Question 3: Consider an oligopoly in which firms choose quantities. The inverse

Question

Question 3: Consider an oligopoly in which firms choose quantities. The inverse market demand curve is given by P = a-b (qi + q), where qi is the quantity produced by Firm 1, and q, is the quantity produced by Firm 2. Each firm has a marginal cost equal to c. 1. What is the equilibrium market quantity if the two firms acted as a cartel (i.e., attempt to set prices and outputs together to maximize total industry profits). How about the equilibrium market price? 2. Instead of cartel, suppose now firm 1 acts as the leader and firm 2 acts as the follower. What is the Stackelberg equilibrium quantities determined by each firm? What is the equilibrium market price? 3. Find using the Stackelberg quantities and price. Whose profit is higher?

Explanation / Answer

1) In the given question, Price(P) = Average Revenue(AR), hence

For Firm 1:

(Average Revenue) AR1 = a - b(q1 + q2)

(Total Revenue) TR1 = q1[a - b(q1 + q2)]

= a*q1 - b*q1^2 - b*q1*q2

(Marginal Revenue) MR1 = a - 2*b*q1 - b*q2

For Firm 2:

(Average Revenue) AR2 = a - b(q1 + q2)

(Total Revenue) TR2 = q2[a - b(q1 + q2)]

= a*q2 - b*q2^2 - b*q1*q2

  (Marginal Revenue) MR2 = a - 2*b*q2 - b*q1

  AT EQUILIBRIUM: MR1 + MR2 = MC(Marginal Cost)

(a - 2*b*q1 - b*q2) + (a - 2*b*q2 + b*q1) = c

2*a - 3*b*q1 - 3*b*q2 = c

(2*a - c)/(3*b) = q1 + q2

EQUILIBRIUM MARKET PRICE, P = a - b(q1 + q2), where q1 + q2 = (2*a - c)/(3*b)

P = a - b[(2*a - c)/(3*b)]

P = a + c

2) P = a - b(q1 + q2)

where Firm 1 is a leader and Firm 2 is a follower

According to Stackelberg, consider Firm 2's reaction function, given by:

q2 = 30 - 1/2*q1

Therefore, P = a - b(q1 + 30 - 1/2*q1)

P = (2*a - b*q1 + 60*q1)/2 = MR

MR(Marginal Revenue) = MC(Marginal Cost)

(2*a - b*q1 + 60*q1)/2 = c

q1 = (2*c - 2*a)/(60 - b)

  Therefore q2 = (1800 - 30*b - c + a)/(60 - b)

   EQUILIBRIUM MARKET PRICE, P = a - b(q1 + q2)

P = a - b((2*c - 2*a)/(60 - b) + (1800 - 30*b - c + a)/(60 - b))

P = a - b[(c - a + 1800 - 30*b)/(60 - b)]

P = 30*b*2 - 1800*b - b*c + 60*a

3) In the Stackelberg Model, a Firm will have a higher profit whose ISOPROFIT CURVE will be lower, hence the one who is the leader will have more share and profit i.e. Firm 1.

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