When both headline and core inflation rose above target after the financial crisis, helped by rising oil prices, the Fed and Bank of England kept their nerve and did not raise interest rates. They saw through what was a temporary episode. The ECB’s judgement was not as good. Even in the UK it was close, with three out of nine MPC members voting for a rate increase for a few months. But it was the outcome that mattered - excess inflation was ignored because it was temporary. To what extent is what we are seeing right now just the mirror image of this period?
In terms of where inflation is and the monetary policy response, the situation today does indeed look like a mirror image. Headline inflation is or is about to be negative, and core inflation has fallen below target. As Tim Duy points out for the US, core inflation seems to be heading lower rather than returning to target. However I think that is where the symmetry ends. While the dangers of letting inflation rise above target because of temporary shocks are small, the dangers in the opposite direction are more serious.
One of the arguments used by the inflation hawks when oil prices were high is that even if the impact of higher oil prices on inflation was itself temporary, there was a danger that inflation expectations would increase, and the central bank would lose its anti-inflation credibility. My response at the time was three-fold: first, the private sector can see the reason that rates are not being raised (the continuing recession), so credibility should not be in danger; second, the best indication that the expectations that matter have shifted is when nominal wage inflation starts to pick up (which it did not), and third when that happens it will be easy to restore credibility and reduce expectations by raising rates. 
None of these arguments apply with deflation today. Then unemployment was clearly too high. Today unemployment is not clearly too low. How far we are from the natural rate is unclear, but no one would argue that we are in a boom that is the mirror image of the recession a few years ago. The second argument is that we could use changes in nominal wages as a clear indicator that the inflation expectations that mattered had shifted. That argument is not symmetrical because of the well known resistance to nominal wage cuts. Finally if credibility does seem about to be lost, the central bank will find it very difficult to take action to restore it because of the Zero Lower Bound (ZLB).
Please allow me to get technical for just one paragraph, which can be safely skipped. As has often been pointed out, the ZLB means that there are two steady states in the economy associated with a given real interest rate: the ‘intended’ equilibrium with target inflation, and the ‘ZLB equilibrium’ when inflation is negative. I recently discussed a paper that treated agents views about which equilibrium was appropriate as a ‘belief’ and that perhaps the liquidity trap could be a manifestation that agents believed we were heading for the ZLB steady state. The controversial aspect of this analysis is the suggestion that this belief could be shifted by the monetary authorities raising rates. I find that very unconvincing, but it would be a mistake to dismiss the exercise completely on that account. In that post I did suggest an alternative rationalisation for why we might be heading for the ZLB equilibrium: agents no longer believed that the monetary authority had the means to stop it happening.
This last asymmetry is the one that troubles me the most, and why I am not as relaxed as monetary policy makers appear to be about deflation. There are three interpretations of this relaxed attitude to negative headline inflation. The first is the one I suspect monetary policy makers actually hold, which is that the beneficial impact of lower prices on demand will with a year or so push inflation back to target, so there is no reason for concern.  I think the probability is that they are correct, but good policy does not just think about the most likely outcome, but should also be robust to risks, particularly risks with large consequences.
The second interpretation that the private sector could give for the relaxed attitude by central banks in the US and UK is that deviations above and below 2% are not treated symmetrically. In theory this should be more of a concern in the US than the UK, because in the UK asymmetry is against the central bank’s mandate. However I’m not sure the private sector thinks that is as important as MPC members do. There is one obvious additional asymmetry between now and a few years ago: many of those calling for higher rates back then are still pushing for higher rates today.
The third interpretation about why central banks are doing nothing is there is nothing they can do. Quantitative Easing seems to have come to a permanent halt either because it has stopped having a useful effect, or because policy makers fear it is having undesirable consequences. Under this interpretation the inflation target loses credibility not because the private sector no longer believes policy makers’ stated objectives, but because they no longer believe they have the means to achieve them.
This possibility is the one that should really be worrying central banks right now. It is a scenario that is quite consistent with what is currently happening, and it puts at risk central bank credibility in a most fundamental way. Quite simply, central bank credibility is destroyed because people believe they have lost the ability (rather than the will) to do their job, and there is very little central banks can do to get it back because of the ZLB. This is what should be giving central banks nightmares. Strangely, however, they seem to be sleeping just fine.
 A footnote for macroeconomists: this is why I have never been convinced by Cochrane’s worries about using the inflation target as a transversality condition.
 Resistance to nominal wage cuts may also dampen any the deflationary path, giving time for these positive effects to come through. However resistance to nominal wage cuts does not mean the ZLB equilibrium will never occur – in the paper I discuss in this post, it is the reason why that equilibrium is associated with high unemployment.