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Given this market demand curve and cost structure

# Given this market demand curve and cost structure - 2 Q 1...

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Given this market demand curve and cost structure, we want to find the reaction curve  for Firm 1. In the Cournot model, we assume  Q   2  is fixed and proceed. Firm 1's reaction  curve will satisfy its profit maximizing condition,  MR  =  MC  . In order to find Firm 1's  marginal revenue, we first determine its total revenue, which can be described as  follows  Total Revenue = P * Q1 = (100 - Q) * Q1 = (100 - (Q1 + Q2)) * Q1 = 100Q1 - Q1 ^ 2 - Q2 * Q1  The marginal revenue is simply the first derivative of the total revenue with respect to  1  (recall that we assume  Q   2  is fixed). The marginal revenue for Firm 1 is thus:  MR1 = 100 - 2 * Q1 - Q2\  Imposing the profit maximizing condition of  MR  =  MC  , we conclude that Firm 1's  reaction curve is:  100 - 2 * Q1* - Q2 = 10 => Q1* = 45 - Q2/2  That is, for every choice of  Q
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Unformatted text preview: 2 , Q 1 * is Firm 1's optimal choice of output. We can perform analogous analysis for Firm 2 (which differs only in that its marginal costs are 12 rather than 10) to determine its reaction curve, but we leave the process as a simple exercise for the reader. We find Firm 2's reaction curve to be: Q2* = 44 - Q1/2 The solution to the Cournot model lies at the intersection of the two reaction curves. We solve now for Q 1 * . Note that we substitute Q 2 * for Q 2 because we are looking for a point which lies on Firm 2's reaction curve as well. Q1* = 45 - Q2*/2 = 45 - (44 - Q1*/2)/2 = 45 - 22 + Q1*/4 = 23 + Q1*/4 => Q1* = 92/3 By the same logic, we find: Q2* = 86/3 Again, we leave the actual computation of Q 2 * as an exercise for the reader....
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