<p>To get started note that there are three main types of utility functions/production functions in economics. for a multi-variable case I'll illustrate it below.</p>

<p><strong>Complements</strong>
$$U(\mathbf{X})=\min\{x_i,...,x_n\}$$
<strong>Subsitutes</strong>
$$U(\mathbf{X)}=\sum_{i=1}^{n}a_1x_i$$</p>

<p>where $a_i$ is a constant greater than zero (for normal goods).</p>

<p><strong>Cobb-douglas</strong> (as Matthew Gunn has already said).
$$u(\mathbf{X}) = \prod_i^n x_i^{\lambda_i} $$ </p>

<p>where each $\lambda_i$ is greater than zero (for normal goods).</p>

<p>Hope this helps.</p>

CC BY-SA 3.0
2017-11-16T16:56:39.797Z
The complements and substitutes forms might not work too well if the essay is using Lagrangian maximisation.
Ubiquitous
CC BY-SA 3.0
2017-11-16T17:01:53.117Z
@Ubiquitous agreed. still its a useful reference.
EconJohn
CC BY-SA 3.0
2017-11-16T15:18:09.500Z
2017-11-16T15:18:09.500Z
EconJohn
<p>The word "utility" is in brackets in the body of your question, so I am not sure if it means that you prefer or that you only want utility maximization problems. Anyway, here is an optimization problem that is not about utility.</p>

<p>Suppose a company produces output $y$ using inputs $x_1$ and $x_2$. The technology used is described by the concave production function $f(x_1,x_2)$. (For example $\sqrt{x_1x_2}$.) The company wishes to minimize its costs, that is given the input prices $w_1$ and $w_2$ it wishes to produce $y$ units as cheaply as possible.</p>

<p>The optimization problem is
\begin{align*}
\min_{x_1,x_2} \ &amp; w_1x_1 + w_2x_2 \\
\\
\mbox{s.t. } &amp; f(x_1,x_2) = y \\
\\
&amp; x_1,x_2 \geq 0.
\end{align*}
Assuming the solution is an interior solution, that is $x_1^*, x_2^* &gt; 0$, the Lagrange-multiplier of the production function's condition has a rather nice economic meaning. It is the marginal cost of production.</p>

<p>If you insist on three or more variables, you can easily adjust the number of inputs, e.g. $f(x_1,x_2,x_3) = \sqrt[4]{x_1x_2x_3}$.</p>

CC BY-SA 3.0
2017-11-17T22:51:37.597Z
+1 Understanding the typical utility maximization problem subject to a budget constraint is nice, but understanding more broadly the mathematics of optimization is better! Optimization is fantastically useful and shows up *everywhere.*
Matthew Gunn
CC BY-SA 3.0
2017-11-16T18:57:30.763Z
2017-11-16T18:57:30.763Z
Giskard
<p>Let's say you have $k$ goods $x_1, x_2, \ldots, x_k$. Let $\mathbf{x}$ denote the vector $\mathbf{x} = \begin{bmatrix}x_1 \\ \ldots \\  x_k \end{bmatrix}$.</p>

<h3>Some common workhorses to get you started:</h3>

<ul>
<li><a href="https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function" rel="nofollow noreferrer">Multivariate Cobb-Douglas</a>: $u(\mathbf{x}) = \prod_{i=1}^k x_i^{\lambda_i} $. 

<ul>
<li>Note that $\max_\mathbf{x} u(\mathbf{x})$ has the same solution as $\max_\mathbf{x} \log u(\mathbf{x})$ because the logarithm is a monotonically increasing function. Hence maximizing the utility function $u(\mathbf{x}) = \sum_{i=1}^k \lambda_i \log x_i$ will give the same solution and also is referred to as Cobb-Douglas.</li>
</ul></li>
<li><a href="https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution" rel="nofollow noreferrer">Constant elasticity of substitution</a> $u(\mathbf{x}) = \left( \sum_i \alpha_i ^\frac{1}{s} x_i ^\frac{s - 1}{s}\right)^\frac{s}{s-1}$</li>
</ul>

CC BY-SA 4.0
2017-11-16T14:36:13.747Z
2018-07-28T03:25:36.930Z
2018-07-28T03:25:36.930Z
Matthew Gunn
<p>For a highschool economics/mathematics interdisciplinary essay I will use the Lagrange multipliers and deriving formulas that find the maximum. Could some one maybe suggest any (utility) functions that have 3 (also 4) variables (doesn't matter if it has constants).
Thanks in advance!</p>

<microeconomics><mathematical-economics><utility>
Multivariable Utility functions
Sven van Holten
# Multivariable Utility functions

(Asked by: username_0 on 2017-11-16)

For a highschool economics/mathematics interdisciplinary essay I will use the Lagrange multipliers and deriving formulas that find the maximum. Could some one maybe suggest any (utility) functions that have 3 (also 4) variables (doesn't matter if it has constants).
Thanks in advance!

**Tags:** microeconomics, mathematical-economics, utility


---

## 3 answers

### Answer by username_4 on 2017-11-16. Score: 5 | Accepted answer

Let's say you have $k$ goods $x\_1, x\_2, \ldots, x\_k$. Let $\mathbf{x}$ denote the vector $\mathbf{x} = \begin{bmatrix}x\_1 \\ \ldots \\ x\_k \end{bmatrix}$.

### Some common workhorses to get you started:

* [Multivariate Cobb-Douglas](https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function): $u(\mathbf{x}) = \prod\_{i=1}^k x\_i^{\lambda\_i} $.
  + Note that $\max\_\mathbf{x} u(\mathbf{x})$ has the same solution as $\max\_\mathbf{x} \log u(\mathbf{x})$ because the logarithm is a monotonically increasing function. Hence maximizing the utility function $u(\mathbf{x}) = \sum\_{i=1}^k \lambda\_i \log x\_i$ will give the same solution and also is referred to as Cobb-Douglas.
* [Constant elasticity of substitution](https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution) $u(\mathbf{x}) = \left( \sum\_i \alpha\_i ^\frac{1}{s} x\_i ^\frac{s - 1}{s}\right)^\frac{s}{s-1}$


---

### Answer by username_1 on 2017-11-16. Score: 1

To get started note that there are three main types of utility functions/production functions in economics. for a multi-variable case I'll illustrate it below.

**Complements**
$$U(\mathbf{X})=\min\{x\_i,...,x\_n\}$$
**Subsitutes**
$$U(\mathbf{X)}=\sum\_{i=1}^{n}a\_1x\_i$$

where $a\_i$ is a constant greater than zero (for normal goods).

**Cobb-douglas** (as username_4 has already said).
$$u(\mathbf{X}) = \prod\_i^n x\_i^{\lambda\_i} $$

where each $\lambda\_i$ is greater than zero (for normal goods).

Hope this helps.

> **Comments:**
> **username_2**: The complements and substitutes forms might not work too well if the essay is using Lagrangian maximisation.  
> **username_1**: @username_2 agreed. still its a useful reference.  

---

### Answer by username_3 on 2017-11-16. Score: 1

The word "utility" is in brackets in the body of your question, so I am not sure if it means that you prefer or that you only want utility maximization problems. Anyway, here is an optimization problem that is not about utility.

Suppose a company produces output $y$ using inputs $x\_1$ and $x\_2$. The technology used is described by the concave production function $f(x\_1,x\_2)$. (For example $\sqrt{x\_1x\_2}$.) The company wishes to minimize its costs, that is given the input prices $w\_1$ and $w\_2$ it wishes to produce $y$ units as cheaply as possible.

The optimization problem is
\begin{align\*}
\min\_{x\_1,x\_2} \ & w\_1x\_1 + w\_2x\_2 \\
\\
\mbox{s.t. } & f(x\_1,x\_2) = y \\
\\
& x\_1,x\_2 \geq 0.
\end{align\*}
Assuming the solution is an interior solution, that is $x\_1^\*, x\_2^\* > 0$, the Lagrange-multiplier of the production function's condition has a rather nice economic meaning. It is the marginal cost of production.

If you insist on three or more variables, you can easily adjust the number of inputs, e.g. $f(x\_1,x\_2,x\_3) = \sqrt[4]{x\_1x\_2x\_3}$.

> **Comments:**
> **username_4**: +1 Understanding the typical utility maximization problem subject to a budget constraint is nice, but understanding more broadly the mathematics of optimization is better! Optimization is fantastically useful and shows up *everywhere.*  

---

