Question

Suppose a certain contractor wants to maximize
$\pi$
from building one bridge. The contractor uses both labor and capital, and efficient combinations of Labor and capital that are sufficient to make a bridge is by the function 0.25 L 2 1 K 2 1 , if the prices of labor (w) and capital (r) are $ 5 and $ 10 respectively. Find the least cost combination of L and K​

Answer 1

Therefore, efficient combination (least cost combination) of L and K are$\frac{16}{\sqrt{2}}$ and $\frac{8}{\sqrt{2}}$respectively.

What is the least cost combination of L and K​?

The principle of least cost combination declares that if two-factor inputs are assessed for a given output the least cost combination will be such where their inverse price ratio stands equivalent to their marginal rate of substitution.

The optimum combination of inputs that are needed to produce output at the least possible cost is named the least cost combination.

The contractor wants to build one bridge. Thus, the constraint equation can be written as

$0.25 \mathrm{~L}^{\frac{1}{2}}$

$&k^{\frac{1}{2}}=1$

$&\mathrm{MPL}=0.125 \mathrm{~L}^{\frac{-1}{2}} \mathrm{~K}^{\frac{1}{2}}$

$&\mathrm{MPK}=0.125 \mathrm{~L}^{\frac{1}{2}} \mathrm{~K}^{\frac{-1}{2}}$

The equilibrium condition is$\frac{M P L}{M P K}=\frac{W}{r}$

$K^{\frac{-1}{2}}}=\frac{\ 5}{\ 10}$

$&\frac{K}{L}=\frac{1}{2} \Rightarrow L=2 K$

Substituting$\mathrm{L}=2 \mathrm{~K}$in the constraint equation we obtain

$

$&0.125(2 \mathrm{~K})^{\frac{-1}{2}} \mathrm{~K}^{\frac{1}{2}}=1$

$&0.125 \sqrt{2} \cdot \mathrm{K}=1$

$&\mathrm{~K}=\frac{1}{0.125 \sqrt{2}}=K=\frac{8}{\sqrt{2}}$

$&\mathrm{~L}=2 \mathrm{~K} \Rightarrow \frac{16}{\sqrt{2}}$

Therefore, efficient combination (least cost combination) of L and K are$\frac{16}{\sqrt{2}}$ and $\frac{8}{\sqrt{2}}$respectively.

The least cost is $\mathrm{C}=5\left(\frac{16}{\sqrt{2}}\right)+10\left(\frac{8}{\sqrt{2}}\right)=\ \frac{160}{\sqrt{2}}$

To learn more about least cost combination refer to:

https://brainly.com/question/13897740

#SPJ9