What I've read recently though, is that workers' expectations of inflation also play a big role in inflation. [1] The idea is that low unemployment means workers are confident and have leverage to demand pay raises that meet their inflation expectations. We may have broken inflation expectations. The Brazilians did an amazing psychological hack with the creation of the Real to fix their inflation expectations to get their inflation in check. (Though, they had the opposite problem, workers expecting very high inflation.)

[0] https://www.investopedia.com/articles/investing/080415/true-...

[1] https://www.economist.com/schools-brief/2020/08/22/why-does-...

The most common number is the U3, which measures all the people who want to get back to work. That number is a good way to track all the labor that could easily be pulled into the market if the right jobs were available. Shorter term shocks might move this number around, as people get laid off and find new employment.

There is also the U6 rate, which includes a lot of people that have fallen out of the market, and those who have taken up part time work because they couldn’t find full time work.

You have to look at both to get a good picture of the employment situation. There are a lot of cases where the U3 might drop, a good thing in normal times, because the economy is so bad that a lot of workers have either given up on finding work, or have taken up part time work since they can’t find a full time job. Both rates are useful, but focusing on only one will certainly lead you astray.

I think you're confusing the "Dual Mandate" of the Fed to minimize inflation and maximize employment with some misplaced belief in economics. Every American economist of the last century has realized these are often competing ideals. It's their raison d'etre, without it the Fed doesn't exist.