<blockquote>
<p>I was wondering which assumption of standard existence theorems this violates.</p>
</blockquote>
<p>None. An equilibrium exists (defined by the starting allocation), it is merely not in the interior of the set. The existence theorems I know make no such guarantees.</p>
<blockquote>
<p>I guess this is the assumption that preferences are convex?</p>
</blockquote>
<p>The preferences in the example are convex.</p>
<hr />
<p>Suggestion: <br>
Heavily edit this question or post a new one based on your third question:</p>
<blockquote>
<p>I was also wondering whether anyone can cast some light on why this type of counterexample must fail (if it indeed must) if endowments are sufficiently high to allow for an interior equilibrium. Does this illustrate anything general and interesting?</p>
</blockquote>
<p>I recommend rephrasing this as well, as the final line seems clearly opinion-based. Suggested text: <br>
&quot;In general equilibrium are there any existence theorems that guarantee the existence of an interior equilibrium? By interior I mean [explain if you mean only the good space or if you also wish to restrict the price vector.]&quot;</p>

CC BY-SA 4.0
2021-10-20T11:46:58.820Z
Can you explain why an equilibrium exists? In the question, I provide what seems to be a valid proof that there is no equilibrium price. (Though I also understand your point that, in some sense, we would expect individuals to just consume their endowment -- for example, there are no mutually beneficial trades from this starting point.)
afreelunch
CC BY-SA 4.0
2021-10-20T11:51:24.917Z
@afreelunch I did not read your detailed calculations because I trust the theorems (: But it seems like $p=1/3$ would do the trick; everyone's inital endowment is an optimal choice given this price of good $y$. So I guess the error is in your final paragraph.
Giskard
CC BY-SA 4.0
2021-10-20T11:53:43.203Z
Seems to like I have made a very stupid error! Indeed $p = 1/3$ does work, alas I failed to correctly solve the equation $3 + 3p = 4$....
afreelunch
CC BY-SA 4.0
2021-10-20T11:54:25.550Z
I guess I should delete the question? (Thanks anyway for clearing this up for me!)
afreelunch
CC BY-SA 4.0
2021-10-20T11:54:50.680Z
@afreelunch Or you could accept the answer (:
Giskard
CC BY-SA 4.0
2021-10-20T11:55:12.997Z
@afreelunch Third question (conditions for interior equilibria) remains valid, you could repost that.
Giskard
CC BY-SA 4.0
2021-10-20T11:45:32.900Z
2021-10-20T11:53:41.763Z
2021-10-20T11:53:41.763Z
Giskard
<p>I recently came across an interesting example of why general equilibrium need not exist with quasi-linear utility (in case anyone is interested, I'm posting this at the end). To make the example work, you need to assume that one person's endowment is sufficiently low to rule out an equilibrium in which all agents choose an interior bundle.</p>
<p>I was wondering which assumption of standard existence theorems this violates. I guess this is the assumption that preferences are convex?</p>
<p>I was also wondering whether anyone can cast some light on why this type of counterexample must fail (if it indeed must) if endowments are sufficiently high to allow for an interior equilibrium?</p>
<hr />
<p><em>The example</em>. Suppose that there are two individuals <span class="math-container">$A, B$</span>, each with the same utility function <span class="math-container">$u = x + \ln(y)$</span>. Endowments are <span class="math-container">$w^a = (0, 1)$</span>, <span class="math-container">$w^b = (4, 3)$</span> where the first component denotes possession of good <span class="math-container">$x$</span>. Normalise <span class="math-container">$p_x \equiv 1$</span> and define <span class="math-container">$p \equiv p_y$</span>. Solving <span class="math-container">$A$</span>'s optimisation problem reveals that they demand <span class="math-container">$x^a = (p-1, 1/p)$</span> if <span class="math-container">$p \geq 1$</span>; but otherwise demand their endowment so <span class="math-container">$x^a = (0, 1)$</span>. Meanwhile, solving <span class="math-container">$B's$</span> problem reveals that <span class="math-container">$x^b = (3 + 3p, 1/p)$</span>.</p>
<p>To begin, let's check if there is an equilibrium with <span class="math-container">$p \geq 1$</span>. Since there are <span class="math-container">$4$</span> units of <span class="math-container">$y$</span> in total, market clearing would then require that
<span class="math-container">$$ \frac{1}{y} + \frac{1}{y} = 4 \iff p = \frac{1}{2}$$</span>
which contradicts <span class="math-container">$p \geq 1$</span>. So there is no such equilibrium.</p>
<p>Next, let's look for any equilibrium with <span class="math-container">$p &lt; 1$</span>. Since there are <span class="math-container">$4$</span> units of <span class="math-container">$x$</span>, market clearing would require
<span class="math-container">$$ 0 + 3 + 3p = 4 \iff p = 4/3$$</span>
which contradicts <span class="math-container">$p &lt; 1$</span>. So there is no equilibrium with <span class="math-container">$p &lt; 1$</span> either.</p>
<p><strong>UPDATE:</strong> So it turns out I just made a very basic error, and the equilibrium is <span class="math-container">$p = 1/3$</span> (the solution to the equation <span class="math-container">$0 + 3 + 3p = 4$</span>).</p>

<general-equilibrium>
Non-existence of general equilibrium with quasi-linear utility
afreelunch
# Non-existence of general equilibrium with quasi-linear utility

(Asked by: username_0 on 2021-10-20)

I recently came across an interesting example of why general equilibrium need not exist with quasi-linear utility (in case anyone is interested, I'm posting this at the end). To make the example work, you need to assume that one person's endowment is sufficiently low to rule out an equilibrium in which all agents choose an interior bundle.

I was wondering which assumption of standard existence theorems this violates. I guess this is the assumption that preferences are convex?

I was also wondering whether anyone can cast some light on why this type of counterexample must fail (if it indeed must) if endowments are sufficiently high to allow for an interior equilibrium?

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*The example*. Suppose that there are two individuals $A, B$, each with the same utility function $u = x + \ln(y)$. Endowments are $w^a = (0, 1)$, $w^b = (4, 3)$ where the first component denotes possession of good $x$. Normalise $p\_x \equiv 1$ and define $p \equiv p\_y$. Solving $A$'s optimisation problem reveals that they demand $x^a = (p-1, 1/p)$ if $p \geq 1$; but otherwise demand their endowment so $x^a = (0, 1)$. Meanwhile, solving $B's$ problem reveals that $x^b = (3 + 3p, 1/p)$.

To begin, let's check if there is an equilibrium with $p \geq 1$. Since there are $4$ units of $y$ in total, market clearing would then require that
$$ \frac{1}{y} + \frac{1}{y} = 4 \iff p = \frac{1}{2}$$
which contradicts $p \geq 1$. So there is no such equilibrium.

Next, let's look for any equilibrium with $p < 1$. Since there are $4$ units of $x$, market clearing would require
$$ 0 + 3 + 3p = 4 \iff p = 4/3$$
which contradicts $p < 1$. So there is no equilibrium with $p < 1$ either.

**UPDATE:** So it turns out I just made a very basic error, and the equilibrium is $p = 1/3$ (the solution to the equation $0 + 3 + 3p = 4$).

**Tags:** general-equilibrium

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## 1 answers

### Answer by username_1 on 2021-10-20. Score: 3 | Accepted answer

> I was wondering which assumption of standard existence theorems this violates.

None. An equilibrium exists (defined by the starting allocation), it is merely not in the interior of the set. The existence theorems I know make no such guarantees.

> I guess this is the assumption that preferences are convex?

The preferences in the example are convex.

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Suggestion:   
Heavily edit this question or post a new one based on your third question:

> I was also wondering whether anyone can cast some light on why this type of counterexample must fail (if it indeed must) if endowments are sufficiently high to allow for an interior equilibrium. Does this illustrate anything general and interesting?

I recommend rephrasing this as well, as the final line seems clearly opinion-based. Suggested text:   
"In general equilibrium are there any existence theorems that guarantee the existence of an interior equilibrium? By interior I mean [explain if you mean only the good space or if you also wish to restrict the price vector.]"

> **Comments:**
> **username_0**: Can you explain why an equilibrium exists? In the question, I provide what seems to be a valid proof that there is no equilibrium price. (Though I also understand your point that, in some sense, we would expect individuals to just consume their endowment -- for example, there are no mutually beneficial trades from this starting point.)  
> **username_1**: @afreelunch I did not read your detailed calculations because I trust the theorems (: But it seems like $p=1/3$ would do the trick; everyone's inital endowment is an optimal choice given this price of good $y$. So I guess the error is in your final paragraph.  
> **username_0**: Seems to like I have made a very stupid error! Indeed $p = 1/3$ does work, alas I failed to correctly solve the equation $3 + 3p = 4$....  
> **username_0**: I guess I should delete the question? (Thanks anyway for clearing this up for me!)  
> **username_1**: @afreelunch Or you could accept the answer (:  
> **username_1**: @afreelunch Third question (conditions for interior equilibria) remains valid, you could repost that.  

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