Question

Recently, I've posted a question that goes as follows

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.

And the answer that I got went like this

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.

However, the problem with the answer was that even at P=1, Firm 2 would still have an incentive to change their price as P=0.9 under Q=100 for instance will give it 90 as profit, while at P=1, only 50 will be its profit due to the fact that at 1 as firm 1 will also be producing despite him earning 0 profit. Because of this, case 3 above also cannot be a Nash Equilibrium, and I'm just puzzled on what could be one, not to mention that the question also asks for 2 Nash equilibria in this game.

I would really appreciate it if anyone could give help on this problem

Thanks

Answer 1