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Unformatted text preview: Models of Competition A Cournot game with linear demand and constant marginal cost: Two firms, firm 1 and firm 2. Each firm chooses a quantity of a product to produce, q 1 and q 2 . Each unit costs c to produce, so the total cost of producing q units is q · c . They each choose how much quantity to produce simultaneously. The firms take their product to the market where a price is set so that all quantity is sold. The price is a function of the total quantity produced. In particular P ( q 1 + q 2 ) = a q 1 q 2 Firm 1’s payoff from producing q 1 when firm 2 produces q 2 is P ( q 1 + q 2 ) · q 1 c · q 1 ECG 700 Game Theory Lecture 2 1/ 1 Cournot Model In this game, a firm’s strategy is the quantity of product to produce. There are an infinite number of possible actions (q does not have to be an integer), so we can not write down a matrix for this game. We must use best response functions. Firm i solves the following problem to find a best response to q j : max q i P ( q i + q j ) · q i + c · q i ECG 700 Game Theory Lecture 2 2/ 1 Cournot Model BR i ( q j ) = a q j c 2 If firm 1 and 2 are playing mutual best responses, then: q 1 = a q 2 c 2 and q 2 = a q 1 c 2 substituting one into the other yields: q 1 = q 2 = a c 3 you can check that if firm 2 produces a c 3 , then firm 1’s best response is to also produce a c 3 and vice versa. ECG 700 Game Theory Lecture 2 3/ 1 Collusion What if firm 1 and firm 2 colluded? They would produce a combined output of Q M , the monopolist’s output. We know Q M = a c 2 , so each firm produces a c 4 . The total output without collusion is 2 ( a c ) 3 . With perfect competition, price = marginal cost=c, so total output is a c . Therefore: Q Monopoly < Q Cournot < Q perfectcompetition ECG 700 Game Theory Lecture 2 4/ 1 Bertrand Model of Oligopoly Bertrand Model of Oligopoly Cournot introduced game theory in 1838. It was ignored until Bertrand wrote a review of Cournot’s article in 1883....
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 Fall '09
 Morrill
 Game Theory, Game Theory Lecture

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