Question

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.

Answer 1

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.