I am happy to see Paul Krugman address the question. He writes:
So why not forget about open-market operations, and just drop the stuff
from helicopters? Well, remember that at this point cash and short-term
bonds are equivalent. So a helicopter drop is just like a temporary
lump-sum tax cut. And we would expect people to save much or most of
such a tax cut – all of it, if you believe in full Ricardian
I hold a different opinion for two reasons. First, cash and short-term bonds may be near-substitutes but they are not literally, strictly equivalent. The nominal rate on T-bills is not exactly zero and furthermore you can't use a T-bill for every retail purchase. The demand curve for real cash need slope down only slightly for a quantity theory result to hold. After everyone spends the new cash balances, and prices rise, people end up with the quantity of real balances which they initially desired. These equilibria have "knife-edge" properties, where "identical to T-Bills" and "nearly identical to T-Bills" do not bring the same results. Tsiang showed this in a very good JMCB article on Friedman's optimum quantity of money, in the early 1970s and you might regard it as implicit in Bewley's Econometrica article on Friedman.
Second, after a helicopter drop no one need expect future taxes to be raised to retire the money (although maybe a sufficiently credible government could create such an expectation). So there is no Ricardian motive to save the new cash, as Brad DeLong points out. Indeed, if you think there is some chance that others will spend the money, raising the price level, you will want to spend your new cash soon, so as to preserve its value against forthcoming price inflation. The resulting game-theoretic equilibrium, applying dominant strategies, again leads to higher prices, higher aggregate demand, and the desired quantity of real cash balances held.
Those are not the only possible cases (see the work of Fischer Black) but I take them to be the most sensible default cases. Both indicate that a helicopter drop of cash will work fine in boosting aggregate demand.
The most likely scenario for no positive AD effect is simply that the helicopter drop is so small that no one expects a price level rise and thus no one expects an inflationary tax on the new cash, people (for bounded rationality reasons) treat the new cash as a transfer purely to themselves, the precautionary motive for saving is strong, and so the new money is simply held. A larger helicopter drop should overcome that inertia, if need be.
Maybe these arguments are incorrect but they date from a consensus established in the mid- to late 1960s and early 1970s, much of it springing from Patinkin's book on money and the subsequent discussions thereof. Krugman suggests this perspective is wrong, but he hasn't yet given me — or others — a reason to budge from it.