Here is Nick’s reply:
OK, I think I see what you are saying now.
Yes, when people talk about “NAIRU” they implicitly assume a dynamic model. And the dynamic equation is the expectations equation. Because when the term “NAIRU” was invented, the economists who invented it thought in terms of some sort of Adaptive Expectations hypothesis, where people revise their expectations over time based on their past mistakes. Which is a dynamic equation.
And when I get pedantic, it is precisely for that reason that I prefer to talk about the “natural” rate of unemployment, rather than the “NAIRU”, because “NAIRU” presupposes not just adaptive expectations, but a very particular form of adaptive expectations. Let me explain:
Adaptive expectations on X means Xe(t)=Xe(t-1) +c(X(t-1)-Xe(t-1)) where 0<c<1.
But what is “X” in this context? Is X the price level? Or the rate of inflation? Or the rate of change of the rate of inflation (the second time derivative of the price level)? Or the third derivative? Or what? And whether inflation does or does not accelerate if and only if you hold U<U* depends on which of those 3 versions of adaptive expectations you assume. “NAIRU” only makes sense as a name for U* if you assume the second version.
(Though “natural”, as a name, has its own problems too, because it connotes “good”, and “immutable”, etc.)