Suppose a firm pays £500,000 in short-run costs for its capital and unskilled labour.
Its only short-run decision, therefore, is to determine how many high-skilled workers, E, to hire.
The wage for unskilled labour is Ws and the firm's short-run production function is
$Q$ = $f(E)$ = $100E$.
The firm faces a downward sloping demand for its output given by $Q = 12000 - 20P$, where P is the price per unit at which it sells its product.
Derive the firm's short-run labour demand function (you can either make E the dependent variable or show the inverse labour demand curve making Ws the dependent variable). NB; this firm is not a price taker.
I know that for short-run the following is true: profits = TR - TC = $pf(E)$ - $w_s$$E$
and $pMP_L$ = $w_s$ so we have $((12000 - Q) * 100)/20$ = $w_s$ and $Q=100E$
from this $E$ = (60000 - $w_s$)/500
is it right?