Isn't Ben Bernanke what the New Keynesians want?

Paul Krugman made me think of one thing that has always bothered me about New Keynesian economics when he said that Bernanke is being intimidated by inflation hawks the other day. The idea of the conservative central banker.

I have never really liked the claim that you need an overly conservative central banker to maximize social welfare because your normal run of the mill central banker will over inflate. This always bothered me. There always seemed to me to be a fair amount of intellectual path dependence to that finding in the 1999 "Science of Monetary Policy" paper. I'm just not sure how you come to that conclusion without looking for it. Like, it seemed to me the New Keynesians were looking to prove the ad hoc finding that you need a central banker who's DNA tells him to keep inflation low.

Speculation on the history of economic thought aside, I never really understood that proposition in a general equilibrium model. I mean, I know the math works, but I can't help but feel like in a perfectly rational world of perfectly rational agents the perfectly rational central banker will figure out the optimal way to manage inflation. For instance, Ben Bernanke has probably seen and can probably solve the equation that tells him to be overly conservative and he can act on it, whether or not he was beaten with the inflation stick when he was a boy.

Of course, that's the problem. There is a clear penalty for not being conservative but no clear penalty for being too conservative. That's a serious asymmetry and I don't think Ben Bernanke is being forced to do anything, listening to the hawks is welfare maximizing.