Question

consider 2 profit-maximizing firms that behave as quantity setters and setter and supply a homogeneous product . market demand is P=20-Q . Total cost of each firm are

CCqi1= 1/2qi^2

a) calculat the equilibrium quantities and the profit of each firm

b)suppose the two firms play a trigger strategy but firm 1 decides to cheat on the cartel agreement. what quantity would it set ? what would its profits be in the the period that it defects?

c) for what values of the discount factor will the cartel be stable ?

d) now suppose that the two firms play a repeated game and decide to form a cartel . they make a cartel agreement that maximize their joint profits and involves each of them setting the same quantity in each period . calculate the cartel quantity for each firm and the resultinging per-period cartel profit for reach firm

Answer 1

A)firm 1 best reaction function,

P=20-q1-q2

TR=20q1-q1^2-q1q2

MR=20-2q1-q2

MC=q1

MR=MC

20-2q1-q1=q1

20-q2=3q1

Q1=20/3-q2/3

By same method and because both firm have same cost function,

q2=20/3-q1/3{best response of firm 2}

Putting q2 into q1

Q1=20/3-{20/3-q1/3}/3=20/3-(20-q1)/9=(60-20+q1)/9

8q1=40

Q1=5

Q2=20/3-5/3=15/3=5

P=20-10=10

TC for each firm =1/2*25=12.5

Profit for each firm=5*10-12.5=37.5

B) monopoly output (means only single firm means other player output equal to zero) =20/3-0/3=20/3

So in cartel each firm produces =monopoly output /2=(20/3)/2=10/3

But firm 1 cheat and try to maximize his profit by selling more QUANTITY,so

Q1=20/3-q2/3

Q2=10/3

Q1=20/3-(10/3)/3=20/3-10/9=50/9

Q=50/9+10/3=80/9

P=20-80/9=100/9

TR of firm 1=(100/9)*(50/9)=5000/81=

TC of firm 1 =1/2*50/9*50/9=1250/81

Profit=5000/81-1250/81=3750/81=46.2

D)cartel Q of each firm =10/3(solved in previous one)

TC of each firm =1/2*10/3*10/3=50/9

Q=10/3+10/3=20/3

P=20-20/3=40/3

TR=40/3*10/3=400/9

Profit of each firm=400/9-50/9=350/9=38.8

C) the stability condition of cartel,

Cartel profit *1/(1-@)≥ cheating from +cournot profit*@/(1-@)

Where profit are of each firm and @ is discount factor.

So putting values of profit,

38.8*1/(1-@)=46.2+37.5*@/(1-@)

38.8=46.2-46.2@+37.5@

-8.7@=-7.4

@=7.4/8.7=0.85

So discount factor should equal or more than 0.85 to make cartel stable.