Tsukuba Economics Working Papers
No. 2009-007

Variety-skill complementarity: a simple resolutionof the trade-wage inequality anomaly
byYoshinori Kurokawa
March 2009This version: April 2010

UNIVERSITY OF TSUKUBA
Department of Economics
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPAN

Variety-skill complementarity: a simple resolution of the
trade-wage inequality anomaly
Yoshinori Kurokawa
University of Tsukuba
April 30, 2010
(Published in Economic Theory, 46 (2011), 297-325.)

Abstract
The Stolper-Samuelson theorem predicts that the relative wage of high-skilled to
low-skilled labor will increase in the high-skill abundant U.S. but decrease in low-skill
abundant Mexico after trade liberalization, while it actually began to rise in both countries in the late 1980s. We present a simple resolution of this "trade-wage inequality
anomaly" in a model of variety trade. Variety trade increases the variety of intermediate
goods used by the …nal good. If the varieties and high-skilled labor are complements, theskill premium rises in both countries. This linking of imports of new foreign varieties—
the extensive margin— to wage inequality is compatible with evidence. Our numericalexamples illustrate that small amounts of variety trade can produce a signi…cant increase in relative wage.
Keywords Trade, Wage inequality, Variety-skill complementarity, Extensive margin
JEL Classi…cation F12, F16

Tel.&Fax: +81-29-853-7426, E-mail address: kurokawa@dpipe.tsukuba.ac.jp. I am very grateful to
Timothy Kehoe for his invaluable guidance and to Cristina Arellano, Michele Boldrin, and Terry Roe for
their helpful advice. I am also grateful to Pedro Amaral, Winston Chang, Julian Diaz, Koichi Hamada,Katsuhito Iwai, Hiroshi Mukunoki, Michihiro Ohyama, Martin Schneider, Yoshimasa Shirai, the editor
Nicholas Yannelis, and two anonymous referees for their detailed comments and suggestions. I wish to thankseminar participants at the Trade and Development Workshop at Minnesota, the Minneapolis Fed, Tokyo,
Santa Clara, West Virginia, Keio, SUNY-Bu¤alo, Hitotsubashi, the MVEA 43rd Annual Meeting, the MEA
71st Annual Meeting, the Spring 2007 Midwest International Economics Meetings, the 2007 JEA Annual
Spring Meeting, and the 2008 Midwest Macro Meetings for their useful comments. I also thank Andrew
Cassey and Kim Strain for their careful correction of my English and Daniel Chiquiar and Gordon Hanson
for their help with Mexican data. This paper previously circulated under the titles "Trade and Variety-Skill
Complementarity" and "Variety-Skill Complementarity Hypothesis."11Introduction

One of the most well documented empirical facts in recent U.S. economic history is that,as Fig. 1 shows, the relative wage of high-skilled to low-skilled labor began to rise in
manufacturing industries in the late 1980s, and this fact was observed in Mexico as well.1
As can be seen, these two countries showed a surprisingly similar timing of the rise inrelative wage in the late 1980s and early 1990s.2 The data also show that, as in Fig. 2,
U.S.-Mexican trade (as a percent of U.S. GDP) was dramatically increasing during thesame period.3 Hence, this increased trade might have contributed to the recent increase inskill premium in these countries. However, there are a number of criticisms on this line ofthought.
One criticism is based on a "trade-wage inequality anomaly." The standard HeckscherOhlin (H-O) model demonstrates a discrepancy between the model and data. The StolperSamuelson theorem of the H-O model predicts that the relative wage of high-skilled to lowskilled labor will increase in the high-skill abundant U.S. but decrease in low-skill abundant
Mexico after trade liberalization. The H-O model thus generates a positive relationshipbetween the trade and wage inequality in the U.S. but generates a negative relationship in
Mexico. On the other hand, as we have seen in Fig. 1 and 2, the data generated a positive
relationship between the trade and wage inequality in both countries in the late 1980s andearly 1990s. This is a "trade-wage inequality anomaly."
A second criticism is based on price movements. The Stolper-Samuelson theorem predicts the same direction of movement of the relative price of high-skill to low-skill intensive
goods and the relative wage of high-skilled to low-skilled labor since the rise in the relative
wage of high skill should be driven by the rise in the relative price of high-skill intensive
good in the high-skill abundant U.S. However, data show that the relative prices of highskill intensive goods were declining or constant during the 1980s while the relative wage ofhigh skill was increasing in the U.S. (Lawrence and Slaughter 1993).
A third criticism is based on the volume of trade. Trade-based explanations have often
1Here, we use non-production and production workers as an index for high-skilled and low-skilled workers
in the U.S. and Mexican manufacturing industries (Berman et al. 1994; Robertson 2004). We calculate theU.S. relative wage during the period 1980-2000 on the basis of the U.S. Annual Survey of Manufactures(ASM). On the other hand, we calculate the Mexican relative wage on the basis of the Mexican Monthly
Industrial Survey (Encuesta Industrial Mensual, or EIM) by …rst calculating the average monthly wage of
non-production relative to production labor. The annual average is then produced by averaging this monthly
relative wage.
2
As shown in Fig. 1, the non-production/production wage ratio in Mexico reached a plateau after the
North American Free Trade Agreement (NAFTA) was enacted in 1994. Esquivel and Rodríguez-López(2003) also show the same movements of Mexican wages. Robertson (2004) argues, using the Mexican
Industrial Census, that the Mexican skill premium declined from 1994 to 1998. We also note that the U.S.
and Mexican relative wages are shown on di¤erent scales in Fig. 1, for here we want to emphasize thequalitative movements of these series. In Section 4, we will emphasize the quantitative di¤erence of thesame series during the period 1987-1994.
3
Here, U.S.-Mexican trade is de…ned by the sum of U.S. exports to and U.S. imports from Mexico. The
data for trade and GDP are from the International Trade Administration and the Bureau of Economic
Analysis.

2

been criticized due to the small volume of trade as has been shown in Fig. 2. Krugman(1995) provides a theoretical argument to explain why the small volume of trade in the U.S.
makes it unlikely that trade can account for the change in wages.4
Thus, mainly due to these criticisms, trade-based explanations for rising wage inequality
have been minor in economic academia.5 Rather, major explanations have been based ontechnological change. A sharp decline in equipment prices in the 1980s led to an increasein the demand for high-skilled workers, who were complements for this equipment, and a
decline in the demand for low-skilled workers, who were substitutes (Krusell et al. 2000).6
This technology-based explanation is consistent with the decline in the price of high-techgoods and the increase in the wage inequality both in the U.S. and in Mexico.
We now propose a simple theoretical framework to illustrate the possibility of an increasein wage inequality in each of the trading countries as a result of even small amounts of trade,and this can happen without a rise in the relative price of high-skill intensive good.
We …rst present a simple resolution of the trade-wage inequality anomaly. Our resolution is based on a straightforward application of the well-known model of variety trade inintermediate goods due to Ethier (1982).7 Ethier’s model demonstrates that the variety ofintermediate goods, which …nal goods producers can use, increases in both countries after
trade and, therefore, their production increases through the higher productivity caused byincreased number of inputs. Let us emphasize again that Ethier says something increases
in both countries after trade.
Upon application of this logic, we show that the variety trade in di¤erentiated intermediate goods increases the variety of intermediate goods used by the …nal good in bothcountries. The increased variety of inputs then can mean the increased variety of tasks to
be handled and thus corresponds to higher demand for high-skilled labor. Through thisvariety-skill complementarity, the relative wage of high skill— the skill premium— rises inboth countries. Thus our model provides a resolution of the trade-wage inequality anomaly.8 Moreover, our model manages to capture an interesting di¤erence in U.S. and Mexican4

It should be noted that Krugman (2008) argues that, due to the increase in U.S. trade with poor countries
and the growing fragmentation of production, it is no longer safe to assume that the e¤ect of trade on wage
inequality is very minor, although he admits that it is hard to prove the actual e¤ect.
5
Many papers relate trade to wage inequality in the U.S. and Mexico. For example, Borjas and Ramey(1994) show how trade volumes can be linked to U.S. inequality, and Harrigan and Balaban (1999) estimate
an econometric general equilibrium model of U.S. wages as a function of prices, technology, and factor
supplies. Revenga (1997) and Hanson and Harrison (1999) link changes in Mexican inequality to changes intrade policy, and Verhoogen (2008) links quality upgrading for export to Mexican inequality. See Feenstra
and Hanson (2003) for a survey on trade and inequality.
6
Berman et al. (1994), Karni and Zilcha (1995), Berman et al. (1998), and Katz and Autor (1999) also
relate technological change to wage/income inequality. Kremer and Maskin (1996) link technological changeor skill-distribution to both wage inequality and segregation by skill, and their model is extended by McCannand Trokhimtchouk (2010).
7
Ethier’s (1982) model is an intermediate-good version of Krugman’s (1979) variety-trade model.
8
Dinopoulos et al. (2009) also link variety trade to wage inequality. Their model, however, modi…es
the standard one-sector variety-trade model by introducing quasi-homothetic preferences for varieties andnon-homothetic technology in the production of each variety, thus relating an increase in the output of eachvariety— not an increase in the number of variety— to an increase in the relative demand for high-skilled3

wages: the smaller country Mexico shows a much larger increase in the skill premium thanthe larger country U.S. does as shown in Fig. 1.
We next argue that our model, though simple, has the capability of being applied to realworld data, although a de…nitive answer must wait for more serious empirical work. In fact,
the linking of imports of new foreign varieties— the extensive margin— to wage inequality
is compatible with available empirical evidence. The correlation between the growth in theextensive margin and the growth in the relative wage of high-skilled labor was high, over
0.93, in both U.S. and Mexican manufacturing industries during the period 1980-2000. The
variety-skill complementarity appears to be a plausible assumption as shown by the facts
in regards to U.S. production organization. The movements of the relative price of highskill to low-skill intensive goods and the relative wage of high-skilled to low-skilled labor
are also consistent with the observations in the U.S., so our model does not require theStolper-Samuelson price-wage mechanism.
We …nally show several numerical examples with plausible parameters to see if we canobtain a signi…cant increase in skill premium with relatively small amounts of trade. In
fact, our numerical examples illustrate the possibility that increased U.S.-Mexican manufacturing variety trade, which is a small fraction of U.S. manufacturing GDP, is capable ofsigni…cantly contributing to the increase in skill premium in both U.S. and Mexican manufacturing industries from 1987 to 1994. We also show that trade and technological change
are complementary in that they both can contribute to increased skill premium in bothcountries.
Of course, other economists have also been successful in resolving the anomaly on thebasis of trade models. One major explanation is based on foreign direct investment. Feenstra
and Hanson (1996) show that foreign direct investment shifts production activities fromthe North to the South— an endogenous transfer of technology— and thus increases theNorth’s outsourcing the low-skill intensive goods to the South, and these goods are highskill intensive goods by the South standards.9
A second major explanation is based on the Schumpeterian mechanism. Dinopoulos andSegerstrom (1999) show that trade increases the relative price of innovation (the reward forinnovation relative to the current level of R&D di¢ culty), thus encouraging high-skill intensive R&D investment in each country.10 Acemoglu (2003) shows that trade "induces"
skill-biased technological change in the U.S., and this improved technology can be translabor by each variety.
9
Zhu and Tre‡er (2005) also show a mechanism closely related to this mechanism by Feenstra and Hanson(1996). We note that Feenstra and Hanson (1996) resolve the trade-wage inequality anomaly observed during
the 1980s on the basis of a skill intensity reversal: intermediate goods— previously produced in the North but
now produced in the South— are relatively high-skill intensive in the South but relatively low-skill intensivein the North. This assumption, however, poses an empirical challenge since past research has found little
evidence for the so-called "factor intensity reversal" over the period. We, however, resolve the anomaly
without assuming this skill intensity reversal.
10
Dinopoulos and Segerstrom (1999) show that a contemporaneous correlation between an index of therelative price of innovation and an index of the U.S. skill premium was 0.80 during the period 1963-1989.

4

ferred to other countries by spillover e¤ects. Thus these explanations also demonstrate therise in the relative wage of high-skilled labor in each of the trading countries.11
Compared to these past studies, without introducing any foreign direct investment or
dynamic Schumpeterian mechanism, this paper is successful in formulating a simpler trademodel in which trade between two countries can cause an increase in wage inequality inboth countries. Moreover, this paper is the …rst to quantitatively show the possibility that
trade, even small in volume, signi…cantly contributes to the increase in skill premium bothin the U.S. and in Mexico.
The rest of this paper is organized as follows: in Section 2, we formulate a very simple
model of variety trade, and we provide a resolution of the trade-wage inequality anomaly.
Section 3 shows that our model is compatible with available empirical evidence. In Section
4, we present our numerical examples. Finally, we summarize main results and mention
future research in Section 5.

2

The Model

In this section, we …rst formulate our model. Second, we explicitly solve the model and showthat variety trade can increase the skill premium in both countries. Finally, we discuss some
economic implications of the derived results.

2.1

Ingredients of the Model

Consider an economy with a …nal good sector and an intermediate goods sector. There are
two types of skills: high-skilled and low-skilled labor. Their endowments are given by H
and L, respectively. These skills di¤er in that the high-skilled labor can do both high-skill
and low-skill tasks while the low-skilled labor can do only a low-skill task. As will be shown
later, this excludes the possibility that the relative wage of high-skilled to low-skilled labor
is less than one in equilibrium.
The production side is as follows. The …nal good sector is perfectly competitive andnon-traded. It uses a continuum [0; n] of di¤erentiated intermediate goods and the high skill.
The technology is given by the following constant returns to scale production function:

y=

" Z
n

=

x (j) dj

0

+H

#1=

;

where y is the output of …nal good, x (j) and H are the demand for di¤erentiated intermediate good j and high skill, and 0 << 1. In our model, handling a variety of inputs is

11
Acemoglu (2003) might not be successful in explaining the fact that the U.S. and Mexico showed thesurprisingly similar timing of the rise in skill premium. This is because the rise in skill premium in Mexico
should be driven by the spillover e¤ects in his model but this spillover process usually takes many years.

5represented as handling a variety of tasks and thus corresponds to a high-skill task.12 We
thus assume that < 0, that is, the elasticity of substitution between the varieties and highskill is given by= 1= (1) < 1. We de…ne this case< 0 ( < 1) as the case where thevarieties and the high skill are complements.13
On the other hand, the di¤erentiated intermediate goods sector is monopolistically competitive. Firms are symmetric and follow Cournot pricing rules.14 There is also free entry
and exit. The intermediate goods can be traded. Each variety does not require handling avariety of inputs and thus can use the low-skill. The technology of each variety is given by
the following increasing returns to scale production function:x (j) =

1b

max [l (j)f; 0] ; 8j;

where l (j) is the demand for low skill to produce each variety j, f is the …xed cost in termsof low skill, and b is the unit low-skill requirement. We note that the high skill can also do
this low-skill task.
The demand side is as follows. For simplicity, we focus on a representative consumer
who has the endowments of high skill and low skill: H and L. He or she consumes the …nal
good. His or her utility function is given byu(c) = c;where c is the quantity of the …nal good he or she consumes. His or her budget constraint
is given bypy c = wH H S + wL LS ;where py is the price of the …nal good, wH is the wage for the high skill, and wL is thewage for the low skill. H S is the supply of high skill for the …nal sector, and LS is the
supply of low skill for the intermediate sector, which can include the high skill. We assume
HS

0

H, L

LS

L + H, and H S + LS = H + L.

The feasibility conditions for high-skilled labor and low-skilled labor are
H = HS;
12

In some papers, the number of inputs plays a role in a related way. Blanchard and Kremer (1997) de…nethe index of complexity which relates the increased number of inputs to more complexity in production
processes. Kremer (1993) shows that higher skill workers will use more complex technologies that incorporate
more tasks.
13
We note that we can generalize the production function of the …nal good by assuming that the …nal gooduses three factors: varieties, high-skilled labor, and low-skilled labor. The results, however, are unchanged aslong as we assume that the varieties are more complementary to the high-skilled labor than to the low-skilledlabor. We also note that, as will be noted in footnote 15, switching the role of high-skilled and low-skilledlabor but assuming > 0 (the varieties and the low skill are substitutes) gives the same results in this model.
14
In this model with a continuum [0; n] of di¤erentiated intermediate goods, Bertrand pricing rules givethe same results as the Cournot pricing rules do.

6

Z n

l (j) dj = LS :02.2

Explicit Solutions and the Autarky Equilibrium

We explicitly solve our model. First, we derive the solutions in the intermediate goods
sector.
Given an arbitrary n, each producer of a variety facing the indirect demand by the …nal
good sector maximizes the pro…t p (j) x (j)wL bx (j)wL f where p (j) is the price ofintermediate good j. By using the symmetry x (j) = x, each variety’s output x and price p
corresponding to this n can be given by

x=

"

=(1

wL b
py n ( = ) 1

# 1=

)

n=

p=

wL b

H; 8j;; 8j:

Since the price does not depend on the number of varieties n, the price when the pro…tof each variety becomes zero by the free entry and exit is also given by p = wL b= , and thezero pro…t condition px

bx

f = 0 gives the output x of each variety. The equality oflabor demand and supply in intermediate goods sector, n (bx + f ) = LS , gives the numberof varieties n. Thus the price p and output x of each variety and the number of varieties n
are given by
p=

wL b

x=

f
b (1

n=

LS (1
f

; 8j;
)

; 8j;
)

:

We next derive the solutions in the …nal good sector.
In our model with the CES production function, it is not di¢ cult to obtain an explicit
solution for the demand for each variety by the …nal good sector, but we solve the maximization problem for the …nal good sector by means of the following short-cut method.
De…ne a new good
X=

Z n

1=

x (j) dj0and its price pX , and we can show desired results more easily.
The pro…t of the …nal good sector now becomespy (X + H )1=

pX X

wH H:

First, by solving the cost minimization problem for the good X, we …nd that the price7

of X is:
pX =

Z n

(

p (j) =(

1)

dj

1)=

p;

1)=

b

1)=

:

0

By symmetry p (j) = p, this becomes
pX = n (where p = wL b= .
Dividing both sides by wL gives:
pX
= n(
wL

:

(1)

Second, we solve for X. Since the technology of the …nal good shows the constantreturns to scale with X and H, we have the following equality:y=pX X + wH H:
py

On the other hand, the demand for the …nal good is given byc=wH H S + wL LS:
py

The …nal good market clearing y = c and the feasible condition for the high skill H = H S
then give
X=

wL LS
:
pX

(2)

Third, we solve for the relative wage of high-skilled to low-skilled labor wH =wL . The…rst-order conditions with respect to X and H for the …nal sector give
1

X
H

=

pX
:
wH

By using (2) and H = H S , in autarky equilibrium the relative wage of high-skilled labor
wH =wL is given by
wH
=
wL

LS
HS

pX
wL

1

:

(3)

This autarky equilibrium is represented in Fig. 3-a and 3-b. The demand for high skill
and low skill by the production side, H and L, is represented by the isoquant curve of the…nal good: y = [ (wL L=pX ) + H ] 1= which is given by y = (X + H )1= and (2). On
the other hand, the supply of labor for each sector, H S and LS , is represented by AB. The
autarky equilibrium is then achieved at A in Fig. 3-a or C in Fig. 3-b, and thus the relative
wage of high skill wH =wL , given by the slope of the isoquant curve, is greater than or equal8to one before trade.
Since the focus of this paper is on the skill premium, in the following main text weconcentrate on the interesting case as shown in Fig. 3-a, in which the relative wage of highskill given by (3) is greater than one. Thus the high skill and low skill each do their own
task, letting H S = H and LS = L. In Appendix A.1, we brie‡y analyze the case as shown
Fig. 3-b, in which the relative wage of high skill given by (3) is one and the high skill isdoing both high-skill and low-skill tasks.

2.3

Trade Equilibrium and a Resolution of the Trade-Wage Inequality
Anomaly

Consider two countries: country 1 and country 2. They have identical technologies andpreferences. They can be di¤erent in their endowments of high-skilled and low-skilled labor.
We assume that the relative wage of high-skilled to low-skilled labor is greater than one inboth countries before trade as shown in Fig. 3-a. Thus the high skill and low skill each dotheir own task, letting HiS = Hi and LSi = Li in each country i, i = 1; 2. The number ofvarieties is thus given by ni = Li (1)=f in each country i before trade.

Let these two countries trade with each other. Then, from the derived solutions in theintermediate goods, we easily get the following information. The supply of labor for theintermediate goods sector, which is given by LSi = Li before trade, cannot fall below thisLi after trade. This implies that the number of varieties produced within each country i,which is given by ni = Li (1)=f before trade, cannot fall below this autarky level ni after

trade. Thus the total number of varieties which is available to the …nal good sector after
trade, n1 + n2 , is greater than the autarky level ni in each country i.
Given this information, we show the following results. Here, let us focus only on country
1:
First, pX1 now becomes
pX1 =

Z n1

p (j)

=(

1)

dj +

0

Z n1 +n2

(

p (j)

=(

1)

dj

1)=

:

n1

By the symmetry p (j) = p1 for j 2 [0; n1 ] and p (j) = p2 for j 2 [n1 ; n1 + n2 ], thisbecomespX1 = n1 p1

=(

1)

=(

1) (

=(

1)

+ n 2 p21)=;

where p1 = wL1 b= and p2 = wL2 b= .
Dividing both sides by wL1 gives:
pX1
=
wL1

n1 + n2

wL2
wL1

!(

1)=

b

:

(1’)

Thus we see that the trading level of pX1 =wL1 given by (1’) becomes lower than the9(autarky level pX1 =wL1 = n1

1)=

smaller due to n1 + n2 (wL2 =wL1 )b==(given by (1) since the coe¢ cient of b=1)> n1 and (becomes

1) = < 0.

Second, from (2) we see that X1 increases after trade since pX1 =wL1 decreases and LS1 ,which is L1 before trade, does not decrease. This implies that the marginal product ofhigh-skilled labor given by M P H1 = (X1 + H1 )(1= ) 1 H1 1 increases for any H1 . That is,the demand for high skill by the …nal good shifts upward. Since the supply of high skill forthe …nal good, which is H1 before trade, does not increase, this implies that the real wageof high skill wH1 =py1 increases.
Finally, from (3) we see that since< 0 ( < 1), that is, since the varieties and highskill are complements, the relative wage of high skill wH1 =wL1 — the skill premium— increases
after trade. This is because (pX1 =wL1 ) increases and LS1 =H1S

1

, which is L1 =H1

1

before trade, does not decrease.15
Thus it follows that the high skill and low skill each do their own task after trade as well
as before trade. That is, the supply of labor for the …nal and intermediate sectors remains
at H1S = H1 and LS1 = L1 , respectively. Hence, the number of varieties produced withincountry 1 after trade remains at the autarky level n1 = L1 (1) =f .

We note that the above results are also obtained in country 2. Hence, we get thefollowing results:
The variety trade in intermediate goods causes the total number of varieties available
to the …nal good sector to simply increase from ni to n1 + n2 , the sum of the autarky
levels, in each country i. This causes pXi =wLi to decline and thus causes Xi to increase inboth countries. Consequently, the demand for high skill shifts upward, thus increasing thereal wage of high skill wHi =pyi in both countries. Moreover, since the varieties and highskill are complements, the decrease in pXi =wLi also increases the relative wage of high skill
wHi =wLi — the skill premium— in both countries. Thus our model has provided a resolutionof the trade-wage inequality anomaly.
We can derive more results from the above discussion. First, since the number of varieties
before trade is given by ni = Li (1)=f in each country i, the ratio of the number of varieties

produced within each country before trade is given by n1 =n2 = L1 =L2 . This implies thatthe rate of increase in ni is smaller in a country with the larger size of Li , and, therefore,
the rate of decrease in pXi =wLi is also smaller as can be seen in (1’). Hence, the rise in therelative wage of high skill wHi =wLi is smaller in a country with the larger size of Li as canbe seen in (3).16
Second, if= 0 ( = 1), that is, if the production function of the …nal good is given by

the Cobb-Douglas function, from (3) we see that the relative wage of high skill wHi =wLi is15

We note that switching the role of high-skilled and low-skilled labor but assuming > 0 (the varieties
and the low skill are substitutes) gives the same results in this model.
16
In fact, this prediction is compatible with the following observations: the number of production workers
in manufacturing industries was much greater in the U.S. than in Mexico during the period 1980-1994. As
shown in Fig. 1, the U.S. skill premium increased by 12.5 percent from 1980 to 1994, while the Mexican
skill premium increased by 48.9 percent.

10

not a¤ected by the decrease in pXi =wLi and therefore does not change after trade in either
country.

2.4

Economic Implications of the Results

Before moving on to Section 3, we need to consider economic implications of some of theresults which have been shown in Section 2.3 on the basis of the explicit solutions to themodel. First, we explain the economic reason why the good X increases after trade, thatis, why the M P H increases after trade.
As we have seen, the activities in the intermediate goods sector never change at all ineach country after trade. Some changes, however, do occur after trade. The number ofvarieties used by the …nal good sector increases, while the input quantity of each variety
used by the …nal good sector decreases in each country since each variety is shared by two
countries.
Can the e¤ect of increase in the number of varieties be canceled by the e¤ect of decreasein the input quantity of each variety? The answer is no. This is because the e¤ect of increasein the number of varieties is greater than the e¤ect of decrease in the input quantity of eachvariety. This is the crucial e¤ect in the variety-trade models which Ethier (1982) called
the "international returns to scale." That is, the increased number of inputs translates intohigher productivity. Thus the good X increases after trade, that is, the M P H increases
after trade.
We next explain the economic reason why the relative wage of high skill can rise after
trade. Now the …nal good market clearings yi = ci in each country i, i = 1,2, before trade
are given byyi =wHi Hi + wLi Li:
pyi

Since wHi =pyi = M P Hi , this becomes the following:yi = M P H i Hi +

wLi
Li :
pyi

As we have seen, the marginal product of high skill increases in each country after trade.
For the same reason, the output of …nal good also increases in each country after trade.
Since M P Hi = Xi + Hi

(1= ) 1Hi

1and yi = Xi + Hi

1=

, it can be shown thatthe rate of increase in M P Hi is greater than the rate of increase in yi since< 0, thatis, since the varieties and high skill are complements. This relationship and the …nal goodmarket clearing condition yi = M P Hi Hi + wLi =pyi Li imply that the rate of increase in
M P Hi should be greater than the rate of change in wLi =pyi . In other words, the rate ofincrease in the real wage of high skill wHi =pyi is greater than the rate of change in the real
wage of low skill wLi =pyi . Thus the relative wage of high skill wHi =wLi can increase in eachcountry i.

113

Indirect Evidence for Mechanism

In this section, we …rst show that the linking of imports of new foreign varieties— the extensive margin— to wage inequality is compatible with available empirical evidence. Second, we
claim that variety-skill complementarity appears to be a plausible assumption. We …nallyshow that our model demonstrates price movement consistent with observed facts.

3.1

Extensive Margin and the Relative Wage of High-Skilled Labor

Fig. 4-a plots the 1980-2000 data on the growth in what Kehoe and Ruhl (2009) call the"least traded goods" in U.S. manufacturing imports from Mexico and on the relative wageof high-skilled to low-skilled labor in U.S. manufacturing industries. Fig. 4-b, on the other
hand, plots the growth in the least traded goods in Mexican manufacturing imports fromthe U.S and the relative wage in Mexican manufacturing industries during the same period.
Kehoe and Ruhl classify the set of goods which accounts for only 10 percent of tradeas the least traded goods. Here, we use the least traded goods for measuring the extensive
margin. The data for the least traded goods growth are the Standard International Trade
Classi…cation (SITC) (revision 2) 3-digit manufacturing data from the OECD International
Trade by Commodities Statistics (ITCS).17 The source of data for the U.S. and Mexican
relative wages is the same as for Fig. 1.
As can be seen in Fig. 4-a and 4-b, the least traded goods that account for 10 percent
of U.S. manufacturing imports from Mexico in 1980 account for 42.5 percent in 2000, andthe least traded goods that account for 10 percent of Mexican manufacturing imports fromthe U.S. in 1980 account for 17.5 percent in 2000, respectively. This indicates that eachcountry started importing goods that it had not imported before or had imported only insmall quantities. Moreover, these …gures reveal that this growth in the least traded goods
was highly correlated with the growth in the relative wage in each country over 1980-2000.
In fact, the correlation between these two series was high in each country: it was 0.932 and
0.947 in the U.S. and Mexico, respectively. Thus the linking of the extensive margin to theskill premium is compatible with this evidence in both countries.
It is worth noting that the method by Kehoe and Ruhl (2009) used in this paper formeasuring the extensive margin is di¤erent from methods used in the few previous studiesof the extensive margin. Broda and Weinstein (2006), for example, classify a good as nottraded if the value of trade is zero, and Evenett and Venables (2002) classify a good as nottraded if its yearly value of trade is less than or equal to 50,000 1985 U.S. dollars, regardlessof the country to be studied. In Kehoe and Ruhl’s de…nition of a non-traded good, on theother hand, goods with very small but non-zero amounts of trade can also be considered,and the actual dollar value of the cuto¤ can di¤er across countries.
17
See Kehoe and Ruhl (2009) for the detailed procedure used to construct Fig. 4-a and 4-b. We note thatthe manufacturing imports in these …gures include imports of both …nal and intermediate goods. Fortunately,
however, much of the increase in trade has been in intermediate goods (Feenstra 1998).

12

3.2

Variety-Skill Complementarity

In our model, we have represented the variety of inputs as the variety of tasks which workers
need to handle. Thus it is plausible to assume that the increased variety of inputs— theincreased variety of tasks to be handled— translates into higher demand for high-skilledworkers. In fact, this variety-skill complementarity appears to be a plausible assumption
as shown by the historical facts in regards to U.S. production organization emphasized by
Mitchell (2005).
During the …rst half of the 20th century, the spread of mass production, which is characterized by Ford’s factories, led to the larger size of manufacturing plants. On the other
hand, during the second half of the century, ‡exible machine tools have allowed plants tooperate at a smaller scale. The organization of production has changed from mass production with a traditional assembly line to smaller customized batches, thus making the size ofplants smaller.18
Workers on the assembly line have a single routine task to perform; however, workers ineach batch are no longer as highly specialized in a single routine task. Each batch is highly
customizable and requires a worker who can handle a wide variety of tasks depending on thecustom features of the batch. The change in the production organization therefore a¤ected
the number of tasks and therefore a¤ected the importance of skills. As the tasks shiftedfrom a single routine task to a wide variety of tasks, the required skill shifted from low skill
to high skill.
Our assumption of the variety-skill complementarity is thus compatible with these historical facts in regards to U.S. production organization, although a de…nitive answer must
wait for serious empirical work.19

3.3

Relative Price of High-Skill Intensive Goods

The standard H-O model predicts the same direction of movement of the relative price ofhigh-skill to low-skill intensive goods and the relative wage of high-skilled to low-skilledlabor since the rise in the relative wage of high skill should be driven by the rise in therelative price of high-skill intensive good in the high-skill abundant U.S. However, data
show that the relative prices of high-skill intensive goods were declining or constant during
the 1980s while the relative wage of high skill was increasing in the U.S. (Lawrence andSlaughter 1993).
Our model demonstrates price movement consistent with this observed fact whereas the
H-O model cannot. In Section 2.4, it has been shown that the rate of change in wLi =pyi
should be smaller than the rate of increase in M P Hi since < 0. This implies that wLi =pyi
18
Milgrom and Roberts (1990) present the empirical facts on a change in the size of U.S. manufacturing
plants.
19
The above historical observation is compatible with the theoretical result obtained by Kremer (1993)
referred to in footnote 12.

13can rise (but it should rise less than M P Hi ), and therefore, the price of high-skill intensive…nal good relative to the low-skill wage, pyi =wLi , can decline. Here, let us recall that theprice of the low-skill intensive variety relative to the low-skill wage, pi =wLi , is constantat b= before and after trade. Hence, the relative price of high-skill to low-skill intensive
goods, (pyi =wLi )=(pi =wLi ), can decline while the relative wage of high skill rises, letting< 0. Thus the rise in the relative wage of high skill can happen without the rise in therelative price of high-skill intensive good.20

4

Numerical Examples

We have shown that trade— in particular, variety trade— can theoretically cause the increasein skill premium in two countries and that our model is compatible with available empiricalevidence. This section shows several numerical examples with plausible parameters to see if
relatively small amounts of variety trade can produce a signi…cant increase in skill premium
without technological change.21
An increase in variety trade is here represented as a tari¤ reduction, for a tari¤ reductionin each country can mean that each country can use more foreign varieties.22 Technologicalchange, on the other hand, is here represented as a decrease in …xed cost f . A decrease inf can cause an increase in the number of varieties, n = L (1

) =f , without an increase invariety trade and thus can cause an increase in the demand for the high skill.23

4.1

Model with Tari¤s

We introduce tari¤s into our simple model and assume that each country i, i = us; mex,
imposes iceberg tari¤si on imports from the other country, that is, the import quantityof a foreign variety is equal to the sum of the input quantity of the foreign variety used by
the …nal good and the iceberg tari¤s. We also introduce the share parameter , 0 << 1,into the production function of the …nal good:

yi =

"

Z nus +nmex
0=x (j)i dj

+ (1

) Hi

#1=

; i = us; mex:

We note that the de…nition of an equilibrium with tari¤s and all the derivations of
20

We note that the price of …nal good can be constant or increase if << 0:
Our model is very simple and thus cannot perform full-scale calibrations capturing many facts. Here,
we just want to show several numerical examples with plausible parameters.
22
In Section 2, we have looked at the movement from autarky to trade in order to show our idea in thesimplest way. However, here we begin with a trade equilibrium in order to compare our model with actual
trade data. Thus variety trade can increase due to the increased import volumes of existing foreign varieties
as well as the imports of new foreign varieties.
23
We note that in our model "technological change" refers to non-trade-based technological change which
can occur without trade, although it is possible to interpret the increased number of inputs due to trade as
trade-based technological change.
21

14

equations below are shown in Appendix A.2. We also note that our focus is on wHi =wLi > 1,
thus HiS = Hi and LSi = Li , i = us; mex.
The relative wages of high skill are now given by
wHus
1
=
wLus

(1

1
wHmex
=
wLmex

(1

)

=(

b

1)

Lus + Lmex

f
)

b

=(

1)

=(

=(

1)

wLus
(1 + mex )
wLmex

Lus

f

wLmex
(1 + us )
wLus

1)

!! (

+ Lmex

1)=

!! (

Lus
Hus
(4)
1)=

(5)
The volume of U.S.-Mexican variety trade relative to U.S. GDP is given by
2

Lus Lmex (1 +

us )

=(

Lus (wLus =wLmex ) + Lmex (1 +

1)
us )

=(

1)

= (wHus =wLus ) Hus + Lus ;(6)

where wHus =wLus is given by (4).

4.2

Numerical Examples: Variety Trade and the Skill Premium

We simulate our model, "backcasting" from 1994 to 1987, to see what changes in U.S. andMexican skill premium between 1987 and 1994 are predicted by the model.24
We …rst give plausible values to some parameters. The value of= 0:83 (= 1/1.2)

is chosen so that the markup charged by each variety is 1.2. Norrbin (1993) and Basu
(1996) both obtained empirical estimates of 1.05-1.4 for markups for intermediate goods,
indicating that our choice is plausible. We normalize b = 10 and f = 100, the choice of
which leaves our results (percent changes in skill premium) unchanged. We note that bykeeping f constant from 1987 to 1994, we assume that no technological change occurs. The
labor endowments Li and Hi , i = us; mex, are constructed from the OECD Structural
Analysis (STAN), the ASM, and the EIM data. U.S. endowments are …rst chosen fromthe data. We then calculate Lmex so that the ratio Lus =Lmex matches with the observed
ratio wLus Lus =wLmex Lmex in each year (e¤ective labor units). This is because, as will beshown later, the balance of trade holds at wLus =wLmex = 1 in each year under our choiceof parameters. We also calculate Hmex so that the ratio Hmex =Lmex matches with theobserved ratio.25
24

Bergoeing and Kehoe (2003) quantitatively test the "new trade theory" by "backcasting" from 1990 to
1961. Due to data constraint, here we use data from 1987 to 1994. Fortunately, however, Mexico accededto the General Agreement on Tari¤s and Trade (GATT) in 1986 and agreed to a major liberalization ofbilateral trade relations with the U.S. in 1987. Though the time-series movements of the inequality over
1987-1994 are outside the scope of this paper, it is worth noting that recent studies on the dynamics andpersistence of inequality are, for example, Ray (2006) and Alexopoulos and Cavalcanti (2009).
25
U.S. endowments are: Hus;1987 = 6707:6; Lus;1987 = 12242:7; Hus;1994 = 6274:3; Lus;1994 =
11845:3 (in thousands of workers). Mexican endowments are: Hmex;1987 = 94:6; Lmex;1987 = 222:5;
Hmex;1994 = 210:0; Lmex;1994 = 481:2; which satisfy: Lus;1987 =Lmex;1987 = 55:03; Lus;1994 =Lmex;1994 = 24:61;
Hmex;1987 =Lmex;1987 = 0:425; Hmex;1994 =Lmex;1994 = 0:436:

15

1

Lmex
Hmex

;

1

:

We then perform our simulations using the following method:
Step 1: We choose the value of .
Step 2: We simulate our model to 1994 data. We normalize
calculate the values ofandus;1994 = 0:01 and thenmex;1994 so that the U.S. relative wage in 1994 given by(4) matches with the corresponding data, satisfying the balance of trade in 1994 atwLus =wLmex = 1.
Step 3: We "backcast" to 1987. We calculate the values of tari¤sus;1987 andmex;1987 sothat the change in (6) between 1987 and 1994 is the same as the observed change in thevolume of U.S.-Mexican manufacturing variety trade relative to U.S. manufacturingGDP, satisfying the balance of trade in 1987 at wLus =wLmex = 1 as well.
Step 4: We calculate how much the U.S. and Mexican relative wages (4) and (5) increase
from 1987 to 1994.
Table 1-a reports the results of our benchmark numerical example in which(=1= 0:5). Here, the volume of U.S.-Mexican manufacturing variety trade is measured

by multiplying the Grubel-Lloyd (1975) index (a measurement of the variety-trade share)
and the volume of U.S.-Mexican total manufacturing trade. The index for a country is aweighted average over SITC (revision 2) 3-digit manufacturing industries as follows:
1P
jEXk IMk j
Pk;k (EXk + IMk )where EXk and IMk represent exports and imports of industry k. These data are obtained
from the OECD ITCS and the OECD STAN.26 The data for the U.S. and Mexican relative
wages are extracts from Fig. 1. The parameters in the model are listed separately: freeparameters in (a) and calculated parameters in (b).
As can be seen, the U.S. relative wage in 1994 is the same as the observed data, 1.780,
and the volume of U.S.-Mexican manufacturing variety trade relative to U.S. manufacturing
GDP increases by 158.2 percent as do the corresponding data. As a result, the U.S. relative
wage increases by 6.8 percent from 1987 to 1994 while the data show a 9.2 percent increase,and the Mexican relative wage increases by 34.2 percent while the data show a 43.6 percent
increase.
Thus the results indicate that increased variety trade accounts for 73.8 percent of thechange in U.S. skill premium and accounts for 78.5 percent of the change in Mexican skill
premium in the manufacturing industries during the period 1987-1994. We have here illustrated the possibility that U.S.-Mexican manufacturing variety trade, which is a small
26

We note that the manufacturing variety trade includes variety trade in both …nal and intermediate
goods. Fortunately, however, variety trade between the U.S. and Mexico primarily involves intermediate
goods (Ray 1991).

16

fraction of U.S. manufacturing GDP, can signi…cantly contribute to the increase in wage
inequality.27
We note, however, that U.S.-Mexican manufacturing variety trade is not small from
the Mexican viewpoint. In fact, U.S.-Mexican manufacturing variety trade as a fraction of
Mexican manufacturing GDP was 50.2 percent in 1987 and 75.5 percent in 1994 as shown in
the table. The table also shows the corresponding results in the model, but the results arefar from the data in terms of the percent change. This is because much of the ‡uctuationsin the trade to GDP ratio in Mexico were caused by ‡uctuations in GDP and in the realexchange rate. Our model cannot capture these ‡uctuations.
Table 1-b reports the results of the numerical example in which the reduction in f —technological change— occurs together with the tari¤ reduction from 1987 to 1994. The
results indicate that if f decreases by 10.6 percent together with the same tari¤ reduction
as in the previous benchmark example, then this can cause U.S. skill premium to increaseby the same as data and can account for 85.4 percent of the increase in Mexican skill
premium.28
Thus the results indicate that trade and technological change are complementary in thatthey both can contribute to increased skill premium in both countries.

4.3

Sensitivity Analysis

The results obviously depend on the values ofvariety ofand . We present some calculations for aand .

Table 2 reports the results of the numerical examples in whichchanged but =0:5 ( = 0:66) and == 0:83 remains un-

1:5 ( = 0:4), respectively. The results indicatethat a more negative value of — a smaller elasticity of substitution between the varieties
and high skill, — is accompanied by a larger change in skill premium in both countries.
Table 2 also reports the results of the numerical examples in whichunchanged but= 0:7 and=1 remains= 0:9, respectively. The results indicate that a change inthe value of — the elasticity of substitution between varieties— has larger e¤ects on skill
premium in the smaller country, Mexico.

5

Conclusion and Future Research

The main purpose of this paper has been to provide a simple resolution of the trade-wage
inequality anomaly in a model of variety trade compatible with available empirical evidence.
27

It should be noted that even if we use data on trade in intermediate goods instead of data on variety
trade, the numerical results would be little changed. This is because evidence suggests that a considerable
amount of trade in intermediate goods is variety trade (Turkcan 2005).
28
We note that the 10.6 percent decrease in f is equivalent to the 10.6 percent increase in the number of…rms, n = L (1
) =f , in each country.

17Section 2 has presented a simple resolution of the anomaly. We have shown that thevariety trade increases the variety of intermediate goods used by the …nal good in bothcountries; as a result, if the varieties and high skill are complements, the skill premium rises
in both countries after trade. Thus variety trade can stimulate variety-skill complementarity.
Section 3 has shown that our model is compatible with available empirical evidence.
The correlation between the growth in the extensive margin and the growth in the relative
wage of high-skilled labor was high, over 0.93, in both U.S. and Mexican manufacturingindustries during the period 1980-2000. The variety-skill complementarity appears to be aplausible assumption as shown by the facts in regards to U.S. production organization, andthe rise in the relative wage of high-skilled to low-skilled labor can happen without the risein the relative price of high-skill to low-skill intensive goods, which is also consistent withthe observed fact in the U.S.
Section 4 has shown that our model can produce a signi…cant increase in relative wage
with relatively small amounts of trade. In fact, our numerical examples have illustrated
the possibility that increased U.S.-Mexican manufacturing variety trade, which is a small
fraction of U.S. manufacturing GDP, is capable of signi…cantly contributing to the increasein skill premium in both U.S. and Mexican manufacturing industries from 1987 to 1994.
It is true that the standard H-O model is incompatible with data which show a rising
wage inequality in each of the trading countries. However, we can now show, using anelementary model, that trade— in particular, variety trade— can be a possible source of
the increased wage inequality in each country. We note that the result that trade cantheoretically increase wage inequality is not necessarily negative, for our model shows thatthe real wage of both high skill and low skill can rise despite the increase in inequality.29
Of course, room for future research still exists. First, this paper has made a theoreticalcontribution in formulating a simple trade model to illustrate the possibility of an increasein skill premium as a result of variety trade. It, however, has not yet provided a compelling
empirical analysis, although it has shown several numerical examples. This is because (a)trade in …nal goods is ignored in this paper, (b) much of output is services, which are largely
non-traded but ignored in this paper, and (c) trade is not balanced in data.
Thus, in another paper, we formulate a more general version of our variety-trade model
which can resolve problems (a)-(c) and allow us to perform a full-scale calibration.30 In themodel calibrated to the Mexican input-output data for 1987, our numerical experiments
show that the increase in U.S.-Mexican manufacturing variety trade can account for approximately 12 percent of the actual increase in Mexican skill premium over 1987-2000.
The result indicates that our mechanism is an important factor contributing to the increasein skill premium in its own right, but it seems to be quantitatively less important than29

In fact, the real wage of non-production labor has increased, and, further, the real wage of production
workers have slightly increased since the 1980s.
30
See Atolia and Kurokawa (2009).

18

other mechanisms such as technological change.31
Second, intermediate goods are horizontal in nature in our model, but it would beinteresting to modify the model to consider vertical integration as in Yi (2003). Third,
we can analyze the relationship between competition policies and wage inequality. In ourmodel, the change in the number of varieties is related to wage inequality. This implies thatgovernment can a¤ect wage inequality by entry policies which adjust the number of …rms
(see Kurokawa 2010). Finally, our model has been applied to the problems of trade between
the U.S. and Mexico, but we can also directly apply it to the problems of intra-trade among
EU nations.

Appendix
A.1

The Movement of High-Skilled LaborIn Section 2, we have focused on the interesting case in which the relative wage wH =wL
given by (3) is greater than one before trade. Thus the high skill and low skill each dotheir own task, letting H S = H and LS = L. In this appendix, we brie‡y analyze the othercase in which the relative wage wH =wL given by (3) is one and the high skill is doing both
high-skill and low-skill tasks before trade.
In the autarky equilibrium as shown in Fig. 3-b, the relative wage wH =wL given by(3) is one at C, and part of high skill is doing the low-skill task in the intermediate goods
sector. This movement of high skill from A to C maximizes the output of …nal good, thatis, the consumer’s utility.
As we have seen in Section 2.3, the case as shown in Fig. 3-a let us conclude that theskill premium rises after trade. On the other hand, if it is one before trade as shown in Fig.
3-b, it can be shown that the relative wage wH =wL rises or remains after trade, and, in any
case, the number of varieties used by the …nal good surely increases.

A.2

Model with Tari¤s

A.2.1 Equilibrium
De…nition 1 An equilibrium is prices pyus , pymex , p (j), j 2 [0; nus + nmex ], wHus , wHmex ,wLus , wLmex , and quantities cus , cmex , yus , ymex , x (j)us , x (j)mex , x (j), j 2 [0; nus + nmex ],S , H S , LS , LS , and the number of …rms in
Hus , Hmex , l (j), j 2 [0; nus + nmex ], Hus
mex
us
mex

the intermediate sectors nus , nmex , given iceberg tari¤ s

us andmex , such that

1. Final good: Given prices pyi , p (j), and wHi , quantities yi , x (j)i , and Hi solve
31

Berman et al. (1994) argue, using a regression, that skill-biased technological change can account for 40
percent of the shift in demand away from low-skilled and toward high-skilled labor in U.S. manufacturing
during the 1980s. Krusell et al. (2000) …nd, using a calibrated model, that most of the wage inequality shiftof the last 30 years in the U.S. can be explained by the capital-skill complementarity hypothesis.

19(a) U.S.
Z nus

p (j) x (j)us dj

yus =

"

max pyus yus

0

s,t,

Z nus +nmex

Z nus +nmex
0p (j) (1 +nus

us ) x (j)us dj=

x (j)us dj

+ (1

) Hus

#1=

wHus Hus;(b) Mexico
max pymex ymex

Z nus

p (j) (1 +0s,t,ymex ="mex ) x (j)mex dj

Z nus +nmex
0Z nus +nmex
nus

p (j) x (j)mex dj wHmex Hmex=

x (j)mex dj+ (1) Hmex#1=;

2. Intermediate goods: Given price wLi , quantity x (j) solves(a) U.S. j 2 [0; nus ]max p (j) x (j)wLus bx (j)wLus f; where x (j) = x (j)us + (1 +mex ) x (j)mex ;(b) Mexican j 2 [nus ; nus + nmex ]max p (j) x (j)wLmex bx (j)wLmex f; where x (j) = (1 +us ) x (j)us + x (j)mex ;3. Consumer: Given prices pyi , wHi , and wLi , quantities ci , HiS , LSi solve(a) U.S.
S
max cus s,t, pyus cus = wHus Hus+ wLus LSus ;(b) MexicoS
max cmex s,t, pymex cmex = wHmex Hmex
+ wLmex LSmex ;4. Market clearing:cus = yus ; cmex = ymex ;x (j)us + (1 +(1 +mex ) x (j)mex = x (j) ;us ) x (j)us + x (j)mex = x (j) ;j 2 [0; nus ] ;j 2 [nus ; nus + nmex ] ;SS
Hus = Hus
; Hmex = Hmex;
Z nus
Z nus +nmex
l (j) dj = LSmex ; j 2 [nus ; nus + nmex ] :l (j) dj = LSus ; j 2 [0; nus ] ;
0

nus

20

N ote : (1 +and (1 +mex ) x (j)mex , j 2 [0; nus ], means that the imports of a U.S. variety by Mexico,us ) x (j)us , j 2 [nus ; nus + nmex ], means that the imports of a Mexican variety by

the U.S. We note that the U.S. and Mexican …nal good can use only x (j)us and x (j)mex as
input, respectively.
A.2.2 Solutions
Intermediate Goods
Introducing tari¤s does not change the solutions in the intermediate goods sector.
By the symmetry p (j) = pus for j 2 [0; nus ] and p (j) = pmex for j 2 [nus ; nus + nmex ], theprice and output of each variety and the number of varieties in each country are now given
bywLus b

pus =
xus =

f
b (1

; j 2 [0; nus ] ; pmex =)wLmex b; j 2 [0; nus ] ; xmex =nus =Lus (1f); j 2 [nus ; nus + nmex ] ;fb (1; nmex =); j 2 [nus ; nus + nmex ] ;Lmex (1f):

Final Good
The pro…t of the …nal good sector now becomes) Hi )1=pyi ( Xi + (1pXi XiwHi Hi ; i = us; mex:

By solving the cost minimization problem for the good X, we can …nd that the price of X
in each country is:
pXus =

Z nus

p (j)

=(

1)

dj +

pXmex =(((1 +us ) p (j))

=(

1)

1)=

dj

;

nus

0

Z nus

Z nus +nmex((1 +mex ) p (j))

=(

1)

dj +

Z nus +nmex(

p (j) =(

1)

dj

1)=

:

nus

0

By the symmetry p (j) = pus for j 2 [0; nus ] and p (j) = pmex for j 2 [nus ; nus + nmex ],pXus = nus pus=(1)

pXmex = nus ((1 ++ nmex ((1 +mex ) pus )=(us ) pmex )

1)

=(

1)

=( 1)
+ nmex pmex

The good X in each country is now given byXus =wLus LuswLmex Lmex
; Xmex =
:
pXus
pXmex

21

(

1)=

(

1)=

;
:

The relative wage of high-skilled to low-skilled labor in each country is now given by
wHus
1
=
wLus

pXus
wLus

Lus
Hus

1

;

wHmex
1
=
wLmex

pXmex
wLmex

1

Lmex
Hmex

:

By substituting pXi , ni , and pi , i = us; mex, into the above formula, the equilibrium relative
wages are rewritten as follows:
wHus
1
=
wLus

(1

1
wHmex
=
wLmex

(1

)

=(

b

1)

Lus + Lmex

f
)

=(

b

1)

=(

=(

1)

wLus
(1 + mex )
wLmex

Lus

f

wLmex
(1 + us )
wLus

1)

!! (

+ Lmex

1)=

!! (

Lus
Hus
(4)
1)=

1

Lmex
Hmex

(5)
A.2.3 Trade
Step 1: Balance of trade.
The balance of trade (U.S. exports = U.S. imports) is given by the following:nus pus (1 +where (1 +mex ) xus;mex = nmex pmex (1 +us ) xmex;us ;mex ) xus;mex means that the imports of a U.S. variety by Mexico, and (1 +us ) xmex;usmeans that the imports of a Mexican variety by the U.S.
Step 2: The …rst-order conditions for each variety by the …nal good.
The …rst-order conditions for each variety by the …nal good givexmex;us = (1 +

1=(
us )

1)

xus;us ; xus;mex = (1 +

1=(
mex )

1)

xmex;mex :Step 3: The ratio of each country’s share in each variety.
From Steps 1 and 2, we obtain the following:(1 +us ) (1 +

(1 +

1=(
us )

1)

mex ) xus;mex

xus;us=nus pus;nmex pmex

(1 +

(1 +

nus pusus ) xmex;us=:1=( 1)
nmex pmex
xmex;mexmex ) (1 + mex )Thus, by substituting ni and pi , i = us; mex, into the above formula, the ratio of thedemand for a U.S. variety by the U.S. to the demand for a U.S. variety by Mexico becomes
xus;us = (1 +mex ) xus;mex = Lus wLus =Lmex wLmex (1 +

22

us )

=(

1)

:

;

1

:

The ratio of the demand for a Mexican variety by the U.S. to the demand for a Mexican
variety by Mexico also becomes(1 +us ) xmex;us =xmex;mex = Lus wLus (1 +

=(

mex )

1)

=Lmex wLmex :Step 4: Trade.
The volume of U.S.-Mexican variety trade is represented as the sum of U.S. exports andimports:nus pus (1 +mex ) xus;mex + nmex pmex (1 +us ) xmex;us :From Step 3, this becomesnus pus

Lmex wLmex (1 +

=(

us )

Lus wLus + Lmex wLmex (1 +

1)
us )

=(

x +nmex pmex
1) us

Lus wLus (1 +Lus wLus (1 +

mex )

mex )
=( 1)

=(

1)

+ Lmex wLmex

By substituting ni , pi and xi , i = us; mex, into the above formula, the volume of U.S.Mexican variety trade is given by
Lus wLus Lmex (1 +

us )

=(

Lus (wLus =wLmex ) + Lmex (1 +

1)
us )

=(

+
1)

Lmex wLmex Lus (1 +Lus (1 +

mex )

=(

1)

mex )

=(

1)

+ Lmex (wLmex =wLus );where the balance of trade requires the following equality:
Lus wLus Lmex (1 +

us )

=(

Lus (wLus =wLmex ) + Lmex (1 +

1)
us )

=(

=
1)

Lmex wLmex Lus (1 +Lus (1 +

mex )

=(

1)

mex )

=(

1)

+ Lmex (wLmex =wLus ):Accordingly, the volume of U.S.-Mexican variety trade is simply given by
2

=(

1)

Lus (wLus =wLmex ) + Lmex (1 +us )Lus wLus Lmex (1 +

us )

=(

1)

:

Thus the volume of U.S.-Mexican variety trade relative to U.S. GDP is given by
2

Lus Lmex (1 +

us )

=(

Lus (wLus =wLmex ) + Lmex (1 +

1)
us )

23

=(

1)

= (wHus =wLus ) Hus + Lus :(6)xmex :

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26

Table 1-a Results for benchmark numerical example

1987

1994

Change

0.018
0.502
1.630
2.020

0.046
0.755
1.780
2.900

158.2%
50.4%
9.2%
43.6%

0.015
0.912
1.667
1.678

0.038
0.924
1.780
2.252

158.2%
1.4%
6.8%
34.2%

Data
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill PremiumModel(a) ε = − 1, ρ = 0.83, f = 100,τ us,1994 = 0.01(b) α = 0.55,τ m ex,1994 = 0.18,τ us,1987 = 0.05, τ mex,1987 = 0.73
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill Premium

Table 1-b Results for benchmark numerical example with technological change

1987

1994

Change

0.018
0.502
1.630
2.020

0.046
0.755
1.780
2.900

158.2%
50.4%
9.2%
43.6%

0.015
0.920
1.630
1.641

0.038
0.924
1.780
2.252

158.2%
0.4%
9.2%
37.2%

Data
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill PremiumModel(a) ε = −1, ρ = 0.83, f 1994 = 100,τ us,1994 = 0.01(b) α = 0.55, τ mex,1994 = 0.18, τ us,1987 = 0.05,τ mex,1987 = 0.73,
f1987 = 111.82
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill Premium

27

Table 2 Results for numerical examples with di¤erent

and

1987

1994

Change

0.018
0.502
1.630
2.020

0.046
0.755
1.780
2.900

158.2%
50.4%
9.2%
43.6%

0.015
0.864
1.694
1.935

0.038
0.934
1.780
2.206

158.2%
8.1%
5.1%
14.0%

0.015
0.960
1.640
1.448

0.038
0.915
1.780
2.299

158.2%
-4.7%
8.5%
58.7%

0.015
1.129
1.670
0.990

0.039
0.963
1.780
2.179

158.2%-14.6%
6.6%
120.0%

0.014
0.800
1.666
2.066

0.037
0.876
1.780
2.320

158.2%
9.5%
6.9%
12.3%

Data
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill PremiumModel(a) ε = −0.5, ρ = 0.83, f = 100,τ us,1994 = 0.01(b) α = 0.53,τ me x,1994 = 0.18,τ us,1987 = 0.05, τ mex,1987 = 0.72
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill Premium(a) ε = −1.5, ρ = 0.83, f = 100,τ us,1994 = 0.01(b) α = 0.57,τ mex ,1994 = 0.18,τ us ,1987 = 0.05,τ mex ,1987 = 0.74
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill Premium(a) ε = −1, ρ = 0.7, f = 100,τ us,1994 = 0.01(b) α = 0.93,τ me x,1994 = 0.22, τ us,1987 = 0.09,τ mex ,1987 = 2.06
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill Premium(a) ε = −1, ρ = 0.9, f = 100,τ us ,1994 = 0.01(b) α = 0.34,τ mex,1994 = 0.14,τ us ,1987 = 0.03,τ mex ,1987 = 0.38
Manuf. Variety Trade/U.S. Manuf. GDP
Manuf. Variety Trade/Mex. Manuf. GDP
U.S. Skill Premium
Mex. Skill Premium

28

Fig. 1 Relative wage of high-skilled to low-skilled labor in U.S. and Mexican
manufacturing industries, 1980-2000

Fig. 2 U.S.-Mexican trade as percent of U.S. GDP, 1980-2000

29

Fig. 3-a Autarky equilibrium with wH =wL > 1

L, LS

y

The autarky equilibrium is achieved at A,
and the slope at A is w H wL > 1 .
The high skill and low skill each do theirown task.

B

A
L

wH wL > 1
H, H S

H

Fig. 3-b Autarky equilibrium with wH =wL = 1

L, LS

The autarky equilibrium is achieved at C
between A and B, and wH wL = 1 .
Part of the high skill denoted by H x is
doing the low-skill task.

y

B
L+Hx

C
A

L
wH wL = 1

H − HxH, H S

H30

Fig. 4-a Least traded goods growth: U.S. manufacturing imports from
Mexico, 1980-2000

Fig. 4-b Least traded goods growth: Mexican manufacturing imports from
U.S., 1980-2000

31