<p>At the end of the day, the firm seeks to maximize <strong>profits</strong>, 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.</p>

CC BY-SA 3.0
2017-10-26T17:05:13.173Z
You are absolutely right. I didnt think the “economic example” through... the reason is that I just wanted to ask about economic literature of multi objective optimization, and thought that an economic example would make it suitable for the site. Would it be ok if i edit the question accordingly? I guess that the “economic example” would be if the firm wanted to maximize production instead of profits.
thekiciminister
CC BY-SA 3.0
2017-10-26T18:18:03.027Z
@agustromo: Yes, feel free to edit your question to reflect what you really want to ask.
Herr K.
CC BY-SA 3.0
2017-10-26T16:00:14.657Z
2017-10-26T16:18:50.740Z
2017-10-26T16:18:50.740Z
Herr K.
<p>I was just thinking about the question</p>

<blockquote>
  <p>How does a firm choose the optimal input vector as to maximize profits when the firm produces more than one final product</p>
</blockquote>

<p>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. </p>

<p>My question is, how does the firm choose the input vector $x \in \mathbb{R}_{\geq 0}^{\ell}$ as to maximize profits?</p>

<p>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. </p>

<p>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.</p>

<p>I suspect that a treatment of this topic from an economics point of view would involve some discussion of social choice theory.</p>

<p>Any help is most welcomed!   </p>

CC BY-SA 3.0
2017-10-26T23:08:59.690Z
Sounds to me like any notion of equilibrium used in economics is a “multi-objective” optimisation problem.
Theoretical Economist
<microeconomics><reference-request>
Multi-product profit maximization
thekiciminister
# Multi-product profit maximization

(Asked by: username_0 on 2017-10-26)

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!

**Tags:** microeconomics, reference-request

> **Comments:**
> **username_1**: Sounds to me like any notion of equilibrium used in economics is a “multi-objective” optimisation problem.  

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## 1 answers

### Answer by username_2 on 2017-10-26. Score: 3

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.

> **Comments:**
> **username_0**: You are absolutely right. I didnt think the “economic example” through... the reason is that I just wanted to ask about economic literature of multi objective optimization, and thought that an economic example would make it suitable for the site. Would it be ok if i edit the question accordingly? I guess that the “economic example” would be if the firm wanted to maximize production instead of profits.  
> **username_2**: @agustromo: Yes, feel free to edit your question to reflect what you really want to ask.  

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