Question

If the total cost function of a firm is C=1/3Q³+5Q²+30Q+10 and the price under perfect competition is given as birr 6

A.  Find for what values of Q profit will be maximized

B.  Will the firm continue production at that output level? 

Answer 1

Given the cost function, $C(q)={1\over3}q^3+5q^2+30q+10$ and price, $p=6$, the competitive firm’s profit maximization problem can be written as:

$max_{q\ge0}=6q-{1\over3}q^3-5q^2-30q-10$

Differentiating profit with respect to $q$ gives marginal profit curve:

Marginal Profit $(q)=6−q^2+10q−30$

Equating the marginal profit to 0 and solving,

$-q^2-10q-24=0$

Using the quadratic formula to solve,

$q={-b\pm\sqrt{b^2-4ac}\over 2a}$ where, $a=-1,b=-10,c=-24$

So,

$q={10\pm\sqrt{100-96}\over -2}$

Therefore, the values of $q$ are $-4$ and $-6$.

To confirm whether this firm will continue production at this output level, we substitute for the values of $q$ in,the function,

$6q-{1\over3}q^3-5q^2-30q-10$

Now, when q=-4, this function gives, ${82\over3}$

When q=-6, the above function gives, 26

For both values above, the profits are greater than 0, which indicates price minimum. Therefore, the firm should not continue with this output level.