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The most relevant balance of payments for the Euro-zone is, of course, the aggregated Euro-zone balance of payments. However, that is impolitic to make a big deal about, because it would be one more road-block to expanding the Euro-zone to include more of the EU membership, so I guess everybody just continues to publicly pretend that it is the national balance of payments that matter, while in private running up spreadsheets to aggregate the Eurozone nation accounts when they want to look at what is really going on.
When working at the individual accounts, you need to sum the individual accounts and then subtract the sum of the corresponding account for each bilateral flows within a group, to work out a set of balance of payments for a given bloc.
If bilateral flows of each member of the Eurozone have already been grouped to give a figure "to other Eurozone nations", that simplifies that second process substantially. I've been accused of being a Marxist, yet while Harpo's my favourite, it's Groucho I'm always quoting. Odd, that.
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