<p>Let <span class="math-container">$m_i$</span> be the market value of the endowment of player <span class="math-container">$i\in\{1,2\}$</span>.</p>
<p>In optimum player <span class="math-container">$i$</span> spends everything such that <span class="math-container">$p_x x_i + p_y y_i =m_i$</span>.
And because of the preferences in optimum also <span class="math-container">$x_1 = 4 y_1$</span> and <span class="math-container">$x_2=y_2$</span> holds.</p>
<p>You use that to get 2's demand as <span class="math-container">$x^*_2 = \frac{m_2}{p_1+p_2}=y^*_2$</span> and a similar bur slightly different formula for <span class="math-container">$4 y^*_1 = x^*_1$</span>.</p>
<p>The excess demand would be this demand minus the endowment. The sum of excess demands is zero in equilibrium.</p>
<p>In equilibrium, <span class="math-container">$x_1^*+x_2^* =2+4$</span> and <span class="math-container">$y_1^*+y_2^* =2+4$</span>, which you can both use to calculate an equilibrium price ratio. Any price combination with that ratio clears the market.</p>

CC BY-SA 4.0
2020-11-20T15:11:48.587Z
and the rest is homework ;)
Bayesian
CC BY-SA 4.0
2024-08-22T06:39:17.743Z
In this particular economy, at a competitive equilibrium allocation both $x_1=4y_1$ and $x_2=y_2$ cannot happen together. In fact, equilibrium occurs at any prices satisfying $p_Y=0$ and $p_X>0$, and there is a range of allocations that are competitive equilibria. See https://economics.stackexchange.com/a/59033/11824
Amit
CC BY-SA 4.0
2020-11-20T15:10:22.153Z
2020-11-20T15:10:22.153Z
Bayesian
<p>Given a pure-exchange economy with:</p>
<ul>
<li>Utility functions: <span class="math-container">$u_1(x_1,y_1)=\min(x_1,4y_1)$</span>, <span class="math-container">$u_2(x_2,y_2)=\min(x_2,y_2)$</span></li>
<li>Endowment Allocation: <span class="math-container">$\omega = (\omega_1,\omega_2) =((2,4),(4,2))$</span></li>
</ul>
<p>Set of feasible allocations are <span class="math-container">$\mathcal{F}=\{((x_1,y_1),(x_2,y_2))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_1+x_2=y_1+y_2=6\}$</span></p>
<p>As can be observed from the Edgeworth Box, competitive equilibrium prices are <span class="math-container">$(p_X^*,p_Y^*)=(1,0)$</span> and the supported allocations are all feasible allocations <span class="math-container">$((x_1,y_1),(x_2,y_2))$</span> that satisfy <span class="math-container">$x_1=2$</span> and <span class="math-container">$\frac{1}{2}\leq y_1 \leq 2$</span>.</p>
<p><a href="https://i.sstatic.net/BHCZsCkz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BHCZsCkz.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-08-22T06:22:21.527Z
2024-08-22T06:27:38.820Z
2024-08-22T06:27:38.820Z
Amit
<p>Consider an economy with two agents. There are two goods, x and y. Agents' preferences are Leontif ones as follows:
<span class="math-container">$u_1(x,y)=\min(x,4y)$</span> and <span class="math-container">$u_2(x,y)=\min(x,y)$</span>.
Initial endowment for 1 is (2,4), for 2 is (4,2)
The problem asks me to use the excess demand function to find the equilibrium price and show that there are infinitely many equilibria with that price. The problem also asks what is the cause of such multiplicity of equilibria.
I tried this many times, but I always get that the equilibirum price is (1,-1/2), which is impossible.
Any help is appreciated.</p>

CC BY-SA 4.0
2020-11-20T12:47:06.210Z
It looks like you are missing the endowment. Fix some equilibrium prices. What happens when you double both prices?
Bayesian
<microeconomics><mathematical-economics><supply-and-demand><walrasian>
Find equilibrium price using excess demand function
biden
# Find equilibrium price using excess demand function

(Asked by: username_0 on 2020-11-20)

Consider an economy with two agents. There are two goods, x and y. Agents' preferences are Leontif ones as follows:
$u\_1(x,y)=\min(x,4y)$ and $u\_2(x,y)=\min(x,y)$.
Initial endowment for 1 is (2,4), for 2 is (4,2)
The problem asks me to use the excess demand function to find the equilibrium price and show that there are infinitely many equilibria with that price. The problem also asks what is the cause of such multiplicity of equilibria.
I tried this many times, but I always get that the equilibirum price is (1,-1/2), which is impossible.
Any help is appreciated.

**Tags:** microeconomics, mathematical-economics, supply-and-demand, walrasian

> **Comments:**
> **username_1**: It looks like you are missing the endowment. Fix some equilibrium prices. What happens when you double both prices?  

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## 2 answers

### Answer by username_1 on 2020-11-20. Score: 2

Let $m\_i$ be the market value of the endowment of player $i\in\{1,2\}$.

In optimum player $i$ spends everything such that $p\_x x\_i + p\_y y\_i =m\_i$.
And because of the preferences in optimum also $x\_1 = 4 y\_1$ and $x\_2=y\_2$ holds.

You use that to get 2's demand as $x^\*\_2 = \frac{m\_2}{p\_1+p\_2}=y^\*\_2$ and a similar bur slightly different formula for $4 y^\*\_1 = x^\*\_1$.

The excess demand would be this demand minus the endowment. The sum of excess demands is zero in equilibrium.

In equilibrium, $x\_1^\*+x\_2^\* =2+4$ and $y\_1^\*+y\_2^\* =2+4$, which you can both use to calculate an equilibrium price ratio. Any price combination with that ratio clears the market.

> **Comments:**
> **username_1**: and the rest is homework ;)  
> **username_2**: In this particular economy, at a competitive equilibrium allocation both $x_1=4y_1$ and $x_2=y_2$ cannot happen together. In fact, equilibrium occurs at any prices satisfying $p_Y=0$ and $p_X>0$, and there is a range of allocations that are competitive equilibria. See https://economics.stackexchange.com/a/59033/11824  

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### Answer by username_2 on 2024-08-22. Score: 0

Given a pure-exchange economy with:

* Utility functions: $u\_1(x\_1,y\_1)=\min(x\_1,4y\_1)$, $u\_2(x\_2,y\_2)=\min(x\_2,y\_2)$
* Endowment Allocation: $\omega = (\omega\_1,\omega\_2) =((2,4),(4,2))$

Set of feasible allocations are $\mathcal{F}=\{((x\_1,y\_1),(x\_2,y\_2))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_1+x\_2=y\_1+y\_2=6\}$

As can be observed from the Edgeworth Box, competitive equilibrium prices are $(p\_X^\*,p\_Y^\*)=(1,0)$ and the supported allocations are all feasible allocations $((x\_1,y\_1),(x\_2,y\_2))$ that satisfy $x\_1=2$ and $\frac{1}{2}\leq y\_1 \leq 2$.

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


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