Two firms are involved in Bertrand competition. The marginal cost for firm 1 and 2 are mc1=1 and mc2=0. As usual, the consumers purchase only from the firm with a lower price. If p1=p2, then each firm will sell to 50% of the consumers. Find any two Nash Equilibria of the game. And explain why they are Nash Equilibria.
To find the nash equilibrium we will try to figure out if there exists a possible profitable deviation or not.
Case 1: P1 = P2 > mc1.
Now both consumers will earn positive profit but both can deviate by charging a lesser price and get control of the complete market thus is not a nash equilibrium.
Case 2: P1 =P2< mc0.
Still there exists a possible deviation as firms can lower their loss by increasing price equal to their marginal cost.
Case 3: P1=P2 =mc1. This is a nash equilibrium as there is no profitable deviation for any player. If palyer1 lowers price it run into loss. Similarly if firm2 lowers the price it may capture whole market but will reduce revenue thus reducing the profit.
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