A Cournot oligopoly has 2 firms, and inverse market demand $P = 60 - Q$. All firms have marginal cost, $M C = \$ 20$. The equilibrium price in this market will be:
A) $\$ 20.50$
B) $\$ 22$
C) $\$ 33.33$
D) $\$ 40.15$

Question

Quizplus · Accepted Answer

To find the equilibrium output for each firm, we need to solve for the best response function. The profit for each firm is given by:
π₁ = (P - 20)Q₁
π₂ = (P - 20)Q₂

Taking the derivative of each firm's profit with respect to its output, we get:

MR₁ = 60 - 2Q - Q₂
MR₂ = 60 - 2Q - Q₁

Setting each firm's MR equal to its MC, we get:

60 - 2Q - Q₂ = 20
60 - 2Q - Q₁ = 20

Solving for Q₁ and Q₂, we get:

Q₁ = Q₂ = 20

Plugging this into the demand function, we get:

P = 60 - 20 - 20 = 20

Therefore, the equilibrium price in this market is $$ 20$. However, since the question asks for the equilibrium price for the entire market, we need to sum the quantities of each firm to get:

Q = Q₁ + Q₂ = 20 + 20 = 40

And the equilibrium price is:

P = 60 - 40 = $$ 20$

Note that this is a very competitive market, with each firm producing a large quantity and the price being equal to the marginal cost. The quantity demanded in this market is given by:

Q = 60 - P

Each firm's profit function is:

πi = (P - MC) qi

where i = 1, 2, and q1 and q2 are the quantities produced by firms 1 and 2 respectively.

To find the Cournot equilibrium, we need to solve for the best response of each firm given the other's output. That is, firm 1 will set its quantity as:

q1 = (1/2)(60 - q2 - MC)

And firm 2 will set its quantity as:

q2 = (1/2)(60 - q1 - MC)

Simplifying these equations, we get:

q1 = 20 + (1/2)q2 
q2 = 20 + (1/2)q1

Substituting the second equation into the first, we get:

q1 = 20 + (1/2)(20 + (1/2)q1)

Solving for q1, we get:

q1 = 26.67

Substituting this back into the equation for q2, we get:

q2 = 26.67

Therefore, the total quantity produced in this market is:

Q = 53.33

And the equilibrium price is:

P = 60 - Q = 33.33

Therefore, the answer is C.

A Cournot Oligopoly Has 2 Firms, and Inverse Market Demand P=60−QP = 60 - QP=60−Q

A Cournot Oligopoly Has 2 Firms, and Inverse Market Demand $P = 60 - Q$