Question

In a duopoly market firms face the market demand curve P=280-Q where Q=q1+q2.
Each firm has a marginal cost of 40 taka per unit. What is the Cournot equilibrium?

Answer 1

In Cournot duopoly market the firm takes the decision of output simultaneously assuming other firm keeps its quantity constant.

The point at which best response curve of firms intersect is the equilibrium point.

The given demand function is:

$P = 280 − q_1−q_2$

The best response function is determined by intersection of marginal revenue (MR) and marginal cost (MC) curve.

Best function function of firm 1:

Determine the marginal revenue (MR₁) of firm 1:

$TR_1 = Pq_1\\=(280−q_1 −q_2)q_1\\MR_1=\frac{ ∂TR_1}{∂q_1}\\=280 − 2q_1 −q_2\\ Equate\space MR\space and\space MC:\\ MR_1 =MC\\280 − 2q_1 −q_2=40\\ 240 − q_2=2q_1\\ 120 − 0.5q_2=q_1(1)$

Equation (1) is bet response function of firm 1.

Best function function of firm 2:

Determine the marginal revenue (MR₂) of firm 2:

$TR_2 = Pq_2\\=(280−q_1 −q_2)q_2\\MR_2=\frac{ ∂TR_2}{∂q_2}\\=280 − q_1 −2q_2\\ Equate\space MR\space and\space MC:\\ MR_2 =MC\\280 − q_1 −2q_2=40\\ 240 − q_1=2q_2\\ 120 − 0.5q_1=q_2(1)$

Equation (2) is bet response function of firm 1.

Substitute equation (2) in equation (1):

$120 − 0.5q_2=q_1\\ 120 − 0.5(120−0.5q_1) =q_1\\120 − 60 + 0.25q_1=q_1\\ 60 = 0.75q_1\\q_1= 80$

Substitute the value of q₁ in equation 2:

$120 − 0.5q_1=q_2\\ 120 − 0.5(80)=q_2\\ 80 =q_2$

Substitute the value of q₁ and q₂ in demand function to determine P:

$P = 280 − q_1 −q_2\\=280 − 80 − 80\\=120$

Therefore, at equilibrium quantity produced by firm 1 is 80 units, quantity produced by firm 2 is 80 units and price charged is 120 taka per unit.