Bertrand Paradox

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Introduction to the Bertrand Model
The Bertrand model was developed by Joseph Bertrand to challenge Cournot’s work on non-cooperative oligopolies. Cournot’s model dealt with an N number of firms who will choose a specific quantity of output where price is a known decreasing function of total output. (About.com 2011) However, Bertrand’s argument was with regard to the setting of prices. He said the only factors influencing the price in an oligopolistic market were the firms themselves and therefore based his model on the fact that firms set prices rather than output. (Carlton & Perloff 2005) As with the Cournot model, the Bertrand model makes some assumptions. There is no market entry limiting the number of firms to two (duopoly) who produce homogenous products in a single period, have the same demand curve and set prices simultaneously. These two non-cooperative firms are also identical in nature, have the same constant returns to scale signifying that both firms have the same unit cost of production and each firm believes its rival’s price is fixed. (Carlton & Perloff 2005) In order to illustrate the Bertrand model, the following hypothetical situation can be used. There are two firms in an industry, Firm A and Firm B that are in line with the assumptions made above. Firm A charges a price PA while Firm B charges a price PB. If PA is greater than MC, then firm A will make a positive economic profit. However, the fact that both firms produce identical products makes the demand for the good perfectly elastic. Therefore if PB was less than PA, the demand for Firm A’s products would fall to zero whereas the demand for Firm B’s products will be equal to the entire market demand. As the two firms are non-cooperative they will chose the only strategy that will allow them to make a positive economic profit which is lowering their price below the competitor. This strategy will continue until both firms reach the point where they make a normal economic profit where Price = Marginal Cost. It is impossible for either firm to set a price below or above this price as either strategy will result in the firm having to shut down. If a firm charges a price lower than MC it will make a loss and if a firm charges a price higher than MC it will make no sales at all and will still make a loss. (Carlton & Perloff 2005) Therefore, both firms will charge the same price for its homogenous products where PA = PB =MC and market demand will be split between the two firms equally where both firms will earn zero economic profits.

The Bertrand Paradox
As explained earlier, the two firms will undercut each other’s prices until they reach a state of Nash Equilibrium which is the best response for players in a non-cooperative game such that no single one of them would be better off switching strategies unless the other player does. (About.com 2011) The Bertrand Paradox arises from the fact that these two firms reach a unique pure strategy Nash Equilibrium at P=MC despite being a duopoly. (Kaplan & Wettstein 2000)

Implications of the Bertrand Paradox
Allocative efficiency occurs in perfectly competitive markets when there is an optimal distribution of goods and services where firms produce at an output level where P=MC. (Economicshelp.org 2011)However Bertrand’s paradoxical result shows that firms in Oligopolistic markets, in this case a duopoly, will also price as if they were in perfect competition. Usually, a monopoly’s is price is greater than a duopoly whose prices are greater than firms in perfect competition. Firms in perfect competition are most efficient (as they produce where P=MC making zero economic profits) with oligopolies following, while monopolies are the least efficient. The implications of this paradox would be that if the number of firms in a particular industry increases from one to two, the price will decrease from monopoly pricing to the prices of perfect competition and will remain the same way as the number...
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