KEO Discussion Paper No. 154

The Leontief Paradox Redux
Kozo Kiyota
Keio University
and Research Institute of Economy, Trade and Industry

The Leontief Paradox Redux*
Kozo Kiyota†
Keio University and RIETI
October 7, 2020

Abstract
Shortly after Leamer (1980) found that the Leontief Paradox was based ona simple conceptual misunderstanding, Brecher and Choudhri (1982) argued
that the fact that the United States exported labor services was, in itself, paradoxical because it is true if and only if its per-capita consumption is less than
the world average. Surprisingly, however, no formal answer to this paradox
has been provided for nearly four decades. This paper revisits this paradox
and formally shows that the paradox can be resolved if the Heckscher–Ohlin–
Vanek model takes into account technology differences across countries andtrade imbalance. In contrast, the paradox cannot be resolved even if the analysis takes into account quasi-homothetic preferences, the Armington homebias, or offshoring.

Key words: Leontief Paradox, Technology differences, Trade imbalance, Nonhomothetic preferences, Armington home bias, Offshoring
JEL classification code: F14* I gratefully acknowledge the financial support received from the JSPS Grants-in-Aid(JP18H03637). The usual disclaimers apply.
†
2-15-45, Mita, Minato-ku. Tokyo, 108-8345, Keio Economic Observatory, Keio University, Japan.
E-mail: kiyota@sanken.keio.ac.jp

1

“In sum, Leontief’s results about the direction of US trade in labor services cannot bereconciled with Vanek’s version of the Heckscher-Ohlin model. ... the Leontief Paradox isstill with us.” — Brecher and Choudhri (1982, p.823)

1

Introduction

Leontief (1953) found that US imports embodied a higher ratio of capital to labor than US exports, using data for 1947. Under the presumption that the US was
capital-abundant in 1947, this appears to contradict the Heckscher–Ohlin (HO) theorem which predicts that US exports would be relatively capital intensive. Thisfinding came to be called the “Leontief Paradox”, one of the most famous paradoxes in economics. Subsequent empirical research reconsidered the analysis,
while taking into account other primary factors such as land and skills. The general conclusion of these studies is that the paradox remained in several cases.1
Nearly a quarter of a century later, Leamer (1980) provided the definitive critique of the Leontief Paradox. He found that the Leontief Paradox is based ona simple conceptual misunderstanding. While the Leontief Paradox is based on
the assumption of balanced trade, the US had a trade surplus in 1947 and wasexporting both labor and capital services as embodied in trade. Under trade imbalance, Leamer (1980) theoretically showed that the test employed by Leontief(1953) was not valid, based on the factor content version of the HO model, theso-called Heckscher–Ohlin–Vanek (HOV) model.2 He then proposed an alternative test that is robust to trade imbalance and found that, once trade imbalance istaken into account explicitly, US trade patterns in 1947 were consistent with theprediction of the HOV model.3
Shortly after Leamer’s finding, Brecher and Choudhri (1982) found another, butclosely related, paradox. The fact that the US exported labor services was in itself
paradoxical because it can occur if and only if its per-capita consumption is lessthan the world average per-capita consumption. Although this is one of the fun1

For more details about the empirical research on the Leontief Paradox from the 1950s to theearly 1980s, see Deardorff (1984).
2
Because the factor content version of the HO model was developed by Vanek (1968), it is called
the Heckscher–Ohlin–Vanek model.
3
However, Baldwin (2008) pointed out that “Leamer’s reversal of the Leontief paradox failed tohold for other years in the 1950s, 1960s, and 1970s when the US trade balance was adjusted in themanner he specified” (p.86).

2

damental questions in the HOV model, surprisingly, no formal answer has been
provided for nearly four decades. For example, Leamer (1987) argued that “Evenafter adjusting for trade surplus, this is impossible to square with the facts” (p.2),
referring to Brecher and Choudhri’s study. Similarly, Davis and Weinstein (2003)
stated that “this paradox refused to perish” (p.135). Even more recent textbooks
such as Feenstra (2015, p.33) made a similar argument.4 Based on the fact that the
HOV model remains one of the core models of international trade, it is essential toresolve this paradox.5
This paper revisits the paradox outlined by Brecher and Choudhri (1982). The
aim of this paper is to present one possible formal solution to the paradox. To doso, I first generalize the paradox from the case of the US to that of any country in
Section 2. Then, I investigate the paradox, focusing on five possible explanations:
1) technology differences, 2) trade imbalance, 3) quasi-homothetic preferences, 4)Armington home bias, and 5) offshoring. Here, I show that it is important to distinguish the difference between physically-based consumption and efficiency-based
consumption to resolve the paradox. The main results are threefold. First, one
can resolve the paradox if the HOV model takes into account technology differences, while trade imbalance is also important for measuring consumption in a
precise manner. Second, the paradox cannot be resolved even if the analysis takes
into account quasi-homothetic preferences or the Armington home bias. Finally,
although offshoring is another logical explanation of the paradox, it is a relatively
recent phenomenon and, therefore, is not an explanation of the original paradox.
Section 3 discusses the paradox from the view point of the actual data. A summaryof the findings is presented in the final section.
4

Baldwin (2008) and Bowen, Hollander, and Viaene (2012) provided helpful explanations of thedebate on the Leontief Paradox.
5
Although the HOV model is one of the traditional trade models, several studies have improved
the HOV model. For recent development of the HOV model, see Ito, Rotunno, and Vézina (2017)
and Morrow and Trefler (2017).

3

2

Theory2.1

Setup and the paradox

This subsection explains the setup of the standard HOV model and the paradox
pointed out by Brecher and Choudhri (1982). Consider many countries, indexedby i = 1, ..., C; many industries, indexed by j = 1, ..., , N ; and many factors, indexed by k or l = 1, ..., M .6 Suppose that countries have identical and homothetic
preferences. Assume further that technologies are constant returns scale and identical across countries, and factor price equalization prevails under free trade.7 . Assume that both goods and factor markets are perfectly competitive, and zero-profit
and full-employment conditions hold. Trade costs such as transportation costs andtrade barriers are assumed to be negligibly small. Goods are mobile but factors areimmobile between countries.
Denote the M × N matrix A = [ajk ]0 as the amounts of primary factors suchas labor and capital that are needed for one unit of production in each industry,
which is common across countries because of the identical technologies. Let Y i ,Di , and T i be the vectors of outputs, demands, and net exports of each industry
for country i, where T i = Y i − Di . Country i’s GDP is defined as:
GDP i = p0 Y i = p0 Di + p0 T i ,(1)where p is the price vector and, thus, p0 Y i , p0 Di , and p0 T i indicate the values of
output (or income), consumption, and net exports, respectively.
The factor content of trade is defined as:
F i ≡ AT i = AY i − ADi .

(2)

In this equation, AY i equals the demand for factors in country i. Denote the factor
endowment vector in country i as V i . The full employment condition implies:
AY i = V i .
67This follows the notation used by Feenstra (2015) for ease of readability of the paper.
Strictly speaking, I assume no factor intensity reversals and incomplete specialization

4(3)

The identical and homothetic preferences mean that the consumption vectors of
all countries are proportional to each other as follows: Di = si DW , where si is theshare of country i in world consumption and DW denotes the world consumption
vector.8
Under balanced trade, p0 T i = 0. From equation (1), therefore, GDP i = p0 Y i =p0 Di . The share of country i’s GDP relative to world GDP is si = p0 Di /p0 DW =p0 Y i /p0 Y W . Note that, from the market clearing condition, world production must
equal world consumption. Thus, ADi = A(si DW ) = si (AY W ) = si V W , where V W
is the world’s factor endowment vector. Therefore:
F i = AY i − ADi = V i − si V W ,(4)which is called the HOV equation. In terms of each factor k, this is:
Fki = Vki − si VkW ,(5)which is a statement of the HOV theorem. If country i’s endowment of factor k
relative to the world endowment is greater than country i’s share of world GDP(i.e., Vki /VkW > si or Vki > si VkW ), then it is said that country i is abundant in thatfactor and, thus, the factor-content of trade in factor k will be positive, Fki > 0, andvice versa for the scarce factor.
Focusing on two factors, capital K and labor L, equation (5) is written as:
FKi = K i − si K W

and FLi = Li − si LW .

(6)

While Leamer (1980) proposed and conducted an alternative test for the Leontief
Paradox based on equation (6), he also found that the US (i = U S) had a trade
surplus in 1947 and was exporting both capital and labor as embodied in tradeFKU S > 0 and FLU S > 0, respectively.
From equation (6), FLi is rewritten as:
Li LW
i
FL =
W



Y

8

YW
Yi
− i
LW
L



Hereafter, superscript W indicates the world total (e.g., DW =

5

(7)

.
PC
j

Dj ).

From Li , LW , Y W > 0, I have:sign(FLi ) = sign



YW
Yi
−
LW
Li


.

(8)

Therefore, Y W /LW − Y i /Li > 0 means that country i’s per-capita consumption is
below the world average per-capita consumption.
In this context, based on the fact that the US (i = U S) was exporting labor as
embodied in trade in 1947 (i.e., FLU S > 0), Brecher and Choudhri (1982) pointed out
that:
Y US
YW
>
,
(9)
FLU S > 0 ⇔LW
LU S
which contradicts the actual data: Y W /LW < Y U S /LU S . In words, Brecher andChoudhri (1982) argued that the fact that the US exported labor services (i.e., FLU S >0) was in itself paradoxical because US per-capita consumption is above the world
average per-capita consumption (i.e., Y W /LW < Y U S /LU S ). Brecher and Choudhri(1982) concluded that “the Leontief Paradox is still with us” as quoted at the beginning of this article.
Note that their discussion is based on the comparison between a country’s percapita consumption and the world average per-capita consumption. In a similar
fashion, it is also paradoxical if FLi < 0 and country i’s per-capita consumption is
below the world average per-capita consumption (i.e., Y W /LW > Y i /Li ) because
the HOV equation suggests FLi < 0 ⇔ Y W /LW − Y i /Li < 0. Noting that theparadox pointed out by Brecher and Choudhri (1982) comes from the difference insign, it can be generalized as follows:
Paradox. It is paradoxical if the actual data indicate:sign(FLi ) 6= sign



Yi
YW
−
LW
Li


,

because equation (10) contradicts the sign of the HOV equation (i.e., equation (8)).
This is a general statement of the paradox.

6

(10)

2.2

Technology differences

One of the restrictive assumptions in the HOV model is that technology is common
to all countries.9 One way to relax this assumption was developed by Trefler (1993),
which allows all factors in every country to differ in their productivities. Thisadjustment is important because it allows me to measure endowments in terms ofefficiency units rather than physical units.
Let πki (0 < πki ) be the productivity of factor k in country i relative to its productivity in the US (πkU S = 1). In terms of efficiency units, the effective endowmentof factor k in country i is πki . The HOV equation is rewritten in terms of effective
factor endowments as follows:
Fki = πki Vki − siCX

πkj Vkj = Ṽki − si ṼkW ,

(11)

j=1

P
j j
where Ṽki (= πki Vki ) and ṼkW (= C
j=1 πk Vk ) are the effective factor endowments ofcountry i and the world, respectively.
Note that FLi is rewritten in terms of efficiency units as follows:L̃i L̃W
FLi =
W



Y

Yi
YW
−
L̃W
L̃i


(12)

.

Because L̃i , L̃W , Y W > 0, I have:sign(FLi ) = sign



YW
Yi
−
L̃W
L̃i


.

(13)

This equation implies that, if FLi > 0, country i’s per-efficiency-unit consumption is
below the world average per-efficiency-unit consumption: Y W /L̃W > Y i /L̃i . Therefore:10
YW
Yi
FLi > 0 ⇔
>
.
(14)
L̃W
L̃i
Equation (14) is similar to equation (9) but has a different implication. Equation(14) could be consistent with the case when country i’s per-capita consumption
9
In this connection, Helpman (1999) argued that “allowing for differences in techniques of production can dramatically improve the fit of factor content equations” (p.133).
10Equation (14) is rewritten: L̃i > si L̃W . Equation (14) thus, simply states that country i is a laborabundant country in terms of efficiency units.

7

is greater than the world average per-capita consumption: Y W /LW < Y i /Li .11
This is because the paradox comes from physically-based consumption, while the
HOV equation is now built upon efficiency-based consumption. In other words, if
productivity differences are introduced in this manner, the analysis can distinguishefficiency-based consumption from physically-based consumption.
To see this point, let me assume that productivity is the same for all countries
except country i: πkj = πk−i , ∀ j 6= i. Equation (11) can be rewritten as:
FLi = πLi Li − siCX

πL−i Lj = πLi Li − si πL−i LW .

(15)

j=1

If FLi > 0, πLi Li − si πL−i LW > 0. Therefore:
πLi Li > si πL−i LW .

(16)

Noting that si = p0 Di /p0 DW (= p0 Y i /p0 Y W ), I have:12
Yi
YW
>
.
πLi Li
πL−i LW(17)

Unlike the standard HOV equation, this equation could be consistent with the casein which country i’s per-capita consumption is greater than the world average percapita consumption.
Let me give a simple numerical example. Suppose that country i’s per-capita
consumption is twice that of the world average: Y W /LW = 1 < 2 = Y i /Li . However, if country i’s productivity is three times as large as the world average’s productivity: πLi = 1 > πL−i = 1/3, I have:
YW
Yi
=
3
>
2
=
.
πLi Li
πL−i LW(18)

Because both sign(FLi ) 6= sign(Y W /LW − Y i /Li ) and sign(FLi ) = sign(Y W /L̃W −11

When FLi > 0 and Y W /LW < Y i /Li , from equation (14), I have L̃i /L̃W > Y i /Y W > Li /LW .
This relationship is not surprising when the productivity of country i is greater than that of othercountries.
−i
12i
In this particular case, equation (17) can be rewritten as: πL/πL> (Y i /Li )/(Y W /LW ). Thismeans that country i’s relative productivity is greater than its relative per-capita consumption.

8

Y i /L̃i ) hold simultaneously, the sign of relative per-capita consumption does notcontradict the HOV equation. That is, the paradoxical situation disappears.
For a more general case when productivities differ across countries, Trefler(1993) derived the following theorem.
Theorem. Allowing for all factors in all but one country to differ in their productivities
πki , there will be a solution for productivities πki such that equation (12) holds with equality
for i = 1, ..., C and k = 1, ..., M for almost all datasets.
Proof. See Trefler (1993).

This in turn implies that equation (13) holds under certain values of productivities. Noting that the paradox pointed out by Brecher and Choudhri (1982) comesfrom relative per-capita consumption in physical units rather than efficiency units,
I obtain the following proposition.
Proposition. If one can find productivity values that equalize FLi and πLi L−si
the paradox outlined by Brecher and Choudhri (1982) is resolved.

PC

j j
j=1 πL L ,Pj j
Proof. Under the productivity values that equalize FLi and πLi Li − si C
j=1 πL L ,equation (13) holds, regardless of the relationship between sign(FLi ) and sign(Y W /LW −Y i /Li ). This in turn means that sign(FLi ) 6= sign(Y W /LW − Y i /Li ) does not contradict the HOV equation. Therefore, the paradox no longer exists.
This proposition states that the paradox is resolved if equation (13) holds underequation (10). The proposition suggests that, even if the US was a net exporter oflabor services in 1947, it does not necessarily contradict the HOV equation whenthe endowment is measured in efficiency units. Empirically, checking whether
equation (13) holds or not is the same as the sign test employed by Bowen, Leamer,and Sveikauskas (1987). However, the test has an additional restriction as in equation (10). This means that the usual sign test is not adequate in investigating theparadox outlined by Brecher and Choudhri (1982) because it includes both paradoxical and nonparadoxical cases. In other words, in order to resolve the paradox,
the sign test of equation (13) should be conducted, conditional on equation (10).
9In this context, Trefler (1993) estimated the productivity parameters that satisfy
the HOV equation as an identity.13 This in turn means that, under the productivity
estimates by Trefler (1993), the paradox outlined by Brecher and Choudhri (1982) isresolved. One may then ask whether relaxing other assumptions of the HO model
can also resolve the paradox. To address this issue, the following subsections investigate four other possibilities: trade imbalance, quasi-homothetic preferences,
Armington home bias, and offshoring.

2.3

Trade imbalance

Many standard trade models assume balanced trade because the models are generally static and, thus, focus on long-run equilibrium. However, the results andtestable implications sometimes change if one relaxes this assumption, as pointed
out by Leamer (1980).14 Therefore, it is important to note the existence of tradeimbalance when one investigates the empirical validity of trade theories.
When trade is unbalanced, p0 T i 6= 0. Let the consumption share under tradeimbalance be s∗i . Under trade imbalance, the consumption share s∗i is:15s∗i =Y i − Ti
p0 D i=.
p0 D W
YW(19)

Under trade imbalance, FLi is:
Li LW
FLi =
W



Y

YW
Y i − Ti
−
LW
Li


(20)

.

From Li , LW , Y W > 0, I have:sign(FLi ) = signYW
Y i − Ti
−
LW
Li

13


.

(21)

However, Trefler’s (1993) productivity estimates are controversial. For example, Gabaix (1997)
pointed out that the productivity parameters would be mechanically correlated with wages if
wages are correlated with GDP per capita. For more details, see Davis and Weinstein (2003).
14
Kiyota (2011) also pointed out that the test of the law of comparative advantage by Bernhofen
and Brown (2004) is applicable only to the case of balanced trade, and proposed a more general test
that is consistent with both balanced and unbalanced trade.
15
If Y W covers all countries, the sum of trade balances will be zero from the market clearing
PC
PC
PC
condition: j=1 p0 T j = 0 and thus j=1 (p0 Y j − p0 T j ) = j=1 p0 Y j = p0 Y W . Therefore, p0 DW =p0 Y W . See, also, Trefler (1993, p.964).

10

As before, Y W /LW − (Y i − T i )/Li > 0 means that country i’s per-capita consumption (excluding net exports) is below world average per-capita consumption:
Y W /LW > (Y i − T i )/Li . This in turn implies:
FLi > 0⇔YW
Y i − Ti
>
.
LW
Li

(22)

The implication under trade imbalance remains essentially the same as under trade balance. However, equation (22) indicates that some adjustments are
required to measure per-capita consumption correctly when trade is unbalanced
because per-capita consumption is not Y i /Li but (Y i − T i )/Li .16

2.4

Quasi-homothetic preferences

Another restrictive assumption of the HOV model is that preferences are homothetic across countries. For example, based on an empirical model with quasihomothetic preferences, Hunter and Markusen (1988) found statistically significant nonhomothetic preferences in world demand patterns.17 To address this issue,
I follow the model developed by Torstensson (1993). In Torstensson’s model, preferences are nonhomothetic and there are two factors (i.e., capital K and labor L)
of production. All other assumptions are retained as in the standard HOV model,
including balanced trade and identical constant returns-to-scale technology.
Assume that all individuals have identical and quasi-homothetic preferences.
They first have to consume a given subsistence bundle of goods. The remaining
income is allocated proportionally across industries.18 Therefore, both production
and consumption are divided into subsistence and nonsubsistence. The factor content of trade equation (2) becomes:19
AT i = AY i − A(D0i + D1i ),
16

(23)

Trefler and Zhu (2010) also argued that it is important to employ such an adjustment in computing the country’s share of world consumption in testing the prediction of the HOV model.
17
In quasi-homothetic preferences, the Engel curves are linear but not through the origin. Thisis called a linear expenditure system. Recent studies also argued the importance of nonhomothetic
preferences in explaining bilateral trade patterns. See, for example, Fieler (2011).
18
Without loss of generality, I assume that the income is large enough to cover both subsistence
and nonsubsistence consumption.
19
If there is no subsistence, equation (23) reverts to equation (2).

11

where D0i and D1i are the vectors of subsistence and nonsubsistence consumption, respectively. Because all individuals have identical subsistence consumption,
country i’s share of subsistence consumption equals its share of world population:
D0i =

Li W
Y ,
LW 0

(24)

where Y0W is the vector of world subsistence production. In contrast, quasi-homothetic
preferences mean that country i’s vector of nonsubsistence consumption is proportional to its share of world nonsubsistence income, si1 :D1i = si1 Y1W ,(25)

where Y1W is the vector of world nonsubsistence production and si1 = Y1i /Y1W .
Let V W , V0W , and V1W be the vectors of world factor endowment used for total
production, subsistence production, and nonsubsistence production, respectively:
V W = V0W + V1W . The full employment condition is:
AY0W = V0W

and AY1W = V1W .

(26)

From equations (23)–(26), the factor content of trade becomes:iiAT = V −



Li
LW



V0W − si1 V1W .

(27)

Let µL (0 < µL < 1) be the share of total world endowment of labor used forWsubsistence production: µL = LW
0 /L . For labor inputs, equation (27) becomes:
FLi


Li
µL LW − si1 (1 − µL )LW= Li −
LW= (1 − µL )(Li − si1 LW ).
(28)This equation is then:
Li LW
FLi = (1 − µL ) W
Y1

12



Y1W
Y1i
−
LW
Li


.

(29)

Because 0 < µL < 1 and LW , LW , Y1W > 0, I have:sign(FLi ) = sign



Y1W
Y1i
−
LW
Li


.

(30)

As for the previous discussions, Y1W /LW − Y1i /Li > 0 means that country i’s percapita consumption of nonsubsistence goods is below world average per-capita
consumption: Y1W /LW > Y1i /Li . Therefore:
FLi > 0

⇔

Y1W
Y1i
>
.
LW
Li

(31)

Equation (31) states that, if country i is a net exporter of labor, its per-capita
consumption of nonsubsistence goods is less than the world average. The only
difference between equations (9) and (31) is whether consumption is of all goods
or of only nonsubsistence goods. When country i is the US, this equation statesthat the per-capita US consumption of nonsubsistence goods is lower than theworld average consumption, which is clearly unrealistic. This in turn implies that,while quasi-linear preferences may be more realistic than homothetic preferences,the paradox cannot be resolved even when quasi-homothetic preferences are employed.

2.5

Armington home bias

One may be concerned that Armington home bias is another possible way of resolving the paradox. To address this concern, this subsection introduces Armington home bias into the baseline framework in Section 2. Following Trefler (1995),
from Di = si DW = si Y W , I distinguish between domestic and foreign goods asfollows:
D i = si α i Y i + β i Y W − Y i ,(32)where β i < 1 < αi captures home bias.20
Without measurement error, the variant of the HOV equation (6) for labor im20

Theoretically, β i can be negative. For example, if αi = 2.5 and β i = −0.5, Y i /Y W = (1 −β )/(αi − β i ) = 1/2.
i13

plied by the Armington assumption is:
FLi



W
Yi
i Y
i
i W
= L − W (1 − β ) i L + β L
Y
Y
 W

i W
i
Y
Y
iL L
= β
− i .
YW
LW
L
i

(33)
(34)

So long as β i > 0, I have:sign(FLi ) = sign



Yi
YW
−
LW
Li


,

(35)

which is the same as equation (8).
Another possible explanation of the paradox is β i < 0. However, as Trefler(1995) argued, it is difficult to interpret negative β i . Besides, he confirmed only
three out of 33 countries had a significantly negative β i using actual data. Therefore, a negative β i does not seem to explain the paradox. This in turn means thatthe paradox cannot be resolved even if Armington home bias is introduced, although it is important in explaining the missing trade.

2.6

Offshoring

One may be further concerned about the role of offshoring. As technology differences play a prominent role in improving the fit of factor content equations (Helpman, 1999), whether or not a country offshores, which has many similar features totechnology differences, may have a similar effect. Offshoring has two aspects. The
first is international factor movement and the second is increases in trade in intermediate inputs through the fragmentation of production process. The following
subsections examine each of these aspects.
2.6.1

International factor movement

To address the issue of international factor movement in the HOV model, I follow
the framework developed by Gaisford (1995). Suppose that a subset of factors canbe mobile between countries without cost while maintaining the other assumptions. Let Vkh,i be the vector of factors deployed in the home country. Let Vkf,i be
the vector of factors moved between countries in net terms. Therefore, Vkf,i > 0 and
14

Vkf,i < 0 mean a net outflow and net inflow of factor k in country i, respectively.
Define the vector of total factor endowments of country i as: Vki ≡ Vkh,i + Vkf,i .
Equation (5) can be rewritten as:
Fki = Vkh,i − si VkWor Fki + Vkf,i = Vki − si VkW .

(36)

The left-hand-side of the HOV equation now includes the factors that move between countries, while Vki on the right-hand-side indicates the sum of factors athome and abroad. For example, if some of the capital stocks are allocated abroad
through foreign direct investment, then K f,i > 0. This HOV equation implies
that a capital-abundant country could have negative net exports of capital (i.e.,
FKi < 0) if capital stocks deployed abroad (K f,i ) are large. Similarly for labor input, if the home country consumes foreign labor services directly through services
offshoring, Lf,i < 0.21 A labor-abundant country could have negative net exportsof labor (i.e., FLi < 0) if labor services deployed abroad (Lf,i ) are large.
When capital is mobile while labor is immobile internationally, Gaisford (1995)
found that the US exports of goods tended to be labor intensive because of US
capital stocks deployed abroad (i.e., K f,i ). Note, however, that when labor is immobile between countries, no labor is deployed in foreign countries: Lf,i = 0. FromLi ≡ Lf,i + Lh,i , Lf,i = 0 means Li = Lh,i . Therefore, the HOV equation for labor
is not affected and, thus, the paradox cannot be resolved even when the analysis
allows for capital movement between countries if labor is immobile.
When labor is mobile between countries, some of the labor services could be
deployed abroad. Equation (36) for labor is written as:
Li LW
FLi =
W
Y



YW
Yi
−
LW
Li



− Lf,i .

(37)

This in turn implies that the different signs of FLi and (Y W /LW − Y i /Li ) do not necessarily mean the paradoxical situation because of labor services deployed abroad(i.e., Lf,i ). Therefore, the paradox outlined by Brecher and Choudhri (1982) seems
to be resolved if the analysis takes into account international labor movement.
21

Head, Mayer, and Ries (2009) argued that “the key idea of service offshoring is that a firm
can replace the services of domestic workers directly with the services of workers residing in foreign countries (“offshore”). Foreign workers can supply their services via communication or via
temporary visits to the domestic producer’s facility” (p.434).

15

Note, however, that if the US exports labor services (i.e., FLU S > 0), US percapita consumption is above the world average per-capita consumption (i.e., Y W /LW <Y U S /LU S ) only when the net outflow of US labor services is sufficiently large. This
may not be plausible for the following two reasons. First, international labor movement was costly in 1947 when the paradox was confirmed. Second, the outflow oflabor services needs to exceed the inflow of labor services in the US, even though
the US has accepted large numbers of immigrants historically. While international
labor movement could be one logical explanation for the paradox, it does not seemto be a plausible explanation in reality. Therefore, a simple international factor
movement does not seem to resolve the paradox.
2.6.2

Trade in intermediate inputs

In the discussion on offshoring above, I maintain the assumption that factor priceequalization prevails under free trade. When offshoring involves the fragmentation of production processes, one needs to relax this assumption so that the endowments of countries can locate in the different cones of diversification (Deardorff,
2001).22 Indeed, Schott (2003) and Kiyota (2012) presented empirical evidences forthe existence of multiple cones. Besides, the fragmentation of production processes
leads to increases in trade in intermediate inputs.23
Trefler and Zhu (2010) showed that the HOV model could be consistent withboth international technology differences and trade in intermediate inputs. With
the assumption that each country’s consumption of any other country’s good is
a fixed proportion of world consumption for that good, the choice of production
techniques can vary across countries. Their study argued that the HOV equation(5) holds even when the HOV model accounts for both international technology
differences and trade in intermediate inputs. However, the measurement of factor
content of trade is different. The analysis requires knowledge not only of what
intermediate inputs are used in the production of goods, but also of where these
inputs come from. Their empirical analysis is, thus, built upon a world input–
output table.
22

In Section 2.2, Trefler (1993) also assumed productivity-equivalent factor price equalization
under technology differences.
23
For example, Kimura (2006) pointed out that active back-and-forth transactions of machineryparts and components are observed among countries with different income levels in East Asia.

16

In the context of my study, exported labor services from the US in 1947 may
have been estimated incorrectly because the analysis did not take into account international technology differences and trade in intermediate inputs. In order totake these two issues into account, however, a world input–output table is needed,which has only been available in recent years. Besides, fragmentation of the production process is a relatively recent phenomenon. While fragmentation throughoffshoring is another logical explanation of the paradox, it does not seem to be themain reason for the paradox.
In sum, offshoring is another logical explanation of the paradox. However, thegrowth of offshoring is a relatively recent phenomenon. Because simple international factor movement does not seem to explain the paradox, it is not necessarily
clear whether offshoring was regarded as an important factor when the paradox
was confirmed.

3

Discussion

3.1

Does the paradox matter in reality?

Section 1 argued that no formal answer has been provided for nearly four decades.
A possible reason for this is that the US has not been a net exporter of labor services
since the 1960s.24 A concern is that, although the paradox may matter theoretically,
it does not matter in reality.
From equation (21), under trade imbalance, the paradox stems from:sign(FLi ) 6= signY i − Ti
YW
−
LW
Li


.

(38)

We check these signs, using the data used in Trefler (1993) because of its reliability
and accessibility.25 Another advantage of the use of his data is that, because the24See, for example, Bowen, Leamer, and Sveikauskas (1987, Table 1) for the year 1967, andMaskus (1985, Table 2) for the year 1972.
25
The data are obtained from the website for Feenstra’s (2015)
P book. Because Trefler’s data do
not cover all countries, the sum of net exports is not zero (i.e., j p0 T j 6= 0). Therefore, Y W 6= DW .
Therefore, I also adjust world average consumption: DW = Y W − T W to compute DW /LW − (Y i −T i )/Li and DW /L̃W − (Y i − T i )/L̃i . However, I continue to use Y W instead of DW for ease of17

data are based on the year 1983, offshoring was still not common and, thus, it isplausible to assume that its effect is small.
Table 1 presents the results. I highlight two main findings. First, the US was a
net importer of labor services in 1983 (i.e., FLU S < 0). Its per-capita consumption
was also higher than the world average (i.e., Y W /LW − (Y U S − T U S )/LU S > 0).
Because the signs are the same, no paradox was confirmed for the US in 1983.
Second, sign(FLi ) 6= sign(Y W /LW − (Y i − T i )/Li ) is confirmed for 18 out of 33
countries. For example, Japan and West Germany experienced the paradox in 1983
because FLi > 0 and Y W /LW − (Y i − T i )/Li < 0, which is similar to the caseof the US in 1947. According to Brecher and Choudhri (1982), these 18 countries
experienced the paradox. Using more recent data, Kiyota (2013) confirmed thatJapan experienced the paradox in 17 of the 26 years from 1980 to 2005. These
results together imply that the paradox is confirmed for many countries and many
years, suggesting that the paradox matters in reality.

3.2

Does trade imbalance really matter?

One may ask whether trade imbalance really matters because Section 2 indicatesthat the implications under trade imbalance remains essentially the same as thoseunder trade balance. To answer this question, I investigate whether the paradox
can be resolved without adjusting the trade imbalance, while taking into accountthe productivity difference only.
Table 2 indicates that, without adjusting the trade imbalance, the paradox still
exists for 14 of 33 countries. Note that under Trefler’s (1993) productivity estimates, the paradox is completely resolved once the trade imbalance is adjusted
because he estimated productivity parameters such that the HOV equation canhold as an identity. These results suggest that adjustment of the trade imbalancealso matters.

3.3

Alternative productivity estimates

One may also ask whether the paradox can be resolved if we use alternative productivity estimates. In this context, Trefler (1995) estimated country-specific productivity parameters π i instead of country-factor-specific productivity parameters
exposition.

18

Table 1: The Paradox Outlined by Brecher and Choudhri (1982)
Country
Austria
Bangladesh
Belgium
CanadaColombia
Denmark
FinlandFranceGreeceHong Kong
Indonesia
IrelandIsraelItaly
Japan
Netherlands
New Zealand
NorwayPakistan
PanamaPortugal
SingaporeSpain
Sri Lanka
SwedenSwitzerland
ThailandTrinidadUK
Uruguay
USA
West Germany
YugoslaviaFLi

Y W /LW−(Y i − T i )/Li
-71,962-2,716-11,939
19,107-54,717-8,359375,879-8,489-31,908
12,15396,426-5,512
87,115-5,190-126,439-8,367-100,890
6,683
28,988
5,823-21,490
18,091
13,2084,336-62,401-5,488353,051-2,0301,037,299-2,498149,314-9,23058,226-734-8,048-10,121-7,402
18,245-19,420
9,670-34,732
12,709-136,613-3,975-51,474
3,666-11,441
18,337
107,829-6,819-112,607-17,096
20,023
17,922-47,088-2,476-638,258-5,49523,905
12,922-1,570,669-18,816
467,201-8,72336,406
12,326

Note: The paradox = “Yes” if sign(FLi ) 6= sign
Source: Computed from Trefler’s (1993) data.



19

i
i
YW
− Y L−Ti
LW.

Paradox

Yes
Yes
Yes
Yes
Yes
Yes
Yes

Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes

Yes

Table 2: The Paradox without the Adjustment of Trade Imbalance
Country
Austria
Bangladesh
Belgium
CanadaColombia
Denmark
FinlandFranceGreeceHong Kong
Indonesia
IrelandIsraelItaly
Japan
Netherlands
New Zealand
NorwayPakistan
PanamaPortugal
SingaporeSpain
Sri Lanka
SwedenSwitzerland
ThailandTrinidadUK
Uruguay
USA
West Germany
YugoslaviaFLi

Y W /LW−Y i /Li-71,962194-11,9392,304-54,717-464
375,879518-31,908216
96,4262,157
87,1152,019-126,439
265-100,890
764
28,988
2,942-21,490-1,276
13,208
1,703-62,401
2,221
353,051
1,187
1,037,299
136
149,314-12
58,226
2,642-8,048-3,803-7,402
1,891-19,420
2,706-34,732
3,149-136,613
1,908-51,474917-11,441
1,612
107,829
578-112,607-658
20,023
3,463-47,088-10,513-638,258-1,70923,905
2,674-1,570,669-223
467,201-342
36,4061,737

Note: The paradox = “Yes” if sign(FLi ) 6= sign
Source: Computed from Trefler’s (1993) data.



20

i
YW
− YLi
LW



.

Paradox
Yes
Yes

Yes

Yes
Yes

Yes

Yes

Yes
Yes
Yes
Yes
Yes
Yes

Yes

πLi . This subsection investigates how the results change if I use country-specificproductivity parameters.
I first check the correlation between the country-factor-specific productivity estimates πLi and the country-specific productivity estimates π i . The correlation ishigh at 0.950. At first sight, this high correlation suggests that alternative productivity estimates also work well. The results in Table 3, however, indicate that 12of 33 countries are experiencing the paradox. These results together imply thatthe use of the country-specific productivity estimates cannot resolve the paradox.
The results suggest that whether or not the paradox can be resolved empiricallydepends upon the estimates of productivity.26

4Concluding Remarks

Shortly after Leamer (1980) found that the Leontief Paradox was based on a simple conceptual misunderstanding, Brecher and Choudhri (1982) found another, butclosely related, paradox. The fact that the US exported labor services was in itself
paradoxical because it is true if and only if its per-capita consumption is less than
the world average per-capita consumption. Even though this is one of the fundamental questions in the HOV model, surprisingly, no formal answer has been
provided for nearly four decades.
This paper revisits the paradox outlined by Brecher and Choudhri (1982). Building upon the study by Trefler (1993), I showed that the paradox can be resolved
if the HOV model takes into account technology differences, while trade imbalance is also important to the precise measurement of consumption. This is because the introduction of technology differences enables one to distinguish between physically-based and efficiency-based consumption. Besides, a country’s
consumption is under- or over-estimated without the adjustment of the trade imbalance. In addition, this paper showed that the paradox cannot be resolved even if
the analysis takes into account quasi-homothetic preferences, or Armington homebias. While fragmentation through offshoring is another logical way to resolve
the paradox, it does not seem to be the main reason when the paradox was found26

In this context, Fisher and Marshall (2016) pointed out the problem of only using the US technology matrix. However, as mentioned above, Trefler and Zhu (2010) had already presented a moregeneral framework that relaxed this assumption.

21

Table 3: The Paradox under Alternative Productivity Estimates
Country
Austria
Bangladesh
Belgium
CanadaColombia
Denmark
FinlandFranceGreeceHong Kong
Indonesia
IrelandIsraelItaly
Japan
Netherlands
New Zealand
NorwayPakistan
PanamaPortugal
SingaporeSpain
Sri Lanka
SwedenSwitzerland
ThailandTrinidadUK
Uruguay
USA
West Germany
YugoslaviaFLi

Y W /π i LW−(Y i − T i )/π i Li
-71,962-843-11,939
15,079-54,717-6,644
375,879-14,744-31,908-10,890
96,426
1,996
87,115-1,768-126,439-1,395-100,890
7,561
28,988
1,780-21,490
19,991
13,208
8,575-62,401-14,907353,051300
1,037,299
4,818
149,314-3,64958,226-17,306-8,048-6,690-7,402
19,856-19,420595-34,732-13,703-136,613-12,827-51,474-1,697-11,441
20,885
107,829-10,011-112,607-10,039
20,02325,835-47,088-10,689-638,258-6,92623,905-25,492-1,570,669-1,963
467,20197
36,406
10,995 Wii
Note: The paradox = “Yes” if sign(FLi ) 6= sign πYi LW − Yπi−T.
Li
Source: Computed from Trefler’s (1993) and (1995) data.

22

Paradox

Yes
Yes

Yes
Yes
Yes

Yes
Yes
Yes
Yes

Yes
Yes

Yes

because it is a relatively recent phenomenon.
The paradox pointed out by Brecher and Choudhri (1982) was confirmed for
18 of 33 countries in 1983. This suggests that the paradox matters in reality. Adjusting the trade imbalance is also important in explaining the paradox. However,whether or not the paradox can be resolved empirically depends upon the estimates of productivity.
It can be said that, when Trefler (1993) succeeded in introducing country- andfactor-specific productivity into the HOV model, the paradox pointed out by Brecher
and Choudhri (1982) had already been resolved. The contribution of this paper isto generalize the paradox from the case of the US to that of all countries, and toclarify this point formally and explicitly. The bottom line of this paper is that theparadox outlined by Brecher and Choudhri (1982) is now formally resolved. Note
that, as discussed by Feenstra (2015), one limitation of Trefler’s (1993) methodology is that there are many “free” parameters in the sense that the use of πki requires
C × M parameters. Note also that what this paper shows is that there is at least
one possible solution to the paradox. Thus, following Maskus and Nishioka (2009),
it is also important to explore alternative country-factor-specific productivity estimates. Some of these issues will be explored in the next stage of my research.

References
Baldwin, Robert E. (2008) The Development and Testing of Heckscher–Ohlin Trade
Models: A Review, Cambridge, MA: MIT Press.
Bernhofen, Daniel M. and John C. Brown (2004) “A Direct Test of the Theory ofComparative Advantage: The Case of Japan,” Journal of Political Economy,
112(1): 48–67.
Bowen, Harry P., Edward E. Leamer, and Leo Sveikauskas (1987) “Multicountry, Multifactor Tests of the Factor Abundance Theory,” American Economic
Review, 77(5): 791–809.
Bowen, Harry P., Abraham Hollander, and Jean-Marie Viaene (2012) Applied International Trade, 2nd edition, New York, NY: Palgrave Macmillan.

23

Brecher, Richard A. and Ehsan U. Choudhri (1982) “The Leontief Paradox, Continued,” Journal of Political Economy, 90(4): 820–823.
Davis, Donald R. and David E. Weinstein (2001) “An Account of Global Factor
Trade,” American Economic Review, 91(5): 1423–1453.
Davis, Donald R. and David E. Weinstein (2003) “The Factor Content of Trade,”in E. Kwan Choi and James Harrigan (eds.), Handbook of International Trade,
Oxford, UK: Blackwell, 119–145.
Deardorff, Alan V. (1984) “Testing Trade Theories and Predicting Trade Flows,”
in Ronald Jones and Peter Kenen (eds.), Handbook of International Economics,
Vol. 1, Amsterdam: North Holland, 467–517.
Deardorff, Alan V. (2001) “Fragmentation in Simple Trade Models,” North American Journal of Economics and Finance, 12(2): 121–137.
Feenstra, Robert C. (2015) Advanced International Trade: Theory and Evidence, 2ndedition, Princeton and Oxford: Princeton University Press.
Fieler, Ana Cecı́lia (2011) “Nonhomotheticity and Bilateral Trade: Evidence andA Quantitative Explanation,” Econometrica, 79(4): 1069–1101.
Fisher, Eric O’N. and Kathryn G. Marshall (2016) “Leontief Was Not Right After
All,” Journal of Productivity Analysis, 46: 15–24.
Gabaix, Xavier (1997) “The Factor Content of Trade: A Rejection of the Heckscher–
Ohlin–Leontief Hypothesis,” manuscript, Harvard University.
Gaisford, James D. (1995) “International Capital Mobility, the Factor Content ofTrade and Leontief Paradoxes,” Journal of International Economics, 39(1–2):
175–183.
Head, Keith, Thierry Mayer, and John Ries (2009) “How Remote Is the OffshoringThreat?” European Economic Review, 53(4): 429–444.
Helpman, Elhanan (1999) “The Structure of Foreign Trade,” Journal of Economic
Perspectives, 13(2): 121–144.

24

Hunter, Linda C. and James R. Markusen (1988) “Per-capita Income as a Determinant of Trade,” in Robert Feenstra (ed.), Empirical Methods for International
Economics, Cambridge, MA: MIT Press, 89–109.
Ito, Tadashi, Lorenzo Rotunno, and Pierre-Louis Vézina (2017) “Heckscher–Ohlin:
Evidence from Virtual Trade in Value Added,” Review of International Economics, 25(3): 427–446.
Kimura, Fukunari (2006) “International Production and Distribution Networks
in East Asia: Eighteen Facts, Mechanics, and Policy Implications,” Asian Economic Policy Review, 1: 326–344.
Kiyota, Kozo (2011) “A Test of the Law of Comparative Advantage, Revisited,”
Review of World Economics, 147(4): 771–778.
Kiyota, Kozo (2012) “A Many-cone World?” Journal of International Economics,
86(2): 345–354.
Kiyota, Kozo (2013) “Skills and Changing Comparative Advantage,” Japan and theWorld Economy, 28: 33–40.
Leamer, Edward E. (1980) “The Leontief Paradox, Reconsidered,” Journal of Political Economy, 88(3): 495–503.
Leamer, Edward E. (1987) “The Leontief Paradox,” in Palgrave Macmillan (eds.),
The New Palgrave: A Dictionary of Economics, 1st edition, London: Palgrave
Macmillan, 1–3.
Leontief, Wassily W. (1953) “Domestic Production and Foreign Trade: The American Capital Position Re-examined,” Proceedings of the American PhilosophicalSociety, 97: 332–349.
Maskus, Keith E. and Shuichiro Nishioka (2009) “Development-related Biases in
Factor Productivities and the HOV Model of Trade,” Canadian Journal of Economics, 42(2): 519–553.
Morrow, Peter M. and Daniel Trefler (2017) “Endowments, Skill-biased Technology, and Factor Prices,” A Unified Approach to Trade,” NBER Working paper, No. 24078.
25

Schott, Peter K. (2003) “One Size Fits All? Heckscher-Ohlin Specialization in
Global Production,” American Economic Review, 93(3): 686–708.
Torstensson, Johan (1993) “The Heckscher–Ohlin–Vanek Theorem with Quasihomothetic Preferences,” Economics Letters, 41(2): 157–160.
Trefler, Daniel (1993) “International Factor Price Differences: Leontief was Right!”
Journal of Political Economy, 101(6): 961–987.
Trefler, Daniel (1995) “The Case of the Missing Trade and Other Mysteries,” American Economic Review, 85(5): 1029–1046.
Trefler, Daniel and Susan Chun Zhu (2010) “The Structure of Factor Content Predictions,” Journal of International Economics, 82(2): 195–207.
Vanek, Jaroslav (1968) “The Factor Proportions Theory: The N-Factor Case,” Kyklos, 21(4): 749–754.

26