# How do I find the set of pareto optimal allocations?

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?

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:

• Thank you dear Amit! But how do I find the highest and lowest value of $p_y$? Commented Oct 31, 2023 at 0:12
• 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