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Suppose there are 2 individuals and 2 goods. $u_1(x_1,y_1)=x_1+2y_1$, and $u_2(x_2,y_2)=x_2+y_2$. There are 2 units of good $x$ and 1 unit of good $y$ in total.

How do I find the set of Pareto optimal allocations for this economy? Here's what I've got: \begin{align*} \max x_2+y_2\\ \text{s.t. }(2-x_2)+(1-y_2)=\bar{u}\implies x_2+2y_2=4-\bar{u}\\ \implies \max_x x+\frac{4-\bar{u}-x}{2}\\ \text{s.t. }0\leq x\leq 2 \end{align*} Thus, $x_2=2$ is the optimal. Is this correct? Also, suppose at the Walrasian equilibrium $p_x^*=1$. What is the lowest and highest of $p_y^*$ could be?

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In the economy you provided, set of feasible allocations is

$\mathcal{F}=\{((x_1,y_1),(x_2,y_2))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_1+x_2=2 \ \wedge \ y_1+y_2=1\}$

and is represented by points in the Edgeworth Box.

Set of Pareto efficient Allocations is

$\mathcal{PE}=\{((x_1,y_1),(x_2,y_2))\in\mathcal{F}|x_1=0 \ \vee \ y_1=1\}$

and is represented by the left and the top boundaries of the box.

Here is the picture:

enter image description here

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  • $\begingroup$ For additional details on the methodology, please watch youtu.be/oWXHbX9j0mc?feature=shared $\endgroup$
    – Amit
    Oct 31, 2023 at 0:02
  • $\begingroup$ Thank you dear Amit! But how do I find the highest and lowest value of $p_y$? $\endgroup$ Oct 31, 2023 at 0:12
  • $\begingroup$ In this case, Walrasian equilibrium price ratio $\frac{p_X}{p_Y}$ will always lie in the interval $[\frac{1}{2},1]$. Given that $p_X=1$, this means that $p_Y\in [1,2]$. Exact price ratio will, however, depend on the initial endowment allocation. For example: See this youtu.be/wWXLkgwWmgc?feature=shared $\endgroup$
    – Amit
    Oct 31, 2023 at 0:27
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    $\begingroup$ very nice graph, did you made it using tikz? What settings did you use for font, and arrow axes? $\endgroup$
    – 1muflon1
    Oct 31, 2023 at 0:28
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    $\begingroup$ @1muflon1 Thanks. I used Mathcha to plot the graph in the answer : mathcha.io Please try it. I find it very fast and convenient to plot graphs. Let me know if you need any help. $\endgroup$
    – Amit
    Oct 31, 2023 at 0:30

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