Question

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

Answer 1

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

Collusive Model

We have the following information

Demand equation: P = 18 – (Q₁ + Q₂)

Assuming Q = Q₁ + Q₂

Total Cost (TC) equation for Firm 1: C₁ = 0.5Q²₁

Total Cost (TC) equation for Firm 2: C₂ = 0.5Q²₂

The main aim of the central agency running the cartel is to maximize the total profit of the cartel

Π_T = Π₁ + Π₂

Where

Π₁ = TR₁ – TC₁ and Π₂ = TR₂ – TC₂

TR = Total Revenue

TC = Total Cost

Thus

Π_T = (TR₁ + TR₂) – TC₁ – TC₂

Π_T = P(Q₁ + Q₂) – 0.5Q²₁ – 0.5Q²₂

Π_T = (18 – Q₁ – Q₂)(Q₁ + Q₂) – 0.5Q²₁ – 0.5Q²₂

Π_T = 18Q₁ – Q₁² – Q₁Q₂ + 18Q₂ – Q₁Q₂ – Q₂² – 0.5Q²₁ – 0.5Q²₂

Π_T = 18Q₁ + 18Q₂ – 2Q_AQ_B – 1.5Q_B² – 1.5Q_A²

Setting the partial derivatives equal to zero we obtain

∂Π₁/∂Q₁ = 18 – 3Q₁ – 2Q₂ = 0

∂Π₂/∂Q₂ = 18 – 3Q₂ – 2Q₁ = 0

Q₁ = 6 – (2/3)Q₂

Q₂ = 6 – (2/3)Q₁

Solving for Q₁ and Q₂ we obtain

Q₁ = 3.6

Q₂ = 3.6

Total Output: Q = Q₁ + Q₂

Q = 3.6 + 3.6 = 7.2

P = 18 – Q

P = 18 – 7.2

P = 10.8

So, we can see that in Collusive model the price is higher and output is lower as compared to the Cournot model. So, we can say that while consumers will prefer Cournot model, the two firms will prefer Collusive model.