Question

Suppose we have a market demand Q = 18 – P and a cost C(Q) 9) = 3Q?.

Answer 1

First Mover Model (Stackelberg’s Model)

We have the following information

Market the demand function is

P = 18 – (Q₁ + Q₂)

Q = Q₁ + Q₂

and the duopolists' costs are

Total Cost of Firm 1: C₁ = 0.5Q²₁

Total Cost of Firm 2: C₂ = 0.5Q²₂

The reaction functions are found by taking the partial derivatives of the duopolists' profit functions and equating them to zero:

Π₁ = PQ₁ – C₁

Π₁ = [18 – (Q₁ + Q₂)]Q₁ – 0.5Q²₁

Π₁ = 18Q₁ – Q²₁ – Q₁Q₂ – 0.5Q²₁

Π₁ = 18Q₁ – 1.5Q²₁ – Q₁Q₂

Π₂ = PQ₂ – C₂

Π₂ = [18 – (Q₁ + Q₂)]Q₂ – 0.5Q²₂

Π₂ = 18Q₂ – Q²₂ – Q₁Q₂ – 0.5Q²₂

Π₂ = 18Q₂ – 1.5Q²₂ – Q₁Q₂

The partial derivatives are

∂Π₁/∂Q₁ = 18 – 3Q₁ –Q₂

∂Π₂/∂Q₂ = 18 – 3Q₂ –Q₁

The reaction functions are

Q₁ = 6 – (1/3)Q₂ ----- Firm 1 Reaction Curve

Q₂ = 6 – (1/3)Q₁ ------ Firm 2 Reaction Curve

Stackelberg's solution with Firm 1 being the sophisticated leader or first mover

Firm 1 will substitute the reaction function of Firm 2 in its own profit equation, which it will then maximise as if it were a monopolist:

Π₁ = PQ₁ – C₁ = 18Q₁ – 1.5Q²₁ – Q₁Q₂

Substitute, Q₂ = Q₂ = 6 – (1/3)Q₁

Π₁ = 18Q₁ – 1.5Q²₁ – Q₁(6 – (1/3)Q₁)

Maximise:                                                       Π₁ = 12Q₁ – 1.17Q²₁

First-order condition: ∂Π₁/∂Q₁ = 12 – 2.34Q₁ = 0

This yields output:                 Q₁ = 5.1

Π₁ = 12Q₁ – 1.17Q²₁

Π₁ = 12(5.1) – 1.17(5.1)²

Π₁ = 30.77

Firm 2 would be the follower. It would assume that Firm 1 would produce 5.1 units; thus Firm 2 substitutes this amount in its reaction function

Q₂ = 6 – (1/3)Q₁

Q₂ = 6 – (1/3)(5.1)

Q₂ = 4.3

Π₂ = 18Q₂ – 1.5Q²₂ – Q₁Q₂

Π₂ = (18 × 4.3) – 1.5(4.3)² – (5.1 × 4.3)

Π₂ = 77.4 – 27.74 – 21.93

Π₂ = 27.73

Thus, the total output in the market is

Q = Q₁ + Q₂ = 5.1 + 4.3 = 9.4

And the market price

P = 18 – Q

P = 18 – 9.4

P = 8.6

Cournot Model

P = 18 – Q

P = 18 – (Q₁ + Q₂)

Cost function of Firm 1: C₁ = 0.5Q²₁

Cost function of Firm 2: C₂ = 0.5Q²₁

The profits of the duopolists are

Π₁ = PQ₁ – C₁ = [18 – (Q₁ + Q₂)]Q₁ – 0.5Q²₁

Π₁ = 18Q₁ – Q²₁ – Q₁Q₂ – 0.5Q²₁

Π₁ = 18Q₁ – 1.5Q²₁ – Q₁Q₂

Π₂ = PQ₂ – C₂ = [18 – (Q₁ + Q₂)]Q₂ – 0.5Q²₂

Π₂ = 18Q₂ – Q²₂ – Q₁Q₂ – 0.5Q²₂

Π₂ = 18Q₂ – 1.5Q²₂ – Q₁Q₂

For profit maximization under the Cournot assumption we have

∂Π₁/∂Q₁ = 0 = 18 – 3Q₁ – Q₂

∂Π₂/∂Q₂ = 0 = 18 – 3Q₂ – Q₁

The reaction functions are

Q₁ = 6 – (1/3)Q₂

Q₂ = 6 – (1/3)Q₁

Replacing Q₂ into the Q₁ reaction function we get

Q₁ = 6 – 1/3[6 – (1/3)Q₁]

Q₁ = 4.5

And

Q₂ = 6 – (1/3)Q₁

Q₂ = 6 – (1/3)(4.5)

Q₂ = 4.5

Thus, the total output in the market is

Q = Q₁ + Q₂ = 4.5 + 4.5 = 9

And the market price

P = 18 – Q

P = 18 – 9

P = 9

Π₁ = PQ₁ – C₁

Π₁ = (9 × 4.5) – 0.5(4.5)²

Π₁ = 40.5 – 10.125

Π₁ = 30.36

Π₂ = PQ₂ – C₂

Π₂ = (9 × 4.5) – 0.5(4.5)²

Π₂ = 40.5 – 10.125

Π₂ = 30.36

In the First Mover Model the price is lower and output is higher as compared to the Cournot model. So, we can say that while consumers will prefer First Mover Model.

However, for Firm 2 profit is higher under Cournot Model so, Firm 2 will prefer Cournot Model.