Unformatted text preview: notice that QA is a decreasing function of QB. This means if firm A expects firm B to increase output they will reduce QA. Now, we can find firm B's reaction function in a similar manner. B = P QB B = (600 - QA - QB) QB B = 600QB QB2 - QAQB FOCQB 600 QA - 2QB = 0 2QB = 600 QA QB = 300 0.5QA (2) We now have both firm's best responses to each other's output choices. This means that we know what each firm will produce given the output choices of the other firm. For this to be in equilibrium, we must have both firms playing best responses, otherwise one of the firms will have an incentive to change their output. So, let's re-write (1) as QB = 600 2QA and since (2) = (3) we can set them equal and solve for QA. 600 2QA = 300 0.5QA 3/2QA = 300 QA* = 200 We can similarly proceed to solve for QB*, using (1) and re-writing (2) as: QA = 600 2QB (4) (3) 287 and since (1) = (4) we can set them equal and solve for QB. 600 2QB = 300 0.5QB 3/2QB = 300 QB* = 200 and we can find price in equilibrium by subbing QA* = QB* = 200 into the demand equation. P = 600 - QA - QB P* = 200 and further we can solve the firm's profit function in equilibrium. A* = P* QA* = 200 200 = 40,000 B* = P* QB* = 200 200 = 40,000 For the next example, we will consider a Cournot Duopoly Quantity competition with ATC 0 and a situation where one of the firms has a cost advantage. Consider the same two firms as in the example above but this time firm A has a marginal cost of $6 per unit and firm B has a marginal cost of $8 per unit. Let the market demand remain the same... P = 600 - QA - QB The reaction function for firm A is: A = P QA - 6 QA A = (600 - QA - QB) QA 6 QA A = 600QA - QA2 - QAQB 6 QA FOCQA 600 QB - 2QA 6 = 0 2QA = 594 QB QA = 297 0.5QB (1) 288 The reaction function for firm B is: B = P QB - 8 QB B = (600 - QA - QB) QB 8 QB B = 600QB QB2 - QAQB 8 QB FOCQB 600 QA - 2QB 8 = 0 2QB = 592 QA QB = 296 0.5QA Re-write (1) as QB = 594 2QA Re-write (2) as QA = 592 2QB Now, equate (1) and (4) to find QB*... 592 2QB = 297 0...
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This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.
- Spring '10