Derive the labour demand function.

Question

$$Y = 4[(K^α)(L^{(1-α)})]$$

I took the derivative with respect to $L$, and ended up with:

$$Y'= 4[(K^α)(1-α)L^{(-α)}$$

But the correct answer is something like: $$[4(1-a)K^a/w]^{(1/a)}$$

I'm not totally sure where the $w$ came from but my guess is MP of labor $= w$, where $w =$ wages.

Please show me the correct method.

Answer 1

Assuming a unit price $p=1$ the profit function becomes $$G=4K^\alpha L^{1-\alpha}-cK-wL$$ where w is the wage. Maximize the profit with respect to $L$ $$\frac{\partial G}{\partial L}=4K^\alpha (1-\alpha)L^{-\alpha}-w=0$$ and solve for $L$ $$L=\big(4 (1-\alpha) K^\alpha /w \big)^{1/\alpha}$$