What is Bertrand Competition?

Legal Definition
Bertrand competition is a model of competition used in economics, named after Joseph Louis François Bertrand (1822–1900). It describes interactions among firms (sellers) that set prices and their customers (buyers) that choose quantities at the prices set. The model was formulated in 1883 by Bertrand in a review of Antoine Augustin Cournot's book Recherches sur les Principes Mathematiques de la Theorie des Richesses (1838) in which Cournot had put forward the Cournot model. Cournot argued that when firms choose quantities, the equilibrium outcome involves firms pricing above marginal cost and hence the competitive price. In his review, Bertrand argued that if firms chose prices rather than quantities, then the competitive outcome would occur with price equal to marginal cost. The model was not formalized by Bertrand: however, the idea was developed into a mathematical model by Francis Ysidro Edgeworth in 1889.

The model rests on very specific assumptions. There are at least two firms producing a homogeneous (undifferentiated) product and can not cooperate in any way. Firms compete by setting prices simultaneously and consumers want to buy everything from a firm with a lower price (since the product is homogeneous and there are no consumer search costs). If two firms charge the same price, consumers demand is split evenly between them. It is simplest to concentrate on the case of duopoly where there are just two firms, although the results hold for any number of firms greater than 1.

A crucial assumption about the technology is that both firms have the same constant unit cost of production, so that marginal and average costs are the same and equal to the competitive price. This means that as long as the price it sets is above unit cost, the firm is willing to supply any amount that is demanded (it earns profit on each unit sold). If price is equal to unit cost, then it is indifferent to how much it sells, since it earns no profit. Obviously, the firm will never want to set a price below unit cost, but if it did it would not want to sell anything since it would lose money on each unit sold.
-- Wikipedia
Legal Definition
When output and prices are set based on the idea that the competition will not change their prices. Refer to Cournot competition.