Question

The (total) cost function is given by C = 50 + 60Q – 18Q² + 2Q³

e. Write down the average variable cost function AVC(Q). (1) f. Write down the average total cost function AC(Q). (1) g. Find the break-even point (Q and AC). (1) h. Find the shut-down point (Q and AVC). (1) i. Draw a graph to illustrate AC, AVC, and MC functions for quantities Q on the interval between 1 and 10. Make sure you show (put the numbers there) where exactly the MC curve intercepts AVC and AC curves. (1) j. If the price is P = 60, calculate the profit-maximizing firm’s profit. (1)

Answer 1

Cost function is C = 50 + 60Q – 18Q² + 2Q³

MC = derivative of C with respect to Q = 60 - 36Q + 6Q^2

AC = C/Q = 50/Q + 60 - 18Q + 2Q^2

AVC = VC/Q = 60 - 18Q + 2Q^2

At break even point AC = MC

60 - 36Q + 6Q^2 = 50/Q + 60 - 18Q + 2Q^2

50/Q + 18Q - 4Q^2 = 0

4Q^3 - 18Q^2 - 50 = 0

This gives Q = 5

AC = 50/5 + 60 - 18*5 + 2*5^2 = 30

Hence break even quantity is 5 and AC = $30

At shut down point AVC = MC

60 - 18Q + 2Q^2 = 60 - 36Q + 6Q^2

18Q - 4Q^2 = 0

Q = 4.5

AVC = 60 - 18*4.5 + 2*4.5^2 = $19.5

Shut down Q is 4.5 units and AVC = $19.5

When P = MC = 60 we have

60 = 60 - 36Q + 6Q^2

Q = 6 units

AC = 50/6 + 60 - 18*6 + 2*6^2 = 32.33

Profit = (P - AC)*Q = (60 - 32.33)*6 = $166