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D profit1 p1 p2 p1 b1 a11 mc1 2 a11 p1 a12 p2 and we

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Unformatted text preview: 1 doesn't enter into the derivative of profit so has no bearing on the first order condition for firm 1 setting its profit maximizing price. The formula for derivative of profit derived above gives q1 p1, p2 b1 a11 p1 p1 mc1 a11 mc1 D q1 p1, p2 , p1 p1 a12 p2 But we can check these are the same because the equation D profit1 p1, p2 , p1 b1 a11 mc1 2 a11 p1 q1 p1, p2 a12 p2 b1 p1 mc1 D q1 p1, p2 , p1 a11 p1 a11 mc1 p1 a12 p2 simplifies to Simplify True which means that the equation is identically valid. Since the derivative of profit is a nice simple linear function of p1 we can easily solve symbolically for the profit maximizing price for firm 1 given a price p2 as profitmaxp1 p2_ b1 a11 mc1 p1 . Solve D profit1 p1, p2 , p1 0, p1 1 0, p2 1 a12 p2 2 a11 and a profit maximizing price for firm 2 given a price p1 as profitmaxp2 p1_ b2 a22 mc2 p2 . Solve D profit2 p1, p2 , p2 a21 p1 2 a22 The Nash equilibrium prices are determined by solving the two first order conditions simultaneously. First let's write the system of equations to be solved as a list. firstorderconditions b1 a11 mc1 2 a11 p1 D profit1 p1, p2 , p1 a12 p2 0, b2 a22 mc2 0, D profit2 p1, p2 , p2 a21 p1 2 a22 p2 0 0 Since this is a simple system of linear equations in p1 and p2 we can solve symbolically for the Nash equilibrium prices. 4 m256hw03soln.nb nashsolns Solve firstorderconditions, p1, p2 2 a22 b1 a12 b2 2 a11 a22 mc1 a12 a22 mc2 p1 , a21 b1 a12 a21 4 a11 a22 2 a11 b2 a11 a21 mc1 2 a11 a22 mc2 p2 a12 a21 4 a11 a22 We can't really see what's going on from these formulas. If the demand isn't linear, it's much less likely that we will be able to solve symbolically anyway. In order to develop some understanding of what the Nash equilibrium solutions mean, it's useful to have numerical demand functions to work with. What information would we require to determine a numerical linear approximation to a demand function? There are 6 coefficients to determine, a11, a12, a21, a22, b1, and b2 so we need somehow 6 conditions. It seems likely that we at least know (say...
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