Show how price output decision is taken by duopolists, taking into account, their mutual reaction. Under what condition will the duopolistic market be in equilibrium?

Duopoly Models: Price and Output Decisions

There are two primary models used to analyze price and output decisions in a duopoly: the Cournot model and the Bertrand model.

1. Cournot Model (Quantity Competition)

The Cournot model, developed by Antoine Cournot in 1838, assumes that firms compete by choosing the quantity of output to produce. Each firm takes the output of the other firm as given when making its own production decision. The process unfolds as follows:

• Firm 1 chooses its output (Q1), assuming Q2 is fixed.
• Firm 2 chooses its output (Q2), assuming Q1 is fixed.
• This process continues until a Nash Equilibrium is reached, where neither firm has an incentive to change its output level given the output of the other.

The reaction function for each firm shows the optimal quantity to produce for any given quantity produced by the rival. The market price is then determined by the total quantity supplied (Q = Q1 + Q2).

2. Bertrand Model (Price Competition)

The Bertrand model, developed by Joseph Bertrand in 1883, assumes that firms compete by setting prices. Both firms simultaneously choose a price, and consumers purchase from the firm offering the lower price. Key features include:

• Homogeneous Products: Products are identical.
• Price Competition: Firms undercut each other’s prices.
• Nash Equilibrium: The equilibrium occurs when both firms set price equal to marginal cost (P = MC).

In the Bertrand model, even with only two firms, the outcome resembles perfect competition due to intense price rivalry. However, this outcome is sensitive to assumptions like product differentiation or capacity constraints.

Equilibrium Conditions in a Duopolistic Market

The equilibrium in a duopolistic market depends on the model used:

Model	Equilibrium Condition	Price Level	Output Level
Cournot	Each firm maximizes profit given the output of the other.	Higher than marginal cost, but lower than monopoly price.	Output is less than the monopoly output, but greater than under perfect competition.
Bertrand	Both firms set price equal to marginal cost.	Equal to marginal cost (P = MC).	Output is determined by market demand at the marginal cost price.

Mutual Reaction: The core of duopolistic equilibrium lies in the mutual reaction of firms. Each firm anticipates the other’s response to its actions and adjusts its strategy accordingly. This leads to a stable outcome where neither firm can improve its profit by unilaterally changing its price or output.

Factors Affecting Equilibrium: The specific equilibrium outcome is influenced by factors such as demand elasticity, marginal cost, and the nature of product differentiation. If products are differentiated, the Bertrand model’s outcome of P=MC is less likely, as firms can compete on attributes other than price.