Question

Assume that you are the manager of a monopoly that sells MX calculator model. You produce calculators in two plants. The demand function for MX calculators Q = 500 – 5P. The cost of production in plant 1 is C1=100+3 Q_1^2, and the cost of production in plant 2 is C1=80+2 Q_2^2. Find the output level each plant has to produce and the price you have to charge.

Answer 1

Q = 500 - 5P

5P = 500 - Q

P = 100 - 0.2Q

Total revenue (TR) = P x Q = 100Q - 0.2Q²

MR = dTR/dQ = 100 - 0.4Q = 100 - 0.4Q1 - 0.4Q2 [Since Q = Q1 + Q2]

C1 = 100 + 3Q1², therefore MC1 = dC1/dQ1 = 6Q1

C2 = 80 + 2Q2², therefore MC2 = dC2/dQ2 = 4Q2

For a multiplant monopolist, profit is maximized when MC1 = MC2 = MR.

For Plant 1, MR = MC1

100 - 0.4Q1 - 0.4Q2 = 6Q1

6.4Q1 + 0.4Q2 = 100

16Q1 + Q2 = 250 (Dividing by 0.4).......(1)

For Plant 2, MR = MC2

100 - 0.4Q1 - 0.4Q2 = 4Q2

0.4Q1 + 4.4Q2 = 100

Q1 + 11Q2 = 250 (Dividing by 0.4).......(2)

Equilibrium is obtained by solving (1) and (2).

16Q1 + Q2 = 250 ...............(1)

(2) x 16 yields:

16Q1 + 176Q2 = 4,000......(3)

(3) - (1) yields:

175Q2 = 3,750

Q2 = 21.43

Q1 = 250 - 11Q2 [From (2)] = 250 - (11 x 21.43) = 250 - 235.73 = 14.27

Q = 14.27 + 21.43 = 35.7

P = 100 - (0.2 x 35.7) = 100 - 7.14 = 92.86