Bertrand Duopoly: Price Competition & Equilibrium Outcomes

Bertrand competition models price competition where firms compete by setting prices rather than quantities. With just two firms, Bertrand competition can drive prices to marginal cost because any firm raising its price above marginal cost can be undercut by one cent and capture the entire market. The type of competition depends on production characteristics: industries with long production lags (automobiles) tend toward quantity competition, while those with short lags (cereal) tend toward price competition. Firms escape Bertrand competition through product differentiation, which creates imperfect substitutes and allows prices above marginal cost. The breakfast cereal industry evolved from a few identical products to over 150 brands. This creates welfare trade-offs: while differentiation allows firms to charge above marginal cost (reducing consumer surplus), it also creates new products that consumers value. The net welfare effect depends on whether differentiation represents genuine innovation or merely superficial changes.

In Bertrand competition with differentiated goods, firms simultaneously set prices to maximize profits, with each firm's demand depending on both its own price and the competitor's price; the solution involves deriving each firm's reaction function (which expresses optimal price as a function of competitor's price) and solving the system of equations to find the Nash equilibrium prices, where both firms charge $18 and produce 54 units each when marginal costs are zero.

In Bertrand competition with differentiated products, firms compete on prices. For symmetric firms with demand Q1 = 18 - P1 + P2 and cost C1 = 2Q1: Profit π1 = (18 - P1 + P2)P1 - 2(18 - P1 + P2) = 20P1 - P1² + P1P2 - 36. First-order condition: 20 - 2P1 + P2 = 0, so P1 = 10 + P2/2. Solving the system gives P1 = P2 = 20.

This segment introduces differentiated Bertrand competition in oligopolistic markets. Key concepts include: (1) Differentiated goods where consumers perceive products as distinct, creating brand loyalty and substitution effects; (2) Demand functions depend on both own price and competitor's price (QA = 12 - 2PA + 0.4PB); (3) Unlike homogeneous goods where demand drops to zero with any price increase, differentiated goods show dampened substitution effects; (4) Firms compete by choosing prices strategically while facing cross-price dependencies in demand.

In Bertrand competition with differentiated products and constant marginal costs, firms determine their optimal prices by solving their reaction functions simultaneously; when firms have identical constant marginal costs (e.g., MC = $2), the equilibrium prices depend on the cross-price elasticity of demand, and the solution involves taking partial derivatives of the profit functions with respect to each firm's price, setting them to zero, and solving the resulting system of equations to find the Nash equilibrium prices.

In oligopoly markets, firms can sustain tacit collusion by maintaining monopoly prices through trigger strategies, where each firm receives half the monopoly profit as reward but faces punishment of reverting to marginal cost pricing (zero profits) if any firm deviates; this collusion is sustainable when the discount factor (δ) is greater than or equal to 1/2, meaning firms must value future profits sufficiently to resist short-term deviation gains.

In infinitely repeated Bertrand competition with zero marginal costs, firms can sustain the monopoly pricing outcome as a subgame perfect equilibrium using a trigger strategy: both firms charge the monopoly price (5) in every period unless someone deviates, in which case both firms permanently switch to zero pricing; this outcome is sustainable when players are sufficiently patient (discount factor δ ≥ 1/2), as the threat of permanent punishment deters deviation despite the short-term gain of capturing monopoly profits.

In industrial organization, firms can sustain collusive agreements (cartels) in infinitely repeated games when the probability-adjusted discount factor (ρ) exceeds a critical threshold, calculated as (Defection Profit - Monopoly Profit) / (Defection Profit - Nash Profit); this occurs because the threat of permanent punishment deters cheating, unlike finite repeated games where backward induction eliminates cooperation incentives.

Sustaining collusion requires credible punishment mechanisms. The grim trigger strategy provides such a mechanism: firms cooperate initially by setting monopoly prices, continue doing so as long as others cooperate, but switch to permanent punishment (Bertrand price = MC) forever if any deviation occurs. To determine sustainability, compare the present value of always colluding (PV = 75r/(1-r)) against the one-time gain from defecting (450). Collusion is sustainable only when the interest rate r ≤ 20%. Low interest rates make future cooperative payoffs more valuable, reducing the temptation to defect immediately.

This lecture covers dynamic competition theory, explaining how repeated interactions among firms can sustain markups through collusion. The simplest model shows that with n identical firms and discount factor Delta, collusion is possible when Delta ≥ 1 - 1/n, using grim trigger strategies where firms charge monopoly prices until someone deviates, then punish with price wars. However, imperfect monitoring makes perfect collusion impossible—when firms cannot observe rivals' actions directly, they must rely on signals like demand patterns, leading to partial collusion with finite punishment phases. The Green-Porter model demonstrates that even without cheating, suspicious demand patterns trigger price wars, showing that well-functioning cartels deliberately include price wars as designed features. Additional factors affecting collusion include detection lags, lumpiness of demand, market growth, and cost shocks. Modern computational approaches using Q-learning show that algorithms competing online learn to collude, achieving 70-90% of monopoly profits, demonstrating how repeated interaction enables coordination even without explicit communication.

The Bertrand model analyzes oligopoly when firms compete on price rather than quantity. With price competition, even two firms drive the market to perfect competition: if one firm charges above marginal cost, the other can undercut by a small amount and capture the entire market. This continues until price equals marginal cost. The Bertrand model predicts that with price competition, two firms achieve the same outcome as perfect competition, which is radically different from the Cournot quantity competition model. The appropriate model depends on industry characteristics: price competition is more likely in industries with rapid production turnaround (like breakfast cereals), while quantity competition is more likely in industries with production lags (like automobiles).

In oligopoly theory, Cournot competition involves firms choosing quantities, resulting in softer competition where unilateral output increases only marginally affect rivals' revenues, while Bertrand competition involves firms choosing prices, creating extremely stiff competition where even a small price reduction captures the entire market; as more firms enter, Cournot outcomes converge gradually toward perfect competition, whereas Bertrand outcomes can jump discontinuously to perfect competition even with just two firms, making Bertrand more realistic for price-setting industries like restaurants while Cournot better suits quantity-constrained sectors.

Two main oligopoly models describe strategic interaction: Bertrand (price competition) and Cournot (quantity competition). In Bertrand, firms simultaneously set prices, driving prices down to marginal cost even with few firms. In Cournot, firms simultaneously choose quantities, assuming rivals' quantities are fixed, resulting in prices above marginal cost. Both models show how interdependence affects market outcomes compared to perfect competition, with Bertrand yielding more competitive results.

This section provides detailed analysis of three major oligopoly models. Cournot model: firms compete by choosing output quantities simultaneously, assuming competitor's output is fixed. Equilibrium output is n/(n+1) of perfect competition level, with price between monopoly and perfect competition. Individual firm output is 1/(n+1) of total. Bertrand model: firms compete by setting prices, assuming competitor's price is fixed. With homogeneous products and zero MC, equilibrium is P=MC (perfect competition outcome). With differentiated products, equilibrium price equals average cost. Stackelberg model: one leader moves first and knows follower's reaction function, incorporating it into profit maximization. Leader produces more than Cournot, earns higher profits. Comparison: Cournot (quantity competition, downward sloping reaction curves), Bertrand (price competition, upward sloping reaction curves), Stackelberg (leader-follower, first-mover advantage).

In Bertrand competition, firms compete on prices, resulting in equilibrium prices equal to marginal cost (similar to perfect competition). In Cournot competition, firms compete on quantities, resulting in equilibrium prices above marginal cost (similar to monopoly). Under Bertrand competition, equilibrium quantity is higher and price is lower than under Cournot competition. Cournot competition is a strategic interdependence model where each firm's output decision depends on expected output of rival firms.