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There are two people in the economy, A and B. Both have the utility function: U=min{x,y}. A has an initial endowment of (x,y)=(50,100) and B has (x,y)=(100,50).

I know how to draw the Edgeworth box and establish the contract curve, but how do I determine the Walrasian equilibrium ratio of Px/Py? I'm confused by the fact that I can't differentiate U, nor is there money exchanged to establish a budget constraint.

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You do not need money exchanged to establish a budget constraint, most of these standard micro questions work in cashless economies.

Your Px/Py is just the exchange rate between the goods. If it's 3:1, it means that you exchange 3 $y$ for one $x$ - no cash involved.

Not being able to differentiate sucks, but actually requires you to know what you're doing. The point of differentiation is finding a place at which all Pareto improving trade has been exhausted.

Here, that means that, indexing the two players by i:

$$ x_i, y_i \in \arg\max_{x,y} U^i(x, y) s.t. p_x x + p_y y = p_x x^i + p_y y^i \, , \forall i $$

Without first-order conditions, this problem is potentially hard to solve. However, most non-differentiable utility functions give you a hint on how the outcome must look like.

Here, aggregate resources add up: $A=x_1 + x_2 = y_1 + y_2$. Hence, the planner would chose some allocation where x=y for both players, as for any positive $a$ and $\epsilon$ it is inefficient to give one player $(a + \epsilon, a)$ and the other one $(A - a - \epsilon, A - a)$.

However, that leaves a continuum of distributions as potential outcomes: Those where we give players one and two $(a, a)$ and $(A-a, A-a)$ for any $0 \leq a \leq A$.

Which equilibrium we actually get depends on the relative value of the initial endowments. I will leave you with this: What does $p_x/p_y$ necessarily have to be given total endowments and preferences? What does this imply for the unique equilibrium distribution of resources among the players?

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