AS-AD model and labor force increase and natural rate of unemployment

I am trying to analyze the effects on the unemployment rate $u$ as the labor force $L$ increases, by use of the AS-AD model.

I have however some issues I can't wrap my head around:

• If we increase the labor force L, how does it affect the natural rate of output? I am looking at the equation:

$$P = (1+m)P_eF(1 - \frac{Y}{L},z)$$

where $P$ is the price level, $P_e$ is the expected price level, $m$ is the markup, and $F$ is a function of unemployment $u = 1 - \dfrac{Y}{L}$ and a catch-all variable $z$.

• If we are initially at equilibrium with $P = P_e$, that is equilibrium $Y$ is $Y_n$, the natural level, then what happens in the medium-run once we increase the labor force? Here I use the AS-AD model, and figure that an increase in $L$ causes a downwards shift in the $AS$-curve so equilibrium output initially increases, but then reverts back to $Y_n$ due to people changing their expectations of the price level.... but what happens to the unemployment rate in the medium run?