If i'm given a firm's production function of

$ \displaystyle Y=zK^{\alpha}{N}^{1-\alpha}$

Then assuming $ \displaystyle K$ is fixed and cost free, we can get our profit maximzation problem of

$ \displaystyle \max_{{N}}zF(K^{\alpha}{N}^{1-\alpha})-wN$

To find the optimality condition, $ \displaystyle {MP}_{N}=w$ , I take the partial derivative and find

$ \displaystyle z{F}_{N}=z(1-\alpha){K}^{\alpha}{N}^{-\alpha}=w$

Here is where i'm stuck.

I need to show that the optimality condition can be written as $ \displaystyle \alpha=1-\frac{wN}{Y}$

Any help would be appreciated.

Thank you,

Gin