Problem $1$

Suppose that a firm's production function is given by $\backslash ($ Q$=\backslash$left$($K$^\{1/4\}+$L$^\{1/4\}\backslash$right$)^\{\backslash$alpha$\}\backslash ),$ here $\backslash (\backslash$alpha $\backslash )$ is a given parameter.

$($a$)$ Find the marginal product of capital and labor.

$($b$)$ Find the average product of capital and labor.

$($c$)$ Does this production function have increasing, constant or decreasing returns to scale? Show how the returns to scale result depends on the parameter $\backslash (\backslash$alpha $\backslash ).$

$($d$)$ Find the marginal rate of technical substitution $($RTS$)$

$($e$)$ Use your result in $($d$)$ to find the elasticity of substitution.

For the next two questions, suppose $\backslash (\backslash$alpha$=4\backslash ).$

$($f$)$ Assume that capital is fixed in the short run, we know that $\backslash ($ K$=\backslash$bar$\{$K$\}\backslash ).$ Set up the cost minimization problem of the firm and solve for the conditional demand for L$.$ Find the short run cost function.

$($g$)$ Assume that the firm can choose both, capital and labor. Set up the cost minimization problem of the firm and solve for the conditional demand for K and L $.$ Find the cost function.