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.
 Spring '11
 Schantz
 Math

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