**Bilateral Factor
Abundance and Intensity with Many Factors, Products, and Countries **

Dajun Tuo

Auburn University

Whether countries export products that use their
abundant factors intensively depends on how these terms are defined when there
are more than two factors, products, and countries since factor abundance and
intensity are ratios. The present note proposes
bilateral measures of factor abundance and intensity for high dimensional data based
directly on the two dimensional definitions.

Do countries
export products that use their abundant factors intensively? The scientific status of this fundamental proposition
from factor proportions trade theory hinges on data that include various
factors of production, a tremendous number of products, and many countries.

The classic factor
content studies of Leontief (1954) and Baldwin (1971) include no measure of factor
abundance as emphasized by Bowen, Leamer, and Sveikauskas (1987), Leamer
(1994), and Thompson (1999). High
dimensional factor proportions trade theory derives various generalities but
falls short of defining factor abundance or intensity as in Jones (1961), Jones
and Scheinkman (1976), Chipman (1979), Chang (1979), Ethier (1984), Thompson
(1985, 2004), Jones (2004), and Choi (2004).
The empirical factor content literature is focused on the single labor input
as pointed out by Stern (1975), Deardorff (1984), and Leamer and Levinsohn
(1995). There are, however, at least
eight labor skill groups in US manufacturing as shown by Clark, Hofler, and
Thompson (1988) and aggregation bias illustrated by Thompson (2005) raises doubts
about empirical results in the factor content literature.

There are four
measures of high dimensional factor abundance in the factor content literature. *Share
abundance* of Vanek (1968) assumes factor price equalization. *World
abundance* of Leamer (1980) is the portion of the world endowment. Rassekh and Thompson (2002) show *mean weighted factor abundance* better
explains trade than either share or world abundance in the Trefler (1985) data
set. The *Euclidean distance abundance* of Thompson (2003) defines the
abundance of a factor relative to every other factor and performs better than
the other three measures as shown by Kang, Malki, Rassekh, and Thompson (2005).

The present
note proposes an alternative measure of factor abundance and factor intensity based
directly on the two dimensional bilateral concept. Elements of the derived high dimensional abundance
and intensity matrices are bilateral comparisons for every pair of factors with
every pair of products (intensity) and countries (abundance). These measures of *bilateral factor intensity* and *bilateral
factor abundance* provide a framework to analyze the empirical proposition
that countries export products using their abundant factors intensively for
data with any number of factors, products, and countries.

**1. Bilateral factor intensity and factor abundance
for high dimensional data**

Introducing notation,

r number
of productive factors indexed by i, g

n number
of products indexed by j, h

c number
of countries indexed by k, m

q_{jk } revenue from production of product j in country k

v_{ijk }total input of factor i for
product j in country k

v_{ik} ≡ S_{j}v_{ijk }endowment of factor i in
country k

V ≡ (v_{ik})
rxc world endowment matrix

A_{k} ≡ (v_{ijk}/q_{jk})
≡ (a_{ijk}) rxn unit
input matrix in country k

c_{jk} consumption spending on product j in country k

x_{jk} ≡ q_{jk}
- c_{jk} net
export revenue of product j in country k.

Factor i is intensive in
product j relative to factor g and product h if a_{ij}/a_{gj}
> a_{ih}/a_{gh} and the positive determinant

a_{ig}^{jh}
≡ a_{ij}a_{gh }- a_{ih}a_{gj} (1)

of the 2x2 matrix transforms bilateral factor intensity
into a scalar. Collect each of these a_{ig}^{jh}
into a bilateral intensity matrix A_{M}. The elements of A_{M} are bilateral factor
intensities.

The input matrix A has
dimension rxn and the dimension of A_{M} is [r(r - 1)/2] x [n(n - 1)/2]
with pairs of factors in each row and pairs of products in each column. The location of element a_{ig}^{jh}
in A_{M} is (row, column) = (i(r - i - 1)/2 - (r - g), j(n - j - 1)/2 -
(n - h)) where i < g and j < h. As
an example the intensity matrix with three factors and three products is

a_{12}^{12}_{
}a_{12}^{13}_{ }a_{12}^{23}_{}

A_{M}^{3x3} = _{ }a_{13}^{12}_{
}a_{13}^{13}_{ }a_{13}^{23 }_{}

_{ }a_{23}^{12}_{
}a_{23}^{13}_{ }a_{23}^{23} .

A similar bilateral abundance
matrix V_{M} is constructed from the rxc world endowment matrix V composed
of 2x2 sub-determinants. Its dimensions are
[r(r - 1)/2] x [c(c - 1)/2]. Each component
v_{ig}^{km} of V_{M} reflects the abundance between factors
i and g and countries k and m,

v_{ig}^{km}
≡ v_{ik}v_{gm }- v_{im}v_{gk}. (2)

The location of element v_{ig}^{km}
in the bilateral abundance matrix V_{M} is (i(r - i - 1)/2 - (r - g), k(c
- k - 1)/2 - (c - m)) where i < g and k < m.

The bilateral intensity
matrix A_{M} includes a comparison of every pair of factors across
every pair of products, and the abundance matrix V_{M} includes a
comparison of every pair of factors across every pair of countries. The following section introduces the matrix
of net exports and proposes a test of the proposition that countries export
products using their abundant factors intensively.

**2. Testing the bilateral factor content
proposition in high dimensional data **

Suppose factor i is
abundant in country k relative to factor g and country m, in the present
notation v_{ig}^{km}_{ }> 0. Suppose also factor i is intensive in product
j relative to product h, that is a_{ig}^{jh} > 0. The narrow factor content issue for these
factors and products is whether country k takes advantage of its bilateral abundance
in factor i relative to factor g with net exports to country m of product j
relative to product h. If country k
produces a higher ratio of product j to product h than country m, the net
export revenue ranking is x_{jk }> 0 > x_{hk} given the
underlying assumptions of equalized prices by trade and homothetic demand as
developed by Ruffin (1977). This narrow factor
content result is not necessary in the present high dimensional model for
arbitrary factors, products, and countries as Thompson (2001) shows with for three factors, products,
and countries. Nevertheless, factor content might be expected to hold as a correlation.

The two dimensional net export
revenue term

x_{jh}^{km}
≡ (x_{jk }- x_{hk}) - (x_{jm} - x_{hm}) (3)

summarizes net trade in products j and h between
countries k and m. In the 2x2x2 factor
proportions model v_{12}^{12}_{ }> 0 implies x_{12}^{12}
> 0. In the present high dimensional context
a positive v_{ig}^{km} does not imply a corresponding positive
x_{jh}^{km} but positive correlation might be expected across
the data.

Collect net export terms
x_{jh}^{km} into the [n(n - 1)/2] x [c(c - 1)/2] net export
matrix

X
≡ (x_{jh}^{km}) (4)

with pairs of products in rows and pairs of
countries in columns. The x_{jk}
terms can be scaled relative to GDP to eliminate the issue of different units
of measure and make x_{jh}^{km} an index.

Multiply the bilateral intensity
measure a_{ig}^{jh} in (1) by the bilateral abundance measure v_{ig}^{km}
in (2) to derive the bilateral factor proportions scalar

z_{ig}^{jhkm }≡
a_{ig}^{jh}v_{ig}^{km} (5)

that summarizes the relationship between factors
i and g for products j and h between countries k and m. A larger factor proportions term z_{ig}^{jhkm}
indicates either higher intensity of factor i relative to factor g between
products j and h or higher abundance between countries k and m. Either higher intensity or abundance would
encourage exports.

The relationship between
products j, h and countries k, m is the sum of these z_{ig}^{jhkm}
terms across factor pairs,

z_{jh}^{km}
≡ Σ_{ig}z_{ig}^{jhkm} (6)

where Σ_{ig} refers to the sum
across unique product pairs. In the 3x3
model the three terms are ig = 12, 13, 23 and with r factors there are r(r -
1)/2 unique factor pairs.

The bilateral factor
proportions matrix

Z ≡ (z_{jh}^{km}) (7)

has elements from unique factor pairs in each
column. In matrix notation,

Z = A_{M}′V_{M}. (8)

With n products and c countries, the bilateral intensity
abundance matrix Z has dimension [n(n - 1)/2] x [c(c - 1)/2] as does the net
export matrix X in (4).

A Mantel (1967) matrix correlation
or nonparametric sign test between Z and X would provide tests of the
proposition that countries tend to export products using their abundant factors
intensively.

**3. Conclusion**

The present bilateral measures of
factor abundance and intensity for high dimensional data can directly test the
proposition that countries export products using their abundant factors
intensively. While empirical studies of
factor proportions trade theory have focused on two factors, two products, or two
countries, theory provides no prediction of production or exports with as few
as three factors, products, and countries.
The factor content proposition is then an empirical issue and the
present note provides a measure to test it.

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