Question

Suppose two firms are engaged in Stackelberg Competition. The demand curve is P = 56 - 2Q and MC=20. What is the equilibrium market quantity?

Answer 1

Solution to Stackelberg Model

Given : P= 56-2Q, MC = 20

Step 1 :

Q = q1 + q 2 , where q1 = Output of Firm 1; q2 = Output of firm 2

Deriving the inverse demand function we get :

P = 56-2Q = 56 - 2(q1+q2) = 56 - 2q1 - 2q2 ------------------------------- (1)

Step 2 :

Under stackleberg model one of the firm acts as a leader and the other as the follower. The leader decided the output based on the best response function of the follower also known as the reaction function of the follower. Lets consider - Firm 1 as the leader and Firm 2 as the follower.

Deriving the reaction function of Firm 2 :  

• Calculate Total Revenue of firm 2

TR2 = P*q2 = (56 - 2q1 - 2q2)*q2 = 56q2 - 2q1q2 - (2q2)2 ----------------------------- (2)

• Partially Differentiating Equation (2) with respect to q2 we get MR2 of Firm 2 :

= 56 - 2q1 - 4q2 =

• For equilibrium MR2 = MC

56 - 2q1 - 4q2 = 20

56 -20 = 2q1 + 4 q2

36 = 2q1 + 4q2

36 - 2q1 = 4q2

Therefore, q2 = 9 - 1/2q1 which is the reaction function of firm 2 ------------------------------ (3)

Step 3 : Calculating Firm 1 response to Firm 2 output choice. We incooperate, Firm 2 reaction function in the inverse demand function :

P = 56 - 2q1 - 2 ( 9 - 1/2q1) = 56 - 2q1 - 18 + q1 = 38 - q1

P = 38 - q1

TR1 = P * q1 = (38 - q1)q1 = 38q1 - q12  ------------------------------ (4)

Partially Differentiating eqution (4) with respect to q1, we get MR1

= 38 - 2q1

For profit maximization of Firm 1 : MR1 = MC

38 - 2q1 = 20

38 -20 = 2q1

18 = 2q1

q1 = 9

Step 4: Using q1 in the reaction function of q2 we get the output for Firm 2

q2 = 9 - 1/2q1

q2 = 9 - 1/2 * 9 = 9 - 4.5

q2 = 4.5  

Since q1 = 9 & q2 = 4.5, this shows that when 2 profit maximizing firms have have same MC the leader (here Firm ) will have an advantage over the follower (here Firm 2) and will produce more than the follower based on the follwer's best response.

Step 5 :

Equillibrium Market Quantity (Q) = q1 + q2 = 9 + 4.5 = 13.5

Equillibrium Price (P) = 56 - 2Q = 56 - 2 *13.5 = 56 - 27 = 29