Rip van Winkle is put in charge of a New Keynesian central bank. He sets a nominal interest rate that he believes will keep inflation at 0% and the price level constant. (It doesn't matter if I change this to rising at 2% per year.) Then he falls asleep for 70 years, and that nominal interest rate cannot be changed until he awakes. This is clearly not good, because the nominal interest rate cannot be adjusted in response to any shocks that hit the economy over the next 70 years.
But before falling asleep, he has a choice of posting one of two announcements of what he will do when he wakes up:
1 Price level target. He will do what is needed to bring the price level back to what it was when he fell asleep, and keep it there.
2 Inflation target. He will do what is needed to keep the price level constant at whatever new level it is at when he wakes up.
Which is best?
What is the worst that could happen with a price level target? Suppose the natural rate of interest rises by 1% just after he falls asleep, and stays there. If prices are perfectly flexible the equilibrium price level will double, then fall slowly at 1% per year, returning to its original level just as he wakes up. (The 1% deflation means the real interest rate is the same 1% higher as the natural rate of interest.) With sticky prices it won't rise quite as much as that, and will fall a little more slowly.
What is the worst that could happen with an inflation target, if we hit the economy with the same shock? God only knows. If prices are perfectly flexible then anything can happen. The price level doubling then falling at 1% per year is one equilibrium. The price level quadrupling then falling at the same 1% per year is another equilibrium. The price level halving then falling at 1% per year is another equilibrium. And so on. And it could switch from one equilibrium to another just on a whim. What happens depends on what people expect to happen. And there are lots of things they could rationally expect to happen. And if prices are sticky? Well, it's going to depend on the exact nature of that stickiness. As well as on what people expect to happen. I can't answer that question with any confidence at all. I don't think any macroeconomist can.
In both cases we know the inflation rate will be 0% after Rip wakes up. But with a price level target we also know what the price level will be after Rip wakes up. That helps me figure it out. That also helps the people in my story figure it out.
Now you might think that's a daft thought-experiment. Because central banks don't fall asleep for 70 years. But central banks do not and cannot respond immediately to shocks that hit the economy. They don't observe changes in the natural rate of interest (or anything else) in real time. There are always lags between something happening, the bank figuring out what has happened (is it a temporary or permanent shock?), and responding to what has happened, and the economy responding to the bank's response. Sure, it's less than 70 years. But it is exactly as if the central bank fell asleep.
Would your answer change if Rip van Winkle fell asleep for only 10 years, or 1 year, or less? Why?