Pure exchange economy with two consumers and non differentiable utility functions

External question — Economics Stack Exchange Author: Charlie Original post: https://economics.stackexchange.com/questions/333 License: CC BY-SA 3.0 — https://creativecommons.org/licenses/by-sa/3.0/ Adaptation: HTML converted to plain text; contact email addresses removed. We have a pure exchange economy, two consumers $A,B$ and two goods $x,y$. The utility functions are as follows $$u_A=\min\{x_A,y_A\}\qquad u_B=\min\{x_B,\sqrt{y_B}\}$$ The endowments are $$\omega_A=(30,0)\qquad \omega_B=(0,20)$$ I want to derive the equilibrium price and the equilibrium allocation. Now, I can understand that if the vector of prices $\mathbf{p}>>\mathbf{0}$ the equilibrium does not exist because the offer curves do not intersect (and this is evident once you have drawn the Edgeworth Box). My problem is that I don't know how to approach the case in which one of the two prices is zero. How should I study this situation? Should I derive formally the two offer curves as functions of prices?

External answer — Economics Stack Exchange Author: Martin Van der Linden Original post: https://economics.stackexchange.com/a/334 License: CC BY-SA 3.0 — https://creativecommons.org/licenses/by-sa/3.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I would suggest that ask yourself the following questions (hopefully, this should help you figure out how to solve the problem) : If good $z \in (x,y)$ was free, what would be the demand for both agents? Is the conjunction of these demands feasible given the endowments? (this should allow you to rule out one of the cases) If the demands are feasible, which price(s) for the other good would support such demands? What could be the equilibrium(a) allocation(s)? Notice that if good $z$ is free, one of the agents is not able to consume the other good, and she is therefore indifferent between consuming any quantity of good $z$. Hope this helps.

External answer — Economics Stack Exchange Author: Amit Original post: https://economics.stackexchange.com/a/57658 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 an exchange economy: Utility functions: $u_A(x_A,y_A)=\min(x_A,y_A)$ , $u_B(x_B, y_B)=\min(x_B, \sqrt{y_B})$ Endowments: $\omega_A = (30,0)$ , $\omega_B=(0,20)$ Set of feasible allocations is $\mathcal{F} = \{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=30, \ y_A+y_B=20\}$ As we can observe in the picture below that the set of competitive equilibrium allocations is given by: $\mathcal{CE} = \{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|0\leq x_A \leq 30-\sqrt{20}, \ y_A=0\}$ All these allocations are supported by prices $(p_X,p_Y)=(0,1)$