Question

Consider a market where two firms sell an identical product to consumers and face the
following inverse demand function p = 100 - q1 - q2
but the firms face different marginal costs. Firm 1 has a constant marginal cost of MC1 = 10
and firrm 2 has a constant marginal cost of MC2 = 40.

a) What is firm 1s best response function?

b) What is firm 2's best response function?

c) What are the equilibrium quantities, price and profits for both firms?

Answer 1

First firm’s marginal cost function is MC = 10 and second firm's MC = 40 and the market demand function is P = 100 – (q1 + q2) where Q is the sum of each firm’s output q₁ and q_2.

Find the best response functions for both firms:

a) Revenue for firm 1

R₁ = P*q₁ = (100 – (q₁ + q₂))*q₁ = 100q₁ – q₁² – q₁q₂.

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

MR₁ = 100 – 2q₁ – q₂

MC₁ = 10

Profit maximization implies:

MR₁ = MC₁

100 – 2q₁ – q₂ = 10

which gives the best response function:

q₁ = 45 - 0.5q₂.

Find the best response functions for firm 2:

b) MR₂ = 100 – 2q₂ – q₁

MC2 = 40

Profit maximization implies:

MR₂ = MC₂

100 – 2q₂ – q₁= 40

which gives the best response function:

q₂ = 30 - 0.5q₁

c) Solve them to get

q₂ = 30 - 0.5*(45 - 0.5q₂)

q2 = 7.5 + 0.25q2

This gives q2 = 10 and q1 = 45 - 10*0.5 = 40 units Price = 100 - 50 = $50 Profit 1 = (50 - 10)*40 = 1600 and profit 2 = (50 - 40)*10 = 100.