The Bertrand Duopoly Equilibrium
Why is the competitive price a Nash equilibrium in the Bertrand model? First, if both firms set the competitive price with price equal to marginal cost (unit cost). Neither firm will earn any profits. However, if one firm sets price equal to marginal cost, then if the other firm raises its price above unit cost, then it will earn nothing, since all consumers will buy from the firm still setting the competitive price (recall that it is willing to meet unlimited demand at price equals unit cost even though it earns no profit). No other price is an equilibrium. If both firms set the same price above unit cost and share the market, then each firm has an incentive to undercut the other by an arbitrarily small amount and capture the whole market and almost double its profits. So there can be no equilibrium with both firms setting the same price above marginal cost. Also, there can be no equilibrium with firms setting different prices. 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).
Note that the Bertrand equilibrium is a weak Nash-equilibrium. The firms lose nothing by deviating from the competitive price: it is an equilibrium simply because each firm can earn no more than zero profits given that the other firm sets the competitive price and is willing to meet all demand at that price.
Read more about this topic: Bertrand Competition
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