In a duopolist market two firms can produce at a constant average and marginal cost of AC = MC = 2. They face the market demand curve P = 14 – Q, where Q = Q1 + Q2, here Q1 is the output of Firm 1, Q2 is the output of Firm 2. In the Cournot’s model: (i) Find the action-reaction functions of the two firms. (ii) What are the profits of the two firms. (iii) Calculate the profit maximizing levels of output (Q1 and Q2) and price.

Introduction

In the Cournot model of duopoly, two firms compete by choosing quantities simultaneously and independently. Each firm chooses its output assuming the other firm’s output is fixed. The goal is to maximise profit given the market demand and cost conditions. Let’s solve the numerical parts of this problem step by step.

Given:

Market demand: P = 14 – Q, where Q = Q1 + Q2
Costs: AC = MC = 2 for both firms

(i) Action-Reaction Functions

Let’s derive the reaction functions for Firm 1 and Firm 2 using profit maximisation.

Firm 1’s Profit Function:

Revenue: R1 = P × Q1 = (14 – Q1 – Q2) × Q1
Cost: C1 = 2 × Q1
Profit: π1 = R1 – C1 = (14 – Q1 – Q2)Q1 – 2Q1

Simplify:
π1 = 14Q1 – Q1² – Q1Q2 – 2Q1 = 12Q1 – Q1² – Q1Q2

Take derivative of π1 with respect to Q1 and set to 0:
dπ1/dQ1 = 12 – 2Q1 – Q2 = 0

Firm 1’s Reaction Function:
Q1 = (12 – Q2) / 2

Similarly, Firm 2’s Profit Function:

π2 = (14 – Q1 – Q2)Q2 – 2Q2 = 12Q2 – Q2² – Q1Q2

dπ2/dQ2 = 12 – 2Q2 – Q1 = 0

Firm 2’s Reaction Function:
Q2 = (12 – Q1) / 2

(ii) Profit Maximising Outputs (Q1 and Q2)

Now solve the two equations simultaneously:

Q1 = (12 – Q2)/2 → (1)
Q2 = (12 – Q1)/2 → (2)

Substitute (2) into (1):
Q1 = (12 – (12 – Q1)/2) / 2

Simplify:
Q1 = (12 – 6 + Q1/2) / 2 = (6 + Q1/2) / 2

Multiply both sides by 2:
2Q1 = 6 + Q1/2 → Multiply both sides by 2 again:
4Q1 = 12 + Q1 → 4Q1 – Q1 = 12 → 3Q1 = 12 → Q1 = 4

From equation (2): Q2 = (12 – 4)/2 = 4

So, Q1 = Q2 = 4 and Total Output Q = Q1 + Q2 = 8

Price: P = 14 – Q = 14 – 8 = 6

(iii) Profits of the Two Firms

Revenue per firm: R = P × Q = 6 × 4 = 24
Cost per firm: C = 2 × 4 = 8
Profit per firm: π = 24 – 8 = 16

Final Answers:

Reaction functions: Q1 = (12 – Q2)/2 and Q2 = (12 – Q1)/2
Equilibrium outputs: Q1 = Q2 = 4
Equilibrium price: P = 6
Profit of each firm: 16 units

Conclusion

In the Cournot model of duopoly, both firms consider the output of the other while choosing their own. The equilibrium is reached when both firms are producing the best possible output given the other firm’s decision. In this example, both firms produce equal output, the market price is determined accordingly, and both earn equal profits.