Nash Equilibrium of a Bertrand Oligopoly

A Nash equilibrium occurs when no firm has an incentive to change its strategy given its competitor's choice. In Bertrand competition with identical products, both firms set prices equal to marginal cost, eliminating economic profit. At this equilibrium point, neither firm can improve its outcome by adjusting its price alone, making it a stable market outcome.

Firms price at marginal cost because customers always choose the cheaper option when products are identical. If one firm charges above marginal cost, its competitor undercuts it and captures the entire market. This price competition continues until both firms reach marginal cost, where further reductions would create negative profit, eliminating any incentive to lower prices further.

A reaction function shows a firm's optimal price based on its competitor's price. Both firms' reaction functions have positive slopes because when a competitor raises its price, the other firm can benefit by offering a higher price. The intersection of both firms' reaction functions marks the Nash equilibrium, where neither firm has incentive to adjust its price.

Firms in a Bertrand oligopoly earn zero economic profit at equilibrium. Since both firms price at marginal cost with identical products, there is no markup above production costs. This outcome is characteristic of Bertrand competition and contrasts sharply with monopoly or differentiated product markets where firms can maintain positive economic profits.

Use the market demand equation by substituting the equilibrium price. For example, with demand P = 300 − Q and equilibrium price of $30, substituting yields 30 = 300 − Q, so Q = 270 units total. This total quantity represents the combined output supplied by both firms at the Nash equilibrium price level.

Homogeneous products are essential to Bertrand competition because customers always choose the cheaper option when products are identical. This creates intense price competition, as any firm charging above marginal cost loses its entire market share to competitors. The homogeneity of products drives both firms toward marginal cost pricing and zero economic profit.

In Bertrand competition with identical products, firms price at marginal cost and earn zero profit. However, in differentiated product markets, firms can charge prices above marginal cost and maintain positive economic profit because customers have brand preferences. Understanding equilibrium in a differentiated products bertrand market reveals how product differentiation allows firms to escape the zero-profit trap.