Question

Two firms compete in selling identical widgets. They choose their output levels Upper Q1 and Q2 simultaneously and face the demand curve:

P=100? Q, where Q=Q₁+Q₂.

Until recently, both firms had zero marginal costs. Recent environmental regulations have increased Firm 2's marginal cost to $50. Firm 1's marginal cost remains constant at zero.

True or false: As a result, the market price will rise to the monopoly level. As a result of Firm 2's marginal cost rising to $50 , the market price:

a)will rise to the monopoly level because Firm 2 will not produce.

b)will rise to the monopoly level because each firm will produce half the monopoly output level.

c)will rise to the monopoly level because both firms will produce less output.

d) will not rise to the monopoly level because Firm 1 will not be profitable.

e) will not rise to the monopoly level because one firm can produce at lower cost than multiple firms.

I am looking for a detailed answer, so please show your work. It is very important for me to understand and learn how to solve this problem.

Answer 1

Two firms choose their output levels Q1 and Q2 simultaneously and face the demand curve:

P=100 - Q, where Q=Q1+Q2. Hence they compete as Cournot competitors

Until recently, both firms had zero marginal costs. In that case we have to first find the Cournot solution before the change.

Each firm’s marginal cost function is MC = 0 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:

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 = 0

Profit maximization implies:

MR₁ = MC₁

100 – 2q₁ – q₂ = 0

which gives the best response function:

q₁ = 50 - 0.5q₂.

By symmetry, Firm 2’s best response function is:

q₂ = 50 - 0.5q₁.

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

q₁ = 50 - 0.5(50 - 0.5q₁)

q₁ = 25 + 0.25q₁

This gives q₁ = q₂ = 33.33 units This the Cournot solution

Market price is 100 - 33.33 - 33.33 = 33.34

Now with MC1 = 0 and MC2 = 50, we have

Profit maximization implies:

MR₁ = MC₁

100 – 2q₁ – q₂ = 0

which gives the best response function:

q₁ = 50 - 0.5q₂.

Now Firm 2’s best response function is:

MR₂ = 100 – 2q₂ – q₁

But its marginal cost is MC₂ = 50

Profit maximization implies:

MR₂ = MC₂

100 – 2q₂ – q₁ = 50

which gives the best response function:

q₂ = 25 - 0.5q₁.

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

q₂ = 25 - 0.5(50 - 0.5q₂)

q₂ = 0 + 0.25q₂

q₂ = 0

This gives q₁ = 50 and q₂ = 0 units This the Cournot solution

Hence the correct option is a)will rise to the monopoly level because Firm 2 will not produce.