Bertrand's Duopoly Model: Assumptions, Diagram with Explanation

BERTRAND’S DUOPOLY MODEL

Joseph Bertrand, a French mathematician, criticized Cournot’s duopoly solution and put forth a substitute model of duopoly. According to Bertrand, two firms produce a homogenous good and compete in prices. There is no limit to the fall in price since each producer can always lower the price by underbidding the other and increasing his supply of output until the price becomes equal to his unit cost of production [marginal cost]. Theoretically, this competition in prices, ends with the firms selling their goods at marginal costs and thus making zero profits. 

In Bertrand’s model, the producers first set the price of the product and then produce the output which is demanded at that price. Thus, in Bertrand’s adjusting variable is price and not output.

Assumptions:

1- Duopolists produce homogeneous goods and consumers have no preference for either of the firms and, therefore, buy from the firm offering the good at a lower price.

2-Both firms compete with price.

3- Both firms have the same constant unit cost of production and average cost and marginal cost are equal. 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. 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.

4- The productive capacity of the producers is unlimited, that is, there is no limit to their increase in the supply of output up to the maximum requirement of demand.
5-In Bertrand’s model each producer believes that his rival will keep his price constant at the present level whatever price he might himself set.

Now to understand Bertrand’s duopoly model, we assume that there are two producers A and B. Suppose that A is the only producer at present he sets the price at the monopoly level, which is the most profitable for him.

Now, suppose that B also enters into the business and starts producing the same product as produced by A. But B assumes that A will go on charging the same price which he is doing at present, irrespective of whatever price he himself might set.

Producer B sets a price slightly lower than A’s price and capture the whole market and make substantial number of profits. As a result. A’s sales, for the moment, falls to zero. Now, producer A will reconsider his price policy. But while deciding about his new price policy he assumes that B will continue to charge the same price which he is doing at present. There are two alternatives open to him. First, he may charge the same price as B is now charging. In this case, he will secure half the market, the other half going to the producer B. Secondly, he may undercut B and set a slightly lower price than that of B. In this case, A will seize the entire market. Considering second option more profitable, producer A undercuts B and sets a price lower than B’s price. With the above move of A, the product demand of producer B will be zero.

Now Producer B has similarly two alternatives:

He may match A’s price or undercut him. Finding the undercutting more profitable, B will set a bit lower price than A and thus seize the whole market. But again, A will be forced to undercut B. The process of undercutting [price war] will go on until the price falls to the level of unit cost of production. Once the price has fallen to the level of unit cost of production, neither of them will like to cut the price further because in that case total costs would exceed total revenue and will therefore bring losses to the duopolists.

Also, neither of them would like to raise the price, since in doing so either of them would be afraid of losing his entire business, given the belief that the other will go on charging the same lower price. Thus, when the price equals to the level of cost, the equilibrium has been achieved.

The result of the firms' strategies is a  Nash equilibrium, (In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players and no player has anything to gain by changing only their own strategy) that is, a pair of strategies (prices in this case) where neither firm can increase profits by unilaterally changing price. 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). 

The fundamental assumption about the behavior of the duopolists in the Cournot and Bertrand’ s models of duopoly is similar. The duopolists in both models (Cournot’s model, relates to the output policy, and in Bertrand’s model, it relates to the price policy) have erroneous and incorrigible belief that the rival will continue to do what he is presently doing regardless of what he himself might do. According to Cournot’s model, equilibrium output is less than the purely competitive output and, therefore, the price is higher than the purely competitive price. But, according to Bertrand’s model, output and price under duopoly are equal to those under pure or perfect competition.

Dr. Swati Gupta

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