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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 1s 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 firms 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 1s 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 monopolists 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 Cournots article in 1883....
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This note was uploaded on 10/22/2009 for the course ECG 700 taught by Professor Morrill during the Fall '09 term at N.C. State.
 Fall '09
 Morrill

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