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Quiz 27. Cournot Ans

Quiz 27. Cournot Ans - q 1 – q 2 q 2 30 q 2 = 90 q 2 –...

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Shomu Banerjee ECON 201 COURNOT DUOPOLY ANSWERS Suppose two duopolists produce the same identical product. The market demand for their product is given by the inverse demand curve p = 120 – q 1 q 2 , where p is the price, and q 1 and q 2 the output levels of the firms. The cost of producing one unit is zero for firm 1 and 30 for firm 2. (a) How much will each firm produce in Nash equilibrium? Firm 1’s profit is: Π 1 = pq 1 = (120 – q 1 q 2 ) q 1 = 120 q 1 – ( q 1 ) 2 q 1 q 2 . Differentiate this with respect to q 1 and set equal to zero: 120 – 2 q 1 q 2 = 0. Solve for q 1 to get firm 1’s best-response: q 1 = 60 – 0.5 q 2 . Firm 2’s profit is: Π 2 = pq 2 = (120 –
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Unformatted text preview: q 1 – q 2 ) q 2- 30 q 2 = 90 q 2 – q 1 q 2 – ( q 2 ) 2 . Differentiate this with respect to q 2 and set equal to zero: 90 – q 1 – 2 q 2 = 0. Solve for q 2 to get firm 2’s best-response: q 2 = 45 – 0.5 q 1 . Substitute for q 1 from 1’s best-response into 2’s best-response: q 2 = 45 – 0.5(60 – 0.5 q 2 ) q 2 = 45 – 30 + 0.25 q 2 0.75 q 2 = 15 q 2 *= 20. Substitute for q 2 *=20 into 1’s best-response to get q 1 *= 50. (b) Calculate the equilibrium price p * and profits for each firm. From the demand curve, p * = \$50 and 1 *= \$2500 and 2 * = \$400....
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