# COVID-19's Impact on International Trade
## 3 Results
### PageRank-CheiRank Product Trade Balance  
It is interesting to analyze the trade balance not only for countries but also for products performing a summation of \(B_{pc}\) over countries. This peculiarity is not possible in the case of the ImportRank-ExportRank approach, since, by definition, the ImportRank-ExportRank product trade balance is zero, \(\hat{B}_{p}=\sum_{c}\hat{B}_{pc}=0\). Indeed, after summation over countries, the export and import probabilities--i.e., the relative export and import volumes--are equal, \(\hat{P}_{p}^{*}=\hat{P}_{p}\). However, within the PageRank-CheiRank approach, the probabilities \(P_{p}^{*}\) and \(P_{p}\) are different, and thus the product trade balance of some products, \(B_{p}=\sum_{c}B_{pc}\), can be positive and negative for some others. The trade balances for each group of products (Table 1) are presented in Figure 6 for the years 2018 and 2020. The balance \(B_{p}\) is positive for products \(p=0,1,2,3,9\), and hence they are more export oriented, \(P_{p}^{*}>P_{p}\). These products are more central in the inverted WTN than in the direct WTN. Otherwise stated,  
Figure 5: Geographical distributions of the PageRank–CheiRank trade balance sensitivity \(dB_{c}/d\delta\) to _food_ and _live animals_ (\(p=0\), **top** row), _mineral fuels_, _lubricants_ and _related materials_ (\(p=3\), **middle** row), and _machinery_, _transport equipment_ (\(p=7\), **bottom** row) products. The left column presents the PageRank-CheiRank trade balance sensitivities for the year 2018. The right column presents the difference between the years 2020 and 2018, i.e., \(\Delta dB_{c}/d\delta=(dB_{c}/d\delta)^{(2020)}-(dB_{c}/d\delta)^{(2018)}\). the export abilities of these products exceed their import abilities. In contrast, the products \(p=4,5,6,7,8\) have negative balances and are import oriented, \(P_{p}>P_{p}^{*}\). These products' import abilities supersede their export abilities. The PageRank-CheiRank approach allows us to infer a kind of structural law of supply and demand. The strongest variation of product balance from 2018 to 2020 takes place for _mineral fuels_ (\(p=3\)), with an increase of \(B_{p}\) by a factor 2.3 from 2018 to 2020, while the strongest negative decrease of \(B_{p}\) is observed for _manufactured goods_ (\(p=6\)) by a factor of 1.4 from 2018 to 2020. An even stronger negative decrease of \(B_{p}\) takes place for _miscellaneous manufactured articles_ (\(p=8\)) with a factor of 8; however, for this product, the balance \(B_{p}\) is relatively small. The effects of COVID-19 are salient here. The products _manufactured goods_ and _miscellaneous manufactured articles_ lost their export ability during 2020, whereas import abilities were maintained. Conversely, export abilities were maintained for _mineral fuels_, but structural demand declined in 2020. The distribution of product trade balance by countries, \(B_{pc}\), obtained from the PageRank-CheiRank approach, is shown in Figure 7 for 2018. The variations, \(\Delta B_{pc}\), from 2018 to 2020 of the product trade balances per country are also shown (the results obtained from the ImportRank-ExportRank approach are given in the Appendix A, see Figure A3). In 2018, China has a positive trade balance \(B_{pc}\) for products \(p=6\), 7 and 8, with the most negative trade balance \(B_{pc}\) is found for product \(p=3\). The product trade balance variation from 2018 to 2020 is significantly positive for product \(p=9\) (which includes financial transactions) and slightly positive (negative) for \(p=3\) (\(p=2\) and 1). The COVID-19-related variation of the USA product trade balance from 2018 to 2020 is significantly negative for products \(p=4\), 5, 7, 8 and 9 and positive for products \(p=2\) and 3. Among significant positive values of \(B_{pc}\) for the other countries in Figure 7a, we highlight Indonesia and Malaysia for \(p=4\), Russia for \(p=3\) and Brazil for \(p=2\); the negative product trade balances are less pronounced.