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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?

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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.

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    $\begingroup$ Sorry that was not clear. When I wrote "I would suggest that you ask the following questions", I meant : "In order to help you solve the problem, I would suggest that you ask yourself the following questions". I'll edit my answer. My hope was that if you go through answering these questions yourself, you might be able to solve your problem and get more out of it than if someone gave you the final answer right away. Then maybe you can come back and give your own answer here. $\endgroup$
    – Martin Van der Linden
    Nov 26, 2014 at 15:12
  • $\begingroup$ Since $$y^*_A=x^*_A=\frac{30p_x}{(p_x+p_y)}$$ when $p_y=0$ $y^*_A=30$ and this is not feasible as you say. Hence we can neglect $p_y=0$. When instead $p_x=0$ I find that $y^*_A=x^*_A=0$, $x^*_B=\sqrt{20}$ and $y^*_B=20$. So these two demanded bundles are compatible for $y$. But I cannot get why they should be compatible for $x$. Why when one good is free one of the agents is indifferent between consuming any quantity of that good? $\endgroup$
    – Charlie
    Nov 26, 2014 at 15:36
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    $\begingroup$ Is $y_A^* = x^*_A = 0$ really the only possible demand from $A$ given that $p_x = 0$? Given $p_x = 0$, $A$' endowment is worth nothing, so given $p_y > 0$, she can only consume some quantity of $x$ (because it is free). Given $y_A = 0$, how does she "feel" about consuming different quantities of good $x$? $\endgroup$
    – Martin Van der Linden
    Nov 26, 2014 at 15:37

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