Regarding, again, the size of the output gap, this remark is found in the most recent Fed minutes:
However, a couple of participants noted that the rate of inflation over the past year had not fallen as much as would be expected if the gap in resource utilization were large, suggesting that the level of potential output was lower than some current estimates.
I think this has less to do with the size of the output gap and more to do with downward nominal wage rigidities. Note that wages are still rising, although the pace of wage growth for production and nonsupervisory workers is still falling:
Perhaps a better example is the relatively new series, wages for all workers:
Overall private wage growth bottomed out in 2009 and held around 1.75%, perhaps just beginning to rise in recent months.
Despite very high unemployment and underemployment, wage growth is still positive. It tends to be very difficult to induce workers to take wages cuts (think also how the newly unemployed will resist taking new jobs with a substantially lower pay), which in-turn helps put a downside to inflation. In other words, one would expect the relationship between the output gap (or, similarly, high unemployment) and inflation to flatten as inflation rates fall toward zero.
Also note that rising wages doesn’t necessarily imply higher inflation. Between the two is productivity growth. To account for the latter, we can look at unit labor costs:
Not exactly a lot of inflationary pressures stemming from unit labor cost growth. Presumably, high real wages could come by redistributing productivity gains to workers in the context of low inflation. For that to happen, however, I think we will need a lot more upward pressure on the labor market than we are seeing right now.