# How to derive (mathematically and intuitively) the relationship between expected price level and natural rate of employment?

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.