Multi-product profit maximization

Question

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!

Answer 1

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.