Question

(Courtnot competition) Suppose there are two local firms (firm 1 and firm 2) that supply doormats to a particular area. The demand for mats as a function of its price is given by M = 1200 - 20P, where M is the amount demanded when the price is P per mat. The cost of production for firm 1 is C1(m1) = 10m1 and that for firm 2 is C2(h2) = 20m2 respectively.

Does a Nash equilibrium exists? State what are the output choices of each firm in the Nash equilibrium.

Answer 1

The marginal cost of firm 1 is 10 and that of firm 2 is 20. The market demand function is P = 1200/20 – M/20 or P = 60 – 0.05M , Where M is the sum of each firm’s output M1 and M2_.

Find the best response functions for firm 1:

Revenue for firm 1

R₁ = P*M1 = (60 – 0.05(M1 + M2))*M1 = 60M1 – 0.05M1² – 0.05M1M2.

Firm 1 has the following marginal revenue and marginal cost functions:

MR₁= 60 – 0.1M1 – 0.05M2

MC₁ = 10

Profit maximization implies:

MR₁ = MC₁

60 – 0.1M1 – 0.05M2 = 10

which gives the best response function:

M1 = 500 - 0.5M2.

Find the best response functions for firm 2:

Revenue for firm 2

R₂ = P*M2 = (60 – 0.05(M1 + M2))*M2 = 60M2 – 0.05M2² – 0.05M1M2.

Firm 2 has the following marginal revenue and marginal cost functions:

MR₂= 60 – 0.1M2 – 0.05M1

MC₂ = 20

Profit maximization implies:

MR₂ = MC₂

60 – 0.1M2 – 0.05M1 = 20

which gives the best response function:

M2 = 400 - 0.5M1.

Cournot equilibrium is determined at the intersection of these two best response functions

M2 = 400 – 0.5(500 - 0.5M2)

M2 = 150 + 0.25M2

0.75M2 = 150

M2 = 200 units

M1 = 500 – 200*0.5 = 400 units

In the Nash equilibrium output choices are 400 units by firm 1 and 200 units by firm 2.