Bertrand competition is a model of
competitionCompetition is a contest between individuals, groups, nations, animals, etc. for territory, a niche, or allocation of resources. It arises whenever two or more parties strive for a goal which cannot be shared. Competition occurs naturally between living organisms which co-exist in the same...
used in
economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, named after
Joseph Louis François BertrandJoseph Louis François Bertrand was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics....
(1822-1900). Specifically, it is a model of price competition between
duopolyA true duopoly is a specific type of oligopoly where only two producers exist in one market. In reality, this definition is generally used where only two firms have dominant control over a market...
firms which results in each charging the price that would be charged under
perfect competitionIn neoclassical economics and microeconomics, perfect competition describes the perfect being a market in which there are many small firms, all producing homogeneous goods...
, known as
marginal costIn economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. It is the cost of producing one more unit of a good. Mathematically, the marginal cost function is expressed as the first derivative of the total cost function with...
pricing.
The model has the following assumptions:
- There are at least two firms producing homogeneous products;
- Firms do not cooperate;
- Firms have the same marginal cost
In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. It is the cost of producing one more unit of a good. Mathematically, the marginal cost function is expressed as the first derivative of the total cost function with...
(MC);
- Marginal cost is constant;
- Demand
Supply and demand is an economic model based on price, utility and quantity in a market. It concludes that in a competitive market, price will function to equalize the quantity demanded by consumers, and the quantity supplied by producers, resulting in an economic equilibrium of price and...
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.
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Bertrand competition is a model of
competitionCompetition is a contest between individuals, groups, nations, animals, etc. for territory, a niche, or allocation of resources. It arises whenever two or more parties strive for a goal which cannot be shared. Competition occurs naturally between living organisms which co-exist in the same...
used in
economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, named after
Joseph Louis François BertrandJoseph Louis François Bertrand was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics....
(1822-1900). Specifically, it is a model of price competition between
duopolyA true duopoly is a specific type of oligopoly where only two producers exist in one market. In reality, this definition is generally used where only two firms have dominant control over a market...
firms which results in each charging the price that would be charged under
perfect competitionIn neoclassical economics and microeconomics, perfect competition describes the perfect being a market in which there are many small firms, all producing homogeneous goods...
, known as
marginal costIn economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. It is the cost of producing one more unit of a good. Mathematically, the marginal cost function is expressed as the first derivative of the total cost function with...
pricing.
The model has the following assumptions:
- There are at least two firms producing homogeneous products;
- Firms do not cooperate;
- Firms have the same marginal cost
In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. It is the cost of producing one more unit of a good. Mathematically, the marginal cost function is expressed as the first derivative of the total cost function with...
(MC);
- Marginal cost is constant;
- Demand
Supply and demand is an economic model based on price, utility and quantity in a market. It concludes that in a competitive market, price will function to equalize the quantity demanded by consumers, and the quantity supplied by producers, resulting in an economic equilibrium of price and...
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. Some examples of firms that might operate in this way are bars, 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
- pM = 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
In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given...
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 (PM) firm 1 prices at monopoly level, p1=pM.
- 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
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally...
, 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
Collusion is an agreement, usually secretive, which occurs between two or more persons to limit open competition by deceiving, misleading, or defrauding others of their legal rights, or to obtain an objective forbidden by law typically by defrauding or gaining an unfair advantage . It is an...
to charge the monopolyIn economics, a monopoly exists when a specific individual or an enterprise has sufficient control over a particular product or service to determine significantly the terms on which other individuals shall have access to it...
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 costIn economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. It is the cost of producing one more unit of a good. Mathematically, the marginal cost function is expressed as the first derivative of the total cost function with...
, which is the non-cooperative outcome is the only Nash equilibriumIn game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally...
of this model.
- If one firm has lower average cost (a superior production technology
In economics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...
), 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"A limit price is the price set by a monopolist to discourage economic entry into a market, and is illegal in many countries. The limit price is the price that the entrant would face upon entering as long as the incumbent firm did not decrease output...
Bertrand competition versus Cournot competitionCournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot after he observed competition in a...
- Bertrand predicts a duopoly is enough to push prices down to marginal cost level, that duopoly will result in 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 most critical flaw of the model is the assumption that firms compete in one period, the price being chosen and set forever. However, as it is 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
The prisoner's dilemma constitutes a problem in game theory. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...
, 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
In marketing, product differentiation is the process of distinguishing the differences of a product or offering from others, to make it more attractive to a particular target market. This involves differentiating it from competitors' products as well as one's own product offerings.Differentiation...
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.
See also
- Cournot competition
Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot after he observed competition in a...
- Differentiated Bertrand competition
As a solution to the Bertrand paradox in economics, it has been suggested that each firm produces a somewhat differentiated product, and consequently faces a demand curve that is downward-sloping for all levels of the firm's price....
- Stackelberg competition
The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially...
- Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally...
- Game theory
Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology, engineering, political science, international relations, computer science, and philosophy...
- Bertrand paradox (economics)
In economics and commerce, the Bertrand paradox– named after its creator, Joseph Bertrand–describes a situation in which two players reach a state of Nash equilibrium where both firms charge a price equal to marginal cost. The paradox is that in reality, it usually takes a large number...