<blockquote>
<p>Assuming <span class="math-container">$u(x)=u(y)$</span>.</p>
<p>I am to prove or disprove that <span class="math-container">$x \succcurlyeq y$</span>.</p>
</blockquote>
<p><em>Proof:</em></p>
<p>The utility function <span class="math-container">$u$</span> represents your preferences <span class="math-container">$\succcurlyeq$</span> over consumption bundles. By definition of the term &quot;represents&quot;, this means that <span class="math-container">$u(x)\ge u(y)\Leftrightarrow x\succcurlyeq y$</span> for all bundles <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
<p>If we have <span class="math-container">$u(x)=u(y)$</span>, then we have <span class="math-container">$u(x)\ge u(y)$</span>. Thus we get <span class="math-container">$x\succcurlyeq y$</span>. QED</p>
<p><em>Remark</em>: From <span class="math-container">$u(x)=u(y)$</span>, we also get <span class="math-container">$u(y)\ge u(x)$</span> and therefore <span class="math-container">$y\succcurlyeq x$</span>. Together with <span class="math-container">$x\succcurlyeq y$</span> then, by definition of the relation <span class="math-container">$\sim$</span>, we have <span class="math-container">$x\sim y$</span>. However, this is not really needed if you are only asked to prove that <span class="math-container">$x\succcurlyeq y$</span>.</p>

CC BY-SA 4.0
2021-06-07T14:19:14.480Z
Cheers. I'm still unclear about this: Why do we have $x \succcurlyeq y$ if $u(x) = u(y)$ with no further information on $x$ nor $y$? Why do we not know right away that $x \sim y$ or $y \sim x$? Like why is it you're mentioning $x \succcurlyeq y$ first, not $y \succcurlyeq x$
j3141592653589793238
CC BY-SA 4.0
2021-06-08T15:33:29.677Z
@j3141592653589793238, that's because of the way these symbols are defined. We know everything "right away", but for a proof you have to use the exact definition, and the indifference relation is defined via the at-least-as-good-as relation, so that's the sequence of steps you have to follow.
VARulle
CC BY-SA 4.0
2021-06-07T14:15:23.557Z
2021-06-07T14:15:23.557Z
VARulle
<p>Given a utility function <span class="math-container">$u(\cdot)$</span> and two bundles <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
<p>Assuming <span class="math-container">$u(x)=u(y)$</span>.</p>
<p>I am to prove or disprove that <span class="math-container">$x \succcurlyeq y$</span>.</p>
<p>Now I'm confused by this.
We say <span class="math-container">$x$</span> is <em>strictly</em> preferred to <span class="math-container">$y$</span>, or <span class="math-container">$x \succ y$</span>, if <span class="math-container">$x \succcurlyeq y$</span> <em>but not</em> <span class="math-container">$y \succcurlyeq x$</span>.
We say the agent is indifferent between <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, or <span class="math-container">$x \sim y$</span>, if <span class="math-container">$x \succcurlyeq y$</span> and <span class="math-container">$y \succcurlyeq x$</span>.
Thus if <span class="math-container">$x \sim y$</span>, we should be able to say <span class="math-container">$y \sim x$</span>.</p>
<p>So we know that if <span class="math-container">$x \sim y$</span>, <span class="math-container">$y \sim x $</span> and <span class="math-container">$x \succcurlyeq y$</span> and <span class="math-container">$y \succcurlyeq x$</span>.</p>
<p>Now what does <span class="math-container">$\succcurlyeq$</span> say at all? It's either <span class="math-container">$\succ$</span> or <span class="math-container">$\sim$</span>.</p>
<p>So can we say <span class="math-container">$\succcurlyeq$</span> is just a superset of <span class="math-container">$\succ$</span> and <span class="math-container">$\sim$</span> when we're unsure?
Because if <span class="math-container">$x \succcurlyeq y$</span>, <span class="math-container">$x$</span> can either be strictly preferred or indifferent to <span class="math-container">$y$</span>. But if <span class="math-container">$x \sim y$</span> (which could be, because <span class="math-container">$\succcurlyeq$</span> means either indifferent or strict preference), <span class="math-container">$x$</span> can not be strictly preferred to y anymore. This is confusing to me. Aren't these two statements contradictory? It can be strictly preferred first, but under the second assumption, it can't?</p>
<p>And for the example above, this would mean because <span class="math-container">$u(x)=u(y)$</span>, we know <span class="math-container">$x \sim y$</span> and <span class="math-container">$y \sim x$</span> and thus <span class="math-container">$x \succcurlyeq y$</span> (and <span class="math-container">$y \succcurlyeq x$</span>) are correct?</p>
<p>I know this is a very basic question. I just got confused because I think I have a slight disagreement with a fellow student who says that for the example above, we cannot say that <span class="math-container">$x \sim y$</span> (which, obviously, is false).</p>

CC BY-SA 4.0
2021-06-07T12:46:24.973Z
You have $x \sim y$ when $x \succcurlyeq y$ and $y \succcurlyeq x$.
Bayesian
CC BY-SA 4.0
2021-06-07T12:48:15.627Z
@Bayesian Right. So if $u(x) = u(y)$, this means $x \sim y$ and therefore $x \succcurlyeq y$ as well as $y \succcurlyeq x$, correct? Thanks!
j3141592653589793238
<microeconomics><utility><preferences>
How are weak preferences different to strict preferences/indifference?
j3141592653589793238
# How are weak preferences different to strict preferences/indifference?

(Asked by: username_0 on 2021-06-07)

Given a utility function $u(\cdot)$ and two bundles $x$ and $y$.

Assuming $u(x)=u(y)$.

I am to prove or disprove that $x \succcurlyeq y$.

Now I'm confused by this.
We say $x$ is *strictly* preferred to $y$, or $x \succ y$, if $x \succcurlyeq y$ *but not* $y \succcurlyeq x$.
We say the agent is indifferent between $x$ and $y$, or $x \sim y$, if $x \succcurlyeq y$ and $y \succcurlyeq x$.
Thus if $x \sim y$, we should be able to say $y \sim x$.

So we know that if $x \sim y$, $y \sim x $ and $x \succcurlyeq y$ and $y \succcurlyeq x$.

Now what does $\succcurlyeq$ say at all? It's either $\succ$ or $\sim$.

So can we say $\succcurlyeq$ is just a superset of $\succ$ and $\sim$ when we're unsure?
Because if $x \succcurlyeq y$, $x$ can either be strictly preferred or indifferent to $y$. But if $x \sim y$ (which could be, because $\succcurlyeq$ means either indifferent or strict preference), $x$ can not be strictly preferred to y anymore. This is confusing to me. Aren't these two statements contradictory? It can be strictly preferred first, but under the second assumption, it can't?

And for the example above, this would mean because $u(x)=u(y)$, we know $x \sim y$ and $y \sim x$ and thus $x \succcurlyeq y$ (and $y \succcurlyeq x$) are correct?

I know this is a very basic question. I just got confused because I think I have a slight disagreement with a fellow student who says that for the example above, we cannot say that $x \sim y$ (which, obviously, is false).

**Tags:** microeconomics, utility, preferences

> **Comments:**
> **username_1**: You have $x \sim y$ when $x \succcurlyeq y$ and $y \succcurlyeq x$.  
> **username_0**: @username_1 Right. So if $u(x) = u(y)$, this means $x \sim y$ and therefore $x \succcurlyeq y$ as well as $y \succcurlyeq x$, correct? Thanks!  

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## 1 answers

### Answer by username_2 on 2021-06-07. Score: 4 | Accepted answer

> Assuming $u(x)=u(y)$.
>
> I am to prove or disprove that $x \succcurlyeq y$.

*Proof:*

The utility function $u$ represents your preferences $\succcurlyeq$ over consumption bundles. By definition of the term "represents", this means that $u(x)\ge u(y)\Leftrightarrow x\succcurlyeq y$ for all bundles $x$ and $y$.

If we have $u(x)=u(y)$, then we have $u(x)\ge u(y)$. Thus we get $x\succcurlyeq y$. QED

*Remark*: From $u(x)=u(y)$, we also get $u(y)\ge u(x)$ and therefore $y\succcurlyeq x$. Together with $x\succcurlyeq y$ then, by definition of the relation $\sim$, we have $x\sim y$. However, this is not really needed if you are only asked to prove that $x\succcurlyeq y$.

> **Comments:**
> **username_0**: Cheers. I'm still unclear about this: Why do we have $x \succcurlyeq y$ if $u(x) = u(y)$ with no further information on $x$ nor $y$? Why do we not know right away that $x \sim y$ or $y \sim x$? Like why is it you're mentioning $x \succcurlyeq y$ first, not $y \succcurlyeq x$  
> **username_2**: @username_0, that's because of the way these symbols are defined. We know everything "right away", but for a proof you have to use the exact definition, and the indifference relation is defined via the at-least-as-good-as relation, so that's the sequence of steps you have to follow.  

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