Question

3. The inverse market demand for mineral water is P = 200-10Q, where Q is the total market output and P is the market price. Two firms, A and B, have complete control over the supply of mineral water and both have zero costs. a. Operating independently, how would each firm determine the quantity to be produced? Will this quantity maximize the profits of both firms?

Answer 1

We have the following information

P = 200 – 10Q

Q = q_A + q_B

Marginal Cost (MC) = 0

Total revenue of A: TR_A = (200 – 10q_A – 10q_B)q_A

TR_A = 200q_A – 10q²_A – 10q_Bq_A

Marginal Revenue of A: MR_A = ?TR_A/?q_A = 200 – 20q_A – 10q_B

Total revenue of B: TR_B = (200 – 10q_A – 10q_B)q_B

TR_B = 200q_B – 10q²_B – 10q_Bq_A

Marginal Revenue of B: MR_B = ?TR_B/?q_B = 200 – 20q_B – 10q_A

200 – 20q_A – 10q_B = 0

20q_A + 10q_B = 200    ………………………….     (1)

Similarly,

200 – 20q_B – 10q_A = 0

20q_B + 10q_A = 200    ………………………….     (2)

Multiplying equation 2, with 2 we get

40q_B + 20q_A = 400    ………………………….     (3)

Using equation 1 and 3 we get

30q_B = 200

q_B = 6.67

20q_B + 10q_A = 200   

20(6.67) + 10q_A = 200  

133.33 + 10q_A = 200  

q_A = 6.67

Yes, the given quantities will maximize the profits of both the firms.