Question #$2$: Cost Minimization Problem $[30$ Points$]$

Nike produces its sneakers using labor $($L$)$ and capital $($K$).$ Nike has the following production function

Q $=50$K $1/5$ L$1/4.$ The wage rate $($PL$)$ is $$2$ and the price of capital $($PK$)$ is $$5.$ Nike wants to produce $1000$

sneakers at the lowest possible cost.

$($a$)$ Use the Lagrangian Method to find the cost$-$minimizing quantity of capital and labor to produce

$1000$ sneakers? Round your answers to $[12$ Points$]$

$($b$)$ What is the total cost of producing $1000$ sneakers? $[4$ Points$]$

$($F$)$ Suppose that the price of labor doubles to P L $=$ $$4.$ Assume all other variables are held constant.

Find the new cost$-$minimizing quantity of capital and labor.