Find the Pareto Efficient allocations and Competitive Equilibrium when both agents have funky functions

External question — Economics Stack Exchange Author: Nicolas Torres Original post: https://economics.stackexchange.com/questions/59222 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I'm trying to solve this General Equilibrium excercise which I find quite challenging as both agents have funky utility functions. Find the Pareto Efficient allocations and Competitive Equilibra for this pure exchange economy: $U_A(x,y)=\max(5,\min(x,y))$ $U_B(x,y)=\min(x,\frac{3}{2}y)$ $\omega_A = (20,0)$ $\omega_B = (0,15)$ Here I have my Edgeworth box where Red: Indifference curve shape for A (L shape above and to the right of the kink) Blue: Indifference curve shape for B (L shape below and to the left of the kink) Green: Budget line for $\frac{p_1}{p_2}=\frac{1}{3}$ Orange: Leontief expansion path for A: $x=y>5$ Pink: Expansion path for B: $y=\frac{2}{3}x$ Is the Pareto efficient set the portion of the blue indifference curve that intersects the y axis? Now for the competitive equilibria: The set of optimal bundles for B is the entire pink curve. For A, the optimal bundles are: the set below the budget line for $p=\frac{1}{3}$ , which is supported by prices $\frac{p_1}{p_2}\leq \frac{1}{3}$ union the orange curve which is supported by prices $\frac{p_1}{p_2} > \frac{1}{3}$ . So would the competitive equilibria allocations be the interesction of the optimal points for A and B, which would be the portion of the pink segment below the budget line for $\frac{p_1}{p_2} = \frac{1}{3}$ ? If this is correct, how would I write the Competitive Equilibria set in the form $(p,x)$ where $p$ is the price vector and $x$ is the allocation vector, i.e. the quantities each consumer gets of each good. Also, are there any equilibria for $p_1 = 0$ or $p_2 = 0$ ?

External answer — Economics Stack Exchange Author: Amit Original post: https://economics.stackexchange.com/a/59223 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Given the pure-exchange economy: Utility functions: $u_A=\max(5,\min(x_A,y_A))$ , $u_B=\min(x_B,\frac{3}{2}y_B)$ Endowments: $\omega_A=(20,0)$ , $\omega_B=(0,15)$ Set of Feasible allocations is given by: $\mathcal{F}=\{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=20 \ \wedge \ y_A+y_B=15\}$ Set of Pareto optimal Allocations is given by: $\mathcal{PE}=\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|x_A=0 \ \wedge 0\leq y_A\leq \frac{5}{3}\} \ \cup \ \{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|5<\frac{5}{3}+\frac{2}{3}x_A\leq y_A \leq x_A\} $ Set of competitive equilibrium allocations is given by $\mathcal{CE}=\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|y_B=\frac{2}{3}x_B\geq 10\}$ Each allocation in the set above corresponds to exactly one equilibrium price ratio $\frac{p_X}{p_Y}$ in the set $[\frac{1}{12},\frac{1}{3}]$ . To be specific, for each $x_B\in[15,20]$ , the equilibrium allocation is $y_B=\frac{2}{3}x_B$ , $x_A=20-x_B$ , $y_A=15-\frac{2}{3}x_B$ and the corresponding price ratio is $\frac{p_X}{p_Y}=\frac{45-2x_B}{3x_B}$ . There are no equilibria with either $p_X=0$ or $p_Y=0$ because at $p_X=0$ , consumer 2 will demand at least $22.5$ units of $X$ leading to excess demand for it, and at $p_Y=0$ , consumer 1 will demand at least $20$ units of $Y$ causing exceed demand for $Y$ .