Question

Consider a three-firm oligopoly in which the market demand for the homogeneous good is given by q = 24 - p, and costs are zero. Suppose firm 1 and 2 simultaneously pick their output, and then firm 3, observing these choices, picks its output (i.e. two “leaders”, one “follower”). Find the subgame perfect equilibrium quantities produced by these forms.

Answer 1

AR: p=24-q, Where q=q1+q2+q3

For the follower best response curve:

Profit= Total revenue-total cost= p*q3-0= 24q3-q1q3-q2q3-q32

Differentiate profit with respect to q3

dProfit/dq= 24-q1-q2-2q3=0

q3= (24-q1-q2) / 2 Best response curve of firm 3

Now for equilibrium in firm 1 and 2 simultaneous game:

Profit(firm 1)= 24-q12-q1q2-q1q3

Put value of q3 from its BRS

Profit(firm 1)= 24q1-q12-q1q2-q1( (24-q1-q2) / 2 )

Differentiate profit of 1 with respect to q1

dprofit1/dq= 24-2q1-q2-((24-2q1-q2)/2)=0

48-4q1-2q2-24+2q1+q2=0

24= 2q1+q2 equation 1

Profit (firm 2)= 24q2-q22-q1q2-q2( (24-q1-q2) / 2 )

Put value of q3 from its BRS

Profit(firm 1)= 24q2-q22-q1q2-q2( (24-q1-q2) / 2 )

Differentiate wrt q2

dprofit2/dq2= 24-2q2-q1-((24-q1-2q2)/2)=0

q1+2q2=24 equation 2

Solve equation 1 and 2

q1=8

q2=8

Put the value of q1 and q2 in BRS of firm 3

q3= 24-8-8 / 2= 8/2= 4

q= 8+8+4= 20

p= 24-q= 24-20= 4