For Tyler Cowen. I think I see his point, and I've been trying to get my own head around this question.

1. Suppose I live in a world where the central bank targets the price level, as measured by the GDP deflator. And suppose I believe the central bank should target Nominal GDP instead. Given my preferred target, if NGDP fell below target I would say that money is "too tight"; and if NGDP rose above target I would say that money is "too loose". And suppose in 2016 the central bank hits its price level target, but RGDP falls. By definition, NGDP must fall too, since the GDP deflator is defined as: P=NGDP/RGDP.

If I blamed the fall in RGDP on tight money causing NGDP to fall, or on tight money allowing NGDP to fall, would I be making a tautologous statement?

No. I would be making a counterfactual conditional statement. I would be saying: "If the central bank had instead been targeting NGDP, and had prevented NGDP falling below target, the fall in RGDP would not have happened".

2. Now let us imagine a parallel universe where the central bank targets NGDP, and suppose I want the central bank to target the price level, as measured by the GDP deflator. Given my preferred target, if P fell below target I would say that money is "too tight"; and if P rose above target I would say that money is "too loose". And suppose in 2016 in that parallel universe, the central bank hits its NGDP target, but RGDP falls. By definition the price level must rise.

If I blamed the fall in RGDP on loose money causing P to rise, or on loose money allowing P to rise, would I be making a tautologous statement?

No. I would be making a counterfactual conditional statement. I would be saying: "If the central bank had instead been targeting P, and had prevented P rising above target, the fall in RGDP would not have happened".

3. In both cases we have the same accounting identity: RGDP=NGDP/P. Someone who knew nothing whatsoever about economics, except for that accounting identity, would find both claims equally plausible, or equally implausible. To keep the ratio constant, where we can target either the numerator or the denominator but not both, should we try to keep the numerator constant, or try to keep the denominator constant? Actually, if the only thing you knew is that one variable is the ratio of two other variables, and that we can target numerator or denominator but not both, both claims would look utterly implausible.

For someone who knew no economics, and who knew only that RGDP=NGDP/P, it would be a divine coincidence if stabilising P were consistent with stabilising RGDP. And it would be an equally divine coincidence if stabilising NGDP were consistent with stabilising RGDP. And it would be a double divine coincidence that both those divine coincidences were true: so that stabilising either P or NGDP would be consistent with stabilising RGDP.

4. Now what is really weird is that the real world did in fact seem to be a place where double divine coincidence were true. Until it wasn't. An outside observer, who did not know that central banks were targeting inflation (OK, it was CPI not GDP deflator inflation), and who mistakenly thought they were targeting NGDP instead, would infer from the data that stabilising NGDP did indeed seem to be consistent with stabilising RGDP, and that the NGDP version of divine coincidence were true. Furthermore, that outside observer would see no reason to change his mind since the recent recession. It is the P version of divine coincidence that has failed empirically, when we look around the world. The NGDP version of divine coincidence is hanging in there.

[P.S. If it weren't for the accursed Lucas Critique, and the fact that central banks cannot be assumed to be able to hit their targets perfectly, I think it would be easy to test which version of divine coincidence is better. Find which of NPGD and P has the higher variance, and target it.]