Question

5. There are three oligopolists who compete on quantity. Firm 1 has cost function c1(q1) = 100 + 10q1. Firm 2 has cost function c2(q2) = 100 + 15q2. And firm 3 has cost function c3(q3) = 100 + 20q3. These cost functions apply to each period. The market demand function is 100-p.

   a. In the first period, all firms compete. Find the equilibrium price and consumer surplus, as well as the profit of each firm, and the total surplus.
   b. In the second period, firm 1 has bought out Firm 3. It ceases production in Firm 3’s plant, so there are no longer any fixed costs associated with Firm 3.Find the equilibrium price and consumer surplus, as well as the profit of each firm, and the total surplus.
   c. Should this merger be allowed? Explain briefly (100 words or less)

Answer 1

Suppose firm 1 takes firm 2’s output choice 2
y as given. Then firm 1’s problem is to
maximize its profit by choosing its output level 1
y . If firm 1 produces 1
y units and firm 2
produces 2
y units then total quantity supplied is 1 2
y + y . Define .
1 2
y ≡ y + y The
market price will be 120 .
1 2 P = − y − y
Firm 1 has a constant marginal and average cost of $15: c1 = 15.
Firm 2 has a constant marginal and average cost of $30: c2 = 30 .
Firm 1’s profit maximization problem:
max ( ) [ ( )] 1 1 2 120 1 2 1 15 1 π y , y = − y + y y − y
1
y
First order conditions:
120 ( ) ( ) 1 2 1 1 15 0
1
1 = − + + − − =
∂
∂
y y y
y
π

2
105
2 105
Since the profit- maximization problem faced by the two firms are NOT symmetric in
this case, we have to explicitly solve Firm 2’s problem to find its best response function
or reaction function.
Firm 2’s profit maximization problem:
max ( ) [ ( )] 2 1 2 120 1 2 2 30 2 π y , y = − y + y y − y
2
y
First order conditions:
120 ( ) ( ) 1 2 2 1 30 0
2
2 = − + + − − =
∂
∂
y y y
y
π

2
90
2 90
90 2 0
1
2
2 1
1 2
y
y
y y
y y
− ⇒ =
⇒ = −
⇒ − − =
So, Firm 2’s best response to 1
y or Firm 2’s best response or reaction function is
: ( )
2
90 1
2 1
y
y R y
−
= = (2)
To find the Cournot-Nash equilibrium quantities, we have to solve equation (1) and
(2) simultaneously for 1
y and 2
y .
Substituting (1) into (2),





 −
= −
2
105
2
1
2
90 2
2
y
y

25
3 75
4
75
4
3
405 2 0
2
1
1 2
1 2
y
y
y y
y y
− ⇒ =
⇒ = −
⇒ − − =
So, Firm 1’s best response to 2
y or Firm 1’s best response or reaction function is
: ( )
2
105 2
1 2
y
y R y
−
= =