Question

Consider an industry consisting of two firms producing an identical product. The inverse market demand equation is P = 100 − 2Q. The total cost equations for firms 1 and 2 are TC1 = 4Q1 and TC2 = 4Q2, respectively. Firm 1 is the Stackelberg leader and firm 2 is the Stackelberg follower. The profit of the Stackelberg follower is:

• $864.
  
  

• $576.
  
  

• $432.
  
  

• $288.
  
  

$1,152.

Answer 1

In Stackelberg model where firm 1 is a first mover, it must take the reaction function of firm 2 in its computation of marginal revenue.

Derivation of firm 2’s reaction function

Total revenue of firm 2 = P*(q2) = (100 – 2(q1 + q2))q2 = 100q2 – 2q2² – 2q1q2

Marginal revenue = 100 – 4q2 – 2q1

Marginal cost = 4

Solve for the reaction function

100 – 4q2 – 2q1 = 4

This gives q2 = 24 - 0.5q1

Incorporate this in the reaction function o2f firm 1

Total revenue for firm 1 = P*(q1) = (100 – 2(q1 + q2))q1

TR = 100q1 - 2q1^2 - 2q1q2

= 100q1 - 2q1^2 - 2q1*(24 - 0.5q1)

= 100q1 - 2q1^2 - 48q1 + q1^2

= 52q1 - q1^2

MR = MC

52 - 2q1 = 4

q1 = 24 and so q2 = 12 units.

Price = (100 - 2*36) = $28

Profit for follower = (28 - 4)*12 = $288

Select $288