Case: Carterl formation by the two companies A and B

Industry's inverse demand function: P = 400 - 0.1Q
P = 400 - 0.1(Q_A+Q_B)

Industry's total Revenue: TR = PQ = [400 - 0.1(Q_A+Q_B)](Q_A+Q_B)

TR = 400(QA+QB) - 0.1(QA+QB)²

Total cost function of enterprise A: TC_A = 0.1Q_A+ 0.1Q_A²+ 100000

Total cost function of enterprise B: TC_B = 32Q_B+ 0.4Q_B²+ 20000

Joint profit is given by
π = TR - TC = TR - TC_A - TC_B

π = 400(QA+QB) - 0.1(QA+QB)² - (0.1Q_A+ 0.1Q_A²+ 100000) - (32Q_B+ 0.4Q_B²+ 20000)

Maximize π

First order conditions:

∂π/∂Q_A = 0
=>  400 - 0.2(Q_A+Q_B) - 0.1 - 0.2Q_A = 0
=> 399.9 - 0.4Q_A - 0.2Q_B= 0
=>  0.4Q_A + 0.2Q_B= 399.9 ...(i)

∂π/∂Q_B = 0
=>  400 - 0.2(Q_A+Q_B) - 32 - 0.8Q_B = 0
=> 368 - 0.2Q_A - Q_B = 0
=>  0.2Q_A + Q_B = 368 ...(ii)

Subtract equation (i) from 2 * equation (ii)

2Q_B - 0.2Q_B= 2*368 - 399.9
=> Q_B = 186.72

0.2Q_A + Q_B = 368
=> 0.2Q_A = 368 - 186.72
=>  Q_A = 906.39

Q = Q_A + Q_B = 186.72 + 906.39 = 1093.11

Output for enterprise A,
Q_A = 906.39

Output for enterprise B,
Q_B = 186.72

Total output for the industry,
Q = 1093.11

P = 400 - 0.1Q = 400 - 0.1*1093.11 = 290.69

Price, P = 290.69

Profit for enterprise A,
π_A = TR_A - TC_A

π_A = PQ_A - (0.1Q_A+ 0.1Q_A²+ 100000)

π_A = 290.69*906.39 - (0.1*906.39 + 0.1*906.39² + 100000) = 81231.65

π_A = 81231.65

Profit for enterprise B,
π_B = TRB - TC_B

π_B = PQ_B - (32Q_B+ 0.4Q_B²+ 20000)

π_B = 290.39*186.72 - (32*186.72 + 0.4*186.72²+ 20000) = 14300.71

π_B = 14300.71

Total profit for the entire industry,
π = π_{A +} π_B

π = 81231.65 + 14300.71 = 95532.36

π = 95532.36