Model Answer
0 min readIntroduction
Duopoly markets, characterized by two firms, represent a crucial intermediate step in understanding market structures beyond perfect competition and monopoly. The Stackelberg model, a variant of the Cournot model, analyzes scenarios where firms make output decisions sequentially. This contrasts with the simultaneous decision-making in the standard Cournot model. Understanding these dynamics is vital for analyzing industries with dominant firms and their impact on market outcomes. This answer will determine the equilibrium output, price, and profit for both firms in the given duopoly scenario, where Firm 1 acts as the leader and Firm 2 as the follower.
Derivation of Equilibrium
We are given the market demand function: P = 100 – 2Q, where Q = q₁ + q₂. The marginal cost (MC) for both firms is 10.
Step 1: Follower's (Firm 2) Reaction Function
Firm 2, the follower, maximizes its profit given Firm 1’s output (q₁). Firm 2’s profit function is:
π₂ = Pq₂ – MCq₂ = (100 – 2(q₁ + q₂))q₂ – 10q₂ = (100 – 2q₁ – 2q₂)q₂ – 10q₂
To maximize profit, we take the first-order condition (FOC) with respect to q₂ and set it to zero:
∂π₂/∂q₂ = 100 – 2q₁ – 4q₂ – 10 = 0
Solving for q₂, we get Firm 2’s reaction function:
q₂ = (90 – 2q₁) / 4 = 22.5 – 0.5q₁
Step 2: Leader's (Firm 1) Profit Maximization
Firm 1, the leader, anticipates Firm 2’s reaction and incorporates it into its profit maximization problem. Firm 1’s profit function is:
π₁ = Pq₁ – MCq₁ = (100 – 2(q₁ + q₂))q₁ – 10q₁
Substituting Firm 2’s reaction function into Firm 1’s profit function:
π₁ = (100 – 2(q₁ + 22.5 – 0.5q₁))q₁ – 10q₁ = (100 – 2q₁ – 45 + q₁)q₁ – 10q₁ = (55 – q₁)q₁ – 10q₁ = 55q₁ – q₁² – 10q₁ = 45q₁ – q₁²
To maximize profit, we take the FOC with respect to q₁ and set it to zero:
∂π₁/∂q₁ = 45 – 2q₁ = 0
Solving for q₁, we get Firm 1’s optimal output:
q₁ = 45 / 2 = 22.5
Step 3: Determining Firm 2’s Output
Now, we substitute Firm 1’s output (q₁ = 22.5) into Firm 2’s reaction function:
q₂ = 22.5 – 0.5(22.5) = 22.5 – 11.25 = 11.25
Step 4: Calculating Equilibrium Price
The total output (Q) is the sum of both firms’ outputs:
Q = q₁ + q₂ = 22.5 + 11.25 = 33.75
Substituting Q into the market demand function:
P = 100 – 2Q = 100 – 2(33.75) = 100 – 67.5 = 32.5
Step 5: Calculating Profits
Firm 1’s profit:
π₁ = (P – MC)q₁ = (32.5 – 10)(22.5) = 22.5 * 22.5 = 506.25
Firm 2’s profit:
π₂ = (P – MC)q₂ = (32.5 – 10)(11.25) = 22.5 * 11.25 = 253.125
Summary of Equilibrium
| Firm | Output (q) | Price (P) | Profit (π) |
|---|---|---|---|
| Firm 1 (Leader) | 22.5 | 32.5 | 506.25 |
| Firm 2 (Follower) | 11.25 | 32.5 | 253.125 |
Conclusion
In conclusion, the Stackelberg duopoly model demonstrates how sequential decision-making impacts market outcomes. Firm 1, as the leader, enjoys a higher output and profit compared to Firm 2, the follower, due to its ability to commit to an output level first. The equilibrium price is determined by the total output of both firms. This model highlights the strategic advantage of being a first-mover in certain market structures and provides valuable insights into the dynamics of oligopolistic competition.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.