The Divine Coincidence is the idea that by controlling inflation we also bring the output gap to zero, so we do not need separate targets for both. I talked about this in a recent post, but in writing that I realised I could make my point in another (and perhaps more effective) way. The Divine Coincidence essentially works both ways. So imagine a parallel universe where the monetary authority targeted the output gap, and not inflation.   The authority said that by targeting the output gap they also controlled inflation.
Now you may make a practical objection at this point, which is that it is obvious to target inflation rather than the output gap because the latter is so difficult to measure. I think that argument becomes weak once we recognise that by targeting inflation, we are in fact trying to reduce the costs of inflation, and the published inflation rate may be a poor indicator of these costs. In a Woodford type framework, for example, inflation is costly because prices are sticky, so we should focus on those goods (and labour) where prices are sticky. This is one rationale for looking at core inflation, but core inflation is hardly a perfect measure of sticky prices. Arguably our proxies for the output gap, like unemployment, are at least as good at capturing the true costs of a non-zero output gap.
In this parallel universe they too had a Great Recession, and (being parallel and all) their recovery was of a very similar shape to ours. How would the output gap targeting monetary authority in this parallel universe perceive its performance? The story would be one of complete failure. After six years of trying, the output gap had still not been closed. A huge amount of resources had been wasted as a result. In fact, I suspect by now the monetary authority would have said quite explicitly that it just did not have the tools to do its job any more. Furthermore, they probably would have made it clear that one reason they did not have the tools (i.e. why interest rates were still stuck at the Zero Lower Bound) was because of fiscal austerity. If they did not try and blame someone else, they would look utterly incompetent.
I do not think, in our inflation targeting world, the monetary authorities have this view. Instead, based on the limited information I have, and at least outside the Eurozone, they believe performance over the last six years has not been too bad. Inflation was a bit high around 2011, and is maybe a bit low now, but nothing too serious. Yet the data they are looking at is exactly the same as the data in the parallel universe. The only difference is that they are targeting inflation, while in the parallel universe the focus is on the output gap. And by the Divine Coincidence, this difference should not matter!
My parallel universe idea illustrates two points. First, over the last six years, the Divine Coincidence has been distinctly unholy. Second, as a result it is terribly misleading to focus on inflation (and consumer price inflation in particular) rather than the output gap. I suspect in thirty years students will look back on this period with the same disbelief that we look back on the 1930s. How could they have allowed the recession to continue for so long, they will ask, when they had the tools to do much better? Part of the answer will be inflation targeting.
 It’s not quite that simple, because with either a traditional or New Keynesian Phillips curve, getting inflation to target ensures a zero output gap, but a zero output gap alone does not ensure hitting the inflation target. However I do not think this point is crucial here.
 Because of the dual mandate, the US Fed could in principle be in either universe. In practice they seem to focus on inflation.