<p>Consider consumer 1. Let <span class="math-container">$(x,y)$</span> be his optimal consumption bundle in the first equilibrium and let <span class="math-container">$(\bar x, \bar y)$</span> be his optimal consumption bundle in the second equilibrium.</p>
<p>Then the budget restriction in the first equilibrium gives:
<span class="math-container">$$
30 = 5 x + 6 y \leftrightarrow y = 5 - \frac{5}{6} x
$$</span>
The restrictions on the total endowment gives:
<span class="math-container">$$
x \le 2.
$$</span>
From this, it follows that:
<span class="math-container">$$
\begin{align*}
6 x + 5 y &amp;= 6 x + 5 \left(5 - \frac{5}{6} x\right),\\
&amp;=6 x + 25 - \frac{25}{6} x,\\
&amp;=\frac{11}{6} x + 25 \le \frac{22}{6} + 25,\\
&amp;&lt; 30.
\end{align*}
$$</span>
Now, the total income for individual 1 in the second equilibrium is <span class="math-container">$30 (= 6 \times 5)$</span>. With this money, he could have bought the bundle that he consumed in the first period, because:
<span class="math-container">$$
6 x + 5 y &lt; 30.
$$</span>
By a simple revealed preference argument, this means that consumer 1 prefers the chosen bundle <span class="math-container">$(\bar x, \bar y)$</span> over the bundle <span class="math-container">$(x,y)$</span>. (He consumed the bundle <span class="math-container">$(\bar x, \bar y)$</span> but the bundle <span class="math-container">$(x,y)$</span> was cheaper to obtain.)</p>
<p>Now, we we do a similar exercise for the second equilibrium. The budget constraint in the second equilibrium gives:
<span class="math-container">$$
30 = 6 \bar x + 5 \bar y \leftrightarrow \bar x = 5 - \frac{5}{6} \bar y.
$$</span>
Also, the endowment constraint gives:
<span class="math-container">$$
\bar y \le 2.
$$</span>
As such,
<span class="math-container">$$
\begin{align*}
5 \bar x + 6 \bar y &amp;= 5\left(5 - \frac{5}{6} \bar y\right) + 6 \bar y,\\
&amp;= 25 - \frac{25}{6} \bar y + 6 \bar y,\\
&amp;= 25 + \frac{11}{6} \bar y \le 25 + \frac{22}{6},\\
&amp;&lt; 30
\end{align*}
$$</span>
The income for individual 1 in the first equilibrium is <span class="math-container">$30 = (6 \times 5)$</span>. With this money, he could have bought the bundle in the second equilibrium because:
<span class="math-container">$$
5 \bar x + 6 \bar y &lt; 30,
$$</span>
which means that the bundle <span class="math-container">$(x,y)$</span> which he actually purchased is preferred over the bundle <span class="math-container">$(\bar x, \bar y)$</span>.</p>
<p>To conclude, our consumer prefers <span class="math-container">$(x,y)$</span> over <span class="math-container">$(\bar x, \bar y)$</span> and he prefers <span class="math-container">$(\bar x, \bar y)$</span> over <span class="math-container">$(x,y)$</span>, which is a contradiction.</p>

CC BY-SA 4.0
2023-10-26T06:04:36.647Z
2023-10-26T06:04:36.647Z
tdm
<p>I'm trying to understand page 6-7 of the <a href="http://www.econ.ucla.edu/rmatzkin/research/Brown_Matzkin.pdf" rel="nofollow noreferrer">Brown &amp; Matzkin paper</a>. Here is an example:</p>
<p>Consider a 2-person, 2-good pure exchange economy <span class="math-container">$E=(u_i,w_i)_{i=1}^2$</span> where both utility functions are continuous, strictly quasi-concave and strictly monotone. The endowments are <span class="math-container">$w_1=(0,5)$</span> and <span class="math-container">$w_2=(2,1)$</span>. Let <span class="math-container">$\bar{E}=(u_i,\bar{w}_i)_{i=1}^2$</span> be another exchange economy, with endowments <span class="math-container">$\bar{w}_1=(5,0)$</span> and <span class="math-container">$\bar{w}_2=(1,2)$</span>.</p>
<p>Now I want to show that if <span class="math-container">$p=(5,6)$</span> is a Walrasian equilibrium price vector for economy <span class="math-container">$E$</span>, then <span class="math-container">$\bar{p}=(6,5)$</span> cannot be a Walrasian equilibrium price vector for <span class="math-container">$\bar{E}$</span>. The paper mentions using Afriat's theorem, but I'm having a difficult time finding the correct path to show this.</p>

<pure-exchange-economy>
Trying to understand Walrasian equilibrium (Brown & Matzkin 1996)
Ludwig Gershwin
# Trying to understand Walrasian equilibrium (Brown & Matzkin 1996)

(Asked by: username_0 on 2023-10-26)

I'm trying to understand page 6-7 of the [Brown & Matzkin paper](http://www.econ.ucla.edu/rmatzkin/research/Brown_Matzkin.pdf). Here is an example:

Consider a 2-person, 2-good pure exchange economy $E=(u\_i,w\_i)\_{i=1}^2$ where both utility functions are continuous, strictly quasi-concave and strictly monotone. The endowments are $w\_1=(0,5)$ and $w\_2=(2,1)$. Let $\bar{E}=(u\_i,\bar{w}\_i)\_{i=1}^2$ be another exchange economy, with endowments $\bar{w}\_1=(5,0)$ and $\bar{w}\_2=(1,2)$.

Now I want to show that if $p=(5,6)$ is a Walrasian equilibrium price vector for economy $E$, then $\bar{p}=(6,5)$ cannot be a Walrasian equilibrium price vector for $\bar{E}$. The paper mentions using Afriat's theorem, but I'm having a difficult time finding the correct path to show this.

**Tags:** pure-exchange-economy

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## 1 answers

### Answer by username_1 on 2023-10-26. Score: 1

Consider consumer 1. Let $(x,y)$ be his optimal consumption bundle in the first equilibrium and let $(\bar x, \bar y)$ be his optimal consumption bundle in the second equilibrium.

Then the budget restriction in the first equilibrium gives:
$$
30 = 5 x + 6 y \leftrightarrow y = 5 - \frac{5}{6} x
$$
The restrictions on the total endowment gives:
$$
x \le 2.
$$
From this, it follows that:
$$
\begin{align\*}
6 x + 5 y &= 6 x + 5 \left(5 - \frac{5}{6} x\right),\\
&=6 x + 25 - \frac{25}{6} x,\\
&=\frac{11}{6} x + 25 \le \frac{22}{6} + 25,\\
&< 30.
\end{align\*}
$$
Now, the total income for individual 1 in the second equilibrium is $30 (= 6 \times 5)$. With this money, he could have bought the bundle that he consumed in the first period, because:
$$
6 x + 5 y < 30.
$$
By a simple revealed preference argument, this means that consumer 1 prefers the chosen bundle $(\bar x, \bar y)$ over the bundle $(x,y)$. (He consumed the bundle $(\bar x, \bar y)$ but the bundle $(x,y)$ was cheaper to obtain.)

Now, we we do a similar exercise for the second equilibrium. The budget constraint in the second equilibrium gives:
$$
30 = 6 \bar x + 5 \bar y \leftrightarrow \bar x = 5 - \frac{5}{6} \bar y.
$$
Also, the endowment constraint gives:
$$
\bar y \le 2.
$$
As such,
$$
\begin{align\*}
5 \bar x + 6 \bar y &= 5\left(5 - \frac{5}{6} \bar y\right) + 6 \bar y,\\
&= 25 - \frac{25}{6} \bar y + 6 \bar y,\\
&= 25 + \frac{11}{6} \bar y \le 25 + \frac{22}{6},\\
&< 30
\end{align\*}
$$
The income for individual 1 in the first equilibrium is $30 = (6 \times 5)$. With this money, he could have bought the bundle in the second equilibrium because:
$$
5 \bar x + 6 \bar y < 30,
$$
which means that the bundle $(x,y)$ which he actually purchased is preferred over the bundle $(\bar x, \bar y)$.

To conclude, our consumer prefers $(x,y)$ over $(\bar x, \bar y)$ and he prefers $(\bar x, \bar y)$ over $(x,y)$, which is a contradiction.


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