Faculty of Economics & Management Program: Global Economics and Management
-- Research master thesis –
Rejecting simple hypotheses to explain the direction of trade in emissions
Bingqian Yan
b.yan.1@student.rug.nl
Supervisor: Erik Dietzenbacher
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Rejecting simple hypotheses to explain the direction of trade in emissions
Abstract
This paper examines the environmental impact of international trade for 40 countries and explores simple hypotheses to explain the direction of trade in emissions. Using input-output techniques, we empirically test the pollution haven hypothesis and the factor endowment hypothesis for many countries and present a worldwide overview, filling a gap in the literature. Furthermore, the triangularization technique is employed to explore whether any simple hypothesis (whether the direction of trade in emissions depends on the ordering of countries) can explain the direction of trade in emissions. All the bilateral calculations are based on the World Input-Output Table (covering 40 countries).
Neither the pollution haven hypothesis nor the factor endowment hypothesis is widely supported by the empirical results. Moreover, the results of triangularization reject simple hypotheses to explain the direction of trade in emissions (i.e. no ordering of countries can
explain the trade pattern of 𝐶𝑂2 emissions). The outcome is robust to change in the
underlying assumptions.
Keywords: international trade; emissions embodied in trade; pollution haven; factor
endowment; input-output analysis; triangularization
1.
Introduction
International trade facilitates displacement of pollution from one country to another. A country can enjoy a clean environment and still consume pollution-intensive goods through its imports, because another country specializes in pollution-intensive industries (Copeland and Taylor 2004). The debate about the impact of international trade on the environment has gained more importance over the last decade, due to the increasing liberalization of trade, the discussion on the influence of greenhouse emissions on global warming and climate change, and the implementation of the Kyoto Protocol (Wyckoff and Roop 1994; UNEP 2000).
The impact of international trade on the environment is mainly through a scale, a technique and a composition effect, and the last effect is of great importance (Grossman and Krueger 1991). The composition effect reveals that increased trade leads countries to specialize in products, in which they have a comparative advantage. The comparative advantage may lie in lower capital costs, larger resource endowments, the availability of a large amount of cheap labor, lax environmental regulations, or a combination of them. This results in changes in countries’ economic structure and in patterns of trade. For instance, stronger environmental regulations will promote certain countries (such as Japan) to import manufactured goods whose production process is pollution-intensive rather than importing raw materials and producing these goods.
If it is the difference in environmental regulations that determines the comparative advantage, we arrive at the case that is consistent with the Pollution Haven Hypothesis (PHH). According to the hypothesis, freer trade will induce “dirty industries” to migrate from developed countries with strict environmental regulations to developing countries with lax
environmental regulations.1 Strict environmental regulations lead to high production costs for
intensive goods and contribute to countries’ comparative disadvantage in
pollution-intensive industries.2 Therefore, countries with lax environmental regulations may become
havens for pollution-intensive industries. The hypothesis can be interpreted within the Heckscher-Ohlin (HO) theory, which we will discuss in the theory section.
If it is the difference in relative capital endowments that determines the comparative
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Several reasons contribute to strict environmental regulations in developed countries and lax regulations in developing countries. First, higher income in developed countries generates higher demand for clean air. Second, the cost of monitoring and implementing environmental standards is relatively higher in developing countries. Third, growth in developing countries implies a shift from agriculture to manufacturing with rapid urban growth and massive investment in urban infrastructure, which will raise the pollution intensity. (Dietzenbacher and Mukhopadhyay 2007).
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advantage, we arrive at the case that is consistent with the Factor Endowment Hypothesis (FEH). According to the hypothesis, freer trade will induce capital-intensive industries to migrate from developing countries with a low capital-labor ratio to developed countries with a higher capital-labor ratio. Because capital-intensive goods are generally pollution-intensive (Antweiler et al. 1998; Cole and Elliott 2003), countries with high capital endowments tend to have a comparative advantage in pollution-intensive industries. This is the competing theory against the pollution haven hypothesis. The paper will test both of these two competing theories bilaterally for a set of 40 countries.
The main focus of the study is on what determines that countries gain or lose from trade in terms of pollution and answers the following questions. First, do countries with lax environmental regulations export more pollution-intensive goods (i.e., is the pollution haven hypothesis valid)? This issue is investigated through a bilateral comparison of trade in
embodied 𝐶𝑂2 emissions for every pair of countries (40 countries in total). If the comparative
advantage in pollution stems from environmental regulations, we will be able to observe, that the average emission intensity in export bundle is larger than that in import bundle for less developed countries, while the opposite holds for more developed countries. Second, do capital-abundant countries export more pollution-intensive goods (i.e., is the factor endowment hypothesis valid)? Finally, if the answers are no to both questions, is there any
other hypothesis that can explain the pattern of trade in embodied 𝐶𝑂2 emissions? In other
words, is there any ordering of countries that fits the observed pattern of trade in emissions? This study builds on the work by Dietzenbacher and Mukhopadhyay (2007) but goes beyond their work by two major innovations. First, we modify the method in Dietzenbacher and Mukhopadhyay (2007) to bilateral comparisons for every pair of countries (40 countries in total). Although the PHH has been widely studied in the input-output literature, studies are limited to particular countries, for instance, Machado et al. (2001) for Brazil; Dietzenbacher and Mukhopadhyay (2007) for India; Temurshoev (2006) for China and USA, and their conclusions cannot be extended to other countries. Taylor (2004; p24) states: “In fact, no study in the literature provides a compelling many country test of the PHH, and I know of no theory paper detailing what such a test would look like”. By applying the input-output method and the World Input-Output Table (which takes full account of interdependence among sectors and countries), this study fills the gap. Second, the triangularization technique is applied to
find the ordering of countries that can best explain the pattern of trade in embodied 𝐶𝑂2
emissions.
Heckscher-Ohlin-Samuelson (HOS) model. Section 4 introduces the World Input-Output Database and the methodology we applied to test these two hypotheses. The empirical results are summarized in section 5, which shows that neither the PHH nor the FEH is widely supported by the empirical results. Then, the triangularization technique, introduced in section
6, is employed to find the ordering that best explains the trade pattern of 𝐶𝑂2 emissions. The
results indicate that there is no ordering that explains the trade pattern in terms of 𝐶𝑂2
emissions for all the countries in a satisfactory way and thus, the results reject simple hypotheses to explain the direction of trade in emissions. Section 7 concludes.
2.
Pollution haven hypothesis: some relevant studies
Copeland and Taylor (2004) provide a detailed theoretical framework to analyze the relationship among trade, growth and the environment. They discuss both the theoretical and empirical literature on this topic and arrive at three conclusions. First, increasing income affects the environment positively. Although increasing economic activities are associated with environmental damage, rising income can benefit the environment through changes in environmental policy. Second, environmental regulations have an effect on trade and investment flows. The third and more tentative conclusion is that little evidence for the pollution haven hypothesis was found.
The empirical examinations of the impact of environmental regulations on trade can be categorized into two classes, based on the methodology. The first strand of empirical literature examines the PHH using econometric methods. Most of these papers only find the pollution haven effect (which is weaker than the PHH, see Section 3). Antweiler et al. (2001) develop a theoretical model to divide the trade’s impact on the environment into three effects (scale effect, composition effect, and technique effect) and examine these effects on the
concentration of 𝑆𝑂2 (sulfur dioxide) at the country level. They find that the coefficients of
the interaction term between trade openness and the relative capital to labor ratio (FEH) and the interaction term between trade openness and the relative income (PHH) are both consistent with the corresponding hypothesis and significant. Nevertheless, the estimated effects are quite small. Cole and Elliott (2003) apply the same analytical framework and find
the same result for 𝐶𝑂2(carbon dioxide), but not for 𝑁𝑂𝑥(nitrogen oxides) and BOD
(biochemical oxygen demand, a measure of water pollution). Their tentative explanation is that for BOD, there is no statistically significant correlation between the sectors that are
capital-intensive and those that are water-pollution-intensive. For 𝑁𝑂𝑥, it is mainly generated
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Levinson and Taylor (2008) examine the effect of environmental regulations on trade both theoretically and empirically. They use a theoretical model to demonstrate how the sector-specific heterogeneity and endogeneity of the environmental policy may affect the relationship. Later, they apply the data on US regulation and trade with Canada and Mexico from 1977 to 1986 and use the fixed effect panel model with instrumental variable estimator to solve those problems. Their results indicate that net imports increase the most for industries whose pollution abatement costs increase most.
Similarly, Ederington and Minier (2003) find that the effect of environmental regulation on trade will be significantly higher if the level of regulation is treated as an endogenous variable. They use a simultaneous equation model to endogenize both the level of environmental regulation and the level of net imports. Furthermore, Ederington et al. (2005) provide two possible explanations why the PHH receives little evidence. First, industries with high abatement costs are also least geographically mobile (footloose). Second, the abatement costs account only for a small share in total costs. After controlling for these factors, they find a positive relationship between abatement cost and net import.
The second branch of empirical literature applies the input-output technique to address the PHH. Input-output techniques have been widely applied to analyze the impact of international trade on the environment (for example, Fieleke 1974; Machado et al. 2001). Compared to econometric methods, input-output techniques have several advantages in testing the PHH. First, the input-output approach is an economic accounting tool, which can demonstrate the real picture. By using input-output techniques, we can directly compare the emissions embodied in exports with the emissions avoided by imports. The econometric methods can only explore the relationship between two variables, and often suffer from endogeneity and heterogeneity problems. In contrast, the causality is clear in input-output models. Second, output tables describe the interdependence among industries, so input-output analysis can account for indirect effects. If producing a good requires use of pollution-intensive inputs, the production of the good induces more emissions indirectly. With IO analysis, all indirect effects are captured.
Three(Xu et al. 2000; Muradian et al. 2002; Mongelli et al. 2006) out of the thirteen studies employ the downward trend of net export of the emissions embodied in exports for industrialized economies to prove the existence of the PHH. Nevertheless, Copeland and Taylor (2004) show that the factor endowment/economic development can also lead to a declining trend, so the existence of declining trend is not sufficient to validate the PHH. According to the factor endowment/economic development argument, at first, industrialized economies are capital-abundant and have a comparative advantage in pollution-intensive industries and the opposite holds for developing countries. However, developing countries will shift from primary industries to capital-intensive dirty industries with capital accumulation. This will increase competition in world markets and lead developed countries to shift out of these industries, therefore concentrating on clean industries. As a result, the authors arrive at a decreasing trend, although the reasons are totally different.
In fact, in examining the PHH, we are interested in the domestic pollution content in the domestically produced commodities that are substituted by imported goods, because the substitution reduces the production and hence the pollution in the domestic country. Thus, another method (used in Machado et al. 2001 and the last 4 studies) – comparing the emissions embodied in exports and the emissions avoided by imports—is more appropriate to prove the validity of the PHH. However, only examining the balance of avoided emissions (BAE) is inadequate to validate the PHH. This is because the deficit in BAE might be caused by the high volume of net exports, rather than the comparative advantage in pollution-intensive production (as the PHH predicts). Hence, the method employed in Dietzenbacher and Mukhopadhyay (2007)-- comparing the emissions embodied in extra exports and the emissions that are avoided by the same amount of imports (which keeps the balance of current account invariant)-- is the most suitable way to validate the PHH.
Concluding the review of literature above, it is apparent that the empirical tests for the PHH are limited to particular countries and the tests for the FEH are lacking. Therefore, our aim is to test both hypotheses globally and present a worldwide overview.
3. Theoretical framework for the PHH
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regulations, while developed countries are associated with stringent regulations. As a result, free trade allows developed countries to move away from producing pollution-intensive goods and displace pollution to developing countries. As stated in Copeland and Taylor (2004), “the pollution haven effect is so strong that it more than offsets other motives of trade in dirty goods”.
The PHH states that developing countries have a comparative advantage in dirty goods, which can be interpreted within the Heckscher-Ohlin-Samuelson (HOS) model (Dietzenbacher and Mukhopadhyay 2007). In the simplest form – with two countries, two goods, and two factors (capital and labor) – the HOS model predicts that capital-abundant countries will export capital-intensive goods and import labor-intensive goods. When pollution is regulated, it can be viewed that “emission permits” play a role as a third factor, which is the dominant factor of trade in dirty goods under the PHH. This can be illustrated in the Figure 1 below.
Figure 1 The theoretical diagram for the PHH
Note: indicates the pre-trade (autarky) production and consumption; represents the post-trade production. Since we assume the pollution is only generated during production, the post-trade consumption points are not drawn in the diagram.
The left part denotes the situation for the rich/developed country while the right part Dirty(D) Clean (C) Clean (C) Dirty(D) 𝑍 = 𝑒𝑥 𝑍𝑇𝑟 𝑍𝐴𝑟 𝑍𝐴𝑝 𝑍𝑇𝑝 Emission Emission 𝑍 = 𝑒𝑥 𝑋𝐴𝑝 𝑋𝑇𝑝 𝑋𝑇𝑟 𝑋 𝐴𝑟
Emission change for Rich (r) country Emission change for Poor (p) country
𝑆𝑅
𝑆𝑤
𝑆𝑤
represents the situation for the poor/developing country. Denote r, p, D and C as the rich country, the poor country, the dirty good, and the clean good. For the sake of simplicity, we assume the same fixed emission intensity (e) in both countries, thus, the total emission equals 𝑍 = 𝑒𝑋, where 𝑋 is the amount of dirty goods. The poor country is believed to be well endowed with “emission permits”. Thus, in absence of trade, the price of dirty goods in the
poor country is lower than that in the rich country, 𝑃𝐷𝑝 < 𝑃𝐷𝑟. Hence, the price ratio in autarky
for the poor country, 𝑆𝑝 = 𝑃
𝐷𝑝/𝑃𝐶𝑝, is smaller than that (𝑆𝑟= 𝑃𝐷𝑟/𝑃𝐶𝑟) for the rich country, for
a given price of clean goods. As shown in Figure 1, the production possibility frontier for the poor country is flatter. Note that the indifference curves are not drawn in Figure 1. Readers can imagine appropriate indifference curves being tangent to the tangency points of the price ratio and the production possibility frontier. Since the poor country produces more
pollution-intensive goods than the rich country in autarky (𝑋𝐴𝑝 > 𝑋𝐴𝑟), the pollution level is higher for
the poor country than the for the rich country (𝑍𝐴𝑝 > 𝑍𝐴𝑟). With trade, the poor country will
import “clean” goods from the rich country and the developed country will import “dirty”
goods from the developing country, which result in the world price ratio (𝑆𝑤). This expands
(contracts) the dirty goods production in the developing (developed) country: 𝑍𝑇𝑝 > 𝑍𝐴𝑝,
𝑍𝑇𝑅 < 𝑍
𝐴𝑅. As a result, free trade is good (leads to less pollution) for the developed country and
bad (induces more pollution) for the developing country.
Figure 1 is also consistent with the Factor Endowment Hypothesis, which states that capital-abundant countries (usually developed countries) will export capital-intensive goods and capital-intensive goods are, at the same time, pollution-intensive. Hence, free trade is good for developing countries, while bad for developed countries in terms of pollution. In that case, D denotes the capital-intensive good and C denotes the labor-intensive good. And the letters need to be changed: change r for p and vice versa.
4. Methodology
The comparison of the emissions related to bilateral trade is an effective way to test the Pollution Haven Hypothesis (PHH). Because for two selected countries, one country must be relatively rich and the other is relatively poor. It is possible to predict (according to the PHH) which country will benefit (or lose) in terms of pollution if both exports and imports increase by the same amount. Then the PHH can be supported (rejected) if the empirical application confirms (contradicts) the prediction.
To calculate the emissions embodied in bilateral trade, bilateral trade data and 𝐶𝑂2
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sectors and 40 countries, the database covers 85% of world GDP in 2008. The World Input-Output Table (WIOT) describes the production interdependence among industries and countries. Entries in every row i in the WIOT represent the output from industry i used by destination country-industry combinations, while entries in every column j in the WIOT
indicate the input sourced from country-industry combinations to industry j.3 Hence, bilateral
trade data can be obtained from the WIOT. In addition, the WIOD provides data on labor and
capital inputs and pollution indicators at the industry level. The data of 𝐶𝑂2 emissions (Genty
et al. 2012) facilitate the calculation of 𝐶𝑂2 emissions embodied in trade. The database is publicly available at www.wiod.org.
4.1 World Input-Output Table
Table 1 illustrates a simplified WIOT with N countries and every country has S industries. Every row in the input-output table indicates the use of outputs, including intermediate use (in the blocks labeled Z) and final use (in the blocks labeled F). And every column indicates the inputs of production, including intermediate inputs (in the blocks labeled Z) and value added inputs (in the blocks labeled w).
Table 1 A simplified WIOT with N countries and S sectors per country Intermediate use
(S columns per country)
Final use
(h columns per country) Total
1 … N 1 … N S industries, country 1 𝒁11 𝒁1. 𝒁1𝑁 𝑭11 𝑭1. 𝑭1𝑁 𝒙1 … 𝒁.1 𝒁.. 𝒁.𝑁 𝑭.1 𝑭.. 𝑭.𝑁 𝒙. S industries, country N 𝒁𝑁1 𝒁𝑁. 𝒁𝑁𝑁 𝑭𝑁1 𝑭𝑁. 𝑭𝑁𝑁 𝒙𝑁 Value added (𝒘1)′ (𝒘.)′ (𝒘𝑁)′ Output (𝒙1)′ (𝒙.)′ (𝒙𝑁)′ 𝐶𝑂2 emissions (𝒆1)′ (𝒆.)′ (𝒆𝑁)′
The superscript and subscript denote country and industry, respectively. For example,
𝒁𝑀𝑁 is an S × S matrix and its element 𝑧
𝑖𝑗𝑀𝑁 indicates the inputs from industry i in country M
to industry j in country N. Note i, j = 1, 2, … , S where S is the number of industries. 𝑭𝑀𝑁 is an
S × h matrix and its element 𝑓𝑖ℎ𝑀𝑁 indicates the final demand of commodity i from country M
required by type h in country N, where h indicates the final demand type, covering household
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and government consumption, consumption of non-profit organizations, gross fixed capital
formation, and changes in inventories (h = 1, 2, 3, 4, 5). Finally, 𝒘𝑁 is an S-element vector
and its typical element 𝑤𝑖𝑁 gives the value added in industry i in country N.
𝐶𝑂2 emissions at the industry level are provided in the environmental accounts. 𝒆𝑀 is an
S × 1 vector and its element 𝑒𝑗𝑀 indicates the 𝐶𝑂
2 emissions by industry j in country M, and
the unit is 1000 tons (kt). Note that in a real WIOT, S = 35, N = 40.
4.2 Emissions embodied in bilateral trade
There are two methods to calculate the emissions embodied in trade: the national input-output model (Emissions Embodied in Bilateral Trade approach, EEBT) and the world input-output model (Multi-Regional Input Output approach, MRIO). The difference between the two approaches is that the EEBT only calculates the domestic emissions embodied in bilateral trade and does not count the domestic emissions embodied in the inputs from other countries, while MRIO gives the world emissions embodied in final goods. Which method to choose depends on the question we are studying. Take a simple example to illustrate the hypothesis we want to test. Suppose India imports steel from China to produce engines, and exports engines to US. According to the PHH, US increases imports of engines from India not because the Chinese steel industry is pollution-intensive and India imports steel from China, but because the US engine industry is pollution-intensive and therefore US imports engines from India. The emissions embodied in the inputs that India imports from other countries are not of importance. Therefore, the emissions embodied in bilateral trade method is more suitable in the context of the pollution haven hypothesis. Thus, the national input-output model and the cross border gross trade are employed to test the pollution haven hypothesis. The methodology is explained in detail below.
First, for a single country R, the input-output model is as follows (Miller and Blair 2005). The output of country R in the WIOT context can be expressed as
𝒙𝑅 = 𝑨𝑅𝑅𝒙𝑅 + 𝒇𝑅 (1)
Where 𝒙𝑅 is the 35 × 1 vector of output in country R. 𝑨𝑅𝑅 is the 35*35 matrix of input
coefficients in country R, obtained as 𝑨𝑅𝑅 = 𝒁𝑅𝑅(𝒙̂)𝑅 −1, where a hat indicates a diagonal
matrix. In the World Input-Output Table, the unit of gross output and final demand is million
US dollars (mUS$). Thus, the input coefficient 𝑎𝑖𝑗𝑅𝑅 (𝑎
𝑖𝑗 𝑅𝑅 = 𝑧
𝑖𝑗𝑅𝑅/𝑥𝑗𝑅) can be interpreted as the
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𝒇𝑅 is the 35 × 1 vector of final demands, including exports of intermediate goods and
final goods, which can be expressed as 𝒇𝑅 = ∑ 𝒁𝑅𝐾𝝉
35+ ∑𝑁𝐾=1𝑭𝑅𝐾𝝉5 𝑁
𝐾=1,𝐾≠𝑅 , where 𝝉35 and
𝝉5 are the 35 × 1 and 5 × 1 column summation vectors, respectively.
The solution for equation (1) is 𝒙𝑅 = (𝑰 − 𝑨𝑅𝑅)−1𝒇𝑅, where 𝑰 is the identity matrix. In
this expression, (𝑰 − 𝑨𝑅𝑅)−1is the Leontief inverse of country R, which we will denote by
𝑳𝑅𝑅. The element 𝑙
𝑖𝑗𝑅𝑅 indicates the total outputs from industry i to satisfy one unit of extra
final demand in industry j. And the jth column sum of 𝑳𝑅𝑅 can be interpreted as the total
output from all industries needed to satisfy one unit of extra final demand in industry j. Using
the assumption of fixed input coefficients, the amount of output 𝒙𝑅𝑃 needed to satisfy the
exports 𝒚𝑅𝑃 from country R to country P (where 𝒚𝑅𝑃 = 𝒁𝑅𝑃𝝉
35+ 𝑭𝑅𝑃𝝉5) is calculated as
𝒙𝑅𝑃 = (𝑰 − 𝑨𝑅𝑅)−1𝒚𝑅𝑃 = 𝑳𝑅𝑅𝒚𝑅𝑃.
The second step is to calculate how much 𝐶𝑂2 emissions are embodied in output for
satisfying the specified export vector. 𝝆𝑅 denotes the 𝐶𝑂
2 emission coefficient vector of
country R, the jth element in which is defined as 𝜌𝑗𝑅 = 𝑒
𝑗𝑅/𝑥𝑗𝑅 and indicates the amount of
𝐶𝑂2 emission in kt (1000 tons) that are emitted into air per mUS$ output by industry j in
country R. Then the 𝐶𝑂2 emissions that are required for the exports 𝒚𝑅𝑃 from country R to
country P is obtained as
𝐸𝑅𝑃= 𝝆𝑅′(𝑰 − 𝑨𝑅𝑅)−1𝒚𝑅𝑃 = 𝝆𝑅′𝑳𝑅𝑅𝒚𝑅𝑃 (2)
4.3 Testing the Pollution Haven Hypothesis
For any two arbitrarily selected countries from the 40 countries included in the WIOT, we denote the rich one as R and the poor one as P, based on the level of GDP per capita. To test
the pollution haven hypothesis, we will evaluate the effect of increased trade on 𝐶𝑂2
emissions. Assume that both exports and imports for each country increase by the same amount of money, for instance, 1 billion US$, to keep the balance of trade invariant. For each country, we will calculate and compare the pollution content of 1 billion US$ of exports and the pollution content that would have been emitted in producing 1 billion US$ of imports at home. We use the same notation as in Dietzenbacher and Mukhopadhyay (2007) -- ∆𝒆 and ∆𝒎 represent the vector of change in exports and imports, respectively. For bilateral trade,
exports of Rich (Poor) equal to the imports of Poor (Rich), thus, we have ∆𝒆𝑅 = ∆𝒎𝑃(∆𝒆𝑃 =
∆𝒎𝑅). The total value of changes in exports and imports are assumed to be the same, so
∑ (∆𝑒𝑖 𝑘)𝑖 = ∑ (∆𝑚𝑖 𝑘)𝑖 (k = R, P).
vector 𝝆𝑅′𝑳𝑅𝑅 (𝝆𝑃′𝑳𝑃𝑃) indicates the emission of 𝐶𝑂
2 (in kt) that is accompanied with the
output to satisfy one unit of export from industry j in Rich (Poor). Therefore, the increase in
exports by Rich (Poor) will raise the 𝐶𝑂2 emissions by 𝝆𝑅′𝑳𝑅𝑅 [∆𝒆
𝑅] (𝝆𝑃′𝑳𝑃𝑃[∆𝒆𝑃]) at home.
The increase in imports by Rich (Poor) will reduce the 𝐶𝑂2 emissions by 𝝆𝑅′𝑳𝑅𝑅[∆𝒎𝑅]
(𝝆𝑃′𝑳𝑃𝑃(∆𝒎
𝑃)) at home, since these goods are no longer produced at home. Let ∆𝜋𝑅
represent the net extra emissions in Rich caused by increased trade, then
∆𝜋𝑅 = 𝝆𝑅′𝑳𝑅𝑅(∆𝒆𝑅 − ∆𝒎𝑅).
According to the pollution haven hypothesis, with trade liberalization, countries with stringent environmental regulations tend to displace pollution to countries with lax environmental regulations. In general, rich countries have more stringent environmental
regulations. Hence, trade is good for Rich and bad for Poor in terms of 𝐶𝑂2 emissions. We
have ∆𝜋𝑅 < 0 and ∆𝜋𝑃 > 0.
From a global perspective, the increased trade is good for the environment if the total
amount of net extra emissions decreases, that is, ∆𝜋𝑅+ ∆𝜋𝑃 = (𝝆𝑅′𝑳𝑅𝑅− 𝝆𝑃′𝑳𝑃𝑃)(∆𝒆𝑅−
∆𝒎𝑅) < 0. If technologies are the same in both countries, then 𝝆𝑅′ = 𝝆𝑃′ and 𝑳𝑅𝑅 = 𝑳𝑃𝑃.
Then it is obvious that the change of pollution at a global level is zero, which implies that the
gain in terms of 𝐶𝑂2 emissions in the one is exactly offset by the loss of the other. If
technologies are different for the two countries, there are four possible outcomes, (1) both
countries benefit from trade in terms of pollution (∆𝜋𝑅 < 0 𝑎𝑛𝑑 ∆𝜋𝑅 < 0); (2) both countries
lose from trade (∆𝜋𝑅 > 0 𝑎𝑛𝑑 ∆𝜋𝑅 > 0); (3) Rich benefits from trade while Poor loses from
trade ((∆𝜋𝑅 < 0 𝑎𝑛𝑑 ∆𝜋𝑅 > 0), which reflects the pollution haven hypothesis; (4) Rich loses
from trade while Poor benefits from trade ((∆𝜋𝑅 > 0 𝑎𝑛𝑑 ∆𝜋𝑅 < 0). At the world level, it is
clear that the first case is beneficial while the second is not. For the third and fourth case, it is
not clear whether they are good (< 0) or bad (> 0), because it depends on the sum (∆𝜋𝑅+
∆𝜋𝑃). We will explore the net effect of increased trade on pollution at the world level in the
empirical section.
We will calculate the net extra emissions for every country in every bilateral trade situation and get 40 × 39 results. For the convenience of comparison and analysis, all the
results are summarized into a 40 × 40 matrix 𝑾. The element 𝑤𝑖𝑗 in the matrix 𝑾 is the
comparison of the emissions embodied in 1 billion US$ of extra exports from country i to
country j (𝐸𝐸𝑋𝑖𝑗) with country i’s emissions in the domestic replacement of 1 billion US$ of
extra imports from country j (𝐸𝑅𝐼𝑖𝑗).
13
country i is – according to the PHH – a “pollution haven” for country j and country i will
export more emissions than are involved in replacing its imports (𝐸𝐸𝑋𝑖𝑗 > 𝐸𝑅𝐼𝑖𝑗); if country j
is located to the right of the main diagonal (i.e, j > i), it is relatively poorer than country i and
country i will export less emissions than are involved in replacing its imports (𝐸𝐸𝑋𝑖𝑗 <
𝐸𝑅𝐼𝑖𝑗). Hence, we have When i < j, 𝑤𝑖𝑗 = { 1, 𝐸𝐸𝑋𝑖𝑗 < 𝐸𝑅𝐼𝑖𝑗 0, 𝐸𝐸𝑋𝑖𝑗 > 𝐸𝑅𝐼𝑖𝑗 (3) When i > j 𝑤𝑖𝑗 = {1, 𝐸𝐸𝑋0, 𝐸𝐸𝑋𝑖𝑗 > 𝐸𝑅𝐼𝑖𝑗 𝑖𝑗 < 𝐸𝑅𝐼𝑖𝑗 (4)
Note that 𝑤𝑖𝑗 evaluates the situation of country i that exports to j and evaluates its
imports from j as replacing domestic production (and thus emissions). 𝑤𝑖𝑗 = 1 indicates that
the PHH is corroborated, and 𝑤𝑖𝑗 = 0 indicates rejection. The valuation is illustrated below in
Table 2.
Table 2 Valuation of 𝑤𝑖𝑗
5. Empirical results
5.1 Empirical results of bilateral comparisons
Since the data of 𝐶𝑂2 emissions in 2009 are the most up-to-date in the environmental
accounts, we apply the World Input-Output Table for that year (2009) in our analysis.
One example of the calculation process for the bilateral comparison (Luxembourg and India) is given in Table 3. Luxembourg has a higher GDP per capita than India, thus in this case, Luxembourg is the rich (R) country and India is the poor (P) country. Columns (2) and
(3) indicate ∆𝒆𝑅 and ∆𝒎𝑅, respectively and column (7) and (8) indicate ∆𝒆𝑃 and ∆𝒎𝑃,
1 2 3 4 … 37 38 39 40 1 2 3 4 ⋮ 37 38 39 40 ⋮ i > j, thus the entry
equals 1 if
𝐸𝐸𝑋39,1> 𝐸𝑅𝐼39,1; otherwise it equals 0.
i < j, thus the entry equals 1 if
respectively. We have calculated the extra pollution if the bilateral trade increases by 1 billion US$, keeping the current account balance invariant. For example, the extra export vector from
Luxembourg to India is calculated as ∆𝒆𝑅 = 1000𝒆𝑅/𝝉35′ 𝒆𝑅, where 𝒆𝑅 (𝒆𝑅 = 𝒁𝐿𝐼𝝉35+
𝑭𝐿𝐼𝝉
5) is the actual export vector in 2009 from Luxembourg (L) to India (I) and the
denominator(𝝉35′ 𝒆
𝑅) is the total amount of exports form Luxembourg to India. The vector ∆𝒆𝑅
in column (2) in Table 3 gives the extra exports of each commodity if the total exports increase by 1 billion US$. It is obvious from Table 3 that the change in Luxembourg exports (imports) in column (2) is equal to the change in Indian imports (exports) in column (8), as mentioned in Section 4.
The multipliers are given in column (4) and column (9). For instance, column (4) gives
the multipliers for Luxembourg 𝝆𝐿′𝑳𝐿𝐿 and the j th element in the vector indicates the
emission of 𝐶𝑂2 (in kt) that is required per million US$ of exports in production j in
Luxembourg. For each production, multiplying the multiplier with the amount of extra exports (imports) gives the extra emissions in column 5(6), corresponding to the vector 𝝆𝐿′𝑳𝐿𝐿∆𝒆
𝑅 (𝝆𝐿′𝑳𝐿𝐿∆𝒎𝑅).
The total results in column (5) and (6) show that for Luxembourg, 𝐸𝐸𝑋𝐿𝑈𝑋,𝐼𝑁𝐷>
𝐸𝑅𝐼𝐿𝑈𝑋,𝐼𝑁𝐷. Hence, according to Table 2, 𝑤𝐿𝑈𝑋,𝐼𝑁𝐷 (which is located in the upper triangular part of the matrix 𝑾) equals to 0. Similarly, the total results in column (10) and (11) show that
for India, 𝐸𝐸𝑋𝐼𝑁𝐷,𝐿𝑈𝑋 < 𝐸𝑅𝐼𝐼𝑁𝐷,𝐿𝑈𝑋. Hence, 𝑤𝐼𝑁𝐷,𝐿𝑈𝑋 (which is located in the lower
triangular part of the matrix 𝑾) equals to 0.
The most important conclusion from Table 3 is that Luxembourg is the “pollution haven”: Luxembourg exports more emissions than are involved in replacing the same amount of imports, while India imports more emissions than are embodied in the same amount of exports, which is contradicted with the pollution haven hypothesis.
All the empirical results for every pair of countries are summarized in Table 4. Countries are ordered according to the GDP per capita from high to low. For the ease of exposition, we color the entries with a value of 1 green (when the PHH is supported) and color the entries with a value of 0 red (when the PHH is violated).
Please note that the matrix in Table 4 is not symmetric. Every element in the same row is from the same country’s (in the row) viewpoint, while every element in the same column is
from different country’s view point. For instance, although both the elements wUSA,AUS and
15
Table 3 Emissions from 1 billion US$ of extra exports and imports between Luxembourg and India, 2009
20 2.16 0.01 0.01 0.02 0.00 0.01 2.16 0.01 0.00 0.03 21 29.60 0.09 0.02 0.66 0.00 0.09 29.60 0.03 0.00 1.00 22 0.38 285.65 0.02 0.01 5.37 285.65 0.38 0.43 123.85 0.17 23 31.65 0.55 0 0.00 0 0.55 31.65 0.17 0.09 5.40 24 0.03 0 0 0.00 0 0 0.03 1.75 0 0.05 25 0.97 0 0 0.00 0 0 0.97 0.83 0 0.80 26 0.76 0.00 0 0.00 0 0.00 0.76 0.29 0.00 0.22 27 0.42 0 0 0.00 0 0 0.42 0.08 0 0.03 28 241.96 20.02 0.00 1.01 0.08 20.02 241.96 0.01 0.14 1.71 29 0.01 0 0.00 0.00 0 0 0.01 0.00 0 0.00 30 51.72 0 0.00 0.24 0 0 51.72 0.05 0 2.42 31 0.75 0 0.01 0.00 0 0 0.75 0.01 0 0.01 32 0.00 0 0.01 0.00 0 0 0.00 0.03 0 0.00 33 0.03 0 0.01 0.00 0 0 0.03 0.02 0 0.00 34 0.48 0.43 0.06 0.03 0.03 0.43 0.48 0.12 0.05 0.06 35 0 0 0 0.00 0 0 0.00 0.00 0 0.00 total 1000 1000 85.32 7.72 1000 1000.00 222.95 358.29
17
and AUS, the former is from USA’s viewpoint and the calculation of EEXUSA,AUS and
ERIUSA,AUS employ USA’s technology matrix and the industrial emission coefficient vector of
USA, while the latter is from AUS’s viewpoint and the calculation of EEXAUS,USA and
ERIAUS,USA employ AUS’s technology matrix and the industrial emission coefficient vector of AUS.
Table 4 Empirical Results of testing pollution haven hypothesis
Note: The ISO country code for all the countries in WIOD is given in Appendix D. Each entry ij in Table 4 indicates the comparison between the emissions embodied in the exports from country i to country j (𝐸𝐸𝑋𝑖𝑗) and
country i’s emissions that are replaced by importing from country j (𝐸𝑅𝐼𝑖𝑗). Entries which are consistent with the
PHH are colored green, while entries which contradict the PHH are colored red.
As shown in Table 4, out of the 1560 empirical results, 718 entries have a value of 1 and 842 entries have a value of 0. Less than half of the results (46.0%) are consistent with the
PHH.4 This shows that the trade pattern in 𝐶𝑂
2 emissions is not primarily determined by the
environmental regulations. In fact, the pollution haven hypothesis can be accepted only if
both requirements are fulfilled (i.e. ∆𝜋𝑅 < 0 and ∆𝜋𝑃 > 0). That is, the pollution haven
hypothesis is accepted only if the symmetric elements in Table 4 are both green. Hence, strictly speaking, the acceptance is only 31.7%.
4
Cole and Elliott (2013) state that only manufacturing industries are subject to the pollution haven hypothesis, we have also calculated the 𝐸𝐸𝑋 and 𝐸𝑅𝐼 by only including the manufacturing industries and made the bilateral comparison for every pair of countries again as a robustness check. The calculation process and results are summarized in Appendix B. The results do not change much and only 41.5% of the results are consistent with the PHH. Again, strictly speaking, the acceptance is only 21.2%.
LUX USA NLD IRL AUT AUS SWE DNK CAN BEL FIN DEU UK FRA ITA TWN ESP JPN CYP GRC KOR SVN CZE PRT MLT SVK HUN EST RUS POL LTU LVA ROU TUR BGR MEX BRA CHN IDN IND
A possible explanation of the results might be given by the factor endowment hypothesis
(FEH). The hypothesis maintains that pollution-intensive industries are also capital-intensive.5
According to the Heckscher-Ohlin theory, relatively capital-abundant countries (i.e. typically rich, developed countries) will export capital-intensive goods, and relatively labor-abundant countries will export labor-intensive goods. Thus, capital-abundant countries have a comparative advantage in pollution-intensive goods, which they will export. As a result, we get the following hypothesis:
Hypothesis: In the bilateral trade setting, the net extra emissions due to the increased trade will be positive (𝐸𝐸𝑋 > 𝐸𝑅𝐼) for the country with the higher capital-to-labor ratio, while the net extra emissions will be negative (𝐸𝐸𝑋 < 𝐸𝑅𝐼) for the country with the lower capital-to-labor.
To test this hypothesis, we reorder countries based on their capital-to-labor ratio from low to high, to keep the same indicator function for elements as in the matrix 𝑾. The socio-economic accounts in the WIOD contain industry-level data on employment, social capital and value added. The complete and most up-to-date data are for 2007 and data for 2009 are not complete. Thus, we use the “employee” variable (number of persons engaged, in millions) and “capital at constant price” variable (capital stock at current PPPs, in million US$, which is suitable to compare across countries) in 2009 in the latest Penn World Tables (PWT 8.1) to calculate the capital-to-labor ratio. The results are given in Table 5.
Table 5 Ordering of countries by capital-to-labor ratio from low to high
Ranking Country Employee
(1)
Capital stock (2)
Capital labor ratio (3)=(2)/(1) 1 IND 475.31 8526102 17938.0 2 IDN 104.90 2942223 28048.5 3 CHN 777.38 35744012 45980.4 4 BGR 3.90 245655 62992.6 5 BRA 96.23 6093020 63316.4 6 MEX 43.66 3959214 90673.5 7 POL 15.71 1473756 93825.4 8 LTU 1.33 140327 105518.3 9 RUS 67.75 7160576 105693.8 10 TUR 20.37 2235154 109723.1 11 ROU 8.98 1062058 118229.2 12 LVA 0.99 123535 124479.1 13 EST 0.60 83642 140029.9 14 SVK 2.20 338490 153947.7 15 TWN 10.34 1842257 178183.4 16 CZE 5.23 970037 185652.0 5
19 17 SWE 4.58 851936 186079.4 18 HUN 4.04 759875 188040.7 19 SVN 0.99 189597 190991.8 20 CAN 16.97 3289840 193839.6 21 GBR 28.80 5673085 196981.9 22 PRT 4.99 1061109 212628.5 23 MLT 0.17 36239 216675.8 24 GRC 5.10 1139686 223399.2 25 KOR 23.24 5279380 227153.2 26 AUS 11.28 2780861 246480.9 27 DEU 40.52 10345106 255308.3 28 NLD 8.72 2238130 256808.5 29 DNK 2.89 789181 272623.6 30 IRL 1.86 512700 275837.2 31 CYP 0.32 88181 279829.4 32 USA 141.60 41246380 291297.7 33 AUT 4.12 1221433 296813.5 34 JPN 62.61 18613806 297317.5 35 BEL 4.56 1373586 301327.8 36 FRA 26.74 8265503 309160.9 37 FIN 2.53 810645 320783.3 38 LUX 0.36 116336 325314.0 39 ESP 19.06 6588276 345597.9 40 ITA 24.74 9153038 369981.7
Note: Countries are ranked by their capital-to-labor ratio from low to high. The capital and labor data (for the year of 2009) are collected from the latest Penn World Table, PWT 8.1. The employee variable (in millions) shows the number of people engaged. The capital stock variable (in million US$) gives each country’s capital stock adjusted by current PPP.6
Again, we summarize the empirical results in a matrix. Each row and column indicates a particular country. When countries are ranked by capital-to-labor ratio from low to high, then for each row i: if country j is located right to the main diagonal (i.e, j > i), it is relatively more capital-abundant than country i – country j has a comparative advantage in capital-intensive (pollution-intensive) industries and i in labor-intensive (clean) industries. Then country i – according to FEH – will export clean labor-intensive goods and imports will replace
pollution- and capital-intensive goods. Hence, 𝐸𝐸𝑋𝑖𝑗 < 𝐸𝑅𝐼𝑖𝑗; if country j is located left to
the main diagonal (i.e, j < i), it is relatively less capital-abundant than country i, then country i has the comparative advantage in capital-intensive (pollution-intensive) and j in labor-intensive (clean) industries. Then country i will export pollution- and capital-labor-intensive goods
and imports will replace clean labor-intensive goods. Hence, 𝐸𝐸𝑋𝑖𝑗 > 𝐸𝑅𝐼𝑖𝑗. It is obvious that
the indicator functions are exactly the same as in equation (3) and (4).
6
We again arrive at an asymmetric matrix, in which entries with the a value of 1 indicating that the empirical results are consistent with the factor endowment hypothesis and entries with a value of 0 indicating that the hypothesis is rejected by the corresponding results.
Table 6 The empirical results of testing the Factor Endowment Hypothesis
Note: The ISO country code for all the countries in WIOD is given in Appendix D. Each entry ij indicates the comparison between the emissions embodied in exports from country i to country j (𝐸𝐸𝑋𝑖𝑗) and country i’s
emissions that are avoided by importing from country j(𝐸𝑅𝐼𝑖𝑗). Entries which are consistent with the FEH are
colored green, while entries which contradict with the FEH are colored red.
The results are given in Table 6. Out of the 1560 empirical results, 804 entries have a value of 1 and 756 entries have a value of 0. Only 51.5% of the results are consistent with the factor endowment hypothesis, which is slightly more than the concordance rate of the pollution haven hypothesis in Table 4. In fact, the factor endowment hypothesis can be
accepted only if both requirements are fulfilled (i.e., ∆𝑒𝑅 > 0 and ∆𝑒𝑃 < 0) – symmetric
elements are both green. Hence, strictly speaking, the acceptance rate is only 36.7%.
Until now, both the PHH and the FEH are not widely supported by the empirical results. Using the sample of 40 countries, which are included in the World Input-Output Table, we find a significant correlation of 0.81 between countries’ GDP per capita and capital-to-labor ratio. This shows that high-income countries have both a high GDP per capita and a high capita-to-labor ratio. According to the FEH (Heckscher-Ohlin theory), high-income countries will specialize in pollution-intensive industries. However, the PHH suggests that high-income countries will specialize in clean industries and replace pollution to low-income countries. Similarly, low-income countries have both a low GDP per capita and a low capital-to-labor ratio. The FEH argues that developing countries import more pollution, while the PHH argues
IND IDN CHN BGR BRA MEX POL ROU LTU RUS TUR LVA EST SVK TWN CZE SWE HUN SVN CAN UK POT MLT GRC KOR AUS DEU NLD DNK IRL CYP USA AUT JPN BEL FRA FIN LUX ESP ITA
21
that developing countries export more pollution. Because the correlation between GDP and capital-to-labor ratio is large, the ordering in Table 4 and Table 6 more or less switches. Hence, if the PHH holds, Table 4 would be all green while Table 6 would be all red. If FEH holds, Table 4 would be all red and Table 6 would be all green. However, in both Table 4 and Table 6, approximately 50% of the entries are green and approximately 50% of the entries are red. This motivates us to raise the question: what ordering can make the table almost full of green cells? To put it another way, can we find a hierarchy of countries such that those whose 𝐸𝐸𝑋 is smaller than its 𝐸𝑅𝐼 (when comparing with the lower ranked countries) appear first;
while those whose 𝐸𝐸𝑋 is larger than its 𝐸𝑅𝐼 (when comparing with the higher ranked
countries) appear last. This can be viewed as a triangularization problem of input-output tables (Chenery and Watanabe 1958; Dietzenbacher 1996), the original objective of which was to find a hierarchy of sectors such that those who are mainly producing for consumers appear last, while those who are predominantly primary producers appear first. We will discuss the triangularization problem in detail in the next section.
5.2 Is trade good for the environment at the global level?
In the previous part, we explored whether trade is good (𝐸𝐸𝑋𝑖𝑗<𝐸𝑅𝐼𝑖𝑗) or not for the
environment for country i. Since 𝐶𝑂2 is a purely global externality, it is also necessary to look
into the effect of trade on 𝐶𝑂2 emissions at the global level. That is, take also the change in
the emissions in country j into account.
As mentioned in Section 3, increased trade is good for the environment if the total
amount of the net extra emissions is negative, that is ∆𝜋𝑅+ ∆𝜋𝑃 = (𝐸𝐸𝑋𝑖𝑗− 𝐸𝑅𝐼𝑖𝑗) +
(𝐸𝐸𝑋𝑗𝑖− 𝐸𝑅𝐼𝑗𝑖) < 0 . Again, we make the calculation for every pair of countries and
summarize the results in a matrix (Table 7). Countries are alphabetically ordered. Each row and each column of the matrix indicates a particular country and each off-diagonal element indicates whether the total amount of the net extra emissions due to increased trade between a pair of countries increase or decrease. That is, 1 if the total amount is negative (indicating that increased trade is good for the environment); 0 if the total amount is positive (revealing that increased trade is bad for the environment). In addition, we color the element with a value of 1 yellow and color the elements with a value of 0 red. Note that the matrix is symmetric (for
instance, 𝑤𝑈𝑆𝐴,𝐴𝑈𝑆 = 𝑤𝐴𝑈𝑆,𝑈𝑆𝐴), because the indicator function is the same for any two
symmetric elements.
Table 7 Evaluation of emissions at the global level
Note: The ISO country code for all the countries in WIOD is given in Appendix D. Each entry ij indicates whether the bilateral trade between country i and country j is good for the environment (or not) at the global level. Entries with a value of 1 indicate that bilateral trade is good for the environment and are colored yellow, while entries with a value of 0 indicate the bilateral trade is bad for the environment and are colored red.
One interesting finding from Table 7 is that the top five countries that have the highest number of 1s are Ireland, Mexico, India, Korea, and Luxembourg– more than 31 out of 39 entries are colored yellow in each row. For these countries, more than 75% of the bilateral trade is good for the environment at the global level.
6. Triangularization
Since both the PHH and FEH hypotheses are not widely supported by the empirical results, in this part we will look for a potential alternative hypothesis that can explain the trade pattern of
𝐶𝑂2 emissions. As stated in Section 5, the question boils down to finding the ordering of
countries that makes the matrix 𝑾 as green as possible. It can be restated as a
triangularization problem. For the ease of exposition, we define another indicator function
𝑞𝑖𝑗 = 1{𝐸𝐸𝑋𝑖𝑗 > 𝐸𝑅𝐼𝑖𝑗} and get the matrix 𝑸. It is apparent that to make the matrix 𝑾
perfectly green is equivalent to making the matrix 𝑸 triangular. That is, to make all entries above the main diagonal zero in the matrix 𝑸.
6.1 Triangularization problem in the input-output literature
In the input-output literature, triangularization has been widely discussed and applied to the technical coefficient matrix. The aim of triangularizing the technical coefficient matrix is to find certain characteristics of the industrial structure of the economy. The idea of triangularization was first proposed by Chenery and Watanabe (1958), who compared the
23
hierarchical structure of sectors in United States, Japan, Norway and Italy. The hierarchical structure of these four countries turned out to be quite similar.
Triangularization of the input-output table is a linear ordering problem, which is an
NP-hard combinatorial optimization problem (Grötschel, Jünger and Reinelt 1984). If there are n
sectors, the number of feasible solutions is n!. To solve the problem, one can enumerate all the n! permutations of sectors and choose the optimal permutation which maximizes the sum of the entries in the lower triangular part of the matrix. However, such a brute force search is not feasible since it cannot find the result in polynomial time even when n is only moderately
large. For example, in our case, 𝑛 = 40, which yields 𝑛! = 8 ∗ 1047.
Some algorithms have been specifically designed for the triangularization problem in the literature. Simpson and Tsukui(1965) and Fukui(1986), for example, provide heuristic algorithms in which interchange of sectors, called ringshift permutations, is iterated. However, the optimal ordering is not necessarily found by implementing the algorithms.
Later, Grötschel, Jünger and Reinelt(1984), Chiarini, Chaovalitwongse and Pardalos(2004)
and Kondo(2014) rewrite the triangularization problem as an integer linear program and propose more efficient algorithms. Based on their studies, we will first define the triangularization problem and then transform the problem we are studying into an integer linear program. Finally, we present the empirical results of the triangularization in section 6.4.
6.2 Definition of the triangularization problem
First, we denote the set of countries (which are alphabetically ordered) by natural numbers as
𝑵 = {1, 2, 3, ⋯ , 𝑛} . Then we define the permutation of countries by
𝝅 = {𝜋(1), 𝜋(2), ⋯ , 𝜋(𝑛)} and the set of all permutation of countries by 𝚷, where π(p) denotes the ranking of country p (e.g. according to GDP per capita). For a given
permutation𝛑 , we define the matrix after permuting sectors according to 𝛑 as 𝐐(𝝅) =
(𝑄𝑖𝑗(𝛑)), that is
𝑄𝑖𝑗(𝜋) = 𝑄𝜋−1(𝑖)𝜋−1(𝑗) (𝑖, 𝑗 ∈ 𝑁) (5)
Where 𝜋−1(𝑖) represents the country that ranks (for instance, in terms of GDP per capita) at
the 𝑖th position.
To illustrate this, let’s take three countries, Brazil, Canada and UK, as example. According the alphabetical ordering, we denote these three countries as 1, 2, and 3, respectively. In the meantime, in terms of GDP per capita, Canada>UK>Brazil. Hence, we have 𝜋(1) = 3, 𝜋(2) = 1, and 𝜋(3) = 2.
We define the permutation matrix 𝑷 = (𝑃𝑖𝑗), in which
𝑃𝑖𝑗 = 1{𝑖 = 𝜋(𝑗)} (𝑖, 𝑗 ∈ 𝑁) (6)
𝐏 = [0 1 00 0 1 1 0 0
] (7)
Therefore, the matrix after permutation (𝐐(𝝅)), can be expressed as
𝐐(𝛑) = 𝐏𝐐𝑷𝑇 (8)
The original 𝑸 matrix is
𝑸 = [𝐵𝐵 𝐵𝐶 𝐵𝑈𝐶𝐵 𝐶𝐶 𝐶𝑈
𝑈𝐵 𝑈𝐶 𝑈𝑈
] (9)
where we use the first letter of the country name to indicate the country. The entries in 𝑸 represent the comparison of the emissions embodied in exports from the first country to the second country with first country’s emissions avoided by importing from the second country. For instance, BU indicates the comparison of Brazilian emissions exported to UK with Brazilian emissions avoided due to imports from UK.
Then the matrix after permutation is
𝑸(𝝅) = [𝐶𝐶 𝐶𝑈 𝐶𝐵𝑈𝐶 𝑈𝑈 𝑈𝐵
𝐵𝐶 𝐵𝑈 𝐵𝐵
] (10)
Then the triangularization problem, aimed at finding the permutation of countries that maximizes the sum of the elements in the lower triangular part of the matrix, can be formulated as a combinatorial optimization problem
|𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 ∑ ∑ 𝑄𝑖𝑗(𝜋) 𝑖−1 𝑗=1 𝑛 𝑖=2 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝝅 ∈ 𝚷 (11)
Note that the relationship between the entries in the matrix 𝑾 and those in the matrix 𝑸 is as follows:
𝑤𝑖𝑗 = 𝑔𝑟𝑒𝑒𝑛 𝑖𝑓 {𝑄𝑄𝑖𝑗 = 1 𝑓𝑜𝑟 𝑖𝑗 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑜𝑤𝑒𝑟 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥
𝑖𝑗 = 0 𝑓𝑜𝑟 𝑖𝑗 𝑖𝑛 𝑡ℎ𝑒 𝑢𝑝𝑝𝑒𝑟 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥
Hence, the number of green entries in the matrix 𝑾 equals to the number of entries with a value of 1 in the lower triangular part of the matrix 𝑸 plus the number of entries with a value of 0 in the upper triangular part of the matrix 𝑸, which can be expressed as:
𝜆(𝑸(𝝅)) =∑𝑖>𝑗𝑄𝑖𝑗(𝝅)+ (780 − ∑𝑖<𝑗𝑄𝑖𝑗(𝝅))
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Note that the second part of the numerator in equation (12) means that the number of entries with a value of 0 in the upper triangular part equals the total number of entries in the upper triangular part (780) minus the number of entries with a value 1 in the upper triangular
part. Since the total number of off-diagonal entries with a value of 1 is ∑𝑖≠𝑗𝑄𝑖𝑗(𝝅), the
number of entries with a value of 1 in the upper triangular part is ∑𝑖≠𝑗𝑄𝑖𝑗(𝝅)− ∑𝑖>𝑗𝑄𝑖𝑗(𝝅).
Hence, equation (12) can be rewritten as
𝜆(𝑸(𝝅)) =∑𝑖>𝑗𝑄𝑖𝑗(𝝅)+ {780 − [∑𝑖≠𝑗𝑄𝑖𝑗(𝝅)− ∑𝑖>𝑗𝑄𝑖𝑗(𝝅)]} 1560 =780 + 2 ∑𝑖>𝑗𝑄𝑖𝑗(𝝅)− ∑𝑖≠𝑗𝑄𝑖𝑗(𝝅) 1560 =1 2+ ∑𝑖>𝑗𝑄𝑖𝑗(𝝅) 780 − ∑𝑖≠𝑗𝑄𝑖𝑗(𝝅) 1560 (13)
Note that the numerator of the second part in the last equation is the same as the objective function of the triangularization problem defined in (11). The third part, the number of off-diagonal elements with a value of 1, is constant and does not depend on 𝝅. Therefore, the permutation 𝝅, which maximizes the objective function of the triangularization problem in
(11), also maximizes 𝜆(𝑸(𝝅)) the percentage of green entries in the matrix 𝑾. Hence,
𝜆(𝑸(𝝅)) also reveals the degree of triangularization. The more 𝜆(𝑸(𝝅)) is close to unity, the better the ordering of countries explain the direction of trade in emissions.
6.3 Rewriting the triangularization problem as an integer linear program
The optimization problem (11) can be solved by enumerating all the permutations and selecting the optimal one if n is very small. However, this brute force search is not feasible even in case of a moderate number of countries. In this part, we describe how to rewrite the triangularization problem as an integer linear program, which can be easily solved.
Define a {0,1}-matrix 𝑴 = (M𝑖𝑗) as
𝑀𝑖𝑗 = 1{𝑖 > 𝑗} (𝑖, 𝑗 ∈ 𝑁) (14)
Hence, the objective function can be written as
∑ ∑ 𝑄𝑖𝑗(𝜋) 𝑖−1 𝑗=1 𝑛 𝑖=2 = ∑ ∑ 𝑀𝑖𝑗𝑄𝑖𝑗(𝜋) 𝑛 𝑗=1 𝑛 𝑖=1 = ∑ ∑(𝑴 ∘ 𝑸(𝝅))𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 = 𝝉𝑛𝑇(𝑴 ∘ 𝑸(𝝅))𝝉 𝑛 = 𝝉𝑛𝑇(𝑴 ∘ 𝑷𝑸𝑷𝑇)𝝉𝑛 (15)
Where (⋅)𝑖𝑗 donates the (𝑖, 𝑗)-element of the matrix in parentheses, 𝜏𝑛 is the 𝑛 × 1 summation
vector and ∘ refers to the Hadamard product of two matrix, e.g., (𝑩 ∘ 𝑪)𝑖𝑗 = 𝐵𝑖𝑗𝐶𝑖𝑗 for two
The objective function of the triangularization problem can be stated as (15), based on the one-to-one correspondence between the permutation and the permutation matrix. However, it is still not linear and it is likely difficult to solve. Thus, an alternative, linear statement is preferred.
The objective function can be further rewritten as
𝝉𝑛𝑇(𝑴 ∘ 𝑷𝑸𝑷𝑇)𝝉 𝑛 = 𝑡𝑟(𝑷𝑸𝑷𝑇𝑴𝑇) = 𝑡𝑟(𝑸𝑷𝑇𝑴𝑇𝑷) = 𝑡𝑟(𝑸(𝑷𝑇𝑴𝑷)𝑇) = 𝝉 𝑛 𝑇(𝑷𝑇𝑴𝑷 ∘ 𝑸)𝝉 𝑛 (16)
Where 𝑡𝑟(⋅) is the trace of a square matrix. The first and last equation hold because
𝑡𝑟(𝑫𝑪𝑇) = ∑(𝑫𝑪𝑇) 𝑖𝑖 𝑛 𝑖=1 = ∑ ∑ 𝐷𝑖𝑘𝐶𝑖𝑘 𝑛 𝑘=1 𝑛 𝑖=1 = 𝝉𝑛𝑇(𝑪 ∘ 𝑫)𝝉𝑛 (17)
And the second equation holds because 𝑡𝑟(𝑪𝑫) = 𝑡𝑟(𝑫𝑪). It is interesting to note that the left- and right-hand-side of (16) reflect two different approaches to calculate the objective function. The left hand side is the sum of the elements in the lower triangular part of the matrix after permutation. This is equivalent to the expression on the right hand side, which picks up the elements from the original matrix and sums up.
If we define a new variable 𝑿 and 𝑿 = 𝑷𝑇𝑴𝑷, then the objective function can be
represented linear in 𝑿: ∑ ∑ 𝑞𝑖𝑗(𝜋) 𝑖−1 𝑗=1 𝑛 𝑖=2 = 𝝉𝑛𝑇(𝑿 ∘ 𝑸)𝝉 𝑛 = ∑ ∑ 𝑋𝑖𝑗𝑄𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 (18) The set of 𝑿 is 𝚷𝑿 = {𝑿 ∈ ℝ𝑛×𝑛|𝑿 = 𝑷𝑇𝑴𝑷, 𝑷 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥} (19)
Until now, the objective function can be expressed as linear in {0,1}-variables, 𝑋𝑖𝑗, in
(18). However, the constraints of 𝑿 (which is one-to-one correspondence with the permutation
matrix) in (19) are nonlinear. Next, we need to derive another expression of 𝚷𝑿, which is
conditioned on linear inequalities, in order to represent the triangularization problem as an integer linear program. Grötschel, Junger and Reinelt(1984, (10)-(12), page 1202), Chiarini, Chaovalitwongse and Pardalos(2004, (2.4)-(2.5) page 8) show that the following inequalities
can properly define 𝚷𝑿.
(E1) 𝑋𝑖𝑖 = 0(i ∈ N);
(E2)𝑋𝑖𝑗 + 𝑋𝑗𝑖 = 1 (𝑖 < 𝑗; 𝑖, 𝑗 ∈ 𝑁); and
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The constraint (E2) shows that there is one and only one entry with a value of 1 for any two symmetric entries – either countries i is ranked before country j or countries i is ranked after country j. The constraint (E3) indicates the relationship of any three elements, any two of which are in the same row or in the same column – for instance, if country i is ranked after country j and country j is ranked after country k, then country i must be ranked after country k.
Therefore, the triangularization problem can be expressed as the integer linear program:
| |maximize ∑ ∑ 𝑋𝑖𝑗𝑄𝑖𝑗 41 𝑗=1 41 𝑖=1 subject to 𝑋𝑖𝑗 ∈ {0,1}(𝑖, 𝑗 ∈ 𝑁) 𝑋𝑖𝑖 = 0(i ∈ N) 𝑋𝑖𝑗 + 𝑋𝑗𝑖 = 1 (𝑖 < 𝑗; 𝑖, 𝑗 ∈ 𝑁) 0 ≤ 𝑋𝑖𝑗+ 𝑋𝑗𝑘− 𝑋𝑖𝑘 ≤ 1 (𝑖 < 𝑗 < 𝑘; 𝑖, 𝑗, 𝑘 ∈ 𝑁) (20)
After solving the triangularization problem, we get the optimal matrix 𝑋 . The
permutation 𝛑 can be recovered from the corresponding matrix 𝑿 as
∑ 𝑋𝑖𝑗 𝑛 𝑗=1 = ∑ 1{π(i) > π(j)} = π(i) 𝑛 𝑗=1 (21)
Where the first equation holds because the matrix 𝑿 only picks up the elements in the lower
triangular part of the matrix after permutation. Therefore, 𝑋𝑖𝑗 equals 1 only if country i is
placed after country j.
Hence, triangularizing the matrix 𝑸 (i.e., making the matrix 𝑾 as green as possible) is rewritten as an integer linear program. The integer linear program will be solved with the Xpress Mosel language version 3.4.2 and the algorithm based on Konda (2014). The results are summarized in Section 6.4.
6.4 Empirical result of triangularization problem
The triangularization result is displayed in Table 8. 𝜆(𝑸(𝝅)) , the degree of
triangularization, is 74.7%, indicating that 74.7% of the results are in compliance with the hypothesis according to the ordering. Strictly speaking, only if the entry in the lower triangular part of the matrix is colored green and its symmetric entry is colored red, the
ordering of countries can explain the trade pattern of 𝐶𝑂2 emissions. Then the percentage is
Table 8 Triangularization of the matrix 𝐐
Note: The ISO country code for all the countries in WIOD is given in Appendix D. The element 𝑞𝑖𝑗 compares the
emissions embodied in exports from country i to country j (𝐸𝐸𝑋𝑖𝑗) with the avoided emissions by country i due
to imports replacing domestic production (𝐸𝑅𝐼𝑖𝑗) with 𝑞𝑖𝑗= 1{𝐸𝐸𝑋𝑖𝑗> 𝐸𝑅𝐼𝑖𝑗} (𝑖, 𝑗 = 1,2, … ,40). We color the
entries with a value of 1 green and color the entries with a value of 0 red.
To check the robustness of the result, we also solve the triangularization program (20) for the actual differences matrix (instead of only 0-1 indicators): a new matrix 𝐕 with its elements
𝑣𝑖𝑗 as the difference between the emissions embodied in exports from country i to country j
(𝐸𝐸𝑋𝑖𝑗) and the avoided emissions due to imports from j replacing domestic production
(𝐸𝑅𝐼𝑖𝑗), that is 𝑣𝑖𝑗 = 𝐸𝐸𝑋𝑖𝑗 − 𝐸𝑅𝐼𝑖𝑗, (𝑖, 𝑗 = 1,2, … , 𝑛).
The results are summarized in Table 9, which is similar as Table 8. λ(V(π)) is 74.1%, indicating that 74.1% of the results are in compliance with the hypothesis according to the ordering in Table 9. Again, strictly speaking, only if the entry in the lower triangular part of the matrix is colored green and its symmetric entry is colored red, the ordering of countries
can explain the trade pattern of 𝐶𝑂2 emissions. Then the percentage is only 60.1%. The
degrees of triangularization of the optimal triangularized matrix 𝐐(𝛑) and 𝐕(𝛑) are not so
close to unity, implying that there is no ordering of countries that can explain the trade pattern
in terms of 𝐶𝑂2 emissions for all the countries.
For some countries (like Russia), no matter which country it is trading with, its 𝐸𝐸𝑋 is always larger than 𝐸𝑅𝐼 (as shown in Table 9, most entries in the row of Russia are colored green); while for other countries (for example, Ireland, Korea and Sweden), the opposite holds.
KOR AUT SWE MLT CYP CHN HUN IND ROU MEX IDN SVK JPN USA UK ITA TWN BRA TUR IRL BEL EST POL CZE SVN FIN DEU FRA AUS LVA NLD PRT CAN GRC BGR ESP DNK LTU RUS LUX
