Question

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?

Answer 1

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