We know that $ W = \mathbb{E}(P)f(u, z) $ , so that nominal wage $ W / P = (\mathbb{E}(P) / P)f(u, z) $
From Blanchard.
Furthermore,
$ W / P = 1/(1+m) $ (where m is defined as the markup on wages to prices $ P = (1+m)W $ )
By definition, if $ \mathbb{E}(P) = P $, then $ u = u_n $, so $ Y = Y_n $, as where $Y$ is some production function $Q$ of total workers $N$ (ignoring capital), $Y = Q(N), N = Q^{-1}(Y) $, and $ u = 1 - (Q^{-1}(Y) / L)$, so $u$ and therefore $f(u, \bar{z})$ map directly to a $Y$
But how do changes in $ \mathbb{E}(P) $ affect things?
If $ \mathbb{E}(P) < P $, then apparently $ Y_n < Y $
Where do we get this from?
I understand that if $ \mathbb{E}(P) < P $, then nominal wage becomes $ W / P = (\mathbb{E}(P) / P)f(u, z) < f(u, z) $, but how does this mean that output decreases? How do we get from lowering the expected value of P to a lower Y_natural, and causally (intuitively) how does this work?
Sidenote: I understand this has something to do with the aggregate supply curve, and since it's upward sloping, if P decreases, Y must also decrease, but I don't get that mathematically or causally.
