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A bit for sure by who we vote in but monetary policy is mostly set by the central bank isn't it?
Yes - Brown removed the ability for the Treasury to set interest rates and gave it to the BoE very early on when he was chancellor. I'm not sure the reason for it though.
I think (not sure as I was quite disinterested at the time...) that prior to that in theory at least, the Treasury could change rates at any time, on any day
Yes.