More and more central banks are facing a situation in which the output gap they are looking at looks close to zero, but inflation is below their target. This is arguably the case for Japan, Sweden and the US, for example. Even the eurozone is getting close to this situation. Sometimes journalists discuss a zero output gap combined with too-low inflation as if such a situation were strange, but a range of different macroeconomic theories all have the property that a zero output gap is consistent with any constant inflation rate. (This is an aspect of "monetary superneutrality.")

Older "sacrifice ratio" models predict that a positive output gap will tend to make inflation gradually rise, while a negative output gap will tend to make inflation gradually fall. Newer models that have a simple Calvo pricing mechanism predict that that an increase in expected future output gaps will make inflation jump up, while a current positive output gap is associated with inflation gradually falling, with the direction reversed for negative output gaps.

I think recent history is easiest to understand by thinking in terms of one of the older "sacrifice ratio" models combined with the idea that the output gap has some immediate effect on inflation has some immediate effect on inflation. I don't have a good microfoundation, but suppose that

gap = log(GDP) - log(natural GDP)

d inflation/dt = a d(gap)/dt + b gap

where both a and b are positive. With this equation, if the output gap is constant at zero, inflation remains unchanged. So the economy can be completely recovered in terms of the output gap without any tendency at all for inflation to go back to target. The term "b gap" that relates the level of the gap to the rate of change of inflation would have dragged down inflation during persistent negative output gaps during the Great Recession and the slow recovery from the great recession. To bring inflation back up to target would require a period of time with a positive output gap. 

Since positive output gaps are pleasant given all the distortions that make the natural level of output lower than the level at which (other than the effects on inflation) the marginal benefit of output equals the marginal cost, why don't central banks shoot for a positive output gap until inflation returns to the target of 2% per year? I think the answer is:

1. Central bankers are not used to trying to have positive output gaps. The low unemployment rates associated with positive output gaps seem dangerous.
2. Central bankers don't really like having an inflation target above zero. They feel in their bones that their job is to keep inflation down, close to zero, not to push it up. 

When there is a clear need to raise inflation (say in a situation where inflation is negative), I don't have much sympathy for an aversion to temporarily positive output gaps. But I have a lot of sympathy for the idea that a 2% inflation target is too high. As I argued in "The Costs and Benefits of Repealing the Zero Lower Bound...and Then Lowering the Long-Run Inflation Target," in the absence of any lower bound on interest rates, the inflation target can be lower than if there is a lower bound on interest rates. 

Any central bank that is genuinely committed to eliminating lower bounds on interest rates should reevaluate its inflation target accordingly. Even a willingness to use mildly negative rates should bring down the appropriate inflation target. Both the Bank of Canada and Ben Bernanke have taken the position that planning to use negative interest rates when necessary is better than raising the inflation target above 2%. (See "Bank of Canada would consider setting interest rate below zero: Poloz," "Renewing Canada’s Inflation-Control Agreement," and "Ben Bernanke: Negative Interest Rates are Better than a Higher Inflation Target.") To the extent that position envisions only mildly negative rates, a commitment to use deep negative rates when called for might well reduce the optimal inflation target below 2%. If the optimal inflation target fell, say, as far as 1.5%, the current concern for raising the inflation rate—a goal central bankers feel half-hearted about—would go away. 

If monetary policy were being designed from scratch, I doubt that central bankers would design in a lower bound on interest rates. The tradition of a lower bound on interest rates is simply a bad historical carryover, much like the gold standard was. (Good riddance!) At the moment, that tradition of a lower bound on interest rates is pushing inflation higher than it would otherwise have to be. Or maybe the tradition of a lower bound on interest rates is simply making central bankers who can't bear to have a higher inflation rate feel guilty about not being willing to raise inflation, without actually getting them to raise inflation. 

Further Reading: "How and Why to Eliminate the Zero Lower Bound: A Reader’s Guide."