A client very helpfully pointed out this research paper from the NY Fed and suggested it is in line with my 2 Stage Disinflation hypothesis. There is an intellectual (for lack of a better word) sense in which that is untrue, but then a practical sense in which it seems very true. And practical usually matters more.
I could not help noticing that I do not accept the main premise of the paper and associated modeling work. Like most research from the conventional school (no disparagement implied), it views the spike of inflation after the Covid shock as arising from supply shocks and labor market conditions. And its contribution is to present a couple novel measures of how the labor market tightened after 2021.
Specifically, the natural rate of unemployment rose because of an increase of matching frictions, which I hasten to add is not measured via vacancies in the paper (and because of a decline in the willingness to work, which is corroborated by but not estimated with an observed increase of reservation wages. The rise of what the authors call “underlying” inflation (as opposed to the inflation component caused by supply shocks), reflected a rise in the natural rate of unemployment, a decline in the actual unemployment rate, and an expectation among wage setters that it would take policy makers time to reduce the gap between the actual and natural unemployment rates. There seems to have been a tendency in modeling recently to measure labor market tightness with both the current unemployment rate and its expected path, because wages are sticky and wage setters look forward.)
Where I am skeptical (as opposed to confident I am right) is on the basic assumption that the recent (and ongoing) inflation episode needs to be seen exclusively through the conventional model. I suspect that there was a demand component of the inflation spike that did not largely even involve the labor market. Fiscal policy poured in an excess of nominal aggregate demand, but even desired output and therefore even the ex ante demand for labor responded sluggishly. So, most of the excess demand went into price without even involving the labor market. However, the authors of the paper, quite typically, do not even entertain that as a possibility. They think the only questions are the timing and effects of the supply shocks and then the correct way to model how labor market conditions (taken as the exclusive influence) affect underlying inflation. Maybe they and every other conventional economist, seemingly, is right to think of it in those terms. My contention is that it is possible they are not.
But as a practical matter, they draw conclusions that are consistent with the possibilities I draw from the 2 Stage Disinflation view. Indeed, those conclusions are presented in much stronger terms. They describe the labor market as “very” tight because the unemployment rate has fallen, and the natural rate of unemployment has risen. And they interpret the recent pace of wage inflation as strongly corroborating that take, just as I do, although more humbly.
The main source of the decline of even underlying inflation in recent months can be attributed to expectations that the Fed will deliver a higher unemployment rate, even (I presume) as the natural rate reverts lower. But if those expectations were to be disappointed, then underlying inflation would hook higher, according to their model.
Accordingly, the so-called “last mile” (a term they use approvingly) of disinflation will be tough, in their view. Great timing of the paper, incidentally, just as the client pointed out to me. First comes the policy, and then the research! Kidding.

In the top left panel, the natural unemployment rate is shown in black and its secular trend is shown in red. The blue dashed line is the actual unemployment rate. The bottom left panel depicts how tight the labor market is. That tightness along with inflation expectations largely determines underlying inflation, within the model. Supply shocks affect the gap between actual inflation and underlying.
So should we buy this. Weighing in on the theoretical and econometric issues involved here is far beyond my competence. I do think it is helpful to consider arguments involving the idea that recent wage disinflation is not necessarily evidence that the labor market has eased much. This paper is consistent with that take, and newly so. A consideration of this possibility seems to me to have been a while coming. And bless them for specifically rejecting the unemployment vacancy gap, despite it having worked once, post hoc, as they directly point out.