The Production function $of$ a firm $is$ given $byy=\sqrt[\phantom{2}]{x_1{x}_2},$ where $yis$ the output

level, $x_{1}is$ the amount $of$ factor $1$ used, and $x_{2}is$ the amount $of$ factor $2$ used.

Assume that factor $2is$ fixed $at$ some level ${\stackrel{}{x}}_2$ and factor $1is$ a variable input.

Let $w_1$ represent the price $of$ factor $1,$ let $w_2$ represent the price $of$ factor $2,$ and

let $p$ represent the price $of$ output.

$(1)$ Write down the firm's short$-$run profit$-$maximization problem. $(3$ points$)$

$(2)$ Find the marginal product $M{P}_1(x_1,{\stackrel{}{x}}_2)of$ factor $1.(5$ points$)$

$(3)$ Write the condition for the optimal $(or$ profit$-$maximizing$)$ choice $x_1^{*}of$ factor $1.$

$(5$ points$)$

$(4)$ Find the optimal choice $x_1^{*}of$ factor $1.(5$ points$)$

$(5)$ Write the firm's short$-$run factor demand function for factor $1.(2$ points$)$

$(6)$ Assume that ${\stackrel{}{x}}_2=100,w_1=2,w_2=4,$ and $p=8.$

Find the optimal choice $x_1^{*}of$ factor $1.(5$ points$)$

What $is$ the output $y^{*}$ supplied $by$ the firm? $(2$ points$)$

What are the resulting profits ${}^{*}of$ the firn? $(3$ points$)$

The production function $of$ a firm $is$ given $byy=10\sqrt[\phantom{2}]{x_1{x}_2},$ where $yis$ the output

level, $x_{1}is$ the amount $of$ factor $1$ used, and $x_{2}is$ the amount $of$ factor $2$ used.

Assume that factor $2is$ fixed $at$ some level ${\stackrel{}{x}}_2$ and factor $1is$ a variable input.

Let $w_1$ represent the price $of$ factor $1$ and let $w_2$ represent the price $of$ factor $2.$

$(1)$ Given that the firm wants $to$ produce $y$ units $of$ output, write the firm's short$-$run

$cost-$minimization problem. $(2$ points$)$

$(2)$ Find the optimal $(orcost-$minimizing$)$ choice $x_1^{*}of$ factor $1.(5$ points$)$

$(3)$ Write the short$-$run conditional factor demand function for factor $1.(2$ points$)$

$(4)$ Write the short$-$run cost function $of$ the firm. $(3$ points$)$

$(5)$ Assume that ${\stackrel{}{x}}_2=100,w_1=2,w_2=4,$ and $y=1,000.$

Find the optimal $(orcost-$minimizing$)$ choice $x_1^{*}of$ factor $1.(5$ points$)$

What are the minimum costs $of$ producing $1,000$ units $of$ output? $(3$ points$)$