# 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 = $$pf(E)$$ - $$w_sE$$

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?

• I haven't gone through all the calculations, but in the profit function there's a problem. You said that $w_s$ is the wage for the unskilled workers, but you multiply it by $E$, the number of high-skilled workers. Also you should consider the fixed costs. Take a look again – RScrlli Apr 11 at 21:11

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.

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

• why in the cost side you are taking 100000 as fixed costs ? – Amalya Apr 11 at 21:16
• Sorry 500,000 fixed costs. It is irrelevant to the final answer though. – Doug M Apr 11 at 21:17
• so we do not really need first part of calculations derivative of revenue with respect to price, right? – Amalya Apr 11 at 21:18
• so in the final answer w is the wage of high-skilled workers? – Amalya Apr 11 at 21:22
• I didn't use it here. But, in many cases (and micro economics questions) calculating revenue as a function of price is more common, and what students are accustomed to seeing. – Doug M Apr 11 at 21:23