Question

Graphically illustrate the best response functions and best response equilibrium for the case of a duopoly, if each of the firms faces a horizontal marginal cost function.

Answer 1

Let us assume the production function is given as following

P = 31 – 2(Q_A + Q_B)

Assuming that cost functions are given as following

C_A = Q_A

C_B = 2Q_B

The above cost functions show horizontal marginal costs.

The profits of the duopolists are

Π_A = PQ_A – C_A = [31 – 2(Q_A + Q_B)]Q_A – Q_A

Π_A = 31Q_A – 2Q²_A – 2Q_AQ_B – Q_A

Π_A = 30Q_A – 2Q²_A – 2Q_AQ_B

Π_B = PQ_B – C_B = [31 – 2(Q_A + Q_B)]Q_B – 2Q_B

Π_B = 31Q_B – 2Q_AQ_B – 2Q²_B – 2Q_B

Π_B = 29Q_B – 2Q_AQ_B – 2Q²_B

For profit maximization under the Cournot assumption we have

∂Π_A/∂Q_A = 0 = 30 – 4Q_A – 2Q_B

∂Π_B/∂Q_B = 0 = 29 – 4Q_B – 2Q_A

The reaction functions are

Q_A = 7.5 – 0.5Q_B

Q_B = 7.25 – 0.5Q_A

Replacing Q_B into the Q_A reaction function we get

Q_A = 7.5 – 0.5(7.25 – 0.5Q_A)

Q_A = 5.2

And

Q_B = 7.25 – 0.5Q_A

Q_B = 7.25 – 0.5(5.2)

Q_B = 4.7

Thus, the total output in the market is

Q = Q_A + Q_B = 5.2 + 4.7 = 9.9

And the market price

P = 31 – 2Q

P = 31 – 2(9.9)

P = 11.2

Total Revenue (TR) = Price × Quantity = P × Q

Marginal revenue of firm A (MR_A) = ∂TR_A/∂Q_A = ∂(PQ_A)/∂Q_A = P + Q_A(∂P/∂Q)

MR_A = 11.2 + 5.2(– 2)

MR_A = 0.8

MR_B = 11.2 + 4.7(– 2)

MR_B = 1.8

The above calculation shows that the firm with the larger output has the smaller marginal revenue. The profits of the duopolists are

Π_A = PQ_A – C_A

Π_A = (11.2 × 5.2) – 5.2

Π_A = 58.24 – 5.2

Π_A = 53.04

And

Π_B = PQ_B – C_B

Π_B = (11.2 × 4.7) – (2 × 4.7)

Π_B = 52.64 – 9.4

Π_B = 43.24