Explain the Cournot model of duopoly using reaction functions and interpret it as a Nash equilibrium.

The Cournot Model: A Detailed Explanation

The Cournot model assumes two firms, Firm 1 and Firm 2, producing a homogeneous product. Both firms aim to maximize their profits. Key assumptions include:

• Homogeneous Product: Both firms produce an identical product.
• Simultaneous Decision-Making: Firms choose their output levels simultaneously.
• Perfect Information: Each firm knows the market demand curve.
• Independent Decision-Making: Firms act independently without collusion.

Deriving Reaction Functions

A reaction function shows the optimal output of one firm given the output of the other firm. To derive the reaction function for Firm 1, we need to find its profit-maximizing output level (Q₁) for any given output level of Firm 2 (Q₂). Let's assume the inverse demand function is P = a - b(Q₁ + Q₂), where 'a' and 'b' are positive constants. The profit function for Firm 1 is:

π₁ = P * Q₁ - C₁(Q₁)

Where C₁(Q₁) is the cost function for Firm 1. Assuming a constant marginal cost 'c', C₁(Q₁) = cQ₁. Therefore:

π₁ = (a - b(Q₁ + Q₂)) * Q₁ - cQ₁

Taking the first-order condition (FOC) with respect to Q₁ and setting it to zero gives us:

a - 2bQ₁ - bQ₂ - c = 0

Solving for Q₁, we get Firm 1’s reaction function:

Q₁ = (a - c - bQ₂) / 2b

Similarly, the reaction function for Firm 2 is:

Q₂ = (a - c - bQ₁) / 2b

The Cournot Equilibrium

The Cournot equilibrium is found at the intersection of the two reaction functions. This is the point where each firm is producing its best response to the other firm’s output. To find the equilibrium, we can substitute Firm 2’s reaction function into Firm 1’s reaction function:

Q₁ = (a - c - b((a - c - bQ₁) / 2b)) / 2b

Simplifying this equation, we get:

Q₁ = (a - c) / (3b)

Since both firms are identical, Q₂ will also be equal to (a - c) / (3b). Therefore, the equilibrium output for each firm is (a - c) / (3b). The total market output is Q₁ + Q₂ = 2(a - c) / (3b). Substituting this into the inverse demand function, we get the equilibrium price:

P = a - b(2(a - c) / 3b) = (a + 2c) / 3

Nash Equilibrium Interpretation

The Cournot equilibrium is a Nash equilibrium because neither firm has an incentive to unilaterally deviate from its chosen output level, given the output of the other firm. In other words, each firm is playing its best response strategy. If Firm 1 were to increase its output, given Firm 2’s output, it would lower the market price and reduce its profits. Similarly, if Firm 1 were to decrease its output, Firm 2 would increase its output to capture a larger market share, again reducing Firm 1’s profits. The same logic applies to Firm 2. Therefore, the Cournot equilibrium represents a stable outcome where both firms are maximizing their profits, given the strategic behavior of the other.

Firm	Equilibrium Output	Equilibrium Price
Firm 1	(a - c) / 3b	(a + 2c) / 3
Firm 2	(a - c) / 3b	(a + 2c) / 3