The inverse market demand for mineral water is P = 200 - 10Q. where Q is total m

Question

The inverse market demand for mineral water is P = 200 - 10Q. where Q is total market output and P is the market price. Two firms have complete control of the supply of mineral water and both have zero costs. Find the Cournot quantity, price, and each firm's profit. Calculate the DWL from Cournot oligopoly. Denote the Cournot quantity for each firm by q_a, and denote half of the monopoly quantity by q_a Suppose that the two firms interact with each other for infinite periods, and in each period they choose quantities simultaneously. Consider the following collusive strategy, the same as discussed in class: produce q_b only if no one has cheated so far. and produce q_a forever if some has cheated before. Assume each firm acts to maximize its sum of discounted profits where the interest rate is r. For what values of r can such collusion be sustained?

Explanation / Answer

(1) Cournot

P = 200 - 10Q where Q = q1 + q2

P = 200 – 10q1 – 10q2

So,

Total revenue of firm 1, TR1 = P x q1 = 200q1 – 10q12 – 10q1q2

Total revenue of firm 2, TR2 = P x q2 = 200q2 – 10q1q2 – 10q22

So,

Marginal revenue of firm 1, MR1 = dTR1 / dq1 = 200 - 20q1 - 10q2

Equating with MC1:

200 - 20q1 - 10q2 = 0

20q1 + 10q2 = 200

2q1 + q2 = 20 ......(1) [Reaction function, firm 1]

Marginal revenue of firm 2, MR2 = dTR2 / dq2 = 200 – 10q1 – 20q2

MC2 = 0

Equating MR2 = MC2,

200 – 10q1 – 20q2 = 0

Or,

10q1 + 20q2 = 200

q1 + 2q2 = 20 .....(2) [Reaction function, firm 2]

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

2q1 + q2 = 20 ......(1)

(2) x 2:

2q1 + 4q2 = 40 .....(3)

(3) - (1): 3q2 = 20

q2 = 6.67

q1 = 20 - 2q2 = 20 - (2 x 6.67) = 20 - 13.33 = 6.67

Q = q1 + q2 = 6.67 + 6.67 = 13.34

P = 200 - 10Q = 200 - (10 x 13.34) = 200 - 133.4 = 66.6

(2) DWL

If the market were perfectly competitive, profit would be maximized by equating Price with MC.

200 - 10Q = 0

10Q = 200

Q = 20

P = MC = 0

SWL from Cournot = (1/2) x Difference in price x Difference in quantity

= (1/2) x (66.6 - 0) x (20 - 13.34) = (1/2) x 66.6 x 6.66 = 221.78

Note: First 2 questions are answered.

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