<p>An equilibrium is usually defined as the combination of a price(s) and quantity(/quantities/allocation).</p>
<p>Here <span class="math-container">$p_y = 1$</span>, <span class="math-container">$x_A=0$</span>, <span class="math-container">$y_A = 1$</span>, <span class="math-container">$x_B=1$</span>, <span class="math-container">$y_B=0$</span> is an equilibrium, <span class="math-container">$p_y = 1$</span> is an equilibrium price, the consumptions define an equilibrium allocation.</p>
<p>This is true even though given <span class="math-container">$p_y = 1$</span> the consumers can plan individually optimal consumptions that would not constitute an equilibrium in the economy (i.e.; <span class="math-container">$x_A = x_B = 1$</span>, which is not feasible).</p>

CC BY-SA 4.0
2023-03-17T00:20:08.243Z
Thanks! I was confused as the definition I got for a Walrasian Equilibrium was a price vector (or relative prices).
Nicolas Torres
CC BY-SA 4.0
2023-03-15T16:33:39.533Z
2023-03-16T09:02:05.273Z
2023-03-16T09:02:05.273Z
Giskard
<p>let us first write the demand functions of individual <span class="math-container">$A$</span> and <span class="math-container">$B$</span></p>
<p><span class="math-container">$$(x_A^d,y_A^d)(p_x,p_y,m_A)\in\left\{\begin{matrix}
(\frac{m_A}{p_x},0) &amp; ,\frac{p_x}{p_y}&lt;1\\ 
(0,\frac{m_A}{p_y}) &amp; , \frac{p_x}{p_y}&gt;1 \\ 
BL_A &amp; , \frac{p_x}{p_y}=1
\end{matrix}\right. \\
where, \; BL_A: Budget \; Line  \; of \; A$$</span></p>
<p>similarly,
<span class="math-container">$$(x_B^d,y_B^d)(p_x,p_y,m_B)\in\left\{\begin{matrix} (\frac{m_B}{p_x},0) &amp; ,\frac{p_x}{p_y}&lt;2\\ (0,\frac{m_B}{p_y}) &amp;, \frac{p_x}{p_y}&gt;2 \\ BL_B &amp; ,\frac{p_x}{p_y}=2 \end{matrix}\right.$$</span></p>
<p>In order to simplify our analysis, let us consider <span class="math-container">$p_y=1$</span> i.e., let good <span class="math-container">$Y$</span> be the numeraire. Using the value of the endowments we also get that <span class="math-container">$m_A=m_B=\frac{p_x}{2}+\frac{1}{2}$</span></p>
<p>now both demand functions become functions of <span class="math-container">$p_x$</span> only, so we need to use these demand functions and total endowments for <span class="math-container">$X$</span> to find <span class="math-container">$p_x^*$</span> such that demand for good <span class="math-container">$X$</span> equals the total endowment of good <span class="math-container">$X$</span> (supply of good <span class="math-container">$X$</span>).</p>
<p>New demand functions are:</p>
<p><span class="math-container">$(x_A^d,y_A^d)(p_x)\in\left\{\begin{matrix}
(0.5+\frac{0.5}{p_x},0) &amp; ,p_x&lt;1\\ 
(0,0.5p_x+0.5) &amp; , p_x&gt;1 \\ 
BL_A &amp; , p_x=1
\end{matrix}\right.
\; (x_B^d,y_B^d)(p_x)\in\left\{\begin{matrix} (0.5+\frac{0.5}{p_x},0) &amp; ,p_x&lt;2\\ (0,0.5p_x+0.5) &amp;, p_x&gt;2 \\ BL_B &amp; ,p_x=2 \end{matrix}\right. $</span></p>
<p>In market for good <span class="math-container">$X$</span>:</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th>Price</th>
<th>Demand for <span class="math-container">$X$</span></th>
<th>Supply for <span class="math-container">$X$</span></th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="math-container">$p_x&lt;1$</span></td>
<td><span class="math-container">$1+\frac{1}{p_x}(&gt;2)$</span></td>
<td><span class="math-container">$1$</span></td>
</tr>
<tr>
<td><span class="math-container">$p_x=1$</span></td>
<td><span class="math-container">$[1,2] (\ni 1)$</span></td>
<td><span class="math-container">$1$</span></td>
</tr>
<tr>
<td><span class="math-container">$p_x \in (1,2)$</span></td>
<td><span class="math-container">$0.5+\frac{0.5}{p_x}(&lt;1)$</span></td>
<td><span class="math-container">$1$</span></td>
</tr>
<tr>
<td><span class="math-container">$p_x=2$</span></td>
<td><span class="math-container">$[0,0.75]   $</span></td>
<td><span class="math-container">$1$</span></td>
</tr>
<tr>
<td><span class="math-container">$p_x&gt;2$</span></td>
<td><span class="math-container">$0$</span></td>
<td><span class="math-container">$1$</span></td>
</tr>
</tbody>
</table>
</div>
<p>Therefore, <span class="math-container">$\boxed{\frac{p_x^*}{p_y^*}=1}$</span> is the equilibrium price ratio supporting the allocation <span class="math-container">$\boxed{((0,1),(1,0))}$</span></p>

CC BY-SA 4.0
2023-03-15T19:55:25.097Z
2023-03-15T20:21:45.193Z
2023-03-15T20:21:45.193Z
mynameparv
<p>I'm trying to solve the following excercise:</p>
<p>Find the Walrasian equilibria for a pure exchange economy where agents' (<span class="math-container">$A$</span> and <span class="math-container">$B$</span>) preferences and endowments are given by:</p>
<p><span class="math-container">$u_A = x_A + y_A$</span></p>
<p><span class="math-container">$u_B = 2 x_A + y_A$</span></p>
<p><span class="math-container">$(\omega_{xA},\omega_{yA}) = (\frac{1}{2},\frac{1}{2})$</span></p>
<p><span class="math-container">$(\omega_{xB},\omega_{yB}) = (\frac{1}{2},\frac{1}{2})$</span></p>
<p>I think the specific parameters for the linear functions and endowments are needed for my issue regarding non-unique demands, that’s why I included them.</p>
<p>I computed the demands for both utilities as usual for linear functions, having set <span class="math-container">$p_x = 1$</span>.</p>
<p>The demands I got (after replacing the endowments) are:</p>
<p>Agent <span class="math-container">$A$</span>:</p>
<ul>
<li><span class="math-container">$p_y &gt; 1$</span></li>
</ul>
<p><span class="math-container">$x_A^\star = \frac{1 + p_y}{2}, y_A^\star = 0$</span></p>
<ul>
<li><span class="math-container">$p_y &lt; 1$</span></li>
</ul>
<p><span class="math-container">$x_A^\star = 0, y_A^\star = \frac{1 + p_y}{2 p_y}$</span></p>
<ul>
<li><span class="math-container">$p_y = 1$</span></li>
</ul>
<p><span class="math-container">$x_A^\star \in [0,1], y_A^\star = 1 - x_A^\star$</span></p>
<p>Agent <span class="math-container">$B$</span>:</p>
<ul>
<li><span class="math-container">$p_y &gt; \frac{1}{2}$</span></li>
</ul>
<p><span class="math-container">$x_B^\star = \frac{1+p_y}{2}, y_B^\star = 0$</span></p>
<ul>
<li><span class="math-container">$p_y &lt; \frac{1}{2}$</span></li>
</ul>
<p><span class="math-container">$x_B^\star = 0, y_B^\star = \frac{1+p_y}{2 p_y}$</span></p>
<ul>
<li><span class="math-container">$p_y = \frac{1}{2}$</span></li>
</ul>
<p><span class="math-container">$x_B^\star \in [0,\frac{3}{4}], y_B^\star = \frac{3}{2} - 2 x_B^\star$</span></p>
<p>I then plot a <span class="math-container">$\mathbb{R}^+$</span> ray for <span class="math-container">$p_y$</span> with the demands by cases, taking into account the two different partitions generated by <span class="math-container">$A$</span> and <span class="math-container">$B$</span>.</p>
<p><a href="https://i.sstatic.net/ID5uR.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ID5uR.png" alt="enter image description here" /></a></p>
<p>In all cases where <span class="math-container">$p_y \neq 1$</span>, I get that there cannot be any Walrasian equilibrium.</p>
<p>However, in the case where <span class="math-container">$p_y = 1$</span>, <span class="math-container">$A$</span>'s demands are not unique and I'd get that both markets would be in equilibrium if I chose <span class="math-container">$x_A^\star = 0$</span>, but the markets wouldn't be in equilibrium if instead I chose any <span class="math-container">$x_A^\star \in (0,1]$</span>.</p>
<p>So would <span class="math-container">$p_y^\star = 1$</span> count as a Walrasian equilibrium or not?</p>
<p>I know there is a way to graphically show the equilibrium situations but I don't understand it with linear functions where <span class="math-container">$MRS$</span> arguments don't work.</p>
<p>I would appreciate it very much if someone could support their answer with an Edgeworth box graph.</p>

<microeconomics><general-equilibrium><pure-exchange-economy><edgeworth-box><perfect-substitutes>
Finding Walrasian equilibria when Walrasian demands are not unique
Nicolas Torres
# Finding Walrasian equilibria when Walrasian demands are not unique

(Asked by: username_0 on 2023-03-15)

I'm trying to solve the following excercise:

Find the Walrasian equilibria for a pure exchange economy where agents' ($A$ and $B$) preferences and endowments are given by:

$u\_A = x\_A + y\_A$

$u\_B = 2 x\_A + y\_A$

$(\omega\_{xA},\omega\_{yA}) = (\frac{1}{2},\frac{1}{2})$

$(\omega\_{xB},\omega\_{yB}) = (\frac{1}{2},\frac{1}{2})$

I think the specific parameters for the linear functions and endowments are needed for my issue regarding non-unique demands, that’s why I included them.

I computed the demands for both utilities as usual for linear functions, having set $p\_x = 1$.

The demands I got (after replacing the endowments) are:

Agent $A$:

* $p\_y > 1$

$x\_A^\star = \frac{1 + p\_y}{2}, y\_A^\star = 0$

* $p\_y < 1$

$x\_A^\star = 0, y\_A^\star = \frac{1 + p\_y}{2 p\_y}$

* $p\_y = 1$

$x\_A^\star \in [0,1], y\_A^\star = 1 - x\_A^\star$

Agent $B$:

* $p\_y > \frac{1}{2}$

$x\_B^\star = \frac{1+p\_y}{2}, y\_B^\star = 0$

* $p\_y < \frac{1}{2}$

$x\_B^\star = 0, y\_B^\star = \frac{1+p\_y}{2 p\_y}$

* $p\_y = \frac{1}{2}$

$x\_B^\star \in [0,\frac{3}{4}], y\_B^\star = \frac{3}{2} - 2 x\_B^\star$

I then plot a $\mathbb{R}^+$ ray for $p\_y$ with the demands by cases, taking into account the two different partitions generated by $A$ and $B$.

[![enter image description here](https://i.sstatic.net/ID5uR.png)](https://i.sstatic.net/ID5uR.png)

In all cases where $p\_y \neq 1$, I get that there cannot be any Walrasian equilibrium.

However, in the case where $p\_y = 1$, $A$'s demands are not unique and I'd get that both markets would be in equilibrium if I chose $x\_A^\star = 0$, but the markets wouldn't be in equilibrium if instead I chose any $x\_A^\star \in (0,1]$.

So would $p\_y^\star = 1$ count as a Walrasian equilibrium or not?

I know there is a way to graphically show the equilibrium situations but I don't understand it with linear functions where $MRS$ arguments don't work.

I would appreciate it very much if someone could support their answer with an Edgeworth box graph.

**Tags:** microeconomics, general-equilibrium, pure-exchange-economy, edgeworth-box, perfect-substitutes

---

## 2 answers

### Answer by username_1 on 2023-03-15. Score: 0 | Accepted answer

An equilibrium is usually defined as the combination of a price(s) and quantity(/quantities/allocation).

Here $p\_y = 1$, $x\_A=0$, $y\_A = 1$, $x\_B=1$, $y\_B=0$ is an equilibrium, $p\_y = 1$ is an equilibrium price, the consumptions define an equilibrium allocation.

This is true even though given $p\_y = 1$ the consumers can plan individually optimal consumptions that would not constitute an equilibrium in the economy (i.e.; $x\_A = x\_B = 1$, which is not feasible).

> **Comments:**
> **username_0**: Thanks! I was confused as the definition I got for a Walrasian Equilibrium was a price vector (or relative prices).  

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### Answer by username_2 on 2023-03-15. Score: 2

let us first write the demand functions of individual $A$ and $B$

$$(x\_A^d,y\_A^d)(p\_x,p\_y,m\_A)\in\left\{\begin{matrix}
(\frac{m\_A}{p\_x},0) & ,\frac{p\_x}{p\_y}<1\\
(0,\frac{m\_A}{p\_y}) & , \frac{p\_x}{p\_y}>1 \\
BL\_A & , \frac{p\_x}{p\_y}=1
\end{matrix}\right. \\
where, \; BL\_A: Budget \; Line \; of \; A$$

similarly,
$$(x\_B^d,y\_B^d)(p\_x,p\_y,m\_B)\in\left\{\begin{matrix} (\frac{m\_B}{p\_x},0) & ,\frac{p\_x}{p\_y}<2\\ (0,\frac{m\_B}{p\_y}) &, \frac{p\_x}{p\_y}>2 \\ BL\_B & ,\frac{p\_x}{p\_y}=2 \end{matrix}\right.$$

In order to simplify our analysis, let us consider $p\_y=1$ i.e., let good $Y$ be the numeraire. Using the value of the endowments we also get that $m\_A=m\_B=\frac{p\_x}{2}+\frac{1}{2}$

now both demand functions become functions of $p\_x$ only, so we need to use these demand functions and total endowments for $X$ to find $p\_x^\*$ such that demand for good $X$ equals the total endowment of good $X$ (supply of good $X$).

New demand functions are:

$(x\_A^d,y\_A^d)(p\_x)\in\left\{\begin{matrix}
(0.5+\frac{0.5}{p\_x},0) & ,p\_x<1\\
(0,0.5p\_x+0.5) & , p\_x>1 \\
BL\_A & , p\_x=1
\end{matrix}\right.
\; (x\_B^d,y\_B^d)(p\_x)\in\left\{\begin{matrix} (0.5+\frac{0.5}{p\_x},0) & ,p\_x<2\\ (0,0.5p\_x+0.5) &, p\_x>2 \\ BL\_B & ,p\_x=2 \end{matrix}\right. $

In market for good $X$:

| Price | Demand for $X$ | Supply for $X$ |
| --- | --- | --- |
| $p\_x<1$ | $1+\frac{1}{p\_x}(>2)$ | $1$ |
| $p\_x=1$ | $[1,2] (\ni 1)$ | $1$ |
| $p\_x \in (1,2)$ | $0.5+\frac{0.5}{p\_x}(<1)$ | $1$ |
| $p\_x=2$ | $[0,0.75] $ | $1$ |
| $p\_x>2$ | $0$ | $1$ |

Therefore, $\boxed{\frac{p\_x^\*}{p\_y^\*}=1}$ is the equilibrium price ratio supporting the allocation $\boxed{((0,1),(1,0))}$


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