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Bertrand Competition with Linear or Quadratic Costs

This Demonstration offers a quick way of analyzing Bertrand (i.e. price) competition among firms, assuming they are identical and have costs that are linear (no capacity constraints) or quadratic (capacity constraints). You can choose the number of firms, the type of costs involved, and the parameters governing a linear demand.
The graph shows, for a hypothetical firm 1:
• Profits earned when the firm acts as a monopolist (brown),
• Profits earned when the firm charges the same price as the competitors (green),
• Profits that the firm can earn for any price it charges, given that the competitors are all charging a certain price (blue).
The behavioral assumption is that, when firm 1 charges less than , the firm sells to the entire demand; when firm 1 charges , each of the firms sells of the entire demand (the consumer buys equally from all the firms); when firm 1 charges more than , the firm sells nothing (consumers buy from the other firms because they are cheaper). The dynamics of price competition involve each firm acting in its best interest given the prices set by the other firms (). As the model is symmetric, the Nash equilibrium of the game is symmetric and involves all the firms charging the same price. You can discover the Nash equilibria of this game by varying and observing how firm 1's profits (blue) change according to the choice of . The red dots indicate the range of prices that generate a symmetric Nash equilibrium of the game. To facilitate comprehension, shown in the gray box turns red when it reaches a Nash equilibrium price.

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DETAILS

This Demonstration offers a quick way to find the Nash equilibria of the Bertrand oligopoly model of price competition. Focus your attention on the blue piecewise curve, which shows firm 1's profits as a function of (the price this firm charges) and (the price all the other firms are charging). The blue curve has three parts: the first expresses the profits when (it follows the monopolist's profits curve, because when charging the lowest price, firm 1 sells to the entire market); the second expresses the profits when (the blue dot); it follows the green curve, which represents firm 1's profits when sharing the market with all the other firms (since they are charging the same price, they sell of the market demand each); the third expresses the profits when , which are zero since consumers will buy from the firms charging a cheaper price.
You can vary because it is endogenous to the problem. Since all the firms are identical, we just look at the symmetric Nash equilibria of this game (where all the firms charge the same price). They are given by the prices at which the blue dot (corresponding to firm 1's profits when matching the other firms' prices) is not below the blue curve (corresponding to firm 1's profits when charging a price different from the other firms). After getting some acquaintance with this Demonstration, by changing , you should see how the Nash equilibrium concept applies in this oligopolistic market.
When costs are linear, there is only one Nash equilibrium, where each firm charges a price equal to the marginal cost, irrespective of the number of firms. Profits are zero.
When costs are quadratic, there are many Nash equilibria, most of which induce positive profits for the firms. Price competition does not necessarily drive profits down to zero.
The equations for the model are:
demand: ,
total cost for each firm: ,
monopoly profits: ,
matching price profits: ,
where (the costs are linear or quadratic) and indicates the number of firms.
Firm 1's profits =
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