When the Federal Reserve "prints money" by increasing the size of the monetary base, no money is actually printing. The creation of physical currency is a separate process that's largely unrelated to monetary policy. Physical currency is a relatively small share of the money supply (which is mostly composed of checking, savings, and money market accounts) and most of the physical currency doesn't circulate in the United States. But money does need to be printed, and as this cool Planet Money graphic Miki Meek posted today shows it's largely coming in the form of $100 bills:
That $400 million in $100 bills represents the vast bulk of the monetary value of new physical currency. Mostly new money printing reflects old bills becoming damaged and taken out of circulation but "the government has been printing lots more $100 dollar bills in the past few years, because people overseas have taken to holding onto them." This strikes me as a not-so-hot idea. A healthy share of foreign demand for large denomination physical currency comes from criminals. The rise in demand in recent years probably reflects a growing lack of confidence in €100 bills for these purposes. The only legitimate use of large quantities of $100 bills I can think of is that you sometimes need them in order to stage fictional depictions of criminals stockpiling cash.
Meanwhile, there's been a recent resurgence of interest in the fact that economic conditions during the Great Recession would have warranted substantially negative interest rates from the Fed. An important additional consideration is that if rates could have gone negative and people knew they could have gone negative, the economy probably wouldn't have deteriorated so much as to require them to be sharply negative. And the view that nominal rates "can't" be negative strikes me as unproven.
The theory is that negative rates will prompt people to horde physical currency instead. But this is logistically difficult. Microsoft isn't going to keep $63 billion in cash laying around the office in Redmond just because rates go slightly negative. And the supply of physical bills is in fact limited. The Fed makes a habit of trying to fully accommodate demand for physical currency, but insofar as this hampers monetary policy in order to facilitate foreigners' tax evasion it doesn't seem like a particularly wise policy.
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