<p>To find efficient allocations in this economy, we can first determine the production possibility frontier (PPF) which is given by the line segment <span class="math-container">$\dfrac{x}{2}+y=100$</span> where <span class="math-container">$x\in[0,200]$</span>. Pareto efficient allocation in this economy is a pair of points <span class="math-container">$(\beta, \alpha)$</span> where <span class="math-container">$\beta$</span> is a point on the PPF and <span class="math-container">$\alpha$</span> is the corresponding consumption allocation of the two consumers determined in the following way:
<a href="https://i.sstatic.net/zgMeM.png" rel="noreferrer"><img src="https://i.sstatic.net/zgMeM.png" alt="enter image description here" /></a>
For a point <span class="math-container">$\beta = (x, y)$</span> on the PPF such that it satisfy <span class="math-container">$\frac{x}{2}\leq y\leq 2x$</span>, the consumption allocation <span class="math-container">$\alpha = ((x_1, y_1), (x_2, y_2))$</span> that satisfy the following is Pareto efficient:</p>
<ul>
<li><span class="math-container">$x_1+x_2=x$</span> and <span class="math-container">$y_1+y_2=y$</span></li>
<li><span class="math-container">$y_1=2x_1$</span></li>
<li><span class="math-container">$\text{MRS}_2=\text{MRT}$</span> i.e. <span class="math-container">$\dfrac{y_2}{x_2}=\dfrac{1}{2}$</span></li>
</ul>
<p>Solving the above gives the following :</p>
<ul>
<li><span class="math-container">$x_1 = \dfrac{1}{3}(2y-x)$</span> and <span class="math-container">$y_1=\dfrac{2}{3}(2y-x)$</span></li>
<li><span class="math-container">$x_2 = \dfrac{2}{3}(2x-y)$</span> and <span class="math-container">$y_2=\dfrac{1}{3}(2x-y)$</span></li>
</ul>

CC BY-SA 4.0
2022-12-14T23:27:15.233Z
great answer, what software did you use for that picture? I really like it!
csilvia
CC BY-SA 4.0
2022-12-15T03:19:36.407Z
@csilvia Thank you. I have used LaTeX Tikz.
Amit
CC BY-SA 4.0
2022-12-14T15:13:44.853Z
2022-12-14T15:13:44.853Z
Amit
<p>Total endowments</p>
<p><span class="math-container">$e_x=2l=2*40=80$</span> and <span class="math-container">$e_y = 60$</span>
<a href="https://i.sstatic.net/Yv0Kk.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Yv0Kk.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-10-15T18:48:31.350Z
2022-10-15T18:48:31.350Z
studentp
<p>On the production economy which have 3 good <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$l$</span> and <span class="math-container">$2$</span> consumers(called <span class="math-container">$1$</span> and <span class="math-container">$2$</span>) and two firms(called <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>). Firm <span class="math-container">$1$</span> is owned by <span class="math-container">$1$</span> and produces only <span class="math-container">$x$</span> in function <span class="math-container">$x=2l$</span> and firm <span class="math-container">$2$</span> is owned by <span class="math-container">$Y$</span> and produces only <span class="math-container">$y$</span> in function <span class="math-container">$y=l$</span>. <span class="math-container">$1$</span> owns <span class="math-container">$40$</span> units of <span class="math-container">$l$</span> and <span class="math-container">$2$</span> owns <span class="math-container">$60$</span> units of <span class="math-container">$l$</span>. Each consumers utility function are <span class="math-container">$u_1(x_1,y_1) =  \min(2x_1,y_1)$</span> and <span class="math-container">$u_2(x_2,y_2) = \sqrt{x_2y_2}$</span>. Given price of <span class="math-container">$x,y,l$</span> is <span class="math-container">$p_x, p_y, w$</span>.
Now the problem is to find the set of pareto efficient allocations and we have to get MRS of <span class="math-container">$x,y$</span> for <span class="math-container">$1$</span> and <span class="math-container">$2$</span>, and MRT of <span class="math-container">$x,y$</span> and let all of them are same. But it is impossible to make MRS of 1 and MRT because MRS of 1 is 0 or infinite and MRT is 1/2. I can't understand how we can find P.E of this economey in this problem.</p>

<microeconomics><general-equilibrium><pareto-efficiency><leontief>
Question for general equilibrium
user1097146
# Question for general equilibrium

(Asked by: username_0 on 2022-10-15)

On the production economy which have 3 good $x$, $y$, $l$ and $2$ consumers(called $1$ and $2$) and two firms(called $X$ and $Y$). Firm $1$ is owned by $1$ and produces only $x$ in function $x=2l$ and firm $2$ is owned by $Y$ and produces only $y$ in function $y=l$. $1$ owns $40$ units of $l$ and $2$ owns $60$ units of $l$. Each consumers utility function are $u\_1(x\_1,y\_1) = \min(2x\_1,y\_1)$ and $u\_2(x\_2,y\_2) = \sqrt{x\_2y\_2}$. Given price of $x,y,l$ is $p\_x, p\_y, w$.
Now the problem is to find the set of pareto efficient allocations and we have to get MRS of $x,y$ for $1$ and $2$, and MRT of $x,y$ and let all of them are same. But it is impossible to make MRS of 1 and MRT because MRS of 1 is 0 or infinite and MRT is 1/2. I can't understand how we can find P.E of this economey in this problem.

**Tags:** microeconomics, general-equilibrium, pareto-efficiency, leontief

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## 2 answers

### Answer by username_1 on 2022-12-14. Score: 5

To find efficient allocations in this economy, we can first determine the production possibility frontier (PPF) which is given by the line segment $\dfrac{x}{2}+y=100$ where $x\in[0,200]$. Pareto efficient allocation in this economy is a pair of points $(\beta, \alpha)$ where $\beta$ is a point on the PPF and $\alpha$ is the corresponding consumption allocation of the two consumers determined in the following way:
[![enter image description here](https://i.sstatic.net/zgMeM.png)](https://i.sstatic.net/zgMeM.png)
For a point $\beta = (x, y)$ on the PPF such that it satisfy $\frac{x}{2}\leq y\leq 2x$, the consumption allocation $\alpha = ((x\_1, y\_1), (x\_2, y\_2))$ that satisfy the following is Pareto efficient:

* $x\_1+x\_2=x$ and $y\_1+y\_2=y$
* $y\_1=2x\_1$
* $\text{MRS}\_2=\text{MRT}$ i.e. $\dfrac{y\_2}{x\_2}=\dfrac{1}{2}$

Solving the above gives the following :

* $x\_1 = \dfrac{1}{3}(2y-x)$ and $y\_1=\dfrac{2}{3}(2y-x)$
* $x\_2 = \dfrac{2}{3}(2x-y)$ and $y\_2=\dfrac{1}{3}(2x-y)$

> **Comments:**
> **username_2**: great answer, what software did you use for that picture? I really like it!  
> **username_1**: @csilvia Thank you. I have used LaTeX Tikz.  

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### Answer by username_3 on 2022-10-15. Score: 0

Total endowments

$e\_x=2l=2\*40=80$ and $e\_y = 60$
[![enter image description here](https://i.sstatic.net/Yv0Kk.jpg)](https://i.sstatic.net/Yv0Kk.jpg)


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