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# HW4 - Microeconomic Theory II Assignment 4 Using Game...

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Microeconomic Theory II Assignment 4: Using Game Theory Solutions February 25, 2007 Problem 1. Consider the simple duopoly model with linear demand: Two firms produce a homogeneous good for sale in a market with the inverse demand curve P = Max { d Q, 0 } , where P is the price and Q represents total industry output. Let q i be the output of firm i = 1 , 2; d > 0 is a parameter measuring the intensity of demand. Suppose each firm produces at constant marginal cost, c i > 0 represent the unit cost of firm i = 1 , 2, with d > c 1 c 2 0 . (a) If the two firms could conclude a binding agreement to arrange production so as to maximize industry profits, what levels of output would that agreement specify for the two firms, and how much profit would they share? (b) For a fixed level of output for firm 1, q 1 0, find the output level for firm two which maximizes its profits, q 2 = R 2 ( q 1 ). (This is firm 2’s “reaction function.”) Likewise, find firm 1’s reaction function, q 1 = R 1 ( q 2 ). Depict these functions graphically, on a common diagram. Find the Cournot-Nash equilibrium outcome for this industry, and the level of profits associated with it for each firm. When will this Cournot equilibrium coincide with the industry’s profit-maximizing outcome? (c) Let d = 1, and c 1 = c 2 = 0, and consider the following three period game: In the first period firm 1 moves, selecting an output level which he must maintain unchanged for two periods. Firm 2’s output is zero in the first period. In the second period firm 2 enters, and selects an output which he must maintain until the end of the game (i.e., for the second * Prepared by ¨ Omer ¨ Ozak ([email protected]), Department of Economics, Brown University. If you find any typos or mistakes please let me know, so that it can be fixed. 1

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Microeconomic Theory II Solutions and third periods). He knows what output level firm one has selected before him, and firm 1 knows that he will know this. In the third period firm 1 selects his output level for the final time, knowing the level that firm 2 chose in the second period. [Thus, production involves being committed for at least two periods; firm 1 is an encumbent who faces the possibility of entry, and can react to that entry with a one period lag.] Each firm wants to maximize the undiscounted sum of profits over the three periods. Draw an extensive form game tree that diagrams the order of moves and strategic relations obtaining in this situation. Assume that the rationality of both firms is common knowledge between them, and use the logic of backward induction to determine each firm’s level of production in each period.
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