Question

An industry comprising only two firms produces a homogenous product where the market demand function is given by P = 200 – 2(q₁ +q₂) where q₁ and q₂ are the output levels of firm 1 and firm 2, respectively. The individual firm's cost functions are TC₁ = 4q₁ and TC₂ = 4q₂, where TC₁ and TC₂ are total costs of firm 1 and 2, respectively. If firm 2 is a Stackelberg Leader, and firm 1 is a Follower, then the profit of the Stackelberg Leader will be ________ (rounded off to two decimal places).

Answer 1

Correct Answer: 2390

To solve this problem, we need to determine the profit of firm 2 (the Stackelberg Leader) in a duopoly where firm 1 follows the quantity decision of firm 2.

Step 1: Identify Functions

• Market Demand: P = 200 - 2(q₁ + q₂)
• Cost Functions:
  • Firm 1: TC₁ = 4q₁
  • Firm 2: TC₂ = 4q₂

Step 2: Determine the Follower's Reaction Function

• Firm 1 (Follower) maximizes its profit π₁ = Pq₁ - TC₁.
• Replace P with the demand function: π₁ = (200 - 2(q₁ + q₂))q₁ - 4q₁.
• π₁ = 200q₁ - 2q₁² - 2q₁q₂ - 4q₁.
• To find the reaction function, take the derivative w.r.t. q₁ and set to 0:
• dπ₁/dq₁ = 200 - 4q₁ - 2q₂ - 4 = 0.
• Therefore, q₁ = 49 - 0.5q₂.

Step 3: Stackelberg Leader Maximizes Profit

• Firm 2 maximizes π₂ = Pq₂ - TC₂.
• Substitute q₁ = 49 - 0.5q₂ in the demand function: P = 200 - 2((49 - 0.5q₂) + q₂).
• P = 200 - 98 + q₂ = 102 - q₂.
• π₂ = (102 - q₂)q₂ - 4q₂.
• π₂ = 102q₂ - q₂² - 4q₂.
• To maximize, take the derivative and set it to zero: dπ₂/dq₂ = 102 - 2q₂ - 4 = 0.
• 98 = 2q₂, so q₂ = 49.

Step 4: Calculate Profits

• Substitute q₂ = 49 into q₁'s reaction function: q₁ = 49 - 0.5(49) = 24.5.
• Find P: P = 102 - 49 = 53.
• π₂ = 53(49) - 4(49) = 2597 - 196 = 2401.

Verification

• The profit of firm 2 is 2401, which falls within the given range of (2390,2390).