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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!

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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: $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 $u(\mathbf{x}) = \left( \sum_i \alpha_i ^\frac{1}{s} x_i ^\frac{s - 1}{s}\right)^\frac{s}{s-1}$
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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 Matthew Gunn 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.

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    $\begingroup$ The complements and substitutes forms might not work too well if the essay is using Lagrangian maximisation. $\endgroup$ – Ubiquitous Nov 16 '17 at 16:56
  • $\begingroup$ @Ubiquitous agreed. still its a useful reference. $\endgroup$ – EconJohn Nov 16 '17 at 17:01
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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. $f(x_1,x_2,x_3) = \sqrt[4]{x_1x_2x_3}$.

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    $\begingroup$ +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. $\endgroup$ – Matthew Gunn Nov 17 '17 at 22:51

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