DSE2010 (Ques#58, Option A)

[vikram]

A Bertrand duopoly with 2 firms producing homogeneous good and set prices p1 and p2 respectively. p1 and p2 can only be positive integers. If p1<p2, firm 1 sells 5-p1 and firm 2 sells nothing and if p1>p2, firm 2 sells 5-p2 and firm 1 sells nothing. If p1=p2, each firm sells (5-p1)/2. Avg cost of firm 1 = 5/2 and avg cost of firm 2 = 3/2. In equilibrium

(a) p1=p2=2
(b) p1=p2=3
(c) p1=3 and p2=2
(d) p1=3 and p2=2 or 3

How can the answer be (d) to this question? Shouldn't prices be equal in equilibrium?

[vikram]

Please let me know how to answer this question.

[Chocolate Frog]

Construct a pay-off matrix for both players and find the Nash equilibrium/equilibria. The pay-offs will, of course, be their respective profits.

[vikram]

You mean only at the price values of 2 and 3 for each firm? i.e. p1=2,3 and p2=2,3? Aren't then we supposing already that the rest of the price values are not feasible (p1 or p2 = 0,1,4,5)

[Chocolate Frog]

In Bertrand competition with firms producing homogeneous products and competing by setting prices, prices will get driven down to marginal cost levels. If one firm has a cost advantage (lower average cost), it will charge a price slightly lower than the marginal cost of the other firm, and capture the entire market.

Now, prices 0 and 1 are obviously unfeasible, because both firms make negative profits; they'd rather not produce anything at all. The lowest price firm 1 can bear is 2.5 and the lowest price firm 2 can bear is 1.5. Since only integral prices are allowed, p1 = 3 and p2 = 2, 3 are the only interesting cases. Now, if you construct a pay-off matrix, you'll find that firm 2 is indifferent between choosing p2 = 2 and p2 = 3, earning a profit of 3/2 in either case.