Question

Question:Consider a market with the following demand: P=56?2Q

P=56?2Q

If the market is served by two duopolists with the same cost structure, no fixed cost but $20 cost per unit, each firm's total cost is $20Q.

Find each firm's reaction function.

Determine the profit-maximizing output for each seller.

Determine the equilibrium price.

Calculate each firm's profit.

How much total profit is earned in the market?

Answer 1

Each firm’s marginal cost function is MC= 20 and the market demand function is p = 56 – 2Q 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 = (56 – 2(q₁ + q₂))*q₁ = 56₁ – 2q₁² – 2q₁q₂.

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

MR₁ = 56 – 4q₁ – 2q₂

MC₁ = 20

Profit maximization implies:

MR₁ = MC₁

56 – 4q₁ – 2q₂ = 20

which gives the best response function:

q₁ = 9 - 0.5q₂.

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

q₂ = 9 - 0.5q₁.

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

q₂ = 9 - 0.5(9 - 0.5q₂)

q₂ = 9 - 4.5+ 0.25q₂

0.75q₂ = 4.5

q₁ = q₂ = 6

Thus,

Q = q₁ + q₂ = 6 + 6 = 12

P = 56 - 2*12 = 32

Profit by each firm = (32 - 20)*6 = 72 and total profit is 144

Find each firm's reaction function.

q₁ = 9 - 0.5q₂ and q₂ = 9 - 0.5q₁

Determine the profit-maximizing output for each seller.

q₁ = q₂ = 6

Determine the equilibrium price.

$32 per unit

Calculate each firm's profit.

$72

How much total profit is earned in the market

$144