"Next, conduct a thought experiment. What happens if there is an increase in the aggregate stock of liquid assets, say because the Treasury issues more debt? This will in general reduce liquidity premia on all assets, including money and short term debt. But we're in a liquidity trap, and the rates of return on money and short-term government debt are both minus the rate of inflation. Since the liquidity payoffs on money and short-term government debt have gone down, in order to induce asset-holders to hold the money and the short-term government debt, the rates of return on money and short-term government debt must go up. That is, the inflation rate must go down. Going in the other direction, a reduction in the aggregate stock of liquid assets makes the inflation rate go up."
Start in equilibrium. Suppose the helicopters drop more money on the population. People do not wish to hold more money. Because there are diminishing marginal benefits to liquidity, and those marginal benefits are now below the opportunity cost of holding money. So people will try to spend their excess supply of money. This creates an excess demand for goods. This causes the price level to rise. In the new equilibrium, the real value of the total stock of money is the same as it was in the old equilibrium. If the money supply doubles, the equilibrium price level doubles too. And so the marginal benefits of holding money are the same as they were in the old equilibrium.
Update: Bob Murphy has a fine reductio ad absurdum (assume zero real income growth to make Bob's argument more clear):
"...suppose we weren’t talking about a sudden surprise QE announcement. Instead, imagine we are in the midst of a Friedmanite rule where the quantity of money grows at a constant rate–let’s say it’s 2%–year after year, regardless of circumstances. Further suppose this rule has been in place for 10 years, and the public is convinced it will continue to be the rule for as far as the mind can forecast.
Thus, there is no question that we are bouncing from one equilibrium to another. We are in the long-run equilibrium (in terms of any standard way you are going to model such a situation). Now, within this long-run equilibrium path, Williamson could still make his claim: Each year, the rate of inflation has to perpetually fall, because each year you have to convince the public to hold more cash than they held the previous year. Thus, for any positive growth in the quantity of money, the rate of price inflation must constantly fall."
Update2: which explains why Zimbabwe had hyperdeflation.
Can somebody resurrect David Hume?