m256hw03soln

A11 110 a12 b1 elasticityi 2000 100 a21 100 a11 110

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Unformatted text preview: 100. a21 110. a22 2.5 b2 which amount to 6 separate conditions. Now at first sight, this might seem not so easy to solve, but in fact clearing fractions in the elasticity matrix entry equations reduces this to a system of linear equations. In general, such calibration conditions might need to be solved numerically for the appropriate parameters. The best procedure is to first see if Mathematica can solve the given conditions on its own, then perhaps to try and simplify the conditions enough to get a solution, and lastly to use the numerical root finding function of Mathematica with an initial approximation to (or guess of) a solution. calibrationsolns Solve calibrationconditions, a11, a12, a21, a22, b1, b2 Solve::ratnz : Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. a11 40., a12 21.8182, a21 18., a22 40.9091, b1 3600., b2 4500. Since we are done with all the symbolic work with these coefficients we can make a simultaneous assignment according to this solution. a11, a12, a21, a22, b1, b2 40., 21.8182, 18., a11, a12, a21, a22, b1, b2 . calibrationsolns 1 40.9091, 3600., 4500. We can check then q1 p1, p2 3600. 40. p1 21.8182 p2 q2 p1, p2 4500. 18. p1 40.9091 p2 q1 p1i, p2i 2000. q2 p1i, p2i 1800. MatrixForm elasticitymatrix p1, p2 40. p1 21.8182 p2 3600. 40. p1 21.8182 p2 3600. 40. p1 21.8182 p2 18. p1 40.9091 p2 4500. 18. p1 40.9091 p2 4500. 18. p1 40.9091 p2 MatrixForm elasticitymatrix p1i, p2i 2. 1. 1.2 2.5 Next 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 6 m256hw03soln.nb N...
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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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