Fed announcements as "cheap talk"
As any semi-loyal KPC reader knows, I find the "just do your job" critique of the Fed baffling. In our current institutional framework, the Fed cannot credibly commit to future actions that will conflict with their period by period utility function.
That is to say (more or less accurately), the Fed cannot credibly promise to tolerate higher inflation than it prefers after the economy recovers because, when the economy recovers the Fed will still not like inflation and there is nothing to prevent them from not tolerating it. Knowing this, people will not believe the initial announcement.
So, why does the Fed make announcements about the future at all? Could they ever "work".
I believe the relevant economic theory here is the literature on "cheap talk" in games. A readable approach can be found in Ferrell & Rabin.
For cheap talk to be effective, Ferrell & Rabin argue that it must be self-signalling meaning that the sender only wants to send the message if it is true (or if it is at least correlated with the truth). They also argue that it must be self-committing, meaning if the receiver believes the message, the sender has incentives to fulfill it.
I think that the messages people want the Fed to send are not self-committing, so that such "cheap talk" won't work.
However, there is a famous paper by Jeremy Stein on cheap talk and the Fed that argues that the Fed can make imprecise announcements which will have some effect on expectations. I haven't fully figured that paper out yet.