Question

Consider the Stackelberg model with demand function p(q_1, q_2)=10-q_1-q_2p(q1​,q2​)=10−q1​−q2​ and cost functions c_1(q_1)=q_1c1​(q1​)=q1​, c_2(q_2)=2q_2c2​(q2​)=2q2​.

Draw the game tree and solve for the pure SPE.

Write down each firm's strategy in the pure SPE here: Firm 1:, Firm 2:

Answer 1

p=10-q1-q2
c1(q1)=q1
c2​(q2​)=2q2​
(Game tree in the picture attached)
Pure SPE:
Using backward induction
F2 takes q1 as given and maximises profit
(Max wrt to q2)[ (10-q1-q2)q2 – 2q2 ]
FOC(q2)=0
10-q1-2q2-2=0
q2(q1)=(8-q1)/2
This is the best response function of F2.
F1 knows how F2 will react.
F1 maximises own profit, given the reaction curve:
(max wrt to q1)[ (10-q1-q2)q1 – q1 ]
subject to q2=(8-q1)/2
(max wrt to q1)[ (10-q1-(8-q1)/2)q1 – q1 ]
FOC(q1)=0
Solving gives q1s= 5 ------ This is the strategy of F1
So q2s= (8-5)/2 = 1.5
P = 10-1.5-5 = 3.5