Question

Suppose that a firm’s fixed proportion production function is given by:

q = min{5k, 10l}

Please calculate the firm’s long-run total, average, and marginal cost functions.

Answer 1

Given

Production function is given as 

q = min{5k,10l}

calculation of long run total cost

Total Cost is minimized where $5K=10l$

$k=2l$  or $l=\frac{k}{2}$

Now we got the value of k put it in production function 

q=min{5k,5k}

q=5k

$k=\frac{q}{5}$

Now put $k=\frac{q}{5}$  in total cost function  

$l=\frac{k}{2}$

$l=\frac{q}{\frac{5}{2}}$   

$l=\frac{q}{10}$

therefore Total Cost function is   TC = w*l + r*k  

put the values of l and k  

TC = $w*\frac{q}{10}+r*\frac{q}{5}$

Calculation of average Cost

$AC=\frac{TC}{q}$

where, TC = $w*\frac{q}{10}+r*\frac{q}{5}$

 $AC=\frac{w}{10}+\frac{r}{5}$

Calculation of Marginal Cost

MC =$\frac{\delta TC}{\delta q}$  

where  TC = $w*\frac{q}{10}+r*\frac{q}{5}$

MC =  $\frac{w}{10}+\frac{r}{10}$