Since central banks can print money at no cost to them, and they can replace their balance sheet losses with money printing, does their losses matter at all? If they write off their losses, then what happens to inflation, or does it cause deflation?
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In the US, the central bank (Federal Reserve) has to pas through losses and gains to the US government. So losses and gains do affect the budget of the United states.
The central bank has its own account. The stabilization policy can be loss and gain. It has to pay interest as well. But it cannot use printing money to write off debts. It has to keep financial discipline. It has to report to the Congress and get approved.