Question 1: Suppose that two firms produce differentiated products and compete in prices. As in class, the two firms are located at two ends of a line one mile apart. Consumers are evenly distributed along the line. The firms have identical marginal cost, $60.
Firm B produces a product with value $110 to consumers.
Firm A (located at 0 on the unit line) produces a higher quality product with value $120 to consumers.
The cost of travel are directly related to the distance a consumer travels to purchase a good. If a consumer has to travel a mile to purchase a good, they incur a cost of $20. If they have to travel x fraction of a mile, they incur a cost of $20x.
(a) Write down the expressions for how much a consumer at location d would value the products sold by firms A and B, if they set prices Pa and Pb?
(b) Based on your expressions in (a), how much will be demanded from each firm if prices Pa and Pb are set?
(c) What are the Nash equilibrium prices?
Question 2: Consider two firms selling identical products and competing in prices (Bertrand game). The firms compete against each other in a repeated game. Market demand is given by Q = 500 - P. Firms compete in prices and either face zero demand, half of market demand or all of market demand depending on if their price is greater than, equal to, or less than their competitor's price. Marginal cost is constant and equal to 100.
Suppose that each firm sets the monopoly price if their competitor has never cheated. If their competitor has cheated, the firm will play the Nash equilibrium in the next period and forever more.
(a) If the discount factor is d = 80%, will the firms tacitly collude?
(b) Suppose in the previous problem the firms tacitly collude. When they are tacitly colluding, are they are setting MR = MC? Briefly explain.
(c) Suppose that firms only chose prices every other period. Would that make it easier or harder for firms to cooperate? Briefly explain.
(d) Finally, briefly explain the costs and benefits of a strategy that "forgives" a competitor for cheating after a certain amount of time.