Trying to understand Walrasian equilibrium (Brown & Matzkin 1996)
Trying to understand Walrasian equilibrium (Brown & Matzkin 1996)
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Ludwig Gershwin · External communityPost link
External question — Economics Stack Exchange
Author: Ludwig Gershwin
Original post: https://economics.stackexchange.com/questions/56783
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
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I'm trying to understand page 6-7 of the
Brown & Matzkin paper
. 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.
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tdm · External communityPost link
External answer — Economics Stack Exchange
Author: tdm
Original post: https://economics.stackexchange.com/a/56785
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
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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Quoted from Forex.com.bd-Editorial External answer — Economics Stack Exchange Author: tdm Source score (net votes, not local likes): 1 Original post: https://economics.stackexchange.com/a/56785 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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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