Model Answer
0 min readIntroduction
In the realm of industrial organization, a duopoly represents a market structure dominated by two firms. When products are differentiated, these firms compete not solely on price but also on features, branding, and quality, leading to distinct demand curves for each. This scenario often results in strategic interdependence, where each firm's optimal decision depends on the actions of its rival. Understanding such market dynamics is crucial for firms to formulate effective pricing and production strategies, ultimately impacting market equilibrium and consumer welfare.
Understanding the Duopoly Model with Product Differentiation
In a duopoly with product differentiation, firms produce similar but not identical products. This allows each firm to have some market power and face a downward-sloping demand curve for its specific product. The demand for one firm's product is influenced by its own price and quantity, as well as the price and quantity of the rival firm.
Step-by-Step Calculation for Duopolist I
1. Duopolist I's Demand and Cost Functions:
Given demand function for Duopolist I:
P₁ = 200 - 4q₁ - 2q₂
Given cost function for Duopolist I:
C₁ = 5q₁²
2. Derive Duopolist I's Total Revenue (TR₁) and Marginal Revenue (MR₁):
Total Revenue (TR₁) = P₁ * q₁
TR₁ = (200 - 4q₁ - 2q₂) * q₁
TR₁ = 200q₁ - 4q₁² - 2q₁q₂
Marginal Revenue (MR₁) is the derivative of TR₁ with respect to q₁:
MR₁ = ∂TR₁/∂q₁
MR₁ = 200 - 8q₁ - 2q₂
3. Derive Duopolist I's Marginal Cost (MC₁):
Marginal Cost (MC₁) is the derivative of C₁ with respect to q₁:
MC₁ = ∂C₁/∂q₁
MC₁ = 10q₁
4. Profit Maximization for Duopolist I:
To maximize profit, Duopolist I sets MR₁ = MC₁:
200 - 8q₁ - 2q₂ = 10q₁
200 - 2q₂ = 18q₁
This gives Duopolist I's reaction function (how q₁ changes based on q₂):
18q₁ = 200 - 2q₂
q₁ = (200 - 2q₂) / 18
q₁ = 100/9 - (1/9)q₂ (Equation 1)5. Incorporate Duopolist II's Market Share:
Assume Duopolist II has ⅓ rd share of the whole market. This implies that q₂ = (1/3) * (q₁ + q₂), but this interpretation is not directly applicable in a Cournot-style model where quantities are set independently, though influencing each other. A more practical interpretation in such a problem, especially without the explicit total market size, is that Duopolist II's output is some fraction of the total market as perceived or an assumption about relative production. If the question implies that Duopolist II's output is 1/3 of the total market output (Q = q₁ + q₂), then q₂ = (1/3)(q₁ + q₂). This simplifies to 3q₂ = q₁ + q₂, so 2q₂ = q₁, or q₂ = (1/2)q₁.
Let's consider the interpretation that Duopolist II's output is related to Duopolist I's output in a fixed ratio, such that if Duopolist I has ⅔ share, then Duopolist II has ⅓ share. This would mean q₂ = (1/2)q₁. However, this is usually given directly. Given the phrasing "Duopolist II has ⅓ rd share of the whole market," and without information on total market demand function, the most direct interpretation for solving this specific problem structure is often that q₂ is some constant fraction of q₁. Let's re-evaluate based on the demand function. A more common simplification in such problems (if not explicitly stated as a reaction function) is to consider the relative output. If Duopolist II has 1/3 share of the whole market, it usually implies that its output (q₂) is 1/3 of the total market output (Q = q₁+q₂). So, q₂ = (1/3)(q₁+q₂), which simplifies to 3q₂ = q₁+q₂, or 2q₂ = q₁. Therefore, q₂ = (1/2)q₁.
Substitute q₂ = (1/2)q₁ into Equation 1:
q₁ = 100/9 - (1/9)((1/2)q₁)
q₁ = 100/9 - (1/18)q₁
q₁ + (1/18)q₁ = 100/9
(18q₁ + q₁) / 18 = 100/9
19q₁ / 18 = 100/9
q₁ = (100/9) * (18/19)
q₁ = 200/19
6. Calculate Optimal Output for Duopolist I (q₁):
q₁ ≈ 10.526 units
7. Calculate Output for Duopolist II (q₂):
Using q₂ = (1/2)q₁:
q₂ = (1/2) * (200/19)
q₂ = 100/19
q₂ ≈ 5.263 units
8. Calculate Optimal Price for Duopolist I (P₁):
Substitute q₁ and q₂ into Duopolist I's demand function:
P₁ = 200 - 4q₁ - 2q₂
P₁ = 200 - 4(200/19) - 2(100/19)
P₁ = 200 - 800/19 - 200/19
P₁ = 200 - 1000/19
P₁ = (3800 - 1000) / 19
P₁ = 2800 / 19
P₁ ≈ 147.368
9. Calculate Optimal Profit for Duopolist I (π₁):
Profit (π₁) = TR₁ - C₁
TR₁ = P₁ * q₁ = (2800/19) * (200/19)
TR₁ = 560000 / 361 ≈ 1551.246
C₁ = 5q₁² = 5 * (200/19)²
C₁ = 5 * (40000 / 361)
C₁ = 200000 / 361 ≈ 554.017
π₁ = (560000 / 361) - (200000 / 361)
π₁ = 360000 / 361
π₁ ≈ 997.229
Summary of Optimal Values for Duopolist I and Duopolist II
| Parameter | Value (Exact) | Value (Approximate) |
|---|---|---|
| Optimal Output for Duopolist I (q₁) | 200/19 | 10.53 units |
| Optimal Price for Duopolist I (P₁) | 2800/19 | 147.37 |
| Optimal Profit for Duopolist I (π₁) | 360000/361 | 997.23 |
| Output for Duopolist II (q₂) | 100/19 | 5.26 units |
Conclusion
The analysis of a duopoly with product differentiation, as demonstrated by the calculation for Duopolist I, highlights the strategic interdependence inherent in such market structures. By understanding its own demand and cost functions, and making assumptions about the rival's behavior (in this case, market share), a firm can determine its profit-maximizing output and price. These calculations are fundamental in microeconomics for understanding firm behavior, market equilibrium, and the implications of various competitive strategies, ultimately contributing to a robust understanding of market dynamics.
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.