m256hw03soln

On the other hand if we couldnt estimate the marginal

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Unformatted text preview: ext we need to define the cost functions. The fixed costs won't enter into the derivatives of profits so we can set these to zero. The marginal costs might be given to us, and in terms of the marginal costs we can compute Nash equilibrium prices. (On the other hand, if we couldn't estimate the marginal costs maybe we would assume that initial prices given we already a Nash equilibrium and use this information to infer what the marginal costs must be.) Suppose fc1 fc2 mc1 mc2 0.; 0.; 40.; 45.; Then the cost and profit functions become cost1 q1 0. 40. q1 cost2 q2 0. 45. q2 cost1 q1 p1, p2 0. 40. 3600. 40. p1 21.8182 p2 18. p1 40.9091 p2 40. p1 21.8182 p2 18. p1 40.9091 p2 cost2 q2 p1, p2 0. 45. 4500. profit1 p1, p2 0. 40. 3600. p1 3600. 40. p1 21.8182 p2 profit2 p1, p2 0. 45. 4500. 4500. 18. p1 40.9091 p2 p2 Suppose firm 1 sets its price to p1i. Then the profit function for firm 2 is profit2 p1i, p2 0. 45. 6300. 40.9091 p2 6300. 40.9091 p2 p2 Plot profit2 p1i, p2 , p2, 0, 150. 100 000 20 40 60 80 100 120 140 100 000 200 000 looks to have a maximum closer to 100. than 110., i.e., firm 2 can lower price and make more. The exact price for 2 given p1=p1i is p2a p2 . Solve D profit2 p1i, p2 , p2 0, p2 1 99.5 If firm 2 charges this price then the profit function for firm 1 becomes m256hw03soln.nb 7 profit1 p1, p2a 0. 40. 5770.91 40. p1 5770.91 40. p1 p1 Plot profit1 p1, p2a , p1, 0, 150. 100 000 50 000 20 40 60 80 100 120 140 50 000 100 000 150 000 200 000 with a maximum closer to 95., or more precisely, p1b p1 . Solve D profit1 p1, p2a , p1 0, p1 1 0, p2 1 0, p1 1 92.1364 But then firm 2 would want a price of p2c p2 . Solve D profit2 p1b, p2 , p2 97.77 and firm 1 would want p1d p1 . Solve D profit1 p1, p2c , p1 91.6645 Each firm lowers its price in response to the other firm lowering price. Initially firms 1 and 2 were making respectively profit1 p1i, p2i , profit2 p1i, p2i 120 000., 117 000. Then firm 2 lowered price and firms were making profit1 p1i, p2a , profit2 p1i, p2a 106 255., 121 510....
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