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The profit of a firm $i$ is given by: $$ \pi_i(p) = p q_i - C_i(q_i) $$ where $p$ is the price, $q$_i is the output of firm $i$ and $C_i(.)$ is the cost function which differs across firms. The first order condition gives: $$ p = \frac{\partial C_i(q_i)}{\partial q_i} = MC_i(q_i^\ast) $$ This shows how to obtain the optimal supply of firm $i$, i.e. where $MC(...

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