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The Production Problem

$$ \underset{q_i \in \mathbb{N}}{\max} \ F(p, q_i, q_{-i}),\quad \text{where} \ F(p, q_i, q_{-i}) = p\Big(\sum_{j=1}^n q_j\Big)q_i - c(q_i) $$

First Order Necessary Condition

Note $q = \sum q_j$

$$ \partial F_{q_i}(p, q_i, q_{-i}) = p'(q)q_i + p(q) - c'(q_j) $$