Question

Q6

(9 marks) Suppose the market for cotton is competitive. A typical cotton farmer has a total cost function of: C= 100 + 159 - 6q? +q?. The prevailing market price is $15. Find the profit-maximizing output level of this farmer. Calculate the corresponding profit at this output level. Show your steps. (7 marks) Suppose all the fixed cost is unavoidable. Explain whether this farmer should shut down its production in the short run. (2 marks)

Answer 1

For a competitive market, P = MC ;

MC = dC/dq = 15 -12q + 3 q​​​​​​2   and P= 15

P = MC ==> 15 -12q + 3q2 = 15

3q2 - 12q = 0 ==> 3q(q-4) = 0

==> q = 4

Hence, the profit maximising output is q= 4 and at this level profit will be TR-TC

TR = PQ= 15(4) = 60

TC = 100 + 15(4) - 6(16) + 64 = 100 + 60 - 96 + 64 = 128

B) in short run, the shut down point of a firm is the output at the minimum of the average variable cost. At this point, firm is indifferent between producing or not producing. If firm produces or doesn't , it is not able to cover its fixed cost. If it produces at a lesser q , then it won't even cover its variable cost and therefore should shut down and if it produces more q , then it will cover a part of its fixed cost as well and should produce more.

Here , variable cost ; VC = 15q - 6q2 + q​​​​​​3

And AVC = VC/q = 15 - 6q + q​​​​​​2

dAVC/dq = -6 + 2q = 0

==> q= 3

Therefore, shut down point of firm is q = 3 and profit maximising output produced by firm is q= 4 .

Therefore at q= 4 , firm should not shut down.

Firm will shut down at q<3.