Update: Selgin points out in correspondence and Sumner points out in comments below that, the below discussion is implicitly using variables in per capita terms.
This post continues the discussion from Scott Sumner's thoughtful reply to my critique of NGDP targeting from 2015.
I. The benefit of NGDP targeting is that inflation can fluctuate in the short run. But can NGDP targeting achieve a long-run optimal inflation rate?
Targeting NGDP rather than targeting inflation allows inflation to fluctuate in the short run. This is the major benefit of NGDP targeting, since it makes sense to have higher inflation in the short run when there is a cyclical growth slowdown and lower inflation when there is a growth boom, (see Selgin, Sumner, Sheedy, myself).
This is an argument about the short or medium run, at the frequency of business cycles (say 2-5 years).
Separately, you could imagine – whether or not inflation is allowed to vary in the short run, as it would be under NGDP targeting – that there is a long-run rate of inflation which is optimal. That is, is there a "best" inflation rate at which the economy should ideally settle, at a 10+ year horizon?
If there is an optimal long-run inflation rate, you would hope that this could be achieved under NGDP targeting in the long-run, even while inflation is allowed to fluctuate in the short run.
II. The optimal long-run inflation rate
Economists have thought a lot about the question of what the long-run optimal inflation rate is. There are two competing answers :
1. No inflation: One strand of literature argues that the optimal long-run inflation rate is precisely zero, based on price stickiness. The argument goes: by keeping the price level stable, sticky prices cannot distort relative prices.
2. Friedman rule: Alternatively, another strand of the literature going back to Milton Friedman argues that the optimal inflation rate is the negative of the short-term risk-free real interest rate (i.e. slight deflation). The argument here is that this would set the nominal risk-free interest rate to zero. In this world, there would be no opportunity cost to holding money, since both cash and risk-free bonds would pay zero interest, and the economy could be flush with liquidity and the optimum quantity of money achieved.
These two schools of thought clearly contradict each other. We will consider each separately.
What we want to know is this: could NGDP targeting achieve the optimal inflation rate in the long run (even while allowing beneficial short-run fluctuations in inflation)?
III. NGDP targeting and zero long-run inflation
In a previous essays post, I critiqued NGDP targeting by pointing out that NGDP targeting could not achieve zero inflation in long-run, unless the central bank could discretionarily change the NGDP target. In other words, I was arguing based on the first strand of literature that NGDP targeting was deficient in this respect.
The accounting is simple: NGDP growth = real growth + inflation. Under NGDP targeting without discretion, the growth rate of NGDP is fixed. But, real growth varies in the long run due to changing productivity growth – for example, real growth was higher in the 1960s than it has been in recent decades. As a result, the long-run inflation rate must vary and thus is unanchored.
Zero inflation can be achieved in the long run, but only at the cost of trusting the central bank to act discretionarily and appropriately modify the long-run NGDP target.
I think that such discretion would be problematic, for reasons I outline in the original post. I'll note, however, that I (now) assess that the benefits of NGDP targeting in preventing short-run recessions outweigh this smaller long-run cost.
IV. NGDP targeting and the Friedman rule
On the other hand – and I haven't seen this result discussed elsewhere before – NGDP targeting can achieve the Friedman rule for the optimal inflation rate in the long run without discretion. That is, under the logic of the second strand of literature, NGDP targeting can achieve the optimum. Here's the accounting logic:
The Friedman rule prescribes that the optimal inflation rate, pi*, be set equal to the negative of the real interest rate r so that the nominal interest rate is zero:
pi* = -r
Here's the kicker: Under a wide class of models (with log utility), the long-run real interest rate equals the rate of technological progress g plus the rate of time preference b. See Baker, DeLong, and Krugman (2005) for a nice overview. As a result, the optimal inflation rate under the Friedman rule can be written:
pi* = -r = -(b+g)
This can be achieved under NGDP targeting without discretion! Here's how.
Suppose that the central bank targets a nominal GDP growth rate of -b, that is, an NGDP path that declines at the rate of time preference. Recall again, under NGDP targeting, NGDP growth = g + pi. Since the central bank is targeting an NGDP growth rate of -b, if we rearrange to solve for inflation, we get that
pi = NGDP growth - g = -b - g
That's the optimal inflation rate implied by the Friedman rule shown above. This result holds even if the long-run rate of productivity growth (g) changes.
Thus, we have shown that if the central bank targets an NGDP path that declines at the rate of time preference, then in the long run the Friedman rule will be achieved.
To summarize, under such a regime, the economy would get the short-run benefits of flexible inflation for which NGDP targeting is rightfully acclaimed; while still achieving the optimal long-run inflation rate.
This is a novel point in support of NGDP targeting, albeit a very specific version of NGDP targeting: an NGDP target of negative the rate of time preference.
V. Summing up
There's still the tricky problem that economists can't even agree on whether the Friedman rule or no-inflation is superior.
So, to sum up once more:
To close this out, I'll note that an alternative middle ground exists... an NGDP target of 0%. This would see a long-run inflation rate of -g: not as low as -g-b as prescribed by the Friedman rule; but not as high as 0% as prescribed by no-inflationistas.
Such a policy is also known as a "productivity norm," (since long-run inflation is negative of productivity growth), advocated prominently by George Selgin (1997).
 I ignore ZLB considerations, which typically imply a higher optimal inflation rate, since many advocates of NGDP targeting do not see the ZLB as a true policy constraint (myself included).