Bertrand competition is a model of competition used in economics, named after Joseph Louis François Bertrand (1822-1900). Specifically, it is a model of price competition between duopoly firms which results in each charging the price that would be charged under perfect competition, known as marginal cost pricing.

The model has the following assumptions:

• There are two firms producing homogeneous products;
• Firms do not cooperate;
• Firms have the same marginal cost (MC);
• Marginal cost is constant;
• Demand is linear;
• Firms compete in price, and choose their respective prices simultaneously;
• There is strategic behaviour by both firms;
• Both firms compete solely on price and then supply the quantity demanded;
• Consumers buy everything from the cheaper firm or half at each, if the price is equal.

Competing in price means that firms can easily change the quantity they supply, but once they have chosen a certain price, it is very hard, if not impossible, to change it, for example bars or shops or other companies that publish non-negotiable prices.

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 1s optimum price depends on what it believes firm 2 will set prices at. Pricing just below the other firm will obtain full market demand (D), while maximising profits. If firm 1 expects firm 2 to price below marginal cost, then its best strategy is to price higher, at marginal cost. In general terms, firm 1s best response (its reaction 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 firms 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

• There are two plausible outcomes: colluding to charge the monopoly price and supplying one half of the market each, or not colluding and charging marginal cost, which is the non-cooperative Nash equilibrium outcome.

• 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.

Bertrand competition versus Cournot competition

• Although both models have similar assumptions, both have very different implications.
• Bertrand predicts a duopoly is enough to push prices down to marginal cost level, that duopoly will result in perfect competition.
• Neither model is ‘better’, as it depends on the industry as to which is more accurate.
• 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.

Critical analysis of the Bertrand model

• Examining the assumptions reveals some inadequacies of the model: it 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?
• There are rarely just two firms in a market.
• The model assumes firms compete in one period, that price is chosen and set for ever. However, in reality lowering price will often cause a rival to also immediately match or even beat that price, meaning no market share is gained. Firms will not undercut each other for fear of retaliation.

• If a firm does undercut a rival and get full market share, it now has to supply the whole market; many firms would not have the capacity to do this. In general, the greater the overall capacity constraints, the higher the price is than marginal cost.