Question

Problem IV: A perfectly competitive firm has a total cost function given by T C(Q) = 2Q3 − 20Q2 + 100Q. 

a) What will be the optimal quantity produced by the firm if the market price is P = 300? What will be the profit? (Q=10, Profit=2000) 

b) How about if the market price is P = 45? What will be the optimal quantity and profit in this case? (Q=0, Profit=0) 

c) Find the break even point and the shut down point.

d) Assuming the initial market price of $300, what will be the optimal quantity and profit if a new fixed cost of $1000 has to be incurred by the firm? How about if this FC=$3000?

Answer 1

TC = 2Q³ - 20Q² + 100Q

Marginal cost (MC) = dTC/dQ = 6Q² - 40Q + 100

Average variable cost (AVC) = TC / Q = 2Q² - 20Q + 100

(a) Profit is maximized when Price = MC. When Price = 300,

6Q² - 40Q + 100 = 300

6Q² - 40Q - 200 = 0

3Q² - 20Q - 100 = 0

3Q² - 30Q + 10Q - 100 = 0

3Q(Q - 10) + 10(Q - 10) = 0

(Q - 10) (3Q + 10) = 0

Q = 10 or Q = - 10/3 (Inadmissible since Q is non-zero)

Total revenue (TR) = P x Q = 300 x 10 = 3,000

TC = 2 x (10)³ - 20 x (10)² + 100 x 10 = 2 x 1,000 - 20 x 100 + 1,000 = 2,000 - 2,000 + 1,000 = 1,000

Profit = TR - TC = 3,000 - 1,000 = 2,000

(b) When P = 45,

6Q² - 40Q + 100 = 45

6Q² - 40Q + 55 = 0

Solving this quadratic equation using online solver,

Q = 4.73 or Q = 1.94.

When Q = 4.73, AVC = 2 x (4.73)² - 20 x 4.73 + 100 = 2 x 22.37 - 94.6 + 100 = 44.74 - 94.6 + 100 = 50.14 < Price

When Q = 1.94 < 4.73, AVC will be less than Price too.

Since P < AVC, the firm will shut down. Therefore Q = 0 and Profit = 0.

(c) In breakeven, P = MC = ATC

ATC = TC / Q = 2Q² - 20Q + 100

Equating MC and ATC,

6Q² - 40Q + 100 = 2Q² - 20Q + 100

4Q² = 20Q

Q = 5 (Assuming Q is non-zero, dividing by 4Q)

Break-even P = ATC = (2 x 5 x 5) - (20 x 5) + 100 = 50 - 100 + 100 = 50

A firm shuts down when AVC is minimized. AVC is minimized when

dAVC/dQ = 0

4Q - 20 = 0

4Q = 20

Q = 5

Shut down price = (Minimum) AVC = (2 x 5 x 5) - (20 x 5) + 100 = 50 - 100 + 100 = 50

(Since there are no fixed costs, TVC = TC and AVC = ATC, so Breakeven point = Shutdown point)

(d) When P = $300,

(i) The fixed cost of $1,000 will not affect MC, therefore optimal quantity will remain unchanged at Q = 10

Profit will decrease by amount of fixed cost, i.e. new Profit = $(2,000 - 1,000) = $1,000

(ii) The fixed cost of $3,000 will not affect MC, therefore optimal quantity will remain unchanged at Q = 10

Profit will decrease by amount of fixed cost, i.e. new Profit = $(2,000 - 3,000) = -$1,000 (Firm will make a loss).