Ex $8\mathrm{.}2$: Profit maximization and elasticity

Consider a competitive profit$-$maximizing firm with the production function F $($L$,$ K$)=3$L $1$

$6$ K $1$

$6.$ It is

a price taker in both input and output markets, paying w for each unit of labor and r for each unit of

capital, and selling each unit of its output at price p$.$ It faces no costs other than those generated by

labor and capital; it can freely choose labor $($L$),$ but capital is fixed in the short run at K $=64.($Note

that since $641$

$6=2,$ this means its short$-$run production function is F $($L$)=6$L $1$

$6.)$

$($a$)$ Solve the firm$$s short$-$run profit maximization problem to derive its short$-$run profit$-$maximizing

labor demand function and its short$-$run supply function. For this problem, use the method

of writing profit solely as a function of L$,$ finding L$($w$,$ r$,$ p$),$ and then plugging that into the

production function to find q$($w$,$ r$,$ p$).$

$($b$)$ In the short run, what is the firm$$s wage elasticity of labor demand $($elasticity of L with

respect to w$)?$ Its cross$-$price elasticity of labor demand $($elasticity of L with respect to r$)?$

Its price elasticity of supply $($elasticity of q with respect to p$)?$