Question

The inverse market demand is P=75 – 2Q. There are 2 plants with cost functions
TC1 = 15+6q1+q1²
TC2 = 18+2q2+2q2²

a. How much quantity should each plant produce?

b. Determine the single plant monopoly profit-maximizing output. (Assume plant 2 is closed then assume plant 1 is closed.)

Answer 1

(a)

P = 64 - (Q / 7)

Total revenue, TR = P x Q = 64Q - (Q² / 7)

Marginal revenue, MR = dTR / dQ = 64 - (2Q / 7)

4Q1 = MC1, or Q1 = MC1 / 4 = 0.25MC1

2+ 2Q2 = MC2, or Q2 = (MC2 - 2) / 2 = 0.5MC2 - 1

6 + Q3 = MC3, or Q3 = MC3 - 6

Adding them up,

Q = Q1 + Q2 + Q3 = 025MCT + 0.5MCT - 1 + MCT - 6

Q = 1.75MCT - 7

Or,

1.75MCT = Q + 7

MCT = (Q + 7) / 1.75 [MCT = 0.57Q + 4]

The graph is as follows.

Equating MCT with MR:

(Q + 7) / 1.75 = 64 - (2Q / 7)

Q + 7 = 112 - 0.5Q

1.5Q = 105

Q = 105 / 1.5 = 70

P = 64 - (Q / 7) = 64 - (70 / 7) = 64 - 10 = 54

(b)

MCT = (Q + 7) / 1.75

= (70 + 7) / 1.75 = 77 / 1.75 = 44

Q1 = MCT / 4 = 44 / 4 = 11

Q2 = 0.5MC2 - 1 = 22 - 1 = 21

Q3 = MCT - 6 = 44 - 6 = 38