Blanchard analyzed US unemployment and inflation with a model including time varying parameters and concluded two things. First it appears that inflation expectations are anchored. What this really means is that recent inflation has a small effect on current inflation (Blanchard and Blanchard et al don't attempt to directly measure expectations). Second, the slope of the Phillips curve has declined with large changes in unemployment followed by small changes in inflation. Blanchard stressed that the computer is convinced that the slope declined in the early 90s and that this is not a new great-recession pattern.
I will present some very simple graphs assuming expectations are completely anchored. That is, I will do what old Keynesians are often (incorrectly) accused of doing and ignore fluctuations in expected inflation entirely. The point, as noted by Bunker following Ekaterina V. Peneva and Jeremy B. Rudd, is that the change from the 60s to around now is a reduction of the pass through of labor costs to price inflation. Like Blanchard and BCS, they use a sophisticated time varying parameter model, but the point is simple -- in recent decades wage and price inflation have not moved together.
I think it is worth a blog post to check whether the relationship between unemployment and the increase of nominal labor costs (roughly wage inflation) has changed too. My impression is that it hasn't. I look at two series from FRED
HCOMPBS Business Sector: Compensation Per Hour, Index 2009=100, Quarterly, Seasonally Adjusted
and UNRATE "Civilian Unemployment Rate (UNRATE), Percent, Quarterly, Seasonally Adjusted" with 1950s econometrics, that is scatter plots. awinf is the % rate of increase of HCOMBS over 4 quarters (so the points on the scatter are not independent observations. I pool data from before 1973q1 and after 1985q1, that is back in the good old days and after the Volcker deflation. A 0 next to the dot means data from after 1985q1 and a 1 means data from before 1973q1.
Here is another scatter using only data from after 1953 so 1953-1972 and then 1985-2015.
After 1985 the (still statistically signficant) correlation was much reduced