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Consider a $2$-good, $2$-person pure-exchange economy where $A$ is endowed with $(0,5)$ and $B$ is endowed with $(5,0)$. If the utility functions are $u_A = xy$ and $u_B = \min\{x,y\}$, what are all the competitive equilibrium allocations?

I think all the points on the $y=x$ axis (under the constraint that $x, y \in [0,10]$). I don't have an explanation except for an observation. Look at the following Edgeworth box diagram for example:

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The point $5 \cdot ((0.3,0.3), (0.7,0.7)) = ((1.5, 1.5), (3.5, 3.5))$ seems like a competitive equilibrium with the green line as the budget constraint. I feel like every point on $y=x$ will be a CE in a similar manner.

I could be wrong. How do I find the CE allocations in a more systematic manner especially when I can't differentiate the functions?

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  • $\begingroup$ If you're looking for sample problems to learn the process of solving such problems, you can watch videos in this playlist and then try: youtube.com/playlist?list=PLUJGfL_499TKsujAH6aeObLCw5VvSjzAx $\endgroup$
    – Amit
    Jan 20, 2023 at 17:35
  • $\begingroup$ @Amit Thanks, that playlist is indeed helpful. There's probably no trick as I thought. Can you verify if my conjecture that allocations along the $y=x$ line are CE allocations is correct? $\endgroup$
    – user43302
    Jan 20, 2023 at 18:41

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Here is a lazy tricks-based approach that requires almost no calculations: First, we know that every equilibrium allocation must be Pareto efficient by the first welfare theorem. If $B$ consumes different amounts of both goods, say $x>y$, then $\min\{x,y\}=\min\{y,y\}$. Unless $A$ consumes nothing of the second good, $A$ will be better off if they receive an additional amount of $x-y>0$ of the first good. This would then be a Pareto improvement. In equilibrium, this is not possible, so $B$ must consume the same amount of both goods.

Since the total endowment of both goods is the same, $A$ too must consume the same amount of both goods. But $A's$ utility function is differentiable (outside the boundary), and we can conclude from the condition that the ratio of marginal utilities equals the ratio of prices that both goods must have the same price.

From what we had before, we know that $B$ must consume the same amount of both goods. Since both goods have the same price, $B$ must spend the same amount on both goods. So $B$ will consume $(5/2,5/2)$. Then $A$ will consume the remainder, which too is $(5/2,5/2)$. So there is a unique equilibrium up to the normalization of prices.

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