11.8 : Nash Equilibrium of a Bertrand Oligopoly

A Bertrand oligopoly occurs when a few firms compete by strategically setting prices rather than lowering them indefinitely. In this model, firms sell homogeneous (identical) products; thus, customers always choose the cheaper option. As a result, firms set prices at marginal cost, eliminating any economic profit.

For example, imagine two coffee stands at a busy train station, both selling identical coffee. If one stand sets its price at $5 per cup, the other will undercut it by pricing at $4.90 to capture the entire market. The first stand, in response, lowers its price to $4.80. This process continues until the price reaches marginal cost, say $4 per cup. At this point, further price reductions would result in negative profit, so neither firm has an incentive to lower prices further.

In a pure Bertrand model, price competition happens simultaneously, not sequentially. If a firm tries to charge above marginal cost, its competitor will undercut it and capture the entire market, forcing both firms to remain at marginal cost. Consequently, firms in a Bertrand oligopoly earn zero economic profit in equilibrium.

A Nash equilibrium occurs when neither firm can improve its outcome by changing its strategy alone. Let's assume two companies, Firm A and Firm B, are operating within a market where demand is represented by the equation, P = 160 - Q. Here, Q = Q₁ + Q₂, where Q₁ and Q₂ represent the quantity of the goods produced by Firm A and Firm B, respectively. Both Firm A and Firm B have the same marginal cost of $40.

If either firm sets a price above $40, the competitor captures the entire market by pricing just below. As a result, both firms set their prices at $40. Substituting P = 40 into the demand equation, 40 = 160 - Q, gives Q = 120 units. Therefore, the total market supply is 120 units. At this equilibrium, both firms price at marginal cost, leading to zero economic profit for both, which is characteristic of Bertrand competition.