Bertrand competition

Bertrand competition is a model of competition used in economics, named after Joseph Louis François Bertrand^[1] (1822-1900). It describes interactions among firms (sellers) that set prices and their customers (buyers) that choose quantities at that price.

The model rests on the following assumptions:

• There are at least two firms producing homogeneous (undifferentiated) products;
• Firms do not cooperate;
• Firms compete by setting prices simultaneously;
• Consumers buy everything from a firm with a lower price. If all firms charge the same price, consumers randomly select among them.

Calculating the classic Bertrand model

• MC = Marginal cost
• p₁ = firm 1’s price level
• p₂ = firm 2’s price level
• p_M = monopoly price level

• Firm 1's optimum price depends on where it believes firm 2 will set its prices. Pricing just below the other firm will obtain full market demand (D), though this is not optimal if the other firm is pricing below marginal cost as that would entail negative profits. In general terms, firm 1's best response function is p₁’’(p₂), this gives firm 1 optimal price for each price set by firm 2.

• Diagram 1 shows firm 1’s reaction function p₁’’(p₂), with each firm's strategy on each axis. It shows that when P₂ is less than marginal cost (firm 2 pricing below MC) firm 1 prices at marginal cost, p₁=MC. When firm 2 prices above MC but below monopoly prices, then firm 1 prices just below firm 2. When firm 2 prices above monopoly prices (P^M) firm 1 prices at monopoly level, p₁=p^M.

• Because firm 2 has the same marginal cost as firm 1, its reaction function is symmetrical with respect to the 45 degree line. Diagram 2 shows both reaction functions.

• The result of the firms' strategies is a Nash equilibrium, that is, a pair of strategies (prices in this case) where neither firm can increase profits by unilaterally changing price. This is given by the intersection of the reaction curves, Point N on the diagram. At this point p₁=p₁’’(p₂), and p₂=p₂’’(p₁). As you can see, point N on the diagram is where both firms are pricing at marginal cost.

Another way of thinking about it, a simpler way, is to imagine if both firms set equal prices above marginal cost, firms would get half the market at a higher than MC price. However, by lowering prices just slightly, a firm could gain the whole market, so both firms are tempted to lower prices as much as they can. It would be irrational to price below marginal cost, because the firm would make a loss. Therefore, both firms will lower prices until they reach the MC limit.

Implications

• Note that colluding to charge the monopoly price and supplying one half of the market each is the best that the firms could do in this set up. However not colluding and charging marginal cost, which is the non-cooperative outcome is the only Nash equilibrium of this model.

• If one firm has lower average cost (a superior production technology), it will charge the highest price that is lower than the average cost of the other one (i.e. a price just below the lowest price the other firm can manage) and take all the business. This is known as "limit pricing"

Bertrand competition versus Cournot competition

• Bertrand predicts a duopoly is enough to push prices down to marginal cost level; a duopoly will result in an outcome exactly equivalent to what prevails under perfect competition.

• Neither model is necessarily "better." The accuracy of the predictions of each model will vary from industry to industry, depending on the closeness of each model to the industry situation.

• If capacity and output can be easily changed, Bertrand is generally a better model of duopoly competition. Or, if output and capacity are difficult to adjust, then Cournot is generally a better model.

• Under some conditions the Cournot model can be recast as a two stage model, where in the first stage firms choose capacities, and in the second they compete in Bertrand fashion.

Critical analysis of the Bertrand model

• The classic Bertrand model assumes firms compete purely on price, ignoring non-price competition. Firms can differentiate their products and charge a higher price. For example, would someone travel twice as far to save 1% on the price of their vegetables? The Bertrand model can be extended to include product or location differentiation but then the main result - that price is driven down to marginal cost - no longer holds.

• The model ignores capacity constraints. If a single firm does not have the capacity to supply the whole market then the "price equals marginal cost" result may not hold. The analysis of this case was started by Francis Ysidro Edgeworth and has become known as the Bertrand-Edgeworth Model. With capacity constraints, there may not exist any pure strategy Nash equilibrium, the so-called Edgeworth paradox. However, in general there will exist a mixed-strategy Nash equilibrium as shown by Huw Dixon^[2]

• The classical model only focuses on the pure strategy Nash equilibrium. There are mixed-strategy Nash equilibrium with positive economic profits (see Kaplan & Wettstein, 2000, and Baye & Morgan, 1999).

• The classical model ignores search cost of consumers. If the consumer does not know the price of the product before visiting a firm and each visit is costly (however small), then a Nash equilibrium price will not arise. This creates the possibility that firms will randomly price at some point between marginal cost and the monopoly price.

See also

• Cournot competition
• Differentiated Bertrand competition
• Stackelberg competition
• Nash equilibrium
• Game theory
• Bertrand paradox (economics)
• Bertrand-Edgeworth Model
• Edgeworth paradox

References

• Bertrand, J. (1883) "Book review of theorie mathematique de la richesse sociale and of recherches sur les principles mathematiques de la theorie des richesses", Journal de Savants 67: 499–508.
• Oligoply Theory made Simple, Chapter 6 of Surfing Economics by Huw Dixon.