In: Economics
Question 2: Firm Competitions
Two firms, Firm 1 and Firm 2, produce the same goods and are competing in the same market. Firm 1 has a cost function of c1 = 20q1 and Firm 2 has a cost function of
c2 = 10q2.
The market price is determined by the inverse demand function p=100−q1 −q2
(a) Suppose Firm 1 and Firm 2 compete in a quantity competition. And suppose both firms decidesimultaneously. What are the Cournot-Nash equilibrium quantities of Firm 1 and Firm 2? Show your work.
(b) Suppose Firm 1 and Firm 2 compete in a quantity competition. And suppose Firm 1 moves first. Firm 2 observes Firm 1’s choice and moves second. What are the Stackelberg-Nash equilibrium quantities of Firm 1 and Firm 2? Show your work.
(c) Are the answers in (a) and (b) the same? Why or why not? Briefly explain in your own words.
(d) Now assume that the firms engage in a price competition and that the firms decide on their prices simultaneously. What are the Bertrand-Nash equilibrium prices? You do not need to show any calculation, but you need to explain your logic. Note: You can assume a price increment of $0.01 if needed.
Cournot Equilibrium
We have the following information
p = 100 – (q1 + q2)
C1 = 20q1
C2 = 10q2
The profits of the duopolists are
Π1 = pq1 – C1 = [100 – (q1 + q2)]q1 – 20q1
Π1 = 100q1 – q21 – q1q2 – 20q1
Π1 = 80q1 – q21 – q1q2
Π2 = pq2 – C2 = [100 – (q1 + q2)]q2 – 10q2
Π2 = 100q2 – q1q2 – q22 – 10q2
Π2 = 90q2 – q1q2 – q22
For profit maximization under the Cournot assumption we have
∂Π1/∂q1 = 0 = 80 – 2q1 – q2
∂Π2/∂q2 = 0 = 90 – 2q2 – q1
The reaction functions are
q1 = 40 – 0.5q2
q2 = 45 – 0.5q1
Replacing q2 into the q1 reaction function we get
q1 = 40 – 0.5(45 – 0.5q1)
q1 = 40 – 22.5 + 0.25q1
0.75q1 = 17.5
q1 = 23.33 or 23
And
q2 = 45 – 0.5q1
q2 = 45 – 0.5(23.3)
q2 = 33.33 or 33
Thus, the total output in the market is
q = q1 + q2 = 23 + 33 = 56
And the market price
p = 100 – (q1 + q2)
p = 100 – (23 + 33)
p = 100 – 56
p = 44
Total Revenue (TR) = Price × Quantity = p × q
Marginal revenue of firm 1 (MR1) = ∂TR1/∂q1 = ∂(pq1)/∂q1 = p + q1(∂p/∂q1)
MR1 = 44 + 23(– 1)
MR1 = 44 – 23
MR1 = 21
MR2 = 44 + 33(– 1)
MR2 = 44 – 33
MR2 = 11
The above calculation shows that the firm with the larger output has the smaller marginal revenue. The profits of the duopolists are
Π1 = pq1 – C1
Π1 = (44 × 23) – 20(23)
Π1 = 1012 – 460
Π1 = 552
And
Π2 = pq2 – C2
Π2 = (44 × 33) – 10(33)
Π2 = 1452 – 330
Π2 = 1122
The second order condition is satisfied for both the duopolists
∂Π1/∂q1 = 0 = 80 – 2q1 – q2
∂2Π1/∂q21 = – 2 < 0
∂Π2/∂q2 = 0 = 90 – 2q2 – q1
∂2Π2/∂q22 = – 2 < 0
Stackelberg Equilibrium
We have the following information
p = 100 – (q1 + q2)
and the duopolists' costs are
C1 = 20q1
C2 = 10q2
The reaction functions are found by taking the partial derivatives of the duopolists' profit functions and equating them to zero:
The profits of the duopolists are
Π1 = pq1 – C1 = [100 – (q1 + q2)]q1 – 20q1
Π1 = 100q1 – q21 – q1q2 – 20q1
Π1 = 80q1 – q21 – q1q2
Π2 = pq2 – C2 = [100 – (q1 + q2)]q2 – 10q2
Π2 = 100q2 – q1q2 – q22 – 10q2
Π2 = 90q2 – q1q2 – q22
∂Π1/∂q1 = 0 = 80 – 2q1 – q2
∂Π2/∂q2 = 0 = 90 – 2q2 – q1
The reaction functions are
q1 = 40 – 0.5q2----- Reaction Curve of Firm 1
q2 = 45 – 0.5q1------ Reaction Curve of Firm 2
(I) Stackelberg's solution with Firm 1 being the sophisticated leader
Firm 1 will substitute reaction function of Firm 2 in its own profit equation, which it will then maximise as if it were a monopolist:
Π1 = 80q1 – q21 – q1q2
Substitute, q2 = 45 – 0.5q1
Π1 = 80q1 – q21 – q1(45 – 0.5q1)
Π1 = 35q1 – 0.5q21
Maximise: Π1 = 35q1 – 0.5q21
First-order condition: ∂Π1/∂q1 = 35 – q1 = 0
This yields output: q1 = 35
Π1 = 35q1 – 0.5q21
Π1 = 35(35) – 0.5(35)21
Π1 = 1225 – 612.5
Π1 = 612.5
Second order condition: ∂2Π1/∂q21 = – 1 < 0
So, the second-order condition for profit maximisation is fulfilled
Firm 2 would be the follower. It would assume that Firm 1 would produce 35 units; thus Firm 2 substitutes this amount in its reaction function
q2 = 45 – 0.5q1
q2 = 45 – 0.5(35)
q2 = 27.5
Π2 = 90q2 – q1q2 – q22
Π2 = 90(27.5) – (35 × 27.5) – (27.5)22
Π2 = 756.25
p = 100 – (q1 + q2)
p = 100 – (35 + 27.5)
p = 37.5
Bertrand-Nash Equilibrium
The model assumes that the firms compete with each other on the basis of price. So unlike the Cournot model, in the Bertrand model the firms choose prices conditional on what they expect their rivals’ price to be. So, the Nash equilibrium in such a situation is similar to the competitive outcome. In other words, both the firms will set price equal to their respective marginal costs.
C1 = 20q1
C2 = 10q2
Marginal cost of Firm 1 (MC1) = ∂C1/∂q1 = 20
Marginal cost of Firm 2 (MC2) = ∂C2/∂q2 = 10
So, Firm 1 will charge a price of 20, and Firm 2 will charge a price of 10.