Derive short-run demand function? 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?

Demand for the product

$r = qp\\\ q = 12000 - 20 p\\\ r = 12000 p - 20 p^2\\\ \frac {dr}{dp} = 12000 - 40 p$

or

$p = 600 - \frac {q}{20}\\\ r = 600 q - \frac {q^2}{20}\\\ \frac {dr}{dq} = 600 - \frac {q}{10}$

On the costs side

$c = 500,000 + wE\\\ E = \frac {q}{100}\\\ c = 100,000 + \frac {wq}{100}\\\ \frac {dc}{dq} = \frac {w}{100}$

Profits are maximized when marginal revenue equals marginal costs.

$600 - \frac{q}{10} = \frac {w}{100}\\\ 6000 - \frac {w}{10} = q\\\ q = 100e\\\ e = 60 - \frac {w}{1000}\\\ $