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Quiz 28. CourStack Ans - quiz Now solve firm 1’s problem...

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Shomu Banerjee ECON 201 COURNOT STACKELBERG 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 $10 for firm 1 and $20 for firm 2. Firm 1 is the market leader, firm 2 the follower. How much will each firm produce in a Stackelberg equilibrium? Solve the problem backwards: assume that firm 1 has already chosen q 1 . Then firm 2 maximizes Π 2 = pq 2 - 20 q 2 = (120 – q 1 q 2 ) q 2 - 20 q 2 = 100 q 2 q 1 q 2 – ( q 2 ) 2 . Differentiate this with respect to q 2 and set equal to zero: 100 – q 1 – 2 q 2 = 0. Solve for q 2 to get q 2 = 50 – 0.5 q 1 which is the same as this firm’s best-response calculated in the Cournot duopoly
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Unformatted text preview: quiz. Now solve firm 1’s problem. Firm 1’s profit is: 1 = pq 1- 10 q 1 = (120 – q 1 – q 2 ) q 1- 10 q 1 = 110 q 1 – ( q 1 ) 2 – q 1 q 2 . But because it is a leader, it can take into account how the follower is going to choose its output in figuring out the best level of output to produce. Therefore substitute firm 2’s best-response into 1 : 1 = 110 q 1 – ( q 1 ) 2 – q 1 (50 – 0.5 q 1 ) = 60 q 1 – 0.5( q 1 ) 2 . Differentiate this with respect to q 1 and set equal to zero: 60 – q 1 = 0, or q 1 * = 60. Substitute in firm 2’s best-response to find q 2 * = 20....
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