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You can use Karush-Kuhn-Tucker conditions. As you are considering a minmisation problem, this gives: $$ \frac{\partial F'}{\partial L_i} \ge \lambda \text{ with equality if $L_i > 0$}\\ \frac{\partial F'}{\partial C_i} \ge \mu \text{ with equality if $C_i > 0$} $$ where $\lambda$ and $\mu$ are the Lagrange multipliers for the adding up constraintes. ...


Maybe the following is useful: Sato (1970), The Estimation of Biased Technical Progress and the Production Function, International Economic Review, 11, 179-208 JSTOR link


Your eq (2.10) is not more general than (2.9), but corresponds to an alternative specification. A more general version would be: $$ C_i(Q_i)=FC_i+a_{1,i}Q_i+a_{2,i}Q^2_i. $$ This specification allows marginal cost to be constant as in (2.9) if $a_{2,i}=0$ or nonconstant as in (2.10) for $a_{1,i}=0,a_{2,i} \neq 0.$ It is more general because it is compatible ...

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