CESSA WP 2016-01

Decomposition of gross exports into value-added:
an alternative accounting system

Masaaki Kuboniwa
Hitotsubashi University
October 2016

Center for Economic and Social Studies in Asia (CESSA) Working Paper
Downloadable from:http://www.econ.ynu.ac.jp/cessa/publication/workingpaper.html

Center for Economic and Social Studies in Asia, Department of Economics
Yokohama National University

Decomposition of gross exports into valueadded: an alternative accounting system
Abstract
We demonstrate that we can trace decomposition processes of gross exports into valueadded or factor contents across countries in a straightforward manner at both the nonbilateral and bilateral levels, if we redefine value-added exports, based on Trefler and Zhu(2010). For a bilateral trade system, we witness double counts as well as additive counts.
We also provide a numerical example for a three-country case (China, the USA, and therest of the world).
JEL E01, E16, F14, F23, L14
Keywords: Value-added exports, gross exports, decomposition, bilateral trade

1. Introduction
In light of the growing intermediate goods trade, major internationalorganizations including the World Trade Organization (WTO) have emphasized theimportance of a new concept of trade in value-added (TiVA) in place of traditional grosstrade in order to capture global value chains through foreign trades across countries. In
this paper we present an alternative accounting system of gross exports at both the nonbilateral level and the bilateral level1 in place of that of Koopman et al. (2014). First, we

The non-bilateral level focuses on a country’s exports from, and imports to, the world, while thebilateral level considers exports and imports between two countries (if one is the origin/source
country, another is the destination country). For a two-country case, we do not have to distinguishthese two levels.
1

re-define non-bilateral and bilateral value-added exports by using elements of the valueadded ratios, international Leontief inverse, and gross exports/imports based on Trefler
and Zhu (2010). Second, we prove that our definition of bilateral value-added exports ismathematically equivalent to Johnson and Noguera (2012)’s value-added exports definedby using elements of the value-added ratios, international Leontief inverse, anddestination country’s final demand. Third, we demonstrate that we can trace gross exports
through the decomposition of gross exports into value-added, double and additive counts
in a straightforward manner if employing our representation of value-added exports inplace of Johnson and Noguera’s definition. For the non-bilateral trade system with morethan three countries we can clearly witness two parts of return-home and no-return-homeas double counts in the foreign content of gross exports. Particularly, we can clearly seethat the latter, no-return-home as the double count, means return-home as the additive
count at the bilateral level. We also provide a numerical example for a three-country case
(China, the USA and the rest of the world), using aggregated World Input-Output Data,in order to enhance our alternative gross exports accounting system.

2. ModelFollowing Isard (1951), Trefler and Zhu (2010) and Johnson and Noguera (2012),
we reproduce an inter-country multi-sector model in a general framework.
We assume that there are r, s =1, 2, …, R countries (areas or regions), each ofwhich produces and inputs r(i), s(j) = 1, 2, .., n products. We further assume the classical
Leontief open input–output model with fixed input coefficients and final demand for eachcountry. In this model, each sector produces a single commodity without joint production.
We regard the last country R as ROW (the rest of the world). We consider an international

input–output system not in physical terms but in value terms. Table 1 shows the basic data
structure of the system.

[Table 1 here]

We denote the following:Ars = (ar(i)s(j)) (n×n): country r’s export coefficient matrix to country s or country s’simport coefficient matrix from country r if r ≠ s, and country r’s input coefficientmatrix of domestically produced intermediate goods if s = r;Yrs = [Yr(i)s] (n × 1): country s’s final demand vector for country r (n×1) or country
r’s final goods export vector to country s if r ≠ s;

Yr = [Yr(i)] (n×1): origin country r’s final demand vector in an international input–
output table (origin-wise final demand vector); Yr =Σ𝑠 Yrs ;̃ 𝑟 = [𝒀
̃ 𝑟(𝑖) ] (n × 1): country r’s final demand vector, including exports of𝒀intermediate goods, in each country’s input–output system;Fs = [Yrs] ((n × R)×1): destination country s’s final demand vector for all countries(destination-wise final demand vector);Xrs = [Xr(i)s] (n × 1): origin country r’s output vector induced by destination country
s’s final demand;

Xr = [Xr(i)] (n × 1): origin country r’s output vector (origin-wise output vector); Xr=Σ𝑠 Xrs ;

X = [Xr] ((n × R)×1): an overall output vector;𝑿∗𝑠 = [Xrs] ((n × R)×1): an overall output vector induced by destination country s’s
final demand (destination-wise output vector);

I: an (n × R) dimensional identity matrix; and

In: an n-dimensional identity matrix.
We assume that non-negative matrixes A and Arr are productive.
Denoting X* as the equilibrium output vector, the global equilibrium (market

clearing) condition for an Isard type of non-competitive inter-country multi-sector input–
output table in value terms can be written as:
𝑿∗ = 𝑨𝑿∗ + 𝒀; 𝑿∗ = 𝑩𝒀, where 𝑩 = (𝑰 − 𝑨)−𝟏 ,

(1)

where
𝑨11
…
𝑨
𝑨 = 𝑟1
…
[𝑨𝑅1

𝑨12
…
𝑨𝑟1
…
𝑨𝑅2

𝑩11
…
−𝟏
𝑩 = (𝑰 − 𝑨) = 𝑩𝑟1…
[𝑩𝑅1… 𝑨1𝑠 …
…
…
… 𝑨𝑟𝑠 …
…
…
… 𝑨𝑅𝑠 …
𝑩12
…
𝑩𝑟1
…
𝑩𝑅2

…
…
…
…
…

𝐴1𝑅
…
𝑨𝑟𝑅 ,
…
𝑨𝑅𝑅 ]
𝑩1𝑠 …
…
𝑩𝑟𝑠 …
…
𝑩𝑅𝑠 …

𝑩1𝑅
…
𝑩𝑟𝑅 , and
…
𝑩𝑅𝑅 ]

𝒀1
𝒀11
𝒀1𝑠
𝒀1𝑅
…
…
…
…
𝒀 = 𝒀𝑟 = 𝒀𝑟1 + ⋯ + 𝒀𝑟𝑠 + ⋯ + 𝒀𝑟𝑅 = 𝑭1 + ⋯ + 𝑭𝑠 + ⋯ + 𝑭𝑅 ;…
…
…
…
[𝒀𝑅 ] [𝒀𝑅1 ][𝒀𝑅𝑠 ][𝒀𝑅𝑅 ]

𝑿1
𝑿11
𝑿1𝑠
𝑿1𝑅
…
…
…
…
𝑿 = 𝑿𝑟 = 𝑿𝑟1 + ⋯ + 𝑿𝑟𝑠 + ⋯ + 𝑿𝑟𝑅 = 𝑿∗1 + ⋯ + 𝑿∗𝑠 + ⋯ + 𝑿∗𝑅 .
…
…
…
…
[𝑿𝑅 ]
[𝑿𝑅1 ][𝑿𝑅𝑠 ][𝑿𝑅𝑅 ]Overall output 𝑿∗∗𝑠 and origin country r’s output 𝑿∗𝑟𝑠 , induced by a fixed destination
country s’s final demand 𝑭𝑠 , are given by∗𝑿∗∗𝑠 = 𝑨𝑿∗∗𝑠 + 𝑭𝑠 = 𝑩𝑭𝑠 ; 𝑿∗∗𝑠 = ( 𝑿1𝑠… 𝑿∗𝑟𝑠 … 𝑿∗𝑅𝑠 )′ ; 𝑿∗𝑟𝑠 = Σ𝑘 𝑨𝑟𝑘 𝑿∗𝑘𝑠 + 𝒀𝑟𝑠 . (2)k =1,2, .., R also denotes country k, while (′) denotes the transpose of a matrix or a vector.
This equation is essential for Johnson and Noguera’s definition of value-added exports.
∗∗
By the given definitions of 𝑭𝑠 and 𝒀𝑟𝑠 , we have 𝑿∗ = Σ𝑠 𝑿∗∗𝑠 ; ; 𝑋𝑟𝑠= Σ𝑖 𝑋𝑟(𝑖)𝑠. Notethat the origin-wise output 𝑿𝑟 or 𝑿𝑠 is different from the respective destination-wise
output 𝑿∗𝑟 or 𝑿∗𝑠 .
In view of intermediate gross exports (imports), Ers = 𝑨𝑟𝑠 𝑿∗ 𝑠 in Table 1,
country r’s gross exports to country s, 𝑬𝑟𝑠 and its total gross exports, 𝑬𝑟 are given by
𝑬𝑟𝑠 = 𝑨𝑟𝑠 𝑿∗ 𝑠 + 𝒀𝑟𝑠 (𝑠 ≠ 𝑟); 𝑬𝑟 = Σ𝑠≠𝑟 𝑬𝑟𝑠 , where 𝑿∗𝑠 = Σ𝑘 𝑿∗𝑠𝑘 .

(3)

Hence, the local equilibrium (market clearing) condition that each country
must satisfy is given bỹ 𝑟 , = (𝑰𝑛 − 𝑨𝒓𝒓 )−𝟏 (𝑬𝑟 +𝒀𝑟𝑟 ).
𝑿∗𝑟 = (𝑰𝑛 − 𝑨𝒓𝒓 )−𝟏 𝒀(4)̃ 𝑟 , where 𝑩𝑟 = (𝑰𝑛 − 𝑨𝒓𝒓 )−𝟏. Generally, 𝑩 𝑟 ≠ 𝑩𝑟𝑟 . It is
This can also be written as 𝑿∗𝑟 = 𝑩𝑟 𝒀
noteworthy that the global equilibrium and local equilibria are simultaneously satisfied for theinternational input–output system.

Let us define country r’s i-th value-added ratio as 𝑣𝑟(𝑖) = 𝑉𝑟(𝑖) /𝑋𝑟(𝑖) , where𝑉𝑟(𝑖) is country r’s i-th value-added. Country r’s value-added ratio vector and the overall
vector are 𝒗𝑟 = (𝑣𝑟(𝑖) )(1 × 𝑛) and 𝒗 = (𝒗𝑟 )(1 × (𝑛 × 𝑅)), respectively. Then, by virtueof definitions of input coefficients and value-added ratios, we have𝒖 = 𝒖𝑨 + 𝒗; 𝒖𝑛 = 𝒖𝑛 Σ𝑘 𝑨𝑘𝑟 + 𝒗𝑟 .
Therefore, value-added ratios are given by(5)𝒗 = 𝒖(𝑰 − 𝑨); 𝒗𝑟 = 𝒖𝑛 (𝑰𝑛 − Σ𝑘 𝑨𝑘𝑟 ) .

(6)

where 𝒖 = (1,1,…,1) (1×(n×R)) and 𝒖𝑛 = (1,1, … ,1) (1×n) are aggregation vectors ofunities. That is to say, the price vector associated with an input–output system in value
terms always equals an aggregation vector.

3. Definitions and theorems for value-added trade
The new concept of value-added trade is defined as follows:
Definition 1. (Johnson and Noguera 2012; Kuboniwa 2015). The new concept of valueadded exports and TiVA:̂ 𝑟 𝑿∗𝑟𝑠 ,Using equation (2), country r’s value-added exports to country s are defined as 𝑽
̂ 𝑟 = 𝑑𝑖𝑎𝑔 {𝑣𝑟 , … , 𝑣𝑟 } (n×n). The total value-added exports of origin country
where 𝑽(1)(𝑛)𝑣𝑎∗
̂ 𝑟 𝑿∗𝑟𝑠 = 𝒗𝑟 𝑿∗𝑟𝑠 . At the non-bilateral level,r to destination country s amount to 𝐸𝑟𝑠= 𝒖𝑛 𝑽the overall value-added exports of origin country r to all destination countries account for
𝐸𝑟𝑣𝑎∗ = 𝒗𝑟 ∑𝑠 𝑿∗𝑟𝑠 .

Corresponding to Definition 1, we can define a country’s factor content of trade
employed worldwide to produce the country’s net trade vector as follows:

Definition 2. (Trefler and Zhu 2010). The factor content of gross exports in the case withmany countries and sectors:

We consider the following equation for country s’s gross output vector 𝒁𝑠 employed to
produce the country’s net trade vector 𝑬𝑛𝑒𝑡for s = 1, 2, …,R (s ≠ r) in the case that𝑠country s exports to and imports from countries 1, …, r,…, R (r ≠ s):
𝒁𝑠 = 𝑨𝒁𝑠 +𝑬𝑛𝑒𝑡= 𝑩𝑬𝑛𝑒𝑡𝑠𝑠 ,(7)where, assuming s > r without loss of generality,
∗∗
−𝑬1𝑠
−𝑿1𝑠
…
…
∗∗
−𝑿𝑟𝑠
−𝑬𝑟𝑠
…
…
𝒁𝑠 =
and 𝑬𝑛𝑒𝑡
=
.
𝑠
∗∗
∗∗
∗∗
𝑬𝑠1 + ⋯ +𝑬𝑠𝑟 + ⋯ + 𝑬𝑠𝑅
𝑿𝑠1 + ⋯ + 𝑿𝑠𝑟 + ⋯ + 𝑿𝑠𝑅
…
…
∗∗
(
−𝑬𝑅𝑠
)
(
−𝑿𝑅𝑠
)

𝑿∗∗
𝑟s (𝑛 × 1) is country r’s gross output vector employed worldwide to produce the grossexports of country r to country s or the gross imports of country s from country r. Then,the factor content of gross exports from origin country r to destination country s is defined̂ 𝑟 𝑿∗∗
as 𝑽
𝑟𝑠 . The total factor content of gross exports from origin country r to destination∗∗𝑣𝑎∗∗
country s amounts to 𝐸𝑟𝑠= 𝒗𝑟 𝑿𝑟𝑠 . Equation (7) simultaneously defines the overallfactor content of origin country r to all destination countries, at the non-bilateral, which
accounts for 𝐸𝑟𝑣𝑎∗∗ = 𝒗𝑟 ∑𝑠 𝑿∗∗
𝑟𝑠 .
We arrive at the following equivalence theorem:
Theorem 2. Bilateral equivalence theorem (Kuboniwa 2014b2, 2015)
Definition 1 is bilaterally equivalent to Definition 2.
∗
∗∗
∗
𝑣𝑎∗∗
𝑣𝑎∗
̂𝑟 𝑿∗∗
̂ ∗
𝑿∗∗
= 𝐸𝑟𝑠
for r, s=1, 2,…., R (s≠r).
𝑟𝑠 = 𝑿𝑟𝑠 , 𝑽
𝑟𝑠 = 𝑽𝑟 𝑿𝑟𝑠 and 𝒗𝑟 𝑿𝑟𝑠 = 𝒗𝑟 𝑿𝑟𝑠 ; 𝐸𝑟𝑠

2

Kuboniwa (2014b) proved the theorem for a 3-country world with many sectors.

Proof
In view of equations (2) and (3), we have𝑿∗𝑟𝑠 = Σ𝑘 𝑨𝑟𝑘 𝑿∗𝑘𝑠 + 𝒀𝑟𝑠 = (Σ𝑘≠𝑠 𝑨𝑟𝑘 𝑿∗𝑘𝑠 + 𝑨𝑟𝑠 𝑿∗𝑠𝑠 ) + ( 𝑬𝑟𝑠 − 𝑨𝑟𝑠 𝑿∗𝑠 )= Σ𝑘≠𝑠 𝑨𝑟𝑘 𝑿∗𝑘𝑠 − 𝑨𝑟𝑠 Σ𝑘≠𝑠 𝑿∗𝑠𝑘 + 𝑬𝑟𝑠 .

(8)

On the other hand, it immediately follows from equation (7) that
∗∗
∗∗
𝑿∗∗
𝑟𝑠 = Σ𝑘≠𝑠 𝑨𝑟𝑘 𝑿𝑘𝑠 − 𝑨𝑟𝑠 Σ𝑘≠𝑠 𝑿𝑠𝑘 + 𝑬𝑟𝑠 .

(9)

Let us compare equations (8) and (9). The solution for the equation system (2), 𝑿∗𝑟𝑠 canbe a solution for the equation system (7), 𝑿∗∗
𝑟𝑠 , and vice versa. In view of the existenceof the unique B due to productiveness of A, we learn that the solution of equation (2) is
unique and that the solution of equation (7) is also unique. Therefore, we can conclude
𝑿∗𝑟𝑠 ≡ 𝑿∗∗
𝑟𝑠 for all r, s =1, 2,…, R (s≠r).

(10)

∗∗
∗
̂ 𝑟 𝑿∗∗
̂ ∗This implies 𝑽
𝑟𝑠 = 𝑽𝑟 𝑿𝑟𝑠 and 𝒗𝑟 𝑿𝑟𝑠 = 𝒗𝑟 𝑿𝑟𝑠 for r, s=1, 2,…., R (s≠r).

Q.E.D.
TiVA, which is based on value-added exports proposed by Johnson and Noguera(2012) and international organizations such as OECD and WTO, measures an origin(source) country’s value-added employed worldwide to produce a destination country’s
final demand, excluding intermediates. The factor (value-added) content of trade,
proposed by Trefler and Zhu (2010), measures a country’s value-added employedworldwide to produce the country’s net trade vector, which appropriately arranges grossexports and imports, including intermediates, as positive and negative elements,
respectively. At a glance, these two measures may look quite different. However, we
proved that in the world with many countries and sectors, these two measures of TiVAand the factor (value-added) content of trade are bilaterally equivalent. 3
3

Johnson and Noguera (2012) implicitly suggest this equivalence. Foster-McGregor,4. Decomposition of gross exports
4.1.

The first stage:First, we employ Koopman et al. (2010)’s decomposition of gross exports into

̂ , where
̂ B𝑬
domestic and foreign value-added terms. Making a matrix calculation of 𝑽
̂ =diag{ 𝑬𝑟 }, its diagonal blocks and off-diagonal blocks yield̂ =diag{ 𝒗𝑟 } and 𝑬
𝑽
domestic value-added and foreign value-added induced by gross exports, respectively.
The domestic content (DV) and foreign content (FV) of value-added of country r are
defined respectively as
DVr = 𝒗𝑟 𝑩𝑟𝑟 𝑬𝑟 and FVr =Σ𝑠≠𝑟 𝒗𝑠 𝑩𝑠𝑟 𝑬𝑟 .

(11)

A country’s gross exports generate its value-added. They also induce othercountries’ (foreign) value-added through the country’s gross imports from many countries(many countries’ exports to the country) to produce its gross exports. By virtue of the priceequation (5) or (6), we have the following macro identity:
DVr + FVr = 𝒖𝑛 𝑬𝑟 = 𝐸𝑟.

(12)

Thus a country’s gross exports are fully decomposed into its domestic and foreignvalue-added contents induced by its gross exports. From equation (12) and definitions wealways have
DVr ≤ 𝐸𝑟 for all r.

(13)

Origin country’s domestic content is always less than its gross exports by theforeign content. For convenience in the following discussion, we define country r’s
domestic content induced by its exports to destination s as
and Stehrer (2013) showed the non-bilateral equivalence, whereas they did not refer tothe bilateral equivalence.

𝐷𝑉𝑟𝑠 = 𝒗𝑟 𝑩𝑟𝑟 𝑬𝑟𝑠 .
We also define country r’s foreign content induced by its exports to destination s as
𝐹𝑉𝑟𝑠 = Σ𝑘≠𝑟 𝒗𝑘 𝑩𝑘𝑟 𝑬𝑟𝑠 .
In a similar way, we have

4.2.

DVrs + FVrs= 𝒖𝑛 𝑬𝑟𝑠 = 𝐸𝑟𝑠.

(12’)

DVrs ≤ 𝐸𝑟𝑠 for all r and s (r ≠ s).

(13’)

The second stage
The concept of value-added exports or TIVA clarified that a country’s domestic

DVr should further be decomposed into value-added exports and part of double counts
due to complicated foreign trade across countries. Koopman et al. (2010, 2014) linked
Johnson and Noguera’s definition of value-added exports with their concepts of domestic
and foreign value-added. This implies that they linked “gross trade approach” to equation(11) with “final demand approach” to value-added exports (Definition1). Instead, we link
“gross net trade approach” to equation (11) with our defined “gross net exports-approach”
to value-added exports (Definition 2) so that we may directly and explicitly capture therelationships between value-added exports, domestic value-added and foreign valueadded at the bilateral level by Definition 2 in a straightforward manner.

4.3.

Two-country caseLet us consider two-country case (r,s =1,2) with many sectors. Our

decomposition result on the first stage and the second stage is shown by Figure 1.

[Figure 1 here]

The first stage:
For country 1: In view of equation (11) we have the following results.
The total value of gross exports is E1＝E12 ＝unE12.
The domestic content is DV1=v1B11E12.
The foreign content is FV1= v2B21E12.
This foreign content returns to country 2.
For country 2: Similarly, we have the following.
The total value of gross exports is E2 = E21 =unE21.
The domestic content is DV2= v2B22E21.
The foreign content is FV2= v1B12E21.
This foreign content returns to country 1.
The second stage:
For country 1: By virtue of equation (7), we have the following result.
𝑣𝑎
The total value of value-added exports is 𝐸1𝑣𝑎 = 𝐸12=v1(B11E12－B12E21) .

The difference between DV1 and 𝐸1𝑣𝑎 equals the value-added which returns home fromcountry 2, v1B12E21. This means that the double count in the domestic content equals thevalue-added which returns home.
For country 2: Similarly, we have the following.
𝑣𝑎
The total value of value-added exports is 𝐸2𝑣𝑎 = 𝐸21=v2(B22E21－B21E12) .

The difference between DV2 and 𝐸2𝑣𝑎 equals the value-added which returns home fromcountry 1, v2B21E12. This implies that the double account in the domestic content equals
the value-added which returns home.
In a 2-country case with many sectors, first, we do not have to distinguish thebilateral exports from each country’s total exports. Second, the double counting in thehome domestic content of gross exports is merely the total value-added of the foreigncontent, which is produced in the corresponding destination partner (the rest of the world)
for origin’s gross exports and returns home. That is to say, all of home value-added
demanded by the partner for home’s gross exports should constitute the double counting
in the home domestic content. Third, the bilateral total value of origin country’s valueadded exports is always less than its domestic content by the double count returned home.
Fourth, in our two-country case, if country 1 is a specific country such as China, country
2 should be the rest of the world, including all other countries such as the USA, Japan andothers. We cannot use our two-country case assuming that country 1 is China and country
2 is only the USA. 4

Accounting of gross exports for a two-country case
Summarizing the above discussions, gross exports are decomposed as follows:
E12 = DV1+FV1 = v1B11E12 + v2B21E12𝑣𝑎= 𝐸12+v1B12E21+ FV1 = v1(B11E12－B12E21) +[v1B12E21+ v2B21E12 ].

(14)

E21 = DV2+FV2 = v2B22E21 + v1B12E21𝑣𝑎= 𝐸21+v2B21E12+ FV2 = v2(B22E21－B21E12) +[v2B21E12+ v1B12E21] .

(15)

Terms of [.] show total double counts in the gross exports. Equations (14) and (15) areessentially equivalent to Koopman et al. (2014)’s equations 11, 13 and 14 whereas ourequations straightforwardly present double counted elements and their sources bydefinition without any toil and tear. Equation 11 in Koopman et al. (2014) can bearranged as

E12 = VT12 + v1B12(A21 X1+ Y21)+ v2B21(A12 X2+ Y12),(16)

where VT12 = v1B11Y12 + v1B12Y22 is country 1’s value-added exports by Johnson andIn this context, the numerical example of two-country case (China and the USA) in
Koopman et al. (2014) may be inappropriate.
4𝑣𝑎Noguera’s definition (Definition1). By virtue of Theorem 1, 𝑉𝑇12 = 𝐸12, and E21 =A21 X1+

Y21 and E12=A12 X2+ Y12. Thus, the above equation comes down to our equation (14) .
However, as the double counting part is hided in Definition 1, we may toil up to equation(16).
In the two-country case, from equations (14) and (15), a country’s grosstrade balance with the rest of the world or net ‘gross exports’ results in
𝑔

𝑣𝑎
𝑣𝑎
𝑣𝑎
𝑇12 = 𝐸12 − 𝐸21 = 𝐸12
− 𝐸21
= 𝑇12
;
𝑔

𝑣𝑎
𝑣𝑎
𝑣𝑎
𝑇21 = 𝐸12 − 𝐸21 = 𝐸12
− 𝐸21
= 𝑇21
.

At the macro-economy level, generally, a country’s gross trade balance is identical to its
value-added trade balance (Kuboniwa 2015, Theorem 1 and Theorem 1’). That is to say,double accounting parts of gross exports vanish in GDP on the expenditure side.5
Country r’s GDPr at the expenditure side is as follows:
GDP1= 𝒖𝑛 (𝒀11 + 𝒀21 + 𝑬12 − 𝑬21 ),GDP2= 𝒖𝑛 (𝒀12 + 𝒀22 + 𝑬21 − 𝑬12 ).
By virtue of well-known Leontief’s duality equation with a price system p = 𝒖𝑛 , GDP at
the production side is identical to GDP at the expenditure side.6 In our framework, we
can represent the following:

5

Using local equilibrium equation (4), Koopman et al. further decompose intermediate imports, Asr

Xr (country r) of equation (16) into Asr𝑩𝑟 𝒀𝑟𝑟 and Asr𝑩𝑟 𝑬𝑟𝑠 . Then they expand country r’s GDP(GDPr=VTr+vrXrr) based on equation (2) and inversion by 4-block partitioning. Comparing theirdecomposition and GDP concept, they find a ‘pure double-counted term’ of double-counted terms,
vrBrsAsr𝑩𝑟 𝑬𝑟𝑠 for country r. Their fact-finding may need further investigation. However, in view ofour discussion based on GDP on the expenditure side, their finding may not be significant for grossexports accounting.
6

For example, see Miller and Blair (2009, p.53).

𝑣𝑎
GDP1=𝒗1 𝑿1 = 𝒗1 (𝑿11 + 𝑿12 ) = 𝐸12+ 𝒗1 𝑿11 ,𝑣𝑎
GDP2=𝒗2 𝑿2 = 𝒗2 (𝑿21 + 𝑿22 ) = 𝐸21+ 𝒗2 𝑿22 .

4.4.

A three-country case
Next, we examine a three-country case (r,s =1, 2, 3) with many sectors. Our

decomposition results at the first and the second stages are shown by Figure 2.

[Figure 2 here]

The first stage:
For country 1: It follows from equation (18) that:
The total value of gross exports is E1=un(E12 + E13).
The domestic content is DV1 = v1B11(E12+ E13).
The foreign content is FV1= v2B21(E12+ E13)+v3B31(E12+ E13).
Unlike two-country case, a part of this foreign content does not return to countries 2 and3. This reason will be clarified when we examine the bilateral level.
For country 2: Similarly, we have:
The total value of gross exports is E2 = un(E21 + E23).
The domestic content is DV2 = v2B22(E21+ E23).
The foreign content is FV2 = v1B12(E21+ E23)+v3B32(E21+ E23) .
A part of this foreign content does not return to countries 1 and 3.
For country 3: We also have:
The total value of gross exports is E3 = un(E31 + E32). .
The domestic content is DV3 = v3B33(E31+ E32).
The foreign content is FV3 = v1B13(E31+ E32)+v2B23(E31+ E32) .
A part of this foreign content does not appear to return to countries 1 and 2.
The second stage:

For country 1:
At the non-bilateral level:
From equation (11), the total value of value-added exports is
𝐸1𝑣𝑎 =v1{B11(E12+ E13)－B12E21－B13E31}.
The difference between DV1 and 𝐸1𝑣𝑎 equals the value-added which returns home fromcountry 2 and 3, v1(B12E21+ B13E31). This difference shows the double count in thedomestic content. Unlike 2-country case, capturing double counts for 3-country case israther complicated. At the non-bilateral level, a part of foreign contents (FV2 and FV3 )generated by country 1 and country 2, v1B12E23 + v1B13E32, does not return home as
double counts (no-return as double counts). However, as will be shown soon, with anadverse adjustment, this part also returns home at the bilateral level. By definitions, wehave always DV1 ≥ 𝐸1𝑣𝑎 . Origin country’s domestic content is always greater than its
value-added exports by the double count returned home.

At the bilateral level:
In view of equation (7), the value of country 1’s value-added exports to country 2 isdefined as𝑣𝑎
𝐸12= v1{B11E12－B12(E21+ E23) + B13E32}.
𝑣𝑎
A part of the difference between DV12 = v1B11E12 of DV1 and 𝐸12
equals the foreignvalue-added which returns home from country 2, v1B12(E21+ E23) as the double counts inthe domestic content. The term of v1B13E32, which is country 1’s value-added generated
by non-destination country 3’s exports to destination country 2 (country 2’s imports fromcountry 3), is a part of the foreign content of non-destination country 3 for origin country
1. At the bilateral level, all the foreign content of the destination country for the origin
country returns home as double counts. However, this needs an adverse adjustment of theforeign content of the non-destination country for the origin country. The adjustment term
returns home not as the double accounting but as the additive count for the value-addedexports (no return as double counts but return as additive counts).
It is noteworthy to learn the following fact:𝑣𝑎
DV12 ⋛ 𝐸12
if v1B12(E21+ E23) ⋛ v1B13E32.

The domestic content of origin country’s value-added induced by its exports to thedestination country can be greater (less) than its value-added exports if the double countreturned home is greater (less) than the additive term returned home.
The value of country 1’s value-added exports to country 3 is defined as𝑣𝑎
𝐸13= v1{B11E13－B13(E31+ E32)+ B12E23}].
𝑣𝑎
A part of the difference between the term DV13 = v1B11E13 of DV1 and 𝐸13
equals thevalue-added which returns home from country 3, v1B13(E31+ E32), which is the foreigncontent of country 3 for country 1and the double count part in the domestic content at thebilateral level. The term of v1B12E23, which is country 1’s value-added generated by nondestination country 2’s exports to destination country 3 (country 3’s imports from country2), which is a part of the foreign content of country 2 for country 1. This part of the foreigncontent returns home as not the double count but the additive count to the bilateral valueadded exports.
As can easily be verified, we have
𝑣𝑎
𝑣𝑎
𝐸12
+ 𝐸13
=v1B11E12－B12(E21+ E23) +B13E32} + v1{B11E13－B13(E31+ E32)+ B12E23}=v1{B11(E12+ E13)－B12E21－B13E31} = 𝐸1𝑣𝑎 .
In 𝐸1𝑣𝑎 , terms across countries 2 and 3 for adjustments, v1B13E32 and v1B12E23 in
𝑣𝑎
𝑣𝑎
𝐸12
and 𝐸13are cancelled out.

For country 2:
At the non-bilateral level:
Similarly, the total value of value-added exports is
𝐸2𝑣𝑎 = v2{B22(E21+ E23)－B21E12－B23E32}
The difference between DV2 and 𝐸2𝑣𝑎 equals the value-added which returns home fromcountry 2 and 3, v2(B21E12+ B23E32). This difference demonstrates the double count in thedomestic content.
At the bilateral level:
The value of country 2’s value-added exports to country 1 is calculated by𝑣𝑎
𝐸21= v2{B22E21－B21(E12+ E13)+B23E31}.
𝑣𝑎
A part of difference between the term DV21=v2B22 E21 of DV2 and 𝐸12
equals the value-

added which returns home from country 1, v2{B21(E12+ E13), which constitutes the double
count in the domestic content of country 2 for country 1. The term of v2B23E31, which iscountry 2’s value-added generated by non-destination country 3’s exports to destination
country 1 (country 1’s imports from country 3), is a part of the foreign content of country
3 for country 2. This term returns home as the additive count.
The value of country 2’s value-added exports to country 3 is𝑣𝑎
𝐸23
= v2{B22E23－B23(E31+ E32)+B21E13}.
𝑣𝑎
A part of the difference between the term v2B22E23 of DV2 and 𝐸23
equals the value-

added which returns home from country 3 with an adjustment, v2B23(E31+ E32), whichconstitutes the double count in the domestic content of country 2 for country 3. The termof v2B21E13, is a part of the foreign content of country 1 for country 2.
𝑣𝑎𝑣𝑎
In a similar way, we have 𝐸2𝑣𝑎 = 𝐸21+ 𝐸23.

For country 3:
At the non-bilateral level:
The total value of value-added exports is
𝐸3𝑣𝑎 = v3{B33(E31+ E32)－B31E13－B32E23}.
The difference between DV3and 𝐸3𝑣𝑎 equals the value-added which returns home fromcountry 1 and 2, v3(B31E13+ B32E23). This difference provides the double count in thedomestic content.
At the bilateral level:
The value of country 3’s value-added exports to country 1 is𝑣𝑎
𝐸31
= v3{B33E31－B31(E12+ E13)+B32E21}.
𝑣𝑎
The difference between the term DV31= v3B33E31 of DV3 and 𝐸31
equals the value-added

which returns home from country 1 with an adjustment, v3{B31(E12+ E13)－B32E21}
which yields the double count part in the domestic content of country 1 for country 3. The
term of v3B31(E12+ E13) is the foreign content of country 1 for country 3. The term of
v3B32E21, which is country 3’s value-added generated by country 2’s exports to country 1(country 1’s imports from country 2), is a part of the foreign content of country 2 forcountry 3.
The value of country 3’s value-added exports to country 2 is𝑣𝑎
𝐸32
= v3{B33E32－B32(E21+ E23) +B31E12}.
𝑣𝑎
The difference between the term DV32= v3B33E32 of DV3 and 𝐸32
equals the value-added

which returns home from country 2 with an adjustment, v3{B32(E21+ E23)－B31E12}, whichis the double count part in the domestic content of country 3 for country 2. The term of
v3B32(E21+ E23) is the foreign content of country 2 for country 3. The term of v3B31E12,
which is country 3’s value-added generated by non-destination country 1’s exports todestination country 2 (country 2’s imports from country 1), is a part of the foreign content
of country 1 for country 3.
𝑣𝑎𝑣𝑎
Similarly, we have 𝐸3𝑣𝑎 = 𝐸31+ 𝐸32.

Accounting of gross exports for three-country case
Summarizing the above discussions, gross exports are decomposed as follows:
At the non-bilateral level:
E1=E12 + E13 = DV1+FV1 = v1B11E1 + (v2B21+ v3B31)E1= 𝐸1𝑣𝑎 +v1(B12E21+ B13E31) +FV1
= v1(B11E1−B12E21−B13E31) +[v1(B12E21+ B13E31)+ (v2B21 + v3B31)E1].
E2 = E21 + E23 =DV2+FV2 = v2B22E2 + (v1B12+ v3B32)E2= 𝐸2𝑣𝑎 +v2(B21E12+B21E32) +FV2
= v2(B22E2−B21E12−B21E32) +[v2(B21E12+B21E32) + (v1B12+ v3B32)E2] .
E3 = E31 + E32 =DV3+FV3 = v3B33E3 + (v1B13+ v2B23)E3= 𝐸3𝑣𝑎 +v3(B31E13+B32E23) +FV3
= v3(B33E3−B31E13−B32E23) +[v3(B31E13+B32E23) +(v1B13+ v2B23)E3] .
Terms of [.] show total double counts in the gross exports.
At the bilateral level:
Assume that origin/source is country 1, the destination country is 2 or 3 and the nondestination/third party country is 3 or 2.
E12= DV12+FV12 = v1B11E12 + (v2B21+ v3B31)E12𝑣𝑎= 𝐸12+v1{B12(E21+ E23)−B13E32}+FV12= v1{B11E12−B12(E21+ E23) + B13E32}+[v1{B12(E21+E23)−B13E32}+ (v2B21 + v3B31)E12].
E13= DV13+FV13 = v1B11E13 + (v2B21+ v3B31)E13

𝑣𝑎
= 𝐸13
+v1{B13(E31+ E32)−B12E23}+FV13= v1{B11E13−B13(E31+ E32) + B12E23}+[v1{B13(E31+E32)−B12E23}+ (v2B21 + v3B31)E12].
Suppose that origin/source is country 2, the destination country is 1 or 3 and the nondestination/third party country is 3 or 1.
E21= DV21+FV21 = v2B22E21 + (v1B12+ v3B32)E21𝑣𝑎= 𝐸21+v2{B21(E12+ E13)−B23E31}+FV21= v2{B22E21−B21(E12+ E13) + B23E31}+[v2{B21(E12+E13)−B23E31}+ (v1B12 + v3B32)E21].
E23= DV23+FV23 = v2B22E23 + (v1B21+ v3B31)E23𝑣𝑎= 𝐸23+v2{B23(E31+ E32)−B21E13}+FV23= v2{B22E23−B23(E31+ E32) + B21E13}+[v2{B13(E31+E32)−B12E23}+ (v1B12 + v3B32)E23].
Let origin/source be country 3, the destination country be 1 or 2 and the nondestination/third party country be 2 or 1.
E31= DV31+FV32 = v3B33E31 + (v1B13+ v3B31)E31𝑣𝑎= 𝐸31+v3{B31(E12+ E13)−B32E21}+FV31= v3{B33E31−B31(E12+ E13) + B32E21}+[v3{B31(E12+E13)−B32E21}+ (v1B13 + v3B31)E31].
E32= DV32+FV32 = v3B33E32 + (v1B13+ v3B31)E32
𝑣𝑎= 𝐸32+v3{B32(E21+ E23)−B31E12}+FV32= v3{B33E32−B32(E21+ E23) + B31E12}+[v3{B32(E21+E23)−B31E12}+ (v1B13 + v3B31)E32].
+ v3 B32E21+ v3 B31E12
At the bilateral level, the overall sum of return home from each destination
country as double accounts, R(-) which should be deducted from value-added exports,
amount to:
RETURN(-) = v1B12E1+ v1B13E3+ v2B21E1+ v2B23E3+ v3B31E1+v3B32E2 = FV1+FV2+FV3.
At the bilateral level, all foreign contents return origin home as double counts deductedfrom the origin’s value-added exports. However, at the same time, additive counts returnhome from a non-destination or third party, the total of which accounts for
RETURN(+) = v1B13E32 + v1B12E23+ v2 B23E31+ v2 B21E13+ v3 B32E21+ v3 B31E12.
These are part of foreign contents. At the non-bilateral level, this results in the total of
double accounts returning home less than the total foreign content.

4.5.

The general caseWe are now in a position to present the general case with many countries andmany sectors as follows:
The first stage:
For an arbitrary country, r (1,2,3,…,R): It follows from equation (11) that:
Gross exports are given by equation (3) as 𝑬𝑟 = Σ𝑠≠𝑟 𝑬𝑟𝑠 .
The total value of gross exports is 𝐸𝑟 = un𝑬𝑟 .
The domestic content is DVr = vrBrr𝑬𝑟 .
The foreign content is FVr = Σ𝑠≠𝑟 vsBsr𝑬𝑟
The second stage:
For country r:
At the non-bilateral level:
By virtue of equation (7), the total value of country r’s value-added exports is
𝐸𝑟𝑣𝑎 =vr (Brr𝑬𝑟 －Σ𝑠≠𝑟 𝑩𝑟𝑠 𝑬𝑠𝑟 ).

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The difference between DVr and 𝐸𝑟𝑣𝑎 equals the value-added which returns home fromthe world (countries s ≠ ｒ), vrΣ𝑠≠𝑟 𝑩𝑟𝑠 𝑬𝑠𝑟 . This difference shows the double accounting
in the domestic content. The foreign content of each partner s is defined as

FVs = Σ𝑟≠𝑠 vrBrs𝑬𝑠 = Σ𝑟≠𝑠 vrBrs(𝑬𝑠1 +𝑬𝑠2 + 𝑬𝑠3 + ⋯ +𝑬𝑠𝑟 + ⋯ + 𝑬𝑠𝑅 ).
As can be seen, only one part of r’s value-added generated by foreign partner s’s exports,
vrBrs𝑬𝑠𝑟 returns home r as a double accounting term.

At the bilateral level:
In view of equation (7), the value of origin country r’s value-added exports to destination
country s ≠ｒ is defined as𝑣𝑎𝐸𝑟𝑠= vr{𝑩𝑟𝑟 𝑬𝑟𝑠 －𝑩𝑟𝑠 𝑬𝑠 +Σ𝑘≠𝑟,𝑠 𝑩𝑟𝑘 𝑬𝑘𝑠 }.

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𝑣𝑎
A part of the difference between DVrs = vrBrrErs of DVr and 𝐸𝑟𝑠
equals the value-added

which returns home from country s, vr𝑩𝑟𝑠 𝑬𝑠 . This difference constitutes the double countin the domestic content. The term of vr𝑩𝑟𝑠 𝑬𝑠 is clearly country r’s foreign value-added
generated by destination country’s gross exports to the world. The term of vr𝛴𝑘≠𝑟,𝑠 𝑩𝑟𝑘 𝑬𝑘𝑠 ,which is origin country r’s value-added generated by the third party non-destination
country k’s exports to destination country s (country s’s imports from country k), is a partof the foreign content of country k for country r. At the bilateral level, all the foreigncontent of the destination country for the origin country returns home. However, thisneeds an adverse adjustment. The adverse adjustment term is the foreign content of thethird party non-destination countries for the origin country which returns origin home notas the double count deducted from the value-added exports but as the positive addition to
the value-added exports.
𝑣𝑎
As can easily be seen, we have 𝐸𝑟𝑣𝑎 = 𝛴𝑠≠𝑟 𝐸𝑟𝑠. Clearly, in view of equation(21) and (22), and definitions, we also have the following theorem.

Theorem 2.

For r = 1, 2, 3,…, R
𝐷𝑉𝑟 ≥ 𝐸𝑟𝑣𝑎 ;𝑣𝑎𝐷𝑉𝑟𝑠 ⋛ 𝐸𝑟𝑠
if 𝑣𝑟 𝑩𝑟𝑠 𝑬𝑠 ⋛ 𝑣𝑟 Σ𝑘≠𝑟,𝑠 𝑩𝑟𝑘 𝑬𝑘𝑠 .(s = 1, 2, 3,…, R; R≧3;s≠ r)(19)

Origin country r’s domestic value-added content induced by its exports to allother destination countries is always greater than its value-added exports to all othercountries by the double count, vrΣ𝑠≠𝑟 𝑩𝑟𝑠 𝑬𝑠𝑟 , which is the sum of r’s value-added induced
by destination country s’s exports to origin country r, over s. Origin country r’s domestic
value-added content induced by its exports to a destination country s can be greater (less)than its value-added exports if the double count returned home r from the destination s isgreater (less) than the additive count returned home r from all other non-destination
countries k.

Remark:It would be an easy exercise to extend our discussions by dividing gross exports into twoparts, that is to say, exports for intermediate demand and for final demand.

Accounting of gross exports for the general case

𝐸𝑟 = Σ𝑠≠𝑟 𝐸𝑟𝑠 =DVr+FVr = vrBrrEr + (Σ𝑠≠𝑟 vsBsr)𝑬𝑟= 𝐸𝑟𝑣𝑎 + 𝒗𝑟 Σ𝑠≠𝑟 𝑩𝑟𝑠 𝑬𝑠𝑟 +FVr
= vr (Brr𝑬𝑟 －Σ𝑠≠𝑟 𝑩𝑟𝑠 𝑬𝑠𝑟 ) + [𝒗𝑟 Σ𝑠≠𝑟 𝑩𝑟𝑠 𝑬𝑠𝑟 + (Σ𝑠≠𝑟 vsBsr)𝑬𝑟 ] .

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Terms of [.] show total double counts in the gross exports.

As 𝑬𝑠𝑟 = 𝑨𝑠𝑟 𝑿𝑟 + 𝒀𝑠𝑟 (𝑠 ≠ 𝑟), 𝑬𝑟 = Σ𝑠≠𝑟 𝑬𝑟𝑠 = Σ𝑠≠𝑟 (𝑨𝑟𝑠 𝑿𝑠 + 𝒀𝑟𝑠 ) and 𝐸𝑟𝑣𝑎 =VTr, theabove equation can be written as
𝐸𝑟 = VTr +[𝒗𝑟 Σ𝑠≠𝑟 𝑩𝑟𝑠 (𝑨𝑠𝑟 𝑿𝑟 + 𝒀𝑠𝑟 ) + (Σ𝑠≠𝑟 vsBsr){Σ𝑠≠𝑟 (𝑨𝑟𝑠 𝑿𝑠 + 𝒀𝑟𝑠 )}].
This is exactly equation 34 in Koopman et al. (2015, p.480).

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𝐸𝑟𝑠 =DVrs+FVrs = vrBrrErs + (Σ𝑠≠𝑟 vsBsr)𝑬𝑟𝑠𝑣𝑎= 𝐸𝑟𝑠+ 𝒗𝑟 (𝑩𝑟𝑠 𝑬𝑠 +Σ𝑘≠𝑟,𝑠 𝑩𝑟𝑘 𝑬𝑘𝑠 ) +FVr=(vr𝑩𝑟𝑟 𝑬𝑟 − 𝑩𝑟𝑠 𝑬𝑠 +Σ𝑘≠𝑟,𝑠 𝑩𝑟𝑘 𝑬𝑘𝑠 ) + [𝒗𝑟 (𝑩𝑟𝑠 𝑬𝑠 − Σ𝑘≠𝑟,𝑠 𝑩𝑟𝑘 𝑬𝑘𝑠 ) + (Σ𝑠≠𝑟 vsBsr)𝑬𝑟 ] .
(22)

At the bilateral level, the overall sum of return home from each destination
country as double accounts, RETURN(-) which should be deducted from value-addedexports, amount to:
RETURN(-) = Σ𝑟,𝑠,𝑠≠𝑟 𝒗𝑟 (𝑩𝑟𝑠 𝑬𝑠 ) = Σ𝑟 FVr.

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At the bilateral level, all foreign contents return origin home as double counts deductedfrom the origin’s value-added exports. However, at the same time, additive counts, R(+)
return home from non-destinations or third parties which are part of foreign contents than
the total foreign content:
RETURN(+) = Σ𝑟,𝑠,𝑠≠𝑟 𝒗𝑟 (Σ 𝑘≠𝑟,𝑠 𝑩𝑟𝑘 𝑬𝑘𝑠 ).

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5. Numerical example
We provide a numerical example using an aggregated 3-country data shown by
Table 2. 3-country consists of China (country 1), the USA (country 2) and the ROW (therest of the world; country 3).

[Table 2 here]

Using the data of Table 2, the international Leontief inverse B= (Brs) and valueadded ratio vector v=(vr) are given by0.5703 0.0036 0.0060
2.3323 0.0161 0.0268
𝐀 = (0.0059 0.4071 0.0131), 𝑩 = (0.0309 1.6897 0.0422)，
0.3471 0.1356 1.9019
0.0780 0.0416 0.4722
v = (0.3458 0.5477 0.5088).

Destination-wise final demand is defined as
267.3
1,968.1
127.4
𝑭1 = ( 11.9 ), 𝑭2 = (12,492.9), 𝑭3 = ( 352.4 ) .
29,423.9
110.5
664.1

Using equations (2), we have
∗
∗
∗
𝑋12
𝑋11
𝑋13
1,418.4515.7
4,593.4∗∗∗( 𝑋22 ) = 𝑩𝑭2=(21,141.6 ) ; ( 𝑋21 ) = 𝑩𝑭1 = ( 85.5 ) ; ( 𝑋23 ) = 𝑩𝑭3 = ( 1,845.2 ).
∗∗∗3,001.3
56,101.1
𝑋32
𝑋31
𝑋33
894.9

It follows from Definition 1 that
∗
∗
∗
∗
∗
∗
𝑣1 𝑋12=178.3; 𝑣2 𝑋21=46.8; 𝑣1 𝑋13=490.5; 𝑣3 𝑋31= 455.3; 𝑣2 𝑋23=1,010.6; 𝑣3 𝑋32= 1,527.0.

On the other hand, from Table 2 and Definition 2 we have the following results of bilateralvalue-added exports:

∗∗
∗∗
𝑋12
+ 𝑋13
𝐸12 + 𝐸13836.7
1,934.1∗∗( −𝑋21 ) =𝑩 ( −𝐸21 )=𝑩 ( −50.4 )=( −85.5 );
∗∗
−𝐸31
−𝑋31
−619.7−894.9∗∗−𝑋12−𝐸12−210.6−515.7∗∗∗∗(𝑋21 + 𝑋23 ) =𝑩 (𝐸21 + 𝐸23 ) = 𝑩 ( 1,187.0 )=( 1,930.7 );∗∗−𝐸32−1,624.3−𝑋32−3001.3
∗∗
−𝑋13
−𝐸13
−626.1−1,418.4∗∗( −𝑋23 ) = 𝑩 ( −𝐸23 ) = 𝑩 ( −1136.6 )=(−1,845.2 ).
∗∗∗∗𝐸31 + 𝐸32
2,243.9
3,896.2
𝑋31+ 𝑋32

Clearly,
∗∗

∗∗

∗∗

∗
∗∗
∗
∗∗
𝑋∗12 = 𝑋12 , 𝑋∗13 = 𝑋13 , 𝑋∗21 = 𝑋21 , 𝑋∗23 = 𝑋∗∗
23 , 𝑋31 = 𝑋31 , 𝑋32 = 𝑋32 .

Hence
∗
∗∗
∗
∗∗
∗
∗∗
∗
∗∗
∗
∗∗
∗
∗∗
𝑣1 𝑋12= 𝑣1 𝑋12, 𝑣1 𝑋13= 𝑣1 𝑋13, 𝑣2 𝑋21= 𝑣2 𝑋21, 𝑣2𝑋23= 𝑣2 𝑋23, 𝑣3 𝑋31= 𝑣3 𝑋31, 𝑣3 𝑋32= 𝑣3𝑋32.

Theorem 1 is exemplified.
The results at the non-bilateral level are summarized by Tables 3 and 4.
Table 3. Decomposition of gross exports at non-bilateral level for 2005 in bln US$
Table 4. Decomposition of gross exports at non-bilateral level for 2005 in % of grossexports[Table 3 and Table 4 here]
From Tables 3 and 4 we can witness the facts: the share of China’s domestic
content in its gross exports, 81% shows the smallest value of the three countries as its shareand the amount of the foreign content show the largest values. At the non-bilateral level,
the difference between the domestic content and the value-added exports is positive for all
countries. This difference for China, 0.7% of its gross exports, is rather small because thedouble counts, which returns from the USA and the ROW to China, are very small.
Particularly, the double count returning home from the USA, 0.03% of China’s grossexports, appears to be negligible. The difference for the USA, 3.5% of its gross exports, isnot so large due to the small double count returning home from China, 0.3% of USA’s
gross exports. The difference for the ROW show the largest share, 8.4% of its gross exports
because both of double counts returning home from China and the USA are enough large.
However, it is noteworthy that China’s share of value-added exports, 80% is much smaller
than USA’s 89% and ROW’s 88%.

[Table 5 and Table 6 here]

Tables 5 and 6 present our results at the bilateral level which illustrate Theorem
2. These tables show an interesting evidence that China’s value-added exports to the USA
is greater than China’s domestic content induced by its gross exports to the US because
the double count returning from the USA, 3.1% of share of gross exports, is smaller than
the additive count returning from the third party non-destination ROW, 7.2%. The US
value-added exports to China is also slightly larger than the US domestic content, despite
the US large share of double count, 28.1% returning from China, because the US share of
the additive count returning from the third party, non-destination ROW is also large as
28.4%, and greater than the double count returning from China. All other bilateral valueadded exports are smaller than respective domestic content. In particular, ROW’s share ofvalue-added exports in its gross exports, 73.5%, is much less than its domestic content
share, 97% due to the large double count returning from China. Table 5 also shows theidentity between the total sum of double counts at the bilateral level (US$ billion 323) andthat of double counts returning home (US$ billion 236.1) and no return part as double
counts (US$ billion 86.9) at the non-bilateral level. As shown by Table 5, this no return
part consists of return part as additive counts at the bilateral level.

6. Concluding remarks
We have presented a full description of an alternative accounting system of grossexports at both the non-bilateral level and bilateral level in place of that of Koopman etal. (2014) by using our definition of value-added exports. We demonstrated that we could
trace value added and double accounting in gross exports across countries in astraightforward manner when we employ our representation of value-added exports(Definition 2). As was shown, tracing value-added and double counting in gross exportsbecame dramatically more difficult and complicated when moving from a two-country to
three-country or multi-country world. We demonstrated that the value of bilateral valueadded exports can be greater than the respective domestic content due to the additive
count returning from the third parties, which can be larger than the double count returningfrom the destination country. Koopman et al. (2014) rightly provided the decomposition
into the domestic and foreign content of gross exports in the first stage. However, theyfailed to present an easily understandable exposition of further decomposition into valueadded exports, terms returning home as double or additive counts at both non-bilateraland bilateral levels. This is simply due to their employment of Johnson and Noguera’s
definition of value-added trade, where double accounting terms are not explicitly orvisually shown. In contrast, our definition (Definition 2) straightforwardly shows logic
and elements of decompositions. We also provided a numerical and empirical example
for a three-country case (China, the USA, and the ROW), using aggregated World InputOutput Data. Needless to say, we need further theoretical empirical investigation.

Appendix

1. Traditional value-added trade on the country baseLet us consider a 2-country case with many sectors where country 1 is anarbitrary country, such as the USA, China, or Japan, and country 2 is the rest of the world(ROW).
From equation (4), the local equilibrium condition for country 1 is given bỹ1 , = (𝑰𝑛 − 𝑨11 )−𝟏 (𝑬12 +𝒀11 );𝑿1∗ = (𝑰𝑛 − 𝑨11 )−𝟏 𝒀(4’)

̃1 = 𝑩1 (𝑬12 +𝒀11 ), where 𝑩1 = (𝑰𝑛 − 𝑨11 )−𝟏 .
𝑿1∗ = 𝑩1 𝒀
𝑬12 = 𝑬1 is a column vector of country 1’s gross exports to the world. 𝒀11 is a column
vector of country 1’s final demand for its own domestic products. Generally, 𝑩 1 ≠ 𝑩11 ,while the global equilibrium and local equilibria yield the same solution in theinternational input–output system.
Country 1’s traditional domestic contents of value-added, induced by its exports tocountry 2 or the world, are defined as
𝐷𝑉𝟏 = 𝒗1 (𝑰𝑛 − 𝑨11 )−1 𝑬12 .

(A1)This definition, given by Miyazawa (1975), has widely been utilized in Japan andincluded in the Japanese Statistics Bureau’s official reports (the latest version can be seenon the website).7
Country 1’s exports need its imports from country 2, which amount to
𝐹𝑀𝑋1 = 𝒖𝑛 𝑨21 (𝑰𝑛 − 𝑨11 )−1 𝑬12 .

An Excel file of “Gross Value Added Induced by Individual Final demand Items”
http://www.soumu.go.jp/english/dgpp_ss/data/io/io11.htm (Access on July 1, 2016).
7(A2)

In view of our price equation, 𝒗1 + 𝒖𝑛 𝑨21 = 𝒖𝑛 (𝑰𝑛 − 𝑨11 ), we have𝒗1 (𝑰𝑛 − 𝑨11 )−1 𝑬12 + 𝒖𝑛 𝑨21 (𝑰𝑛 − 𝑨11 )−1 𝑬12 .
= 𝒖𝑛 (𝑰𝑛 − 𝑨11 )(𝐈𝑛 − 𝑨11 )−1 𝑬12 = 𝒖𝑛 𝑬12 ;𝐸1 = 𝐷𝑉𝟏 + 𝐹𝑀𝑋1 .

(A3)Country 1’s total traditional value-added exports to the world, 𝐷𝑉𝟏 , plus
imports induced by exports, 𝐹𝑀𝑋1 , should equal its total gross exports to the world, 𝐸1(Miyazawa, 1975). This identity implies that gross exports can be decomposed into value
added and imports induced by gross exports on the country base. This is a corollary ofidentity between the GDP on the production side and the GDP on the expenditure side. In
the definition of the traditional value-added exports on the country base, the exports ofintermediates are exogenously given. When we measure traditional value-added trade on
the country base, we do not have to be concerned with endogeneity issues of the ROW.
Given the information of a country’s non-competitive input-output table and gross exports
by country and sector, we can easily compute the country’s traditional domestic valueadded. However, this approach is insufficient for tracing international transfers of value
added and outputs in the world.
Imports of country 1, which are exports of country 2, generated by its exports
also, in turn, induce value added in country 2, that is to say, foreign value added on the
country base, FV1. When we apply the country base equation to country 2, we have
𝐹𝑉1 = 𝒗2 (𝐈𝑛 − 𝑨22 )−1 𝑨21 (𝐈𝑛 − 𝑨11 )−1 𝑬12 .
If 𝒗2 (𝐈𝑛 − 𝑨22 )−1 = 𝒖𝑛 , we can write equation (A1) as
𝐸1 = 𝐷𝑉𝟏 + 𝐹𝑉1 .
If and only if 𝑨12 = 𝑶, generally we can obtain 𝒗2 (𝐈𝑛 − 𝑨22 )−1 = 𝒖𝑛 . This might be acase discussed in Los et al. (2016). However, in general, 𝑨12 ≠ 𝑶. Therefore, when we
employ separately each country’s traditional domestic value added, generally
𝐸1 ≠ 𝐷𝑉𝟏 + 𝐹𝑉1 .
This implies that, in a separate use of traditional domestic value-added, gross exports
cannot be absorbed fully or decomposed into value-added. Only when we use a global
Leontief inverse, we reach the basic identity equation (12) in the text where all grossexports of a country are fully absorbed and decomposed into domestic and foreign valueadded. However, it is noteworthy to mention a possible joint use of traditional and newconcepts of domestic value-added, as Koopman et al. (2014) attempted. By exploring thedomestic value-added and induced imports on the country base into equation (12) on theinternational base, for instance, we have the following additional identity:
𝐸1 = 𝐷𝑉𝟏 + (𝐷𝑉𝟏 − 𝐷𝑉𝟏 ) + 𝐹𝑀𝑋1 +(𝐹𝑉1 − 𝐹𝑀𝑋1 ).

(A4)Equation (A4) means that total exports (LHS) = (term 1 of RHS), the domestic valueadded on the country base + (term 2 of RHS), the difference between the domestic valueadded on the country base and that on the international base + (term 3 of RHS), induced
imports on the country base + (term 4 of RHS), the difference between foreign valueadded on the international base and induced imports on the country base. Obviously, thesmaller term 2 and term 4 of RHS become, the more we can rely upon results on the
country base. Empirically, terms 2 and 4 are rather negligible in comparison with resultson the country base because each country’s input coefficients of domestic intermediates
are much larger than those of imported/exported intermediates.
If we have complete data on international input-output tables (Table 1), inprinciple, we do not have to be concerned with equation (A4) because this equation is notessential for the decomposition of gross exports on the international base. As can be seen,
Koopman et al. (2014) shifted the main focus from 𝐷𝑉𝟏 (in Koopman et al. 2010) to
𝐷𝑉𝟏 without any explanation. This shift makes their decomposition a hard read, while
this would need further research.

2. The double use of intermediate imports/exports on the international baseOn the international base, local domestic value added can be written as
𝐷𝑉𝟏 = 𝒗1 (𝑰𝑛 − 𝑨11 )−1 𝑬12 = (𝑰𝑛 − 𝑨11 )−1 (𝑨12 𝑿∗ 2 + 𝒀12 ).

(A1’)

The term of the export of intermediates, 𝑨12 𝑿∗ 2 , is considered an endogenous variablein the international input-output system. Throughout this paper, on the international base,
we also employ 𝒗𝑟 𝑩𝑟𝑠 𝑬𝑟𝑠 or 𝑩𝑟𝑠 𝑬𝑟𝑠 . 𝑩𝑟𝑠 embodies 𝑨𝑟𝑠 , while 𝑬𝑟𝑠 includes the termof exports/imports of intermediates, 𝑨𝑟𝑠 𝑿∗ 𝑠 . This implies that we doubly use endogenous

parts of the international input-output system as endogenous and exogenous variables.
Therefore, some input-output experts (here called fundamentalists), including Los et al.
(2016), may think this operation ( 𝑩𝑟𝑠 post-multiplied by 𝑬𝑟𝑠 , including exports ofintermediates) is not right because they believe that the Leontief inverse, irrespective of
national or international frameworks, should be post-multiplied by exogenous final
demand terms. For fundamentalists, only Johnson and Noguera (2012) can be accepted
because their paper provides the definition of value-added exports by using 𝑩𝑟𝑠 postmultiplied by the final demand, 𝑭𝑠 , excluding exports of intermediates. For
fundamentalists, all other papers, including those of Trefler and Zhu (2010), Koopman etal. (2010; 2014), and this paper, are not plausible from an economics point of view. Theiridea may be true for the use of Leontief equations for projections. However, given inputoutput structures, all parts of endogenous intermediate transactions are also given. To
analyze the given input-output structure, our double use in the system with fixed andconstant input coefficients and value-added coefficients can be allowed, as mathematics
and Trefler and Zhu (2010) show the feasibility and fruitfulness of our operations.
Let us consider the most orthodox Leontief system with competitive imports on
the country base for country 1:−1

̃ 11 𝑿1 + 𝒀11 + 𝑬1 − 𝑴1 ; 𝑿1 = (𝑰𝑛 − 𝑨̃ 11 ) (𝒀11 + 𝑬1 − 𝑴1 ),𝑿1 = 𝑨̃ 11 is a matrix of input coefficients of intermediates, including domestic andwhere 𝑨
imported intermediates; 𝑴1 is a vector of all imports for intermediate demand and final
̃ 11 )−1 post-multiplieddemand as well. Can input-output fundamentalists allow (𝑰𝑛 − 𝑨̃ 11 , 𝑨̃ 11 𝑿1, and 𝑴1 ?8 If
by 𝑴1 despite the double use of imports of intermediates in 𝑨
the answer is yes, our double use should also be accepted.
It is noteworthy to learn that, through decomposition processes, all the double
uses of intermediate exports or imports are eliminated, which is shown by the identity
between the total amount of a country’s value-added trade balances with other countries

Input-output economists recognize that the exogenous treatment of imports should be
improved because imports largely depend on domestic production activities. Along this line,
there have been several attempts in the input-output literature to make imports endogenous,
including this paper. However, this is the story of the second step in input-output analysis. The
key problem of input-output analysis, as posed by many economists, is its exogenous treatmentof all terms of final demand.
8

and that of its gross trade balances or, in the conventional terminology, net exports(Kuboniwa, 2014a, 2015). Regardless, you should not stop or remain at the entrance of
decompositions. Go further along with decompositions shown in the text, then you would
arrive at a full understanding of movements of the value-added flows, induced by thegross trade, across the world.

Acknowledgements
The author thanks Dr. Jiemin Guo, the U.S. Bureau of Economic Analysis, for his helpfulcomments on an earlier version of this paper with encouragement. He also thanks Dr.
Yoshisada Shida, Hitotsubashi University, for his diligent cooperation. He is grateful for
the JSPS scientific support: Grant-in-Aid (A) numbers 15H01959 and (B) 24330085.

References
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Comprehensive Approach. Economics Letters 120, 354-357.
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Johnson R.C. and G. Noguera (2012) Accounting for Intermediates: Production Sharing
and Trade in Value-added. Journal of International Economics, 86, 224-236.
Koopman, R., W. Powers, Z. Wang and S.-J. Wei (2010, 2011) Give Credit to Where
Credit is due: Tracing Value-added in Global Production Chains. (NBER Working
Papers Series. 16426), (September 2010, revised September 2011).
Koopman, R., Z. Wang, Z., and S. Wei (2014) Tracing Value-added and Double Counting
in Gross Exports. American Economic Review, 104, 459–494

Kuboniwa, M. (2014a) Fundamental Theorem on the Relationship between Trade
Balances in Value-added and Gross Terms: Amendment. (IER Discussion Paper
Series, A.600. Hitotsubashi University).
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University)
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Data. Eastern European Economics 53, 277-308.
Los, B., M. Timmer, and G. de Vries (2016) Tracing Value-Added and Double Counting
in Gross Exports: Comment. American Economic Review 106, 1958–1966.
Miller, R.E., and P.D. Blair (2009) Input-Output Analysis (the second edition), New York:
Cambridge University Press.
Miyazawa, K., ed., 1975. Sangyou-Renkan Bunseki Nyumon (The Introduction of InputOutput Analysis, in Japanese), Tokyo: Nikkei Publisher.
Timmer, M.P., et al. (2012) The World Input-Output Database (WIOD): Contents,
Sources and Methods. (WIOD Working Paper 10)
Trefler, D., and S. Zhu (2010) The Structure of Factor Content Predictions. Journal ofInternational Economics 82, 195-207.
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Value Chains in East Asia: from Trade in Goods to Trade in Tasks. WTO.

Table 1. Data structure of an international input-output table
Intermediate demand/input
Country Country

1

Country 1
Country 2…
Country r
…
…
ROW

R

…

Country

…

r

…

s

Final demand (destination)

… ROW
…

Country Country

R

1

2

…

Country…

Country

…

r

…

s

… ROW Output…

R

X 11 E 12 … E 1r … E 1s … E 1R
E 21 X 22 … E 2r … E 2s … E 2R

F1 F2 … Fr … Fs … FR X
Y 11 Y 12 … Y 1r … Y 1s … Y 1R X 1
Y 21 Y 22 … Y 2r … Y 2s … Y 2R X 2

…

…

…

…

…

…

…

…

…

E r 1 E r 2 … X rr … E rs … E rR…

Country s

2…

Country

…

…

…

…

…

…

…

…

…

…

…

…

…

…

Y r 1 Y r 2 … Y rr … Y rs … Y rR X r

…

…

…

…

…

…

…

…

…

…

E s 1 E s 2 … E sr … X ss … E sR

Y s 1 Y s 2 … Y sr … Y ss … Y sR X s

…

…

…

…

…

…

…

…

…

E R 1 E R 2 … E Rr … X Rs … X RR

Value-added

V1 V2 … Vr … Vs … VR

OutputX1 X2 … Xr … Xs … XR

…

…

…

…

…

…

…

…

Y R 1 Y R 2 … Y Rr … Y Rs … Y RR X R

Notes:
E rs (s ≠r ): country r 's gross export matrix of intermediate goods to country s or country s 's gross import matrix
of intermediate goods from country r .
X rr : country r 's input matrix of domestically produced intermediate goods.
Y rs (s ≠r ): country r 's gross export matrix of final goods to country s or country s 's gross import matrix of final
goods from country r .
Y rr : country r 's input matrix of domestically produced final goods.
V r : country r 's value-added vector.

Table 2. Aggregated 3-country world input-output table (China-USA-ROW): 2005
(in producer prices; current US$ bln)
Intermediate demandcountry 1 country 2

Intermediate
Final demandcountry 3
demandcountry 1 country 2 country 3

country 1 China
country 2 USA
country 3 ROW

China
3,722.7
38.5
509.1

USA
ROW
83.2
358.8
9,392.4
784.2
960.2 28,329.4

Intermediate input

4,270.3 10,435.7 29,472.4

Final
OutputChina
USA
ROW demand
4,164.6 1,968.1
127.4
267.3 2,362.9 6,527.5
10,215.1
11.9 12,492.9
352.4 12,857.2 23,072.3
29,798.7
110.5
664.1 29,423.9 30,198.5 59,997.3
44,178.4

2,090.5 13,284.4 30,043.7 45,418.6 89,597.0

Value-added
2,257.2 12,636.6 30,524.9 45,418.6
Output
6,527.5 23,072.3 59,997.3 89,597.0
Sources: Author's calculations using WIOD.
Notes:
1. Producer prices = basic prices + net taxes on products.
2. For China and USA, WIOD employs data in producer prices, while, for EU countries and some other countries,
included in ROW, WIOD uses data in basic prices. We converted data for ROW into data in producer prices.

Table 3. Decomposition of gross exports at non-bilateral level for 2005 in bln US$
Gross
Origin Destination
exportsrs (s', s")Er

Domestic
fromdouble return
content
country
count (-) home
DV r
s'

from
returncountryhomes"Valueaddedexports

returnno return
Foreigntotohome asascontent
return country return country
doubledouble
FV r
s'
s"
count
count

1 China

2,3

836.7

674.8

6.0

0.3 from 2

5.7 from 3

668.8

161.9

114.1

3.6

to 2

110.6

to 3

47.8

2 USA

1,3

1,187.0

1,098.5

41.1

3.6 from 1

37.5 from 3

1,057.4

88.5

78.7

0.3

to 1

78.4 to 3

9.8

3 ROW

1,2

2,243.9

2,171.3

189.0 110.6 from 1

78.4 from 2

1,982.3

72.7

43.3

5.7

to 1

37.5 to 2

29.4

Table 4. Decomposition of gross exports at non-bilateral level for 2005 in % of gross exports(% of gross exports)
returnValueno return
Gross Domestic
Foreign home
from
from
to
to
Destination
double return
added
returnexports content
content as return country return country as
country
country
Origin r
homedoublecount (-) homes (s', s")exports
s'
s"
s'
s"
Er
DV r
FV r double
count
count

1 China
2 USA
3 ROW

2,3
1,3
1,2

100.0
100.0
100.0

80.7
92.5
96.8

0.72
3.5
8.4

0.03 from 2
0.3 from 1
4.9 from 1

0.69 from 3
3.2 from 3
3.5 from 2

79.9
89.1
88.3

19.3
7.45
3.2

13.6 0.4
6.63 0.02
1.9 0.3

to 2
to 1
to 1

13.2
6.61
1.7

to 3
to 3
to 2

5.7
0.82
1.3

Table 5. Decomposition of gross exports at bilateral level for 2005 in bln US$.
Origin Destination

r

s

Gross
exports
E rs

1 China

2
3

210.6
626.1

2 USA

1

3 ROW

3
1
2

1,624.3

ValueDomestic Double
from Additive from
added
content count (-); country count (+); country
exports
DV rs
returns
return k (k≠r,s )
178.3
490.5

Foreign
Return to country s Return to country kcontent
as double count (-) as addditive count (+)
FV rs

169.8
505.0

6.6 from 2
20.8 from 3

15.1 from 3
6.3 from 2

40.7
3.6
121.2 110.6

to 2(-)
to 3(-)

37.2 to 3(+)
10.6 to 2(+)

50.4

46.6

14.1 from 1

14.3 from 3

46.8

3.8

0.3

to 1(-)

3.5 to 3(+)

1,136.6
619.7

1,051.9
599.6

51.9 from 3
147.8 from 1

10.6 from 1
3.5 from 2

1,010.6
455.3

84.7
20.1

78.4
5.7

to 3(-)
to 1(-)

6.3 to 1(+)
14.3 to 2(+)

1,571.7

81.9 from 2

37.2 from 1

1,527.0

52.6

37.5

to 2(-)

15.1 to 1(+)

Table 6. Decomposition of gross exports at bilateral level for 2005 in % of gross exports
Origin Destinationr
1 China
2 USA
3 ROW

s
2
3
1
3
1
2

ValueGross Domestic Double
from Additive
from
added
exports content count (-); country count (+); country
exports
E rs
DV rs
returns
return k (k≠r,s )
100.0
100.0
100.0
100.0
100.0
100.0

80.7
80.7
92.5
92.5
96.8
96.8

3.1
3.3
28.1
4.6
23.8
5.0

from 2from 3from 1from 3from 1from 2

7.2
1.0
28.4
0.9
0.6
2.3

from 3from 2from 3from 1from 2from 1

84.7
78.3
92.9
88.9
73.5
94.0

Foreign
Return to country s Return to country k as
content
as double count (-)
addditive count (+)
FV rs
19.3
19.3
7.5
7.5
3.2
3.2

1.7
17.7
0.6
6.9
0.9
2.3

to 2(-)
to 3(-)
to 1(-)
to 3(-)
to 1(-)
to 2(-)

17.7
1.7
6.9
0.6
2.3
0.9

to 3(+)
to 2(+)
to 3(+)
to 1(+)
to 2(+)
to 1(+)

Country 1

Gross exports
E 1 = u n E 12

Domestic content
DV 1 = v 1B 11E 12

u n = (1,..,1)

Foreign content
FV 1= v 2B 21E 12Value-added exports

Return home

E va11 = v 1(B 11E 12－B 12E 21)v 1B 12E 21Country 2 (ROW)

Gross exports
E 2 = u n E 21

Domestic content
DV 2 = v 2B 22E 21

Foreign content
FV 2 = v 1B 12E 21

Value-added exports

Return home

E va22 = v 2(B 22E 21－B 21E 12)v 2B 21E 12

Figure 1. Decomposition of gross exports and double counts:
country case with many sectors

a two-

(Panel 1 of Figure 2)
Country 1
Gross exports
E 1 = u n E 12 + u n E 13

Domestic content
DV 1 = v 1B 11(E 12+E 13)

Value-added exports E 1va 1
v 1{B 11(E 12+ E 13)－B 12E 21－B 13E 31}

Return home (-)
v 1(B 12E 21+ B 13E 31)

va

Value-added exports to country 2 E 1212v 1{B 11E 12－B 12(E 21+ E 23) + B 13E 32}

Value-added exports to country 3 E va 13
13
v 1{B 11E 13－B 13(E 31+ E 32) + B 12E 23}

Foreign content
FV 1= v 2B 21(E 12+ E 13) + v 3B 31(E 12+ E 13)

Return to 2&3 as double count (-) No return as double countv 2B 21E 12+ v 3B 31E 13v 2B 21E 13+ v 3B 31E 12from country 2
Return home as double count (-)

from country 3
Return home as addition (+)

v 1B 12(E 21+ E 23)

v 1B 13E 32

from country 3
Return home as double count (-)

from country 2
Return home as addition (+)

v 1B 13(E 31+ E 32)

v 1B 12E 23

(Panel 2 of Figure 2)
Country 2
Gross exports
E 2 = u n E 21 + u n E 32

Domestic content
DV 2 = v 2B 22(E 21+E 23)

Value-added exports E va2 2
v 2 {B 22 (E 21 + E 23 )－B 21 E 12 －B 23 E 32 }Foreign content
FV 2= v 1B 12(E 21+ E 23) + v 3B 32(E 21+ E 23)

Return home (-)

Return to 2&3 as double count (-)

No return as double countv 2 (B 21 E 12 + B 23 E 32 )v 1 B 12 E 21 + v 3 B 32 E 23v 1 B 12 E 23 + v 3 B 32 E 21

from country 1

from country 3

Value-added exports to country 1 E va2121
v 2 {B 22 E 21 －B 21 (E 12 + E 13 ) + B 23 E 31 }Return home as double count (-)

Return home as addition (+)

v 2 B 21 (E 12 + E 13 )v 2 B 23 E 31

from country 3

from country 1

Value-added exports to country 3 E va2323
v 2 {B 22 E 23 －B 23 (E 31 + E 32 ) + B 21 E 13 }Return home as double count (-)

Return home as addition (+)

v 2 B 23 (E 31 + E 32 )v 2 B 21 E 13

(Panel 3 of Figure 2)
Country 3
Gross exports
E 3 = u n E 31 + u n E 32

Domestic content
DV 3 = v 3B 33(E 31+E 32)

Value-added exports E va3 2
v 3 {B 33 (E 31 + E 32 )－B 31 E 13 －B 32 E 23 }Foreign content
FV 3= v 1B 13(E 31+ E 32) + v 2B 23(E 31+ E 32)

Return home (-)

Return to 2&3 as double count (-)

No return as double countv 3 (B 31 E 13 + B 32 E 23 )v 1 B 13 E 31 + v 2 B 23 E 32v 1 B 13 E 32 + v 2 B 23 E 31

Value-added exports to country 1 E va3131
v 3 {B 33 E 31 －B 31 (E 12 + E 13 ) + B 32 E 21 }va

Value-added exports to country 2 E 3232
v 3 {B 33 E 32 －B 32 (E 21 + E 23 ) + B 31 E 12 }from country 1

from country 2

Return home as double count (-)

Return home as addition (+)

v 3 B 31 (E 12 + E 13 )v 3 B 32 E 21

from country 2

from country 1

Return home as double count (-)

Return home as addition (+)

v 3 B 32 (E 21 + E 23 )v 3 B 31 E 12

Figure 2. Decomposition of gross exports, double counts and additive
counts: a 3-country case with many sectors