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.

A Nash equilibrium occurs when no firm has an incentive to change its strategy given its competitor’s choice.

Consider two firms, TrueTech and  BestCo, producing identical products.

Their market demand function is P = 300 − Q and P = 300−(𝑄₁+𝑄₂), where Q1 and Q2 are the quantities produced by TrueTech and BestCo, respectively.

Both firms have the same marginal cost, which is MC1 = MC2 = $30.

BestCo sets its price first, fearing undercutting by TrueTech. Both firms eventually set prices equal to their marginal cost at $30 to avoid losing market share.

Using the demand equation P = 300 − Q, substituting the price of thirty dollars yields 30 = 300−Q and Q = 270 units total units supplied to the market by both firms.

The reaction function indicates a firm's optimal price based on its competitor's price.

The positive slope for both firms arises because when a competing firm raises its price, the other firm can benefit from offering a higher price as well.

The intersection of TrueTech's and BestCo's reaction functions marks the Nash equilibrium, where neither firm has an incentive to adjust its price.