Clarification On Carry Trade
Since that is the scenario usually presented in international economics textbooks and classes, I still think it was right of me to focus on that. However, in hindsight I think I should have also clarified that the incompatibility of the purchasing power parity theory and the uncovered interest parity theory in a world where real interest rates differ does not depend on the initial adjustment taking place.
Assuming that the initial adjustment process, where the exchange rate of the high interest rate currency initially becomes overvalued from a goods market perspective, initiates, then the increased demand for goods from low interest rate countries will also increase the demand for their currencies, thus preventing the full initial appreciation of high interest rate currencies needed to create the expectation of future depreciation of high interest rate currencies needed to equalize return.
Suppose however, that no initial adjustment takes place, can the incompatibility of the purchasing power parity theory and the uncovered interest parity theory still be relevant? Yes, because in order for return to be equal, countries with higher real interest rates must still see their currencies depreciate in real terms. If the high interest rate currency wasn't overvalued to begin with, this means that it will become more and more undervalued as the years pass by. As a result, demand for goods in high interest rate countries will increase as the years pass by, which also means that demand for that currency gradually increases. That in turn will prevent or at least limit the gradual depreciation needed to equalize return.
This clarification has little theoretical importance except in one aspect, namely that it doesn't necessarily mean that high interest rate countries will empirically see net exports decline (see trade deficit increase or trade surplus drop). If the initial adjustment does not take place, it could in fact mean that net exports increases.