Extension of the Cournot Model

# Extension of the Cournot Model - => Q1 = 90(1 n By...

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Extension of the Cournot Model 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 n (which are identical to firm 1) to determine their reaction curves. Because the firms are identical and because no firm has a strategic advantage over the others (as in the Stackelberg model), we can safely assume all would output the same quantity. Set Q 1 * = Q 2 * = . .. = Q n * . Substituting, we can solve for Q 1 * . Q1* = 45 - (Q1*)*(n-1)/2 => Q1* ((2 + n - 1)/2) = 45
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Unformatted text preview: => Q1* = 90/(1+n) By symmetry, we conclude: Qi* = 90/(1+n) for all firms I In our model of perfect competition, we know that the total market output Q = 90 , the zero profit quantity. In the n firm case, Q is simply the sum of all Q i * . Because all Q i * are equal due to symmetry: Q = n * 90/(1+n) As n gets larger, Q gets closer to 90, the perfect competition output. The limit of Q as n approaches infinity is 90 as expected. Extending the Cournot model to the n firm case gives us some confidence in our model of perfect competition. As the number of firms grow, the total market quantity supplied approaches the socially optimal quantity....
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