Bertrand competition

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.

Why is the competitive price a Nash equilibrium in the Bertrand model? First, if both firms set the competitive price with price equal to marginal cost (unit cost), neither firm will earn any profits. However, if one firm sets price equal to marginal cost, then if the other firm raises its price above unit cost, then it will earn nothing, since all consumers will buy from the firm still setting the competitive price (recall that it is willing to meet unlimited demand at price equals unit cost even though it earns no profit). No other price is an equilibrium. If both firms set the same price above unit cost and share the market, then each firm has an incentive to undercut the other by an arbitrarily small amount and capture the whole market and almost double its profits. So there can be no equilibrium with both firms setting the same price above marginal cost. Also, there can be no equilibrium with firms setting different prices. The firms setting the higher price will earn nothing (the lower priced firm serves all of the customers). Hence the higher priced firm will want to lower its price to undercut the lower-priced firm. Hence the only equilibrium in the Bertrand model occurs when both firms set price equal to unit cost (the competitive price).

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