The Bertrand equilibrium is simply the no profit equilibrium. First, we will demonstrate that the Bertrand outcome is indeed an equilibrium. Imagine a market in which two identical firms sell at market price P, the competitive price at which neither firm earns profits. Implicit in our argument is our assumption that at equal price, each firm will sell to half the market. If Firm 1 were to raise its price above the market price P, Firm 1 would lose all its sales to Firm 2 and would have to exit the market. If Firm 1 were to lower its price below P, it would be operating below cost and therefore at a loss overall. At the competitive outcome, Firm 1 cannot increase profits by changing its price in either direction. By the same logic, Firm 2 has no incentive to change prices. Therefore, the no profit outcome is an equilibrium, in fact a Nash equilibrium, in the Bertrand model.
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