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Suppose there is an oligopoly industry consisting of two companies A and B
Suppose there is an oligopoly industry consisting of two companies A and B. The demand curve of the industry and the total cost function of the two companies are:
Industry demand equation: P=400-0.1Q,
The total cost function of enterprise A: TCA=0.1QA+0.1QA2+100000
The total cost function of enterprise B: TCB=0.4QB2+32QB+20000
If these two companies unite to form a cartel, ask: What price should be unified to maximize the total profit of the industry? What is the total output at this time? How should this output be distributed among the companies? Each company will get its own income. What is the profit? What is the total profit of the entire industry?
Expert Solution
Case: Carterl formation by the two companies A and B
Industry's inverse demand function: P = 400 - 0.1Q
P = 400 - 0.1(QA+QB)
Industry's total Revenue: TR = PQ = [400 - 0.1(QA+QB)](QA+QB)
TR = 400(QA+QB) - 0.1(QA+QB)2
Total cost function of enterprise A: TCA = 0.1QA+ 0.1QA2+ 100000
Total cost function of enterprise B: TCB = 32QB+ 0.4QB2+ 20000
Joint profit is given by
π = TR - TC = TR - TCA - TCB
π = 400(QA+QB) - 0.1(QA+QB)2 - (0.1QA+ 0.1QA2+ 100000) - (32QB+ 0.4QB2+ 20000)
Maximize π
First order conditions:
∂π/∂QA = 0
=> 400 - 0.2(QA+QB) - 0.1 - 0.2QA = 0
=> 399.9 - 0.4QA - 0.2QB= 0
=> 0.4QA + 0.2QB= 399.9 ...(i)
∂π/∂QB = 0
=> 400 - 0.2(QA+QB) - 32 - 0.8QB = 0
=> 368 - 0.2QA - QB = 0
=> 0.2QA + QB = 368 ...(ii)
Subtract equation (i) from 2 * equation (ii)
2QB - 0.2QB= 2*368 - 399.9
=> QB = 186.72
0.2QA + QB = 368
=> 0.2QA = 368 - 186.72
=> QA = 906.39
Q = QA + QB = 186.72 + 906.39 = 1093.11
Output for enterprise A,
QA = 906.39
Output for enterprise B,
QB = 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,
πA = TRA - TCA
πA = PQA - (0.1QA+ 0.1QA2+ 100000)
πA = 290.69*906.39 - (0.1*906.39 + 0.1*906.392 + 100000) = 81231.65
πA = 81231.65
Profit for enterprise B,
πB = TRB - TCB
πB = PQB - (32QB+ 0.4QB2+ 20000)
πB = 290.39*186.72 - (32*186.72 + 0.4*186.722+ 20000) = 14300.71
πB = 14300.71
Total profit for the entire industry,
π = πA + πB
π = 81231.65 + 14300.71 = 95532.36
π = 95532.36
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