Consider the Bertrand model studied in class. When solving the game, we took a d

Question

Consider the Bertrand model studied in class. When solving the game, we took a different approach than usual: first, come up a conjecture and then show that it is actually true. This approach, however, may not be so fruitful in some cases because when one struggles to prove a conjecture, he cannot be sure whether the struggle is due to his lack of ability to prove it or to an incorrect conjecture. Now we take the usual approach to identifying Nash equilibrium. Suppose that the inverse demand function is p = a by. For each i {1, 2}, let pi the price charged by firm i. Let c > 0 be the common marginal cost. (a) When choosing p1, firm 1 should take p2 into account. Find the best response correspondence b1(p2) by distinguishing the following cases: (i) p2 < c; (ii) p2 = c; (iii) p2 > c. Is b1(p2) well-defined for p2 > c? 1 (b) Using your answer to part (a), find all Nash equilibria of the game.

Explanation / Answer

a. If firm 2 sells at less than marginal cost of the firm, then firm 1 will also lower its price below the marginal cost as firm 2 will capture the entire market.

If prices charged are equal to marginal cost, then firm 1 will also charge price equal to marginal cost of production of the firm.

If price of firm 2 is greater than c, then firm 1 will charge price less than p2 in order to capture the entire market.

b. The Nash equilibrium in case of Bertrand price competition lies when both firms charge price equal to marginal cost of production of both the firms.

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