3. [12 points] Let us consider a Bertrand model with three firms: 1, 2, and 3. The products of the three firms are 3. (continued) identical from the consumers' point of view, and they supply their products to the same market whose demand curve is Q = 100 - P. (e) Is there any Nash Equilibrium in which (at least some) consumers buy the product from Firm 3? Circle the right answer and then explain. Each firm i sets price Pi for its own product, simultaneously and independently of one another. The price Pi must be Yes, (P1, P2, P3) = ( ) is such an equilibrium. / No. Why not? an integer multiple of 1 cent, and it cannot be negative. Thus the allowable prices are $0, $0.01, $0.02, ..., $0.99, $1, $1.01, etc. Consumers will buy from the firm with the lowest price. If the lowest price is set by two or more firms, these firms will share the market equally. Firm I has constant marginal cost $10 per unit and no fixed cost; Firm 2 has constant marginal cost $20 per unit and no fixed cost; Firm 3 has constant marginal cost $30 per unit and no fixed cost. As usual, firms are assumed to be profit-maximizers. We will regard this as a game in strategic form. Answer the following questions. You only need to report one correct answer if you think there may be more than one correct answers. (a) Consider the profile of strategies (P1 , P2, P3) = (25, 20, 15). Given the prices of the other two firms, is Firm 1's price (i.e. P1 = 25) a best response? Yes/ No. Given the prices of the other two firms, is Firm 2's price (i.e. P2 = 20) a best response? Yes/ No. Given the prices of the other two firms, is Firm 3's price (i.e. P3 = 15) a best response? Yes/ No. Is this a Nash Equilibrium? Yes/ No. (b) Consider the profile of strategies (P1 , P2 , P3) = (1 1 , 12, 13). (f) Is there any Nash Equilibrium in which (at least some) consumers pay a price higher than 20? Circle the right answer and then explain. Given the prices of the other two firms, Firm 1's best response should be to set P1 = Yes, (P1, P2, P3) = ( ) is such an equilibrium. / No. Why not? Is this a Nash Equilibrium? Yes/ No. (circle the right answer) (c) Consider the profile of strategies (P1 , P2, P3) = (11.99, 12, 30). Is this a Nash Equilibrium? Yes/ No. One firm is using a strategy that is weakly dominated. It is Firm and its strategy is weakly dominated by the (alternative) strategy (d) Consider the profile of strategies (P1 , P2, P3) = (19.99 , 30, 20). Is this a Nash Equilibrium? Yes/ No. One firm is using a strategy that is weakly dominated. It is Firm and (g) Is setting P1 = 10 a maximin strategy for Firm 1? Yes/ No. its strategy is weakly dominated by the (alternative) strategy Is setting P1 = 19.99 a maximin strategy for Firm 1? Yes/ No. Is setting P2 = 19.99 a maximin strategy for Firm 2? Yes/ No. Is setting P3 = 100 a maximin strategy for Firm 3? Yes No. (Question 3 continues on next page.) Page 6 of 12 Page 7 of 12