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. Competition is the act of striving against another force for the purpose of achieving dominance or attaining a reward or goal, or out of a biological imperative such as survival. ... Economics (from the Greek οίκος [oikos], house, and νέμω [nemo], rules, hence household management) is the social science that studies the production, distribution, trade and consumption of goods and services in the context of the competing alternative allocations of goods and courses of action. ... Joseph Louis François Bertrand (March 11, 1822 - April 5, 1900, born and died in Paris) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, and thermodynamics. ... A duopoly is a form of oligopoly where only two producers are present in a given market. ... Perfect competition is a model in economic theory. ... In economics and finance, marginal cost is the cost of increasing the quantity produced (or purchased) by one unit. ...

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. Homogeneous is an adjective that has several meanings. ... In economics and finance, marginal cost is the cost of increasing the quantity produced (or purchased) by one unit. ... The supply and demand model describes how prices vary as a result of a balance between product availability at each price (supply) and the desires of those with purchasing power at each price (demand). ...

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. In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ...

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.

In the study of economics, collusion takes place within an industry when rival companies cooperate for their mutual benefit. ... In economics, a monopoly (from the Greek monos, one + polein, to sell) is defined as a persistent market situation where there is only one provider of a kind of product or service. ... In economics and finance, marginal cost is the cost of increasing the quantity produced (or purchased) by one unit. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ... In microeconomics, a production function expresses the relationship between an organizations inputs and its outputs. ...

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.

Cournot competition is an economics model used to describe industry structure. ...

Critical analysis of the Bertrand model

• The most critical flaw of the model is the assumption that firms compete in one period, the price being chosen and set for ever. However, as it's unreasonable to expect the other firm to indefinitely keep higher prices and sell nothing, each firm must expect that lowering the price will almost immediately be met with the same move by the other firm, thus no firm can expect to get bigger market share by cutting price, and the preferred strategy is keeping prices at monopoly price level. The situation is analogous to the prisoner's dilemma, single-period version of which has completely opposite implications than the iterated version.

• 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.

• 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.

Will the two prisoners cooperate to minimise total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? The prisoners dilemma is a type of non-zero-sum game (game in the sense of Game Theory). ... In marketing, product differentiation is the modification of a product to make it more attractive to the target market. ...

See also

• Cournot competition
• Stackelberg competition
• Nash equilibrium
• game theory
• Bertrand paradox (economics)