Model Answer
0 min readIntroduction
Cournot duopoly, developed by Antoine Augustin Cournot in 1838, is a model of imperfect competition where two or more firms produce homogeneous products. Unlike perfect competition, firms in a Cournot duopoly have some market power and make decisions about quantity produced, assuming the output of their rivals is fixed. This contrasts with Bertrand competition, where firms compete on price. The equilibrium in a Cournot model is achieved when each firm’s output level is optimal, given the output level of the other firm, resulting in a stable market outcome. This question requires us to determine the market equilibrium price and quantity, and the resulting profits for two firms operating as Cournot duopolists in the carbonated water market.
Deriving the Reaction Functions
Let Q1 and Q2 be the quantities produced by firm 1 and firm 2, respectively. The market demand curve is given by P = 20 - 9Q, where Q = Q1 + Q2. Each firm has a constant marginal cost of c = 2.
The profit function for firm 1 is:
π1 = P * Q1 - c * Q1 = (20 - 9(Q1 + Q2)) * Q1 - 2 * Q1
To maximize profit, firm 1 takes the derivative of π1 with respect to Q1 and sets it equal to zero:
dπ1/dQ1 = 20 - 9Q1 - 9Q2 - 2 - 9Q1 = 0
Simplifying, we get the reaction function for firm 1:
Q1 = (18 - 9Q2) / 18 = 1 - 0.5Q2
Similarly, the profit function for firm 2 is:
π2 = P * Q2 - c * Q2 = (20 - 9(Q1 + Q2)) * Q2 - 2 * Q2
Taking the derivative of π2 with respect to Q2 and setting it equal to zero:
dπ2/dQ2 = 20 - 9Q1 - 9Q2 - 2 - 9Q2 = 0
Simplifying, we get the reaction function for firm 2:
Q2 = (18 - 9Q1) / 18 = 1 - 0.5Q1
Solving for Equilibrium Quantities
To find the equilibrium quantities, we solve the two reaction functions simultaneously. Substituting the reaction function of firm 1 into the reaction function of firm 2:
Q2 = 1 - 0.5(1 - 0.5Q2)
Q2 = 1 - 0.5 + 0.25Q2
0.75Q2 = 0.5
Q2 = 0.5 / 0.75 = 2/3
Now, substitute Q2 back into the reaction function of firm 1:
Q1 = 1 - 0.5 * (2/3) = 1 - 1/3 = 2/3
Therefore, the equilibrium quantities are Q1 = 2/3 and Q2 = 2/3.
Determining the Market Equilibrium Price
The total market quantity is Q = Q1 + Q2 = 2/3 + 2/3 = 4/3.
Substituting this into the market demand curve:
P = 20 - 9 * (4/3) = 20 - 12 = 8
Thus, the market equilibrium price is INR 8.
Calculating Firms' Profits
The profit for firm 1 is:
π1 = (P - c) * Q1 = (8 - 2) * (2/3) = 6 * (2/3) = 4
The profit for firm 2 is:
π2 = (P - c) * Q2 = (8 - 2) * (2/3) = 6 * (2/3) = 4
Therefore, each firm earns a profit of INR 4.
Conclusion
In conclusion, when each firm behaves as a Cournot duopolist, the market equilibrium price for carbonated water is INR 8, and the market quantity is 4/3 units. Each firm produces 2/3 units and earns a profit of INR 4. This demonstrates how firms strategically adjust their output levels based on their expectations of their rival’s behavior, leading to a stable, albeit imperfectly competitive, market outcome. The Cournot model provides a valuable framework for understanding oligopolistic markets and the strategic interactions between firms.
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.