Question

The market inverse demand curve is P = 85 – Q.
There are i firms in the market with all firms (plants) that have cost function
TC_i = 20 + q_i + q_i^2.

Find the market profit for a maximizing multiplant-monopoly assuming two plants.

Answer 1

TC1 = 20 + q1 + q1², therefore MC1 = dTC1/dq1 = 1 + 2q1, and q1 = (MC1 - 1)/2

TC2 = 20 + q2 + q2², therefore MC2 = dTC2/dq2 = 1 + 2q2, and q2 = (MC2 - 1)/2

P = 85 - Q

Total revenue (TR) = P x Q = 85Q - Q²

Marginal revenue (MR) = dTR/dQ = 85 - 2Q

Since Q1 = q1 + q2,

Q = [(MC1 - 1)/2] + [(MC2 - 1)/2]

Setting MC1 = MC2 = MC,

Q = [(MC - 1)/2] + [(MC - 1)/2]

Q = MC - 1

MC = Q + 1

Profit is maximized by equating MR and MC.

85 - 2Q = Q + 1

3Q = 84

Q = 28

P = 85 - 28 = 57

Total revenue (TR) = P x Q = 57 x 28 = 1,596

MC = 28 + 1 = 29

q1 = (29 - 1)/2 = 28/2 = 14

q2 = (29 - 1)/2 = 28/2 = 14

TC1 = 20 + 14 + (14 x 14) = 34 + 196 = 230

TC2 = 20 + 14 + (14 x 14) = 34 + 196 = 230

Aggregate total cost (TC) = TC1 + TC2 = 230 + 230 = 460

Market Profit = TR - TC = 1,596 - 460 = 1,136