In: Economics
Consider three firms that face market demand P= 99-Q . The cost functions are C1(q1)=6q1*2for firm 1, C2=2q2*2( C2 = (2q2)square ) for firm 2, andC3(q3)=2q3*2 for firm 3. Firm 1 is the Stackelberg leader and firms 2 and 3 are the followers. What is firm 1's equilibrium output q1?
P= 99-Q
Q= q1+q2+q3
C1= 6q12
C2= 2q22
C3= 2q32
Firm 1 is the Stackelberg leader and firms 2 and 3 are the followers.
Calculate the best response functions of followers:
For the best response function of firm 2:
Profit of firm 2= Pr2= P x q2 - C2
Pr2= (99-q1-q2-q3)q2- 2q22
Differentiate it with respect to q2:
dPr2/dq2= 99-q1-2q2-q3-4q2= 0
6q2= 99-q1-q3
q2= (99-q1-q3)/6 Best response function of firm 2
For the best response function of firm 3:
Profit of firm 3= Pr3= P x q3 - C3
Pr3= (99-q1-q2-q3)q3- 2q32
Differentiate it with respect to q3:
dPr3/dq3= 99-q1-q2-2q3-4q3= 0
6q3= 99-q1-q2
q3= (99-q1-q2)/6 Best response function of firm 3
Use the value of q3= (99-q1-q2)/6 into Best response function of firm 2
q2= (99-q1-q3)/6
6q2= 99-q1-(99-q1-q2)/6
Take LCM
36q2= 594-6q1-99+q1+q2
35q2= 495-5q1
q2= (495-5q1)/35
q2= (99-q1)/7 Equation 1
Use equation 1 into the best response function of firm 3:
q3= (99-q1-q2)/6
6q3= 99-q1-(99-q1)/7
42q3= 693-7q1-99+q1
42q3= 594-6q1
q3= (594-6q1)/42
q3= (99-q1)/7 Equation 2
Profit of firm 1= Pr1= P x q1 - C1
Pr1= (99-q1-q2-q3)q1 - 6q12
Use Equation 1 and Equation 2 in profit function of firm 1:
Pr1= [99-q1-(99-q1)/7 -(99-q1)/7]q1 - 6q12
Pr1= [99q1-q12-(99q1-q12)/7 -(99q1-q12)/7] - 6q12
Pr1= [ 693q1-7q12-99q1+q12-99q1+q12]/7 - 6q12
Pr1= [495q1-5q12]/7 - 6q12
Differentiate with respect to q1
dPr1/dq1= [495-10q1]/7-12q1= 0
495/7= 12q1+(10q1/7)
495= 94q1
q1= 5.26 Equilibrium quantity q1