<p><strong>hint:</strong></p>
<p>This is constrained optimization problem. The constraints your objective functions are subject to are <span class="math-container">$x_1≥0,x_2≥0,x_3∈\{0,1\}$</span> and <span class="math-container">$1 \cdot x_1 + 1 \cdot x_2+1000 \cdot x_3≤w$</span>. Hence consumption here of any of the good here can't reach infinity, <span class="math-container">$x_3$</span> is explicitly constrained to be between 0 and 1 and consumption of <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> multiplied by their prices cannot exceed the budget constraint given the wealth of individual, which will be some finite constant.</p>
<p>The way how you can solve this problem is to set up a Lagrangian with multiple constraints. Some of the constraints will be inequalities so you will have to check  Karush–Kuhn–Tucker conditions afterwards. You should make an attempt at setting and solving the Lagrangian yourself.</p>

CC BY-SA 4.0
2021-10-10T16:32:28.953Z
Thank you for your help, I was quite confused about the FOC that I forget about the finite wealth.
Garen
CC BY-SA 4.0
2021-10-10T10:01:38.467Z
2021-10-10T10:01:38.467Z
1muflon1
<p>Consider the following utility function.</p>
<p><span class="math-container">$U(w) = \max x_{1}^{0.25}x_{2}^{0.25}(x_{3} +1)^{0.5} $</span>
<span class="math-container">$s.t. x_{1} \geq 0, x_{2} \geq 0, x_{3} \in \{0,1\}$</span> and <span class="math-container">$1 \times x_{1} +1 \times x_{2} + 1000 \times x_{3} \leq w$</span>, where <span class="math-container">$w$</span> is the wealth and <span class="math-container">$x_{1}$</span>, <span class="math-container">$x_{2}$</span> and <span class="math-container">$x_{3}$</span> denote the three goods with \$1, \$1 and \$1000 as their respective prices.</p>
<p>My problem with this function is that I am unable to see where the optimiser is. If I try to solve the FOC of this function, I simply yield the result that both the first goods tend to infinity, and this is obvious, because the consumer would prefer more. However, is my method wrong? Certainly the consumer would purchase more yet I cannot see how I would try to yield an optimiser.</p>

CC BY-SA 4.0
2021-10-10T09:04:53.353Z
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
Community
<utility>
Utility Maximisation With Infinity
Garen
# Utility Maximisation With Infinity

(Asked by: username_0 on 2021-10-10)

Consider the following utility function.

$U(w) = \max x\_{1}^{0.25}x\_{2}^{0.25}(x\_{3} +1)^{0.5} $
$s.t. x\_{1} \geq 0, x\_{2} \geq 0, x\_{3} \in \{0,1\}$ and $1 \times x\_{1} +1 \times x\_{2} + 1000 \times x\_{3} \leq w$, where $w$ is the wealth and $x\_{1}$, $x\_{2}$ and $x\_{3}$ denote the three goods with \$1, \$1 and \$1000 as their respective prices.

My problem with this function is that I am unable to see where the optimiser is. If I try to solve the FOC of this function, I simply yield the result that both the first goods tend to infinity, and this is obvious, because the consumer would prefer more. However, is my method wrong? Certainly the consumer would purchase more yet I cannot see how I would try to yield an optimiser.

**Tags:** utility

> **Comments:**
> **username_1**: Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.  

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## 1 answers

### Answer by username_2 on 2021-10-10. Score: 0 | Accepted answer

**hint:**

This is constrained optimization problem. The constraints your objective functions are subject to are $x\_1≥0,x\_2≥0,x\_3∈\{0,1\}$ and $1 \cdot x\_1 + 1 \cdot x\_2+1000 \cdot x\_3≤w$. Hence consumption here of any of the good here can't reach infinity, $x\_3$ is explicitly constrained to be between 0 and 1 and consumption of $x\_1$, $x\_2$ and $x\_3$ multiplied by their prices cannot exceed the budget constraint given the wealth of individual, which will be some finite constant.

The way how you can solve this problem is to set up a Lagrangian with multiple constraints. Some of the constraints will be inequalities so you will have to check Karush–Kuhn–Tucker conditions afterwards. You should make an attempt at setting and solving the Lagrangian yourself.

> **Comments:**
> **username_0**: Thank you for your help, I was quite confused about the FOC that I forget about the finite wealth.  

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