What is the Optimality Condition for a Firm's Profit Maximization Problem?

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Ginnee
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If I'm given a firm's production function of

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

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

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

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

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

Here is where I'm stuck.

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

Any help would be appreciated.

Thank you,

Gin
 

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Ginnee said:
If I'm given a firm's production function of

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

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

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

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

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

Here is where I'm stuck.

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

Any help would be appreciated.

Thank you,

Gin

Dont get the notation. What is z and F and also F with a subscript N