m256hw03soln

What we really want are nash equilibrium prices we

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Unformatted text preview: p2 , p2 21.8182 p2 0 0, 6340.91 18. p1 81.8182 p2 0 0, 6340.91 18. p1 81.8182 p2 0 firstorderconditions 5200. Solve p1 80. p1 21.8182 p2 D profit1 p1, p2 , p1 91.6344, p2 0, D profit2 p1, p2 , p2 0 , p1, p2 97.6596 nashsolns p1 91.6344, p2 p1nash, p2nash 97.6596 p1, p2 . nashsolns 1 91.6344, 97.6596 So after two rounds of price cuts we were already close to the Nash equilibrium prices. This was also evident from the fact that profits weren't improved much in successive rounds of price cutting. We can graphically see that this is the best each firm can do given that the other firm sets its Nash equilibrium price. Plot profit1 p1, p2nash , p1, 0, 150. 100 000 50 000 20 40 60 80 100 120 140 50 000 100 000 150 000 200 000 Plot profit1 p1, p2nash , p1, 91., 92. 106 640 106 635 106 630 91.2 91.4 91.6 91.8 92.0 m256hw03soln.nb 9 Plot profit2 p1nash, p2 , p2, 0, 150. 100 000 20 40 60 80 100 120 140 100 000 200 000 Plot profit2 p1nash, p2 , p2, 97., 98. 113 440 113 435 113 430 113 425 The final Nash equilibrium demands are q1 p1nash, p2nash 2065.38 q2 p1nash, p2nash 2154.26 so sales have increased with the price cuts. The elasticity matrix at the final prices is given by MatrixForm elasticitymatrix p1nash, p2nash 1.77468 0.765657 1.03165 1.85455 so demand is slightly less elastic at the lower prices, reducing the motivation for firms to lower prices even more. The final profits of the two firms are profit1 p1nash, p2nash , profit2 p1nash, p2nash 106 645., 113 442. whereas their profits at the initial prices were significantly greater profit1 p1i, p2i , profit2 p1i, p2i 120 000., 117 000. If the firms merge or could somehow get together and collude to set prices higher, they could both make more money. However, even if they could come to an agreement on higher prices, including how much less firm 1 should charge than firm 2, as soon as one firm sets its price the other would improve its profit by cutting price. A merger of the firms would mean that the combined firm would maximize the total profit on the two products instead of trying to maximize each separate profit on a product at the e...
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This note was uploaded on 07/12/2013 for the course MATH 256 taught by Professor Schantz during the Spring '11 term at Vanderbilt.

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