Processing Raw Data
The raw data used for this analysis is World Input-Output Database 2016 Release from World Input-Output Database (https://www.rug.nl/ggdc/valuechain/wiod/?lang=en), which comprises intermediates, final consumption, investment by 56 sectors of 43 countries and rest of the world from 2000 to 2014. Summing up all the sectoral intermediates by country in the raw data gives a 44ⅹ44 matrix for each year to each subject including 43 countries and rest of the world. Countries in row-wide are producers or suppliers of intermediates while countries in column-wide represent consumers or demanders of the intermediates from the country corresponding to each row. In each year, a single country in row-wide has 44 entities that consume its own intermediates(Refer to the Excel file), including its domestic market in column-wide. There are 15 of data sheets for the respective 15 years from 2000 to 2014. If country in each row is extracted from each of 15 sheets and transposed, it makes a 44ⅹ15 matrix, then 44 of 44 ⅹ 15 for respective 44 subjects. Each row in the 44ⅹ15 matrix for one country represents countries that consume the country ’s intermediates during the period (Refer to sheets 1-44), that is, how much the country supplied their products to the other countries and the domestic market. If you sum this matrix in column-wise, the 1ⅹ15 matrix means the sum of the 44 countries that source their intermediates from the country, that is, how much the country supplied their intermediates to the world. Then, summing up all the 44 matrices in this way gives one 44 ⅹ 15 matrix for worldwide total during the time series.
Defining Inequality
Now, 44 subjects are comprised in the dataset for each year. If the values are evenly distributed across the subjects, they should proportionately increase as the number of the subjects increases. The cumulative value should be the n multiplied by 1/N for the nth subject when the ratio is normalized to 1 and N is the number of observations. The zigzagged gray line in <Graph 1> shows this trend.
<Graph 1> Inequality in Intermediates sourced by Australia
If there’s any deviation from the zigzagged baseline, which can be either upward or downward, that means the sourced intermediates are unevenly distributed across subjects. The orange graph shows the real cumulative ratio of intermediates supplied by Australia to all over the world, including itself. The countries Australia supplies its intermediates to were lined up by the size of the quantity from the smallest to the largest. <Graph 1> shows both of the baseline and the real cumulative ratio of the intermediates supplied by Australia to all over the world, including itself. Then, the difference between the baseline and the actual cumulative line can represent the inequality how less Australia is evenly diversifying its sourcers of intermediates.
Model
When there are \(M\) subjects in the data set and the domain measure is normalized to one, the equality baseline for subject \(s\), \(a_s\) is s accumulations of \({1/M}\).
$$
\begin{align}
\tag{1} a_s &= \ {{s}\over{M}} \ \\
\end{align}
$$
The area under the base line can be indicated as the summation of the arithmetical progression Then, the summation of the baseline values of perfect equality up to subject \(s\) is
$$
\begin{align}
\tag{2.1} L_s= \ {{a_s(a_s+1)}\over{2}} \ \\
\end{align}
$$
The index of intermediate inputs for country \(h\) from country \(s\), at period \(t\), \(m_{s,t}^h\) is given in (3). The sources of the intermediates for the country, \(h\) are arranged from the smallest to the largest in order of the magnitudes they supply to the country \(h\).
$$
\tag{3} m_{1,t}^h<{m_{1,t}^h}<\cdots<{m_{3,t}^h}.
$$
Cumulative quantity of intermediates by source \(n\) is
$$
\begin{align}
\tag{4} M_{n,t}^h&= \sum_{s=1}^n m_{s,t}^h \\
\end{align}
$$
\(B_{h,t}^M\) denotes the set of country indices that supply intermediates to country \(h\), the input “buy-from” set for country \(h\). Then, \(R_{s,t}^h\) denotes the ratio of intermediate input from source \(s\) to the total intermediates \(M_{M,t}^h\).
$$
\begin{align}
\tag{5.1} R_{s,t}^h &= \ {{m_{s,t}^h}\over{M_{M,t}^h}}. \ \\
\end{align}
$$
$$
\begin{align}
\tag{2.2} L_n &= \ {{1}\over{M}}{\cdot}{{a_n(a_n+1)}\over{2}} \ \\
\end{align}
$$
\begin{align}
\tag{5.2} R_{n,t}^h &= {{1}\over{M}}{\cdot}{{\sum_{s\in B_{h,t}^M}^n m_{s,t}^h}\over{M_{M,t}^h}} \\\
\end{align}
$$
where \(L_n\) is the area under the equality baseline by subject \(n\) among \(M\) subjects. Over the domain from \(0\) to subject \(M\), the disparity between \(L_s\) and \(R_{s,t}^h\) represents the degree of inequality over the \(M\) subjects.
$$
\begin{align}
\tag{6} I_M&= \sum_{s=1}^M L_s-{R_{s,t}^h}. \\
\end{align}
$$
We can assume that the date set is exhaustive and no economic trading activities are out of the analysis. Then, the perfect inequality is where the sourcing or supplying is from or to a single nation respectively. Then, the perfect inequality is
$$
\begin{align}
\tag{7} I_p&= \sum_{s=1}^M L_s-{{1}\over{M}} \\
\end{align}
$$
where \(1/M\) means the area under \(R_{M,t}^h\) with the same \({\sum_{s\in B_{h,t}^M}^M m_{s,t}^h}\) and \(M_{M,t}^h\). In this current dataset, there are 44 subjects, then \(M\) is 44. If the degree of inequality \(I_M\) compared to \(I_p\), \(I_r\) is,
$$
\begin{align}
\tag{8} I_r&= {{I_M}\over{I_p}}. \\
\end{align}
$$
The inequality is measured for the world, and respective 44 subjects including 43 countries and rest of the world. Also, each nation is both a source and supplier so that inequalities for the respective \(N\) countries as well as the inequality of the world can be measured in dimensions of their sourcing and supplying. The world’s inequality indicators are estimated for each year from 2000 to 2014.
< Inequality Index in World Intermediate Sourced>
| Year | Inequality in Intermediates Sourced by the World Ir(IM/IP) |
World Total Intermediates Sourced |
| Perfect Inequality=1 | Unit: in Billions of Constant 2015 US$ |
|
| 2000 | 0.7507 | 42592.8651 |
| 2001 | 0.7472 | 43070.5513 |
| 2002 | 0.7344 | 43519.7198 |
| 2003 | 0.7218 | 45636.0006 |
| 2004 | 0.7155 | 48916.5001 |
| 2005 | 0.7167 | 52943.2777 |
| 2006 | 0.7608 | 68961.4926 |
| 2007 | 0.7056 | 60927.0586 |
| 2008 | 0.6718 | 52216.7438 |
| 2009 | 0.7156 | 60321.0375 |
| 2010 | 0.7219 | 64992.7757 |
| 2011 | 0.7268 | 69743.2785 |
| 2012 | 0.7425 | 72260.4085 |
| 2013 | 0.7481 | 75551.2743 |
| 2014 | 0.7544 | 78276.8129 |
<Graph 2> Inequality in Intermediates sourced by the World
<Graph 3> World Total Intermediates Sourced
※ This work is worth as reference only because of inaccuracy.