<p>Set of Pareto efficient allocations is given by the dashed line in the Edgeworth Box. It is the set of feasible allocations satisfying <span class="math-container">$y_1 = x_1$</span> and <span class="math-container">$x_1y_1 \geq 9$</span>
.
<a href="https://i.sstatic.net/pIKp7.png" rel="noreferrer"><img src="https://i.sstatic.net/pIKp7.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2020-12-31T23:45:23.343Z
Where did $x_1y_1 \ge 9$ come from? Also, the dashed line doesn't work, it decreases agent 2's utility, so none of those trades could ever occur. The contract curve is the line $x_1=y_1$ in the intersection of the shaded region you have and to the right of the blue indifference curve through the endowment point.
Amaan M
CC BY-SA 4.0
2021-01-01T02:15:30.003Z
@AmaanM Contract curve $\neq$ Pareto set. The dashed line is the Pareto set, not the contract curve
Brennan
CC BY-SA 4.0
2020-12-31T12:03:50.523Z
2020-12-31T12:10:55.363Z
2020-12-31T12:10:55.363Z
Amit
<p>Consumer 1: <span class="math-container">$U_1(x_1,y_1)=x_1y_1$</span></p>
<p>Consumer 2:<span class="math-container">$ u_2(x_2,y_2)=min\{x_2y_2 , 4\} $</span></p>
<p>Initial endowments e1=(1,4) and e2=(4,1)</p>
<p>I want to find Pareto optimal allocations and show its edgeworth box</p>
<p>My solution
First I solve utility maximization
For consumer 1,</p>
<p><span class="math-container">$MRS=P_x/P_y$</span> And budget constraint <span class="math-container">$x_1P_x + y_1P_y= P_x+4P_y$</span></p>
<p><span class="math-container">$x_1^*=(P_x+4P_y)/2P_x$</span></p>
<p><span class="math-container">$y_1^*=(P_x+4P_y)/2P_y$</span></p>
<p>For consumer 2</p>
<p>At optimum, <span class="math-container">$x_2y_2=4$</span></p>
<p>Budget constraint <span class="math-container">$x_2P_x+y_2P_y= 4P_x+P_y$</span></p>
<p>Then <span class="math-container">$x_2[(4P_x+P_y)/P_y -P_x/P_y x_2]=4$</span></p>
<p>Feasibility constraint</p>
<p><span class="math-container">$$x_2=5-x_1$$</span></p>
<p><span class="math-container">$$y_2=5-y_1$$</span></p>
<p>Then I solve together</p>
<p><span class="math-container">$$(5-x_1)(5-y_1)=4$$</span></p>
<p>When I insert <span class="math-container">$x_1$</span> and <span class="math-container">$y_1$</span> I will obtain</p>
<p><span class="math-container">$$(9P_x-4P_y)(6P_y-P_x)=16P_xP_y$$</span></p>
<p>From this point I cannot proceed the solution</p>
<p>How can I find the Pareto optimal allocations ?</p>

CC BY-SA 4.0
2020-12-19T23:54:44.020Z
What do budget constraints have to do with Pareto optimal allocations?
Michael Greinecker
CC BY-SA 4.0
2020-12-20T08:36:19.957Z
I don’t understand what you mean. I wrote the budget constraint @MichaelGreinecker
studentp
CC BY-SA 4.0
2020-12-20T08:54:02.540Z
The budget constraint relates to who owns what. Pareto efficiency does not. Using something clearly unrelated is a non-starter.
Michael Greinecker
CC BY-SA 4.0
2020-12-20T10:15:34.737Z
For example, for consumer 1 $xP_x +yP_y= P_x e_x + P_y e_y$ where e represents endowment for a good. @MichaelGreinecker
studentp
CC BY-SA 4.0
2020-12-20T11:56:13.833Z
What has that to do with Pareto efficiency? Do you know the definition of Pareto efficiency?
Michael Greinecker
<microeconomics><self-study><general-equilibrium>
Find pareto optimal allocations
studentp
# Find pareto optimal allocations

(Asked by: username_0 on 2020-12-19)

Consumer 1: $U\_1(x\_1,y\_1)=x\_1y\_1$

Consumer 2:$ u\_2(x\_2,y\_2)=min\{x\_2y\_2 , 4\} $

Initial endowments e1=(1,4) and e2=(4,1)

I want to find Pareto optimal allocations and show its edgeworth box

My solution
First I solve utility maximization
For consumer 1,

$MRS=P\_x/P\_y$ And budget constraint $x\_1P\_x + y\_1P\_y= P\_x+4P\_y$

$x\_1^\*=(P\_x+4P\_y)/2P\_x$

$y\_1^\*=(P\_x+4P\_y)/2P\_y$

For consumer 2

At optimum, $x\_2y\_2=4$

Budget constraint $x\_2P\_x+y\_2P\_y= 4P\_x+P\_y$

Then $x\_2[(4P\_x+P\_y)/P\_y -P\_x/P\_y x\_2]=4$

Feasibility constraint

$$x\_2=5-x\_1$$

$$y\_2=5-y\_1$$

Then I solve together

$$(5-x\_1)(5-y\_1)=4$$

When I insert $x\_1$ and $y\_1$ I will obtain

$$(9P\_x-4P\_y)(6P\_y-P\_x)=16P\_xP\_y$$

From this point I cannot proceed the solution

How can I find the Pareto optimal allocations ?

**Tags:** microeconomics, self-study, general-equilibrium
> **Comments:**
> **username_1**: What do budget constraints have to do with Pareto optimal allocations?  
> **username_0**: I don’t understand what you mean. I wrote the budget constraint @MichaelGreinecker  
> **username_1**: The budget constraint relates to who owns what. Pareto efficiency does not. Using something clearly unrelated is a non-starter.  
> **username_0**: For example, for consumer 1 $xP_x +yP_y= P_x e_x + P_y e_y$ where e represents endowment for a good. @MichaelGreinecker  
> **username_1**: What has that to do with Pareto efficiency? Do you know the definition of Pareto efficiency?  

---

## 1 answers

### Answer by username_2 on 2020-12-31. Score: 5 | Accepted answer

Set of Pareto efficient allocations is given by the dashed line in the Edgeworth Box. It is the set of feasible allocations satisfying $y\_1 = x\_1$ and $x\_1y\_1 \geq 9$
.
[![enter image description here](https://i.sstatic.net/pIKp7.png)](https://i.sstatic.net/pIKp7.png)

> **Comments:**
> **username_3**: Where did $x_1y_1 \ge 9$ come from? Also, the dashed line doesn't work, it decreases agent 2's utility, so none of those trades could ever occur. The contract curve is the line $x_1=y_1$ in the intersection of the shaded region you have and to the right of the blue indifference curve through the endowment point.  
> **username_4**: @AmaanM Contract curve $\neq$ Pareto set. The dashed line is the Pareto set, not the contract curve  

---

