS2 � FS1 )�T (t)dt#

+Y TSY TC

"Z 2�tT�t̂T

t̂T�F (t)dt�

Z 2�tT�t̂T

t̂T�H(t)dt� 2(�tH � �tF )�T

�2�tT � t̂T

�#:

Note that the following relationship is satis�ed:Z t̂T

tHmin

t(FS2 � FS1 )�T (t)dt =Z t̂T

tHmin

FS�T (t)dt+

Z t̂T

tHmin

tFS(�T )0(t)dt��tFS�T

�t̂TtHmin

=

Z t̂T

tHmin

FS�T (t)dt+

Z t̂T

tHmin

tFS(�T )0(t)dt

� �C�tT t̂T�T (t̂T )pT + tHminFS�tHmin; 2�t

T � tHmin��T (tHmin):

Using this relationship, A becomes

A = �2�T (t̂T )t̂T��tH � �tF

��tT

pT +2

�C�tT

"Z t̂T

tHmin

FS�H(t)dt�Z t̂T

tHmin

FS�F (t)dt

#

� 2

�C�tT

"Z tHmin

tFmin

FS�F (t)dt��tH � �tF�tT

Z tHmin

tFmin

t(FS2 � FS1 )�T (t)dt#

+2

�C�tT

�tH � �tF�tT

"Z t̂T

tHmin

FS�T (t)dt+

Z t̂T

tHmin

tFS(�T )0(t)dt+ tHminFS�tHmin; 2�t

T � tHmin��T (tHmin)

#

+Y TSY TC

"Z 2�tT�t̂T

t̂T�F (t)dt�

Z 2�tT�t̂T

t̂T�H(t)dt� 2(�tH � �tF )�T

�2�tT � t̂T

�#:

In addition, using the following property:Z 2�tT�t̂T

t̂T�T (t)dt = 2�tT�T

�2�tT � t̂T

�� t̂T

��T�2�tT � t̂T

�+ �T (t̂T )

��Z 2�tT�t̂T

t̂Tt(�T )0(t)dt;

28

A can be written as

A = �2�T (t̂T ) t̂T (�tH � �tF )

�tT

pT +

�T�2�tT � t̂T

�+ �T (t̂T )

2�T (t̂T )

Y TSY TC

!

� 2

�C�tT

�tH � �tF�tT

"Z tHmin

tFmin

FS�T (t)dt�Z tHmin

tFmin

t(FS2 � FS1 )�T (t)dt#

� 2

�C�tT

"Z tHmin

tFmin

FS�F (t)dt��tH � �tF�tT

tHminFS�tHmin; 2�t

T � tHmin��T (tHmin)

#

+2

�C�tT

"Z t̂T

tHmin

FS�H(t)dt�Z t̂T

tHmin

FS�F (t)dt+�tH � �tF�tT

Z t̂T

tHmin

tFS(�T )0(t)dt

#

+Y TSY TC

"Z 2�tT�t̂T

t̂T�F (t)dt�

Z 2�tT�t̂T

t̂T�H(t)dt�

�tH � �tF�tT

Z 2�tT�t̂T

t̂Tt(�T )0(t)dt

#:

Therefore, we have@K@t̂T

@t̂T

@ +@K@

=1Z 2�tT�t̂T

t̂T�T (t)dt

� B;

where B is given by

B = � 2

�C�tT

�tH � �tF�tT

"Z tHmin

tFmin

FS�T (t)dt�Z tHmin

tFmin

t(FS2 � FS1 )�T (t)dt#

� 2

�C�tT

"Z tHmin

tFmin

FS�F (t)dt��tH � �tF�tT

tHminFS�tHmin; 2�t

T � tHmin��T (tHmin)

#

+2

�C�tT

"Z t̂T

tHmin

FS�H(t)dt�Z t̂T

tHmin

FS�F (t)dt+�tH � �tF�tT

Z t̂T

tHmin

tFS(�T )0(t)dt

#

+Y TSY TC

"Z 2�tT�t̂T

t̂T�F (t)dt�

Z 2�tT�t̂T

t̂T�H(t)dt�

�tH � �tF�tT

Z 2�tT�t̂T

t̂Tt(�T )0(t)dt

#:

The �rst two terms are negative, and if both �H and �F are uniform, last two terms are equalto 0. Thus, we have

@K@t̂T

@t̂T

@ +@K@

< 0:

Appendix F: The Comparison of Wage Schedules

29

The comparison of the wages in the home country with those in the foreign country isshown as follows:

wF (t) � wH(t); if t 2 [tHmin; t̂F ) and t 2 [t̂F ; t̂H);wF (t) = wH(t); if t 2 [t̂H ; 2�tF � t̂F ];wF (t) � wH(t); if t 2 (2�tF � t̂F ; 2�tH � t̂H) and t 2 [2�tH � t̂H ; tFmax]:

To do so, we �rst derive the wage schedules of both countries under autarky. Note that (7),(9), and (10) hold for each country. Substituting the average skills of both countries intothe above equations, we obtain the nominal wage of workers in the home country, wH(t), fort 2 [tHmin; tHmax], and that of Foreign, wF (t), for t 2 [tFmin; tFmax] as follows:

wH(t) =

8>>>>>><>>>>>>:

p�C t̂H

2�Z t̂H

tFS1��; 2�tH � �

�d�; t 2 [tHmin; t̂H)

p�Ct

2; t 2 [t̂H ; 2�tH � t̂H ]

FS�2�tH � t; t

�� p�C t̂

H

2+

Z t̂H

2�tH�tFS1��; 2�tH � �

�d�; t 2 (2�tH � t̂H ; tmax]

;

wF (t) =

8>>>>>><>>>>>>:

p�C t̂F

2�Z t̂F

tFS1��; 2�tF � �

�d�; t 2 [tFmin; t̂F )

p�Ct

2; t 2 [t̂F ; 2�tF � t̂F ]

FS�2�tF � t; t

�� p�C t̂

F

2+

Z t̂F

2�tF�tFS1��; 2�tF � �

�d�; t 2 (2�tF � t̂F ; tFmax]

:

The wage schedule is linear in the range [t̂H ; 2�tH � t̂H ] for the home country ([t̂F ; 2�tF � t̂F ]for the foreign country) and convex outside this range.

Proof:(1)t 2 [tHmin; t̂F )

If t 2 [tHmin; t̂F ), we have

wF (t)� wH(t)

=p�C t̂

F

2�Z t̂F

tFS1��; 2�tF � �

�d� �

"p�C t̂

H

2�Z t̂H

tFS1��; 2�tH � �

�d�

#

=(1� �)p�C t̂F

2+

Z �t̂F

t̂FFS1��; 2��tF � �

�d� +

Z t̂F

t

�FS1��; 2��tF � �

�� FS1

��; 2�tF � �

��d�

Because of the submodularity of FS , we have FS12 5 0. It follows that

FS1��; 2��tF � �

�� FS1

��; 2�tF � �

�5 0

30

for all � 2 [t; t̂F ). Next, we examine the sign of

(1� �)p�C t̂F2

+

Z �t̂F

t̂FFS1��; 2��tF � �

�d�

as follows:

(1� �)p�C t̂F2

+

Z �t̂F

t̂FFS1��; 2��tF � �

�d�

� (1� �)p�C t̂F2

+��t̂F � t̂F

�FS1��t̂F ; 2��tF � �t̂F

�=(1� �)t̂F

2

�p�C � 2FS1

�t̂F ; 2�tF � t̂F

��=(1� �)t̂F2�tF

�FS�t̂F ; 2�tF � t̂F

�� 2�tFFS1

�t̂F ; 2�tF � t̂F

��=(1� �)t̂F

�2�tF � t̂F

�2�tF

�FS2�t̂F ; 2�tF � t̂F

�� FS1

�t̂F ; 2�tF � t̂F

��5 0:

We substitute (5) into the third equality to obtain the fourth equality. The �fth equality isobtained because of the assumption of constant returns to skill of FS .

(2) t 2 [t̂F ; t̂H)If t 2 [t̂F ; t̂H), we have

wF (t)� wH(t) = p�Ct

2�"p�C t̂

H

2�Z t̂H

tFS1��; 2�tH � �

�d�

#

=p�C

�t� t̂H

�2

+

Z t̂H

tFS1��; 2�tH � �

�d�

5p�C

�t� t̂H

�2

+�t̂H � t

�FS1�t̂H ; 2�tH � t̂H

�=�t� t̂H

� �p�C2� FS1

�t̂H ; 2�tH � t̂H

��=t� t̂H2�tH

�FS�t̂H ; 2�tH � t̂H

�� 2�tHFS1

�t̂H ; 2�tH � t̂H

��=

�t� t̂H

� �2�tH � t̂H

�2�tH

�FS2�t̂H ; 2�tH � t̂H

�� FS1

�t̂H ; 2�tH � t̂H

��5 0:

(3) t 2 (2�tF � t̂F ; 2�tH � t̂H)If t 2 (2�tF � t̂F ; 2�tH � t̂H), we have

wF (t)� wH(t) = FS�2�tF � t; t

�� p�C t̂

F

2+

Z t̂F

2�tF�tFS1��; 2�tF � �

�d� � p�Ct

2

= FS�2�tF � t; t

�+

Z t̂F

2�tF�tFS1��; 2�tF � �

�d� �

p�C�t+ t̂F

�2

:

31

Converging t to 2�tF � t̂F yields limt!2�tF�t̂F�wF (t)� wH(t)

�= 0. In addition, we have

@�wF (t)� wH(t)

�@t

= FS2�2�tF � t; t

�� FS1

�2�tF � t; t

�+ FS1

�2�tF � t; t

�� p�C

2

= FS2�2�tF � t; t

�� 1

2�tFFS�t̂F ; 2�tF � t̂F

�=�FS2�2�tF � t; t

�� FS2

�t̂F ; 2�tF � t̂F

��+t̂F

2�tF�FS2�t̂F ; 2�tF � t̂F

�� FS1

�t̂F ; 2�tF � t̂F

��:

We have FS2�t̂F ; 2�tF � t̂F

�� FS1

�t̂F ; 2�tF � t̂F

�due to Appendix A. Due to the submodularity

of FS and the assumption of constant returns to skill, we obtain FS22 � 0 � FS21. It follows thatFS2�2�tF � t; t

�� FS2

�t̂F ; 2�tF � t̂F

�for t > 2�tF�t̂F . Thus, we have @

�wF (t)� wH(t)

�=@t = 0.

Therefore, we have wF (t) � wH(t) for t 2�2�tF � t̂F ; 2�tH � t̂H

�.

(4) t 2 [2�tH � t̂H ; tFmax]If t 2 [2�tH � t̂H ; tFmax], we have

wF (t)� wH(t)

= FS�2�tF � t; t

�+p�C(t̂

H � t̂F )2

+

Z t̂F

2�tF�tFS1��; 2�tF � �

�d� � FS

�2��tF � t; t

��Z t̂H

2�tH�tFS1��; 2�tH � �

�d�

= �(1� �)p�C t̂F

2+�FS�2�tF � t; t

�� FS

�2��tF � t; t

��+

Z t̂F

2�tF�tFS1��; 2�tF � �

�d�

�Z �t̂F

2��tF�tFS1��; 2��tF � �

�d�:

Clearly, lim�!1�wF (t)� wH(t)

�= 0. In addition, we have

@�wF (t)� wH(t)

�@�

=p�C t̂

F

2� t̂FFS1

��t̂F ; 2��tF � �t̂F

�� 2�tF

Z �t̂F

2��tF�tFS12

��; 2��tF � �

�d�

=t̂F

2�tF

"FS��t̂F ; 2��tF � �t̂F

��

� 2�tFFS1��t̂F ; 2��tF � �t̂F

�#� 2�tF

Z �t̂F

2��tF�tFS12

��; 2��tF � �

�d�

=t̂F

2�tF�FS�t̂F ; 2�tF � t̂F

�� 2�tFFS1

�t̂F ; 2�tF � t̂F

��� 2�tF

Z �t̂F

2��tF�tFS12

��; 2��tF � �

�d�

=t̂F�2�tF � t̂F

�2�tF

�FS2�t̂F ; 2�tF � t̂F

�� FS1

�t̂F ; 2�tF � t̂F

��� 2�tF

Z �t̂F

2��tF�tFS12

��; 2��tF � �

�d�:

32

We have FS1�t̂F ; 2�tF � t̂F

�� FS2

�t̂F ; 2�tF � t̂F

�due to Appendix A. Because of the supermod-

ularity of FS , we have F12 5 0. It follows that @�wF (t)� wH(t)

�=@� � 0. Therefore, we

have wF (t) = wH(t) for t 2 [2��tF � t̂H ; tFmax] when � > 1.

References

[1] Antras, Pol and Davin Chor (2013), �Organizing the Global Value Chain�, Econometrica,Vol. 81, No. 6, pp. 2127�2204.

[2] Antras, Pol and Esteban Rossi-Hansberg (2009), �Organizations and Trade�, AnnualReview of Economics, Vol. 1, pp. 43-64.

[3] Antras, Pol, Luis Garicano, and Esteban Rossi-Hansberg (2006), �O¤shoring in a Knowl-edge Economy�, Quarterly Journal of Economics, Vol. 121, Issue 1, pp. 31-77.

[4] Baldwin, Richard and Anthony J. Venables (2013), �Spiders and snakes: O¤shoring andagglomeration in the global economy�, Journal of International Economics, Vol. 90, Issue2, pp. 245-254.

[5] Bombardini, Matilde, Gallipoli, Giovanni, and Pupato, Germán (2012) �Skill Dispersionand Trade Flows,�American Economic Review, Vol. 102, No. 5, pp. 2327-2348.

[6] Bombardini, Matilde, Gallipoli, Giovanni, and Pupato, Germán (2014) �UnobservableSkill Dispersion and Comparative Advantage,�Journal of International Economics, Vol.92, No. 2, pp. 317-329.

[7] Bougheas, Spiros and Raymond Riezman (2007), �Trade, and the Distribution of HumanCapital�, Journal of International Economics, Vol. 73, Issue 2, pp. 421-433.

[8] Chang, Pao li and Fali Huang (2014) �Trade and Divergence in Education Systems,�International Economic Review, Vol. 55, No. 4, pp. 1251-1280.

[9] Costinot, Arnaud (2009), �An Elementary Theory of Comparative Advantage�, Econo-metrica, Vol. 77, No. 4, pp. 1165-1192.

[10] Costinot, Arnaud and Jonathan Vogel (2010), �Matching and Inequality in the WorldEconomy�, Journal of Political Economy, Vol. 118, Issue 4, pp. 747-786.

[11] Costinot, Arnaud, Jonathan Vogel, and Su Wang (2013), �An Elementary Theory ofGlobal Supply Chain�, Review of Economic Studies, Vol. 80, Issue 1, pp. 109-144.

[12] Feenstra, Robert C and Gordon H, Hanson (1996), �Globalization, Outsourcing, andWage Inequality,�American Economic Review, vol. 86, No. 2, pp. 240-245.

33

[13] Feenstra, Robert C and Gordon H, Hanson (1997), �Foreign direct investment and relativewages: Evidence from Mexico�s maquiladoras�, Journal of International Economics, Vol.42, Issue 3-4, pp. 371-393.

[14] Feenstra, Robert C and Gordon H, Hanson (1999), �The Impact of Outsourcing and High-Technology Capital on Wages: Estimates For the United States, 1979�1990�, QuarterlyJournal of Economics, Vol. 114, Issue 3, pp. 907-940.

[15] Grossman, Gene (2004) �The Distribution of Talent and the Pattern and Consequencesof International Trade,�Journal of Political Economy, Vol. 112, No. 1, pp. 209-239.

[16] Grossman, Gene and Esteban Rossi-Hansberg (2008), �Trading Task: A Simple Theoryof O¤shoring�, American Economic Review, Vol. 98, No. 5, pp. 1978-1997.

[17] Grossman, Gene and Rossi-Hansberg (2012) �Task Trade between Similar Countries�,Econometrica, Vol. 80, No. 2, pp. 593�629.

[18] Grossman, Gene and Maggi, Giovanni (2000) �Diversity and Trade,�American EconomicReview, Vol. 90, No. 5, pp.1255-1275.

[19] Hummels, David, Jun Ishii, and Kei-Mu Yi (2001), �The Nature and Growth of VerticalSpecialization in World Trade�, Journal of International Economics, Vol. 54, Issue 1, pp.75-96.

[20] Hummels, David, Rasmus Jorgensen, Jakob Munch, and Chong Xiang (2014), �The WageE¤ects of O¤shoring: Evidence from Danish Matched Worker-Firm Data�, AmericanEconomic Review, Vol. 104, No. 6, pp. 1597-1629.

[21] Jones, Ronld W. and Henryk Kierzkowski (1990) �The Role of Services in Productionand International Trade: A Theoretical Framework�, ch. 3 in Jones and Krueger, (eds.),The Political Economy of International Trade (Blackwell�s), pp. 31-48.

[22] Ngienthi, Wanida, Yan Ma, and Fumio Dei (2011), �Supermodularity and Global SupplyChains without the South�, Review of International Economics, Vol. 21, Issue 3, pp.562-567.

[23] Nocke, Volker and Stephen Yeaple (2008), �An Assignment Theory of Foreign DirectInvestment�, Review of Economic Studies, Vol. 75, Issue 2, pp. 529-557.

[24] Ohnsorge, Franziska and Daniel Tre�er (2007), �Sorting It Out: International Trade withHeterogeneous Workers�, Journal of Political Economy, Vol. 115, No. 5, pp. 868-892.

[25] Yeaple, Stephen (2005), �A Simple Model of Firm Heterogeneity, International Trade,and Wages�, Journal of International Economics, Vol. 65,Issue 1, pp. 1-20.

[26] Yi, Kei-Mu (2003), �Can vertical specialization explain the growth of world trade?�,Journal of Political Economy, Vol. 111, No. 1, pp. 52�102.

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