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 i is fixed for all firms i not equal to 1. Firm 1's reaction curve will satisfy its profit maximizing condition, MR 1 = MC 1 . 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 +. ..+ Qn)) * Q1 = 100 * Q1 - Q1 ^ 2 - (Q2 +. ..+ Qn)* Q1 The marginal revenue is simply the first derivative of the total revenue with respect to Q 1 (recall that we assume Q i for i not equal to 1 is fixed). The marginal revenue for firm 1 is thus: MR1 = 100 - 2 * Q1 - (Q2 +. ..+ Qn) Imposing the profit maximizing condition of MR = MC , we conclude that Firm 1's reaction curve is: 100 - 2 * Q1* - (Q2 +. ..+ Qn) = 10 => Q1* = 45 - (Q2 +. ..+ Qn)/2 Q 1 * is Firm 1's optimal choice of output for all choices of Q 2 to Q n . We can perform analogous analysis for Firms 2 through
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This note was uploaded on 12/13/2011 for the course ECO 1320 taught by Professor Staff during the Fall '11 term at Texas State.