Question

1. Consider a firm whose total cost function is q³ – 6q² + 13q.
   1. If the price of a unit of output is $13, what is the profit-maximizing value of q?
   2. Find the average cost function of the firm? What is the firm’s minimum average cost?
   3. What is the equilibrium price of output? How much does each firm produce at equilibrium price? What is each firm’s profit at the equilibrium price?

Answer 1

a)

Given the total cost function q³-6q²+13q and price per unit output or AR=13

Profit maximizing condition is given by MC=MR

Therefore we find the MC by differentiating total cost function with respect to q

MC=3q²-12q+13

MR=AR in this case since price is given to be constant

Therefore, equating the two we get,

3q²-12q+13=13

3q²-12q=0

q=4

b)

The AC function is given by simply dividing the TC function with quantity. Therefore,

AC = (q³-6q²+13q)/q

AC= q²-6q+13

The lowest point on the AC curve will be the point where it intersects the MC curve. Therefore, lowest AC will be given by the condition AC = MC

q²-6q+13=3q²-12q+13

2q²-6q=0

q-3=0

q=3

At this level of q, AC will be

3²-6(3)+13=4

AC = 4

c)

At the long run equilibrium level MC=AC=P, from the above subpart we already know that when AC=MC, q=3 and AC=4 Therefore, at the equilibrium level price will be equal to 4. At the equilibrium price the firm produces at q=3.

Each firm's profit at this price can be seen by taking TR-TC,

TR = pq = 4*3= 12

TC = q³-6q²+13q = (3)³-6(3)²+13(3) = 12

Therefore TR-TC=0, so at this equilibrium level each firm's profits are 0.