Multi-product profit maximization 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!
At the end of the day, the firm seeks to maximize profits, which is a one-dimensional object. So the firm simply solves 
\begin{equation}
\max_{\mathbf x}\;\mathbf{p}\cdot\mathbf f(\mathbf x)-c(\mathbf x)
\end{equation}
where $\mathbf x\in\mathbb R_+^\ell$ is a vector of $\ell$ inputs, $\mathbf p\in\mathbb R_+^n$ is a vector of $n$ prices for the final output, $\mathbf f:\mathbb R_+^\ell\to\mathbb R_+^n$ is the production function mapping $\ell$ inputs to $n$ outputs, and $c:\mathbb R_+^\ell\to \mathbb R_+$ is the cost function. This is a standard unconstrained optimization exercise that any first year grad student in economics should know how to solve. The standard textbooks --- MWG, Jehle and Reny, Simon and Blume --- have sections that cover the techniques.