Answer to Question #283163 in Microeconomics for lilly

Question

In a duopoly market, two firms produce the identical products, the cost function of firm 1 is: C₁=20q₁, the cost function of firm 2 is: C₂=20q₂ , the market demand function is: P=500-2Q , here Q = q₁+q₂

In a Bertrand model, the two firms set their price simultaneously, assume both firms do not have production capacity limits, and there is no collusion. What is the market equilibrium price and quantity?

If the two firms decide to form a Cartel, i.e. they want to maximize the profit of the whole industry, and then split the production and profit evenly. What is the market price? What is the industry’s total quantity produced? What is the quantity produced and profit of each firm?

Answer 1

Solution:

The market equilibrium price and quantity in a Bertrand model:

TR₁ = P $\times$ Q₁ = (500 – 2Q_{1 +} Q₂) $\times$ Q₁ = 500Q₁ – 2Q₁² + Q₂Q₁

MR₁ = $\frac{\partial TR_{1} } {\partial Q_{1} }$ = 500 – 4Q₁ + Q₂

Set MR₁ to MC₁:

500 – 4Q₁ + Q₂ = 20

500 – 20 - 4Q₁ + Q₂ = 0

480 - 4Q₁ + Q₂ = 0

Q₁ = 120 + 0.25Q₂

 

TR₂ = P $\times$ Q₂ = (500 – 2Q_{1 +} Q₂) $\times$ Q₂ = 500Q₂ – 2Q₁Q₂ + Q₂²

MR₂ = $\frac{\partial TR_{2} } {\partial Q_{2} }$ = 500 – 2Q₁ + 2Q₂

Set MR₂ to MC₂:

500 – 2Q₁ + 2Q₂ = 20

500 – 20 - 2Q₁ + 2Q₂ = 0

480 - 2Q₁ + 2Q₂ = 0

240 – Q₁ + Q₂ = 0

Q₂ = 240 + Q₁

 

Q₁ = 120 + 0.25Q₂

Q₁ = 120 + 0.25(240 + Q₁)

Q₁ = 120 + 60 + 0.25Q₁

Q₁ – 0.25Q₁ = 180

0.75Q₁ = 180

Q₁ = 240

 

Q₂ = 240 + Q₁ = 240 + 240 = 480

Market equilibrium quantity = 240 + 480 = 720

Substitute to derive the market equilibrium price:

P=500-2Q = 500 – 2(720) = 500 – 1440 = -940