Question $1(25$ points$)$

Consider a firm uses labor $(L)$ and capital $(K)to$ produce one output. The firm's production

technology can $be$ represented $by$ the production function

$y=f(L,K)=\sqrt[\phantom{2}]{L*K}=L^{\frac{1}{2}}*K^{\frac{1}{2}}.$

Suppose the input prices are

$w=16$

per unit $of$ labor and

$r=\frac{1}{4}=0\mathrm{.}25$

per unit $of$ capital.

Find the firm's conditional factor demands $(i.e.cost-$minimizing input choices$)$

$L^{*}(y$;$16,0\mathrm{.}25)$

and

$K^{*}(y$;$16,0\mathrm{.}25)$

$as$ functions $of$ the target output level $y>0.$

Use your answer $to$ find the firm's long$-$run total cost function $C(y)at$ the given input

prices.

$In$ the short run, the firm's capital $is$ fixed $at$

$\frac{?}{\text{b}\text{a}\text{r}}(K)=2,500.$

Find the firm's short$-$run conditional factor demand for labor,

$L^{*}(y\frac{\text{;}}{b}ar(K)=2500)$

and the short$-$run total cost function

$STC(y\frac{\text{;}}{b}ar(K)=2500)$

Identify the producer's short$-$run variable cost $(SVC)$ and short$-$run sunk cost $(SSC).$