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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?

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  • $\begingroup$ 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 $\endgroup$ – RScrlli Apr 11 at 21:11
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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}\\ $

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  • $\begingroup$ why in the cost side you are taking 100000 as fixed costs ? $\endgroup$ – Amalya Apr 11 at 21:16
  • $\begingroup$ Sorry 500,000 fixed costs. It is irrelevant to the final answer though. $\endgroup$ – Doug M Apr 11 at 21:17
  • $\begingroup$ so we do not really need first part of calculations derivative of revenue with respect to price, right? $\endgroup$ – Amalya Apr 11 at 21:18
  • $\begingroup$ so in the final answer w is the wage of high-skilled workers? $\endgroup$ – Amalya Apr 11 at 21:22
  • $\begingroup$ 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. $\endgroup$ – Doug M Apr 11 at 21:23

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