Final answer:
When the US Treasury sells dollars and buys euros to weaken the dollar, the monetary base increases. If U.S. interest rates fall relative to the world, there would be less demand for and a greater supply of dollars, causing the dollar exchange rate to potentially depreciate against the euro. Central banks are cautious when increasing foreign currency reserves due to the associated opportunity costs.
Step-by-step explanation:
If the US Treasury engages in a foreign exchange intervention to lower the value of the dollar relative to the euro by selling dollars and buying euros, the monetary base will increase. This happens because when the Federal Reserve buys euros (or any foreign currency) in exchange for dollars, it effectively adds more dollars into circulation, increasing the supply of money. In contrast, if the Federal Reserve were selling euros and buying dollars, the monetary base would decline.
Considering a scenario where U.S. interest rates decline compared to the rest of the world, the likely impact would be a decrease in demand for dollars, an increase in the supply of dollars, and a potential depreciation in the exchange rate of the dollar compared to the euro. Lower interest rates make dollar-denominated investments less attractive, leading to investors seeking higher returns in other currencies. As a consequence, the value of the dollar may fall against other currencies.
However, it is important to note that central banks will not want to boost their reserves of foreign currencies indefinitely due to the opportunity cost associated with holding large reserves.