<p>The concepts of feasible set and utility should be thought of separately.</p>
<p>Non-satiation just means that there are no &quot;thick&quot; indifference curves - this guarantees that an agent is not indifferent between two close, but different bundles of goods.</p>

CC BY-SA 4.0
2021-03-05T03:35:13.687Z
I am not sure I understand how this fixes the math they have. Also, I do not think your interpretation of non-satiation is correct. Additionally, uniqueness is never guaranteed by non-satiation, strict concavity of utility is required for that (among other things).
Denizalp
CC BY-SA 4.0
2021-03-05T03:54:26.967Z
Yes, it is a requirement, not the only requirement, for well-behaved consumer theory - I'll edit above to make sure this is clear for future readers. What I mean to stress is that it is simply a property of the utility function, the feasible set is only relevant insofar as it determines what x or y you could have. If you look online, you will see some definitions of non-satiation involve a ball of arbitrary radius, if you think about it this way in your compact space, you can always shrink the distance arbitrarily small - hence no "thick" indifference lines.
qwerty
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2021-03-05T04:33:21.377Z
I normally think of non-satiation independently of the consumption sets as well but that is not how non-satiation is stated in the paper. This is a seminal paper and the math does not suggest that you can separate the definition of non-satiated from the consumption sets, so there must be something else going on, which I would like to figure out.
Denizalp
CC BY-SA 4.0
2021-03-05T03:20:20.280Z
2021-03-05T03:54:49.113Z
2021-03-05T03:54:49.113Z
qwerty
<p><em>Converting my comments into an answer:</em></p>
<p>At the bottom of p.268, the authors say:</p>
<blockquote>
<p><em>The set of consumption vectors <span class="math-container">$X_i$</span> available to individual <span class="math-container">$i$</span> <span class="math-container">$(=1,\cdots,m)$</span> is a closed convex subset of <span class="math-container">$R^l$</span> which is <strong>bounded from below</strong></em>.</p>
</blockquote>
<p>[Emphasis added.]</p>
<p>Since the <a href="https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem" rel="nofollow noreferrer">Heine-Borel Theorem</a> establishes that <span class="math-container">$S\subset R^n$</span> is compact if and only if <span class="math-container">$S$</span> is both closed <em>and bounded</em> (from below <em>and above</em>), one cannot conclude necessarily that <span class="math-container">$X_i$</span> is compact. (A counterexample is that <span class="math-container">$[0,\infty)$</span> is closed but not compact.)</p>

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2021-03-05T05:15:00.960Z
2021-03-05T05:15:00.960Z
Herr K.
<p>Let <span class="math-container">$n$</span> be the number of consumers and <span class="math-container">$m$</span> be the number of commodities.</p>
<p>The Arrow-Debreu theorem requires closed and convex consumption sets <span class="math-container">$X_i \subset \mathbb{R}^m$</span> for all buyers <span class="math-container">$i \in [n]$</span>. Additionally, it requires the utility function of any consumer <span class="math-container">$i$</span>, <span class="math-container">$u_i: \mathbb{R}^m \to \mathbb{R}$</span> to be continuous, quasi-concave and non-satiated over the consumption set <span class="math-container">$X_i$</span>, where non satitation is defined as <span class="math-container">$\forall \mathbf{x} \in X_i, \exists \mathbf{y} \in X_i$</span> such that <span class="math-container">$u_i(\mathbf{y}) &gt; u_i(\mathbf{x})$</span> (<a href="https://web.stanford.edu/class/msande311/arrow-debreu.pdf" rel="noreferrer">pages 268-269</a>).</p>
<p>It seems to me like these assumptions are contradictory. How can the utility function be non-satiated if the consumption set is closed, i.e., compact since it is a closed subset in <span class="math-container">$R^m$</span>. Doesn't compactness and continuity of the utility function guarantee that there exists a bundle within the consumption set that maximizes utility which implies that non-satiation cannot hold?</p>

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2021-03-05T03:19:10.160Z
Looking at the reference, the definition of nonsatiation is that $\forall x \in \hat{X}_i$ ... . This set $\hat{X}_i$ is defined (at 3.3.0) as a subset of $X_i$, which I believe resolves the contradiction.
Max
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2021-03-05T03:30:09.640Z
You're referring to the modified assumption III.b' they make later. But they state it in claim III.b the way I have it above and do actually use it.
Denizalp
CC BY-SA 4.0
2021-03-05T04:26:50.557Z
At the bottom of p.268, the authors say "*The set of consumption vectors $X_i$ available to individual $i$ $(=1,\cdots,m)$ is a closed convex subset of $R^l$ which is **bounded from below***". So $X_i$ isn't necessarily compact...
Herr K.
CC BY-SA 4.0
2021-03-05T04:29:49.923Z
I am not sure, bounded from below implies that the set is not compact, to the opposite it suggests it is. Also, any closed subset of $R^l$ is compact, so it must necessarily be compact.
Denizalp
CC BY-SA 4.0
2021-03-05T04:36:59.727Z
The [Heine-Borel Theorem](https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem) establishes that $S\subset \mathbb R^n$ is compact if and only if $S$ is both closed **and bounded** (from below *and* above). For example, $[0,\infty)$ is closed but not compact.
Herr K.
CC BY-SA 4.0
2021-03-05T04:45:58.950Z
Oh makes sense!!! For some reason, I was thinking that bounded from below implied bounded. Thank you!!
Denizalp
<mathematical-economics><utility><optimization><general-equilibrium>
Arrow-Debreu Theorem of Existence: Non satiation
Denizalp
# Arrow-Debreu Theorem of Existence: Non satiation

(Asked by: username_0 on 2021-03-05)

Let $n$ be the number of consumers and $m$ be the number of commodities.

The Arrow-Debreu theorem requires closed and convex consumption sets $X\_i \subset \mathbb{R}^m$ for all buyers $i \in [n]$. Additionally, it requires the utility function of any consumer $i$, $u\_i: \mathbb{R}^m \to \mathbb{R}$ to be continuous, quasi-concave and non-satiated over the consumption set $X\_i$, where non satitation is defined as $\forall \mathbf{x} \in X\_i, \exists \mathbf{y} \in X\_i$ such that $u\_i(\mathbf{y}) > u\_i(\mathbf{x})$ ([pages 268-269](https://web.stanford.edu/class/msande311/arrow-debreu.pdf)).

It seems to me like these assumptions are contradictory. How can the utility function be non-satiated if the consumption set is closed, i.e., compact since it is a closed subset in $R^m$. Doesn't compactness and continuity of the utility function guarantee that there exists a bundle within the consumption set that maximizes utility which implies that non-satiation cannot hold?

**Tags:** mathematical-economics, utility, optimization, general-equilibrium

> **Comments:**
> **username_1**: Looking at the reference, the definition of nonsatiation is that $\forall x \in \hat{X}_i$ ... . This set $\hat{X}_i$ is defined (at 3.3.0) as a subset of $X_i$, which I believe resolves the contradiction.  
> **username_0**: You're referring to the modified assumption III.b' they make later. But they state it in claim III.b the way I have it above and do actually use it.  
> **username_2**: At the bottom of p.268, the authors say "*The set of consumption vectors $X_i$ available to individual $i$ $(=1,\cdots,m)$ is a closed convex subset of $R^l$ which is **bounded from below***". So $X_i$ isn't necessarily compact...  
> **username_0**: I am not sure, bounded from below implies that the set is not compact, to the opposite it suggests it is. Also, any closed subset of $R^l$ is compact, so it must necessarily be compact.  
> **username_2**: The [Heine-Borel Theorem](https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem) establishes that $S\subset \mathbb R^n$ is compact if and only if $S$ is both closed **and bounded** (from below *and* above). For example, $[0,\infty)$ is closed but not compact.  
> **username_0**: Oh makes sense!!! For some reason, I was thinking that bounded from below implied bounded. Thank you!!  

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## 2 answers

### Answer by username_2 on 2021-03-05. Score: 4 | Accepted answer

*Converting my comments into an answer:*

At the bottom of p.268, the authors say:

> *The set of consumption vectors $X\_i$ available to individual $i$ $(=1,\cdots,m)$ is a closed convex subset of $R^l$ which is **bounded from below***.

[Emphasis added.]

Since the [Heine-Borel Theorem](https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem) establishes that $S\subset R^n$ is compact if and only if $S$ is both closed *and bounded* (from below *and above*), one cannot conclude necessarily that $X\_i$ is compact. (A counterexample is that $[0,\infty)$ is closed but not compact.)


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### Answer by username_3 on 2021-03-05. Score: 1

The concepts of feasible set and utility should be thought of separately.

Non-satiation just means that there are no "thick" indifference curves - this guarantees that an agent is not indifferent between two close, but different bundles of goods.

> **Comments:**
> **username_0**: I am not sure I understand how this fixes the math they have. Also, I do not think your interpretation of non-satiation is correct. Additionally, uniqueness is never guaranteed by non-satiation, strict concavity of utility is required for that (among other things).  
> **username_3**: Yes, it is a requirement, not the only requirement, for well-behaved consumer theory - I'll edit above to make sure this is clear for future readers. What I mean to stress is that it is simply a property of the utility function, the feasible set is only relevant insofar as it determines what x or y you could have. If you look online, you will see some definitions of non-satiation involve a ball of arbitrary radius, if you think about it this way in your compact space, you can always shrink the distance arbitrarily small - hence no "thick" indifference lines.  
> **username_0**: I normally think of non-satiation independently of the consumption sets as well but that is not how non-satiation is stated in the paper. This is a seminal paper and the math does not suggest that you can separate the definition of non-satiated from the consumption sets, so there must be something else going on, which I would like to figure out.  

---

