Question

3) Assuming that firms compete a la Cournot, that all firms have the same marginal cost, and

that demand is linear, when is price most sensitive to changes in marginal cost: in a market

with very few firms or in a market with many firms? Show this formally. [Hint: assume

demand 𝑝 = 𝑎 – 𝑏𝑄]

b) Consider a Bertrand duopoly with differentiated products. Demand curves are given by

𝑝1 = 600 − 2𝑞1 − 𝑞2

𝑝2 = 600 − 𝑞1 − 2𝑞2

Suppose that the cost functions are given by (𝑞i) = 60𝑞i , for 𝑖 = 1, 2. Find the equilibrium

outputs, the prices and the profits.

Answer 1

Solution:

a.). The price is most sensitive to changes in marginal cost when the number of firms increase.

If the oligopoly is symmetric, meaning that all firms have identical products and cost conditions, the degree to which price exceeds marginal cost is inversely related to the number of firms.

 

b.). TR₁ = (600 – 2q₁ – q₂) q₁ = 600q₁ – 2q₁² – q₂q₁

Profit₁ = (600q₁ – 2q₁² – q₂q₁) – 60q₁

$\frac{\partial \pi 1} {\partial q1}$ = 600 – 4q₁ – q₂ – 60

600 – 4q₁ – q₂ – 60 = 0

4q₁ = 540 – q₂

q₁ = 135 – 0.25q₂

TR₂ = (600 – q₁ – 2q₂) q₂ = 600q₂ – q₁q₂ – 2q₂²

Profit₁ = (600q₂ – q₁q₂ – 2q₂²) – 60q₂

$\frac{\partial \pi 2} {\partial q2}$ = 600 – q₁ – 4q₂ – 60

600 – q₁ – 4q₂ – 60 = 0

4q₂ = 540 – q₁

q₂ = 135 – 0.25q₁

 

q₁ = 135 – 0.25q₂

q₁ = 135 – 0.25(135 – 0.25q₁)

q₁ = 135 – 33.75 + 0.0625q₁

q₁ = 101.25 + 0.0625q₁

q₁ – 0.025q₁ = 101.25

0.9375q₁ = 101.25

q₁ = 108

q₂ = 108

Equilibrium quantity = 108

P₁ = 600 – 2q₁ – q₂ = 600 – 2(108) – 108 = 276

Equilibrium price = 276

 

Profit = TR – TC

Profit = (P $\times$ Q) – (60q₁)

Profit = (276$\times$108) – (60 $\times$108)

Profit = 29,808 – 6,480 = 23,328

Profit = 23,328