But if domestic interest rates do change, as they likely would in either a closed economy or a large open economy, private investment and saving would also likely change, and any strict link between budget and trade deficits would be broken. One could spin any number of scenarios, but these are enough to make the basic point.
Because of the underlying relationship between saving and investment, budget and trade deficits could be strictly linked. But in a large open economy like the United States, it is easy to imagine plenty of scenarios in which they are imperfectly linked, and even some scenarios in which they move in opposite directions.