Let me start out with an extreme (and very silly) assumption, just so I can explain something simply. Assume that the demand for currency does not depend on the price level, nor on real income, nor on interest rates, nor on anything. It's just fixed. Every individual wants to hold exactly $100 in currency, no more no less, regardless of anything.
And let's start out with a very simple monetary system. There's a central bank that issues currency, that people use as their sole medium of exchange, to buy and sell everything else. No other banks at all. So it's a simple money supply function, and a totally degenerate money demand function Md = $100 x population.
And let's start out in monetary equilibrium, where the money supply (the stock of currency in circulation) is exactly $100 per person.
Is Helicopter Money impossible? If the central bank prints more currency, and drops it out of a helicopter, will the people refuse to pick it up, and leave the newly-printed notes lying on the sidewalk?
No. That's silly. They will pick it up, and spend it. Each individual knows they he can get rid of any excess money, even though it is impossible for individuals in aggregate to get rid of excess money. What is true for each individual is false for the whole. It's a fallacy of composition to assume otherwise.
But this extreme (and very silly) example shows that money, as medium of exchange, is very different from any other consumer durable. If the central bank were giving away free fridges, and everyone already had one fridge, and only wanted one fridge, everybody would leave the free fridges lying on the sidewalk.
Money is weird, because even though economists talk about people demanding a stock of money, that desired stock of money represents an average (over time) level of inventory in my pocket, where money is both flowing into and out of my pocket, so the stock of money in my pocket at any time fluctuates above and below its average level. Money circulates. And paper currency is intrinsically worthless money, which means that if I offered you a $20 note, but made you promise you would keep that note in your pocket always and never spend or give it away under any circumstances, you would refuse my offer.
In this extreme (and very silly) example, Helicopter Money would not be impossible; instead it would be extremely powerful. If I had made the more sensible assumption that the demand for currency is proportional to nominal income, then doubling the supply of currency by Helicopter Money would double the equilibrium level of NGDP. But with my silly assumption the multiplier is infinite. The Helicopter Money is a Hot Potato that never cools down no matter how much the price level or real income rise.
Now let me assume that the central bank also offers savings accounts, as well as issuing currency. Savings accounts may pay interest (at a rate set by the central bank), but cannot be used as a medium of exchange.
Start in equilibrium where the stock of currency is exactly $100 per person. What happens if the central bank prints more currency and drops it out of a helicopter, holding constant the nominal rate of interest it pays on savings accounts?
I know what you are thinking. I know how most economists would be thinking. (At least, I think I do.) "Aha! This time it's different! Because now people can get rid of the excess currency, by depositing it in their savings accounts at the central bank, so Helicopter Money won't work." You are implicitly invoking the Law of Reflux to say that an excess supply of money must return to the bank that issued that money.
And you are thinking wrong. You are making exactly the same fallacy of composition as you would have been making if you said that people would leave the excess currency lying on the sidewalk. "People in aggregate can only get rid of the excess currency by depositing it in their savings accounts (or throwing it away) therefore each individual will get rid of his excess currency by depositing it in his savings account (since it's better than throwing it away)."
There are 1,001 different ways an individual can get rid of excess currency, and depositing it in his savings account is only one of those 1,001 ways. Why should an individual care if depositing it in his savings account is the only way that works for the aggregate? (If people always thought like that, littering would never be a problem.) And if individuals do spend any portion of their excess currency, so that NGDP rises, and is expected to keep in rising, then the (assumed fixed) nominal interest rate offered on savings accounts at the central bank will start to look less attractive, and people will actually withdraw money from their savings accounts. Not because they want to hold extra currency, but because they plan to spend it.
Money is weird like that. Money is not like marriage "to have and to hold". We want to get our hands on the intrinsically worthless medium of exchange only because we plan to get rid of it again. If we could never get rid of it, we wouldn't want to have it, even for free.
Now let me bring commercial banks into the story. The central bank issues currency, and also offers accounts at which central banks can keep "reserves". People use both central bank currency and commercial bank chequing accounts as their media of exchange; commercial banks use their reserve accounts at the central bank as the medium of exchange they use for transactions between themselves. And the central bank allows commercial banks to swap currency for reserves in either direction, and reserves pay a nominal rate of interest set by the central bank.
My story now (as best as I can tell) matches the (implicit) model in "Helicopter Money: the Illusion of a Free Lunch" by Claudio Borio, Piti Disyatat, and Anna Zabai. (HT Giles Wilkes.) They argue that Helicopter Money will be unwanted and must Reflux to the central bank to be held as central bank reserves, where those reserves pay interest and so are just like (very short-term) government bonds, or savings accounts at the central bank. Their argument rests on a fallacy of composition. Individuals in aggregate can only get rid of unwanted currency that way, but this does not mean that individuals will choose to get rid of unwanted currency that way.
We know, empirically, that helicopter money (in moderation of course) does not lead to bizarre consequences. Helicopter money is perfectly normal; central banks do it (almost) all the time. They print currency, the stock of currency grows over time, and since that currency pays no interest this is a profitable business for central banks and the governments that own them. And the governments that own central banks sooner or later spend those profits, by having transfer payments higher, or taxes lower, than they otherwise would be for the same level of government expenditure. Aside from the political question of the government deciding who the helicopter flies over, and when it flies, it is exactly as if central banks were doing helicopter money as a normal part of their operations.
The demand for central bank currency seems to rise roughly in proportion to NGDP (the US is maybe an exception, since much is held abroad), so countries with rising NGDP are normally doing helicopter money. And doing helicopter money, just once, does not empirically lead to central banks being forced to set nominal interest rates at zero forever. And it would be utterly bizarre if it did; what else are governments supposed to do with the profits central banks earn from printing paper currency?
The lesson we learn from all this is that the Law of Reflux will prevent Helicopter Money from working only if the central bank refuses to let NGDP rise at the same time. Which is like saying that pressing down on the gas pedal won't work if you press the brake pedal down hard enough so the car can't accelerate.