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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 $MC(... 2 Premise of your question is simply false. You state (emphasis mine): Price elasticity of demand$\mathrm{e_{D,P} = \dfrac{dD}{dP}. \dfrac{P}{Q^*}}$clearly depends on the levels of price and quantity. Incorrect. Consider trivial counter example. A perfectly reasonable demand can be given by: $$Q = A p^{-\epsilon}$$ where$Q$is quantity,$p$price and$...