I was just thinking about the question
How does a firm choose the optimal input vector as to maximize profits when the firm produces more than one final product
More precisely if $\mathbb{R}_{\geq 0}^{\ell}=\{x \in \mathbb{R}^{\ell} : x \geq 0\}$, the production function would be $$ f:\mathbb{R}_{\geq 0}^{\ell}\rightarrow \mathbb{R}_{\geq 0}^{n}, \quad n\geq 2 $$ For simplicity we can assume the firm operates in a competitive market so that the price is given.
My question is, how does the firm choose the input vector $x \in \mathbb{R}_{\geq 0}^{\ell}$ as to maximize profits?
I did some research and apparently the type of problem this is a “multi-objective” optimization problem, and it appears to be an active field in engineering. Some of the methods to solve this kind of problems use pareto-dominance style arguments.
I wanted to know if anyone could clarify the specific question of multi product profit maximization, and more generally I would also appreciate any literature or books on this topic (multi objective optimization), written for economists, as I have been searching in google and every book I find appears to be written for engineers.
I suspect that a treatment of this topic from an economics point of view would involve some discussion of social choice theory.
Any help is most welcomed!