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The US has the largest gross trade deficit with China. But some economists have argued that this is misleading, e.g. using the iPhone as an example in which most parts don't orginate in China.

There's actually a more academic version of this analysis called "trade in value-added", which can be roughly summarized as:

Global imbalances: accounting for trade in value-added (specifically accounting for trade in intermediate parts and components) and taking into account “trade in tasks” does not change the overall trade balance of a country with the rest of the world – it redistributes the surpluses and deficits across partner countries. When bilateral trade balances are measured in gross terms, the deficit with final goods producers (or the surplus of exporters of final products) is exaggerated because it incorporates the value of foreign inputs. The underlying imbalance is in fact with the countries that supplied inputs to the final producer. As pressure for rebalancing increases in the context of persistent deficits, there is a risk of protectionist responses that target countries at the end of global value chains on the basis of an inaccurate perception of the origin of trade imbalances.

This does look like quite an involved method to follow for all imports though. My question for here is: with whom would the US have the largest trade imbalance if one uses the "trade in value-added" (possibly recursively) to track US imports to their components?

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Found an answer in the same WTO book, although it's for a decade ago:

China’s bilateral trade: Global value chains in a changing world surplus with the United States was over US$ 40 billion, or 25 per cent smaller in value-added terms in 2009 and 30 per cent smaller in 2005. This partly reflects the higher share of US value-added imports in Chinese final demand but also the fact that a significant share (one-third) of China’s exports reflect foreign content – the “factory asia” phenomenon. The data illustrate that significant exports of value-added from the Republic of Korea and Japan pass through China on their way to final consumers, resulting in significantly smaller Chinese trade deficits with these countries but also typically higher Japanese and the Republic of Korea’s trade surpluses with other countries. Similarly, the database shows that the Republic of Korea’s significant trade deficit with Japan in gross terms almost disappears when measured in value-added terms.

So there's some 25% reduction for US deficit with China in 2009 when using TiVA, but it still looks like China is the most significant source of US trade deficit, even calculated this way. There's also a graph in the book from China's perspective:

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I also found a slighly more recent (2011) TiVA data:

Indeed, economists at the Organization for Economic Cooperation and Development (OECD) and the World Trade Organization have been trying to develop a so-called “value-added” methodology for tallying trade balances. It can make a big difference. In 2011, the most recent year with available value-added data, the United States ran a \$275.1 billion trade deficit with China; under the value-added approach, that was cut to \$178.7 billion. The contrast is starker in electronics. The 2011 U.S. trade deficit in that sector amounted to \$136.3 billion; a “value-added” deficit was just \$54.2 billion.

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