PS1F07

# PS1F07 - Fall 2007 GAME THEORY IN THE SOCIAL SCIENCES...

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Fall 2007 G AME T HEORY IN THE S OCIAL S CIENCES Problem Set 1 (Due in Lecture on Tuesday, September 18) 1. In Bertrand price competition the firms decide on the price that they will charge and the consumers buy from the firm with the lowest price. Another way to think about oligopolistic markets is Cournot competition. In this model, the firms decide how much to produce. Consider the following problem: Two firms simultaneously decide how much to produce. Each firm's factory is capable of producing outputs of 0, 2, 4, or 6. The cost of producing each unit is 1. The price that prevails in the market depends on the total output from the two firms. In particular, the price for each unit is given by where q 1 is firm 1's output and q 2 is firm 2's output. So, for example, if firm 1 produces 2 and firm 2 produces 4, the total output is 6 and the market price is 13-(2+4)=7 per unit. This gives firm 1 revenues of (7)(2)=14; costs of (2)(1)=2; and profits of 14-2=12. Firm 2's revenues are 7(4)=28; its costs are 4; and its profits are 24. pq =− + 13 12 () q (a) Specify the strategic form for this game and solve it by iterated deletion of dominated strategies. (b) Suppose that there was only one firm in the market. Perhaps it owns both factories. How much would this monopolist produce to maximize its profits and what would these profits be? 2. In the Bertrand model we discussed in class, we assumed that two firms were selling to a

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PS1F07 - Fall 2007 GAME THEORY IN THE SOCIAL SCIENCES...

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