Question

Suppose the market for cotton is competitive. A typical cotton farmer has a total cost function of: C = 100 + 15q – 6q^2 + q^3 . The prevailing market price is $15.

a. Find the profit-maximizing output level of this farmer. Calculate the corresponding profit at this output level. Show your steps.

b. Suppose all the fixed cost is unavoidable. Explain whether this farmer should shut down its production in the short run.

Answer 1

a)

A firm in a competitive market is a price taker and profit maximization occurs at an output level at which marginal cost equals price.

The cost equation is given as

C = 100 + 15Q - 6Q2 + Q3

Marginal cost is obtained by taking the first derivative of the cost function

dC/dQ = 15 - 12Q + 3Q2

Equating the marginal cost equation to price, we get

15 - 12Q + 3Q2   = $ 15

3Q2 -12Q = 0

Q(3Q - 12) = 0

3Q = 12

Q = 4 units

The profit maximizing output level for this farmer is 4 units

Profit = Revenue - total cost

Profit = $ 15 4 - [ 100 + 15 4 - 6 42 + 43 ]

Profit = $ 60 - 100 - 60 + 96 - 64

Profit = - $ 68

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b)

The farmer must shut down the production in the short run because the profit is negative and the fixed cost is unavoidable in the short run. In the short run, fixed factors of input cannot be changed whereas in the long run all the fixed input factors can be changed. Therefore the farmer must shut down in the short run to avoid facing loss.