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Unformatted text preview: Econ 121 – Fall 2010 UC Berkeley Professor Cristian Santesteban Problem Set 5 Due: Thursday, October 14th by 11:15am Problem 1 We know that the single shot prisoner’s dilemma game results in a dominant Nash equilibrium strategy that is Pareto inefficient. Suppose we allow the two prisoners to retaliate after their respective prison terms. Formally, what aspect of the game would this affect? Could a Pareto efficient outcome result? Problem 2 Consider the following game: Player 1/ Player 2 T B L 2, 1 0, 0 R 0, 0 2, 1 Derive the best response correspondences for both players, depict them graphically, and find all Nash equilibria of this game. Problem 3 Goliath Software is a large software company that sells an internet browser called X. Each year a new startup comes along. The startup can either produce another browser that is comparable to X or a search engine. Seeing what the startup produced, Goliath Software either updates X or not. At the end of that year, independent of the outcome, the start up disappears, leaving its place to the next start up. The annual profits for each contingency are as in the following table, where the first entry is the profit of Goliath Software: (The above is just a table, not a game.) If Goliath Software gets annual profit of π0 in year 0, π1 in year 1, . . . , πt in year t, . . ., then its overall payoff is π0 + .9π1 + (.9)2 π2 + · · · + (.9)t πt + · · · . The payoff of each startup is its own annual profit. The entire history is observable, and the game never ends. For the strategy profile below, check whether it is a subgame perfect equilibrium, and identify the observed outcome of the strategy profile. • • • • Problem 4 Suppose that demand is given by P = A ‐ Q and marginal costs are constant and equal to zero. Assume that there are two firms that are Cournot competitors, and punishments are “grim.” A grim strategy is defined as follows: • • Play the collusive output in each period as long as all other firms have done so in the past. If any firm has ever deviated from playing its collusive output, play the Cournot output. If the startup produces a browser and Goliath Software has updated X every time a startup produced a browser in the past, then Goliath Software updates its product; otherwise, it does not update. A startup produces a search engine if Goliath Software has updated X every time a startup produced a browser in the past; otherwise, the startup produces a browser. (a) As a function of the discount factor, find the minimum industry output that is sustainable—assuming the two firms agree to produce the same amount. (b) As the discount factor increases, what happens to the extent of collusion or the exercise of market power? Why? Problem 5 An industry consists of two firms. The demand function for the product of firm i is qi (pi , pj ) = a − bpi + cpj . The marginal cost of production for each firm is zero. (a) Find the critical value of the discount factor that supports joint profit maximization with grim punishment strategies. [Hint: Let r = c/b, where 0 < r < 1.] (b) How does the critical value of the discount factor depend on the degree of product differentiation r? What does r = 1 imply about the relationship between the two goods? r = 0? ...
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