Lagranges multiplier to minimize function of two variables with two constraints

Question

I have a Cobb Douglas type production function with $K$ and $L$ as inputs; $\alpha$ and $1-\alpha$ as output elasticities and $C$ as efficiency parameter. Now I have to minimize cost $=wL+rK$ w.r.t two constraints

1. Cobb Douglas output constraint and
2. $L=1$.

Please help me how to do this. Thanks.

Answer 1

You have to minimize:

$\mathcal L(K|L=1)= w\cdot 1+rK+ \lambda \left(\overline Y - C \cdot K^{\alpha} \cdot 1^{1-\alpha} \right )=w+rK+ \lambda \left(\overline Y - C \cdot K^{\alpha} \right )$

$\overline Y$ is the given output (output constraint).

Now calulate $\frac{\partial \mathcal L}{\partial K}=0,\frac{\partial \mathcal L}{\partial \lambda}=0$ Then solve this two equations for $K^*$.