Robert M.
La Follette School of Public Affairs
at the University of Wisconsin-Madison
Working Paper Series
La Follette School Working Paper No. 2013-009
http://www.lafollette.wisc.edu/publications/workingpapers
A Consideration of Patterns of Intra-Industry Trade:
Why Do Countries Export and Import More/Less
in the Same Industry?
Isao Kamata
La Follette School of Public Affairs, University of Wisconsin–Madison
ikamata@lafollette.wisc.edu
1225 Observatory Drive, Madison, Wisconsin 53706
608-262-3581 / www.lafollette.wisc.edu
The La Follette School takes no stand on policy issues; opinions expressed
in this paper reflect the views of individual researchers and authors.
A Consideration of Patterns of Intra-Industry Trade:
Why Do Countries Export and Import More/Less in the
Same Industry?
Isao Kamata §
University of Wisconsin–Madison
This version: March 15, 2013
Abstract
It is observed in data that in most manufacturing industries a country’s exports and imports,
measured in values per GDP within an industry, are positively correlated. This indicates that
the countries exporting more (less) in an industry tend to import more (less) in the same
industry, relative to the size of their economy. In this paper I employ the model of
intra-industry trade that has been widely used in the literature, also allowing transport costs
and price asymmetry across countries, to examine whether and how the model can explain
such positive correlations between a country’s export and import in the same industry. The
theoretical prediction of the model is that a country’s export and import per GDP could both be
smaller (greater) in the same industry if the country‘s production share in the world total in that
industry and its GDP share in the world are both greater (smaller). The result of the data
analysis shows, however, that the model can predict only smaller or negative correlations in
many industries, while countries’ production sizes and GDPs are almost perfectly positively
correlated. This result seems to imply that an alternative mechanism missed in the
conventional model is needed to explain the pattern of intra-industry trade.
Keywords: Intra-industry trade; Export-import correlation; New Trade Theory
JEL classification: F10, F12, F14,
I am particularly grateful to Alan Deardorff and Juan Carlos Hallak for extensive discussions and also thank Gary
Saxonhouse for valuable and helpful comments. I am solely responsible for all remaining errors.
§
1. Introduction
The law of comparative advantage is the oldest and the most prevailing proposition
in the theory of international trade. The law states that a country will specialize in
production and export of the goods in which it has an advantage in relative production
costs due to difference in technology (in the Ricardian sense) or difference in relative factor
abundance (in the sense of Heckscher-Ohlin (H-O)); and will import the other goods.1 This
provides a sharp prediction of patterns of international trade in the case in which goods are
homogeneous: a country will not import the goods that it exports, or vice versa, and thus
there is no intra-industry trade. The “new theory” of international trade such as the work by
Krugman (1979, 1980) introduced the market structure of monopolistic competition to
explain the intra-industry trade that was observed in reality. Helpman and Krugman (1985)
incorporated the monopolistic competition model with the traditional H-O framework, but
the original prediction by the law of comparative advantage is still valid in their model, in a
slightly weaker manner: that is, a country is expected to be a net exporter of the industry in
which it has comparative advantage. The expected cross-country pattern of trade within an
industry, then, would be such that, if a country exports more in an industry, the country
should import less in that industry.
What do data tell about the pattern? Table 1 shows the correlations between the
values of exports and imports across countries within each of the manufacturing industries
classified by the 3-digit ISIC (Revision 2).2 The values of exports and imports are divided
by GDP of each country in order to adjust for the size of the economy. The result is
surprising: in most of the manufacturing industries, exports and imports of the countries are
positively correlated, and the correlations are fairly strong in some industries. This implies
that the countries that export more (relative to the size of the economy) in an industry tend
to import more in the same industry, and vice versa. This seems to be the opposite to what
is expected according to the law of comparative advantage.3
The model of intra-industry trade with monopolistic competition of differentiated
products has been widely used in the literature, such as Romalis (2004). The purpose of this
1 Indeed in some cases countries will specialize incompletely and the law of comparative advantage cannot
predict the exact patterns of countries’ specialization. Deardorff (1980) has a great summary on this.
2 Description of the data is in Section 5 of this paper.
3 The correlation cannot be positive if products are homogeneous and thus there is no intra-industry trade.
That is, when a country exports in an industry (Xci>0), it does not import in that industry at all (Mci=0), or vice
versa. Thus, the covariance Cov(Xci,Mci) ≡ E[XciMci]-E[Xci]·E[Mci] is definitely non-positive since the first
term is zero but E[Xci]≥0 and E[Mci]≥0 so that the second term is negative. Note that the signs of covariance
and correlation coefficient are the same.
1
1. Introduction
The law of comparative advantage is the oldest and the most prevailing proposition
in the theory of international trade. The law states that a country will specialize in
production and export of the goods in which it has an advantage in relative production
costs due to difference in technology (in the Ricardian sense) or difference in relative factor
abundance (in the sense of Heckscher-Ohlin (H-O)); and will import the other goods.1 This
provides a sharp prediction of patterns of international trade in the case in which goods are
homogeneous: a country will not import the goods that it exports, or vice versa, and thus
there is no intra-industry trade. The “new theory” of international trade such as the work by
Krugman (1979, 1980) introduced the market structure of monopolistic competition to
explain the intra-industry trade that was observed in reality. Helpman and Krugman (1985)
incorporated the monopolistic competition model with the traditional H-O framework, but
the original prediction by the law of comparative advantage is still valid in their model, in a
slightly weaker manner: that is, a country is expected to be a net exporter of the industry in
which it has comparative advantage. The expected cross-country pattern of trade within an
industry, then, would be such that, if a country exports more in an industry, the country
should import less in that industry.
What do data tell about the pattern? Table 1 shows the correlations between the
values of exports and imports across countries within each of the manufacturing industries
classified by the 3-digit ISIC (Revision 2).2 The values of exports and imports are divided
by GDP of each country in order to adjust for the size of the economy. The result is
surprising: in most of the manufacturing industries, exports and imports of the countries are
positively correlated, and the correlations are fairly strong in some industries. This implies
that the countries that export more (relative to the size of the economy) in an industry tend
to import more in the same industry, and vice versa. This seems to be the opposite to what
is expected according to the law of comparative advantage.3
The model of intra-industry trade with monopolistic competition of differentiated
products has been widely used in the literature, such as Romalis (2004). The purpose of this
1
Indeed in some cases countries will specialize incompletely and the law of comparative advantage cannot
predict the exact patterns of countries’ specialization. Deardorff (1980) has a great summary on this.
2 Description of the data is in Section 5 of this paper.
3 The correlation cannot be positive if products are homogeneous and thus there is no intra-industry trade.
That is, when a country exports in an industry (Xci>0), it does not import in that industry at all (Mci=0), or vice
versa. Thus, the covariance Cov(Xci,Mci) ≡ E[XciMci]-E[Xci]∙E[Mci] is definitely non-positive since the first
term is zero but E[Xci]≥0 and E[Mci]≥0 so that the second term is negative. Note that the signs of covariance
and correlation coefficient are the same.
1
paper is to examine whether this conventional model can explain the positive correlation
between a country’s exports and imports in the same industry that is observed in data. I start
with the model under a conventional (but restrictive) set of assumptions such as free trade
and symmetric prices, and later introduce a more general form by allowing transport costs
and price differences across countries in order to see whether such relaxing affects the
prediction of the model. The restrictive version of the model is then tested using
cross-country data on manufacturing production, trade, and GDP by comparing the
predicted correlations to the actual. The relaxed version is not tested because it involves
variables that are not observable in the data. Instead, a possible procedure for testing the
relaxed version of the model without having those unobservable variables is proposed.
The key finding of this paper is the following: The theoretical results from the
comparative statics of the model indicate that a country’s exports and imports per GDP
could both be smaller (greater) in the same industry if the country‘s production share in the
world total in that industry and the country’s GDP share in the world are both greater
(smaller). The data analysis shows, however, that the model can predict only smaller or
negative correlations in many manufacturing industries, while the sizes of the countries’
production and their GDPs are almost perfectly positively correlated. This result seems to
imply that an alternative mechanism that is missed in the conventional model is needed to
explain the pattern of intra-industry trade.
The organization of the paper is as follows: In Section 2 a piece of evidence is
presented to show that the positive correlation between exports and imports is robust
regardless the level of industry aggregation. Section 3 presents the model and derives
expressions of key variables. Section 4 shows comparative statics of the key variables to
show how the model predicts ‘co-movement’ of a country’s exports and imports. Section 5
performs a data analysis for the restrictive version of the model, together with description
of the data employed. Concluding Section 6 discusses the results.
2. Is It Due to Aggregation?
Finger (1975) claimed that intra-industry trade would disappear if commodities
were better-classified at a more disaggregated level. Likewise, one might ask whether the
positive cross-country correlation between exports per GDP and imports per GDP within
the same industry shown in Table 1 is simply due to the aggregation of industries, and
whether, at a less aggregated level, countries do indeed import different goods from those
2
they export?
In order to check whether this is the case or not, let us see the correlations at as
disaggregated level as possible. Although my analysis overall in this paper is based on the
industry classification (the International Standard Industrial Classification: ISIC), the data
at a disaggregated-level ISIC are not available for multiple countries. Therefore, in this
section I employ United Nations data on world trade classified by the Standard
International Trade Classification (SITC: Revision 2)4 and calculate the correlation
coefficient between exports per GDP and imports per GDP within each commodity by 3-,
4-, and 5-digit SITC.5 In addition, I draw on the data for the 6-digit Harmonized System
(HS: 1992) from the same database for further disaggregating. The number of commodities
included in the data for SITC at each digit level is 239, 786, and 1,464 respectively, and the
number of countries is 97, 95, and 94, respectively. For the 6-digit HS, the number of
commodities is 5,035 and the number of countries is 31.6
Figure 1 presents the histograms showing the distribution of the number of
commodities over the level of correlations between exports and imports per GDP for each
level of the aggregation. As shown in the figure, the distributions of the correlation
coefficients do not change much by the level of aggregation, and the correlations are
positive in most commodities. This shows that, at least for any of the aggregation levels that
the data allow, the positive correlation between exports and imports (both per GDP) is
robust. The aggregation may not matter for this pattern.
3. The Model
The model that I present here is a version of the monopolistic competition model
of intra-industry trade employed by Romalis (2004). I first present the common
assumptions that the model is based on throughout this paper, and derive common elements
of the model. I then consider three cases, ranging from the most restrictive assumptions to
the least restrictive assumptions, to derive three versions of the model.
(1) Common Assumptions
The assumptions that are common to all the versions of the model are the
4
5
6
The source of the data is United Nations Commodity Trade Statistics Database, or UN Comtrade (on-line).
5-digit is the most disaggregated level for SITC (Rev.2) available in UN Comtrade.
The data for HS are available only for a limited number of countries, which are mainly the OECD members.
3
following:
1. There are M (>1) countries (c = 1, 2, …, M) in the world (the multi-country setting).
(Note, however, that in the following parts of this paper I also apply the two-country
setting (country c and “the rest of the world”) in order to examine the model in a simpler
manner.)
2. All consumers in all countries have identical Cobb-Douglas preferences over industries,
so that consumers’ expenditure shares on each industry are constant for all prices and
incomes. bi shows the expenditure share for industry i. That is;
U   bi ln Q i ;
(1)
b
(2)
i
i
 1.
i
3. Each industry is monopolistically competitive. That is, each industry consists of a
number of varieties that are imperfect substitutes for each other, and each firm produces
one variety. nci denotes the number of firms in industry i in country c, and Ni denotes the
world total number of firms in the industry: that is;
M
N i   nci .
(3)
c
4. Consumers have identical preferences with a constant elasticity of substitution (CES)
over varieties within an industry. By interpreting Qi in Equation (1) above as consumers’
sub-utility, this is shown by:
Ni
Q  [ (qcD ( ))i ]1/ i ;
i
(4)
i
where qcD(ω) is the quantity of product (variety) ω in industry i demanded by consumers,
θi = 1 - 1/σi where σi is the elasticity of substitution among varieties in the industry (σi >
1).
5. There may be transport costs for international trade. For modeling convenience, I use the
“iceberg” form of transport costs that may vary across industries, according to which, for
each industry i, τi ≥1 units of output must be shipped from one country for one unit to
reach any other country.
4
Under these assumptions, the quantity of variety ω in industry i that is demanded
by consumers in country c is:
qcD ( ) 
pˆ c ( )  i
Eci ;
1 i
ˆ
p
(

')
 c
(5)
 'i
where pˆ c ( ) is the price of variety ω (in industry i) paid by consumers in country c. Eci
denotes the total expenditure of consumers in country c on products in industry i: that is;
Eci  b iYc ;
(6)
where Yc is total income, or GDP, of country c.
It is convenient to define the ideal price index for industry i in country c:
1
Gci  [ pˆ c ( )1 i ]1 i .
(7)
i
Note that, letting p(ω) be the price charged by the firm producing variety ω, the
consumers’ price is pˆ ( ) = τip(ω) for foreign consumers due to the trade cost, while
pˆ ( ) = p(ω) for domestic consumers. Hence, the ideal price index defined by Equation (7)
becomes:
nci
G  [ ( p( )
i
c

1 i
nci '
)  { ( p( '))
i
c ' c
1 i
1
1 i
}]
.
(8)
'
In equilibrium, the firm producing variety ω must supply the quantity that is
demanded (qS = qD) to domestic consumers, but must supply and ship τi times the quantity
demanded (qS = τqD) to consumers in any other country due to the “iceberg” trade cost.
Hence, the value of output produced by firm ω (located in country c), or the revenue of the
firm, is:
p ( )q S ( )  p( )[biYc
 biYc (
p ( )  i
( i p ( ))  i
i i

{

b
Y
]

c'
(Gci )1 i c ' c
(Gci ' )1 i
p ( ) 1 i
 i p( ) 1 i
i
)

{
b
Y
(
) }.

c'
Gci
Gci '
c ' c
(2) Case I
In this first case, I add the following set of assumptions to the common
assumptions presented above:
5
(9)
I-6. Production technologies of all varieties in an industry are identical in the world.
I-7. Free and frictionless international trade: countries costlessly trade their products each
other. That is, τi = 1 for any industry i.
I-8. In equilibrium, the producers’ prices of all varieties in an industry are equal to each
other. That is, p( )  p i   i .
Note that, under these assumptions, the output of each variety (or the size of each firm) in
an industry is identical: qS(ω) = qiS   i .
With these additional assumptions, Equations (8) and (9) above are modified to the
following version for this Case I (from now on, for simplicity, the output quantity qS is
denoted by q without the script S):
i 1 i
G  G  [n ( p )
i
c
i
i
c
p i q i  bi
1
i 1 i 1 i
 n ( p )
c ' c
i
c'
1 M
( Yc ) .
Ni c
]
1
i 1 i
 p (N )
i
c ;
(8-I)
(9-I)
That is, all firms in an industry have the same size and revenue.
Now I derive the expressions for the values of production, exports, and imports of
a country in an industry for this Case I. Since my interest is in comparison of values of
exports and imports in an industry, I suppress the script i denoting an industry from the
expressions below. Let VQc, VXc, and VMc be the values of production, exports, and imports
of country c in an industry, respectively; and VQw, VXw, and VMw be those of the world
(total). Then;
nc M
( Yc ) ;
N c
VQc  nc pq  b
VX c  b
(10-I)
nc
( Yc ' ) ;
N c ' c
VM c  bYc
n
c ' c
N
c'
 bYc
(11-I)
( N  nc )
.
N
(12-I)
Equation (11-I) is so because VXc is the total value of the varieties produced in and shipped
from country c to all other countries. Equation (12-I) is because VMc is the total value of the
foreign varieties (for country c) bought by consumers in country c.
6
Therefore, for each country c, the values of exports and imports in an industry, as
ratios to the country’s GDP, are, respectively:7
VX c VQw 1  yc
(13-I)

xc ;
Yc
Yw yc
VM c VQw

(1  xc ) ;
Yc
Yw
(14-I)
where xc ≡ VQc/VQw is country c’s share of world production in the industry, and yc ≡ Yc/Yw
is country c’s share of the world income (GDP). Note that, in this Case I, since the values of
output (or revenues) of all firms in an industry are identical, xc is equal to country c’s share
of the number of firms in the industry to the world total number of firms: xc = ncpq/Npq =
nc/N.
Correlation Coefficient
In this Case I, the correlation coefficient between the values of exports and imports
per GDP within an industry, ρi(VXci/Yci, VMci/Yci), is derived as follows:8
VX i VM ci
1  yc i
1  yc i i
i ( c ,
)  i(
xc ,1  xci )    i (
xc , xc ) .
(15-I)
Yc
Yc
yc
yc
(3) Case II
In this second case, I keep the same set of assumptions as in Case I above, except
one: I introduce a positive trade cost in the model rather than free trade. That is, assumption
I-7 in Case I is replaced with the following:
II-7. There is a transport cost for international trade in industry i: i.e., τi > 1.
The other two assumptions I-6 and I-8 are kept in this Case II.9
With this second set of assumptions added to the common assumptions 1 through 5,
Equations (8) and (9) are modified into the following version for this Case II (suppressing
the script i for the industry):10
1
Gc  p[nc  T ( N  nc )]1 where T   1 ;
7
See Appendix C-1 for derivation of Equations (13-I) and (14-I).
See Appendix C-2 for the details of the derivation of the correlation coefficient.
9 Note that the values of output (pq) are identical for all firms in an industry also in this Case II.
10 See Appendix C-1 for derivation of the following equations (8-II) through (14-II).
8
7
(8-II)
pq  b[Yc (
 b[
p 1
p
)  {Yc ' ( )1 }]
Gc
Gc '
c ' c
Yc
Yc '
 T {
}] .
nc  T ( N  nc )
c ' c nc '  T ( N  nc ' )
(9-II)
Then, using the same notations as in Case I, the value of production in country c in
the industry is:
Yc
Yc '
(10-II)
VQc  nc pq  ncb[
 T {
}] .
nc  T ( N  nc )
c ' c nc '  T ( N  nc ' )
The first term in the equation shows how much value is consumed domestically in country
c, and the second shows how much is shipped to other countries. Hence, the value of
exports from country c in an industry is:
Yc '
(11-II)
VX c  ncbT {
}.
c ' c nc '  T ( N  nc ' )
On the other hand, the value of country c’s imports in an industry is:
( N  nc )bTYc
.
VM c 
nc  T ( N  nc )
(12-II)
Thus, for each country c, the values of exports and imports in the industry as ratios
to the country’s GDP are, respectively;
VX c VQw xc
yc '
(13-II)

T {
};
Yc
Yw yc c 'c (1  T ) xc '  T
VM c VQw
T  Txc

{
};
Yc
Yw T  (1  T ) xc
(14-II)
where xc ≡ VQc/VQw and yc ≡ Yc/Yw as defined in Case I. Since also in this Case II the
values of output (or revenues) of all firms in an industry are identical, it holds that xc =
nc/N .
On the other hand, in order for both prices and quantities to be the same for all
varieties/firms in one industry in equilibrium, as assumed in this Case II, the following
condition must be satisfied for any pair of countries:11
yc
x  T (1  xc )
c '  c
 c
yc ' xc '  T (1  xc ' )
11
(II-A)
See Appendix C-3 for details.
8

yc '
yc
.

xc '  T (1  xc ' ) xc  T (1  xc )
Substituting this into Equation (13-II) yields the following expression of a country’s export
value per GDP:
VX c VQw xc
yc
VQw
xc
.
(13-II-2)

T {
}
T ( M  1)
Yc
Yw yc c 'c xc  T (1  xc )
Yw
xc  T (1  xc )
Correlation Coefficient
The correlation coefficient ρi(VXci/Yci, VMci/Yci) in this Case II is the following:12
i (
VX ci VM ci
xci
xci
,
)  i ( i
,1

)  1 .
Yc
Yc
xc  T (1  xci )
xci  T (1  xci )
(15-II)
Therefore, in this Case II (with all the assumptions holding strictly), exports and imports
per GDP within an industry must be perfectly negatively correlated. The correlation cannot
be positive at all, which means that this Case II model cannot explain the positive
correlations that are observed in the data. I thus omit this Case II for further analysis in the
rest of this paper.
(4) Case III
In Case III, I also relax the other two assumptions that have been kept in Cases I
and II. Now instead of the assumptions I-6 and I-8 in Case I,13 the following assumptions
are introduced:
III-6. Production technologies of varieties (or firms) in an industry are identical within a
country, but not necessarily identical across countries.
III-8. In equilibrium, the prices of varieties charged by firms (excluding trade cost) in an
industry are the same within a country, but different in general across countries.14
That is, p(ω) = pci if ω is a variety in industry i produced in country c, but pc’i ≠ pci for
c’ ≠ c.
Thus, in this Case III, the quantities produced (or the sizes of firms) are the same for
12
See Appendix C-2 for the details of the derivation of the correlation coefficient.
Remember that these two assumptions were kept also in Case II.
14 This may be because of technology differences across countries by the assumption III-6. Or, this can be
interpreted as the case in which factor prices are different across countries, while production technologies are
identical for all firms in the world.
13
9
varieties in a country (q(ω) = qci if ω is a variety in industry i produced in country c), but
not across countries in general (qc’i ≠ qci for c’ ≠ c).
For the trade cost, the assumption II-7 holds here as in Case II: that is, a transport
cost τi >1 is incurred for trade of products in industry i between any pair of two countries.
With this third set of assumptions, the expressions of the ideal price index and the
value of output of a single variety/firm are modified to the following version for Case III
(suppressing the script i for an industry):15
1
Gc  [nc pc1  T  (nc ' pc1'  )]1 where T   1 ;
(8-III)
c ' c
pc qc  pc1 b[
Yc
Yc '
 T {
}] .
1
1
1
 T  (nc ' pc ' )
c ' c nc ' pc '  T  (nc " pc " )
1
c
nc p
c ' c
(9-III)
c " c '
Accordingly, country c’s values of production, exports, and imports in an industry
are as follows:
VQc  nc pc1 b[
1
c
nc p
VX c  nc pc1 bT {
c ' c
VM c 
Yc
Yc '
 T {
}] ; (10-III)
1
1
1
 T  (nc ' pc ' )
c ' c nc ' pc '  T  (nc " pc " )
c ' c
1
c'
nc ' p
c " c '
Yc '
};
 T  (nc" pc1"  )
(11-III)
c " c '
bYcT  (nc ' pc1'  )
1
c
nc p
c ' c
 T  (nc ' pc1'  )
.
(12-III)
c ' c
Hence, for each country c, the values of exports and imports in an industry, as
ratios to the country’s GDP, become as follows:
VX c VQw nc pc1
yc '

[
T {
}] ;
1
Yc
Yw
yc
 T  (nc " pc1"  )
c ' c nc ' pc '
(13-III)
c " c '
T  (nc ' pc1'  )
VM c VQw
c ' c

[
].
1
Yc
Yw nc pc  T  (nc ' pc1'  )
(14-III)
c ' c
yc ≡ Yc/Yw as defined. Note, however, that in this Case III VXc/Yc and VMc/Yc can no longer
15
See Appendix C-1 for derivation of the following equations (8-III) through (14-III).
10
M
usefully be expressed using xc  VQc / VQw  nc pc qc /  nc pc qc since the values of output of
c
firms in an industry are different across countries (pc’qc’ ≠ pcqc for c’ ≠ c), so that xc is not
equal to the share in the number of firms, nc/N, in this case.
Correlation Coefficient
The correlation coefficient ρi(VXci/Yci, VMci/Yci) in this Case III is as follows:16
VX i VM ci
ni p i1
i ( c ,
)  i ( c c
Yc
Yc
yc
{ n
c ' c
i
c'
 (n
i
c'
yc '
pci1'  )
c ' c
},
)
pci1'   T i  (nci " pci1" ) nci pci1  T i  (nci ' pci1'  )
c " c '
c ' c
……
(15-III)
Two-country setting
Here I also consider the two-country version (i.e., country c and ‘the rest of the
world’) of the model, in addition to the multi-country version that is shown above. In the
two-country setting, since xc’ = 1-xc and yc’ = 1-yc for all c’≠c, Equations (13-III) and
(14-III) are modified into the following expressions, respectively:17, 18
VX c VQw (1  yc )
Tnc pc1

[
];
(13’-III)
Yc
Yw
yc {( N  nc )  Tnc pc1 }
VM c VQw
T ( N  nc )

[
].
1
Yc
Yw nc pc  T ( N  nc )
(14’-III)
The corresponding correlation coefficient is:19
VX i VM ci
(1  yc )
nci pci1
N i  nci
i ( c ,
)  i (
,
)
Yc
Yc
yc {N i  (1  T i pci1 )nci } T i N i  ( pci1  T i )nci
  i (
(1  yc )
T i nci pci1
nci pci1
,
)
yc {N i  (1  T i pci1 )nci } T i N i  ( pci1  T i )nci
……
16
(15’-III)
See Appendix C-2 for the details of the derivation of the correlation coefficient.
In these expressions for the two-country version, the price of foreign varieties is normalized to 1 (p-c = 1),
so pc denotes the relative price of home (country c’s) varieties to that of foreign varieties.
18 See Appendix C-1 also for derivation of these two equations (13’-III) and (14’-III).
19 See Appendix C-2 for derivation.
17
11
4. How Does The Model Predict ‘Co-movement’ of Exports and Imports?:
Comparative Statics
The expressions for the correlation coefficients between exports and imports per
GDP in each case, which are (15-I), (15-II), (15-III) or (15’-III) derived in the previous
section, are not directly informative by themselves in order to examine in what
circumstances the correlation becomes positive or negative. Therefore, in this section, I
perform comparative-static analyses on the expressions for the values of exports and
imports per GDP, VXc/Yc and VMc/Yc, of a country in an industry in order to give some
intuition about how VXc/Yc and VMc/Yc within an industry will ‘co-move’ according to the
theoretical model.20 Each of Cases I and III presented in the previous section are analyzed
below in order.21
(1) Case I
I first examine the effects of change in a country’s share of production value in an
industry, xc = VQc/VQw, on the country’s values of exports and imports per GDP in that
industry. From Equations (13-I) and (14-I) above;
(VX c / Yc ) VQw (1  yc )
(16-I)

 0;
xc
Yw
yc
(VM c / Yc )
VQ
  w  0.
xc
Yw
(17-I)
That is, if the shares of countries’ GDPs are fixed, the country that has a larger production
share in an industry has a larger value of exports per GDP in that industry, and on the other
hand a smaller value of imports per GDP in the same industry.
Next I examine the effects of change in a country’s GDP share, yc = Yc/Yw. Again,
from Equations (13-I) and (14-I);
 (VX c / Yc )
VQ 1
  w 2 xc  0 ;
(18-I)
y c
Yw y c
(VM c / Yc )
0.
yc
(19-I)
20
The drawback of comparative statics should be noted, however: it treats the variables in the expressions of
VXc/Yc and VMc/Yc (i.e., production share (xc) and the GDP share (yc) for Case I; the price of varieties (pc), the
number of varieties (nc), and yc for Case III) as all exogenous, so that it disregards the possibility of
endogeneity among these variables. A direct analysis of the derived correlation coefficient is more desirable,
and I also perform it though only for Case I. See Appendix A.
21 As shown in the previous section, the Case II model cannot explain the positive correlation between
exports and imports at all, so Case II is not examined here.
12
That is, if the sizes of the production of countries are all equal in an industry, the country
that has a larger GDP share in the world has a smaller value of exports per GDP in an
industry, but the values of imports per GDP do not vary across countries.
The mechanism generating these results is as follows. Under the assumptions for
this Case I, each firm in an industry has an identical value of production (pq), and, out of
the production value of each variety, the value exported to (the value consumed by
consumers in) all other countries (or the rest of the world) is equal for each variety, which is
determined by the size of incomes of all other countries. Thus, after adjusting by a
country’s income size (GDP), the value of exports of the country in an industry is larger
when (i) the country has a larger number of varieties/firms in that industry, and thus a
higher share of production in the industry; or (ii) the income size of all other countries
relative to that country ((1-yc)/yc) is larger: i.e., that country’s income share is smaller. On
the other hand, the value of imports of the country in an industry is determined by (i) the
number of varieties produced in all other countries; and (ii) that country’s income size. The
latter does not matter for the value adjusted by GDP of the country, so the value of imports
depends only on the number of varieties produced in the rest of the world: i.e., a country’s
import value is larger when more varieties are produced out of the country, which means
that the country has a smaller production share in that industry.
The following table summarizes the prediction of the change in exports and
imports per GDP according to the Case I model.
xc ↑
yc ↑
VXc/Yc
↑
↓
VMc/Yc
↓
--
Therefore, the Case I model provides the following prediction: if a country’s share in the
production value in an industry and its GDP share are both smaller (greater),22 then that
country’s exports and imports in values per GDP in the same industry could both be higher
(lower). Otherwise, the export value (per GDP) in an industry is higher for the country
whose import value (per GDP) in that industry is lower.
(2) Case III
Next, I perform comparative-static analysis for Case III. Note that, as shown in the
22
Note that this is a necessary condition for the values of exports and imports to be both greater or both
smaller.
13
previous section, the expressions for VXc/Yc and VMc/Yc derived from the Case III model do
not directly include the production share xc, unlike Case I. Therefore, the comparative
statics here are with respect to three variables: (i) the number of varieties/firms in country c
(nc); (ii) the (common) price of varieties produced in country c (pc); and (iii) the GDP share
of country c in the world (yc = Yc/Yw).
I first examine the effects of change in the number of varieties in a country in the
industry, nc. In the multi-country setting, from Equations (13-III) and (14-III) above;
 (VX c / Yc ) VQw Tpc1

nc
Yw yc
[
 (n

 T  (n p
yc '{nc ' pc1'   T
1
c'
{nc ' p
c ' c
c " c , c '
c " c '
c"
pc1"  )}
1
c"
c"
)}2
]0;
Tp  (nc ' p )
 (VM c / Yc )
VQw
c ' c

0.
1
nc
Yw {nc pc  T  (nc ' pc1'  )}2
1
c
(16-III)
1
c'
(17-III)
c ' c
23
In the two-country setting, from Equations (13’-III) and (14’-III);
 (VX c / Yc ) VQw (1  yc )
TNpc1

 0;
nc
Yw
yc {( N  nc )  Tnc pc1 }2
(16’-III)
 (VM c / Yc )
VQ
TNpc1
 w
 0.
nc
Yw {nc pc1  T ( N  nc )}2
(17’-III)
That is, if the prices of products in an industry and GDPs are the same across countries, the
country that produces more varieties in an industry has a larger value of exports per GDP in
that industry, and on the other hand a smaller value of imports per GDP in the same
industry.
Next I examine the effects of change in the (common) price of varieties produced
in a country, pc. In the multi-country setting, from Equations (13-III) and (14-III);
 (VX c / Yc ) VQw (1   )Tnc pc

pc
Yw
yc
 (n

 T  (n p
yc '{nc ' pc1'   T
[
1
c'
{nc ' p
c ' c
T { (nc ' p
1
c'
c " c , c '
c " c '
c"
c"
pc1"  )}
1
c"
)}2
]0;
(18-III)
)}(1   )nc pc
 (VM c / Yc )
VQ
  w c 'c 1
 0.
pc
Yw {nc pc  T  (nc ' pc1'  )}2
(19-III)
c ' c
(   1  1    0)
23
In the two-country version of the model, pc denotes the (common) price of varieties produced in country c
relative to the price of varieties produced in the rest of the world (the price in the rest of the world is
normalized to 1).
14
In the two-country setting, from Equations (13’-III) and (14’-III);
 (VX c / Yc ) VQw (1  yc ) (1   )Tnc ( N  nc ) pc

 0;
pc
Yw
yc {( N  nc )  Tnc pc1 }2
(18’-III)
 (VM c / Yc )
VQ (1   )Tnc ( N  nc ) pc
 w
 0.
pc
Yw {nc pc1  T ( N  nc )}2
(19’-III)
These imply that, if the numbers of product varieties in an industry and GDPs are the same
across countries, the country that produces more expensive varieties (in the two-country
version, relative to varieties produced in the rest of the world) in an industry has a smaller
value of exports per GDP in that industry, and a larger value of imports per GDP in the
same industry.
I lastly examine the effects of change in a country’s GDP share in the world, yc.
In the multi-country setting, from Equations (13-III) and (14-III);
 (VX c / Yc )
VQ Tn p1
  w c 2c
yc
Yw
yc
{n
c ' c
c'
1
c'
p
yc '
 T  (nc " pc1"  )
} 0;
(20-III)
c " c '
(VM c / Yc )
0.
yc
(21-III)
In the two-country setting, from Equations (13’-III) and (14’-III);
 (VX c / Yc )
VQ 1
Tnc pc1
 w 2
0;
yc
Yw yc {( N  nc )  Tnc pc1 }
(20’-III)
(VM c / Yc )
0.
yc
(21’-III)
Thus, if the prices of products and the numbers of varieties in an industry are the same
across countries, the country that has a larger GDP share in the world has a smaller value of
exports per GDP in an industry, but the values of imports per GDP in the industry do not
vary across countries.
These results are summarized in the following table.
nc ↑
pc ↑
yc ↑
VXc/Yc
↑
↓
↓
VMc/Yc
↓
↑
--
According to this summary table, one can derive the following predictions for Case III:
VXc/Yc and VMc/Yc could be both higher (lower) when:
15
(i) nc is lower (higher) as yc is lower (higher);
(ii) pc is higher (lower) as yc is lower (higher); or
(iii) pc is higher (lower) and nc is lower (higher) as yc is lower (higher).
Can production share (xc) be a ‘predictor’ in Case III, too?
As shown above, in Case III, VXc/Yc and VMc/Yc depend on the three variables: pc,
nc, and yc, while they depend only on the two variables, xc and yc, in Case I. However,
prices (pc’s) and the numbers of firms/varieties (nc’s) are in general much more difficult to
observe in data24 than countries’ shares of production in an industry (xc = VQc/VQw). Then,
isn’t it possible to give similar predictions of the ‘co-movement’ of VXc/Yc and VMc/Yc only
from the observable variables xc and yc without having the information on prices and the
numbers of varieties? My answer is YES: it is possible. The reason is as follows:25
Using the technique of total differentiation, the following conditions are derived for a
change in the value of a country’s total production in an industry, VQc:
dVQc  0  dnc  BQ  dpc
dVQc  0  dnc  BQ  dpc ;
where BQ  (
VQc / pc
).
VQc / nc
In the same manner, for the values of exports and imports per GDP;
VX
(VX c / Yc ) / pc
d ( c )  ()0  dnc  () BX  dpc where BX  (
);
Yc
(VX c / Yc ) / nc
d(
VM c
(VM c / Yc ) / pc
)  ()0  dnc  () BM  dpc where BM  (
).
Yc
(VM c / Yc ) / nc
However, it is derived that:
BQ  BX  BM  (1   )nc / pc  0 (for the multi-country version)
 (1   )nc ( N  nc ) / Npc  0 (for the two-country version)
In addition, sign(dVQc )  sign(dxc ) 26.
24
The price information may be observable if data are detailed enough. However, the number of firms (nc) is
generally unobservable because this variable comes from the assumptions of the model, which does not mean
the real number of firms.
25 For the technical details of the following part, see Appendix C-4.
26
See Appendix C-4.
16
Therefore, one can conclude that, if countries’ shares of GDP (yc’s) are fixed;
dxc  0  d (VX c / Yc )  0  d (VM c / Yc )  0
dxc  0  d (VX c / Yc )  0  d (VM c / Yc )  0
That is, the country that has a larger production share in an industry has a larger value of
exports per GDP, and a smaller value of imports per GDP.
One can thus give a prediction of the ‘co-movement’ of VXc/Yc and VMc/Yc from
the two observables xc and yc also in this Case III such as in the table below:27
xc ↑
yc ↑
VXc/Yc
↑
↓
VMc/Yc
↓
--
This prediction of the model is thus the same as that in Case I: if a country’s share in the
production value in an industry and its GDP share are both smaller (greater)28, then that
country’s exports and imports in values per GDP in the same industry could both be higher
(lower). Otherwise, the export value (per GDP) in an industry is higher for the country
whose import value (per GDP) in that industry is lower.
5. Data Analysis of Correlation (Case I only)
Previous sections have shown that, according to the model, the exports and imports
in an industry per GDP, VXci/Yci and VMci/Yci, and their ‘co-movement’ can be predicted by
‘predictors,’ which are each country’s share of production in that industry (xc) and its share
of GDP (yc) for Case I; the (common) price of each country’s product varieties in an
industry (pc), the number of varieties in each country (nc), and yc for Case III. In this section,
I use data to examine how close the predicted measure of the correlation coefficient
between VXci/Yci and VMci/Yci (ρi(VXci/Yci, VMci/Yci)) based on these ‘predictors’ is to the
direct measure of the correlation coefficient. This is a test of the extent to which the
standard model (and its variation) can explain the observed positive correlation between
exports and imports within the same industry. Cross-country data on production and
27
Note that, however, for an accurate analysis of the correlation coefficients with data, one still needs the
variables pc and nc rather than xc. I discuss this in the next section and propose a procedure for a data analysis
in Appendix B.
28 See Footnote 22.
17
international trade in manufacturing industries and GDP are employed.
Here I examine only the Case I model, for which all the necessary ‘predictors’ are
available in the data, but not the Case III model that requires the prices and the numbers of
varieties as ‘predictors’, both of which are not directly observable in the data. Instead, in
Appendix B, I propose a possible procedure for a data analysis of Case III without having
these unobserved variables.
In the following parts of this section, I first re-present the expressions for the
correlation coefficient for each of Cases I and III,29 then describe the data, and finally show
the results for Case I.
(1) Predicted Measure of Correlation Coefficient
Let us recall the following expression for the coefficient of correlation across
countries, ρi(VXci/Yci, VMci/Yci), which has been derived in Section 3 for each of Case I and
Case III.
Case I:
(15-I):  i (
VX ci VM ci
1  yc i
1  yc i i
,
)  i(
xc ,1  xci )    i (
xc , xc ) .
Yc
Yc
yc
yc
Case III:
For the multi-country version, (15-III):
VX i VM ci
ni p i1
i ( c ,
)  i ( c c
Yc
Yc
yc
{ n
c ' c
i
c'
yc '
 (n
i
c'
pci1'  )
c ' c
},
)
pci1'   T i  (nci " pci1" ) nci pci1  T i  (nci ' pci1'  )
c " c '
c ' c
For the two-country version, (15’-III):
i
i
nci pci1
N i  nci
i VX c VM c
i (1  yc )
(
,
) (
,
)
Yc
Yc
yc {N i  (1  T i pci1 )nci } T i N i  ( pci1  T i )nci
  i (
(1  yc )
T i nci pci1
nci pci1
,
)
yc {N i  (1  T i pci1 )nci } T i N i  ( pci1  T i )nci
The right-hand side of each expression above is the predicted measure of the correlation
coefficient. That is, if the model correctly predicts countries’ export and import values per
GDP, the cross-country correlation coefficients calculated by the formula above showed
29
Case II has been omitted for further examination, as mentioned in Section 3.
18
equal the actual values of ρi(VXci/Yci, VMci/Yci). Note, however, that the predicted measure
of the correlation coefficient for Case III requires the price information for pci, which is not
available in the data used for this paper, as well as the number of product varieties, nci,
which is unobservable in reality.30 On the other hand, the Case I model requires only the
data on countries’ GDPs and production, both of which are available in the dataset
described below.
(2) Data
The data I employ for this paper contain the variables of production, exports,
imports, and GDP of various countries. The data on production are from UNIDO (2003).
The values of the (gross) output in manufacturing industries in U.S. dollars (USD) were
selected. The data on exports and imports are from Feenstra, Lipsey, and Bowen (1997) and
Feenstra (2000). The original datasets provide the values of exports and imports for each
bilateral pair, and I summed up these bilateral trade values for each origin country (for
exports) and for each destination country (for imports). The trade data are in thousands of
USD. The GDP data in current USD were taken from the World Bank (2002).
There is an issue concerning the industry classifications employed by the data
sources. UNIDO’s production dataset categorizes manufacturing industries according to the
International Standard Industrial Classification (ISIC: Revision 2) at the 3-digit level, while
Feenstra et al.’s trade data are categorized by the Standard International Trade
Classification (SITC: Revision 2) at the 4-digit level. I transformed the original trade data
into ISIC utilizing the concordance information provided by OECD, which is made
available on the web page of Jon Haveman’s Industry Concordances.31
I selected the data for the years of 1970, 1975, 1980, 1985, 1990, and 1995.32
Since availability of the data is different across industries, countries and years, I used for
each industry only the countries for which all the variables of production, exports, imports,
and GDP are available in order to calculate the across-country correlation coefficient for
each industry in each year.33 The sample size thus varies from 22 countries to 77 countries
30
See Footnote 24.
URL: http://www.macalester.edu/research/economics/PAGE/HAVEMAN/Trade.Resources/
tradeconcordances.html
32 UNIDO’s dataset on production covers the period of 1963-2001, Feenstra et al.’s (1997) trade data covers
1970-1992, which is supplemented by Feenstra (2000) through the year of 1997. The GDP data from the
World Bank (2002) covers the period of 1960-2000.
33 To obtain x i = VQ i/VQ i and y = Y /Y , I needed to calculate the world total value of production in each
c
c
w
c
c w
31
i
industry ( VQW
  c VQci ) and world GDP ( YW  c Yc ). For each summation I include the countries for
which the figure to be summed (VQci or Yc) is available even if the countries are not included in the sample for
19
by industry and year. Table 2 shows the number of countries included in the sample for
each industry in each year.
(3) Result (for Case I)
The actual ρi(VXci/Yci, VMci/Yci) and the correlation coefficient predicted by the
Case I model are compared for each industry and each period in Table 3. Although the
actual correlations are significantly positive in many industries and years, the correlations
predicted by the model are mostly small or negative (see Figure 2). Although there are
several industries in which the predicted correlations are fairly close to, or even greater than,
the actual, the model predicts much lower values for the correlation coefficients than the
actual coefficients in most industries (see Figure 3). That is, at least in its Case I version,
the model cannot explain the strong positive correlations found in the data.
In Section 4, I used comparative statics to show that the necessary condition for
VXci/Yci and VMci/Yci to ‘move together’ (i.e., both be higher or lower) is that a country with
a larger (smaller) production share (xci) has a larger (smaller) GDP share (yc). Indeed, as
shown in Table 4, in the data xci and yc are almost perfectly correlated: that is, in any
industry, a country with a larger GDP share almost always has a larger production share.34
However, the more direct term that is included in the expression of VXci/Yci derived from
the model is (1-yc)/yc: that is, GDP of ‘the rest of the world’ relative to a country’s own GDP,
rather than the country’s own GDP itself. In other words, a country’s exports and imports
(in values per GDP) become both smaller (greater) when the country has a larger (smaller)
production share but the GDPs of the other countries are smaller (greater) relative to that
country’s. This mechanism also applies to Case III. Therefore, I also list the correlations
between xci and (1-yc)/yc in Table 4. The signs of the correlations are indeed all negative, as
the model requires for positive correlations between exports and imports, but the values are
small: they are all between -0.1 and -0.3. It might thus be necessary to have more strongly
negative correlations between xci and (1-yc)/yc to predict the strongly positive correlations
between exports and imports such as found in the data. This may be so also in Case III.35
each industry/year. Therefore, the sum of xci’s and that of yc’s are not necessarily equal to one. (However, I
should note that, when I include only the countries in the sample, in order for the sum of xci’s and that of yc’s
to be one, the results presented below do not change.)
34 Moreover, the values of x i and y are almost equal to each other for many countries in most industries and
c
c
years: the two variables are almost on a ’45-degree line’.
35 Appendix B also presents the result of the data analysis for Case III by the proposed ‘indirect’ method.
20
6. Concluding Discussion
In this paper I employed the monopolistic competition model of intra-industry
trade that is widely used in the trade literature, first with the assumptions of free trade and
identical production technologies in an industry; and then with the more relaxed
assumptions allowing existence of transport costs and price asymmetry within an industry
across countries, in order to see whether such a model can explain the positive correlation
between countries’ exports and imports in values per GDP within one industry that is
observed in the data for most manufacturing industries. The theoretical model tells, in both
its restricted and relaxed versions, that a country’s exports and imports in value per GDP in
an industry could both be higher (lower) if the country’s share of the production value in
that industry in the world and its GDP share in the world are both smaller (greater);
otherwise the export value per GDP in an industry is higher for the country whose import
value per GDP in that industry is lower. However, the cross-country correlations between
exports and imports predicted by the model from the data on manufacturing production and
GDP of various countries showed that the model cannot explain, at least in its restricted
form, such positive correlations between exports and imports as observed in the data, while
the correlations between the countries’ production shares and GDPs are almost equal to one
in most of the industries. For the relaxed case allowing iceberg transport cost and price
asymmetry across countries, a data analysis is not straightforward because of data
limitation, but inferring from the result of the comparative statics in Section 4, this relaxing
of the assumptions may not be enough to make the conventional model capable of
explaining the fact.36
What can explain the positive correlation between exports and imports within the
same industry? What can be an alternative model to the conventional monopolistic
competition model of trade? Harrigan (1994) pointed out that the “love of varieties”
assumption for the consumer’s preference, which is represented by a CES utility function
over all available varieties, is the driving force of intra-industry trade in this type of model,
and similar models with the preference assumption such as Deardorff and Stern (1986) also
generate intra-industry trade. In fact, the analysis in this paper relies on the demand side of
the conventional model, so that the result here implies that the assumption on the
consumers’ preferences may need to be reconsidered. Or, being a little less ambitious, the
model with further-relaxed assumptions on transport costs could explain the fact. Hummels
36
Indeed, as shown in Appendix B, the data analysis performed according to the proposed ‘indirect’ method
concludes that the model does not explain the positive correlation well even under the relaxed assumptions.
21
and Levinsohn (1995) presented evidence of a significant effect of distance between two
countries on the volume of bilateral intra-industry trade. Haveman and Hummels (2004)
pointed out that countries may be choosing their sources of import (or trading partners)
taking account of bilateral transport costs.37 Therefore, the monopolistic competition
model might be able to explain the positive correlation between exports and imports if it
incorporates transport cost that differs for each pair of countries. These could be directions
for future research.
A considerable amount of research has been conducted on the volume of
intra-industry trade, including a series of studies on the gravity equation. All of them,
however, focused on the total volume of trade without paying much attention to the
direction of the trade: how much is exported or imported. By pointing out the ‘fact’ of the
positive correlation between exports and imports, which is inconsistent with conventional
models based only on comparative advantage, and which may be puzzling according to the
conventional monopolistic competition model, this study presents a new issue to be
considered when examining what is a plausible theory of international trade.
37
It should be noted that Haveman and Hummels discussed this in the context of the model with imperfect
specialization of countries’ production, while the model analyzed in this paper features countries’ perfect
specialization in certain varieties of a good.
22
Appendix A
Section 4 has performed comparative-static analysis to provide some intuition on the
‘co-movement’ of countries’ export and import values per GDP within the same industry
(VXci/Yc and VMci/Yc), but it does not necessarily directly tell what the correlation
coefficient of these two variables will be in various circumstances. In this appendix, I
present a direct examination of the correlation coefficient. Specifically, limiting the analysis
to the Case I model and focusing on five potential circumstances on the patterns of
countries production sizes and GDPs, I show whether the model predicts the correlation to
be positive or negative in each circumstance.38
Preliminaries
Recall Equation (15-I):
VX i VM ci
1  yc i
1  yc i i
i ( c ,
)  i(
xc ,1  xci )    i (
xc , xc ) .
Yc
Yc
yc
yc
Then;
i (
VX ci VM ci
1  yc i i
,
)  ()0   i (
xc , xc )  ()0
Yc
Yc
yc
1  yc i i
 Cov(
xc , xc )  ()0
yc
 Var ( xci )  ()Cov (
xci i
, xc ) .
yc
…… (*)
Therefore, in the following parts of this appendix, with which inequality (*) holds is
examined in each case.
Five Potential Cases and Expected Correlations
Case (i): yc is the same for all countries
The first case to be examined is when all countries are in an equal size: i.e., the
countries all have the same GDP regardless of their industrial production compositions. In
this case, each country’s share of GDP in the world, yc, is equal to 1/N where N(≥2) is the
number of countries in the world. Therefore;
38
The detailed derivation or the proof of the expressions or (in)equalities that appear in this appendix are
suppressed to avoid lengthiness, but these can be provided by the author upon request.
23
Cov(
xci i
, xc )  Cov( Nxci , xci )  N  Var ( xci )  Var ( xci ) ;
yc
and thus the model predicts the correlation between export and import per GDP to be
negative.
More specifically, in this case, the correlation coefficient is:
i (
VX ci VM ci
1  yc i i
,
)   i (
xc , xc )    i (( N  1) xci , xci )  1 :
Yc
Yc
yc
that is, a perfect negative correlation is predicted.
Case (ii): xci is constant for all countries
The next case is when, in a certain industry, all the countries produce the same
value: i.e., the countries’ production sizes are all equal, regardless of the sizes of their
economies or GDPs. In this case, each country’s share of production in that industry in the
world, xci, is equal to 1/N. Therefore;
xi
1
1
Var ( xci )  Var (1/ N )  Cov ( c , xci )  Cov (
, )  0.
yc
N  yc N
That is, in this case, the correlation coefficient is undefined, while the covariance between
exports and imports per GDP becomes zero.
It is natural to think that the larger a country’s GDP is, the larger the country’s
production size in an industry is.39 The following three cases are all in such a circumstance.
Case (iii): xci = yc
This case is when a country’s share of production in an industry equals its GDP
share: i.e., the production size of each country in an industry is proportional to the country’s
GDP. In this case;
xi
Cov( c , xci )  Cov(1, xci )  0  Var ( xci ) ;
yc
and thus the correlation between export and import per GDP is expected to be positive.
More specifically, the correlation coefficient in this case is:
VX i VM ci
1  yc i i
i ( c ,
)   i (
xc , xc )    i (1  yc , yc )  1 :
Yc
Yc
yc
that is, exports and imports are predicted to be perfectly correlated.
39
Indeed, as shown in Table 4, countries’ production shares and GDP shares are almost perfectly correlated
in almost all manufacturing industries.
24
Case (iv): xci = (1-a)/N + a∙yc; 0<a<1
This case implies that countries’ GDPs are more diversified than their sizes of
production in an industry, or in other words, the degree of countries’ specialization in that
industry is not very large.40 In this case;
Cov(
xci i
(1  a )a
1
, xc ) 
 Cov( , yc )  0  Var ( xci ) ;
yc
N
yc
and thus the correlation between export and import per GDP is expected to be positive.
Case (v): yc = (1-a)/N + a∙xci; 0<a<1
This case contrasts with Case (iv) above: that is, this case implies that countries’
sizes of production in an industry are more diversified than their GDPs. In other words, the
degree of countries’ specialization in that industry is very large.41 In this case;
1
Var ( yc )
a
xi
(1  a)
1
Cov( c , xci ) 
 Cov( , yc )
2
yc
N a
yc
Var ( xci ) 
In fact, the former value becomes greater than the latter value ( Var ( xci )  Cov ( xci / yc , xci ) )
only when the value of a is very close to one. That is, in this case, the correlation between
export and import per GDP can be positive when the production of each country in an
industry is almost proportional to its GDP, but the correlation is predicted to be negative
otherwise.42
Summary: When Will Correlation Be Positive?
By summarizing the five cases examined above, the following conclusion is
derived:
The Case I model predicts that the correlation between countries’ export and import
values per GDP within the same industry will be positive when the following
40
Being more accurate, this is a special linear case of such a circumstance. Note that, in this case, it holds

that
41
N
c
xci   c yci  1 .
N
This is a special linear case of such a circumstance. Note that, also in this case, it holds that

N
c
xci   c yci  1 .
N
42
In fact, in the limit as N rises to infinity, any a such that 0<a<1 cannot generate a positive correlation,
which means that the correlation is expected to be always negative.
25
conditions are all satisfied: (i) countries vary both in their shares of production in
the industry and in their shares of GDP; (ii) the larger a country’s GDP, the larger
the country’s production share in the industry; and (iii) a country’s production share
is almost equal to its GDP share, or the dispersion of countries’ GDP shares is
greater than that of their production shares (i.e., countries are not very specialized in
production).
26
Appendix B
As mentioned in Section 5, direct calculation of the predicted correlation coefficient
between exports and imports per GDP, ρi(VXci/Yci, VMci/Yci), is not possible for Case III due
to lack of data availability on the ‘predictors’: the price of varieties produced in each
country (pci), which is not available in the data currently used; and the numbers of varieties
in each country (nci), which is unobservable in data. In this appendix, I propose an ‘indirect’
way of computing the predicted correlation coefficient for Case III using only variables that
are available in the data but without relying on the unavailable ‘predictors.’
It has turned out, however, that the proposed method needs to rely not only on each
country’s production share in an industry (xci) and GDP share (yc), which are the
‘predictors’ for Case I and observable in the data, but also on the actual value of each
country’s export in an industry per GDP, VXci/Yc, rather than having the prediction of it.
Therefore, the correlation coefficient obtained by this method is between the actual export
per GDP (VXci/Yc) and the predicted import per GDP (VMci/Yc). Moreover, it should be
noted that this method also requires knowledge of the values of the ‘iceberg’ transport cost,
τi, and the elasticity of substitution among varieties, σi, for each industry. I use plausible
figures for these values, which are estimated by Hummels (2001)43.
In the rest of this appendix, I present the procedure for the computation of the
predicted correlation coefficient and the result of the comparison of the predicted
correlation to the actual correlation between the observed values of exports and imports per
GDP.
Preliminaries
(The script i for an industry is suppressed below.) Let us recall, for Case III:
(10-III):
(11-III):
VQc  nc pc1 b[
1
c
nc p
VX c  nc pc1 bT {
c ' c
Yc
Yc '
 T {
}] ;
1
1
1
 T  (nc ' pc ' )
c ' c nc ' pc '  T  (nc " pc " )
c ' c
1
c'
nc ' p
c " c '
Yc '
};
 T  (nc" pc1"  )
c " c '
43
Another approach is estimating these values from the data employed for this analysis, rather than
borrowing figures from other sources. For example, the values of τi and σi could be determined to minimize
the difference between the actual and the predicted correlation coefficient. Such a method might be an
interesting exercise, but it does not guarantee that the estimated figures are plausible.
27
(12-III):
VM c 
bYcT  (nc ' pc1'  )
1
c
nc p
c ' c
 T  (nc ' pc1'  )
.
c ' c
Define:
zc  nc pc1 ;
wc  1/( zc  T  zc ' ) .
c ' c
Then, Equations (10-III) through (12-III) above can be expressed as follows:
VQc  zcb[ wcYc  T  wc 'Yc ' ] ;
(i)
VX c  zcbT  wc 'Yc '  VQc  zcbwcYc ;
(ii)
VM c  wcYcbT  zc ' .
(iii)
c ' c
c ' c
c ' c
Equation (ii) implies:
zcbwcYc  VQc  VX c  wc 
VQc  VX c
.
bzcYc
(ii)’
Then, by the definition of wc, the equation below follows:
(1/ wc ) zc  T  zc '  bzcYc /(VQc  VX c ) .
(iv)
c ' c
Equation (iv) holds for any country c. Also note that T  z c '  T ( z c '  z c ) . Thus,
c ' c
c'
considering Equation (iv) for two countries c and c’ (c ≠ c’), and subtracting one from
another on both sides, yields:
zcYc
zc 'Yc '
(1  T )( zc  zc ' )  b(

).
VQc  VX c VQc '  VX c '
Define Rc  Yc /(VQc  VX c ) . Then, since 0  T   1  1, the equation above implies:
{1 
b
b
Rc }zc  {1 
Rc ' }zc ' .
1 T
1 T
Here, define the “benchmark” country c = 0 such that z0  n0 p01  1 . Then, the equation
above can be expressed as follows:
28
zc  {1 
b
b
R0 }/{1 
Rc } where R0  Y0 /(VQ0  VX 0 ) .
1 T
1 T
(v)
Now zc is the (unobservable) variable measuring some combination of a country’s product
price and its number of varieties relative to that of the benchmark country44.
Proposed Procedure for a Predicted Measure of ρi(VXci/Yci, VMci/Yci)
Now I propose steps to calculate the measure of ρi(VXci/Yci, VMci/Yci) predicted
from the Case III model.
 Step 1: Choose a country in the sample as the “benchmark” country c =0 (e.g., USA).

Consider z0 =1 for the benchmark country.
Step 2: Using the data on production, export, and GDP, and according to Equation (v),
calculate zci for every country in the sample other than the benchmark country
with the parameter Ti: i.e.;
bi
bi
i
R
}/{1

Rci }
0
i
i
1 T
1 T
i
where
R0 = Y0/(VQ0i - VX0i)
Rci = Yc/(VQci – VXci)
bi = VQwi/Yw (by market clearing in industry i).
Note here that all the variables other than zci are observable. The value for the
parameter Ti ≡ (τi)1-σ for each industry, where τi is the “iceberg” transport cost and
σi is the CES preference parameter, needs to be estimated separately, or obtained
from other sources. I use the estimates by Hummels (2001).45
Step 3: Obtain the import per GDP predicted from the model: i.e., from Equation (14-III)
and using the calculated zci ;46
zci  {1 

44
zci can be considered as an indirect measure of a country’s production size in an industry relative to that of
the benchmark country. Indeed, xc / zc  0 , so that a country with larger zci has a larger production share
i
i
in that industry (xci = VQci/VQwi).
45 Hummels (2001) has measured transport costs by the freight rate relative to value of imports for seven
countries, and also estimated the CES preference parameter for each commodity using the data on bilateral
trade flows among those countries. For the transport cost, since the model of this paper assumes that the
transport cost in an industry is identical for every pair of countries in the world, I have taken the
trade-value-weighted average of the values that Hummels has estimated for the seven countries. In addition,
since the model needs the parameter value for each 3-digit ISIC industry but Hummels’ estimation is for each
2-digit SITC (Rev.2) commodity, I have converted the figures of the transport cost and the CES preference
parameter estimated by Hummels into those for each ISIC industry based on concordance information.
46 The predicted export could be calculated from Equation (13-III) and using z i, but it is tautological since
c
the actual value of export has been used to estimate the value of zci.
29
T i  (nci ' pc1'  )
i
i
c
i
w
VM
VQ
c ' c

[
i
i ].
Yc
Yw nci pc1  T i  (nci ' pc1'  )
c ' c

Step 4: Calculate the correlation coefficient, ρi(VXci/Yci, VMci/Yci), using the actual value
of VXci/Yci and the estimated value of VMci/Yci obtained in Step 3 above.
Result
Table A1 compares the actual and predicted correlation coefficients for each
3-digit ISIC industry in each year for Case III, which is similar to Table 3 for Case I.
Although the actual correlation coefficients are positive in most of the industries and years,
the predicted correlation coefficients by this method are mostly negative, as shown in
Figure A1 (similar histograms to Figure 2 for Case I). This result seems to imply that, even
in Case III that allows transport cost and price asymmetry across countries, the standard
monopolistic competition model of intra-industry trade does not explain the observed
positive correlation between export and import per GDP within the same industry.
Figure A2, which is similar to Figure 3 for Case I, shows how the correlation
predicted by this method over- or under-estimates the actual correlation. What is striking is
that the distribution of the estimation gaps is not continuous: there are two clusters and the
group of the industry-year observations for which the correlations are over-estimated is
clearly separated from the group of the under-estimated observations. About a quarter of all
the industry-year cells (41 out of 168: 28 industries in 6 year periods) obtain the predicted
correlation over-estimating the actual, but I do not find a systematic pattern between the
over-estimation and certain industries or years. This estimation method seems to be very
sensitive to data, and it may be caused by the discontinuity of the estimated zci 47.
In fact, the discontinuity of the estimated zci seems to imply that the model’s assumption of identical and
constant expenditure shares in each industry is inconsistent with reality. The method of estimating zci
presented here relies on this assumption, while the expenditure share on a certain industry seems to vary
across countries, which may result in inappropriate estimation of the figure of zci in some cases.
47
30
References
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Journal of Political Economy, 88(5), pp.941-957.
Deardorff, Alan V., and Stern, Robert M. (1986), The Michigan Model of World Production
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1970-1992, with Production and Tariff Data”, NBER Working Paper No.5910.
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Institute of Governmental Affairs, University of California, Davis.
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pp.581-589.
Harrigan, James (1994), “Scale Economies and The Volume of Trade”, Review of
Economics and Statistics, 76(2), pp.321-328.
Haveman, Jon, and Hummels, David (2004), “Alternative Hypotheses and The Volume of
Trade: The Gravity Equation and The Extent of Specialization”, Canadian Journal of
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Helpman, Elhanan, and Krugman, Paul R. (1985), Market Structure and Foreign Trade,
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Hummels, David (2001), “Toward a Geography of Trade Costs”, manuscript.
Hummels, David, and Levinsohn, James (1995), “Monopolistic Competition and
International Trade: Reconsidering The Evidence”, Quarterly Journal of Economics, 110,
pp.799-836.
Krugman, Paul R. (1979), “Increasing Returns, Monopolistic Competition, and
International Trade”, Journal of International Economics, 9, pp.469-479.
31
Krugman, Paul R. (1980), “Scale Economies, Product Differentiation, and the Patterns of
Trade”, American Economic Review, 70(5), pp.950-959.
Romalis, John (2004), “Factor Proportions and the Structure of Commodity Trade”,
American Economic Review, 94(1), pp.67-97.
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World Bank (2002), World Development Indicators 2002.
32
Table 1: Correlations between Volumes of Exports and Imports per GDP across Countries
Industry*
311 (Food products)
313 (Beverages)
314 (Tobacco)
321 (Textiles)
322 (Wearing apparel)
323 (Leather products)
324 (Footwear)
331 (Wood products)
332 (Furniture)
341 (Paper and products)
342 (Printing and publishing)
351 (Industrial chemicals)
352 (Other chemicals)
353 (Petroleum refineries)
354 (Misc. petroleum & coal products)
355 (Rubber products)
356 (Plastic products)
361 (Pottery, china, earthenware)
362 (Glass and products)
369 (Other non-metallic mineral products)
371 (Iron and steel)
372 (Non-ferrous metals)
381 (Fabricated metal products)
382 (Non-electric machinery)
383 (Electric machinery)
384 (Transport equipment)
385 (Professional & scientific equipment)
390 (Other manufactured products)
1970
.183
.646
.431
.084
.432
.378
.162
-.006
.418
-.166
.555
.397
.320
.127
.508
.982
.652
.052
.100
.053
.247
-.046
.167
.161
.322
.077
.340
.507
1975
.304
.041
.498
.494
.208
.136
.124
-.241
.598
-.136
.312
.234
.381
.346
.235
-.008
.584
.107
-.020
-.022
-.013
.371
.191
.036
.785
.005
.242
.555
Year
1980
1985
.085
.202
.058
.384
.513
.670
.424
.123
.375
.470
.336
.232
.215
.202
-.189
-.122
.329
.483
-.091
-.041
.459
.570
.381
.567
.492
.494
-.032
-.042
.382
.121
.084
.013
.533
.762
-.030
-.094
.163
.238
.143
.084
-.189
-.056
-.092
.001
.242
.278
.141
.297
.543
.951
-.083
-.041
.056
.330
.697
.187
1990
.314
.381
.248
.131
.077
.184
-.019
-.136
.099
.009
.443
.359
.231
.086
-.029
.047
.749
-.013
.118
.049
-.107
-.060
.433
.137
.939
-.007
.090
.281
1995
.184
.269
.185
.412
.165
.235
-.092
-.207
.329
-.056
.495
.286
.494
-.023
.051
-.016
.618
-.050
.105
-.005
.243
-.075
.343
.057
.963
.185
.665
.694
* Industries are classified by the International Standard Industry Classification (ISIC: Revision 2) at the
3-digit level.
33
Table 2: Sample Sizes: Number of Countries included in Each Sample
Year
Industry
311 (Food products)
313 (Beverages)
314 (Tobacco)
321 (Textiles)
322 (Wearing apparel)
323 (Leather products)
324 (Footwear)
331 (Wood products)
332 (Furniture)
341 (Paper and products)
342 (Printing and publishing)
351 (Industrial chemicals)
352 (Other chemicals)
353 (Petroleum refineries)
354 (Misc. petroleum & coal products)
355 (Rubber products)
356 (Plastic products)
361 (Pottery, china, earthenware)
362 (Glass and products)
369 (Other non-metallic mineral products)
371 (Iron and steel)
372 (Non-ferrous metals)
381 (Fabricated metal products)
382 (Non-electric machinery)
383 (Electric machinery)
384 (Transport equipment)
385 (Professional & scientific equipment)
390 (Other manufactured products)
1970
60
59
56
59
57
59
54
59
62
55
61
57
63
43
30
51
51
42
50
55
43
43
61
47
57
54
39
55
1975
71
77
69
68
66
66
61
70
75
70
74
62
72
51
40
63
67
51
60
72
50
50
72
58
68
65
40
62
1980
68
75
68
68
63
63
58
71
72
68
74
69
73
49
34
61
64
51
55
73
55
47
77
63
67
67
37
61
1985
71
69
62
64
54
54
54
71
66
68
68
64
69
47
33
61
68
49
53
63
47
43
75
56
63
61
36
54
1990
76
68
63
70
60
53
53
66
69
72
74
60
71
45
34
58
65
51
56
65
50
42
71
50
65
67
33
61
Note: The names of the countries included in each sample can be provided upon request.
34
1995
63
67
56
61
50
47
50
56
63
66
69
51
64
44
22
57
61
49
55
68
50
40
66
48
59
52
33
46
Table 3: Correlations between VXci/Yc and VMci/Yc: Actual and Predicted (Case I Model)
Industry
311
313
314
321
322
323
324
331
332
341
342
351
352
353
354
355
356
361
362
369
371
372
381
382
383
384
385
390
Note:
1970
Act. Prd.
.183 .108
.646 .118
.431 .153
.084 -.009
.432 -.024
.378 -.009
.162 .043
-.006 -.136
.418 -.024
-.166 -.122
.555 -.199
.397 -.221
.320 -.071
.127 -.013
.508 -.082
.982 .004
.652 -.188
.052 -.276
.100 -.114
.053 .008
.247 -.228
-.046 -.069
.167 -.185
.161 -.266
.322 -.278
.077 -.249
.340 -.164
.507 -.072
1975
Act. Prd.
.304 .092
.041 .116
.498 .092
.494 .027
.208 -.034
.136 -.022
.124 -.038
-.241 -.172
.598 -.139
-.136 -.188
.312 -.271
.234 -.205
.381 -.069
.346 .042
.235 -.182
-.008 -.030
.584 -.175
.107 -.366
-.020 -.143
-.022 -.043
-.013 -.247
.371 -.096
.191 -.212
.036 -.335
.785 -.213
.005 -.310
.242 -.165
.555 -.104
Year
1980
1985
Act. Prd. Act. Prd.
.085 .099 .202 .142
.058 .134 .384 .125
.513 -.021 .670 .012
.424 -.190 .123 -.109
.375 -.020 .470 .071
.336 -.179 .232 -.038
.215 .079 .202 .064
-.189 -.160 -.122 -.044
.329 -.122 .483 -.019
-.091 -.190 -.041 -.098
.459 -.293 .570 -.192
.381 -.285 .567 -.168
.492 -.101 .494 .030
-.032 .022 -.042 .127
.382 -.062 .121 -.094
.084 -.081 .013 -.040
.533 -.207 .762 -.129
-.030 -.516 -.094 -.389
.163 -.150 .238 -.056
.143 -.078 .084 .027
-.189 -.309 -.056 -.197
-.092 -.038 .001 .036
.242 -.230 .278 -.106
.141 -.444 .297 -.398
.543 -.314 .951 -.193
-.083 -.452 -.041 -.339
.056 -.296 .330 -.279
.697 -.106 .187 -.070
i
Act. = Actual correlation (direct measure from the data) of  (
VX ci VM ci
,
);
Yc
Yc
Prd. = Predicted correlation from the Case I version of the model:
i (
VX ci VM ci
1  yc i i
,
)   i (
xc , xc ) .
Yc
Yc
yc
35
1990
Act. Prd.
.314 .203
.381 .140
.248 .060
.131 -.195
.077 .081
.184 -.149
-.019 -.028
-.136 -.020
.099 -.068
.009 -.130
.443 -.260
.359 -.113
.231 .014
.086 .185
-.029 -.034
.047 -.068
.749 -.182
-.013 -.186
.118 -.072
.049 .034
-.107 -.100
-.060 .023
.433 -.195
.137 -.479
.939 -.212
-.007 -.465
.090 -.512
.281 -.052
1995
Act. Prd.
.184 .219
.269 .184
.185 .095
.412 -.084
.165 .053
.235 -.370
-.092 -.124
-.207 -.078
.329 -.142
-.056 -.121
.495 -.258
.286 -.024
.494 .019
-.023 .144
.051 .009
-.016 -.024
.618 -.199
-.050 -.053
.105 -.082
-.005 -.071
.243 -.139
-.075 .012
.343 -.159
.057 -.523
.963 -.182
.185 -.373
.665 -.520
.694 -.034
Table 4: Correlations of Country’s Production Share (xci) with Own GDP Share (yc) or
with Rest of the World’s GDP Share Relative to Own GDP ((1-yc)/yc).
Year
Industry
311
313
314
321
322
323
324
331
332
341
342
351
352
353
354
355
356
361
362
369
371
372
381
382
383
384
385
390
Note:
1970
1975
1980
1985
1990
1995
O
R
O
R
O
R
O
R
O
R
O
R
.996
.981
.988
.962
.995
.965
.983
.936
.994
.990
.996
.982
.999
.988
.948
.997
.947
.770
.993
.974
.934
.984
.994
.987
.970
.995
.994
.991
-.149
-.156
-.168
-.177
-.128
-.197
-.157
-.164
-.150
-.143
-.128
-.145
-.137
-.224
-.300
-.141
-.156
-.213
-.132
-.171
-.258
-.219
-.137
-.130
-.133
-.133
-.104
-.131
.989
.970
.945
.958
.992
.938
.918
.931
.970
.981
.984
.986
.995
.986
.884
.994
.952
.686
.994
.962
.942
.986
.992
.990
.972
.990
.982
.967
-.125
-.149
-.206
-.227
-.183
-.248
-.237
-.197
-.212
-.174
-.113
-.191
-.195
-.218
-.158
-.114
-.190
-.220
-.171
-.211
-.281
-.243
-.182
-.167
-.182
-.167
-.113
-.160
.990
.969
.871
.900
.982
.850
.899
.938
.973
.977
.981
.985
.991
.990
.930
.966
.945
.617
.988
.931
.906
.976
.992
.987
.960
.987
.974
.960
-.159
-.228
-.232
-.164
-.196
-.266
-.256
-.214
-.151
-.183
-.119
-.200
-.213
-.232
-.242
-.234
-.205
-.214
-.193
-.160
-.220
-.251
-.102
-.178
-.179
-.178
-.124
-.116
.992
.982
.928
.907
.986
.773
.809
.978
.989
.991
.990
.975
.994
.995
.974
.970
.953
.629
.979
.946
.829
.980
.988
.973
.926
.986
.988
.951
-.189
-.190
-.212
-.223
-.168
-.276
-.253
-.169
-.170
-.157
-.136
-.193
-.176
-.217
-.118
-.196
-.167
-.194
-.191
-.207
-.275
-.272
-.091
-.156
-.150
-.144
-.105
-.161
.994
.972
.943
.892
.969
.747
.683
.975
.975
.987
.982
.987
.989
.979
.962
.947
.961
.634
.969
.919
.849
.981
.957
.940
.856
.973
.946
.945
-.108
-.143
-.151
-.266
-.235
-.276
-.249
-.124
-.093
-.110
-.076
-.198
-.091
-.235
-.195
-.231
-.186
-.243
-.241
-.240
-.246
-.254
-.145
-.214
-.109
-.103
-.060
-.113
.994
.954
.936
.922
.963
.577
.755
.984
.960
.980
.993
.986
.997
.964
.957
.975
.984
.800
.981
.905
.911
.988
.970
.963
.928
.988
.970
.952
-.192
-.225
-.249
-.197
-.195
-.190
-.272
-.168
-.249
-.214
-.145
-.176
-.174
-.244
-.186
-.217
-.195
-.258
-.243
-.194
-.234
-.232
-.164
-.201
-.151
-.209
-.167
-.144
O: Correlation of a country’s share in production value (xci) with that country’s own GDP
share (yc);
R: Correlation of xci with the share in GDP of the rest of the world (all countries other than
country c) relative to country c’s own GDP share ((1-yc)/yc);
where: xci  VQci / VQci ; VQci = the value of production of country c in industry i;

c
yc  Yc /  Yc ; Yci = GDP of country c.
c
36
Table A1: Correlations between VXci/Yc and VMci/Yc: Actual and Predicted (Case III Model)
Industry
311
313
314
321
322
323
324
331
332
341
342
351
352
353
354
355
356
361
362
369
371
372
381
382
383
384
385
390
Note:
1970
Act. Prd.
.183 -.047
.646 -.211
.431 -.064
.084 .070
.432 -.192
.378 .059
.162 .057
-.006 -.130
.418 -.394
-.166 -.685
.555 -.395
.397 -.187
.320 -.099
.127 -.105
.508 .025
.982 -.842
.652 -.135
.052 -.272
.100 -.242
.053 -.140
.247 -.600
-.046 -.039
.167 -.464
.161 -.576
.322 -.603
.077 -.618
.340 -.888
.507 .107
1975
Act. Prd.
.304 -.031
.041 -.021
.498 -.003
.494 -.396
.208 -.069
.136 .184
.124 .179
-.241 -.281
.598 -.499
-.136 -.776
.312 -.342
.234 -.104
.381 -.210
.346 .204
.235 -.165
-.008 -.009
.584 -.609
.107 -.339
-.020 -.323
-.022 -.116
-.013 -.519
.371 -.318
.191 -.462
.036 -.680
.785 -.370
.005 -.684
.242 -.753
.555 .163
Year
1980
1985
Act. Prd. Act. Prd.
.085 -.002 .202 -.098
.058 -.065 .384 .042
.513 .070 .670 .108
.424 -.064 .123 -.089
.375 -.097 .470 .122
.336 -.559 .232 .064
.215 .316 .202 -.496
-.189 -.350 -.122 -.149
.329 -.182 .483 .018
-.091 -.784 -.041 -.517
.459 -.150 .570 -.126
.381 -.196 .567 -.126
.492 .110 .494 .105
-.032 .117 -.042 .028
.382 -.050 .121 .073
.084 .074 .013 .174
.533 -.339 .762 -.321
-.030 -.366 -.094 -.304
.163 -.202 .238 -.088
.143 -.103 .084 -.021
-.189 -.381 -.056 -.283
-.092 -.240 .001 -.381
.242 -.379 .278 -.210
.141 -.204 .297 -.219
.543 -.331 .951 -.156
-.083 -.586 -.041 -.199
.056 -.507 .330 -.600
.697 -.037 .187 -.202
i
Act. = Actual correlation (direct measure from the data) of  (
1990
Act. Prd.
.314 .070
.381 .067
.248 .042
.131 -.041
.077 .211
.184 -.097
-.019 .161
-.136 .090
.099 -.414
.009 -.254
.443 -.194
.359 -.024
.231 .091
.086 .093
-.029 -.385
.047 -.673
.749 .138
-.013 -.087
.118 -.138
.049 -.001
-.107 -.253
-.060 -.691
.433 -.211
.137 -.328
.939 -.047
-.007 -.336
.090 -.470
.281 -.209
1995
Act. Prd.
.184 .049
.269 .016
.185 .056
.412 -.403
.165 .114
.235 -.411
-.092 .749
-.207 -.359
.329 -.083
-.056 -.643
.495 -.253
.286 -.094
.494 .124
-.023 .019
.051 .075
-.016 -.749
.618 -.142
-.050 .048
.105 -.383
-.005 -.373
.243 -.104
-.075 -.026
.343 -.021
.057 -.420
.963 -.416
.185 -.221
.665 -.305
.694 -.231
VX ci VM ci
,
);
Yc
Yc
Prd. = Correlation between the actual VXci/Yc and the predicted VMci/Yc (based on Case III
version of the model).
37
Figure 1: Cross-Country Correlations between Exports/GDP and Imports/GDP:
Histograms (Number of Sectors) at Each Level of Aggregation
4-digit SITC (Rev.2)
#sectors = 786;
#countries = 95(max);
Mean=0.219; median=0.132
100
Frequency
0
0
50
20
Frequency
40
150
60
200
3-digit SITC (Rev.2)
#sectors = 239;
#countries = 97(max);
Mean=0.232; median=0.183
-1
-.5
0
corr(VX/Y, VM/Y)
.5
1
-1
.5
1
6-digit HS (1992)
#sectors = 5,035;
#countries = 31(max);
Mean=0.209; median=0.116
800
400
400
Frequency
600
300
200
0
200
100
0
Frequency
0
corr(VX/Y, VM/Y)
1000
5-digit SITC (Rev.2)
#sectors = 1,464;
#countries = 94(max);
Mean=0.223; median=0.132
-.5
-1
-.5
0
corr(VX/Y, VM/Y)
.5
1
-1
38
-.5
0
corr(VX/Y, VM/Y)
.5
1
Figure 2: Histogram of ρ(VXci/Yc, VMci/Yc): Actual and Predicted (Case I Model)
(Frequency = numbers of 3-digit ISIC industries)
Predicted: mean = -0.109, median = -0.097
30
20
0
0
10
10
Frequency
Frequency
20
40
50
30
Actual: mean = 0.233, median = 0.189
-1
-.5
0
corr(VX/Y, VM/Y): Actual
.5
1
-1
-.5
0
corr(VX/Y, VM/Y): Case I Pred
.5
Figure 3: Histogram of Gap (Predicted Correlation - Actual Correlation) (Case I Model)
(Frequency = numbers of 3-digit ISIC industries)
15
10
5
0
Frequency
20
25
mean = -0.342, median = -0.326
-1.5
-1
-.5
rho_Gap: Case I Pred - Actual
39
0
.5
Figure A1: Histogram of ρ(VXci/Yc, VMci/Yc): Actual and Predicted (Case III Model)
(Frequency = numbers of 3-digit ISIC industries)
20
0
10
Frequency
0
10
Frequency
20
30
Predicted: mean = -0.194, median = -0.153
30
Actual: mean = 0.233, median = 0.189
-1
-.5
0
corr(VX/Y, VM/Y): Actual
.5
1
-1
-.5
0
.5
corr(VX/Y, VM/Y): Case III Pred
1
Figure A2: Histogram of Gap (Predicted Correlation - Actual Correlation) (Case III Model)
(Frequency = numbers of 3-digit ISIC industries)
10
5
0
Frequency
15
20
mean = -0.745, median = -1.080
-2
-1
0
rho_Gap: Case III Pred - Actual
40
1
Appendix C-1
This appendix shows how the sets of equations in Section 3 are derived for each of the three
cases (Case I, II and III). Derivation of the common equations (Equations (1) through (9)) to all
the cases is as explained in Section 3.
Case I
Derivation of Equations (8-I) through (12-I) is shown in Section 3.
For Equation (13-I), from Equation (11-I);
VX c
1 n
 b c ( Yc ' ) .
Yc
Yc N c ' c
Define yc ≡ Yc/Yw. Also note that  Yc '  Yw  Yc . Hence;
c ' c
VX c
n
(1  y c ) nc
1

b c (1  y c )Yw 
b .
Yc
y cYw N
yc
N
Define xc ≡ VQc/VQw. In this Case I, since all firms (in an industry) in the world charge the same
price and produce the same amount, VQc = ncpq and VQw = Npq, so that xc = nc/N. Therefore;
VX c (1  y c )

bxc .
Yc
yc
Finally, by market clearing in an industry, the value of the total world production must equal the
world total expenditure. That is;
VQw  bYw  b  VQw / Yw .
Hence;
VX c VQw (1  y c )

xc .
Yc
Yw
yc
(13-I)
For Equation (14-I), from Equation (12-I);
VM c
( N  nc )
n
1
 bYc
 b(1  c ) .
Yc
Yc
N
N
Hence, since xc = nc/N and b = VQw/Yw as shown above;
VX c VQw

(1  xc ) .
Yc
Yw
(14-I)
Case II
Equations (8-II) through (14-II) for Case II are derived as follows:
In this case, since the prices of all the varieties charged by firms (excluding trade cost)
are the same, p(ω) = p for all ω. Thus, from Equation (8);
i
Gc  [ n c p
1
  {nc ' (p)
1
}]
1
1
c ' c
 p[nc  
1
n
c ' c
c'
]
1
1
1
 p[nc   1 ( N  nc )]1
Thus, by defining T ≡ τ1-σ;
1
1
Gc  p[nc  T ( N  nc )]
.
(8-II)
Equation (9-II) follows by substituting p(ω) = p and Equation (8-II) into Equation (9). Equations
(10-II) and (11-II) are then self-explaining.
The value of imports of a country is equal to the expenditure of consumers in that
country to all the foreign varieties in an industry. Since p(ω) = p for all ω in this model;
p
VM c  {nc ' bYc ( )1 } .
Gc
c ' c
Then, by substituting T = τ1-σ and Equation (8-II) for Gc into the expression above;
nc '
( N  nc )bTYc
VM c  bYc T  {
}
.
(12-II)
nc  T ( N  nc )
c ' c n c  T ( N  n c )
The last term of the equation is because  nc '  N  nc .
c ' c
Now for the values of exports and imports per GDP: from Equations (11-II) and (12-II);
VX c nc b
Yc '

T {
};
Yc
Yc c ' c nc '  T ( N  nc ' )
VM c
( N  nc )bT

.
Yc
nc  T ( N  nc )
Note that Yc = ycYw for all c. Also, since xc = nc/N in this case, nc = xcN for all c. Therefore;
VX c
x N
y c 'Yw
x
yc'
 c bT {
}  c bT {
};
Yc
y cYw
yc
c ' c x c ' N  TN (1  x c ' )
c ' c x c '  T (1  x c ' )
VM c
N (1  xc )bT
(1  xc )bT


.
Yc
xc N  TN (1  xc ) xc  T (1  xc )
Thus, Equations (13-II) and (14-II) follow by substituting b = VQw/Yw into the two equations
above.
Case III
Equations (8-III) through (14-III) for Case III are derived as follows:
In this case, since there is a single price for all the varieties produced in each country,
p(ω) = pc for all ω produced in c. Thus, Equation (8) becomes:
ii
Gc  [nc p
1
c
  {nc ' (p c ' )
1
}]
1
1
.
c ' c
Then, Equation (8-III) follows by defining T ≡ τ1-σ. Thus, each firm in country c earns the same
revenue (or produces the same value of output), which is, by substituting Equation (8-III) into
Equation (9):
pc qc  bYc (
 bYc
pc 1
p
)  {bYc ' ( c )1 }
Gc
Gc '
c ' c
{nc pc1
.
pc1
 1 pc1

b
[
{
Y
}
]

c'
 T  (nc ' pc1'  )}
{nc ' pc1'   T  (nc" pc1"  )}
c ' c
c ' c
c" c '
1-σ
Equation (9-III) then follows by substituting T = τ into the expression above. Equation (10-III)
also follows by substituting (9-III) into the expression of the value of the total output in country
c: VQc = ncpcqc. The expression for the value of total exports from country c, Equation (11-III), is
obvious since the second term of RHS of Equation (10-III) is the value exported abroad, while
the first term is the value consumed domestically. On the other hand, since the value of imports
is equal to the expenditure of consumers in country c to all the foreign varieties in an industry;
pc '
VM c  {nc ' bYc (
c ' c
Gc
bYc 1 {nc ' ( pc ' )1 }
)1 } 
c ' c
Gc1
.
Then, Equation (12-III) is obtained by substituting T = τ1-σ and Equation (8-III) for Gc into the
expression above.
Now the expressions for the values of exports and imports per GDP are derived as
follows. From Equations (11-III) and (12-III);
VX c nc p c1
Yc '

bT  {
}
1
Yc
Yc
 T  (nc" p c1"  )
c ' c n c ' p c '
c " c '
1
c
nc p
y c 'Yw
nc p c1
yc'

bT  {
}
bT  {
}
1
1
1
y c Yw
yc
 T  ( nc" p c" )
 T  (nc" p c1"  )
c ' c nc ' p c '
c ' c nc ' p c '
c " c '
c " c '
bT  (nc ' p )
VM c
c ' c
.

Yc
nc pc1  T  (nc ' pc1'  )
1
c'
c ' c
The second equality in the former expression (VXc/Yc) is because Yc = ycYw for all c. Then,
Equations (13-III) and (14-III) are derived by substituting b = VQw/Yw into the two expressions
above.
Finally, consider the two-country version of the model for this Case III. In the
two-country setting, there is only one foreign country for country c (home), which is ‘the rest of
iii
the world.’ Having the home country and ‘the rest of the world’ denoted by c and –c, respectively,
Equations (13-III) and (14-III) are modified as follows:
VX c VQw nc p c1
y c

[
T{
}] ;
1
Yc
Yw
yc
n c p c  Tn c p c1
VM c VQw
Tn c p 1c

[
].
Yc
Yw nc p c1  Tn c p 1c
By normalizing the price of varieties produced in ‘the rest of the world’ to one (i.e., p-c = 1) and
substituting n-c = N - nc and y-c = 1 - yc, the expressions above become Equations (13’-III) and
(14’-III), respectively.
iv
Appendix C-2
This appendix shows derivation of the expressions of the correlation between VXci/Yci and
VMci/Yci (ρi(VXci/Yci, VMci/Yci)) for the three cases of the model that are presented in Section 3.
This appendix also examines whether the possibility that the correlation becomes positive or
negative can be rejected theoretically or not. Since the correlation is across countries within an
industry, the script i for an industry is suppressed below.
Case I:
From Equations (13-I) and (14-I) in Section 3;
VX VM c
VQ (1  yc ) VQw
( c ,
)  ( w
xc ,
(1  xc )) .
Yc
Yc
Yw
yc
Yw
Since VQw/Yw is constant (both VQw and Yw do not vary across countries);
VX VM c
(1  yc )
(1  yc )
( c ,
)  (
xc , (1  xc ))    (
xc , xc ) a;
Yc
Yc
yc
yc
and Equation (15-I) follows.
Let us check whether this correlation could be either positive or negative: first, note
that:
(1  yc )
(1  yc )
(
xc , xc )  0  Cov(
xc , xc ) .
yc
yc
Thus one can see the sign of the correlation of the two variables from the sign of their covariance.
By definition;
(1  yc )
(1  yc ) 2
(1  yc )
Cov(
xc , xc )  E[
xc ]  E[
xc ]  E[ xc ] .
yc
yc
yc
Since 0<xc ≡VQc/VQw <1, 0< xc2 < xc <1 for any c. Thus;
(1  yc ) 2
(1  yc )
E[
xc ]  E[
xc ] .
yc
yc
However, since 0< E[xc] <1;
(1  yc )
(1  yc )
E[
xc ]  E[
xc ]  E[ xc ] .
yc
yc
Then, it is ambiguous which of the first term or the second term of RHS of the expression of the
covariance above is greater, and thus the covariance could be positive or negative, depending on
a
Note that, for random variables x and y and constants a and b;
ρ(ax, by) = ρ(x, y).
Also;
ρ(x, 1-y) = -ρ(x, y).
v
the data on xc and yc. Therefore, one cannot reject either possibility that the correlation between
VXc/Yc and VMc/Yc is positive or negative.
Case II:
From Equations (13-II-2) and (14-II) in Section 3;
VX VM c
VQ
xc
VQ
1  xc
( c ,
)   ( w  T ( M  1) 
, w T 
).
Yc
Yc
Yw
xc  T (1  xc ) Yw
xc  T (1  xc )
Since VQw/Yw, T, and M-1 are all constant;
VX VM c
xc
T (1  xc )
( c ,
)  (
,
)
Yc
Yc
xc  T (1  xc ) xc  T (1  xc )
  (
xc
xc
,
)b
xc  T (1  xc ) xc  T (1  xc )
= -1.
The second equality is because
(15-II)
T (1  xc )
xc
.
 1
xc  T (1  xc )
xc  T (1  xc )
Case III:
Here I first show the two-country version, and then the multi-country version.
(Two-country Version)
From Equations (13’-III) and (14’-III) in Section 3;
VX VM c
VQ (1  yc )
Tnc pc1
VQ
T ( N  nc )
( c ,
)  ( w
, w
)
1
1
Yc
Yc
Yw
yc {( N  nc )  Tnc pc } Yw {nc pc  T ( N  nc )}
VQ (1  yc )
Tnc pc1
VQ
T ( N  nc )
 ( w
, w
)
1
Yw
yc {N  (1  Tpc )nc } Yw {TN  ( pc1  T )nc }
Since VQw/Yw is constant;
VX c VM c
(1  yc )
Tnc pc1
T ( N  nc )
,
)  (
,
).
(i):  (
1
Yc
Yc
yc {N  (1  Tpc )nc } {TN  ( pc1  T )nc }
Consider the second argument of the equation (i) above. Note that:
T ( N  nc )
nc pc1

1

.
TN  ( pc1  T )nc
TN  ( pc1  T )nc
Thus;
b
See the previous footnote a.
vi
.
(1  yc )
Tnc pc1
T ( N  nc )
(
,
)
1
yc {N  (1  Tpc )nc } {TN  ( pc1  T )nc }
(ii):
  (
(1  yc )
Tnc pc1
nc pc1
,
)
yc {N  (1  Tpc1 )nc } TN  ( pc1  T )nc
On the other hand;
(1  yc )
Tnc pc1
T ( N  nc )
(
,
)
1
yc {N  (1  Tpc )nc } {TN  ( pc1  T )nc }
(iii):
(1  yc )
nc pc1
N  nc
 (
,
)
1
yc {N  (1  Tpc )nc } TN  ( pc1  T )nc
since T ≡ τ1-σ is constant.
Therefore, by combining (i), (ii), and (iii) above, Equation (15’-III) is derived as follows:
VX VM c
(1  yc )
nc pc1
N  nc
( c ,
)  (
,
)
1
Yc
Yc
yc {N  (1  Tpc )nc } TN  ( pc1  T )nc
(15’-III)
(1  yc )
Tnc pc1
nc pc1
  (
,
)
yc {N  (1  Tpc1 )nc } TN  ( pc1  T )nc
Now check whether this correlation could be either positive or negative. Since:
(1  yc )
Tnc pc1
nc pc1
(
,
)0
yc {N  (1  Tpc1 )nc } TN  ( pc1  T )nc
(1  y c )
Tn c p c1
nc p c1
 Cov(
,
)  0;
y c {N  (1  Tp c1 )nc } TN  ( p c1  T )nc
the sign of the correlation of the two variables is the same as that of their covariance. By
definition;
Cov(
 E[
 E[
(1  yc )
Tnc pc1
nc pc1
,
)
yc {N  (1  Tpc1 )nc } TN  ( pc1  T )nc
(1  yc )
Tnc pc1
nc pc1
]
yc {N  (1  Tpc1 )nc } {TN  ( pc1  T )nc }
(1  yc )
Tnc pc1
nc pc1
]

E
[
]
yc {N  (1  Tpc1 )nc }
TN  ( pc1  T )nc
Note that:
nc pc1
nc pc1

 1 for any nc, pc
TN  ( pc1  T )nc nc pc1  ( N  nc )T
since the term (N – nc)T in the denominator above is positive. Hence;
(1  yc )
Tnc pc1
nc pc1
(1  yc )
Tnc pc1

c
yc {N  (1  Tpc1 )nc } {TN  ( pc1  T )nc }
yc {N  (1  Tpc1 )nc }
vii
(1  yc )
Tnc pc1
nc pc1
(1  yc )
Tnc pc1
 E[
]  E[
].
yc {N  (1  Tpc1 )nc } {TN  ( pc1  T )nc }
yc {N  (1  Tpc1 )nc }
nc pc1
] 1;
However, since E[
TN  ( pc1  T )nc
(1  yc )
Tnc pc1
nc pc1
(1  yc )
Tnc pc1
E[
]  E[
]  E[
].
yc {N  (1  Tpc1 )nc }
TN  ( pc1  T )nc
yc {N  (1  Tpc1 )nc }
It is thus ambiguous which of the first term or the second term of RHS of the expression of the
covariance above is greater, so that one cannot reject either possibility that the correlation
between VXc/Yc and VMc/Yc is positive or negative.
(Multi-country Version)
From Equations (13-III) and (14-III);
T  (nc ' pc1'  )
VX c VM c
VQw nc pc1
yc '
VQw
c ' c
(
,
)  (
T {
},
) Si
1
i
1
1
Yc
Yc
Yw
yc
n
p

T
(
n
p
)
Y
{
n
p
 T i  (nc ' pc1'  )}
c ' c
 c" c"
c' c'
w
c c
c " c '
c ' c
i
nce VQw/Yw and T are both constant;
VX VM c
n p1
( c ,
)  ( c c
Yc
Yc
yc
{ n
c ' c
 (n
c'
yc '
pc1'  )
c ' c
},
)
1
i
1
1
p

T
(
n
p
)
n
p
 T i  (nc ' pc1'  )
 c" c"
c' c'
c c
c " c '
c ' c
and thus Equation (15-III) follows.
This correlation could be either positive or negative, depending on the data for xc and yc, by the
same token as the two-country version.
viii
Appendix C-3
In the Case II model, all firms (or varieties) in an industry charge the same price and produce the
same quantity across countries in equilibrium, by assumption. When such an equilibrium holds,
the following conditions that are presented in Section 3 must be satisfied among xc ≡ VQc/VQw,
yc ≡ Yc/Yw, and T ≡ τ1-σ where τ >1 is the ‘iceberg’ transport cost.
yc
x  T (1  xc )
c '  c
(II-A)
 c
yc ' xc '  T (1  xc ' )
In this appendix, I derive the condition above for each version.
From Equation (9-II), the value of output from a single firm in country c is:
Yc
Yc '
pqc  b[
 T {
}] .
nc  T ( N  nc )
c ' c nc '  T ( N  nc ' )
In the same manner, a typical firm in country c’≠ c produces:
Yc '
Yc"
pqc '  b[
T {
}] .
nc '  T ( N  nc ' )
c " c ' nc "  T ( N  nc " )
Define zc 
(D1)
(D2)
bYc
bYc
. Then, (D1) and (D2) above are expressed as

nc  T ( N  nc ) (1  T )nc  TN
follows:
pqc  z c  T ( z c '  z c ) ;
(D1’)
c'
pqc '  zc '  T ( zc  zc ' ) .
(D2’)
c
In equilibrium, pqc  pqc ' ( pq) for any pairs of c and c’; c ≠ c’. Thus, subtracting each side of
(D1’) from that of (D2’) yields:
0  ( zc '  zc )  T ( zc  zc ' )  (1  T )( zc '  zc ) .
In order for this equality to hold, it must be that zc = zc’ (since T ≡ τ1-σ ≠ 1 by assumptionc).
Hence;
bYc
bYc '
.
(D3)
zc  z c ' 

(1  T )nc  TN (1  T )nc '  TN
b in both sides cancel each other. Moreover, by substituting nc = xcN and Yc = ycYw;
ycYw
yc 'Yw
(D3) 
.
(D3’)

(1  T ) xc N  TN (1  T ) xc ' N  TN
Yw/N in both sides cancel, so that:
c
On the other hand, if T = 1 (i.e., free international trade: Case I), zc does not have to equal zc’, so that the presented
condition does not have to hold for equal-price equal-quantity equilibrium in Case I.
ix
(D3’) 
yc
yc '
.

(1  T ) xc  T (1  T ) xc '  T
Or, this is equivalent to:
yc
x  T (1  xc )
.
 c
yc ' xc '  T (1  xc ' )
This is the condition presented above, and this must hold for any pairs of countries.
What does this condition Imply?
Note that the codition (II-A) implies:
y
yc  yc ' ( c  1)  xc  xc '
yc '
That is, if prices and quantities of all varieties in an industry are equal in equilibrium, then the
country that has larger income (GDP) must have a larger world production share in that industry.
x
Appendix C-4
This appendix is to provide technical details of the discussion in Section 4 about why the
observable xc = VQc/VQw can be the ‘predictor’ of the ‘co-movement’ of VXc/Yc and VMc/Yc
without having pc and nc. I first show for the multi-country version, and then show for the
two-country version.
Multi-country Setting
(Value of production: VQc)
By totally differentiating (with respect to the variables for the home country c) Equation (10-III);
VQc
VQc
VQc
dVQc  (
)  dnc  (
)  dpc  (
)  dYc .
nc
pc
Yc
Fixing the country’s GDP: i.e., setting dYc = 0;
VQc
VQc
dVQc  (
)  dnc  (
)  dpc .
nc
pc
Thus;
dVQc  0  (
VQc
VQc
)  dnc  (
)  dpc  0  dnc  BQ  dpc
nc
pc
VQc
VQc
dVQc  0  (
)  dnc  (
)  dpc  0  dnc  BQ  dpc
nc
pc
by defining BQ  (
;
(E-1)
VQc / pc
).
VQc / nc
From Equation (10-III);
 (n p  )}
0

 T  (n p  )}
Yc  (nc ' pc1'  )
Yc '{nc ' pc1'   T
VQc
c ' c
 (1   )bTnc pc [
 [
1
pc
{nc pc  T  (nc ' pc1'  )}2 c ' c {nc ' pc1'
c ' c
c"
c " c , c '
c " c '
c"
1
c"
1
c"
(   1  1    0) ;
Yc  (nc ' p
1
c'
)
 (n p  )}
 0.

 T  (n p  )}
1
c'
Yc '{nc ' p
VQc
c ' c
 bTpc1 [
 [
1
nc
{nc pc  T  (nc ' pc1'  )}2 c ' c {nc ' pc1'
c ' c
T
c"
c " c , c '
c " c '
c"
1
c"
1
c"
2
Therefore;
BQ  (
VQc / pc
(1   )bTnc pc (1   )Tnc
)

 0.
VQc / nc
bTpc1
pc
(Value of exports per GDP: VXc/Yc)
By the same token as for VQc;
xi
(E-2)
2
d(
VX c
 (VX c / Yc )
 (VX c / Yc )
)0(
)  dnc  (
)  dpc  0  dnc  BX  dpc
Yc
nc
pc
VX
 (VX c / Yc )
 (VX c / Yc )
d( c )  0  (
)  dnc  (
)  dpc  0  dnc  BX  dpc
Yc
nc
pc
by defining BX  (
(E-3)
(VX c / Yc ) / pc
).
(VX c / Yc ) / nc
Note that:
(17-III):
 (VX c / Yc ) VQW (1   )Tnc p

pc
YW
yc

c
 (VX c / Yc ) VQW Tpc1

(15-III):
nc
YW
yc
[
c ' c
[
c ' c
1
c'
{nc ' p
{nc ' pc1'
T
c " c , c '
c " c '
c " c , c '
c " c '
 (n

 T  (n p
1
c'
yc '{nc ' p
 (n

 T  (n p
yc '{nc ' pc1'   T
c"
c"
c"
c"
pc1"  )}
1
c"
)}2
 0;
pc1"  )}
1
c"
)}2
 0.
Therefore;
VQw
  (VX c / Yc ) 
(1   )Tnc pc


pc
Yw yc
(1   )Tnc

BX   

 0 . (E-4)
VQw 1
p
  (VX c / Yc ) 
c
Tpc


nc
Yw yc


(Value of imports per GDP: VMc/Yc)
By the same token;
VM c
 (VM c / Yc )
 (VM c / Yc )
d(
)0(
)  dnc  (
)  dpc  0  dnc  BM  dpc
Yc
nc
pc
VM c
 (VM c / Yc )
 (VM c / Yc )
d(
)0(
)  dnc  (
)  dpc  0  dnc  BM  dpc
Yc
nc
pc
by defining BM  (
(VM c / Yc ) / pc
) . Notice the sign of the last inequality in each of the two
(VM c / Yc ) / nc
lines above: this is because (VM c / Yc ) / nc  0 as shown below.
Note that:
T (1   )nc pc { (nc ' pc1'  )}
(19-III):
 (VM c / Yc )
VQ
c ' c
 w
 0;
1
pc
Yw {nc pc  T  (nc ' pc1'  )}2
c ' c
Tpc1  (nc ' pc1'  )
(17-III):
(E-5)
 (VM c / Yc )
VQ
c ' c
 w
0.
1
nc
Yw {nc pc  T  (nc ' pc1'  )}2
c ' c
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Therefore;
VQ
  (VM c / Yc ) 
 w (1   )Tnc pc


pc
Yw
(1   )Tnc

BM   

 0 . (E-6)
VQw 1
p
  (VM c / Yc ) 
c

Tpc


nc
Yw


From (E-2), (E-4), and (E-6) above;
(1   )Tnc
BQ  BX  BM 
0.
pc
Thus, from (E-1), (E-3), and (E-5) above;
(1   )Tnc
dnc 
 dpc  dVQc  0  d (VX c / Yc )  0  d (VM c / Yc )  0 ;
pc
dnc 
(1   )Tnc
 dpc  dVQc  0  d (VX c / Yc )  0  d (VM c / Yc )  0 .
pc
Finally, since:
xc  VQc / VQW  VQc /(VQc '  VQc ) ;
c ' c
then:
dxc / dVQc  (VQW  VQc ) / VQW2  0 ;
which implies sign(dxc )  sign(dVQc ) .
Hence, in conclusion;
dxc  0  d (VX c / Yc )  0  d (VM c / Yc )  0
dxc  0  d (VX c / Yc )  0  d (VM c / Yc )  0 .
Two-country Setting
(Value of production: VQc)
By the same token as for the multi-country setting shown above;
VQc
VQc
dVQc  0  (
)  dnc  (
)  dpc  0  dnc  BQ  dpc
nc
pc
VQc
VQc
dVQc  0  (
)  dnc  (
)  dpc  0  dnc  BQ  dpc
nc
pc
by defining BQ  (
VQc / pc
).
VQc / nc
xiii
;
By modifying Equation (10-III) for the two-country version;
Yc
(YW  Yc )
VQc  nc pc1 b[
T
].
1
nc pc  T ( N  nc )
( N  nc )  Tnc pc1
Thus;
VQc
Yc
YW  Yc
 (1   )bTnc ( N  nc ) pc [

]0
1
2
pc
{nc pc  T ( N  nc )} {( N  nc )  Tnc pc1 }2
VQc
Yc
YW  Yc
 bTNpc1 [

]0
1
2
nc
{nc pc  T ( N  nc )} {( N  nc )  Tnc pc1 }2
Therefore;
BQ  (
VQc / pc
(1   )bTnc ( N  nc ) pc (1   )nc ( N  nc )
)

 0.
VQc / nc
bTNpc1
Npc
For the value of exports per GDP (VXc/Yc) and that of imports (VMc/Yc), all the steps are the same
as those for the multi-country setting shown above, except using Equations (16’-III) through
(19’-III) instead of (16-III) through (19-III), respectively. Then, one obtains the following:
(1   )nc ( N  nc )
BX  BM 
 BQ  0 ;
Npc
which implies:
dnc 
(1   )nc ( N  nc )
 dpc  dVQc  0  d (VX c / Yc )  0  d (VM c / Yc )  0 ;
Npc
dnc 
(1   )nc ( N  nc )
 dpc  dVQc  0  d (VX c / Yc )  0  d (VM c / Yc )  0 .
Npc
Furthermore, since sign(dxc )  sign(dVQc ) as shown above, one reaches the same conclusion
as that for the multi-country setting: i.e.;
dxc  0  d (VX c / Yc )  0  d (VM c / Yc )  0
dxc  0  d (VX c / Yc )  0  d (VM c / Yc )  0 .
xiv