# Optimal production?

• Oct 1st 2010, 05:15 PM
Quixotic
Optimal production?
I need to find the optimal level of inputs, the minimum total cost, and the average cost given the following conditions:

Min 25K + 5L
s.t.$\displaystyle 100=L^{.75}K^{.25}$

The second equation looks like a standard Cobb-Douglas production function, but I don't understand how they interact.
• Oct 1st 2010, 09:28 PM
CaptainBlack
Quote:

Originally Posted by Quixotic
I need to find the optimal level of inputs, the minimum total cost, and the average cost given the following conditions:

Min 25K + 5L
s.t.$\displaystyle 100=L^{.75}K^{.25}$

The second equation looks like a standard Cobb-Douglas production function, but I don't understand how they interact.

The constraint condition is:

$\displaystyle 100=L^{3/4}K^{4}$

raise this to the fourth power:

$\displaystyle 10^8=L^3K$

so

$\displaystyle K=\dfrac{10^8}{L^3}$

Substituting this into the objective gives:

$\displaystyle Ob(L)=\dfrac{25 \times 10^8}{L^3}+5L$

Now the minimum of the objective occurs when:

$\displaystyle \dfrac{d}{dL}Ob(L)=0$

etc.

CB