In some cases, competition in terms of price changes seems more logical than quantity competition, especially in the short run. Besides, one of the assumptions of Cournot’s duopoly model is that firms supply a homogeneous product. Considering this, Bertrand proposed an alternative to Cournot. Considering Bertrand’s model from a game theory perspective, it can be analysed as a simultaneous game where the strategic choice is on prices, rather than quantities.
In this model, consumers will buy from the firm that offers the lowest price, so we can easily have the intuition that the Nash equilibrium is going to be the two firms setting the same price.
Assuming equal and constant cost functions, the demand for each firm is as follows:
The logic is simple: if the price set by both firms is the same but the marginal cost is lower, there will be an incentive for both firms to lower their prices and seize the market. Therefore, the only equilibrium in which none of the firms will be willing to deviate is when price equals marginal cost.
The result of the model creates a paradox, known as Bertrand’s paradox: in a case of imperfect competition (here, a duopoly), where there is a strong incentive to collude, we end up with the same outcome as in perfect competition. The equilibrium does not hold with asymmetric cost functions since the firm with the lowest marginal cost would seize the entire market and become a monopoly. A great analysis of this paradox, known as Edgeworth duopoly model or Bertrand-Edgeworth duopoly, was developed by Francis Y. Edgeworth in his paper “The Pure Theory of Monopoly”, 1897.