There are two firms 1 and 2 in an industry, each producing output Q1 and Q2 respectively and facing the industry demand given by P=50-2Q, where P is the market price and Q represents the total industry output, that is Q= Q1 + Q2. Assume that the cost function is C = 10 + 2q. Solve for the Cournot equilibrium in such an industry.

Introduction

In an oligopolistic market structure, a few firms dominate and make decisions strategically, keeping in mind the reactions of their competitors. The Cournot model is a foundational model of duopoly (two-firm competition) where firms choose output levels simultaneously to maximize their profits, assuming the other firm’s output is fixed. This question involves solving for Cournot equilibrium where two firms face a linear demand curve and identical cost functions.

Given Data

Market Demand: P = 50 – 2Q
Where Q = Q₁ + Q₂

Cost Function for both firms: C = 10 + 2q

Each firm chooses output (Q₁ and Q₂) to maximize its profit. We will find the Cournot equilibrium by solving the reaction functions of both firms.

Step 1: Total Revenue and Profit for Firm 1

Firm 1’s revenue:

R₁ = P × Q₁ = (50 – 2(Q₁ + Q₂)) × Q₁
= (50 – 2Q₁ – 2Q₂) × Q₁

Expand:

R₁ = 50Q₁ – 2Q₁² – 2Q₁Q₂

Cost: C₁ = 10 + 2Q₁

Profit: π₁ = R₁ – C₁
= (50Q₁ – 2Q₁² – 2Q₁Q₂) – (10 + 2Q₁)

π₁ = 48Q₁ – 2Q₁² – 2Q₁Q₂ – 10

Step 2: Maximize Firm 1’s Profit

Take the derivative of π₁ with respect to Q₁ and set it to zero (First Order Condition):

dπ₁/dQ₁ = 48 – 4Q₁ – 2Q₂ = 0

Reaction function for Firm 1:
Q₁ = (48 – 2Q₂) / 4 = 12 – 0.5Q₂

Step 3: Do the Same for Firm 2

Following the same steps, since Firm 2 faces the same cost and demand:

π₂ = 48Q₂ – 2Q₂² – 2Q₁Q₂ – 10

dπ₂/dQ₂ = 48 – 4Q₂ – 2Q₁ = 0

Reaction function for Firm 2:
Q₂ = 12 – 0.5Q₁

Step 4: Solve the Two Reaction Functions Simultaneously

We have:

Q₁ = 12 – 0.5Q₂
Q₂ = 12 – 0.5Q₁

Substitute (2) into (1):

Q₁ = 12 – 0.5(12 – 0.5Q₁)

= 12 – 6 + 0.25Q₁

= 6 + 0.25Q₁

0.75Q₁ = 6 → Q₁ = 8

Substitute Q₁ = 8 into Q₂‘s equation:

Q₂ = 12 – 0.5×8 = 12 – 4 = 8

Step 5: Market Price and Profit

Total quantity: Q = Q₁ + Q₂ = 8 + 8 = 16

Price: P = 50 – 2Q = 50 – 2×16 = 18

Firm Profits

Revenue = P × Q = 18 × 8 = ₹144
Cost = 10 + 2×8 = 10 + 16 = ₹26
Profit = 144 – 26 = ₹118 (per firm)

Conclusion

The Cournot equilibrium occurs when both firms produce 8 units each, leading to a total market output of 16 units. The market price becomes ₹18. Each firm earns a profit of ₹118. This analysis shows how competing firms in an oligopoly take into account their rival’s output decisions, resulting in an equilibrium where neither firm wants to change its strategy unilaterally.