<p>As said in the comments, the question lacks of details and should be edited.</p>
<p>However,  I try to give an answer based on the most plausible interpretation of the question, as its overall meaning is enough clear to outline an answer. I hope it can help you, to clarify your ideas and also to edit the question in an appropriate way.</p>
<p>First of all, it should be specified what  <span class="math-container">$X$</span> is, in particular which set it is.</p>
<p>I suppose it is <span class="math-container">$X \subset \mathbb{R^n}$</span>. In principle,  <span class="math-container">$X$</span> could be any metric space, that is a set on which a <em>distance</em> is defined. But, in your case I take, as it is usual,  <span class="math-container">$X \subset \mathbb{R^n}$</span>.</p>
<p><strong>Question 1.</strong> In this case, <span class="math-container">$x$</span> is a vector in <span class="math-container">$\mathbb{R^n}$</span>, <span class="math-container">$x=(x_1, x_2,..., x_n)$</span>.</p>
<p><strong>Question 2</strong>. No, the difference (no matter what this 'difference' could be, that is, which difference you refer to) has <strong>nothing to do</strong> with the minus sign of the utility function <strong><span class="math-container">$u(a)= -d(a,x)$</span>.</strong>
To understand this point we have to know what is a <em>distance</em> in mathematics (your exercise speaks of <em>distance function)</em>.</p>
<p>I put the rigorous definition of <em>distance</em>, if you don't know it already,  separately at the end of this answer. For now, think of the everyday notion of distance, e.g. the distance between two cities or streets.</p>
<p>The consumer has an ideal point, the most preferred bundle, <span class="math-container">$x$</span>, so they prefers a bundle of goods as close as possible to the ideal <span class="math-container">$x$</span>, that is at the smallest possible distance from <span class="math-container">$x$</span>.</p>
<p>Why the minus sign? Because, in mathematics (as in everyday life) a distance is <strong>always positive</strong>, so the minus sign says that utility is <strong>greater if the distance <span class="math-container">$d(a, x)$</span> is smaller</strong>.</p>
<p><strong>Question 3</strong> This question is not clear, what is <span class="math-container">$y$</span>? You should correct it.
If you are asking what is the utility of a bundle <span class="math-container">$b$</span>, you have already the answer, by the definition of the utility function: <span class="math-container">$u(b)= -d(b,x)$</span>.</p>
<hr />
<p><strong>Appendix. Definition of <em>distance</em> in mathematics.</strong></p>
<p>The notion of <em>distance</em> in mathematics is a very important notion, together with the notion of <em>metric space</em>. The notion of distance is an abstraction and a generalization of the intuitive, everyday life, idea of the distance between two objects.</p>
<p><em>Definition</em>. Given a set <span class="math-container">$Y$</span> of <strong>any</strong> nature (so not only   <span class="math-container">$\mathbb{R^n}$</span>), a <em>distance</em> or <em>metric</em> <span class="math-container">$d$</span> is a function from <span class="math-container">$Y  \times Y$</span> to <span class="math-container">$\mathbb{R}$</span> such that, for all <span class="math-container">$x, y, z \in Y$</span>,  the following properties hold:</p>
<ol>
<li><p><span class="math-container">$d \geq 0$</span>. (<em>positivity</em>)</p>
</li>
<li><p><span class="math-container">$d(x,y)=0$</span> if and only if <span class="math-container">$x=y$</span>.</p>
</li>
<li><p><span class="math-container">$d(x,y)=d(y,x)$</span>   <em>(Symmetry)</em></p>
</li>
<li><p><span class="math-container">$d(x,y)\leq d(x,z)+d(z,y)$</span> <em>(Triangle inequality)</em>.</p>
</li>
</ol>
<p><em>Definition.</em> A <em>metric space</em> is a set <span class="math-container">$Y$</span> on which a distance <span class="math-container">$d$</span> is defined, so a metric space is a pair <span class="math-container">$(Y,d)$</span>.</p>
<p>It must be underlined that on a set <span class="math-container">$Y$</span> can be defined various distances, also in <span class="math-container">$\mathbb{R^n}$</span> there could be many different distances.</p>
<p>But in <span class="math-container">$\mathbb{R^n}$</span> there is a usual distance, the <strong>Euclidean distance</strong>, which, given two vectors <span class="math-container">$x=(x_1,...,x_n)$</span> and <span class="math-container">$y=(y_1,...,y_n)$</span>, is defined as</p>
<p><span class="math-container">$$\sqrt {(x_1-y_1)^2+ (x_2-y_2)^2+…+ (x_n-y_n)^2}$$</span>.</p>
<p>Notice that in <span class="math-container">$\mathbb{R}$</span> the Euclidean  distance between two points <span class="math-container">$x$</span> and <span class="math-container">$y$</span> is reduced to <span class="math-container">$d(x,y)=|x-y|$</span>, that is it is reduced to the absolute value of the difference.</p>

CC BY-SA 4.0
2022-12-02T13:37:32.037Z
2022-12-02T13:37:32.037Z
BakerStreet
<p><a href="https://i.sstatic.net/zDzzK.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/zDzzK.png" alt="enter image description here" /></a></p>
<p>The question is from Ariel Rubinstein's book.
1-My first question is that is the ideal point x something like (x1,x2)?</p>
<p>2-Is the reason the utility function is expressed in negative form we are taking the difference between them?</p>
<p>3- In that case, is u(b) = -d(a,y)?</p>

CC BY-SA 4.0
2022-11-30T12:26:41.207Z
"Ariel Rubinstein's book"? Rubinstein has written 7 books...
VARulle
CC BY-SA 4.0
2022-11-30T12:29:25.817Z
Q1 cannot be answered without the full context. In Q2 it is unclear what "them" refers to. In Q3, it is unclear where this $y$ comes from. Please edit.
VARulle
<microeconomics>
ideal point and distance function
Disintegrators
# ideal point and distance function

(Asked by: username_0 on 2022-11-30)

[![enter image description here](https://i.sstatic.net/zDzzK.png)](https://i.sstatic.net/zDzzK.png)

The question is from Ariel Rubinstein's book.
1-My first question is that is the ideal point x something like (x1,x2)?

2-Is the reason the utility function is expressed in negative form we are taking the difference between them?

3- In that case, is u(b) = -d(a,y)?

**Tags:** microeconomics

> **Comments:**
> **username_1**: "Ariel Rubinstein's book"? Rubinstein has written 7 books...  
> **username_1**: Q1 cannot be answered without the full context. In Q2 it is unclear what "them" refers to. In Q3, it is unclear where this $y$ comes from. Please edit.  

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## 1 answers

### Answer by username_2 on 2022-12-02. Score: 2

As said in the comments, the question lacks of details and should be edited.

However, I try to give an answer based on the most plausible interpretation of the question, as its overall meaning is enough clear to outline an answer. I hope it can help you, to clarify your ideas and also to edit the question in an appropriate way.

First of all, it should be specified what $X$ is, in particular which set it is.

I suppose it is $X \subset \mathbb{R^n}$. In principle, $X$ could be any metric space, that is a set on which a *distance* is defined. But, in your case I take, as it is usual, $X \subset \mathbb{R^n}$.

**Question 1.** In this case, $x$ is a vector in $\mathbb{R^n}$, $x=(x\_1, x\_2,..., x\_n)$.

**Question 2**. No, the difference (no matter what this 'difference' could be, that is, which difference you refer to) has **nothing to do** with the minus sign of the utility function **$u(a)= -d(a,x)$.**
To understand this point we have to know what is a *distance* in mathematics (your exercise speaks of *distance function)*.

I put the rigorous definition of *distance*, if you don't know it already, separately at the end of this answer. For now, think of the everyday notion of distance, e.g. the distance between two cities or streets.

The consumer has an ideal point, the most preferred bundle, $x$, so they prefers a bundle of goods as close as possible to the ideal $x$, that is at the smallest possible distance from $x$.

Why the minus sign? Because, in mathematics (as in everyday life) a distance is **always positive**, so the minus sign says that utility is **greater if the distance $d(a, x)$ is smaller**.

**Question 3** This question is not clear, what is $y$? You should correct it.
If you are asking what is the utility of a bundle $b$, you have already the answer, by the definition of the utility function: $u(b)= -d(b,x)$.

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**Appendix. Definition of *distance* in mathematics.**

The notion of *distance* in mathematics is a very important notion, together with the notion of *metric space*. The notion of distance is an abstraction and a generalization of the intuitive, everyday life, idea of the distance between two objects.

*Definition*. Given a set $Y$ of **any** nature (so not only $\mathbb{R^n}$), a *distance* or *metric* $d$ is a function from $Y \times Y$ to $\mathbb{R}$ such that, for all $x, y, z \in Y$, the following properties hold:

1. $d \geq 0$. (*positivity*)
2. $d(x,y)=0$ if and only if $x=y$.
3. $d(x,y)=d(y,x)$ *(Symmetry)*
4. $d(x,y)\leq d(x,z)+d(z,y)$ *(Triangle inequality)*.

*Definition.* A *metric space* is a set $Y$ on which a distance $d$ is defined, so a metric space is a pair $(Y,d)$.

It must be underlined that on a set $Y$ can be defined various distances, also in $\mathbb{R^n}$ there could be many different distances.

But in $\mathbb{R^n}$ there is a usual distance, the **Euclidean distance**, which, given two vectors $x=(x\_1,...,x\_n)$ and $y=(y\_1,...,y\_n)$, is defined as

$$\sqrt {(x\_1-y\_1)^2+ (x\_2-y\_2)^2+…+ (x\_n-y\_n)^2}$$.

Notice that in $\mathbb{R}$ the Euclidean distance between two points $x$ and $y$ is reduced to $d(x,y)=|x-y|$, that is it is reduced to the absolute value of the difference.


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