I have a couple of questions regarding divine coincidence.

1) As Blanchard and Gali (2006) stated NKPC illustrates that stabilising inflation (make inflation constant) achieves output gap = 0. If you do the simple calculation if π = E( π ) then the output will not equal to the natural level of output?- or does he implicitly assumes that stabilising inflation means that π = E( π ) = 0?

2) Followed by their literature, divine coincident happens whenever there is a constant gap between the natural level of output and efficient level of output.

And I don't quite understand how the stabilising the output gap (setting output - natural level) is equivalent to stabilising the welfare-relevant output gap ??? (output - the efficient level of output)

- since there exists a gap between natural level and efficient level of output, shouldn't it be the case that setting output greater than natural level (=nd hence closer to efficient level ) better policy?