New research from the IMF suggests that the negative correlation (inverse relationship) between unemployment and inflation no longer exists (since 1995). The main reason is that central banks seem to have become more skilled at targeting inflation and managing inflationary expectations.
If this is the case, then maybe the long-run Phillips curve is horizontal (not vertical!) at the targeted rate of inflation. It would also imply that governments could increase aggregate demand and reduce unempolyment with no increase in the price level. However, the issue is whether the unemployment is cyclical or “voluntary “(mainly structural where there is a skills mismatch between the unemployed and the job vacancies on offer). If the unemployment is cyclical (there is spare capacity) then the AS curve could be effectively horizontal and an expansionary fiscal or monetary policy would not raise prices. But if the unemployment is mainly structural (or seasonal/frictional/residual) then higher AD would be inflationary.
This video is a very simple review of the differences between Keynesian and (New) Classical Economics:
and this one explains the Phillips curve from the beginning, but notice that it makes no reference to the possibility of a long-run Phillips curve:
However PAJ Holden does develop the theoretical concept: (but do not worry about the NAIRU…the non-accelerating inflation rate of unemployment; you do not need to know about it)