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If those B borrowers had to repay that debt in A currency, then they probably would have to exchange large amounts of B currency for A currency, thus driving up the value of A currency with respect to B currency. Is that correct, so far?
Correct, in the givens of your scenario.
On the other hand, if those B borrowers were allowed to repay their debts in B currency, then the exchange rate would not be affected, or at least, not as drastically. Is that correct?
Yes, correct again. And I argue that this is effectively the case in the real world today because of highly liquid markets for currency and virtually costless and instantaneous exchange of currency. People therefore are in possession of a particular currency at any moment because they want it, not because they're required to.
The dollar did not go up in order to repay debts. That was my point. It went up because people wanted to hold dollars relative to either other currencies or other assets. However, we know that people were actually selling other assets -- real estate and stocks -- so that means that the dollar must have increased because people, for some reason, wanted to hold cash in US dollars.
Let's go to Switzerland:
What if Switzerland, not the U.S., were the source of all the mad financial shenanigans and most of the debt were denominated in Swiss francs and had to be paid back in Swiss francs (again, another fantasy, but...) In this scenario, wouldn't we have seen a rush for Swiss francs and a corresponding rise in the value of the CHF?
No, and the reason is because repaying debt is NOT what people do when a country goes into mad financial shenanigans such as occurred in the US (or in Sweden in the early 1990's, for a closer comparison to your Swiss example). What people do is sell assets to hold currency, paying some debt in the process only incidentally. (No one is clamoring to repay debt right now and creditors have no way of making people accelerate repayment either, so the driving phenomenon is assets sales not debt repayment.) And if safer currencies then the precarious domestic one were available, people, including lenders, will buy and hold those currencies. In your Swiss example, people would sell Swiss real estate and other assets at fire sale prices and use the proceeds buy Euros or Dollars. If they could, they would even borrow MORE CHF denominated debt to buy euros or dollars.
The question then, is why didn't that happen when US financial system collapsed last year and why isn't it happening now? Your graphs tell a story of a surge in the dollar's value followed by a return to pre-crisis prices. That's not a declining dollar story -- that's a stable currency story. The dollar's fall since the beginning of the year indicates a willingness to invest in productive assets, possibly including foreign assets, instead of holding cash, and that's good news, not bad.
So, if the dollar were perceived to be a weak security for wealth compared to other currencies, we should never have seen any surge in its price at all. Foreigners were not ones selling foreign assets to repay US creditors last year, which is your Swiss story. Foreigners were the net creditors, in fact, as shown by the US current account deficit. Rather, both foreigners and Americans were selling US assets (causing creditors to collect some it). And they all tended to choose to hold dollars in cash while this was going on instead of immediately converting it to foreign currency or foreign assets. That's evidence that political power trumps everything else when it comes to money.
The Yen? Following my narrative, Japan is the world's second largest economy, Tokyo is the financial center of Asia, and Japan is therefore the only nation-state comparable in power to the US in a crisis. People believe that the Japanese have the power to make good on their claims and debts, through coercion if not through growth. If people wanted to take some wealth out of Europe and America and go to Asia for protection, Japanese political authority to protect that wealth was the best bet.
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